On Thermally Interacting Multiple Boreholes with Variable Heating Strength: Comparison between Analytical and Numerical Approaches

Transcription

1 Sustanablty 0, 4, ; do:0.3390/su Artcle OPEN ACCESS sustanablty ISSN On Thermally Interactng Multple Boreholes wth Varable eatng Strength: Comparson between Analytcal and Numercal Approaches Seama Kooh-Fayegh and Marc A. osen * Faculty of Engneerng and Appled Scence, Unversty of Ontaro Insttute of Technology, 000 Smcoe Street North, Oshawa, Ontaro L 7K4, Canada; E-Mal: Sma.Kouh@uot.ca * Author to whom correspondence should be addressed; E-Mal: Marc.osen@uot.ca; Tel.: ; Fax: eceved: 0 July 0; n revsed form: 3 July 0 / Accepted: 9 August 0 / Publshed: 6 August 0 Abstract: The temperature response n the sol surroundng multple boreholes s evaluated analytcally and numercally. The assumpton of constant heat flux along the borehole wall s examned by couplng the problem to the heat transfer problem nsde the borehole and presentng a model wth varable heat flux along the borehole length. In the analytcal approach, a lne source of heat wth a fnte length s used to model the conducton of heat n the sol surroundng the boreholes. In the numercal method, a fnte volume method n a three dmensonal meshed doman s used. In order to determne the heat flux boundary condton, the analytcal quas-three-dmensonal soluton to the heat transfer problem of the U-tube confguraton nsde the borehole s used. Ths soluton takes nto account the varaton n heatng strength along the borehole length due to the temperature varaton of the flud runnng n the U-tube. Thus, crtcal depths at whch thermal nteracton occurs can be determned. Fnally, n order to examne the valdty of the numercal method, a comparson s made wth the results of lne source method. Keywords: geothermal energy; vertcal ground heat exchangers; numercal analyss; analytcal analyss; varable heat flux

2 Sustanablty 0, Introducton Geothermal energy systems are ncreasngly utlzed recently, but questons exst regardng the sustanablty and mpact of these systems on the envronment [ 3]. The use of geothermal energy s often benefcal, especally due to ts effcency. Lttle research s avalable to help regulatory agences and ndustry develop desgns and nstallatons that have good sustanablty aspects. One potental factor that detracts from the sustanablty of these systems at ther desgn effcency s the system thermal loss, whch can nteract wth adjacent systems and the surroundng ground [4]. Interference effects are present n some nstalled geothermal systems, suggestng that these systems may have a spacng below the threshold spacng for such systems to avod thermal nteractons. Thus, there may be a lmt to the densty of geothermal development n a gven regon. Many geothermal energy studes have focused on modelng sngle ground boreholes, usually based on analytcal [5 ] and numercal approaches [ 3]. The models vary n regardng how () heat conducton n the sol s solved and () heat transfer outsde of the boreholes s coupled to the heat transfer nsde of the borehole, and (3) the numercal methods are accelerated. The performance of borehole heat exchangers (BEs) has been smulated wth a three-dmensonal fnte-dfference method n rectangular coordnates [9]. Each borehole s approxmated by a square column to avod usng fne grds nsde the borehole, and heat transfer n the borehole s evaluated for quas-steady state condtons, allowng varable temperature and loadng along the borehole. A three-dmensonal unstructured fnte-volume numercal model s proposed of a vertcal U-tube ground heat exchanger (GE) [30]. The sol s dvded nto layers n the axal drecton to account for axal temperature varatons. A three-dmensonal numercal model that smulates flud transport n a ppe loop and heat transfer wth the ground has been developed to address the effect of the thermal mass of the crculatng flud and the dynamcs of flud transport through the loop [3]. An mportant lmtaton n most prevous reports s the assumpton of constant and unform heat nput from the borehole nto the ground, whether the borehole s assumed cylndrcal or a lne source of heat. Transent heat conducton from a bured power transmsson lne tower has been nvestgated by formulatng the problem of ground heat transfer wth a lne source of heat wth varyng heatng strength along ts length [33,34]. owever, that problem nvolves conducton along a bured rod n the ground whch dfferentates t somewhat from borehole analyss. Madan et al. [35] evaluate the effect of mass flow rate of the flud runnng n the borehole heat exchanger on pumpng power, effcency of the pump, heat dstrbuton n the borehole, heat pump heat capacty and overall coeffcent of performance. They present measured temperatures and calculated specfc heat extracted along the U-ppes for seven sectons along the borehole. They show that the temperature rse (n case of heat extracton) along the U-ppes becomes weaker as the flud travels between the nlet and outlet ppes. Acuna et al. [36] propose a Dstrbuted Thermal esponse Test by measurng temperatures at dfferent depths along the borehole whle runnng the conventonal thermal response test and showng local varatons of the ground thermal conductvty and borehole thermal resstance along the borehole. Marcotte and Pasquer [37] use a 3D fnte element model of the borehole to show that the average of nlet and outlet flud temperatures n the borehole that s used to estmate the thermal parameters n the nterpretaton of n stu thermal response tests s only vald for the unrealstc assumpton of constant heat flux along the borehole and does not correspond to the flud mean temperature wthn the

