If the flux we want to suck out of the ground in winter is much bigger

than these natural fluxes then we know that our sucking is going to signif-

icantly alter ground temperatures, and may thus not be feasible. For this

calculation, I’ll assume the ground just below the surface is held, by the

combined influence of sun, air, cloud, and night sky, at a temperature that

varies slowly up and down during the year (figure E.16).

Working out how the temperature inside the ground responds, and what

the flux in or out is, requires some advanced mathematics, which I’ve

cordoned off in box E.19 (p306).

The payoff from this calculation is a rather beautiful diagram (figure

E.17) that shows how the temperature varies in time at each depth.

This diagram shows the answer for any material in terms of the *character-
istic length-scale*

and heat capacity

temperature variations. (We can choose to look at either daily and

yearly variations using the same theory.) At a depth of 2

in temperature are one seventh of those at the surface, and lag them by

about one third of a cycle (figure E.17). At a depth of 3

in temperature are one twentieth of those at the surface, and lag them by

half a cycle.

For the case of daily variations and solid granite, the characteristic

length-scale is *z*_{0} = 0.16 m. (So 32 cm of rock is the thickness you need

to ride out external daily temperature oscillations.) For yearly variations

and solid granite, the characteristic length-scale is *z*_{0} = 3 m.

Let’s focus on annual variations and discuss a few other materials.

Characteristic length-scales for various materials are in the third column

of table E.18. For damp sandy soils or concrete, the characteristic length-

scale *z*_{0} is similar to that of granite – about 2.6 m. In dry or peaty soils, the

length-scale *z*_{0} is shorter – about 1.3 m. That’s perhaps good news because

it means you don’t have to dig so deep to find ground with a stable tem-

perature. But it’s also coupled with some bad news: the natural fluxes are

smaller in dry soils.

The natural flux varies during the year and has a peak value (equation

(E.9)) that is smaller, the smaller the conductivity.

For the case of solid granite, the peak flux is 8 W/m^{2}. For dry soils,

the peak flux ranges from 0.7 W/m^{2} to 2.3 W/m^{2}. For damp soils, the peak

flux ranges from 3 W/m^{2} to 8 W/m^{2}.

What does this mean? I suggest we take a flux in the middle of these

numbers, 5 W/m^{2}, as a useful benchmark, giving guidance about what

sort of power we could expect to extract, per unit area, with a ground-

source heat pump. If we suck a flux significantly smaller than 5 W/m^{2},

the perturbation we introduce to the natural flows will be small. If on the