MASS DIFFUSION INSIDE PROLATE SPHERICAL SOLIDS: AN ANALYTICAL SOLUTION

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1 Revista Basileia e Poutos Agoiustiais, Campia Gae, v.4,., p.4-5, 4 ASS DIFFUSION INSIDE PROATE SPHERICA SOIDS: AN ANAYTICA SOUTION Vital Aaújo Babosa e Oliveia, Atoio Gilso Babosa e ima ABSTRACT The aaltial solutio of the tasiet mass iffusio equatio i polate spheial ooiates b osieig ostat taspot oeffiiet a ovetive boua oitios is pesete. The solutio is obtaie b the vaiables sepaatio metho. The fomal solutio is applie to peit the aveage moistue otet a moistue otet istibutio of a polate spheial soli (ellipsoi of evolutio) uig the ig poess. Aaltial esults ae ompae with umeial esults that ae epote i the liteatue a goo ageemet was obtaie. Kewos: ig, fomal solutio, mass, iffusio, elliptial geomet DIFUSÃO DE ASSA NO INTERIOR DE SÓIDOS ESFEROIDAIS PROATOS: UA SOUÇÃO ANAÍTICA RESUO A solução aalítia a equação e ifusão e massa em ooeaas esfeoiais polata osieao oefiiete e ifusão ostate e oição e otoo ovetiva é apesetaa. A solução é obtia usao o métoo a sepaação e vaiáveis. A metoologia é apliaa paa peie o teo e umiae méio e a istibuição o teo e umiae, e um sólio esfeoial polato (elipsóie e evolução), uate o poesso e seagem. Resultaos aalítios são ompaaos om esultaos uméios epotaos a liteatua e uma boa ooâia foi obtia. Palavas-have: seagem, solução exata, massa, ifusão, geometia elíptia INTRODUCTION The fomal solutio of the iffusio equatio has bee obtaie fom vaious boua oitios with ostat o vaiable iffusio oeffiiet, i homogeeous o heteogeeous a isotopi o asotopi boies, a i stea o ustea ases. The patial iffeetial equatio fo o stea-state mass iffusio has bee solve to mass tasfe i boies with sigle geomet, like plates, lies a sphees (uikov, 968; Skella, 974 a Cak, 99). Nomito s a Blakwell s (964), Haji-Sheikh s & Spaow (966), Alassa s (999) a imass et al. (999) woks a be ite, fo example, to omplex geomet a ostat boua oitios. Nomito & Blakwell (964) peset a aaltial solutio to peit the heat flow i the half-spae aou of a isothemal thi iula isk.. este em Egehaia eâia, Depatameto e Egehaia eâia, CCT, Uivesiae Feeal a Paaíba (UFPB), CEP , Cx. Postal 69, Campia Gae-PB, Basil. Foe (8) -7. Pofesso Douto o Depatameto e Egehaia eâia, CCT, Uivesiae Feeal a Paaíba (UFPB), CEP , Cx. Postal 69, Campia Gae-PB, Basil. Foe (8) -7, gilso@em.ufpb.b

2 4 Haji-Sheikh & Spaow (966) gave a aaltial solutio to the heat tasfe equatio i a polate spheoi bo with ostat tempeatue at the sufae, usig a elliptial ooiate sstem i two-imesioal ases, but the esults of the tempeatue i the ete a foal poit ae give aloe. ima et al. (999), pesete a aaltial solutio to peit the mass tasfe isie a polate spheoi. The osiee ostat popeties a equilibium boua oitios at the soli sufae. As appliatio, esults of the moistue otet istibutio isie of soli as well as of the aveage moistue otet fo a aspet atio is pesete. The objetive of this wok is to evelop a aaltial solutio to esibe the moistue taspot i a otiuous meium, b utiliig the polate spheoi ooiate sstem i twoimesioal ases, osieig ovetive boua oitios at the sufae of the soli. ATHEATICA ODE This mass iffusio equatio i the shot fom is give b: t D () whee D is the iffusio oeffiiet, is the moistue otet a t is the time. Depeig o the geometial fom of the bo, a ooiate sstem, aequate to esibe the omai i stu, must be selete. I the speifi ase of ellipsoi of evolutio, the aequate oe is the polate spheial sstem. The elatios betwee the atesia (x,, ) a the polate spheial (,, ) o-oiate sstems ae give b Haji-Sheikh & Spaow (966): x= sih si os = sih si si (a-) = osh os whee is the foal legth equal to ( - ) /. A ellipsoi of evolutio sheme is show i Figue. x Figue - Chaateistis of a polate spheial soli Defiig =osh, =os a = os, the metis oeffiiet a the aplaia to the ew ooiate sstem a be obtaie usig the mathematial elatios whih ae give b Abamowit & Stegu (97). Utiliig the metis oeffiiets, the vaiables, a a the iffeetiatio s ules, the mass iffusio equatio a be witte: t D D () Fo a situatio with smmet aou the axis, it is: t D D D (4) Aoig to Figue, the = o (ostat), o> is a elogate ellipsoi of evolutio with bigge axis of legth a smalle legth axis ( -) /. The sufaes ostats ae a polate spheois ofoal famil a the have thei Revista Basileia e Poutos Agoiustiais, Campia Gae, v.4,., p.4-5,

