S.A. Klein and G.F. Nellis Cambridge University Press, 2011

Transcription

1 A. Entropy Balances 6.A-1 The two places that thermal energy is transferred from a car are through the engine block (to the coolant) and through the exhaust pipe. A company is marketing a device that scavenges heat from your car in order to produce some auxiliary power. The device is shown schematically in Figure 6.A-1. W = 1.2 kw T ex = 225 C Q = 0.7 kw ex power plant Q C T C = 32 C Q = 1.5 kw eb T eb = 92 C Figure 6.A-1: Device for scavenging heat from an engine to produce auxiliary power. You have been hired to assess the feasibility of this device for some potential investors. The company claims that the device receives Q eb = 1.5 kw from the engine block and the engine block temperature is T eb = 92ºC. The device also receives Q ex = 0.7 kw from the exhaust pipe and the exhaust pipe temperature is T ex = 225ºC. The device rejects heat to surrounding air and is supposed to continue to function as advertised even in climates where the air temperature reaches T C = 32ºC. The company claims that they have measured a rate of work transfer equal to W = 1.2 kw under these conditions. The device operates at steady state. a.) Assess the company's claim using the Second Law of Thermodynamics - would you suggest that the investors invest money or not? b.) What is the maximum rate at which the engine can produce power under these conditions?

2 6.A-2 Whenever there is a temperature difference, there exists the potential to produce power. One innovative solution that has been proposed for our energy crisis involves towing an iceberg into the harbor of a coastal city and making use of the temperature difference that will exist between the water in the harbor and the ice. The relatively warm water in the harbor will provide heat transfer to a heat engine that will reject heat to the ice (causing it to slowly melt), as shown in Figure 6.A-2. The result is electricity generated without consuming coal, natural gas, or any fossil fuel. electrical power W iceberg M = 1000 ton Δh fus = kj/kg Q C T C =0 C Q H TH =15 C heat engine Figure 6.A-2: A power plant operating between the water in the harbor and an iceberg. You have been asked to assess this idea from an engineering perspective. Your research suggests that it will be possible to tow one of the smaller icebergs that naturally occur, with mass M ice = 1000 tons, into a harbor where the water temperature is relatively high, T H = 15ºC. The temperature of the ice is T C = 0ºC. The latent heat of fusion for water is Δh fus = kj/kg (this is the amount of heat required to melt ice per unit mass). You would like to determine the absolute largest amount of electrical work transfer that can be produced using this technique and the value of that electricity. A reasonable average electricity cost is about $0.08/kW-hr but you think that you can get a premium for the electricity produced by this technique since it is a "renewable energy technology". Therefore, assume that the cost of electricity is ec = $0.12/kWhr. a.) Determine the maximum amount of money that can be made using this technique and comment on whether this seems like it is an idea that is worth pursuing.

3 6.A-3 Figure 6.A-3 illustrates a device advertised on the internet for producing power. W = 25kW. T H =80 C Q H = 3kW Device for producing power T C =20 C Q C Figure 6.A-3: Device for producing power. The manufacturer of the device claims that it accepts a heat transfer at a rate of Q H = 3.0 kw from a low-grade source of geothermal heat at T H = 80ºC and rejects heat at rate Q C to the atmosphere at T C = 20ºC. The device operates at steady state and produced work at a rate of W = 2.5 kw. There are no other heat or work transfers from the device. a.) What is the rate at which the device rejects heat to the atmosphere, Q C? b.) Is this device possible? Justify your answer using an entropy balance.

4 6.A-4 One method for storing hydrogen involves cooling it so that it liquefies. Figure 6.A-4(a) illustrates a single-stage refrigerator that is used for this purpose. A refrigerator that operates at very low temperature is referred to as a cryocooler. hydrogen m = kg/s Tin = 20 C P = 1atm Q H, ss T in W ss hydrogen m = kg/s Tin = 20 C P = 1atm mid-stage heat exchanger Q H Q mid T in T mid W mid-stage cold-head heat exchanger Q Css, single-stage cryocooler cold-head low-stage heat exchanger T mid Q C low-stage cold-head T sat T sat saturated liquid saturated liquid (a) (b) Figure 6.A-4: (a) Single-stage refrigerator and (b) multi-stage refrigerator used to liquefy hydrogen. A stream of gaseous hydrogen at m = kg/s, T in = 20ºC, and P = 1 atm is fed to the device. The gaseous hydrogen is liquefied by flowing through a heat exchanger that interfaces with the cold-head of the cryocooler. The cold head is held at the saturation temperature of the hydrogen, T sat, and hydrogen exits the heat exchanger as saturated liquid. Assume that the cryocooler operates at steady state, is internally reversible, and rejects heat at temperature T in. Assume that there is no pressure loss associated with the flow of hydrogen. a.) Determine the rate of heat transfer to the cold head at T sat, Q Css,. b.) Determine the rate at which heat is rejected from the cryocooler to T in, Q H, ss. c.) Determine the rate of power consumed by the cryocooler, W ss. What is the coefficient of performance of the cryocooler? Figure 6.A-4(b) illustrates an alternative, two-stage cryocooler used for liquefaction. A two-stage cryocooler can provide refrigeration at two separate temperatures. The hydrogen is cooled to an intermediate temperature, T mid = 100 K, by flowing through the mid-stage heat exchanger that interfaces with the mid-stage cold head of the cryocooler. The mid-stage cold head is maintained at T mid. The hydrogen leaving the mid-stage heat exchanger is further cooled by flowing through the low-stage heat exchanger that interfaces with the low-stage cold-head of the cryocooler. The low-stage cold head is held at the saturation temperature of the hydrogen, T sat, and hydrogen exits the low-stage heat exchanger as saturated liquid. Assume that the two-stage cryocooler operates at steady state, is internally reversible, and rejects heat at temperature T in. Assume that there is no pressure loss associated with the flow of hydrogen. d.) Determine the rate of heat transfer to the mid-stage cold head at T mid, Q mid. Q. e.) Determine the rate of heat transfer to the low-stage cold head at T sat, C f.) Determine the rate at which heat is rejected from the two-stage cryocooler to T in, g.) Determine the rate of power consumed by the two-stage cryocooler, W. Q H.

5 h.) Plot the rate of power consumed by the two-stage cryocooler, W, as a function of the midstage temperature, T mid. i.) Your plot from (h) should show that there is an optimal mid-stage temperature. Explain why this optimal value exists. j.) Use the optimization capability in EES to precisely determine the optimal value of T mid.

6 6.A-5 You are designing the power plant shown in Figure 6.A-5. Q = 250 MW H hot heat exchanger T H = 1000 C T H,c Q H power plant W Q C T C,c cold heat exchanger T C =20 C Q C Figure 6.A-5: Power plant. The power plant receives heat at a rate of Q H = 250 MW from a heat source at T H = 1000ºC and rejects heat at a rate Q C to an environment at T C = 20ºC. The power plant requires two heat exchangers. The hot heat exchanger is used to transfer Q H from the thermal reservoir at T H into the cycle and the cold heat exchanger is used to transfer Q C out of the cycle to the thermal reservoir at T C. The size of a heat exchanger is indicated by its total conductance, UA. A heat exchanger with a large value of UA will be physically very large but will allow the heat transfer to pass into or out of the system with a small temperature drop. Therefore, a large hot heat exchanger will allow Q H to enter the cycle at a temperature, T H,c, that is close to T H. Q = T T UA (, ) H H H c A large cold heat exchanger will allow Q C to leave the cycle with a small temperature drop - therefore the cold temperature that is actually seen by the power plant (T C,c in Figure 6.A-5) will be very close to T C. Q C = ( TC, c TC) UA Assume that the hot and cold heat exchangers have the same conductance, UA = 2x10 6 W/K. Also, assume that the power cycle has an efficiency that is 50% of the efficiency that a reversible power plant would exhibit if it were operating between T H,c and T C,c (this is sometimes called the Second Law efficiency of the power plant). The system operates at steady state. a.) Determine the efficiency of the power plant and the rate of power production, W. b.) Determine the rate of entropy generation in the hot heat exchanger, the rate of entropy generation in the cold heat exchanger, and the rate of entropy generation in the power plant itself. c.) Plot the efficiency of the power plant as a function of UA (for 750x10 3 W/K < UA < 1x10 7 W/K). Provide a brief explanation for the shape of your plot. You would like to optimize your power plant design by correctly sizing your heat exchangers. The result from (c) should have shown that the efficiency of you plant increases as UA increases. However, heat exchangers with very large conductance are expensive. You estimate that the cost of the heat exchanger per unit of conductance is UAc = 10$-K/W. The capital cost of the

7 mechanical components in the power cycle is fixed and equal to EngineCost = $35x10 6. You can sell the electrical power that is produced at a rate of ec = 0.065$/kW-hr. d.) Plot the revenue (i.e., the money you receive by selling power) over a t = 5 year period (assume the plant operates continuously) as a function of UA. Ignore the time value of money. Explain the shape of your plot. e.) Overlay on your plot from (d) the capital cost (i.e., the initial investment required to buy the power plant and the heat exchangers) as a function of UA. Explain the shape of your plot. f.) Overlay on your plot from (d) the profit for t = 5 years of operation (i.e., the revenue less the capital cost). You should see that there is a value of UA that maximizes your profit - explain why this is the case. g.) What is the economically optimal value of UA obtained using the Min/Max function in EES? What is the corresponding value of profit?

8 6.A-6 Figure 6.A-6 illustrates a rigid, insulated tank that is divided into two parts by an interior partition. rigid insulated tank V 2 =0.25m 3 x 2 =1.0 P 2 = 200 kpa V 3 =0.5m 3 V 1 =0.25m 3 T 1 = 250 C P 1 =1.0MPa partition T 3 and P 3 Figure 6.A-6: Rigid tank divided into two parts by a partition. The volume below the partition is V 1 = 0.25 m 3 and contains water at T 1 = 250ºC and P 1 = 1.0 MPa. The volume above the partition is V 2 = 0.25 m 3 and contains saturated water vapor at P 2 = 200 kpa. At some time, the partition is ruptured allowing the water below the partition to mix with the water above the partition. Eventually, all of the water in the tank comes to new equilibrium at a uniform temperature, T 3, and pressure, P 3. a.) What is the final temperature and pressure in the tank, T 3 and P 3? b.) Determine S gen, the entropy generated by this process. c.) Use EES to prepare a T-v and a T-s diagram for water and overlay your states on these diagrams.

9 6.A-7 Figure 6.A-7(a) illustrates a block of metal that is initially at T 1 = 500ºC in surroundings at T o = 20ºC. The mass of the block is m = 10 kg and the specific heat capacity is c = 1500 J/kg-K. You may model the block as being incompressible. block of metal m =10kg Q a c = 1500 J/kg-K T 1 = 500 C surroundings at T o =20 C reversible steady state heat engine Q b W b block of metal m =10kg c =1500J/kg-K T 1 = 500 C surroundings at T o =20 C (a) (b) Figure 6.A-7: Block of metal (a) coming to equilibrium with the environment and (b) transferring energy to the environment through a reversible, steady state heat engine. In Figure 6.A-7(a), the block of metal transfers heat directly to the atmosphere. This process continues until the metal reaches T 2 = T o. a.) What is the entropy generated by this process? Figure 6.A-7(b) illustrates the same block of metal at the same initial temperature. In this case, though, the metal transfers heat to a reversible heat engine that is operating at steady state. The heat engine rejects heat to the atmosphere at T o and produces a work transfer. The equilibration process shown in Figure 6.A-7(b) is reversible because there is never any heat transfer through a temperature gradient. b.) Determine the work transfer produced by the heat engine. Entropy generation in some sense represents the lost opportunity to do work. The entropy generation calculated in part (a) represents the fact that you could have produced the work transfer calculated in part (b). c.) If you've done the problem correctly, the product of the temperature of the surroundings and the entropy generation from (a) should be equal to the work transfer from (b). Show that this is so.

10 6.A-8 In order to measure the loss associated with eddy currents in a motor stator designed to operate at low temperature you have developed the experiment shown in Figure 6.A-8. The stator core is placed in a V = 1.5 liter tank of two-phase nitrogen that is placed on a sensitive scale. The tank is equipped with a relief valve that maintains the pressure at P rv = 2 atm by allowing only saturated vapor to escape to ambient at P amb = 1 atm. You have separately measured the mass of the tank and stator alone; therefore, you know that at the start of the experiment the mass of nitrogen in the tank is m 1 = 119 g. The initial pressure in the tank is P 1 = P rv. You energize the core with an oscillating voltage and allow it to operate for t = 50 min. During this time, the mass of nitrogen decreases by m out = 50 g due to the boil off induced by the eddy current losses in the stator as well as heat transfer to the tank from surroundings at T amb = 20ºC. You have separately measured the rate of heat transfer to the tank to be Q = 0.92 W. The eddy current losses manifest themselves as an electrical work transfer to the core, to the nitrogen. two-phase nitrogen m 1 =119g amb W e, that is subsequently transferred as heat pressure relief valve opens at P rv =2atm allows only vapor to escape P 1 =2atm m out =50g tank, V = 1.5 liter stator core W e P amb =1atm Q 0.92 W T amb =20 C amb = Figure 6.A-8: Experiment to measure eddy current losses in a motor stator. a.) What is the rate of eddy current loss suggested by the experimental results (i.e., what is W e )? b.) Determine the entropy generated (J/K) due to the conversion of the electrical work to heat transfer in the stator. Clearly draw the system that you use to accomplish this calculation. c.) Determine the entropy generated (J/K) due to the heat transfer from ambient through the tank wall to the liquid nitrogen. Clearly draw the system that you use to accomplish this calculation. d.) Determine the entropy generated (J/K) due to the flow through the valve. Clearly draw the system that you use to accomplish this calculation. e.) Determine the entropy generated (J/K) due to the re-equilibration of the cold nitrogen leaving the valve with the atmosphere. Clearly draw the system that you use to accomplish this calculation. f.) Draw a system that encompasses all of the sources of entropy generation that were separately considered in parts (b) through (e). Carry out an entropy balance on this system and show that the total entropy generation is the sum of what you calculated in parts (b) through (e).

11 6.A-9 A tank with volume V = 0.2 m 3 contains refrigerant R22 at T tank = -20 C, as shown in Figure 6.A- 9. vapor R22 T tank = -20 C tank, V =0.2m 3 reversible, adiabatic compressor m = 0.1 kg/s for t = 10min P = 18 bar cout, liquid R22 V f,1 =0.05m 3 T tank = -20 C surroundings at W c T = 10 C Figure 6.A-9: Tank of R22 with compressor. Initially the tank contains V f,1 = 0.05 m 3 of liquid R22 and the rest of the tank contains vapor R22. A reversible, adiabatic compressor operating at steady state pulls only saturated vapor from the tank and compresses it to P c,out = 18 bar. The mass flow rate through the compressor is m = 0.1 kg/s. The compressor operates for t = 10 min. During this process, the tank is maintained at a constant temperature by heat transfer from surroundings at T = -10 C. The R22 in the tank remains in a two-phase state throughout the process. a.) What is the temperature of the R22 leaving the compressor (ºC)? b.) What is the power required by the compressor (kw)? c.) What is the initial quality of the R22 in the tank? d.) What is the quality of the R22 that remains in the tank? e.) What is the heat transfer to the tank required to maintain its temperature (kj)? f.) Is this process reversible? If not, explain how entropy is generated by the process and calculate the amount of entropy generated by the process (kj/k).

12 6.A-10 Two air streams enter an industrial heating unit, as shown in Figure 6.A-10. Stream 1 enters at T 1 = 40ºC, P 1 = 150 kpa with a volumetric flow rate of V 1 = 0.30 m 3 /s and stream 2 enters at T 2 = 25ºC, P 1 = 100 kpa with a volumetric flow rate of V 2 = 0.10 m 3 /s. The heating unit is equipped with an electrical resistance heater that draws W e = 11.5 kw. The air exits the unit at T 3 = 55ºC, P 3 = 105 kpa. The heating unit experiences a heat transfer with the surroundings at T = 25 ºC. The heating unit operates at steady state. Model air as an ideal gas with constant specific heat capacities c V = kj/kg-k and c P = kj/kg-k. air T2 = 25 C P2 = 100 kpa V = m /s (2) surroundings at T = 25 C air T1 = 40 C P1 = 150 kpa 3 V = 0.30 m /s 1 (1) (3) air T3 = 55 C P = 105 kpa 3 W = 11.5 kw e Figure 6.A-10: Heating unit. a.) Determine the mass flow rate of air exiting the heating unit at state (3), b.) Determine the rate of heat transfer FROM the device to the surroundings, and. c.) Determine the total rate of entropy generation resulting from operating the device.

