Convective drying kinetics of osmotically pretreated papaya cubes

Cinética de secagem convectiva de cubos de mamão pré-tratados osmoticamente

ABSTRACT

This study assessed the fitting of mathematical models to the convective drying kinetics of osmotically pre-dehydrated papaya cubes. Papaya cubes were subjected to osmotic dehydration in sucrose solutions at 40 and 50 ºBrix, at temperatures of 50 and 60 ºC, followed by complementary convective drying in forced air circulation oven under three temperatures (50, 60 and 70 °C) and constant air velocity of 1.0 m s^-1. Ten thin-layer drying mathematical models were fitted to the experimental data. The increase in air temperature and the decrease in osmotic solution concentration resulted in increased water removal rate. Based on the statistical indices, the Two Terms model was the one that best described the drying kinetics of the samples for all evaluated conditions. The effective diffusion coefficients increased with the elevation of air temperature, ranging from 1.766 x 10^-10 to 3.910 x 10^-6 m² s^-1, whereas the convective mass transfer coefficients ranged from 3.910 x 10^-7 to 1.201 x 10^-6 m s^-1 with Biot number from 0.001 to 12.500.

Key words:
Carica papaya; osmotic dehydration; effective diffusivity

RESUMO

Este estudo avaliou o ajuste de modelos matemáticos na cinética de secagem convectiva de cubos de mamão pré-desidratados osmoticamente. Os cubos de mamão foram submetidos à desidratação osmótica em soluções de sacarose a 40 e 50 ºBrix, em temperaturas de 50 e 60 ºC, seguida de secagem convectiva complementar em estufa com circulação forçada de ar sob três temperaturas (50, 60 e 70 °C) e velocidade do ar constante de 1,0 m s^-1. Aos dados experimentais foram ajustados dez modelos matemáticos de secagem em camada fina. O aumento da temperatura do ar e a diminuição da concentração da solução osmótica resultou em aumento da taxa de remoção de água. Com base nos índices estatísticos, o modelo de Dois Termos foi o que melhor descreveu a cinética de secagem das amostras para todas as condições avaliadas. Os coeficientes de difusão efetivos aumentaram com a elevação da temperatura do ar, variando de 1,766 x 10^-10 a 3,910 x 10^-6 m² s^-1, enquanto os coeficientes convectivos de transferência de massa variaram entre 3,910 x 10^-7 a 1,201 x 10^-6 m s^-1 com número de Biot de 0,001 a 12,500.

Palavras-chave:
Carica papaya; desidratação osmótica; difusividade efetiva

Introduction

Brazil is the second largest producer of papaya in the world, having produced a total of 1.42 million tons in 2016 (FAOSTAT, 2018). Papaya has a fast ripening, which manifests itself immediately as a structural softening that, associated with its high moisture content and water activity, makes the product highly perishable, resulting in post-harvest losses throughout its chain (Kandasamy et al., 2012).

Convective drying, for its simplicity and low cost, compared to other drying methods such as lyophilization, is one of the most used technologies for the conservation of agricultural products. However, this method causes alterations in sensory and nutritional properties and in the bioactive compounds of the dry products (Gava et al., 2008; Orikasa et al., 2014). Such alterations can be minimized using combined drying methods, as proposed in the osmo-convective drying (Prosapio & Norton, 2017; Dermesonlouoglou et al., 2018).

Mathematical modeling of the drying process is fundamental for understanding and providing information about the behavior of certain parameters that describe heat and mass transfer mechanisms (Silva et al., 2014a; Tzempelikos et al., 2015; Pacheco-Angulo et al., 2016), which can provide a solid basis for optimizing the process.

The quality of fit of the models to the experimental data can be assessed with different statistical indices; however, according to Kucuk et al. (2014), the best model to describe the drying curve of the product is the one with highest values of correlation coefficient, coefficient of determination, modeling efficiency and/or adjusted R² and the lowest values of chi-square, mean squared deviation, relative mean percentage error, mean polarization error, standard error of estimation, residual sum of squares, reduced sum of squared errors and/or residuals.

