It has been suggested by Maroulis e

It has been suggested by Maroulis et al. (1988) that a direct nonlinear
regression analysis and curve fitting of the data were more
Table 2
Predicted parameters of the fitted models to the experimental data for the moisture sorption isotherm of spray-dried pure orange juice powder at 20, 30, 40 and 50 C.
Goodness of fit parameters Model coefficients
Model type Temperature (C) SSE* Adj-R-sq RMSE** P (%) AB C D
GAB 20 0.003 0.99 0.014 3.6 7.613, c 0.953, k 0.126, xm e
30 0.005 0.99 0.0164 5.8 8.956, c 0.969, k 0.109, xm e
40 0.0023 0.99 0.012 7.0 9.485, c 0.9867, k 0.102, xm e
50 0.0044 0.99 0.0167 6.2 6.885, c 0.994, k 0.098, xm e
Peleg 20 0.0022 0.99 0.0124 6.3 0.279 0.6321 0.9557 5.508
30 0.0058 0.99 0.0179 5.2 0.2293 0.5571 0.8103 4.878
40 0.0103 0.97 0.0262 7.3 0.253 0.6499 0.7472 5.225
50 0.0065 0.98 0.0209 6.3 0.2295 0.6733 0.7581 5.052
Halsey 20 0.0032 0.99 0.0142 6.4 0.09431 1.265 e e
30 0.0085 0.99 0.0206 7.9 0.08495 1.257 e e
40 0.0112 0.97 0.0257 7.9 0.08181 1.252 e e
50 0.0075 0.98 0.0209 7.9 0.08675 1.163 e e
Oswin 20 0.0035 0.99 0.0148 7.1 0.2119 0.6357 e e
30 0.0087 0.99 0.0208 9.1 0.1966 0.6241 e e
40 0.0111 0.97 0.0255 8.7 0.1905 0.615 e e
50 0.0072 0.98 0.0206 7.4 0.1763 0.6589 e e
Caurie 20 0.0086 0.99 0.0232 9.8 3.289 3.31 e e
30 0.0107 0.98 0.0231 10.2 3.192 3.064 e e
40 0.0126 0.97 0.0273 7.9 3.139 2.92 e e
50 0.0082 0.98 0.0219 7.0 3.286 3.054 e e
BET 20 0.0009 0.98 0.0113 8.0 9.755, c 0.1134, Xm e e
30 0.0011 0.96 0.0098 6.1 12.85, c 0.0981, xm e e
40 0.0011 0.95 0.0108 8.6 8.806, c 0.1031, Xm e e
50 0.0004 0.97 0.0069 6.2 7.274, c 0.09624, xm e e
Kuhn 20 0.0104 0.98 0.0256 11.9 0.1002 0.05083 e e
30 0.0123 0.98 0.0248 11.1 0.0932 0.04306 e e
40 0.0031 0.99 0.0138 9.1 0.0991 0.0304 e e
50 0.0091 0.98 0.0231 9.6 0.0946 0.02359 e e
Iglesias and Chirife 20 0.013 0.98 0.0285 13.8 0.08946 0.1025 e e
30 0.0158 0.98 0.0281 13.0 0.07881 0.09543 e e
40 0.0159 0.96 0.0306 13.1 0.07425 0.0943 e e
50 0.0102 0.97 0.0245 11.6 0.0595 0.09728 e e
White and Eiring 20 0.013 0.98 0.0285 16.5 9.644 9.546 e e
30 0.0175 0.97 0.0296 16.5 10.88 10.87 e e
40 0.0179 0.96 0.0324 17.2 11.45 11.51 e e
50 0.0146 0.96 0.0293 18.4 12.78 13.15 e e
Smith 20 0.0153 0.98 0.0309 11.6 0.01256 0.3113 e e
30 0.0154 0.98 0.0277 13.0 0.0157 0.2804 e e
40 0.0142 0.97 0.0289 10.2 0.01933 0.262 e e
50 0.0107 0.97 0.0251 10.7 0.0086 0.2613 e e
Henderson 20 0.0308 0.98 0.0439 7.0 1.171 3.835 e e
30 0.0454 0.97 0.0477 14.4 1.15 4.146 e e
40 0.0287 0.97 0.0423 12.0 1.162 4.391 e e
50 0.0243 0.98 0.0378 11.5 1.096 4.294 e e
* Sum of squares due to error of the fit, ** root mean squared error (standard error), P: mean absolute percentage error.
M. Edrisi Sormoli, T.A.G. Langrish / LWT - Food Science and Technology 62 (2015) 875e882 879
accurate for fitting the GAB parameters. There are three methods to
determine the parameters of GAB equation. In the first method, the
GAB equation is rearranged, and a polynomial regression analysis is
performed. This method has been suggested to have two disadvantages.
Firstly the transformation results in incorrect weighing of
data and secondly the confidence intervals for the GAB parameters
cannot be determined directly (Samaniego-Esguerra et al., 1991).
Hence this method will not be discussed in this paper. In the second
method, non-linear regression analysis of the standard threeparameter
GAB equation (Eq. (9)) is performed, and the three parameters
of the GAB equation (Xm, c and k) are determined at each
temperature. Then the parameters DH1, DH2, c0, k0 are estimated by
using a successive regression of equations (10) and (11). This second
method is referred to here as the indirect regression method. In the
third method, the definition of the parameters of the GAB equation
c, k and Xm, (Eqs (10)e(12)) are substituted into the standard GAB
equation (Eq. (9)). This substitution results in a form of GAB
equation where the unknown parameters to be determined are
DH1, DH2, DHx, c0, k0 and Xm0. A non-linear regression has been
performed for this six-parameter GAB equation (Maroulis et al.,
1988; Samaniego-Esguerra et al., 1991). This third method has
been described as direct regression and has been recommended for
predicting the GAB parameters (Samaniego-Esguerra et al., 1991).
Direct regression considers and incorporates the temperature as a
variable in the GAB equation. Hence, once these six unknowns are
calculated, it is possible to calculate and predict the parameters of
the GAB equation and consequently the moisture sorption behavior
at other temperatures. However, the determination of six unknown
parameters using regression analysis is quite challenging. Quirijns,
van Boxtel, van Loon, and van Straten (2005a) argued that the
physical meaning of the parameters of the GAB equation may get
lost during the direct regression process, and these parameters are
difficult to determine with high precision and within a narrow
confidence interval. Both direct and indirect regression methods
were used in this study to fit the experimental data to the GAB
equation.
Table 3 shows the calculated parameters for the GAB equation
using the indirect regression methods. As found for the indirect
regression method, the values of the monolayer moisture contents
(Xm) decreased slightly as the temperature increased, whereas the
parameter k slightly increased, and the parameter c also increased
up to 40 C. A slight decrease in parameter c was observed at 50 C.
The same increasing and decreasing trend for the parameter c has
been reported in the literature for onions, apricots and green beans
(Djendoubi Mrad et al., 2012; Samaniego-Esguerra et al., 1991)
using the indirect regression method. All the calculated parameters
in this study fall within the range found for c, k and Xm values for
food products. It has been suggested that, in order to represent a
sigmoid type of sorption isotherm, the parameters of the GAB
equations should be kept in the following ranges: c > 5.67 and
0 < k < 1 (Lewicki, 1997). The monolayer moisture content has also
been found to be comparable with the data found in the literature
for similar fruits such as 12% (w/w) for dried apple, 17% (w/w) for
dried sour cherry and 12% (w/w) for dried black currants (Klewicki
et al., 2009).