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Varennes,al, C.PKeywords:arevari, shoreffects. In one example considered, it is shown that the borehole length is 15% shorter when axialC211 2009 Elsevier Ltd. All rights reserved.coupledincreasinglis presenteivelyrejecter or extractor is used at peak conditions to reduce the lengthof the ground heat exchanger.they necessitate a high level of expertise. Furthermore, it is noteasily possible to obtain ground temperature distributions like theones shown later in this paper. In this paper hourly simulations areperformed using the so-called finite and infinite line sourceapproximations where the borehole is approximated by a line witha constant heat transfer rate per unit length. These approximationspresent, 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. Departement des genies civil, geologique et des mines,Ecole Polytechnique de Montreal, C.P. 6079 Succ. Centre-ville, Montreal, (Qc),H3C 3A7 Canada. Tel.: 1 514 340 4711x4620; fax: 1 514 340 3970.Contents lists availableRenewablejournal homepage: www.elseRenewable Energy 35 (2010) 763770E-mail address: denis.marcottepolymtl.ca (D. Marcotte).lates a heat transfer fluid in a closed circuit from the GLHE to a heatpump (or a series of heat pumps). Typically, GLHE consistsof boreholes that are 100150 m deep and have a diameter of1015 cm. The number of boreholes in the borefield can range fromone, for a residence, to several dozens, in commercial applications.Furthermore, several borehole configurations (square, rectangular,L-shaped) are possible. Typically, a borehole consists of two pipesforming a U-tube (Fig.1). The volume between these pipes and theborehole wall is usually filled with grout to enhance heat transferfrom the fluid to the ground. In some situations it is advantageousto design so-called hybrid systems in which a supplementary heattion and minimum/maximum heat pump entering water temper-ature 8,3. There are design software programs that perform thesecalculations. Some use the concept of the g-functions developed byEskilson 5. The g-functions are derived from a numerical modelthat, by construction, includes the axial effects. The other approachis to perform hourly simulation. This last approach is essential fordesign of hybrid systems in which supplemental heat rejection/injection is used. There are several software packages that canperform hourly borehole simulations. For example, TRNSYS 9 andEnergyPlus 4 use the DST 6 and the short-time step model 5,respectively. Even though these packages account for axial effects,Hybrid systemsUnderground water freezing1. IntroductionGeothermal systems using ground-exchangers (GLHE) are becominggrowing energy costs. Such a systemThe operation of the system is relat0960-1481/$ see front matter C211 2009 Elsevier Ltd.doi:10.1016/j.renene.2009.09.015closed-loop heaty popular due tod in Fig. 1.simple: a pump circu-Given the relatively high cost of GLHE, it is important to designthem properly. Among the number of parameters that can bevaried, the length and configuration of the borefield are important.There are basically two ways to design a borefield. The first methodinvolves using successive thermal pulses (typically 10-years1month6 h) to determine the length based on a given configura-Finite line sourceGround loop heat exchangersare considered.Infinite line sourceconduction effects are considered. In another example dealing with underground water freezing, theamount of energy that has to be removed to freeze the ground is three times higher when axial effectsThe importance of axial effects for boreholeheat-pump systemsD. Marcottea,b,c,*, P. Pasquiera, F. Sheriffb, M. BernieraGolder Associates, 9200 lAcadie, Montreal, (Qc), H4N 2T2 CanadabCANMET Energy Technology Centre-Varennes, 1615 Lionel-Boulet Blvd., P.O. Box 4800,cDepartement des genies civil, Geologique et des mines, Ecole Polytechnique de Montrearticle infoArticle history:Received 13 May 2008Accepted 18 September 2009Available online 23 October 2009abstractThis paper studies the effectssystems. The axial effectsline source methods. Usingimportant. UnsurprisinglyAll rights reserved.design of geothermalc(QC), J3X 1S6 Canada. 