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Use of the Wilson equation for the prediction of the sorptional equilibrium of sugar-based foodstuffs
Fluid Phase Equilibria, 78 (1992) 191-207 Elsevier Science Publishers B.V., Amsterdam
191
Use of the Wilson equation for the prediction of the sorptional equilibrium of sugar-based foodstuffs G.K. Vagenas and D. Marinos-Kouris Department of Chemical Engineering, National Technical University, GR-157 73 Athens (Greece) (Received April 4, 1991; accepted in final form April 14, 1992)
ABSTRACT Vagenas, G.K. and Marinos-Kouris, D., 1992. Use of the Wilson equation for the prediction of the sorptional equilibrium of sugar-based foodstuffs. Fluid Phase Equilibria, 78: 191-207. The concepts of phase equilibrium were applied for the representation of the water activity of five dried fruits characterized by high sugar content (raisins, currants, figs, prunes and apricots) as a function of moisture content, using several thermodynamic equations for the calculation uf the activity coefficients. The results showed that the Wilson equation, modified to include the effect of large cyclic molecules (sugars), can be used with considerable accuracy. The estimated values of the unknown parameters provide an insight into the nature of the interaction between the molecules of sugars and water, and its variation with temperature.
INTRODUCTION
The importance of water activity in relation to the quality of food products and their stability is widely recognized (Van Den Berg and Bruin, 1981). It is defined as the ratio of the fugacity of water in a solution to that of pure water. Fugacity may be approximated by partial vapor pressure under normal conditions of temperature and pressure, and this justifies the commonly used definition:
Correspondence to: D. Marino-Kouris, Department Technical University, Athens GR-157 73, Greece. 0378-3812/92/$05.00
of Chemical
Engineering,
0 1992 Elsevier Science Publishers B.V. All rights reserved
National
192
G.K Vagenas and D. Marinos-Kouris /Fluid Phase Equilibria 78 (1992) 191-207
where a, is the water activity, P, is the vapor pressure of water over the solution, and Pi is the vapor pressure of pure water at the temperature of the solution. The system foodstuff-moist air in equilibrium can be studied as a two-phase equilibrium system, consisting of a “solution” phase, containing solid constituents and water, and a gas phase, containing water vapor and air. Depending on the amount of water in the solution phase, the foodstuff can be considered as a concentrated solution with more or less “solid” appearance (low moisture content, sorbed water) or as a dilute solution of “liquid” appearance (high moisture content, dissolved solid constituents). The main difference between a true solution and a condensed phase made up of an adsorbent and the adsorbate attached to it, is in the size of the adsorbent particles, which is finite. Such a “solution” cannot be reduced to the infinitesimal size required for the application of the thermodynamic equations. However, similar approximations have been made in normal solutions and we assume here that the physical process of adding or removing one particle is equivalent to the infinitesimal mathematical increment. The second difficulty has to do with the homogeneity criterion, i.e. that all particles making up the “solution” have the same specific properties. We will assume this assumption to be met hereafter. These points have been analyzed very successfully by Hill (1950), and Le Maquer (1985). The development of mathematical correlations for the prediction of the water activity as a function of moisture content, is essential for many practical design problems in the food industry, such as dehydration, packaging and storage. However, most attempts have been confined to sorption models, which are strictly applicable only at very low moisture content, where the food cells are no longer acting as water reservoirs (Rotstein and Cornish, 1978). In contrast, thermodynamic correlations based on the solution concept are scarce in the literature, although they seem very promising because they have a more or less theoretical origin; moreover, some of these equations can be used for the prediction of the sorptional equilibrium of a foodstuff, using only parameters obtained from sorption characteristics of the constituents. Bruin and Prausnitz (1971) and Gmehling and Onken (1977) calculated the water activity of water in liquid organic mixtures. Chandrasekaran and King (1972) and Marinos-Kouris and Saravacos (1975) studied the vapor-liquid equilibria of organic components in aqueous sugar solutions. However, their studies were confined to very high moisture contents of the system, where the solution phase is liquid. The objective of this work was to apply the concept of phase equilibria for the calculation of the water activity of five dried fruits with high sugar content, namely raisins, currants, figs, prunes and apricots, as a function of moisture content, using the Wilson equation for the calculation of the
G.K Vagenas and D. Marinas-Kouti /Fluid
Phase Equilibria 78 (1992) 191-207
193
activity coefficients. In addition, other thermodynamic relationships described in the literature were also tested as to their ability to predict water activity values for the entire moisture range, from the bone-dry fruit to full turgor.
EXPERIMENTAL
DATA
The experimental data of the water activity as a function of moisture content were taken from the recent literature (Maroulis et al., 1988; Tsami et al., 1990). They comprised measurements for the five dried fruits (raisins, currants, figs, prunes and apricots) at four temperatures (15, 30, 45 and 60°C) and at atmospheric pressure. The method used was the static gravimetric method developed by the European Cooperative Project COST-90bis on Water Activity of Foods (Spiess and Wolf, 1983).
MATERIALS
The five dried fruits used in this study are characterized by high sugar content, especially monosaccharides (glucose and fructose) and, to a lesser degree, disaccharides (sucrose). Other constituents are polysaccharides (mainly cellulose), proteins and minerals, but their concentration is very low compared to that of the sugars. Typical compositions of the five dried fruits, according to Young (19751, are given in Table 1.
