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Evaluation of semi-empirical and non-linear drying models by Bayesian inference/Avaliacao de modelos semi-empiricos para secagam usando inferencia Bayesiana.


Passion fruit (Passiflora edulis), a Brazilian native plant, is a popular tropical fruit known worldwide. In the juice industry, the passionfruit produces thousands of tons of seeds as agricultural byproducts during juice extraction. The seeds, which contain large amounts of oil, are generally discarded. Since they total several metric tons, the byproduct is an asset of great economical, scientific and technological interest.

The basic objective in drying food products is the removal of water from the solids up to a certain level at which microbial spoilage and deterioration of chemical reactions are greatly minimized (KRODIDA et al. 2003). Convective air-drying is the most frequently used dehydration method in the food and chemical industry, although a considerable amount of data has been reported in the literature on thin-layer drying of different agricultural products such as eggplant (ERTEKIN; YALDIZ, 2004), green bean (ROSSELLO et al., 1997; SENADEERA et al., 2003), green peas (SIMAL et al., 1996), okra (ADOM et al., 1997; DOYMAZ, 2005; GOGUS; MASKAN, 1999), red pepper and red chili (DOYMAZ; PALA, 2002; GUPTA et al. 2002), sweet potato (DIAMANTE; MUNRO, 1993); black tea (PANCHARIYA et al., 2002), carrots (DOYMAS, 2004a), white mulberry (DOYMAZ, 2004b) and kiwi fruit (SIMAL et al., 2005).

The choice of an appropriate statistical method for modeling non-linear curves used for the drying processes is of paramount importance in obtaining results. In fact, Bayesian inference is a good option as opposed to frequentist methods (least squares), because tab is based on iterative processes, or alternatively, means are used to linearize the model by logarithmic transformation (SILVA et al., 2005; ROSSI, 2011).

Current study reports the effect of drying temperature and air flow rate on the drying characteristics of passion fruit seeds to evaluate a suitable drying model by Bayesian inference, in order to describe the drying process and compute effective moisture diffusivity and activation energy of passion fruit seeds while drying.

Mathematical modeling

In current study, the falling rate period was not observed in any of the experiments. In most studies on drying, diffusion is generally accepted to be the main mechanism during the transport of moisture to the surface to be evaporated. The solution of Fick's equation is presented below, with the assumptions of moisture migration by diffusion, negligible shrinkage, constant diffusion coefficients and temperature and for a sphere (CRANK, 1975; DOYMAZ, 2004a; PALA et al., 1996):

MR = M--[M.sub.e]/[M.sub.o]--[M.sub.e] = 6/[[pi].sup.2] [[infinity].summation over (n=1)] 1/[n.sup.2] exp (- [n.sup.2][[pi].sup.2] [D.sub.eff]t/[r.sup.2]) (1)


M, [M.sub.o] and [M.sub.e] are content, initial and equilibrium moisture content, respectively; r is the radius; t is the time; MR is the moisture ratio; [D.sub.eff] is the effective diffusivity.

The semi-theoretical models are generally derived by simplifying general series solutions of Fick's second law or the modification of simplified models and valid within the temperature, relative humidity, air flow velocity and moisture content range for which they were developed (FORTES; OKOS, 1981). These models require a shorter time when compared to theoretical thin-layer models and do not need assumptions of geometry of a typical food, its mass diffusivity and conductivity (PARRY, 1985). There are several semi-theoretical thin-layer drying models, although in current study Lewis's model has been employed: Lewis's model with two parameters and Lewis's model with three parameters. Models are shown in Table 1.

Bayesian inference

In general, the drying seeds have been carried out from a frequentist approach, adjusting nonlinear models which aim at synthesizing pieces of information into parameter estimates to be interpreted. According to Oliveira et al. (2012a and b; 2013), estimation is based on iterative processes such as those by Gauss-Newton, DUD and Marquardt algorithm, due to the nonlinearity of the variables. These procedures minimize the sum of residue squares. However, when individual adjustments are considered, i.e. adjustments for many experimental units of mathematically complex models or few possible longitudinal observations are extant, interactive methods frequently provide negative estimates for parameters which may cause the formation of atypical curves.

Furthermore, when dealing with comparisons of curves derived from different treatments, the distribution of nonlinear model parameter estimators do not usually follow Gaussian distribution. Therefore, the process to formulate statistical tests becomes complex once presuppositions related to the asymptotic theory are not attended (OLIVEIRA et al., 2012a and b). Bayesian inference, involving the adjustment of linear and nonlinear regression models, has been successfully used in recent years since it reduced the number of biased estimations even when little information was used (OLIVEIRA et al., 2012a and b; 2013).

