Acessibilidade / Reportar erro

Natural convection drying kinetics of ‘Prata’ and ‘D’água’ banana cultivars (Musa ssp) by nonlinear regression models

Cinética de secagem em convecção natural de bananas Prata e D’água (Musa ssp) por modelos de regressão não linear

Abstract

Banana is among fruits most planted in tropical countries and belongs to the fruit group most consumed in the world; however, banana needs proper conservation techniques. The aim of this study was to describe the drying kinetics of ‘Prata’ and ‘D’água’ banana cultivars at temperatures of 40 and 70°C, comparing the Henderson, STPE, Lewis, Page and Fick regression models, estimating the Absolute Drying Rate (ADR). Parameters were estimated with R and SAS Studio softwares, using for comparison and selection models of the R²aj, RSD and corrected Akaike Information Criteria. The Page and Fick models did not adjust, and the others presented good adjustment to data. The Henderson model was the most suitable to describe data of ‘Prata’ banana at both temperatures and ‘D’água’ banana at 70°C and Lewis at 40°C for this cultivar. The drying rate of ‘Prata’ banana at temperatures of 40 and 70°C were 0.00079 g of water/ min and 0.00400 g of water/min respectively and for ‘D’água’ banana, drying rates were 0.00111 g of water/min. and 0.00495 g of water/min., respectively. Using ADR, it was observed that in one third of the drying period, there was 70% of moisture loss at 70°C.

Index terms
Food preservation; Absolute Drying Rate; Estimation; Statistical models

Resumo

A bananeira está entre as frutíferas mais plantadas em países tropicais e pertence ao grupo de frutos mais consumidos do mundo; entretanto, necessitam de técnicas de conservação adequadas. Objetivou-se descrever a cinética de secagem de bananas Prata e D’água nas temperaturas de 40 e 70°C, comparando os modelos de regressão Henderson, ESTP, Lewis, Page e Fick, estimando-se a Taxa de Secagem Absoluta (TSA). Os parâmetros foram estimados com os softwares R e SAS Studio, utilizando-se para comparação e seleção de modelos R²aj, DPR e Critério de Informação de Akaike corrigido. Os modelos de Page e Fick não se ajustaram, e os demais apresentaram bom ajuste aos dados. O modelo de Henderson foi o mais indicado para descrever os dados de banana-Prata em ambas as temperaturas e banana D’água em 70°C e o de Lewis para a temperatura de 40°C desta cultivar. A taxa de secagem da banana-Prata, nas temperaturas de 40 e 70°C, foram de 0,00079 g de água/min e 0,00400 g de água/min, respectivamente, e para banana D’água foram de 0,00111 g de água/min e 0,00495 g de água/min, respectivamente. Com a TSA , observou-se que em um terço do período de secagem houve 70% da perda de umidade aos 70°C.

Termos para indexação
Conservação de alimentos; Taxa de Secagem Absoluta; Estimação; Modelos estatísticos

Introduction

Most banana cultivars originated in Asia, with some cultivars originating in Africa and Pacific Ocean islands.

They are typically tropical plants that require constant heat and high humidity for their natural development (BORGES et al., 2006 BORGES, A.L.; OLIVEIRA, A.M.G.; RITZINGER, C.H.S.P.; ALMEIDA, C.O. de; COELHO, E.F.; SANTOS-SEREJO, J.A. dos; SOUZA, L .da S.; LIMA, M.B.; FANCELLI, M.; FOLEGATTI, M.I. da S.; MEISSNER FILHO, P.E.; SILVA, S. de O.; MEDINA, V.M.; CORDEIRO, Z.J.M. A cultura da banana. 3.ed. Brasília, DF: Embrapa Informação Tecnológica, 2006. ). According to Salomão et al. (2016) SALOMÃO, C.; SIQUEIRA, D. L. de; LINS, L. D. L. de; CECON, P.R. Crescimento e produção da bananeira (Musa spp.AAB) ‘Prata-Anã ´, oriunda de rizoma e micropropagada. Revista Ceres, Viçosa, MG, v.63, n.3, p.340–347, 2016. , banana (Musa spp) is among the fruit trees most planted in tropical countries and in Brazil, its cultivation generates significant number of jobs, representing an important source of income for producers, positively affecting the country’s economy.

The countries with the largest banana production in the order are India, China, Brazil and Indonesia. The annual Brazilian banana production is over 6.7 million tonnes in an area of approximately 470 thousand hectares, destined for domestic supply, considering the high per capita consumption of 60 kg / person / year (FAOSTAT, 2018 FAOSTAT. Crops and trade: crops and livestok products. Rome: FAO, 2018. Disponível em: http://www.fao.org/economic/est/est-commodities/bananas/en/. Acesso em: 27 Nov. 2018.
http://www.fao.org/economic/est/est-comm...
). Brazil also exports its excess production of 41 thousand tons mainly to Argentina and Uruguay. The Brazilian states with the highest production are in the order SP, BA, SC and MG, with emphasis on ‘Nanica’, ‘Prata’ and ‘Maçã’ cultivars, among others (IBGE, 2018 IBGE - Instituto Brasileiro de Geografia e Estatística. Produção agrícola aunicipal 1974 – 2017. Rio de Janeiro, 2018. Disponível em: https://sidra.ibge.gov.br/tabela/1613#resultado. Acesso em: 11 dez. 2018.
https://sidra.ibge.gov.br/tabela/1613#re...
).

Bananas are among the fruits most consumed in the world, whether processed, fried, cooked or even fresh (NOMURA et al., 2013 NOMURA, E.S.; DAMATTO JUNIOR, E.R., FUZITANI, E.J., AMORIM, E.P., SILVA, S. de O. Avaliação agronômica de genótipos de bananeiras em condições subtropicais, Vale do Ribeira, São Paulo - Brasil. Revista Brasileira de Fruticultura, Jaboticabal, v.35, n.1, p.112–122, 2013. ), which has great food importance because it is a fruit rich in carbohydrates, calcium, magnesium, potassium, vitamin A and other minerals (BORGES et al., 2006 BORGES, A.L.; OLIVEIRA, A.M.G.; RITZINGER, C.H.S.P.; ALMEIDA, C.O. de; COELHO, E.F.; SANTOS-SEREJO, J.A. dos; SOUZA, L .da S.; LIMA, M.B.; FANCELLI, M.; FOLEGATTI, M.I. da S.; MEISSNER FILHO, P.E.; SILVA, S. de O.; MEDINA, V.M.; CORDEIRO, Z.J.M. A cultura da banana. 3.ed. Brasília, DF: Embrapa Informação Tecnológica, 2006. ; TBCA, 2018 TABELA BRASILEIRA DE COMPOSIÇÃO DE ALIMENTOS (TBCA). Universidade de São Paulo (USP). Food Research Center (FoRC). Versão 6.0. São Paulo, 2017. Available in: http://www.fcf.usp.br/tbca/. Access on: Nov., 27, 2018.
http://www.fcf.usp.br/tbca/...
). In the fresh form, it is an extremely perishable fruit and its marketing requires adequate logistics to be performed quickly and rationally. Post-harvest fruit management requires special care to minimize losses and maintain product quality so that it reaches the consumer perfectly. Banana quality is extremely important for both export and domestic market, considering that product pricing takes this aspect into account (COSTA, AUGUSTO, REGO, 2014 COSTA, B.P.; AUGUSTO, C.; REGO, R.D.M. As várias cultivares de banana e a problemática de sua comercialização no município de Olinda Nova do Maranhão The various cultivars of banana and the problem of marketing in the municipality of Olinda Nova do Maranhão. Agropecuária Científica no Semiárido, Campina Grande, v.10, n.4, p.1–4, 2014. ; SALOMÃO et al., 2016 SALOMÃO, C.; SIQUEIRA, D. L. de; LINS, L. D. L. de; CECON, P.R. Crescimento e produção da bananeira (Musa spp.AAB) ‘Prata-Anã ´, oriunda de rizoma e micropropagada. Revista Ceres, Viçosa, MG, v.63, n.3, p.340–347, 2016. ). Therefore, the use of harvest and post-harvest banana conservation technologies are strategies to assist producers, enabling the expansion of production volume and increasing competitiveness for both domestic and external markets (SANTOS et al., 2017 SANTOS, L.O.; MARTINS, R.N.; CASTRICINI, A.; RODRIGUES, M.G.V.; DIAS, M.S.C. “Prata-Anã” banana conservation at 12°C and 14°C under controlled atmosphere. Científica, Jaboticabal, v.45, n.1, p.57–63, 2017. ).

