mathematical model for prediction of moistur

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MATHEMATICAL MODEL FOR PREDICTION OFMOISTURE CONTENT IN JALAPEO PEPPER (Capsifrutescens) MODELO MATEMTICO PARA LAPREDICCIN DEL CONTENIDO DE HUMEDAD ENCHILE JALAPEO (Capsicum frutescens) MODELOMATEMTICO PARA A PREDICCIN DO CONTIDODE HUMIDADE EN CHILE JALAPEO (Capsicumfrutescens)L. A. Ochoa-Martneza , J. A. Gallegos-Infantea , J Morales-Castroa , H. Medrano-Roldna & N. E. Rocha-Guzmnaa Instituto Tecnolgico de Durango, Departamento de Ingeniera Qumica y Bioqumica. ,Blvd. Felipe Pescador 1830 Ote., Durango, Dgo. C.P., 34080, MxicoPublished online: 14 Oct 2009.

To cite this article: L. A. Ochoa-Martnez , J. A. Gallegos-Infante , J Morales-Castro , H. Medrano-Roldn & N. E.Rocha-Guzmn (2004) MATHEMATICAL MODEL FOR PREDICTION OF MOISTURE CONTENT IN JALAPEO PEPPfrutescens) MODELO MATEMTICO PARA LA PREDICCIN DEL CONTENIDO DE HUMEDAD EN CHILE JALAPE

frutescens) MODELO MATEMTICO PARA A PREDICCIN DO CONTIDO DE HUMIDADE EN CHILE JALAPEO (Cfrutescens), Ciencia y Tecnologia Alimentaria, 4:3, 154-157, DOI:10.1080/11358120409487754

To link to this article: http://dx.doi.org/10.1080/11358120409487754

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Cienc. Tecnol. Aliment. Vol. 4, No. 3, pp 154-157, 2004 www.altaga.org/cytaCopyright 2004 Asociacin de Licenciados en Ciencia y Tecnologa de los Alimentos de Galicia (ALTAGA). ISSN 1135-8122

MATHEMATICAL MODEL FOR PREDICTION OF MOISTURE CONTENTIN JALAPEO PEPPER (Capsicum f rutescens )

AbstractA model to predict the moisture content in jalapeo peppers ( Capsicum frutescens ) has been presented. The

product was dried under five drying temperatures (50, 60, 70, 80 and 95C), and these experiments provided data for model formulation. Analysis of the simulated results have indicated that there are two possible mechanisms for jalapeo

peppers drying, both of which are a function of the temperature. It was found that one mechanism occurs below 60C andthe other at temperature above 80C. 2004 Altaga. All rights reserved.

Keywords: Capsicum, drying mechanisms, model

ResumenEn este trabajo, se presenta un modelo para predecir el contenido de humedad en chile jalapeo ( Capsicum

frutescens ). El producto se deshidrat a cinco diferentes temperaturas (50, 60, 70, 80 y 95C), cuyos datos experimentalesse utilizaron en la formulacin del modelo. El anlisis de resultados, indic que existen dos posibles mecanismos en elsecado del chile jalapeo, los cuales estn en funcin de la temperatura. Se encontr que un mecanismo ocurre por debajode 60C y el otro a temperatura por encima de 80C. 2004 Altaga. Todos los derechos reservados.

Palabras clave: Capsicum, mecanismos de secado, modelo

Resumo Neste traballo,presntase un modelo para predeci-lo contido de humidade en chile jalapeo ( Capsicum frutescens ).

O producto deshidratouse a cinco diferentes temperaturas (50, 60, 70, 80 y 95C), cuios datos experimentais utilizronsena formulacin do modelo. A anlise de resultados, indicou que existen dous posibles mecanismos no secado do chile

jalapeo, os cales estn en funcin da temperatura. Encontrouse que un mecanismo ocorre por debaixo de 60C e o outroa temperatura por riba de 80C. 2004 Altaga. Tdolos dereitos reservados.

Palabras chave: Capsicum, mecanismos de secado, modelo

MODELO MATEMTICO PARA LA PREDICCIN DEL CONTENIDO DE HUMEDADEN CHILE JALAPEO (Capsicu m f r utescens )

MODELO MATEMTICO PARA A PREDICCIN DO CONTIDO DE HUMIDADEEN CHILE JALAPEO (Capsicum frutescens )

Ochoa-Martnez, L. A.; Gallegos-Infante, J. A.*; Morales-Castro, J; Medrano-Roldn, H.; Rocha-

Guzmn, N. E.

Instituto Tecnolgico de Durango, Departamento de Ingeniera Qumica y Bioqumica. Blvd. Felipe Pescador 1830 Ote.Durango, Dgo. C.P. 34080 Mxico.