TABLE 1 Composition
(weight %o)of the five dried fruits (Young, 1975)
Constituent
Raisins
Currants
Water Total sugars glucose fructose sucrose Proteins Fat Cellulose Minerals Total
14 69 36 32 1 3 3 6 5 100
1.5 70 36 34 0 2 1 1 10 100
Figs
Prunes 23 55
24 57 31 26 0 3 1 5 10 100
39 14 2 2 1 2 16 100
Apricots 22 57 26 16 15 3 1 4 13 100
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G.K Vagenas and D. Marinas-Kouris /Fluid Phase Equilibria 78 (I 992) 191-207
MATHEMATICAL ANALYSIS
For an ideal dilute solution, Raoult’s law of water activity may be written as
where x1 is the mole fraction of water in the solution. However, foodstuffs deviate greatly from ideality (Van Den Berg and Bruin, 1981) and require the use of an activity coefficient. Thus, eqn. (2) should be replaced by %
=
(3)
YlXl
where y1 is the activity coefficient of water. The mole fraction of water is calculated from the relationship X
(4)
x1 =
where X is the moisture content of the foodstuff (mass of water/mass of dry matter), N is the total number of constituents, m, is the mass of constituent i, d is the total mass of dry matter, and M,, Mi are the molecular weights of water and constituent i, respectively. The activity coefficient y1 can be calculated using one of several thermodynamic relationships. Wilson (1964) proposed the following equation for the calculation of the excess Gibbs energy of mixing for a binary system: gE/RT
=. -x1
ln(x, + f&n,)
-x2 ln(&x,
+x2)
(5)
where gE is the excess Gibbs energy of mixing, T is absolute temperature, xi is the mole fraction, and h12, A21 are adjustable parameters related to the pure component molar volumes and to characteristic energy differences:
R is the gas constant,
A12=
$ q( -
A’;TA1l)
(64
A,,=: exp(-A12~~22)
WI
where ui is the molar volume of component i and the A’s are energies of interaction between the molecules designated in the subscripts. The activity coefficient y1 is given by In y1 = -ln(x,
+A12x2) +x2 x +Ar x - A t2i, 1 12 2 211
2
(7)
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Phase Equilibria 78 (1992) 191-207
195
Although the Wilson equation has been applied to a variety of mixtures (Orye and Prausnitz, 1965; Silverman and Tassios, 1977), its application has been limited to systems whose components do not differ greatly in molecular size. The system solid-foodstuff-water could be effectively considered as a binary mixture of sugars and water, where the sugars (mainly glucose and fructose and, to a lesser degree, sucrose> have an effective molecular size between that of the monosaccharide (C,H,,O,) and the disaccharide (C,,H,,O,,). However, the equilibrium properties of the system cannot be predicted by the corresponding properties of the sugars alone, because the existence of other components, such as proteins and minerals, even at very low concentrations, can affect the properties of the solution to an unpredictable degree. For this reason, the unknown parameters of the Wilson equation will be estimated by fitting the calculated values to the experimental data. The system sugars-water is a mixture of high complexity because both constituents have peculiar characteristics. The monosaccharides (glucose and fructose) are aliphatic compounds. However, in pure form, as well as in solutions, the prevailing form is of a cyclic type (semi-acetal). The adsorption of water by these molecules can be described with considerable accuracy by the assumption of a number of possible hydration sites, equal to the number of oxygen atoms in the molecule. These are polar groups capable of hydrogen bonding with water molecules. Thus, glucose and fructose have six possible hydration sites in their molecule (Stokes and Robinson, 1966). The sucrose molecule consists of two monosaccharide molecules and can be considered to have eleven possible solvation sites. Two more effects are of importance for the system sugars-water. One is the wide disparity in size between the molecules of water and the sugars which are our chief objects of interest. The other is the “structured” nature of liquid water. On the basis of the above considerations, it was necessary to redefine the local volume fractions introduced by Wilson (1964). If we consider the water as the solvent (1) and the sugars as the solute (2), then the local volume fraction of solvent molecules about a central solvent molecule is
511=
ew( -A,,/RT) exp(-All/W + uox2 ew( -A,,/RT) 4x1
WI
0%
and the local volume fraction of solute molecules about a central solvent molecule is
521 =
ulxl
uox2
exp(-A12/W
exp( -All/IV)
+
uox2
exp(
-A,,/RT)
(3
G.X. Vagenas and D. Marinas-Kouti /Fluid Phase Equilibria 78 (I 992) 191-207
196
Here we surrounded by a number of hydration sites, rather than whole solute Thus, 511 + 521 = 1. The local volume fractions about a central solute molecule, however, are those defined by Wilson (1964), because the solute molecule is by molecules of water. Thus u2x2
‘22 =
ulxl
exP(-A,,/W
exp( -h,,/RT)
+
u2x2
exp( -h22/RT)
(94
+x1 exp( -A,,/RT) ‘12 = ulxl exp( -A,,/RT)
+ u2x2 exp( -A,,/RT)
and 522 + 5r2 = 1. The introduction of these new volume fractions, eqns. (8) and (9), into the analysis of Wilson does not change the form of eqn. (7), but the parameter Al2 given by eqn. (6a) now becomes
A,,=$ exp(-A1:tll) The molar volume of the polar group of the solutes can be calculated by
uo= u2/q
(11)
where q is the number of polar groups (hydration sites) of the solute molecule. For the purpose of our analysis, this is estimated for every dried fruit as a weighted average of the hydration sites for monosaccharides and disaccharides. The energy parameters A12, A,, and A,, are determined by the intermolecular forces between sugars and water. The differences A,, - A,, and A,, - A,, are usually considered independent of temperature, at least over modest temperature intervals. However, this is unlikely to occur in the case of the strongly hydrogen-bonded system sugars-water over the temperature interval of 15-60°C used in this study. There are two methods of regression analysis for estimating the temperature dependence of these parameters from the experimental data of moisture content versus water activity: the indirect and the direct method. In the indirect (or successive) method, the parameters A,, - A,, and A,, - A,, are estimated at each temperature by regression analysis of eqn. (3) and the calculated values are correlated with temperature using a second regression analysis. However, this approach may introduce significant errors if there is some uncertainty in the estimated values of the unknown parameters.
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Phase Equilibria 78 (1992) 191-207
197
In the direct regression method, a specified temperature dependence of the parameters is assumed, usually in the form of eqn. (13), see below, and the constants b,, b,, b,, b4 are estimated by substituting eqn. (13) into the analysis. All the experimental data are used directly with the same weight. In both methods, the unknown parameters were estimated by fitting the mathematical model to the experimental data. This procedure minimizes the sum of the squares of the residuals between the experimental and the predicted values, using the method of non-linear regression (Draper and Smith, 1981). For the requirements of this study, we used a modified Marquardt-Levenberg method, which ensures fast convergence, even when the initial guesses of the parameters are poor.
RESULTS AND DISCUSSION
The experimental data of the moisture content versus water activity for the five dried fruits are in exact agreement with the corresponding data of Norrish (1966) for sucrose solutions. This is an indication that the sorption behavior of the dried fruits is mainly controlled by the sugar constituents and, hence, the assumption that they can be represented as a mixture of sugars and water is justified. The results of the indirect (successive) regression for the five dried fruits are given in Table 2. The standard deviation due to regression sR is very close to the pure error standard deviation sn and, therefore, the predictions of the Wilson equation are satisfactory. The estimated parameters A,, - A,, and A,, - A,, show a strong temperature dependence which must be attributed to the fact that the cohesive forces between the two molecules depend on temperature. For low temperatures (15-3O”C), A,, - A,, < 0, which means that the hydrogen bonds between sugars and water are stronger than the hydrogen bonds between water molecules. Water molecules would “prefer” sugar molecules as neighbors rather than alike molecules. In this way, the adsorption of water molecules on the food surface takes place spontaneously and is favored. However, A,, - A,, > 0, which means that the cohesive forces between sugar molecules are stronger than those between dissimilar molecules. Dissolution cannot take place spontaneously. As a result, at low temperatures, the adsorption of water molecules is favored over the dissolution of solute molecules in water. However, at higher temperatures (30-6O”C), this changes drastically. The difference A,, - A,, becomes greater than zero, which means that the water molecules “prefer” water molecules as neighbors. However, A,, - A, < 0, which means that the cohesive forces between sugar molecules and water are stronger than those between sugar molecules. Dissolution is
198
G.K Vagenas and D. Marinos-Kouris /Fluid Phase Equilibria 78 (1992) 19X-207
TABLE 2 Results of the indirect regression analysis of the Wilson model for the five dried fruits Product
Temperature (“0
An-Jh,
(kJ mol-‘)
A,,-A22
0c.f mol-‘)
Standard deviation (kg kg-‘) SR
sE
a
Raisins
15 30 45 60
-
4.20 2.05 1.83 0.93
15.29 - 2.43 - 2.85 - 4.04
0.035 0.023 0.019 0.026
0.014 0.018 0.013 0.017
Currants
15 30 45 60
-3.19 - 1.25 3.86 7.62
7.30 - 2.85 - 6.74 -7.17
0.039 0.033 0.021 0.018
0.015 0.011 0.013 0.014
Figs
15 30 45 60
- 3.54 - 2.63 - 1.45 0.91
12.38 - 1.22 - 2.87 - 4.56
0.034 0.021 0.015 0.026
0.012 0.006 0.007 0.011
Prunes
15 30 45 60
- 4.27 -2.31 - 1.34 -0.23
11.21 - 1.03 - 2.34 - 6.51
0.034 0.016 0.011 0.019
0.010 0.005 0.004 0.008
Apricots
15 30 45 60
- 3.67 - 1.29 1.34 3.28
9.25 0.15 - 3.56 - 5.72
0.023 0.017 0.016 0.021
0.011 0.005 0.005 0.008
a Maroulis et al. (1988).
