Simulation of Fluid Flow and Heat Transfer in Porous Medium Using Lattice Boltzmann Method

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Journl of Physics: Conference Series PAPER OPEN ACCESS Simultion of Fluid Flow nd Het Trnsfer in Porous Medium Using Lttice Boltzmnn Method To cite this rticle: Imm Wijy nd Acep Purqon 7 J. Phys.: Conf. Ser. 877 56 View the rticle online for updtes nd enhncements.

1 Journl of Physics: Conference Series PAPER OPEN ACCESS Simultion of Fluid Flow nd Het Trnsfer in Porous Medium Using Lttice Boltzmnn Method To cite this rticle: Imm Wijy nd Acep Purqon 7 J. Phys.: Conf. Ser View the rticle online for updtes nd enhncements. Relted content - Modelling of fluid flow nd het trnsfer in reciprocting compressor J Tuhovck, J Hejcik nd M Jich - Single bubble rising dynmics in porous medi using lttice Boltzmnn method Xi Ji, Jun Shi nd Zhenqin Chen - A Lttice Boltzmnn Model for Fluid-Solid Coupling Het Trnsfer in Frctl Porous Medi Ci Jun nd Hui Xiu-Ln This content ws downloded from IP ddress on 7//9 t 8:7

2 Interntionl Conference on Energy Sciences (ICES 6) IOP Conf. Series: Journl of Physics: Conf. Series (7) 56 doi :.88/ /877//56 Simultion of Fluid Flow nd Het Trnsfer in Porous Medium Using Lttice Boltzmnn Method Imm Wijy,), Acep Purqon, Erth Physics nd complex system Reserch Division, Institut Teknologi Bndung, Jln Gnes, Bndung 43, Indonesi ) robih.wijy39@gmil.com Abstrct. Fluid flow nd het trnsfer in porous medium re n interesting phenomen to study. One kind exmple of porous medium is geotherml reservoir. By understnding the fluid flow nd het trnsfer in porous medium, it help us to understnd the phenomen in geotherml reservoir, such s therml chnge becuse of injection process. Therml chnge in the reservoir is the most importnt physicl property to known since it hs correltion with performnce of the reservoir, such s the electricl energy produced by reservoir. In this simultion, we investigte the fluid flow nd het trnsfer in geotherml reservoir s simple flow in porous medium cnl using Lttice Boltzmnn Method. In this simultion, we worked on dimension with nine vectors velocity (DQ9). To understnd the fluid flow nd het trnsfer in reservoir, we vried the fluid temperture tht inject into the reservoir nd set the het source constnt t 4 C. The first vrition we set the fluid temperture 45 C, second.5 C, nd the lst 37.5 C. Furthermore, we lso set the prmeter of reservoir such s porosity, density, nd injected fluid velocity re constnt. Our results show tht for the first temperture vrition distribution between experiment nd simultion is 9.86% mtch. From second vrition shows tht there is one pick of therml distribution nd one of turbulence zone, nd from the lst vrition show tht there re two pick of therml distribution nd two of turbulence zone.. Introduction Fluid flow nd convection het trnsfer of porous medium hve been studied by mny reserchers in different field of science nd engineering including geotherml engineering, civil engineering nd mechnics, chemistry nd petroleum engineering, hydrology, nd nucler especilly in cooling mngement. In geotherml engineering, het trnsfer nd fluid flow re very importnt thing to understnd, becuse it hs correltion with electricl energy produced by the reservoir. The property of fluid flow nd het trnsfer in porous medium hd been studied nd modeled it with some numericl methods. The numericl method used to simulte is divided into three groups i.e. mcroscopic modeling (FDM, NS, FEM), microscopic modelling (MD, DSMC), nd meso-scopic modelling ( LBM, LGA, SPH). The meso-scopic modeling is reltively new method tht could bridge between two pproches mcro-scle nd micro-scle. In the meso-scopic modelling (LBM) stted tht the behvior of group prticles into the behvior of single prticle. Therefore, we do not need to declre the behvior of ech prticle. The behvior of group prticle is stted by distribution function. The distribution function fulfilled for this cse is Boltzmnn Distribution. LMB hs severl dvntges, two of these re it does not need computer with high of RAM nd it does not need to solve Poisson eqution for ech itertion, therefore the running process is more fster. In this study, we simplify the process of fluid flow nd het trnsfer in geotherml reservoir s simple Content from this work my be used under the terms of the Cretive Commons Attribution 3. licence. Any further distribution of this work must mintin ttribution to the uthor(s) nd the title of the work, journl cittion nd DOI. Published under licence by Ltd

Interntionl Conference on Energy Sciences (ICES 6) IOP Conf. Series: Journl of Physics: Conf. Series 3456789 877 (7) 56 doi :.

