Application of Fractional Calculus in Food Rheology E. Vozáry, Gy. Csima, L. Csapó, F. Mohos

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1 Applicain f Facinal Calculus in F Rhelgy Szen Isán Uniesiy, Depamen f Physics an Cnl Vzay.sze@ek.szie.hu Keyws: facinal calculus, iscelasic, f, ceep, ecey Absac. In facinal calculus he e (β) f iffeeniain inegain is n an inege numbe, geneally β is a facinal numbe beween an. he f facinal calculus ges back he 8h cenuy, an his calculus is inensiely eelping nwaays,. Applicain f facinal calculus can be fun in helgy, in elecical impeance specscpy, in physilgical escipin. I is ineesing, ha he escipin f iscelasic ppeies f bilgical maeial is much me accuae wih facinal calculus han wih inay iffeeniain an inegain. In his wk he ceeping an ecey cues f a simple swee, - gum cany an bea slice, wee appache wih inay an facinal calculus. he helgical paamees f gum cany wee eemine. he facinal calculus gae bee fi n he measue ceep ecey cue pins, han classical helgy mels cnaining iscee elasic an iscus elemens. INRODUCION In many cases he expeimenally bsee elaxain funcin exhibi a seche (Khlausch) expnenial ecay F() F exp( ( / τ ) α ), whee F enes he elaxing physical quaniy (f example ligh inensiy, sess elaxain in iscelasic maeial, ielecic elaxain, ec.), is he ime, τ is a cnsan an < α < is a numbe (Schiessel e al., 995). An apppiae l escibe hese elaxain pcesses is he facinal calculus (Süli, ). he s calle Facinal Calculus was bn me han 3 yeas ag. In a lee ae Sepembe 3h, 695 L'Hpial we Leibniz asking him abu a paicula nain he ha use in his publicains f he nh-eiaie f D n x he linea funcin f(x) x, Dx n L'Hpial's pse he quesin Leibniz, wha wul he esul be if n /. Leibniz's espnse: "An appaen paax, fm which ne ay useful cnsequences will be awn." Wihin he h cenuy especially numeus applicains an physical manifesains f facinal calculus hae been fun (Mainai an Spaa, ; Schiessel e al., 995). While he physical meaning is ifficul gasp, he efiniins hemseles ae n me igus han hse f hei inege e cunepas. Wihin he h cenuy especially numeus applicains an physical manifesains f facinal calculus hae been fun (Mainai an Spaa, ;

2 Schiessel e al., 995). While he physical meaning is ifficul gasp, he efiniins hemseles ae n me igus han hse f hei inege e cunepas. α e facinal inegal accing Riemann-Liuille f a f() eal cmplex funcin can be gien by he fmula (J α x f )(x) Γ ( α ) ( x ) α f () whee is eal aiable, x>, α > is a eal numbe an Γ ( α ) y α e y y is he Gamma-funcin. hee ae seeal efiniins f α e facinal eiaie, pacically as many as many mahemaicians eal wih facinal calculus. A efiniin in which he Riemann-Liuille facinal inegal is use can be gien by he nex expessin: f () τ ( τ ) m τ () τ Γ ( α ) α α m m m m < α < m ( D ) f ( ) :, m α m f ( ) m whee m-< α <m an m is an inege numbe, α is a eal numbe (Le, 4). In his wk a seche expnenial funcin was fie n ceep an ecey cues f aius f maeials. MARIALS AND MHODS he inesigae maeials: bea an gum cany wee puchase in he lcal make. he ceep-ecey es (CR) cues wee measue wih a exue analyse A-X fm Sable Mic Sysem (Galming, Unie Kingm). he bea slices wee pesse wih a plexi cyline f 36 mm iamee, an gum canies wee pesse a meal cyline f 75 mm iamee. he CR es cnsiss f fu segmens. he fis segmen is laing he sample wih cnsan spee f measuing hea unil a pe-se fce is eache. In he secn segmen he efmain is ceeping une he cnsan fce uing a pe-se pei. In he hi segmen (unlaing) he pbe is aise unil he fce n hea becmes ze. In he fuh segmen - in ecey - he elaxain f sample cninues s, ha he measuing hea is aise when he elaxe sample eaches he pbe. In u measuemens he pe-se ime was 6 s f bh ceeping an ecey pei. he fce in ceeping pei was 5 N an,5 N f gum cany an bea, especiely. he ceeping an ecey pa f CR ae suiable f he eeminain f helgical paamees f sample maeial wih mel funcins. he fu-elemen Buges mel (Fig. ) can escibe bh he ceeping an ecey pcesses

