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Chapter 4. Entropy > Homework Problems

4.18. Homework Problems

4.1Extending Example 4.2 on page 141 from solids to gases is straightforward if you recall the development of Eqn. 1.13 on page 19. Consider N2 for example. Being diatomic, we should expect that Uig = 2(3NAkT/2) = 6RT/2 in the limit of classical vibrations. Vibrational energy means that heat can be absorbed in the vibration of a bond. Since N2 has only one bond, it can only absorb energy in one way, removing one degree of freedom. We show in Eqn. 6.49 on page 240 that the change in energy due to vibration is ΔUvib = ϵ/[exp(–β ϵ) – 1]. For now, without concern for the proof, assume ΔUvib as given. Adapting Example 4.2 for N2 then gives: Uig = 5RT/2 + ϵ/[exp(–β ϵ) – 1].
  1. Use the NIST WebBook to plot data for CV of N2 at 0.1 MPa and T = [150, 2000 K].

  2. Derive an expression for CV based on above dissussion. Evaluate your expression at 1000K assuming ϵ/k = 1000K.

  3. Regress an optimal value for ϵ/k of N2 and plot a comparison of the calculated results to experimental data. Show the calculated results as a curve with no points.

4.2An ideal gas, with temperature-independent CP = (7/2)R, at 15°C and having an initial volume of 60 m3, is heated at constant pressure (P = 0.1013 MPa) to 30°C by transfer of heat from a reservoir at 50°C. Calculate ΔSgas, ΔSheat reservoir, ΔSuniverse. What is the irreversible feature of this process?
4.3Steam undergoes a state change from 450°C and 3.5 MPa to 150°C and 0.3 MPa. Determine ΔH and ΔS using the following:
  1. Steam table data

  2. Ideal gas assumptions (be sure to use the ideal gas heat capacity for water)

4.4The following problems involve one mole of an ideal monatomic gas, CP = 5R/2, in a variable volume piston/cylinder with a stirring paddle, an electric heater, and a cooling coil through which refrigerant can flow (see figure). The piston is perfectly insulated. The piston contains 1 gmole of gas. Unless specified, the initial conditions are: Ti = 25°C, Pi = 5 bar.

  1. Status: Heater on; cooler off; paddle off; piston fixed. Five kJ are added by the heater. Find ΔU, ΔS, ΔP, and ΔT.

  2. Status: Heater off; cooler off; paddle off; piston moveable. What reversible volume change will give the same temperature rise as in part (a)? Also find ΔU, ΔS, and ΔP.

  3. Status: Heater off; cooler off; paddle on; piston fixed. What shaft work will give the same ΔU, ΔS as part (a)?

  4. Status: Heater off; cooler off; paddle on; piston fixed. The stirring motor is consuming 55 watts and is 70% efficient. What rate is the temperature changing? At what initial rates are U and S changing?

  5. Status: Heater unknown; cooler unknown; paddle off; piston free. We wish to perform a reversible isothermal compression until the volume is half of the initial volume. If the volume is decreasing at 2.0 cm3/s, at what rate should we heat or cool? Express your answer in terms of the instantaneous volume. What is the total heat transfer necessary?

4.5When a compressed gas storage tank fails, the resultant explosion occurs so rapidly that the gas cloud can be considered adiabatic and assumed to not mix appreciably with the surrounding atmosphere. Consider the failure of a 2.5-m3 air storage tank initially at 15 bar. Atmospheric pressure is 1 bar, CP = 7R/2. Provide an estimate by assuming reversibility.
  1. Calculate the work done on the atmosphere. Does the reversibility approximation overestimate or under-estimate the actual work?

  2. A detonation of 1 kg of TNT releases about 4.5 MJ of work. Calculate the equivalent mass of TNT that performs the same work as in part (a).

