#
Explicit Multipole Formulas for Calculating Thermal Resistance of Single U-Tube Ground Heat Exchangers^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

_{b}) are the two principal parameters that govern the heat transfer mechanism of a borehole heat exchanger, and thus influence the sizing and performance of the overall GSHP system [2]. The heat transfer outside the borehole boundary is dictated by the thermal conductivity of the ground, whereas the heat transfer inside the borehole is characterized by the borehole thermal resistance between the heat carrier fluid and the borehole wall. A high ground thermal conductivity is beneficial for the ground heat transfer. However, being an intrinsic property of the ground, ground thermal conductivity cannot be controlled in practice. On the other hand, a low borehole thermal resistance is desirable for better heat transfer inside the borehole heat exchanger. The borehole thermal resistance depends upon the physical arrangement and the thermal properties of borehole components including grouting, heat exchanger pipes, and the heat carrier fluid. Its value can be engineered to a certain extent by optimizing the geometry and layout of the borehole heat exchanger and by choosing appropriate materials for the borehole components.

## 2. Borehole Thermal Resistance

_{b}) is the ratio of the temperature difference between the heat carrier fluid and the borehole wall to the heat transfer rate per unit length of the borehole. It is defined locally at a specific depth in the borehole as represented in Equation (1), where T

_{f,loc}is the local mean fluid temperature, T

_{b}is the average borehole wall temperature and q

_{b}is the heat transfer rate per unit length of the borehole.

_{b}) i.e., fluid-to-ground resistance is considered to be made up of two major parts: pipe resistance (R

_{p}) and grout resistance (R

_{g}). The pipe resistance includes both conductive resistance of the pipe (R

_{pc}) and convective resistance of the fluid (R

_{pic}). The grout resistance constitutes the thermal resistance between the outer pipe wall of the U-tube and the borehole wall. Its value depends on the pipe resistance and the ground thermal conductivity [13]. As the individual pipe resistances are in parallel with each other, the pipe resistance is generally calculated for a single pipe and is then divided by the total number (N) of pipes.

_{1}and q

_{2}and fluid temperatures T

_{f1}and T

_{f2}.

_{1b}and R

_{2b}are thermal resistances between pipe 1 or 2 and the borehole wall, respectively, and R

_{12}is the fluid-to-fluid thermal resistance between pipes 1 and 2. These resistances include the thermal resistances of the fluid and the pipe. Using the Δ network of Figure 1, the total borehole resistance, R

_{b}, can be determined by setting the fluid temperatures T

_{f1}= T

_{f2}and solving for two parallel resistances R

_{1b}and R

_{2b}:

_{a}by setting the heat flows q

_{1}= −q

_{2}. The resistance, R

_{a}, is the total internal resistance between the upward and downward flowing legs of the U-tube when there is no net heat flow from the borehole. It is related to the fluid-to-fluid internal resistance R

_{12}by the thermal network of Figure 1. The resistance R

_{12}, which is sometimes also referred to as the direct coupling resistance, is the effect of the network representation and is not a directly measurable physical resistance. Resistance R

_{12}is in parallel with the series resistances R

_{1b}and R

_{2b}.

_{a}is critical to the understanding of the effective borehole thermal resistance ${R}_{\mathrm{b}}^{*}$, which is the thermal resistance between the heat carrier fluid, characterized by a simple mean T

_{f}of the inlet and outlet temperatures, and the average borehole wall temperature T

_{b}. The effective borehole thermal resistance is mathematically expressed as Equation (7).

_{b}, defined by Equation (1), and effective borehole thermal resistance ${R}_{\mathrm{b}}^{*}$, defined by Equation (7), is that the borehole thermal resistance applies locally at a specific depth in the borehole, whereas effective borehole thermal resistance applies to the entire borehole. The effective borehole thermal resistance is what is measured in an experiment, for example an in-situ thermal response test. The experimental measurements of mean fluid temperature taken at the top of the borehole include the effects of thermal short-circuiting between the upward and downward flows in the U-tube. Consequently, the effective borehole thermal resistance is often higher than the borehole thermal resistance due to the higher fluid temperature caused by the thermal short-circuiting between the U-tube pipes.