favored over adsorption. These results confirm that adsorption is an exothermic reaction and, therefore, is favored at low temperatures, while dissolution of sugars is an endothermic reaction and is favored at high temperatures. Although the above results seem plausible and indicate a dependence of the parameters on temperature, there is still the question of the accuracy of the estimation of the unknown parameters. Silverman and Tassios (1977) reported that several sets of parameters, widely different in values, can give small values of the standard deviation and are, therefore, equally possible as the result of the regression analysis, when the binary activity coefficients yi < 1. (Actually, this is the case for the system foodstuff-water, as will be mentioned below.) Tassios (1979) reported similar results for the NRTL and LEMF equations. The best way to elucidate this question is to calculate the joint confidence regions of the unknown parameters. A confidence region is the region in the parameter space where the real
G.K Vagenas and D. Marinos-Kouti / Fluid Phase Equilibria 78 (1992) 191-207
B 1
z
ll-
199
60 C
Q.
4.-
7-
I
15 c
D 5 -10
-I5
d
I
10
$
lb
$0
A, 2 - hz2,
Fig. 1. Joint confidence regions of the parameters tures; + , estimated values.
A&,
2’s
kJoule/moie
and A,,-&,, at several tempera-
values of the parameters are most probably located. In the case of non-linear regression, these regions are calculated from the equation (Draper and Smith, 1981)
NW*) -wp R/b -P)
=F
_
la
n
(p
7
_p)
(14 .
where S(b,, bJ is the total sum of the squares of the residuals, bi are the unknown parameters, R is the minimum S, n is the number of experimental points, p is the number of parameters (here p = 2) and f’,_,(p, n -p> is the value of the F distribution for p and n -p degrees of freedom and probability 1 - a. Figure 1 presents the confidence regions of the unknown parameters for all temperatures in the case of raisins. The central points of the contours (+) represent the estimated values of the parameters and the three concentric curves refer to 90, 95 and 99% probability levels. The examination of these contours gives a deep insight into the physical significance of the parameters. Firstly, it is clear that the experimental data at 15°C give a very accurate estimate of Ah,,-A,, (within lo%), but they cannot be used . . for the estimatron of A,, -A,, beyond the obvious result that this difference is always less than zero. Secondly, the experimental data at 30,45 and 60°C give identical values for the parameter A,, - A,, and slightly different
200
G.K Vagenas and D. Marirws-Kouti /Fluid Phase Equilibria 78 (1992) 191-207
values for the parameter A,, -A,,. Thirdly, these data (30, 45 and 60°C) cannot be used for the accurate estimation of A,, - An, which can take any value close to or less than zero, while they give a more or less reliable estimate of A,, -A,,. What these results mean in physical terms is that at low temperatures, the adsorption prevails and the parameter corresponding to dissolution (A,, -A,,) cannot be ,estimated from the experimental data. In contrast, at higher temperatures, dissolution of sugars prevails and the parameter A,, -A,, corresponding to adsorption cannot be estimated accurately. In this way, the conclusions of the previous paragraph are confirmed, but also the limitations of this method of indirect estimation at different temperatures are revealed. Therefore, the second stage of this method was not attempted. Once the dependence of the parameters on temperature has been justified, we can proceed to the direct regression analysis of the proposed model, using all the experimental data. We made the assumption of a linear change of the parameters with temperature, which is the most obvious choice. Although a linear relation is unlikely to occur in a physical problem, it is a good approximation for the temperature interval 1%60°C; besides, the four values of temperature used for the experiments do not permit the use of more complex relationships. Thus:
Pa) Pb)
A,, - A,, = b, + b,T A,, -A,,
= b, + b,T
results of the direct regression analysis for the five dried fruits are given in Table 3. The standard deviation sa is found to be about 3 times greater than sE and, therefore, the Wilson equation predicts the experimental data well. The accuracy of the estimation of the parameters b,, b,, b, and b, is satisfactory, as indicated by the joint confidence regions of the constants b, and b,, which are presented in Fig. 2. Moreover, the conclu-
TABLE 3 Results of the direct regression analysis of the Wilson model for the five dried fruits Product
Constants
Raisins Currants Figs Prunes Apricots
b, - 6.38 - 6.25 -5.72 - 4.27 - 5.92
a Maroulis et al. (1988).
b, 0.154 0.206 0.146 0.138 0.168
b, 18.02 17.36 18.53 16.23 16.89
b, -
0.591 0.625 0.733 0.642 0.698
Standard deviation a SR sE 0.050 0.016 0.040 0.014 0.028 0.009 0.024 0.007 0.022 0.007
G.K. Vagenas and D. Marinas-Kouris /Fluid Phase Equilibria 78 (1992) 191-207
201
-5.0 z
-5.5 1 -6.O-
-6.5 -
-7.0 -
-7.5 -
-6.0 14.0
lJ.0
16:O
I 17.0
.I. 18.0
I. 19.0
2010
2110
2210
b3 Fig. 2. Joint confidence regions of the constants b, and b, of eqn. (13); + , estimated values.
sions drawn so far, regarding the physical significance of the estimated parameters, are confirmed. Although the above results show that the Wilson equation appears especially suitable for this problem, we also attempted to correlate the experimental data for the five dried fruits using several thermodynamic relationships in the literature. The results are compared in Table 4, using the standard deviation between the experimental and the predicted values TABLE 4 Standard deviations sr (kg H,O/kg d.s.) between experimental and predicted values of the moisture content for raisins, using several models in the literature Model (parameters)
Temperature
PC)
15
30
45
60
Margules a (1) Margules b (2) Van Laar (2) Flory-Huggins (1) Wilson (2) NRTL ’ (2) UNIQUAC (2)
0.052 0.038 0.028 0.058 0.036 0.054 0.048
0.035 0.034 0.033 0.040 0.024 0.037 0.034
0.026 0.024 0.022 0.034 0.020 0.029 0.024
0.036 0.032 0.032 0.051 0.027 0.048 0.032
a Two-suffix Margules equation. b Three-suffix Margules equation. c The NRTL equation has three parameters, but only two were fitted. The parameter of the non-randomness in the mixture was given a medium value (0.35).
