A Multiphase Model to Analyze Transport Phenomena in Convective Drying

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1 Excerpt from the Proceedings of the COMSOL Conference 010 Pris A Multiphse Model to Anlyze rnsport Phenomen in Convective Drying S. Curcio 1, M. Avers *,1 1 Deprtment of Engineering Modeling, Ponte P.Bucci - Cubo 39/c, University of Clbri, Rende (CS) 87030, IALY. *Corresponding uthor: mri.vers@unicl.it Abstrct: he im of the present work ws the formultion of theoreticl describing the trnsport phenomen involved in food drying process. he simultneous trnsfer of momentum, het nd wter, both s liquid nd s vpor, were described ccounting for the vrition of food smple volume. he volume vrition, generlly neglected by the vilble theoreticl s but ctully occurring during rel processes, ws clculted by proper structurl mechnics nlysis tht llowed determining the nisotropic displcements (shrinkge) of food due to wter evportion. hese displcements, described by the Arbitrry Lgrngin Eulerin (ALE) method, determined modifiction of both the integrtion domins (ir nd food), ccounted for by the present. Keywords: Food, Finite Elements Method, Shrinkge List of symbols [A] Mtrix defined by eq. 19 1/m C + Universl constnt for - smooth wlls C Wter concentrtion in ir mol/m 3 C p Air specific het J/(kg K) C ps Food specific het J/(kg K) C v Vpor concentrtion in food mol/m 3 C w Liquid wter concentrtion mol/m 3 in food du otl displcement m dε otl strin - dε ο Shrinkge strin - dε s Mechnicl strin - dσ Mechnicl stess P [D] Stress-strin mtrix P D Diffusion coefficient of m /s wter in ir D v Effective diffusion m /s coefficient of vpor in food D w Cpillry diffusion m /s coefficient of wter in food I Identity mtrix - Volumetric rte of mol/(m 3 I evportion s) k urbulent kinetic energy m /s k Air therml conductivity W/(m K) k eff Effective therml W/(m K) conductivity of food n Unity vector norml to the - surfce p Pressure within the drying P chmber p v Vpor pressure of wter P p vs Sturted vpor pressure of P wter R Gs constnt J/(mol K) r Sptil coordinte m Food temperture K t ime s Air temperture K ir Air temperture t the drier K inlet u Averged velocity field m/s u Fluctuting prt of velocity m/s field u 0 Air velocity t the drier inlet m/s u τ Friction velocity m/s U r Reltive humidity of ir - V Food volume m 3 <Xb> Averge moisture contentdry kg w /kg ds bsis X b Moisture content on dry kg w /kg ds bsis z Sptil coordinte m Greek Symbols α Constnt compring in k-ω - β Constnt compring in k-ω - β k Constnt compring in k-ω - δ w Distnce from the wll m κ von Krmn s constnt - η Air viscosity P s η t Air turbulent viscosity P s

