Performance evaluation of coaxial borehole heat exchangers considering ground non-uniformity based on analytical solutions

Highlights

• A new analytical model is proposed for deep geothermal systems.

• Heat exchange efficiency is evaluated based on reliability theory.

• New observations are provided on heat exchange efficiency in non-uniform ground.

• A multi-factor evaluation of a coaxial borehole heat exchanger is performed.

Abstract

The coaxial borehole heat exchangers (CBHEs) are efficient geothermal energy utilization method, which has recently attracted increasing attention. However, this method still has no suitable or efficient analysis method, which limits the wide application of CBHEs. Moreover, the non-uniformity of the ground has less been considered in related analyses, which will lead to estimation errors in CBHEs heat exchange efficiency. Hence, an analytical solution for coaxial borehole exchangers considering non-uniformity of the ground with high calculation efficiency is introduced in this paper. By comparing the CBHEs analytical model with typical experimental results, the reliability of this model is verified. Next, using reliability theory, we further investigate the heat exchange efficiency of CBHEs considering the non-uniformity of the ground, the temperature gradient, the fluid inlet direction, and the working temperature difference between the inlet and initial ground. The results show that the changes (±10%) of basic thermal conductivity will decrease the heat exchange efficiency by 4.5% and increase by 4.1%, respectively. With an increase in the ground's non-uniformity (from 0.2 to 0.6 W/m·K), the heat exchange efficiency of the CBHEs will decrease (from 0.4% to 1.4%). Compared to the results without considering the temperature gradient, the heat exchange efficiency of the CBHEs increased by 0.3%, 0.9%, and 1.3% (corresponding to 1, 2, and 3 °C temperature gradients at a 200 m depth). The CBHEs with the annular region as the inlet presented better heat exchange efficiency (by about 0.3%) than CBHEs with the round region as the inlet. the increase in the working temperature difference between the inlet and initial ground was found to greatly improve the heat exchange efficiency of the CBHEs. Moreover, according to the experimental and design cases, the necessity of considering soil non-uniformity has been proved. This study will provide a useful analytical simulation tool and an important guide to optimize CBHEs design considering the ground non-uniformity in the horizontal and vertical direction.

Introduction

In recent years, with the shortage of fossil fuels and the increasing awareness surrounding environmental protection, energy utilization has been developing in the direction of high efficiency, environmental protection, and sustainability [[1], [2], [3]]. Geothermal energy, as a highly efficient form of clean energy, has been widely utilized in China, Japan, and other countries for decades and has obvious commercial potential [2,4]. Compared to traditional air-source energy, geothermal energy is more efficient and stable, and some studies [5] have shown that the soil temperature remains almost constant throughout the year. This constant temperature of the underground soil is usually higher in winter and lower in summer compared to the ambient air temperature. To make efficient use of geothermal energy, ground source heat pumps (GSHPs) have been promoted [4,6].

Ground source heat pump systems (GSHPs) [7] consist of four main parts, including the borehole heat exchanger (BHE), heat pumps, control system, and heating network for users, in which the BHE is the most important part. To fully improve the heating/cooling performance of the BHE, several section types have been designed, such as a U-type, double U-type [8], and coaxial type. Owing to its greater heat exchange section, a BHE with coaxial pipes is recognized to have the best heat exchange performance. With a focus on coaxial borehole heat exchangers (CBHEs), Iry et al. [9] studied the impact of the diameter ratio of coaxial pipes on heat exchange performance and found that a suitable diameter ratio improved effectiveness by 22% (obtained according to the borehole depth reducing ratio). Li et al. [10] designed different sectional shapes of the inner pipe (smooth pipe, spiral pipe, and corrugated pipe) and concluded that a change of the inner pipe shape has less of an effect on heat exchange performance than fluid speed. For the full application of geothermal energy, improving CBHEs is attracting increasing attention from researchers and designers.

