MATHEMATICAL SIMULATION OF GEOTHERMAL HEAT TRANSFER IN HOT DRY ROCK UNDERGROUND HEAT EXCHANGERS

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1 SCIENIFIC PROCEEDINGS 009, Fculty of Mechnicl Engineering, SU in Brtislv MAHEMAICAL SIMULAION OF GEOHERMAL HEA RANSFER IN HO DRY ROCK UNDERGROUND HEA EXCHANGERS Kristín KÁZMÉROVÁ, Michl MASARYK Slovk University of echnology Brtislv, Slovki, ABSRAC he rticle describes clcultion method for non- stedy stte het trnsfer in Hot Dry Rock mssive his method is used in geotherml pplictions focused on the eploittion of stored het in deep hot erth underground which is quntittively much lrger source of therml energy thn common geotherml underground wters Obtining of this kind of therml energy requires the building of underground het echngers, drilled directly into rock mssive Het crrying fluid which is usully river wter or crbon dioide pumped from the erth s surfce in to the underground het echnger vi drillings (wells) tkes out the het from the surrounding rock hus non-stedy stte het trnsfer occurs in tht surrounding rock mssive which is continully sub cooled On the other hnd the sme surrounding rock mssive is continully heted from deeper lyers of rock mssive he clcultion of these het trnsfer processes nd eperimentl verifiction of this clcultion is presented below Keywords: geotherml energy, het trnsfer in hot dry rock, non-stedy stte het trnsfer INRODUCION Geotherml energy belongs mongst the most promising lterntive energy sources, especilly Hot Dry Rock systems in which the het is stored in rock mssive directly, nd thus it is not bound to reltively scrce geotherml underground wters Becuse quntittively, this het source is prcticlly unlimited, it hs specil importnce for future energy supplies he reson tht this het source is not utilized industrilly on lrger scle yet, is to be found, bove ll, in the reltively high costs for wells nd in the uncertinties of mking lrge volume underground het echngers (even one kilometre in dimeter) which hve to be creted directly in underground rock mssive However the drilling technologies re progressing quickly nd the first dozens of geotherml power plnts bsed on Hot Dry Rock het utiliztion re lredy functioning worldwide, including in the EU Additionlly, s Slovki nd the Crpthin region hve generlly prticulrly suitble geologicl conditions for such pplictions, the respective technologies, know how nd design skills re becoming more nd more importnt [] UNDERGROUND HEA SORED IN ROCK In generl, the higher the temperture difference tht is vilble in thermodynmic cycle, the higher the qulity (ie energy) the respective het source hs herefore, bove ll, het sources with high temperture levels enble n effective echnge of het into mechnicl work Low temperture het sources under 30 C re not considered s suitble for energy trnsformtion into mechnicl work, nd such low potentil het is utilized directly nd only for purposes such s district heting, or heting of greenhouses or swimming pools

2 SCIENIFIC PROCEEDINGS 009, Fculty of Mechnicl Engineering, SU in Brtislv herefore, if there is rock underground with tempertures bove 50 C t resonble depths (ie up to 4000 m), geotherml power plnt, bsed on the utiliztion of Hot Dry Rock het becomes rel possibility his is the cse of some res of Slovki A Hot Dry Rock bsed geotherml system consists bsiclly of geotherml power plnt bsed on the Orgnic Rnkine cycle (ORC), coupled to injection nd eploittion deep wells nd underground het echngers [] here is no doubt tht the most risky prt of building such system is the underground het echnger One of the very importnt prts t the design of the bove mentioned het echnger is the correct estimte of the necessry dimensions of het echnging surfces between the hot rock nd the het crrying fluid, which is usully wter he wter (obviously river wter) is pumped into n underground het echnger where it is heted by the surrounding hot rock herefore the rock mssive is grdully cooled round the pipes of the echnger, in which the het crrying wter flows On the other hnd, it is obvious, tht sub cooled rock prts re heted from the more distnt, unffected surrounding hot rock mssive hus, n interesting sitution ppers in the het flows inside the underground rock het echnger Principlly, this thermo kinetic sitution cn be formlly clssified s non stedy stte het trnsfer in n unlimited mssive or hlf limited mssive CALCULAION OF HEA RANSFER Logiclly, we need to know the het trnsfer reltions for determining the underground het echnger dimensions t required energy demnds he determintion of this key prmeter ie het trnsfer in the het echnger ws crried out by following method: We used the following simplifictions in our modelling: we considered pipe drilled into rock mssive in which cold wter flows s geometric model All clcultions were crried out on this geometric model he conductivity het trnsfer coefficient α ws clculted for 40 C wter flowing turbulently through pipe in 00 C rock mssive Grnite ws tken s the surrounding rock mssive, with n verge therml conductivity of,5 W(m K) - he physicl properties of rock nd wter re indicted in b nd b b Rock properties Density ρ H kgm Specific het cpcity c ph J(kg K) - 84 emperture of unffected rock H C 00 herml conductivity λ H W(m K) -,5 Depth of echnger h m 3000 he het trnsport medium (het crrier) is considered wter in liquid phse (b )