The shelf-life of foods in general can be increased by using drying techniques (GONÇALVES et al., 2016 GONÇALVES, J.Q.; SILVA, M.A.P. da; PLÁCIDO, G.R.; CALIARI, M.; SILVA, R.M.; MOURA, L.C.; SOUZA, D.G. Secagem da casca e polpa da banana verde (Musa acuminata): Propriedades físicas e funcionais da farinha. Global Science and Technology, Rio Verde, v.9, n.3, p.62–72, 2016. ).

Drying is a practice with several advantages such as low cost and simplicity of use, which allow transforming foods with high water content into dehydrated foods while preserving their main physical and chemical characteristics. The drying process involves the removal of water or other liquids from the food. Among its benefits, reduction of post-harvest loss stands out (CELESTINO, 2010 CELESTINO, S. M. C. Princípios de secagem de alimentos. Planaltina: Embrapa Cerrados, 2010. p.51. (Documentos, 276) ).

Dried banana or raisin banana has high sugar content, being considered a product with high food value and of easy absorption. Among the various forms of consumption, banana raisin consumed pure or used as an ingredient for cakes are preferred by consumers.

Acceptance is mainly due to taste, which is much appreciated. However, dry fruit consumption is still small due to the lack of quality control of production processes (MOTA, 2005). In this sense, it is necessary to know and explore drying techniques to improve the quality of the product and consequently promote its consumption.

The study of fruit drying curves can be performed based on the construction of mathematical models that relate moisture content with time. According to Furtado et al. (2019) FURTADO, T. D. R.; MUNIZ, J. A.; SILVA, E. M.; FERNANDES, J. G. Kinetics of drying of jabuticaba pulp by regression models. Revista Brasileira de Fruticultura, Jaboticabal, v.41, n.1, 2019. Disponível em: http://www.scielo.br/scielo.php?script=sci_arttextepid=S0100-29452019000100903.
http://www.scielo.br/scielo.php?script=s...
, drying curves are sigmoid and can be described by nonlinear regression models. The authors compared nonlinear Henderson, Lewis, Fick and Thompson models in describing the drying of jabuticaba pulp and concluded that the Lewis model is the most adequate to describe data. Silva et al. (2017) SILVA, D. C.; LIMA, M. F.; VÉLEZ, H. A. V.; SANTANA, A. A. Study of modeling cupuaçu pulp drying kinetic in biopolymers production. Revista Brasileira de Iniciação Científica, Itapetininga, v.4, n.3, p. 50-572017. described the cupuassu pulp drying kinetics behavior in biopolymer production using different models that characterize a decreasing growth curve. Araujo et al. (2017) ARAUJO, W.D.; GONELI, A.L.D.; CORRÊA, P.C.; HARTMANN FILHO, C.P.; MARTINS, E.A.S. Modelagem matemática da secagem dos frutos de amendoim em camada delgada. Revista Ciencia Agronomica, Fortaleza, v.48, n.3, p.448–457, 2017. evaluated nonlinear models for the adjustment to drying data of peanuts submitted to different temperatures. In general, nonlinear regression models describe well the drying kinetics processes and obtain parameters with biological interpretation.

The aim of this study was to evaluate the adjustment quality of nonlinear Page, Lewis, Henderson, Simple Three-Parameter Exponential and Fick models in the description of the drying kinetics of ‘Prata’ and ‘D’água’ banana cultivars at different temperatures and to estimate the absolute pulp drying rate for the most suitable model.

Material and methods

Data used in this study were obtained from Borges et al. (2011) BORGES, S.V.; MANCINI, M.C.; CORRÊA, J.L.G.; LEITE, J.B. Drying kinetics of bananas by natural convection: influence of temperature, shape, blanching and cultivar. Ciência e Agrotecnologia, Lavras, v.35, n.2, p.368-376, 2011. . ‘Prata’ and ‘D’água’ bananas used in the research were purchased at the local market of Lavras-MG and selected considering the degree of uniform ripeness.

The raw material was cleaned, peeled and cut into disc shape (0.005 cm in thickness and average fruit diameter of 0.035 ± 0.003 m). The material was submitted to the bleaching process and drying through natural convection at temperatures equal to 40ºC and 70ºC. Evaluations were performed in triplicate in the same sample composed of the homogenized material. The moisture ratio was observed in the first 8 hours, thereafter every 3 hours and 12 minutes until approximately 24 hours, totaling 14 measurements over time, considering that at time 0, the moisture ratio is equal to 1 for both temperatures and cultivars.

Nonlinear Lewis (L), Page (P), Henderson (H), Simple Three-Parameter Exponential (STPE) and Fick (F) regression models were adjusted to the set of data, described by respective equations:

1 M R i = exp - k 1 t i + u i

2 M R i = exp - k 1 t i n + u i

3 M R i = k 0 exp - k 1 t i + u i

4 M R i = k 0 exp - k 1 t i + k 2 + u i

5 M R i = 6 π 2 n = 1 i 1 n 2 e x p ( - n 2 π 2 D e f 1 t i ) + u i

In the expressions of models ui = f1ui-1 +...+ fpui-p +ei with i = 1,2,...,14, being ui the adjustment residue at the i-th time; fi the autoregressive parameter of order i; ui -1 the time adjustment residue immediately prior to the i-th measurement; fp the autoregressive parameter of order p; ui -1 the adjustment residue in times prior to the i-th measurement;ei the white noise residue with normal distribution . In the case of independent residues, parameters fi will be null, and thus, ui = e1 (GUEDES et al., 2004 GUEDES, M.H.P.;MUNIZ, J.A.; PEREZ, J.R.O.; SILVA, F.F.; AQUINO, L.H.de; SANTOS, C.L. dos. Estudo das curvas de crescimento de cordeiros das raças santa Inês e bergamácia considerando heterogeneidade de variâncias. Ciência e Agrotecnologia, Lavras, v.28, n.2, p.381-388, 2004. ; MAZZINI et al., 2003 MAZZINI, A.R.de A.; MUNIZ, J.A.; AQUINO, L.H.de; SILVA, F.F. Análise da curva de crescimento de machos Hereford. Ciência e Agrotecnologia, Lavras, v.27, n.5, p.1105-1112, 2003. ; MUNIZ et al., 2017 MUNIZ, J.A.; NASCIMENTO, M da S.; FERNANDES, T.J. Nonlinear models for description of cacao fruit growth with assumption violations. Revista Caatinga, Mossoró, v.30, n.1, p.250-257, 2017. ; PRADO; SAVIAN; MUNIZ, 2013 PRADO, T.K.L.do; SAVIAN, T.V.; MUNIZ, J.A. Ajuste dos modelos Gompertz e Logístico aos dados de crescimento de frutos de coqueiro anão verde. Ciência Rural, Santa Maria, v.43, n.5, p.803-809, 2013. ; RIBEIRO et al., 2018b RIBEIRO, T.D.; PACOPAHYBA, R.W. de M.; MORAIS, A.R. de; MUNIZ, J.A. Description of the growth of pequi fruits by nonlinear models. Revista Brasileira de Fruticultura, Jaboticabal, v.40, n.3, p.e949, 2018b. ; SOUZA et al., 2010 SOUZA, E. M. DE; MUNIZ, J. A.; MARCHI, G.; GUILHERME, L. R. G. Modelagem não-linear da extração de zinco de um lodo de esgoto. Acta Scientiarum. Technology (Impresso), v. 32, p. 193-199, 2010. ; SILVEIRA et al., 2011 SILVEIRA, F. G.DA; SILVA, F. F. E; CARNEIRO, P.L. S.; MALHADO, C. H. M.; MUNIZ, J. A.. Análise de agrupamento na seleção de modelos de regressão não-lineares para curvas de crescimento de ovinos cruzados. Ciência Rural (UFSM. Impresso), v. 41, p. 692-698, 2011. ).