For optimizing the design of CBHEs, four typical methods are usually considered, including (in-situ or in-door) experiments [11], numerical analysis [12], empirical software [13], and analytical solutions [14]. In-situ experiments [15] are recognized as the best method, as they occur in the real working environments of CBHEs. However, their higher experimental investments limit the wider use of such experiments. Owing to the progress of computers and calculation theory, numerical analysis is gradually developing [16]. By building three dimensional models, the heat exchange performance of CBHEs can be studied. However, the model setting process is complex, and the choice of analytical parameters is difficult, which limits the widespread promotion of numerical analysis. Empirical software [13] is certainly a good choice. However, in empirical software, the studying parameters are limited, which cannot fully meet the requirements for optimizing CBHEs. Analytical solutions [17,18] represent a calculation method based on the balance equation, initial conditions, and boundary conditions. These methods can not only achieve a higher calculation efficiency but also have great engineering reliability. According to the comparison between the advantages and disadvantages of the typical calculation methods in above analysis, the analytical method of CBHEs was ultimately recognized as the most suitable method for optimizing the design [19,20].

In the analytical model, CBHEs was divided into the region inside the borehole wall and the region outside the borehole wall. Thermal resistance theory is recognized as a suitable method to analyze the heat exchange between the fluid, pipes, and backfill soil. After years of development, thermal resistance theory [21] has developed from one-dimensional (linear model), to two-dimensional (plane model), to three-dimensional (entity model). To estimate the transient heat transfer inside the borehole wall, Javed and Spitler [22] proposed a thermal resistance model with three dimensions and achieved good results. To analyze heat transfer outside the borehole wall, Ingersoll et al. [23] demonstrated that the line-source model has high efficiency and reliability. After that, to better consider the geometric shape of the borehole, the cylinder source model [24] was put forward. Based on the above studies, the limited line-source [25] and finite cylinder source models [14, 26] were developed to better consider the vertical heat transfer of the ground.

By combining the separate models inside and outside the borehole wall, a three-dimensional model of CBHEs was established. Based on convolution theory, Pan et al. [17] built an analytical model of a CBHEs that can accurately calculate the heat exchange characteristics of the CBHEs. By comparing the calculation results with previous studies, the reliability of Pan's model was verified. After that, based on the segmented method of the line-source mode, Luo et al. [14] put forward a new analytical model that can better consider the non-uniformity of the CBHEs along the depth. To consider the geometry of the borehole, Luo et al. [26] created an improved method based on the cylinder source model. This analytical method can accurately calculate the heat performance of the CBHEs, but the solution process is inefficient due to the existence of error functions in the calculations, because the error function cannot be directly solved and only can be calculated by approximate formulas [21]. Hence, it was valuable and meaningful to build an efficient and reliable analytical model of CBHEs to optimize CBHEs design.

To build a three-dimensional analytical model of CBHEs, some simplifications were considered, such as 1) assuming that the temperature of the borehole wall along the depth is uniform and 2) assuming that the thermal physical parameters [27] of the soil are uniform. Assumption 1z will lead to underestimating or overestimating the heat exchange performance of the CBHEs and may also lead to errors in the calculation results. Luo et al. [14] proved that, when the uniform temperature assumption is employed in a deep borehole heat exchanger, the greater the depth is, the greater the calculation error is. For assumption 2, the uniform assumption of the soil layers is different from the real working situation of CBHEs, which can lead to errors in the calculation results. Based on random functions, Luo et al. [5] studied heat exchange performance, considering the non-uniformity of thermal insulation in CBHEs and found that the CBHEs had greater cooling efficiency when considering non-uniformity, which was different from the uniform assumption. According to statistical data of ground, bean et al. [28] and Zhang et al. [29] concluded that, the physical and mechanical properties of ground was non-uniformity and met certain distribution rules, such as normal distribution, spatial distribution, etc. Moreover, they point out that, the influence of non-uniformity properties should be applied to the analysis to obtain more realistic results. Therefore, it is necessary to consider the temperature non-uniformity of the borehole wall and the ground non-uniformity outside the borehole according to the corresponding distribution rules when estimating the heat exchange efficiency of the CBHEs.

Based on the above analysis and the segment method, an analytical solution for a coaxial borehole exchanger considering non-uniformity of the ground with a high calculation efficiency was put forward. The process for establishing the model is detailed, and the reliability of the analytical results is verified compared to the typical experimental results. Moreover, a random variable generation method satisfying normal distribution is proposed for considering ground non-uniformity. For further optimizing the CBHEs design, on the basis of 95% engineering reliability, a multi-factor evaluation of CBHEs was conducted, including the ground non-uniformity, temperature gradient, fluid inlet direction, and working temperature difference between the inlet and initial ground. This study will provide a useful analytical simulation tool and an important guide to optimize CBHEs design considering the ground non-uniformity in the horizontal and vertical direction.