In this context, the objective was to assess the mathematical modeling of convective drying kinetics, at temperatures of 50, 60 and 70 °C, of papaya cubes osmotically pre-dehydrated in sucrose solutions and to obtain the effective diffusivity coefficients and convective mass transfer coefficients.

Material and Methods

To conduct this study, the raw material used was ripe papaya fruits (Carica papaya L.) cv. Formosa, 2017 Season, purchased at the local market of the city of Campina Grande, PB, Brazil. Papaya fruits were washed with neutral detergent and subsequently sanitized with sodium hypochlorite solution (100 ppm) for 15 min. The peel was removed with a stainless-steel knife, and the seeds were discarded. The pulp was cut into cubes with dimensions of 20 mm, measured with digital caliper (Absolute model, Mitutoyo, Brazil) with resolution of 0.01 mm. The cubes were osmotically pre-dehydrated in sucrose solution (syrup) with 40 and 50 °Brix, in a cubes:syrup proportion of 1:6 (g:g), at temperatures of 50 and 60 °C. The osmotic dehydration (OD) process was carried out in a BOD chamber and lasted 4 h, considering the maximum rate of water removal from the papaya cubes during the OD. The cubes were removed from the sucrose solution with plastic sieves and left on the bench to drain excess solution from the surface.

About 25 g of the osmotically dehydrated cubes were arranged, in a single layer, in stainless-steel rectangular baskets (15 x 12 cm) and dried, in triplicate, in a forced air circulation oven (320/5 model, Foneman, Brazil) at temperatures of 50, 60, 70 °C and air velocity of 1.0 m s^-1, determined by means of a digital anemometer (ITTHAL-300 model, Instrutemp, Brazil). Water loss was monitored by weighing on an electronic scale (AS5500C model, Marte, Brazil) with a resolution of ± 0.01 g, at regular times of 5, 10, 20, 30 and 60 min, until the samples reached constant mass. The data of drying kinetics were used to calculate the drying rates (Eq. 1) (Özdemira et al., 2017) and moisture content ratios (Eq. 2) (Galaz et al., 2017).

$$DR=\frac{M_{t_0}-M_{t_0+\Delta t}}{\Delta t}$$ (1)

where:

DR - drying rate, kg kg^-1 h^-1;

M_t0 - moisture content at previous time, kg kg^-1 d.b.;

M_{t0 + Δt} - moisture content at current time, kg kg^-1 d.b.; and,

Δt - difference between the current time (t_i) and previous time (t₀) of drying, min.

$$MR=\frac{M-M_e}{M_i-M_e}$$ (2)

where:

MR - moisture content ratio, dimensionless;

M - moisture content at a specific time, d.b.;

M_e - equilibrium moisture content, d.b.; and,

M_i - initial moisture content, d.b.

Different mathematical models were fitted to the experimental data of drying kinetics (Eqs. 3 to 12), using the computer program Statistica®, version 7.0, through non-linear regression, by the Quasi-Newton method (Statsoft, 2007).

- Newton - Lewis (1921):

$$MR=\exp \left(-kt\right)$$ (3)

- Page - Page (1949):

$$MR=\exp \left(-k{t}^n\right)$$ (4)

- Henderson & Pabis - Henderson & Pabis (1961):

$$MR=a\exp \left(-kt\right)$$ (5)

- Two-Term Exponential - Sharaf-Eldeen et al. (1980):

$$MR=a\exp \left(-kt\right)+\left(1-a\right)\exp \left(-kat\right)$$ (6)

- Thompson - Thompson et al. (1968):

$$MR=\exp \left[-a{\left(a^2+4bt\right)}^{0.5}\right]/2b$$ (7)

- Logarithmic - Yagcioglu et al. (1999):

$$MR=a\exp \left(-kt\right)+c$$ (8)

- Approximation of Diffusion - Sharaf-Elden et al. (1980):

$$MR=a\exp \left(-kt\right)+\left(1-a\right)\exp \left(-kbt\right)$$ (9)

- Modified Henderson & Pabis - Karathanos (1999):

$$MR=a\exp \left(-kt\right)+b\exp \left(-kt\right)+c\exp \left(-kt\right)$$ (10)

- Two Terms - Henderson (1974):

$$MR=a\exp \left(-k_{0}t\right)+b\exp \left(-k_{1}t\right)$$ (11)

- Midilli - Midilli et al. (2002):

$$MR=a\exp \left(-k{t}^n\right)+bt$$ (12)

where:

MR - moisture content ratio, dimensionless;

a, b, c, k, k₀, k₁, n - coefficients of the models; and,

t - drying time, min.

