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Fundamentals of Engineering Thermodynamics SI VERSION

Michael J. Moran, Howard N. Shapiro

Chapter 5

The Second Lav of Thermodynamics - all with Video Answers

Educators

Section 1

Exercises: Things Engineers- Thinks about

01:09

Problem 1

A heat pump receives energy by heat transfer from the outside air at $0^{\circ} \mathrm{C}$ and discharges energy by heat transfer to a dwelling at $20^{\circ} \mathrm{C}$. Is this in violation of the Clausius statement of the second law of thermodynamics? Explain.

Vipender Yadav
Vipender Yadav
Numerade Educator
01:42

Problem 1

Explain how work might be developed when (a) $T_{i}$ is less than $T_{0}$ in Fig. $5.1 \mathrm{a}$, (b) $p_{i}$ is less than $p_{0}$ in Fig. 5.1b.

Manik Pulyani
Manik Pulyani
Numerade Educator
03:00

Problem 2

Air as an ideal gas expands isothermally at $20^{\circ} \mathrm{C}$ from a volume of $1 \mathrm{~m}^{3}$ to $2 \mathrm{~m}^{3} .$ During this process there is heat transfer to the air from the surrounding atmosphere, modeled as a thermal reservoir, and the air does work. Evaluate the work and heat transfer for the process, in $\mathrm{kJ} / \mathrm{kg}$. Is this process in violation of the second law of thermodynamics? Explain.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:36

Problem 2

A system consists of an ice cube in a cup of tap water. The ice cube melts and eventually equilibrium is attained. How might work be developed as the ice and water come to equilibrium?

Manik Pulyani
Manik Pulyani
Numerade Educator
01:44

Problem 3

Complete the demonstration of the equivalence of the Clausius and Kelvin-Planck statements of the second law given in Sec. $5.1$ by showing that a violation of the Kelvin-Planck statement implies a violation of the Clausius statement.

Vipender Yadav
Vipender Yadav
Numerade Educator
01:48

Problem 3

Describe a process that would satisfy the conservation of energy principle, but does not actually occur in nature.

Ma Ednelyn Lim
Ma Ednelyn Lim
Numerade Educator
06:28

Problem 4

An inventor claims to have developed a device that undergoes a thermodynamic cycle while communicating thermally with two reservoirs. The system receives energy $Q_{C}$ from the cold reservoir and discharges energy $Q_{\mathrm{H}}$ to the hot reservoir while delivering a net amount of work to its surroundings. There are no other energy transfers between the device and its surroundings. Using the second law of thermodynamics, evaluate the inventor's claim.

Shital Rijal
Shital Rijal
Numerade Educator
01:47

Problem 4

Referring to Fig. 2.3, identify internal irreversibilities associated with system A. Repeat for system B.

Manik Pulyani
Manik Pulyani
Numerade Educator
03:48

Problem 5

A hot thermal reservoir is separated from a cold thermal reservoir by a cylindrical rod insulated on its lateral surface. Energy transfer by conduction between the two reservoirs takes place through the rod, which remains at steady state. Using the Kelvin-Planck statement of the second law, demonstrate that such a process is irreversible.

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
01:34

Problem 5

What are some of the principal irreversibilities present during operation of (a) an automobile engine, (b) a household refrigerator, (c) a gas-fired water heater, (d) an electric water heater?

Manik Pulyani
Manik Pulyani
Numerade Educator
00:35

Problem 6

Methane gas within a piston-cylinder assembly is compressed in a quasiequilibrium process. Is this process internally reversible? Is this process reversible?

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:51

Problem 6

For the gearbox of Example $2.4$, list the principal irreversibilities and classify them as internal or external.

Manik Pulyani
Manik Pulyani
Numerade Educator
06:04

Problem 7

Water within a piston-cylinder assembly cools isothermally at $120^{\circ} \mathrm{C}$ from saturated vapor to saturated liquid while interacting thermally with its surroundings at $20^{\circ} \mathrm{C}$. Is the process internally reversible? Is it reversible? Discuss.