3 Sustanablty 0, borehole. Usng the smulaton results, they propose a new estmator that closely fts the average flud temperature. Another key lmtaton of past studes s that the potental exstence of thermal nteracton among multple boreholes, although dentfed, s not formulated, and the relevant parameters have not been assessed. To model nteractng borehole systems, Kooh-Fayegh and osen [38] evaluate numercally the temperature response n the sol surroundng multple boreholes n a study, assumng the heat flux from the borehole wall s constant and, therefore, that heat conducton n the drecton of the borehole length s neglgble for a major part of the soluton doman. Also, Kooh-Fayegh and osen perform analytcal and numercal analyss to study the thermal nteracton among boreholes wth varyng heat nput nto the ground, to extend pror work [39,40]. The analytcal quas-three-dmensonal soluton to the heat transfer problem of the U-tube confguraton nsde the borehole s used to determne the heat flux from the borehole wall [5], showng that thermal nteracton at the top of the boreholes s at ts hghest value due to the hgher heatng strength at ths depth. The current study valdates and compares the two approaches n terms of the sol temperature rse and the borehole wall heat flux [39,40].. Methods To examne the exstence of thermal nteracton among multple boreholes and ther possble negatve effects on the desgn performance of the exstng nearby boreholes, the transent conducton of heat n the sol surroundng these systems needs to be studed n order to evaluate the temperature rse and the heat flows n the sol surroundng the boreholes. epresentaton of heat flows to and from the system based n ths smulaton can serve as nputs nto large scale ground water models... Insde Borehole: Varable eat Flux Model The authors used a quas-three-dmensonal model proposed by eng et al. [5,4] n order to derve the heat flux dstrbuton along the borehole depth [39,40]. The heat flow rate per unt length of the borehole ( q ) transferred to the sol, calculated from the temperature dfference between the borehole wall and the flud n each of the tubes n the borehole, can be obtaned from T f ( z) Tb T f ( z) q( z) where T f, T f and T b are the temperatures of the flud runnng downwards, the flud runnng upwards and borehole wall, respectvely, and T b () ere, and are the thermal resstance between the crculatng flud and the borehole wall, and s the resstance between the tubes (Fgure ) obtaned from the relatons derved by detal by ellström [9]. r dfferent numbers of ppes n any poston n the borehole, Claesson and ellström [0] present a method to calculate the thermal resstances between the heat carrer flud n the ppes of the borehole and the mmedate vcnty of the surroundng ground. In most engneerng applcatons, the confguraton of the U-tube n the borehole may be assumed symmetrc, and here t s assumed that the thermal resstance between the crculatng flud n each of the tubes and the borehole wall s equal. ()

4 Sustanablty 0, 4 85 Fgure. Thermal resstances n the borehole. The heat flow rate per unt length of the borehole can be formulated [39,40] as ( ) q( z) T f Tb (3) where the dmensonless parameters are defned as z z T f Tb T T f b, z (4) where T f s the temperature of the flud enterng the U-tube. The temperature profles of the fluds flowng n the U-tubes n the boreholes ( and ) are formulated by eng et al. [5]. Assumng that the heat s dsspated symmetrcally n the sol around each borehole, Equaton (3) can be wrtten n the followng form: q ( z) T T f r b b ( ) Fgure. Dstrbuton of heat flux along the borehole length. (5)

