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The use of nonlinearity measures to discriminate the equilibrium moisture equations for Bixa orellana seeds
Journal of Food Engineering 66 (2005) 63–68 www.elsevier.com/locate/jfoodeng
The use of nonlinearity measures to discriminate the equilibrium moisture equations for Bixa orellana seeds J.A. Ribeiro a, D.T. Oliveira a, M.L. Passos b, M.A.S. Barrozo b
a,*
a Faculty of Chemical Engineering, Federal University of Uberl^andia, 38400-902 Uberl^andia-MG, Brazil Department of Chemical Engineering, Federal University of Minas Gerais, 30160-030 Belo Horizonte-MG, Brazil
Received 29 January 2003; received in revised form 2 January 2004; accepted 23 February 2004
Abstract In this work, equilibrium moisture data for Bixa orellana seeds at different temperatures and relative humidities were determined using the Thermoconstanter Novasina instrument. Since the majority of the sorption equilibrium equations presented in the literature are nonlinear, care should be taken in estimating their parameters, because in some situations, the least squares estimators may not be appropriate. Therefore, some procedures are available in the literature to validate the statistical properties of the least squares estimators of nonlinear models. In this work, the nonlinear measures are used to discriminate between five model equations that represent the sorption equilibrium isotherms of Bixa orellana seeds. Results show that the Halsey modified equation is the best one to describe the experimental data because it has nonsignificance bias and nonlinear measurement parameters. 2004 Elsevier Ltd. All rights reserved. Keywords: Bixa orellana; Equilibrium moisture; Nonlinearity measures; Drying
1. Introduction Annatto is a natural reddish-yellow extract obtained from Bixa orellana seeds. This extract is noteworthy because of its lack of toxicity, its intense coloring capacity and its range of color, comprising red, orange and yellow hues. Annatto is widely used for coloring cosmetic and pharmaceutical products, wax polishes and foodstuffs such as cheese, butter, chocolate, ice cream, cereals, salad dressing, snack food, among others (Massarani, Passos, & Barreto, 1992). Bixa orellana is a tropical shrub, which grows quickly in Brazil, the Caribbean, India, and East Africa. Its seeds are composed of an ‘‘inner seed’’ with a shelled kernel containing oils, waxy substances, mineral ash and alkaloid compounds, a peel comprised of cellulose and tannins, and an outer cover containing pigments, moisture, and a small amount of oils. About 90% of the total pigments in this outer cover are the red oil-soluble carotenoid bixin. The bixin must be removed and concentrated to form annatto. Recently, a new mechanical
technique has been developed to easily extract bixin from these seeds (Massarani et al., 1992). However, this technique requires seed drying as the first step. To avoid thermal degradation of bixin, this drying step must be well controlled. Therefore, drying kinetic parameters, as well as, the moisture sorption isotherm for Bixa orellana seeds should be well known. Since the majority of the sorption equilibrium equations in the literature are nonlinear, care should be taken when estimating their parameters from experimental data. Note that, in some situations, the estimators (especially, confidence intervals) may not be appropriate. Therefore, some procedures are available in the literature to validate the statistical properties of the least squares estimators of nonlinear models. In this work, the nonlinear measures (Bates & Watts, 1980) are used to select, from five model equations, the best one to represent the sorption equilibrium isotherms of Bixa orellana seeds.
2. Equilibrium moisture equations *
Corresponding author. Fax: +55-34-239-4188. E-mail addresses: [email protected] (M.L. Passos), [email protected] (M.A.S. Barrozo). 0260-8774/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2004.02.040
A great number of equations have been proposed in the literature to describe the equilibrium moisture
64
J.A. Ribeiro et al. / Journal of Food Engineering 66 (2005) 63–68
content, Meq , of biological materials as a function of the air relative humidity, RH, and the solid material temperature, Ts . Among them, the theoretical equations are based on well-known sorption kinetic theories, such as the Kelvin, the Langmuir or the BET one. However, in practical applications, these theoretical equations cannot accurately predict the equilibrium moisture content of seeds over a wide range of temperatures and air relative humidities. This has led some researchers to develop empirical or semi-empirical models in order to improve the accuracy of the predicted Meq values. Table 1 presents some usual equations reported in the literature for estimating Meq ¼ f ðRH; Ts Þ of biological materials. The a, b, c and d coefficients are the model parameters to be fitted to experimental data. These models have been tested and their parameters evaluated for many agricultural crops (Ajibola, 1989; Duggal, Muir, & Brooker, 1982; Flood, Koon, Trombul, & Brower, 1987; White, Bridges, McNeill, & Overhults, 1985). In Table 1, the semi-empirical Henderson equation (Henderson, 1952) is based on the Gibbs sorption model. Although this equation is commonly used to predict Meq of many biological products, it cannot describe well the grain sorption equilibrium. To improve its application, Thompson et al. (1968) and Chen and Clayton (1971) have modified this Henderson equation by introducing a new fitted parameter (Eqs. (2) and (4)). Eq. (3), developed by Chung and Pfost (1967), is derived from the potential theory associated with the simplified equations of thermodynamic states. Eq. (5) is an empirical version of the Halsey equation (Halsey, 1948), which is based on the BET model. The Chung–Pfost (Eq. (3)), Henderson–Thompson (Eq. (2)) and modified Halsey (Eq. (5)) equations, among others, have been adopted as standard equations by the American Society of Agricultural Engineers for
describing sorption isotherms (ASAE, 1995). Mazza and Jayas (1991) described equilibrium moisture content data of pea seeds at four temperatures and proposed the Chung–Pfost equation (Eq. (3)) as the best model among the other four models studied. Chen (2003) showed that the Henderson–Thompson (Eq. (2)) equation was the best model to describe the relationship between relative humidity, equilibrium moisture and temperature of the pea sorption data. Ajibola (1989) showed that the modified Halsey equation (Eq. (5)) was the best model (among the equations from Table 1) to represent the equilibrium data of melon seed. The choice of the best equations in the works above mentioned was based on R2 analysis and in some situations, on residuals analysis. In the present work the statistical discrimination between these equations was based on nonlinearity measures.