The criteria for the fit of the mathematical models to the experimental data were the coefficient of determination (R²) (Eq. 13), mean squared deviation (MSD) (Eq. 14), mean relative error (P) (Eq. 15), mean estimated error (SE) (Eq. 16) and the chi-square (χ²) (Eq. 17) (Costa et al., 2016; Haas et al., 2017; Rabha et al., 2017).

$${\text{R}}^2=\frac{{{\displaystyle \sum }}_{\text{i}=1}^{\text{N}}{\left[\left({\text{MR}}_{\text{exp},\text{i}}-{\overline{\text{MR}}}_{\text{exp},\text{i}}\right)\left({\text{MR}}_{\text{pred},\text{i}}-{\overline{\text{MR}}}_{\text{pred},\text{i}}\right)\right]}^2}{{{\displaystyle \sum }}_{\text{i}=1}^{\text{N}}{\left({\text{MR}}_{\text{exp},\text{i}}-{\overline{\text{MR}}}_{\text{exp},\text{i}}\right)}^2{{\displaystyle \sum }}_{\text{i}=1}^{\text{N}}{\left({\text{MR}}_{\text{pred},\text{i}}-{\overline{\text{MR}}}_{\text{pred},\text{i}}\right)}^2}$$ (13)

$$\text{MSD}={\left[\frac{1}{\text{N}}{{\displaystyle \sum }}_{\text{i}=1}^{\text{N}}{\left({\text{MR}}_{\text{pred}\text{,i}}-{\text{MR}}_{\text{exp}\text{,i}}\right)}^2\right]}^{\frac{1}{2}}$$ (14)

$$\text{P}=\frac{100}{\text{N}}{{\displaystyle \sum }}_{\text{i}=1}^{\text{N}}\frac{\left|{\text{MR}}_{\text{exp}\text{,i}}-{\text{MR}}_{\text{pred}\text{,i}}\right|}{{\text{MR}}_{\text{exp}\text{,i}}}$$ (15)

$$\text{SE}={\left[\frac{1}{\text{N}-\text{n}}{{\displaystyle \sum }}_{\text{i}=1}^{\text{N}}{\left({\text{MR}}_{\text{exp}\text{,i}}-{\text{MR}}_{\text{pred}\text{,i}}\right)}^2\right]}^{\frac{1}{2}}$$ (16)

$${\chi }^2=\frac{1}{\text{N}-\text{n}}{{\displaystyle \sum }}_{\text{i}=1}^{\text{N}}{\left({\text{MR}}_{\text{exp}\text{,i}}-{\text{MR}}_{\text{pred}\text{,i}}\right)}^2$$ (17)

where:

MR_exp - experimental moisture content ratio;

MR_exp,i - average experimental moisture content ratio;

MR_pred - moisture content ratio predicted by the model;

MR_pred,i - average moisture content ratio predicted by the model;

N - number of observations; and,

n - number of coefficients of the model.

The geometric shape of the samples was assumed to be that of a cube (parallelepiped with equal sides), and the analytical solution of the second Fick’s law for this geometry, considering internal diffusive mass flow equal to the external convective flow in the vicinity of the samples (convective boundary condition) (Eq. 18) (Silva et al., 2014b), was fitted to the experimental data of drying kinetics for determining the effective diffusion coefficients (D_ef) and convective mass transfer coefficients (h_w), using 16 x 16 x 16 terms of the analytical solution referring to the three summations of Eq. (18), employing the program Convective Adsorption - Desorption, version 3.2 (Silva & Silva, 2018).