5 Sustanablty 0, 4 85 Ths s the spatal dstrbuton of the heatng strength along the rod. In contrast to past studes, ths heatng strength vares along the rod and s not constant (Fgure ). Note that the varable heat source (VS) model has made certan smplfyng assumptons, such as constant ground temperature. In order to compare the results ganed by constant heat flux model wth the results ganed by the VS model, an equvalent nlet temperature ( T 90.6 K ) for the VS model, resultng n the same total heat conducton n the sol, s assumed... Outsde Borehole f A three-dmensonal model of transent conducton of heat n the sol around multple ground heat exchangers s presented n ths secton. A doman consstng of two vertcal borehole heat exchangers havng a dstance of h from each other s consdered (Fgure 3). Fgure 3. orzontal cross sectons (xy) of the soluton doman at the borehole md-length (z = 0 m). It s assumed that the domnant mode of heat transfer n the sol s conducton. The general heat conducton equaton n cylndrcal coordnates appears n the followng form: where t s the tme from the start of operaton, α s the thermal dffusvty of sol, and T s the temperature of the ground. The frst two terms on the left sde of Equaton (6) are the heat flux components n the radal (r) drecton, the thrd and the fourth terms are related to the crcumferental (φ) and axal (z) drectons, respectvely, and the ffth term relates to the heat generated n the control volume. The rght sde of Equaton (6) represents the transent effects of heat conducton. In the analyss, an analytcal and a numercal approach are used to calculate the temperature profles of the sol around the boreholes.... Analytcal Approach T r T r r r T T z q k gen T t The model of eng et al. [6] establshes the transent response at any pont n the ground, subject to a constant lne heat source n the rod. owever, the prevous analyss has shown that the heatng strength vares wth depth. Thus, the model by eng et al. [6] model can be extended to ths case by ntegratng the heatng strength over the depth of the rod. The temperature response at any pont n the (6)

6 Sustanablty 0, sem-nfnte medum wll be calculated for a pont, P(q,z), n the medum. Duan et al. [33] and Duan and Naterer [34] extended eng et al. s model [6] for the case of a bured rod. They formulated the temperature profle n the sol around the rod. Based on ther study, t can be shown that (7) where 0 T T, h z, r, t and q s the heatng strength per unt length whch vares along the borehole depth. Usng ths procedure for the case of a vertcal borehole contanng U-tube wth runnng flud, the heatng strength formulated for the case of varable heatng strength (Equaton (3)) s substtuted n Equaton (7) to obtan the temperature rse n the sol surroundng a borehole. r the case of multple boreholes, snce the conducton equaton s lnear, the temperature response n the sol can be calculated by superposng the temperature rse n the sol caused by each sngle borehole. Kooh-Fayegh and osen [38] examne the valdty of superposton method n thermal response n the sol surroundng multple boreholes by comparng the superposton results of the lne source theory wth results obtaned by a fnte volume numercal method. It was shown that the results of the two methods agree well and the effect of the temperature rse due to one borehole on the thermal performance of other boreholes can be neglected. Therefore, the temperature response n the sol surroundng a borehole system of n boreholes can be calculated by superposng the temperature response evaluated by each borehole n Equaton (7): (8) where q s the heat flow rate per unt length of Borehole (Fgure 4), n s the number of boreholes and (9) where l and w are dstances of boreholes along x and y drectons, respectvely. r the case of multple boreholes shown n Fgure (4), Equaton (8) can be smplfed to: 0 4,, d erfc erfc k q n d erfc erfc k q 0 4,, w y l x

7 Sustanablty 0, (0) where, as seen n Fgure 4, () Fgure 4. System geometrc parameters for two boreholes at dstances and from a desred pont (x,y) n the surroundng sol.... Numercal Approach In the numercal approach, the transent governng ntegral equatons for the conservaton of energy s solved wth a control volume method n FLUENT. Unlke many of the studes on the heat transfer around multple boreholes, the current three-dmensonal numercal soluton takes nto account the temperature gradents n the drecton adjacent to the borehole length correspondng to the axal heat transfer effects n the sol. The heat transfer symmetry about the two vertcal planes shown n Fgure 3 s utlzed. Therefore, only one fourth of the borehole feld s modelled and the soluton doman (sol) s enclosed by the far-feld, the ground surface and two symmetry planes. In Fgure 5, the gray area s the soluton doman, the results of whch can be replcated to the other areas drawn wth dashed lnes due to ther symmetry ,, d erfc erfc k q d erfc erfc k q y h x and y h x

