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Brazilian Journal of Chemical Engineering

Print version ISSN 0104-6632On-line version ISSN 1678-4383

Braz. J. Chem. Eng. vol.17 n.1 São Paulo Mar. 2000 

A study of the desorption isotherms of lentils



M.A.S. Barrozo, D.T. Oliveira, G.P. Sancineti and M. V. Rodrigues
Universidade Federal de Uberlândia, Departamento de Engenharia Química, Bloco K,
Campus Santa Mônica, CEP 38400-902, Phone (55-34) 239-4189,
Fax (55-34) 239-4188, Uberlândia - MG, Brazil.
E - mail:



Received: June 1, 1999; Accepted: August 17, 1999)



Abstract - The aim of this work was to analyze the main equilibrium equations used for grains to find the best way to represent the equilibrium conditions between lentil and air. The experimental study was based on the static method using saturated salt solutions. We developed criteria for distinguish between some existing equations used for grains. To distinguishing between these equations we explored some nonlinearity measures. The results obtained showed that the Halsey modified equation was the best model in terms of nonsignificance for bias and nonlinearity measures.
Keywords: desorption, lentils, equilibrium equations.




Lentils (Lens culinaris Medik) have been produced for centuries in certain areas of Asia as a major source of protein in the human diet. Because of their relatively stable price on the world market and increasing domestic demand, the worldwide production of lentils has increased steadily in recent years. Lentils require artificial drying during wet seasons, and to study the drying process a knowledge of the moisture equilibrium is fundamental.

The two classic procedures for obtaining moisture equilibrium are by static and dynamic methods. In the first approach, motionless air surrounds the material and in the second approach, the fluid is moved mechanically. The advantage of the static method is that constant conditions can be more easily achieved (Barrozo et al., 1994).

Most of the moisture equilibrium equations presented in the literature are nonlinear, and in this case some care should be taken when we want to estimate parameters, since in some situations, the estimators (especially, confidence intervals or hypotheses tests) could be inappropriate. Thus, there are some procedures to validate the statistical properties of the least squares estimators of nonlinear models in the literature.

In this paper, we distinguish between the most important existing equilibrium equations, examining an experimental data set by nonlinear measures (see for example, Bates and Watts, 1980) to find the best model to represent the moisture equilibrium of lentils.



A large number of theoretical equations, both empirical and semi-empirical, have been proposed to estimate the moisture equilibrium of biological materials. The most frequently utilized equations presented in the literature to predict the moisture equilibrium for grains are given in Table 1, where Meq is the moisture equilibrium for grains, UR is the relative humidity of the air, Ts is the solid temperature, and a, b, c and d are the parameters involved.


a10t01.gif (6151 bytes)




The lentil is lens-shaped. Cotyledons, which constitute more than 94% of the total weight, are surrounded by a seedcoat except at the narrow hilum. The surface of the seedcoat is covered with a cuticle layer, consisting of waxy, water-resistant excretions. This layer is believed to provide the major moisture barrier (Tang and Sokhansanj, 1994).

Experimental Methodology

The technique used to obtain the equilibrium date was based on the static method with the use of saturated salt solutions. The salt was chosen in order to get a larger range of relative humidity. The procedures used were described elsewhere by Greenspan (1977).

The cylinder reservoirs used in the experiments were 6x10-2 m in diameter and 7x10-2 m in height. The grains were placed on a perfurated tray that was held in place by a support 3x10-2m from the base, which kept the grains from coming into contact with the liquid. The initial lentil mass given in each reservoir was of about 5.0x10-3 kg. The initial lentil moisture content was always higher than the equilibrium value to guarantee the desorption process.

After each sample was weighed on an analytical scale with a precision of 1x10-7 kg , it was put into reservoirs. The reservoirs were hermetically closed and placed in an incubator with internal temperature control (maximum variation of 0.5° C). It was assumed that the equilibrium conditions had been attained when three subsequent measurements of the sample mass gave identical results. Measurements were taken at 48-hour intervals. The moisture content of the material was determined (stove method, 105± 2° C) as soon as the equilibrium conditions were reached.

For the experimental data, we observed a small variance for the measured moisture equilibrium (replicates), where the average standard deviation was equal to 0.30. Table 2 presents the saturated salt solutions used and the respective relative humidity for the four temperature levels (20, 30, 40 and 50° C) with the experimental data for moisture equilibrium.


a10t02.gif (8415 bytes)



Nonlinearity Measures

Nonlinear regression models differ in general from linear regression models because the least squares (LS) estimators of the parameters are biased, do not fall on a normal distribution curve and have variances that exceed the minimum variances possible. The extent of the bias, the deviation from normal distribution and the excess variance differ greatly from model to model. Box (1971) presented a useful formula for estimating the bias in the LS estimators; Bates and Watts (1980) developed new measures of nonlinearity based on the geometric concept of curvature.

The Curvature Measures of Nonlinearity of Bates and Watts

Let us assume a nonlinear model given by

Yi = f ( Xi , q ) + ei,      i = 1,2...n (6)

where ei are independent, identically distributed random errors with normal distribution N(0,d 2).

If a10i02.gif (83 bytes) represents the LS estimator, the curvature of the solution locus in the vicinity of a10i02.gif (83 bytes) is a measure of intrinsic (IN) nonlinearity, while the unequal spacing and lack of parallelism of parameter lines projected onto the plane tangent to the solution locus is a measure of parameter-effects (PE) nonlinearity. These authors quantified these measures by employing three-dimensional acceleration arrays. See the theoretical development and their formulas in Bates and Watts, 1980. For intrinsic nonlinearity, the acceleration array is of dimension (n-p)p2, where p is the dimension of parameter q , and these (n-p)p2 elements reduce to a single measure represented by the maximum intrinsic curvature, denoted by IN. For the parameter-effects nonlinearity, the corresponding acceleration array is of dimension p3, and these p3 elements reduce to a single measure represented by the maximum parameter-effects curvature, denoted by PE.