3. Nonlinearity measures In general, the technique used for estimating the unknown parameters in linear or nonlinear equations is the least squares (LS) method. This method has some optimum properties when certain conditions are met. The LS estimators of the unknown parameters in linear models are unbiased, normally distributed and have the property of being minimum variance estimators (Seber & Wild, 1989). Given that the assumptions noted above are satisfied, the criterion of least squares thus provides the best available estimates in practice (Seber, 1977). An important point in the LS estimators for the nonlinear models is that these estimators do not have the properties of the linear models estimators. Only ‘‘asymptotically’’, that is, as the sample size increases to infinity, do the properties of the estimators in a nonlinear model approach the properties of a linear model
Table 1 Equations for predicting the equilibrium moisture content of biological materials Name Henderson
Henderson–Thompson
Equation 1=b lnð1 URÞ Meq ¼ aTs Meq ¼
lnð1 URÞ aðTs þ cÞ
Chung–Pfost
Meq ¼
1 ln b
Chen–Clayton
Meq ¼
1 ln cTsd
Halsey modified
Meq ¼
Reference ð1Þ
1=b ð2Þ
ðTs þ cÞ lnðURÞ a
lnðURÞ aTsb
Thompson, Peart, and Foster (1968)
ð3Þ
Chung and Pfost (1967)
ð4Þ
Chen and Clayton (1971)
expðaTs þ cÞ lnðURÞ
Henderson (1952)
1=b ð5Þ
Osborn, White, Sulaiman, and Welton (1989)
J.A. Ribeiro et al. / Journal of Food Engineering 66 (2005) 63–68
(Jennrich, 1969). For finite sample size, a LS estimator of a parameter in a nonlinear model has essentially unknown properties. So, in general, the nonlinear regression models differed from linear regression models in that the LS estimators of the parameters are biased, nonnormally distributed, and have variances exceeding the minimum possible variance. The extent of the bias, the nonnormality and the excess variance differs widely from model to model. Beale (1960) made the first serious attempt to measure nonlinearity. Box (1971) presented a formula for estimating the bias in the LS estimators, and Gillis and Ratkowsky (1978), using simulation studies, found that this formula not only predicted bias to the correct order of magnitude but also gave a good indication of the extent of nonlinear behavior of the model. Bates and Watts (1980) developed new measures of nonlinearity based on the geometric concept of curvature. They showed that the nonlinearity of a model can be separated into two components: (a) an ‘‘intrinsic’’ nonlinearity (IN) associated with the curvature of the solution locus, and (b) a ‘‘parameter-effects’’ nonlinearity (PE) associated with the fact that the projections of the parameter lines on the tangent plane to the solution locus are, in general, neither straight, parallel, nor equispaced. They demonstrated the relationship between their measures and those of Beale (1960), and explained why Beale’s measures generally tend to underestimate the true nonlinearity. In addition, they showed that the bias measure of Box (1971) is closely related to the their measure of parameter-effects nonlinearity (PE).
