A semi-analytical temperature solution for multi-segment deep coaxial borehole heat exchangers

Abstract

A semi-analytical and a finite-difference scheme are presented for the simulation of temperature and the heat transfer in a multi-segment coaxial borehole heat exchanger. The single-segment solution on closed-form is extended to a semi-analytical multi-segment solution, where each segment may have unique properties. These properties are such as different casings, widths of the annulus, radius of the inner tubing, material properties, rock properties and geothermal gradients. The multi-segment model is a simple and powerful alternative to numerical methods for simulating a complex coaxial borehole heat exchanger with a constant flow rate. It is demonstrated with a deep coaxial borehole heat exchanger made of three different segments. The analytical and semi-analytical models are validated by comparison with numerical solutions obtained with an upstream finite difference scheme. The match between the solutions is excellent. The solution on a closed-form is used to study the temperature difference between the outlet and the inlet regarding two dimensionless numbers. It is found that the maximum temperature difference occurs when the dimensionless heat transfer coefficient for the casing-rock is much larger than one. A second necessary condition is that the dimensionless heat transfer coefficient for the insulator between the inner tube and the annulus must be much less than one. The power leakage from the inner tubing to the annulus is also at a maximum under these conditions.