6079 Succ. Centre-ville, Montreal, (Qc), H3C 3A7 Canadaof axial heat conduction in boreholes used in geothermal heat pumpexamined by comparing the results obtained using the finite and infiniteous practical design problems, it is shown that axial effects are relativelyt boreholes and unbalanced yearly ground loads lead to stronger axialat ScienceDirectE/locate/renenerarely used in routine design due to the perceived computational axial effects on the GLHE design. Our main finding is that formodels is the change in temperature felt at a given location andNomenclaturea Thermal diffusivity (m2sC01)A, B, C, D Synthetic load model parameters (kW)b r/HCsGround volumetric heat capacity (JmC03KC01)erfc (x) Complementary error function(erfcx12ppRNxeC0t2dtEWT Temperature of fluid entering the heat pump (K orC14C)FoFourier number, Foat/r2ksVolumetric ground thermal conductivity (WmC01KC01)H Borehole length (m)HP Heat Pumpq0Radial heat transfer rate (W)q Radial heat transfer rate per unit length (WmC01)S Borehole spacing (m)r Distance to borehole (m)rbBorehole radius (m)RbBorehole effective thermal resistance (KmWC01)t TimeDT (r, t) Ground temperature variation at time t and distance rfrom the borehole (K orC14C)TfFluid temperature (K orC14C)TgUndisturbed ground temperature (K orC14C)TwTemperature at borehole wall (K orC14C)uH2atpx, y Spatial coordinates (m)z Elevation (m)D. Marcotte et al. / Renewable Energy 35 (2010) 763770764burden.The major difference between the finite and infinite line sourcelies in the treatment of axial conduction (at the bottom and top ofthe borehole) which is only accounted for in the former. Thetheoretical basis of the finite line source, although more involvedthan for the infinite line source, was first established by Ingersollet al. 7. It has been rediscovered recently by Zeng et al. 15 whoimproved the model by imposing a constant temperature at theground surface. Lamarche and Beauchamp 11 have made a usefulcontribution to speed up the computation of Zengs model. Finally,Sheriff 13 extended Zengs model by permitting the borehole topto be located at some distance below the ground surface. She alsodid a detailed comparison of the finite and infinite line sourceresponses, but did not examine the repercussion on borefielddesign.At first glance, the axial heat-diffusion is likely to decrease(increase) the borehole wall temperature in cooling (heating)modes respectively. Therefore, designing without consideringaxial effects appears to provide a safety factor for the design. But,is it really always the case? Moreover, are the borehole designsincorporating axial effects significantly different from thoseneglecting it? Under which circumstances are we expected tohave significant design differences? These are the main questionswe seek to answer. The main contribution of this research is todescribe, using synthetic case studies, the impact of consideringFig. 1. Sketch of atime due tothe effectof a constant pointsourcereleasing q0units ofheat per second 7:DTr;tq04pksrerfcC18r2atpC19(1)where erfc is the complementary error function, r the distance tothe point heat source, and a is 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,many realistic circumstances the axial effects cannot be neglec-ted. Therefore, design practices should be revised accordingly toinclude the axial effects.We first review briefly the theory for infinite and finite linesource models. Then, we present three different design situations.The first two situations involve the sizing of geothermal systemswith and without the hybrid option, under three different hourlyground load scenarios. The last design problem examines theenergy required and ground temperature evolution in the contextof ground freezing for environmental purposes.2. Theoretical backgroundThe basic building block of both infinite and finite line sourceGLHE system.the combined effect felt at distance r from the source is obtained byintegration along the line.2.1. Infinite line sourceIn an infinite medium, the line-integration gives the so-called(infinite) line source model 7:DTr;tq4pksZNr2=4ateC0uudu (2)2.2. Finite line sourceIn the case of a finite line source, the upper boundary isconsidered at constant temperature, taken as the undisturbedground temperature 15. This condition is represented by addinga mirror image finite line source with the same load, but oppositesign, as the real finite line. Then, integrating between the limits ofthe