Decomposition and coupling of soil domain for modeling vertical ground heat exchangers using the state model size reduction technique

Highlights

• A numerical model of vertical GHXs is proposed using the reduction technique.

• A domain decomposition method is developed proposing a new type of boundaries.

• A coupling method to assemble the decomposed sub-zones is developed.

• CPU time of the proposed model is more than 300 times faster than a reference model.

Abstract

Numerical modeling of the ground heat exchangers requires important computational resources to accurately reproduce the short-time thermal responses over a long-time period. Considering geometries of the vertical ground heat exchanger, domain decomposition is proposed in the vertical and horizontal directions. As the principal direction of the heat transfer in such a system is horizontal and here the cumbersome finite element method in treating the domain decomposition is adopted, only the horizontal soil domain case is investigated in this work. A new type of boundaries to decomposed sub-zones is proposed, and then a restitution method is developed to keep all the matrix terms inherent in initial matrices ahead of decomposition. The state model size reduction technique is used for each sub-zone model to reduce the model size, and then the reduced models are regrouped into a single state model using the new proposed coupling method. Results show that the developed decomposition and coupling methods are pertinent with negligible errors (less than 0.005 °C for the maximum), and the final reduced model gives acceptable results for typical purposes of geothermal simulations. From some sensitivity tests, it is found that accuracy of the reduced model can be systematically improved by increasing the reduced order or the number of sub-zones. The proposed model is remarkably faster than the reference model (more than 300 times), implying that a detailed 3-D numerical model can be used for long-time simulations.

Introduction

Accurate prediction of the thermal responses of ground heat exchangers (GHX) is crucial to determine size of GHXs, to optimize the operation of coupled heat pump systems, or to evaluate economical gain of novel system designs [1]. A short-time thermal response to the rapid variation of heat pump operation is of order of minutes. An accurate mathematical description of geometries of boreholes and U-tubes of the vertical GHXs is important to evaluate this short-time response. Importance awareness of accurate modeling of such response is recently rising particularly for obtaining a precise system control [2]. As periodic heat injections or extractions over years slowly but continually increase thermal gradients from the U-tubes to the far-field soil, the modeling of the large-scale soil domain influences the long-time response. A quasi steady state of the soil region can be reached in at least 10 years of system operation. Thus, sizing of GHXs must be evaluated by simulations over such a long time period. Modeling of these different time-scales responses occurring in the whole ground domain is one of the main research topic in the community.

Early studies began with some analytical solutions. One of them is the infinite line source (ILS) model proposed by Ingersoll et al., in 1954 [3]. The model uses a constant heat transfer rate from an infinite line source. Similarly, the cylindrical heat source solution (CHS) has been developed to evaluate the heat transfer rate at the borehole wall [4]. While these models are one-dimensional, the finite line source (FLS) method [5] has been proposed to predict two-dimensional (radial and axial) heat transfer from a finite line source. These simplified solutions result in different validity ranges of the models, so attention must be paid for the model selection [6]. In addition, they are usually coupled to a thermal resistance model for modeling of the borehole region. Since the thermal resistance model doesn't take into account the thermal capacity effect of boreholes, the short-time thermal variation cannot be accurately evaluated. Lamarche et al. [7] have estimated the validity of the resistance model at 3–6 h required to reach the steady-state after a step modification of the input.

On the other hand, numerical models have also been used to assess GHXs performances. One of the main contributions of the numerical models was to provide thermal response factors for specific GHXs configurations. Eskilson [8] developed the so-called g-function obtained from a 2-D numerical model using a coarse mesh. Since the g-function does not provide the short-time response, researches concerning the extension of the g-function to short-time scales have been conducted using numerical detailed models [9]. These g-functions are also used for the purpose of benchmark of existing models. Bertagnolio et al. [10] compared the CHS models using different load aggregation schemes with the g-functions while Fossa [11] tested the FLS based models.

With increasing interest on 3-dimensional models [12] or innovative geometries of GHXs, several numerical models have recently been developed. Most of the models used commercial simulation tools using detailed unstructured mesh [13] to investigate the thermal behavior of GHXs for short-time scales [14] while some developed their own programs using unstructured mesh [15] or structured grids [16] for the whole soil domain. On the other hand, Wang et al. [17] developed a dynamic thermal boundary model to reduce the ground volume size and consequently decrease the number of nodes. As detailed numerical models require a large amount of computational resources, they are still not adequate for conventional simulation purposes such as sizing or long-term economical evaluation.

To reduce the calculation time of numerical detailed models, the state model size reduction technique is employed in this paper. Performance and relevance of the method are well known for linear-invariant systems and the current ground domain can be assumed as such a system. However, application of the method to a large system exposed to a fast varying solicitation cannot correctly reproduce the short time response. To adequately use the reduction method for the GHX problem, a domain decomposition method is proposed to separately apply the reduction method to each decomposed sub-model. Then, a coupling method to assemble the sub-zone models is also proposed in this work.

Before presenting the methodologies, the problem statement section describes some specific characteristics in the ground modeling and presents our previous works that uses a similar state model size reduction technique with their shortcomings. Sub-structuring of the soil volume is presented in the following section. A brief presentation of the model reduction process is also given in the same section. Then, the decomposition process is detailed using matrix formulations in Section 4. Then, Section 5 discusses on the coupling method. The proposed methods and the reduced models are numerically tested in the following section. Finally, conclusions are given in the last section.