Heat and Mass Transfer Modelling During Freezing ... - COMSOL.com

Heat and Mass Transfer Modelling During Freezing of FoodstuffsO. Rouaud *1 , Q.T. Pham 21 LUNAM Université, ONIRIS, CNRS, GEPEA, UMR 6144, Nantes, France, 2 University of New SouthWales, Sydney, Australia*Corresponding author: ONIRIS, site de la Géraudière, BP 82225, 44322 Nantes cedex 3, France –olivier.rouaud@oniris-nantes.frAbstract: The objective of this study is todevelop a mathematical model to determine theweight loss and the freezing rate during thefreezing of unwrapped foodstuffs in a very coldenvironment. The model allows comparing twofreezing processes applied to methylcellulosegel; the first uses nitrogen gas at -80°C and thesecond is the classical method using cold air. Inorder to take into account the phase change, themodel includes thermophysical properties thatare temperature dependent and a model based onSchwartzberg’s model is used for the iceformation. The numerical study reveals that thetime step has to be chosen carefully in order topreserve the heat balance during the simulationof freezing and that the mesh grid has to berefined near the surface to take into accountstrong gradients of water content. Numericalresults agree well with experimental data andconfirm that the weight loss and freezing timecan be reduced if low temperatures are used.Keywords: freezing, cryogenic, weight loss,apparent heat capacity.1. IntroductionFreezing is a preservation method widelyused in the food industry and frozen foods can beobtained by several techniques. Among themcryogenic freezing is useful for increasing thefreezing rate thanks to low temperatures. Thewater to ice transition induces mechanical stressdue to expansion during phase change andexperimental results confirm that freezingproduces structural damages regardless of thetype of freezing [1,2]. However numerous andsmaller ice crystals result in reduced freezedamage. Because the number of ice crystals isknown to be inversely dependent to the freezingrate, freezing in cryogenic conditions canimprove the quality of food [1,2].A rapid reduction of the surface temperatureof unwrapped food results in a rapid reduction ofpartial vapour pressure at the surface of food.Hence in case of rapid freezing, a reduction ofthe mass loss by evaporation is expected incomparison with slow freezing conditions [1,3].There are many models developed forcoupled heat and mass transfer in foodprocessing but few of them are yet available forfreezing, especially if mass transfer isconsidered. Efforts remain to be made inmodelling if we want to understand and predictthe behaviour of food quality factors [4,5]. Themain objective of this study is to develop ageneral model that can predict the weight lossand the freezing time according to freezingconditions, especially in very cold environments.2. Governing equationsThe study concerns the freezing of a plate ina cryogenic system. For non porous food, waterevaporates from the surface and is replaced bywater diffusing from the centre to the surfaceuntil freezing occurs. Thereafter there is nosignificant water diffusion inside the food andsublimation will occur at the surface [5]. Weassume that moisture exists as liquid water andice and that the moisture flux is only due toliquid phase diffusion. If we define X as the drybasis moisture content, Fick's diffusion law canbe written as:ρs,app∂X= ∇∂t.( ρ D ∇X) ( 1)s,appeffWhere ρ s,app is the apparent density of drymatter, and D eff is the effective diffusivity ofliquid moisture. X l is the liquid part of themoisture and the quantity of ice is assumed to bedependent on the temperature and the watercontent as follows:Xiceand=X = X − Xl( X − X ).f( T) ( 2)icebl( 3)