3 Interntionl Conference on Energy Sciences (ICES 6) IOP Conf. Series: Journl of Physics: Conf. Series (7) 56 doi :.88/ /877//56 flow in porous medium cnl with vrying of fluid temperture injected, then we compre the result of simultion with experiment results by Olimpi Bnete (4).. Lttice Boltzmnn Method.. Bsic Concept of Lttice Boltzmnn Method LBM is kind of numericl method tht cn be used to simulte fluid flow. This method cn bridge between two pproches mcro-scle nd micro-scle. In this method the behvior of group prticles is stted into the behvior of single prticle. Therefore, we do not need to declre the behvior of ech prticle. The behvior of group prticle stted by distribution function. The distribution function fulfilled for this cse is Boltzmnn Distribution. The function is expressed by eqution: t eq f( x et, t t) f( x, t) ( f( x, t) f ( x, t)) () where f ( x, t ) is Boltzmnn distribution function to clculte density nd velocity field, is time relxtion tht hs correlted with kinemtic viscosity v, f ( x e t, t t) is velocity distribution eq function t intervls time t t, nd f ( x, t ) is distribution function in equilibrium stte. eq eq eq eq e. u ( e. u ) ( u ) f ( x, t) w 4 cs cs cs In this simultion, we used lttice model DQ9, the figure of model s follow. () Figure. Lttice DQ9 structure (Sukop,7) For this model, we define e s : e (,),, c cos,sin 4 4 With,,,3,4, 5,6,7,8 (3) Density nd velocity define s f ui f ciui (4).. LBM for het trnsfer in pore medium Nithirsu et ll. (997) ssumed if the Boussinesq in limit vlue re fulfilled nd there is therml equilibrium ner fluid nd solid, then the equtions used to describe the behvior of incompressible fluid flow nd convection proses in porous medium s follow:

4 Interntionl Conference on Energy Sciences (ICES 6) IOP Conf. Series: Journl of Physics: Conf. Series (7) 56 doi :.88/ /877//56. u ' (5) u u u. p ve u F t f (6) T u T m T t.. (7) where coefficient is rtio of therml cpcity for solid nd liquid, u is dischrge, p is pressure, v c is effective viscosity, F is totl externl force experienced by porous medium. Guo nd Zho (5) defined the totl of externl force experienced by the medium s follow: v F F u u u G K K (8) where v is fluid viscosity, G is grvittionl force. The grvittionl force G is defined s follow : G g T T (9) where g is grvittionl ccelertion, is therml expnsion coefficient, T initil temperture, ccelertion becuse of externl force. Ergum (95) developed geometry function F nd permebility K s function of porosity bsed on their experiment. The function is defined s follow: F nd K 3 d p 5( ) ().3. LBM for velocity field Guo nd Zho () developed the eqution for fluid flow in porous medium defined s follow: eq f ( x e t, t t ) f ( x, t) f ( x, t) f ( x, t) t F v () Where tfis totl force experienced by porous medium or totl force experienced in the system. Guo nd Zho () defined the distribution eqution in equilibrium stte for DnQb model s follow: e. F uf : ( ee cs I) F w 4 v cs cs () Bsed on the equtions bove (Guo nd Zho), F hs non-liner correltion with u. Guo nd Zho () lso defined the density nd fluid velocity s follow: v u f c c c v nd (3) where v is t v e f G (4) c nd c re defined s F c t K F c t nd K (5) 3

Interntionl Conference on Energy Sciences (ICES 6) IOP Conf. Series: Journl of Physics: Conf. Series 3456789 877 (7) 56 doi :.88/74-6596/877//56.4. Temperture Field Eqution He et ll 998 found the distribution function for equilibrium stte.

