A building at a temperature T (in K ) is heated by an ideal...

A building at a temperature $T$ (in K ) is heated by an ideal heat pump which uses the atmosphere at $T_0(\mathrm{~K})$ as heat source. The pump consumes power $W$ and the building loses heat at a rate $\alpha\left(T-T_0\right)$. What is the equilibrium temperature of the building?

Key Concepts

Thermodynamic Equilibrium

This concept refers to the condition in which the energy entering a system exactly balances the energy leaving it, leading to a steady state where the system’s temperature remains constant. In the context of the building, the energy delivered by the heat pump must match the rate of heat loss to achieve equilibrium.

Energy Balance

An energy balance involves accounting for all energy flows into and out of a system. Here, it means equating the heat delivered to the building (via the heat pump) to the heat lost (proportional to the temperature difference between the building and the ambient environment). This balance is central to determining the equilibrium temperature.

Heat Pump Operation

A heat pump is a device that transfers thermal energy from a colder region to a warmer region by consuming work. Its operation is governed by thermodynamic principles and is an efficient way to heat a space if the temperature difference between the heat source and the destination is not too large.

Coefficient of Performance (COP)

The COP is a measure of a heat pump's efficiency, defined as the ratio of the heat delivered to the work input. For an ideal or reversible heat pump, the COP is given by the ratio of the hot reservoir temperature to the temperature difference between the hot and cold reservoirs. This relationship is crucial in determining the amount of work required to counterbalance heat losses.

Carnot Efficiency

Carnot efficiency represents the theoretical maximum efficiency any heat engine or heat pump can achieve when operating between two thermal reservoirs. For a heat pump, this concept underpins the ideal COP formula and helps establish the limits on performance when transferring heat from a colder source to a warmer destination.

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