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The importance of axial effects for borehole design of geothermal heat pump systems D Marcotte a b c P Pasquiera F Sheriffb M Bernierc aGolder Associates 9200 l Acadie Montreal Qc H4N 2T2 Canada bCANMET Energy Technology Centre Varennes 1615 Lionel Boulet Blvd P O Box 4800 Varennes QC J3X 1S6 Canada cDe partement des ge nies civil Ge ologique et des mines E cole Polytechnique de Montre al C P 6079 Succ Centre ville Montre al Qc H3C 3A7 Canada a r t i c l e i n f o Article history Received 13 May 2008 Accepted 18 September 2009 Available online 23 October 2009 Keywords Infi nite line source Finite line source Ground loop heat exchangers Hybrid systems Underground water freezing a b s t r a c t This paper studies the effects of axial heat conduction in boreholes used in geothermal heat pump systems The axial effects are examined by comparing the results obtained using the fi nite and infi nite line source methods Using various practical design problems it is shown that axial effects are relatively important Unsurprisingly short boreholes and unbalanced yearly ground loads lead to stronger axial effects In one example considered it is shown that the borehole length is 15 shorter when axial conduction effects are considered In another example dealing with underground water freezing the amount of energy that has to be removed to freeze the ground is three times higher when axial effects are considered 2009 Elsevier Ltd All rights reserved 1 Introduction Geothermal systems using ground coupled closed loop heat exchangers GLHE are becoming increasingly popular due to growing energy costs Such a system is presented in Fig 1 The operation of the system is relatively simple a pump circu lates a heat transfer fl uid in a closed circuit from the GLHE to a heat pump or a series of heat pumps Typically GLHE consists of boreholes that are 100 150 m deep and have a diameter of 10 15 cm The number of boreholes in the borefi eld can range from one for a residence to several dozens in commercial applications Furthermore several borehole confi gurations square rectangular L shaped are possible Typically a borehole consists of two pipes forming a U tube Fig 1 The volume between these pipes and the borehole wall is usually fi lled with grout to enhance heat transfer from the fl uid to the ground In some situations it is advantageous to design so called hybrid systems in which a supplementary heat rejecter or extractor is used at peak conditions to reduce the length of the ground heat exchanger Given the relatively high cost of GLHE it is important to design them properly Among the number of parameters that can be varied the length and confi guration of the borefi eld are important There are basically two ways to design a borefi eld The fi rst method involves using successive thermal pulses typically 10 years 1 month 6 h to determine the length based on a given confi gura tion and minimum maximum heat pump entering water temper ature 8 3 There are design software programs that perform these calculations Some use the concept of the g functions developed by Eskilson 5 The g functions are derived from a numerical model that by construction includes the axial effects The other approach is to perform hourly simulation This last approach is essential for design of hybrid systems in which supplemental heat rejection injection is used There are several software packages that can perform hourly borehole simulations For example TRNSYS 9 and EnergyPlus 4 use the DST 6 and the short time step model 5 respectively Even though these packages account for axial effects they necessitate a high level of expertise Furthermore it is not easily possible to obtain ground temperature distributions like the ones shown later in this paper In this paper hourly simulations are performed using the so called fi nite and infi nite line source approximations where the borehole is approximated by a line with a constant heat transfer rate per unit length These approximations present in a convenient analytical form the solution to the tran sient 2 D heat conduction problem Despite their advantages hourly simulations based on the line source approximation are Corresponding author De partement des ge nies civil ge ologique et des mines E cole Polytechnique de Montre al C P 6079 Succ Centre ville Montre al Qc H3C 3A7 Canada Tel 1 514 340 4711x4620 fax 1 514 340 3970 E mail address denis marcotte polymtl ca D Marcotte Contents lists available at ScienceDirect Renewable Energy journal homepage 0960 1481 see front matter 2009 Elsevier Ltd All rights reserved doi 10 1016 j renene 2009 09 015 Renewable Energy 35 2010 763 770 rarely used in routine design due to the perceived computational burden The major difference between the fi nite and infi nite line source lies in the treatment of axial conduction at the bottom and top of the borehole which is only accounted for in the former The theoretical basis of the fi nite line source although more involved than for the infi nite line source was fi rst established by Ingersoll et al 7 It has been rediscovered recently by Zeng et al 15 who improved the model by imposing a constant temperature at the ground surface Lamarche and Beauchamp 11 have made a useful contribution to speed up the computation of Zeng s model Finally Sheriff 13 extended Zeng s model by permitting the borehole top to be located at some distance below the ground surface She also did a detailed comparison of the fi nite and infi nite line source responses but did not examine the repercussion on borefi eld design At fi rst glance the axial heat diffusion is likely to decrease increase the borehole wall temperature in cooling heating modes respectively Therefore designing without considering axial effects appears to provide a safety