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In Ground Source Heat Pump Systems (GSHPS), distance between boreholes is a very important parameter for reliability, long life time and performance of the whole system. In large scale applications of GSHPS, more than one borehole is needed and determination of the optimal distance between boreholes becomes an important issue. In this study, the effect of distance between boreholes on heat transfer rate per unit borehole length (unit HTR value) is computationally investigated. Four different configurations consisting of 2, 3, 5 and 9 boreholes are considered. 3 and 6 months averaged unit HTR value of the most critical borehole in each configuration is compared with that of single borehole to determine the performance loss. Variations of performance loss due to thermal interactions of boreholes with both time and distance are analyzed. Furthermore, the effects of thermal conductivity of ground on temperature distributions around borehole is also examined. Results can be used to determine the optimal borehole distance for various applications.

Abstract
A one-dimensional transient ground heat exchanger model is proposed to account for fluid and grout thermal capacities in borehole ground heat exchangers with the objective of predicting the outlet fluid temperature for varying inlet temperature and flow rate. The standard two-pipe configuration is replaced with an equivalent geometry consisting of a single pipe and a cylinder core filled with grout. Transient radial heat transfer in the grout is solved numerically while the ground outside the borehole is treated analytically using the cylindrical heat source method. The proposed model is validated successfully against analytical solutions, experimental data, a three-dimensional transient numerical model, and TRNSYS's Type 451.
For a typical two-pipe configuration, it is shown that the fluid outlet temperature predicted with and without borehole thermal capacity differs by 1.4, 0.35, and 0.23 °C after 0.1, 0.2 and 1 h, respectively. Annual simulations are also performed over an entire heating season (5600 h) with a 6 min time step. Results show that the outlet fluid temperature is always higher when borehole thermal capacity is included. Furthermore, the difference in fluid outlet temperature prediction with and without borehole thermal capacity increases when the heat pump operates infrequently. The end result is that the annual COP predicted is approximately 4.5% higher when borehole thermal capacity is included.

Knowledge of borehole exit fluid temperature is required to optimize the design and performance of ground source heat pump systems. The borehole exit fluid temperature depends upon the prescribed heat injection and extraction rates. This paper presents a method to determine the fluid temperature of a single or a multiple borehole heat exchanger for any prescribed heat injection or extraction rate. The fluid temperature, from minutes to decades, is determined using step response functions. An analytical radial solution is used for shorter times. A finite line-source solution is used for longer times. The line-source response function has been reduced to one integral only. The derivative, the weighting function, is given by an explicit formula both for single boreholes and any configuration of vertical boreholes.

The ability to predict both the long-term and short-term behavior of ground loop heat exchangers is critical to the design and energy analysis of ground source heat pump systems. A numerical model for the simulation of transient heat transfer in vertical ground loop heat exchangers is presented. The model is based on a two-dimensional fully implicit finite volume formulation. Numerical grids have been generated for different pipe sizes, shank spacing and borehole sizes using an automated parametric grid generation algorithm. The numerical method and grid generation techniques have been validated against an analytical model. The model has been developed with two main purposes in mind. The first application is used in a parameter estimation technique used to find the borehole thermal properties from short time scale test data. The second application is the calculation of nondimensional temperature response factors for short time scales that can be used in annual energy simulation.

An analytical solution of the transient temperature response in a semi-infinite medium with a line source of finite length has been derived, which is a more appropriate model for boreholes in geothermal heat exchangers, especially for their long-duration operation. The steady-state temperature distribution has also been obtained as a limit of this solution. An erratic approach to this problem that appears in some handbooks and textbooks is indicated. Two representative steady-state borehole wall temperatures, the middle point temperature and the integral mean temperature, are defined. Differences between them are compared, and concise expressions for both are presented for engineering applications. On this basis the influence of the annual imbalance between heating and cooling loads of the geothermal heat exchangers is discussed regarding their long-term performance. © 2002 Wiley Periodicals, Inc. Heat Trans Asian Res, 31(7): 558–567, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/htj.10057

