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7.10: Problems

  • Page ID
    84367
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    Problem \(7.1\)

    The lunar module is designed to make a safe landing on the moon if its vertical velocity at impact is \(V_{\max } \leq 5 \mathrm{~m} / \mathrm{s}\). The acceleration of gravity on the moon is \(1 / 6\) the value on Earth, \(g_{\text { moon}}=g_{\text { Earth}} / 6\).

    (a) Using the work-energy principle (conservation of energy), determine the maximum height \(h\) above the surface of the moon at which the pilot can safely shut off the engine when the velocity of the lunar module relative to the moon's surface is (i) zero; (ii) \(3 \mathrm{~m} / \mathrm{s}\) up; and (iii) \(3 \mathrm{~m} / \mathrm{s}\) down.

    (b) Repeat Part (a), only this time using conservation of linear momentum. Is one approach easier than the other?

    Problem \(7.2\)

    The collar \(A\) slides on a smooth horizontal bar and is attached to the hanging mass \(B\) by a cord as shown in the figure. The collar \(A\) has mass \(m_{A}=30 \mathrm{~kg}\), and the hanging mass \(B\) has mass \(m_{B}=60 \mathrm{~kg}\). Assume that the mass of the cord and the mass of the frictionless pulley are negligible. If the collar is initially stationary, determine the velocity of collar \(A\) after it has moved \(0.5\) meters to the right.

    A collar A slides along a horizontal bar. A cord that passes over a pulley attached to the right end of the bar connects the bottom of A to a hanging mass B.

    Figure \(\PageIndex{1}\): A sliding collar is connected to a hanging mass by a cord and frictionless pulley.

    Problem \(7.3\)

    Repeat Problem \(7.2\) assuming that the coefficients of static and sliding friction between collar \(A\) and the horizontal bar are \(0.4\) and \(0.3\), respectively.

    A careful examination of how the cable is connected to collar \(A\) indicates that there are extra normal forces between the collar and the bar. These extra forces keep the collar from rotating and form a force couple. For our analysis, you may neglect the normal forces due to this couple. However, consider how you could estimate these forces and how they would change your answer.

    Problem \(7.4\)

    Repeat Problem \(8.2\) assuming that the mass \(B\) is replaced by a constant force \(T = 600 \mathrm{~N}\).

    Problem \(7.5\)

    The collar \(A\) slides on a smooth horizontal bar and is attached to the hanging mass \(B\) by a cord as shown in the figure. The collar \(A\) has mass \(m_A = 14 \mathrm{~kg}\), and the hanging mass \(B\) has mass \(m_B = 18 \mathrm{~kg}\). The spring constant is \(k = 700 \mathrm{~N}/\mathrm{m}\). Assume that the mass of the cord and the mass of the frictionless pulley are negligible.

    A collar A slides along a horizontal bar. A spring connects the top of A to the left support of the bar, and a cable passing over a pulley attached to the right support of the bar connects the bottom of A to a hanging mass B.

    Figure \(\PageIndex{2}\): A sliding collar is connected to a hanging mass by a cable and to a support by a spring.

    (a) If the collar is initially stationary and the spring unstretched, determine the velocity of collar \(A\) after it has moved 0.2 meters to the right.

    (b) If the mass \(B\) is replaced by a constant force \(F_B\) with a magnitude equal to the weight of mass \(B\), e.g. \(F_{\mathrm{B}} = m_{\mathrm{B}} g\), and the collar is initially stationary and the spring unstretched, determine the velocity of collar \(A\) after it has moved 0.2 meters to the right. Is the value larger or smaller than the result from Part (a)? Explain

    Problem\(7.6\)

    A collar is attached to a linear spring as shown in the figure and moves freely on the vertical rod. The mass of the collar is \(m=2.0 \mathrm{~kg}\), and the spring constant for the spring is \(k=30 \mathrm{~N} / \mathrm{m}\). The unstretched length of the spring is \(1.5 \mathrm{~m}\). The collar is released from rest at \(A\) and slides up the smooth rod under the action of a constant force \(F=50 \mathrm{~N}\) applied at \(30^{\circ}\) from the vertical as shown in the figure.

    A sliding collar on a vertical bar is connected to the left end of a horizontal spring, with the right end of the spring fixed in place. In this position the spring is 2.0 m long. A force of 50 N upwards and to the left, 30 degrees from the vertical, is applied to the collar, moving it to point B 1.5 meters above the original position.

    Figure \(\PageIndex{3}\): A spring connects a support and a sliding collar.

    (a) Determine the work done on the collar by the force \(F\) in moving it from point \(A\) to point \(B\), in \(\mathrm{N} \cdot \mathrm{m}\).

    (b) Determine the velocity \(V\) of the collar as it passes point \(B\), in \(\mathrm{m} / \mathrm{s}\).

    Problem \(7.7\)

    In a preliminary design for a mail-sorting machine, parcels move down an inclined, smooth ramp and are brought to rest by a linear spring. The ramp is inclined at \(30^{\circ}\) with the horizontal. A typical packages weighs \(10 \mathrm{~lbf}\). The initial package velocity on the ramp is \(2 \mathrm{~ft} / \mathrm{s}\) and the package travels a distance of \(10 \mathrm{~ft}\) down the ramp before it contacts the spring. The spring is designed to bring the packages to a full stop in a distance of 8 inches.

    (a) Determine the spring constant \(k\), in \(\mathrm{lbf} / \mathrm{in}\), required to stop the package in the distance desired. Does your answer depend on the mass of the package?

    (b) Determine the maximum deceleration for the typical package in units of standard gravitational acceleration, e.g. \(3 \ g\)’s.

    Problem \(7.8\)

    In an alternate design to the one in Problem \(7.7\), the spring is eliminated and the package is brought to rest solely due to a friction coating over the last 5 feet of the ramp. All variables are the same as in Problem \(7.7\) except the spring has been replaced by a 5-foot-long friction strip.

    (a) Determine the minimum value of the coefficient of kinetic friction between the package and the friction strip material that will bring the package to rest on the friction strip. Does your answer depend on the mass of the package?

    (b) Determine the maximum deceleration for the typical package in units of standard gravitational acceleration, e.g. \(3 g\)’s.

    Problem \(7.9\)

    The work done by an internal combustion engine can be modeled by considering the work done by a closed system containing a gas. The gas in the system executes a three-stage process that returns the gas to its initial state.

    State \(1\): \(\quad p_{1}=100 \mathrm{~kPa} ; \ V_{1}=0.80 \mathrm{~m}^{3}\)
    Process \(1 \text{-} 2\): Compressed along a process where \(p V=\) constant.
    State \(2\): \(\quad V_{2}=0.2 V_{1}\)
    Process \(2 \text{-} 3\): Heated and expanded at constant pressure, e.g. \(p=\) constant
    State \(3\): \(\quad V_{3}=V_{1}\)
    Process \(3 \text{-} 1\): Cooled at constant volume until \(p=p_{1}\).

    (a) Sketch a generic piston-cylinder device and identify the closed system used for your analysis.

    (b) Calculate the work done on the gas during each process.

    (c) Sketch the three processes and their end states on a \(p \text{-} V\) diagram.

    Label your axes and end states. Accurately show the path of each process on the diagram. Indicate the area on the diagram that represents the magnitude of the work done on the system during each process.

    (d) Calculate the net work done on the system during this cycle and indicate the corresponding area on the \(p \text{-} V\) diagram.

    Problem \(7.10\)

    A closed system of mass \(5 \mathrm{~lbm}\) undergoes a process in which there is a heat transfer of \(200 \mathrm{~ft} \cdot \mathrm{lbf}\) from the system to the surrounding. There is no work during the process. The velocity of the system increases from \(10 \mathrm{~ft} / \mathrm{s}\) to \(50 \mathrm{~ft} / \mathrm{s}\), and the elevation decreases by \(150 \mathrm{~ft}\). The acceleration of gravity at this particular geographical location is \(32.0 \mathrm{~ft} / \mathrm{s}^{2} .\)

    Please sketch your system and show all of your work to determine the change in (a) kinetic energy of the system, in \(\mathrm{ft} \cdot \mathrm{lbf}\),

    (b) gravitational potential energy of the system, in \(\mathrm{ft} \cdot \mathrm{lbf}\), and

    (c) internal energy of the system, in \(\mathrm{ft} \cdot \mathrm{lbf}\) and \(\mathrm{Btu}\).

    Problem \(7.11\)

    A closed system undergoes a process during which there is heat transfer to the system at a constant rate of \(5 \mathrm{~kW}\), and the power out of the system varies with time according to \[\dot{W}_{\text {out}} = \begin{cases}+2.5 t & 0<t \leq 2.0 \mathrm{~h} \\ +5.0 & t>2.0 \mathrm{~h}\end{cases} \nonumber \] where \(t\) is in hours and \(W_{\text {out}}\) is in \(\mathrm{kW}\).

    (a) Sketch the system and label the energy flows.

    (b) Determine the time rate of change of system energy at \(t=1.2 \mathrm{~h}\) and \(2.4 \mathrm{~h}\), in \(\mathrm{kW}\)

    (c) Determine the change in system energy after \(3 \mathrm{~h}\), in \(\mathrm{kW} \cdot \mathrm{h}\) and in \(\mathrm{kJ}\).

    Problem \(7.12\)

    A piston-cylinder assemby is equipped with a paddle wheel driven by an external motor and is filled with \(30 \mathrm{~g}\) of a gas. The walls of the cylinder are well insulated, and the friction between the piston and the cylinder wall is negligible. Initially the gas is in state 1 (see table). The paddle wheel is then operated, but the piston is allowed to move to keep the pressure in the gas constant. When the paddle wheel is stopped, the system is in state 2 . Determine the work transfer, in joules, along the paddle-wheel shaft.

    State \(P\), bars \(v, \mathrm{~cm}^{3} / \mathrm{g}\) \(u, \mathrm{~J} / \mathrm{g}\)
    1 15 \(7.11\) \(22.75\)
    2 15 \(19.16\) \(97.63\)
    Problem \(7.13\)

    An insulated piston-cylinder assembly containing a fluid has a stirring device operated externally. The piston is frictionless, and the force holding it against the fluid is due to standard atmospheric pressure (\(101.3 \mathrm{~kPa}\)) and a coil spring with a spring constant of \(7200 \mathrm{~N} / \mathrm{m}\). The stirring device is turned 100 revolutions with an average torque of \(0.68 \mathrm{~N} \cdot \mathrm{m}\). As a result, the gas expands and the piston moves outward \(0.10 \mathrm{~m}\). The diameter of the piston is \(0.10 \mathrm{~m}\). Assume changes in kinetic energy and potential energy are negligible.

    A cylinder-piston device filled with fluid has the stirring end of an externally operated stirrer moving through the fluid. The piston is attached to one end of a spring, whose opposite end is attached to a support.

    Figure \(\PageIndex{4}\): Fluid is stirred in a piston-cylinder device whose piston is held in place by a spring.

    (a) Determine the work done by the gas during its expansion, in \(\mathrm{kJ}\). In addition, determine how much of the work done by the gas is against the force of the atmosphere and how much is done against the spring. Assume that the spring initially exerts no force on the piston.

