PROBLEMAS: 1. Un gas ideal, inicialmente a 600 K Y 10 bar, se somete a un ciclo mecánicamente reversible de cuatro etapas en un sistema cerrado. En la etapa 1-2, la presión decrece isotérmicamente a 3 bar; en la etapa 2-3, la presión disminuye a volumen constante a 2 bar; en la etapa 3-4, el volumen disminuye a presión constante, y en la etapa 4-1, el gas regresa en forma adiabática a su estado inicial. Considere: Cp = (7/2)R y Cv = (5/2)R. a) Dibuje el ciclo en un diagrama PV. b) Determine (donde no se tienen) tanto a T como a P para los estados 1, 2, 3 y 4. c) Calcule W, Q, ∆H y ∆U para cada etapa del ciclo. 2. Un gas ideal, con Cp = (5/2)R y Cv = (3/2)R, cambia de P = 1 bar y v1t = 12 m3 a P2 = 12 bar y v2t = 1 m3 mediante los procesos mecánicamente reversibles siguientes: a) Compresión isotérmica. b) Compresión adiabática seguida por enfriamiento a presión constante. c) Compresión adiabática seguida por enfriamiento a volumen constante. d) Calentamiento a volumen constante seguido por enfriamiento a presión constante. e) Enfriamiento a presión constante seguido por calentamiento a volumen constante. Calcule Q, W, ∆Ht y ∆Ut para cada uno de los procesos, y dibuje las trayectorias de todos los procesos en un solo diagrama PV. 3. Un gas ideal, inicialmente a 30°C y 100 kPa, experimenta los siguientes procesos cíclicos en un sistema cerrado: a) En un proceso mecánicamente reversible, primero hay una compresión adiabática a 500 kPa, a continuación un enfriamiento a presión constante de 500 kPa hasta 30°C, y al final una expansión isotérmica hasta su estado original. b) El ciclo experimenta los mismos cambios de estado, pero cada etapa es irreversible con una eficiencia de 80% en comparación con la del correspondiente proceso mecánicamente reversible. Nota: la etapa inicial puede no ser más larga que el adiabático. Calcule W, Q, ∆H y ∆U para cada etapa del proceso y para todo el ciclo. Considere Cp = (7/2)R y Cv= (5/2)R. 4. Un mol de gas ideal, inicialmente a 30°C y 1 bar, cambia a 130°C y 10 bar mediante tres distintos procesos mecánicamente reversibles: a) El gas se calienta primero a volumen constante hasta que su temperatura es de 130°C; a continuación, se comprime isotérmicamente hasta que su presión es de 10 bar. b) El gas se calienta primero a presión constante hasta que su temperatura es de 130°C; a continuación, se comprime isotérmicamente hasta 10 bar. c) El gas se comprime primero isotérmicamente hasta 10 bar; a continuación, se calienta a presión constante hasta 130°C.
Calcule W, Q, ∆H y ∆U en cada caso. Considere Cp = (7/2)R Y Cv = (5/2)R. En otro caso, considere Cp = (5/2)R y Cv = (3/2)R. 5. Un gas ideal, inicialmente a 25°C y 1 bar, se somete al siguiente proceso cíclico en un sistema cerrado: a) En procesos mecánicamente reversibles primero se comprime de manera adiabática a 5 bar, a continuación se enfría a presión constante de 5 bar hasta 25°C y por último se expande isotérmicamente a su presión original. b) El ciclo es irreversible y cada etapa tiene una eficiencia de 80% en comparación con el correspondiente proceso que es mecánicamente reversible. El ciclo completo consiste en una etapa de compresión adiabática, una etapa de enfriamiento isobárico y una expansión isotérmica. Calcule W, Q, ∆H y ∆U para cada etapa del proceso y para el ciclo. Considere que Cp = (7/2)R y Cv= (5/2)R.
Use the property information from the tables in the appendix to do the problems in this section. 1.