In equations (1), (2), (3), (4) and (5), MRi corresponds to the dependent variable, indicating the average value of the banana pulp moisture ratio (dimensionless) in time i in days; k0 refers to the initial fruit condition, being close to 1 because it represents 100% of the initial pulp moisture; k1 refers to the drying rate, which can also be obtained by differentiation and refers to the moisture loss variation over the process time, with values between 0 and 1 because moisture loss is always smaller than the drying period; k2 corresponds to the dimensionless parameter of the model adjustment; exp is the basis of the neperian logarithm.

The parameter estimation process for the adjustment of nonlinear models generally considers minimizing the sum of squares of residues that leads to a system of normal nonlinear equations requiring the use of iterative methods. Among methods, the most used is the one by Gauss-Newton (SAVIAN; MUNIZ, 2007 SAVIAN T.V.; MUNIZ, J.A. A Study of in situ degradability: heterogeneity of variances and correlated errors. Scientia Agrícola, Piracicaba, v.64, p.548-554, 2007. ; ZEVIANI et al., 2012 ZEVIANI, W.M.; SILVA, C.A.; CARNEIRO, W.J.DE O.; MUNIZ, J.A. Modelos não lineares para a liberação de potássio de estercos animais em latossolos. Ciência Rural, Santa Maria, v.42, n.10, p.1789-1796, 2012. ; CARNEIRO et al., 2014 CARNEIRO, A.P.S.; MUNIZ, J.A.; CARNEIRO, P.L.S.; MALHADO, C.H.M.; MARTINS-FILHO, R.; SILVA, F.F. Identidade de modelos não lineares para comparar curvas de crescimento de bovinos da raça Tabapuã. Pesquisa Agropecuária Brasileira, Brasília, DF, v.49, n.1, p.57-62, 2014. ; FERNANDES; PEREIRA; MUNIZ, 2015 FERNANDES, T.J.; MUNIZ, J.A.; PEREIRA, A.A.; MUNIZ, F.R.; MUIANGA, C.A. Parameterization effects in nonlinear models to describe growth curves. Acta Scentiarum Technology, Maringá, v.37, n.4, p.397-402, 2015 ; MUIANGA et al., 2016 MUIANGA, C.A.; MUNIZ, J.A.; NASCIMENTO, M.da S.; FERNANDES, T.J.; SAVIAN, T.V. Descrição da curva de crescimento de frutos do cajueiro por modelos não lineares. Revista Brasileira de Fruticultura, Jaboticabal, v.38, n.1, p. 22-32, 2016. ; MUNIZ et al. al., 2017 MUNIZ, J.A.; NASCIMENTO, M da S.; FERNANDES, T.J. Nonlinear models for description of cacao fruit growth with assumption violations. Revista Caatinga, Mossoró, v.30, n.1, p.250-257, 2017. ; FERNANDES et al., 2017 FERNANDES. T.J.; PEREIRA, A.A., MUNIZ, J.A. Double sigmoidal models describing the growth of coffee berries. Ciência Rural, Santa Maria, v.47, n.8, e20160646, 2017. Disponível em: http://www.scielo.br/scielo.php?script=sci_arttextepid=S0103-84782017000800401. Acesso em 11 dez. 2018.
http://www.scielo.br/scielo.php?script=s...
; RIBEIRO et al., 2018 RIBEIRO, T.D.SAVIAN, T.V.; FERNANDES, T.J.; MUNIZ, J.A. The use of the nonlinear models in the growth of pears of ‘Shinseiki’ cultivar. Ciência Rural, Santa Maria, v.48, n.1, e20161097, 2018a. Disponível em: http://www.scielo.br/scielo.php?script=sci_arttextepid=S0103-84782018000100202. Acesso em: 18 jan. 2018.
http://www.scielo.br/scielo.php?script=s...
a, b RIBEIRO, T.D.; PACOPAHYBA, R.W. de M.; MORAIS, A.R. de; MUNIZ, J.A. Description of the growth of pequi fruits by nonlinear models. Revista Brasileira de Fruticultura, Jaboticabal, v.40, n.3, p.e949, 2018b. ; SOUZA et al., 2014 SOUZA, I.F. ; KUNZLE NETO, J.E. ; MUNIZ, J. A.; GUIMARÃES, R.M. ; SAVIAN, T. V.; MUNIZ, F.R.. Fitting nonlinear autoregressive models to describe coffee seed germination. Ciência Rural (UFSM. Impresso), v. 44, p. 2016-2021, 2014 ; SILVEIRA et al., 2018 SILVEIRA, S.C. ; MUNIZ, J. A.; SOUSA, F.A. ; CAMPOS, A.T. . Modelos não lineares ajustados à produção acumulada de biogás provenientes de camas sobrepostas de suínos. REVISTA AGROGEOAMBIENTAL, v. 10, p. 91-103, 2018. ). Model adjustments were made using the R software (R DEVELOPMENT CORE TEAM, 2017 R DEVELOPMENT CORE TEAM. R: a language and environment for statistical computing. Vienna: R Foundation for Statistical Computing, 2012. ), which uses the Gauss-Newton iterative method and the free-version Statistical Analysis System (SAS STUDIO, 2018 SAS - Statistical Analysis System Institute. SAS procedures guide for computers. 6th ed. Cary, 2018. v.3, 373 p. ).

To validate inferences made, based on the adjusted models, statistical tests and graphs were used to verify the assumptions of independence, normality and homoscedasticity of residues. The Durbin-Watson (DW) test (MORETTIN; TOLOI, 2006 MORETTIN, P.; TOLOI, C. Análise de séries temporais. São Paulo: Editora Edgard Blücher, 2004. 535 p. ) was used to assess the existence of residual autocorrelation, while the Shapiro- Wilk (SW) (SHAPIRO; WILK, 1965 SHAPIRO, S.S.; WILK, M.B. An analysis of variance test for normality. Biometrika, Cambridge, v.52, n.3, p.591-611, 1965. ) and Breusch-Pagan (BP) tests (BREUSCH; PAGAN, 1979 BREUSCH, T.S.; PAGAN, A.R. A Simple test for heteroscedasticity and random coefficient variation. Econometrica, Chicago, v.47, n.5, p.1287-1294, 1979. ) were applied to verify the normality and homogeneity of residues, respectively.