3 Figue. he Buges mel. an, epesens he elasic mulus f he w sping elemens an he an epesens he iscsiy f he w ashps he iffeenial equain f fu-elemen Buges mel (Sikei, 98) f sess an f sain in nmal calculus an he sluin f i is f ceeping sain ( ) in ime,, when he sess is cnsan cns : () e, () whee, he eaain ime an he sluin f i f ecey sain () in ime,, afe ime (ime elapse beginning f ecey) when he sess becmes ze ( ): () e e () he iffeenial equain f Buges mel in facinal calculus: ν ν ν ν

4 an he sluin is f ceeping an ecey pas: β ( ) () e (3) an β β ( ) ( ) () e e (4) especiely, whee < ν < an < β < he () an (3) expessins wee fie n ceeping cues an he () an (4) expessins n he ecey cues. he elasic mulus, iscsiies an β paamee wee eemine. RSULS AND DISCUSSION ypical ceep-ecey cue f bea can be seen n Fig.. Simila CR cue was measue n gum canies,. he elaie quick la pa is fllwe by much lnge ceeping pa. he ceeping pa f all CR cues was appache wih Buges mel f nmal an facinal calculus, wih equain () an (3). Afe he ceeping pa hee is a elaie quick unla segmen which is fllwe by ecey cue. he ecey pa f CR cues was fie by Buges mel bh wih nmal an facinal calculus, wih expessins () an (4). Figue. A ypical ceep-ecey cue f a bea slice. he esul f cue fiing is emnsae n Fig. 3. he seche expnenial funcin gae bee appach f measue pins especially in he beginning f bh ceeping an ecey pa. he alue f paamees fm ceeping pei is ey simila he alues fm ecey pei (able.) f bh inesigae bjecs. Geneally paamee alues ae lwe fm appaching he ecey pa accing paamee alues fm ceeping pa.

5 elaie efmain,4,,6, A measue Buges mel Buges mel wih seche expnen 4 6 ceeping ime, s elaie efmain,6,,8,4 B measue Buges mel Buges mel wih seche expnen 4 6 ecey ime, s elaie efmain,34,3,3,8 C measue Buges mel Buges mel wih seche expnen 4 6 ceeping ime, s elaie efmain,6,,8,4 D measue Buges mel Buges mel wih seche expnen 4 6 ecey ime, s Figue 3 he fiing f ceeping an ecey pas f CR cue f bea (A,B) an f gum cany (C,D)

6 able : he paamees f Buges mel fm appaching he measue cues Maeial Calculus pa, kpa, kpa, MPas bea nmal ceeping 6,,8 5,3,,39, ecey 34,6,4,37,4 facinal ceeping 8,4, 3,83,5,5,6 ecey 6,,,4,4 gum cany nmal ceeping 47,8,6 5,73, 3,, ecey 7,8,,57,9 facinal ceeping 48,3, 33,78, 3,3,3 ecey 88,7,5,7, Maeial Calculus pa, MPas β bea nmal ceeping 8,73,7 ecey 5,,64 facinal ceeping 6,39,98,5 ecey 5,45,57,68 gum cany nmal ceeping 5,738,57 ecey 6,3,4 facinal ceeping 33,6335,97,6 ecey 58,739,76,65 I can be explaine by he fac, ha in he ecey pei he sess is aleay ze, bu in ceeping pei is abu -3 kpa. Ou ealie inesigain shwe, ha bh elasic muli an iscsiies f Buges mel f gum cany linealy incease, if he sess n he sample incease (Csima, 5). his incease can be cause by sucue changes une sess. he lwe paamee alues f bea accing paamee alues f gum cany can be explaine wih he lwe elasiciy an haness f bea cmpae gum cany. CONCLUSIONS he appach f ceep an ecey pa f CR cues was pe me pecise wih facinal calculus han wih nmal calculus. I seems ha he seche expnenial funcin bee escibes especially he quick pcesses f bh ceeping an ecey cues. RFRNCS Csima Gy. (5) Zselain alapú éesipai emék elógiájának jellemzése, PhD. isszeáció, Buapesi Cinus gyeem. Le, A. (4): Facinal Calculus: Hisy, Definiins an Applicains f nginee hp://www3.n.eu/ msen/eaching/un eres/fac.calc.pf Mainai, F. an Spaa, G. (): Ceep, elaxain an iscsiy ppeies f basic facinal mels in helgy.

7 he upian Physical Junal Special pics, 93:33-6. Schiessel, H., Mezle, R., Blumen, A., Nnnenmache,.F. (995): Genealize iscelasic mels: hei facinal equains wih sluins. Junal f Physics A: Mahemaical an Geneal, 8: SIKI, GY. (98): A mezőgazasági anyagk mechanikája. Akaémiai Kiaó, Buapes Süli B. M. (): Mi legyen egy függény -eik eiálja?. Szaklgza, öös Lán umányegyeem, emészeumányi ka, Analízis anszék