Practice Problems on Conservation of Energy. heat loss of 50,000 kj/hr. house maintained at 22 C

Transcription

1 COE_10 A passive solar house that is losing heat to the outdoors at an average rate of 50,000 kj/hr is maintained at 22 C at all times during a winter night for 10 hr. The house is to be heated by 50 glass containers each containing 20 L of water that is heated to 80 C during the day by absorbing solar energy. A thermostat-controlled, 15 kw back-up electric resistance heater turns on whenever necessary to keep the house at 22 C. a. How long will the electric heating system need to run during the night? b. How long would the electric heater run during the night if the house did not incorporate solar heating? heat loss of 50,000 kj/hr house maintained at 22 C 50 glass containers filled with 20 L of water each at an initial temperature of 80 C t = 4.8 hrs t = 9.3 hrs C. Wassgren, Purdue University Page 1 of 20 Last Updated: 2010 Sep 03

2 COE_12 A steel rivet of mass 2 lb m, initially at 1000 F, is placed in a large tank containing 5 ft 3 of liquid water initially at 70 F. Eventually, the rivet and water cool back to 70 F as a result of heat transfer to the surroundings. Taking the rivet and water as the system, determine the heat transfer, in Btu, to the surroundings. The specific heat for steel is 0.11 Btu/(lb m R). Q into system = -205 Btu C. Wassgren, Purdue University Page 2 of 20 Last Updated: 2010 Sep 03

3 COE_13 Consider a large classroom on a hot summer day with 150 students, each dissipating 60 W of sensible heat. All the lights, with 4.0 kw of rated power, are kept on. The room has no external walls, and thus heat gain through the walls and the roof is negligible. Chilled air is available at 15 C and the temperature of the return air is not to exceed 25 C. Determine the required flow rate of air, in kg/s, that needs to be supplied to the room to keep the average temperature of the room constant. m 1.30 kg s C. Wassgren, Purdue University Page 3 of 20 Last Updated: 2010 Sep 03

4 COE_14 Re-visit the hydraulic jump problem. Estimate the temperature change in the fluid across the jump if heat transfer to the surroundings is negligible. Express your answer in terms of the given variables (H 1, U 1, g) and any thermodynamic properties that may be required. U H gh H T2 T c H2 c H1 C. Wassgren, Purdue University Page 4 of 20 Last Updated: 2010 Sep 03

5 COE_16 One kilogram of air, initially at 5 bars (abs) and 77 C, and 3 kg of carbon dioxide (CO 2 ), initially at 2 bars (abs) and 177 C, are confined to opposite sides of a rigid, well-insulated container. The partition is free to move and allows conduction from one gas to the other without energy storage in the partition itself. Determine: a) the final equilibrium temperature b) and the final pressure You may assume that the specific heats for both the air and CO 2 remain constant over the range of temperatures: c v,air =0.726 kj/(kg K); c v,co2 =0.750 kj/(kg K); c p,air =1.013 kj/(kg K); c p,co2 =0.939 kj/(kg K) CO 2 3 kg initially at 2 bars 177 C air 1 kg initially at 5 bars 77 C well-insulated, rigid tank movable, conductive partition m c T m c T Tf m c m c tank CO v,co CO, i air v,air air, i CO 2 v,co2 air v,air Tf pf m R m R V CO2 CO2 air air m m R T 2 air 2 2 i R T Vtank mco m 2 air CO 2, i air, i pco 2, i pair, i CO CO CO, air air, i VCO 2, i Vair, i C. Wassgren, Purdue University Page 5 of 20 Last Updated: 2010 Sep 03

6 COE_17 In a proposed jet propulsion system for an automobile, air is drawn in vertically through a large intake in the roof at a rate of 3 kg/s, the velocity through this intake being small. Ambient pressure and temperature are 100 kpa (abs) and 30 C. This air is compressed, heated, and then discharged horizontally out of a nozzle at the rear of the automobile at a velocity of 500 m/s and a pressure of 140 kpa (abs). If the rate of heat addition to the air stream is 600 kw, find the nozzle discharge area and the thrust developed by the system. 0 = 4.6*10-3 m 2 F = 1670 N C. Wassgren, Purdue University Page 6 of 20 Last Updated: 2010 Sep 03

