Environmental Engineering Reference

In-Depth Information

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).

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