PARAMETRIC STUDY OF FLUID DYNAMICS IN PEM FUEL CELLS

Transcription

1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY Seres A OF THE ROMANIAN ACADEMY Volume Number /9 pp PARAMETRIC STUDY OF FLUID DYNAMICS IN PEM FUEL CELLS Anca CAPATINA* Hora ENE* Dan POLIŞEVSCHI* Ruanra STAVRE* *Insttute o Mathematcs Smon Stolow o Romana Unversty o Ptest Romana Corresponng author: Dan POLIŞEVSCHI e-mal: DanPolsevsch@marro Ths paper s a research n the rame o the Grant CE 89/6 Water an thermc management or PEM uel cells We stuy here the low n the Gas Duson Layer o a PEM uel cell In ths regon the ynamc o lus s governe by capllary an uson phenomenon The usonconvecton process s stue nvolvng the oygen an water vapors Our moel s smlar wth the Hele-Shaw moel whch escrbes the splacement o two mmscble lus wth erent vscostes between two parallel plates at a small stance (see [9]) The concentratons o the lus are nvertble n terms o the vscostes I the splacng lu s less vscous the nterace between the lus s unstable We conser a uson regon between the two lus where the vscosty s a parameter We stuy the lnear stablty o a basc soluton n terms o the uson coecent We get an upper estmate o the growth constant o perturbaton A large enough uson coecent gves us a sgncant mprovement o the low stablty Key wors: Proton echange membrane Fuell cells Numercal methos Engenvalue problems INTRODUCTION The water transport n a PEM uel cells (PEMFC) s governe by the mass transer an uson appearng n the Gas Duson Layer (GDL) regon Some prevous an nterestng results are gven n [3] [4] [5] [3] A capllary pressure ests on the nterace gas-lqu an the surace tenson s nvolve In [3] the capllary pressure s gven by Genuchten s uncton escrbe also n [4] The uson process n characterze by the uson coecent η In [4] are gven some methos to n η At the Workshop Moelng an Smulaton o PEM Fuel Cells Freburg Unverstät 6 are stue some uson an transport phenomena n PEMFC In [5] s stue the transport n the catalze layer; n [7] s stue the b-phasc low n comple porous mea; n [] are estmate the eectve usvty coecents The homogenzaton metho s use n [8] to obtan a reuce moel o GDL A trphazc moel s stue n [] All these papers are relate to our approach Some qualtatve an numercal aspects relate to the transer processes PEM are stue n [6]-[7] an [8]-[] In ths paper we conser a quas-one mensonal moel or the GDL The length s much longer compare wth the thckness In [3] at the ens are use perocty contons; then n act our meum can be consere o nnte length n the splacement recton On ths wa we can use the Hele-Shaw moel I splacng lu (gas) s less vscous the nterace wth the splace lu (water) s unstable Between the two lus we conser a Transton Zone (TZ) where the vscosty s an ncreasng uncton We have two nteraces wth two suraces tensons; here we have small jumps o the vscostes In TZ we allow a uson process A basc steay soluton wth straght ntal nteraces s consere We stuy the lnear stablty o ths basc soluton We obtan a qualtatve result concernng the mprovement o the stablty an an upper estmate o the growth constant A large enough uson coecent η gves us an mprove stablty We thanks to proessor G Pascha rom the Insttute o Mathematcs Smon Stolow or useul uscutons concernng the uson moel n the Gas Duson Layer

