Transport Phenomena and Shrinkage Modeling During Convective Drying of Vegetables.

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1 Excerp from he Proceedings of he COMSOL Conference 2009 Miln Trnspor Phenomen nd Shringe Modeling During Convecive Drying of Vegebles. Sefno Curcio *1 nd Mri Avers 1 1 Deprmen of Engineering Modelling Universiy of Clbri. *Corresponding uhor: Pone P. Bucci cubo 9/C Rende (CS) - ITALY, sefno.curcio@unicl.i Absrc: The im of he presen or is he formulion of heoreicl model describing he rnspor phenomen involved in food drying process. The enion hs been focused on he simulneous rnsfer of momenum, he nd mss occurring in convecive drier here ho dry ir flos, in urbulen condiions, round he food smple. Shringe, s ell s ll he rnspor phenomen occurring in boh ir nd food domins, hve been described. The proposed model does no rely on he specificion of inerfcil he nd mss rnsfer coefficiens nd, herefore, represens generl ool cpble of describing he behvior of rel driers over ide rnge of process nd fluiddynmic condiions. The sysem of non-liner unsedy-se pril differenil equions modelling he process hs been solved by mens of he Finie Elemens Mehod coupled o he ALE (Arbirry Lgrngin Eulerin) procedure h, by proper modificion of inegrion domin, ccouns for shringe effecs. In order o describe shringe phenomenon, he bovemenioned rnspor equions hve been coupled ih srucurl mechnics nlysis performed on he food smple. Keyords: Drying, shringe, srucurl mechnics, Trnspor Phenomen, ALE. 1. Inroducion In previous pper (Curcio, Avers, Clbrò, Iorio 2008) he uhors of he presen or formuled heoreicl model describing he rnspor phenomen involved in food drying process. The enion s focused on he simulneous rnsfer of momenum, he nd mss occurring in convecive drier here dry nd ho ir floed, in urbulen condiions, round e nd cold food smple ih lo porosiy. Moisure rnspor inside food ih lo porosiy, here inner evporion cn be negleced, (My & Perré 2002), s modelled by n effecive diffusion coefficien of er in he food, hus no disinguishing beeen he cul rnspor of boh liquid er nd vpour ihin he food srucure. Porous foods re hygroscopic merils h conin boh bound nd free er (D 2007 I), in fc hey re chrcerized by level of moisure conen belo hich he inernl vpour pressure is funcion of moisure conen nd emperure (his is chrcerisic of bound er) nd is loer hn h of pure er. Above his moisure level he vpour pressure is funcion of emperure only (free er). During drying, due o he porosiy of food mer, er evporion es plce inside he food s ell s he food exernl surfce. The he necessry for er evporion nd for food heing is supplied, in convecive drying process, by drying ir. Wer hus exiss boh s liquid nd s vpour in food for hich porosiy cnno be negleced. Acully, o describe he rnspor of boh liquid er nd vpour o differen mss blnce equions re needed since i is necessry o ccoun for he differen rnspor mechnisms, i.e. cpillriy nd moleculr diffusion, ing plce in he food mer (D 2007 I). Moreover, lhough evporion es plce inside he food, he rnsfer res occurring ir/food inerfce re srongly dependen on he drying ir velociy field exising in he drying chmber nd, priculrly, in he boundry lyers developing close o he food surfces exposed o ir. For his reson, o improve he precision of he model, in priculr close o he solid surfces, he -ω model (Wilcox 1998) hs been used in he presen pper o clcule drying ir velociy field nd o describe he momenum rnspor in urbulen condiions. The -ω s chosen becuse of is min feures, for insnce he higher ccurcy in boundry lyer modelling (ih boh dverse nd fvourble pressure grdien). The im of he presen or is o dop conservion-bsed pproch o develop muliphse rnspor model so o describe convecive drying process. The model is bsed on conservion of liquid er, vpour nd energy in boh ir nd food domin. Moreover,

