HEAT TRANSFER ANALYSIS OF GEOTHERMAL HEAT EXCHANGERS IN GROUND-COUPLED HEAT PUMP SYSTEMS. Abstract. 1. Introduction

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1 HEX-01 HEAT TRANSFER ANALYSIS OF GEOTHERMAL HEAT EXCHANGERS IN GROUND-COUPLED HEAT PUMP SYSTEMS Z Fang, N Diao, M Yu, P Cui The Ground Source Heat Pump Research Center Shandong Institute o Architecture and Engineering, Jinan, China Abstract Utilizing the ground as a heat source/sink, ground-coupled heat pump (GCHP systems have been gaining increasing popularity or space conditioning in residential and commercial buildings. The geothermal heat exchanger (GHE is devised or extraction or injection o thermal energy rom/into the ground. This paper summarizes the authors studies on heat transer in groundcoupled heat pump systems. Taking the luid axial convective heat transer and thermal shortcircuiting among U-tube legs into account, a quasi-3-d model has been solved or heat transer inside boreholes. The transient -D temperature response in a semi-ininite medium with a linesource o inite length has also been derived or heat conduction outside boreholes. In order to investigate the impact o groundwater advection on the perormance o ground heat exchangers, an analytical solution is obtained or a line heat source in an ininite porous medium with groundwater advection. These explicit expressions have more solid theoretical basis, and can be easily incorporated into computer programs or thermal analysis and engineering design o the ground heat exchangers. 1. Introduction Due to their reduced energy and maintenance costs ground-coupled heat pump (GCHP systems, which use the ground as a heat source/sink, have been gaining increasing popularity or space conditioning in buildings. The eiciency o GCHP systems is inherently higher than that o air source heat pumps because the ground maintains a relatively stable temperature. This system is environment-riendly, causing less CO emission than their conventional alternatives. The ground heat exchanger (GHE is devised or extraction or injection o heat rom/into the ground. These systems consist o a sealed loop o pipe, buried in the ground and connected to a heat pump through which water/antireeze is circulated. In vertical borehole systems the GHE consists o a number o boreholes, each containing single or double U-tube pipes. The borehole annulus should be grouted with backilling materials that provide thermal contact between the pipe and the soil/rock and to protect groundwater rom possible contamination. Despite all the advantages o the GCHP system, commercial growth o the technology has been hindered by higher capital cost o the system, o which a signiicant portion is attributed to the GHEs. Heat transer between a GHE and its surrounding soil/rock is diicult to model or the purpose o sizing the exchanger or energy analysis o the system. Thus, it is crucial to work out appropriate and validated tools, by which the thermal behaviour o GCHP systems can be assessed and then, optimised in technical and economical aspects. There are roughly two categories o approaches in dealing with the thermal analysis and design o the GHEs. Empirical or semi-empirical ormulations are recommended in textbooks and monographs or GHE design purposes, e.g. by Bose et al. (1985 and Caneta Research Inc. (1995. Containing ew parameters o the GHE structure and physical properties, these approaches are relatively simple, and may be manipulated easily by design engineers. However, they do not reveal in detail the impacts o complicated actors on the GHE perormance such as coniguration o the borehole ields and imbalance in the annual heating and cooling loads. The second kind o approaches involves numerical simulation o the heat transer in the GHEs. With ast development o computers and computational technologies, numerical simulation has become a routine in heat transer studies, capable o taking into account details which are ignored in the empirical engineering approaches. Nevertheless, the heat transer in GHEs is normally transient threedimensional (3-D problems, dealing with complicated domains and long durations o tens o years.

