CHAPTER 10. The problems that are labeled advanced starts at number 53.

Transcription

1 10-1 CHAPTER 10 The correspondence between the new problem set and the previous 4th edition chapter 8 problem set. New Old New Old New Old 1 new 21 new new 22 new new new new new new new The problems that are labeled advanced starts at number 53. The English unit problems are: New Old New Old New Old 61 new new new new 78 new

2 Calculate the reversible work and irreversibility for the process described in Problem 5.18, assuming that the heat transfer is with the surroundings at 20 C. C.V.: A + B. This is a control mass. Continuity equation: m 2 - (m A1 + m B1 ) = 0 ; Energy: m 2 u 2 - m A1 u A1 - m B1 u B1 = 1 Q 2-1 W 2 System: if V B 0 piston floats P B = P B1 = const. if V B = 0 then P 2 < P B1 and v = V A /m tot see P-V diagram State A1: Table B.1.1, x = 1 v A1 = m 3 /kg, u A1 = kj/kg m A1 = V A /v A1 = kg State B1: Table B.1.2 sup. vapor v B1 = m 3 /kg, u B1 = kj/kg P P B1 a m B1 = V B1 /v B1 = kg => m 2 = m TOT = 1.56 kg At (T 2, P B1 ) v 2 = > v a = V A /m tot = so V B2 > 0 so now state 2: P 2 = P B1 = 300 kpa, T 2 = 200 C => u 2 = kj/kg and V 2 = m 2 v 2 = = m 3 2 V 2 (we could also have checked T a at: 300 kpa, m 3 /kg => T = 155 C) 1 Wac 2 = P B dv B = P B1 (V 2 - V 1 ) B = P B1 (V 2 - V 1 ) tot = kj 1 Q 2 = m 2 u 2 - m A1 u A1 - m B1 u B1 + 1 W 2 = kj From the results above we have : s A1 = , s B1 = , s 2 = kj/kg K 1 Wrev 2 = T o (S 2 - S 1 ) - (U 2 - U 1 ) + 1 Q 2 (1 - T o /T H ) = T o (m 2 s 2 - m A1 s A1 - m B1 s B1 ) + 1 W ac 2-1 Q 2 T o /T H = ( ) + ( ) - (-484.7) / = = 6.6 kj 1 I 2 = 1 Wrev 2-1 Wac 2 = ( ) = kj

3 Calculate the reversible work and irreversibility for the process described in Problem 5.65, assuming that the heat transfer is with the surroundings at 20 C. P 2 1 Linear spring gives 1 W 2 = PdV = 1 2 (P 1 + P 2 )(V 2 - V 1 ) v 1 Q 2 = m(u 2 - u 1 ) + 1 W 2 Equation of state: PV = mrt State 1: V 1 = mrt 1 /P 1 = 2 x x /500 = m 3 State 2: V 2 = mrt 2 /P 2 = 2 x x /300 = m 3 1 W 2 = 1 ( )( ) = kj 2 From Figure 5.10: C p (T avg ) = 45/44 = C v = 0.83 = C p - R For comparison the value from Table A.5 at 300 K is C v = kj/kg K 1 Q 2 = mc v (T 2 - T 1 ) + 1 W 2 = 2 x 0.83(40-400) = kj 1 Wrev 2 = T o (S 2 - S 1 ) - (U 2 - U 1 ) + 1 Q 2 (1 - T o /T H ) = T o m(s 2 - s 1 )+ 1 W ac 2-1 Q 2 T o /T o = T o m[c P ln(t 2 / T 1 ) R ln(p 2 / P 1 )] + 1 W ac 2-1 Q 2 = x 2 [ ln(313/673) ln(300/500)] = = kj 1 I 2 = 1 Wrev 2-1 Wac 2 = (-45.72) = kj 10.3 The compressor in a refrigerator takes refrigerant R-134a in at 100 kpa, 20 C and compresses it to 1 MPa, 40 C. With the room at 20 C find the minimum compressor work. Solution: C.V. Compressor out to ambient. Minimum work in is the reversible work. SSSF, 1 inlet and 2 exit energy Eq.: w c = h 1 - h 2 + q rev Entropy Eq.: s 2 = s 1 + dq/t + s gen = s 1 + q rev /T => q rev = T 0 (s 2 - s 1 ) w c min = h 1 - h 2 + T 0 (s 2 - s 1 ) = ( ) = kj/kg

4 Calculate the reversible work out of the two-stage turbine shown in Problem 6.41, assuming the ambient is at 25 C. Compare this to the actual work which was found to be MW. C.V. Turbine. SSSF, 1 inlet and 2 exits. Use Eq for each flow stream with q = 0 for adiabatic turbine. Table B.1: h 1 = , h 2 = , h 3 = x = s 1 = , s 2 = , s 3 = x = Ẇ rev = T 0 (ṁ 2 s 2 + ṁ 3 s 3 - ṁ 1 s 1 ) - (ṁ 2 h 2 + ṁ 3 h 3 - ṁ 1 h 1 ) = ( ) ( ) = MW = Ẇ ac + Q. rev = kw kw 10.5 A household refrigerator has a freezer at T F and a cold space at T C from which energy is removed and rejected to the ambient at T A as shown in Fig. P10.5. Assume that the rate of heat transfer from the cold space, Q. C, is the same as from the freezer, Q. F, find an expression for the minimum power into the heat pump. Evaluate this power when T A = 20 C, T C = 5 C, T F = 10 C, and Q. F = 3 kw. C.V. Refrigerator (heat pump), SSSF, no external flows except heat transfer. Energy Eq.: Q. F + Q. c + Ẇ = Q. A (amount rejected to ambient) Reversible gives minimum work in as from Eq or 10.9 on rate form. Ẇ = Q. F 1 T A T + Q. c F 1 T A T = C = kw (negative so work goes in) 10.6 An air compressor takes air in at the state of the surroundings 100 kpa, 300 K. The air exits at 400 kpa, 200 C at the rate of 2 kg/s. Determine the minimum compressor work input. C.V. Compressor, SSSF, minimum work in is reversible work. ψ 1 = 0 at ambient conditions s 0 - s 2 = s T0 - s T2 - R ln(p 0 /P 2 ) = ln(100/400) = ψ 2 = h 2 - h 0 + T 0 (s 0 - s 2 ) = (s 0 - s 2 ) ψ 2 = kj/kg -Ẇ REV = ṁ(ψ 2 -ψ 1 ) = kw = Ẇ c

5 A supply of steam at 100 kpa, 150 C is needed in a hospital for cleaning purposes at a rate of 15 kg/s. A supply of steam at 150 kpa, 250 C is available from a boiler and tap water at 100 kpa, 15 C is also available. The two sources are then mixed in an SSSF mixing chamber to generate the desired state as output. Determine the rate of irreversibility of the mixing process. 1 2 B.1.1: h 1 = s 1 = B.1.3: h 2 = s 2 = B.1.3: h 3 = s 3 = C.V. Mixing chamber, SSSF Cont.: ṁ 1 + ṁ 2 = ṁ 3 Energy: ṁ 1 h 1 + ṁ 2 h 2 = ṁ 3 h ṁ 2 /ṁ 3 = (h 3 - h 1 )/(h 2 - h 1 ) = = ṁ 2 = kg/s, ṁ 1 = kg/s. I = T0 Ṡ gen = T 0 (ṁ 3 s 3 - ṁ 1 s 1 - ṁ 2 s 2 ) Entropy: ṁ 1 s 1 + ṁ 2 s 2 + Ṡ gen = ṁ 3 s 3 = ( ) = 1269 kw 10.8 Two flows of air both at 200 kpa of equal flow rates mix in an insulated mixing chamber. One flow is at 1500 K and the other is at 300 K. Find the irreversibility in the process per kilogram of air flowing out C.V. Mixing chamber Cont.: ṁ 1 + ṁ 2 = ṁ 3 = 2ṁ 1 Energy: ṁ 1 h 1 + ṁ 1 h 2 = 2ṁ 1 h 3 Properties from Table A.7 h 3 = (h 1 + h 2 )/2 = ( )/2 = kj/kg s T3 = Entropy Eq.: ṁ 1 s 1 + ṁ 1 s 2 + Ṡ gen = 2ṁ 1 s 3 Ṡ gen /2ṁ 1 = s 3 - (s 1 + s 2 )/2 = ( )/2 = kj/kg K

6 A steam turbine receives steam at 6 MPa, 800 C. It has a heat loss of 49.7 kj/kg and an isentropic efficiency of 90%. For an exit pressure of 15 kpa and surroundings at 20 C, find the actual work and the reversible work between the inlet and the exit. C.V. Reversible adiabatic turbine (isentropic) w T = h i - h e,s ; s e,s = s i = kj/kg K, h i = kj/kg x e,s = ( )/ = , h e,s = = w T,s = = kj/kg C.V. Actual turbine w T,ac = ηw T,s = kj/kg = h i - h e,ac - q loss h e,ac = h i - q loss - w T,ac = = Actual exit state: P,h sat. vap., s e,ac = C.V. Reversible process, work from Eq q R = T 0 (s e,ac - s i ) = ( ) = kj kg w R = h i - h e,ac + q R = = kj/kg A 2-kg piece of iron is heated from room temperature 25 C to 400 C by a heat source at 600 C. What is the irreversibility in the process? C.V. Iron out to 600 C source, which is a control mass. Energy Eq.: m Fe (u 2 - u 1 ) = 1 Q 2-1 W 2 Process: Constant pressure => 1 W 2 = Pm Fe (v 2 - v 1 ) 1 Q 2 = m Fe (h 2 - h 1 ) = m Fe C(T 2 - T 1 ) 1 Q 2 = (400-25) = 315 kj m Fe (s 2 - s 1 ) = 1 Q 2 /T res + 1 S 2 gen 1S 2 gen = m Fe (s 2 - s 1 ) - 1 Q 2 /T res = m Fe C ln (T 2 /T 1 ) - 1 Q 2 /T res = ln = kj/k I 2 = T o ( 1 S 2 gen ) = = 96.4 kj

7 A 2-kg/s flow of steam at 1 MPa, 700 C should be brought to 500 C by spraying in liquid water at 1 MPa, 20 C in an SSSF setup. Find the rate of irreversibility, assuming that surroundings are at 20 C. C.V. Mixing chamber, SSSF. State 1 is superheated vapor in, state 2 is compressed liquid in, and state 3 is flow out. No work or heat transfer. Cont.: m 3 = m 1 + m Energy: m 3h 3 = m 1h 1 + m 2h 2 Entropy: m 3s 3 = m 1s 1 + m 2s 2 + S gen Table B.1.3: h 1 = , s 1 = , h 3 = , s 3 = , For state 2 interpolate between, saturated liquid 20 C table B.1.1 and, compressed liquid 5 MPa, 20 C from Table B.1.4: h 2 = 84.9, s 2 = x = m 2/m 1 = (h 3 - h 1 )/(h 2 - h 3 ) = m 2 = = kg/s ; m 3 = = kg/s S gen = m 3s 3 - m 1s 1 - m 2s 2 = kw/k. I = Ẇ rev - Ẇ ac = Ẇ rev = To S gen = = kw

8 Fresh water can be produced from saltwater by evaporation and subsequent condensation. An example is shown in Fig. P10.12 where 150-kg/s saltwater, state 1, comes from the condenser in a large power plant. The water is throttled to the saturated pressure in the flash evaporator and the vapor, state 2, is then condensed by cooling with sea water. As the evaporation takes place below atmospheric pressure, pumps must bring the liquid water flows back up to P 0. Assume that the saltwater has the same properties as pure water, the ambient is at 20 C and that there are no external heat transfers. With the states as shown in the table below find the irreversibility in the throttling valve and in the condenser. State T [ C] C.V. Valve. ṁ 1 = m ex = ṁ 2 + ṁ 3 Energy Eq.: h 1 = h e ; Entropy Eq.: s 1 + s gen = s e h 1 = , s 1 = , P 2 = P sat (T 2 = T 3 ) = kpa h e = h 1 x e = ( )/ = s e = = s gen = s e - s 1 = = I = ṁt0 s gen = = kw C.V. Condenser. h 2 = , s 2 = 8.558, h 5 = 96.5, s 5 = ṁ 2 = (1 - x e )ṁ 1 = kg/s h 7 = 71.37, s 7 = , h 8 = 83.96, s 8 = ṁ 2 h 2 + ṁ 7 h 7 = ṁ 2 h 5 + ṁ 7 h 8 ṁ 7 = ṁ 2 (h 2 - h 5 )/(h 8 - h 7 ) = kg = s ṁ 2 s 2 + ṁ 7 s 7 + Ṡ gen = ṁ 2 s 5 + ṁ 7 s 8. I = T0 Ṡ gen = T 0 [ ṁ 2 (s 5 - s 2 ) + ṁ 7 (s 8 - s 7 ) ] = [ ( ) ( )] = = 7444 kw

9 An air compressor receives atmospheric air at T 0 = 17 C, 100 kpa, and compresses it up to 1400 kpa. The compressor has an isentropic efficiency of 88% and it loses energy by heat transfer to the atmosphere as 10% of the isentropic work. Find the actual exit temperature and the reversible work. C.V. Compressor Isentropic: w c,in,s = h e,s - h i ; s e,s = s i Table A.7: P r,e,s = P r,i (P e /P i ) = = h e,s = w c,in,s = = Actual: w c,in,ac = w c,in,s /η c = ; q loss = w c,in,ac + h i = h e,ac + q loss => h e,ac = = => T e,ac = 621 K Reversible: w rev = h i - h e,ac + T 0 (s e,ac - s i ) = ( ) = = kj/kg Since q loss is also to the atmosphere it is not included as it will not be reversible A piston/cylinder has forces on the piston so it keeps constant pressure. It contains 2 kg of ammonia at 1 MPa, 40 C and is now heated to 100 C by a reversible heat engine that r eceives heat fr om a 200 C sour ce. Find the w or k out of the heat engine. C.V. Ammonia plus heat engine Energy: m am (u 2 - u 1 ) = 1 Q 2,200 - W H.E. - 1 W 2,pist Entropy: m am (s 2 - s 1 ) = 1 Q 2 /T res + 0 => 1 Q 2 = m am (s 2 - s 1 )T res Process: P = const. 1 W 2 = P(v 2 - v 1 )m am W H.E. = m am (s 2 - s 1 )T res - m am (h 2 - h 1 ) NH 3 Q L H.E. Q 200 C H W Table B.2.2: h 1 = , s 1 = , h 2 = , s 2 = W H.E. = 2 [( ) ( )] = kj

10 Air flows through a constant pressure heating device, shown in Fig. P It is heated up in a reversible process with a work input of 200 kj/kg air flowing. The device exchanges heat with the ambient at 300 K. The air enters at 300 K, 400 kpa. Assuming constant specific heat develop an expression for the exit temperature and solve for it. C.V. Total out to T 0 h 1 + q 0 rev - w rev = h 2 s 1 + q 0 rev /T 0 = s 2 q 0 rev = T 0 (s 2 - s 1 ) h 2 - h 1 = T 0 (s 2 - s 1 ) - w rev (same as Eq ) Constant C p gives: C p (T 2 - T 1 ) = T 0 C p ln (T 2 /T 1 ) T 2 - T 0 ln(t 2 /T 1 ) = T /C p T 1 = 300, C p = 1.004, T 0 = 300 T ln (T 2 /300) = (200/1.004) = At 600 K LHS = 392 (too low) At 800 K LHS = Linear interpolation gives T 2 = 790 K (LHS = OK) Air enters the turbocharger compressor (see Fig. P10.16), of an automotive engine at 100 kpa, 30 C, and exits at 170 kpa. The air is cooled by 50 C in an intercooler before entering the engine. The isentropic efficiency of the compressor is 75%. Determine the temperature of the air entering the engine and the irreversibility of the compression-cooling process. a) Compressor. First ideal which is reversible adiabatic, constant s: T 2S = T 1 ( P 2 P 1 ) k-1 k = 303.2( )0.286 = K w S = C P0 (T 1 - T 2S ) = 1.004( ) = w = w S /η S = -49.9/0.75 = kj/kg = C P (T 1 - T 2 ) T 2 = K T 3 (to engine) = T 2 - T INTERCOOLER = = K = 46.3 C b) Irreversibility from Eq with rev. work from Eq.10.12, (q = 0 at T H ) s 3 - s 1 = ln ln = kj kg K i = T(s 3 - s 1 ) - (h 3 - h 1 ) - w = T(s 3 - s 1 ) - C P (T 3 - T 1 ) - C P (T 1 - T 2 ) = 303.2( ) (-50) = kj/kg

11 A car air-conditioning unit has a 0.5-kg aluminum storage cylinder that is sealed with a valve and it contains 2 L of refrigerant R-134a at 500 kpa and both are at room temperature 20 C. It is now installed in a car sitting outside where the whole system cools down to ambient temperature at 10 C. What is the irreversibility of this process? C.V. Aluminum and R-134a Energy Eq.: m Al (u 2 - u 1 ) Al + m R (u 2 - u 1 ) R = 1 Q 2-1 W 2 ( 1 W 2 = 0) Entropy Eq.: m AL (s 2 - s 1 ) Al + m R (s 2 - s 1 ) R = 1 Q 2 /T S 2 gen (u 2 - u 1 ) Al = C v,al (T 2 - T 1 ) = 0.9(-10-20) = - 27 kj/kg (s 2 - s 1 ) Al = C p,al ln(t 2 /T 1 ) = 0.9 ln(263.15/293.15) = Table B.5.2: v 1 = , u 1 = = kj/kg, s 1 = kj/kg K, m R134a = V/v 1 = kg v 2 = v 1 = & T 2 => x 2 = ( )/ = u 2 = = 264.9, s 2 = = Q 2 1 S 2 gen = 0.5 (-27) ( ) = kj = 0.5 ( ) ( ) + = kj/k I 2 = T 0 ( 1 S 2 gen ) = = 1.0 kj A steady combustion of natural gas yields 0.15 kg/s of products (having approximately the same properties as air) at 1100 C, 100 kpa. The products are passed through a heat exchanger and exit at 550 C. What is the maximum theoretical power output from a cyclic heat engine operating on the heat rejected from the combustion products, assuming that the ambient temperature is 20 C? Heat PROD.e T i Exchanger o o ( C). 550 C QH e Heat. Engine W NET. S Q T L T = 20 o L L C. Q H = ṁ PROD C P0 (T L - T e ) PROD = ( ) = PROD.i 1100 o C s Ṡ L = - Ṡ i-e = ṁ PROD C P0 ln(t i /T e) PROD = ln( /823.15) = kw/k. Q L = T L Ṡ L = = kw. W NET = Q. H - Q. L = = kw

12 A counterflowing heat exchanger cools air at 600 K, 400 kpa to 320 K using a supply of water at 20 C, 200 kpa. The water flow rate is 0.1 kg/s and the air flow rate is 1 kg/s. Assume this can be done in a reversible process by the use of heat engines and neglect kinetic energy changes. Find the water exit temperature and the power out of the heat engine(s). H O 2 2 W AIR C.V. Total m a h 1 + m H2O h 3 = m a h 2 + m H2O h 4 + W m a s 1 + m H2O s 3 = m a s 2 + m H2O s 4 (s gen = 0) Table A.7: h 1 = , s T1 = Table A.7: h 2 = , s T2 = , B.1.1: h 3 = 83.96, s 3 = s 4 = (m a /m H2O )(s 1 - s 2 ) + s 3 = (1/0.1)( ) = : P 4 = P 3, s 4 Table B.1.2: x 4 = ( )/5.597 = , h 4 = = kj/kg, T 4 = C W = m a (h 1 - h 2 ) + m H2O (h 3 - h 4 ) = 1( ) + 0.1( ) = kw Water as saturated liquid at 200 kpa goes through a constant pressure heat exchanger as shown in Fig. P The heat input is supplied from a reversible heat pump extracting heat from the surroundings at 17 C. The water flow rate is 2 kg/min and the whole process is reversible, that is, there is no overall net entropy change. If the heat pump receives 40 kw of work find the water exit state and the increase in availability of the water. C.V. Heat exchanger + heat pump. ṁ 1 = ṁ 2 = 2 kg/min, ṁ 1 h 1 + Q. 0 + Ẇ in = ṁ 1 h 2, ṁ 1 s 1 + Q. /T 0 = ṁ 1 s 2 Substitute Q. 0 into energy equation and divide by ṁ 1 h 1 - T 0 s 1 + w in = h 2 - T 0 s 2 LHS = /2 = kj/kg State 2: P 2, h 2 - T 0 s 2 = kj/kg At sat. vap. h g - T 0 s g = so state 2 is superheated vapor at 200 kpa. At 600 C: h 2 - T 0 s 2 = = kj/kg At 700 C: h 2 - T 0 s 2 = = kj/kg Linear interpolation T 2 = 667 C ψ = (h 2 - T 0 s 2 ) - (h 1 - T 0 s 1 ) = w in = 1200 kj/kg = kj/kg

13 Calculate the irreversibility for the process described in Problem 6.63, assuming that heat transfer is with the surroundings at 17 C. I = T o S gen so apply 2nd law out to T o = 17 o C m 2 s 2 - m 1 s 1 = m i s i + 1 Q 2 / T o + 1 S 2gen T o S gen = T o ( m 2 s 2 - m 1 s 1 - m i s i ) - 1 Q 2 m 1 = 0.90 kg m i = kg m 2 =3.982 kg T o S gen = I = T o [ m 1 (s 2 - s 1 ) + m i (s 2 - s i )] - 1 Q 2 = [0.9(C p ln R ln ) (C pln - R ln )] - ( kj) = ( ) = kj The high-temperature heat source for a cyclic heat engine is a SSSF heat exchanger where R-134a enters at 80 C, saturated vapor, and exits at 80 C, saturated liquid at a flow rate of 5 kg/s. Heat is rejected from the heat engine to a SSSF heat exchanger where air enters at 150 kpa and ambient temperature 20 C, and exits at 125 kpa, 70 C. The rate of irreversibility for the overall process is 175 kw. Calculate the mass flow rate of the air and the thermal efficiency of the heat engine. C.V. R-134a Heat Exchanger, ṁ R134a = 5 kg/s Inlet: T 1 = 80 o C, sat. vap. x 1 = 1.0, Table B.5.1 h 1 = h g = kj/kg, s 1 = s g = kj/kg-k Exit: T 2 = 80 o C, sat. liq. x 2 = 0.0 h 2 = h f = kj/kg, s 2 = s f = kj/kg-k C.V. Air Heat Exchanger, C p = kj/kg-k, R = kj/kg-k Inlet: T 3 = 20 o C, P 3 = 150 kpa Exit: T 4 = 70 o C, P 4 = 125 kpa 2 nd Law: İ = To Ṡ net = ṁ R134a (s 2 - s 1 ) + ṁ air (s 4 - s 3 ) s 4 - s 3 = C p ln(t 4 /T 3 ) - Rln(P 4 /P 3 ) = kj/kg-k ṁ air = [İ - ṁ R134a (s 2 - s 1 )]/(s 4 - s 3 ) = 10.0 kg/s 1 st Law for each line: Q. + ṁh in = ṁh ex + Ẇ; Ẇ = 0 R-134a: 1 Q. 2 = ṁ R134a (h 2 - h 1 ) = -532 kw; Q. H = - 1 Q. 2 Air: 3 Q. 4 = ṁ air (h 4 - h 3 ) = ṁ air (T 4 - T 3 ) = kw; Q. L = 3 Q. 4 Ẇ net = Q. H - Q. L = 30.2 kw; η th = Ẇ net / Q. H = 0.057, or 5.7%

14 A control mass gives out 10 kj of energy in the form of a. Electrical work from a battery b. Mechanical work from a spring c. Heat transfer at 500 C Find the change in availability of the control mass for each of the three cases. a) φ = -W el = -10 kj b) φ = -W spring = -10 kj c) φ = -[1 - (T 0 /T H )] Q out = = kj Calculate the availability of the water at the initial and final states of Problem 8.32, and the irreversibility of the process. 1: u 1 = s 1 = Wac 2 = 203 kj v 1 = : u 2 = s 2 = Qac 2 = v 2 = : u o = , s o = v o = , T H = K φ = (u - T o s) - (u o - T o s o ) + P o ( v - v o ) φ 1 = ( ) - ( ) ( ) = kj/kg φ 2 = ( ) - ( ) ( ) = 850 kj/kg 1 I 2 = m(φ 1 - φ 2 ) + [1-(T 0 /T H )] 1 Qac 2-1 Wac 2 + P o ( V 2 - V 1 ) = ( ) ( ) = = kj [(S gen = 3.75 kj/k T o S gen = 1118 kj so OK] A steady stream of R-22 at ambient temperature, 10 C, and at 750 kpa enters a solar collector. The stream exits at 80 C, 700 kpa. Calculate the change in availability of the R-22 between these two states. i R-22 Solar Collector e Inlet (T,P) Table B.4.1 (liquid) h i = kj/kg, s i = kj/kg K Exit (T,P) Table B.4.2 (sup. vap.) h e = kj/kg, s e = kj/kg K ψ ie = ψ e - ψ i = (h e - h i ) - T 0 (s e - s i ) = ( ) ( ) = kj/kg

15 Nitrogen flows in a pipe with velocity 300 m/s at 500 kpa, 300 C. What is its availability with respect to an ambient at 100 kpa, 20 C? ψ = h 1 - h 0 + (1/2)V T 0 (s 1 - s 0 ) = C p (T 1 - T 0 ) + (1/2)V T 0 (C p ln(t 1 /T 0 ) - R ln(p 1 /P 0 )) = (300-20) + (300 2 /2000) ln ln = kj/kg A 10-kg iron disk brake on a car is initially at 10 C. Suddenly the brake pad hangs up, increasing the brake temperature by friction to 110 C while the car maintains constant speed. Find the change in availability of the disk and the energy depletion of the car s gas tank due to this process alone. Assume that the engine has a thermal efficiency of 35%. All the friction work is turned into internal energy of the disk brake. m(u 2 - u 1 ) = 1 Q 2-1 W 2 1 Q 2 = m Fe C Fe (T 2 - T 1 ) 1 Q 2 = (110-10) = 450 kj Neglect the work to the surroundings at P 0 φ = m(u 2 - u 1 ) - T 0 m(s 2 - s 1 ) m(s 2 -s 1 ) = mc ln(t 2 /T 1 ) = ln = kj/k φ = = kj W engine = η th Q gas = 1 Q 2 = Friction work Q gas = 1 Q 2 /η th = 450/0.35 = kj

16 A 1 kg block of copper at 350 C is quenched in a 10 kg oil bath initially at ambient temperature of 20 C. Calculate the final uniform temperature (no heat transfer to/from ambient) and the change of availability of the system (copper and oil). C.V. Copper and oil. C co = 0.42 C oil = 1.8 m 2 u 2 - m 1 u 1 = 1 Q 2-1 W 2 = 0 = m co C co (T 2 - T 1 ) co + (mc) oil (T 2 - T 1 ) oil ( T 2-350) (T 2-20) = T 2 = 507 => T = 27.5 C = K For each mass copper and oil, we neglect work term (v = C) so Eq is (φ 2 - φ 1 ) = u 2 - u 1 - T o (s 2 - s 1 ) = mc [(T 2 - T 1 ) - T o ln (T 2 / T 1 ) ] m cv (φ 2 - φ 1 ) cv + m oil (φ 2 - φ 1 ) oil = = 0.42 [(-322.5) ln ] [ ln ] = = kj Calculate the availability of the system (aluminum plus gas) at the initial and final states of Problem 8.74, and also the process irreversibility. State 1: T 1 = 200 o C, v 1 = V 1 / m = 0.05 / = State 2: v 2 = v 1 (2 / 1.5) ( / ) = The metal does not change volume, so the combined is using Eq as φ 1 = m gas φ gas + m Al φ Al = m gas [u 1 -u o -T o (s 1 - s o )] cv + m gas P o (v 1 -v o ) + m Al [u 1 -u o -T o (s 1 -s o )] Al = m gas C v (T 1 - T o ) - m gas T o [C p ln T 1 - R ln P 1 ] + m T o P gas P o (v 1 - v o ) o + m Al [C (T 1 - T o ) - T o C ln (T 1 /T o ) ] Al φ 1 = [ 0.653(200-25) (0.842 ln ln ) ( ) ] [ ln ] = = kj φ 2 = [ 0.653(25-25) (0.842 ln ln ) ( ) ] [ ln ] = = kj 1 I 2 = φ 1 - φ 2 + [1-(T 0 /T H )] 1 Q 2-1 W 2 AC + P o m( V 2 - V 1 ) = (-14) ( ) = kj [(S gen = T o S gen = so OK ]

17 Consider the springtime melting of ice in the mountains, which gives cold water running in a river at 2 C while the air temperature is 20 C. What is the availability of the water (SSSF) relative to the temperature of the ambient? ψ = h 1 - h 0 - T 0 (s 1 - s 0 ) flow availability from Eq Approximate both states as saturated liquid ψ = ( ) = kj/kg Why is it positive? As the water is brought to 20 C it can be heated with q L from a heat engine using q H from atmosphere T H = T 0 thus giving out work Refrigerant R-12 at 30 C, 0.75 MPa enters a SSSF device and exits at 30 C, 100 kpa. Assume the process is isothermal and reversible. Find the change in availability of the refrigerant. h i = , s i = , h e = , s e = ψ = h e - h i - T 0 (s e - s i ) = ( ) = kj/kg A geothermal source provides 10 kg/s of hot water at 500 kpa, 150 C flowing into a flash evaporator that separates vapor and liquid at 200 kpa. Find the three fluxes of availability (inlet and two outlets) and the irreversibility rate. C.V. Flash evaporator chamber. SSSF with no work or heat transfer. Cont. Eq.: ṁ 1 = ṁ 2 + ṁ 3 ; Energy Eq.: ṁ 1 h 1 = ṁ 2 h 2 + ṁ 3 h 3 Entropy Eq.: ṁ 1 s 1 + Ṡ gen = ṁ 2 s 2 + ṁ 3 s 3 1 Vap. 2 Liq. 3 B.1.1: h o = , s o = , h 1 = , s 1 = B.1.2: h 2 = , s 2 = , h 3 = , s 3 = h 1 = xh 2 + (1 - x) h 3 => x = ṁ 2 /ṁ 1 = h 1 - h 3 = h 2 - h 3 ṁ 2 = xṁ 1 = kg/s ṁ 3 = (1-x)ṁ 1 = kg/s Ṡ gen = = kw/k Flow availability Eq.10.22: ψ = (h - T o s) - (h o - T o s o ) = h - h o - T o (s - s o ) ψ 1 = ( ) = ψ 2 = ( ) = ψ 3 = ( ) = ṁ 1 ψ 1 = kw ṁ 2 ψ 2 = kw ṁ 3 ψ 3 = kw. I = ṁ1 ψ 1 - ṁ 2 ψ 2 - ṁ 3 ψ 3 = 37 kw

18 Air flows at 1500 K, 100 kpa through a constant pressure heat exchanger giving energy to a heat engine and comes out at 500 K. What is the constant temperature the same heat transfer should be delivered at to provide the same availability? C.V. Heat exchanger. SSSF, no work term. q out = h 1 - h 2 = = kj/kg ψ = (1 - T o T H ) q out = ψ 1 - ψ 2 ψ 1 - ψ 2 = h 1 - h 2 - T o ( s 1 - s 2 ) Heat exch 1. 2 Q H W. H.E.. Q L = ( ) = = kj/kg 1 - T o = (ψ T 1 - ψ 2 ) / q out = / = H T o = => T T H = 924 K H A wooden bucket (2 kg) with 10 kg hot liquid water, both at 85 C, is lowered 400 m down into a mineshaft. What is the availability of the bucket and water with respect to the surface ambient at 20 C? C.V. Bucket and water. Both thermal availability and potential energy terms. v 1 v 0 for both wood and water so work to atm. is zero. Use constant heat capacity table A.3 for wood and table B.1.1 (sat. liq.) for water. From Eq φ 1 - φ 0 = m wood [u 1 - u 0 - T 0 (s 1 - s 0 )] + m H2 O [u 1 - u 0 - T 0 (s 1 - s 0 )] + m tot g(z 1 - z 0 ) = 2[1.26(85-20) ln ] + 10[ ( )] (-400) /1000 = = kj

19 A rigid container with volume 200 L is divided into two equal volumes by a partition. Both sides contains nitrogen, one side is at 2 MPa, 300 C, and the other at 1 MPa, 50 C. The partition ruptures, and the nitrogen comes to a uniform state at 100 C. Assuming the surroundings are at 25 C find the actual heat transfer and the irreversibility in the process. m 2 = m A + m B = P A1 V A + P B1 V B = RT A1 RT B = = kg P 2 = m 2 RT 2 /V tot = /0.2 = kpa Energy Eq.: m A (u 2 - u 1 ) A + m B (u 2 - u 1 ) B = 1 Q 2-0/ 1 Q 2 = m A C v (T 2 - T 1 ) A + m B C v (T 2 - T 1 ) B = ( ) (100-50) = kj m A (s 2 - s 1 ) A + m B (s 2 - s 1 ) B = 1 Q 2 /T sur + 1 S s gen (s 2 - s 1 ) A = ln ln = (s 2 - s 1 ) B = ln ln = = ( ) / = kj/k 1 S 2,gen I = (T 0 )( 1 S 2,gen ) = kj An air compressor is used to charge an initially empty 200-L tank with air up to 5 MPa. The air inlet to the compressor is at 100 kpa, 17 C and the compressor isentropic efficiency is 80%. Find the total compressor work and the change in availability of the air. C.V. Tank + compressor (constant inlet conditions) USUF, no heat transfer. Continuity: m 2 - m 1 = m in ( m 1 = 0 ) Energy: m 2 u 2 = m in h in - 1 W 2 Entropy: m 2 s 2 = m in s in + 1 S 2 gen Reversible compressor: 1 S 2 GEN = 0 s 2 = s in P r2 = P rin (P 2 /P in ) = T 2,s = 855, u 2,s = w 2,s = h in - u 2,s = Actual compressor: 1 w 2,AC = 1 w 2,s /η c = u 2,AC = h in - 1 w 2,AC = T 2,AC = 958 K v 2 = RT 2 /P 2 = State 2 u, P m 2 = V 2 /v 2 = W 2 = m 2 ( 1 w 2,AC ) = kj m 2 (φ 2 - φ 1 ) = m 2 [u 2 - u 1 + P 0 (v 2 - v 1 ) - T 0 (s 2 - s 1 )] = [ ( ) - 290[ ln(5000/100)] = kj

20 Air enters a compressor at ambient conditions, 100 kpa, 300 K, and exits at 800 kpa. If the isentropic compressor efficiency is 85%, what is the second-law efficiency of the compressor process? 800 kpa T 2s = 300(8) = K 2s 2 -W 100 kpa s = 1.004( ) = kj/kg -w = -W s η = s 0.85 = K K s T 2 = T 1 + -w = C P = K s 2 - s 1 = ln(586.8/300) ln 8 = ψ 2 - ψ 1 = (h 2 - h 1 ) - T 0 (s 2 - s 1 ) = ( ) = kj/kg 2nd law efficiency: η 2nd Law = (ψ 2 - ψ 1 )/(-w) = 264.9/287.8 = Steam enters a turbine at 25 MPa, 550 C and exits at 5 MPa, 325 C at a flow rate of 70 kg/s. Determine the total power output of the turbine, its isentropic efficiency and the second law efficiency. h i = , s i = , h e = , s e = Actual turbine: w T,ac = h i - h e = kj/kg Isentropic turbine: s e,s = s i h e,s = w T,s = h i - h e,s = 429 kj/kg Rev. turbine: w rev = w T,ac + T 0 (s e - s i ) = = kj/kg η T = w T,ac /w T,s = 339.1/429 = 0.79 η II = w T,ac /w rev = 339.1/ = A compressor is used to bring saturated water vapor at 1 MPa up to 17.5 MPa, where the actual exit temperature is 650 C. Find the irreversibility and the second-law efficiency. Inlet state: Table B.1.2 h i = , s i = Actual compressor Table B.1.3 h e,ac = , s e,ac = Energy Eq. Actual compressor: -w c,ac = h e,ac - h i = kj/kg From Eq.10.14: i = T 0 (s e,ac - s i ) = ( ) = kj/kg From Eq.10.13: w rev = i + w c,ac = = η II = -w rev /w c,ac = /915.8 = 0.951

21 A flow of steam at 10 MPa, 550 C goes through a two-stage turbine. The pressure between the stages is 2 MPa and the second stage has an exit at 50 kpa. Assume both stages have an isentropic efficiency of 85%. Find the second law efficiencies for both stages of the turbine CV: T1, h 1 = kj/kg, s 1 = kj/kg K Isentropic s 2s = s 1 h 2s = T1 T2 w T1,s = h 1 - h 2s = 483 kj/kg Actual T1: w T1,ac = η T1 w T1,s = = h 1 - h 2ac h 2ac = h 1 - w T1,ac = , s 2ac = CV: T2, s 3s = s 2ac = x 3s = ( )/ = , h 3s = = kj/kg w T2,s = h 2ac - h 3s = w T2,ac = η T2 w T2,s = kj/kg h 3ac = , x 3ac = ( )/ =0.9354, s 3ac = = kj/kg K Actual T1: i T1,ac = T 0 (s 2ac -s 1 ) = ( ) = 36.4 kj/kg w R T1 = w T1,ac + i = 447 kj/kg, η II = w T1,ac /w R T1 = Actual T2: i T2,ac = T 0 (s 3ac -s 2ac ) = ( ) = kj/kg w R T2 = w T2,ac + i T2,ac = 681.5, η II = w T2,ac /w R T2 = Consider the two-stage turbine in the previous problem as a single turbine from inlet to final actual exit and find its second-law efficiency. From solution to Problem we have w T,ac = w T1,ac + w T2,ac = h 1 - h 3ac = 1004 kj/kg The actual compared to the reversible turbine has, Eq.10.14, i = q R = T 0 (s 3 - s i ) = q R T1 + qr T2 = kj/kg w R = w T,ac + i = kj/kg η II = w T,ac /w R = 0.89

22 The simple steam power plant shown in Problem 6.39 has a turbine with given inlet and exit states. Find the availability at the turbine exit, state 6. Find the second law efficiency for the turbine, neglecting kinetic energy at state 5. h 6 = , s 6 = , h 5 = , s 5 = ψ 6 = h 6 - h 0 - T 0 (s 6 - s 0 ) = ( ) = kj/kg w rev = ψ 5 - ψ 6 = h 5 - h 6 - T 0 (s 5 - s 6 ) = kj/kg. w ac = h 5 - h 6 = ; η II = w ac /w rev = A compressor takes in saturated vapor R-134a at 20 C and delivers it at 30 C, 0.4 MPa. Assuming that the compression is adiabatic, find the isentropic efficiency and the second law efficiency. Table B.5 h i = , s i = , h e,ac = , s e,ac = Ideal exit: P e, s e,s = s i h e,s = Isentropic compressor w c,s = h e,s - h i = Actual compressor w c,ac = h e,ac - h i = Reversible between inlet and actual exit -w c,rev = h i - h e,ac - T 0 (s i - s e,ac ) = ( ) = η TH,c = (w c,s /w c,ac ) = (22.43/37.14) = η II = (w c,rev /w c,ac ) = (22.23/37.14) = Steam is supplied in a line at 3 MPa, 700 C. A turbine with an isentropic efficiency of 85% is connected to the line by a valve and it exhausts to the atmosphere at 100 kpa. If the steam is throttled down to 2 MPa before entering the turbine find the actual turbine specific work. Find the change in availability through the valve and the second law efficiency of the turbine. Take C.V. as valve and a C.V. as the turbine. Valve: h 2 = h 1 = , s 2 > s 1 = , h 2, P 2 s 2 = ψ 1 - ψ 2 = h 1 h 2 T 0 (s 1 -s 2 ) = ( ) = 55.3 kj/kg So some potential work is lost in the throttling process. Ideal turbine: s 3 = s 2 h 3s = w T,s = kj/kg w T,ac = h 2 - h 3ac = ηw T,s = kj/kg h 3ac = = s 3ac = kj/kg K w rev = h 2 - h 3ac - T 0 (s 2 - s 3ac ) = ( ) = kj/kg η II = 835.2/ = 0.91

23 The condenser in a refrigerator receives R-134a at 700 kpa, 50 C and it exits as saturated liquid at 25 C. The flowrate is 0.1 kg/s and the condenser has air flowing in at ambient 15 C and leaving at 35 C. Find the minimum flow rate of air and the heat exchanger second-law efficiency. 3 4 AIR C.V. Total heat exchanger. 2 R-134a 1 m 1 h 1 + m a h 3 = m 1 h 2 + m a h 4 m a = m 1 h 1 - h = 0.1 = kg/s h 4 - h (35-15) ψ 1 - ψ 2 = h 1 - h 2 - T 0 (s 1 - s 2 ) = ψ 4 - ψ 3 = h 4 - h 3 - T 0 (s 4 - s 3 ) ( ) = = 1.004(35-15) ln = η II = m a (ψ 4 - ψ 3 )/m 1 (ψ 1 - ψ 2 ) = 1.007(0.666) 0.1(8.7208) = A piston/cylinder arrangement has a load on the piston so it maintains constant pressure. It contains 1 kg of steam at 500 kpa, 50% quality. Heat from a reservoir at 700 C brings the steam to 600 C. Find the second-law efficiency for this process. Note that no formula is given for this particular case so determine a reasonable expression for it. 1: Table B.1.2 P 1, x 1 v 1 = = 0.188, h 1 = = , s 1 = = : P 2 = P 1,T 2 v 2 = , h 2 = , s 2 = m(u 2 - u 1 ) = 1 Q 2-1 W 2 = 1 Q 2 - P(V 2 - V 1 ) 1 Q 2 = m(u 2 - u 1 ) + Pm(v 2 - v 1 ) = m(h 2 - h 1 ) = kj 1 W 2 = Pm(v 2 - v 1 ) = kj 1 W 2 to atm = P 0 m(v 2 - v 1 ) = kj Useful work out = 1 W 2-1 W 2 to atm = φ reservoir = (1 - T 0 /T res ) 1 Q 2 = = η II = W net / φ = 0.177

24 Air flows into a heat engine at ambient conditions 100 kpa, 300 K, as shown in Fig. P Energy is supplied as 1200 kj per kg air from a 1500 K source and in some part of the process a heat transfer loss of 300 kj/kg air happens at 750 K. The air leaves the engine at 100 kpa, 800 K. Find the first and the second law efficiencies. C.V. Engine out to reservoirs h i + q 1500 = q h e + w w ac = = kj/kg η TH = w/q 1500 = For second law efficiency also a q to/from ambient s i + (q 1500 /T H ) + (q 0 /T 0 ) = (q 750 /T m ) + s e q 0 = T 0 (s e - s i ) + (T 0 /T m )q (T 0 /T H )q 1500 = ln (300/1500) 1200 = kj/kg w rev = h i - h e + q q q 0 = w ac + q 0 = kj/kg η II = w ac /w rev = / = Consider the high-pressure closed feedwater heater in the nuclear power plant described in Problem Determine its second-law efficiency. For this case with no work the second law efficiency is from Eq : η II = m 16 (ψ 18 - ψ 16 )/m 17 (ψ 17 - ψ 15 ) Properties (taken from computer software): h 15 = 585 h 16 = 565 h 17 = 2593 h 18 = 688 s 15 = s 16 = s 17 = s 18 = ψ 18 - ψ 16 = h 18 - h 16 - T 0 (s 18 - s 16 ) = ψ 17 - ψ 15 = h 17 - h 15 - T 0 (s 17 - s 15 ) = η II = ( )/( ) = 0.85

25 Consider a gasoline engine for a car as an SSSF device where air and fuel enters at the surrounding conditions 25 C, 100 kpa and leaves the engine exhaust manifold at 1000 K, 100 kpa as products assumed to be air. The engine cooling system removes 750 kj/kg air through the engine to the ambient. For the analysis take the fuel as air where the extra energy of 2200 kj/kg of air released in the combustion process, is added as heat transfer from a 1800 K reservoir. Find the work out of the engine, the irreversibility per kilogram of air, and the first- and second-law efficiencies. 1 2 AIR T 0 Q out Q H T H W C.V. Total out to reservoirs m a h 1 + Q H = m a h 2 + W + Q out m a s 1 + Q H/T H + S gen = m a s s + Q out/t 0 h 1 = s T1 = h 2 = s T1 = w ac = W /m a = h 1 - h 2 + q H - q out = = kj/kg η TH = w/q H = 702.4/2200 = s gen = s 2 - s 1 + q out - q H 750 = T 0 T H = kj/kg K 1800 i tot = (T 0 )s gen = kj/kg For reversible case have s gen = 0 and q R 0 from T 0, no q out q R 0,in = T 0 (s 2 - s 1 ) - (T 0 /T H )q H = kj/kg w rev = h 1 - h 2 + q H + q R 0,in = w ac + i tot = kj/kg η II = w ac /w rev = Air enters a steady-flow turbine at 1600 K and exhausts to the atmosphere at 1000 K. The second law efficiency is 85%. What is the turbine inlet pressure? C.V.: Turbine, exits to atmosphere so assume P e = 100 kpa Inlet: T i = 1600 K, Table A.7: h i = kj/kg, s o i = kj/kg K Exit: T e = 1000 K, h e = kj/kg, s o e = kj/kg K 1 st Law: q + h i = h e + w; q = 0 => w = (h i - h e ) = kj/kg 2 nd Law: ψ i - ψ e = w/η 2ndLaw = 711.1/0.85 = kj/kg ψ i - ψ e = (h i - h e ) - T o (s i - s e ) = kj/kg h i - h e = w = kj/kg, assume T o = 25 o C s i - s e = kj/kg-k s i - s e = s o e - so i - R ln(p i /P e ) = => P e /P i = 30.03; P i = 3003 kpa

26 Consider the two turbines in Problem What is the second-law efficiency for the combined system? From the solution to Problem 9.72 we have s 1 = , s 2 = , s 5 = , m T1 /m tot = m T1 /m tot = => w ac = W net /m tot = kj/kg C.V. Total system out to T 0 m T1 /m tot h 1 + m T2 /m tot h 2 + q 0 rev = h 5 + w rev w qc + q rev 0 = w rev m T1 /m tot s 1 + m T2 /m tot s 2 + q 0 rev = s 5 q 0 rev = [ ] = kj/kg w rev = w ac + q 0 rev = = 332 kj/kg η II = w ac /w rev = /332 = Air in a piston/cylinder arrangement is at 110 kpa, 25 C, with a volume of 50 L. It goes through a reversible polytropic process to a final state of 700 kpa, 500 K, and exchanges heat with the ambient at 25 C through a reversible device. Find the total work (including the external device) and the heat transfer from the ambient. C.V. Total out to ambient m a (u 2 - u 1 ) = 1 Q 2-1 W 2,tot, m a (s 2 - s 1 ) = 1 Q 2 /T 0 m a = / = kg 1 Q 2 = T 0 m a (s 2 - s 1 ) = [ W 2,tot = 1 Q 2 - m a (u 2 - u 1 ) ln (700/110)] = kj = ( ) = kj

27 10-27 Advanced Problems Refrigerant-22 is flowing in a pipeline at 10 C, 600 kpa, with a velocity of 200 m/s, at a steady flowrate of 0.1 kg/s. It is desired to decelerate the fluid and increase its pressure by installing a diffuser in the line (a diffuser is basically the opposite of a nozzle in this respect). The R-22 exits the diffuser at 30 C, with a velocity of 100 m/s. It may be assumed that the diffuser process is SSSF, polytropic, and internally reversible. Determine the diffuser exit pressure and the rate of irreversibility for the process. C.V. Diffuser out to T 0, Int. Rev. flow s gen R-22 = 0/ m 1 = m 2, h 1 + (1/2)V q = h 2 + (1/2)V 2 2 s 1 + dq/t + s gen = s 2 = s 1 + q/t 0 + s gen Rev. Polytropic Process: w 12 = 0/ = - v dp + (V V 2 2 )/2 LHS = v dp = n n-1 (P 2 v 2 - P 1 v 1 ) = 1 2 (V2 1 - V 2 2) = 1 2 ( )/1000 = 15 kj/kg P 1 v 1 n = P 2 v 2 n n = ln(p 2 /P 1 )/ln(v 1 /v 2 ) Inlet state: v 1 = , s 1 = , h 1 = Exit state: T 2,V 2,? only P 2 unknown. v 2 = v(p 2 ), n = n(p 2 ) Trial and error on P 2 to give LHS = 15: P 2 = 1 MPa v 2 = , n = ln 1000 / ln = LHS = (1.0412/0.0412)( ) = LHS too small so P 2 > 1.0 MPa Assume sat. 30 C, P 2 = MPa v 2 = , n = ln /ln = LHS = ( ) = > P 2 = 1.1 MPa v 2 = , n = , LHS = Interpolate to get the final state: P 2 = MPa, v 2 = , n = , LHS = OK s 2 = h 2 = Find q from energy equation (Assume T 0 = T 1 = 10 C) q = h 2 - h (V V 1 2 ) = = kj/kg I = ṁ[t 0 (s 2 - s 1 ) - q ]= 0.1[283.15( ) ] = kw

28 A piston/cylinder contains ammonia at 20 C, quality 80%, and a volume of 10 L. A force is now applied to the piston so it compresses the ammonia in an adiabatic process to a volume of 5 L, where the piston is locked. Now heat transfer with the ambient takes place so the ammonia reaches the temperature of the ambient at 20 C. Find the work and heat transfer. If it is done in a reversible process, how much work and heat transfer would be involved? P Process : v 1 = m 3 /kg, s 1 = kj/kg K u 1 = = kj/kg V m = V 1 /v 1 = 0.010/ = 0.02 kg V 2 V 1 Assume reversible adiabatic compression m(u 2 - u 1 ) = - 1 W 2 ; m(s 2 - s 1 ) = 0, v 2 = v 1 V 2 /V 1 = : s 2 = s 1, v 1 Trial and error on T 2. For 0 C, v 2 s = , For -2 C, v 2 s = T 2 = -1 C, x = 0.832, u 2 = 1125 kj/kg 1 W 2 = -m(u 2 - u 1 ) = kj, ( 1 Q 2 = 0) 3: T 3, v 3 = v 2 P 3 = 537 kpa, u 3 = , s 3 = Q 3 = m(u 3 - u 2 ) = 0.02( ) = kj Q rev = mt 0 (s 3 - s 1 ) = ( ) = kj W rev = Q rev - m(u 3 - u 1 ) = ( ) = -1.1 kj

29 Consider the irreversible process in Problem Assume that the process could be done reversibly by adding heat engines/pumps between tanks A and B and the cylinder. The total system is insulated, so there is no heat transfer to or from the ambient. Find the final state, the work given out to the piston and the total work to or from the heat engines/pumps. C.V. Water m A + m B + heat engines. No Qexternal, only 1 W 2,cyl + W HE m 2 = m A1 + m B1 = 6 kg, m 2 u 2 - m A1 u A1 - m B1 u B1 = - 1 W 2,cyl - W HE m 2 s 2 - m A1 s A1 - m B1 s B1 = 0/ + 0/ v A1 = u A1 = s A1 = V A = m 3 v B1 = u B1 = s B1 = V B = m 3 m 2 s 2 = = s 2 = kj/kg K If P 2 < P lift = 1.4 MPa then V 2 = V A + V B = m 3, v 2 = m 3 /kg (P lift, s 2 ) v 2 = V 2 = m 3 > V 2 OK P 2 = P lift = 1.4 MPa u 2 = kj/kg 1 W 2,cyl = P lift (V 2 - V A - V B ) = 1400 ( ) = kj W HE = m A1 u A1 + m B1 u B1 - m 2 u 2-1 W 2,cyl = = kj

30 Water in a piston/cylinder is at 100 kpa, 34 C, shown in Fig. P The cylinder has stops mounted so V min = 0.01 m 3 and V max = 0.5 m 3. The piston is loaded with a mass and outside P 0, so a pressure inside of 5 MPa will float it. Heat of kj from a 400 C source is added. Find the total change in availability of the water and the total irreversibility. P 1a CV water plus cyl. wall out to reservoir m 2 = m 1 = m, m(u 2 - u 1 ) = 1 Q 2-1 W 2 m(s 2 - s 1 ) = 1 Q 2 /T res + 1 S 2 gen System eq: States must be on the 3 lines 1 2 1b v Here P eq = 5 MPa, so since P 1 < P eq V 1 = V min = 0.01 m 3 1: v 1 = m 3 /kg, u 1 = kj/kg, s 1 = kj/kg K m = V 1 /v 1 = kg Heat added gives q = Q/m = kj/kg so check if we go from 1 past 1a (P eq,v 1 ) and past 1b (P eq,v max ) 1a: T = 40 C, u 1a = q 1a = u 1a - u 1 = 24.5 kj/kg 1b: T = C, u 1b = as 1 w 1b = P eq (v 1b - v 1 ) = q 1b = h 1b - u 1 - P eq v 1 = The state is seen to fall between 1a and 1b so P 2 = P eq. 1 w 2 = P eq (v 2 - v 1 ), u 2 - u 1 = 1 q 2-1 w 2 u 2 + P eq v 2 = h 2 = 1 q 2 + u 1 + P eq v 1 = < h g (P 2 ) => T 2 = 264 C x 2 = ( )/ = , v 2 = = , s 2 = = The non-flow availability change of the water (Eq.10.22) m(φ 2 - φ 1 ) = m[u 2 - u 1 - T 0 (s 2 - s 1 ) + P 0 (v 2 - v 1 )] = 9.944[ ( ) + 100( )] = = 4448 kj I = T( 1 S 2 gen ) = T [m(s 2 - s 1 ) - 1 Q 2 /T res ]= [ 9.944( ) /673.15] = 3326 kj (Piston and atm. have an increase in availability)

31 A rock bed consists of 6000 kg granite and is at 70 C. A small house with lumped mass of kg wood and 1000 kg iron is at 15 C. They are now brought to a uniform final temperature with no external heat transfer. a. For a reversible process, find the final temperature and the work done in the process. b. If they are connected thermally by circulating water between the rock bed and the house, find the final temperature and the irreversibility of the process, assuming that surroundings are at 15 C. a) For a reversible process a heat engine is installed between the rockbed and the house. Take C.V. Total m rock (u 2 - u 1 ) + m wood (u 2 - u 1 ) + m Fe (u 2 - u 1 ) = - 1 W 2 m rock (s 2 - s 1 ) + m wood (s 2 - s 1 ) + m Fe (s 2 - s 1 ) = 0/ (mc) rock ln T 2 T 1 + (mc) wood ln T 2 T 1 + (mc) Fe ln T 2 T 1 = 0/ ln (T 2 /343.15) ln (T 2 /288.15) => T 2 = K - 1 W 2 = ( ) 1 W 2 = kj ln (T 2 /288.15) = 0/ b) No work, no Q irreversible process. (mc) rock (T 2-70) + (mc wood + mc Fe )(T 2-15) = 0/ T 2 = 29.0 C = K + ( )( ) S gen = m i (s 2 - s 1 ) i = 5340 ln ln = kj/k I 2 = (T 0 ) 1 S 2,gen = = kj

32 Consider the heat engine in Problem The exit temperature was given as 800 K, but what are the theoretical limits for this temperature? Find the lowest and the highest, assuming that the heat transfers are as given. For an exit temperature that is the average of the highest and lowest possible, find the firstand second-law efficiencies for the heat engine. The lowest exhaust temperature will occur when the maxumum amount of work is delivered which is a reversible process. Assume no other heat transfers then 2nd law: s i + q H /T H + 0/ = s e + q m /T m s e - s i = q H /T H - q m /T m = s Te - s Ti - R ln(p e /P i ) s Te = s Ti + R ln(p e /P i ) + q H /T H - q m /T m = ln(100/100) / /750 = T e,min = 446 K The maximum exhaust temperature occurs with no work out h i + q H = q m + h e h e = = T e,max = 1134 K T e = (1/2)(T e,min + T e,max ) = 790 K see solution for to follow calculations h e = , s Te = w ac = = , η TH = w ac /q H = 0.32 q 0 = 300[ ln ] = kj/kg 1500 w rev = w ac + q 0 = , η II = w ac /w rev = / = 0.683

33 Air in a piston/cylinder arrangement, shown in Fig. P10.59, is at 200 kpa, 300 K with a volume of 0.5 m 3. If the piston is at the stops, the volume is 1 m 3 and a pressure of 400 kpa is required to raise the piston. The air is then heated from the initial state to 1500 K by a 1900 K reservoir. Find the total irreversibility in the process assuming surroundings are at 20 C. P stop P 1 P 1 2 W 1 2 m 2 = m 1 = m = P 1 V 1 /RT 1 = / = kg m(u 2 - u 1 ) = 1 Q 2-1 W 2 m(s 2 - s 1 ) = dq/t + 1 S 2 gen V 1 V stop Process: P = P 0 + α(v-v 0 ) if V V stop Information: P stop = P 0 + α(v stop -V 0 ) Eq. of state T stop = T 1 P stop V stop /P 1 V 1 = 1200 < T 2 So the piston will hit the stops => V 2 = V stop 1W 2 = 1 2 (P 1 + P stop )(V stop - V 1 ) = 1 ( )(1-0.5) = 150 kj 2 1 Q 2 = M(u 2 - u 1 ) + 1 W 2 = 1.161( ) = 1301 kj s 2 - s 1 = s o T2 - s o T1 -R ln(p 2 /P 1 ) = ln 2.5 = 1.48 kj/kg K Take control volume as total out to reservoir at T RES 1S 2 gen tot = m(s 2 - s 2 ) - 1 Q 2 /T RES = kj/k 1 I 2 = T 0( 1 S 2 gen ) = = 303 kj

34 Consider two rigid containers each of volume 1 m 3 containing air at 100 kpa, 400 K. An internally reversible Carnot heat pump is then thermally connected between them so it heats one up and cools the other down. In order to transfer heat at a reasonable rate, the temperature difference between the working substance inside the heat pump and the air in the containers is set to 20 C. The process stops when the air in the coldest tank reaches 300 K. Find the final temperature of the air that is heated up, the work input to the heat pump, and the overall second-law efficiency. A Q A H.P. W H.P. Q B B The high and the low temperatures in the heat pump are T A +20 and T B -20, respectively. Since T A and T B change during the process, the coefficient of performance changes, and so it must be integrated. du A = m A du A = m A C v dt A = dq A du B = m B du B = m B c v dt B = -dq B Carnot heat pump: dq A = T A + 20 dq B T B - 20 = -m A C v dt A = -dt A m B C v dt B dt B (dt A )/(T A + 20) = - (dt B )/(T B - 20) dt A T A + 20 = ln T A T A = - dt B T B - 20 = -ln T B2-20 T B1-20 T A1 = T B1 = 400 K, T B2 = 300 K ln T A = -ln T A2 = 550 K m A = m B = P 1 V 1 /RT 1 = (100 1)/( ) = kg W H.P. = Q A - Q B = mc v (T A2 - T A1 ) + mc v (T B2 - T B1 ) = ( ) = 31.2 kj Second law efficiency is total increase in air availability over the work input. φ = mc v (T 2 - T 1 ) + P 0 (V 2 - V 1 ) - T 0 (s 2 - s 1 ) φ A = Q A - T 0 m(s 2 - s 1 ) A φ B = -Q B - T 0 m(s 2 - s 1 ) B φ A + φ B = W H.P. - mt 0 [(s 2 - s 1 ) A + (s 2 - s 1 ) B ] To find entropies we need the pressures (P 2 /P 1 ) = T 2 /T 1 for both A and B s 2 - s 1 = C p ln(t 2 /T 1 ) - R ln(t 2 /T 1 ) = C v ln(t 2 /T 1 ) φ tot = W H.P. - mt 0 C v [ ln(t 2 /T 1 ) A + ln(t 2 /T 1 ) B ] = ln ln = η II = φ tot /W H.P. = 25.47/31.2 = 0.816

35 10-35 ENGLISH UNIT PROBLEMS 10.61ECalculate the reversible work and irreversibility for the process described in Problem 5.122, assuming that the heat transfer is with the surroundings at 68 F. P 2 1 Linear spring gives 1 W 2 = PdV = 1 2 (P 1 + P 2 )(V 2 - V 1 ) v 1 Q 2 = m(u 2 - u 1 ) + 1 W 2 Equation of state: PV = mrt ( ) State 1: V 1 = mrt 1 /P 1 = = ft ( ) State 2: V 2 = mrt 2 /P 2 = = ft W 2 = 1 ( )( ) x144/778 = Btu 2 From Table C.7 C p (T avg ) = [(6927-0)/( )]/M = 10.45/44.01 = Btu/lbm R C V = C p R = /778 = Q 2 = mc V (T 2 - T 1 ) + 1 W 2 = 4x (75-750) = Btu 1 Wrev 2 = T o (S 2 - S 1 ) - (U 2 - U 1 ) + 1 Q 2 (1 - T o /T H ) = T o m(s 2 - s 1 )+ 1 Wac 2-1 Q 2 T o /T o = T o m[c P ln(t 2 / T 1 ) R ln(p 2 / P 1 )] + 1 W ac 2-1 Q 2 = x 4 [ ln(535/1210) ln(45/70)] = = Btu 1 I 2 = 1 Wrev 2-1 Wac 2 = (-55.98) = Btu 10.62EThe compressor in a refrigerator takes refrigerant R-134a in at 15 lbf/in. 2, 0 F and compresses it to 125 lbf/in. 2, 100 F. With the room at 70 F find the reversible heat transfer and the minimum compressor work. C.V. Compressor out to ambient. Minimum work in is the reversible work. SSSF, 1 inlet and 2 exit energy Eq.: w c = h 1 - h 2 + q rev Entropy Eq.: s 2 = s 1 + dq/t + s gen = s 1 + q rev /T => q rev = T 0 (s 2 - s 1 ) q rev = ( ) = Btu/lbm w c min = h 1 - h 2 + T 0 (s 2 - s 1 ) = = Btu/lbm

36 EA supply of steam at 14.7 lbf/in. 2, 320 F is needed in a hospital for cleaning purposes at a rate of 30 lbm/s. A supply of steam at 20 lbf/in. 2, 500 F is available from a boiler and tap water at 14.7 lbf/in. 2, 60 F is also available. The two sources are then mixed in an SSSF mixing chamber to generate the desired state as output. Determine the rate of irreversibility of the mixing process. 1 2 C.8.1 h 1 = 28.08, s 1 = C.8.2 h 2 = , s 2 = C.8.2 h 3 = , s 3 = C.V. Mixing chamber ṁ 1 + ṁ 2 = ṁ 3 ṁ 1 h 1 + ṁ 2 h 2 = ṁ 3 h ṁ 2 /ṁ 3 = (h 3 - h 1 )/(h 2 - h 1 ) = = ṁ 2 = lbm/s, ṁ 1 = lbm/s. I = T0 Ṡ gen = T 0 (ṁ 3 s 3 - ṁ 1 s 1 - ṁ 2 s 2 ) ṁ 1 s 1 + ṁ 2 s 2 + Ṡ gen = ṁ 3 s 3 = ( ) = 961 Btu/s 10.64EA 4-lbm piece of iron is heated from room temperature 77 F to 750 F by a heat source at 1100 F. What is the irreversibility in the process? C.V. Iron out to 1100 F source, which is a control mass. Energy Eq.: m Fe (u 2 - u 1 ) = 1 Q 2-1 W 2 Process: Constant pressure => 1 W 2 = Pm Fe (v 2 - v 1 ) 1 Q 2 = m Fe (h 2 - h 1 ) = m Fe C(T 2 - T 1 ) 1 Q 2 = (750-77) = Btu m Fe (s 2 - s 1 ) = 1 Q 2 /T res + 1 S 2 gen 1S 2 gen = m Fe (s 2 - s 1 ) - 1 Q 2 /T res = m Fe C ln (T 2 /T 1 ) - 1 Q 2 /T res = ln = Btu/R I 2 = T o ( 1 S 2 gen ) = = Btu

37 EFresh water can be produced from saltwater by evaporation and subsequent condensation. An example is shown in Fig. P10.12 where 300-lbm/s saltwater, state 1, comes from the condenser in a large power plant. The water is throttled to the saturated pressure in the flash evaporator and the vapor, state 2, is then condensed by cooling with sea water. As the evaporation takes place below atmospheric pressure, pumps must bring the liquid water flows back up to P 0. Assume that the saltwater has the same properties as pure water, the ambient is at 68 F, and that there are no external heat transfers. With the states as shown in the table below find the irreversibility in the throttling valve and in the condenser. State T [F] C.V. Valve: ṁ 1 = m ex = ṁ 2 + ṁ 3, Energy Eq.: h 1 = h e ; Entropy Eq.: s i + s gen = s e h 1 = s i = P 2 = P sat (T 2 = T 3 ) = h e = h 1 x e = ( )/ = s e = = s gen = s e - s i = = I = ṁt0 s gen = = Btu/s C.V. Condenser. h 2 = , s 2 = 2.044, h 5 = 42.09, s 5 = ṁ 2 = (1 - x e )ṁ i = lbm/s h 7 = 31.08, s 7 = , h 8 = 36.09, s 8 = Energy Eq.: ṁ 2 h 2 + ṁ 7 h 7 = ṁ 2 h 5 + ṁ 7 h 8 ṁ 7 = ṁ 2 (h 2 - h 5 )/(h 8 - h 7 ) = = lbm s Entropy Eq.: ṁ 2 s 2 + ṁ 7 s 7 + Ṡ gen = ṁ 2 s 5 + ṁ 7 s 8. I = T0 Ṡ gen = T 0 [ṁ 2 (s 5 - s 2 ) + ṁ 7 (s 8 - s 7 )] = 528[297.44( ) ( )] = = 5467 Btu/s

38 EAir flows through a constant pressure heating device as shown in Fig. P It is heated up in a reversible process with a work input of 85 Btu/lbm air flowing. The device exchanges heat with the ambient at 540 R. The air enters at 540 R, 60 lbf/in. 2. Assuming constant specific heat develop an expression for the exit temperature and solve for it. C.V. Total out to T 0 Energy Eq.: h 1 + q 0 rev - w rev = h 2 Entropy Eq.: s 1 + q 0 rev /T 0 = s 2 q 0 rev = T 0 (s 2 - s 1 ) h 2 - h 1 = T 0 (s 2 - s 1 ) - w rev Constant C p gives: C p (T 2 - T 1 ) = T 0 C p ln (T 2 /T 1 ) + 85 T 2 - T 0 ln(t 2 /T 1 ) = T /C p T 1 = 540 C p = 0.24 T 0 = 540 T ln (T 2 /540) = (85/0.24) = At 1400 R LHS = (too low) At 1420 R LHS = Interpolate to get T 2 = 1414 R (LHS = OK) 10.67EAir enters the turbocharger compressor of an automotive engine at 14.7 lbf/in 2, 90 F, and exits at 25 lbf/in 2, as shown in Fig. P The air is cooled by 90 F in an intercooler before entering the engine. The isentropic efficiency of the compressor is 75%. Determine the temperature of the air entering the engine and the irreversibility of the compression-cooling process. 1 Comp. Air 2 3 Cooler. -W c -Q. c T 3 = T 2-90 F, P 2 = 25 lbf/in 2 η s COMP = 0.75, P 1 = 14.7 lbf/in 2 T 1 = 90 F = 550 R -w s = C P0 (T 2s - T 1 ) = 0.24( ) = T 2s = T 1 ( P 2 P 1 ) k-1 k = 550( )0.286 = R -w = - w s /η s = 21.65/0.75 = = C P0 (T 2 - T 1 ) = 0.24(T 2-550) => T 2 = R T 3 = = R b) Irreversibility from Eq with rev. work from Eq.10.12, (q = 0 at T H ) s 3 - s 1 = 0.24 ln ln = Btu/lbm R i = T(s 3 - s 1 ) - (h 3 - h 1 ) - w = T(s 3 - s 1 ) - C P (T 3 - T 1 ) - C P (T 1 - T 2 ) = 550( ) (-90) = +8.7 Btu/lbm

39 ECalculate the irreversibility for the process described in Problem 6.101, assuming that the heat transfer is with the surroundings at 61 F. I = T o S gen so apply 2nd law out to T o = 61 F m 2 s 2 - m 1 s 1 = m i s i + 1 Q 2 / T o + 1 S 2gen T o S gen = T o ( m 2 s 2 - m 1 s 1 - m i s i ) - 1 Q 2 m 1 = lbm m i = lbm m 2 =9.256 lbm Use from table C.4: C p = 0.24 R = / 778 = T o S gen = I = T o [ m 1 (s 2 - s 1 ) + m i (s 2 - s i )] - 1 Q = 520.7[2.104(C p ln - R ln ) (C pln - R ln )] - ( ) = ( ) = Btu 10.69EA control mass gives out 1000 Btu of energy in the form of a. Electrical work from a battery b. Mechanical work from a spring c. Heat transfer at 700 F Find the change in availability of the control mass for each of the three cases. a) φ = -W el = Btu b) φ = -W spring = Btu c) φ = T 0 T = Q out = H = -537 Btu 10.70EA steady stream of R-22 at ambient temperature, 50 F, and at 110 lbf/in. 2 enters a solar collector. The stream exits at 180 F, 100 lbf/in. 2. Calculate the change in availability of the R-22 between these two states. i R-22 Solar Collector 50 F 180 F 110 lbf/in lbf/in 2 h i = h e = s i = s e = e ψ ie = ψ e - ψ i = (h e - h i ) - T 0 (s e - s i ) = ( ) -510( ) = 2.6 Btu/lbm

40 EA 20-lbm iron disk brake on a car is at 50 F. Suddenly the brake pad hangs up, increasing the brake temperature by friction to 230 F while the car maintains constant speed. Find the change in availability of the disk and the energy depletion of the car s gas tank due to this process alone. Assume that the engine has a thermal efficiency of 35%. All the friction work is turned into internal energy of the disk brake. m(u 2 - u 1 ) = 1 Q 2-1 W 2 1 Q 2 = m Fe C Fe (T 2 - T 1 ) 1 Q 2 = (230-50) = Btu Neglect the work to the surroundings at P 0 m(s 2 - s 1 ) = mc ln(t 2 / T 1 ) = ln = φ = m(u 2 - u 1 ) - T 0 m(s 2 - s 1 ) = = Btu W engine = η th Q gas = 1 Q 2 = Friction work Q gas = 1 Q 2 /η th = 385.2/0.35 = 1100 Btu 10.72ECalculate the availability of the system (aluminum plus gas) at the initial and final states of Problem 8.113, and also the irreversibility. State 1: T 1 = 400 F v 1 = 2/2.862 = P 1 = 300 psi Ideal gas v 2 = v 1 (300 / 220)(537 / 860) = ; v o = = RT o / P o The metal does not change volume so the terms as Eq are added φ 1 = m gas φ gas + m Al φ Al = m gas C v (T 1 - T o ) - m gas T o [C p ln T 1 - R ln P 1 ] + m T o P gas P o (v 1 - v o ) o + m Al [C (T 1 - T o ) - T o C ln (T 1 / T o ) ] Al φ 1 = 2.862[0.156(400-77) - 537(0.201 ln ln ) ( )( ) ] [ ln ] = = Btu φ 2 = 2.862[ 0.156(77-77) (0.201 ln ln ) ( )( )] [ ln ] = = Btu 1 I 2 = φ 1 - φ 2 + (1 - T o /T H ) 1 Q 2-1 W 2 AC + P o m( V 2 - V 1 ) = (-14.29) ( ) 778 = Btu [(S gen = Btu/R T o S gen = so OK ]

41 EConsider the springtime melting of ice in the mountains, which gives cold water running in a river at 34 F while the air temperature is 68 F. What is the availability of the water (SSSF) relative to the temperature of the ambient? ψ = h 1 - h 0 - T 0 (s 1 - s 0 ) flow availability from Eq Approximate both states as saturated liquid ψ = ( ) = Btu/lbm Why is it positive? As the water is brought to 68 F it can be heated with q L from a heat engine using q H from atmosphere T H = T 0 thus giving out work EA geothermal source provides 20 lbm/s of hot water at 80 lbf/in. 2, 300 F flowing into a flash evaporator that separates vapor and liquid at 30 lbf/in. 2. Find the three fluxes of availability (inlet and two outlets) and the irreversibility rate. C.V. Flash evaporator chamber. SSSF with no work or heat transfer. Cont. Eq.: ṁ 1 = ṁ 2 + ṁ 3 ; Energy Eq.: ṁ 1 h 1 = ṁ 2 h 2 + ṁ 3 h 3 Entropy Eq.: ṁ 1 s 1 + Ṡ gen = ṁ 2 s 2 + ṁ 3 s 3 1 Vap. 2 Liq. 3 C.8.1: h o = 45.08, s o = , h 1 = , s 1 = C.8.1: h 2 = , s 2 = , h 3 = 218.9, s 3 = h 1 = xh 2 + (1 - x) h 3 => x = ṁ 2 /ṁ 1 = h 1 - h 3 = h 2 - h 3 ṁ 2 = xṁ 1 = lbm/s ṁ 3 = (1-x)ṁ 1 = lbm/s Ṡ gen = = kw/k Flow availability Eq.10.22: ψ = (h - T o s) - (h o - T o s o ) = h - h o - T o (s - s o ) ψ 1 = ( ) = ψ 2 = ( ) = ψ 3 = ( ) = ṁ 1 ψ 1 = Btu/s ṁ 2 ψ 2 = Btu/s ṁ 3 ψ 3 = Btu/s. I = ṁ1 ψ 1 - ṁ 2 ψ 2 - ṁ 3 ψ 3 = 27.4 Btu/s

42 EA wood bucket (4 lbm) with 20 lbm hot liquid water, both at 180 F, is lowered 1300 ft down into a mineshaft. What is the availability of the bucket and water with respect to the surface ambient at 70 F? v 1 v 0 for both wood and water φ 1 - φ 0 = m wood [u 1 - u 0 -T 0 (s 1 - s 0 )] + m H2 O [u 1 - u 0 -T 0 (s 1 - s 0 )] + m tot g(z 1 -z 0 ) = 4[0.3(180-70) ln 640 ] + 20[ ( )] + 24 (g/g c ) (-1300 / 778) = = Btu 10.76EAn air compressor is used to charge an initially empty 7-ft 3 tank with air up to 750 lbf/in. 2. The air inlet to the compressor is at 14.7 lbf/in. 2, 60 F and the compressor isentropic efficiency is 80%. Find the total compressor work and the change in energy of the air. C.V. Tank + compressor (constant inlet conditions) Continuity: m 2-0 = m in Energy: m 2 u 2 = m in h in - 1 W 2 Entropy: m 2 s 2 = m in s in + 1 S 2 GEN To use isentropic efficiency we must calc. ideal device Reversible compressor: 1 S 2 GEN = 0 s 2 = s in P r2 = P rin (P 2 /P in ) = (750/14.7) = T 2,s = 1541 R u 2,s = Btu/lbm 1 w 2,s = h in - u 2,s = = Actual compressor: 1 w 2,AC = 1 w 2,s /η c = Btu/lbm u 2,AC = h in - 1 w 2,AC = 312 T 2,AC = 1729 R Final state 2 u, P v 2 = RT 2 /P 2 = ft 3 /lbm m 2 = V 2 /v 2 = 8.2 lbm 1 W 2 = m 2 ( 1 W 2,AC ) = Btu m 2 (φ 2 - φ 1 ) = m 2 [u 2 - u 1 + P 0 (v 2 - v 1 ) - T 0 (s 2 - s 1 ) ] = 8.2 [ ( ) ( ln750 )] = = Btu 14.7

43 EA compressor is used to bring saturated water vapor at 150 lbf/in. 2 up to 2500 lbf/in. 2, where the actual exit temperature is 1200 F. Find the irreversibility and the second law efficiency. Inlet state: Table C.8.1 h i = Btu/lbm, s i = Btu/lbm R Actual compressor C.8.2: h e = Btu/lbm, s e = Btu/lbm R Energy Eq. actual compressor: -w c,ac = h e - h i = Btu/lbm Eq.10.14: i = T 0 (s e - s i ) = ( ) = 21.3 Btu/lbm Eq.10.13: w rev = i + w c,ac = = Btu/lbm η II = -w rev /(-w c,ac ) = / = EThe simple steam power plant in Problem 6.91, shown in Fig P6.39 has a turbine with given inlet and exit states. Find the availability at the turbine exit, state 6. Find the second law efficiency for the turbine, neglecting kinetic energy at state 5. Turbine from 6.91 C.8: h 6 = s 6 = h 5 = s 5 = h o = s o = ψ 6 = h 6 - h o - T o ( s 6 - s o ) = Btu/lbm w rev = ψ 5 - ψ 6 = h 5 - h 6 - T o ( s 5 - s 6 ) = Btu/lbm w AC = h 5 - h 6 = η II = w ac / w rev = / = ESteam is supplied in a line at 450 lbf/in. 2, 1200 F. A turbine with an isentropic efficiency of 85% is connected to the line by a valve and it exhausts to the atmosphere at 14.7 lbf/in. 2. If the steam is throttled down to 300 lbf/in. 2 before entering the turbine find the actual turbine specific work. Find the change in availability through the valve and the second law efficiency of the turbine. C.V. Valve: h 2 = h 1 = , s 2 > s 1, h 2, P 2 s 2 = Ideal turbine: s 3 = s 2 h 3s = w T,s = h 2 -h 3s = Actual turbine: w T,ac = η T w T,s = Btu/lbm ψ 2 - ψ 1 = h 2 - h 1 - T 0 (s 2 - s 1 ) = ( ) = Btu/lbm h 3ac = h 2 - w Tac = s 3ac = w rev = ψ 2 - ψ 3 = ( ) = η II = w ac /w rev = / = 0.905

44 EA piston/cylinder arrangement has a load on the piston so it maintains constant pressure. It contains 1 lbm of steam at 80 lbf/in. 2, 50% quality. Heat from a reservoir at 1300 F brings the steam to 1000 F. Find the second-law efficiency for this process. Note that no formula is given for this particular case, so determine a reasonable expression for it. 1: P 1, x 1 v 1 = h 1 = s 1 = : P 2 = P 1,T 2 v 2 = h 2 = s 2 = m(u 2 - u 1 ) = 1 Q 2-1 W 2 = 1 Q 2 - P(V 2 - V 1 ) 1 Q 2 = m(u 2 - u 1 ) + Pm(v 2 - v 1 ) = m(h 2 - h 1 ) = Btu 1 W 2 = Pm(v 2 - v 1 ) = Btu 1 W 2 to atm = P 0 m(v 2 - v 1 ) = 22 Btu Useful work out = 1 W 2-1 W 2 to atm = = Btu φ reservoir = 1 - T 0 T 1 Q 2 = res = 556 Btu n II = W net / φ = EAir flows into a heat engine at ambient conditions 14.7 lbf/in. 2, 540 R, as shown in Fig. P Energy is supplied as 540 Btu per lbm air from a 2700 R source and in some part of the process a heat transfer loss of 135 Btu per lbm air happens at 1350 R. The air leaves the engine at 14.7 lbf/in. 2, 1440 R. Find the first- and the second-law efficiencies. C.V. Engine out to reservoirs h i + q H = q L + h e + w w ac = = η TH = w/q H = 180.7/540 = For second law efficiency also a q to/from ambient s i + (q H /T H ) + (q 0 /T 0 ) = (q loss /T m ) + s e q 0 = T 0 [ s e - s i + (q loss /T m ) - (q H /T H )] = = w rev = h i - h e + q H - q loss + q 0 = w ac + q 0 = η II = w ac /w rev = 180.7/257.7 = 0.70

45 EConsider a gasoline engine for a car as an SSSF device where air and fuel enters at the surrounding conditions 77 F, 14.7 lbf/in. 2 and leaves the engine exhaust manifold at 1800 R, 14.7 lbf/in. 2 as products assumed to be air. The engine cooling system removes 320 Btu/lbm air through the engine to the ambient. For the analysis take the fuel as air where the extra energy of 950 Btu/lbm of air released in the combustion process, is added as heat transfer from a 3240 R reservoir. Find the work out of the engine, the irreversibility per pound-mass of air, and the first- and second-law efficiencies. 1 2 C.V. Total out to reservoirs AIR m a h 1 + Q H = m a h 2 + W + Q out W m a s 1 + Q H/T H + S gen = m a s s + Q out/t 0 Q out Q h H 1 = s T1 = T 0 T H h 2 = s T1 = w ac = W /m a = h 1 - h 2 + q H - q out = = Btu/lbm η TH = w/q H = 308.6/950 = i tot = (T 0 )s gen = T 0 (s 2 - s 1 ) + q out - q H T 0 /T H = ( ) = Btu/lbm 3240 For reversible case have s gen = 0 and q R 0 from T 0, no q out q R 0,in = T 0 (s 2 - s 1 ) - (T 0 /T H )q H = i tot - q out = 5.67 Btu/lbm w rev = h 1 - h 2 + q H + q R 0,in = w ac + i tot = Btu/lbm η II = w ac /w rev = EConsider the two turbines in Problem 9.119, shown in Fig. P9.72. What is the second-law efficiency of the combined system? From the solution to we have s 1 = s 2 = s 5 = m T1 /m tot = m T1 /m tot = w ac = W net /m tot = Btu/lbm C.V. Total system out to T 0 m T1 /m tot h 1 + m T2 /m tot h 2 + q 0 rev = h 5 + w rev w qc + q rev 0 = w rev m T1 /m tot s 1 + m T2 /m tot s 2 + q 0 rev = s 5 q 0 rev = [ ] = Btu/lbm w rev = w ac + q 0 rev = = η II = w ac /w rev = / = 0.911

46 E(Adv.) Refrigerant-22 is flowing in a pipeline at 40 F, 80 lbf/in. 2, with a velocity of 650 ft/s, at a steady flowrate of 0.2 lbm/s. It is desired to decelerate the fluid and increase its pressure by installing a diffuser in the line (a diffuser is basically the opposite of a nozzle in this respect). The R-22 exits the diffuser at 80 F, with a velocity of 160 ft/s. It may be assumed that the diffuser process is SSSF, polytropic, and internally reversible. Determine the diffuser exit pressure and the rate of irreversibility for the process. Note: The exit velocity should be 320 ft/s at 80 F C.V. Diffuser out to T 0, Int. Rev. flow s gen R-22 = 0/ m 1 = m 2, h 1 + (1/2)V q = h 2 + (1/2)V 2 2 s 1 + dq/t + s gen = s 2 = s 1 + q/t 0 + s gen Rev. Process: w 12 = 0/ = - v dp + (V V 2 2 )/2 Polytropic process: LHS = v dp = n n - 1 (P 2 v 2 - P 1 v 1 ) = (V V 2 2 )/2 = (1/2)( )/ = ft lbf P 1 v 1 n = P 2 v 2 n n = ln(p 2 /P 1 )/ln(v 1 /v 2 ) Inlet state: v 1 = , s 1 = , h 1 = Exit state: T 2,V 2,? only P 2 unknown. v 2 = v(p 2 ), n = n(p 2 ) Trial and error on P 2 to give LHS = ft lbf: P 2 = 125 psi v 2 = n = ln(125/80)/ln( / ) = LHS = ( ) 144 = P 2 = 150 psi v 2 = , n = , LHS = 5006 Interpolate to get P 2 = psi. Take 150 psi as OK. P 2 = 150 psi v 2 = , h 2 = s 2 = Find q from energy equation (Assume T 0 = T 1 = 40 F) q = h 2 - h 1 + (V V 1 2 )/2 = (4974.5/778) = Btu/lbm i = I /m = T 0 (s 2 - s 1 ) - q = ( ) = Btu/lbm

47 E(Adv.) Water in a piston/cylinder is at 14.7 lbf/in. 2, 90 F, as shown in Fig. P The cylinder has stops mounted so that V min = 0.36 ft 3 and V max = 18 ft 3. The piston is loaded with a mass and outside P 0, so a pressure inside of 700 lbf/in. 2 will float it. Heat of Btu from a 750 F source is added. Find the total change in availability of the water and the total irreversibility. P 1a CV water plus cyl. wall out to reservoir m 2 = m 1 = m, m(u 2 - u 1 ) = 1 Q 2-1 W 2 m(s 2 - s 1 ) = 1 Q 2 /T res + 1 S 2 gen System eq: States must be on the 3 lines 1 2 1b v Here P eq = 700 lbf/in 2, since P 1 < P eq => V 1 = V min = 0.36 ft 3 1: v 1 = , u 1 = 58.07, m = V 1 /v 1 = lbm Heat added gives q = Q/m = Btu/lbm so check if we go from 1 past 1a (P eq,v 1 ) and past 1b (P eq,v max ) 1a: T 100 F, u 1a q 1a = u 1a - u b: T 600 F, u 1b = 1180 => 1 q 1b = h 1b - u 1 - P eq v 1 = The state is seen to fall between 1a and 1b so P 2 = P eq. 1 w 2 = P eq (v 2 - v 1 ), u 2 - u 1 = 1 q 2-1 w 2 u 2 + P eq v 2 = h 2 = 1 q 2 + u 1 + P eq v 1 = < h g (P 2 ) x 2 = , v 2 = , s 2 = T 2 = 467 F, V 2 = S 2 gen = m(s 2 - s 1 ) - 1 Q 2 /T res = ( ) /1210 = Btu/R 1 I 2 = T 0 ( 1 S 2,gen ) = 3304 Btu The non-flow availability change of the water, Eq φ 2 - φ 1 = u 2 - u 1 + P 0 (v 2 - v 1 ) - T 0 (s 2 - s 1 ) = ( ) ( ) = Btu/lbm m φ = 3669 Btu

48 EA rock bed consists of lbm granite and is at 160 F. A small house with lumped mass of lbm wood and 2000 lbm iron is at 60 F. They are now brought to a uniform final temperature with no external heat transfer. a. For a reversible process, find the final temperature and the work done in the process. b. If they are connected thermally by circulating water between the rock bed and the house, find the final temperature and the irreversibility of the process assuming that surroundings are at 60 F. a) For a reversible process a heat engine is installed between the rockbed and the house. Take C.V. Total m rock (u 2 - u 1 ) + m wood (u 2 - u 1 ) + m Fe (u 2 - u 1 ) = - 1 W 2 m rock (s 2 - s 1 ) + m wood (s 2 - s 1 ) + m Fe (s 2 - s 1 ) = 0/ (mc) rock ln T 2 T 1 + (mc) wood ln T 2 T 1 + (mc) Fe ln T 2 T 1 = 0/ ln (T 2 /619.67) ln (T 2 /519.67) T 2 = R - 1 W 2 = ( ) 1 W 2 = Btu ln (T 2 /519.67) = 0/ + ( )( ) b) No work, no Q irreversible process. (mc) rock (T 2-160) + (mc wood + mc Fe )(T 2-60) = 0/ T 2 = 82 F = R S gen = m i (s 2 - s 1 ) i = 2916 ln ln = Btu/R I 2 = T 0 S gen = = Btu

49 EAir in a piston/cylinder arrangement, shown in Fig. P10.59, is at 30 lbf/in. 2, 540 R with a volume of 20 ft 3. If the piston is at the stops the volume is 40 ft 3 and a pressure of 60 lbf/in. 2 is required. The air is then heated from the initial state to 2700 R by a 3400 R reservoir. Find the total irreversibility in the process assuming surroundings are at 70 F. P stop P 1 P 1 2 W 1 2 m 2 = m 1 = m = P 1 V 1 /RT 1 = /( ) = 3.0 lbm m(u 2 - u 1 ) = 1 Q 2-1 W 2 m(s 2 - s 1 ) = dq/t + 1 S 2 gen V 1 V stop Process: P = P 0 + α(v-v 0 ) if V V stop Information: P stop = P 0 + α(v stop -V 0 ) Eq. of state T stop = T 1 P stop V stop /P 1 V 1 = 2160 < T 2 So the piston hits the stops => V 2 = V stop => P 2 = 2.5 P 1 1 W 2 = 1 2 (P 1 + P stop )(V stop - V 1 ) = 1 ( )(40-20) = Btu 2 1 Q 2 = m(u 2 - u 1 ) + 1 W 2 = 3( ) = Btu s 2 - s 1 = s o T2 - s o T1 -R ln(p 2 /P 1 ) = ln (2.5)/778 = Btu/lbm R Take control volume as total out to reservoir at T RES 1S 2 gen tot = m(s 2 - s 2 ) - 1 Q 2 /T RES = Btu/R 1 I 2 = T 0( 1 S 2 gen ) = = 337 Btu

