Transactions on Engineering Sciences vol 5, 1994 WIT Press, ISSN

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1 Modelling and numerical simulation of the simultaneous heat and mass transfer during food freezing and storage A.M. Tocci & R.H. Mascheroni Centra de Investigation y Desarrollo en Criotecnologia de Alimentos - CIDCA, ABSTRACT A numerical model is developed for the prediction of simultaneous heat and mass transfer during food freezing and storage. The resulting system of coupled partial differential equations, with variable coefficients was solved by two explicit finite difference methods, one with constant mesh size and the other of equal volume elements. The prediction method was applied to the calculation of temperature and water concentration profiles, freezing time and weight loss of meat balls and hamburgers under industrial freezing conditions. INTRODUCTION Weight loss due to sublimation during the freezing and storage of foodstuffs is an important economic and quality factor. Water loss produces changes in food overall appearance, colour, texture and taste. Besides weight loss implies the equivalent economic loss. In this regard it is important to be able to predict the influence of process conditions over water vapour transfer between the foodstuff and the surrounding medium (air). Notwithstanding this subject has seldom been modelled in deep. Chau et al' and Chau and Gaffney* worked over simultaneous heat and mass transfer in refrigeration of respiring vegetables; Sukhwal and Aguirre-Puente* and Aguirre-Puente and Sukhwal* measured and modelled ice sublimation from frozen dispersed media. Most of the remaining published reports are based only on experimental data or on semiempirical models.the main difficulties for a detailed modelling are the coupling of mass and heat balances and the strong dependence of food thermophysical properties (thermal conductivity k, apparent heat capacity Cp, density p and water diffusion coefficient D) with temperature in the freezing zone. An additional problem are the values for k, Cp and D in the partially dried surface layer of the frozen food.

2 580 Heat Transfer MATHEMATICAL FORMULATION Two usual regular geometries were studied: Spheres as occurring in meat balls, whole fruits and melon portions, and finite cylinders as in hamburgers, pizzas, tarts or fruit and vegetable slices. The energy balance to be solved is p Cp -fr = V(JcVT) dt (l) in which T is temperature, t time and where, as stated, p, Cp and k are functions of temperature and composition (Sanz et ap). The apparent heat capacity Cp includes the true heat capacity and the enthalpy of fusion of ice. The last is released over a wide temperature range, as most foods behave as a solution, having a freezing point depression and an equilibrium temperatureconcentration curve. The boundary conditions, usual for this situation, are "57 for the centre of the sphere and cylinder, "a^ '"- * for the centre of the cylinder, i, dt,, for the surface of the sphere and the lateral face of hamburger, and for the circular surface of hamburger. In these equations r and z are the coordinates, C compositions, Ta and Ca refer to ambient air conditions, Lv is the heat of sublimation of ice and h and K the heat and mass transfer coefficients. Their values were taken from own data (Tocci and Mascheroni*) for belt freezers and "corrected" for the presence of a dehydrated layer of depth 5 (6 depends on weight loss and increases with time) as:

3 Heat Transfer 581 Here ho and Ko correspond to the "nondehydrated" food; kd is the thermal conductivity of the dehydrated food and Deff the diffusion coefficient of water vapour in the dehydrated solid matrix (Pham and Willix^). The mass balance to be solved is: =V(DVC) (2) dt Similar assumptions to those used in the development of the heat balance were done. The boundary conditions at food surfaces are n8c for sphere and the lateral face of hamburger, and for the circular surface of hamburger. NUMERICAL METHODS Cleland and Earle* have shown that during the modelling of food freezing through the numerical solution of Equation (1) similar results are obtained using either explicit finite-diferences approaches or more elaborate two- or three-level implicit finite-differences schemes, provided sufficiently small time and space grid increments are used. By far, the main source of error are the values of thermal properties of foods used in the numerical method. Based on those conclusions two explicit methods were tested, between which the main differences arise from the spatial distribution of volume elements (and, consequently, of grid points). Method A: Equal thickness space increments If Ar, Az and At are the radial, axial and time increments and defined as Ar = R/(I-1) and Az = L/(J-1) and T^ and Tj/ refer to temperatures calculated at r = (i-l)ar, z = (j-l)az and t = nat, the general formulation of energy balances for sphere and finite cylinder are given by Equations (3) and (4). After rearrangement, the general formulation valid for interior points of the food is obtained (Equations (5) and (6)). For these equations is

4 582 Heat Transfer REL1 =At / Ar\ REL2 = At / Az* and a = k / (pcp). Similar relations to Equations (5) and (6) are obtained for C*+'. Specific relations for system boundaries (1 = 1, 1=1, j = l and j=j) are developed through the use of boundary conditions. <!-!) Ar» (3) (4) REL1 l j^n ill \ /^J] _rr,zl \ (5) ~ (6) Method B: Equal volume increments Chau and Gaffney^ present a detailed explanation of the development of the method. In brief, it consists in dividing the body (sphere or cylinder) in I concentric shells of equal volume AV, where AV = V / I, being V the total volume of the body. The main advantage of this method is that, for spheres and cylinders, a higher "concentration" of grid points appears near the surface, permitting a more precise calculation of temperature and concentration profiles. Up to now, we have used this method only for the case of sphere. In it

