A building is maintained at a temperature T by means of an...

A building is maintained at a temperature $T$ by means of an ideal heat pump which uses a river at temperature $T_0$ as a source of heat. The heat pump consumes power $W$, and the building loses heat to its surroundings at a rate $\alpha\left(T-T_0\right)$. Show that $T$ is given by $$ T=T_0+\frac{W}{2 \alpha}\left(1+\sqrt{1+4 \alpha T_0 / W}\right) . $$

A building is maintained at a temperature $T$ by means of an ideal heat pump which uses a river at temperature $T_0$ as a source of heat. The heat pump consumes power $W$, and the building loses heat to its surroundings at a rate $\alpha\left(T-T_0\right)$. Show that $T$ is given by
$$
T=T_0+\frac{W}{2 \alpha}\left(1+\sqrt{1+4 \alpha T_0 / W}\right) .
$$

Key Concepts

Energy Balance

This concept involves applying the first law of thermodynamics to a steady?state system, where the energy entering the building (in the form of heat delivered by the heat pump) must equal the energy leaving the building (through heat loss to the surroundings). Establishing this balance is essential for determining the conditions under which the building’s temperature can be maintained constant.

Ideal Heat Pump

An ideal heat pump is a theoretical device that functions with maximum efficiency, often modeled on a reversible Carnot cycle. It transfers heat from a colder reservoir to a warmer one by consuming work. In this context, it moves heat from a river at a lower temperature to a building, illustrating the fundamental operation principles of heat pumps in thermodynamics.

Coefficient of Performance (COP)

The coefficient of performance (COP) is a key efficiency metric for heat pumps, defined as the ratio of heat delivered to the high-temperature reservoir (the building) to the work input required. For an ideal or Carnot heat pump operating between two temperatures, the COP is given by a function of the temperatures of the hot and cold reservoirs. This concept is critical for linking the work input to the amount of heat transferred.

Heat Transfer Rate

This concept refers to the rate at which thermal energy is lost from the building to its surroundings, described by a proportionality constant multiplied by the temperature difference. This linear relation, typically encountered in conduction or convection scenarios, plays a fundamental role in setting up the energy balance in the problem and determining the required performance of the heat pump.

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