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## Food Science and Technology

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*Print version* ISSN 0101-2061*On-line version* ISSN 1678-457X

### Ciênc. Tecnol. Aliment. vol.20 no.2 Campinas May/Aug. 2000

#### http://dx.doi.org/10.1590/S0101-20612000000200017

**MATHEMATICAL MODELLING OF THE OSMOTIC DEHYDRATION OF CHERRY TOMATO ( Lycopersicon esculentum var. cerasiforme)^{1}**

Patricia Moreira AZOUBEL^{2}, Fernanda E. Xidieh MURR^{2,*}

**SUMMARY**

Osmotic dehydration of cherry tomato as influenced by osmotic agent (sodium chloride and a mixed sodium chloride and sucrose solutions) and solution concentration (10 and 25% w/w) at room temperature (25°C) was studied. Kinetics of water loss and solids uptake were determined by a two parameter model, based on Fick's second law and applied to spherical geometry. The water apparent diffusivity coefficients obtained ranged from 2.17x10^{-10} to 11.69x10^{-10} m^{2}/s.

**Keywords:** tomat; osmotic dehydration; diffusivity.

**RESUMO**

MODELAGEM MATEMÁTICA DA DESIDRATAÇÃO OSMÓTICA DE TOMATE CEREJA (*Lycopersicon esculentum* var. *cerasiforme*). O presente trabalho teve como objetivo o estudo da influência do agente osmótico (cloreto de sódio e a mistura cloreto de sódio-sacarose) e da concentração da solução (10 e 25% p/p) na desidratação osmótica de tomate cereja na temperatura ambiente (25^{o}C). A cinética de perda de água foi determinada através de uma equação de dois parâmetros, baseada na segunda lei de Fick e aplicada para uma geometria esférica. Os coeficientes de difusividade aparente obtidos ficaram na faixa de 2,17x10^{-10} a 11,69x10^{-10} m^{2}/s.

**Palavras-chave:** tomate; desidratação osmótica; difusividade.

1 — INTRODUCTION

There are many cultivated and consumed tomato species in Brazil, standing out the cherry tomato (*L. esculentum* var. *cerasiforme*), considered an ancestral form of tomato, based on the fact that the size and the shape of the fruits of *L. cerasiforme* (2 to 2,5 cm of diameter) be intermediary between the wild and the cultivated tomato [3]. This tomato type is used in salads and as supplement in the astronauts' diet during long periods of permanence in the space, due to its fast "in vitro" production [5].

The tomato is very sensitive to cold environment, but is tolerant to high temperatures. Its nutrition requests special care and the fruits are very perishable requiring the removal of water for its preservation through treatments as the osmotic dehydration (dewatering and impregnation soaking process). This process is based on the immersion of foods, whole or in pieces, in hypertonic solutions (sugars, sodium chloride, sorbitol, glycerol), originating two simultaneous counter-current flows: an exit of the water from the product to the solution and a migration of solutes from the solution into the solid. A third flow also involved consists of the loss of some natural solids, as sugars, organic acids, mineral salts, among other nutritious that, although insignificant proporcionally to the two main flows, can be important for the organoleptic (flavor, color and texture) and nutritional (vitaminic, mineral) qualities of the product [11]. This treatment allows to accomplish dehydration and direct formulation of the food [6].

In this work the effect of osmotic agent (sodium chloride and mixed sodium chloride-sugar solutions) and solution concentration (10 and 25% w/w) on the osmotic dehydration kinetics of cherry tomato (*L. esculentum* var. *cerasiforme)* was studied. The apparent diffusion coefficients of water were determined.

