# On Thermally Interacting Multiple Boreholes with Variable Heating Strength: Comparison between Analytical and Numerical Approaches

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Inside Borehole: Variable Heat Flux Model

_{f1}, T

_{f2}and T

_{b}are the temperatures of the fluid running downwards, the fluid running upwards and borehole wall, respectively, and

_{11}and R

_{22}are the thermal resistance between the circulating fluid and the borehole wall, and R

_{12}is the resistance between the tubes (Figure 1) obtained from the relations derived by detail by Hellström [9]. For different numbers of pipes in any position in the borehole, Claesson and Hellström [10] present a method to calculate the thermal resistances between the heat carrier fluid in the pipes of the borehole and the immediate vicinity of the surrounding ground. In most engineering applications, the configuration of the U-tube in the borehole may be assumed symmetric, and here it is assumed that the thermal resistance between the circulating fluid in each of the tubes and the borehole wall is equal.

_{f}

^{'}is the temperature of the fluid entering the U-tube. The temperature profiles of the fluids flowing in the U-tubes in the boreholes (Θ

_{1}(Z) and Θ

_{2}(Z)) are formulated by Zeng et al. [5].

_{f}

^{'}= 290.6 K) for the VHS model, resulting in the same total heat conduction in the soil, is assumed.

#### 2.2 Outside Borehole

**Figure 3.**Horizontal cross sections (xy) of the solution domain at the borehole mid-length (z = 0 m).

#### 2.2.1. Analytical Approach

_{i}and w

_{i}are distances of boreholes i along x and y directions, respectively. For the case of multiple boreholes shown in Figure (4), Equation (8) can be simplified to:

**Figure 4.**System geometric parameters for two boreholes at distances R

_{1}and R

_{2}from a desired point (x,y) in the surrounding soil.

#### 2.2.2. Numerical Approach

**Figure 6.**Simulation model for horizontal cross sections (xy) at (

**a**) borehole mid-length (z = 0 m), and (

**b**) the ground surface (z = 110 m).

_{p}, a

_{p}

^{0}and a

_{nb}are temperature coefficients which are calculated based on the geometric characteristics of each control volume and the time step in the numerical solution.

#### 2.2.3. Initial and Boundary Conditions

^{2}on the borehole wall can be assumed since in order to study the thermal interaction between multiple boreholes, their inner dynamic heat exchange process can be of second priority compared to the heat dissipation in the soil surrounding them. As a second approach, a variable heat flux (VHF) along the borehole is calculated by defining the temperature profiles of the fluid running along the tubes in the borehole. It should be noted that the current article focuses only on the variation of heating strength along the borehole length. Since only the existence of such a variation is intended to be discussed, the current article does not provide typical values for the borehole spacing and the heat flux on the borehole wall and lower values are chosen in order to keep the solution domain size smaller in the numerical solution. It should also be noted that the temperature of the soil in the current problem is assumed to be constant throughout the whole operation time and therefore the current solution is only valid for low temperature variations in the soil surrounding the boreholes which is only gained by assuming lower heat flux values on the borehole wall. Modifying the current problem to one with typical industrial values for ground heat pump systems will need the soil temperature to be assumed variable and is subject of ongoing research by the authors.

## 3. Results and Discussion

(a) Inside the borehole (U-tube and grout region). | |||||||

H(m) | r_{b}(m) | r_{p}(m) | D(m) | D_{b}(m) | k_{b}(W/mK) | (kg/s) | c(J/kgK) |

200 | 0.05 | 0.02 | 0.02 | 2 | 1 | 0.2 | 4187 |

(b) Outside the borehole (soil region). | |||||||

k(W/mK) | c(J/kgK) | ρ (kg/m ^{3}) | |||||

1.5 | 1381 | 1200 |

**Figure 8.**Soil temperature (K) around multiple boreholes in xz plane in t = 6 months, at various distances from borehole wall for variable heat flux (VHF) model.

**Figure 9.**Soil temperature (K) around multiple boreholes in t = 6 months, at various borehole depths for (

**a**) VHF model, and (

**b**) constant heat flux model.

**Figure 10.**Comparison of soil temperature (K) around multiple boreholes at t = 6 months for VHF and constant heat flux models, at (

**a**) z = 95 m and z = −95 m, and (

**b**) z = 0 m.

**Figure 11.**Soil temperature (K) around multiple boreholes in t = 6 months for line source and numerical models at various borehole depths.

