ASHRAE Journal - January 2022 - 20

TECHNICAL FEATURE
than those for one segment, especially for large
values of ln(tts
/ ), where the difference can
reach more than 6%. It should be noted, however,
that for typical boreholes, g-function values
for ln(tts
/ ) >2 are seldom reached in practice
because they correspond to very long operating
times (Figure 3). An example given in the companion
article1 shows a reduction of 2.6% in the
required borehole length for typical conditions
when 12 segments are used instead of 1. It can be
concluded that bore field sizing based on g-functions
evaluated with a large number of segments
will lead to slightly shorter boreholes. In other
words, bore field sizing using g-functions generated
with fewer segments is more conservative.
Finally, if steady-state heat transfer is assumed
in the borehole, then the mean fluid temperature,
Tm , can be obtained using a steady-state
borehole thermal resistance, Rb :
T T qR q
mb bb b
=+ =+



g
2πkg
RT (5)
bg


 +
Values of Rb are tabulated in the ASHRAE Handbook12
or can be evaluated using the GHXSizing spreadsheet
discussed in the companion article.1 They typically
range from 0.05 m·K/W to 0.2 m·K/W (0.09 h·ft·°F/Btu
to 0.34 h·ft·°F/Btu). The value of Rb has a significant
impact on sizing; small values of Rb lead to shorter
boreholes.
Equation 5 is based on " long-term " g-functions, which
neglect the unsteady short-term effects linked to borehole
thermal capacity. There are several ways to account
for unsteady heat transfer in the borehole from the
mean fluid temperature to the borehold wall temperature
(i.e., from Tm to Tb in Figure 1). One of these ways is
through what are called short-term g-functions because
they are a natural extension of g-functions. The reader is
referred to the works of Yavuzturk and Spitler,3 Xu and
Spitler,4 and Brussieux and Bernier5 for more details on
short-term g-functions.
Temperature Penalty
When the classic ASHRAE sizing equation is used, the
ground temperature needs to be corrected with a temperature
penalty, Tp, to account for borehole thermal
interaction. In GHXSizing, the temperature penalty can
20
ASHRAE JOURNAL ashrae.o rg
J A N U A RY 2022
8
6
4
2
-12
40
35
30
25
20
15
10
5
-12
-10
-8
-6
-10
-8
-6
-4
-2
Number of Segments = 1
Number of Segments = 12
2
4
FIGURE 6 g-function curves of 5 × 5 bore field evaluated with 1 and 12 segments.
-4
ln(t/ts)
be evaluated using g-functions.11
The temperature penalty for a field of Nb boreholes is
evaluated based on the following equation:
T () (6)
p =
q g t gt
k
b
yN ()− 1 ()
2π g
where qy is the yearly annual ground load divided
by the boreholes' overall length (Nb × H), gNb and g1 are,
respectively, the g-functions of the entire bore field and
of one single borehole, and t is the design period.
As an example, GHXSizing determines a Tp value
of 2.6°C (4.7°F) for a 3×1 bore field with geometry A
(Table 1), a yearly annual ground thermal imbalance of
6.28 W/m and a design period of 10 years. This means
that the borehole wall temperatures of the 3×1 bore
field increases by 2.6°C (4.7°F) more than the borehole
wall temperature of one single borehole after 10 years of
operation. This value can also be estimated using Figure
2 by subtracting the g-function (g1= 6.1) for the single
borehole (BH/ = ∞) at ln tts
()/ = -1.1 from the g-function
(g3 = 8.7) for BH/ = 0.05.
Concluding Remarks
This article reviews how thermal response factors,
also known as g-functions, are calculated from nondimensional
parameters and used for sizing VHGEs.
-2
2
4
g-function
Difference (%)
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ASHRAE Journal - January 2022

