Three mathematical models for osmotic dehydration pre-treatment are presented in the literature and can be used to model ultrasound-assisted osmotic dehydration: Fickian, Peleg’s and Mass transfer model.
The Fickian model (Fito and Chiralt, 1997) assumes that the main mass transfer mechanism during osmotic dehydration is diffusional in nature and sample deformation and shrinkage during drying can be considered negligible. The diffusional water and solute coefficients are considered effective parameters, as a result of all the transport phenomena that could take place during the process. The model assumes that the diffusivity coefficients are effective parameters, constant and uniform, and also assumes an isotropic behavior of the solid with regard to water and solute diffusion. As such the mass transports can be described by Fick’s law in an unsteady state mass balance.
The Fickian model can be solved analytically assuming that the sample volume remains constant throughout the process and that the initial soluble solids content of the samples is also negligible. For a parallelepipedic shape sample:
( ) ( )
where, C is the soluble solids concentration [g/m3]; C0 is the initial soluble solids concentration [g/m3]; Ce is the equilibrium soluble solids concentration [g/m3]; DS is the effective diffusivity of the solids [m2/s]; DW is the effective water diffusivity [m2/s]; H is the moisture content of the sample, H0 is the initial moisture content; He is the equilibrium moisture content; L is the height of the sample [m]; and t is the time [s].
Peleg’s model (1988) is used to describe the osmotic dehydration curves in some published papers (Azoubel and Murr, 2004; El-Aouar, Azoubel, and Murr, 2003; Park et al., 2002). Peleg’s equation presents a satisfactory fitting for water loss. This two parameter model describes most of the published curves which approach equilibrium asymptotically (Palou, Lopez-Malo, Argaiz, and Welti, 1993):
t
where, MC is the water or solids’ amount at instant t [g]; MC0 is the initial water or solids amount [g]; k1 and k2 are the Peleg’s parameters; and t is the time [s].
The Mass transfer model takes into account the mass transfer between the fruit or vegetable and the liquid medium (distilled water or osmotic solution). The mass balance for the sample (fruit or vegetable) includes the mass transfer of water from the sample to the
osmotic solution and the mass transfer of sugar and/or salt from the osmotic solution to the
S is the soluble solids concentration of the fruit [g/m3], CFR
W is the water concentration of the fruit [g/m3], COS
S is the soluble solids concentration of the osmotic solution [g/m3], COS
W is the water concentration of the osmotic solution [g/m3], Km
S is the mass transfer coefficient of soluble solids [1/m2.s], Km
W is the mass transfer coefficient of water [1/m2.s], MFR
S is the mass of soluble solid of the fruit [g], MFR
W is the mass of water of the fruit [g], and VFR is the volume of the fruit [m3].
During the dehydration process the sample may shrink and this phenomenon has to be considered by the mathematical model to increase the accuracy of the mass transfer coefficients. In the model, the shrinkage effect is set to be proportional to the water mass change in the sample, according to equation 6. The sample superficial area is assumed to decrease at a proportional rate following the decrease in volume of the sample.
dt
The mass balance for the liquid medium includes the gain of water that is removed from the sample and the loss of sugar to the sample. As the material balances are based on mass balances, the amount of water leaving the sample is equal to the amount of water entering the liquid medium. The opposite occurs with the mass balance of sugar, where the amount of solids entering the sample is equal to the amount of solids leaving the liquid medium.
dt
is the mass of soluble solid of the osmotic solution [g], MOSW
is the mass of water of the osmotic solution [g], and VOS is the volume of the osmotic solution [m3].
To estimate the mass transfer coefficients of the ultrasound-assisted osmotic dehydration experimental data should be gathered and used with a parameter estimation procedure based on the minimization of the error sum of squares. The model is solved by numerical integration using the Runge-Kutta method or similar integration technique.
The mass transfer coefficients obtained for the experiments carried out with pineapples are presented in table 7. Except for the treatment carried out with distilled water as the liquid medium, the Mass transfer model represents very well the data points as shown in figure 11 and through the regression errors. The F-values calculated for all experiments were above the listed F-values confirming the validity of this model within a 95% level of confidence.
Table 7. Mass transfer coefficients for the ultrasound-assisted osmotic dehydration of pineapples
Liquid medium Mass Transfer Coefficient for Water [1/h.m2]
Mass Transfer Coefficient for Solids [1/h.m2]
R2
F-test*
Distilled water Sucrose solution (35ºBrix) Sucrose solution (70ºBrix)
12.8 435.6 824.4
87.0 84.6 210.6
0.752 0.991 0.995
2.6 106.6 96.3
*Listed F-test: 18.5 for 95% level of confidence.
Figure 11. Normalized water and sugar content for pineapples submitted to ultrasound pre-treatment at 30ºC.
The mass transfer coefficient of water for pineapples presented an almost linear relationship with the concentration of sucrose in the liquid medium going from 12 h-1.m-2 when distilled water was used to 800 h-1.m-2 when an osmotic solution of 70ºBrix was
employed, which was expected based on the larger osmotic pressure gradient between the fruit and the osmotic solution. The mass transfer coefficient of sucrose did not show significant change when distilled water and an osmotic solution of 35ºBrix was employed, but increased 2.5 fold when an osmotic solution of 70ºBrix was used.