202
G.K Vagenas and D. Marinas-Kouris / Fluid Phase Equilibria 78 (1992) 191-207
0
T=lS
C
+
T=30
C
aw
Fig. 3. Experimental values (discrete (continuous line) for raisins.
points)
and predictions
of the Wilson
equation
of moisture content sT as a validation criterion. These results showed that the lowest standard deviation was obtained in all cases with the Wilson equation. The Van Laar equation and the UNIQUAC equation also gave satisfactory results, but the Wilson equation is preferable because it is simpler than the UNIQUAC equation and its parameters possess a more obvious physical meaning, compared to those of the Van Laar equation which are strictly empirical. The predictions of the Wilson equation are given in Fig. 3 for raisins and in Fig. 4 for currants. The agreement is satisfactory over the entire range of water activity. The largest deviation occurs for very low water activities (first experimental point), where there is localized adsorption on active sites (monomolecular layer) and a physical adsorption model would be more appropriate. It is worth mentioning that the Wilson equation predicts the effect of temperature on the sorption isotherm of these sugar-based products; the temperature has the expected negative effect on equilibrium moisture content at low water activities, but the reverse effect is observed at higher water activities, and this behavior is well represented by the Wilson equation. An important criterion for the suitability of the Wilson equation in the case of foodstuffs is its ability to predict the experimental data using values of parameters estimated from a limited set of experimental data. This was done in the case of raisins. We estimated the unknown parameters using the experimental data in the water activity range O-0.5, and then we used
G.X. Vagenas and D. Marinas-Kouris/Fluid Phase Equilibria 78 (1992) 191-207
+
T=SO
203
C
Fig. 4. Experimental values (discrete (continuous line) for currants.
points)
and predictions
of the Wilson
equation
these values of parameters to predict the experimental data over the entire range of water activities. Typical results are shown in Fig. 5 and we can see that the predictions are also satisfactory. 1.0 -
0.8 -
0
T=lS
C
+
T=30
C
03
0.5
0.6 -
0.4 -
0.2 -
0.0
0.0
O.‘l
0.i
0.3
oh
0.v
oh
oh
Fig. 5. Experimental and predicted values for raisins using parameters in the water activity range O-0.5.
1 .b
estimated from data
G.K Vagenas and D. Maritws-Kouris /Fluid Phase Equilibria 78 (1992) I91 -207
204
&
m-11-;
‘t;
3
-0.2
1
-0.3 -
-0.4
0.0
O.‘l
03
0.5
0.2
0.3
0.6
o.‘r
oh
oh
X, kg H20/kg
da.
, 1.0
Fig. 6. Predicted values of the excess Gibbs energy of mixing for raisins using the Wilson equation: 1, T = 15°C; 2, T = 30°C; 3, T = 45°C; 4, T = 60°C.
1.0
0.9
>-’ 0.6
0.7
0.6
oh
O.‘l
oh
0.5
0.2
0.3
0.6
0.f
oh
oh
X, kg H2O/kg
d.s.
1.b
Fig. 7. Predicted values of the activity coefficients of water in raisins using the Wilson equation: 1, T = 15°C; 2, T = 30°C; 3, T = 45°C; 4, T = 60°C.
G.R Vagenas and D. Marinas-Kouris /Fluid Phase Equilibria 78 (1992) 191-207
-2.5
’ oh
0.7
I 0.2
0.5
0.14
0.b
0.6
oh
0.b
X, kg H2O/kg
0.i
d.s.
205
1.b
Fig. 8. Predicted values of the excess enthalpy of mixing in raisins using the Wilson equation: 1, T = 15°C; 2, T = 30°C; 3, T = 45°C; 4, T = 60°C.
The predicted excess Gibbs energy of mixing and the activity coefficients of water and sugars are shown in Figs. 6 and 7 respectively for currants, as a function of temperature. The results show that the system foodstuff-water exhibits large negative deviations from ideality at 15°C but it approaches an ideal behavior at higher temperatures. The activity coefficients of water are always less than unity, in agreement with the results of Taylor and Rowlinson (1955) for glucose solutions, Stokes and Robinson (1966) for sucrose and glucose solutions and Norrish (1966) for sugars in general. However, the above authors were confined to high water contents and this prohibits a quantitative comparison. Chen (1987) reported that the activity coefficient of water is very close to unity for most sugar solutions with concentrations up to 40% by weight. The activity coefficients of sugars exhibit a similar behavior, approaching unity at high temperatures. The excess heat of mixing can be calculated from the Wilson equation: hE=
-T2
(14)
Typical plots of the heat of mixing as a function of water content are given in Fig. 8 for several temperatures. Although the results are reasonable (Tsami et al., 1990), the estimation of the heat of sorption is rough,
206
G.K. Vagenas and D. Marinos-Kouti /Fluid Phase Equilibria 78 (1992) 191-207
especially for the low moisture-content region, where the localized adsorption (monomolecular layer) causes extremely high (absolute) values of the heat of sorption.
CONCLUSIONS
The Wilson equation can be used successfully for the prediction of phase equilibria between foodstuffs and moist air. The estimated parameters of this equation, A,, - A,, and A,, - A,, are temperature dependent and reflect the nature of the interaction between the molecules of sugars and water. At low temperatures, the adsorption of moisture on the solid surface takes place spontaneously and is favored over the dissolution of sugars, while at higher temperatures the reverse is true, that is, dissolution of sugars is favored over adsorption.
LIST OF SYMBOLS
water activity mass of dry material (kg) excess free enthalpy of mixing (k3 mol-‘1 excess enthalpy of mixing 0~3 mol-‘) mass of constituent i (kg) molecular weight of constituent i (kg mol-‘1 number of constituents vapor pressure of water over the solution (Pa) vapor pressure of pure water (Pa) number of polar groups in the solute molecule gas constant, 0.008317 (W mol-’ K-‘) absolute temperature (K) molar volume (m3 mol-‘) molar volume of the polar group in the sugar molecule (m3 mol-‘) mole fraction moisture content of the sample (kg H,O per kg dry material) Greek
letters
Aij energies of interaction between molecules i and j (kJ mol-‘) Aij energy parameters of the Wilson model 5 local volume fraction
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Phase Equilibria 78 (1992) 191-207
207
REFERENCES Bruin, S. and Prausnitz, J.M., 1971. Ind. Eng. Chem. Process Des. Dev., 10: 562. (In Van Den Berg and Bruin, 1981.) Chandrasekaran, S.K. and King, C.J., 1972. Multicomponent diffusion and vapor-liquid equilibria of dilute organic components in aqueous sugar solutions. AIChE J., 18: 513-520. Chen, C.S., 1987. Calculation of water activity and activity coefficient of sugar solutions and some liquid foods. Lebensm. Wiss. Technol., 20: 64-67. Draper, N.R. and Smith, H., 1981. Applied Regression Analysis, 2nd edn. Wiley, New York. Gmehling, J. and Onken, U., 1977. Vapor-liquid equilibrium data collection. Dechema Chem. Data Ser. 1, Part 1. (In Van Den Berg and Bruin, 1981). Hill, T.L., 1950. Statistical mechanics of adsorption. Thermodynamics and heat of adsorption. J. Phys. Chem., 18: 246-256. Le Maguer, M., 1985. Solution thermodynamics and the starch-water system. In: D. Simatos and J.L. Multon (Eds.), Properties of Water in Foods. Academic Press, New York, pp. 133-151. Marinos-Kouris, D. and Saravacos, G.D. 1975. Volatility of organic compounds in aqueous sucrose solutions. 5th International Congress of Chemical Engineering, Equipment Design and Automation, SHISA ‘75, Prague. Maroulis, Z.B., Tsami, E. and Marinos-Kouris, D., 1988. Application of the GAB model to the moisture sorption isotherms for dried fruits. J. Food Eng., 7: 63-78. Norrish, R.S., 1966. An equation for the activity coefficients and equilibrium relative humidities of water in confectionary syrups. J. Food Technol., 1: 25-34. Orye, R.V. and Prausnitz, J.M., 1965. Multicomponent equilibria with the Wilson equation. Ind. Eng. Chem., 57: 18-26. Rotstein, E. and Cornish, A.R.H., 1978. Prediction of the sorptional equilibrium relationship for the drying of foodstuffs. AIChE J., 24: 956-966. Silverman, N. and Tassios, D., 1977. The number of roots in the Wilson equation and its effect on vapor-liquid equilibrium calculations. Ind. Eng. Chem. Process Des. Dev., 16: 13-20. Spiess, W.E.L. and Wolf, W.R., 1983. The results of the COST 90 project on Water Activity. In: R. Jowitt, F. Escher, B. Hallstrom, H.F.Th. Meffert, W.E.L. Spiess and G. Vos (Eds.), Physical Properties of Foods. Elsevier, London, pp. 65-91. Stokes, R.H. and Robinson, R.A., 1966. Interactions in aqueous nonelectrolyte solutions. I. Solute-solvent equilibria. J. Phys. Chem., 7: 2126-2131. Tassios, D., 1979. The number of roots in the NRTL and LEMF equations and the effect on their performance. Ind. Eng. Chem. Process Des. Dev., 18: 182-186. Taylor, J.B. and Rowlinson, J.S., 1955. The thermodynamics of solutions of glucose. Trans. Faraday Sot., 51: 1186-1190. Tsami, E., Marinos-Kouris, D. and Maroulis, Z.B., 1990. Water sorption isotherms of raisins, currants, figs, prunes and apricots. J. Food Sci., 55: 1594-1597. Van Den Berg, C. and Bruin, S., 1981. Water activity and its estimation in food systems: Theoretical aspects. In: L.B. Rockland and G.F. Stewart (Eds.), Water Activity: Influences on Food Quality. Academic Press, New York, pp. 1-61. Wilson, G.M., 1964. A new expression for the excess free energy of mixing. J. Am. Chem. Sot., 86: 127-130. Young, C.T., 1975. Composition and nutritive value of raw and processed fruits. In: J.G. Woodroof and B.S. Luh (Eds.), Commercial Fruit Processing. Avl Publishing, Westport, CT, pp. 539-579.
