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Equilibrium sorption isotherms and thermodynamic properties of starch and gluten
Journal of Food Engineering 40 (1999) 287±292
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Equilibrium sorption isotherms and thermodynamic properties of starch and gluten Pascual E. Viollaz a,*, Clara O. Rovedo a,b a
Facultad de Ciencias Exactas y Naturales (UBA), Departamento de Industrias, Ciudad Universitaria, 1428 Buenos Aires, Argentina b Department of Biological and Agricultural Engineering, University of California, Davis, CA, 95616, USA Received 14 September 1998; accepted 16 March 1999
Abstract A modi®cation of the GAB isotherm (Guggenheim±Anderson±De Boer) is proposed in order to correlate the sorption data for water activities higher than 0.9. The proposed isotherm retains the desirable properties of the GAB isotherm, i.e. good ®tting in the range of aw between 0.05 and 0.80, and also provides a noticeable improvement in the ®tting quality for high values of aw , by introducing a new term with an additional constant. Sorption data for native potato starch and gluten at dierent temperatures (2°C, 20°C, 40°C and 67°C) were well correlated for the whole aw range. For starch at 67°C, the values of the constants of the proposed equation do not follow the same tendency obtained for lower temperatures, suggesting that the structure of the material could be changed due to the high temperature. The proposed isotherm can be of interest in the area of drying given that there are few isotherms that accurately represent sorption data at dierent temperatures in the zone of high aw . Also, it can be useful to predict other thermodynamic functions. In addition, a new procedure is proposed to determine the isosteric heat by using a second order polynomial for representing the variation of moisture as a function of temperature at a ®xed water activity. Ó 1999 Elsevier Science Ltd. All rights reserved.
Nomenclature aw water activity, dimensionless constants in Eqs. (14)±(17) a1 ±a8 a, b, c constants in Eq. (8) C Guggenheim constant k constant in Eq. (1) constant in Eq. (2) k2 N number of data points degrees of freedom np R gas constant (J/mole K) net isosteric heat (J/mole) qst T temperature (°C) u moisture content (g water/g dry mass) monolayer moisture content (g water/g dry mass) um
1. Introduction Knowledge of the sorption properties of foods is important for calculating food stability during storage, which depends mainly on the water activity of the product. The sorption isotherm can also aect the dry-
*
Corresponding author. E-mail: [email protected]
ing kinetics, changing the boundary conditions or altering the value of the diusion coecient. Drying kinetics can be predicted by numerical integration of the partial dierential mass and energy balances. The sorption isotherm of the material is used in the boundary condition of such equations to predict the interface conditions. For this reason, it is necessary to have a desorption isotherm that can represent with reasonable accuracy the sorption data for the whole range of aw values. Furthermore, in some models of diusion, the diusion coecients are a function of the slope of the desorption isotherm. In recent years, there have been important contributions in the ®eld of sorption. Sorption isotherms for foodstus were summarized by van den Berg and Bruin (1978), van den Berg (1981) and Chirife (1983). However, there is no isotherm with a simple mathematical structure capable of representing the sorption properties of foods in the whole range of water activities. One of the best isotherms for foods is the GAB isotherm. But this equation cannot accurately represent the equilibrium data for high aw , especially for high values of the constant C (type II in the BET classi®cation of isotherms), which correspond to foods with high initial moisture content (potato, cassava).
