2D analytical model of a direct methanol fuel cell

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1 Electrocheistr Counications 6 (4) D analtical oel of a irect ethanol fuel cell Hang Guo a,b, *, Chong-fang Ma a,b a School of Energ an Power Engineering, XiÕan Jiaotong Universit, XiÕan 749, China b The Ke Laborator of Enhance Heat Transfer an Energ Conservation, College of Environental an Energ Engineering, Beijing Universit of Technolog, o. Pingleuan, Chaoang District, Beijing, China Receive 6 Deceber 3; receive in revise for 5 Januar 4; cepte 5 Januar 4 Publishe online: 8 Januar 4 Abstrt Analtical oel is a valuable tool for unerstaning the effect of basic paraeters on fuel cell perforance. In this paper, twoiensional (D) full cell oel of the irect ethanol fuel cell was evelope to escribe not onl electrocheical retions on the anoe an cathoe electroes, but also transport phenoena in fuel cell. Those phenoena inclue ethanol crossover, iffusion of retants in porous eia laers, flui flow in retants istributor, etc. The analtical solution of the oel was erive. The theoretical preiction of the analtical solution was in agreeent with the eriental ata. The D analtical oel provie ceptable cur to preict the perforance of irect ethanol fuel cells. With the ai of the analtical solution, it was foun that the ethanol concentration ecreases alost linearl along the flow irection in the anoe channel. Methanol concentration istribution in the anoe catalst laer showe siilar variation. The perforance of fuel cell at high current ensit will benefit fro enhancing ass transfer of the ethanol fro the anoe channel to the catalst laer. Ó 4 Elsevier B.V. All rights reserve. Kewors: Direct ethanol fuel cells; D oel; Analtical solution; Polarization curve; Concentration istribution. Introuction Direct ethanol fuel cells (DMFC) have rawn increasing attention in recent ears an are consiere as a proising choice of priar or auxiliar power evice in the next generation for stationar, obile, portable an urban transportation sstes. In recent ears, ost research efforts of DMFC focuse on eriental investigations an nuerical stuies, however, analtical oels of DMFC are still lking in the literature. B now, just a one-iensional (D) full cell analtical oel of DMFC was publishe []. Consiering its vantage in unerstaning the effect of basic paraeters on cell perforance, a two-iensional (D) full cell analtical oel of DMFC was evelope to ress the paraetric analsis an preiction in this paper. * Corresponing author. Tel.: ; fax: E-ail ress: hangguo@sohu.co (H. Guo).. The oel.. Assuptions The sste of coorinates for the oel is sketche in Fig.. The ajor assuptions opte in the oel are as follows.. The flui flow in the fuel cell is consiere as ste.. The cell teperature is assue to be unifor an the electrocheical retion occurs uner the conition of constant teperature. 3. Due to the thickness of the catalst laer is two orers of agnitue saller than its size along the channel irection, transport of retants in the catalst laer along the -irection is ignore. 4. Due to the thickness of the iffusion laer is one orer of agnitue larger than that of catalst laer, the retants concentration change ross the catalst laer is not consiere /$ - see front atter Ó 4 Elsevier B.V. All rights reserve. oi:.6/j.eleco.4..5

