# 8.6: Problems

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In the following problems, all references to an ideal gas and an incompressible substance refer to the models with room-temperature specific heats.

A gearbox operating at steady-state conditions receives \(2 \mathrm{~hp}\) along the input shaft and delivers \(1.9 \mathrm{~hp}\) along the output shaft. The outer surface of the gearbox is at \(105^{\circ} \mathrm{F}\). The temperature of the air in the room is \(70^{\circ} \mathrm{F}\).

(a) Determine the rate of heat transfer, in \(\mathrm{Btu} / \mathrm{h}\). Indicate the direction

(b) Determine the rate of entropy production, in \(\mathrm{Btu} /\left(\mathrm{h} \cdot{ }^{\circ} \mathrm{R}\right)\), within the system consisting of the gearbox and the shafts. (Sketch the system.)

(c) Consider the layer of air immediately adjacent to the gearbox. It receives energy by heat transfer at \(105^{\circ} \mathrm{F}\) and loses energy by heat transfer at \(70^{\circ} \mathrm{F}\). Determine the steady-state rate of entropy production, in \(\mathrm{Btu} /\left(\mathrm{h} \cdot{ }^{\circ} \mathrm{R}\right)\), within the air layer. (Sketch the system.)

(d) Now consider an enlarged system that consists of both the gearbox and shafts and the air layer. This system loses energy by heat transfer to the surroundings at \(70^{\circ} \mathrm{F}\). Determine the steady-state rate of entropy production, in \(\mathrm{Btu} /\left(\mathrm{h} \cdot{ }^{\circ} \mathrm{R}\right)\), for this combined system. (Sketch the system.)

(e) Discuss how your result for Part (d) compares with your answers for Part (b) and Part (c).

An inventor claims to have developed a new device that operates at steady-state conditions and produces both shaft power and electrical power. A schematic of the device is shown in the figure with the known operating conditions and proposed energy transfers.

Figure \(\PageIndex{1}\): Device with heat input that produces shaft power and electrical power.

(a) Determine the electrical power output from the device, in \(\mathrm{kW}\).

(b) Determine the entropy production rate for the device, in \(\mathrm{kW} / \mathrm{K}\).

(c) Based upon your answer to part (b), do believe that this device is possible? Explain the rationale for your answer.

An electric transformer is used to step down the voltage from 220 to 110 volts (AC). The current on the high-voltage side is \(23 \mathrm{~A}\) and on the low voltage side it is \(43 \mathrm{~A}\). The power factor is one for both sides of the transformer. The transformer operates under steady-state conditions with a surface temperature of \(40^{\circ} \mathrm{C}\). Determine (a) the heat-transfer rate for the device, in watts, and (b) the entropy production rate, in \(\mathrm{W} / \mathrm{K}\).

An electric motor operates under steady-state conditions and draws \(3 \mathrm{~kW}\) of electric power. Ten percent of the electrical power supplied to the motor is lost to the surroundings by heat transfer. The surface temperature of the motor is \(45^{\circ} \mathrm{C}\). Determine (a) the shaft power delivered by the motor in \(\mathrm{kW}\) and (b) the entropy production rate for the motor, in \(\mathrm{kW} / \mathrm{K}\).

A soldering iron draws \(0.10 \mathrm{~A}\) from a \(110 \text{-V}\) circuit at steady-state conditions. The operating temperature of the soldering iron is \(105^{\circ} \mathrm{C}\). Determine the entropy production rate for the soldering iron in \(\mathrm{W}/\mathrm{K}\).

A transmission consists of two gearboxes connected by an intermediate shaft. A torque of \(220 \mathrm{~ft} \cdot \mathrm{lbf}\) is applied to the input shaft which rotates at \(200 \mathrm{~rpm}\). The intermediate shaft and the output shaft rotate at \(160 \mathrm{~rpm}\) and \(128 \mathrm{~rpm}\), respectively. *Each* gearbox transmits only \(95 \%\) of the shaft power supplied to it. The remainder of the energy is lost to the surroundings by heat transfer. The surface temperature of each gearbox is measured to be \(120^{\circ} \mathrm{F}\), and the ambient air temperature is \(70^{\circ} \mathrm{F}\).

Determine (a) the torque for the intermediate and output shafts, in \(\mathrm{ft} \cdot \mathrm{lbf}\), (b) the entropy production rate for each gearbox individually and for the overall transmission, in \(\mathrm{Btu} / \left(h \cdot { }^{\circ} \mathrm{R}\right)\).

