Modeling of simultaneous heat and mass transfer during convective oven ring cake bakingMelike Sakin-Yilmazer a ,⇑,Figen Kaymak-Ertekin a ,Coskan Ilicali ba Ege University,Faculty of Engineering,Department of Food Engineering,35100Bornova,Izmir,TurkeybKyrgyzstan-Turkey Manas University,Faculty of Engineering,Department of Food Engineering,720044Bishkek,Kyrgyzstana r t i c l e i n f o Article history:Received 11November 2011Received in revised form 2February 2012Accepted 11February 2012Available online 20February 2012Keywords:Ring cakeBaking modeling SimulationFinite difference method Heat and mass transfera b s t r a c tThe convective oven ring cake baking process was investigated experimentally and numerically as a simultaneous heat and mass transfer process.The mathematical model described previously by the authors for cup cake baking was modified to simulate ring cake baking.The heat and mass transfer mech-anisms were defined by Fourier’s and Fick’s second laws,respectively.The implicit alternating direction finite difference technique was used for the numerical solution of the representative model.Prior to the utilization of the developed model in predicting the temperature and moisture profiles for ring cake bak-ing,the results of the numerical model were compared with analytical results involving only heat or mass transfer with constant thermo-physical properties.Excellent agreement was observed.The numerical temperature and moisture contents predicted by the model were compared with the experimental pro-files.They agreed generally reasonably well with the experimental temperature and moisture profiles.Ó2012Elsevier Ltd.All rights reserved.1.IntroductionBaking is a complex process since the physical,chemical and biochemical changes taking place are of complex nature and these changes are simultaneous and coupled.Quantitative understand-ing of processes like volume expansion,evaporation of water,for-mation of a porous structure,denaturation of protein,gelatinization of starch,crust formation and browning reaction are still limited (Sablani et al.,1998;Zhang and Datta,2006).Sim-plified models for the baking process can be developed by treating baking as simultaneous heat and moisture transfer.Even for such simplified models,the estimation of the thermo-physical and transport properties,and the transfer coefficients is a challenging task.Nevertheless,mathematical models supply valuable informa-tion on baking process which may be utilized to optimize the bak-ing conditions to reduce energy consumption.As a traditional process,optimization and process control to ob-tain the desired final product quality is still largely based on expe-rience and good craftsmanship rather than on predictive calculations (Feyissa et al.,2011).Experimental and theoretical studies for baking in general and cake baking in particular were gi-ven in details in Sakin (2005)and Sakin et al.(2007b)and will not be repeated here.Mathematical models representing the baking process reduce the need for the tedious trial and error baking experiments to achieve a high quality product and they serve as a quick and prac-tical tool for pre-design,optimization and process control (Sakin et al.,2007b ).Ring cake is a popular cake geometry all over the world.Selecting this geometry as the test geometry and developing accurate mathematical models will have practical implications to-wards optimizing the baking conditions for better quality and en-ergy saving.Since there is no published work in literature about ring cake baking and its modeling,a two dimensional simplified mathemat-ical model for simultaneous heat and moisture transfer will be developed.Volume rise during baking will be considered.The pre-dictions of the numerical model will be compared with experimen-tal temperature and moisture content profiles.2.Material and experimentalThe cake batter was prepared by thoroughly mixing of 49.4%(of total weight)ready to bake dry cake mix (Dr.Oetker,containing wheat flour,sugar,corn starch,and baking powder),24.7%pasteur-ized whole liquid egg,16.2%vegetable margarine and 9.7%water.Its preparation was given in details in Sakin et al.