In order to evaluate the adjustment quality of models, the adjusted determination coefficient (R²aj) and the residual standard deviation (RSD) were used.

The higher the R2aj value, the lower the RSD, the better the model adjustment to data. The corrected Akaike Information Criterion (AIC) was used as model selection criterion (AKAIKE, 1974 AKAIKE, H. A New look at the statistical model identification. IEEE Transactions on Automatic Control, New York, v.19, n.6, p.716–723, 1974. ).

The significance of estimates obtained for the model parameters was tested by applying the Student t-test and then, 95% confidence intervals were calculated. Finally, the drying curve was studied based on the Absolute Drying Rate (ADR), calculating the first and second derivatives of the most appropriate model, according to Furtado et al. (2019) FURTADO, T. D. R.; MUNIZ, J. A.; SILVA, E. M.; FERNANDES, J. G. Kinetics of drying of jabuticaba pulp by regression models. Revista Brasileira de Fruticultura, Jaboticabal, v.41, n.1, 2019. Disponível em: http://www.scielo.br/scielo.php?script=sci_arttextepid=S0100-29452019000100903.
http://www.scielo.br/scielo.php?script=s...
.

Results and discussion

Firstly, residual analyses were presented for the model adjustments and evaluation of the adjustment quality of models with their practical interpretations, when applicable. The Page and Fick models did not converge at both temperatures and cultivars, as occurred with the Simple Three-Parameter Exponential Model (STPE) for the ‘D’água’ Banana data at 70 ° C, so the results for these models do not appear in tables and figures. Table 1 presents the Shapiro-Wilk, Breusch-Pagan and Durbin-Watson tests, with their respective significance, and Figures 1 and 2 show the residual analysis graphs for the Lewis (L), Henderson (H) and Simple Three-Parameter Exponential Model (STPE) models to the drying kinetic data of ‘Prata’ and ‘D’água’ banana cultivars at 40 ° C and 70 ° C except for STPE for ‘D’água’ cultivar.

Table 1
Values for the Shapiro-Wilk (SW), Breuch-Pagan (BP) and Durbin-Watson (DW) tests with their respective significance in the analysis of estimated residues after the adjustment of the Lewis (L), Henderson (H) and Simple Three-Parameter Exponential (STPE) models to the drying kinetic data of ‘Prata’ and ‘D'água’ banana cultivars at temperatures of 40ºC and 70°C.

Figure 1
Residual analysis for the drying kinetics of ‘Prata’ and ‘D’agua’ bananas at 40°C, where (a), (b), (c), (d), (e) and (f) represent the adjusted values for residues, and in (g), (h), (i), (j), (k) and (l), the residual values relative to theoretical quantiles for the Henderson (H), Lewis (L) and Simple Three-Parameter Exponential (STPE) models.

Figure 2
Residual analysis for the drying kinetics of ‘Prata’ and ‘D’agua’ bananas at 70°C, where (a), (b), (c), (d), (e) and (f) represent the adjusted values for the residues, and in (g), (h), (i), (j), (k) and (l), the residual values relative to theoretical quantiles for the Henderson (H), Lewis (L) and Simple Three-Parameter Exponential (STPE) models.

The Durbin-Watson test was generally significant, indicating residual autocorrelation at both temperatures for ‘Prata’ and ‘D’água’ cultivars, i.e., there was residual dependence over time. This result was expected, as measurements were performed on the same ‘D’água’ and ‘Prata’ banana material, respectively, over time. In fruit growth studies, similar results were found by PRADO; SAVIAN; MUNIZ, (2013) PRADO, T.K.L.do; SAVIAN, T.V.; MUNIZ, J.A. Ajuste dos modelos Gompertz e Logístico aos dados de crescimento de frutos de coqueiro anão verde. Ciência Rural, Santa Maria, v.43, n.5, p.803-809, 2013. , Muniz et al. (2017) MUNIZ, J.A.; NASCIMENTO, M da S.; FERNANDES, T.J. Nonlinear models for description of cacao fruit growth with assumption violations. Revista Caatinga, Mossoró, v.30, n.1, p.250-257, 2017. , Muianga et al. (2016) MUIANGA, C.A.; MUNIZ, J.A.; NASCIMENTO, M.da S.; FERNANDES, T.J.; SAVIAN, T.V. Descrição da curva de crescimento de frutos do cajueiro por modelos não lineares. Revista Brasileira de Fruticultura, Jaboticabal, v.38, n.1, p. 22-32, 2016. and Ribeiro et al. (2018a) RIBEIRO, T.D.SAVIAN, T.V.; FERNANDES, T.J.; MUNIZ, J.A. The use of the nonlinear models in the growth of pears of ‘Shinseiki’ cultivar. Ciência Rural, Santa Maria, v.48, n.1, e20161097, 2018a. Disponível em: http://www.scielo.br/scielo.php?script=sci_arttextepid=S0103-84782018000100202. Acesso em: 18 jan. 2018.
http://www.scielo.br/scielo.php?script=s...
, who observed the presence of autocorrelated errors in the adjustment of nonlinear regression models to describe the growth of coconut, cacao, cashew and pear fruits, respectively.

According to these authors, whenever residual dependence is observed, this correlation should be modeled by an autoregressive process. Different results were found by Ribeiro et al. (2018b) RIBEIRO, T.D.; PACOPAHYBA, R.W. de M.; MORAIS, A.R. de; MUNIZ, J.A. Description of the growth of pequi fruits by nonlinear models. Revista Brasileira de Fruticultura, Jaboticabal, v.40, n.3, p.e949, 2018b. with the growth of pequi fruits, as residual dependence was not observed because biometric measurements were made on different fruits at each age.

After the study on residual dependence, considering the autoregressive errors of orders 1 and / or 2, it was verified that the Shapiro-Wilk and Breusch-Pagan tests were not significant in both temperatures and cultivars, and it could be inferred that the residual values are usually distributed and with homogeneous variance. Similarly, Figures 1 and 2 show that there is no pattern in figures, corroborating results of Table 1. The verification of assumptions for the regression models is important for the validation of estimates obtained, since if not met, the model is said to be unsuitable.

Estimates with 95% confidence intervals for parameters of the Henderson, Lewis and Simple Three- Parameter Exponential models at temperatures of 40 and 70 ° C for ‘Prata’ and ‘D’água’ cultivars are presented in Table 2. Based on the estimates of parameter k0 presented in Table 2, it was observed that the values obtained were close to 1 at both temperatures, as expected, as they refer to 100% of the initial moisture in the banana material. The STPE model at temperature of 40 ° C for ‘Prata’ cultivar overestimated the expected k0 value and still presented wide confidence interval, indicating that the estimate is not reliable. Leite et al. (2015) LEITE, A.L.M.P.; SILVA, F.S. da; PORTO, A.G.; PIASSON, D.; SANTOS, P. dos. Contração volumétrica e cinética de secagem de fatias de banana variedade Terra. Pesquisa Agropecuária Tropical , Goiânia, v.45, n.2, p.155-162, 2015. studied the drying kinetics of slices of ‘Terra’ banana cultivar and found values close to 1 for the initial moisture content, which in their study, they named “a” for the Midilli model.

Table 2
Estimates of parameters with their respective confidence intervals (LI and LS), Lewis (L), Henderson (H) and Simple Three-parameter Exponential (STPE) to the drying kinetics data of ‘Prata’ and ‘D'água’ banana cultivars at temperatures of 40ºC and 70°C, considering autoregressive errors of orders 1 and 2.

