Kinetics of drying Bactris gasipaes Kunth sub-products: comparison of mathematical models

Cinética de secado de subproductos del Bactris gasipaes Kunth: una comparación de modelos

matemáticos

Cinética secando subprodutos de Bactris gasipaes Kunth: uma comparação de modelos

matemáticos

Jorge Ruperto Velasquez-Rivera

Jesus R. Melendez

José L. Rodríguez-Sánchez

Manuel G. Roca-Argüelles

Rev. Fac. Agron. (LUZ). 2022, 39(1): e223901

ISSN 2477-9407

DOI: https://doi.org/10.47280/RevFacAgron(LUZ).v39.n1.01

Food Technology

Associate editor: Dra. Gretty Ettiene

Universidad Católica de Santiago de Guayaquil,

Facultad de Educación Técnica para el Desarrollo,

Carrera de Ingeniería Agroindustrial, C.P. 090615,

Guayaquil, Ecuador

Instituto de Farmacia y Alimentos-Universidad de La

Habana. C.P. 17100, Cuba.

Instituto de Investigaciones para la Industria

Alimenticia-Cuba, La Habana. C.P. 17100, Cuba

Received: 01-02-2020

Accepted: 10-07-2021

Published: 16-12-2021

Abstract

The industrialization of the heart of palm (palm heart), obtained

from the sprout of a palm known as pejibaye, chontaduro, or peach

palm (Bactris gasipaes Kunth), generates two main products:

the fruit and the heart of the stem. The stem produces a highly

perishable residue due to its high humidity, making drying an

alternative to increase its useful life. The main objective of this

study was to describe which of the selected mathematical models

conform to better drying kinetics in samples (by-product) of palm

heart (palm heart), according to the selected statistical criteria.

Mathematical Modeling of the by-product drying curves (the heart

of palm) was performed at two working temperatures (70 and 80

°C) and two groups, one minced and the other ground. The results

of the water content were statistically processed to nd the most

convenient model among those proposed by other researchers.

The calculation of the parameters of the different drying models

was carried out with the STATISTICA version 8.0 program, using

the non-linear estimation tool, according to the quasi-Newton

algorithm estimation method. The results show that the models

MR = exp(-k.t

) and MR = exp(-(k.t)

), called Page and modied

Page respectively, were the best t to the experimental data in all

cases. Therefore, the models named Page and modied Page best

t the innovative information and the most suitable model.

Keywords:

Kinetics of drying

Chopped

Crushed

Mathematical model

Waste

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Resumen

La industrialización del palmito (corazón de palma),

obtenido del brote de una palma conocida como pejibaye,

chontaduro o palma de melocotón (Bactris gasipaes Kunth),

genera dos productos principales: la fruta y el corazón del

tallo. El tallo genera un residuo que por su alta humedad es

muy perecedero, por lo que el secado es una alternativa para

aumentar su vida útil. El objetivo principal de este estudio

fue describir cuál de los modelos matemáticos seleccionados

se ajusta a una mejor cinética de secado en muestras

(subproducto) de palmito (corazón de palma), de acuerdo

con los criterios estadísticos seleccionados. El modelado

matemático de las curvas de secado del subproducto (palmito)

se realizó a dos temperaturas de trabajo (70 y 80 °C) y en

dos grupos, uno troceado y el otro triturado. Los resultados

del contenido de agua se procesaron estadísticamente para

encontrar el modelo más conveniente entre los propuestos

por otros investigadores. El cálculo de los parámetros de

los diferentes modelos de secado se realizó con el programa

STATISTICA versión 8.0, utilizando la herramienta de

estimación no lineal, de acuerdo con el método de estimación

del algoritmo cuasi-Newton. Los resultados muestran que los

modelos MR = exp(-k.t

) y MR = exp(- (k.t)

), denominados

Page and modied Page respectivamente, fueron los de

mejor ajuste a los datos experimentales en todos los casos.

