A Refrigeration Cycle Operating as Continuously and Remove Energy Determine Net Power
Problem 1
An automobile has a mass of $1200 \mathrm{~kg}$. What is its kinetic energy, in $\mathrm{kJ}$, relative to the road when traveling at a velocity of $50 \mathrm{~km} / \mathrm{h}$ ? If the vehicle accelerates to $100 \mathrm{~km} / \mathrm{h}$, what is the change in kinetic energy, in $\mathrm{kJ}$ ?
Narayan Hari
Numerade Educator
Problem 2
An object whose mass is $400 \mathrm{~kg}$ is located at an elevation of $25 \mathrm{~m}$ above the surface of the earth. For $g=9.78 \mathrm{~m} / \mathrm{s}^{2}$, determine the gravitational potential energy of the object, in $\mathrm{kJ}$, relative to the surface of the earth.
Narayan Hari
Numerade Educator
Problem 3
An object of mass $1000 \mathrm{~kg}$, initially having a velocity of $100 \mathrm{~m} / \mathrm{s}$, decelerates to a final velocity of $20 \mathrm{~m} / \mathrm{s}$. What is the change in kinetic energy of the object, in $\mathrm{kJ}$ ?
Narayan Hari
Numerade Educator
Problem 4
An airplane whose mass is $5000 \mathrm{~kg}$ is flying with a velocity of $150 \mathrm{~m} / \mathrm{s}$ at an altitude of $10,000 \mathrm{~m}$, both measured relative to the surface of the earth. The acceleration of gravity can be taken as constant at $g=9.78 \mathrm{~m} / \mathrm{s}^{2}$.
(a) Calculate the kinetic and potential energies of the airplane, both in $\mathrm{kJ}$.
(b) If the kinetic energy increased by $10,000 \mathrm{~kJ}$ with no change in elevation, what would be the final velocity, in $\mathrm{m} / \mathrm{s}$ ?
Narayan Hari
Numerade Educator
Problem 5
An object whose mass is $0.5 \mathrm{~kg}$ has a velocity of $30 \mathrm{~m} / \mathrm{s}$. Determine
(a) the final velocity, in $\mathrm{m} / \mathrm{s}$, if the kinetic energy of the object decreases by $130 \mathrm{~J}$.
(b) the change in elevation, in $\mathrm{ft}$, associated with a $130 \mathrm{~J}$ change in potential energy. Let $g=9.81 \mathrm{~m} / \mathrm{s}^{2}$.
Narayan Hari
Numerade Educator
Problem 6
An object whose mass is $2 \mathrm{~kg}$ is accelerated from a velocity of $200 \mathrm{~m} / \mathrm{s}$ to a final velocity of $500 \mathrm{~m} / \mathrm{s}$ by the action of a resultant force. Determine the work done by the resultant force, in $\mathrm{kJ}$, if there are no other interactions between the object and its surroundings.
Narayan Hari
Numerade Educator
Problem 7
A disk-shaped flywheel, of uniform density $\rho$, outer radius $R$, and thickness $w$, rotates with an angular velocity $\omega$, in $\mathrm{rad} / \mathrm{s}$.
(a) Show that the moment of inertia, $I=\int_{\text {vol }} \rho r^{2} d V$, can be expressed as $I=\pi \rho w R^{4} / 2$ and the kinetic energy can be expressed as $\mathrm{KE}=I \omega^{2} / 2$. (b) For a steel flywheel rotating at 3000 RPM, determine the kinetic energy, in $\mathrm{N} \cdot \mathrm{m}$, and the mass, in $\mathrm{kg}$, if $R=0.38 \mathrm{~m}$ and $w=0.025 \mathrm{~m}$.
(c) Determine the radius, in $\mathrm{m}$, and the mass, in $\mathrm{kg}$, of an aluminum flywheel having the same width, angular velocity, and kinetic energy as in part (b).
Problem 8
Two objects having different masses fall freely under the influence of gravity from rest and the same initial elevation. Ignoring the effect of air resistance, show that the magnitudes of the velocities of the objects are equal at the moment just before they strike the earth.