0260-8774/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 0 - 8 7 7 4 ( 9 9 ) 0 0 0 6 6 - 7
288
P.E. Viollaz, C.O. Rovedo / Journal of Food Engineering 40 (1999) 287±292
The objective of this work is to extend the range of application of the GAB isotherm in order to obtain a sorption isotherm that can accurately represent the whole range of aw values. The proposed isotherm allows one to calculate in a precise way the value of the slope of the sorption isotherm, which is important in some diffusion models (Krishna, 1987). Also a new method for calculating the isosteric heat at dierent temperatures is presented. 2. Theory The GAB sorption isotherm is (Sch ar & Ruegg, 1985) u
um Ckaw ; 1 ÿ kaw 1 ÿ kaw Ckaw
1
where u is the moisture content of the solid on a dry basis, aw the water activity, um the monolayer moisture content, C the Guggenheim constant and k a constant whose value generally lies between 0.7 and 1. Expression (1) will be modi®ed in an empirical way by adding a term, as follows u
um Ckaw um Ckk2 a2w : 1 ÿ kaw 1 ÿ kaw Ckaw 1 ÿ kaw 1 ÿ aw 2
The second term of the RHS allows the necessary ¯exibility to obtain a good ®tting for high aw values. That term has a very low weight for low values of aw , so the values of um , C and k are not severely aected by the addition of this new term. It can be observed that if the value of k2 is equal to zero, the GAB isotherm (Eq. (1)) can be obtained. The predictive capacity of Eqs. (1) and (2) was evaluated using the sorption data for native potato starch (van den Berg, 1981) and for gluten (De Jong, van den Berg & Kokelaar, 1996), that cover the range of high aw values. The parameters of Eqs. (1) and (2) were obtained by non-linear regression, using as objective function the minimization of the relative deviations s P 0 2 ui ÿ ui =u0i 100; 3a RMS % N where u0 and u are the experimental and predicted values, respectively, using Eqs. (1) or (2), and N is the number of data points. De Jong et al. (1996) proposed that a sorption isotherm is useful when the RMS is lower than 4%. Given that an increase in the number of parameters automatically leads to a better ®tting of the experimental data. Eq. (3a) was calculated by replacing the number of data N, by the number of degrees of freedom (Maroulis, Tsami, Marino-Kouris & Saravacos, 1988), as follows
sP 2 u0i ÿ ui =u0i 100; RMS % N ÿ np
3b
where np is the number of parameters. 2.1. Calculation of the sorption heat The net isosteric heat is usually calculated by means of the following expression (Rizvi & Benado, 1984) d ln aw qst 4 ÿ ; R d 1=T u where R is the universal gas constant and qst the net isosteric heat. The derivation process must be performed at constant moisture content. To calculate the isosteric heat, it is necessary to use a sorption isotherm with an adequate temperature function, or to plot ln(aw ) vs 1/T using sorption data at two dierent temperatures, evaluating the sorption heat at an intermediate one. On the other hand, an alternative expression can be derived from Eq. (4), for calculation of the net isosteric heat, as follows. Eq. (4) can be written as RT 2 daw : 5 qst aw dT u
The values of daw /dT|u in Eq. (5) are calculated using the sorption isotherm (Eq. (2)), which can be represented in the form u f aw ; T :
6
Dierentiating Eq. (6) and imposing the condition that u constant, we get ÿof =oT jaw daw : 7 dT u of =oaw jT The derivative that appears in the denominator of Eq. (7) is the slope of the sorption isotherm. This slope can be easily calculated from the isotherm (Eq. (2)), because the derivation process is made at constant temperature. To evaluate the derivative that appears in the numerator of Eq. (7), it was assumed that the moisture values as a function of temperature, for a ®xed value of aw , can be represented by a polynomial expression of second degree as follows u a bT cT 2 :
8
Applying Eq. (8) to three values of moisture, corresponding to three dierent temperatures for a ®xed value of aw , we get u1 a bT1 cT12 ; u2 a bT2 cT22 ; u3 a bT3
cT32 :
9
P.E. Viollaz, C.O. Rovedo / Journal of Food Engineering 40 (1999) 287±292
289
The equation system (9) can be expressed in a matrix array to obtain the values of the coecients a, b and c as follows 0
0
a b c Aÿ1 u1 u2 u3 ;
10
0
where the indicates the transpose of the ®le vector and Aÿ1 is the inverse of the matrix A, de®ned as 2 3 1 T1 T12 A 4 1 T2 T22 5: 11 1 T3 T32 Coecients a, b and c are the function of aw . But the matrix A and its inverse are not a function of aw but only depend on temperature, so they must be evaluated only once. Finally, the expression of the derivative that appears in the numerator of Eq. (7), is the following of ou b 2cT : 12 oT aw oT
Fig. 1. Comparison between experimental and predicted sorption isotherms for potato starch. (D) T 2.7°C; () T 20.2°C; (h) T 40.2°C; (¨) T 67.2°C; (A) predicted by Eq. (1).
aw
Replacing Eq. (12) in Eq. (7), the derivative of aw with respect to the temperature at constant moisture can be obtained oaw ÿ b 2cT : 13 oT ou=oa j u
w T
This derivative can be replaced in Eq. (5) to calculate the isosteric heat. The values calculated by this method were compared with those obtained using the following procedures: (a) The traditional method, i.e., by plotting the values of the logarithm of the water activity versus the inverse of the absolute temperature, for ®xed values of moisture contents. (b) By numerical dierentiation, using Eqs. (5) and (7), assuming a temperature dependence of the four constants of Eq. (2) as follows um a 1 a 2 T ;
14
ln k a3 a4 =T ;
15
ln C a5 a6 =T ;
16
ln k2 a7 a8 =T :
17
The same sorption data of Fig. 1 were ®tted by nonlinear regression, using Eq. (2) (Fig. 2). It can be observed that this equation can correctly represent the sorption data with a very low deviation, even for water activities up to 0.97. The values of the parameters of Eqs. (1) and (2), obtained by non-linear regression, and the values of RMS(%) calculated with Eqs. (3a) and (3b), are shown in Tables 1 and 2 for potato starch. An example of the parameters' correlation matrix for Eq. (2) at 40°C is shown in Table 3. Less than 10 iterations were needed to obtain the values of the parameters that minimized Eq. (3a). Only the regression of the data corresponding to 67°C was particularly dicult and more than 60 iterations were needed in this case. It is possible that at this temperature
3. Results and discussion Experimental data of native potato starch were compared with the isotherms predicted by the GAB Eq. (1) and the proposed Eq. (2). In Fig. 1, sorption data for potato starch at four temperatures (2.7°C, 20.2°C, 40.2°C and 67.2°C) were plotted, together with the predictions using Eq. (1). It can be observed in that ®gure, that the ®tting was not satisfactory for water activities higher than 0.75.