Pure water is held in a container. The temperature of the water is T1 = 520°C and the pressure is P1 =800 kPa. a) Sketch a T-v diagram and locate the state of the water on the diagram. Your sketch should be qualitatively correct, but it does not have to be to scale. b) Determine the specific volume of the water (m3/kg) and the density of the water (kg/m3). c) c) If the mass of water in the container is m=7.2 kg then what is the volume of the container? 2. a) b) c)
Determine the boiling temperature (°F) of water: at normal atmospheric pressure (Patm = 14.7 psi, which corresponds to sea-level), in Denver, where the pressure is P=24.58 inch Hg, and at the summit of Mount Everest, where the pressure is P=30 kPa.
A rigid tank with volume V=8000cm3 is filled with water with quality x1 =0.05 and temperature T1 =140°C. a) What is the specific volume (m3/kg) and the pressure (kPa) of the water. b) What is the total mass of water in the tank (kg)? What is the mass of liquid (kg) and the mass of vapor (kg) in the tank? c) What are the volumes of liquid and vapor in the tank (m3)? 3.
The water in the tank is heated to T2 =200°C. The tank is rigid (i.e., its volume doesn’t change) and leak tight. a) What is the pressure (kPa) and quality of the water in the tank at state 2? b) What is the mass of liquid (kg) in the tank at state 2?
4.
Figure 2.A-7 illustrates a pressure cooker.
The pressure cooker has an internal volume of V = 2.5 liter and contains m = 0.25 kg of pure water. The pressure relief valve consists of a spring loaded disk that is positioned over a hole in the top of the cooker. The disk has diameter Drd =0.88 inch and mass mrd =0.05 kg. The spring is compressed x=0.25 inch and has a spring constant of K=200 lbf/inch. The atmospheric pressure is Patm = 101.3 kPa. a) Determine the internal pressure in the pressure cooker that is required to open the pressure relief valve. b) The pressure relief valve allows vapor to escape in order to maintain the pressure that you calculated in (a). What is the temperature of the water remaining in the pressure cooker after the relief valve opens, assuming that some liquid remains? You have been asked to examine the possibility that the relief valve fails to open. In this case, no water can escape and therefore the temperature and pressure of the contents of the pressure cooker will continue to rise until the device fails. Assume that the pressure cooker is rigid (i.e., the volume does not change). a) c) What is the temperature in the pressure cooker when the pressure reaches P2 =24 MPa?
5. a)
b) c) d) e)
Refrigerant R134a is contained in a small recharge tank with volume V=2liter. Initially, the tank is filled with m1 =2.0 kg of R134a at T1 =15°C. Sketch a T-v diagram and locate the state of the R134a on your sketch. The sketch can be approximate, but it should clearly show the intersection of the two property lines that define the state. What is the pressure in the tank at state 1? What is the quality of the R134a in the tank? What is the mass of liquid R134a in the tank? What is the mass of vapor R134a in the tank? What is the volume of liquid R134a in the tank? What is the volume of vapor R134a in the tank?
The recharge tank is equipped with a pressure relief valve that allows refrigerant to escape when the pressure within the tank reaches P2 = 9.0 bar. The tank is accidentally transported in an un-conditioned truck where the temperature is Ttruck = 40°C. As a result, the temperature of the R134a in the tank slowly starts to rise causing the pressure to rise. State 2 is defined to be the state of the refrigerant where the pressure relief valve just opens. f) On your T-v diagram from (a) overlay state 2. Indicate on your diagram what two properties define state 2. g) What is the temperature of the R134a at the time that it reaches state 2? The refrigerant continues to increase in temperature until finally it reaches T3 =Ttruck. During this time, the pressure relief valve vents refrigerant in order to maintain the pressure in the tank always at P3 =P2 =9.0 bar. h) On your T-v diagram from (a) overlay state 3. Indicate on your diagram what two properties define state 3. i) What mass of refrigerant is vented from the tank during the time that it goes from state 2 to state 3? Eventually, the tank is unloaded from the truck and cooled to T4 =20°C. j) On your T-v diagram from (a) overlay state 4. It should be clear on your diagram what two properties define state 4. k) k) What is the volume of liquid refrigerant in the tank at state 4? 6.