2 H. Guo, C.-F. Ma / Electrocheistr Counications 6 (4) oenclature b with (c) c olar concentration (ol c 3 ) D iffusion coefficient (c s ) E electrootive force (V) F FaraÕs constant (96,487 A s ol ) J current ensit (A c ) j current ensit per unit volue (A c 3 ) l length (c) M olecular weight (g ol ) n electro-osotic rag coefficient P pressure (Pa) R universal gas constant (8.34 J ol K ) T teperature (K) V cell voltage (V) _V volue flow rate (c 3 in ) v velocit (c s ) x olar frtion (ol ol ); horizontal coorinate () longituinal coorinate () Greek sbols a transfer coefficient v nuber of flow channels thickness or epth (c) g overpotential (V) r proton conuctivit (S c ) Superscript H O water MEOH ethanol O oxgen stanar conition Subscript a anoe anoe catalst laer anoe iffusion laer ah anoe flow channel inlet of anoe flow channel c cathoe cc cathoe catalst laer cathoe iffusion laer cell fuel cell ch cathoe flow channel inlet of cathoe flow channel proton exchange ebrane erence state Anoe flow channel Anoe iffusion laer Anoe catalst laer Proton exchange ebrane Cathoe catalst laer Cathoe iffusion laer Cathoe flow channel x Fig.. Scheatic iagra of the coorinate sste. 5. Due to the thickness of the iffusion laer is uch saller than its length along the channel irection, iffusion of retants in this laer along the -irection is neglecte. 6. Retant iffusion in ebrane along the -irection is not consiere because the thickness of the ebrane is uch saller than its length along the channel irection. 7. Due to the epth of the channel is over one orer of agnitue saller than its length, the retants concentration change through over the epth of the channel is assue to be zero. 8. Pressure rop in the flow channels is ignore. 9. The velocit of the flui flow in channel is assue to be constant an unifor.. The variation of overpotential ross the catalst laer is ver sall. The overpotential is also assue to be constant along the irection of channel... Basic equations The retion equation of DMFC is CH 3 OH þ 3 O H O þ CO ðþ The ethanol concentration in anoe flow channel,, is escribe b following equation: ah ah ah v ah ¼ ; ðþ where is ethanol flux fro the anoe channel to the iffusion laer. It can be eterine b the FickÕs law ¼ o ox ð3þ The anoe retion is CH 3 OH þ H O 6e þ 6H þ þ CO ð4þ Accoring to above equation, consuption of ol ethanol can prouce 6 ol electrons. On the other

3 38 H. Guo, C. Ma / Electrocheistr Counications 6 (4) 36 3 han, the ethanol crossover les to a part of ethanol pereate through proton exchange ebrane (PEM) to cathoe. Thus, ¼ 6F J a þ ; ð5þ where J a is current ensit J a ¼ J MEOH ca a af RT g a ; ð6þ where g a is anoe overpotential. In (5), is ethanol crossover flux, an can be calculate b following equation: ¼ o þ n MEOH J ox F : ð7þ In cathoe catalst laer, one olecule of oxgen, four electrons an four protons cobine an prouce two olecules of water: 3 O þ 6e þ 6H þ 3H O ð8þ So,, the oxgen flux fro the cathoe iffusion laer to cathoe catalst laer coul be eterine: ¼ 4F J c þ 3 ; ð9þ where J c is current ensit J c ¼ J O c O cc cc c F O RT g c ; ðþ where g c is cathoe overpotential. In cathoe iffusion laer, the FickÕs law can be written as ¼ ch o ox : ðþ The oxgen concentration in cathoe flow channel,, is escribe b following equation: ch ch v ch ¼ : ðþ Finall, V, the cell voltage is eterine b following equation: V ¼ E cell g c g a J JR contt ; ð3þ r where J is average current ensit of the cell, is thickness of PEM, r is ionic conuctivit of PEM, is contt resistance, E cell is ieal electrootive R contt force of the cell: E cell ¼ E cell þðt T Þ oe cell ot Dn RT F ln P ð4þ The eanings of the other sbols in above equations are liste in the oenclature. P : 3. Analtical solutions Assuing that the ethanol concentration in the PEM changes linearl because of ver thin thickness of PEM an neglecting ethanol concentration in the cathoe catalst laer as copare to that in anoe catalst laer, Eq. (7) can be reuce as ¼ þ n MEOH J F : ð5þ In orer to siplif the analsis, let the average values substitute the local values in (5), we have ¼ þ n MEOH J F ; ð6þ where the average values of ethanol concentration an current ensit were obtaine b averaging local values over the channel length. Substituting (9) into () an integrating, we obtain the oxgen concentration in the cathoe catalst laer cc ¼ co ch 4F J c : ð7þ Letting the cathoe retion orer, c c ¼ [] an substituting () into (7), we can write cc ¼ þ n ch ð þ n Þ where iensionless paraeter n ¼ 4F J O ; ð8þ a cf RT g c : ð9þ Using (9), (), () an (8), the following equation can be erive: ch ¼ n ch þ 3 n ; ðþ where iensionless paraeter n ¼ J O a cf ch v ch 4F ð þ n Þ RT g c : ðþ Solving () with the bounar conition ch j ¼ ¼ co the oxgen concentration in the cathoe channel can be obtaine ch ¼ co þ 3 n : ðþ Substituting () into (8), the oxgen concentration in the cathoe catalst laer can be erive