3 4 ommo ete at the oigi. The egeeate sufae = is the uve that liks the ete (=) to the foal poit (=). The sufae = o (ostat), o<, is a asmptoti oe whose two sheets hpeboloi of evolutio geeatig b lie passes though the oigi a it is ilie at the agle =os - to the -axis. The egeeate sufae = is pat of the axis >. The iitial a boua oitios of equatio (4) ae: (,,)= o = ostat (5a) D h m f,, t f e (5b) whee f= / at the sufae of the soli, h m is the mass tasfe oeffiiet a e is the equilibium moistue otet. Usig the sepaatio of vaiables solutio metho we a wite (,,t)= (, ) (t). The solutio of the equatio (4) is the, (Haji-Sheikh a Spaow, 966): = (, )exp(- Dt/ ) (6) whee is ostat. Assumig that the iffusio oeffiiet is ostat a applig the Equatio (6) to Equatio (4), we have: whih a be witte i a shot fom as follows: (7) (8) Assumig that (, )= ( ). ( ), puttig it ito Equatio (8), a sepaatig the vaiables, two oia iffeetial equatio ae obtaie: b b (9) () I Equatios (9) a (), b is the sepaatio ostat o eigevalues. These two equatios ae exatl i the same fom, as a futio of, vaies betwee a the sigula poit +, while as a futio of, vaies betwee the sigula poit + a /. The solutio of the agula futio ( ) is expesse i tems of a egee futio seies of the fist ki (P ), while the aial futio ( ) is obtaie fom a spheial Bessel futios seies of the fist ki of oe (j ). The solutio of the Equatios (9) a () ae give b: ostat ostat Rotatio axis to agle Figue Polate spheial ooiate sstem Revista Basileia e Poutos Agoiustiais, Campia Gae, v.4,., p.4-5,

4 44 m (, ).. j m, m, m (, ) ( ). P ( ) m, m m b m m (a-b),,, () whee: ; 5 with =,, ; (a-) The b values ae give b the taseetal equatios as follows: U (b ) = U (b ) + U (b ) = (4) with: U b b U a b b 4 ; b 4 b... (5a) 4 4 b... (5b) The tehique utilie i the Equatio (4) to etemie the b oeffiiets is alle otiue fatio tehique (Statto et al., 94; Statto et al., 956). This tehique has bee use to etemie the eigevalues to 8.. Whe., the eigevalues ae obtaie though a asmptoti expasio. The asmptoti evelopmet of b is ; give b the suessive appoximatios metho, as follows: b whee O ( -6 ) epesets the eo oe O 6 (6) A oveget seies fo,m a be obtaie to a isete set of values of the eigevalues b. Thee ae two sets of fiite solutios, oe fo eve values of, the othe fo o values. The lowest value of b oespos to =, the ext to =, (ose & Feshbah, 95), so, the set that oespos to eve value of was use i this wok. The values of the oeffiiets,m ae iffeet, epeig o the omaliatio aopte sheme. The utilie iteio b the authos is pesete below:!!!!, (7)!! m fo =,,... a =,,.... The Equatio (7) togethe with the Equatios (6) o (), allows the omplete etemiatio of the oeffiiets,m. The iex is ito all the ases. We have P ( )= Fo < iiatig that the seies eall begi at =. The oitio that estits the values of the b i the iffeetial equatios is eflete i Equatio () as a equiemet that the atio of the oeffiiets,m / -,m, whe (Statto et al.,956). Obsevig that the oeffiiets, b a, the must be obtaie a satisf the Equatio (5b) at the sufae of the polate spheial ( = /). This oitio is give b: f m m (8) Bi / With the etemie,m oeffiiets, the fomal solutio of the poblem is give b: f Revista Basileia e Poutos Agoiustiais, Campia Gae, v.4,., p.4-5,