7 COE_20 A 12 cm tall, 6 cm diameter cylindrical can of cold water with a temperature of 5 C is contained within a 1.5 cm thick insulating foam jacket (a huggie ) which has a thermal conductivity of W/(m K). The air temperature at the outer surface of the foam remains constant at 24 C. Neglecting heat transfer from the top and bottom surfaces of the can, determine the temperature of the water after five hours. You may also assume that the fluid properties within the can remain uniform. T i (t = 5 min) = 15.1 C C. Wassgren, Purdue University Page 7 of 20 Last Updated: 2010 Sep 03

8 COE_21 A homeowner is considering replacing their tank hot water heater with a tankless one. Two models are under consideration: one that heats the water with natural gas and one that uses electricity. The incoming water stream has a temperature T i, the desired outgoing water temperature is T o, and the maximum expected flow rate is m. a. Determine the power required to heat the water if the incoming water temperature is 47 F, the desired outgoing temperature is 110 F, and the volumetric flow rate is 2.2 gpm (corresponding to a typical shower flow rate). b. If the cost of natural gas is $0.5245/therm (where 1 therm = 100,000 Btu) and the efficiency of the water heater is 82%, determine the cost of heating the water for each minute of operation. c. If the cost of electricity is $0.1036/kWh and the efficiency of the water heater is 91%, determine the cost of heating the water for each minute of operation. P = 1160 Btu/min C gas = $0.0074/min C elec = $0.0386/min C. Wassgren, Purdue University Page 8 of 20 Last Updated: 2010 Sep 03

9 COE_cv03 Air at 10 C and 80 kpa (abs) enters the diffuser of a jet engine steadily with a velocity of 200 m/s. The inlet area of the diffuser is 0.4 m 2. The air leaves the diffuser with a velocity that is very small compared with the inlet velocity. Determine a. the mass flow rate of the air and b. the temperature of the air leaving the diffuser. You may assume adiabatic flow through the diffuser. m = 78.8 kg/s T 2 = K 30 ºC C. Wassgren, Purdue University Page 9 of 20 Last Updated: 2010 Sep 03

10 COE_cv04 A well-insulated valve is used to throttle steam from 8 MPa and 500 C to 6 MPa. Determine the final temperature of the steam. T o = 490 C C. Wassgren, Purdue University Page 10 of 20 Last Updated: 2010 Sep 03

11 COE_cv05 Air flows through a nozzle with an inlet diameter of 200 mm, velocity of 400 m/s, pressure of 7 kpa, and temperature of 420 C. The nozzle exit diameter is adjusted such that the exiting velocity is 700 m/s. Determine: a. the exit temperature, and b. the mass flow rate through the nozzle T o = 264 C m = 0.44 kg/s C. Wassgren, Purdue University Page 11 of 20 Last Updated: 2010 Sep 03

12 COE_cv06 Air at 100 kpa and 280 K is compressed steadily to 600 kpa and 400 K. The mass flow rate of the air is 0.02 kg/s and a heat loss of 16 kj/kg occurs during the process. Assuming the changes in kinetic and potential energies are negligible, determine the necessary power input to the compressor. W = 2.74 kw shaft, on CV C. Wassgren, Purdue University Page 12 of 20 Last Updated: 2010 Sep 03

13 COE_cv07 Consider an ordinary shower where hot water at 140 F is mixed with cold water at 50 F. If it is desired that a steady stream of warm water at 110 F be supplied, determine the ratio of the mass flow rates of the hot to cold water. Assume the heat losses from the mixing chamber to be negligible and the mixing to take place at a pressure of 20 psia. m H 2.0 m C C. Wassgren, Purdue University Page 13 of 20 Last Updated: 2010 Sep 03

14 COE_cv09 Air, treated as an ideal gas, flows through the turbine and heat exchanger arrangement shown in the figure with the data for the flow streams also indicated. Heat transfer to the surroundings can be neglected, as can all kinetic and potential energy effects. Determine the temperature T 3, in K, and the power output of the second turbine, in kw, at steady state. T1 power out = 10,000 kw T 2 = 1100 K p 2 = 4 bars T 3 =? p 3 = p 2 T2 power out =? air T 1 = 1400 K p 1 = 12 bars heat exchanger T 4 = 980 K p 4 = 1 bar T 6 = 1200 K p 6 = p 5 air T 5 = 1480 K p 5 = 1 bar mass flow rate = 1200 kg/min T 3 = 1300 K W on 10,570 kw CV C. Wassgren, Purdue University Page 14 of 20 Last Updated: 2010 Sep 03