Uma Kumari
Uma Kumari
Numerade Educator
01:20

Problem 7

Steam at a given state enters a turbine operating at steady state and expands adiabatically to a specified lower pressure. Would you expect the power output to be greater in an internally reversible expansion or an actual expansion?

Manik Pulyani
Manik Pulyani
Numerade Educator
02:05

Problem 8

Complete the discussion of the Kelvin-Planck statement of the second law in Sec. 5.3.1 by showing that if a system undergoes a thermodynamic cycle reversibly while communicating thermally with a single reservoir, the equality in Eq. $5.1$ applies.

Vipender Yadav
Vipender Yadav
Numerade Educator
00:58

Problem 8

Air at a given state enters a compressor operating at steady state and is compressed adiabatically to a specified higher pressure.
Would you expect the power input to the compressor to be greater in an internally reversible compression or an actual compression?

Manik Pulyani
Manik Pulyani
Numerade Educator
01:58

Problem 9

If a window air conditioner were placed on a table in a room and operated, would the room temperature increase, decrease, or remain the same?

Manik Pulyani
Manik Pulyani
Numerade Educator
01:53

Problem 9

A power cycle $I$ and a reversible power cycle $R$ operate between the same two reservoirs, as shown in Fig. 5.6. Cycle I has a thermal efficiency equal to two-thirds of that for cycle $R$. Using the Kelvin-Planck statement of the second law, prove that cycle I must be irreversible.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:57

Problem 10

To increase the thermal efficiency of a reversible power cycle operating between thermal reservoirs at temperatures $T_{\mathrm{H}}$ and $T_{C}$, would it be better to increase $T_{\mathrm{H}}$ or decrease $T_{\mathrm{C}}$ by equal amounts?

Manik Pulyani
Manik Pulyani
Numerade Educator
01:53

Problem 10

A reversible power cycle $\mathrm{R}$ and an irreversible power cycle I operate between the same two reservoirs.
(a) If each cycle receives the same amount of energy $Q_{\mathrm{H}}$ from the hot reservoir, show that cycle I necessarily discharges more energy $Q_{c}$ to the cold reservoir than cycle R. Discuss the implications of this for actual power cycles.
(b) If each cycle develops the same net work, show that cycle I necessarily receives more energy $Q_{\mathrm{H}}$ from the hot reservoir than cycle R. Discuss the implications of this for actual power cycles.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:20

Problem 11

Electric power plants typically reject energy by heat transfer to a body of water or the atmosphere. Would it be advisable to reject heat instead to large blocks of ice maintained by a refrigeration system?

Manik Pulyani
Manik Pulyani
Numerade Educator
01:26

Problem 11

Provide the details left to the reader in the demonstration of the second Carnot corollary given in Sec. 5.3.2.

Michelle Pribek
Michelle Pribek
Numerade Educator
02:01

Problem 12

Referring to Eqs. $5.9$ and 5.10, how might the coefficients of performance of refrigeration cycles and heat pumps be increased?

Manik Pulyani
Manik Pulyani
Numerade Educator
07:04

Problem 12

Using the Kelvin-Planck statement of the second law of thermodynamics, demonstrate the following corollaries:
(a) The coefficient of performance of an irreversible refrigeration cycle is always less than the coefficient of performance of a reversible refrigeration cycle when both exchange energy by heat transfer with the same two reservoirs.
(b) All reversible refrigeration cycles operating between the same two reservoirs have the same coefficient of performance.
(c) The coefficient of performance of an irreversible heat pump cycle is always less than the coefficient of performance of a reversible heat pump cycle when both exchange energy by heat transfer with the same two reservoirs.
(d) All reversible heat pump cycles operating between the same two reservoirs have the same coefficient of performance.

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
Numerade Educator
01:31

Problem 13

Is it possible for the coefficient of performance of a refrigeration cycle to be less than one? To be greater than one? Answer the same questions for a heat pump cycle.