Where X b is the bound water (ie unfreezablewater), T f the initial freezing temperature andf(T) a temperature dependent function.Introducing value of X l eq(1) can be modified tobe in a convenient form for modelling inComsol:ρs,app⎛∇.⎜ρ⎝∂X=∂ts,appDeff⎡⎢⎣dfdT( 1 − f ( T)) ∇X+ ( X − X) ∇T⎟ ( 4)b⎤ ⎞⎥ ⎟⎦ ⎠The PDE interface in coefficient form is used toimplement eq(4) in Comsol 4.2a. Symmetry andwater flux are considered at the boundaries:( ρs,appDeff∇X) = 0 at x = 0( 5)( ρ D ∇X) = −h( P .a − P RH)s,appeffmsatwsat _ ambat x = L / 2( 6)The mass transfer coefficient h m is obtained fromthe Lewis analogy, P sat and P sat_amb are thesaturated vapour pressures at the surface andambient temperatures respectively, a w is thewater activity and RH is the ambient relativehumidity (RH = 0 in the N 2 environment).The heat transfer equation takes into accountthe phase change as follows:where h is the heat transfer coefficient, T amb theexternal temperature, F w the flux of water at thesurface, H vl the latent heat of evaporation andf ice =1 when ice is present.The Heat Transfer in Solids module was used forimplementing the aforementioned equations inComsol 4.2a.3. Results and discussion3.1 ExperimentsSeveral experiments were carried out tomeasure the weight loss during freezing in acryogenic freezer (Figure 1). Samples are madeof Tylose®, a gel composed of water (85%),methylcellulose powder (14%) and salt (1%).This model food is widely used in food-relatedrelated research because of its thermophysicalproperties that are close to those of meat. Flatslabs 2 cm thick were built. The otherdimensions (20 and 10 cm) are sufficiently largeto assume 1D heat and mass transfer.ρs,appCp,app∂T= ∇.∂t( k ∇T) ( 7)effcryogenicchamberbalanceThe apparent heat capacity depends on the heatcapacities of dry matter, water and ice andincludes the latent heat of freezing H li :Cp,app= C + X C + X Cslliceice− Hli∂X∂TEvaporation and sublimation occur at surfaceand the boundary conditions are:ice( 8)( − keff∇T) = 0 at x = 0( 9)( − keff∇T) = h( T − Tamb) + Fw( Hvl+ ficeHli)at x = L / 2( 10)Figure 1. Experimental set-up3.2 Thermophysical properties of foodKnowledge of the thermophysical propertiesof Tylose is necessary to predict heat and masstransfer during freezing. The density of the foodis deduced from measurement carried out in theunfrozen part and from the quantity ratio ofcomponents for the frozen part (dry matter, iceand liquid water). The thermal conductivity iscalculated by means of a Maxwell type model,including measurement of the thermalconductivity of unfrozen Tylose. The apparentheat capacity and the bound water are deducedfrom thermal analysis carried out using

Differential Scanning Calorimetry (DSC). Thewater activity was measured for several watercontents and modelled by a GAB equation andthe effective diffusivity comes from the literature[6].The results from the DSC have to becorrected to include the ice formation beginningat the initial freezing temperature. According toLe Bail et al. [7], a simple relation can be usedfor the function f(T):to 30 W.m -2 K -1 , value estimated in the cryogenicchamber, according to the gas velocities.Tf− 273.15f (T) = 1−T − 273.15f (T) = 0if T ≤ Tif T > Tff( 11)This relation remains an approximation. It can beadjusted knowing the final temperature offreezing T ff for which it is assumed that 100% ofthe freezable water is frozen:⎛ Tf− 273.15 ⎞⎛T − Tff⎞f (T) = 1−⎜ ⎟T 273.15⎜TfT⎟⎝ − ⎠⎝−ff ⎠eq(12) is used between T f and T ff and f(T) = 1 forT

The weight loss predicted by the model iscompared with experimental data in Figure 4.Numerical results agree reasonably well with theexperimental weight loss. However there is alsoan influence of the heat transfer coefficient. A10% higher h leads to 8% lower weight lossbecause the chilling time decreases (Figure 5).more conventional technique using cold air at -30°C. Results shown in Figure 6 present thetemperature of Tylose sample at severallocations with a heat transfer coefficient set to 30W.m -2 K -1 . The results indicate that the effectivefreezing time is approximately 3 times higherthan in cryogenic freezing.Figure 4. Numerical and experimental resultsconcerning the weight loss with T amb = -80°C and heattransfer coefficient of 30 W.m -2 K -1 .Figure 6. Temperature distribution inside the foodsample during freezing with air at -30°C and heattransfer coefficient of 30 W.m -2 K -1 .The total weight loss at the end of freezing isclose to 1.1%. This result is in agreement withexperimental results obtained from experimentscarried out in an air blast freezer (total weightloss of 1.15%).The weight loss predicted by the numericalmodel with different heat transfer coefficient ispresented in Figure 7.Figure 5. Influence of the heat transfer coefficient onthe weight loss.3.5 Cryogenic freezing vs air-blast freezingAccording to the previous results, we canassume that the model is able to predicttemperature and weight loss during freezing,despite the uncertainties concerning the heattransfer coefficient. We use the model tocompare two freezing methods, the cryogenicmethod described in the previous section and theFigure 7. Weight loss during freezing with air at-30°C according to different heat transfer coefficients.Results show that increasing the heat transfercoefficient increases the weight loss during thechilling. However because the initial freezing