2 he rnsfer of momenum in ir, in urbulen condiions, is lso modelled by -ω model. Wer in ir hs been considered s vpour only, heres, in food, conemporry presence of liquid er nd vpour hs been considered. To properly describe shringe phenomenon (food volume vriion during drying), he bove-menioned rnspor equions hve been coupled ih srucurl mechnics nlysis performed on he food smple. The sysem of non-liner unsedy-se pril differenil equions ere solved by mens of he Finie elemens mehod nd by Comsol Muliphysics Theory The food under sudy s poo smple, dried in convecive oven s shon in Fig. 1. To develop he presen rnspor model, i is necessry o nlyse he cul rnspor phenomen occurring in he boh he food nd he ir domins. Liquid er rnspor in food sysem is promoed by pressure nd cpillry pressure grdiens hile vpour rnsfer is promoed by pressure nd concenrion grdiens. For liquid er rnspor he cpillry pressure previls over he effec of exernl pressure if he hygroscopic meril conins n moun of bound er highly greer hn free er: in food sysem his is rue in he mjoriy of he cses nd lso in he cse of pooes. Wih respec o vpour rnsfer s promoed by exernl pressure, i is generlly described by he Drcy s equion h is vlid ih Reynolds number rnging from 1 o 10. In ypicl cse of microve heing of poo smples, hoever, i s found h he vlue of Reynolds number s equl o10-5 (D 2007 I). Generlly, in he cse of convecive drying he inner evporion is considerbly loer hn h of microve process; so, i cn be ssumed h pressure driven flo is negligible nd vpour moleculr diffusion cn be considered s he previling mechnism. Definiively, cpillry flo s used o model he liquid er rnsfer nd moleculr diffusion s used o model he vpour flo. The mss blnce referred o he liquid er nd vpour in he food smple leds, respecively, o he unsedy-se mss rnsfer equions (Bird, Ser, Lighfoo, 1960, Wely, Wics, Wilson, Rorrer, 2001): C D C I 0 (1) Cv D C I 0 (2) v v here C is he er concenrion in food, C v is he vpour concenrion in food, D is he cpillry diffusiviy in food nd D v is he diffusion coefficien of vpour in food nd I is he evporion re. The energy blnce in he food meril leds, ccording o he Fourier s l, o he unsedy-se he rnsfer equion (Bird, Ser, Lighfoo, 1960, Wely, Wics, Wilson, Rorrer, 2001). T sc p s T I 0 eff () here T is he food emperure, s is he densiy of food smple, C p s is specific he, eff is he food effecive herml conduciviy ccouning for combinion of differen rnspor mechnisms. The energy blnce conins he evporion erm since i is ssumed h evporion phenomen e plce no only on he food exernl surfces, bu lso ihin is srucure. Figure 1. Schemic of he sysem under considerion. The non-isoherml urbulen flo of ir ihin he drying chmber s modelled by mens of he ell-non -ω model (Curcio, Avers, Clbrò, Iorio 2008), h is bsed on o ddiionl semi-empiricl rnspor equions for he vribles nd ω i.e. he

3 urbulen ineic energy nd he dissipion for uni of urbulen ineic energy respecively. The unsedy-se momenum blnce coupled o he coninuiy equion rien for he drying ir leded o (Bird, Ser, Lighfoo, 1960, Verboven, Scheerlinc, De Berdemeer, Nicolï, 2001): u 0 (4) u u u pi T u ( u) (2 ) ui (2 ) I (5) u 2 P u ( ) u (6) u 2 2 P u u ( )[ ( ) ] (7) here ρ is he ir densiy, η is is viscosiy, boh expressed in erms of he locl vlues of emperure nd of er conen, p is he pressure ihin he drying chmber, u is he velociy vecor. β, σ, σ ω,α, β re consns (Wilcox 1998). The erm, P(u), conins he conribuion of he sher sresses: P T 2 u 2 u u u u (8) The folloing definiion for η, i.e. he urbulen viscosiy inroduced in he bove Eqs. 4-7, holds: (9) The energy blnce in he drying