Fig. 2. Comparison between experimental and predicted sorption isotherms for potato starch. (D) T 2.7°C; () T 20.2°C; (h) T 40.2°C; (¨) T 67.2°C; (A) predicted by Eq. (2).
290
P.E. Viollaz, C.O. Rovedo / Journal of Food Engineering 40 (1999) 287±292
Table 1 Parameters of the GAB isotherm (Eq. (1)) for potato starch Temperature (°C)
um
k
C
RMS(%) (Eq. (3a))
RMS(%) (Eq. (3b))
2 20 40 67
0.10221 0.09828 0.08573 0.08738
0.78696 0.78848 0.79277 0.71900
28.4147 18.9935 18.0858 11.3984
3.275 3.777 5.667 3.427
3.695 4.515 6.462 4.534
Table 2 Parameters of the proposed isotherm (Eq. (2)) for potato starch Temperature (°C)
um
k
C
k2
RMS(%) (Eq. (3a))
RMS(%) (Eq. (3b))
2 20 40 67
0.10729 0.10709 0.09770 0.20002
0.75207 0.72015 0.67030 0.07340
25.6484 17.1920 16.1790 31.2587
0.00014 0.00055 0.00186 0.03041
1.63 1.855 2.450 1.57
1.937 2.394 2.945 2.40
Table 3 Correlation matrix for the parameters of Eq. (2), for 40°C
um k C k2
um
k
C
k2
1 ÿ0.9515 ÿ0.8181 0.9394
1 0.6769 ÿ0.9411
1 ÿ0.8530
1
the starch was gelatinized and the material was very dierent from the original native starch. In order to analyze the ¯exibility of the proposed Eq. (2) in ®tting the experimental data in the high aw range, the predicted values using the parameters um , C and k, for 20°C (see Table 1) were plotted in Fig. 3, for dierent values of k2 . It can be observed that the variation of k2 aects mainly the moisture values predicted for high aw , but has a small in¯uence for low aw .
Fig. 4. Sorption isotherm slopes for four dierent temperatures, as a function of water activity.
The slopes of the isotherms calculated using sorption data for starch at four dierent temperatures (2°C, 20°C, 40°C and 67°C), were plotted in Fig. 4. In this ®gure, a plateau for aw values between 0.1 and 0.8 can be observed. Out of this range, the value of the slope markedly increases. Sorption data for gluten of wheat Spring Original (De Jong et al., 1996), at 25°C and 40°C, was ®tted using Eq. (1) (Fig. 5) and Eq. (2) (Fig. 6). It can be observed in these ®gures that the dierence in the quality of the ®tting is not signi®cant. This eect can be attributed to the fact that the maximum moisture content in gluten (0.38 kg/kg, (d.b.)) is lower than the maximum moisture content of the starch. (0.49 kg/kg (d.b.)). 3.1. Calculation of the isosteric heat
Fig. 3. In¯uence of parameter k2 (Eq. 2) on the ®tting of experimental sorption data in the high aw range.
The calculation process of the isosteric heat using Eq. (5) can be easily implemented using a spreadsheet,
P.E. Viollaz, C.O. Rovedo / Journal of Food Engineering 40 (1999) 287±292
Fig. 5. Comparison between experimental and predicted sorption isotherms for gluten. (n) T 25°C; (h) T 40°C; (A) predicted by Eq. (1).
Fig. 6. Comparison between experimental and predicted sorption isotherms for gluten. (n) T 25° C ; (h) T 40°C; (A) predicted by Eq. (2).
291
Fig. 8. Calculated isosteric heat as a function of moisture content. - - - - predicted by polynomial approximation, Eqs. (5), (7) and (13) - - - - x- - - - - predicted by the traditional method (Eq. (4) in semilog plot) - - - - predicted by numerical dierentiation of Eq. (2) with temperature dependence given by Eqs. (14)±(17).