Narayan Hari
Numerade Educator
Problem 9
An object whose mass is $25 \mathrm{~kg}$ is projected upward from the surface of the earth with an initial velocity of $60 \mathrm{~m} / \mathrm{s}$. The only force acting on the object is the force of gravity. Plot the velocity of the object versus elevation. Determine the elevation of the object, in $\mathrm{ft}$, when its velocity reaches zero. The acceleration of gravity is $g=9.8 \mathrm{~m} / \mathrm{s}^{2}$.
Anuraj Sunda
Numerade Educator
Problem 10
A block of mass $10 \mathrm{~kg}$ moves along a surface inclined $30^{\circ}$ relative to the horizontal. The center of gravity of the block is elevated by $3.0 \mathrm{~m}$ and the kinetic energy of the block decreases by $50 \mathrm{~J}$. The block is acted upon by a constant force $\mathbf{R}$ parallel to the incline and by the force of gravity. Assume frictionless surfaces and let $g=9.81 \mathrm{~m} / \mathrm{s}^{2}$. Determine the magnitude and direction of the constant force $\mathbf{R}$, in $\mathrm{N}$.
Narayan Hari
Numerade Educator
Problem 11
Beginning from rest, an object of mass $200 \mathrm{~kg}$ slides down a $10-\mathrm{m}$-long ramp. The ramp is inclined at an angle of $40^{\circ}$ from the horizontal. If air resistance and friction between the object and the ramp are negligible, determine the velocity of the object, in $\mathrm{m} / \mathrm{s}$, at the bottom of the ramp. Let $g=$ $9.81 \mathrm{~m} / \mathrm{s}^{2}$
Narayan Hari
Numerade Educator
Problem 12
A system with a mass of $5 \mathrm{~kg}$, initially moving horizontally with a velocity of $40 \mathrm{~m} / \mathrm{s}$, experiences a constant horizontal deceleration of $2 \mathrm{~m} / \mathrm{s}^{2}$ due to the action of a resultant force. As a result, the system comes to rest. Determine the length of time, in s, the force is applied and the amount of energy transfer by work, in $\mathrm{kJ}$.
Narayan Hari
Numerade Educator
Problem 13
The drag force, $F_{\mathrm{d}}$, imposed by the surrounding air on a vehicle moving with velocity $\mathrm{V}$ is given by
$$
F_{\mathrm{d}}=C_{\mathrm{d}} \mathrm{A}_{2}^{\frac{1}{2}} \rho \mathrm{V}^{2}
$$
where $C_{\mathrm{d}}$ is a constant called the drag coefficient, $\mathrm{A}$ is the projected frontal area of the vehicle, and $\rho$ is the air density. Determine the power, in $\mathrm{kW}$, required to overcome aerodynamic drag for a truck moving at $110 \mathrm{~km} / \mathrm{h}$, if $C_{\mathrm{d}}=0.65, \mathrm{~A}=10 \mathrm{~m}^{2}$ and $\rho=1.1 \mathrm{~kg} / \mathrm{m}^{3}$.
Narayan Hari
Numerade Educator
Problem 14
A major force opposing the motion of a vehicle is the rolling resistance of the tires, $F_{r}$, given by
$$
F_{\mathrm{r}}=f^{\circ} W
$$
where $f$ is a constant called the rolling resistance coefficient and $W$ is the vehicle weight. Determine the power, in $\mathrm{kW}$, required to overcome rolling resistance for a truck weighing $322.5 \mathrm{kN}$ that is moving at $110 \mathrm{~km} / \mathrm{h}$. Let $f=0.0069$.
Narayan Hari
Numerade Educator
Problem 15
Measured data for pressure versus volume during the expansion of gases within the cylinder of an internal combustion engine are given in the table below. Using data from the table, complete the following:
(a) Determine a value of $n$ such that the data are fit by an equation of the form, $p V^{m}=$ constant.
(b) Evaluate analytically the work done by the gases, in $\mathrm{kJ}$, using Eq. $2.17$ along with the result of part (a).
(c) Using graphical or numerical integration of the data, evaluate the work done by the gases, in $\mathrm{kJ}$.
(d) Compare the different methods for estimating the work used in parts (b) and (c). Why are they estimates?