4.6Work problem 4.5 but consider a steam boiler that fails. The boiler is 250 L in size, operating at 4 MPa, and half full of liquid.
4.7An isolated chamber with rigid walls is divided into two equal compartments, one containing gas at 600 K and 1 MPa and the other evacuated. The partition between the two compartments ruptures. Compute the final T, P, and ΔS for the following:
  1. An ideal gas with CP/R = 7/2

  2. Steam.

4.8An isolated chamber is divided into two equal compartments, one containing gas and the other evacuated. The partition between the two compartments ruptures. At the end of the process, the temperature and pressure are uniform throughout the chamber.
  1. If the filled compartment initially contains an ideal gas at 25 MPa and 650 K, what is the final temperature and pressure in the chamber? What is ΔS for the process? Assume a constant heat capacity of CP/R = 4.041.

  2. If the filled chamber initially contains steam at 25 MPa and 650 K, what is the final temperature and pressure in the chamber? What is ΔS for the process? (Use the steam tables.)

4.9Airplanes are launched from aircraft carriers by means of a steam catapult. The catapult is a well-insulated cylinder that contains steam, and is fitted with a frictionless piston. The piston is connected to the airplane by a cable. As the steam expands, the movement of the piston causes movement of the plane. A catapult design calls for 270 kg of steam at 15 MPa and 450°C to be expanded to 0.4 MPa. How much work can this catapult generate during a single stroke? Compare this to the energy required to accelerate a 30,000 kg aircraft from rest to 350 km per hour.
4.10We have considered heat and work to be path-dependent. However, if all heat transfer with surroundings is performed using a reversible heat transfer device (some type of reversible Carnot-type device), work can be performed by the heat transfer device during heat transfer to the surroundings. The net heat transferred to the surroundings and the net work done will be independent of the path. Demonstrate this by calculating the work and heat interactions for the system, the heat transfer device, and the sum for each of the following paths where the surroundings are at Tsurr = 273 K. The state change is from state 1, P1 = 0.1 MPa, T1 = 298 K and state 2, P2 = 0.5 MPa and T2 which will be found in part (a). CP = 7R/2.
  1. Consider a state change for an ideal gas in a piston/cylinder. Find T2 by an adiabatic reversible path. Find the heat and work such that no entropy is generated in the universe. This is path a. Sketch path a qualitatively on a P-V diagram.

  2. Now consider a path consisting of step b, an isothermal step at T1, and step c, an isobaric step at P2. Sketch and label the step on the same P-V diagram created in (a). To avoid generation of entropy in the universe, use heat engines/pumps to transfer heat during the steps. Calculate the WEC and WS as well as the heat transfer with the surroundings for each of the steps and overall. Compare to part (a) the total heat and work interactions with the surroundings.

  3. Now consider a path consisting of step d, an isobaric step at P1, and step e, an isothermal step at T2. Calculate the WEC and WS as well as the heat transfer with the surroundings for each of the steps and overall. Compare to part (a) using this pathway and provide the same documentation as in (b).

4.11Consider the wintertime heating of a house with a furnace compared to addition of Carnot heat engines/pumps. To compensate for heat losses to the surroundings, the house is maintained at a constant temperature Thouse by a constant rate of heat transfer, . The furnace operates at a constant temperature TF, and with direct heat transfer, the heat required from the furnace, is equal to .
  1. Instead of direct heat transfer, if we utilize the surroundings, at TS, as an additional heat source and include heat pump technology, may be reduced by generating work from a heat engine operating between TF and TS, then applying that work energy to a heat pump operating between TS and Thouse. Given that TF = 800 K, T = 293 K, TS = 265 K, and = 40 kJ/h, determine utilizing heat pump technology. No other sources of energy may be used.

  2. Another option is to run a heat engine between TF and Thouse and the heat pump between TS and Thouse. Compare this method with part (a).

4.12An ideal gas enters a valve at 500 K and 3 MPa at a steady-state rate of 3 mol/min. It is throttled to 0.5 MPa. What is the rate of entropy generation? Is the process irreversible?
4.13SO2 vapor enters a heat exchanger at 100°C and at a flowrate of 45 mole/h. If heat is transferred to the SO2 at a rate of 1,300 kJ/h, what is the rate of entropy transport in the gas at the outlet relative to the inlet in kJ/K/h given by ?
4.14An ideal gas stream (Stream A), CP = 5R/2, 50 mole/h, is heated by a steady-state heat exchanger from 20°C to 100°C by another stream (Stream B) of another ideal gas, CP = 7R/2, 45 mole/h, which enters at 180°C. Heat losses from the exchanger are negligible.
  1. For concurrent flow in the heat exchanger, calculate the molar entropy changes (SoutSin) for each stream, and for the heat exchanger.

