Ground Source Heat Pump

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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.

Early Adopter

Mass uptake

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 Ti°C. Since Ti > 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:

TDequation.png


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

erf() is the Gauss error function; conveniently, this function is supported for MS Excel spreadsheets as =ERF()

and for this case study the following values were used:

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

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

α = 0.0647 m²/day the Thermal Diffusivity for "Coal Measures Group" geology (GeoReports, British Geological Survey)

Additionally, the house is on a 12 metre x 18 metre plot and has a heating requirement of 12,000 kW-h per year equivalent to the OFGEM Typical Domestic Consumption Value (TDCV) for gas, for a medium size household (the GSHP is providing space heating and domestic hot water in place of a gas boiler). The case study covers an eight month 'heating season' from October to May. It is a fairly trivial exercise to create a spreadsheet from scratch using the information given above. Here is a graph of the results: Sub-surface temperature distribution, over time

Heat flux at the surface

A second equation gives the heat flux, that is, the rate of heat that can be extracted from the ground directly below the householder's property boundary:

Qequation.png


where Qsurface is heat flux in Watts (W)

k = 1.79 W/m/K, the Thermal Conductivity for "Coal Measures Group" geology (GeoReports, British Geological Survey)

A = 216 m², the area of the property (12m by 18m)

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

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

π = 3.1415…

α = 0.0647 m²/day the Thermal Diffusivity for "Coal Measures Group" geology (GeoReports, British Geological Survey)