Programa de planificación específica

The volume change or shrinkage during thin layer convective air drying of plums is described as a linear function of moisture content by the following equation:

V/Vo= A + BX (10.23)

Table 10.1 shows the constants A and B at different temperatures and the coefficient of determination (r²), which presented satisfactory values. This equation is used to estimate the particle radius of the plum in the numerical solution described previously.

FIGURE 10.3 Experimental drying apparatus: (1) centrifugal fan, (2) frequency modulator, (3) orifice plate, (4) pressure transmitter, (5) dry bulb temperature, (6) wet bulb temperature, (7) electric resistance heaters, (8) power converter, (9) beehive, (10) square metal trays, (11) thermocouple before the trays, (12) thermocouple at the center of the plum, (13) thermo-couple at the surface of the plum, (14) thermothermo-couple after the trays, (15) current converter, (16) computer.

14 16 10

13 12 11

9 15

6 5 4

2 7

8

3

1

Figure 10.4 shows the volume change of the plums at 50–80°C, where considerable shrinkage is observed, although with little influence of temperature. This can be attributed to the viscoelastic nature of the material, which suffers a structural collapse during drying. Similar results were obtained for thin layer drying of Thompson seedless grapes by Rhagavan et al. (1995), who found a linear relationship of volume change as a function of the moisture content of the berries. Simal et al. (1996) employed a diffusivity model with moving boundary conditions to simulate the drying kinetics of seedless grapes of the Flame variety, with the volume change represented by plotting V/V0 vs. X/X0 and by fitting a linear equation with similar constants.

The initial moisture content was similar for all samples of plums. The influence of air drying temperature on the drying curves can be observed in Figure 10.5. As expected, increases in the air temperature substantially shortened the drying time in comparison with that observed at 50^oC. The drying time required to evaporate 90%

of water was shorter by 39.8, 57.7, and 70.5% at 60, 70, and 80^oC, respectively, compared to that at 50^oC. Therefore, it is evident that at the same water content, the

TABLE 10.1

Constants of Equation (23), Which Describes the Volume Change during Drying of Plums at Different Temperatures

T (^oC) A^a B^a r²

50 0.174 0.172 0.998

60 0.181 0.174 0.988

70 0.191 0.175 0.997

80 0.193 0.178 0.998

a All fits had a significance level less than 5%.

FIGURE 10.4 Shrinkage during drying of plums.

0 2 4

0.0 0.2 0.4 0.6 0.8 1.0

T = 80°C T = 70°C T = 60°C T = 50°C

V/Vo

Moisture content (dry basis)

1 3 5

drying rate is dependent on air temperature. Induction periods or constant drying rate periods were not observed, and the drying was characterized only by the falling rate period. Barbanti et al. (1994), writing about air drying kinetics of cultivar plums, also observed that there was no constant drying rate period in this process. The existence of only the falling rate period could be indicative of a diffusion-controlled mechanism of drying.

It can be also seen in Figure 10.5 that the analytical model of Fick’s Law, consid-ering radius and effective diffusivity constants, is not applicable to fit the experi-mental data in the final drying period. However, during the drying period fitted by this model, the plums had not reached the commercially desirable moisture content (approximately 0.15 kg water/kg dry matter). In addition, according to Figure 10.4, the radius of the plum is a function of moisture content, showing that shrinkage has to be taken into account during the drying process. The apparent agreement between the analytical solution of Equation (10.2) applied to plums and the experimental data in a great part of the drying curves (Figure 10.5) could be explained by sup-posing a simultaneous decrease in the effective diffusivity and in the radius, in such a way that these effects compensated for each other. The numerical solution fitted to the experimental data can be also seen in this figure. As expected, a good adjustment was obtained for all drying curves, including for the final drying period.

From the numerical solution of Fick’s second law, the effective diffusivity as a function of simultaneous shrinkage and moisture content can be calculated. Figure 10.6 presents the effective diffusivity behavior with decreasing moisture content of the plums, and it can be observed that the effective diffusivity parameter is far from being constant during the process. The effective diffusivity is more influenced by moisture content at 80^oC than at other temperatures. On the other hand, neglecting the initial inductive period, the effective diffusivity at 60 and 50^oC changes almost FIGURE 10.5 Drying curves of plums at different temperatures; symbols refer to experimental data; (— —) analytical solution from Equation (10.2); () numerical solution from Equation (10.15).

0 500 1000 1500 2000 2500

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

T = 50°C T = 60°C T = 70°C T = 80°C

Moisture content (dry basis)

Drying time (min)

linearly with decreasing moisture content. The increase of effective diffusivity in this initial period could be attributed to the time necessary for the temperature of the plum to increase until it reaches the drying process temperature. According to Figure 10.7, in general, the temperature of the fruit begins to reach equilibrium with the drying air at a moisture content of approximately 3.0 kg/kg dry mass. Sabarez et al. (1997) reported that the resistance to moisture transfer through the skin layer is important during drying of plums, particularly in the early stages of drying. These authors concluded that the skin layer limits the maximum effective evaporation rate from the plum surface and provides significant resistance to initial moisture loss.

