Toni Alchofino ( ) Physics Department, Bandung Institute of Technology, Indonesia,

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1 Study of Numercal alyss for Coductve System Usg Fte Dfferece lgortms for Smle Oe Dmesoal Heat-Coducto Problems Wt Costat Termal Dffusvtes To lcofo (53 Pyscs Deartmet, Badug Isttute of Tecology, Idoesa, 8 E-mal: To_lcofo@Studets.tb.ac.d Suervsed by : Dr. Eg. lamta Sgarmbu bstract Heat moves by coducto, covecto, radato ad advecto. Beeat te eart s teral structure, eat trasfer majorly by covecto ad coducto rocess. Wt geotermal reservor, eat trasferred from magma to a ermeable rocs system called bedroc. It s eat eergy te absorbed by water from aqufer system wc located above te bedroc. Tere, ot oly coductve rocess but also covectve occur. Ts eat coductve trasfer s oe of mortat rocess wc establs te geotermal reservor system. Most of roblems volvg eat ad mass trasfer are reducble to te soluto of artal dfferetal equato. To solve te dfferetal equato tat gover real yscal rocess are geerally very comlcated. Terefore a aroxmate codto smlest cases s very useful. To solve te dfferetal equato o a comuter, a roblem must be formulated umercally terms of some sutable artmetc oeratos. Heat-Coducto Problems wt geotermal reservor ca be formulated umercally. I. Itroducto te late 96s, may dfferetal equato are ossble to formulate umercally. umercal tecques allow us to determe a table of aroxmato values of te desred soluto. May metods ca be used to aroxmate some yscal rocess, suc aroaces as te fte-dfferece metod, stragt le metod, large-artcle metod, ad Mote Carlo metod. Heat ad mass trasfer are domat rocess occur wt te geotermal reservor. Some of te rocess ca be aroaced by artal dfferetal roblem wc formulated to umerc soluto. Wt umercally aroacmet, te roblems ca be solved wt el of a comuter. Nowadays, some geotermal modelers ave used reservor flud cotag varous cemcals ad oters ave cluded extra features ad more comlcated. Ts aer s a reelmary study of yscal rocess of geotermal. Te urose of ts aer s to revew umercal soluto to coduct a model smle reresetatve case study of eat-coductve trasfer eomea. Te goverg soluto of eat-coductve eomea s smle roblems ad t wll be used to descrbed te eomea troug a model buld based o te umercal soluto. Metod wc s used to gover te soluto s fte-dfferece wt costat termal dffusvtes. II. Te Defto of Fte-Dfferece Metod ad Classfcato of Geotermal system Fte-dfferece metod s a metod of soluto of dfferetal roblems wc dfferetal oerators are relaced by ter aroxmate values exressed terms of fucto at dvdual dscrete ots. Ts metod s also called te grd metod. (Nogotov 978. Tere are some mortat ts we we use fte-dfferece tecque. fter some dfferetal roblems relaced wt oter form, we sould test te stablty of t, weter ts covergece or ot. We ca aalze te stablty troug stablty aalyss by Fourer trasformato or by maxmum rcle. Tere are two geotermal system (Bowe 989 :. Covectve geotermal system. Coductve geotermal system Covectve geotermal systems are caracterzed by atural crculato of te worg flud, te majorty of te eat beg trasferred by crculatg fluds ad ot by coducto. Te covectve rocess romotes creased temeratures te uer art of te crculato system ad corresodg lowerg of temerature occurs te lower art. Covectve geotermal system ca be subdvded to ydrotermal ad crculato systems.