The statistical significance of IN and PE may be assessed by comparing these values with a10i03.gif (150 bytes), where F=F(p,n-p,a ) is obtained from a table of the F-distribution (significance level a ).

The Bias Measure of Box

The formula derived by Box (1971) for calculating the bias in the LS estimates of the parameters in nonlinear regression models having a single response variate is

a10i04.gif (1072 bytes)

where Fi (=Fu) is the px1 vector of first derivatives of f ( Xi , q ) and Hu is the pxp matrix of second derivatives with respect to each of the elements of q , evaluated at Xi, where i=1,2...,n. In practice, a10i02.gif (83 bytes) and a10i06.gif (111 bytes) are usually used in place of the unknown quantities.

We usually present the percentage bias, given by

a10i05.gif (515 bytes)

where an absolute value in excess of 1% indicates nonlinear behavior (see Ratkowsky,1983).



The parameters of the equations given in Table 1 were estimated by the least squares method, where all 37 experimental observations were used as the data set. The curvature measures (Bates and Watts, 1980) and bias (Box, 1971) were obtained by using a Fortran program introduced by Ratkowsky (1983), adapted to this case.

In Table 3, we show the results of least squares estimation in Meq in dry base (g of water/100g of dry solid), T in ºC and UR as a decimal fraction, with the respective values for the correlation coefficient (R2), the intrinsic curvature measure (IN), the parameter-effects measure (PE) and the bias measures for all models considered.


a10t03.gif (8599 bytes)


We notice that all the intrinsic curvature measures (IN) given in Table 3, when compared to the a10i07.gif (154 bytes)value, are not significant for the nonlinear models considered (IN<a10i07.gif (154 bytes)), which indicates little nonlinearity for the solution locus. We also observe that the parameters-effects curvature measures (PE) are significant for the first four equations (PE>a10i07.gif (154 bytes)), showing that at least one parameter for these equations has a highly nonlinear behavior. In the Halsey modified equation, we observe better results, since the curvature measures (PE) are smaller than a10i07.gif (154 bytes), and therefore, are not significant. Thus, for this equation, the nonlinearity due to parametrization is little, and inference results for the least squares estimators are valid.

Since Equations (1) to (4) present significant nonlinear parameter effects (PE), the bias measures could show which parameters are responsible for this behavior (% bias > 1%). We also observe that only the Halsey equation presents a nonsignificant bias.

The results of the Bates and Watts (1980) curvature and bias measures study show that the only equation that gives good inference results based on least squares estimators is the modified Halsey equation. The value for the R2 of this equation is also the largest, so this equation with the parameters estimated for lentil data is given by (Ts, ° C and UR, decimal)

a10i08.gif (786 bytes)



From the results obtained from the analysis, we can conclude that:

(a) from the discrimination approach based on nonlinear measures, it was possible to get the best equation to represent the equilibrium data on lentils; these analyses could be extended to other nonlinear models;

(b) the modified Halsey equation was the only one that presented curvature measures and nonsignificant bias; therefore, this equation is the most appropriate to represent equilibrium data for lentils.



Barrozo, M.A.S., Sartori, D.J.M. and Freire, J.T., Analysis of the Kinetic and Equilibrium Equations in Soybeans Drying, 9th International Drying Symposium, Vol. 2, pp. 1053-1060, Gold Coast, Australia (1994).

Bates, D.M. and Watts, D.G., Relative Curvature Measures of Nonlinearity, J. R. Statist. Soc. B, 42, pp. 1-25 (1980).

Box, M.J., Bias in Nonlinear Estimation. J. R. Statist. Soc. B, 33, pp. 171-201 (1971).

Chen, C.S. and Clayton, J.T., The Effect of Temperature on Sorption Isotherms of Biological Materials, Transactions of the ASAE, 14(5), pp. 927-929 (1971).

Chung, D.S. and Pfost, H.B., Adsorption and Desorption of Water Vapour by Cereal Grains and Their Products Part II, Transactions of the ASAE, 10(4), pp. 549-551 (1967).

Greenspan, L., Humidity Fixed Points of Binary Saturated Aqueous Solutions, Journal Res. Natl. Bureau of Standards, 81(A), pp. 89-93 (1977).

Halsey, G., Physical Adsorption on Non-uniform Surfaces, Journal of Chem. Phys. 16(10), pp. 931-937 (1948).

Henderson, S.M., A Basic Concept of Equilibrium Moisture Content, Agricultural Engineering, 33(2), pp. 29-31 (1952).

Osborn, G.S., White, G.M., Sulaiman, A.H. and Welton, L.R., Predicting Equilibrium Moisture Proportions of Soybeans, Transactions of the ASAE, 32(6), pp. 2109-2113 (1989).

Ratkowsky, D.A., Nonlinear Regression Modeling, Marcel Dekker Inc., New York, 276 pp. (1983).

Tang, J. and Sokhansanj, S., A Model for the Thin-layer Drying of Lentils. Drying Technology, 12(4), pp. 849-867 (1994).

Thompson, T.L., Peart, R.M. and Foster, G.H., Mathematical Simulation of Corn Drying - A New Model, Transactions of the ASAE, 11, pp. 582-586 (1968).

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