$$\overline{M}\left(t\right)=M_{eq}+\left(M_0-M_{eq}\right)\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }\sum _{k=1}^{\infty }{B}_n{B}_m{B}_k·exp\left[-\left(\frac{{\mu }_{n^2}}{{\left(L_1/2\right)}^2}+\frac{{\mu }_{m^2}}{{\left(L_2/2\right)}^2}\frac{{\mu }_{k^2}}{{\left(L_3/2\right)}^2}\right)D_{ef}t\right]$$ (18)

where:

M(t) - average moisture content at time t, d.b.;

M_eq - equilibrium moisture content, d.b.;

M₀ - initial moisture content, d.b.;

µ_n, µ_m and µ_k - roots of the characteristic equation, obtained through Eq. 19;

B_n, B_m and B_k - solution parameters calculated according to Eq. 21;

D_ef - effective diffusion coefficient, m² s^-1;

L₁, L₂ and L₃ - length, height and thickness, respectively, m; and,

t - time, s.

$$\text{cot}{\mu }_{\text{j}}=\frac{{\mu }_{\text{j}}}{\text{Bi}}$$ (19)

With j equal to the indices n, m and k. The parameter Bi is the mass transfer Biot number and is given by Eq. 20:

$$\text{Bi}=\frac{{\text{h}}_{\text{w}}\text{L}}{{\text{D}}_{\text{ef}}}$$ (20)

where:

h_w - convective mass transfer coefficient, m s^-1;

L - characteristics length, m; and,

D_ef - effective diffusion coefficient, m² s^-1.

$${\text{B}}_{\text{j}}=\frac{2{\text{Bi}}^2}{{\mu }_{{\text{j}}^2}\left({\text{Bi}}^2+\text{Bi}+{\mu }_{{\text{j}}^2}\right)}$$ (21)

where j is equal to the indices n, m and k.

Results and Discussion

In the drying curves of the osmotically dehydrated papaya cubes (Figures 1A to D), it can be observed that, as the concentration and temperature of the solution increase, the drying curves (Figures 1C and D) become less distant from one another, denoting less relative influence of drying temperature on water removal dynamics. Although the increase in concentration and temperature of the solution causes a greater water removal during the OD process (Germer et al., 2011; Souraki et al., 2014), due to the elevation of the concentration gradient of soluble solids between the fruit and the solution, it also results in the incorporation of solute in the sample (Garcia-Noguera et al., 2010; Mendes et al., 2013), which could lead to a higher resistance to heat and mass transfers during convective drying, resulting in lower drying effectiveness. Fernandes et al. (2008) demonstrated, in experiments with pineapple, that OD in solutions with high concentrations (> 35 °Brix) results in high gain of solids by the samples, which may cause a reduction in the water removal rate during convective drying.

Figure 1
Drying curves of papaya cubes subjected to osmotic dehydration at: (A) 40 °Brix/50 °C; (B) 40 °Brix/60 °C; (C) 50 °Brix/50 °C; and (D) 50 °Brix/60 °C followed by convective drying at temperatures of 50, 60 and 70 °C

The drying rates (Figures 2A to D), for the same concentration of the solution, in general, were higher in samples subjected to solutions at lower temperatures, ranging from 0.85 to 1.64 kg kg^-1 h^-1 for the samples subjected to OD pre-treatment of 50 °Brix/60 °C and 50 °Brix/50 °C, respectively, dried at 50 °C.

Figure 2
Drying rates of papaya cubes subjected to osmotic dehydration at: (A) 40 °Brix/50 °C; (B) 40 °Brix/60 °C; (C) 50 °Brix/50 °C; and (D) 50 °Brix/60 °C followed by convective drying at temperatures of 50, 60 and 70 °C

The values of the drying rates changed over time, gradually increasing to the maximum value and then decreasing rapidly. This occurs because, at the beginning of the drying, liquid diffusion is the main mechanism of water transport and, as the drying progresses, vapor diffusion becomes the dominant mode, so the drying rate increases. However, with the continuity of the process, the samples become unsaturated with moisture, the vapor diffusion decreases and, consequently, the drying rate also decreases (Chen et al., 2017). In addition, drying occurred mainly in the falling rate period, and no constant rate period was observed (Figures 2A to D), indicating that the internal resistance to water movement is greater than the rate of removal from the sample surface (Pilatti et al., 2016). Similar behavior was observed by Kaushal & Sharma (2016) during convective drying at different temperatures (50-70 ºC) of osmo-dehydrated jackfruit pulp.