5 45,, t e A mk Dt m, k e mk,, (9) m mk m mk A mk t e mk m, k o e m mk, m mk, (5) The oeffiiets A mk ae obtaie fom the othogoal oitios. Substitutig the iitial oitio Equatio (5a) i Eq. (9), it is obtaie: A,, o e mk m mk m mk m, k () ultiplig both sies of Equatio () b p( pk, ) p ( pk, )( - ) a itegatig i a quate of the ellipsoi volume, it is obtaie:,, m, k p pk p pk o e p pk, Amk m mk, m mk, () whee the itegatio a the sum opeatios wee exhage. Cosieig that the itegatio i Equatio () a be mae tem b tem, a the othogoalit of the futios, the uique tem i the ight sie that suppl a itegal that s iffeet fom eo, is the tem with m=p. Fo m=p, the esult is: A mk p m mk, m mk, o e,, m mk m mk pk, () whee the eomiato is the om of ( m m )( - ). Defiig the followig imesioless paametes: * e ; o e Dt t ; Bi hm D () It a be wite the Equatio (8) as follows A mk mkt e o e m m, k, (4) mk, m mk The aveage moistue otet of the soli a be alulate as follows: whee the eomiato is the total soli volume i the polate spheial ooiate sstem. RESUTS AND DISCUSSIONS As appliatio, the methoolog was use to peit the ig kietis a moistue otet istibutio of a polate spheoi with aspet atio / =. a Bi=.. Table pesets the values, oots of the aial spheial futio, fo = /, the eigevalues b of the expasio oeffiiets,m a oeffiiets A m.k a, fiall, the obtaie values fo the othogoalit iteio to aial a agula futios. Thee omputatioal oe wee implemete, utiliig the softwae athematia. To obtai the values of the, b,,m, a A m,k oeffiiets a the othogoalit oitios of the futio of the fial pesete solutio i the Equatio (9). It a be obseve that the aquiemet of this aaltial solutio equies a ve ha wok a a exessive umbe of omputatioal wok hous, besies its ompaiso with the umeial solutio that s give b ima (999) a ima & Neba (). Some obtaie esults with the omputatioal oe fo give oitios wee exhaustivel ompae with the supplie esults i woks of Flamme (957), Haji-Sheikh & Spaow (966), Statto et al. (94), a Abamowit & Stegu (97). The give values i the Table a be use to epoue the esults that ae show i this wok a to help ivestigatos to valiate omputatioal oes i futue woks. Results of this wok wee ompae with umeial esults fo a ellipsoi ( / =.), with Bi ifiite give b Haji-Sheikh & Spaow (966) to valiate the mathematial moel Figue shows the ompaiso betwee the oetatio atio at the ete a foal poit of a polate spheoi as a futio of Fo that s efie as Fo=Dt/. Almost omplete ooae exists betwee the esults, like a be obseve. Figue 4 shows the ompaiso betwee the aveage moistue otet as a futio of Fouie umbe uig the ig poess fo a polate spheoi with aspet atio / =. a Bi=., whih was obtaie i this wok a umeial esults that wee epote b ima (999) a ima & Neba (). Revista Basileia e Poutos Agoiustiais, Campia Gae, v.4,., p.4-5,

6 46 Figue 5 illustates the moistue otet istibutio isie the polate spheoi though the use of iffeet tos fo thee Fouie umbes. The moistue otet hages with the hagig agula a aial ooiates. The ompaiso of the gaphs iiates that the ieasig of the values of auses the moistue otet eease fo a at a Fo. The moistue otet pofile eease i a poit fo ieasig values of Fo (, ), what iiates iiatig that the moistue flux ous fom ete to the sufae. The stog moistue otet epeee with the aial a agula ooiates it a be also obseve. The oetatio epeee with the agula ooiate is slightl lage tha its epeee with the aial ooiate. I this ase, the imesioless moistue otet is eeasig with. the iease of, fo all values of Fo. It is veifie that the moistue otet gaiets ae high, exept the oes fo the egios ea the ete of the bo. It is veifie that the sufaes of a ostat, ae ot spheial, but the peset appoximatel a elliptial behaviou. It is veifie that the oetatio atio eeases faste i the extemit of the -axis (= ). This effet eas to the e of -axis (= ). This behaviou ous i all tpes of ellipsois, a it ieases popotioall to the aspet atio. I oe, the behaviou of the moistue otet with the agula ooiate is iffeet fom a sphee ( / =.), whee smmet exists elative to this ooiate. This iffeee will iease with the iease /..8 /=. (-e)/(o-e).6.4. This wok Cete Foal poit Haji-Sheikh a Spaow (966) Cete Foal poit... Fo=Dt/^ Figue - Compaiso betwee the moistue otet atio i the ete a foal poit of a spheoi with / =., that s give b the authos a Haji-Sheikh & Spaow (966)..8 Bi=. /=. (Aaltial, uikov, 968) /=. (Aaltial, This wok) /=. (Numeial, ima, 999).6 * Fo=Dt/ Figue 4 - Dimesioless aveage moistue otet as a futio of Fouie umbe of a spheoi with / =. a Bi=., give b the authos a ima (999), a uikov (968) to sphee. Revista Basileia e Poutos Agoiustiais, Campia Gae, v.4,., p.4-5,