Material and methods

The laboratory dryer

The experimental apparatus used for convective drying of passionfruit seeds has been described by Fiorentin et al. (2010).

Drying experiments

A local factory provided the passionfruit seeds which were washed to remove the juice remaining on them. The cleaned seeds were then pre-dried under sunlight at day temperature, about 32 [+ or -] 2[degrees]C, for 5 hours. The seeds were later dried on thin-layer.

Experiments were performed to determine the effect of process variables on thin-layer drying characteristics of passionfruit seeds. Four temperatures (40, 50, 60, and 65[degrees]C) were studied, with constant air flow rate at three points (0.6, 0.9, and 1.2 m [s.sup.-1]), at constant air temperature. Twenty-four drying runs were performed in a systematic way, constituting two replicates. An anemometer with a reading accuracy of [+ or -] 0.05 m [s.sup.-1] measured air velocity, with measurement location being 60 cm above the plenum of the test chamber. The temperature of air entering the plenum chamber was measured by thermocouple.

The initial moisture content of passionfruit seeds was measured according to Brazilian rules for seeds (BRASIL, 1992). The 500 g sample was uniformly spread in a basket in a single layer after the desired drying conditions had stabilized. Water losses were measured by weighing the basket and its content at 3-minute intervals until constant weight. These tests were performed in duplicate and the averages were reported and used at the simulation modeling.

Mathematical modeling by Bayesian inference

For the first stage it was assumed that Y:MR has normal distribution truncated within the range [0,1], i.e:

[Y.sub.i] ~ Normal [([mu]([t.sub.i]), [[sigma].sub.e.sup.2]).sub.[0,1]] (2)

[[sigma].sup.2.sub.e] = [[tau].sup.-1.sub.e] OpenBugs parameterization, E(Y) = [mu](t) = proposed model. The models are shown in Table 1.

For all model parameters the prior was considered non-informative Gamma distribution (default

OpenBugs), for strictly positive quantities, i.e.:

parameters ~ Gamma([10.sup.-3], [10.sup.-3])

The posterior distributions of the parameters were obtained by BRugs on R program (R


million samples were obtained by Monte Carlo Markov Chain (MCMC), where 100,000 were discarded ("burn-in samples") to eliminate the effect of the initial values. The final samples were taken with steps of 100 which contained 9,000 values. The convergence chains were verified by Convergence Diagnosis and Output Analysis--CODA program--(BEST et al., 1995) by Geweke (1992) and Heidelberger and Welch (1983) criteria.

Deviance Information Criterion (DIC) may be used as a comparison, as well as the selection of (co)variables in models. Speigelhalter et al. (2002) suggested the use of difference criterion module between DIC values of two models, A and B, analyzed. This criterion is shown in Equation 3.

D = [absolute value of ([DIC.sub.A] - [DIC.sub.B])] (3)

If D < 5, it may be concluded that there was no significant difference; if 5 [less than or equal to] D [less than or equal to] 10, it may be concluded that there was significant difference; if D > 10, it may be concluded that there was high significant difference.

Results and discussion

Influence of process parameters

Table 2 shows the initial and final moisture conditions of passionfruit seeds.

Drying kinetics of the curves do not present constant drying period, or rather, the time spent during the drying process was only for removing internal moisture. The constant rate was reported by various authors (DOYMAZ 2005; 2004a and b). When there is an absence of constant period, the diffusion is the physical mechanism that governs the drying process. Mandamba et al. (1996); Gouveia et al. (2003); Doymaz (2004a and b; 2005); Ertekin and Yaldiz (2004); Simal et al. (2005) also noted this behavior.

Increase in the drying temperature caused an important increase in the drying rate; consequently, drying time is decreased. The effect of air flow rate was more sensitive when compared to the air flow rate of 0.6 and 0.9 m [s.sup.-1]. In these conditions, the increase of air flow rate caused a decrease on drying time.

For a speed of drying air at 0.6 m [s.sup.-1], the time necessary for the establishment of a constant mass condition in the system was approximately 64% higher for the temperatures 40, 50, and 60[degrees]C, when they were compared to a temperature of 65[degrees]C. In the case of speed at 0.9 m [s.sup.-1], the time required for the system tray/seed to reach constant mass condition was about 73% higher for the temperature 40[degrees]C when the latter is compared to the temperatures 50, 60, and 65[degrees]C. At 1.2 m [s.sup.-1], the time necessary for reaching the constant mass condition at the system decreased 44% when the temperature of 40[degrees]C was compared to the temperature 65[degrees]C.