13 6.A-11 You have access to steam from a geothermal resource. The steam can be extracted from the ground at a rate of m = 0.25 kg/s and leaves the ground at temperature T 1 = 310 C and atmospheric pressure, P 1 = P o. Atmospheric conditions at this location are T 0 = 20 C and P 0 = 1 atm. The maximum possible rate of work is extracted from the resource when reversible equipment operating at steady state is used to bring the steam to the environmental conditions exchanging heat only with the environment, as shown in Figure 6.A-11. W max steam P 1 =1atm T 1 =310 C reversible equipment steam P 0 =1atm T 0 =20 C Q 0 Figure 6.A-11: Reversible equipment used to extract the maximum rate of power from the flow. a.) What is the maximum rate of work that can be produced using this steam? b.) If you can sell the power for ec = 0.25 $/kw-hr then how much can you afford to pay for the equipment required to generate this power? Assume that you need to payback (i.e., earn enough money to pay for the equipment) in time = 5 years. Neglect the time value of money for your analysis. T 0

14 6.A-12 Figure 6.A-12 illustrates a "vortex tube". inlet T 1 = 305 K P 1 = 650 kpa cold end f =0.25 P 3 = 100 kpa vortex tube hot end T 2 =325K P 2 = 100 kpa Figure 6.A-12: A vortex tube. According to the company that makes the vortex tube, the device takes in high pressure air at pressure P 1 = 650 kpa and temperature T 1 = 305 K and splits it into two streams of air that both leave at lower pressure, P 2 = P 3 = 100 kpa. One of the streams exits through the cold end of the device at a low temperature, T 3, and the other exits through the warm end at a high temperature, T 2 = 325 K. The fraction of the entering mass that leaves through the cold end is f = The vortex tube operates at steady-state, it is adiabatic and there is no work done on or by the device. Model air as an ideal gas with R = 287 J/kg-K and constant c P = 1004 J/kg-K. a.) What is the temperature of the air leaving through the cold end, T 3? b.) Is this device possible? Justify your answer.

15 6.A-13 A well-insulated piston-cylinder device is shown in Figure 6.A-13. gas m g =1kg c V = 620 J/kg-K R =410J/kg-K T 1 =295K P 1 = 200 kpa block m b =2kg c b = 390 J/kg-K Figure 6.A-13: Piston-cylinder device with a block of metal. The piston-cylinder contains m g = 1 kg of gas with c V = 620 J/kg-K (assume constant) and R = 410 J/kg-K. The initial pressure of the gas is P 1 = 200 kpa. The piston-cylinder also contains a metal block that can be modeled as incompressible. The metal block has mass m b = 2 kg and specific heat capacity c b = 390 J/kg-K. The piston is pushed in until the temperature of the gas reaches T g,2 = 422 K. This process happens rapidly so that there is no heat transfer between the gas and the block of metal. The gas undergoes a reversible process. a.) Determine the final pressure of the gas. The piston is locked in place and the gas and the block come to thermal equilibrium. b.) What is the final temperature of the gas and the block? c.) Determine the entropy generated by the thermal equilibration.

16 6.A-14 A rigid tank with volume V tank = 4.0 m 3 is initially filled with saturated liquid water at T 1 = 250 C and quality x 1 = 0.0. a.) What is the initial pressure of the water in the tank, P 1 (MPa)? b.) What is the initial mass of water in the tank, m 1 (kg)? A valve at the bottom of the tank is opened, allowing liquid water to leave the tank (note that the valve is placed at the bottom of the tank so ONLY liquid water passes through the valve). The tank is not insulated, therefore the water in the tank is always maintained at 250 C due to heat transfer with surroundings at T o = 250 C. The valve is shut off at the point where only saturated vapor remains in the tank. State 2 corresponds to the water in the tank when the valve is turned off: x 2 = 1.0 and T 2 = T 1 = 250 C. c.) Locate states 1 and 2 on a qualitative T-v diagram. d.) What is the mass of water that leaves the tank and passes through the valve during the emptying process (kg)? e.) What is the total heat transfer to the tank during the emptying process, from state 1 to state 2 (kj)? The pressure of the water at the exit of the valve is P v,out = 100 kpa throughout the emptying process. State v,out is defined as the state of the water leaving the valve. f.) Locate state v,out on the T-v diagram from part (c). g.) What is the temperature of the water leaving the valve, T v,out ( C)? h.) What is the quality of the water leaving the valve, x v,out? Carry out an entropy balance on the system composed of the tank and the valve during the emptying process. g.) What is the amount of entropy produced by the emptying process (kj/k)?

17 6.A-15 The device shown in Figure 6.A-15 has two chambers that are separated from each other by a piston that is frictionless and free to move in the cylinder. The mass of the piston can be neglected. Therefore, the pressure in the two chambers must always be the same. Chamber A, air V A,1 = m 3 P 1 =100kPa T A,1 =300K piston massless, frictionless, allows no heat transfer electrical heater Chamber B, water V B,1 =0.001m 3 P 1 =100kPa x B,1 =0.2 Figure 6.A-15: Device with two chambers separated by a piston. The chambers are insulated from each other as well as from the environment. Initially, the volumes in the two chambers are equal, V A,1 = V B,1 = m 3 and they are both at pressure P 1 = 100 kpa. The top chamber (Chamber A) is filled with air that is initially at temperature T A,1 = 300 K. Model the air as an ideal gas with constant specific heat capacities, R = 287 J/kg-K, c v = 718 J/kg-K, and c P = 1005 J/kg-K. The bottom chamber (Chamber B) is filled with water with an initial quality, x B,1 = 0.2. An electrical heater is installed in Chamber B. Electrical work, W e, is provided to the heater in Chamber B until the pressure in Chamber A reaches P 2 = 300 kpa. The process is sufficiently slow that the air in Chamber A undergoes a reversible process. a.) What is the mass of air in Chamber A? b.) What is the mass of water in Chamber B? c.) What is the volume of Chamber A at the conclusion of this process? d.) How much work does the piston do on the air during this process? e.) How much electrical work is provided to the heater during this process? f.) Determine the entropy generated by the process.

18 6.A-16 A piston cylinder device is shown in Figure 6.A-16. P atm = 100 kpa T amb =300K g =9.81m/s 2 free floating piston A p = m 2 m p =10kg cylinder initially filled with helium V 1 =1.0m 3 T 1 = 300 K source of helium at T s =400K,P s = 500 kpa Figure 6.A-16: Piston cylinder device. The piston is free floating with area A p = m 2 and mass m p = 10 kg. The acceleration of gravity is g = 9.81 m/s 2. The ambient pressure is P atm = 100 kpa and the ambient temperature is T amb = 300 K. The cylinder volume is initially V 1 = 1.0 m 3. The cylinder is initially filled with helium at T 1 = T amb. The cylinder is connected to a source of high pressure helium at P s = 500 kpa and T s = 400 K through a valve. The valve is opened, allowing m in = 0.5 kg of helium to enter the cylinder. Then the valve is closed. Eventually the temperature of the helium in the cylinder returns to T 2 = T amb due to heat transfer with the environment. Model helium as an ideal gas with R = 2077 J/kg-K and constant c P = 5200 J/kg-K. a.) What is the heat transfer to the surroundings? b.) What is the total entropy generated by the process?

19 6.A-17 An R e = 30-ohm electrical resistor is located in an air duct through which air is flowing with a steady rate. A current of i = 14.5 amp passes through the resistor. The surface temperature of the resistor remains at a constant temperature of T s = 43 C. Air from the surroundings enters the duct at T in = 15 C with a volumetric flow rate of V in = m 3 /s. Air exits the duct at T out = 32 C. The air is at atmospheric pressure. a.) Determine the rate of heat transfer from the duct to the surroundings. b.) Determine the First-Law efficiency of this heating process. c.) Choosing the resistor to be the system, determine the rate of entropy generation in this system. d.) For a control volume that includes the resistor and the air duct, determine the rate of entropy generation. e.) If your answers for parts (c) and (d) differ, provide an explanation for the difference.

20 6.A-18 A steady electrical current of 6 A is passed through a rod of ceramic material having a diameter of 0.3 m and length of 4.5 m. Due to the electrical resistance of the material, the rod experiences a uniform "heat generation". Heat transfer from the surface of the rod occurs to the surrounding air at 25ºC with a convection coefficient of 25 W/m 2 -K. Data for ceramic material at 25ºC: Electrical resistivity = 14 ohm-cm; Density = 4000 kg/m 3 Specific heat = kj/kg-k; Thermal conductivity = 5 W/m-K a. What are the surface and center temperatures of the rod? b. What is the rate of entropy generation in this process? c. If the current were stopped and the rod was suddenly quenched to a uniform temperature of 25ºC, what would be the resulting change in energy and entropy? What is the total entropy generation for this quenching process?

21 6.A-19 A bar of aluminum is placed in a large bath of ice and water. Electrical current is provided at a rate that results in a steady state power dissipation of 1000 W. A thermocouple placed on the surface of the bar reads 338 C. Film boiling occurs at the surface of the bar. The process continues for 2 minutes after steady-state conditions have been established. It is noted that some ice remains in the bath at the end of this period. a) What is the entropy change of the bar during the two minute period? b) What is the entropy change of the water during the two minute period? c) What is the total entropy change during the two minute period?

22 6.A-20 An insulated vessel having a volume of 0.18 m 3 and initially containing air at 5 bar and 25 C is connected through a valve to a piston-cylinder device, as shown in Figure 6.A-20. An 850 kg weight rests on the piston. The vessel and cylinder both have an internal cross-sectional area of 0.42 m 2. Initially, the piston rests on the bottom of the cylinder with negligible air mass trapped between valve and the piston. The valve is opened so that gas flows from the vessel into the cylinder raising the piston until the pressures in the cylinder and vessel are equalized. The cylinder, vessel and piston are all made of stainless-steel (density = 8000 kg/m3, specific heat = 0.48 kj/kg-k). The piston mass is 50 kg. The mass of metal cylinder walls is 150 kg, not including the piston. The entire apparatus is well-insulated on its outside surfaces from the surroundings which are at 100 kpa, 25 C throughout this process. P atm =100 kpa T atm =25 C 50 kg piston A=0.42 m kg metal 850 kg Cylinder Valve Vessel Air Figure 6.A-20: Piston-cylinder apparatus with separate air vessel a) Estimate the work done by the air and the temperature of the air in the cylinder shortly after the pressures in the vessel and cylinder have equalized. In your calculation, assume that the time period is sufficiently short such that heat transfer to the metal components can be neglected. b) The valve is left often and after a sufficient period of time, it can be expected that heat transfer between the air and the metal containing walls has occurred to the extent possible and further, conduction in the metal has resulted in it being at a uniform temperature. What is your estimate of the temperature of the air in the cylinder and the work done by the air at this point?

23 6.A-21 Steam flows in a pipeline at 1.5 MPa and 320 C. Connected to the pipeline is a cylinder with a piston as shown in figure 6.A-21. The portion of the cylinder to the left of the piston contains 0.05 kg of saturated steam at 1 atm while the other side contains 0.05 kg of air at 1 atm and 100 C. The valve is opened allowing steam from the pipeline to quickly enter the cylinder until the pressure reaches 1.5 MPa, at which time, the valve is closed. Using your engineering training, estimate the mass of steam added, and the states of air and steam in the cylinder directly after the valve is closed. Since the time required for this process is very short, you can assume that the air and water are adiabatic. State any other assumptions you employ. Steam 1.5 MPa, 320 C Valve Water Air Figure 6.A-21: Cylinder with piston separating steam and air

24 6.A-22 You want to build a potato gun using a 4.0 inch schedule 80 pipe that you've located; the schedule 48 pipe has an outer diameter of 4.50 inch and a wall thickness inch; therefore, the inner diameter is about D in = 3.79 inch. The pipe has a length of L = 6 ft and you have insulated its external surface. The potato is jammed into the barrel so that it sits a distance a 1 = 10 inch from the bottom of the pipe, as shown in Figure 6.A-22(a). The potato is held in place by a pin which is removed in order to initiate the launch process. The potato has mass M p = 0.75 lbm. atmospheric pressure, P atm =1 atm insulation D in =3.79inch potato a 1 =10inch pin L =6ft charge volume valve steam at P s = 85 psig with superheat ΔT sh,s =10 C Figure 6.A-22(a): Potato gun. You are evaluating various options for launching the potato. One option is to inject some lighter fluid into the space between the potato and the bottom of the pipe and light it, causing an explosion that launches the potato. You could also launch the potato using a source of compressed air. (All of these fall into the category of don't try this at home.) For this problem, evaluate the option of connecting the pipe to a steam line that runs through the radiators in your building. The bottom cap of the pipe is connected to the steam line in your building through a valve. According to the building maintenance man, the steam in your building is at pressure P s = 85 psig and has a superheat of ΔT sh,s = 10ºC (i.e., the temperature of the steam, T s, is 10ºC higher than the saturation temperature at P s ). Process 1: Charging the gun The first part of the launch process involves filling the charge volume (i.e., the volume between the bottom of the potato and the bottom of the pipe) with steam. Initially, the charge volume contains some air, but for the purposes of this problem you can ignore this air and assume that the charge volume is begins the process completely empty (i.e., there is no mass at all in the charge volume at the start of process 1). You open the valve and allow the charge volume to fill with steam from the source, as shown in Figure 6.A-22(b).

25 empty (P =0) (1) filled with steam at P 1 = P s steam at P s =85psig with superheat ΔT sh,s = 10 C Figure 6.A-22(b): Process 1, charging the gun. Process 1 concludes when the steam reaches state 1, where the pressure in the charge volume is equal to the source pressure, P 1 = P s ; at this point you close the valve. You may assume that there is no heat transfer from the charge volume to the surroundings. You are using a big potato, so you can assume that it forms a tight seal between the inner diameter and the potato. a.) What is the temperature of the steam in the charge volume at the end of process 1? b.) What is the mass of steam in charge volume at the end of process 1? Process 2: Launching the potato Once the gun is charged, the valve is closed and the pin is removed in order to launch the potato. The launch process is over once the potato reaches the end of the gun (i.e., a 2 = L = 6 ft) and the steam reaches state 2, as shown in Figure 6.A-22(c). potato launch velocity, v (2) a 2 =L filled with steam at P 1 = P s (1) Figure 6.A-22(c): Process 2, launching the potato. Again, you may assume that none of the steam leaks out of the gun during the launch. Further, you may assume that the steam is always internally in equilibrium during the launch process so that there is no entropy generated in the steam as it expands from state 1 to state 2; note that this provides a reasonable upper bound on the performance of your gun. Assume that there is no heat transfer from the steam during the launch. These assumptions together imply that the specific entropy of state 2 is equal to the specific entropy of state 1. c.) What is the specific volume of the steam at state 2? d.) What are the temperature and pressure of the steam at state 2?

26 e.) What is the work transfer from the steam to the potato during process 2? Note that the pressure in the gun chamber is not constant because the potato is being accelerated throughout the process; however, you should be able to compute the work transfer using an energy balance. f.) What is the velocity of the potato as it is launched from the gun ( v )? You may neglect friction between the potato and the bore of the gun. g.) If the potato experiences no air resistance after it is launched, then what is the maximum height that it will travel? h.) You have some freedom to vary the position of the pin; you can stuff the potato way down into the barrel so that a 1 (see Figure 1) is small or keep it near the mouth of the barrel so that a 1 is large. Prepare a plot showing the velocity of the potato as it is launched from the gun as a function of a 1 for 10 inch < a 1 < 70 inch. You should see that there is an optimal pin location; explain why this is the case.

