DIFFUSION COEFFICIENTS OF WATER AND SUCROSE IN OSMOTIC DEHYDRATION OF PAPAYA

ABSTRACT

Mass transfer during osmotic dehydration of papaya in sucrose solution has been studied. A mathematical model based on Fick's law for unsteady state mass transfer, considering shrinkage, has been used to calculate the moisture and sugars diffusion coefficients. Also, a diffusional model with simultaneous shrinkage was developed using a finite difference method to solve differential equations, assuming moisture content-dependent diffusivities for moisture and sugars. The validity of both proposed models was verified by the good fit between the experimental and calculated curves. The mean percent relative errors were 2.48 and 9.92 for moisture content, and 4.26 and 22.51 for total sugars content, for Model 1 and Model 2, respectively.

INTRODUCTION

Papaya (Carica Papaya L.) is a widely acclaimed temperate zone fruit with commercial production today in Hawaii, tropical Africa, Asia and Australia. There is a widespread but smaller scale production in South Africa and Latin America.[1] One of the main reasons for low production and commercialisation of tropical fruits products like papaya, is the lack of technical processes and adequate equipments for their preservation and transformation into good quality and attractive products.[2]

Osmotic dehydration is a process leading to “intermediate” products. Further processing (i.e., drying, pasteurisation or freezing) are an interesting way of valuation for papaya.[3]

This technique is based on placing foods, such as pieces of fruits or"/>in a hypertonic solution. The solution has a high osmotic pressure, and hence, lowers water activity (a_w); so the driving force for water removal is the activity gradient between the osmotic solution and the food, with natural cells acting as a semi-permeable membrane. During this process there is an important flow of water from the inside of the food to the solution, and also solute enters from the solution into the product. Simultaneously, there is some loss of product's solutes affecting its organoleptic properties; but this loss is quantitatively negligible.4-6

Most of the models for mass transfer are either based on the mathematic solutions of Fick's second law for several shapes and boundary conditions (assuming a simple transfer of water), or empirical.7-9 In addition, most of the published models do not often consider the volume reduction and the variation of the diffusion coefficients with moisture content.

The aim of this research was to study the osmotic dehydration of papaya in saturated sucrose solutions and to model the mass transfer applying Fick's model to obtain the diffusion coefficients for water and sugars. Also a finite difference method was applied considering both diffusion coefficients as a function of moisture content.

MATERIALS AND METHODS

Materials

Fresh, ripe papaya fruits obtained from the same plantation were used for processing. The fruits were selected according to the total area of yellow-orange peel colour percentage.[3]

Papayas with 80% yellow-orange peel coloration were hand-peeled (seeds were removed) and cut into slabs (5×3×1 cm). The slabs were then covered with a plastic material on to the major faces, so the diffusion took place only at the 3×1 faces. Saturated solutions were prepared using commercial sucrose, and the concentration was monitored in each experiment. The solution:fruit volume ratio was 10:1 to avoid a medium dilution. Sodium benzoate at a concentration of 700 ppm was added to avoid microbial growth.

Osmotic Dehydration

Samples were weighed and then immersed in the osmotic medium for different time periods (0, 2, 6, 9 and 15 h) at room temperature. Fluid movement was achieved using a metallic stirrer at 60 rpm. The slabs were removed from the solution, wiped with tissue paper, and ten cuts of 5 mm each were then made. Of these, two sets of five were considered. Cuts 1 to 5 were used for moisture content determination and cuts 6 to 10 were taken for sugar content determination.

Shrinkage

Shrinkage coefficient is defined as the relation between the volume at any time and the initial volume.[10] Assuming unidirectional diffusion, shrinkage coefficient was considered as the relation between the larger length at time t and the initial one of the slab. At any time, the length of the slabs was measured before and after each treatment, to obtain a correlation with moisture content. The other dimensions were not considered (they were covered with plastic material) because one direction mass transfer was assumed.