    (b) Determine the work done by the stirring device on the gas, in \(\mathrm{kJ}\), during

    (c) Determine the change in internal energy, in \(\mathrm{kJ}\), of the fluid.

    Problem \(7.14\)

    An electric motor drives an air compressor that delivers air with inlet and outlet conditions as shown on the figure. The motor-compressor set operates at steady-state conditions. The electric power supplied to the motor is 25 kilowatts at 220 volts ac, and the measured heat transfer rate from the combined motor and compressor is \(4.4\) kilowatts. Experience shows that changes in kinetic and potential energy are negligible for this system.

    An electric motor supplied with 220 volts AC runs an air compressor. The compressor takes in air at pressure 1 bar, temperature 300 K, and specific enthalpy 302 kJ/kg. Air leaves the compressor at pressure 8 bar, temperature 500 K, and specific enthalpy 503 kJ/kg.

    Figure \(\PageIndex{5}\): An air compressor and the electric motor driving it form a system.

    a) Determine the mass flow rate through the compressor in kilograms per second.

    (b) Assuming a power factor of 1 (purely resistive circuit), calculate the electric current AC drawn by the motor, in amps

    Problem \(7.15\)

    Air enters a compressor at \(100 \mathrm{~kPa}\) and \(280 \mathrm{~K}\) and leaves the compressor at \(600 \mathrm{~kPa}\) and \(400 \mathrm{~K}\). The compressor operates at steady-state conditions with an entering mass flow rate of \(0.02 \mathrm{~kg} / \mathrm{s}\). In addition, the heat transfer rate from the compressor is \(0.32 \mathrm{~kW}\). The specific enthalpy of the entering and leaving air streams are found to be \(h_{1}=280.13 \mathrm{~kJ} / \mathrm{kg}\) and \(h_{2}=400.98 \mathrm{~kJ} / \mathrm{kg}\), respectively. Assume that changes in kinetic and potential energy for the streams flowing through the system are negligible.

    A steady-state air compressor is powered by a drive shaft.

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

    (a) Determine the mass flow rate of the air leaving the compressor, in \(\mathrm{kg} / \mathrm{s}\). Show your reasoning.

    (b) Determine the ratio of the entering volumetric flow rate to the leaving volumetric flow rate, i.e. \(\dot{V}_{1} / \dot{V}_{2}\). Assume ideal gas behavior for air. (Hint: Solve for the volumetric flow rate in terms of the mass flow rate and the density.)

    (c) Determine the power required to operate the compressor, in \(\mathrm{kW}\) and in \(\mathrm{hp}\).

    Answer
    Problem \(7.16\)

    Methane gas \(\left(\mathrm{CH}_{4}\right)\) is burned with the stoichiometric amount of oxygen \(\left(\mathrm{O}_{2}\right)\) in a steady-state combustion process. The methane gas enters the burner with a mass flow rate of \(16.0 \mathrm{~kg} / \mathrm{h}\) and a specific enthalpy \(h=4,778 \mathrm{~kJ} / \mathrm{kg}\) at \(1 \mathrm{~atm}\) and \(25^{\circ} \mathrm{C}\). The oxygen enters the burner with a mass flow rate of \(64.0 \mathrm{~kg} / \mathrm{h}\) and a specific enthalpy \(h=100 \mathrm{~kJ} / \mathrm{kg}\) at \(1 \mathrm{~atm}\) and \(25^{\circ} \mathrm{C}\). The combustion products exit the burner at \(1 \mathrm{~atm}\) and \(25^{\circ} \mathrm{C}\) and a specific enthalpy \(h=-10,865 \mathrm{~kJ} / \mathrm{kg}\). The burner operates with negligible work at non-flow boundaries and negligible changes in kinetic and gravitational potential energy.

    Methane and oxygen enter a burner, which produces exhaust products.

    Figure \(\PageIndex{7}\): Steady-state burner for methane gas.

    (a) Determine the mass flow rate of combustion products leaving the burner, in \(\mathrm{kg} / \mathrm{s}\).

    (b) Determine the heat transfer rate for the process, in \(\mathrm{kW}\). Be sure and indicate both the magnitude and the direction of the heat transfer.

    (c) If the velocity of the oxygen entering the burner is approximately \(1 \mathrm{~m} / \mathrm{s}\), determine the cross-sectional area of the inlet duct. Assume that oxygen can be modeled as an ideal gas.

    Problem \(7.17\)

    An dc-electric generator is attached directly to a steam turbine as shown in the figure. Steam flows into the turbine with a mass flow rate of \(360 \mathrm{~lbm} / \mathrm{h}\) and the turbine-generator set operates at steady-state conditions. The specific enthalpy of the entering and leaving steam is shown on the figure. The heat transfer rate from the turbine is \(3000 \mathrm{~Btu} / \mathrm{h}\) and from the generator is \(1000 \mathrm{~Btu} / \mathrm{h}\). Assume changes in kinetic and gravitational potential energy for the steam flowing through the generator are negligible.

    A generator with a 220-volt DC output is attached to a steam turbine. Steam enters the turbine with a specific enthalpy of 2000 Btu/lbm and exits with a specific enthalpy of 1450 Btu/lbm.

    Figure \(\PageIndex{8}\): System consisting of a d electric generator and steam turbine.

    (a) Determine the electrical power output from the generator, in \(\mathrm{Btu} / \mathrm{h}\).

    (b) Determine the de current supplied by the generator, in amps.

    (c) Determine the torque, in \(\mathrm{lbf} \cdot \mathrm{ft}\), in the 220-volt dc shaft connecting the turbine to the output generator assuming it rotates at \(3600 \mathrm{~rpm}\).

    Problem \(7.18\)

    Steam flows through a heat exchanger and then through a steam turbine as shown in the figure. Steam leaves the turbine at two different points, State 3 and State 4. The shaft power output from the turbine is \(675 \mathrm{~MW}\). All known state information is shown in the table. All devices operate at steady-state conditions and changes in kinetic and gravitational potential energy are negligible. By design, heat exchangers have no work transfer of energy. In addition, the steam turbine operates adiabatically.

    Steam at state 1 enters a heat exchanger and leaves the exchanger at state 2 to enter a steam turbine. The steam turbine turns a shaft, and steam exits it at two points, one in state 3 and one in state 4.

    Figure \(\PageIndex{9}\): Steam flows through a heat exchanger and steam turbine in steady-state conditions.

    State \(h \, (\mathrm{kJ} / \mathrm{kg})\) \(P \,(\mathrm{kPa})\) \(T \, \left({ }^{\circ} \mathrm{C}\right)\) \(v \, \left(\mathrm{~m}^{3} / \mathrm{kg}\right)\) \(\dot{\mathrm{m}} \, (\mathrm{kg} / \mathrm{s})\)
    1 \(562.0\) 300 133 \(0.001\) 1000
    2 \(\cdots\) 300 400 1032 \(-\)
    3 3114 150 320 1819 200
    4 3034 100 280 2546 \(-\)
    Note: You need not complete this table to work the problem.

    (a) Determine the mass flow rate leaving the turbine at State \(4\).

    (b) Determine the heat transfer rate to the heat exchanger.

    (c) Determine the torque transferred by the turbine shaft if the turbine rotates at \(3600 \mathrm{~rpm}\).

    Problem \(7.19\)

    One means for generating electricity is to use a gas-turbine engine connected to an electric generator. Although the gas-turbine engine, as shown in the figure, consists of three components—a compressor, a heat exchanger, and a turbine—connected together with air flowing through each sequentially, the engine can be analyzed using the open system indicated. The compressor and turbine have a common shaft and are directly connected to the generator. Operating information is shown on the figure. The mass flow rate of air through the gas turbine is \(2.0 \mathrm{~lbm} / \mathrm{s}\). For purposes of analysis you may assume that all systems operate at steady-state conditions and that changes in kinetic and gravitational potential energy are negligible.

    An electric generator that outputs heat at rate 60 Btu/s is connected to a shaft that is shared by an air compressor and a turbine. Air enters the compressor at mass flow rate 2.0 lbm/s, in state 1 with specific enthalpy 130 Btu/lbm, pressure 14.7 psia, and temperature 70 degrees F. The air exits the compressor at state 2 before entering a heat exchange, where 1000 Btu/s of heat is inputted. Air exits the heat exchange in state 3 and enters the turbine, where it exits at state 4: specific enthalpy 230 Btu/lbm, pressure 14.7 psia, and temperature 500 degrees F.

    Figure \(\PageIndex{10}\): Electric generator is powered by a system consisting of a compressor, heat exchanger, and turbine.

    (a) Determine the shaft power delivered to the electric generator, in \(\mathrm{Btu} / \mathrm{s}\), and the electrical power output of the generator, in kilowatts.

    (b) Determine the shaft torque in the shaft supplying power to the electric generator, in \(\mathrm{ft} \cdot \mathrm{lbf}\), if the shaft speed is \(1800 \mathrm{~rpm}\).

    Problem \(7.20\)

    A small steam turbine is connected to an air compressor through a gear reducer as shown in the figure. A gear reducer is a device used to change the shaft rotation speed when two devices must be connected but operate at different speeds.

    Steam enters the turbine at \(110^{\circ} \mathrm{C}\) with a specific enthalpy \(h_{1}=2691.5 \mathrm{~kJ} / \mathrm{kg}\) and exits the turbine at a pressure of \(100 \mathrm{~kPa}\) and a specific enthalpy \(h_{2} = 2675.5 \mathrm{~kJ} / \mathrm{kg}\). The turbine shaft rotates at \(2000 \mathrm{~rpm}\).

    Air enters the compressor at a mass flow rate of \(70 \mathrm{~kg} / \mathrm{min}\) at \(100 \mathrm{~kPa}\) and \(300 \mathrm{~K}\) and exits the compressor at \(500 \mathrm{~kPa}\) and \(460 \mathrm{~K}\). The compressor shaft rotates at \(600 \mathrm{~rpm}\). Assume that air can be modeled as an ideal gas with room temperature specific heats.

    Assume all devices shown in the figure—turbine, compressor and gear reducer—operate adiabatically at steady-state conditions with negligible changes in kinetic and gravitational potential energy.

    A steam turbine turns a shaft at 2000 rpm; the shaft passes through a gear reducer, which changes its rotation speed to 600 rpm, and powers an air compressor at that new speed. Air enters and exits the steam turbine and the compressor at the described pressures and specific enthalpies.

    Figure \(\PageIndex{11}\): A steam turbine and air compressor connected by a common shaft that passes through a gear reducer.

    (a) Determine the mass flow rate of steam into the turbine, in \(\mathrm{kg} / \mathrm{min}\).

    (b) Determine the shaft power required by the air compressor, in \(\mathrm{kW}\).

    (c) Determine the torque, in \(\mathrm{N} \cdot \mathrm{m}\), transmitted by the air compressor shaft.

    Problem \(7.21\)

    The compression stroke of an air compressor can be modeled as a closed system. Initially the air inside the chamber occupies a volume of \(100 \mathrm{~cm}^{3}\) and has a pressure of \(100 \mathrm{~kPa}\) and a temperature of \(25^{\circ} \mathrm{C}\). During the compression process the gas follows a process where the product of pressure and volume remain constant, e.g. \(P V\kern-1.0em\raise0.3ex- = C\). After the compression process, the gas occupies a volume of \(12.5 \mathrm{~cm}^{3}\). Assume air can be modeled as an ideal gas with room-temperature specific heats.