7 47 Table - Values of the oeffiiets a othogoalit iteio of the spheial futios fo / =. a Bi=. m k b A / mk o e m m p p,9549,854,4848 -, , ,6989 -,86645,7, ,55454,47,956 4,96689,96 5,468 -, , ,9-6,6, , ,644 9,56 -,55 7,95,59-4,59,697 8,886,799-9,59 -,955 5,89 9,76,9559 -,6486,67 6,46,775, , ,799 7,8, , ,65 -,7898 -,4797 5,7,7,457,749 6,86 5,8-4,8797 -,86 7 6,586 8,869 8,486,859,665 8,89,57958,464 5,6948 9,67,855,4774 8, ,969,586,7 4 4,98 78,686-6,84 -,8 5,889,5,98, ,544 7,5-7,66 -, ,8485 5,,84,798,5867 7,59,567,4789 8,,65 -,894 -,9767,49 5,55,86, ,8 86,8-9,7495 -,45 5 6,5788,55 7,4446,54 6 9,96 56,679 7,, ,968 4,96-7,699 -,4578 5,68,,97,87496,4,76 -, ,499, ,79,879, ,5474 4,966 -,677 4,4768,E-7 5 9, ,64,474,54 6 5, ,,9675, ,54 6,9 -,6 -,874 7,499 97,57,4766,64,5 4,7 -, -, , ,8 -,9447 -, ,85 55,598-5,8569 -, , ,9-4,879 -, ,94 767,6,6966, , ,97 -,6956 -,77, 99,69 4,854,997 4,446 56,96 -,5 -,766 9,767 64,774 6,5,57 4 5,5 795,59,897, ,576 9,959-4,554 -, ,86 996,54,747,5678, ,67 5,47,575 6, ,4 -,888 -,9797 4,76 766,5-5,959 -, ,496 96,88 -,7847 -, ,9675,,65-5, ,8976 4,6,4665,68, m p Figue 6 illustates the moistue otet istibutio isie the soli with aspet atios / =.5 a / =. i Fo=. a Bi ifiit. The highest moistue gaiets ae fou as expete i ompaiso with the ase to Bi=... Figues 7 a 8 show the moistue otet as a futio of the aial a agula ooiates fo vaious Fouie s umbes uig the ig poess. Revista Basileia e Poutos Agoiustiais, Campia Gae, v.4,., p.4-5,

8 (a) Fo=. (b) Fo=.66 () Fo=.7 Figue 5 oistue otet istibutio isie the polate spheoi with aspet atio / =. a Bi=. uig the ig poess. Fo=Dt/ (a).5. Figue 6 oistue otet istibutio isie the polate spheoi with aspet atios (a) / =.5 a (b) / =. to Fo=. a Bi ifiit. (b) Revista Basileia e Poutos Agoiustiais, Campia Gae, v.4,., p.4-5,