50 11-1 CHAPTER 11 The correspondence between the new problem set and the previous 4th edition chapter 9 problem set. New Old New Old New Old 1 New New New new a 36 New New 6 4b 37 New New mod New mod New mod New 73 New New mod New mod New mod New New 80 New New New 22 New New mod mod mod mod New New mod 59 New a,b New c,d mod New New New

51 11-2 The problems that are labeled advanced are: New Old New Old New Old New The English unit problems are: New Old New Old New Old 117 New New a b New mod mod mod mod New mod New

52 A steam power plant as shown in Fig operating in a Rankine cycle has saturated vapor at 3.5 MPa leaving the boiler. The turbine exhausts to the condenser operating at 10 kpa. Find the specific work and heat transfer in each of the ideal components and the cycle efficiency. C.V. Pump Rev adiabatic -w P = h 2 - h 1 ; s 2 = s 1 since incompressible it is easier to find work as -w p = v dp = v 1 (P 2 - P 1 ) = ( ) = kj/kg => h 2 = h 1 - w p = = C.V. Boiler : q h = h 3 - h 2 = = kj/kg C.V. Turbine : w t = h 3 - h 4 ; s 4 = s 3 s 4 = s 3 = = x 4 (7.501) => x 4 = 0.73 => h 4 = ( ) = w t = = kj/kg C.V. Condenser : q L = h 4 - h 1 = = kj/kg η cycle = w net / q H = (w t + w p ) / q H = ( ) / = Consider a solar-energy-powered ideal Rankine cycle that uses water as the working fluid. Saturated vapor leaves the solar collector at 175 C, and the condenser pressure is 10 kpa. Determine the thermal efficiency of this cycle. H 2 O ideal Rankine cycle T T 3 = 175 C P 3 = P G 175 C = 892 kpa CV: turbine s 4 = s 3 = = x x 4 = h 4 = = s w T = h 3 - h 4 = = kj/kg -w P = v 1 (P 2 - P 1 ) = (892-10) = 0.89 w NET = w T + w P = = kj/kg h 2 = h 1 - w P = = q H = h 3 - h 2 = = kj/kg η TH = w NET /q H = 674.4/ = 0.261

53 A utility runs a Rankine cycle with a water boiler at 3.5 MPa and the cycle has the highest and lowest temperatures of 450 C and 45 C respectively. Find the plant efficiency and the efficiency of a Carnot cycle with the same temperatures. Solution: 1: 45 o C, x = 0 => h 1 = , v 1 = , P sat = 9.6 kpa 3: 3.5 MPa, 450 o C => h 3 = , s 3 = C.V. Pump Rev adiabatic -w P = h 2 - h 1 ; s 2 = s 1 since incompressible it is easier to find work as -w p = v dp = v 1 (P 2 - P 1 ) = ( ) = => h 2 = h 1 - w p = = C.V. Boiler : q h = h 3 - h 2 = = C.V. Turbine : w t = h 3 - h 4 ; s 4 = s 3 s 4 = s 3 = = x 4 (7.5261) => x 4 = => h 4 = ( ) = w t = = 1123 kj/kg C.V. Condenser : q L = h 4 - h 1 = = kj/kg η cycle = w net / q H = (w t + w p ) / q H = ( ) / = η carnot = 1 - T L / T H = = A steam power plant operating in an ideal Rankine cycle has a high pressure of 5 MPa and a low pressure of 15 kpa. The turbine exhaust state should have a quality of at least 95% and the turbine power generated should be 7.5 MW. Find the necessary boiler exit temperature and the total mass flow rate. C.V. Turbine w T = h 3 - h 4 ; s 4 = s 3 4: 15 kpa, x 4 = 0.95 => s 4 = , h 4 = : s 3 = s 4, P 3 h 3 = , T 3 = 758 C w T = h 3 - h 4 = = ṁ = Ẇ T /w T = / = 4.82 kg/s

54 A supply of geothermal hot water is to be used as the energy source in an ideal Rankine cycle, with R-134a as the cycle working fluid. Saturated vapor R-134a leaves the boiler at a temperature of 85 C, and the condenser temperature is 40 C. Calculate the thermal efficiency of this cycle. CV: Pump (use R-134a Table B.5) 2 -w P = h 2 - h 1 = vdp v 1 (P 2 -P 1 ) 1 = ( ) = 1.67 kj/kg h 2 = h 1 - w P = = kj/kg CV: Boiler q H = h 3 - h 2 = = kj/kg T D 2 1 CV: Turbine s 4 = s 3 = = x => x 4 = h 4 = = Energy Eq.: w T = h 3 - h 4 = = kj/kg w NET = w T + w P = = kj/kg η TH = w NET /q H = 17.29/ = o 85 C o 40 C s 11.6 Do Problem 11.5 with R-22 as the working fluid. Same T-s diagram and analysis as in problem 11.5 with R-22 CV: Pump (use R-22 Table B.4) T 2 -w P = h 2 - h 1 = vdp v 1 (P 2 -P 1 ) 1 = ( ) = 2.21 kj/kg h 2 = h 1 - w P = = CV: Boiler: q H = h 3 - h 2 = = kj/kg CV: Turbine s 4 = s 3 = = x , => x 4 = h 4 = = w T = h 3 - h 4 = = kj/kg η TH = w NET /q H = ( )/ = D o 85 C o 40 C s

55 Do Problem 11.5 with ammonia as the working fluid. Solution: CV: Pump (use Ammonia Table B.2) -w P = h 2 - h 1 = 2 1 vdp = v1 (P 2 -P 1 ) = ( ) = 5.27 kj/kg h 2 = h 1 - w P = = kj/kg CV: Boiler q H = h 3 - h 2 = = kj/kg T D 2 1 CV: Turbine s 4 = s 3 = = x => x 4 = h 4 = = Energy Eq.: w T = h 3 - h 4 = = kj/kg w NET = w T + w P = = kj/kg η TH = w NET /q H = 121.4/ = o 85 C o 40 C s

56 Consider the ammonia Rankine-cycle power plant shown in Fig. P11.8, a plant that was designed to operate in a location where the ocean water temperature is 25 C near the surface and 5 C at some greater depth. a. Determine the turbine power output and the pump power input for the cycle. b. Determine the mass flow rate of water through each heat exchanger. c. What is the thermal efficiency of this power plant? a) Turbine s 2S = s 1 = = x 2S x 2S = h 2S = = w ST = h 1 - h 2S = = w T = η S w ST = = kj/kg Ẇ T = ṁw T = = kw T 4 4s 1 3 2s 2 o 20 C o 10 C s Pump: w SP -v 3 (P 4 - P 3 ) = ( ) = w P = w SP /η S = /0.80 = kj/kg Ẇ P = ṁw P = 1000(-0.484) = -484 kw b) h 2 = h 1 - w T = = Q. to low T H2O = 1000( ) = kw ṁ low T H2O = = kg/s h 4 = h 3 - w P = = Q. from high T H2O = 1000( ) = kw ṁ low T H2O = = kg/s c) η TH = Ẇ NET /Q H = = 0.027

57 Do Problem 11.8 with R-134a as the working fluid in the Rankine cycle. Solution: a) Turbine s 2S = s 1 = = x 2S x 2S = h 2S = = w ST = h 1 - h 2S = = 6.6 kj/kg w T = η S w ST = = 5.28 kj/kg Ẇ T = ṁw T = 5280 kw T 4 4s 1 3 2s 2 o 20 C o 10 C s Pump: w SP -v 3 (P 4 - P 3 ) = ( ) = kj/kg w P = w SP /η S = => Ẇ P = ṁw P = -156 kw b) h 2 = h 1 - w T = = Q. to low T H2O = 1000( ) = kw ṁ low T H2O = = kg/s h 4 = h 3 - w P = = Q. from high T H2O = 1000( ) = kw ṁ low T H2O = = kg/s c) η TH = Ẇ NET /Q H = = 0.026

58 Consider the boiler in Problem 11.5 where the geothermal hot water brings the R- 134a to saturated vapor. Assume a counter flowing heat exchanger arrangement. The geothermal water temperature should be equal to or greater than the R-134a temperature at any location inside the heat exchanger. The point with the smallest temperature difference between the source and the working fluid is called the pinch point. If 2 kg/s of geothermal water is available at 95 C, what is the maximum power output of this cycle for R-134a as the working fluid? (hint: split the heat exchanger C.V. into two so the pinch point with T = 0, T = 85 C appears) 2 kg/s of water is available at 95 o C for the boiler. The restrictive factor is the boiling temperature of 85 C. Therefore, break the process up from 2-3 into two parts as shown in the diagram. liquid R-134a 2 D 3 LIQUID HEATER. -Q BC sat liq at 85 o C BOILER -Q. AB sat. vap R-134a 85 o C liquid H2O out C B liq H2O at 85 o C Write the enrgy equation for the first section A-B and D-3: -Q. AB = ṁ H2O (h A - h B ) = 2( ) = kw = ṁ R134A ( ) ṁ R134A = kg/s A liquid H2O 95 o C To be sure that the boiling temp. is the restrictive factor, calculate T C from the energy equation for the remaining section: -Q. AC = ( ) = kw = 2( h C ) CV Pump: h C = 323.1, T C = 77.2 C > T 2 OK -w P = v 1 (P 2 -P 1 ) = ( ) = 1.67 kj/kg CV: Turbine s 4 = s 3 = = x => x 4 = h 4 = = Energy Eq.: w T = h 3 - h 4 = = kj/kg Cycle: w NET = w T + w P = = kj/kg Ẇ NET = ṁ R134A w NET = = kw

59 Do the previous problem with R-22 as the working fluid. A flow with 2 kg/s of water is available at 95 o C for the boiler. The restrictive factor is the boiling temperature of 85 o C. Therefore, break the process up from 2-3 into two parts as shown in the diagram. liquid R-22 2 D 3 LIQUID HEATER. -Q BC sat liq at 85 o C BOILER -Q. AB sat. vap R o C liquid H2O out C B liq H2O at 85 o C -Q. AB = ṁ H2O (h A - h B ) = 2( ) = kw A liquid H2O 95 o C = ṁ R-22 ( ) ṁ R-22 = kg/s To verify that T D = T 3 is the restrictive factor, find T C. -Q. AC = 0.949( ) = = 2.0( h C ) h C = T C = 77.2 o C OK From Problem 11.6 then: Ẇ net = ( ) = kw The power plant in Problem 11.1 is modified to have a superheater section following the boiler so the steam leaves the super heater at 3.5 MPa, 400 C. Find the specific work and heat transfer in each of the ideal components and the cycle efficiency. C.V. Tubine: Energy: w T,s = h 3 - h 4s ; Entropy: s 4s = s 3 = x 4s = ( )/7.501 = ; h 4s = = kj/kg w T,s = = kj/kg C.V. Pump: -w P = v dp = v 1 (P 2 - P 1 ) = ( ) = kj/kg h 2 = h 1 - w P = = C.V. Condenser: -q C = h 4 - h 1 = = 1975 C.V. Boiler: q H = h 3 - h 2 = = 3027 η CYCLE = w NET /q H = ( )/3027 =

60 A steam power plant has a steam generator exit at 4 MPa, 500 C and a condenser exit temperature of 45 C. Assume all components are ideal and find the cycle efficiency and the specific work and heat transfer in the components. T From the Rankine cycle we have the states: 3 1: 45 C x = 0/, v 1 = , h 1 = : 4 MPa 2 3: 4 MPa, 500 C, h 3 = , s 3 = : P sat (45 C) = s C.V. Turbine: s 4 = s 3 x 4 = ( )/ = , h 4 = = w T = h 3 - h 4 = = 1204 kj/kg C.V. Pump: -w P = v 1 (P 2 - P 1 ) = ( ) = 4.03 kj/kg -w P = h 2 - h 1 h 2 = = kj/kg C.V. Boiler: q H = h 3 - h 2 = = kj/kg C.V. Condenser: q L,out = h 4 - h 1 = = kj/kg η TH = w net /q H = (w T + w P )/q H = ( )/ = Consider an ideal Rankine cycle using water with a high-pressure side of the cycle at a supercritical pressure. Such a cycle has a potential advantage of minimizing local temperature differences between the fluids in the steam generator, such as the instance in which the high-temperature energy source is the hot exhaust gas from a gas-turbine engine. Calculate the thermal efficiency of the cycle if the state entering the turbine is 25 MPa, 500 C, and the condenser pressure is 5 kpa. What is the steam quality at the turbine exit? s 4 = s 3 = = x x 4 = Very low for a turbine exhaust s 1 = , h 1 = h 4 = 1816, h 3 = s 2 = s 1 => h 2 = w NET = h 3 - h 4 - (h 2 - h 1 ) = q H = h 3 - h 2 = , η = w NET /q H = 0.44 T MPa o 500 C 5 kpa 4 s

61 Steam enters the turbine of a power plant at 5 MPa and 400 C, and exhausts to the condenser at 10 kpa. The turbine produces a power output of kw with an isentropic efficiency of 85%. What is the mass flow rate of steam around the cycle and the rate of heat rejection in the condenser? Find the thermal efficiency of the power plant and how does this compare with a Carnot cycle. Solution: Ẇ T = kw and η Ts = 85 % State 1 : T 1 = 400 o C, P 1 = 5 MPa, superheated h 1 = kj/kg, s 1 = kj/kgk State 3 : P 3 = P 2 = 10 kpa, sat liq, x 3 = 0 T 3 = 45.8 o C, h 3 = h f = kj/kg, v 3 = v f = m 3 /kg C.V Turbine : 1st Law: q T + h 1 = h 2 + w T ; q T = 0 w T = h 1 - h 2, Assume Turbine is isentropic s 2s = s 1 = kj/kgk, s 2s = s f + x 2s s fg, solve for x 2s = h 2s = h f + x 2s h fg = kj/kg w Ts = h 1 - h 2s = 1091 kj/kg, w T = η Ts w Ts = kj/kg ṁ = Ẇ T = kg/s, h w 2 = h 1 - w T = kj/kg T C.V. Condenser: 1st Law : q c + h 2 = h 3 + w c ; w c = 0 q c = h 3 - h 2 = kj/kg, Q. c = ṁ q c = kw C.V. Pump: Assume Isentropic w ps = - v dp = -v 3 (P 4 - P 3 ) = kj/kg 1st Law : q p + h 3 = h 4 + w p ; q p = 0 h 4 = h 3 - w p = kj/kg C.V Boiler : 1st Law : q B + h 4 = h 1 + w B ; w B = 0 q B = h 1 - h 4 = kj/kg (or q B = w T + w B - q c ) w net = w T + w P = kj/kg η th = w net / q B = Carnot cycle : T H = T 1 = 400 o C, T L = T 3 = 45.8 o C η th = T H - T L T H = 0.526

62 Consider an ideal steam reheat cycle where steam enters the high-pressure turbine at 3.5 MPa, 400 C, and then expands to 0.8 MPa. It is then reheated to 400 C and expands to 10 kpa in the low-pressure turbine. Calculate the cycle thermal efficiency and the moisture content of the steam leaving the low-pressure turbine. P 3 = 3.5 MPa, T 3 = T 5 = 400 o C P 4 = P 5 = 0.8 MPa, P 6 = 10 kpa From solution 11.12: -w P = kj/kg, h 2 = h 3 = , s 3 = q H1 = h 3 -h 2 = = s 4 = s 3 => h 4 = ; h 5 = q H = q H1 + h 5 - h 4 = = kj/kg T MPa o C kpa s 6 = s 5 = => x 6 = ( )/7.501 = ; h 6 = 2400 w T,tot = h 3 - h 4 + h 5 - h 6 = = kj/kg η CYCLE = ( )/ = s The reheat pressure effect the operating variables and thus turbine performance. Repeat Problem twice, using 0.6 and 1.0 MPa for the reheat pressure. P 3 = 3.5 MPa, T 3 = T 5 = 400 o C P 4 = P 5 = 0.6 MPa, P 6 = 10 kpa From solution 11.12: -w P = kj/kg, h 2 = T MPa o C 4 10 kpa 6 h 3 = , s 3 = q H1 = h 3 -h 2 = = s w T,tot = h 3 - h 4 + h 5 - h 6, q H = q H1 + h 5 - h 4, s 4 = s 3, s 6 = s 5 For P 4 =P 5 = 1 MPa: s 5 = 7.465, state 4 is superheated vapor For P 4 =P 5 = 0.6 MPa: s 5 = , state 4 is superheated vapor x 6 P 4 =P 5 h 4 h 5 h 6 w T q H η CYCLE Notice the very small changes in efficiency.

63 The effect of a number of reheat stages on the ideal steam reheat cycle is to be studied. Repeat Problem using two reheat stages, one stage at 1.2 MPa and the second at 0.2 MPa, instead of the single reheat stage at 0.8 MPa. P 4 = P 5 = 1.2 MPa, P 6 = P 7 = 0.2 MPa From solution to 11.12, -w P = kj/kg, h 2 = : h 3 = , s 3 = : P 4, s 4 = s 3 sup. vap. h 4 = : h 5 = , s 5 = : P 6, s 6 = s 5 sup. vap. h 6 = T 3.5 MPa o C kpa 1 8 s 7: h 7 = , s 7 = : P 8, s 8 = s 7 sup. vap. h 8 = w T = (h 3 - h 4 ) + (h 5 - h 6 ) + (h 7 - h 8 ) = = kj/kg q H = (h 3 - h 2 ) + (h 5 - h 4 ) + (h 7 - h 6 ) = = kj/kg η TH = (w T + w P )/q H = ( )/ = A closed feedwater heater in a regenerative steam power cycle heats 20 kg/s of water from 100 C, 20 MPa to 250 C, 20 MPa. The extraction steam from the turbine enters the heater at 4 MPa, 275 C, and leaves as saturated liquid. What is the required mass flow rate of the extraction steam? From table B.1 B.1.4: 100 C, 20 MPa h 1 = C.V. Feedwater Heater B.1.4: 250 C, 20 MPa h 2 = B.1.3: 4 MPa, 275 C h 3 = B.1.2: 4 MPa, sat. liq. h 4 = Energy Eq.: ṁ 1 h 1 + ṁ 3 h 3 = ṁ 1 h 2 + ṁ 3 h 4 ṁ 3 = ṁ 1 (h 1 -h 2 )/(h 4 -h 3 ) = kg/s

64 An open feedwater heater in a regenerative steam power cycle receives 20 kg/s of water at 100 C, 2 MPa. The extraction steam from the turbine enters the heater at 2 MPa, 275 C, and all the feedwater leaves as saturated liquid. What is the required mass flow rate of the extraction steam? Solution: C.V Feedwater heater 2 ṁ 1 + ṁ 2 = ṁ 3 3 ṁ 1 h 1 + ṁ 2 h 2 = ṁ 3 h 3 = (ṁ 1 + ṁ 2 ) h 3 1 Table B.1.1 and Table B.1.4 at 100 C interpolate to 2 MPa: h 1 = Table B.1.3: h 2 = 2963, Table B.1.2: h 3 = ṁ 2 = ṁ h 3 - h = 20 = kg/s h 2 - h Remark: h 1 was interpolated between sat liq and compressed liquid at 5 MPa both at 100 o C. The saturated liquid value could have been used with very small error A power plant with one closed feedwater heater has a condenser temperature of 45 C, a maximum pressure of 5 MPa, and boiler exit temperature of 900 C. Extraction steam at 1 MPa to the feedwater heater condenses and is pumped up to the 5 MPa feedwater line where all the water goes to the boiler at 200 C. Find the fraction of extraction steam flow and the two specific pump work inputs. s 1 = , h 1 = v 1 = s 4 = , h 4 = T 6 => h 6 = From turbine Pump From condenser Pump 2 4 C.V. Turbine: s 3 = s IN = , T 3 = 573.8, h 3 = C.V. P1: w P1 = -v 1 (P 2 - P 1 ) = h 2 = h 1 - w P1 = C.V. P2: w P2 = -v 4 (P 7 - P 4 ) = h 7 = h 4 - w P2 = C.V. Tot: ṁ 6 = ṁ 1 + ṁ 3, ṁ 6 /ṁ 1 = 1 + x, ṁ 3 /ṁ 1 = x (1 + x)h 6 = xh 3 - xw P2 - w P1 + h 1 = xh 3 - xw P2 + h 2 x = (h 6 -h 2 )/(h 3 + w P2 - h 6 ) = , ṁ 3 /ṁ 6 =

65 A power plant with one open feedwater heater has a condenser temperature of 45 C, a maximum pressure of 5 MPa, and boiler exit temperature of 900 C. Extraction steam at 1 MPa to the feedwater heater is mixed with the feedwater line so the exit is saturated liquid into the second pump. Find the fraction of extraction steam flow and the two specific pump work inputs. Solution: From turbine State out of boiler 5 6 Pump 1 To boiler 1 h 5 = , s 5 = From C.V. Turbine: s 7 = s 6 = s condenser 2 P 6, s 6 => h 6 = , Pump 2 T 6 = 574 o C C.V Pump P1 -w P1 = h 2 - h 1 = v 1 (P 2 - P 1 ) = ( ) = 1.0 kj/kg => h 2 = h 1 - w P1 = = kj/kg C.V. Feedwater heater: Call ṁ 6 / ṁ tot = x Energy Eq.: (1 - x) h 2 + x h 6 = 1 h 3 x = h 3 - h 2 h 6 - h 2 = = C.V Pump P2 -w P2 = h 4 - h 3 = v 3 (P 4 - P 3 ) = ( ) = 4.5 kj/kg

66 A steam power plant operates with a boiler output of 20 kg/s steam at 2 MPa, 600 C. The condenser operates at 50 C dumping energy to a river that has an average temperature of 20 C. There is one open feedwater heater with extraction from the turbine at 600 kpa and its exit is saturated liquid. Find the mass flow rate of the extraction flow. If the river water should not be heated more than 5 C how much water should be pumped from the river to the heat exchanger (condenser)? Solution: The setup is as shown in Fig , Condenser: 1: 50 o C sat liq. v 1 = , h 1 = : 600 kpa s 2 = s 1 3: 600 kpa sat liq. h 3 = h f = : P, T h 5 = s 5 = : 600 kpa s 6 = s 5 => h 6 = CV P1 C.V FWH To river To pump 1 w P1 = -v 1 (P 2 - P 1 ) = ( ) = h 2 = h 1 - w P1 = x h 6 + (1 -x) h 2 = h 3 x = h 3 - h = h 6 - h = Ex turbine From river CV Turbine s 7 = s 6 = s 5 => x 7 = , h 7 = kj/kg q L = h 7 - h 1 = = kj/kg Q. L = (1 - x) ṁq L = = kw = ṁ H2O h H2O = ṁ (20.93) = kw ṁ = 1836 kg/s CV Condenser + Heat Exchanger 0 = ṁ H2O (s 7 - s 1 ) (1 - x) - Q. L T L + Ṡ g Ṡ g = ( ) = kw/k

67 Consider an ideal steam regenerative cycle in which steam enters the turbine at 3.5 MPa, 400 C, and exhausts to the condenser at 10 kpa. Steam is extracted from the turbine at 0.8 MPa for an open feedwater heater. The feedwater leaves the heater as saturated liquid. The appropriate pumps are used for the water leaving the condenser and the feedwater heater. Calculate the thermal efficiency of the cycle and the net work per kilogram of steam. Solution: C.V. Turbine 2nd Law s 7 = s 6 = s 5 = kj/kg K P 6, s 6 => h 6 = 2851 (sup vap) C.V Pump P1 -w P1 = h 2 - h 1 = v 1 (P 2 - P 1 ) = (800-10) = kj/kg => h 2 = h 1 - w P1 = = Call ṁ 6 / ṁ tot = x (1 - x) h 2 + x h 6 = 1 h 3 x = h 3 - h 2 = h 6 - h 2 C.V Pump P2 CV Boiler = w P2 = h 4 - h 3 = v 3 (P 4 - P 3 ) = ( ) = 3.01 h 4 = h 3 - w P2 = = q H = h 5 - w 4 = = CV Turbine Ẇ T / ṁ 5 = h 5 - h 6 + (1 - x) (h 6 - h 7 ) s 7 = s 6 = s 5 = => x 7 = x 7 = => h 7 = x = w T = ( ) ( ) = w net = w T + (1 - x) w P1 + w P2 = ( )(-0.798) = kj/kg η cycle = w net / q H = / = 0.366

68 Repeat Problem 11.24, but assume a closed instead of an open feedwater heater. A single pump is used to pump the water leaving the condenser up to the boiler pressure of 3.5 MPa. Condensate from the feedwater heater is drained through a trap to the condenser. Solution: C.V. Turbine, 2nd law: BOILER s 4 = s 5 = s 6 = kj/kg K 4 3 h 4 = , h 5 = 2851 TURBINE. FW HTR => x 6 = ( )/ = h 6 = x = P Trap 7 Assume feedwater heater exit at the T of the condensing steam C.V Pump -w P = h 2 - h 1 = v 1 (P 2 - P 1 ) = ( ) = h 2 = h 1 - w P = = T 3 = T sat (P 5 ) = , h 3 = h f = h 7 = C.V FWH ṁ 5 / ṁ 3 = x, Energy Eq.: h 2 + x h 5 = h 3 + h 7 x 1 COND. 6 x = h 3 - h = h 5 - h f = w T = h 4 - h 5 + (1 - x)(h 5 - h 6 ) = ( ) = w net = w T + w P = = 883 q H = h 4 - h 3 = = η cycle = w net / q H = 883 / = 0.353

69 A steam power plant has high and low pressures of 25 MPa and 10 kpa, and one open feedwater heater operating at 1 MPa with the exit as saturated liquid. The maximum temperature is 800 C and the turbine has a total power output of 5 MW. Find the fraction of the flow for extraction to the feedwater and the total condenser heat transfer rate. The physical components and the T-s diagram is as shown in Fig in the main text for one open feedwater heater. The same state numbering is used. From the Steam Tables: h 5 = , s 5 = , h 1 = , v 1 = h 3 = v 3 = Pump P1: -w P1 = -v 1 (P 2 - P 1 ) = = 1 kj/kg h 2 = h 1 - w P1 = kj/kg Turbine 5-6: s 6 = s 5 h 6 = 2948 kj/kg w T56 = h 5 - h 6 = kj/kg Feedwater Heater (ṁ TOT = ṁ 5 ): xṁ 5 h 6 + (1 - x)ṁ 5 h 2 = ṁ 5 h 3 x = h 3 - h = h 6 - h = To get state 7 into condenser consider turbine. s 7 = s 6 = s 5 x 7 = ( )/ = h 7 = = kj/kg Find specific turbine work to get total flow rate Ẇ T = ṁ TOT h 5 - xṁ TOT h 6 - (1 - x)ṁ TOT h 7 = ṁ TOT ṁ TOT = 5000/ = 2.95 kg/s Q. L = ṁ TOT (1-x) (h 7 -h 1 ) = ( ) = 4691 kw

70 Do Problem with a closed feedwater heater instead of an open and a drip pump to add the extraction flow to the feed water line at 25 MPa. Assume the temperature is 175 C after the drip pump flow is added to the line. One main pump brings the water to 25 MPa from the condenser. Solution: v 1 = , h 1 = T 4 = 175 o C ; h 4 = h 6 = , s 6 = h 6a = h f 1MPa = , v 6a = From turbine 6 4 6b Pump 2 3 6a From condenser 2 1 Pump 1 C.V Pump 1 -w P1 = h 2 - h 1 = v 1 (P 2 - P 1 ) = ( ) = kj/kg => h 2 = h 1 - w P1 = = C.V Pump 2 -w P2 = h 6b - h 6a = v 6a (P 6b - P 6a ) = ( ) = kj/kg Turbine section 1: s 6 = s 5 P 6 = 1 MPa => h 6 = 2948 kj/kg C.V FWH + P2 ṁ 6 / ṁ 4 = x Turbine: x h 6 + (1 - x) h 2 + x (-w P2 )= h 4 x = h 4 - h = h 6 - h 2 - w P = 0.19 s 7 = s 6 = s 5 & P 7 = 10 kpa => x 7 = = Ẇ T = ṁ 5 [ h 5 - h 6 + (1 - x) (h 6 - h 7 ) ] = ṁ 5 [ ( )] = ṁ Ẇ T = 5000 kw = ṁ => ṁ 5 = kg/s Q. L = ṁ 5 (1 - x) (h 7 - h 1 ) = ( ) = 4755 kw

71 Consider an ideal combined reheat and regenerative cycle in which steam enters the high-pressure turbine at 3.5 MPa, 400 C, and is extracted to an open feedwater heater at 0.8 MPa with exit as saturated liquid. The remainder of the steam is reheated to 400 C at this pressure, 0.8 MPa, and is fed to the lowpressure turbine. The condenser pressure is 10 kpa. Calculate the thermal efficiency of the cycle and the net work per kilogram of steam. 5: h 5 = , s 5 = : h 7 = , s 7 = : h 3 = h f = T1 T2 C.V. T1 s 5 = s 6 => h 6 = x x 6 8 w T1 = h 5 - h 6 = = C.V. Pump 1 -w P1 = h 2 - h 1 = v 1 (P 2 - P 1 ) P 4 HTR 3 1-x 2 P COND. 1 = (800-10) = => h 2 = h 1 - w P1 = = C.V. FWH T o 400 C 5 7 x h 6 + (1 - x) h 2 = h 3 x = h 3 - h = h 6 - h = C.V. Pump w P2 = h 4 - h 3 = v 3 (P 4 - P 3 ) = ( ) = 3.01 => h 4 = h 3 - w P2 = = q H = h 5 - h 4 + (1 - x)(h 7 - h 6 ) = = C.V. Turbine 2 s 7 = s 8 => x 8 = ( )/7.501 = h 8 = h f + x 8 h fg = = w T2 = h 7 - h 8 = = w net = w T1 + (1 - x) w T2 + (1 - x) w P1 + w P2 = = kj/kg η cycle = w net / q H = / = kpa s

72 An ideal steam power plant is designed to operate on the combined reheat and regenerative cycle and to produce a net power output of 10 MW. Steam enters the high-pressure turbine at 8 MPa, 550 C, and is expanded to 0.6 MPa, at which pressure some of the steam is fed to an open feedwater heater, and the remainder is reheated to 550 C. The reheated steam is then expanded in the low-pressure turbine to 10 kpa. Determine the steam flow rate to the high-pressure turbine and the power required to drive each of the pumps. a) 7 T o C 5 7 HI P LOW P T1 T kpa a 1 8 s HTR COND P P b) -w P12 = (600-10) = 0.6 kj/kg h 2 = h 1 - w P12 = = w P34 = ( ) = 8.1 kj/kg h 4 = h 3 - w P34 = = ; h 5 = , s 6 = s 5 = T 6 = o C h 6 = , h 7 = , s 8 = s 7 = = x x 8 = h 8 = = CV: heater Cont: m 6a + m 2 = m 3 = 1 kg, 1st law: m 6a h 6 + m 2 h 2 = m 3 h m 6a = = , m 2 = m 7 = 1 - m 6a = CV: turbine w T = (h 5 - h 6 ) + (1 - m 6a )(h 7 - h 8 ) = ( ) = CV: pumps w P = m 2 w P12 + m 4 w P34 = (-0.6) + 1 (-8.1) = -8.6 kj/kg w N = = kj/kg (m 5 ) ṁ 5 = Ẇ N /w N = 10000/ = 6.53 kg/s

73 The low pressure turbine in a reheat and regenerative cycle receives 10 kg/s steam at 600 kpa, 550 C. The turbine exhausts to a condenser operating at 10 kpa. The condenser cooling water temperature is restricted to a maximum of 10 C increase so what is the needed flow rate of the cooling water? The steam velocity in the turbine-condenser connecting pipe is restricted to a maximum of 100 m/s, what is the diameter of the connecting pipe? Solution: a) 7 T o C 7 5 HI P LOW P 4 T1 T kpa a COND. T 1 8 s HTR 4 V 8 = 100 m/s, T COOL H2 O = 10 o 2 C 3 1 P P h 7 = , s 7 = CV T2 => s 8 = s 7 = = x => x 8 = v 8 = v f + x 8 v fg = x = h 8 = h f + x 8 h fg = x = CV CONDENSER: Q. L = ṁ(h 8 - h 1 ) = 10( ) = kw h H2 O = h 30 - h 20 = = (=C P liq ( T) = ) ṁ H2 O = Q. L / h H 2 O = / = 571 kg/s Turbine - condenser connecting pipe: ṁ = ρav = AV/v => A = ṁv/v = /100 = m 2 D = (4A/π) 1/2 = 1.37 m

74 A steam power plant has a high pressure of 5 MPa and maintains 50 C in the condenser. The boiler exit temperature is 600 C. All the components are ideal except the turbine which has an actual exit state of saturated vapor at 50 C. Find the cycle efficiency with the actual turbine and the turbine isentropic efficiency. Simple Rankine cycle. Boiler exit: h 3 = , s 3 = Ideal Turbine: 4s: 50 C, s = s 3 => x = ( )/ = , h 4s = = => w Ts = h 3 - h 4s = Condenser exit: h 1 = , Actual turbine exit: h 4ac = h g = Actual turbine: w Tac = h 3 - h 4ac = η T = w Tac / w Ts = 0.803: Isentropic Efficiency Pump: -w P = v 1 ( P 2 - P 1 ) = ( ) = 5.05 kj/kg h 2 = h 1 - w P = = kj/kg q H = h 3 - h 2 = = kj/kg η cycle = (w Tac + w P ) / q H = 0.31: Cycle Efficiency A steam power cycle has a high pressure of 3.5 MPa and a condenser exit temperature of 45 C. The turbine efficiency is 85%, and other cycle components are ideal. If the boiler superheats to 800 C, find the cycle thermal efficiency. Basic Rankine cycle as shown in Figure 11.3 in the main text. C.V. Turb.: w T = h 3 - h 4, s 4 = s 3 + s T,GEN Ideal: s 4 = s 3 = => h 4S = , w T,S = Actual: w T,AC = η w T,S = C.V. Pump: -w P = v dp v 1 (P 2 - P 1 ) = 3.52 kj/kg C.V. Boiler: q H = h 3 - h 2 = h 3 - h 1 + w P = η = (w T,AC + w P )/q H = ( )/ = 0.352

75 A steam power plant operates with with a high pressure of 5 MPa and has a boiler exit temperature of of 600 C receiving heat from a 700 C source. The ambient at 20 C provides cooling for the condenser so it can maintain 45 C inside. All the components are ideal except for the turbine which has an exit state with a quality of 97%. Find the work and heat transfer in all components per kg water and the turbine isentropic efficiency. Find the rate of entropy generation per kg water in the boiler/heat source setup. Take CV around each component (all SSSF) in standard Rankine Cycle. 1: v = ; h = , s = (saturated liquid at 45 C). 3: h = , s = superheated vapor 4 ac : h = = kj/kg CV Turbine: no heat transfer q = 0 w ac = h 3 - h 4ac = = kj/kg Ideal turbine: s 4 = s 3 = => x 4s = 0.88, h 4s = 2295 w s = h 3 - h 4s = = , Eff = w ac / w s = / = CV Condenser: no shaft work w = 0 q Out = h 4ac - h 1 = = 2323 kj/kg CV Pump: no heat transfer, q = 0 incompressible flow so v = constant -w = v(p 2 - P 1 ) = ( ) = 5.04 kj/kg CV Boiler: no shaft work, w = 0 q H = h 3 - h 2 = h 3 - h 1 + w P = = 3473 kj/kg s 2 + (q H / T H ) + s Gen = s 3 and s 2 = s 1 (from pump analysis) s Gen = /( ) = 3.05 kj/kg K

76 Repeat Problem assuming the turbine has an isentropic efficiency of 85%. The physical components and the T-s diagram is as shown in Fig in the main text for one open feedwater heater. The same state numbering is used. From the Steam Tables: h 5 = , s 5 = , h 1 = , v 1 = h 3 = 762.8, v 3 = Pump P1: -w P1 = -v 1 (P 2 - P 1 ) = = 1 kj/kg h 2 = h 1 - w P1 = kj/kg Turbine 5-6: s 6 = s 5 h 6 = 2948 kj/kg w T56,s = h 5 - h 6 = kj/kg w T56,AC = = w T56,AC = h 5 - h 6AC h 6AC = h 5 - w T56,AC = = Feedwater Heater (ṁ TOT = ṁ 5 ): xṁ 5 h 6AC + (1 - x)ṁ 5 h 2 = ṁ 5 h 3 x = h 3 - h = h 6 - h = To get the turbine work apply the efficiency to the whole turbine. (i.e. the first section should be slightly different). s 7s = s 6s = s 5 x 7s = , h 7s = w T57,s = h 5 - h 7s = = w T57,AC = w T57,s η T = = h 5 - h 7AC => h 7AC = Find specific turbine work to get total flow rate ṁ TOT = Ẇ T 5000 = = kg/s xw T56 + (1-x)w T Q. L = ṁ TOT (1 - x)(h 7 - h 1 ) = ( ) = 6343 kw

77 Steam leaves a power plant steam generator at 3.5 MPa, 400 C, and enters the turbine at 3.4 MPa, 375 C. The isentropic turbine efficiency is 88%, and the turbine exhaust pressure is 10 kpa. Condensate leaves the condenser and enters the pump at 35 C, 10 kpa. The isentropic pump efficiency is 80%, and the discharge pressure is 3.7 MPa. The feedwater enters the steam generator at 3.6 MPa, 30 C. Calculate the the thermal efficiency of the cycle and the entropy generation for the process in the line between the steam generator exit and the turbine inlet, assuming an ambient temperature of 25 C. 1 ST. GEN TURBINE. η = 0.88 P 4 3 COND. T 5 5S MPa MPa o 400 C o 375 C 2 10 kpa 3S 3 s 1: h 1 = , s 1 = , 2: h 2 = , s 2 = s: s 3S = s 2 x 3S = , h 3S = w T,S = h 2 - h 3S = = w T,AC = ηw T,S = kj/kg, 3ac: h 3 = h 2 - w T,AC = w P,S = v f (P 5 - P 4 ) = ( ) = 3.7 kj/kg -w P,AC = -w P,S /η P = 4.6 kj/kg q H = h 1 - h 6 = = kj/kg η = w NET /q H = ( )/ = C.V. Line from 1 to 2: w = /0, Energy Eq.: q = h 2 - h 1 = = kj/kg Entropy Eq.: s 1 + s gen + q/t 0 = s 2 => s gen = s 2 - s 1 -q/t 0 = (-56.6/298.15) = kj/kg K

78 For the steam power plant described in Problem 11.1, assume the isentropic efficiencies of the turbine and pump are 85% and 80%, respectively. Find the component specific work and heat transfers and the cycle efficiency. CV Pump, Rev & Adiabatic: -w Ps = h 2s - h 1 = v 1 (P 2 - P 1 ) = ( ) = ; s 2s = s 1 -w Pac = -w Ps / η P = 3.525/0.8 = = h 2a - h 1 h 2a = -w Pac + h 1 = = kj/kg CV Boiler: CV Turbine: q H = h 3 - h 2a = = kj/kg w Ts = = kj/kg w Tac = w Ts η T = = h 3 - h 4a h 4a = h 3 - w Tac = = CV Condensor: q L = h 4a - h 1 = = η cycle = (w Tac + w Pac ) / q H = ( ) / = 0.28 This compares to 0.33 for the ideal case A small steam power plant has a boiler exit of 3 MPa, 400 C while it maintains 50 kpa in the condenser. All the components are ideal except the turbine which has an isentropic efficiency of 80% and it should deliver a shaft power of 9.0 MW to an electric generator. Find the specific turbine work, the needed flow rate of steam and the cycle efficiency. CV Turbine (Ideal): s 4s = s 3 = , x 4s = ( )/ = h 4s = , h 3 = => w Ts = h 3 - h 4s = CV Turbine (Actual): w Tac = η T w Ts = = h 3 - h 4ac, => h 4ac = 2572 ṁ = Ẇ / w Tac = 9000/ = kg/s C.V. Pump: -w P = h 2 - h 1 = v 1 (P 2 - P 1 ) = ( ) = 3.04 kj/kg => h 2 = h 1 -w P = = kj/kg C.V. Boiler: q H = h 3 - h 2 = = kj/kg η cycle = (w Tac + w P ) / q H = ( ) / = 0.227

79 In a particular reheat-cycle power plant, steam enters the high-pressure turbine at 5 MPa, 450 C and expands to 0.5 MPa, after which it is reheated to 450 C. The steam is then expanded through the low-pressure turbine to 7.5 kpa. Liquid water leaves the condenser at 30 C, is pumped to 5 MPa, and then returned to the steam generator. Each turbine is adiabatic with an isentropic efficiency of 87% and the pump efficiency is 82%. If the total power output of the turbines is 10 MW, determine the mass flow rate of steam, the pump power input and the thermal efficiency of the power plant. T 2 2S 1 5 MPa 0.5 MPa 3 o 450 C kpa 4S 4 6S 6 s P HP η η SP = 0.82 ST1 = η 5 LP TURBINE. ST2 = COND. a) s 4S = s 3 = = x 4S => x 4S = h 4S = = w T1,S = h 3 - h 4S = = kj/kg w T1 = η T1,S w T1,S = = kj/kg h 4ac = = s 6S = s 5 = = x 6S x 6S = h 6S = = kj/kg w T2,S = h 5 - h 6S = = w T2 = = w T1+T2 = = 1279 kj/kg ṁ = Ẇ T /w T1+T2 = 10000/1279 = 7.82 kg/s b) -w P,S = ( )( ) = 5.01 kj/kg -w P = -w SP /η SP = 5.01/0.82 = 6.11 kj/kg Ẇ P = w P ṁ = = kw c) q H = (h 3 - h 2 ) + (h 5 - h 4 ) = ( ) = kj/kg w N = = kj/kg η TH = w N /q H = / = 0.34

80 A supercritical steam power plant has a high pressure of 30 MPa and an exit condenser temperature of 50 C. The maximum temperature in the boiler is 1000 C and the turbine exhaust is saturated vapor There is one open feedwater heater receiving extraction from the turbine at 1MPa, and its exit is saturated liquid flowing to pump 2. The isentropic efficiency for the first section and the overall turbine are both 88.5%. Find the ratio of the extraction mass flow to total flow into turbine. What is the boiler inlet temperature with and without the feedwater heater? Basically a Rankine Cycle 1: 50 C, kpa, h = , s = : 30 MPa 3: 30 MPa, 1000 C, h = , s = AC: 50 C, x = 1, h = a) C.V. Turbine Ideal: s 4S = s 3 x 4S = , T 2b 2 1a h 4S = => w T,S = h 3 - h 4S = b 3 4s 4ac 30 MPa 1000 C 1 MPa 3b 3a 50 C s Actual: w T,AC = h 3 -h 4AC = , η = w T,AC /w T,S = b) P2 1b 2b m 1b: Sat liq C, h = tot 3a: 1 MPa, s = s 3 -> h 3a = , 3b T 3a = > w T1s = b: 1 MPa, w m T1ac = ηw T1s = w T1ac = h 3 -h 3b => h 3b = a: w P1 = -v 1 (P 1a -P 1 ) -1 1a P1 1 h 1a = h 1 - w P1 = C.V. Feedwater Heater: ṁ TOT h 1b = ṁ 1 h 3b + (ṁ TOT - ṁ 1 )h 1a ṁ 1 /ṁ TOT = x = (h 1b - h 1a )/(h 3b - h 1a ) = c) C.V. Turbine: (ṁ TOT ) 3 = (ṁ 1 ) 3b + (ṁ TOT - ṁ 1 ) 4AC W _ T = ṁ TOT h 3 - ṁ 1 h 3b - (ṁ TOT - ṁ 1 )h 4AC = 25 MW = ṁ TOT w T w T = h 3 -xh 3b - (1-x)h 4AC = => ṁ TOT = kg/s d) C.V. No FWH, Pump Ideal: -w P = h 2S - h 1, s 2S = s 1 Steam table h 2S = 240.1, T 2S = 51.2 C 1 FWH, CV: P2. s 2b = s 1b = => T 2b = C

81 In one type of nuclear power plant, heat is transferred in the nuclear reactor to liquid sodium. The liquid sodium is then pumped through a heat exchanger where heat is transferred to boiling water. Saturated vapor steam at 5 MPa exits this heat exchanger and is then superheated to 600 C in an external gas-fired superheater. The steam enters the turbine, which has one (open-type) feedwater extraction at 0.4 MPa. The isentropic turbine efficiency is 87%, and the condenser pressure is 7.5 kpa. Determine the heat transfer in the reactor and in the superheater to produce a net power output of 1 MW. 5 MPa 6 T 6 o 600 C SUP. HT. TURBINE. 0.4 MPa Q s 7.5 kpa REACT. HTR P P COND. -w P12 = ( ) = 0.4 kj/kg h 2 = h 1 - w P12 = = w P34 = ( ) = 5.0 kj/kg h 4 = h 3 - w P34 = = s 8 s Ẇ NET = 1 MW, η ST = 0.87 s 7S = s 6 = , P 7 =0.4 MPa => T 7S = o C, h 7S = h 6 - h 7 = η ST (h 6 - h 7S ) h 7 = 0.87( ) = h 7 = s 8S = s 6 = = x 8S x 8S = h 8S = = h 6 - h 8 = η ST (h 6 - h 8S ) h 8 = 0.87( ) = h 8 = CV: heater cont: m 2 + m 7 = m 3 = 1.0 kg, Energy Eq.: m 2 h 2 + m 7 h 7 = m 3 h 3 m 7 = ( )/( ) = CV: turbine w T = (h 6 - h 7 ) + (1 - m 7 )(h 7 - h 8 ) = ( ) = kj/kg

82 11-33 CV: pumps w P = m 1 w P12 + m 3 w P34 = (-0.4) + 1(-5.0) = -5.3 kj/kg w NET = = => ṁ = 1000/ = kg/s CV: reactor Q. REACT = ṁ(h 5 - h 4 ) = 0.885( ) = 1933 kw CV: superheater Q. SUP = 0.885(h 6 - h 5 ) = 0.885( ) = 746 kw A cogenerating steam power plant, as in Fig , operates with a boiler output of 25 kg/s steam at 7 MPa, 500 C. The condenser operates at 7.5 kpa and the process heat is extracted as 5 kg/s from the turbine at 500 kpa, state 6 and after use is returned as saturated liquid at 100 kpa, state 8. Assume all components are ideal and find the temperature after pump 1, the total turbine output and the total process heat transfer. Pump 1: Inlet state is saturated liquid: h 1 = , v 1 = w P1 = - v dp = -v 1 ( P 2 - P 1 ) = ( ) = kj/kg - w P1 = h 2 - h 1 => h 2 = h 1 - w P1 = , T 5 = 42 C Turbine: h 5 = , s 5 = P 6, s 6 = s 5 => x 6 = , h 6 = P 7, s 7 = s 5 => x 7 = , h 7 = Ẇ T = ṁ 5 ( h 5 - h 6 ) ṁ 5 ( h 6 - h 7 ) = 25 ( ) + 20 ( ) = = MW Q. proc = ṁ 6 (h 6 - h ) = 5( ) = MW 8

83 A 10 kg/s steady supply of saturated-vapor steam at 500 kpa is required for drying a wood pulp slurry in a paper mill. It is decided to supply this steam by cogeneration, that is, the steam supply will be the exhaust from a steam turbine. Water at 20 C, 100 kpa, is pumped to a pressure of 5 MPa and then fed to a steam generator. It may be assumed that the isentropic efficiency of the pump is 75%, and that of the turbine is 85%. What is the steam temperature exiting the steam generator? What is the additional heat transfer rate to the steam generator beyond what would have been required to produce only the desired steam supply? What is the difference in net power? State 3: P 3 = 5 MPa State 4: P 4 = 500 kpa, sat. vap. -> x 4 = 1.0, T 4 = C h 4 = h g = kj/kg, s 4 = s g = kj/kg-k State 1: T 1 = 20 C, P 1 = 100 kpa h 1 = h f = kj/kg, v 1 = v f = m 3 /kg State 2: P 2 = 5 MPa (a) C.V.: Turbine, Assume reversible & adiabatic, s 3 = s 4s s 3 = s 4s = s f + x 4s s fg = x 4s h 4s = h f + x 4s h fg = x 4s η ts = w t = h 3 - h 4 = 0.85 w ts h 3 - h 4s Trial and Error to find T P 3 = 5 MPa Assume T 3 = 400 C -> s 3 = kj/kg-k, h 3 = kj/kg s 4s = s 3 -> x 4s = , h 4s = h f + x 4s h fg = kj/kg h 3 - h 4 = η h 3 - h ts ; T 4s 3 = 400 C (b) With Cogeneration; C.V. Pump w Ps = - vdp = -v 1 ( P 2 - P 1 ) = kj/kg w Pw = w Ps / η Ps = kj/kg 1 st Law: q + h 1 = h 2 + w; q = 0; h 2 = h 1 - w Pw = 90.5 kj/kg C.V. Steam Generator: q w = h 3 - h 2 = kj/kg Without Cogeneration; C.V. Pump: w Ps = -v 1 ( P 4 - P 1 ) = -0.4 kj/kg w Pw/o = w Ps / η Ps = kj/kg; h 2 = h 1 - w Pw/o = 84.5 kj/kg C.V. Steam Generator: q w/o = h 4 - h 2 = kj/kg Additional Heat Transfer: q w - q w/o = kj/kg; Q. extra = 4409 kw Difference in Net Power: w diff = (w t + w Pw ) - w Pw/o, w t = h 3 - h 4 = kj/kg w diff = kj/kg, Ẇ diff = 4409 kw

84 An industrial application has the following steam requirement: one 10-kg/s stream at a pressure of 0.5 MPa and one 5-kg/s stream at 1.4 MPa (both saturated or slightly superheated vapor). It is obtained by cogeneration, whereby a highpressure boiler supplies steam at 10 MPa, 500 C to a turbine. The required amount is withdrawn at 1.4 MPa, and the remainder is expanded in the lowpressure end of the turbine to 0.5 MPa providing the second required steam flow. Assuming both turbine sections have an isentropic efficiency of 85%, determine the following. a. The power output of the turbine and the heat transfer rate in the boiler. b. Compute the rates needed were the steam generated in a low-pressure boiler without cogeneration. Assume that for each, 20 C liquid water is pumped to the required pressure and fed to a boiler. -Ẇ P BOILER Q P H 1 20 o C H 2 O IN η s = 0.85 o 10 MPa, 500 C HP TURB. η s = W HPT LP TURB. { 1.4 MPa 5 kg/s. W LPT STEAM a) high-pressure turbine s 4S = s 3 = T 4S = o C, h 4S = MPa { 10 kg/s STEAM w S HPT = h 3 - h 4S = = w HPT = η S w S HPT = = h 4 = h 3 - w = = T 4 = C, s 4 = low-pressure turbine s 5S = s 4 = = x 5S , x 5S = h 5S = = w S LPT = h 4 - h 5S = = w LPT = η S w S LPT = = kj/kg

85 11-36 h 5 = h 4 - w = = > h G OK Ẇ TURB = = 8438 kw b) -ẇ P = 15[ ( )] = kw h 2 = h 1 - w P = = 94.0 Q. H = ṁ 1 (h 3 - h ) = 15( ) = kw 2 This is to be compared to the amount of heat required to supply 5 kg/s of 1.4 MPa sat. vap. plus 10 kg/s of 0.5 MPa sat. vap. from 20 o C water. 5 kg/s 20 o C 10 kg/s 20 o C P P. Q 56 5 kg/s SAT. VAP. AT 1.4 MPa kg/s. -W P2 SAT. VAP. AT 0.5 MPa h 2 = h 1 - w P = ( ) = Q. 3 = ṁ 1 (h 3 - h ) = 5( ) = kw 2 -Ẇ P1 = = 7 kw h 5 = h 4 - w P2 = ( ) = Q. 6 = ṁ 4 (h 6 - h ) = 10( ) = kw 5 -Ẇ P2 = = 5 kw Total Q. = = kw H

86 In a cogenerating steam power plant the turbine receives steam from a highpressure steam drum and a low-pressure steam drum as shown in Fig. P The condenser is made as two closed heat exchangers used to heat water running in a separate loop for district heating. The high-temperature heater adds 30 MW and the low-temperature heaters adds 31 MW to the district heating water flow. Find the power cogenerated by the turbine and the temperature in the return line to the deaerator. Assume a reversible turbine ṁ 1 h 1 + ṁ 2 h 2 = ṁ 3 h 3 + ṁ 4 h 4 + Ẇ T ṁ 1 s 1 + ṁ 2 s 2 = ṁ TOT s mix h 1 = s 1 = h 2 = s 2 = ṁ TOT = 27 kg/s s MIX = s 3 = s MIX h 3 = x 3 = s 4 = s MIX h 4 = x 4 = Turbine 3 4. W T Ẇ T = = kw = 22 MW District heating line Q. TOT = ṁ(h 95 - h ) = kw 60 OK, this matches close enough C.V. Both heaters: ṁ 3 h 3 + ṁ 4 h 4 - Q. TOT = ṁ TOT h EX = = 27 h EX h EX = 262 h f T EX = 62.5 C

87 A boiler delivers steam at 10 MPa, 550 C to a two-stage turbine as shown in Fig After the first stage, 25% of the steam is extracted at 1.4 MPa for a process application and returned at 1 MPa, 90 C to the feedwater line. The remainder of the steam continues through the low-pressure turbine stage, which exhausts to the condenser at 10 kpa. One pump brings the feedwater to 1 MPa and a second pump brings it to 10 MPa. Assume the first and second stages in the steam turbine have isentropic efficiencies of 85% and 80% and that both pumps are ideal. If the process application requires 5 MW of power, how much power can then be cogenerated by the turbine? h 5 = , s 5 = s: s 6S = s 5 h 6S = w T1,S = h 5 - h 6S = w T1,AC = h 6AC = h 5 - w T1,AC = ac: P 6, h 6AC s 6AC = s: s 7S = s 6AC h 7S = w T2,S = h 6AC - h 7S = P2 Boiler P1 T1 6 Process heat 5 MW 1 T2 7 C w T2,AC = 622 = h 6AC - h 7AC h 7AC = : h 8 = q PROC = h 6AC - h 8 = ṁ 6 = Q. /q PROC = 5000/ = kg/s = 0.25 ṁ TOT ṁ TOT = ṁ 5 = kg/s, ṁ 7 = ṁ 5 - ṁ 6 = kg/s Ẇ T = ṁ 5 h 5 - ṁ 6 h 6AC - ṁ 7 h 7AC = 7424 kw

88 Consider an ideal air-standard Brayton cycle in which the air into the compressor is at 100 kpa, 20 C, and the pressure ratio across the compressor is 12:1. The maximum temperature in the cycle is 1100 C, and the air flow rate is 10 kg/s. Assume constant specific heat for the air, value from Table A.5. Determine the compressor work, the turbine work, and the thermal efficiency of the cycle. P T 2 P 3 3 s s s P P 4 s P 1 v s k-1 P 2 a) T 2 = T 1 k = 293.2(12) = K P 1 P 1 = 100 kpa T 1 = 20 o C P 2 = 12 P 1 T 3 = 1100 o C ṁ = 10 kg/s -w C = - 1 w 2 = C P0 (T 2 - T 1 ) = 1.004( ) = kj/kg T 4 = T 3 k-1 P 4 P 3 k = = K w T = C P0 (T 3 - T 4 ) = 1.004( ) = kj/kg Ẇ C = ṁw C = kw, Ẇ T = ṁw T = 7013 kw q H = C P0 (T 3 - T 2 ) = 1.004( ) = kj/kg η TH = w NET /q H = ( )/779.5 = b) v 4 = RT 4 /P 4 = = m 3 /kg v 2 = RT 2 /P 2 = = mep = w NET = = 221 kpa v 4 - v Too low for a reciprocating machine (compared with corresponding values for Otto and Diesel cycles.)

89 Repeat Problem 11.46, but assume variable specific heat for the air, table A.7. a) From A.7: h 1 = 293.6, P r1 = s 2 = s 1 P r2 = P r1 (P 2 /P 1 ) = = T 2 = 590 K, h 2 = kj/kg -w C = - 1 w 2 = h 2 - h 1 = = kj/kg From A.7: h 3 = , P r3 = s 4 = s 3 P r4 = P r3 (P 4 /P 3 ) = (1/12) = 27.8 T 4 = 735 K, h 4 = kj/kg w T = h 3 - h 4 = = 732 kj/kg Ẇ C = ṁw C = kw, Ẇ T = ṁw T = 7320 kw q H = h 3 - h 2 = = kj/kg w NET = = kj/kg η TH = w NET /q H = 428.3/885.9 = b) v 4 = ( )/100 = m 3 /kg v 2 = ( )/1200 = mep = 428.3/( ) = kpa An ideal regenerator is incorporated into the ideal air-standard Brayton cycle of Problem Find the thermal efficiency of the cycle with this modification. T 2 s x y 3 s 4 Problem ideal regen. From 11.46: w T = 701.3, w C = kj/kg w NET = kj/kg Ideal regen.: T X = T 4 = K 1 q H = h 3 - h X = 1.004( ) s = kj/kg = w T η TH = w NET /q H = 396.5/701.3 = 0.565

90 A Brayton cycle inlet is at 300 K, 100 kpa and the combustion adds 670 kj/kg. The maximum temperature is 1200 K due to material considerations. What is the maximum allowed compression ratio? For this calculate the net work and cycle efficiency assuming variable specific heat for the air, table A.7. Combustion: h 3 = h 2 + q H ; 2 w 3 = 0 and T max = T 3 = 1200 K h 2 = h 3 - q H = = T K; P r2 = ; T 1 = 300 K; P r1 = Ideal Compression: P 2 / P 1 = P r2 / P r1 = Ideal Expansion: P r4 = P r3 / (P 3 / P 4 ) = / = T K, h 4 = linear interpolation w T = h 3 - h 4 = = w C = h 2 - h 1 = = w net = w T + w C = = η = w net / q H = / 670 = A large stationary Brayton cycle gas-turbine power plant delivers a power output of 100 MW to an electric generator. The minimum temperature in the cycle is 300 K, and the maximum temperature is 1600 K. The minimum pressure in the cycle is 100 kpa, and the compressor pressure ratio is 14 to 1. Calculate the power output of the turbine. What fraction of the turbine output is required to drive the compressor? What is the thermal efficiency of the cycle? T 3 T 1 = 300 K, P 2 /P 1 = 14, T 3 = 1600 K s a) Assume const C P0 : s 2 = s 1 k-1 T 2 = T 1 (P 2 /P 1 ) k = 300(14) = K -w C = -w 12 = h 2 - h 1 = C P0 (T 2 - T 1 ) = ( ) = kj/kg k-1 Also, s 4 = s 3 T 4 = T 3 (P 4 /P 3 ) k = 1600 (1/14) = K w T = w 34 = h 3 h 4 = C P0 (T 3 T 4 ) = ( ) = kj/kg w NET = = kj/kg ṁ = Ẇ NET /w NET = /511.7 = kg/s Ẇ T = ṁw T = = MW -w C /w T = 339.5/851.2 = b) q H = C P0 (T 3 - T 2 ) = ( ) = kj/kg η TH = w NET /q H = 511.7/965.7 = s 1 4 s

91 Repeat Problem 11.50, but assume that the compressor has an isentropic efficiency of 85% and the turbine an isentropic efficiency of 88%. Same as problem 11.50, except η SC = 0.85 & η ST = 0.88 P 1 = 100 kpa, T 1 = 300 K, P 2 = 1400 kpa, T 3 = 1600 K a) From solution 11.50: T 2S = K, w SC = kj/kg -w C = -w SC /η SC = 339.5/0.85 = kj/kg = C P0 (T 2 -T 1 ) T 2 = T 1 - w c /C P0 = = K From solution 11.50: T 4S = K, w ST = kj/kg w T = η ST w ST = = kj/kg = C P0 (T 3 -T 4 ) T 4 = T 3 - w T /C P0 = = K w NET = = kj/kg ṁ = Ẇ NET /w NET = /349.7 = kg/s ẇ T = ṁw T = = MW -w C /w T = 399.4/749.1 = b) q H = C P0 (T 3 - T 2 ) = 1.004( ) = kj/kg η TH = w N /q H = 349.7/905.8 = Repeat Problem 11.51, but include a regenerator with 75% efficiency in the cycle. T 2s 2 x x' 3 4s 4 Same as 11.51, but with a regenerator h X - h 2 T X - T 2 T X η REG = 0.75 = h ' = = X - h T 4 - T T X = K 1 s a) Turbine and compressor work not affected by regenerator. b) q H = C P0 (T 3 - T X ) = 1.004( ) = kj/kg η TH = w NET /q H = 349.7/788.2 = 0.444

92 A gas turbine with air as the working fluid has two ideal turbine sections, as shown in Fig. P11.53, the first of which drives the ideal compressor, with the second producing the power output. The compressor input is at 290 K, 100 kpa, and the exit is at 450 kpa. A fraction of flow, x, bypasses the burner and the rest (1 x) goes through the burner where 1200 kj/kg is added by combustion. The two flows then mix before entering the first turbine and continue through the second turbine, with exhaust at 100 kpa. If the mixing should result in a temperature of 1000 K into the first turbine find the fraction x. Find the required pressure and temperature into the second turbine and its specific power output. C.V.Comp.: -w C = h 2 - h 1 ; s 2 = s 1 P r2 = P r1 (P 2 /P 1 ) = (450/100) = , T r2 = 445 h 2 = , -w C = = C.V.Burner: h 3 = h 2 + q H = = kj/kg T 3 = 1509 K C.V.Mixing chamber: (1 - x)h 3 + xh 2 = h MIX = kj/kg x = h 3 - h MIX = h 3 - h = 0.50 Ẇ T1 = Ẇ C,in ẇ T1 = -w C = = h 3 - h 4 h 4 = = T 4 = 861 K P 4 = (P r4 /P rmix )P MIX = (51/91.65) 450 = kpa s 4 = s 5 P r5 = P r4 (P 5 /P 4 ) = 51(100/250.4) = h 5 = T 5 = 676 K w T2 = h 4 - h 5 = = kj/kg

93 The gas-turbine cycle shown in Fig. P11.54 is used as an automotive engine. In the first turbine, the gas expands to pressure P 5, just low enough for this turbine to drive the compressor. The gas is then expanded through the second turbine connected to the drive wheels. The data for the engine are shown in the figure and assume that all processes are ideal. Determine the intermediate pressure P 5, the net specific work output of the engine, and the mass flow rate through the engine. Find also the air temperature entering the burner T 3, and the thermal efficiency of the engine. k-1 P 2 a) s 2 = s 1 T 2 = T 1 k = 300(6) = K P 1 -w C = -w 12 = C P0 (T 2 - T 1 ) = 1.004( ) = kj/kg w T1 = -w C = = C P0 (T 4 - T 5 ) = 1.004( T 5 ) T 5 = K s 5 = s 4 P 5 = P 4 k-1 T 5 T 4 k-1 P 6 k = = 375 kpa b) s 6 = s 5 T 6 =T 5 k = = K P 5 w T2 = C P0 (T 5 - T 6 ) = 1.004( ) = kj/kg ṁ = Ẇ NET /w T2 = 150/442.2 = kg/s c) Ideal regenerator T 3 = T 6 = K q H = C P0 (T 4 - T 3 ) = 1.004( ) = kj/kg η TH = w NET /q H = 442.2/643.8 = 0.687

94 Repeat Problem 11.54, but assume that the compressor has an efficiency of 82%, that both turbines have efficiencies of 87%, and that the regenerator efficiency is 70%. k-1 P 2 a) From solution 11.54: T 2 = T 1 k = 300(6) = K P 1 -w C = -w 12 = C P0 (T 2 - T 1 ) = 1.004( ) = kj/kg -w C = -w SC /η SC = 201.6/0.82 = kj/kg = w T1 = C P0 (T 4 - T 5 ) = 1.004( T 5 ) T 5 = K w ST1 = w T1 /η ST1 = 245.8/0.87 = = C P0 (T 4 - T 5S ) = 1.004( T 5S ) T 5S = K s 5S = s 4 P 5 = P 4 (T 5S /T 4 ) k-1 = 600( )3.5 = kpa b) P 6 = 100 kpa, s 6S = s 5 k-1 P 6 k T 6S = T 5 P k = = 985.2K w ST2 = C P0 (T 5 -T 6S ) = 1.004( ) = kj/kg w T2 = η ST2 w ST2 = = kj/kg = C P0 (T 5 -T 6 ) = 1.004( T 6 ) T 6 = K ṁ = Ẇ NET /w NET = 150/323.2 = kg/s c) w C = = C P0 (T 2 - T 1 ) = 1.004(T 2 300) T 2 = K η REG = h 3 - h 2 T 3 - T 2 T = = h 6 - h 2 T 6 - T = 0.7 T 3 = K q H = C P0 (T 4 - T 3 ) = 1.004( ) = 716 kj/kg η TH = w NET /q H = 323.2/716 = 0.451

95 Repeat the questions in Problem when we assume that friction causes pressure drops in the burner and on both sides of the regenerator. In each case, the pressure drop is estimated to be 2% of the inlet pressure to that component of the system, so P 3 = 588 kpa, P 4 = 0.98 P 3 and P 6 = 102 kpa. P k-1 2 a) From solution 11.54: T 2 = T 1 k = 300(6) = K P 1 -w C = -w 12 = C P0 (T 2 - T 1 ) = 1.004( ) = kj/kg P 3 = = 588 kpa, P 4 = = kpa k s 5 = s 4 P 5 = P 4 (T 5S /T 4 ) k-1 = 576.2( )3.5 = kpa b) P 6 = 100/0.98 = 102 kpa, s 6S = s 5 T 6 = T 5 P k-1 6 P 5 k = = 975.2K w ST2 = C P0 (T 5 -T 6 ) = 1.004( ) = kj/kg ṁ = Ẇ NET /w NET = 150/425.7 = kg/s c) T 3 = T 6 = K q H = C P0 (T 4 - T 3 ) = ( ) = kj/kg η TH = w NET /q H = 425.7/627.3 = 0.678

96 Consider an ideal gas-turbine cycle with two stages of compression and two stages of expansion. The pressure ratio across each compressor stage and each turbine stage is 8 to 1. The pressure at the entrance to the first compressor is 100 kpa, the temperature entering each compressor is 20 C, and the temperature entering each turbine is 1100 C. An ideal regenerator is also incorporated into the cycle. Determine the compressor work, the turbine work, and the thermal efficiency of the cycle. 10 REG 1 I.C CC 6 9 COMP COMP TURB TURB 7 8 CC P 2 /P 1 = P 4 /P 3 = P 6 /P 7 = P 8 /P 9 = 8.0 P 1 = 100 kpa T 1 = T 3 = 20 o C, T 6 = T 8 = 1100 o C Assume const. specific heat s 2 = s 1 and s 4 = s 3 T 4 = T 2 = T 1 P k-1 2 P k = 293.2(8) = K 1 T Total -w C = 2 (-w 12 ) = 2C P0 (T 2 - T 1 ) = ( ) = kj/kg Also s 6 = s 7 and s 8 = s 9 : T 7 = T 9 = T 6 P 7 P k = = K Total w T = 2 w 67 = 2C P0 (T 6 - T 7 ) = ( ) = kj/kg w NET = = kj/kg Ideal regenerator: T 5 = T 9, T 10 = T 4 k-1 q H = (h 6 - h 5 ) + (h 8 - h 7 ) = 2C P0 (T 6 - T 5 ) = ( ) = kj/kg η TH = w NET /q H = 757.4/ = s

97 Repeat Problem 11.57, but assume that each compressor stage and each turbine stage has an isentropic efficiency of 85%. Also assume that the regenerator has an efficiency of 70%. T 1 = T 3 = 20 o C, T 6 = T 8 = 1100 o C T 6 8 P 2 /P 1 = P 4 /P 3 = P 6 /P 7 = P 8 P 9 = From solution 11.57: 5 7 7S 4 9S T 4S = T 2S = K, w SC = kj/kg 4S 2S 2 T 7S = T 9S = K, w ST = kj/kg 3 1 -w C = -w SC /η SC = 478.1/0.85 = kj/kg s -w 12 = -w 34 = 562.5/2 = kj/kg w T = η ST w ST = = kj/kg w 67 = w 89 = /2 = kj/kg T 2 = T 4 = T 1 + (-w 12 /C P0 ) = = K T 7 = T 9 = T 6 - (+w 67 /C P0 ) = = K η REG = h 5 - h 4 T 5 - T 4 = = h 9 - h 4 T 9 - T 4 q H = C P0 (T 6 - T 5 ) + C P0 (T 8 - T 7 ) T = 0.7 T 5 = 767 K = 1.004( ) ( ) = kj/kg w NET = w T + w C = = η TH = w NET /q H = 487.7/ = 0.430

98 A gas turbine cycle has two stages of compression, with an intercooler between the stages. Air enters the first stage at 100 kpa, 300 K. The pressure ratio across each compressor stage is 5 to 1, and each stage has an isentropic efficiency of 82%. Air exits the intercooler at 330 K. The maximum cycle temperature is 1500 K, and the cycle has a single turbine stage with an isentropic efficiency of 86%. The cycle also includes a regenerator with an efficiency of 80%. Calculate the temperature at the exit of each compressor stage, the second-law efficiency of the turbine and the cycle thermal efficiency. State 1: P 1 = 100 kpa, T 1 = 300 K State 7: P 7 = P o = 100 kpa State 3: T 3 = 330 K; State 6: T 6 = 1500 K, P 6 = P 4 P 2 = 5 P 1 = 500 kpa; P 4 = 5 P 3 = 2500 kpa Ideal compression T 2s = T 1 (P 2 /P 1 ) (k-1)/k = K 1 st Law: q + h i = h e + w; q = 0 => w c1 = h 1 - h 2 = C P (T 1 - T 2 ) w c1 s = C P (T 1 - T 2s ) = kj/kg, w c1 = w c1 s / η = T 2 = T 1 - w c1 /C P = K T 4s = T 3 (P 4 /P 3 ) (k-1)/k = K w c2 s = C P (T 3 - T 4s ) = kj/kg; w c2 = kj/kg T 4 = T 3 - w c2 / C P = K Ideal Turbine (reversible and adiabatic) T 7s = T 6 (P 7 /P 6 ) (k-1)/k = K => w Ts = C P (T 6 - T 7s ) = kj/kg 1 st Law Turbine: q + h 6 = h 7 + w; q = 0 w T = h 6 - h 7 = C P (T 6 - T 7 ) = η Ts w Ts = = kj/kg T 7 = T 6 - w T / C P = /1.004 = K s 6 - s 7 = C P ln T 6 - R ln P 6 = kj/kg K T 7 P 7 ψ 6 - ψ 7 = (h 6 - h 7 ) - T o (s 6 - s 7 ) = ( ) = kj/kg w T η 2nd Law = = / = ψ 6 -ψ 7 d) η th = q H / w net ; w net = w T + w c1 + w c2 = kj/kg 1 st Law Combustor: q + h i = h e + w; w = 0 q c = h 6 - h 5 = C P (T 6 - T 5 ) Regenerator: η reg = T 5 - T 4 = 0.8 T 7 - T 4 -> T 5 = K q H = q c = kj/kg; η th = 0.405

99 A two-stage air compressor has an intercooler between the two stages as shown in Fig. P The inlet state is 100 kpa, 290 K, and the final exit pressure is 1.6 MPa. Assume that the constant pressure intercooler cools the air to the inlet temperature, T 3 = T 1. It can be shown, see Problem 9.130, that the optimal pressure, P 2 = (P 1 P 4 ) 1/2, for minimum total compressor work. Find the specific compressor works and the intercooler heat transfer for the optimal P 2. Optimal intercooler pressure P 2 = = 400 kpa 1: h 1 = , P r1 = C.V.: C1 -w C1 = h 2 - h 1, s 2 = s 1 P r2 = P r1 (P 2 /P 1 ) = , T 2 = 430, h 2 = w C1 = = kj/kg C.V.Cooler: T 3 = T 1 h 3 = h 1 q OUT = h 2 - h 3 = h 2 - h 1 = -w C1 = kj/kg C.V.: C2 T 3 = T 1, s 4 = s 3 P r4 = P r3 (P 4 /P 3 ) = P r1 (P 2 /P 1 ) = , T 4 = T 2 Thus we get -w C2 = -w C1 = kj/kg Repeat Problem when the intercooler brings the air to T = 320 K. The 3 corrected formula for the optimal pressure is P 2 =[ P 1 P 4 (T 3 /T 1 ) n/(n-1) ] 1/2 see Problem 9.131, where n is the exponent in the assumed polytropic process. The polytropic process has n = k (isentropic) so n/(n - 1) = 1.4/0.4 = 3.5 P 2 = 400 (320/290) 3.5 = kpa C.V. C1: s 2 = s 1 P r2 = P r1 (P 2 /P 1 ) = (475.2/100) = T 2 = 452 K, h 2 = w C1 = h 2 - h 1 = = kj/kg C.V. Cooler: q OUT = h 2 - h 3 = = kj/kg C.V. C2: s 4 = s 3 P r4 = P r3 (P 4 /P 3 ) = (1600/475.2) = T 4 = T 2 = 452 K, h 4 = w C2 = h 4 - h 3 = = kj/kg

100 Consider an ideal air-standard Ericsson cycle that has an ideal regenerator as shown in Fig. P The high pressure is 1 MPa and the cycle efficiency is 70%. Heat is rejected in the cycle at a temperature of 300 K, and the cycle pressure at the beginning of the isothermal compression process is 100 kpa. Determine the high temperature, the compressor work, and the turbine work per kilogram of air. P T P 2 = P 3 = 1 MPa 2 P 3 3 T 4 T 1 = T 2 = 300 K T P T 1 = 100 kpa P 1 P 4 P 2 q 3 = - 4 q (ideal reg.) 1 2 q T 1 H = 3 q 4 & w T = q H r v s p = P 2 /P 1 = 10 η TH = η CARNOT TH. = 1 - T L /T H = 0.7 T 3 = T 4 = T H = 1000 K q L = -w C = v dp = RT 1 ln P 2 P = ln = w T = q H = - v dp = -RT 3 ln(p 4 /P 3 ) = kj/kg An air-standard Ericsson cycle has an ideal regenerator. Heat is supplied at 1000 C and heat is rejected at 20 C. Pressure at the beginning of the isothermal compression process is 70 kpa. The heat added is 600 kj/kg. Find the compressor work, the turbine work, and the cycle efficiency. See the cycle diagrams in solution to 11.62: T 3 = T 4 = K, P 1 = 70 kpa, T 1 = T 2 = K, q H = 600 kj/kg Ideal regenerator: 2 q 3 = - 4 q 1, w T = q H = 600 kj/kg η TH = η CARNOT = / = w NET = η TH q H = = q L = -w C = = 138.2

101 Consider an ideal air-standard cycle for a gas-turbine, jet propulsion unit, such as that shown in Fig The pressure and temperature entering the compressor are 90 kpa, 290 K. The pressure ratio across the compressor is 14 to 1, and the turbine inlet temperature is 1500 K. When the air leaves the turbine, it enters the nozzle and expands to 90 kpa. Determine the pressure at the nozzle inlet and the velocity of the air leaving the nozzle. 2 3 T 3 1 COMP BURN TURB 4 NOZ P s 4 s 5 P = 90 kpa s C.V. Comp.: s 2 = s 1 P r2 = P r1 (P 2 /P 1 ) = = h 2 = 617.2, T 2 = 609 K -w C = h 2 - h 1 = = kj/kg C.V. Turb.: w T = h 3 - h 4 = -w C h 4 = h 3 + w C = = 1309 P r4 = 209.1, T 4 = 1227 P 4 = P r4 (P 3 /P r3 ) = 209.1(1260/ ) = kpa C.V. Nozzle: s 2 = s 1 P r5 = P r4 (P 5 /P 4 ) = 209.1(90/545.3) = => T 5 = 778 K, h 5 = kj/kg (1/2)V 2 5 = h 4 - h 5 = V 5 = = m/s

102 The turbine in a jet engine receives air at 1250 K, 1.5 MPa. It exhausts to a nozzle at 250 kpa, which in turn exhausts to the atmosphere at 100 kpa. The isentropic efficiency of the turbine is 85% and the nozzle efficiency is 95%. Find the nozzle inlet temperature and the nozzle exit velocity. Assume negligible kinetic energy out of the turbine. C.V. Turb.: h i = , P ri = 226, s e = s i P re = P ri (P e /P i ) = 226(250/1500) = T e = 796, h e = 817.9, w T,s = = w T,AC = w T,s η T = 441 = h e,ac - h i h e,ac = T e,ac = 866 K, P re,ac = C.V. Nozzle: (1/2)V e 2 = h i - h e ; s e = s i P re = P ri (P e /P i ) = 52.21(100/250) = T e,s = 681 K, h e,s = kj/kg (1/2)V 2 e,s = h i - h e,s = = kj/kg 2 (1/2)V e,ac = (1/2)V 2 e,s η NOZ = kj/kg V e,ac = = 620 m/s

103 Repeat Problem 11.64, but assume that the isentropic compressor efficiency is 87%, the isentropic turbine efficiency is 89%, and the isentropic nozzle efficiency is 96%. Same as except η SC = 0.87, η ST = 0.89, η SN = 0.96 Assume const. specific heat P 1 = P 5 = 90 kpa, T 1 = 290 K P 2 /P 1 = 14, T 3 = 1500 K From solution 11.64: T 2S = 609 K, w SC = kj/kg T 2S S s w C = w SC /η SC = 326.8/0.87 = kj/kg w T = w C = = h 3 - h 4 h 4 = = , T 4 = 1185 K w ST = w T /η ST = 375.6/0.89 = kj/kg = h 4s h 4s = , T 4s = 1145, P r4s = s 4S = s 3 P 4 = P 3 P r4s /P r3 = / = 412 kpa s 5S = s 4 P r5 = P r4 P 5 /P 4 = /412 = T 5S = 807, h 5S = 829.8, V 2 5S /2000 = h 4 h 5S = = => V 2 5 /2000 = η SN V 2 /2000 = = kj/kg 5s V 5 = 2000*413.2 = 909 m/s

104 Consider an air standard jet engine cycle operating in a 280K, 100 kpa environment. The compressor requires a shaft power input of 4000 kw. Air enters the turbine state 3 at 1600 K, 2 MPa, at the rate of 9 kg/s, and the isentropic efficiency of the turbine is 85%. Determine the pressure and temperature entering the nozzle at state 4. If the nozzle efficiency is 95%, determine the temperature and velocity exiting the nozzle at state 5. C.V. Shaft: Ẇ T = ṁ(h 3 - h 4 ) = Ẇ C h 3 - h 4 = Ẇ C / ṁ = 4000/9 = h 4 = = w Ta = w C = w Ts = w Ta / η = = h 3 - h 4s h 4s = P r = , T 4s 1163 K P 4 = (P r4 / P r3 ) P 3 = (168.28/ )2000 = 530 kpa 4a: 530 kpa, h = , T 1230 K, P r4 = : 100 kpa P r5s = P r4 (100/530) = 40 h 5s = V 2 5s = h 4a - h = kj/kg 5s 0.5V 2 5a = η(0.5v2 5s ) = V = 957 m/s 5a h 5a = h 4-0.5V 2 5a = = 855 T 5a 830 K

105 A jet aircraft is flying at an altitude of 4900 m, where the ambient pressure is approximately 55 kpa and the ambient temperature is 18 C. The velocity of the aircraft is 280 m/s, the pressure ratio across the compressor is 14:1 and the cycle maximum temperature is 1450 K. Assume the inlet flow goes through a diffuser to zero relative velocity at state 1. Find the temperature and pressure at state 1 and the velocity (relative to the aircraft) of the air leaving the engine at 55 kpa. T 2 1 X P amb s Ambient T X = -18 o C = K, P X = 55 kpa = P 5 also V X = 280 m/s Assume that the air at this state is reversibly decelerated to zero velocity and then enters the comp. at 1. P 2 /P 1 = 14 & T 3 = 1450 K T 1 = T X + P 1 = P X T 1 T X V 2 X = (280) = K k k-1 = = 90.5 kpa k-1 Then, T 2 = T 1 (P 2 /P 1 ) k = 294.3(14) = K -w C = -w 12 = C P0 (T 2 -T 1 ) = 1.004( T 4 ) T 4 = K P 3 = P 2 = = 1267 kpa k P 4 = P 3 (T 4 /T 3 ) k-1 = 1267(1118.3/1450) 3.5 = 510 kpa k-1 T 5 = T 4 (P 5 /P 4 ) k = (55/510) = K V = C P0 (T 4 - T ) = 1.004( ) = kj/kg 5 V 5 = 1028 m/s

106 Air flows into a gasoline engine at 95 kpa, 300 K. The air is then compressed with a volumetric compression ratio of 8 1. In the combustion process 1300 kj/kg of energy is released as the fuel burns. Find the temperature and pressure after combustion. Compression 1 to 2: s 2 = s 1 v r2 = v r1 /8 = /8 = , T 2 = 673 K, u 2 = 491.5, P r2 = 20 P 2 = P r2 (P 1 /P r1 ) = 20(95/1.1146) = 1705 kpa Compression 2 to 3: u 3 = u 2 + q H = = T 3 = 2118 K P 3 = P 2 (T 3 /T 2 ) = 1705(2118/673) = 5366 kpa A gasoline engine has a volumetric compression ratio of 9. The state before compression is 290 K, 90 kpa, and the peak cycle temperature is 1800 K. Find the pressure after expansion, the cycle net work and the cycle efficiency using properties from Table A.7. Use table A.7 and interpolation. Compression 1 to 2: s 2 = s 1 v r2 = v r1 (v 2 /v 1 ) v r2 = /9 = T K, P r , u 2 = P 2 = P 1 (P r2 /P r1 ) = 90(20.784/0.995) = 1880 kpa 1 w 2 = u 1 - u = = kj/kg 2 Combustion 2 to 3: q H = u 3 - u 2 = = kj/kg P 3 = P 2 (T 3 /T 2 ) = 1880(1800/680) = 4976 kpa Expansion 3 to 4: s 4 = s 3 v r4 = v r3 9 = = T 4 = 889 K, P r4 = , u 4 = P 4 = P 3 (P r4 /P r3 ) = 4976(57.773/1051) = kpa 3 w 4 = u 3 - u = = kj/kg 4 w NET = 3 w w 2 = = kj/kg η = w NET /q H = 530.8/ = 0.536

107 To approximate an actual spark-ignition engine consider an air-standard Otto cycle that has a heat addition of 1800 kj/kg of air, a compression ratio of 7, and a pressure and temperature at the beginning of the compression process of 90 kpa, 10 C. Assuming constant specific heat, with the value from Table A.5, determine the maximum pressure and temperature of the cycle, the thermal efficiency of the cycle and the mean effective pressure. P 3 T 3 q H = 1800 kj v 2 s s 4 v 1 2 s 1 v v s 4 v 1 /v 2 = 7 P 1 = 90 kpa T 1 = 10 o C v s a) P 2 = P 1 (v 1 /v 2 ) k = 90(7) 1.4 = 1372 kpa T 2 = T 1 (P 2 /P 1 )(v 2 /v 1 )= = K T 3 = T 2 + q H /C V0 = /0.717 = 3127 K P 3 = P 2 T 3 /T 2 = / = 6958 kpa b) η TH = 1 - T 1 /T 2 = /616.5 = c) w NET = η TH q H = = kj/kg v 1 = RT 1 /P 1 = ( )/90 = v 2 = (1/7) v 1 = m 3 /kg mep = w NET = = 1265 kpa v 1 -v

108 Repeat Problem 11.71, but assume variable specific heat. The ideal gas air tables, Table A.7, are recommended for this calculation (and the specific heat from Fig at high temperature). Table A.7 is used with interpolation. a) T 1 = K, u 1 = 202.3, v r1 = s 2 = s 1 v r2 = v r1 (v 2 /v 1 ) = (1/7) = T 2 = K, u 2 = => -w 12 = u 2 - u 1 = 237.9, u 3 = = => T 3 = K, v r3 = P 3 = / = 5730 kpa b) v r4 = v r3 7 = => T 4 = 1437 K; u 4 = w 4 = u 3 - u = = Ë w net = = kj/kg η TH = w net / q H = / 1800 = (c) mep = / ( ) = 1105 kpa A gasoline engine takes air in at 290 K, 90 kpa and then compresses it. The combustion adds 1000 kj/kg to the air after which the temperature is 2050 K. Use the cold air properties (i.e. constant heat capacities at 300 K) and find the compression ratio, the compression specific work and the highest pressure in the cycle. Standard Otto Cycle T 3 = 2050 K u 2 = u 3 - q H T 2 = T 3 - q H / C vo = / = K P 2 = P 1 (T 2 / T 1 ) k/(k-1) = 90(655.3/290) 3.5 = 1561 kpa CR = v 1 / v 2 = (T 2 / T 1 ) 1/(k-1) = (655.3 / 290) 2.5 = w 2 = u 2 - u 1 = C vo ( T 2 - T 1 ) = 0.717( ) = 262 kj / kg P 3 = P 2 T 3 / T 2 = / = 4883 kpa

109 Answer the same three questions for the previous problem, but use variable heat capacities (use table A.7). T 3 = 2050 K u 3 = u 2 = u 3 - q H = = T 2 = v r2 = v r1 = => v 1 / v 2 = v r1 / v r2 = / = w 2 = u 2 - u 1 = = kj/kg P 3 =P 2 T 3 / T 2 = P 1 ( T 3 / T 1 )( v 1 / v 3 ) = 90 (2050 / 290) = kpa When methanol produced from coal is considered as an alternative fuel to gasoline for automotive engines, it is recognized that the engine can be designed with a higher compression ratio, say 10 instead of 7, but that the energy release with combustion for a stoichiometric mixture with air is slightly smaller, about 1700 kj/kg. Repeat Problem using these values. same as except v 1 /v 2 = 10 & q H = 1700 kj a) P 2 = P 1 (v 1 /v 2 ) k = 90(10) 1.4 = kpa T 2 = T 1 (v 1 /v 2 ) k-1 = (10) 0.4 = K T 3 = T 2 + q H / C vo = / = 3084 K P 3 = P 2 (T 3 / T 2 ) = / = 9803 kpa η TH = 1 - T 1 /T 2 = /711.2 = w net = η TH q H = = kj/kg v 1 = RT 1 /P 1 = /90= , v 2 = v 1 /10= mep = w NET /(v 1 - v 2 )= / ( ) = 1255 kpa

110 It is found experimentally that the power stroke expansion in an internal combustion engine can be approximated with a polytropic process with a value of the polytropic exponent n somewhat larger than the specific heat ratio k. Repeat Problem but assume that the expansion process is reversible and polytropic (instead of the isentropic expansion in the Otto cycle) with n equal to See solution to except for process 3 to 4. T 3 = 3127 K, P 3 = MPa v 3 = RT 3 /P 3 = v 2 = m 3 /kg, v 4 = v 1 = Process: Pv 1.5 = constant. P 4 = P 3 (v 3 /v 4 ) 1.5 = 6958 (1/7) 1.5 = kpa T 4 = T 3 (v 3 /v 4 ) 0.5 = 3127(1/7) 0.5 = k 1 w 2 = R Pdv = (T 2 - T 1 ) = ( )= w 4 = Pdv = R(T 4 - T 3 )/(1-1.5) = ( )/0.5 = w NET = = η CYCLE = w NET /q H = 877.2/1800 = mep = w NET /(v 1 - v 2 ) = 877.2/( ) = 1133 kpa Note a smaller w NET, η CYCLE, mep compared to an ideal cycle.