An inventor claims to have a invented a device that takes in \(10 \mathrm{~kW}\) by heat transfer at \(500 \mathrm{~K}\), rejectes energy by heat transfer at \(300 \mathrm{~K}\) and produces \(5 \mathrm{~kW}\) of power. As a U.S. Patent Examiner, you must determine if this device is possible or a hoax. Based on the description in the patent application, it appears that it is a closed, steady-state device. What do you think? Is it possible? Explain your reasoning.

A steady-state heat pump is designed to reject energy by heat transfer at a rate of \(20,000 \mathrm{~Btu} / \mathrm{h}\) at a temperature of \(90^{\circ} \mathrm{F}\) and requires an electrical power input equivalent to \(5,000 \mathrm{~Btu} / \mathrm{h}\). Heat transfer into the system occurs at a temperature of \(40^{\circ} \mathrm{F}\).

Determine (a) the COP for the heat pump and (b) the entropy generation rate for the heat pump, in \(\mathrm{Btu} /\left(\mathrm{h} \cdot{ }^{\circ} \mathrm{R}\right)\). (c) Would you describe the heat pump as operating reversibly or irreversibly, or is it impossible to operate as specified?

A heat pump with a COP of 3 receives energy from the outdoors at \(30^{\circ} \mathrm{F}\) and rejects energy to the air inside the house at \(72^{\circ} \mathrm{F}\). The heat pump rejects energy to the air inside the house at the rate of \(100,000 \mathrm{~Btu} / \mathrm{h}\).

(a) Determine the following:

- the power of the motor required to operate the heat pump, in horsepower,
- the rate of heat transfer from the outdoors, in \(\mathrm{Btu} / \mathrm{h}\), and
- the rate of entropy production for the heat pump, in \(\mathrm{Btu} / \left( \mathrm{h} \cdot { }^{\circ} \mathrm{R}\right)\).

(b) Determine the maximum possible COP for a heat pump operating between these temperatures and the power of the motor, in \(\mathrm{hp}\), required to operate this *ideal* heat pump.

(c) Some people would consider the "extra" electrical energy required to run the real heat pump when compared with the power required to operate the *ideal* heat pump as being wasted, since it can't be used for anything else. If the ideal or best possible performance is associated with an internally reversible cycle and this cycle produces no entropy, entropy production may be a measure of energy waste. To check this out, investigate the validity of the following equation using your results from parts (a) and (b): \[\frac{\left(\dot{W}_{\text {actual}} - \dot{W}_{\text {ideal}}\right)}{T_{\text {outdoors}}} = \dot{S}_{\text {production}} \nonumber \] where all power values are in \(\mathrm{Btu} / \mathrm{h}\), temperatures are in \({ }^{\circ} \mathrm{R}\), and the entropy production rate is what you calculated in part (a). Is this result correct?

A system executes a power cycle. During each cycle, the system receives \(2000 \mathrm{~kJ}\) of energy by heat transfer at a temperature of \(500 \mathrm{~K}\) and discharges energy by heat transfer at a temperature of \(300 \mathrm{~K}\). There is no other heat transfer of energy.

(a) Assuming that the cycle has a thermal efficiency of \(25 \%\), determine the work out per cycle, in \(\mathrm{kJ}\), and the amount of entropy produced per cycle, in \(\mathrm{kJ} / \mathrm{K}\).

(b) Assuming that the cycle rejects \(900 \mathrm{~kJ}\) of energy by heat transfer, determine the work out per cycle, in \(\mathrm{kJ}\), the amount of entropy produced per cycle, in \(\mathrm{kJ} / \mathrm{K}\), and the thermal efficiency.

(c) Assuming that the cycle is internally reversible, i.e. rate of entropy production is zero, calculate the work out per cycle, in \(\mathrm{kJ}\), and the thermal efficiency for this cycle.

(d) Compare your answers to Parts (a), (b), and (c). What does this tell you about the three cycles? Is it possible to build a power cycle that operates between the same two temperatures and is more efficient than the one you examined in Part (c)?

A geothermal power plant utilizes an underground source of hot water at \(160^{\circ} \mathrm{C}\) as the heat source for a power cycle. The power plant boiler receives energy by heat transfer at a rate of \(100 \mathrm{~MW}\) from the hot water source at \(T_{H, \text { Source}} = 160^{\circ} \mathrm{C}\) The power plant condenser rejects energy by heat transfer at the rate of \(78 \mathrm{~MW}\) to the ambient air at \(T_{L, \text{ sink}}=15^{\circ} \mathrm{C}\).

Figure \(\PageIndex{2}\): Geothermal power plant consisting of a boiler and condenser, with an output of shaft work.

(a) Determine the net power output, in \(\mathrm{MW}\), and the thermal efficiency for this power cycle (heat engine) expressed as a percent.

