The rate of heat transfer from the condensing steam to the cooling water is given by. (h 4 h 3 )

Transcription

1 Capter 4 Exaple Stea enters te condenser of a vapor power plant at 0. bar wit a quality of 0.95 and condensate exits at 0. bar and 45 o C. Cooling water enters te condenser in a separate strea as a liquid at 0 o C and exits as a liquid at 5 o C wit no cange in pressure. Heat transfer fro te outside of te condenser and canges in te kinetic and potential energies of te flowing streas can be ignored. For steady-state operation, deterine (a) te ratio of te ass flow rate of cooling water to te ass flow rate of te condensing stea. (b) te rate of energy transfer fro te condensing stea to te cooling water, in kj per kg of stea passing troug te condenser. Solution (a) Deterine te ratio of te ass flow rate of cooling water to te ass flow rate of te condensing stea. Te rate of eat transfer fro te condensing stea to te cooling water is given by In tis equation, Q ( ) c ( 4 ) is te ass flow rate of te ot strea (or condensing stea) and c is te ass flow rate of te cold strea (or cooling water). Te specific entalpies of te inlet and exit streas are listed in Table E4.4-. For strea (), (), and (4) te entalpies are taken as saturated liquid at te listed teperature so tat f (T ), f (T ), and 4 f (T 4 ). 6 Moran, M. J. and Sapiro H. N., Fundaentals of Engineering Terodynaics, Wiley, 008, pg

2 Table E4.4- Stea properties fro CATT Specific Tep Pressure Entalpy Quality Pase C MPa kj/kg Liquid Vapor Mixture Saturated Liquid Saturated Liquid Saturated Liquid Solving for te ratio of te ass flow rate gives c (b) Deterine te rate of energy transfer fro te condensing stea to te cooling water, in kj per kg of stea passing troug te condenser. Q ( ) Q Q kj/kg Exaple A supply line carries a two-pase liquid-vapor ixture of stea at 00 lbf/in. A sall fraction of te flow in te line is diverted troug a trottling calorieter and exausted to te atospere at 4.7 lbf/in. Te teperature of te exaust stea is easured as 50 o F. Deterine te quality of te stea in te supply line. Solution A trottling calorieter is a device used to reduce te pressure of a gas or liquid strea. Tis can be siply done by eans of a partially opened valve or a porous plus as sown in Figure E4.4-4a. Figure E4.4-4a Exaples of trottling devices. 7 Moran, M. J. and Sapiro H. N., Fundaentals of Engineering Terodynaics, Wiley, 008, pg

3 Trottling ay be used as a eans of controlling te flow rate (valves and flow regulators), aintaining a constant downstrea pressure (pressure regulator), or easuring te flow rate (flow etering orifices). For trottling devices we usually ake te following assuptions: * Steady state steady flow (SSSF) * No work or eat transfer * Potential and kinetic energy are negligible relative to oter energy ters Table E4.4-4 Stea properties fro CATT Specific Tep Pressure Entalpy Quality Pase F psia Btu/lb Saturated Liquid Saturated Vapor Supereated Vapor Making energy balance between () and () we obtain V V Q W s + g(z z ) + ( ) V V For g(z z ) 0, ( ) 0, Q W 0, and s 0, we ave ( x ) f + x g f + x ( g f ) Te quality of te stea in te supply line is ten x f g f