    (a) Determine the mass of air inside the piston in \(\mathrm{kg}\).

    (b) Determine the work done on the gas by the piston during this compression process, in kilojoules.

    (c) Determine the heat transfer for the process, in kilojoules.

    Problem \(7.22\)

    Repeat Problem 8.21 and this time assume that the air follows a process where \(P V\kern-1.0em\raise0.3ex-^{1.3} = C\).

    Problem \(7.23\)

    The water faucet in your bathroom produces a stream of warm water by mixing a hot stream and a cold stream. You adjust the temperature by adjusting the flow rates of the two entering streams. On a cold day, city water enters the house at \(50^{\circ} \mathrm{F~} \left(510^{\circ} \mathrm{R}\right)\). Some of this water is diverted to the water heater and heated to a temperature of \(140^{\circ} \mathrm{F~} \left(600^{\circ} \mathrm{R}\right)\).

    Assume that the faucet can be modeled as a mixing tee as shown in the figure. In addition, you may assume that water can be modeled as an incompressible substance with room-temperature specific heats. Experience has also shown that for this type of problem, changes in pressure, kinetic energy, and potential energy have a negligible effect on the answer; thus, you may neglect them.

    A mixing tee where hot water enters one of the T's arms, cold water enters the other, and the mixed water exits through the upright.

    Figure \(\PageIndex{12}\): Mixing tee for hot and cold streams of water.

    (a) Determine the steady-state volumetric flow rate of cold water that produces a warm water temperature of \(100^{\circ} \mathrm{F}\) if the hot water flows at \(0.100\) gallons per minute. [Hint: When applying the energy balance, replace the specific enthalpy \(h\) in the energy balance with its definition, \(h=u+p v=u+p / \rho\)]

    (b) Revisit your analysis for part (a) and rewrite the results as a function of the ratio of the flow rates, \(\dot{m}_{\text{cold}} / \dot{m}_{\text{hot}}.\) Plot your results in two ways:

    • Plot A: \(T_{\text {warm}}\) vs. \(\dot{m}_{\text {cold}} / \dot{m}_{\text {hot}}\)
    • Plot B: \(\left(T_{\text {warm}} - T_{\text {cold}}\right) / \left(T_{\text {hot}} - T_{\text {cold}}\right)\) vs. \(\dot{m}_{\text {cold}} / \dot{m}_{\text {hot}}\)

    What, if any, is the advantage of Plot \(\mathrm{B}\) over Plot \(\mathrm{A}\) ?

    Problem \(7.24\)

    An on-demand water heater is designed to supply hot water almost instantaneously without having to keep a tank of hot water available at all times. One design for an electrically powered on-demand water heater is shown in the figure below. For purposes of analysis, we can assume that the water heater operates at steady-state conditions immediately after the power is switched on.

    Water enters one opening of a tank through which a resistance heating element passes, and exits the tank through a second opening.

    Figure \(\PageIndex{13}\): Water flows through a tank containing a resistance-heating element.

    This design is supposed to supply 3 gallons of water per minute on a steady basis. It is also designed to operate at 220-volt ac power. The entering water temperature is \(60^{\circ} \mathrm{F}\) and the leaving water temperature is \(140^{\circ} \mathrm{F}\). Water can be modeled as an incompressible substance. Changes in kinetic and potential energy are negligible.

    (a) Determine the mass flow rate of the water, in \(\mathrm{lbm} / \mathrm{s}\).

    (b) Determine the electric power requirements to operate the water heater as designed, in \(\mathrm{kW}\).

    (c) Determine the resistance of the resistance-heating element, in ohms.

    Problem \(7.25\)

    An electric motor is used to power a mixer in a chemical process plant. The electric motor operates at \(1800 \mathrm{~rpm}\) (revolutions per minute). To reduce the speed, the motor shaft is connected to a speed reducer followed by a \(90^{\circ}\) drive. The following information is known about all of the components:

    • Electric Motor: Adiabatic, steady-state operation; \(1800 \mathrm{~rpm}\) shaft speed.
    • Speed Reducer: \(15: 1\) speed reduction; steady-state operation; heat transfer rate from the system is \(10 \%\) of the power supplied by the input shaft.
    • \(90^{\circ}\)-Drive: Adiabatic, steady-state operation; \(6: 5\) speed reduction.
    • Mixer Shaft: Required torque is \(500 \mathrm{~ft} \cdot \mathrm{lbf}\).

    In addition, the tank contains \(1000 \mathrm{~ft}^{3}\) of a liquid with a density of \(70 \mathrm{~lbm} / \mathrm{ft}^{3}\). The specific internal energy of the liquid in the tank can be calculated using the equation \(u=c \cdot T\) where temperature is in degrees Rankine \(\left({ }^{\circ} \mathrm{R}\right)\) and \(c=1.5 \mathrm{Btu} / left(\mathrm{lbm} \cdot { }^{\circ} \mathrm{R}\right)\).

    An electric motor turns a horizontal shaft, which runs through a speed reducer and then a 90-degree drive that redirects its rotation to a vertical shaft. The end of the vertical shaft mixes a tank of liquid.

    Figure \(\PageIndex{14}\): Mixer powered by electric motor equipped with speed reducer.

    (a) Determine the rotational speed of the speed reducer output shaft and the mixer shaft, in \(\mathrm{rpm}\).

    (b) Determine the shaft power required to turn the mixer shaft, in \(\mathrm{ft} \cdot \mathrm{lbf} / \mathrm{s}\) and \(\mathrm{hp}\).

    (c) Determine the power supplied by the input shaft to the \(90^{\circ}\)-drive, in \(\mathrm{ft} \cdot \mathrm{lbf} / \mathrm{s}\) and \(\mathrm{hp}\).

    (d) Determine the heat transfer rate from the speed reducer and the power supplied to the speed reducer by the motor shaft, in \(\mathrm{ft} \cdot \mathrm{lbf} / \mathrm{s}\) and \(\mathrm{hp}\).

    (e) Determine the electric power that must be supplied to the motor, in \(\mathrm{ft} \cdot \mathrm{lbf}\), \(\mathrm{hp}\), and \(\mathrm{kW}\).

    (f) Determine the torque, in \(\mathrm{ft} \cdot \mathrm{lbf}\), supplied by the motor.

    (g) If the mixer runs for one hour (60 minutes) and the tank is essentially adiabatic, determine the increase in temperature of the liquid in the tank. (You may neglect changes in kinetic and potential energy.)

    Problem \(7.26\)

    Two streams of air are mixed in a steady-flow, mixing process. Stream one enters at a temperature of \(36^{\circ} \mathrm{C}\) and a mass flow rate of \(5 \mathrm{~kg} / \mathrm{min}\). Stream two enters at \(10^{\circ} \mathrm{C}\) and \(15 \mathrm{~kg} / \mathrm{min}\). The entire mixing process occurs in a heavily insulated sheet-metal mixing tee and occurs at a pressure of \(110 \mathrm{~kPa}\). An electric resistance heater element is built into the tee, so that the outlet temperature can be controlled. Assume that changes in kinetic and gravitational potential energy are negligible and that air can be modeled as an ideal gas with room temperature specific heats.

    Two separate streams of air enter a mixing tee, which has a resistance-heater element built into it. A single mixed air stream exits the tee.

    Figure \(\PageIndex{15}\): Streams of air are mixed in a mixing tee that contains a heating element.

    (a) If the electric resistance heater is turned off, determine the temperature of the air leaving the mixing tee, in \({ }^{\circ} \mathrm{C}\), and the volumetric flow rate of the air leaving the mixing tee, in \(\mathrm{m}^{3} / \mathrm{s}\).

    (b) If the temperature of the low temperature stream drops from \(10^{\circ} \mathrm{C}\) to \(3^{\circ} \mathrm{C}\), how much power must be supplied by the electric heater to maintain the same outlet temperature as you found in Part (a)?

    Problem \(7.27\)

    A centrifugal pump is driven by an electric motor as shown in the figure. Water flows steadily through the pump with the inlet and outlet conditions shown in the figure. The 440-ac-volt electric motor receives \(42 \mathrm{~kW}\) of electrical power and delivers \(40 \mathrm{~kW}\) of shaft power to the pump under steady-state conditions. The motor rotates at \(1750 \mathrm{~rpm}\) and has a power factor of unity. Assume that water can be modeled as an incompressible substance with constant specific heats, and assume changes in gravitational potential energy are negligible.

    An electric motor powers a centrifugal pump. At the pump's inlet pipe, water enters at a temperature of 20 degrees C, pressure of 100 kPa, area of 180 square centimeters, and velocity of 5 m/s. Water exits the pump outlet pipe at a temperature of 20 degrees C, pressure of 500 kPa, and area 125 square centimeters.

    Figure \(\PageIndex{16}\): An electric motor drives a centrifugal water pump.

    (a) Determine the direction and magnitude of the heat transfer rate for the pump, in kilowatts.

    (b) Determine the torque transmitted by the motor shaft to the pump, in \(\mathrm{N} \cdot \mathrm{m}\).

    (c) Determine the electric current supplied to the motor, in amps.

    Problem \(7.28\)

    A piston-cylinder device as shown in the figure contains helium gas. Initially, the gas has a pressure of \(70 \mathrm{~psia}\), a temperature of \(600^{\circ} \mathrm{R}\), and a volume of \(7 \mathrm{~ft}^{3}\). During a process where \(P V=C\), a constant, the helium is expanded to a final volume of \(28 \mathrm{~ft}^{3}\). Assume that helium gas can be modeled as an ideal gas with constant specific heats and assume that changes in kinetic and potential energy are negligible.

    Determine the direction and the magnitude of the work and the heat transfer for the helium gas, in \(\mathrm{ft} \cdot \mathrm{lbf}\).

    A piston-cylinder device contains helium gas.

    Figure \(\PageIndex{17}\): A piston-cylinder device contains helium gas.

    Problem \(7.29\)

    A piston-cylinder device contains carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) gas initially occupying the volume \(V_{1}\) at the pressure \(P_{1}\) and temperature \(T_{1}\) indicated below. The gas undergoes the process described below: \[\begin{array}{l} & \text { State } 1: P_{1}=150 \mathrm{~kPa} ; \quad T_{1}=400 \mathrm{~K} ; \quad V_{1}=0.5 \mathrm{~m}^{3} \\ & \text { Process 1-2: Quasistatic process where } P=\left(300 \mathrm{~kPa} / \mathrm{m}^{3}\right) V \\ &\text { State } 2: V_{2}=1.0 \mathrm{~m}^{3} \end{array} \nonumber \] Assume that carbon dioxide can be modeled as an ideal gas with constant specific heats and that changes in kinetic and gravitational potential energy are negligible for the process.

    Determine the work and heat transfer of energy for process \(1 \text{-} 2\), in \(\mathrm{kJ}\). Be sure to also clearly indicate the direction of each energy transfer.