(b) Determine the theoretical maximum thermal efficiency for a power cycle (heat engine) that operates between these two temperatures: \(T_{\text {boiler}} = T_{H, \text { source}}\) and \(T_{\text {condenser}} = T_{L, \text { sink}}\). Give your answer as a percent and compare it to your result from part (a).

(c) In reality, the rate of heat transfer is proportional to the temperature difference available to "drive" the heat transfer, i.e. \(Q \propto \Delta T\). Practically this means to receive energy by heat transfer from a thermal source at temperature \(T_{H, \text { source}}\), the surface temperature of the boiler \(T_{\text{boiler}}\) must be less than the source temperature. Similarly to reject energy by heat transfer to a thermal sink at temperature \(T_{L, \text{ sink}}\), the surface temperature of the condenser \(T_{\text {condenser}}\) must be greater than the sink temperature.

As a first guess, assume that a temperature difference of \(5^{\circ} \mathrm{C}\) is required, and determine the theoretical maximum thermal efficiency for a power cycle that operates between these new more realistic temperatures: \[T_{\text {boiler }}=T_{H, \text { Source}}-5^{\circ} \mathrm{C} \quad \text { and } \quad T_{\text {condenser}}=T_{L, \text { sink}}+5^{\circ} \mathrm{C}. \nonumber \] How does the efficiency of this more realistic cycle compare with your answers to part (a) and part (b)?

A reversed power cycle operates between a high temperature of \(50^{\circ} \mathrm{C}\) and a low temperature of \(5^{\circ} \mathrm{C}\). Determine the best possible coefficient for this reversed power cycle (a) if it is operated as a heat pump cycle and (b) if it operated as a refrigeration cycle.

A heat pump receives energy by heat transfer from outside air at \(T_{\text {outdoors}}\) and rejects energy to a dwelling at \(T_{\text {room}}\). Starting with the conservation of energy and entropy accounting equation for the steady-state heat pump, develop an expression for the COP of this heat pump similar to Eq. \(8.4.7\). Show your work.

A refrigeration cycle receives energy by heat transfer from a freezer compartment at \(T_{\text {freezer}}\) and rejects energy to the kitchen at \(T_{\text {room}}\). Starting with the conservation of energy and entropy accounting equation for the steady-state refrigeration cycle, develop an expression for the COP of this cycle similar to Eq. \(8.4.7\). Show your work.

Liquid water flows steadily through a small centrifugal pump at a volumetric flow rate of \(6.0 \mathrm{~m}^{3} / \mathrm{min}\). The water enters the pump at \(100 \mathrm{~kPa}\) and \(27^{\circ} \mathrm{C}\) and leaves the pump at a pressure of \(400 \mathrm{~kPa}\). Inlet and outlet areas are identical and changes in potential energy are negligible. Assume that water can be modeled as an incompressible substance.

Figure \(\PageIndex{3}\): Water flows steadily through a centrifugal pump.

(a) If the pump is adiabatic and internally reversible:

- Determine the change in specific entropy, \(s_{2}-s_{1}\), for the water as it flows through the pump, in \(\mathrm{kJ} /(\mathrm{kg}-\mathrm{K})\). [Hint: Apply the entropy accounting equation to the pump, make the appropriate modeling assumptions, and solve for \(s_{2}-s_{1}\).]
- Determine the shaft power input under these conditions in \(\mathrm{kW}\). [Hint: After applying the conservation of energy equation and appropriate modeling assumptions, don’t forget to see what your result for \(\Delta s\) from above along with the incompressible substance model tells you about how the pressure and/or temperature of the water may change.]

(b) Now assume the pump operates adiabatically and that the temperature of the water increases as it flows through the pump, \(T_{2}-T_{1}=0.05^{\circ} \mathrm{C}\) :

- Determine the entropy production rate for the pump, in \(\mathrm{kW} / \mathrm{K}\). Is this process internally reversible or internally irreversible? How can you tell?
- Determine the shaft power input under these conditions, in \(\mathrm{kW}\).

(c) Compare your answers from Part (a) and (b).

- Which operating condition requires the larger power input? Why?
- Do you think it would be possible to reduce the shaft power further by operating this same pump under steady-state, adiabatic conditions so that the water temperature would decrease as it flows through the pump, e.g. \(T_{2}-T_{1}=-0.05^{\circ} \mathrm{C}\)?