8 Sustanablty 0, Fgure 5. Vertcal cross secton (xz) of the soluton doman. Fgure 6. Smulaton model for horzontal cross sectons (xy) at (a) borehole md-length (z = 0 m), and (b) the ground surface (z = 0 m). (a) (b) A control-volume-based technque s used that dvdes the doman nto dscrete control volumes usng unstructured computatonal trangular grds, as shown n Fgure 6. The temperature gradent n the doman between the borehole wall and the far-feld changes gradually from large to small ones. Therefore, to reduce computer memory and computatonal tme, the sze of the mesh cells s chosen based on ths gradual change. Fgure 6(a,b) refer to two xy cross sectons shown n Fgure 5 to show

9 Sustanablty 0, the mesh n the cross sectons contanng the borehole and above or below t. The vertcal secton doman may be dscretzed usng structured grds due to relatvely smple geometrc structure, as shown n Fgure 7. Fgure 7. Smulaton model for multple boreholes n vertcal cross secton (xz). The dscretzaton n unstructured meshes can be developed from the basc control volume technque where the ntegral form of the energy conservaton equaton s used as the startng pont: CV dv gradt dv CV T dv a t () ere, V s the volume. Integraton of Equaton () over a tme nterval from t to t t gves: tt tcv dv gradt dvdt Usng a fully mplct formulaton, Equaton (3) s dscretzed n the followng form: a P T P a nb all surfaces tt T t nb CV a T dvdt a t 0 P T 0 P (3) (4) where a P, 0 a P and a nb are temperature coeffcents whch are calculated based on the geometrc characterstcs of each control volume and the tme step n the numercal soluton. Equaton (4) s solved teratvely at each tme level before movng to the next tme step to yeld updated values of temperature. The purpose of performng the numercal smulaton s to gan a degree of confdence n the analytcal soluton. In order to prevent the errors assocated wth the quass-three-dmensonal soluton, the soluton to ths model s used n the numercal soluton as a boundary condton. Thus, any

10 Sustanablty 0, dfference between sol temperature profles of analytcal and numercal solutons would be caused by the used superposton method and the lne source theory. A future extenson to the current study s planned to compare the numercal method wth the analytcal one ncludng the numercal soluton of the nsde of the borehole as well...3. Intal and Boundary Condtons A unform ntal temperature of 88 K (equal to the undsturbed ground temperature) s assumed to be effectve over the entre borefeld. The ground surface s assumed to be sothermal and equal to the ground ntal temperature. At the outer edge of the doman, a constant far-feld temperature condton equal to the ntal temperature s appled (88 K) n the numercal soluton. The temperature and heat flux dstrbutons on the borehole wall cannot be decded due to the dynamc nature of the heat exchange process between the tubes n the borehole and the borehole wall. owever, to smplfy the current model, a constant heat flux of 0 W/m on the borehole wall can be assumed snce n order to study the thermal nteracton between multple boreholes, ther nner dynamc heat exchange process can be of second prorty compared to the heat dsspaton n the sol surroundng them. As a second approach, a varable heat flux (VF) along the borehole s calculated by defnng the temperature profles of the flud runnng along the tubes n the borehole. It should be noted that the current artcle focuses only on the varaton of heatng strength along the borehole length. Snce only the exstence of such a varaton s ntended to be dscussed, the current artcle does not provde typcal values for the borehole spacng and the heat flux on the borehole wall and lower values are chosen n order to keep the soluton doman sze smaller n the numercal soluton. It should also be noted that the temperature of the sol n the current problem s assumed to be constant throughout the whole operaton tme and therefore the current soluton s only vald for low temperature varatons n the sol surroundng the boreholes whch s only ganed by assumng lower heat flux values on the borehole wall. Modfyng the current problem to one wth typcal ndustral values for ground heat pump systems wll need the sol temperature to be assumed varable and s subject of ongong research by the authors. In order to account for the transent term n Equaton (6), the tme s subdvded nto 400 tme steps of 3600 s whch equals a tme perod of 6 months. 3. esults and Dscusson In the current study, typcal geometrcal and thermal characterstcs for the borehole and the surroundng sol are assumed (Table ). Note that the propertes of sol are approxmate values for dry clay. The temperature responses of the sol around multple boreholes evaluated by the VF model at varous borehole depths are compared n Fgures 8 and 9a. It s shown n Fgure 8 that the maxmum temperature rse due to thermal nteracton of multple boreholes n a sx-month perod of heat transfer from the borehole nto the sol occurs at the top 3% heatng length of the borehole and t decreases along the borehole length as the heat flux from the borehole wall nto the sol decreases. Therefore, wth the objectve of lmtng boreholes operatons and szes n order to prevent ther thermal nteracton, the top length of the boreholes (about 3% total length) s the crtcal area. Also, as expected the maxmum temperature rse n the sol occurs at the borehole wall (x = 0.95 m and x =.05 m). Snce the current study s not usng typcal condtons such as typcal values for borehole spacng, heat

