Geothermal Cooling - Low Energy Cooling for Sustainable Buildings - page 86

The main task of the design process is to determine the required length of tubing for
the required thermal energy to be extracted or rejected. Modelling the thermal response
of the surrounding soil requires information on the ground temperature distribution,
the moisture content, groundwater movement, freezing or thawing of the soil and the
geometry of the heat exchanger. Furthermore, experimental results showed that the
temperature distribution around soil tubes is usually not symmetric (Bi et al ., 2001).
The soil surrounding the borehole is usually considered as homogeneous with a mean
thermal conductivity λ and mean diffusivity α λ/ρ c . Simulation models are often

separated in 'inner' solutions for the heat transfer between the fluid and the perimeter
of the borehole, including the mutual influence of the U-tubes and the outer region
between the edge of the borehole and the ground.
The simplest model for the inner borehole is to calculate the heat flux per unit length
of the borehole q b from the resistance R b , the mean fluid temperature T f ( t ) and a mean
borehole temperature T b ( t ). The thermal resistance contains the convective resistance
between the fluid and pipe wall, the conductive resistance of the wall and borehole
filling material:

=

T f ( t ) − T b ( t )

q b = (4.1)

R b

As the steepest temperature gradients occur at the pipe-soil or backfill material
interface, several authors developed detailed models for the near temperature field.
Zeng et al . (2003) developed analytical models for a range of tube configurations
within the borehole, Piechowski (1999) solved heat and moisture transport equations
for horizontal pipes embedded in the soil. Piechowski states that the soil temperature
around a pipe drops by 30-40% within a few centimetres, indicating the influence of
precise near-field models.
To obtain the mean temperature of the borehole and to calculate the complete soil
temperature field T ( r, z, t ), the heat conduction equation has to be solved as a function
of time t , depth z and distance from the borehole r :

δ 2 T
δz 2

δ 2 T
δr 2

1
α

δT
δt =

1
r

δT
δr

+ + (4.2)

The height dependence z of the temperature is often ignored and the simplest solu-
tion is obtained if a step function heat input is applied at the origin r =

0. The solution
is known as Kelvin's infinite line source theory and is used to analyse thermal response
test data. The temperature at the boundary of the borehole is given by

ln 4 αt
r b

∞

e − β 2
β

q
4 πλ

q
4 πλ

T ( r b ,t ) − T ( t = 0) = dβ = − γ (4.3)

r 2
4 at

The solution is valid for a constant heat flux per metre borehole q , assuming a con-
stant temperature and an infinite length of the borehole. γ is Euler's constant (0.5772).