4 H. Guo, C.-F. Ma / Electrocheistr Counications 6 (4) cc ¼ þ n þ 3 n ð3þ Substituting into () with assuption of first orer retion an integrating the average current ensit is written as Z lch J ¼ J c l " ch ¼ J O a # " cf RT g c ðn Þ ð þ n Þn þ 3 #: ð4þ Siilarl, the following ression can be erive: The ethanol concentration in the anoe channel ah ¼ þ n 4 n 3 l ah n 3 : ð5þ The istribution of ethanol concentration in the anoe catalst laer ¼ þ þ n 3 n 3 n 4 : ð6þ l ah n 3 The ean of ethanol concentration in the anoe catalst laer ¼ ðn 4Þ þ ðn 4 Þ ð þ n 3 Þn 4 n 5 n 3 n 5 ð þ n 3 Þn 4 n MEOH J: F The average current ensit J ¼ Z lah J a l ah ¼ J MEOH a af RT g a þ þ n 3 n 3 ðn 4Þ n 4 n 3 where iensionless paraeters n 3 ¼ J MEOH 6F a af RT g a ð7þ ; ð8þ ; ð9þ l ah n n 4 ¼ 3 ; ð3þ ah v ah þ n 3 n 5 ¼ ðn 4 Þ : ð3þ n 3 ð þ n 3 Þn 4 Using (6), (7) an (8) the following equation can be erive to escribe relationship between anoe overpotential an current ensit: nmeoh ðn 4 Þ J n 3 F 6 n 5 ð þ n 3 Þn 4 ðn 4Þ ¼ : ð3þ ð þ n 3 Þn 4 n 5 Using (6), (4) an (7) the relationship between cathoe overpotential an current ensit can be written as n F 4 ðn Þ n 5 ð þ n Þn n MEOH ðn Þ ð þ n Þn ¼ 3 ðn 4Þ ð þ n 3 Þn 4 n 5 J ðn Þ ð þ n Þn : ð33þ The analtical solutions for the anoe an cathoe overpotential are given in (3) an (33), respectivel. Cobining the with (3) an (4) analtical solution for the cell voltage an current ensit response can be obtaine. 4. Results an iscussion 4.. Valiation of the analtical solution The eriental ata publishe in [3] were use to valiate the analtical solution. Most of paraeters in ressions of the analtical solution coul be foun in literatures, but a few nee to be assue (Table ). To obtain the assue paraeters, the fitting of the polarization curve for ethanol concentration of.5 M was perfore using trial an error etho. The fitting result is shown in Fig.. Once eterine, those paraeters, which are liste in Table, are not change in valiation cases. The onl ifference between fitting case an valiation cases was inlet concentration of the ethanol solution. In valiation cases, the ethanol concentration is.5,.375,.5 an.65 M, respectivel. Results of valiation are also shown in Fig.. Those results showe that the analtical oel agrees quite well with eriental ata except that the preicte precision of the D analtical oel is unsatisftor in the region of tivation polarization. Fig. 3 shows anoic an cathoic overpotential curves preicte b D analtical oel uner sae

5 3 H. Guo, C. Ma / Electrocheistr Counications 6 (4) 36 3 Table Paraeters Paraeter/Sbol (unit) Value Ref. Teperature of the fuel cell/t (K) [3] Pressure of air in cathoe/p (Pa) [3] Length of anoe channel/l ah (c) 3. [3] Length of cathoe channel/ (c) 3. [3] uber of anoe channels/v ah [3] uber of cathoe channels/v ch [3] With of anoe channel/b ah (c). [3] With of cathoe channel/b ch (c). [3] Depth of anoe channel/ ah (c). [4] Depth of cathoe channel/ ch (c). [4] Inlet flow rate of anoe/ _V ah (c 3 in ). [3] Inlet flow rate of cathoe/ _V ch (c 3 in ) 8 [3] Velocit of flui in anoe channel/v ah (c s ) v ah ¼ _V ah =ð6v ah b ah ah Þ Velocit of flui in cathoe channel/v ch (c s ) v ch ¼ _V ch =ð6v ch b ch ch Þ Methanol concentration at anoe inlet/ (ol c 3 ).5 3 [3] Oxgen concentration at cathoe inlet/c O (ol c 3 ) c O ¼ : 6 P=ðRT Þ [3] Thickness of anoe iffusion laer/ (c).3 [3] Thickness of cathoe iffusion laer/ (c).3 [3] Thickness of anoe catalst laer/ (c).5 Assue