The indices of the models fitted to the experimental data of drying kinetics of the samples (Table 1), at different temperatures, demonstrate that the Two Terms model had the highest values of the coefficients of determination (R²) (0.997-0.999) and the lowest mean squared deviations (MSD) (0.008-0.014), mean relative error (P) (1.306-6.039%), mean estimated error (SE) (0.009-0.017) and chi-square (χ²) (1.0 x 10^-4-3.0 x 10^-4), so it better represents the drying process of the samples under the studied conditions. However, it should be pointed out that the models of Page, Approximation of Diffusion and Midilli, also had high R² values, above 0.995, and low MSD, P, SE and χ², below 0.021, 8.199%, 0.023 and 5.0 x 10^-4, respectively, indicating their adequacy to represent the drying kinetics of osmotically pre-dehydrated papaya cubes.

Table 1
Values of the coefficient of determination (R²), mean squared deviation (MSD), mean relative error (P), mean estimated error (SE) and chi-square (χ²) of the models fitted to the experimental drying data of papaya cubes subjected to osmotic dehydration (OD)

Model	OD	Temperature
		R2	MSD	P (%)	SE	χ2 (×10-4)
		50 ºC
Newton	40 °Brix/50 °C	0.976	0.047	13.536	0.049	23.0
	40 °Brix/60 °C	0.978	0.046	19.320	0.048	23.0
	50 °Brix/50 °C	0.964	0.058	26.586	0.060	36.0
	50 °Brix/60 °C	0.985	0.038	19.136	0.040	16.0
Page	40 °Brix/50 °C	0.995	0.021	8.199	0.023	5.0
	40 °Brix/60 °C	0.996	0.018	4.622	0.020	4.0
	50 °Brix/50 °C	0.995	0.021	7.488	0.022	5.0
	50 °Brix/60 °C	0.997	0.016	6.684	0.017	3.0
Henderson & Pabis	40 °Brix/50 °C	0.986	0.036	9.661	0.039	15.0
	40 °Brix/60 °C	0.984	0.039	15.061	0.041	17.0
	50 °Brix/50 °C	0.977	0.046	18.852	0.049	24.0
	50 °Brix/60 °C	0.989	0.033	15.009	0.035	13.0
Two-Term Exponential	40 °Brix/50 °C	0.995	0.021	7.670	0.023	5.0
	40 °Brix/60 °C	0.995	0.020	8.260	0.022	5.0
	50 °Brix/50 °C	0.988	0.033	14.118	0.035	12.0
	50 °Brix/60 °C	0.998	0.014	6.554	0.015	2.0
Thompson	40 °Brix/50 °C	0.993	0.025	11.478	0.027	7.0
	40 °Brix/60 °C	0.997	0.015	4.746	0.016	3.0
	50 °Brix/50 °C	0.995	0.021	13.451	0.023	5.0
	50 °Brix/60 °C	0.998	0.013	10.520	0.014	2.0
Logarithmic	40 °Brix/50 °C	0.987	0.035	11.583	0.038	15.0
	40 °Brix/60 °C	0.990	0.031	9.606	0.034	1.2
	50 °Brix/50 °C	0.982	0.040	18.276	0.044	20.0
	50 °Brix/60 °C	0.992	0.027	15.998	0.030	9.0
