COMPARING THE THERMAL PERFORMANCE OF GROUND HEAT EXCHANGERS OF VARIOUS LENGTHS

Transcription

1 COMPARING THE THERMAL PERFORMANCE OF GROUND HEAT EXCHANGERS OF VARIOUS LENGTHS Marco Fossa Diptem, University of Genova, Via Opera pia 15a, Genova, ITALY Odile Cauret EDF R&D, "Energy in Buildings and Territories" Department, Moret-sur-Loing, France Michel Bernier École Polytechnique de Montréal, Département de génie mécanique, Montréal H3C 3A7, Canada ABSTRACT In many ground-coupled heat pump systems, borehole heat exchangers (BHE) represent the most important cost item. Therefore, alternative BHE construction should be investigated. In this paper the possibility of using short BHE (drilling depth lower than 50 m) is considered, based on the premise that shorther BHE drilling technologies and procedures could result in a lower cost per unit length of ground heat exchanger. The objective of this study is to compare the thermal performances of standard depth BHE with short ones, for a given building load. The thermal response of several BHE arrangements, expressed in the form of g-functions, are obtained using the so-called finite line source method with appropriate spatial superposition. The results are discussed with reference to literature data and also examined with respect to annual hourly simulations, in order to assess advantages of short BHE arrangements compared to current drilling solutions. 1. INTRODUCTION Ground-coupled heat pump (GCHP) systems are now routinely installed to provide space conditioning in a wide range of applications from small residences to large commercial buildings. In closed systems, a fluid is circulated in the ground heat exchanger and then in the evaporator of the heat pump (winter operating mode). Typically, the ground heat exchanger consists of vertical boreholes heat exchangers (BHE) that are approximately 100 m deep and have a diameter of about 100 mm. The number of boreholes in the bore field can range from one for a residence to several dozens in commercial applications. GCHP systems equipped with vertical BHE are characterised by relatively low operating costs. However, they require a high initial investment, mainly due to borehole drilling and installation. Therefore, in order to insure a durable development of GCHP systems, drilling and installation costs have to be reduced. In that context, this study explores the possibility of using short boreholes. Reducing drilling depth below 50 m could permit the use of geotechnical drilling materials and techniques that are less expensive and better adapted than standard deep vertical boreholes, in particular in Europe. However, using shorter boreholes can have consequences in term of thermal performances will the possible result that more

2 boreholes may be required. The objective of this study is to compare the thermal performances of standard depth BHE with short BHE, for a given building load.. THEORETICAL BACKGROUND AND ANALYSIS The thermal interaction between the ground and a BHE arrangement, when underground water circulation can be neglected, is governed by the three-dimensional time-dependent conduction equation. Due to its complexity this equation is usually solved numerically. However, a number of one-dimensional (in the radial direction) and two-dimensional (radial and axial) analytical solutions exists. Combined with spatial superposition, these solutions can be used to obtain a quasi three-dimensional solution to heat transfer from BHE leading to relatively short computational time even for multi- annual hourly simulations. Analytical approaches can be divided into models based on the line source theory and models based on the cylinder source method (CSM). Both methods give the radial temperature distribution as a function of time. The line source theory, introduced by Kelvin in 188 (cited by Ingersoll et al., 1954), approximates the BHE as an infinitely long line in an infinite medium subjected to a constant heat transfer rate per unit length. The medium is assumed to be initially at a uniform temperature and the thermal properties are assumed constant, The cylindrical source method was first introduced by Carslaw and Jaeger (1947). This method is similar to the line source method except that the constant heat transfer rate condition per unit length ( Q ) is applied to a cylindrical surface of radius r cyl. The ground temperature excess, with respect to far field temperature (T gr, ), is expressed in terms of Bessel s functions. Carslaw and Jaeger (1947) have also presented an abbreviated expression using a G function: Q T ( r) Tgr, = G( Fo, r / r cyl ) (1) k where: α t Fo = () r cyl and k and α are the ground thermal conductivity and thermal diffusivity, respectively. Tabulated values of G are presented by Carslaw and Jaeger (1947). Recently Bernier (001) provided a correlation to approximate the G-function A number of authors, including Deerman and Kavanaugh (1991) and Bernier et al. (004) have employed the CSM solution with success. A new insight to the linear source theory is given by the analysis of the finite linear source model (FLS). Even if the problem was already tackled by Eskilson (1987), a major contribution to the FSL solution was given first by Zeng et al. (00) and later by Lamarche and Beauchamp (007). The Lamarche and Beauchamp solution in particular provides the averaged (along the depth) borehole temperature in terms of the complementary error function (erfc) according to the expression: T ave ( r) T gr, Q = π k β + 1 β erfc z β + 4 ( γ z) erfc( γ z) β dz D A β + 1 z β dz D B (3) where β is the radial distance made dimensionless by the BHE depth H, and γ is related to a Fourier number based on the BHE depth, Fo H :