  2. For countercurrent flow in the heat exchanger, calculate the molar entropy changes (SoutSin) for each stream, and for the heat exchanger. Comment on the comparison of results from parts (a) and (b).

4.15An inventor has applied for a patent on a device that is claimed to utilize 1 mole/min of air (assumed to be an ideal gas) with temperature independent CP = (7/2)R which enters at 500 K and 2 bar, and leaves at 350 K and 1 bar. The process is claimed to produce 2000 J/min of work and to require an undisclosed amount of heat transfer with a heat reservoir at 300 K. Should the inventor be issued a patent on this device?
4.16Two streams of air are mixed in a steady-state process shown below. Assume air is an ideal gas with a constant heat capacity CP = 7R/2.
  1. What is the temperature of the stream leaving the tank if the process is adiabatic?

  2. What is the rate of entropy generation within the tank if the process is adiabatic?

  3. If we duplicated the stream conditions (temperatures, pressures, and flowrates) with an internally reversible process, what is the maximum rate at which work could be obtained? If desirable, you are permitted to transfer heat to the surroundings at the surroundings’ temperature of 295 K.

4.17Air is flowing at steady state through a 5 cm diameter pipe at a flow rate of 0.35 mole/min at P = 5 bar and T = 500 K. It flows through a throttle valve and exits at 1 bar. Assume air is an ideal gas with CP = 29.1 J/mol-K. If the throttle valve was replaced by a reversible steady-state flow device to permit exactly the same state change for the air in this steady-state process, at what rate could work could be obtained? Heat transfer, if desired, can occur with the surroundings at 298 K, which may be considered a reservoir.
4.18A common problem in the design of chemical processes is the steady-state compression of gases from a low pressure P1 to a much higher pressure P2. We can gain some insight about optimal design of this process by considering adiabatic reversible compression of ideal gases with stage-wise intercooling. If the compression is to be done in two stages, first compressing the gas from P1 to P*, then cooling the gas at constant pressure down to the compressor inlet temperature T1, and then compressing the gas to P2, what should the value of the intermediate pressure be to accomplish the compression with minimum work?
4.19Steam flowing at steady state enters a turbine at 400°C and 7 MPa. The exit is at 0.275 MPa. The turbine is 85% efficient. What is the quality of the exiting stream? How much work is generated per kg of steam?
4.20An adiabatic steam turbine inlet is to be 4 MPa. The outlet of the turbine is to operate at 0.01 MPa, and provide saturated steam. The turbine has an efficiency of 85%. Determine the superheat which is required on the turbine inlet, and the work produced by the turbine.
4.21Steam is fed to an adiabatic turbine at 4 MPa and 500°C. It exits at 0.1 MPa.
  1. If the turbine is reversible, how much work is produced per kg of steam?

  2. If the turbine is 80% efficient, how much work is produced per kg of steam?

4.22Methane is compressed in a steady-state adiabatic compressor (87% efficient) to 0.4 MPa. What is the required work per mole of methane in kJ? If the flow is to be 17.5 kmol/h, how much work must be furnished by the compressor (in kW)? What is the rate of entropy generation (in kJ/K/h)? (a) the inlet is at 0.1013 MPa and –240°F; (b) the inlet is 0.1013 MPa and 200 K.
4.23Methane is to be compressed from 0.05 MPa and –120°F to 5 MPa in a two-stage compressor. In between adiabatic, reversible stages, a heat exchanger returns the temperature to – 120°F. The intermediate pressure is 1.5 MPa.
  1. What is the work required (kJ/kg) in the first compressor of methane?