real and image line, one obtains 15,13:DTr;t;zqZH0erfcC16du2atpC17C0erfcC16d0u2atpC1701Adu (3)In hourly simulations, the fluid temperature (Tfin Fig. 1)isrequired. This necessitates knowledge of the borehole thermalresistance Rb(i.e. from the fluid to the borehole wall), and of theborehole wall temperature (Twin Fig. 1) 2. The average boreholewall temperature it obtained by integrating Equation (3) along z.However, this is computationally intensive due to the doubleintegration. Lamarche and Beauchamp 11 have shown, using anappropriate change of variables, how to simplify Equation (3) toa single integration. Accounting for small typos in 11 and 15 asnoted by Sheriff 13, the average temperature difference, betweena point located at distance r from the borehole and the undisturbedground temperature, is given by:DTr;tq2pks0BBBZb21pberfcuzz2C0b2q dzC0DAC0Zb24pb21perfcuzz2C0b2q dzC0DB1CCCA(4)D. Marcotte et al. / Renewable Energy 35 (2010) 763770 7654pks0du d uwhere dur2zC0u2qand d0ur2zu2q, z is theelevation of the point where the computation is done. The left partof the integrand in Equation (3) represents the contribution by thereal finite line, the right part, the contribution of the image line.Fig. 2 shows the vertical temperature profile obtained withEquation (3) at radial distance r2 m, after 200 days, and atr1 m, after 2000 days of heat injection. The correspondinginfinite lines-source temperature is indicated as a reference. In thisexample,theboreholeis50 mlong,thegroundthermalparametersare ks2.1 WmC01KC01and Cs2e06 JmC03KC01. The ground is inti-tially at 10oC. The applied load is 60 W per m for a total heatingpower of 3000 W. As expected, the importance of axial effects andthe discrepancy between infinite and finite models increases withthe Fourier number (at/r24.54 and 181.4 for these two cases).10 12 14 16 18 20 22 240102030405060Temperature ( oC)Depth (m)Vertical temperature profileInfline, r=2, t=200 dFline, r=2, t=200 dFline average, r=2, t=200 dInfline, r=1, t=2000 dFline, r=1, t=2000 dFline average, r=1, t=2000 dFig. 2. Vertical ground temperature profile at radial distances r1mandr2mafterrespectively 2000 days and 200 days, Fo(r1, t2000)181.4 and Fo(r2,C01 C01t200)4.54.Constantheatinjectionof3000 W.Thermalparameters:ks2.1 Wm K ,Cs2e06 JmC03KC01.where br/H, r is the radial distance from the borehole center,u H2atpand DA, and DBare given by:DAb21qerfcC18ub21q C19C0b erfcubC0eC0u2b21C0eC0u2b2upp!andDBb21qerfcC18ub21q C19C00:5C18b erfcubb24qerfcC18ub24q C19C19C0eC0u2C0b21C1C00:5C16eC0u2b2eC0u2C0b24C1C17upp!0 1000 2000 3000 4000 5000024681012DaysT (oC)InfiniteFiniteFEMFig. 3. Comparison of Finite and Infinite line source model with finite element model(FEM) for a 30 m borehole. Average temperature variation computed at 0.5 m from theborehole axis, over the borehole length. Constant heat transfer rate of 1000 W.Thermal parameters: ks2.1 WmC01KC01, Cs2e06 JmC03KC01.and 50 m radius cylinder. The borehole is represented by a 30 m0.001 0.01 0.1 1 10 100 10001020304050607080r=0.075 mr=1 mGround temperatureTime (y)Temperature (oC)Fig. 4. Comparison of Finite (solid) and Infinite (broken) line source model, computedat distance 1 m and 0.075 m from the borehole. Constant heat transfer rate per unit1 2 3 4 5 61000100Cooling (+) Heating () load Time (h)Load (kw)1 2 3 4 5 62001000100Load decompositionLoad (kw)Fig. 6. Principle of temporal superposition for variable loads.D. Marcotte et al. / Renewable Energy 35 (2010) 763770766long and 0.075 m radius cylinder delivering 1000 W. The axis ofrevolution is located at the borehole center and constitutesThe particular case rrbin Equation (4) gives the borehole walltemperature.2.3. Numerical validationFig. 3 compares the variation in temperature over timecomputed with finite and infinite line source to the numericalresults of a finite element model (FEM) constructed withinCOMSOLC211. The finite element model is 2-D with axial symmetryaround the borehole axis. The ground is represented bya 50 m longlength of 100 W/m. Thermal parameters: ks2.1 WmC01KC01, Cs2e06 JmC03KC01.a thermal insulation boundary whereas all external boundaries areset to the undisturbed ground temperature. Over 6000 triangular0 100 200 300 400 500 600 700 800 900 10001212.51313.5Borehole length (m)Temperature (oC)Average temperature vs borehole lengthInfinite linesourceFinite linesourceFig. 