3 0.5 α t 0.5 γ = 0.5 = 0.5Fo H H (4) In Eq. (3), D A and D B are also expressed as a function of erfc, and they are constants at given time and depth H. The first integral is improper and has to be handled with suitable numerical techniques. Readers are referred to the work of Lamarche and Beauchamp (007) and Cauret and Bernier (009) for more details. In this paper the FLS solution expressed by Eq. (3) is iteratively calculated until a given degree of accuracy is obtained in either the proper or improper integral by employing the extended midpoint algorithm (Press et al., 199).Of practical interest is the calculation of the time varying temperature at the borehole boundary, i.e. for r = r b. A hybrid method, which combines the advantages of numerical and analytical techniques, is used to account for borehole thermal interaction.it exploits the linear properties of the conduction equation to perform a solution superposition. The superposition of single solutions can be made either with respect to similar thermal pulses applied in different BHE in a bore field (spatial superposition ) or to a series of different thermal pulses applied in a single BHE (superposition in time). The spatial superposition can be applied to calculate the response of a BHE field to a heat step pulse, resulting in a mean borehole wall temperature for the whole bore field. The superposition in time was first suggested by Eskilson (1987) and then refined as a numerical technique by Yavuzturk and Spitler (1999) and by Bernier et al. (004). The best known hybrid approach is the one introduced by Eskilson (1987). It is based on dimensionless temperature response factors, called g-functions. They were obtained using a - D numerical calculation using the transient finite-difference equations on the radial-axial coordinate system with a given borehole wall temperature. Eskilson (1987) established that the thermal response is a function of four parameters: i) the Fourier number, ii) r b /H, iii) B/H where B is the distance between boreholes, iv) the bore field configuration. Therefore, the thermal response of a given bore field geometry can be expressed as: Q ave Tave( rb ) Tgr, = g ( ln(9 FoH ), rb / H, B / H, borefield geometry) π k (5) where T ave (r b ) is the average borehole wall temperature for the whole borefield, and Q ave is the average heat transfer rate per unit length. Eskilson (1987) has evaluated g-functions for a number of configurations. However, in the case of the present study, the g-functions available in the literature did not cover the full range of bore field configurations. Therefore, g- functions were generated using the FLS solution combined with spatial superposition. The procedure is as follows: first, the temperature at each borehole wall (at r=r b ) is evaluated. This temperature is the result of the heat transfer rate (constant at all BHE) applied to the borehole itself and of the thermal influence of the other boreholes in the bore field. Mathematically, for N boreholes and a given heat transfer rate per unit length, this can be expressed as: N T i ( rb ) = T j ( rj i, t) (6) where r j-i represents the distance between borehole i and j with r j-i equal to r b for i=j. The FLS solution (Eq. 3) is thus applied N times to obtain the average borehole temperature of borehole i. Finally the average borehole temperature of the whole bore field, T ave, is obtained as: T ave ( r ) 1 b = N N T ( r i b ) (7)

4 By repeating the calculation at different time steps (i.e. various Fo), it is possible to infer the thermal response of the whole bore field and, using Eq. (5), to calculate the g-function of the BHE system under investigation. Finally, by varying the field geometry and depth, it is possible to compare the performance of different BHE arrangements, including short and long BHE arrays. Once a g-function is available, the time superposition can be applied to investigate the ground response of complex heat loads made by a series of basic heat pulses. Once the g-functions are known, it is possible to use them in cases of time varying heat pulses. This can be accomplished using temporal superposition (see for example, Yavuzturk and Spitler, 1999). One other method that has been suggested is the one developed by Bernier (Bernier et al., 004, Bernier et al., 008). In this method, the mean borehole fluid temperature at time t, T f,t, can be obtained using the following relationship: 1 T f,t = Tgr, + Tp,t + Q' ave Rb ( MLAA ) (8) kh where, T p,t is the temperature penalty at time t, R b is the steady-state borehole thermal resistance per unit length of bore, H is the borehole length, and MLAA is a multiple load aggregation algorithm. The temperature penalty accounts for thermal interaction among boreholes. It uses g-functions (generated as described in this paper by the FLS method) to obtain the average temperature increase (decrease) at the borehole wall caused by the thermal interaction from the other boreholes. It is evaluated using the average heat transfer rate over the period from time =0 to the time=t. The MLAA term on the right hand side of equation 1 account for heat transfer at the borehole level (excluding borehole interaction). This term represents the temperature difference between the borehole wall and the undisturbed ground. This difference is evaluated using the cylinder source model combined with temporal superposition as outlined by Bernier et al. (004). 3. RESULTS AND DISCUSSION The objective of this study is to use g-functions to compare the thermal performance of short and standard BHE depths. The standard reference configurations are 100 m deep while the short BHEs have depths of 60, 50, 40 and 30 m. Results concerning 50 and 40 m are here discussed. Before performing the comparisons described above, a number of preliminary runs have been made in order to assess the reliability of the numerical technique and of the FLS method itself, with respect to the numerical solutions obtained by Eskilson. The configurations that were examined included single boreholes, x8 in line, and square bore fields, up to 8x8. Figures 1 and report some results of this preliminary analysis with 8x1 and 3x3 geometries. As indicated in equation 5, g-functions are a function of time (expressed here by the natural logarithm of 9 times the Fourier number). They are also a function of r b /H ( in this case), of the bore field configuration (8x1 and 3x3 in this case), and of the borehole spacing ratio B/H. The FLS solutions are here overlapped to the original Eskilson graphical solutions for different borehole spacing ratio B/H, where the single BHE case is represented by B/H=. As can be observed in both figures, the agreement among the FLS and the Eskilson results is very good. For the sake of brevity it is not possible to show all the results obtained, but it must be outlined that the FLS solutions sometimes overestimated the Eskilson g-function for small values of the parameter B/H (as in Figure ), while in other cases the behaviour was opposite, with the FLS solution underestimating the Eskilson g-function for large values of the B/H parameter (as in Figure 1). This behaviour was also observed by Sherif (007).

