HUNGARIAN JOURNAL OF INDUSTRIAL CHEMISTRY VESZPRÉM Vol. 33(1-2). pp. 43-48. (2005) MATHEMATICAL MODELLING IN OSMOTIC DEHYDRATION A. Matusek1, B. Czukor1, P. Merész2, F. Örsi2 1Central Food Research Institute, Unit of Technology H-1022 Budapest, Herman O. út 15., Hungary 2Budapest University of Technology and Economics, Department of Biochemistry and Food Technology H-1111 Budapest, Műegyetem rkp. 3., Hungary e-mail: a.matusek@cfri.hu This paper was presented at the 10th International Workshop on Chemical Engineering Mathematics, Budapest, Hungary, August 18-20 2005 Mass transfer parameters during osmotic dehydration of apple cv. Idared has been studied. The solution of Fick’s law for unsteady state mass transfer in rectangular configuration has been used to calculate the apparent diffusion coefficient of fructo-oligosaccharides (FOS) - used as osmotic agent - and water. Apple cubes were blanched and osmotically treated (0-60 min) in 60 % FOS solution at 40 °C, sample-to-solution ratio was 1:10. Total solid (TS) content of the samples was determined by gravimetric method, total soluble solid (TSS) content was measured by refractometer and FOS content was measured by polarimeter. The apparent diffusion coefficient of FOS and water were found to be of the order of 10-9 m2/s. The model of the osmotic dehydration process is better for the water than for the FOS. Keywords: mathematical modelling, osmotic dehydration, diffusion coefficient, apple 1. Introduction Osmotic dehydration is a water removal process by immersing the fruit or vegetable in a hypertonic solution. Two simultaneous counter current flows take place because of the semipermeable property of the plant cell structure: a water flow from the tissues into the solution and a simultaneous transfer of solute from the solution into the food (Salvatori et al., 1999a,b). Since the membrane is not perfectly selective, other solutes (minerals, vitamins, and flavours) present in the food are also leached into the osmotic solution (Taiwo et al., 2002). The driving force of diffusion is the gradient of chemical potential. In practice osmotic dehydration is used for partial dehydration of foods, especially fruits and vegetables as a pre-treatment step before different drying methods or freezing (Torreggiani, 1993). The rate of mass transport depends upon many factors such as concentration, temperature and the kind of osmotic solution, time of treatment, level of agitation, sample-to-solution ratio, vacuum level and different pre-treatments if applied. Several studies in the literature describe the effect of these variables (Lerici et al., 1985; Beristain et al., 1990; Lazarides et al., 1997; Park et al., 2002). The conventional used osmotic agents are sucrose and sodium chloride. The purpose of present work was to study mass transfer parameters during osmotic dehydration of apple and to develop a model based on Fick’s second law describing the transport phenomena in longitudinal direction. 2. Mathematical modelling Most of the available models proposed are based on Fick’s law of diffusion and use the particular solution given by Crank (1975) for one-dimensional unsteady state mass transport (Salvatori et al., 1999a): 2 2 x C D t C ∂ ∂ = ∂ ∂ (1) where C is the concentration of the diffusing component, D is the diffusion coefficient, t is the time coordinate, x is the space coordinate, the diffusion path. The double mass transfer is reduced this way to only one transport process, either of water or solute. Lenart & Flink (1984) proposed a model that was able to predict osmotic mass transport data and water activity data for short, non equilibrium osmosis time. Solution of Fick’s law for short contact time diffusion was used by Hawkes & Flink (1978) and Magee et al. (1983). Kaymak-Ertekin & Sultanoglu (2000) developed a model for OD of fruits with limited sample-to-solution ratio applying Fick’s law. In order to analyse the data 44 and indicate the overall exchange of solutes and water between the apples and the osmotic solution, different parameters were determined for each sample, as water loss (WL), solid gain (SG), weight reduction (WR), soluble solid concentration (SSC). They investigated apple slices with thickness 5 mm. They used the following assumptions in the development: 1. Apple slices are infinite slabs of width = 2r. 2. Initial water and SSC in the apple are uniform. 3. Apparent diffusion coefficient is constant (D ≠ f (c)). 4. The process is isothermal. 5. Simultaneous counter-current flows; the diffusion of water from the apple and the diffusion of sugar into the apples are only considered. 6. A sugar solution film is thought to be at the surface of the apple as a boundary layer. Being a part of the apple, this film is assumed to the equilibrium concentration and the process is directly proceeded by diffusion. 