2 — MATERIAL AND METHODS

Fresh cherry tomatoes (*L. esculentum* var. *cerasiforme*), obtained from a local market, were sorted visually for color (completely red), size (average diameter of 2.8 cm) and physical damage. As the cherry tomato waxy skin represents a high resistance to mass transfer, the fruits were perforated with needles (1 mm of diameter) [13] to pin hole density of 16 holes/cm^{2}, immersed in NaCl and NaCl-sucrose (3:2) solutions of different concentrations (10 and 25% w/w) at room temperature (25°C) and agitation of 70 rpm, mantained in a temperature-agitation controlled shaker (Tecnal TE-421). The fruits (16g) were placed in 250 mL beakers, containing 160g of osmotic solution. An excess of osmosis solution (fruit to solution ratio 1:10) was used to limit concentration changes due to uptake of water from the cherry tomato and loss of solute to the fruit. Samples were removed from the osmosis solution at 0.5, 1.0, 1.5, 2.0 and 3.0 h of immersion, drained and the excess of solution at the surface was removed with absorbent paper for posterior weight. The water loss was determined by gravimetric measurement and the salt content by Mohr's titration method [10]. All determinations were conducted in triplicate.

**2.1 – Osmotic Kinetics Analysis**

The determination of the apparent diffusion coefficients of cherry tomato was based on a two-parameter equation from mass balance, adapted to a spherical geometry, using data obtained during a relatively short period of time, as developed by AZUARA *et al* [1]:

where is the fraction of water (g water/100g of sample) lost by the foodstuff at time is the fraction of water (g water/100g of sample) lost by the foodstuff at equilibrium; is the fraction of water that can diffuse out, but remains inside the foodstuff at time .

The value of is fixed for the established conditions of temperature, concentration and fruit to solution ratio, and the values of and are functions of the rate of water loss and time. increases as these variables increase, and can be calculated as:

where and are the initial water (g) and solids contents (g), respectively; is the total wet weight (g).

The value of decreases as the rate of water loss and time increase, suggesting a relationship between and , represented by a parameter . This parameter is in turn a function of time and the rate of water loss:

The rate of water loss is a function of time, temperature and initial concentration of the osmotic solution. As the osmotic dehydration experiments have been carried out at a given initial concentration and at constant temperature, we assume that the rate of water loss is only a function of time. Based on this, Azuara *et al* [1] proposed a function for in terms of time and a constant related to the water loss:

Substituting equations (4) and (1) into equation (3), and rearranging the terms, we get an equation that associates the water lost () with time ():

When (at equilibrium), equation (5) becomes asymptotic at a value corresponding to . To predict the fraction of water lost by the foodstuff () at time in equation (5), it is necessary to know the values for and . These can be calculated by linear regression , using experimental data () obtained during a short time period and equation (6), which is the linear form of equation (5):

where can be measured from the slope of the plot of against *t*.

Based on Fick's second law, Crank [2] proposed an equation for the diffusion of solutes in spheres in contact with an infinite amount of solution. Its simplified equation for small values of is:

where is the amount of water leaving the foodstuff at time is the amount of water that left the foodstuff after infinite time; is the water apparent diffusion coefficient; is the radius of the sample. This equation is only applicable in the early stages of adsorption, when diffusion is assumed to occur in a semi-infinite medium, and the amount of water leaving the foodstuff is directly proportional to the square root of time.

Relating equation (7) with equation (5), a simple expression from which can be calculated at different times:

where is the apparent diffusion coefficient at time is the value at equilibrium obtained from equation (5); is the value at equilibrium obtained from experimentation. When is unknown, by assuming that it equals , equation (8) may be used to obtain a good estimation for as long as the kinetic data are adequately fitted by equations (5).

The average apparent diffusion coefficient is calculated by:

where n is the number of data points used.

The mean relative deviation modulus was used as a criterion to evaluate the fit of the tested model, as applied to the experimental data. Values of were calculated using the following equation:

where and are the observed and the predicted values, respectively. Values of less than or equal to 10% are considered to fit the experimental data satisfactorily.

3 — RESULTS AND DISCUSSION

**Cherry tomato characterization**

The physico-chemical characteristics of the cherry tomatoes used in the experiments are shown on *Table 1*.