65
and Watts (1980) work. The statistical significance of these measures can be by comparing the IN pffiffiffievaluated ffi and PE values with 12 F , where F ¼ F ða; n p; pÞ is the inverse of the probability Fisher distribution obtained at pffiffiffiffi the significance level a. The value 12 F may be regarded as the radius of curvature of the 100ð1 aÞ% confidence region. So, the solution locus may be considered to be sufficiently linearpover an approximate 95% confidence ffiffiffiffi pffiffiffiffi region if IN < 12 F (a ¼ 0:05). Similarly, if PE < 12 F , the projected parameter lines may be regarded as being sufficiently parallel and uniformly spaced, i.e., the LS estimates of the parameter do not depend on the user being able to supply a good initial prevision and the tests of parameters invariance will be adequate. For carrying out all calculations to determine the IN and PE values, a computer program in Fortran language, similar to one proposed by Ratkowsky (1983), has been developed for the present work. 3.2. The bias measure proposed by Box (1971) The formula for calculating the bias of the LS estimates in a nonlinear regression model is found in Box (1971). The bias expressed as a percentage of the LS estimate, is a useful quantity as an absolute value in excess of 1% appears to be a good rule of thumb for indicating nonlinear behavior (Ratkowsky, 1983). The percentage bias is given by %biasð^hÞ ¼
100 biasð^hÞ : ^h
ð6Þ
3.1. Curvature measures of nonlinearity proposed by Bates and Watts (1980)
4. Materials and methods
A question that has occupied the attention of researchers is that of how well some specified model fits the data. Extensive methodology has been developed for investigating whether a proposed model provides a good description of the data. Comparisons of the R2 values and residuals analysis can be insufficient to discriminate between nonlinear regression models (Ratkowsky, 1983). Other properties are desirable for nonlinear models, such as, the LS estimators of its parameters are almost unbiased, normally distributed, and whose variances are close to the minimum variance. Such a model should have both a low intrinsic nonlinearity (IN) and a low parameter-effects nonlinearity (PE). A negligible intrinsic nonlinearity (IN) will mean negligible bias in the predicted values of response; if, simultaneously, the parameter-effects nonlinearity (PE) is also negligible, the more valid will be statistical tests of the consistency of the adjusted parameters (Bates & Watts, 1980). Details about the development, procedure and equations for determining IN and PE are found in Bates
Bixa orellana seeds used in this work were obtained from plantations in the Uberl^andia region located in the southeast Brazilian State of Minas Gerais. Their physical properties are presented in Table 2 (details of the measuring methods can be found in Barrozo et al., 2000). Experimental data on the seed equilibrium moisture content (desorption) have been obtained using the Thermoconstant Novasina instrument, model TH200 (made in Switzerland), which is specially designed for temperature-controlled measurements of water activity. This instrument is composed of a temperature regulator, a sealed measuring chamber, a moisture-temperature sensor and a transmitter. The chamber temperature is regulated to a set point by a controller with accuracy of 0.2 C. This temperature control creates a stable climatic condition for precise and replicable measurements. The state of equilibrium between the moisture-temperature sensor (inserted into the head of the sample cell) and the sample (to be measured) is then quickly reached. The
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Table 2 Physical properties of the Bixa orellana seeds (Barrozo, Ataıde, Tavares, Dias, & Passos, 2000) Seed properties
Experimental data
Technique used
Particle density, qs (kg/m3 ) Moisture content, m (% in dry basis) Particle diameter, dp (m) Sphericity, / (–)
1587.58 7.07 3.16 · 103 0.68
Pycnometric method using n-hexane as a liquid Weighing samples dried in an oven at 105 C until mass constant Pycnometric method using n-hexane as a liquid ¼ dp /(maximum particle length)
transmitter converts the signals from the sensor into measured values of relative humidity and temperature that can be read directly on the instrument display. The temperature range corresponding to 0–50 C and the relative humidity range is 0–100%. To ensure troublefree calibration of the humidity sensor, it has been periodically immersed into sealed units of saturated salt solutions (pure organic salts and distilled water) for checking its precision and, when necessary, for a new calibration. For recording the desorption curve the product must always be more humid than the first measuring point of the sorption isotherm. The equilibrium conditions can be obtained, for each saturated salt solution at a determined temperature, when the measured value of relative humidity remains unchanged for several minutes and the temperature indicated corresponds to the preselected value. So, after this time, the material is removed and weighed in an analytical balance. After this, a new salt solution is placed in measuring chamber and the process is repeated. At the end the product is dried to determine the dry weight and therefore to determine the equilibrium moisture content for each salt solution (RH). This procedure is repeated for another temperature.
5. Results and discussion Table 3 presents the results obtained by the least squares parameter estimation for the five model equations listed in Table 1. These results include the estimated parameter values (for Meq expressed in % of dry basis, temperature in C and RH in the decimal fraction), as well as, the respective values of the regression coefficient (R2 ), the intrinsic curvature measure (IN), the parameter-effects measure (PE) and the % bias. As mentioned previously, the higher R2 value for the Halsey equation is not an enough to guarantee the statistical validity of the parameters obtained in a nonlinear regression. It can be observed in Table 3 that all calculated values of the intrinsic measures, IN, are not signifipffiffifficurvature ffi cant (IN < 12 F ), inferring a small deviation of the model solution locus from linearity (Ratkowsky, 1983). On the other hand, the calculated values of PE (the parameters-effects curvature measure) are significant for pffiffiffiffi Eqs. (1)–(4) (PE > 12 F ), meaning that, at least, one parameter of these equations has a strong nonlinear behavior. The best results are obtained for the p Halsey ffiffiffiffi modified model equation (Eq. (5)), since PE < 12 F for
Table 3 Statistical results of the least-squares estimation Parameter
Henderson, Eq. (1)a
a b
1.05 · 104 1.68
IN ¼ 0.067 PE ¼ 5.78
7.76 0.34
Henderson–Thompson, Eq. (2)b
a b c
1.05 · 104 1.71 3.38
IN ¼ 0.102 PE ¼ 7.02
3.65 0.30 77.6
0.905
IN ¼ 0.116 PE ¼ 1.06
8.08 0.35 15.60
0.906
Chung–Pfost, Eq. (3)b
Chen–Clayton, Eq. (4)c
Halsey modified, Eq. (5)b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a1 F ð0:95; 31; 2Þ ¼ 0:275. 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b1 F ð0:95; 30; 3Þ ¼ 0:293. 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c1 F ð0:95; 29; 4Þ ¼ 0:304. 2
a b c a b c d a b c
Estimated value
124.57 0.08 )10.69 0.992 0.486 0.004 0.816 )4.19 · 102 2.575 9.00
Curvature measures
IN ¼ 0.217 PE ¼ 135.82 IN ¼ 0.026 PE ¼ 0.096
% Bias measure
R2
Equation
304.7 1.69 40.46 0.19 0.123 0.055 0.073
0.900
0.931
0.983
J.A. Ribeiro et al. / Journal of Food Engineering 66 (2005) 63–68
67
Fig. 1. Experimental equilibrium moisture content of Bixa orellana seeds as a function of RH and Ts and the response surface predicted by the Halsey modified equation (Eq. (5) in Table 1).