Los modelos nombrados como Page y modied Page, son los

que mejor se ajustaron a la información experimental y al

modelo más adecuado.

Palabras clave: cinética de secado, picado, triturado,

modelo matemático, residuos

Resumo

A industrialização do palmito (coração de palma), obtido

a partir de um broto de uma palma conhecida como Pejibaye,

chontaduro ou palma de pêssego (Bactris gasipaes Kunth),

gera dois produtos principais: a fruta e o coração do caule.

O caule gera um resíduo que por sua alta umidade é muito

perecível, pelo que a secagem é uma alternativa para

aumentar sua vida útil. O objetivo principal deste estudo

foi descrever qual dos modelos matemáticos selecionados

corresponde a uma melhor cinética de secagem em amostras

(subproduto) de palmito (coração de palma), de acordo

com os critérios estatísticos selecionados. O modelamento

matemático das curvas de secagem do subproduto (palmito)

foi realizado a duas temperaturas de trabalho (70 e 80 °C) e

em dois grupos, um picado e o outro triturado. Os resultados

do teor de água foram estatisticamente processados para

encontrar o modelo mais conveniente entre os propostos

por outros pesquisadores. O cálculo dos parâmetros dos

diferentes modelos de secagem foi realizado com o programa

STATISTICA versão 8.0, utilizando a ferramenta de

estimativa não linear, de acordo com o método de estimativa

do algoritmo quase-Newton. Os resultados mostram que os

modelos MR = exp(-k.t

) e MR = exp(-(k.t)

), denominados

Page and modied Page respectivamente, foram os de

melhor ajuste aos dados experimentais em todos os casos.

Os modelos nomeados como Page and Modied Page são os

que melhor se ajustaram à informação experimental e ao

modelo mais adequado.

Palavras-chave: cinética secando, picado, moído, modelo

matemático, residuos

Introduction

The generation and bad disposition of by-products of

agricultural and agro-industrial activities cause damage to

the environment, however, these by-products can contain

compounds of high economic value that nowadays are little

used (Santacruz et al., 2020). Palmito is a domesticated

palm commercially crucial for producing fruits and hearts of

palm (Padilha et al., 2021). The industrialization of palmito

(heart-of-palm) obtained from the head of a palm known

as pejibaye, chontaduro, or peach palm (Bactris gasipaes),

is an essential crop in the Amazon (Schroth et al., 2002).

The palmito generates two main products: the fruit and the

heart of the stem, which have various applications in the

agroindustry.

In an industrial context, the use of waste materials from

specic food manufacturing processes and other derivatives

represents new alternatives for producing products with

high nutritional value and environmentally sustainable

biofuels. There are alternatives for waste materials with a

large amount of lignocellulosic content, such as sugar cane,

fruit peels, hearts of palm, which are within this group

(Melendez et al., 2021). The production and consumption

of palm hearts, especially from the Bactris gasipaes Kunth,

generate many lignocellulosic by-products (Vieira et al.,

2021). Palm sub-products can be found in ours, rich in

dietary ber (62-71 %), which can be used in other mixing

processes and ingredients in food technologies (Bolanho et

al., 2015). Other specic applications in different research

elds related to the use of bers in the microbiota of the

human intestine (Cantu-Jungles et al., 2017) have been

studied to improve health and prepare functional foods

(Waldron et al., 2003). The large amount of residues

generated by the peach palm agroindustry and its cellulose

content (34 g.100 g

−1

) (Franco et al., 2019) had motivated

different studies. In addition, ofcial data show that the

heart of palm generates approximately 478,751.39 t of waste

per year (Instituto Nacional de Preinversión, 2014), which

could be used in other applications. The residue obtained

has a high content of dietary ber; however, due to its high

humidity, it is a highly perishable product and, therefore,

studies on extending its shelf life are reasonably necessary

(Ribeiro et al., 2021), so drying is an alternative to increase

its useful life.

