Journal ofFood Engineering 7 (1988) 63-78

Application of the GAB Model to the Moisture Sorption Isotherms for Dried Fruits

Z. B. Maroulis, E. Tsarni, D. Marinos-Kouris

Department of Chemical Engineering, National Technical University, GR-10682 Athens, Greece

and

G. D. Saravacos

Department of Food Science, Rutgers University, New Brunswick, NJ 08903, USA

(Received 9 March 1987; revised version received and accepted 10 July 1987)

ABSTRACT

The moisture adsorption isotherms for dried raisins, figs, prunes, and apricots were determined at 15, 30, 4.5, and WC, using the gravimetric static equilibrium method. The applicability of the GAB (Gug- genheim-Anderson-de Boer) model to the experimental results was found to depend on the regression method employed. So,’ using direct non-linear regression analysis, five constants of the GAB model were estimated which enables the equilibrium moisture content of each driedhit in the temperature range 15-60°C to be predicted with reasonable accuracy. The indirect (successive) non-linear regression method is not recommended since the basic three constants of the GAB equation are interrelated.

INTRODUCTION

Several empirical and semi-empirical equations have been proposed for the correlation of the equilibrium moisture content with the water activity of food products. Among them the GAB (Guggen- heim-Anderson-de Boer) equation has been applied successfully to var- ious foods (van den Berg, 1984), and it is recommended by the European project COST 90 on Physical Properties of Foods (Wolf et aZ.,

63

Journal of Food Engineering 0260-8774/88/$03.50 - 0 Elsevier Applied Science Publishers Ltd, England, 1988. Printed in Great Britain

64 Z. B. Mmmlis, E. l‘sami. D. Murinos-Kouris, G. D. Saravacos

1984). The GAB model contains three constants which have physical meaning, two of them being functions of temperature.

The GAB model was applied with some difficulty to the water adsorption isotherms for dried raisins, using linear regression analysis (Saravacos et al., 1986). Dried fruits contain a high percentage of sugars (monosaccharides), which give the characteristic shape of their sorption isotherms, with low moisture contents at low water activities and sharp increases in moisture at high water activities. In an investigation of the storage stability of dried fruits, the adsorption isotherms for dried raisins, figs, prunes, and apricots were determined at four temperatures, 15, 30, 45, and 60°C (Saravacos et&., 1987).

Linear regression of the GAB model has been found less accurate than non-linear regression, because the transformation of the GAB model into a linear equation (in respect of the three constants), results in large mean relative errors between the experimental and the predicted values of the equilibrium moisture content (Schar and Riiegg, 1985).

The GAB model has been applied accurately to the sorption iso- therms for ground coffee in the range of 20 to 80°C (Weisser, 1985). However, the GAB model was less satisfactory in predicting the effect of temperature on the adsorption isotherms for dried raisins, presumably because of the dissolving of fruit sugars at high water activities (Sara- vacos et al., 1986).

The purpose of the present investigation was to apply a non-linear regression technique to the GAB model, using experimental moisture adsorption data for four dried fruits at four temperature levels, and to estimate the joint confidence regions of the constants estimation.

NON-LINEAR REGRESSION OF THE GAB MODEL

Mathematical model

The GAB equation is normally written in the following form,

CK W&,

w=(I-KLI,)(l-Ka,+c~u~,) (1)

where W is the moisture content of the material on a dry basis (kg water/ kg dry solids), u, is the water activity, and W,,,, C, K are constants, defined as follows: W, is the moisture content (dry basis) corresponding to an adsorbed monolayer (BET ) and C and K are constants related to the temperature effect.

Application of the GAB model 6.5

C= C,, exp(AH,/RT) (2)

K = K,, exp(AH,/RT) (3)

where T is the absolute temperature (K), R is gas constant (R = 8.31 kJ kmoll ’ K- I), and AH,, AH, are functions of the heat of sorption of water:

AH,=H,-H,,, AH2=H1-H,,.

Here, H,,, and H,, are the heats of sorption of the monolayer and the multilayer of water respectively, and H, is the heat of condensation of water vapor at the given temperature (kJ kmol- I). CO and K, are adjusted constants for the temperature effect.

Regression analysis

There are two methods of regression analysis for estimating the con- stants of the GAB equation from experimental moisture sorption data; the direct and the indirect method. In the direct regression method the five GAB constants IV,,,, CO, K,, AH, and AH, are estimated by substi- tuting eqns (2) and (3) into eqn (1). All the experimental data are used directly with the same weight.

