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4.1  Extending 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 N_{2} for example. Being diatomic, we should expect that U^{ig} = 2(3N_{A}kT/2) = 6RT/2 in the limit of classical vibrations. Vibrational energy means that heat can be absorbed in the vibration of a bond. Since N_{2} 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 ΔU^{vib} = ϵ/[exp(–β ϵ) – 1]. For now, without concern for the proof, assume ΔU^{vib} as given. Adapting Example 4.2 for N_{2} then gives: U^{ig} = 5RT/2 + ϵ/[exp(–β ϵ) – 1].

4.2  An ideal gas, with temperatureindependent C_{P} = (7/2)R, at 15°C and having an initial volume of 60 m^{3}, is heated at constant pressure (P = 0.1013 MPa) to 30°C by transfer of heat from a reservoir at 50°C. Calculate ΔS_{gas}, ΔS_{heat reservoir}, ΔS_{universe}. What is the irreversible feature of this process? 
4.3  Steam 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:

4.4  The following problems involve one mole of an ideal monatomic gas, C_{P} = 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: T^{i} = 25°C, P^{i} = 5 bar.

4.5  When 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 2.5m^{3} air storage tank initially at 15 bar. Atmospheric pressure is 1 bar, C_{P} = 7R/2. Provide an estimate by assuming reversibility.

4.6  Work 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. 
4.7  An 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:

4.8  An 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.

4.9  Airplanes are launched from aircraft carriers by means of a steam catapult. The catapult is a wellinsulated 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. 
4.10  We have considered heat and work to be pathdependent. However, if all heat transfer with surroundings is performed using a reversible heat transfer device (some type of reversible Carnottype 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 T_{surr} = 273 K. The state change is from state 1, P_{1} = 0.1 MPa, T_{1} = 298 K and state 2, P_{2} = 0.5 MPa and T_{2} which will be found in part (a). C_{P} = 7R/2.

4.11  Consider 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 T_{house} by a constant rate of heat transfer, . The furnace operates at a constant temperature T_{F}, and with direct heat transfer, the heat required from the furnace, is equal to .

4.12  An ideal gas enters a valve at 500 K and 3 MPa at a steadystate rate of 3 mol/min. It is throttled to 0.5 MPa. What is the rate of entropy generation? Is the process irreversible? 
4.13  SO_{2} vapor enters a heat exchanger at 100°C and at a flowrate of 45 mole/h. If heat is transferred to the SO_{2} 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 ? 
4.14  An ideal gas stream (Stream A), C_{P} = 5R/2, 50 mole/h, is heated by a steadystate heat exchanger from 20°C to 100°C by another stream (Stream B) of another ideal gas, C_{P} = 7R/2, 45 mole/h, which enters at 180°C. Heat losses from the exchanger are negligible.

4.15  An 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 C_{P} = (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? 
4.16  Two streams of air are mixed in a steadystate process shown below. Assume air is an ideal gas with a constant heat capacity C_{P} = 7R/2.

4.17  Air 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 C_{P} = 29.1 J/molK. If the throttle valve was replaced by a reversible steadystate flow device to permit exactly the same state change for the air in this steadystate 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. 
4.18  A common problem in the design of chemical processes is the steadystate compression of gases from a low pressure P_{1} to a much higher pressure P_{2}. We can gain some insight about optimal design of this process by considering adiabatic reversible compression of ideal gases with stagewise intercooling. If the compression is to be done in two stages, first compressing the gas from P_{1} to P*, then cooling the gas at constant pressure down to the compressor inlet temperature T_{1}, and then compressing the gas to P_{2}, what should the value of the intermediate pressure be to accomplish the compression with minimum work? 
4.19  Steam 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? 
4.20  An 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. 
4.21  Steam is fed to an adiabatic turbine at 4 MPa and 500°C. It exits at 0.1 MPa.

4.22  Methane is compressed in a steadystate 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. 
4.23  Methane is to be compressed from 0.05 MPa and –120°F to 5 MPa in a twostage compressor. In between adiabatic, reversible stages, a heat exchanger returns the temperature to – 120°F. The intermediate pressure is 1.5 MPa.

4.24  A 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. 
4.25  Propane is to be compressed from 0.4 MPa and 360 K to 4 MPa using a twostage 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%.

4.26 

4.27  Steam 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.

4.28  Liquid 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.

4.29  Propane flows into a steadystate 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. 
4.30  Propane (1000 kg/hr) is to be liquefied following a twostage 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%.

4.31  A 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.

4.32  Presently, 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.

4.33  A 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 highpressure 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 C_{P} = 7R/2.

4.34  Methane gas is contained in a 0.65m^{3} 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:

4.35  A thermodynamically interesting problem is to analyze the fundamentals behind the product called “fixaflat.” 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 C_{P}/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 cm^{3}. We will reconsider this problem with liquid contents, after discussing phase equilibrium in a pure fluid. 
4.36  Wouldn’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? 
4.37  A 1 m^{3} tank is to be filled using N_{2} at 300 K and 20 MPa. Instead of throttling the N_{2} 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.) 
4.38  Two wellinsulated tanks are attached as shown in the figure below. The tank volumes are given in the figure. There is a massflow 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 C_{P} = 7/2·R may be assumed to be independent of T.

4.39  Two storage tanks (0.1 m^{3} 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 C_{P} = 29.3 J/molK.

4.40  A 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, C_{P} = 29.3 J/(molK). Consider the system adiabatic.

4.41  The 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.

4.42  A 2m^{3} 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 C_{P} = 29 J/(molK).

4.43  Two gas storage tanks are interconnected through an isothermal expander. Tank 1 (V = 1 m^{3}) is initially at 298 K and 30 bar. Tank 2 (V = 1 m^{3}) 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? 