The nozzle in a turbojet engine receives air at \(180 \mathrm{~kPa}\) and \(707^{\circ} \mathrm{C}\) with a velocity of \(70 \mathrm{~m} / \mathrm{s}\). The air expands adiabatically in a steady-state process to an outlet pressure of \(70 \mathrm{~kPa}\). The mass flow rate of air is \(3.0 \mathrm{~kg} / \mathrm{s}\). Assume that air can be modeled as an ideal gas with room temperature specific heats.

(a) If the expansion process is internally *reversible*, determine the outlet air temperature in \({ }^{\circ} \mathrm{C}\) and the outlet velocity of the air, in \(\mathrm{m} / \mathrm{s}\). [Hint: Apply the entropy accounting equation along with the ideal gas model.]

(b) If the expansion process is internally *irreversible* and \(T_{2}\) is \(527^{\circ} \mathrm{C}\), determine the entropy production rate for the nozzle, in \(\mathrm{kW} / \mathrm{K}\), and determine the outlet velocity, in \(\mathrm{m} / \mathrm{s}\).

(c) Compare and discuss your results, especially \(T_{2}\) and \(V_{2}\), in terms of the entropy production rate for each process.

An electric water heater having a 100-liter capacity employs an electric resistor to heat the water from \(18^{\circ} \mathrm{C}\) to \(60^{\circ} \mathrm{C}\). The outer surface of the resistor remains at an average temperature of \(97^{\circ} \mathrm{C}\) during the heating process. Heat transfer from the outside of the water heater is negligible, and the energy and entropy storage in the resistor and the tank holding the water are insignificant. Model the water as an incompressible substance.

(a) Determine the amount of electrical energy, in \(\mathrm{kJ}\), required to heat the water.

(b) Determine the amount of entropy produced, in \(\mathrm{kJ} / \mathrm{K}\), within the water only, i.e. take the water as the system.

(c) Determine the amount of entropy produced, in \(\mathrm{kJ} / \mathrm{K}\), within the overall water heater including the resistor, i.e. take the overall water heater including the resistor as the system.

(d) Why do the results of (b) and (c) differ? What is within the system for (c) that was excluded in (b)?

Air enters a shop air compressor with a steady flow rate of \(0.7 \mathrm{~m}^{3} / \mathrm{s}\) at \(32^{\circ} \mathrm{C}\) and \(0.95 \mathrm{~bars}\). The air leaves the compressor at a pressure of \(15 \mathrm{~bars}\). Assume air can be modeled as an ideal gas and that changes in kinetic and gravitational potential energy are negligible. Determine the minimum power requirement to drive the adiabatic compressor, in \(\mathrm{kW}\). [Hint: How does varying the entropy generation rate affect the power input to the compressor? What are the limiting values on the entropy generation rate?]

A rigid air tank has a volume of \(1.0 \mathrm{~m}^{3}\) and contains air at \(27^{\circ} \mathrm{C}\) and \(3400 \mathrm{~kPa}\). Should the tank wall fail catastrophically, the tank would explode and cause considerable damage. To estimate the amount of energy that could be transferred from the air to the surroundings in an explosion, we will estimate the work done by the expanding gas. To model the expansion process, assume that the gas acts like a closed system and expands adiabatically and reversibly until the gas pressure matches the ambient air pressure of \(100 \mathrm{~kPa}\).

(a) Determine the work done by the gas on the surroundings during this expansion process, in kilojoules.

(b) Determine the temperature of the air after this hypothetical expansion process.

(c) How conservative are your results from Part (a)? Would you expect the actual blast to transfer more or less energy to the surroundings?

A short pipe and valve connect two heavily insulated tanks. Tank A has a volume of \(1.0 \mathrm{~m}^{3}\) and Tank B has a volume of \(2.0 \mathrm{~m}^{3}\). Tank A initially contains carbon dioxide at \(400 \mathrm{~K}\) and \(300 \mathrm{~kPa}\). Tank B is initially evacuated. Once the value is opened, the carbon dioxide expands into Tank B.

Determine (a) the final equilibrium pressure and temperature of the carbon dioxide and (b) the entropy produced within the gas during this expansion process, in \(\mathrm{kJ} / \mathrm{K}\). [Assume carbon dioxide can be modeled as an ideal gas.]

The figure below shows a steady-state gas turbine power plant consisting of a compressor, a heat exchanger, and a turbine. Air enters the compressor with a mass flow rate of \(3.9 \mathrm{~kg} / \mathrm{s}\) at \(0.95 \mathrm{bar}, 22^{\circ} \mathrm{C}\) and exits the turbine at \(0.95\) bar, \(421^{\circ} \mathrm{C}\). Heat transfer to the air as it flows through the heat exchanger occurs at an average temperature of \(488^{\circ} \mathrm{C}\). The compressor and turbine operate adiabatically. Assume that air behaves like an ideal gas and assume changes in kinetic and gravitational potential energy are negligible.