In the indirect (successive) method, the three GAB constants W,, C and K are estimated at each temperature by regression analysis of eqn (1 ), and then the constants CO, K,, AH, and A H2 are estimated by regression analysis of eqns (2) and (3). The experimental data are grouped at various temperatures for the first regression, and they are subsequently represented by derived values. The derived values may introduce significant uncertainties into the second and third regressions. Thus, the application of the indirect (successive) regression method depends very much on the confidence limits and regions of the constants IV,,,, C and K, which are obtained from the first approximation.

In both linear and non-linear parameter estimation the sum of squares of the residuals between the experimental and the predicted values (SST ) is given by the following equation (Beck and Arnold 1977):

(4)

where vV is the experimental value of the moisture content of the jth replicate of the ith experiment, yi is the predicted value of the model for the ith experiment, Al; is the number of replicates in the ith experiment, N is the total number of experiments, and /3,, & . . ., /3p are the (p) parameters of the model.

66 Z. B. Maroulis, E. Tsami, D. Marinas-Kouris, G. D. Saravacos

The sum of squares of the residuals (SST ) can be considered to consist of the lack of fit of the regression line and the pure error, as given by the equation,

SST = ss, + ss, (5) where

S&= 5 n;(QV;-y,)’ (6) ,=I n; I#,

S&= c c (Vi,-,;)? (7) i= 1 j= 1

Vi= C V,ln, (8) /= I

SS, is the sum of squares of the local mean about the regression line (lack of fit sum of squares), and SS, is the sum of squares within the data sets (pure error sum of squares).

The mean square deviations of the local sum of squares (Sk) and the error sum of squares (Si) can be calculated from the following equations:

S;=SS,/(N-p) (9)

S;=SS,/(M- N) (10)

M=?n, (11) I=1

where (N-p) and (M -N) are the degrees of freedom for S, and S, respectively. M is the total number of experimental measurements (including the replicates), N is the total number of experiments, p is the number of parameters, and H, is the number of replicates of the ith experiment.

In a good mathematical model, the mean deviation between predicted and real values (S,) should be close to the mean experimental error (S,).

Confidence intervals and regions

Since the three constants of the GAB equation ( W,, C, K) for a given food have physical significance, it is essential to know the precision of their estimation, if they are to be used in correlations and predictions of the equilibrium moisture content of a given food product at various temperatures. A good measure of the precision of the estimated con- stants is given by the joint confidence intervals and regions (Beck and Arnold, 1977).

Application of the GAB model 67

A joint confidence region of the three GAB constants is the region in the three-dimensional space where the real values are most probably located. For non-linear models, the joint confidence regions can be estimated from the equation,

KS,-= &(min) 1 +p F, -ai,~,,2,_,,~ M-P 1 I (12)

where SS,,i,, is the minimum of SS,, and F, _ a(p, M _p) is the value of F- distribution for p and (M-p) degrees of freedom and probability (1 -a).

The minimum of SS, was obtained using the non-linear method of Levenberg-Marquart in the form of the algorithm of subroutine ZXSSQ/IMSL (International Mathematical and Statistical Library).

EXPERIMENTAL PROCEDURE

The adsorption isotherms for four dried fruits were determined using the static gravimetric method, developed and standardized in the European COST 90 project (Wolf et al., 1984). Ten saturated salt solutions were used to maintain constant water activities at four temperatures, 15, 30, 45 and 60°C (Greenspan, 1977).

Commercial samples of the following Greek dried fruits were used in these experiments: Sultana (seedless) raisins, figs, prunes and apricots. Samples of about 400 mg of each dried fruit were prepared by cutting thin slices of the fruit approximately 1 mm thick and predrying in a desiccator over P,O, for 15 days. Five dry samples were placed in each of 10 hygrostats each at different constant relative humidity and the equilibrium weight was measured (after 15 days at the constant tempera- ture). Three replications were made at each relative humidity, using five samples per replication. The moisture content of each sample (W, kg water kg dry solids - ’ ) was based on the dry weight, which was deter- mined after each adsorption experiment by the vacuum oven method at 50 Torr and 70°C until constant weight (Saravacos et al., 1986).

