Journal of Power Sources

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1 Joural of Power Sources 196 (211) Cotets lists available at ScieceDirect Joural of Power Sources joural homepage: Semiaalytical metho of solutio for soli phase iffusio i lithium io battery electroes: Variable iffusio coefficiet Sihuja Regaatha,1, Ralph E. White Departmet of Chemical Egieerig, Uiversity of South Carolia, Columbia, SC 2928, USA article ifo abstract Article history: Receive 22 Jue 21 Accepte 23 Jue 21 Available olie 3 Jue 21 Keywors: Itegral trasform techique Semiaalytical metho Lithium io battery electroe Noliear iffusio Spherical cooriate A semiaalytical methoology base o the itegral trasform techique is propose to solve the iffusio equatio with cocetratio epeet iffusio coefficiet i a spherical itercalatio electroe particle. The metho makes use of a itegral trasform pair to trasform the oliear partial ifferetial equatio ito a set of oriary ifferetial equatios, which is solve with less computatioal efforts. A geeral solutio proceure is presete a two illustrative examples are use to emostrate the usefuless of this metho for moelig of iffusio process i lithium io battery electroe. The solutios obtaie usig the metho presete i this stuy are compare to the umerical solutios. 21 Elsevier B.V. All rights reserve. 1. Itrouctio Lithium io batteries typically cosist of itercalatio type materials as electroes. Stuies have show that iffusio coefficiet of lithium i the host materials (like carbo for example) is a fuctio of cocetratio or state of charge (SOC) 1 3. Whe the effects of thermoyamic variatios 4 7 or the mechaical stress 7 9 o the iffusio process isie the soli phase are take ito cosieratio, the iffusio equatio becomes oliear. I most of the literature pertaiig to mathematical moelig of lithium io batteries, iffusio isie the soli phase is treate as a liear problem with costat iffusio coefficiet Botte a White usig a carbo base electroe as the moelig system, emostrate the importace of cosierig the oliear effects i the soli phase 5. I spite of several stuies iicatig the importace of icluig the oliear effects for the soli phase of the battery electroe, very few moels have iclue these effects because of the ae complexity 4 9. The use of these moels for the estimatio of parameters a cycle life stuies is limite as it ivolves aitioal computatioal cost. There are certai successful efforts take i the past to simplify the rigorous physics base moels with reasoable accuracy Most of these stuies have bee evelope for liear iffusio equatio with costat coefficiet i the electroe particle. Correspoig author. Tel.: ; fax: aresses: SRegaatha@lbl.gov (S. Regaatha), white@cec.sc.eu (R.E. White). 1 Preset aress: Evirometal Eergy Techologies Divisio, Lawrece Berkeley Natioal Laboratory, Berkeley, CA 9476, USA. The objective of this paper is to exte these efforts a evelop a methoology to solve the oliear iffusio equatio i a spherical itercalatio electroe. The approach use i the preset stuy is base o the fiite itegral trasform techique that has bee previously use to solve oliear bouary value problems i heat trasfer 16,17 a solute trasport i porous meia 18. The attractive feature of this metho is the flexibility to hale most of the oliear equatios a ease of extesio to icorporate oliear bouary coitios a ifferet geometries (icluig cylirical geometry). The geeral methoology is presete i the followig sectio. Two simple illustrative examples are iscusse to emostrate the potetial of this techique as a competitive mathematical tool for aressig oliear iffusio processes i the battery electroe. The Eige fuctio expasio metho presete by Tsag a Hammarstrom 19 is use to further simplify the problem for the two cases cosiere i this stuy. 2. Moel escriptio For the purpose of this stuy the electroe particle is cosiere to be a sphere a the ischarge process i the electroe is escribe usig the sigle particle moel 2. Diffusio of lithium isie the particle is escribe by the followig equatio: C t = 1 r 2 r D eff r 2 C r where D eff is the cocetratio (or SOC) epeet iffusio coefficiet arisig ue to icorporatio of the thermoyamic variatio or icluig the effect of mechaical stress o the iffusio process (1) /$ see frot matter 21 Elsevier B.V. All rights reserve. oi:1.116/j.jpowsour