Approximation of Diffusion	40 °Brix/50 °C	0.997	0.015	6.169	0.017	3.0
	40 °Brix/60 °C	0.998	0.010	2.540	0.012	1.0
	50 °Brix/50 °C	0.997	0.014	4.743	0.016	3.0
	50 °Brix/60 °C	0.999	0.009	5.833	0.010	1.0
Modified Henderson & Pabis	40 °Brix/50 °C	0.986	0.036	9.661	0.042	17.0
	40 °Brix/60 °C	0.984	0.039	15.059	0.044	19.0
	50 °Brix/50 °C	0.977	0.046	18.852	0.052	27.0
	50 °Brix/60 °C	0.989	0.033	15.009	0.037	14.0
Two Terms	40 °Brix/50 °C	0.997	0.014	6.039	0.017	3.0
	40 °Brix/60 °C	0.999	0.009	1.306	0.011	1.0
	50 °Brix/50 °C	0.998	0.013	4.283	0.015	2.0
	50 °Brix/60 °C	0.999	0.008	5.392	0.009	1.0
Midilli	40 °Brix/50 °C	0.997	0.016	4.889	0.018	3.0
	40 °Brix/60 °C	0.997	0.016	3.869	0.018	3.0
	50 °Brix/50 °C	0.996	0.019	5.068	0.022	5.0
	50 °Brix/60 °C	0.998	0.014	5.481	0.016	3.0
		60 ºC
Newton	40 °Brix/50 °C	0.981	0.044	18.651	0.045	3.0
	40 °Brix/60 °C	0.984	0.040	26.478	0.041	17.0
	50 °Brix/50 °C	0.970	0.054	26.015	0.055	30.0
	50 °Brix/60 °C	0.988	0.035	27.045	0.036	13.0
Page	40 °Brix/50 °C	0.997	0.016	8.094	0.017	3.0
	40 °Brix/60 °C	0.998	0.010	4.408	0.011	1.0
	50 °Brix/50 °C	0.997	0.016	8.239	0.017	3.0
	50 °Brix/60 °C	0.999	0.006	4.720	0.012	0.4
Henderson & Pabis	40 °Brix/50 °C	0.988	0.034	12.643	0.036	1.3
	40 °Brix/60 °C	0.989	0.032	20.612	0.034	12.0
	50 °Brix/50 °C	0.983	0.040	17.405	0.042	18.0
	50 °Brix/60 °C	0.991	0.029	22.856	0.031	10.0
Two-Term Exponential	40 °Brix/50 °C	0.996	0.018	6.444	0.019	4.0
	40 °Brix/60 °C	0.997	0.015	10.980	0.016	2.0
	50 °Brix/50 °C	0.992	0.027	12.461	0.029	8.0
	50 °Brix/60 °C	0.998	0.011	11.170	0.012	31.0
Thompson	40 °Brix/50 °C	0.996	0.019	14.230	0.020	4.0
	40 °Brix/60 °C	0.999	0.009	12.484	0.010	1.0
	50 °Brix/50 °C	0.996	0.019	16.449	0.021	4.0
	50 °Brix/60 °C	0.999	0.008	8.103	0.009	1.0
Logarithmic	40 °Brix/50 °C	0.990	0.031	13.519	0.034	12.0
	40 °Brix/60 °C	0.993	0.026	21.517	0.029	8.0
	50 °Brix/50 °C	0.986	0.035	17.578	0.039	1.5
	50 °Brix/60 °C	0.994	0.023	20.213	0.026	7.0
Approximation of Diffusion	40 °Brix/50 °C	0.998	0.012	5.500	0.013	2.0
	40 °Brix/60 °C	0.999	0.003	2.196	0.004	10.0
	50 °Brix/50 °C	0.998	0.010	3.716	0.011	1.0
	50 °Brix/60 °C	0.999	0.006	1.702	0.007	0.5
Modified Henderson & Pabis	40 °Brix/50 °C	0.988	0.034	12.647	0.039	15.0
	40 °Brix/60 °C	0.989	0.032	20.613	0.036	1.3
	50 °Brix/50 °C	0.983	0.040	17.407	0.045	20.0
