TY - JOUR

T1 - A spectral model for heat transfer with friction heat gain in geothermal borehole heat exchangers

AU - BniLam, Noori

AU - Al-Khoury, Rafid

PY - 2016/8/1

Y1 - 2016/8/1

N2 - This paper introduces a semi-analytical model for the simulation of transient heat transfer with friction heat gain in a single U-tube geothermal borehole heat exchanger subjected to an arbitrary heat flux signal. The friction effect appears as a nonhomogeneous term in the governing equations, which constitutes a set of coupled partial differential equations describing heat flow in the three components of the borehole; pipe-in, pipe-out and grout. We utilize the spectral analysis for discretizing the time domain, and the eigenfunction expansion for discretizing the spatial domain to solve the governing initial and boundary value problem. The proposed model combines the exactness of the analytical methods with an important extent of generality in describing the geometry and boundary conditions of the numerical methods. The model is verified analytically against a simplified one-dimensional solution. A numerical example is given to illustrate the effect of friction on heat transfer in the borehole heat exchanger for different fluid velocities and viscosities. The analysis shows; for the geometry, materials fluid velocities and viscosities, typically utilized in shallow geothermal systems; the friction is not really significant. However, the main advantage of this work is on the solution technique that can be useful for many other applications, including fluid flow in narrow pipes, high fluid velocities, high fluid viscosities, and pipes made of composite materials and of complex geometry. Also, the method can be useful for solving other nonhomogeneous coupled partial differential equations.

AB - This paper introduces a semi-analytical model for the simulation of transient heat transfer with friction heat gain in a single U-tube geothermal borehole heat exchanger subjected to an arbitrary heat flux signal. The friction effect appears as a nonhomogeneous term in the governing equations, which constitutes a set of coupled partial differential equations describing heat flow in the three components of the borehole; pipe-in, pipe-out and grout. We utilize the spectral analysis for discretizing the time domain, and the eigenfunction expansion for discretizing the spatial domain to solve the governing initial and boundary value problem. The proposed model combines the exactness of the analytical methods with an important extent of generality in describing the geometry and boundary conditions of the numerical methods. The model is verified analytically against a simplified one-dimensional solution. A numerical example is given to illustrate the effect of friction on heat transfer in the borehole heat exchanger for different fluid velocities and viscosities. The analysis shows; for the geometry, materials fluid velocities and viscosities, typically utilized in shallow geothermal systems; the friction is not really significant. However, the main advantage of this work is on the solution technique that can be useful for many other applications, including fluid flow in narrow pipes, high fluid velocities, high fluid viscosities, and pipes made of composite materials and of complex geometry. Also, the method can be useful for solving other nonhomogeneous coupled partial differential equations.

KW - Borehole heat exchanger, BHE

KW - FFT

KW - Friction in pipes

KW - GHP

KW - GSHP

KW - Spectral analysis

UR - http://www.scopus.com/inward/record.url?scp=84959907383&partnerID=8YFLogxK

UR - http://resolver.tudelft.nl/uuid:8f74226d-1fee-48b6-a2ff-fa29bb879904

U2 - 10.1016/j.apm.2016.02.031

DO - 10.1016/j.apm.2016.02.031

M3 - Article

AN - SCOPUS:84959907383

VL - 40

SP - 7410

EP - 7421

JO - Applied Mathematical Modelling: simulation and computation for engineering and environmental systems

JF - Applied Mathematical Modelling: simulation and computation for engineering and environmental systems

SN - 0307-904X

IS - 15-16

ER -