Multivariate analysis in mathematical model selection to describe <i>Croton urucurana</i> Baill drying kinetics

LOPES ALVES, Jáliston Júlio; RESENDE, Osvaldo; RIBEIRO NETO, Francisco de Araújo; RIBEIRO AGUIAR, Ana Carolina; FERREIRA VIEIRA BESSA, Jaqueline; QUEQUETO, Wellytton Darci

doi:10.1590/fst.12821

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Food Science and Technology

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Original Article • Food Sci. Technol (Campinas) 42 • 2022 • https://doi.org/10.1590/fst.12821 copy

Multivariate analysis in mathematical model selection to describe Croton urucurana Baill drying kinetics

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

The Croton urucurana Baill species is known in Brazil as “sangra d’água” and is popular due to its medicinal properties. For better processing of herbal medicines, it is essential that efficient drying and storage techniques are developed and that compounds are preserved. Therefore, this study aimed to select models through multivariate cluster analysis applying Akaike (AIC) and Bayesian information criteria (BIC) to describe Croton urucurana leaves drying kinetics at different temperatures (40-70 °C). The initial moisture content in Croton urucurana leaves was 1.791, 1.841, 2.196 and 2.144 kg water kg dry matter^-1, and 8.25, 7.75, 4.25 and 2 hours were required to reach hygroscopic equilibrium, with a final moisture content of 0.134, 0.105, 0.065 and 0.0601 kg water kg dry matter^-1, at 40, 50, 60 and 70 °C, respectively. The models with the greatest similarity to the experimental data were Diffusion Approximation; Cavalcanti Mata; Two-term; Two-term Exponential; Modified Henderson & Pabis; Logarithmic; Midilli; Page and Verma. The multivariate cluster technique associated with AIC and BIC criteria during model selection is a great applicability tool to help decision-making when evaluating the drying plant leaves. The Cavalcanti Mata mathematical model was selected to represent the drying kinetics.

Keywords:
mathematical modeling; AIC and BIC; plant products; post-harvest

Model	Model description
Diffusion Approximation (Kassem, 1998Kassem, A. S. (1998). Comparative studies on thin layer drying models for wheat. In E. H. Bartali (Ed.), 13th International Congress on Agricultural Engineering. Moroco: ANAFID.)	$MR = a exp (-k t) + (1-a) exp(-k b t)$	((2)
Cavalcanti Mata (Cavalcanti Mata et al., 2006Cavalcanti Mata, M. E. R. M., Almeida, F. A. C., & Duarte, M. E. M. (2006). Secagem de sementes. In: F. A. C. Almeida, M. E. M. Duarte, & M. E. R. M Cavalcanti Mata. (Ed.), Tecnologia de armazenamento em sementes (pp. 271-370). Campina Grande: UFCG.)	${MR = a}_{1} exp ({-k}_{1} {[t]}^{n_{1}}) {+a}_{2} exp ({-k}_{1} {[t]}^{n_{2}}) {+a}_{3}$	((3)
Two-term (Henderson, 1974Henderson, S. M. (1974). Progress in developing the thin layer drying equation. Transactions of the ASAE. American Society of Agricultural Engineers, 17(6), 1167-1168. http://dx.doi.org/10.13031/2013.37052. http://dx.doi.org/10.13031/2013.37052... )	$MR = a exp ({-k}_{0} t) {+b exp(-k}_{1} t)$	((4)
Two-term Exponential (Sharaf-Eldeen et al., 1980Sharaf-Eldeen, Y. I., Blaisdell, J. L., & Hamdy, M. Y. (1980). A model for ear corn drying. Transactions of the ASAE. American Society of Agricultural Engineers, 23(5), 1261-1265. http://dx.doi.org/10.13031/2013.34757. http://dx.doi.org/10.13031/2013.34757... )	$MR = a exp (-k t) + (1-a) exp(-k a t)$	((5)
Modified Henderson & Pabis (Karathanos, 1999Karathanos, V. T. (1999). Determination of water content of dried fruits by drying kinetics. Journal of Food Engineering, 39(4), 337-344. http://dx.doi.org/10.1016/S0260-8774(98)00132-0. http://dx.doi.org/10.1016/S0260-8774(98)... )	$MR = a exp ({-k}_{0} t) +b exp ({-k}_{1} t) +cexp ({-k}_{2} t)$	((6)
Henderson & Pabis (Henderson & Pabis, 1961Henderson, S. M., & Pabis, S. (1961). Grain drying theory I: temperature effect on drying coefficient. Journal of Agricultural Engineering Research, 6(3), 169-174.)	$MR = a exp (-k t)$	((7)
Logarithmic (Yagcioglu et al., 1999Yagcioglu, A., Degirmencioglu, A., & Cagatay, F. (1999). Drying characteristics of laurel leaves under different drying conditions. In Proceedings of the 7th International Congress on Agricultural Mechanization and Energy (pp. 565-569). Adana, Turkey.)	$MR = a exp (-k t) +c$	((8)
Logistic (Chandra & Singh, 1995Chandra, P. K., & Singh, R. P. (1995). Applied numerical methods for food and agricultural engineers (pp. 163-167). Boca Raton: CRC Press.)	$MR = \frac{a_{0}}{1 + a exp(k t)}$	((9)
Midilli (Midilli et al., 2002Midilli, A., Kucuk, H., & Yapar, Z. (2002). A new model for single-layer drying. Drying Technology, 20(7), 1503-1513. http://dx.doi.org/10.1081/DRT-120005864. http://dx.doi.org/10.1081/DRT-120005864... )	$MR = aexp ({-k t}^{n}) +b t$	((10)
Newton (Lewis, 1921Lewis, W. K. (1921). The rate of drying of solid materials. Journal of Industrial and Engineering Chemistry, 13(5), 427-432. http://dx.doi.org/10.1021/ie50137a021. http://dx.doi.org/10.1021/ie50137a021... )	$MR =exp (-k t)$	((11)
Page (Page, 1949Page, G. E. (1949). Factors influencing the maximum rate of air drying shelled corn in thin-layers (Msc thesis). Purdue University, West Lafayette.)	$MR =exp ({-k t}^{n})$	((12)
Modified Page (Overhults et al., 1973Overhults, D. G., White, G. M., Hamilton, H. E., & Ross, I. J. (1973). Drying soybeans with heated air. Transactions of the ASAE. American Society of Agricultural Engineers, 16(1), 112-113. http://dx.doi.org/10.13031/2013.37459. http://dx.doi.org/10.13031/2013.37459... )	${MR = exp(-k t)}^{n}$	((13)
Verma (Verma et al., 1985Verma, L. R., Bucklin, R. A., Ednan, J. B., & Wratten, F. T. (1985). Effects of drying air parameters on rice drying models. Transactions of the ASAE. American Society of Agricultural Engineers, 28(1), 296-301. http://dx.doi.org/10.13031/2013.32245. http://dx.doi.org/10.13031/2013.32245... )	$MR = a exp (-k t) + (1-a) exp(-g t)$	((14)
Wang & Sing (Wang & Singh, 1978Wang, C. Y., & Singh, R. P. (1978). A single layer drying equation for rough rice. ASAE Paper No: 78-3001, St. Joseph: ASAE.)	${MR = 1+a t+b t}^{2}$	((15)