27 6.A-23 You have the air-powered water rocket toy shown in Figure 6.A-23(a). A simplified schematic of the toy is shown in Figure 6.A-23(b). L =12inch T o =20 C D =1.5inch air P 1 =80psi water v w = m 3 /kg (a) (b) Figure 6.A-23: (a) A picture and (b) a schematic of an air-powered water rocket. The rocket operates using water and air. The rocket itself is a hollow plastic tube with an inner diameter D = 1.5 inch and a length L = 12 inch. In a typical launch, the rocket is filled partway with liquid water and then connected to a hand-powered air pump that pressurizes the air and water to about P 1 = 80 psi. The fraction of the rocket that is filled with water is f = 0.5. The surroundings are at T o = 20ºC and the air and water are initially at this temperature. When the trigger is released, the valve is opened and the pressurized water shoots of the nozzle and propels the rocket upwards. This process continues until all of the water has been pushed out, at which time the rocket is spent and falls back to the ground. The total time associated with the flight is time = 2 s. Model air as an ideal gas with R = J/kg-K and constant c v = J/kg-K. The specific volume of liquid water is constant and equal v w = m 3 /kg. a.) Assume that the air in the rocket undergoes a reversible and adiabatic expansion as the water is pushed out. Determine the work done by the air onto the water (J). The thrust produced by the water varies with time and requires the consideration of the details of the nozzle and the instantaneous pressure in the rocket. However, a very simple and approximate model assumes that the work done by the air on the water, W, goes entirely to increasing the kinetic energy of the water and pushing the atmosphere out of the way. With this model, the average velocity of the water leaving the rocket can be approximately determined according to: 2 V W = mw + Patm Vw 2 where m w and V w are the mass and volume, respectively, of water initially in the rocket, V is the velocity of the water leaving the rocket, and P atm is atmospheric pressure. b.) Use the equation above to estimate the velocity of the water leaving the rocket. c.) If we ignore the velocity of the rocket itself, then the thrust force produced by the rocket is the product of the velocity and mass flow rate of the water; estimate the thrust force produced by the rocket (N).

28 d.) Plot the thrust as a function of f, the fraction of the rocket volume initially filled with water. You should see that an optimal value of f exists. Explain clearly why there is an optimal value for f. An alternative model for the rocket assumes that the air in the rocket is not adiabatic, but rather isothermal. Rather than having no heat transfer during the launch, the isothermal model assumes that the heat transfer is sufficient that the air in the rocket is always at T o. The adiabatic and isothermal models will bound the actual performance of the rocket. e.) Repeat your calculations using an isothermal model of the launch. Overlay the thrust force as a function of f for the isothermal model onto the plot generated in (d).

29 6.A-24 A cold fluid cannot be stored for long periods since thermal gains inevitably occur, even in a Dewar. An alternative is to store a high pressure gas, e.g., air and release it in an adiabatic expansion process through a nozzle to atmospheric pressure. In a particular gas, air at 100 atm is stored at 25 C in a 15 liter tank. When cold temperatures are needed the valve is opened an air is expanded through the nozzle until the pressure in the tank is 2 atm. a) What is the lowest temperature that can be obtained with this equipment? b) What is the cooling capacity (in J) that can be provided at temperatures lower than 0 C? c) What is the total entropy production associated with this process?

30 6.A-25 A 4 m 3 storage tank contains 2 m 3 of a combustible liquid that has a density of 750 kg/m 3. The tank is to be pressurized with nitrogen from a large high-pressure reservoir through a valve at the top of the tank, in order to permit rapid ejection of the liquid. The nitrogen in the reservoir is maintained at 8 bar (800 kpa) and 10 C. The gas space above the liquid in the tank is initially at 1 bar (100 kpa) and 10 C. The valve connecting the nitrogen reservoir is now opened slowly. The liquid transfer valve is opened when the pressure in the tank reaches 5 bar (500 kpa) and then liquid is ejected at a rate of 0.2 m 3 /min while the tank pressure remains constant at 5 bar. Neglect heat interactions at the gas-liquid and gas-wall boundaries. Figure 6.A-25: Liquid tank with nitrogen gas a) Determine the temperature of the nitrogen in the tank when the pressure in the tank first reaches 5 bar (before any liquid is drained). b) Determine the temperature of the nitrogen in the tank when all of the liquid has been drained and the pressure is at 5 bar. c) Determine the total change in entropy for this process.

31 6.A-26 R134a is contained in a 0.05 m 3 tank as shown in the figure. The R134a is initially 90% liquid and 10% vapor by volume at 30 C. The valve is opened and R134a vapor exits through the nozzle. At the nozzle outlet, the pressure is atmospheric and the velocity is 115 m/s. During this process, an electrical heater within the tank is energized so as to maintain the contents of the tank at 30 C. When submerged within liquid R134a, the surface of the electrical heater is 128 C. The valve remains open until only saturated vapor remains in the tank. Figure 6.A-26: Tank containing R134a equipped with heater and nozzle a. Determine the quality in the tank at the start of this process. b. Determine the electrical energy provided to the heater during this process. c. Determine the total rate of entropy production resulting from this process

32 6.A-27 An industrial plant has a compressed air stream at P 1 = 2.5 atm and T 1 = 35 C that is presently vented to the atmosphere. Due to increased fuel costs, the management is very interested in a device which is claimed to separate the air stream into two equal mass flow rates: a hot stream at T 2 = 230 C and a cold stream at T 3 = -160 C, both at atmospheric pressure. These two streams could be used to satisfy some of the heating and cooling requirements in other parts of the plant. The device is also claimed to be self-sustaining, requiring no additional heat or work. Is such a device possible? Explain why or why not. If the device is possible, suggest a process (or series of processes) which would produce the same effect as this device.

33 6.A-28 An industrial plant has an old insulated tank that has not been in use for many years. Unfortunately, it is in an inaccessible location and covered with insulation so that its volume cannot be determined. The tank currently contains air at atmospheric pressure and 25 C. One of the engineers in the plant has suggested a way to determine the volume. The suggestion is to connect the tank to a compressed air tank which has a known volume of 13.7 m 3 with air initially at 5.2 atm and 25 C. The valve between the two tanks is opened until the pressures equilibrate and then quickly closed. The pressure is measured at this time to be 2.8 atm. Can this information be used to determine the volume of the tank? If so, what is its volume?

34 6.A-29 A company advertises a heat-driven refrigerator, shown in Figure 6.A-29. heat driven refrigerator T H = 250 C Q 200 W H = heat from burner Q amb Q = 350 W T amb = 20 C T C =-10 C heat rejected to ambient heat extracted from refrigerated space Figure 6.A-29: Heat-driven refrigerator. The refrigerator is driven by a heat transfer from a burner at T H = 250ºC at a rate of Q H = 200 W. The refrigerator removes Q C = 350 W from a refrigerated space at T C = -10ºC and rejects heat to the surroundings at T amb = 20ºC. The refrigerator operates at steady state and requires no work transfer. a.) Is the device possible? Does it violate the Second Law of Thermodynamics? b.) A reversible, heat-driven refrigerator obtains Q H = 200 W from a thermal reservoir at T H = 250ºC and provides refrigeration at T C = -10ºC with heat rejection to ambient at T amb = 20ºC. What is the rate of refrigeration provided by this reversible device? c.) What is the Coefficient of Performance (COP) of the reversible device that you examined in (b)? The Coefficient of Performance for this system is defined as the ratio of what you want ( Q C ) to what you pay for ( Q H ) and the COP of the reversible device will be the maximum possible COP for this type of device operating with these temperatures. d.) Plot the Coefficient of Performance of the reversible device as a function of T H. C

35 6.A-30 On a typical winter day, your house is maintained at T house = 70ºF and the outdoor temperature is T amb = 10ºF. Due to this temperature difference, there is a heat transfer from your house to the outdoors at a rate of Q loss = 2500 W. In order to keep your house at a comfortable temperature, it is necessary to provide heat at a rate that balances this heat loss. One technique for providing heat is to use electrical resistance heaters, as shown in Figure 6.A-30(a). Electrical work is transferred to resistive elements within the house at a rate W eh that is sufficient to keep the house at steady state. Q = 2500 W Q loss = outdoors 2500 W house T house =70 F resistance heater T amb = 10 F Q loss = outdoors 2500 W house T house =70 F T amb =10 F outdoors heat pump loss house T house = 70 F Q hp, out T amb =10 F W hp T furnace = 1600 C outdoors T amb =10 F W eh Q furnace Q hp, in (a) (b) (c) Figure 6.A-30: (a) Electrical heating system, (b) natural gas heating system, and (c) heat pump based heating system. a.) Determine the rate of electrical work transfer to the house, W eh. b.) Determine the rate of entropy generation associated with the use of electrical heating. The heating season lasts time = 110 days and electricity costs ec = 0.12 $/kw-hr. c.) Determine the cost associated with heating your house using electrical resistance heaters over an entire heating season. An alternative heating system burns natural gas in order to generate the required heat transfer, as shown in Figure 6.A-30(b). Assume that the heat transfer from the natural gas combustion is obtained at a temperature of T furnace = 1600ºC and that the cost of natural gas is gc = 1.5 $/therm. d.) What is the rate of entropy generation associated with the use of natural gas heating? e.) What is the cost associated with heating your house using natural gas over an entire heating system? A third alternative heating system utilizes a heat pump, as shown in Figure 6.A-30(c). The heat pump receives heat from the outdoor air at T amb at a rate of Q and provides heat to the house at T house at a rate of Q hp,out. The heat pump requires electrical power at a rate of W hp in order to operate. Assume that the heat pump is operating at steady state and is reversible (the heat pump system is reversible - this system is enclosed in the dashed lines in Figure 6.A-30(c)). f.) Determine the rate of heat transferred from the heat pump to the house, Q. g.) Determine the rate at which electrical energy is consumed by the heat pump, hp,in hp,out W hp.

36 h.) What is the rate of entropy generation associated with using the heat pump heating system? i.) What is the cost associated with heating your house using the heat pump over an entire heating system?

37 6.A-31 A piston-cylinder device is shown in Figure 6.A-31. P atm =100kPa T amb =150 C water T 1 =350 C P 1 =400kPa piston A c =0.01m 2 m p =100kg z 1 =0.5m Figure 6.A-31: Piston-cylinder device. The initial position of the piston is z 1 = 0.5 m and the piston cross-sectional area is A c = 0.01 m 2. The mass of the piston is m p = 100 kg. The cylinder contains water that is initially at T 1 = 350ºC and P 1 = 400 kpa. The surroundings are at P atm = 100 kpa and T amb = 150ºC. The piston is initially held in place by a pin to prevent it from moving due to the internal pressure. At some time, the pin is removed and the piston quickly and violently shoots upward under the action of the internal pressure. The piston motion continues for some time until eventually the oscillations are damped out and the piston obtains a new equilibrium position at state 2 where it is in mechanical equilibrium with the surroundings (i.e., a force balance on the piston can be used to provide the internal pressure at state 2). There is no heat transfer between the contents of the piston and the surroundings during the time required by the equilibration. Note: this is an irreversible mechanical equilibration process. You do not know, nor is there any way to determine, the pressure of the water acting on the lower surface of the cylinder during this process. However, you do know the pressure of the atmosphere acting on the upper surface of the piston during the process. Your system selection should be informed by these facts. a.) Determine the position of the piston at state 2, z 2. b.) Determine the temperature of the water at state 2, T 2. c.) What is the entropy generated by the process of moving from state 1 to state 2, S gen,1-2? d.) What is the work transfer from the water to the piston during the process of going from state 1 to state 2, W out,1-2? After some time has passed, heat transfer between the water to the surroundings causes the water to come to a final temperature that is equal to the temperature of the surroundings, T amb. This is an irreversible thermal equilibration process that must result in entropy generation because heat is being transferred through a temperature gradient. The piston is allowed to move freely during this process. e.) Determine the position of the piston at state 3, z 3. f.) Determine the heat transferred from the water to the surroundings during this process, Q out,2-3. g.) Determine the entropy generated by the process of moving from state 2 to state 3, S gen,2-3. h.) Using EES, generate a temperature-entropy diagram that shows states 1, 2, and 3. i.) Plot the entropy generated by the process of moving from state 1 to state 2 as a function of P 1 for 100 < P 1 < 500 kpa. You should see that there is an optimal pressure at which the entropy generated by this process is minimized. Explain why this is the case. j.) What initial pressure and temperature should you use if you want to minimize the total entropy generated by the equilibration processes (i.e., you want to minimize S gen = S gen,12 + S gen,23 ). Why?

38

39 6.A-32 Figure 6.A-32 illustrates a system that is used to produce a high velocity flow of R22. The R22 stream is directed to a cleaning bench where it is used to clean dust and dirt from electronic components. reversible P 2 = P reg =250kPa nozzle D n,out =0.0625inch pressure regulator T 3 = T amb cleaning bench connecting tube R22 bottle V bottle =20liter x ini =0.02 T=T amb =20 C 3 Q in,ct Q in,eq cleaning bench exhaust T 5 = T amb P 5 = P atm Q in,tank Figure 6.A-32: System used to clean components in a clearning bench. The R22 used for the process is contained in a bottle with volume V bottle = 20 liter. Heat transfer from the atmosphere at T amb = 20ºC always maintains the temperature of the R22 in the bottle at T amb. The bottle is initially in a two-phase state that is mostly liquid. The initial quality is x ini = Saturated vapor leaves the bottle at state 1 and passes through a pressure regulator. The pressure regulator is a valve that maintains a constant pressure at state 2, P reg = 250 kpa. The flow of gas leaving the valve passes through a long connecting tube as it travels to the cleaning bench. Heat transfer from the atmosphere causes the temperature of the R22 to increase to T amb at state 3. Assume that there is no loss in pressure as the R22 passes through the connecting tube. The R22 passes through a nozzle in order to produce the high velocity flow. The nozzle is reversible. You may neglect the kinetic energy of the flow entering the nozzle. The nozzle exit, state 4, is at atmospheric pressure, P atm = 1 atm. The diameter of the nozzle exit passage is D n,out = inch. Finally, the R22 is recovered at the cleaning bench exhaust, state 5. The pressure and temperature of the R22 at this point are ambient, P atm and T amb, respectively, and the velocity is very low so that the kinetic energy can be neglected at state 5. The valve, connecting tube, and nozzle all operate at steady state. The system must be shut off when the bottle contains only saturated vapor. a.) Determine the temperature of the R22 leaving the valve. b.) Determine the exit velocity of the nozzle and the temperature of the R22 leaving the nozzle. c.) Determine the mass flow rate through the nozzle. d.) How long will the system operate before the bottle must be replaced? e.) Determine the heat transfer to the tank during the process. f.) Determine the entropy generated in the tank. g.) Determine the entropy generated in the valve. h.) Determine the heat transfer to the connecting line. i.) Determine the entropy generated in the connecting line.

40 j.) Determine the heat transferred to the R22 as it flows from the exit of the nozzle to the bench exhaust. k.) Determine the entropy generated as the R22 flows from the exit of the nozzle and the bench exhaust. l.) Use an overall energy balance to check your answer. That is, define a system that will allow you to calculate the total heat transfer from ambient and verify that this answer agrees with the sum of parts (e), (h), and (j). m.) Use an overall entropy balance to check your answer. That is, define a system that will allow you to calculate the total entropy generated by the process and verify that this answer agrees with the sum of parts (f), (g), (i), and (k).

