A New Model for Predicting Sorption Isotherm of Water in Foods

Volume 7, Issue 2 2011 Article 16

International Journal of FoodEngineering

A New Model for Predicting SorptionIsotherm of Water in Foods

Farid B. Cortés, Universidad Nacional de ColombiaFarid Chejne, Universidad Nacional de Colombia

Benjamin Rojano, Universidad Nacional de Colombia

Recommended Citation:Cortés, Farid B.; Chejne, Farid; and Rojano, Benjamin (2011) "A New Model for PredictingSorption Isotherm of Water in Foods," International Journal of Food Engineering: Vol. 7: Iss. 2,Article 16.DOI: 10.2202/1556-3758.2254

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A New Model for Predicting SorptionIsotherm of Water in Foods

Farid B. Cortés, Farid Chejne, and Benjamin Rojano

Abstract

A new model for predicting sorption isotherms of type II and III based on the Polanyi theoryis proposed. This model allows the prediction of the sorption isotherms at different temperaturesfrom one experimental isotherm. The theoretical predictions of isotherms and isosteric heat werevalidated successfully using data from literature for twenty one foods. This method allows totalexperimental time and operation costs to be reduced.

KEYWORDS: foods, sorption, isotherm, isosteric heat, Polanyi theory

Author Notes: The authors thank COLCIENCIAS and Universidad Nacional de Colombia sedeMedellín for logistical and financial support.


INTRODUCTION

The influence of water on food reactions and food quality is very important because the majority of biological reactions take place in aqueous media. Water content alone does not allow the evolution of many food phenomena to be predicted; and the state of the water molecules in the food matrix are a better indicator (Clemente et al., 2009).

Water activity in food products is equivalent to the relative humidity of the air in equilibrium with the product. Knowing the state of the sorption-thermodynamic equilibria between the relative humidity of the surrounding air and the moisture content of the solid matter is a basic prerequisite for understanding mass transfer processes (Comaposada et al., 2000; Borgognoni et al. 2008). Therefore, sorption isotherms and isosteric heats of sorption are important parameters in the design and modeling of drying processes, food engineering and industrial quality control, just as it is for any similar mass transfer situation (Borgognoni et al. 2008; Orikasa et al., 2010; Cortés & Chejne, 2010). Their correct determination represents a challenge to find an effective method to evaluate those parameters due to the intense development of processes.

The most widely used experimental method of obtaining the sorption equilibrium data of foods is a static method using a gravimeter (Wolf et al.,1985; Comaposada et al., 2000; Borgognoni et al. 2008). In this method, the food sample is exposed to several atmospheres of known relative humidities that are controlled with different salts according to the method proposed by Chirife & Resnik (1984) and Wolf et al. (1985) until there is no discernible weight change, indicating that the samples have reached equilibrium. The time required to acquire one experimental point, that is, the time required for the sample to equilibrate with the atmosphere, varied from 2 to 8 weeks based on the change in weight expressed on a dry basis (Clemente et al., 2009; Singh, 2006). Several experimental points are needed to construct one isotherm, requiring high experimental acquisition times (Saurel et al., 2004). Complete knowledge of sorption isotherms is essential, and isotherms require experimental determination because current theoretical techniques are not able to accurately simulate systems as complex as food (Kaymak-Ertekin & Saltanoglu, 2001).

Another important parameter is the isosteric heat of sorption, which is useful in assessing the energy needed for the drying process. Two different methods can be used to determinate the isosteric heat of sorption (Clemente et al., 2009): an indirect technique based on the sorption equilibrium data and estimation using calorimetric techniques and the Riedel equation (Reidel, 1977). The first method uses sorption experimental data (isotherms at different temperatures) to construct the sorption isosteres using a Clapeyron diagram and the Clausius–Clapeyron equation. The second technique requires accurate experimental work

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Cortés et al.: A New Model for Predicting Sorption Isotherm of Water in Foods


and expensive equipment. Therefore, the indirect method based on the Clausius-Clapeyron equation is the most widely used (Mulet, 1999; Comaposada et al., 2000; Delgado and Sun, 2002a; Delgado and Sun, 2002b; Mulet et al., 2002; Saurel et al., 2004; Clemente et al., 2009; Cortés et al., 2009; Orikasa et al., 2010).

