Name: Carnot Cycle : Third and Fourth Stage
Uploaded: 2019-07-19
Duration: 7 min 49 s
Description: Work is done on the system during the third and fourth stages of the Carnot cycle. The video teaches us how to calculate the total work done in a Carnot cycl...

Question

What is Carnot's ideal refrigerator? Find an expressions for its coefficient of performance.

Accepted Answer

A refrigerator is a Carnot's heat engine working in the reverse direction in a refrigerator, the working substances absorbs an amount of heat $Q_{2}$ from the cold reservoir at the temperature $T_{2}$ . An amount of work $w$ is done on it by some external agencies $($ a compressor pump $)$ and rejects a larger quantity of heat $Q$ , to the source at temperature, $T$ , as shown in figure below.

The working substances is carried through the cycle of the following four steps $:$

(I) The gas is allowed to expand suddenly $($ adiabatically $)$ from high to low pressure This cools it and converts into a vapour-liquid mixture.

(II) The cold fluid is allowed to absorb heat $Q_{2}$ isothermally from the cold reservoir. This converts the mixture into vapour.

(III) Then the vapour is adiabatically compressed till it heats up to the temperature of the surroundings.

(IV) Finally the vapour is compressed isothermally in contact with the surrounding. The vapour release heat $Q_{1} (= Q_{2} + W)$ to the surroundings and returns to the initial state. Here is the work done on the gas per cycle.

Coefficient of performance $(K)$ is defined as the ratio of heat taken in from the cold body to the work needed to run the refrigerator. Thus purpose of the refrigerator is to remove as much heat $(Q_{2})$ as possible from the cold body with expenditure of as little work $(w)$ as possible.

$∴ K = \frac{Q_{2}}{W}$ $— (1)$

By first law of thermodynamics, the heat released to the hot reservoir is:

$Q_{1} = Q_{2} + W$

$\Rightarrow W = Q_{1} - Q_{2}$ $— (2)$

From $(1)$ and $(2)$

$K = \frac{Q_{2}}{Q_{1} - Q_{2}}$

$\Rightarrow K = \frac{1}{\frac{Q_{1}}{Q_{2}} - 1}$

Since, $(\frac{Q_{1}}{Q_{2}} = \frac{T_{1}}{T_{2}})$

$\Rightarrow K = \frac{1}{\frac{T_{1}}{T_{2}} - 1}$ or

$K = \frac{T_{2}}{T_{1} - T_{2}}$

This is an expression for the coefficient of performance.

What is Carnot's ideal refrigerator? Find an expressions for its coefficient of performance.

Important Questions on Carnot's Heat Engine : Second Law of Thermodynamics

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