Here are the formulae for the ideal efficiency of a heat pump, that is, the

electrical energy required per unit of heat pumped. If we are pumping heat

from an outside place at temperature *T*_{1} into a place at higher temperature

*T*_{2}, both temperatures being expressed relative to absolute zero (that is, *T*_{2},

in kelvin, is given in terms of the Celsius temperature *T*_{in}, by 273.15 + *T*_{in}),

the ideal efficiency is:

If we are pumping heat out from a place at temperature *T*_{2} to a warmer

exterior at temperature *T*_{1}, the ideal efficiency is:

These theoretical limits could only be achieved by systems that pump heat

infinitely slowly. Notice that the ideal efficiency is bigger, the closer the

inside temperature *T*_{2} is to the outside temperature *T*_{1}.

While in theory ground-source heat pumps might have better perfor-

mance than air-source, because the ground temperature is usually closer

than the air temperature to the indoor temperature, in practice an air-

source heat pump might be the best and simplest choice. In cities, there

may be uncertainty about the future effectiveness of ground-source heat

pumps, because the more people use them in winter, the colder the ground

gets; this thermal fly-tipping problem may also show up in the summer

in cities where too many buildings use ground-source (or should I say

“ground-sink”?) heat pumps for air-conditioning.

Here’s an interesting calculation to do. Imagine having solar heating pan-

els on your roof, and, whenever the water in the panels gets above 50 °C,

pumping the water through a large rock under your house. When a dreary

grey cold month comes along, you could then use the heat in the rock to

warm your house. Roughly how big a 50 °C rock would you need to hold

enough energy to heat a house for a whole month? Let’s assume we’re

after 24 kWh per day for 30 days and that the house is at 16 °C. The heat

capacity of granite is 0.195 × 4200 J/kg/K = 820 J/kg/K. The mass of

granite required is:

100 tonnes, which corresponds to a cuboid of rock of size 6 m × 6 m × 1 m.

Heat capacity: | C = 820 J/kg/K |

Conductivity: | κ = 2.1 W/m/K |

Density: | ρ = 2750 kg/m^{3} |

Heat capacity per unit volume: | |

C_{V} = 2.3 MJ/m^{3}/K |