*Corresponding author. e-mail: [email protected]

Recibido: 26 de Junio de 2003; recibida versin revisada: 14 de Octubre de 2003; aceptado: 16 de Octubre de 2003Received: 26 June 2003; revised version received: 14 October 2003; accepted: 16 October 2003


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INTRODUCTION

The change in product moisture during thedehydration of high moisture material like fruits andvegetables is a complex problem. The interaction betweenthe moisture in the food and its surrounding atmosphereis very important in many food processing operations like

drying, packaging and transportation (Raghavan andVenkatachalapathy, 1999). Process modeling is of greatsignificance in the analysis of design and optimization of dryers (Vagenas and Marinos-Kouris, 1992). The mostessential part of process model development involvesmoisture removal from the system. Empirical kineticmodels involve parameters of a phenomenological nature,although these parameters have no physical meaning.Empirical models can be deduced from detailedmechanistic ones under certain assumptions, or can beformulated from experiments. Various kinds of modelshave been proposed, such as physical dynamic models,

based on the diffusion mechanism, semi-empirical modelsand empirical models. The model prediction accuracy isan important issue in control, because the optimal valuemay be very sensitive to modeling errors.

Mexico is a large producer of peppers, in largevarieties. Jalapeo pepper ( Capsicum frutescens ) is animportant condiment for everyday cooking to contribute

pungency and taste to foods. There is a great need toimprove the drying, handling and storage system in order to enhance quality of processed jalapeo pepper.

The objetive of this work was to evaluate the predictability of several drying models for the moisturetransfer, and to obtain the model parameter for this

product.

MATERIALS AND METHODS

The dimensions of the whole jalapeo pepper usedin this study were 8 cm large, 3.3 cm upper width and 2.1cm lower width. The slices had 5-7 mm upper width, 2-3mm lower width and 2-3 mm thickness. The dryingexperiments were carried out in an air tray dryer at 50,60, 70, 80 and 95C. The air velocity was kept constant at2.5 m/s. The drying process last until a final moisturecontent of 6% was reached. The initial and final moisturecontent was determined by the oven method. Theequilibrium moisture content (X e) was determined bymeasuring the weight changes of jalapeo peppers at fivetemperatures until equilibrium moisture content wasattained. The relative humidity was set by the saturationof sodium chloride. Three samples of jalapeo (2 g eachone) were placed in glass bottles of 1 l with saturated saltsolution, the sealed bottle was then placed in atemperature-controlled chamber. At a certain time interval,the sample was removed for weight change determination.

RESULTS AND DISCUSSION

The experimental results of equilibrium moisturecontent are shown in Table 1.

From these data no clear relation as a function of temperature in the low range, can be identify. However,at high temperature, an equilibrium moisture lower in the

jalapeo pepper is observed; these data were then used inthe analytical solution of equation 1.

Model developmentThe empirical models chosen to describe the moisture

transfer within dried jalapeo can take the form of a generallinear ordinary differential equation, in which the right sideof the equation contains an empirical mass transfer coefficientmultiplied by the corresponding driving force.

-dX/dt = (K X) + X e (1)

where X is the material moisture content and K isthe drying constant. The equilibrium moisture content isthe ordinate (X e), and t is the time.

The analytical solution of the above model whenthe conditions remain constant is:

X = X o exp (-K t) + X e (2)

When X o is the initial material moisture = 88%The effect of other variables can be embodied into

the empirical equation of Arrhenius-type:

1/K = Ao e Ea/RT (3)

Where Ao is a parameter of the model(dimensionless), Ea is the activation energy (J/mol), R isthe gas constant (8.3 J/mol K), T is the temperature(Kelvin) and K is the reaction rate.

In the case of one-variable problems, the mostfrequently used method for estimating the parameters of a model is by minimizing the mean standard deviation:

SR

2 = eij

2/ N (4)

Where e ij is the residual created when comparingthe model calculations for the experimental point i andfor the replicate j and N is the number of experimental

points.The mean standard experimental error for all

experiments is given by the following expression:

SE2 = n i S i/N (5)

Where n i is the number of replicates, S i is thestandard deviation of the experimental points with eachexperiment.

Another form of equation 1 is as follows:

dX/dt = -K (X X e) (6)

Table 1.- The equilibrium moisture content from desorptionexperiments of jalapeo pepper (Average of three replicates).

Temperature (C) 50 60 70 80 95

X e 9.37 8.41 7.82 7.50 7.20

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The results obtained from equation 1 are shown inTable 2. From this Table it is clear that the drying constant

increases when the drying temperature increases. Theestimates in all cases were significant, in all cases the valueof R was high. The problem was present in the residuals,especially in the medium zone, but there was no problemin both, low and high moisture regions.

For the Arrhenius model (Equation 3), the resultswere as follows:

Ao = 0.0265 (SE=0.037)Ea = 24368 J/mol (SE = 3120.4)R = 0.923R for residuals = 0.156

The value of residual did not show a random behavior, thus, this model was not effec tive in this predic tion. As a result , the same matrix of data wasused, excluding those corresponding to each

temperature (one temperature each time). The resultsare shown in Table 3.

