The coefficient of performance measures the effectiveness of a reversed heat engine

Engineering Physics | AQA A-Level Physics

What is the coefficient of performance?

The efficiency of a reversed heat engine depends on its purpose. A coefficient of performance (COP)A measure of how effective a reversed heat engine is at transferring heat per unit of work done. It is not the same as efficiency, because COP can be greater than 1. is used to measure effectiveness instead of efficiency.
The COP is not a measure of efficiency, as it can be greater than 1.
The COP is a measure of how effective a reversed heat engine is at transferring heat per unit of work done. For example, a COP of 7 means 7 J of heat energy is transferred per 1 J of work done.
Because the COP is a ratio, it has no units.

COP of a refrigerator

A refrigerator aims to extract as much heat as possible from the cold space. The COP of a refrigerator is:

$$\text{COP}_{\text{ref}} = \frac{Q_C}{W} = \frac{Q_C}{Q_H - Q_C} = \frac{T_C}{T_H - T_C}$$

Where:
- $Q_C$ = energy extracted from the cold space (J)
- $Q_H$ = energy delivered to the hot space (J)
- $W$ = work inputted (J)
- $T_H$ = temperature of hot space (K)
- $T_C$ = temperature of the cold space (K)

COP of a heat pump

A heat pump aims to deliver as much heat as possible to the hot space. The COP of a heat pump is:

$$\text{COP}_{\text{hp}} = \frac{Q_H}{W} = \frac{Q_H}{Q_H - Q_C} = \frac{T_H}{T_H - T_C}$$

Relationship between the two COPs

Since $Q_H = Q_C + W$, the two COPs can be written in terms of each other. Multiplying both sides by $W$:

$$W \cdot \text{COP}_{\text{hp}} = W \cdot \text{COP}_{\text{ref}} + W$$

Dividing through by $W$ gives the key relationship:

$$\text{COP}_{\text{hp}} = \text{COP}_{\text{ref}} + 1$$

Since $Q_H > Q_C$, this means:

$$\text{COP}_{\text{hp}} > \text{COP}_{\text{ref}}$$

Why heat pumps are used

Heat pumps are used instead of conventional electric or gas heaters for large-scale buildings because the energy transferred by a heat pump exceeds the work done on the pump.
An electric or gas heater will at most convert 1 J of energy per 1 J of work done, so would be far more expensive to run on large scales.

Common Mistake

The COP equations in terms of $Q$ and $W$ are included in your data sheet, but not the equations with $T_H$ and $T_C$. For those temperature-based values, you can assume the engine is running at maximum theoretical efficiency. Also, when defining the COP, make sure you specify where the heat is being input to. Simply writing "heat input" without clarification is too vague and will not be accepted by the examiner.