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Johan Claesson is a professor at Chalmers University of Technology and Lund University of Technology, Sweden. Saqib Javed is a
graduate student at Chalmers University of Technology, Sweden.
An Analytical Method to Calculate Borehole
Fluid Temperatures for Time-scales from
Minutes to Decades
Johan Claesson, Ph.D. Saqib Javed, P.E.
Student Member ASHRAE
ABSTRACT
Knowledge of borehole exit fluid temperature is required to optimize the design and performance of ground source heat
pump systems. The borehole exit fluid temperature depends upon the prescribed heat injection and extraction rates. This
paper presents a method to determine the fluid temperature of a single or a multiple borehole heat exchanger for any
prescribed heat injection or extraction rate. The fluid temperature, from minutes to decades, is determined using step
response functions. An analytical radial solution is used for shorter times. A finite line-source solution is used for longer
times. The line-source response function has been reduced to one integral only. The derivative, the weighting function, is
given by an explicit formula both for single boreholes and any configuration of vertical boreholes.
INTRODUCTION
Optimizing the design and performance of ground source heat pump (GSHP) system requires accurate knowledge of the
fluid temperatures exiting the borehole heat exchanger. The fluid temperature exiting a borehole heat exchanger depends
upon the short-term and the long-term thermal response of the borehole and the ground surrounding the borehole,
respectively. For a multiple borehole heat exchanger, the exiting fluid temperature also depends upon the thermal interactions
between the boreholes. The development of the thermal response of the ground surrounding the borehole field is a slow
process and depends upon the injections and extractions of ground heat, over time. Because both the thermal mass and the
thermal capacity of the ground surrounding a borehole field are very large, the changes in ground temperatures are very slow.
A time resolution of months or years is typically used to study the temperature development of the ground. On the other
hand, the borehole heat exchanger itself has limited thermal mass and capacity and, consequently, the heat transfer inside the
borehole is more sensitive to any changes in the required injection or extraction rates. As a result, the thermal response of the
borehole is quite rapid and, therefore, is studied using a time resolution ranging from minutes to hours. Development of
thermal interactions between different boreholes is again a slow and long-term process and, thus, requires monthly or yearly
time resolution. Determining the accurate borehole fluid temperatures is an intricate procedure as it involves thermal
processes that vary from short- to long-term intervals, with time resolutions ranging from minutes to years. At present, no
single model exists that can effectively calculate both the short-term thermal response of the borehole and the long-term
development of surrounding ground temperatures.
EXISTING SOLUTIONS
Traditionally, the focus of borehole heat transfer related research has been to determine the long-term response of the
borehole heat exchanger. A number of analytical and numerical methods, including the classical line and cylindrical source
solutions (Ingersoll et al., 1954), have been developed to model the development of the ground temperature surrounding the
borehole. The classical line and cylindrical source methods provide solutions to the radial transient heat transfer problem in
the ground, assuming the borehole to be a line or a cylindrical heat source of infinite length. Various discrepancies occur
when applying these two solutions to model the borehole heat transfer. These solutions not only ignore the end effects of
their heat sources, they also ignore the thermal properties of the borehole elements. Moreover, these solutions are inaccurate
when determining the short-term response of the borehole because of their underlying assumptions regarding geometry and
the length of their heat sources. Some of these issues were addressed by Eskilson (1987), who used the finite line-source
approach to develop the non-dimensional thermal response solutions, also known as g-functions. The g-functions were
developed using a numerical approach that considered the transient radial-axial heat transfer in the borehole heat exchanger.
The g-functions are valid for times longer than 200 hours (Yavuzturk, 1999). Eskilson also determined the thermal
interactions between boreholes using intricate superposition of numerical solutions for each borehole. The use of g-functions
to determine the borehole fluid temperature is somewhat restricted by the fact that these functions need to be computed
numerically, which is a time-consuming and computationally-intensive task. Hence, these functions are pre-computed for
different borehole heat exchanger geometries and configurations and are stored as databases in ground loop design software.
Lately, several researchers have also attempted to develop analytical and semi-analytical g-functions to address the
flexibility issues of numerically-developed g-functions. Zeng et al. (2002) developed an analytical g-function expression
using a constant value of borehole wall temperature, taken at the middle of the finite line-source. Lamarche and Beauchamp
(2007) developed another expression for analytical g-function using the integral mean temperature along the finite line-
source. The authors compared their analytical g-function to numerically obtained g-functions for different cases. They
concluded that using the integral mean temperature along the borehole length, instead of the temperature at the middle of the
borehole, gives more accurate results. Bandos et al. (2009) have developed simple approximate solutions for the cases
considered by Zeng and Lamarche and Beauchamp.
