Efficiency of Carnot Refrigerator

Question

In which case will your refrigerator work more efficiently?

a. When you put a glass of hot milk inside.
b. When you put a glass of milk at room temperature inside.
c. Efficiency is same in both the cases.

Solution

A Carnot refrigerator is basically a Carnot engine cycle operating in reverse. In a refrigerator, heat is extracted from a low temperature source and rejected to a high temperature sink. Since heat is flowing from a cold region to hot region (against the natural flow), according to the principle of conservation of energy, work need to be done on the system.
The efficiency of the refrigerator is calculated in terms of the Coefficient of Performance, !!COP!! given by:

!!COP = ("Heat extracted from cold region")/("Work done on the system")!!

Let,
The temperature of cold region be !!T_1!!
The temperature of hot region be !!T_2!!
Heat extracted from cold region be !!Q_1!!
Heat rejected to hot region be !!Q_2!!
Work done on the system be !!W!!

Therefore,
!!COP = (Q_1)/(-W)!!

Work done is negative since it is done on the system.

According to the First Law of Thermodynamics, work done on the system is equal to the difference between the heat extracted and heat rejected by the system,i.e,

!!W = Q_1 - Q_2!!

Therefore,
!!COP = (Q_1)/(Q_2 - Q_1)!!

Since we are talking about a Carnot refrigerator, the process is reversible and efficiency is the maximum possible. Therefore,

!!(Q_1)/(T_1) = (Q_2)/(T_2) = const!!

Therefore, we can write the Coefficient of Performance as:
!!COP = (T_1)/(T_2 - T_1)!!

Therefore, the Coefficient of Performance (or efficiency) of a Carnot refrigerator is inversely proportional to the temperature difference between the cold and hot regions, i.e., greater the temperature difference, lower will be the efficiency.

Therefore, the answer to this question will be (b), i.e., the refrigerator will work more efficiently if we put a glass of milk at room temperature inside it.

Application

All cooling appliances like the refrigerator and the Air Conditioner work on this principle. So, if you want to lower your electricity bills, it will be better if you pre-cool food items to the room temperature before kepping them in the fridge.

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