    Problem \(7.30\)

    An energy-recovery system reclaims energy from hot oil using a liquid-air heat exchanger to heat the air. Light oil flows through the heat exchanger at a flow rate of \(100 \mathrm{~lbm} / \mathrm{min}\). The oil enters at \(80 \mathrm{~psia}\) and \(200^{\circ} \mathrm{F}\), and leaves at \(70 \mathrm{~psia}\) and \(100^{\circ} \mathrm{F}\). The air enters the heat exchanger at \(14.8 \mathrm{~psia}\) and \(80 \mathrm{~F}\) and a volumetric flow rate of \(17,550 \mathrm{~ft}^{3} / \mathrm{min}\). The exit pressure of the air is \(14.6 \mathrm{~psia}\). Assume changes in kinetic and gravitational potential energy are negligible. Also assume that air can be modeled as an ideal gas with constant specific heats and liquid water can be modeled as an incompressible substance with constant specific heats. The density of light oil is \(62.4 \mathrm{~lbm} / \mathrm{ft}^{3}\).

    Oil and air flow through a heat exchanger in separated streams.

    Figure \(\PageIndex{18}\): A heat exchanger for hot oil and cool air.

    (a) Determine the outlet temperature of the air, in \({ }^{\circ} \mathrm{F}\).

    (b) Determine the inlet flow area for the air, if the inlet velocity is \(50 \mathrm{~ft} / \mathrm{s}\).

    (c) Determine the inlet flow area for the oil, if the inlet velocity is \(10 \mathrm{~ft} / \mathrm{s}\)

    Problem \(7.31\)

    A rigid storage tank for water in a home has a volume of \(0.40 \mathrm{~m}^{3}\). The tank initially contains \(0.30 \mathrm{~m}^{3}\) of water at \(20^{\circ} \mathrm{C}\) and \(240 \mathrm{~kPa}\). The space above the water contains air at the same temperature and pressure as the water. An additional \(0.05 \mathrm{~m}^{3}\) of water is slowly pumped into the tank so that the temperature of the air remains constant during the entire filling process. Assume that air behaves as an ideal gas with constant specific heats and changes in the kinetic and potential energies of the gas trapped above the water are negligible.

    A tall tank has water pumped into it from the bottom, with a pocket of gas remaining between the water surface and the tank top.

    Figure \(\PageIndex{19}\): Water is pumped into a tall tank from the bottom.

    (a) Determine the final pressure of the air in the tank, in \(\mathrm{kPa}\).

    (b) Determine the work and heat transfer for the gas during this isothermal compression process. Report both the direction and the magnitude of these energy transfers in kilojoules.

    Problem \(7.32\)

    Liquid water enters a steady-state pump at \(1.0 \mathrm{~bar}\) and \(20^{\circ} \mathrm{C}\) with a velocity of \(2.6 \mathrm{~m} / \mathrm{s}\) through an opening of \(22.0 \mathrm{~cm}^{2}\). The water leaves the pump at \(6.0 \mathrm{~bars}\) and \(7.8 \mathrm{~m} / \mathrm{s}\). The elevation of the pump exit is \(0.5\) meters above the elevation of the pump inlet. Assume water can be modeled as an incompressible substance with constant specific heats and a density of \(1000 \mathrm{~kg} / \mathrm{m}^{3}\).

    Water enters a pump through a single inlet, and is raised in elevation and pressure before exiting the pump through a single outlet.

    Figure \(\PageIndex{20}\): Water is raised in pressure and elevation by a steady-state pump.

    (a) Determine the ratio of the diameter of the outlet pipe to the diameter of the inlet pipe, e.g. \(D_{2} / D_{1}\).

    (b) Assuming that the pump operates adiabatically and the temperature of the water does not change, i.e. \(T_{1}=T_{2}\), determine the shaft power required to run the pump in kilowatts and horsepower.

    (c) Repeat part (b), only this time assume that the water temperature increases by \(0.10^{\circ} \mathrm{C}\). How does this change the shaft power input to the pump?

    Problem \(7.33\)

    Air flows through a simple nozzle as shown in the figure. A nozzle is a rigid hollow tube whose sides are contoured as shown in the figure. If operating correctly, the velocity of the fluid leaving the device is greater than the velocity of the fluid entering the device. The walls of the nozzle are rigid and impermeable, except for the inlet and outlet areas. Typically, heat transfer from the nozzle is negligible.

    Air enters the steady-state nozzle at State 1 at \(200 \mathrm{~kPa}\) and \(300 \mathrm{~K}\) and a velocity of \(48 \mathrm{~m} / \mathrm{s}\), and leaves the nozzle at State 2 with a pressure of \(100 \mathrm{~kPa}\) and the temperature of \(246 \mathrm{~K}\). Changes in gravitational potential energy are negligible. Assume that air acts like an ideal gas with room-temperature specific heats.

    Air enters a nozzle through a small inlet opening, and exits the nozzle through a large outlet opening.

    Figure \(\PageIndex{21}\): Air flows through a steady-state nozzle.

    (a) Determine the velocity of the air leaving the nozzle, in \(\mathrm{m} / \mathrm{s}\). Use the system shown in the figure.

    (b) Determine the ratio of the exit area to the inlet area, \(A_{2} / A_{1}\).

    (c) How would your analysis change if the system was enlarged so that it included the sidewall of the nozzle? An explanation is all that is required. No numbers are required.

    Problem \(7.34\)

    Liquid water flows steadily through an adiabatic nozzle. The water enters the nozzle with negligible velocity at \(600 \mathrm{~kPa}\) and \(100^{\circ} \mathrm{C}\) and leaves the nozzle at a pressure of \(400 \mathrm{~kPa}\). Assume that liquid water can be modeled as an incompressible substance with room-temperature specific heats.

    (a) Determine the exit velocity of the liquid water, in \(\mathrm{m} / \mathrm{s}\), it the exit temperature of the liquid water is \(100^{\circ} \mathrm{C}\).

    (b) Repeat Part (a) only this time assume the outlet temperature is \(100.01^{\circ} \mathrm{C}\).

    Problem \(7.35\)

    A \(10,000 \Omega\) resistor is connected across the terminals of a 12-volt car battery. Heat transfer occurs from the surface of the resistor by natural convection heat transfer according to the relationship \[\dot{Q}_{\text {out}}=h_{\text {conv }} A_{\text {surface}}\left(T_{\text {surface}}-T_{\text {amb}}\right) \nonumber \] where

    \(h_{\text {conv}}=5 \mathrm{~W} /\left(\mathrm{m}^{2} \cdot \mathrm{K}\right)\), the convection heat transfer coefficient.

    \(A_{\text{surface}} = 1.8 \mathrm{~cm}^{2}\), the surface area of the resistor.

    \(T_{\text {surface}}=\) the surface temperature of the resistor, in degrees kelvin.

    \(T_{\text{amb}}=\) the temperature of the ambient air surrounding the resistor, say \(300 \mathrm{~K}\).

    (a) Sketch the system showing the resistor and the battery.

    (b) Calculate the steady-state current through the resistor, in amps, and the electric power supplied to the resistor, in watts.

    (c) If the battery is connected for 30 minutes and heat transfer from the battery is negligible, determine the change in the internal energy of the battery.

    (d) Determine the surface temperature of the resistor assuming steady-state conditions, in \(\mathrm{K}\).

    (e) If a second and identical resistor was placed in parallel across the terminals of the battery and new steady-state conditions established, would the surface temperature of the resistors increase, decrease, or remain unchanged? Explain and support your answer.

    Problem \(7.36\)

    An ac transformer is supplied with electric power at 230 watts with an input voltage of 220 volts ac and a power factor of unity. The transformer output is \(1.9 \mathrm{~A}\) at \(110 \mathrm{~V}\) ac with a power factor of unity. The heat transfer surface area for the transformer can be modeled as a \(10 \mathrm{~cm} \times 10 \mathrm{~cm} \times 10 \mathrm{~cm}\) cube. The convection heat transfer coefficient for the transformer is \(\mathrm{h}_{\text {conv}}=6 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\).

    (a) Determine the input ac current to the transformer, in amps.

    (b) Determine the rate of heat transfer from the transformer at steady-state operation conditions, in \(\mathrm{W}\).

    (c) Determine the steady-state surface temperature of the transformer if the ambient air temperature is \(25^{\circ} \mathrm{C}\).

    Problem \(7.37\)

    A simple piston-cylinder device contains helium. The closed system formed by the helium in the device executes a four-process cycle with negligible changes in kinetic and potential energy. Assume helium can be modeled as an ideal gas with room-temperature specific heats. The available process and state information is detailed in the table below:

    \(T\) \(P\) \(V\) \(Q_{\text{in}}\) \((\mathrm{kJ})\) \(W_{\text {in}}\) \((\mathrm{kJ})\) \(\Delta U\) \((\mathrm{kJ})\)
    State 1 \(400 \mathrm{~K}\) \(500 \mathrm{~kPa}\) \(0.5 \mathrm{~m}^{3}\)      
    Process 1-2 Constant Pressure Compression      
    State 2 \(160 \mathrm{~K}\) \(500 \mathrm{kPa}\) \(0.2 \mathrm{~m}^{3}\)      
    Process 2-3 Expansion with \(p V=\) constant      
    State 3 \(160 \mathrm{~K}\) \(100 \mathrm{kPa}\) \(1.0 \mathrm{~m}^{3}\)      
    Process 3-4 Constant Volume Heating      
    State 4 252 K \(158 \mathrm{kPa}\) \(1.0 \mathrm{~m}^{3}\)      
    Process 4-1 Adiabatic Compression        

    (a) Calculate the work and the heat transfer for each process in the cycle. Clearly show your work

    (b) Based on your analysis in Part (a), is the cycle a power cycle (heat engine) or a refrigerator? Clearly indicate your reasoning.

    (c) Based on your answer to Part (b), calculate the appropriate measure of performance, e.g. power cycle thermal efficiency or the refrigerator coefficient of performance.

    Problem \(7.38\)

    A gas is contained in a simple piston-cylinder device. Changes in kinetic and potential energy are negligible for this closed system. The following information is known about the four processes that make up the cycle:

    Process Description \(Q_{\mathrm{in}} / \mathrm{m}\)
    \((\mathrm{kJ} / \mathrm{kg})\)
    \(W_{\mathrm{in}} / \mathrm{m}\)
    \((\mathrm{kJ} / \mathrm{kg})\)
    \(\Delta u\)
    \((\mathrm{kJ} / \mathrm{kg})\)
    \(1 \cdots 2\) Adiabatic compression \(0) \(458.73\)  
    \(2 \cdots 3\) Isobaric heating \(+1038.12\)   \(811.20\)
    \(3 \cdots 4\) Adiabatic expansion \(0\)   \(-819.65\)
    \(4 \cdots 1\) Constant volume cooling \(-450.23\) \(0\)  
    Total \(\cdots \cdots\)      

    (a) Complete the table by providing numerical values for the unknown heat transfers and work transfers.

    (b) Is this a power cycle or a refrigerator? Explain how you made this decision

    (c) Based upon your answer for Part (b), calculate the appropriate measure of performance: thermal efficiency for a power cycle or coefficient of performance for a refrigerator.