In the last decade or so, the calculation of short-term response to optimize the design and performance of a borehole
heat exchanger has also attracted the interest of many researchers. Yavuzturk (1999) extended the work of Eskilson and
developed g-functions for times between 2.5 min and 200 hours using a numerical approach. Xu and Spitler (2006) developed
a numerical model with variable convective resistance and the thermal mass of the fluid to determine short-term borehole
response. Beier and Smith (2003) and Bandyopadhyay et al. (2008) developed semi-analytical solutions based on Laplace
transforms. With regard to long-term response, the numerical and semi-analytical solutions used to determine the short-term
response of a borehole are also computationally intensive. Recently, Javed and Claesson (2011) developed an analytical
approach to determine the short-term response of borehole heat exchangers.
PROBLEM STATEMENT AND SOLUTION METHODOLOGY
The performance optimization of a GSHP system requires knowledge of fluid temperature for any prescribed heat
injection or extraction rate. The fluid temperatures can be simulated using a short-term response solution. However, at
present, the use of short-term borehole response solutions to determine fluid temperature is largely limited to a few software
programs used for ground loop design. These programs use short-term solutions to determine the minimum and maximum
fluid temperatures under peak load conditions when calculating the required length of the borehole heat exchanger. This
approach, though adequate to design a ground heat exchanger, is not well-suited to determining the resulting fluid
temperatures for a prescribed heat injection rate. This paper presents a simple, but accurate, method to calculate the borehole
fluid temperature for any prescribed heat injection rate q(t) (W/m). Both single and multiple borehole heat exchangers are
considered. The required fluid temperature, at any time t, depends upon the value of the injection rate, at time t, and on the
preceding sequence of heat injection.
In this analysis, the so-called step response solution becomes an important tool. This step response solution helps
determine the required fluid temperature for a constant injection rate q0. Next, the fluid temperature for any q(t) is given by
an integral of q(t-
), multiplied by the time derivative of the step-response solution taken at time . The integration in is
taken from zero to sufficiently large values. This means that the time derivative of the step response shows how the preceding
extraction rates influence the current fluid temperature; it is a weighting function for the preceding injection rates.
This paper provides a methodology to calculate the response function from very short times (minutes) to very long
times (years, or longer). For short times, up to 100 hours, an analytical radial solution is used. After this point, a solution
based on the finite line-source is used. It is important to note that the line-source response function has been reduced to one
integral only. The derivative, the weighting function, is given by an explicit formula both for single boreholes and any
configuration of vertical boreholes.
SHORT-TERM RESPONSE
Javed and Claesson (2011) developed a new analytical solution, which they used to calculate the short-term response of
the borehole. The solution models the two legs of the U-tube as a single equivalent-diameter pipe and uses a single average
value to represent the fluid temperatures entering and exiting the U-tube. The resulting radial heat transfer problem is shown
in Figure 1. The heat flux q0 is injected into the circulating fluid with temperature Tf (t). The fluid has a thermal capacity of
Cp. The pipe thermal resistance is Rp, and the pipe’s outer boundary temperature is Tp(t). The heat flux qp(t) flows through the
pipe wall to the grout. The thermal conductivity and the thermal diffusivity of the grout are λg and ag, respectively. The heat
flux qb(t) flows across the borehole boundary to the surrounding ground (soil). The borehole boundary temperature is Tb(t).
The thermal conductivity and the thermal diffusivity of the ground (soil) are λs and as, respectively. The heat transfer
problem, shown in Figure 1a, can be represented by means of the thermal network shown in Figure 1b. The network involves
a sequence of composite resistances. The Laplace transform for the fluid temperature,
, is readily obtained from the
thermal network. Finally, the fluid temperatures in time domain are obtained from
using an inversion formula. The
short-term response solution has been fully validated using both simulated and experimental data. Further details of the
solution can be found elsewhere (Javed and Claesson, 2011).
