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Diffusion mechanisms during the osmotic dehydration of Granny Smith apples subjected to a moderate electric field
Journal of Food Engineering 166 (2015) 204–211
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Journal of Food Engineering journal homepage: www.elsevier.com/locate/jfoodeng
Diffusion mechanisms during the osmotic dehydration of Granny Smith apples subjected to a moderate electric field R. Simpson a,b,⇑, C. Ramírez a, V. Birchmeier a, A. Almonacid a,b, J. Moreno c, H. Nuñez a, A. Jaques a a
Departamento de Ingeniería Química y Ambiental, Universidad Técnica Federico Santa María, P.O. Box 110-V, Valparaíso, Chile Centro regional de estudios en alimentos saludables (CREAS) Conicyt-Regional R06I1004, Blanco 1623 Room 1402, Valparaíso, Chile c Departamento de Ingeniería en Alimentos, Universidad del Bío-Bío, P.O. Box 447, Chillán, Chile b
a r t i c l e
i n f o
Article history: Received 27 January 2015 Received in revised form 19 May 2015 Accepted 24 May 2015 Available online 27 May 2015 Keywords: Diffusion mechanisms Osmotic dehydration Moderate electric field
a b s t r a c t Osmotic dehydration is a process wherein foods are partially dehydrated by immersion in an aqueous hypertonic solution. Osmotic dehydration reduces the water activity of the food, thereby minimizing the potential growth of microorganisms and extending the shelf life of food products. A major disadvantage of the osmotic dehydration process is the long time required to reduce the water activity, which makes its industrial implementation impractical. The aim of this research is to determine the diffusion mechanisms in Granny Smith apples exposed to different electric fields varying from 0 to 17 V/cm. Osmotic dehydration was performed at 40 °C for both the conventional treatment and for the MEF treatment. The fruit/solution ratio used was 1:11 w/w to prevent and minimize the change in concentration of the solution during the experiment. The osmotic solution was exposed to electric fields of 0, 9, 13 or 17 V/cm, respectively. To determine the diffusion mechanisms two phenomenological models were tested: Fick’s second law and Anomalous diffusion model. Sucrose diffusion in Granny Smith apples is highly influenced by the application of a MEF, meaning that as the application of the electric field increases, the higher the effective diffusion coefficient (Deff) becomes. The exponential decay from the Fick law, in the tail of the diffusion profile, does not represent the shape of the data. However, as the application of the electric field increases the fit of Fick’s model improves, and specifically for an electric field of 17 V/cm it is observed that the behavior of the experimental data resembles the behavior predicted by Fick’s second law. The empirical parameter a for the anomalous diffusion model was always greater than one, but as the MEF increased, a was monotonically tending to one. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Osmotic dehydration is a process wherein foods are partially dehydrated by immersion in an aqueous hypertonic solution. Osmotic dehydration reduces the water activity of the food, thereby minimizing the potential growth of microorganisms and extending the shelf life of food products. Osmotic dehydration is a technology with broad applications in food processing. Nevertheless, the osmotic dehydration process can achieve only partial dehydration of food. Another disadvantage of the osmotic dehydration process is the long time required to reduce the water activity, which makes its industrial implementation impractical (Simpson et al., 2007). The application of a moderate electric field (MEF) means that an electric current passes through a food material. When applied, an ⇑ Corresponding author at: Departamento de Ingeniería Química y Ambiental, Universidad Técnica Federico Santa María, P.O. Box 110-V, Valparaíso, Chile. http://dx.doi.org/10.1016/j.jfoodeng.2015.05.027 0260-8774/Ó 2015 Elsevier Ltd. All rights reserved.
electric field can cause changes in the permeability of cell membranes of plant tissue at lower temperatures at which these membranes are permeabilized by thermal effects, a phenomenon known as electroporation (An and King, 2007; Lima and Sastry, 1999). The resulting effect is that the diffusion process increases, the electrical conductivity changes and the moisture more easily migrates out of the plant tissue (Leizerson and Shimoni, 2005). In general, structural changes of cells such as cell disruption, electroporation, and cell damage tend to improve the diffusion rate because water movement through the cell membrane and cell wall is easier. For example, Ramírez et al. (2011) evaluated some pre-treatments from a microstructural point of view and their effect on water diffusion rate, finding that those pre-treatments that induced more damage to the cellular structure produced an increase in diffusion rate during air drying. The case of electroporation is the same concept, with cell damage favoring the diffusion of water during the dehydration process. Moreno et al. (2013)
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showed the effects at the microstructural level in strawberries of applying an electric field of 130 V/cm, where the cells presented changes in shape and thickness of lamellae and an increase in cell breakage. The phenomenon of osmotic dehydration can be modeled by the fundamentals of mass transfer that describe the origin of the diffusive forces that are involved in and control these processes (Spiazzi and Mascheroni, 1997). In recent years, numerous studies have been conducted to improve knowledge regarding the internal mass transfer occurring during osmotic dehydration of foods and to model the mechanism of the relevant process (Spiazzi and Mascheroni, 1997; Kaymak-Ertekin and Sultanoglu, 2000). This process is usually represented by Fick’s second law. The best known phenomenological model to represent the diffusional mechanism is the model of Crank (Crank, 1975), consisting of a set of solutions of Fick’s law of diffusion for different geometries, boundary conditions and initial conditions. This model has been used by many authors because it is the best known phenomenological model for representing diffusional mechanisms (Ochoa and Ayala, 2005). To these ends, researchers have developed several mathematical models (Barat et al., 2001; Fito, 1994; Salvatori et al., 1999; Toupin et al., 1989; Agnelli et al., 2005) and determined effective diffusion coefficient values in different fruits and geometries using Fick’s second law (Rastogi et al., 2002; Rastogi and Raghavarao, 2004; Azuara et al., 1992). However, the complexity of the mass transfer process due to the complex nature of plant tissues, their strongly heterogeneous cell structure, shrinkage and volume changes during dehydration process due to moisture removing makes accurate prediction challenging, so that the exact analytical solutions cannot normally be applied because several of the assumptions of Fick’s second law do not fit heterogeneous tissues such as apple (Varea and Hernández, 2010; Zaritzky and Califano, 1999). These structural changes are also present in osmotic dehydration, however is less significant than in traditional dehydration which can reach values close to 60–70% (Sturm et al., 2014). For example, Souraki et al. (2014) and Nieto et al. (2004) have reported that the effect of osmotic dehydration on shrinkage of apple could be around 10% up to 30%. In the present study the effect of shrinkage will be neglected, because the fractional diffusion formulation for shrinking geometries has not been developed yet. Anomalous or non-Fickian diffusion has been proven to be a useful tool for quantitatively describing diffusion during drying of foods because it considers many of the changes that occur in the food material and its microstructure during the process such as shrinkage and porosity. Fractional calculus modeling is a new and innovative approach in the food processing field that requires rethinking the diffusion process in food materials (Simpson et al., 2013). Fractional calculus is a mathematical tool for mathematically representing the phenomena of real anomalous diffusion of solutes whose movement can be faster or slower than postulated in Fick’s second law due to the influence of the cellular structure of the food material (Simpson et al., 2013). Descriptions of mass transfer behavior can be modeled with differential equations of non-integer order as fractional numbers. The aim of this research is to determine the diffusion mechanisms in Granny Smith apples exposed to different electric fields varying from 0 to 17 V/cm by application of fractional diffusional formulation using fractional calculus tool.
and were subsequently immersed in 1% ascorbic acid solution and 2% citric acid solution to prevent enzymatic browning. Osmotic solutions of 45, 55 and 65 °Brix were prepared using sucrose (commercial sugar). The osmotic solutions contained 7 mg/L of potassium sorbate (C6H7KO2) as a preservative, inhibiting any unwanted microbiological activity and 1.27, 1.33 or 1.13 g/L of calcium chloride (CaCl2) to increase the conductivity of the solution.