41 6.A-33 Figure 6.A-33 illustrates a cryogenic refrigerator that is used to maintain a detector at a very low temperature. The refrigerator utilizes the reverse-brayton cycle. reversible turbine W t T H = 20 C 3 aftercooler ΔT ac =10K Q rec Q H P high = 532 kpa 2 1 recuperative heat exchanger ΔT rec =5K reversible compressor W c P low = 152 kpa m = kg/s 4 5 load heat exchanger 6 ΔT LHX =2K detector T C =70K Q C Figure 6.A-33: Reverse-Brayton refrigeration cycle. The reverse-brayton refrigeration cycle utilizes a gas working fluid so that it can continue to operate even at very low temperatures where more conventional refrigerants will freeze. The device shown in Figure 6.A-33 utilizes neon. Model neon as an ideal gas with R = 412 J/kg-K and c P = 1030 J/kg-K. Neon enters the compressor at state 1 with pressure P low = 152 kpa and mass flow rate m = kg/s. The compressor is reversible and adiabatic. The neon leaves the compressor at state 2 with pressure P high = 532 kpa. The neon leaving the compressor is cooled in an aftercooler that rejects heat at rate Q H to the ambient temperature at T H = 20ºC. The aftercooler approach temperature difference is ΔT ac = 10 K; this means that the neon leaving the aftercooler at state 3 has been cooled to within ΔT ac of T H (i.e., T 3 = T H + ΔT ac ). Neon then flows through the recuperative heat exchanger where it is cooled by heat transfer with the cold neon returning from the cold end of the device. Cold neon leaves the recuperator at state 4 and enters the turbine. Assume that the turbine is reversible and adiabatic. The neon exits the turbine at state 5 with pressure P low. The neon leaving the turbine is quite cold (this is the coldest point in the cycle). This neon enters the load heat exchanger where it is warmed by a heat transfer from the detector which is at temperature T C = 70 K. The heat transfer rate from the detector to the neon is Q C and the approach temperature difference for the load heat exchanger is ΔT LHX = 2 K. That is, the neon passing through the load heat exchanger is warmed to within ΔT LHX of T C (T 6 = T C - ΔT LHX ). Finally, the neon leaving the load heat exchanger at state 6 passes through the recuperator where it is warmed by heat transfer with the high pressure neon flowing in the opposite direction. The recuperator has an approach temperature difference of ΔT rec = 5 K. Therefore, the neon leaving the recuperator at state 1 has been warmed to within ΔT rec of T 3 (T 1 =

42 T 3 - ΔT rec ). Assume that there is no loss of pressure associated with flow through any of the heat exchangers. a.) Determine the temperature and pressure at each of the six numbered states in Figure 6.A-33. b.) Determine the power required by the compressor ( W c ), the power produced by the turbine ( W t ), the rate of heat transfer from the neon in the aftercooler ( Q H ), and the rate of heat transfer to the neon in the load heat exchanger ( Q C ). c.) Check your solution by drawing a system that encompasses the entire cryocooler. Verify that energy balances for this overall system. d.) Assuming that the turbine power is used to help drive the compressor, determine the Coefficient of Performance (COP) of the cryocooler. e.) What is the maximum possible COP for a refrigerator operating between T C and T H. f.) Determine the total rate of entropy generation in the cryocooler. g.) Plot the refrigeration power ( Q C ) as a function of the load temperature (T C ). You should see that your refrigerator can provide less refrigeration as the temperature is reduced. h.) The detector, however, will require more refrigeration as its temperature is reduced since, the the rate of heat transfer is proporation to the temperature difference between the detector and its surroundings. Assume that the refrigeration required by the detector is given by: Q Creq, = 2W [ ] + 0.1W/K [ ]( TH TC) Overlay the required refrigeration as a function of T C onto your plot of available refrigeration from part (g). Based on your plot - what is the lowest possible temperature that the system can achieve?

43 6.A-34 A piston-cylinder device that contains m = 2 kg of water. The water in the cylinder is initially at T 1 = 120ºC and P 1 = 1.5 bar (150 kpa). The piston-cylinder apparatus is well-insulated and the piston is pushed in until the water in the cylinder is saturated vapor (x 2 = 1) at T 2 = 150ºC. a.) Determine the volume occupied by the water at state 1 and state 2. b.) What is the work done on the water in going from state 1 to state 2? c.) Is the process possible? Justify your answer with a calculation. The piston is locked in place so that it cannot move and the contents of the cylinder are allowed to come thermal equilibrium with the surroundings at T amb = 50 C. Therefore, T 3 = T amb and V 3 = V 2. d.) Determine the volume of liquid water that exists at the final state. e.) What is the entropy generated by the thermal equilibration process?

44 6.A-35 A refrigeration unit operates at steady-state between two thermal reservoirs at constant temperatures of 10 C and 60 C, respectively, as shown in Figure 6.A-35. The cold part of the refrigeration unit that receives thermal energy from the 10 C thermal reservoir maintains a steady temperature of 5 C. The hot part of the refrigeration units that transfers thermal energy to the 60 C reservoir maintains a steady temperature of 70 C. The coefficient of performance of the refrigeration unit is 2.0. The steady-state rate of heat transfer to the 60 C thermal reservoir, Q H, is 425 W. Thermal losses from the equipment are negligible. Refrigerator with COP = 2 10 C 60 C Q Q L 5 C 70 C H Figure 6.A-35: Refrigeration system operating between two thermal reservoirs W a.) Determine the steady-state heat transfer rate from the 10 C, Q L, and the steady-state rate at which power, W. b.) Determine the steady-state rate of entropy change of the 10 C thermal reservoir. c.) Determine the steady-state rate of entropy change of the refrigeration unit. d.) Determine the total rate of entropy generation resulting from the overall process. e.) What would the power input and the total rate of entropy generation resulting from the overall process be if the refrigeration unit operated reversibly between 5 C and 70 with Q H = 425 W?

45 6.A-36 On a day in which the outdoor temperature is 10 F, an energy input of 22 kw is needed to maintain the indoor temperature of a house constant at a comfortable 70 F. This energy is supplied by an electric resistance furnace, as shown in Figure 6.A-36. Air steadily enters the heating unit at 65 F, 15.1 psia and exits at 105 F, 14.8 psia after passing over the heating element. The surface temperature of the heating element is measured to be 360 F. Assume the house is air-tight. Figure 6.A-36: Schematic of an electrical house heating system a.) Determine the rate of entropy change of the air as it passes through the heating unit during steady-state operation. b.) Determine the rate of entropy change of the heating element during steady-state operation. c.) What is the total rate of entropy production for the house heating operation? d.) What electrical power would be needed for the house heating application if a heat pump having a coefficient of performance that is 50% of a reversible heat pump operating between the indoor and outdoor temperatures were used in place of the electrical resistance heater? e.) What would the total rate of entropy production for the house heating operation be if the heat pump in part d) were used instead of the electrical resistance heater?

46 6.A-37 A well-insulated cylinder fitted with a piston has an initial volume of 5 ft 3 and it contains steam at 60 psia, 400 F. The steam is expanded adiabatically to a final state that is measured to be 14.7 psia and 11.7 ft 3. a.) Determine the final temperature and quality of the steam, if applicable. b.) Determine the work done in this process. c.) Comment on whether you believe the measurements.

47 6.A-38 Steady-state test data are reported for the device shown in Figure 6.A-38. Compressed air enters (state 1) at 0.65 m 3 /sec, 25 C and 300 kpa. The device has two exit streams. Air exits at state 2 at 0.65 m 3 /s, 62.4 C, 100 kpa. At the other outlet (state 3), air exits at -2.7 C and 100 kpa. There are no moving parts internal to this device. The surroundings are at 25 C, 100 kpa C, 100 kpa 0.65 m 3 /s -2.7 C, 100 kpa 1 25 C, 300 kpa, 0.65 m 3 /s Figure 6.A.38: Steady-state test results for a device a.) Determine the net heat transfer rate to/from this device. Indicate the direction of the heat flow b.) What is the total rate of entropy generation resulting from operation of this device? c.) Do you believe that a device can operate as indicated in the problem statement? Provide a sentence of two of explanation.

48 6.A-39 A perfectly-insulated tank contains m 1 = 25 kg of refrigerant R134a that is initially at P tank = 300 kpa with a quality (vapor mass fraction) of x 1 = The tank pressure is maintained constant by nitrogen gas acting against a flexible bladder, as shown in Figure 6.A-39. The valve is opened between the tank and the supply line that carries R134a at T s = 120 C and P s = 1 MPa, allowing R134a to enter the tank. The tank pressure remains at P tank during this process. The valve is closed when all of the R134a in the tank has been vaporized. pressure regulator flexible bladder nitrogen nitrogen supply tank inlet valve R134a m 1 =25kg P tank = 300 kpa x 1 =0.8 R134a at T s = 120 C and P s =1.0MPa Figure 6.A-39: Tank holding pressurized R134a and nitrogen. a.) What is the mass of R134a that enters the tank? b.) Calculate the entropy produced in this process.

49 6.A-40 A tank of fixed volume is to be evacuated. The tank has a volume of V = 1.4 m 3 and it initially contains air at T amb = 25 C and P atm = 1 atm. Evacuation is carried out slowly using a small vacuum pump which, like all real devices, does not operate in a reversible manner. Because the process occurs slowly, the air that remains in the tank stays at T amb as a result of heat transfer through the tank walls. a.) Calculate the total heat transfer to the contents of the tank during the time required to reduce the pressure in the tank to P 2 = 0.1 atm. b.) Determine the minimum possible work required to accomplish the process in part (a). c.) Is it possible to completely evacuate the tank? If you believe that this is not possible, indicate what physical law is violated if this were to happen. Otherwise, determine the minimum work that would be needed for complete evacuation.

50 6.A-41 Thermal energy storage units that use low cost off-peak electrical energy to meet space heating requirements at a later time are gaining popularity. One design consists of an electrical heating element embedded in a masonry material. Air channels are provided to facilitate convection to the circulating air in the discharging mode. The specific heat of the masonry material is 1.1 kj/kg-k. The thermal capacity of the heating element is negligible compared to that of the masonry. Consider the charging process. The storage unit is initially at a uniform temperature of 30 C. An electrical current is steadily passed through the heating element dissipating 20,000 W for 3.5 hours, at which time the masonry material is at a uniform temperature of 625 C. A thermocouple positioned on the surface of the heating element indicates a temperature of 915 C which varies only slightly during the charging process. Heat losses are negligible during the charging process. During the discharging process, air at 20 C from inside the building is circulated through the storage unit until the temperature of the unit is again uniformly at 30 C. The process occurs at a sufficient to maintain the indoor space at 20 C while the outdoor temperature is -5 C. a.) Determine the entropy change of the storage unit during the charging process. b.) Determine the total entropy generation during the charging process. c.) Determine the energy change of the storage unit during the discharging process. d.) What is the entropy change of the air in the building during the discharging process? e.) What is the total rate of entropy generation (for the storage unit and the building) during the discharging process? f.) What is the total entropy generation for the complete cycle in which the storage unit is charged and discharged, returning to its initial state?

51 6.A-42 The process of inflating a balloon can be modeled, approximately, by assuming that the elastic material behaves like a spring that opposes the expansion. An experiment has been developed to test this assumption as shown in Figure 6.A-42. Air is slowly admitted into the cylinder from source at 70 F, 100 psia, causing the spring to compress in an approximately adiabatic process. The cylinder walls and piston are mode of a nonconductive material with very low mass. The piston has a diameter of 6.0 in. The spring is characterized by a spring constant of 576 lb f /ft. There is no force exerted on the spring when L=0.5 ft, where L is shown in Figure 6.A-42. Initially, L=0.5 ft and the air in the cylinder is at 70 F, 14.7 psia. Air, 70 F, 100 psia L diam. = 6 in Figure 6.A-42: Piston-cylinder with spring 14.7 psia 70 F a.) What is the temperature and pressure of the air to the left of the piston when L=2 ft? b.) Calculate the total entropy production for this process. Is this process reversible? If not, indicate why.

52 6.A-43 A pot of water at 1 atm is boiling on an electric stove. Careful measurements indicate that the heating element surface remains at a steady temperature of 457 F during a 5 minute period in which the boiling takes place. The power is steady at 2.23 kw during this process. Thermal losses from the pot and heating element are negligible. a.) Determine the mass of water vaporized during the 5 minute period. b.) Determine the change in entropy of the heating element during the 5 minute period. c.) Determine the change in entropy of the water during the 5 minute period. d.) Determine the total change in entropy for this process during the 5 minute period.

53 6.A-44 A piston-cylinder apparatus contains 0.5 m 3 of air at 25 C, 1.5 bar. A process occurs and the final pressure of the air is 2.75 bar. Measurements taken during this process indicate that 8.2 kj of work are done on the air and the total heat from the air in the cylinder to the surroundings is 3.5 kj. The surroundings are at 25 C, 1.01 bar. Do you believe the measurements? Please provide a complete justification of your answer.

54 6.A-45 The following short note appeared in Chemical & Engineering News (3/11/96) based on a letter to the editor form Howard Guest of Charleston, W. Va. During the waning days of World War II, a U.S. Navy technical group was assigned to follow in the footsteps of Allied troops racing through German industrial areas and liberate scientific/technological reports from the ruins. The picked up documents by the truckload and rushed them to the U.S. for appraisal. Union Carbide s Research Library in South Charleston received a generous share of this material. Guess, then employed at Carbone, says the Carbiders who could read the German text eagerly volunteered to work night to search for useful information. Pickings were disappointing slim, but one report was more than interesting It was incredible. The report, Guest says, minutely described a device that challenged the common knowledge of thermodynamics. The Carbibe people, intrigued, built a laboratory model of the device in their machine shop. The heart of the device is a machined block of steel, ¾ in square and ½ in deep. It is machined to accept compressed air into face A (see Figure 6.A-45) through a control valve and ¼ in tube, 5 ½ in long. Two exit ports (B and C) are directly opposite each other and 90 from intake A. Cylinder air at room temperature and 90 psig is introduced through a stopcock at A and enters the block. Exit air flows from port B without hindrance through a ¼ in tube, 1 ½ in long. Exit air from port C through a ¼ in tube, 14 in long and through a stopcock. When the cylinder air is flowing, Guest says, and the stopcocks at A and C are properly positioned, the exit air at C becomes hot. The exit air at B becomes so cold that frost forms on the exit tube. Guest claims that he has yet to see a satisfactory explanation of the behavior of this device. 1/4intube B cold air exhaust 1.5 in 1/4 in tube 14 in C hot air exhaust 1/2intube A air at 90 psig 70 F Figure 6.A-45: Device described by Mr. Guest Perform a thermodynamic analysis of this device and indicated whether or not it is possible to operate as described my Mr. Guest. If it is possible, can you provide a satisfactory explanation for the behavior?

55 6.A-46 An insulated piston-cylinder apparatus is fitted with an electrical heating element. The cylinder initially contains m s,1 = 4.0 kg of ice and m f,1 = 1.0 kg of liquid water, both at T m = 0 C. A thermocouple on the surface of the heating element registers at steady temperature of T htr = 168 C. After t = 400 seconds of operation there are m s,2 = 2.6 kg of ice remaining in the cylinder and m f,2 = 2.4 kg of water, both at T m. The cylinder contents remain at constant pressure during this process. The temperature and pressure of the surroundings are T amb = 25 C and P atm = kpa, respectively. a.) Determine the steady power dissipation in the heating element. b.) Determine the change in entropy of the heating element during the process. c.) Determine the change in entropy of the cylinder contents (i.e., the ice and liquid water) during the process. d.) Determine the total entropy generation during the process.

56 6.A-47 A 5.0 m 3 tank contains 4.0 m 3 of hydraulic fluid and 1.0 m 3 of air, both at 25 C, 100 kpa as shown in Figure 6.A-47. The tank is connected to a compressed air line so that the hydraulic fluid in the tank can be rapidly ejected. The compressed air in the line is at maintained at 1 MPa and 25 C. When it is necessary to eject the liquid, a valve connecting the air line to the tank is opened and air enters the tank, increasing the pressure. When the pressure reaches 500 kpa, the liquid transfer valve opens and hydraulic fluid exits the tank at a rate of 0.25 m 3 /min while the tank pressure is maintained at 500 kpa. The liquid transfer valve is closed when all of the liquid exits the tank. However, the valve on the air pipeline remains open so that air continues to flow into the tank until the pressure in the tank is 1 MPa. After some time, the temperature of the air in the tank equilibrates with the surroundings which are at 25 C. Figure 6.A-47: Tank with hydraulic fluid Air at 25 C, 1.0 MPa air inlet valve Air Tank Volume = 5 m 3 Hydraulic Fluid liquid transfer valve a.) The plot below shows the pressure in the tank as a function of time. Indicate qualitatively how you expect the temperature of the air in the tank to vary with time P [kpa] Temperature [ C] 100 Time b.) What is the total heat transfer to the air in the tank during the entire process? c.) Determine the total entropy production occurring during the entire process.

