STUDY ON TWO PHASE FLOW IN MICRO CHANNEL BASED ON EXPERI- MENTS AND NUMERICAL EXAMINATIONS

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1 Blucher Mechancal Engneerng Proceedngs May 0, vol., num. STUDY ON TWO PHASE FLOW IN MICRO CHANNEL BASED ON EXPERI- MENTS AND NUMERICAL EXAMINATIONS Takahko Kurahash, Reka Hkch and Hdeo Koguch Department of Mechancal Engneerng, Nagaoka Unversty of Technology Graduate school of Nagaoka Unversty of Technology Abstract. We dscuss on two phase flow n mcro channel based on numercal and expermental studes. The fnte element method usng stablzed bubble functon element s appled to analyze two phase flow. In addton, CFS model (contnuum surface model) s ncluded n numercal computaton. In ths study, two phase flow conssts of pure water and Ethyl acetate s treated, and we nvestgate valdty of numercal soluton based on expermental results. Keywords: Fnte element method, Stablzed bubble functon element, Mcro-channel flow, Two phase flow, CSF model.. INTRODUCTION In ths study, we focused on macro channel flow n mcro chemcal chp (See Fgure.) []. It s consdered that ths chp s wdely applcable when we compound a medcne, or carry out envronmental analyss, or make emulsons for cosmetcs. In case that envronmental analyss s carred out by usng mcro chemcal chp, t s necessary to keep two phase flow n mcro channel. Hence, the purpose of ths study s to construct computatonal program that s applcable to smulate two phase flow n mcro-channel approprately. Fgure. Mcro chemcal chp In ths study, we employ two fluds, water and ethyl acetate, and temperature of those flud s 0 degree. Measurement of nterface poston s carred out by dgtal mcroscope to compare results by numercal and expermental studes. In numercal study, ncompressble

2 Naver-Stokes equaton wth CFS model [] and contnuty equaton are employed to express flow felds, and advecton equaton s appled to compute the nterface poston. The fnte element method s appled to compute those equatons, and stablzed bubble functon element [] s employed for dscretzaton n space. In addton, fractonal step method s appled to compute flow feld. In ths study, we focus on Y-juncton part for upstream and downstream sdes n mcro chemcal chp, and numercal smulaton of two phase flow s carred out. In addton, comparson wth measurement result s also carred out to confrm the valdty of result of numercal smulaton.. GOERNING EQUATION In ths study, Naver-Stokes equatons shown n Equatons () and () are appled to compute flow feld. f j, j P,, j j,, j, (), 0, () where, P,, and f ndcate velocty components, pressure, densty, vscosty coeffcent and body force per unt volume. In two phase flow, nterface poston s gven by soluton of the followng advecton equaton,, 0, () and two knd of fluds are separated by value of ndex functon. Flud propertes, are (n) (n) expressed by those of two fluds, (n=,) and ndex functon such as () () () (),. Here, n case of 0 and, flud propertes, () () () () ndcate, and,, respectvely. In addton, the CFS model [] s employed as the nterface tenson model. Interface or surface tenson s gven by Equaton (). f, [ ], () where and ndcates nterface or surface tenson coeffcent and curvature, and curvature s expressed as Equaton (5). n n n n, n, (5)

3 where n ndcates drecton cosne, and s gven by the gradent of ndex functon,.. DISCRETIZATION BY STABILIZED BUBBLE FUNCTION FINITE ELEMENT METHOD In ths study, mxed nterpolaton s appled to flow feld, and bubble functon and lnear trangle elements are employed for velocty and pressure, respectvely. Interpolaton functons for velocty and pressure are wrtten as Equatons (6) and (7). N N N ~ N, (6) P N P NP NP, (7) where N ndcates shape functon, and the shape functon s expressed by area coordnate,, such as N, N, N, N 7. It s well known that numercal vscosty can t be gven enough, even f bubble functon element s appled. Therefore, the Mom. stablzaton parameter Bubble for bubble functon element shown n Equaton (8) s proposed [], and ths parameter s derved such that stablzaton parameter n the SUPG method shown n Equaton (9) s equvalent to that n the FEM usng bubble functon element. Mom. Bubble N d e N, N, dae e, (8) SUPG U h h, (9) where and ndcate numercal vscosty parameter and area of element, and the stablzaton parameter s only operated at bubble functon pont n element,.e., ths parameter s ncluded n, component n vscosty matrx of fnte element equaton n case of D problem. The equaton expressed by, component n vscosty matrx s expressed as Equaton (0). U N, N, d N d e A e e h h, (0) where U and h ndcate magntude of velocty n each element and mean element length, and h s gven by Ae. In addton, nterpolaton functons for ndex functon s wrtten as Equaton (). ~ N N N N, ()

