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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

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