Moisture and Sugar Content

Moisture content of the samples was determined by the gravimetric method in a vacuum oven at 70±1°C for 6 h.[11] Approximately a 2.5 g sample was blended with the same weight of water and placed in a 50 mL centrifuge tube. Then, 30 mL of a heated mixture 80% ethanol and 20% water was added and centrifuged for 15 min at 2500 rpm. Supernatant was decanted and retained, and the extraction process was repeated three times. Resulting supernatants were brought to a certain volume with the ethanol-water mixture and held for analysis by HPLC. A Shimadzu LC-6A liquid chromatograph equipped with Shimadzu SCL–6A controller and a Komla KWK-024–751 differential refraction index detector, was used for the determination of sucrose, fructose and glucose. The mobile phase was aceto nitrile-water (85:15) and the flow rate was 2 mL/min. Also, a NH₂ Phenomenex IB-SIL5NH₂ column (4.6×250 mm) with a 5 µm particle size was used. All chromatograms were recorded with a C-R3A integrator.12-14

Mathematical Models

Model 1

The first approximation for the mass transfer was based on Fick's second law of unsteady state diffusion.[15] The following assumptions were made: mass transfers take place only in one direction; in the interphase the fruit and the solution are in equilibrium; initial moisture and sugars concentrations are uniform; and shrinkage is linear dependent on moisture content and calculated with Eq. 6

Water and sugars diffusivities were calculated using the analytical solution for a slab, in terms of an infinite series given by:[16]

Model 2

Our second mathematical model considered that mass transfer could be calculated in terms of Fick's law and the microscopic mass balance. The differential Equation 2 was solved using a finite difference method and the assumptions mentioned earlier.

The half of slab was subdivided into five equal parts (five nodal points); each nodal thickness was Δz, except for the external one, which had a thickness Δz/2, and the mass balances were applied at these points. As the mass transfer proceeded, the slab length and the nodes thickness decreased due to water losses, thus moving the boundary of each element. Local moisture and sugar contents were determined by mass balance on each node in a time interval of Δt=1 min.[17]

A computer program was written to estimate the transient moisture and sugar contents at each experimental run. Both diffusivities were considered dependent on moisture content according to Eqs. 3 and 4:

On the first step, the experimental values for moisture and total sugars contents were introduced in the program and values of D_W0, D_A0, B_W and B_A were assumed for each treatment. Using that data, shrinkage, moisture and sugars contents were calculated for each node. Therefore, the final moisture and sugar diffusivities were the values that best fit the transient moisture and sugars distribution at the nodal points, considering the mean percent relative error (Eq. 5) as a fit parameter.

RESULTS AND DISCUSSION

Shrinkage data were analysed using linear regression and fitted with Eq. 6,

with an E=1.75. In Fig. 1 the experimental average moisture contents and the predicted values obtained using Fick's model, as a function of time are shown, with an E value of 2.48; while Fig. 2 shows the agreement between the experimental and predicted values of average sugars contents for different times, with E=4.26.

Figure 1. Experimental and predicted moisture contents during osmotic dehydration (Model 1).

Figure 2. Experimental and predicted sugar contents during osmotic dehydration.

The diffusivity coefficients found using Eq. 1 (Fick's model) were 1.3×10⁻⁹ m²/s and 3.47×10⁻⁹ m²/s for moisture and sugars, respectively. E values obtained indicate the adequacy of the model for reasonably describing the osmotic dehydration of papaya in these experimental conditions. Also, D_W and D_A obtained are similar to those found for other fruits, in sucrose and with other conditions.18-19

Table 1 shows the moisture contents distribution inside the papayas, as a function of position and time, compared to those values predicted with the finite difference model, involving an average E of 9.92. Table 2 shows the experimental concentration profile of total sugars (fructose, glucose and sucrose) inside the papayas, compared to the calculated values, with average E=22.51. Parameters D_W0, D_A0, B_W and B_A of Eqs. 3 and 4 were found from the fitting of the data, giving:

Values of the diffusion coefficients and E for both models are shown in Table 3. Comparing the E values obtained for the two models (E=2.48 with the Fick's model and average E=9.92 with the finite difference method for moisture, and E=4.26 with Model 1 and average E=22.51 with Model 2 for total sugars) it is found that Fick's model, considering shrinkage, could pro vide enough approximation to describe the mass transfer. Also, it should be mentioned that Fick's model is simpler to apply than the finite difference one.