2. Materials and methods
C ðtÞ ¼
2.1. Sample preparation Granny Smith apples were acquired in a supermarket in the city of Valparaiso, Chile and kept under refrigeration at 2 °C. The apples were peeled and cut into sheets 40 mm in diameter and 7 mm thick
2.2. Osmotic dehydration Osmotic dehydration was performed at 40 °C for both the conventional treatment and for the MEF treatment. The fruit/solution ratio used was 1:11 w/w to prevent and minimize the change in concentration of the solution during the experiment. The samples were immersed in a cylindrical cell made of stainless steel with a plastic bottom. The cell was composed of two concentric cylinders (3.7 cm and 19 cm diameter) connected to a generator by means of two electrodes (Moreno et al., 2011). The osmotic solution was exposed to an alternating current with voltage of 70, 100 or 130 V, generating electric fields of 9, 13 or 17 V/cm, respectively. The cell was immersed in a thermo-regulated bath (BS-21 Jeio Tech, Korea) with the aim of maintaining the medium temperature at 40 °C during the experiment. In addition, constant and gentle stirring was provided to attain a homogeneous solution without damaging the fruit. To monitor the temperature of the solution and the applied electric field, two T type thermocouples (copper– constantan) were used. Data on temperature, voltage and current were recorded every three seconds with a data logger (Omega 220, USA) having a connection via modem and connection port to a computer. The data logger includes HiperWareTM software version 4.77 (1996–2005), which communicates with and gathers data from the program online. Each experimental run was performed for 12 h with sampling at different times (optimal sampling for diffusion experiments) to measure the concentration of sugar in the apple. 2.3. Analytical determinations Moisture was determined by drying the samples for 24 h at 60 °C in a vacuum oven to constant weight according to the method defined by the Association of Official Analytical Chemists (AOAC) (Association of Official Analytical Chemists, 2000) for fruits rich in sugar. The determination of soluble solids was performed, after homogenization of the samples, using a digital refractometer (Hanna Instruments, model HI 96811, USA). 2.4. Mathematical models 2.4.1. Fick’s second law of diffusion To develop the model, the following assumptions were considered: (a) the apple slices are assumed to be infinite slabs, (b) the initial soluble solids are evenly distributed, (c) the process is isothermal (40 °C), (d) the diffusion coefficient is assumed to be constant, (e) simultaneous counter-current flows: diffusion of water from the fruit and diffusion of sugar to the fruit are considered only, and (f) other transfer mechanisms and shrinkage of the sample are neglected. Therefore, the analytical solution to Fick’s second law for solid diffusion in one dimension is given by Eq. (1).
¼
CðtÞ C e C0 Ce 1 8 X 1
p2 n¼0 ð2n þ 1Þ2
exp Deff
2 ! ð2n þ 1Þp t L
ð1Þ
where C*(t) is the dimensionless solid concentration of the sample at instant t, C(t) the concentration of the product at instant t, C0 is
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the initial soluble solids concentration (t = 0), Ce is the solids concentration at equilibrium, Deff is the effective diffusion coefficient, L is the material thickness and t is time. In the case that the analysis is made for long times, Eq. (1) can be simplified using only the first term of the series, as shown in Eq. (2). Although it is less rigorous in the calculation, it is a good approximation of reality (n = 0) (Giraldo et al., 2010).
p2 CðtÞ C e 8 C ðtÞ ¼ ¼ 2 exp Deff t C0 Ce p L
ð2Þ
2.4.2. Anomalous diffusion model The anomalous diffusion equation is represented by Simpson et al. (2013).
@aC @bC a ¼D @t @xb
ð3Þ
Considering only the fractional order derivative for time and the order of the spatial derivative of order 2 as in the model of Fick’s second law, then the anomalous diffusion equation (3) can be presented as shown in Eq. (4).
@aC @2C a ¼D @t @x2
ð4Þ
The use of fractional time derivatives has been substantiated by studies indicating that porous materials can be well characterized with respect to both super-diffusive and sub-diffusive behavior using only temporary fractional orders. These studies also suggest that the use of fractional spatial derivatives means that sub-diffusive anomalous phenomena cannot occur. It has also been observed that fractional spatial derivatives fail to satisfactorily describe the displacement distributions of anomalous diffusive processes of porous materials (Berkowitz et al., 2002). Thus the solution for the anomalous diffusion (Eq. (4)) is shown as follows:
C ðtÞ ¼ ¼
CðtÞ C e C0 Ce 1 8 X 1
p2 n¼0 ð2n þ 1Þ
E Deff 2 a
2 ! ð2n þ 1Þp ta L
ð5Þ
where Ea corresponds to the Mittag–Leffler function. This function is important and recurrent in solving equations such as fractional order derivatives, and it has the form presented in the following equation:
Ea ðxÞ ¼
1 X
xk
ð6Þ
Cð1 þ akÞ k¼0
If a = 1, the Mittag–Leffler equation converges to exponential function. This transforms the fractional equation model (4) into a Fickian model as reported by Crank, which is consistent with the fact that if the fractional factor a is one, the fundamental equation of anomalous diffusion becomes the fundamental equation of Fick’s diffusion. In this model, the same approaches to long times are met, as shown in Eq. (7) (Giraldo et al., 2010; Simpson et al., 2013).
C ðtÞ ¼
p2 CðtÞ C e 8 ¼ 2 exp Deff ta C0 Ce p L
Showing that if a = 1, Eq. (7) converges to Eq. (2).
Table 1 Effective diffusion coefficients (Deff) of Fick’s model for different treatments.
Fig. 1. Sucrose concentration change over time at different electric fields (9, 13 and 17 V/cm) for osmotic solutions of: (a) 45 ° Brix, (b) 55 ° Brix and (c) 65 ° Brix.
Sucrose concentration (°Brix)
Electric field (V/cm)
Effective diffusion coefficients, Deff 1010 (m2/s)
45 45 45 45 55 55 55 55 65 65 65 65
0 9 13 17 0 9 13 17 0 9 13 17
1.484 ± 0.163 2.290 ± 0.207 2.906 ± 0.028 5.085 ± 0.496 1.594 ± 0.062 1.627 ± 0.115 2.229 ± 0.039 5.140 ± 0.444 1.813 ± 0.118 1.910 ± 0.029 2.012 ± 0.002 3.408 ± 0.300
ð7Þ
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tetraoxide solution for 1–2 h in 0.1 M phosphate buffer at 4 °C. The samples were dehydrated by immersion in ethanol solutions (30%, 50%, 70%, 85% and 95%) for 15 min each. The samples were then immersed in pure ethanol for 20 min, then dried in a critical point dryer (Balzers-Union, Balzers, Liechtenstein), covered with a thin silver layer, vacuum evaporated and covered with a gold coat. The samples were examined at an acceleration voltage of 20 kV. 2.6. Statistical analysis Tests of significance of the results were performed using analysis of variance (ANOVA). Statistical significance was set at a probability level of 0.05 (p < 0.05) with a confidence level of 95% and was determined using the Minitab 16 software.
Fig. 2. Effective diffusion coefficients (Deff) at different electric fields (0, 9, 13 and 17 V/cm).
2.5. Microstructural analysis Structural analyses were conducted using a scanning electron microscope (SEM; Jeol JSM-6380LV) and a transmission electron microscope (TEM; Jeol JEM 12000EX-II) (Oxford Instruments, UK). Both fresh and treated samples were analyzed from the surface to the center. Sample fixation was accomplished by immersion in glutaraldehyde (2–4%) for 4–24 h in 0.1 M phosphate buffer (pH = 7.2– 7.4) at 4 °C. A second fixation was performed with 1% osmium
3. Results and discussion The variation of sugar concentration in apple samples for both the standard process and for the MEF process is shown in Fig. 1. In the experiments, osmotic solutions of 45, 55 and 65 °Brix were used. Fig. 1a–c shows that the higher the MEF treatment (from 0 to 17 V/cm) the higher was the solids content. This result can be explained due to the mechanism of mass transfer in combination with the effect of electroporation, which together promote changes in the structure of the treated samples (Chiralt and Talens, 2005; Moreno et al., 2011, 2013). Furthermore, the sugar concentration of the product increased with osmotic solutions of increasing concentration, being highest in solutions of 65 °Brix. This is due to the
Fig. 3. Fick’s and anomalous diffusion models for an osmotic solution of 55 °Brix and the application of different moderate electric fields: (a) 0 V/cm, (b) 9 V/cm, (c) 13 V/cm and (d) 17 V/cm.
