Home air conditioners in the United States have their power specified in the truly obscure units of tons, where 1 ton is the power needed to melt 1 ton (2000 lb or 910 kg) of ice in 24 hours. A modest-size house typically has a 4.0 ton air conditioner. If a 4.0 ton air conditioner has a coefficient of performance of 2.5, a typical value, at what rate in kW is heat energy removed from the house?

검증된 단계별 안내

Understand the problem: The air conditioner's power is given in tons, and we need to calculate the rate at which heat energy is removed from the house in kilowatts (kW). The coefficient of performance (COP) is also provided, which relates the heat energy removed to the work input.

Convert the power from tons to watts. Recall that 1 ton of cooling is equivalent to the power required to melt 1 ton of ice in 24 hours. This is equal to 3.517 kW. Therefore, a 4.0 ton air conditioner corresponds to \( 4.0 \times 3.517 \) kW.

Use the definition of the coefficient of performance (COP). The COP is defined as \( \text{COP} = \frac{Q_c}{W} \), where \( Q_c \) is the heat energy removed from the house (in watts) and \( W \) is the work input (in watts). Rearrange this equation to find \( Q_c \): \( Q_c = \text{COP} \times W \).

Relate the work input \( W \) to the cooling power of the air conditioner. The cooling power of the air conditioner (in watts) is equal to the work input \( W \) because the air conditioner operates at the given COP. Substitute \( W \) into the equation for \( Q_c \).

Finally, calculate \( Q_c \) in kilowatts by multiplying the COP (2.5) by the cooling power of the air conditioner (in kW). This gives the rate at which heat energy is removed from the house in kW.

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주요 개념

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Coefficient of Performance (COP)

The Coefficient of Performance (COP) is a measure of the efficiency of a heating or cooling system. It is defined as the ratio of useful heating or cooling provided to the work input required. For air conditioners, a higher COP indicates better efficiency, meaning more heat is removed per unit of energy consumed. In this context, a COP of 2.5 means that for every unit of energy consumed, the air conditioner removes 2.5 units of heat.

Tons of Refrigeration

Tons of refrigeration is a unit of power used to describe the cooling capacity of air conditioning systems. One ton is equivalent to the amount of heat required to melt one ton of ice in 24 hours, which is approximately 3.517 kW. This unit helps quantify the cooling power of air conditioners, making it easier to compare their capacities. A 4.0 ton air conditioner can thus remove heat at a rate of about 14.068 kW.

Heat Transfer

Heat transfer is the process of thermal energy moving from one object or substance to another due to a temperature difference. In the context of air conditioning, heat is removed from the indoor environment and transferred outside, effectively cooling the space. Understanding the principles of heat transfer, including conduction, convection, and radiation, is essential for analyzing how air conditioners operate and how efficiently they can maintain desired indoor temperatures.