In general, the increase on the air speed results in a drying rate increase, which is represented by the declivity of the line tangent to the drying curve. If the temperature is kept constant and the drying air speed varies between 0.6 and 1.2 m [s.sup.-1], there is a reduction of 39% on the drying time. A similar behavior may be observed for the temperatures 50, 60 and 65[degrees]C, with reductions approximately equal to 64, 50 and 22%, respectively.

These results are only valid for drying in thin layer. They cannot be applied to drying in thicker layers, in which other variables may have an influence on the time the seed is dried.

Evaluation of the models

So that moisture content as a function of drying time could be determined, the models of Table 1 were fitted according to Bayesian inference. The Bayesian estimations (posterior) to the model parameters to drying (mean, standard deviation SD- and DIC) are given in Tables 3 to 5 for different conditions.

The comparison of the DICs obtained, listed in Tables from 3 to 5, at the diverse experimental conditions, by means of Equation 3, showed there was no significant difference among the models evaluated. As Lewis's model presented only one parameter to be adjusted, it should be used for describing the drying kinetics of passionfruit seed in a thin layer.

It may be observed in Tables 3 to 5 that the mean obtained was close to the median for all the parameters of the models studied. It showed a symmetrical distribution. It may also be verified that zero did not belong to the credibility intervals of the parameters of the models studied, in all the conditions. This means that all parameters studied were statically not null.

When Lewis's model ([M.sub.1], [M.sub.2] and [M.sub.3]) and the three air drying speeds are taken into account, it may be verified that the drying coefficient ([k.sub.1], [k.sub.2] and [k.sub.3]) varied with the variation of absolute temperature (in Kelvin). The variation from 40 to 50[degrees]C was a significant temperature increase, whereas the increases from 50 to 60, and to 65[degrees]C were not so significant. Henderson and Pabis (1961) demonstrated that the drying coefficient k is not constant, but dependent on the drying air temperature and on the mechanisms of water diffusion or water vapor inside the seeds.

Figure 1 shows the drying coefficient variation (k's) with variations of absolute temperature (in Kelvin) for the speeds of drying air under analysis. The adjustment equations and R2 (determination coefficients) are presented in Table 6. In their experiments on drying kiwi fruit, Simal et al. (2005) observed that variation of k's with temperature exhibited a linear trend.

Evaluation of moisture diffusivity and action energy

Finally, effective diffusivity coefficient was determined to establish diffusion model. The solution of Fick's second law (Equation 1) out of spheres was shown. According to several authors (NUH, BRINKWORTH, 1997; PALA et al, 1996; RIVA; PERI, 1986; DOYMAZ, 2005) Equation (1) assumes that the effective diffusivity (D) is constant and shrinkage of the samples is negligible. For long drying time (setting n= 1), Equation (1) could be further simplified to a straight-line equation as:

ln(MR) = ln (6/[[pi].sup.2]) - ([[pi].sup.2] [D.sub.eff]/[r.sup.2] t) (4)

The effective diffusivity was calculated by Equation (4) with slopes derived from linear regression of ln(MR) against time data shown in Figure 2. Table 7 shows rates of [D.sub.eff] for different temperatures and air flow rate.

Increase of temperature means an increase on effective diffusivity ([D.sub.eff]) at constant air velocity.

Consequently, there is a reduction of internal resistance of drying with temperature increase. The figures for diffusivity coefficient proposed by different authors and reported by Pavon-Melendez et al. (2002) varied between 2.2 x [10.sup.-10] and 9.4 x [10.sup.-10] [m.sup.2] [s.sup.-1] for different fruits and vegetables such as grapes, potatoes, apples and carrots. Similar behaviors were reported by Panchariya et al. (2002) with regard to black tea; by Doymaz (2004a) on carrots; by Doymaz (2004b) on white mulberry; by Doymaz (2005) on okra and by Simal et al. (2005) on kiwi fruit.

According to several authors, temperature dependence on diffusivity coefficient follows the Arrhenius equation (Equation 5), where Ea is the activation energy (kJ mo[L.sup.-1]); [D.sub.0] is the pre-exponential factor ([m.sup.2] [s.sup.-1]); T is the absolute temperature (K); R is the gas constant (kJ moL [K.sup.-1]). To obtain the temperature influence on effective diffusivity, the rates of ln([D.sub.eff]) are plotted versus 1/T, as shown in Figure 5.