Ma Ednelyn Lim
Ma Ednelyn Lim
Numerade Educator
05:02

Problem 13

Before introducing the temperature scale now known as the Kelvin scale, Kelvin suggested a logarithmic scale in which the function $\psi$ of Eq. $5.5$ takes the form
$$
\psi=\exp \theta_{\mathrm{c}} / \exp \theta_{\mathrm{H}}
$$
where $\theta_{\mathrm{H}}$ and $\theta_{\mathrm{C}}$ denote, respectively, the temperatures of the hot and cold reservoirs on this scale.
(a) Show that the relation between the Kelvin temperature $T$ and the temperature $\theta$ on the logarithmic scale is
$$
\theta=\ln T+\mathrm{C}
$$
where $C$ is a constant.
(b) On the Kelvin scale, temperatures vary from 0 to $+\infty$. Determine the range of temperature values on the logarithmic scale.
(c) Obtain an expression for the thermal efficiency of any system undergoing a reversible power cycle while operating between reservoirs at temperatures $\theta_{\mathrm{H}}$ and $\theta_{\mathrm{C}}$ on the logarithmic scale.

Morgan Cheatham
Morgan Cheatham
Numerade Educator
04:33

Problem 14

Demonstrate that the gas temperature scale (Sec. 1.6) is identical to the Kelvin temperature scale.

Dading Chen
Dading Chen
Numerade Educator
01:57

Problem 15

To increase the thermal efficiency of a reversible power cycle operating between reservoirs at $T_{\mathrm{H}}$ and $T_{\mathrm{C}}$, would you increase $T_{\mathrm{H}}$ while keeping $T_{\mathrm{C}}$ constant, or decrease $T_{\mathrm{C}}$ while keeping $T_{\mathrm{H}}$ constant? Are there any natural limits on the increase in thermal efficiency that might be achieved by such means?

Manik Pulyani
Manik Pulyani
Numerade Educator
01:33

Problem 16

Two reversible power cycles are arranged in series. The first cycle receives energy by heat transfer from a reservoir at temperature $T_{\mathrm{H}}$ and rejects energy to a reservoir at an intermediate temperature $T$. The second cycle receives the energy rejected by the first cycle from the reservoir at temperature $T$ and rejects energy to a reservoir at temperature $T_{\mathrm{C}}$ lower than $T$. Derive an expression for the intermediate temperature $T$ in terms of $T_{\mathrm{H}}$ and $T_{\mathrm{C}}$ when
(a) the net work of the two power cycles is equal.
(b) the thermal efficiencies of the two power cycles are equal.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:20

Problem 17

If the thermal efficiency of a reversible power cycle operating between two reservoirs is denoted by $\eta_{\max }$, develop an expression in terms of $\eta_{\max }$ for the coefficient of performance of
(a) a reversible refrigeration cycle operating between the same two reservoirs.
(b) a reversible heat pump operating between the same two reservoirs.

Manik Pulyani
Manik Pulyani
Numerade Educator
03:11

Problem 18

The data listed below are claimed for a power cycle operating between reservoirs at $527^{\circ} \mathrm{C}$ and $27^{\circ} \mathrm{C}$. For each case, determine if any principles of thermodynamics would be violated.
(a) $Q_{\mathrm{H}}=700 \mathrm{~kJ}, W_{\text {cycle }}=400 \mathrm{~kJ}, Q_{\mathrm{C}}=300 \mathrm{~kJ}$
(b) $Q_{\mathrm{H}}=640 \mathrm{~kJ}, W_{\text {cyck }}=400 \mathrm{~kJ}, Q_{\mathrm{C}}=240 \mathrm{~kJ}$
(c) $Q_{\mathrm{H}}=640 \mathrm{~kJ}, W_{\text {cycle }}=400 \mathrm{~kJ}, Q_{\mathrm{C}}=200 \mathrm{~kJ}$