2°C, 20°C and 40°C. Data corresponding to 67°C were not used because at that temperature the starch can be gelatinized. The sorption heat values shown in Fig. 8 are reasonable for moisture contents greater than 0.05 kg/kg. For lower moisture contents, the sudden fall of the isosteric heat at 40°C is due to the absence of experimental sorption values in that range. The isosteric heat values calculated using Eq. (5) (full line curves) were compared with the values obtained using the classical Eq. (4) (continuous line with x curve, i.e. - - - - x - - - -), giving an excellent agreement for 20°C. The value of constants a1 ±a8 in Eqs. (14)±(17) were obtained using the data of Table 2. The isotherms at the three temperatures were predicted using those constants and compared with the experimental data (van den Berg, 1981). A good agreement between predicted and experimental data was observed as shown in Fig. 9. The
Fig. 7. Isosteric heat calculated using Eq. (5), as a function of water activity.
after doing the calculation of the inverse matrix Aÿ1 . The results were represented as a function of water activity (Fig. 7) and as a function of moisture (Fig. 8), for dierent temperatures. The data used, correspond to
Fig. 9. Comparison between experimental and predicted sorption isotherms for potato starch. (D) T 2.7°C; () T 20.2°C; (h) T 40.2°C; (¨) T 67.2°C; (A) predicted assuming a temperature dependence of the four constants in Eq. (2).
292
P.E. Viollaz, C.O. Rovedo / Journal of Food Engineering 40 (1999) 287±292
Fig. 10. Isosteric heat calculated obtained assuming a temperature dependence of the four constants in Eq. (2), as a function of moisture content.
isosteric heat values obtained by this last procedure for 20°C were in good agreement with the values obtained by the other two methods, as is shown in Fig. 8 (dashed line). It is interesting to note that a good prediction of the isotherms at dierent temperatures can be obtained, using Eqs. (2), (14)±(17), assuming that the monolayer moisture content is constant. But, the prediction of the isosteric heat obtained in this way is nearly half the value obtained taking into account the temperature dependence of the monolayer moisture. Therefore, it is essential to take into account the temperature dependence of the four constants in Eq. (2) in order to obtain values of the isosteric heat that are consistent with values obtained by means of the traditional method using Eq. (4). In Fig. 10 are shown the values of the isosteric heat calculated using the proposed isotherm i.e., Eq. (2), taking into account the temperature functionality given by Eqs. (14)±(17) is shown. The values of the isosteric heat were obtained by numerical dierentiation of Eq. (2), taking into account Eqs. (5), (7), (14)±(17). The obtained results appear very reasonable, and in Fig. 8 is observed a notable agreement with the curve for 20°C calculated by the two other methods. 4. Conclusions The proposed equation satisfactorily correlates the experimental sorption data for starch and gluten for a
very wide range of water activities. The functional nature of the proposed equation allows performing the ®tting process without problems of numerical instability. The additional constant k2 allows a higher ¯exibility for high aw values, without jeopardizing the good qualities of the GAB equation in its application range. The proposed methodology allows calculating the isosteric heat values in good agreement with those obtained by the traditional method, at intermediate temperatures. A good prediction of moisture content and isosteric heat values at dierent temperatures is obtained, assuming temperature dependence for the four parameters of the proposed isotherm. In this way, this equation can be useful for modeling drying kinetics. Also, the calculation can be extended to other thermodynamic functions. Acknowledgements This work was supported by the Universidad de Buenos Aires and by Consejo Nacional de Investigaciones Cienti®cas y Tecnicas (CONICET, Argentina). References Chirife, J. (1983). A survey of existing sorption data. In Physical properties of foods (pp. 55±64). London and New York: Applied Science Publishers. Jong, G. I. W., Berg, C., & Kokelaar, A. J. (1996). De van den water sorption behaviour of original and defatted wheat gluten. International Journal of Food Science and Technology, 31, 519±526. Krishna, R. (1987). A uni®ed theory of separation processes based on irreversible thermodynamics. Chemical Engineering Communications, 59, 33±64. Maroulis, Z. B., Tsami, E., Marino-Kouris, D., & Saravacos, G. D. (1988). Application of the GAB model to the moisture sorption isotherms for dried fruits. Journal of Food Engineering, 7, 63±78. Rizvi, S. S. H., & Benado, A. L. (1984). Thermodynamic properties of dehydrated foods. Food Technology, 83±92. Sch ar,W., & Ruegg, M. (1985). The evaluation of GAB constants from water sorption data. Lebbensmittel Wiss und Technology, 18, 225± 229. van den Berg, C. (1981). Vapor sorption equilibria and other water± starch interactions: A physico-chemical approach. PhD Dissertation, Agricultural University Wageningen, The Netherlands. van den Berg, C., & Bruin, S. (1978). Water activity and its estimation in food systems: Theoretical aspects. Second International Symposium on properties of water in relation to food quality and stability, Isopow-II, Osaka, Japan.