[D.sub.eff] = [D.sub.0] exp (-[E.sub.a]/RT) (5)

So that temperature influence on effective diffusivity could be evaluated, ln ([D.sub.eff]) rates were plotted versus the reciprocal of absolute temperature, as shown in Figure 3. When obtaining the curves presented in Figure 3, the experimental points for speeds 0.9 and 1.2 m [s.sup.-1] referring to 65[degrees]C were neglected because they did not present Arrhenius-type dependence relationship.

The rates from the lines in Figure 3 and in linearized Equation 5 were used to determine the rates of activation energy at distinct conditions of drying air speed. Rates obtained were 24.36, 35.24 and 13.86 kJ mo[L.sup.-1] for drying air speeds equal to 0.6, 0.9 and 1.2 m [s.sup.-1], respectively. Gupta et al. (2002) reported activation energy of 41.95 kJ mo[L.sup.-1] for chili peppers; Simal et al. (1996) found activation energy of 28.40 kJ mo[L.sup.-1], whereas Doymaz (2005) reported an activation energy of 51.26 kJ mo[L.sup.-1].


The effect of temperature and air flow rate on the drying of passion fruit seeds was analyzed. The drying process took place only in the falling rate period for passionfruit seeds. Semi-theoretical Lewis, Lewis with two parameters and Lewis with three parameters models were fitted according to Bayesian inference. Although all of them may be used to describe drying behavior, Lewis's model was employed to investigate the dependence relationship of k parameter with temperature. The values of effective diffusivity calculated ranged between 1.70 x [10.sup.-10] and 4.68 x [10.sup.-10] [m.sup.2] [s.sup.-1]. In fact, the effective diffusivity increases as temperature increases. Temperature dependence on diffusivity coefficients was described by Arrhenius relationship and activation energy was found as 24.36, 35.24, and 13.86 kJ mo[L.sup.-1] for drying air speeds equal to 0.6, 0.9, and 1.2 m [s.sup.-1], respectively.

Doi: 10.4025/actascitechnol.v36i4.12691


The authors would like to thank the Coordination for the Upgrading of Higher Education Personnel (Capes) for its financial support.


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Received on March 4, 2011.

Accepted on March 26, 2014.

License information: This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Ricardo Cardoso de Oliveira (1) *, Robson Marcelo Rossi (2), Marcelino Luiz Gimenes (1) and Sueli Teresa Davantel de Barros (1)

(1) Departamento de Engenharia Qufmica, Universidade Estadual de Maringa, Av. Colombo, 5790, 87020-900, Maringa, Parana, Brazil. (2) Departamento de Estatlstica, Universidade Estadual de Maringa, Maringa, Parana, Brazil. * Author for correspondence. E-mail:

Table 1. Thin-layer drying models.

                 Model                    Equation

[M.sub.1]        Lewis             MR = exp(-[k.sub.1]t)

[M.sub.2]    Lewis with two    MR = [a.sub.2]exp(-[k.sub.2]t)

[M.sub.3]   Lewis with three         MR = [a.sub.3]exp
               parameters        (-[k.sub.3]t) + [b.sub.3]

Where [k.sub.1], [k.sub.2] and [k.sub.3] are diffusivity and
[a.sub.2], [a.sub.3] and [b.sub.3] are model constants.

Table 2. Initial and final moisture conditions of passionfruit
seeds in different conditions, on dry base.

Velocity         Temperature    Initial moisture   Final moisture
(m [s.sup.-1])   ([degrees]C)         (%)               (%)

0.6                   40              11.1              8.51
                      50              12.9              11.0
                      60              14.0              12.6
                      65              14.0              8.46
0.9                   40              16.4              6.34
                      50              12.3              7.39
                      60              10.1              6.95
                      65              10.1              7.11
1.2                   40              8.20              5.11
                      50              8.20              5.73
                      60              8.00              5.13
                      65              8.00              5.41

Table 3. Bayesian estimations model parameters on drying of
passionfruit seeds at 0.6 m s-1 at different temperatures.