The total solids content, the aciditty and the soluble solids content were similar to the values obtained by Gould [4] for tomato, but the NaCl content was higher and the reducing sugars content was lower. Sugars and organic acids were the majority of the total dry matter content of tomato fruit. The same result for cherry tomato was obtained by Picha [9]. The pH value was similar to the result of Petro-Turza [8] for tomato and the density to tomato pulp [10].

**Osmotic dehydration**

The limiting factor for water removing from whole tomato is water penetration through the skin, as dehydration kinetics is governed predominantly by the skin permeability. The thick epicuticular waxy layers are present on the surface of tomatoes and have a high resistance to mass transfer. The tightly packed cellular structure of the epicarp under the skin is composed of small cells with thick cell walls and a large amount of middle lamella components. The fruit puncturing method created small holes per tomato fruit, allowing more water removal [13].

The osmotic dehydration process was studied in terms of moisture and salt contents *(Figure 1*). A strong effect of the osmotic solution concentration in the water lost was observed. An increase in the concentration results in an increase in osmotic pressure gradient (and, hence, a decrease in the water activity), increasing the driving force for water removal between solution and food, and thereby higher mass transfer rates and water apparent diffusion coefficients, which is in agreement with the results of Rastogi and Raghavarao [12] for osmotic dehydration of carrots in sucrose solutions and Vijayanand *et al* [15] for cauliflower in salt solutions. The rate of water loss is faster within the first two hours of the process, decreasing gradually while approaching the end of the experiment, which were not sufficient to reach the equilibrium. The rapid water loss in the beginning is apparently due to the large osmotic driving force between the dilute sap of the fresh fruit and the hypertonic solution.

*Figure 1 (b)* shows the salt gain during the osmotic dehydration of cherry tomato. The salt gain is smaller in relation to the water lost, due to the cellular membrane freely allow the solvent to pass through, but they also allow, to a lesser degree, the passage of solute molecules. This type of membrane could be classified differentially permeable, rather than semi-permeable [14]. An increase in the concentration of the osmotic solution gave higher salt gain within the 6h of process was observed. In this period, only the solutions with 10% w/w concentration reached equilibrium. Fixing the concentration of the solution, the effect of the osmotic agent could be evaluated, observing that the salt gain was lower in the mixed NaCl-sucrose solution.

The experimental water loss results were used to estimate the water apparent diffusion coefficients. The time interval used for prediction by the proposed model was 180 min*. Figures 2* to *5* show some of the kinetics run studied and their corresponding linearizations, using equation (6). *Table 2* presents the calculated diffusivities, and values. It was observed that higher values for indicated a higher diffusion of the material per unit of time. The model was able to predict the whole osmotic dehydration process up to equilibrium, using data obtained from a short period of time, with satisfactory mean relative modulus values (<10%). The proposed model did not take into account the puncturing treatment. The limiting factor for removing water from whole tomato is water penetration through the skin, as dehydration kinetics is governed predominantly by skin permeability. Whole tomato, with its waxy skin, was treated with needles to improve the water permeability, but the un-damaged skin after the treatment puncturing is still an effective barrier to mass transfer.

The solution composition is an important parameter for the osmotic dehydration process. High concentrated solutions gave higher water loss and apparent diffusion coefficients than the lower concentrated solutions, as the osmotic pressure gradient is the driving force for osmotic mass transfer. The same result was observed by Moy *et al* [7] for the osmotic dehydration of mango and papaya in sucrose solutions. When salt is combined with sucrose, the water loss was lower than using salt alone, probably due to its low molecular weigh that allows a higher rate of penetration and to its ionization in the solution. NaCl in osmotic solutions increases the driving force for dehydration owing to the water activity lowering capacity of the salt, however, its use is limited due to its saltiness. In addition, sucrose allows the formation of a sugar subsurface layer, which interferes with the concentration gradients across the product-medium interface and act as a barrier against removal of water and solid uptake.

