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Chapter 15: Q26P (page 412)

Question:(II) A heat engine exhausts its heat at 340掳C and has a Carnot efficiency of 36%. What exhaust temperature would enable it to achieve a Carnot efficiency of 42%?

Short Answer

Expert verified

The exhaust temperature is \(555.5\;{\rm{K}}\).

Step by step solution

01

Given Data

The temperature is \({T_0} = 340^\circ {\rm{C}}\).

The Carnot efficiency is \({\eta _{\rm{C}}} = 36\% \).

The new Carnot efficiency is \({\eta '_{\rm{C}}} = 42\% \).

02

Understanding adiabatic and isothermal process

In this problem, firstly find the intake temperature by using the initial Carnot efficiency. Now, evaluate the exhaust temperature for new Carnot efficiency.

03

Calculation of intake temperature from Carnot efficiency

The relation of efficiency is given by,

\({\eta _{\rm{C}}} = 1 - \frac{{{T_0}}}{{{T_{\rm{i}}}}}\)

Here, \({T_{\rm{i}}}\) is the intake temperature.

On plugging the values in the above relation.

\(\begin{aligned}{c}0.36 &= 1 - \left( {\frac{{\left( {340^\circ {\rm{C}} + 273} \right)\;{\rm{K}}}}{{{T_{\rm{i}}}}}} \right)\\{T_{\rm{i}}} &= 957.8\;{\rm{K}}\end{aligned}\)

04

Calculation of exhaust temperature from new Carnot efficiency

The relation of efficiency is given by,

\({\eta '_{\rm{C}}} = 1 - \frac{{{T_0}^\prime }}{{{T_{\rm{i}}}}}\)

Here, \({T'_0}\) is the exhaust temperature.

On plugging the values in the above relation.

\(\begin{aligned}{c}0.42 &= 1 - \left( {\frac{{{T_{\rm{0}}}^\prime }}{{957.8\;{\rm{K}}}}} \right)\\{T_{\rm{0}}}^\prime &= 555.5\;{\rm{K}}\end{aligned}\)

Thus, \({T_{\rm{0}}}^\prime = 555.5\;{\rm{K}}\) is the required temperature.

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Most popular questions from this chapter

An ideal monatomic gas expands slowly to twice its volume (1) isothermally; (2) adiabatically; (3) isobarically. Plot each on a PV diagram. In which process \(\Delta U\) is the greatest, and in which is \(\Delta U\) the least? In which is W the greatest and the least? In which is Q the greatest and the least?

Question: (II) A 1.0-L volume of air initially at 3.5 atm of (gauge) pressure is allowed to expand isothermally until the pressure is 1.0 atm. It is then compressed at constant pressure to its initial volume, and lastly is brought back to its original pressure by heating at constant volume. Draw the process on a PV diagram, including numbers and labels for the axes.

Question: An ideal heat pump is used to maintain the inside temperature of a house at \({T_{{\rm{in}}}} = 22{\rm{^\circ C}}\) when the outside temperature is \({T_{{\rm{out}}}}\). Assume that when it is operating, the heat pump does work at a rate of 1500 W. Also assume that the house loses heat via conduction through its walls and other surfaces at a rate given by \(\left( {650\;{{\rm{W}} \mathord{\left/

{\vphantom {{\rm{W}} {{\rm{^\circ C}}}}} \right.} {{\rm{^\circ C}}}}} \right)\left( {{T_{{\rm{in}}}} - {T_{{\rm{out}}}}} \right)\). (a) For what outside temperature would the heat pump have to operate all the time in order to maintain the house at an inside temperature of 22掳C? (b) If the outside temperature is 8掳C, what percentage of the time does the heat pump have to operate in order to maintain the house at an inside temperature of 22掳C?

Calculate the work done by an ideal gas while going from state A to state C in Fig. 15鈥�28 for each of the following processes:

(a) ADC,

(b) ABC, and

(c) AC directly.

FIGURE 15鈥�28

Problem 68

(II) Energy may be stored by pumping water to a high reservoir when demand is low and then releasing it to drive turbines during peak demand. Suppose water is pumped to a lake 115 m above the turbines at a rate of\({\bf{1}}{\bf{.00 \times 1}}{{\bf{0}}{\bf{5}}}\;{\bf{kg/s}}\)for 10.0 h at night. (a) How much energy (kWh) is needed to do this each night? (b) If all this energy is released during a 14-h day, at 75% efficiency, what is the average power output?
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