Elsevier

Geothermics

Volume 46, April 2013, Pages 1-13
Geothermics

A three-dimensional numerical model of borehole heat exchanger heat transfer and fluid flow

https://doi.org/10.1016/j.geothermics.2012.10.004Get rights and content

Abstract

Common approaches to the simulation of borehole heat exchangers assume heat transfer within the circulating fluid and grout to be in a quasi-steady state and ignore axial conduction heat transfer. This paper presents a numerical model that is three-dimensional, includes explicit representations of the circulating fluid and other borehole components, and so allows calculation of dynamic behaviours over short and long timescales. The model is formulated using a finite volume approach using multi-block meshes to represent the ground, pipes, fluid and grout in a geometrically correct manner. Validation and verification exercises are presented that use both short timescale data to identify transport delay effects, and long timescale data to examine the modelling of seasonal heat transfer and show the model is capable of predicting outlet temperatures and heat transfer rates accurately. At long timescales borehole heat transfer seems well characterized by the mean fluid and borehole wall temperature if the fluid circulating velocity is reasonably high but at lower flow rates this is not the case. Study of the short timescale dynamics has shown that nonlinearities in the temperature and heat flux profiles are noticeable over the whole velocity range of practical interest. The importance of representing the thermal mass of the grout and the dynamic variations in temperature gradient as well as the fluid transport within the borehole has been highlighted. Implications for simplified modelling approaches are also discussed.

Highlights

► A three-dimensional borehole heat exchanger model has been developed. ► The model represents the borehole components and circulating fluid explicitly. ► Experimental and analytical validation at short and long timescales is presented. ► A numerical study of vertical temperature and heat flux profiles is presented. ► The importance of the interaction between the fluid and borehole has been shown.

Introduction

Single pairs of pipes formed in a ‘U’ loop and grouted into vertical boreholes are probably the commonest form of ground heat exchanger found in ground source heat pump systems, and are known as borehole heat exchangers (BHEs). The components of such a heat exchanger are illustrated in Fig. 1. BHEs of this type are not only used in building heating and cooling systems but in large thermal storage schemes also. The primary physical phenomena of interest in the study of heat exchanger performance are the dynamic conduction in the pipe, grout and surrounding ground as well as convection at the pipe wall. In reality, the heat transfer in the surrounding ground may be enhanced by groundwater flow through porous and possibly fractured rock. If interaction with the heat-pump system and its controls is to be considered then it becomes necessary to consider the physics of variable flow and diffusion of heat in the circulating fluid.

It is not common, nor always necessary, to include representation of all these physical processes in BHE models. This may be partly a practical consideration of what physical parameter data are available (or measurable) as well as the level of detail required to meet the modelling objective. Models of BHEs have three principle applications namely (i) design of BHEs – determining the required borehole depth, number of boreholes, etc.; (ii) analysis of in situ ground thermal response test (TRT) data; and (iii) integrated building and system simulation i.e. with the model coupled to HVAC and building thermal models to study overall system performance.

A number of analytical, numerical and hybrid models exist and the features of a number are reviewed here. These models differ mostly according to whether they consider three spatial dimensions, multiple boreholes, groundwater convection and buoyancy effects, heterogeneous thermal properties, grout and pipe thermal capacity and explicit representation of transport of heat by the circulating fluid.

The question of dimensionality and to what level of detail the gout, pipe and fluid components are represented bears a relationship to both the timescales and length scales that have to be considered. At short timescales, being able to resolve the dynamic changes in temperature gradient within the borehole is essential to determining fluid temperatures. At long timescales (a number of years), it is necessary to consider conduction in the surrounding ground in the axial (third) dimension. This is because – particularly if an array of boreholes is considered – conduction below the borehole, and towards the ground surface further from the borehole, become more significant after a number of years of operation. This was demonstrated in the early work of Eskilson (1987) who applied an axial-radial 2D numerical model to capture axial conduction effects and an analytical superposition method to consider interaction between neighbouring boreholes in the horizontal direction. The importance of axial heat transfer has been commented upon by Marcotte et al. (2010) and has also been recognised in more recent application of analytical finite line source models (Zeng et al., 2002, Molina-Giraldo et al., 2011) although in these models interaction between neighbouring boreholes is neglected.

If only medium and long timescales are considered – as they are in the ‘g-function’ response factor models of Eskilson (1987) and Hellström (1991) – then the borehole can be considered a single resistive element. This can be argued to be sufficient for applications of the model for design purposes where it is more important to consider long-term responses, particularly where annual heating and cooling demands are not well balanced. Eskilson stated that g-function data derived using his approach should only be applied at timescales such that t>5rb2/α. This limit may amount to a number of days. If shorter timescales are to be considered – as they need to be where system simulation is the objective – it may be sufficient to consider heat transfer in two-dimensions, and possibly only the radial direction. At shorter time scales, behaviour is strongly dependent on the dynamic behaviour of the borehole pipe, grout and fluid components. Hybrid approaches whereby different models are applied depending on time scale can be devised to treat the whole range of timescales. For example, a one-dimensional numerical model was added to the response factor approach in the DST model (Hellström, 1991) to allow simulation in the TRNSYS simulation environment (SEL, 1997).