    Problem \(7.39\)

    A piston-cylinder assembly contains a gas that undergoes a series of processes that make up a cycle. Assume that changes in kinetic and gravitational potential energy are negligible. The following state and process information is known about the cycle:

    State/Process \(P\) \((\mathrm{kPa})\) \(V\) \((\mathrm{m}^{3})\) \(T\) \(({ }^{\circ} \mathrm{C})\) \(U\) \((\mathrm{kJ})\)
    \(1\) \(95\) \(0.00570\) \(20\) \(1.47\)
    \(1 \rightarrow 2\) Adiabatic Compression
    \(2\) \(2390\) \(0.00057\) \(465\) \(3.67\)
    \(2 \rightarrow 3\) Constant Pressure
    \(3\) \(2390\) \(0.00171\) \(1940\) \(11.02\)
    \(3 \rightarrow 4\) Adiabatic Expansion
    \(4\) \(445\) \(0.00570\) \(1095\) \(6.79\)
    \(4 \rightarrow 1\) Constant Volume

    (a) Determine the heat transfer and work for each process of the cycle.

    (b) Is this a power cycle or a refrigerator? Explain how you made your decision!

    (c) Calculate the appropriate Measure of Performance (MOP) for this cycle based on Part (b).

    Problem \(7.40\)

    A heat pump cycle delivers energy by heat transfer to a dwelling at a rate of \(60,000 \mathrm{~Btu} / \mathrm{h}\). The power input to the cycle is \(7.8 \mathrm{~hp}\).

    (a) Determine the coefficient of performance for the cycle.

    (b) If the heat pumps runs for 12 hours a day, how much electrical energy is required to run the heat pump for one day?

    (c) If electricity costs \(\$ 0.08\) per kilowatt-hour, how much does it cost per month to run the heat pump?

    (d) If all of the heat transfer to the house is supplied by an electric-resistance furnace, the furnace would require an electrical power input of \(60,000 \mathrm{~Btu} / \mathrm{~h}\). How much would it cost per month to heat the house with an electric resistance furnace?

    (e) Which system would you recommend to a home buyer based on your operating cost information - heat pump or electric furnace?

    Problem \(7.41\)

    A power cycle generates electricity and has a thermal efficiency of \(33 \%\). The electricity from the power cycle is used to run a refrigeration cycle which has a COP of \(4\). The refrigeration cycle receives \(4,000 \mathrm{~Btu} / \mathrm{h}\) of energy by heat transfer at a low temperature. [Hint: It may help you to sketch both systems showing all pertinent heat transfer and work transfers of energy.]

    (a) Determine the electrical power required to run the refrigeration cycle, in \(\mathrm{Btu} / \mathrm{s}\) and \(\mathrm{hp}\).

    (b) Determine the total heat transfer rate of energy into the power cycle if all of its power output is going to drive the refrigeration cycle, in \(\mathrm{Btu} / \mathrm{s}\) and \(\mathrm{hp}\).

    Problem \(7.42\)

    In an iron ore mixing operation, a bucket full of ore is suspended from a traveling crane which moves along a stationary bridge. The bucket is to swing no more than \(4 \mathrm{~m}\) horizontally when the crane is brought to a sudden stop. Determine the maximum allowable speed \(v\) of the crane.

    A traveling crane moves to the right at speed v, along a stationary horizontal bridge. A bucket is attached to a pivot at the bottom of the crane by a 10-meter-long cable.

    Figure \(\PageIndex{22}\): A bucket hangs from a traveling crane via a 10-meter cable.

    Problem \(7.43\)

    Packages are thrown down an incline at \(A\) with a velocity of \(4 \mathrm{~ft} / \mathrm{s}\). The packages slide along the surface \(ABC\) to a conveyor belt which moves with a velocity of \(\8 mathrm{~ft} / \mathrm{s}.\) Knowing that \(\mu_{\mathrm{k}}=0.25\) between the packages and the surface \(\\) determine the distance \(d\) if the packages are to arrive at \(C\) with a velocity of 8 \(\mathrm{ft} / \mathrm{s}\).

    Point A is the top of a 30-degree incline. At the bottom of the ramp, point B, a horizontal surface 20 feet long connects B to point C. A package slides from point A down a distance d along the ramp, then from B to C, where it slides onto a horizontal conveyor belt moving away from point A.

    Figure \(\PageIndex{23}\): A package slides down an incline whose base connects to a horizontal platform.

    Problem \(7.44\)

    An elastic cable is to be designed for bungee jumping from a tower \(130 \mathrm{~ft}\) high. The specifications call for the cable to be \(85 \mathrm{~ft}\) long when unstretched, and to stretch to a total length of \(100 \mathrm{~ft}\) when a \(600 \text{-lb}\) weight is attached to it and dropped from the tower. Determine:

    (a) the required spring constant \(\mathrm{k}\) of the cable,

    (b) how close to the ground a \(185 \text{-lb}\) man will come if he uses this cable to jump from the tower, and

    (c) the maximum acceleration experienced by the man.

    A bungee cable dangles from the top of a 130-meter-tall tower.

    Figure \(\PageIndex{24}\): A bungee cable is attached to the top of a \(130 \text{-ft}\) tower.

    Problem \(7.45\)

    The \(4 \mathrm{~lbf}\) object is dropped 5 feet onto the \(20 \mathrm{~lbf}\) block that is initially at rest on two springs, each with a stiffness \(\mathrm{k}=5 \mathrm{~lb} / \mathrm{in}\). Calculate the maximum deflection of the springs assuming the two objects stick together after the impact.

    Two vertical springs stand on a flat horizontal surface, holding up a 20-lbf object. A 4-lbf object is dropped onto this object, from a distance of 5 feet above.

    Figure \(\PageIndex{25}\): One object is dropped onto another supported by springs.

    Problem \(7.46\)

    The system is at rest in the position shown, with the \(10 \mathrm{~kg}\) collar \(A\) resting on the spring \((\mathrm{k}=500 \mathrm{~N} / \mathrm{m})\), when a constant \(0.5 \mathrm{~kN}\) force is applied to the cable. What is the velocity of the collar when it has risen \(0.2 \mathrm{~m}\) ? Assume there is no friction between the vertical shaft and \(A\).

    A collar A slides on a vertical shaft,, with a vertical spring connected to the bottom support of the shaft. At rest, A rests on top of the spring, and a cable connected to the right side of A runs over a pulley located 0.4 meters to the right of and 0.5 meters above A with a tension of 0.5 kN applied to it.

    Figure \(\PageIndex{26}\): Sliding collar with an applied tension rests on top of a spring.

    Problem \(7.47\)

    Two kg of air (assume ideal gas) in a piston-cylinder device undergoes a thermodynamic cycle consisting of three processes, each with negligible kinetic and gravitational potential energy.

    Process:

    \(1 \rightarrow 2\) Constant Volume

    \(2 \rightarrow 3\) : Compression with \(P=-\left(250 \mathrm{~kPa} / \mathrm{m}^{3}\right) V + 550 \mathrm{~kPa}\)

    \(3 \rightarrow 1\) : Adiabatic Expansion

    A graph of pressure in kPa vs volume in cubic meters. State 1 is at V=1.0 and P=52.5, state 2 is at V=1.0 and P=300, and state 3 is at V=0.2 and P=500. States 1 and 2 are related by a constant-volume pressure change, the process to get from state 2 to 3 is given by P=-250V+550, and the process to get from state 3 to 1 is adiabatic expansion.

    Figure \(\PageIndex{27}\): Pressure-volume relationships between three states of an ideal gas.

    a) Complete the table given (Show work for full credit)

    b) Is it a power cycle or a refrigeration cycle? Be sure to explain your answer.

    Process \(Q_{\text{in}}\) \((\mathrm{kJ})\) \(W_{\text{in}}\) \((\mathrm{kJ})\) \(U_{\text{final}} - U_{\text{initial}}\) \((\mathrm{kJ})\)
    \(1->2\)   \(0\)  
    \(2->3\)      
    \(3->1\)      
    NET      
    Problem \(7.48\)

    Block \(A\) with mass \(m_{A}=10 \mathrm{~kg}\) slides to the right on a horizontal frictionless surface with an initial velocity of \(10 \mathrm{~m} / \mathrm{s}\) until it hits a Bumper \(B\) with mass \(m_{B}\). The Bumper \(B\) is attached to a linear, massless spring with a spring constant \(k=1000 \mathrm{~N} / \mathrm{m}\). The spring is initially unloaded and uncompressed. The bumper is designed so that the spring can only be compressed. Consider the distance required to stop Block \(A\) for two different cases.

    Block A rests on a horizontal surface, some distance to the right of block B. B also rests on the horizontal surface, and it is connected to the left end of a spring whose right end is connected to a support.

    Figure \(\PageIndex{28}\): Two blocks on a horizontal surface, one connected to a support via spring.

    (a) Case I: Determine how far Block \(A\) travels after it impacts the bumper if the impact is purely plastic (perfectly inelastic), i.e. Block \(A\) and Bumper \(B\) stick together after they contact, and the bumper is massless, i.e. \(m_{B}=0 \mathrm{~kg}\).

    (b) Case II: Repeat Part (a), only this time assume that the bumper has mass \(m_{B}=5 \mathrm{~kg}\). For this part assume that the motion of the Bumper \(B\) is negligible, i.e. spring force is negligible, during the impact between Block \(A\) and the Bumper \(B\).

    Problem \(7.49\)

    A light rod with a fixed collar of mass \(m=10 \mathrm{~kg}\) is initially at rest in the inclined position shown in the figure. It is then rotated counter-clockwise about the pivot \(B\) from rest by the constant force \(\mathbf{P}\) until it is brought to rest in the horizontal position by the linear spring which has a spring constant of \(k=40 \mathrm{~kN} / \mathrm{m}\). The spring is compressed from its free (uncompressed) length by \(50 \mathrm{~mm}\) when the rod comes to rest. Assume the mass of the rod is negligible.

    A 2-meter-long rod has its left endpoint B fixed to a pivot; it is currently at a 30-degree angle below the horizontal. A collar is fixed on the rod, 0.8 meters from point B. A constant upwards force P is applied to the current position of the rod's right endpoint, C_1, to pivot the rod horizontal so its right endpoint rests on position C_2. A vertical spring is located 1.2 meters to the right of point B, with its tip in its uncompressed state extending 50 mm below the horizontal.

    Figure \(\PageIndex{29}\): A rod with an attached mass is pivoted by application of a constant force.

    (a) Determine the work done by the force \(\mathbf{P}\) to rotate the rod from \(C_{1}\) to \(C_{2}\), in joules.

    (b) Determine the magnitude of force \(\mathbf{P}\), in newtons.

    Problem \(7.50\)

    A centrifugal pump is driven by an electric motor as shown in the figure. Water flows steadily through the pump with the inlet and outlet conditions shown in the figure. The 440-ac-volt electric motor receives \(42 \mathrm{~kW}\) of electrical power and delivers \(40 \mathrm{~kW}\) of shaft power to the pump under steady-state conditions. The motor rotates at \(1750 \mathrm{~rpm}\) and has a power factor of unity. Assume that water can be modeled as an incompressible substance with constant specific heats, and assume changes in gravitational potential energy are negligible.

