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Chapter 21: Q. 58 (page 597)

A heat engine using a monatomic gas follows the cycle shown in FIGURE P $21.58$ .
a. Find $W_{s}, Q$ , and ${螖贰}_{th}$ for each process in the cycle. Display your results in a table.
b. What is the thermal efficiency of this heat engine?

Short Answer

Expert verified

a. Table for process of cycle is .

b. The thermal efficiency is $15 %$ .

Step by step solution

01

Calculation for work done in first process (part a)

(a) .

The third process is adiabatic,

So the maximal pressure is,

$p_{1} V_{1}^{纬} = p_{3} V_{3}^{纬}$

$鈬� p_{1} = {(\frac{V_{3}}{V_{1}})}^{纬} p_{3}$

So,

$p_{1} = 6^{1.67} 脳 100 atm$

$= 1993 kPa$

The first process is isobaric,

so,

$T_{1} = \frac{V_{2}}{V_{1}} T_{2}$

$= 100 K$

The heat is,

$Q = \frac{p_{2} V_{2}}{{RT}_{2}} C_{p} (600 - 100) m^{3}$

$鈮� 8300 J$

The work done is,

role="math" localid="1650269695241" $W = 1993 脳 10^{3} J 脳 500 脳 10^{- 6}$

$鈮� 1000 J$

02

Calculation for work in second process (part a) solution

a.

For second process the work done is null,

$Q = \frac{p_{2} V_{2}}{{RT}_{2}} C_{p} 螖罢$

$= \frac{p_{2} V_{2}}{{RT}_{2}} C_{p} \frac{p_{2} - p_{3}}{p_{2}} T_{2}$

$= \frac{C_{p} V_{2} (p_{2} - p_{3})}{R}$

So,

$Q = \frac{12.5 脳 6 脳 10^{- 4} m^{3} 脳 1893 脳 10^{3} J}{8.314 惟}$

$鈮� 2850 J$

The third point is,

$T_{3} = \frac{p_{3}}{p_{2}} T_{2}$

role="math" localid="1650270106364" $= \frac{100 atm}{1993 kPa} 脳 600 m^{3}$

$= 30.1 K$

The heat is,

$Q = \frac{p_{2} V_{2}}{{RT}_{2}} C_{v} (T_{3} - T_{2})$

$= - 210 J$

03

Result for process in cycle part(a) solution

a.

04

Calculation for thermal efficiency (part b)

b.

From the table,

The total heat input is $5240 J$ ,

The work done is $790 J$ .

The efficiency is,

$畏 = \frac{790 J}{5240 J}$

$= 15 %$

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Most popular questions from this chapter

A heat engine using a diatomic gas follows the cycle shown in FIGURE P $21.56$ . Its temperature at point $1$ is $20 掳 C$ .

a. Determine $W_{s}, Q$ , and ${螖贰}_{th}$ for each of the three processes in this cycle. Display your results in a table.

b. What is the thermal efficiency of this heat engine?

c. What is the power output of the engine if it runs at $500 rpm$ ?

A uniform rod of length L oscillates as a pendulum about a ALC pivot that is a distance x from the center.

a. For what value of x, in terms of L, is the oscillation period a minimum?

b. What is the minimum oscillation period of a $15 kg, 1.0$ -m-long steel bar?

a. A heat engine does $200 J$ of work per cycle while exhausting $600 J$ of heat to the cold reservoir. What is the engine's thermal efficiency?

b. A Carnot engine with a hot-reservoir temperature of $400 掳 C$ has the same thermal efficiency. What is the cold-reservoir temperature in ${}^{鈭�}C$ $?$

A heat engine operating between energy reservoirs at $20 掳 C$ and $600 掳 C$ has $30 %$ of the maximum possible efficiency. How much energy must this engine extract from the hot reservoir to do $1000 J$ of work?

A typical coal-fired power plant burns $300$ metric tons of coal every hour to generate $750 M W$ of electricity. $1$ metric ton = $1000 kg$ . The density of coal is 1500 kg/m3 and its heat of combustion is $28 M J / k g$ . Assume that all heat is transferred from the fuel to the boiler and that all the work done in spinning the turbine is transformed into electric energy.

a. Suppose the coal is piled up in a $10 m * 10 m$ room. How tall must the pile be to operate the plant for one day?

b. What is the power plant鈥檚 thermal efficiency?

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