Shahab Ullah
Shahab Ullah
Numerade Educator
01:37

Problem 19

A refrigeration cycle operating between two reservoirs receives energy $Q_{c}$ from a cold reservoir at $T_{C}=280 \mathrm{~K}$ and rejects energy $Q_{H}$ to a hot reservoir at $T_{\mathrm{H}}=320 \mathrm{~K}$. For each of the following cases determine whether the cycle operates reversibly, irreversibly, or is impossible:
(a) $Q_{\mathrm{C}}=1500 \mathrm{~kJ}, W_{\text {cycle }}=150 \mathrm{~kJ}$
(b) $Q_{C}=1400 \mathrm{~kJ}, Q_{\mathrm{H}}=1600 \mathrm{~kJ}$
(c) $Q_{\mathrm{H}}=1600 \mathrm{~kJ}, W_{\text {cycle }}=400 \mathrm{~kJ}$
(d) $\beta=5$

Manik Pulyani
Manik Pulyani
Numerade Educator
01:20

Problem 20

A reversible power cycle receives $Q_{\mathrm{H}}$ from a hot reservoir at temperature $T_{\mathrm{H}}$ and rejects energy by heat transfer to the surroundings at temperature $T_{0} .$ The work developed by the power cycle is used to drive a refrigeration cycle that removes $Q_{C}$ from a cold reservoir at temperature $T_{C}$ and discharges energy by heat transfer to the same surroundings at $T_{0}$ -
(a) Develop an expression for the ratio $Q_{\mathrm{C}} / Q_{\mathrm{H}}$ in terms of the temperature ratios $T_{\mathrm{H}} / T_{0}$ and $T_{\mathrm{C}} / T_{0-}$
(b) Plot $Q_{\mathrm{c}} / Q_{\mathrm{H}}$ versus $T_{\mathrm{H}} / T_{0}$ for $T_{\mathrm{C}} / T_{0}=0.85,0.9$, and $0.95$, and versus $T_{\mathrm{c}} / T_{0}$ for $T_{\mathrm{H}} / T_{0}=2,3$, and 4.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:20

Problem 21

A reversible power cycle receives energy $Q_{\mathrm{H}}$ from a reservoir at temperature $T_{\mathrm{H}}$ and rejects $Q_{\mathrm{c}}$ to a reservoir at temperature $T_{\mathrm{C}}$. The work developed by the power cycle is used to drive a reversible heat pump that removes energy $Q_{\mathrm{C}}^{\prime}$ from a reservoir at temperature $T_{\mathrm{C}}^{\prime}$ and rejects energy $Q_{\mathrm{H}}^{\prime}$ to a reservoir at temperature $T_{\mathrm{H}}^{\prime}$
(a) Develop an expression for the ratio $Q_{H}^{t} / Q_{H}$ in terms of the temperatures of the four reservoirs.
(b) What must be the relationship of the temperatures $T_{\mathrm{H}}, T_{\mathrm{C}}$, $T_{\mathrm{C}}^{\prime}$ and $T_{\mathrm{H}}^{\prime}$ for $Q_{H}^{\prime} / Q_{\mathrm{H}}$ to exceed a value of unity?
(c) Letting $T_{\mathrm{H}}^{\prime}=T_{\mathrm{C}}=T_{0}$, plot $Q_{\mathrm{H}}^{\prime} / Q_{\mathrm{H}}$ versus $T_{\mathrm{H}} / T_{0}$ for $T_{\mathrm{C}}^{\prime} / T_{0}=0.85,0.9$, and $0.95$, and versus $T_{\mathrm{C}}^{\prime} / T_{0}$ for $T_{\mathrm{H}} / T_{0}$ $=2,3$, and 4.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:35