  2. What is the temperature at the exit of the first compressor (°C)?

  3. What is the cooling requirement (kJ/kg) in the interstage cooler?

4.24A steady stream (1000 kg/hr) of air flows through a compressor, entering at (300 K, 0.1 MPa) and leaving at (425 K, 1 MPa). The compressor has a cooling jacket where water flows at 1500 kg/hr and undergoes a 20 K temperature rise. Assuming air is an ideal gas, calculate the work furnished by the compressor, and also determine the minimum work required for the same state change of air.
4.25Propane is to be compressed from 0.4 MPa and 360 K to 4 MPa using a two-stage compressor. An interstage cooler returns the temperature of the propane to 360 K before it enters the second compressor. The intermediate pressure is 1.2 MPa. Both adiabatic compressors have a compressor efficiency of 80%.
  1. What is the work required in the first compressor per kg of propane?

  2. What is the temperature at the exit of the first compressor?

  3. What is the cooling requirement in the interstage cooler per kg of propane?

  1. A steam turbine in a small electric power plant is designed to accept 5000 kg/h of steam at 60 bar and 500°C and exhaust the steam at 1 bar. Assuming that the turbine is adiabatic and reversible, compute the exit temperature of the steam and the power generated by the turbine.

  2. If the turbine in part (a) is adiabatic but only 80% efficient, what would be the exit temperature of the steam? At what rate would entropy be generated within the turbine?

  3. One simple way to reduce the power output of the turbine in part (a) (100% efficient) is by adjusting a throttling valve that reduces the turbine inlet steam pressure to 30 bar. Compute the steam temperature to the turbine, the rate of entropy generation, and the power output of the turbine for this case. Is this a thermodynamically efficient way of reducing the power output? Can you think of a better way?

4.27Steam is used in the following adiabatic turbine system to generate electricity; 15% of the mass flow from the first turbine is diverted for other use.

  1. How much work (in kJ/h) is generated by the first turbine which is 80% efficient?

  2. How much work (in kJ/h) is generated by the second turbine which is 80% efficient?

  3. Steam for the turbines is generated by a boiler. How much heat must be supplied to the boiler (not shown) which has 300 kg/h of flow? The stream entering the boiler is T = 170°C, P = 8 MPa. The stream exiting the boiler matches the inlet to the first turbine.

4.28Liquid nitrogen is useful for medical purposes and for research laboratories. Determine the minimum shaft work needed to liquefy nitrogen initially at 298 K and 0.1013 MPa and ending with saturated liquid at the normal boiling point, 77.4 K and 0.1013 MPa. The heat of vaporization at the normal boiling point is 5.577 kJ/mol, and the surroundings are at 298 K. The constant pressure heat capacity of gaseous nitrogen can be assumed to be independent of temperature at 7/2R for the purpose of this calculation.
  1. Consider nitrogen entering a flow device at 1 mol/min. Give shaft work in kW.

  2. Consider nitrogen in a piston/cylinder device. Give the work in kJ per mole liquefied.

  3. Compare the minimum shaft work for the two processes. Is one of the processes more advantageous than the other on a molar basis?

4.29Propane flows into a steady-state process at 0.2 MPa and 280 K. The final product is to be saturated liquid propane at 300 K. Liquid propane is to be produced at 1000 kg/h. The surroundings are at 295 K. Using a propane property chart, determine the rate of heat transfer and minimum work requirement if the process is to operate reversibly.
4.30Propane (1000 kg/hr) is to be liquefied following a two-stage compression. The inlet gas is to be at 300 K and 0.1 MPa. The outlet of the adiabatic compressor I is 0.65 MPa, and the propane enters the interstage cooler where it exits at 320 K, then adiabatic compressor II raises the propane pressure to 4.5 MPa. The final cooler lowers the temperature to 320 K before it is throttled adiabatically to 0.1 MPa. The adiabatic compressors have an efficiency of 80%.
  1. Determine the work required by each compressor.

  2. If the drive motors and linkages are 80% efficient (taken together), what size motors are required?

  3. What cooling is required in the interstage cooler and the final cooler?

  4. What percentage of propane is liquefied, and what is the final temperature of the propane liquid?

4.31A heat exchanger operates with the following streams: Water in at 20°C, 30 kg/hr; water out at 70°C; Organic in at 100°C, 41.8 kg/hr; organic out at 40°C.
  1. What is the maximum work that could be obtained if the flow rates and temperatures of the streams remain the same, but heat transfer is permitted with the surroundings at 298 K? (CP water = 4.184 kJ/(kgK), CP organic = 2.5 kJ/(kgK).)

