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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

A heat engine using 1.0 mol of a monatomic gas follows the cycle shown in FIGURE P21.55. 3750 J of heat energy is transferred to the gas during process 1 -> 2.
a. Determine Ws, Q, and 螖贰_th for each of the four processes in this cycle. Display your results in a table.
b. What is the thermal efficiency of this heat engine?

A heat engine uses a diatomic gas that follows the $pV$ cycle in FIGURE P $21.59$ .

a. Determine the pressure, volume, and temperature at point $2 .$

b. Determine ${螖贰}_{th}, W_{s}$ , and

$p (kPa)$ $Q$ for each of the three processes. Put your results in a FIGURE P $21.59$ table for easy reading.

c. How much work does this engine do per cycle and what is its thermal efficiency?

There has long been an interest in using the vast quantities of thermal energy in the oceans to run heat engines. A heat engine needs a temperature difference, a hot side and a cold side. Conveniently, the ocean surface waters are warmer than the deep ocean waters. Suppose you build a floating power plant in the tropics where the surface water temperature is $鈮� 30 掳 C$ . This would be the hot reservoir of the engine. For the cold reservoir, water would be pumped up from the ocean bottom where it is always $鈮� 5 掳 C$ . What is the maximum possible efficiency of such a power plant?

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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?

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