	50 °Brix/60 °C	0.991	0.029	22.856	0.033	11.0
Two Terms	40 °Brix/50 °C	0.998	0.011	5.369	0.013	2.0
	40 °Brix/60 °C	0.999	0.003	1.942	0.004	0.1
	50 °Brix/50 °C	0.999	0.010	3.635	0.011	1.0
	50 °Brix/60 °C	0.999	0.005	2.038	0.006	0.1
Midilli	40 °Brix/50 °C	0.998	0.014	4.634	0.016	3.0
	40 °Brix/60 °C	0.999	0.009	3.234	0.011	1.0
	50 °Brix/50 °C	0.997	0.014	3.934	0.016	3.0
	50 °Brix/60 °C	0.999	0.010	2.899	0.012	1.0
		70 ºC
Newton	40 °Brix/50 °C	0.997	0.016	7.957	0.017	3.0
	40 °Brix/60 °C	0.997	0.016	14.011	0.017	3.0
	50 °Brix/50 °C	0.996	0.019	8.749	0.020	4.0
	50 °Brix/60 °C	0.993	0.027	14.204	0.028	8.0
Page	40 °Brix/50 °C	0.997	0.016	8.646	0.017	3.0
	40 °Brix/60 °C	0.998	0.011	5.748	0.012	1.0
	50 °Brix/50 °C	0.999	0.007	10.705	0.008	1.0
	50 °Brix/60 °C	0.999	0.010	7.352	0.011	1.0
Henderson & Pabis	40 °Brix/50 °C	0.998	0.015	9.638	0.016	3.0
	40 °Brix/60 °C	0.998	0.015	12.944	0.016	3.0
	50 °Brix/50 °C	0.998	0.014	7.317	0.015	2.0
	50 °Brix/60 °C	0.995	0.023	10.214	0.024	6.0
Two-Term Exponential	40 °Brix/50 °C	0.997	0.017	7.664	0.018	3.0
	40 °Brix/60 °C	0.998	0.015	12.490	0.016	3.0
	50 °Brix/50 °C	0.999	0.008	6.161	0.009	1.0
	50 °Brix/60 °C	0.999	0.006	6.212	0.007	1.0
Thompson	40 °Brix/50 °C	0.997	0.015	7.137	0.017	3.0
	40 °Brix/60 °C	0.999	0.008	6.037	0.009	1.0
	50 °Brix/50 °C	0.999	0.011	16.197	0.012	2.0
	50 °Brix/60 °C	0.998	0.010	13.652	0.012	1.0
Logarithmic	40 °Brix/50 °C	0.998	0.013	14.501	0.015	2.0
	40 °Brix/60 °C	0.998	0.012	15.520	0.014	2.0
	50 °Brix/50 °C	0.998	0.013	10.994	0.015	2.0
	50 °Brix/60 °C	0.996	0.019	16.281	0.022	5.0
Approximation of Diffusion	40 °Brix/50 °C	0.998	0.015	8.900	0.017	4.0
	40 °Brix/60 °C	0.999	0.007	5.498	0.008	1.0
	50 °Brix/50 °C	0.999	0.005	7.349	0.006	0.1
	50 °Brix/60 °C	0.999	0.006	6.037	0.007	1.0
Modified Henderson & Pabis	40 °Brix/50 °C	0.998	0.015	9.638	0.017	3.0
	40 °Brix/60 °C	0.998	0.015	12.944	0.017	3.0
	50 °Brix/50 °C	0.998	0.014	7.317	0.016	3.0
	50 °Brix/60 °C	0.995	0.022	10.217	0.026	7.0
Two Terms	40 °Brix/50 °C	0.998	0.011	8.802	0.013	2.0
	40 °Brix/60 °C	0.999	0.006	4.541	0.007	1.0
	50 °Brix/50 °C	0.999	0.006	7.319	0.007	0.1
	50 °Brix/60 °C	0.999	0.006	5.760	0.007	0.1
Midilli	40 °Brix/50 °C	0.998	0.014	10.848	0.016	2.0
	40 °Brix/60 °C	0.999	0.009	6.712	0.011	1.0
	50 °Brix/50 °C	0.999	0.005	2.141	0.006	0.1
	50 °Brix/60 °C	0.999	0.008	3.954	0.010	1.0