temperature is reached sooner, the total weightloss is lower if high heat transfer coefficients areused. After 4000s, the mass loss represents 0.9,1.1 and 1.3% of the initial mass for h equal to 40,30 and 20 W.m -2 K -1 respectively. Please alsonote that for the last heat transfer coefficientvalue, the sample is not totally frozen at the endof the simulated process with a centraltemperature of -6°C. Thus the evaporationcontinues in this case and the final weight loss ishigher than 1.3%.The influence of the external temperaturecan also be studied with the numerical model.Weight losses for different ambient airtemperatures are presented in Figure 8. Asexpected, the lower the external temperature is,the lower the weight loss will be.Figure 8. Weight loss during freezing according todifferent ambient temperatures (h = 30 W.m -2 K -1 ).3.6 DiscussionWeight loss during freezing is mainly due tomoisture evaporation and a high degree ofdehydration takes place in the beginning of thefreezing process. Numerical results confirm thatthe moisture evaporation is slowed downconsiderably after passing the latent heat zone.Thus using a cryogenic freezer can reduce theweight loss. The distribution of the water contentin the sample at the end of freezing is presentedin Figure 9 for both ambiances with the sameheat transfer coefficient. The strong watercontent gradient close to the surface involves theuse of finer grid in this region.Figure 9. Moisture content distribution in the food atthe end of freezing according to the ambiance (h = 30W.m -2 .K -1 ).If numerical results agree well withexperimental data, some efforts have to be doneto improve the model. From a numerical point ofview, the function f(T) could be modifiedbecause some discontinuities occur at T = T ff .The function also assumes the knowledge of thefinal freezing temperature that may be difficult todetermine. For the experiments, special attentionhas to be paid to the experimental conditionssuch as the heat transfer coefficient. It is alsodifficult to accurately measure the centraltemperature of the product and thus comparisonscan be biased.4. ConclusionA mathematical model was developed topredict the freezing time and the weight loss offood during freezing in very low ambienttemperature. Such freezing processes arecharacterised by high heat and mass transfercompared with conventional freezing in cold air.A modified function f(T) was proposed to takeinto account the ice formation during freezing.Numerical results confirm that using very coldmedium for freezing can reduce the freezing timeand the weight loss, the latter being mainly dueto moisture evaporation before the latent heatzone.

5. References1. D. F. Olivera, V. O. Salvadori, Effect offreezing rate in textural and rheologicalcharacteristics of frozen cooked organic pasta,Journal of Food Engineering, 90, 271-276(2009)2. S. Chassagne-Berces, F. Fonseca, M. Citeau,M. Marin, Freezing protocol effect on qualityproperties of fruit tissue according to the fruit,the variety and the stage of maturity, LWT –Food Science and Technology, 43, 1441-1449(2010)3. S. Boonsumrej, S. Chaiwanichsiri, S.Tantratian, T. Suzuki, R. Takai, Effects offreezing and thawing on the quality changes oftiger shrimp (Penaeus monodon) frozen by airblastand cryogenic freezing, Journal of FoodEngineering, 80, 292-299, (2006)4. A. E. Delgado, D-W. Sun, Heat and masstransfer models for predicting freezing processes– a review, Journal of Food Engineering, 47,157-174 (2001)5. Q. T. Pham, Modelling heat and mass transferin frozen foods: a review, International Journalof Refrigeration, 29, 876-888 (2006)6. B. A. Anderson, R. P. Singh, MoistureDiffusivity in Tylose Gel (Karlsruhe TestMaterial), Journal of Food Science E: FoodEngineering and Physical Properties, 70, 331-337 (2005)7. A. Le Bail, N. Chapleau, M. Anton-deLamballerie, M. Vignolle, Evaluation of themean ice ratio as a function of temperature in aheterogeneous food: Application to thedetermination of the target temperature at the endof freezing, International Journal ofRefrigeration, 31, 816-821 (2008)6. AcknowledgementsThe authors would like to thank the SOLCompany for its financial support and CristopheCouëdel and Lotfi Boudjema for theirexperimental help.