Refrigerator efficiency is not typically measured as a percentage efficiency like a heat engine. Instead, it is quantified by its Coefficient of Performance (COP), often denoted by the symbol β. The COP represents how much heat is removed from the cold space (the refrigerator interior) for each unit of work energy consumed by the appliance.
Understanding the Coefficient of Performance (COP)
The fundamental way to calculate a refrigerator's efficiency (its COP) is by comparing the useful heat removed to the energy input required.
According to the reference, the efficiency here is called the coefficient of performance, β.
The basic formula is:
β = (Heat Removed from the Cold Reservoir) / (Work Input)
In thermodynamic terms, this is expressed as:
β = Q₂ / W
- Q₂: This represents the amount of heat energy absorbed from the cold source, which is the interior of the refrigerator (also called the sink in thermodynamics). The reference states Q₂ IS least taken from sink, which in standard thermodynamic context refers to the heat taken from the cold sink.
- W: This is the work energy input required by the refrigerator's compressor and other components to move the heat from the cold interior to the warmer surroundings.
Relating Work Input to Heat Transfer
The work input (W) is the difference between the heat expelled to the warm surroundings (the hot reservoir, Q₁) and the heat absorbed from the cold interior (Q₂).
So, the work done can be expressed as:
W = Q₁ - Q₂
Where:
- Q₁: The amount of heat energy expelled to the hot reservoir (the environment outside the refrigerator).
- Q₂: The amount of heat energy absorbed from the cold reservoir (inside the refrigerator).
Substituting this into the basic COP formula gives the most common expression for refrigerator COP:
β = Q₂ / (Q₁ - Q₂)
This formula shows that a higher COP is achieved when a larger amount of heat (Q₂) is removed from the cold space for a given amount of work (W) or when the difference between heat rejected (Q₁) and heat absorbed (Q₂) is smaller.
Components of the Calculation
Here's a quick overview of the terms in the formula:
| Term | Description | Unit (typically) |
|---|---|---|
| β | Coefficient of Performance (dimensionless) | N/A |
| Q₂ | Heat absorbed from the cold reservoir (inside) | Joules (J) |
| Q₁ | Heat expelled to the hot reservoir (outside) | Joules (J) |
| W | Work input required (by compressor, etc.) | Joules (J) |
Note that Q₁, Q₂, and W must be measured in the same units (e.g., Joules or BTU) or rates (e.g., Watts or BTU/hr) for the calculation to be valid.
Ideal Refrigerator and Relation to Carnot Efficiency
The reference mentions that an ideal refrigerator is a carnot engine working in opposite direction. This refers to a theoretical refrigerator operating on the reverse Carnot cycle, which represents the maximum possible efficiency for a refrigerator operating between two given temperatures.
For such an ideal (Carnot) refrigerator operating between a cold reservoir temperature T₂ (in Kelvin) and a hot reservoir temperature T₁ (in Kelvin), the COP is also given by:
β_Carnot = T₂ / (T₁ - T₂)
The reference also provides a relationship between the refrigerator's COP (β) and the efficiency (μ) of the corresponding Carnot engine working in reverse: β = (1 - μ) / μ. Here, μ represents the efficiency of the equivalent forward Carnot engine operating between the same temperature limits (μ = W/Q₁ or 1 - T₂/T₁). While this relationship is valid for ideal cycles, the most practical way to calculate the efficiency (COP) of a real refrigerator is using the heat transferred and work input as described above.
In summary, calculating refrigerator efficiency involves determining its Coefficient of Performance (COP), which is the ratio of the heat removed from the cold interior to the work input required by the system.
