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Chapter 21: Q. 25 (page 595)

A heat engine does $10 J$ of work and exhausts $15 J$ of waste heat during each cycle.
a. What is the engine's thermal efficiency?
b. If the cold-reservoir temperature is $20 掳 C$ , what is the minimum possible temperature in ${}^{鈭�}C$ of the hot reservoir?

Short Answer

Expert verified

a.Engine thermal efficiency is $40 %$ .

b.The minimum temperature of the heat reservoir $T_{h, \min}$ is $215 掳 C$ .

Step by step solution

01

Calculation for engine thermal efficiency (part a)

a.

The efficiency of a heat engine,

$畏 = \frac{W}{Q_{h}}$

W is work done , $畏$ is efficiency and $Q_{h}$ is heat used up by machine.

$Q_{h} = W + Q_{c}$

$Q_{c}$ is heat dumped to cold reservoir.

So, we can write our expression for the efficiency as.

$畏 = \frac{W}{W + Q_{c}}$

$畏 = \frac{10}{10 + 15} = 40 %$

02

Calculation for the minimum temperature of the heat reservoir (part b)

b.

The maximum efficiency of a Carnot engine is calculated as a function of the minimum and maximum temperatures as follows:

$畏_{m} = 1 - \frac{T_{c}}{T_{h}}$

The actual efficiency is calculated as a function of the amount of work done and the amount of heat lost.

$畏 = \frac{W}{W + Q_{c}}$

We can write since our engine's efficiency cannot be higher than Carnot's,

$畏鈮� 畏_{m}$

$\frac{W}{W + Q_{c}} 鈮� 1 - \frac{T_{c}}{T_{h}}$

$\frac{T_{c}}{T_{h}} 鈮� 1 - \frac{W}{W + Q_{c}} = \frac{Q_{c}}{W + Q_{c}}$

The temperature of the heat reservoir,

$T_{h} 鈮� \frac{W + Q_{c}}{Q_{c}} T_{c}$

$T_{h} 鈮� \frac{10 + 15}{15} 路 (20 + 273) = 488 K$

$T_{h, \min} = 215 掳 C$

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Most popular questions from this chapter

The first and second laws of thermodynamics are sometimes stated as "You can't win" and "You can't even break even." Do these sayings accurately characterize the laws of thermodynamics as applied to heat engines? Why or why not?

A refrigerator using helium gas operates on the reversed cycle shown in FIGURE. What are the refrigerator鈥檚 ( $a$ ) coefficient of performance and ( $b$ ) power input if it operates at $60$ cycles per second?

In Problems $65$ through $68$ you are given the equation(s) used to solve a problem. For each of these, you are to

a. Write a realistic problem for which this is the correct equation(s).

b. Finish the solution of the problem.

$0.20 = 1 - Q_{C} / Q_{H}$

$W_{out} = Q_{H} - Q_{C} = 20 J$

How much work is done per cycle by a gas following the $P_{V}$ trajectory of FIGURE EX $21.11$ ?

$(T_{2} + 273) K = \frac{200 k P a}{500 k P a} 脳 1 脳 (400 + 273) K$

In Problems 67 through 70 you are given the equation(s) used to solve a problem. For each of these, you are to

a. Write a realistic problem for which this is the correct equation(s).

b. Draw a pVdiagram.

c. Finish the solution of the problem.

See all solutions

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