Problem 16
One-fourth kg of a gas contained within a piston-cylinder assembly undergoes a constant-pressure process at 5 bar beginning at $v_{1}=0.20 \mathrm{~m}^{3} / \mathrm{kg}$. For the gas as the system, the work is $-15 \mathrm{~kJ}$. Determine the final volume of the gas, in $\mathrm{m}^{3}$.
Narayan Hari
Numerade Educator
Problem 17
A gas is compressed from $V_{1}=0.3 \mathrm{~m}^{3}, p_{1}=1$ bar to $V_{2}=0.1 \mathrm{~m}^{3}, p_{2}=3$ bar. Pressure and volume are related linearly during the process. For the gas, find the work, in $\mathrm{kJ}$.
Narayan Hari
Numerade Educator
Problem 18
A gas expands from an initial state where $p_{1}=500 \mathrm{kPa}$ and $V_{1}=0.1 \mathrm{~m}^{3}$ to a final state where $p_{2}=100 \mathrm{kPa}$. The relationship between pressure and volume during the process is $p V=$ constant. Sketch the process on a $p-V$ diagram and determine the work, in $\mathrm{kJ}$.
Narayan Hari
Numerade Educator
Problem 19
Warm air is contained in a piston-cylinder assembly oriented horizontally as shown in Fig. P2.19. The air cools slowly from an initial volume of $0.003 \mathrm{~m}^{3}$ to a final volume of $0.002 \mathrm{~m}^{3} .$ During the process, the spring exerts a force that
Narayan Hari
Numerade Educator
Problem 20
Air undergoes two processes in series:
Process 1-2: polytropic compression, with $n=1.3$, from $p_{1}=$ $100 \mathrm{kPa}, v_{1}=0.04 \mathrm{~m}^{3} / \mathrm{kg}$ to $v_{2}=0.02 \mathrm{~m}^{3} / \mathrm{kg}$
Process 2-3: constant-pressure process to $v_{3}=v_{1}$
Sketch the processes on a $p-v$ diagram and determine the work per unit mass of air, in $\mathrm{kJ} / \mathrm{kg}$.
Problem 21
For the cycle of Problem $1.25$, determine the work for each process and the net work for the cycle, each in $\mathrm{kJ}$.
Problem 22
The driveshaft of a building's air-handling fan is turned at 300 RPM by a belt running on a $0.3$-m-diameter pulley. The net force applied by the belt on the pulley is $2000 \mathrm{~N}$. Determine the torque applied by the belt on the pulley, in $\mathrm{N} \cdot \mathrm{m}$, and the power transmitted, in $\mathrm{kW}$.
Narayan Hari
Numerade Educator
Problem 23
An electric motor draws a current of 10 amp with a voltage of $110 \mathrm{~V}$. The output shaft develops a torque of $10.2 \mathrm{~N} \cdot \mathrm{m}$ and a rotational speed of 1000 RPM. For operation at steady state, determine
(a) the electric power required by the motor and the power developed by the output shaft, each in $\mathrm{kW}$.
(b) the net power input to the motor, in $\mathrm{kW}$.
(c) the amount of energy transferred to the motor by electrical work and the amount of energy transferred out of the motor by the shaft, in $\mathrm{kW} \cdot \mathrm{h}$ during $2 \mathrm{~h}$ of operation.
Problem 24
A 12 -V automotive storage battery is charged with a constant current of $2 \mathrm{amp}$ for $24 \mathrm{~h}$. If electricity costs $\$ 0.08$ per $\mathrm{kW} \cdot \mathrm{h}$, determine the cost of recharging the battery.
Narayan Hari
Numerade Educator
Problem 25
For your lifestyle, estimate the monthly cost of operating the following household items: microwave oven, refrigerator, electric space heater, personal computer, hand-held hair drier, a $100-\mathrm{W}$ light bulb. Assume the cost of electricity is $\$ 0.08$ per $\mathrm{kW} \cdot \mathrm{h}$.
Narayan Hari
Numerade Educator
Problem 26
A solid cylindrical bar (see Fig. 2.9) of diameter $5 \mathrm{~mm}$ is slowly stretched from an initial length of $10 \mathrm{~cm}$ to a final length of $10.1 \mathrm{~cm}$. The normal stress in the bar varies according to $\sigma=C\left(x-x_{0}\right) / x_{0}$, where $x$ is the length of the bar, $x_{0}$ is the initial length, and $C$ is a material constant (Young's modulus).