all the three parameters. Therefore, among the five equations, this Halsey modified equation behaves closest to the LS linear model approach, meaning valid inference results based on asymptotic approximations assumed in the LS nonlinear estimates. Since Eqs. (1)–(4) present significant nonlinear parameter-effects (PE), the bias measures should be used to identify parameters that are responsible for the nonlinear behavior (% bias > 1%). As shown in Table 3, the highest values of PE and % bias are obtained for the Chen–Clayton equation (Eq. (4)), which is characterized by four model parameters (p ¼ 4). Another important result seen in Table 3 is that the addition of one more parameter into the Henderson equation causes a greater model deviation from the linear behavior. Since this parameter is empirical, a parametric analysis leading to a new formulation (reparameterization) of Eqs. (2) and (3) can bring their behavior closer to the linear one. As already predicted by the PE values, Eqs. (1)–(4) present, at least, one parameter with % bias higher than 1%. The Halsey modified equation, expressed by Eq. (5), is the only one in which % bias values are not significant (% bias < 1% for all the three parameters). Based on the curvature and bias measures, the best model equation to predict the equilibrium moisture content of Bixa orellana seeds is Eq. (5), the modified Halsey equation. The experimental values of the Bixa orellana seed equilibrium moisture content (Meq ) expressed as percent dry basis at the different temperatures and relatives humidities, obtained in a range of 10 < RH (%) < 95 and 25 6 Ts ðCÞ 6 45, are presented in Fig. 1. It was not possible to obtain equilibrium at relative humidities above 90%, for most of the temperatures, due to mould
growth before equilibrium was reached. Conventionally, equilibrium moisture content was found to increase with decrease in temperature at constant relative humidity and to increase with increase in relative humidity of air when temperature was kept constant. The response surface given by the modified Halsey is also presented in Fig. 1. We observe good agreement between the experimental data and the predicted values.
6. Conclusions From results obtained in this present work, the following conclusion can be drawn: • from the discrimination approach based on nonlinear measures, it is possible to select the best model equation for describing the experimental sorption equilibrium data of Bixa orellana seeds; • among five models analyzed in this work, the Halsey modified equation is the only one that for which curvature measures and bias were not significant; therefore, this equation is the most appropriate to represent the sorption equilibrium data of Bixa orellana seeds; • the statistical analysis, as well as, the computer program developed in this work can be easily extended to any other nonlinear regression model estimation.
Acknowledgements The authors thank the Brazilian funding agencies, FAPEMIG and CNPq for the financial support.
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References Ajibola, O. O. (1989). Thin layer drying of melon seed. Journal of Food Engineering, 9, 305–320. ASAE (1995). Moisture relationship of plant-based agricultural products. ASAe standard D245.5, St Joseph, Michigan. Barrozo, M. A. S., Ataıde, C. H., Tavares, L. E., Dias, A. R. C., & Passos, M. L. A. (2000). Bixin powder production in a spouted bed. Recents Progres en Genie des Procedes, 76(14), 467–474. Bates, D. M., & Watts, D. G. (1980). Relative curvature measures of nonlinearity. Journal of the Royal Statistical Society Series B, 42, 1–25. Beale, E. M. L. (1960). Confidence regions in nonlinear estimation. Journal of the Royal Statistical Society Series B, 22, 41–76. Box, M. J. (1971). Bias in nonlinear estimation. Journal of the Royal Statistical Society Series B, 33, 171–201. Chen, C. (2003). Moisture sorption isotherms of pea seeds. Journal of Food Engineering, 58(1), 45–51. Chen, C. S., & Clayton, J. T. (1971). The effect of temperature on sorption isotherms of biological materials. Transactions of the ASAE, 14(5), 927–929. Chung, D. S., & Pfost, H. B. (1967). Adsorption and desorption of water vapour by cereal grains and their products Part II. Transactions of the ASAE, 10(4), 549–551. Duggal, A. K., Muir, W. E., & Brooker, D. B. (1982). Sorption equilibrium moisture contents of wheat kernels and chaff. Transactions of the ASAE, 25(4), 1086–1090. Flood, C. A., Koon, J. L., Trombul, R. D., & Brower, R. N. (1987). Equilibrium moisture relations of pine shavings poultry litter materials. Transactions of the ASAE, 30(3), 848–852.