2 S. Regaatha, R.E. White / Joural of Power Sources 196 (211) Nomeclature C cocetratio of lithium i the soli phase (mol cm 3 ) C imesioless cocetratio of lithium i particle D iffusio coefficiet of lithium i the particle icluig the oliear effects (cm 2 s 1 ) D imesioless iffusio coefficiet of lithium i the particle E Youg s moulus (N cm 2 ) F Faraay s costat, 96,487 (C mol 1 ) i applie curret esity (A cm 2 ) j rate of the electrochemical reactio at the particle surface (mol cm 2 s) K umber of terms cosiere i the expasio M molecular mass (g mol 1 ) umber of electros ivolve i electrochemical reactio. R gas costat, (J mol 1 K) R p raius of the particle (Cm) r raial cooriate (Cm) t time (S) T temperature (K) u isplacemet vector (Cm) V Li partial molar volume of lithium i itercalatio material (cm 3 mol 1 ) V cell potetial (V) Greek Symbols oliear cotributio fuctio i iffusio coefficiet activity coefficiet i the soli phase imesioless flux at the surface of the particle eigevalues Poisso s ratio imesioless cooriate esity of the particle (g cm 3 ) h hyrostatic stress (N cm 2 ) r raial compoet of stress (N cm 2 ) t tagetial compoet of stress (N cm 2 ) kerel of the trasform or eigefuctio imesioless time ω oliear cotributio to the iffusio process Subscripts eff effective k umber of terms l umber of terms max maximum umber of terms r raial s separator t tagetial of lithium ito the host material. At the surface of the electroe particle electrochemical reactio takes place a this ictates the flux of lithium ito the particle. The flux at the surface is give by: C D eff r r=r p = j = i F where j is the rate of the electrochemical reactio at the electroe surface which is a proportioal to local curret esity at (2) the particle surface (i ). is the umber of electro ivolve i the electrochemical reactio a F is the Faraay s costat. The flux of lithium at the ceter of the particle is give by: C D eff r r=o = (3) The iitial coitio is escribe by: C(r, ) = C (4) 3. Solutio proceure 3.1. Dimesioless goverig equatios The system of Eqs. (1) (4) is cast ito a more coveiet form by efiig the followig imesioless variables: = r R p ; = D t Rp 2 ; C = C C (5) C max where R p is the raius of the particle, C is the iitial cocetratio of lithium i the particle, C max is the maximum stoichiometric cocetratio of lithium i the host. D is the iffusio coefficiet of lithium i the particle with the iitial cocetratio of lithium C. The goverig equatio (Eq. (1)) is expresse usig the above imesioless variables as give below: C = 1 2 D ( C ) 2 C, <<1 (6) ( where D C ) = D eff /D. A the bouary coitios a iitial coitio escribe i Eqs. (2) (4) are moifie as follows: D C D C =o =1 = (7) = (8) C ( = ) = (9) where is the imesioless flux of lithium at the surface of the particle a is give by the followig expressio: i R P = (1) FC max D The flux ca be a fuctio of cocetratio or a costat base o the problem Solutio methoology The goverig equatio (Eq. (6)) a the bouary coitios escribe i Eqs. (7) (9) ca be reuce to a set of oriary ifferetial equatios by makig use of the itegral trasform a the iverse trasform pair escribe i the followig sectio. The itegral trasform for the spherical cooriate is give below 17: ( ) T (,) = T () = 2, C ( ), ; = 1...K (11) = where (,) is the kerel of the trasform a s are the eigevalues which are obtaie by solvig a auxiliary homogeous eigevalue problem correspoig to the goverig equatio a bouary coitios. For simplicity (,) is represete as () from this poit owars i the text.