LONG-TERM RESPONSE
The long-term step response is obtained from a continuous line heat source with the strength q0 (W/m) along the
borehole x = 0, y = 0, and D < z < D+H. The initial ground temperature is zero and the heat emission starts at t = 0. The
solution is obtained by an integration of a point heat source along the borehole and integration in time from zero to t. The
solution is:
(a)
(b)
Figure 1 (a) Geometry, temperatures, heat fluxes and thermal properties of the borehole. (b) The thermal network
for the radial heat flow process for a borehole in the Laplace domain.
T(r,t)
r
rp
λg , ag
Grout
Ground (soil)
Fluid
rb
Tb(t)
Tp(t)
Rp
qp(t) qb(t)
Tf (t)
Cp
q0
λs , as
q0
sTf (s)
1
Cps
Rp
qp (s)
Tp (s)
Rp (s)
Rs (s)
Rt (s) Tb (s)
qb (s)
Rb (s)
0
0
00
(1)
The temperature is zero at the ground surface z = 0. This is achieved by introducing a mirror sink above the ground surface or
subtracting T(r,-z,t) from the solution obtained above. With the substitution
, the line-source solution may
be written in the following way:
(2)
The second exponential in the second integral represents the mirror sink. The mean temperature over the heat source length
D < z < D+H at any radial distance r is of particular interest.
(3)
Substituting Tls(r,z,t) from Equation 2 into Equation 3 gives:
(4)
Next, the double integral I in the expression for
must be evaluated. Applying the substitutions sz = sD+u and
sz′ = sD+v, results in:
(5)
Equation 5 can be rewritten as:
(6)
When evaluating the double integral Ils(h,d), h = Hs, and d = Ds, the integration in v gives error functions with u in the
argument. The second integration in u gives integrals of the error function, as follows:
(7)
The final expression for the double integral becomes:
(8)
The mean temperature (4) over the borehole length can now be represented as a single integral:
(9)
LONG-TERM STEP RESPONSE FOR SINGLE AND MULTIPLE BOREHOLES
The mean temperature at the borehole radius rb gives the long-term response for a single borehole:
(10)
The time derivative of the response temperature T1(t) is readily obtained since time only occurs in the lower limit of the
integral:
(11)
The last factor involves the derivative of
. It is gratifying that the time derivative, which gives the weighting
functions, is obtained as an explicit formula.
Now, consider N vertical boreholes at the positions (xj ,yj ,z), D < z < D+H, j = 1, 2,…, N. The total temperature
field becomes:
(12)
The mean temperature is needed along the borehole wall (bw) for any borehole i.
(13)
Here ri,j denotes the radial distance between borehole i and j (i ≠ j). The contribution from the own heat source of the
borehole i is obtained for the radial distance rb.
(14)
The mean borehole wall temperature for the entire set of N boreholes is:
(15)
This mean temperature is used as the response function. Using Equation 9, the response function for N boreholes may now be
written in the following way:
(16)
Here, the function Ie(s) involves a double sum in the exponentials:
(17)
The time derivative of the response functions for N boreholes is now obtained in the same way as it was for a single borehole.
Here, Ie and Ils are given by (17) and (7-8), respectively:
(18)
Here, Ie and Ils are given by (17) and (7-8), respectively:
The following examples show how the exponent Ie(s) can be obtained for different configurations of multiple borehole
heat exchangers. The first example considers 3 boreholes in a straight line, separated by the spacing B. The double sum in
Equation 17 involves nine terms. The exponent involves the distances ri,j. Three terms involve rb, four terms involve B, and
two terms involve 2B. Therefore:
(19)
The second example considers 9 boreholes in a square with spacing B. The double sum (17) now involves 9 x 9 = 81
terms. The exponent involves the distances rb, B, , 2B, , and . For example, in Figure 2, the diagonal distance
occurs four times between boreholes: 1 to 9, 3 to 7, 7 to 3, and 9 to 1. Counting the number of occurrences for each
distance gives:
(20)
The sum of the coefficients before the exponentials is 81 in Equation 20.
Figure 2 Radial distances between boreholes: (a) Three boreholes in a straight line; (b) Nine boreholes in a square.
COMBINED STEP RESPONSE
The final step response that accounts for both short and long term is obtained in the following way. Up to a certain time,
the radial short-term response is used. After that time period, the long-term response from the line-source solution is used.
One complication that arises is that the line-source solution does not account for the local thermal processes in the borehole.