Gillis, P. R., & Ratkowsky, D. A. (1978). The behaviour of estimators of the parameters of various yield-density relationships. Biometrics, 34, 191–198. Halsey, G. (1948). Physical adsorption on non-uniform surfaces. Journal of Chemical Physics, 16(10), 931–937. Henderson, S. M. (1952). A basic concept of equilibrium moisture content. Agricultural Engineering, 33(2), 29–31. Jennrich, R. I. (1969). Asymptotic properties of non-linear least squares estimators. Annals of Mathematical Statistics, 40, 633–643. Massarani, G., Passos, M. L., & Barreto, W. (1992). Production of annatto concentrates in spouted beds. Canadian Journal of Chemical Engineering, 70(5), 954–959. Mazza, G., & Jayas, D. S. (1991). Evaluation of four three-parameter equations for the description of the moisture sorption data of Lathyrus pea seeds. Lebensmittel-Wissenschaft und-Technologie, 24, 562–565. Osborn, G. S., White, G. M., Sulaiman, A. H., & Welton, L. R. (1989). Predicting equilibrium moisture proportions of soybeans. Transactions of the ASAE, 32(6), 2109–2113. Ratkowsky, D. A. (1983). Nonlinear regression modeling. New York: Marcel Dekker. Seber, G. A. F. (1977). Linear regression. New York: Wiley. Seber, G. A. F., & Wild, J. (1989). Nonlinear regression. New York: Wiley. Thompson, T. L., Peart, R. M., & Foster, G. H. (1968). Mathematical simulation of corn drying––a new model. Transactions of the ASAE, 11, 582–586. White, G. M., Bridges, T. C., McNeill, S. G., & Overhults, D. G. (1985). Equilibrium moisture properties of corn cobs. Transactions of the ASAE, 28(1), 280–285.
The use of nonlinearity measures to discriminate the equilibrium moisture equations for Bixa orellana seeds J.A. Ribeiro a, D.T. Oliveira a, M.L. Passos b, M.A.S. Barrozo b
a,*
a Faculty of Chemical Engineering, Federal University of Uberl^andia, 38400-902 Uberl^andia-MG, Brazil Department of Chemical Engineering, Federal University of Minas Gerais, 30160-030 Belo Horizonte-MG, Brazil
Received 29 January 2003; received in revised form 2 January 2004; accepted 23 February 2004
Abstract In this work, equilibrium moisture data for Bixa orellana seeds at different temperatures and relative humidities were determined using the Thermoconstanter Novasina instrument. Since the majority of the sorption equilibrium equations presented in the literature are nonlinear, care should be taken in estimating their parameters, because in some situations, the least squares estimators may not be appropriate. Therefore, some procedures are available in the literature to validate the statistical properties of the least squares estimators of nonlinear models. In this work, the nonlinear measures are used to discriminate between five model equations that represent the sorption equilibrium isotherms of Bixa orellana seeds. Results show that the Halsey modified equation is the best one to describe the experimental data because it has nonsignificance bias and nonlinear measurement parameters. 2004 Elsevier Ltd. All rights reserved. Keywords: Bixa orellana; Equilibrium moisture; Nonlinearity measures; Drying
1. Introduction Annatto is a natural reddish-yellow extract obtained from Bixa orellana seeds. This extract is noteworthy because of its lack of toxicity, its intense coloring capacity and its range of color, comprising red, orange and yellow hues. Annatto is widely used for coloring cosmetic and pharmaceutical products, wax polishes and foodstuffs such as cheese, butter, chocolate, ice cream, cereals, salad dressing, snack food, among others (Massarani, Passos, & Barreto, 1992). Bixa orellana is a tropical shrub, which grows quickly in Brazil, the Caribbean, India, and East Africa. Its seeds are composed of an ‘‘inner seed’’ with a shelled kernel containing oils, waxy substances, mineral ash and alkaloid compounds, a peel comprised of cellulose and tannins, and an outer cover containing pigments, moisture, and a small amount of oils. About 90% of the total pigments in this outer cover are the red oil-soluble carotenoid bixin. The bixin must be removed and concentrated to form annatto. Recently, a new mechanical
technique has been developed to easily extract bixin from these seeds (Massarani et al., 1992). However, this technique requires seed drying as the first step. To avoid thermal degradation of bixin, this drying step must be well controlled. Therefore, drying kinetic parameters, as well as, the moisture sorption isotherm for Bixa orellana seeds should be well known. Since the majority of the sorption equilibrium equations in the literature are nonlinear, care should be taken when estimating their parameters from experimental data. Note that, in some situations, the estimators (especially, confidence intervals) may not be appropriate. Therefore, some procedures are available in the literature to validate the statistical properties of the least squares estimators of nonlinear models. In this work, the nonlinear measures (Bates & Watts, 1980) are used to select, from five model equations, the best one to represent the sorption equilibrium isotherms of Bixa orellana seeds.