11 Sustanablty 0, flux on the borehole wall, etc., a mnmum value of spacng s not suggested n ths study. An extenson of the current study to typcal ndustral values may requre the assumptons of constant borehole wall temperature and constant ground surface temperature made n the current model to be modfed to be varable and s subject of ongong research by the authors. In such a case, usng the current soluton method, t s possble to gan a mnmum value of spacng or maxmum amount of heat nput to the ground to avod thermal nteractons between boreholes under typcal condtons. Table. Parameters of the reference borehole. (a) Insde the borehole (U-tube and grout regon). (m) r b (m) r p (m) D (m) D b (m) k b (W/mK) m (kg/s) c (J/kgK) (b) Outsde the borehole (sol regon). k (W/mK) c (J/kgK) ρ (kg/m 3 ) Fgure 8. Sol temperature (K) around multple boreholes n xz plane n t = 6 months, at varous dstances from borehole wall for varable heat flux (VF) model. It s shown n Fgure 9a that the thermal nteracton between the boreholes s at ts mnmum at the bottom of the borehole (z = 99.9 m) where the heat flux to the sol s lowest. Ths s not true for the case of constant heat flux from the borehole wall to the surroundng sol along the borehole length (Fgure 9b). It s seen n Fgure 9b that the greatest thermal nteracton occurs at top of the borehole, but remans at ts maxmum amount along the borehole length. r ths case, the crtcal length of the borehole would be almost 95% of the borehole length. owever, as dscussed earler, the case of constant heat flux s only a smplfcaton to the VF problem and does not present the problem as accurate as the VF problem. Another notable characterstc of Fgures 9a and 9b s the decrease n the thermal nteracton n the lengths of z = 99.9 m when one moves from z = 95 m towards the top end of the borehole. Specfcally for the case of VF (Fgure 9a), there s hgher heat flux as one moves towards the top

12 Sustanablty 0, end and one expects greater thermal nteractons. In both cases, the temperature rse n the sol around the borehole declnes at the very end of borehole length, and ths can be due to axal heat transfer effects whch become notable only at the very ends of borehole lengths. Fgure 9. Sol temperature (K) around multple boreholes n t = 6 months, at varous borehole depths for (a) VF model, and (b) constant heat flux model. (a) The results of the VF model and constant heat flux model are compared n Fgure 0. It s seen n Fgure 0a that the assumpton of constant heat flux on the borehole wall ntroduces numerous naccuraces especally when dealng wth the temperature rses n the sol at the very top and bottom of the borehole. Fgure 0b shows that, by usng varyng heat flux method, the heat flux on the borehole s spread along the borehole n a way that the mddle area remans smlar to ts average amount. It can be concluded that usng the constant heat flux method s only vald for the mddle length of the boreholes and movng any further to the top or bottom of the borehole, the temperature rses evaluated become ncreasngly naccurate. Quas-three-dmensonal models reveal drawbacks of (b)

13 Sustanablty 0, two-dmensonal models and are thus preferred for desgn and analyss of ground heat exchangers, as they provde more accurate nformaton for performance smulaton, analyss and desgn. Fgure 0. Comparson of sol temperature (K) around multple boreholes at t = 6 months for VF and constant heat flux models, at (a) z = 95 m and z = 95 m, and (b) z = 0 m. (a) It should be noted that the effect of temperature rse due to one borehole on the other s neglected by applyng the superposton method. Ths effect has been examned for a two-dmensonal numercal study by Kooh-Fayegh and osen [38]. A comparson of the results of the numercal soluton wth analytcal results of lne source theory where the superposton method s used to account for the temperature rse n the sol surroundng multple boreholes shows that these effects are mnor n comparson to the order of the temperature rse n the sol due to the ndvdual performance of the boreholes. Snce the objectve n the current study s to examne at what depths the thermal nteracton among boreholes creates a crtcal temperature rse, the focus s mostly on ntroducng a heat flow rate (b)