For $C=2 \times 10^{7} \mathrm{kPa}$, determine the work done on the bar, in $\mathrm{J}$, assuming the diameter remains constant.
Problem 27
A wire of cross-sectional area $A$ and initial length $x_{0}$ is stretched. The normal stress $\sigma$ acting in the wire varies linearly with strain, $\varepsilon$, where
$$
\varepsilon=\left(x-x_{0}\right) / x_{0}
$$
and $x$ is the length of the wire. Assuming the cross-sectional area remains constant, derive an expression for the work done on the wire as a function of strain.
Narayan Hari
Numerade Educator
Problem 28
A soap film is suspended on a $5 \mathrm{~cm} \times 5 \mathrm{~cm}$ wire frame, as shown in Fig. 2.10. The movable wire is displaced $1 \mathrm{~cm}$ by an applied force, while the surface tension of the soap film remains constant at $25 \times 10^{-5} \mathrm{~N} / \mathrm{cm}$. Determine the work done in stretching the film, in $J$.
$2.29$ Derive an expression to estimate the work required to inflate a common balloon. List all simplifying assumptions.
Problem 29
Derive an expression to estimate the work required to inflate a common balloon. List all simplifying assumptions.
Problem 30
A $0.2-\mathrm{m}$-thick plane wall is constructed of concrete. At steady state, the energy transfer rate by conduction through a $1-\mathrm{m}^{2}$ area of the wall is $0.15 \mathrm{~kW}$. If the temperature distribution is linear through the wall, what is the temperature difference across the wall, in $\mathrm{K}$ ?
Narayan Hari
Numerade Educator
Problem 31
A 2-cm-diameter surface at $1000 \mathrm{~K}$ emits thermal radiation at a rate of $15 \mathrm{~W}$. What is the emissivity of the surface? Assuming constant emissivity, plot the rate of radiant emission, in $\mathrm{W}$, for surface temperatures ranging from 0 to $2000 \mathrm{~K}$. The Stefan-Boltzmann constant, $\sigma$, is $5.67 \times 10^{-8} \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}^{4}$.
Narayan Hari
Numerade Educator
Problem 32
A flat surface having an area of $2 \mathrm{~m}^{2}$ and a temperature of $350 \mathrm{~K}$ is cooled convectively by a gas at $300 \mathrm{~K}$. Using data from Table 2.1, determine the largest and smallest heat transfer rates, in $\mathrm{kW}$, that might be encountered for (a) free convection, (b) forced convection.
Narayan Hari
Numerade Educator
Problem 33
A flat surface is covered with insulation with a thermal conductivity of $0.08 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The temperature at the interface between the surface and the insulation is $300^{\circ} \mathrm{C}$. The outside of the insulation is exposed to air at $30^{\circ} \mathrm{C}$, and the heat transfer coefficient for convection between the insulation and the air is $10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Ignoring radiation, determine the minimum thickness of insulation, in $\mathrm{m}$, such that the outside of the insulation is no hotter than $60^{\circ} \mathrm{C}$ at steady state.
Narayan Hari
Numerade Educator
Problem 34
Each line in the following table gives information about a process of a closed system. Every entry has the same energy units. Fill in the blank spaces in the table.
Problem 35
A closed system of mass $5 \mathrm{~kg}$ undergoes a process in which there is work of magnitude $9 \mathrm{~kJ}$ to the system from the surroundings. The elevation of the system increases by $700 \mathrm{~m}$ during the process. The specific internal energy of the system decreases by $6 \mathrm{~kJ} / \mathrm{kg}$ and there is no change in kinetic energy of the system. The acceleration of gravity is constant at $g=9.6$ $\mathrm{m} / \mathrm{s}^{2}$. Determine the heat transfer, in $\mathrm{kJ}$.
Narayan Hari
Numerade Educator
Problem 36
A closed system of mass $20 \mathrm{~kg}$ undergoes a process in which there is a heat transfer of $1000 \mathrm{~kJ}$ from the system to the surroundings. The work done on the system is $200 \mathrm{~kJ}$. If the initial specific internal energy of the system is $300 \mathrm{~kJ} / \mathrm{kg}$, what is the final specific internal energy, in $\mathrm{kJ} / \mathrm{kg}$ ? Neglect changes in kinetic and potential energy.