2. Equilibrium moisture equations *
Corresponding author. Fax: +55-34-239-4188. E-mail addresses: [email protected] (M.L. Passos), [email protected] (M.A.S. Barrozo). 0260-8774/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2004.02.040
A great number of equations have been proposed in the literature to describe the equilibrium moisture
64
J.A. Ribeiro et al. / Journal of Food Engineering 66 (2005) 63–68
content, Meq , of biological materials as a function of the air relative humidity, RH, and the solid material temperature, Ts . Among them, the theoretical equations are based on well-known sorption kinetic theories, such as the Kelvin, the Langmuir or the BET one. However, in practical applications, these theoretical equations cannot accurately predict the equilibrium moisture content of seeds over a wide range of temperatures and air relative humidities. This has led some researchers to develop empirical or semi-empirical models in order to improve the accuracy of the predicted Meq values. Table 1 presents some usual equations reported in the literature for estimating Meq ¼ f ðRH; Ts Þ of biological materials. The a, b, c and d coefficients are the model parameters to be fitted to experimental data. These models have been tested and their parameters evaluated for many agricultural crops (Ajibola, 1989; Duggal, Muir, & Brooker, 1982; Flood, Koon, Trombul, & Brower, 1987; White, Bridges, McNeill, & Overhults, 1985). In Table 1, the semi-empirical Henderson equation (Henderson, 1952) is based on the Gibbs sorption model. Although this equation is commonly used to predict Meq of many biological products, it cannot describe well the grain sorption equilibrium. To improve its application, Thompson et al. (1968) and Chen and Clayton (1971) have modified this Henderson equation by introducing a new fitted parameter (Eqs. (2) and (4)). Eq. (3), developed by Chung and Pfost (1967), is derived from the potential theory associated with the simplified equations of thermodynamic states. Eq. (5) is an empirical version of the Halsey equation (Halsey, 1948), which is based on the BET model. The Chung–Pfost (Eq. (3)), Henderson–Thompson (Eq. (2)) and modified Halsey (Eq. (5)) equations, among others, have been adopted as standard equations by the American Society of Agricultural Engineers for
describing sorption isotherms (ASAE, 1995). Mazza and Jayas (1991) described equilibrium moisture content data of pea seeds at four temperatures and proposed the Chung–Pfost equation (Eq. (3)) as the best model among the other four models studied. Chen (2003) showed that the Henderson–Thompson (Eq. (2)) equation was the best model to describe the relationship between relative humidity, equilibrium moisture and temperature of the pea sorption data. Ajibola (1989) showed that the modified Halsey equation (Eq. (5)) was the best model (among the equations from Table 1) to represent the equilibrium data of melon seed. The choice of the best equations in the works above mentioned was based on R2 analysis and in some situations, on residuals analysis. In the present work the statistical discrimination between these equations was based on nonlinearity measures.
3. Nonlinearity measures In general, the technique used for estimating the unknown parameters in linear or nonlinear equations is the least squares (LS) method. This method has some optimum properties when certain conditions are met. The LS estimators of the unknown parameters in linear models are unbiased, normally distributed and have the property of being minimum variance estimators (Seber & Wild, 1989). Given that the assumptions noted above are satisfied, the criterion of least squares thus provides the best available estimates in practice (Seber, 1977). An important point in the LS estimators for the nonlinear models is that these estimators do not have the properties of the linear models estimators. Only ‘‘asymptotically’’, that is, as the sample size increases to infinity, do the properties of the estimators in a nonlinear model approach the properties of a linear model
Table 1 Equations for predicting the equilibrium moisture content of biological materials Name Henderson
Henderson–Thompson
Equation 1=b lnð1 URÞ Meq ¼ aTs Meq ¼
lnð1 URÞ aðTs þ cÞ
Chung–Pfost
Meq ¼
1 ln b
Chen–Clayton
Meq ¼
1 ln cTsd
Halsey modified
Meq ¼
Reference ð1Þ
1=b ð2Þ
ðTs þ cÞ lnðURÞ a
lnðURÞ aTsb
Thompson, Peart, and Foster (1968)
ð3Þ
Chung and Pfost (1967)
ð4Þ
Chen and Clayton (1971)
expðaTs þ cÞ lnðURÞ
Henderson (1952)
1=b ð5Þ
Osborn, White, Sulaiman, and Welton (1989)
J.A. Ribeiro et al. / Journal of Food Engineering 66 (2005) 63–68
(Jennrich, 1969). For finite sample size, a LS estimator of a parameter in a nonlinear model has essentially unknown properties. So, in general, the nonlinear regression models differed from linear regression models in that the LS estimators of the parameters are biased, nonnormally distributed, and have variances exceeding the minimum possible variance. The extent of the bias, the nonnormality and the excess variance differs widely from model to model. Beale (1960) made the first serious attempt to measure nonlinearity. Box (1971) presented a formula for estimating the bias in the LS estimators, and Gillis and Ratkowsky (1978), using simulation studies, found that this formula not only predicted bias to the correct order of magnitude but also gave a good indication of the extent of nonlinear behavior of the model. Bates and Watts (1980) developed new measures of nonlinearity based on the geometric concept of curvature. They showed that the nonlinearity of a model can be separated into two components: (a) an ‘‘intrinsic’’ nonlinearity (IN) associated with the curvature of the solution locus, and (b) a ‘‘parameter-effects’’ nonlinearity (PE) associated with the fact that the projections of the parameter lines on the tangent plane to the solution locus are, in general, neither straight, parallel, nor equispaced. They demonstrated the relationship between their measures and those of Beale (1960), and explained why Beale’s measures generally tend to underestimate the true nonlinearity. In addition, they showed that the bias measure of Box (1971) is closely related to the their measure of parameter-effects nonlinearity (PE).
