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An analytical method to calculate borehole fluid temperatures for time-scales from minutes to decades.


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.


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.


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-[tau]), multiplied by the time derivative of the step-response solution taken at time [tau].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.


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 [q.sub.0] is injected into the circulating fluid with temperature [T.sub.f] (t). The fluid has a thermal capacity of [C.sub.p]. The pipe thermal resistance is [R.sub.p], and the pipe's outer boundary temperature is [T.sub.p](t). The heat flux [q.sub.p](t) flows through the pipe wall to the grout. The thermal conductivity and the thermal diffusivity of the grout are [[lambda].sub.g] and [a.sub.g], respectively. The heat flux [q.sub.b](t) flows across the borehole boundary to the surrounding ground (soil). The borehole boundary temperature is [T.sub.b](t). The thermal conductivity and the thermal diffusivity of the ground (soil) are [lambda].sub.s] and [a.sub.s], 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, [[bar of T].sub.f](S), is readily obtained from the thermal network. Finally, the fluid temperatures in time domain are obtained from [[bar of T].sub.f](S) 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).


The long-term step response is obtained from a continuous line heat source with the strength [q.sub.0] (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:



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 s = 1/[square root of 4a(t - t')], the line-source solution may be written in the following way:


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.


Substituting [](r, t, z) from Equation 2 into Equation 3 gives:


Next, the double integral I in the expression for [[bar of T]](r, t) must be evaluated. Applying the substitutions sz = sD+u and sz'= sD+v, results in:


Equation 5 can be rewritten as:

I = 1/H[s.sup.2] * [](Hs, Ds) (6)

When evaluating the double integral [](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:


The final expression for the double integral becomes:

[](h, d) = 2 * ierf(h) + 2 * ierf(h + 2d) - ierf(2h + 2d) - ierf (2d) (8)

The mean temperature (4) over the borehole length can now be represented as a single integral:



The mean temperature at the borehole radius [r.sub.b] gives the long-term response for a single borehole:

[T.sub.1](t) = [[bar of T]]([r.sub.b], t) (10)

The time derivative of the response temperature [T.sub.1](t) is readily obtained since time only occurs in the lower limit of the integral:


The last factor involves the derivative of 1/[square root of 4at]. 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 ([x.sub.j], [y.sub.j], z), D < z < D+H, j = 1, 2, ..., N. The total temperature field becomes:


The mean temperature is needed along the borehole wall (bw) for any borehole i.


Here [r.sub.i, j] denotes the radial distance between borehole i and j (i [not equal to] j). The contribution from the own heat source of the borehole i is obtained for the radial distance [r.sub.b].

[r.sub.i, j] = [r.sub.b], [r.sub.i, j] = [square root of [([x.sub.i] - [x.sub.j]).sup.2] + [([y.sub.i] - [y.sub.j]).sup.2]] i [not equal to] j (14)

The mean borehole wall temperature for the entire set of N boreholes is:


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:


Here, the function Ie(s) involves a double sum in the exponentials:


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, [I.sub.e] and [] are given by (17) and (7-8), respectively:

(d[T.sub.N])/(dt) = [q.sub.0]/(4[pi][lambda]) x [I.sub.e](1/[square root of 4at]) x [](H/[square root of 4at], D/[square root of 4at]) x 1/H x[square root of a/t] (18)

Here, [I.sub.e] and [] are given by (17) and (7-8), respectively:

The following examples show how the exponent [I.sub.e](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 [r.sub.i, j]. Three terms involve [r.sub.b], four terms involve B, and two terms involve 2B. Therefore:


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 [r.sub.b], B, [square root of 2]B, 2B, [square root of 5]B, and [square root of 8]B. For example, in Figure 2, the diagonal distance [square root of 8]B 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:


The sum of the coefficients before the exponentials is 81 in Equation 20.



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 b reeking time ([]) 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 [] = 100 hours.



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.sup.-2] to [10.sup.6] hours. The three curves are identical below the breaking time. The curves start to deviate from each other after 500 hours.
Table 1. Parameters Considered for Examples

Property                  Value

Heat injection rate     10 W/m (10.4 Btu/h * ft)

Borehole radius         55 cm (22 in)

Pipe radius             28 cm (11 in)

Ground (soil)

  thermal conductivity  ([[lambda].sub.s]) 3.0 W/m * K
                        (1.73 Btu/h * ft * [degrees]F)

  density               2500 kg/[m.sup.3] (156 lb/[ft.sup.3])

  heat capacity         750 J/kg * K (0.18 Btu/lb * [degrees]F)


  thermal conductivity  1.5 W/m * K (0.87 Btu/h *
  ([[lambda].sub.g])    ft * [degrees]F)

  density               1550 kg/[m.sup.3] (97 lb/[ft.sup.3])

  heat capacity         2000 J/kg * K (0.48 Btu/lb * [degrees]F)

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.


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.



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 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.


a = thermal diffusivity ([m.sup.2]/s or [ft.sup.2]/h)

B = spacing between boreholes (m or ft)

C = thermal capacity per unit length (J/m * K or Btu/ft * [degrees]F)

c = specific heat capacity (J/kg * K or Btu/lb * [degrees]F)

D = starting point of active borehole depth (m or ft)

H = active borehole height (m or ft)

[lambda] = thermal conductivity (W/m * K or Btu/h * ft * [degrees]F)

q = rate of heat transfer per unit length (W/m or Btu/h * ft)

R = thermal resistance (m * K/W or h * ft * [degrees]F/Btu)

[bar of R] (s) = thermal resistance in the Laplace domain (m * K/W or h * ft * [degrees]F/Btu)

r = radius (m or ft)

[rho] = density (kg/[m.sup.3] or lb/[ft.sup.3])

s = Laplace transform variable (in the short-term response) and

= 1/[square root of 4a(t - t')] (in the long-term response)

T = temperature (K or [degrees]F)

[bar of T] = mean temperature (K or [degrees]F)

[bar of T] (s) = Laplace transform of T (K * s or [degrees]F * h)

t = time (s or h)

z = vertical coordinate


b = borehole

bw = borehole wall

f = fluid

g = grout

ls = line-source

p = pipe

s = ground (soil)


Bandos, T.V., Montero, A., Fernandez, E., Santander, J., Isidro, J., Perez, J., Cordoba, P. and Urchueguia, 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.

Johan Claesson, Ph.D. Saqib Javed, P.E. Student Member ASHRAE

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.
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Author:Claesson, Johan; Javed, Saqib
Publication:ASHRAE Transactions
Article Type:Report
Geographic Code:4EUSW
Date:Jul 1, 2011
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