_{a}and direct coupling resistance R

_{12}between the two U-tube legs, respectively, in addition to the borehole thermal resistance R

_{b}.

## 3. Multipole Method for N Pipes in a Borehole

_{n},y

_{n}). The thermal conductivity of the grout region outside the pipes is λ

_{b}(W/m-K), and of the ground outside the boreholes is λ. The borehole radius is r

_{b}and the outer radius of the pipes is r

_{p}. The thermal resistance from the fluid in the pipe to the grout adjacent to the pipe is R

_{p}(m-K/W).

_{n}, n = 1, …, N, and any prescribed average temperature T

_{b,av}around the borehole wall. The fluid temperature in pipe n is T

_{fn}, n = 1, …, N. From this the borehole thermal resistances are determined. The temperature field consists of a linear combination of line heat source solutions and so-called multipole solutions at the center of each pipe:

_{0}(z,z

_{n}) gives the temperature field for a line heat source at z = z

_{n}with the strength ${q}_{n}=2\pi {\lambda}_{\mathrm{b}}$(W/m):

_{b,av}around the borehole wall is zero.

_{n}:

_{n,j}are complex-valued numbers. This means the expression (10) can account for a Fourier expansion of any temperatures around the pipes with Cosine and Sine terms up to order j = J.

_{fn}and the temperature outside the pipe, which varies around the pipe. Let as in Equation (14), $\rho $ be the radial distance from the center of pipe n and ψ the polar angle. The temperature in the grout at the pipe wall and the radial heat flux multiplied by the pipe resistance per unit pipe area are given by the two left-hand terms below:

_{fn}to obtain the prescribed heat fluxes q

_{n}.

_{n,j}, is inserted in Equation (15). It is possible by quite lengthy calculation involving complex-valued formulations and Taylor expansions to separate the expressions in Fourier exponentials of any order j. The Fourier coefficient of order j for any pipe n depends on the heat fluxes and multipole factors:

**q**represents the N prescribed heat fluxes q

_{n}, and the matrix

**P**represents the N·J complex-valued multipole factors P

_{n,j}. The multipole factors are now chosen so that the Fourier coefficients up to order J become zero. This gives N·J equations for the multipole factors:

_{fn}, and the terms above j = J represent the error for the heat balance over the pipe wall as a function of the polar angle ψ:

## 4. Two Symmetrically Placed Pipes in a Borehole

_{p},0) and (−x

_{p},0), respectively. The prescribed heat fluxes from the pipes are q

_{1}and q

_{2}(W/m). The corresponding fluid temperatures are T

_{f1}and T

_{f2}. The main objective is to present a very precise method to calculate the relations between heat flows and fluid temperatures.

_{n,j}turn out to be real-valued numbers. The number of unknowns and equations are reduced to 2·J for the two sets of multipoles P

_{1,j}and P

_{2,j}.

#### 4.1. Multipole Relations for Even and Odd Solutions

_{b,av}= 0.

#### 4.2. Formulas for Even and Odd Thermal Resistances

**V**and

_{J}**Vb**, and a matrix

_{J}**M**for the even case s = +1 and the odd case s = −1:

_{J}**M**matrices appear in Equation (30). The components of the vectors

**V**and

_{J}**Vb**are given by:

_{J}**M**matrices are given by:

**A**

**and**

^{+}**A**

**, which account for various interactions between line sources and multipoles in the boundary conditions at the two pipes, are fairly complicated:**

^{−}**A**matrices are symmetric.