Yavuzturk and Spitler (1999) used a two-dimensional numerical model of a borehole (Yavuzturk et al., 1999) to calculate short timescale responses and subsequently extend g-function response data to allow short simulation time steps (as short as a few minutes) to be simulated. Several studies have been carried out using this ‘short time step g-function’ model (Gentry et al., 2006, Sankaranarayanan, 2005) and the model has been implemented in the EnergyPlus simulation environment (Fisher et al., 2006). Yavuzturk's numerical model (1999) represented the pipes as ‘pie sector’ shapes and did not include an explicit representation of the circulating fluid. Young (2004) sought to address this by applying a ‘buried cable’ analogy to include the effect of the fluid's thermal capacity. Xu and Spitler (2006) sought to simplify the derivation of short timescale response data further by developing an equivalent one-dimensional numerical model that includes the thermal mass of the fluid.

As variation in fluid temperature according to depth cannot be considered explicitly in two-dimensional models such as those discussed above, some assumption has to be made about the fluid temperatures associated with the two pipes and their relationship to the inlet and outlet temperatures. For example, both pipes could be assumed to be at a temperature equivalent to the average of the inlet and outlet temperatures. An alternative is to assume one pipe temperature is the same as that of the inlet and the other is at the outlet temperature. These assumptions can be avoided in a three-dimensional numerical model where temperature variation with depth can be considered explicitly.

In some ground source heat pump systems, depending on the dynamic load profile, the minimum and maximum operating fluid temperatures are a dominant consideration in the design and control of the system. These extreme temperatures can occur on very short timescales, for example, where capacity control is by switching the heat pump cyclically on and off. Extreme temperatures may be exhibited on timescales of a few hours in intermittently occupied buildings such as churches. The importance of such short timescale effects, and their impact on the operation of the control system, were demonstrated in a simulation study of a hybrid domestic system by Kummert and Bernier (2008) and measurements of a larger non-residential system by Naiker and Rees (2011). It is apparent that at such short timescales, or generally at higher frequencies of inlet temperature variation, the response at the BHE outlet is far from instantaneous and peaks in outlet temperature are both damped and delayed. This is also indicated in the experimental data presented later in this paper. Accurately predicting peak or minimum temperatures is therefore a significant modelling issue in some important applications.

Damping of the inlet temperature fluctuations can be accounted for partly by the observation that, particularly in non-residential systems with larger diameter distribution pipes, the total thermal capacity of the fluid in the U-tubes and interconnecting pipes is relatively large – probably of the same order as that of the grout in all the boreholes. The physical process that has a further effect on the short timescale response is the dynamic transport of the circulating fluid and thermal diffusion along the pipes. This could be expected to be important at short timescales if one considers that the nominal transit time of the fluid travelling through the U-tube could be of the order of a few minutes with typical BHE depths and pipe velocities. Variations in inlet temperature are diffused because fluid does not circulate in a ‘plug’ with uniform velocity but fluid at the centre of the pipe travels at higher velocity than the fluid near the pipe wall. Hence, fluid at the outlet will generally have been mixed with fluid in the pipe that entered the heat exchanger at an earlier time and probably at a different temperature. Both the thermal mass of the fluid and the diffusive transport process mean that swings in inlet temperature tend to be damped. Such effects can also be expected to be more noticeable in systems with variable flow. In such systems, the transport delay could be several minutes when the flow is reduced to minimum levels during part load conditions.

A number of BHE models include an explicit representation of fluid circulation but stop short of complete three-dimensional discretization. Some models make other simplifications in order to limit the total number of equations to be solved. For example, Wetter and Huber's EWS model (1997) is discretized vertically so that a series of fluid nodes are included but the heat transfer in the grout is represented by a single lumped capacitance and conduction is assumed radial only. Oppelt et al. (2010) have sought to address this limitation of the EWS model by dividing the grout into sectors so that each vertical layer of a double U-tube was represented by five lumped thermal capacitances. De Carli et al. (2010) developed a so-called capacity resistance model (CaRM) and discretized the borehole – including the circulating fluid – into several slices along its depth with each slice also discretized in the radial direction. They also proposed modifying the outer boundary conditions to allow whole borehole arrays to be modelled.

Bauer et al. (2011) used a simplified representation of the borehole components in the form of a network of resistances and capacitances in the TRCM model and discretized the borehole in the vertical direction in a similar way to the EWS and CaRM models. Fluid responses and vertical temperature gradients calculated over short timescales using this model compared favourably with those from a fully discretized finite element model. These network or simplified finite difference models could be thought of as quasi-three-dimensional and have the advantage that fluid transport is explicitly represented but are limited in not being able to calculate axial heat transfer outside the borehole and hence not able to represent all long timescale effects.