57 6.A-48 A flat-bottom pan containing boiling water at atmospheric pressure (101.3 kpa) is steadily heated by a 500 W electric resistance element. The temperature of the heating element is 173 C. The bottom outside surface of the pan is at 112 C whereas the inside surface of the pan in contact with the liquid water is at 104 C. The air in the room is at 25 C, 50% relative humidity. The outer surface of the pan not in contact with the heating element is insulated so thermal losses may be neglected. a.) What is the mass flow rate of water vapor out of the pan? b.) Determine the rate of entropy generation in the liquid water remaining in the pan. c.) Determine the rate of entropy generation within in the material composing the pan d.) What is the rate of entropy generation in the electric element? e.) Determine the total rate of entropy generation for this process.

58 6.A-49 Water can be cooled by pulling a vacuum over liquid water in a Dewar flask using the equipment that is shown in Figure 6.A-49. In this case, a V = 5 liter Dewar flask initially contains V f,1 = 2.5 liters of saturated liquid water at T 1 = 25 C with the remainder of the volume occupied by saturated water vapor. The vacuum pump is turned on causing water vapor to exit the top of the flask. The temperature of the water remaining in the flask decreases as the process proceeds and at some time later, the water temperature is T 2 = 2 C. Measurements indicate that, the volume of liquid water remaining in the flask at this time is 92% of the original liquid volume. The surroundings are at T amb = 25 C and P atm = kpa. vacuum pump saturated water vapor T 1 =25 C tank V = 5 liter V f,1 = 2.5 liter of saturated liquid water Figure 6.A-49: Pulling a vacuum over water in a Dewar flask. a.) Determine the mass and quality of water in the flask at the start of the experiment. b.) Determine the mass and quality of water in the flask at the end the experiment. c.) The Dewar flask is very well insulated so heat transfer during this experiment can be considered to be negligible. However, the flask is made of a material that has some thermal capacitance. Estimate the effective mass specific-heat product of the flask. d.) Determine the total entropy generation resulting from this process.

59 6.A-50 Measurements are taken during the steady-state operation of a hand-held hair dryer. Air enters the hair dryer at 25 C, atmospheric pressure and 3.7 m/s. Air exits at 79 C, atmospheric pressure, and 9.1 m/s. The surface temperature of the heating element is 158 C. The exit nozzle of the hair dryer has a cross-sectional area of 15 cm 2. There is no significant heat transfer between the surfaces of the hair dryer and the surrounding air due to short time period that the air is in contact with the surfaces. a.) The mass flow rate of air through the dryer is kg/s. b.) The electrical power required is kw. c.) The rate of entropy change of the air is kw/k. d.) The rate of entropy change of the heating element is kw/k. e.) The total rate of entropy production is kw/k.

60 6.A-51 A heat engine of a novel design uses hot air as a heat source and water as a heat sink. The air enters the device at 510 C, 1.50 bar and exits at 42 C, 1 bar. The water enters at 25 C, 1.25 bar and exits at 32 C, 1 bar. The jacket of the engine is covered with a thick blanket of insulation. The inventor of this engine claims that, under these operating conditions, the engine produces a steady 3.5 kw of mechanical power with a thermal efficiency of Calculate the maximum possible efficiency for this engine. Is the inventor s claim possible?

61 B. Isentropic Efficiencies and Heat Exchangers 6.B-1 Figure 6.B-1 illustrates a compressor that is providing air to a tank. The tank is leaking air through a valve; the air that is leaking re-enters the atmosphere and eventually returns to atmospheric conditions. Q tank atmosphericair 3 V, m /s cin = T 20 C amb = P = 100 kpa atm compressor tank P tank = 600 kpa T tank =70 C valve W c leak to atmosphere Figure 6.B-1: Compressor providing air to a tank that is leaking through a valve. The compressor is adiabatic and operating at steady state. The compressor efficiency is η c = The compressor draws in atmospheric air at T amb = 20 C and P atm = 100 kpa and compresses it to the tank pressure, P tank = 600 kpa. The volumetric flow rate of air at the compressor inlet is V c,in = m 3 /s. The temperature of the air in the tank is T tank = 70 C. Both the pressure and temperature of the air in the tank are constant with time (i.e., the tank is at steady state). Model air as an ideal gas with constant specific heat capacities; R = 287 J/kg-K, c P = 1005 J/kg-K, and c v = 718 J/kg-K. a.) What is the mass flow rate provided by the compressor (kg/s)? b.) What is the power required by the compressor (kw)? c.) What is the temperature of the air leaving the compressor? d.) What is the rate of entropy generation in the compressor? e.) Determine the rate of heat transfer from the tank. f.) List and describe as many of the specific sources of entropy generation as you can for this problem. g.) Calculate the total rate that entropy is generated by the system, the sum of all of the sources of entropy generation from (f).

62 6.B-2 Figure 6.B-2 illustrates a heat exchanger in which hot air is used to generate steam. Air enters the heat exchanger at T a,in = 320 C, P a = 100 kpa, and m = 0.5 kg/s. Model air as an ideal gas with constant specific heat capacity (R a = kj/kg-k, c P,a = 1.01 kj/kg-k). Water enters the heat exchanger at T w,in = 20 C and P w = 100 kpa. The mass flow rate of the water is m = kg/s. The pinch point associated with the heat exchanger is on the hot side (i.e., where the air enters and water leaves). The approach temperature difference is ΔT = 160 K. Neglect pressure drop on the air and the water sides of the heat exchanger. a w air T, 320 C ain = P 100 kpa a = m = 0.5 kg/s a heat exchanger Q Figure 6.B-2 Heat exchanger. water T, 20 C win = P 100 kpa w = m = kg/s w a.) Determine the rate of heat transfer from the air to the water in the heat exchanger. b.) Determine the total rate entropy generation in the heat exchanger. c.) Determine the effectiveness of the heat exchanger. d.) If the approach temperature difference of the heat exchanger were reduced to ΔT = 0 K (i.e., the heat exchanger became "perfect") then would the entropy generation in the heat exchanger be reduced to zero? Justify your answer.

63 6.B-3 A concentric tube heat exchanger is used to cool oil at a flow rate of 0.1 kg/sec from 100 C to 60 C using water at 0.1 kg/s that enters at 30 C. The oil flows in the inner tube which has an inner diameter of 2.5 cm. The water flows in the annulus. The tubes are smooth. The inner diameter of the outer tube is 5.0 cm. The tube wall is made of copper with a thickness of 1 mm. The heat transfer coefficient per unit area, U, is believed to be about 40 W/m 2 -K based on the inner tube area. a) Determine the outlet temperature of the water with the given value of U. b) Determine the length of the heat exchanger using the provided heat transfer coefficient with the given value of U. c) Use heat transfer relations to estimate the heat transfer coefficient between the oil and the water and then use this estimate of the heat transfer coefficient to determine an improved estimate of the heat exchanger length.

64 6.B-4 The gas turbine engines that you see attached to the wing of a jet air plane consists of a diffuser, compressor, combustor, turbine, and nozzle that are arranged in series as shown in Figure 6.B- 4(a). Q cb combustor P 4 = P 3 compressor P 3 /P 2 =5.5 η c =0.87 (3) W c W t (4) T 4 = 1120 C turbine η t =0.87 (1) (2) (5) (6) inlet air P 6 = P 1 m = kg/s diffuser V 1 = 350 mph (reversible) nozzle P1 = 75 kpa η n =0.87 T1 = 5C Figure 6.B-4(a): Schematic of a gas turbine jet engine. Figure 6.B-4(b) shows the Goblin II turbojet engine (used as the primary engine on the F80 airplane) sectioned so that the compressor, combustion chambers, and turbine are all visible. This problem analyzes the jet engine shown in Figure 6.B-4(a) under conditions that are approximately consistent with the Goblin II jet engine mounted on a plane flying at high altitude and a velocity of 350 mph. The analysis will be carried out on a component-by-component basis. Model the air using the substance 'Air' in EES which assumes that air is an ideal gas, but do not assume that it has constant specific heat capacities. Diffuser Air drawn into the engine first encounters the diffuser. The purpose of the diffuser is to reduce the velocity of the air (with respect the engine) which causes an increase in its pressure. The diffuser is the reverse of a nozzle; it is a duct with a varying cross-sectional area that slows the air down. The air enters the diffuser with a velocity that is equal to the velocity of the jet, V 1 = 350 mph. The jet is flying at an altitude where the inlet air pressure is P 1 = 75 kpa and the temperature is T 1 = 5ºC. Assume that the diffuser is reversible (i.e., it generates no entropy), operates at steady state, and is adiabatic. Further assume that the exit velocity of the diffuser is small enough that the kinetic energy of the air leaving the diffuser is negligible. You may ignore the potential energy of the flow entering and leaving the diffuser. a.) What are the pressure and temperature of the air leaving the diffuser, P 2 and T 2?

65 Figure 6.B-4(b): The Goblin II jet engine. Compressor The compressor shown in Figure 6.B-4(b) for the Goblin II engine is a single stage centrifugal compressor which has a pressure ratio PR c = 5.5; that is, the pressure at the compressor exit (P 3 ) is 5.5 times larger than the pressure at the compressor inlet (P 2 ). The compressor efficiency is η c = You may assume that the compressor operates at steady state and is adiabatic. You may neglect the kinetic and potential energy of the flow entering and leaving the compressor. b.) What is the power required by the compressor, W c? c.) What is the temperature of the air leaving the compressor, T 3? d.) What is the rate of entropy production in the compressor, S gen, c? Combustor As the air passes through the combustor, it is mixed with fuel which is ignited, releasing heat. Here, we will model this combustion as a heat addition, Q cb, which brings the temperature of the air leaving the combustor to temperature T 4 = 1120ºC. You may assume that there is no pressure loss in the combustor so that P 4 = P 3. The combustor operates at steady state and you can ignore the kinetic and potential energy of the flow entering and leaving the combustor. e.) What is the rate of heat transfer that must be provided to the combustor, Q? Turbine The turbine provides sufficient mechanical power to drive the compressor (sometime additional power is extracted to run auxiliary systems like electronics, etc. but we'll neglect that here); therefore, W t = W c. The turbine efficiency is η t = You may assume that the turbine operates at steady state and is adiabatic. You may neglect the kinetic and potential energy of the flow entering and leaving the turbine. cb

66 Hint: tackle the analysis of the turbine using the following steps: (1) carry out an energy balance on the actual turbine to get h 5 and T 5, (2) use the definition of the turbine efficiency to get W ts,, the power that would be produced by a reversible turbine, (3) carry out an entropy and enthalpy balance on the reversible turbine to get h 5,s and s 5,s, (4) determine the exit pressure P 5 from h 5,s and s 5,s. f.) What is the temperature of the air leaving the turbine (T 5 )? g.) What is the pressure of the air leaving the turbine, P 5? h.) What is the rate of entropy production in the turbine, S gen, t? Nozzle The nozzle accelerates the fluid to a high velocity so that the air can be propelled out the back of the turbojet engine at a velocity that is faster than the plane itself is flying; this produces thrust (like a rocket engine). The nozzle efficiency is η n = The nozzle exit pressure is the same as the diffuser inlet pressure, P 6 = P 1. You may assume that the nozzle operates at steady state and is adiabatic. You may neglect the kinetic energy of the flow entering (but not leaving) the nozzle and the potential energy of the flow entering and leaving the nozzle. i.) What is the velocity of the air leaving the nozzle, V 6? j.) What is the rate of entropy production in the nozzle, S gen, n? k.) Prepare an h-s plot for air and overlay your state points onto the plot. Make sure that the states look correct to you and that you understand why enthalpy and entropy are changing as the air flows through each component. The whole point of a turbojet engine is to produce a thrust force that pushes the jet forward. The thrust force is given by: F = mv V thrust ( 6 1) l.) What is the thrust force produced by this engine? (Note that the thrust rating for the Goblin II engine was about 14 kn so your answer shouldn't be too different from that). m.) The thrust capacity of an engine is reduced with engine velocity (because the velocity difference in the thrust equation is reduced); plot the thrust capacity of the engine as a function of jet velocity (i.e., V 1) for a range of speeds from 0 to 1000 mile/hr. n.) The thrust required to move a plane forwards increases with velocity due to air friction and form drag; assume that the drag force for an F80 jet is related to velocity according to: N 1.5 Fdrag = 0.9 V mph Overlay the drag force equation onto your plot from (m) and use it to predict the maximum velocity of the F80 equipped with a single Goblin II jet engine. The heat provided to the combustor is obtained by burning fuel. The amount of fuel required can be estimated from the fuel's heating value, HV, which is the ratio of the heat provided to the mass of the fuel burned. The heating value of jet fuel is approximately HV = 4.2x10 7 J/kg. o.) Determine the rate at which the engine consumes fuel, m. There are two important figures of merit for a jet engine; one is the total thrust produced and the second is called the specific thrust, which is defined as the amount of thrust you get divided by the rate at which the engine consumes fuel. p.) Determine the specific thrust of the engine. fuel

67 q.) Plot the specific thrust of your engine as a function of the compressor pressure ratio, for pressure ratios from 2 to 100 (with V 1 = 350 mph). If you were trying to optimize the specific thrust of your engine then what pressure ratio would you try to achieve? r.) Plot the thrust force produced by your engine as a function of the compressor pressure ratio, for pressure ratios from 2 to 100; overlay on your plot curves that correspond to several values of turbine inlet temperature including T 5 = 1000ºC, 1200ºC, 1400ºC, and 1600ºC. If you were trying to optimize the thrust provided by your engine, then what pressure ratio would you try to achieve? Your plots should indicate that the performance of your engine gets much better as T 5 increases; this trend explains the intense effort towards developing high temperature materials for gas turbine engines. Figure 6.B-4(c) illustrates the turbine inlet temperature for various types of gas turbine engines as a function of the year and shows clearly that, over time, we are coming up with better materials that lead to higher efficiency gas turbine engines. Figure 6.B-4(c): Gas turbine temperature trends.

68 6.B-5 You have access to heat that is the byproduct of an industrial operation. The heat transfer rate is Q H = 5 kw and the temperature of the heat source is T H = 500ºC. Currently, the heat is transferred directly to ambient temperature at T o = 20ºC, as shown in Figure 6.B-5(a). T o = 20 C W max T o =20 C Q = 5kW H Q o Q = 5kW H T H =500 C (a) reversible, steady state heat engine (b) T H = 500 C P o =1atm turbine η t = W p W t water at P o =1atm T o =20 C 1 tank P tank = 120 psi 2 vapor liquid 3 Q = 5kW H T H =500 C Q amb (c) Figure 6.B-5: (a) Heat transferred directly to ambient, (b) heat source being used by a steady-state, reversible heat engine, and (c) heat source being used by the heat engine that you've designed. a.) Determine the rate at which entropy is generated by rejecting the heat to ambient (W/K). Because you have taken thermodynamics you know that where entropy generation occurs there is a lost potential to produce work. b.) Evaluate the lost potential for power production by determining the power that would be produced (W) if the same heat source were used in a reversible, steady state heat engine, as shown in Figure 6.B-5(b). You have designed the energy scavenging system shown in Figure 6.B-5(c) in order to produce some power using the heat source. The system consists of a tank of water that contains a twophase liquid/vapor condition. The tank is heated by the heat source. The saturated vapor that is produced is fed to a turbine with efficiency η t = The turbine exhausts to atmospheric

69 pressure, P o = 1 atm. In order to maintain the tank at a steady operating condition, it is fed with liquid that is provided by a reversible pump. The pump takes in liquid at atmospheric conditions, T o and P o, and increases its pressure it to the tank pressure, P tank = 120 psi. The flow rate of the pump is controlled so that the tank pressure never changes and the level of the liquid water in the tank never varies. The tank is not perfectly insulated. There is a heat transfer from the tank to the surroundings given by: Q = UA T T ( ) amb tank o where UA = 8.2 W/K and T tank is the temperature of the contents of the tank. c.) Determine the net power produced by your system (W); this is the turbine output power less the power required by the pump. d.) Determine the total entropy generation rate associated with operating the system (W/K). The lost power associated with the system in Figure 6.B-5(c) is equal to the maximum power that could have been obtained, calculated in (b), less the net power that was obtained, calculated in (c). e.) Determine the lost power associated with the system (W). f.) If you've done the problem right, then the product of the entropy generation rate, from (d), and the ambient temperature should equal the lost power, from (e). Show that this is so. This is the most direct and intuitive meaning of entropy generation - it is the lost power divided by the ambient temperature. g.) Determine the total efficiency of the system. h.) Plot the efficiency of the system as a function of the tank pressure. You should see that an optimal pressure exists. Clearly explain why this is so.