4 4.5 Energy balance on Integrated or Transient Syste In real life applications, we usually encounter integrated systes consisting of any coponents discussed in previous sections. Exaple An industrial process discarges 0 5 ft /in of gaseous cobustion products at 400 o F, at. As sown in Figure E4.5-, a proposed syste for utilizing te cobustion products cobined a eat-recovery stea generator wit a turbine. At steady state, cobustion products exit te stea generator at 60 o F, at and a separate strea of water enters at 40 psia, 0 o F wit a ass flow rate of 75 lb/in. At te exit of te turbine, te pressure is psia and te quality is 9%. Heat transfer fro te outer surfaces of te strea generator and turbine can be ignored, as can te canges in kinetic and potential energies of te flowing streas. Tere is no significant pressure drop for te water flowing troug te stea generator. Te cobustion products can be odeled as air as an ideal gas. (a) Deterine te power developed by te turbine, in Btu/in. (b) Deterine te turbine inlet teperature, in o F. Solution (a) Deterine te power developed by te turbine, in Btu/in. Applying te steady state energy balance on te syste consisting of te stea generator and te turbine we obtain wit negligible canges in kinetic and potential energies Q W s 8 Moran, M. J. and Sapiro H. N., Fundaentals of Engineering Terodynaics, Wiley, 008, pg

5 Since te gas and water streas do not ix and eat transfer is negligible, we ave, 5, and W s ( ) + ( 5 ) Te ass flow rate is given by law ( ) AV v, were te specific volue v can be obtained fro ideal gas RT v p R / M T ( ) p v 545 ft lbf o 8.97 lb R 4.7 psia o (860 R) ft 44 in.667 lb/ft Te ass flow rate is ten 5 0 ft / in.667 lb/ft 90.6 lb/in Te air properties are listed in Table E4.5-a and stea properties are listed in Table E4.5-b Table E4.5-a Air properties fro CATT Specific Entalpy Tep Pressure (Mass) F psia Btu/lb Table E4.5-b Stea properties fro CATT Tep Pressure Entalpy Quality Pase F psia Btu/lb Copressed Liquid Liquid Vapor Mixture Te power developed by te turbine is 4-

6 W s ( ) + ( 5 ) W s (90.6 lb/in)( ) Btu/lb + (75 lb/in)(70.4 0) Btu/lb W s 49,980 Btu/in (a) Deterine te turbine inlet teperature, in o F. We need to know properties at (4). Since pressure drop is negligible, p 4 40 psia, te entalpy at (4) can be deterine fro te energy balance around te stea generator: 0 ( ) + ( 4 ) 4 + ( ) Btu/lb ( ) Btu/lb 4.7 Btu/lb Te inlet teperature at te turbine inlet is 55.8 o F fro Table E4.5-c. Table E4.5-c Stea properties fro CATT Specific Tep Pressure Entalpy Quality Pase F psia Btu/lb Supereated Vapor 4-4

7 Exaple A tank aving a volue of 0.85 initially contains water as a two-pase liquid-vapor ixture at 60 o C and a quality of 0.7. Saturated water vapor at 60 o C is slowly witdrawn troug a pressure-regulating valve at te top of te tank as energy is transferred by eat to aintain te pressure constant in te tank. Tis continues until te tank is filled wit saturated vapor at 60 o C. Deterine te aount of eat transfer, in kj. Neglect all kinetic and potential energy effects. Solution Te ass balance for te content, cv, in te tank is given by d cv dt e Te energy balance for te tank can be written as du cv dt Q cv e e Q cv + e dcv dt Integrating te energy equation wit e constant We obtain du cv Q cvdt + edcv U cv Q cv + e cv Solving for te eat transfer gives Q cv U cv e cv u u e ( ) 9 Moran, M. J. and Sapiro H. N., Fundaentals of Engineering Terodynaics, Wiley, 008, pg

8 In tis equation, and are te initial and final aounts of ass witin te tank, respectively. Te stea properties are listed intable Table E4.5- Stea properties fro CATT Specific Internal Specific Tep Pressure Volue Energy Entalpy Quality Pase C MPa /kg kj/kg kj/kg Liquid Vapor Mixture Saturated Vapor Te ass initially contained in te tank is V v /kg 8.4 kg Te ass finally contained in te tank is V v /kg 0.4 kg Te eat transfer is ten Q cv u u e ( ) Q cv (0.4)(599) (8.4)(58) (797)( ) 4,60 kj 4-6