5 Interntionl Conference on Energy Sciences (ICES 6) IOP Conf. Series: Journl of Physics: Conf. Series (7) 56 doi :.88/ /877//56.4. Temperture Field Eqution He et ll 998 found the distribution function for equilibrium stte. The eqution is defined g eq e RT D / e e exp RT DRT e DRT D RT e. u e. u ( RT) u RT e e e D 4 e. u e D u exp D / RT RT DRT D ( RT) DRT D RT (6) For DQ9 model, Set et l 6 used eqution 7 until 9 to describe the distribution of density in equilibrium stte which is the vlue of the function is discrete. eu g eq (=) (7) 3c g e 3 3 e. u 9 ( e. u) 3 u 4 9 c c c eq 4 (=,,3,4) (8) eq e 6e. u 9 ( e. u) 3 u g (=5,6,7,8,9) (9) 36 c c c Internl energy hs correltion with temperture, e 3RT / where R is idel gs constnt (R=8.34 J/moll-K)..5. Boundry Condition In this simultion, we used mny kind of boundry condition, such s bounce bck boundry condition nd inlet velocity boundry condition. The prticles bounce when those hit the wll in bck boundry condition. This is simple boundry condition to mke solid formtion in the system. Figure. Bounce Bck boundry Condition (Muhmmed,) By used inlet velocity boundry condition this boundry condition, we could tke some ssumption tht there is no velocity in y-direction. The velocity in x-direction equl to when the prticle hit the wll. After the streming process, there re unknown distribution function f, f5, f 8 nd lso the density. To get the distribution function we used non-equilibrium eqution (4). Temperture distribution function for lttice model DQ9 defined 8 T g g g g g g 3 g 4 g 5 g 6 g 7 g8 () 4

6 Interntionl Conference on Energy Sciences (ICES 6) IOP Conf. Series: Journl of Physics: Conf. Series (7) 56 doi :.88/ /877//56 Figure 3. velocity Boundry Condition Figure 4. Temperture Boundry Condition 3. Model nd Simultion Algorithm The simultions were performed in this study divided in to two ctegories, the first ctegories is single phse fluid flow simultion for poissellie flow nd het convection, nd second is simultion of fluid flow nd het trnsfer in porous medium. 3.. Single Phse Fluid Flow In the first ctegory, we designed the simultion which the fluid flows in pipe with both ends open to ech other, otherwise it is ssumed tht there is no friction between the fluid nd the pipe nd there is no interction between fluid prticles. The fluid in the system is non-viscous nd incompressible fluid. In this simultion we used Lttice Boltzmnn Method DQ9 to described the fluid flow in the cylinder. We used some ssumption i.e. the interction between prticles only collision, there is no externl force, nd flow occurs just becuse of the differences pressure between inlet nd outlet. We used bounce bck boundry condition on the cylinder wll nd ssumed no slip. While in inlet nd outlet of the cylinder we used periodic boundry conditions, therefore the fluid cn flow continuously. The dimension of the models we used 4x lttice unit. The Reynolds number is, density is, nd the inlet pressure is.5. The time relxtion is defined by this formul 3 v / with v v mks / Re. From this simultion we plot the velocity profile, then we compre the result to the theory. 3.. Nturl Convection in the Wter The second ctegory is simultion of nturl convection in the wter. In this simultion, we used two distribution function, there re distribution function for liquid prticle nd distribution function for temperture, nd we lso used two kind of lttice model; there re DQ9 for prticle nd DQ5 for temperture. Both of distribution function we stcked together in the system. We ssumed tht the interction between prticles on the system is only collision, no externl forces except grvity, nd flow occurs becuse of the differences density between cold nd hoot wter. We used three kind of boundry conditions; there re no-slip boundry condition, periodic boundry condition, nd streming boundry condition. In this simultion we composed the model into dimension x lttice unit, the Ryleight number. In this condition, het source comes from the bottom wll of the system. We compred the result of this simultion with nother simultion used FDM tht hs been done by Septin Setyoko to see the pttern of distribution temperture Fluid Flow nd Hert Trnsfer in Porous Medium In this simultion, firstly we simulte the fluid flow nd het trnsfer in porous medium with het source comes from left wll, then we simulte fluid flow nd het trnsfer with vrious fluid temperture injected into the system nd we set the het source (comes from bottom wll) t constnt temperture. From this simultion we plot the profile of temperture distribution. 5