Temperature                                  40

Model          Parameter     Mean      DIC             SD

[M.sub.1]      [k.sub.1]   0.000641   -57.67   5.09 x [10.sup.-5]

[M.sub.2]      [a.sub.2]   0.862800   -54.96   4.62 x [10.sup.-2]
               [k.sub.2]   0.000517            5.47 x [10.sup.-5]

[M.sub.3]      [a.sub.3]   0.859300   -56.62   3.43 x [10.sup.-2]
               [b.sub.3]   0.018130            1.91 x [10.sup.-2]
               [k.sub.3]   0.000565            6.31 x [10.sup.-5]

Temperature                                    50

Model          Parameter     Mean      DIC             SD

[M.sub.1]      [k.sub.1]   0.001525   -33.20   2.04 x [10.sup.-4]

[M.sub.2]      [a.sub.2]   0.896200   -33.04   7.79 x [10.sup.-2]
               [k.sub.2]   0.001319            2.31 x [10.sup.-4]

[M.sub.3]      [a.sub.3]   0.882700   -35.11   6.59 x [10.sup.-2]
               [b.sub.3]   0.031710            2.83 x [10.sup.-2]
               [k.sub.3]   0.001545            3.64 x [10.sup.-4]

Temperature                                  60

Model          Parameter     Mean      DIC             SD

[M.sub.1]      [k.sub.1]   0.00144    -29.40   2.87 x [10.sup.-4]

[M.sub.2]      [a.sub.2]   0.871400   -29.44   8.73 x [10.sup.-2]
               [k.sub.2]   0.001194            2.48 x [10.sup.-4]

[M.sub.3]      [a.sub.3]   0.862200   -32.67   6.90 x [10.sup.-2]
               [b.sub.3]   0.045670            3.54 x [10.sup.-2]
               [k.sub.3]   0.001528            3.94 x [10.sup.-4]

Temperature                                  65

Model          Parameter     Mean       DIC             SD

[M.sub.1]      [k.sub.1]   0.001476    -23.52   1.31 x [10.sup.-3]

[M.sub.2]      [a.sub.2]    0.86980    -23.70   9.82 x [10.sup.-2]
               [k.sub.2]    0.00120             2.90 x [10.sup.-4]

[M.sub.3]      [a.sub.3]   0.847100    -27.57   6.95 x [10.sup.-2]
               [b.sub.3]   0.095110             4.27 x [10.sup.-2]
               [k.sub.3]   0.0019880            7.87 x [10.sup.-4]

Table 4. Bayesian estimations model parameters on drying of
passionfruit seeds at 0.29 m [s.sup.-1] at different temperatures.

Temperature                               40

Model       Parameter     Mean       DIC             SD

[M.sub.1]   [k.sub.1]   0.0009819   -32.5    1.18 x [10.sup.-4]

[M.sub.2]   [a.sub.2]   0.8523000   -34.81   6.49 x [10.sup.-2]
            [k.sub.2]   0.0007791            1.24 x [10.sup.-4]

[M.sub.3]   [a.sub.3]   0.847000    -37.2    6.27 x [10.sup.-2]
            [b.sub.3]   0.0394300            3.45 x [10.sup.-2]
            [k.sub.3]   0.0009542            2.13 x [10.sup.-4]

Temperature                                 50

Model       Parameter     Mean      DIC             SD

[M.sub.1]   [k.sub.1]   0.001771   -30.68   2.08 x [10.sup.-4]

[M.sub.2]   [a.sub.2]   0.926700   -29.91   6.20 x [10.sup.-2]
            [k.sub.2]   0.001611            2.17 x [10.sup.-4]

[M.sub.3]   [a.sub.3]   0.925000   -30.68   6.22 x [10.sup.-2]
            [b.sub.3]   0.021290            2.62 x [10.sup.-2]
            [k.sub.3]   0.001799            3.23 x [10.sup.-4]

Temperature                               60

Model       Parameter     Mean      DIC             SD

[M.sub.1]   [k.sub.1]   0.00199    -31.96   2.33 x [10.sup.-4]

[M.sub.2]   [a.sub.2]   0.941400   -30.26   6.66 x [10.sup.-2]
            [k.sub.2]   0.001857            2.75  x [10.sup.-4]

[M.sub.3]   [a.sub.3]   0.929000   -31.63   6.07  x [10.sup.-2]
            [b.sub.3]   0.021880            2.46  x [10.sup.-2]
            [k.sub.3]   0.002049            4.37  x [10.sup.-4]

Temperature                               65

Model       Parameter     Mean      DIC             SD

[M.sub.1]   [k.sub.1]   0.00184    -31.35   2.15 x [10.sup.-4]

[M.sub.2]   [a.sub.2]   0.935900   -29.72   6.49  x [10.sup.-2]
            [k.sub.2]   0.001705            2.52  x [10.sup.-4]

[M.sub.3]   [a.sub.3]   0.928300   -33.36   5.69  x [10.sup.-2]
            [b.sub.3]   0.036680            2.90  x [10.sup.-2]
            [k.sub.3]   0.002028            3.52  x [10.sup.-4]

Table 5. Bayesian estimations model parameters on drying of
passionfruit seeds at 1.29 m s-1 at different temperatures.