191
Use of the Wilson equation for the prediction of the sorptional equilibrium of sugar-based foodstuffs G.K. Vagenas and D. Marinos-Kouris Department of Chemical Engineering, National Technical University, GR-157 73 Athens (Greece) (Received April 4, 1991; accepted in final form April 14, 1992)
ABSTRACT Vagenas, G.K. and Marinos-Kouris, D., 1992. Use of the Wilson equation for the prediction of the sorptional equilibrium of sugar-based foodstuffs. Fluid Phase Equilibria, 78: 191-207. The concepts of phase equilibrium were applied for the representation of the water activity of five dried fruits characterized by high sugar content (raisins, currants, figs, prunes and apricots) as a function of moisture content, using several thermodynamic equations for the calculation uf the activity coefficients. The results showed that the Wilson equation, modified to include the effect of large cyclic molecules (sugars), can be used with considerable accuracy. The estimated values of the unknown parameters provide an insight into the nature of the interaction between the molecules of sugars and water, and its variation with temperature.
INTRODUCTION
The importance of water activity in relation to the quality of food products and their stability is widely recognized (Van Den Berg and Bruin, 1981). It is defined as the ratio of the fugacity of water in a solution to that of pure water. Fugacity may be approximated by partial vapor pressure under normal conditions of temperature and pressure, and this justifies the commonly used definition:
Correspondence to: D. Marino-Kouris, Department Technical University, Athens GR-157 73, Greece. 0378-3812/92/$05.00
of Chemical
Engineering,
0 1992 Elsevier Science Publishers B.V. All rights reserved
National
192
G.K Vagenas and D. Marinos-Kouris /Fluid Phase Equilibria 78 (1992) 191-207
where a, is the water activity, P, is the vapor pressure of water over the solution, and Pi is the vapor pressure of pure water at the temperature of the solution. The system foodstuff-moist air in equilibrium can be studied as a two-phase equilibrium system, consisting of a “solution” phase, containing solid constituents and water, and a gas phase, containing water vapor and air. Depending on the amount of water in the solution phase, the foodstuff can be considered as a concentrated solution with more or less “solid” appearance (low moisture content, sorbed water) or as a dilute solution of “liquid” appearance (high moisture content, dissolved solid constituents). The main difference between a true solution and a condensed phase made up of an adsorbent and the adsorbate attached to it, is in the size of the adsorbent particles, which is finite. Such a “solution” cannot be reduced to the infinitesimal size required for the application of the thermodynamic equations. However, similar approximations have been made in normal solutions and we assume here that the physical process of adding or removing one particle is equivalent to the infinitesimal mathematical increment. The second difficulty has to do with the homogeneity criterion, i.e. that all particles making up the “solution” have the same specific properties. We will assume this assumption to be met hereafter. These points have been analyzed very successfully by Hill (1950), and Le Maquer (1985). The development of mathematical correlations for the prediction of the water activity as a function of moisture content, is essential for many practical design problems in the food industry, such as dehydration, packaging and storage. However, most attempts have been confined to sorption models, which are strictly applicable only at very low moisture content, where the food cells are no longer acting as water reservoirs (Rotstein and Cornish, 1978). In contrast, thermodynamic correlations based on the solution concept are scarce in the literature, although they seem very promising because they have a more or less theoretical origin; moreover, some of these equations can be used for the prediction of the sorptional equilibrium of a foodstuff, using only parameters obtained from sorption characteristics of the constituents. Bruin and Prausnitz (1971) and Gmehling and Onken (1977) calculated the water activity of water in liquid organic mixtures. Chandrasekaran and King (1972) and Marinos-Kouris and Saravacos (1975) studied the vapor-liquid equilibria of organic components in aqueous sugar solutions. However, their studies were confined to very high moisture contents of the system, where the solution phase is liquid. The objective of this work was to apply the concept of phase equilibria for the calculation of the water activity of five dried fruits with high sugar content, namely raisins, currants, figs, prunes and apricots, as a function of moisture content, using the Wilson equation for the calculation of the
G.K Vagenas and D. Marinas-Kouti /Fluid
Phase Equilibria 78 (1992) 191-207
193
activity coefficients. In addition, other thermodynamic relationships described in the literature were also tested as to their ability to predict water activity values for the entire moisture range, from the bone-dry fruit to full turgor.
EXPERIMENTAL
DATA
The experimental data of the water activity as a function of moisture content were taken from the recent literature (Maroulis et al., 1988; Tsami et al., 1990). They comprised measurements for the five dried fruits (raisins, currants, figs, prunes and apricots) at four temperatures (15, 30, 45 and 60°C) and at atmospheric pressure. The method used was the static gravimetric method developed by the European Cooperative Project COST-90bis on Water Activity of Foods (Spiess and Wolf, 1983).
MATERIALS
The five dried fruits used in this study are characterized by high sugar content, especially monosaccharides (glucose and fructose) and, to a lesser degree, disaccharides (sucrose). Other constituents are polysaccharides (mainly cellulose), proteins and minerals, but their concentration is very low compared to that of the sugars. Typical compositions of the five dried fruits, according to Young (19751, are given in Table 1.