70 6.B-6 Supercritical carbon dioxide power cycles are being considered for the next generation of nuclear power plants. These cycles operate with carbon dioxide in the vicinity of the critical point. There are some potential advantages associated with this cycle including high efficiency and small equipment. However, there are several engineering challenges that must be overcome as well. The leakage of carbon dioxide through the seals in the turbomachinery required by the cycle must be well-understood in order to design the system. Your company has received a contract from the Dept. of Energy to build a test facility into which sub-scale seal designs can be tested. You need to design an experiment that is capable of applying a specific inlet condition and exit pressure to a seal and measure the leakage flow rate through various seal geometries. This problem addresses the design of the equipment required for this facility. Your preliminary system design is shown in Figure 6.B-6. The seal (labeled test facility in the figure) acts like a valve - it is a restriction to the flow and the purpose of the system is to characterize this restriction. The nominal design calls for an inlet pressure and temperature (at state 1) of P in = 9 MPa and T in = 310 K and an exit pressure (at state 2) of P out = 2 MPa. In order to design the system you have estimated the flow through the test facility to be m = 0.12 kg/s. Eventually, this is the quantity to be measured. test The carbon dioxide leaving the test facility at state 2 lies under the vapor dome and therefore is a mixture of liquid and vapor. The liquid cannot be allowed to enter the compressors that energize the system. Therefore, the carbon dioxide at state 2 is captured in a large insulated tank equipped with an electrical heater. The vapor from the top of the tank at state 3 passes through another electrical heater in order to elevate its temperature by ΔT htr = 10 K and to therefore be absolutely sure that no liquid enters the compressor. The low pressure compressor has a suction volumetric V = 10.2 ft 3 /min and an efficiency of η LPc flow rate (i.e., the volumetric flow rate at state 4) of LPc = The carbon dioxide leaving the low pressure compressor at state 5 is intercooled by rejecting heat to the atmosphere at T o = 20 C. The intercooler has an approach temperature difference of ΔT ic = 5 K; that is, the temperature of the carbon dioxide leaving the intercooler has been cooled to within ΔT ic of T o (T 6 = T o + ΔT ic ). The carbon dioxide leaving the intercooler is compressed again in the high pressure compressor. The high pressure compressor has a suction volumetric flow rate (i.e., the volumetric flow rate at state 6) of V HPc = 4.5 ft 3 /min and an efficiency of η HPc = The carbon dioxide leaving the high pressure compressor at state 7 is divided into two streams. The mass flow that is not required by the test facility passes through a throttle valve and returns to the tank at state 8. The remaining mass flow rate passes through a cooler that rejects heat to T o in order to reduce its temperature to the inlet temperature required by the test facility. Assume that there is no loss of pressure due to flow through the tank, heater, intercooler, or cooler (i.e., P 2 = P 8 = P 3 = P 4, P 5 = P 6, and P 7 = P 1 ). Also assume that the system operates at steady state and each component is adiabatic except the intercooler and cooler.

71 cooler Q clr 7 6 high pressure compressor η = ft /min W HPc HPc 3 V HPc = P in =9MPa T in =310K 1 Q ic intercooler ΔT ic =5K test facility P out =2MPa 2 bypass flow m b low pressurecompressor η = ft /min W LPc LPc 3 V LPc = m = 0.12 kg/s test 3 W htr heater ΔT htr =10K m c insulated tank W tank Figure 6.B-6: System to measure leakage of carbon dioxide through sub-scale turbomachinery seals. a.) Determine the properties at states 1, 2, 3, and 4. b.) Determine the mass flow rate that is passing through the compressors, m c. c.) Determine the specific enthalpy, pressure, specific entropy, temperature, and specific volume at state 6. Hint, the specific volume at state 6 is related to the mass flow rate and suction volumetric flow rate of the high pressure compressor. d.) You should now have the inlet conditions and exit pressure for both compressors. Use this information together with the compressor efficiency in order to determine states 5 and 7. What is the compressor power required by the low pressure and high pressure compressors? e.) Determine state 8 in your arrays table. f.) Prepare a temperature-entropy plot that shows each of the states. g.) Determine the heater power required by the tank ( W tank ) and the heater power required by the heater immediately before the low pressure compressor ( W htr ). h.) Determine cooling required in the intercooler ( Q ic ) and the cooler ( Q clr ). i.) Check your solution by carrying out an overall energy balance on the system. j.) Determine the total rate of entropy generation in the system. k.) Plot the bypass mass flow rate as a function of the exit pressure from the test facility, P out (assume that the mass flow rate through the test facility does not change). Based on this graph, what is the lowest pressure that your system can achieve?

72 l.) Plot the total electrical power required (compressors and heaters) as a function of the test facility exit pressure. If your laboratory electrical service can only provide 60 kw of electrical power then what is the maximum value of the test facility exit pressure that you can achieve?

73 6.B-7 You own a swimming pool that must be heated to T pool = 80ºF using the system shown in Figure 6.B-7. SF =950W/m 2 Q loss T c,out T o =70 F Q = 9.5 kw pool collector A col =6m 2 pump η p =0.40 T pool =80 F Q furnace furnace T p,out T pool W p Figure 6.B-7: Swimming pool heating system. The system uses both a conventional natural gas fired boiler as well as a low temperature, unglazed solar collector. On this day, the outdoor air temperature is T o = 70ºF and the solar flux is SF = 950 W/m 2. The total heat loss from the pool under these conditions is Q pool = 9.5 kw. The collector has a total surface area A col = 6 m 2. A pump pulls water out of the pool and pumps it through the solar collector where it is heated to T c,out and returned to the pool. The collector absorbs all of the incident solar radiation. The rate at which the collector transfers heat to the surroundings is given by: Q = UA T T (, ) loss c out o where UA = 120 W/K is the loss coefficient for the collector. The pressure drop associated with flow through the collector has been characterized by the manufacturer and is given by: 2 12 Pa-s 2 Δ Pcol = 5x10 V 6 m where V is the volumetric flow rate of water through the collector (in m 3 /s) and ΔP col is the pressure drop across the collector (in Pa). The pump is currently operating at N = 1000 rev/min. The dead-head pressure rise produced by the pump (i.e., the pressure rise with no flow) depends on the rotational speed and is given by: Pa-min Δ Pdh, pump = 150 N rev where ΔP dh,pump is the dead head pressure rise (in Pa) and N is the rotational speed (in rev/min). The open-circuit volumetric flow rate produced by the pump (i.e., the flow rate produced with no pressure rise) also depends on rotational speed and is given by: 3 7 m- min V oc, pump = 5x10 N s-rev where V oc, pump is the open circuit flow rate (in m 3 /s) and N is the rotational speed (in rev/min). Assume that the pump curve is linear between Δ P dh, pump at zero flow and V oc, pump at zero pressure

74 rise. That is, the pressure rise produced by the pump (ΔP pump ) as a function of flow rate (V ) is given by: V Δ Ppump =ΔPdh, pump 1 V oc, pump The pump efficiency is relatively constant and equal to η p = The cost of the electricity required to run the pump is ec = 0.12 $/kw-hr. The portion of the total pool heating load that is not met by the solar collector is met using a conventional natural gas-fired furnace ( Q ). The cost of natural gas is ngc = 1.25$/therm. Model the pool water as an incompressible substance with v w = m 3 /kg and c w = 4200 J/kg-K. a.) Determine the volumetric flow rate of water pumped through the collector (V ). b.) Determine the pump power required ( W ). p c.) Determine the temperature of the water leaving the pump (T p,out ). d.) Determine the temperature of the water leaving the solar collector (T c,out ). e.) What is the rate at which energy is provided to the pool from the solar collector system? What is the cost ($/hr) associated with providing this energy (i.e., the cost of running the pump)? f.) What is the remaining rate of heat transfer required by the pool that is provided by the furnace ( Q furnace )? What is the cost ($/hr) associated with providing this energy (i.e., the cost of burning natural gas)? g.) What is the total cost ($/hr) associated with running the pool heating system. h.) You have purchased a variable speed drive for the pump so that you can adjust the rotational speed, N, in order to optimally run the system. Plot the total cost associated with running the system as a function of the rotational speed of the pump. You should see that an optimal pump exists - clearly explain why this is so. furnace

75 6.B-8 Many detectors must be maintained at cryogenic temperatures in order to operate. One method of keeping space-borne detectors cold is to launch them immersed in a tank of liquid cryogen (e.g., liquid nitrogen, liquid neon, etc.) that slowly boils off due to unavoidable but hopefully small, parasitic heat transfer from the surroundings. As long as the cryogenic tank (referred to as a Dewar) contains some liquid, the detector can operate. Eventually, the tank runs out of liquid and the mission must be terminated. Figure 6.B-8(a) illustrates a Dewar of liquid oxygen that is vented to space. Model space as being at pressure P out = 0.1 atm. relief valve vented to space P out =0.1atm Q = 0.25 W sur V = 50 liter P d =3atm x 1 =0 T v,out relief valve Q HX, B T tank Q = 0.25 W sur V = 50 liter P d =3atm x 1 =0 dewar (a) turbine η t =0.75 (b) Q = 0.25 W sur T HX,out vented to space P out =0.1atm W t T t,out Q HX, C T tank V =50liter P d =3atm x 1 =0 vented to space T HX,out P out =0.1atm (c) Figure 6.B-8: (a) Dewar vented to space, (b) Dewar with heat exchanger, and (c) Dewar with turbine and heat exchanger. The volume of the Dewar is V = 50 liter and the Dewar is initially filled with saturated liquid oxygen at P d = 3 atm. A relief valve maintains the pressure within the Dewar at P d by venting saturated vapor to space. The mission is terminated when all of the liquid is gone and the Dewar contains saturated vapor. The Dewar experiences a parasitic heat transfer rate from the surroundings of Q sur = 0.25 W. a.) Determine the mission duration (days) for the vented Dewar shown in Figure 6.B-8(a). An engineer has proposed the alternative system shown in Figure 6.B-8(b). He noticed that the temperature of the oxygen leaving the Dewar drops somewhat as it passes through the valve. Therefore, rather than vent the oxygen directly to space it could be passed through a heat exchanger that is interfaced with the tank in order to extract some heat from the tank as it warmed back up to near the tank temperature. The approach temperature difference for the heat exchanger is ΔT HX = 0.5 K. That is, the temperature of the vapor leaving the heat exchanger is T HX,out = T tank - ΔT HX where T tank is the temperature of the contents of the tank. Assume that the

76 pressure downstream of the valve (at state v,out) and downstream of the heat exchanger (at state HX,out) are the same and are equal to the pressure of space, P out. b.) Determine the temperature of the oxygen leaving the valve, T v,out. c.) Determine the heat transfer per unit mass of oxygen extracted from the tank by the heat exchanger ( Q, / m ). HX B d.) What is the effectiveness of the heat exchanger? e.) Determine the mission duration (in units of day) for the system shown in Figure 6.B-8(b). The engineer is disappointed with the results of his first idea and therefore has proposed the system shown in Figure 6.B-8(c). The relief valve is replaced by a turbine with efficiency η t = 0.75 in order to provide a larger temperature reduction prior to entering the heat exchanger and therefore provide more cooling effect to the Dewar. The pressure in the Dewar is maintained at P d by appropriately metering the flow through the turbine. All of the other conditions for the problem remain the same. f.) Determine the temperature of the oxygen leaving the turbine, T t,out. g.) Determine the rate of heat transfer to the heat exchanger per mass of oxygen passing through the heat exchanger. h.) Determine the mission duration (day) for the system shown in Figure 6.B-8(c). i.) Plot the mission duration for the system shown in Figure 6.B-8(c) as a function of the turbine efficiency.

77 6.B-9 Figure 6.B-9 illustrates a two-stage helium compressor system that is used to recompress low pressure helium that is captured in a recovery system so that it can be re-liquefied. Q ac,1 Q ac,2 T amb T amb aftercooler 1, ΔT ac,1 =10K aftercooler 2, ΔT ac,2 =7K helium T = 20 C amb P = 100 kpa amb m = 0.05 kg/s 1 2 P int = 500 kpa first stage compressor η c,1 = P discharge = 3500 kpa second stage compressor η c,2 = to liquefier W c,1 W c,2 Figure 6.B-9: Helium compression system. Low pressure helium at T amb = 20 C and P amb = 100 kpa enters the first stage compressor with mass flow rate m = 0.05 kg/s. The efficiency of the first stage compressor is η c,1 = The first stage compressor increases the pressure of the helium to P int = 500 kpa. The hot helium leaving the first stage compressor is cooled in aftercooler 1 prior to entering the second stage compressor. There is no loss of pressure due to the flow through the aftercooler. Aftercooler 1 has an approach temperature difference ΔT ac,1 = 10 K and rejects heat to T amb ; therefore, the temperature of the helium leaving aftercooler 1 will be T amb + ΔT ac,1. The helium next enters the second stage compressor where it is compressed to the final pressure, P discharge = 3500 kpa. The efficiency of the second stage compressor is η c,2 = The helium leaving the second stage compressor is cooled in aftercooler 2 before it enters the liquefier. Aftercooler 2 has an approach temperature difference ΔT ac,2 = 7 K and rejects heat to T amb ; therefore, the temperature of the helium leaving aftercooler 2 will be T amb + ΔT ac,2. a.) Determine the power required by the first stage compressor. b.) Determine the rate of entropy generation in the first stage compressor. c.) Determine the rate of heat transfer from aftercooler 1. d.) Determine the rate of entropy generation in aftercooler 1. e.) Determine the power required by the second stage compressor. f.) Determine the rate of entropy generation in the second stage compressor. g.) Determine the rate of heat transfer from aftercooler 2. h.) Determine the rate of entropy generation in aftercooler 2. i.) Check your answer by carrying out an energy balance around the entire two-stage compressor/aftercooler system. Verify that energy balances for this system. j.) Check your answer by carrying out an entropy balance around the entire two-stage compressor/aftercooler system. Verify that entropy balances for this system. k.) If the volumetric efficiency of the compressors is η vol = 0.85 then determine the displacement rate of each compressor. Recall that the displacement rate determines the overall size of the compressor and is defined as the ratio of the suction volumetric flow rate to the volumetric efficiency of the compressor. l.) Plot the total compressor power and the total rate of entropy generation as a function of the intermediate pressure, P int. You should see that the same intermediate pressure minimizes both the power required and the entropy generation rate.

78 m.) Use the optimization capability of EES to identify the optimal intermediate pressure. Comment out the intermediate pressure that you specified in the Equations Window and then select Min/Max from the Calculate menu. Minimize the total power (check the radio button minimize and select W total from the list of variables in the left window) by adjusting the intermediate pressure (select P int from the list of independent variables in the right window). Hit the Bounds button and select reasonable limits for P int.