factor for the design But is it really always the case Moreover are the borehole designs incorporating axial effects signifi cantly different from those neglecting it Under which circumstances are we expected to have signifi cant design differences These are the main questions we seek to answer The main contribution of this research is to describe using synthetic case studies the impact of considering axial effects on the GLHE design Our main fi nding is that for many realistic circumstances the axial effects cannot be neglec ted Therefore design practices should be revised accordingly to include the axial effects We fi rst review briefl y the theory for infi nite and fi nite line source models Then we present three different design situations The fi rst two situations involve the sizing of geothermal systems with and without the hybrid option under three different hourly ground load scenarios The last design problem examines the energy required and ground temperature evolution in the context of ground freezing for environmental purposes 2 Theoretical background The basic building block of both infi nite and fi nite line source models is the change in temperature felt at a given location and time due to the effect of a constant point source releasing q0units of heat per second 7 DT r t q0 4pksr erfc r 2 ffi ffi ffi ffi ffi at p 1 where erfc is the complementary error function r the distance to the point heat source andais the ground thermal diffusivity The line is then represented as a series of points equally spaced In the limit when the distance between point sources goes to zero Fig 1 Sketch of a GLHE system Nomenclature aThermal diffusivity m2s 1 A B C D Synthetic load model parameters kW br H CsGround volumetric heat capacity Jm 3K 1 erfc x Complementary error function erfc x 1 2 ffi ffiffi p p RN x e t 2dt EWT Temperature of fl uid entering the heat pump K or C FoFourier number Fo at r2 ksVolumetric ground thermal conductivity Wm 1K 1 HBorehole length m HPHeat Pump q0Radial heat transfer rate W qRadial heat transfer rate per unit length Wm 1 SBorehole spacing m rDistance to borehole m rbBorehole radius m RbBorehole effective thermal resistance KmW 1 tTime DT r t Ground temperature variation at time t and distance r from the borehole K or C TfFluid temperature K or C TgUndisturbed ground temperature K or C TwTemperature at borehole wall K or C u H 2 ffi ffi ffiffi at p x ySpatial coordinates m zElevation m D Marcotte et al Renewable Energy 35 2010 763 770764 the combined effect felt at distance r from the source is obtained by integration along the line 2 1 Infi nite line source In an infi nite medium the line integration gives the so called infi nite line source model 7 DT r t q 4pks ZN r2 4at e u u du 2 2 2 Finite line source In the case of a fi nite line source the upper boundary is considered at constant temperature taken as the undisturbed ground temperature 15 This condition is represented by adding a mirror image fi nite line source with the same load but opposite sign as the real fi nite line Then integrating between the limits of the real and image line one obtains 15 13 DT r t z q 4pks ZH 0 0 erfc d u 2 ffi ffi ffiffi at p d u erfc d0 u 2 ffi ffi ffiffi at p d0 u 1 Adu 3 where d u ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi r2 z u 2 q and d0 u ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi r2 z u 2 q z is the elevation of the point where the computation is done The left part of the integrand in Equation 3 represents the contribution by the real fi nite line the right part the contribution of the image line Fig 2 shows the vertical temperature profi le obtained with Equation 3 at radial distance r 2 m after 200 days and at r 1 m after 2000 days of heat injection The corresponding infi nite lines source temperature is indicated as a reference In this example the borehole is 50 m long the groundthermal parameters are ks 2 1 Wm 1K 1and Cs 2e06 Jm 3K 1 The ground is inti tially at 10 oC The applied load is 60 W per m for a total heating power of 3000 W As expected the importance of axial effects and the discrepancy between infi nite and fi nite models increases with the Fourier number at r2 4 54 and 181 4 for these two cases In hourly simulations the fl uid temperature Tfin Fig 1 is required This necessitates knowledge of the borehole thermal resistance Rb i e from the fl uid to the borehole wall and of the borehole wall temperature Twin Fig 1 2 The average borehole wall temperature it obtained by integrating Equation 3 along z However this is computationally intensive due to the double integration Lamarche and Beauchamp 11 have shown using an appropriate change of variables how to simplify Equation 3 to a single integration Accounting for small typos in 11 and 15 as noted by Sheriff 13 the average temperature difference between a point located at distance r from the borehole and the undisturbed ground temperature is given by DT r t q 2pks 0 B B B Z ffiffi ffiffi ffiffi ffiffi ffi b2 1 p b erfc uz ffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffi z2 b2 qdz DA Z ffi ffi ffi ffi ffi ffi ffi ffiffi b2 4 p ffi ffi ffi ffi ffi ffi ffi ffiffi b2 1 p erfc uz ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi z2 b2 qdz DB 1 C C C A 4 whereb r H r is the radial distance from the borehole center u H 2 ffi ffi ffiffi at p and DA and DBare given by DA ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi b2 1 q erfc u ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi b2 1 q berfc ub e u 2 b2 1 e u 2b2 u ffi ffi ffi p p and DB ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi b2 1 q erfc u ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi b2 1 q 0 5 berfc ub ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi b2 4 q erfc u ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi b2 4 q e u2 b2 1 0 5 e u 2b2 e u 2 b2 4 u ffi ffi ffi p p 1012141618202224 0 10 20 30 40 50 60 Temperature oC Depth m Vertical temperature profile Inf line r 2 t 200 d F line r 2 t 200 d F line average r 2 t 200 d Inf line r 1 t 2000 d F line r 1 t 2000 d F line average r 1 t 2000 d Fig 2 Vertical ground temperature profi le at radial distances r 1 m and r 2 m after respectively 2000 days and 200 days Fo r 1 t 2000 181 4 and Fo r 2 t 200 4 54 Constantheatinjectionof3000 W Thermalparameters ks 2 1 Wm 1K 1 Cs 2e06 Jm 3K 1 010002000300040005000 0 2 4 6 8 10 12 Days T oC Infinite Finite FEM Fig 3 Comparison of Finite and Infi nite line source model with fi nite element model FEM for a 30 m borehole Average temperature variation computed at 0 5 m from the borehole axis over the borehole length Constant heat transfer rate of 1000 W Thermal parameters ks 2 1 Wm 1K 1 Cs 2e06 Jm 3K 1 D Marcotte et al Renewable Energy 35 2010 763 770765 The particular case r rbin Equation 4 gives the borehole wall temperature 2 3 Numerical validation Fig 3 compares the variation in temperature over time computed with fi nite and infi nite line source to the numerical results of a fi nite element model FEM constructed within COMSOL The fi nite element model is 2 D with axial symmetry around the borehole axis The ground is represented bya 50 m long and 50 m radius cylinder The borehole is represented by a 30 m long and 0 075 m radius cylinder delivering 1000 W The axis of revolution is located at the borehole center and constitutes a thermal insulation boundary whereas all external boundaries are set to the undisturbed ground temperature Over 6000 triangular elements equipped with quadratic interpolating functions are used to discretize the model The agreement between the FEM model and the fi nite line source is almost perfect the maximum absolute difference in temperature over the 5000 days period being only 0 019 oC Fig 4 compares the temperature obtained with the infi nite and fi nite line source models at r 1 m and r 0 075 m a typical value for rb with the thermal parameters specifi ed above A 1 o C temperature difference between the infi nite and fi nite models is obtained after 2 5 y and 2 y at 1 m and 0 075 m respectively Note that the temperature reaches a plateau for the fi nite line source model indicating that a steady state condition has been reached In contrast the infi nite line source model exhibits a linear behavior Fig 5 shows the ground temperature computed at a distance of 1 m from the borehole for increasing values of the borehole length As expected the fi nite line source solution reaches the infi nite line source solution for long boreholes 0 001 0 01 0 11101001000 10 20 30 40 50 60 70 80 r 0 075 m r 1 m Ground temperature Time y Temperature oC Fig 4 Comparison of Finite solid and Infi nite broken line source model computed at distance 1 m and 0 075 m from the borehole Constant heat transfer rate per unit length of 100 W m Thermal parameters ks 2 1 Wm 1K 1 Cs 2e06 Jm 3K 1 01002003004005006007008009001000 12 12 5 13 13 5 Borehole length m Temperature oC Average temperature vs borehole length Infinite line source Finite line source Fig 5 Infi nite vs fi nite line source average temperature along a vertical profi le The load is 20 W m thermal parameters ks 2 1 Wm 1K 1 Cs 2e06 Jm 3K 1 Temperature computed after one year at r 1 m from the borehole 123456 100 0 100 Cooling Heating load Time h Load kw 123456 200 100 0 100 Load decomposition Load kw Fig 6 Principle of temporal superposition for variable loads 05101520253035 3 3 5 4 4 5 5 5 5 6 6 5 7 7 5 COP vs EWT EWT COP Cooling Heating Fig 7 COP as a function of EWT D Marcotte et al Renewable Energy 35 2010 763 770766 3 Design of complete geothermal systems In this section we compare the design length of borefi elds obtained with the fi nite and infi nite line source models for given hourly ground load scenarios These calculations imply that single borehole solutions will need to be superimposed spatially We have already seen an instance of this principle of superposition while computing the line source solution from a series of constant point sources along a line 7 see Equations 1 and 2 The additivity of effects variation in temperature stems from the linear relation between q andDT and the fact that energy is an extensive and additive variable The temporal superposition also follows the same general principle of addition of effects as described by Yavuzturk and Spitler 14 and illustrated by Fig 6 When the load is varying hourly a new pulse is applied each hour It is simply the difference between the load for two consecutive hours More formally for the infi nite line source as an example with a single borehole we have DT r t X i ti t q i 4pk ZN r2 4a t ti e u u du 5 where q 1 q1 and q i qi qi 1 i 2 I tI t is the incremental load between two successive hours With multiple boreholes DT x0 t X n j 1 X i ti t q0i 4pk ZN kxj x0k2 4a t ti e u u du 6 where n is the number of boreholes xjand x0are the coordinate vectors of borehole j and point where temperature is computed respectively Note that for long simulation periods the computa tional burden becomes important In the test cases that follow we assume that all of the building heating and cooling loads are to be provided by the GLHE system i e there is no supplementary heat rejection injection Synthetic building loads are used to enhance the reproducibility of our results These building loads are simulated using Q t A B cos t 8760 2p C cos t 24 2p D cos t 24 2p cos 2t 8760 2p 7 In Equation 7 t is in hours A controls the annual load unbalance B the half amplitude of annual load variation C and D 40 30 20 10010203040 40 30 20 10 0 10 20 30 40 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 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