Estimates for parameter k1 were between 0 and 1,as expected, as they represent the product’s drying rate, which is given by the ratio between the moisture variation and the drying time variation. Considering the values observed in Table 2, the drying rates at temperatures 40 ° C and 70 ° C for ‘Prata’ cultivar averaged 0.00079 g water / min and 0.00400 g water / min respectively, and 0.00111 g water/min and 0.00495 g water/min for ‘D’água’ cultivar. As it can be observed, with temperature increase, k1 increases, that is, the moisture loss rate over time is greater with the increase in the drying temperature, corroborating results obtained by Gouveia et al. (2003) GOUVEIA, J. P.G.; ALMEIDA, F. A. C.; FARIAS, E. S.; SILVA, M. M.; CHAVES, M. C. V.; REIS, L. S. Determinação das curvas de secagem em frutos de cajá. Revista Brasileira de Produtos Agroindustriais, Campina Grande, n.1, p.65-68, 2003. Volume especial with the drying curve of cajá fruits. Temperature is one of the most influential factors in the drying process, and the higher the temperature, the shorter the drying time and, consequently, the faster the equilibrium moisture content (Ue) of the product is reached. Similar results were found by Borges et al. (2008) BORGES, S.V.; MANCINI, M.C.; CORRÊA, J.L.G.; NASCIMENTO, D.A. Secagem de fatias de abóboras (Cucurbita moschata, L.) por convecção natural e forçada. Ciência e Tecnologia de Alimentos, Campinas, v.28, p.245-251, 2008. with the drying kinetics of pumpkin slices and Leite et al. (2015) LEITE, A.L.M.P.; SILVA, F.S. da; PORTO, A.G.; PIASSON, D.; SANTOS, P. dos. Contração volumétrica e cinética de secagem de fatias de banana variedade Terra. Pesquisa Agropecuária Tropical , Goiânia, v.45, n.2, p.155-162, 2015. with the drying of ‘Terra’ banana cultivar at temperatures of 40 to 60 ° C.

Confidence Intervals (CI) for parameter k2 of the STPE model in both temperatures and banana cultivars involved the zero value, so this model was considered inadequate to describe data. Parameters k0 and k1 presented small amplitude CI with only positive values in all temperatures and cultivars, which according to Ribeiro et al. (2018a) RIBEIRO, T.D.SAVIAN, T.V.; FERNANDES, T.J.; MUNIZ, J.A. The use of the nonlinear models in the growth of pears of ‘Shinseiki’ cultivar. Ciência Rural, Santa Maria, v.48, n.1, e20161097, 2018a. Disponível em: http://www.scielo.br/scielo.php?script=sci_arttextepid=S0103-84782018000100202. Acesso em: 18 jan. 2018.
http://www.scielo.br/scielo.php?script=s...
, very small amplitude intervals indicate adjustment quality of models to data.

According to Muniz et al. (2017) MUNIZ, J.A.; NASCIMENTO, M da S.; FERNANDES, T.J. Nonlinear models for description of cacao fruit growth with assumption violations. Revista Caatinga, Mossoró, v.30, n.1, p.250-257, 2017. and Ribeiro et al. (2018a) RIBEIRO, T.D.SAVIAN, T.V.; FERNANDES, T.J.; MUNIZ, J.A. The use of the nonlinear models in the growth of pears of ‘Shinseiki’ cultivar. Ciência Rural, Santa Maria, v.48, n.1, e20161097, 2018a. Disponível em: http://www.scielo.br/scielo.php?script=sci_arttextepid=S0103-84782018000100202. Acesso em: 18 jan. 2018.
http://www.scielo.br/scielo.php?script=s...
, the use of autoregressive errors, when there is residual dependence, has the characteristic of improving the adjustment quality of models, making the estimated values to have greater reliability with more accurate results. For ‘Prata’ and ‘D’água’ banana data at 40°C, the autoregressive components of orders 1 and 2 were considered. Regarding the values estimated for ∅1 and ∅2, values were greater than 1 and ∅2 values were within the [-1;1] range. Observing that f2 - f1 < 1 and f1 - f2 <1, there are indications that the values are in the unitary circle, which is justified by Morettin and Toloi (2006) MORETTIN, P.; TOLOI, C. Análise de séries temporais. São Paulo: Editora Edgard Blücher, 2004. 535 p. , because under these conditions, there are guarantees of stationarity AR2 data series.

Based on the above, it could be inferred that the drying kinetics data of ‘Prata’ and ‘D’água’ banana cultivars are highly correlated at temperature of 40°C.

At 70°C, data for both cultivars presented residual dependence of order 1, with confidence intervals free from zero, indicating reliability in the result presented in Table 2, with high correlation. In general, the results indicated a high positive correlation among residues at both temperatures and cultivars, which corresponds to the strong correlation of observations over time, confirming the need to be considered when adjusting models to data.

Table 3 presents the adjustment quality evaluators R2aj, RSD, and the AIC selection criterion for models that adjusted to the drying kinetics data of ‘Prata’ and ‘D’água’ banana cultivar at temperatures of 40 and 70 °C. The estimated values for the adjusted determination coefficient at temperatures of 40 and 70 ° C were above 0.97 for Henderson models in both cultivars, and those of RSD were low, in the order of 10-2, indicating satisfactory adjustments; however, the Lewis models presented low R2aj values and high RSD values, suggesting lack of adjustment, except at 40 ° C for ‘D’água’ cultivar. Leite et al. (2015) LEITE, A.L.M.P.; SILVA, F.S. da; PORTO, A.G.; PIASSON, D.; SANTOS, P. dos. Contração volumétrica e cinética de secagem de fatias de banana variedade Terra. Pesquisa Agropecuária Tropical , Goiânia, v.45, n.2, p.155-162, 2015. , in studies with the drying kinetic of ‘Terra’ banana observed R2 values above 0.97 and standard error of 10-2. Gonçalves et al. (2016) GONÇALVES, J.Q.; SILVA, M.A.P. da; PLÁCIDO, G.R.; CALIARI, M.; SILVA, R.M.; MOURA, L.C.; SOUZA, D.G. Secagem da casca e polpa da banana verde (Musa acuminata): Propriedades físicas e funcionais da farinha. Global Science and Technology, Rio Verde, v.9, n.3, p.62–72, 2016. when studying the drying of green banana pulp found R² values above 0.95 for the Lewis, Page, Henderson and Midilli models. Sari et al. (2019) SARI, B. G.; LÚCIO, A. D.; SANTANA, C. S.; SAVIAN, T. V.Describing tomato plant production using growth models. Scientia Horticulturae, Amsterdam, v.246, p.146–154, 2019. , in studies with tomato growth, found R2 values above 0.96 for nonlinear regression models, and considered adjustment quality.

Table 3
Adjustment quality evaluators of the Lewis (L) and Henderson (H) models to the drying kinetic data of ‘Prata’ and ‘D'água’ banana cultivars at temperatures of 40ºC and 70°C, considering autoregressive errors of orders 1 and 2.

Considering the AIC selection criteria presented in Table 3, the most suitable model to describe ‘Prata’ banana kinetic data at 40 and 70°C and ‘D’água’ banana at 70°C was the Henderson model, and the Lewis model at 40°C for ‘D’água’ cultivar, with lower criterion values. Muniz et al. (2017) MUNIZ, J.A.; NASCIMENTO, M da S.; FERNANDES, T.J. Nonlinear models for description of cacao fruit growth with assumption violations. Revista Caatinga, Mossoró, v.30, n.1, p.250-257, 2017. and Ribeiro et al. (2018a) RIBEIRO, T.D.SAVIAN, T.V.; FERNANDES, T.J.; MUNIZ, J.A. The use of the nonlinear models in the growth of pears of ‘Shinseiki’ cultivar. Ciência Rural, Santa Maria, v.48, n.1, e20161097, 2018a. Disponível em: http://www.scielo.br/scielo.php?script=sci_arttextepid=S0103-84782018000100202. Acesso em: 18 jan. 2018.
http://www.scielo.br/scielo.php?script=s...
used the AIC selection criteria to compare nonlinear regression models in the description of cocoa and pear fruit growth, respectively, and found that lower AIC values indicated the most suitable nonlinear regression model to describe data.

