# Numerical Modeling of Vertical Geothermal Heat Exchangers Using Finite Difference and Finite Element Techniques

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## Abstract

This paper presents the development of a 2D finite difference modelling approach and a 3D finite element numerical model for simulating vertical geothermal heat exchangers (GHEs), explaining the theory governing the thermal processes, element discretization and the selection of the appropriate boundary conditions. Both of these models provide fully coupled solutions for the fluid flow in the circulation pipes and the thermal processes between the fluid and solid domains (pipes, grout and soil). The numerical models are verified with a field test and subsequently they are utilized to simulate the thermal performance of a borehole heat exchanger integrated with a single U-tube. Two different thermal operation cases are analyzed; a constant rate heat injection and a fluid injection at a constant temperature. A model validation study is also carried out for the constant rate heat injection case by comparing the numerical results with the available analytical solution for a finite line source. Furthermore, effective thermal conductivity of the ground back-calculated from the results of the numerical analyses is compared with the value used in the numerical models. Comparison of the results obtained from both numerical models and validating model predictions with the analytical solution confirms that both FE and FD models can accurately simulate the heat transfer mechanisms governing the thermal performance of GHE systems.

## Keywords

Geothermal heat exchanger Numerical modeling Finite difference analysis Finite element analysis Finite line source## List of symbols

*A*Cross-section area (m

^{2})*c*_{p}Specific heat capacity (J kg

^{−1}K^{−1})*d*Diameter (m)

*d*_{h}Hydraulic diameter of pipe (m)

*f*_{D}Darcy friction factor

*h*Heat transfer coefficient (W m

^{−2}K^{−1})*H*Length of the heat exchanger (m)

*k*Thermal conductivity (W m

^{−1}K^{−1})*K*Slope of the temperature versus log time curve

- Nu
Nusselt number

*q*_{wall}External heat exchange through pipe wall (W)

*q*Heat (W)

*r*Radial coordinate (m)

*r*_{p}Pipe radius (m)

*s*Shank spacing between two pipe legs (m)

*T*Temperature (K)

*t*Time (s)

*v*Flow velocity (m s

^{−1})- \(\dot{V}\)
Volumetric flow rate (m

^{3}s^{−1})**u**Velocity field (m s

^{−1})*z*Depth (m)

## Greek symbols

- α
Thermal diffusivity (m

^{2}s^{−1})- Δ
*T*(*t*) Applied temperature difference at time

*t*(K)*ρ*Density (kg m

^{−3})

## Subscripts

- eff
Effective

- ext
External

- f
Fluid

*i, j*Node index

- in
Inlet

- out
Outlet

- p
Pipe

## Notes

### Acknowledgments

The first and third authors would like to express their gratitude for the support by the National Science Foundation under Grants No. CMMI-0928807 and CMMI-1100752. The authors would also like to gratefully acknowledge the financial support by the Mid-Atlantic Universities Transportation Center (MAUTC) under Grant No. 415-77 76R20. The first author is funded as a visiting scholar by the Turkish Council on Higher Education and Istanbul Technical University. These funding supports are greatly appreciated.

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