3 SCIENIFIC PROCEEDINGS 009, Fculty of Mechnicl Engineering, SU in Brtislv b Physicl properties of H O Density ρ kgm , Specific het cpcity c p J(kg K) emperture t 0 C 40 herml conductivity λ W(m K) - 0,65 Conductivity Het rnsfer coefficient α W m - K MEHOD GAUSS DIVERGENCE INEGRAL he rock will be considered s hlf-limited mssive influenced by sudden subcooling he limiting condition is tht the finl temperture is equl to zero (Θ 0) t the time 0s on the pipe surfce 0m herefore, in the solution of the differentil eqution for non-stedy het trnsfer []: ε (, ) Θ(, ) [ A( ε ) cos( ε ) B( ε ) sin( ε ) ] e i + S i i i i () i it is necessry to put the coefficient A( ε i ) s equl to zero nd consider the solution in the following simplified epression: ε Θ (, ) B( εi ) sin( εi ) e i () i After severl modifictions, we cn rech reltionship for the clcultion of the het flow s function of time qs λh ρ H ch ( S ) W/m (3) π If we input time step of 0 h, the specific het flow fter the first ten hours will be [3]:, q S ( 00 40) 443,85 W/m (3) π36000 his method considers therml ccumultion potentil insted of incresing therml resistnce herml ccumultion potentil hs n equivlent reltionship with the convection coefficient of het trnsfer on the flowing het crrier side

SCIENIFIC PROCEEDINGS 009, Fculty of Mechnicl Engineering, SU in Brtislv Fig Het flow from the rock mssive into het crrying medium As cn be seen from the grph bove, the specific het flow will be

4 SCIENIFIC PROCEEDINGS 009, Fculty of Mechnicl Engineering, SU in Brtislv Fig Het flow from the rock mssive into het crrying medium As cn be seen from the grph bove, the specific het flow will be stbilized t vlue of 60 Wm - he constnt temperture fields in continully subcooled rock mssive re determined by using of Guss divergence integrl, where the beginning of process is given [3]: 0 0 ψ (4) And for the unfinite time: ψ (5) hen the temperture t time nd t dimeter distnce r from the is of the pipe will be: ( ) ψ, t S ( ) ( ) S r, ψ [ C] (6) Where is the temperture of rock mssive in the unffected re S is the temperture of the het crrying medium (ie wter) t the point of inlet into the het echnger pipe

5 SCIENIFIC PROCEEDINGS 009, Fculty of Mechnicl Engineering, SU in Brtislv Fig Hot rock mssive temperture drop depending on time Fig 3 Fields of constnt tempertures in grdully sub cooled rock mssive he clcultion ws performed using the MAHEMAICA progrm nd its ccurcy ws verified eperimentlly in lbortory tests t the Institute of herml Power Engineering, SU he eperiments were crried out in n eperimentl modelling stnd with smller dimeter, thus the verifiction of the bove described clcultion could be confirmed only for smller dimeters of solid rock mteril

6 SCIENIFIC PROCEEDINGS 009, Fculty of Mechnicl Engineering, SU in Brtislv In the eperiment, wter t 80 C ws used s het source for solid rock he cooling medium, simulting the role of the het crrying medium in the underground het echnger ws wter t 7 C he comprison of the bove mentioned clcultion nd eperimentl results re shown in fig 4 Fig 4 Comprison of eperiment nd clcultion (6: R0,5m, 7:Rm) MEHOD FORCHHEIMER MEHOD Due the fct tht n eperimentl verifiction of the thermo kinetic behvior for sub cooled rock mssive cn be crried out only for limited dimeters, we used nother clcultion method to verify the clculted results Both clcultion methods, the Guss divergence integrl nd the Forchheimer, method were compred For the clcultion using the Forchheimer method, the sme limiting conditions, ie unffected rock mssive temperture of 00 C ( H ) nd temperture of the het crrying medium of 40 C( s ) were epected A horizontl pipe with dimeter r 0,m ws considered s geometricl givens he surfce temperture of the het echnger pipe ws tken s equl to the wter temperture, ie 40 C In the literture [4] the het flow density is: q hor λ Hor ( ) π H h ln + d S d h [Wm - ] (7)

7 SCIENIFIC PROCEEDINGS 009, Fculty of Mechnicl Engineering, SU in Brtislv he curvture of isotherms is epressed in the numertor of the eqution, nd het resistnce is epressed in the denomintor of the frction, where λhor is therml conductivity of the rock, d is the dimeter of the pipe nd h is depth of echnger hus, the het flow density t the inlet into the het echnger pipe (wter 40 C, rock 00 C) is: q hor,5 π 500 ln + 0, ( 00 40) 500 0, 43,85 [Wm - ] Of course, the het flow density depends on the temperture difference between the wter nd the rock, nd therefore is not constnt long the whole pipe COMPARISON OF RESULS he results from both clcultions methods were compred After re-clcultion of the first method, the Guss divergence integrl for pipe of dimeter d 0,m nd t different temperture grdients, we obtined the following result vlues: b 3 Comprison of results Men medi temperture [ C] emperture grdient [K] Het flow [W/m] method method ,8703 4, ,84 40, , , , , ,907 3, ,608 9, ,489 6, ,6770 4, ,935, ,933 8, ,454 6, ,7095 3, ,9676 0, ,57 8, ,4838 5, ,749,683758

8 SCIENIFIC PROCEEDINGS 009, Fculty of Mechnicl Engineering, SU in Brtislv CONCLUSION As we cn see in the ble 3, both clculting methods gve similr results he correltion between clculted vlues nd the limited eperimentl results is lso stisfctory, therefore both presented methods for the thermo kinetic behvoiur of solid rock mssive nd het underground het echnger cn be considered s suitble bses for the design clcultion of geotherml Hot- Dry-Rock power systems REFERENCES [] Gróf, G: Muszki hőtn, hőközlés, BME Budpest, 007 [] Šorin, SN: Sdílení tepl, SNL-Nkldtelství technické litertury, Prh, 968 [3] Msryk M: Energetický potenciál geotermálneho tepl suchých hornín n území Slovensk jeho využitie pre výrobu elektrickej energie, ézy hbilitčnej prednášky, SjF SU 009, ISBN: [4] Knížt, B, Rjzinger, J, óth, P: Prenos tepl medzi plynom okolím, Slovgs /005

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