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Fig. 4. Error distribution for Fick’s model fitted to diffusional process coupled with MEF at (A) 0 V/cm, (B) 9 V/cm, (C) 13 V/cm and (D) 17 V/cm.
increased osmotic strength existing between the food and the osmotic solution in the early stages of the process (Bianchi et al., 2009). 3.1. Fitting the experimental data to Fick’s second law Fick’s second law was fitted to the experimental data using Microsoft Excel Solver tool 2010. The effective diffusion coefficient (Deff) obtained for different electric fields is presented in Table 1. The values of the effective diffusion coefficient for apple are between 1484 1010 m2/s and 5140 1010 m2/s. This order of magnitude is comparable with results reported in the literature. The values reported for the diffusion coefficient range from 4.0 1012 to 6.4 109 m2/s (Saravacos and Maroulis, 2001). The highest value was obtained for MEF of 17 V/cm with an osmotic solution concentration of 55 °Brix. However, if this result is compared with the concentration of 45 °Brix, there is no statistically significant difference (p > 0.05); that is, the diffusion coefficients are not affected by the concentration of the osmotic solution. The lowest value of Deff was obtained without the application of an electric field and an osmotic concentration of 45 °Brix. If this result (Deff) is compared with that obtained for a concentration of 55 °Brix without the application of an electric field, no significant difference (p > 0.05) is found, which confirms the above, meaning that the concentration of the osmotic solution has no significant effect on the diffusion coefficient. The differences among the diffusion coefficients can be attributed to the different experimental conditions, to certain violations of the assumptions of the Crank model (Spiazzi and Mascheroni, 1997), or to the potential existence of non-Fickian mechanisms. Thus, the use of
the Crank model becomes an empirical procedure to fit the experimental data, and the effective diffusion coefficient is a kinetic parameter strongly dependent on experimental conditions (Salvatori et al., 1999; Shi and Le Maguer, 2002). Graphically, the diffusion coefficients for different electric fields are shown in Fig. 2. It is possible to observe that for a constant osmotic solution concentration, the diffusion of sucrose in the apple sample is faster as the applied electric field increases. Given a constant electric field, the diffusion coefficient does not change for different concentrations of the osmotic solution. According to statistical analysis, the most significant factor (p < 0.05) for the change in Deff is the application of an electric field. Increasing the electric field (from 0 to 17 V/cm) increases the diffusion coefficient with a 95% degree of confidence, which may be explained by the phenomenon of electroporation, which alters the structure of the apple. This is consistent with previous studies that have determined that the application of an electric field increases diffusion through the food material, favoring the gain of solids (Kemp and Fryer, 2007; Moreno et al., 2012). However, as the concentration of the osmotic solution increases, the diffusion coefficient in most cases is constant, which can be explained by the fact that this parameter does not depend on the conditions during the process of osmotic dehydration, but rather it depends on the product that is dehydrated (Vega et al., 2007), i.e., the microstructure of the apple. Fig. 3 shows the fitting of Fick’s second law for each set of experimental data. These experiments were performed at an osmotic concentration of 55 °Brix with different electric fields (0, 9, 13 and 17 V/cm). Although they are not shown, the figures corresponding to 45 and 65 °Brix depict the same trend as shown in Fig. 3.
R. Simpson et al. / Journal of Food Engineering 166 (2015) 204–211
In general, Fig. 3 shows that the diffusion model from Fick’s second law does not fit the experimental data well. However, as the electric field is increased, the fit of Fick’s second law improves. This statement is substantiated by observing Fig. 4. In fact, the error distribution is much more randomly distributed as the electric field approaches 17 V/cm (Fig. 4D), indicating that only at 17 V/cm can Fick’s second law be considered an adequate diffusion model. This can be explained because of the application of an electric field that causes changes in cellular structure. These changes can be explained and justified by the electroporation phenomenon. When cells are permeabilized, channels are opened that allow better and more homogeneous diffusion into the tissue (Kulshrestha and Sastry, 2006). Thus, with the path of the molecules made less tortuous, the assumptions of Fick’s second law are more closely met, demonstrating the influence of food structure on models of diffusion mechanisms.
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deformation (Fig. 5C and D). The plasmalemma and tonoplast contraction is correlated with the cellular reduction and deformation. These changes result in the loss of cellular cohesion, fruit firmness and fruit textural quality (Moreno et al., 2011). Osmodehydrated apple samples treated with moderate electric field (OD/MEF) had altered and perforated cell walls (Fig. 5E and F). This alteration may be explained by the mechanism of electropermeabilization. When cells are permeabilized, channels are opened to allow diffusion into the tissue (Kulshrestha and Sastry, 2006). This effect is shown in Fig. 5E and F, with treatments at 13 V/cm (100 V) resulting in cell wall perforations, this effect is more pronounced with an electric field at 17 V/cm (130 V). Therefore, the electropermeabilization effect induced by the moderate electric filed (MEF) promoted the increased mass transfer process as compared to conventional osmotic treatment. 3.3. Fitting the experimental data to an anomalous diffusion model
3.2. Structural changes The tissue of fresh apples was composed of numerous cells, such cell walls, intercellular space, and protoplasts (Fig. 5a and b). In these cells, a large vacuole occupied most of the protoplast, and the tonoplast and plasmalemma were close to the cellular wall (Fig. 5b). The osmotic treatments of the apples resulted in the following effects: cellular collapse, intracellular space contraction and
Because the fit of Fick’s second law is not adequate for treatments under 17 V/cm, an anomalous diffusion model was attempted. In Fig. 3, a fractional model was fitted for each set of experimental data (electric fields from 0 to 17 V/cm) using an osmotic solution of 55 °Brix. Furthermore, it is observed that the fractional-time order a indicates that the nature of the anomalous diffusion process is super-diffusive with fractional orders greater than one for all treatments. In addition, the fractional-time order
Fig. 5. SEM and TEM micrographs of parenchyma tissue from fresh and treated apples at 13 V/cm (100 V). (a and b) Fresh control, (c and d) OD, and (e and f) OD/MEF. CW: cell wall. IS: intercellular space. TN: tonoplast. PT: protoplast. CR: cell rupture.
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Table 2 Comparison between coefficients of determination (R2) for effective diffusion coefficients (Deff) obtained by Fick’s and anomalous diffusion models. Sucrose concentration (°Brix)
Electric field (V/cm)
R2 Fick
R2 fractional
45 45 45 45 55 55 55 55 65 65 65 65
0 9 13 17 0 9 13 17 0 9 13 17
0.926 0.995 0.975 0.991 0.948 0.959 0.984 0.949 0.957 0.927 0.982 0.946
0.947 0.997 0.994 0.998 0.955 0.961 0.989 0.951 0.972 0.938 0.987 0.962
monotonically becomes closer to one as the electric field becomes higher. Strictly, a statistical analysis indicates that the fractional-time order at 17 V/cm (a = 1.06 ± 0.056) is not different from one, indicating that at the latter electric field Fick’s second law is a good model to represent the diffusion process. The fact that Fick’s second law underestimates the effective diffusion coefficient indicates that the process can take less time than that predicted by Fick’s law (see Fig. 3). As an example, in Fig. 3c, to get a C⁄ = 0.1 the anomalous diffusion model predicts a processing time just over 400 min and the Fick’s model almost 600 min. Nevertheless, because the speed of the diffusion process is significantly accelerated with an electric field of 17 V/cm, then it will be reasonable to use Fick’s second law to represent the diffusion phenomena under this optimum experimental condition. The latter is considered an optimum experimental condition in the way that the diffusion process is significantly accelerated. A comparison between the coefficients of determination (R2) for both models is presented in Table 2, being higher for the fractional model in all treatments, which proves that the anomalous diffusion model is a better representation of the process of sucrose diffusion in apple slices. Even though with the application of a higher electric field such as 17 V/cm the experimental data tend to show Fickian behavior, the anomalous diffusion model always fit the experimental data better.