Narayan Hari
Numerade Educator
Problem 37
As shown in Fig. P2.37, $5 \mathrm{~kg}$ of steam contained within a piston-cylinder assembly undergoes an expansion from state 1 , where the specific internal energy is $u_{1}=2709.9 \mathrm{~kJ} / \mathrm{kg}$, to state 2 , where $u_{2}=2659.6 \mathrm{~kJ} / \mathrm{kg}$. During the process, there is heat transfer to the steam with a magnitude of $80 \mathrm{~kJ}$. Also, a paddle wheel transfers energy to the steam by work in the amount of $18.5 \mathrm{~kJ}$. There is no significant change in the kinetic or potential energy of the steam. Determine the energy transfer by work from the steam to the piston during - the Qrocet in kJ.
Problem 38
An electric generator coupled to a windmill produces an average electric power output of $15 \mathrm{~kW}$. The power is used to charge a storage battery. Heat transfer from the battery to the surroundings occurs at a constant rate of $1.8 \mathrm{~kW}$. Determine, for $8 \mathrm{~h}$ of operation
(a) the total amount of energy stored in the battery, in $\mathrm{kJ}$.
(b) the value of the stored energy, in $\$$ if electricity is valued at $\$ 0.08$ per $\mathrm{kW} \cdot \mathrm{h}$.
Anuraj Sunda
Numerade Educator
Problem 39
A closed system undergoes a process during which there is energy transfer from the system by heat at a constant rate of $10 \mathrm{~kW}$, and the power varies with time according to
$$
\dot{W}= \begin{cases}-8 t & 0<t \leq 1 \mathrm{~h} \\ -8 & t>1 \mathrm{~h}\end{cases}
$$
where $t$ is time, in $\mathrm{h}$, and $\dot{W}$ is in $\mathrm{kW}$. (a) What is the time rate of change of system energy at $t=$ $0.6 \mathrm{~h}$, in $\mathrm{kW}$ ?
(b) Determine the change in system energy after $2 \mathrm{~h}$, in $\mathrm{kJ}$.
Narayan Hari
Numerade Educator
Problem 40
A storage battery develops a power output of
$$
\dot{W}=1.2 \exp (-t / 60)
$$
where $\dot{W}$ is power, in $\mathrm{kW}$, and $t$ is time, in s. Ignoring heat transfer
(a) plot the power output, in $\mathrm{kW}$, and the change in energy of the battery, in $\mathrm{kJ}$, each as a function of time.
(b) What are the limiting values for the power output and the change in energy of the battery as $t \rightarrow \infty$ ? Discuss.
Problem 41
A gas expands in a piston-cylinder assembly from $p_{1}=8$ bar, $V_{1}=0.02 \mathrm{~m}^{3}$ to $p_{2}=2$ bar in a process during which the relation between pressure and volume is $p V^{1.2}=$ constant. The mass of the gas is $0.25 \mathrm{~kg}$. If the specific internal energy of the gas decreases by $55 \mathrm{~kJ} / \mathrm{kg}$ during the process, determine the heat transfer, in kJ. Kinetic and potential energy effects are negligible.
Narayan Hari
Numerade Educator
Problem 42
Two kilograms of air is contained in a rigid well-insulated tank with a volume of $0.6 \mathrm{~m}^{3}$. The tank is fitted with a paddle wheel that transfers energy to the air at a constant rate of $10 \mathrm{~W}$ for $1 \mathrm{~h}$. If no changes in kinetic or potential energy occur, determine
(a) the specific volume at the final state, in $\mathrm{m}^{3} / \mathrm{kg}$.
(b) the energy transfer by work, in $\mathrm{kJ}$.
(c) the change in specific internal energy of the air, in $\mathrm{kJ} / \mathrm{kg}$.
Narayan Hari
Numerade Educator
Problem 43
A gas is contained in a closed rigid tank. An electric resistor in the tank transfers energy to the gas at a constant rate of $1000 \mathrm{~W}$. Heat transfer between the gas and the surroundings occurs at a rate of $\dot{Q}=-50 t$, where $\dot{Q}$ is in watts, and $t$ is time, in min.