111 In the Otto cycle all the heat transfer q H occurs at constant volume. It is more realistic to assume that part of q H occurs after the piston has started its downward motion in the expansion stroke. Therefore, consider a cycle identical to the Otto cycle, except that the first two-thirds of the total q H occurs at constant volume and the last one-third occurs at constant pressure. Assume that the total q H is 2100 kj/kg, that the state at the beginning of the compression process is 90 kpa, 20 C, and that the compression ratio is 9. Calculate the maximum pressure and temperature and the thermal efficiency of this cycle. Compare the results with those of a conventional Otto cycle having the same given variables. P s s 5 1 T 2 s 1 v 3 v 4 s 5 P 1 = 90 kpa, T 1 = 20 o C r V = v 1 /v 2 = 7 a) q 23 = (2/3) 2100 = 1400 kj/kg; v b) P 2 = P 1 (v 1 /v 2 ) k = 90(9) 1.4 = 1951 kpa T 2 = T 1 (v 1 /v 2 ) k-1 = (9) 0.4 = 706 K s q 34 = 2100/3 = 700 T 3 = T 2 + q 23 /C V0 = /0.717 = 2660 K P 3 = P 2 T 3 /T 2 = 1951(2660/706) = kpa = P 4 T 4 = T 3 + q 34 /C P0 = /1.004 = 3357 K v 5 v 1 P 4 = = v 4 v 4 P 1 T 1 = T = T 5 = T 4 (v 4 /v 5 ) k-1 = 3357(1/7.131) 0.4 = 1530 K q L = C V0 (T 5 -T 1 ) = 0.717( ) = kj/kg η TH = 1 - q L /q H = /2100 = Std. Otto Cycle: η TH = 1 - (9) -0.4 = 0.585, small difference

112 A diesel engine has a compression ratio of 20:1 with an inlet of 95 kpa, 290 K, state 1, with volume 0.5 L. The maximum cycle temperature is 1800 K. Find the maximum pressure, the net specific work and the thermal efficiency. P 2 = 95 (20) 1.4 = kpa T 2 = T 1 (v 1 / v 2 ) k-1 = = 961 K P 3 = P 2 (T 3 / T 2 ) = (1800 / 961) = kpa - 1 w 2 = u 2 - u 1 C vo ( T 2 - T 1 ) = 0.717( ) = kj / kg T 4 = T 3 ( v 3 / v 4 ) 0.4 = 1800(1 / 20) 0.4 = 543 K 3 w 4 = u 3 - u 4 C vo (T 3 - T ) = 0.717( ) = kj / kg 2 w net = 3 w w 2 = = kj / kg η = w net / q H = 1- (1 / 20) 0.4 = ( = / 0.717( )) A diesel engine has a bore of 0.1 m, a stroke of 0.11 m and a compression ratio of 19:1 running at 2000 RPM (revolutions per minute). Each cycle takes two revolutions and has a mean effective pressure of 1400 kpa. With a total of 6 cylinders find the engine power in kw and horsepower, hp. Power from mean effective pressure. mep = w net / (v max - v min ) -> w net = mep(v max - v min ) V = πbore S = π = m 3 W = mep(v max - V min ) = 1400 π = Work per Cylinders per Power Stroke Ẇ = W n RPM 0.5 (cycles / min) (min / 60 s) (kj / cycle) = (1/60) = 121 kw = 162 hp

113 At the beginning of compression in a diesel cycle T = 300 K, P = 200 kpa and after combustion (heat addition) is complete T = 1500 K and P = 7.0 MPa. Find the compression ratio, the thermal efficiency and the mean effective pressure. P 2 = P 3 = 7000 kpa => v 1 / v 2 = (P 2 /P 1 ) 1/ k = (7000 / 200) = T 2 = T 1 (P 2 / P 1 ) (k-1) / k = 300(7000 / 200) = v 3 / v 2 = T 3 / T 2 = 1500 / = 1.81 v 4 / v 3 = v 1 / v 3 = (v 1 / v 2 )( v 2 / v 3 ) = / 1.81 = 7 T 4 = T 3 (v 3 / v 4 ) k-1 = (1500 / 7) 0.4 = q L = C vo (T 4 - T 1 ) = 0.717( ) = q H = h 3 - h 2 C po (T 3 - T 2 ) = 1.004( ) = 674 η = 1 - q L / q H = / 674 = w net = q net = = v max = v 1 = R T 1 / P 1 = / 200 = m 3 / kg v min = v max / (v 1 / v 2 ) = / = m 3 / kg mep = w net / (v max - v min ) = / ( ) = 997 kpa Consider an ideal air-standard diesel cycle in which the state before the compression process is 95 kpa, 290 K, and the compression ratio is 20. Find the maximum temperature (by iteration) in the cycle to have a thermal efficiency of 60%? Diesel cycle: P 1 = 95 kpa, T 1 = 290 K, v 1 /v 2 = 20, η TH = 0.6 T 2 = T 1 (v 1 /v 2 ) k-1 = 290(20) 0.4 = K v 1 = /95 = = v 4, v 2 = / 20 = v 3 = v 2 (T 3 /T 2 ) = (T 3 /961.2) = T 3 T 3 = T 4 (v 4 /v 3 ) k = ( ) T T 4 = T 3 3 T 4 - T T 1.4 η TH = 0.60 = 1 - k(t 3 - T 2 ) = (T ) T T = : LHS = : LHS = , T 3 = 3040 K

114 Consider an ideal Stirling-cycle engine in which the state at the beginning of the isothermal compression process is 100 kpa, 25 C, the compression ratio is 6, and the maximum temperature in the cycle is 1100 C. Calculate the maximum cycle pressure and the thermal efficiency of the cycle with and without regenerators. P T 3 v 2 T T 4 v 1 v s a) T 1 = T 2 P 2 = P 1 (v 1 /v 2 ) = = 600 kpa 2 T v 1 3 T 4 v Ideal Stirling cycle T 1 = T 2 = 25 o C P 1 = 100 kpa v 1 /v 2 = 6 T 3 = T 4 = 1100 o C V 2 = V 3 P 3 = P 2 T 3 /T 2 = /298.2 = 2763 kpa b) w 34 = q 34 = RT 3 ln(v 4 /v 3 ) = ln6 = kj/kg q 23 = C V0 (T 3 - T 2 ) = ( ) = kj/kg w 12 = q 12 = -RT 1 ln(v 1 /v 2 ) = ln(6) = kj/kg w NET = = kj/kg η NO REGEN = = 0.374, η WITH REGEN = = An air-standard Stirling cycle uses helium as the working fluid. The isothermal compression brings helium from 100 kpa, 37 C to 600 kpa. The expansion takes place at 1200 K and there is no regenerator. Find the work and heat transfer in all of the 4 processes per kg helium and the thermal cycle efficiency. Helium table A.5: R = 2.077, C vo = Compression/expansion: v 4 / v 3 = v 1 / v 2 = P 2 / P 1 = 600 / 100 = 6 1 -> 2-1 w 2 = -q 12 = P dv = R T 1 ln(v 1 / v 2 ) = RT 1 ln (P 2 /P 1 ) = ln6 = kj/kg 2 -> 3 : 2 w 3 = 0; q 23 = C vo (T 3 - T ) = ( ) = 2773 kj/kg 2 3 -> 4: 3 w 4 = q 34 = R T 3 lnv 4 = ln6 = kj/kg v 3 4 -> 1 4 w 1 = 0; q 41 = C vo (T 4 - T 1 ) = kj/kg η cycle = ( 1 w w 4 )/(q 23 +q 34 ) = ( ) /( ) = 0.458

115 Consider an ideal air-standard Stirling cycle with an ideal regenerator. The minimum pressure and temperature in the cycle are 100 kpa, 25 C, the compression ratio is 10, and the maximum temperature in the cycle is 1000 C. Analyze each of the four processes in this cycle for work and heat transfer, and determine the overall performance of the engine. Ideal Stirling cycle diagram as in solution 11.82, with P 1 = 100 kpa, T 1 = T 2 = 25 o C, v 1 /v 2 = 10, T 3 = T 4 = 1000 o C From 1-2 at const T: 1 w 2 = 1 q 2 = T 1 (s 2 - s 1 ) = -RT 1 ln(v 1 /v 2 ) = ln(10) = kj/kg From 2-3 at const V: 2 w 3 = 0/ q 23 = C V0 (T 3 - T 2 ) = ( ) = From 3-4 at const T; 3 w 4 = 3 q 4 = T 3 (s 4 - s 3 ) v 4 = +RT 3 ln = ln(10) = kj/kg v 3 From 4-1 at const V; 4 w 1 = 0/ q 41 = C V0 (T 1 - T 4 ) = ( ) = kj/kg w NET = = kj/kg Since q 23 is supplied by -q 41 (regenerator) q h = q 34 = kj/kg, η TH = w NET q H = = NOTE: q H = q 34 = RT 3 ln(10), q L = -q 12 = RT 1 ln(10) η TH = (q H - q L )/q H = (T 3 - T 1 )/T 3 = (975/1273.2) = = Carnot efficiency

116 The air-standard Carnot cycle was not shown in the text; show the T s diagram for this cycle. In an air-standard Carnot cycle the low temperature is 280 K and the efficiency is 60%. If the pressure before compression and after heat rejection is 100 kpa, find the high temperature and the pressure just before heat addition. η = 0.6 = 1 - T H /T L T H = T L /0.4 = 700 K P 1 = 100 kpa 1 P 2 = P 1 (T H /T L ) k-1 = 2.47 MPa [or P 2 = P 1 (P r2 /P r1 ) = MPa] T H T L T s Air in a piston/cylinder goes through a Carnot cycle in which T = 26.8 C and the L total cycle efficiency is η = 2/3. Find T, the specific work and volume ratio in H the adiabatic expansion for constant C, C. Repeat the calculation for variable p v heat capacities. Carnot cycle: Same T-s diagram as in previous problem. η = 1 - T L /T H = 2/3 T H = 3 T L = = 900 K Adiabatic expansion 3 to 4: Pv k = constant 3 w 4 = (P 4 v 4 - P 3 v R /(1 - k) = k (T 4 - T 3 ) = u 3 - u 4 = C v (T 3 - T 4 ) = ( ) = kj/kg v 4 /v 3 = (T 3 /T 4 ) 1/(k - 1) = = 15.6 For variable C p, C v we get, T H = 3 T L = 900 K 3 w 4 = u 3 - u = = kj/kg 4 v 4 /v 3 = v r4 /v r3 = / = 18.1

117 Consider an ideal refrigeration cycle that has a condenser temperature of 45 C and an evaporator temperature of 15 C. Determine the coefficient of performance of this refrigerator for the working fluids R-12 and R-22. T Ideal Ref. Cycle T cond = 45 o 2 C = T 3 3 T evap = -15 o C 4 1 s R-12 R-22 h 1, kj/kg s 2 = s P 2, MPa T 2, o C h 2, kj/kg h 3 =h 4, kj/kg w C = h 2 -h q L = h 1 -h β =q L /w C The environmentally safe refrigerant R-134a is one of the replacements for R-12 in refrigeration systems. Repeat Problem using R-134a and compare the result with that for R-12. h 1 = 389.2, s 2 = s 1 = T h 3 = , P 3 = P 2 = 1.16 MPa 2 At 1 MPa, T 2 = 45.9, h 2 = o C At 1.2 MPa,T 2 = 53.3, h 2 = o -15 C T 2 = 51.8, h 2 = w C = h 2 - h 1 = s = 40.7 kj/kg q L = h 1 - h 4 = h 1 - h 3 = = kj/kg β = q L /(-w C ) = 125.1/40.7 = 3.07

118 A refrigerator using R-22 is powered by a small natural gas fired heat engine with a thermal efficiency of 25%. The R-22 condenses at 40 C and it evaporates at - 20 C and the cycle is standard. Find the two specific heat transfers in the refrigeration cycle. What is the overall coefficient of performance as Q L /Q 1? Evaporator: Inlet State is sat. liq-vap with h 4 = h 3 = kj / kg The exit state is sat. vap. with h = kj / kg q L = h 1 - h 4 = h 1 - h 3 = kj / kg Compressor: Inlet State 1 and Exit State 2 about 1.6 MPa w C = h 2 - h 1 ; s 2 = s 1 = kj / kg K 2: T 2 70 C h 2 = kj / kg w C = h 2 - h 1 = kj / kg Condenser: Brings it to sat liq at state 3 q H = h 2 - h 3 = = kj / kg Overall Refrigerator: β = q L / w C = / = Heat Engine: Ẇ HE = η HE Q. 1 = Ẇ C = Q. L / β Q. L / Q. 1 = ηβ = = A refrigerator with R-12 as the working fluid has a minimum temperature of 10 C and a maximum pressure of 1 MPa. Assume an ideal refrigeration cycle as in Fig Find the specific heat transfer from the cold space and that to the hot space, and the coefficient of performance. h 1 = , s 1 = , h 3 = s 2 = s 1 & P 2 h q L = h 1 - h 4 = h 1 - h 3 = = kj/kg q H = h 2 - h 3 = = 134 kj/kg b = q L /(-w c ) = q L /(q H - q L ) = 3.945

119 A refrigerator in a meat warehouse must keep a low temperature of -15 C and the outside temperature is 20 C. It uses R-12 as the refrigerant which must remove 5 kw from the cold space. Find the flow rate of the R-12 needed assuming a standard vapor compression refrigeration cycle with a condenser at 20 C. Basic refrigeration cycle: T 1 = T 4 = -15 C, T 3 = 20 C Table B.3: h 4 = h 3 = ; h 1 = h g = Q. L = ṁ R-12 q L = ṁ R-12 (h 1 - h 4 ) q L = = kj/kg ṁ R-12 = 5.0 / = kg / s A refrigerator with R-12 as the working fluid has a minimum temperature of 10 C and a maximum pressure of 1 MPa. The actual adiabatic compressor exit temperature is 60 C. Assume no pressure loss in the heat exchangers. Find the specific heat transfer from the cold space and that to the hot space, the coefficient of performance and the isentropic efficiency of the compressor. State 1: Inlet to compressor, sat. vap. -10 C, h 1 = , s 1 = State 2: Actual compressor exit, h 2AC = State 3: Exit condenser, sat. liq. 1MPa, h 3 = State 4: Exit valve, h 4 = h 3 C.V. Evaporator: q L = h 1 - h 4 = h 1 - h 3 = kj/kg C.V. Ideal Compressor: w C,S = h 2,S - h 1, s 2,S = s 1 State 2s: T 2,S = 49.66, h 2,S = 209.9, w C,S = C.V. Actual Compressor: w C = h 2,AC - h 1 = kj/kg β = q L /w C = 3.076, η C = w C,S /w C = C.V. Condenser: q H = h 2,AC - h 3 = kj/kg

120 Consider an ideal heat pump that has a condenser temperature of 50 C and an evaporator temperature of 0 C. Determine the coefficient of performance of this heat pump for the working fluids R-12, R-22, and ammonia. Ideal Heat Pump: T cond = 50 o C = T 3 T 3 2 T evap = 0 o C 4 1 s R-12 R-22 NH 3 h 1, kj/kg s 2 = s P 2, MPa T 2, o C h 2, kj/kg h 3 =h 4, kj/kg w C = h 2 -h q H = h 2 -h β =q H /w C The air conditioner in a car uses R-134a and the compressor power input is 1.5 kw bringing the R-134a from kpa to 1200 kpa by compression. The cold space is a heat exchanger that cools atmospheric air from the outside down to 10 C and blows it into the car. What is the mass flow rate of the R-134a and what is the low temperature heat transfer rate. How much is the mass flow rate of air at 10 C? Standard Refrigeration Cycle Table B.5 h 1 = ; s 1 = ; h 4 = h 3 = 266 C.V. Compressor (assume ideal) ṁ 1 = ṁ 2 w C = h 2 - h 1 ; s 2 = s 1 + s gen P 2, s = s 1 -> h 2 = > w C = 37.2 ṁ w C = Ẇ C -> ṁ = 1.5 / 37.2 = kg /s C.V. Evaporator Q. L = ṁ(h 1 - h ) = ( ) = 5.21 kw 4 C.V. Air Cooler ṁ air h air = Q. L ṁ air C p T ṁ air = Q. L / (C T) = 5.21 / ( ) = 0.26 kg / s p

121 A refrigerator using R-134a is located in a 20 C room. Consider the cycle to be ideal, except that the compressor is neither adiabatic nor reversible. Saturated vapor at -20 C enters the compressor, and the R-134a exits the compressor at 50 C. The condenser temperature is 40 C. The mass flow rate of refrigerant around the cycle is 0.2 kg/s, and the coefficient of performance is measured and found to be 2.3. Find the power input to the compressor and the rate of entropy generation in the compressor process. Table B.5: P 2 = P 3 = P sat 40C = 1017 kpa, h 4 = h 3 = kj/kg s , h ; s 1 = , h 1 = β = q L / w C -> w C = q L / β = (h 1 - h 4 ) / β = ( ) / 2.3 = Ẇ C = ṁ w C = kw C.V. Compressor h 1 + w C + q = h 2 -> q in = h 2 - h 1 - w C = = i.e. a heat loss s 1 + dq/t + s gen = s 2 s gen = s 2 - s 1 - q / T o = (11.53 / ) = Ṡ gen = ṁ s gen = = kw / K A small heat pump unit is used to heat water for a hot-water supply. Assume that the unit uses R-22 and operates on the ideal refrigeration cycle. The evaporator temperature is 15 C and the condenser temperature is 60 C. If the amount of hot water needed is 0.1 kg/s, determine the amount of energy saved by using the heat pump instead of directly heating the water from 15 to 60 C. Ideal R-22 heat pump T T 1 = 15 o C, T 3 = 60 o C 2 h 1 = , s 2 = s 1 = P 2 = P 3 = MPa, h 3 = T 2 = 78.4 o C, h 2 = w C = h 2 - h 1 = q H = h 2 - h 3 = To heat 0.1 kg/s of water from 15 o C to 60 o C, Q. = ṁ( h) = 0.1( ) = kw H2O Using the heat pump Ẇ IN = Q. H2O (w C /q ) = 18.81(27.84/160.68) = 3.26 kw H a saving of kw 4 1 s

122 The refrigerant R-22 is used as the working fluid in a conventional heat pump cycle. Saturated vapor enters the compressor of this unit at 10 C; its exit temperature from the compressor is measured and found to be 85 C. If the isentropic efficiency of the compressor is estimated to be 70%, what is the coefficient of performance of the heat pump? R-22 heat pump: T T EVAP = 10 o 2 C, η S COMP = S T 2 = 85 o C 3 Isentropic compressor: s 2S = s 1 = but P 2 unknown. Trial & error. s Assume P 2 = 2.11 MPa At P 2,s 2S : T 2S = 72.1 o C, h 2S = , At P 2,T 2 : h 2 = calculate η S COMP = (h 2S - h 1 )/(h 2 - h 1 ) = OK P 2 = 2.11 MPa = P 3 T 3 = 53.7 o C, h 3 = , w C = h 2 - h 1 = q H = h 2 - h 3 = , β = q H /w C = =

123 In an actual refrigeration cycle using R-12 as the working fluid, the refrigerant flow rate is 0.05 kg/s. Vapor enters the compressor at 150 kpa, 10 C, and leaves at 1.2 MPa, 75 C. The power input to the compressor is measured and found be 2.4 kw. The refrigerant enters the expansion valve at 1.15 MPa, 40 C, and leaves the evaporator at 175 kpa, 15 C. Determine the entropy generation in the compression process, the refrigeration capacity and the coefficient of performance for this cycle. Actual ref. cycle T 2 P 1 = 0.15 MPa, P 2 = 1.2 MPa 1: comp. inlet T 1 = -10 o C, T 2 = 75 o C P 3 = 1.15 MPa, P 5 = MPa T 3 = 40 o C, T 5 = -15 o C 5: evap. exit Ẇ COMP = 2.4 kw in, ṁ = 0.05 kg/s Table B.3 h 1 = , s 1 = , h 2 = , s 2 = CV Compressor: h 1 + q COMP = h 2 + w COMP ; s 1 + dq/t + s gen = s 2 w COMP = -2.4/0.05 = kj/kg q COMP = h 2 + w COMP - h 1 = = -6.1 kj/kg s gen = s 2 - s 1 - q / T o = / = kj / kg K b) q L = h 5 - h 4 = = kj/kg Q. L = ṁq = = kw L c) β = q L /-w COMP = 106.5/48.0 = s Consider a small ammonia absorption refrigeration cycle that is powered by solar energy and is to be used as an air conditioner. Saturated vapor ammonia leaves the generator at 50 C, and saturated vapor leaves the evaporator at 10 C. If 7000 kj of heat is required in the generator (solar collector) per kilogram of ammonia vapor generated, determine the overall performance of this system. T NH 3 absorption cycle: sat. vapor at 50 o C exits the generator sat. vapor at 10 o C exits the evaporator q H = q GEN = 7000 kj/kg NH 3 out of gen. q L = h 2 - h 1 = h o G 10 C - h o F 50 C = = kj/kg q L /q H = /7000 = C o 10 o C 1 2 GEN. EXIT EVAP EXIT s

124 The performance of an ammonia absorption cycle refrigerator is to be compared with that of a similar vapor-compression system. Consider an absorption system having an evaporator temperature of 10 C and a condenser temperature of 50 C. The generator temperature in this system is 150 C. In this cycle 0.42 kj is transferred to the ammonia in the evaporator for each kilojoule transferred from the high-temperature source to the ammonia solution in the generator. To make the comparison, assume that a reservoir is available at 150 C, and that heat is transferred from this reservoir to a reversible engine that rejects heat to the surroundings at 25 C. This work is then used to drive an ideal vapor-compression system with ammonia as the refrigerant. Compare the amount of refrigeration that can be achieved per kilojoule from the high-temperature source with the 0.42 kj that can be achieved in the absorption system. o T = 50 C T H T ' o = 150 C Q H 2 H 3 Q ' COND. 3 H = 1 kj 2 REV. W C 4 1 H.E. COMP. s Q ' L EVAP. T 1 = -10 o C o 1 T = 25 C Q 4 h L ' L 1 = , s 1 = o h T L = -10 C 4 = h 3 = For the rev. heat engine: η TH = 1 - T L /T H = = W C = η TH Q = kj H For the NH 3 refrig. cycle: P 2 = P 3 = 2033 kpa, Use 2000 kpa Table s 2 = s 1 = => T C h w C = h 2 - h 1 = = q L = h 1 - h 4 = = β = q L /w C = / = 3.44 Q L = βw C = = kj based on assumption of ideal heat engine & refrig. cycle.

125 A heat exchanger is incorporated into an ideal air-standard refrigeration cycle, as shown in Fig. P It may be assumed that both the compression and the expansion are reversible adiabatic processes in this ideal case. Determine the coefficient of performance for the cycle. q T L COMP 2 q H 3 4 EXP s T 1 = T 3 = 15 o C = K, P 1 = 100 kpa, P 2 = 1.4 MPa T 4 = T 6 = -50 o C = K, s 2 = s 1 k-1 T 2 = T 1 (P 2 /P 1 ) k = 288.2(1400/100) = 613 K w C = -w 12 = C P0 (T 2 - T 1 ) = ( ) = 326 kj/kg k-1 s 5 = s 4 T 5 = T 4 (P 5 /P 4 ) k = 223.2(100/1400) = K w E = C P0 (T 4 - T 5 ) = ( ) = kj/kg w NET = = kj/kg q L = C P0 (T 6 - T 5 ) = ( ) = kj/kg β = q L /w NET = 118.7/207.3 = Repeat Problem , but assume an isentropic efficiency of 75% for both the compressor and the expander. From solution : 2 T 2S = 613 K, w SC = 326 kj/kg T 5S = K, w SE = T 2S w C = w SC / η SC = 326/0.75 = kj/kg w E = η SE w SE = = 89.0 kj/kg = C P0 (T 4 -T 5 ) = 1.004( T 5 ) T 5 = K w NET = = kj/kg q L = C P0 (T 6 - T 5 ) = 1.004( ) = 89.0 kj/kg β = q L /(-w NET ) = 89.0/345.6 = S s

126 Repeat Problems and , but assume that helium is the cycle working fluid instead of air. Discuss the significance of the results. a) Problem , except for Helium T 2 = = K, T5 = = 77.7 K w C = -w 12 = ( ) = kj/kg w E = ( ) = kj/kg w NET = = kj/kg q L = ( ) = kj/kg β = 755.5/ = b) Problem , except for Helium From part a): T 2S = K, w SC = T 5S = 77.7 K, w SE = w C = w SC /η SC = /0.75 = kj/kg w E = η SE w SE = = kj/kg w NET = = kj/kg w E = = ( T 5 ) T 5 = K q L = ( ) = kj/kg β = 566.6/ = 0.179

127 A binary system power plant uses mercury for the high-temperature cycle and water for the low-temperature cycle, as shown in Fig The temperatures and pressures are shown in the corresponding T s diagram. The maximum temperature in the steam cycle is where the steam leaves the superheater at point 4 where it is 500 C. Determine the ratio of the mass flow rate of mercury to the mass flow rate of water in the heat exchanger that condenses mercury and boils the water and the thermal efficiency of this ideal cycle. The following saturation properties for mercury are known P, MPa T g, C h f, kj/kg h g, kj/kg s f kj/kgk s g, kj/kgk a) For the mercury cycle: s d = s c = = x d , x d = h b = h a - w P HG h a ( since v F is very small) q H = h c - h a = = kj/kg q L = h d - h a = = kj/kg For the steam cycle: s 5 = s 4 = = x , x 5 = h 5 = = w P v 1 (P 2 - P 1 ) = ( ) = 4.7 kj/kg h 2 = h 1 - w P = = q H (from Hg) = h 3 - h 2 = = q H (ext. source) = h 4 - h 3 = = CV: Hg condenser - H 2 O boiler: 1st law: m Hg (h d - h a ) = m H2O (h 3 - h 2 ) m Hg /m H2O = = b) q H TOTAL = (m Hg /m H2O )(h c - h b ) + (h 4 - h 3 ) (for 1 kg H 2 O) = = kj All q L is from the H 2 O condenser: q L = h 5 - h 1 = = kj w NET = q H - q L = = kj η TH = w NET /q H = / = 0.529

128 A Rankine steam power plant should operate with a high pressure of 3 MPa, a low pressure of 10 kpa, and the boiler exit temperature should be 500 C. The available high-temperature source is the exhaust of 175 kg/s air at 600 C from a gas turbine. If the boiler operates as a counterflowing heat exchanger where the temperature difference at the pinch point is 20 C, find the maximum water mass flow rate possible and the air exit temperature. -w P = h 2 - h 1 = v 1 (P 2 - P 1 ) T = ( ) = 3.02 h 2 = h 1 - w P = = h 3 = , h air,in = a 2 State 2a: T 2a = T SAT = C 1 h 2a = s h air (T 2a + 20) = Air temperature should be C at the point where the water is at state 2a. C.V. Section 2a-3, i-a ṁ H2 O (h 3 - h 2a ) = ṁ air (h i - h a ) e 2 a HEAT EXCH 2a i ṁ H2 = 175 = kg/s O Take C.V. Total: ṁ H2 O (h 3 - h 2 ) = ṁ air (h i - h e ) h e = h i - ṁ H2 O (h 3 - h 2 )/ṁ air = ( )/175 = T e = K = C, T e > T 2 = 46.5 C OK A simple Rankine cycle with R-22 as the working fluid is to be used as a bottoming cycle for an electrical generating facility driven by the exhaust gas from a Diesel engine as the high temperature energy source in the R-22 boiler. Diesel inlet conditions are 100 kpa, 20 C, the compression ratio is 20, and the maximum temperature in the cycle is 2800 C. Saturated vapor R-22 leaves the bottoming cycle boiler at 110 C, and the condenser temperature is 30 C. The power output of the Diesel engine is 1 MW. Assuming ideal cycles throughout, determine a. The flow rate required in the diesel engine. b. The power output of the bottoming cycle, assuming that the diesel exhaust is cooled to 200 C in the R-22 boiler.

129 11-80 T 2 s 1 v v 3 s 4 DIESEL AIR-STD CYCLE s P 1 = 100 kpa, T 1 = 20 o C, T 7 = 110 o C, x 7 = 1.0 T IDEAL R-12 RANKINE BOTTOMING CYCLE s v 1 /v 2 = 20, T 3 = 2800 o C, T 5 = T 8 = 30 o C, Ẇ DIESEL = 1.0 MW a) T 2 = T 1 (v 1 /v 2 ) k-1 = 293.2(20) 0.4 = K P 2 = P 1 (v 1 /v 2 ) k = 100(20) 1.4 = 6629 kpa q H = C P0 (T 3 - T 2 ) = 1.004( ) = kj/kg v 1 = = , v 2 = = v 3 = v 2 (T 3 /T 2 ) = (3073.2/971.8) = T 4 = T 3 v 3 v k-1 = = K q L = 0.717( ) = w NET = = kj/kg ṁ AIR = Ẇ NET /w NET = 1000/ = 0.79 kg/s b) s 8 = s 7 = = x , x 8 = h 8 = = w T = h 7 - h 8 = = 21.9 kj/kg -w P = v 5 (P 6 - P 5 ) = ( ) = 2.50 h 6 = h 5 - w P = = 67.1 q H = h 7 - h 6 = = kj/kg Q. H available from Diesel exhaust cooled to 200 o C: Q. = ( ) = 564 kw H ṁ R-12 = Q. H /q = 564/130.9 = kg/s H Ẇ R-12 = 4.309( ) = 83.6 kw

130 For a cryogenic experiment heat should be removed from a space at 75 K to a reservoir at 180 K. A heat pump is designed to use nitrogen and methane in a cascade arrangement (see Fig ), where the high temperature of the nitrogen condensation is at 10 K higher than the low-temperature evaporation of the methane. The two other phase changes take place at the listed reservoir temperatures. Find the saturation temperatures in the heat exchanger between the two cycles that gives the best coefficient of performance for the overall system. The nitrogen cycle is the bottom cycle and the methane cycle is the top cycle. Both std. refrigeration cycles. T Hm = 180 K = T 3m, T LN = 75 K = T 4N = T 1N T Lm = T 4m = T 1m = T 3N - 10, Trial and error on T 3N or T Lm. For each cycle we have, -w C = h 2 - h 1, s 2 = s 1, -q H = h 2 - h 3, q L = h 1 - h 4 = h 1 - h 3 Nitrogen: T 4 = T 1 = 75 K h 1 = s 1 = N 2 T 3 h 3 P 2 h 2 -w c -q H q L a) b) c) Methane: T 3 = 180 K h 3 = -0.5 P 2 = MPa CH 4 T 4 h 1 s 1 h 2 -w c -q H q L a) b) c) The heat exchanger that connects the cycles transfers a Q Q. Hn = q Hn ṁ n = Q. Lm = q Lm ṁ m => ṁ m /ṁ n = q Hn /q Lm The overall unit then has Q. L 75 K = ṁ n q Ln ; Ẇ tot in = - (ṁ n w cn + ṁ m w cm ) β = Q. L 75 K /Ẇ tot in = q Ln/[-w cn -(ṁ m /ṁ n )w cm ] Case ṁ m /ṁ n w cn +(ṁ m /ṁ n )w cm β a) b) c) A maximum coeff. of performance is between case b) and c).

131 A cascade system is composed of two ideal refrigeration cycles, as shown in Fig The high-temperature cycle uses R-22. Saturated liquid leaves the condenser at 40 C, and saturated vapor leaves the heat exchanger at 20 C. The low-temperature cycle uses a different refrigerant, R-23 (Fig. G.3 or the software). Saturated vapor leaves the evaporator at 80 C, and saturated liquid leaves the heat exchanger at 10 C. Calculate the ratio of the mass flow rates through the two cycles and the coefficient of performance of the system. o COND T 2 ' 3 ' 3 ' = 40 C x = ' C R-22 o T 1 ' = -20 C x = ' T = -80 o C 1 x = ' 2 1 C R-23 EVAP 4 ' 3 4 T = -10 o C 3 x = T T 3 ' 2 ' ' 1 ' 4 1 s s T, o C P h s T, o C P h s a) ṁ/ṁ = h 1 - h = h 2 - h = b) Q. L /ṁ = h 1 - h = = Ẇ TOT /ṁ = (h 2 - h 1 ) + (ṁ /ṁ)(h 2 - h 1 ) = ( ) + (1/0.672)( ) = β = Q L /-Ẇ TOT = 145/144.8 = 1.0

132 Consider an ideal dual-loop heat-powered refrigeration cycle using R-12 as the working fluid, as shown in Fig. P Saturated vapor at 105 C leaves the boiler and expands in the turbine to the condenser pressure. Saturated vapor at 15 C leaves the evaporator and is compressed to the condenser pressure. The ratio of the flows through the two loops is such that the turbine produces just enough power to drive the compressor. The two exiting streams mix together and enter the condenser. Saturated liquid leaving the condenser at 45 C is then separated into two streams in the necessary proportions. Determine the ratio of mass flow rate through the power loop to that through the refrigeration loop. Find also the performance of the cycle, in terms of the ratio Q /Q L H. T 1. Q L 6 BOIL. 5 TURB. 7 P COND. 3 COMP. 2 E V A P T 3 = 45 o C sat. liq. T 1 = -15 o C sat. vap. 2 1 s P 5 = P 6 = MPa; T 6 = 105 o C sat. vap.; P 2 = P 3 = P 7 = MPa h 1 = ; h 3 = h 4 = ; h 6 = s 7 = s 6 = = x ; x 7 = h 7 = = s 2 = s 1 = , P 2 => T 2 = 54.7 o C, h 2 = a) CV: turbine + compressor Cont: ṁ 1 = ṁ 2, ṁ 6 = ṁ 7 ; 1st law: ṁ 1 h 1 + ṁ 6 h 6 = ṁ 2 h 2 + ṁ 7 h 7 ṁ 6 /ṁ 1 = ( )/( ) = b) CV: pump -w P = v 3 (P 5 - P 3 ) = ( ) = kj/kg h 5 = h 3 - w P = CV: evaporator Q. L = ṁ 1 (h 1 - h 4 ) CV: boiler Q. H = ṁ 6 (h 6 - h 5 ) Q. L β = Q. ṁ 1 (h 1 - h 4 ) = ṁ H 6 (h 6 - h 5 ) = ( ) = 0.435

133 Find the availability of the water at all four states in the Rankine cycle described in Problem Assume that the high-temperature source is 500 C and the lowtemperature reservoir is at 25 C. Determine the flow of availability in or out of the reservoirs per kilogram of steam flowing in the cycle. What is the overall cycle second law efficiency? Reference State: 100 kpa, 25 C, s 0 = , h 0 = ψ 1 = h 1 - h 0 - T 0 (s 1 - s 0 ) = ( ) = 2.89 ψ 2 = ( ) = ψ = 6.42 ψ 3 = ( ) = ψ 4 = ψ 3 - w T,s = ψ H = (1 - T 0 /T H )q H = = ψ L = (1 - T 0 /T 0 )q C = /0 η II = w NET / ψ H = ( )/ = Notice T H > T 3, T L < T 4 = T 1 so cycle is externally irreversible. Both q H and q C over finite T.

134 The effect of a number of open feedwater heaters on the thermal efficiency of an ideal cycle is to be studied. Steam leaves the steam generator at 20 MPa, 600 C, and the cycle has a condenser pressure of 10 kpa. Determine the thermal efficiency for each of the following cases. A: No feedwater heater. B: One feedwater heater operating at 1 MPa. C: Two feedwater heaters, one operating at 3 MPa and the other at 0.2 MPa. a) no feed water heater 2 3 w P = vdp ( ) = 20.2 kj/kg h 2 = h 1 + w P = = s 4 = s 3 = = x x 4 = h 4 = = w T = h 3 - h 4 = ST. GEN. T 2 TURBINE. 4 COND. 1 P 20 MPa o C = kj/kg w N = w T - w P = = q H = h 3 - h 2 = = kpa s η TH = w N q H = = b) one feedwater heater w P12 = ( ) = 1.0 kj/kg h 2 = h 1 + w P12 = = w P34 = ( ) = 21.4 kj/kg h 4 = h 3 + w P34 = = s 6 = s 5 = = x ST. GEN. 5 6 HTR. 3 P 4 2 TURBINE. COND. P 7 1

135 11-86 x 6 = h 6 = = CV: heater const: m 3 = m 6 + m 2 = 1.0 kg 1st law: m 6 h 6 + m 2 h 2 = m 3 h m 6 = = T 20 MPa o 600 C kpa 7 1 s 5 1 MPa m 2 = , h 7 = ( = h 4 of part a) ) CV: turbine w T = (h 5 - h 6 ) + m 2 (h 6 - h 7 ) = ( ) ( ) = kj/kg CV: pumps w P = m 1 w P12 + m 3 w P34 = (1.0) + 1(21.4) = 22.2 kj/kg w N = = kj/kg CV: steam generator q H = h 5 - h 4 = = kj/kg η TH = w N /q H = / = c) two feedwater heaters w P12 = (200-10) = 0.2 kj/kg h 2 = w P12 + h 1 = = w P34 = ( ) = 3.0 kj/kg h 4 = h 3 + w P34 = = ST. GEN. 6 HP HTR P LP HTR TURBINE. P P 10 COND. 1

136 11-87 w P56 = ( ) = 20.7 kj/kg h 6 = h 5 + w P56 = = s 8 = s 7 = T 8 = o C at P 8 = 3 MPa h 8 = s 9 = s 8 = = x T o 600 C 7 3 MPa MPa 0.2 MPa 10 kpa s x 9 = => h 9 = = CV: high pressure heater cont: m 5 = m 4 + m 8 = 1.0 kg ; 1st law: m 5 h 5 = m 4 h 4 + m 8 h m 8 = = m 4 = CV: low pressure heater cont: m 9 + m 2 = m 3 = m 4 ; 1st law: m 9 h 9 + m 2 h 2 = m 3 h ( ) m 9 = = m 2 = = CV: turbine w T = (h 7 - h 8 ) + (1 - m 8 )(h 8 - h 9 ) + (1 - m 8 - m 9 )(h 9 - h 10 ) = ( ) ( ) ( ) = kj/kg CV: pumps w P = m 1 w P12 + m 3 w P34 + m 5 w P56 = (0.2) (3.0) + 1(20.7) = 23.2 kj/kg w N = = kj/kg CV: steam generator q H = h 7 - h 6 = = kj/kg η TH = w N /q H = / = 0.488

137 Find the availability of the water at all the states in the steam power plant described in Problem Assume the heat source in the boiler is at 600 C and the low-temperature reservoir is at 25 C. Give the second law efficiency of all the components. From solution to 11.1 and : States 1-2a - 3-4a - 1 1: h f = , s f = a: h 2a = 196.2, s 2a : h 3 = , s 3 = a: h 4a = , s 4a = , x 4a = ψ = h - h o - T o (s - s o ) h o = , s o = ψ 1 = ( ) = 2.90 ψ 2a = ( ) = 3.24 ψ 3 = ( ) = ψ 4a = ( ) = η II Pump = (ψ 2a - ψ 1 ) / w p ac = ( ) / = η II Boiler = (ψ 3 - ψ 2a ) / [(1- T o /T H ) q H ] = ( ) / [ ] = 0.57 η II Turbine = w T ac / (ψ 3 - ψ 4a ) = / ( ) = η II Cond = ψ amb / (ψ 4a - ψ 1 ) = 0

138 The power plant shown in Fig combines a gas-turbine cycle and a steamturbine cycle. The following data are known for the gas-turbine cycle. Air enters the compressor at 100 kpa, 25 C, the compressor pressure ratio is 14, and the isentropic compressor efficiency is 87%; the heater input rate is 60 MW; the turbine inlet temperature is 1250 C, the exhaust pressure is 100 kpa, and the isentropic turbine efficiency is 87%; the cycle exhaust temperature from the heat exchanger is 200 C. The following data are known for the steam-turbine cycle. The pump inlet state is saturated liquid at 10 kpa, the pump exit pressure is 12.5 MPa, and the isentropic pump efficiency is 85%; turbine inlet temperature is 500 C and the isentropic turbine efficiency is 87%. Determine a. The mass flow rate of air in the gas-turbine cycle. b. The mass flow rate of water in the steam cycle. c. The overall thermal efficiency of the combined cycle.. Q H = 60 MW P 2 /P = 14 1 η = 0.87 SC HEAT 2 3 COMP GAS TURB o T 3 = 1250 C. W NET CT AIR 1 P 1 = 100 kpa o T = 25 C o 1 T 5 = 200 C 6 5 HEAT EXCH P = P = 12.5 MPa 6 7 H O STEAM TURB P 4 = 100 kpa η = 0.87 ST o T 7 = 500 C. W ST η ST= 0.87 η = 0.85 SP P 9 COND 8 P 8 = P 9 = 10 kpa a) From Air Tables, A.7: P r1 = , h 1 = , h 5 = s 2 = s 1 P r2s = P r1 (P 2 /P 1 ) = = T 2S = 629 K, h 2S = w SC = h 1 - h 2S = = kj/kg w C = w SC /η SC = /0.87 = -386 = h 1 - h 2 h 2 = At T 3 = K: P r3 = , h 3 =

139 11-90 ṁ AIR = Q. H /(h 3 - h 2 ) = = kg/s b) P r4s = P r3 (P 4 /P 3 ) = (1/14) = => T 4S = 791 K, h 4S = w ST = h 3 - h 4S = = w T = η ST w ST = = = h 3 - h 4 => h 4 = kj/kg Steam cycle: -w SP ( ) = w P = - w SP /η SP = /0.85 = h 6 = h 9 - w P = = At 12.5 MPa, 500 o C: h 7 = , s 7 = ṁ H2 O = ṁ h 4 - h = = kg/s AIR h 7 - h c) s 8S = s 7 = = x 8S 7.501, x 8S = h 8S = = w ST = h 7 - h 8S = = w T = η ST w ST = = kj/kg Ẇ NET = ṁ(w T +w C ) AIR + ṁ(w T +w P ) H2 O = 61.27( ) ( ) = = kw = MW η TH = Ẇ NET /Q. = /60 = H

140 For Problem , determine the change of availability of the water flow and that of the air flow. Use these to determine a second law efficiency for the boiler heat exchanger. From solution to : ṁ H2O = kg/s, h 2 = , s 2 = h 3 = , s 3 = , s Ti = , s Te = h i = , h e = ψ 3 - ψ 2 = h 3 - h 2 - T 0 (s 3 - s 2 ) = kj/kg ψ i - ψ e = h i - h e - T 0 (s Ti - s Te ) = kj/kg η II = (ψ 3 - ψ 2 )ṁ H2O = = (ψ i - ψ e )ṁ air

141 One means of improving the performance of a refrigeration system that operates over a wide temperature range is to use a two-stage compressor. Consider an ideal refrigeration system of this type that uses R-12 as the working fluid, as shown in Fig. P Saturated liquid leaves the condenser at 40 C and is throttled to 20 C. The liquid and vapor at this temperature are separated, and the liquid is throttled to the evaporator temperature, 70 C. Vapor leaving the evaporator is compressed to the saturation pressure corresponding to 20 C, after which it is mixed with the vapor leaving the flash chamber. It may be assumed that both the flash chamber and the mixing chamber are well insulated to prevent heat transfer from the ambient. Vapor leaving the mixing chamber is compressed in the second stage of the compressor to the saturation pressure corresponding to the condenser temperature, 40 C. Determine a. The coefficient of performance of the system. b. The coefficient of performance of a simple ideal refrigeration cycle operating over the same condenser and evaporator ranges as those of the two-stage compressor unit studied in this problem. 4 5 COMP. ST.2 MIX.CHAM COMP. ST.1 9 ROOM +Q. H COND SAT.VAP. o -20 C FLASH CHAMBER 6 SAT.LIQ. 40 o C SAT.LIQ. o -20 C T o 40 C o -20 C o -70 C 1 6 R-12 ref. with 2-stage compression s 8 SAT.VAP. o -70 C EVAP. -Q L COLD SPACE 7 CV: expansion valve, upper loop h 2 = h 1 = = x ; x 2 = m 3 = x 2 m 2 = x 2 m 1 = kg ( for m 1 =1 kg) m 6 = m 1 - m 3 = kg CV: expansion valve, lower loop h 7 = h 6 = 17.8 = x , x 7 = 0.242

142 11-93 Q L = m 3 (h 8 - h 7 ) = 0.647( ) q L = 89.1 kj/kg-m 1 CV: 1st stage compressor s 8 = s 9 = , P 9 = P o SAT -20 C = MPa T 9 = 9 o C, h 9 = kj/kg CV: mixing chamber (assume const. press. mixing) 1st law: m 6 h 9 + m 3 h 3 = m 1 h 4 or h 4 = = kj/kg h 4, P 4 = MPa T 4 = -1.0 o C, s 4 = CV: 2nd stage compressor P 4 = MPa = P 9 = P 3 P 5 = P o sat 40 C = MPa, s 5 = s 4 T 5 = 70 o C, h 5 = CV: condenser 1st law: -q H = h 1 - h 5 = = β 2 stage = q L /(q H - q L ) = 89.1/( ) = b) 1 stage compression h 3 = h 4 = T h 1 = q L = h 1 - h 4 = 81.0 kj/kg o 40 C 3 2 s 1 = s 2 = P 2 = MPa T 2 = 80.9 o C, h 2 = o -70 C 4 1 s q H = h 2 - h 3 = = kj/kg β 1 stage = q L /(q H - q L ) = 81.0/( ) = 1.032

143 A jet ejector, a device with no moving parts, functions as the equivalent of a coupled turbine-compressor unit (see Problems 9.82 and 9.90). Thus, the turbinecompressor in the dual-loop cycle of Fig. P could be replaced by a jet ejector. The primary stream of the jet ejector enters from the boiler, the secondary stream enters from the evaporator, and the discharge flows to the condenser. Alternatively, a jet ejector may be used with water as the working fluid. The purpose of the device is to chill water, usually for an air-conditioning system. In this application the physical setup is as shown in Fig. P Using the data given on the diagram, evaluate the performance of this cycle in terms of the ratio Q L /Q H. a. Assume an ideal cycle. b. Assume an ejector efficiency of 20% (see Problem 9.90). 2 T VAP o 150 C. Q H BOIL. 11 JET EJECT. 3 COND. 1 VAP 10 o C 30 o C 4 T 1 = T 7 = 10 o C HP FLASH P. CH. T = 150 o C 9 7 LIQ 10 o T 4 = 30 o C C 20 o C LP T CHILL 9 = 20 o C P.. 8 Assume T 5 = T 10 Q L (from mixing streams 4 & 9). P 3 = P 4 = P 5 = P 8 = P 9 = P 10 = P o G 30 C = kpa , ' 3 1 ' 1 s P 11 = P 2 = P G 150 o C = kpa, P 1 = P 6 = P 7 = P G 10 o C = kpa Cont: ṁ 1 + ṁ 9 = ṁ 5 + ṁ 10, ṁ 5 = ṁ 6 = ṁ 7 + ṁ 1 ṁ 7 = ṁ 8 = ṁ 9, ṁ 10 = ṁ 11 = ṁ 2, ṁ 3 = ṁ 4 a) ṁ 1 + ṁ 2 = ṁ 3 ; ideal jet ejector s 1 = s 1 & s 2 = s 2 (1' & 2' at P 3 = P 4 ) then, ṁ 1 (h 1 - h 1 ) = ṁ 2 (h 2 - h 2 ) From s 2 = s 2 = x ; x 2 = h 2 = =

144 11-95 From s 1 = s 1 = T 1 = 112 C, h 1 = ṁ 1 /ṁ 2 = = Also h 4 = , h 7 = 42.01, h 9 = Mixing of streams 4 & 9 5 & 10: (ṁ 1 + ṁ 2 )h 4 + ṁ 7 h 9 = (ṁ 7 + ṁ 1 + ṁ 2 )h 5 = 10 Flash chamber (since h 6 = h 5 ) : (ṁ 7 +ṁ 1 )h 5 = 10 = ṁ 1 h 1 + ṁ 7 h 1 using the primary stream ṁ 2 = 1 kg/s: ṁ = (ṁ )h 5 & (ṁ )h 5 = ṁ Solving, ṁ 7 = & h 5 = LP pump: -w LP P = ( ) = h 8 = h 7 - w LP P = = Chiller: Q. L = ṁ 7 (h 9 -h 8 ) = ( ) = 8500 kw (for ṁ 2 = 1) HP pump: -w HP P = ( ) = 0.47 h 11 = h 10 - w HP P = = Boiler: Q. 11 = ṁ 11 (h 2 - h ) = 1( ) = kw 11 Q. L /Q. = 8500/ = H b) Jet eject. eff. = (ṁ 1 /ṁ 2 ) ACT /(ṁ 1 /ṁ 2 ) IDEAL = 0.20 (ṁ 1 /ṁ 2 ) ACT = = using ṁ 2 = 1 kg/s: ṁ = (ṁ )h 5 & (ṁ )h 5 = ṁ Solving, ṁ 7 = & h 5 = h 10 = Then, Q. = ( ) = 1668 kw L h 11 = = Q. = 1( ) = kw H & Q. L /Q. = 1668/ = H

145 11-96 English Unit Problems E A steam power plant, as shown in Fig. 11.3, operating in a Rankine cycle has saturated vapor at 600 lbf/in. 2 leaving the boiler. The turbine exhausts to the condenser operating at 2 lbf/in. 2. Find the specific work and heat transfer in each of the ideal components and the cycle efficiency. 1: h 1 = 93.73, v 1 = , 3: h 3 = h g = , s 3 = s g = C.V. Pump: -w P = v dp = v 1 (P 2 - P 1 ) = (600-2) 144 = 1.8 Btu/lbm 778 h 2 = h 1 - w P = = Btu/lbm C.V. Boiler: q H = h 3 - h 2 = = Btu/lbm C.V. Tubine: w T = h 3 - h 4, s 4 = s 3 s 4 = s 3 = = x => x 4 = , h 4 = = w T = = Btu/lbm η CYCLE = (w T + w P )/q H = ( )/ = 0.33 C.V. Condenser: q L = h 4 - h 1 = = Btu/lbm E Consider a solar-energy-powered ideal Rankine cycle that uses water as the working fluid. Saturated vapor leaves the solar collector at 350 F, and the condenser pressure is 1 lbf/in. 2. Determine the thermal efficiency of this cycle. T H 2 O ideal Rankine cycle CV: turbine s 4 = s 3 = = x x 4 = h 4 = = w T = h 3 - h 4 = = Btu/lbm -w P = vdp v 1 (P 2 - P 1 ) = (67-1) 144 / 778 = 0.2 w NET = w T + w P = = 311 h 2 = h 1 - w P = = 69.9 Btu/lbm q H = h 3 - h 2 = = Btu/lbm η TH = w NET /q H = 311/ = s

146 E A supply of geothermal hot water is to be used as the energy source in an ideal Rankine cycle, with R-134a as the cycle working fluid. Saturated vapor R-134a leaves the boiler at a temperature of 180 F, and the condenser temperature is 100 F. Calculate the thermal efficiency of this cycle. T a) From the R-134a tables, h 1 = v 1 = D 3 P 1 = P 2 = P 3 = F h 3 = s 3 = F CV. Pump: -w P = v 1 (P 2 - P 1 ) s = ( ) = Btu/lbm = h 2 - h 1 h 2 = h 1 - w P = = CV: Turbine s 4 = s 3 x 4 = ( )/ = h 4 = , w T = h 3 - h 4 = btu/lbm CV: Boiler q H = h 3 - h 2 = = Btu/lbm η TH = (w T + w 12 )/q H = ( )/74.83 = E Do Problem with R-22 as the working fluid. Same diagram as in problem , now from the R-22 tables, h 1 = , v 1 = , P 1 = 210.6, P 2 = P 3 = 554.8, h 3 = , s 3 = CV: Pump -w P = v 1 (P 2 -P 1 ) = ( ) 144 = Btu/lbm 778 h 2 = h 1 - w P = = CV: Turbine s 4 = s 3 x 4 = ( )/ = h 4 = , w T = h 3 - h 4 = Btu/lbm CV: Boiler q H = h 3 - h 2 = = Btu/lbm η TH = (w T + w 12 )/q H = ( )/ = 0.104

147 E The power plant in Problem is modified to have a superheater section following the boiler so the steam leaves the super heater at 600 lbf/in. 2, 700 F. Find the specific work and heat transfer in each of the ideal components and the cycle efficiency. h 3 = , s 3 = , h 1 = 94.01, v 1 = C.V. Pump: -w P = v dp = v 2 (P 2 - P 1 ) = (600-2)(144 / 778) = 1.8 Btu/lbm h 2 = h 1 - w P = Btu/lbm C.V. Boiler: q H = h 3 - h 2 = = Btu/lbm C.V. Tubine: w T = h 3 - h 4, s 4 = s 3 x 4 = , h 4 = w T = = Btu/lbm η CYCLE = (w T + w P )/q H = ( )/ = C.V. Condenser: q L = h 4 - h 1 = = Btu/lbm E Consider a simple ideal Rankine cycle using water at a supercritical pressure. Such a cycle has a potential advantage of minimizing local temperature differences between the fluids in the steam generator, such as the instance in which the high-temperature energy source is the hot exhaust gas from a gasturbine engine. Calculate the thermal efficiency of the cycle if the state entering the turbine is 3500 lbf/in. 2, 1100 F, and the condenser pressure is 1 lbf/in. 2. What is the steam quality at the turbine exit? s 4 = s 3 = = x x 4 = very low for a turbine exhaust h 4 = 848.3, h 1 = 69.73, h 3 = 1496 s 2 = s 1 = => h 2 = q H = h 3 - h 2 = Btu/lbm w net = h 3 - h 4 - (h 2 - h 1 ) = Btu/lbm η = w net /q H = 637.3/ = 0.45 T lbf/in F 4 1 lbf/in 2 s

148 E Consider an ideal steam reheat cycle in which the steam enters the highpressure turbine at 600 lbf/in. 2, 700 F, and then expands to 120 lbf/in. 2. It is then reheated to 700 F and expands to 2 lbf/in. 2 in the low-pressure turbine. Calculate the thermal efficiency of the cycle and the moisture content of the steam leaving the low-pressure turbine. Basic cycle as in plus reheat. From solution : -w P = 1.8 Btu/lbm, h 2 = h 3 = , s 3 = q H1 = h 3 -h 2 = = CV T1: w T1 = h 3 - h 4 ; s 4 = s 3 => x 4 = , h 4 = 1190 Btu/lbm CV T2: w T2 = h 5 - h 6 ; s 6 = s 5 = , h 5 = => x 6 = , h 6 = , w T2 = = w T,tot = w T1 + w T2 = = Btu/lbm T s q H = q H1 + h 5 - h 4 = = 1443 Btu/lbm η CYCLE = w T,tot /q H = 503.1/1443 = E A closed feedwater heater in a regenerative steam power cycle heats 40 lbm/s of water from 200 F, 2000 lbf/in. 2 to 450 F, 2000 lbf/in. 2. The extraction steam from the turbine enters the heater at 500 lbf/in. 2, 550 F and leaves as saturated liquid. What is the required mass flow rate of the extraction steam? 2 3 From the steam tables: h 1 1 = h 2 = all h 3 = h 4 = Btu/lbm C.V. Feedwater Heater 4 ṁ 3 = ṁ 1 (h 1 - h 2 )/(h 4 - h 3 ) = lbm = s

149 E Consider an ideal steam regenerative cycle in which steam enters the turbine at 600 lbf/in. 2, 700 F, and exhausts to the condenser at 2 lbf/in. 2. Steam is extracted from the turbine at 120 lbf/in. 2 for an open feedwater heater. The feedwater leaves the heater as saturated liquid. The appropriate pumps are used for the water leaving the condenser and the feedwater heater. Calculate the thermal efficiency of the cycle and the net work per pound-mass of steam. From solution : h 5 = , s 5 = h 7 = , h 1 = ST. GEN. 5 TURBINE. 2nd law: s 6 = s 5 => x 6 = h 6 = 1190 Btu/lbm Also: h 3 = w P12 = (120-2)144/778 = 0.35 w P34 = ( )144/778 = 1.59 P 4 T 3 FW HTR COND. P psi 120 psi h 2 = 94.36, h 4 = CV: feedwater heater psi s cont : m 3 = m 6 + m 2 = 1 lbm, 1st law: m 6 h 6 + m 2 h 2 = m 3 h 3 m 6 (1190) + (1 - m 6 )(94.36) = m 6 = , m 2 = CV: turbine w T = (h 5 - h 6 ) + (1 - m 6 )(h 6 - h 7 ) = ( ) ( ) = Btu/lbm CV: pumps w P = m 2 w P12 + m 3 w P34 = = 1.87 Btu/lbm w NET = w T - w P = = 374 Btu/lbm CV: steam generator q H = h 5 - h 4 = = Btu/lbm η TH = w NET /q H = 374/ = 0.361

150 E Consider an ideal combined reheat and regenerative cycle in which steam enters the high-pressure turbine at 500 lbf/in. 2, 700 F, and is extracted to an open feedwater heater at 120 lbf/in. 2 with exit as saturated liquid. The remainder of the steam is reheated to 700 F at this pressure, 120 lbf/in. 2, and is fed to the lowpressure turbine. The condenser pressure is 2 lbf/in. 2. Calculate the thermal efficiency of the cycle and the net work per pound-mass of steam. 5: h 5 = , s 5 = : h 7 = , s 7 = : h 3 = h f = , v 3 = C.V. T1 1-x 6 s 5 = s 6 => h 6 = x w T1 = h 5 - h 6 = HTR = Btu/lbm 3 P C.V. Pump 1 -w P1 = h 2 - h 1 = v 1 (P 2 - P 1 ) = (120-2) = => h 2 = h 1 - w P1 = = Btu/lbm T C.V. FWH 7 T1 T2 8 1-x COND. 2 P F 7 x h 6 + (1 - x) h 2 = h 3 x = h 3 - h = h 6 - h = C.V. Pump psi -w P2 = h 4 - h 3 = v 3 (P 4 - P 3 ) = ( )(144/778) = 1.26 Btu/lbm => h 4 = h 3 - w P2 = = Btu/lbm q H = h 5 - h 4 + (1 - x)(h 7 - h 6 ) = = Btu/lbm C.V. Turbine 2 s 7 = s 8 => x 8 = ( )/1.746 = h 8 = h f + x 8 h fg = = w T2 = h 7 - h 8 = = w net = w T1 + (1 - x) w T2 + (1 - x) w P1 + w P2 = = kj/kg η cycle = w net / q H = / = s

151 E A steam power cycle has a high pressure of 500 lbf/in. 2 and a condenser exit temperature of 110 F. The turbine efficiency is 85%, and other cycle components are ideal. If the boiler superheats to 1400 F, find the cycle thermal efficiency. C.V. Turb.: w T = h 3 - h 4, s 4 = s 3 + s T,GEN Ideal: s 4S = s 3 = => x 4S = , h 4S = Btu/lbm => w T,S = = Btu/lbm Actual: w T,AC = η w T,S = Btu/lbm C.V. Pump: -w P = vdp v 1 (P 2 - P 1 ) C.V. Boiler: = ( )144/778 = 1.49 Btu/lbm q H = h 3 - h 2 = h 3 - h 1 + w P = Btu/lbm C.V. Condenser: q L = h 4ac - h 1 = = η = (w T,AC - w P )/q H = ( )/ = E The steam power cycle in Problem has an isentropic efficiency of the turbine of 85% and that for the pump it is 80%. Find the cycle efficiency and the specific work and heat transfer in the components. States numbered as in fig 9.3 of text. CV Pump: -w P,S = v 1 (P 2 -P 1 ) = (600-2)144/778 = 1.8 Btu/lbm -w P,AC = 1.8/0.8 = Btu/lbm h 2 = h 1 - w P,AC = = CV Turbine: w T,S = h 3 - h 4s, s 4 = s 3 = Btu/lbm R w T,S = = Btu/lbm w T,AC = h 3 - h 4AC = = h 4AC = CV Boiler: q H = h 3 - h 2 = = Btu/lbm q L = h 4AC - h 1 = = Btu/lbm η CYCLE = (w T + w P )/q H = ( )/ = Compared to ( )/ = 0.33 in the ideal case.

152 E Steam leaves a power plant steam generator at 500 lbf/in. 2, 650 F, and enters the turbine at 490 lbf/in. 2, 625 F. The isentropic turbine efficiency is 88%, and the turbine exhaust pressure is 1.7 lbf/in. 2. Condensate leaves the condenser and enters the pump at 110 F, 1.7 lbf/in. 2. The isentropic pump efficiency is 80%, and the discharge pressure is 520 lbf/in. 2. The feedwater enters the steam generator at 510 lbf/in. 2, 100 F. Calculate the thermal efficiency of the cycle and the entropy generation of the flow in the line between the steam generator exit and the turbine inlet, assuming an ambient temperature of 77 F. 2 1 ST. GEN. 6 TURBINE. η st = COND. 5 P 4 T 500 lbf/in F 625 F lbf/in 2 5s 6 4 3s 3 s η ST = 0.88, η SP = 0.80 h 1 = , h 2 = s 3S = s 2 = = x 3S => x 3S = h 3S = = w ST = h 2 - h 3S = = w T = η ST w ST = = Btu/lbm h 3 = h 2 - w T = = w SP = ( ) 144 = 1.55 Btu/lbm 778 -w p = -w SP /η SP = 1.55/0.80 = 1.94 Btu/lbm q H = h 1 - h 6 = = Btu/lbm η TH = w NET /q H = ( )/ = C.V. Line from 1 to 2: w = /0, Energy Eq.: q = h 2 - h 1 = = - 14 Btu/lbm Entropy Eq.: s 1 + s gen + q/t 0 = s 2 => s gen = s 2 - s 1 -q/t 0 = (-14/536.7) = Btu/lbm R

153 E In one type of nuclear power plant, heat is transferred in the nuclear reactor to liquid sodium. The liquid sodium is then pumped through a heat exchanger where heat is transferred to boiling water. Saturated vapor steam at 700 lbf/in. 2 exits this heat exchanger and is then superheated to 1100 F in an external gas-fired superheater. The steam enters the turbine, which has one (open-type) feedwater extraction at 60 lbf/in. 2. The isentropic turbine efficiency is 87%, and the condenser pressure is 1 lbf/in. 2. Determine the heat transfer in the reactor and in the superheater to produce a net power output of 1000 Btu/s. 700 lbf/in 2 6 T F SUP. HT. TURBINE. 60 lbf/in 2 Q s 7 REACT. HTR lbf/in 2 3 COND. 1 8s P 1 s P Ẇ NET = 1000 Btu/s, η ST = w P12 = (60-1)144/778 = 0.18 Btu/lbm h 2 = h 1 - w P12 = = Btu/lbm -w P34 = (700-60)144/778 = 2.06 Btu/lbm h 4 = h 3 - w P34 = = Btu/lbm s 7S =s 6 = , P 7 => T 7S = F, h 7S = h 7 = h 6 - η ST (h 6 - h 7S ) = ( ) = s 8S = s 6 = = x 8S => x 8S = h 8S = = Btu/lbm h 8 = h 6 - η ST (h 6 - h 8S ) = ( ) = CV: heater: cont: m 2 + m 7 = m 3 = 1.0 lbm, 1st law: m 2 h 2 + m 7 h 7 = m 3 h 3 m 7 = ( ) / ( ) = CV: turbine: w T = (h 6 - h 7 ) + (1 - m 7 )(h 7 - h 8 ) = ( ) = Btu/lbm CV pumps: w P = m 1 w P12 + m 3 w P34 = -( ) = -2.2 Btu/lbm w NET = = Btu/lbm => ṁ = 1000/513.5 = lbm/s CV: reactor Q. REACT = ṁ(h 5 -h ) = 1.947( ) = Btu/s 4 CV: superheater Q. SUP = ṁ(h 6 - h ) = 1.947( ) = 825 Btu/s 5

154 E A boiler delivers steam at 1500 lbf/in. 2, 1000 F to a two-stage turbine as shown in Fig After the first stage, 25% of the steam is extracted at 200 lbf/in. 2 for a process application and returned at 150 lbf/in. 2, 190 F to the feedwater line. The remainder of the steam continues through the low-pressure turbine stage, which exhausts to the condenser at 2 lbf/in. 2. One pump brings the feedwater to 150 lbf/in. 2 and a second pump brings it to 1500 lbf/in. 2. Assume the first and second stages in the steam turbine have isentropic efficiencies of 85% and 80% and that both pumps are ideal. If the process application requires 5000 Btu/s of power, how much power can then be cogenerated by the turbine? 3: h 3 = , s 3 = s: s 4S = s 3 h 4S = w T1,S = h 3 - h 4S = w T1,AC = Btu/lbm h 4AC = h 3 - w T1,AC = ac: P 4, h 4AC s 4AC = s: s 5S = s 4AC h 5S = w T2,S = h 4AC - h 5S = P2 B 2 3 T1 4 Proc Btu/s 7 P1 6 T2 5 C w T2,AC = = h 4AC - h 5AC h 5AC = Btu/lbm 7: h 7 = ; q PROC = h 4AC - h 7 = Btu/lbm ṁ 4 = Q. /q PROC = 5000/ = lbm/s = 0.25 ṁ TOT ṁ TOT = ṁ 3 = lbm/s, ṁ 5 = ṁ 3 - ṁ 4 = lbm/s Ẇ T = ṁ 3 h 3 - ṁ 4 h 4AC - ṁ 5 h 5AC = 7221 Btu/s

155 E A large stationary Brayton cycle gas-turbine power plant delivers a power output of hp to an electric generator. The minimum temperature in the cycle is 540 R, and the maximum temperature is 2900 R. The minimum pressure in the cycle is 1 atm, and the compressor pressure ratio is 14 to 1. Calculate the power output of the turbine, the fraction of the turbine output required to drive the compressor and the thermal efficiency of the cycle? Brayton: T ẇ NET = hp P 1 = 1 atm, T 1 = 540 R 2 P 2 /P 1 = 14, T 3 = 2900 R s a) Assume const C P0 : 1 s 2 = s 1 T 2 = T 1 ( P 2 P 1 ) k-1 k = 540(14) = R -w C = -w 12 = h 2 - h 1 = C P0 (T 2 -T 1 ) 3 s 4 s = 0.24( ) = Btu/lbm As s 4 = s 3 T 4 = T 3 ( P 4 P 3 ) k-1 k = 2900( 1 14 )0.286 = R w T = w 34 = h 3 - h 4 = C P0 (T 3 -T 4 ) = 0.24( ) = Btu/lbm w NET = w T + w C = = Btu/lbm ṁ = Ẇ NET /w NET = /222.7 = lbm/h Ẇ T = ṁw T = hp, -w C /w T = b) q H = C P0 (T 3 -T 2 ) = 0.24( ) = Btu/lbm η TH = w NET /q H = 222.7/420.3 = 0.530

156 E An ideal regenerator is incorporated into the ideal air-standard Brayton cycle of Problem Calculate the cycle thermal efficiency with this modification. T 2 s 1 x y 3 s 4 s Problem ideal regen., where : w T = 368.8, w C = Btu/lbm w NET = Btu/lbm Ideal regen.: T X = T 4 = R q H = h 3 - h X = 0.24( ) = Btu/lbm = w T η TH = w NET /q H = 222.7/368.8 = E Consider an ideal gas-turbine cycle with two stages of compression and two stages of expansion. The pressure ratio across each compressor stage and each turbine stage is 8 to 1. The pressure at the entrance to the first compressor is 14 lbf/in. 2, the temperature entering each compressor is 70 F, and the temperature entering each turbine is 2000 F. An ideal regenerator is also incorporated into the cycle. Determine the compressor work, the turbine work, and the thermal efficiency of the cycle. 10 REG 1 I.C CC 6 9 COMP COMP TURB TURB 7 8 CC P 2 /P 1 = P 4 /P 3 = P 6 /P 7 = P 8 /P 9 = 8.0 P 1 = 14 lbf/in 2 T 6 8 T 1 = T 3 = 70 F, T 6 = T 8 = 2000 F Assume const. specific heat s 2 = s 1 and s 4 = s 3 k-1 T 4 = T 2 = T 1 (P 2 /P 1 ) k = (8) = R s Total compressorwork -w C = 2 (-w 12 ) = 2C P0 (T 2 - T 1 ) = ( ) = Btu/lbm

157 Also s 6 = s 7 and s 8 = s 9 T 7 = T 9 = T 6 P k-1 7 P 6 k = = R Total turbinework w T = 2 w 67 = 2C P0 (T 6 - T 7 ) = ( ) = Btu/lbm w NET = = Btu/lbm Ideal regenerator: T 5 = T 9, T 10 = T 4 q H = (h 6 - h 5 ) + (h 8 - h 7 ) = 2C P0 (T 6 - T 5 ) = ( ) = w T = Btu/lbm η TH = w NET /q H = / = E Repeat Problem , but assume that each compressor stage and each turbine stage has an isentropic efficiency of 85%. Also assume that the regenerator has an efficiency of 70%. T 4S = T 2S = R, -w CS = T 7S = T 9S = R, w TS = w C = -w SC /η SC = Btu/lbm -w 12 = -w 34 = 242.7/2 = T 2 = T 4 = T 1 + (-w 12 /C P0 ) = /0.24 = R T 4S S 2S S s w T = η T w TS = T 7 = T 9 = T 6 - (+w 67 /C P0 ) = / = 1523 R η REG = h 5 - h 4 T 5 - T 4 = = h 9 - h 4 T 9 - T 4 q H = C P0 (T 6 - T 5 ) + C P0 (T 8 - T 7 ) T = 0.7 T 5 = R = 0.24( ) ( ) = Btu/lbm w NET = w T + w C = = Btu/lbm η TH = w NET /q H = 206.8/484.7 = 0.427

158 E An air-standard Ericsson cycle has an ideal regenerator as shown in Fig. P Heat is supplied at 1800 F and heat is rejected at 68 F. Pressure at the beginning of the isothermal compression process is 10 lbf/in. 2. The heat added is 275 Btu/lbm. Find the compressor work, the turbine work, and the cycle efficiency. P T T 4 = T 3 = 1800 F 2 P 3 3 T 4 T 1 = T 2 = 68 F = 527.7R T P T 1 = 10 lbf/in 2 P 1 P 4 P 2 q 3 = - 4 q (ideal reg.) 1 2 q T 1 H = 3 q 4 & v s w T = q H = 275 Btu/lbm η TH = η CARNOT TH. = 1 - T L /T H = / = w net = η TH q H = = Btu/lbm q L = -w C = = Btu/lbm E The turbine in a jet engine receives air at 2200 R, 220 lbf/in. 2. It exhausts to a nozzle at 35 lbf/in. 2, which in turn exhausts to the atmosphere at 14.7 lbf/in. 2. The isentropic efficiency of the turbine is 85% and the nozzle efficiency is 95%. Find the nozzle inlet temperature and the nozzle exit velocity. Assume negligible kinetic energy out of the turbine. C.V. Turb.: h i = , P ri = , s e = s i P re = P ri (P e /P i ) = (35/220) = T e = 1381, h e = , w T,s = = w T,AC = w T,s η T = = h e,ac - h i h e,ac = T e,ac = R, P re,ac = C.V. Nozzle: (1/2)V e 2 = h i - h e ; s e = s i P re = P ri (P e /P i ) = (14.7/35) = T e,s = R, h e,s = Btu/lbm (1/2)V 2 e,s = h i - h e,s = = 80.2 Btu/lbm 2 (1/2)V e,ac = (1/2)V 2 e,s η NOZ = Btu/lbm V e,ac = = 1953 ft/s

159 E Air flows into a gasoline engine at 14 lbf/in. 2, 540 R. The air is then compressed with a volumetric compression ratio of 8 1. In the combustion process 560 Btu/lbm of energy is released as the fuel burns. Find the temperature and pressure after combustion. Compression 1 to 2: s 2 = s 1 v r2 = v r1 /8 = /8 = T 2 = 1212 R, u 2 = Btu/lbm, P r2 = 20 P 2 = P r2 (P 1 /P r1 ) = 20(14/1.1146) = lbf/in 2 Compression 2 to 3: u 3 = u 2 + q H = = 771.3, T 3 = 3817 R P 3 = P 2 (T 3 /T 2 ) = 251.2(3817/1212) = 791 lbf/in E To approximate an actual spark-ignition engine consider an air-standard Otto cycle that has a heat addition of 800 Btu/lbm of air, a compression ratio of 7, and a pressure and temperature at the beginning of the compression process of 13 lbf/in. 2, 50 F. Assuming constant specific heat, with the value from Table C.4, determine the maximum pressure and temperature of the cycle, the thermal efficiency of the cycle and the mean effective pressure. P 3 T 3 q H = 800 Btu v 2 s s 4 v 1 2 s 1 v v s 4 v 1 /v 2 = 7 P 1 = 13 lbf/in 2, T 1 = 50 F v 1 = RT 1 /P 1 = / v s = ft 3 /lbm v 2 = v 1 /7 = ft 3 /lbm a) P 2 = P 1 (v 1 /v 2 ) k = 13(7) 1.4 = lbf/in 2 T 2 = T 1 (v 1 /v 2 ) k-1 = 510(7) 0.4 = R T 3 = T 2 + q H /C V0 = /0.171 = 5789 R P 3 = P 2 T 3 /T 2 = / = 1033 lbf/in 2 b) η TH = 1 - (T 1 /T 2 ) = 1-510/ = c) w NET = η TH q H = = Btu mep = w NET v 1 -v 2 = = 188 lbf/in2 ( ) 144

160 E In the Otto cycle all the heat transfer q H occurs at constant volume. It is more realistic to assume that part of q H occurs after the piston has started its downwards motion in the expansion stroke. Therefore consider a cycle identical to the Otto cycle, except that the first two-thirds of the total q H occurs at constant volume and the last one-third occurs at constant pressure. Assume the total q H is 700 Btu/lbm, that the state at the beginning of the compression process is 13 lbf/in. 2, 68 F, and that the compression ratio is 9. Calculate the maximum pressure and temperature and the thermal efficiency of this cycle. Compare the results with those of a conventional Otto cycle having the same given variables. P 3 4 T s 2 5 s 1 v 4 3 s v 2 5 s v 1 s P 2 = P 1 (v 1 /v 2 ) k = 13(9) 1.4 = lbf/in 2 T 2 = T 1 (v 1 /v 2 ) k-1 = (9) 0.4 = R P 1 = 13, T 1 = R r V = v 1 /v 2 = 7 q 23 = 2 Btu 700 = lbm q 34 = 1 Btu 700 = lbm T 3 = T 2 + q 23 /C V0 = /0.171 = 4000 R P 3 = P 2 (T 3 /T 2 ) = / = lbf/in 2 = P 4 T 4 = T 3 + q 34 /C P0 = /0.24 = 4972 R v 5 v 1 = = (P v 4 v 4 /P 1 ) (T 1 /T 4 ) = = T 5 = T 4 (v 4 /v 5 ) k-1 = 4972(1/7.242) 0.4 = 2252 R q L = C V0 (T 5 -T 1 ) = 0.171( ) = Btu/lbm η TH = 1 - q L /q H = /700 = Std Otto cycle: η TH = 1 - (9) -0.4 = 0.585

161 E It is found experimentally that the power stroke expansion in an internal combustion engine can be approximated with a polytropic process with a value of the polytropic exponent n somewhat larger than the specific heat ratio k. Repeat Problem but assume the expansion process is reversible and polytropic (instead of the isentropic expansion in the Otto cycle) with n equal to See solution to : T 3 = 5789 R, P 3 = 1033 lbf/in 2 v 3 = RT 3 /P 3 = v 2 = ft 3 /lbm, v 4 = v 1 = except for process 3 to 4: Pv 1.5 = constant. P 4 = P 3 (v 3 /v 4 ) 1.5 = 1033(1/7) 1.5 = lbf/in 2 T 4 = T 3 (v 3 /v 4 ) 0.5 = 5789(1/7) 0.5 = 2188 R 1 w 2 = P dv = R(T 2 - T 1 )/(1-1.4) = ( )/( ) = Btu/lbm 3 w 4 = P dv = R(T 4 - T 3 )/(1-1.5) = ( )/( ) = Btu/lbm w NET = = Btu/lbm η CYCLE = w NET /q H = /700 = mep = w NET /(v 1 -v 2 ) = /( ) = lbf/in 2 Notice a smaller w NET, η CYCLE, mep compared to ideal cycle E A diesel engine has a bore of 4 in., a stroke of 4.3 in. and a compression ratio of 19:1 running at 2000 RPM (revolutions per minute). Each cycle takes two revolutions and has a mean effective pressure of 200 lbf/in. 2. With a total of 6 cylinders find the engine power in Btu/s and horsepower, hp. Power from mean effective pressure. mep = w net / (v max - v min ) -> w net = mep(v max - v min ) V = πbore S = π = in 3 W = mep(v max - V min ) = /(12 778) = Btu Work per Cylinders per Power Stroke Ẇ = W n RPM 0.5 (cycles / min) (min / 60 s) (Btu / cycle) = (1/60) = Btu/s = / hp = 164 hp

162 E At the beginning of compression in a diesel cycle T = 540 R, P = 30 lbf/in. 2 and the state after combustion (heat addition) is 2600 R and 1000 lbf/in. 2. Find the compression ratio, the thermal efficiency and the mean effective pressure. P 2 = P 3 = 1000 lbf/in 2 T 2 = T 1 (P 2 /P 1 ) (k-1)/k = 540(1000/30) = v 1 /v 2 = (P 2 /P 1 ) 1/k = (1000/30) = v 3 /v 2 = T 3 /T 2 = 2600/ = v 4 /v 3 = v 1 /v 3 = (v 1 /v 2 )(v 2 /v 3 ) = 12.24/1.768 = T 4 = T 3 (v 3 /v 4 ) k-1 = 2600* = 1199 q L = C V (T 4 - T 1 ) = 0.171( ) = q H = h 3 - h 2 = C P (T 3 - T 2 ) = 0.24( ) = η = 1 - q L /q H = / = w net = q net = = v max = v 1 = RT 1 /P 1 = /(30 144) = v min = v max (v 1 /v 2 ) = / = mep = [158.4/( )] (778/144) = lbf/in 2

163 E Consider an ideal air-standard diesel cycle where the state before the compression process is 14 lbf/in. 2, 63 F and the compression ratio is 20. Find the maximum temperature(by iteration) in the cycle to have a thermal efficiency of 60%. Diesel cycle: P 1 = 14, T 1 = R, v 1 /v 2 = 20, η TH = 0.60 T 2 = T 1 (v 1 /v 2 ) k-1 = (20) 0.4 = R v 1 = = , v = = v 3 = v 2 T 3 /T 2 = T 3 / = T 3 T 3 T 4 =( v 4 v 3 ) k-1 =( T 3 ) 0.4 T 4 = T T 4 -T 1 η TH = 0.60 = 1 - k(t 3 -T 2 ) = T (T ) T T = R: LHS = , 5500R: LHS = 5.04, 5450R: LHS = -0.5 Linear interpolation, T 3 = 5455 R 1.4

164 E Consider an ideal Stirling-cycle engine in which the pressure and temperature at the beginning of the isothermal compression process are 14.7 lbf/in. 2, 80 F, the compression ratio is 6, and the maximum temperature in the cycle is 2000 F. Calculate the maximum pressure in the cycle and the thermal efficiency of the cycle with and without regenerators. P 3 v 2 T T 4 v 1 v T 2 T v 1 3 T 4 v s Ideal Stirling cycle T 1 = T 2 = 80 F P 1 = 14.7 lbf/in 2 v 1 = 6 v 2 T 3 = T 4 = 2000 F T 1 = T 2 P 2 = P 1 v 1 /v 2 = = 88.2 V 2 = V 3 P 3 = P 2 T 3 /T 2 = w 34 = q 34 = RT 3 ln (v 4 /v 3 ) = lbf/in2 = (53.34/778) 2460 ln 6 = Btu/lbm q 23 = C V0 (T 3 -T 2 ) = 0.171( ) = Btu/lbm v 1 w 12 = q 12 = -RT 1 ln = ln 6 = Btu/lbm v w NET = = Btu/lbm η NO REGEN = = 0.374, η WITH REGEN = = 0.781

165 E An ideal air-standard Stirling cycle uses helium as working fluid. The isothermal compression brings the helium from 15 lbf/in. 2, 70 F to 90 lbf/in. 2. The exspansion takes place at 2100 R and there is no regenerator. Find the work and heat transfer in all four processes per lbm helium and the cycle efficiency. Substance helium C.4: R = 386 ft-lbf/lbmr C V = v 4 /v 3 = v 1 /v 2 = P 2 /P 1 = 90/15 = 6 1 -> 2: - 1 w 2 = - 1 q 2 = P dv = ln(6)/778 = Btu/lbm 2 -> 3: 2 w 3 = 0; 2 q 3 = C P (T 3 - T 2 ) = 0.753( ) = > 4: 3 w 4 = 3 q 4 = RT 3 ln(v 4 /v 3 ) = ln(6)/778 = > 1: 4 w 1 = 0; 4 q 1 = C P (T 4 - T 1 ) = η Cycle = w net / q H = ( ) / ( ) = E The air-standard Carnot cycle was not shown in the text; show the T s diagram for this cycle. In an air-standard Carnot cycle the low temperature is 500 R and the efficiency is 60%. If the pressure before compression and after heat rejection is 14.7 lbf/in. 2, find the high temperature and the pressure just before heat addition. η = 0.6 = 1 - T H /T L T H = T L /0.4 = 500/0.4 = 1250 R 1 P 2 = P 1 (T H /T L ) k-1 = 14.7( )3.5 = lbf/in 2 T H T L [or P 2 = P 1 (P r2 /P r1 ) = / = 388 lbf/in 2 ] T s

166 E Air in a piston/cylinder goes through a Carnot cycle in which T L = 80.3 F and the total cycle efficiency is η = 2/3. Find T H, the specific work and volume ratio in the adiabatic expansion for constant C p, C v. Repeat the calculation for variable heat capacities. Carnot cycle: Same process diagram as in previous problem. η = 1 - T L /T H = 2/3 T H = 3 T L = = 1620 R Adiabatic expansion 3 to 4: Pv k = constant 3 w 4 = (P 4 v 4 - P 3 v 3 )/(1 - k) = [R/(1-k)](T 4 - T 3 ) = u 3 - u 4 = C v (T 3 - T 4 ) = 0.171( ) = Btu/lbm v 4 /v 3 = (T 3 /T 4 ) 1/(k - 1) = = 15.6 For variable C p, C v we get, T H = 3 T L = 1620 R 3 w 4 = u 3 - u = = Btu/lbm 4 v 4 /v 3 = v r4 /v r3 = / = E Consider an ideal refrigeration cycle that has a condenser temperature of 110 F and an evaporator temperature of 5 F. Determine the coefficient of performance of this refrigerator for the working fluids R-12 and R-22. T Ideal Ref. Cycle 2 T cond = 110 F = T 3 3 T evap = 5 F Use Table C.10 for R-22 Use computer table for R R-12 R-22 h 1, Btu/lbm s 2 = s P 2, lbf/in T 2, F h 2, Btu/lbm h 3 =h 4, Btu/lbm w C = h 2 -h q L = h 1 -h β =q L /(-w C ) s

167 E The environmentally safe refrigerant R-134a is one of the replacements for R-12 in refrigeration systems. Repeat Problem using R-134a and compare the result with that for R-12. T Ideal Ref. Cycle 2 T cond = 110 F 3 T evap = 5 F h 1 = Btu/lbm 4 1 s 2 = s 1 = Btu/lbm R s P 2 = lbf/in 2, T 2 = F, h 2 = Btu/lbm h 3 =h 4 = Btu/lbm -w C = h 2 -h 1 = Btu/lbm ; q L = h 1 -h 4 = Btu/lbm β =q L /(-w C ) = E Consider an ideal heat pump that has a condenser temperature of 120 F and an evaporator temperature of 30 F. Determine the coefficient of performance of this heat pump for the working fluids R-12, R-22, and ammonia. T Ideal Heat Pump 2 T cond = 120 F 3 T evap = 30 F Use Table C.9 for NH Use Table C.10 for R-22 Use computer table for R-12 s R-12 R-22 NH 3 h 1, Btu/lbm s 2 = s P 2, lbf/in T 2, F h 2, Btu/lbm h 3 =h 4, Btu/lbm w C = h 2 -h q H = h 2 -h β =q H /(-w C )

168 E The refrigerant R-22 is used as the working fluid in a conventional heat pump cycle. Saturated vapor enters the compressor of this unit at 50 F; its exit temperature from the compressor is measured and found to be 185 F. If the isentropic efficiency of the compressor is estimated to be 70%, what is the coefficient of performance of the heat pump? T R-22 heat pump: T 2 = 185 F 2 2S T EVAP = 50 F, η S COMP = Isentropic compressor: s 2S = s 1 = but P 2 unknown. Trial & error. Assume P 2 = 307 lbf/in 2 s At P 2,s 2S : T 2S = 162 F, h 2S = , At P 2,T 2 : h 2 = calculate η S COMP = h 2S - h = = h 2 - h OK P 2 = 307 lbf/in 2 = P 3 T 3 = F, h 3 = w C = h 2 - h 1 = 17.29, q H = h 2 - h 3 = 77.58, β = q H /(-w C ) = E Consider a small ammonia absorption refrigeration cycle that is powered by solar energy and is to be used as an air conditioner. Saturated vapor ammonia leaves the generator at 120 F, and saturated vapor leaves the evaporator at 50 F. If 3000 Btu of heat is required in the generator (solar collector) per pound-mass of ammonia vapor generated, determine the overall performance of this system. NH 3 absorption cycle: T sat. vapor at 120 F exits the generator. Sat. vapor at 50 F exits the evaporator q H = q GEN = 3000 Btu/lbm NH 3 out of generator. 120F 50 F q L = h 2 - h 1 = h G 50 F - h F 120 F = GEN. EXIT EVAP EXIT s = Btu/lbm q L /q H = /3000 =

169 E Consider an ideal dual-loop heat-powered refrigeration cycle using R-12 as the working fluid, as shown in Fig. P Saturated vapor at 220 F leaves the boiler and expands in the turbine to the condenser pressure. Saturated vapor at 0 F leaves the evaporator and is compressed to the condenser pressure. The ratio of the flows through the two loops is such that the turbine produces just enough power to drive the compressor. The two exiting streams mix together and enter the condenser. Saturated liquid leaving the condenser at 110 F is then separated into two streams in the necessary proportions. Determine the ratio of mass flow rate through the power loop to that through the refrigeration loop. Find also the performance of the cycle, in terms of the ratio Q L /Q H. TURB. COMP BOIL. COND. 5 3 P E V A P Q L T s T P h s Computer tables for F lbf/in 2 Btu/lbm Btu/lbm R properties P 2 =P 3 =P SAT at 110 F P 5 =P 6 =P SAT at 220 F s 2 =s 1 = h 2 = Pump work: w P = h 5 -h v 5 (P 5 -P 3 ) -w P = ( ) = h 5 = = Btu/lbm (1-x 7 ) = = = h 7 = (54.313) = Btu/lbm

170 CV: turbine + compressor Continuity Eq.: ṁ 1 = ṁ 2, ṁ 6 = ṁ 7 Energy Eq.: ṁ 1 h 1 + ṁ 6 h 6 = ṁ 2 h 2 + ṁ 7 h 7 ṁ 1 /ṁ 6 = = = 0.545, ṁ 6 /ṁ 1 = CV: pump: -w P = v 3 (P 5 -P 3 ), h 5 = h 3 - w P CV evaporator: Q. L = ṁ 1 (h 1 -h 4 ), CV boiler: Q. H = ṁ 6 (h 6 -h 5 ) β = Q. L /Q. H = ṁ 1 (h 1 -h 4 ) ṁ 6 (h 6 -h 5 ) = ( ) = E (Adv.) Find the availability of the water at all four states in the Rankine cycle described in Problem Assume the high-temperature source is 900 F and the low-temperature reservoir is at 65 F. Determine the flow of availability in or out of the reservoirs per pound-mass of steam flowing in the cycle. What is the overall cycle second law efficiency? Ref. state 14.7 lbf/in 2, 77 F, h 0 = Btu/lbm, s 0 = Btu/lbm R From solution to : s 1 = , s 2 = = s 1, s 4 = s 3 = Btu/lbm R h 1 = 94.01, h 2 = 95.81, h 3 = , h 4 = ψ = h - h 0 - T 0 (s - s 0 ) ψ 1 = ( ) = ψ 2 = 3.92, ψ 3 = , ψ 4 = Btu/lbm ψ H = (1 - T 0 /T H )q H = = Btu/lbm ψ L = (1 - T 0 /T 0 )q C = 0/ η II = w NET / ψ H = ( )/ = Notice T H > T 3, T L < T 4 = T 1, so cycle is externally irreversible. Both q H and q C over finite T.

171 12-1 CHAPTER 12 The correspondence between the new problem set and the previous 4th edition chapter 11 problem set. New Old New Old New Old 1 New new 2 New 32 new New 33 new New new 66 new 7 new 37 new new new 39 new new 40 new new new 42 new new 44 new new mod 17 new mod 18 new new 51 new 22 new 52 new a mod 54 new 25 8b mod 55 new new 27 new new new The problems that are labeled advanced start at number 74.