(2007b).The ini-tial moisture content of the batter was %0.50kg water/kg dry solid.Teflon coated Aluminum was chosen as the material for ring cake mould due to Aluminum’s high thermal conductivity value (206W/mK,Geankoplis,2003).The ring baking mould was an annular ring having an annular radius of 0.04m,radius of 0.11and a height of 0.05m.Baking experiments were carried out in an electrical baking oven (Teba High-01Inox)which was0260-8774/$-see front matter Ó2012Elsevier Ltd.All rights reserved.doi:10.1016/j.jfoodeng.2012.02.020⇑Corresponding author.Tel.:+902323113039;fax:+902323427592.E-mail addresses:melike.sakin@.tr (M.Sakin-Yilmazer),figen.ertekin@.tr (F.Kaymak-Ertekin),coskan.ilicali@ (C.Ilicali).0.39Â0.44Â0.35m(LÂWÂH)wide.The batter was baked at three different oven temperatures(165,185and205°C)under forced convection conditions.The oven was preheated before the baking experiment to obtain uniform oven baking conditions.The air was circulated by a fan in-stalled on the back side of the oven at a constant speed(0.56m/s, measured by Airflow anemometer,LCA6000,UK)and fresh air en-tered steadily into the oven cavity,through a hole(/:28mm)up in the oven to prevent moisture accumulation.Moisture and temperature profiles were recorded during baking experiments.The procedures for taking temperature and moisture content data were similar to that given in Sakin et al.(2007b). Moisture profiles were measured at the top,bottom,annular and side surface crusts as two parallel measurements.An infrared moisture analyzer(Ohaus,MB200,USA),with a precision of ±0.007g,was used for the moisture content determination of the samples.The results obtained by the instrument were previously validated by the standard air oven method for total solids and moisture in baked products,as1h at130°C(AOAC,1995).Temperature profiles were recorded at the top(r=0.063m), bottom(r=0.08m)and annular(z=0.015m)surfaces by using j-type thermocouples(wire size,/:1mm;accuracy:1–1.5°C)that were calibrated in water and ice baths against glass mercury ther-mometer,and by a multi-channel data-logger(Hanna Inst.,HI 98804,Portugal)having an accuracy of±0.5°C,excluding probe er-ror.All the temperature measurements were conducted as two parallels.Techniques used in thermocouple positioning were sim-ilar to Sakin et al.(2007b).The change in the height of the cake batter was measured by a digital caliper(Mitutoyo,Digimatic,Japan).3.Mathematical model descriptionSimultaneous heat and moisture transfer mechanisms in a ring cake during baking can be described by the two dimensional(in r and z directions)unsteady state heat and moisture transfer equa-tions with appropriate boundary conditions.The Fourier’s equation for unsteady state heat conduction for solids with constant thermo-physical properties in afinite cylinder geometry can be modified by the addition of a phase change term to account for the evaporation of the product moisture(Sakin, 2005;Sakin et al.,2003,2004,2005,2007b;Tong and Lund, 1993;Thorvaldsson and Janestad,1999;Shilton et al.,2002).@T¼a1Á@Tþ@2Tþ@2T!þkPÁ@Xð1Þwhere,T is the temperature(°C),t is the time(s),a is the thermal diffusivity(a=k/q c P,m2/s),r is the distance in the radial direction (m),z is the distance in the upward direction(m),k is latent heat of vaporization of water(J/kg),c p is the specific heat capacity(J/kgK),X is the moisture content(kg water/kg dry solid),k is the thermal con-ductivity(W/mK)and q is the density(kg/m3).Uniform initial temperature distribution throughout the cake batter was assumed:t¼0;T¼T ini at all r and zð2Þwhere,T ini is the initial batter temperature(°C).The top,bottom,side and annular surfaces boundary conditions for heat transfer were written as Eqs.(3)–(6),respectively as shown below.kÁ@T@zr¼r;z¼H¼hÁT aÀT r;HðÞð3ÞÀkÁ@T@zr¼r;z¼0¼hÁT aÀT r;0ðÞð4ÞkÁ@T@rr¼R;z¼z¼hÁT aÀT R;zðÞð5ÞNomenclatureA(i),B(i),C(i),AA(i),BB(i),CC(i)elements of coefficient matrixBi Biot number for heat transfer,Bi r¼h cÂR=k;Bi z¼h cÂðH=2Þ=kBi m Biot number for moisture transfer,Bi mr¼k cÂRk cÂR=D eff;Bi mz k¼k cÂðH=2ÞD effc p specific heat(J/kgK)D eff effective moisture diffusivity(m2/s)D(i),DD(i)elements of the right-hand side vectorH thickness of cake batter(m)h surface heat transfer coefficient(W/m2K)k thermal conductivity(W/mK)k c convective mass transfer coefficient(m/s)M number of intervals in z directionMP1M+1,the top surface layerN number of intervals in r directionNH number of intervals in the cake in r directionNH1NH+1,annular surfaceNP1N+1,the radial axisr distance in radial direction(m)R the radius(m)T temperature(°C)T r oven surface temperature(°C)t time(s)U b lower surface overall heat transfer coefficient(W/m2K) U l lateral surface overall heat transfer coefficient(W/m2K) X moisture content(kg water/kg dry solid)z distance in upward direction(m) Subscriptsa ambientannulus annular surfaceb batterbottom bottom surfacecup baking cupdc donecakeeff effectiveffinali the i th nodeini initialj the j th noder,z the r and z directionssides side surfacetop top surfaceGreek symbolsa thermal diffusivity(m2/s)k latent heat of vaporization of water(J/kg) D r,D z the spatial increments in r and z directions.D x thickness(m)D t the time step(s)q the product density(kg/m3)290M.Sakin-Yilmazer et al./Journal of Food Engineering111(2012)289–298ÀkÁ@Tr¼R a;z¼z¼hÁT aÀT Ra;zðÞwhere,h is a convective and radiative surface cient(W/m2K),T a is the baking oven thickness of product(m),R is the radius of the R a is the annular radius(m).Fick’s2nd law describes the unsteady statein solids.Infinite cylinder geometry,the change tent with time and position during baking (Crank,1975).