4. Conclusions Sucrose diffusion in Granny Smith apples is highly influenced by the application of a MEF, meaning that as the application of the electric field increases, the higher the effective diffusion coefficient (Deff) becomes. As has been mentioned, this can be explained by electroporation mechanisms that alter the structure of the apple tissue. However, the concentration of the osmotic solution does not significantly affect the effective diffusion coefficient, which as explained is because the effective diffusion coefficient is not dependent on the osmotic solution but rather depends on the characteristics of the dehydrated product. From the above it is observed that the combined effect of osmotic dehydration plus MEF application in Granny Smith apple slices produces an acceleration of the mass transfer that enables a significant reduction in processing time, thereby resulting in an increase in productivity and a decrease in operating costs. The model based on Fick’s second law delivered effective diffusion coefficients for the apple tissue that are within the typical range of the data reported in the literature. The differences in the diffusion coefficients can be attributed to the different experimental conditions established (i.e., the application of the MEF) and because the characteristics of cell structure ensure that some of the assumptions of the Crank model are not met. The exponential
decay from the Fick law, in the tail of the diffusion profile, does not represent the shape of the data. Modeled data showed that Fick’s model over-predicted processing time in most cases. However, as the application of the electric field increases the fit of Fick’s model improves, and specifically for an electric field of 17 V/cm it is observed that the behavior of the experimental data resembles the behavior predicted by Fick’s second law. This is because the application of an electric field causes changes in cell structure, especially in the porosity of the food, making the assumptions established by Fick’s second law adequate. The empirical parameter a for the anomalous diffusion model was always greater than one, but as the MEF increased, a was monotonically tending to one. However, the parameter a for the treatment with an application of an electric field of 17 V/cm is not significantly different from one. This indicates that the diffusion of sucrose in apple slices did not follow Fickian behavior with the exception of treatment with an electric field of 17 V/cm. However future work including shrinkage and volume change during osmotic dehydration process using fractional calculus tools must be performed with the aim to get higher accuracy in the determination of diffusion coefficient. Finally, the application of MEF significantly accelerated the osmotic dehydration process, and the diffusion mechanisms tend to be Fickian at an electric field of 17 V/cm. Acknowledgments Author Dr. Ricardo Simpson is grateful for financial support provided by CONICYT through the FONDECYT project 1121113 and 1090628. References Agnelli, M., Marani, C., Mascheroni, R., 2005. Modeling of heat and mass transfer during (osmo) dehydrofreezing of fruits. J. Food Eng. 69, 415–424. An, H.J., King, J.M., 2007. Thermal characteristics of ohmically heated rice starch and rice flours. J. Food Sci. 72 (1), C84–C88. Association of Official Analytical Chemists (AOAC), 2000. Moisture in Fruits. An Adaptation of Method 934.06, 16th ed. Official Methods of Analysis of Association of Official Analytical Chemists International, Gaithersburg, Maryland, USA. Azuara, E., Cortes, R., Garcia, H.S., Bristain, C., 1992. Kinetic model for osmotic dehydration and its relationship with Fick’s second law. Int. J. Food Sci. Technol. 27, 409–418. Barat, J., Fito, P., Chiralt, A., 2001. Modeling of simultaneous mass transfer and structural changes in fruit juices. J. Food Eng. 49, 77–85. Berkowitz, B., Klafter, J., Metzler, R., Scher, H., 2002. Physical pictures of transport in heterogeneous media: advection–dispersion, random-walk, and fractional derivative formulations. Water Resour. Res. 38 (10), 1191–1202. Bianchi, M., Milisenda, P., Guarnaschelli, A., Mascheroni, R.H., 2009. Transferencia de masa en deshidratación osmótica de frutas. Determinación experimental y simulación. In: XII Congreso Argentino de Ciencia y Tecnología de los Alimentos y 3ª Simposio Internacional de Nuevas Tecnologías – CYTAL XII. Chiralt, A., Talens, P., 2005. Physical and chemical changes induced by osmotic dehydration in plant tissues. J. Food Eng. 65, 167–177. Crank, J., 1975. The Mathematics of Diffusion. Oxford University Press, Great Britain. Fito, P., 1994. Modeling of vacuum osmotic dehydration of foods. J. Food Eng. 22, 313–318. Giraldo, A., Arévalo, A., Ferreira, A., Ferreira, P., Valdés, J.C., 2010. Datos experimentales de la cinética del secado y del modelo matemático para pulpa de cupuaçu (Theobroma grandiflorum) en rodajas. Ciencia y Tecnología de Alimentos, Campinas, Brasil, Marzo, pp. 179–181. Kaymak-Ertekin, F., Sultanoglu, M., 2000. Modelling of mass transfer during osmotic dehydration of apples. J. Food Eng. 46, 243–250. Kemp, M.R., Fryer, P.J., 2007. Enhancement of diffusion through foods using alternating electric fields. Innov. Food Sci. Emerg. Technol. 8, 143–153. Kulshrestha, S.A., Sastry, S., 2006. Low-frequency dielectric changes in cellular food material from ohmic heating: effect of end point temperature. Innov. Food Sci. Emerg. Technol. 7, 257–262. Leizerson, S., Shimoni, E., 2005. Effect of ultrahigh temperature continuous ohmic heating treatment on fresh orange juice. J. Agric. Food Chem. 53, 3519–3524. Lima, M., Sastry, S.K., 1999. The effects of ohmic heating frequency on hot-air drying rate and juice yield. J. Food Eng. 41, 115–119. Moreno, J., Simpson, R., Estrada, D., Lorenzen, S., Moraga, Almonacid, S., 2011. Effect of pulsed-vacuum and ohmic heating on the osmodehydration kinetics, physical properties and microstructure of apples (cv. Granny Smith). Innov. Food Sci. Emerg. Technol. 12, 562–568.
R. Simpson et al. / Journal of Food Engineering 166 (2015) 204–211 Moreno, J., Simpson, R., Pizarro, N., Parada, K., Pinilla, N., Almonacid, S., 2012. Effect of ohmic heating and vacuum impregnation on the quality and microbial stability of osmotically dehydrated strawberries (cv. Camarosa). J. Food Eng. 110, 310–316. Moreno, J., Simpson, R., Pizarro, N., Pavez, C., Dorvil, F., Petzold, G., Bugueño, G., 2013. Influence of ohmic heating/osmotic dehydration treatments on polyphenoloxidase inactivation, physical properties and microbial stability of apples (cv. Granny Smith). Innov. Food Sci. Emerg. Technol. 20, 198–207. Nieto, A.B., Salvatori, D.M., Castro, M.A., Alzamora, S.M., 2004. Structural changes in apple tissue during glucose and sucrose osmotic dehydration: shrinkage, porosity, density and microscopic features. J. Food Eng. 61, 269–278. Ochoa, C., Ayala, A., 2005. Modelos matemáticos de transferencia de masa en deshidratación osmótica. Ciencia y Tecnología Alimentaria 4 (5), 330–342. Ramírez, C., Troncoso, E., Muñoz, J., Aguilera, J.M., 2011. Effect of microstructure of pre-treated apple slices on the kinetics of water loss during air drying. J. Food Eng. 106, 253–261. Rastogi, N., Raghavarao, K., 2004. Mass transfer during osmotic dehydration of pineapple: considering Fickian diffusion in cubical configuration. LWT – Food Sci. Technol. 37, 43–47. Rastogi, N., Raghavarao, K., Niranjan, K., Knorr, D., 2002. Recent developments in osmotic dehydration: methods to enhance mass transfer. Trends Food Sci. Technol. 13, 48–59. Salvatori, D., Andrés, A., Chiralt, A., Fito, P., 1999. Osmotic dehydration progression in apple tissue I: Spatial distribution of solutes and moisture content. J. Food Eng. 42, 125–132. Saravacos, G., Maroulis, Z., 2001. Transport Properties of Foods. Marcel Dekker Inc., New York.