172 12-2 The English unit problems are: New Old New Old New Old new new new mod new new

173 A gas mixture at 20 C, 125 kpa is 50% N 2, 30% H 2 O and 20% O 2 on a mole basis. Find the mass fractions, the mixture gas constant and the volume for 5 kg of mixture. From Eq. 12.3: c i = y i M i / y j M j M MIX = y j M j = = = c N2 = / = , c H2O = / = c O2 = / = , sums to 1 OK R MIX = R /M MIX = / = kj/kg K V = mr MIX T/P = /125 = m A 100 m 3 storage tank with fuel gases is at 20 C, 100 kpa containing a mixture of acetylene C 2 H 2, propane C 3 H 8 and butane C 4 H 10. A test shows the partial pressure of the C 2 H 2 is 15 kpa and that of C 3 H 8 is 65 kpa. How much mass is there of each component? Assume ideal gases, then the ratio of partial to total pressure is the mole fraction, y = P/P tot y C2H2 = 15/100 = 0.15, y C3H8 = 65/100 = 0.65, y C4H10 = 20/100 = 0.20 n tot = PV/R T = / ( ) = kmoles m C2H2 = (nm) C2H2 = y C2H2 n tot M C2H2 = = kg m C3H8 = (nm) C3H8 = y C3H8 n tot M C3H8 = = kg m C4H10 = (nm) C4H10 = y C4H10 n tot M C4H10 = = kg 12.3 A mixture of 60% N 2, 30% Ar and 10% O 2 on a mole basis is in a cylinder at 250kPa, 310 K and volume 0.5 m 3. Find the mass fractions and the mass of argon. From Eq. 12.3: c i = y i M i / y j M j M MIX = y j M j = = c N2 = ( ) / = , c Ar = ( ) / = c O2 = ( ) / = 0.1, sums to 1 OK R MIX = R /M MIX = / = kj/kg K m MIX = PV/(R MIX T) = / = kg m Ar = c Ar m MIX = = kg

174 A carbureted internal combustion engine is converted to run on methane gas (natural gas). The air-fuel ratio in the cylinder is to be 20 to 1 on a mass basis. How many moles of oxygen per mole of methane are there in the cylinder? The mass ratio m AIR /m CH4 = 20, so relate mass and mole n = m/m n AIR n CH4 = ( m AIR m CH4 ) M CH4 /M AIR = /28.97 = n O2 n CH4 = = mole O 2 /mole CH Weighing of masses gives a mixture at 60 C, 225 kpa with 0.5 kg O 2, 1.5 kg N 2 and 0.5 kg CH 4. Find the partial pressures of each component, the mixture specific volume (mass basis), mixture molecular weight and the total volume. From Eq. 12.4: y i = (m i /M i ) / m j /M j n tot = m j /M j = (0.5/31.999) + (1.5/28.013) + (0.5/16.04) = = y O2 = / = , y N2 = / = , y CH4 = / = P O2 = y O2 P tot = = 35 kpa, P N2 = y N2 P tot = = 120 kpa P CH4 = y CH4 P tot = = 70 kpa V tot = n tot R T/P = / 225 = m 3 v = V tot /m tot = / ( ) = m 3 /kg M MIX = y j M j = m tot /n tot = 2.5/ =

175 At a certain point in a coal gasification process, a sample of the gas is taken and stored in a 1-L cylinder. An analysis of the mixture yields the following results: Component H 2 CO CO 2 N 2 Percent by volume Determine the mass fractions and total mass in the cylinder at 100 kpa, 20 C. How much heat transfer must be transferred to heat the sample at constant volume from the initial state to 100 C? Volume fractions same as mole fractions so From Eq. 12.3: c i = y i M i / y j M j y i M i = kg i /kg = c i kmol i /kmol kg i /kmol i = kg i /kmol = kg i /kg H = 0.504/ = CO = / = CO = / = N = / = M MIX = R MIX = R /M MIX = / = kj/kg/k m = PV/RT = / = kg C V0 MIX = c i C V0 i = = kj/kg K Q 12 = U 12 = mc V0 (T 2 -T 1 ) = (100-20) = kj 12.7 A pipe, cross sectional area 0.1 m 2, carries a flow of 75% O 2 and 25% N 2 by mole with a velocity of 25 m/s at 200 kpa, 290 K. To install and operate a mass flow meter it is necessary to know the mixture density and the gas constant. What are they? What mass flow rate should the meter then show? From Eq. 12.3: c i = y i M i / y j M j M MIX = y j M j = = R MIX = R /M MIX = / = kj/kg K v = R MIX T/P = / 200 = m 3 /kg ρ = 1/v = kg/m 3 ṁ = ρav = = kg/s

176 A pipe flows 0.05 kmole a second mixture with mole fractions of 40% CO 2 and 60% N 2 at 400 kpa, 300 K. Heating tape is wrapped around a section of pipe with insulation added and 2 kw electrical power is heating the pipe flow. Find the mixture exit temperature. C.V. Pipe heating section. Assume no heat loss to the outside, ideal gases. Energy Eq.: Q. = ṁ(h e h i ) = ṅ(h - e h- i ) = ṅc P mix (T e T i ) CP mix = y i C i = = kj/kmole T e = T i + Q. / ṅc P mix = /( ) = K 12.9 A rigid insulated vessel contains 0.4 kmol of oxygen at 200 kpa, 280 K separated by a membrane from 0.6 kmol carbon dioxide at 400 kpa, 360 K. The membrane is removed and the mixture comes to a uniform state. Find the final temperature and pressure of the mixture. C.V. Total vessel. Control mass with two different initial states. Mass: n = n O2 + n CO2 = = 1.0 kmole Process: V = constant (rigid) => W = 0, insulated => Q = 0 Energy: U 2 - U 1 = 0-0 = n O2 C V O2 (T 2 - T 1 O2 ) + n CO2 C V CO2 (T 2 - T 1 CO2 ) Initial state from ideal gas Table A.5 CV O2 = = , C V CO2 = = kj/kmole O 2 : V O2 = nr T 1 /P = /200 = m 3, CO 2 : V CO2 = nr T 1 /P = /400 = 4.49 m 3 n O2 C V O2 T 2 + n CO2 C V CO2 T 2 = n O2 C V O2 T 1 O2 + n CO2 C V CO2 T 1 CO2 ( ) T 2 = = kj => T 2 = K P 2 = nr T 2 /V = / = kpa An insulated gas turbine receives a mixture of 10% CO 2, 10% H 2 O and 80% N 2 on a mole basis at 1000 K, 500 kpa. The volume flow rate is 2 m 3 /s and its exhaust is at 700 K, 100 kpa. Find the power output in kw using constant specific heat from A.5 at 300 K. C.V. Turbine, SSSF, 1 inlet, 1 exit flow with an ideal gas mixture, q = 0. Energy Eq.: Ẇ T = ṁ(h i h e ) = ṅ(h - i h- e ) = ṅc P mix (T i T e ) PV = nr T => ṅ = PV. / RT = 500 2/( ) = kmole/s CP mix = y i C i = = kj/kmol K Ẇ T = ( ) = 1098 kw

177 Solve Problem using the values of enthalpy from Table A.8. C.V. Turbine, SSSF, 1 inlet, 1 exit flow with an ideal gas mixture, q = 0. Energy Eq.: Ẇ T = ṁ(h i h e ) = ṅ(h - i h- e ) = ṅ [ y j (h - i h- e ) j ] PV = nr T => ṅ = PV. / RT = 500 2/( ) = kmole/s Ẇ T = [0.1( ) + 0.1( ) + 0.8( )] = 1247 kw Consider Problem and find the value for the mixture heat capacity, mole basis and the mixture ratio of specific heats, k mix, both estimated at 850 K from values (differences) of h in Table A.8. With these values make an estimate for the reversible adiabatic exit temperature of the turbine at 100 kpa. We will find the individual heat capacities by : C P i = (h h- 800 )/( ) CP CO2 = ( )/100 = 52.24; C P H2O = ( )/100 = CP N2 = ( )/100 = CP mix = y i C P i = = kj/kmol CV mix = C P mix - R = = 26.26, k = CP mix /C V mix = Rev. ad. turbine => s = constant. Assume const. avg. heat capacities: k T e = T i (P e /P i ) k = 1000 (100/500) = 1000 (0.2) = 679 K The gas mixture from Problem 12.6 is compressed in a reversible adiabatic process from the initial state in the sample cylinder to a volume of 0.2 L. Determine the final temperature of the mixture and the work done during the process. C P0 MIX = C V0 MIX + R MIX = = k = C P0 /C V0 = / = T 2 = T 1 ( V 1 V 2 ) k-1 = ( ) = K W 12 = - U 12 = -mc V0 (T 2 -T 1 ) = ( ) = kj

178 Three SSSF flows are mixed in an adiabatic chamber at 150 kpa. Flow one is 2 kg/s of O 2 at 340 K, flow two is 4 kg/s of N 2 at 280 K and flow three is 3 kg/s of CO 2 at 310 K. All flows are at 150 kpa the same as the total exit pressure. Find the exit temperature and the rate of entropy generation in the process. C.V. Mixing chamber, no heat transfer, no work. 1 O Continuity: ṁ 1 + ṁ 2 + ṁ 3 = ṁ N2 Energy: ṁ 1 h 1 + ṁ 2 h 2 + ṁ 3 h 3 = ṁ 4 h 4 3 CO mix 2 Entropy: ṁ 1 s 1 + ṁ 2 s 2 + ṁ 3 s 3 + Ṡ gen = ṁ 4 s 4 Assume ideal gases and since T is close to 300 K use heat capacity from A.5 ṁ 1 C P O2 (T 1 - T 4 ) + ṁ 2 C P N2 (T 2 - T 4 ) + ṁ 3 C P CO2 (T 3 - T 4 )= = ( ) T 4 => = T 4 => T 4 = K State 4 is a mixture so the component exit pressure is the partial pressure. For each component s e s i = C P ln(t e / T i ) R ln(p e / P i ) and the pressure ratio is P e / P i = y P 4 / P i = y for each. n = m M = = = y O2 = = , y N2 = = , y CO2 = = Ṡ gen = ṁ 1 (s 4 - s 1 ) + ṁ 2 (s 4 - s 2 ) + ṁ 3 (s 4 - s 3 ) = 2 [ ln(301.83/340) ln(0.2285)] + 4 [ ln(301.83/280) ln(0.5222)] + 3 [ ln(301.83/310) ln(0.2493)] = = kw/k

179 Carbon dioxide gas at 320 K is mixed with nitrogen at 280 K in a SSSF insulated mixing chamber. Both flows are at 100 kpa and the mole ratio of carbon dioxide to nitrogen is 2 1. Find the exit temperature and the total entropy generation per mole of the exit mixture. CV mixing chamber, SSSF. The inlet ratio is ṅ CO2 = 2 ṅ N2 and assume no external heat transfer, no work involved. Continuity: ṅ CO2 + 2ṅ N2 = ṅ ex = 3ṅ N2 ; Energy Eq.: ṅ N2 (h - N 2 + 2h - CO 2 ) = 3ṅ N2 h - mix ex Take 300K as reference and write h - = h C- Pmix (T-300). C - P N 2 (T i N2-300) + 2C - P CO 2 (T i CO2-300) = 3C - P mix (T mix ex -300) C - P mix = y i C - P i = ( )/3 = C - P mix T mix ex = C- P N 2 T i N2 + 2C - P CO 2 T i CO2 = T mix ex = K; P ex N2 = P tot /3; P ex CO2 = 2P tot /3 Ṡ gen = ṅ ex s - ex -(ṅs- ) ico2 -(ṅs - ) in2 = ṅ N2 (s - e -s- i ) N 2 + 2ṅ N2 (s - e -s- i ) CO 2 Ṡ gen /3ṅ N2 = [C - PN 2 ln T ex T in2 - R ln y N2 + 2C - PCO 2 ln T ex T ico2-2 R ln y CO2 ]/3 = [ ]/3 = 5.35 kj/kmol mix K A piston/cylinder contains 0.5 kg argon and 0.5 kg hydrogen at 300 K, 100 kpa. The mixture is compressed in an adiabatic process to 400 kpa by an external force on the piston. Find the final temperature, the work and the heat transfer in the process. C.V. Mixture in cylinder. Control mass with adiabatic process: 1 Q 2 = 0 Cont.Eq.: m 2 = m 1 = m ; Energy Eq.: m(u 2 u 1 ) = 1 W 2 Entropy Eq.: m(s 2 s 1 ) = dq/t + 1 S 2 gen = assume reversible. R mix = c i R i = = C P mix = c i C Pi = = C V = C P - R = = , k = C P / C V = s 2 = s 1 => T 2 = T 1 (P 2 /P 1 ) (k-1)/k = 300 (400/100) = 451 K 1 W 2 = m(u 1 u 2 ) = mc V (T 1 T 2 ) = ( ) = kj

180 Natural gas as a mixture of 75% methane and 25% ethane by volume is flowing to a compressor at 17 C, 100 kpa. The reversible adiabatic compressor brings the flow to 250 kpa. Find the exit temperature and the needed work per kg flow. C.V. Compressor. SSSF, adiabatic q = 0, reversible s gen = 0 Energy Eq.: -w = h ex - h in ; Entropy Eq.: s i + s gen = s i = s e Process: reversible => s gen = 0 => s e = s i Assume ideal gas mixture and constant heat capacity, so we need k and C P C - P = y i C - Pi = = M mix = = C p = C - P / M mix = 2.066, R = R / M mix = / = C v = C p - R = , k = C p / C v = T e = T i (P e / P i ) (k-1)/k = 290 (250/100) = K w c in = C P (T e - T i ) = ( ) = kj/kg Take Problem with inlet temperature of 1400 K for the carbon dioxide and 300 K for the nitrogen. First estimate the exit temperature with the specific heats from Table A.5 and use this to start iterations using A.8 to find the exit temperature. CV mixing chamber, SSSF. The inlet ratio is ṅ CO2 = 2 ṅ N2 and assume no external heat transfer, no work involved. C - P CO 2 = = C - P N 2 = = Cont. Eq.: 0 = Σṅ in - Σṅ ex ; Energy Eq.: 0 = Σṅ in h - in - Σṅ ex h- ex 0 = 2ṅ N2 C - P CO 2 (T in - T ex ) CO2 + ṅ N2 C - P N 2 (T in - T ex ) N2 0 = (1400-T ex ) (300-T ex ) 0 = T ex _ T ex = 1089 K From Table A.8: Σṅ in h - in = ṅ N 2 [ ] = ṅ N2 1000K : Σṅ ex h - ex = ṅ N 2 [ ] = ṅ N2 1100K : Σṅ ex h - ex = ṅ N 2 [ ] = ṅ N2 1200K : Σṅ ex h - ex = ṅ N 2 [ ] = ṅ N Now linear interpolation between 1100 K and 1200 K T ex = = 1164 K

181 A mixture of 60% helium and 40% nitrogen by volume enters a turbine at 1 MPa, 800 K at a rate of 2 kg/s. The adiabatic turbine has an exit pressure of 100 kpa and an isentropic efficiency of 85%. Find the turbine work. Assume ideal gas mixture and take CV as turbine. w T s = h i - h es, s es = s i, T es = T i (P e /P i ) (k-1)/k C - P mix = = (k-1)/k = R/C - P mix = / = M mix = = , C P =C - P /M mix = T es = 800(100/1000) = 362 K, w Ts = C P (T i -T es ) = 777 w T ac = ηw Ts = kj/kg; W = ṁw Ts = 1321 kw A mixture of 50% carbon dioxide and 50% water by mass is brought from 1500 K, 1 MPa to 500 K, 200 kpa in a polytropic process through a SSSF device. Find the necessary heat transfer and work involved using values from Table A.5. Process Pv n = constant leading to n ln(v 2 /v 1 ) = ln(p 1 /P 2 ); v = RT/P n = ln(1000/200)/ln( / ) = R mix = c i R i = = C P mix = c i C Pi = = w = - vdp = - n n-1 (P e v e -P i v nr i )= - n-1 (T e -T i ) = kj/kg q = h e -h i + w = C P (T e -T i ) + w = kj/kg Solve Problem using specific heats C P = h/ T, from Table A.8 at 1000 K. C P CO2 = ( ) = C P H2 O = ( ) = C P mix = Σ c i C Pi = = R mix = Σ c i R i = = Process Pv n = C => n = ln(p 1 /P 2 ) / ln(v 2 /v 1 ) = w = - vdp = - n n-1 (P e v e -P i v nr i )= - n-1 (T e -T i ) = kj/kg q = h e -h i + w = ( ) = kj/kg

182 A 50/50 (by mole) gas mixture of methane CH 4 and ethylene C 2 H 4 is contained in a cylinder/piston at the initial state 480 kpa, 330 K, 1.05 m 3. The piston is now moved, compressing the mixture in a reversible, polytropic process to the final state 260 K, 0.03 m 3. Calculate the final pressure, the polytropic exponent, the work and heat transfer and net entropy change for the process. Ideal gas mixture: CH 4, C 2 H 4, 50% each by mol => y CH4 = y C2 H 4 = 0.5 State 1: n = P 1 V 1 /R T 1 = / ( ) = kmol C - v mix = y i M i C vi = kj/kmol-k State 2: T 2 = 260 K, V 2 = 0.03 m 3, Ideal gas => P 2 = P 1 V 1 V 2 T 2 T 1 = kpa Process: PV n = constant, T 2 = V 1 T 1 V n-1 T 2, ln = (n-1) ln V 1 => n = T 1 V 2 => 1 W 2 = P dv = 1 1-n (P 2 V 2 - P 1 V 1 ) = kj Energy Eq.: 1 Q 2 = n(ū 2 - ū 1 ) + 1 W 2 = n C- v mix (T 2 - T 1 ) + 1 W 2 = ( ) = kj - s2 - s- 1 = C- v mix ln (T 2 / T 1 ) + R- ln (V 2 / V 1 ) = kj/kmol-k 2 nd Law: S net = n(s s- 1 ) - 1 Q 2 /T o with T 0 = 260 K S net = ( ) - ( )/260 = kj/k A mixture of 2 kg oxygen and 2 kg of argon is in an insulated piston cylinder arrangement at 100 kpa, 300 K. The piston now compresses the mixture to half its initial volume. Find the final pressure, temperature and the piston work. C.V. Mixture. Control mass, boundary work and no Q, assume reversible. Energy Eq.: u 2 - u 1 = 1 q 2-1 w 2 = - 1 w 2 ; Entropy Eq.: s 2 -s 1 = = 0 Process: constant s => Pv k = constant, v 2 = v 1 /2, Assume ideal gases (T 1 >> T C ) and use k mix and C v mix for properties. R mix = Σ c i R i = = kj/kg mix C Pmix = Σ c i C Pi = = kj/kg mix C vmix = C Pmix -R mix = 0.487, k mix = C Pmix /C vmix = P 2 = P 1 (v 1 /v 2 ) k = P 1 (2) k = 100(2) = 279 kpa T 2 = T 1 (v 1 /v 2 ) k-1 = T 1 (2) k-1 = 300(2) = K 1 W 2 = m tot (u 1 -u 2 ) = m tot C v (T 1 -T 2 ) = ( ) = -231 kj

183 Two insulated tanks A and B are connected by a valve. Tank A has a volume of 1 m 3 and initially contains argon at 300 kpa, 10 C. Tank B has a volume of 2 m 3 and initially contains ethane at 200 kpa, 50 C. The valve is opened and remains open until the resulting gas mixture comes to a uniform state. Determine the final pressure and temperature. C.V. Tanks A + B. Control mass no W, no Q. Energy Eq.: U 2 -U 1 = 0 = n Ar C - V0 (T 2 -T A1 ) + n C 2 H 6 C - VO (T 2 -T B1 ) n Ar = P A1 V A /R T A1 = (300 1) / ( ) = kmol n C2 H 6 = P B1 V B /R T B1 = (200 2) / ( ) = kmol Continuity Eq.: n 2 = n Ar + n C2 H 6 = kmol Energy Eq.: (T ) (T ) = 0 Solving, T 2 = K P 2 = n 2 R T 2 /(V A +V B ) = /3 = 242 kpa

184 Reconsider the Problem 12.24, but let the tanks have a small amount of heat transfer so the final mixture is at 400 K. Find the final pressure, the heat transfer and the entropy change for the process. C.V. Both tanks. Control mass with mixing and heating of two ideal gases. n Ar = P A1 V A /R T A1 = = kmol n C2 H 6 = P B1 V B /R T B1 = = kmol Continuity Eq.: n 2 = n Ar + n C2 H 6 = kmol Energy Eq.: U 2 -U 1 = n Ar C - V0 (T 2 -T A1 ) + n C 2 H 6 C - VO (T 2 -T B1 ) = 1 Q 2 P 2 = n 2 R T 2 /(V A +V B ) = / 3 = kpa 1 Q 2 = ( ) ( ) = kj S SURR = - 1 Q 2 /T SURR ; S SYS = n Ar S - Ar + n C 2 H 6 S - C 2 H 6 y Ar = / = S - Ar = C- P Ar ln T 2 - R y Ar P 2 ln T A1 P A1 400 = ln ln = kj/kmol K S - C 2 H 6 = C - T 2 C 2 H 6 ln - R y C2 H 6 P 2 ln T B1 P B1 400 = ln ln = kj/kmol K Assume the surroundings are at 400 K (it heats the gas) S NET = n Ar S - Ar + n C 2 H 6 S - C 2 H 6 + S SURR = /400 = kj/k

185 A piston/cylinder contains helium at 110 kpa at ambient temperature 20 C, and initial volume of 20 L as shown in Fig. P The stops are mounted to give a maximum volume of 25 L and the nitrogen line conditions are 300 kpa, 30 C. The valve is now opened which allows nitrogen to flow in and mix with the helium. The valve is closed when the pressure inside reaches 200 kpa, at which point the temperature inside is 40 C. Is this process consistent with the second law of thermodynamics? P 1 = 110 kpa, T 1 = 20 o C, V 1 = 20 L, V max = 25 L = V 2 P 2 = 200 kpa, T 2 = 40 o C, P i = 300 kpa, T i = 30 o C Const. P to stops, then const. V = V max Q cv = U 2 - U 1 + W cv - n i h - i, W cv = P 1 (V 2 - V 1 ) = n 2 h n 1 h- 1 - n i h- i - (P 2 - P 1 )V 2 = n A (h - A2 - h- Ai ) + n B (h- B2 - h- B1 ) - (P 2 - P 1 )V 2 n B = n 1 = P 1 V 1 /R - T 1 = / = kmol n 2 = n A + n B = P 2 V 2 /R - T 2 = / = kmol, n A = n 2 - n B = kmol Q cv = (40-30) (40-20) - ( ) = = kj S gen = n - 2 s2 - n 1 s- 1 - n i s- i - Q cv /T 0 = n A (s - A2 - s- Ai ) + n B (s- B2 - s- B1 ) - Q cv /T 0 y A2 = / = , y B2 = s -A2 - s Ai = ln R- ln *200 = s -B2 - s B1 = ln R- ln *200 = S gen = /293.2 = kj/k > 0 Satisfies 2nd law.

186 Repeat Problem for an isentropic compressor efficiency of 82%. C.V. Compressor. SSSF, adiabatic q = 0, reversible s gen = 0 Energy Eq.: -w = h ex - h in ; Entropy Eq.: s i + s gen = s i = s e Process: For ideal compressor reversible => s gen = 0 => s e = s i Assume ideal gas mixture and constant heat capacity, so we need k and C P C - P = y i C - Pi = = M mix = = C p = C - P / M mix = 2.066, R = R / M mix = / = C v = C p - R = , k = C p / C v = T e = T i (P e / P i ) (k-1)/k = 290 (250/100) = K w c in = C P (T e - T i ) = ( ) = kj/kg w c actual = w c in /η = 124.3/0.82 = kj/kg = C p (T e actual - T i ) => T e actual = T + w c actual /C P = / = K A spherical balloon has an initial diameter of 1 m and contains argon gas at 200 kpa, 40 C. The balloon is connected by a valve to a 500-L rigid tank containing carbon dioxide at 100 kpa, 100 C. The valve is opened, and eventually the balloon and tank reach a uniform state in which the pressure is 185 kpa. The balloon pressure is directly proportional to its diameter. Take the balloon and tank as a control volume, and calculate the final temperature and the heat transfer for the process. A V A1 = π 6 13 = , m A1 = P A1 V A = = kg RT A m B1 = P B1 V B1 /RT B1 = / = kg B CO 2 P 2 V -1/3 A2 = P A1 V -1/3 A1 V A2 = V A1 ( P 2 P A1 ) 3 = ( )3 = m 3 2: Uniform ideal gas mix. : P 2 (V A2 +V B ) = (m A R A +m B R B )T 2 T 2 = 185( ) / ( ) = K W 12 = P 2 V A2 -P A1 V A1 1-(-1/3) = (4/3) = kj Q = m A C V0A (T 2 -T A1 ) + m B C V0B (T 2 -T B1 ) + W 12 = ( ) ( ) = = -2.4 kj

187 A large SSSF air separation plant takes in ambient air (79% N 2, 21% O 2 by volume) at 100 kpa, 20 C, at a rate of 1 kmol/s. It discharges a stream of pure O 2 gas at 200 kpa, 100 C, and a stream of pure N 2 gas at 100 kpa, 20 C. The plant operates on an electrical power input of 2000 kw. Calculate the net rate of entropy change for the process. Air 79 % N 2 21 % O 2 P 1 = 100 kpa T 1 = 20 o C ṅ 1 = 1 kmol/s 1 2 pure O 2 pure N 2 3 -Ẇ IN = 2000 kw P 2 = 200 kpa T 2 = 100 o C P 3 = 100 kpa T 3 = 20 o C Ṡ gen = Σ ṅ i s - i - Q. CV /T 0 = (ṅ 2 s- 2 + ṅ 3 s- 3 - ṅ 1 s- 1 ) - Q. CV /T 0 Q. CV = Σ ṅ h- i + Ẇ CV = ṅ O 2 C - P0 O 2 (T 2 -T 1 ) + ṅ N2 C - P0 N 2 (T 3 -T 1 ) + Ẇ CV = (100-20) = kw Σ ṅ i s i = 0.21[29.49 ln ln ] [ ln (100/79)]= kw/k Ṡ gen = 1505/ = kw/k

188 An insulated vertical cylinder is fitted with a frictionless constant loaded piston of cross sectional area 0.1 m 2 and the initial cylinder height of 1.0 m. The cylinder contains methane gas at 300 K, 150 kpa, and also inside is a 5-L capsule containing neon gas at 300 K, 500 kpa. The capsule now breaks, and the two gases mix together in a constant pressure process. What is the final temperature, final cylinder height and the net entropy change for the process. A p = 0.1 m 2, h = 1.0 m => V tot = V a1 + V b1 = 0.1 m 3 Methane: M = kg/kmol, C p = kj/kg-k, R = kj/kg-k Neon: M = kg/kmol, C p = kj/kg-k, R = kj/kg-k State 1: Methane, T al = 300 K, P a1 = 150 kpa, V a1 = V tot - V b1 = m 3 Neon, T bl = 300 K, P b1 = 500 kpa, V b1 = 5 L m a = P al V al R a T al = kg, n a = m a M a = kmol m b = P bl V bl = kg, n R b T b = m b = kmol bl M b State 2: Mix, P 2mix = P a1 = 150 kpa Energy Eq: 1 Q 2 = m a (u a2 -u a1 ) + m b ( u b2 - u b1 ) + 1 W 2 ; 1 Q 2 = 0 1 W 2 = P dv = P 2 (V 2 -V 1 ) tot ; P 2 V 2 = m tot R mix T 2 = (m a R a + m b R b )T 2 Assume Constant Specific Heat 0 = m a C pa (T 2 -T a1 ) + m b C pb (T 2 -T b1 + (m a R a + m b R b )T 2 - P 2 V 1 tot Solving for T 2 = K V 2 = (m a R a + m b R b )T 2 P 2 = m 3, h 2 = V 2 A p = m Entropy Eq.: S net = m a (s a2 - s a1 ) + m b (s b2 -s b1 ) - 1Q 2 T 0 ; 1 Q 2 = 0 y a = n a n a +n b = 0.851, y b = 1 - y a = s a2 - s a1 = C pa ln T 2 T a1 - R a ln y a P 2 P a1 = kj/kg-k s b2 - s b1 = C pb ln T 2 T b1 - R b ln y b P 2 P b1 = kj/kg-k S net = kj/k

189 The only known sources of helium are the atmosphere (mole fraction approximately ) and natural gas. A large unit is being constructed to separate 100 m 3 /s of natural gas, assumed to be He mole fraction and CH 4. The gas enters the unit at 150 kpa, 10 C. Pure helium exits at 100 kpa, 20 C, and pure methane exits at 150 kpa, 30 C. Any heat transfer is with the surroundings at 20 C. Is an electrical power input of 3000 kw sufficient to drive this unit? 1 2 P 2 = 100 kpa CH He He CH 4 T 2 = 20 o C at 150 kpa, 10 o C V. 3 P 3 = 140 kpa 1 = 100 m3 /s -Ẇ CV = 3000 kw T 3 = 30 o C. ṅ 1 = P 1 V 1/RT1 = /( ) = 6.37 kmol/s => ṅ 2 = 0.001; ṅ 1 = ; ṅ 3 = C - P He = = , C- P CH 4 = = Q. CV = ṅ 2 h- 2 + ṅ 3 h- 3 - ṅ 1 h- 1 + Ẇ CV = ṅ 2 C- P0 He (T 2 -T 1 ) + ṅ 3 C- P0 CH 4 (T 3 -T 1 ) + Ẇ CV = (20-10) (30-10)+(-3000) = kw Ṡ gen = ṅ 2 s ṅ 3 s- 3 - ṅ 1 s- 1 - Q. CV /T 0 = [ ln ln ] [ ln ln 140 ] /293.2 = kw/k > A steady flow of 0.01 kmol/s of 50% carbon dioxide and 50% water at 1200K and 200 kpa is used in a heat exchanger where 300 kw is extracted from the flow. Find the flow exit temperature and the rate of change of entropy using Table A.8. C.V. Heat exchanger, SSSF, 1 inlet, 1 exit, no work. Continuity Eq.: y CO2 = y H2O = 0.5 Energy Eq.: Q. = ṁ(h e h i ) = ṅ(h - e h- i ) => h- e = h- i + Q. /ṅ Inlet state: Table A.8 h - i = = kj/kmol Exit state: h - e = h- i + Q. /ṅ = /0.01 = kj/kmol Trial and error for T with h values from Table K h - e = 0.5( ) = K h - e = 0.5( ) = Interpolate to have the right h: T = K

190 An insulated rigid 2 m 3 tank A contains CO 2 gas at 200 C, 1MPa. An uninsulated rigid 1 m 3 tank B contains ethane, C 2 H 6, gas at 200 kpa, room temperature 20 C. The two are connected by a one-way check valve that will allow gas from A to B, but not from B to A. The valve is opened and gas flows from A to B until the pressure in B reaches 500 kpa and the valve is closed. The mixture in B is kept at room temperature due to heat transfer. Find the total number of moles and the ethane mole fraction at the final state in B. Find the final temperature and pressure in tank A and the heat transfer to/from tank B. Tank A: V A = 2 m 3, state A 1 : CO 2, T A1 = 200 C = K, P A1 = 1 MPa C - v0 CO 2 = = 28.74, C - P0 CO 2 = = Tank B: V B = 1 m 3, state B 1 : C 2 H 6, T B1 = 20 C = K, P B1 = 200 kpa Slow Flow A to B to P B2 = 500 kpa and assume T B2 = T B1 = T 0 P B1 V B = n B1 RT B1, P B2 V B = n B2 mix RT B2 y C2 H 6 B 2 = n B1 n B2 = P B1 P B2 = = n B1 = P B1 V B = = kmol RT B1 R n B2 mix = P B2 V B = = kmol RT B2 R n CO2 B2 = = kmol n A1 = P A1 V A = RT A1 R = kmol ; n A2 = n A1 - n CO 2 B2 = kmol C.V. A: All CO 2 Energy Eq.: Q CV A = 0 = n A2 ū A2 - n A1 ū A1 + n ave h - ave 0 = n A2 C - v0 CO 2 T A2 - n A1 C - v0 CO 2 T A1 + n ave C - P0 CO 2 ( T A1 + T A2 )/2 0 = 28.74( T A ) ( T A2 )/2 => T A2 = K P A2 = n A2 RT A R = = 700 kpa V A 2 C.V. B: Q CV B + n Bi h - Bi ave = (nū) B2 - (nū) B1 = (nū) CO 2 B2 + (nū) C 2 H 6 B2 - (nū) C 2 H 6 B1 Q CV B = ( )/2 = kj

191 A tank has two sides initially separated by a diaphragm. Side A contains 1 kg of water and side B contains 1.2 kg of air, both at 20 C, 100 kpa. The diaphragm is now broken and the whole tank is heated to 600 C by a 700 C reservoir. Find the final total pressure, heat transfer and total entropy generation. C.V. Total tank out to reservoir. Energy Eq.: U 2 -U 1 = m a (u 2 -u 1 ) a + m v (u 2 -u 1 ) v = 1 Q 2 Entropy Eq.: S 2 -S 1 = m a (s 2 -s 1 ) a + m v (s 2 -s 1 ) v = 1 Q 2 /T res + S gen Volume: V 2 = V A + V B = m v v v1 + m a v a1 = = 1.01 m 3 v v2 = V 2 /m v = 1.01, T 2 => P 2v = 400 kpa v a2 = V 2 /m a = , T 2 => P 2a = mrt 2 /V 2 = kpa P 2tot = P 2v + P 2a = kpa Water table B.1: u 1 = 83.95, u 2 = 3300, s 1 = , s 2 = Air table A.7: u 1 = 293, u 2 = 652.3, s T1 = 2.492, s T2 = From energy equation we have = 1( ) + 1.2( ) = kj 1 Q 2 From the entropy equation we have S gen = 1( ) + 1.2[ ln(301.6/100)] / = 5.4 kj/k

192 A 0.2 m 3 insulated, rigid vessel is divided into two equal parts A and B by an insulated partition, as shown in Fig. P The partition will support a pressure difference of 400 kpa before breaking. Side A contains methane and side B contains carbon dioxide. Both sides are initially at 1 MPa, 30 C. A valve on side B is opened, and carbon dioxide flows out. The carbon dioxide that remains in B is assumed to undergo a reversible adiabatic expansion while there is flow out. Eventually the partition breaks, and the valve is closed. Calculate the net entropy change for the process that begins when the valve is closed. A CH 4 B CO 2 P MAX = 400 kpa, P A1 = P B1 = 1 MPa V A1 = V B1 = 0.1 m 3 T A1 = T B1 = 30 o C = K CO 2 inside B: s B2 = s B1 to P B2 = 600 kpa (P A2 = 1000 kpa) For CO 2, k = => T B2 = 303.2( 600) = K 1000 n B2 = P B2 V B2 /R - T B2 = / = n A2 = n A1 = / = kmol Q 23 = 0 = n 3 ū 3 - n i2 ū i2 + 0 = n A2 C - V0 A (T 3 -T A2 ) + n B2 C- V0 B (T 3 -T B2 ) = 0 i (T ) (T ) = 0 Solve T 3 = K P 3 = /0.2 = kpa P A3 = = kpa, s - A3 - s - A2 s - B3 - s - B2 P B3 = P 3 - P A3 = kpa = ln(289.8) ln = kj/kmol K = ln(289.8) ln = kj/kmol K S NET = = kj/k

193 Consider 100 m 3 of atmospheric air which is an air water vapor mixture at 100 kpa, 15 C, and 40% relative humidity. Find the mass of water and the humidity ratio. What is the dew point of the mixture? Air-vap P = 100 kpa, T= 15 o C, φ = 40% P g = P sat15 = kpa => P v = φ P g = = kpa m v = P v V R v T = = kg P a = P tot - P v1 = = m a = P a V R a T = = kg w 1 = m v m a = = T dew is T when P v = P g = kpa; T = 1.4 o C The products of combustion are flowing through a SSSF heat exchanger with 12% CO 2, 13% H 2 O and 75% N 2 on a volume basis at the rate 0.1 kg/s and 100 kpa. What is the dew-point temperature? If the mixture is cooled 10 C below the dewpoint temperature, how long will it take to collect 10 kg of liquid water? y H2 O = 0.13; P H 2 O = = 13 kpa, T DEW = o C Cool to o C P G = kpa y H2 O = 7.805/100 = n H 2 O(v) /(n H 2 O(v) ) n H2 O(v) = per kmol mix in n LIQ = = M MIX IN = = kg/kmol ṅ MIX IN = ṁ TOTAL /M MIX IN = 0.1/28.63 = kmol/s ṅ LIQ COND = = kmol/s or ṁ LIQ COND = = kg/s For 10 kg, takes ~ 47 minutes

194 A new high-efficiency home heating system includes an air-to-air heat exchanger which uses energy from outgoing stale air to heat the fresh incoming air. If the outside ambient temperature is 10 C and the relative humidity is 30%, how much water will have to be added to the incoming air, if it flows in at the rate of 1 m 3 /s and must eventually be conditioned to 20 C and 40% relative humidity? Outside ambient air: P V1 = φ 1 P G1 = = kpa Assuming P 1 = P 2 = 100 kpa, ṁ A = P A1 V. 1 R A T 1 = => P A1 = = kpa = kg/s w 1 Conditioned to : T 2 = 20 o C, φ 2 = 0.40 = = P V2 = φ 2 P G2 = = => w 2 = = ṁ LIQ IN = ṁ A (w 2 -w 1 ) = ( ) = kg/s = 25.6 kg/h Consider 100 m 3 of atmospheric air at 100 kpa, 25 C, and 80% relative humidity. Assume this is brought into a basement room where it cools to 15 C, 100 kpa. How much liquid water will condense out? P g = P sat25 = kpa => P v = φ P g = = kpa m v1 = P v V R v T = = kg P g15 C = < P v1 => State 2 is saturated φ 2 = 100%, P v2 = P g2 = m v2 = P v V R v T = = kg m liq = m v1 - m v2 = = 0.56 kg A flow of 2 kg/s completely dry air at T 1, 100 kpa is cooled down to 10 C by spraying liquid water at 10 C, 100 kpa into it so it becomes saturated moist air at 10 C. The process is SSSF with no external heat transfer or work. Find the exit moist air humidity ratio and the flow rate of liquid water. Find also the dry air inlet temperature T 1. 2: saturated P v = P g = kpa and h fg (10 C)= kj/kg w 2 = / ( ) = C.V. Box ṁ a + ṁ liq = ṁ a (1 + w 2 ) => ṁ liq = w 2 ṁ a = kg/s ṁ a h a1 - ṁ liq h f = ṁ a (h a2 + w 2 h g2 ) h a1 - h a2 = C pa (T 1 - T 2 ) = w 2 h g2 - w 2 h f = w 2 h fg = => T 1 = 29.1 C

195 A piston/cylinder has 100 kg of saturated moist air at 100 kpa, 5 C. If it is heated to 45 C in an isobaric process, find 1 Q 2 and the final relative humidity. If it is compressed from the initial state to 200 kpa in an isothermal process, find the mass of water condensing. Energy Eq.: m(u 2 -u 1 ) = Q 12 - W 12, Initial state 1: φ 1 = 100%, h 1 = 39 kj/kg dry air, w 1 = m a = m tot /(1+ w 1 ) = kg, m v1 = w 1 m a = kg Case a: P = constant => W 12 = mp(v 2 -v 1 ) => Q 12 = m(u 2 -u 1 ) + W 12 = m(h 2 -h 1 ) = m a (h 2 - h 1 ) State 2: w 2 = w 1, T 2 => h 2 = 79, φ 2 = 9.75% Q 12 = (79-39) = 3979 kj. Case b: T = constant & φ 2 = 100% => P v = P g = kpa w 2 = P v2 /P a2 = P g2 /(P tot2 -P g2 ) = m v2 = w 2 m a = kg, m liq = m v1 - m v2 = kg A flow moist air at 100 kpa, 40 C, 40% relative humidity is cooled to 15 C in a constant pressure SSSF device. Find the humidity ratio of the inlet and the exit flow, and the heat transfer in the device per kg dry air. C.V. Cooler. ṁ v1 = ṁ liq + ṁ v2 Psychrometric chart: State 2: T < T dew = 23 C => φ 2 = 100% ṁ v1 /ṁ a = ω 1 = 0.018, h ~ 1 = 106; ṁ v2 /ṁ a = ω 2 = , h 2 = 62 ṁ liq /ṁ a = ω 1 - ω 2 = , h f = ṁ a q - out = ṁ a h 1 - ṁ liq h f - ṁ a h 2 => q - out = h 1 - (ω 1 - ω 2 ) h f - h 2 = = kj/kg-dry air Tables: P g1 = kpa P v1 = ω 1 = P v2 = kpa = P g2 => ω 2 = h v1 = , h v2 = , h f = q - out = C P (T 1 - T 2 ) + ω 1 h v1 - ω 2 h v2 - (ω 1 - ω 2 ) h f = 0.717(40-15) = 38.8 kj/kg dry air

196 Ambient moist air enters a steady-flow air-conditioning unit at 102 kpa, 30 C, with a 60% relative humidity. The volume flow rate entering the unit is 100 L/s. The moist air leaves the unit at 95 kpa, 15 C, with a relative humidity of 100%. Liquid condensate also leaves the unit at 15 C. Determine the rate of heat transfer for this process. State 1: P V1 = φ 1 P G1 = = w 1 = /( ) = ṁ A = P A1 V = = kg/s R A T P V2 = P G2 = 1.705, w 2 = /( ) = Energy Eq.: Q. CV + ṁ A h A1 + ṁ V1 h V1 = ṁ A h A2 + ṁ V2 h A2 + ṁ 3 h L3 Q. CV /ṁ A = C P0A (T 2 -T 1 ) + w 2 h V2 - w 1 h V1 + (w 1 -w 2 )h L3 = 1.004(15-30) = kj/kg air Q. CV = (-26.73) = kw A steady supply of 1.0 m 3 /s air at 25 C, 100 kpa, 50% relative humidity is needed to heat a building in the winter. The outdoor ambient is at 10 C, 100 kpa, 50% relative humidity. What are the required liquid water input and heat transfer rates for this purpose? Air: R = kj/kg K, C p = kj/kg-k State 1: T 1 = 10 C, φ 1 = 50%, P 1 = 100 kpa P g1 = kpa, P v1 = φ 1 P g1 = kpa, P a1 = P 1 - P v1 = kpa => ω 1 = P v1 /P a1 = State 2: T 2 = 25 C, P 2 = 100 kpa, φ 2 = 50%, V. a2 = 1 m3 /s P g2 = kpa, P v2 = φ 2 P g2 = kpa, P a2 = P 2 - P v2 = kpa, ω 2 = P v2 /P a2 = ṁ a2 = P a2 V. a2 /RT a2 = /( ) = 1.15 kg/s Steam tables B.1.1: h v1 = kj/kg, h v2 = kj/kg State 3: Assume: Liq. Water at T 3 = 25 C, h f3 = kj/kg Conservation of Mass: ṁ a1 = ṁ a2, ṁ f3 = ṁ v2 - ṁ v1 ṁ f3 = ṁ a2 (ω 2 - ω 1 ) = = kg/s 1 st Law: Q. + ṁ a1 h a1 + ṁ v1 h v1 + ṁ f3 h f3 = ṁ a2 h a2 +ṁ v2 h v2 Q. = C ṁ p (T 2 - T 1 ) + ω 2 h v2 - ω 1 h v1 - m. f3 h a ṁ f3 => Q. = kw a

197 Consider a 500-L rigid tank containing an air water vapor mixture at 100 kpa, 35 C, with a 70% relative humidity. The system is cooled until the water just begins to condense. Determine the final temperature in the tank and the heat transfer for the process. P v1 = φp G1 = = kpa Since m v = const & V = const & also P v = P G2 : P G2 = P v1 T 2 /T 1 = T 2 /308.2 = T 2 Assume T 2 = 30 o C: Assume T 2 = 25 o C: = =/ = P G 30C = =/ = P G 25C interpolating T 2 = 28.2 o C w 2 = w 1 = = ( ) m a = P a1 V/R a T 1 = ( ) 0.5/( ) = kg 1st law: Q 12 = U 2 -U 1 = m a (u a2 -u a1 ) + m v (u v2 -u v1 ) = 0.717( ) ( ) = kj/kg Q 12 = 0.543(-5.11) = kj Air in a piston/cylinder is at 35 C, 100 kpa and a relative humidity of 80%. It is now compressed to a pressure of 500 kpa in a constant temperature process. Find the final relative and specific humidity and the volume ratio V 2 /V 1. Check to see if the second state is saturated or not. First assume no water is condensed 1: w 1 = : w 2 = P v2 /(P 2 -P v2 ) w 2 = w 1 => P v2 = > P g = kpa Conclusion is state 2 is saturated φ 2 = 100%, w 2 = P g /(P 2 -P g ) = To get the volume ratio, write the ideal gas law for the vapor phases V 2 = V a2 + V v2 + V f2 = (m a R a + m v2 R v )T/P 2 + m liq v f V 1 = V a1 + V v1 = (m a R a + m v1 R v )T/P 1 Take the ratio and devide through with m a R a T/P 2 to get V 2 /V 1 = (P 1 /P 2 )[ w 2 + (w 1 -w 2 )P 2 v f /R a T] / [ w 1 ] = The liquid contribution is nearly zero (= ) in the numerator.

198 A 300-L rigid vessel initially contains moist air at 150 kpa, 40 C, with a relative humidity of 10%. A supply line connected to this vessel by a valve carries steam at 600 kpa, 200 C. The valve is opened, and steam flows into the vessel until the relative humidity of the resultant moist air mixture is 90%. Then the valve is closed. Sufficient heat is transferred from the vessel so the temperature remains at 40 C during the process. Determine the heat transfer for the process, the mass of steam entering the vessel, and the final pressure inside the vessel. i H 2 O P v1 = φ 1 P G1 = = kpa P v2 = = kpa P a2 = P a1 = = kpa AIR + H 2 O w 1 = = w 2 = = m a = /( ) = 0.5 kg P 2 = = kpa m vi = 0.5( ) = kg u v1 = u v2 u G at 40 o C and u a2 = u a1 Q CV = m a (u a2 - u a1 ) + m v2 u v2 - m v1 u v1 - m vi h i = m vi (u G at T - h i ) = ( ) = kj

199 A combination air cooler and dehumidification unit receives outside ambient air at 35 C, 100 kpa, 90% relative humidity. The moist air is first cooled to a low temperature T 2 to condense the proper amount of water, assume all the liquid leaves at T 2. The moist air is then heated and leaves the unit at 20 C, 100 kpa, relative humidity 30% with volume flow rate of 0.01 m 3 /s. Find the temperature T 2, the mass of liquid per kilogram of dry air and the overall heat transfer rate. MIX IN 1 COOL -Q. C 2 LIQ OUT 2' HEAT. Q H CV MIX OUT a) P v1 = φ 1 P G1 = = kpa w 1 = = P v3 = φ 3 P G3 = = kpa w 2 = w 3 = = ṁ LIQ 2 /ṁ a = w 1 - w 2 = = kg/kg air P G2 = P v3 = kpa T 2 = 1.7 o C b) For a C.V. around the entire unit Q. CV = Q. H + Q. C Net heat transfer, 1st law: Q. CV /ṁ a = (h a3 -h a1 ) + w 3 h v3 - w 1 h v1 + ṁ L2 h L2 /ṁ a = 1.004(20-35) = kj/kg air ṁ a = P a3 V. 3 = ( ) 0.01 = kg/s R a T Q. CV = (-88.82) = kw

200 A rigid container, 10 m 3 in volume, contains moist air at 45 C, 100 kpa, φ = 40%. The container is now cooled to 5 C. N eglect the volume of any liquid that might be present and find the final mass of water vapor, final total pressure and the heat transfer. CV container. m 2 = m 1 ; m 2 u 2 - m 1 u 1 = 1 Q 2 1: 45 C, φ = 40% => w 1 = , T dew = 27.7 C Final state T 2 < T dew so condensation, φ 2 = 100% P v1 = 0.4 P g = 0.4*9.593 = kpa, P a1 = P tot - P v1 = kpa m a = P a1 V/RT 1 = kg, m v1 = w 1 m a = kg P v2 = P g2 = kpa, P a2 = P a1 T 2 /T 1 = kpa P 2 = P a2 + P v2 = kpa m v2 = P v2 V/R v T 2 = kg (=V/v g = steam table) m f2 = m v1 - m v2 = kg The heat transfer from the energy equation becomes 1Q 2 = m a (u 2 -u 1 ) a + m v2 u g2 + m f2 u f2 - m v1 u g1 = m a C v (T 2 -T 1 ) + m v m f m v = = kj A saturated air-water vapor mixture at 20 o C, 100 kpa, is contained in a 5-m 3 closed tank in equilibrium with 1 kg of liquid water. The tank is heated to 80 o C. Is there any liquid water in the final state? Find the heat transfer for the process. AIR + VAP LIQ Q 12 a) Since V LIQ = m LIQ v F m 3, V GAS V φ 1 = 1.00 P v1 = P G1 = kpa w 1 = /( ) = m a = P a1 V = R a T = kg => m v1 = w 1 m = kg a At state 2: P a2 = = kpa 5 w MAX 2 = / = But w 2 ACTUAL = = < w MAX 2 No liquid at 2 Q 12 = m a (u a2 -u a1 ) + m v2 u v2 - m v1 u v1 - m liq 1 u liq 1 = (80-20) = = 2655 kj

201 An air-water vapor mixture enters a steady flow heater humidifier unit at state 1: 10 C, 10% relative humidity, at the rate of 1 m 3 /s. A second air-vapor stream enters the unit at state 2: 20 C, 20% relative humidity, at the rate of 2 m 3 /s. Liquid water enters at state 3: 10 C, at the rate of 400 kg per hour. A single airvapor flow exits the unit at state 4: 40 C. Calculate the relative humidity of the exit flow and the rate of heat transfer to the unit. Assume: P = 100 kpa State 1: T 1 = 10 C, φ 1 = 10%, V. a1 = 1 m3 /s P g1 = kpa, P v1 = φ 1 P g1 = kpa, P a1 = P - P v1 = kpa ṁ a1 = P a1 V. a1 = kg/s RT a1 ṁ v1 = ω 1 ṁ a1 = kg/s, h v1 = h g1 = kj/kg ω 1 = P v1 P a1 = , State 2: T 2 = 20 C, φ 2 = 20%, V. a2 = 2 m3 /s P g2 = kpa, P v2 = φ P g2 = kpa, P a2 = P - P v2 = kpa ω 2 = P v2 = , ṁ P a2 = P a2 V. a2 = kg/s a2 RT a2 ṁ v2 = ω 2 ṁ a2 = kg/s, h v2 = h g2 = kj/kg State 3: Liquid. T 3 = 10 C, ṁ f3 = 400 kg/hr = kg/s, h f3 = 42 kj/kg State 4: T 4 = 40 C Continuity Eq. air: ṁ a4 = ṁ a2 + ṁ a1 = , Continuity Eq. water: ṁ v4 = ṁ v1 + ṁ v2 + ṁ f3 = kg/s ω 4 = ṁ v4 ṁ a4 = = P v4 P-P v4 P v4 = kpa P g4 = kpa, φ 4 = P v4 P g4 = 0.684, h v4 = h g4 = kj/kg 1 st Law: Q. + ṁ a1 h a1 + ṁ v1 h v1 + ṁ a2 h a2 +ṁ v2 h v2 + ṁ f3 h f3 = ṁ a4 h a4 +ṁ v4 h v4 Q. = 1.004( ) = 366 kw

202 In a hot and dry climate, air enters an air-conditioner unit at 100 kpa, 40 C, and 5% relative humidity, at the steady rate of 1.0 m 3 /s. Liquid water at 20 C is sprayed into the air in the AC unit at the rate 20 kg/hour, and heat is rejected from the unit at at the rate 20 kw. The exit pressure is 100 kpa. What are the exit temperature and relative humidity? State 1: T 1 = 40 C, P 1 = 100 kpa, φ 1 = 5%, V. a1 = 1 m3 /s P g1 = kpa, P v1 = φ 1 P g1 = kpa, P a1 = P- P v1 = kpa ω 1 = P v1 = , ṁ P a1 = P a1 V. a1 = kg/s, h a1 RT v1 = kj/kg a1 State 2 : Liq. Water. 20 C, ṁ f2 = 20 kg/hr = kg/s, h f2 = 83.9 kj/kg Conservation of Mass: ṁ a1 = ṁ a3, ṁ v1 + ṁ 12 = ṁ v3 ω 3 = (ṁ f2 / ṁ a1 ) + ω 1 = ( /1.108 ) = State 3 : P 3 = 100 kpa and P v3 = P 3 ω 3 /( ω 3 ) = 1.16 kpa 1 st Law: Q. + ṁ a1 h a1 + ṁ v1 h v1 + ṁ f2 h f2 = ṁ a3 h a3 +ṁ v3 h v3 ; Q. = - 20 kw (h a3 -h a1 ) + ω 3 h v3 = ω 1 h v1 + (ṁ f2 h f2 - Q. )/ṁ a1 = 24.39; Unknowns: h a3, h v3 Trial and Error for T 3 ; T 3 = K, P g3 = kpa, φ 3 = P v3 P g3 = A water-filled reactor of 1 m 3 is at 20 MPa, 360 C and located inside an insulated containment room of 100 m 3 that contains air at 100 kpa and 25 C. Due to a failure the reactor ruptures and the water fills the containment room. Find the final pressure. CV Total container. m v (u 2 -u 1 ) + m a (u 2 -u 1 ) = 1 Q 2-1 W 2 = 0 Initial water: v 1 = , u 1 = , m v = V/v = kg Initial air: m a = PV/RT = / = kg Substitute into energy equation (u ) (T 2-25) = 0 u T 2 = & v 2 = V 2 /m v = m 3 /kg Trial and error 2-phase (T guess, v 2 => x 2 => u 2 => LHS) T = 150 LHS = 1546 T = 160 LHS = T = 155 LHS = => T = 156 C LHS = OK x 2 = , P sat = kpa P a2 = P a1 V 1 T 2 /V 2 T 1 = / ( ) = kpa => P 2 = P a2 + P sat = 700 kpa.

203 Use the psychrometric chart to find the missing property of: φ, ω, T wet, T dry a. T dry = 25 C, φ =80% b. T dry =15 C, φ =100% c.t dry = 20 C, and ω = d. T dry = 25 C, T wet = 23 C a. 25 C, φ = 80% => ω = 0.016; T wet = 22.3 C b. 15 C, φ = 100% => ω = ; T wet = 15 C c. 20 C, ω = => φ = 57%; T wet = 14.4 C d. 25 C, T wet = 23 C => ω = 0.017; φ = 86% Use the psychrometric chart to find the missing property of: φ, ω, T wet, T dry a. φ = 50%, ω = b. T wet =15 C, φ = 60%. c. ω = and T wet =17 C d. T dry = 10 C, ω = a. φ = 50%, ω = => T dry = 23.5 C, T wet = 20.6 C b. T wet = 15 C, φ =60% => T dry = 20.2 C, ω = c. ω = 0.008, T wet =17 C => T dry = 27.2 C, φ = 37% d. T dry = 10 C, ω = => φ = 80%, T wet = 8.2 C For each of the states in Problem find the dew point temperature. The dew point is the state with the same humidity ratio (abs humidity ω) and completely saturated φ = 100%. From psychrometric chart: a. T dew = 16.8 C c. T dew = 10.9 C b. T dew = 12 C d. T dew = 6.5 C Finding the solution from the tables is done for cases a,c and d as Eq solve: P v = P g = ωp tot /[ω ] = Psat(T dew ) in B.1.1 For case b use energy Eq to find ω 1 first from T ad sat = T wet.

204 One means of air-conditioning hot summer air is by evaporative cooling, which is a process similar to the SSSF adiabatic saturation process. Consider outdoor ambient air at 35 C, 100 kpa, 30% relative humidity. What is the maximum amount of cooling that can be achieved by such a technique? What disadvantage is there to this approach? Solve the problem using a first law analysis and repeat it using the psychrometric chart, Fig. F.5. Ambient Air Liquid Cooled Air P 1 = P 2 = 100 kpa T 1 = 35 o C, φ 1 = 30% P v1 = φ 1 P g1 = = ω 1 = /98.31 = For adiabatic saturation (Max. cooling is for φ 2 = 1), 1st law, Eq ω 1 (h v1 - h f2 ) = C p (T 2 - T 1 ) + ω 2 h fg2 φ 2 = 1 & ω 2 = P G2 /(P 2 - P G2 ) Only one unknown: T 2. Trial and error on energy equation: C p T 2 + ω 2 h fg2 + ω 1 h f2 = C p T 1 + ω 1 h v1 = = T 2 = 20 o C: P G2 = 2.339, h f2 = 83.94, h fg2 = => ω 2 = / = LHS = = T 2 = 25 o C: P G2 = 3.169, h f2 = , h fg2 = => ω 2 = / = LHS = = linear interpolation: T 2 = 21.4 o C b) chart F.5 : Ad. sat. T WetBulbTemperature 21.5 o C

205 Use the formulas and the steam tables to find the missing property of: φ, ω, and T dry, total pressure is 100 kpa; repeat the answers using the psychrometric chart a. φ = 50%, ω = b. T wet =15 C, φ = 50% c. T dry = 25 C, T wet = 21 C a. From Eq P v = P ω /( ω) = /0.632 = kpa From Eq P g = P v /φ = 1.582/0.5 = kpa => T = 25 C b. Assume Twet is adiabatic saturation T and use energy Eq At 15 C: P g = => ω = /( ) = LHS = ω 1 (h v1 - h f2 ) + C p T 1 = RHS = C p T 2 + ω 2 h fg2 RHS = = kj/kg ω 1 = φp g /(100 - φp g ) where P g is at T 1. Trial and error. LHS 25C = 49.98, LHS 20C = 38.3 => T = 21.4 C, ω 1 = c. At 21 C:: P g = => ω 2 = /( ) = h f2 = and h fg2 = kj/kg, h v1 = From Eq.12.23: ω 1 = [C p (T 2 -T 1 ) + ω 2 h fg2 ]/(h v1 - h f2 ) = P v = P ω /( ω) = 2.247, φ = 2.247/3.169 = 0.71 Using the psychrometric chart F.5: a: T dry = 25.3 C b. T dry = 21.6 C, ω = c: ω = , φ = 71-72% Compare the weather two places where it is cloudy and breezy. At beach A it is 20 C, kpa, relative humidity 90% and beach B has 25 C, 99 kpa, relative humidity 20%. Suppose you just took a swim and came out of the water. Where would you feel more comfortable and why? Your skin being wet and air is flowing over it will feel T wet. With the small difference in pressure from 100 kpa use the psychrometric chart. A: 20 C, φ = 90% => T wet = 18.7 C B: 25 C, φ = 20% => T wet = 12.3 C At beach A it is comfortable, at B it feels chilly.

206 Ambient air at 100 kpa, 30 C, 40% relative humidity goes through a constant pressure heat exchanger in a SSSF process. In one case it is heated to 45 C and in another case it is cooled until it reaches saturation. For both cases find the exit relative humidity and the amount of heat transfer per kilogram dry air. CV heat exchanger: ṁ Ai = ṁ Ae, ṁ vi = ṁ ve, w e = w i (h a + wh v ) i + q = (h a + wh v ) e = h e, q = h e - h i Using the psychrometric chart: i: w i = , h i = 76 Case I) e: T e = 45 o C, w e = w i => h e = 92, φ e = 17%, q = = 16 kj/kg dry air Case II) e: w e = w i, φ e = 100% => h e = 61, T e = 14.5o C q = = -15 kj/kg dry air A flow, 0.2 kg/s dry air, of moist air at 40 C, 50% relative humidity flows from the outside state 1 down into a basement where it cools to 16 C, state 2. Then it flows up to the living room where it is heated to 25 C, state 3. Find the dew point for state 1, any amount of liquid that may appear, the heat transfer that takes place in the basement and the relative humidity in the living room at state 3. Solve using psychrometric chart: a) T dew = 27.2 (w = w 1, φ = 100%) w 1 = , h 1 = b) T 2 < T dew so we have φ 2 = 100% liquid water appear in the basement. => w 2 = h 2 = 64.4 and from steam tbl. h f = ṁ liq = ṁ air (w 1 -w 2 ) = 0.2( ) = kg/s c) Energy equation: ṁ air h 1 = ṁ liq h f + ṁ air h 2 + Q out Q out = 0.2[ ] = 10.6 kw d) w 3 = w 2 = & 25 C => φ 3 = 58%. If you solve by the formula's and the tables the numbers are: P g40 = ; P v1 = = kpa w 1 = / ( ) = P v1 = P g (T dew ) => T dew 1 = 27.5 C 2: φ = 100%, P v2 = P g2 = kpa, w 2 = / = ṁ liq = ṁ air (w 1 -w 2 ) = = kg/s 3: w 3 = w 2 => P v3 = P v2 = & P g3 = φ 3 = P v /P g = 1.832/3.169 = 57.8%

207 A flow of air at 5 C, φ = 90%, is brought into a house, where it is conditioned to 25 C, 60% relative humidity. This is done in a SSSF process with a combined heater-evaporator where any liquid water is at 10 C. Find any flow of liquid, and the necessary heat transfer, both per kilogram dry air flowing. Find the dew point for the final mixture. CV heater and evaporator. Use psychrometric chart. Inlet: w 1 = , h 1 = 37.5, h f = Exit: w 2 = , h 2 = 75, T dew = 16.5 C From these numbers we see that water and heat must be added. Continuity eq. and energy equation give ṁ LIQ IN /ṁ A = w 2 - w 1 = kg/kg dry air q = h 2 - h 1 - (w 2 -w 1 )h f = 37.3 kj/kg dry air Atmospheric air at 35 C, relative humidity of 10%, is too warm and also too dry. An air conditioner should deliver air at 21 C and 50% relative humidity in the amount of 3600 m 3 per hour. Sketch a setup to accomplish this, find any amount of liquid (at 20 C) that is needed or discarded and any heat transfer. CV air conditioner. Check from psychrometric chart. Inlet: w 1 = , h mix,1 = 62.8, h f,20 = Exit: w 2 = , h mix,2 = 60.2 Water must be added. Continuity and energy equations ṁ A (1+w 1 ) + ṁ liq = ṁ A (1+w 2 ); ṁ A h 1mix + ṁ liq h f + Q. CV = ṁ A h 2mix ṁ A = P a V. /RT = ( )1/ = kg/s ṁ liq = ṁ A (w 2 - w 1 ) = kg/s = 19.4 kg/h Q. CV = ( ) = kw Liquid water Cooler Exit Inlet

208 In a car s defrost/defog system atmospheric air, 21 C, relative humidity 80%, is taken in and cooled such that liquid water drips out. The now dryer air is heated to 41 C and then blown onto the windshield, where it should have a maximum of 10% relative humidity to remove water from the windshield. Find the dew point of the atmospheric air, specific humidity of air onto the windshield, the lowest temperature and the specific heat transfer in the cooler. Solve using the psychrometric chart Air inlet: T = 21 C, φ = 80% => w 1 = , T dew = 17.3 C Air exit: T = 41 C, φ = 10% => w 3 = , T dew = 1.9 C To remove enough water we must cool to the exit T dew, followed by heating to T ex. The enthalpies are h 1 = 72, h 2 = 32.5, h f (1.9 C) = 8 CV cooler: ṁ liq /ṁ air = w 1 -w 3 = = kg liq/kg air q = Q/ṁ air = h 2 + (w 1 - w 3 ) h f - h 1 = = kj/kg dry air If the steam and air tables are used the numbers are 1: P g1 = 2.505, P v1 = => w 1 = h g1 = , h a1 = => h 1 = : P g3 = 7.826, P v3 = => w 3 = : w g3 = w 3 => T 2 = T 3dew = 3.3 C, h f2 = h g2 = , h a2 = => h 2 = ṁ liq /ṁ air = , q = = kj/kg air Two moist air streams with 85% relative humidity, both flowing at a rate of 0.1 kg/s of dry air are mixed in a SSSF setup. One inlet flowstream is at 32.5 C and the other at 16 C. Find the exit relative humidity. 1: w 1 = , h 1 = 120 2: w 2 = , h 2 = 60 CV mixing chamber. Continuity Eq. water: ṁ air w 1 + ṁ air w 2 = 2ṁ air w ex ; Energy Eq.: ṁ air h 1 + ṁ air h 2 = 2ṁ air h ex w ex = (w 1 + w 2 )/2 = ; h ex = (h 1 + h 2)/2 = 90 exit: w ex, h ex => T ex = 24.5 C, φ = 94%

209 A flow of moist air at 21 C, 60% relative humidity should be produced from mixing of two different moist air flows. Flow 1 is at 10 C, relative humidity 80% and flow 2 is at 32 C and has T wet = 27 C. The mixing chamber can be followed by a heater or a cooler. No liquid water is added and P = 100 kpa. Find the two controls one is the ratio of the two mass flow rates ma1/ma2 and the other is the heat transfer in the heater/cooler per kg dry air. 1: w 1 = 0.006, h 1 = 45 2: w 2 = , h 2 = 105 4: w 4 = , h 4 = 64, T dew 4 = 12.5 C C.V : Total Setup ṁ a1 w 1 + ṁ a2 w 2 = (ṁ a1 + ṁ a2 ) w 4 ṁ a1 h 1 + ṁ a2 h 2 + Q. a1 = (ṁ a1 + ṁ a2 ) h 4 x = ṁ a1 /ṁ a2 => x w 1 + w 2 = (1+x) w 4 => x = w 4 - w 2 w 1 - w 4 = q = Q. a1 /(ṁ a1 + ṁ a2 ) = h 4 - [x/(1+x)] h 1 - [1/(1+x)] h 2 = = 6.43 kj/kg-dry air Consider two states of atmospheric air. (1) 35 C, T wet = 18 C and (2) 26.5 C, φ = 60%. Suggest a system of devices that will allow air in a SSSF process to change from (1) to (2) and from (2) to (1). Heaters, coolers (de)humidifiers, liquid traps etc. are available and any liquid/solid flowing is assumed to be at the lowest temperature seen in the process. Find the specific and relative humidity for state 1, dew point for state 2 and the heat transfer per kilogram dry air in each component in the systems. Use the psychrometric chart 1: w 1 = 0.006, h 1 = 70.5, φ 1 = 18%, T dew = 6.5 C, h dew = 42 2: w 2 = 0.013, h 2 = 79.4, φ 2 = 60%, T dew = 18 C, h dew = 71 Since w 2 > w 1 water must be added in process I to II and removed in the process II to I. Water can only be removed by cooling below dew point temperature so I to II: Adiab. sat I to Dew,II, then heater from Dew,II to II II to I: Cool to Dew,I then heat Dew,I to I The first one can be done because T dew II = T ad sat I I to II: q = h II - h dewii = = 8.4 kj/kg air II to I: q cool = h II - h dewi - (w 2 -w 1 )h f (at T dewi ) = = 37.2 kj/kg air q heat = h I - h dewi = = 28.5 kj/kg air

210 An insulated tank has an air inlet, ω 1 = , and an outlet, T 2 = 22 C, φ 2 = 90% both at 100 kpa. A third line sprays 0.25 kg/s of water at 80 C, 100 kpa. For a SSSF operation find the outlet specific humidity, the mass flow rate of air needed and the required air inlet temperature, T 1. Take CV tank in SSSF. Continuity and energy equations are: Continuity: ṁ 3 + ṁ a w 1 = ṁ a w 2 ; Energy Eq.: ṁ 3 h f + ṁ a h 1 = ṁ a h 2 All state properties are known except T 1. From the psychrometric chart we get State 2: w 2 = 0.015, h 2 = 79.5 State 3: h f = (steam tbl) ṁ a = ṁ 3 /(w 2 -w 1 ) = 0.25/( ) = kg/s h 1 = h 2 - (w 2 -w 1 )h f = = 77.3 Chart (w 1, h 1) => T 1 = 36.5 C You have just washed your hair and now blow dry it in a room with 23 C, φ = 60%, (1). The dryer, 500 W, heats the air to 49 C, (2), blows it through your hair where the air becomes saturated (3), and then flows on to hit a window where it cools to 15 C (4). Find the relative humidity at state 2, the heat transfer per kilogram of dry air in the dryer, the air flow rate, and the amount of water condensed on the window, if any. The blowdryer heats the air at constant specific humidity to 2 and it then goes through an adiabatic saturation process to state 3, finally cooling to 4. 1: 23 C, 60% rel hum => w 1 = , h 1 = 69 2: w 2 = w 1, T 2 => φ 2 = 15%, h 2 = 95 CV. 1 to 2: w 2 = w 1 ; q = h 2 - h 1 = = 26 kj/kg dry air ṁ a = Q/q = 0.5/26 = kg/s CV. 2 to 3: w 3 - w 2 = ṁ liq /ṁ a ; ṁ a h 2 + ṁ liq h f = ṁ a h 3 3: φ = 100% => T 3 = T wet,2 = 24.8 C, w 3 = : φ = 100%, T 4 => w 4 = ṁ liq = (w 3 -w 4 )ṁ a = ( ) = g/s If the steam tables and formula's are used then we get h g1 = , h g2 = , P g1 = 2.837, P v1 = , P g2 = 11.8, w 1 = , w 2 = w 1, P v2 = P v1 φ 2 = P v2 /P g2 = 14.4%, h f3 = 114, Trial and error for adiabatic saturation temperature. T 3 = 25 C, w 3 = 0.02, P v4 =P g4 = kpa, w 4 = /( ) =

211 A water-cooling tower for a power plant cools 45 C liquid water by evaporation. The tower receives air at 19.5 C, φ = 30%, 100 kpa that is blown through/over the water such that it leaves the tower at 25 C, φ = 70%. The remaining liquid water flows back to the condenser at 30 C having given off 1 MW. Find the mass flow rate of air, and the amount of water that evaporates. CV Total cooling tower, SSSF. Water in air: w in + ṁ evap /ṁ a = w ex Energy: ṁ a h in + ṁ 1 h 45 = ṁ a h ex + (ṁ 1 - ṁ evap ) h 30 Inlet: 19.5 C, 30% rel hum => w in = , h in = 50 Exit : 25 C, 70% rel hum => w ex = , h ex = 80 Take the two water flow difference to mean the 1 MW Q. = ṁ 1 h 45 - (ṁ 1 - ṁ evap ) h 30 = 1 MW ṁ a (h ex-h in) = ṁ a (80-50) = 1000 kw => ṁ a = kg/s ṁ evap = (w ex - w in ) ṁ a = = kg/s The needed make-up water flow could be added to give a slightly different meaning to the 1 MW An indoor pool evaporates kg/h of water, which is removed by a dehumidifier to maintain 21 C, φ = 70% in the room. The dehumidifier, shown in Fig. P12.71, is a refrigeration cycle in which air flowing over the evaporator cools such that liquid water drops out, and the air continues flowing over the condenser. For an air flow rate of 0.1 kg/s the unit requires 1.4 kw input to a motor driving a fan and the compressor and it has a coefficient of performance, β = Q L / W c = 2.0. Find the state of the air as it returns to the room and the compressor work input. The unit must remove kg/h liquid to keep steady state in the room. As water condenses out state 2 is saturated. 1: 21 C, 70% => w 1 = , h 1 = 68.5 CV 1 to 2: ṁ liq = ṁ a (w 1 - w 2 ) => w 2 = w 1 - ṁ liq /ṁ a q L = h 1 - h 2 - (w 1 - w 2 ) h f2 w 2 = / = : w 2, 100% => T 2 = 8 C, h 2 =45, h f2 = 33.6 q L = = kj/kg dry air CV Total system : h 3 = h 1 + Ẇ el /ṁ a - (w 1 -w 2 ) h f = = kj/kg dry air 3: w 3 = w 2, h 3 => T 3 = 46 C, φ 3 = 11-12% Ẇ c = ṁ a q L / β = kw

212 To refresh air in a room, a counterflow heat exchanger, see Fig. P12.72, is mounted in the wall, drawing in outside air at 0.5 C, 80% relative humidity and pushing out room air, 40 C, 50% relative humidity. Assume an exchange of 3 kg/min dry air in a SSSF device, and also that the room air exits the heat exchanger to the atmosphere at 23 C. Find the net amount of water removed from the room, any liquid flow in the heat exchanger and (T, φ) for the fresh air entering the room. State 1: w 1 = , h 1 = 119.2, T dew,1 = 27 C The room air is cooled to 23 C < T dew1 so liquid will form in the exit flow channel and state 2 is saturated. 2: 23 C, φ = 100% => w 2 = , h 2 = 88, h f2 = : 0.5 C, φ = 80% => w 3 = , h 3 = 29.2 CV 1 to 2: ṁ liq,2 = ṁ a (w 1 - w 2 ) = 3 ( ) = kg/min CV room: ṁ v,out = ṁ a (w 1 - w 4 ) = ṁ a (w 1 - w 3 ) = 3( ) = 0.06 kg/min CV Heat exchanger: m a (h 4 - h 3) = m a (h 1 - h 2) - m liq h f2 h 4 = h 3 + h 1 - h 2 - (w 1 -w 2 ) h f2 = = 59.9 kj/kg dry air 4: w 4 = w 3, h 4 => T 4 = 32.5 C, φ = 12%

213 Steam power plants often utilize large cooling towers to cool the condenser cooling water so it can be recirculated; see Fig. P The process is essentially evaporative adiabatic cooling, in which part of the water is lost and must therefore be replenished. Consider the setup shown in Fig. P12.73, in which 1000 kg/s of warm water at 32 C from the condenser enters the top of the cooling tower and the cooled water leaves the bottom at 20 C. The moist ambient air enters the bottom at 100 kpa, dry bulb temperature of 18 C and a wet bulb temperature of 10 C. The moist air leaves the tower at 95 kpa, 30 C, and relative humidity of 85%. Determine the required mass flow rate of dry air, and the fraction of the incoming water that evaporates and is lost. P 4 = 95 kpa Air + vap. T 4 = 30 o C 4 1 T 1 = 32 o C φ 4 = 0.85 P 2 = 100 kpa T 2 = 18 o C WBT 2 = 10 o C 18 o C ' LIQ IN (h a2 -h a2 ) + w 2 h FG2 w 2 = = h v2 - h F2 LIQ H O 2 3 Air + vap. ṁ 1 = 1000 kg/s T 3 = 20 o C T 2 = 10 o C, φ 2 = 1.0 w 2 = = (10-18) P v4 = = 3.609, w 4 = Cons. mass: ṁ a2 = ṁ a4 = ṁ a, = ṁ 1 + ṁ v2 = ṁ 3 + ṁ v4 or ṁ 3 = ṁ 1 + ṁ a (w 2 -w 4 ) and set r = ṁ 1 /ṁ a 1st law: ṁ 1 h 1 + ṁ a h a2 + ṁ v2 h v2 = ṁ 3 h 3 + ṁ a h a4 + ṁ v4 h v4 = or r h 1 + (h a2 -h a4 ) + w 2 h v2 = (r + w 2 - w 4 )h 3 + w 4 h v4 r(h 1 -h 3 ) = h a4 - h a2 + w 4 h v4 - w 2 h v2 - (w 4 -w 2 )h 3 r( ) = 1.004(30-18) r = ṁ 1 /ṁ a = ṁ a = kg/s ṁ 3 = ṁ 1 + ṁ a (w 2 -w 4 ) = = ṁ/ṁ 1 =

214 A semipermeable membrane is used for the partial removal of oxygen from air that is blown through a grain elevator storage facility. Ambient air (79% nitrogen, 21% oxygen on a mole basis) is compressed to an appropriate pressure, cooled to ambient temperature 25 C, and then fed through a bundle of hollow polymer fibers that selectively absorb oxygen, so the mixture leaving at 120 kpa, 25 C, contains only 5% oxygen. The absorbed oxygen is bled off through the fiber walls at 40 kpa, 25 C, to a vacuum pump. Assume the process to be reversible and adiabatic and determine the minimum inlet air pressure to the fiber bundle. A = N 2 ; B = O A B P 1 =? B 0.79 A B (0.79/0.95) MIX 5 % O 2 P 2 = 120 kpa P 3 = 40 kpa All T = 25 o C Let s A1 = s B1 = 0 at T = 25 o C & P 1 s - MIX 1 = y A1 R - ln y A1 - y B1 R - ln y B1 s - MIX 2 = R - ln (P 2 /P 1 ) - y A2 R - ln y A2 - y B2 R - ln y B2 Pure B: s - 3 = 0 - R- ln (P 3 /P 1 ) For ṅ 1 s - 1 = ṅ 2 s- 2 + ṅ 3 s- 3 R - [ ln (P 2 /P 1 ) ln ln ln (P 3 /P 1 ) ln ln 0.21]= ln (P 2 /P 1 ) ln (P 3 /P 1 ) = ln P 1 = P 1 min = 141 kpa For P 1 > P 1 min Ṡ > 0

215 A 100-L insulated tank contains N 2 gas at 200 kpa and ambient temperature 25 C. The tank is connected by a valve to a supply line flowing CO 2 at 1.2 MPa, 90 C. A mixture of 50% N 2, 50% CO 2 by mole should be obtained by opening the valve and allowing CO 2 flow in to an appropriate pressure is reached and close the valve. What is the pressure? The tank eventually cools to ambient temperature. Find the net entropy change for the overall process. CO i 2 V = 100 L, P 1 = 200 kpa, T 1 = T 0 = 25 o C P i = 1.2 MPa, T i = 90 o C At state 2: y N2 = y CO2 = 0.50 N 2 n 2 CO2 = n 2 N2 = n 1 N2 = P 1 V/R - T 1 = 200*0.1/8.3145*298.2 = kmol n 2 = kmol 1st law: n i h - i = n 2 ū 2 - n 1 ū 1, for const specific heats n i C - Poi T i = (n i C- Voi +n 1 C- Vo1 )T 2 - n 1 C- Vo1 T 1 But n i = n 1 C - Poi T i = C- Voi T 2 + C- Vo1 (T 2 -T 1 ) = T (T ) T 2 = K P 2 = n 2 R - T 2 /V = /0.1 = 532 kpa Cool to T 3 = T 0 = K P 3 = P 2 T 3 /T 2 = /396.7 = 400 kpa Q 23 = n 2 C - Vo2 (T 3 -T 2 ) = ( )( ) = kj S NET = n 3 s n 1 s- 1 - n i s- i - Q 23 /T 0 = n i [(s - CO 2 ) 3 - s - i ] + n 1 [(s- N 2 ) 3 - s - 1 ] - Q 23 /T 0 = ( ln ln ) ( ln ) = = kj/k

216 A cylinder/piston loaded with a linear spring contains saturated moist air at 120 kpa, 0.1 m 3 volume and also 0.01 kg of liquid water, all at ambient temperature 20 C. The piston area is 0.2 m 2, and the spring constant is 20 kn/m. This cylinder is attached by a valve to a line flowing dry air at 800 kpa, 80 C. The valve is opened, and air flows into the cylinder until the pressure reaches 200 kpa, at which point the temperature is 40 C. Determine the relative humidity at the final state, the mass of air entering the cylinder and the work done during the process. i A+V DRY AIR P 1 = 120 kpa, T 1 = 20 o C = T 0, V 1 = 0.1 m 3, m LIQ 1 = 0.01 kg, A P = 0.2 m 2, k s = 20 kn/m P i = 800 kpa P 2 = 200 kpa T i = 80 o C T 2 = 40 o C P 2 = P 1 + (k s /A 2 p)(v 2 -V 1 ) 200 = (20/0.2 2 )(V 2-0.1) V 2 = 0.26 m 3 φ 1 = 1.0 (or w 1 = / = ) m v1 = P v1 V = R v T = ( = w 1 m A1 ) Assume no liquid at state 2 m v2 = m v1 + m L1 = kg P v2 = m v R v T = = kpa V a) φ 2 = = b) m A1 = P A1 V 1 R A T 1 = = m A2 = = m Ai = m A2 - m A1 = kg c) W CV = PdV = 1 2 (P 1 +P 2 )(V 2 -V 1 ) = 1 ( )( ) = 25.6 kj 2

217 Consider the previous problem and additionally determine the heat transfer. Show that the process does not violate the second law. a) Q CV = m A2 h A2 - m A1 h A1 - m Ai h Ai + m v2 h v2 -m v1 h v1 - m L1 h L1 - P 2 V 2 + P 1 V 1 + W CV = = kj b) S CV = S 2 - S 1 = m A2 s A2 - m A1 s A1 + m v s v2 - m v1 s v1 - m L1 s L1 S SURR = - Q CV /T 0 - m Ai s Ai S NET = m A2 s A2 - m A1 s A1 - m Ai s Ai + m v s v2 - m v1 s v1 - m L1 s L1 - Q CV /T 0 = m A2 (s A2 -s Ai ) + m A1 (s Ai -s A1 ) + m v s v2 - m v1 s v1 - m L1 s L1 - Q CV /T 0 s A2 - s Ai = ln ln = s Ai - s A1 = ln ln = s v2 = s G2 - R v ln φ 2 = ln = s v1 = s G1 - R v ln φ 1 = = s L1 = s F1 = S NET = ( ) ( ) /293.2 = kj/k

218 The air-conditioning by evaporative cooling in Problem is modified by adding a dehumidification process before the water spray cooling process. This dehumidification is achieved as shown in Fig. P12.78 by using a desiccant material, which absorbs water on one side of a rotating drum heat exchanger. The desiccant is regenerated by heating on the other side of the drum to drive the water out. The pressure is 100 kpa everywhere and other properties are on the diagram. Calculate the relative humidity of the cool air supplied to the room at state 4, and the heat transfer per unit mass of air that needs to be supplied to the heater unit. States as noted on Fig. P12.78, text page 466. At state 1, 35 o C: P V1 = φ 1 P G1 = = w 1 = /98.31 = At T 3 = 25 o C: w 3 = w 2 = w 1 /2 = Evaporative cooling process to state 4, where T 4 = 20 o C As in Eq : w 3 (h V3 -h L4 ) = C P0A (T 4 -T 3 ) + w 4 h FG ( ) = 1.004(20-25) + w w 4 = = P V4 /(100-P V4 ) P V4 = kpa, φ 4 = 1.176/2.339 = At T 5 = 25 o C, w 5 = w 4 = Evaporative cooling process to state 6, where T 6 = 20 o C w 5 (h V5 -h L6 ) = C P0A (T 6 -T 5 ) + w 6 h FG ( ) = 1.004(20-25) + w => w 6 = For adiabatic heat exchanger, ṁ A2 = ṁ A3 = ṁ A6 = ṁ A7 = ṁ A, Also w 2 = w 3, w 6 = w 7 h A2 + w 2 h V2 + h A6 + w 6 h V6 = h A3 + w 3 h V3 + h A7 + w 7 h V7 or C P0A T 7 + w 6 (h V7 -h V6 ) = C P0A (T 2 +T 6 -T 3 ) + w 2 (h V2 -h V3 ) T (h V ) = 1.004( ) ( ) = By trial and error, T 7 = 54.7 o C, h V7 = For the heater 7-8, w 8 = w 7, Q. /ṁ A = C P0A (T 8 -T 7 ) + w 7 (h V8 -h V7 ) = 1.004( ) ( ) = 25.8 kj/kg dry air

219 A vertical cylinder is fitted with a piston held in place by a pin, as shown in Fig. P The initial volume is 200 L and the cylinder contains moist air at 100 kpa, 25 C, with wet-bulb temperature of 15 C. The pin is removed, and at the same time a valve on the bottom of the cylinder is opened, allowing the mixture to flow out. A cylinder pressure of 150 kpa is required to balance the piston. The valve is closed when the cylinder volume reaches 100 L, at which point the temperature is that of the surroundings, 15 C. a. Is there any liquid water in the cylinder at the final state? b. Calculate the heat transfer to the cylinder during the process. c. Take a control volume around the cylinder, calculate the entropy change of the control volume and that of the surroundings. From Fig. F.5, at T 1 = 25 o C & WBT 1 = 15 o C a) w 1 = for saturation at state 2, where P 2 = P EXT = 150 kpa, T 2 = 15 o C; Max P V2 = P G2 = kpa w MAX 2 = /( ) = > w 1 not saturated at 2, no liquid P V1 b) w 1 = = P 100-P V1 = kpa V1 m A1 = ( ) m A2 = ( ) = kg, m V1 = w 1 m A1 = kg = kg, m V2 = w 2 m A2 = kg W CV = P EXT dv = P EXT (V 2 -V 1 ) = 150( ) = -15 kj Q CV = m A2 u A2 - m A1 u A1 + m V2 u V2 - m V1 u V1 + (m AE h AE AVE + m VE h VE AVE ) + W CV = ( )/ ( )/ = kj c) S NET = m A2 (s A2 - s AE AVE ) + m A1 (s AE AVE - s A1 ) + m V2 (s V2 -s VE AVE ) + m V1 (s VE AVE - s V1 ) - Q CV /T 0

220 12-50 Assume that the mixture exiting is throttled across the value to 100 kpa and then discharged. Therefore, since the composition is constant. P AE = P A1 = kpa, P VE = P V1 = kpa Also, P A2 = kpa & P V2 = kpa Using T AE T VE K & T 0 = 15 o C = K S NET = ( ln ln ) ( ln ) ( ln ln ) ( ln ) ( 12.1)/288.2 = kj/k

221 Ambient air is at a condition of 100 kpa, 35 C, 50% relative humidity. A steady stream of air at 100 kpa, 23 C, 70% relative humidity, is to be produced by first cooling one stream to an appropriate temperature to condense out the proper amount of water and then mix this stream adiabatically with the second one at ambient conditions. What is the ratio of the two flow rates? To what temperature must the first stream be cooled? 1 P 5 1 = P 2 = 100 kpa 2 MIX T 1 = T 2 = 35 o C COOL. -Q COOL 3 4 LIQ H O 2. -Q MIX = 0 φ 1 = φ 2 = 0.50, φ 4 = 1.0 P 5 = 100, T 5 = 23 o C φ 5 = 0.70 P v1 = P v2 = = kpa => w 1 = w 2 = = P v5 = = kpa => w 5 = = C.V.: Mixing chamber: Call the mass flow ratio r = m a2 /m a1 cons. mass: w 1 + r w 4 =(1+ r)w 5 Energy Eq.: h a1 + w 1 h v1 + rh a4 + rw 4 h v4 = (1+r)h a5 + (1+r)w 5 h v rw 4 = (1+r) or r = m a2 m a1 = w 4, with w 4 = r T 4 + r w 4 h v4 = (1+r) (1+r) or r[1.004 T 4 + w 4 h G ] = 0 Assume T 4 = 5 o C P G4 = , h G4 = w 4 = /( ) = P G4 100-P G4 r = m a2 /m a1 = = [ ] = OK

222 12-52 ENGLISH UNIT PROBLEMS 12.81EA gas mixture at 70 F, 18 lbf/in. 2 is 50% N 2, 30% H 2 O and 20% O 2 on a mole basis. Find the mass fractions, the mixture gas constant and the volume for 10 lbm of mixture. From Eq. 12.3: c i = y i M i / y j M j M MIX = y j M j = = = c N2 = / = , c H2O = / = c O2 = / = , sums to 1 OK R MIX = R /M MIX = / = lbf ft/lbm R V = mr MIX T/P = / = 164 ft EWeighing of masses gives a mixture at 80 F, 35 lbf/in. 2 with 1 lbm O 2, 3 lbm N 2 and 1 lbm CH 4. Find the partial pressures of each component, the mixture specific volume (mass basis), mixture molecular weight and the total volume. From Eq. 12.4: y i = (m i /M i ) / m j /M j n tot = m j /M j = (1/31.999) + (3/28.013) + (1/16.04) = = y O2 = / = , y N2 = / = , y CH4 = / = P O2 = y O2 P tot = = 5.45 lbf/in. 2, P N2 = y N2 P tot = = lbf/in. 2, P CH4 = y CH4 P tot = = lbf/in. 2 V tot = n tot R T/P = / (35 144) = 33.2 ft 3 v = V tot /m tot = 33.2 / ( ) = 6.64 ft 3 /lbm M MIX = y j M j = m tot /n tot = 5/ =

223 EA pipe flows 0.15 lbmol a second mixture with molefractions of 40% CO 2 and 60% N 2 at 60 lbf/in. 2, 540 R. Heating tape is wrapped around a section of pipe with insulation added and 2 Btu/s electrical power is heating the pipe flow. Find the mixture exit temperature. C.V. Pipe heating section. Assume no heat loss to the outside, ideal gases. Energy Eq.: Q. = ṁ(h e h i ) = ṅ(h - e h- i ) = ṅc P mix (T e T i ) CP mix = y i C i = = Btu/lbmole T e = T i + Q. / ṅc P mix = /( ) = R 12.84EAn insulated gas turbine receives a mixture of 10% CO 2, 10% H 2 O and 80% N 2 on a mole basis at 1800 R, 75 lbf/in. 2. The volume flow rate is 70 ft 3 /s and its exhaust is at 1300 R, 15 lbf/in. 2. Find the power output in Btu/s using constant specific heat from C.4 at 540 R. C.V. Turbine, SSSF, 1 inlet, 1 exit flow with an ideal gas mixture, q = 0. Energy Eq.: Ẇ T = ṁ(h i h e ) = ṅ(h - i h- e ) = ṅc P mix (T i T e ) PV = nr T => ṅ = PV. / RT = / ( ) = lbmole/s CP mix = y i C i = = 7.27 Btu/lbmol R Ẇ T = ( ) = Btu/s 12.85ESolve Problem using the values of enthalpy from Table C.7 C.V. Turbine, SSSF, 1 inlet, 1 exit flow with an ideal gas mixture, q = 0. Energy Eq.: Ẇ T = ṁ(h i h e ) = ṅ(h - i h- e ) PV = nr T => ṅ = PV. / RT = / ( ) = lbmole/s Ẇ T = [0.1( ) + 0.1( ) + 0.8( )] = Btu/s

224 ECarbon dioxide gas at 580 R is mixed with nitrogen at 500 R in a SSSF insulated mixing chamber. Both flows are at 14.7 lbf/in. 2 and the mole ratio of carbon dioxide to nitrogen is 2 1. Find the exit temperature and the total entropy generation per mole of the exit mixture. CV mixing chamber, SSSF. The inlet ratio is ṅ CO2 = 2 ṅ N2 and assume no external heat transfer, no work involved. ṅ CO2 + 2ṅ N2 = ṅ ex = 3ṅ N2 ; ṅ N2 (h - N 2 + 2h - CO 2 ) = 3ṅ N2 h - mix ex Take 540R as reference and write h - = h C- Pmix (T-540). C - P N 2 (T i N2-540) + 2C - P CO 2 (T i CO2-540) = 3C - P mix (T mix ex -540) C - P mix = y i C - = ( )/3 = P i 3C - P mix T mix ex = C- T P N i N2 + 2C - T P CO i CO2 = T mix ex = R; P ex N2 = P tot /3; P ex CO2 = 2P tot /3 Ṡ gen = ṅ ex s - ex -(ṅs- ) ico2 - (ṅs - ) in2 = ṅ N2 (s - e - s- i ) N 2 + 2ṅ N2 (s - e - s- i ) CO 2 T ex T ex Ṡ gen /ṅ N2 = C - ln - R - ln y PN 2 T N2 + 2C - ln - 2 R - ln y PCO in2 2 T CO2 ico2 = = Btu/lbmol N 2 R 12.87EA mixture of 60% helium and 40% nitrogen by volume enters a turbine at 150 lbf/in. 2, 1500 R at a rate of 4 lbm/s. The adiabatic turbine has an exit pressure of 15 lbf/in. 2 and an isentropic efficiency of 85%. Find the turbine work. Assume ideal gas mixture and take CV as turbine. w T s = h i -h es, s es = s i, T es = T i (P e /P i ) (k-1)/k C - = = P mix (k-1)/k = R - /C - = 1545/( ) = P mix M mix = = , C P =C - P /M mix = T es = 1500(15/150) = 680 R, w Ts = C P (T i -T es ) = w T ac = w Ts = Btu/lbm; Ẇ = ṁw Ts = 1184 Btu/s

225 EA mixture of 50% carbon dioxide and 50% water by mass is brought from 2800 R, 150 lbf/in. 2 to 900 R, 30 lbf/in. 2 in a polytropic process through a SSSF device. Find the necessary heat transfer and work involved using values from C.4. Process Pv n = constant leading to n ln(v 2 /v 1 ) = ln(p 1 /P 2 ); v = RT/P n = ln(150/30)/ln( / ) = R mix = Σc i R i = ( )/778 = C P mix = Σc i C Pi = = w = - vdp = - n n-1 (P e v e -P i v i ) = - nr n-1 (T e -T Btu ) = i lbm q = h e -h i + w = C P (T e -T i ) + w = Btu/lbm 12.89EA mixture of 4 lbm oxygen and 4 lbm of argon is in an insulated piston cylinder arrangement at 14.7 lbf/in. 2, 540 R. The piston now compresses the mixture to half its initial volume. Find the final pressure, temperature and the piston work. Since T 1 >> T C assume ideal gases. u 2 -u 1 = 1 q 2-1 w 2 = - 1 w 2 ; s 2 -s 1 = 0 Pv k = constant, v 2 = v 1 /2 P 2 = P 1 (v 1 /v 2 ) k = P 1 (2) k ; T 2 = T 1 (v 1 /v 2 ) k-1 = T 1 (2) k-1 Find k mix to get P 2,T 2 and C v mix for u 2 -u 1 R mix = Σc i R i = ( )/778 = C Pmix = Σc i C Pi = = C vmix = C Pmix -R mix = , k mix = C Pmix /C vmix = P 2 = 14.7(2) = lbf/in 2, T 2 = 540* = R 1 w 2 = u 1 -u 2 = C v (T 1 -T )= ( ) = Btu/lbm 2 1 W 2 = m tot1 w = 8 (-24.82) = Btu 2

226 ETwo insulated tanks A and B are connected by a valve. Tank A has a volume of 30 ft 3 and initially contains argon at 50 lbf/in. 2, 50 F. Tank B has a volume of 60 ft 3 and initially contains ethane at 30 lbf/in. 2, 120 F. The valve is opened and remains open until the resulting gas mixture comes to a uniform state. Find the final pressure and temperature and the entropy change for the process. a) 1st law: U 2 -U 1 = 0 = n Ar C - V0 (T 2 -T A1 ) + n C - C H VO (T 2 -T B1 ) 2 6 n Ar = P A1 V A /R - T A1 = = lbmol n C2 = P H B1 V B /R - T B1 = = lbmol n 2 = n Ar + n C2 = lbmol H (T ) (T ) = 0 Solving, T 2 = R P 2 = n 2 R - T 2 /(V A +V B ) = = 38 lbf/in 2 b) S SURR = 0 S NET = S SYS = n Ar S - Ar + n C 2 H 6 S - C 2 H 6 y Ar = / = S - Ar = C- P Ar ln T 2 - R - y Ar P 2 ln T A1 P A1 = ln ln = Btu/lbmol R 50 S - C 2 H 6 = C - T 2 C 2 H 6 ln - R - y C2 H 6 P 2 ln T B1 P B1 = ln ln = Btu/lbmol R S NET = = Btu/R

227 EA large SSSF air separation plant takes in ambient air (79% N 2, 21% O 2 by volume) at 14.7 lbf/in. 2, 70 F, at a rate of 2 lb mol/s. It discharges a stream of pure O 2 gas at 30 lbf/in. 2, 200 F, and a stream of pure N 2 gas at 14.7 lbf/in. 2, 70 F. The plant operates on an electrical power input of 2000 kw. Calculate the net rate of entropy change for the process. Air 79 % N 2 21 % O 2 P 1 = 14.7 T 1 = 70 F ṅ 1 = 2 lbmol/s 1 2 pure O 2 pure N 2 3 -Ẇ IN = 2000 kw P 2 = 30 T 2 = 200 F P 3 = 14.7 T 3 = 70 F ds NET dt Q. CV = - T 0 + ṅ i s - Q. i = - CV + (ṅ - i T 2 s 2 + ṅ 3 s- 3 - ṅ 1 s- 1 ) 0 Q. CV = Σṅ h- i + Ẇ CV = ṅ O 2 C - P0 O 2 (T 2 -T 1 ) + ṅ N2 C - P0 N 2 (T 3 -T 1 ) + Ẇ CV = [ (200-70)] /3600 = = Btu/s Σṅ i s = [ ln i ln 30 ] ds NET dt [ ln ] = Btu/R s = = Btu/R s

228 EA tank has two sides initially separated by a diaphragm. Side A contains 2 lbm of water and side B contains 2.4 lbm of air, both at 68 F, 14.7 lbf/in. 2. The diaphragm is now broken and the whole tank is heated to 1100 F by a 1300 F reservoir. Find the final total pressure, heat transfer, and total entropy generation. U 2 -U 1 = m a (u 2 -u 1 ) a + m v (u 2 -u 1 ) v = 1 Q 2 S 2 -S 1 = m a (s 2 -s 1 ) a + m v (s 2 -s 1 ) v = 1 Q 2 /T + S gen V 2 = V A + V B = m v v v1 + m a v a1 = = ft 3 v v2 = V 2 /m v = , T 2 => P 2v = 58.7 lbf/in 2 v a2 = V 2 /m a = , T 2 => P 2a = mrt 2 /V 2 = lbf/in 2 P 2tot = P 2v + P 2a = 102 lbf/in 2 Water: u 1 = 36.08, u 2 = , s 1 = , s 2 = Air: u 1 = 90.05, u 2 = , s T1 = , s T2 = Q 2 = 2( ) + 2.4( ) = 3208 Btu S gen = 2( ) + 2.4[ (53.34/778) ln(43.415/14.7)] /1760 = = Btu/R 12.93EConsider a volume of 2000 ft 3 that contains an air-water vapor mixture at 14.7 lbf/in. 2, 60 F, and 40% relative humidity. Find the mass of water and the humidity ratio. What is the dew point of the mixture? Air-vap P = 14.7 lbf/in. 2, T= 60 F, φ = 40% P g = P sat60 = lbf/in. 2 P v = φ P g = = lbf/in. 2 m v1 = P v V = = lbm R v T P a = P tot - P v1 = = m a = P a V = = lbm R a T w 1 = m v = m a = T dew is T when P g (T dew ) = ; T = 35.5 F

229 EConsider a 10-ft 3 rigid tank containing an air-water vapor mixture at 14.7 lbf/in. 2, 90 F, with a 70% relative humidity. The system is cooled until the water just begins to condense. Determine the final temperature in the tank and the heat transfer for the process. P v1 = φp G1 = = lbf/in 2 Since m v = const & V = const & also P v = P G2 : P G2 = P v1 T 2 /T 1 = T 2 /549.7 For T 2 = 80 F: /549.7 = =/ ( = P G at 80 F ) For T 2 = 70 F: /549.7 = =/ ( = P G at 70 F ) interpolating T 2 = 78.0 F w 2 = w 1 = ( ) = m a = P a1 V = = lbm R a T st law: Q 12 = U 2 -U 1 = m a (u a2 -u a1 ) + m v (u v2 -u v1 ) = 0.698[0.171(78-90) ( )] = 0.698( Btu/lbm air) = Btu 12.95EAir in a piston/cylinder is at 95 F, 15 lbf/in. 2 and a relative humidity of 80%. It is now compressed to a pressure of 75 lbf/in. 2 in a constant temperature process. Find the final relative and specific humidity and the volume ratio V 2 /V 1. Check if the second state is saturated or not. First assume no water is condensed 1: P v1 = φ 1 P G1 = 0.66, w 1 = /14.34 = : w 2 = P v2 /(P 2 -P v2 ) = w 1 => P v2 = > P g = lbf/in 2 Conclusion is state 2 is saturated φ 2 = 100%, w 2 = P g /(P 2 -P g ) = To get the volume ratio, write the ideal gas law for the vapor phases V 2 = V a2 + V v2 + V f2 = (m a R a + m v2 R v )T/P 2 + m liq v f V 1 = V a1 + V v1 = (m a R a + m v1 R v )T/P 1 Take the ratio and devide through with m a R a T/P 2 to get V 2 /V 1 = (P 1 /P 2 )[ w 2 + (w 1 -w 2 )P 2 v f /R a T]/( w 1 ) = The liquid contribution is nearly zero (= ) in the numerator.