    An electric motor drives a centrifugal pump. Water enters the pump at 20 degrees C and 100 kPa, through an opening of area 180 square cm. Water exits the pump at 20 degrees C and 500 kPa, through an opening of area 125 square cm.

    Figure \(\PageIndex{30}\): An electric motor powers a centrifugal pump for water.

    (a) Determine the direction and magnitude of the heat transfer rate for the pump, in kilowatts.

    (b) Determine the torque transmitted by the motor shaft to the pump, in \(\mathrm{N}-\mathrm{m}\).

    (c) Determine the electric current supplied to the motor, in amps.

    Problem \(7.51\)

    A piston-cylinder device contains helium gas. Initially, the gas has a pressure of \(70 \mathrm{~psia}\), a temperature of \(600^{\circ} \mathrm{R}\), and a volume of \(7 \mathrm{~ft}^{3}\). During a process where \(P V=C\), a constant, the helium is expanded to a final volume of \(28 \mathrm{~ft}^{3}\). Assume that helium gas can be modeled as an ideal gas with constant specific heats and assume that changes in kinetic and potential energy are negligible.

    Determine the direction and the magnitude of the work and the heat transfer for the helium gas, in \(\mathrm{ft} \cdot \mathrm{lbf}\).

    Problem \(7.52\)

    Air is contained inside of a piston-cylinder device that also contains an electric resistance heating element (see the figure). The cylinder walls and piston are heavily insulated giving an adiabatic boundary. The air expands from State 1 to State 2 in a constant pressure (isobaric) process.

    During the expansion process electrical energy is supplied to the resistance heating element. For purposes of this analysis, you may assume that the heating element has negligible mass. Assume that air can be modeled as an ideal gas with room temperature specific heats. Also assume changes in kinetic and gravitational potential energy are negligible.

    A piston-cylinder device filled with air contains an electric resistance heating element of negligible mass. In state 1, the air is at pressure 150 kPa, temperature 300 K, and volume 1 cubic meter. In state 2, the air is at pressure 150 kPa and volume 3 cubic meters.

    Figure \(\PageIndex{31}\): State information of air in a piston-cylinder device containing a resistance heating element.

    (a) Determine the temperature of the gas in State 2.

    (b) Determine the direction and magnitude of the transfer of energy by electric work for the process, in \(\mathrm{kJ}\).

    Problem \(7.53\)

    The Collar \(A\) is released from rest at the position shown in the figure and slides up the fixed rod under the action of a constant force \(P\) applied to the cable. The rod is inclined at \(30^{\circ}\) from the horizontal as shown in the figure, and the position of the small pulley \(B\) is fixed. When the collar has traveled 40 inches along the rod to position \(D\), the spring is compressed 6 inches, the cable makes a \(90^{\circ}\) angle with the rod (see dashed line), and the collar is still moving with an unknown velocity.

    The mass of the collar is \(30 \mathrm{~lbm}\) and the constant force \(P=50 \mathrm{~lbf}\). The spring has a stiffness \(\mathrm{k}=200 \mathrm{~lbf} / \mathrm{ft}\). Assume that friction between the collar and rod is negligible.

    Determine the speed of the collar, in \(\mathrm{ft} / \mathrm{s}\), when the collar reaches Point \(D\), i.e. the cable is coincident with the dashed line in the figure.

    A rod is fixed in a diagonal position, 30 degrees above the horizontal. Collar A, 12 inches tall, currently rests at the bottom of the rod. A spring is attached to the upper end of the rod, and ends at point D, 40 inches from the current midpoint of A. A pulley B is located perpendicular to D, 30 inches from D. A dashed line connects points B and D. A cable attached to the midpoint of A runs over the pulley and is pulled with a tension of P.

    Figure \(\PageIndex{32}\): System consisting of a rod, spring, and collar with attached cable.

    Problem \(7.54\)

    A hydroelectric turbine-generator produces an electric power output of \(20 \mathrm{~MW}\) (megawatts). Water enters the turbine penstock at Point 1 and exits the turbine at Point 2 as shown in the figure. The known information at the inlet and exit are shown in the figure. The turbine-generator operates adiabatically at steady-state conditions. Do not neglect kinetic or gravitational potential energy unless you can substantiate your assumption.

    Assume water can be modeled as an incompressible substance with room-temperature specific heats.

    A dam containing a turbine penstock leads into a turbine-generator. Water enters the penstock at opening 1, which has an area of 20 square meters, with an elevation above the riverbed of 20 m, pressure of 1100 kPa, and temperature of 25 degrees C. Water exits the turbine-generator at opening 2, with an area of 20 square meters, at elevation 5 meters, pressure 100 kPa, and temperature 25 degrees C.

    Figure \(\PageIndex{33}\): Water powers a turbine-generator attached to a dam.

    (a) Determine the mass flow rate of the water through the turbine-generator, in \(\mathrm{kg} / \mathrm{s}\).

    (b) If a shaft inside the turbine-generator system transmits \(22 \mathrm{~MW}\) of power at a rotational speed of \(100 \mathrm{~rpm}\), determine the torque in the shaft.

    Problem \(7.55\)

    An ideal gas is contained in a simple piston-cylinder device and executes the three-step process shown in the table.

    State \(1\) \(P_{1} = 100 \mathrm{~kPa}; \ V\kern-1.0em\raise0.3ex-_{1} = 1.00 \mathrm{~m}^3; \ T_{1}=300 \mathrm{~K}\)
    Process \(1 \rightarrow 2\) Constant-pressure (isobaric) expansion
    State \(2\) \(V\kern-1.0em\raise0.3ex-_{2} = 2.00 \mathrm{~m}^3\)
    Process \(2 \rightarrow 3\) Constant-temperature (isothermal) expansion where \(P V\kern-1.0em\raise0.3ex- = C\).
    State \(3\) \(V\kern-1.0em\raise0.3ex-_{3} = 3.00 \mathrm{~m}^3\)
    Process \(3 \rightarrow 4\) Constant-pressure (isobaric) compression
    State \(4\) \(V\kern-1.0em\raise0.3ex-_{4} = V\kern-1.0em\raise0.3ex-_{1} = 1.00 \mathrm{~m}^3\)
    (a) Determine the temperature of the gas at state \(2\).

    (b) Sketch the process on a \(P-V\) diagram. Clearly label the four states, \(1\), \(2\), \(3\), and \(4\) and the connecting processes

    (c) Using your sketch from part (b) above, identify the area on the diagram that represents the work done during process \(2 \rightarrow 3\) by shading or cross-hatching the area.

    Problem \(7.56\)

    A 455 cubic inch Pontiac engine (A) is connected to a TH400 automatic transmission (B) which sends power out to the rear wheels (C). A transmission cooler heat exchanger (D) is used to remove heat generated by frictional losses within the transmission. A liquid coolant circulates through a closed loop to transfer the energy from the transmission to the cooler heat exchanger. Steady-state performance data is measured in the lab and the results are shown in the table:

    Power Measurements  
    Engine output shaft @ \(3800 \mathrm{~rpm}\) \(390.0 \mathrm{~hp}\)
    Transmission output shaft \(350.4 \mathrm{~hp}\)
    Coolant Temperatures  
    Cooler inlet / transmission outlet \(300^{\circ} \mathrm{F}\)
    Cooler outlet / transmission inlet \(237^{\circ} \mathrm{F}\)

    An engine A turns a shaft connected to a transmission B. The transmission turns an output shaft connected to the rear wheels (C), and a coolant flows in a closed loop between B and a transmission cooler heat exchanger (D).

    Figure \(\PageIndex{34}\): Engine is connected to a transmission and trasnmission coolant system.

    If needed, you may assume the transmission coolant fluid can be modeled as an incompressible substance with a density of \(56.8 \mathrm{~lb}_{\mathrm{m}} / \mathrm{ft}^{3}\) and a specific heat \(0.4286 \mathrm{~Btu} /\left(\mathrm{lb}_{\mathrm{m}} \cdot { }^{\circ} \mathrm{F}\right)\).

    a) Determine the torque produced by the motor at the engine output shaft, in \(\mathrm{ft} \cdot \mathrm{lb}_{\mathrm{f}}\).

    b) Determine the heat transfer rate out of the transmission cooler heat exchanger, in \(\mathrm{Btu} / \mathrm{s}\). Assume that there is negligible heat transfer from the surfaces of the transmission casing and the coolant lines.

    c) Determine the mass flow rate of coolant through the coolant lines, in \(\mathrm{lb}_{\mathrm{m}} / \mathrm{s}\). Assume that pressure drops inside the coolant loop circuit are negligible, i.e. pressure inside the coolant loop is uniform.

    d) Experience has shown that the heat transfer from the transmission casing may not be negligible. To check this out, estimate the heat transfer rate from the surface of the transmission by convection. Assume that the surface area of the casing is \(8.68 \mathrm{~ft}^{2}\), the surface temperature is \(80^{\circ} \mathrm{F}\), the temperature of the surrounding air is \(70^{\circ} \mathrm{F}\), and the convective heat transfer coefficient is \(0.0144 \mathrm{~Btu} /\left(\mathrm{ft}^{2} \cdot \mathrm{s} \cdot { }^{\circ} \mathrm{F}\right)\).

    Problem \(7.57\)

    In a belated move to surpass the U.S. space program, chipmunks have decided to place a chipmunk in space. The planned launch vehicle for the chipnaut is a potato slingshot as shown in the figure.

    For the launch, the chipnaut first climbs out on a limb and takes a position immediately above the potato slingshot. Then his launch crew pulls back the slingshot into the firing position as shown. Once the slingshot is fired, the chipnaut awaits the potato "booster". When the potato booster arrives, it picks up (impacts) the chipnaut without touching the tree launch platform and carries the chipnaut upward to the heavens.

    Your job is to predict how the flight will proceed. You may assume that all motion is in the vertical direction. Additional information is provided below:

    Mass of the potato \(=1.0 \mathrm{~kg}\) Elastic Band: Spring Constant \(k=500 \mathrm{~N} / \mathrm{m}\)
    Mass of the chipnaut \(=2.0 \mathrm{~kg}\) \(\quad\) Total length (unstretched) = 1.0 m
      \(\quad\) Total length installed (with initial stretch) = 1.3 m
      \(\quad\) Total length stretched for firing = 2.0 m

    A chipmunk wearing a parachute sits on a branch 3.5 meters above the ground. An elastic band is stretched across a frame on the ground, poised to fire a potato booster directly upwards to impact the chipnaut.

    Figure \(\PageIndex{35}\): Chipnaut and booster prior to launch.

    a) Determine the velocity, in \(\mathrm{m} /\mathrm{s}\), of the potato "booster" immediately before it picks up the chipnaut.

    b) Determine the maximum elevation, in meters, that can be achieved by the "booster" carrying the chipnaut. (Please note that some chipnauts have been concerned about remaining conscious during the flight. Future test flights will investigate this potential problem.)

    (Please note that some chipnauts have been concerned about remaining conscious during the flight. Future test flights will investigate this potential problem.)