**Extension of results to systems of boreholes:**The idea of using line source theory for calculating the temperature profiles in the soil around two boreholes can also be applied to two systems of vertical GHEs. For example, if an area of 40 m × 40 m × 200 m in the soil is occupied for one system of vertical GHEs, the ratio of system depth to its initial size is large enough to be accounted as one cylinder or line source of heat when system interactions and temperature excess around a system with larger distances are to be accounted for. The study of variable heating strength along the borehole length also accounts for the system of boreholes as well. Therefore, the parametric study on two interacting boreholes likely exhibits the same affecting parameters and results as those for two interacting systems of boreholes. However, a more detailed comparison of the results of modeling the two systems must be performed in order to show similarities between the two problems. Furthermore, certain assumptions such as the assumption of constant ground temperature as well as constant ground surface temperature must be examined further in order to improve the accuracy of the proposed method.

## 4. Conclusions

## Nomenclature

a | temperature coefficient |

c_{p} | specific heat at constant pressure [J/kgK] |

D | distance between the tubes in the borehole [m] |

D_{b} | distance between the boreholes [m] |

Fo | Fourier number |

h | borehole distance from the coordinate centre [m] |

h_{z} | integration variable [m] |

H | heating length, [m] |

dimensionless integration variable | |

k | soil thermal conductivity [W/mK] |

k_{b} | grout thermal conductivity [W/mK] |

mass flow rate [kg/s] | |

generated heat per unit volume [W/m ^{3}] | |

heat flow rate per unit length [W/m] | |

heat flux at borehole wall [W/m ^{2}] | |

r | radial coordinate [m] |

r_{b} | borehole radius [m] |

dimensionless distance of Borehole i to a given point (x,y) in the solution domain | |

distance of Borehole 1 to a given point (x,y) in the solution domain [m] | |

distance of Borehole 2 to a given (x,y) point in the solution domain [m] | |

R_{11} | thermal resistance between the inlet circulating fluid and the borehole wall [mK/W] |

R_{12} | thermal resistance between the inlet and outlet tubes [mK/W] |

R_{22} | thermal resistance between the outlet circulating fluid and the borehole wall [mK/W] |

thermal resistance [mK/W] | |

thermal resistance [mK/W] | |

T | temperature [K] |

inlet circulating fluid temperature at z = 100 m | |

t | time [s] |

V | volume [m ^{3}] |

Z | dimensionless parameter |

z | axial coordinate [m] |

Greek Letters | |
---|---|

α | thermal diffusivity [m ^{2}/s] |

Θ | dimensionless temperature |

θ | temperature rise [K] |

φ | circumferential coordinate [rad] |

ρ | density [kg/m ^{3}] |

Subscripts | |
---|---|

b | borehole |

f1 | inlet circulating fluid |

f2 | outlet circulating fluid |

nb | node number of the adjacent cell |

P | centroid P |

Superscripts | |
---|---|

0 | previous time step |

f1 | inlet circulating fluid |

f2 | outlet circulating fluid |

n | discretization step designation in time |

## Acknowledgments

## Conflict of Interest

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**MDPI and ACS Style**

Koohi-Fayegh, S.; Rosen, M.A.
On Thermally Interacting Multiple Boreholes with Variable Heating Strength: Comparison between Analytical and Numerical Approaches. *Sustainability* **2012**, *4*, 1848-1866.
https://doi.org/10.3390/su4081848

**AMA Style**

Koohi-Fayegh S, Rosen MA.
On Thermally Interacting Multiple Boreholes with Variable Heating Strength: Comparison between Analytical and Numerical Approaches. *Sustainability*. 2012; 4(8):1848-1866.
https://doi.org/10.3390/su4081848

**Chicago/Turabian Style**

Koohi-Fayegh, Seama, and Marc A. Rosen.
2012. "On Thermally Interacting Multiple Boreholes with Variable Heating Strength: Comparison between Analytical and Numerical Approaches" *Sustainability* 4, no. 8: 1848-1866.
https://doi.org/10.3390/su4081848