  2. What is the maximum work that could be obtained by replacing the heat exchanger with a reversible heat transfer device, where the inlet flowrates and temperatures are to remain the same, the organic outlet temperature remains the same, and no heat transfer with the surroundings occurs?

4.32Presently, benzene vapors are condensed in a heat exchanger using cooling water. The benzene (100 kmol/h) enters at 0.1013 MPa and 120°C, and exits at 0.1013 MPa and 50°C. Cooling water enters at 10°C and exits at 40°C.
  1. What is the current demand for water (kg/h)?

  2. To what flowrate could the water demand be lowered by introducing a reversible heat transfer device that is adiabatic with the surroundings? The temperature rise of water is to remain the same. What work could be obtained from the new heat transfer device?

4.33A Hilsch vortex tube is an unusual device that takes an inlet gas stream and produces a hot stream and a cold stream without moving parts. A high-pressure inlet stream (A) enters towards one end of the tube. The cold gas exits at outlet B on the end of the tube near the inlet where the port is centered in the end cap. The hot stream exits at outlet C on the other end of the tube where the exit is a series of holes or openings around the outer edge of the end cap.

The tube works in the following way. The inlet stream A enters tangent to the edge of the tube, and swirls as it cools by expansion. Some of the cool fluid exits at port B. The remainder of the fluid has high kinetic energy produced by the volume change during expansion, and the swirling motion dissipates the kinetic energy back into internal energy, so the temperature rises before the gas exits at port C.

Inlet A is at 5 bar and 310 K and 3.2 mol/min. Outlet B is at 1 bar and 260 K. Outlet C is at 1 bar and 315 K. The tube is insulated and the fluid is air with CP = 7R/2.

  1. Determine the flowrates of streams B and C.

  2. Determine for the Hilsch tube.

  3. Suppose a reversible heat engine is connected between the outlet streams B and C which is run to produce the maximum work possible. The proposed heat engine may only exchange heat between the streams and not with the surroundings as shown. The final temperature of streams B and C will be TB′ as they exit the apparatus. What is TB′?

  4. What work output is possible in W? What is for the entire system including the tube plus the heat engine?

  5. Suppose that instead of using the heat engine, streams B and C were mixed directly with one another to form a single outlet stream. What would this temperature be, and how does it compare with TB′ from part (c)? Calculate and compare it with from part (c). What do you conclude from the comparison?

4.34Methane gas is contained in a 0.65-m3 gas cylinder at 6.9 MPa and 300 K. The cylinder is vented rapidly until the pressure falls to 0.5 MPa. The venting occurs rapidly enough that heat transfer between the cylinder walls and the gas can be neglected, as well as between the cylinder and the surroundings. What is the final temperature and the final number of moles of gas in the tank immediately after depressurization? Assume the expansion within the tank is reversible, and the following:
  1. Methane is considered to be an ideal gas with CP/R = 4.298