2 It takes too substantial computing time to conduct such simulations or practical engineering applications. A notable example o such simulations was carried out by Mei and Baxter (1986, in which the geometry inside the borehole was simpliied and, then, a -D model in the cylindrical coordinates was solved by the inite dierence method. On the other hand, Yavuzturk et al. (1999 analysed the transient -D heat transer by the inite element method in the cross-section perpendicular to the borehole axis, ignoring the heat low along the borehole depth. While having provided important understandings on GHE heat transer, these studies o numerical simulation have not yet been suitable to design and/or energy analysis o ull scale engineering projects. Because o all the complications o this problem and its long time scale, the heat transer process may usually be analysed in two separated regions. One is the solid soil/rock outside the borehole, where the heat transer has to be treated as a transient process. The region inside the borehole may also be isolated or analysis, including the backilling, the U-tube and circulating luid inside the pipes. Compared with the ininite ground outside it, both the dimensional scale and thermal mass o the borehole are much smaller. Thus, it is a common practice that the heat transer in this region is approximated as a steady-state process. Such simpliication has been proved appropriate or most engineering practices except or analysis dealing with dynamic responses within a ew hours. The concept o thermal resistances and the principle o superimposition have been used in a new approach or GHE analysis initiated by Eskilson (1987. Here the temperature response and, then, the resistance o a single borehole experiencing a constant heating rate are cited repeatedly to obtain the actual GHE transient perormance. It is more adequate and accurate than the empirical approaches and yet much more convenient or computations than the numerical simulations. In this regard, better understanding o every thermal resistances o the single-borehole GHE is crucial, and their analytical solutions are especially preerred to acilitate the computation. This paper summarizes the authors work in this direction in recent years. Three important analytical solutions have been derived or heat transer processes both inside and outside the boreholes, which can be easily incorporated into computer programs or thermal analysis and engineering design o the GHEs while providing better insight into inluences o various actors on the GHE perormance.. Heat Transer inside Boreholes The thermal resistance inside the borehole bears strong impact on GHE perormance, deined by the thermal properties o the grouting materials and the arrangement o low channels o the borehole. The main objective o this analysis is to determine the entering and leaving temperatures o the circulating luid in the exchanger according to the borehole wall temperature and its heat low. A ew models o varying complexity have been established to describe the heat transer inside the GHE boreholes. Models or practical engineering designs are oten oversimpliied in dealing with the complicated geometry inside the boreholes. One-dimensional models have been recommended by Bose et al. (1985 and Caneta Research Inc. (1995 or engineering design, conceiving the U- tube pipes as a single equivalent pipe. Such an approach seems to be inadequate and unsatisactory in spite o its simplicity. By a dierent approach Hellstrom (1991 has derived analytical -D solutions o the thermal resistances among pipes in the cross-section perpendicular to the borehole axis, which are superior to empirical expressions. On assumptions o identical temperatures and heat luxes o all the pipes in it the borehole resistance can then be worked out. However, the luid circulating through dierent legs o the U-tubes is, in act, o varying temperatures. As a result, thermal intererence, or thermal short-circuiting, among U-tube legs is inevitable, which degrades the eective heat transer in the GHEs. With the assumption o identical temperature o all the pipes, it is impossible or these models to reveal impact o this thermal intererence on GHE perormances. Taking the luid axial convective heat transer and thermal short-circuiting among U-tube legs into account, a quasi-3-d model or boreholes in GHEs has been established, and analytical solutions o the luid temperature proiles along the borehole depth have been obtained or both single and double U-tube boreholes (Zeng and Fang 00 and Zeng et al The schematic diagram o a borehole with U-tubes in vertical GHEs is illustrated in Figure 1. In the case o a single U-tube in the borehole the energy equilibrium equations can be written or up-low and down-low o the circulating luid. That is