2 λ Wter ltent het of J/mol vporiztion Air density kg/m 3 s Food Density kg/m 3 σ k Constnt compring in k-ω - σ ω Constnt compring in k-ω - ω Dissiption per unit of turbulent kinetic energy subscripts 0 Initil condition (t=0) tm Atmospheric conditions ds On dry bsis (dry solid) r Sptil coordinte w On wet bsis z Sptil coordinte superscripts rnspose 1. Introduction Wter removl from hygroscopic mterils (like foods) is rther complex phenomenon, since both unbound nd bound wter re ctully to be trnsported. Generlly, bound wter is removed by progressive vporiztion within the solid mtrix, followed by diffusion nd pressure driven trnsport of wter vpor through the solid [Myor nd Sereno 004]. During drying, due to the porosity of food mtter, wter evportion tkes plce inside the food s well s t the food externl surfce. An exhustive nlysis of ll the complex trnsport phenomen involved in the drying process ws regrded s being too onerous nd time consuming for prcticl purposes [Rtti 1994]. he simultneous presence of both liquid wter nd vpor within the solid mkes drying ing even more difficult [Chen 007]. he trnsport phenomen occurring t the food-ir interfces were often described in terms of het nd mss trnsfer coefficients, estimted from the vilble literture dt. his prctice, however, limited the ccurcy since even smll errors in the estimtion of the trnsfer coefficients could led to lrge devitions between predicted nd rel vlues, thus determining inpproprite equipment design or severe processing problems [Kondjoyn nd Boisson 1997; Verboven, Nicolï et l. 1997]. Moreover, the estimtion of interfcil trnsport rtes by het/mss trnsport coefficients is ctully unrelible when food shpe is not regulr or it even chnges with time, due to shrinkge [Myor nd Sereno 004; Rtti 1994]. As drying proceeds, in fct, the initil mechnicl equilibrium, which determines the food size nd its shpe, fils due to the wter trnsport from the solid mtrix; progressive modifiction of the food structure is, therefore, observed. he rpid removl of wter during drying does not llow the ttinment of new equilibrium stte within the solid mtrix, whose min chrcteristics (mechnicl properties, porosity, shpe, dimension, etc.) re not constnt nd vry with time. On the bsis of the previous discussion, it is evident tht proper ing of shrinkge cnnot leve structurl nlysis of food smple out of considertion. A proper chrcteriztion of food smple deformtions, however, is strongly dependent on the ctul het nd wter rtes, which, in turn, re ffected by the velocity field of drying ir flowing round the food smple. he im of the present work ws the formultion of multiphse trnsport imed t the exhustive description of convective drying process. he ws bsed on the conservtion of liquid wter, vpor nd energy, both in ir nd in food domin. he turbulent momentum trnsfer of drying ir, considered s gs mixture owing to its reltive humidity lwys different from zero, ws lso considered nd described by the k-ω [Wilcox, 004], which s compred to the simpler k-ε llowed improving the predictions, especilly in the boundry lyer developing close to the food surfces. o properly describe the shrinkge, the clssicl trnsport equtions were coupled with structurl mechnics nlysis performed on the food smple, i.e. cylindricl shped potto whose mechnicl properties, depending on the locl vlue of food moisture content, were tken from the literture [Yng nd Ski, 001]. he clculted displcements of food smple were necessry to estimte, by n ALE method, the ctul deformtion of the mesh nd, therefore, the vritions of both the integrtion domins (food nd ir) involved in the present process ing. he system of non-liner unstedy-stte prtil differentil equtions resulting from the formultion of the present were solved by

the Finite Elements Method implemented by Comsol Multiphysics 3.4.. heory he food under study ws cylindricl potto smple dried in convective oven.