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Shi, J., Le Maguer, M., 2002. Osmotic dehydration of foods: mass transfer and modeling aspects. Food Rev. Int. 18, 305–335. Simpson, R., Jiménez, M., Carevic, E., Grancelli, R., 2007. Aceleración de la Deshidratación Osmótica de Frambuesas (Rubus idaeus) por medio de Calentamiento Óhmico. Revista Archivos Latinoamericanos de Nutrición (ALAN) 57 (2), 192–195. Simpson, R., Jaques, A., Nuñez, H., Ramirez, C., Almonacid, A., 2013. Fractional calculus as a mathematical tool to improve the modeling of mass transfer phenomena in food processing. Food Eng. Rev. 5 (1), 45–55. Souraki, B.A., Ghavami, M., Tondro, H., 2014. Correction of moisture and sucrose effective diffusivities for shrinkage during osmotic dehydration of apple in sucrose solution. Food Bioprod. Process. 92, 1–8. Spiazzi, E., Mascheroni, R., 1997. Mass transfer model for osmotic dehydration of fruits and vegetables. I. Development of the simulation model. J. Food Eng. 34, 387–410. Sturm, B., Nunez, A.M., Hofacker, W.C., 2014. Influence of process control strategies on drying kinetics, color and shrinkage of air dried apples. Appl. Therm. Eng. 62, 455–460. Toupin, C.J., Marcotte, M., Le Maguer, M., 1989. Osmotically-induced mass transfer in plant storage tissues: a mathematical model. Part I. J. Food Eng. 10, 13–38. Varea, C., Hernández, D., 2010. Difusión Anómala en Sistema Complejos. Revista Digital Universitaria – UNAM 11 (6), 1–11. Vega, A., Tello, C., Lemus, R., 2007. Simulación matemática del proceso de secado de la Gracilaria Chilena (Gracilaria Chilensis), Ingeniare. Revista Chilena de Ingeniería 15 (1), 55–64. Zaritzky, N., Califano, A., 1999. Effective diffusion coefficients of chemical preservatives in food tissue. Trends Heat Mass Moment. Transf. 5, 127–139.
Contents lists available at ScienceDirect
Journal of Food Engineering journal homepage: www.elsevier.com/locate/jfoodeng
Diffusion mechanisms during the osmotic dehydration of Granny Smith apples subjected to a moderate electric field R. Simpson a,b,⇑, C. Ramírez a, V. Birchmeier a, A. Almonacid a,b, J. Moreno c, H. Nuñez a, A. Jaques a a
Departamento de Ingeniería Química y Ambiental, Universidad Técnica Federico Santa María, P.O. Box 110-V, Valparaíso, Chile Centro regional de estudios en alimentos saludables (CREAS) Conicyt-Regional R06I1004, Blanco 1623 Room 1402, Valparaíso, Chile c Departamento de Ingeniería en Alimentos, Universidad del Bío-Bío, P.O. Box 447, Chillán, Chile b
a r t i c l e
i n f o
Article history: Received 27 January 2015 Received in revised form 19 May 2015 Accepted 24 May 2015 Available online 27 May 2015 Keywords: Diffusion mechanisms Osmotic dehydration Moderate electric field
a b s t r a c t Osmotic dehydration is a process wherein foods are partially dehydrated by immersion in an aqueous hypertonic solution. Osmotic dehydration reduces the water activity of the food, thereby minimizing the potential growth of microorganisms and extending the shelf life of food products. A major disadvantage of the osmotic dehydration process is the long time required to reduce the water activity, which makes its industrial implementation impractical. The aim of this research is to determine the diffusion mechanisms in Granny Smith apples exposed to different electric fields varying from 0 to 17 V/cm. Osmotic dehydration was performed at 40 °C for both the conventional treatment and for the MEF treatment. The fruit/solution ratio used was 1:11 w/w to prevent and minimize the change in concentration of the solution during the experiment. The osmotic solution was exposed to electric fields of 0, 9, 13 or 17 V/cm, respectively. To determine the diffusion mechanisms two phenomenological models were tested: Fick’s second law and Anomalous diffusion model. Sucrose diffusion in Granny Smith apples is highly influenced by the application of a MEF, meaning that as the application of the electric field increases, the higher the effective diffusion coefficient (Deff) becomes. The exponential decay from the Fick law, in the tail of the diffusion profile, does not represent the shape of the data. However, as the application of the electric field increases the fit of Fick’s model improves, and specifically for an electric field of 17 V/cm it is observed that the behavior of the experimental data resembles the behavior predicted by Fick’s second law. The empirical parameter a for the anomalous diffusion model was always greater than one, but as the MEF increased, a was monotonically tending to one. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Osmotic dehydration is a process wherein foods are partially dehydrated by immersion in an aqueous hypertonic solution. Osmotic dehydration reduces the water activity of the food, thereby minimizing the potential growth of microorganisms and extending the shelf life of food products. Osmotic dehydration is a technology with broad applications in food processing. Nevertheless, the osmotic dehydration process can achieve only partial dehydration of food. Another disadvantage of the osmotic dehydration process is the long time required to reduce the water activity, which makes its industrial implementation impractical (Simpson et al., 2007). The application of a moderate electric field (MEF) means that an electric current passes through a food material. When applied, an ⇑ Corresponding author at: Departamento de Ingeniería Química y Ambiental, Universidad Técnica Federico Santa María, P.O. Box 110-V, Valparaíso, Chile. http://dx.doi.org/10.1016/j.jfoodeng.2015.05.027 0260-8774/Ó 2015 Elsevier Ltd. All rights reserved.
electric field can cause changes in the permeability of cell membranes of plant tissue at lower temperatures at which these membranes are permeabilized by thermal effects, a phenomenon known as electroporation (An and King, 2007; Lima and Sastry, 1999). The resulting effect is that the diffusion process increases, the electrical conductivity changes and the moisture more easily migrates out of the plant tissue (Leizerson and Shimoni, 2005). In general, structural changes of cells such as cell disruption, electroporation, and cell damage tend to improve the diffusion rate because water movement through the cell membrane and cell wall is easier. For example, Ramírez et al. (2011) evaluated some pre-treatments from a microstructural point of view and their effect on water diffusion rate, finding that those pre-treatments that induced more damage to the cellular structure produced an increase in diffusion rate during air drying. The case of electroporation is the same concept, with cell damage favoring the diffusion of water during the dehydration process. Moreno et al. (2013)
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showed the effects at the microstructural level in strawberries of applying an electric field of 130 V/cm, where the cells presented changes in shape and thickness of lamellae and an increase in cell breakage. The phenomenon of osmotic dehydration can be modeled by the fundamentals of mass transfer that describe the origin of the diffusive forces that are involved in and control these processes (Spiazzi and Mascheroni, 1997). In recent years, numerous studies have been conducted to improve knowledge regarding the internal mass transfer occurring during osmotic dehydration of foods and to model the mechanism of the relevant process (Spiazzi and Mascheroni, 1997; Kaymak-Ertekin and Sultanoglu, 2000). This process is usually represented by Fick’s second law. The best known phenomenological model to represent the diffusional mechanism is the model of Crank (Crank, 1975), consisting of a set of solutions of Fick’s law of diffusion for different geometries, boundary conditions and initial conditions. This model has been used by many authors because it is the best known phenomenological model for representing diffusional mechanisms (Ochoa and Ayala, 2005). To these ends, researchers have developed several mathematical models (Barat et al., 2001; Fito, 1994; Salvatori et al., 1999; Toupin et al., 1989; Agnelli et al., 2005) and determined effective diffusion coefficient values in different fruits and geometries using Fick’s second law (Rastogi et al., 2002; Rastogi and Raghavarao, 2004; Azuara et al., 1992). However, the complexity of the mass transfer process due to the complex nature of plant tissues, their strongly heterogeneous cell structure, shrinkage and volume changes during dehydration process due to moisture removing makes accurate prediction challenging, so that the exact analytical solutions cannot normally be applied because several of the assumptions of Fick’s second law do not fit heterogeneous tissues such as apple (Varea and Hernández, 2010; Zaritzky and Califano, 1999). These structural changes are also present in osmotic dehydration, however is less significant than in traditional dehydration which can reach values close to 60–70% (Sturm et al., 2014). For example, Souraki et al. (2014) and Nieto et al. (2004) have reported that the effect of osmotic dehydration on shrinkage of apple could be around 10% up to 30%. In the present study the effect of shrinkage will be neglected, because the fractional diffusion formulation for shrinking geometries has not been developed yet. Anomalous or non-Fickian diffusion has been proven to be a useful tool for quantitatively describing diffusion during drying of foods because it considers many of the changes that occur in the food material and its microstructure during the process such as shrinkage and porosity. Fractional calculus modeling is a new and innovative approach in the food processing field that requires rethinking the diffusion process in food materials (Simpson et al., 2013). Fractional calculus is a mathematical tool for mathematically representing the phenomena of real anomalous diffusion of solutes whose movement can be faster or slower than postulated in Fick’s second law due to the influence of the cellular structure of the food material (Simpson et al., 2013). Descriptions of mass transfer behavior can be modeled with differential equations of non-integer order as fractional numbers. The aim of this research is to determine the diffusion mechanisms in Granny Smith apples exposed to different electric fields varying from 0 to 17 V/cm by application of fractional diffusional formulation using fractional calculus tool.
