A building at absolute temperature T is heated by means of a...

A building at absolute temperature $T$ is heated by means of a heat pump which uses a river at absolute temperature $T_0$ as a source of heat. The heat pump has an ideal performance and consumes power $W$. The building loses heat at a rate $\alpha\left(T-T_0\right)$, where $\alpha$ is a constant. (a) Show that the equilibrium temperature $T_e$ of the building is given by $$ T_e=T_0+\frac{W}{2 \alpha}\left[1+\left(1+\frac{4 \alpha T_0}{W}\right)^{\frac{1}{2}}\right] . $$ (b) Suppose that the heat pump is replaced by a simple heater which also consumes a constant power $W$ and which converts this into heat with $100 \%$ efficiency. Show explicitly why this is less desirable than a heat pump.

A building at absolute temperature $T$ is heated by means of a heat pump which uses a river at absolute temperature $T_0$ as a source of heat. The heat pump has an ideal performance and consumes power $W$. The building loses heat at a rate $\alpha\left(T-T_0\right)$, where $\alpha$ is a constant.
(a) Show that the equilibrium temperature $T_e$ of the building is given by
$$
T_e=T_0+\frac{W}{2 \alpha}\left[1+\left(1+\frac{4 \alpha T_0}{W}\right)^{\frac{1}{2}}\right] .
$$
(b) Suppose that the heat pump is replaced by a simple heater which also consumes a constant power $W$ and which converts this into heat with $100 \%$ efficiency. Show explicitly why this is less desirable than a heat pump.

Key Concepts

Heat Pump

A heat pump is a device that transfers heat from a lower-temperature source to a higher-temperature sink using work. Its performance is typically measured by its coefficient of performance (COP), which, in an ideal scenario, can be greater than 1. This means that for every unit of work input, the heat pump can transfer multiple units of heat, making it more energy efficient compared to direct electric heating.

Energy Balance

In thermal systems, energy balance involves equating the heat inputs and outputs to determine system behavior over time. For a system in thermal equilibrium, the rate at which energy is added must match the rate at which energy is lost. This concept is crucial in setting up equations that balance the work done by heating devices with the heat loss from the system, leading to an expression for the equilibrium temperature.

Steady-State Equilibrium

Steady-state equilibrium refers to a condition where a system's properties remain constant over time because the energy being added equals the energy being lost. In the context of building heating, establishing a steady-state involves solving for the temperature at which incoming heat (from a heat pump or heater) balances the heat lost through processes such as conduction, convection, or radiation.

Comparison of Heating Devices

The comparison between a heat pump and a direct electric heater revolves around efficiency. While both can deliver the same amount of energy input (measured in power), a heat pump is more desirable because it uses ambient thermal energy in addition to the work input, effectively providing more heating than the electrical energy consumed. This enhanced efficiency can be explicitly shown through the different energy balances and resulting equilibrium temperatures in the system.

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