Geothermal Cooling - Low Energy Cooling for Sustainable Buildings - page 87

As the mean fluid temperature is measured from the inlet and outlet temperature dur-
ing the thermal response experiment, the borehole resistance R b is used to calculate
the fluid temperature as a function of time:

ln 4 αt
r b

γ

q
4 πλ

T f ( t ) − T ( t = 0) = − + qR b (4.4)

The measured fluid temperature is then plotted against the logarithm of time to
obtain the mean conductivity of the soil from the gradient K:

q
4 πK

λ =

Knowing the thermal conductivity and the heat injected permetre of borehole length,
the borehole resistance can also be calculated. For a valid analysis of the results, a
minimum duration for a thermal response test has to be given: t> 5 r b /α . Measured
typical borehole resistances are around 0 . 1mKW − 1 .
If the heat flux is applied at the perimeter of the borehole r b , the solution to the
so-called cylindrical heat source method is given by Carslaw and Jaeger (1947). Both
solutions are similar for αt/r b > 20 . For a sandy soil with a thermal diffusivity of
4 . 74

10 − 7 m 2 s − 1 and a radius of 0 . 1m, this corresponds to nearly 5 days. However,
neither the line source nor the cylindrical source method saturate at large timescales,
which is a major drawback. Also, parameters like heat capacity or groundwater flow
cannot be quantified.
An extension of the line source approach has been proposed by Eskilson and
Cleasson (1988). This includes the height dependence of the temperature field. The
integrals in the solution of the temperature field are called g -functions, which have
been mostly computed numerically and then used in tabulated form. Recently analyt-
ical solutions have been developed for the g -functions by Lamarche and Beauchamp
(2007). If the g -function is known, the temperature at the perimeter of the borehole
can be easily calculated:

×

2 πλ g t

q

t s , r b

− = =

T b ( t ) T ( t 0) (4.5)

H

H 2 / 9 α and H as the depth of the borehole; t s is the time when the transient
process ends and the temperature field becomes stationary. The power injected into
the borehole then equals the heat given off to the ground. The timescale is very long:
for the sandy soil described above and a borehole 100 m deep, it takes 74 years to
reach steady-state conditions at constant heat injection rate.
Using the g -function approach, Lamarche and Beauchamp (2007) calculated the
time when the height dependence (so-called axial effect) becomes important for a
100 m borehole length as t ∼ t s / 20. For the wet sandy soil described above, this takes

with t s =