Full-scale temperature response function (G-function) for heat transfer by borehole ground heat exchangers (GHEs) from sub-hour to decades

Highlights

• A composite full-scale is built and verified for ground heat exchangers.

• Full-scale model is simplified as multi-stage models to reduce computational cost.

• These models are applicable for time from several minutes to decades.

• Parametric study is performed for ground heat exchangers with single U-tube.

Abstract

Heat transfer by borehole ground heat exchangers involves diverse time–space scales and thus imposes a significant challenge to geothermal engineers. In order to overcome this challenge, this paper develops an analytical full-scale model from the idea of matched asymptotic expansion. The full-scale model is a composite expression consisting of a composite-medium line-source solution (inner solution), a finite line-source solution (outer solution), and an infinite line-source solution. The full-scale model is first verified by a frequency-decomposition method. Furthermore, the full-scale model is reformulated as a multi-stage model based on Duhamel’s theorem to reduce the computational cost. The multi-stage model combines the three separate solutions in a sequential way, i.e., the inner solution for the short-time scale, the conventional infinite line-source solution for the intermediate time scale, and the outer solution for the long-time scale. Finally, we perform a parametric study on a ground heat exchanger with single U-shaped tube, by which the spacing between U-tube legs, the length-to-radius ratio of borehole, the ratios of thermal diffusivities and conductivities of the ground and backfilling material are analyzed.

Introduction

Borehole ground heat exchangers (GHEs) are known as critical components in borehole ground heat storage (GHS) and ground-coupled heat pumps (GCHPs) [1], [2], [3]. The purpose of GHS requires that the GHEs must be installed relatively compact for reducing heat loss from the ground storage volume, and the efficiency of GHS will increase with the volume of the GHE cluster; thus GHS prefers large scale applications. In a GCHP system, however, GHEs should be installed as sparsely as possible in order to maximize heat interaction between GHEs and the surrounding ground. Recently, GCHPs have been used increasingly in large buildings. For example, there are GCHPs installed for heating buildings of more than 1,000,000 m² in China; therefore a very large cluster of borehole GHEs is also required for such a GCHP system.

The practical use of the large clusters of GHEs has highlighted the need for an accurate and efficient approach to calculating the thermal process in the ground. From the perspective of accuracy, the heat transfer calculation should use a time resolution ranging from sub-hour to decades, corresponding to a space range from several centimeters to more than one hundred meter (Fig. 1). As a result, an accurate calculation inevitably requires vast computational resources in order to tackle the complete spectrum of the broad time-length scales [4].

The long-term thermal response in the underground process determines the overall feasibility of the GHS and GCHP systems during their application lifetime. The long-time heat build-up in the ground is influenced by the end effects of GHEs and is a result of a net imbalance between the extraction and the injection of heat. A GHE matrix may have large vertical and horizontal dimensions (∼100 m); in consequent, the transient thermal process in the ground can be as long as the lifecycle of GHEs (∼decades) because of the huge heat capacity involved (Fig. 1). Much work has been performed to solve the long-term thermal process underground, including numerical and analytical methods. Numerical methods [5], [6], [7], [8], [9], [10] are effective methods of modeling all the underlying thermal processes, but they are computationally too intense for large-scale applications. Thus various analytical models have been proposed and widely used [11], [12], [13], [14], [15], [16], [17], among which conventional finite line-source models are the most suitable and efficient for this purpose [11], [12], [13].

The thermal response within a borehole can react quickly to variations in thermal loads because of the limited space dimension and heat capacity in the borehole. Scale analysis has shown that the calculation of the temperature within the borehole with a radius r_b ∼ 5 cm requires a time resolution in the order of magnitude of an hour [18]. The hourly temperature response is crucial for predicting peak temperatures and thus is vital for the hourly energy analysis and the optimum control of GCHP systems [19]. Predicting the short-term response, however, is more difficult than predicting the long-time process because it is associated with transient heat conduction in a composite medium, together with taking various installations of U-shaped pipes into consideration.

On way to deal with the challenge as mentioned above is to simplify the geometry arrangement in the borehole. A widely used simplification is the equivalent-diameter assumption [20], [21], [22], [23], [24], [25], [26], [27], [28], which assumes the U-shaped pipes in a borehole to be a pipe of “equivalent” diameter, thereby reducing the two-dimensional geometry to a one-dimensional hollow cylindrical composite region. The simplified problem can be solved by either generalized orthogonal expansion techniques [20] or the Laplace transform method [22], [28]. However, the empirical equivalent-diameter assumption fails to address the thermal influence between legs of the U-shaped pipe and thus leads to an empirical parameter for the design method of borehole GHEs proposed by the ASHRAE [1].

Our group has recently developed an alternative strategy for modeling the short-time response of a GHE based on Jaeger’s instantaneous line-source solution for a cylindrical composite medium [18], [29], [30]. This approach enables the removal of the equivalent-diameter assumption and can tackle the difficulties associated with both the composite medium and the geometric installations of U-shaped pipes, including single and double U-shaped tubes, W-shaped channels, and spiral-coils. However, the composite-medium line-source model may be one of the most computationally-intensive analytical models because of its complicated mathematical formulation. Moreover, this approach ignores the end effects of GHEs; it is unsuitable for predicting the long-term thermal process.

Thus an efficient analytical model that can address the entire time–space spectrum of the thermal response of borehole GHEs will be beneficial. Claesson and Javed attempted to develop a heat transfer model covering time scales from minutes to decades [25]; but they used the equilibrium-diameter assumption so that a temperature difference exists between the used short- and long-term solutions and an arbitrary temperature shift is proposed at a so-called breaking time. Moreover, the breaking time is empirically determined, without being related to any established theory.

The objective of this paper is to overcome a major obstacle in the calculation of the diverse-scale thermal problem. We build in this paper a full-scale model for the temperature responses of borehole GHEs based on the idea of matched asymptotic expansions. This model can analytically tackle the complexities of time–space characteristic of vertical GHEs with U-shaped tubes. An equivalent form of the full-scale model, called multi-stage model for reducing the computational cost, is also provided. To the best of our knowledge, these two models should be the first two attempts to tackle the effect of composite medium, the time range from sub-hour to decades, as well as the intricate installations of U-shaped pipes in boreholes. More importantly, the two models have a solid theoretical basis; their analytical forms provide an excellent starting point for improving the analysis, design, and simulation of borehole GHEs.