@X¼D eff 1Á@Xþ@2Xþ@2X!Initially a uniform moisture distribution waspositionst¼0;X¼X ini at all r and zwhere,X ini is the initial moisture content of cakedry solid).The experimental data obtained in thisthe moisture contents in the top,bottom,sidedropped sharply at the start of the baking process.Although the top surface moisture content was the lowest,the moisture con-tents for the bottom,side and annular crusts were of comparable values.As the cake batter expands in the upward direction,sharp decrease in the moisture contents and sharp increase in tempera-tures were observed in the layers of the batter in contact with the cake mould.When crust formation begins on the surfaces in contact with the mould surfaces,it is believed that the crust sepa-rates slightly from the hot Aluminum surfaces enabling the loss of moisture also from these surfaces.Therefore,to account for the moisture loss in surface layers effective mass transfer coefficients were assigned to these surfaces.The top surface moisture transfer boundary condition was defined as below,by Eq.(9).ÀD effÁ@X@zr¼r;z¼H¼k c;topÁX r;HÀX1ðÞð9Þwhere,k c,top is an effective convective mass transfer coefficient(m/ s),X1is the equilibrium moisture content in the oven medium.The bottom,side and annulus boundary conditions were de-fined similarly as follows by Eqs.(10)–(12),respectively.D effÁ@X@zr¼r;z¼0¼k c;bottomÁX r;0ÀX1ðÞð10ÞÀD effÁ@X@rr¼R;z¼z¼k c;sidesÁX R;zÀX1ðÞð11ÞD effÁ@Xr¼R a;z¼z¼k c;annulusÁX Ra;zÀX1ðÞð12Þ4.Numerical solution4.1.Grid systemThe cake batter in a ring shaped baking mould was a ring of ra-dius R,annular radius R a and height H.A schematic diagram of the coordinate system used is given in Fig.1.Any rectangular cross-section from the annular surface represents the whole product.A grid system where terms i and j were used to index the nodes in r and z directions was formed(Sakin(2005)).D r and D z were spa-tial increments,and N and M were numbers of intervals in the r and z directions.M was taken as20and N was taken as44.The number of intervals within the cake in the radial direction NH was taken as 28.The volume increase and the subsequent decrease during bak-ing were taken into account as linear changes in height.In the numerical solution,the height of the baking product was refreshed at each time step by using the regression equations obtained from experimental height versus time data.4.2.Solution methodThe governing partial differential equations,defining heat and moisture transfer have been approximated byfinite differences. The implicit alternating direction method was used for the solu-tion.The tri-diagonal matrix systems were formed and solved by the Gaussian Elimination method for the temperature profile,T i,j and moisture profile,X i,j at all points from i=1to NH+1,j=1to M+1,with a time step(D t)of2s(Carnahan et al.,1969;Chandra and Singh,1995).The elements of the coefficients matrix and the right-hand side vector in tri-diagonal matrix system were defined as A(i,j),B(i,j), C(i,j)and D(i,j)for heat transfer;AA(i,j),BB(i,j),CC(i,j)and DD(i,j) for moisture transfer,respectively by Sakin(2005).These coeffi-cients were modified for ring cake baking.The simulation of the model has been done by a computer pro-gram written on Fortran95.Theflow schema of the program is similar to the one given in(Sakin et al.,2007b).4.3.Verification of numerical method against analytical solutions for simplified geometries and materialsThe mathematical model developed for ring cake baking and its numerical results for temperature and concentration profiles must be verified against well known analytical solutions before ring cake baking simulation.The solution of the given coupled partial differ-ential equations,defining the simultaneous heat and mass transfer case of baking,by analytical methods is essentially impossible (Franks,1972).Therefore,the developed model was compared with possible analytical solutions and independently for heat and mass transfer.For heat transfer verification Eq.