Drying kinetics of jabuticaba pulp by regression models

Jabuticaba tree is native to the Atlantic Forest in Southern Brazil, and its fruit is widely consumed in the fresh form, but it is highly perishable, requiring conservation techniques. The aim of this study was to describe the drying kinetics of jabuticaba pulp at temperatures of 50 and 60°C, comparing the Henderson, Simple Three-Parameter Exponential, Lewis, Thompson, Fick and Wang and Sing regression models and estimating the Absolute Drying Rate (ADR) for the best model. Parameters were estimated using the SAS software. The evaluation of the quality in the adjustment and selection of models was made based on the adjusted determination coefficient, Residual Standard Deviation and Akaike Information Criterion. Models presented good adjustment to data, and the Lewis model was the most suitable to describe the drying kinetics of jabuticaba pulp at temperatures of 50 and 60°C, with drying rate of 0.000063 and 0.000082 g of water/s respectively. ADR indicated that in one third of the drying time, 70% of moisture loss occurred at both temperatures and after this period, there was a deceleration of moisture loss until stabilization, when equilibrium moisture content is reached.

Index terms
food preservation; moisture ratio; nonlinear regression; statistical models

Authorship

Thais Destefani Ribeiro Furtado

PhD student in Agricultural Statistics and Experimentation, Federal University of Lavras/ Lavras-MG, Brazil. E-mail: t.destefani.ribeiro@gmail.com Federal University of LavrasBrasilLavras, MG, Brasil PhD student in Agricultural Statistics and Experimentation, Federal University of Lavras/ Lavras-MG, Brazil. E-mail: t.destefani.ribeiro@gmail.com

Joel Augusto Muniz

PhD in Agricultural Statistics and Experimentation, Full Professor in Federal University of Lavras/Lavras-MG, Brazil. E-mail: joamuniz@des.ufla.br Federal University of LavrasBrasilLavras, MG, Brasil PhD in Agricultural Statistics and Experimentation, Full Professor in Federal University of Lavras/Lavras-MG, Brazil. E-mail: joamuniz@des.ufla.br

https://orcid.org/0000-0002-1069-4136

Edilson Marcelino Silva

PhD student in Agricultural Statistics and Experimentation, Federal University of Lavras/ Lavras-MG, Brazil .E-mail: edilsonmg3@hotmail.com Federal University of LavrasBrasilLavras, MG, Brasil PhD student in Agricultural Statistics and Experimentation, Federal University of Lavras/ Lavras-MG, Brazil .E-mail: edilsonmg3@hotmail.com

https://orcid.org/0000-0002-2800-3495

Jaqueline Gonçalves Fernandes

PhD student in Agricultural Statistics and Experimentation, Federal University of Lavras/Lavras-MG, Brazil. E-mail: jaqgonfer@gmail.com Federal University of LavrasBrasilLavras, MG, Brasil PhD student in Agricultural Statistics and Experimentation, Federal University of Lavras/Lavras-MG, Brazil. E-mail: jaqgonfer@gmail.com

https://orcid.org/0000-0002-7236-9718

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables | Formulas

Figures (4)
Tables (3)
Formulas (6)

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Figure 1
Analysis of residuals for the drying kinetics of jabuticaba pulp at 50 ° C, where (a), (b), (c), (d) and (e) represent the values adjusted for residuals, and in (f), (g), (h), (i) and (j), the residual values in relation to theoretical quantis for the Henderson, Lewis, Simple Three-Parameter Exponential (ESTP), Thompson and Wang and Sing models, respectively.

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Figure 2
Analysis of residuals for the drying kinetics of jabuticaba pulp at 60 ° C, where (a), (b), (c), (d) and (e) represent the values adjusted for residuals, and in (f), (g), (h), (i) and (j), the residual values in relation to theoretical quantis for the Henderson, Lewis, Simple Three-Parameter Exponential (ESTP), Thompson and Wang and Sing models, respectively.