3 a double U-tube b single U-tube Figure1 Schematic diagram o boreholes in the vertical GHE exchanger dt Mc dz dt Mc dz 1 ( T 1 Tb ( T 1 T + R R 1 ( T Tb ( T T 1 + R R 1 1 ( 0 z H (1 where M is the low rate o the circulating luid and c its speciic heat, and and are deined R11R R1 R11R R1 R11R R1 as R1 R R1. R 11 and R are thermal resistance R R1 R11 R1 R1 between the pipe and borehole wall, R 1 is the resistance between the two pipes, ollowing Hellstrom s (1991 approach and denotation. The ull solution o this problem can be ound elsewhere (Zeng and Fang, 00. At the instance o symmetric disposal o the U-tube inside the borehole, the dimensionless temperature proiles in the two pipes are reduced as R 1 R Θ Θ 1 ( Z cosh( βz ( Z cosh cosh ( βz sinh ( β sinh( β ( β + sinh( β cosh 1 P cosh cosh ( β sinh( βz ( β + sinh( β ( βz ( βz sinh + cosh cosh ( β sinh( β P ( β + sinh( β (

4 T z T where the dimensionless parameters are deined as Θ T T H β. Mc ( R + R ( R R1 ( b b, Z z/h, P R 1 /R 11, and Typical dimensionless temperature proiles o the single U-tube pipes are plotted in Figure. It is noticed that the temperature proiles in the two illustrated examples are o dierent eatures due to dierence in their pipe-to-pipe resistance, or the dimensionless parameter P, which, o course, results in dierent exiting temperatures with other conditions unchanged. This demonstrates clearly the impact o thermal intererence between the U-tube pipes. Figure : Fluid temperature proiles along the borehole depth Diering rom the single U-tube boreholes, the double U-tube boreholes may have dierent low conigurations. The two sets o U-tubes may be arranged in parallel or series low circuits, and each o them includes a ew connecting patterns. The luid temperature proiles in the low channels and, then, the borehole resistance is aected by borehole conigurations. All these options have been analysed separately. Although the temperature proiles or the double U-tube boreholes are even more complicated, these explicit solutions can be computed readily on a computer. Fortunately, the temperature proiles or double U-tubes in parallel are similar to those o the single U-tube coniguration. Detailed results about double U-tube borehole heat transer can be ound elsewhere (Zeng et al., 003. The concept o eective thermal resistance acilitates heat transer analysis. The eective borehole thermal resistance deines the proportional relationship between the heat low rate transerred by the borehole and the temperature dierence between the circulating luid and the borehole wall. It can be determined explicitly according to the analytical solutions o luid temperature proiles in the borehole. For the single U-tube borehole and double U-tube in series the borehole thermal resistance can be obtained as R b H Mc 1+ Θ 1 Θ 3 The expressions o the borehole thermal resistance presented here take into account more actors than previous models ever did beore, including the geometrical parameters (borehole and pipe sizes and pipe disposal in the borehole and physical parameters (thermal conductivity o the materials, low rate and luid properties as well as the low circuit conigurations. The solutions have provided a reliable tool or GHE sizing and perormance simulation and a solid basis or technical and economic assessment o dierent borehole conigurations.

5 3. Heat Conduction outside Boreholes In GHEs the borehole diameter is much smaller compared to their depth, and the ground can be treated as a semi-ininite medium. Then, the one-dimensional (1-D line-source model in ininite medium is oten used in GHE analysis (Bose et al., 1985, oten reerred to as Kelvin s model. While being simple, this model is inadequate or analysis o long-term perormance o the GHEs. In many GCHP applications the heating loads are not in balance with the cooling loads in a year round basis. In case that the heat injected in summer cannot be extracted in winter, the redundant heat will accumulate in the ground and thus lead to increase in the annual mean temperature in adjacent soil. With the eect o the heat transer on the ground surace taken into account, the inluence o the imbalanced heat will approach a relatively steady state ater the ground heat exchanger operates or a long enough period. This process normally takes ten years or even more, depending mainly on the depth o the boreholes. The variation in the annual mean temperature o the ambient soil around the GHE will aect its long-term behaviour; and thus it must be taken into account when the ground loop is designed. The transient heat conduction around boreholes o the GHEs is analysed here in a -D model. An analytical solution o the transient temperature response has been derived in a semiininite medium with a line-source o inite length. This solution is suitable or sizing the ground loop o GCHP systems because it describes the GHE perormance more adequately than the 1-D