Table of Contents for the Digital Edition of ASHRAE Journal - January 2022

Contents
ASHRAE Journal - January 2022 - Intro
ASHRAE Journal - January 2022 - Cover1
ASHRAE Journal - January 2022 - Cover2
ASHRAE Journal - January 2022 - Cover2A
ASHRAE Journal - January 2022 - Cover2B
ASHRAE Journal - January 2022 - Cover2C
ASHRAE Journal - January 2022 - Contents
ASHRAE Journal - January 2022 - 3
ASHRAE Journal - January 2022 - 4
ASHRAE Journal - January 2022 - 5
ASHRAE Journal - January 2022 - 6
ASHRAE Journal - January 2022 - 7
ASHRAE Journal - January 2022 - 8
ASHRAE Journal - January 2022 - 9
ASHRAE Journal - January 2022 - 10
ASHRAE Journal - January 2022 - 11
ASHRAE Journal - January 2022 - 12
ASHRAE Journal - January 2022 - 13
ASHRAE Journal - January 2022 - 14
ASHRAE Journal - January 2022 - 15
ASHRAE Journal - January 2022 - 16
ASHRAE Journal - January 2022 - 17
ASHRAE Journal - January 2022 - 18
ASHRAE Journal - January 2022 - 19
ASHRAE Journal - January 2022 - 20
ASHRAE Journal - January 2022 - 21
ASHRAE Journal - January 2022 - 22
ASHRAE Journal - January 2022 - 23
ASHRAE Journal - January 2022 - 24
ASHRAE Journal - January 2022 - 25
ASHRAE Journal - January 2022 - 26
ASHRAE Journal - January 2022 - 27
ASHRAE Journal - January 2022 - 28
ASHRAE Journal - January 2022 - 29
ASHRAE Journal - January 2022 - 30
ASHRAE Journal - January 2022 - 31
ASHRAE Journal - January 2022 - 32
ASHRAE Journal - January 2022 - 33
ASHRAE Journal - January 2022 - 34
ASHRAE Journal - January 2022 - 35
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ASHRAE Journal - January 2022 - 48
ASHRAE Journal - January 2022 - S1
ASHRAE Journal - January 2022 - S2
ASHRAE Journal - January 2022 - S3
ASHRAE Journal - January 2022 - S4
ASHRAE Journal - January 2022 - S5
ASHRAE Journal - January 2022 - S6
ASHRAE Journal - January 2022 - S7
ASHRAE Journal - January 2022 - S8
ASHRAE Journal - January 2022 - S9
ASHRAE Journal - January 2022 - S10
ASHRAE Journal - January 2022 - S11
ASHRAE Journal - January 2022 - S12
ASHRAE Journal - January 2022 - S13
ASHRAE Journal - January 2022 - S14
ASHRAE Journal - January 2022 - S15
ASHRAE Journal - January 2022 - S16
ASHRAE Journal - January 2022 - S17
ASHRAE Journal - January 2022 - S18
ASHRAE Journal - January 2022 - S19
ASHRAE Journal - January 2022 - S20
ASHRAE Journal - January 2022 - S21
ASHRAE Journal - January 2022 - S22
ASHRAE Journal - January 2022 - S23
ASHRAE Journal - January 2022 - S24
ASHRAE Journal - January 2022 - S25
ASHRAE Journal - January 2022 - S26
ASHRAE Journal - January 2022 - S27
ASHRAE Journal - January 2022 - S28
ASHRAE Journal - January 2022 - S29
ASHRAE Journal - January 2022 - S30
ASHRAE Journal - January 2022 - S31
ASHRAE Journal - January 2022 - S32
ASHRAE Journal - January 2022 - S32A
ASHRAE Journal - January 2022 - S32B
ASHRAE Journal - January 2022 - S32C
ASHRAE Journal - January 2022 - S32D
ASHRAE Journal - January 2022 - S33
ASHRAE Journal - January 2022 - S34
ASHRAE Journal - January 2022 - S35
ASHRAE Journal - January 2022 - S36
ASHRAE Journal - January 2022 - S37
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ASHRAE Journal - January 2022 - S40
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ASHRAE Journal - January 2022 - S53
ASHRAE Journal - January 2022 - S54
ASHRAE Journal - January 2022 - S55
ASHRAE Journal - January 2022 - S56
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ASHRAE Journal - January 2022 - S58
ASHRAE Journal - January 2022 - S59
ASHRAE Journal - January 2022 - S60
ASHRAE Journal - January 2022 - S61
ASHRAE Journal - January 2022 - S62
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ASHRAE Journal - January 2022 - S64
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ASHRAE Journal - January 2022 - S66
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ASHRAE Journal - January 2022 - S80
ASHRAE Journal - January 2022 - S81
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ASHRAE Journal - January 2022 - S85
ASHRAE Journal - January 2022 - S86
ASHRAE Journal - January 2022 - S87
ASHRAE Journal - January 2022 - S88
ASHRAE Journal - January 2022 - S89
ASHRAE Journal - January 2022 - S90
ASHRAE Journal - January 2022 - S91
ASHRAE Journal - January 2022 - S92
ASHRAE Journal - January 2022 - S93
ASHRAE Journal - January 2022 - S94
ASHRAE Journal - January 2022 - S95
ASHRAE Journal - January 2022 - S96
ASHRAE Journal - January 2022 - S97
ASHRAE Journal - January 2022 - S98
ASHRAE Journal - January 2022 - S99
ASHRAE Journal - January 2022 - S100
ASHRAE Journal - January 2022 - S101
ASHRAE Journal - January 2022 - S102
ASHRAE Journal - January 2022 - S103
ASHRAE Journal - January 2022 - S104
ASHRAE Journal - January 2022 - S105
ASHRAE Journal - January 2022 - S106
ASHRAE Journal - January 2022 - S107
ASHRAE Journal - January 2022 - S108
ASHRAE Journal - January 2022 - S109
ASHRAE Journal - January 2022 - S110
ASHRAE Journal - January 2022 - S111
ASHRAE Journal - January 2022 - S112
ASHRAE Journal - January 2022 - S113
ASHRAE Journal - January 2022 - S114
ASHRAE Journal - January 2022 - S115
ASHRAE Journal - January 2022 - S116
ASHRAE Journal - January 2022 - S117
ASHRAE Journal - January 2022 - S118
ASHRAE Journal - January 2022 - S119
ASHRAE Journal - January 2022 - S120
ASHRAE Journal - January 2022 - S121
ASHRAE Journal - January 2022 - S122
ASHRAE Journal - January 2022 - S123
ASHRAE Journal - January 2022 - S124
ASHRAE Journal - January 2022 - S125
ASHRAE Journal - January 2022 - S126
ASHRAE Journal - January 2022 - S127
ASHRAE Journal - January 2022 - S128
ASHRAE Journal - January 2022 - S129
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ASHRAE Journal - January 2022 - S136
ASHRAE Journal - January 2022 - S137
ASHRAE Journal - January 2022 - S138
ASHRAE Journal - January 2022 - S139
ASHRAE Journal - January 2022 - S140
ASHRAE Journal - January 2022 - S141
ASHRAE Journal - January 2022 - S141A
ASHRAE Journal - January 2022 - A141B
ASHRAE Journal - January 2022 - S142
ASHRAE Journal - January 2022 - S143
ASHRAE Journal - January 2022 - S144
ASHRAE Journal - January 2022 - 49
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ASHRAE Journal - January 2022 - 64
ASHRAE Journal - January 2022 - Cover3
ASHRAE Journal - January 2022 - Cover4
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