Drying is one of the widely used methods of grain, fruit,

and vegetable preservation. The critical aim of drying is to

reduce the moisture content and increase the lifetime of

products by limiting enzymatic and oxidative degradation.

In addition, reducing the amount of water, drying reduces

crop losses, improves the quality of dried products, and

facilitates its transportation, handling, and storage

requirements. Drying is a process comprising simultaneous

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Velasquez et al. Rev. Fac. Agron. (LUZ). 2022, 39(1): e223901

heat and mass transfer within the material and between the

surface of the material and the surrounding media (Ertekin

& Firat, 2017).

Generally, the temperatures used for drying some

agricultural products are between 40 to 60 °C. However,

some researchers have used high temperatures (70, 80, and

90 °C), and they do not report extreme changes in physicals

and chemicals characteristics (Rojas-Garbanzo et al., 2012;

Helm et al., 2014; Rajoriya et al., 2021; Ayetigbo et al., 2021).

The ours obtained from palmito by-products show high

values for the total dietary ber, 59.1 to 65.5 g 100 g

-1

, almost

entirely represented by the insoluble ber, as well as a low

proportion of calories 96.1 to 101.1 kcal or 408.2-429.5 kJ

per 100 g of product when compared to wheat bran. These

results highlight this type of our as a potential source of

ber for human nutrition, particularly as a brous ingredient

of formulated food and functional supplements (Helm et al.,

2014). Furthermore, Vieira et al. (2021) report the obtainment

of xylooligosaccharides (XOS) from xylans extracted from

these residues, which could be applied as natural antioxidants

in functional food and pharmaceutical preparations. However,

although these wastes contain substances with high ber and

protein content, which may be of interest in the food industry,

the main destinations are landlls, which represent major

environmental problems worldwide, related to the affectation

to the water bodies, production of high CO

contents, for which

the fertility of the soils should be protected as a fundamental

objective of sustainable development (Mendoza et al., 2020a).

The recovery of this waste has become one of the most critical

challenges in the food industry since it not only allows them

to be used again as raw material but also allows lower costs

and helps reduce environmental impact (Rezzadori et al.,

2012) and sometimes the composition of the soil exerted by

the presence of available minerals, which allows better crops

(Mendoza et al., 2020b).

In this scenario, research in the Agroindustrial sector

has developed unit drying processes to evaluate organic

raw materials’ behavior when experimenting with different

temperatures. Currently, mathematical models allow us

to know which models are more efcient (Omolola et al.,

2019). In this sense, the determination of the ideal time

and temperature are goals of the current industry to lower

costs and obtain adequate physical, chemical, rheological,

microbiological, and sensory parameters.

Many models have been used to describe the drying

process for different agricultural products. These models are

used to estimate the drying time of several products under

different drying conditions, and how to increase the drying

process efciency, and also to generalize drying curves for

the design and operation of dryers. Several investigators

have proposed numerous mathematical models for the thin-

layer drying of many agricultural products (Ertekin & Firat,

2017). Nowadays, mathematical simulation and Modeling of

drying curves are valuable instruments in order to improve

control systems for nal product quality under various

conditions. These approaches are usually applied to study

the factors present in the process, optimize the conditions

and working factors, and predict the drying kinetics of

products (Mokhtarian et al., 2021).

There are mathematical models such as the one proposed

in Page’s equation, which is one of the empirical equations

that allow us to describe water migration through food

drying processes (Simpson et al., 2017). Other authors have

worked with some models such as Newton, Modied Page,

Henderson & Pabis, Wang & Singh, Logarithmic, and others

in some agricultural products (Sadaka, 2020; Zhao et al.,

2014; Taghian-Dinani et al., 2014). Therefore, this research

was oriented to selecting the efcient palmito drying model,

as an alternative that allows handling, preservation, and

utilization of this by-product in the Agroindustrial and

industrial sectors. To carry out the experimental process, we

consider the quantitative methodological perspective, with

innovative design, of the quasi-experimental type and the

level of multiple treatments (Hernández et al., 2014).