The effects of the geothermal load on the ground temperature recovery in a ground heat exchanger (GHE) were investigated. A three-dimensional equivalent transient GHE analysis model was developed and validated against measured thermal response test (TRT) data and sandbox reference dataset. The effects of amount of geothermal load, duration of the recovery time per day, and daily geothermal load pattern on the ground temperature recovery were examined. The results showed that decreasing the amount of geothermal load and increasing the recovery time can improve the ground temperature recovery. However, there is little correlation between the daily geothermal load pattern and ground temperature recovery. The effects of the geothermal load on ground temperature recovery were also analyzed under different soil thermal conductivity conditions. The duration of the recovery time significantly influences the ground temperature recovery at low soil thermal conductivity. These results demonstrate the importance of considering the recovery time in the GHEs design stage to reduce the borehole length.

Laboratory studies were conducted to determine the effect of grout thermal conductivity, borehole diameter, pipe size, and pipe configuration on the total thermal resistance in the borehole. Borehole thermal resistance decreased with increasing grout thermal conductivity, but increased grout thermal conductivity above 1.0 Btu/h·ft·°F provided very small additional reduction. The studies resulted in a set of relationships for borehole thermal resistance. A series of independent field tests verified that the borehole wall conservatively accounted for the thermal conductivity of the backfill or grout material. The effect of increasing grout thermal conductivity resulted in overall reductions in thermal resistance between the circulating fluid and the earth.

A simplified model to simulate single U-tube ground heat exchangers is presented in this paper. The model is based on the electrical analogy to simulate heat transfer within the borehole and the use of thermal response factors (g-functions) to estimate injection–extraction heat flow to surrounding ground. The inclusion of two capacities for the heat carrier fluid and the grout material makes the model suitable for short time step simulations. The thermal resistance of the borehole has been changed considering different heat conduction paths. Several variants of the model have been validated through comparison with a refined computational fluid dynamic (CFD) reference model. Good results have been obtained for the different variables analysed (outlet temperature and surface borehole temperature). RMSE (Root Mean Squared Error) values were smaller than 1 °C while relative errors were under 5%.

This paper introduces a new methodology for the generation of thermal response factors of geothermal bore fields using the concept of g-functions introduced by Eskilson. Boreholes are divided into segments to consider the variation of the heat extraction rates along the length of the boreholes and the analytical finite line source (FLS) solution is used to calculate the temperature variations at the wall of each borehole segment along the axial direction. The proposed methodology accounts for the time variation of the heat extraction rates among boreholes and along the length of individual boreholes to obtain a uniform borehole wall temperature equal for all boreholes in accordance with the original boundary conditions proposed by Eskilson. In addition, the methodology is generalized to account for boreholes of different lengths and buried depths. g-Functions calculated with the proposed methodology are compared to the numerical technique used by Eskilson to derive the g-functions for fields of 1 to 12 x 12 boreholes. The difference between the two models is within 5% for all studied bore fields, except for fields of boreholes located on a single row. The variation of the heat extraction rates of individual boreholes along their length as well as in time also showed good agreement with the numerical model. It is shown that using 12 borehole segments is adequate to calculate the g-functions in most practical cases. For instance, the error on the g-function of a 10 x 10 bore field calculated using 12 borehole segments is 2.2% after 20 years and 4.7% at steady-state.

The fluid extracts or rejects heat with subsurface by downward leg of pipe (DLP) and upward leg of pipe (ULP) inside the vertical borehole heat exchanger (BHE). As the borehole diameter is only 0.11 m to 0.2 m, the temperature difference between DLP and ULP inevitably leads to thermal short-circuiting. In order to discuss how different geometrical characteristics influence on short-circuiting, the heat transfer between the two legs was investigated by a 2-D model, and then a best-fit expression of short-circuiting thermal resistance was presented in dimensionless form. A 3-D equivalent rectangular numerical model was established to evaluate the fluid temperature variations along the pipe, how the flow velocity and grout conductivity and borehole depth influence on the outlet temperature and average heat flux per unit length and short-circuiting loss rate were analyzed. By comparing the arithmetic average fluid temperature and integral average fluid temperature, it was found that the lager short-circuiting loss rate would lead to greater error for effective subsurface conductivity estimation. The experiment done in NanJing, China also validated that the smaller flow velocity and larger borehole depth would bring about the smaller measured effective subsurface conductivity during TRT.