Models	40 °C				50 °C				60 °C				70 °C
Models	R²	P(%)	SE	χ²	R²	P(%)	SE	χ²	R²	P(%)	SE	χ²	R²	P(%)	SE	χ²
Diffusion Approximation	99.40	21.43	0.000	0.001	99.86	6.18	0.000	0.000	99.96	12.47	0.000	0.000	99.77	9.45	0.000	0.000
Cavalcanti Mata	99.94	3.75	0.000	0.000	99.94	3.30	0.000	0.000	99.96	7.55	0.000	0.000	99.67	9.33	0.001	0.001
Two-term	99.49	19.58	0.000	0.001	99.86	6.16	0.000	0.000	99.71	35.18	0.000	0.000	99.38	11.74	0.001	0.001
Two-term Exponential	99.24	25.39	0.000	0.001	99.75	6.67	0.000	0.000	99.96	10.21	0.000	0.000	99.61	8.76	0.000	0.001
Modified Henderson & Pabis	99.85	6.95	0.000	0.000	99.74	6.50	0.000	0.000	98.98	53.87	0.001	0.002	99.83	10.02	0.000	0.001
Logarithmic	99.30	22.48	0.000	0.001	99.61	5.72	0.000	0.000	99.35	31.50	0.000	0.001	99.33	13.30	0.001	0.001
Midilli	99.83	5.51	0.000	0.000	99.81	8.09	0.000	0.000	99.94	10.60	0.000	0.000	99.63	8.39	0.000	0.001
Page	99.62	12.29	0.000	0.000	99.77	6.36	0.000	0.000	99.94	1.59	0.000	0.000	99.61	8.66	0.000	0.001
Verma	99.16	25.47	0.001	0.001	99.12	12.21	0.001	0.001	99.24	38.17	0.000	0.001	99.77	9.45	0.000	0.000

Models	40 °C		50 °C		60 °C		70 °C
Models	AIC	BIC	AIC	BIC	AIC	BIC	AIC	BIC
Diffusion Approximation	-84.45	-80.08	-91.53	-87.55	-53.10	-50.55	-31.32	-30.53
Cavalcanti Mata	-137.83	-130.20	-124.76	-117.79	-96.24	-91.77	-96.24	-91.77
Two-term Exponential	-75.14	-71.86	-83.27	-80.28	-47.26	-45.35	-33.60	-33.01
Logarithmic	-86.81	-82.44	-92.16	-88.17	-54.62	-52.06	-54.62	-52.06
Midilli	-116.49	-111.03	-110.05	-105.07	-87.63	-84.44	-87.63	-84.44
Page	-102.34	-99.06	-104.70	-101.71	-90.52	-88.61	-38.68	-38.09
Verma	-75.71	-72.44	-83.28	-80.29	-47.27	-45.35	-31.21	-30.62

Parameter	Temperature (°C)
Parameter	40	50	60	70
$a_{1}$	-0.2653^ Significant at 0.01 probability level by Tukey’s test.	0.2900^**	0.0643 ^ns	-0.0400^**
$k_{1}$	0.1652^**	0.2320^**	0.7973^**	1.6435^**
$n_{1}$	2.4233^**	2.2213^**	4.0458 ^ns	175.1356^**
$a_{2}$	1.2651^**	0.8690^**	0.9410^**	1.0386^**
$n_{2}$	1.4740^**	0.9038^**	1.2148^**	1.2497^**
$a_{3}$	-0.0096 ^ns	-0.1589 ^ns	-0.0020 ^ns	0.0062^**

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[1] *Corresponding author: wellytton_quequeto@hotmail.com