Temperature                                  40

Model          Parameter     Mean      DIC             SD

[M.sub.1]      [k.sub.1]   0.001155   -36.51   9.29 x [10.sup.-5]

[M.sub.2]      [a.sub.2]   0.920600   -37.16   4.84 x [10.sup.-2]
               [k.sub.2]   0.001039            1.05 x [10.sup.-4]

[M.sub.3]      [a.sub.3]   0.92080    -37.19   4.70 x [10.sup.-2]
               [b.sub.3]   0.01330             2.08 x [10.sup.-2]
               [k.sub.3]   0.00110             1.43 x [10.sup.-4]

Temperature                                    50

Model          Parameter     Mean      DIC             SD

[M.sub.1]      [k.sub.1]   0.001553   -33.02   1.49 x [10.sup.-4]

[M.sub.2]      [a.sub.2]   0.930800            5.82 x [10.sup.-4]
               [k.sub.2]   0.001430   -32.13   1.78 x [10.sup.-4]

[M.sub.3]      [a.sub.3]   0.925700            5.03 x [10.sup.-2]
               [b.sub.3]   0.015160            2.27 x [10.sup.-2]
               [k.sub.3]   0.001523   -32.76   2.30 x [10.sup.-4]

Temperature                                  60

Model          Parameter     Mean      DIC             SD

[M.sub.1]      [k.sub.1]   0.001498   -34.88   1.11 x [10.sup.-4]

[M.sub.2]      [a.sub.2]   0.943000   -33.83   5.03 x [10.sup.-2]
               [k.sub.2]   0.001405            1.32 x [10.sup.-4]

[M.sub.3]      [a.sub.3]   0.933700   -34.35   4.75 x [10.sup.-2]
               [b.sub.3]   0.009098            1.66 x [10.sup.-2]
               [k.sub.3]   0.001453            1.62 x [10.sup.-4]

Temperature                                  65

Model          Parameter     Mean      DIC             SD

[M.sub.1]      [k.sub.1]   0.001747   -27.22   2.11 x [10.sup.-4]

[M.sub.2]      [a.sub.2]   0.932400   -26.10   6.57 x [10.sup.-2]
               [k.sub.2]   0.001616            2.43 x [10.sup.-4]

[M.sub.3]      [a.sub.3]   0.925300   -27.26   5.66 x [10.sup.-2]
               [b.sub.3]   0.020580            2.92 x [10.sup.-2]
               [k.sub.3]   0.001749            4.06 x [10.sup.-4]

Table 6. Adjustment equations of kt, k and k parameters.

Velocity                   Adjustment equation             [R.sup.2]
(m [s.sup.-1])

0.6              [k.sub.1] = 2 x [10.sup.-7] [T.sup.3] -     0.999
                   0.0002[T.sup.2] + 0.0077T - 8.4623

0.9                   [k.sub.2] = -4 x [10.sup.-7]           0.999
                 [T.sup.3] + 0.0004[T.sup.2] - 0.1153T +

1.2              [k.sub.3] = 2 x [10.sup.-7][T.sup.3] -      0.999
                   0.0002[T.sup.2] + 0.0076T - 8.2664

Table 7. Effective diffusivity at different conditions.

Velocity         Temperature            Deg
(m [s.sup.-1])   ([degrees]C)     (m [s.sup.-1])

0.6                   40        1.70 x [10.sup.-10]
                      50        2.55 x [10.sup.-10]
                      60        2.12 x [10.sup.-10]
                      65        3.40 x [10.sup.-10]
0.9                   40        1.70 x [10.sup.-10]
                      50        3.40 x [10.sup.-10]
                      60        3.83 x [10.sup.-10]
                      65        3.40 x [10.sup.-10]
1.2                   40        3.41 x [10.sup.-10]
                      50        4.26 x [10.sup.-10]
                      60        4.68 x [10.sup.-10]
                      65        4.26 x [10.sup.-10]
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Author:de Oliveira, Ricardo Cardoso; Rossi, Robson Marcelo; Gimenes, Marcelino Luiz; de Barros, Sueli Teres
Publication:Acta Scientiarum. Technology (UEM)
Article Type:Report
Date:Oct 1, 2014
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