喜歡就充值下載吧。。資源目錄里展示的文件全都有，，請(qǐng)放心下載，，有疑問(wèn)咨詢(xún)QQ:414951605或者1304139763 ======================== 喜歡就充值下載吧。。資源目錄里展示的文件全都有，，請(qǐng)放心下載，，有疑問(wèn)咨詢(xún)QQ:414951605或者1304139763 ======================== The importance of axial effects for borehole design of geothermalheat-pump systemsD. Marcottea,b,c,*, P. Pasquiera, F. Sheriffb, M. BerniercaGolder Associates, 9200 lAcadie, Montreal, (Qc), H4N 2T2 CanadabCANMET Energy Technology Centre-Varennes, 1615 Lionel-Boulet Blvd., P.O. Box 4800, Varennes, (QC), J3X 1S6 CanadacDe partement des ge nies civil, Ge ologique et des mines, Ecole Polytechnique de Montre al, C.P. 6079 Succ. Centre-ville, Montre al, (Qc), H3C 3A7 Canadaa r t i c l e i n f oArticle history:Received 13 May 2008Accepted 18 September 2009Available online 23 October 2009Keywords:Infinite line sourceFinite line sourceGround loop heat exchangersHybrid systemsUnderground water freezinga b s t r a c tThis paper studies the effects of axial heat conduction in boreholes used in geothermal heat pumpsystems. The axial effects are examined by comparing the results obtained using the finite and infiniteline source methods. Using various practical design problems, it is shown that axial effects are relativelyimportant. Unsurprisingly, short boreholes and unbalanced yearly ground loads lead to stronger axialeffects. In one example considered, it is shown that the borehole length is 15% shorter when axialconduction 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 effectsare considered.? 2009 Elsevier Ltd. All rights reserved.1. IntroductionGeothermal systems using ground-coupled closed-loop heatexchangers (GLHE) are becoming increasingly popular due togrowing 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 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 heatrejecter or extractor is used at peak conditions to reduce the lengthof the ground heat exchanger.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-tion 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,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. De partement des ge nies civil, ge ologique et des mines,Ecole 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.marcottepolymtl.ca (D. Marcotte).Contents lists available at ScienceDirectRenewable Energyjournal homepage: see front matter ? 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.renene.2009.09.015Renewable Energy 35 (2010) 763770rarely used in routine design due to the perceived computationalburden.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 consideringaxial effects on the GLHE design. Our main finding is that formany 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 sourcemodels is the change in temperature felt at a given location andtime due to the effect of a constant point source releasing q0units ofheat per second 7:DTr;t q04pksrerfc?r2ffiffiffiffiffiatp?(1)where erfc is the complementary error function, r the distance tothe 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.NomenclatureaThermal diffusivity (m2s?1)A, B, C, D Synthetic load model parameters (kW)br/HCsGround volumetric heat capacity (Jm?3K?1)erfc (x)Complementary error function(erfcx 12ffiffiffippRNxe?t2dtEWTTemperature of fluid entering the heat pump (K or?C)FoFourier number, Foat/r2ksVolumetric ground thermal conductivity (Wm?1K?1)HBorehole length (m)HPHeat Pumpq0Radial 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)tTimeDT (r, t)Ground temperature variation at time t and distance rfrom the borehole (K or?C)TfFluid temperature (K or?C)TgUndisturbed ground temperature (K or?C)TwTemperature at borehole wall (K or?C)uH2ffiffiffiffiatpx, ySpatial coordinates (m)zElevation (m)D. Marcotte et al. / Renewable Energy 35 (2010) 763770764the 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;t q4pksZNr2=4ate?uudu(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;z q4pksZH00erfc?du2ffiffiffiffiatp?du?erfc?d0u2ffiffiffiffiatp?d0u1Adu(3)where du ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 z ? u2qand d0u ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 z u2q, 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, the borehole is 50 m long, the groundthermal parametersare ks2.1 Wm?1K?1and Cs2e06 Jm?3K?1. 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).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;t q2pks0BBBZffiffiffiffiffiffiffiffiffib21pberfcuzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2?b2qdz ? DA?Zffiffiffiffiffiffiffiffiffib24pffiffiffiffiffiffiffiffiffib21perfcuzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2?b2qdz ? DB1CCCA(4)wherebr/H, r is the radial distance from the borehole center,uH2ffiffiffiffiatpand DA, and DBare given by:DAffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 1qerfc?uffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 1q?berfcub? e?u2b21? e?u2b2uffiffiffipp!andDBffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 1qerfc?uffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 1q? 0:5?berfcubffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 4qerfc?uffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 4q? e?u2?b21? 