The thermal performances of several types of vertical ground heat exchangers (GHEs) for ground source heat pump system have been investigated with different operation mode. Short time period of operation, discontinuous of 6 and 12 h operations in a day, and continuous operation modes were applied in the GHE system. The short time period of operation includes discontinuous 2 h operation in cooling mode and alternative operation mode with operating the GHE in cooling process and heating process to provide hot water supply. The models of three types of vertical GHEs, including U-tube, double-tube, and multi-tube GHEs, were built and simulated using the commercial computational fluid dynamics software FLUENT. The heat exchange rates of the GHEs have been investigated. The numerical results show the reasonable agreement with the experimental results. The off-time period in the discontinuous operation and extracting heat from the ground in the heating process in the alternative operation mode contributed significantly to the increasing the heat exchange rate. Operating the GHEs with different operation mode shows the different characteristic in their heat exchange rates. It can be constructive information for design of the GHE system in practical engineering.

An improvement to the thermal resistance capacity model (TRCM) used to model borehole heat exchangers is presented. Here, the original model is extended to integrate the thermal capacities of the heat carrier fluid and pipe and to better account for the spacing between the pipes. Model results are compared to results provided by numerical models and show very good agreement. It is shown that the improved model brings a significant improvement for short times over the original model, allowing a rapid computation of the temperature response function at virtually any time and distance from a single borehole.

Sizing of ground-coupled loop heat exchangers (GLHE) depends on the ground thermal conductivity and capacity, and the borehole thermal resistance. One popular method to estimate the thermal parameters is the interpretation of in situ thermal response tests. The modeled response is Tm=(Tin+Tout)/2Tm=(Tin+Tout)/2, the average temperature of the fluid entering and leaving the ground. The TmTm response corresponds to the physically unrealistic hypothesis of constant heat flux along a borehole. Using a 3D finite element model of the borehole, we show that TmTm does not correspond to the fluid mean temperature within the borehole. Accordingly, with TmTm, an overestimation of the borehole thermal resistance results. The resistance overestimation has a noticeable economic impact. We propose instead a new estimator we name “p-linear” average of TinTin and ToutTout with parameter p→-1p→-1, as determined by numerical simulations. We show that the p-linear average closely fits the average fluid temperature computed with the numerical model, hence avoiding bias in estimation of borehole thermal resistance. Finally, we discuss the problem of collinearity arising in the estimation of thermal parameters.

Better energy efficiency of ground coupled heat pump systems in comparison with traditional applications leads to continued growth in the number of installations for space conditioning. The solution with vertical heat exchangers is the most widespread. The design of the borehole heat exchangers (BHEs) is a nodal point, both from an energy efficiency and an economic point of view.In literature, several methods to design these systems are available. However, to promote their application, easy-to-use procedures are required; furthermore designers very often are discouraged from the use of not-open or complicated computational tools. Among the literature models, the ASHRAE method is surely the simplest procedure and, as a consequence, it is suitable for this goal. In this approach, sizing of the BHE length is strongly affected by the parameter named penalty temperature, which is an index to evaluate the long-term behaviour of the borehole field.In this paper, a review of this index is reported and in addition a new approach for its evaluation is presented. Furthermore, a detailed analysis of the proposed method is performed on a real case-study building with only heating conditioning.

In this paper an improvement of the model CaRM (CApacity Resistance Model) is presented to consider the borehole thermal capacitance, both of the filling material of the borehole and of the heat carrier fluid inside the ground heat exchanger. Several models, numerical and analytical, are available in literature for short time step analyses of ground-coupled heat pump systems. According to the modelling for the surrounding ground, the new approach for the inside of the borehole is based on electrical analogy. In this study the double U-tube ground heat exchanger is analyzed. The new model has been validated by means of a commercial software based on the finite elements method as well as measurements of ground response test, using a suitable plant system. In this last comparison, the contribution of the thermal capacitance of the circulating fluid is investigated, since it is frequently neglected in short time step simulations. In both cases, there is agreement between the CaRM results and data from numerical simulations and measurements as well.