3 444 S. Regaatha, R.E. White / Joural of Power Sources 196 (211) The iverse of the trasform is give by the followig expressio 1 17: C = ( ) K =1 N T () (12) where K is a fiite umber of terms cosiere i the expasio a the orm of the itegral trasformatio, N is give by the followig expressio 17: ( ) N = 2 2 ; = 1...K (13) The homogeeous eigevalue problem correspoig to iffusio i a spherical particle is give by: ( ) () = (14) The homogeeous bouary coitios are expresse as: = ; = (15) =o =1 Solutio of Eq. (14) subjecte to the bouary coitios escribe i Eq. (15) yiels the kerel or the eigefuctios of the system, which are give by: ( ) ( ) si = (16) A the eigevalues are the roots of the followig equatio: cot( ) = 1 (17) (1 + C ) 2 C = { 2 C } { } C + C + S=,1 { S ( C ) 2 C } (22) Makig use of Eqs. (14) (16) a the itegral trasform efie i Eqs. (11) a (22) is rewritte as give below: (1 + C ) 2 C = 2 T () (1) + ( C) 2 C (23) Substitutio of Eq. (23) ito Eq. (2) yiels the followig set of ifferetial equatios: T () + 2 T () = ( C) 2 C (1); = 1...K (24) The iverse trasform escribe i Eq. (12) for the epeet variable C is substitute i Eq. (24) a the resultig equatio ca be rewritte i the matrix form as give below: T() + A(T, ) T() = G(T, ) (25) Usig the trasformatio give i Eq. (12) o the goverig equatio (Eq. (6)) the followig equatio is obtaie: ( ) 1 2 C = D 2 C (18) The expressio o the left-ha sie of the Eq. (18) ca be rewritte usig the trasform escribe i Eq. (12) as show below: 2 () C = 2 () C = T () (19) Substitutio of Eq. (19) ito Eq. (18) yiels the followig moifie goverig equatio: T () = D 2 C ; = 1...K (2) The cocetratio epeet iffusio coefficiet i Eq. (2) ca be expresse as a polyomial expressio i terms of epeet variable ( C) usig simple arithmetic maipulatios for most of the problems ecoutere i battery moelig. I this stuy the followig geeral fuctioal form of the expressio is cosiere: D = 1 + C (21) where ca either be a costat or a fuctio of C. The fuctioal form of the iffusio coefficiet Eq. (21) is substitute i the rightha sie expressio of Eq. (2). The itegral o the right sie of Eq. (2) is evaluate usig Gree s itegral theorem 21 as show below: where the matrices A a G epe o the fuctios a respectively. The set of couple oliear oriary ifferetial equatios (ODEs) escribe i Eq. (25) is solve umerically to obtai the trasform fuctio T () a the cocetratio profile is obtaie usig the iverse fuctio give by Eq. (12). The solutio to the iffusio problem is therefore aalytical with respect to the spatial cooriate, but umerical with respect to time. For certai simple cases complete aalytical or approximate solutio is possible as emostrate i Ref. 19. Two examples are iscusse i the followig sectio alog with a proceure similar to that presete i Ref. 19, to further simplify the complexity of the ifferetial equatios escribe i Eq. (25). 4. Discussio 4.1. Examples: case A Let us cosier the effect of mechaical stress o the iffusio of lithium ito the itercalatio material urig the galvaostatic ischarge process. The itercalatio material is treate as a biary solutio a the iffusio of lithium ito the host material is escribe by 7 9: C t = 1 r 2 r Dr 2 { C r + C RT ( V M ) } h r (26) where V is the partial molar volume of lithium i the host, M is the molecular mass of the biary solutio, is the esity of the solutio a h is the hyrostatic stress or pressure.

4 S. Regaatha, R.E. White / Joural of Power Sources 196 (211) For a isotropic material Eq. (26) is rewritte as follows see Appeix for erivatio: C t = 1 r 2 D eff r 2 C (27) r r The effective iffusio coefficiet is give by the expressio: ( C C ) D eff = D 1 + ω (28) C max where ω is give by the followig expressio: ω = 2 ( VEC max V M ) (29) 9RT(1 ) I Eq. (29), E is the Youg s moulus a is the Poisso s ratio of the material. For the galvaostatic ischarge process the flux at the surface is give by: C D eff r r=r p = i F (3) where i is the applie curret esity, is the umber of electro ivolve i the electrochemical reactio a F is the Faraay s Costat. Use of imesioless variables efie earlier by Eq. (5) reuces the goverig Eq. (27) a bouary coitios