RESULTS AND DISCUSSION

Equilibrium moisture contents

The results of the experimental measurements of the equilibrium moisture contents of dried raisins, figs, prunes and apricots, at four

68 Z. 3. Marouiis, E. Tsami, D. Marinos-Konris, G. D. Saravacos

TABLE 1 Equilibrium Moisture Content of Dried Fruits

Temperature Water

(“U rrcti\?Q Raisins

Moisture coruent (% dr?/ basis)

Figs Prunes Apricots

15

30

45

60

r 0.113 0.234

0.333 0432 1bSS9

0.607 0.741 0.756 0.859

\ O.Y20

I- 0.113

0.216 0.324 o--I32 0.5 14

< 0~561) O-69 1 0.75 1

0.836 . 0wK~

c 0~112 0.195 0.3 1 1 Cl.432

0.469 0.5 20 0.640

0.745 0,817

\ 0.880

’ 0~110 0.160 0.293 0.432

1 0440 0.497 0~580 074s o.tio3

c 0.840

64 5.7 3.6 6.7 6.4 3.8 7.4 7.2 4.9

104 9.2 6.0 I -1.Y 14.5 11.2 17% 17.2 14.7 29.9 31.1 25.7 33.1 32.3 26.X 54.7 53.0 49.7 7 1.2 72.7 69.8

3%) 3.1 2.4

3.6 3.6 3.8 4.7 S.3 3.7 x.3 8.2 5.6

11.7 11.8 X.6 I-bz? 14.3 11.3 75.1 25.2 20.5 334 34.1 2x.4 51.x 53.3 47.1 73.6 75.4 706

2.0 09 1.1 2.x 1.5 1.2 -1.0 3.5 2. 1 7.5 6.7 4.8 0.2 7.8 5.6

1 I.5 93 X.0 10.2 18.4 15.1 309 34.0 2Y.Y 44.2 43.1 44.5 7-l. I 74.6 71.7

I.1 0.4 0.2 I.9 1.1 0.9 2.1 2.6 1.4 6.5 54 4.0 7.0 6.2 3.1 Y.1 8.3 6.0

124 12.3 0.2 26.Y 2Y.O 25.3 35.6 3W 37.’ .__ 73.2 69.8 73.3

Application of the GAB model 69

temperatures, 15, 30, 45 and 60°C are shown in Table 1. The equilib- rium moisture content at each water activity represents the mean value of the three replications, each replicate being the mean of five samples.

The results in Table 1 (see Figs 1, 3 and 4) show the typical rela- tionship of high-sugar foods, with equilibrium moisture contents increas- ing sharply at high water activities, due to the dissolving of fruit sugars. Temperature had the expected negative effect on equilibrium moisture content at low water activities, but the reverse effect was observed at higher water activities. Similar results have been found for raisins in the temperature range of 20 to 35°C (Saravacos ef al., 1986).

The mean standard deviations of the equilibrium moisture contents of the dried fruits for the four temperatures were, raisins 1.216, figs O-778, prunes 0.524, and apricots 0.604. The standard deviation varied signifi- cantly with the water activity of the dried fruit. Large deviations were found at high water activities because of the difficulty of determining a constant equilibrium moisture content, due to the gradual dissolving of fruit sugars.

The high moisture contents and the corresponding large standard deviations at high water activities are related to the sugar content of the dried fruits. The sugar content of the four dried fruits is (Oh dry basis), raisins 82%, figs 64%, prunes 52%, and apricots 5 1% (Young, 1975).

Non-linear regression

The results of the indirect (successive) non-linear regression analysis of the experimental results for the raisins are shown in Table 2 and Fig. 1. In this analysis the following numerical values were used: Number of experiments N= 10, replications nj = 3, total experimental measure-

TABLE 2 Indirect Regression Analysis of GAB Model for Sultana Raisins

Temperature

(“C) will

GAB comtatIts Standard deviation

C K SR S,

15 16.7 1.02 0.891 2.02 1.37 30 26.5 0.42 0.897 0.80 1.81 45 10.2 1.30 1 .o 1s I.08 1.26 60 8.1 1.54 lW8 1.61 1.71

Root mean square value 1.46 1.55

70 Z. B. Maroulis, E. Tsami, D. Marinos-Kouris, G. D. Saravacos

.o .2 . r. .6 .8 1.3

%

%W i 3o"c

Fig. 1. Experimental values and predicted isotherms for moisture sorption for sultana

raisins at 15°C. WC, 45°C and 6o”C, using the GAB equation: (+). experimental points: ( 1). indirect; and t.21, direct non-linear regression.

1

%W

.QO.

80.

60.

40.

20.

Application of the GAB model 71

.o .2 .4 .6 .B 1. 0

Fig. 1 - contd.

72 Z. B. Maroulis, E. Tsami, D. Marinos-Kouris, G. D. Saravacos

ments M = 30, and parameters p = 3. Each of the three replications represents the mean value of five samples. The standard deviations S, were found to be lower than the pure error standard deviations S,, except for the temperature of 15°C. The GAB model can therefore be considered statistically satisfactory. A similar conclusion was reached using linear regression analysis (Saravacos et al., 1986).

