### Theory of heat pumps

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 T1 into a place at higher temperature
T2, both temperatures being expressed relative to absolute zero (that is, T2,
in kelvin, is given in terms of the Celsius temperature Tin, by 273.15 + Tin),
the ideal efficiency is:

If we are pumping heat out from a place at temperature T2 to a warmer
exterior at temperature T1, 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 T2 is to the outside temperature T1.

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.

## Heating and the ground

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/m3 Heat capacity per unit volume: CV = 2.3 MJ/m3/K
Table E.14. Vital statistics for granite. (I use granite as an example of a typical rock.)
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