5. Infinite vs finite line source average temperature along a vertical profile. The loadis 20 W/m, thermal parameters: ks2.1 WmC01KC01, Cs2e06 JmC03KC01. Temperaturecomputed after one year at r1 m from the borehole.elements equipped with quadratic interpolating functions are usedto discretize the model. The agreement between the FEM modeland the finite line source is almost perfect, the maximum absolutedifference in temperature over the 5000 days period being only0.019oC.Fig. 4 compares the temperature obtained with the infiniteand finite line source models, at r1m and r0.075 m (atypical value for rb), with the thermal parameters specifiedabove. A 1oC temperature difference between the infinite andfinite models is obtained after 2.5 y and 2 y,at1mand0.075mrespectively. Note that the temperature reaches a plateau for thefinite line source model indicating that a steady-state conditionhas been reached. In contrast, the infinite line source modelexhibits a linear behavior.Fig. 5 shows the ground temperature, computed at a distance of1 m from the borehole, for increasing values of the borehole length.As expected, the finite line source solution reaches the infinite linesource solution for long boreholes.0 5 10 15 20 25 30 3533.544.555.566.577.5COP vs EWTEWTCOPCoolingHeatingFig. 7. COP as a function of EWT.010203040123 456 78 911 1213141718 1920 21 24 2527 2829303334 3536 3742 4344 4547 4849505354 5556 5760 6166 6768 6975 7677 7879 8081828586 8788 8994 9596 97100 101106 107108 109111 112113114117118 119120 121126 127128 129134 135136 137142 143144 145147 148149150155156 157158 159160 161166 167168 169174 175176 177182 183184 185190 191192 193198 199200 201204 205210 211212 213218 219220 221 224 225Borehole location and priority numberD. Marcotte et al. / Renewable Energy 35 (2010) 763770 7673. Design of complete geothermal systemsIn this section we compare the design length of borefieldsobtained with the finite and infinite line source models for givenhourly ground load scenarios. These calculations imply that singleborehole solutions will need to be superimposed spatially. We havealready seen an instance of this principle of superposition whilecomputing the line source solution from a series of constant pointsources along a line 7, see Equations (1 and 2). The additivity of40 30 20 1040302010162232384052586264717384909298102104116122124130132138140152154162164170172178180186188194196202206208214216222Coord. x (m)Coord. y (m)Fig. 8. Borehole grid and priority number. Number indicateseffects (variation in temperature) stems from the linear relationbetween q and DT, and the fact that energy is an extensive andadditive variable. The temporal superposition also followsthe samegeneral principle of addition of effects as described by Yavuzturkand Spitler 14 and illustrated by Fig. 6. When the load is varyinghourly, a new pulse is applied each hour. It is simply the differencebetween the load for two consecutive hours. More formally, for theinfinite line source as an example, with a single borehole, we have:DTr;tXi; tiC20tqC3i4pkZNr2=4atC0tieC0uudu (5)where: q*1q1, and q*iqiC0qiC01, i2.I, tIC20t, is the incrementalload between two successive hours. With multiple boreholes,Table 1Number of boreholes required, complete geothermal system. Constant T assumesa constant ground surface temperature of 10oC, Periodic T assumes a periodicground surface temperature with an amplitude of C620oC in phase with the heatload.Scenario Borehole length Infinite line Finite lineConstant T Periodic TBalanced (AC017) 100 m 33 33 34Balanced 50 m 76 74 80Cooling dominant (A17) 100 m 39 36 37Cooling dominant 50 m 93 79 81Heating dominant(AC030)100 m 57 53 56Heating dominant 50 m 134 115 124DTx0;tXnj1Xi; tiC20tq0i4pkZNkxjC0x0k2=4atC0tieC0uudu (6)where: n is the number of boreholes, xjand x0are the coordinatevectors of borehole j and point where temperature is computed,respectively. Note that for long simulation periods, the computa-tional burden becomes important.0 10 20 30 4010 15 2326 31 394146 5159636570727483 919399103105110 115 123125131133139141146 151153163165171173179181187189195197203207209215217223order of inclusion in the design when required.In the test cases that follow we assume that all of the buildingheating and cooling loads are to be provided