to the same form as escribe by Eqs. (6) (9). The iffusio coefficiet for this example epes liearly o the cocetratio of lithium a escribe i Eq. (21) is a costat parameter ( ω). D = 1 + C = 1 + ω C (31) ω is the measure of the cotributio of stress or mechaical eergy towars the iffusio process. The itegral trasform approach escribe i the previous sectio is use to reuce the oliear iffusio equatio to a set of ODEs which is give below: ( ( ) ) 2 k Dl T () = K k=1 l=1 K T k () ; = 1...K (32) As a further simplificatio the cocetratio epeet iffusio coefficiet is expae usig the first eigefuctio. This has bee emostrate earlier for a plaar geometry i Ref. 19: D = 1 + ω C = 1 + ω l ()T l (); l = 1 (33) Fig. 1. Compariso of cocetratio profiles withi the particle obtaie usig the semiaalytical metho (SA) a umerical solutio (N) for =.1. correspoig full umerical solutios obtaie usig the fiite elemet package, COMSOL Multiphysics 23. It ca be observe from Eqs. (27) (31) that the cocetratio profile epes o two importat parameters: (i) the reactio rate or imesioless flux at the surface of the particle () a (ii) the oliear cotributio to iffusio process ( ω). For a give itercalatio material, ω is costat (assumig all mechaical properties are costat). Therefore the flux at the surface of the particle etermies the cocetratio profile withi the particle. The first set of simulatios was carrie out for a low value of ω of.1 a two ifferet values of (.3, 2). For a particle of size 8.5 m a a costat iffusio coefficiet value of cm 2 s 1 9,1 at the iitial cocetratio, the value of =.3 correspos to 1 C rate of ischarge. The solutios are compare at ifferet time urig the ischarge process a the results for the two flux values, are show i Figs. 1 a 2 respectively. It ca be observe that the results agree well with each other of low value of (.3) for all time. At high value of (2), the surface cocetratio varies oticeably urig the begiig of ischarge but the eviatio is reuce towars the e of ischarge. Therefore for a low value of ω, the results preicte by the metho presete i the stuy are vali eve at high rates of ischarge a the percetage of error i preictio is less tha 1%. After substitutio of Eq. (33) ito Eq. (32) a cosierig the first elemet ( = 1), the pricipal iagoal elemets ( = k) of the matrix, the oliear ODEs give by Eq. (32) are simplifie as follows: T () where + ( 2 + B 1)T () = B 1 = ωt 1 () (1) ; 1 + (1)T 1 () = 1...K (34) 2 1 () () (35) The set of equatios escribe i Eq. (34) is oliear with respect to T whe = 1 a liear whe =2...K. The Eq. (34) is solve for the case whe = 1 a the vector B give i Eq. (35) is upate for each time step (). This solutio is the use i the subsequet calculatios of T for =2...K. Hece the set of oliear ODEs ( =1...K) are ecouple to a sigle oliear ODE ( = 1) a a set of liear ODEs ( =2...K). I this example K is 9 a the set of ODEs is solve usig gear s umerical metho i the Mathematical package, Maple 22. The results obtaie are compare to the Fig. 2. Compariso of cocetratio profiles withi the particle obtaie usig the semiaalytical metho (SA) a umerical solutio (N) =2.

5 446 S. Regaatha, R.E. White / Joural of Power Sources 196 (211) Fig. 3. Error i preictio of surface cocetratio relative to the umerical solutio for three ifferet oliear parameter (ω =.1, 1 a 3). Fig. 3 compares the percetage of the relative error i the surface cocetratio for three ifferet values of oliear cotributio ω. Sice the electrochemical reactio at the iterface epes o the cocetratio at the surface of the particle, preictio of it with reasoable accuracy etermies valiity of the approximatio use i this stuy. The values of ω cosiere i this stuy are:.1, 1 a 3. It ca be observe from Fig. 3 that up to a value of = 1.8, the error i surface cocetratio preictios are less tha 5% for all the cases presete i this stuy. This value of correspos to 6 C for the particles cosiere i this stuy. The eviatio of the solutio obtaie from the metho presete i this stuy from the umerical solutio icrease with a icrease i the value of ω. This is expecte as the oliear terms are approximate base o the first