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Equilibrium sorption isotherms and thermodynamic properties of starch and gluten Pascual E. Viollaz a,*, Clara O. Rovedo a,b a
Facultad de Ciencias Exactas y Naturales (UBA), Departamento de Industrias, Ciudad Universitaria, 1428 Buenos Aires, Argentina b Department of Biological and Agricultural Engineering, University of California, Davis, CA, 95616, USA Received 14 September 1998; accepted 16 March 1999
Abstract A modi®cation of the GAB isotherm (Guggenheim±Anderson±De Boer) is proposed in order to correlate the sorption data for water activities higher than 0.9. The proposed isotherm retains the desirable properties of the GAB isotherm, i.e. good ®tting in the range of aw between 0.05 and 0.80, and also provides a noticeable improvement in the ®tting quality for high values of aw , by introducing a new term with an additional constant. Sorption data for native potato starch and gluten at dierent temperatures (2°C, 20°C, 40°C and 67°C) were well correlated for the whole aw range. For starch at 67°C, the values of the constants of the proposed equation do not follow the same tendency obtained for lower temperatures, suggesting that the structure of the material could be changed due to the high temperature. The proposed isotherm can be of interest in the area of drying given that there are few isotherms that accurately represent sorption data at dierent temperatures in the zone of high aw . Also, it can be useful to predict other thermodynamic functions. In addition, a new procedure is proposed to determine the isosteric heat by using a second order polynomial for representing the variation of moisture as a function of temperature at a ®xed water activity. Ó 1999 Elsevier Science Ltd. All rights reserved.
Nomenclature aw water activity, dimensionless constants in Eqs. (14)±(17) a1 ±a8 a, b, c constants in Eq. (8) C Guggenheim constant k constant in Eq. (1) constant in Eq. (2) k2 N number of data points degrees of freedom np R gas constant (J/mole K) net isosteric heat (J/mole) qst T temperature (°C) u moisture content (g water/g dry mass) monolayer moisture content (g water/g dry mass) um
1. Introduction Knowledge of the sorption properties of foods is important for calculating food stability during storage, which depends mainly on the water activity of the product. The sorption isotherm can also aect the dry-
*
Corresponding author. E-mail: [email protected]
ing kinetics, changing the boundary conditions or altering the value of the diusion coecient. Drying kinetics can be predicted by numerical integration of the partial dierential mass and energy balances. The sorption isotherm of the material is used in the boundary condition of such equations to predict the interface conditions. For this reason, it is necessary to have a desorption isotherm that can represent with reasonable accuracy the sorption data for the whole range of aw values. Furthermore, in some models of diusion, the diusion coecients are a function of the slope of the desorption isotherm. In recent years, there have been important contributions in the ®eld of sorption. Sorption isotherms for foodstus were summarized by van den Berg and Bruin (1978), van den Berg (1981) and Chirife (1983). However, there is no isotherm with a simple mathematical structure capable of representing the sorption properties of foods in the whole range of water activities. One of the best isotherms for foods is the GAB isotherm. But this equation cannot accurately represent the equilibrium data for high aw , especially for high values of the constant C (type II in the BET classi®cation of isotherms), which correspond to foods with high initial moisture content (potato, cassava).