TABLE 1 Composition
(weight %o)of the five dried fruits (Young, 1975)
Constituent
Raisins
Currants
Water Total sugars glucose fructose sucrose Proteins Fat Cellulose Minerals Total
14 69 36 32 1 3 3 6 5 100
1.5 70 36 34 0 2 1 1 10 100
Figs
Prunes 23 55
24 57 31 26 0 3 1 5 10 100
39 14 2 2 1 2 16 100
Apricots 22 57 26 16 15 3 1 4 13 100
194
G.K Vagenas and D. Marinas-Kouris /Fluid Phase Equilibria 78 (I 992) 191-207
MATHEMATICAL ANALYSIS
For an ideal dilute solution, Raoult’s law of water activity may be written as
where x1 is the mole fraction of water in the solution. However, foodstuffs deviate greatly from ideality (Van Den Berg and Bruin, 1981) and require the use of an activity coefficient. Thus, eqn. (2) should be replaced by %
=
(3)
YlXl
where y1 is the activity coefficient of water. The mole fraction of water is calculated from the relationship X
(4)
x1 =
where X is the moisture content of the foodstuff (mass of water/mass of dry matter), N is the total number of constituents, m, is the mass of constituent i, d is the total mass of dry matter, and M,, Mi are the molecular weights of water and constituent i, respectively. The activity coefficient y1 can be calculated using one of several thermodynamic relationships. Wilson (1964) proposed the following equation for the calculation of the excess Gibbs energy of mixing for a binary system: gE/RT
=. -x1
ln(x, + f&n,)
-x2 ln(&x,
+x2)
(5)
where gE is the excess Gibbs energy of mixing, T is absolute temperature, xi is the mole fraction, and h12, A21 are adjustable parameters related to the pure component molar volumes and to characteristic energy differences:
R is the gas constant,
A12=
$ q( -
A’;TA1l)
(64
A,,=: exp(-A12~~22)
WI
where ui is the molar volume of component i and the A’s are energies of interaction between the molecules designated in the subscripts. The activity coefficient y1 is given by In y1 = -ln(x,
+A12x2) +x2 x +Ar x - A t2i, 1 12 2 211
2
(7)
G.K. Vagenas and D. Marinas-Kouris/Fluid
Phase Equilibria 78 (1992) 191-207
195
Although the Wilson equation has been applied to a variety of mixtures (Orye and Prausnitz, 1965; Silverman and Tassios, 1977), its application has been limited to systems whose components do not differ greatly in molecular size. The system solid-foodstuff-water could be effectively considered as a binary mixture of sugars and water, where the sugars (mainly glucose and fructose and, to a lesser degree, sucrose> have an effective molecular size between that of the monosaccharide (C,H,,O,) and the disaccharide (C,,H,,O,,). However, the equilibrium properties of the system cannot be predicted by the corresponding properties of the sugars alone, because the existence of other components, such as proteins and minerals, even at very low concentrations, can affect the properties of the solution to an unpredictable degree. For this reason, the unknown parameters of the Wilson equation will be estimated by fitting the calculated values to the experimental data. The system sugars-water is a mixture of high complexity because both constituents have peculiar characteristics. The monosaccharides (glucose and fructose) are aliphatic compounds. However, in pure form, as well as in solutions, the prevailing form is of a cyclic type (semi-acetal). The adsorption of water by these molecules can be described with considerable accuracy by the assumption of a number of possible hydration sites, equal to the number of oxygen atoms in the molecule. These are polar groups capable of hydrogen bonding with water molecules. Thus, glucose and fructose have six possible hydration sites in their molecule (Stokes and Robinson, 1966). The sucrose molecule consists of two monosaccharide molecules and can be considered to have eleven possible solvation sites. Two more effects are of importance for the system sugars-water. One is the wide disparity in size between the molecules of water and the sugars which are our chief objects of interest. The other is the “structured” nature of liquid water. On the basis of the above considerations, it was necessary to redefine the local volume fractions introduced by Wilson (1964). If we consider the water as the solvent (1) and the sugars as the solute (2), then the local volume fraction of solvent molecules about a central solvent molecule is
511=
ew( -A,,/RT) exp(-All/W + uox2 ew( -A,,/RT) 4x1
WI
0%
and the local volume fraction of solute molecules about a central solvent molecule is
521 =
ulxl
uox2
exp(-A12/W
exp( -All/IV)
+
uox2
exp(
-A,,/RT)
(3
G.X. Vagenas and D. Marinas-Kouti /Fluid Phase Equilibria 78 (I 992) 191-207
196
Here we surrounded by a number of hydration sites, rather than whole solute Thus, 511 + 521 = 1. The local volume fractions about a central solute molecule, however, are those defined by Wilson (1964), because the solute molecule is by molecules of water. Thus u2x2
‘22 =
ulxl
exP(-A,,/W
exp( -h,,/RT)
+
u2x2
exp( -h22/RT)
(94
+x1 exp( -A,,/RT) ‘12 = ulxl exp( -A,,/RT)
+ u2x2 exp( -A,,/RT)
and 522 + 5r2 = 1. The introduction of these new volume fractions, eqns. (8) and (9), into the analysis of Wilson does not change the form of eqn. (7), but the parameter Al2 given by eqn. (6a) now becomes
A,,=$ exp(-A1:tll) The molar volume of the polar group of the solutes can be calculated by
uo= u2/q
(11)
where q is the number of polar groups (hydration sites) of the solute molecule. For the purpose of our analysis, this is estimated for every dried fruit as a weighted average of the hydration sites for monosaccharides and disaccharides. The energy parameters A12, A,, and A,, are determined by the intermolecular forces between sugars and water. The differences A,, - A,, and A,, - A,, are usually considered independent of temperature, at least over modest temperature intervals. However, this is unlikely to occur in the case of the strongly hydrogen-bonded system sugars-water over the temperature interval of 15-60°C used in this study. There are two methods of regression analysis for estimating the temperature dependence of these parameters from the experimental data of moisture content versus water activity: the indirect and the direct method. In the indirect (or successive) method, the parameters A,, - A,, and A,, - A,, are estimated at each temperature by regression analysis of eqn. (3) and the calculated values are correlated with temperature using a second regression analysis. However, this approach may introduce significant errors if there is some uncertainty in the estimated values of the unknown parameters.
G.K Vagenas and D. Marinas-Kouris/Fluid
Phase Equilibria 78 (1992) 191-207
197
In the direct regression method, a specified temperature dependence of the parameters is assumed, usually in the form of eqn. (13), see below, and the constants b,, b,, b,, b4 are estimated by substituting eqn. (13) into the analysis. All the experimental data are used directly with the same weight. In both methods, the unknown parameters were estimated by fitting the mathematical model to the experimental data. This procedure minimizes the sum of the squares of the residuals between the experimental and the predicted values, using the method of non-linear regression (Draper and Smith, 1981). For the requirements of this study, we used a modified Marquardt-Levenberg method, which ensures fast convergence, even when the initial guesses of the parameters are poor.