model does with an ininite line-source. The -D model assumes that the ground is regarded as a homogeneous semi-ininite medium with a uniorm initial temperature, and that a line-source stretching vertically rom the boundary to a certain depth, H, releases heat at a constant rate per length, q l. Due to the central symmetry o the problem the temperature distribution is two-dimensional in the cylindrical coordinates. The solution o the temperature excess has been obtained by Zeng et al. (00 as ( z h ( z + h r + r + H erc erc ql aτ aτ θ ( r, z, τ dh (4 4kπ 0 r + ( z h r + ( z + h where k and a denote thermal conductivity and diusivity o the ground, respectively. In the design o the GHEs in the GCHP system, special attention is paid to the borehole wall temperature rise. In engineering applications a common practice is to use the temperature at the middle o the borehole wall as its representative temperature. The temperature responses obtained with equation (4 o the -D model are plotted in Figure 3 together with that o the Kelvin s 1-D model or comparison. It can be seen in Figure 3 that according to the -D model the temperature Figure 3: The temperature responses o a line source in 1-D and -D models

6 excess rises rapidly at the early period o heating and, then, turns to a rather gentle increase. Finally the temperature reaches a steady state when the time approaches ininity. In contrast, the temperature response indicated by the 1-D model keeps rising even when time approaches ininity. Similarly, equation (4 deines the thermal resistance o the soil outside the borehole, and can be used in thermal analysis and design o the GHEs. 4. Ground Heat Exchangers with Groundwater Advection Researchers and engineers in the GCHP arena realize that groundwater iltration may exert impact, large or small, on perormance o GHEs. Few studies on this problem, however, can be seen in literature except some qualitative discussions. All o the GHE design tools, thereore, are based simply on principles o heat conduction, and do not consider the implications o groundwater low in carrying away heat. The reasons or such simpliication are the diiculties encountered both in modelling and computing the convective heat transer and in learning about the actual groundwater low in engineering practice. In general, groundwater low is beneicial to the thermal perormance o GHEs. A moderate groundwater advection is expected to make notable dierence in alleviating the possible heat build-up around the borehole over time. As a result, it is desirable to consider the groundwater low in the heat transer analysis to avoid over-sizing o the GHEs. Among the ew reports on the eect o the groundwater low ound in literature Eskilson (1987 discussed the problem based on a steady-state analytical solution. Chiasson et al. (000 made a preliminary investigation o the eects o groundwater low by means o a numerical inite element scheme. No correlations among the inluencing parameters were obtained due to the discrete nature o the approach. The authors (Diao et al., 003 have solved the combined heat transer o conduction and advection in the vertical GHEs by an analytical approach, and an explicit expression o the temperature response has been derived describing correlation among various actors, which impact on this process. In this study the ground around the boreholes is assumed to be a homogeneous porous medium saturated by groundwater. Heat is transported through the saturated porous medium in a combined mechanism: by conduction through its solid matrix and liquid in its pores as well as by convection o the moving liquid. Following the Eskilson s (1987 model, a urther approximation is accepted that the groundwater velocity is uniorm in the whole domain concerned and parallel to the ground surace. The term advection is oten used to describe such a low. Deine that the velocity u is in the direction o the x-coordinate. Then, on the assumption o constant thermal properties the governing equation o the heat transer can be written as t t + U a t τ x (5 where U uρ w c ( ρc, and the eective thermal diusivity a k ( ρc w This problem has been solved under ollowing assumptions:. 1. The ground is regarded as homogeneous and semi-ininite medium;. The heat transer along the borehole axis is neglected. Then, the problem may be simpliied as two-dimensional; 3. The borehole is approximated by a line heat source; 4. The medium has a uniorm initial temperature, t 0 ; 5. The heating rate per length o the source, q l, is constant since a starting instant, τ 0. The authors have derived the solution on transient conditions. When a single line-source is deployed at the origin o the coordinates, the temperature rise in the medium, θ t t, o the 0 discussed case can be written as