2 A Capatna H Ene D Polşevsch R Stavre THE HELE-SHAW MODEL In [6] s consere the low o a Stokes lu between two parallel plates at a small stance δ n the plane Oy; Oz s orthogonal on the plane In the ollowng (uvw) p are the velocty an the pressure In our case w = then p s not epenng on z The y ervatves o the velocty are neglecte compare wth the z ervatve Thereore we get the system u z p = v p = z where s the vscosty A no-slp conton s mpose at z= z= δ then δ p δ p u = v = δ u = δ u( z) z v = δ v( z) z The above equaton s obtane by usng Taylor epanson o secon orer or u v The relaton (A) s qute smlar wth the Darcy law whch governs the low n a porous two-mensonal meum wth the permeablty δ / The ltraton velocty s gven by u an v We emphasze that only the permeablty an velocty are ctve whle the vscosty n (A) s the same as n the Darcy law We conclue that n thn plane regons (n our case the oman between two plates) the low o a vscous lu s governe wth a goo appromaton by the Darcy law n a ctve porous meum The uel cells are compose by some thn plane regons where the low an transer processes are perorme The above conseratons allow us to appromate the low n these regons by the Darcy law n the rame o the Hele-Shaw moel δ (A) 3 THE MATHEMATIC MODEL OF DIFFUSION IN GDL In the e plane Oy we conser the low o three lus n a homogeneous porous meum usng the Hele-Shaw moel The meum s saturate wth ollowng three lus: oygen-vapors a mture orme by oygen-vapors an water an the thr part lle wth water We have three vscostes: oygen vapors wth constant vscosty the mture wth varable vscosty an water wth constant vscosty All regons are movng by oygen velocty U ar upstream We have also two nteraces enote by Γ ( ) The low s governe by the Darcy law the contnuty equaton or veloctes an a conservaton law or the vscostes n the mture (here the vscostes replace the concentratons as we mentone above) Let (u v) an p be the velocty an the pressure The temperature s consere constant an we stuy only the hyroynamcal aspects The case o a varable temperature was consere n a prevous stu relate to the same Grant We neglect the asorpton an sperson phenomena Thereore the low s governe by the ollowng equatons: u v p p = = - u = v Γ ( ) Γ ( ); () u v = η ( Γ ( )) = ( Γ ( )) = < ; = < Γ ( ); = > Γ ( ) ( Γ ( ) Γ ( )) (3) (4) On Γ ( ) we use the Laplace law: the pressure jump s balance by the surace tenson multple wth the nterace's curvature The normal component o the velocty on the nteraces Γ ( ) s contnuous Far rom Γ ( ) we have the conton ()

3 3 Flu ynamcs n PEM uel cells u = U v = << Γ ( ) or >> Γ ( ) (5) The ollowng basc soluton (6)--() ests or the problem ()--(5): P P u = U v = = U = v Γ ( ) Γ ( ) (6) ( ) : = Ut l Γ ( ) : = = η ( Γ ( ) Γ ( a( U b ( Γ ( ) Γ ( )) Γ ( ) = Γ ( ) = Γ Ut (7) U )); (8) = (9) (9) The vscosty n the mture s a lnear uncton n terms o ( -U We have two straght materal nteraces then the ollowng conton hols: P s contnous on Γ ( ) () Our task s to stuy the lnear stablty o the basc soluton (6)--() 4 THE STABILITY SYSTEM We conser the perturbatons u' v' p' ' o the basc veloct pressure an vscosty an get the system ( U u') v' = Γ ( ) Γ ( ); ( P p') ( P p') = ( ')( U u') = ( ') v' Γ ( ) Γ ( ) ; ( ') ( ') ( ') ( U u') = η ( Γ ( ) Γ ( )) wth the ollowng bounary contons a) the Laplace's law on Γ ( ); b) u' v' = ar rom Γ ( ); c) ' = on Γ ( ) () We use the moble coornate system ( τ ) : = Ut τ = t thereore or the arbtrary uncton F( y) we get the relaton F F U F = τ However n the ollowng τ s also enote by t In the moble system the perturbatons are governe by the problem u' v' = { - l} () p' = u' ' U { -l} (3) p' = v' { -l} (4) { ' / ' / } ( l) '/ u' / = η (5)

4 A Capatna H Ene D Polşevsch R Stavre 4 = a b ( l) = () = < ( ) (5) l = < l = > l η a b (5) (5v) The parameters o the problem are The basc vscosty s epenng on a b an s Γ contnuous on where the surace tenson s zero Here epens only on but the perturbaton ' s uncton o ( The proo s as ollowng The problem ()--(5) s lnear The rst step s to conser the perturbaton o basc velocty u' ( = ( )ep( α y (6) where α s the wave number n recton orthogonal on the splacng recton O; s the growth constant (n tme) o the perturbatons From ()--(4) we get v' = ep( αy p' = ep( y α α α Cross erentatng the pressure we obtan [ u' ' U ] = [ v' ]; ' = α ep( αy y U α α The above relatons gve us '( = h( )ep( αy ; h( ) = { ( ) } α (7) α U thereore the secon orer partal ervatves o ' are gven by: = h( ) ' ' ep( αy ; = α h( )ep( αy We use (5) an get the ollowng stablty system where (-l ): ( ) α = α Uh; (8) η h ( ηα ) h = α (9) In the ollowng we erve the bounary contons or (8)--(9) The perturbe nteraces are escrbe by the relatons () = g( gt = u' g( = ep( αy ; ( l) = l g( l gt = u' g( l = ep( αy thereore the lmt values o the pressure on the nterace = can be appromate by: P p ( g P( g p'( P( ( g( p'( () p ( ) P() () U ep( αy () () ep( y α α () p ( ) P() () U ep( αy () () ep( y α α Smlar relatons are obtane or lmt values o the pressure on =-l We use the above relatons to obtan the bounary contons or stablty system (8)--(9) as ollows:

5 5 Flu ynamcs n PEM uel cells a) Bounary contons or The vscosty s contnuous on =-l an the surace tenson s zero On = we have the vscosty jump [ () ] an the surace tenson T On =-l = we conser the rst appromaton o Laplace's law p ( ) p () Tg ( ; () p ( l) p ( l) () We erentate the pressure an get the equaton o out o the mture: α = The perturbatons must be zero ar rom the nteraces (that means u'=) thereore the above equaton gves us the epresson o out o the mture: α ( l) ( ) = e ( l) < l; α ( ) = e () > () From here we obtan the ollowng lmt values o : ( l) = α ( l) () = α () We use the above relatons an rom ()--() we get ( l) = α ( l); () 4 ( ) = ( Uα ( ()) α T ) α () = ( eλ q) (); () (3) e = { Uα ( ()) α T} ; λ = ; () (4) α q = () (5) Here ( l) ( l) are the 'rght' lmts n =-l an () () are the 'let' lmts n = b) Bounary contons or h We use (c) an get: h() = h( l) = (6) yy 5 THE APPROXIMATION PROCEDURE We have to estmate the growth constant appearng n (8)--(9) wth bounary contons ()--(6) The system (8)--(9) can be reuce to a sngle ourth orer equaton or the egenuncton But as we can see above we have only two bounary contons or In ths stuaton t s more useul to use two equatons o secon orer both wth two bounary contons but couple In ollowng sectons we obtan an upper estmate or by usng ths proceure In ths secton we erve an appromate orm o our stablty system Conser ( M ) ponts n the segment [-l ]: M = l < M < M < < < = wth the constant scretzaton step = For the lateral ervatves appearng n ()-(3) we use the appromatons q ( ) / = ( eλ q) ; = λ ; (7) e e e ( M M ) / = α M ; M = M ; (8) α wth 4 Uα ( ()) α T α e = q = (9) () ()

6 A Capatna H Ene D Polşevsch R Stavre 6 The values M ; h h hm (the values o unctons n the consere nteror ponts) are unknown an we use the ollowng appromatons h( y / ) h( y / ) h( y ) h( y) h( y ) h( y) = ; h( y) = Two kns o nces wll be use: j k = ( M ); m = ( M ) then (7)--(9) gve us the scretze orm o our problem: A A k A = λ ; (3) k = α Uh ; (3) { B ( α η) } h = a η m m (3) where A an B are two tragonal matrces gven by the ollowng epressons: q A = A = A = > e e e (33) / / / A = A = A = α M (34) n / AM M = n / n / n / AM M = α n α (35) B = B = B = M (36) We get only one equaton or by usng (3)-(3): a h = ηb α η α η m m a Ak k = α Uh = ( α U ) ηb mhm α η α η From (3) we obtan Amk k = α Uhm We replace hm wth the epresson n the last term o the rght han se o the above ormula The nal orm o our scretze system s (recall m M an k M ): A A = λ; (37) Ak k = ( aα U ) ηb m Amk k (38) α η α η The above system contans only the egenuncton Then the use appromaton s useul or avong a ourth orer erental equaton or whch appears n the eact orm There are (M-) equatons an M unknowns The prouct BA s appearng thereore the soluton s not obvous The k culty s gven by the matr A appearng n (38) Ths matr s not quaratc thereore not nvertble The growth constant an the egenvalues λ are comple numbers We use the notaton = λ = λ λ; λ R; λ = h