7 Interntionl Conference on Energy Sciences (ICES 6) IOP Conf. Series: Journl of Physics: Conf. Series (7) 56 doi :.88/ /877// Simultion Algorithm Figure 5. Simultion Algorithm 4. Results nd Discussion 4.. Single Phse Fluid Flow From the simultions for the Poiseuille flow, we got the result s follows. Figure 6. Velocity profile for poiseuille flow in the pipe (nlytic nd simultion). In Figure 5 we known tht from simultion nd the nlytic result re the sme. The fluid velocity profile hs the shpe of hyperbolic functions where the highest rte velocity locted in the center of the pipe. Further vlidtion process by compring the visuliztion result from the simultion using LBM nd Finite Difference method tht hs been done by Septin Setyoko (). The pttern of the temperture distribution results cn be seen in the picture below. 6

Interntionl Conference on Energy Sciences (ICES 6) IOP Conf. Series: Journl of Physics: Conf. Series 3456789 877 (7) 56 doi :.88/74-6596/877//56 Figure 7.

The big difference is likely due to differences in physicl prmeters used in the simultion. Even though we cn use the LBM to pproch het convection process in porous medium. 4.

simulte het trnsfer in porous medium with vrition temperture fluid injected where the source comes from the bottom.

8 Interntionl Conference on Energy Sciences (ICES 6) IOP Conf. Series: Journl of Physics: Conf. Series (7) 56 doi :.88/ /877//56 Figure 7. Temperture distribution using FDM Figure 7b. Temperture distribution using LBM From the simultion, using FDM nd LBM we got similr profile of temperture distribution. The big difference is likely due to differences in physicl prmeters used in the simultion. Even though we cn use the LBM to pproch het convection process in porous medium. 4.. Fluid Flow nd Hert Trnsfer in Porous Medium After the simultions with nlyticl vlidtion ws done, then we simulte het trnsfer in porous medium when the het source comes from left wll nd lso simulte het trnsfer in porous medium with vrition temperture fluid injected where the source comes from the bottom. Firstly we simulte fluid flow nd het trnsfer where the het source comes from the left of the wll. The temperture on the left wll we set t 45 C, while the upper nd lower wlls we set nd we mintined into C. Figure 8. Temperture distribution profile in porous medium s convection procces used LBM with temperture fluid flow 45 C nd upper bottom wlls temperture C 7

Interntionl Conference on Energy Sciences (ICES 6) IOP Conf. Series: Journl of Physics: Conf. Series 3456789 877 (7) 56 doi :.88/74-6596/877//56 Figure 9.

9 Interntionl Conference on Energy Sciences (ICES 6) IOP Conf. Series: Journl of Physics: Conf. Series (7) 56 doi :.88/ /877//56 Figure 9. Experiment result of therml temperture distribution profile in porous medium s convection process by Olimpi Bnete. From the simultions nd experiments conducted by Olimpi Bnete, we hve the similr profile. The profile hs prbolic shpe with pek on the right. If we compre it, the pek hs vlue lmost the sme,.5 C for the experiments nd C for the simultion, but there is striking difference round the wlls. In the simultion, temperture distribution round the wll rise rpidly, while the experimentl reltively slow to rise. Secondly, we simulte fluid flow nd het trnsfer where the het source comes from the bottom wll, in this cse, there is hot fluid injection from the left wll. We set the temperture t bottom wll constnt t 4 C nd injection fluid temperture t.5 C with the rte injection.m /s. Their rte of injection in the left wll intended to pproximte the behvior of the temperture distribution t the time of the geotherml reservoir injection process. This process is very importnt since it hs reltionship to the energy tht would be generted by the reservoir. From the simultion, temperture distribution profile is obtined s follows. 8

Interntionl Conference on Energy Sciences (ICES 6) IOP Conf. Series: Journl of Physics: Conf. Series 3456789 877 (7) 56 doi :.88/74-6596/877//56 Figure.

From figure bove, in number of itertion, the temperture does not rise significntly from the bottom nd sides.

The results show tht the pek temperture distribution getting higher nd shifting towrd to the right. The high pek temperture distribution ffects the pttern of fluid flow in the system.

it could be seen from the velocity vector of fluid flow in the following figure. Figure. Vector velocity in low temperture Figure b.

