Eng Thermodynamics I - Examples 1

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1 Eng Thermodynamics I - Examples 1 1 pdv Work 1. Air is contained in a vertical frictionless piston-cylinder. The mass of the piston is 500 kg. The area of the piston is m 2. The air initially occupies m 3. Heat is added to the air until its volume doubles. Atmospheric pressure is 100 kpa. (a) Draw a p-v diagram for the process. (b) How much work is done by the air? 2. Consider problem 1, but with a linear spring attached to the piston-cylinder such that the spring is compressed as the piston rises. The spring constant is k s = 20 kn/m. Assume the spring is initially uncompressed. (a) Draw a p-v diagram for the process. (b) How much work is done by the air? 3. Air is contained in a vertical frictionless piston-cylinder that has two sets of stops limiting the maximum and minimum volume of the air. The mass of the piston is 500 kg. The area of the piston is m 2. The maximum and minimum volumes of the air are limited to 0.01 m 3 and m 3, respectively. The atmospheric pressure outside the cylinder is 100 kpa. The initial absolute pressure of the air is 80 kpa. Heat is added to the air until its pressure is 2 MPa. (a) Draw a p-v diagram for the process. (b) How much work is done by the air? 4. Consider problem 3, but with a linear spring attached to the piston-cylinder such that the spring is compressed as the piston rises. The spring constant is k s = 20 kn/m. (a) Draw a p-v diagram for the process. (b) How much work is done by the air? 2 Closed System 1st Law 1. A gas undergoes a thermodynamic cycle consisting of three processes: 1 2: Compression with pv = const, p 1 = 1 bar, V 1 = 1.6 m 3, V 2 = 0.2 m 3, U 2 U 1 = 0 2 3: Constant pressure, V 3 = V 1 3 1: Constant volume, U 1 U 3 = 3549 kj (a) Draw a p-v diagram for the cycle. (b) Determine the heat transfer and work for each process. (c) Determine the thermal efficiency, η th, of the cycle.

2 Eng Thermodynamics I - Examples 2 3 Properties 1. A vertical, frictionless piston-cylinder contains water. The cylinder has a set of stops which limits the maximum volume to 0.05 m 3. The atmospheric pressure is 100 kpa. The mass of the piston is 500 kg and the area of the piston is 0.05 m 2. A linear spring (k s = kn/m) is attached to piston such that the spring is compressed as the piston rises. Initially the spring is uncompressed, and the water temperature is 20 C and it displaces m 3. Heat is added to the water and the piston rises until it reaches the stops. Heat addition continues until the pressure in the water is 18 MPa. State all assumptions when answering the following questions. (a) Draw a neat p-v diagram for the process. (b) How much work is done by the water during the expansion process? (c) How much heat is added to the water from the initial state until the piston reaches the stops? (d) What is the total amount of heat transferred to the water, i.e. from the initial to final states? 2. A vertical, frictionless piston-cylinder has two sets of stops which limit the maximum and minimum volume to 0.1 m 3 and m 3, respectively. The mass and area of the piston are 100 kg and m 2, respectively. The atmospheric pressure is 130 kpa. Initially, the piston rests on the lower set of stops, and the temperature and absolute pressure of the water are 20 C and 100 kpa, respectively. Heat is added to the water and the piston rises to the upper set of stops. Heat addition continues until the temperature of the water is 360 C. State all assumptions when answering the following questions. (a) Draw a neat p-v diagram for the process. (b) How much work is done by the water during the process? (c) How much heat is added to the water from the beginning of the heat transfer process until the piston reaches the upper set of stops? (d) What is the total amount of heat that must be added to the water during the process? 3. Two kilograms of air with initial absolute pressure and temperature of 100 kpa and 300K, respectively, are compressed in a polytropic process (n = k = c p /c v ) until the volume is half the initial value. The air then expands in a constant pressure process until its final volume is equivalent to the initial volume. Using the air properties given below, answer the following questions. State all assumptions. (a) Draw a (neat) p-v diagram for the process. (b) What are the final pressure and temperature of the air? (c) How much work is done by the air? (d) What is the total amount of heat transfer to the air? Air properties: c p = 1005 J/k/gK, c v = 718 J/kg/K, R = 287 J/kg/K