Interestingly, the stabili ty of parametersindicates that, excluding the low temperature, thenumerical values are significantly different. This factindicates that two mechanisms are present in the system,

because in other cases the pa rameters were stable .However, the model was not effective (c.f. the valuesof R residual) as it is shown in Figure 1. From this figureit follows that the Arrhenius equation cannot fit the datato linear behavior at 50 and 95 C but in the mediumrange, the Arrhenius model was very good (Figure 2);this fact indicates that the regions of drying wasdifferent in the zone of lower temperature and in thezone of highest temperature tested in this experiment.

The empirical model to describe the moisturecontent, can take the form of a linear algebraic equation.

H = dt + W + Td + E (7)

Table 2.- Parameter of the model of Equation 1.

Table 3.-Parameters of Arrhenius model excluding one temperature each time.

Table 4.-Summary of the model (Equation 7). *No convergence was reached.

1/T (K)

L n

K m

-5.6

-5.2

-4.8

-4.4

-4

-3.6

-3.2

0.00265 0.00275 0 .00285 0.00295 0.00305 0 .00315

Figure 1. The Arrhenius model of jalapeo pepper drying .

1/T (K)

L n

K m

-5.4

-5.3

-5.2

-5.1

-5-4.9

-4.8

-4.7

-4.6

-4.5

0.00282 0.00286 0.0029 0.00294 0.00298 0.00302

Figure 2. The Arrhenius model for jalapeo pepper drying with thelowest and highest temperature excluded.

Dryingtemperatures

Associatedparameter

to the model

Associatedcoefficient

to drying time ( )

Associatedcoefficient

to weight ( )

Associatedcoefficient to drying

temperature ( )50, 60, 70, 80, 95 90.92 -0.41 -0.65 *60, 70, 80, 95 114.56 0.07 -0.96 -0.08

50, 70, 80, 95 87.89 -0.42 -0.68 *50, 60, 80, 95 91.59 -0.44 -0.61 *50, 60, 70, 95 88.58 -0.47 -0.58 *50, 60, 70, 80 79.33 -0.35 -0.67 *

Temperatureexcluded (C)

Ao Ea (J/mol) R R of residual

50 0.000469* 358349* 0.978 0.17460 0.015 25794.8 0.967 0.13370 0.024 24604.9 0.916 0.17380 0.035 23583.3 0.901 0.17995 0.064 21948.1 0.939 0.093

50C 60C 70C 80C 95CEstimate (K) 0.0045 0.00513 0.00726 0.00989 0.02799 Std. Err. 0.00016 0.00021 0.00035 0.00061 0.00262 R 0.9740 0.9705 0.9726 0.9717 0.9842 R(of residuals) 0.7212 0.7287 0.7130 0.649 0.635


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This model where H = moisture content (%),has only three variables (drying time (dt, min), weight(W, g), drying temperature (Td, C) and three associated

parameters . The model features high determinationcoefficient, low associated error, and no co-linearity.

The stability of the parameters was evaluatedeliminating one drying temperature each time

(Montgomery and Peck, 1992). The results are presented in Table 4.From the above results, a good stability of the

associated parameters to the model can be seen, withthe exception of one group. This group may then beeliminated for the lowest temperature drying, as thesame of Arrehnius equation (See Table 3).

The model was verified by cross validationmethodology from Cochran and Cox (1982) modified

by Gallegos-Infante and Rico-Martnez (1999). A groupof data not taken into account during the modelconstruction, was proved in the model and the resultsshowed high confidence in the prediction of themoisture content.

CONCLUSIONS

The Arrhenius model was not effective to cover all range of drying tempratures for jalapeo pepper,

but shows good agreement in the med ium ran ge of temperature (60, 70 and 80 C). As expected, the

pr es en ce of an in ve rt ed re la ti on sh ip be tw ee n th emoisture content in jalapeo peppers and the drying

time was found. Unexpectedly however, dryingtemperature was not a significant variable, though itaffected the mechanisms under which the drying was

performed.

BIBLIOGRAFA

Cochran, W. G.; Cox, G. M. 1992. DiseosExperimentales. Pp. 163-164. Ed. Trillas.Mxico.

Gallegos-Infante, A.; Rico-Martnez, R. 1999. Thekinetics of the solidification of highlysupersaturated solutions of palmitic acid in oleicacid: a comparison between two models. Journal of the Serbian Chemical Society 64(7-8) 471 -481.

Montgomery, D. C.; Peck, E. A. 1992. pp 223.Introduction to l inear regression analysis.Second edition. John Wiley & Sons, Inc.

Raghavan, G. S. V.; Venkatachalapathy, K. 1999.Shrinkage of Strawberries During MicrowaveDrying. Drying Technology 17(10), 2309 - 2322.

Sawhney, R. L.; Pangavhane, D. R.; Sarsavadia, P.N.1999. Drying kinetics of single layer Thompsonseedless grapes under heated ambient a i r conditions. Drying Technology 17, 215-236.

Vagenas, G. K.; Marinos-Kouris, D. 1992. Use of theWilson equation for the prediction of thesorptional equilibrium of sugar-based foodstuffs.

Fluid phase equi libr ia 78, 191-207.

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mathematical model for prediction of moistur

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