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and Watts (1980) work. The statistical significance of these measures can be by comparing the IN pffiffiffievaluated ffi and PE values with 12 F , where F ¼ F ða; n p; pÞ is the inverse of the probability Fisher distribution obtained at pffiffiffiffi the significance level a. The value 12 F may be regarded as the radius of curvature of the 100ð1 aÞ% confidence region. So, the solution locus may be considered to be sufficiently linearpover an approximate 95% confidence ffiffiffiffi pffiffiffiffi region if IN < 12 F (a ¼ 0:05). Similarly, if PE < 12 F , the projected parameter lines may be regarded as being sufficiently parallel and uniformly spaced, i.e., the LS estimates of the parameter do not depend on the user being able to supply a good initial prevision and the tests of parameters invariance will be adequate. For carrying out all calculations to determine the IN and PE values, a computer program in Fortran language, similar to one proposed by Ratkowsky (1983), has been developed for the present work. 3.2. The bias measure proposed by Box (1971) The formula for calculating the bias of the LS estimates in a nonlinear regression model is found in Box (1971). The bias expressed as a percentage of the LS estimate, is a useful quantity as an absolute value in excess of 1% appears to be a good rule of thumb for indicating nonlinear behavior (Ratkowsky, 1983). The percentage bias is given by %biasð^hÞ ¼
100 biasð^hÞ : ^h
ð6Þ
3.1. Curvature measures of nonlinearity proposed by Bates and Watts (1980)
4. Materials and methods
A question that has occupied the attention of researchers is that of how well some specified model fits the data. Extensive methodology has been developed for investigating whether a proposed model provides a good description of the data. Comparisons of the R2 values and residuals analysis can be insufficient to discriminate between nonlinear regression models (Ratkowsky, 1983). Other properties are desirable for nonlinear models, such as, the LS estimators of its parameters are almost unbiased, normally distributed, and whose variances are close to the minimum variance. Such a model should have both a low intrinsic nonlinearity (IN) and a low parameter-effects nonlinearity (PE). A negligible intrinsic nonlinearity (IN) will mean negligible bias in the predicted values of response; if, simultaneously, the parameter-effects nonlinearity (PE) is also negligible, the more valid will be statistical tests of the consistency of the adjusted parameters (Bates & Watts, 1980). Details about the development, procedure and equations for determining IN and PE are found in Bates
Bixa orellana seeds used in this work were obtained from plantations in the Uberl^andia region located in the southeast Brazilian State of Minas Gerais. Their physical properties are presented in Table 2 (details of the measuring methods can be found in Barrozo et al., 2000). Experimental data on the seed equilibrium moisture content (desorption) have been obtained using the Thermoconstant Novasina instrument, model TH200 (made in Switzerland), which is specially designed for temperature-controlled measurements of water activity. This instrument is composed of a temperature regulator, a sealed measuring chamber, a moisture-temperature sensor and a transmitter. The chamber temperature is regulated to a set point by a controller with accuracy of 0.2 C. This temperature control creates a stable climatic condition for precise and replicable measurements. The state of equilibrium between the moisture-temperature sensor (inserted into the head of the sample cell) and the sample (to be measured) is then quickly reached. The
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Table 2 Physical properties of the Bixa orellana seeds (Barrozo, Ataıde, Tavares, Dias, & Passos, 2000) Seed properties
Experimental data
Technique used
Particle density, qs (kg/m3 ) Moisture content, m (% in dry basis) Particle diameter, dp (m) Sphericity, / (–)
1587.58 7.07 3.16 · 103 0.68
Pycnometric method using n-hexane as a liquid Weighing samples dried in an oven at 105 C until mass constant Pycnometric method using n-hexane as a liquid ¼ dp /(maximum particle length)
transmitter converts the signals from the sensor into measured values of relative humidity and temperature that can be read directly on the instrument display. The temperature range corresponding to 0–50 C and the relative humidity range is 0–100%. To ensure troublefree calibration of the humidity sensor, it has been periodically immersed into sealed units of saturated salt solutions (pure organic salts and distilled water) for checking its precision and, when necessary, for a new calibration. For recording the desorption curve the product must always be more humid than the first measuring point of the sorption isotherm. The equilibrium conditions can be obtained, for each saturated salt solution at a determined temperature, when the measured value of relative humidity remains unchanged for several minutes and the temperature indicated corresponds to the preselected value. So, after this time, the material is removed and weighed in an analytical balance. After this, a new salt solution is placed in measuring chamber and the process is repeated. At the end the product is dried to determine the dry weight and therefore to determine the equilibrium moisture content for each salt solution (RH). This procedure is repeated for another temperature.