Table 2 presents the effective diffusion coefficients (D_ef) and convective mass transfer coefficients (h_w) obtained for the drying of the samples, subjected to the temperatures of 50, 60 and 70 °C. The increase in convective drying temperature (50-70 °C) causes the increment in D_ef values. In addition, it was observed that samples subjected to the solutions with the same temperature, but with higher sucrose concentration, in particular for the drying temperature of 70 ºC, offer greater resistance to external mass transfer, which in turn may be related to the reduction of h_w. This behavior may be associated with increased concentration of soluble solids during the OD, on the surface of the sample (Rodríguez et al., 2015; Sangeeta & Hathan, 2016; Goula et al., 2017), capable of forming, at high temperatures (≥ 70 ºC), a dense and poorly permeable layer, increasing the resistance to heat transfer to the samples and establishing an additional barrier to the water exit from its interior (Munhoz et al., 2014; Corrêa et al., 2017). Similar results have been reported in strawberry (Garcia-Noguera et al., 2010) and plum (Dehghannya et al., 2016).

Table 2
Effective diffusion coefficients (D_ef) and convective mass transfer coefficients (h_w) obtained in convective drying, at temperatures (Temp) of 50, 60 and 70 °C, of osmotically pre-dehydrated (OD) papaya cubes

OD	Temp (°C)	hw (m s-1)	Def (m2 s-1)	Bi	R2	χ2
40 °Brix/50 °C	50	5.697 × 10-7	8.440 × 10-10	6.750	0.996	6.571 × 10-3
	60	7.411 × 10-7	1.097 × 10-9	6.750	0.998	3.944 × 10-3
	70	3.910 × 10-7	3.910 × 10-6	0.001	0.998	4.885 × 10-3
40 °Brix/60 °C	50	6.553 × 10-7	9.709 × 10-10	6.750	0.997	5.746 × 10-3
	60	9.742 × 10-7	1.443 × 10-9	6.750	0.999	2.050 × 10-3
	70	5.845 × 10-7	4.031 × 10-9	1.450	0.998	2.850 × 10-3
50 °Brix/50 °C	50	1.201 × 10-6	9.611 × 10-10	12.500	0.996	7.161 × 10-3
	60	1.138 × 10-6	1.059 × 10-9	10.750	0.998	3.577 × 10-3
	70	5.622 × 10-7	2.444 × 10-9	2.300	0.999	6.693 × 10-4
50 °Brix/60 °C	50	6.661 × 10-7	1.268 × 10-9	5.250	0.997	4.561 × 10-4
	60	9.273 × 10-7	1.766 × 10-10	5.250	0.998	2.811 × 10-4
	70	6.680 × 10-7	1.979 × 10-9	3.375	0.999	1.605 × 10-4

• Bi - Biot number; χ² - Chi-square

It is observed that the Biot number (Bi) was within the range from 0.001 to 12.500 (Table 2), which, according to Kaya et al. (2010), is indicative of the existence of internal and external resistances to water transfer, being considered the most realistic case in practical applications. It should be pointed out that Bi tended to decrease with the elevation in the drying temperature, especially at 70 °C, indicating that there is a higher resistance to mass flow on the surface of the samples (Silva et al., 2013).

The solution of the Fick’s second law equation (Eq. 18), for all OD treatments, considering convective boundary condition, showed, even in Biot number << 1 (Bi = 0.001; R² = 0.998; χ² = 4.885 x 10^-3), adequate fit to the experimental data of drying kinetics of the samples (R² > 0.996 and χ² < 7.161 x 10^-3) (Table 2), which ensures the physical representativeness of the values of D_ef and h_w.

Conclusions

1. Among the fitted mathematical models, the Two Terms model was selected as the most adequate for drying kinetics of osmo-dehydrated papaya cubes.

2. The effective diffusivity in the samples increased with the increase of air temperature, whereas the convective mass transfer coefficient showed a less defined trend.

Acknowledgments

The authors thank the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for the financial support.

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Publication Dates

• Publication in this collection
  02 Mar 2020
• Date of issue
  Mar 2020

History

• Received
  29 Oct 2018
• Accepted
  31 Dec 2019
• Published
  12 Feb 2020