According to results of Table 3, the Henderson model was the most suitable for ‘Prata’ cultivar at both temperatures and at 70 ° C for ‘D’água’ cultivar, while the Lewis model was the most suitable for ‘D’água’ cultivar at 40°C. Figures 3 and 4 illustrate the drying process of cultivars at both temperatures. At 70°C, curves were generally steeper, clearly characterizing the four phases involved in the drying process (GOUVEIA et al., 2003 GOUVEIA, J. P.G.; ALMEIDA, F. A. C.; FARIAS, E. S.; SILVA, M. M.; CHAVES, M. C. V.; REIS, L. S. Determinação das curvas de secagem em frutos de cajá. Revista Brasileira de Produtos Agroindustriais, Campina Grande, n.1, p.65-68, 2003. Volume especial ). At 40°C, curves were smoother, impairing the identification of phases, suggesting that further measurements would be needed over time to better visualize stabilization at the end of the last drying phase, reaching the equilibrium moisture content.

Figure 3
Adjustment curves of the Henderson model for Moisture Ratio (MR) as a function of Time (s) to ‘Prata’ Banana drying kinetics data at 40°C and 70°C in (a) and (b) respectively.

Figure 4
Adjustment curves of the Lewis and Henderson models for the Moisture Ratio (MR) as a function of Time (s) to ‘Prata’ Banana drying kinetics data at 40°C and 70°C in (a) and (b) respectively.

The drying processes presented in Figures 3 (a) and 4 (a) show that the moisture loss for ‘Prata’ and ‘D’água’ cultivars at 40 ° C, respectively, decreases more sharply until approximately 600 minutes. From this point, fruits undergo constant and decreasing moisture loss phases until they approach equilibrium moisture stabilization at 1450 minutes. For Cano-Chauca et al. (2004) CANO-CHAUCA, M.; RAMOS, A.M.; STRINGHETA, P.C.; MARQUES, J.A.; SILVA, P.I. Curvas de secagem e avaliação da atividade de água da banana passa. Boletim CEPPA, Curitiba, v.22, n.1, p.121-132. , in studies with raising bananas at the beginning of drying processes, significant part of the moisture is easily removed, and after a certain period, there is greater internal resistance to moisture removal. The authors also consider that, adequate evaluation regarding the moisture content allows the prediction of the ideal times for the drying process.

The drying curves at 70°C for ‘Prata’ and ‘D’água’ cultivars are shown in Figures 3 (b) and 4 (b), respectively. In general, both cultivars reach equilibrium moisture content within approximately 200 minutes, following the other drying process stages until reaching equilibrium moisture content within 900 minutes, and are observed up to 1450 minutes without significant changes. It is noteworthy that the drying process for ‘Prata’ cultivar obtained lower mean moisture contents compared to those of ‘D’água’ cultivar at the end of the process, but even so, presented similar characteristics at both temperatures evaluated and even with an increase of 30°C in temperature, there was a considerable decrease in moisture content, as can be seen in Figures 3 and 4. Similar results were obtained by Madureira et al. (2011) MADUREIRA, I. A.; FIGUEIRÊDO, R. M. F.; QUEIROZ, A. J. M.; SILVA FILHO, E. D. Cinética de secagem da polpa do figo-da-índia. Revista Brasileira de Produtos Agroindustriais, Campina Grande, v.13, p.345-354, 2011. N. especial with the drying of fig pulp with the addition of modified starch, since with a 10°C increase in the drying temperature, there was a decrease of about 30% in the drying process.

Gonçalves et al. (2016) GONÇALVES, J.Q.; SILVA, M.A.P. da; PLÁCIDO, G.R.; CALIARI, M.; SILVA, R.M.; MOURA, L.C.; SOUZA, D.G. Secagem da casca e polpa da banana verde (Musa acuminata): Propriedades físicas e funcionais da farinha. Global Science and Technology, Rio Verde, v.9, n.3, p.62–72, 2016. with the drying of green banana pulp, observed that the higher the temperature, the shorter the drying time.

According to Furtado et al. (2019) FURTADO, T. D. R.; MUNIZ, J. A.; SILVA, E. M.; FERNANDES, J. G. Kinetics of drying of jabuticaba pulp by regression models. Revista Brasileira de Fruticultura, Jaboticabal, v.41, n.1, 2019. Disponível em: http://www.scielo.br/scielo.php?script=sci_arttextepid=S0100-29452019000100903.
http://www.scielo.br/scielo.php?script=s...
in the drying phases with constant moisture loss, it was observed that the increase in temperature reduces this phase, indicating that a significant amount of water is in the free form in jabuticaba pulp and the other portion interacts with soluble solids. Similar results were observed in the present study, with the drying kinetics of ‘Prata’ and ‘D’água’ banana cultivars, because by raising temperature, fruits reached the equilibrium moisture content faster, optimizing the drying process. However, chemical, physical and biological analyses are required to decide the best drying temperature.

After choosing the most suitable models, Absolute Drying Rates (ADR) were obtained, which can be seen in Figures 5 and 6. It is generally found that at 40°C for both cultivars, Figures 5 (a) and 6 (a), there was greater moisture loss in the range from 0 to 700 minutes, even if it occurred mildly, in which during this period, 55% of the drying process already occurred. After this period, moisture loss decelerated until it approached stabilization at the end of the evaluation period. In (b) of Figures 5 and 6, ADR indicated accelerated moisture loss from 0 to 410 minutes, which represents about one third of the drying period, which represented 72% of the moisture loss occurred throughout the process, as in (a), there was deceleration in moisture loss after this period until stabilization was reached. Furtado et al. (2019) FURTADO, T. D. R.; MUNIZ, J. A.; SILVA, E. M.; FERNANDES, J. G. Kinetics of drying of jabuticaba pulp by regression models. Revista Brasileira de Fruticultura, Jaboticabal, v.41, n.1, 2019. Disponível em: http://www.scielo.br/scielo.php?script=sci_arttextepid=S0100-29452019000100903.
http://www.scielo.br/scielo.php?script=s...
found similar results with the drying of jabuticaba pulp, where there is a process deceleration until stabilization after the most severe moisture loss. Leite et al. (2015) LEITE, A.L.M.P.; SILVA, F.S. da; PORTO, A.G.; PIASSON, D.; SANTOS, P. dos. Contração volumétrica e cinética de secagem de fatias de banana variedade Terra. Pesquisa Agropecuária Tropical , Goiânia, v.45, n.2, p.155-162, 2015. found that after the final stabilization reaching equilibrium moisture content, moisture loss can only be changed if the process conditions are changed, such as drying air temperature and velocity; the same was verified by Cano-Chauca et al. (2004) CANO-CHAUCA, M.; RAMOS, A.M.; STRINGHETA, P.C.; MARQUES, J.A.; SILVA, P.I. Curvas de secagem e avaliação da atividade de água da banana passa. Boletim CEPPA, Curitiba, v.22, n.1, p.121-132. when studying banana raisings.