15 COE_cv12 The velocity profile for a particular pipe flow is linear from zero at the wall to a maximum of u c at the centerline. Determine the average velocity and the kinetic energy correction factor. u c r R 1 u u 3 C C. Wassgren, Purdue University Page 15 of 20 Last Updated: 2010 Sep 03

16 COE_cv13 The figure below shows a solar collector panel with a surface area of 32 ft 2. The panel receives energy from the sun at a rate of 150 Btu/hr per ft 2 of collector surface. Forty percent of the incoming energy is lost to the surroundings. The remainder is used to warm liquid water from 130 to 160 F. The water passes through the solar collector with a negligible pressure drop. Neglecting kinetic and potential energy effects, determine at steady state the mass flow rate of the water in lb m /min. How many solar collectors would be needed to provide a total of 40 gal of 160 F water in 30 min? 150 Btu/(hr ft 2 ) water in, T=130 F 40% loss solar collector panel, A=32 ft 2 water out, T=160 F m = 96.1 lb m /hr = 1.60 lb m /min Seven solar collectors would be required. C. Wassgren, Purdue University Page 16 of 20 Last Updated: 2010 Sep 03

17 COE_cv14 Carbon dioxide flows through a constant area duct. At the inlet to the duct, the velocity is 120 m/s and the temperature and pressure are 200 C and 700 kpa (abs), respectively. Heat is added to the flow in the duct and at the exit of the duct the velocity is 240 m/s and the temperature is 450 C. Find the amount of heat being added to the carbon dioxide per unit mass of gas and the mass flow rate through the duct per unit cross-sectional area of the duct. Assume that the specific heat ratio for carbon dioxide is 1.3 and the gas constant is 189 J/(kg K). q into CV = 226 kj/kg m kg/s/m A C. Wassgren, Purdue University Page 17 of 20 Last Updated: 2010 Sep 03

18 COE_cv15 If the water in a well insulated, 50 gal electric water heater initially has the same temperature as the inlet water temperature (55 F), determine how long it will take for the water at the outlet to reach a comfortable shower temperature of 105 F if the water flows continuously in the shower at a rate of 2 gal/min (a typical flow rate for a shower). What will be the steady state temperature in the water heater for these conditions? This particular water heater can provide 4500 W of power to the heating element. The inlet supply line has a pressure of 50 psi and the pressure in the tank is 70 psi. Using the given parameters, it is not possible to reach the desired shower temperature. In fact, the steady state temperature is only 70 F. In order to reach the desired temperature, one would need to decrease the flow rate. A flow rate of approximately 0.6 gpm will give a steady state temperature of 105 F, but it will take a long time to get there and you ll waste a lot of water! C. Wassgren, Purdue University Page 18 of 20 Last Updated: 2010 Sep 03

19 COE_unsteady_cv01 Steam at 15 bars and 320 C is contained in a large tank. Connected to the tank through a valve is a turbine followed by a small initially evacuated vessel with a volume of 0.6 m 3. The valve is opened and the vessel fills with steam until the pressure is 15 bars and the temperature is 400 C. The valve is then closed. The filling process takes place adiabatically and kinetic and potential energy effects are negligible. Determine the amount of work developed by the turbine, in kj. Steam reservoir 15 bars 320 C valve T work out =? initially evacuated vessel V = 0.6 m 3 W kj on CV C. Wassgren, Purdue University Page 19 of 20 Last Updated: 2010 Sep 03

20 COE_unsteady_cv02 A rigid, insulated tank that is initially evacuated is connected through a valve to a supply line that carries steam at 1 MPa and 300 C. The valve is opened and steam is allowed to flow slowly into the tank until the pressure reaches 1 MPa at which point the valve is closed. Determine the final temperature of the steam in the tank. steam at 1 MPa and 300 C insulated, rigid tank; initially empty; after filling, p=1 MPa T CV,f = C C. Wassgren, Purdue University Page 20 of 20 Last Updated: 2010 Sep 03