10 Interntionl Conference on Energy Sciences (ICES 6) IOP Conf. Series: Journl of Physics: Conf. Series (7) 56 doi :.88/ /877//56 Figure. temperture distribution profile with het source 4 C nd injection fluid temperture.5 C. y-xis nd x-xis is dimension of system. From figure bove, in number of itertion, the temperture does not rise significntly from the bottom nd sides. In 5 number of itertion, there is pek of temperture distribution due to the injection process nd the heting continuously from the bottom wlls. The results show tht the pek temperture distribution getting higher nd shifting towrd to the right. The high pek temperture distribution ffects the pttern of fluid flow in the system. At low temperture, fluid flows in lminr pttern, while t the high tempertures the fluids flow would be turning into turbulent pttern. it could be seen from the velocity vector of fluid flow in the following figure. Figure. Vector velocity in low temperture Figure b. Vector velocity in low temperture For the third simultion we set the bottom temperture into 4 C nd injection fluid temperture 37.5 C with the rte of injection.m/s. The simultion process obtined the temperture distribution profile s show in figure. From the simultion, the temperture increses rpidly nd the profile lmost different from the previews doe to the differences fluid temperture injection with the previews simultion is significnt, round degrees. At 5 number of itertion, there re two pek temperture. The first pek occurs due to combintion temperture from het source nd fluid temperture injection, while the second pek temperture comes from wrming the fluid tht hs higher temperture thn the initil temperture. 9

11 Interntionl Conference on Energy Sciences (ICES 6) IOP Conf. Series: Journl of Physics: Conf. Series (7) 56 doi :.88/ /877//56 Figure. Temperture distribution profile with het source 4 C nd injection fluid temperture.5 C. 5. Conclusion From this simultion, we could be conclude tht the model composed by LBM cn be used to solve fluid flow nd het trnsfer in porous medium problems. The first simultion simulte fluid flow in pipe, from the simultion tht there re similrities with the theory of fluid flow profile for Poiseuille flow. The second simultion simulte nturl wter convection which is the het source comes from the bottom of the wll. The results using LBM hve the sme profile with the Finite Difference Method, it cn be concluded tht LBM could lso be used to simulte the het flow in the fluid. In the simultion of fluid flow nd het trnsfer in porous medium provides the results of the temperture distribution profile similr to the experiment, with the rte reching 9.8%, it show in the temperture distribution profile. References [] Nobovti,Aydin, nd W.L.Edwrd., 9, Generl Model for the Permebility of Fibrous Porous Medi Bsed on Fluid Flow Simultion using the Lttice Boltzmnn Method. Comp: prt A [] Rmstd, Thoms.,, Simultion of Two Phse Fluid Flow in Reservoir Rock Using Lttice Boltzmnn Method. J.SPE 467 [3] Pn.C, Hilpert.M, Miller.C.T.,4, Lttice Boltzmnn Simultion of Two Phse Flow in Porous Medi. Wter Resource Reserch, Vol.4, W5,doi:.9/3WR. [4] Ltr-Koko M, Rothmn DH.,5, Sttic Contct Angle in Lttice Boltzmnn Models of Immicible Fluid.Physicl Review E.5; 7(4):467. [5] Huber, Christin.,7, Lttice Boltzmnn Modeling for Melting with Nturl Convection. [6] Sukop,M.C, Thorne,D.T, Lttice Boltzmnn Modelling An Introduction for Geoscientists nd Engineers, Mimi, Florid USA. [7] Wolf,G.D.A.,5, Lttice Gs Cellulr Automt nd Lttice Boltzmnn Models n Introduction, Alfred Wegener Institute for polr nd mrine reserch, Germny. [8] Hung, Hibo, Li, Zhico, Lio,Shuishui.,8, Shn-nd-Chen Type Multiphse Lttice Boltzmnn Study of Viscous Couppling Effect for Two-Phse Flow in Porous Medi. Int.J. Numer.Meth.Fluids. doi:./fld.97. [9] Q.Zou, nd X.He, Pressure nd Velocity Boundry Condition for the Lttice Boltzmnn, J.Phys.Fluids 9, (996). [] S.Chen,D.Mrtirez, nd R.Met, on Boundry Condition in Lttice Boltzmnn Methods, J.Phys.Fluid 8,57-536(996). [] Altoslmi,Urpo., 5, Fluid Flow in Porous Medi With the Lttice Boltzmnn Method, University of Jyvshyl, Finlnd(Thesis). [] Bnete, Olimpi.,4, Towrds Modeling Het Trnsfer Using Ltic Boltzmnn Method For Porous Medium, Lurentin University,Cnd.

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