ir, ccouning for boh convecive nd conducive conribuions, leded o (Bird, Ser, Lighfoo, 1960, Wely, Wics, Wilson, Rorrer, 2001): T C 2 T C u T 0 (10) p 2 p 2 here T 2 is he ir emperure, C p is is specific he nd is herml conduciviy. The mss blnce in he drying ir, referred o he er vpour nd ccouning for boh convecive nd diffusive conribuions, leded o (Bird, Ser, Lighfoo, 1960, Wely, Wics, Wilson, Rorrer, 2001): C D C u C (11) here C 2 is he er concenrion in he ir nd D is he diffusion coefficien of er in ir. The rnspor nd physicl properies of poo hs been obined by Sriiden, Robers (2008), Zhng, D (2004) nd Mroulis, Srvcos, Kroid, Pngioou (2002). As fr s he boundry condiions ere concerned, he coninuiy of boh he nd mss fluxes nd of emperure s formuled he food/ir inerfces. Moreover n equilibrium relionship beeen he liquid er nd vpour inside food s used (Smih, Vn Ness, Abbo, 1987): x f y p (12) ^ Where γ, he civiy coefficien of er nd f, he fugciy of er, refer o he liquid ^ phse,, he fugciy coefficien of er, refers insed o he vpour phse; x nd y re he molr frcions of er in food nd in ir, respecively, p is he pressure ihin he drying chmber. A lo pressures, vpour phse usully pproximes idel gses nd llos simplifying eq. 12 o: s x P y p (1) here P s is he vpour pressure of er. I should be observed h in hygroscopic merils, lie mos of he foods, γ ccouns for he effecs reled o he moun of physiclly bound er so i is usully expressed s funcion of boh food moisure conen nd of is emperure (D 2007 Pr II, Ruiz-López, Córdov, Rodríguez-Jimenes, Grcí-Alvrdo,

4 2004). Once he civiy coefficien is non for he priculr food under exminion, eq. 1 permis clculing he molr frcion of er occurring in he vpour phse. Young Modulus [P] 6.0E E E+07.0E E E E Poo moisure conen on dry bsis, Xb[-] Figure 2. Young Modulus nd Yield Sress during pooes drying. A he oven oule secion, conducion nd diffusion phenomen ere negleced ih respec o convecion (Dncers condiions). As fr s he boundry condiions referred o he momenum blnce ere concerned, ovelociy scle logrihmic ll funcion s used on he solid surfces, (Lcsse, Turgeon nd Pelleier, 2004). Food shringe s modelled defining he locl ol srins {dε} s funcion of chnges in mechnicl srins {dε s } (consrined deformion due o food mechnicl properies) nd in shringe srins {dε 0 } (he sum of free deformion due o moisure loss): d d sd (14) Tol srin {dε} is cully funcion of ol displcemen {du} d AdU (15) Chnges in sresses {dσ} re funcion of chnges in mechnicl srins {dε s } d Dd s 0 (16) Where [D] is he sress-srin mrix conining he Young Modulus. Yng nd Si (2001) repored is vlue during pooes drying (Fig. 2). To express he free drying shringe srins {dε 0 }, i s ssumed h he free deformion due o moisure loss s proporionl o he er conen vriion, hrough consn (he hydrous compressibiliy fcor). The consn s esimed from he experimenl d shoing drying shringe vs. eigh loss. Virul or principle s formuled o obin he equilibrium equion. By ssuming h zero body nd surfce forces re pplied o food, i cn be rien: T d d dv 0 V (17) As fr s he boundry condiions re concerned, one side of he food ress on he drier ne (fixed posiion) heres he oher hree re free o move. Since physicl nd rnspor properies of boh ir nd food re expressed in erms of he locl vlues of emperure nd moisure conen, he rnspor equions for boh ir nd food, ogeher ih he virul or principle represen sysem of unsedy-se, non-liner, pril differenil equions h cn be solved only by mens of numericl mehod. Equions ere rien referring o ime dependen deformed mesh h ccouned for food volume vriion due o er rnspor. An Arbirry Lgrngin-Eulerin (ALE) descripion, implemened by Comsol Muliphysics, s doped. I is orhhile o remr h he ALE mehod cn be considered s inermedie beeen he Lgrngin nd Eulerin pproches since i combines he bes feures of boh of hem nd llos describing moving boundries ihou he need for he mesh movemen o follo he meril. The moion of he deformed mesh s modeled using Lplce smoohing. The boundry condiions conrol he displcemen of he moving mesh ih respec o he iniil geomery. A he boundries of food smple, his displcemen is cully h clculed by solving he srucurl mechnic problem. A he exerior boundries of he fluid domin, i is se o zero in ll direcions.. Use of COMSOL Muliphysics