7 r 4aτ ql Ux 1 1 U r η θ ( x, y, τ exp exp dη 4π (6 k a η η 0 16a where r + x y. Equation (6 reduces to the solution o the steady condition while the time approaches ininity, that is ql Ux Ur θ s( x, y exp K0 k (7 π a a where K 0 (z is the modiied Bessel unction o the second kind o order zero. On the other hand, or the speciic instance when the advection velocity u, and then, U uρ w cw ( ρc, equals zero, Equation (6 reduces to the common conductive temperature response o the line-source in an ininite medium: q l r θ τ 0 ( r, Ei 4πk 4a τ (8 where Ei(z is the exponential integral unction. As this expression indicates, the temperature in the medium will keep rising and no steady condition will be reached when the time lasts to ininity. The obtained temperature response to a single line-source heating with water advection considered can be used to compute the response o a GHE with multiple boreholes by superimposition o all temperature rises caused by individual boreholes. Figure 4 demonstrates such isotherms o a sixborehole GHE with groundwater advection. It has been shown that such simulations take much less computing time and assure more reliable precision than numerical solutions do with the inite element or inite dierence methods. The inluence o the groundwater advection on GHE perormance is discussed in more details elsewhere (Diao et al., 004. Figure 4: Isotherms o a GHE ield o six boreholes with groundwater advection

8 5. Conclusion Describing key processes in GHE heat transer, analytical expressions play an important role in engineering application o the GCHP technology. This paper presents three analytical solutions derived rom more sophisticated models, i.e. the quasi-3-d model or heat transer inside the borehole, the -D model or heat conduction o a line-source o limited length and the combined conduction/advection model considering the groundwater low in porous media. These explicit expressions are helpul to improve design procedures recommended by available literature. The models mentioned above deal with the temperature response o a single borehole to a step heating. Variation in load and on-o cycling o the GHEs can be considered by superimposition o a series o heating pulses (Fang et al., 00. For a borehole ield o varying conigurations its temperature response and thermal resistance can be determined also by the superimposition principle. According to these improved models computer sotware has been developed or engineering design and perormance simulation o the vertical GHEs, and has been proven appropriate or engineering applications (Yu et al., 00. Acknowledgement: The authors wish to thank the Natural Science Foundation o China or its inancial support (Project No Reerences Bose, J. E., Parker, J. D. and McQuiston, F. C., 1985, Design/data manual or closed-loop groundcoupled heat pump systems, ASHRAE, Atlanta, USA. Caneta Research Inc., 1995, Commercial/institutional ground-source heat pump engineering manual, ASHRAE, Atlanta, USA. Chiasson, A. D., Rees, S. J. and Spitler, J. D., 000, A preliminary assessment o the eects o groundwater low on closed-loop ground-source heat pump systems, ASHRAE Transactions, 106, Diao, N., Li, Q. and Fang, Z., 003, An analytical solution o the temperature response in geothermal heat exchangers with groundwater advection, Journal o Shandong Institute o Architecture and Engineering, 18(3, 1 5. Diao, N., Li, Q. and Fang, Z., 004, Heat transer in ground heat exchangers with groundwater advection, International Journal o Thermal Sciences, 43(1, Eskilson, P., 1987, Thermal analysis o heat extraction boreholes, Doctoral thesis, Department o Mathematical Physics, University o Lund, Sweden. Fang, Z., Diao, N., and Cui, P., 00, Discontinuous operation o geothermal heat exchangers, Tsinghua Science and Technology. 7(, Hellstrom, G., 1991, Ground heat storage -- Thermal analysis o duct storage systems, Doctoral thesis, Department o Mathematical Physics, University o Lund, Sweden. Mei, V. C. and Baxter, V. D., 1986, Perormance o a ground-coupled heat pump with multiple dissimilar U-tube coils in series, ASHRAE Transactions, 9 Part, -5. Yavuzturk, C., Spitler, J. D. and Rees, S. J., 1999, A transient two-dimensional inite volume model or the simulation o vertical U-tube ground heat exchangers, ASHRAE Transactions, 105 Part, Yu, M., Diao, N., Su, D. and Fang, Z., 00, A pilot project o the closed-loop ground-source heat pump system in China, Proceeding o IEA 7th Heat Pump Conerence, Zeng, H. and Fang, Z., 00, A luid temperature model or vertical U-tube geothermal heat exchangers, Journal o Shandong Institute o Architecture and Engineering, 17(1, Zeng, H., Diao, N., Fang, Z., 00, A inite line-source model or boreholes in geothermal heat exchangers, Heat Transer Asian Research, 31(7, Zeng, H., Diao, N. and Fang, Z., 003, Heat Transer Analysis o boreholes in vertical ground heat exchangers, International Journal o Heat and Mass Transer, 46(3,