eigefuctio. Value of ω epes o the extet of chage i molar volume of the material urig itercalatio ( V). Larger the chage i volume, the larger is the value of ω. Therefore more terms ca be iclue to improve the accuracy of the solutio for materials with larger values of ω. The moel presete here provies a reasoably accurate preictio over a wie rage of parameter values a ca be use with reasoable accuracy for materials with low to meium egree of volume expasio ( ω = 3 beig highest i the preset stuy) Case B I the seco example cosiere i this stuy, lithium a the itercalatio host material (carbo i this preset case) are cosiere as a biary oieal soli solutio. The iterestig a uique ature of the itercalatio process i carbo makes it a ieal caiate for this stuy. Itercalatio of lithium i carbo ivolves stagig. Diffusio coefficiet of lithium i each of the stages varies sigificatly resultig i a cocetratio epeet iffusio coefficiet. Moelig of stagig pheomeo with all the etails is beyo the scope of this stuy, hece oly the approach to iclue the cocetratio epeet iffusio coefficiet is presete below. The thermoyamic variatio ue to the oieal ature of the itercalatio electroe is relate to the iffusio coefficiet by the followig expressio 4 6: D D = 1 + l l C (36) where 1 + ( l / l C) represets the lithium io-io iteractio. The activity coefficiet () is replace by a iteractio potetial term 4 a cocetratio epeet iffusio coeffi- Fig. 4. Compariso of potetial as a fuctio of ischarge time obtaie usig the semiaalytical metho (SA) a the umerical solutio (N) for =1. ciet Eq. (36) is moifie as show below: D = 1 + l l C = 1 + C = 1 + C(1 + C) F V (37) RT C Compariso of Eqs. (37) a (21) shows that for this case is a fuctio of the cocetratio C. The iteractio potetial (V) as a fuctio of cocetratio C use i Eq. (37) is obtaie from Ref. 4 a i Eq. (37) is give by the followig expressio: = (1 + C) F a1 + a 2 C + a 3 C 2 + a 4 C 3 + a 5 C 4 + a 6 C 5 (38) RT The values of the parameters 4 a 1 a 6 respectively are , , 67.56, 171.9, a Similar to the example case A, the cocetratio epeet iffusio coefficiet is expae usig the first Eige fuctio. The first elemet ( = 1) a the iagoal elemets are cosiere. The set of ifferetial equatios is give by: T () ( + 2 1) + B T () = (1) 6 m=1 a mt m 1 1(1) m { (1)T 1 () } ; = 1...K (39) The oliear term B is give below: 6 B 1 = a m T m 1 2 m 1 (1 + 1T 1 ()) () (4) m=1 The set of ODEs i Eq. (39) are solve usig gear s umerical metho i the Mathematical package, Maple usig K = 9. The cocetratio at the surface of the particle is use alog with a liear kietic expressio 4 to calculate the potetial at the iterface. The solutio obtaie usig this approach is compare with umerical solutio for a imesioless flux =.2 (equivalet to 1 C for this simulatio) as a fuctio of imesioless ischarge time (). The results are epicte i Fig. 4. The ifferece i eviatio of results from that preicte usig umerical solutio ecreases with icrease i time a beyo a imesioless ischarge time >.35 the results are i close agreemet. The relative error i surface cocetratio preicte usig the preset approach a that of the umerical solutio are show i Fig. 5 for ifferet rates of ischarge (or ). The relative error is less tha 1.2% for all the parameter values cosiere i this stuy (up to 4 C or a value of 1). With the help of two illustrative cases the usefuless of this metho to moel iffusio process i the itercalatio elec-

6 S. Regaatha, R.E. White / Joural of Power Sources 196 (211) expresse as show below 25: h = r + 2 t (A-2) 3 The graiet of hyrostatic stress isie the particle is expresse as: h r = ( r ) + 2 t (A-3) r 3 where the raial compoet of the stress is give by the followig expressio 25: r = E (1 + )(1 2) (1 ) u r + 2 u V (1 + ) r 3 (C C ) (A-4) The tagetial compoet of the stress is give by 25: E t = u (1 + )(1 2) r + u V (1 + ) r 3 (C C ) (A-5) Fig. 5. Error i preictio of surface cocetratio relative to the umerical solutio for various imesioless flux values at the surface of the particle. troes has bee emostrate. Though these simple examples oly cosier a sigle spherical a isotropic particle as the moel geometry, this methoology ca be extee to materials with aisotropic properties a uergoig phase trasformatio 18,24. This methoology ca be effectively use as a moel reuctio/reformulatio techique for moel ivolvig multiple electroes or battery stacks. 