FIGURE 10.6 Variation of effective diffusivity with moisture content of plums at different temperatures.

FIGURE 10.7 Evolution of temperature inside plums during drying.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

2.00E-010 4.00E-010 6.00E-010 8.00E-010 1.00E-009

T = 50°C T = 60°C T = 70°C T = 80°C

Effective diffusivity (m2 /s)

Moisture content (dry basis)

0 3

20 30 40 50 60 70 80 90

T = 50°C T = 60°C T = 70°C T = 80°C

Temperature (°C)

Moisture content (dry basis)

1 2 4

Raghavan et al. (1995) obtained effective diffusivity values varying with moisture content under convective and microwave drying of grapes, and these values were found to decrease with decreasing water content. Effective diffusivity coefficients considering shrinkage but not varying with moisture content were reported by Hawlader et al.

(1991) for tomatoes and by Simal et al. (1996), who obtained diffusivities of around 4.7× 10^–10 to 1 × 10^–9 m² sec^–1 for grapes. Sabarez and Price (1999) studied the drying of plums of the d’Agen variety and obtained D_effvalues in the range of 4.3 to 7.6× 10^–10m²sec^–1at 70, 75, and 80°C.

The temperature dependence of the diffusivity can be represented by an Arrhe-nius type equation (Simal et al., 1996; Lewick et al., 1998; Simal et al., 2000). The activation energy for diffusion is calculated by taking a log plot of effective diffu-sivity at constant moisture content, against the reverse of the absolute air drying temperature. The relationship between activation energy and moisture content is presented in Figure 10.8. The tendency of increasing activation energy with decreas-ing moisture content is expected, and the majority of researchers have recognized this effect, especially for the low moisture content range. Satisfactory values of the determination coefficient (0.99) were obtained during the fitting procedure.

10.5 CONCLUSIONS

The volumetric shrinkage of plums results in a linear correlation with respect to the moisture content. The finite difference method can be used for more accurate pre-dictions and simulation of the drying process, and effective diffusivity values can be obtained as a function of moisture content, taking into account the shrinkage of the material. In general, these values are found to decrease with decreasing moisture content. The reported moisture diffusivity data at different temperatures falls between FIGURE 10.8 Relationship between activation energy for water diffusion and moisture con-tent in plums undergoing drying

0.0 0.5 1.0 1.5 2.0 2.5 3.0

32 33 34 35 36 37 38 39

Activation energy (kJ/mol)

Moisture content (dry basis)

1.2× 10^–10 and 8.9 × 10^–10 m²/sec. These results lead to the conclusion that shrinkage and moisture content cannot be neglected in establishing reliable values of effective diffusivity. The activation energy for diffusion is also dependent on the moisture content. It increases with decreasing water content in plums undergoing drying.

NOMENCLATURE

A Constant in Equation (10.23) a_w Water activity

B Constant in Equation (10.23) Bi_m Biot mass number ([k_c⋅Keq⋅rpo]/D_eff) d Sphere diameter (m)

D_AB Diffusion coefficient (m²/sec) D_eff Effective diffusivity (m²/sec)

D_vap Water vapor diffusivity in the air at a temperature T_a (m²/h) Ea Activation energy (kJ/mol)

Fo Fourier mass number ([D_eff⋅t]/[rpo]²) K_eq Average partition constant

k_c Mass transfer coefficient (m/sec) m Mass (kg)

M Residual moisture content

M_s Dimensionless moisture content at the surface of the plums N Number of nodal points

n Arbitrary nodal point P Total pressure (atm)

P_vap Vapor pressure of water (atm) q_A Chemical reaction in Equation (10.1) R Universal gas constant (8.314 J/mol K) Re_p Modified Reynolds number ([ρ⋅v⋅d]/µ)

r_s Radius of the stone (kernel + shell) component (m) r_p Radius of the whole plum (m)

r_po Radius of the plum at t = 0 (m) Sc Schmidt number (µ/ρ⋅Dvap) Sh Sherwood number ([k_c⋅d]/Dvap) T Absolute temperature (K) T_a Air temperature (°C) t Time (sec)

v elocity (m/sec) V Volume (m³) V_o Initial volume (m³)

X Moisture content dry basis (kg/kg dry matter) X_o Initial moisture content (dry basis)

X_e Equilibrium moisture content (dry basis) Y^∗ Molar fraction at equilibrium

GREEK SYMBOLS

ξ Radial coordinate (m)

ξp Dimensionless particle radius (r_p(t)/r_po) ρ Density (kg/m³)

µ Viscosity (kg/m sec)

ACKNOWLEDGMENT

The authors acknowledge financial support from São Paulo State Research Fund Agency, FAPESP (Processes: 98/12283–5; 98/08738–7 and 98/05130–8).

REFERENCES

AOAC, Official Methods of Analysis, 15th ed., Association of Official Analytical Chemists, Washington, DC, 1990.

Arrouz, S., Jomaa, W., and Belghith, A., Drying Kinetic Equation of Single Layer of Grapes, Drying ’98 — Proceedings of the 11th International Drying Symposium, August 19–22, 1998, B, pp. 988–997.

