8.10 The Carnot Refrigerator

A Carnot Refrigerator is a theoretical thermodynamic device that operates on a reversed Carnot cycle. It is an idealized model for a refrigerator or heat pump. Unlike a heat engine that uses a temperature difference to produce work, a refrigerator uses work input to transfer heat from a low-temperature reservoir to a high-temperature reservoir.

Working Principle

The Carnot Refrigerator is essentially a Carnot heat engine running in reverse. It follows the Clausius statement of the Second Law of Thermodynamics, which states that heat will not spontaneously flow from a cold body to a hot body; external work must be done.

• Heat Absorption: It absorbs heat ($Q_L$ or $Q_2$) from a cold reservoir at a low temperature ($T_L$ or $T_2$).
• Work Input: An external agent (like a compressor) performs work ($W$) on the system.
• Heat Rejection: It rejects a larger amount of heat ($Q_H$ or $Q_1$) to a hot reservoir at a high temperature ($T_H$ or $T_1$).

From the First Law of Thermodynamics, the heat rejected is the sum of the heat absorbed and the work done:

$Q_H=Q_L+W$

$W=Q_H-Q_L$

The Reversed Carnot Cycle

The refrigerator operates on a four-stage, reversible cycle, which is the exact reverse of the Carnot heat engine cycle.

1. Adiabatic Expansion: The working substance (gas) expands without heat exchange. It does work on its surroundings, and its temperature drops from $T_H$ to $T_L$.
2. Isothermal Expansion: The gas is placed in thermal contact with the cold reservoir ($T_L$). It expands at constant temperature, absorbing heat ($Q_L$) from the cold reservoir. This is the cooling step.
3. Adiabatic Compression: The gas is compressed without heat exchange. Work is done on the gas, raising its temperature from $T_L$ back to $T_H$.
4. Isothermal Compression: The gas is placed in contact with the hot reservoir ($T_H$). It is compressed at constant temperature, rejecting heat ($Q_H$) to the hot reservoir. The cycle is complete.

PV Diagram for the Carnot Refrigerator

On a Pressure-Volume (PV) diagram, the Carnot refrigeration cycle traces the same path as the engine cycle but in the counter-clockwise direction. The area enclosed by the loop represents the net work done on the system ($W$) per cycle.

Coefficient of Performance (COP)

The efficiency of a refrigerator is measured by its Coefficient of Performance (COP). The COP is the ratio of the desired effect (heat removed from the cold reservoir) to the required input (work done on the system).

$\text{COP}=\frac{\text{Desired Heat Removed}}{\text{Work Input}}=\frac{Q_L}{W}$

Since $W=Q_H-Q_L$:

$\text{COP}=\frac{Q_L}{Q_H-Q_L}$

For an ideal Carnot refrigerator, since $Q_H/Q_L=T_H/T_L$, the COP in terms of absolute temperatures (in Kelvin) is:

$\text{COP}=\frac{T_L}{T_H-T_L}$

A higher COP indicates a more efficient refrigerator — it removes more heat for a given work input.

Key Points

Can the COP of a refrigerator be greater than 1? Yes. For most practical refrigerators, COP > 1. This means the heat removed from the cold space is greater than the work used. This does not violate energy conservation — energy is being moved, not created. The total energy rejected to the hot reservoir is always $Q_H=Q_L+W$.

Refrigerator vs. Heat Pump: They are physically the same device but differ in purpose. A refrigerator aims to cool the cold reservoir (desired effect = $Q_L$). A heat pump aims to heat the hot reservoir (desired effect = $Q_H$).