5 B/H= g-function B/H=0.1 B/H=0.15 B/H=0. B/H= g-function B/H=0.05 B/H=0.15 B/H=0.3 B/H=0.1 B/H=0. 1 B/H= ln(9fo H ) ln(9fo H ) Calculated g-function values and comparison with graphical results by Eskilson. Figure 1: 8 boreholes in line Figure : 9 boreholes in a regular square 0 As a general finding, the FLS solution agreed with the original Eskilson solution within ±10%. This discrepancy could be ascribed to the different boundary condition employed in the FLS method (constant heat flux along BHE) with respect to boundary condition employed in the original work by Eskilson (constant temperature along BHEs). In order to assess the effect of the boundary conditions on the solution in terms of g-function, a single FLS calculation (8x4 array, B/H=0.05) was made by modifying the numerical routine in order to iteratively change the heat flux at each BHE until a common BHE temperature was obtained. This single attempt revealed that the FLS solutions obtained with both boundary conditions are very similar: both the constant flux and the (almost) constant temperature solutions obtained with the FLS method overestimate (8% for ln(9fo H )=3) the Eskilson values. These evidences deserve an in-depth examination and the matter will be discussed in a future paper. Given that the g-functions generated by the FLS are reliable, attention will now be turned to the thermal performance comparison between shorth and long boreholes. In order to perform this comparison, the FLS equations were solved iteratively by changing the borehole-toborehole distance B, until the asymptotic value of the g-function of the short field had attained approximately the same values as the standard ones. The asymptotic value of the g- function indicates that the bore field as reached a somewhat steady-state condition. It must be noticed that the analysis was carried out by imposing that the overall length of standard and short BHE arrays were the same. Furthermore, the borehole radius, r b, was constant and equal 0.05m. Figures 3a and 3b show the calculated g-function values of equivalent BHE configurations with respect to a reference one, constituted by 4 ground heat exchangers in line, having different spacings. The equivalent short BHE configurations considered here are x4 and 1x10 with corresponding depths of 50 and 40 m, respectively. The examination of Figure 3a reveals that it is possible to obtain shorth boreholes that have the same steady-state g-function as longer boreholes. For example, in Figure 3a the shorth x4 configuration with B=7 m and H=50 m is equivalent, at steady-state, to the standard 100 deep configuration with B/H=0.05. A

6 g-function x4 B/H=0.05 x4 B=7m 1x4 B/H=0.1 x4 B=10m 1x4 B/H=0.15 x4 B=13m 1x4 B/H=0. x4 B=16m 1x4 B/H=0.3 x4 B=0m g-function x4 B/H=0.05 1x10 B=4m 1x4 B/H=0.1 1x10 B=6m 1x4 B/H=0.15 1x10 B=7m 1x4 B/H=0. 1x10 B=9m 1x4 B/H=0.3 1x10 B=11m H=100 H=50 4 H=40 H= ln(9fo H ) similar conclusion can be drawn in Figure 3b with 1x10, B=4 m and H=40 m configuration having the same steady-state value as the 1x4, 100 m deep configuration. Even though short and long boreholes may have the same g-functions at steady-state, they do not have the same performance during their life time. To illustrate this, the two top curves in Figure 3a will be used. Assuming that α=0.1 m /day and t=3650 days (10 years), then the value of ln(9fo H ) is equal to 0.73 and for the short and long boreholes respectively. Reading the g-functions from Figures 3a, one obtains values of 10.8 and 9.4 for the short and long boreholes, respectively. This indicates that, for a given heat pulse, the long borehole will have a lower borehole wall temperature. Thus, in a heat rejection situation the long borehole performs better than the shorther borehole with B=7 m. In order to have a short borehole perform better than the longer one, the borehole spacing has to be increased. For example, if B=10m (fourth line from the top) the g-function is then 8.7 giving the short borehole configuration a thermal performance advantage. The thermal performance of short and long borehole can also be compared by performing annual hourly simulations. Following the example provided above, a 0 year hourly simulation is performed for a 1x4 standard 100m deep borehole (B=5m) and a 4x short 50 m deep borehole (B=8m). A typical medium size building was used. Its building load is shown in Figure ln(9fo H ) Figures 3a and 3b: Calculated g-function values and possible equivalent configurations. Reference case is 4 boreholes in line with H=100m Figure 4: Hourly building load for simulation of long and short BHE configurations.