3 the Finite Elements Method implemented by Comsol Multiphysics heory he food under study ws cylindricl potto smple dried in convective oven. It ws ssumed tht drying ir ws continuously supplied to the oven inlet section nd flowed round the food smple in the xil direction, prllel to its length (Fig. 1). he portion of drier under investigtion hd length of 0.15 m nd rdius of 0.08 m. he food smple, ssumed to be cylinder 0.05 m long nd m rdius, ws plced 0.04 m fr from the drier inlet. Figure 1. Schemtic representtion of the drying chmber. he present theoreticl ctully represented n improvement of previous study [Curcio, 010]. he following min ssumptions were formulted: ) het nd mss trnsfer resistnce were ssumed negligible cross the mesh on which food ws plced [horvldsson nd Jnestd 1999; Viollz nd Rovedo 00]. he system under investigtion ws symmetric nd only hlf of the originl domin ws considered. Moreover, the vrition occurring in the θ direction (i.e. the ngulr coordinte) ws neglected, thus restricting the nlysis to D geometry. Ech dependent vrible ws, therefore, function of the rdil nd xil coordintes, r nd z, respectively, nd of time, t. b) All the phses, i.e. solid, liquid, nd gs, were continuous. c) All the phses were in locl therml equilibrium. d) Vpor pressure ws function of both temperture nd moisture content. e) he trnsport of liquid wter ws due to both gs pressure nd cpillry pressure grdients; cpillry pressure, however, previled over gs pressure ssuming tht the hygroscopic mteril under study ws highly unsturted, like in most drying pplictions. he trnsport of liquid wter, therefore, occurred essentilly by cpillry pressure. f) Vpor ws trnsferred by pressure nd concentrtion grdients; the term contining pressure driven flow, however, ws neglected nd moleculr diffusion ws considered s the previling mechnism. It is worthwhile remrking tht the previous ssumptions e) nd f) my describe well, for instnce, lower temperture drying processes, for which pressure driven flow is not s importnt [Dtt, 007]. g) In the food smple, the contribution of convection to het trnsport ws considered negligible s compred to conduction (Peclèt number in rdil direction tending to nil) nd, therefore, ws ignored; the conservtion eqution referred to ir trnsport in the food smple ws neglected, too. h) he food smple ws multiphse hygroscopic porous medium; yet, it ws ssumed to be fictitious continuum. i) Evportion occurred over the entire food domin nd lso t food outer surfces. j) Moisture removl from the surfce took plce by vpor trnsport, diffusing into the boundry lyer developing in the drying ir, nd by liquid wter trnsport, evporting t the outer food surfces. Both vpor nd liquid wter were convected wy by the drying ir, whose velocity field is expected to ffect strongly the interfcil rtes of het nd mss trnsfer s it ws found, for instnce, in the cse of tngentil flow long immersed bodies with simultneous het nd mss trnsfer into the fluid strem [Bird et l. 00]. k) he continuity of both het nd mss fluxes occurred t the food/ir interfces. On the bsis of the bove ssumptions, the unstedy-stte mss blnce equtions referred to the trnsport of liquid wter nd vpor in the food smple led, respectively, to [Bird et l. 00, Zhng nd Dtt, 004] C w + ( D C ) + I = 0 (1) w w Cv + ( D C ) I = 0 () v v he energy blnce in the food mteril led, ccording to the Fourier s lw, to the unstedystte het trnsfer eqution [Bird et l. 00, Zhng nd Dtt, 004]: sc p s ( k ) + I = 0 t eff λ (3)

4 Obtining I from Eq. nd replcing it in Eqs. 1 nd 3, the following expressions were derived: C w C + v + ( D ) + ( ) = 0 w Cw Dv C (4) t t v + ( ) Cw + ( D C ) (5) C k λ = 0 s p s eff w w he vpor concentrtion, Cv, ws defined using the idel gs ssumption, thus its timederivtive nd grdient red s: C p R p + v = 1 v p C v v w (6) t Cw p = v p v p C 1 + v v c (7) R c w w where pv ws the vpor pressure of wter expressed, through n equilibrium reltionship, s function of locl vlues of both temperture nd moisture content [Rtti et l. 1987]: pv ln pvs = X b exp( 1.87 X b) X b ln( pvs ( )) ( ) (8) It is worthwhile remrking tht the physicl nd the trnsport properties of food, s introduced in the bove system of PDEs, were expressed in terms of the locl vlues of temperture nd of moisture content, s suggested by the vilble literture reltionships. As fr s drying ir ws concerned, the unstedystte momentum blnce, i.e. the so-clled Reynolds-verged Nvier-Stokes equtions, coupled to both the continuity eqution nd the trnsport equtions for k nd ω led to [Wilkox 004]: + = 0 t u (9) (10) u + u u + ' ' u u = p+ [ η ( u + ( u) )] ( u + u ) β kω η k t + u k [( η σ η )( k) ] ( ) t = + k t + (11) [( )( )] ( αω η + σ η ω + ) η ( u + ( u) ) β ω ω + u ω = t ω t k t (1) he deviting prt of the Reynolds stress ws expressed s: ' ' u u ' ' trce u u I = η u + ( u) (13) 3 [ ( )] η t, i.e. the turbulent or eddy viscosity, ws defined s: η k t = ω (14) t k nd the sphericl prt, introduced in Eq. 13, ws written in terms of the turbulent kinetic energy, in order to ttin proper closure for the turbulence : trce ' ' u u I = k 3 (15) 3 he energy blnce in the drying ir, led to [Bird et l. 00]: C p ( k ) + C u = 0 t (16) p Finlly, the mss blnce on wter, s vpor, in the drying ir, led to [Bird et l. 00]: C + ( D C ) + u C = 0 (17) t Eqs nd represented system of unstedy, non-liner, prtil differentil equtions. A set of initil conditions, typiclly previling in n industril drying process, ws necessry to perform the numericl simultions. As fr s ir ws concerned, it ws ssumed tht the temperture in the drying chmber, 0, hd the vlue of 98 K nd tht the wter concentrtion, C 0, ws equl to 0.64 mol/m 3 (corresponding to reltive humidity, U r0, of 50%). It ws lso ssumed tht, before the drying process occurred, the ir ws sttionry nd tht the pressure in the drying chmber, p 0, ws 1 tm. he initil vlues of food temperture, 0, nd of its moisture content, C 0, were set equl to 83 K nd to mol/m 3, respectively. he dopted C 0 vlue referred to n verge initil moisture content typicl of fresh pottoes nd corresponding to 0.8 kg H O/kg w (on wet bsis) or to 4.06 kg H O/kg ds (on dry bsis). he boundry conditions were summrized in Fig. where ech different boundry ws identified by n integer rnging from 1 to 9. Figure. Boundry conditions used to formulte the theoreticl.