and were subsequently immersed in 1% ascorbic acid solution and 2% citric acid solution to prevent enzymatic browning. Osmotic solutions of 45, 55 and 65 °Brix were prepared using sucrose (commercial sugar). The osmotic solutions contained 7 mg/L of potassium sorbate (C6H7KO2) as a preservative, inhibiting any unwanted microbiological activity and 1.27, 1.33 or 1.13 g/L of calcium chloride (CaCl2) to increase the conductivity of the solution.
2. Materials and methods
C ðtÞ ¼
2.1. Sample preparation Granny Smith apples were acquired in a supermarket in the city of Valparaiso, Chile and kept under refrigeration at 2 °C. The apples were peeled and cut into sheets 40 mm in diameter and 7 mm thick
2.2. Osmotic dehydration Osmotic dehydration was performed at 40 °C for both the conventional treatment and for the MEF treatment. The fruit/solution ratio used was 1:11 w/w to prevent and minimize the change in concentration of the solution during the experiment. The samples were immersed in a cylindrical cell made of stainless steel with a plastic bottom. The cell was composed of two concentric cylinders (3.7 cm and 19 cm diameter) connected to a generator by means of two electrodes (Moreno et al., 2011). The osmotic solution was exposed to an alternating current with voltage of 70, 100 or 130 V, generating electric fields of 9, 13 or 17 V/cm, respectively. The cell was immersed in a thermo-regulated bath (BS-21 Jeio Tech, Korea) with the aim of maintaining the medium temperature at 40 °C during the experiment. In addition, constant and gentle stirring was provided to attain a homogeneous solution without damaging the fruit. To monitor the temperature of the solution and the applied electric field, two T type thermocouples (copper– constantan) were used. Data on temperature, voltage and current were recorded every three seconds with a data logger (Omega 220, USA) having a connection via modem and connection port to a computer. The data logger includes HiperWareTM software version 4.77 (1996–2005), which communicates with and gathers data from the program online. Each experimental run was performed for 12 h with sampling at different times (optimal sampling for diffusion experiments) to measure the concentration of sugar in the apple. 2.3. Analytical determinations Moisture was determined by drying the samples for 24 h at 60 °C in a vacuum oven to constant weight according to the method defined by the Association of Official Analytical Chemists (AOAC) (Association of Official Analytical Chemists, 2000) for fruits rich in sugar. The determination of soluble solids was performed, after homogenization of the samples, using a digital refractometer (Hanna Instruments, model HI 96811, USA). 2.4. Mathematical models 2.4.1. Fick’s second law of diffusion To develop the model, the following assumptions were considered: (a) the apple slices are assumed to be infinite slabs, (b) the initial soluble solids are evenly distributed, (c) the process is isothermal (40 °C), (d) the diffusion coefficient is assumed to be constant, (e) simultaneous counter-current flows: diffusion of water from the fruit and diffusion of sugar to the fruit are considered only, and (f) other transfer mechanisms and shrinkage of the sample are neglected. Therefore, the analytical solution to Fick’s second law for solid diffusion in one dimension is given by Eq. (1).
¼
CðtÞ C e C0 Ce 1 8 X 1
p2 n¼0 ð2n þ 1Þ2
exp Deff
2 ! ð2n þ 1Þp t L
ð1Þ
where C*(t) is the dimensionless solid concentration of the sample at instant t, C(t) the concentration of the product at instant t, C0 is
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the initial soluble solids concentration (t = 0), Ce is the solids concentration at equilibrium, Deff is the effective diffusion coefficient, L is the material thickness and t is time. In the case that the analysis is made for long times, Eq. (1) can be simplified using only the first term of the series, as shown in Eq. (2). Although it is less rigorous in the calculation, it is a good approximation of reality (n = 0) (Giraldo et al., 2010).
p2 CðtÞ C e 8 C ðtÞ ¼ ¼ 2 exp Deff t C0 Ce p L
ð2Þ
2.4.2. Anomalous diffusion model The anomalous diffusion equation is represented by Simpson et al. (2013).
@aC @bC a ¼D @t @xb
ð3Þ
Considering only the fractional order derivative for time and the order of the spatial derivative of order 2 as in the model of Fick’s second law, then the anomalous diffusion equation (3) can be presented as shown in Eq. (4).
@aC @2C a ¼D @t @x2
ð4Þ
The use of fractional time derivatives has been substantiated by studies indicating that porous materials can be well characterized with respect to both super-diffusive and sub-diffusive behavior using only temporary fractional orders. These studies also suggest that the use of fractional spatial derivatives means that sub-diffusive anomalous phenomena cannot occur. It has also been observed that fractional spatial derivatives fail to satisfactorily describe the displacement distributions of anomalous diffusive processes of porous materials (Berkowitz et al., 2002). Thus the solution for the anomalous diffusion (Eq. (4)) is shown as follows:
C ðtÞ ¼ ¼
CðtÞ C e C0 Ce 1 8 X 1
p2 n¼0 ð2n þ 1Þ
E Deff 2 a
2 ! ð2n þ 1Þp ta L
ð5Þ
where Ea corresponds to the Mittag–Leffler function. This function is important and recurrent in solving equations such as fractional order derivatives, and it has the form presented in the following equation:
Ea ðxÞ ¼
1 X
xk
ð6Þ
Cð1 þ akÞ k¼0
If a = 1, the Mittag–Leffler equation converges to exponential function. This transforms the fractional equation model (4) into a Fickian model as reported by Crank, which is consistent with the fact that if the fractional factor a is one, the fundamental equation of anomalous diffusion becomes the fundamental equation of Fick’s diffusion. In this model, the same approaches to long times are met, as shown in Eq. (7) (Giraldo et al., 2010; Simpson et al., 2013).
C ðtÞ ¼
p2 CðtÞ C e 8 ¼ 2 exp Deff ta C0 Ce p L
Showing that if a = 1, Eq. (7) converges to Eq. (2).
Table 1 Effective diffusion coefficients (Deff) of Fick’s model for different treatments.
Fig. 1. Sucrose concentration change over time at different electric fields (9, 13 and 17 V/cm) for osmotic solutions of: (a) 45 ° Brix, (b) 55 ° Brix and (c) 65 ° Brix.
