ASHRAE Journal - January 2022 - 19

TECHNICAL FEATURE
wall temperature, Tb. This is a reasonable
assumption considering that all boreholes are
typically fed in parallel with the same inlet
temperature. There is a further requirement
in the g-function defi nition that Tb should be
uniform along the borehole length. Figure 4
shows that the use of one uniform value of qb
over the entire borehole length does not lead
to a uniform Tb. One way to obtain a uniform
Tb is to discretize the borehole into several
segments with different values of qb . This
is shown in Figure 5, where the borehole wall
temperature ( rb = 0.05 m [2 in.]) is evaluated
for two equal-length borehole segments, S1
(top) and S2 (bottom). The heat transfer rate
is assumed uniform along each segment. The
two curves represent the Tb profi les resulting from an
injection of 6.28 W/m (6.53 Btu/h·ft) in the S1 and S2 segments
of the borehole.
As indicated in Figure 5, the average temperatures of
these profi les over S1 and S2 are 5.77°C (42.39°F) and
5.92°C (42.66°F), respectively. The temperature profi le
resulting from heat injection for S2 is slightly different
because the segment is farther away from the surface
than S1 is. Also shown in Figure 5, the infl uences of segment
S2 on S1 and S1 on S2 are nonnegligible, resulting
in an average temperature increase of 0.21°C (0.38°F).
When the infl uences of S1 on S1 and S2 on S1 are superimposed,
the resulting average temperature for segment
S1 is 5.98°C (42.77°F). Similarly, the resulting average
borehole wall temperature for segment S2 is 6.13°C
(43.04°F) after superposition. The results show different
segment temperatures for the same heat transfer rate.
To follow the g-function assumptions, the segment
S2
temperatures are assumed to be identical, giving different
heat transfer rates for each segment. Thus, since
these increases in temperature are directly proportional
to the applied heat transfer rate, the borehole wall
temperatures for both segments can be obtained using
Equation 4a.
T qh
bb rb r
bb r
1
1 11 0 05
,, .
,, .
T qh q hq=+
2 = 
1 1 2 0 05 ++ qh2 2 2 0 05br q /
==
=
=
,, .
where indices 1 and 2 refer here to S1 and S2, respectively.
Because the borehole wall temperature over the entire
2 2 1 0 05 /
,, .
(4a)
FIGURE 5 Temperature profile at r = 0.05 m for a two-segment borehole. The heat transfer rate is
6.28 W/m for each segment.
S1
-20
-40
-60
-80
-100
-120
-140
-160
0.21°C
0.21°C
5.77°C
5.92°C
1
2
3
Temperature (°C)
length has to be uniform, the average heat transfer rate is
given by the average of both heat transfer rates:
TT
bb
=
12
qq q
=+
Solving Equations 4a and 4b leads to qb1
6.37 W/m (6.63 Btu/h·ft), qb2
() =/bb 2 6 28. W/m
=
12
(4b)
= 6.19 W/m (6.44 Btu/h·ft)
and Tb1= Tb2 = 6.06°C (42.91°F). In other words, to maintain
a uniform borehole wall temperature of 6.06°C
(42.91°F), heat transfer rates of 6.37 W/m and 6.19 W/m
(6.63 Btu/h·ft and 6.44 Btu/h·ft) are required in segments
S1 and S2, respectively. The g-function for this
single borehole calculated with two segments is then
6.06, compared to a g-function of 6.2 calculated for the
same borehole with one segment (Figure 4). Thus, segmentation
of the borehole infl uences the calculated value of
the g-function. In this example, only one borehole is considered.
If the 3 × 1 geometry used earlier were calculated
with two segments, Equations 4a and 4b would contain six
different wall temperatures and heat transfer rates.
It is not hard to imagine that the problem becomes
computationally intensive when the number of boreholes
and segments per borehole is large. The complexity
is further increased by the fact that when the bore
fi eld is sized, the length is unknown a priori, and, as
shown in the companion article,1 g-functions need to be
calculated at each iteration.
Figure 6 shows the g-functions of a 5 × 5 borehole
fi eld, evaluated with 1 and 12 segments using the input
parameters reported in Table 1. It can be seen that
g-function values evaluated with 12 segments are lower
J A N U A RY 2 0 2 2 ashrae.org ASHRAE JOURNAL
19
4
5
6
Depth (m)
Borehole Segments
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