**M**matrix:

#### 4.3. Explicit Multipole Formulas for First, Second and Third-Order

**M**matrices and their inverses are:

**A**matrices and

**V**vectors are from Equations (32) and (34):

**M**matrices. A more explicit formula cannot be given. The third-order ${B}_{3}^{\pm}$ corrections are given by Equation (30) for J = 3:

**M**matrices have the form:

#### 4.4. Thermal Resistances R_{1b}, R_{12}, R_{b} and R_{a}

_{1b}, R

_{12}, R

_{b}and R

_{a}can now be obtained by again considering the Δ network of Figure 1. For the case of equal pipes in symmetric positions, the thermal resistances R

_{1b}is equal to R

_{2b}, and thus the heat fluxes of Equations (3) and (4) become:

_{2}= q

_{1}) and odd (q

_{2}= −q

_{1}) cases can be drawn as shown in Figure 4.

_{b}between the fluid in pipes and the borehole wall consists of two parallel equal resistances, each of value R

_{1b}. On the other hand, from the odd case of Figure 4, it can be noticed that the total internal resistance R

_{a}between the two pipes consists of a pair of equal series resistances, each of value 0.5 R

_{12}. Hence, Equations (5) and (6) can now be simplified to:

_{1b}and R

_{12}are obtained in the form of even and odd thermal resistance ${R}_{J}^{+}$ and ${R}_{J}^{-}$ using Figure 4 and Equations (24) and (25).

_{b}and R

_{a}are obtained in the form of even and odd thermal resistance ${R}_{J}^{+}$ and ${R}_{J}^{-}$ using Equations (47) and (48).

## 5. Comparison with Existing Multipole Solutions

_{p}also remains constant at 0.05 m-K/W. For each borehole diameter, three shank spacing configurations, i.e., close, moderate and wide, corresponding, respectively, to Paul’s [8] Configuration A, Configuration B and Configuration C, are considered. Four levels of ground thermal conductivity ranging from 1–4 W/m-K, and six levels of grout thermal conductivity ranging from 0.6–3.6 W/m-K are used. Given the existing and reasonably foreseeable values of design parameters, the 216 cases used for the comparison bracket almost all real-world single U-tube borehole heat exchangers.

_{g}) values obtained from the second-order and third-order multipole formulas are, respectively, within 0.5% and 0.2% of the tenth-order multipole method for all 216 cases. Also, the mean absolute percentage error of the results obtained from the second-order and third-order multipole formula are, respectively, smaller than 0.2% and 0.1%. In comparison, the mean and maximum absolute percentage errors for the zeroth-order multipole formula are as high as 9% and 30%, respectively. The first-order multipole formula has smaller errors than the zeroth-order formula. The mean and maximum absolute percentage errors for the first-order multipole formula are 0.7% and 2.2%, respectively.

_{a}) values calculated from the second order and third-order multipole formulas are, respectively, within 1% and 0.1% of the tenth-order multipole method for all 216 cases. The mean absolute percentage error of the results obtained from the second-order and third-order multipole formulas never exceed 0.4% and 0.1%, respectively. In comparison, the zeroth-order and the first-order multipole formulas give maximum absolute percentage errors of approximately 38% and 6%, respectively. The mean absolute percentage errors of the zeroth-order and the first-order multipole expressions are as high as 23% and 3%, respectively.

_{g}values, and the right-side ones show the total internal thermal resistance R

_{a}values, plotted against the grout thermal conductivity. Each figure presents three curves corresponding to close, moderate and wide shank spacings, shown in black, blue and red colors, respectively. The exact value of the shank spacing for each case is provided in Table 1. It must be pointed out that multipole formulas presented in the previous section, calculate the borehole thermal resistance and not the grout thermal resistance. However, in order to be consistent with the dataset provided by [13], the values of grout thermal resistance have been calculated and presented in Figure 5, Figure 6 and Figure 7. The grout thermal resistance values have been determined from Equation (2), by subtracting half of the fixed pipe resistance of 0.05 m-K/W from the corresponding borehole thermal resistance values obtained from the multipole formulas. Computing the grout thermal resistance directly by disregarding the pipe resistance gives erroneous results for all but zeroth-order multipole calculations.