Thickness of cathoe catalst laer/ cc (c).3 Assue Thickness of ebrane/ (c).6 [5] Electro-osotic rag coefficient of water/n H O :9 ð9ð ÞÞ 333 T Fitte fro [6] Electro-osotic rag coefficient of ethanol/n MEOH n MEOH ¼ x MEOH n HO [] Conuctivit of ebrane/r (S c ) :73 ð68ð ÞÞ 98 T [5] (c s ) Diffusion coefficient of ethanol in ebrane/ Þ 333 T 4:9 6 ð [5] Diffusion coefficient of ethanol in anoe iffusion laer/ (c s ) ¼ e HO [5] Diffusion coefficient of ethanol in water/ HO (c s ) :8 5 ð T Correction ftor of iffusion coefficient/e.55 Assue Diffusion coefficient of oxgen in cathoe iffusion laer/ (c s ) 3 Assue Ieal electrootive force of the fuel cell uner stanar conition/ecell (V).3 [7] Rate of change of electrootive force/ oe cell (V K ) :4 4 [5] ot Contt resistance/r contt (Xc ).8 Assue Transfer coefficient of anoe/a a.8 Assue Transfer coefficient of cathoe/a c.8 Assue Reference concentration of ethanol/ (ol c 3 ) 3 [5] Reference concentration of oxgen/ (ol c 3 ) ¼ : 6 P=ðRT Þ [5] Reference exchange current ensit of anoe/j MEOH (A c ) J MEOH ¼ j a [] Reference exchange current ensit per unit volue of anoe/j a (A c 3 ) : [] Reference exchange current ensit of cathoe/j O (A c ) J O ¼ cc j c [] Reference exchange current ensit per unit volue of cathoe/j c (A c 3 ) : [] conitions for Fig.. The overpotential curves inicate that the anoe sie of the fuel cell liits the current ensit. That is wh ethanol concentration significantl influences liiting current ensit in Fig Minial inlet flow rate of oxgen Eq. (3) showe that when the increases, the oxgen concentration in cathoe catalst laer, cc, onotonousl ecreases. The inial concentration can be obtaine at ¼. So, we have cc iniu ¼ þ n þ 3 ðn Þ : ð34þ Cell voltage / V M.5M.5M.5M.375M.375M.5M.5M.65M.65M Current ensit / A c - Fig.. Experiental ata [3] (ots) an polarization curves of D analtical oel (lines).

6 H. Guo, C.-F. Ma / Electrocheistr Counications 6 (4) Overpotential / V Current ensit / A c - Using (9), (), an (34), the following equation can be erive: cc iniu ¼ þ n þ 3.5M, anoe.5m, cathoe.65m, anoe.65m, cathoe Fig. 3. Overpotential curves of D analtical oel. v ch ch n þ n ð35þ Eq. (35) shows that ue to ethanol crossover oxgen concentration in the catalst laer coul be zero. When the inlet velocit of oxgen, v ch, ecreases, the ð cc Þ iniu ecreases, too. The inial inlet flow rate of oxgen can be eterine b ð cc Þ iniu ¼. D ð _V O ch Þ iniu ¼ 6v ch b ch ch ch 3ðþn n Þ ln A þ n þ : ð36þ c / M x /.4.5 c / M to.5.38 to to.38.5 to c.3 / M..9 to.5.3 to.9.6 to.3. to / Fig. 4. D istribution of the ethanol concentration in the ebrane electroe assebl of a self e DMFC with 5 c tive area. The sste of coorinates is given in Fig.. x ¼ :.3 : anoe iffusion laer; x ¼ :3.33 : anoe catalst laer; x ¼ : : proton exchange ebrane. Cell voltage / V Experiental ata,.5m Analtical solution,.5m Current ensit / A c - Fig. 5. Polarization curve of analtical solution an eriental ata [8]. Methanol concentration:.5 M Methanol concentration istribution In ost erients, it is quite ifficult to easure the retants istribution. The D analtical oel provies a tool for loring the D istribution of retants in the fuel cell or D istribution along the flow irection in channel. Fig. 4 shows D concentration istribution of ethanol in the ebrane electroe assebl of a self e DMFC with 5 c tive area. Before plotting that figure, the preicte curve was fitte with the eriental ata [8] to eterine paraeters of the analtical oel (Fig. 5). Although the change of ethanol concentration along the channel is nonlinear coring to Eq. (5), Fig. 6 shows that the concentration of ethanol ecreases alost linearl along the flow irection in the anoe channel. This near-linear regular pattern was Methanol concentration / M / Fig. 6. Axial profile of the ethanol concentration in the anoe channel of a self e DMFC at 5 A c (soli line) an A c (ashe line).