230 EA 10-ft 3 rigid vessel initially contains moist air at 20 lbf/in. 2, 100 F, with a relative humidity of 10%. A supply line connected to this vessel by a valve carries steam at 100 lbf/in. 2, 400 F. The valve is opened, and steam flows into the vessel until the relative humidity of the resultant moist air mixture is 90%. Then the valve is closed. Sufficient heat is transferred from the vessel so the temperature remains at 100 F during the process. Determine the heat transfer for the process, the mass of steam entering the vessel, and the final pressure inside the vessel. i Air-vap mix: P 1 = 20 lbf/in 2, T 1 = 560 R H 2 O φ 1 = 0.10, T 2 = 560 R, φ 2 = 0.90 P v1 = φ 1 P G1 = = lbf/in 2 AIR P + v2 = = lbf/in 2 H 2 O P a2 = P a1 = P 1 - P v1 = = w 1 = / = w 2 = / = w = m v m m vi = m a (w 2 -w 1 ), m a = = 0.96 lbm a P 2 = = lbf/in 2 m vi = 0.96( ) = lbm CV: vessel Q CV = m a (u a2 -u a1 ) + m v2 u v2 - m v1 u v1 - m vi h i u v u G at T u v1 = u v2 = u G at 100 F, u a2 = u a1 Q CV = m vi (u G at T - h i ) = ( ) = Btu

231 EA water-filled reactor of 50 ft 3 is at 2000 lbf/in. 2, 550 F and located inside an insulated containment room of 5000 ft 3 that has air at 1 atm. and 77 F. Due to a failure the reactor ruptures and the water fills the containment room. Find the final pressure. CV Total container. Energy: m v (u 2 -u 1 ) + m a (u 2 -u 1 ) = 1 Q 2-1 W 2 = 0 Initial water: v 1 = , u 1 = , m v = V/v = lbm Initial air: m a = PV/RT = / = lbm Substitute into energy equation (u ) (T 2-77) = 0 u T 2 = & v 2 = V 2 /m v = ft 3 /lbm Trial and error 2-phase (T guess, v 2 => x 2 => u 2 => LHS) T = 300 LHS = T = 290 LHS = T 2 = 298 F, x 2 = , P sat = 65 lbf/in 2, LHS=541.5 OK P a2 = P a1 V 1 T 2 /V 2 T 1 = / = lbf/in 2 => P 2 = P a2 + P sat = lbf/in EAtmospheric air at 95 F, relative humidity of 10%, is too warm and also too dry. An air conditioner should deliver air at 70 F and 50% relative humidity in the amount of 3600 ft 3 per hour. Sketch a setup to accomplish this, find any amount of liquid (at 68 F) that is needed or discarded and any heat transfer. CV air conditioner. Check from psychrometric chart, inlet 1, exit 2. In: w 1 = , h mix,1 = 27, h f,68 = 36.08, Ex: w 2 = , h mix,2 = 25.4 Water must be added. Continuity and energy equations ṁ A (1 + w 1 ) + ṁ liq = ṁ A (1 + w 2 ) & ṁ A h 1mix + ṁ liq h f + Q. CV = ṁ A h 2mix ṁ tot = PV. tot/rt = / = 270 lbm/h ṁ A = ṁ tot /(1+w 2 ) = lbm/h ṁ liq = ṁ A (w 2 - w 1 ) = ( ) = lbm/h Q. CV = ( ) = Btu/h If the properties are calculated from the tables then P g1 = , h g1 = , P g2 = , h g2 = 1092 P v1 = φ 1 P g1 = , P v2 = φ 2 P g2 = w 1 = 0.622P v1 /(P tot -P v1 ) = , w 2 = h 2-h 1 = 0.24(70-95) = ṁ liq = ( ) = lbm/h Q. CV = ( ) = Btu/h

232 ETwo moist air streams with 85% relative humidity, both flowing at a rate of 0.2 lbm/s of dry air are mixed in a SSSF setup. One inlet flowstream is at 90 F and the other at 61 F. Find the exit relative humidity. CV mixing chamber. ṁ a w 1 + ṁ a w 2 = 2ṁ a w ex ; ṁ a h 1 + ṁ a h 2 = 2ṁ a h ex 1: w 1 = , h 1 = : w 2 = , h 2 = 25.2 w ex = (w 1 + w 2 )/2 = ; h ex = (h 1 + h 2 )/2 = 37.8 Exit: w ex, h ex => T ex = 75 F, φ = 95% E An indoor pool evaporates 3 lbm/h of water, which is removed by a dehumidifier to maintain 70 F, Φ = 70% in the room. The dehumidifier is a refrigeration cycle in which air flowing over the evaporator cools such that liquid water drops out, and the air continues flowing over the condenser, as shown in Fig. P For an air flow rate of 0.2 lbm/s the unit requires 1.2 Btu/s input to a motor driving a fan and the compressor and it has a coefficient of performance, β = Q_L /W_c = 2.0. Find the state of the air after the evaporator, T 2, ω 2, Φ 2 and the heat rejected. Find the state of the air as it returns to the room and the compressor work input. The unit must remove 3 lbm/h liquid to keep steady state in the room. As water condenses out state 2 is saturated. 1: 70 F, 70% => w 1 = 0.011, h 1 = 28.8 CV 1 to 2: ṁ liq = ṁ a (w 1 - w 2 ) => w 2 = w 1 - ṁ liq /ṁ a q L = h 1 - h 2 - (w 1 - w 2 ) h f2 w 2 = /( ) = : w 2, 100% => T 2 = 47 F, h 2 = 18.8, h f2 = q L = = 9.94 Btu/lbm dry air CV Total system : h 3 = h 1 + Ẇ el /ṁ a - (w 1 -w 2 ) h f = / = Btu/lbm dry air 3: w 3 = w 2, h 3 => T 3 = 112 F, φ 3 = 12% Ẇ c = ṁ a q L / β = 1 Btu/s

233 E To refresh air in a room, a counterflow heat exchanger is mounted in the wall, as shown in Fig. P It draws in outside air at 33 F, 80% relative humidity and draws room air, 104 F, 50% relative humidity, out. Assume an exchange of 6 lbm/min dry air in a SSSF device, and also that the room air exits the heat exchanger to the atmosphere at 72 F. Find the net amount of water removed from room, any liquid flow in the heat exchanger and (T, φ) for the fresh air entering the room. State 1: w 1 = , h 1 = 51, T dew,1 = 81.5 F The room air is cooled to 72 F < T dew1 so liquid will form in the exit flow channel and state 2 is saturated. 2: 72 F, φ = 100% => w 2 = 0.017, h 2 = 36, h f2 = : 33 F, φ = 80% => w 3 = , h 3 = 11.3 CV 1 to 2: ṁ liq,2 = ṁ a (w 1 - w 2 ) = 6 ( ) = 0.04 lbm/min CV room: ṁ v,out = ṁ a (w 1 - w 4 ) = ṁ a (w 1 - w 3 ) = 6( ) = lbm/min CV Heat exchanger: ṁ a (h 4 - h 3) = ṁ a (h 1 - h 2) - ṁ liq h f2 h 4 = h 3 + h 1 - h 2 - (w 1 -w 2 ) h f2 = = 26 Btu/lbm dry air 4: w 4 = w 3, h 4 => T 4 = 94 F, φ = 9%

234 E A 4-ft 3 insulated tank contains nitrogen gas at 30 lbf/in. 2 and ambient temperature 77 F. The tank is connected by a valve to a supply line flowing carbon dioxide at 180 lbf/in. 2, 190 F. A mixture of 50 mole percent nitrogen and 50 mole percent carbon dioxide is to be obtained by opening the valve and allowing flow into the tank until an appropriate pressure is reached and the valve is closed. What is the pressure? The tank eventually cools to ambient temperature. Calculate the net entropy change for the overall process. CO 2 i V = 4 ft 3, P 1 = 30 lbf/in 2, T 1 = T 0 = 77 F P i = 180 lbf/in. 2, T i = 190 F At state 2: y N2 = y CO2 = 0.50 N 2 n 2 CO2 = n 2 N2 = n 1 N2 = P 1 V/R - T 1 = /( ) = lbmol n 2 = lbmol 1st law: n i h - i = n 2 ū 2 - n 1 ū, for const specific heats 1 n i C - Poi T i = (n i C- Voi +n 1 C- Vo1 )T 2 - n 1 C- Vo1 T 1 But n i = n 1 C - Poi T i = C- Voi T 2 + C- Vo1 (T 2 -T 1 ) = T (T ) T 2 = R P 2 = n 2 R - T 2 /V = /4 144 = lbf/in 2 Cool to T 3 = T 0 = 77 F = R P 3 = P 2 T 3 /T 2 = /710.9 = 60 lbf/in 2 Q 23 = n 2 C - Vo2 (T 3 -T 2 ) = ( )( ) = Btu S NET = n - 3 s 3 - n 1 s- 1 - n i s- i - Q 23 /T 0 = n i [(s - CO 2 ) 3 - s - i ] + n 1 [(s- N 2 ) 3 - s - 1 ] - Q 23 /T 0 = ( ln ln ) ( ln ) = = Btu/R

235 E Ambient air is at a condition of 14.7 lbf/in. 2, 95 F, 50% relative humidity. A steady stream of air at 14.7 lbf/in. 2, 73 F, 70% relative humidity, is to be produced by first cooling one stream to an appropriate temperature to condense out the proper amount of water and then mix this stream adiabatically with the second one at ambient conditions. What is the ratio of the two flow rates? To what temperature must the first stream be cooled? 1 P 5 1 = P 2 = P 5 = 14.7 lbf/in 2 2 MIX T 1 = T 2 = 95 F COOL φ 1 = φ 2 = 0.50, φ 4 = 1.0 -Q MIX = 0 -Q COOL T 5 = 73 F, φ 5 = 0.70 LIQ H 2 O P v1 = P v2 = = w 1 = w 2 = = P v5 = = => w 5 = = MIX: Call the mass flow ratio r = m a2 /m a1 Conservation of water mass: w 1 + r w 4 =(1+ r)w 5 Energy Eq.: h a1 + w 1 h v1 + rh a4 + rw 4 h v4 = (1+r)h a5 + (1+r)w 5 h v rw 4 = (1+r) or r = w 4, with w 4 = P G P G r 0.24 T 4 + rw 4 h v4 = (1+r) (1+r) or r[0.24 T 4 + w 4 h G ] = 0 Assume T 4 = 40 F P G4 = , h G4 = w 4 = = m a2 = m a = [ ] = OK => T 4 = 40 F

236 13-1 CHAPTER 13 The correspondence between the new problem set and the previous 4th edition chapter 10 problem set: New Old New Old New Old new 44 new mod new new new mod new new new new new new new The problems that are labeled advanced are: New Old New Old New Old new mod 73 new new

237 13-2 The English-unit problems are: New Old New Old New Old mod mod indicates a modification from the previous problem that changes the solution but otherwise is the same type problem.

238 13-3 The following table gives the values for the compressibility, enthalpy departure and the entropy departure along the saturated liquid-vapor boundary. These are used for all the problems using generalized charts as the figures are very difficult to read accurately (consistently) along the saturated liquid line. It is suggested that the instructor hands out copies of this page or let the students use the computer for homework solutions. T r P r Z f Z g d(h/rt) f d(h/rt) g d(s/r) f d(s/r) g

239 A special application requires R-12 at 140 C. It is known that the triple-point temperature is 157 C. Find the pressure and specific volume of the saturated vapor at the required condition. The lowest temperature in Table B.3 for R-12 is -90 o C, so it must be extended to -140 o C using the Clapeyron Eq integrated as in example 13.1 Table B.3: at T 1 = -90 o C = K, P 1 = 2.8 kpa. R = = ln P P 1 = h fg R (T - T 1 ) = ( ) T T = P = 2.8 exp( ) = kpa

240 In a Carnot heat engine, the heat addition changes the working fluid from saturated liquid to saturated vapor at T, P. The heat rejection process occurs at lower temperature and pressure (T T), (P P). The cycle takes place in a piston cylinder arrangement where the work is boundary work. Apply both the first and second law with simple approximations for the integral equal to work. Then show that the relation between P and T results in the Clapeyron equation in the limit T dt. T T 1 2 P P- P P P 1 2 T T T 4 3 P- P 4 3 T- T s fg at T s v fg at T v q H = Ts FG ; q L = (T- T)s FG ; w NET = q H - q L = Ts FG Problem similar to development in section 13.1 for shaft work, here boundary movement work, w = Pdv 3 w NET = P(v 2 -v 1 ) + 2 Pdv + (P- P)(v 4 -v 3 ) Pdv Approximating, 3 Pdv (P - P 4 ) 2 (v 3 -v 2 ); Pdv (P - P ) 2 (v 1 -v ) & collecting terms, w NET P[( v 2 +v 3 2 ) - (v 1 +v 4 2 )] (the smaller the P, the better the approximation) P T s FG ( v 2 +v 3 2 ) - (v 1 +v 4 2 ) In the limit as T 0: v 3 v 2 = v G, & lim T 0 P dp T = SAT dt = s FG v FG v 4 v 1 = v F

241 Ice (solid water) at 3 C, 100 kpa is compressed isothermally until it becomes liquid. Find the required pressure. Water, triple point T = 0.01 o C, P = kpa Table B.1.1: v F = 0.001, h F = 0.01 kj/kg, Tabel B.1.5: v I = , h I = kj/kg Clapeyron dp IF h dt = F -h I (v F -v I )T = = dp IF P T = ( ) = kpa dt P = P tp + P = kpa 13.4 Calculate the values h FG and s FG for nitrogen at 70 K and at 110 K from the Clapeyron equation, using the necessary pressure and specific volume values from Table B.6.1. Clapeyron equation Eq.13.7: dp G h dt = FG s FG = Tv FG v FG For N 2 at 70 K, using values for P G from Table B.6 at 75 K and 65 K, and also v FG at 70 K, P G h FG T(v G -v F ) Τ s FG = h FG /T = kj/kg K (2.97) = 70( )( ) = kj/kg (207.8) Comparison not very close because P G not linear function of T. Using 71 K & 69 K, h FG = 70( )( ) = At 110 K, h FG 110( )( ) = kj/kg (134.17) s FG = = kj/kg K (1.22) 110

242 Using thermodynamic data for water from Tables B.1.1 and B.1.5, estimate the freezing temperature of liquid water at a pressure of 30 Mpa. P 30 MPa T.P. dp IF h H 2 O dt = IF const Tv IF At the triple point, v IF = v F - v I = T = m 3 /kg h IF = h F - h I = ( ) = kj/kg dp IF dt = = kpa/k ( ) at P = 30 MPa, T ( ) ( ) = -2.2 o C 13.6 Helium boils at 4.22 K at atmospheric pressure, kpa, with h FG = 83.3 kj/kmol. By pumping a vacuum over liquid helium, the pressure can be lowered and it may then boil at a lower temperature. Estimate the necessary pressure to produce a boiling temperature of 1 K and one of 0.5 K. Helium at 4.22 K: P 1 = MPa, - h = 83.3 kj/kmol FG dp SAT dt h = FG h FG P SAT P 2 Tv FG RT 2 ln = P 1 For T 2 = 1.0 K: ln P 2 For T 2 = 0.5 K: ln h FG R [ 1 T 1 1 T 2 ] = [ ] => P 2 P = 83.3 [ ] 0.5 P 2 = kpa = Pa = kpa = 48 Pa

243 A certain refrigerant vapor enters an SSSF constant pressure condenser at 150 kpa, 70 C, at a rate of 1.5 kg/s, and it exits as saturated liquid. Calculate the rate of heat transfer from the condenser. It may be assumed that the vapor is an ideal gas, and also that at saturation, v f v g. The following quantities are known for this refrigerant: ln P g = /T C P = 0.7 kj/kg K with pressure in kpa and temperature in K. The molecular weight is 100. Refrigerant: State 1 T 1 = 70 o C P 1 = 150 kpa State 2 P 2 = 150 kpa x 2 = 1.0 State 3 P 3 = 150 kpa x 3 = 0.0 ln (150) = /T 2 => T 2 = K = 45.3 o C = T 3 q 13 = h 3 -h 1 = (h 3 -h 2 ) + (h 2 -h 1 ) = - h FG T3 + C P0 (T 2 -T 1 ) dp G dt = h FG Tv FG, v FG v G = RT P G, dp G d ln P G dt = P G dt h FG = RT 2 P G d ln P G = +1000/T dt 2 = h FG /RT 2 h FG = 1000 R = /100 = q 13 = ( ) = kj/kg Q. COND = 1.5( ) = kw 13.8 A container has a double wall where the wall cavity is filled with carbon dioxide at room temperature and pressure. When the container is filled with a cryogenic liquid at 100 K the carbon dioxide will freeze so the wall cavity has a mixture of solid and vapor carbon dioxide at the sublimation pressure. Assume that we do not have data for CO 2 at 100 K, but it is known that at 90 C: P sat = 38.1 kpa, h IG = kj/kg. Estimate the pressure in the wall cavity at 100 K. For CO 2 space: at T 1 = -90 o C = K, P 1 = 38.1 kpa, h IG = kj/kg For T 2 = T co2 = 100 K: Clapeyron dp SUB dt = h IG h IG P SUB Tv IG RT 2 P 2 ln = h IG [ 1 P 1 R ] 100 = [ ] 100 = or P 2 = P P 2 = kpa = Pa

244 Small solid particles formed in combustion should be investigated. We would like to know the sublimation pressure as a function of temperature. The only information available is T, h FG for boiling at kpa and T, h IF for melting at kpa. Develop a procedure that will allow a determination of the sublimation pressure, P sat (T). T NBP = normal boiling pt T. T NMP = normal melting pt T. T TP = triple point T. 1) T TP T NMP P kpa P TP Solid Liquid Vap. T T T T TP NMP NBP T TP P TP 2) (1/P SAT ) dp SAT h FG RT 2dT MPa T NMP Since h FG const h FG NBP we ca perform the integral over temperature P TP h ln FG NBP [ ] get P R T NBP T TP TP 3) h IG at TP = h G - h I = (h G - h F ) + (h F - h I ) h FG NBP + h IF NMP Assume h IG const. again we can evaluate the integral P P SUB SUB ln = P (1/P TP P TP or P SUB = fn(t) T SUB ) dp SUB T TP h IG h RT 2dT IG [ 1 1 ] R T TP T

245 Derive expressions for ( T/ v) u and for ( h/ s) v that do not contain the properties h, u, or s. ( T ) v u = -( u) v T/( u P ) - T( P ) T v = T v (see Eqs and 13.34) C v As dh = Tds + vdp => ( h s ) v = T + v( P s ) v = T - v( T v ) s But ( T v ) s = - ( s v ) T/( s T ) v = - ( h s ) v = T + vt C v ( P T ) v T( P T ) v C v (Eq.13.22) (Eq.13.20) Derive expressions for ( h/ v) T and for ( h/ T ) v that do not contain the properties h, u, or s. Find ( h v ) T dh = Tds + vdp and ( h T ) v and use Eq ( h v ) T = T( s v ) T + v( P v ) T = T( P T ) v + v( P v ) T Also for the second first derivative use Eq ( h T ) v = T( s T ) v + v( P T ) v = C v + v( P T ) v Develop an expression for the variation in temperature with pressure in a constant entropy process, ( T/ P) s, that only includes the properties P v T and the specific heat, C p. ( T P ) s = - ( s ) P T = - ( s T ) P -( v T ) P (C P /T) = T C P ( v T ) P {( s ) P T = -( v) T P, Maxwell relation Eq and the other is Eq.13.27}

246 Determine the volume expansivity, α P, and the isothermal compressibility, β T, for water at 20 C, 5 MPa and at 300 C, and 15 MPa using the steam tables. Water at 20 o C, 5 MPa (compressed liquid) α P = 1 v ( v T ) P 1 v ( v T ) P Using values at 0 o C, 20 o C and 40 o C, Estimate by finite difference. α P = / o C β T = - 1 v ( v P ) T - 1 v ( v P ) T Using values at saturation, 5 MPa and 10 MPa, β T Water at 300 o C, 15 MPa (compressed liquid) = MPa α P β T = / o C = MPa Sound waves propagate through a media as pressure waves that cause the media to go through isentropic compression and expansion processes. The speed of sound c is defined by c 2 = ( P/ ρ) s and it can be related to the adiabatic compressibility, which for liquid ethanol at 20 C is 940 µm 2 /N. Find the speed of sound at this temperature. c 2 = ( P ρ ) s = v2 ( P v ) s = 1-1 v ( v P ) s ρ = 1 β s ρ From Table A.4 for ethanol, ρ = 783 kg/m 3 c = ( )1/2 = 1166 m/s

247 Consider the speed of sound as defined in Problem Calculate the speed of sound for liquid water at 20 C, 2.5 MPa and for water vapor at 200 C, 300 kpa using the steam tables. From problem : c 2 = ( P ρ ) s = -v2 ( P v ) s Liquid water at 20 o C, 2.5 MPa, assume ( P v ) s ( P v ) T Using saturated liquid at 20 o C and compressed liquid at 20 o C, 5 MPa, c = -( ) ( ) = => c = 1415 m/s Superheated vapor water at 200 o C, 300 kpa v = , s = At P = 200 kpa & s = : T = 157 o C, v = At P = 400 kpa & s = : T = o C, v = c 2 = -(0.7163) 2 ( ) = => c = 506 m/s Find the speed of sound for air at 20 C, 100 kpa using the definition in Problem and relations for polytropic processes in ideal gases. From problem : c 2 = ( P ρ ) s = -v2 ( P v ) s For ideal gas and isentropic process, Pv k = const P = Cv -k P v = -kcv-k-1 = -kpv -1 c 2 = -v 2 (-kpv -1 ) = kpv = krt c = krt = = m/s

248 A cylinder fitted with a piston contains liquid methanol at 20 C, 100 kpa and volume 10 L. The piston is moved, compressing the methanol to 20 MPa at constant temperature. Calculate the work required for this process. The isothermal compressibility of liquid methanol at 20 C is 1220 (µm) 2 /N. 2 Pdv = P( v ) 2 P T dp T = - vβ T PdP T 1 w 12 = 1 For v constant & β T constant the integral can be evaluated vβ T w 12 = - 2 (P2 2 - P 2 1) For liquid methanol, from Table A.4: ρ = 787 m 3 /kg V 1 = 10 L, m = = 7.87 kg W 12 = [(20) 2 - (0.1) 2 ] = 2440 J = 2.44 kj A piston/cylinder contains 5 kg of butane gas at 500 K, 5 MPa. The butane expands in a reversible polytropic process with polytropic exponent, n = 1.05, until the final pressure is 3 MPa. Determine the final temperature and the work done during the process. C 4 H 10 m = 5 kg T 1 = 500 K P 1 = 5 MPa Rev. polytropic process (n=1.05), P 2 = 3 MPa T r1 = = 1.176, P r1 = = From Fig. D.1: Z 1 = 0.68 V 1 = = m P 1 V n 1 = P 2 V n 2 => V 2 = (5/3) 1.05 = m 3 Z 2 T r2 = P 2 V = mrt C = at P r2 = 3/3.8 = Trial & error: T r2 = 1.062, Z 2 = => T 2 = K 2 W 12 = 1 PdV = P 2 V 2 - P 1 V = 1-n = 114 kj

249 Show that the two expressions for the Joule Thomson coefficient µ J given by Eq are valid. µ j = ( T P ) h = - ( h P ) T/( h T ) P = [T( v T ) P - v]/c P Also µ j = [ ZRT P v = ZRT P, ( v T ) P = ZR P + RT P ( Z T ) P + RT2 P ( Z T ) P - v]/c P = RT2 C P P ( Z T ) P A 200-L rigid tank contains propane at 9 MPa, 280 C. The propane is then allowed to cool to 50 C as heat is transferred with the surroundings. Determine the quality at the final state and the mass of liquid in the tank, using the generalized compressibility chart, Fig. D.1. Propane C 3 H 8 : V = 0.2 m 3, P 1 = 9 MPa, T 1 = 280 o C = K cool to T 2 = 50 o C = K From Table A.2: T C = K, P C = 4.25 MPa P r1 = v 2 = v 1 = = 2.118, T r1 = = From Fig. D.1: Z 1 = Z 1 RT = = P From Fig. D.1 at T r2 = 0.874, P G2 = = 1912 kpa v G2 = /1912 = v F2 = /1912 = = x 2 ( ) => x 2 = m LIQ 2 = ( ) 0.2/ = kg

250 A rigid tank contains 5 kg of ethylene at 3 MPa, 30 C. It is cooled until the ethylene reaches the saturated vapor curve. What is the final temperature? T C 2 H V = const m = 5 kg P 1 = 3 MPa T 1 = 30 o C = K cool to x 2 = 1.0 P r1 = = 0.595, T r1 = = v Fig. D.1: Z 1 = 0.82 Z 2 T r2 Z G2 T r2 P r2 = P r1 = Z 1 T r = Z G2 T r2 Trial & error: T r2 Z G2 P r2 P r2 CALC ~ OK => T 2 = K Two uninsulated tanks of equal volume are connected by a valve. One tank contains a gas at a moderate pressure P 1, and the other tank is evacuated. The valve is opened and remains open for a long time. Is the final pressure P 2 greater than, equal to, or less than P 1 /2? V A = V B V 2 = 2V 1, T 2 = T 1 = T P 2 V 1 Z 2 mrt = P 1 V 2 Z 1 mrt = 1 2 If T < T B, Z 2 > Z 1 If T > T B, Z 2 < Z 1 Z 2 A B Z 1 GAS EVAC. P 2 P 1 > 1 2 P 2 P 1 < 1 2 Z P 2 1 T > T B T < T 1 B P 1 P

251 Show that van der Waals equation can be written as a cubic equation in the compressibility factor involving the reduced pressure and reduced temperature as Z 3 ( P r 8T r + 1) Z 2 + van der Waals equation, Eq.13.55: 2 27 P r 27 P r 64 T 2 Z T = 0 r r P = RT v-b - a v 2 a = 27 R 2 2 T C RT C b = 64 P C 8P C multiply equation by v2 (v-b) P Get: Multiply by v 3 - (b + RT P ) v2 + ( a P ) v - ab P = 0 P 3 Pv R 3 3 and substitute Z = T RT Get: Z 3 ( bp RT + 1) Z2 + ( ap abp2 R 2 T 2) Z ( R 3 T 3) = 0 Substitute for a and b, get: Z 3 ( P r 8T r + 1) Z P r 64 T 2 r Z 27 P r T r 3 = 0 Where P r = P P c, T r = T T c

252 Develop expressions for isothermal changes in enthalpy and in entropy for both van der Waals equation and Redlich-Kwong equation of state. van der Waals equation of state: P = RT v-b a v 2 ( P ) T v = R v-b ( u ) v T = T( P) T v - P = RT v-b RT v-b + a v 2 2 (u 2 -u 1 ) T = 1 2 [T( P ) T v - P]dv = a v 2dv = a( 1 1 ) v 1 v 2 1 (h 2 -h 1 ) T = (u 2 -u 1 ) T + P 2 v 2 - P 1 v 1 = P 2 v 2 P 1 v 1 + a( 1 v 1 1 v 2 ) 2 (s 2 -s 1 ) T = 1 2 ( P ) T v dv = 1 Redlich-Kwong equation of state: ( P T ) v = R v-b + 2 (u 2 -u 1 ) T = 1 a 2v(v+b)T 3/2 R v-b dv = R ln(v 2 -b ) v 1 -b P = RT v-b - a v(v+b)t 1/2 3a 2v(v+b)T 1/2dv = -3a 2 +b 2bT 1/2ln[(v )( v 2 )] v 2 v 1 +b 3a 2 (h 2 -h 1 ) T = P 2 v 2 - P 1 v 1 - +b 2bT 1/2ln[(v 2 (s 2 -s 1 ) T = 1 [ R v-b + a/2 v(v+b)t 3/2 ]dv v 2 )( v 2 = R ln( v 2 -b v 1 -b )- a 2 +b 2bT 3/2ln[(v v 1 +b )] v 2 )( v 1 v 1 +b )]

253 Determine the reduced Boyle temperature as predicted by an equation of state (the experimentally observed value for most substances is about 2.5), using the van der Waals equation and the Redlich Kwong equation. Note: It is helpful to use Eqs and in addition to Eq The Boyle temp. is that T at which But lim P 0 ( Z) P T = lim Z-1 P 0 P-0 = 1 lim RT (v - ) RT P 0 P van der Waals: P = RT v-b a v 2 lim P 0 ( Z P ) T = 0 multiply by v-b P, get v-b = RT P - a(v-b) RT 2 or v - Pv P = b a(1-b/v) Pv & RT lim P 0 ( Z) P T = b a(1-0) RT = 0 only at T Boyle or T Boyle = a Rb = 27 8 T C = T C Redlich-Kwong: P = RT v-b a v(v+b)t 1/2 as in the first part, get v - RT P = b a(1-b/v) Pv(1+b/v)T 1/2 & RT lim P 0 ( Z) P T = b a(1-0) Pv(1+0)T 1/2 = 0 or T 3/2 Boyle = a Rb = R2 T 5/2 P C C RP C R T C T Boyle = ( )2/3 T C = 2.9 T C only at T Boyle

254 Consider a straight line connecting the point P = 0, Z = 1 to the critical point P = P C, Z = Z C on a Z versus P compressibility diagram. This straight line will be tangent to one particular isotherm at low pressure. The experimentally determined value is about 0.8 T C. Determine what value of reduced temperature is predicted by an equation of state, using the van der Waals equation and the Redlich Kwong equation. See also note for Problem Z C - 1 slope = P C - 0 But also equals lim P 0 ( Z) P T From solution for T = T lim P 0 ( Z) P T = lim Z-1 P 0 P-0 = 1 lim RT (v ) RT P 0 P VDW: using solution 13.25: 1.0 Z Z C 0 P C C.P. lim P 0 ( Z) P T = Z C = P C RT [b a ] RT P or ( 1-Z C P C )(RT ) 2 + brt a = 0 Substituting Z C = 3 8, a = 27 R 2 T 2 C, b = RT C 64 P C 8P C 40 T 2 r + 8 T r 27 = 0 solving, T r = Redlich-Kwong: using solution 13.25, lim P 0 ( Z) P T = Z C -1 1 = P C RT [b - a RT 3/2 ] or ( 1-Z C P C )R 2 T 5/2 + brt 3/2 - a = 0 Substitute Z C = 1 3, a = R 2 T 5/2 C RT C, b = P C P C get 2 5/2 T 3 r T 3/2 r = 0 solving, T r = 0.787

255 Determine the 2nd virial coefficient B(T) using the van der Waals equation and the Redlich Kwong equation of state. Find also its value at the critical point (the experimentally observed value is about R T C /P C ). From Eq.13.51: B(T) = - lim P 0 α where Eq.13.47: α = R T P v van der Waals: P = R T - a v -b v 2 which we can multiply by v -b P, get v - b = R T P a(v -b) Pv 2 or v R T P = b a(1-b/v ) Pv Taking the limit for P -> 0 then we get : B(T) = b a/r T = R T C P C ( T C 64 T ) where a,b are from Eq At T = T C then we have B(T C ) = R T C ( - 19 P C 64 ) = R TC For Redlich Kwong the result becomes P C v R T P = b a(1- b/v ) Pv (1 + b/v ) T 1/2 => B(T) = b Now substitute Eqs and for a and b, B(T) = R T C [ T C P C T 3/2 and evaluated at T C it becomes B(T C ) = R T C P C [ ] = R ] TC P C a R T 3/2

256 One early attempt to improve on the van der Waals equation of state was an expression of the form P = RT v-b - a v 2 T Solve for the constants a, b, and v using the same procedure as for the van der c Waals equation. From the equation of state take the first two derivatives of P with v: ( P v ) T = - RT (v-b) 2 + 2a v 3 T and ( 2 P v 2 ) T = - 2RT (v-b) 3-6a v 4 T Since both these derivatives are zero at the critical point: RT - (v-b) 2 + 2a 2RT v 3 = 0 and - T (v-b) 3-6a v 4 T = 0 Also, P C = RT C v C -b a v 2 C T C solving these three equations: v C = 3b, a = 27 R 2 T 3 RT C C, b = 64 P C 8P C Use the equation of state from the previous problem and determine the Boyle temperature. P = RT v-b a v 2 T Multiplying by v-b P gives: Using solution from for T Boyle : lim P 0 v b = RT P a(1-b/v) PvT RT a(1-0) (v )= b P RT T = b a RT 2 = 0 at T Boyle or T Boyle = a Rb = 27 R 2 T 3 C 1 8P C = 64 P C R RT C 27 8 T C

257 Calculate the difference in internal energy of the ideal-gas value and the real-gas value for carbon dioxide at the state 20 C, 1 MPa, as determined using the virial equation of state, including second virial coefficient terms. For carbon dioxide we have: B = m 3 /kmol, T(dB/dT) = m 3 /kmol, both at 20 C. virial eq.: P = RT v + BRT v 2 ; ( P T ) v = R v + BR v 2 + RT v 2 ( db dt ) u-u * = - v [ P ( ) v T v - P]dv = - [ RT 2 ( db RT v 2 )]dv = - [T (db )] dt v dt Solution of virial equation (quadratic formula): 1 R T v = 2 P [ BP/R T ]where: R T P = = v = [ (-0.128)/ ] = m3 /kmol Using the minus-sign root of the quadratic formula results in a compressibility factor < 0.5, which is not consistent with such a truncated equation of state. u-u * = [0.266] = kj/kmol Refrigerant-123, dichlorotrifluoroethane, which is currently under development as a potential replacement for environmentally hazardous refrigerants, undergoes an isothermal SSSF process in which the R-123 enters a heat exchanger as saturated liquid at 40 C and exits at 100 kpa. Calculate the heat transfer per kilogram of R- 123, using the generalized charts, Fig. D.2 R-123: M = , T C = K, P C = 3.67 MPa 1 2 T 1 = T 2 = 40 o C, x 1 = 0 Heat Exchanger P 2 = 100 kpa SSSF From D.1: P 1 = = 308 kpa T r1 = T r2 = 313.2/456.9 = 0.685, P r2 = 0.1/3.67 = From Fig. D.2: P r1 = 0.084, (h * h) 1 /RT C = 4.9 P 2 < P 1 with no work done, so process is irreversibel. Energy Eq.: q + h 1 = h 2, Entropy Eq.: s 1 + dq/t + s gen = s 2, s gen > 0 From Fig. D.2: (h * -h) 2 /RT C = h 2 - h 1 = [ ]/152.93= kj/kg

258 Calculate the heat transfer during the process described in Problem From solution 13.18, V 1 = m 3, V 2 = m 3, W 12 = 114 kj T r1 = 1.176, P r1 = ; T r2 = 1.062, P r2 = 0.789, T 2 = K From Fig. D.2: (h * - h) 1 = 1.30 RT C, (h * - h) 2 = 0.90 RT C h * 2 - h * 1 = ( ) = kj/kg h 2 - h 1 = ( ) = kj/kg U 2 - U 1 = m(h 2 - h 1 ) - P 2 V 2 + P 1 V 1 = 5(-58.8) = kj Q 12 = U 2 - U 1 + W 12 = kj Saturated vapor R-22 at 30 C is throttled to 200 kpa in an SSSF process. Calculate the exit temperature assuming no changes in the kinetic energy, using the generalized charts, Fig. D.2 and the R-22 tables, Table B.4 R-22 throttling process 1st law: h 2 -h 1 = (h 2 -h * 2) + (h * 2-h * 1) + (h * 1-h 1 ) = 0 a) Generalized Chart, Fig. D.2, R = / = T r1 = = => (h* 1-h 1 ) = (0.53) = For C P0, use h values from Table B.4 at low pressure. C P ) / (30-20) = kj/kg K Substituting: (h 2 -h * 2) (T 2-30) = 0 at P r2 = 200/4970 = Assume T 2 = 5.0 o C => T r2 =278.2/369.3 = (h * 2-h 2 ) = RT 0.07 = (0.07) = 2.49 Substituting : (5.0-30) = T 2 = 5.0 o C b) R-22 tables, B.4: at T 1 = 30 o C, x 1 = 1.0 => h 1 = h 2 = h 1 = , P 2 = 0.2 MPa => T 2 = 4.7 o C

259 L tank contains propane at 30 C, 90% quality. The tank is heated to 300 C. Calculate the heat transfer during the process. T C H 3 8 T r1 = v Fig. D.1: V = 250 L = 0.25 m 3 T 1 = 30 o C = K, x 1 = 0.90 Heat to T 2 = 300 o C = K M = , T C = K, P C = 4.25 MPa R = , C P0 = Z 1 = (1- x 1 ) Z f1 + x 1 Z g1 = = Fig D.2: h * 1-h 1 RT c = = P SAT r = 0.30 P SAT 1 = MPa m = = kg Z Z 2 P r2 = = at T r2 = 1.55 Trial and error on P r2 P r2 = => P 2 = MPa, Z 2 = 0.94, (h * - h) 2 = 0.35 RT C (h * 2-h * 1) = (300-30) = kj/kg (h * 1-h 1 ) = = 63.5 kj/kg (h * 2-h 2 ) = = 24.4 kj/kg Q 12 = m(h 2 -h 1 ) - (P 2 -P 1 )V = 7.842( ) - ( ) 0.25 = = 3391 kj

260 The new refrigerant fluid R-123 (see Table A.2) is used in a refrigeration system that operates in the ideal refrigeration cycle, except the compressor is neither reversible nor adiabatic. Saturated vapor at C enters the compressor and superheated vapor exits at 65 C. Heat is rejected from the compressor as 1 kw and the R-123 flow rate is 0.1 kg/s. Saturated liquid exits the condenser at 37.5 C. Specific heat for R-123 is C p =0.6 kj/kg. Find the coefficient of performance. R-123: T c = K, P c = 3.67 MPa, M = kg/kmol, R = kj/kg K State 1: T 1 = o C = K, sat vap., x 1 = 1.0 T r1 = 0.54, Fig D.1, P r1 = 0.01, P 1 = P r1 P c = 37 kpa Fig. D.2, h * 1 -h 1 = 0.03 RT C = 0.8 kj/kg State 2: T 2 = 65 o C = K State 3: T 3 = 37.5 o C = K, sat. liq., x 3 = 0 T r3 = 0.68, Fig. D.1: P r3 = 0.08, P 3 = P r3 P c = 294 kpa P 2 = P 3 = 294 kpa, P r2 = 0.080, T r2 = 0.74, Fig. D.2: h * 2 -h 2 = 0.25 RT C = 6.2 kj/kg h * 3 -h 3 = 4.92 RT C = kj/kg State 4: T 4 = T 1 = K, h 4 = h 3 1 st Law Evaporator: q L + h 4 = h 1 + w; w = 0, h 4 = h 3 q L = h 1 - h 3 = (h 1 h * 1 ) + (h* 1 h* 3 ) + (h* 3 h 3) h * 1 h* 3 = C p (T 1 - T 3 ) = kj/kg, q L = = 83.0 kj/kg 1 st Law Compressor: q + h 1 = h 2 + w c ; Q. = -1.0 kw, ṁ = 0.1 kg/s w c = h 1 - h 2 + q; h 1 - h 2 = (h 1 h * 1 ) + (h* 1 h* 2 ) + (h* 2 h 2) h * 1 h* 2 = C P (T 1 - T 2 ) = kj/kg, w c = = kj/kg β = q L /w c = 83.0/59.5 = 1.395

261 A cylinder contains ethylene, C 2 H 4, at MPa, 13 C. It is now compressed in a reversible isobaric (constant P) process to saturated liquid. Find the specific work and heat transfer. Ethylene C 2 H 4 P 1 = MPa = P 2, T 1 = -13 o C = K State 2: saturated liquid, x 2 = 0.0 T r1 = = P r1 = P r2 = = From Figs. D.1, D.2: Z 1 = 0.85, (h * 1-h 1 )/RT c = 0.40 v 1 = Z 1 RT 1 = = P (h * 1-h 1 ) = = 33.5 From Figs. D.1, D.2: T 2 = = K Z 2 = 0.05, (h * 2-h 2 )/RT c = 4.42 v 2 = Z 2 RT 2 = = P (h * 2-h 2 ) = = (h * 2-h * 1) = C P0 (T 2 -T 1 ) = ( ) = w 12 = Pdv = P(v 2 -v 1 ) = 1536( ) = kj/kg q 12 = (u 2 -u 1 ) + w 12 = (h 2 -h 1 ) = = -379 kj/kg

262 A piston/cylinder initially contains propane at T = -7 C, quality 50%, and volume 10L. A valve connecting the cylinder to a line flowing nitrogen gas at T = 20 C, P = 1 MPa is opened and nitrogen flows in. When the valve is closed the cylinder contains a gas mixture of 50% nitrogen, 50% propane on a mole basis at T = 20 C, P = 500 kpa. What is the cylinder volume at the final state and how much heat transfer took place? State 1: Propane, T 1 = -7 o C, x 1 = 0.5, V 1 = 10 L T cp = K, P cp = 4.25 kpa, C pp = kj/kg-k, M = kg/kmol Fig. D.1: T r1 = 0.72, P r1 = 0.12, P 1 = P r1 P c = 510 kpa Fig. D.1: Z f1 = 0.020, Z g1 = 0.88, Z 1 = (1 - x 1 )Z f + x 1 Z g = 0.45 n 1 = P 1 V 1 /(Z 1 R T 1 ) = /( ) = kmol h 1 = h * 1o + C PP (T 1 - T o ) + (h 1 - h * 1 ) ; h * 1o = 0, (h * 1 -h 1 ) f /R T c = 4.79, (h * 1 -h 1 ) g /R T c = 0.25 * h1 - h 1 = (1 - x 1 ) (h * 1 - h 1 ) f + x 1 (h * 1 - h 1 ) g = 7748 kj/kmol h 1 = (-7-20) = kj/kmol Inlet: Nitrogen, T i = 20 o C, P i = 1.0 MPa, T cn = K, P cn = 3.39 MPa, C pn = kj/kg-k, M = kg/kmol T ri = 2.323, P ri = 0.295, h * i -h i = = kj/kmol h i = h * io + C Pn (T i - T o ) + (h i - h * i ) ; h * io = 0, T i - T o = 0 State 2: 50% Propane, 50% Nitrogen by mol, T 2 = 20 o C, P 2 = 500 kpa T cmix = y i T ci = 248 K, P cmix = y i P ci = 3.82 MPa T r2 = 1.182, P r2 = 0.131, Z 2 = 0.97, (h * 2 -h 2 )/R T c = 0.06 h 2 = h * 2o + C Pmix (T 2 - T o ) + (h 2 - h * 2 ) ;; h * 2o = 0, T 2 - T o = 0 a) n i = n 1 => n 2 = n 1 + n i = , V 2 = n 2 Z 2 RT 2 /P 2 = m 3 b) 1 st Law: Q cv + n i h - i = n 2 ū 2 - n 21 ū 21 + W cv ; ū = h- - Pv - W cv = (P 1 + P 2 )(V 2 - V 1 )/2 = kj Q cv = n 2 h n 1 h- 1 - n i h- i - P 2 V 2 + P 1 V 1 + W cv h i = kj/kmol, h2 = kj/kmol, Q cv = kj

263 An ordinary lighter is nearly full of liquid propane with a small amount of vapor, the volume is 5 cm 3 and temperature is 23 C. The propane is now discharged slowly such that heat transfer keeps the propane and valve flow at 23 C. Find the initial pressure and mass of propane and the total heat transfer to empty the lighter. Propane C 3 H 8 T 1 = 23 o C = K = constant, x 1 = 0.0 V 1 = 5 cm 3 = m 3, T r1 = 296.2/369.8 = From Figs. D.1 and D.2, P 1 = P G T1 = = MPa, Z 1 = 0.04 (h * 1-h 1 ) = = P 1 V m 1 = = = kg Z 1 RT State 2: Assume vapor at 100 kpa, 23 o C Therefore, m 2 much smaller than m 1 ( kg) Q CV = m 2 u 2 - m 1 u 1 + m e h e = m 2 h 2 - m 1 h 1 - (P 2 -P 1 )V + (m 1 -m 2 )h e = m 2 (h 2 -h e ) + m 1 (h e -h 1 ) - (P 2 -P 1 )V (h e - h 1 ) = Q CV = (314.5) - ( ) = kj

264 An uninsulated piston/cylinder contains propene, C 3 H 6, at ambient temperature, 19 C, with a quality of 50% and a volume of 10 L. The propene now expands very slowly until the pressure in the cylinder drops to 460 kpa. Calculate the mass of propene, the work and heat transfer for this process. Propene C 3 H 6 : T 1 = 19 o C = K, x 1 = 0.50, V 1 = 10 L From Fig. D.1: T r1 = 292.2/364.9 = 0.80, P r1 = P r sat = 0.25, P 1 = = 1.15 MPa From D.1: Z 1 = = P 1 V m = = = kg Z 1 RT Assume reversible and isothermal process (slow, no friction, not insulated) Q 12 = m(u 2 -u 1 ) + W 12 2 W 12 = PdV (cannot integrate); Q 12 = TdS = Tm(s 2 -s 1 ) 1 1 From Figs. D.2 and D.3: h * 1-h 1 = ( ) = (s * 1-s 1 ) = ( ) = (h * 2-h * 1) = 0 and (s * 2-s * 1) = ln = At T r2 = 0.80, P r2 = 0.10, from D.1, D.2 and D.3, Z 2 = 0.93 (h * 2-h 2 ) = = 11.5 (s * 2-s 2 ) = = (s 2 -s 1 ) = = q 12 = = kj/kg => Q 12 = m q 12 = kj (h 2 -h 1 ) = = u 2 -u 1 = (h 2 -h 1 ) + RT(Z 1 -Z 2 ) = ( ) = w 12 = q 12 - (u 2 -u 1 ) = = 76.4 kj/kg W 12 = m w 12 = 36.0 kj 2

265 A 200-L rigid tank contains propane at 400 K, 3.5 MPa. A valve is opened, and propane flows out until half the initial mass has escaped, at which point the valve is closed. During this process the mass remaining inside the tank expands according to the relation Pv 1.4 = constant. Calculate the heat transfer to the tank during the process. C 3 H 8 : V = 200 L, T 1 = 400 K, P 1 = 3.5 MPa Flow out to m 2 = m 1 /2 ; Pv 1.4 = const inside T r1 = = 1.082, P r1 = = Fig D.1: Z 1 = v 1 = = , v = 2v 1 = m 1 = = kg, m 2 = 1 2 m 1 = kg, P 2 = P 1 ( v 1 v 2 ) = 1.4 = 1326 kpa 2 P r2 = = P 2 v 2 = Z 2 RT 2 Trial & error: saturated with T 2 = = K & Z 2 = = Z 2 = Z F2 + x 2 (Z G2 - Z F2 ) = = x 2 ( ) => x 2 = (h * 1-h 1 ) = (0.9) = 62.8 (h * 2-h * 1) = ( ) = (h * 2-h 2 ) = (h * 2-h F2 ) - x 2 h FG2 = [ ( )] = st law: Q CV = m 2 h 2 - m 1 h 1 + (P 1 -P 2 )V + m e h e AVE Let h * 1 = 0 then h 1 = 0 + (h 1 -h * 1) = h 2 = h * 1 + (h * 2-h * 1) + (h 2 -h * 2) = = h e AVE = (h 1 +h 2 )/2 = Q CV = 6.275(-214.0) (-62.8) + ( ) (-138.4) = kj

266 A newly developed compound is being considered for use as the working fluid in a small Rankine-cycle power plant driven by a supply of waste heat. Assume the cycle is ideal, with saturated vapor at 200 C entering the turbine and saturated liquid at 20 C exiting the condenser. The only properties known for this compound are molecular weight of 80 kg/kmol, ideal gas heat capacity C PO = 0.80 kj/kg K and T C = 500 K, P C = 5 MPa. Calculate the work input, per kilogram, to the pump and the cycle thermal efficiency.. Q H 1 Ht. Exch 4 Turbine P 3. -W P From Fig. D.1, Cond 2. W T T 1 = 200 o C = K, x 1 = 1.0 T 3 = 20 o C = K, x 3 = 0.0 Properties known: M = 80, C PO = 0.8 kj/kg K T C = 500 K, P C = 5.0 MPa T r1 = = 0.946, T r3 = = P r1 = 0.72, P 1 = = 3.6 MPa = P 4 P r3 = 0.023, P 3 = MPa = P 2, Z F3 = v F3 = Z F3 RT 3 = = P w P = - 3 vdp v F3 (P 4 -P 3 ) = ( ) = -3.7 kj/kg q H + h 4 = h 1, but h 3 = h 4 + w P => q H = (h 1 -h 3 ) + w P From Fig. D.2: (h * 1-h 1 ) = = 64.9 (h * 3-h 3 ) = = (h * 1-h * 3) = C P0 (T 1 -T 3 ) = 0.80(200-20) = (h 1 -h 3 ) = = 349.3

267 13-32 q H = (-3.7) = kj/kg Turbine, (s 2 - s 1 ) = 0 = -(s * 2 - s 2 )+(s * 2 - s * 1) + (s * 1 - s 1 ) From Fig. D.3, (s * 1-s 1 ) = = (s * 2-s * 1) = 0.80 ln ln = Substituting, s * 2-s 2 = = = (s * 2-s F2 ) - x 2 s FG = x ( ) => x 2 = (h * 2-h 2 ) = (h * 2-h F2 ) - x 2 h FG2 From Fig. D.2, h FG2 = ( ) = (h * 2-h 2 ) = = 25.0 w T = (h 1 -h 2 ) = = kj/kg η TH = w NET = = 0.29 q H 345.6

268 A geothermal power plant on the Raft river uses isobutane as the working fluid. The fluid enters the reversible adiabatic turbine, as shown in Fig. P13.42, at 160 C, MPa and the condenser exit condition is saturated liquid at 33 C. Isobutane has the properties T c = K, P c = 3.65 MPa, C P0 = kj/kg K and ratio of specific heats k = with a molecular weight as Find the specific turbine work and the specific pump work. Turbine inlet: T 1 = 160 o C, P 1 = MPa Condenser exit: T 3 = 33 o C, x 3 = 0.0, T r3 = / = 0.75 From Fig. D.1: P r3 = 0.16, Z 3 = 0.03 => P 2 = P 3 = = MPa T r1 = / = 1.061, P r1 = / 3.65 = 1.50 From Fig. D.2 & D.3: (h * 1-h 1 ) = = (s * 1-s 1 ) = = (s * 2-s * 1) = ln ln = (s * 2-s 2 ) = (s 2-s * F2 ) - x 2 s FG2 = x ( ) = x (s 2 -s 1 ) = 0 = x => x 2 = 0.99 (h * 2-h * 1) = C P0 (T 2 -T 1 ) = 1.664( ) = From Fig. D.2:, (h * 2-h 2 ) = (h * 2-h F2 ) - x 2 h FG2 = [ ( )] = = 21.3 Turbine: w T = (h 1 -h 2 ) = = 66.8 kj/kg Pump: v F3 = Z F3 RT = = P w P = - v dp v F3 (P 4 -P 3 ) = ( ) = kj/kg

269 Carbon dioxide collected from a fermentation process at 5 C, 100 kpa should be brought to 243 K, 4 MPa in an SSSF process. Find the minimum amount of work required and the heat transfer. What devices are needed to accomplish this change of state? T ri = = P ri = = From D.2 and D.3 : (h * -h) /RT ri C = 0.02, (s * -s) ri /R = 0.01 T re = = 0.80, P re = = From D.2 and D.3: (h * -h) /RT re C = 4.5 (s * -s) re /R = 4.74 (h i -h e ) = - (h * i-h i ) + (h * i-h * e) + (h * e-h e ) = ( ) = kj/kg (s i -s e ) = - (s * i-s i ) + (s * i -s * e) + (s * e-s e ) = ln(278.2/243) ln(0.1/4) = kj/kg K w rev = (h i -h e ) -T 0 (s i -s e ) = (1.7044) = kj/kg q rev = (h e -h i ) + w rev = = kj/kg

270 An insulated cylinder fitted with a frictionless piston contains saturated-vapor carbon dioxide at 0 o C, at which point the cylinder volume is 20 L. The external force on the piston is now slowly decreased, allowing the carbon dioxide to expand until the temperature reaches - 30 o C. Calculate the work done by the CO 2 during this process. CO 2 : T c = K, P c = 7.38 MPa, C p = kj/kg-k, R = kj/kg K State 1: T 1 = 0 o C, sat. vap., x 1 = 1.0, V 1 = 20 L T r1 = 0.9, P 1 = P r1 P c = = 3911 kpa, Z 1 = Z g = 0.67 (h * 1 h 1) g = 0.9 RT C, (s * 1 s 1) g /R = 0.72, m = P 1 V 1 Z 1 RT 1 = kg State 2: T 2 = -30 o C T r2 = 0.8, P 2 = P r2 P c = = 1845 kpa 2 nd Law: S net =m(s 2 s 1 ) 1 Q 2 /T ; 1 Q 2 = 0, S net = 0 s 2 - s 1 = (s 2 s * 2 ) + (s* 2 s* 1 ) + (s* 1 s 1) = 0 s * 2 s* 1 = C P ln T 2 R ln P 2 = kj/kg-k, s * T 1 P 1 s 1 = kj/kg-k 1 s * 2 - s 2 = kj/kg K, (s* 2 s 2) f = 5.46 R, (s * 2 s 2) g = 0.39 R (s * 2 s 2) = (1-x 2 )(s * 2 s 2) f + x 2 (s * 2 s 2) g x 2 = st Law: 1 Q 2 = m(u 2 u 1 ) + 1 W 2 ; 1 Q 2 = 0, u = h - Pv Z 2 = (1 - x 2 )Z f + x 2 Z g = = 0.725; (h 2 - h 1 ) = (h 2 h * 2 ) + (h* 2 h* 1 ) + (h* 1 h 1) h * 2 h* 1 = C p (T 2 - T 1 ) = kj/kg, (h* 1 h 1) = 51.7 kj/kg (h * 2 h 2) f = 4.51 RT C, (h * 2 h 2) g = 0.46 RT C (h * 2 h 2) = (1 - x 2 )(h * 2 h 2) f + x 2 (h * 2 h 2) g = 52.2 kj/kg h 2 - h 1 = = kj/kg u 2 - u 1 = (h 2 - h 1 ) - Z 2 RT 2 + Z 1 RT 1 = = kj/kg 1 W 2 = 55.4 kj

271 An evacuated 100-L rigid tank is connected to a line flowing R-142b gas, chlorodifluoroethane, at 2 MPa, 100 C. The valve is opened, allowing the gas to flow into the tank for a period of time and then it is closed. Eventually, the tank cools to ambient temperature, 20 C, at which point it contains 50% liquid, 50% vapor, by volume. Calculate the quality at the final state and the heat transfer for the process. The ideal-gas specific heat of R-142b is C = kj/kg K. Rigid tank V = 100 L, m 1 = 0 Line: R-142b CH 3 CClF 2 M = , T C = K, P C = 4.25 MPa, C P0 = kj/kg K R = R /M = / = kj/kg K Line P i = 2 MPa, T i = 100 o C, Flow in to T 2 = T 0 = 20 o C V LIQ 2 = V VAP 2 = 50 L Continuity: m i = m 2 ; Energy: Q CV + m i h i = m 2 u 2 = m 2 h 2 - P 2 V From D.2 at i: P ri = 2 / 4.25 = 0.471, T ri = / = 0.91 (h * i-h i ) = = 24.4 (h * 2-h * i) = C P0 (T 2 -T i ) = 0.787(20-100) = From D.2: T r2 = = => P = = 489 kpa 2 sat. liq.: Z F = 0.02, (h * -h F ) = RT C 4.85 = sat. vap.: Z G = 0.88, (h * -h G ) = RT C 0.25 = 8.5 m LIQ 2 = P 2 V LIQ 2 Z F RT 2 = = 50.4 kg m VAP 2 = P 2 V VAP 2 = 1.15 kg, m Z G RT 2 = kg 2 x 2 = m VAP 2 /m 2 = (h * 2-h 2 ) = (1-x 2 )(h * 2-h F2 ) + x 2 (h * 2-h G2 ) = = Q CV = m 2 (h 2 -h i ) - P 2 V = 51.55( ) = kj

272 A cylinder fitted with a movable piston contains propane, initially at 67 o C and 50 % quality, at which point the volume is 2 L. The piston has a cross-sectional area of 0.2 m 2. The external force on the piston is now gradually reduced to a final value of 85 kn, during which process the propane expands to ambient temperature, 4 o C. Any heat transfer to the propane during this process comes from a constant-temperature reservoir at 67 o C, while any heat transfer from the propane goes to the ambient. It is claimed that the propane does 30 kj of work during the process. Does this violate the second law? C H 3 8 F ext +Q from T res = 67 o C -Q to Environment T o = 4 o C F ext 2 = 85 kn Propane: T c = K, P c = 4.25 MPa, R = kj/kg K, C p = kj/kg K State 1: T 1 = 67 o C = K, x 1 = 0.5, V 1 = 2.0 L T r1 = 0.92, Fig D.1, P r1 = 0.61, P 1 = P r1 P c = MPa Z f1 = 0.10, Z g1 = 0.64, Z 1 = (1 - x 1 )Z f1 + x 1 Z g1 = 0.37 m = P 1 V 1 Z 1 RT 1 = kg, (h * 1 h 1) f = 3.95 RT c, (h * 1 h 1) g = 1.03 RT c (s * 1 s 1) f = 4.0 R, (s * 1 s 1) g = 0.82 R State 2: T 2 = 4 o C = K, F ext 2 = 85 kn T r2 = 0.75, P sat 2 = Psat r2 P c = = 701 kpa P 2 = F ext 2 /A p = 425 kpa, P 2 < P sat 2 State 2 is a vapor P r2 = 0.10, Z 2 = 0.92, V 2 = mz 2 RT 2 /P 2 = m 3 h * 2 h 2 = 0.18 RT c =12.6 kj/kg, s* 2 s 2 = 0.16 R = kj/kg K 1 st Law: 1 Q 2 = m(u 2 - u 1 ) + 1 W 2 ; 1 W 2 = 30 kj, u = h - Pv 1 Q 2 = m(h 2 - h 1 ) - P 2 V 2 + P 1 V W 2 (h 2 - h 1 ) = (h 2 h * 2 ) + (h* 2 h* 1 ) + (h* 1 h 1) (h * 1 h 1) = (1 - x 1 )(h * 1 h 1) f + x 1 (h * 1 h 1) g = kj/kg h * 2 h* 1 = C p (T 2 - T 1 ) = kj/kg 1 Q 2 = ( ) = 36.7 kj

273 nd Law: S net = m(s 2 s 1 ) 1Q 2 T ; T res = 67o C = K s 2 - s 1 = (s 2 s * 2 ) + (s* 2 s* 1 ) + (s* 1 s 1) s * 1 s 1 = (1 - x 1 )(s* 1 s 1) f + x 1 (s * 1 s 1) g = kj/kg-k s * 2 s* 1 = C p ln T 2 - R ln P 2 = kj/kg K T 1 P 1 S net = ( ) 36.7/340.2 = kj/k; S net < 0 Process is Impossible Consider the following equation of state, expressed in terms of reduced pressure and temperature: P r 6 Z = 1 + (1 - ) 14 T 2 r T r What does this equation predict for enthalpy departure from the ideal gas value at the state P r = 0.4, T r = 0.9? What does this equation predict for the reduced Boyle temperature? a) Z = Pv RT = 1 + v = RT P + RT c (1-6T 2 c ) 14P c T 2 ; P r 14 T r (1-6 T r 2) v - T v T p = RT c - 18RT 3 c 14P c 14P c T 3 P h - h * = 0 [v - T v ] T p dp = RT c 14 3 c v T p = R P + 12RT 14P c T 3 (1 18 T r 2) P r = RT c b) Z P T = 1 (1-6 14P c T r T r 2) => (1-6 T r 2) = 0 T r = 6 = 2.45 Lim P 0 Z P T = 0 at T boyle

274 Saturated liquid ethane at 2.44 MPa enters (SSSF) a heat exchanger and is brought to 611 K at constant pressure, after which it enters a reversible adiabatic turbine where it expands to 100 kpa. Find the heat transfer in the heat exchanger, the turbine exit temperature and turbine work. From D.2, P r1 = 2.44/4.88 = 0.50, T r1 = 0.89, T 1 = = K (h * 1-h 1 ) = = (h * 2-h * 1) = ( ) = P r2 = 0.50, T r2 = 611/305.4 = 2.00 From D.2: (h * 2-h 2 ) = RT c 0.14 = = 11.8 q = (h 2 -h 1 ) = = kj/kg From D.3, (s * 2-s 2 ) = = (s * 3-s * 2) = ln T ln Assume T 3 = 368 K, T r3 = at P r3 = (s * 3-s * 2) = = From D.3, (s * 3-s 3 ) = = (s 3 -s 2 ) = ΟΚ Therefore, T 3 = 368 K From D.2, (h * 3-h 3 ) = = 0.8 w = (h 2 -h 3 ) = ( ) = kj/kg

275 A flow of oxygen at 230 K, 5 MPa is throttled to 100 kpa in an SSSF process. Find the exit temperature and the entropy generation. T ri = = 1.488, P ri = = 0.992, P re = = 0.02 From D.2: (h * i-h i ) = = 20.1 First law: (h e -h i ) = 0 = - (h * e-h e ) + (h * e-h * i) + (h * i-h i ) Assume T e = 208 K, T re = 1.345: (h * e-h * i) = ( ) = From D.2: (h * e-h e ) = = 0.4 Check first law (h e -h i ) = OK => T e = 208 K From D.3, (s * i-s i ) = = and (s * e-s e ) = = (s * e-s * i) = ln ln = (s e -s i ) = = kj/kg K A cylinder contains ethylene, C 2 H 4, at MPa, 13 C. It is now compressed isothermally in a reversible process to 5.12 MPa. Find the specific work and heat transfer. Ethylene C 2 H 4 P 1 = MPa, T 2 = T 1 = -13 o C = K T r2 = T r1 = / = 0.921, P r1 = / 5.04 = From D.1, D.2 and D.3: Z 1 = 0.85 (h * 1-h 1 ) = = 33.5 and (s * 1-s 1 ) = = From D.1, D.2 and D.3: Z 2 = 0.17, P r2 = 5.12/5.04 = (comp. liquid) (h * 2-h 2 ) = = and (s * 2-s 2 ) = = Ideal gas: (h 2-h * * 1) = 0 and (s * 2-s * 1) = ln = q 12 = T(s 2 -s 1 ) = 260.2( ) = kj/kg (h 2 -h 1 ) = = (u 2 -u 1 ) = (h 2 -h 1 ) - RT(Z 2 -Z 1 ) = ( ) = w 12 = q 12 - (u 2 -u 1 ) = = kj/kg

276 A control mass of 10 kg butane gas initially at 80 C, 500 kpa, is compressed in a reversible isothermal process to one-fifth of its initial volume. What is the heat transfer in the process? Butane C 4 H 10 : m = 10 kg, T 1 = 80 o C, P 1 = 500 kpa Compressed, reversible T = const, to V 2 = V 1 /5 T r1 = = 0.831, P r1 = = From D.1 and D.3: Z 1 = 0.92, (s * 1- s 1 ) = = v 1 = Z 1 RT 1 = = m P 3 /kg v 2 = v 1 /5 = m 3 /kg At T r2 = T r1 = From D.1: P G = = 1235 kpa sat. liq.: Z F = 0.05, (s * -s F ) = R 5.08 = sat. vap.: Z G = 0.775, (s * -s G ) = R = Therefore v F = = v G = = Since v F < v 2 < v G x 2 = (v 2 -v F )/(v G -v F ) = (s * 2-s 2 ) = (1-x 2 )(s * 2-s F2 ) + x 2 (s * 2-s G2 ) = = (s * 2-s * 1) = C P0 ln (T 2 /T 1 ) - R ln (P 2 /P 1 ) = ln (1235/500) = (s 2 -s 1 ) = = Q 12 = Tm(s 2 -s 1 ) = ( ) = kj

277 An uninsulated compressor delivers ethylene, C 2 H 4, to a pipe, D = 10 cm, at MPa, 94 C and velocity 30 m/s. The ethylene enters the compressor at 6.4 MPa, 20.5 C and the work input required is 300 kj/kg. Find the mass flow rate, the total heat transfer and entropy generation, assuming the surroundings are at 25 C. T ri = = 1.040, P ri = = From D.2 and D.3, (h * i-h i ) = = (s * i-s i ) = = T re = = 1.30, P re = = => From D.1: Z e = 0.69 v e = Z e RT e = = m P 3 /kg e A e = π 4 D2 e = m 2 => ṁ = From D.2 and D.3, (h * e-h e ) = = (s * e-s e ) = = (h * e-h * i) = ( ) = (s * e-s * i) = ln ln = (h e -h i ) = = kj/kg (s e -s i ) = = kj/kg K A e V e = = kg/s v e First law: 30 2 q = (h e -h i ) + KE e + w = = kj/kg Q. = ṁq = 32.26(-97.9) = kw cv Q. cv Ṡ gen = + ṁ(s T e - s i ) = (0.5562) = kw/k

278 A distributor of bottled propane, C 3 H 8, needs to bring propane from 350 K, 100 kpa to saturated liquid at 290 K in an SSSF process. If this should be accomplished in a reversible setup given the surroundings at 300 K, find the ratio of the volume flow rates V. in /V. out, the heat transfer and the work involved in the process. From Table A.2: T ri = = 0.946, P ri = = From D.1, D.2 and D.3, Z i = 0.99 (h * i-h i ) = = 2.1 (s * i-s i ) = = T re = = 0.784, From D.1, D.2 and D.3, P re = 0.22, P e = = MPa and Z e = (h * e-h e ) = = (s * e-s e ) = = (h * e-h * i) = 1.679( ) = (s * e-s * i) = ln (h e -h i ) = = ln = (s e -s i ) = = V. in V. Z i T i /P i = = 0.99 Z out e T e /P e = w rev = (h i -h e ) -T 0 (s i -s e ) = (1.8007) = kj/kg q rev = (h e -h i ) + w rev = = kj/kg

279 Saturated-liquid ethane at T 1 = 14 C is throttled into a SSSF mixing chamber at the rate of 0.25 kmol/s. Argon gas at T 2 = 25 C, P 2 = 800 kpa, enters the chamber at the rate of 0.75 kmol/s. Heat is transferred to the chamber from a heat source at a constant temperature of 150 o C at a rate such that a gas mixture exits the chamber at T 3 = 120 o C, P 3 = 800 kpa. Find the rate of heat transfer and the rate of entropy generation. Argon, T a2 = 25 o C, P 2 = 800 kpa, ṅ 2 = 0.75 kmol/s T ca = 150 K, P ca = 4.87 MPa, M a = kg/kmol, C pa = 0.52 kj/kg K h - a3 - h- a2 = M a C pa (T 3 - T a2 ) = kj/kmol Inlet: Ethane, T b1 = 14 o C, sat. liq., x b1 = 0, ṅ 1 = 0.25 kmol/s T cb = K, P cb = 4.88 MPa, M b = kg/kmol, C pb = kj/kg-k T r1 = 0.94, P b1 = P r1 P cb = = 3367 kpa h b1 h b1 = 3.81 R T cb = kj/kmol, - s b1 s- b1 = 3.74 R = 31.1 h b3 - h b1 = M b C pb (T 3 - T b1 ) = kj/kmol Exit: Mix, T 3 = 120 o C, P 3 = 800 kpa consider this an ideal gas mixture. SSSF Energy Eq.: ṅ 1 h - b1 + ṅ 2 h- a2 +Q. = ṅ 3 h - 3 = ṅ 1 h- b3 + ṅ 2 h- a3 Q. = ṅ 1 (h - b3 - h- b1 ) + ṅ 2 (h- a3 - h- a2 ) = 0.25 ( ) (1973.4) = 5306 kw Entropy Eq.: Ṡ gen = ṅ 1 (s - b3 s- b1 ) + ṅ 2 (s- a3 s- a2 ) - Q. /T H ; T H = 150 o C y a = ṅ 2 /ṅ tot = 0.75; y b = ṅ 1 /ṅ tot = 0.25 s - a3 s - a2 = M a C pa lnt 3 - R ln y a P 3 = 8.14 kj/kmol-k T a2 P a2 s - b3 s - b1 = M b C pb lnt 3 - R ln y b P 3 + s - T b1 P b1 b1 s- b1 = = = kj/kmol K Ṡ gen = / 423 = kw/k

280 One kilogram per second water enters a solar collector at 40 C and exits at 190 C, as shown in Fig. P The hot water is sprayed into a direct-contact heat exchanger (no mixing of the two fluids) used to boil the liquid butane. Pure saturated-vapor butane exits at the top at 80 C and is fed to the turbine. If the butane condenser temperature is 30 C and the turbine and pump isentropic efficiencies are each 80%, determine the net power output of the cycle. H 2 O cycle: solar energy input raises 1 kg/s of liquid H 2 O from 40 o C to 190 o C. Therefore, corresponding heat input to the butane in the heat exchanger is Q. H = ṁ(h F 190 C -h F 40 C ) = 1( ) = kw H2O. Q H 1 Ht. Exch 4 Turbine P 3. -W P Cond 2. W T C 4 H 10 cycle T 1 = 80 o C, x 1 = 1.0 ; T 3 = 30 o C, x 3 = 0.0 η ST = η SP = 0.80 T r1 = = From D.1, D.2 and D.3: P 1 = = 1235 kpa (h * 1-h 1 ) = = 34.1 (s * 1-s 1 ) = = T r3 = = From D.1, D.2 and D.3: P 3 = = 429 kpa sat. liq.: (h * -h F ) = RT C 4.81 = ; (s * -s F ) = R 6.64 = sat. vap.: (h * -h G ) = RT C = 14.3 ; (s * -s G ) = R 0.22 = Because of the combination of properties of C 4 H 10 (particularly the large C P0 /R), s 1 is larger than s G at T 3. To demonstrate, (s * 1-s * G3) = ln ln = (s 1 -s G3 ) = = kj/kg K

281 13-46 T s so that T 2S will be > T 3, as shown in the T-s diagram. A number of other heavy hydrocarbons also exhibit this behavior. Assume T 2S = 315 K, T r2s = s From D.2 and D.3: (h * 2S-h 2S ) = RT C 0.21 = 12.8 and (s * 2S-s 2S ) = R 0.19 = (s * 1-s * 2S) = ln (s 1 -s 2S ) = T 2S = 315 K (h * 1-h * 2S) = ( ) = ln = w ST = h 1 -h 2S = = 44.3 kj/kg w T = η S w ST = = 35.4 kj/kg At state 3, v 3 = = m 3 /kg 429 -w SP v 3 (P 4 -P 3 ) = ( ) = 1.55 kj/kg -w P = -w SP = 1.55 = 1.94 kj/kg η 0.8 SP w NET = w T + w P = = kj/kg For the heat exchanger, Q. H = = ṁ C4H10 (h 1 -h 4 ) But h 1 -h 4 = h 1 -h 3 +w P h 1 -h 3 = (h 1 -h * 1) + (h * 1-h * 3) + (h * 3-h 3 ) = (80-30) = kj/kg Therefore, ṁ C4H10 = = 1.87 kg/s Ẇ NET = ṁ C4H10 w NET = = kw

282 A line with a steady supply of octane, C 8 H 18, is at 400 C, 3 MPa. What is your best estimate for the availability in an SSSF setup where changes in potential and kinetic energies may be neglected? Availability of Octane at T i = 400 o C, P i = 3 MPa 3 P ri = 2.49 = 1.205, T ri = = From D.2 and D.3, (h * 1-h 1 ) = = 46.8 ; (s * 1-s 1 ) = = 0.05 Exit state in equilibrium with the surroundings, assume T 0 = K, P 0 = 100 kpa T r0 = = 0.524, P r0 = = From D.2 and D.3, (h * 0-h 0 ) = RT C 5.4 = and (s * 0-s 0 ) = R = (h * i-h * 0) = ( ) = (s * i-s * 0) = ln ln = (h i -h 0 ) = = (s i -s 0 ) = = ϕ i = w rev = (h i -h 0 ) - T 0 (s i -s 0 ) = (1.8509) = kj/kg

283 A piston/cylinder contains ethane gas, initially at 500 kpa, 100 L, and at ambient temperature, 0 C. The piston is now moved, compressing the ethane until it is at 20 C, with a quality of 50%. The work required is 25% more than would have been required for a reversible polytropic process between the same initial and final states. Calculate the heat transfer and the net entropy change for the process. Ethane: T c = K, P c = 4.88 MPa, R = kj/kg-k, C p = kj/kg K State 1: T r1 = 0.895, P r1 = Z 1 = 0.95 v 1 = Z 1 RT 1 /P 1 = m 3 /kg, m 1 = V 1 /v 1 = kg (h * 1 h 1) = 0.13RT c = 11.0 kj/kg, (s * 1 s 1) = 0.09 R = kj/kg K State 2: T 2 = 20 o C, x 2 = 0.5, 1 W 2 = 1.25W rev T r2 = 0.96, P r2 = 0.78, P 2 = P r2 P c = 3806 kpa Z f2 = 0.14, Z g2 = 0.54, Z 2 = (1 - x 2 )Z f + x 2 Z g = 0.34 (h * 2 h 2) = (1 - x 2 ) 3.65 RT c + x 2 (1.39 RT c ) = kj/kg (s * 2 s 2) = (1 - x 2 ) 3.45 R + x R = kj/kg K v 2 = Z 2 RT 2 /P 2 = m 3 /kg, V 2 = mv 2 = m 3 P 1 V n 1 = P 2 Vn 2, ln P 2 P 1 = n ln V 1 V 2 n = W rev = P dv = P 2 V 2 - P 1 V 1 = kj, 1 - n 1 W 2 = 1.25W rev = kj a) 1 st Law: 1 Q 2 = m(u 2 - u 1 ) + 1 W 2 ; u = h - Pv u 2 - u 1 = (h 2 - h 1 ) - (P 2 v 2 - P 1 v 1 ) h 2 - h 1 = = (h 2 h * 2 ) + (h* 2 h* 1 ) + (h* 1 h 1) (h * 2 h* 1 ) = C p (T 2 - T 1 ) = 35.3 kj/kg, h 2 - h 1 = = kj/kg, u 2 - u 1 = kj/kg 1 Q 2 = 0.697(-122.2) = kj b) 2 nd Law: S net = m(s 2 - s 1 ) - 1 Q 2 /T o ; T o = 0 o C s 2 - s 1 = (s 2 s * 2 ) + (s* 2 s* 1 ) + (s* 1 s 1) (s * 2 s* 1 ) = C p ln(t 2 / T 1 ) R ln(p 2 / P 1 ) = kj/kg K, S net = 0.697( ) = kj/k 273.2

284 The environmentally safe refrigerant R-152a (see Problem 13.65) is to be evaluated as the working fluid for a heat pump system that will heat winter households in two different climates. In the colder climate the cycle evaporator temperature is 20 C, and the more moderate climate the evaporator temperature is 0 C. In both climates the cycle condenser temperature is 30 C. For this study assume all processes are ideal. Determine the cycle coefficient of performance for the two climates. R152a difluoroethane. From 13.65: C P0 = Ideal Heat Pump T H = 30 o C From A.2: M = 66.05, R = , T C = K, P C = 4.52 MPa T CASE I: T 1 = -20 o C CASE II: T 1 = 0 o C 2 T 3 r3 = = P 4 1 r3 = P r2 = 0.22 => P 3 = P 2 = 994 kpa Sat.liq.: h * 3 - h v 3 = 4.56 RT C = CASE I) T 1 = -20 o C = K, T r1 = 0.655, P r1 = P 1 = 262 kpa h * 1 - h 1 = 0.14 RT C = 6.8 and s * 1 - s 1 = 0.14 R = Assume T 2 = 307 K, T r2 = given P r2 = 0.22 From D.2, D.3: s * 2 - s 2 = 0.34 R = ; h * 2 - h 2 = 0.40 RT c = 19.5 s * 2 - s * 1 = ln ln = s 2 - s 1 = = OK h 2 - h 1 = ( ) = 40.9 h 2 - h 3 = ( ) = β = q H w IN = h 2 - h 3 h 2 - h 1 = = 5.04

285 13-50 case II) T 1 = 0 o C = K, T r1 = => P r1 = 0.106, P 1 = 479 kpa h * 1 - h 1 = 0.22 RT C = 10.7 and s * 1 - s 1 = 0.21 R = Assume T 2 = 305 K, T r2 = s * 2 - s 2 = 0.35 R = and h * 2 - h 2 = 0.38 RT C = 18.5 s * 2 - s * 1 = ln ln = s 2 - s 1 = = OK h 2 - h 1 = ( ) = 23.9 h 2 - h 3 = ( ) = β = h 2 - h 3 h 2 - h 1 = = Repeat the calculation for the coefficient of performance of the heat pump in the two climates as described in Problem using R-12 as the working fluid. Compare the two results. Problem the same as 13.58, exept working fluid is R-12 For R-12: At T 3 = 30 o C: P 3 = P 2 = MPa, h 3 = 64.6 CASE I) T 1 = -20 o C, h 1 = 178.7, s 1 = At s 2 = s 1 & P 2 T 2 = 39.8 o C h 2 = β = h 2 - h = h 2 - h = = 5.06 CASE II: T 1 = 0 o C, h 1 = 187.5, s 1 = At s 2 = s 1 & P 2 T = 34.6 o C, h = β = = = 8.93

286 One kmol/s of saturated liquid methane, CH 4, at 1 MPa and 2 kmol/s of ethane, C 2 H 6, at 250 C, 1 MPa are fed to a mixing chamber with the resultant mixture exiting at 50 C, 1 MPa. Assume that Kay s rule applies to the mixture and determine the heat transfer in the process. Control volume the mixing chamber, inlet CH 4 is 1, inlet C 2 H 6 is 2 and the exit state is 3. Energy equation is Q. CV = ṅ 3 h- 3 - ṅ 1 h ṅ 2 h - 2 Select the ideal gas reference temperature to be T 3 and use the generalized charts for all three states. P r1 = P rsat = 1/4.60 = => T rsat = 0.783, T 1 = = K, h 1 = 4.57 P r2 = 1/4.88 = 0.205, T r2 = 523/305.4 = h - 1 = C - 1(T 1 - T3) - h 1 R - T c = 36.15( ) = kj/kmol h - 2 = C - 2(T 2 - T 3 ) - h 2 R - T c = 53.11(250-50) = kj/kmol T cmix = ( )/3 = K P cmix = ( )/3 = 4.79 MPa T r3 = 323.2/267.1 = 1.21, P r3 = 1/4.79 = 0.21 h - 3 = = kj/kmol Q. = 3(-333) - 1(-13528) - 2(10419) = kw CV Consider the following reference state conditions: the entropy of real saturated liquid methane at 100 C is to be taken as 100 kj/kmol K, and the entropy of hypothetical ideal gas ethane at 100 C is to be taken as 200 kj/kmol K. Calculate the entropy per kmol of a real gas mixture of 50% methane, 50% ethane (mole basis) at 20 C, 4 MPa, in terms of the specified reference state values, and assuming Kay s rule for the real mixture behavior. CH 4 : T 0 = -100 o C, - s = 100 kj/kmol K LIQ 0 C 2 H 6 : T 0 = -100 o C, P 0 = 1 MPa, s -* 0 = 200 kj/kmol K Also for CH 4 : T C = K, P C = 4.60 MPa For a 50% mixture: T cmix = = K P cmix = = 4.74 MPa

287 13-52 IG MIX at T 0 (=-100 o C), P 0 (=1 MPa): CH 4 : T r0 = 0.91, P G = = MPa s - * 0 CH4 = s - + (s -* -s - LIQ 0 P LIQ ) - R - ln (P at P 0 /P G ) G G = ln (1/2.622) = s - * = (0.5 ln ln 0.5) = MIX C - = = P0 MIX s - * = ln TP MIX ln 4 1 For the mixture at T,P: T r = 1.183, P r = = kj/kmol K s - * TP MIX - s- TP MIX = = 3.63 kj/kmol K Therefore, s - = = kj/kmol K TP MIX An alternative is to form the ideal gas mixture at T,P instead of at T 0,P 0 : s - * TP CH4 = s- LIQ 0 + (s-* -s - LIQ ) + C- P0 CH4 ln T - R - ln P T 0 P G P G, T 0 at P G, T 0 = ln ln = = s - * = ln TP C2H ln 4 1 = = s - * TP MIX = (0.5 ln ln 0.5) = s - = = kj/kmol K TP MIX

288 13-53 Advanced Problems An experiment is conducted at 100 C inside a rigid sealed tank containing liquid R-22 with a small amount of vapor at the top. When the experiment is done the container and the R-22 warms up to room temperature of 20 C. What is the pressure inside the tank during the experiment? If the pressure at room temperature should not exceed 1 MPa, what is the maximum percent of liquid by volume that can be used during the experiment? R-22 tables Go to -70 o C T a) For h FG const & v FG v G RT/P G P G1 h FG ln [ 1-1 P G0 R T 0 T 1 ] extrapolating from -70 o C (Table B.4.1) to T AVE = -85 o C, h FG Also R = = For T 0 = K & T 1 = K ln( P G )= [ ] P G1 = kpa b) Extrapolating v F from -70 o C to T 1 = -100 o C v F Also v G1 RT 1 /P G1 = Since v 1 = v 2 v F2 = = C o = x => x 1 = V LIQ 1 V m = (1-x 1 )v F1 = , VAP 1 m = x 1 v = G1 % LIQ, by vol. = = 76.9 % o C v

289 Determine the low-pressure Joule Thomson inversion temperature from the condition in Eq as predicted by an equation of state, using the van der Waals equation and the Redlich Kwong equation. µ j = ( T P ) n = T( v T ) P - v T ) v C P, But ( v T ) P = - ( P a) van der Waal VDW: ( v ) ( R ) T P = - v-b RT [- (v-b) 2 + 2a ] v 3 ( P v ) T Substitute into µ j eq'n & multiply by v2 2 and then rearranging, v µ j = 1 C P [ -RT( 1 )( b )+ 2a 1-b/v 1-b/v Then or RT( 1 1-b/v )2-2a v lim P 0 µ j = 1 [-b + 2a C P RT ]= 0 only at T INV ] T INV = 2a Rb = 27 4 T C = 6.75 T C b) R-K: ( P T ) v = R v-b + ( P v ) T = -RT (v-b) 2 + Substituting as in part a) then or a(2v+b) v 2 (v+b) 2 T 1/2 brt µ j = 1 (1-b/v) C P [ 2 - RT (1-b/v) a v(v+b)t 3/2 1 2 a (1+b/v) 2 T 1/2 ] a(2+b/v) v(1+b/v) 2 T 1/2 lim P 0 µ j = 1 [b - 5 a C P 2 RT 3/2 ]= 0 only at T INV T INV = ( 5a 2Rb )2/3 = 5.3 T C

290 Suppose the following information is available for a given pure substance: T v x Known: P SAT = fn(t) s vapor P = fn(t,v) values of v F, P C, & T C vapor C V at v x Outline the procedure that should be followed to develop a table of thermodynamic properties comparable to Tables 1,2 and 3 of the steam tables. 1) from vapor pressure curve, get P SAT at each T. 2) v F values are known. 3) reference state: T 0 & sat. liquid. Set h 0 = 0, s 0 = 0, u 0 = 0 - P 0 v 0 4) from eq'n of state, find v 1 = v G at T 0 then v FG(T0) = v G(T0) - v F(T0) 5) from clapeyron eq'n, using slope of vapor pressure curve & v FG, find h FG(T0) then h 1 = h G(T0) = 0 + h FG(T0) 6) u 1 = u G(T0) = h 1 - P 1 v 1 ; u FG(T0) = u 1 - u 0 7) s FG(T0) = h FG(T0) T 0 ; s 1 = s G(T0) = 0 + s FG(T0) 8) find P 2 from eq'n of state ( T 2 = T 0, v 2 = v x ) 2 9) u 2 - u 1 = 1 [T( P T ) v - P]dv T evaluate from eq'n of state then h 2 = u 2 + P 2 v ) s 2 - s 1 = 1 ( P T ) v dv T, find s ) u 3 - u 2 = C vx dt, find u ) find P 3 from eq'n of state ( T 3, v 3 = v x ) 13) h 3 = u 3 + P 3 v 3

291 ) s 3 - s 2 = (C vx /T) dt, find s ) at any point 4 (T 4 =T 3 ) given by desired P 4, find v 4 from eq'n of state. 16) find P 5 = P SAT at T 5 = T 3 from vap. pressure. 17) find v 5 = v G(T5) from eq'n of state. 18) find u 5, h 5, s 5 as in steps 9 & ) find h FG(T5) from Clapeyron eq'n as in step 5. 20) h 6 = h F(T5) = h 5 - h FG(T5) 21) s FG(T5) = h FG(T5) T 5, s 6 = s F(T5) = s F - s FG(T5) 22) u 6 = h 6 - P 6 v 6, u FG(T5) = u 5 - u 6 23) pick a different T (instead of T 3 ) and repeat steps 11 to 22. For T T 5 steps 16 to 22 are eliminated.