RESULTS AND DISCUSSION
The experimental data of the moisture content versus water activity for the five dried fruits are in exact agreement with the corresponding data of Norrish (1966) for sucrose solutions. This is an indication that the sorption behavior of the dried fruits is mainly controlled by the sugar constituents and, hence, the assumption that they can be represented as a mixture of sugars and water is justified. The results of the indirect (successive) regression for the five dried fruits are given in Table 2. The standard deviation due to regression sR is very close to the pure error standard deviation sn and, therefore, the predictions of the Wilson equation are satisfactory. The estimated parameters A,, - A,, and A,, - A,, show a strong temperature dependence which must be attributed to the fact that the cohesive forces between the two molecules depend on temperature. For low temperatures (15-3O”C), A,, - A,, < 0, which means that the hydrogen bonds between sugars and water are stronger than the hydrogen bonds between water molecules. Water molecules would “prefer” sugar molecules as neighbors rather than alike molecules. In this way, the adsorption of water molecules on the food surface takes place spontaneously and is favored. However, A,, - A,, > 0, which means that the cohesive forces between sugar molecules are stronger than those between dissimilar molecules. Dissolution cannot take place spontaneously. As a result, at low temperatures, the adsorption of water molecules is favored over the dissolution of solute molecules in water. However, at higher temperatures (30-6O”C), this changes drastically. The difference A,, - A,, becomes greater than zero, which means that the water molecules “prefer” water molecules as neighbors. However, A,, - A, < 0, which means that the cohesive forces between sugar molecules and water are stronger than those between sugar molecules. Dissolution is
198
G.K Vagenas and D. Marinos-Kouris /Fluid Phase Equilibria 78 (1992) 19X-207
TABLE 2 Results of the indirect regression analysis of the Wilson model for the five dried fruits Product
Temperature (“0
An-Jh,
(kJ mol-‘)
A,,-A22
0c.f mol-‘)
Standard deviation (kg kg-‘) SR
sE
a
Raisins
15 30 45 60
-
4.20 2.05 1.83 0.93
15.29 - 2.43 - 2.85 - 4.04
0.035 0.023 0.019 0.026
0.014 0.018 0.013 0.017
Currants
15 30 45 60
-3.19 - 1.25 3.86 7.62
7.30 - 2.85 - 6.74 -7.17
0.039 0.033 0.021 0.018
0.015 0.011 0.013 0.014
Figs
15 30 45 60
- 3.54 - 2.63 - 1.45 0.91
12.38 - 1.22 - 2.87 - 4.56
0.034 0.021 0.015 0.026
0.012 0.006 0.007 0.011
Prunes
15 30 45 60
- 4.27 -2.31 - 1.34 -0.23
11.21 - 1.03 - 2.34 - 6.51
0.034 0.016 0.011 0.019
0.010 0.005 0.004 0.008
Apricots
15 30 45 60
- 3.67 - 1.29 1.34 3.28
9.25 0.15 - 3.56 - 5.72
0.023 0.017 0.016 0.021
0.011 0.005 0.005 0.008
a Maroulis et al. (1988).
favored over adsorption. These results confirm that adsorption is an exothermic reaction and, therefore, is favored at low temperatures, while dissolution of sugars is an endothermic reaction and is favored at high temperatures. Although the above results seem plausible and indicate a dependence of the parameters on temperature, there is still the question of the accuracy of the estimation of the unknown parameters. Silverman and Tassios (1977) reported that several sets of parameters, widely different in values, can give small values of the standard deviation and are, therefore, equally possible as the result of the regression analysis, when the binary activity coefficients yi < 1. (Actually, this is the case for the system foodstuff-water, as will be mentioned below.) Tassios (1979) reported similar results for the NRTL and LEMF equations. The best way to elucidate this question is to calculate the joint confidence regions of the unknown parameters. A confidence region is the region in the parameter space where the real
G.K Vagenas and D. Marinos-Kouti / Fluid Phase Equilibria 78 (1992) 191-207
B 1
z
ll-
199
60 C
Q.
4.-
7-
I
15 c
D 5 -10
-I5
d
I
10
$
lb
$0
A, 2 - hz2,
Fig. 1. Joint confidence regions of the parameters tures; + , estimated values.
A&,
2’s
kJoule/moie
and A,,-&,, at several tempera-
values of the parameters are most probably located. In the case of non-linear regression, these regions are calculated from the equation (Draper and Smith, 1981)
NW*) -wp R/b -P)
=F
_
la
n
(p
7
_p)
(14 .
where S(b,, bJ is the total sum of the squares of the residuals, bi are the unknown parameters, R is the minimum S, n is the number of experimental points, p is the number of parameters (here p = 2) and f’,_,(p, n -p> is the value of the F distribution for p and n -p degrees of freedom and probability 1 - a. Figure 1 presents the confidence regions of the unknown parameters for all temperatures in the case of raisins. The central points of the contours (+) represent the estimated values of the parameters and the three concentric curves refer to 90, 95 and 99% probability levels. The examination of these contours gives a deep insight into the physical significance of the parameters. Firstly, it is clear that the experimental data at 15°C give a very accurate estimate of Ah,,-A,, (within lo%), but they cannot be used . . for the estimatron of A,, -A,, beyond the obvious result that this difference is always less than zero. Secondly, the experimental data at 30,45 and 60°C give identical values for the parameter A,, - A,, and slightly different
200
G.K Vagenas and D. Marirws-Kouti /Fluid Phase Equilibria 78 (1992) 191-207
values for the parameter A,, -A,,. Thirdly, these data (30, 45 and 60°C) cannot be used for the accurate estimation of A,, - An, which can take any value close to or less than zero, while they give a more or less reliable estimate of A,, -A,,. What these results mean in physical terms is that at low temperatures, the adsorption prevails and the parameter corresponding to dissolution (A,, -A,,) cannot be ,estimated from the experimental data. In contrast, at higher temperatures, dissolution of sugars prevails and the parameter A,, -A,, corresponding to adsorption cannot be estimated accurately. In this way, the conclusions of the previous paragraph are confirmed, but also the limitations of this method of indirect estimation at different temperatures are revealed. Therefore, the second stage of this method was not attempted. Once the dependence of the parameters on temperature has been justified, we can proceed to the direct regression analysis of the proposed model, using all the experimental data. We made the assumption of a linear change of the parameters with temperature, which is the most obvious choice. Although a linear relation is unlikely to occur in a physical problem, it is a good approximation for the temperature interval 1%60°C; besides, the four values of temperature used for the experiments do not permit the use of more complex relationships. Thus:
Pa) Pb)
A,, - A,, = b, + b,T A,, -A,,
= b, + b,T
results of the direct regression analysis for the five dried fruits are given in Table 3. The standard deviation sa is found to be about 3 times greater than sE and, therefore, the Wilson equation predicts the experimental data well. The accuracy of the estimation of the parameters b,, b,, b, and b, is satisfactory, as indicated by the joint confidence regions of the constants b, and b,, which are presented in Fig. 2. Moreover, the conclu-
TABLE 3 Results of the direct regression analysis of the Wilson model for the five dried fruits Product
Constants
Raisins Currants Figs Prunes Apricots
b, - 6.38 - 6.25 -5.72 - 4.27 - 5.92
a Maroulis et al. (1988).
b, 0.154 0.206 0.146 0.138 0.168
b, 18.02 17.36 18.53 16.23 16.89
b, -
0.591 0.625 0.733 0.642 0.698
Standard deviation a SR sE 0.050 0.016 0.040 0.014 0.028 0.009 0.024 0.007 0.022 0.007
G.K. Vagenas and D. Marinas-Kouris /Fluid Phase Equilibria 78 (1992) 191-207
201
-5.0 z
-5.5 1 -6.O-
-6.5 -
-7.0 -
-7.5 -
-6.0 14.0
lJ.0
16:O
I 17.0
.I. 18.0
I. 19.0
2010
2110
2210
b3 Fig. 2. Joint confidence regions of the constants b, and b, of eqn. (13); + , estimated values.
sions drawn so far, regarding the physical significance of the estimated parameters, are confirmed. Although the above results show that the Wilson equation appears especially suitable for this problem, we also attempted to correlate the experimental data for the five dried fruits using several thermodynamic relationships in the literature. The results are compared in Table 4, using the standard deviation between the experimental and the predicted values TABLE 4 Standard deviations sr (kg H,O/kg d.s.) between experimental and predicted values of the moisture content for raisins, using several models in the literature Model (parameters)
Temperature
PC)
15
30
45
60
Margules a (1) Margules b (2) Van Laar (2) Flory-Huggins (1) Wilson (2) NRTL ’ (2) UNIQUAC (2)
0.052 0.038 0.028 0.058 0.036 0.054 0.048
0.035 0.034 0.033 0.040 0.024 0.037 0.034
0.026 0.024 0.022 0.034 0.020 0.029 0.024
0.036 0.032 0.032 0.051 0.027 0.048 0.032
a Two-suffix Margules equation. b Three-suffix Margules equation. c The NRTL equation has three parameters, but only two were fitted. The parameter of the non-randomness in the mixture was given a medium value (0.35).