    Problem \(7.58\)

    The ion sputtering process creates a new surface layer of material on an object by bombarding the surface with ions of the desired material. The top half of the device in the figure is the vacuum chamber and the lower half is the piston-cylinder device used to raise or lower the target. The position of the piston is altered by adding energy to the gas using an electric resistance heating element or removing energy by heat transfer using a cold plate in the cylinder wall. The known information is shown below.

    A cylinder consists of a vacuum chamber on top, with a target material lying on top of a piston, with a piston-cylinder device consisting of a gas in the bottom half. A cold plate and an electric resistance heating element are both located in the portion of the chamber containing gas.

    Figure \(\PageIndex{36}\): A cylinder consisting of a vacuum chamber and a cylinder-piston device of gas, aligned vertically.

    Given Information: \[\begin{aligned} m_{\mathrm{P}} &=\text { Mass of the piston } \\ m_{\mathrm{T}} &=\text { Mass of the target } \\ m_{\mathrm{G}} &=\text { Mass of the gas } \\ A_{\mathrm{P}} &=\text { Area of the piston } \\[4pt] T_{1} &=\text { Initial temperature of the gas } \\ T_{2} &=\text { Final temperature of the gas } \\ P_{1} &= \left(m_{T}+m_{P}\right) g / A_{P}, \text { the initial pressure in the gas. } \\ z_{1} &=\text { Initial elevation of the piston } \\[4pt] g &=\text { Acceleration of gravity } \\ c_{\mathrm{p}} \text{ & } c_{v} &=\text { Specific heats of the gas } \\[4pt] W_{\text {elect, in}} &=\text { Electric work into the gas } \\ Q_{\text {cold, out}} &=\text { Heat transfer out of the gas and into the cold plate. } \end{aligned} \nonumber \]

    Select an appropriate system or systems and determine the change in elevation of the target, \(\Delta z=z_{2}-z_{1}\), in terms of some or all of the given information.

    Assume that the piston is frictionless and initially stationary. The change in elevation occurs very slowly with negligible change in kinetic energy of the piston. In addition, the piston, the cylinder wall, and the vacuum chamber wall (all the cross-hatched regions) are made of material that provides an adiabatic boundary and does not change temperature. The gas can be modeled as an ideal gas with room-temperature specific heats.

    Problem \(7.59\)

    The piston cylinder device shown below contains nitrogen. The nitrogen undergoes a volume-change process where \(P V\kern-1.0em\raise0.3ex- = C\). Other information about the process is shown below. Assume nitrogen can be modeled as ideal gas with room-temperature specific heats.

    \[ \text{Nitrogen} \left(\mathrm{N}_2\right): \nonumber \] \[\begin{aligned} c_{v} &= 0.743 \mathrm{~kJ} / (\mathrm{kg} \cdot \mathrm{K}) \\ c_{p} &= 1.04 \mathrm{~kJ} / (\mathrm{kg} \cdot \mathrm{K}) \\ R &= 0.298 \mathrm{~kJ} / (\mathrm{kg} \cdot \mathrm{K}) \end{aligned} \nonumber \]

    A gas in a piston-cylinder device is initially at state 1, with temperature 127 degrees C, pressure 100 kPa, and volume 0.250 cubic meters. After a process during which the quotient of pressure and volume remains constant, the gas arrives at state 2 where the temperature is 303 degrees C and the volume is 0.3 cubic meters.

    Figure \(\PageIndex{37}\): Initial and final states of gas in a piston-cylinder device.

    (a) Determine the mass of nitrogen in the piston-cylinder device, in \(\mathrm{kg}\).

    (b) Determine the work transfer of energy for the gas during this process, in \(\mathrm{kJ}\).

    (c) Determine the heat transfer of energy for the gas during this process, in \(\mathrm{kJ}\).

    (d) Sketch the process on a \(P \text{-} V\kern-1.0em\raise0.3ex-\) (pressure-volume) diagram and show the work for the process.

    Problem \(7.60\)

    A hot-water heating system is shown in the figure below. The circulating pump is located in the basement of the building and the hot-water radiator is located on an upper floor. Under steadystate conditions, the radiator delivers \(3.0 \mathrm{~kW}\) by heat transfer to the surroundings.

    Pertinent operating conditions are shown in the table. Assume that water can be treated as an incompressible substance with room-temperature specific heats. \[\text { [ Liquid Water Properties: } \quad c_{\mathrm{p}}=4.18 \mathrm{~kJ} /(\mathrm{kg} \cdot \mathrm{K}) \text { and } \rho=997 \mathrm{~kg} / \mathrm{m}^{3} \text { ] } \nonumber \]

    Water enters a circulating pump in state 1. The pump moves the water upwards, until it enters a hot-water heater in state 2. It exits the heater in state 3.

    Figure \(\PageIndex{38}\): Water is pumped upwards before being heated.

    Operating Conditions
    State \(T\) \(({ }^{\circ} \mathrm{C})\) \(P\) \((\mathrm{kPa})\) \(z\) \((\mathrm{m})\) \(A\) \((\mathrm{m}^2)\)
    \(1\) \(60\) \(100\) \(10.0\) \(0.0020\)
    \(2\) \(60\) \(125\) \(30.0\) \(0.0020\)
    \(3\) \(40\) \(125\) \(30.0\) \(0.0020\)
    Heat transfer from the pipes and the pump is negligible.    
    The only significant heat transfer occurs from the radiator.    

    (a) Determine the mass flow rate through the radiator, in \(\mathrm{kg} / \mathrm{s}\).

    (b) Determine the shaft power supplied to the pump, in \(\mathrm{kW}\), to move the water up to the radiator.

    (c) Estimate the surface area of the radiator. Assume that convection heat transfer is the primary mechanism, the convection heat transfer coefficient \(h=50 \mathrm{~W} /\left(\mathrm{m}^{2} \cdot { }^{\circ} \mathrm{C}\right)\), the room temperature is \(22^{\circ} \mathrm{C}\) and the average radiator temperature is \(50^{\circ} \mathrm{C}\).

    Problem \(7.61\)

    A common safety device utilized in mountainous areas is a "runaway truck" ramp used to stop a truck without functional brakes. This device consists of a long, upward-sloped ramp covered in gravel and a bumper attached to a spring. (See the figure below.).

    A runaway truck weighing \(45,000 \mathrm{~lb}_{\mathrm{f}}\) pulls onto the ramp at \(100 \mathrm{~ft} / \mathrm{s}\) (just over \(68 \mathrm{~mph}\) ). The ramp is \(250 \mathrm{~ft}\) long. The bumper weighs \(500 \mathrm{~lb}_{\mathrm{f}}\). The spring is initially undeflected. As a last resort, the spring is attached to an immense, immovable concrete barrier.

    A truck located at the bottom of a 20-degree ramp is facing upslope. There is a distance of 250 feet between the truck and a bumper near the top of the ramp; the bumper is mounted on a 25-foot-long spring whose opposite end is attached to an immovable concrete barrier at the very top of the ramp.

    Figure \(\PageIndex{39}\): A truck faces upslope on a long ramp with a spring-mounted bumper at the top.

    (a) Determine the average friction force the ramp exerts on the truck, in newtons, as it climbs the ramp if the truck velocity is \(30 \mathrm{~ft} / \mathrm{s}\) just before it strikes the bumper.

    (b) Determine the value of the spring constant \(k\), in \(\mathrm{lb}_{\mathrm{f}} / \mathrm{ft}\), necessary to bring the truck to rest in \(25 \mathrm{~ft}\) after the truck strikes and sticks to the bumper. Neglect frictional effects.

    Problem \(7.62\)

    Block \(A\) with mass \(m_{\mathrm{A}}\) is released from rest in the position shown. It slides a distance \(L\) down a smooth incline before hitting and sticking to Block \(B\). Block \(B\) is initially at rest and has mass \(m_{\mathrm{B}}\).

    Determine the equations necessary to find the maximum distance the spring deflects, \(d\).

    Assume \(m_{\mathrm{A}}\), \(m_{\mathrm{B}}\), \(k\), \(L\) and \(\theta\) are known.

    Do not solve these equations. Your solution should consist of a list of equations and unknowns.

    A smooth surface forms an incline of angle theta. Block A rests at the top of the ramp, a distance L from block B. Block B is held in place near the bottom of the ramp by a spring of spring constant k, whose opposite end is fastened to a support at the very bottom of the ramp.

    Figure \(\PageIndex{40}\): A ramp with a spring-mounted block at the bottom and a free-sliding block at the top.

    Problem \(7.63\)

    The \(10 \text{-kg}\) collar is attached to two identical springs and slides on the smooth vertical rod as shown in the figure. The spring constant for each spring is \(k=800 \mathrm{~N} / \mathrm{m}\), and the unstretched length of each spring is \(0.3 \mathrm{~m}\). In position \(A\), the collar has a velocity \(V_{1}=2 \mathrm{~m} / \mathrm{s}\) in the direction shown.

    It is desired to modify this device by applying a constant force \(\mathbf{F}\) to the collar as it moves from \(A\) to \(B\) so that the velocity of the collar at position \(B\) will be zero, i.e. \(V_{2}=0\).

    A 10-kg collar in position A slides on a vertical rod. The left and right sides of A each connect to a horizontal spring, 0.4 meters long and with a spring constant of 800 N/m. The free ends of both springs are connected to supports. The collar is moving downwards at 2 m/s, and it is desired to move the collar to position B, 0.3 meters below position A.

    Figure \(\PageIndex{41}\): A sliding collar is attached to two springs.

    (a) Determine the direction and magnitude, in newtons, of the constant force \(\mathbf{F}\) that must be applied to the collar as it moves from \(A\) to \(B\) so that \(V_{2}=0\).

    (b) Will the collar stop moving once it reaches position \(B\) even though \(V_{2}=0\)? Explain the basis for your answer. (Even without a numerical answer, full credit will be given for part (b) if a clear explanation of how you would determine the answer is given.)

    Problem \(7.64\)

    A small steam turbine is connected to an air compressor through a gear reducer as shown in the figure. A gear reducer is a device used to change the shaft rotation speed when two devices must be connected but operate at different speeds.

    • Steam enters the turbine at \(110^{\circ} \mathrm{C}\) with a specific enthalpy \(h_{1}=2691.5 \mathrm{~kJ} / \mathrm{kg}\) and exits the turbine at a pressure of \(100 \mathrm{~kPa}\) and a specific enthalpy \(h_{2}=2675.5 \mathrm{~kJ} / \mathrm{kg}\). The turbine shaft rotates at \(2000 \mathrm{~rpm}\).
    • Air enters the compressor at a mass flow rate of \(70 \mathrm{~kg} / \mathrm{min}\) at \(P_{3}=100 \mathrm{~kPa}\) and \(T_{3}=300 \mathrm{~K}\) and exits the compressor at \(P_{4}=500 \mathrm{~kPa}\) and \(T_{4}=460 \mathrm{~K}\). The compressor shaft rotates at \(600 \mathrm{~rpm}\). A ssume that air can be modeled as an ideal gas with constant specific heats.

    Assume all devices shown in the figure—turbine, compressor and gear reducer—operate adiabatically at steady-state conditions with negligible changes in kinetic and gravitational potential energy.