0260-8774/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 0 - 8 7 7 4 ( 9 9 ) 0 0 0 6 6 - 7
288
P.E. Viollaz, C.O. Rovedo / Journal of Food Engineering 40 (1999) 287±292
The objective of this work is to extend the range of application of the GAB isotherm in order to obtain a sorption isotherm that can accurately represent the whole range of aw values. The proposed isotherm allows one to calculate in a precise way the value of the slope of the sorption isotherm, which is important in some diffusion models (Krishna, 1987). Also a new method for calculating the isosteric heat at dierent temperatures is presented. 2. Theory The GAB sorption isotherm is (Sch ar & Ruegg, 1985) u
um Ckaw ; 1 ÿ kaw 1 ÿ kaw Ckaw
1
where u is the moisture content of the solid on a dry basis, aw the water activity, um the monolayer moisture content, C the Guggenheim constant and k a constant whose value generally lies between 0.7 and 1. Expression (1) will be modi®ed in an empirical way by adding a term, as follows u
um Ckaw um Ckk2 a2w : 1 ÿ kaw 1 ÿ kaw Ckaw 1 ÿ kaw 1 ÿ aw 2
The second term of the RHS allows the necessary ¯exibility to obtain a good ®tting for high aw values. That term has a very low weight for low values of aw , so the values of um , C and k are not severely aected by the addition of this new term. It can be observed that if the value of k2 is equal to zero, the GAB isotherm (Eq. (1)) can be obtained. The predictive capacity of Eqs. (1) and (2) was evaluated using the sorption data for native potato starch (van den Berg, 1981) and for gluten (De Jong, van den Berg & Kokelaar, 1996), that cover the range of high aw values. The parameters of Eqs. (1) and (2) were obtained by non-linear regression, using as objective function the minimization of the relative deviations s P 0 2 ui ÿ ui =u0i 100; 3a RMS % N where u0 and u are the experimental and predicted values, respectively, using Eqs. (1) or (2), and N is the number of data points. De Jong et al. (1996) proposed that a sorption isotherm is useful when the RMS is lower than 4%. Given that an increase in the number of parameters automatically leads to a better ®tting of the experimental data. Eq. (3a) was calculated by replacing the number of data N, by the number of degrees of freedom (Maroulis, Tsami, Marino-Kouris & Saravacos, 1988), as follows
sP 2 u0i ÿ ui =u0i 100; RMS % N ÿ np
3b
where np is the number of parameters. 2.1. Calculation of the sorption heat The net isosteric heat is usually calculated by means of the following expression (Rizvi & Benado, 1984) d ln aw qst 4 ÿ ; R d 1=T u where R is the universal gas constant and qst the net isosteric heat. The derivation process must be performed at constant moisture content. To calculate the isosteric heat, it is necessary to use a sorption isotherm with an adequate temperature function, or to plot ln(aw ) vs 1/T using sorption data at two dierent temperatures, evaluating the sorption heat at an intermediate one. On the other hand, an alternative expression can be derived from Eq. (4), for calculation of the net isosteric heat, as follows. Eq. (4) can be written as RT 2 daw : 5 qst aw dT u
The values of daw /dT|u in Eq. (5) are calculated using the sorption isotherm (Eq. (2)), which can be represented in the form u f aw ; T :
6
Dierentiating Eq. (6) and imposing the condition that u constant, we get ÿof =oT jaw daw : 7 dT u of =oaw jT The derivative that appears in the denominator of Eq. (7) is the slope of the sorption isotherm. This slope can be easily calculated from the isotherm (Eq. (2)), because the derivation process is made at constant temperature. To evaluate the derivative that appears in the numerator of Eq. (7), it was assumed that the moisture values as a function of temperature, for a ®xed value of aw , can be represented by a polynomial expression of second degree as follows u a bT cT 2 :
8
Applying Eq. (8) to three values of moisture, corresponding to three dierent temperatures for a ®xed value of aw , we get u1 a bT1 cT12 ; u2 a bT2 cT22 ; u3 a bT3
cT32 :
9
P.E. Viollaz, C.O. Rovedo / Journal of Food Engineering 40 (1999) 287±292
289
The equation system (9) can be expressed in a matrix array to obtain the values of the coecients a, b and c as follows 0
0
a b c Aÿ1 u1 u2 u3 ;
10
0
where the indicates the transpose of the ®le vector and Aÿ1 is the inverse of the matrix A, de®ned as 2 3 1 T1 T12 A 4 1 T2 T22 5: 11 1 T3 T32 Coecients a, b and c are the function of aw . But the matrix A and its inverse are not a function of aw but only depend on temperature, so they must be evaluated only once. Finally, the expression of the derivative that appears in the numerator of Eq. (7), is the following of ou b 2cT : 12 oT aw oT
Fig. 1. Comparison between experimental and predicted sorption isotherms for potato starch. (D) T 2.7°C; () T 20.2°C; (h) T 40.2°C; (¨) T 67.2°C; (A) predicted by Eq. (1).
aw
Replacing Eq. (12) in Eq. (7), the derivative of aw with respect to the temperature at constant moisture can be obtained oaw ÿ b 2cT : 13 oT ou=oa j u
w T
This derivative can be replaced in Eq. (5) to calculate the isosteric heat. The values calculated by this method were compared with those obtained using the following procedures: (a) The traditional method, i.e., by plotting the values of the logarithm of the water activity versus the inverse of the absolute temperature, for ®xed values of moisture contents. (b) By numerical dierentiation, using Eqs. (5) and (7), assuming a temperature dependence of the four constants of Eq. (2) as follows um a 1 a 2 T ;
14
ln k a3 a4 =T ;
15
ln C a5 a6 =T ;
16
ln k2 a7 a8 =T :
17
The same sorption data of Fig. 1 were ®tted by nonlinear regression, using Eq. (2) (Fig. 2). It can be observed that this equation can correctly represent the sorption data with a very low deviation, even for water activities up to 0.97. The values of the parameters of Eqs. (1) and (2), obtained by non-linear regression, and the values of RMS(%) calculated with Eqs. (3a) and (3b), are shown in Tables 1 and 2 for potato starch. An example of the parameters' correlation matrix for Eq. (2) at 40°C is shown in Table 3. Less than 10 iterations were needed to obtain the values of the parameters that minimized Eq. (3a). Only the regression of the data corresponding to 67°C was particularly dicult and more than 60 iterations were needed in this case. It is possible that at this temperature
3. Results and discussion Experimental data of native potato starch were compared with the isotherms predicted by the GAB Eq. (1) and the proposed Eq. (2). In Fig. 1, sorption data for potato starch at four temperatures (2.7°C, 20.2°C, 40.2°C and 67.2°C) were plotted, together with the predictions using Eq. (1). It can be observed in that ®gure, that the ®tting was not satisfactory for water activities higher than 0.75.
