ASHRAE Journal - December 2021 - 28

TECHNICAL FEATURE
They are assumed to prevail from t0 = 0 to
t1, from t1 to t2 and from t2 to t3, respectively.
Typically, t1, t2 and t3 correspond to 10 years,
10 years + 1 month, and 10 years + 1 month +
6 hours, respectively. In practice, the three
ground loads determined in the early design
phase may differ from those defined with the
final hourly values.
The following paragraphs will give some
theoretical background on the origin of L2
methods. The bore field sizing methods presented
here use the principle of temporal
superposition of loads.6 With reference to
Figure 3, this principle can be illustrated by
first examining the borehole wall temperature,
Tb, at times t1 and t3 and resulting from
heat injection, qy/L, of 10 W/m (10.4 Btu/h·ft)
for the first 3,650 days (0 to t1) followed by
no heat injection (qy/L = 0) for the next 30.25
days from t1 to t3. Using typical borehole and ground
conditions:
qy
ty
t1
tm
th
t2
t3
hourly (from t2 to t3) ground loads, the borehole wall
temperature at time t3 is then given by:
at t1 TT
bg
−=
at t3 TT
q tt
L
bg
y Γ()−
10
30
=× =°C
10 ..0 3809 3 81
−= ()− −−()yy 31
= ×−.
ΓΓ
q tt q tt
L
10 0 3812 0 2286 1 53..
()=°C
where Tg is the ground temperature, Γ, is a ground thermal
response (the so-called infinite cylindrical source
analytical solution is used is used here but other methods
could be used as well) evaluated over a certain time
period and L is the overall bore field length (H) times
the number of boreholes). Thus, Tb is 3.81°C and 1.53°C
(7.0°F and 2.74°F) higher than Tg after 3,650 days and
3,680.25 days, respectively. This shows that the ground
" recovers " when heat injection is stopped from t1 to t3.
For Equation 1b, the influence of qy is first calculated for
the full duration (t3 - t0) and then its influence over the
period (t3 - t1) is subtracted according to the principle of
superposition.
Equation 1b determines Tb at time t3 for qy = 0 from t1 to
t3, ignoring the effects of the other two ground pulses
(qm and qh) on the final borehole temperature at t3. In
practice, these ground loads are nonzero from t1 to t3.
Introducing the monthly average (from t1 to t2) and peak
28
ASHRAE JOURNAL ashrae.o rg D ECEM BER 2 0 2 1
(1a)
TT
(1b)
Equation 2 can be reformulated to give the bore field
length as a function of the mean fluid temperature:
L =
q tt tt q tt
y
mh hb
()+−()(
ΓΓ
()− −−31 32
q tt tt()− −−
()
TT
−
mg
where Rb is the borehole thermal resistance (between
Tb and Tm) and Tm is the mean fluid temperature in the
borehole. When hourly loads are used, it is more convenient
to transform Equation 3 into the following form:
L ∑ () −
N
i=1 11 hb
=
qq tt qR
TT
i − ()i−
Γ()+()i−
−
n
mg
where qi and qi -1 are the i th and (i - 1)th hourly ground
loads, tn is the total time duration of all loads from t0
to the last load, qN, and ti-1 is the time duration of loads
from t0 to the end of qi -1.
(4)
ΓΓ Γ
30 31
()+
()
32
))+ qR
(3)
bg
−=
q tt tt q tt22mh 3()
L
()+−
ΓΓ
ΓΓ Γ
30 31
q tt tt()− −−
()− −−31 32
y
()
()+
() ()
(2)
FIGURE 3 Hourly load profile decomposed into monthly loads and three load pulses.
Monthly
Peak Load (qh)
Monthly
Average Load (qm)
Yearly
Average Load (qy)
Time
qm
qh
Ground Loads
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ASHRAE Journal - December 2021

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