(a) Plot the time rate of change of energy of the gas for $0 \leq t \leq 20 \mathrm{~min}$, in watts.
(b) Determine the net change in energy of the gas after 20 min, in kJ.
(c) If electricity is valued at $\$ 0.08$ per $\mathrm{kW} \cdot \mathrm{h}$, what is the cost of the electrical input to the resistor for $20 \mathrm{~min}$ of operation?
Problem 44
Steam in a piston-cylinder assembly undergoes a polytropic process, with $n=2$, from an initial state where $p_{1}=3.45$ $\mathrm{MPa}, v_{1}=.106 \mathrm{~m}^{3} / \mathrm{kg}, u_{1}=3171 \mathrm{~kJ} / \mathrm{kg}$ to a final state where $u_{2}=2304 \mathrm{~kJ} / \mathrm{kg}$. During the process, there is a heat transfer from the steam of magnitude $361.8$. The mass of steam is $544 \mathrm{~kg}$. Neglecting changes in kinetic and potential energy, determine the work, in $\mathrm{kJ}$.
Problem 45
Air is contained in a vertical piston-cylinder assembly by a piston of mass $50 \mathrm{~kg}$ and having a face area of $0.01 \mathrm{~m}^{2}$. The mass of the air is $5 \mathrm{~g}$, and initially the air occupies a volume of 5 liters. The atmosphere exerts a pressure of 100 $\mathrm{kPa}$ on the top of the piston. The volume of the air slowly decreases to $0.002 \mathrm{~m}^{3}$ as the specific internal energy of the air decreases by $260 \mathrm{~kJ} / \mathrm{kg}$. Neglecting friction between the piston and the cylinder wall, determine the heat transfer to the air, in $\mathrm{kJ}$.
Narayan Hari
Numerade Educator
Problem 46
A gas contained within a piston-cylinder assembly is shown in Fig. P2.46. Initially, the piston face is at $x=0$, and the spring exerts no force on the piston. As a result of heat transfer, the gas expands, raising the piston until it hits the stops. At this point the piston face is located at $x=$ $0.06 \mathrm{~m}$, and the heat transfer ceases. The force exerted by the spring on the piston as the gas expands varies linearly with $x$ according to
$$
F_{\text {spring }}=k x
$$
where $k=9,000 \mathrm{~N} / \mathrm{m}$. Friction between the piston and the cylinder wall can be neglected. The acceleration of gravity is $g=9.81 \mathrm{~m} / \mathrm{s}^{2}$. Additional information is given on Fig. P2.70. (a) What is the initial pressure of the gas, in $\mathrm{kPa}$ ?
(b) Determine the work done by the gas on the piston, in $\mathrm{J}$.
(c) If the specific internal energies of the gas at the initial and final states are 210 and $335 \mathrm{~kJ} / \mathrm{kg}$, respectively, calculate the heat transfer, in $J$.
Analyzing Thermodynamic Cycles
Problem 47
The following table gives data, in $\mathrm{kJ}$, for a system undergoing a thermodynamic cycle consisting of four processes in series. For the cycle, kinetic and potential energy effects can be neglected. Determine
(a) the missing table entries, each in $\mathrm{kJ}$.
(b) whether the cycle is a power cycle or a refrigeration cycle.
Problem 48
A gas undergoes a thermodynamic cycle consisting of three processes:
Process 1-2: compression with $p V=$ constant, from $p_{1}=1$ bar, $V_{1}=1.6 \mathrm{~m}^{3}$ to $V_{2}=0.2 \mathrm{~m}^{3}, U_{2}-U_{1}=0$
Process 2-3: constant pressure to $V_{3}=V_{1}$
Process 3-1: constant volume, $U_{1}-U_{3}=-3549 \mathrm{~kJ}$
There are no significant changes in kinetic or potential energy. Determine the heat transfer and work for Process 2-3, in kJ. Is this a power cycle or a refrigeration cycle?
Problem 49
A gas undergoes a thermodynamic cycle consisting of three processes:
Process 1-2: constant volume, $V=0.028 \mathrm{~m}^{3}, U_{2}-U_{1}=$ $26.4 \mathrm{~kJ}$
Process 2-3: expansion with $p V=$ constant, $U_{3}=U_{2}$
Process 3-1: constant pressure, $p=1.4 \mathrm{bar}, W_{31}=-10.5 \mathrm{~kJ}$
There are no significant changes in kinetic or potential energy.