Figure \(\PageIndex{4}\): Turbine power plant consisting of a compressor, heat exchanger, and turbine, with an output of shaft work.

Determine the maximum theoretical value for the net power that can be developed by the power plant, in \(\mathrm{MW}\).

*[Hint: Consider the following steps.*

*(1) Apply the conservation of energy equation to develop an equation that relates the net power out to the heat transfer rate into the system.*

*(2) Now apply the entropy accounting equation to find a relationship between the heat transfer rate into the system and the entropy production rate for the system.*

*(3) Now vary the entropy production rate over its possible values and examine how it changes the net power out of the system.*

*(4) Determine the maximum value of the net power out of the system. Clearly indicate why your result is the maximum value.]*

A short pipe and valve connect two heavily insulated tanks. Each tank has a volume of \(0.5 \mathrm{~m}^{3}\). Tank A initially contains nitrogen at \(150 \mathrm{~kPa}\) and \(300 \mathrm{~K}\). Tank B initially contains nitrogen at \(50 \mathrm{~kPa}\) and \(300 \mathrm{~K}\). Suddenly, the valve is opened and the two gases are allowed to mix.

Determine (a) the final pressure and temperature of the mixture and (b) the entropy produced during this mixing process. [Assume nitrogen can be modeled as an ideal gas. ]

The air trapped in the piston cylinder of an air compressor occupies an initial volume of \(42 \mathrm{~in}^{3}\) (cubic inches) when the piston is at the bottom of its stroke. The air has a temperature and pressure of \(70^{\circ} \mathrm{F}\) and \(15 \mathrm{~psi} \ \left(\mathrm{lbf} / \mathrm{in}^{2}\right)\), respectively.

When the piston moves to the top of its stroke, the air is compressed to a volume of \(7.0 \mathrm{~in.}^{3}\). The compression process occurs so fast that heat transfer during the compression process is negligible. If necessary, assume that air can be modeled as an ideal gas and that changes in kinetic and gravitational potential energy for the gas are negligible.

Figure \(\PageIndex{5}\): Air in a piston cylinder.

(a) If the compression process is reversible, determine the temperature and the pressure of the gas after the compression. In addition, calculate the work done on the gas during the compression process, in \(\mathrm{Btu}\).

(b) Now assume that the compression process is irreversible, how will this change the final temperature and pressure of the gas after the compression and the work done on the gas. Clearly indicate whether the values increase or decrease as compared to the values for the reversible process?

(c) Why would an engineer care about the values for a reversible process?

An \(18 \text{-kg}\) lead casting at \(200^{\circ} \mathrm{C}\) is quenched in a tank containing \(0.03 \mathrm{~m}^{3}\) of liquid water initially at \(25^{\circ} \mathrm{C}\). The water tank is insulated immediately after the casting is dropped into the water. Determine (a) the final equilibrium temperature of the lead, in \(\mathrm{K}\), and (b) the entropy generation for the lead-water system, in \(\mathrm{kJ} / \mathrm{K}\). (c) Is this process reversible, irreversible, or impossible? [Assume lead and liquid water can both be modeled as incompressible substances.

A new device is proposed as a steady-state air heater (see figure). Air enters the heater (1) at \(400 \mathrm{~K}\) and \(200 \mathrm{~kPa}\) with a volumetric flow rate of \(1000 \mathrm{~m}^{3} / \mathrm{min}\). It leaves the heater (2) at \(500 \mathrm{~K}\) and \(190 \mathrm{~kPa}\). The heater is powered by electricity and has two different operating modes. Electricity costs \(\$ 0.08\) per kilowatt-hour.

Mode I - Steady-state, adiabatic operation with no heat transfer on the surface of the device, \(Q_{o, \text{ in}}=0\).

Mode II - Steady-state, internally reversible operation with heat transfer on the surface at a boundary temperature of \(T_{o}=300 \mathrm{~K}\).

Figure \(\PageIndex{6}\): A steady-state electric air heater.

(a) For Mode I, determine the electric power required to operate the heater, in \(\mathrm{kW}\), and the entropy generation rate, in \(\mathrm{kW} / \mathrm{K}\).

(b) For Mode II, determine the electric power required to operate the heater, in \(\mathrm{kW}\), the entropy generation rate, in \(\mathrm{kW} / \mathrm{K}\), and the heat transfer rate, in \(\mathrm{kW}\).

(c) For an 8-hour day, how much would it cost to operate the air heater in each mode? Any advice to the plant engineer about which mode of operation should be used?