Ground source heat pump systems often use vertical boreholes to exchange heat with the ground. Two areas of active research are the development of models to predict the thermal performance of vertical boreholes and improved procedures for analysis of in situ thermal conductivity tests, commonly known as thermal response tests (TRT). Both the models and analysis procedures ultimately need to be validated by comparing them to actual borehole data sets. This paper describes reference data sets for researchers to test their borehole models. The data sets are from a large laboratory “sandbox” containing a borehole with a U-tube. The tests are made under more controlled conditions than can be obtained in field tests. Thermal response tests on the borehole include temperature measurements on the borehole wall and within the surrounding soil, which are not usually available in field tests. The test data provide independent values of soil thermal conductivity and borehole thermal resistance for verifying borehole models and TRT analysis procedures. As an illustration, several borehole models are compared with one of the thermal response tests.

This paper presents the development and application of a three-dimensional (3D) numerical simulation model for U-tube borehole heat exchangers (BHEs). The proposed model includes the thermal capacities of the borehole components, viz., the fluid inside the tubes, as well as the grouting material, making it possible to consider the transient effects of heat and mass transports inside the borehole. In this approach, the use of simplified thermal resistance and capacity models (TRCMs) provides accurate results while substantially reducing the number of nodes and the computation time compared with fully discretized computations such as finite element (FE) models. The model is compared with a fully discretized FE model which serves as a reference. Furthermore, the model is used to evaluate thermal response test (TRT) data by the parameter estimation technique. Comparison of the model results with the results of an analytical model based on the line-source theory further establishes the advantage of the developed 3D transient model, as the test duration can be shortened and results are more accurate.

The effective pipe-to-borehole thermal resistance of a vertical ground heat exchanger is investigated numerically. An analysis is carried out to determine the dimensionless geometrical parameters affecting such resistance. The heat transfer rates between the U-shaped pipes and the borehole are determined numerically and compared with some well-known limiting analytical solutions. A best-fit correlation for the effective pipe-to-borehole thermal resistance is presented in dimensionless form. The results are compared against approximate analytical solutions that represent the U-shaped pipes as a single pipe of equivalent diameter and against experimental data available in the literature. It is found that the available models do not accurately represent the effective pipe-to-borehole thermal resistance.

In the design of a ground-source heat pump (GSHP) system, the heat transfer from the fluid to the ground is influenced by the thermal borehole resistance between the fluid and the borehole surface and also by the interference resistance between the two (or four) pipes inside the borehole. Several authors have proposed empirical and theoretical relations to evaluate these resistances as well as methods to evaluate them experimentally. The paper compares the different approaches and proposes good practice to evaluate the resistances. The impact of the different approaches on the design of heat exchanger is also examined. Two-dimensional and fully three-dimensional numerical simulations are used to evaluate the different methods. A new method is also proposed to evaluate the borehole resistances from in situ tests.

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This paper presents the development of a computationally efficient finite element tool for the analysis of 3D steady state heat flow in geothermal heating systems. Emphasis is placed on the development of finite elements for vertical borehole heat exchangers and the surrounding soil layers. Three factors have contributed to the computational efficiency: the proposed mathematical model for the heat exchanger, the discretization of the spatial domain using the Petrov–Galerkin method and the sequential numerical algorithm for solving the resulting system of non-linear equations. These have contributed in reducing significantly the required number of finite elements necessary for describing the involved systems. Details of the mathematical derivations and some numerical examples are presented. Copyright © 2005 John Wiley & Sons, Ltd.

A finite element numerical model has been developed for the simulation of the ground heat exchangers (GHEs) in alternative operation modes over a short time period for ground-coupled heat pump applications. Comparisons between the numerical and analytical results show that the finite line-source model is not capable of modeling the GHEs within a few hours because of the line-source assumption. On the other hand, the experiments with respect to the alternative cooling and heating modes have been undertaken during a short-time period. The comparisons show a reasonable agreement between the numerical and the measured data. The results illustrate that the finite element numerical model can be used to simulate the heat transfer behavior of the GHEs in short time scales instead of the typical finite line-source model. Finally, the variation of the U-tube pipe wall temperatures demonstrates that the discontinuous operation mode and the alternative cooling/heating modes can effectively alleviate the heat buildup in the surrounding soil.

