ASHRAE Journal - December 2021 - 29

TECHNICAL FEATURE
Classic ASHRAE Sizing Equation
The classic ASHRAE sizing equation (Equation 5)
fi rst appeared in the 1995 ASHRAE Handbook-HVAC
Applications.7,8 It is based on the approach described in
Equation 3, and it can be used for either heating or cooling
applications (with proper signs for the loads). Tin,hp is the
heat pump inlet design temperature limit in heating or
cooling, and Tout,hp is determined from an energy balance
on the bore fi eld at peak conditions.
L =
()
−
R PLF G G
y
bm tt


((
+−
()

31
g
qG G k qW
t tt gh/ +−
3 31
− ()−
tt


32
T ()−Tp
TT
in hp
,,
2
In this equation, the annual, monthly and peak load
pulses are given by (1) qy, the net annual average heat
transfer to the ground; (2) (qh - W)PLFm, the monthly
average heat transfer to the ground; and (3) (qh - W ), the
peak hourly heat transfer rate to the ground. Note that
qy is a ground load evaluated using the building block
load, heating and cooling EFLHs and heat pump COPs.
The value of qh is a peak building load converted into a
ground load by subtracting the compressor power, W,
which is obtained using the heat pump COP. PLFm is
the part-load factor during the design month, kg is the
ground thermal conductivity and fi nally Fsc is the short
circuit heat loss factor between the upward and downward
pipes in the borehole.
This last value, which is typically very close to 1, is tabulated
in the 2019 ASHRAE Handbook-HVAC Applications.9
The three terms in brackets correspond to Ry, Rm and Rh
(i.e., the yearly, monthly and hourly effective ground
thermal resistances). They are evaluated using the infi -
nite cylindrical source (ICS) analytical solution, which
is often referred to as a G-factor. The temperature penalty,
Tp, can be regarded as the increase/decrease of the
borehole wall temperature caused by the annual ground
thermal imbalance, qy, and borehole interaction. Values
of Tp are tabulated in the 2015 ASHRAE Handbook-HVAC
Applications10 for a limited number of bore fi eld confi
gurations and annual ground thermal imbalances, qy.
The 2019 ASHRAE Handbook-HVAC Applications11 suggests
an iterative method for the determination of Tp. This
method is implemented in GHXSizing to determine Tp
for rectangular bore fi elds.
)// g
out hp
g + sc  ()−


tt
32
() ()
()−
()− kF Gk()
+


(5)
Example 1 used in the 2019 ASHRAE Handbook-HVAC
Applications12 will now be solved with GHXSizing to
illustrate the use of the classic ASHRAE sizing equation.
Input values and results are shown in Table 1. The
input parameters can be entered in GHXSizing in both
SI and I-P units, and results are calculated based on the
selected unit system. In this article, examples are solved
with SI units.
As explained in the GHXSizing instructions tab, yellow
cells are input values typically used in all sizing methods.
For example, cells D36 and D37 in Table 1 indicate
that a 10 × 2 bore fi eld has been selected. Orange cells
are values used for a particular method, and blue cells
present results. Green buttons are used to calculate the
borehole thermal resistance and the borehole length.
As shown in cells D83 and C91 in Table 1, the borehole
length obtained after fi ve iterations is 85.0 m (279 ft)
(for a total of 1700 m [5,580 ft] for the entire bore fi eld),
and Tp = 1.58°C (2.8°F). These results are the same as
those of Example 1 in the 2019 ASHRAE Handbook-HVAC
Applications.12 In some cases, there may be slight differences
because GHXSizing calculates the effective
ground thermal resistances using the Cooper relationship,13
while the Handbook uses interpolated values on a
G-factor curve.
In this example, the borehole thermal resistance, Rb, is
set to 0.11 m·K/W (0.19 h·ft·°F/Btu) in cell D30. However,
it is possible to let GHXSizing calculate using the socalled
fi rst order multipole method.14 This can be done
by fi lling cells D18 to D29 and clicking the " Estimate
Rb " button in cell A30. If thermal short circuit between
the downward and upward pipes is suspected (for long
boreholes and/or low fl ow rates), then it is suggested
to calculate the effective borehole thermal resistance
Rb*14 by selecting the second option in cell A31. For these
cases, it may also be interesting to calculate the fl uid
temperature profi le inside the borehole by checking the
box near the " Calculate Length " button in cell A73. It is
worth noting that the method to calculate Tp is relatively
sensitive to the number of hollow cylinders, Npc, and
their increment spacing, Bc.
Modifi ed ASHRAE Sizing Equation
Given the uncertainty in the determination of Tp and
Fsc with the classic ASHRAE equation, a modifi ed version
of this equation has been proposed1 (Equation 6). As in
the classic ASHRAE equation (Equation 5), the three terms
D E C E M B E R 2 0 2 1 ashrae.o rg ASHRAE JOURNAL
29
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ASHRAE Journal - December 2021

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