The predicted values of the constant IV, at 15°C and 30°C (16.7% and 26*5%, respectively) are considered too high to correspond to the ‘monolayer’ value for a dried fruit. There is no clear correlation of the value of C with the temperature, as would be expected from eqn (2). The value of K appears to increase slightly with temperature. Weisser (1985) found that for ground coffee in the range 20-8O”C, the value of C decreased with temperature, whereas IV, and K did not vary signifi- cantly.

The results of the direct non-linear regression analysis for the raisins are shown in Table 3 and Fig. 1. The standard deviation S, was found to be about 1.5 times greater than the pure error standard deviation S,, and the GAB model predicts well the moisture content (Fig. 1). The only significant disparity is the first experimental point on the 15°C isotherm.

TABLE 3 Direct Regression Analysis of GAB Model for Sultana Raisins

Temperature PC)

W,,

GAB constams

C‘ K

Statdard deviatiotls

15 12.5 1.966 0.933 30 12.5 1.237 0.963 s, = 2.47 45 12.5 0.813 0.99 1 s, = 1.55 60 12.5 0~555 1+)17

C,,= 1.7 x lWJ, K,,= 1.77, AH, = 22.4. AH? = - I.53

The direct regression method resulted in a constant value for the ‘monolayer’ of the raisins ( W,,, = 12*5%), which is relatively high for a dried fruit. The value of C decreased gradually with temperature, while K increased slightly, in agreement with the results of Weisser (1985) for ground coffee.

The values AH, = 22.4 kJ mol- ’ and AH? = - 1.53 kJ mol- ’ re- present the mean values of the heats of sorption of water on the dried fruit. AH, is the difference in enthalpy between monolayer and multi-

Application of the GAB model 73

layer sorption, which is expected to have a large positive value, due to the strong exothermic interaction of water vapor with the primary sorption sites of the raisins. The small negative value of AH, corresponds to a heat of sorption of the multilayer slightly greater than the heat of con- densation of water. It should be noted that, at high moisture contents, AH, may reach a slightly positive value, due to the endothermic dissolu- tion of fruit sugars (Saravacos et al., 1986).

Figure 1 shows the experimental adsorption results and the predicted isotherm by the indirect (1) and the direct (2) regression methods for raisins at four temperatures, 15, 30, 45 and 60°C. Both regression methods gave equally satisfactory results.

Joint confidence regions

The uncertainty of the values of constants, obtained by the indirect (successive) regression method makes it necessary to examine the joint confidence regions. Figure 2 shows the joint confidence regions of the three constants W,,,, C, and K of the GAB equation for the raisins at 15, 30, 45 and 60°C. The values of the constants in Table 2 were used, ;;d(FeCesults are presented in pairs: (a), ( IV,.,,, C); (b), ( IV,, K); and c, 3 *

The central points ( -t ) of the elliptical curves (contours) of Fig. 2 represent the estimated values of the constants (Table 2) and the three concentric curves refer to 50, 80 and 99% probability levels, cor- responding to ratios SSr/SSTl,i,, equal to 1.05, 1.20 and 1.50.

From Fig. 2 it is clear that C is estimated with great uncertainty, particularly at high temperatures. C is strongly correlated with the two other constants W,,, and K and therefore the effect of temperature on C cannot be estimated precisely. K is estimated with relatively high preci- sion, but is correlated best with W,,, at high temperature and C at low temperature. Accordingly, the slight increase in K with temperature should be considered as uncertain. W, is estimated with moderate precision as compared with C and K and it is correlated with C and K at low temperatures.

The uncertainties of the three constants of the GAB model indicate that the second stage of the indirect regression method should not be attempted. The direct regression method is therefore recommended.

Table 4 shows the results of the direct non-linear regression of the GAB model for the dried raisins, figs, prunes, and apricots in the tem- perature range 15-60°C. In addition, the GAB constants for an ‘average’ dried fruit are given, estimated by direct regression of all the experi- mental data of the four dried fruits. It will be noted that dried figs and

74 Z. B. Maroulis, E. Tsami, D. Marinos-Kouris. G. D. Saravacos

3G. , I

%wm li_l SST (min)

w =1.50 ,.20

1.05 12.

45OC r -= 6OOC

36.

1

;I2 Fig. 2. Joint confidence regions of the GAB constants W,, C, and K.

Application of the GAB model 75

I. lo/T------ 60°C

-

_ 801 I I I I

.O .8 1.6 2. C

Fig. 2 - corztd.