_{g}and the total internal thermal resistance R

_{a}. The thermal resistance values calculated from the second-order multiple formulas are always within 1% of the original tenth-order multipole method over the entire range of parameters. Hence, it can be concluded that due to their excellent accuracy and relative ease of implementation, the second-order multipole formulas are recommended for calculation of borehole thermal resistance and total internal thermal resistance for all cases where the two legs of the U-tube are placed symmetrically in the borehole.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

${B}_{J}^{\pm}$ | = | Multipole correction for multipole order J for even (+) and odd (−) cases, See Equation (30) |

b_{k} | = | Dimensionless parameter, see Equation (29) |

c_{f} | = | Specific heat of the circulating fluid in the U-tube, J/kg-K |

H | = | Depth of the borehole, m |

J | = | Number of multipoles, j = 0, 1, 2, …, J |

N | = | Number of pipes in the borehole; N = 2 for single U-tube |

P_{n,j} | = | Multipole factor of order j for pipe n |

${P}_{n,j}^{\pm}$ | = | Multipole factor of order j for pipe n for even (+) and odd (−) cases |

p_{0} | = | Dimensionless parameter, see Equation (29) |

p_{1} | = | Dimensionless parameter, see Equation (29) |

p_{2} | = | Dimensionless parameter, see Equation (29) |

q_{b} | = | Heat rejection rate per unit length of borehole, W/m |

q_{n} | = | Heat rejection rate per unit length of pipe n, W/m |

R_{a} | = | Total internal borehole thermal resistance, m-K/W |

R_{b} | = | Local borehole thermal resistance between fluid in the U-tube to borehole wall, m-K/W |

${R}_{\mathrm{b}}^{*}$ | = | Effective borehole thermal resistance, m-K/W |

R_{g} | = | Grout thermal resistance, m-K/W |

${R}_{J}^{\pm}$ | = | Thermal resistance for even (+) and odd (−) cases for J multipoles, m-K/W |

R_{n}_{b} | = | Thermal resistance between U-tube leg n and borehole wall, m-K/W |

R_{p} | = | Total fluid-to-pipe resistance for a single pipe i.e., one leg of the U-tube, m-K/W |

R_{12} | = | Thermal resistance between U-tube legs 1 and 2, m-K/W |

r_{b} | = | Radius of the borehole, m |

r_{p} | = | Outer radius of the pipe making up the U-tube, m |

T_{b} | = | Borehole wall temperature, °C |

T_{b,av} | = | Average temperature at the borehole wall, °C |

T_{f} | = | Mean fluid temperature inside the U-tube, °C |

T_{f,loc} | = | Local mean fluid temperature, °C |

T_{fn} | = | Fluid temperature in U-tube leg n, °C |

${T}_{\mathrm{f}n}^{\pm}$ | = | Fluid temperature in U-tube leg n for even and odd cases, °C |

V_{f} | = | Volume flow rate of the circulating fluid in the U-tube, m^{3}/s |

x_{p} | = | Half shank spacing; half of center-to-center distance between U-tube legs, m, See Figure 3 |

β | = | Dimensionless thermal resistance of one U-tube leg, see Equation (20) |

λ | = | Thermal conductivity of the ground, W/m-K |

λ_{b} | = | Thermal conductivity of the borehole/grout, W/m-K |

${\rho}_{\mathrm{f}}$ | = | Density of the circulating fluid in the U-tube, kg/m^{3} |