A rigid tank with volume Vtank = 5.0 gallon is maintained at T1 = 300°C and initially contains m1 = 0.08 kg of water, as shown in Figure 2.A-11. At some time, a valve is opened allowing Vin = 0.5 gallons of water at Tin = 20°C and Pin=20.0MPa to enter the tank. The valve is shut and eventually all of the water in the tank comes to T2 =300°C.
a) Locate state 1and state in, the states of the water initially in the tank and the water added to the tank, respectively, on a sketch of a T-v diagram. b) What is the initial pressure in the tank (MPa)? c) What is the mass of water added to the tank (kg)? d) Locate state 2, the final state of the water in the tank, on the T-v diagram from (a). e) What is the final pressure in the tank (MPa)?
7.
Figure 2.A13(a) illustrates a pressure cooker with the pressure relief valve removed. The pressure cooker has an internal volume of V=2liter. Because the
relief valve is removed, the contents are initially at atmospheric pressure, Patm = 100 kPa. The pressure cooker contains water in a two-phase state. The bottom
5% of the volume of the vessel is filled with liquid water while the remainder of the vessel contains water vapor. a) Determine the initial temperature of the water in the cooker. b) Determine the quality of the water initially in the cooker. c) Locate the initial state of the water (state 1) on a sketch of a T-v diagram. The pressure relief valve is installed on the pressure cooker, as shown in Figure 2.A-13(b).
The pressure relief valve consists of a spring loaded disk that is positioned over a hole in the top of the cooker. The disk has diameter Drv =0.5 inch and mass mrv =0.1kg. The spring is compressed cs=0.1694 inch and has a spring constant of K=150 lbf/inch. d) Determine the internal pressure that is required to open the pressure relief valve. e) Heat is added to the pressure cooker with the relief valve in place. The pressure relief valve opens when the pressure reaches the value calculated in (d) and allows vapor to escape in order to maintain the pressure at this value. What are the temperature and quality of the water in the pressure cooker at the instant that the pressure relief valve opens at state2? Add state 2 to your T-v sketch from (c). f) What is the fraction of the volume of the pressure cooker that is filled with liquid at the instant that the pressure relief valve opens (at state 2).
g) Heat continues to be added to the pressure cooker until all of the liquid disappears at state 3. What is the mass of water that has passed through the pressure relief valve at this instant? Locate state 3 on your sketch from (c). h) Heat continues to be added to the pressure cooker until the temperature reaches T4 =400°C. Locate state 4 on your sketch from (c). What is the mass of water that passes through the pressure relief valve between the time that the pressure cooker is at state 3 and at state 4? i) The pressure cooker is removed from the source of heat and begins to cool. The pressure begins to drop and therefore the pressure relief valve closes. At what temperature does liquid water begin to form in the vessel? Locate this state 5 in your sketch from part (c). j) The pressure cooker is cooled until the temperature reaches T6 = 20°C. Determine the pressure and quality of the water at state 6 and locate state 6 in your sketch from part (c). 8. Refrigerant R134a is contained at its critical point in a piston-cylinder device. a) What is the temperature and the pressure of the R134a? b) If the fluid is cooled slightly at constant volume, how many phases will be present in the cylinder? Provide a sketch to show your result. c) Repeat part (b) assuming that pressure rather than the volume remains constant as the fluid is cooled. 9.
At state 1, superheated water vapor is contained in a sealed glass vial at T1 = 200°C. You would like to know the pressure at this state, but have no means of measuring it directly. However, when the vial is slowly cooled to state 2, T2 = 120°C, you notice that droplets of liquid begin to form on the glass walls. a) Use this information to determine the pressure at state 1. b) Sketch the process on a temperature-volume diagram.
A. Heat and Work 1.
Cryogenic liquids (e.g., liquid helium or liquid neon) are sometimes used to keep instruments at cryogenic (i.e., very cold) temperatures for space science missions. As the liquid boils off due to heat transfer it is vented to space. When all of the cryogenic liquid is gone, the temperature of the instrument increases and the mission is over. Flight operations engineers need to be able to check the amount of liquid that is left in the tank from the ground while the spacecraft is in orbit. In microgravity, the mixture of liquid and vapor in the tank is not stratified by gravity in the same way that it is on earth. Therefore, traditional liquid level measurement techniques do not work. One alternative technique that has been used by NASA is referred to as mass gauging. In order to accomplish mass gauging, a heater is activated for a short time and the temperature rise of the neon and the tank material is measured. The magnitude of the temperature rise can be used to calculate the mass of liquid that is left in the tank. Consider the spherical, aluminum cryogenic tank shown in Figure 3.A-1.