4 — CONCLUSION

The rate of moisture removal and solids uptake in osmotic dehydration of cherry tomato was directly related to the concentration of the solution. When a mixed NaCl-sucrose solution was used, these rates were lower. The obtained water apparent diffusion coefficients calculated from the proposed model ranged from 2.17x10^{-10} a 11.69x10^{-10} m^{2}/s.

5 — REFERENCES

[1] Azuara, E.; Berinstain, C.I.; Garcia, H.S. Development of a mathematical model to predict kinetics of osmotic dehydration. **Journal of** **Food Science Technology**, v. 29, p. 239-242, 1992. [ Links ]

[2] Crank, J. (1975). **Mathematics of diffusion**. Clarendon Press Oxford. [ Links ]

[3] Folquer, F. (1976). **El tomate: estudio de la planta y su producion comercial.** Editorial Hemisferio Sur, Buenos Aires. [ Links ]

[4] Gould, W.A., 1974, **Tomato production**, processing and quality evaluation, Westport: The Avi Publishing Company, 445pp. [ Links ]

[5] Kaur-Sawhney, R.; Applewite, P.B.; Galston, A.W. Formation in vitro of ripe tomato fruits from thin layer explants of flower pedicels. **Journal of Fruits and Nuts**, v. 18, n. 3, p. 191-199, 1996. [ Links ]

[6] Miguel, M.H.; Kieckbusch, T.G. Desidratação osmótica de frutas: influência da combinação de solutos. **Anales del I Congresso de Ingenería de Alimentos**. Campinas, v. 2, p. 255-266, 1995. [ Links ]

[7] Moy, J.H.; Lau, N.B.H.; Dollar, A.M. Effect of sucrose and acids on osmovac-dehydration of tropical fruits. **Journal of Food Processing and Preservation**, v. 2, pp. 135-135, 1978. [ Links ]

[8] Petro-Turza, M. (1987). Flavour of tomato and tomato products, **Food Ver. Int**., 2, pp. 309-351. [ Links ]

[9] Picha, D.H., 1987, Sugar and organic acid content of cherry tomato fruit at different ripening stages, **Hort Science**, 22, pp. 94-96. [ Links ]

[10] Ranganna, S. (1977) **Manual of analysis of fruit and vegetables products.** New Delhi, [ Links ]

[11] Raoult-Wack, A.L.; Lafont, F.; Rios, G.; Saurel, R.; Guilbert, S. Osmotis dehydration: study of mass transfer in terms of engineering properties. In: Mujumdar, A.S.; Roques, M. A. **Drying of solids**. p. 487-495, 1989. [ Links ]

[12] Rastogi, N.K.; Raghavarao, K.S.M.S. Effect of temperature and concentration on osmotic dehydration of coconut. **Food Science and Technology-Lebensmittel-Wissenschaft & Technologie**, v. 27, p. 564-567, 1994. [ Links ]

[13] Shi, J.X.; Le Maguer, M.; Wang, S.L.; Liptay, A. Application of osmotic treatment in tomato processing-effect of skin treatments on mass transfer in osmotic dehydration of tomatoes. **Food Research International**, v. 30, n. 9, p. 669-674, 1997. [ Links ]

[14] Torreggiani, D. Osmotic dehydration in fruit and vegetable processing. **Food Research International**, v. 26, p. 59-68, 1993. [ Links ]

[15] Vijayanand, P.; Nagin, C.; Eipeson, W.E. Optimization of osmotic dehydration of cauliflower. **Journal of Food Processing and Preservation**, v. 14, n. 2, p. 391-413, 1995. [ Links ]

6 — ACKNOWLEDGEMENT

The authors gratefully acknowledge CAPES for the financial support.

^{1} Recebido para publicação em 22/12/99. Aceito para publicação em 18/10/00.

^{2} Depto. DEA/FEA **–** Unicamp. Cx. Postal 6121, CEP 13083-970, Campinas-SP.

** A quem a correspondência deve ser enviada. *