    Steam in state 1 enters a steam turbine, and exits the turbine in state 2. The turbine turns a shaft at 2000 rpm, which feeds into a gear reducer that changes its speed to 600 rpm. This shaft is connected to an air compressor, which air enters in state 3 and exits in state 4.

    Figure \(\PageIndex{42}\): System consisting of a steam turbine, gear reducer, and air compressor, all sharing a common shaft.

    (a) Determine the mass flow rate of steam into the turbine, in \(\mathrm{kg} / \mathrm{min}\).

    (b) Determine the shaft power required by the air compressor, in \(\mathrm{kW}\).

    (c) Determine the torque, in \(\mathrm{N} \cdot \mathrm{m}\), transmitted by the air-compressor shaft.

    Problem \(7.65\)

    A piston-cylinder device contains carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) gas initially occupying the volume \(V_{1}\) at the pressure \(P_{1}\) and temperature \(T_{1}\) indicated below. The gas undergoes the process described below:

    State \(1\): \(P_{1}=150 \mathrm{~kPa} ; \quad T_{1}=400 \mathrm{~K} ; \quad V_{1}=0.5 \mathrm{~m}^{3}\)
    Process \(1 \rightarrow 2\): Quasistatic process where \(P=\left(300 \mathrm{kPa} / \mathrm{m}^{3}\right) V\)
    State \(2\): \(V_{2}=1.0 \mathrm{~m}^{3}\)

    Assume that carbon dioxide can be modeled as an ideal gas with constant specific heats and that changes in kinetic and gravitational potential energy are negligible for the process.

    Determine the work and heat transfer of energy for process \(1 \rightarrow 2\). Indicate both the direction and the magnitude (in kilowatts) of each.

    Problem \(7.66\)

    The collar \(C\) slides on the curved rod in the vertical plane under the action of a constant force \(F\) in the cord guided by the small pulleys at \(D\). The collar has a mass of \(0.70 \mathrm{~kg}\) and slides without friction.

    If the collar is released from rest at \(A\), determine the value of the constant force \(F\) that will result in the collar striking the stop at \(B\) with a velocity of \(4 \mathrm{~m} / \mathrm{s}\).

    A concave-down curved rod stretches between point A and point B, which is 600 mm to the right of and 400 mm above point A. A collar C slides along this rod, with a cable attached to the collar that passes over a pulley at point D, 200 mm to the right of and 200 mm below point B. A tension of F is applied to this cable.

    Figure \(\PageIndex{43}\): A collar with an attached cable that passes over a pulley slides along a curved rod.

    Problem \(7.67\)

    The system shown at right is the back end of a jet aircraft engine. Operating information about the system is shown in the table and figure. A ir flows steadily through the system. Assume changes in gravitational potential energy are negligible and air can be modeled as an ideal gas with room temperature specific heats.

    State \(T\) \(\left({ }^{\circ} \mathrm{C}\right)\) \(P\) \((\mathrm{kPa})\) \(V\) \((\mathrm{~m} / \mathrm{s})\) \(A_{c}\) \(\left(\mathrm{~m}^{2}\right)\)
    \(1\) 600 800 \(V_{1} \approx V_{2}\) \(\cdots\)
    \(2\) \(? ? ?\) 800 \(V_{2} \approx V_{3}\) \(\cdots\)
    3 1300 600 \(V_{3} << V_{4}\) \(\cdots\)
    4 950 100 \(? ? ?\) \(? ? ?\)

    Turbine: steady-state and adiabatic

    Nozzle: steady-state and adiabatic

    Heat exchanger: steady-state

    370 kg/s of air in state 1 enters a heat exchanger, where the rate of heat transfer into the exchanger is 300,000 kilowatts. Air in state 2 exits the exchanger and enters a turbine, which turns a shaft. Air in state 3 exits the turbine and enters the nozzle. Air in state 4 exits the nozzle.

    Figure \(\PageIndex{44}\): System consisting of a heat exchanger, turbine, and nozzle.

    (a) Determine the velocity of the air leaving the nozzle, in \(\mathrm{m} / \mathrm{s}\).

    (b) Determine the cross-sectional area \(A_{\mathrm{c}}\) at the nozzle outlet, in \(\mathrm{m}^{2}\)

    (c) Determine the shaft power out of the turbine, in \(\mathrm{kW}\).

    Problem \(7.68\)

    A well-insulated copper tank of mass \(13 \mathrm{~kg}\) contains \(4 \mathrm{~kg}\) of liquid water. Initially, the temperature of the copper is \(27^{\circ} \mathrm{C}\) and the temperature of the water is \(50^{\circ} \mathrm{C}\). As the tank and its contents come to equilibrium, an electrical resistor of negligible mass transfers \(100 \mathrm{~kJ}\) of energy to the contents of the tank. Assume copper and liquid water can be modeled as incompressible substances.

    A copper tank is filled with water and covered with insulation. An electrical resistor of negligible mass is located in the water and has leads passing through the tank wall and insulating layer.

    Figure \(\PageIndex{45}\): Insulated tank contains water and an electrical resistor.

    (a) Determine the final temperature of the tank and water.

    (b) If current through the resistor is \(0.5\) amps and the applied voltage is 110 volts, determine (i) the electrical power supplied to the resistor and (ii) how long the resistor was "on" to deliver \(100 \mathrm{~kJ}\) of electrical energy.

    Problem \(7.69\)

    A spring-loaded boot-on-a-stick kicks a marble as shown in the figure. Initially both the boot and marble are stationary. To load the device, the boot is swung up to the position shown and the uncompressed spring on the ceiling is compressed a distance \(d\). The stationary boot is then released, swinging down and to the left before kicking the marble. The mass of the boot and marble are \(m_{b}\) and \(m_{m}\), respectively, and the spring has a stiffness \(k\). The stick of length \(L\) has negligible mass and is hinged to a frictionless pin at \(A\).

    A support attached to the ceiling has the left end of a rod of length L attached to it, at pivot A. The opposite end of the rod is attached to a boot. Initially, the rod is horizontal and the boot is compressing a vertical spring attached to the ceiling by a distance d. It is desired to release the boot, in a downwards gravity field of 1 g, so that it swings down and kicks a marble at rest on the floor.

    Figure \(\PageIndex{46}\): A boot on a pivoting rod is loaded via spring before it swings down to kick a marble.

    (a) Find an expression for the velocity of the boot just before it kicks the marble.

    (b) Assuming the boot and the marble stick together, find an expression for the velocity of the marble immediately after it has been kicked.

    (c) If the spring was initially compressed a distance \(d / 3\) before the device was loaded, i.e. before it was compressed a distance \(d\) as described above, would the velocity found in part (a) increase, decrease or remain the same? Why? [A clear, concise, correct explanation without equations is acceptable.]

    Problem \(7.70\)

    A typical cylinder for a Cummins Model H diesel engine is shown in the figure at right. Details of the compression process are shown below. The piston-cylinder volume contains air. For modeling purposes, you may assume that the air can be modeled as an ideal gas with room-temperature specific heats.

    State \(1\): \(P_{1} = 100 \mathrm{~kPa};\) \(T_{1}=320 \mathrm{~K};\) \(V\kern-1.0em\raise0.3ex-_{1} = 300 \mathrm{~cm}^{3}\)
    Process \(1 \rightarrow 2\): Compression process with \(P V\kern-1.0em\raise0.3ex-^{1.3} = C\)
    State \(2\): \(V\kern-1.0em\raise0.3ex-_{2} = (1/16) \ V\kern-1.0em\raise0.3ex-_{1}\)

    A crankshaft rotates counterclockwise, with a connecting rod attached at one end to a point on the shaft and attached at the other end to a piston. As the shaft rotates, the connecting rod causes the piston to move up and down in a vertical cylinder containing air.

    Figure \(\PageIndex{47}\): A piston connected by a rod to a crankshaft moves up and down in a cylinder of air.

    (a) Determine the final pressure and temperature.

    (b) Determine the heat transfer and work transfer of energy for the air during the compression process.

    (c) Sketch the process on a \(P \text{-} V\kern-1.0em\raise0.3ex-\) diagram. What, if anything, is the significance of the area under the process curve?

    Problem \(7.71\)

    High-pressure hot water is mixed with \(0.20 \mathrm{~ft}^{3} / \mathrm{min}\) of high-pressure cold water in a showerhead as shown in order to produce a comfortable shower temperature of \(110^{\circ} \mathrm{F}\). The mixing process can be modeled as adiabatic with negligible kinetic and potential energies of the fluid streams. Assume liquid water can be modeled as an incompressible substance with room-temperature specific heats.

    Find the required flow rate of hot water in \(\mathrm{ft}^{3} / \mathrm{min}\).

    A mixing tee consists of hot water at 140 degrees F and pressure 4320 lbf/sq. ft entering from one arm and 0.20 cubic feet per minute of cold water at 50 degrees F and pressure 4320 lbf/sq. ft entering from the other arm. The mixed water exits through the showerhead at 110 degrees F and pressure 2117 lbf/sq. ft.

    Figure \(\PageIndex{48}\): Mixing tee of hot and cold water for a showerhead.

    Problem \(7.72\)

    A hydraulic power system operates at steady-state conditions and consists of an electrically driven hydraulic pump connected to a hydraulic motor by a two pipes carrying the hydraulic fluid. (See figure below.) The electric power input to the hydraulic pump is \(9.0 \mathrm{~kW}\).

    For purposes of analysis, assume that changes in potential energy are negligible, the hydraulic fluid lines are well insulated, and the fluid can be modeled as an incompressible substance with the properties of liquid water. \[\text { [ Liquid Water Properties: } c_{\mathrm{p}}=4.18 \mathrm{~kJ} /(\mathrm{kg} \cdot \mathrm{K}) \text { and } \rho=997 \mathrm{~kg} / \mathrm{m}^{3} \text { ] } \nonumber \]

    Fluid in state 1 enters a hydraulic pump, exits it in state 2, enters a hydraulic motor in state 3, and exits the motor in state 4 in a fluid line that connects back to the state-1 hydraulic pump. Pressure 1 = Pressure 4 = 700 kPa, and Pressure 2 = Pressure 3 = 4100 kPa. Temperature throughout is 44 degrees C, and area of the fluid lines is constant throughout. Velocity at state 1 is 5 m/s.

    Figure \(\PageIndex{49}\): Two hydraulic fluid lines connect a hydraulic pump and a hydraulic motor.

    (a) Assuming that the pump operates adiabatically, determine the mass flow rate of fluid through the pump, in \(\mathrm{kg} / \mathrm{s}\).

    (b) Assuming the hydraulic motor loses \(1.0 \mathrm{~kW}\) by heat transfer, determine the shaft power out of the hydraulic motor, in \(\mathrm{kW}\).

    (c) Estimate the convection heat transfer coefficient, \(h_{\text {conv}}\), in \(\mathrm{W} /\left(\mathrm{m}^{2} \cdot \mathrm{K}\right)\) for the motor. The motor heat transfer is \(1.0 \mathrm{~kW}\), the room air temperature is \(24^{\circ} \mathrm{C}\), the motor surface area is \(0.22 \mathrm{~m}^{2}\), and the motor surface temperature is \(44^{\circ} \mathrm{C}\).


    7.10: Problems is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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