5 The sysem of unsedy, non-liner PDEs resuling from he presen sudy s solved by he Finie Elemens Mehod using Comsol Muliphysics.4. Boh food nd ir domins ere discreized ino ol number of 9195 ringulr finie elemens leding o bou 9000 degrees of freedom (Fig. ). In priculr, he mesh consised of 151 elemens (ih minimum elemen quliy of ) nd 6044 elemens (ih minimum elemen quliy of ) ihin food nd ir domins, respecively. The considered mesh provided sisfcory spil resoluion for he sysem under sudy. The soluion s, in fc, independen of he grid size, even ih furher refinemens. Lgrnge finie elemens of order o ere chosen for he componens of ir velociy vecor u, for he urbulen ineic energy, he dissipion for uni of urbulen ineic energy nd for he pressure disribuion ihin he drying chmber bu, lso for er concenrion nd emperure in, boh, ir nd food. The ime-dependen problem s solved by n implici ime-sepping scheme, leding o non-liner sysem of equions for ech ime sep. Neon s mehod s used o solve ech non-liner sysem of equions, heres direc liner sysem solver s doped o solve he resuling sysems of liner equions. The relive nd bsolue olernces ere se o nd , respecively. On dul-core compuer running under Linux, ypicl drying ime of 5 hours s ypiclly simuled in bou 50 minues. convecive oven (Memmer Universl Oven model UFP 400) monioring, ih respec o ime, food eigh, by precision blnce (Meler AE 160, ccurcy of ±10-4g) nd food dimensions by vernier cliper. To perform he presen experimenl nlysis idenicl poo smple, hving n iniil side of 0mm nd n iniil hicness of 15mm, ere used. The lb-scle oven lloed monioring, by Dosmnn elecronic Precision Mesuring Insrumen P 655, ir emperure nd is humidiy (by rh probe) nd ir velociy, by H probe. Air velociy s equl o 2.2 m/s nd ir emperure, T, s chosen equl o 50 C nd o 70 C. Ech food smple s plced on ide-mesh perfored ry. Experimenl d ere used lso o obin some imporn prmeers regrding food shringe modelling. Food eigh nd food dimensions ere, indeed, used o evlue food moisure conen nd food volume respecively hich hve been ploed in figures 4 nd 5: A liner fiing procedures of he foremenioned experimenl d lloed esiming he hydrous compressibiliy fcor h, s shon in figures 4 nd 5, ere equl o nd o , in he cse of drying emperure of 50 C nd 70 C respecively. Figure 4. Esimion of hydrous compressibiliy fcor from experimenl d 50 C. Figure. Domin discreizion. 4. Experimenl Poo smples ere ir-dried by ir in convecive oven. The pooes ere cu in slb shpe. Ech smple s dried in lb-scle

6 greemen beeen model predicions nd experimenl d in ll he considered cses. Figure 5. Esimion of hydrous compressibiliy fcor from experimenl d 70 C. 5. Resuls The proposed model s ble o provide orhhile informion bou er, emperure nd velociy profiles in boh he considered domins. This is minly ineresing in he cse of food mer here, he locl vlue of er conen nd emperure is n index of he condiions h give rise o deeriorion recion due o micro-orgnism civiy. Some of he mos represenive resuls re shon in he folloing Fig. 6, here he ime evoluion of poo moisure conen (on e bsis) is presened for poo smple hose shpe nd dimension chnge ih respec o ime becuse of he shringe. Figure 7. Comprison beeen experimenl nd model resuls in erms of ol volume evoluion of poo smples during drying. Figure 8. Comprison beeen experimenl nd model resuls in erms of slb hicness evoluion of poo smples during drying. 