  2. Methane is considered to be a real gas with properties given by a property chart.

4.35A thermodynamically interesting problem is to analyze the fundamentals behind the product called “fix-a-flat.” In reality, this product is a 500 mL can that contains a volatile compound under pressure, such that most of it is liquid. Nevertheless, we can make an initial approximation of this process by treating the contents of the can as an ideal gas. If the initial temperature of both the compressed air and the air in the tire is 300 K, estimate the initial pressure in the compressed air can necessary to reinflate one tire from 1 bar to 3 bar. Also, estimate the final air temperature in the tire and in the can. For the purposes of this calculation you may assume: air is an ideal gas with CP/R = 7/2, the tire does not change its size or shape during the inflation process, and the inner tube of the tire has a volume of 40,000 cm3. We will reconsider this problem with liquid contents, after discussing phase equilibrium in a pure fluid.
4.36Wouldn’t it be great if a turbine could be put in place of the throttle in problem 4.35? Then you could light a small bulb during the inflation to see what you were doing at night. How much energy (J) could possibly be generated by such a turbine if the other conditions were the same as in problem 4.35?
4.37A 1 m3 tank is to be filled using N2 at 300 K and 20 MPa. Instead of throttling the N2 into the tank, a reversible turbine is put in line to get some work out of the pressure drop. If the pressure in the tank is initially zero, and the final pressure is 20 MPa, what will be the final temperature in the tank? How much work will be accomplished over the course of the entire process? (Hint: Consider the entropy balance carefully.)
4.38Two well-insulated tanks are attached as shown in the figure below. The tank volumes are given in the figure. There is a mass-flow controller between the two tanks. Initially, the flow controller is closed. At t = 0, the mass flow controller is opened to a flow of 0.1 mol/s. After a time of 500 seconds, what are the temperatures of the two tanks? Neglect the heat capacity of the tanks and piping. No heat transfer occurs between the two tanks. (After 500 seconds, the pressure in the left tank is still higher than the pressure in the right tank.) The working fluid is nitrogen and the ideal gas law may be assumed. The ideal gas heat capacity CP = 7/2·R may be assumed to be independent of T.

4.39Two storage tanks (0.1 m3 each) contain air at 2 bar. They are connected across a small reversible compressor. The tanks, connecting lines, and compressor are immersed in a constant temperature bath at 280 K. The compressor will take suction from one tank, compress the gas, and discharge it to the other tank. The gas is at 280 K at all times. Assume that air is an ideal gas with CP = 29.3 J/mol-K.
  1. What is the minimum work interaction required to compress the gas in one tank to 3 bar?

  2. What is the heat interaction with the constant temperature bath?

4.40A constant pressure air supply is connected to a small tank (A) as shown in the figure below. With valves B and C, the tank can be pressurized or depressurized. The initial conditions are T = 300 K, P = 1.013 bar, CP = 29.3 J/(mol-K). Consider the system adiabatic.
  1. The tank is pressurized with valve B open and valve C closed. What is the final temperature of the tank? Neglect the heat capacity of the tank and valves.

  2. Taking the system as the tank plus the valves, what is the entropy change of the system due to pressurization? What is the entropy change of the air supply reservoir? What is the entropy change of the universe? Use a reference state of 300 K and 1.013 bar.

  3. During depressurization with valve B closed and valve C open, how does the molar entropy entering valve C compare with the molar entropy leaving? What is the temperature of the tank following depressurization?

4.41The pressurization of problem 4.40 is performed by replacing the inlet valve with a reversible device that permits pressurization that is internally reversible. The system is to remain adiabatic with respect to heat transfer of the surroundings.
  1. What is the final temperature of the tank?

  2. How much work could be obtained?

4.42A 2m3 tank is at 292 K and 0.1 MPa and it is desired to pressurize the tank to 3 MPa. The gas is available from an infinite supply at 350 K and 5 MPa connected to the tank via a throttle valve. Assume that the gas follows the ideal gas law with a constant heat capacity of CP = 29 J/(mol-K).
  1. Modeling the pressurization as adiabatic, what is the final temperature in the tank and the final number of moles when the pressure equals 3 MPa?

  2. Identify factors included in the idealized calculation of part (a) that contribute to irreversibilities.

  3. Identify factors neglected in the analysis of part (a) that would contribute to irreversibilities in a real process.

  4. If the pressurization could be performed reversibly, the final temperature might be different from that found in part (a). Clearly outline a procedure to calculate the temperature indicating that enough equations are provided for all unknowns. Also clearly state how you would use the equations. Additional equipment is permissible provided that the process remains adiabatic with regard to heat transfer to the surroundings.

  5. In part (d), would work be added, removed, or not involved in making the process reversible? Provide equations to calculate the work interaction.

4.43Two gas storage tanks are interconnected through an isothermal expander. Tank 1 (V = 1 m3) is initially at 298 K and 30 bar. Tank 2 (V = 1 m3) is initially at 298 K and 1 bar. Reversible heat transfer is provided between the tanks, the expander, and the surroundings at 298 K. What is the maximum work that can be obtained from the expander when isothermal flow occurs from tank 1 to tank 2?


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