Where three-dimensional numerical models have been applied, the interest has mostly been in studying long time and spatial scales, for example, interaction in larger borehole arrays, heterogeneous ground properties or the effects of groundwater flow. In view of the computational demands of such methods, a number of approaches have been proposed to reduce the discretization of the borehole components. Al-Khoury et al. (2005) and Al-Khoury and Bonnier (2006) developed a special 1D heat pipe finite element that considered the pipe flows and conduction in the grout material using a single element. The borehole fields of interest were discretized using a vertical line of such elements coupled to surrounding 3D elements. A very similar approach was developed by Diersch et al. (2011) to integrate a BHE in a commercial finite element software package and similarly, Signorelli et al. (2007). Mottaghy and Dijkshoorn (2012) have taken a similar approach except that the borehole is represented by a finite difference model coupled to the 3D finite element solver. Cui et al. (2008) used a commercial finite element solver but discretized the grout and pipes with a relatively fine mesh of 3D elements. Although their treatment of the fluid is not fully described the predictions of heat transfer rates and pipe wall temperature compared favourably with short timescale experimental measurements.

One approach to modelling BHE with the aim of capturing all the physical effects noted earlier is to use a three-dimensional numerical model that discretizes the borehole components and includes a discrete dynamic model of the circulating fluid. This is the approach taken in the work reported here. Three-dimensional models have the advantage that dynamic fluid transport along the pipe loop can be represented explicitly and temperature variations according to depth can be modelled. In addition, different layers of rock and soil can be explicitly represented and climate dependent boundary conditions at the surface can be applied. Furthermore, heat transfer below the borehole array can be explicitly considered and initial vertical ground temperature gradients can be imposed.

Three-dimensional models offer most generality and potentially most accurate representation of heat transfer but have the disadvantage that considerable computing resources are required for transient simulation over extended timescales. The model presented here has consequently been used to make generic studies of BHE behaviour and to calculate response function data over the full range of short and long timescales. Although the model has been used in annual simulation this would not be practical for most users. The model has been presented in briefer form, and for particular applications, in He et al. (2010) and more fully in He (2012). The intention has also been to use the model as a reference so that the limitations of simpler two-dimensional models can be better understood and an improved model developed. This is reported elsewhere.

In this paper, we describe the underlying numerical method and the approach taken to discretize the BHE and surrounding ground. The model has been validated in a number of ways. The model's ability to accurately calculate the conduction around the pipes within the borehole, and generally to deal with non-orthogonal meshes, is verified with reference to analytical conduction heat transfer solutions. The limitations of the approach taken to model the transport of fluid along the pipe are also studied with reference to an analytical solution. More than one year of experimental data has been used to validate the ability of the model to calculate seasonal heat balances and high frequency data to study the significance of the fluid circulation on short timescale dynamic response. Later in this paper, we present a numerical study of borehole vertical temperature and heat flux profiles under a range of fluid flow conditions.

Section snippets

Model development

A dynamic three-dimensional numerical model of BHEs has been developed, built upon a finite volume solver known as GEMS3D (General Elliptical Multi-block Solver 3D) which is an in-house code implemented in Fortran 90. The GEMS3D solver has been used to model ground heat exchanger problems in a number of earlier projects (Deng et al., 2005, Rees et al., 2002) and further details are given in He (2012). Model verification and validation of this solver has been reported elsewhere by Young (2004)

Model validation

The ability of the model to calculate transient heat transfer rates over both short and long timescales has been validated by a combination of analytical and experimental analysis. The accuracy of the fluid transport model has been evaluated by reference to analytical solutions for adiabatic pipe flow. Calculation of conduction within the borehole has been verified by reference to analytical solutions of the steady-state borehole thermal resistance. Finally, experimental data has been used to

Vertical temperature and heat flux profiles

One of the features and advantages of a three-dimensional model such as this, is that temperature and heat flux variations with depth are explicitly calculated. We have, further to the validation exercises reported above, made a numerical study of the behaviour of a BHE with respect to temperature and heat flux variation within the borehole and their variation with depth. Of particular interest have been the relationships between fluid temperatures and the inlet and outlet temperatures,

Conclusions

Using a multi-block mesh to represent each component of the borehole heat exchanger in three-dimensions and applying a finite volume numerical method it has been possible to develop a borehole heat exchanger model that can represent both conduction and fluid circulation processes over both short and long timescales. The model's main purpose, in light of the computational demands of three-dimensional numerical calculations, is to derive step response data for other modelling approaches, to act

Acknowledgements

The authors would like to thank J.D. Spitler and S. Hern of the Building, Thermal and Environmental Systems Research Group at Oklahoma State University for provision of the experimental data.

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