3 Eng Thermodynamics I - Examples kg of air initially at an absolute pressure of 400 kpa and temperature of 300 K expands in a polytropic process (n = 1.2) to a volume of 0.2 m 3. It is then compressed back to its original volume in an isothermal process. Use the properties given below, and state all assumptions when answering the following questions. (a) Draw a (neat) p-v diagram for the process. (b) What are the final pressure and temperature of the air? (c) What is the total amount of work done by the air? (d) What is the total amount of heat added to the air? Air properties: c p = kj/kg K, c v = kj/kg K, R = kj/kg K 4 Steady-State, Open System, 1st Law Analysis 1. Air enters a water-cooled air compressor at 100 kpa, 300K at a rate of 10 kg/s. The compressor is designed to deliver two air streams: stream 2 (ṁ 2 = 8 kg/s) exits the compressor at 500 kpa; and stream 3 (ṁ 3 = 8 kg/s) exits the compressor at 1 MPa. The power input to the compressor is 2 MW. The liquid cooling water enters the compressor at 10 C with mass flow rate 15 kg/s. The process in the air is assumed to be polytropic (n = 1.2). Determine the exit temperature of the water. Air properties: c p = kj/kg K, c v = kj/kg K, R = kj/kg K Water properties: c p,w = kj/kg K 2. A model of a gas turbine consists of a compressor, heat exchanger and turbine, as shown below. Air enters the compressor at state 1 (ṁ 1 = 2 kg/s, p 1 = 100 kpa, T 1 = 300K) and is compressed to state 2 (p 2 = 1 MPa). Energy is then added to the air in the heat exchanger in a constant pressure process. The temperature of the air entering the turbine is limited to a maximum value of T 3 = 1800K. The air exits the turbine at state 4 (p 4 = 100 kpa). Assume the processes in the turbine and compressor are adiabatic and polytropic with n = k = c p /c v. Use the air properties given below. (a) What is the power input to the compressor, Ẇ C? (b) What is the net power output of the gas turbine (i.e. the whole device), Ẇ net =? ẆT ẆC (c) What is the heat transfer rate to the air in the heat exchanger, Q in? Air properties: c p = kj/kg K, c v = kj/kg K, R = kj/kg K

4 Eng Thermodynamics I - Examples 4 3. A steam turbine and reheater are shown below. Steam enters the turbine at state 1 (ṁ 1 = 40 kg/s, p 1 = 2 MPa, T 1 = 400 C), and exits at state 2 (p 2 = 1 MPa, T 2 = 200 C) and state 3 (p 3 = 500 kpa, x 3 = 0.94). The water is then reheated at constant pressure in the reheater by combustion gases and exits at state 4 (T 4 = 220 C). The desired power output of the turbine is ẆT = MW. The combustion gases are modelled as air, enter the reheater at state 5 and exit at state 6. Use the properties for air given below, and tabular data for the water. (a) What is the mass flow rate of the steam entering the reheater, ṁ 3? (b) If the temperature drop in the air is limited to 200 C, what is the required mass flow rate of the air, ṁ 5? Air properties: c p = kj/kg K, c v = kj/kg K, R = kj/kg K

5 Eng Thermodynamics I - Examples 5 5 Cycles 1. A simple steam power plant consists of a boiler, a two-stage turbine, a condenser, a pump, a closed feedwater heater, and a throttling valve as shown below. Steam enters the highpressure (H.P.) turbine at state 1. Some of the steam is bled from the H.P. turbine at state 2 to be used in the closed feedwater heater. This stream exits the closed feedwater heater at state 8 and is throttled to the condenser pressure (state 9). The remaining steam exits the H.P. turbine at state 3 and enters the low-pressure (L.P.) turbine. The steam exits the L.P. turbine at state 4 and enters the condenser where it mixes with the water from the feedwater heater (state 9), and exits the condenser at state 5. The water is pressurized to state 6 in the pump, and then it enters the closed feedwater heater. The preheated water then exits the feedwater heater at state 7 and enters the boiler. Determine the thermal efficiency of the power plant. m 1 = 18 kg/s p 3 = 500 kpa p 6 = 5 MPa m 2 = 2 kg/s T 3 = 210 C p 7 = 5 MPa p 1 = 5 MPa p 4 = 10 kpa T 7 = 80 C T 1 = 500 C x 4 = 0.92 p 8 = 1 MPa p 2 = 1 MPa p 5 = 10 kpa x 8 = 0 T 2 = 280 C T 5 = 20 C p 9 = 10 kpa

6 Eng Thermodynamics I - Examples 6 2. A simple refrigeration plant consists of an evaporator, condenser, compressor three throttling valves, a heat exchanger and a direct contact heat exchanger as shown below. The working fluid in the system is R-134a. The refrigerant exits the compressor at state 3 and enters the condenser. At the exit of the condenser, state 4, some of the refrigerant is bled off, throttled to state 7, enters the heat exchanger where it is heated to exit state 8, and is then throttled to state 9. The remaining refrigerant enters the heat exchanger, is cooled to exit state 5, is throttled to state 6 and then flows through the evaporator, emerging at state 1. The two fluid streams, 1 and 9, mix in the well-insulated direct contact heat exchanger and exit at state 2. The refrigerant at state 2 is compressed to state 3 in the compressor. Given the information below, determine the coefficient of performance for the refrigeration cycle. m 1 = 4 kg/s p 2 = 140 kpa p 4 = 1 MPa T 5 = 20 C p 8 = 320 kpa p 1 = 140 kpa p 3 = 1 MPa x 4 = 0 p 6 = 140 kpa x 8 = 1 x 1 = 1 T 3 = 55 C p 5 = 1 MPa p 7 = 320 kpa p 9 = 140 kpa