Problem 22

Figure P5.22 shows a system consisting of a power cycle driving a heat pump. At steady state, the power cycle receives $\dot{Q}_{s}$ by heat transfer at $T_{\mathrm{s}}$ from the high-temperature source and delivers $\dot{Q}_{1}$ to a dwelling at $T_{d}$. The heat pump receives $\dot{Q}_{0}$ from the outdoors at $T_{0}$, and delivers $\dot{Q}_{2}$ to the dwelling. (a) Obtain an expression for the maximum theoretical value of the performance parameter $\left(\dot{Q}_{1}+\dot{Q}_{2}\right) / \dot{Q}_{s}$ in terms of the temperature ratios $T_{\mathrm{s}} / T_{\mathrm{d}}$ and $T_{0} / T_{\mathrm{d}}$.
(b) Plot the result of part (a) versus $T_{\mathrm{s}} / T_{\mathrm{d}}$ ranging from 2 to 4 for $T_{0} / T_{d}=0.85,0.9$, and $0.95$

Manik Pulyani
Manik Pulyani
Numerade Educator
01:27

Problem 23

A power cycle operates between a reservoir at temperature $T$ and a lower-temperature reservoir at $280 \mathrm{~K}$. At steady state, the cycle develops $40 \mathrm{~kW}$ of power while rejecting 1000 $\mathrm{kJ} / \mathrm{min}$ of energy by heat transfer to the cold reservoir. Determine the minimum theoretical value for $T$, in $\mathrm{K}$.

Manik Pulyani
Manik Pulyani
Numerade Educator
05:01

Problem 24

A certain reversible power cycle has the same thermal efficiency for hot and cold reservoirs at 1000 and $500 \mathrm{~K}$, respectively, as for hot and cold reservoirs at temperature $T$ and $1000 \mathrm{~K}$. Determine $T$, in $\mathrm{K}$.

Km Neeraj
Km Neeraj
Numerade Educator
04:03

Problem 25

A reversible power cycle whose thermal efficiency is $50 \%$, operates between a reservoir at $1800 \mathrm{~K}$ and a reservoir at a lower temperature $T$. Determine $T$, in $\mathrm{K}$.

Shahab Ullah
Shahab Ullah
Numerade Educator
01:40

Problem 26

An inventor claims to have developed a device that executes a power cycle while operating between reservoirs at 800 and $350 \mathrm{~K}$ that has a thermal efficiency of (a) $56 \%$, (b) $40 \%$. Evaluate the claim for each case.

Dheeraj Sharma
Dheeraj Sharma
Numerade Educator
01:27

Problem 27

At steady state, a cycle develops a power output of $10 \mathrm{~kW}$ for heat addition at a rate of $10 \mathrm{~kJ}$ per cycle of operation from a source at $1500 \mathrm{~K}$. Energy is rejected by heat transfer to cooling water at $300 \mathrm{~K}$. Determine the minimum theoretical number of cycles required per minute.

Manik Pulyani
Manik Pulyani
Numerade Educator
09:56

Problem 28

At steady state, a power cycle having a thermal efficiency of $38 \%$ generates $100 \mathrm{MW}$ of electricity while discharging energy by heat transfer to cooling water at an average temperature of $70^{\circ} \mathrm{F}$. The average temperature of the steam passing through the boiler is $900^{\circ} \mathrm{F}$. Determine
(a) the rate at which energy is discharged to the cooling water, in Btu/h.
(b) the minimum theoretical rate at which energy could be discharged to the cooling water, in $\mathrm{Btw} / \mathrm{h}$. Compare with the actual rate and discuss.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:21

Problem 29

Ocean temperature energy conversion (OTEC) power plants generate power by utilizing the naturally occurring decrease with depth of the temperature of ocean water. Near Florida, the ocean surface temperature is $27^{\circ} \mathrm{C}$, while at a depth of $700 \mathrm{~m}$ the temperature is $7^{\circ} \mathrm{C}$.
(a) Determine the maximum thermal efficiency for any power cycle operating between these temperatures.
(b) The thermal efficiency of existing OTEC plants is approximately 2 percent. Compare this with the result of part (a) and comment.