σ | = | Thermal conductivity ratio, dimensionless, see Equation (20) |

## References

- Lund, J.W.; Boyd, T.L. Direct utilization of geothermal energy 2015 worldwide review. Geothermics
**2016**, 60, 66–93. [Google Scholar] [CrossRef] - Vieira, A.; Alberdi-Pagola, M.; Christodoulides, P.; Javed, S.; Loveridge, F.; Nguyen, F.; Cecinato, F.; Maranha, J.; Florides, G.; Prodan, I.; et al. Characterisation of ground thermal and thermo-mechanical behaviour for shallow geothermal energy applications. Energies
**2017**, 10, 2044. [Google Scholar] [CrossRef] - Javed, S.; Spitler, J.D. Calculation of borehole thermal resistance. In Advances in Ground-Source Heat Pump Systems, 1st ed.; Rees, S.J., Ed.; Woodhead Publishing: Cambridge, UK, 2016; pp. 63–95. ISBN 978-0-08-100311-4. [Google Scholar]
- Spitler, J.D.; Gehlin, S.E. Thermal response testing for ground source heat pump systems—An historical review. Renew. Sustain. Energy Rev.
**2015**, 50, 1125–1137. [Google Scholar] [CrossRef] - Kavanaugh, S.P. Simulation and Experimental Verification of Vertical Ground-Coupled Heat Pump Systems. Ph.D. Thesis, Oklahoma State University, Stillwater, OK, USA, May 1985. [Google Scholar]
- Gu, Y.; O’Neal, D.L. Development of an equivalent diameter expression for vertical U-tubes used in ground-coupled heat pumps. ASHRAE Trans.
**1998**, 104, 347–355. [Google Scholar] - Shonder, J.A.; Beck, J.V. Field test of a new method for determining soil formation thermal conductivity and borehole resistance. ASHRAE Trans.
**2000**, 106, 843–850. [Google Scholar] - Paul, N.D. The Effect of Grout Thermal Conductivity on Vertical Geothermal Heat Exchanger Design and Performance. Master’s Thesis, South Dakota University, Brookings, SD, USA, July 1996. [Google Scholar]
- Sharqawy, M.H.; Mokheimer, E.M.; Badr, H.M. Effective pipe-to-borehole thermal resistance for vertical ground heat exchangers. Geothermics
**2009**, 38, 271–277. [Google Scholar] [CrossRef] - Bauer, D.; Heidemann, W.; Müller-Steinhagen, H.; Diersch, H.J. Thermal resistance and capacity models for borehole heat exchangers. Int. J. Energy Res.
**2011**, 35, 312–320. [Google Scholar] [CrossRef] - Liao, Q.; Zhou, C.; Cui, W.; Jen, T.C. New correlations for thermal resistances of vertical single U-Tube ground heat exchanger. J. Therm. Sci. Eng. Appl.
**2012**, 4, 031010. [Google Scholar] [CrossRef] - Lamarche, L.; Kajl, S.; Beauchamp, B. A review of methods to evaluate borehole thermal resistances in geothermal heat-pump systems. Geothermics
**2010**, 39, 187–200. [Google Scholar] [CrossRef] - Javed, S.; Spitler, J.D. Accuracy of borehole thermal resistance calculation methods for grouted single U-tube ground heat exchangers. Appl. Energy
**2017**, 187, 790–806. [Google Scholar] [CrossRef] - Bennet, J.; Claesson, J.; Hellström, G. Multipole Method to Compute the Conductive Heat Flows to and between Pipes in a Composite Cylinder. Notes on Heat Transfer 3; University of Lund: Lund, Sweden, 1987. [Google Scholar]
- Claesson, J.; Bennet, J. Multipole Method to Compute the Conductive Heat Flows to and between Pipes in a Cylinder. Notes on Heat Transfer 2; University of Lund: Lund, Sweden, 1987. [Google Scholar]
- Claesson, J.; Hellström, G. Multipole method to calculate borehole thermal resistances in a borehole heat exchanger. HVAC R Res.
**2011**, 17, 895–911. [Google Scholar] - Young, T.R. Development, Verification, and Design Analysis of the Borehole Fluid Thermal Mass Model for Approximating Short Term Borehole Thermal Response. Ph.D. Thesis, Oklahoma State University, Stillwater, OK, USA, December 2004. [Google Scholar]
- He, M. Numerical Modelling of Geothermal Borehole Heat Exchanger Systems. Ph.D. Thesis, De Montfort University, Leicester, UK, February 2012. [Google Scholar]
- Al-Chalabi, R. Thermal Resistance of U-Tube Borehole Heat Exchanger System: Numerical Study. Master’s Thesis, University of Manchester, Manchester, UK, September 2013. [Google Scholar]
- Earth Energy Designer (EED), v3.2; BLOCON: Lund, Sweden, 2015.
- Spitler, J.D. GLHEPRO—A Design Tool for Commercial Building Ground Loop Heat Exchangers. In Proceedings of the Fourth International Heat Pumps in Cold Climates Conference, Aylmer, QC, Canada, 17–18 August 2000. [Google Scholar]
- Claesson, J. Multipole Method to Calculate Borehole Thermal Resistances; Mathematical Report; Chalmers University of Technology: Gothenburg, Sweden, 2012. [Google Scholar]
- Hellström, G. Ground Heat Storage—Thermal Analyses of Duct Storage Systems—Theory. Ph.D. Thesis, University of Lund, Lund, Sweden, April 1991. [Google Scholar]
- Spitler, J.D.; Javed, S.; Ramstad, R.K. Natural convection in groundwater-filled boreholes used as ground heat exchangers. Appl. Energy
**2016**, 164, 352–365. [Google Scholar] [CrossRef] - Zeng, H.; Diao, N.; Fang, Z. Heat transfer analysis of boreholes in vertical ground heat exchangers. Int. J. Heat Mass Transf.
**2003**, 46, 4467–4481. [Google Scholar] [CrossRef] - Ma, W.; Li, M.; Li, P.; Lai, A.C. New quasi-3D model for heat transfer in U-shaped GHEs (ground heat exchangers): Effective overall thermal resistance. Energy
**2015**, 90, 578–587. [Google Scholar] [CrossRef]

