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Chapter 21: Q. 33 (page 596)

The coefficient of performance of a refrigerator is $5.0$ . The compressor uses $10 J$ of energy per cycle.
a. How much heat energy is exhausted per cycle
b. If the hot-reservoir temperature is $27 掳 C$ , what is the lowest possible temperature in ${}^{鈭�}C$ of the cold reservoir?

Short Answer

Expert verified

a.Heat energy is exhausted per cycle is $60 J$ .

b.Lowest possible temperature $T_{c}$ is $- 23 掳 C$ .

Step by step solution

01

Calculation for heat energy is exhausted per cycle (part a)

a.

As a result of the first law that was applied to refrigerators, we now have

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

$K = \frac{Q_{c}}{W} 鈬� Q_{c} = KW$

Substituting this we get,

$Q_{c} = Q_{h} - W = W (K + 1)$

$Q_{h} = 10 路 (5 + 1) = 60 J$

02

Calculation for lowest possible temperature (part b)

b.

When the coefficient of performance is that of a Carnot refrigerator, the lowest value of the temperature of the cold reservoir is reached.

That is,

$K = \frac{T_{c}}{T_{h} - T_{c}} = \frac{T_{h}}{T_{h} - T_{c}} - 1$

The value of the lower temperature can be separated as follows:

$T_{h} = (K + 1) (T_{h} - T_{c}) 鈬� - T_{h} K = - (K + 1) T_{c}$

T_{c} = T_{h} \frac{K}{K + 1}

The minimum temperature ,

$T_{c} = \frac{5}{5 + 1} 路 (273 + 27) = 250 K$

In degrees centigrade,

$T_{c} = - 23 掳 C$

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Most popular questions from this chapter

A heat engine extracts $55 K J$ of heat from the hot reservoir each cycle and exhausts $40 K J$ of heat. What are (a) the thermal efficiency and (b) the work done per cycle?

$F i g u r e P 21.46$ shows a Carnot heat engine driving a Carnot refrigerator.

a. Determine $Q_{2}, Q_{3}$ and $Q_{4}$ .

b. Is $Q_{3}$ greater than, less than, or equal to $Q_{1}$ ?

c. Do these two devices, when operated together in this way, violate the second law?

FIGURE CP $21.70$ shows two insulated compartments separated by a thin wall. The left side contains $0.060 mol$ of helium at an initial temperature of $600 K$ and the right side contains $0.030 mol$ of helium at an initial temperature of $300 K$ . The compartment on the right is attached to a vertical cylinder, above which the air pressure is $1.0 atm$ . A $10 - cm$ -diameter, $2.0 kg$ piston can slide without friction up and down the cylinder. Neither the cylinder diameter nor the volumes of the compartments are known.

a. What is the final temperature?

b. How much heat is transferred from the left side to the right side?

c. How high is the piston lifted due to this heat transfer?

d. What fraction of the heat is converted into work?

The FIGURE shows the Diesel cycle. It is similar to the Otto cycle (see Problem), but there are two important differences. First, the fuel is not admitted until the air is fully compressed at point $2$ . Because of the high temperature at the end of adiabatic compression, the fuel begins to burn spontaneously. (There are no spark plugs in a diesel engine!) Second, combustion takes place more slowly, with fuel continuing to be injected. This makes the ignition stage a constant-pressure process. The cycle shown, for one cylinder of a diesel engine, has a displacement localid="1650278379201" $V_{\max} - V_{\min}$ of localid="1650278390412" $1000 {cm}^{3}$ and a compression ratio localid="1650278397535" $r = V_{\max} / V_{\min} = 21$ These are typical values for a diesel truck. The engine operates with intake air localid="1650278425123" $(纬 = 1.40)$ at localid="1650278430754" $25 掳 C$ and localid="1650278435051" $1.0 a t m$ pressure. The quantity of fuel injected into the cylinder has a heat of combustion of localid="1650278440305" $1000 J$ .

$a$ . Find $p$ , $V$ , and $T$ at each of the four corners of the cycle. Display your results in a table.

$b$ . What is the network done by the cylinder during one full cycle?

$c$ . What is the thermal efficiency of this engine?

$d$ . What is the power output in kW and horsepower $(1 h p = 746 W)$ of an eight-cylinder diesel engine running at $2400 r p m$ ?

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?

See all solutions

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