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Figure 3
Curves of Lewis model adjustments for the Moisture Ratio (MR) as a function of Time (s), the drying kinetic data of jabuticaba pulp at temperatures of 50°C and 60°C in (a) and (b) respectively.

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Figure 4
Absolute drying rate relative to the Lewis model for the Moisture Ratio (MR) as a function of Time (s), for the drying kinetics data of jabuticaba pulp at temperatures of 50°C (a) and 60°C (b).

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Table 1
P-values for the Shapiro-Wilk (SW), Breuch-Pagan (BP) and Durbin-Watson (DW) tests with their respective significance in the analysis of estimated residuals after adjustment of the Henderson (H) models, (ESTP), Lewis (L), Thompson (T) and Wang and Sing (WS) models to the drying kinetics data of jabuticaba pulp at temperatures of 50 and 60°C.

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Table 2
Estimations of parameters, with their respective confidence intervals (UL and LL), for the Henderson (H), Simple Three-Parameter Exponential (ESTP), Lewis (L), Thompson and Wang and Sing models for the drying kinetics data of jabuticaba pulp at temperatures of 50 and 60°C, considering first-order autoregressive errors.

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Table 3
Performance quality evaluators of the Henderson (H), Lewis (L), Thompson (T) and Wang and Sing (WS) models with Ø1 component, referring to the drying kinetics data of jabuticaba pulp at temperatures of 50 and 60°C.

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Figure 1 Analysis of residuals for the drying kinetics of jabuticaba pulp at 50 ° C, where (a), (b), (c), (d) and (e) represent the values adjusted for residuals, and in (f), (g), (h), (i) and (j), the residual values in relation to theoretical quantis for the Henderson, Lewis, Simple Three-Parameter Exponential (ESTP), Thompson and Wang and Sing models, respectively.

Figure 2 Analysis of residuals for the drying kinetics of jabuticaba pulp at 60 ° C, where (a), (b), (c), (d) and (e) represent the values adjusted for residuals, and in (f), (g), (h), (i) and (j), the residual values in relation to theoretical quantis for the Henderson, Lewis, Simple Three-Parameter Exponential (ESTP), Thompson and Wang and Sing models, respectively.

Figure 3 Curves of Lewis model adjustments for the Moisture Ratio (MR) as a function of Time (s), the drying kinetic data of jabuticaba pulp at temperatures of 50°C and 60°C in (a) and (b) respectively.

Figure 4 Absolute drying rate relative to the Lewis model for the Moisture Ratio (MR) as a function of Time (s), for the drying kinetics data of jabuticaba pulp at temperatures of 50°C (a) and 60°C (b).

Table 1 P-values for the Shapiro-Wilk (SW), Breuch-Pagan (BP) and Durbin-Watson (DW) tests with their respective significance in the analysis of estimated residuals after adjustment of the Henderson (H) models, (ESTP), Lewis (L), Thompson (T) and Wang and Sing (WS) models to the drying kinetics data of jabuticaba pulp at temperatures of 50 and 60°C.

Models	SW	BP	DW
Temperature of 50 °C
H	0.4639 ^ns	0.1374 ^ns	0.0010*
L	0.6054 ^ns	0.5101 ^ns	0.0010*
ESTP	0.3370 ^ns	0.3138 ^ns	0.0010*
T	0.4635 ^ns	0.1378 ^ns	0.0010*
WS	0.8187 ^ns	0.0012*	0.0010*
Temperature of 60 °C
H	0.1713 ^ns	0.0894 ^ns	0.0010*
L	0.1504 ^ns	0.8505 ^ns	0.0010*
ESTP	0.2816 ^ns	0.4299 ^ns	0.0010 ^ns
T	0.1717 ^ns	0.0104*	0.0010 ^ns
WS	0.3553 ^ns	0.9981 ^ns	0.0010 ^ns

^*
Significant at 5% probability level by respective SW, BP and DW tests.
^ns
Not significant in relation to respective SW, BP and DW tests.