Fig. 2. Comparison between experimental and predicted sorption isotherms for potato starch. (D) T 2.7°C; () T 20.2°C; (h) T 40.2°C; (¨) T 67.2°C; (A) predicted by Eq. (2).
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Table 1 Parameters of the GAB isotherm (Eq. (1)) for potato starch Temperature (°C)
um
k
C
RMS(%) (Eq. (3a))
RMS(%) (Eq. (3b))
2 20 40 67
0.10221 0.09828 0.08573 0.08738
0.78696 0.78848 0.79277 0.71900
28.4147 18.9935 18.0858 11.3984
3.275 3.777 5.667 3.427
3.695 4.515 6.462 4.534
Table 2 Parameters of the proposed isotherm (Eq. (2)) for potato starch Temperature (°C)
um
k
C
k2
RMS(%) (Eq. (3a))
RMS(%) (Eq. (3b))
2 20 40 67
0.10729 0.10709 0.09770 0.20002
0.75207 0.72015 0.67030 0.07340
25.6484 17.1920 16.1790 31.2587
0.00014 0.00055 0.00186 0.03041
1.63 1.855 2.450 1.57
1.937 2.394 2.945 2.40
Table 3 Correlation matrix for the parameters of Eq. (2), for 40°C
um k C k2
um
k
C
k2
1 ÿ0.9515 ÿ0.8181 0.9394
1 0.6769 ÿ0.9411
1 ÿ0.8530
1
the starch was gelatinized and the material was very dierent from the original native starch. In order to analyze the ¯exibility of the proposed Eq. (2) in ®tting the experimental data in the high aw range, the predicted values using the parameters um , C and k, for 20°C (see Table 1) were plotted in Fig. 3, for dierent values of k2 . It can be observed that the variation of k2 aects mainly the moisture values predicted for high aw , but has a small in¯uence for low aw .
Fig. 4. Sorption isotherm slopes for four dierent temperatures, as a function of water activity.
The slopes of the isotherms calculated using sorption data for starch at four dierent temperatures (2°C, 20°C, 40°C and 67°C), were plotted in Fig. 4. In this ®gure, a plateau for aw values between 0.1 and 0.8 can be observed. Out of this range, the value of the slope markedly increases. Sorption data for gluten of wheat Spring Original (De Jong et al., 1996), at 25°C and 40°C, was ®tted using Eq. (1) (Fig. 5) and Eq. (2) (Fig. 6). It can be observed in these ®gures that the dierence in the quality of the ®tting is not signi®cant. This eect can be attributed to the fact that the maximum moisture content in gluten (0.38 kg/kg, (d.b.)) is lower than the maximum moisture content of the starch. (0.49 kg/kg (d.b.)). 3.1. Calculation of the isosteric heat
Fig. 3. In¯uence of parameter k2 (Eq. 2) on the ®tting of experimental sorption data in the high aw range.
The calculation process of the isosteric heat using Eq. (5) can be easily implemented using a spreadsheet,
P.E. Viollaz, C.O. Rovedo / Journal of Food Engineering 40 (1999) 287±292
Fig. 5. Comparison between experimental and predicted sorption isotherms for gluten. (n) T 25°C; (h) T 40°C; (A) predicted by Eq. (1).
Fig. 6. Comparison between experimental and predicted sorption isotherms for gluten. (n) T 25° C ; (h) T 40°C; (A) predicted by Eq. (2).
291
Fig. 8. Calculated isosteric heat as a function of moisture content. - - - - predicted by polynomial approximation, Eqs. (5), (7) and (13) - - - - x- - - - - predicted by the traditional method (Eq. (4) in semilog plot) - - - - predicted by numerical dierentiation of Eq. (2) with temperature dependence given by Eqs. (14)±(17).