Ground source heat pump (GSHP) systems exchange heat with the ground, often through a vertical, U-tube, borehole heat exchanger (BHE). The performance of this U-tube BHE depends on the thermal properties of the ground formation, as well as grout or backfill in the borehole. The design and economic probability of GSHP systems need the thermal conductivity of geological structure and thermal resistance of BHE. Thermal response test (TRT) method allows the in-situ determination of the thermal conductivity (λ) of the ground formation in the vicinity of a BHE, as well as the effective thermal resistance (Rb) of this latter. Thermal properties measured in laboratory experiments do not comply with data of in-situ conditions. The main goal has been to determine same in-situ ground type of BHE, including the effect of borehole's depths (60 m: VB2; 90 m: VB3). As shown in these results, λ and Rb of the VB2/VB3 boreholes are determined as 1.70/1.70 W m−1 K−1and 0.05/0.03 K W−1 m, respectively.

The imbalance of heat extracted from the earth by the underground heat exchangers in winter and ejected into it in summer is expected to affect the long term performance of conventional ground source heat pump (GSHP) in territories with a cold winter and a warm summer such as the middle and downstream areas of the Yangtze River in China. This paper presents a new multi-function ground source heat pump (MFGSHP) system which supplies hot water as well as space cooling/heating to mitigate the soil imbalance of the extracted and ejected heat by a ground source heat pump system. The heat transfer characteristic is studied and the soil temperature around the underground heat exchangers are simulated under a typical climatic condition of the Yangtze River. A three-dimensional model was constructed with the commercial computational fluid dynamics software FLUENT based on the inner heat source theory. Temperature distribution and variation trend of a tube cluster of the underground heat exchanger are simulated for the long term performance. The results show that the soil temperature around the underground tube keeps increasing due to the surplus heat ejected into the earth in summer, which deteriorates the system performance and may lead to the eventual system deterioration. The simulation shows that MFGSHP can effectively alleviate the temperature rise by balancing the heat ejected to/extracted from underground by the conventional ground source heat pump system. The new system also improves the energy efficiency.

Heat transfer around vertical ground heat exchangers is a common problem for the design and simulation of ground-coupled heat pump (GCHP) systems. Most models are based on step response of the heat transfer rate, and the superposition principle allows the final solution to be in the form of the convolution of these contributions. The step response is thus a very important tool. Some authors propose numerical tabulated values while others propose analytical solutions for purely radial problem as well as axisymmetric problems. In this paper we propose a new analytical model that yields results very similar to the tabulated numerical ones proposed in the literature. Analytical modeling offers better flexibility for a parameterized design.

Single borehole heat exchanger (BHE) and arrays of BHE are modeled by using the finite element method. Applying BHE in regional discretizations optimal conditions of mesh spacing around singular BHE nodes are derived. Optimal meshes have shown superior to such discretizations which are either too fine or too coarse. The numerical methods are benchmarked against analytical and numerical reference solutions. Practical application to a borehole thermal energy store (BTES) consisting of 80 BHE is given for the real-site BTES Crailsheim, Germany. The simulations are controlled by the specifically developed FEFLOW–TRNSYS coupling module. Scenarios indicate the effect of the groundwater flow regime on efficiency and reliability of the subsurface heat storage system.

Applications of a dynamic threedimensional numerical model for borehole heat exchangers

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Seasonal storage of solar energy in borehole heat exchangers

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Chapuis, S., & Bernier, M. (2009). Seasonal storage of solar energy in borehole
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Multipole method to calculate borehole thermal resistances in a borehole heat exchanger

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Claesson, J., & Hellström, G. (2011). Multipole method to calculate borehole
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Development of new ground loop sizing tools for domestic GSHP installations in the UK

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