7 Eng Thermodynamics I - Examples 7 6 Transient Analysis 1. A vertical well-insulated piston-cylinder assembly has two sets of stops that limit the minimum and maximum volumes to m 3 and m 3, respectively. The atmospheric pressure is 100 kpa. The mass of the piston is 500 kg and the area of the piston is 0.05 m 2. A linear spring, k s = 40 kn/m, is attached to the cylinder such that the spring is compressed as the piston rises, however, the piston does not contact the spring until the volume of air is 0.02 m 3. Initially, the mass of air in the cylinder is 5 g, and it has absolute pressure 80 kpa. The cylinder is connected by a valve to a constant air supply line at 1 MPa (abs) and 500 K. The valve is opened and air is allowed to flow slowly into the cylinder until the pressure of the air inside the cylinder reaches 600 kpa (abs). Use the air properties given below. State all assumptions when answering the following questions. (a) Sketch a (neat) p -V diagram for the process, from initial to final states. (b) Determine the final mass of air in the cylinder. Air properties: c p = kj/kg K, c v = kj/kg K, R = kj/kg K 2. A 0.01 m 3 pressure cooker has an operating pressure of 200 kpa. When the cooker first begins to release steam (saturated vapour) it contains 0.95 kg of water (liquid and vapour). If the cooker is on a 200 W stove element, how much time will elapse before the cooker boils dry, i.e. all of the liquid water is evaporated? State all assumptions. 7 Second Law 1. Consider a power cycle (heat engine) operating between reservoirs at T H = 1000 K and T C = 300 K. Determine if the cycle is irreversible, reversible or impossible when: (a) Q H = 600 kj and W = 400 kj; (b) Q H = 600 kj and Q C = 300 kj; (c) W = 500 kj and Q C = 200 kj. 2. A heat pump salesman tries to sell you an air source heat pump system that will supply your house with 10 kw of heat input at a cost of 500 W power input to the compressor when your house is maintained at 20 C and the outside air is at 0 C. Should you sign on the dotted line? 3. Determine if a tray of ice cubes can remain frozen when placed in a freezer having a coefficient of performance of β = 9 operating in a room where the temperature is 32 C. 4. Air enters a turbine at 6 bar, 1100 K and expands to an exit pressure of 1 bar. Determine the maximum possible work from the turbine. 5. An inventor tells you he has a device that can intake saturated water vapour at state 1 (ṁ 1 = 1 kg/s, p 1 = 100 kpa) and air at state 3 (ṁ 3 = 5 kg/s, p 3 = 100 kpa, T 3 = 300 K), and output saturated liquid water at state 2 (p 2 = 100 kpa) and air at state 4 (p 4 = 400 kpa, T 4 = 400 K). Typical of inventors, his device is inside a black box and it is top-secret. The only other information available is that the air and water streams do not

8 Eng Thermodynamics I - Examples 8 mix, the device has no power requirements, and the temperature of the outer casing of the black box does not exceed 50 C. Using tabular data for the water, and the air properties given below, determine if the device is possible. Justify your answer. Air properties: c p = kj/kg K, c v = kj/kg K, R = kj/kg K 6. Let s modify question 4: (a) Air enters a turbine at 6 bar, 1100 K and expands to an exit pressure of 1 bar. If the turbine has an isentropic efficiency of η t = 0.9 what is the power output of the turbine? (b) Air enters a turbine at 6 bar, 1100 K and expands to an exit pressure of 1 bar. If the power output of the turbine is 400 kw, what is the isentropic efficiency of the turbine? 7. The specifications of a steam turbine with one inlet and two exits suggest that it can generate 23 MW. The turbine is well insulated and requires inlet flow conditions ṁ 1 = 20 kg/s, p 1 = 5 MPa, T 1 = 500 C. The turbine allows 15% of the inlet mass flow rate to be bled from the turbine at p 2 = 2 MPa and T 2 = 350 C, to be used in another process. The remainder of the steam exits the turbine at a pressure of 10 kpa. Are these specifications possible? Why? 8. An air compressor is driven by a steam turbine as shown below. The compressor compresses 2 kg/s of air from state 1 (100 kpa (abs), 300 K) to state 2 (400 kpa). The steam enters the turbine as saturated vapour at state 3 (300 kpa) and exits at state 4 (100 kpa). If the isentropic efficiencies of the compressor and turbine are both 0.9, determine the required mass flow rate of the steam. Use the air properties given below, and the attached steam tables. Air properties: c p = kj/kg K, c v = kj/kg K, R = kj/kg K 9. An air-driven turbine has one inlet and two exits. Air enters the turbine at state 1 (ṁ 1 = 10 kg/s, p 1 = 1 MPa, T 1 = 800 K), and exits at state 2 (ṁ 2 = 2 kg/s, p 2 = 500 kpa, T 2 = 670 K) and state 3 (p 3 = 100 kps, T 3 = 440 K). It is proposed that the turbine have a power output of 3 MW. The turbine is not perfectly insulated but the turbine casing temperature is not expected to exceed 350 K. Are these specifications possible? Air properties: c p = kj/kg K, c v = kj/kg K, R = kj/kg K