Table 2 Estimations of parameters, with their respective confidence intervals (UL and LL), for the Henderson (H), Simple Three-Parameter Exponential (ESTP), Lewis (L), Thompson and Wang and Sing models for the drying kinetics data of jabuticaba pulp at temperatures of 50 and 60°C, considering first-order autoregressive errors.

Temperature	Models	Parameters	Estimates	UL	LL
50°C	H	k₀	0.9995**	0.9500	1.0491
		k₁	0.000063**	0.000042	0.000085
		∅₁	0.9186**	0.7316	1.1057
	L	k₁	0.000063**	0.000043	0.000083
	L	∅₁	0.9182**	0.7351	1.1014
	ESTP	k₀	0.9807**	0.8068	1.1546
		k₁	0.000065**	0.000032	0.000097
		k₂	0.0189 ^ns	-0.1501	0.1879
		∅₁	0.9256**	0.7045	1.1467
	T	a	2.3242**	2.1291	2.5194
		b	0.000080**	0.000048	0.000117
		∅₁	0.9575**	0.8289	1.0861
60°C	H	k₀	1.0032**	0.9606	1.0458
		k₁	0.000082**	0.000052	0.000117
		∅₁	0.9315**	0.7097	1.1533
	L	k₁	0.000082**	0.000050	0.000113
	L	∅₁	0.9333**	0.7130	1.1536
	ESTP	k₀	1.0371**	0.8603	1.2139
		k₁	0.000078**	0.000037	0.000123
		k₂	-0.0341 ^ns	-0.2094	0.1412
		∅₁	0.9203**	0.6315	1.2092
	WS	a	-0.000064**	-0.000072	-0.000060
		b	1.04 x10^-9**	9.9 x10^-10	1.1x10^-9
		∅₁	0.8657**	0.6104	1.1210

^**
**Highly significant in relation to Student t-test (1% alpha); ns Not significant in relation to t-Studant test. UL-Upper Limit; LL-Lower Limit.

Table 3 Performance quality evaluators of the Henderson (H), Lewis (L), Thompson (T) and Wang and Sing (WS) models with Ø1 component, referring to the drying kinetics data of jabuticaba pulp at temperatures of 50 and 60°C.

Model	R²_aj		RSD			AIC
	50°C	60°C		50°C	60°C		50°C	60°C
H	0.9953	0.9964		0.0007	0.0005		-121.3	-105.7
L	0.9955	0.9966		0.0006	0.0004		-123.3	-107.7
T	0.9946	-		0.0007	-		-117.2	-
WS	-	0.9969		-	0.0044		-	-103.5

1

{M R}_{i} = k_{0} \exp (- k_{1} t_{i}) + u_{i}

2

{M R}_{i} = \exp (- k_{1} t_{i}) + u_{i}

3

{M R}_{i} = k_{0} \exp (- k_{1} t_{i}) + k_{2} + u_{i}

4

MR i = (\exp (- a - {(a^{2} + 4 b t_{i})}^{0.5}) / 2 b) + u_{i}

5

{M R}_{i} = \frac{6}{π^{2}} \sum_{n = 1}^{i} \frac{1}{n^{2}} \exp (- n^{2} π^{2} D_{e f} \frac{1}{t_{i}}) + u_{i}

6

{M R}_{i} = 1 + a t_{i} + {b t_{i}}^{2} + u_{i}

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Sociedade Brasileira de Fruticultura Via de acesso Prof. Paulo Donato Castellane, s/n , 14884-900 Jaboticabal SP Brazil, Tel.: +55 16 3209-7188/3209-7609 - Jaboticabal - SP - Brazil
E-mail: rbf@fcav.unesp.br

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Drying kinetics of jabuticaba pulp by regression models

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[TFN0101] ^*
Significant at 5% probability level by respective SW, BP and DW tests.

[TFN0102] ^ns
Not significant in relation to respective SW, BP and DW tests.