Finally, the main objective of this article is to describe

which of the selected mathematical models was adjusted to

better kinetics of drying in palmito (palm hearts) samples,

according to the selected statistical criteria.

Materials and methods

Methodological approach. The research is framed

within the quantitative methodological perspective, with

experimental design of the quasi-experimental type and the

level of multiple treatments (Hernández et al., 2014).

For the drying study, palmito by-products were worked

under a factorial design 2x2. Two types of cutting (chopped

and crushed) and two drying temperatures (70 and 80 °C)

were taken into account in the experiments with three

repetitions. Each of the twelve experimental units consisted

of 1.46 kg with a total of 17.52 kg.

The experimental values of the different tests were

processed by InfoStat Group (2015), where the variance

analysis for differences was applied as statistical techniques,

comparing means through Duncan’s multi-range test at

probability levels p ≤ 0.05 for the variable Performance

drying (Percentage).

Preparation of material. The samples used for

ocessing are residues of the palmito (heart-of-palm)

obtained from the palm (Bactris gasipaes Kunth). Palm

heart by-products were purchased from FACUNDO

Agroindustrial enterprise (Babahoyo, Ecuador). The drying

process and the kinetic comparisons of the by-product

started with a treatment of the raw material in one phase.

This phase or stage of preparation considered the selected

material and subjected it to two preparation processes.

The rst condition consisted of the chopping of palmito

samples, these sub-products were chopped into rings, under

the following dimensions, and cylinders of 5 mm maximum

height, a length of 8 cm and 0.5 cm wide were formed. The

second condition consisted of the grinding of the samples,

using an electric disk mill with 8 mm diameter holes, Torrey

brand (made in Mexico).

In both cases, the by-products were weighed and grouped

into two groups as sub-products to be used in the drying

process under experimental conditions.

Drying experiment: design. The drying was

carried out in the Unitary Operations Laboratory of the

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Chemical Faculty of the University of Guayaquil with the

collaboration of the Research Institute of the Food Industry

of Cuba and the Laboratory of Bromatology of the Faculty

of Medicine Catholic University of Santiago de Guayaquil,

Ecuador.

The dried was carried out by a tunnel dryer, with an

airow of 8.4 m

-1

, constant pressure of 68.95 kPa, area

of the drying tunnel of 4 893 cm

of surface, to constant

temperatures of 70 and 80 °C, until reaching a moisture

scales of 0.1 kg.kg

-1

of solid dry. These data correspond to

the drying performance and treatment method (chopped

and crushed) of the by-products of palmito. The building

the curves of dried, the loss of mass was proceeded to

system. The results were expressed in tables of contents as

contain moisture in the dry base, expressed in kg (mass of

water/mass of solid).

Mathematical Modeling of drying curves. The

results of the water content on a dry base are statistically

processed to nd the most convenient model. In this stage,

the four mathematical models selected for the calculation

of the drying unit operation were compared; the table 1

shows the selected models.

The quotient of moisture, MR, was calculated according

to the following expression:

MR = (X - X

) / (X

- X

)

Where X is the contained of moisture in dry base to a

time t; X

contained moisture in scale in dry base, and X

contained initial moisture in dry base.

Statistical criteria for the selection of the

mathematical model. According to the method of

estimation of the Newton quasi- algorithm, the calculation

of the parameters of the different models of dried was

made by the STATISTICA 8.0 software (StatSoft Inc.,

2007) using the tool Odd number linear assessment.

Furthermore, to evaluate the adjustment of the

mathematical models with the experimental information

was used the following statistical criteria: coefcient of

correlation (r), Chi-square test reduced (χ

red

), and the

square root of the residual ones to the square (RRC).