Timothy James
Timothy James
Numerade Educator
04:00

Problem 30

During January, at a location in Alaska winds at $-30^{\circ} \mathrm{C}$ can be observed. Several meters below ground the temperature remains at $13^{\circ} \mathrm{C}$, however. An inventor claims to have devised a power cycle exploiting this situation that has a thermal efficiency of $10 \%$. Discuss this claim.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
04:53

Problem 31

Figure P5.31 shows a system for collecting solar radiation and utilizing it for the production of electricity by a power cycle. The solar collector receives solar radiation at the rate of $0.315 \mathrm{~kW}$ per $\mathrm{m}^{2}$ of area and provides energy to a storage unit whose temperature remains constant at $220^{\circ} \mathrm{C}$. The power cycle receives energy by heat transfer from the storage unit, generates electricity at the rate $0.5 \mathrm{MW}$, and discharges energy by heat transfer to the surroundings at $20^{\circ} \mathrm{C}$. For operation at steady state, (a) determine the minimum theoretical collector area required, in $\mathrm{m}^{2}$
(b) determine the collector area required, in $\mathrm{m}^{2}$, as a function of the thermal efficiency $\eta$ and the collector efficiency, defined as the fraction of the incident energy that is stored. Plot the collector area versus $\eta$ for collector efficiencies equal to $1.0,0.75$, and $0.5$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
19:12

Problem 32

The preliminary design of a space station calls for a power cycle that at steady state receives energy by heat transfer at $T_{\mathrm{H}}=600 \mathrm{~K}$ from a nuclear source and rejects energy to space by thermal radiation according to Eq. 2.33. For the radiative surface, the temperature is $T_{C}$, the emissivity is $0.6$, and the surface receives no radiation from any source. The thermal efficiency of the power cycle is one-half that of a reversible power cycle operating between reservoirs at $T_{\mathrm{H}}$ and $T_{\mathrm{C}}$.
(a) For $T_{C}=400 \mathrm{~K}$, determine $\dot{W}_{\text {cycle }} / \mathrm{A}$, the net power developed per unit of radiator surface area, in $\mathrm{kW} / \mathrm{m}^{2}$, and the thermal efficiency.
(b) Plot $\dot{W}_{\text {cycle }} / \mathrm{A}$ and the thermal efficiency versus $T_{\mathrm{C}}$, and determine the maximum value of $\dot{W}_{\text {cycle }} / \mathrm{A}$.
(c) Determine the range of temperatures $T_{C}$ in $\mathrm{K}$, for which $\dot{W}_{\text {cycle }} / \mathrm{A}$ is within 2 percent of the maximum value obtained in part (b).
The Stefan-Boltzmann constant is $5.67 \times 10^{-8} \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}^{4}$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
04:00

Problem 33

An inventor claims to have developed a refrigeration cycle that requires a net power input of $1.2 \mathrm{~kW}$ to remove $25,000 \mathrm{~kJ} / \mathrm{h}$ of energy by heat transfer from a reservoir at $-30^{\circ} \mathrm{C}$ and discharge energy by heat transfer to a reservoir at $20^{\circ} \mathrm{C}$. There are no other energy transfers with the surroundings. Evaluate this claim.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:50

Problem 34

Determine if a tray of ice cubes could remain frozen when placed in a food freezer having a coefficient of performance of 9 operating in a room where the temperature is $32^{\circ} \mathrm{C}\left(90^{\circ} \mathrm{F}\right)$.

Ashar Tanveer
Ashar Tanveer
Numerade Educator
05:03

Problem 35

The refrigerator shown in Fig. P5.35 operates at steady state with a coefficient of performance of $4.5$ and a power input of $0.8 \mathrm{~kW}$. Energy is rejected from the refrigerator to the surroundings at $20^{\circ} \mathrm{C}$ by heat transfer from metal coils whose average surface temperature is $28^{\circ} \mathrm{C}$. Determine
(a) the rate energy is rejected, in $\mathrm{kW}$.
(b) the lowest theoretical temperature inside the refrigerator, in $\mathrm{K}$.
(c) the maximum theoretical power, in $\mathrm{kW}$, that could be developed by a power cycle operating between the coils and the surroundings. Would you recommend making use of this opportunity for developing power?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:26

Problem 36

Determine the minimum theoretical power, in $\mathrm{W}$, required at steady state by a refrigeration system to maintain a cryogenic sample at $-126^{\circ} \mathrm{C}$ in a laboratory at $21^{\circ} \mathrm{C}$, if energy leaks by heat transfer to the sample from its surroundings at a rate of $900 \mathrm{~W}$.