2 Coductve geotermal systems are cosst of lowtemerature, low-etaly aqufers, georessurzed zoes, ad ot dry roc. III. Smle Heat-Coducto Problems Heat flows from a ot body to a cold body, te rate wc eat s coducted troug a sold s roortoal to te temerature gradet. T T<T Heat er ut of tme eterg te elemet across ts face at s aq(, wereas te eat er ut tme leavg te elemet across ts face at δ s aq(δ. Exadg Q(δ a Taylor seres gves : ( Q Q ( Q(... ( I Taylor seres, te (δ ad tose of ger order are very small ad ca be gored. From eq. te et ga of eat er ut tme s Heat eterg across eat leavg across δ T aq( aq(δ -aδ(δq/δ (5 Pc.. Heat Flows Te rate of eat flow er ut area dow troug te late s roortoal to : T T Δ ( Te rate of eat flow er ut area u troug te late, Q, s roortoal to : Suose eat s geerated ts volume eemet at a rate er ut volume er ut tme. Te total amout of eat geerated er ut tme s te : aδ (6 Combg Eq (5 da (6, gves te total ga eat er ut tme : a a (7 T T Q Δ ( If te materal as desty ad secfc eat C, ad udergoes a temerature crease δt tme δt, te rate at wc eat s gaed s : : Termal Coductvty. Equato ( ca be wrtte to a dfferetal equato : Q (3 I cotext of te eart, s a det beeat te surface. s creases dowwards, a ostve temerature. Cosder te cture below : aq(z zδ aq(zδ Pc. Luas Permuaa a C, Ca (8 Te total ga eat er ut tme s equlable wt te rate at wc eat s gaed. From Eq (7 ad (8 we get : C (9 z Substtute te Eq (3 to Eq (9, gves : C C ( Eq. ( s a smle Heat-Coducto dfferetal equato roblem. I 3 dmeso, te equato becomes :

3 T C T ( For steady-state stuato, we tere s o cage temerature wt tme,eq becomes : We ca wrte Eq. ( as : a f (6 T ( IV. Numercal alyss For Steady State Codto Eq ( s dffuao eat flow equato steady state codto. For oe dmeso, equato ( becomes : (3 s te dstrbuto fucto of te eat sources. s te termal coductvty of materal. (,t, steady state codto, oly deeds o ot t. Terefore, to get te umercal soluto, we ca relaces te dervatves of Eq. (3 wt some dfferece relatos. Ts relacemet s based o te dervatve as te lmt T ( T ( lm If s fxed te above qualty, te aroxmate formula for te frst dervatve exressed terms of fte dfferece may be obtaed : T ( T ( ( T( T( T( T( T ( T ( T ( (5 V.Oe Dmesoal Heat Coducto Problems Wt Costat Termal Dffusvtes. Eq. ( s a smle coducto roblems oe codtoal roblems. C Were a f. s te termal dffusvty, ad Te, we sould formulate te dfferetal equato roblems to a dmesoless temerature, ( T T ΔT. T s fxed temerature, ad T s caracterstc temerature dfferece. If te rage of temerature roblem s ow, te : ( T T ( T T m. max m If te det L s over L, te wt troducto wt ew varables /L ad t t/l, te rego of te roblem may be reduced to terval. Eq. (6 may ow wrte as : a F. Were FL f (7 To solve te Eq. (7, te dfferetal equato (7 must be sulemeted wt boudary ad tal codtos. t eac boudares, te temerature (te frst-d boudary codto, or eat flux (te secod-d codto, or a combato of tese quattes (te trd boudary codto may be gve. I geeral case, te boudary codtos may be wrtte as : α β γ Tus, we α ad β, te boudary codto s of te frst d; α ad β s a secod boudary codto; ad α ad β s a trdd codto roblem. For smlcty, frst-d boudary codtos are assumed at ad : (,ψ( (,t ψ (t (,t ψ (t (8 VI. Fte Dfferece lgortms For Oe Dmesoal Heat Coducto Problems Wt Costat Termal Dffusvtes. Wt fte-dfferece tecque, we ca obta te soluto of eat coducto roblems (7 - (8. To troduce a uform dfferecg grd, te terval s dvded to J equal arts by,