The best adjustments were those models that presented

the higher values of r and the minor of (χ

red

) and RRC

(Yaldiz & Ertekin, 2001; Togrul & Pehlivan, 2002).

The following equations were used to calculate the

statistics mentioned before:

Table 1. Mathematical models to describe drying

kinetics.

Empirical

Expression

Name Reference

MR = exp(-k.t) Newton

O’Callaghan (1971); Liu &

Bakker-Arkema (1997); Togrul

(2006).

MR = exp(-k.t

) Page

Agrawal & Singh (1977); Zhang

& Litcheld (1991); Sozzi et al.

(2021).

MR = exp[-(k.t)

]

Modied

Page

Overhults et al. (1973); White et

al. (1981); Sharma et al. (2005).

MR = a-exp(-k.t)

Henderson

& Pabis

Westerman et al. (1973); Chh-

ninman (1984); Shi et al. (2008).

MR: Quotient of moisture.

Table 2. Performance of drying and grinding of

palmito (palm-heart) by-products.

Type of cut Drying temperature

(°C)

Performance drying

(%)

Chopped 70 9.23

(0.004)

Crushed 70 9.91

(0.04)

Chopped 80 10.14

(0.01)

Crushed 80 11.10

(0.00)

Means with a letter, a, b, c, d are not significantly

different (p ≤ 0.05).

Where MR

exp, i ;

is the experimental value of the

quotient of dampness in a certain time and MR

pre, I

; is

the corresponding value predicted by the model. N and n

are the numbers of observations made and the number of

parameters of the model, respectively.

Results and discussion

Drying performance of Palmito by-products. The

percentages of drying yields of palm-heart by-products are

shown below, with the highest value of 11.10 % for crushed

palm hearts and drying at 80 °C. A linear increase in the

yield of the dry palmito by-products was observed for the

treatment that used the highest temperature. In this

case, the process of crushed formed a mass. The kinetics

of the loss of water in the process of dehydration of some

food materials was recorded by the loss of water from its

surface and the formation of a semipermeable crust during

drying, known as surface crusts (Velásquez et al., 2017).

The table 2 shows a higher yield at a higher temperature

in the sample with the more exposed surface. This

phenomenon occurs to a lesser extent in the by-product

with less surface exposure to the drying temperature. This

kinetic effect of water preservation occurs in the kernels of

the palmito by-product of this experiment.

Effect of the temperature in the process of drying.

Changes in the moisture in a dry base of the sub-product of

palmito with time during the dried to 70 and 80 °C appear

in the gure 1 and 2, respectively.

The characteristic feature of the curves is its similarity

with the typical exponential behavior of the process of the

dried materials by way of warm airow, which is more

pronounced by the increase of the temperature (Ertekin &

Yaldiz, 2004).

The table 3 presents the values of the adjustment

parameters of the selected models.

RRC =

(MR

exp,i

pre,i)

∑

[

]

χ =

red

i=1

∑

(MR

(exp,i)

(pre,i)

(N-n)

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Figure 1. Curves of dried: chopped palmito to 70 °C

(∆) and 80 °C (•).

Table 3. Results of the statistical analysis in the Modeling of the quotient of Moisture vs. the time of dried.

Material presented- °C Model Correlation coefficient (r) x

red

RRC

Chopped: 70 °C MR = exp(-0,00882-t) 0.968 7.34E-03 0.0843

MR = exp(-0,000533-t

1,576

) 0.999 1.80E-04 0.0130

MR = exp(-(0,00834-t)

1,678

) 0.999 7.54E-05 0.0084

MR = 1,1697-exp(-0,01016-t) 0.982 2.12E-01 0.4460

Chopped: 80 °C MR = exp(-0,01224-t) 0.965 7.68E-03 0.0850

MR = exp(-0,000688-t

1,648

) 0.997 6.59E-04 0.0241

MR = exp[-(0,01205-t)