(1)without the phase change term was used.To eliminate the effect of mass trans-fer,all the mass transfer coefficients were taken as zero.To trans-form the ring cake to an infinite slab,the height of the cake was taken to be much smaller than the radius.When this is the case,M.the ring cake becomes an infinite slab with thickness being equal to the height of the ring cake.Predictions of the numerical model were compared with analytical solutions (Cengel,2003)for a wide range of Biot numbers.Excellent agreement was observed.To the knowledge of the authors,no analytical solution exists for unstea-dy state heat conduction in an annulus,with convective boundary conditions.Therefore,for radial testing of the model,the height of the ring cake was taken to be much larger than the radius and the number of radial intervals in the ring cake,NH,was taken to be equal to the total number of intervals À1,N À1.Furthermore,the heat transfer coefficient for the annular surface was taken to be zero (adiabatic axis).Under these conditions,the resulting geome-try will closely simulate an infinite cylinder geometry.The predic-tions of the ring cake numerical model were compared with the analytical solutions for infinite cylinder.A very high level of agree-ment between analytical and numerical models was obtained.A fi-nal check for the heat transfer section was performed by comparing the predictions of the numerical model with analytical solutions for a finite cylinder.The method of superposition was used for the analytical solution of the two dimensional partial dif-ferential equations.Input data used in the verifications for a finite cylinder are given in Table parison of the predictions of the numerical model for the center temperature with analytical solu-tions for a finite cylinder at Bi r =Bi z =1and Bi r =Bi z =10are shown in Fig.2.Excellent agreements were observed.For mass transfer verifications of the model,different cases were considered:In all these cases,the ambient temperature was as-sumed to be equal to the initial temperature and the latent heat was taken to be zero to eliminate heat transfer effects.Initially,the radius of the cake was taken to be much larger than the height and only convective mass transfer on the top surface was consid-ered.Under these conditions,the ring cake becomes an infinite slabwith half thickness being equal to the height of cake.For the second case used in mass transfer verifications,the radius of the cake was taken to be much larger than the height and convective mass trans-fer was considered for the top and bottom surfaces.The same mass transfer coefficients were used for the both surfaces.For this case,the ring cake becomes an infinite slab with thickness equal to height of the cake.Numerical predictions for the surface,symmetry axis and average moisture content were compared with the analyt-ical solutions for a wide range of mass transfer Biot numbers (Bi m ).Very good agreements were observed for all cases.For radial testing of the model,the height of the ring cake was taken to be much lar-ger than the radius and the number of radial intervals in the ring cake,NH,was taken to be equal to the total number of intervals À1,N À1,similar to the heat transfer case.Furthermore,the mass transfer coefficient for the annular surface was taken to be zero.Un-der these conditions,the resulting geometry will closely simulate an infinite cylinder geometry.Again,very good results were ob-tained for these cases for center,surface and average moisture con-tents.Final verification of the model for mass transfer was carried out for an impermeable hollow short cylinder where the annular ra-dius was 1/40times the cylindrical radius.This geometry is practi-cally a short cylinder.The input parameters used for mass transfer verifications are also given in Table 1.Constant transport properties and constant product thickness were parison of the predictions of the numerical model with analytical solutions for Bi mr =Bi mz =1and Bi mr =Bi mz =10for center and average moisture contents are shown in Figs.3and 4,respectively.The average moisture content was calculated by the following equation;X ave ¼2R 2ÀR 2aZRR aXrdr ð13ÞAs can be observed from Figs.2–4,the numerical model devel-oped predicts accurately the analytical temperature and moistureprofiles.The slight differences in the average moisture contents in Fig.4were attributed to numerical integration error in Eq.(13).Therefore,it was concluded that the numerical model devel-oped for ring cake baking can be used to simulate experimental temperature and moisture content profiles.4.4.Input parameters to ring cake baking simulation and method of validation of simulation resultsThe ring cake baking process parameters used during the numerical solution were the product dimensions,thermo-physicalTable 1Input parameters used in the verification of the numerical model.Cake dimensions (m)H :0.0462,R =0.0231Initial temperature,(°C)T ini =25.7Ambient temperature,(°C)T a =185Heat transfer coefficients (W/m 2K)h :10.39,103.9Thermal conductivity (W/mK)k :0.24Density (kg/m 3)q =945Specific heat capacity (J/kg K)Cp =2650Initial moisture content (kg water/kg