Figure 5
Absolute drying rate for the adjustment of the Henderson's Moisture Ratio (UK) model as a function of Time (s), to ‘Prata’ banana drying kinetics at 40 ° C (a) and 70 ° C (b).

Figure 6
Absolute drying rate referring to the adjustment of the Henderson model for the Moisture Ratio (MR) as a function of Time (s) to ‘Prata’ Banana drying kinetics data at 40°C and 70°C in (a) and (b) respectively.

Conclusions

The Lewis, Henderson and Simple Three-Parameter Exponential models adjusted adequately to ‘Prata’ and ‘D’água’ banana drying data at temperatures of 40 and 70°C, while the Fick and Page models did not. The most suitable model to describe ‘Prata’ banana data was the Henderson model, considering first and second order autocorrelation (∅1 and ∅2) among residues at 40°C and first order at 70°C. For ‘D’água’ cultivar, the best models were Lewis considering first and second order autocorrelation (∅1 and ∅2) among residues at 40°C and Henderson considering first order autocorrelation at 70°C.

The drying rates of ‘Prata’ bananas at temperatures of 40°C and 70°C were 0.00079 g of water/min and 0.00400 g of water/min respectively and for ‘D’água’ bananas, drying rates were 0.00111 g of water/min and 0.00495 g of water/min, respectively. With the Absolute Drying Rate, it was observed that in one third of the drying period, there was 70% of moisture loss at 70°C.

Acknowledgments

To the Federal University of Lavras (UFLA) for the development of the research and to CAPES and CNPq for the scholarship.