Sucrose concentration (°Brix)
Electric field (V/cm)
Effective diffusion coefficients, Deff 1010 (m2/s)
45 45 45 45 55 55 55 55 65 65 65 65
0 9 13 17 0 9 13 17 0 9 13 17
1.484 ± 0.163 2.290 ± 0.207 2.906 ± 0.028 5.085 ± 0.496 1.594 ± 0.062 1.627 ± 0.115 2.229 ± 0.039 5.140 ± 0.444 1.813 ± 0.118 1.910 ± 0.029 2.012 ± 0.002 3.408 ± 0.300
ð7Þ
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tetraoxide solution for 1–2 h in 0.1 M phosphate buffer at 4 °C. The samples were dehydrated by immersion in ethanol solutions (30%, 50%, 70%, 85% and 95%) for 15 min each. The samples were then immersed in pure ethanol for 20 min, then dried in a critical point dryer (Balzers-Union, Balzers, Liechtenstein), covered with a thin silver layer, vacuum evaporated and covered with a gold coat. The samples were examined at an acceleration voltage of 20 kV. 2.6. Statistical analysis Tests of significance of the results were performed using analysis of variance (ANOVA). Statistical significance was set at a probability level of 0.05 (p < 0.05) with a confidence level of 95% and was determined using the Minitab 16 software.
Fig. 2. Effective diffusion coefficients (Deff) at different electric fields (0, 9, 13 and 17 V/cm).
2.5. Microstructural analysis Structural analyses were conducted using a scanning electron microscope (SEM; Jeol JSM-6380LV) and a transmission electron microscope (TEM; Jeol JEM 12000EX-II) (Oxford Instruments, UK). Both fresh and treated samples were analyzed from the surface to the center. Sample fixation was accomplished by immersion in glutaraldehyde (2–4%) for 4–24 h in 0.1 M phosphate buffer (pH = 7.2– 7.4) at 4 °C. A second fixation was performed with 1% osmium
3. Results and discussion The variation of sugar concentration in apple samples for both the standard process and for the MEF process is shown in Fig. 1. In the experiments, osmotic solutions of 45, 55 and 65 °Brix were used. Fig. 1a–c shows that the higher the MEF treatment (from 0 to 17 V/cm) the higher was the solids content. This result can be explained due to the mechanism of mass transfer in combination with the effect of electroporation, which together promote changes in the structure of the treated samples (Chiralt and Talens, 2005; Moreno et al., 2011, 2013). Furthermore, the sugar concentration of the product increased with osmotic solutions of increasing concentration, being highest in solutions of 65 °Brix. This is due to the
Fig. 3. Fick’s and anomalous diffusion models for an osmotic solution of 55 °Brix and the application of different moderate electric fields: (a) 0 V/cm, (b) 9 V/cm, (c) 13 V/cm and (d) 17 V/cm.
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Fig. 4. Error distribution for Fick’s model fitted to diffusional process coupled with MEF at (A) 0 V/cm, (B) 9 V/cm, (C) 13 V/cm and (D) 17 V/cm.
increased osmotic strength existing between the food and the osmotic solution in the early stages of the process (Bianchi et al., 2009). 3.1. Fitting the experimental data to Fick’s second law Fick’s second law was fitted to the experimental data using Microsoft Excel Solver tool 2010. The effective diffusion coefficient (Deff) obtained for different electric fields is presented in Table 1. The values of the effective diffusion coefficient for apple are between 1484 1010 m2/s and 5140 1010 m2/s. This order of magnitude is comparable with results reported in the literature. The values reported for the diffusion coefficient range from 4.0 1012 to 6.4 109 m2/s (Saravacos and Maroulis, 2001). The highest value was obtained for MEF of 17 V/cm with an osmotic solution concentration of 55 °Brix. However, if this result is compared with the concentration of 45 °Brix, there is no statistically significant difference (p > 0.05); that is, the diffusion coefficients are not affected by the concentration of the osmotic solution. The lowest value of Deff was obtained without the application of an electric field and an osmotic concentration of 45 °Brix. If this result (Deff) is compared with that obtained for a concentration of 55 °Brix without the application of an electric field, no significant difference (p > 0.05) is found, which confirms the above, meaning that the concentration of the osmotic solution has no significant effect on the diffusion coefficient. The differences among the diffusion coefficients can be attributed to the different experimental conditions, to certain violations of the assumptions of the Crank model (Spiazzi and Mascheroni, 1997), or to the potential existence of non-Fickian mechanisms. Thus, the use of
the Crank model becomes an empirical procedure to fit the experimental data, and the effective diffusion coefficient is a kinetic parameter strongly dependent on experimental conditions (Salvatori et al., 1999; Shi and Le Maguer, 2002). Graphically, the diffusion coefficients for different electric fields are shown in Fig. 2. It is possible to observe that for a constant osmotic solution concentration, the diffusion of sucrose in the apple sample is faster as the applied electric field increases. Given a constant electric field, the diffusion coefficient does not change for different concentrations of the osmotic solution. According to statistical analysis, the most significant factor (p < 0.05) for the change in Deff is the application of an electric field. Increasing the electric field (from 0 to 17 V/cm) increases the diffusion coefficient with a 95% degree of confidence, which may be explained by the phenomenon of electroporation, which alters the structure of the apple. This is consistent with previous studies that have determined that the application of an electric field increases diffusion through the food material, favoring the gain of solids (Kemp and Fryer, 2007; Moreno et al., 2012). However, as the concentration of the osmotic solution increases, the diffusion coefficient in most cases is constant, which can be explained by the fact that this parameter does not depend on the conditions during the process of osmotic dehydration, but rather it depends on the product that is dehydrated (Vega et al., 2007), i.e., the microstructure of the apple. Fig. 3 shows the fitting of Fick’s second law for each set of experimental data. These experiments were performed at an osmotic concentration of 55 °Brix with different electric fields (0, 9, 13 and 17 V/cm). Although they are not shown, the figures corresponding to 45 and 65 °Brix depict the same trend as shown in Fig. 3.
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In general, Fig. 3 shows that the diffusion model from Fick’s second law does not fit the experimental data well. However, as the electric field is increased, the fit of Fick’s second law improves. This statement is substantiated by observing Fig. 4. In fact, the error distribution is much more randomly distributed as the electric field approaches 17 V/cm (Fig. 4D), indicating that only at 17 V/cm can Fick’s second law be considered an adequate diffusion model. This can be explained because of the application of an electric field that causes changes in cellular structure. These changes can be explained and justified by the electroporation phenomenon. When cells are permeabilized, channels are opened that allow better and more homogeneous diffusion into the tissue (Kulshrestha and Sastry, 2006). Thus, with the path of the molecules made less tortuous, the assumptions of Fick’s second law are more closely met, demonstrating the influence of food structure on models of diffusion mechanisms.
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deformation (Fig. 5C and D). The plasmalemma and tonoplast contraction is correlated with the cellular reduction and deformation. These changes result in the loss of cellular cohesion, fruit firmness and fruit textural quality (Moreno et al., 2011). Osmodehydrated apple samples treated with moderate electric field (OD/MEF) had altered and perforated cell walls (Fig. 5E and F). This alteration may be explained by the mechanism of electropermeabilization. When cells are permeabilized, channels are opened to allow diffusion into the tissue (Kulshrestha and Sastry, 2006). This effect is shown in Fig. 5E and F, with treatments at 13 V/cm (100 V) resulting in cell wall perforations, this effect is more pronounced with an electric field at 17 V/cm (130 V). Therefore, the electropermeabilization effect induced by the moderate electric filed (MEF) promoted the increased mass transfer process as compared to conventional osmotic treatment. 3.3. Fitting the experimental data to an anomalous diffusion model
3.2. Structural changes The tissue of fresh apples was composed of numerous cells, such cell walls, intercellular space, and protoplasts (Fig. 5a and b). In these cells, a large vacuole occupied most of the protoplast, and the tonoplast and plasmalemma were close to the cellular wall (Fig. 5b). The osmotic treatments of the apples resulted in the following effects: cellular collapse, intracellular space contraction and
Because the fit of Fick’s second law is not adequate for treatments under 17 V/cm, an anomalous diffusion model was attempted. In Fig. 3, a fractional model was fitted for each set of experimental data (electric fields from 0 to 17 V/cm) using an osmotic solution of 55 °Brix. Furthermore, it is observed that the fractional-time order a indicates that the nature of the anomalous diffusion process is super-diffusive with fractional orders greater than one for all treatments. In addition, the fractional-time order
Fig. 5. SEM and TEM micrographs of parenchyma tissue from fresh and treated apples at 13 V/cm (100 V). (a and b) Fresh control, (c and d) OD, and (e and f) OD/MEF. CW: cell wall. IS: intercellular space. TN: tonoplast. PT: protoplast. CR: cell rupture.