Figure 3 illustrates this problem. The top curve shows the radial solution for a single borehole and the lower curve shows the
corresponding line-source solution. As shown, the slope of the two curves is very similar between 10 and 1000 hours.
The borehole has thermal resistances over the pipe and the grout. These resistances cause an increase in the fluid
temperature. This means that the line-source solution should be shifted upwards to account for this temperature increase. The
temperature difference at a suitable breaking time (tbt) is added to the line-source solution so that the radial and the line-
source solutions coincide at the breaking time. In the final step response, the radial solution is used, up to the breaking time.
After that, the line-source solution, including the upward shift, is used. The choice of the breaking time is not critical since
the two curves are parallel over a large time span. A reasonable choice is tbt = 100 hours.
B
12 3
2 B
B
2 B
√2 B
√8 B √5 B
12 3
45 6
78 9
(a)
(b)
Figure 3 Short- and medium-term fluid temperatures using the radial solution and the finite line-source solution.
EXAMPLES
In this study, three examples are considered: 1 borehole, 3 boreholes in a line, and 9 boreholes in a square (Figure 2).
Table 1 presents the parameters used for the examples. Figure 4 shows the response functions for the three cases with the
logarithm of time on the horizontal axis. The time span ranges from 10-2 to 106 hours. The three curves are identical below
the breaking time. The curves start to deviate from each other after 500 hours.
Figure 4 also presents a comparison of the long-term and the short-term fluid temperatures predicted by the new method
with those predicted by Eskilson’s g-functions (1987) and a numerical model (Javed & Claesson, 2011), respectively. For the
first two cases of a single borehole and for the 3 boreholes in a straight line, the long-term fluid temperatures, predicted by
the new method and Eskilson’s g-functions, are in very good agreement up to 25 years. For the 9 boreholes in a square, the
agreement is very good up to 10 years and reasonably good afterwards. The difference between the fluid temperatures that
are predicted by the new method and Eskilson’s g-functions increases with time and with the number of boreholes. However,
the difference is relatively small for up to 25 years. For all three cases, the short-term fluid temperature predicted by the new
model is identical to the short-term fluid temperature predicted by the numerical solution.
Table 1. Parameters Considered for Examples
Property
Value
Heat injection rate (q0)
10 W/m (10.4 Btu/h∙ft)
Borehole radius (Rb)
55 cm (22 in)
Pipe radius (Rb)
28 cm (11 in)
Ground (soil)
thermal conductivity (λs)
3.0 W/m∙K (1.73 Btu/h∙ft∙°F)
density (ρs)
2500 kg/m3 (156 lb/ft3)
heat capacity (cs)
750 J/kg∙K (0.18 Btu/lb∙°F)
Grout
thermal conductivity (λg)
1.5 W/m∙K (0.87 Btu/h∙ft∙°F)
density (ρg)
1550 kg/m3 (97 lb/ft3)
heat capacity (cg)
2000 J/kg∙K (0.48 Btu/lb∙°F)
0
1
2
3
4
110 100 1000
Response Temperature (K)
Time (hours)
Radial solution
Finite line-source
Figure 4 Response functions for 1, 3, and 9 boreholes using a combination of radial and finite line-source solutions.
Figure 5 shows the time derivative of the response function, which gives the weighting functions. As discussed in the
problem statement, these weighting functions are the key element in determining the fluid temperature for the prescribed heat
injection or heat extraction rate. Figure 5a shows the weighting function during the first two hours (from the radial solution).
It can be seen that, during these two hours, the function falls by the factor 10. While it will continue to fall strongly with time,
the function is still needed when applied to very long times. Therefore, Figure 5b shows the function multiplied by time t.
(a)
(b)
Figure 5 Time derivatives of response functions (weighting functions) for 1, 3 and 9 boreholes.