7 (a) (b) Figure 5a and 5b: Hourly response of long (a) and short (b) BHE fields in terms of mean fluid temperature In order to simplify the analysis a constant COP of 4 was used in both heating and cooling and the borehole resistance was artificially set to zero. The ground thermal conductivity was set to.5 W/mK and T gr, =10 C. Equation 8 was used to obtain the mean fluid temperature. Results are shown in Figures 5a and 5b for the standard and the short boreholes, respectively. As can be seen both bore fields experience the same thermal performance with the mean fluid temperature being almost equal during every hour of a 0 year simulation. 4. CONCLUSIONS In this paper, the thermal response of a number of BHE arrangements, expressed in the form of g-functions, are obtained using the so-called finite line source method with appropriate spatial superposition. The generated g-functions are generally in good agreement with those of Eskilson. However, an in-depth study is needed to clearly understand some minor differences. These new g-functions have been used to compare the thermal performances of short boreholes (H< 50 m) with standard ones (H=100m) for a constant overall length. This study shows that shorter and standard boreholes can have similar performances during a great part of their working time, provided that shorter boreholes are spaced further apart than

8 standard ones. In this sense either a long terms analysis, based on the g-function approach, or a more detailed one based on the BHE response to short term thermal pulses (hourly loads managed by an aggregation algorithm) are necessary to perform a correct comparison of various BHE length, to infer the right spacings in case of short borehole fields and to draw some general sizing rules. This last aspect will be the subject of a future investigation. REFERENCES Bernier, M.A. (001). Ground Coupled Heat Pump System Simulation. ASHRAE Transactions, 106(1), Bernier, M.A., Pinel, P., Labib, R. Paillot, R. (004). A Multiple Load Aggregation Algorithm for Annual Hourly Simulations of GCHP Systems. International Journal of Heating, Ventilating, Air-Conditioning and Refrigeration Research, 10(4), Bernier, M.A., Chahla, A., Pinel, P Long-term Ground Temperature Changes in Geo-Exchange Systems, ASHRAE Transactions, 114() pp Zeng, H.Y., Diao, N.R., Fang, Z.H. (00). A Finite Line-Source Model for Boreholes in Geothermal Heat Exchangers. Heat Transfer-Asian Research, 31(7), Lamarche, L. Beauchamp, B. (007). A New Contribution to the Finite Line-Source Model for Geothermal Boreholes. Energy and Buildings, 39, Yavuzturk, C., Spitler, J.D. (1999). A Short Time Step Response Factor Model for Vertical Ground Loop Heat Exchangers. ASHRAE Transactions. 105(), Press, W.H., Flannery, B.P., Teukolsky, S.A. (199), Numerical Recipes in Fortran 77, Cambridge Press, Carslaw, H.S. and Jaeger, J.C Conduction of Heat in Solids. Oxford, U.K., Claremore Press. Ingersoll, L.R., Zobel, O.J. and Ingersoll, A.C. (1954). Heat Conduction with Engineering, Geological, and other Applications. New York, McGraw-Hill. Deerman, J.D. and Kavanaugh, S.P. (1991) Simulation of Vertical U-tube Ground Coupled Heat Pump Systems using the Cylindrical Heat Source Solution. ASHRAE Transactions 97(1), Eskilson, P. (1987) Thermal Analysis of Heat Extraction Boreholes. Ph.D. thesis, Lund University of Technology, Sweden. Cauret, O., Bernier, M. (009) Experimental Validation of an Underground Compact Collector Model, Proc. Effstock Conference, Stockholm, June 009. Sheriff, F. (007), Génération de facteurs de réponse pour champs de puits géothermiques verticaux, M.Sc.A., Département de Génie Mécanique. École Polytechnique de Montréal, Québec, Canada.