202
G.K Vagenas and D. Marinas-Kouris / Fluid Phase Equilibria 78 (1992) 191-207
0
T=lS
C
+
T=30
C
aw
Fig. 3. Experimental values (discrete (continuous line) for raisins.
points)
and predictions
of the Wilson
equation
of moisture content sT as a validation criterion. These results showed that the lowest standard deviation was obtained in all cases with the Wilson equation. The Van Laar equation and the UNIQUAC equation also gave satisfactory results, but the Wilson equation is preferable because it is simpler than the UNIQUAC equation and its parameters possess a more obvious physical meaning, compared to those of the Van Laar equation which are strictly empirical. The predictions of the Wilson equation are given in Fig. 3 for raisins and in Fig. 4 for currants. The agreement is satisfactory over the entire range of water activity. The largest deviation occurs for very low water activities (first experimental point), where there is localized adsorption on active sites (monomolecular layer) and a physical adsorption model would be more appropriate. It is worth mentioning that the Wilson equation predicts the effect of temperature on the sorption isotherm of these sugar-based products; the temperature has the expected negative effect on equilibrium moisture content at low water activities, but the reverse effect is observed at higher water activities, and this behavior is well represented by the Wilson equation. An important criterion for the suitability of the Wilson equation in the case of foodstuffs is its ability to predict the experimental data using values of parameters estimated from a limited set of experimental data. This was done in the case of raisins. We estimated the unknown parameters using the experimental data in the water activity range O-0.5, and then we used
G.X. Vagenas and D. Marinas-Kouris/Fluid Phase Equilibria 78 (1992) 191-207
+
T=SO
203
C
Fig. 4. Experimental values (discrete (continuous line) for currants.
points)
and predictions
of the Wilson
equation
these values of parameters to predict the experimental data over the entire range of water activities. Typical results are shown in Fig. 5 and we can see that the predictions are also satisfactory. 1.0 -
0.8 -
0
T=lS
C
+
T=30
C
03
0.5
0.6 -
0.4 -
0.2 -
0.0
0.0
O.‘l
0.i
0.3
oh
0.v
oh
oh
Fig. 5. Experimental and predicted values for raisins using parameters in the water activity range O-0.5.
1 .b
estimated from data
G.K Vagenas and D. Maritws-Kouris /Fluid Phase Equilibria 78 (1992) I91 -207
204
&
m-11-;
‘t;
3
-0.2
1
-0.3 -
-0.4
0.0
O.‘l
03
0.5
0.2
0.3
0.6
o.‘r
oh
oh
X, kg H20/kg
da.
, 1.0
Fig. 6. Predicted values of the excess Gibbs energy of mixing for raisins using the Wilson equation: 1, T = 15°C; 2, T = 30°C; 3, T = 45°C; 4, T = 60°C.
1.0
0.9
>-’ 0.6
0.7
0.6
oh
O.‘l
oh
0.5
0.2
0.3
0.6
0.f
oh
oh
X, kg H2O/kg
d.s.
1.b
Fig. 7. Predicted values of the activity coefficients of water in raisins using the Wilson equation: 1, T = 15°C; 2, T = 30°C; 3, T = 45°C; 4, T = 60°C.
G.R Vagenas and D. Marinas-Kouris /Fluid Phase Equilibria 78 (1992) 191-207
-2.5
’ oh
0.7
I 0.2
0.5
0.14
0.b
0.6
oh
0.b
X, kg H2O/kg
0.i
d.s.
205
1.b
Fig. 8. Predicted values of the excess enthalpy of mixing in raisins using the Wilson equation: 1, T = 15°C; 2, T = 30°C; 3, T = 45°C; 4, T = 60°C.
The predicted excess Gibbs energy of mixing and the activity coefficients of water and sugars are shown in Figs. 6 and 7 respectively for currants, as a function of temperature. The results show that the system foodstuff-water exhibits large negative deviations from ideality at 15°C but it approaches an ideal behavior at higher temperatures. The activity coefficients of water are always less than unity, in agreement with the results of Taylor and Rowlinson (1955) for glucose solutions, Stokes and Robinson (1966) for sucrose and glucose solutions and Norrish (1966) for sugars in general. However, the above authors were confined to high water contents and this prohibits a quantitative comparison. Chen (1987) reported that the activity coefficient of water is very close to unity for most sugar solutions with concentrations up to 40% by weight. The activity coefficients of sugars exhibit a similar behavior, approaching unity at high temperatures. The excess heat of mixing can be calculated from the Wilson equation: hE=
-T2
(14)
Typical plots of the heat of mixing as a function of water content are given in Fig. 8 for several temperatures. Although the results are reasonable (Tsami et al., 1990), the estimation of the heat of sorption is rough,
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G.K. Vagenas and D. Marinos-Kouti /Fluid Phase Equilibria 78 (1992) 191-207
especially for the low moisture-content region, where the localized adsorption (monomolecular layer) causes extremely high (absolute) values of the heat of sorption.
CONCLUSIONS
The Wilson equation can be used successfully for the prediction of phase equilibria between foodstuffs and moist air. The estimated parameters of this equation, A,, - A,, and A,, - A,, are temperature dependent and reflect the nature of the interaction between the molecules of sugars and water. At low temperatures, the adsorption of moisture on the solid surface takes place spontaneously and is favored over the dissolution of sugars, while at higher temperatures the reverse is true, that is, dissolution of sugars is favored over adsorption.
LIST OF SYMBOLS
water activity mass of dry material (kg) excess free enthalpy of mixing (k3 mol-‘1 excess enthalpy of mixing 0~3 mol-‘) mass of constituent i (kg) molecular weight of constituent i (kg mol-‘1 number of constituents vapor pressure of water over the solution (Pa) vapor pressure of pure water (Pa) number of polar groups in the solute molecule gas constant, 0.008317 (W mol-’ K-‘) absolute temperature (K) molar volume (m3 mol-‘) molar volume of the polar group in the sugar molecule (m3 mol-‘) mole fraction moisture content of the sample (kg H,O per kg dry material) Greek
letters
Aij energies of interaction between molecules i and j (kJ mol-‘) Aij energy parameters of the Wilson model 5 local volume fraction
G.R Vagenas and D. Marinas-Kouti/Fluid
Phase Equilibria 78 (1992) 191-207
207
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