Table 1. Experimental and Predicted Moisture Contents as a Function of Time and Distance Within the Fruit During Osmotic Dehydration (Model 2) D_W=9.3×10⁻¹⁰(1+0.29x_w)(m²/s)

Time (h)	Distance from the Edge (cm)	Experimental Moisture Contents (w.b.)	Predicted Moisture Contents (w.b.)	E
2	2.25	0.8181	0.8671	6.22
	1.75	0.8165	0.8669
	1.25	0.8142	0.864
	0.75	0.8061	0.8366
	0.25	0.7373	0.6708
6	2.25	0.832	0.8829	18.54
	1.75	0.8309	0.8821
	1.25	0.8156	0.8515
	0.75	0.7709	0.7568
	0.25	0.6646	0.5705
9	2.25	0.7593	0.8912	7.85
	1.75	0.7572	0.8737
	1.25	0.7412	0.8213
	0.75	0.7109	0.711
	0.25	0.6081	0.5497
15	2.25	0.7818	0.829	7.09
	1.75	0.7723	0.8013
	1.25	0.7397	0.7401
	0.75	0.7003	0.6441
	0.25	0.56	0.525

Table 2. Experimental and Predicted Sugars Contents as a Function of Time and Distance Within the Fruit During Osmotic Dehydration (Model 2) D_A=4.3×10⁻⁹(1+0.20x_w)(m²/s)

Time (h)	Distance from the Edge (cm)	Experimental Sugars Contents (w.b.)	Predicted Sugars Contents (w.b.)	E
2	2.25	0.1278	0.09411	32.98
	1.75	0.1558	0.1032
	1.25	0.1353	0.1326
	0.75	0.1636	0.2027
	0.25	0.1808	0.3236
6	2.25	0.1443	0.142	23.89
	1.75	0.1593	0.1653
	1.25	0.1593	0.2114
	0.75	0.2085	0.2767
	0.25	0.3624	0.3541
9	2.25	0.2405	0.2071	18.26
	1.75	0.2487	0.2261
	1.25	0.2743	0.262
	0.75	0.2922	0.3103
	0.25	0.3576	0.3649
15	2.25	0.2753	0.2746	14.91
	1.75	0.268	0.2871
	1.25	0.3014	0.3102
	0.75	0.3143	0.3408
	0.25	0.3974	0.3753

Table 3. Values of Moisture and Total Sugars Diffusion Coefficients (m²/s) for Model 1 and Model 2

	Model 1	Model 2
DW	1.3×10^−9	9.3×10^−10 (1 + 0.29 xW)
E	2.48	9.92
DA	3.47×10^−9	4.3×10^−9 (1 + 0.20 xW)
E	4.26	22.51

CONCLUSIONS

The Fickian unsteady state diffusion model, considering a linear variation of the length of a slab with moisture content, was found to fit well for the osmotic dehydration of papaya. Also, a model that considers the variation of the moisture and sugars diffusion coefficients with the local moisture content was developed. So, the moisture and sugars contents during osmotic dehydration could be predicted with enough accuracy from the equations. These predictions are useful for the application of this process to obtain a good quality fruit product.

NOMENCLATURE

BA	=	constant
BW	=	constant
DA	=	sugars diffusivity (m2/s)
DA0	=	constant
DW	=	moisture diffusivity (m2/s)
DW0	=	constant
i	=	node number
L′	=	half thickness of the slab at time t=L/2 (cm)
L	=	length of the slab at time t (cm)
L0	=	initial length of the slab (cm)
M	=	average moisture content (% dry basis)
t	=	time (min)
xa	=	sugars average content of the slab at time t (wet basis)
xa0	=	sugars initial content of the slab (wet basis)
xae	=	sugars equilibrium content (wet basis)
xexp	=	experimental moisture content (wet basis)
xfit	=	moisture content calculated with the model (wet basis)
xw	=	moisture average content of the slab at time t (wet basis)
xw(i,t)	=	moisture content at node i at time t (wet basis)
xw0	=	moisture initial content of the slab at time t (wet basis)
xwe	=	moisture equilibrium content (wet basis)
z	=	variable distance in the diffusion direction (cm)

Acknowledgments