The tank has an inner radius of Rin = 10 inch and a wall thickness of th = 0.125 inch. The density and specific heat capacity of aluminum (at the cryogenic temperatures associated with this problem) are ρAl = 0.098 lbm/in3 and cAl = 24 J/kg-K, respectively. The tank contains a mixture of saturated liquid and saturated vapor neon at T1 =27 K. The mass of neon in the tank is m=40 kg. For these calculations you may assume that no mass leaves the tank during the short time that it takes to complete the mass gauging process. A heater in the tank is activated and provides Q=1 kJ of heat in order to accomplish the mass gauging. You may assume that the temperature of the neon is uniform and that the tank and the neon are at the same temperature. There is no work done on or by the tank or its contents during this process. a) What is the mass of the aluminum? b) What is the quality of the neon initially in the tank? What is the pressure of the neon initially in the tank? c) What is the temperature of the neon and aluminum at the conclusion of the mass gauging process (T2)? What is the increase in temperature detected by the operators? d) What are the quality and pressure of the neon in the tank at the conclusion of the mass gauging process? e) Generate a calibration curve that an operator could use for the mass gauging process. That is, generate a plot showing the mass of neon in the tank (m) as a function of the temperature rise (∆T=T2 – T1) for neon mass ranging from 1 kg to 80 kg. f) If your temperature sensors are capable of resolving a temperature difference of approximately 1 mK (that is, the uncertainty in your measurement of the temperature rise is δ∆T = 0.001 K) then estimate the resolution of the mass gauging process (that is, how well can you measure the mass of neon in the tank?). Use your engineering judgment to answer this question; you may want to refer to the calibration curve generated in part (e). 2.
Figure 3.A-2 is the pressure-volume diagram for a thermodynamic cycle that is executed by m=18 lbm of nitrogen gas. Assume that nitrogen behaves according to the ideal gas law. a) Determine the temperature at each of the states shown in Figure 3.A-2.
b) Determine the work done by the nitrogen gas for each of the processes shown in Figure 3.A-2 (i.e., process 1 to 2, 2 to 3, 3 to 4, and 4 to 5). What is the net work done by the nitrogen during cycle?
3.
Figure 3.A-4 illustrates a small motor that is being used to lift an m=10 kg mass.
The motor is operating at steady state. The surface area of the motor is As =0.05 m2 and the motor is surrounded by air at T∞ = 70◦F. The surface of the motor has an emissivity of ε =0.75.The heat transfer coefficient between the motor and the air is approximately h=1.5 Btu/hr-ft2-R. The motor radiates to surroundings that are also at T∞. The motor is provided electrical input power with a voltage of Ein =24 V and a current iin =2.5 amp. The motor shaft is rotating at N=350 rev/min and the torque on the shaft is τ =12.8 inch-lbf. a) Carry out an energy balance on a system that consists of the motor. What is the rate at which the energy in the system is changing? What is the rate of heat transfer from the motor? b) What is the surface temperature of the motor? c) Assume that all of the mechanical power carried by the shaft is being used to increase the potential energy of the mass. What is the velocity at which the mass is rising?
4. Water is contained in a piston-cylinder device, as shown in Figure 3.A-6. You may neglect friction between the piston and the wall and assume that the piston does not leak.
The mass of the water ism=0.05 kg and the area of the piston face is Ac =0.2 m2. Initially, the water is at T1 =125°C with a quality of x1 =0.90. The water is heated and the piston begins to rise; as this occurs, the spring is compressed and so the pressure in the cylinder begins to rise. The pressure rise is proportional to the amount that the spring is compressed, according to:
where K=5000 N/cm is the spring constant and z2 and z1 are the final and initial positions of the piston, respectively, as shown in Figure 3.A-2. The heating stops when the water temperature reaches T2 =250◦C. a) What is the initial pressure in the piston? What is the initial position of the piston, z1? b) What is the pressure in the piston at the end of the heating process? c) Using EES, prepare a T-v diagram for water using the Property Plot option from the Plots menu and overlay states 1 and 2 on this plot.