5 At the food-ir interfce, where no ccumultion occurred, the continuity of het nd wter fluxes, both s liquid nd vpor, ws imposed. Moreover, thermodynmic equilibrium t the ir-food interfce ws formulted nd expressed in terms of wter ctivity (eq. 8), thus ccounting lso for the effects of physiclly bound wter. On boundry, it ws ssumed tht ir temperture nd its moisture concentrtion were equl to the vlues mesured t the drier inlet. hese two conditions re ctully vlid under the ssumption tht the temperture nd concentrtion profiles were confined to two very thin regions which developed close to the foodir interfce. At the drier outlet (boundry 3), conduction nd diffusion were neglected in fvour of convection, which previled. Finlly, the boundry conditions for momentum blnce t the solid surfces were expressed in terms of logrithmic wll function. Food shrinkge ws ed defining the locl totl strins {dε} s function of chnges in mechnicl strins {dε s } (constrined deformtion due to food mechnicl properties) nd in shrinkge strins {dε 0 }: { dε} = { dε s } + { dε 0 } (18) otl strin {dε} ws ctully function of the totl displcement {du} { d ε} = [ A]{ du} (19) he chnges in stresses {dσ} were function of the vritions of mechnicl strins {dε s } { dσ } = [ D]{ dε s } (0) o express the free drying shrinkge strins {dε 0 }, it ws ssumed tht the free deformtion due to moisture loss ws proportionl to the time vrition of wter content, through constnt (the hydrous compressibility fctor) [Benboudjem, 005]. his constnt ws preliminrily estimted from the experimentl dt showing drying shrinkge vs. weight loss. he virtul work principle ws formulted to determine the equilibrium eqution. Assuming tht zero body nd surfce forces were pplied to the food smple, it ws obtined tht: δ { dε } { dσ} dv = 0 V (1) As fr s the boundry conditions were concerned, one side of the food rested on the drier net nd ws chrcterized by null displcement in z direction, wheres the other exposed food surfces were free to move. he clculted displcements were used to determine the ctul modifictions of integrtion domins nd, therefore, to solve the present moving boundry problem represented by the nisotropic nd progressive reduction of food smple volume. 3. Use of COMSOL Multiphysics he solution of the present ws performed by Comsol Multiphysics 3.4 nd ctully consisted of the following sequentil steps: 1) the coupled trnsfer of momentum (for ir only) of het nd of wter (for both food nd ir), together with the virtul work principle, ws initilly solved with reference to fixed domins; ) the obtined solutions, in prticulr, the ctul displcements occurring in the food smple s consequence of wter removl nd depending on food mechnicl properties were, then, used s inputs to clculte the mesh deformtion, ccording to n ALE description; 3) the complete trnsport ws solved once gin with reference to the deformed domins so s to clculte the ctul time evolutions of food volume vrition, together with the profiles of ll the dependent vribles (temperture, moisture content, velocity field, etc.) hving physicl significnce. he motion of the deformed mesh ws ed using Winslow smoothing. Both the domins were discretized by tringulr grids, which resulted into totl number of 1080 tringulr finite elements, leding to bout degrees of freedom. he mesh ws thicker close to the boundries ctully exposed to the drying ir nd consisted of 5449 elements nd 5353 elements within food nd ir domins, respectively. he considered mesh provided stisfctory sptil resolution for the system under study; the solution ws indeed independent on the grid size, even with further refinements. Lgrnge finite elements of order two were chosen for the components, u r nd u z, of drying ir velocity vector u, for the turbulent kinetic energy, the dissiption for unit of turbulent kinetic energy, for the pressure distribution within the drying chmber nd for the displcements occurring in food smple. As regrds wter concentrtion nd temperture, for both ir nd food,, Lgrnge finite elements of order three were chosen. he reltive nd bsolute tolernces were set to nd , respectively.