292 The refrigerant R-152a, difluoroethane, is tested by the following procedure. A 10-L evacuated tank is connected to a line flowing saturated-vapor R-152a at 40 C. The valve is then opened, and the fluid flows in rapidly, so that the process is essentially adiabatic. The valve is to be closed when the pressure reaches a certain value P 2, and the tank will then be disconnected from the line. After a period of time, the temperature inside the tank will return to ambient temperature, 25 C, through heat transfer with the surroundings. At this time, the pressure inside the tank must be 500 kpa. What is the pressure P 2 at which the valve should be closed during the filling process? The ideal gas specific heat of R-152a is C P0 = kj/kg K. R-152a CHF 2 CH 3 : A.2: M = 66.05, T C = K, P C = 4.52 MPa, T 3 = T 0 = 25 o C, P 3 = 500 kpa, R = R - /M = /66.05 = T r3 = 298.2/386.4 = 0.772, P r3 = 500/4520 = From D.1 and D.2 at 3: Z 3 = 0.92, (h * -h) 3 = 0.19 RT C m 3 = m 2 = m i = P 3 V = = kg Z 3 RT Filling process: Energy Eq.: h i = u 2 = h 2 - Z 2 RT 2 or (h 2 -h * 2) + C P0 (T 2 -T i ) + (h * i-h i ) - P 2 V/m 2 = 0 From D.2 with T ri = 313.2/386.4 = 0.811, (h * i-h i ) = = 23.8 ; P i = = 1248 kpa Assume P 2 = 575 kpa, P r2 = Now assume T 2 = 339 K, T r2 = => From D.1: Z 2 = 0.93 Z 2 T 2 = = Z 3 T 3 = = P P T 2 = 339 K is the correct T 2 for the assumed P 2 of 575 kpa. Now check the 1st law to see if 575 kpa is the correct P 2. From D.2, h * 2-h 2 = = 8.3 Substituting into 1st law, ( ) P 2 = 575 kpa = (Note: for P 2 = 580 kpa, T 2 = 342 K, 1st law sum = +4.2)

293 An insulated cylinder has a piston loaded with a linear spring (spring constant of 600 kn/m) and held by a pin, as shown in Fig. P The cylinder crosssectional area is 0.2 m 2, the initial volume is 0.1 m 3, and it contains carbon dioxide at 2.5 MPa, 0 C. The piston mass and outside atmosphere add a force per unit area of 250 kpa, and the spring force would be zero at a cylinder volume of 0.05 m 3. Now the pin is pulled out; what is the final pressure inside the cylinder? A P = 0.2 m 3, V Fs = 0 = 0.05 m 3, k s = 600 kn/m Q = 0 CO 2 P 0 P 0 = 250 kpa, P 1 = 2.5 MPa, V 1 = 0.1 m 3, T 1 = 0 o C R = , C = Any point: P EXT = P 0 + k s (V-V 0 ) A 2 p Pin pulled: P EXT = ( ) = 1000 kpa But P 1 = 2500 kpa Irr. process, expansion to 1000 < P 2 < 2500 At 2: P 2 = (V ) = V W 12 = P EXT dv = 1 2 (P EXT 1 + P EXT 2 )(V 2 -V 1 ) 1 Q 12 = 0 = m(h 2 -h 1 ) - P 2 V 2 + P 1 V 1 + W 12 h 2 -h 1 = (h 2 -h * 2) + C P0 (T 2 -T 1 ) + (h * 1-h 1 ) T r1 = = 0.898, P r1 = = 0.339, From Fig. D.1, Z 1 = Also P 1 V 1 = mz 1 RT 1 => m = = 5.91 kg From D.2, (h * 1-h 1 ) = = 28.2 Assume P 2 = 1655 kpa => V 2 = = m3 P 2 V Z 2 T r2 = = mrt C = At T r2 = and P r2 = => Z 2 = 0.850, ZT r = ~ OK T 2 = K

294 13-59 (h * 2-h 2 ) = = 21.3 (h * 2-h * 1) = 0.842( ) = W 12 = 0.5 ( ) ( ) = 58.0 kj Q 12 = 5.91( ) = OK (Note: for P 2 = 1660 kpa, Q 12 = +2.7) P 2 = 1655 kpa A 10- m 3 storage tank contains methane at low temperature. The pressure inside is 700 kpa, and the tank contains 25% liquid and 75% vapor, on a volume basis. The tank warms very slowly because heat is transferred from the ambient. a. What is the temperature of the methane when the pressure reaches 10 MPa? b. Calculate the heat transferred in the process, using the generalized charts. c. Repeat parts (a) and (b), using the methane tables, Table B.7. Discuss the differences in the results. CH 4 : V = 10 m 3, P 1 = 700 kpa V LIQ 1 = 2.5 m 3, V VAP 1 = 7.5 m 3 a) P r1 = = 0.152, P r2 = = From D.1: Z F1 = 0.025, Z G1 = 0.87 & v F1 = v G1 = T 1 = = K = = m LIQ 1 = = kg, m VAP 1 = 7.5 = 82.6 kg Total m = kg v 2 = v 1 = V m = = = Z T r or Z 2 T r2 = at P r2 = By trial and error

295 13-60 T r2 = & Z 2 = 0.73, T 2 = = K b) 1st law: Q 12 = m(u 2 -u 1 ) = m(h 2 -h 1 ) - V(P 2 -P 1 ) Using D.2 & x 1 = = (h * 1-h 1 ) = (h * 1-h F1 ) - x 1 h FG1 = [ ( )] = (h * 2-h * 1) = ( ) = (h * 2-h 2 ) = (1.47) = (h 2 -h 1 ) = = kj/kg Q 12 = (540.9) - 10( ) = kj c) Using Table B.7 for CH 4 T 1 = T SAT 1 = K, v F1 = , u F1 = v G1 = , u G1 = m LIQ 1 = = 934.6, m VAP 1 = = 82.9 Total mass m = kg and v 2 = At v 2 & P 2 = 10 MPa T 2 = K u 2 = = m3 /kg Q 12 = m(u 2 -u 1 ) = ( ) (199.84) = kj

296 Calculate the difference in entropy of the ideal-gas value and the real-gas value for carbon dioxide at the state 20 C, 1 MPa, as determined using the virial equation of state. Use numerical values given in Problem CO 2 at T = 20 o C, P = 1 MPa RT/P* s * P* - s P = ( P ) T v dv ; ID Gas, s* v(p) RT/P* P* - s P = RT/P* Therefore, at P: s * P - s P = -R ln P P * + ( P ) T v dv virial: P = RT v + BRT v 2 and Integrating, v(p) v(p) R v dv = R ln P P * ( P T ) v = R v + BR v 2 + RT v 2 ( db dt ) s * P - s P = -R ln P RT * + R ln P P * + R[B + T(dB v dt )](1 v - P* ) RT = R[ln RT + (B + T(dB Pv dt ))1] v Using values for CO 2 from solution 13.30, s - * P - s = [ln +( ) ] P = kj/kmol K Carbon dioxide gas enters a turbine at 5 MPa, 100 C, and exits at 1 MPa. If the isentropic efficiency of the turbine is 75%, determine the exit temperature and the second-law efficiency. CO 2 turbine: η S = w/w S = 0.75 inlet: T 1 = 100 o C, P 1 = 5 MPa, exhaust: P 2 = 1 MPa 5 a) P r1 = 7.38 = 0.678, T r1 = = 1.227, P r2 = = From D.2 and D.3, (h * 1-h 1 ) = = 29.9 (s * 1-s 1 ) = =

297 13-62 Assume T 2S = 253 K, T r2s = From D.2 and D.3: (h * 2S-h 2S ) = RT C 0.20 = 11.5 (s * 2S-s 2S ) = R 0.17 = (s * 2S-s * 1) = ln ln 1 5 = (s 2S -s 1 ) = T 2S = 253 K (h * 2S-h * 1) = ( ) = w S = (h 1 -h 2S ) = = 82.8 kj/kg w = η S w S = = 62.1 kj/kg = (h 1 -h * 1) + (h * 1-h * 2) + (h * 2-h 2 ) Assume T 2 = 275 K, T r2 = (h * 1-h * 2) = ( ) = 82.7 From D.2 and D.3, (h * 2-h 2 ) = RT C 0.17 = 9.8 ; (s * 2-s 2 ) = R 0.13 = Substituting, w = = T 2 = 275 K b) (s * 2-s * 1) = ln ln 1 5 = (s 2 -s 1 ) = = Assuming T 0 = 25 o C, (ϕ 1 -ϕ 2 ) = (h 1 - h 2 ) - T 0 (s 1 - s 2 ) = (0.0792) = 85.7 kj/kg η 2nd Law = w ϕ 1 -ϕ 2 = = 0.725

298 A 4- m 3 uninsulated storage tank, initially evacuated, is connected to a line flowing ethane gas at 10 MPa, 100 C. The valve is opened, and ethane flows into the tank for a period of time, after which the valve is closed. Eventually, the whole system cools to ambient temperature, 0 C, at which time the it contains one-fourth liquid and three-fourths vapor, by volume. For the overall process, calculate the heat transfer from the tank and the net change of entropy. Rigid tank V = 4 m 3, m 1 = 0 Line: C 2 H 6 at P i = 10 MPa, T i = 100 o C Flow in, then cool to T 2 = T 0 = 0 o C, V LIQ 2 = 1 m 3 & V VAP 2 = 3 m 3 M = 30.07, R = , C P0 = P ri = = 2.049, T ri = = From D.2 and D.3, (h * i-h i ) = = and (s * i-s i ) = = T r2 = = From D.1, D.2 and D.3, P 2 = P G = = 2489 kpa sat. liq.: Z F = ; (h * -h F ) = RT C 4.09 = ; (s * -s F ) = R 4.3 = sat. vap. : Z G = 0.68 ; (h * -h G ) = RT C 0.87 = 73.5 ; (s * -s G ) = R 0.70 = m LIQ 2 = = kg m VAP 2 = = kg m 2 = kg => x 2 = = st law: Q CV = m 2 u 2 - m i h i = m 2 (h 2 -h i ) - P 2 V = m 2 [(h 2 -h * 2) + (h * 2-h * i) + (h * i-h i )]- P 2 V (h * 2-h * i) = (0-100) = (h * 2-h 2 ) = (1-x 2 )(h * 2-h F2 ) + x 2 (h * 2-h G2 ) = = Q CV = 524.1[ ] = kj

299 13-64 S NET = m 2 (s 2 -s i ) - Q CV /T 0 (s 2 -s i ) = (s 2 -s * 2) + (s * 2-s * i) + (s * i -s i ) (s * 2-s * i) = ln ln = (s * 2-s 2 ) = (1-x 2 )(s * 2-s F2 ) + x 2 (s * 2-s G2 ) = = (s 2 -s i ) = = S NET = 524.1( ) = kj/k The environmentally safe refrigerant R-142b (see Problem 13.45) is to be evaluated as the working fluid in a portable, closed-cycle power plant, as shown in Fig. P The air-cooled condenser temperature is fixed at 50 C, and the maximum cycle temperature is fixed at 180 C, because of concerns about thermal stability. The isentropic efficiency of the expansion engine is estimated to be 80%, and the minimum allowable quality of the fluid exiting the expansion engine is 90%. Calculate the heat transfer from the condenser, assuming saturated liquid at the exit. Determine the maximum cycle pressure, based on the specifications listed T 1 R-142b CH 3 CClF 2 M = , T C = K, P C = 4.25 MPa 4 C 3 P0 = kj/kg K 2S2 R = kj/kg K s T 1 = 180 o C, T 2 = T 3 = 50 o C, x 2 = 0.90, η S EXP ENG = 0.80 a) T r2 = = From D.1, D.2 and D.3: P 2 = P 3 = = 978 kpa sat. liq.: (h * -h F ) = RT C 4.55 = ; (s * -s F ) = R 5.62 = sat. vap.: (h * -h G ) = RT C 0.42 = 14.3 ; (s * -s G ) = R 0.37 = Condenser, 1st law q COND = h 3 - h 2 = h F - (h F + x 2 h FG ) = -x 2 h FG = -0.90( ) = kj/kg

300 13-65 b) expansion engine, efficiency η S EXP ENGINE = w 12 w 12S 1st law & 2nd law, ideal engine: w 12S = h 1 - h 2S, s 2S - s 1 = 0 1st law & 2nd law, real engine: w 12 = h 1 - h 2, s 2 - s 1 > 0 (h * 1-h * 2) = (h * 1-h * 2S) = 0.787(180-50) = (s * 1-s * 2) = (s * 1-s * 2S) = ln ln P P 1 = ln 978 (h * 2-h 2 ) = (1-x 2 )(h * 2-h F2 ) + x 2S (h * 2-h G2 ) = = 28.3 (h * 2S-h 2S ) = (1-x 2S )(h * 2S-h F2 ) + x 2S (h * 2S-h G2 ) = x 2S (s * 2S-s 2S ) = (1-x 2S )(s * 2S-s F2 ) + x 2S (s * 2S-s G2 ) = x 2S Substituting, P 1 (s 1 -s * 1) ln x = 0 (Eq.1) 2S Also (h 1 -h * 1) (h 1 -h * = 0.80 (Eq.2) 1) x 2S P 1 (kpa) Both P r1 = 4250, T r1 = = Trial & Error solution: Assume P 1 = 9300 kpa, P r1 = From D.2 and D.3: (h * 1-h 1 ) = RT C 3.05 = and (s * 1-s 1 ) = R 2.10 = Substituting into eq. 1, x 2S = 0 => x 2S = Substituting into eq. 2, = 27.1 = P 1 = 9300 kpa

301 The refrigerant fluid R-21 (see Table A.2) is to be used as the working fluid in a solar energy powered Rankine-cycle type power plant. Saturated liquid R-21 enters the pump, state 1, at 25 o C, and saturated vapor enters the turbine, state 3, at 88 o C. For R-21 : C P0 = kj/kg K and find the boiler heat transfer q 23 and the thermal efficiency of the cycle. R-21: T c = K, P c = 5.18 MPa, M = kg/kmol, R = kj/kg-k State 1: T 1 = 25 o C, x 1 = 0 From D.1 and D.2: T r1 = 0.66, P r1 = 0.06, P 1 = P r1 P c = 311 kpa Z 1 = 0.01, (h * 1 h 1) f = 4.98 RT c, v 1 = Z 1 RT 1 /P 1 = m 3 /kg State 3: T 3 = 88 o C, x 3 = 1.0 T r3 = 0.8, P r3 = 0.25, P 3 = P r3 P c = 1295 kpa (h * 3 h 3) g = 0.46 RT c, (s * 3 s 3) g = 0.39 R a) 1 st Law Boiler: q H + h 2 = h 3 + w; w = 0 => q H = h 3 - h 2 1 st Law Pump: q + h 1 = h 2 + w p ; q = 0 w p = - v dp = -v 1 (P 2 - P 1 ) = ( ) = kj/kg h 3 - h 1 = (h 3 h * 3 ) + (h* 3 h* 1 ) + (h* 1 h 1) h * 3 h* 1 = C P (T 3 - T 1 ) = 0.582(88-25) = 36.7 kj/kg, h 3 - h 1 = RT c ( ) = kj/kg h 2 = h 1 - w p ; q H = h 3 - h 1 + w p = = kj/kg b) η th = w net /q H ; w net = w t + w p 1 st Law Turbine: q + h 3 = h 4 + w t ; q = 0 w t = h 3 - h 4 = (h 3 h * 3 ) + (h* 3 h* 4 ) + (h* 4 h 4) h * 3 h* 4 = C p (T 3 - T 4 ) = 36.7 kj/kg; T 4 = T 1 h * 4 h 4 = (1 - x 4 )(h* 4 h 4) f + x 4 (h * 4 h 4) g ; x 4 =? 2 nd Law Turbine: S net = m(s 3 - s 4 ) - Q/T; Assume Isentropic s 3 - s 4 = (s 3 s * 3 ) + (s* 3 s* 4 ) + (s* 4 s 4) = 0 (s * 3 s* 4 ) = C P ln T 3 T 4 R ln P 3 P 4 = kj/kg K

302 13-67 s * 4 s 4 = kj/kg-k, s * 4 s 4 = (1 - x 4 )(s* 4 s 4) f + x 4 (s * 4 s 4) g = 7.47 R - x 4 ( ) R T r4 = T r1 = 0.66, (s * 4 s 4) f = 7.47 R, (s * 4 s 4) g = 0.15 R = 7.47 R - x 4 ( ) R => x 4 = (h * 4 h 4 ) f = 4.98 RT c, (h* 4 h 4 ) g = 0.15 RT c h * 4 h 4 = 11.5, w t = h 3 - h 4 = 31.5 kj/kg; w net = 30.7 kj/kg η th = w net /q H = A cylinder/piston contains a gas mixture, 50% CO2 and 50% C2H6 (mole basis) at 700 kpa, 35 C, at which point the cylinder volume is 5 L. The mixture is now compressed to 5.5 MPa in a reversible isothermal process. Calculate the heat transfer and work for the process, using the following model for the gas mixture: a. Ideal gas mixture. b. Kay s rule and the generalized charts. c. The van der Waals equation of state. a) Ideal gas mixture U 2 - U 1 = mc v mix (T 2 - T 1 ) = 0 Q 12 = W 12 = P dv = P 1 V 1 ln(v 2 /V 1 ) = - P 1 V 1 ln(p 2 /P 1 ) = ln(5500/700) = kj b) Kay's rule T cmix = = K P cmix = = 6.13 MPa T r1 = / = 1.011, P r1 = 0.7/6.13 = Z 1 = 0.96, h 1 = 0.12, s 1 = 0.08 n = P 1 V 1 /Z 1 R - 700*0.005 T 1 = = kmol 0.962*8.3145* T r2 = T r1, P r2 = 5.5/6.13 = 0.897, Z 2 = 0.58, h 2 = 1.35, s 2 = 1.0 h h - 1 = (h h - 1) - R - T c ( h 2 - h 1 ) = ( ) = ū 2 - ū 1 = h h R - T(Z 1 - Z 2 ) =

303 ( ) = kj/kmol Q 12 = nt(s s - 1) T = [ 0 - ln(5.5/0.7) ] = kj W 12 = Q 12 - n(ū 2 - ū 1 ) = (-2143) = kj c) van der waal's equation For CO 2 : b = R - T c /8P c = / = a = 27 P c b 2 = = For C 2 H 6 : b = R - T c /8P c = / = a = 27 P c b 2 = = a mix = ( ) 2 = b mix = = *308.2 v v = 0 By trial and error: v - 1 = m 3 /kmol *308.2 v v = 0 By trial and error: v - 2 = m 3 /kmol n = V 1 /v - 1 = 0.005/ = Q 12 = nt(s s - 1) T = n R - T ln v- 2 - b v b = ln U 2 -U 1 = ( ) = kj Q 12 = U 2 -U 1 + W 12 => W 12 = (-2.12) = kj = kj

304 A gas mixture of a known composition is frequently required for different purposes, e.g., in the calibration of gas analyzers. It is desired to prepare a gas mixture of 80% ethylene and 20% carbon dioxide (mole basis) at 10 MPa, 25 C in an uninsulated, rigid 50-L tank. The tank is initially to contain CO2 at 25 C and some pressure P1. The valve to a line flowing C2H4 at 25 C, 10 MPa, is now opened slightly, and remains open until the tank reaches 10 MPa, at which point the temperature can be assumed to be 25 C. Assume that the gas mixture so prepared can be represented by Kay s rule and the generalized charts. Given the desired final state, what is the initial pressure of the carbon dioxide, P1? Determine the heat transfer and the net entropy change for the process of charging ethylene into the tank. A = C 2 H 4, B = CO 2 T 1 = 25 o C P 2 = 10 MPa, T 2 = 25 o C y A2 = 0.8, y B2 = 0.2 a) Mixture at 2 : P C2 = = MPa T C2 = = K P =10 MPa i T = 25 C o i V=0.05 m 3 T r2 = /286.7 = 1.040; P r2 = 10/5.508 = D.1 : Z 2 = 0.32 n 2 = P 2 V Z 2 R - T 2 = = kmol n A2 = n i = 0.8 n 2 = kmol C 2 H 4 n B2 = n 1 = 0.2 n 2 = kmol CO 2 T r1 = = P r1 = n 1 Z B1 R - T Z P CB V = B = Z B1 By trial & error: P r1 = & Z B1 = 0.73 P 1 = = 4.56 MPa ` b) 1st law: Q CV + n i h - i = n 2 ū 2 - n 1 ū 1 = n 2 h- 2 - n 1 h- 1 - (P 2 -P 1 )V or Q CV = n 2 (h - 2 -h- * 2) - n 1 (h - 1 -h- * 1) - n i (h - i -h- * i) - (P 2 -P 1 )V A B

305 13-70 (since T i = T 1 = T 2, h - * i = h - * 1 = h - * 2) (h - * 1-h - ) = = 2099 kj/kmol 1 (h - * 2-h - ) = = 8105 kj/kmol 2 T ri = = 1.056, P ri = = (h - * i-h - ) = = 7866 kj/kmol i Q CV = (-8105) (-2099) (-7866) - ( ) 0.05 = kj S CV = n 2 s n 1 s- 1, S SURR = - Q CV /T 0 - n i s- i S NET = n 2 s n 1 s- 1 - Q CV /T 0 - n i s- i Let s -* A0 = s -* B0 = 0 at T 0 = 25 o C, P 0 = 0.1 MPa Then s -* MIX 0 = (0.8 ln ln 0.2) = kj/kmol K s - 1 = s-* B0 + (s -* P1 T1-s -* P0 T0) B + (s - 1 -s- * P1 T1) B = 0 + ( ln 4.56 ) = kj/kmol K 0.1 s - i = s-* A0 + (s -* Pi Ti-s -* P0 T0) A + (s - i -s-* Pi Ti) A = 0 + ( ln 10 ) = kj/kmol K 0.1 s - 2 = s-* MIX 0 + (s -* P2 T2-s -* P0 T0) MIX + (s - 2 -s- * P2 T2) MIX = ( ln 10 ) = kj/kmol K 0.1 S NET = (-55.33) (-36.75) (-58.58) /298.2 = kj/k

306 13-71 English Unit Problems 13.75EA special application requires R-22 at 150 F. It is known that the triple-point temperature is less than 150 F. Find the pressure and specific volume of the saturated vapor at the required condition. The lowest temperature in Table C.10 for R-22 is -100 F, so it must be extended to -150 F using the Clapeyron eqn. At T 1 = -100 F = R, P 1 = lbf/in. 2 and R = = Btu/lbm R ln P P 1 = h fg R (T-T 1 ) T T 1 = ( ) = P = lbf/in. 2 v g = RT = = 132 ft P 3 /lbm g EIce (solid water) at 27 F, 1 atm is compressed isothermally until it becomes liquid. Find the required pressure. Water, triple point T = F = R, P = lbf/in. 2 v F = v I = h F = 0.00 h I = dp IF dt = h F -h I (v F -v I )T = = dp IF P T = ( ) = lbf/in.2 dt P = P tp + P = 5451 lbf/in. 2

307 E Using thermodynamic data for water from Tables C.8.1 and C.8.3, estimate the freezing temperature of liquid water at a pressure of 5000 lbf/in. 2. P H 2 O dp IF dt = h IF Tv IF constant At the triple point, T.P. v IF = v F - v I = T = ft 3 /lbm h IF = h F - h I = ( ) = Btu/lbm dp IF dt = ( ) = lbf/in.2 R at P = 5000 lbf/in 2, T ( ) = 27.4 F ( ) 13.78E Determine the volume expansivity, α P, and the isothermal compressibility, β T, for water at 50 F, 500 lbf/in. 2 and at 500 F, 1500 lbf/in. 2 using the steam tables. Water at 50 F, 500 lbf/in. 2 (compressed liquid) α P = 1 v ( v T ) P 1 v ( v T ) P Using values at 32 F, 50 F and 100 F α P = F -1 β T = - 1 v ( v P ) T - 1 v ( v P ) T Using values at saturation, 500 and 1000 lbf/in β T - = in /lbf Water at 500 F, 1500 lbf/in. 2 (compressed liquid) α P = F β T - = in /lbf

308 E Sound waves propagate through a media as pressure waves that causes the media to go through isentropic compression and expansion processes. The speed of sound c is defined by c 2 = ( P/ ρ) s and it can be related to the adiabatic compressibility, which for liquid ethanol at 70 F is 6.4 in. 2 /lbf. Find the speed of sound at this temperature. c 2 =( P ρ ) s = -v2 ( P v ) s = 1-1 v ( v P ) s ρ = 1 β s ρ From Table C.3 for ethanol, ρ = 48.9 lbm/ft 3 c =( )1/2 = 3848 ft/s 13.80E Consider the speed of sound as defined in Problem Calculate the speed of sound for liquid water at 50 F, 250 lbf/in. 2 and for water vapor at 400 F, 80 lbf/in. 2 using the steam tables. From problem : Liquid water at 50 F, 250 lbf/in. 2 Assume ( P v ) s ( P v ) T c 2 =( P ρ ) s = -v2 ( P v ) s Using saturated liquid at 50 F and compressed liquid at 50 F, 500 lbf/in. 2, c 2 = -( ) 2 ( ( ) ) = c = 4778 ft/s Superheated vapor water at 400 F, 80 lbf/in. 2 v = 6.217, s = At P = 60 lbf/in. 2 & s = : T = F, v = At P = 100 lbf/in. 2 & s = : T = F, v = c 2 = -(6.217) 2 ( (100-60) ) = c = 1690 ft/s

309 EA cylinder fitted with a piston contains liquid methanol at 70 F, 15 lbf/in. 2 and volume 1 ft 3. The piston is moved, compressing the methanol to 3000 lbf/in. 2 at constant temperature. Calculate the work required for this process. The isothermal compressibility of liquid methanol at 70 F is in. 2 /lbf. 2 w 12 = 1 Pdv = P( v ) 2 P T dp T = - vβ T PdP T 1 vβ T For v const & β T const. => w 12 = - 2 (P2 2 - P 2 1) For liquid methanol, from Table C.3 : ρ = 49.1 lbm/ft 3 V 1 = 1.0 ft 3, m = = 49.1 lbm W 12 = [(3000) 2 - (15) 2 ] 144 = ft lbf = 48.0 Btu 13.82EA piston/cylinder contains 10 lbm of butane gas at 900 R, 750 lbf/in. 2. The butane expands in a reversible polytropic process with polytropic exponent, n = 1.05, until the final pressure is 450 lbf/in. 2. Determine the final temperature and the work done during the process. C 4 H 10, m = 10 lbm, T 1 = 900 R, P 1 = 750 lbf/in. 2 Rev. polytropic process (n=1.05), P 2 = 450 lbf/in. 2 T r1 = = 1.176, P r1 = = => From Fig. D.1 : Z 1 = V 1 = = ft P 1 V n 1 = P 2 V n 2 V 2 = ( ) = ft 3 P 2 V Z 2 T r2 = = mrt C at P r2 = 450/551 = = Trial & error: T r2 = 1.068, Z 2 = 0.72 => T 2 = R 2 W 12 = P 2 V 2 - P 1 V 1 PdV = 1-n = ( ) = 98.8 Btu

310 EA 7-ft 3 rigid tank contains propane at 1300 lbf/in. 2, 540 F. The propane is then allowed to cool to 120 F as heat is transferred with the surroundings. Determine the quality at the final state and the mass of liquid in the tank, using the generalized compressibility charts. Propane C 3 H 8 : V = 7.0 ft 3, P 1 = 1300 lbf/in. 2, T 1 = 540 F = 1000 R cool to T 2 = 120 F = 580 R From Table C.1 : T C = R, P C = 616 lbf/in. 2 P r1 = = 2.110, T r1 = = From D.1: Z 1 = 0.83 v 2 = v 1 = Z 1 RT = P = ft 3 /lbm From D.1 at T r2 = 0.871, saturated => P G2 = = 265 lbf/in v G2 = = v F2 = = = x 2 ( ) => x 2 = m LIQ 2 = ( ) 7.0 = 29.8 lbm

311 E A rigid tank contains 5 lbm of ethylene at 450 lbf/in. 2, 90 F. It is cooled until the ethylene reaches the saturated vapor curve. What is the final temperature? T C H v C 2 H 4 m = 5 lbm P 1 = 450 lbf/in 2, T 1 = 90 F = R P r1 = = 0.616, T r1 = = Fig. D.1 Z 1 = 0.82 Z 2 T r2 Z G2 T r2 P r2 = P r1 = Z 1 T r = Z G2 T r2 Trial & error: T r2 Z G2 P r2 P r2 CALC ~ OK => T 2 = R 13.85E Calculate the difference in internal energy of the ideal-gas value and the real-gas value for carbon dioxide at the state 70 F, 150 lbf/in. 2, as determined using the virial equation of state. For carbon dioxide at 70 F, B = ft3/lb mol, T(dB/dT) = ft3/lb mol Solution: CO 2 at 70 F, 150 lbf/in 2 virial: u-u * = - v P = RT v + BRT v 2 ; [T( P ( P T ) v = R v + BR v 2 + RT v 2 ( db dt ) ) T v - P]dv = - RT 2 ( db RT2 ) v v 2 dv = - dt v B = ft 3 /lbmol T( db dt )= ft3 /lbmol v - = 1 R - T 2 P [ BP/R- T] But R- T P = = db dt v - = [ (-2.036)/ ] = ft 3 /lbmol ū-ū * = = Btu/lbmol

312 E Calculate the heat transfer during the process described in Problem From solution 13.82, V 1 = ft 3, V 2 = ft 3, W 12 = 98.8 Btu T r1 = 1.176, P r1 = 1.361, T r2 = 1.068, P r2 = 0.817, T 2 = R From D.1: ( h* -h RT C ) 1 = 1.36, ( h* -h RT C ) 2 = 0.95 h * 2 - h * 1 = ( ) = Btu/lbm h 2 - h 1 = ( ) = Btu/lbm 778 U 2 -U 1 = m(h 2 -h 1 ) - P 2 V 2 + P 1 V 1 = 10(-23.6) Q 12 = U 2 -U 1 + W 12 = Btu = Btu

313 E Saturated vapor R-22 at 90 F is throttled to 30 lbf/in. 2 in a SSSF process. Calculate the exit temperature assuming no changes in the kinetic energy, using the generalized charts, Fig. D.2 and repeat using the R-22 tables, Table C.10. R-22 throttling process T 1 = 90 F, x 1 = 1.00, P 2 = 30 lbf/in. 2 Energy Eq.: h 2 -h 1 = (h 2 -h * 2) + (h * 2-h * 1) + (h * 1-h 1 ) = 0 Generalized charts, T r1 = = From D.2: (h * 1-h 1 ) = (0.55) = To get C P0, use h values from Table C.10 at low pressure. C P = Substituting into energy Eq.: (h 2 -h * 2) (T 2-30) = 0 at P r2 = = Assume T 2 = 43.4 F = R => T r2 = = (h * 2-h 2 ) = (0.07) = Substituting, ( ) = T 2 = 43.4 F b) R-22 tables, C.10: T 1 = 90 F, x 1 = 1.0 => h 1 = h 2 = h 1 = , P 2 = 30 lbf/in. 2 => T 2 = 42.1 F

314 EA 10-ft3 tank contains propane at 90 F, 90% quality. The tank is heated to 600 F. Calculate the heat transfer during the process. C H 3 8 V = 10 ft 3 T 1 = 90 F = R, x 1 = 0.90 T 2 1 v Heat to T 2 = 600 F = R M = , T C = R P C = 616 lbf/in. 2 R = 35.04, C P0 = T r1 = Figs. D.1 and D.2 Z 1 = = 0.707, h * 1-h 1 RT c = = P SAT r = 0.31 ; P SAT 1 = = 191 lbf/in m = = 20.2 lbm Z P r2 = at T r2 = = Z P r2 = 0.79 P = 490 lbf/in.2 2 Trial & error: h * 2-h 2 Z 2 = 0.94 = 0.36 RT c (h * 2-h * 1) = 0.407(600-90) = Btu/lbm (h * 1-h 1 ) = / 778 = 28.0 (h * 2-h 2 ) = / 778 = 10.8 Q 12 = m(h 2 -h 1 ) - (P 2 -P 1 )V = 20.2 ( ) - ( ) = = 3988 Btu

315 EA newly developed compound is being considered for use as the working fluid in a small Rankine-cycle power plant driven by a supply of waste heat. Assume the cycle is ideal, with saturated vapor at 400 F entering the turbine and saturated liquid at 70 F exiting the condenser. The only properties known for this compound are molecular weight of 80 lbm/lbmol, ideal gas heat capacity Cpo = 0.20 Btu/lbm R and Tc = 900 R, Pc = 750 lbf/in. 2. Calculate the work input, per lbm, to the pump and the cycle thermal efficiency.. Q H 1 Ht. Exch 4 Turbine P 3. -W P Cond 2. W T T 1 = 400 F = 860 R x 1 = 1.0 T 3 = 70 F = 530 R x 3 = 0.0 Properties known: M = 80, C PO = 0.2 Btu/lbm R T C = 900 R, P C = 750 lbf/in. 2 T r1 = = 0.956, T r3 = = From D.1: P r1 = 0.76, P 1 = = 570 lbf/in. 2 = P 4 P r3 = 0.025, P 3 = 19 lbf/in. 2 = P 2, Z F3 = v F3 = Z F3 RT 3 /P 3 = 4 w P = = vdp v F3 (P 4 -P 3 ) = (570-19) = Btu/lbm q H + h 4 = h 1, but h 3 = h 4 + w P => q H = (h 1 -h 3 ) + w P From D.2, (h * 1-h 1 ) = (1.9859/80) = 30.0 (h * 3-h 3 ) = (1.9859/80) = (h * 1-h * 3) = C P0 (T 1 -T 3 ) = 0.2(400-70) = 66.0 (h 1 -h 3 ) = = q H = (-1.71) = Btu/lbm

316 13-81 Turbine, (s 2 -s 1 ) = 0 = -(s * 2-s 2 )+(s * 2-s * 1)+(s * 1-s 1 ) From D.3, (s * 1-s 1 ) = (1.9859/80) 1.06 = (s * 2-s * 1) = 0.20 ln ln = Substituting, (s * 2-s 2 ) = = = (s * 2-s F2 ) - x 2 s FG = x ( ) => x 2 = (h * 2-h 2 ) = (h * 2-h F2 ) - x 2 h FG2 From D.2, h fg2 = ( ) = (h * 2-h 2 ) = = 7.9 w T = (h 1 -h 2 ) = = 43.9 Btu/lbm η TH = w NET = = q H 150.4

317 EA 7-ft3 rigid tank contains propane at 730 R, 500 lbf/in. 2. A valve is opened, and propane flows out until half the initial mass has escaped, at which point the valve is closed. During this process the mass remaining inside the tank expands according to the relation Pv1.4 = constant. Calculate the heat transfer to the tank during the process. C 3 H 8 : V = 7.0 ft 3, T 1 = 400 K, P 1 = 500 lbf/in. 2 Flow out to m 2 = m 1 /2; Pv 1.4 = const inside T r1 = = 1.097, P r1 = = => From D.1: Z 1 = v 1 = = 0.270, v = 2v 1 = 0.54 m 1 = 7.0/0.270 = lbm, m 2 = m 1 /2 = lbm, P 2 = P 1 ( v 1 v 2 ) = 1.4 = lbf/in.2 2 P r2 = = P 2 v 2 = Z 2 RT 2 Z 2 = Z F2 + x 2 (Z G2 - Z F2 ) Trial & error: saturated with T 2 = = R & Z 2 = = x 2 ( ) => x 2 = (h * 1-h 1 ) = / 778 = 25.5 (h * 2-h * 1) = ( ) = = (h * 2-h 2 ) = (h * h F2 ) - x 2 h FG2 = [ ( )] = st law: Q CV = m 2 h 2 - m 1 h 1 + (P 1 -P 2 )V + m e h e AVE Let h * 1 = 0 then: h 1 = 0 + (h 1 -h * 1) = h 2 = h * 1 + (h * 2-h * 1) + (h 2 -h * 2) = = h e AVE = (h 1 + h 2 )/2 = Q CV = (-92.1) (-25.5) + ( ) (-58.8) = -892 Btu

318 E A geothermal power plant on the Raft river uses isobutane as the working fluid as shown in Fig. P The fluid enters the reversible adiabatic turbine at 320 F, 805 lbf/in. 2 and the condenser exit condition is saturated liquid at 91 F. Isobutane has the properties Tc = R, Pc = 537 lbf/in. 2, Cpo = Btu/lbm R and ratio of specific heats k = with a molecular weight as Find the specific turbine work and the specific pump work. R = ft lbf/lbm R = Btu/lbm R Turbine inlet: T 1 = 320 F, P 1 = 805 lbf/in. 2 Condenser exit: T 3 = 91 F, x 3 = 0.0 ; T r3 = / = 0.75 From D.1 : P r3 = 0.165, Z 3 = P 2 = P 3 = = 88.6 lbf/in. 2 T r1 = / = 1.061, P r1 = 805 / 537 = From D.2 and D.3, (h * 1-h 1 ) = = 71.5 (s * 1-s 1 ) = = (s * 2-s * 1) = ln ln = (s * 2-s 2 ) = (s * 2-s F2 ) - x 2 s FG2 = x ( ) = x (s 2 -s 1 ) = 0 = x => x 2 = (h * 2-h * 1) = C P0 (T 2 -T 1 ) = ( ) = From D.2, (h * 2-h 2 ) = (h * 2-h F2 ) - x 2 h FG2 = [ ( )] Turbine: = = 8.5 w T = (h 1 -h 2 ) = = 28.0 Btu/lbm Pump v F3 = 4 w P = - 3 Z F3 RT = = P vdp v F3 (P 4 -P 3 ) = ( ) 144 = -4.2 Btu/lbm 778

319 E Carbon dioxide collected from a fermentation process at 40 F, 15 lbf/in. 2 should be brought to 438 R, 590 lbf/in. 2 in a SSSF process. Find the minimum amount of work required and the heat transfer. What devices are needed to accomplish this change of state? R = 35.1 = Btu/lbm R 778 T ri = = 0.913, P ri = = From D.2 and D.3: ( h* -h RT C ) = 0.02, ( s* -s R ) = 0.01 R T re = = 0.80, P re = = From D.2 and D.3: h * e-h e = 4.50 RT c, s * e-s e = 4.70 R (h i -h e ) = - (h * i-h i ) + (h * i-h * e) + (h * e-h e ) = ( ) = Btu/lbm (s i -s e ) = - (s * i-s i ) + (s * i -s * e) + (s * e-s e ) = ln ln = Btu/lbm R w rev = (h i -h e ) -T 0 (s i -s e ) = (0.4042) = Btu/lbm q rev = (h e -h i ) + w rev = = Btu/lbm

320 E A control mass of 10 lbm butane gas initially at 180 F, 75 lbf/in. 2, is compressed in a reversible isothermal process to one-fifth of its initial volume. What is the heat transfer in the process? Butane C 4 H 10 : m = 10 lbm, T 1 = 180 F, P 1 = 75 lbf/in. 2 Compressed, reversible T = const, to V 2 = V 1 /5 T r1 = = 0.836, P r1 = = => From D.1 and D.3: Z 1 = 0.92 (s * 1-s 1 ) = = v 1 = Z 1 RT = = ft P 3 /lbm v 2 = v 1 /5 = ft 3 /lbm At T r2 = T r1 = From D.1 and D.3: P G = =187 lbf/in. 2 sat. liq.: Z F = ; (s * -s F ) = R 5.02 = sat. vap.: Z G = ; (s * -s G ) = R 0.49 = Therefore v F = = v G = = v 2 -v F Since v F < v 2 < v G two-phase x 2 = = v G -v F (s * 2-s 2 ) = (1-x 2 )(s * 2-s F2 ) + x 2 (s * 2-s G2 ) = = (s * 2-s * T 2 1) = C P0 ln T 1 - R ln P 2 = P ln = (s 2 -s 1 ) = = Btu/lbm R Q 12 = Tm(s 2 -s 1 ) = (-0.110) = -704 Btu

321 E A cylinder contains ethylene, C 2 H 4, at lbf/in. 2, 8 F. It is now compressed isothermally in a reversible process to 742 lbf/in. 2. Find the specific work and heat transfer. Ethylene C 2 H 4 P 1 = lbf/in. 2, T 2 = T 1 = 8 F = R State 2: P 2 = 742 lbf/in. 2 R = ft lbf/lbm R = Btu/lbm R T r2 = T r1 = = P r1 = = From D.1, D.2 and D.3: Z 1 = 0.85, ( h* -h RT C ) 1 = 0.40 (h * 1-h 1 ) = = 14.4 (s * 1-s 1 ) = = P r2 = 742 = (comp. liquid) 731 From D.1, D.2 and D.3: Z 2 = 0.17 (h * 2-h 2 ) = = (s * 2-s 2 ) = = (h * 2-h * 1) = 0 (s * 2-s * 1) = ln = q 12 = T(s 2 -s 1 ) = 467.7( ) = Btu/lbm (h 2 -h 1 ) = = (u 2 -u 1 ) = (h 2 -h 1 ) - RT(Z 2 -Z 1 ) = ( ) = w 12 = q 12 - (u 2 -u 1 ) = = Btu/lbm

322 EA cylinder contains ethylene, C 2 H 4, at lbf/in. 2, 8 F. It is now compressed in a reversible isobaric (constant P) process to saturated liquid. Find the specific work and heat transfer. Ethylene C 2 H 4 P 1 = lbf/in. 2 = P 2, T 1 = 8 F = R State 2: saturated liquid, x 2 = 0.0 R = ft lbf/lbm R = Btu/lbm R T r1 = = P r1 = P r2 = = From D.1 and D.2: Z 1 = 0.85, ( h* -h RT C ) 1 = 0.40 v 1 = Z 1 RT 1 P 1 = = (h * 1-h 1 ) = = 14.4 From D.1 and D.2: T 2 = = R Z 2 = 0.05, ( h* -h RT C ) 2 = 4.42 v 2 = Z 2 RT 2 P 2 = = (h * 2-h 2 ) = = (h * 2-h * 1) = C P0 (T 2 -T 1 ) = 0.411( ) = w 12 = Pdv = P(v 2 -v 1 ) = 222.6( ) 144 = Btu/lbm q 12 = (u 2 - u 1 ) + w 12 = (h 2 -h 1 ) = = Btu/lbm

323 E A distributor of bottled propane, C 3 H 8, needs to bring propane from 630 R, 14.7 lbf/in. 2 to saturated liquid at 520 R in a SSSF process. If this should be accomplished in a reversible setup given the surroundings at 540 R, find the ratio of the volume flow rates V.in / V.out, the heat transfer and the work involved in the process. R = 35.04/778 = Btu/lbm R T ri = = P ri = = From D.1, D.2 and D.3 : Z i = 0.99 (h * i-h i ) = = 0.9 (s * i-s i ) = = T re = 520/665.6 = 0.781, From D.1, D.2 and D.3 : P re = = 0.21, P e = = 129 lbf/in. 2 Z e = (h * e-h e ) = = (s * e-s e ) = = (h * e-h * i) = ( ) = (s * e-s * i) = ln ln = (h e -h i ) = = (s e -s i ) = = V. in V. Z i T i /P i = = 0.99 Z out e T e /P e = w rev = (h i -h e ) -T 0 (s i -s e ) = (0.4326) = Btu/lbm q rev = (h e -h i ) + w rev = = Btu/lbm

324 E A line with a steady supply of octane, C 8 H 18, is at 750 F, 440 lbf/in. 2. What is your best estimate for the availability in an SSSF setup where changes in potential and kinetic energies may be neglected? Availability of Octane at T i = 750 F, P i = 440 lbf/in. 2 R = ft lbf/lbm R = Btu/lbm R P ri = = 1.219, T ri = = From D.2 and D.3: (h * 1-h 1 ) = = 20.5 (s * 1-s 1 ) = = Exit state in equilibrium with the surroundings Assume T 0 = 77 F, P 0 = 14.7 lbf/in. 2 T r0 = = 0.524, P r0 = = From D.2 and D.3: (h * 0-h 0 ) = RT C 5.41 = 96.3 and (s * 0-s 0 ) = R = (h * i-h * 0) = 0.409( ) = (s * i-s * 0) = ln (h i -h 0 ) = = ln = (s i -s 0 ) = = ψ i = w rev = (h i -h 0 ) - T 0 (s i -s 0 ) = (0.4415) = Btu/lbm

325 14-1 CHAPTER 14 The correspondence between the new problem set and the previous 4th edition chapter 12 problem set. New Old New Old New Old 1 new new new new new mod new new mod new new new The problems that are labeled advanced start at number 66.

326 14-2 The English unit problems are: New Old New Old New Old mod

327 Calculate the theoretical air-fuel ratio on a mass and mole basis for the combustion of ethanol, C 2 H 5 OH. Reaction Eq.: C 2 H 5 OH + ν O2 (O N 2 ) => aco 2 + bh 2 O + cn 2 Balance C: 2 = a Balance H: 6 = 2b => b = 3 Balance O: 1 + 2ν O2 = 2a + b = = 7 => ν O2 = 3 (air/fuel) mol = ν O2 ( )/1 = = (air/fuel) mass = (ν O2 M O2 + ν N2 M N2 )/M Fuel = ( )/ = In a combustion process with decane, C 10 H 22, and air, the dry product mole fractions are 86.9% N 2, 1.163% O 2, % CO 2 and 0.954% CO. Find the equivalence ratio and the percent theoretical air of the reactants. x C 10 H 22 + φ ν O 2 (O N 2 ) a H 2 O + b CO 2 + c CO + d N 2 + e O 2 Stoichiometric combustion: φ = 1, c = 0, e = 0, C balance: b = 10x H balance: a = 22x/2 = 11x, O balance: 2 ν O = a + 2b = 11x + 20x = 31x 2 ν = 15.5x, ν O 2 N2 = 58.28x => (A/F) s = (ν O2 + ν )/x = N2 Actual combustion: d = 86.9 φ ν O = 86.9 φ ν O2 = C balance: 10x = = => x = (A/F) ac = φ ν O 4.76/ = φ = (A/F) ac / (A/F) s = / = 1.25 or 125% 14.3 A certain fuel oil has the composition C 10 H 22. If this fuel is burned with 150% theoretical air, what is the composition of the products of combustion? C 10 H 22 + φ ν (O O N ) 2 a H 2 O + b CO 2 + c N 2 + d O 2 Stoichiometric combustion: φ = 1, d = 0, C balance: b = 10 H balance: a = 22/2 = 11, O balance: 2 ν O = a + 2b = = 31 2 => ν O2 = 15.5 Actual case: φ = 1.5 => ν O2 = = H balance: a = 11, C balance: b = 10, N balance: c = = O 2 balance: d = /2 = 7.75 (excess oxygen)

328 Natural gas B from Table 14.2 is burned with 20% excess air. Determine the composition of the products. The reaction equation (stoich. complete comb.) with the fuel composition is: 60.1 CH C 2 H C 3 H C 4 H N 2 + ν O2 (O N 2 ) a H 2 O + b CO 2 + c N 2 C balance: = b = H balance: = 2a = => a = O balance: 2 ν O2 = a + 2b = => ν O2 = N 2 balance: ν O2 = c = % excess air: ν O2 = = so now more O 2 and N 2 Extra oxygen: d = = 53.26, c = = 1209 Products: H 2 O CO N O A Pennsylvania coal contains 74.2% C, 5.1% H, 6.7% O, (dry basis, mass percent) plus ash and small percentages of N and S. This coal is fed into a gasifier along with oxygen and steam, as shown in Fig. P14.5. The exiting product gas composition is measured on a mole basis to: 39.9% CO, 30.8% H 2, 11.4% CO 2, 16.4% H 2 O plus small percentages of CH 4, N 2, and H 2 S. How many kilograms of coal are required to produce 100 kmol of product gas? How much oxygen and steam are required? Number of kmol per 100 kg coal: C : n = 74.2/12.01 = H 2 : n = 5.1/2.016 = O 2 : n = 6.7/ = (6.178 C H O 2 )x + y H 2 O + z O 2 in and 39.9 CO H CO H 2 O out in 100 kmol of mix out C balance: x = x = H 2 balance: y = y = O 2 balance: z = Therefore, for 100 kmol of mixture out require: kg of coal kmol of steam kmol of oxygen z =

329 Repeat Problem 14.5 for a certain Utah coal that contains, according to the coal analysis, 68.2% C, 4.8% H, 15.7% O on a mass basis. The exiting product gas contains 30.9% CO, 26.7% H 2, 15.9% CO 2 and 25.7% H 2 O on a mole basis. Number of kmol per 100 kg coal: C : 68.2/12.01 = H 2 : 4.8/2.016 = O 2 : 15.7/32.00 = (5.679 C H O 2 )x + y H 2 O + z O 2 in 30.9 CO H CO H 2 O out in 100 kmol of mix out C : 5.679x = x = H 2 : y = y = O 2 : z = z = Therefore, for 100 kmol of mixture out, require: kg of coal kmol of steam kmol of oxygen A sample of pine bark has the following ultimate analysis on a dry basis, percent by mass: 5.6% H, 53.4% C, 0.1% S, 0.1% N, 37.9% O and 2.9% ash. This bark will be used as a fuel by burning it with 100% theoretical air in a furnace. Determine the air fuel ratio on a mass basis. Converting the Bark Analysis from a mass basis: Substance S H 2 C O 2 N 2 c/m = 0.1/32 5.6/2 53.4/ /32 0.1/28 kmol / 100 kg coal Product SO 2 H 2 O CO 2 oxygen required Combustion requires: = kmol O 2 there is in the bark kmol O 2 so the net from air is kmol O 2 AF = ( ) = 6.44 kg air kg bark

330 Liquid propane is burned with dry air. A volumetric analysis of the products of combustion yields the following volume percent composition on a dry basis: 8.6% CO 2, 0.6% CO, 7.2% O 2 and 83.6% N 2. Determine the percent of theoretical air used in this combustion process. a C 3 H 8 + b O 2 + c N CO CO + d H 2 O O N 2 C balance: 3a = = 9.2 a = H 2 balance: 4a = d d = N 2 balance: c = 83.6 O 2 balance: b = = Air-Fuel ratio = = Theoretical: C 3 H O N 2 3 CO H 2 O N 2 theo. A-F ratio = = % theoretical air = % = 145 % A fuel, C x H y, is burned with dry air and the product composition is measured on a dry basis to be: 9.6% CO 2, 7.3% O 2 and 83.1% N 2. Find the fuel composition (x/y) and the percent theoretical air used. ν Fu C x H y + ν O O ν O2 N 2 9.6CO O N 2 + ν H2O H 2 O N 2 balance: 3.76ν = 83.1 ν O 2 O2 = O 2 balance: ν O 2 = ν H2O ν H2O = H balance: ν Fu y = 2 ν H 2O = C balance: ν Fu x = 9.6 Fuel composition ratio = x/y = 9.6/ = ν O 2AC Theoretical air = = ν O 2stoich =

331 Many coals from the western United States have a high moisture content. Consider the following sample of Wyoming coal, for which the ultimate analysis on an as-received basis is, by mass: Component Moisture H C S N O Ash % mass This coal is burned in the steam generator of a large power plant with 150% theoretical air. Determine the air fuel ratio on a mass basis. Converting from mass analysis: Substance S H 2 C O 2 N 2 c/m = 0.5/32 3.5/2 4.86/12 12/32 0.7/28 kmol / 100 kg coal Product SO 2 H 2 O CO 2 oxygen required Combustion requires then oxygen as: = The coal does include O 2 so only O 2 from air/100 kg coal AF = 1.5 ( ) 28.97/100 = kg air/kg coal Pentane is burned with 120% theoretical air in a constant pressure process at 100 kpa. The products are cooled to ambient temperature, 20 C. How much mass of water is condensed per kilogram of fuel? Repeat the answer, assuming that the air used in the combustion has a relative humidity of 90%. C 5 H (O N 2 ) 5 CO H 2 O O N 2 Products cooled to 20 o C, 100 kpa, so for H 2 O at 20 C: P g = kpa y H2O MAX = P g /P = = n H2O MAX n H2O MAX => n H2O MAX = < n H2O Therefore, n H2O VAP = 1.007, n H2O LIQ = = m H2O LIQ = = kg/kg fuel P v1 = = kpa => w 1 = = n H2O IN = ( ) = kmol n H2O OUT = = => n H2O LIQ = = kmol n H2O LIQ = = kg/kg fuel

332 The coal gasifier in an integrated gasification combined cycle (IGCC) power plant produces a gas mixture with the following volumetric percent composition: Product CH 4 H 2 CO CO 2 N 2 H 2 O H 2 S NH 3 % vol This gas is cooled to 40 C, 3 MPa, and the H 2 S and NH are removed in 3 water scrubbers. Assuming that the resulting mixture, which is sent to the combustors, is saturated with water, determine the mixture composition and the theoretical air fuel ratio in the combustors. CH 4 H 2 CO CO 2 N 2 n y H2O = n V n V +81.7, where n = number of moles of water vapor V Cool to 40 C P G = 7.384, P = 3000 kpa y H2O MAX = = n V n V n V = a) Mixture composition: CH 4 H 2 CO CO 2 N 2 H 2 O(v) 0.3 kmol kmol (from 100 kmol of the original gas mixture) 0.3 CH O CO H 2 O 29.6 H O H 2 O 41 CO O 2 41 CO 2 Number of moles of O 2 = = 35.9 Number of moles of air = (N 2 ) Air Fuel ratio = 28.97( (35.9)) 0.3(16) (2) + 41(28) + 10(44) + 0.8(28) (18) = 2.95 kg air/kg fuel

333 The hot exhaust gas from an internal combustion engine is analyzed and found to have the following percent composition on a volumetric basis at the engine exhaust manifold. 10% CO 2, 2% CO, 13% H 2 O, 3% O 2 and 72% N 2. This gas is fed to an exhaust gas reactor and mixed with a certain amount of air to eliminate the carbon monoxide, as shown in Fig. P It has been determined that a mole fraction of 10% oxygen in the mixture at state 3 will ensure that no CO remains. What must the ratio of flows be entering the reactor? Exhaust gas at state 1: CO 2 10 %, H 2 O 13%, CO 2%, O 2 3%, N 2 72% Exhaust gas at state 3: CO = 0 %, O 2 = 10 % 1 Exh. gas Reactor 2 Air 3 gas out 0.02 CO + x O x N CO 2 + (x-0.01) O x N 2 At 3: CO 2 = = 0.12, H 2 O = 0.13 O 2 = (x-0.01) x N 2 = x or n TOT = x x = x y O2 = 0.16 = x x x = or air 2 Exh. Gas 1 = 4.76x kmol air = kmol Exh. gas Methanol, CH 3 OH, is burned with 200% theoretical air in an engine and the products are brought to 100 kpa, 30 C. How much water is condensed per kilogram of fuel? CH 3 OH + ν {O O N 2 } CO H 2 O ν O2 N 2 Stoichiometric ν O 2 S = 1.5 ν O2 AC = 3 Actual products: CO H 2 O O N 2 P sat (30 C) = kpa ν H2O y H2O = = 1 + ν H2O ν H2O = ν H2O cond = = M Fu = M H2O = = kg H 2 O M Fu kg fuel

334 The output gas mixture of a certain air blown coal gasifier has the composition of producer gas as listed in Table Consider the combustion of this gas with 120% theoretical air at 100 kpa pressure. Determine the dew point of the products and find how many kilograms of water will be condensed per kilogram of fuel if the products are cooled 10 C below the dew-point temperature. {3 CH H N O CO CO 2 } O N CO H 2 O O N 2 Products: 20 y H2O = y H2O MAX = P G /100 = P G = 8.79 kpa At T = 33.2 C, P G = 5.13 kpa T DEW PT = 43.2 C y H2O = = n H2O n H2O n H2O = m H2O LIQ = 8.78(18) 3(16) +14(2) +50.9(28) +0.6(32) +27(28) +4.5(44) = kg/kg fuel Pentene, C 5 H 10 is burned with pure oxygen in an SSSF process. The products at one point are brought to 700 K and used in a heat exchanger, where they are cooled to 25 C. Find the specific heat transfer in the heat exchanger. C 5 H 10 + ν O 2 O 2 5 CO H 2 O ν O2 = ṅ F h - CO2 + 5 ṅ F h- H2O + Q. = 5 ṅ F h - f CO 2 + (5 - x) ṅ F h - liq H 2O + (x) ṅ F h- vap H 2O P Find x: y H 2O max = g (25 ) x = = P tot 5 + x x = Out of the 5 H 2 O only still vapor. Q. n = -5 h - CO2,700 + (5-x)(h- f liq - h - f vap - h ) + x(h- f vap - h - f vap - h ) F = -5(17761) ( ) (14184) = kj/kmol Fu

335 Butane gas and 200% theoretical air, both at 25C, enter a SSSF combustor. The products of combustion exits at 1000 K. Calculate the heat transfer from the combustor per kmol of butane burned. C 4 H 10 + (1/φ) ν O2 (O N 2 ) a CO 2 + b H 2 O + c N 2 + d O 2 First we need to find the stoichiometric air ( φ = 1, d = 0 ) C balance: 4 = a, H balance: 10 = 2b => b = 5 O balance: 2ν O2 = 2a + b = = 13 => ν O2 = 6.5 Now we can do the actual air: (1/φ) = 2 => ν O2 = = 13 N balance: c = 3.76 ν O2 = 48.88, O balance: d = = 6.5 q = H R - H P = H o R - Ho P - H P = M(-Ho RP ) - H P Table 14.3: H o RP = , The rest of the values are from Table A.8 h CO2 = 33397, h N2 = 21463, h O2 = 22703, h H2O = kj/kmol q = (45714) - ( ) = = kj/kmol butane Liquid pentane is burned with dry air and the products are measured on a dry basis as: 10.1% CO 2, 0.2% CO, 5.9% O 2 remainder N 2. Find the enthalpy of formation for the fuel and the actual equivalence ratio. ν Fu C 5 H 12 + ν O O ν O2 N 2 x H 2 O CO CO O N 2 Balance of C: 5 ν Fu = ν Fu = 2.06 Balance of H: 12 ν Fu = 2 x x = 6 ν Fu = Balance of O: 2 ν = x ν O 2 O2 = Balance of N: ν = ν O = OK 2 O2 ν O 2 for 1 kmol fuel = φ = 1, C 5 H O N 2 6 H 2 O + 5 CO N 2 H RP = H P - H R = 6 h - f H2O + 5 h - f CO2 - h- f fuel 14.3: H RP = h - f fuel = kj/kmol φ = ν O 2 stoich /ν O2 AC = 8/ = 0.74

336 A rigid vessel initially contains 2 kmol of carbon and 2 kmol of oxygen at 25 C, 200 kpa. Combustion occurs, and the resulting products consist of 1 kmol of carbon dioxide, 1 kmol of carbon monoxide, and excess oxygen at a temperature of 1000 K. Determine the final pressure in the vessel and the heat transfer from the vessel during the process. 2 C + 2 O 2 1 CO CO O 2 Process V = constant, C: solid, n 1(GAS) = 2, n 2(GAS) = 2.5 P 2 = P 1 n 2 T = 200 = kpa n 1 T H 1 = 0 H 2 = 1( ) + 1( ) + (1/2)( ) = kj 1 Q 2 = (U 2 - U 1 ) = (H 2 - H 1 ) - n 2 R- T 2 + n 1 R - T 1 = ( ) ( ) = kj In a test of rocket propellant performance, liquid hydrazine (N 2 H 4 ) at 100 kpa, 25 C, and oxygen gas at 100 kpa, 25 C, are fed to a combustion chamber in the ratio of 0.5 kg O 2 /kg N 2 H 4. The heat transfer from the chamber to the surroundings is estimated to be 100 kj/kg N 2 H 4. Determine the temperature of the products exiting the chamber. Assume that only H 2 O, H 2, and N 2 are present. The enthalpy of formation of liquid hydrazine is kj/kmol. 1 Liq. N 2 H 4 : 100 kpa, 25 o C Comb. 3 Gas O 2 : 100 kpa, 25 o Chamber Products C 2 ṁ O2 /ṁ N2H4 = 0.5 = 32ṅ O2 /32ṅ N2H4 and Q. /ṁ N2H4 = -100 kj/kg Energy Eq.: Q CV = H P - H R = = kj/kmol fuel 1 N 2 H O 2 H 2 O + H 2 + N 2 H R = 1(50417) + 1 (0) = kj 2 H P = h - H2O + h- H2 + h- N2 H P = h - H2O + h- H2 + h- = = N2 Table A.8 : 2800 K H P = K H P = Interpolate to get T P = 2854 K

337 Repeat the previous problem, but assume that saturated-liquid oxygen at 90 K is used instead of 25 C oxygen gas in the combustion process. Use the generalized charts to determine the properties of liquid oxygen. Problem same as 14.20, except oxygen enters at 2 as saturated liquid at 90 K. ṁ O2 /ṁ N2H4 = 0.5 = 32ṅ O2 /32ṅ N2H4 and Q. /ṁ N2H4 = -100 kj/kg Energy Eq.: Q CV = H P - H R = = kj/kmol fuel Reaction equation: 1 N 2 H O 2 H 2 O + H 2 + N 2 At 90 K, T r2 = 90/154.6 = h ~ f = 5.2 Figure D.2, (h - * - h - ) = = 6684 kj/kmol h - = ( ) = kj/kmol AT 2 H R = ( ) = kj, H P = st law: h - P = h- H2O + h- H2 + h- N2 = Q cv + H R - H P = From Table A.8, H P 2800K = , H P 3000K = Therefore, T P = 2804 K The combustion of heptane C 7 H 16 takes place in a SSSF burner where fuel and air are added as gases at P, T. The mixture has 125% theoretical air and the o o products are going through a heat exchanger where they are cooled to 600 K.Find the heat transfer from the heat exchanger per kmol of heptane burned. The reaction equation for stiochiometric ratio is: C 7 H 16 + v O2 (O N 2 ) => 7CO H 2 O + v O N 2 So the balance (C and H was done in equation) of oxygen gives v O2 = = 11, and actual one is = Now the actual reaction equation is: C 7 H O N 2 => 7CO H 2 O N O 2 To find the heat transfer take control volume as combustion chamber and heat exchanger H R + Q = H P => Q = H Po + H P - H Ro = H RPo + H P Q = (-44922) + 7(12906) + 8(10499) (8894) (9245) = = kj/kmol fuel

338 Ethene, C 2 H 4, and propane, C 3 H 8, in a 1 1 mole ratio as gases are burned with 120% theoretical air in a gas turbine. Fuel is added at 25 C, 1 MPa and the air comes from the atmosphere, 25 C, 100 kpa through a compressor to 1 MPa and mixed with the fuel. The turbine work is such that the exit temperature is 800 K with an exit pressure of 100 kpa. Find the mixture temperature before combustion, and also the work, assuming an adiabatic turbine. φ = 1 C 2 H 4 + C 3 H O N 2 5 CO H 2 O N 2 φ = 1.2 C 2 H 4 + C 3 H O N 2 5 CO H 2 O O N kmol air per 2 kmol fuel C.V. Compressor (air flow) Energy Eq.: w c = h 2 - h 1 Entropy Eq.: s 2 = s 1 P r 2 = P r1 P 2 /P 1 = T 2 air = K w c = = kj/kg = kj/kmol air C.V. Mixing Chanber (no change in composition) ṅ air h - air in + ṅ Fu1 h- 1 in + ṅ Fu2 h- 2 in = (SAME) exit C - P F1 + C- P F2 T exit - T 0 = C- P air T 2 air - T exit C 2 H 4 : C - P F1 = 43.43, C 3 H 8 : C- P F2 = 74.06, C- P air = T exit = C- P air T 2 + C- P F1 + C- P F2 T 0 C - P F1 + C- P F C- = K P air Dew Point Products: y H 2O = = P H 2O = y H2O P tot = kpa T dew = C C.V. Turb. + combustor + mixer + compressor (no Q) w net = H in - H out = H R - H (800 K out so no liquid H2O) P 800 = h - fc 2 H4 + h- fc 3 H8-5 h- CO2-6 h- H2O h- O h- N2 = kj 2 kmol Fu w T = w net + w comp = kj 2 kmol Fu

339 One alternative to using petroleum or natural gas as fuels is ethanol (C 2 H 5 OH), which is commonly produced from grain by fermentation. Consider a combustion process in which liquid ethanol is burned with 120% theoretical air in an SSSF process. The reactants enter the combustion chamber at 25 C, and the products exit at 60 C, 100 kpa. Calculate the heat transfer per kilomole of ethanol, using the enthalpy of formation of ethanol gas plus the generalized charts. C 2 H 5 OH (O N 2 ) 2CO 2 + 3H 2 O + 0.6O N 2 Fuel: h - 0 f = kj/kmol for IG Products at 60 C, 100 kpa y H2O MIX = = n V MAX n V MAX => n V MAX = 4.0 > 3 No liq. T r = / = 0.58 h f = 5.23 from Figure D.2 (h - * - h - )LIQ = = kj/kmol H R = 1( ) = kj H P = 2( ) + 3( ) + 0.6(0+1032) (0+1020) = kj Q CV = H P - H R = kj Another alternative to using petroleum or natural gas as fuels is methanol, CH 3 OH, which can be produced from coal. Both methanol and ethanol have been used in automotive engines. Repeat the previous problem using liquid methanol as the fuel instead of ethanol. CH 3 OH (O N 2 ) 1 CO H 2 O O N 2 React at 25 o C, Prod at 60 o C = K, 100 kpa y H2O MAX = = n V MAX n V MAX => n V MAX = 2.0 > 2 No liq. CH 3 OH: h - o f = kj, T C = K T r = / = h ~ = 5.22 Figure D.2 f (h -* -h - ) LIQ = = H R = 1 h - LIQ = = kj H P = 1( ) + 2( ) + 0.3(1032) (1020) = kj Q = H P - H R = kj

340 Another alternative fuel to be seriously considered is hydrogen. It can be produced from water by various techniques that are under extensive study. Its biggest problem at the present time is cost, storage, and safety. Repeat Problem using hydrogen gas as the fuel instead of ethanol. H O N 2 1 H 2 O O N 2 Products at 60 C, 100 kpa y H2O MAX = = n V MAX n V MAX Solving, n V MAX = < 1 => n V = 0.587, n LIQ = H R = = 0 H P = 0.413( ( )) ( ) + 0.1( ) ( ) = kj Q CV = H P - H R = kj Hydrogen peroxide, H 2 O 2, enters a gas generator at 25 C, 500 kpa at the rate of 0.1 kg/s and is decomposed to steam and oxygen exiting at 800 K, 500 kpa. The resulting mixture is expanded through a turbine to atmospheric pressure, 100 kpa, as shown in Fig. P Determine the power output of the turbine, and the heat transfer rate in the gas generator. The enthalpy of formation of liquid H 2 O 2 is kj/kmol. H 2 O 2 H 2 O O 2 ṅ H2O2 = ṁ H2O2 M = 0.1 = kmol/s ṅ MIX = ṅ H2O2 1.5 = kmol/s C - P0 MIX = = C - V0 MIX = = 24.0 => k MIX = /24.0 = CV: turbine. Assume reversible s 3 = s 2 T 3 = T 2 ( P 3 P 2 ) k-1 k = 800( ) = K w = C - P0 (T 2 - T 3 ) = ( ) = 8765 kj/kmol Ẇ CV = = kw CV: Gas Generator Ḣ 1 = ( ) = Ḣ 2 = ( ) ( ) = Q. CV = Ḣ 2 - Ḣ 1 = = kw

341 In a new high-efficiency furnace, natural gas, assumed to be 90% methane and 10% ethane (by volume) and 110% theoretical air each enter at 25 C, 100 kpa, and the products (assumed to be 100% gaseous) exit the furnace at 40 C, 100 kpa. What is the heat transfer for this process? Compare this to an older furnace where the products exit at 250 C, 100 kpa. 0.90CH C H o C 110% Air Furnace Prod. 40 o C 100 kpa 0.9 CH C 2 H O N CO H 2 O O N 2 H R = 0.9(-74873) + 0.1(-84740) = kj H P = 1.1( ) + 2.1( ) (441) (437) = kj assuming all gas Q CV = H P - H R = kj/kmol fuel b) T P = 250 o C H P = 1.1( ) + 2.1( ) (6808) (6597) = kj Q CV = H P - H R = kj/kmol fuel Repeat the previous problem, but take into account the actual phase behavior of the products exiting the furnace. Same as 14.28, except check products for saturation at 40 o C, 100 kpa y V MAX = = n V MAX n V MAX => Solving, n V MAX = n V = 0.814, n LIQ = = H LIQ = 1.286[ ( )] = kj H GAS = 1.1( ) ( ) (441) (437) = kj Q CV = H P - H R = ( ) = kj/kmol fuel b) no liquid, answer the same as

342 Methane, CH 4, is burned in an SSSF process with two different oxidizers: Case A: Pure oxygen, O 2 and case B: A mixture of O 2 + x Ar. The reactants are supplied at T 0, P 0 and the products should for both cases be at 1800 K. Find the required equivalence ratio in case (A) and the amount of Argon, x, for a stoichiometric ratio in case (B). a) Stoichiometric has ν = 2, actual has: CH 4 + νo 2 CO 2 + 2H 2 O + (ν - 2)O 2 H R = H P + H P 1800 => H P 1800 = - H RP = = h - CO2 = 79432, h- H2O = 62693, h- O2 = H P 1800 = ν = ν = 13.56, φ = 6.78 b) CH O 2 + 2x Ar CO 2 + 2H 2 O + 2x Ar H P 1800 = x ( ) = x = x x = Butane gas at 25 C is mixed with 150% theoretical air at 600 K and is burned in an adiabatic SSSF combustor. What is the temperature of the products exiting the combustor? o 25 C GAS C 4 H % Air 600 K Adiab. Comb. Q = 0 CV Prod. at T p C 4 H (O N 2 ) 4 CO H 2 O O N 2 Reactants: h - AIR = 9.75(9245) (8894) = kj ; h - C4H10 = h- o f IG = kj => H R = kj H P = 4( h - * CO2) + 5( h - * H2O) h - * O h - * N2 Energy Eq.: H P - H R = 0 4 h - * CO2 + 5 h - * H2O h - * O h - * N2 = Trial and Error: LHS 2000 K = , LHS 2200 K = Linear interpolation to match RHS => T P = 2048 K

343 In a rocket, hydrogen is burned with air, both reactants supplied as gases at P o, T o. The combustion is adiabatic and the mixture is stoichiometeric (100% theoretical air). Find the products dew point and the adiabatic flame temperature (~2500 K). The reaction equation is: H 2 + v O2 (O N 2 ) => H 2 O v O2 N 2 The balance of hydrogen is done, now for oxygen we need v O2 = 0.5 and thus we have 1.88 for nitrogen. y v = 1/(1+1.88) = => P v = = kpa = P g Table B.1.2: T dew = 72.6 C. H R = H P => 0 = h water h nitrogen Find now from table A.13 the two enthalpy terms At 2400 K : H P = = At 2600 K : H P = = Then interpolate to hit to give T = K Liquid butane at 25 C is mixed with 150% theoretical air at 600 K and is burned in an adiabatic SSSF combustor. Use the generalized charts for the liquid fuel and find the temperature of the products exiting the combustor. o 25 C LIQ. C 4 H % Air 600 K Adiab. Comb. Q = 0 CV Prod. at T p h - C4H10 = h- o f IG + (h - LIQ - h-* ) see Fig. D.2 = ( ) = kj C 4 H O N 2 T C = K T R = CO H 2 O O N 2 h - = 9.75(9245) (8894) = kj AIR H R = = kj H P = 4( h - * CO2) + 5( h - * H2O) h - * O h - * N2 Energy Eq.: H P - H R = 0 4 h - * CO2 + 5 h - * H2O h - * O h - * N2 = Trial and Error: LHS 2000 K = , LHS 2200 K = Linear interpolation to match RHS => T P = 2039 K

344 A stoichiometric mixture of benzene, C 6 H 6, and air is mixed from the reactants flowing at 25 C, 100 kpa. Find the adiabatic flame temperature. What is the error if constant specific heat at T 0 for the products from Table A.5 are used? C 6 H 6 + ν O 2 O ν O2 N 2 6CO 2 + 3H 2 O ν O 2 N 2 ν = 6 + 3/2 = 7.5 ν O 2 N2 = 28.2 H P =H P + H P = H R = H R H P = -H RP = = H P 2600K = , H P 2400K = T AD = 2529 K ν i C - = = Pi T = H P / ν i C - Pi = 2765 T = 3063 K, 21% high AD Liquid n-butane at T 0, is sprayed into a gas turbine with primary air flowing at 1.0 MPa, 400 K in a stoichiometric ratio. After complete combustion, the products are at the adiabatic flame temperature, which is too high, so secondary air at 1.0 MPa, 400 K is added, with the resulting mixture being at 1400 K. Show that T ad > 1400 K and find the ratio of secondary to primary air flow. C.V. Combustion Chamber. C 4 H O N 2 5 H 2 O + 4 CO N 2 Primary Air Fuel Combustion Chamber T AD Energy Eq.: H air + H fuel = H R = H P Secondary Air Mixing To turbine 1400 K H P + H P = H R + H R H P = H R + H R - H P H P = ( ) = H P 1400 = = < H P Try T AD > 1400: H P = K, H P = K C.V. Mixing Ch. Air Second: ν O2 s O N 2 Η P + ν O2 second H air = H P ν O2 second H air 1400 ν O2 second = H P - H P = H air H air = 9.3 ratio = ν O2 sec /ν O2 prim = 9.3/6.5 = 1.43

345 Consider the gas mixture fed to the combustors in the integrated gasification combined cycle power plant, as described in Problem If the adiabatic flame temperature should be limited to 1500 K, what percent theoretical air should be used in the combustors? Product CH 4 H 2 CO CO 2 N 2 H 2 O H 2 S NH 3 % vol Mixture may be saturated with water so the gases are ( H 2 S and NH 3 out) CH 4 H 2 CO CO 2 N 2 n y V MAX = 7.384/3000 = n V /(n V ) Solving, n V = 0.2 kmol, rest condensed {0.3 CH H CO CO N H 2 O x O x N 2 } 51.3 CO H 2 O (x - 1) O 2 + (135.0x + 0.8) N 2 For the fuel gas mixture, nc - = P0 MIX = nh - 0 f MIX = 0.3(-74873) (0) ( ) ( ) + 0.8(0) + 0.2( ) = kj At 40 C, for the fuel mixture: H MIX = (40-25) = kj Assume air enters at 25 C: h - AIR = 0 Products at 1500 K: H P = 51.3( ) ( ) (x - 1)( ) + (135x + 0.8)( ) = x 1st law: H P = H R = H MIX x = = or 238 % theo. air

346 Acetylene gas at 25 C, 100 kpa is fed to the head of a cutting torch. Calculate the adiabatic flame temperature if the acetylene is burned with a. 100% theoretical air at 25 C. b. 100% theoretical oxygen at 25 C. a) C 2 H O N 2 2 CO H 2 O N 2 H R = h - o f C2H2 = kj H P = 2( h - * CO2) + 1( h - * H2O) h - * N2 Q CV = H P - H R = 0 2 h - * CO2 + 1 h - * H2O h - * N2 = kj Trial and Error A.8: LHS 2800 = , LHS 3000 = Linear interpolation: T PROD = 2909 K b) C 2 H O 2 2 CO 2 + H 2 O H R = kj ; H P = 2( h - * CO2) + 1( h - * H2O) 2 h - * CO2 + 1 h - * H2O = At 6000 K (limit of A.8) = At 5900 K = or 19040/100 K change Difference, extrapolating T PROD / K Ethene, C 2 H 4, burns with 150% theoretical air in an SSSF constant-pressure process with reactants entering at P 0, T 0. Find the adiabatic flame temperature. C 2 H 4 + 3(O N 2 ) 2CO 2 + 2H 2 O N 2 C 2 H (O N 2 ) 2CO 2 + 2H 2 O O N 2 y H 2O = 2/( ) = => P v = T DEW = 43.8 C H P = H P + 2 h - CO2 + 2 h- H2O h- O h- N2 H R = h - f Fu H P + H P = H R H P = -H RP = = Initial guess based on N 2 from A.8 kj kmol Fu T 1 = 2100 K H P (2000) = , H P (1900) = => T AD 1950 K

347 Solid carbon is burned with stoichiometric air in an SSSF process. The reactants at T 0, P 0 are heated in a preheater to T 2 = 500 K as shown in Fig. P14.39, with the energy given by the product gases before flowing to a second heat exchanger, which they leave at T 0. Find the temperature of the products T 4, and the heat transfer per kmol of fuel (4 to 5) in the second heat exchanger. Control volume: Total minus last heat exchanger. Energy Eq.: C + O N 2 CO N 2 H R = H R = H P 3 = H P + H P 3 = h- f CO2 + h- CO h- N2 h - f CO2 = , H P = , H = P T 3 = T ad.flame = 2461 K Control volume: Total. Then energy equation: H R + Q = H P Q = H - RP = h - kj f CO2-0 = kmol fuel

348 A study is to be made using liquid ammonia as the fuel in a gas-turbine engine. Consider the compression and combustion processes of this engine. a. Air enters the compressor at 100 kpa, 25 C, and is compressed to 1600 kpa, where the isentropic compressor efficiency is 87%. Determine the exit temperature and the work input per kilomole. b. Two kilomoles of liquid ammonia at 25 C and x times theoretical air from the compressor enter the combustion chamber. What is x if the adiabatic flame temperature is to be fixed at 1600 K? Air P 1 = 100 kpa 1 2 COMP. P 2 = 1600 kpa T 1 = 25 o C η -W S COMP = 0.87 a) ideal compressor process (adiabatic reversible): s 2S = s 1 T 2S = T 1 ( P k-1 2 k ) = P 298.2( )0.286 = 659 K -w S = C P0 (T 2S - T 1 ) = 1.004( ) = Real process: -w = -w S /η S = 362.2/0.87 = kj/kg T 2 = T 1 - w/c P0 = /1.004 = 713 K b) Also -w = = kj/kmol 2 liq NH 3, 25 o C Air 100 kpa, 25 o C COMP COMB. CHAMBER -W -Q = 0 Prod. P P = 1600 kpa T P = 1600 K 2 NH x O x N 2 3 H 2 O + 1.5(x - 1) O 2 + (5.64x + 1) N 2 Using Tables A.16 and A.2, h - = ( ) = kj/kmol NH3 H R = 2(-66697) + 0 = kj -W = x = x kj H P = 3( ) + 1.5(x - 1)( ) + (5.64x + 1)( ) = x st law: H R = H P + W = = x x => x = 2.11

349 A closed, insulated container is charged with a stoichiometric ratio of oxygen and hydrogen at 25 C and 150 kpa. After combustion, liquid water at 25 C is sprayed in such that the final temperature is 1200 K. What is the final pressure? H O 2 H 2 O P: 1 H 2 O + x i H 2 O U 2 - U 1 = x i h - i = x i h f liq = 1 + x i H P - H R x i R - T P R- T R H R = φ, H P = = , h f liq= Substitute x i ( ) = = x i = P 1 V 1 = n R R - T 1, P 2 V 1 = n p R - T p P x i T P P 2 = 3 = 2 T 1 3 = 1657 kpa Wet biomass waste from a food-processing plant is fed to a catalytic reactor, where in an SSSF process it is converted into a low-energy fuel gas suitable for firing the processing plant boilers. The fuel gas has a composition of 50% methane, 45% carbon dioxide, and 5% hydrogen on a volumetric basis. Determine the lower heating value of this fuel gas mixture per unit volume. For 1 kmol fuel gas, 0.5 CH CO H O 2 ( ) CO H 2 O The lower heating value is with water vapor in the products. Since the 0.45 CO 2 cancels, h - = 0.5( ) ( ) - 0.5(-74873) (0) RP With = kj/kmol fuel gas n V = P/R- 100 T = = kmol/m LHV = = kj/m 3

350 Determine the lower heating value of the gas generated from coal as described in Problem Do not include the components removed by the water scrubbers. The gas from problem is saturated with water vapor. Lower heating value LHV has water as vapor. LHV = -H RP = H P - H R Only CH 4, H 2 and CO contributes. From the gas mixture after the scrubbers has ν i = 81.9 of composition: 0.3CH H CO + 10CO N H 2 O LHV = -[0.3H - RPCH H- RPH2 + 41H- RPCO ]/81.9 = -[0.3( ) ( ) + 41( )]/81.9 = kj kmol gas Propylbenzene, C 9 H 12, is listed in Table 14.3, but not in table A.9. No molecular weight is listed in the book. Find the molecular weight, the enthalpy of formation for the liquid fuel and the enthalpy of evaporation. M^ = = H fg = 384 kj/kg C 9 H O 2 9 CO H 2 O H - RP = P ν i h - fi - h- ffu h- ffu = ν i h - fi - h- P h - ffu = 9h- fco 2 + 6h - fh 2 O g + M^ (-41219) liq = kj/kmol Fu H2O vap RP Determine the higher heating value of the sample Wyoming coal as specified in Problem Note: For SO 2, h - 0 f = kj/kmol From solution 14.10, for 100 kg of coal there are 4.05 kmol CO 2, 1.75 kmol H 2 O, kmol SO 2 in the products. Therefore, h - RP0 = 4.05( ) ( ) ( ) = kj/100 kg coal So that HHV = kj/kg coal

351 Consider natural gas A and natural gas D, both of which are listed in Table Calculate the enthalpy of combustion of each gas at 25 C, assuming that the products include vapor water. Repeat the answer for liquid water in the products. Natural Gas A CH C 2 H C 3 H C 4 H O N CO H 2 O N 2 H R = 0.939(-74878) (-84740) ( ) ( ) = kj a) vapor H 2 O H P = 1.099( ) ( ) = h - RP = H P - H = kj/kmol R b) Liq. H 2 O H P = 1.099( ) ( ) = h - = kj/kmol RP Natural Gas D: CH C 2 H C 3 H C 4 H N 2 + O 2 + N CO H 2 O + N 2 H R = 0.543(-74873) (-84740) ( ) ( ) = kj a) vapor H 2 O H P = 1.651( ) ( ) = kj h - = kj/kmol RP b) Liq. H 2 O H P = 1.651( ) ( ) = kj h - = kj/kmol RP

352 Blast furnace gas in a steel mill is available at 250 C to be burned for the generation of steam. The composition of this gas is, on a volumetric basis, Component CH 4 H 2 CO CO 2 N 2 H 2 O Percent by volume Find the lower heating value (kj/m 3 ) of this gas at 250 C and ambient pressure. Of the six components in the gas mixture, only the first 3 contribute to the heating value. These are, per kmol of mixture: H 2, CH 4, CO For these components, H CH CO O H 2 O CO 2 The remainder need not be included in the calculation, as the contributions to reactants and products cancel. For the lower HV(water as vapor) at 250 C h - = 0.026( ) ( ) (0+6558) RP ( (250-25)) ( ) (0+6810) = kj kmol fuel v - 0 = R T o /P o = /100 = m 3 /kmol LHV = / = 1668 kj/m The enthalpy of formation of magnesium oxide, MgO(s), is kj/kmol at 25 C. The melting point of magnesium oxide is approximately 3000 K, and the increase in enthalpy between 298 and 3000 K is kj/kmol. The enthalpy of sublimation at 3000 K is estimated at kj/kmol, and the specific heat of magnesium oxide vapor above 3000 K is estimated at kj/kmol K. a. Determine the enthalpy of combustion per kilogram of magnesium. b. Estimate the adiabatic flame temperature when magnesium is burned with theoretical oxygen. a) Mg O 2 MgO(s) h COMB = h - COMB /M = h- f /M = /24.32 = kj/kg b) assume T R = 25 C and also that T P > 3000 K, (MgO = vapor phase) 1st law: Q CV = H P - H R = 0, but H R = 0 H P = h - f + (h h- 298 ) SOL + h- SUB + C- P VAP (T P ) = (T P ) = 0 Solving, T P = 4487 K

353 A rigid container is charged with butene, C 4 H 8, and air in a stoichiometric ratio at P 0, T 0. The charge burns in a short time with no heat transfer to state 2. The products then cool with time to 1200 K, state 3. Find the final pressure, P 3, the total heat transfer, 1 Q 3, and the temperature immediately after combustion, T 2. The reaction equation is, having used C and H atom balances: C 4 H 8 + ν O 2 O N 2 4 CO H 2 O ν O2 N 2 Counting now the oxygen atoms we get ν O 2 = 6. C.V. analysis gives: U 2 - U 1 = Q - W = Q = H 2 - H 1 - P 2 V 2 + P 1 V 1 = H 2 - H 1 - R - n 2 T 2 - n 1 T 1 H 2 - H 1 = H P H R = H P - H R + H P = M^ H RP + H P = = Where M^ = and n 1 = = 29.56, n 2 = = 30.56, Τable A.8 at 1200 K: h CO 2 = 44473, h H2O =34506, h N2 = Now solving for the heat transfer: Q = ( ) = To get the pressure, assume ideal gases: P 2 = n 2 R - T 2 V 2 = P 1 n 2 T 2 n 1 T 1 = kpa Before heat transfer takes place we have constant U so: U - U 1 a 1 = 0 = H 1a - H 1 - n 2 R- T + n 1 a 1 R- T 1 Now split the enthalpy H 1 a = H P + H P unknowns on LHS and knowns on RHS: T 1a and arrange things with the H P - n 2 R - T = H R - H P - n 1 R- T 1 = = Trial and error leads to: LHS (3000 K) = = LHS (3000 K) = = linear interpolation T = 3021 K kj kmol fuel

354 In an engine a mixture of liquid octane and ethanol, mole ratio 9 1, and stoichiometric air are taken in at T 0, P 0. In the engine the enthalpy of combustion is used so that 30% goes out as work, 30% goes out as heat loss and the rest goes out the exhaust. Find the work and heat transfer per kilogram of fuel mixture and also the exhaust temperature. 0.9 C 8 H C 2 H 5 OH O N 2 For 0.9 octane ethanol 8.4 H 2 O CO N 2 H - RP mix= 0.9H RP C8H H RP C2H5OH = kj kmol M^ mix = 0.9 M^ oct M^ alc = Energy: h - in + q in = h - ex + ω ex = h- ex + h - ex + ω ex h - ex - h - in = H - RP mix ω ex + h - ex - q in = -H- RP mix ω ex = -q in = 0.3 -H - RP = kj kmol = kj h - prod = h- ex = 0.4 -H - RP = kj kmol Fu kg Fu h - prod = 8.4 h- H2O h- CO h- N2 satisfied for T = 1216 K Consider the same situation as in the previous problem. Find the dew point temperature of the products. If the products in the exhaust are cooled to 10 C, find the mass of water condensed per kilogram of fuel mixture. Reaction equation with 0.9 octane and 0.1 ethanol is 0.9 C 8 H C 2 H 5 OH O N H 2 O CO N 2 y H 2O = = P H 2O = y H2O P tot = 14.3 kpa T = 52.9 C dew 10 C P H 2O = y H2O = = x x x = ν H 2O = m H 2O cond = - ν H 2O = kmol kmol Fu mix kmol kmol Fu mix

355 Calculate the irreversibility for the process described in Problem From solution 14.19, Reactants: 2 C(s) + 2 O 2 at 25 C, 200 kpa Product mixture: 1 CO CO O at 1000 K, kpa 2 & 1 Q 2 = kj Reactants: S R = 2(5.740) + 2( ln 200 ) = kj/k 100 Products: n i y i s - i -R - lny i P CO CO O S P = 1.0( ) + 1.0( ) + 0.5( ) = kj/k I = T 0 (S P - S R ) - 1 Q 2 = ( ) - ( ) = kj P 0 S - i

356 Pentane gas at 25 C, 150 kpa enters an insulated SSSF combustion chamber. Sufficient excess air to hold the combustion products temperature to 1800 K enters separately at 500 K, 150 kpa. Calculate the percent theoretical air required and the irreversibility of the process per kmol of pentane burned. C 5 H XO XN 2 5CO 2 + 6H 2 O + 8(X-1)O XN 2 1 st Law: Q cv + H R = H P + W cv ; W cv = 0, Q cv = 0 => H R = H P Reactants: C 5 H 12 : h o f from A.9 and h 500 for O 2 and N 2 from A.8 H R =(h o f) C5H12 + 8X h O X h N2 = X X 5911 = X H P = 5(h o f + h ) CO2 + 6(h o f + h ) H2O + 8(X-1) h O X h N2 = 5( ) + 6( ) + 8(X-1) X = X Energy Eq. solve for X; => X = b) Reactants: P i = 150 kpa, P o = 100 kpa, o s f n i y i s o f / s o R ln y i P i P o S - i kj kmol K C 5 H O 2 8X N X S R = n i S - = kj/k i Products: P e = 150 kpa, P o = 100 kpa n i y i s o R ln y i P e P o S - i kj kmol K CO H 2 O O 2 8(X-1) N X S P = n i S - = kj/k; i I = T o (S P - S R ) = ( ) = 990 MJ

357 Consider the combustion of methanol, CH 3 OH, with 25% excess air. The combustion products are passed through a heat exchanger and exit at 200 kpa, 40 C. Calculate the absolute entropy of the products exiting the heat exchanger per kilomole of methanol burned, using appropriate reference states as needed. CH 3 OH (O N 2 ) CO H 2 O O N 2 Products exit at 40 o C, 200 kpa, check for saturation: y V MAX = P G P = = n V MAX n V MAX n V = n V MAX = n LIQ = Gas mixture: n i y i s - i -R - ln y i P CO H 2 O O N S GAS MIX = n i S - = kj/k i s - = ( ) = LIQ S LIQ = = kj/k S PROD = = kj/k P 0 - S i

358 The turbine in Problem is adiabatic. Is it reversible, irreversible, or impossible? Inlet to the turbine is the exit from the mixing of air and fuel at 1 MPa. From solution to 14.23, we have: C - P C2H2 = 43.43, C- P C3H8 = 74.06, T turbine,in = K C 2 H 4 + C 3 H O N 2 S ex - S in = dq T + S gen = S gen φ 5 CO 2 +6 H 2 O O N 2 Inlet: 1 MPa, K S - Fu = S- i + C - P Fu ln T/T 0 n i y i s - i -R - ln y i P C 2 H C 3 H O N P 0 - S i S in = = n i y i s - i -R - ln y i P S - i CO H 2 O O N P 0 S ex = S gen = S ex - S in = = kj 2kmol Fu K > 0 kj 2kmol Fu K Possible, but one should check the state after combustion to account for generation by combustion alone and then the turbine expansion separately.