Barbanti, D., Mastrocola, D., and Severine, C., Air drying of plums: a comparison among twelve cultivars, Sci. Aliment., 14, 61–73, 1994.

Bird, R.B., Stewart, W.E., and Lightfoot, E.N., Transport Phenomena, John Wiley & Sons, Inc., New York, 1960, pp. 554–560.

Córdova-Quiroz, A.V., Ruiz-Cabrera, M.A., and García-Alvarado, M.A., Analytical solution of mass transfer equation with interfacial resistance in food drying, Drying Technol., 14, 1815–1826, 1996.

Dãscãlescu, A., Le Sechage et ses Applications Industrielles, Dunod, Paris, 1969.

Gabas, A.L., Menegalli, F.C., and Telis-Romero, J., Effect of chemical pretreatment on the physical properties of dehydrated grapes, Drying Technol., 17, 1215–1226, 1999.

Gabas, A.L., Menegalli, F.C., and Telis-Romero, J., Water sorption enthalpy-entropy com-pensation based on isotherms of plum skin and pulp, J. Food Sci., 65, 680–684, 2000.

Gabitto, J.F. and Aguerre, R.J., Solucion numerica del processo de secado com cambio de volumen, Revista Latinoam. Transf. Cal. Mat., 9, 231–240, 1985.

Hawlader, M.N.A., Uddin, M.S., Ho, J.C., and Teng, A.B.W., Drying characteristics of tomatoes, J. Food Eng., 14, 259–268, 1991.

Hernández, J.A., Pavón, G., and García, M.A., Analytical solution of mass transfer equation considering shrinkage for modeling food-drying kinetics, J. Food Eng., 45, 1–10, 2000.

Karathanos, V.T., Determination of water content of dried fruits by drying kinetics, J. Food Eng., 39, 337–344, 1999.

Lewick, P.P., Witrowa-Rajchert, D., and Nowak, D., Effect of drying mode on drying kinetics of onion, Drying Technol., 16, 59–81, 1998.

Madamba, P.S., Optimisation of the drying process: an application to the drying of garlic, Drying Technol., 15, 117–136, 1997.

Madamba, P.S., Driscoll R.H., and Buckle, K.A., Shrinkage, density and porosity of garlic during drying, J. Food Eng., 23, 309–319, 1994.

Murray, W.D. and Landis, F.L., Numerical and machine solutions of transient heat conduction problems involving melting or freezing, J. Heat Transfer, Trans. ASME, May 1959, 106–112.

Newman, G.M., Price, W.E., and Woolf, L.A., Factors influencing the drying of prunes 1.

Effects of temperature upon the kinetics of moisture loss during drying, Food Chem., 57, 241–244, 1996.

Price W.E., Sabarez, H.T., Laajoki, L.G., and Woolf, L.A., Dehydration of prunes: kinetic aspects, Agro-Food-Industry Hi-Tech, 8, 29–33, 1997.

Raghavan, G.S.V., Tulasidas, T.N., Sablani, S.S., and Ramaswamy, H.S., A method of deter-mination of concentration dependent effective moisture diffusivity, Drying Technol., 13, 1477–1488, 1995.

Sabarez, H.T. and Price W.E., A diffusion model for prune dehydration, J. Food Eng., 42, 167–172, 1999.

Sabarez, H.T., Price W.E., Back, P.J., and Woolf, L.A., Modelling the kinetics of drying of d’Agen plums (Prunus domestica), Food Chem., 60, 371–382, 1997.

Sanjuán, N., Bermejo, M.V., Vivanco, D., Cañellas, J., and Mulet, A., Drying Kinetics of Moscatel Cultivar Grapes, Drying ’96 — Proceedings of the 10th International Drying Symposium, July 30 – August 2, 1996, B, pp. 1069–1076.

Saravacos, G.D., Mass transfer properties of foods, in Engineering Properties of Foods, Rao, M.A. and Rizvi, S.S.H., Eds., Marcel Dekker, New York, 1986, pp. 169–221.

Simal, S., Femenía, A., Llull, P., and Rosselló, C., Dehydration of aloe vera: simulation of drying curves and evaluation of functional properties, J. Food Eng., 43, 109–114, 2000.

Simal, S., Mulet, A., Catalá, P.J., Cañellas, J., and Rosselló, C., Moving boundary model for simulating moisture movement in grapes, J. Food Sci., 61, 157–160, 1996.

Sobral, P.J.A., Secagem de sangue bovino incorporado a proteína texturizada de soja, em leito fluidizado e em leito fixo, thesis, FEA, UNICAMP, 1987.

Telis, V.R.N., Gabas, A.L., Menegalli, F.C., and Telis-Romero, J., Water sorption thermody-namic properties applied to persimmon skin and pulp, Thermochim. Acta, 343, 49–56, 2000.

Zogzas, N.P., Maroulis, Z.B., and Marinos-Kouris, D., Densities, shrinkage and porosity of some vegetables during air drying, Drying Technol., 12, 1653–1666, 1994.

Modeling Dehydration