On qud-core computer running under Windows Vist 64 bit, drying time of 4 hours ws typiclly simulted in bout 40 minutes when the system of PDEs ws referred to fixed mesh, much longer time of bout 6

Results nd Discussion he proposed provided helpful indictions tht were used to clculte the moisture nd the temperture distributions within both food nd ir nd to nlyze the effect of the operting

3 showed the time evolution of potto moisture content (on dry bsis) referred to domin whose shpe nd dimension chnged with time s consequence of shrinkge.

6 On qud-core computer running under Windows Vist 64 bit, drying time of 4 hours ws typiclly simulted in bout 40 minutes when the system of PDEs ws referred to fixed mesh, much longer time of bout 6 hours ws insted necessry to chieve the solution in the cse of deformed mesh. 4. Results nd Discussion he proposed provided helpful indictions tht were used to clculte the moisture nd the temperture distributions within both food nd ir nd to nlyze the effect of the operting conditions on drying behvior. For the ske of brevity only some of the most representtive results were herefter presented. Fig. 3 showed the time evolution of potto moisture content (on dry bsis) referred to domin whose shpe nd dimension chnged with time s consequence of shrinkge. It could be observed tht, s expected, externl surfces got dry more rpidly thn the inner regions. When, initilly, food moisture content ws high nd, consequently, cpillry diffusivity hd lrge vlues, the moisture exhibited slight difference between the core nd the outer surfce. hese differences tended progressively to enlrge due to both decrese in cpillry diffusivity nd n increse of surfce temperture tht strted rising bove the wet bulb temperture (dt not shown). During the flling rte period, the evportion rte previled over the cpillry flux inside the food; the mount of vpor in the internl holes ws significnt nd the moleculr diffusion of vpor took plce to significnt extent. Moreover, s shown in Fig. 3, food domin hd not n isotropic deformtion; in fct, on those sides (where drying ir ctully impinged) chrcterized by higher drying rte, much lrger (nd fster) deformtion ws observed. he importnce of considering food shrinkge so s to ttin n improved prediction of the ctul vlues of interfcil het nd wter fluxes ws confirmed by Fig. 4, showing ir velocity profiles s function of process time. It is worthwhile remrking tht the present trnsport did not rely on the exploittion of ny semi-empiricl correltions to clculte the het nd mss trnsfers t the food/ir interfces; rther, the ctul trnsport rtes strongly depended of the ir velocity field tht, therefore, hd to be exctly clculted, time by time, ccounting for the vritions of food shpe nd of its dimension. Figure 3. ime evolution of potto moisture content during drying ccounting for shrinkge effect ( ir = 33 K, U r = 50%, u 0 =. m/s). Figure 4. ime evolution of ir velocity field nd of the stremlines clculted ccounting for food shrinkge ( ir = 33 K, U r = 50%, u 0 =. m/s). Fig. 5 showed, in typicl cse, comprison between the verge moisture content, <Xb>, clculted, respectively with reference to constnt nd vrible volume of potto smple. It could be observed tht, by the proposed