79 6.B-10 Figure 6.B-10 illustrates a proposed design for a very small refrigeration system that can be used to provide cooling at low temperatures with no moving parts in the cold portion of the system. W c T amb 2 1 P 1atm amb = T 20 C amb = m = kg/s Q ac compressor PR = 1.8 η c = 0.85 aftercooler ΔT ac =5K 3 nozzle η n = Q C diffuser η d = 0.5 T C =-40 C Figure 6.B-10: Proposed cooling system. load heat exchanger ΔT LHX =2K to ambient Ambient air is drawn into the compressor at P amb = 1 atm and T amb = 20ºC with mass flow rate m = kg/s. The compressor has a pressure ratio PR = 1.8 (that is, the ratio of the discharge pressure to the suction pressure is PR). The efficiency of the compressor is η c = The hot air leaving the compressor passes through an aftercooler that has an approach temperature difference of ΔT ac = 5 K. Assume that there is no pressure loss in the aftercooler. The air passes through a nozzle, causing its velocity to increase and its temperature to drop. The nozzle efficiency is η n = The cold, high velocity air passes through the load heat exchanger that accepts heat from the refrigeration load at T C = -40ºC. The load heat exchanger has an approach temperature difference of ΔT LHX = 2 K. Assume that neither the velocity nor the pressure of the air passing through the load heat exchanger change. Finally, the high velocity air passes through a diffuser with efficiency η d = 0.5. The air leaving the diffuser is exhausted to the ambient. Neglect kinetic energy everywhere in the cycle except at states 4 and 5. Model air as an ideal gas but do not assume that it has a constant specific heat capacity. a.) Determine the power required by the compressor and the temperature of the air leaving the compressor. b.) Determine the rate of heat transfer from the aftercooler. It is next necessary to analyze the nozzle, load heat exchanger, and diffuser. This process is made more complicated by the fact that the pressure at state 4 is not known and the velocity at state 5 is not known. Therefore, it is not really possible to directly and sequentially analyze the nozzle, load heat exchanger, and then finally the diffuser. The best way to proceed is to guess a value of P 4 and analyze the nozzle - this guess will eventually have to be removed in order to complete the problem. Set the guess for P 4 = Pa and proceed with the problem. c.) Using P 4 = Pa, determine the exit velocity and temperature of the nozzle. d.) Determine the rate of heat transfer from T C provided in the load heat exchanger. e.) Determine the pressure leaving the diffuser. Note that the actual pressure leaving the diffuser must be atmospheric pressure. However, it won't be for your assumed value of P 4 from part (c). We will force P 6 to be equal to P amb in part (f). f.) The pressure that you calculated in part (e) was not equal to atmospheric pressure and yet the diffuser exhausts to the atmosphere. You need to adjust the assumed value of P 4 from part (c) until P 6 = P amb. You can do this manually but EES makes the process much easier.

80 Update your guess values (select Update Guesses from the Calculate menu) and then comment out (i.e., remove) your guessed value for P 4. Add the specification that P 6 = P amb and solve the problem. What is the value of Q C and the temperature of the air leaving the diffuser? g.) What is COP of the refrigerator? Compare this to the COP of a reversible refrigerator providing refrigeration at T C and rejecting heat to T amb. h.) Use EES to draw a T-s diagram of the cycle. Label each of the points and include isobars for each of the pressures in the cycle. i.) Plot the COP and refrigeration power ( Q C ) of the cycle as a function of the compressor pressure ratio. You should see that there is a pressure ratio that optimizes the COP. j.) What is the highest COP that the cycle shown in Figure P6.B-10 can have? That is, if each of the components were performing at their maximum possible level, what would the COP be? Note that the cycle would still not be reversible because the heat exchangers cannot be reversible devices.

81 6.B-11 Figure 6.B-11(a) illustrates a solar water heating system. water tank T w = 50 C Q HX heat exchanger ΔT HX =8K 1 pump solar collector A c =3m 2 3 Q amb W pump 2 SF = 700 W/m 2 PV panel η PV =0.1 ethylene glycol/water solution r eg = 1039 kg/m 3 c eg = 3737 J/kg-K SF = 700 W/m 2 A PV =0.4m 2 Figure 6.B-11(a): A solar water heating system. A 30% ethylene glycol/water solution is circulated through the solar collector where it is heated by the solar energy. The hot water passes through a heat exchanger in the domestic water tank. The ethylene glycol/water solution is pumped using a small pump that is energized by a solar photovoltaic (PV) panel that produces electrical energy. This approach has the advantage that the pump will automatically turn off when the sun stops shining because the PV panel will stop providing power to the pump. Figure P6.B-11(b) illustrates the pump curves for one pump that would be appropriate for this application. Notice that there is a different pump curve for each value of power. Figure 6.B-11(b): Pump curves. The water in the tank is at T w = 50ºC and the approach temperature difference for the heat exchanger in the ΔT HX = 8 K. The ethylene glycol should be modeled as an incompressible fluid

82 with ρ eg = 1039 kg/m 3 and c eg = 3737 J/kg-K. We will use a simple model of the pump curves shown in Figure 6.B-11(b). The dead head pressure produced by the pump is given by: Δ Pdh = AW pump where A = 1.2 psi/w and W pump is the rate of power provided to the pump (from the PV panel). The open circuit flow produced by the pump is given by: V oc = BW pump where B = 0.15 gpm/w. The pump curve is assumed to be linear between these two limits; therefore the pressure rise produced by the pump is given by: V Δ Ppump =ΔPdh 1 V oc The pump is powered by a PV panel with area A PV = 0.4 m 2 and efficiency η PV = 0.1 (10%). The solar collector has area A c = 3 m 2. The solar flux (on both the PV panel and the collector) is SF = 700 W/m 2. The solar energy on the collector is either transferred to the fluid passing through the collector or lost as a heat transfer to the ambient. The ambient temperature is T amb = 0ºC. The heat transfer between the solar collector and the ambient can be modeled according to: Q = UA T T amb ( ) 3 where UA = 14 W/K is the loss coefficient for the collector and T 3 is the temperature of the ethylene glycol solution leaving the collector. There is no loss of pressure in the fluid as it passes through the collector. However, there is a significant pressure drop associated with the heat exchanger. The pressure loss in the heat exchanger can be modeled as: 2 Δ PHX = CV where C = 50 psi/gpm 2. a.) Determine the volumetric flow rate of the ethylene glycol/water solution that is circulated through the system (in gpm). b.) What is the efficiency of the pump at the operating condition you found in part (a)? c.) Determine the temperature of the ethylene glycol solution leaving the pump at state 2. d.) Determine the temperature of the solution leaving the solar collector at state 3. e.) Determine the rate of heat transfer provided to the water tank, Q HX. f.) What is the efficiency of the solar heating system? The efficiency of such a system is typically defined as the ratio of the heat transfer delivered to the water to the rate of solar energy that is incident on the collector. g.) Plot the rate of heat transfer provided to the tank as a function of the solar flux (note that the solar flux can vary from 0 W/m 2 at night to about 1000 W/m 2 on a really sunny day in summer). Explain why your plot behaves as it does. At what value of solar flux should you deactivate the system? What value of PV power does this correspond to? amb

83 6.B-12 Figure 6.B-12 illustrates a compressor that is being used to fill a tank. Q HX T amb P c,out = 700 kpa heat exchanger ΔT HX =10K W c T P V amb amb cin, = 20 C = 100 kpa 3 = 0.1 m /s compressor η c =0.7 Q tank Figure 6.B-12: Compressor filling a tank. tank V =1m 3 T 1 = T amb, P 1 = P amb T 2 = 70 C, P 2 = P c,out Ambient air is drawn into the compressor at P amb = 100 kpa and T amb = 20ºC. The volumetric flow rate at the compressor inlet is V cin, = 0.1 m 3 /s. The compressor exit pressure is P c,out = 700 kpa and the compressor efficiency is η c = 0.7. The air leaving the compressor passes through a heat exchanger where it is cooled by transferring heat to the atmosphere at T amb. The approach temperature difference for the heat exchanger is ΔT HX = 10 K. The air passes through a valve and enters the tank. The tank is rigid and has volume V = 1 m 3. The initial temperature and pressure of the air in the tank are T 1 = T amb and P 1 = P amb, respectively. The final temperature and pressure of the air in the tank are T 2 = 70ºC and P 2 = P c,out. During the filling process, the tank transfers heat to the surroundings at T amb. Model air as an ideal gas with R = 287 J/kg-K, c P = 1005 J/kg- K, and c v = 718 J/kg-K. a.) Determine the mass flow rate of air entering the compressor, m (kg/s). b.) Determine the rate of power consumed by the compressor, W c (W). c.) Determine the rate of heat transfer from the heat exchanger to the atmosphere, Q HX (W). d.) Determine the time that is required to fill the tank (s). e.) Determine the heat transfer from the air in the tank to the surroundings that occurs during this process, Q tank (J). f.) List as many of the specific sources of entropy generation associated with this process. g.) Calculate the total entropy generation associated with the process (J/K). Clearly show in the figure a dashed line that indicates the system that you are using to accomplish this calculation.

84 6.B-13 Figure 6.B-13 illustrates the cooling system for an engine. T, 95 C af in = V = 30 gpm af antifreeze engine block ΔT eb =10K radiator ε rad =0.65 Q eb T engine air pump V 3200 cfm air = T 20 C amb = P 100 kpa amb = Figure 6.B-13: Cooling system for an engine. Antifreeze is circulated through the engine block at a rate of V af = 30 gpm where it receives heat at a rate Q eb. Model antifreeze as an incompressible substance with ρ af = 1039 kg/m 3 and c af = 3740 J/kg-K. The antifreeze leaves the engine block and enters the radiator at T af,in = 95 C. The antifreeze is cooled in the radiator by a flow of ambient air that enters at P amb = 100 kpa and T amb = 20 C with an inlet volumetric flow rate V air, in = 3200 cfm (cubic feet per minute). Model air as an ideal gas with R air = N-m/kg-K and c P,air = 1005 J/kg-K. The effectiveness of the radiator is ε rad = 0.65 and the pinch point is at the warm end (i.e., at the end where the coolant enters). The approach temperature difference for the engine block is ΔT eb = 10 K. Neglect the effect of the pump on the state of the antifreeze (i.e., assume that the state of the antifreeze entering and leaving the pump is essentially the same). a.) Determine the rate of heat from the antifreeze to the air in the radiator. b.) Determine the temperature of the air and the antifreeze leaving the engine (in ºC). c.) What is the approach temperature difference associated with the radiator? d.) Determine the engine temperature (T engine in C) and the effectiveness of the engine block. e.) Plot the engine temperature and the temperature of the coolant entering the radiator as a function of the rate of heat transfer from the engine block (on the same plot). If the maximum temperature that the coolant can reach before the engine overheats is 100 C then what is the maximum rate of heat transfer from the engine that can be accommodated? f.) Cars tend to overheat more readily in high altitudes where the atmospheric pressure is lower than its value at sea level. Plot Q eb as a function of P amb for T af,in = 95ºC for P amb between 80 kpa and 100 kpa to show how the cooling capacity of the system is affected.

85 6.B-14 You have access to a flow of saturated steam (i.e., saturated vapor) that is continuously vented from an open feedwater heater in a coal-fired heating plant. The mass flow rate of the steam is m stm = 0.35 kg/s and the steam is at P stm = 50 psi. The environment is at pressure P amb = 1 atm and T amb = 20ºC. You are interested in trying to utilize the steam to provide some power. Initially, you would like to estimate the maximum possible rate at which power could be produced using this resource in this environment. The maximum possible rate of power that could be produced occurs if the steam is provided to a set of reversible equipment that exchanges heat with the atmosphere. The steam must be brought to the temperature and pressure of the environment (called the dead state) as it exhausts from the equipment. This hypothetical system is shown in Figure 6.B-14(a). W req steam m 0.35 kg/s stm = P 50 psi stm = x = 1 stm reversible equipment dead state P amb =1atm T amb =20 C T amb steam m 0.35 kg/s stm = P 50 psi stm = x = 1 stm Q req (a) turbine η t =0.82 Q eq P t,out =1atm T amb dead state P amb =1atm T amb = 20 C W t (b) Figure 6.B-14: Steam being (a) used by a reversible set of equipment to produce the maximum possible amount of power and (b) used with a real turbine to produce some power. a.) Analyze the system shown in Figure 6.B-14(a) in order to determine the maximum possible rate at which power can be produced using the steam. In order to use the steam to provide some power, you have decided to feed it to a turbine, which exhausts to the atmosphere, as shown in Figure 6.B-14(b). The efficiency of the turbine is η t = Note that the steam leaving the turbine is at atmospheric pressure but not atmospheric temperature. Therefore, the steam will eventually re-equilibrate thermally with the atmosphere and finally come to the dead state temperature. b.) Analyze the system shown in Figure 6.B-14(b) and determine the power produced by the turbine. c.) Your answer from (b) should have been much less than your answer from (a). The difference is the lost power related to entropy generation. Determine the lost power by subtracting your answer from (b) from your answer from (a).

86 d.) The rate of lost power is related to the entropy generation in the system. Determine the total rate of entropy generation for the system shown in Figure 6.B-14(b) as well as the rate of entropy generation that can be attributed to the irreversible turbine and the rate of entropy generation that can be attributed to the re-equilibration process. e.) If you've done the problem correctly, the product of the total rate of entropy generation and the dead state temperature should be equal to the value of the lost power computed in (c). Verify that this is true. f.) Based on your calculations in (d), what modification to your actual cycle would make the most sense in order to capture more of the power producing potential of the steam?

87 6.B-15 Steam enters a nozzle at T 1 = 520 C and P 1 = 100 bar with a velocity V 1 = 13 m/s. It leaves the nozzle at P 2 = 20 bar and the speed of sound at this condition. Steam does not obey the ideal gas law at these conditions. a.) Determine the temperature of the steam exiting the nozzle b.) Determine the nozzle efficiency c.) Determine the pressure, temperature and velocity at the nozzle exit that would result in a nozzle efficiency of 100%.

88 6.B-16. An industrial process exhausts 9.0 kg/sec of spent steam at 200 C, 1 bar. Currently this water is recovered as a condensate by cooling it to 25 C, 1 bar with cooling water in a counter-current heat exchanger, as shown in Figure 6.B-16a. Cooling water enters at 15 C, 1 bar and exits at 30 C, 1 bar. Thermal losses from the heat exchanger jacket are negligible. Figure 6.B-16a: Heat exchanger for recovering condensate a.) Calculate the effectiveness of the heat exchanger. b.) Determine the rate of entropy production resulting from this steady-state process. c.). An employee has indicated that the current process is wasteful since power could be generated with the steam. She has proposed a system modification in which power would be produced with an efficiency of 10% as shown Figure 6.B-16b. The temperatures and mass flow rates at states 1, 2, and 3 are the same as before. What is temperature of the water at state 4 and the power if the thermal efficiency is 10%, as claimed? Figure 6.B-16b: Proposed equipment for recovering condensate d.) Another employee has questioned the 10% efficiency. What is the maximum efficiency that could be attained in this process (with the same temperatures and mass flow rates at states 1, 2, and 3 as in part b?

89 6.B-17 A steady-state system for producing power is shown in Figure 6.B-17. Water from a lake at 1 atm and 20 C (state 1) is adiabatically pumped to 10 bar (state 2). The pump draws 110 kw of power and the mass flow rate of water is 45 kg/s. The water is heated at constant pressure to 400 C (state 3) using combustion gas that enters at 500 C and exits at 182 C. The combustion gas as an average constant pressure specific heat of 1.1 kj/kg- K. The steam is adiabatically expanded in a turbine having an isentropic efficiency of The turbine exhausts (state 4) to the surroundings at atmospheric pressure and 20 C, 1 bar. Figure 6.B-17: Power production system a.) Determine the isentropic efficiency of the pump. b.) What is the heat exchanger effectiveness of the heater and the rate at which heat must be supplied from the heat source to the water to bring it to 400 C? c.) Determine the power produced by the turbine d.) What is total rate of entropy production resulting from this power production process? e.) What is thermal efficiency of this power production process? f.) If the pump and turbine efficiencies were both 1.0, would the efficiency of this system equal the maximum possible efficiency? (Answer Yes or NO and provide a short explanation.)

90 6.B-18 You have been asked to size a pump that will deliver 11,000 lbm/hr of water form an elevation that is 30 ft below the pump to a job site that is 50 ft above the pump. At the lower location (state 1), the pressure is atmospheric and the temperature is 60 F. The diameter of the inlet pipe is 1.5 in. At the point where the water is delivered (state 2) the pressure must be 80 psia and the pipe diameter is 1.0 in. Estimate the required pump power in hp, assuming the pump efficiency is 0.5.

91 6.B-19 A flow rate of m 3 /s of nitrogen gas at a temperature of 50 C and a pressure of 175 kpa is to be compressed to a final pressure of 17,500 kpa. A system that uses two compressors with an intercooling heat exchanger has been proposed, as shown in Figure 6.B-19. A centrifugal compressor having an isentropic efficiency of 0.80 is used for the low pressure gas as it can efficiently handle high-volume flow rates with relatively low pressure ratios. A reciprocating compressor is chosen for the high pressure gas since this type of compressor is well-suited to low volume flow rates at high pressure ratios. The intercooler is a cross-flow heat exchanger that has a high volume rate of air at 25 C as the coolant. You may assume that the nitrogen represents the minimum capacitance rate in this heat exchanger and that the pressure drop in the nitrogen across the intercooler is negligible. Also assume that nitrogen obeys the ideal gas law. air at 25 C intercooler 4 17,500 kpa 2 3 centrifugal compressor reciprocating compressor m 3 /s 1 50 C, 175 kpa Figure 6.B-19: Compressor system for nitrogen with intercooler a.) Determine the temperature at all states if the pressure at state 2 is set to 10,000 kpa. b.) Plot the total power required with both compressors versus the non-dimensional pressure at state 2 defined as P2 P 1 PR P4 P1 c.) Determine the pressure at state 2 that will result in the minimum power.