**Figure 2.**Steady-state heat conduction in a composite circular region with heat flows between N number of pipes and the adjacent ground region.

**Figure 4.**Thermal resistance networks for equal pipes in symmetric positions for even (

**left**) and odd (

**right**) cases.

**Figure 5.**Grout thermal resistance (R

_{g}) and total internal resistance (R

_{a}) for close (2x

_{p}= 32 mm), moderate (2x

_{p}= 43 mm) and wide (2x

_{p}= 64 mm) configurations with 2r

_{b}= 96 mm and λ = 4 W/m-K.

**Figure 6.**Grout thermal resistance (R

_{g}) and total internal resistance (R

_{a}) for close (2x

_{p}= 32 mm), moderate (2x

_{p}= 75 mm) and wide (2x

_{p}= 160 mm) configurations with 2r

_{b}= 192 mm and λ = 4 W/m-K.

**Figure 7.**Grout thermal resistance (R

_{g}) and total internal resistance (R

_{a}) for close (2x

_{p}= 32 mm), moderate (2x

_{p}= 107 mm) and wide (2x

_{p}= 256 mm) configurations with 2r

_{b}= 288 mm and λ = 4 W/m-K.

**Table 1.**Summary of comparison cases provided by [13].

Parameters | Levels | No. of Levels |
---|---|---|

Ratio of borehole diameter to outer pipe diameter ^{1}, 2r_{b}/2r_{p} | 3, 6, 9 | 3 |

Shank spacing configuration, 2x_{p} (mm) | Close, Moderate, Wide ^{2}For r _{b}/r_{p} = 3, 2x_{p} = 32, 43, 64For r _{b}/r_{p} = 6, 2x_{p} = 32, 75, 160For r _{b}/r_{p} = 9, 2x_{p} = 32, 107, 256 | 3 |

Ground thermal conductivity, λ (W/m-K) | 1, 2, 3, 4 | 4 |

^{1}Pipe outer diameter (2r

_{p}) is fixed at 32 mm, borehole diameters (2r

_{b}) are 96 mm, 192 mm, and 288 mm.

^{2}Corresponding to Paul’s [8] A–C configurations.