dry solids)X ini =0.56Ambient moisture content (kg water/kg dry solids)X 1:0Mass transfer coefficient (m/s)k c =1.3Â10À5Effective diffusivity (m 2/s)D ef =3Â10À7,3Â10À8292M.Sakin-Yilmazer et al./Journal of Food Engineering 111(2012)289–298and mass transport properties,baking oven temperatures,the sur-face heat and mass transfer coefficients,the initial temperatureand moisture content of the cake batter which are illustrated by Table 2.They were either experimentally determined,estimated or taken from the literature to simulate the experimental ring cake baking process.The mass transfer coefficients for oven tempera-ture of 205°C were taken to be 20%higher than the corresponding values at 165and 185°C since the baking time to approximately the same final moisture content was 50min at 205°C,compared to 60min at 165and 185°C.The product density and thermal con-ductivity was assumed to change linearly with cake height accord-ing to the following equations:q ¼q b þðq dc Àq b ÞðH ÀH ini ÞðH f ÀH ini Þð14Þk ¼k b þðk dc Àk b ÞðH ÀH ini ÞðH f ÀH ini Þð15ÞThe specific heat capacity was also evaluated separately and then the thermal diffusivities were calculated.The cake height was measured experimentally.The change in cake height with time was represented by regression straight lines.The %absolute deviation (p ,%)between the experimental and numerical results,for temperature and moisture profiles,were cal-culated by Eq.(16)to validate the simulation results (Boquet,Chi-rife,Iglesias,1978).Table 2Baking process parameters and input variables.Ring cake dimensions (m)H a :0.025,R a a :0.04,R a :0.11Baking temperatures (o C)T a a :165,185,205Initial moisture content (kg moisture/kg solids)X ini a :0.543–0.567Initial temperature (o C)T ini a :23.7–25.7Batter density (kg/m 3)q b a :947.8Donecake density (kg/m 3)q dc a :320Batter thermal conductivity(W/mK)k b b :0.24k b b :0.24Donecake thermalconductivity(W/mK)k dc b :0.121Specific heat capacity (J/kg K)C p b :2650Heat transfer coefficients (W/m 2K)h a :16.3Mass transfer coefficients (m/s)k c ,top a ,c :2.40Â10À5,k c ,bottom a :1.68Â10À5,k c ,sides a :1.56Â10À5,k c ,annulus a :1.56Â10À5k c ,205a =1.2k c,165Effective moisture diffusivity (m 2/s)D eff a ,d :1.7Â10À8a Experimentally measured or estimated values.b Baik et al.(1999).c Demirkol et al.(2006).dSakin et al.(2007a).M.Sakin-Yilmazer et al./Journal of Food Engineering 111(2012)289–298293p ð%Þ¼100N ÁX Ni ¼1X exp ÀXnum j jX expð16Þ5.Results and DiscussionTemperature and moisture profiles at certain positions of thering cake during baking were determined.The difference between the experimental and numerical moisture profiles were compared by p%values;where,a value below 10%indicates the good fit be-tween both results (Boquet,Chirife,Iglesias,1978).The experimentally observed and numerically predicted tem-perature profiles at the positions of the product top surfaceaccuracy of the predictions.For this reason,the experimental data for the center temperature and model predictions were not re-ported in this research.Althoughmeasurements were performed at three oven temperatures,165,185and 205°C,only the results at 165and 205°C were shown to make the figures less crowded.The top surface numerical temperature profile predictions were in good agreement with the experimental results,especially to-wards the completion of baking period (Fig.5).The p %values were calculated as 6.5%and 3.8%,for 165and 205°C,respectively.The specification of the heat transfer coefficient is crucial in heat trans-fer calculations especially at small Biot numbers.When we have a ring cake,the characteristic dimension to be used in the Biot num-ber is ambigous,since we have an annular geometry and the height of the cake also changes.However,if we take (R –R a )is the charac-294M.Sakin-Yilmazer et al./Journal of Food Engineering 111(2012)289–298h¼_mWkAðT aÀTÞtð17Þwhere_m W is the total moisture evaporated(kg)during a com-plete baking run,A is the total heat transfer area(m2),T is the sur-face temperature(o C)and t is the baking time(s);3600s for165°C and3000s for205°C.The surface temperature was assumed to be the mean of ambient temperature and100°C.A single heat trans-fer coefficient was used in all runs.Radiative contribution to heat transfer was not considered.Sakin et al.