5. Results and discussion Table 3 presents the results obtained by the least squares parameter estimation for the five model equations listed in Table 1. These results include the estimated parameter values (for Meq expressed in % of dry basis, temperature in C and RH in the decimal fraction), as well as, the respective values of the regression coefficient (R2 ), the intrinsic curvature measure (IN), the parameter-effects measure (PE) and the % bias. As mentioned previously, the higher R2 value for the Halsey equation is not an enough to guarantee the statistical validity of the parameters obtained in a nonlinear regression. It can be observed in Table 3 that all calculated values of the intrinsic measures, IN, are not signifipffiffifficurvature ffi cant (IN < 12 F ), inferring a small deviation of the model solution locus from linearity (Ratkowsky, 1983). On the other hand, the calculated values of PE (the parameters-effects curvature measure) are significant for pffiffiffiffi Eqs. (1)–(4) (PE > 12 F ), meaning that, at least, one parameter of these equations has a strong nonlinear behavior. The best results are obtained for the p Halsey ffiffiffiffi modified model equation (Eq. (5)), since PE < 12 F for
Table 3 Statistical results of the least-squares estimation Parameter
Henderson, Eq. (1)a
a b
1.05 · 104 1.68
IN ¼ 0.067 PE ¼ 5.78
7.76 0.34
Henderson–Thompson, Eq. (2)b
a b c
1.05 · 104 1.71 3.38
IN ¼ 0.102 PE ¼ 7.02
3.65 0.30 77.6
0.905
IN ¼ 0.116 PE ¼ 1.06
8.08 0.35 15.60
0.906
Chung–Pfost, Eq. (3)b
Chen–Clayton, Eq. (4)c
Halsey modified, Eq. (5)b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a1 F ð0:95; 31; 2Þ ¼ 0:275. 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b1 F ð0:95; 30; 3Þ ¼ 0:293. 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c1 F ð0:95; 29; 4Þ ¼ 0:304. 2
a b c a b c d a b c
Estimated value
124.57 0.08 )10.69 0.992 0.486 0.004 0.816 )4.19 · 102 2.575 9.00
Curvature measures
IN ¼ 0.217 PE ¼ 135.82 IN ¼ 0.026 PE ¼ 0.096
% Bias measure
R2
Equation
304.7 1.69 40.46 0.19 0.123 0.055 0.073
0.900
0.931
0.983
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Fig. 1. Experimental equilibrium moisture content of Bixa orellana seeds as a function of RH and Ts and the response surface predicted by the Halsey modified equation (Eq. (5) in Table 1).
all the three parameters. Therefore, among the five equations, this Halsey modified equation behaves closest to the LS linear model approach, meaning valid inference results based on asymptotic approximations assumed in the LS nonlinear estimates. Since Eqs. (1)–(4) present significant nonlinear parameter-effects (PE), the bias measures should be used to identify parameters that are responsible for the nonlinear behavior (% bias > 1%). As shown in Table 3, the highest values of PE and % bias are obtained for the Chen–Clayton equation (Eq. (4)), which is characterized by four model parameters (p ¼ 4). Another important result seen in Table 3 is that the addition of one more parameter into the Henderson equation causes a greater model deviation from the linear behavior. Since this parameter is empirical, a parametric analysis leading to a new formulation (reparameterization) of Eqs. (2) and (3) can bring their behavior closer to the linear one. As already predicted by the PE values, Eqs. (1)–(4) present, at least, one parameter with % bias higher than 1%. The Halsey modified equation, expressed by Eq. (5), is the only one in which % bias values are not significant (% bias < 1% for all the three parameters). Based on the curvature and bias measures, the best model equation to predict the equilibrium moisture content of Bixa orellana seeds is Eq. (5), the modified Halsey equation. The experimental values of the Bixa orellana seed equilibrium moisture content (Meq ) expressed as percent dry basis at the different temperatures and relatives humidities, obtained in a range of 10 < RH (%) < 95 and 25 6 Ts ðCÞ 6 45, are presented in Fig. 1. It was not possible to obtain equilibrium at relative humidities above 90%, for most of the temperatures, due to mould
growth before equilibrium was reached. Conventionally, equilibrium moisture content was found to increase with decrease in temperature at constant relative humidity and to increase with increase in relative humidity of air when temperature was kept constant. The response surface given by the modified Halsey is also presented in Fig. 1. We observe good agreement between the experimental data and the predicted values.
6. Conclusions From results obtained in this present work, the following conclusion can be drawn: • from the discrimination approach based on nonlinear measures, it is possible to select the best model equation for describing the experimental sorption equilibrium data of Bixa orellana seeds; • among five models analyzed in this work, the Halsey modified equation is the only one that for which curvature measures and bias were not significant; therefore, this equation is the most appropriate to represent the sorption equilibrium data of Bixa orellana seeds; • the statistical analysis, as well as, the computer program developed in this work can be easily extended to any other nonlinear regression model estimation.
Acknowledgements The authors thank the Brazilian funding agencies, FAPEMIG and CNPq for the financial support.
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