0:5?e?u2b2 e?u2?b24?uffiffiffipp!10121416182022240102030405060Temperature ( oC)Depth (m)Vertical temperature profile Infline, 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 r1 m and r2 m afterrespectively 2000 days and 200 days, Fo(r 1, t2000)181.4 and Fo(r 2,t200)4.54.Constantheatinjectionof3000 W.Thermalparameters:ks2.1 Wm?1K?1,Cs 2e06 Jm?3K?1.010002000300040005000024681012Days T (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 Wm?1K?1, Cs2e06 Jm?3K?1.D. Marcotte et al. / Renewable Energy 35 (2010) 763770765The 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 withinCOMSOL?. The finite element model is 2-D with axial symmetryaround the borehole axis. The ground is represented bya 50 m longand 50 m radius cylinder. The borehole is represented by a 30 mlong and 0.075 m radius cylinder delivering 1000 W. The axis ofrevolution is located at the borehole center and constitutesa thermal insulation boundary whereas all external boundaries areset to the undisturbed ground temperature. Over 6000 triangularelements 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 r 1 m and r 0.075 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, at 1 m and 0.075 mrespectively. 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.0.001 0.01 0.111010010001020304050607080r=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 unitlength of 100 W/m. Thermal parameters: ks2.1 Wm?1K?1, Cs2e06 Jm?3K?1.010020030040050060070080090010001212.51313.5Borehole length (m)Temperature (oC)Average temperature vs borehole length Infinite linesourceFinite linesourceFig. 5. Infinite vs finite line source average temperature along a vertical profile. The loadis 20 W/m, thermal parameters: ks2.1 Wm?1K?1, Cs2e06 Jm?3K?1. Temperaturecomputed after one year at r1 m from the borehole.1234561000100Cooling (+) Heating () load Time (h)Load (kw)1234562001000100Load decompositionLoad (kw)Fig. 6. Principle of temporal superposition for variable loads.0510152025303533.544.555.566.577.5COP vs EWTEWTCOP CoolingHeatingFig. 7. COP as a function of EWT.D. Marcotte et al. / Renewable Energy 35 (2010) 7637707663. 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 ofeffects (variation in temperature) stems from the linear relationbetween q andDT, and the fact that energy is an extensive andadditive variable. The temporal superposition also follows the 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;t Xi; ti?tq?i4pkZNr2=4at?tie?uudu(5)where: q*1q1, and q*iqi?qi?1, i2.I, tI?t, is the incrementalload between two successive hours. With multiple boreholes,DTx0;t Xnj1Xi; ti?tq0i4pkZNkxj?x0k2=4at?tie?uudu(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.In the test cases that follow we assume that all of the buildingheating and cooling loads are to be provided by the GLHE system,i.e. there is no supplementary heat rejection/injection. Syntheticbuilding loads are used to enhance the reproducibility of ourresults. These building loads are simulated using:Qt A ? B cos?t87602p? C cos?t242p? D cos?t242p?cos?2t87602p?(7)In Equation (7), t is in hours, A controls the annual loadunbalance, B the half-amplitude of annual load variation, C and D4030201001020304040302010010203040 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225Borehole location and priority numberCoord. x (m)Coord. y (m)Fig. 8. Borehole grid and priority number. Number indicates order of inclusion in the design when required.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 ?20oC in phase with the heatload.ScenarioBorehole length Infinite line Finite lineConstant T Periodic TBalanced (A?17)100 m333334Balanced50 m767480Cooling dominant (A17) 100 m393637Cooling dominant50 m937981Heating dominant(A?30)100 m575356Heating dominant50 m134115124Table 2Number of boreholes required, hybrid system. HP capacity represents 40% ofmaximum building load. The last two column represent the percentage of thebuilding load supplied by the HP for each mode.ScenarioBoreholelengthNumber ofboreholes% EnergyInfiniteFiniteCooling Inf.(Fin.)Heating Inf.(Fin.)Balanced100 m191969 (69)77 (78)Balanced50 m373767 (67)72 (73)Cooling dominant100 m242469 (69)86 (86)Cooling dominant50 m413970 (69)90 (88)Heating dominant100 m373767 (67)83 (86)Heating dominant50 m555372 (70)70 (73)D. Marcotte et al. / Renewable Energy 35 (2010) 763770767the half-amplitude of daily load fluctuations. D/C controls therelative importance of the damped component used to simulatelarger daily fluctuations in winter and summer. Coefficients A to Dare in kW.We consider three different load scenarios, each with B100,C50, and D25. One is approximately balanced (A?17), one isa cooling dominated load (A17) and the other is a heating domi-natedload(A?30).Conversionofbuildingloadstogroundloadsisdone with: qgroundqbuilding(1?1/COP). The heat pump COP variesas a function of entering water temperature as depicted in Fig. 7.For all scenarios, we consider a unique set of possible locationsfor the boreholes. The locations are at the nodes of a regular grid ofmesh S6 m. The boreholes are assigned a priority number (lowernumber / highest priority), moving excentrically from the gridcenter to the fringes (see Fig. 8).