A new contribution to the finite line-source model for geothermal boreholes
Introduction
In recent years, ground source heat pumps (GSHP) have been recognized as being among the cleanest, most energy efficient and cost effective systems for space heating and cooling in residential and commercial buildings. A GSHP system consists of a conventional heat pump coupled with a ground heat exchanger. The main advantage of using the ground as the system's source or sink is that this environment benefits from a relatively constant mean temperature when compared with ambient air. This results in an overall improvement of the thermal performance of the system and therefore reduces operating costs. The heat pump used in residential and commercial buildings is usually a water-to-air unit where the water (or the water-antifreeze mixture) exchanges heat with the ground. In common configurations, the ground heat exchanger consists of loops installed in a horizontal or vertical configuration. Horizontal arrangements are often less expensive but it requires a larger ground area and is normally limited to commercial usage. Currently, most residential and commercial units use a vertical ground heat exchanger which usually offers better performance than the horizontal due to smaller seasonal swing in the ground mean temperature. Depending on the size of the heat pump units and the properties of the ground, more than one borehole may be required and various geometrical configurations of these boreholes can be used.
Recent research in the GSHP field has contributed to reducing the life cycle cost and to broadening the applicability of this technology. One important research area is modeling, which allows system simulations to be performed. For GSHP systems, simulation is an important tool for system design purposes as well as for investigating long-term system performance. The various approaches used for modeling the ground heat exchanger can be split into two major categories. Some models mainly deal in short time-step (less than an hour) simulations, which are used as design tools or in whole-building analyses. The so-called DST model, proposed by Hellström [1] and integrated into the TRNSYS simulation package, is one of the best-known models in this category. Some other models are used for designing heat exchangers, and simplifications are carried out to make them computationally efficient for performing long-term performance (usually up to 20 years) evaluations of GSHP systems. Of course, some models can be used for both purposes. This paper is concerned with the second problem.
For large time values, two phenomena become important: axial effects, which are very small for short and medium time scales, and the thermal interference between boreholes in cases where more than borehole is used. The work of Eskilson [2] is one of the major references for treating these issues. Eskilson presents a method based on non-dimensional thermal responses called “g-functions”. These functions are computed numerically however, and values are given for various bore field configurations. A lack of flexibility is thus observed. Zeng et al. [3] propose an analytical method for generating “g-functions”. Their solution was also suggested initially by Eskilson in his thesis; he does not seem to use it afterwards however, perhaps due to discrepancies with his numerical values. In this paper we present a modified approach to evaluating analytical “g-functions” for the medium and long-time analyses of borefield configurations. Our results are very close to those obtained by Eskilson and can be used to any kind of configuration.
We believe that design engineers may find our results valuable in evaluating lengths for ground heat exchangers.
Section snippets
Physical model
A single geothermal borehole is a system in which a borehole is used to exchange heat between a calorimetric fluid circulating in the borehole and the ground, which acts as a sink or source of energy. The heat-transport phenomenon in the ground operates mainly via heat conduction when the groundwater's movement is nonexistent. The thermal response of a geothermal borehole system is represented by a temperature change in the borehole and in the ground surrounding it as a function of heat
Mathematical model
The heat exchange problem in a buried vertical borehole can be formulated with respect to Fig. 1. If we ignore the angular dependence of the temperature, the basic problem is to find the temperature distribution T(r, z, t) that satisfies the heat conduction equationfor the domain r > rb, t > 0, 0 ≤ z < ∞ and the following boundary conditions:where is the heat flow per unit length. The axial dependence of the
Analytical g-function
As mentioned by Diao et al. [15], using an analytical expression instead of the usual tabulated values for the borehole model is quite attractive, especially in design and optimization, when parameters may have to change often. For analyzing an existing system, both approaches are similar and the more precise approach is preferred. The idea proposed here is simple. It uses (23) as was suggested by Zeng et al. [3], but the expression is modified to make it numerically efficient. Let us start by
Simulation results
To validate and evaluate the analytical form of the g-function, we made the following comparisons:
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Test #1: Comparison between (44) and (23).
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Test #2: Comparison between (44) and (17) and the Eskilson's tabulated g-function [2]. In the latter case, since we did not have the tabulated values, we took values from the published plots. These were difficult to evaluate with precision.
Interacting boreholes
As stated in Section 1, a typical real installation consists of more than one borehole. For a long time period, temperature variations in the various boreholes will interact. Once again, treating the real physical problem is complicated. A simplified procedure [2] consists in computing the final temperature field by superposing different isolated boreholes. Let us call Ti(ri, zi, t), the temperature variation due to a single borehole i, with respect to time. This is in fact the solution of (11a)
Application of the new method
In order to have a feeling for the flexibility advantage of the new method, we will apply the method in order to evaluate the penalty temperature described in Eqs. (9) and (10) for a given physical problem. We will not describe the different design philosophy associated with each of these equation's terms in detail. The interested readers are referred to the references by Kavanaugh and Rafferty [10] and Bernier [11]. Here we will focus only on evaluating the penalty temperature, which is
Extremum of the g-function
Diao et al. [15] report that the evaluation of the analytical g-function based on the temperature at the middle of the borehole gives an extremum for a given time. This surprising result was never observed in our calculations. In fact, calculating the time where we reach a possible extremum yields infinity, as follows:with In the first
Conclusion
Modeling of the heat transfer problem between the vertical boreholes of a ground-coupled heat pump has been done for many years. Most long-term modeling has been carried out using numerical solutions given in the form of tabulated or graphic values. In this paper, an analytical solution is shown to give very similar results to the numerical values reported in the literature. The analytical solution has the advantage of being more flexible, especially in designing nonexistent systems where a
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