4 In the fnte element equaton for the advecton equaton, numercal vscosty shown n Equaton () s added at the bubble functon pont. e N, N, d A e U N d e h, (). DISCRETIZATION BY STABILIZED BUBBLE FUNCTION FINITE ELEMENT METHOD.. Examnatons of nterface poston at juncton pont for upstream sde In ths study, comparson of numercal and expermental results s carred out. Observaton area n mcro chemcal chp s shown n Fgure. The fnte element mesh shown n Fgures and and computatonal condtons shown n Table are employed n numercal study. As the boundary condton for Naver-Stokes equaton, nflow boundary condton,.e., ˆ ˆ x( ) x(), y() 0, x() x(), and y() 0, s gven on, wall boundary condton,.e., x( ) 0, y() 0, x() 0, and y() 0, s gven on, and condton on outlet boundary.e., P 0 s gven on. As the boundary condton for advecton equaton, boundary condton of ndex functon,.e., () ˆ ˆ () and () s gven on (). In addton, observaton of nterface poston by dgtal probe mcro scope s carred out. Magnfcaton s set 50 dameters. Ethyl acetate (Q ) and pure water (Q ) are njected from left sde of the chp. Water quantty of ethyl acetate s constant, 0μl/mn, and that of pure water s changed 0(Case), 0(Case), 50 μl/mn(case ). Results of nterface poston between numercal and expermental examnatons are shown n Fgure 5 and Table. It s found that nterface poston s approprately obtaned n numercal studes. Q Measurement area Q Fgure. Observaton area n mcro chemcal chp

5 Fgure. Fnte element mesh and defnton of boundary condton Fgure. Magnfed fgure of fgure at juncton pont Table. Computatonal condton Condton alue Tme ncrement Δt sec Number of nodes,99 Number of elements,00 Q Ethyl acetate: Densty ρ ,kg/mm Q Ethyl acetate : Knematc eddy vscosty ν 0.557, mm /s Q Pure water : Densty ρ ,kg/mm Q Pure water : Knematc eddy vscosty ν.00, mm /s Interface tenson coeffcent σ N/mm Table. Comparson of b () / H between experment and fnte element analyss Experment b () / H Fnte element analyss b () / H Case : Q 0μl/mn, Q 0μl/mn Case : Q 0μl/mn, Q 0μl/mn Case : Q 0μl/mn, Q 50μl/mn

6 H b () Case : Measurement result (Q 0μl/mn, Q 0μl/mn) Case : Numercal result (Q 0μl/mn, Q 0μl/mn) Case : Measurement result (Q 0μl/mn, Q 0μl/mn) Case : Numercal result (Q 0μl/mn, Q 0μl/mn) Case : Measurement result (Q 0μl/mn, Q 50μl/mn) Case : Numercal result (Q 0μl/mn, Q 50μl/mn) Fgure 5. Comparson of observaton and numercal results

7 .. Examnatons of nterface poston at juncton pont for downstream sde At the downstream sde, t s not usual that complete separate nterface poston s obtaned f same flow quantty for two fluds s gven. Fgure 7 shows the measurement results n case A, Q (Ethyl acetate ) 0μl/mn - Q (Pure water) 0μl/mn, and case B, Q 0μl/mn - Q μl/mn. It s found that though two phase flow s separated at Y-juncton part for downstream sde n case B, that s not clearly separated n case A. When flud analyss at only downstream sde s carred out, t s necessary to know nterface poston n parallel flow to gve boundary condton on nflow boundary. Therefore, estmaton equaton of nterface poston wth respect to flud condtons s derved under an assumpton of Couette-Poseulle flow (See appendx A). Fnally, the followng estmaton equaton s derved. q q b H q q q b () () () () () () () () () () () () () () H q q q b q H b q H 0 () () () () () () () () () () () () () (), () where lower subscrpt,, H, b and q ndcate ndex of two fluds, knematc eddy vscosty, wdth of mcro channel, nterface heght from wall boundary and water quantty per unt length. When Equaton () s solved, four solutons wth respect to nterface poston b are obtaned. If the channel wdth H s consdered, the soluton can be unquely determned. Comparson between measurement results, numercal solutons and exact solutons by Equaton () s shown n Fgure 6. It s seen that exact solutons by Equaton () are good agreement wth measurement results and numercal solutons at upstream sde. Here, numercal analyss s carred out at downstream sde usng result of Equaton (). Computatonal condtons,.e., physcal propertes and so on, are same as those shown n prevous secton. In ths case, nflow condton s gven as Q (Ethyl acetate ) 0μl/mn - Q (Pure water) 0μl/mn. Frst of all, examnaton for mesh dvson s carred out. Two type meshes are shown n Fgure 8. One s a unform dvson mesh n x-drecton, and the other s partally fne mesh near juncton pont n x -drecton. Fgure 9 shows the numercal results around Y-juncton pont n the two cases. In ths analyss, nterface poston on nflow boundary s estmated based on the result of the Equaton (). It s seen that dfference result s obtaned n two cases, better result s obtaned n case of type. From ths result, t can be sad that fne mesh n x-drecton should be prepared around Y-juncton pont. Next, n case of mesh of type, dstrbuton of ndex functon n whole doman s shown n Fgure 0. In ths result, another case of nflow condton, Q (Ethyl acetate ) 0μl/mn - Q (Pure water) μl/mn, s also shown. It s seen that smlar soluton s obtaned comparng to the Fgure 7. Therefore, t can be sad that f two phase flud analyss ncludng contnuous duraton of lamnar flow n mcro channel s carred out, nterface poston s frstly estmated based on the Equaton () and numercal analyss at downstream sde should be carred out based on the boundary condtons ncludng the nterface poston obtaned by the estmaton equaton.