Abhishek Kumar
Abhishek Kumar
Numerade Educator
01:28

Problem 37

For each $\mathrm{kW}$ of power input to an ice maker at steady state, determine the maximum rate that ice can be produced, in $\mathrm{kg} / \mathrm{h}$, from liquid water at $0^{\circ} \mathrm{C}$. Assume that $333 \mathrm{~kJ} / \mathrm{kg}$ of energy must be removed by heat transfer to freeze water at $0^{\circ} \mathrm{C}$, and that the surroundings are at $20^{\circ} \mathrm{C}$.

Narayan Hari
Narayan Hari
Numerade Educator
03:54

Problem 38

At steady state, a refrigeration cycle removes $18,000 \mathrm{~kJ} / \mathrm{h}$ of energy by heat transfer from a space maintained at $-40^{\circ} \mathrm{C}$ and discharges energy by heat transfer to surroundings at $20^{\circ} \mathrm{C}$. If the coefficient of performance of the cycle is 25 percent of that of a reversible refrigeration cycle operating between thermal reservoirs at these two temperatures, determine the power input to the cycle, in $\mathrm{kW}$.

Abhishek Kumar
Abhishek Kumar
Numerade Educator
16:41

Problem 39

A refrigeration cycle having a coefficient of performance of 3 maintains a computer laboratory at $18^{\circ} \mathrm{C}$ on a day when the outside temperature is $30^{\circ} \mathrm{C}$. The thermal load at steady state consists of energy entering through the walls and windows at a rate of $30,000 \mathrm{~kJ} / \mathrm{h}$ and from the occupants, computers, and lighting at a rate of $6000 \mathrm{~kJ} / \mathrm{h}$. Determine the power required by this cycle and compare with the minimum theoretical power required for any refrigeration cycle operating under these conditions, each in $\mathrm{kW}$.

Gordon Ayadju
Gordon Ayadju
Numerade Educator
03:15

Problem 40

A heat pump operating at steady state is driven by a $1-\mathrm{kW}$ electric motor and provides heating for a building whose interior is to be kept at $20^{\circ} \mathrm{C}$. On a day when the outside temperature is $0^{\circ} \mathrm{C}$ and energy is lost through the walls and roof at a rate of $60,000 \mathrm{~kJ} / \mathrm{h}$, would the heat pump suffice?

Vipender Yadav
Vipender Yadav
Numerade Educator
02:39

Problem 41

A heat pump maintains a dwelling at $20^{\circ} \mathrm{C}$ when the outside temperature is $0^{\circ} \mathrm{C}$. The heat transfer rate through the walls and roof is $3000 \mathrm{~kJ} / \mathrm{h}$ per degree temperature difference between the inside and outside. Determine the minimum theoretical power required to drive the heat pump, in $\mathrm{kW}$.

RZ
Rubeena Zulfiqar
Numerade Educator
02:25

Problem 42

A building for which the heat transfer rate through the walls and roof is $400 \mathrm{~W}$ per degree temperature difference between the inside and outside is to be maintained at $20^{\circ} \mathrm{C}$. For a day when the outside temperature is $4^{\circ} \mathrm{C}$, determine the power required at steady state, $\mathrm{kW}$, to heat the building using electrical resistance elements and compare with the minimum theoretical power that would be required by a heat pump. Repeat the comparison using typical manufacturer's data for the heat pump coefficient of performance.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:53

Problem 43

Plot the coefficient of performance $\beta_{\max }$ given by Eq. $5.9$ for $T_{\mathrm{H}}=298 \mathrm{~K}$ versus $T_{\mathrm{C}}$ ranging between 235 and $298 \mathrm{~K}$. Discuss the practical implications of the decrease in the coefficient of performance with decreasing temperature $T_{\mathrm{C}}$.