7. Shrinkage is neglected. In this model, using a mass balance on water movement inside the food, the rate of water loss was obtained as a function of time (Azuara et al., 1992): tS WLtS WL 1 1 1 )( + = ∞ (2) where WL is the fraction of water loss at time t, WL∞ the fraction of water loss at equilibrium, S1 the model constant related to the rate of water loss and t is the time. 100 0 00 M XMXM WL W tt W − = (g/100g fresh sample), where M0 is the initial mass of sample, M1 the mass of the sample at time t, X0 w the initial water concentration of sample, Xt w the water concentration of sample at time t. Similarly for solid gain, it can be written as: tS SGtS SG 2 2 1 )( + = ∞ (3) where SG is the fraction of solid gain at time t, SG∞ the fraction of solid gain at equilibrium, S2 the model constant related to the rate of SG. 100 0 00 M XMXM SG tsts tt −= (g/100g fresh sample), where X0 ts the initial total solids concentration of the sample and Xt ts is the total solids concentration of the sample at time t. Using WL and SG values calculated from the experimental data at different processing times, the WL∞, S1 and SG∞, S2 can be estimated from the slope and intercept of the plot t/WL or t/SG vs t of the linearized form of eq.(2) and eq.(3) respectively. Therefore the equilibrium solute concentration can be estimated from a mass balance equation derived as: ∞ ∞ ∞ +− − = WLMWS SGMWS W s s s 000 000 )1( (4) where S0 is the mass of the initial osmotic solution, W0 s the initial SSC and W∞ s is the SSC at equilibrium. Solving eq.(1) numerically by the Crank-Nicholson method – at each time step the concentration profiles can be calculated numerically without considering the other component. Then concentration at each time step could be adjusted considering the other component changes in the same step calculated numerically from the eq.(1). If t and x are the subscripts indicating the time and position, at time t or at time t+1 the concentrations can be given as: S xt W xt W xtW xt MM M C ,, , , + = S xt W xt W xtW xt MM M C ,1,1 ,1 ,1 ++ + + + = (5) S xt W xt S xtS xt MM M C ,, , , + = S xt W xt S xtS xt MM M C ,1,1 ,1 ,1 ++ + + + = (6) Since the concentrations can be calculated and the masses of water and soluble solids at each position are known, M values at time t+1 can be calculated from eqn.(5) and (6); total mass of water and soluble solids of fruit are estimated by sum of the masses at different positions. The WL and SG values were calculated from the experimental data. The concentration at time t is obtained from a mass balance equation derived as SGWL XSG X ts ts t +− + = 1 0 (7) where Xt ts is the total solids concentration at time t and X0 ts is the initial total solids concentration. A model fitted to the experimental values and the apparent diffusivities of solute and water can be obtained by the trial and error method. Both diffusion coefficient representing water loss and solid gain are found to be in the range of 10-10 – 10-11 m2/s. Beristain et al. (1990) developed a model based on Crank’s equation. They also used the solution of eqn.(1) gave by Crank (1975): ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − ++ + Σ−= ∞ = ∞ 2 2 221 exp 1 )1(2 1 l tDq qWL WL n n n t αα αα (8) where WLt is the weight of water loss at time t; WL∞ is the water loss at time ∞; α is the volumetric ratio of syrup to sample; qn positive roots other than zero of equation: tan qn = -αqn. The maximum fraction of water that a sample (pineapple ring) can lose, for given concentration of sucrose solution was determined; diffusion coefficients were calculated using Park’s method (Crank, 1975). The main limit of models based on Fick’s law of diffusion is that the resulting diffusivities are a combination of the respective water and solute diffusivities and the probably interaction of the flows. Furthermore these methods neglect spatial distribution of the osmotic effect (Salvatori, 1999a). Salvatori developed a model in terms of the advancing rate of 45 disturbance front in the osmosed plant tissues. Diffusion coefficient was calculated by using a non simplified Fick’s equation in terms of concentration profiles. They described the changes in plant tissues with the “advancing disturbance” front (ADF), later they (Salvatori et al., 1999b) developed generalised equations – on the base of the ADF – to describe the concentration profiles and the average concentration during OD process. Regarding to the geometry of the food the mass transport phenomena occur during osmotic dehydration can be analysed in rectangular and in cylindrical configurations. Rastogi et al. (1997) studied the effect of temperature and concentration of the osmotic solution on the OD of banana, they determined the effective diffusion coefficient of water by the use of Fick’s II law in terms of cylindrical coordinates (r, θ, z, Crank, 1975): ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ +⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ +⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ = ∂ ∂ z C rD z C r D r C rD rrt C θθ 1 (9) For long cylinders (∂C/∂z=0), assuming the diffusion to be radial (∂C/∂θ=0) and concentration (C) to be a function of radius (r) and time (t) only, eqn.(9) reduces to: ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ = ∂ ∂ r C rD rrt C 1 (10 ) Considering boundary condition for t>0, C=C0, at r=a and initial condition for t=0, C=C1, 0