1,641

] 0.997 6.58E-04 0.0241

MR = 1,1207-exp(-0,01376-t) 0.981 2.84E-01 0.5006

Crushed: 70 °C MR = exp(-0,00728-t) 0.951 1.16E-03 0.1060

MR = exp(-0,000388-t

1,616

) 0.996 8.74E-04 0.0312

MR = exp[-(0,00708-t)

1,871

] 0.998 3.89E-04 0.0190

MR = 1,1788-exp(-0,00855-t) 0.969 3.87E-01 0.0984

Crushed: 80 °C MR = exp(-0,00930-t) 0.960 9.41E-04 0.0952

MR = exp(-0,000525-t

1,576

) 0.998 4.76E-04 0.0209

MR = exp[-(0,00889-t)

1,678

] 0.998 1.38E-04 0.0113

MR = 1,1622-exp(-0,01070-t) 0.974 7.88E-03 0.0704

Square root of the residual ones to the square.

Figure 2. Curves of dried: crushed palmito to 70 °C

(∆) and 80 °C (•).

In general, with all models proposed by the quotient of

moisture (MR), good results were reached, but the models

MR = exp(-k.t

) and MR = exp[-(k.t)

], named as Page and

modied Page respectively, those are which better t the

experimental information data, in all cases, in agreement

with the selected statistical criteria. The Page is the best

model to describe the drying kinetics in some plant raw

materials (Sozzi et al., 2021; Chouaibi et al., 2021). The

values of the correlation coefcient were considered, test

and the square root of the residual to the square taken

as statistical criteria in the selection of the most suitable

model.

A graphical representation of the quality of the

adjustment of these models appears in the gure 3 that

corresponds to the dried to 80 °C of the sub-product of

crushed palmito.

When examining the constant k in the different models

shown in table 3, it appreciates that it is dependent on the

temperature. While higher it is, his value goes increasing;

from the anomalous diffusion model and also from Fick’s

model, the correct use of an exponential equation such as

Page’s equation indicates that the equation will have a

possible phenomenological meaning when relevant data at

long drying times (Simpson et al., 2017).

Drying kinetics: Presentation of the palmito

(heart-of-palm) sample.

Treatments for the sample: chopped vs.

crushed. The table 3 shows, when examining the

values of the constants n and k, the most convenient

models are determined: Page and modied Page. In this

sense, it was veried that the previous treatment of the

by-product of the heart of palm (chopped and crushed) also

inuences the kinetics of drying because of the constant n

of the corresponding model increases or decreases when the

treatments were compared: chopped vs. crushed.

According to Karathanos & Belessiotis (1999), the increase

of n was associated with an increase in the thickness of the

dry external cape of the product, which affects the speed of

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Figure 3. Curves of dried to 80 °C experimentally

(

ººº

) and the predicted one for the Model of

Pages modified (line) for the sub-product of

crushed palmito.

dried. In this direction, a decrease of the value k was associated

when comparing the treatments chopped vs. crushed due to the

formation of the outer crust.

The value of the drying rate constant k decreases with the

increase in initial material load, indicating slower drying of the

product (Kaleta & Górnicki, 2010).

All this carries out as a consequence of a decrease in the speed

of dried, a phenomenon that was possible to estimate in gures 4

and 5.

Figure 4. Effect of the previous treatment of the sub-

products of palmito in the Kinetic of dried

to 70 °C.

Conclusions

Models named as Page and modied Page respectively,

are the ones that better t the experimental information in

all the cases, in agreement with the values of the coefcient

of correlation; the test χ

and the square root of the residual

to the square taken as statistical criteria in the selection of

the most suitable model.

Acknowledgment

A special thanks to the manager to the Unitary Operations

Laboratory of the Chemical Faculty of the University of

Guayaquil, the Research Institute of the Food Industry of Cuba,

and the Laboratory of Bromatology of the Faculty of Medicine

of the Catholic University of Santiago de Guayaquil, Ecuador,

which allowed a happy ending with this study.

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