359 Saturated liquid butane enters an insulated constant pressure combustion chamber at 25 C, and x times theoretical oxygen gas enters at the same P and T. The combustion products exit at 3400 K. With complete combustion find x. What is the pressure at the chamber exit? and what is the irreversibility of the process? Butane: T 1 = T o = 25 o C, sat liq., x 1 = 0, T c = K, P c = 3.8 MPa Fig. D.1: T r1 = 0.7, P r1 = 0.1, P 1 = P r1 P c = 380 kpa * Figs. D.2 and D.3: h1 h 1 f = 4.85 R T c, (s * 1 - s 1 ) f = 6.8 R Oxygen: T 2 = T o = 25 o C, X - Theoretical O 2 Products: T 3 = 3400 K, Assumes complete combustion C 4 H XO 2 4CO 2 + 5H 2 O + 6.5(X-1)O 2 1 st Law: Q cv + H R = H P + W cv ; Q cv = 0, W cv = 0 H R = n(h o f + h ) C4H10 = 1( ) = kj Products: CO 2 H 2 O O 2 n(h o f + h ) CO2 = 4( ) = kj n(h o f + h ) H2O = 5( ) = kj n(h o f + h ) O2 = 6.5(X-1)( ) = (X-1) kj H P = n i (h o f + h ) i = X H P = H R solve for X; X = Assume that the exit pressure equals the inlet pressure: P e = P i = 380 kpa s C4 H 10 = [s o f - R ln P 1 P o - (s * 1 - s 1 ) f] ; s O 2 = [s o - R ln P 1 P o ] S R = S C4H10 + S O2 = [ ] + [ ] = kj/k Products: n i y i s o i R ln y i P e P o S - i kj kmol K CO H 2 O O S P = n i S - = kj/k; i I = T o (S P - S R ) = ( ) = kj

360 An inventor claims to have built a device that will take kg/s of water from the faucet at 10 C, 100 kpa, and produce separate streams of hydrogen and oxygen gas, each at 400 K, 175 kpa. It is stated that this device operates in a 25 C room on 10-kW electrical power input. How do you evaluate this claim? Liq H 2 O 10 o C, 100 kpa kg/s -Ẇ CV = 10 kw H 2 gas O 2 gas each at 400 K 175 kpa T 0 = 25 o C H 2 O H O 2 H i - H e = [ ( )] (3027) = kj (S i - S e ) = [ ( )] - ( ln 1.75) ( ln 1.75) = W REV = (H i - H e ) - T 0 (S i - S e ) = ( ) = kj Ẇ REV = (0.001/18.015)( ) = kw. I = ẆREV - Ẇ CV = (-10) < 0 Impossible Two kilomoles of ammonia are burned in an SSSF process with x kmol of oxygen. The products, consisting of H 2 O, N 2, and the excess O 2, exit at 200 C, 7 MPa. a. Calculate x if half the water in the products is condensed. b. Calculate the absolute entropy of the products at the exit conditions. 2NH 3 + xo 2 3H 2 O + N 2 + (x - 1.5)O 2 Products at 200 o C, 7 MPa with n H2O LIQ = n H2O VAP = 1.5 a) y H2O VAP = P G /P = = x => x = b) S PROD = S GAS MIX + S H2O LIQ Gas mixture: n i y i s - i -R - ln(y i P/P 0 ) S - i H 2 O O N S GAS MIX = 1.5( ) ( ) + 1.0(185.67) = kj/k S H2O LIQ = 1.5[ ( )] = kj/k S PROD = = kj/k

361 Consider the SSSF combustion of propane at 25 C with air at 400 K. The products exit the combustion chamber at 1200 K. It may be assumed that the combustion efficiency is 90%, and that 95% of the carbon in the propane burns to form carbon dioxide; the remaining 5% forms carbon monoxide. Determine the ideal fuel air ratio and the heat transfer from the combustion chamber. Ideal combustion process, assumed adiabatic, excess air to keep 1200 K out. C 3 H 8 + 5x O x N 2 3 CO H 2 O + 5(x - 1) O x N 2 H R = x( ) x( ) = x H P = 3( ) + 4( ) + 5(x - 1)( ) x( ) = x 1st law: H P - H R = 0 Solving, x = FA IDEAL = 1/( ) = b) FA ACTUAL = /0.90 = C 3 H O N CO CO + 4 H 2 O O N 2 H R = ( ) ( ) = kj H P = 2.85( ) ( ) + 4( ) ( ) ( ) = kj Q CV = H P - H R = kj

362 Graphite, C, at P 0, T 0 is burned with air coming in at P 0, 500 K in a ratio so the products exit at P 0, 1200 K. Find the equivalence ratio, the percent theoretical air, and the total irreversibility. C + φb(o N 2 ) CO 2 + ( φ - 1) O φ N 2 H P = H R H P H R = H R - H P ( φ - 1) φ φ( ) S gen = S P - S R = ν - s - R - ln( y) P-R R: y O 2 = 0.21, y N2 = 0.79 = 0 - ( ) φ = P: y O 2 = , y N2 = 0.79, y CO2 = S P = = S R = ( ) = For the pressure correction the term with the nitrogen drops out (same y). R - -ν ln( y ) = ( ) = P-R S gen = = I = T 0 S gen = kj kmol C

363 A gasoline engine is converted to run on propane. Assume the propane enters the engine at 25 C, at the rate 40 kg/h. Only 90% theoretical air enters at 25 C such that 90% of the C burns to form CO 2, and 10% of the C burns to form CO. The combustion products also include H 2 O, H 2 and N 2, exit the exhaust at 1000 K. Heat loss from the engine (primarily to the cooling water) is 120 kw. What is the power output of the engine? What is the thermal efficiency? Propane: T 1 = 25 o C, ṁ = 40 kg/hr, M = kg/kmol Air: T 2 = 25 o C, 90% theoretical Air produces 90% CO 2, 10% CO Products: T 3 = 1000 K, CO 2, CO, H 2 O, H 2, N 2 C 3 H O N CO CO + 3.3H 2 O + 0.7H N 2 ṅ C3H8 = ṁ/(m 3600) = kmol/s 1 st Law: Q. + H R = H P + Ẇ ; Q. = -120 kw H R = n C3H8 h o f = kj Products: CO 2 - n CO2 (h o f + h ) = 2.7( ) = kj CO - n CO (h o f + h ) = 0.3( ) = kj H 2 O - n H2O (h o f + h ) = 3.3( ) = kj H 2 - n H2 (h o f + h ) = 0.7( ) = kj N 2 - n N2 (h o f + h ) = 16.92( ) = kj H P = n i (h o f + h ) i = kj Ẇ = Q. + ṅ(h R - H P ) = kw C 3 H 8 : Table 14.3 H RPo = kj/kg HHV = ṅ C3H8 M(-H RPo ) = kw η th = Ẇ/HHV = 0.339

364 A small air-cooled gasoline engine is tested, and the output is found to be 1.0 kw. The temperature of the products is measured to 600 K. The products are analyzed on a dry volumetric basis, with the result: 11.4% CO 2, 2.9% CO, 1.6% O 2 and 84.1% N 2. The fuel may be considered to be liquid octane. The fuel and air enter the engine at 25 C, and the flow rate of fuel to the engine is kg/s. Determine the rate of heat transfer from the engine and its thermal efficiency. a C 8 H 18 + b O b N CO CO + c H 2 O O N 2 b = = 22.37, a = 1 ( ) = c = 9a = C 8 H O N CO CO + 9 H 2 O O N 2 H R = h -0 f C8H18 = kj H P = 6.38( ) ( ) + 9( ) ( ) ( ) = kj H P - H R = ( ) = kj Ḣ P - Ḣ R = ( /114.23)( ) = kw Q. = = kw CV Q. = (47893) = kw H η TH = Ẇ NET /Q. = 1.0/7.184 = H A gasoline engine uses liquid octane and air, both supplied at P 0, T 0, in a stoichiometric ratio. The products (complete combustion) flow out of the exhaust valve at 1100 K. Assume that the heat loss carried away by the cooling water, at 100 C, is equal to the work output. Find the efficiency of the engine expressed as (work/lower heating value) and the second law efficiency. C 8 H 18 + ν O 2 O N 2 8 CO H 2 O + 47 N 2 LHV = ν O 2 = ν O2 = 12.5 kj kg fuel LHV = kj kmol fuel H P 1100 = = C.V. Total engine

365 14-41 H in = H ex + W + Q loss = H ex + 2 W 2 W = H in - H ex = H R - H ν = -H RP + H R - H P 1100 W = = = kj kmol fuel W η th = LHV = = Find entropies in and out: inlet: S - Fu = S - O 2 = ln = S N = ln = S - in = = exit: S - CO2 = ln 8 64 = S - H2O = ln 9 64 = S - 47 N = ln 2 64 = S - ex = = Assume the same Q loss out to 100 C reservoir in the reversible case and compute Q rev 0 : S - in + Qrev 0 /T 0 = S - ex + Q loss /T res Q rev 0 = T 0 ( S - ex - S- in ) + Q loss T 0 /T res = ( ) / = kj kmol fuel H in + Q rev 0 = H ex + W rev + Q loss W rev = H in - H ex - Q loss + Q rev 0 = W ac + Q rev 0 = kj kmol fuel η II = W ac / W rev = / = 0.414

366 In Example 14.16, a basic hydrogen oxygen fuel cell reaction was analyzed at 25 C, 100 kpa. Repeat this calculation, assuming that the fuel cell operates on air at 25 C, 100 kpa, instead of on pure oxygen at this state. Anode: 2 H 2 4 e H + Cathode: 4 H e - 1 O H 2 O Overall: 2 H O 2 2 H 2 O Example 14.16: G 25 C = kj (for pure O 2 ) For P O2 = : S - = ln0.21 = O2 S = 2(69.950) 2( ) 1( ) = kj/k G 25 C = ( ) = kj E = /( ) = V

367 Consider a methane-oxygen fuel cell in which the reaction at the anode is CH 4 + 2H 2 O CO 2 + 8e- + 8H+ The electrons produced by the reaction flow through the external load, and the positive ions migrate through the electrolyte to the cathode, where the reaction is 8 e- + 8 H+ + 2 O 2 4 H 2 O a. Calculate the reversible work and the reversible EMS for the fuel cell operating at 25 C, 100 kpa. b. Repeat part (a), but assume that the fuel cell operates at 600 K instead of at room temperature. CH 4 + 2H 2 O CO 2 + 8e - + 8H + and 8e - + 8H + + 2CO 2 4H 2 O Overall CH 4 + 2O 2 CO 2 + 2H 2 O a) 25 o C assume all liquid H 2 O and all comp. at 100 kpa H 0 25 C = ( ) (-74873) 0 = kj S 0 25 C = (69.950) ( ) = kj/k G 0 25 C = ( ) = kj W rev = - Ḡ 0 = kj E 0 - Ḡ 0 = = = 1.06 V b) 600 K assume all gas H 2 O and all comp. at 100 kpa H K = 1( ) + 2( ) - 2( ) - 1[ ( )] = kj S K = 1( ) + 2( ) - 1( ln ) - 2( ) = kj/k G K = H K - T S K = (4.9606) = kj W rev = kj E 0 = = V

368 14-44 Advanced Problems A gas mixture of 50% ethane and 50% propane by volume enters a combustion chamber at 350 K, 10 MPa. Determine the enthalpy per kilomole of this mixture relative to the thermochemical base of enthalpy using Kay s rule. h -* = 0.5(-84740) + 0.5( ) = kj/kmol MIX O C - = = P0 MIX h - * h - * 298 = ( ) = 3294 kj/kmol Kay s rule: T C MIX = = K P C MIX = = MPa T r = 350/337.6 = 1.037, P r = 10/4.565 = 2.19 From Fig. D.2: h - * - h - = = 9909 kj/kmol h - = = kj/kmol MIX 350K,10MPa

369 A mixture of 80% ethane and 20% methane on a mole basis is throttled from 10 MPa, 65 C, to 100 kpa and is fed to a combustion chamber where it undergoes complete combustion with air, which enters at 100 kpa, 600 K. The amount of air is such that the products of combustion exit at 100 kpa, 1200 K. Assume that the combustion process is adiabatic and that all components behave as ideal gases except the fuel mixture, which behaves according to the generalized charts, with Kay s rule for the pseudocritical constants. Determine the percentage of theoretical air used in the process and the dew-point temperature of the products. Reaction equation: Fuel mix: h - 0 f FUEL = 0.2(-74873) + 0.8(-84740) = kj/kmol C - P0 FUEL = = h - * FUEL = (65-25) = 1989 kj/kmol T CA = K, T CB = K T c mix =282.4 K P CA = 4.88, P CB =4.60 P c mix = MPa T r = 338.2/282.4 = 1.198, P r = 10/4.824 = (h -* - h - ) FUEL IN = = 5119 h - kj FUEL IN = = kmol 1st law: 1.8( ) + 2.8( ) + 3.2(x - 1)(29761) + (12.03x)(28109) (3.2x)(9245) - (12.03x)(8894) = 0 a) x = or % b) n P = ( ) = y H2O = 2.8/ = P V /100 ; P V = 4.38 kpa, T = 30.5 C

370 Gaseous propane mixes with air, both supplied at 500 K, 0.1 MPa. The mixture goes into a combustion chamber and products of combustion exit at 1300 K, 0.1 MPa. The products analyzed on a dry basis are 11.42% CO 2, 0.79% CO, 2.68% O 2, and 85.11% N 2 on a volume basis. Find the equivalence ratio and the heat transfer per kmol of fuel. C 3 H 8 + α O α N 2 β CO 2 + γ H 2 O α N 2 β = 3, γ = 4, α = β + γ 2 = 5 ( A/F) S = 4.76α = 23.8 φ = ( A/F )/( A/F) S = % Theoretical Air = % h P = h P + ν j h(1300 K) q = h P - h R = h P - h R + ν j h(1300 K) = - H + ν j h(1300 K) ν j h(1300 K) = q = kj kmol Fu kj kmol Fu

371 A closed rigid container is charged with propene, C 3 H 6, and 150% theoretical air at 100 kpa, 298 K. The mixture is ignited and burns with complete combustion. Heat is transferred to a reservoir at 500 K so the final temperature of the products is 700 K. Find the final pressure, the heat transfer per kmole fuel and the total entropy generated per kmol fuel in the process. C 3 H 6 + ν O 2 O N 2 3 CO H 2 O + x N 2 Oxygen, O 2, balance: 2 ν O 2 = = 9 ν O2 = 4.5 Actual Combustion: φ = 1.5 ν O2 = = 6.75 ac C 3 H O N 2 3 CO H 2 O N O 2 P 2 = P 1 n p T 2 n R T 1 = = kpa H P 700 = = U 2 - U 1 = 1 Q 2-0 = H 2 - H 1 - n 2 R T 2 + n 1 R T 1 1 Q 2 = H RP + H P n P R T 2 + n 1 R T 1 = = kj kmol fuel Reactants: n i y i s - i -R - ln(y i P/P 0 ) S - i C 3 H O N S 1 = = 6577 kj kmol fuel K Products: n i y i s - i -R - ln(y i P/P 0 ) S - i CO H 2 O O N S 2 = 3( ) = 7421 kj/kmol fuel K 1 S 2 gen = S 2 - S 1-1 Q 2 /T res = = kj kmol fuel K kj kmol fuel

372 Consider one cylinder of a spark-ignition, internal-combustion engine. Before the compression stroke, the cylinder is filled with a mixture of air and methane. Assume that 110% theoretical air has been used, that the state before compression is 100 kpa, 25 C. The compression ratio of the engine is 9 to 1. a. Determine the pressure and temperature after compression, assuming a reversible adiabatic process. b. Assume that complete combustion takes place while the piston is at top dead center (at minimum volume) in an adiabatic process. Determine the temperature and pressure after combustion, and the increase in entropy during the combustion process. c. What is the irreversibility for this process? 1 CH O N 2 1 CO H 2 O O H 2 O P 1 = 100 kpa, T 1 = K, V 2 /V 1 = 1/8, Rev. Ad. s 2 = s 1 Assume T 2 ~ 650 K T AVE ~ 475 K Table A.6: C - P0 CH4 = , C- P0 O2 = , C- P0 N2 = C - = ( )/11.47 = P0 MIX C - V0 MIX = C- P0 - R- = , k = C - P0 /C- V0 = a) T 2 = T 1 (V 1 /V 2 ) k-1 = (9) = K (avg OK) P 2 = P 1 (V 1 /V 2 ) k = 100 (9) = 2011 kpa b) comb. 2-3 const. vol., Q = 0 2 Q 3 = 0 = (H 3 - H 2 ) - R- (n 3 T 3 - n 2 T 2 ) H 2 = 1 h - 0 f CH4 + n 2 C - P0 MIX (T 2 - T 1 ) H 2 = ( ) = kj H 3 = 1( h - * CO2) + 2( h - * H2O) h - * O h - * N2 Substituting, 1 h - * CO2 + 2 h - * H2O h - * O h - * N T = 0 Trial & error: T 3 = 2907 K OK n 3 T 3 T 3 P 3 = P 2 = P n 2 T 2 = = 8772 kpa 2 T

373 14-49 c) state 1 REAC n i y i s - i - R - ln(y i P/P 0 ) S - i CH O N S 2 = S 1 = n i S - = kj/k i state 3 PROD n i y i s - i - R - ln(y i P/P 0 ) S - i CO H 2 O O N S 3 = n i S - = kj/k i I = T 0 (S 3 - S 2 ) = 298.2( ) = kj

374 Consider the combustion process described in Problem a. Calculate the absolute entropy of the fuel mixture before it is throttled into the combustion chamber. b. Calculate the irreversibility for the overall process. From solution 14.67, fuel mixture 0.8 C 2 H CH 4 at 65 C, 10 MPa C - P0 FUEL = kj/kmol K. Using Kay s rule: T r1 = 1.198, P r1 = and x = % theoretical air or O N 2 in at 600 K, 100 kpa and 1.8CO H 2 O O N 2 out at 100 kpa, 1200 K a) s -* 0 FUEL = 0.2( ) + 0.8( ) (0.2 ln ln 0.8) = s * TP = ln ln = From Fig. D.3: (s -* -s - ) FUEL = = s - = = kj/kmol K FUEL b) Air at 600 K, 100 kpa n i y i s - i -R - ln(y i P/P 0 ) S - i O N S AIR = n i S - = kj/k i S R = = kj/k Products at 1200 K, 100 kpa PROD n i y i s -o i -R - ln(y i P/P 0 ) S - i CO H 2 O O N S P = n i S - = kj/k i I = T 0 (S P - S R ) - Q CV = ( ) + 0 = kj

375 Liquid acetylene, C 2 H 2, is stored in a high-pressure storage tank at ambient temperature, 25 C. The liquid is fed to an insulated combustor/steam boiler at the steady rate of 1 kg/s, along with 140% theoretical oxygen, O 2, which enters at 500 K, as shown in Fig. P The combustion products exit the unit at 500 kpa, 350 K. Liquid water enters the boiler at 10 C, at the rate of 15 kg/s, and superheated steam exits at 200 kpa. a.calculate the abs olute entropy, per kmol, of liquid acetylene at the s tor age tank state. b. Determine the phase(s) of the combustion products exiting the combustor boiler unit, and the amount of each, if more than one. c. Determine the temperature of the steam at the boiler exit. a) C 2 H 2 : S - IG 25 C = T R 1 = 298.2/308.3 = => From Fig. D.1: P R1 = 0.82 P 1 = = 5.03 MPa, ( S -* - S - ) 1 = 3.33R- = S - liq T1 P1 = S- T0 P0 + T - R- ln P 1 /P + ( S - - S - * ) P1 T1 = b) 1 C 2 H O 2 2 CO H 2 O + 1 O 2 H 1 = (-3.56 R ) = kj H 2 = 3.5( ) = kj Products T 3 = 350 K = 76.8 C P G = 41.8 kpa kj kmol K y V max = P G P = = = n V max n V max n V max = = n V gas mix n liq = = Gas Mix = 2 CO H 2 O + 1 O 2 c) H liq = ( ( )) = kj 3 H gas mix = 2( ) ( ) = kj H 3 = H + H liq = = kj 3 gas mix3 H 3 - H 1 - H 2 = kj or Ḣ 3 - Ḣ 1 - Ḣ 2 = / = kw = ṁ H 2O h 4 - h 5 h 5 = = T = C 5

376 Natural gas (approximate it as methane) at a ratio of 0.3 kg/s is burned with 250% theoretical air in a combustor at 1 MPa where the reactants are supplied at T 0. Steam at 1 MPa, 450 C at a rate of 2.5 kg/s is added to the products before they enter an adiabatic turbine with an exhaust pressure of 150 kpa. Determine the turbine inlet temperature and the turbine work assuming the turbine is reversible. CH 4 + ν O 2 O N 2 CO H 2 O N 2 2 ν O 2 = ν O2 = 2 => Actual ν O2 = = 5 CH O N 2 CO H 2 O + 3 O N 2 C.V. combustor and mixing chamber H R + n H 2O h- H2O in = H P ex n H 2O = ṅ H 2O ṅ Fu = Energy equation becomes ṁ H 2O M Fu = ṁ Fu M H 2O = kmol steam kmol fuel n H 2O h- ex - h- in H2O + h- CO2 + 2 h- H2O + 3 h- O h- N2 ex = -H RP = = h - ex - h- in H2O = h , so then: H2O ex h - CO h- H2O + 3 h- O h- N2 ex = Trial and error on T ex kj kmol fuel T ex = 1000 K LHS = ; T ex = 1100 K LHS = T ex = 1200 K LHS = => T ex 1139 K = T in turbine If air then T ex turbine 700 K and T avg 920 K. Find C - P mix between 900 and 1000K. From Table A.8: C - P mix = n i C- / Pi n (40.63) + 3(34.62) (32.4) i = = kj/kmol K C - V mix = C- P mix - R- = , k mix = T ex turbine = 1139 (150 / 1000) = 732 K H 732 = (15410) + 3(13567) (12932) = w T = H in - H ex = H in - H ex = = kj kmol fuel Ẇ T = ṅ Fu w T = m Fu w T /M^ Fu = ( ) = 8710 kw

377 Liquid hexane enters a combustion chamber at 31 C, 200 kpa, at the rate 1 kmol/s 200% theoretical air enters separately at 500 K, 200 kpa, and the combustion products exit at 1000 K, 200 kpa. The specific heat of ideal gas hexane is C = p 143 kj/kmol K. Hexane: T c = K, P c = 3010 kpa T r1 = 0.6, Fig. D.1: P rg = 0.028, P g1 = P r1 P c = kpa Figs D.2 and D.3: (h * 1 - h 1 ) f = 5.16 R T c, (s * 1 - s 1 ) f = 8.56 R Air: T 2 = 500 K, P 2 = 200 kpa, 200% theoretical air Products: T 3 = 1000 K, P 3 = 200 kpa a) h C6H14 = h o f - (h * 1 - h 1 ) f + (h * 1 - h * 0 ) + (h * 0 - h 0 ) * h0 - h 0 = 0, h * 1 - h * 0 = C P (T 1 - T o ) = 858 kj/kmol, h o f = kj/kmol * h1 - h 1 = = kj/kmol, h C6H14 = kj/kmol s C6H14 = s o T o + C P ln T 1 T o - R ln P 1 P o + (s 1 - s * 1 ) s o T o + C P ln T 1 T o - R ln P 1 P o = = kj/kmol-k s * 1 - s 1 = = kj/kmol-k, s C6H14 = kj/kmol-k b) C 6 H O N 2 6CO 2 + 7H 2 O + 9.5O N 2 T 3 T c prod = y i T ci = K, T r3 = = 5.58 Ideal Gas T c prod c) 1 st Law: Q + H R = H P + W; W = 0 => Q = H P - H R H R = (h ) C6H h O h N2 = = kj/kmol fuel H P = 6(h o f + h ) CO2 = 7(h o f + h ) H2O (h o f + h ) O (h o f + h ) N2 CO 2 - (h o f + h ) = ( ) = kj/kmol H 2 O - (h o f + h ) = ( ) = kj/kmol O 2 - (h o f + h ) = ( ) = kj/kmol

378 14-54 N 2 - (h o f + h ) = ( ) = kj/kmol H P = kj; Q. = kw d) İ = T o ṅ (S P - S R ) - Q. ; T o = 25 o C S R = (s ) C6H (s o R ln y O2 P 2 ) P O (s o R ln y N2 P 2 ) o P N2 o (s ) C6H14 = kj/kmol K, (s o 500 ) O2 = kj/kmol K (s o 500 ) N2 = kj/kmol K, y O2 = 0.21, y N2 = 0.79 Ṡ R = kw/k Products: n i y i s o i R ln y i P e P o - S i (kj/kmol-k) CO H 2 O O N S P = n i s i = kj/k;. I = To ṅ (S P - S R ) - Q. = kw

379 14-55 English unit problems 14.75E Pentane is burned with 120% theoretical air in a constant pressure process at 14.7 lbf/in 2. The products are cooled to ambient temperature, 70 F. How much mass of water is condensed per pound-mass of fuel? Repeat the answer, assuming that the air used in the combustion has a relative humidity of 90%. C 5 H (O N 2 ) 5 CO H 2 O O N 2 Products cooled to 70 F, 14.7 lbf/in 2 a) for H 2 O at 70 F: P G = lbf/in 2 y H2O MAX = P G P = = n H2O MAX n H2O MAX Solving, n H2O MAX = < n H2O Therefore, n H2O VAP = 1.066, n H2O LIQ = = m H2O LIQ = = lbm/lbm fuel b) P v1 = = lbf/in 2 w 1 = = n H2O IN = ( ) = lbmol n H2O OUT = = 7.04 n H2O LIQ = = lb mol n H2O LIQ = = lbm/lbm fuel

380 E The output gas mixture of a certain air-blown coal gasifier has the composition of producer gas as listed in Table Consider the combustion of this gas with 120% theoretical air at 14.7 lbf/in. 2 pressure. Find the dew point of the products and the mass of water condensed per pound-mass of fuel if the products are cooled 20 F below the dew point temperature? a) {3 CH H N O CO CO 2 } O N CO H 2 O O N 2 Products: y H2O = y H2O MAX = P G 14.7 = P G = lbf/in 2 T DEW PT = F b) At T = 90.4 F, P G = lbf/in 2 y H2O = = n H2O n H2O => n H2O = n H2O LIQ = = 9.49 lb mol 9.49(18) m H2O LIQ = 3(16)+14(2)+50.9(28)+0.6(32)+27(28)+4.5(44) = lbm/lbm fuel 14.77E Pentene, C 5 H 10 is burned with pure oxygen in an SSSF process. The products at one point are brought to 1300 R and used in a heat exchanger, where they are cooled to 77 F. Find the specific heat transfer in the heat exchanger. C 5 H 10 + ν O O CO H 2 O, stoichiometric: ν O2 = 7.5 Heat exchanger in at 1300 R, out at 77 F, so some water will condense. 5 H 2 O (5 - x)h 2 O liq + x H 2 O vap P g 77 y H = = O max P tot = = x 5 + x x = q = Q = 5 ṅ fuel h - ex - h- in CO2 + 5 h- ex - h- in - (5 -x)h - H2O vap fg H2O = Btu lb mol fuel

381 E A rigid vessel initially contains 2 pound mole of carbon and 2 pound mole of oxygen at 77 F, 30 lbf/in. 2. Combustion occurs, and the resulting products consist of 1 pound mole of carbon dioxide, 1 pound mole of carbon monoxide, and excess oxygen at a temperature of 1800 R. Determine the final pressure in the vessel and the heat transfer from the vessel during the process. 2 C + 2 O 2 1 CO CO O 2 V = constant, C: solid, n 1(GAS) = 2, n 2(GAS) = 2.5 n 2 T lbf P 2 = P 1 = 30 = n 1 T in2 H 1 = 0 H 2 = 1( ) + 1( ) + 1 ( ) = Btu 2 1 Q 2 = (U 2 -U 1 ) = (H 2 -H 1 ) - n 2 R- T 2 + n 1 R - T 1 = ( ) ( ) = Btu 14.79E In a test of rocket propellant performance, liquid hydrazine (N 2 H 4 ) at 14.7 lbf/in. 2, 77 F, and oxygen gas at 14.7 lbf/in. 2, 77 F, are fed to a combustion chamber in the ratio of 0.5 lbm O 2 /lbm N 2 H 4. The heat transfer from the chamber to the surroundings is estimated to be 45 Btu/lbm N 2 H 4. Determine the temperature of the products exiting the chamber. Assume that only H 2 O, H 2, and N 2 are present. The enthalpy of formation of liquid hydrazine is Btu/lb mole. 1 N 2 H 4 O 2 2 Comb. Chamber 3 Products N 2 H O 2 H 2 O + H 2 + N 2 n O2 /n Fu (m O2 32)/(m Fu 32) = 0.5 ; Q. CV /m Fu = 45 Q CV = = Btu lb mol fu C.V. combustion chamber: n Fu h- 1 + n O2 h- 2 + Q. CV = n tot h- 3 - or - H 1 + H 2 + Q CV = H P 3 => H R + Q CV = H P + H P 3 H P 3 = H R - H P + Q CV = = Trial and error on T 3 : T 3 = 5000 H P = , Btu lb mol fuel T 3 = 5200 H P = T 3 = 5133 R

382 E Repeat the previous problem, but assume that saturated-liquid oxygen at 170 R is used instead of 77 F oxygen gas in the combustion process. Use the generalized charts to determine the properties of liquid oxygen. Problem the same as 14.79, except oxygen enters at 2 as saturated liquid at 170 R. At 170 R, T r2 = = 0.61 h f = 5.1 From Fig. D.2: (h - * - h - ) = = 2822 Btu/lbmol H P 3 = H R + H R - H P + Q CV = (2822) + 0.5(0.219)( )(32) = With H P R = , H P R = T 3 = 5045 R

383 E Ethene, C 2 H 4, and propane, C 3 H 8, in a 1 1 mole ratio as gases are burned with 120% theoretical air in a gas turbine. Fuel is added at 77 F, 150 lbf/in. 2 and the air comes from the atmosphere, 77 F, 15 lbf/in. 2 through a compressor to 150 lbf/in. 2 and mixed with the fuel. The turbine work is such that the exit temperature is 1500 R with an exit pressure of 14.7 lbf/in. 2. Find the mixture temperature before combustion, and also the work, assuming an adiabatic turbine. C 2 H 4 + C 3 H 8 + ν O2 (O N 2 ) 5CO 2 + 6H 2 O + ν N2 N 2 φ = 1 ν O2 = 8 φ = 1.2 ν O2 = 9.6 so we have lbmol air per 2 lbmol fuel C 2 H 4 + C 3 H (O N 2 ) 5CO 2 + 6H 2 O O N 2 C.V. Compressor (air flow) w c,in = h 2 - h 1 s 2 = s 1 P r1 = P r2 = P r1 P 2 /P 1 = T 2 air = R w c,in = = Btu/lbm = Btu/lbmol air C.V. Mixing chamber n air h- air in + n fu1 h- fu1 + n fu2 h- fu2 = (same) exit (C - PF1 + C- PF2 )(T ex - T 0 ) = C- P air (T 2 air - T ex ) C - PF1 = 11.53, C- PF2 = 17.95, C- P air = C - P air T 2 + (C- PF1 + C- PF2 )T 0 T ex = C - PF1 + C- PF C- = R = T in combust P air Turbine work: take C.V. total and subtract compressor work. W total = H in - H out = H R - H P,1500 = h - f F1 + h - f F2-5h - CO2-6h- H2O h- N2-1.6h- O2 = (-44669) - 5( ) - 6( ) = Btu/2 lbmol Fuel w T = w tot + w c,in = = Btu/2 lbmol fuel

384 E One alternative to using petroleum or natural gas as fuels is ethanol (C 2 H 5 OH), which is commonly produced from grain by fermentation. Consider a combustion process in which liquid ethanol is burned with 120% theoretical air in an SSSF process. The reactants enter the combustion chamber at 77 F, and the products exit at 140 F, 14.7 lbf/in. 2. Calculate the heat transfer per pound mole of ethanol, using the enthalpy of formation of ethanol gas plus the generalized tables or charts. C 2 H 5 OH (O N 2 ) 2CO 2 + 3H 2 O + 0.6O N 2 Products at 140 F, 14.7 lbf/in 2, y H2O = 2.892/14.7 = n v /( n v ) n v = > 3 no condensation h - f = Btu/lbmol as gas T r = /925 = 0.58 D.2: h/rt c = 5.23 H R = = Btu/ lbmol fuel H P = 2( ) + 3( ) + 0.6(443.7) (438.4) = Btu/ lbmol fuel Q CV = H P - H R = Btu/lbmol fuel

385 E Hydrogen peroxide, H 2 O 2, enters a gas generator at 77 F, 75 lbf/in. 2 at the rate of 0.2 lbm/s and is decomposed to steam and oxygen exiting at 1500 R, 75 lbf/in. 2. The resulting mixture is expanded through a turbine to atmospheric pressure, 14.7 lbf/in. 2, as shown in Fig. P Determine the power output of the turbine, and the heat transfer rate in the gas generator. The enthalpy of formation of liquid H 2 O 2 is Btu/lb mol. H 2 O 2 H 2 O O 2 n Fu = m Fu /M = 0.2/ = lbmol/s Fu n ex,mix = 1.5 n = lbmol/s Fu C - p mix = = C - v mix = C- p mix = ; k mix = C- p mix /C- v mix = Reversible turbine T 3 = T 2 (P 3 /P 2 ) (k-1)/k = 1500 (14.7/75) = R w - = C - p (T 2 - T ) = ( ) = Btu/lbmol 3 W CV = n mix w- = = 34.9 Btu/s C.V. Gas generator Q CV = H 2 - H = ( ) (7297.5) (-80541) = Btu/s

386 E In a new high-efficiency furnace, natural gas, assumed to be 90% methane and 10% ethane (by volume) and 110% theoretical air each enter at 77 F, 14.7 lbf/in. 2, and the products (assumed to be 100% gaseous) exit the furnace at 100 F, 14.7 lbf/in. 2. What is the heat transfer for this process? Compare this to an older furnace where the products exit at 450 F, 14.7 lbf/in CH C H 2 6 Prod. 77 F 110% Air Furnace 100 F 14.7 lbf/in 2 H R = 0.9(-32190) + 0.1(-36432) = Btu 0.9 CH C 2 H O N CO H 2 O O N 2 a) T P = 100 F H P = 1.1( ) + 2.1( ) (162) (160) = Btu, assuming all gas Q CV = H P - H R = Btu/lb mol fuel b) T P = 450 F H P = 1.1( ) + 2.1( ) (2688) (2610) = Btu Q CV = H P - H R = Btu/lb mol fuel 14.85E Repeat the previous problem, but take into account the actual phase behavior of the products exiting the furnace. Same as 14.84, except possible condensation. a) 100 F, 14.7 lbf/in 2 y v max = / 14.7 = n v max /[n v max ] n v max = n v = 0.705; n liq = = H liq = 1.395[ ( )] = H gas = 1.1( ) ( ) (162) (160) = H P = H liq + H gas = Q CV = H P - H R = Btu/lbmol fuel b) Older furnace has no condensation, so same as in

387 E Methane, CH 4, is burned in an SSSF process with two different oxidizers: A. Pure oxygen, O 2 and B a mixture of O 2 + x Ar. The reactants are supplied at T 0, P 0 and the products in are at 3200 R both cases. Find the required equivalence ratio in case A and the amount of Argon, x, for a stoichiometric ratio in case B. a) CH 4 + νo 2 CO 2 + 2H 2 O + (ν - 2)O 2 ν s = 2 for stoichiometric mixture. H P 3200 = H R = H P + H P 3200 h - CO2 = 33579, h- H2O = 26479, h- O2 = H P = H R - H P = -H = (50010/2.326) = RP = (ν - 2)21860 = ν ν = φ = ν/ν s = 6.91 b) CH 4 + 2O 2 + 2xAr CO 2 + 2H 2 O + 2xAr H P = ( ) = x = x = E Butane gas at 77 F is mixed with 150% theoretical air at 1000 R and is burned in an adiabatic SSSF combustor. What is the temperature of the products exiting the combustor? C 4 H (O N 2 ) 4CO 2 + 5H 2 O O N 2 H R = H R + H air,in H P = H P + 4 h- CO2 + 5 h- H2O h- O h- N2 H P = H R H P = H R - H P + H air,in H P = -H RP + H air,in = = Btu/lbmol fuel = 4 h - CO2 + 5 h- H2O h- O h- N2 at T ad H P,3600R = H P,3800R = T ad = 3628 R

388 E Liquid n-butane at T, is sprayed into a gas turbine with primary air flowing at 150 lbf/in. 2, 700 R 0 in a stoichiometric ratio. After complete combustion, the products are at the adiabatic flame temperature, which is too high so secondary air at 150 lbf/in. 2, 700 R is added, with the resulting mixture being at 2500 R. Show that T > 2500 R and find the ratio of secondary to primary air flow. ad C 4 H (O N 2 ) 4CO 2 + 5H 2 O N 2 Fuel 2nd.air Primary T ad air COMBUSTOR MIXING 2500 R C.V. Combustor H R = H air + H Fu = H P = H P + H P = H R + H R H P = H R H P + H R = -H RP + H R = / ( ) = Btu/lbmol fuel H P,2500R = = H P > H P,2500R T ad > 2500 R (If iteration T ad 4400 R) C.V. Mixing chamber H P + ν O2 2nd H air,700 = H P,2500R + ν O2 2nd H air,2500r ν O2 2nd = H P - H P, = = H air, H air,700 Ratio = ν O2 2nd /ν O2 Prim. = 9.344/6.5 = 1.44

389 E Acetylene gas at 77 F, 14.7 lbf/in. 2 is fed to the head of a cutting torch. Calculate the adiabatic flame temperature if the acetylene is burned with 100% theoretical air at 77 F. Repeat the answer for 100% theoretical oxygen at 77 F. a) C 2 H O N 2 2 CO H 2 O N 2 H R = h -o f C2H2 = Btu H P = 2( h -* CO2 ) + 1( h-* H2O) h -* N2 Q CV = H P - H R = 0 2 h -* CO2 + 1 h-* H2O h -* N2 = Btu Trial and Error: T PROD = 5236 R = OK b) C 2 H O 2 2 CO 2 + H 2 O H R = Btu H P = 2( h - * CO2) + 1( h - * H2O) 2 h - * CO2 + 1 h - * H2O = At R (limit of C.7): = At 9500 R: = or 4507/100 R change, Difference, extrapolating T PROD R 14.90E Ethene, C 2 H 4, burns with 150% theoretical air in an SSSF constant-pressure process with reactants entering at P, T. Find the adiabatic flame temperature. 0 0 C 2 H (O N 2 ) 2CO 2 + 2H 2 O + 1.5O N 2 H P = H R = H P + H P = H R H P = H R - H P = -H = /2.326 = RP H P = 2 h - CO2 + 2 h- H2O h- O h- N2 Trial and error on T ad... H P,3400R = H P,3600R = T ad = 3510 R

390 E Solid carbon is burned with stoichiometric air in an SSSF process, as shown in Fig. P The reactants at T, P are heated in a preheater to T = 900 R with the energy given by the products 0 0 before flowing to a second heat 2 exchanger, which they leave at T. Find the temperature of the products T, and the heat transfer per lb mol of fuel 0 (4 to 5) in the second heat exchanger. 4 a) Following the flow we have: Inlet T 1, after preheater T 2, after mixing and combustion chamber T 3, after preheater T 4, after last heat exchanger T 5 = T 1. b) Products out of preheater T 4. Control volume: Total minus last heat exchanger. C + O N 2 CO N 2 Energy Eq.: H R = H R = H = P 3 H P + H = h- P 3 f CO2 + h- CO h- N2 h - = , H f CO 2 P = , H P = T 3 = T ad.flame = 4430 R c) Control volume total. Then energy equation: H R + q - = H P q - = H RP = h - f CO2-0 = Btu lbmol fuel 14.92E A closed, insulated container is charged with a stoichiometric ratio of oxygen and hydrogen at 77 F and 20 lbf/in. 2. After combustion, liquid water at 77 F is sprayed in such that the final temperature is 2100 R. What is the final pressure? H O 2 H 2 O ; P: 1 H 2 O + x i H 2 O U 2 - U 1 = x i h - i = x i h- f liq = 1 + x i H P - H R x i R- T P R- T R H R = φ, H P = = , h - f liq = Substitute x i ( ) = ( )= x i = P 1 V 1 = n R R - T 1, P 2 V 2 = n p R - T p P 2 = P 1 (1 + x i )T P = 20(4.187)(2100) = lbf/in 1.5 T 1 1.5(536.67) 2

391 E Blast furnace gas in a steel mill is available at 500 F to be burned for the generation of steam. The composition of this gas is, on a volumetric basis, Component CH 4 H 2 CO CO 2 N 2 H 2 O Percent by volume Find the lower heating value (Btu/ft 3 ) of this gas at 500 F and P. 0 Of the six components in the gas mixture, only the first 3 contribute to the heating value. These are, per lb mol of mixture: H 2, CH 4, CO For these components, H CH CO O H CO 2 The remainder need not be included in the calculation, as the contributions to reactants and products cancel. For the lower HV(water vapor) at 500 F h - = 0.026( ) ( ) ( ) RP ( (500-77)) ( ) ( ) = Btu/lb mol fuel v - R - T 0 = = P = ft3 /lb mol LHV = / = Btu/ft 3

392 E Two pound moles of ammonia are burned in an SSSF process with x lb mol of oxygen. The products, consisting of H 2 O, N 2, and the excess O 2, exit at 400 F, 1000 lbf/in. 2. a. Calculate x if half the water in the products is condensed. b. Calculate the absolute entropy of the products at the exit conditions. 2NH 3 + xo 2 3H 2 O + N 2 + (x - 1.5)O 2 Products at 400 F, 1000 lbf/in 2 with n H2O LIQ = n H2O VAP = 1.5 P G a) y H2O VAP = P = = x x = b) S PROD = S GAS MIX + S H2O LIQ Gas mixture: n i y i s - i -R - i P lny - S i H 2 O O N S GAS MIX = 1.5(43.335) (45.040) + 1.0(44.249) = Btu/R S H2O LIQ = 1.5[ ( )] = Btu/R S PROD = = Btu/R P 0

393 E Graphite, C, at P, T is burned with air coming in at P, 900 R in a ratio so the products exit at P 0, R. Find the equivalence ratio, the 0 percent theoretical air and the total irreversibility. 0 C + φ(o N 2 ) CO 2 + (φ - 1)O φN 2 H P = H R H P, H R = H R - H P = -H RP (φ - 1) φ φ( ) = 0 - ( ) φ = φ = or 342 % theoretical air Equivalence ratio = 1/φ = S gen = s P - s R = P ν i (s i - R - ln y i ) - R ν i (s i - R - ln y i ) R: y = 0.21 y O 2 N2 = 0.79 P: y = y O 2 N2 = 0.79 y CO2 = S gen = ( ) ( ) ( ( )) = Btu/lbmol C R I = T 0 S gen = Btu/lbmol C

394 E A small air-cooled gasoline engine is tested, and the output is found to be 2.0 hp. The temperature of the products is measured and found to be 730 F. The products are analyzed on a dry volumetric basis, with the following result 11.4% CO 2, 2.9% CO, 1.6% O 2 and 84.1% N 2. The fuel may be considered to be liquid octane. The fuel and air enter the engine at 77 F, and the flow rate of fuel to the engine is 1.8 lbm/h. Determine the rate of heat transfer from the engine and its thermal efficiency. a C 8 H 18 + b O b N CO CO + c H 2 O O N 2 b = 84.1/3.76 = 22.37, a = (1/8)( ) = 1.788, c = 9a = C 8 H O N CO CO + 9 H 2 O O N 2 a) H R = h - = Btu f C8H18 H P = 6.38( ) ( ) + 9( ) (0+4822) ( ) = Btu H P -H R = ( ) = Btu 1.8 Ḣ P -Ḣ R = ( ) = Btu/h Q. = (2544) = Btu/h CV b) Q. = = H Ẇ NET = = 5088 ; η TH = = 0.137

395 E A gasoline engine uses liquid octane and air, both supplied at P, T, in a stoichiometric ratio. The products (complete combustion) flow out of 0 the exhaust 0 valve at 2000 R. Assume that the heat loss carried away by the cooling water, at 200 F, is equal to the work output. Find the efficiency of the engine expressed as (work/lower heating value) and the second law efficiency. C 8 H (O N 2 ) 8CO 2 + 9H 2 O + 47N 2 LHV = /2.326 = Btu/lbmol fuel H P,2000 = = C.V. Total engine: H in = H ex + W + Q loss = H ex + 2W W = (H in - H ex )/2 = (H R - H P )/2 = (-H RP - H P,2000 )/2 = ( )/2 = Btu/lbmol fuel η TH = W/LHV = / = For 2nd law efficiency we must find reversible work S - in = s- fuel (s- O s- N2 ) = [ ln (1/4.76)] = Btu/lbmol fuel R + 47[ ln (3.76/4.76)] S - ex = 8s- CO2 + 9s- H2O + 47s- = 8[ ln (8/64)] N2 = Btu/lbmol fuel R + 9[ ln (9/64)] + 47[ ln (47/64)] Assume the same Q loss out to 200 F = R reservoir and compute Q 0 rev : S - in + Q rev /T 0 0 = S - ex + Q loss /T res Q rev 0 = T 0 (S - ex - S- in ) + Q loss T 0 /T res = ( ) *536.67/ = Btu/lbmol fuel W rev = H in - H ex - Q loss + Q rev rev 0 = W ac + Q 0 = = η II = W ac /W rev = / = 0.419

396 EIn Example 14.16, a basic hydrogen oxygen fuel cell reaction was analyzed at 25 C, 100 kpa. Repeat this calculation, assuming that the fuel cell operates on air at 77 F, 14.7 lbf/in. 2, instead of on pure oxygen at this state. Anode: 2 H 2 4 e H + Cathode: 4 H e - 1 O H 2 O Overall: 2 H O 2 2 H 2 O Example : G 25 C = kj/kmol Or G 77 F = Btu/lbmol P O2 = y O2 P = = lbf/in 2 s - O = ln 0.21 = S = 2(16.707) 2(31.186) 1(52.072) = Btu/R H = 2( ) 2(0) 1(0) = Btu G 77 F = (-81.03) = Btu E = - G/N 0 e n e = /( ) = V

397 15-1 CHAPTER 15 The correspondence between the new problem set and the previous 4th edition chapter 13 problem set. New Old New Old New Old 1 1 mod mod new new new new mod 10 new new new 32 new new 14 new new new 16 new new 19 new mod The problems that are labeled advanced start at number 53. The English unit problems are: New Old New Old New Old new new new

398 Carbon dioxide at 15 MPa is injected into the top of a 5-km deep well in connection with an enhanced oil-recovery process. The fluid column standing in the well is at a uniform temperature of 40 C. What is the pressure at the bottom of the well assuming ideal gas behavior? Z 1 Z 2 1 CO 2 2 (Z 1 -Z 2 ) = 5000 m, P 1 = 15 MPa T = 40 o C = const Equilibrium at constant T -w REV = 0 = g + PE = RT ln (P 2 /P 1 ) + g(z 2 -Z 1 ) = ln (P 2 /P 1 ) = = P 2 = 15 exp(0.8287) = MPa 15.2 Consider a 2-km-deep gas well containing a gas mixture of methane and ethane at a uniform temperature of 30 C. The pressure at the top of the well is 14 MPa, and the composition on a mole basis is 90% methane, 10% ethane. Determine the pressure and composition at the bottom of the well, assuming an ideal gas mixture. Z 1 1 (Z 1 -Z 2 ) = 2000 m, Let A = CH 4, B = C 2 H 6 P 1 = 14 MPa, y A1 = 0.90, y B1 = 0.10 GAS MIX A+B T = 30 o C = const Z 2 2 From section 15.1, for A to be at equilibrium between 1 and 2: W REV = 0 = n A (Ḡ A1 -Ḡ A2 ) + n A M A g(z 1 -Z 2 ) Similarly, for B: W REV = 0 = n B (Ḡ B1 -Ḡ B2 ) + n B M B g(z 1 -Z 2 ) Using eq for A: R - T ln (P A2 /P A1 ) = M A g(z 1 -Z 2 ) with a similar expression for B. Now, ideal gas mixture, P A1 = y A1 P, etc. Substituting: y A2 P 2 ln = M A g(z 1 -Z 2 ) y A1 P 1 R - T and ln (y A2 P 2 ) = ln(0.9 14) (2000) = => y A2 P 2 = y B2 P 2 ln = M B g(z 1 -Z 2 ) y B1 P 1 R - T ln (y B2 P 2 ) = ln(0.1 14) (2000) => y B2 P 2 = (1-y A2 )P 2 = Solving: P 2 = MPa & y A2 = =

399 Using the same assumptions as those in developing Eq. d in Example 15.1: Develop an expression for pressure at the bottom of a deep column of liquid in terms of the isothermal compressibility, β T. For liquid water at 20 C, β T = [1/MPa]. Use the result of the first question to estimate the pressure in the Pacific ocean at the depth of 3 km. d g T = v (1-β T P) dp T d g T + g dz = 0 v (1-β T P) dp T + g dz = 0 and integrate v (1-β T P) dp T = - g dz P (1-βT P) dp T = + g v +H 1 0 dz => P - P0 - β T 2 [P2 - P 2 0 ] = g v H P 0 P (1-1 2 β T P) = P β T P 02 + g v H v = v f 20 C = ; H = 3000 m, g = m/s 2 ; β T = /MPa P ( P) = [ / ] 10-6 = MPa, which is close to P P = MPa 15.4 Calculate the equilibrium constant for the reaction O 2 <=> 2O at temperatures of 298 K and 6000 K. Reaction O 2 <=> 2O At 25 o C ( K): H 0 = 2h - 0 f O - 1h - 0 f O2 = 2( ) - 1(0) = kj/kmol S 0 = 2s -0 O - 1s -0 O2 = 2( ) - 1( ) = kj/kmol K G 0 = H 0 - T S 0 = = kj/kmol G 0 ln K = - R - T = = At 6000 K: H 0 = 2( ) - ( ) = kj/kmol S 0 = 2( ) -1( ) = kj/kmol K G 0 = = kj/kmol ln K = =

400 Calculate the equilibrium constant for the reaction H 2 <=> 2H at a temperature of 2000 K, using properties from Table A.8. Compare the result with the value listed in Table A.10. h - H 2 = s - H 2 = h o f = 0 h - H = s- H = h o f = G 0 = H - T S = H RHS - H LHS T (S 0 RHS - S0 LHS ) = 2 ( ) ( ) = ln K = - G 0 /R - T = / ( ) = Table A.10 ln K = OK 15.6 Plot to scale the values of lnk versus 1/T for the reaction 2CO 2 <=> 2CO + O 2. Write an equation for lnk as a function of temperature. 2 CO 2 2 CO + 1 O 2 T(K) T ln K T(K) T ln K For the range below ~ 5000 K, ln K A + B/T almost linear Using values at 2000 K & 5000 K A = B = K x T _

401 Calculate the equilibrium constant for the reaction:2co 2 <=> 2CO + O 2 at 3000 K using values from Table A.8 and compare the result to Table A.10. h - CO = h- 0 f CO = s - CO = h - CO 2 = h -0 f CO 2 = s - CO 2 = h - O 2 = s - O 2 = G 0 = H - T S = 2 H CO + H O2 2 H CO2 - T (2s - CO + s- O 2-2s - CO 2 ) = 2 ( ) ( ) -3000( ) = ln K = - G 0 /R - T = / ( ) = Table A.10 ln K = OK 15.8 Pure oxygen is heated from 25 C to 3200 K in an SSSF process at a constant pressure of 200 kpa. Find the exit composition and the heat transfer. The only reaction will be the dissociation of the oxygen O 2 2O ; K(3200) = Look at initially 1 mol Oxygen and shift reaction with x n O2 = 1 - x; no = 2x; n tot = 1 + x; y i = n i /n tot y O 2 K = ( P ) y 02 P 2-1 4x 2 = 1 + x o (1 + x) x 2 = 8x2 1 - x 2 x 2 K/8 = x = ; y 1 + K/8 02 = 0.859; y 0 = q - = n 02 ex h- 0 2 ex + n 0ex h- Oex - h- 0 2 in = (1 + x)(y h - + y h - O ) - 0 h = ; h - O = q- = kj/kmol O 2 q = q - /32 = 4532 kj/kg (= if no reaction)

402 Pure oxygen is heated from 25 C, 100 kpa to 3200 K in a constant volume container. Find the final pressure, composition, and the heat transfer. As oxygen is heated it dissociates O 2 2O ln K eq = C. V. Heater: U 2 - U 1 = 1 Q 2 = H 2 - H 1 - P 2 v + P 1 v Per mole O 2 : 1 q- 2 = h- 2 - h- 1 + R- (T 1 - (n 2 /n 1 )T 2 ) Shift x in reaction 1 to have final composition: (1 - x)o 2 + 2xO n 1 = 1 n 2 = 1 - x + 2x = 1 + x y O2 2 = (1 - x)/(1 + x) ; y = 2x/(1 + x) O2 Ideal gas and V 2 = V 1 P 2 = P 1 n 2 T 2 /n 1 T 1 P 2 /P o = (1 + x)t 2 /T 1 K eq = e = y O 2 y 02 ( P 2 P o ) = ( 2x 1 + x )2 ( 1 + x 1 - x ) (1 + x 1 ) (T 2 T 1 ) 4x x = T 1 e T = x= (n O2 ) 2 = , (n O ) 2 = , n 2 = q- 2 = (106022) ( ) - Ø ( (3200)) = kj/kmolo 2 y O2 2 =0.9676/ = 0.937; y =0.0648/ = O Nitrogen gas, N 2, is heated to 4000 K, 10 kpa. What fraction of the N 2 is dissociated to N at this state? N 2 <=> 2 T = 4000 K, lnk = Initial 1 0 K = 3.14x10-6 Change -x 2x Equil. 1-x 2x n tot = 1 - x + 2x = 1 + x y N2 = 1 - x 1 + x, y N = 2x 1 + x y 2 N K = P y N2 P 2-1 ; => 3.14x10-6 = 4x2 o 1 - x => x = n N2 = , n N = , y N2 = , y N =

403 Hydrogen gas is heated from room temperature to 4000 K, 500 kpa, at which state the diatomic species has partially dissociated to the monatomic form. Determine the equilibrium composition at this state. H 2 2H Equil. n H2 = 1 - x -x +2x n H = 0 + 2x K = n = 1 + x (2x) 2 (1-x)(1+x) ( P P 0 ) 2-1 at 4000 K: ln K = => K = (500/100) x2 = = 2 Solving, x = x n H2 = 0.664, n H = 0.672, n tot = y H2 = 0.497, y H = Consider the chemical equilibrium involving H 2 O, H 2, CO, and CO 2, and no other substances. Show that the equilibrium constant at any temperature can be found using values from Table A.10 only. The reaction between the species is (1) 1 CO H 2 1 CO + 1 H 2 O From A.10 we find: (2) 2 H 2 O 2 H O 2, (3) 2 CO 2 2 CO + 1 O 2 Form (1) from (2) and (3) as: (1) = 1 2 (3) (2) => ln K = (ln K 3 - ln K 2 )/2 From A.10 ln K at any T is listed for reactions (2) and (3)

404 One kilomole Ar and one kilomole O 2 is heated up at a constant pressure of 100 kpa to 3200 K, where it comes to equilibrium. Find the final mole fractions for Ar, O 2, and O. The only equilibrium reaction listed in the book is dissociation of O 2. So assuming that we find in Table A.10: ln(k) = Ar + O 2 Ar + (1 - x)o 2 + 2x O The atom balance already shown in above equation can also be done as Species Ar O 2 O Start Change 0 -x 2x Total 1 1-x 2x The total number of moles is n tot = x + 2x = 2 + x y Ar = 1/(2 + x); y O2 = 1 - x/(2 + x); y O = 2x/(2 + x) and the definition of the equilibrium constant (P tot = P o ) becomes K = e = = y O 2 4x 2 = y 02 (2 + x)(1 - x) The equation to solve becomes from the last expression (K + 4)x 2 + Kx - 2K = 0 If that is solved we get x = ± = ; x must be positive y O = ; y 02 = ; y Ar = so

405 A piston/cylinder contains 0.1 kmol hydrogen and 0.1 kmol Ar gas at 25 C, 200 kpa. It is heated up in a constant pressure process so the mole fraction of atomic hydrogen is 10%. Find the final temperature and the heat transfer needed. When gas is heated up H 2 splits partly into H as H 2 2H and the gas is diluted with Ar Component H 2 Ar H Initial Shift -x 0 2x Final 0.1-x 0.1 2x Total = x y H = 0.1 = 2x/(0.2+x) 2x = x x = n tot = y H2 = = [(0.1-x)/(0.2+x)]; y Ar = 1 rest = Do the equilibrium constant: K(T) = y2 H y H2 ( P P 0 ) 2-1 = ( ) ( ) = ln (K) = so from Table A.10 interpolate to get T = 3110 K To do the energy eq., we look up the enthalpies in Table A.8 at 3110K h H2 = ; h H = = (= h f + h) h Ar = C P ( ) = ( ) = (same as h for H) Now get the total number of moles to get n H = ; n H2 = n tot 1-x 2+x = ; n Ar = 0.1 Since pressure is constant W = P V and Q becomes differences in h Q = n h = = kj

406 Air (assumed to be 79% nitrogen and 21% oxygen) is heated in an SSSF process at a constant pressure of 100 kpa, and some NO is formed. At what temperature will the mole fraction of N.O be 0.001? 0.79 N O 2 heated at 100 kpa, forms NO N 2 + O 2 2 NO n N2 = x -x -x +2x n O2 = x n NO = 0 + 2x At exit, y NO = = 2x 1.0 n N2 = , n O2 = K = y 2 NO y N2 y O2 ( P P 0 ) 0 = x = n tot = = or ln K = From Table A.10, T = 1444 K

407 Saturated liquid butane enters an insulated constant pressure combustion chamber at 25 C, and x times theoretical oxygen gas enters at the same pressure and temperature. The combustion products exit at 3400 K. Assuming that the products are a chemical equilibrium gas mixture that includes CO, what is x? Butane: T 1 = 25 o C, sat. liq., x 1 = 0, T c = K, P c = 3.8 Mpa T r1 = 0.7, Figs. D.1and D.2, P r1 = 0.10, P 1 = P r1 P c = 380 kpa * h1 h 1 f = 4.85 RT c Oxygen: T 2 = 25 o C, X * theoretical air Products: T 3 = 3400 K C 4 H XO 2 =>?CO 2 + 5H 2 O +?O 2 2CO 2 <=> 2CO + O 2 Initial (X-1) Change -2a 2a a Equil. 4-2a 2a 6.5(X-1) + a n CO2 = 4-2a, n CO = 2a, n O2 = 6.5(X-1) + a, n H2O = 5, n tot = a + 6.5X y CO = 2a a +6.5X, y CO2 = 4-2a a +6.5X, y 6.5(x - 1) + a O2 = a +6.5X y 2 CO y O K = 2 y 2 P = a P o 2 6.5X a P 1 2-a 6.5X a CO P o T 3 = 3400 K, ln(k) = 0.346, K = = a 2-a 2 6.5X a (3.76) Equation X a Need a second equation: 1 st Law: Q cv + H R = H P + W cv ; Q cv = 0, W cv = K: H R = (h o f + h ) C4 H 10 = ( ) = kj H P = n(h o f + h ) CO2 + n(h o f + h ) CO + n(h o f + h ) O2 + n(h o f + h ) H2O = (4-2a)( ) + 2a( ) + [6.5(X - 1) + a]( ) + 5( ) = kj/kmol H P = H R => = a X Equation 2. Two equations and two unknowns, solve for X and a. a 0.87, X 1.96

408 The combustion products from burning pentane, C 5 H 12, with pure oxygen in a stoichiometric ratio exists at 2400 K, 100 kpa. Consider the dissociation of only CO 2 and find the equilibrium mole fraction of CO. C 5 H O 2 5 CO H 2 O At 2400K, 2 CO 2 2 CO + 1 O 2 ln K = Initial K = 4.461x10-4 Change -2z +2z +z Equil. 5-2z 2z z Assuming P = P o = 0.1 MPa, and n tot = 5 + z + 6 = 11 + z y 2 CO y O2 K = 2 ( P P 0) = 2z 5-2z 2 z 11 + z (1) = 4.461x10-4 ; y CO2 Trial & Error (compute LHS for various values of z): z = n CO2 = 4.418; n CO = 0.582; n O2 = => y CO = Find the equilibrium constant for the reaction 2NO + O 2NO from the 2 2 elementary reactions in Table A.10 to answer which of the nitrogen oxides, NO or NO, is the more stable at ambient conditions? What about at 2000 K? 2 2 NO + O 2 2 NO 2 (1) But N 2 + O 2 2 NO (2) N O 2 2 NO 2 (3) Reaction 1 = Reaction 3 - Reaction 2 G 0 1 = G G 0 2 => ln K 1 = ln K 3 - ln K 2 At 25 o C, from Table A.10: ln K 1 = ( ) = or K 1 = an extremely large number, which means reaction 1 tends to go very strongly from left to right. At 2000 K: ln K 1 = (-7.825) = or K 1 = meaning that reaction 1 tends to go quite strongly from right to left.

409 Methane in equilibrium with carbon and hydrogen as: CH 4 C + 2H 2. has lnk = at 800 K. For a mixture at 100 kpa, 800 K find the equilibrium mole fractions of all components (CH 4, C, and H 2 neglect hydrogen dissociation).redo the molefractions for a mixture state of 200 kpa, 800 K. CH 4 C + 2H 2 ln K = n CH4 = 1-x; n C = x; n H2 = 2x; n tot = 1+2x y 2 y C H 2 y CH4 P P = K = e = o x 1+2x 4x 2 (1+2x) 2 1-x 1+2x x 3 = (1+2x) 2 (1-x) = = Solve by trial and error or successive substitutions x = 0.5 LHS = x = 0.6 LHS = x = 0.7 LHS = x 0.67 LHS = Interpolate x = LHS = x = y CH4 = 1-x 1+2x = y C = x 1+2x = y H2 = 2x 1+2x = P = 200 P o 100 = 2 x 3 (1+2x) 2 (1-x) = = x = 0.4 LHS = x = 0.45 LHS = x = 0.44 LHS = x = y CH4 = 1-x 1+2x = 0.293; y C = x 1+2x = 0.236;y = 2x H 2 1+2x =

410 A mixture of 1 kmol carbon dioxide, 2 kmol carbon monoxide, and 2 kmol oxygen, at 25 C, 150 kpa, is heated in a constant pressure SSSF process to 3000 K. Assuming that only these same substances are present in the exiting chemical equilibrium mixture, determine the composition of that mixture. initial mix: 1 CO 2, 2 CO, 2 O 2 Const. Pressure Q Equil. mix: CO 2, CO, O 2 at T = 3000 K, P = 150 kpa Reaction 2 CO 2 2 CO + O 2 initial change -2x +2x +x equil. (1-2x) (2+2x) (2+x) From A.10 at 3000 K: K = For each n > 0-1 < x < K = y 2 COy O2 y 2 CO2 ( P P 0 ) 1 = 4( 1+x 1-2x )2 ( 2+x 5+x )( ) or ( 1+x 1-2x )2 ( 2+x )= , Trial & error: x = x n CO2 = n CO = n O2 = n TOT = CO2 = y CO = y O2 = y Repeat the previous problem for an initial mixture that also includes 2 kmol of nitrogen, which does not dissociate during the process. Same as Prob except also 2 kmol of inert N 2 Equil.: n CO2 = (1-2x), n CO = (2+2x), n O2 = (2+x), n N2 = 2 K = y 2 COy O2 y 2 CO2 ( P P 0 ) 1 = 4( 1+x 1-2x )2 ( 2+x 7+x )( ) or ( 1+x 1-2x )2 ( 2+x )= Trial & error: x = x n n CO2 = O2 = n n CO = N2 = 2.0 n TOT = y CO2 = y O2 = y CO = y N2 = 0.306

411 Complete combustion of hydrogen and pure oxygen in a stoichiometric ratio at P, T to form water would result in a computed adiabatic flame temperature of K for an SSSF setup. a. How should the adiabatic flame temperature be found if the equilibrium reaction 2H + O H O is considered? Disregard all other possible reactions (dissociations) and show the final equation(s) to be solved. b. F ind the equilibr ium composition at 3800 K, again dis r egar ding all other r eactions. c. Which other reactions should be considered and which components will be present in the final mixture? a) 2H 2 + O 2 2H 2 O Species H 2 O 2 H 2 O H P = H R = H P o + H P = H R o = Ø Initial 2 1 Ø Keq = 2 y H2 O Shift -2x -x 2x Final 2-2x 1-x 2x y 2 ( P H2 y O2 P 0)-1, n tot = 2-2x + 1-x + 2x = 3-x H p = (2-2x) h - + (1-x) h - + 2x(h - o H 2 O 2 fh2o + h - H2O ) = Ø (1) 4x 2 Keq = (3-x)2 3-x (3-x) 2 (2-2x) 2 1-x = x2 (3-x) 3 = Keq(T) (2) (1-x) h - o = ; h - (T), h - (T), h - fh2o H 2 O 2 H 2 O (T) Trial and Error (solve for x,t) using Eqs. (1) and (2). y O2 = 0.15; y H2 = 0.29; y H2 O = 0.56] b) At 3800 K Keq = e (Reaction is times -1 of table) x 2 (3-x)(1-x) -3 = e = x y H2 O = 2x 3-x = 0.43; y = 1-x O 2 3-x = 0.19; y = 2-2x H 2 3-x = 0.38 c) Other possible reactions from table A.10 H 2 2 H O 2 2 O 2 H 2 O H OH

412 Gasification of char (primarily carbon) with steam following coal pyrolysis yields a gas mixture of 1 kmol CO and 1 kmol H 2. We wish to upgrade the hydrogen content of this syngas fuel mixture, so it is fed to an appropriate catalytic reactor along with 1 kmol of H 2 O. Exiting the reactor is a chemical equilibrium gas mixture of CO, H 2, H 2 O, and CO 2 at 600 K, 500 kpa. Determine the equilibrium composition. Note: see Problem CO + 1 H 2 1 H O 2 Chem. Equil. Mix CO, H 2, H 2 O, CO K 500 kpa (1) 1 CO + 1 H 2 O 1 CO H 2 -x -x +x +x (2) 2 H 2 O 2 H O 2 (3) 2 CO 2 2 CO + 1 O 2 (1) = 1 2 (2) (3) ln K 1 = 1 2 [ (-92.49)]= +3.35, K 1 = Equilibrium: n CO = 1-x, n H2O = 1-x, n CO2 = 0 + x, n H2 = 1 + x n = 3, K = y CO2 y H2 y CO y H2O ( P P 0 ) 0 = y CO2 y H2 y CO y H2O = x(1+x) (1-x) 2 x = n y % CO H 2 O CO H

413 A gas mixture of 1 kmol carbon monoxide, 1 kmol nitrogen, and 1 kmol oxygen at 25 C, 150 kpa, is heated in a constant pressure SSSF process. The exit mixture can be as s umed to be in chemical equilibr ium w ith CO 2, CO, O 2, and N 2 pres ent. The mole fraction of CO 2 at this point is Calculate the heat transfer for the process. initial mix: 1 CO, 1 O 2, 1 N 2 Const. Pressure Q Equil. mix: CO 2, CO, O 2, N 2 y CO2 = P = 150 kpa reaction 2 CO 2 2 CO + O 2 also, N 2 initial change +2x -2x -x 0 equil. 2x (1-2x) (1-x) 1 y CO2 = = 2x 3-x n CO2 = n CO = K = y 2 COy O2 y 2 CO2 x = n O2 = y CO2 = n N2 = 1 y CO = y O2 = ( P P 0 ) 1 = ( ) = From A.10, T PROD = 3213 K From A.9, H R = kj H P = ( ) ( ) ( ) + 1( ) = kj Q CV = H P - H R = ( ) = kj

414 A rigid container initially contains 2 kmol of carbon monoxide and 2 kmol of oxygen at 25 C, 100 kpa. The content is then heated to 3000 K at which point an equilibrium mixture of CO 2, CO, and O 2 exists. Disregard other possible species and determine the final pressure, the equilibrium composition and the heat transfer for the process. 2 CO + 2 O 2 2 CO 2 + O 2 Species: CO O 2 CO 2 Initial Shift -2x -2x+x 2x Final 2-2x 2-x 2x : n tot = 2-2x + 2-x + 2x = 4-x y CO = 2-2x 4-x ; y O 2 = 2-x 4-x ; y CO 2 = 2x 4-x U 2 - U 1 = 1 Q 2 = H 2 - H 1 - P 2 v + P 1 v = (2-2x)h - CO 2 + (2 - x)h- O xh- - 2h - o CO fco 2 2-2h - o fo 2 - R - (4 - x)t 2 + 4R - T 1 K eq = e = 2 y CO2 P 2 y 02 y ( 2 ) P -1 = CO o 4x 2 4-x 4(1-x) 2 2-x 4T 1 (4-x)T 2 ( x 1 1-x )2 2-x = 1 T 2 e 4 T = x = ; y CO = 0.102; y O2 = 0.368; y CO2 = 0.53 P 2 = P 1 (4-x)T 2 /4T 1 = 100(3.1618)(3000/4)(298.15) = kpa 1 Q 2 = ( ) (98013) ( ) - 2( ) - 2(Ø) (4(298.15) (3.1618)) = kj

415 One approach to using hydrocarbon fuels in a fuel cell is to reform the hydrocarbon to obtain hydrogen, which is then fed to the fuel cell. As a part of the analysis of such a procedure, consider the reforming section a. Determine the equilibrium constant for this reaction at a temperature of 800 K. b. One kilomole each of methane and water are fed to a catalytic reformer. A mixture of CH 4, H 2 O, H 2, and CO exits in chemical equilibrium at 800 K, 100 kpa; determine the equilibrium composition of this mixture. For CH 4, use C P0 at ave. temp., 550 K. Table A.6, C - = kj/kmol K P0 a) h K = h - 0 f + C - T = ( ) = kj/kmol P0 s K = ln 800 = kj/kmol K For CH 4 + H 2 O 3H 2 + CO H K = 3( ) + 1( ) - 1( ) - 1( ) = kj S K = 3( ) + 1( ) - 1( ) - 1( ) = kj/k G 0 = H 0 - T S 0 = ( ) = kj G 0 ln K = - R - T = = => K = b) CH 4 +H 2 O 3 H 2 + CO initial change -x -x +3x +x equil. (1-x) (1-x) 3x x n TOTAL = 2 + 2x K = or or y 3 H2y CO y CH4 y H2O ( P P 0 ) 2 = (3x) 3 x ( 100 (1-x)(1-x)(2+2x) )2 ( x 1-x )2 ( x 1+x )2 = = x 2 2 = = Solving, x = x n CH4 = n H2O = n H2 = n CO = n TOT = y CH4 = y H2O = y H2 = y CO =

416 In a test of a gas-turbine combustor, saturated-liquid methane at 115 K is to be burned with excess air to hold the adiabatic flame temperature to 1600 K. It is assumed that the products consist of a mixture of CO, H O, N, O, and NO in chemical equilibrium. Determine the percent excess air used in the combustion, and the percentage of NO in the products. CH 4 + 2x O x N 2 1 CO H 2 O + (2x-2) O x N 2 Then N 2 + O 2 = 2 NO Also CO 2 H 2 O init 7.52x 2x ch. -a -a +2a 0 0 equil. (7.52x-a) (2x-2-a) 2a 1 2 n TOT = x 1600 K: ln K = , K = K = From A.9 and B.7, y 2 NO y N2 y O2 ( P P 0 ) 0 = y 2 NO y N2 y O2 = 4a 2 (7.52x-a)(2x-2-a) H R = 1[ ( )] = kj (Air assumed 25 o C) H P = 1( ) + 2( ) + (7.52x-a)(41 904) + (2x-2-a)(44 267) + 2a( ) = x a Assume a ~ 0, then from H P - H R = 0 x = Subst. a 2 ( a)(1.483-a) = , get a Use this a in 1st law x = a 2 = ( a)( a) = , a = x = % excess air = 73.7 % % NO = = %

417 The van't Hoff equation d ln K = Ho R T 2 dt P o relates the chemical equilibrium constant K to the enthalpy of reaction H o. From the value of K in Table A.10 for the dissociation of hydrogen at 2000 K and the value of H o calculated from Table A.8 at 2000 K use van t Hoff equation to predict the constant at 2400 K. H 2 2H H = 2 ( ) = lnk 2000 = ; Assume H is constant and integrate the Van t Hoff equation 2000 lnk lnk 2000 = ( H /R - T 2 )dt = - H R - ( ) T T lnk 2400 = lnk H ( - ) / R - T 2400 T 2000 = ( 6-5 ) / = = Table A.10 lists ( H not exactly constant)

418 Catalytic gas generators are frequently used to decompose a liquid, providing a desired gas mixture (spacecraft control systems, fuel cell gas supply, and so forth). Consider feeding pure liquid hydrazine, N H, to a gas generator, from 2 4 which exits a gas mixture of N, H, and NH in chemical equilibrium at 100 C, kpa. Calculate the mole fractions of the species in the equilibrium mixture. Initially, 2 N 2 H 4 1 N H NH 3 then, N H 2 2 NH 3 initial change -x -3x +2x equil. (1-x) (1-3x) (2+2x) n TOTAL = (4-2x) K = y2 NH3 ( P ) -2 y N2 y 3 P 0 = (2+2x)2 (4-2x) 2 ( 350 (1-x)(1-3x) )-2 H2 At 100 o C = K, for NH 3 use A.5 C - = = P0 h - 0 NH3 = ( ) = kj/kmol s - 0 NH3 = ln = kj/kmol K Using A.8, H C = 2( ) - 1(0+2188) - 3(0+2179) = kj S C = 2( ) - 1( ) - 3( ) = kj/k G C = H 0 - T S 0 = ( ) = kj G 0 ln K = - R - T = = => K = Therefore, [ (1+x)(2-x) (1-3x) ] 2 1 (1-x)(1-3x) = = By trial and error, x = n N2 = n NH3 = n H2 = n TOT = y N2 = y H2 = y NH3 =

419 Acetylene gas at 25 C is burned with 140% theoretical air, which enters the burner at 25 C, 100 kpa, 80% relative humidity. The combustion products form a mixture of CO 2, H 2 O, N 2, O 2, and NO in chemical equilibrium at 2200 K, 100 kpa. This mixture is then cooled to 1000 K very rapidly, so that the composition does not change. Determine the mole fraction of NO in the products and the heat transfer for the overall process. C 2 H O N 2 + water 2 CO H 2 O + 1 O N 2 + water water: P V = = kpa n V = n A P V /P A = ( ) 2.535/ = So, total H 2 O in products is a) reaction: N 2 + O 2 <-> 2 NO change : -x -x +2x at 2200 K, from A.10: K = Equil. products: n CO2 = 2, n H2O = 1.433, n O2 = 1-x, K = n N2 = x, n NO = 0+2x, n TOT = (2x) 2 = => x = (1-x)(13.16-x) y NO = = b) Final products (same composition) at 1000 K H R = 1( ) ( ) = kj H P = 2( ) ( ) ( ) ( ) ( ) = kj Q CV = H P - H R = kj

420 The equilibrium reaction as: CH 4 C + 2H 2. has ln K = at 800 K and lnk = at 600 K. By noting the relation of K to temperature show how you would interpolate ln K in (1/T) to find K at 700 K and compare that to a linear interpolation. A.10: ln K = at 800K ln K = at 600K lnk 700 = lnk = Linear interpolation: lnk 700 = lnk ( ) = (lnk lnk 600 ) = ( ) = ( ) Use the information in Problem to estimate the enthalpy of reaction, H o, at 700 K using Van t Hoff equation with finite differences for the derivatives. dlnk = [ H /R - T 2 ]dt or solve for H H = R - T 2 dlnk dt = R- T 2 lnk T = [Remark: compare this to A.8 values + A.5, A.9, = kj/kmol H = H C + 2H H2 - H CH4 = ( ) ( ) - (-74873) = ]

421 A step in the production of a synthetic liquid fuel from organic waste matter is the following conversion process: 1 kmol of ethylene gas (converted from the waste) at 25 C, 5 MPa, and 2 kmol of steam at 300 C, 5 MPa, enter a catalytic reactor. An ideal gas mixture of ethanol, ethylene, and water in chemical equilibrium leaves the reactor at 700 K, 5 MPa. Determine the composition of the mixture and the heat transfer for the reactor. 25 o C, 5 MPa 1 C H 2 4 IG chem. equil. mixture C 2 H 5 OH, C 2 H 4, H 2 O 300 o C, 5 MPa 2 H 2 O 700 K, 5 MPa 1 C 2 H H 2 O 1 C 2 H 5 OH A.6 at ~ 500 K: init C - P0 C2H4 = 62.3 ch. -x -x +x equil. (1-x) (2-x) x a) H K = 1( ( )) - 1( ( )) - 1( ) = kj S K = 1( ln ) - 1( ln ) - 1( ) = kj/k G K = H 0 - T S 0 = kj ln K = - G0 R - T = => K = = y C2H5OH y C2H4 y H2O ( P P 0 ) -1 ( x 1-x )(3-x )= x 0.1 = By trial and error: x = => C 2 H 5 OH: n = , y = C 2 H 4 : n = , y = , H 2 O: n = , y = b) Reactants: C 2 H 4 : T r = 298.2/282.4 = 1.056, P r = 5/5.04 = H 2 O, LIQ Ref. + St. Table: A.15: (h -* -h - ) = = 3062 kj H C2H4 = 1( ) = kj H H2O = 2( ( )) = kj H PROD = ( ( )) ( ( )) ( ) = kj Q CV = H P - H R = kj

422 Methane at 25 C, 100 kpa, is burned with 200% theoretical oxygen at 400 K, 100 kpa, in an adiabatic SSSF process, and the products of combustion exit at 100 kpa. Assume that the only significant dissociation reaction in the products is that of carbon dioxide going to carbon monoxide and oxygen. Determine the equilibrium composition of the products and also their temperature at the combustor exit. Combustion: CH 4 + 4O 2 CO 2 + 2H 2 O + 2O 2 Dissociation: 2 CO 2 2 CO + O 2,H 2 O inert initial change -2x +2x +x 0 equil. 1-2x 2x 2+x 2 n TOT = 5+x Equil. Eq'n: K = or y 2 COy O2 y 2 CO2 ( x 0.5-x )2 ( 2+x 5+x )= K (P/P 0 ) 1st law: H P - H R = 0 ( P P 0 )= ( x 0.5-x )2 ( 2+x )( P ) 5+x P 0 (1) (1-2x)( h - CO2 ) + 2x( h- CO ) + 2( h - H2O ) + (2+x) h- O2-1( ) - 4(3027) = 0 or (1-2x) h - CO2 + 2x h- CO + 2 h- H2O + (2+x) h x = 0 O2 Assume T P = 3256 K. From A.10: K = Solving (1) by trial & error, x = Substituting x and the h - values from A.8 (at 3256 K) into (2) OK T P = 3256 K & x = n CO2 = , n CO = , n H2O = 2.0, n O2 =

423 Calculate the irreversibility for the adiabatic combustion process described in the previous problem. From solution of Prob , it is found that the product mixture consists of CO 2, CO, 2.0 H 2 O & O 2 at 3256 K, 100 kpa. The reactants include 1 CH 4 at 25 o C, 100 kpa and 4 O 2 at 400 K, 100 kpa. Reactants: S R = 1( ) + 4( ) = kj/k Products: n i y i s -0 i -R - y i P ln S - * i P 0 CO CO H 2 O O S P = ( ) ( ) + 2.0( ) ( ) = kj/k I = T 0 (S P -S R ) - Q CV = ( ) - 0 = kj In rich (too much fuel) combustion the excess fuel may be broken down to give H 2 and CO may form. In the products at 1200 K, 200 kpa the reaction called the water gas reaction may take place: CO 2 + H 2 H 2 O + CO Find the equilibrium constant for this reaction from the elementary reactions. (1) 1 CO H 2 1 CO + 1 H 2 O (2) 2 H 2 O 2 H O 2, (3) 2 CO 2 2 CO + 1 O 2 (1) = 1 2 (3) (2) => ln K = (ln K 3 - ln K 2 )/2 From A.10 at 1200 K for reactions (2) and (3) ln K 2 = , ln K 3 = => ln K = => K =

424 An important step in the manufacture of chemical fertilizer is the production of ammonia, according to the reaction: N 2 + 3H 2 2NH 3 a. Calculate the equilibrium constant for this reaction at 150 C. b. For an initial composition of 25% nitrogen, 75% hydrogen, on a mole basis, calculate the equilibrium composition at 150 C, 5 MPa. 1N 2 + 3H 2 <=> 2NH 3 at 150 o C h - o NH3 150 C = (150-25) = s - o NH3 150 C = ln = H o 150 C = 2( ) - 1(0+3649) - 3(0+3636) = kj S C = 2( ) - 1( ) - 3( ) = kj/k G C = ( ) = kj/kmol ln K = = , K = b) n NH3 = 2x, n N2 = 1-x, n H2 = 3-3x K = y2 NH3 ( P ) -2 y N2 y 3 P 0 = (2x)2 2 2 (2-x) 2 ( P ) (1-x) 4 P 0 H2 or ( x 1-x )2 ( 2-x 1-x )2 = ( )2 = or ( x 1-x )(2-x) 1-x = n y Trial & Error: NH x = N H

425 A space heating unit in Alaska uses propane combustion is the heat supply. Liquid propane comes from an outside tank at -44 C and the air supply is also taken in from the outside at -44 C. The airflow regulator is misadjusted, such that only 90% of the theoretical air enters the combustion chamber resulting in incomplete combustion. The products exit at 1000 K as a chemical equilibrium gas mixture including only CO 2, CO, H 2 O, H 2, and N 2. Find the composition of the products. Propane: Liquid, T 1 = -44 o C = K Air: T 2 = -44 o C = K, 90% Theoretical Air Products: T 3 = 1000 K, CO 2, CO, H 2 O, H 2, N 2 Theoretical Air: C 3 H 8 + 5O N 2 => 3CO 2 + 4H 2 O N 2 90% Theoretical Air: C 3 H O N 2 => aco 2 + bco + ch 2 O + dh N 2 Carbon: a + b = 3 Oxygen: 2a + b + c = 9 Where: 2 a 3 Hydrogen: c + d = 4 Reaction: CO + H 2 O CO 2 + H 2 Initial: b c a d Change: -x -x x x Equil: b - x c - x a + x d + x Chose an Initial guess such as: a = 2, b = 1, c = 4, d = 0 Note: A different initial choice of constants will produce a different value for x, but will result in the same number of moles for each product. n CO2 = 2 + x, n CO = 1 - x, n H2O = 4 - x, n H2 = x, n N2 = The reaction can be broken down into two known reactions to find K (1) 2CO 2 2CO + O 1000 K ln(k 1 ) = (2) 2H 2 O 2H 2 + O 1000 K ln(k 2 ) = For the overall reaction: lnk = (ln(k 2 ) - ln(k 1 ))/2 = ; K = K = y CO2 y H2 P y CO y H2O P = y CO2 y H2 (2 + x)x = = o y CO y H2O (1 4)(4 x) => x = n CO2 = n CO = n N2 = n H2O = n H2 =

426 One kilomole of carbon dioxide, CO 2, and 1 kmol of hydrogen, H 2 at room temperature, 200 kpa is heated to 1200 K at 200 kpa. Use the water gas reaction, see problem 15.36, to determine the mole fraction of CO. Neglect dissociations of H 2 and O 2. 1 CO H 2 1 CO + 1 H 2 O Initial Shift -x -x +x +x Total 1-x 1-x x x; n tot = 2 y H2O = y CO = x/2, y H2 = y CO2 = (1-x)/2 From solution to problem 15.36, K = (x/2)(x/2) x ( 1-x = K = 2 2 )(1-x 2 ) (1-x) 2 => x 1-x = x = / = y H2O = y CO = x/2 = 0.27, y H2 = y CO2 = (1-x)/2 = 0.23

427 Consider the production of a synthetic fuel (methanol) from coal. A gas mixture of 50% CO and 50% H 2 leaves a coal gasifier at 500 K, 1 MPa, and enters a catalytic converter. A gas mixture of methanol, CO and H 2 in chemical equilibrium with the reaction: CO + 2H 2 <=> CH 3 OH leaves the converter at the same temperature and pressure, where it is known that ln K = a. Calculate the equilibrium composition of the mixture leaving the converter. b. Would it be more desirable to operate the converter at ambient pressure? 1 CO 1 H 2 CONVERTER Equil. Mix CH 3 OH, CO, H K, 1 MPa Reaction: CO + 2 H 2 CH 3 OH initial change -x -2x +x equil. (1-x) (1-2x) x y CH3OH a) K = y CO y 2 H2 ( P P 0 ) -2 = ( x 1-x )(2-2x 1-2x )2 ( P P 0 ) -2 => x(1-x) (1-2x) 2 = K 4 ( P P 0 ) 2 ln K = , K = x(1-x) = ( 1 (1-2x) )2 = => x = n CH3OH = x = , n CO = 1-x = , n H2 = 1-2x = y CH3OH = , y CO = , y H2 = b) For P = 0.1 MPa x(1-x) = ( 0.1 (1-2x) )2 = x is much smaller (~ ) not good

428 Consider the following coal gasifier proposed for supplying a syngas fuel to a gas turbine power plant. Fifty kilograms per second of dry coal (represented as 48 kg C plus 2 kg H) enter the gasifier, along with 4.76 kmol/s of air and 2 kmol/s of steam. The output stream from this unit is a gas mixture containing H 2, CO, N 2, CH 4, and CO 2 in chemical equilibrium at 900 K, 1 MPa. a. Set up the reaction and equilibrium equation(s) for this system, and calculate the appropriate equilibrium constant(s). b. Determine the composition of the gas mixture leaving the gasifier. a) Entering the gasifier: 4 C + 1 H O N H 2 O Since the chem. equil. outlet mixture contains no C, O 2 or H 2 O, we must first consider preliminary reaction (or reactions) to eliminate those substances in terms of substances that are assumed to be present at equilibrium. One possibility is 4 C + 1 O H 2 O 4 CO + 2 H 2 such that the "initial" composition for the equilibrium reaction is 4 CO + 3 H N 2 (or convert equal amounts of CO and H 2 to half of CH 4 and CO 2 - also present at equilibrium. The final answer will be the same.) reaction 2 CO + 2 H 2 CH 4 + CO 2 also N 2 initial change -2x -2x +x +x 0 equil. (4-2x) (3-2x) x x 3.76 n TOT = x For CH 4 at 600 K (formula in Table A.6), C - P0 = At 900 K h - 0 CH4 = ( ) = kj/kmol s - 0 CH4 = ln (900 / 298.2) = kj/kmol K (The integrated-equation values are and ) H K = 1( ) + 1( ) - 2( ) - 2( ) = kj S K = 1( ) + 1( ) - 2( ) - 2( ) = kj/k G K = ( ) = kj

429 15-33 ln K = = , K = y CH4 y CO2 b) K = y 2 COy 2 H2 ( P P 0 ) -2 = x x ( x)2 (4-2x) 2 (3-2x) 2 ( P P 0 ) -2 x( x) or (4-2x)(3-2x) = P P 0 K = = By trial & error, x = n CO = 1.444, n H2 = 0.444, n CH4 = n CO2 = 1.278, n N2 = 3.76 y CO = 0.176, y H2 = 0.054, y CH4 = y CO2 = 0.156, y N2 = 0.458

430 Ethane is burned with 150% theoretical air in a gas turbine combustor. The products exiting consist of a mixture of CO, H O, O, N, and NO in chemical equilibrium at 1800 K, 1 MPa. Determine the mole fraction of NO in the products. Is it reasonable to ignore CO in the products? Combustion: C 2 H O N 2 2 CO H 2 O O N 2 Products at 1800 K, 1 MPa Equilibrium mixture: CO 2, H 2 O, O 2, N 2, NO N 2 + O 2 2 NO initial change -x -x +2x equil x 1.75-x 2x Equil. comp. n CO2 = 2, n O2 = 1.75-x, n NO = 2x, n H2O = 3, n N2 = x K = = y 2 NO y N2 y O2 ( P P 0 ) 0 = Solving, x = y NO = = x 2 (19.74-x)(1.75-x) b) 2 CO 2 2 CO + O 2 initial change -2a +2a +2x equil. 2-2a 2a 2x K = = y 2 COy O2 y 2 CO2 ( P P 0 ) 1 =( 2a 2-2a )2 ( 1.75-x+a a ) This equation should be solved simultaneously with the equation solved in part a) (modified to include the unknown a). Since x was found to be small and also a will be very small, the two are practically independent. Therefore, use the value x = in the equation above, and solve for a. ( a a 1-a )2 ( ) = ( a 1.0 ) Solving, a = or y CO = negligible for most applications.