7 3 H. Guo, C. Ma / Electrocheistr Counications 6 (4) 36 3 Methanol concentration / M siilar to the results fro coputational flui naics etho []. Methanol concentration istribution in the anoe catalst laer showe siilar variation (Fig. 7). Fig. 6 shows that in anoe channel, the arithetic ean value of the ethanol concentration at A c was just less % than that at 5 A c. But in anoe catalst laer, that average value at A c was lower 84% than that at 5 A c (Fig. 7). The ethanol solution in anoe catalst laer was ver ilute at 5 A c. It eans that concentration polarization has alre occurre. The voltage current ensit curve (Fig. 5) can help us to unerstan this result clearl. In this case, if the retant transfer fro the flow channel to catalst laer can be enhance, the perforance of the fuel cell will be iprove. 5. Conclusions / Fig. 7. Distribution along the channel irection of ethanol concentration in the anoe catalst laer of a self e DMFC at 5 A c (soli line) an A c (ashe line).. A D full cell oel of the irect ethanol fuel cell was establishe to escribe not onl electrocheical retions on the anoe an cathoe electroes, but also ethanol crossover, flui flow in channels, ass transport of retants in iffusion laers, etc.. The analtical solution of the oel was erive. It can be use to preict the cell voltage vs. current ensit response of the fuel cell. The theoretical preiction of the analtical solution was in agreeent with the eriental ata. 3. The oel can be use to analze an preict paraeters of the fuel cell. With the ai of the analtical solution, it was foun that the ethanol concentration ecreases alost linearl along the flow irection in the anoe channel. Methanol concentration istribution in the anoe catalst laer showe siilar variation. If ass transfer of the ethanol fro the anoe channel to the anoe catalst surfe can be enhance, the perforance of the fuel cell will be iprove at high current ensit. Further analsis will be reporte subsequentl. Acknowlegeents The authors are grateful to the ational atural Science Founation of China (Grant os. 536, 5865), Sino-Geran Center for Science Prootion of DFG an SFC (Grant os. GZ7(/7)) an the Youthful Researcher Founation of Beijing Universit of Technolog (Grant o. Z54) for the financial support an Prof. C.Y. WAG (Pennslvania State Universit, USA) for the helpful iscussion. References [] A.A. Kulikovsk, Electroche. Coun. 4 () 939. [] Z.H. Wang, C.Y. Wang, J. Electroche. Soc. 5 (3) A58. [3] K. Scott, W.M. Taaa, S. Kraer, P. Argropoulos, K. Sunher, Electrochi. Acta 45 (999) 945. [4] K. Scott, W.M. Taaa, P. Argropoulos, J. Power Sources 79 (999) 43. [5] K. Scott, W. Taaa, J. Cruickshank, J. Power Sources 65 (997) 59. [6] X.M. Ren, W. Henerson, S. Gottesfel, J. Electroche. Soc. 44 (997) L67. [7] O. Zerbinati, A. Maran, M.M. Richter, J. Che. Euc. 79 () 89. [8] H. Guo, C.F. Ma, M.H. Wang, F. Ye, Y. Wang, J. Yu, C.Y. Wang, J. Eng. Therophs. 5 (4) 48.