92 6.B-20 A portable power system consists of an un-insulated evacuated tank made of steel with a volume of V = 0.75 m 3. The tank is attached to the exhaust of an air turbine-driven grinding wheel. The plan is to drive the air turbine with the pressure difference between the air at atmospheric pressure and T amb = 20 C and the (lower) pressure in the tank. As air enters the tank, the pressure in the tank will rise and the process will eventually stop when the pressure in the tank is atmospheric pressure. The turbine operates adiabatically with an isentropic efficiency of η t = 0.7. The air in the tank is always at T amb due to heat transfer with the tank walls. The mass flow rate of air through the turbine varies with the pressure in the tank according to m = 0.01 [kg/s] ( P P ) atm where P atm is atmospheric pressure and P tank is the pressure in the tank. a.) How many seconds of operation will be provided by this system? b.) Determine the total mechanical work that can be obtained from the turbine during the operation period. c.) Calculate the total entropy generated by the process. P atm tank

93 6.B-21 The coefficient of pressure recovery is a valid metric for characterizing the performance of a diffuser regardless of the working fluid or operating conditions. However, it will not always limit to unity even for a reversible diffuser. a.) Show that the coefficient of pressure recovery will limit to unity for a reversible adiabatic diffuser that is operating on an incompressible fluid if the kinetic energy at the exit is negligible.

94 6.C-1 A 1.25 m 3 externally-insulated vessel originally contains a mixture of saturated liquid water and water vapor at 3450 kpa with 20% liquid on a volumetric basis. The liquid in this vessel is to be drained through a valve located at the bottom of the vessel. The instantaneous mass flow rate of liquid out of the vessel can be represented by: m C P- P 1/2 o where P o is the atmospheric pressure, P is the pressure in the vessel at time t and C is a constant equal to kg/s-kpa 0.5. The valve stays open until time all the liquid is ejected. We wish to know how much time is required for this process. The actual process is quite complicated since both heat and mass transfer take place between the vapor and liquid. It is possible, however, to bind the time required to discharge all the liquid by considering two cases. Case A: The interface between the vapor and liquid is adiabatic and impermeable Case B: The overall process is adiabatic and the liquid and vapor within the tank are always in thermodynamic equilibrium. a) Plot the temperature and pressure of the vapor remaining in the tank as a function of time for the time period between the start of the ejection and the point at which no more liquid remains in the tank for cases A and B. Compare the results. b) Determine the total entropy production resulting from this process, assuming that the ejected liquid equilibrates with the environment at 25 C, 1 atm for cases A and B.

95 6.C-2 Solar electric generation systems (SEGS) are becoming increasingly popular in southwestern U.S. because of the abundant solar resource and high electricity prices in this area. However, one problem that needs to be addressed with these systems is energy storage. The times of peak electrical demand often extend hours beyond the time when direct solar-generated electricity is available. The purpose of this problem is to provide a thermodynamic analysis of a simplified version of a storage concept that uses steam. The underground steam storage tank has a volume of 28,000 m 3. At the start of a charge cycle, the tank contains water at P i =4.1 MPa and 90% quality. The steam inlet valve is opened and steam enters the valve at P c =12.5 MPa, 338 C at rate Pc P m 114 [kg/s] Pc Pi where P is the current pressure in the tank in MPa. The charging process continues for 4.5 hours. We will neglect thermal interactions between the steam and tank walls in this analysis because the ratio of the tank volume to surface area is large. The discharge process is initiated by opening the valve that supplies steam from the tank to the low pressure turbine. The steam exits the tank rate P Pd m 86 [kg/s] Pc Pd where P d =3.2 MPa is the discharge pressure, The discharge process continues until the mass of steam remaining in the tank is equal to the mass of steam in the tank at the start of charging process. Please analyze this storage system by doing the following. a) Plot the mass, temperature and pressure of the water as a function of time for the charging process. b) Plot the mass, temperature and pressure of the water as a function of time for the discharging process c) Determine the amount of time needed during discharging to return the tank to the same conditions it had at the start of the charging process. d) How much energy is stored and retrieved during a charge/discharge cycle in MWhrs? e) If storage were not required, steam could have been supplied a constant rate at P c =12.5 MPa, 338 C. However, because of the storage, the steam is supplied at the conditions is emerges from the storage tank during the discharge process. Determine the total entropy production per charge/discharge cycle.

96 6.C-3 Compressed air at T s = 30 C and P s = 500 kpa is available in a factory that is not air conditioned. One of the engineers is tired of working in an uncomfortably hot environment and he has suggested that an air-conditioning effect can be achieved using this compressed air supply. His plan is illustrated in Figure 6.C-3. P s =500kPa T s =30 C valve A P atm =100kPa T amb =30 C receiver tank V tank =30m 3 valve C valve B building T indoors =20 C Figure 6.C-3: Proposed air-conditioning system using compressed air. A receiver tank with volume V tank = 30 m 3 is placed on the roof of the building. The tank is initially filled with ambient air at T amb = 30 C and P atm = 100 kpa. During the initial filling process (state 1 to 2), valves B and C are closed and valve A is opened so that air from the compressed air line rushes into the receiver tank. The filling process continues until the pressure in the tank reaches P s. The filling process occurs very quickly and therefore can be approximated as being adiabatic. a.) Determine the temperature of the air in the tank at the conclusion of the filling process and the mass of air that enters the tank during the process. During the constant pressure cooling process (state 2 to state 3), valve A remains open while the air in the tank cools to T 3 = 35 due to heat transfer with the surroundings at T amb. The pressure remains constant during this process. b.) Determine the mass of air that enters the tank and the heat transfer from the tank during the constant pressure cooling process. At the conclusion of the cooling process, the air in the receiver tank is still warmer than the building. In order to avoid having this warm air enter the building, valve A is closed, valve C is opened and air is vented from the tank. The first venting process (state 3 to 4) continues until the temperature of air remaining in the tank reaches T indoors = 20 C. The venting process occurs very quickly and therefore it is appropriate to neglect heat transfer during this process. c.) Determine the mass of air in the tank at the conclusion of the first venting process.

97 Finally, valve C is closed and valve B is opened so that air from the receiver tank rushes into the building. The second venting process (state 4 to 5) continues until the pressure in the receiver tank reaches P atm. The venting process is rapid and therefore the tank can be modeled as being adiabatic. The air passing through the building is warmed and leaves the building at T indoors, providing cooling to the building. d.) Determine the cooling provided to the building during the second venting process. e.) The compressor installed in the building has an isentropic efficiency of η c = Determine the work transfer required by the compressor in order to accomplish this cycle and the coefficient of performance (defined as the ratio of cooling provided to work required) associated with this cycle.

98 6.C-4 My son received a rather interesting toy rocket for his birthday. The rocket operates using only water and air and is claimed to shoot about 75 ft up into the air. The rocket itself is a hollow plastic projectile with a mass of 16 grams and an internal volume of 88 ml as shown in Figure 6.C-4. In a typical operation, the rocket is filled about halfway with liquid water and then connected to a hand air pump which pressurizes the air and water to about 4.5 bars. The air in the rocket is 25 C at this time due to heat transfer with the water and atmosphere. When the trigger is released, the pressurized water shoots out of the nozzle and propels the rocket upwards. As a first approximation, we can assume that the nozzle efficiency is 1 and the water temperature is constant at 25 C. The air trapped above the rocket expands until, when no water is left, air exits the nozzle until the pressure in the rocket is atmospheric. (It seems that the thrust resulting from the exiting air can be neglected.) The temperature of the air in the rocket during the expansion process is difficult to calculate. However, the performance of the rocket can be bracketed by calculating its performance assuming that the air expansion process is isothermal at the atmospheric temperature (25 C) at one extreme and adiabatic at the other. Air Water Rocket Data: Internal volume = 88 ml Mass (empty) = 16 grams Rocket nozzle diameter = 5.56E-3 m Total length of rocket = 14 cm Maximum diameter of rocket = 3.25 cm Aerodynamic drag coefficiency = 0.25 (estimate) Air pump Figure 6.C-4 Air-powered rocket toy a) Plot the height of the rocket as a function of time for a 5 sec period for both the isothermal and adiabatic cases assuming that the rocket is initially charged with 44 ml of water at 25 C. Which case results in the rocket achieving the higher height and why? b) A question of real interest to my son is how much water should be used to maximize the height the rocket will achieve. Estimate the optimum water charge, assuming the air is isothermal at 300 K. Be sure to state the assumptions in your analysis.

99 6.C-5 A steady flow of 0.05 kg/sec of air at 25 C, 7 bar is normally provided to an industrial process through a central pipeline. There are, however, times when the industrial process must be shut down for short intervals. It is not possible to stop the air flow in the pipeline. Rather than vent the high pressure air to the environment (which is wasteful), it is diverted to a 4 m 3 spherical storage vessel by closing valve A and opening valve B (see Figure 6.C-5). The vessel initially contains air at the ambient conditions of 25 C, bar. The storage vessel is fitted with a safety valve that opens when the pressure in the vessel exceeds 6 bar. Once opened, the contents of the vessel remain at a constant pressure of 6 bar. In your analyses, assume ideal gas behavior and neglect any heat transfer interaction between the tank wall and the air. State any other assumptions. Air 25 C, 7 bar Valve A To industrial process Valve B 4 m 3 Spherical Vessel Safety Valve Figure 6.C-5: Industrial process of compressed air with storage vessel a. What are the temperature and mass of the air in the vessel when the pressure valve first opens? How many minutes of operation are possible before the safety valve opens? b. Air flow continues to the vessel for an additional 4 minutes after the safety valve opens while the pressure in the vessel remains at 6 bar. Air exiting the safety valve is vented to the surroundings. Plot the temperature and mass of air in the vessel as a function of time for this four-minute period c. Calculate the total entropy change for the processes in parts a and b. What are the causes of the entropy production in these processes? Compare this value with the total entropy production that would have occurred if the air were vented directly to the surroundings for the same time period in which it was diverted.

100 6.C-6 Ammonia (NH 3 ) has been suggested as a possible fuel for transportation vehicles. The hydrogen in ammonia could be used to power fuel cells. However, ammonia has a number of disadvantages, toxicity being one of them. The experiment in Figure 6.C-6 has been developed to investigate possible consequences of a leak from an ammonia tank. A heavy-walled uninsulated 0.35 m 3 steel vessel initially contains ammonia at 5 bar and 25 C. The vessel is located in a well-insulated constant temperature enclosure containing ammonia vapor that is maintained at 25 C, 1 atm by an electrical heater, as shown in the figure. You may assume the heat loss through the enclosure walls to be negligible. When the electric heater is energized, its surface temperature quickly reaches a temperature of 288 C. It is cycled on and off as needed to maintain the ammonia around the tank at 25 C. The exit pipe is provided on the vessel is equipped with a pressure regulation valve adjusted to 3.2 bar and a conventional stopcock valve. Insulation - Ammonia 1 bar, 25 C Pressure valve to maintain 3.2 bar Stopcock valve Ammonia 1 bar, 25 C C Steel vessel Ammonia Initially at 25 C, 5 bar Figure 6.C-6: Equipment to test consequences of ammonia leak The stopcock is opened and gaseous ammonia is rapidly expelled outside of the insulated enclosure to a large reservoir of ammonia maintained at 25 C, 1 atm. The process continues until the pressure reaches 3.2 bar. The tank contents remain at 3.2 bar thereafter. a. Shortly after the ammonia pressure reaches 3.2 bar, the temperature of the ammonia remaining in the vessel is measured to be -7 C. Do you believe the experimental measurement of this temperature to be correct? Why or why not? If you do not believe that the temperature of the ammonia remaining in the vessel reaches -7 C, what is the lowest temperature you think it can reach? b. At some time later, the ammonia remaining in the vessel at 3.2 bar returns to 25 C. Estimate the total energy supplied to the electrical heater c. Estimate the total change in entropy resulting from the venting process.

101 6.C-7 A camp stove uses a 0.54 liter canister of n-butane as the fuel. Initially, the canister contains 0.22 kg of n-butane at 25 C. The stove produces 2.5 kw of heat with a burner efficiency of 0.85 based on the higher heating value of the butane vapor (49,130 kj/kg) when it is fed with saturated butane vapor at 25 C. Assume that the mass flow rate is constant for this problem. a) Determine the initial pressure, quality and volume vapor fraction of the canister. b) The stove can be configured such that liquid or vapor is extracted from the canister to feed the burner. The fuel line can be assumed to be adiabatic. The burner rate (2.5 kw) is the same in either case. Calculate and compare the required fuel flow rates for vapor and liquid feed to the burner. c) Assuming heat transfer between the fuel and the canister wall is negligible, plot the temperature and pressure of the butane in the canister for a period of 10 minutes (600 sec) assuming that saturated vapor is extracted from the canister and fed to the burner. d) Repeat part c) assuming saturated liquid, rather than saturated vapor, is extracted from the canister and fed to the burner. e) A slow extraction of a fluid from an adiabatic tank is often modeled by assuming that the fluid remaining in the canister has constant specific entropy. Is this model applicable for the process in part c) or d)? f) Calculate the entropy generation in the canister for parts c) and d). g) Is there an advantage to extracting vapor or liquid? Which do you recommend and why?

102 6.C-8 A small toy is advertised that will send up a signal flare. The toy consists of a metal tube 2.0 meters long and 2.5 cm in diameter as shown in Figure 6.C-8. A 0.5 kg piston fits into the tube and a mechanical trigger holds it in place 0.65 m above the bottom of the tube. To operate the device, the volume below the piston is slowly pumped up to a pressure of 4 bar with a small hand pump. The temperature of the air in the cylinder at this point is 38 C. Then, the trigger is depressed and the piston flies out of the top. The drag coefficient for aerodynamic drag on the piston is estimated to be 1.1. The process occurs rapidly and so it can be assumed to be adiabatic. State any other assumptions that you employ. Piston shoots out of cylinder m diameter tube 2.0 m 0.50 kg piston Air at 4 bar after pumping Figure 6.C-8: Toy that uses compressed air to send up a signal flare a. Estimate the velocity of the piston as it exits the top of the cylinder. b. How high would you expect the piston to go? The atmospheric temperature and pressure are 25 C, and 100 kpa, respectively. c. The design of this toy can perhaps be improved by securing the piston-trigger device at different position. What is the piston location during the pumping process which will result in the highest flight?

103 6.C-9 An annular fin (Figure 6.C-9)used in an evaporator is made of aluminum and bonded to a tube that has a 2 cm outer diameter. The outer diameter of the fin is 8 cm and its thickness is 0.25 mm. Air at 10 C flows on the outside surface of the fin. The convection coefficient between the fin and the air is 12.5 W/m 2 -K. Refrigerant evaporates inside of the tube at a rate sufficient to maintain the outer tube surface at a steady-state temperature of C. The steady-state temperature distribution in the fin is known to be a function involving modified Bessel functions of the first and second kind: I 0 mr K1 mr2 K 0 mr I1 mr2 b I 0 mr1 K1 mr2 K 0 mr1 I1 mr2 where = (T T air ) = (T base T air ) b r 1 inner radius of fin r 2 outer radius of fin 2h m kt h = convection coefficient k = thermal conductivity of fin material w= fin thickness 2h m Figure 6.C-9: Annular Fin kw Evaluate the properties of aluminum at the average of the air and base temperatures. a. Derive the governing differential equation that results in the steady-state temperature distribution in the fin. Plot the steady state temperature distribution of the annular fin as a function of its radius. b. Calculate the change in internal energy when the refrigerant flowing through the tube is stopped and the annular fin returns to a uniform temperature of 10 C.