4 ,,,3...,J, wt /J. Te soluto s sougt at te ots for tme stes t σ,,,... Te smlest ad most atural metod for roblems(7-(8 s based o a exlct aroxmato of te orgal dfferetal equato : a σ ψ ( ψ F J ψ (9 Te algortm for umercal soluto s exressed by ξ( ( ξ Wt Wt, aτ ξ,... τf ( ow, successve determato of s ot dffcult. Te comutatoal stablty requres te tme stes to be coformable wt τ ( a Substtuto of te exact soluto to ((9, followed by Taylor-seres exaso of (, t τ, (, t, (, t about te ot (, t redly gves te error exresso for te metod : τ a R (, t O( τ Te error s tus of te order O ( τ. It may easly be demostrated tat oe may select τ suc tat te order of te error s O(. Ideed, because of te form of te dfferetal equato (6, we may wrte F a Substtuto of ts exresso to te error formula gves aτ R (, t a O( τ ( Wc mles tat f τ s cose ad te 6a rgt-ad sde of Eq. (9 F τ F R O( F s relaced by Te olds. Te order of aroxmato s obvously s ot dsturbed uo te substtuto F ( F τ F F F F (3 a Sce stablty codto ( s satsfed wt τ, te ger accuracy sceme s stable. 6a VII. Coclusos ad Dscussos For steady state codto, te eat coducto roblems are oly trasformed to dfferece of dfferetal equato form. Ts umercal soluto s a smlest soluto for eat flow. I ts codto, te eat trasfer oly deeds o te det (, te temerature wll varey for every det ( value. Wt ts oter form sowed Eq.(5, we ca ow create a smulato a comuter to calculate ow te temerature varey for every det ( or eat source temerature. I o steady state codto, we frst smlyfy te Eq. (. Because of te eat flow deeds ot oly te tme but also te det ad te oters, ts roblems s a artal dfferetal equato wc we eed to solve t by some metod. I ts aer, t sows ow te roblem s aroaced by fte dfferetal metod. For costat dffusvtes were te coductvty factor, secfc eat ad te desty of materal are costat. d also we eed to comute we s te soluto form, Eq ( stabl. Ts s called comutatoal stablty. Usg ts aroacmet, we wll get te error value from eq (. Te error value sows ow te smulato aroac te real codto. Tese umercal aalyss oly gves te soluto a very smlfed stuato. I realty, we ow tat te dffusvtes factor are ot costat. It s because eart materals are uqe, tey ave varetes of desty ad coductvty value. Te geometry of te eart s also more comlcated, ad ts ot tat smle. We eed to ow te boudary codto of geometry we used.

5 VIII. Refereces Bowe, Robert Geotermal Resources d edto. Elsever Scece Publsg. Boas, Mary.L.983. Matematcal Metods Te Pyscal Sceces. Jo Wley&Sos. Fowler,C.M.R.99.Te Sold Eart : Itroducto to Global Geoyscs.d ed. Cambrdge Uoversty Press. Fser, Mcael E.988. Itroductory Numercal Metods wt Te NG Software Lbrary. Deartmet Matematcs Uversty of Wester ustrala. Fauz,Umar, Prad S. bdoessoe. Fsa Utu Geolog. ITB O Sullva, M.J., Pruess, K., ad Lma, M.J. (.Geotermal reservor smulato: Te state-of-te-ractce ad emergg treds. Lawrece Bereley Natoal Laboratory reort LBNL-699. Nogotov,E.F.978.lcato of Numercal Heat Trasfer. McGraw-Hll Boo Comay. Mattax, C. Calv.,Robert L. Dalto. 99. Reservor Smulato. Socety of Petroleum. Egeers. Putra,V.Doddy.5. las Metode P/ Utu alsa Cadaga Reservor Paas Bum Domas Ua. Setawa,gus.6. Pegatar Metode Numer. Yogyaarta : Peerbt d. emasy,m.w.,ad Dttma,R.H., Kalor da Termodama,eterjema Log,T.H.,986, Badug :ITB.