TABLE 4 GAB Constants by Direct Regression for Dried Fruits

Dried fruit

W,

GAB constants

Cox IO4 AH, K, AH? & &

Raisins 125 1.7 22.4 1.77 - 1.53 2-47 1.55 Figs 11.7 0.5 25.2 1.77 - 1.55 2.08 0.92 Prunes 13.3 0.7 23.9 1.82 - 1.65 1.75 0.70 Apricots 15.1 1.1 21.1 2.13 - 2.05 1.88 0.74 ‘Average’ dried fruit 12.4 0.9 23.4 1.83 - 1.63 3.02 1.04

prunes have effectively the same isotherm as the ‘average’ dried fruit. Figure 3 shows the moisture sorption isotherms of the ‘average’ fruit at the four temperatures, as predicted by the GAB model.

A comparison of the moisture sorption isotherms of the four dried fruits and the ‘average’ fruit at 3O”C, as predicted by the GAB model, is shown in Fig. 4. Raisins sorb the greatest amount of moisture, followed

76 Z. B. Marouiis, E. Tsami, D. Marinos-Kouris, G. D. Saravacos

80.

60.

40.

20.

0.

0 .2 .4 .6 .a I. 0

"W

Fig. 3. lsotherms for moisture sorption by an ‘average’ dried fruit, predicted by the GAB model.

by figs, prunes, and apricots in turn, corresponding to the respective sugar contents of the four dried fruits (Young, 1975).

CONCLUSIONS

The successful application of the GAB equation to the moisture sorption isotherms for these dried fruits depends on the regression method employed on the experimental results. Direct non-linear regression of the GAB model yields for each fruit five constants, which describe adequately the experimental measurements with a precision comparable to the experimental error.

The estimated constants of the GAB model W,,,, C,,, K,,, AH,, and A Hz have relatively low precision and they are correlated among themselves. The constants AH, and A H2 have physical significance (heats of sorption) and they are acceptable. The estimated values of W, are relatively high for the ‘monolayer’ moisture content of the dried fruits. It is concluded that the GAB model can be used adequately for the

Application of the GAB model 77

80.

%W

60.

40.

20.

.2 .4 .6 .a 1. 0

aW

Fig. 4. Comparison of the isotherms for moisture sorption by dried fruits at 30°C. predicted by the GAB model.

representation of the sorption isotherms for the four dried fruits, raisins, figs, prunes, and apricots.

REFERENCES

Beck, J. V. and Arnold, K. J. (1977). Parameter Estimation in Engineering and Science, John Wiley, New York, pp. 17%80,360~86.

Greenspan, L. (1977). Humidity fixed points of binary saturated aqueous solutions. 1. Rex Nat. Bur. ofstandards, 81A (l), 89-96.

Saravacos, G. D., Tsiourvas, D. A. and Tsami, E. (1986). Effect of temperature on the water adsorption isotherms of Sultana raisins. 1. Food Sci., 5 1,38 1-3.

Saravacos, G. D., Tsami, E. and Marinos-Kouris, D. (1987). Effect of water activity on the loss of volatiles of dried fruits. 5th Znt. Flavor Con& Port0 Carras, Greece, July 1-3, 1987, G. Charalambous, ed., St Louis, Missouri.

Schar, W. and Riiegg, M. (1985). The evaluation of GAB constants from water vapour sorption data. Lebemm- Wiss. u. Technol., 18,225-9.

van den Berg, C. (1984). Description of water activity of foods for engineering purposes by means of the GAB model of sorption. In: Engineering and Food, Vol. 1, B. M. McKenna, ed., Elsevier Applied Science, London, pp. 3 11-2 1.

78 Z. R. Maroulis, E. Tsami, D. Marinos-Kouris, G. D. Saravacos

Weisser, H. (1985). Influence of temperature on sorption isotherms. In: Food Engineering and Process Applications, Vol. 1, M. LeMaguer and P. Jelen, eds, Elsevier Applied Science, London, pp. 189-200.

Wolf, W., Spiess, W. E. L., Jung, G., Weisser, H., Bizot, H. and Duckworth, R. B. ( 1984). The water-vapor sorption isotherms of microcrystalline cellulose and purified potato starch. Results of a collaborative study. J. Food Engineering, 3,5 l-73.

Young, C. T. (1975). Composition and nutritive value of raw and processed fruits. In: Commercial Fruit Processing, J. G. Woodroof and B. S. Luh, eds, Avi Publishing, Westport, Connecticut, 539-79.