An air conditioner operating between 93^∘ F and 70^∘ F is rated at 4000 Btu/h cooling capacity.

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An air conditioner operating between 93^∘ F and 70^∘ F is...

An air conditioner operating between $93^{\circ} \mathrm{F}$ and $70^{\circ} \mathrm{F}$ is rated at 4000 Btu/h cooling capacity. Its coefficient of performance is $27 \%$ of that of a Carnot refrigerator operating between the same two temperatures. What horsepower is required of the air conditioner motor?

Key Concepts

Coefficient of Performance (COP)

The COP of a refrigeration or air-conditioning device is the ratio of the cooling effect achieved to the work input required. It is a key performance metric which indicates efficiency—a higher COP means more cooling per unit of work. In practical applications, the actual COP is compared to the ideal (Carnot) COP to evaluate device performance.

Carnot Refrigerator Efficiency

The Carnot cycle represents the idealized limit of efficiency for any refrigeration cycle operating between two temperature reservoirs. The Carnot COP is derived from the temperatures of the cold and hot reservoirs in absolute units (Kelvin) and provides the maximum theoretical COP that any real system can achieve, serving as a benchmark for comparing actual performance.

Temperature Conversion

Absolute temperature scales (Kelvin) are essential when applying thermodynamic equations, such as those for calculating the Carnot efficiency, because these formulas require temperatures to be measured from absolute zero. Converting from the Fahrenheit or Celsius scales to Kelvin ensures the correct application of these thermodynamic relationships.

Unit Conversions

In engineering and physics problems, it is often necessary to convert between different unit systems. This includes converting cooling capacity from British Thermal Units per hour (Btu/h) to Watts, and working out power requirements in horsepower. Accurate conversions are crucial for ensuring that all terms in the energy and work equations are compatible.

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