14 Sustanablty 0, 4 86 profle along the borehole length whch can be coupled to the numercal or lne source model outsde the borehole to show the effect of varyng heat flux along borehole length on temperature rse n the sol. In Fgure, comparson s made between the two methods and t s shown that the temperature rse n the sol caused by both methods agree well. Therefore, t can be concluded that for such problems where the heat nput nto the ground does not vary wth tme, the analytcal method presented n ths paper can present as accurate results as a numercal method. Fgure. Sol temperature (K) around multple boreholes n t = 6 months for lne source and numercal models at varous borehole depths. Extenson of results to systems of boreholes: The dea of usng lne source theory for calculatng the temperature profles n the sol around two boreholes can also be appled to two systems of vertcal GEs. r example, f an area of 40 m 40 m 00 m n the sol s occuped for one system of vertcal GEs, the rato of system depth to ts ntal sze s large enough to be accounted as one cylnder or lne source of heat when system nteractons and temperature excess around a system wth larger dstances are to be accounted for. The study of varable heatng strength along the borehole length also accounts for the system of boreholes as well. Therefore, the parametrc study on two nteractng boreholes lkely exhbts the same affectng parameters and results as those for two nteractng systems of boreholes. owever, a more detaled comparson of the results of modelng the two systems must be performed n order to show smlartes between the two problems. Furthermore, certan assumptons such as the assumpton of constant ground temperature as well as constant ground surface temperature must be examned further n order to mprove the accuracy of the proposed method.

15 Sustanablty 0, Conclusons The performance of multple boreholes or neghbourng borehole systems and ther possble thermal nteractons are dscussed va analytcal and numercal methods. The effect of spatal varaton of borehole heat flux on the transent response of multple ground heat exchangers and ther thermal nteracton s descrbed. A quas-three-dmensonal model for heat transfer nsde the borehole s utlsed as the boundary condton for the three-dmensonal transent heat transfer analyss outsde the borehole n order to evaluate the temperature rse n the sol surroundng multple boreholes and ther nteracton. It s shown that the maxmum temperature rse due to thermal nteracton of multple boreholes n a sx-month perod of heat transfer from the borehole nto the sol occurs rght after the begnnng of the borehole (about 3% total length) and t decreases along the borehole length as the heat flux from the borehole wall nto the sol decreases. Therefore, wth the objectve of lmtng boreholes operatons and szes n order to prevent ther thermal nteracton, the top length of the boreholes s the crtcal area. It can be concluded that usng the constant heat flux method s only vald for the mddle length of the boreholes and movng any further to the top or bottom of the borehole, the temperature rse evaluatons become ncreasngly naccurate. Furthermore, a comparson between the results of lne source method and numercal fnte volume method shows that the temperature rse n the sol caused by both methods agree well. Therefore, t can be concluded that for such problems where the heat nput nto the ground does not vary wth tme, the analytcal method presented n ths paper can result n as accurate results as a numercal method. Nomenclature a c p D D b h h z k k b m q q q r r b temperature coeffcent specfc heat at constant pressure [J/kgK] dstance between the tubes n the borehole [m] dstance between the boreholes [m] urer number borehole dstance from the coordnate centre [m] ntegraton varable [m] heatng length, [m] dmensonless ntegraton varable sol thermal conductvty [W/mK] grout thermal conductvty [W/mK] mass flow rate [kg/s] generated heat per unt volume [W/m 3 ] heat flow rate per unt length [W/m] heat flux at borehole wall [W/m ] radal coordnate [m] borehole radus [m] dmensonless dstance of Borehole to a gven pont (x,y) n the soluton doman dstance of Borehole to a gven pont (x,y) n the soluton doman [m]

16 Sustanablty 0, T T t f dstance of Borehole to a gven (x,y) pont n the soluton doman [m] thermal resstance between the nlet crculatng flud and the borehole wall [mk/w] thermal resstance between the nlet and outlet tubes [mk/w] thermal resstance between the outlet crculatng flud and the borehole wall [mk/w] thermal resstance [mk/w] thermal resstance [mk/w] temperature [K] nlet crculatng flud temperature at z=00 m tme [s] V volume [m 3 ] dmensonless parameter z axal coordnate [m] Greek Letters α thermal dffusvty [m /s] Θ dmensonless temperature θ temperature rse [K] φ crcumferental coordnate [rad] ρ densty [kg/m 3 ] Subscrpts b borehole f nlet crculatng flud f outlet crculatng flud nb node number of the adjacent cell P centrod P Superscrpts 0 prevous tme step f nlet crculatng flud f outlet crculatng flud n dscretzaton step desgnaton n tme Acknowledgments The support provded by the Ontaro Mnstry of Envronment through ts Best n Scence program s gratefully acknowledged. Conflct of Interest The authors declare no conflct of nterest.

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