9 49. * /=. Bi=. Fo=.66 Fo=.7 Fo=.464 Fo=.44 Fo=.66 Fo=.549 Fo=.79 Fo=. Fo=.586 Figue 7 Dimesioless moistue otet as a futio of umbes, / =, a Bi=,. i =, ( ) a vaious Fouie s. * /=. Bi=. Fo=.66 Fo=.7 Fo=.464 Fo=.44 Fo=.66 Fo=.549 Fo=.79 Fo=. Fo=.586 Figue 8 - Dimesioless moistue otet as a futio of umbes, / =, a Bi=,. i =, ( ) a vaious Fouie s The esults fo ellipsoi, alulate with the oeffiiets show i Table, peset a small eo fo a Fouie s umbe. This iffeee a be attibute to the istabilit of the Bessel futios fo small Fouie s umbes, a to the suessive appoximatio of the pesete metho hee. This last poblem a be solve usig a highe umbe of tems i the etemiatio of the oeffiiets. Some othes esults ae ote: the moistue otet stogl epes o the Fo a the equilibium moistue otet is ahieve at Fo 5.. CONCUSIONS The peset theo was applie to a ellipsoi with / =. teate hee as a appliatio of the geeal metho. It iiates that the metho a be solve ietl b use of the Equatio (4) a (5), with etemie the eigevalues a spheial oeffiiets. The metho oes ot equie a patiula geomet fom i stu hagig fom sphee to lie besies ellipsois of evolutio. The aaltial solutio pesete hee a be use to obtai esults that esibe the tasiet pheomea, i patiula, moistue otet istibutio a ig kietis, i boies with spheial, liial a elliptial geometies, osieig the iffusio oeffiiet ostat a the mass iffusio as sole mehaism of moistue migatio. As obtaie the solutio is efee to the ase with ovetive boua oitio at the sufae, it a be use to valiate umeial solutios, whih a be extee to ases with less estitive oitios. The mea value of moistue otet is patiulal useful whe the moel is use to Revista Basileia e Poutos Agoiustiais, Campia Gae, v.4,., p.4-5,

10 5 etemie the iffusio oeffiiet fom ig kieti expeimetal ata. The use imesioless ooiates, moistue atio a Fouie s umbe, wee aequate to get geeal esults, to be applie to a ase of heat o mass tasfe. The moistue otet is stogl ifluee b the Fouie s umbe i a positio i the iteio of the spheoi. The equilibium moistue otet is appoahe, at a poit of the bo, to Fo=Dt/ 5. ( / =. a Bi=.) aoig to the esults epote i the liteatue. The imesioless moistue otet eeases faste i the extemit of the axis (= ) what a eas to the e of axis (= ) iiates that the egios ea the = fist. ACKNOWEDGEENTS The authos woul like to expess thei thaks to CAPES (Cooeação e Apefeiçoameto e Pessoal e Nível Supeio, Bail) a CNPq (Coselho Naioal e Desevolvimeto Cietífio e Teológio), fo its fiaial suppot º /-7 to this wok. BIBIOGRAPHIC REFERENCES Alassa, R. S., Heat outio fom spheois. Joual of Heat Tasfe, V.,., p , 999. Abamowit,.; Stegu, I. A., Habook of mathematial futios, New Yok: Dove Publiatios, I., p , 97. Cak, J., The athematis of Diffusio, New Yok: Oxfo Siee Publiatios, p. Flamme, C., Spheoial Wave Futios. Stafo: Stafo Uivesit Pess, 957. Haji-Sheikh, A.; Spaow, E.., Tasiet heat outio i a polate spheoial soli, Tasatios of the ASE: Joual of Heat Tasfe, v. 88,., p. -, 966. to ellipsoi of evolutio. I: Bailia Cogess of ehaial Egieeig (COBE). Poeeigs Águas e ióia, Bail, CD-Rom. ima, A. G. B., 999, Diffusio pheomeo i polate spheoial solis. Case stuie: ig of baaa. State Uivesit of Campias (UNICAP), p. (Doto Thesis). (I potuguese). ima, A. G. B.; Neba, S. A., Theoetial aalsis of the iffusio poess isie polate spheoial solis. Dig Teholog, v. 8,. -, p uikov, A. V., Aaltial heat iffusio theo, oo: Aaemi Pess, I. t, p. ose, P..; Feshbah, H., ethos of Theoetial Phsis, Pat II, New Yok: Gaw-Hill Book Compa, I., p. 5-5, 956. Nomito, E. J., Blakwell, J. H. Tasiet heat flow fom ostat tempeatue spheois a the thi iula isk. The Quatel Jou. of eh. a Applie athematis, v. XVII, Pat, p. 65-7, 964. Robi,. Fotios Sphéiques e egee et Fotios Sphéoiales, Tome III, Pais: Gauthie-Villaes, p Skella, A. H. P., Diffusioal mass tasfe, New Yok: Joh Wile & Sos, 974. Statto, J. A.; ose, P..; Chu,. J.; ittle, J. D. C.; Cobató, F. J., Spheoial wave futios, New Yok: The Teh. Pess of. I. T. a Joh Wile & Sos, I., 956. Statto, J. A.; ose, P..; Chu,. J.; ittle, J. D. C.; Hute, R. A. Ellipti lie a spheoial wave futios, New Yok: The Teh. Pess of. I. T. a Joh Wile & Sos, I., 94. ima, A. G. B., Neba, S. A., 999, Aaltial solutio of the mass iffusio equatio applie Revista Basileia e Poutos Agoiustiais, Campia Gae, v.4,., p.4-5,

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