(2009)have shown that the oven wall temperature and the oven medium temperature dif-fer appreciably depending on oven medium temperature.For an oven medium temperature of160°C,the oven wall average tem-perature was181.6°C in the forced convection oven.At higher oven temperatures,the wall and medium temperature approaches to each other.In the present work,lack of experimental data on heat transfer coefficients will lead to inaccuracies in temperature predictions.The predicted bottom temperatures(Fig.6)were in good agree-ment with the experimental temperatures especially at an oven medium temperature of205°C.The p%values were calculated as 6.1%and3.9%,for165and205°C,respectively.The same heat transfer coefficient was used for all surfaces.However,the convec-tive and radiative heat transfer coefficients will be different for dif-ferent surfaces.Accurate determination of the heat transfer coefficients will be essential for more accurate predictions.Annular surface temperature predictions by the present model were com-pared with experimental values in Fig.7.Satisfactory agreement was observed for oven medium temperature of165°C,where the p%value was calculated as5.5%.However,experimental tempera-tures were greater than the predicted temperatures for205°C.The p%value was calculated as10.9%.This larger deviation was again attributed to the use of the same heat transfer coefficient for all surfaces.The estimation of thermo-physical properties of the ring cake and the transfer coefficients were very important in the numerical solution.They were mostly taken from the related liter-ature(Table2).The differences between the experimental temper-ature profiles and the numerical predictions can also be attributed to these inaccuracies in the thermo-physical properties.More com-prehensive studies on the thermo-physical property estimation of a baking cake batter and the estimation of heat transfer coefficients at different surfaces on a ring cake will result in more accurate numerical models.Similarly,the effective moisture diffusivity term and the mass transfer coefficients have a strong effect on the moisture profile prediction.The relative importance of the estimation of these two parameters will depend on the mass transfer Biot number.It may be shown from the data in Table2that mass transfer Biot numbers vary between11and66.For mass transfer,the internal resistance controls the diffusion of moisture.Therefore,accurate estimation of the effective diffusivity is essential for accurate mass transfer modeling.The available literature for this property for cake baking was limited mostly to the model equations generated for a thin layer of cake batter during drying/baking(Baik and Marcotte,2002;Sakin et al.,2005,2007a)and cup cake baking (Sakin et al.,2007b).As pointed out by Sakin et al.(2007b)the use of the proposed effective moisture diffusivity expressions by Baik and Marcotte(2002)and Sakin(2005)resulted in higher moisture profiles than experimental results.Although the effective diffusivity does not depend on the system geometry,for the same material,the geometry chosen in the estimation of the diffusivity affects the diffusion behaviour.Thus,the effective diffusivities cal-culated from experimental moisture data obtained from the drying or baking of a model system,will be indirectly affected by the geometry,in this case the ring cake.Rather than modifying these expressions,a constant effective diffusivity value was used for the sake of simplicity.The effective moisture diffusivity was taken as1.7Â10À8m2/s in line with the observations of Sakin(2005).When the moisture contents of different crusts of the ring cake were examined,it was observed that generally the top crust had the lowest moisture content and the bottom crust had the second lowest moisture.Moisture profiles for the side and annular crusts were higher than the top and bottom moisture contents and were very similar.Considering the moisture profiles,the top surface con-vective mass transfer coefficient was taken as 1.3Â10À5m/s (Demirkol et al.,2006).The effective convective mass transfer coef-ficients for the other surfaces were taken as shown in Table2.The experimentally observed and numerically predicted moisture pro-files at the top,bottom,side and annular surface positions for oven temperatures of165and205°C were shown in Figs.8–11.In almost all cases,thefinal numerical moisture contents were in close agreement with the experimental values.For the top crust moisture content(Fig.8)the numerical and experimental moisture profiles agreed quite well at165°C(p%value is9.1%).For oven temperature of205°C,the experimental profiles were lower than the numerical profiles;with a p%value larger than10%,indicating that the temperature dependence of the mass transfer coefficient and the effective diffusivity should be considered for this surface. For the bottom surface(Fig.9),the experimental and numerical moisture profiles agreed very well at205°C with a p%8.2.How-ever,at165°C,experimental profiles were higher than the numer-ical ones(the p%value equals to10.0).This can again beattributed M.Sakin-Yilmazer et al./Journal of Food Engineering111(2012)289–298295。