7 7 Flu ynamcs n PEM uel cells In the ollowng we gve an upper estmate o the real part o the growth constant enote by Conser the relaton (38) n the equvalent orm M A k k = ( aα U ) η Bm k = M m= M ( α η) A ; = M (39) We use the ollowng notaton then (39) becomes M k = k = mk k y = A (4) k k M m= ( α η) y = aα U η B y (4) Let VM = ma{ y = M } Then we have the ollowng possble relatons or or VM α η ηb aα U y m VM = y η B (4) < < s s M α η ηb ( ) ss aα U η Bs s Bs (43) y = ys s s VM m n n = M α η ηbn n aα U η Bn (44) y = yn n n Here α B η are real then we have the nequalty α η ηb j j α η ηb j j In the above equatons (4)--(44) we use the obvous relatons B B = / / ; B B B = / / / = ss s s s s B M M M M B = / / The last relaton an the estmates (4)--(44) gve us the nal upper boun or : s α η aα U ys = ma{ y M } (45) ys where y are gven by the enton (4) Remark The proceure use n ths secton s qute smlar wth the Gerschgorn's localzaton result or the egenvalues o the matr equaton A = λ However n our case the new term y s appearng Remark The estmate (45) allows us the ollowng concluson: a large enough uson coecent can mprove the low stablty Ths means that the growth constant becomes negatve or large uson coecent η REFERENCES BAKER D R BERNING T FELL S GU W WIESER C On the Eectve Dusvty n PEFC Duson Mea Proc o Moelng an Smulaton o PEM Fuel Cells Freburg Unverst German pp BERG P NOVRUZI A VOLKOV O Trple-phase Bounary n PEM Fuel Cell Catalyst layers Proc o Moelng an Smulaton o PEM Fuel Cells Freburg Unverst German pp 5-3 6

8 A Capatna H Ene D Polşevsch R Stavre 8 3 BERG P PROMISLOW K STOKIE J WETTON B Mathematcal Moelng o Water Management n PEM Fuel Cells Techncal Proceengs o the Nanotechnology Conerence an Trae Show Manchester Unversty UK pp5-3 4 BERG P PROMISLOW K Moelng Water Uptake o Proton Echange Membranes Techncal Proceengs o the Nanotechnology Conerence an Trae Show Manchesterr Unverst UK pp BERNING T LU D DJIALI N Three mensonal computatonal analyss o transport phenomena n PEM uel cell Proc Seventh Grave Symposum pp 5-6 CARCADEA E ENE H INGHAM D LAZAR R POURKASHANIAN M STEFANESCU I Numercal smulaton o mass an charge transer or PEM uel cell IntComm Heat an Mass Transer 3 pp CARCADEA E ENE H INGHAM D LAZAR R POURKASHANIAN M STEFANESCU I A computatonal lu ynamcs analyss o a PEM uel cell system or power generaton Int J Numer Math Heat an Flu Flow 7 pp CARCADEA E STEFANESCU I ENE H INGHAM D LAZAR R Computatonal Moel o A PEM Fuel Cell wth Serpentne Gas Flow Channels Progress o Cryogencs an Isotopes Separaton nr 7-8 pp CARCADEA E STEFANESCU I ENE H INGHAM D LAZAR R A 3D transport phenomena analyss or PEM uel cells Proceengs Romanan Fluent Users Meetng Snaa Romana E Prntech pp CARCADEA E STEFANESCU I ENE H INGHAM D LAZAR R A revew o water an heat management or PEM uel cells Conerence on Moelng Flu Flow (CMFF-6) The 3 th Internatonal Conerence on Flu Flow Technologes Buapest Hungar pp CARCADEA E ENE H INGHAM D LAZAR MA L POURKASHANIAN M STEFANESCU I A computatonal lu ynamcs analyss o a PEM uel cell system or power generaton Proceengs Conerence Fluent Users Nothngham Englan pp CARCADEA E ENE H INGHAM D LAZAR R MA L POURKASHANIAN M STEFANESCU I A computatonal lu ynamcs analyss o a PEM uel cell system or power generaton n Progress n Computatonal Heat an Transer Conerence Pars France pp R CRISTOPHER 3 Mathematcal Moelng or a PEM Fuel Cell Scentc Report New Jersey Inttute o Technolog USA 3 4 GENUCHTEN MT A close orm equaton or prectng the hyraulc conuctvty o unsaturate sols Sol Sc Soc o Amer 46 pp GURAU V ZAWODINSKI T MANN J Two-phase transport n PEM uel cells cathoes Proc o Moelng an Smulaton o PEM Fuel Cells Unversty o Freburg German pp LAMB Sr H Hyroynamcs Cambrge Unversty Press UK 93 7 HASSANIZADEH S Two-phase Flow n Comple Porous Mea Proc o Moelng an Smulaton o PEM Fuel Cells Unversty o Freburg German pp MIHAILOVICI M SCHWEIZER B Reuce moels or the cathoe catalyst layer n PEM uel cells Proc o Moelng an Smulaton o PEM Fuel Cells Unversty o Freburg German pp SAFFMAN P TAYLOR Sr G I The penetraton o a lu n a porous meum or Helle-Shaw cell contanng a more vscous lu Proc Roy Soc A 45 pp Receve October 8