CONCLUSION
Knowledge of the borehole exit fluid temperature is critical to the design and the performance optimization of GSHP
systems. The exit fluid temperature depends upon both the short-term response of the borehole and the long-term response of
the surrounding ground. This paper presents a simple analytical method to calculate fluid temperatures for times ranging from
minutes to decades. The short-term borehole response is calculated using a recently developed and well-validated analytical
0
3
6
9
12
15
1.E-02 1.E+00 1.E+02 1.E+04 1.E+06
Fluid Temperature (K)
Time (hours)
1 Borehole (new method)
g-functions (Eskilson)
3 Boreholes (new method)
Numerical Model
9 Boreholes (new method)
Breaking time
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0 0.5 1 1.5 2
dTf/dt
Time (hours)
1 Borehole
3 Boreholes
9 Boreholes
0.0
0.5
1.0
1.5
2.0
1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07
t∙ dTf/dt(K)
Time (hours)
1 Borehole
3 Boreholes
9 Boreholes
Breaking time
solution. The long-term response is calculated using a finite line-source solution. For a single borehole, a closed form
formula (Equations 10 and 7-9) has been developed to determine the long-term step response. For multiple boreholes, a
simple and systematic approach (Equations 16, 17 and 7-9) is introduced to calculate the long-term response. The long-term
response predicted by the new method is in good agreement with the response obtained from Eskilson’s g-functions. The total
response from minutes to decades is obtained by joining the long-term response to the short-term response at a suitable
breaking time. The choice of breaking time is not critical and any time between 10 and 1000 hours may be selected. Finally,
the time derivative of the step response is given as an explicit expression to be used for modelling. This expression shows the
effect of the preceding extraction rates on the current fluid temperature.
NOMENCLATURE
a = thermal diffusivity (m2/s or ft2/h)
B = spacing between boreholes (m or ft)
C = thermal capacity per unit length (J/m∙K or Btu/ft∙°F)
c = specific heat capacity (J/kg∙K or Btu/lb∙°F)
D = starting point of active borehole depth (m or ft)
H = active borehole height (m or ft)
= thermal conductivity (W/m∙K or Btu/h∙ft∙°F)
q = rate of heat transfer per unit length (W/m or Btu/h∙ft)
R = thermal resistance (m∙K/W or h∙ft∙°F/Btu)
= thermal resistance in the Laplace domain (m∙K/W or h∙ft∙°F/Btu)
r = radius (m or ft)
= density (kg/m3 or lb/ft3)
s = Laplace transform variable (in the short-term response) and
=
(in the long-term response)
T = temperature (K or °F)
= mean temperature (K or °F)
= Laplace transform of T (K∙s or °F∙h)
t = time (s or h)
z = vertical coordinate
Subscripts
b = borehole
bw = borehole wall
f = fluid
g = grout
ls = line-source
p = pipe
s = ground (soil)
REFERENCES
Bandos, T.V., Montero, Á., Fernández, E., Santander, J., Isidro, J., Pérez, J., Córdoba, P. and Urchueguía, J, 2009. Finite
line-source model for borehole heat exchangers: effect of vertical temperature variations. Geothermics, 38(2): 263-270.
Bandyopadhyay, G., Gosnold, W., and Mann, M. 2008. Analytical and semi-analytical solutions for short-time transient
response of ground heat exchangers. Energy and Buildings, 40(10): 1816-1824.
Beier, R.A. and Smith, M.D. 2003. Minimum duration of in-situ tests on vertical boreholes. ASHRAE Transactions, 109(2):
475-486.
Eskilson, P. 1987. Thermal analysis of heat extraction boreholes. Department of Mathematical Physics, PhD Thesis, (Lund
University.) Sweden.
Ingersoll, L.R., Zobel, O.J. and Ingersoll, A.C. 1954. Heat conduction with engineering, geological and other applications.
McGraw-Hill, New York.
Javed, S. and Claesson, J. 2011. New analytical and numerical solutions for the short-term analysis of vertical ground heat
exchangers. ASHRAE Transactions, 117(1).
Lamarche, L. and Beauchamp, B. 2007. A new contribution to the finite line-source model for geothermal boreholes. Energy
and Buildings, 39(2): 188-198.
Xu, X. and Spitler, J.D. 2006. Modeling of vertical ground loop heat exchangers with variable convective resistance and
tehrmal mass of the fluid. Proceedings of the 10th international conference on thermal energy storage: Ecostock 2006.
Pomona, NJ, May 31-June 2.
Yavuzturk, C. 1999. Modelling of vertical ground loop heat exchangers for ground source heat pump systems. Building and
Environmental Thermal Systems Research Group, PhD Thesis, (Oklahoma State University.) USA.
Zeng, H.Y., Diao, N.R. and Fang, Z.H. 2002. A finite line-source model for boreholes in geothermal heat exchangers. Heat
Transfer - Asian Research, 31(7): 558-567.