5. Coclusios A simple a straightforwar methoology to solve oliear iffusio equatio i spherical itercalatio electroe is presete. Two ifferet case stuies icorporatig oliear effects o iffusio equatio are presete. The first case cosiere iclues the effect of mechaical stress o the iffusio process a i this case the iffusio coefficiet ha a liear variatio with compositio. The seco case stuy cosiere the oliear effect ue to lithium io-io iteractio withi the soli phase a i this case the iffusio coefficiet was a polyomial fuctio of cocetratio. I both the cases, results obtaie from the semiaalytical approach presete i this stuy were fou to be i goo agreemet with the umerical solutio. The average error i the preictio was fou to be 1 5%. The sigificat ecrease i the computatioal efforts combie with a reasoable egree of accuracy i results obtaie makes the approach presete i this paper a goo alterative for complete umerical solutio. This techique ca also be use as a tool for parameter estimatio. Ackowlegemet The authors are grateful for the fiacial support of this project provie by the Natioal Recoaissace Office (NRO) uer cotract # NRO--3-C-122. Appeix A. Diffusio of lithium ito the host material is escribe by 7 9: { C t = 1 C r 2 Dr 2 r r + C ( V M ) } h (A-1) RT r The hyrostatic stress ( h ) is the average of the three priciple compoets of the stress tesor a for a spherical particle it ca be where C is the stress free cocetratio i the electroe. The graiet of the hyrostatic stress isie the particle is moifie usig Eqs. (A-4) a (A-5) as follows: h r = E 3(1 2) 1 r 2 r ( r 2 u r ) V C r (A-6) The equatio of mometum isie the spherical particle which is i mechaical equilibrium is as follows 22: r r + 2 r ( r t) = (A-7) I terms of the raial isplacemet u, Eq. (A-7) ca be re-writte usig the Eqs. (A-4) a (A-5) as follows: 1 r 2 r ( r 2 u r ) = 1 + V C 1 3 r (A-8) The graiet of the hyrostatic stress isie the particle ca be expresse usig Eqs. (A-6) a (A-8) as follows: h r = 2 VE C 9(1 ) r Therefore the iffusio equatio is moifie as give below: C t = 1 r 2 r D eff r 2 C r (A-9) (A-1) The effective iffusio coefficiet is give by the followig expressio: D eff = D 1 2 ( VEC max V M )( C C ) (A-11) 9RT(1 ) C max Refereces 1 J.M. Tarasco, D. Guyomar, J. Electrochem. Soc. 139 (1993) M. Morita, N. Hishimura, Y. Matsua, Electrochim. Acta 38 (1993) (1721). 3 M.D. Levi, D. Aurbach, J. Phys. Chem. B 11 (1997) M.W. Verbrugge, B.J. Koch, J. Electrochem. Soc. 146 (1999) G.G. Botte, R.E. White, J. Electrochem. Soc. 148 (21) A54. 6 D.K. Karthikeya, G. Sikha, R.E. White, J. Power Sources 185 (28) J. Christese, J. Newma, J. Soli State Electrochem. 1 (26) J. Christese, J. Newma, J. Electrochem. Soc. 153 (26) A S. Regaatha, G. Sikha, S. Sathaagopala, R.E. White, J. Electrochem. Soc. 157 (21) A P. Ramaass, Ph.D. Thesis, Uiversity of South Carolia, Q. Zhag, R.E. White, J. Electrochem. Soc. 154 (27) A M. Doyle, T.F. Fuller, J. Newma, J. Electrochem. Soc. 14 (1993) V.R. Subramaia, R.E. White, J. Power Sources 96 (21) V.R. Subramaia, R.E. White, J. Electrochem. Soc. 148 (21) E W.B. Gu, C.Y. Wag, B.Y. Liaw, J. Electrochem. Soc. 145 (1998) M.B. Ab-el-Malek, M.M. Helal, J. Comp. Appl. Math. 193 (26) M.N. Ozisik, Bouary Value Problems of Heat Couctio, Dover Publicatios Ic., NY, 22.

7 448 S. Regaatha, R.E. White / Joural of Power Sources 196 (211) C. Liu, J.E. Szecsoy, J.M. Zachara, W.P. Ball, Av. Water Res. 23 (2) T. Tsag, C.A. Hammarstrom, I. Eg. Chem. Res. 26 (1987) B.S. Hara, B.N. Popov, R.E. White, J. Power Sources 75 (1998) J.C. Slattery, Avace Trasport Pheomea, Cambrige Uiversity Press, NY, Maple, Maplesoft software, available from: 23 COMSOL Multiphysics simulatio package, available from: comsol.com. 24 M.S. Selim, R.C. Seagrave, I. Eg. Chem. Fuam. 12 (1973) S. Timosheko, J.N. Gooier, Theory of Elasticity, 2 e., McGraw-Hill Book Compay Ic., NY, 1951.