5. Conclusions Figure 6. Time evoluion of poo moisure conen (on e bsis) during drying ccouning for shringe effec. (Air emperure of 50 C, ir velociy of 2.2m/s) Comprison beeen mesured nd clculed food dimensions in erms of ol volume nd food hicness re shon in figures 7 nd 8 respecively, shoing remrble The rnspor phenomen involved in food drying process hve been nlyzed. A generl predicive model, i.e. no bsed on ny semiempiricl correlion for esiming he nd mss fluxes food-ir inerfce, hs been formuled. The proposed model lso prediced he spil moisure profiles ll imes, hus lloing deecing he regions ihin he food core, here high vlues of moisure conen cn promoe microbil spoilge. The model hs been lso improved o predic food shringe by coupling rnspor equions o virul or principle, rien ih reference o deformed mesh hose movemen hs been

7 described by ALE mehod. The obined resuls re promising. I is inended o improve he rnspor model by considering lso he influence of convecion inside he food. Also shringe descripion needs o be improved, for insnce by formuling differen ssumpion reling free deformion due o moisure loss o er conen vriion, or by ing ino ccoun he influence of body nd surfce forces pplied o food. 6. References Bird, R.B., Ser, W.E., Lighfoo, E.N., Trnspor Phenomen. John Wiley & Sons, London, UK, (1960). Curcio S., Avers M., Clbro' V., Iorio G., Simulion of food drying: FEM nlysis nd experimenl vlidion, Journl of Food Engineering, Vol 87 (4), pges (2008). D A.K., Porous medi pproches o sudying simulneous he nd mss rnsfer in food processes. I: Problem formulions. Journl of Food Engineering, Vol 80 (1), pges (2007). D A.K., Porous medi pproches o sudying simulneous he nd mss rnsfer in food processes. II: Propery d nd represenive resuls, Journl of Food Engineering, Vol 80 (1), pges (2007). Lcsse D., Turgeon É. nd Pelleier D., On he judicious use of he -e psilon model, ll funcions nd dpiviy, Inernionl Journl of Therml Sciences, Vol 4 (10), pges (2004). Mroulis Z.B., Srvcos G.D., Kroid M.K., Pngioou N.M., Therml conduciviy predicion for foodsuffs: effec of moisure conen nd emperure, In J Food Proper, Vol,5 pges21 45 (2002). Ruiz-López I.I., Córdov A.V., Rodríguez- Jimenes G.C, Grcí-Alvrdo M.A., Moisure nd Temperure Evoluion during Food Drying: Effec of Vrible Properies, Journl of Food Engineering, Vol 6, pges (2004). Smih, J.M., Vn Ness, H.C., Abbo, M.M. Chemicl Engineering Thermodynmics (IV ediion). Ne Yor, McGr-Hill, (1987). Sriiden J., Robers J.S., Predicing moisure profiles in poo nd crro during convecive ho ir drying using isohermlly mesured effecive diffusiviy, Journl of Food Engineering Vol 84, pges (2008) Verboven P., Scheerlinc N., De Berdemeer J., Nicolï B. M., Sensiiviy of he Food Cenre Temperure ih Respec o he Air Velociy nd he Turbulence Kineic Energy, Journl of Food Engineering Vol 48, pges 5-60 (2001). Wely, J., Wics, C., Wilson, R., Rorrer, G., Fundmenls of Momenum, He, nd Mss Trnsfer. Ne Yor: John Wiley nd Sons (2001). Wilcox, D. C., Turbulence Modeling for CFD, DCW Indusries, Inc., L Cnd, CA (1998). Yng, H., Si, N. nd Wnbe, M., Drying model ih non-isoropic shringe deformion undergoing simulneous he nd mss rnsfer, Drying Technology, Vol 19 (7), pges (2001). Zhng, J. nd D, A. K., Some considerions in modeling of moisure rnspor in heing of hygroscopic merils, Drying Technology, Vol 22 (8), pges (2004). My, B.K., Perrè, P., The impornce of considering exchnge surfce re reducion o exhibi consn drying flux period in foodsuffs, Journl of Food Engineering, Vol 54, pges (2002).