5 Heat Transfer 583 each grid point is placed in the centre of the corresponding volume increment. The general formulation based on an energy balance in a volume element AVj is: Tr" A,_I '-1 lr" A,+i ' Ar, (7) which, rearranged, leads to k" i4-i Ar 1- ~2 AF)p *Q?r Ar,_i ^ k i^-ici -" ' 1 v4 i T" i-»-l Af _^ + L (8) with: = 4 T r, = (1 + (1.5 - i) / 1) for 2 < i < I-hl ; Ar^ = r^ - r^ Point 1 is on the surface and Point 1+2 is on the centre. A relation similar to Equation (8) is obtained for C^\ For i = 1 (surface) a specific relation is deduced using the boundary condition. Point 1+2 is not calculated and may be obtained by polynomial extrapolation. RESULTS The numeric methods were implemented in QuickBASIC and run on a PC AT 486. In all calculations thermal properties of minced beef were used. Spheres (meat balls) A unique size was tested: R = 0.019m. Three air temperatures Ta = -25, -30 and -35 *C and air velocities ranging from va = 1 m s"* to va = 10 m s"* were used. In all cases the relative humidity of air was 100% (saturation). For Method A 16 grid points were used in most calculations, although 31 and 46 grid points were also tested, without meaningful differences in results. For 16 grid points At = 0.50 s gave stable results. For Method B 14 volume elements were used in calculations, but also 29 and 44 elements were tested without significative differences in calculated freezing times ft or weight loss wl. For I = 14, At = 0.10 s gave stable results.

6 584 Heat Transfer There were no differences between predicted ft or wl between the two methods. The highest variation in ft was 0.22 min for Ta = -35 C and va = 10 m s'\ For wl the highest difference was 0.026% for Ta = -35 C and va = 7 m s~* /D - [ 70 - ^\ Ā k \ ixr ^2,. -^ c ^^ 55 1, CD 50 ; 1 Ta = -25 C 45-2 Ta = -30 C 0> 3 Ta C C 'N 40-00) 35 LL Ē 30 - ^^- -* ~^ 25 > L 0 20^/ ! va (m/s) 7 8 _-- -^ : ii ~^ ( r\ V Figure 1: Predicted freezing times and relation between freezing times calculated with and without considering weight loos for meat balls in belt freezers Relation Figures 1 and 2 present the main results obtained. Figure 1 shows the variation in predicted ft (the time to reach -18 C in the centre) as a function of air temperature and speed. As expected ft decreases with the decrease of Ta and the increase of va. The same Figure presents the relation between freezing times calculated considering and neglecting water evaporation. As can be seen an error range of 5 to 15% is obtained when neglecting the effect of ice sublimation on freezing time. Figure 2 presents the data of wl for the same operating conditions. Predicted wl range between 1.04 and 2.07% for the extreme ranges of Ta and va. These results show the possibility of reducing water losses through an adequate selection of freezing conditions. Finite cylinders (hamburgers) Only Method A was used. The freezing of beef hamburgers with R = 0.05 m and L = m was simulated. An 11x11 point grid was used for most calculations, but also 16 x 16 and 21 x 21 points grids were tested. For 11 x

7 Heat Transfer 585 To = -35 C va (m/s) Figure 2: Predicted weight loss during meat balls freezing in belt freezers 10 Figure 3: Predicted freezing times for hamburgers in belt freezers

586 Heat Transfer 11 points a At of 0.075 s was sufficiently small to secure stability.

8 586 Heat Transfer 11 points a At of s was sufficiently small to secure stability. As an example of the results obtained, Figure 3 presents ft calculated for Ta = -30, -35 and -40 C and va ranging from 1 to 10 m st As expected the trends in tf variation with Ta and va are similar to those of spheres, but there is a higher variation rate at low air speeds compared to the case of spheres. DISCUSSION AND CONCLUSIONS - It is possible to predict simultaneous heat and mass transfer during food freezing and storage through the use of a theoretical model, with no simplifying assumptions, but employing explicit finite difference-methods. This facilitates the development and use of calculation programs. - The developed model enables to calculate temperature and concentration profiles, freezing times and weight loss. - There were no significative differences between results generated for both numerical schemes tested. REFERENCES 1. Chau, K.V., Gaffney, J.J. and Romero, R.A. 'A mathematical model for the transpiration from fruits and vegetables'. ASHRAE Trans. 94 (1): pp , Chau, K.V. and Gaffney, J.J. A finite- difference model for heat and mass transfer in products with internal heat generation and transpiration'. /. Food Sci. 55: pp , Sukhwal, R.N. and Aguirre-Puente, J. 'Sublimation des milieux disperses. Considerations theoriques et experimentation'. Rev. Gen. Therm. Fr. (262), pp , Aguirre-Puente, J. and Sukhwal, R.N. 'Sublimation of ice in frozen dispersed media', in Proceedings of The Third International Offshore Mechanics and Arctic Engineering Symposium (Ed. V.J. Lunardini), Vol. 3, pp , Sanz, P.D., Dominguez Alonso, M. and Mascheroni, R.H. 'Equations for the prediction of thermophysical properties of meat products', Lat. Am. Appl. Res. 19: pp , Tocci, A.M. and Mascheroni, R.H. 'Heat and mass transfer coefficients during the refrigeration, freezing and storage of meats, meat products and analogues' /. Food Engng. (in press), Pham, Q.T. and Willix, J. 'A model for food desiccation in frozen storage', /. Food Sci. 49: pp & 1294, Cleland, A.C. and Earle, R.L. 'Assessment of freezing time prediction methods', /. Food Sci. 49: , 1984.

Monitoring of Weight Losses in Meat Products during Freezing and Frozen Storage

Monitoring of Weight Losses in Meat Products during Freezing and Frozen Storage L.A. Campan one, 1,2 L.A. Roche, 1,2 V.O. Salvadori 1,2 and R.H. Mascheroni 1,2, * 1 Centro de Investigacio n y Desarrollo