7 pproch, significnt difference of <Xb> occurred throughout the considered time horizon. he predicted time-evolution of food volume exhibited plteu vlue fter bout 140 min. Averge moisture content, <Xb> [/] <Xb> with shrinkge <Xb> without shrinkge Volume [m^3] ime [min] Figure 5. Comprison between the predicted time evolutions of verge moisture content (on dry bsis) nd food volume vrition during drying ( ir = 33 K, U r = 50%, u 0 =. m/s). 7. Conclusions he trnsport phenomen involved in food drying process were nlyzed. A generl predictive, i.e. not bsed on ny semiempiricl correltion for estimting het nd mss fluxes t food-ir interfce, ws formulted. he llowed predicting food shrinkge by coupling the clssicl trnsport equtions to the virtul work principle, written with reference to deformed mesh whose movement ws described by the ALE method. he obtined results confirmed tht, even if much more complicted nd time consuming, it is crucil to develop complete cpble of predicting the ctul vrition of food shpe nd to hve n insight into the clcultion of the mechnicl stresses occurring in the smple during drying. his informtion, in fct, llowed clculting the ctul deformtion nd might lso be used s qulity index for dried foods. 8. References Bird, R.B., Stewrt, W.E., Lightfoot, E.N. rnsport Phenomen, nd ed., John Wiley & Sons, London (00). Benboudjem F., Mefth F., orrenti J.M., Interction between drying, shrinkge, creep nd crcking phenomen in concrete, Engineering Structures, 7, (005) Curcio, S., A Multiphse Model to Anlyze rnsport Phenomen in Food Drying Processes, Drying echnology, 8 (6), (010) Smple volume [m 3 ] Chen, X. D., Moisture Diffusivity in Food nd Biologicl Mterils, Drying echnology, 5, (007). Dtt, A. K. Porous medi pproches to studying simultneous het nd mss trnsfer in food processes. I: Problem formultions. J. Food Eng., 80, (007). Kondjoyn, A., Boisson, H.C., Comprison of Clculted nd Experimentl Het trnsfer Coefficients t the Surfce of Circulr Cylinders Plced in urbulent Cross-flow of Air, J. Food Eng., 34, (1997). Myor, L., Sereno A. M., Modelling shrinkge during convective drying of food mterils: review, J. Food Eng., 61 (3), , (004). Rtti C., Shrinkge During Drying of Foodstuffs, J. Food Eng., 3, (1994). Rtti, C., Crpiste, G. H., Rotstein, E., A new wter sorption equilibrium expression for solid foods bsed on thermodynmic considertions, J. of Food Science, 54(3), (1987). horvldsson, K., Jnestd, H., A for simultneous het, wter nd vpour diffusion, Journl of Food Engineering, 40, (1999). Wilcox, D.C., urbulence Modeling for CFD, nd Ed, DCW Industries Inc., (004). Verboven, P., Nicolï, B. M., Scheerlinck, N., De Berdemeker, J., he Locl Surfce Het rnsfer Coefficient in herml Food Process Clcultions: A CFD Approch, J. Food Eng., 33, 15-35(1997). Viollz, P.E.; Rovedo, C.O. A drying for three-dimensionl shrinking bodies.j. Food Eng., 5, (00). Yng, H., Ski, N. nd Wtnbe, M., Drying with non-isotropic shrinkge deformtion undergoing simultneous het nd mss trnsfer, Drying echnology, 19 (7), (001). Zhng J., Dtt A. K., Some considertions in ing of moisture trnsport in heting of hygroscopic mterils, Drying echnology, (8), (004).

First Law of Thermodynamics. Control Mass (Closed System) Conservation of Mass. Conservation of Energy

First Law of Thermodynamics. Control Mass (Closed System) Conservation of Mass. Conservation of Energy First w of hermodynmics Reding Problems 3-3-7 3-0, 3-5, 3-05 5-5- 5-8, 5-5, 5-9, 5-37, 5-0, 5-, 5-63, 5-7, 5-8, 5-09 6-6-5 6-, 6-5, 6-60, 6-80, 6-9, 6-, 6-68, 6-73 Control Mss (Closed System) In this section