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Chapter 15: Q60P (page 412)

A 鈥淐arnot鈥� refrigerator (the reverse of a Carnot engine) absorbs heat from the freezer compartment at a temperature of -17掳C and exhausts it into the room at 25掳C.

(a) How much work would the refrigerator do to change 0.65 kg of water at 25掳C into ice at -17掳C.

(b) If the compressor output is 105 W and runs 25% of the time, how long will this take?

Short Answer

Expert verified

(a) The work done by the refrigerator is\(5.03 \times {104}\;{\rm{J}}\).

(b) The time taken to do the work is\(32\;\min \).

Step by step solution

01

Heat removed from the water

In this problem, there are three parts in which the heat rejected is evaluated. First, the cooling of water up to the freezing point; second, freezing the liquid water; and last, cooling the ice.

02

Given data

The temperature of the freezer is \({T_{\rm{1}}} = - 17\circ {\rm{C}}\).

The temperature of the room is \({T_{\rm{2}}} = 25\circ {\rm{C}}\).

The mass is \(m = 0.65\,{\rm{kg}}\).

The output power is \(P = 105\,{\rm{W}}\).

03

Evaluation of heat rejected from the water

The relation to find the heat rejected is given by:

\(\begin{array}{l}Q = mc\Delta T + mL + mc'\Delta T'\\Q = m\left( {c\Delta T + L + c'\Delta T'} \right)\end{array}\)

Here, \(c\)and \(c'\) are the specific heat for water and ice, \(\Delta T\) and \(\Delta T'\) are the change in temperature for water and ice, and Lis the latent heat of water.

Substitute the values in the above expression.

\(\begin{array}{l}Q = \left( {0.65\;{\rm{kg}}} \right)\left( {\left( {4186\;{\rm{J/kg}} \cdot \circ {\rm{C}}} \right)\left( {25\circ {\rm{C}}} \right) + \left( {3.33 \times {{10}5}\;{\rm{J/kg}}} \right) + \left( {2100\;{\rm{J/kg}} \cdot \circ {\rm{C}}} \right)\left( {17\circ {\rm{C}}} \right)} \right)\\Q = 3.07 \times {105}\;{\rm{J}}\end{array}\)

04

(a) Evaluation of the work done by the refrigerator

The relation of efficiency is given by:

\(\begin{array}{c}\eta = \frac{W}{{W + Q}}\\1 - \frac{{{T_{\rm{1}}}}}{{{T_{\rm{2}}}}} = \frac{W}{{W + Q}}\\W = Q\left( {\frac{{{T_2}}}{{{T_1}}} - 1} \right)\end{array}\)

Substitute the values in the above expression.

\(\begin{array}{l}W = \left( {3.07 \times {{10}5}\;{\rm{J}}} \right)\left( {\frac{{\left( {25\circ {\rm{C}} + 273} \right)\;{\rm{K}}}}{{\left( { - 17\circ {\rm{C}} + 273} \right)\;{\rm{K}}}} - 1} \right)\\W = 5.03 \times {104}\;{\rm{J}}\end{array}\)

Thus, the work done by the refrigerator is \(5.03 \times {104}\;{\rm{J}}\).

05

(b) Evaluation of the time consumed by using compressor power

The relation to find the time is given by:

\(t = \frac{W}{P}\)

Substitute the values in the above expression.

\(\begin{array}{l}t = \left( {\frac{{5.03 \times {{10}4}\;{\rm{J}}}}{{105\,{\rm{W}}\left( {0.25} \right)}}} \right)\\t = 1916.1\;{\rm{s}}\\t \approx 32\;\min \end{array}\)

Thus, the time taken to do the work is \(32\;\min \).

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Most popular questions from this chapter

An ideal gas undergoes an adiabatic expansion, a process in which no heat flows into or out of the gas. As a result,

(a) the temperature of the gas remains constant and the pressure decreases.

(b) both the temperature and pressure of the gas decrease.

(c) the temperature of the gas decreases and the pressure increases.

(d) both the temperature and volume of the gas increase.

(e) both the temperature and pressure of the gas increase

What is the change in entropy of\({\bf{1}}{\bf{.00}}\;{{\bf{m}}{\bf{3}}}\)water at 0掳C when it is frozen to ice at 0掳C?

(I) An ideal gas expands isothermally, performing\({\bf{4}}{\bf{.30 \times 1}}{{\bf{0}}^{\bf{3}}}\;{\bf{J}}\) of work in the process. Calculate (a) the change in internal energy of the gas, and (b) the heat absorbed during this expansion.

Question: (III) The PV diagram in Fig. 15鈥�23 shows two possible states of a system containing 1.75 moles of a monatomic ideal gas. \(\left( {{P_1} = {P_2} = {\bf{425}}\;{{\bf{N}} \mathord{\left/{\vphantom {{\bf{N}} {{{\bf{m}}^{\bf{2}}}}}} \right.} {{{\bf{m}}^{\bf{2}}}}},\;{V_1} = {\bf{2}}{\bf{.00}}\;{{\bf{m}}^{\bf{3}}},\;{V_2} = {\bf{8}}{\bf{.00}}\;{{\bf{m}}^{\bf{3}}}.} \right)\) (a) Draw the process which depicts an isobaric expansion from state 1 to state 2, and label this process A. (b) Find the work done by the gas and the change in internal energy of the gas in process A. (c) Draw the two-step process which depicts an isothermal expansion from state 1 to the volume \({V_2}\), followed by an isovolumetric increase in temperature to state 2, and label this process B. (d) Find the change in internal energy of the gas for the two-step process B.

Question: (a) At a steam power plant, steam engines work in pairs, the heat output of the first one being the approximate heat input of the second. The operating temperatures of the first are 750掳C and 440掳C, and of the second 415掳C and 270掳C. If the heat of combustion of coal is \({\bf{2}}{\bf{.8 \times 1}}{{\bf{0}}^{\bf{7}}}\;{{\bf{J}} \mathord{\left/{\vphantom {{\bf{J}} {{\bf{kg}}}}} \right.} {{\bf{kg}}}}\) at what rate must coal be burned if the plant is to put out 950 MW of power? Assume the efficiency of the engines is 65% of the ideal (Carnot) efficiency. (b) Water is used to cool the power plant. If the water temperature is allowed to increase by no more than 4.5 C掳, estimate how much water must pass through the plant per hour.
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