**Table 2.**Mean and maximum absolute percentage errors in calculation of the grout thermal resistance (R

_{g}) for the 216 cases provided by [13].

Method | Shank Spacing Configuration | Ground Conductivity | |||||
---|---|---|---|---|---|---|---|

Low (0.6–1.2 W/m-K) | Moderate (1.2–2.4 W/m-K) | High (2.4–3.6 W/m-K) | |||||

Mean | Max | Mean | Max | Mean | Max | ||

Zeroth-order multipole | Close | 6.1 | 12.4 | 2.9 | 8.0 | 1.0 | 2.5 |

Moderate | 3.3 | 10.8 | 1.5 | 6.5 | 0.4 | 1.6 | |

Wide | 8.9 | 30.4 | 1.9 | 11.2 | 0.3 | 1.8 | |

First-order multipole | Close | 0.2 | 0.4 | 0.2 | 0.6 | 0.7 | 1.5 |

Moderate | 0.0 | 0.2 | 0.0 | 0.2 | 0.1 | 0.5 | |

Wide | 0.5 | 2.2 | 0.0 | 0.2 | 0.1 | 0.6 | |

Second-order multipole | Close | 0.0 | 0.0 | 0.1 | 0.2 | 0.2 | 0.5 |

Moderate | 0.0 | 0.0 | 0.0 | 0.1 | 0.0 | 0.1 | |

Wide | 0.1 | 0.3 | 0.0 | 0.0 | 0.0 | 0.1 | |

Third-order multipole | Close | 0.0 | 0.0 | 0.0 | 0.1 | 0.1 | 0.2 |

Moderate | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | |

Wide | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

**Table 3.**Mean and maximum absolute percentage errors in calculation of the total internal thermal resistance (R

_{a}) for the 216 cases provided by [13].

Method | Shank Spacing Configuration | Ground Conductivity | |||||
---|---|---|---|---|---|---|---|

Low (0.6–1.2 W/m-K) | Moderate (1.2–2.4 W/m-K) | High (2.4–3.6 W/m-K) | |||||

Mean | Max | Mean | Max | Mean | Max | ||

Zeroth-order multipole | Close | 23.3 | 37.6 | 6.9 | 15.0 | 1.2 | 2.7 |

Moderate | 1.8 | 7.8 | 1.1 | 6.0 | 0.3 | 1.8 | |

Wide | 1.9 | 8.5 | 0.4 | 2.3 | 0.2 | 1.3 | |

First-order multipole | Close | 3.2 | 5.9 | 0.6 | 0.7 | 0.7 | 0.7 |

Moderate | 0.2 | 0.8 | 0.0 | 0.2 | 0.1 | 0.2 | |

Wide | 0.3 | 1.2 | 0.0 | 0.1 | 0.0 | 0.1 | |

Second-order multipole | Close | 0.4 | 1.0 | 0.2 | 0.2 | 0.1 | 0.2 |

Moderate | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | |

Wide | 0.0 | 0.2 | 0.0 | 0.0 | 0.0 | 0.0 | |

Third-order multipole | Close | 0.0 | 0.1 | 0.1 | 0.1 | 0.0 | 0.0 |

Moderate | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | |

Wide | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Claesson, J.; Javed, S.
Explicit Multipole Formulas for Calculating Thermal Resistance of Single U-Tube Ground Heat Exchangers. *Energies* **2018**, *11*, 214.
https://doi.org/10.3390/en11010214

**AMA Style**

Claesson J, Javed S.
Explicit Multipole Formulas for Calculating Thermal Resistance of Single U-Tube Ground Heat Exchangers. *Energies*. 2018; 11(1):214.
https://doi.org/10.3390/en11010214

**Chicago/Turabian Style**

Claesson, Johan, and Saqib Javed.
2018. "Explicit Multipole Formulas for Calculating Thermal Resistance of Single U-Tube Ground Heat Exchangers" *Energies* 11, no. 1: 214.
https://doi.org/10.3390/en11010214