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Table 2 Comparison between coefficients of determination (R2) for effective diffusion coefficients (Deff) obtained by Fick’s and anomalous diffusion models. Sucrose concentration (°Brix)
Electric field (V/cm)
R2 Fick
R2 fractional
45 45 45 45 55 55 55 55 65 65 65 65
0 9 13 17 0 9 13 17 0 9 13 17
0.926 0.995 0.975 0.991 0.948 0.959 0.984 0.949 0.957 0.927 0.982 0.946
0.947 0.997 0.994 0.998 0.955 0.961 0.989 0.951 0.972 0.938 0.987 0.962
monotonically becomes closer to one as the electric field becomes higher. Strictly, a statistical analysis indicates that the fractional-time order at 17 V/cm (a = 1.06 ± 0.056) is not different from one, indicating that at the latter electric field Fick’s second law is a good model to represent the diffusion process. The fact that Fick’s second law underestimates the effective diffusion coefficient indicates that the process can take less time than that predicted by Fick’s law (see Fig. 3). As an example, in Fig. 3c, to get a C⁄ = 0.1 the anomalous diffusion model predicts a processing time just over 400 min and the Fick’s model almost 600 min. Nevertheless, because the speed of the diffusion process is significantly accelerated with an electric field of 17 V/cm, then it will be reasonable to use Fick’s second law to represent the diffusion phenomena under this optimum experimental condition. The latter is considered an optimum experimental condition in the way that the diffusion process is significantly accelerated. A comparison between the coefficients of determination (R2) for both models is presented in Table 2, being higher for the fractional model in all treatments, which proves that the anomalous diffusion model is a better representation of the process of sucrose diffusion in apple slices. Even though with the application of a higher electric field such as 17 V/cm the experimental data tend to show Fickian behavior, the anomalous diffusion model always fit the experimental data better.
4. Conclusions Sucrose diffusion in Granny Smith apples is highly influenced by the application of a MEF, meaning that as the application of the electric field increases, the higher the effective diffusion coefficient (Deff) becomes. As has been mentioned, this can be explained by electroporation mechanisms that alter the structure of the apple tissue. However, the concentration of the osmotic solution does not significantly affect the effective diffusion coefficient, which as explained is because the effective diffusion coefficient is not dependent on the osmotic solution but rather depends on the characteristics of the dehydrated product. From the above it is observed that the combined effect of osmotic dehydration plus MEF application in Granny Smith apple slices produces an acceleration of the mass transfer that enables a significant reduction in processing time, thereby resulting in an increase in productivity and a decrease in operating costs. The model based on Fick’s second law delivered effective diffusion coefficients for the apple tissue that are within the typical range of the data reported in the literature. The differences in the diffusion coefficients can be attributed to the different experimental conditions established (i.e., the application of the MEF) and because the characteristics of cell structure ensure that some of the assumptions of the Crank model are not met. The exponential
decay from the Fick law, in the tail of the diffusion profile, does not represent the shape of the data. Modeled data showed that Fick’s model over-predicted processing time in most cases. However, as the application of the electric field increases the fit of Fick’s model improves, and specifically for an electric field of 17 V/cm it is observed that the behavior of the experimental data resembles the behavior predicted by Fick’s second law. This is because the application of an electric field causes changes in cell structure, especially in the porosity of the food, making the assumptions established by Fick’s second law adequate. The empirical parameter a for the anomalous diffusion model was always greater than one, but as the MEF increased, a was monotonically tending to one. However, the parameter a for the treatment with an application of an electric field of 17 V/cm is not significantly different from one. This indicates that the diffusion of sucrose in apple slices did not follow Fickian behavior with the exception of treatment with an electric field of 17 V/cm. However future work including shrinkage and volume change during osmotic dehydration process using fractional calculus tools must be performed with the aim to get higher accuracy in the determination of diffusion coefficient. Finally, the application of MEF significantly accelerated the osmotic dehydration process, and the diffusion mechanisms tend to be Fickian at an electric field of 17 V/cm. Acknowledgments Author Dr. Ricardo Simpson is grateful for financial support provided by CONICYT through the FONDECYT project 1121113 and 1090628. References Agnelli, M., Marani, C., Mascheroni, R., 2005. Modeling of heat and mass transfer during (osmo) dehydrofreezing of fruits. J. Food Eng. 69, 415–424. An, H.J., King, J.M., 2007. Thermal characteristics of ohmically heated rice starch and rice flours. J. Food Sci. 72 (1), C84–C88. Association of Official Analytical Chemists (AOAC), 2000. Moisture in Fruits. An Adaptation of Method 934.06, 16th ed. Official Methods of Analysis of Association of Official Analytical Chemists International, Gaithersburg, Maryland, USA. Azuara, E., Cortes, R., Garcia, H.S., Bristain, C., 1992. Kinetic model for osmotic dehydration and its relationship with Fick’s second law. Int. J. Food Sci. Technol. 27, 409–418. Barat, J., Fito, P., Chiralt, A., 2001. Modeling of simultaneous mass transfer and structural changes in fruit juices. J. Food Eng. 49, 77–85. Berkowitz, B., Klafter, J., Metzler, R., Scher, H., 2002. Physical pictures of transport in heterogeneous media: advection–dispersion, random-walk, and fractional derivative formulations. Water Resour. Res. 38 (10), 1191–1202. Bianchi, M., Milisenda, P., Guarnaschelli, A., Mascheroni, R.H., 2009. Transferencia de masa en deshidratación osmótica de frutas. Determinación experimental y simulación. In: XII Congreso Argentino de Ciencia y Tecnología de los Alimentos y 3ª Simposio Internacional de Nuevas Tecnologías – CYTAL XII. Chiralt, A., Talens, P., 2005. Physical and chemical changes induced by osmotic dehydration in plant tissues. J. Food Eng. 65, 167–177. Crank, J., 1975. The Mathematics of Diffusion. Oxford University Press, Great Britain. Fito, P., 1994. Modeling of vacuum osmotic dehydration of foods. J. Food Eng. 22, 313–318. Giraldo, A., Arévalo, A., Ferreira, A., Ferreira, P., Valdés, J.C., 2010. Datos experimentales de la cinética del secado y del modelo matemático para pulpa de cupuaçu (Theobroma grandiflorum) en rodajas. Ciencia y Tecnología de Alimentos, Campinas, Brasil, Marzo, pp. 179–181. Kaymak-Ertekin, F., Sultanoglu, M., 2000. Modelling of mass transfer during osmotic dehydration of apples. J. Food Eng. 46, 243–250. Kemp, M.R., Fryer, P.J., 2007. Enhancement of diffusion through foods using alternating electric fields. Innov. Food Sci. Emerg. Technol. 8, 143–153. Kulshrestha, S.A., Sastry, S., 2006. Low-frequency dielectric changes in cellular food material from ohmic heating: effect of end point temperature. Innov. Food Sci. Emerg. Technol. 7, 257–262. Leizerson, S., Shimoni, E., 2005. Effect of ultrahigh temperature continuous ohmic heating treatment on fresh orange juice. J. Agric. Food Chem. 53, 3519–3524. Lima, M., Sastry, S.K., 1999. The effects of ohmic heating frequency on hot-air drying rate and juice yield. J. Food Eng. 41, 115–119. Moreno, J., Simpson, R., Estrada, D., Lorenzen, S., Moraga, Almonacid, S., 2011. Effect of pulsed-vacuum and ohmic heating on the osmodehydration kinetics, physical properties and microstructure of apples (cv. Granny Smith). Innov. Food Sci. Emerg. Technol. 12, 562–568.
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