Ground Source Heat Pump

GSHP in a suburban setting - a case study

In the following scenario an early adopter installs a Ground Source Heat Pump (GSHP) and initially enjoys uncontended use of the sensible heat contained within a hemispherical volume of sub-soil beneath her - and her neighbours' properties. Later, under a policy of mass adoption of GSHP technology, all the surrounding properties are competing for the same heat and the effective source volume is bounded by the area of property, straight down.

This latter case can be mathematically modelled as "Transient Heat Conduction in a Semi-infinite Solid". The housing estate is an infinite surface plane, held at the heat extraction temperature T₀°C. The ground extends to an infinite distance in all horizontal directions - and downward (hence semi-infinite solid) and is initially at a uniform temperature T_i°C. Since T_i > T₀ heat flows from below ground towards the surface where it is collected by the slinky coils and used to heat the house. The mathematical solution (proof not given here) provides equations for the temperature distribution within the sub-soil, as a function of time and depth and the heat flux, that is the quantity of heat that can be extracted at the surface, a function of time alone

Temperature distribution

The governing equation is:

where T(x,τ) is the sub-soil temperature at depth x metres and after τ days

and for this case study the following values were used:

T₀ = 5°C, the temperature at which heat is being extracted

T_i = 10°C, the initial, uniform temperature throughout the ground

α = 0.0647 m²/day the Thermal Diffusivity for "Coal Measures Group" geology