Mayukh Banik
Mayukh Banik
Numerade Educator
01:43

Problem 44

At steady state, a refrigerator whose coefficient of performance is 3 removes energy by heat transfer from a freezer compartment at $0^{\circ} \mathrm{C}$ at the rate of $6000 \mathrm{~kJ} / \mathrm{h}$ and discharges energy by heat transfer to the surroundings, which are at $20^{\circ} \mathrm{C}$. Determine the power input to the refrigerator and compare with the power input required by a reversible refrigeration cycle operating between reservoirs at these two temperatures.

Saman Zulfiqar
Saman Zulfiqar
Numerade Educator
02:39

Problem 45

By supplying energy to a dwelling at a rate of $25,000 \mathrm{~kJ} / \mathrm{h}$, a heat pump maintains the temperature of the dwelling at $20^{\circ} \mathrm{C}$ when the outside air is at $-10^{\circ} \mathrm{C}$. If electricity costs 8 cents per $\mathrm{kW} \cdot \mathrm{h}$, determine the minimum theoretical operating cost for each day of operation.

RZ
Rubeena Zulfiqar
Numerade Educator
01:00

Problem 46

Two kilograms of water execute a Carnot power cycle. During the isothermal expansion, the water is heated until it is a saturated vapor from an initial state where the pressure is 40 bar and the quality is $15 \%$. The vapor then expands adiabatically to a pressure of $1.5$ bar while doing $491.5 \mathrm{~kJ} / \mathrm{kg}$ of work.
(a) Sketch the cycle on $p-v$ coordinates.
(b) Evaluate the heat and work for each process, in $\mathrm{kJ}$.
(c) Evaluate the thermal efficiency.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:31

Problem 47

One kilogram of air as an ideal gas executes a Carnot power cycle having a thermal efficiency of $60 \%$. The heat transfer to the air during the isothermal expansion is $40 \mathrm{~kJ}$. At the end of the isothermal expansion, the pressure is $5.6$ bar and the volume is $0.3 \mathrm{~m}^{3}$. Determine
(a) the maximum and minimum temperatures for the cycle, in $\mathrm{K} .$
(b) the pressure and volume at the beginning of the isothermal expansion in bar and $\mathrm{m}^{3}$, respectively.
(c) the work and heat transfer for each of the four processes, in $\mathrm{kJ}$.
(d) Sketch the cycle on $p-v$ coordinates.

Manik Pulyani
Manik Pulyani
Numerade Educator
03:56

Problem 48

The pressure-volume diagram of a Carnot power cycle executed by an ideal gas with constant specific heat ratio $k$ is shown in Fig. P5.48. Demonstrate that
(a) $V_{4} V_{2}=V_{1} V_{3}$.
(b) $T_{2} / T_{3}=\left(p_{2} / p_{3}\right)^{(k-1) / k} .$
(c) $T_{2} / T_{3}=\left(V_{3} / V_{2}\right)^{k-1}$.

Ashok Prajapati
Ashok Prajapati
Numerade Educator
04:47

Problem 49

One-tenth kilogram of air as an ideal gas with $k=1.4$ executes a Carnot refrigeration cycle, as shown in Fig. 5.13. The isothermal expansion occurs at $-23^{\circ} \mathrm{C}$ with a heat transfer to the air of $3.4 \mathrm{~kJ}$. The isothermal compression occurs at $27^{\circ} \mathrm{C}$ to a final volume of $0.01 \mathrm{~m}^{3}$. Using the results of Prob. $5.64$ as needed, determine
(a) the pressure, in $\mathrm{kPa}$, at each of the four principal states.
(b) the work, in $\mathrm{kJ}$, for each of the four processes.

Eduard Sanchez
Eduard Sanchez
Numerade Educator