2°C, 20°C and 40°C. Data corresponding to 67°C were not used because at that temperature the starch can be gelatinized. The sorption heat values shown in Fig. 8 are reasonable for moisture contents greater than 0.05 kg/kg. For lower moisture contents, the sudden fall of the isosteric heat at 40°C is due to the absence of experimental sorption values in that range. The isosteric heat values calculated using Eq. (5) (full line curves) were compared with the values obtained using the classical Eq. (4) (continuous line with x curve, i.e. - - - - x - - - -), giving an excellent agreement for 20°C. The value of constants a1 ±a8 in Eqs. (14)±(17) were obtained using the data of Table 2. The isotherms at the three temperatures were predicted using those constants and compared with the experimental data (van den Berg, 1981). A good agreement between predicted and experimental data was observed as shown in Fig. 9. The
Fig. 7. Isosteric heat calculated using Eq. (5), as a function of water activity.
after doing the calculation of the inverse matrix Aÿ1 . The results were represented as a function of water activity (Fig. 7) and as a function of moisture (Fig. 8), for dierent temperatures. The data used, correspond to
Fig. 9. Comparison between experimental and predicted sorption isotherms for potato starch. (D) T 2.7°C; () T 20.2°C; (h) T 40.2°C; (¨) T 67.2°C; (A) predicted assuming a temperature dependence of the four constants in Eq. (2).
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P.E. Viollaz, C.O. Rovedo / Journal of Food Engineering 40 (1999) 287±292
Fig. 10. Isosteric heat calculated obtained assuming a temperature dependence of the four constants in Eq. (2), as a function of moisture content.
isosteric heat values obtained by this last procedure for 20°C were in good agreement with the values obtained by the other two methods, as is shown in Fig. 8 (dashed line). It is interesting to note that a good prediction of the isotherms at dierent temperatures can be obtained, using Eqs. (2), (14)±(17), assuming that the monolayer moisture content is constant. But, the prediction of the isosteric heat obtained in this way is nearly half the value obtained taking into account the temperature dependence of the monolayer moisture. Therefore, it is essential to take into account the temperature dependence of the four constants in Eq. (2) in order to obtain values of the isosteric heat that are consistent with values obtained by means of the traditional method using Eq. (4). In Fig. 10 are shown the values of the isosteric heat calculated using the proposed isotherm i.e., Eq. (2), taking into account the temperature functionality given by Eqs. (14)±(17) is shown. The values of the isosteric heat were obtained by numerical dierentiation of Eq. (2), taking into account Eqs. (5), (7), (14)±(17). The obtained results appear very reasonable, and in Fig. 8 is observed a notable agreement with the curve for 20°C calculated by the two other methods. 4. Conclusions The proposed equation satisfactorily correlates the experimental sorption data for starch and gluten for a
very wide range of water activities. The functional nature of the proposed equation allows performing the ®tting process without problems of numerical instability. The additional constant k2 allows a higher ¯exibility for high aw values, without jeopardizing the good qualities of the GAB equation in its application range. The proposed methodology allows calculating the isosteric heat values in good agreement with those obtained by the traditional method, at intermediate temperatures. A good prediction of moisture content and isosteric heat values at dierent temperatures is obtained, assuming temperature dependence for the four parameters of the proposed isotherm. In this way, this equation can be useful for modeling drying kinetics. Also, the calculation can be extended to other thermodynamic functions. Acknowledgements This work was supported by the Universidad de Buenos Aires and by Consejo Nacional de Investigaciones Cienti®cas y Tecnicas (CONICET, Argentina). References Chirife, J. (1983). A survey of existing sorption data. In Physical properties of foods (pp. 55±64). London and New York: Applied Science Publishers. Jong, G. I. W., Berg, C., & Kokelaar, A. J. (1996). De van den water sorption behaviour of original and defatted wheat gluten. International Journal of Food Science and Technology, 31, 519±526. Krishna, R. (1987). A uni®ed theory of separation processes based on irreversible thermodynamics. Chemical Engineering Communications, 59, 33±64. Maroulis, Z. B., Tsami, E., Marino-Kouris, D., & Saravacos, G. D. (1988). Application of the GAB model to the moisture sorption isotherms for dried fruits. Journal of Food Engineering, 7, 63±78. Rizvi, S. S. H., & Benado, A. L. (1984). Thermodynamic properties of dehydrated foods. Food Technology, 83±92. Sch ar,W., & Ruegg, M. (1985). The evaluation of GAB constants from water sorption data. Lebbensmittel Wiss und Technology, 18, 225± 229. van den Berg, C. (1981). Vapor sorption equilibria and other water± starch interactions: A physico-chemical approach. PhD Dissertation, Agricultural University Wageningen, The Netherlands. van den Berg, C., & Bruin, S. (1978). Water activity and its estimation in food systems: Theoretical aspects. Second International Symposium on properties of water in relation to food quality and stability, Isopow-II, Osaka, Japan.