431 One kilomole of liquid oxygen, O 2, at 93 K, and x kmol of gaseous hydrogen, H, 2 at 25 C, are fed to an SSSF combustion chamber. x is greater than 2, such that there is excess hydrogen for the combustion process. There is a heat loss from the chamber of 1000 kj per kmol of reactants. Products exit the chamber at chemical equilibrium at 3800 K, 400 kpa, and are assumed to include only H 2 O, H 2, and O. a. Determine the equilibrium composition of the products and also x, the amount of H 2 entering the combustion chamber. b. Should another substance(s) have been included in part (a) as being present in the products? Justify your answer. x H O 2 2 H 2 O + (x-2) H 2 (1) 1 H 2 O 1 H O shift -a +a +a and a > 0 Equil 2-a x-2+a a a < 2 and n tot = x + a (2) 2 H 2 O 2 H O 2 ln K 2 = (3) 1 O 2 2 O ln K 3 = ln K 1 = 0.5( ln K 2 + ln K 3 ) = => K 1 = Equil.: K 1 (P/P o ) 1 = (x-2+a)a (2-a)(x+a) = = st law: Q + H R = H P, Q = (1+x)(-1000) kj Table A.8: h - * IG = kj/kmol [or = ( ) = kj/kmol ] Fig. D.2: T r = 93/154.6 = 0.601, h - f = R = H R = x(0 + 0) + 1(0 + h - * IG + h - f) = 1( ) = kj H P = (2-a)( ) + (x-2+a)( ) + a( ) = x a = Q + H R = x Rearrange eq. to: x a = Substitute it into the equilibrium eq.: ( a) a (2-a)( a) = Solve a = 0.198, LHS = , x = y H2O = 2-a x+a = 0.755, y H2 = x-2+a x+a = 0.162, y O = a x+a = Other substances and reactions: 2 H 2 O <=> H OH, ln K = , H 2 <=> 2 H, : ln K = 0.201, O 2 <=> 2 O, : ln K = All are significant as K's are of order 1.

432 Butane is burned with 200% theoretical air, and the products of combustion, an equilibrium mixture containing only CO, H O, O, N, NO, and NO, exit from the combustion chamber at 1400 K, 2 MPa. Determine the equilibrium composition at this state. Combustion: C 4 H O N 2 4 CO H 2 O O N 2 Dissociation: 1) N 2 + O 2 2 NO 2) N 2 + 2O 2 2 NO 2 change -a -a +2a change -b -2b +2b At equilibrium: n H2O = 5 n N2 = 48.9-a-b n NO = 2a n CO2 = 4 n O2 = 6.5-a-2b n NO2 = 2b n TOT = 64.4-b At 1400 K, from A.10: K 1 = , K 2 = K 1 = (2a) 2 (48.9-a-b)(6.5-a-2b) ; K 2 = (2b) 2 (64.4-b) (6.5-a-2b) 2 (48.9-a-b) ( P P 0 ) -1 As K 1 and K 2 are both very small, with K 2 << K 1, the unknowns a & b will both be very small, with b << a. From the equilibrium eq.s, for a first trial a ~ 1 K ~ ;b ~ K ~ Then by trial & error, a 2 (48.9-a-b)(6.5-a-2b) = = b 2 (64.4-b) (6.5-a-2b) 2 (48.9-a-b) = ( ) 4 = Solving, a = , b = n CO2 = 4, n H2O = 5, n N2 = , n O2 = 6.482, y CO2 = , y H2O = , y N2 = , y O2 = n NO = , n NO2 = y NO = , y NO2 =

433 A mixture of 1 kmol water and 1 kmol oxygen at 400 K is heated to 3000 K, 200 kpa, in an SSSF process. Determine the equilibrium composition at the outlet of the heat exchanger, assuming that the mixture consists of H 2 O, H 2, O 2, and OH. Reactions and equilibrium eq'ns the same as in example 15.8 (but different initial composition). At equil.: n H2O = 1-2a-2b, n H2 = 2a+b, n O2 = 1+a n OH = 2b, n TOT = 2+a+b Since T = 3000 K is the same, the two equilibrium constants are the same: K 1 = , K 2 = The two equilibrium equations are K 1 = 2a+b 1-2a-2b 1+a 2+a+b ( P P 0 ); K 2 = 2a+b 2+a+b ( 2b 1-2a-2b )2 ( P P 0 ) which must be solved simultaneously for a & b. If solving manually, it simplifies the solution to divide the first by the second, which leaves a quadratic equation in a & b - can solve for one in terms of the other using the quadratic formula (with the root that gives all positive moles). This reduces the problem to solving one equation in one unknown, by trial & error. Solving => b = 0.116, a = => n H2O = 0.844, n H2 = , n O2 = 0.962, n OH = 0.232, n TOT = y H2O = , y H2 = , y O2 = , y OH =

434 One kilomole of air (assumed to be 78% nitrogen, 21% oxygen, and 1% argon) at room temperature is heated to 4000 K, 200 kpa. Find the equilibrium composition at this state, assuming that only N 2, O 2, NO, O, and Ar are present. 1 kmol air (0.78 N 2, 0.21 O 2, 0.01 Ar) heated to 4000 K, 200 kpa. Equil.: 1) N 2 + O 2 2 NO n N2 = 0.78-a change -a -a +2a n O2 = 0.21-a-b n Ar = ) O 2 2 O n O = 2b change -b +2b n NO = 2a n tot = 1+b K 1 = = 4a 2 (0.78-a)(0.21-a-b) ( )0 4b 2 K 2 = = (1+b)(0.21-a-b) (200) 100 Divide 1st eq'n by 2nd and solve for a as function(b), using Get Also X = a = K 1 K 2 ( P P 0 )= Xb 2 2(1+b) [ (1+b) Xb 2 ] b 2 (1+b)(0.21-a-b) = K 2 4(P/P 0 ) = (2) Assume b = From (1), get a = Then, check a & b in (2) OK Therefore, n N2 = n O = y N2 = y O = n O2 = n NO = y O2 = y NO = n Ar = 0.01 y Ar = (1)

435 Acetylene gas and x times theoretical air (x > 1) at room temperature and 500 kpa are burned at constant pressure in an adiabatic SSSF process. The flame temperature is 2600 K, and the combustion products are assumed to consist of N 2, O 2, CO 2, H 2 O, CO, and NO. Determine the value of x. Combustion: C 2 H x O x N 2 2 CO 2 + H 2 O + 2.5(x-1)O x N 2 Eq. products 2600 K, 500 kpa: N 2, O 2, CO 2, H 2 O, CO & NO 2 Reactions: 1) 2 CO 2 2 CO + O 2 2) N 2 + O 2 2 NO change -2a +2a +a change -b -b +2b Equil. Comp.: n N2 = 9.4x-b, n H2O = 1, n CO = 2a, n NO = 2b n O2 = 2.5x a - b, n CO2 = 2-2a, n TOT = 11.9x a At 2600 K, from A.10: K 1 = , K 2 = K 1 (P/P 0 ) = = ( a 5 1-a )2 ( 2.5x-2.5+a-b ) 11.9x+0.5+a K 2 = = (2b) 2 (9.4-b)(2.5x-2.5+a-b) Also, from the 1st law: H P - H R = 0 where H R = 1( ) = kj H P = (9.4x-b)( ) + (2.5x-2.5+a-b)( ) + (2-2a)( ) + 1( ) + 2a( ) + 2b( ) Substituting, EQ1: x a b = 0 which results in a set of 3 equations in the 3 unknowns x, a, b. Assume x = 1.07 Then EQ2: = ( a 1-a )2 ( a-b a ) EQ3: b 2 = 4 ( b)(0.175+a+b) Solving, a = , b = Then checking in EQ1, Therefore, x = 1.07

436 One kilomole of water vapor at 100 kpa, 400 K, is heated to 3000 K in a constant pressure SSSF process. Determine the final composition, assuming that H 2 O, H 2, H, O 2, and OH are present at equilibrium. Reactions: 1) 2 H 2 O 2 H 2 + O 2 2) 2 H 2 O H OH change -2a +2a +a change -2b +b +2b 3) H 2 2 H change -c +2c At equilibrium (3000 K, 100 kpa) n H2O = 1-2a-2b n O2 = a n H = 2c n H2 = 2a+b-c n OH = 2b n TOT = 1+a+b+c K 1 (P/P 0 ) = = ( 2a+b-c a 1 1-2a-2b )2 ( ) 1+a+b+c K 2 (P/P 0 ) = = ( 2a+b-c )( 2b 1 1+a+b+c 1-2a-2b )2 K 3 (P/P 0 ) = (2a) 2 = 1 (2a+b-c)(1+a+b+c) These three equations must be solved simultaneously for a, b & c: a = , b = , c = and n H2O = y H2O = n H2 = y H2 = n O2 = y O2 = n OH = y OH = n H = y H =

437 Operation of an MHD converter requires an electrically conducting gas. It is proposed to use helium gas seeded with 1.0 mole percent cesium, as shown in Fig. P The cesium is partly ionized (Cs Cs` + e ) by heating the mixture to 1800 K, 1 MPa, in a nuclear reactor to provide free electrons. No helium is ionized in this process, so that the mixture entering the converter consists of He, Cs, Cs`, and e. Determine the mole fraction of electrons in the mixture at 1800 K, where ln K = for the cesium ionization reaction described. Reaction: C S C + S + e -, Also He ln K = init => K = change -x +x +x 0 Equil (0.01-x) x x 0.99 ; total: 1 + x K = y e- y Cs+ y Cs or ( P P 0 )=( x 0.01-x )( x 1+x )( P P 0 ) ( x )( x )= / (1/0.1) = x 1+x Quadratic equation: x = y e- = x/(1+x) = One kilomole of argon gas at room temperature is heated to K, 100 kpa. Assume that the plasma in this condition consists of an equilibrium mixture of Ar, Ar`, Ar``, and e according to the simultaneous reactions 1) Ar Ar + + e - 2) Ar + Ar ++ + e - The ionization equilibrium constants for these reactions at K have been calculated from spectroscopic data as ln K 1 = 3.11 and ln K 2 = Determine the equilibrium composition of the plasma. 1) Ar Ar + + e - 2) Ar + Ar ++ + e - ch. -a +a +a ch. -b +b +b Equil. Comp.: n Ar = 1-a, n Ar+ = a-b, n Ar++ = b, n e- = a+b, n TOT = 1+a+b K 1 = y Ar+ y e- y Ar ( P (a-b)(a+b) P 0 )= (1) = (1-a)(1+a+b) K 2 = y Ar++ y e- y Ar+ ( P b(a+b) P 0 )= (1) = (a-b)(1+a+b) By trial & error: a = , b = n Ar = , n Ar+ = , n Ar++ = , n e- = a y Ar = , y Ar+ = 0.484, y Ar++ = , y e- =

438 Plot to scale the equilibrium composition of nitrogen at 10 kpa over the temperature range 5000 K to K, assuming that N 2, N, N`, and e are present. For the ionization reaction N N` + e, the ionization equilibrium constant K has been calculated from spectroscopic data as T [K] K ) N 2 2N 2) N N + + e - change -a +2a change -b +b +b Equil. Comp.: n N2 = 1-a, n N = 2a-b, n N+ = b, n e- = b EQ1: K 1 = y2 N y N2 ( P P 0 )= (2a-b) ( P ) (1-a)(1+a+b) P 0 EQ2: K 2 = y N+ y e- y N ( P P 0 )= b 2 ( P ) (2a-b)(1+a+b) P 0 For T < K: b ~ 0 so neglect EQ2: K 1 = 4a2 (1-a 2 ) ( ) To extrapolate K 1 above 6000 K: ln K (from values at 5000 K & 6000 K) T T(K) K 1 a y N y N For T > K: a 1.0 => K 2 = b 2 (2-b)(2+b) ( ) = b 2 (4-b 2 ) 0.1 T(K) K 2 b y N y N Note that b 0 is not a very good approximation in the vicinity of K. In this region, it would be better to solve the original set simultaneously for a & b. The answer would be approximately the same.

439 Hydrides are rare earth metals, M, that have the ability to react with hydrogen to form a different substance MH with a release of energy. The hydrogen can then x be released, the reaction reversed, by heat addition to the MH. In this reaction x only the hydrogen is a gas so the formula developed for the chemical equilibrium is inappropriate. Show that the proper expression to be used instead of Eq is ln (P H2 /P o ) = G 0 /RT when the reaction is scaled to 1 kmol of H 2. M x H 2 <=> MH x At equilibrium G P = G R, assume g of the solid is a function of T only. ḡ MHx = h - 0 MHx - Ts -0 MHx = ḡ 0 MHx, ḡ M = h - 0 M - Ts -0 M = ḡ 0 M ḡ H2 = h - 0 H2 - Ts -0 H2 + R - T ln(p H2 /P o ) = ḡ 0 H2 + R - T ln(p H2 /P o ) G P = G R : ḡ MHx = ḡ M x ḡ H2 = ḡ0 M x[ḡ0 H2 + R - T ln(p H2 /P o )] Ḡ 0 = ḡ 0 MHx - ḡ 0 M - x ḡ 0 H2/2 = ḡ 0 MHx - ḡ 0 M Scale to 1 mole of hydrogen G ~ 0 = (ḡ 0 MHx - ḡ 0 M)/(x/2) = R - T ln(p H2 /P o ) which is the desired result.

440 15-44 Advanced Problems Repeat Problem 15.1 using the generalized charts, instead of ideal gas behavior. (Z Z 1 -Z 2 ) = 5000 m, P 1 = 15 MPa 1 1 T = 40 o C = const CO 2 T Z 2 2 r = = 1.03, P r1 = = Equilibrium: -w REV = 0 = g + PE g 2 - g 1 = h 2 - h 1 -T(s 2 - s 1 ) = g(z 1 - Z 2 ) = From Figures D.2 and D.3, = kj/kg h * 1 - h 1 = RT c 3.54 = kj/kg ; s* 1 - s 1 = R 2.61 = kj/kg K h * 2 - h* 1 = 0 ; s* 2 - s* 1 = 0 - R ln(p 2 /P 1 ) = ln( P 2 /15) Trial and error. Assume P 2 = 55 MPa (P r2 = 55/7.38 = 7.45) h * 2 - h 2 = RT c 3.60 = kj/kg ; s* 2 - s 2 = R 2.14 = kj/kg K g = [ ln(55/15) ] = 45.7 Too low so assume P 2 = 60 MPa (P r2 = 60/7.38 = 8.13) h * 2 - h 2 = RT c 3.57 = kj/kg ; s* 2 - s 2 = R 2.11 = kj/kg K g = [ ln(60/15) ] = 50.7 Make linear interpolation P 2 = 58 MPa Derive the van t Hoff equation given in problem 15.28, using Eqs and Note: the d(ḡ/t) at constant P for each component can be expressed using the relations in Eqs and Eq : G 0 = v C ḡ 0 C + v D ḡ 0 D - v A ḡ 0 A - v B ḡ 0 B Eq : lnk = G 0 /RT Eq : G 0 = Η - T S 0 dlnk dt = - d dt ( G0 R - T ) = - 1 dg 0 RT dt + G0 R - T 2 = 1 R - T 2 [ G0 T dg0 dt ] 1 = R - T 2 [ G0 + T S 0 ] = 1 R - T 2 Η0 used Eq dḡ dt = - s-

441 A coal gasifier produces a mixture of 1 CO and 2H 2 that is then fed to a catalytic converter to produce methane. A chemical-equilibrium gas mixture containing CH 4, CO, H 2, and H 2 O exits the reactor at 600 K, 600 kpa. Determine the mole fraction of methane in the mixture. CO + 3H 2 CH 4 + H 2 O Initial Change -x -3x x x Equil. 1-x 2-3x x x n = (1 - x) + (2-3x) + x + x = 3-2x K = y CH4 y H2O y CO y 3 ( ) = -2 x 2 = P o (1-x)(2-3x) 3-2 P o H2 lnk = - G o /R ; G o = H o - T S o H P = n CH4 [h o f + C P (T - T o )] + n H2O (h o f + h ) = [ ( )] + ( ) = H R = n CO (h o f + h ) + n H2 (h o f + h ) = 1( ) + 3( ) = kj Η o 600 = H P - H R = (-75188) = kj (s o T ) CH4 = s o To + C P ln(t/t o ) = ln(600/298.2) = (s o T ) H2O = kj/kmol-k; S P = kj/k (s o T ) CO = kj/kmol-k, (s o T ) H2 = kj/kmol-k S o 600 = S P - S R = (ns o T ) CH4 + (ns o T ) H2O - (ns o T ) CO - (ns o T ) H2 = ( ) - ( ) = kj/k G o = H o - T S o = ( ) = kj, lnk = -(-71915)/( ) = => K = Solve for x, x = , n tot = , y CH4 = 0.4

442 Dry air is heated from 25 C to 4000 K in a 100-kPa constant-pressure process. List the possible reactions that may take place and determine the equilibrium composition. Find the required heat transfer. Air assumed to be 21% oxygen and 79% nitrogen by volume. From the elementary reactions we have at 4000 K (A.10) (1) O 2 <=> 2 O K 1 = = y 2 O/y O2 (2) N 2 <=> 2 N K 2 = = y 2 N/y N2 (3) N 2 + O 2 <=> 2 NO K 3 = = y 2 NO/y N2 y O2 Call the shifts a,b,c respectively so we get n O2 = 0.21-a-c, n O = 2a, n N2 = 0.79-b-c, n N = 2b, n NO = 2c, n tot = 1+a+b From which the molefractions are formed and substituted into the three equilibrium equations. The result is K 1 = = y 2 O/y O2 = 4a 2 /[(1+a+b)(0.21-a-c)] K 2 = = y 2 N/y N2 = 4b 2 /[(1+a+b)(0.79-b-c)] K 3 = = y 2 NO/y N2 y O2 = 4c 2 /[(0.79-b-c)(0.21-a-c)] which gives 3 eqs. for the unknowns (a,b,c). Trial and error assume b = c = 0 solve for a from K 1 then for c from K 3 and finally given the (a,c) solve for b from K 2. The order chosen according to expected magnitude K 1 >K 3 >K 2 a = 0.15, b = , c = => n O2 = , n O = 0.3, n N2 = 0.765, n N = , n NO = Q = H ex - H in = n O2 h - O2 + n N2 h - N2 + n O (h - fo + h - O) + n N (h - fn + h - N) + n NO (h - fno + h - NO) - 0 = ( ) ( ) ( ) = kj/kmol air [If no reac. Q = n O2 h - O2 + n N2 h - N2 = kj/kmol air]

443 Methane is burned with theoretical oxygen in an SSSF process, and the products exit the combustion chamber at 3200 K, 700 kpa. Calculate the equilibrium composition at this state, assuming that only CO, CO, H O, H, O, and OH are present. Combustion: CH O 2 CO H 2 O Dissociation reactions: 1) 2 H 2 O 2 H 2 + O 2 2) 2 H 2 O H OH change -2a +2a +a change -2b +b +2b 3) 2 CO 2 2 CO + O 2 change -2c +2c +c At equilibrium: N H2O = 2-2a-2b n O2 = a+c n CO2 = 1-2c N H2 = 2a+b n OH = 2b n CO = 2c n TOT = 3+a+b+c Products at 3200 K, 700 kpa K 1 = = ( 2a+b a+c 2-2a-2b )2 ( 3+a+b+c )(700) 100 2b K 2 = = ( 2-2a-2b )2 ( 2a+b 3+a+b+c )(700) 100 K 3 = = ( 2c a+c 1-2c )2 ( 3+a+b+c )(700) 100 These 3 equations must be solved simultaneously for a, b, & c. If solving by hand divide the first equation by the second, and solve for c = fn(a,b). This reduces the solution to 2 equations in 2 unknowns. Solving, a = 0.024, b = , c = Substance: H 2 O H 2 O 2 OH CO 2 CO n y

444 15-48 ENGLISH UNIT PROBLEMS 15.58ECarbon dioxide at 2200 lbf/in. 2 is injected into the top of a 3-mi deep well in connection with an enhanced oil recovery process. The fluid column standing in the well is at a uniform temperature of 100 F. What is the pressure at the bottom of the well assuming ideal gas behavior? Z 1 Z 2 1 CO 2 2 (Z 1 -Z 2 ) = 3 miles = ft P 1 = 2200 lbf/in 2, T = 100 F = const Equilibrium and ideal gas beahvior -w REV = 0 = g + PE = RT ln (P 2 /P 1 ) + g(z 2 -Z 1 ) = ln (P 2 /P 1 ) = = P 2 = 2200 exp(0.8063) = 4927 lbf/in E Calculate the equilibrium constant for the reaction O 2 <=> 2O at temperatures of 537 R and R. Find the change in Gibbs function at the two T s from Table C.7: H 0 = 2( ) - ( ) = kj/kmol S 0 = 2( ) -1( ) = kj/kmol K G 0 = = kj/kmol ln K = = R: H 0 = 2h - 0 f O - 1h - 0 f O2 = = ; S 0 = 2s -0 O - 1s -0 O2 = = G 0 = H 0 - T S 0 = = G 0 ln K = - R - T = = R: H 0 = 2h - 0 f O - 1h - 0 f O2 = 2 ( ) = ; S 0 = 2s -0 O - 1s -0 O2 = = G 0 = H 0 - T S 0 = = ln K = - G 0 /RT = / ( ) =

445 E Pure oxygen is heated from 77 F to 5300 F in an SSSF process at a constant pressure of 30 lbf/in. 2. Find the exit composition and the heat transfer. The only reaction will be the dissociation of the oxygen O 2 2O ; K(5300 F) = K(3200 K) = Look at initially 1 mol Oxygen and shift the above reaction with x n O2 = 1 - x; n O = 2x; n tot = 1 + x; y i = n i /n tot K = y O 2 ( P ) y 02 P 2-1 4x 2 = 1 + x o (1 + x) x 2 = 8x2 1 - x 2 x 2 K/8 = 1 + K/8 x = ; y 02 = 0.859; y 0 = q - = n 02ex h - 02ex + n 0ex h- Oex - h- 02in = (1 + x)(y 02 h y 0 h- O ) - 0 h - 02 = ; h- O = = q - = 1.076( ) = Btu/lbmol O 2 q = q - /32 = 1948 Btu/lbm (=1424 if no dissociation) 15.61E Pure oxygen is heated from 77 F, 14.7 lbf/in. 2 to 5300 F in a constant volume container. Find the final pressure, composition, and the heat transfer As oxygen is heated it dissociates O 2 2O ln K eq = C. V. Heater: U 2 - U 1 = 1 Q 2 = H 2 - H 1 - P 2 v + P 1 v Per mole O 2 : 1 q- 2 = h- 2 - h- 1 + R- (T 1 - (n 2 /n 1 )T 2 ) Shift x in reaction final composition: (1 - x)o 2 + 2xO n 1 = 1 n 2 = 1 - x + 2x = 1 + x y O2 = (1 - x)/(1 + x) ; y O = 2x/(1 + x) Ideal gas and V 2 = V 1 P 2 = P 1 n 2 T 2 /n 1 T 1 P 2 /P o = (1 + x)t 2 /T 1 K eq = e = y O 2 y 02 ( P 2 P o ) = ( 2x 1 + x )2 ( 1 + x 1 - x ) (1 + x 1 ) (T 2 T 1 ) 4x x = T 1 e T = x= (n O2 ) 2 = , (n O ) 2 = , n 2 = q- 2 = (45581) ( ) - Ø ( ) = Btu/lbmol O 2 y O2 = / = 0.937; y O =0.0648/ =

446 E Air (assumed to be 79% nitrogen and 21% oxygen) is heated in an SSSF process at a constant pressure of 14.7 lbf/in. 2, and some NO is formed. At what temperature will the mole fraction of NO be 0.001? 0.79N O 2 heated at 14.7 lbf/in 2, forms NO At exit, y NO = N 2 + O 2 2 NO n N2 = x -x -x +2x n O2 = x n NO = 0 + 2x n = 1.0 y NO = = 2x 1.0 x = n N2 = , n O2 = K = y 2 NO y N2 y O2 ( P P 0 ) 0 = = or ln K = From Table A.10, T = 1444 K = 2600 R 15.63E The combustion products from burning pentane, C 5 H 12, with pure oxygen in a stoichiometric ratio exists at 4400 R. Consider the dissociation of only CO 2 and find the equilibrium mole fraction of CO. C 5 H O 2 5 CO H 2 O At 4400 R, 2 CO 2 2 CO + 1 O 2 ln K = Initial K = 7.272x10-4 Change -2z +2z +z Equil. 5-2z 2z z Assuming P = P o = 0.1 MPa, K = y 2 CO y O2 2 ( P y CO2 P 0) = ( 2z z 5-2z )2 ( )(1); T & E : z = z n CO2 = ; n CO = ; n O2 = y CO =

447 E A gas mixture of 1 pound mol carbon monoxide, 1 pound mol nitrogen, and 1 pound mol oxygen at 77 F, 20 lbf/in. 2, is heated in a constant pressure SSSF process. The exit mixture can be assumed to be in chemical equilibrium with CO 2, CO, O 2, and N 2 present. The mole fraction of CO 2 at this point is Calculate the heat transfer for the process. initial mix: 1 CO, 1 O 2, 1 N 2 Const. Pressure Q Equil. mix: CO 2, CO, O 2, N 2 y CO2 = P = 20 lbf/in 2 reaction 2 CO 2 2 CO + O 2 also, N 2 initial change +2x -2x -x 0 equil. 2x (1-2x) (1-x) 1 y CO2 = = 2x 3-x n CO2 = n CO = K = y 2 COy O2 y 2 CO2 x = n O2 = y CO2 = n N2 = 1 y CO = y O2 = ( P P 0 ) 1 = ( )= Since Table A.10 corresponds to a pressure P 0 of 100 kpa, which is lbf/in 2. Then, from A.10, T PROD = 3200 K = 5760 R H R = Btu H P = ( ) ( ) ( ) + 1( ) = Btu Q CV = H P - H R = ( ) = Btu

448 EIn a test of a gas-turbine combustor, saturated-liquid methane at 210 R is to be burned with excess air to hold the adiabatic flame temperature to 2880 R. It is assumed that the products consist of a mixture of CO 2, H 2 O, N 2, O 2, and NO in chemical equilibrium. Determine the percent excess air used in the combustion, and the percentage of NO in the products. CH 4 + 2x O x N 2 1 CO H 2 O + (2x-2) O x N 2 Then N 2 + O 2 2 NO Also CO 2 H 2 O init 7.52x 2x ch. -a -a +2a 0 0 equil. (7.52x-a) (2x-2-a) 2a 1 2 n TOT = x 2880 R: ln K = , K = K = y 2 NO y N2 y O2 ( P P 0 ) 0 = y 2 NO y N2 y O2 = 4a 2 (7.52x-a)(2x-2-a) H R = 1[ ( )] (assume 77 F) = Btu H P = 1( ) + 2( ) + (7.52x-a)(18 015) + (2x-2-a)(19 031) + 2a( ) = x a Assume a ~ 0, then from H P - H R = 0 x = Subst. a 2 (13.1-a)(1.484-a) = , 4 get a Use this a in 1st law x = a 2 = ( a)(1.474-a) = , a = x = % excess air = 73.7 % % NO = = %

449 E Acetylene gas at 77 F is burned with 140% theoretical air, which enters the burner at 77 F, 14.7 lbf/in. 2, 80% relative humidity. The combustion products form a mixture of CO 2, H 2 O, N 2, O 2, and NO in chemical equilibrium at 3500 F, 14.7 lbf/in. 2. This mixture is then cooled to 1340 F very rapidly, so that the composition does not change. Determine the mole fraction of NO in the products and the heat transfer for the overall process. C 2 H O N 2 + water 2 CO H 2 O + 1 O N 2 + water Water: P V = = lbf/in 2 P V n V = n A = ( ) P A = So, total H 2 O in products is : 1 + n V = a) reaction: N 2 + O 2 2 NO change : -x -x +2x at 3500F = 3960 R (=2200 K), from A.10: K = Equilibrium products: n CO2 = 2, n H2O = 1.428, n O2 = 1-x, n N2 = x, n NO = 0+2x, n TOT = (2x) 2 K = = (1-x)(13.16-x) By trial and error, x = y NO = = b) Final products (same composition) at 1340 F = 1800 R H R = 1(97 476) ( ) = Btu H P = 2( ) ( ) (0+9761) (0+9227) ( ) = Btu Q CV = H P - H R = Btu

450 E The equilibrium reaction with methane as CH 4 C + 2H 2 has ln K = at 1440 R and ln K = at 1080 R. By noting the relation of K to temperature, show how you would interpolate lnk in (1/T) to find K at 1260 R and compare that to a linear interpolation. ln K = at 1440 R ln K = at 1080 R lnk 1260 = lnk ( ) = Linear interpolation: lnk 1260 = lnk ( ) = (lnk lnk 1080 ) = ( ) = E Use the information in problem to estimate the enthalpy of reaction, H o, at 1260 R using the van t Hoff equation (see problem 15.28) with finite differences for the derivatives. dlnk = [ H /R - T 2 ]dt or solve for H H = R - T 2 dlnk dt = R- T 2 lnk T = = Btu/lb mol [Remark: compare this to C.7 values + C.4, C.12, H = H C + 2H H2 - H CH4 = ( ) ( ) -(-32190) = ]

451 E An important step in the manufacture of chemical fertilizer is the production of ammonia, according to the reaction: N 2 + 3H 2 2NH 3 a. Calculate the equilibrium constant for this reaction at 300 F. b. For an initial composition of 25% nitrogen, 75% hydrogen, on a mole basis, calculate the equilibrium composition at 300 F, 750 lbf/in. 2. 1N 2 + 3H 2 <=> 2NH 3 at 300 F a) h - o NH3 300 F = (300-77) = s - o NH3 300 F = ln = H o 300 F = 2(-17723) - 1( ) - 3( ) = Btu S F = 2(48.98) - 1(48.164) - 3(33.60) = Btu/R G F = (-51.0) = Btu ln K = = , K = b) n NH3 = 2x, n N2 = 1-x, n H2 = 3-3x K = y2 NH3 ( P ) -2 y N2 y 3 P 0 = (2x)2 2 2 (2-x) 2 ( P ) (1-x) 4 P 0 H2 or ( x 1-x )2 ( 2-x 1-x )2 = ( )2 = or ( x 1-x )(2-x 1-x )= n y Trial & Error: NH x = N H

452 E Ethane is burned with 150% theoretical air in a gas turbine combustor. The products exiting consist of a mixture of CO 2, H 2 O, O 2, N 2, and NO in chemical equilibrium at 2800 F, 150 lbf/in. 2. Determine the mole fraction of NO in the products. Is it reasonable to ignore CO in the products? Combustion: C 2 H O N 2 2 CO H 2 O O N 2 Products at 2800 F, 150 lbf/in 2 a) Equilibrium mixture: CO 2, H 2 O, O 2, N 2, NO N 2 + O 2 2 NO initial change -x -x +2x equil x 1.75-x 2x Equil. comp. n CO2 = 2, n H2O = 3, n O2 = 1.75-x, n N2 = x, n NO = 2x K = = y 2 NO y N2 y O2 ( P P 0 ) 0 = 4x 2 (19.74-x)(1.75-x) Solving, x = y NO = = b) 2 CO 2 2 CO + O 2 initial change -2a +2a +2x equil. 2-2a 2a 2x K = = y 2 COy O2 y 2 CO2 ( P P 0 ) 1 = ( 2a 2-2a )2 ( 1.75-x+a 150 )( ) a Since Table A.10 corresponds to a pressure P 0 of 100 kpa, which is lbf/in 2. This equation should be solved simultaneously with the equation solved in part a) (modified to include the unknown a). Since x was found to be small and also a will be very small, the two are practically independent. Therefore, use the value x = in the equation above, and solve for a. ( a a 1-a )2 ( )=( a 150 ) Solving, a = or y CO = negligible for most applications.

453 E One pound mole of air (assumed to be 78% nitrogen, 21% oxygen, and 1% argon) at room temperature is heated to 7200 R, 30 lbf/in. 2. Find the equilibrium composition at this state, assuming that only N 2, O 2, NO, O, and Ar are present. 1 lbmol air (0.78 N 2, 0.21 O 2, 0.01 Ar) heated to 7200 R, 30 lbf/in 2. 1) N 2 + O 2 2 NO 2) O 2 2 O change -a -a +2a change -b +2b Equil.: n N2 = 0.78-a n Ar = 0.01 n NO = 2a n O2 = 0.21-a-b n O = 2b n = 1+b K 1 = = K 2 = = 4a 2 (0.78-a)(0.21-a-b) ( )0 4b 2 (1+b)(0.21-a-b) ( ) Divide 1st eq'n by 2nd and solve for a as function(b), using Get X = K 1 K 2 ( P P 0 ) = Xb 2 a = 2(1+b) [-1 + Also (1+b) Xb 2 ] b 2 (1+b)(0.21-a-b) = K 2 4(P/P 0 ) = (2) Assume b = From (1), get a = Then, check a & b in (2) OK Therefore, Subst. N2 O2 Ar O NO n y (1)

454 E Dry air is heated from 77 F to 7200 R in a 14.7 lbf/in. 2 constant-pressure process. List the possible reactions that may take place and determine the equilibrium composition. Find the required heat transfer. Air assumed to be 21% oxygen and 79% nitrogen by volume. From the elementary reactions at 4000 K = 7200 R (A.10): (1) O 2 <=> 2 O K 1 = = y 2 O/y O2 (2) N 2 <=> 2 N K 2 = = y 2 N/y N2 (3) N 2 + O 2 <=> 2 NO K 3 = = y 2 NO/y N2 y O2 Call the shifts a,b,c respectively so we get n O2 = 0.21-a-c, n O = 2a, n N2 = 0.79-b-c, n N = 2b, n NO = 2c, n tot = 1+a+b From which the molefractions are formed and substituted into the three equilibrium equations. The result is corrected for 1 atm = 14.7 lbf/in 2 = kpa versus the tables 100 kpa K 1 = = y 2 O/y O2 = 4a 2 /[(1+a+b)(0.21-a-c)] K 2 = = y 2 N/y N2 = 4b 2 /[(1+a+b)(0.79-b-c)] K 3 = = y 2 NO/y N2 y O2 = 4c 2 /[(0.79-b-c)(0.21-a-c)] which give 3 eqs. for the unknowns (a,b,c). Trial and error assume b = c = 0 solve for a from K 1 then for c from K 3 and finally given the (a,c) solve for b from K 2. The order chosen according to expected magnitude K 1 >K 3 >K 2 a = 0.15, b = , c = => n O2 = , n O = 0.3, n N2 = 0.765, n N = , n NO = Q = H ex - H in = n O2 h - O2 + n N2 h - N2 + n O (h - fo + h - O) + n N (h - fn + h - N) + n NO (h - fno + h - NO) - 0 = ( ) ( ) ( ) = Btu/lbmol air [If no reaction: Q = n O2 h - O2 + n N2 h - N2 = Btu/lbmol air]

455 E Acetylene gas and x times theoretical air (x > 1) at room temperature and 75 lbf/in. 2 are burned at constant pressure in an adiabatic SSSF process. The flame temperature is 4600 R, and the combustion products are assumed to consist of N 2, O 2, CO 2, H 2 O, CO, and NO. Determine the value of x. Combustion: C 2 H x O x N 2 2 CO 2 + H 2 O + 2.5(x-1)O x N 2 Eq. products 4600 R, 75 lbf/in 2 : N 2, O 2, CO 2, H 2 O, CO, NO 2 Reactions: 1) 2 CO 2 2 CO + O 2 2) N 2 + O 2 2 NO change -2a +2a +a change -b -b +2b Equil. Comp.: n N2 = 9.4x-b n CO2 = 2-2a n O2 = 2.5x-2.5+a-b n H2O = 1 n CO = 2a n NO = 2b n TOT = 11.9x+0.5+a At 4600 R, from A.17: K 1 = , K 2 = K 1 (P/P 0 ) = = =( a 1-a )2 ( 2.5x-2.5+a-b ) 11.9x+0.5+a K 2 = = (2b) 2 (9.4-b)(2.5x-2.5+a-b) Also, from the 1st law: H P - H R = 0 where H R = 1( ) = Btu H P = (9.4x-b)( ) + (2.5x-2.5+a-b)( ) + (2-2a)( ) + 1( ) + 2a( ) + 2b( ) Substituting, x a b = which results in a set of 3 equations in the 3 unknowns x,a,b. Trial and error solution from the last eq. and the ones for K 1 and K 2. The result is x = 1.12, a = , b =

456 E One pound mole of water vapor at 14.7 lbf/in. 2, 720 R, is heated to 5400 R in a constant pressure SSSF process. Determine the final composition, assuming that H 2 O, H 2, H, O 2, and OH are present at equilibrium. Reactions: 1) 2 H 2 O 2 H 2 + O 2 2) 2 H 2 O H OH change -2a +2a +a change -2b +b +2b 3) H 2 2 H change -c +2c At equilibrium (5400 R, 14.7 lbf/in 2 ) n H2O = 1-2a-2b n OH = 2b n H2 = 2a+b-c n H = 2c n O2 = a n TOT = 1+a+b+c K 1 (P/P 0 ) = = ( 2a+b-c a a-2b )2 ( ) 1+a+b+c K 2 (P/P 0 ) = = ( 2a+b-c )( 2b a+b+c 1-2a-2b )2 K 3 (P/P 0 ) = (2a) 2 = 1.03 (2a+b-c)(1+a+b+c) These three equations must be solved simultaneously for a, b & c: a = , b = , c = and n H2O = y H2O = n H2 = y H2 = n O2 = y O2 =

457 E Methane is burned with theoretical oxygen in an SSSF process, and the products exit the combustion chamber at 5300 F, 100 lbf/in. 2. Calculate the equilibrium composition at this state, assuming that only CO 2, CO, H 2 O, H 2, O 2, and OH are present. Combustion: CH O 2 CO H 2 O Dissociation reactions: At equilibrium: 1) 2 H 2 O 2 H 2 + O 2 n H2O = 2-2a-2b change -2a +2a +a n H2 = 2a+b 2) 2 H 2 O H OH n O2 = a+c change -2b +b +2b n OH = 2b 3) 2 CO 2 2 CO + O 2 n CO2 = 1-2c change -2c +2c +c n CO = 2c n TOT = 3+a+b+c Products at 5300F, 100 lbf/in 2 K 1 = = ( 2a+b a+c a-2b )2 ( )( ) 3+a+b+c b K 2 = = ( 2-2a-2b )2 ( 2a+b 100 )( ) 3+a+b+c K 3 = = ( 2c a+c c )2 ( )( ) 3+a+b+c These 3 equations must be solved simultaneously for a, b, & c. If solving by hand divide the first equation by the second, and solve for c = fn(a,b). This reduces the solution to 2 equations in 2 unknowns. Solving, a = , b = , c = Substance: H2O H2 O2 OH CO2 CO n y

458 16-1 CHAPTER 16 Notice that most of the solutions are done using the computer tables, which includes the steam tables, air table, compressible flow table and the normal shock table. This significantly reduces the amount of time it will take to solve a problem, so this should be considered in problem assignments and exams. Changes of problems from the 4th edition Chapter 14 to the new Chapter 16 are: New Old New Old New Old new new new E 39 5 new E E new 7 new E new 8 new E E new E new E new 12 new E new E new E new E new 56E mod skip a) E new new New Old New Old New Old

459 Steam leaves a nozzle with a pressure of 500 kpa, a temperature of 350 C, and a velocity of 250 m/s. What is the isentropic stagnation pressure and temperature? h 0 = h 1 + V 1 2 /2 = /2000 = kj/kg s 0 = s 1 = kj/kg K Computer software: (h o, s o ) T o = 365 C, P o = 556 kpa 16.2 An object from space enters the earth s upper atmosphere at 5 kpa, 100 K with a relative velocity of 2000 m/s or more. Estimate the object`s surface temperature. h o1 - h 1 = V 1 2 /2 = /2000 = 2000 kj/kg h o1 = h = = 2100 kj/kg => T = 1875 K The value for h from ideal gas table A.7 was estimated since the lowest T in the table is 200 K The products of combustion of a jet engine leave the engine with a velocity relative to the plane of 400 m/s, a temperature of 480 C, and a pressure of 75 kpa. Assuming that k = 1.32, C p = 1.15 kj/kg K for the products, determine the stagnation pressure and temperature of the products relative to the airplane. h o1 - h 1 = V 2 1 /2 = /2000 = 80 kj/kg T o1 - T 1 = (h o1 - h 1 )/C p = 80/1.15 = 69.6 K T o1 = = 823 K P o1 = P 1 (T o1 /T ) k/(k-1) 1 = 75(823/753.15) = 108 kpa 16.4 A meteorite melts and burn up at temperatures of 3000 K. If it hits air at 5 kpa, 50 K how high a velocity should it have to experience such a temperature? Assume we have a stagnation T = 3000 K h 1 + V 1 2 /2 = h stagn. Use table A.7, h stagn. = , h 1 = 50 V 1 2 /2 = = kj/kg (remember convert to J/kg = m 2 /s 2 ) V 1 = = 2636 m/s

460 16.5 I drive down the highway at 110 km/h on a day with 25 C, kpa. I put my hand, cross sectional area 0.01 m 2, flat out the window. What is the force on my hand and what temperature do I feel? 16-3 The air stagnates on the hand surface : h 1 + V 1 2 /2 = h stagn. Use constant heat capacity V 2 1 /2 T stagn. = T + 1 = (1000/3600) 2 = C C p 1004 Assume a reversible adiabatic compression P stagn. = P 1 (T stagn. /T 1 ) k/(k-1) = ( /298.15) 3.5 = kpa 16.6 Air leaves a compressor in a pipe with a stagnation temperature and pressure of 150 C, 300 kpa, and a velocity of 125 m/s. The pipe has a cross-sectional area of 0.02 m 2. Determine the static temperature and pressure and the mass flow rate. h o1 - h 1 = V 1 2 /2 = /2000 = kj/kg T o1 - T 1 = (h o1 - h 1 )/C p = /1.004 = 7.8 K T 1 = T o1 - T = = C = K P 1 = P o1 (T 1 /T o1 ) k/(k-1) = 300(415.4/423.15) 3.5 = 281 kpa ṁ = ρav = AV P v = 1 AV 1 = 281.2(0.02)(125) = 5.9 kg/s RT (415.4) 16.7 A stagnation pressure of 108 kpa is measured for an air flow where the pressure is 100 kpa and 20 C in the approach flow. What is the incomming velocity? Assume a reversible adiabatic compression T o1 = T 1 (P o1 /P 1 ) (k-1)/k = ( ) = K V 1 2 /2 = h o1 - h 1 = C p (T o1 - T 1 ) = V 1 = = m/s

461 A jet engine receives a flow of 150 m/s air at 75 kpa, 5 C across an area of 0.6 m 2 with an exit flow at 450 m/s, 75 kpa, 600 K. Find the mass flow rate and thrust. ṁ = ρav; ideal gas ρ = P/RT 75 ṁ = (P/RT)AV = ( ) = = kg/s F net = ṁ (V ex -V in ) = ( ) = N 16.9 A water cannon sprays 1 kg/s liquid water at a velocity of 100 m/s horizontally out from a nozzle. It is driven by a pump that receives the water from a tank at 15 C, 100 kpa. Neglect elevation differences and the kinetic energy of the water flow in the pump and hose to the nozzle. Find the nozzle exit area, the required pressure out of the pump and the horizontal force needed to hold the cannon. ṁ = ρav = AV/v A= ṁv/v = = m 2 W _ p = ṁw p = ṁv(p ex - P in ) = ṁv 2 /2 ex P ex = P in + V ex 2 /2v = / = 150 kpa F = ṁv ex = = 100 N An irrigation pump takes water from a lake and discharges it through a nozzle as shown in Fig. P At the pump exit the pressure is 700 kpa, and the temperature is 20 C. The nozzle is located 10 m above the pump and the atmospheric pressure is 100 kpa. Assuming reversible flow through the system determine the velocity of the water leaving the nozzle. Assume we can neglect kinetic energy in the pipe in and out of the pump. Incompressible flow so Bernoulli's equation applies (V 1 V 2 V 3 0) v(p 3 - P 2 )+ (V V 2 2 )/2 + g(z 3 - Z 2 )= 0 g(z 3 - Z 2 ) 9.807(10) P 3 = P 2 - = = 602 kpa v 1000( ) V 4 2 /2 = v(p 3 - P 4 ) V 4 = 2v(P 3 - P 4 ) = = m/s

462 A water turbine using nozzles is located at the bottom of Hoover Dam 175 m below the surface of Lake Mead. The water enters the nozzles at a stagnation pressure corresponding to the column of water above it minus 20% due to friction. The temperature is 15 C and the water leaves at standard atmospheric pressure. If the flow through the nozzle is reversible and adiabatic, determine the velocity and kinetic energy per kilogram of water leaving the nozzle. P = ρg Z = g Z v = = kpa P ac = 0.8 P = kpa v P = V ex 2 /2 V ex = 2v P V ex = = 62.4 m/s V ex 2 /2 = v P = kj/kg A water tower on a farm holds 1 m 3 liquid water at 20 C, 100 kpa in a tank on top of a 5 m tall tower. A pipe leads to the ground level with a tap that can open a 1.5 cm diameter hole. Neglect friction and pipe losses and estimate the time it will take to empty the tank for water. Bernoulli Equation: P e = P i ; V i = 0; Z e = 0; Z i = H V e 2 /2 = gz i V e = 2gZ = = 9.9 m/s ṁ = ρav e = AV e /v = m/ t m = V/v; A = πd 2 /4 = π / 4 = m 2 t = mv/av e = V/AV e t = = sec = 9.53 min

463 Find the speed of sound for air at 100 kpa at the two temperatures 0 C and 30 C. Repeat the answer for carbon dioxide and argon gases. From eq we have c 0 = krt = = 331 m/s c 30 = = 349 m/s For Carbon Dioxide: R = , k = c 0 = c 30 = = m/s = m/s For Argon: R = , k = c 0 = c 30 = = m/s = m/s If the sound of thunder is heard 5 seconds after seing the lightning and the weather is 20 C how far away is the lightning taking place? The sound travels with the speed of sound in air (ideal gas). L = c t = krt t = = 1716 m Estimate the speed of sound for steam directly from Eq and the steam tables for a state of 6 MPa, 400 C. Use table values at 5 and 7 MPa at the same entropy as the wanted state. Eq is then done by finite difference. Find also the answer for the speed of sound assuming steam is an ideal gas. c 2 = ( δp δρ ) s = ( P ρ ) s State 6 MPa, 400 C s = MPa, s v = ; ρ = 1/v = MPa, s v = ; ρ = 1/v = c = = c = m/s C p = = 36.91; C 100 v = C p - R = = k = C p /C v = ; R = c = krt = = m/s

464 16.16 A convergent-divergent nozzle has a throat diameter of 0.05 m and an exit diameter of 0.1 m. The inlet stagnation state is 500 kpa, 500 K. Find the back pressure that will lead to the maximum possible flow rate and the mass flow rate for three different gases as: air; hydrogen or carbon dioxide There is a maximum possible flow when M=1 at the throat, T * = 2 k+1 T o ; P* = P o ( 2 k+1 ) k-1; k ρ * =ρ o ( 2 1 k+1 ) k-1 ṁ = ρ * A * V = ρ * A * c = P * A * k/rt * A * = πd 2 /4 = m 2 k T * P * c ρ * ṁ a) b) c) A E /A * = (D E /D * ) 2 = 4. There are 2 possible solutions corresponding to points c and d in Fig and Fig For these we have M E P E /P o M E P E /P o a) b) c) P B = P E = kpa all cases point c P B = P E = a) 14.9 b) c) kpa, point d

465 Air is expanded in a nozzle from 2 MPa, 600 K, to 200 kpa. The mass flow rate through the nozzle is 5 kg/s. Assume the flow is reversible and adiabatic and determine the throat and exit areas for the nozzle. Mach # Velocity P * = P 2 o k+1 k k-1 = = MPa Area Density 2.0 MPa P 0.2 MPa T * = T o 2/(k+1) = = 500 K v * = RT * /P * = /1056 = m 3 /kg c * = krt * = = m/s A * = ṁv * /c * = /448.2 = m 2 P 2 /P o = 200/2000 = 0.1 M * 2 = = V 2 /c* V 2 = = m/s T 2 = = K v 2 = RT 2 /P 2 = /200 = m 3 /kg A 2 = ṁv 2 /V 2 = /762.4 = m Consider the nozzle of Problem and determine what back pressure will cause a normal shock to stand in the exit plane of the nozzle. This is case g in Fig What is the mass flow rate under these conditions? P E /P o = 200/2000 = 0.1 ; M E = = M x M y = ; P y /P x = P y = = 1055 kpa ṁ = 5 kg/s same as in Problem 16.9

466 At what Mach number will the normal shock occur in the nozzle of Problem if the back pressure is 1.4 MPa? (trial and error on M x ) Relate the inlet and exit conditions to the shock conditions with reversible flow before and after the shock. It becomes trial and error. Assume M x = 1.8 M y = ; P oy /P ox = A E /A * x = A 2 /A* = / = A x /A * x = ; A x /A* y = A E /A * y = (A E /A* x )(A x /A* y )/(A x /A* x ) = (1.1694) M E = ; P E /P oy = = P E = (P E /P oy )(P oy /P ox )P ox = = 1353 kpa So select the mach number a little less M x = 1.7 M y = ; P oy /P ox = A x /A * x = ; A x /A* y = A E /A * y = (A E /A* x )(A x /A* y )/(A x /A* x ) = (1.1446) M E = ; P E /P oy = = P E = (P E /P oy )(P oy /P ox )P ox = = kpa Now interpolate between the two M x = M y = ; P oy /P ox = A x /A * x = ; A x /A* y = A E /A * = / = y M E = 0.5 ; P E /P oy = P E = = kpa OK

467 Consider the nozzle of Problem What back pressure will be required to cause subsonic flow throughout the entire nozzle with M = 1 at the throat? A E /A * = / = M E = ; P E /P o = P E = = 1796 kpa Determine the mass flow rate through the nozzle of Problem for a back pressure of 1.9 MPa. P E /P ox = 1.9/2.0 = 0.95 M E = T E = (T/T o )T o = = K c E = krt E = = m/s V E = M E c E = = m/s v E = RT/P = /1900 = m 3 /kg ṁ = A E V E /v E = / = kg/s At what Mach number will the normal shock occur in the nozzle of Problem flowing with air if the back pressure is halfway between the pressures at c and d in Fig ? First find the two pressure that will give exit at c and d. See solution to 16.8 a) A E /A * = (D E /D * ) 2 = 4 P E = (c) 16.9 (d) P E = ( )/2 = kpa Assume M E = 2.4 M y = ; P oy /P ox = A x /A * x = ; A x /A* y = A E /A * y = (A E /A* x )(A x /A* y ) / (A x /A* ) = / = x M E = ; P E /P oy = P E = (P E /P oy )(P oy /P ox )P ox = = kpa Repeat if M x = 2.5 P E = kpa Interpolate to match the desired pressure => M x = 2.41

468 16.23 A convergent nozzle has minimum area of 0.1 m 2 and receives air at 175 kpa, 1000 K flowing with 100 m/s. What is the back pressure that will produce the maximum flow rate and find that flow rate? P * = ( 2 P o k+1 ) k k-1 = Critical Pressure Ratio Find P o : h 0 = h 1 + V 1 2 /2 = /2000 = T 0 = T i = from table A.7 P 0 = P i (T 0 /T i ) k/(k-1) = 175 (1004.4/1000) 3.5 = P * = P o = = kpa T * = T o = K ρ * = P* RT * = = V = c = krt * = = m/s ṁ = ρav = = kg/s A convergent-divergent nozzle has a throat area of 100 mm 2 and an exit area of 175 mm 2. The inlet flow is helium at a total pressure of 1 MPa, stagnation temperature of 375 K. What is the back pressure that will give sonic condition at the throat, but subsonic everywhere else? A E /A * = 175/100 = 1.75 ; k He = M E = ; P E /P O = P E = = 906 kpa

469 A nozzle is designed assuming reversible adiabatic flow with an exit Mach number of 2.6 while flowing air with a stagnation pressure and temperature of 2 MPa and 150 C, respectively. The mass flow rate is 5 kg/s, and k may be assumed to be 1.40 and constant. a. Determine the exit pressure, temperature, and area, and the throat area. b. Suppose that the back pressure at the nozzle exit is raised to 1.4 MPa, and that the flow remains isentropic except for a normal shock wave. Determine the exit Mach number and temperature, and the mass flow rate through the nozzle. (a) From Table A.11: M E = 2.6 P E = = MPa T * = = K P * = = MPa c * = = m/s v * = /1057 = m 3 /kg A * = /376.5 = m 2 A E = = m 2 T E = = K Assume M x = 2 then M y = , P oy /P ox = , A E /A * x = A x /A * x = , A x /A * y = , A E /A * y = / = M E = 0.293, P E /P oy = P E = = MPa, OK T E = = 416 K, ṁ = 5 kg/s

470 A jet plane travels through the air with a speed of 1000 km/h at an altitude of 6 km, where the pressure is 40 kpa and the temperature is 12 C. Consider the inlet diffuser of the engine where air leaves with a velocity of 100 m/s. Determine the pressure and temperature leaving the diffuser, and the ratio of inlet to exit area of the diffuser, assuming the flow to be reversible and adiabatic. V = 1000 km/h = m/s, v 1 = /40 = h 1 = , P r1 = h o1 = /2000 = T o1 = K, P ro1 = P o1 = / = kpa h 2 = /2000 = T 2 = K, P r2 = P 2 = / = 61 kpa v 2 = /61 = A 1 /A 2 = (1.874/1.386)(100/277.8) = A 1-m 3 insulated tank contains air at 1 MPa, 560 K. The tank is now discharged through a small convergent nozzle to the atmosphere at 100 kpa. The nozzle has an exit area of m 2. a. Find the initial mass flow rate out of the tank. b. Find the mass flow rate when half the mass has been discharged. c. Find the mass of air in the tank and the mass flow rate out of the tank when the nozzle flow changes to become subsonic. a. The back pressure ratio: P B /P o1 = 100/1000 = 0.1 < (P * /P o ) crit = so the initial flow is choked with the maximum possible flow rate. M E = 1 ; P E = = kpa T E = T * = = K V E = c = krt * = = 433 m/s v E = RT * /P E = /528.3 = m 3 /kg

471 16-14 ṁ 1 = AV E /v E = / = kg/s b. The initial mass is m 1 = P 1 V/RT 1 = / = kg with a mass at state 2 as m 2 = m 1 /2 = kg. Assume an adiabatic reversible expansion of the mass that remains in the tank. P 2 = P 1 (v 1 /v ) k 2 = = kpa T 2 = T 1 (v 1 /v ) k-1 2 = = 424 K The pressure ratio is still less than critical and the flow thus choked. P B /P o2 = 100/378.9 = < (P * /P ) o crit M E = 1 ; P E = = kpa T E = T * = = K V E = c = krt * = = 377 m/s ṁ 2 = AV E P E /RT E = (377)(200.2) = kg/s 0.287(353.7) c. The flow changes to subsonic when the pressure ratio reaches critical. P B /P o3 = P o3 = kpa v 1 /v 3 = (P o3 /P 1 ) 1/k = (189.3/1000) = m 3 = m 1 v 1 /v 3 = kg T 3 = T 1 (v 1 /v 3 ) k-1 = = 348 K P E = P B = 100 kpa ; M E = 1 T E = = 290 K ; V E = krt E = m/s ṁ 3 = AV E P E /RT E = (341.4)(100) 0.287(290) = kg/s

472 A 1-m 3 uninsulated tank contains air at 1 MPa, 560 K. The tank is now discharged through a small convergent nozzle to the atmosphere at 100 kpa while heat transfer from some source keeps the air temperature in the tank at 560 K. The nozzle has an exit area of m 2. a. Find the initial mass flow rate out of the tank. b. Find the mass flow rate when half the mass has been discharged. c. Find the mass of air in the tank and the mass flow rate out of the tank when the nozzle flow changes to become subsonic. a. Same solution as in a) b. From solution b) we have m 2 = m 1 /2 = kg P 2 = P 1 /2 = 500 kpa ; T 2 = T 1 ; P B /P 2 = 100/500 = 0.2 < (P * /P o ) crit The flow is choked and the velocity is: V E = c = krt * = 433 m/s from a) P E = = kpa ; M E = 1 ṁ 2 = AV E P E /RT E = (433)(264.2) = kg/s 0.287(466.7) c. Flow changes to subsonic when the pressure ratio reaches critical. P B /P o = ; P 3 = kpa m 3 = m 1 P 3 /P 1 = kg ; T 3 = T 1 V E = 433 m/s ṁ 3 = AV E P E /RT E = (433)(189.3) 0.287(466.7) = kg/s The products of combustion enter a convergent nozzle of a jet engine at a total pressure of 125 kpa, and a total temperature of 650 C. The atmospheric pressure is 45 kpa and the flow is adiabatic with a rate of 25 kg/s. Determine the exit area of the nozzle. The critical pressure: P crit = P 2 = = 66 kpa > P amb The flow is then choked. T 2 = = K V 2 = c 2 = = 556 m/s v 2 = /66 = A 2 = /556 = m 2

473 Air is expanded in a nozzle from 700 kpa, 200 C, to 150 kpa in a nozzle having an efficiency of 90%. The mass flow rate is 4 kg/s. Determine the exit area of the nozzle, the exit velocity, and the increase of entropy per kilogram of air. Compare these results with those of a reversible adiabatic nozzle. T 2s = T 1 (P 2 /P 1 ) (k-1)/k = (150/700) = K V 2s 2 = ( ) = V 2 2 = V 2 = 552 m/s h 2 + V 2 2 /2 = h 1 T 2 = T 1 - V 2 2 /2C p T 2 = /( ) = K ; v 2 = /150 = m 3 /kg A 2 = /552 = m 2 = 4460 mm 2 s 2 - s 1 = ln ln = kj/kg K Repeat Problem assuming a diffuser efficiency of 80%. Same as problem 16.26, except η D = We thus have from h 3 - h 1 h = h o1 - h = 0.8 h h 3 = , P r3 = P o2 = P 3 = / = kpa s P ro2 = P ro1 = h 2 = /2000 = T 2 = K, P r2 = P 2 = / = 55.6 kpa v 2 = /55.6 = A 1 /A 2 = (1.874/1.521)(100/277.8) = 0.444

474 Consider the diffuser of a supersonic aircraft flying at M = 1.4 at such an altitude that the temperature is 20 C, and the atmospheric pressure is 50 kpa. Consider two possible ways in which the diffuser might operate, and for each case calculate the throat area required for a flow of 50 kg/s. a. The diffuser operates as reversible adiabatic with subsonic exit velocity. b. A normal shock stands at the entrance to the diffuser. Except for the normal shock the flow is reversible and adiabatic, and the exit velocity is subsonic. This is shown in Fig. P a. Assume a convergent-divergent diffuser with M = 1 at the throat. Relate the inlet state to the sonic state P 1 /P o = ; P * /P o1 = P * = = 84 kpa ; T* = ; = K c * = krt * = = m/s v * = RT * /P * = /84 = A * = ṁv * /c * = /343.5 = m 2 b. Across the shock we have M y = ; P y = = 106 kpa ; T y = = K P * = = 80.6 kpa T * = = K, c* = m/s v * = /80.6 = A * = /343.5 = m 2

475 Air enters a diffuser with a velocity of 200 m/s, a static pressure of 70 kpa, and a temperature of 6 C. The velocity leaving the diffuser is 60 m/s and the static pressure at the diffuser exit is 80 kpa. Determine the static temperature at the diffuser exit and the diffuser efficiency. Compare the stagnation pressures at the inlet and the exit. T o1 = T 1 + V 1 2 /2C p = /( ) = K T o2 = T o1 T 2 = T o2 - V 2 2 /2C p = /( ) = K T o1 - T 1 = k-1 P o1 - P 1 T 1 k P 1 T o2 - T 2 = k - 1 P o2 - P 2 T 2 k P 2 P o1 - P 1 = P o1 = 88.3 kpa P o2 - P 2 = 1.77 P o2 = 81.8 kpa T ex s = T 1 (P o2 /P 1 ) k-1/k = = K T ex s - T η D = = T o1 - T = Steam at a pressure of 1 MPa and temperature of 400 C expands in a nozzle to a pressure of 200 kpa. The nozzle efficiency is 90% and the mass flow rate is 10 kg/s. Determine the nozzle exit area and the exit velocity. First do the ideal reversible adiabatic nozzle s 2s = s 1 = T 2s = C ; h 2s = 2851, h 1 = Now the actual nozzle can be calculated h 1 - h 2ac = η D (h 1 - h 2s ) = 0.9( ) = h 2ac = , T 2 = C, v 2 = V 2 = 2000( ) = 862 m/s A 2 = ṁv 2 /V 2 = /862 = m 2

476 Steam at 800 kpa, 350 C flows through a convergent-divergent nozzle that has a throat area of 350 mm 2. The pressure at the exit plane is 150 kpa and the exit velocity is 800 m/s. The flow from the nozzle entrance to the throat is reversible and adiabatic. Determine the exit area of the nozzle, the overall nozzle efficiency, and the entropy generation in the process. h o1 = , s o1 = P * /P o1 = (2/(k+1)) k/(k-1) = P * = kpa At *: (P *,s * = s o1 ) h * = , v * = h = V 2 /2 V * = 2000( ) = m/s ṁ = AV * /v * = / = kg/s h e = h o1 - V e 2 /2 = / = Exit: P e, h e : v e = 1.395, s e =7.576 A e = ṁv e /V e = /800 = m 2 s gen = s e - s o1 = = kj/kg K Air at 150 kpa, 290 K expands to the atmosphere at 100 kpa through a convergent nozzle with exit area of 0.01 m 2. Assume an ideal nozzle. What is the percent error in mass flow rate if the flow is assumed incompressible? T e = T i ( P e P i ) k-1 k = K V e 2 /2 = h i - h e = C p (T i - T e ) = ( ) = V e = m/s; v e = RT e = = P e ṁ = AV e / v e = = 3.4 kg/s Incompressible Flow: v i = RT/P = /150 = m 3 /kg V e 2 /2 = v P = v i (P i - P e ) = ( ) = kj/kg => V e = 235 m/s => ṁ = AV e / v i = / = 4.23 kg/s ṁ incompressible = 4.23 = 1.25 about 25% overestimation. ṁ compressible 3.4

477 A sharp-edged orifice is used to measure the flow of air in a pipe. The pipe diameter is 100 mm and the diameter of the orifice is 25 mm. Upstream of the orifice, the absolute pressure is 150 kpa and the temperature is 35 C. The pressure drop across the orifice is 15 kpa, and the coefficient of discharge is Determine the mass flow rate in the pipeline. T = T i k-1 k P = P i = 8.8 v i = RT i /P i = P e = 135 kpa, T e = , v e = ṁ i = ṁ e V i = V e (D e /D i ) 2 v i /v e = h i - h e = V e 2 ( )/2 = C p (T i - T e ) V e s = /( ) 2 = m/s ṁ = 0.62 (π/4) (0.025) (2133.1/0.6364) = kg/s A critical nozzle is used for the accurate measurement of the flow rate of air. Exhaust from a car engine is diluted with air so its temperature is 50 C at a total pressure of 100 kpa. It flows through the nozzle with throat area of 700 mm 2 by suction from a blower. Find the needed suction pressure that will lead to critical flow in the nozzle, the mass flow rate and the blower work, assuming the blower exit is at atmospheric pressure 100 kpa. P * = P o = kpa, T * = T o = K v * = RT * /P * = /52.83 = c * = krt * = = m/s ṁ = Ac * /v * = /1.463 = kg/s

478 A convergent nozzle is used to measure the flow of air to an engine. The atmosphere is at 100 kpa, 25 C. The nozzle used has a minimum area of 2000 mm 2 and the coefficient of discharge is A pressure difference across the nozzle is measured to 2.5 kpa. Find the mass flow rate assuming incompressible flow. Also find the mass flow rate assuming compressible adiabatic flow. Assume V i 0, v i = RT i /P i = /100 = V e,s 2 /2 = h i - h e,s = v i (P i - P e ) = kj/kg V e,s = = m/s ṁ s = AV e,s /v i = / = kg/s ṁ a = C D ṁ s = kg/s T e,s = T i (P e /P ) (k-1)/k i = (97.5/100) = 296 K h = C p T = = = V 2 e,s /2 V e,s = = m/s v e,s = /97.5 = ṁ s = AV e,s /v e,s = / = kg/s ṁ a = C D ṁ s = kg/s

479 A convergent nozzle with exit diameter of 2 cm has an air inlet flow of 20 C, 101 kpa (stagnation conditions). The nozzle has an isentropic efficiency of 95% and the pressure drop is measured to 50 cm water column. Find the mass flow rate assuming compressible adiabatic flow. Repeat calculation for incompressible flow. Convert P to kpa: P = 50 cm H 2 O = = kpa T 0 = 20 C = K Assume inlet V i = 0 P 0 = 101 kpa P e = P 0 - P = = kpa P e T e = T 0 ( ) k-1 k = ( P ) = V e 2 /2 = h i - h e = C p (T i - T e ) = ( ) = kj/kg = J/kg => V e = m/s V 2 e ac /2 = η V 2 e s /2 = = V e ac = m/s T e ac = T i - ρ e ac = V 2 e ac /2 = C p = P e = = kg/m3 RT p ṁ = ρav = π = kg/s Steam at 600 kpa, 300 C is fed to a set of convergent nozzles in a steam turbine. The total nozzle exit area is m 2 and they have a discharge coefficient of The mass flow rate should be estimated from the measurement of the pressure drop across the nozzles, which is measured to be 200 kpa. Determine the mass flow rate. s e,s = s i = kj/kg K ; (P e, s e,s ) h e,s = 2961 kj/kg v e,s = , h i = V e,s = ( ) = m/s ṁ = / = kg/s

480 The coefficient of discharge of a sharp-edged orifice is determined at one set of conditions by use of an accurately calibrated gasometer. The orifice has a diameter of 20 mm and the pipe diameter is 50 mm. The absolute upstream pressure is 200 kpa and the pressure drop across the orifice is 82 mm of mercury. The temperature of the air entering the orifice is 25 C and the mass flow rate measured with the gasometer is 2.4 kg/min. What is the coefficient of discharge of the orifice at these conditions? P = /760 = kpa T = T i k-1 k P/P 0.4 = /200 = 4.66 i 1.4 v i = RT i /P i = , v e = RT e /P e = V i = V e A e v i /A i v e = V e (V e 2 - V i 2 )/2 = V e 2 ( )/2 = h i - h e = C p T V e = /( ) = 97.9 m/s ṁ = A e V e /v e = π / = kg/s C D = 2.4/ = 0.58

481 (Adv.) Atmospheric air is at 20 C, 100 kpa with zero velocity. An adiabatic reversible compressor takes atmospheric air in through a pipe with cross-sectional area of 0.1 m 2 at a rate of 1 kg/s. It is compressed up to a measured stagnation pressure of 500 kpa and leaves through a pipe with cross-sectional area of 0.01 m 2. What is the required compressor work and the air velocity, static pressure and temperature in the exit pipeline? C.V. compressor out to standing air and exit to stagnation point. ṁ h o1 + W _ c = ṁ(h + V2 /2) ex = ṁh o,ex ṁs o1 = ṁs o ex P r,o,ex = P r,o1 (P st,ex /P o1 ) = 1.028(500/100) = 5.14 W _ T o,ex = 463, h o,ex = , h o1 = c = ṁ(h o,ex - h ) = 1( ) = kw o1 P ex = P o,ex (T ex /T o,ex ) k/(k-1) T ex = T o,ex - V 2 ex /2C p ṁ = 1 kg/s = (ρav) ex = P ex AV ex /RT ex Now select 1 unknown amongst P ex, T ex, V ex and write the continuity eq. ṁ and solve the nonlinear equation. Say, use T ex then V ex = 2C p (T o,ex - T ex ) ṁ = 1 kg/s = P o ex (T ex /T o,ex )k/k-1 A 2C p (T o,ex - T ex )/RT ex solve for T ex /T o,ex (close to 1) T ex = K V ex = 28.3 m/s, P ex = kpa

482 16-25 English Unit Problems 16.44E Steam leaves a nozzle with a velocity of 800 ft/s. The stagnation pressure is 100 lbf/in 2, and the stagnation temperature is 500 F. What is the static pressure and temperature? h 1 = h o1 - V 1 2 /2g c = s 1 = s 0 = Btu/lbm R Btu 778 = lbm (h, s) Computer table P 1 = 88 lbf/in. 2, T = 466 F 16.45E Air leaves the compressor of a jet engine at a temperature of 300 F, a pressure of 45 lbf/in 2, and a velocity of 400 ft/s. Determine the isentropic stagnation temperature and pressure. h o1 - h 1 = V 1 2 /2g c = / = 3.2 Btu/lbm T o1 - T - 1 = (h o1 - h 1 )/C p = 3.2/0.24 = 13.3 T o1 = T + T = = F = 773 R k P o1 = P 1 T o1 /T 1 k-1 = 45(773/759.67) 3.5 = lbf/in E A meteorite melts and burn up at temperatures of 5500 R. If it hits air at 0.75 lbf/in. 2, 90 R how high a velocity should it have to reach such temperature? Assume we have a stagnation T = 5500 R h 1 + V 1 2 /2 = h stagn. Extrapolating from table C.6, h stagn. = , h 1 = 21.4 V 1 2 /2 = = Btu/lbm V 1 = = 8738 ft/s

483 E A jet engine receives a flow of 500 ft/s air at 10 lbf/in. 2, 40 F inlet area of 7 ft 2 with an exit at 1500 ft/s, 10 lbf/in. 2, 1100 R. Find the mass flow rate and thrust. ṁ = ρav; ideal gas ρ = P/RT ṁ = (P/RT)AV = = lbm/s F net = ṁ (V ex - V in ) = ( ) / = 5877 lbf 16.48E A water turbine using nozzles is located at the bottom of Hoover Dam 575 ft below the surface of Lake Mead. The water enters the nozzles at a stagnation pressure corresponding to the column of water above it minus 20% due to friction. The temperature is 60 F and the water leaves at standard atmospheric pressure. If the flow through the nozzle is reversible and adiabatic, determine the velocity and kinetic energy per kilogram of water leaving the nozzle. P = ρg Z g c = g( Z/v)/g c = 575/( ) = 249 lbf/in. 2 P ac = 0.8 P = lbf/in. 2 and Bernoulli v P = V ex 2 /2 V ex = 2v P = 2g Z = = ft/s V ex 2 /2 = v P = g Z/g c = 575/778 = Btu/lbm 16.49E Find the speed of sound for air at 15 lbf/in.2, at the two temperatures of 32 F and 90 F. Repeat the answer for carbon dioxide and argon gases. From eq we have c 32 = krt = = 1087 ft/s c 90 = = 1149 ft/s For Carbon Dioxide: R = 35.1, k = c 32 = c 90 = = 846 ft/s = ft/s For Argon: R = 38.68, k = c 32 = = 1010 ft/s c 90 = = 1068 ft/s

484 E Air is expanded in a nozzle from 300 lbf/in.2, 1100 R to 30 lbf/in. 2. The mass flow rate through the nozzle is 10 lbm/s. Assume the flow is reversible and adiabatic and determine the throat and exit areas for the nozzle. Mach # Velocity P * = P 2 o k+1 k k-1 = = lbf/in. 2. Area Density 300 psia P 30 psia T * = T o 2/(k+1) = = R v * = RT * /P * = /( ) = ft 3 /lbm c * = krt * = = 1484 ft/s A * = ṁv * /c * = /1484 = ft 2 P 2 /P o = 30/300 = 0.1 Table A.11 M * 2 = = V 2 /c* V 2 = = 2524 ft/s T 2 = = R v 2 = RT 2 /P 2 = /(30 144) = ft 3 /lbm A 2 = ṁv 2 /V 2 = / 2524 = ft 2

485 E A convergent nozzle has a minimum area of 1 ft 2 and receives air at 25 lbf/in. 2, 1800 R flowing with 330 ft/s. What is the back pressure that will produce the maximum flow rate and find that flow rate? P * = ( 2 P o k+1 ) k k-1 = Critical Pressure Ratio Find P o : C p = ( )/50 = from table C.6 h 0 = h 1 + V 1 2 /2 T 0 = T i + V 2 /2C p T 0 = / = => T* = T o = R P 0 = P i (T 0 /T i ) k/(k-1) = 25 ( /1800) 3.5 = P * = P o = = lbf/in 2 ρ * = P* * = RT = V = c = krt * = = ft/s ṁ = ρav = = lbm/s 16.52E A jet plane travels through the air with a speed of 600 mi/h at an altitude of ft, where the pressure is 5.75 lbf/in. 2 and the temperature is 25 F. Consider the diffuser of the engine where air leaves at with a velocity of 300 ft/s. Determine the pressure and temperature leaving the diffuser, and the ratio of inlet to exit area of the diffuser, assuming the flow to be reversible and adiabatic. V = 600 mi/h = 880 ft/s v 1 = /( ) = , h 1 = , P r1 = h o1 = /( ) = Btu/lbm P ro1 = P o1 = = 8.9 lbf/in. 2 h 2 = /( ) = T 2 = 542 R, P r2 = => P 2 = / = 8.48 lbf/in. 2 v 2 = /( ) = ft 3 /lbm A 1 /A 2 = (31.223/23.67)(300/880) = 0.45

486 E The products of combustion enter a nozzle of a jet engine at a total pressure of 18 lbf/in. 2, and a total temperature of 1200 F. The atmospheric pressure is 6.75 lbf/in. 2. The nozzle is convergent, and the mass flow rate is 50 lbm/s. Assume the flow is adiabatic. Determine the exit area of the nozzle. P crit = P 2 = = 9.5 lbf/in. 2 > P amb The flow is then choked. T 2 = = 1382 R V 2 = c 2 = = 1822 ft/s v 2 = / = 53.9 A 2 = /1822 = ft E Repeat Problem assuming a diffuser efficiency of 80%. From solution to h 1 = , h o1 = , P r1 = , v 1 = h 2 = , P r2 = , P ro1 = η D = (h 3 - h 1 )/(h o1 - h 1 ) = 0.8 h 3 = , P r3 = P o2 =P 3 = / = lbf/in. 2 P 2 = P o2 P r2 /P ro1 = / = lbf/in T 2 = 542 R v 2 = 144 = ft 3 /lbm A 1 /A 2 = v 1 V 2 /v 2 V 1 = /( ) = 0.414

487 E A 50-ft 3 uninsulated tank contains air at 150 lbf/in. 2, 1000 R. The tank is now discharged through a small convergent nozzle to the atmosphere at 14.7 lbf/in. 2 while heat transfer from some source keeps the air temperature in the tank at 1000 R. The nozzle has an exit area of ft 2. a. Find the initial mass flow rate out of the tank. b. Find the mass flow rate when half the mass has been discharged. c. Find the mass of air in the tank and the mass flow rate out of the tank when the nozzle flow changes to become subsonic. P B /P o = 14.7/150 = < (P * /P o ) crit = a. The flow is choked, max possible flow rate M E =1 ; P E = = lbf/in. 2 T E = T * = = R V E = c = krt * = = 1415 ft/s v E = RT * /P E = /( ) = ft 3 /lbm Mass flow rate is : ṁ 1 = AV E /v E = /3.895 = lbm/s b. m 1 = P 1 V/RT 1 = / = lbm m 2 = m 1 /2 = lbm, P 2 =P 1 /2 = 75 lbf/in. 2 ; T 2 = T 1 P B /P 2 = 14.7/75 = < (P * /P o ) crit The flow is choked and the velocity is the same as in 1) P E = = lbf/in. 2 ; M E =1 ṁ 2 = AV E P E /RT E = = lbm/s c. Flow changes to subsonic when the pressure ratio reaches critical. P B /P o = P 3 = lbf/in. 2 m 3 = m 1 P 3 /P 1 = lbm ; T 3 = T 1 V E = 1415 ft/s ṁ 3 = AV E P E /RT E = = lbm/s

488 E Air enters a diffuser with a velocity of 600 ft/s, a static pressure of 10 lbf/in. 2, and a temperature of 20 F. The velocity leaving the diffuser is 200 ft/s and the static pressure at the diffuser exit is 11.7 lbf/in. 2. Determine the static temperature at the diffuser exit and the diffuser efficiency. Compare the stagnation pressures at the inlet and the exit. V 1 2 T o1 = T 1 + = g c C 2 /( ) = 510 R p T o2 = T o1 T 2 = T o2 - V 2 2 /2C p = /( ) = R T o2 - T 2 = k-1 P o2 - P 2 P T 2 k P o2 - P 2 = P o2 = lbf/in. 2 2 T ex,s = T 1 (P o2 /P 1 ) (k-1)/k = = R η D = T ex,s - T 1 T o1 - T 1 = = 0.844

489 E A convergent nozzle with exit diameter of 1 in. has an air inlet flow of 68 F, 14.7 lbf/in. 2 (stagnation conditions). The nozzle has an isentropic efficiency of 95% and the pressure drop is measured to 20 in. water column. Find the mass flow rate assuming compressible adiabatic flow. Repeat calculation for incompressible flow. Convert P to lbf/in 2 P = 20 in H 2 O = = lbf/in 2 T 0 = 68 F = R P 0 = 14.7 lbf/in 2 Assume inlet V i = 0 P e = P 0 - P = = lbf/in 2 P e T e = T 0 ( ) k-1 k = ( P ) = V e 2 /2 = h i - h e = C p (T i - T e ) = 0.24 ( ) = Btu/lbm V 2 e ac /2 = η V 2 e /2 = = Btu/lbm V e ac = = ft/s T e ac = T i - ρ e ac = P e RT e ac = V 2 e ac /2 = C p 0.24 = = lbm/ft ṁ = ρav = π 4 ( 1 12 ) = lbm/s Incompressible: ρ i = P 0 RT 0 = = lbm/ft V 2 e /2 = v i (P i - P e ) = P = = Btu/lbm ρ i V 2 e ac /2 = η V 2 e /2 = = Btu/lbm V e ac = = ft/s ṁ = ρav = π 4 ( 1 12 ) = lbm/s