(a) Sketch the cycle on a $p-V$ diagram.
(b) Calculate the net work for the cycle, in $\mathrm{kJ}$.
(c) Calculate the heat transfer for process 2-3, in kJ.
(d) Calculate the heat transfer for process 3-1, in $\mathrm{kJ}$.
Is this a power cycle or a refrigeration cycle?
Problem 50
For a power cycle operating as in Fig. $2.15 a$, the heat transfers are $Q_{\text {in }}=50 \mathrm{~kJ}$ and $Q_{\text {out }}=35 \mathrm{~kJ}$. Determine the net work, in $\mathrm{kJ}$, and the thermal efficiency.
Problem 51
The thermal efficiency of a power cycle operating as shown in Fig. 2.15a is $35 \%$, and $Q_{\text {out }}=40 \mathrm{MJ}$. Determine the net work developed and the heat transfer $O$.meach in MJ
Problem 52
A power cycle receives energy by heat transfer from the combustion of fuel at a rate of $300 \mathrm{MW}$. The thermal efficiency of the cycle is $33.3 \%$.
(a) Determine the net rate power is developed, in MW.
(b) For 8000 hours of operation annually, determine the net work output, in $\mathrm{kW} \cdot \mathrm{h}$ per year.
(c) Evaluating the net work output at $\$ 0.08$ per $\mathrm{kW} \cdot \mathrm{h}$, determine the value of the net work, in \$/year.
Problem 53
A power cycle has a thermal efficiency of $35 \%$ and generates electricity at a rate of $100 \mathrm{MW}$. The electricity is valued at $\$ 0.08$ per $\mathrm{kW} \cdot \mathrm{h}$. Based on the cost of fuel, the cost to supply $\dot{Q}_{\text {in }}$ is $\$ 4.50$ per GJ. For 8000 hours of operation annually, determine, in \$,
(a) the value of the electricity generated per year.
(b) the annual fuel cost.
Problem 54
For each of the following, what plays the roles of the hot body and the cold body of the appropriate Fig. $2.15$ schematic?
(a) Window air conditioner
(b) Nuclear submarine power plant
(c) Ground-source heat pump
Problem 55
In what ways do automobile engines operate analogously to the power cycle shown in Fig. $2.15 a ?$ How are they different? Discuss.
Problem 56
A refrigeration cycle operating as shown in Fig. $2.15 b$ has heat transfer $Q_{\text {out }}=2530 \mathrm{~kJ}$ and net work of $W_{\text {cycle }}=844 \mathrm{~kJ}$. Determine the coefficient of performance for the cycle.
Problem 57
A refrigeration cycle operates as shown in Fig. $2.15 b$ with a coefficient of performance $\beta=1.5 .$ For the cycle, $Q_{\text {out }}=$ $500 \mathrm{~kJ}$. Determine $Q_{\text {in }}$ and $W_{\text {cycle }}$, each in $\mathrm{kJ}$.
Problem 58
A refrigeration cycle operates continuously and removes energy from the refrigerated space at a rate of $3.5 \mathrm{~kW}$. For a coefficient of performance of $2.6$, determine the net power required.
Problem 59
A heat pump cycle whose coefficient of performance is $2.5$ delivers energy by heat transfer to a dwelling at a rate of $20 \mathrm{~kW}$.
(a) Determine the net power required to operate the heat pump, in $\mathrm{kW}$.
(b) Evaluating electricity at $\$ 0.08$ per $\mathrm{kW} \cdot \mathrm{h}$, determine the cost of electricity in a month when the heat pump operates for 200 hours.
Narayan Hari
Numerade Educator
Problem 60
A household refrigerator with a coefficient of performance of $2.4$ removes energy from the refrigerated space at a rate of $200 \mathrm{~W}$. Evaluating electricity at $\$ 0.08$ per $\mathrm{kW} \cdot \mathrm{h}$, determine the cost of electricity in a month when the refrigerator operates for 360 hours.
Source: https://www.numerade.com/books/chapter/energy-and-the-first-law-of-thermodynamics/?section=60774
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