8 Fgure 6. Relatonshp between b () /H and q () /q () Q : Pure water 0 μl/mn Q : Pure water μl/mn Q : Ethyl acetate 0 μl/mn Q : Ethyl acetate 0 μl/mn (Case A) Q 0μl/mn, Q 0μl/mn (Case B) Q 0μl/mn, Q μl/mn Fgure 7. Measurement results

9 Type Type Type Type Fgure 8. Comparson of mesh dvson Type Type Fgure 9. Comparson of dstrbuton of ndex functon

10 (Case A) Q 0μl/mn, Q 0μl/mn (Case B) Q 0μl/mn, Q μl/mn Fgure 0. Dstrbuton of ndex functon usng mesh type 5. CONCLUSIONS In ths study, computatonal process n whch flow computaton of two phase fluds at Y- juncton part for downstream sde can be only carred out was constructed. In numercal study, stablzed bubble functon fnte element method was appled to spatal dscretzaton. In addton, the CFS model was ntroduced n the governng equaton to express nterface tenson effect. In expermental study, measurement of nterface poston by dgtal mcroscope was carred out. Moreover, estmaton equaton of nterface poston was derved, and valdty of the result of nterface heght was confrmed by comparng results between soluton by estmaton equaton, measurement and numercal results. Conclusons are wrtten as follows. < Y-juncton pont for upstream sde > Result of fnte element analyss was good agreement wth measurement result. < Y-juncton pont for downstream sde > It was found that even f nterface poston s center lne n mddle of channel, t s not usual that two knd of flud can be dvded nto two channels, respectvely. It was found that t s necessary to prepare fne mesh near Y-juncton pont to obtan hgh accurate result. Approprate soluton could be obtaned by settng boundary condton based on the estmaton equaton of nterface poston derved n ths study.

11 APPENDIX Momentum equaton for Couette-Poseulle flow s wrtten as Equaton (A.). d dp x( ) ( ) ( ) (,) dy dx ( ), (A.) where, P, and ndcate velocty components, pressure, densty and vscosty coeffcent. Fgure A. shows Defnton of two fluds and coordnate system. Boundary condtons for each flud are wrtten as Equatons (A.) and (A.). Fgure A.. Defnton of two fluds and coordnate system x() 0 Int. x() x Fa. on on y b () y 0. (A.) Int. x() x 0 x() Fa. on on y 0 y b (). (A.) General soluton for velocty can be represented as Equatons (A.) and (A.5). P y Int.. y b y () Fa x( ) () x () x b (), (A.) P y Int.. y b y () Fa x( ) () x () x b (). (A.5) Integratng Equatons (A.) and (A.5), Equatons (A.6) and (A.7) are obtaned. q () 0 b() x() () dy dp () dx b () b() Int. Fa x, (A.6)

12 q () b( ) dy x() 0 dp() b dx () () dp () dx () b H b () () Int. Fa. x H b () Int. Fa. x, (A.7) () () Here, f the equlbrum condton for shearng force on nterface,.e., yx yx, s consdered, the equaton (A.8) s obtaned. d dy d x () x() ( ) () dy. (A.8) In addton, ntroducng a condton that pressure gradent for a flud s equlbrum to that for the other flud, the equaton (A.9) s obtaned by the equatons (A.), (A.5) and (A.8). P P H b() ( ) () () () Int. Fa. x x x b (). (A.9) Substtutng the equaton (A.9) to the equatons (A.6) and (A.7), and arrangng by nterface Int.Fa. velocty, the equaton (A.7) s fnally obtaned. x q q b H q q q b () () () () () () () () () () () () () () H q q q b q H b q H 0 () () () () () () () () () () () () () (). (A.0) Interface poston b() n two fluds can be estmated by solvng the equaton (A.0) Acknowledgements Ths work was supported by Grant-n-Ad for Young Scentsts (B) (No. 7609). REFERENCES [] Arata Aota, Akhde Hbara, and Takehko Ktamor, Pressure Balance at the Lqud- Lqud Interface of Mcro Countercurrent Flows n Mcrochps, Anal. Chem., 79, pp.99-9, 007. [] J.U. Brackbll, D.B.Kothe and C.Zemach, Contnuum method for modelng surface tenson, Journal of computatonal physcs. 00, pp.5-5, 99.

13 [] J.Matsumoto, A.A.Khan, S.S.Y.Wang and M.Kawahara, Shallow Water Flow Analyss wth Movng Boundary Technque Usng Least-squares Bubble Functon, Internatonal Journal of Computatonal Flud Dynamcs, 6, (), pp.9-, 00. [] U.Gha,K.N.Gha and C.T.Shn, Hgh-Re Solutons for Incompressble Flow Usng the Naver-Stokes Equatons and a Multgrd Method, J. Compt. Phy., 8, 87-, 98.