  • AKAIKE, H. A New look at the statistical model identification. IEEE Transactions on Automatic Control, New York, v.19, n.6, p.716–723, 1974.
  • ARAUJO, W.D.; GONELI, A.L.D.; CORRÊA, P.C.; HARTMANN FILHO, C.P.; MARTINS, E.A.S. Modelagem matemática da secagem dos frutos de amendoim em camada delgada. Revista Ciencia Agronomica, Fortaleza, v.48, n.3, p.448–457, 2017.
  • BORGES, A.L.; OLIVEIRA, A.M.G.; RITZINGER, C.H.S.P.; ALMEIDA, C.O. de; COELHO, E.F.; SANTOS-SEREJO, J.A. dos; SOUZA, L .da S.; LIMA, M.B.; FANCELLI, M.; FOLEGATTI, M.I. da S.; MEISSNER FILHO, P.E.; SILVA, S. de O.; MEDINA, V.M.; CORDEIRO, Z.J.M. A cultura da banana. 3.ed. Brasília, DF: Embrapa Informação Tecnológica, 2006.
  • BORGES, S.V.; MANCINI, M.C.; CORRÊA, J.L.G.; LEITE, J.B. Drying kinetics of bananas by natural convection: influence of temperature, shape, blanching and cultivar. Ciência e Agrotecnologia, Lavras, v.35, n.2, p.368-376, 2011.
  • BORGES, S.V.; MANCINI, M.C.; CORRÊA, J.L.G.; NASCIMENTO, D.A. Secagem de fatias de abóboras (Cucurbita moschata, L.) por convecção natural e forçada. Ciência e Tecnologia de Alimentos, Campinas, v.28, p.245-251, 2008.
  • BREUSCH, T.S.; PAGAN, A.R. A Simple test for heteroscedasticity and random coefficient variation. Econometrica, Chicago, v.47, n.5, p.1287-1294, 1979.
  • CANO-CHAUCA, M.; RAMOS, A.M.; STRINGHETA, P.C.; MARQUES, J.A.; SILVA, P.I. Curvas de secagem e avaliação da atividade de água da banana passa. Boletim CEPPA, Curitiba, v.22, n.1, p.121-132.
  • CARNEIRO, A.P.S.; MUNIZ, J.A.; CARNEIRO, P.L.S.; MALHADO, C.H.M.; MARTINS-FILHO, R.; SILVA, F.F. Identidade de modelos não lineares para comparar curvas de crescimento de bovinos da raça Tabapuã. Pesquisa Agropecuária Brasileira, Brasília, DF, v.49, n.1, p.57-62, 2014.
  • CELESTINO, S. M. C. Princípios de secagem de alimentos. Planaltina: Embrapa Cerrados, 2010. p.51. (Documentos, 276)
  • COSTA, B.P.; AUGUSTO, C.; REGO, R.D.M. As várias cultivares de banana e a problemática de sua comercialização no município de Olinda Nova do Maranhão The various cultivars of banana and the problem of marketing in the municipality of Olinda Nova do Maranhão. Agropecuária Científica no Semiárido, Campina Grande, v.10, n.4, p.1–4, 2014.
  • FAOSTAT. Crops and trade: crops and livestok products. Rome: FAO, 2018. Disponível em: http://www.fao.org/economic/est/est-commodities/bananas/en/ Acesso em: 27 Nov. 2018.
    » http://www.fao.org/economic/est/est-commodities/bananas/en/
  • FERNANDES, T.J.; MUNIZ, J.A.; PEREIRA, A.A.; MUNIZ, F.R.; MUIANGA, C.A. Parameterization effects in nonlinear models to describe growth curves. Acta Scentiarum Technology, Maringá, v.37, n.4, p.397-402, 2015
  • FERNANDES. T.J.; PEREIRA, A.A., MUNIZ, J.A. Double sigmoidal models describing the growth of coffee berries. Ciência Rural, Santa Maria, v.47, n.8, e20160646, 2017. Disponível em: http://www.scielo.br/scielo.php?script=sci_arttextepid=S0103-84782017000800401. Acesso em 11 dez. 2018.
    » http://www.scielo.br/scielo.php?script=sci_arttextepid=S0103-84782017000800401. Acesso em 11 dez. 2018.
  • FURTADO, T. D. R.; MUNIZ, J. A.; SILVA, E. M.; FERNANDES, J. G. Kinetics of drying of jabuticaba pulp by regression models. Revista Brasileira de Fruticultura, Jaboticabal, v.41, n.1, 2019. Disponível em: http://www.scielo.br/scielo.php?script=sci_arttextepid=S0100-29452019000100903.
    » http://www.scielo.br/scielo.php?script=sci_arttextepid=S0100-29452019000100903.
  • GONÇALVES, J.Q.; SILVA, M.A.P. da; PLÁCIDO, G.R.; CALIARI, M.; SILVA, R.M.; MOURA, L.C.; SOUZA, D.G. Secagem da casca e polpa da banana verde (Musa acuminata): Propriedades físicas e funcionais da farinha. Global Science and Technology, Rio Verde, v.9, n.3, p.62–72, 2016.
  • GOUVEIA, J. P.G.; ALMEIDA, F. A. C.; FARIAS, E. S.; SILVA, M. M.; CHAVES, M. C. V.; REIS, L. S. Determinação das curvas de secagem em frutos de cajá. Revista Brasileira de Produtos Agroindustriais, Campina Grande, n.1, p.65-68, 2003. Volume especial
  • GUEDES, M.H.P.;MUNIZ, J.A.; PEREZ, J.R.O.; SILVA, F.F.; AQUINO, L.H.de; SANTOS, C.L. dos. Estudo das curvas de crescimento de cordeiros das raças santa Inês e bergamácia considerando heterogeneidade de variâncias. Ciência e Agrotecnologia, Lavras, v.28, n.2, p.381-388, 2004.
  • IBGE - Instituto Brasileiro de Geografia e Estatística. Produção agrícola aunicipal 1974 – 2017. Rio de Janeiro, 2018. Disponível em: https://sidra.ibge.gov.br/tabela/1613#resultado Acesso em: 11 dez. 2018.
    » https://sidra.ibge.gov.br/tabela/1613#resultado
  • LEITE, A.L.M.P.; SILVA, F.S. da; PORTO, A.G.; PIASSON, D.; SANTOS, P. dos. Contração volumétrica e cinética de secagem de fatias de banana variedade Terra. Pesquisa Agropecuária Tropical , Goiânia, v.45, n.2, p.155-162, 2015.
  • MADUREIRA, I. A.; FIGUEIRÊDO, R. M. F.; QUEIROZ, A. J. M.; SILVA FILHO, E. D. Cinética de secagem da polpa do figo-da-índia. Revista Brasileira de Produtos Agroindustriais, Campina Grande, v.13, p.345-354, 2011. N. especial
  • MAZZINI, A.R.de A.; MUNIZ, J.A.; AQUINO, L.H.de; SILVA, F.F. Análise da curva de crescimento de machos Hereford. Ciência e Agrotecnologia, Lavras, v.27, n.5, p.1105-1112, 2003.
  • MORETTIN, P.; TOLOI, C. Análise de séries temporais. São Paulo: Editora Edgard Blücher, 2004. 535 p.
  • MOTA, R.V. da. Avaliação da qualidade de banana passa elaborada a partir de seis cultivares. Ciência e Tecnologia de Alimentos, Campinas, v.25, n.3, p.560–563, 2005.
  • MUIANGA, C.A.; MUNIZ, J.A.; NASCIMENTO, M.da S.; FERNANDES, T.J.; SAVIAN, T.V. Descrição da curva de crescimento de frutos do cajueiro por modelos não lineares. Revista Brasileira de Fruticultura, Jaboticabal, v.38, n.1, p. 22-32, 2016.
  • MUNIZ, J.A.; NASCIMENTO, M da S.; FERNANDES, T.J. Nonlinear models for description of cacao fruit growth with assumption violations. Revista Caatinga, Mossoró, v.30, n.1, p.250-257, 2017.
  • NOMURA, E.S.; DAMATTO JUNIOR, E.R., FUZITANI, E.J., AMORIM, E.P., SILVA, S. de O. Avaliação agronômica de genótipos de bananeiras em condições subtropicais, Vale do Ribeira, São Paulo - Brasil. Revista Brasileira de Fruticultura, Jaboticabal, v.35, n.1, p.112–122, 2013.
  • PRADO, T.K.L.do; SAVIAN, T.V.; MUNIZ, J.A. Ajuste dos modelos Gompertz e Logístico aos dados de crescimento de frutos de coqueiro anão verde. Ciência Rural, Santa Maria, v.43, n.5, p.803-809, 2013.
  • R DEVELOPMENT CORE TEAM. R: a language and environment for statistical computing. Vienna: R Foundation for Statistical Computing, 2012.
  • RIBEIRO, T.D.; PACOPAHYBA, R.W. de M.; MORAIS, A.R. de; MUNIZ, J.A. Description of the growth of pequi fruits by nonlinear models. Revista Brasileira de Fruticultura, Jaboticabal, v.40, n.3, p.e949, 2018b.
  • RIBEIRO, T.D.SAVIAN, T.V.; FERNANDES, T.J.; MUNIZ, J.A. The use of the nonlinear models in the growth of pears of ‘Shinseiki’ cultivar. Ciência Rural, Santa Maria, v.48, n.1, e20161097, 2018a. Disponível em: http://www.scielo.br/scielo.php?script=sci_arttextepid=S0103-84782018000100202. Acesso em: 18 jan. 2018.
    » http://www.scielo.br/scielo.php?script=sci_arttextepid=S0103-84782018000100202. Acesso em: 18 jan. 2018.
  • SALOMÃO, C.; SIQUEIRA, D. L. de; LINS, L. D. L. de; CECON, P.R. Crescimento e produção da bananeira (Musa spp.AAB) ‘Prata-Anã ´, oriunda de rizoma e micropropagada. Revista Ceres, Viçosa, MG, v.63, n.3, p.340–347, 2016.
  • SANTOS, L.O.; MARTINS, R.N.; CASTRICINI, A.; RODRIGUES, M.G.V.; DIAS, M.S.C. “Prata-Anã” banana conservation at 12°C and 14°C under controlled atmosphere. Científica, Jaboticabal, v.45, n.1, p.57–63, 2017.
  • SARI, B. G.; LÚCIO, A. D.; SANTANA, C. S.; SAVIAN, T. V.Describing tomato plant production using growth models. Scientia Horticulturae, Amsterdam, v.246, p.146–154, 2019.
  • SAS - Statistical Analysis System Institute. SAS procedures guide for computers. 6th ed. Cary, 2018. v.3, 373 p.
  • SAVIAN T.V.; MUNIZ, J.A. A Study of in situ degradability: heterogeneity of variances and correlated errors. Scientia Agrícola, Piracicaba, v.64, p.548-554, 2007.
  • SHAPIRO, S.S.; WILK, M.B. An analysis of variance test for normality. Biometrika, Cambridge, v.52, n.3, p.591-611, 1965.
  • SILVA, D. C.; LIMA, M. F.; VÉLEZ, H. A. V.; SANTANA, A. A. Study of modeling cupuaçu pulp drying kinetic in biopolymers production. Revista Brasileira de Iniciação Científica, Itapetininga, v.4, n.3, p. 50-572017.
  • TABELA BRASILEIRA DE COMPOSIÇÃO DE ALIMENTOS (TBCA). Universidade de São Paulo (USP). Food Research Center (FoRC). Versão 6.0. São Paulo, 2017. Available in: http://www.fcf.usp.br/tbca/ Access on: Nov., 27, 2018.
    » http://www.fcf.usp.br/tbca/
  • ZEVIANI, W.M.; SILVA, C.A.; CARNEIRO, W.J.DE O.; MUNIZ, J.A. Modelos não lineares para a liberação de potássio de estercos animais em latossolos. Ciência Rural, Santa Maria, v.42, n.10, p.1789-1796, 2012.
  • SILVEIRA, S.C. ; MUNIZ, J. A.; SOUSA, F.A. ; CAMPOS, A.T. . Modelos não lineares ajustados à produção acumulada de biogás provenientes de camas sobrepostas de suínos. REVISTA AGROGEOAMBIENTAL, v. 10, p. 91-103, 2018.
  • SILVEIRA, F. G.DA; SILVA, F. F. E; CARNEIRO, P.L. S.; MALHADO, C. H. M.; MUNIZ, J. A.. Análise de agrupamento na seleção de modelos de regressão não-lineares para curvas de crescimento de ovinos cruzados. Ciência Rural (UFSM. Impresso), v. 41, p. 692-698, 2011.
  • SOUZA, E. M. DE; MUNIZ, J. A.; MARCHI, G.; GUILHERME, L. R. G. Modelagem não-linear da extração de zinco de um lodo de esgoto. Acta Scientiarum Technology (Impresso), v. 32, p. 193-199, 2010.
  • SOUZA, I.F. ; KUNZLE NETO, J.E. ; MUNIZ, J. A.; GUIMARÃES, R.M. ; SAVIAN, T. V.; MUNIZ, F.R.. Fitting nonlinear autoregressive models to describe coffee seed germination. Ciência Rural (UFSM. Impresso), v. 44, p. 2016-2021, 2014

Publication Dates

  • Publication in this collection
    14 Nov 2019
  • Date of issue
    2019

History

  • Received
    23 Jan 2019
  • Accepted
    02 Aug 2019
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