Thus, in this work, a new model for predicting the experimental food sorption isotherms based on the Polanyi theory (Polanyi 1914; 1916) is proposed. We computed the isosteric heats of sorption using the Clausius-Clapeyron equation from the estimated equilibrium data. The main objective of this work was to predict sorption properties on moisture sorption isotherms of twenty one foods at different temperatures from one isotherm, considerably reducing the experimental time and operation costs.

THEORY

In 1914 a theory of adsorption was proposed to describe adsorption phenomena. This description is associated with the name Polanyi (Polanyi 1914; 1916). The assumptions underlying the so-called potential theory of adsorption, commonly called the Polanyi theory, originate from earlier work by de Saussure (Dabrowski, 2001). According to this theory, the adsorbed layer is considered to be a thick multilayer film of decreasing density as the distance from the solid surface increases. The basic concepts of the Polanyi theory include the adsorption potential and the characteristic adsorption curve. This characteristic curve presents a simple relationship between the adsorption potential and the distance from the solid surface (Figure 1).

Figure 1. Schematic of an adsorption system in which the shaded molecules are the “captured” molecules under the influence of interaction forces near the pore surfaces.

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International Journal of Food Engineering, Vol. 7 [2011], Iss. 2, Art. 16

DOI: 10.2202/1556-3758.2254


The aforementioned distance may be expressed in terms of volume units of the adsorbed phase. Polanyi (Polanyi 1914; 1916) assumed that the adsorption potential is also independent of temperature, meaning that the characteristic curve of adsorption of nonpolar molecules is also temperature independent. Such a statement follows from the fact that the van der Waals forces are also independent of temperature. The principal assumption of the theory is given by:

where, T is temperature, X is moisture and A is the adsorption potential. Equation (1) expresses the temperature invariance of the adsorption of

different pairs as a function of the adsorption potential, a direct consequence of the Polanyi theory (Polanyi 1914 and 1916).

The adsorption potential is given by:

where, is the steam activity considered as ideal gas, is the universal gas constant, is the saturation pressure at a given temperature and is the equilibrium pressure of water.

Having calculated the characteristic curve for a given temperature, it is possible to determine sorption isotherms at other temperatures. However, the Polanyi theory [15, 16] does not give a definite equation for sorption isotherms that, to some extent, replaces the characteristic sorption equation (Dubinin, 1967; Dabrowski, 2001).

Our method is based on the Polanyi theory (Polanyi 1914; 1916). Taking Eq. (2) under isosteric conditions (X = constant), we can construct the sorption isosteres using only one experimental data set. The latter occurs because the isotherms at any value of temperature show the same trend respect to the characteristic curve for any food (Figure 2). Particularly, Figure 2 shows a characteristic curve for lemmon peel at different temperatures, and hence it is possible to see the same trend. Therefore, it is possibly to claim that for any isosteric condition the adsorption potential (differential molar work of sorption) is independent of temperature, i.e., for a defined value of moisture, the value of adsorption potential is always the same for any temperature.

0 1

2

This relationship is the characteristic adsorption equation (Eq. (1)).

3



Figure 2. Characteristic curve for lemmon peel at 20, 30, 40 and 50°C (García-Peréz et al., 2008). Points denote the experimental data.

To obtain the model is necessary to apply an isosteric condition in Eq. (2), which demands that each equilibrium point i (i.e. for each moisture value) has the same value of differential molar work of sorption at any temperature, thus:

3

ATi is the differential molar work evaluated at temperature Ti.

Replacing Eq. (2) in Eq. (3), the new expression obtained is represented by:

4

The Eq. (4) represents the equality between the differential molar work of sorption at different temperatures , , . Therefore, it is possible to calculate the steam activities at different temperatures starting from a previous given temperature. In other words, the steam activity at temperature i can be obtained from adsorption isotherm (experimental data of sorption equilibrium) for each moisture. This procedure is repeated several times at other temperatures and moisture, which are determined experimentally at which is given as:

0

0,4

0,8

1,2

1,6

2

0 50 100 150 200 250 300 350

X

A (J/mol)

20°C

30°C

40°C

50°C

4


DOI: 10.2202/1556-3758.2254


exp 5

The expression in Eq. (5) allows us to construct sorption isosteres (PT model), considerably reducing the experimental time needed for any experimental method. From isosteres is possible obtain the sorption isotherms. The isosteric heat Q can be obtained using the Clausius–Clapeyron equation (Cortés et al., 2010):

ln1

16

where P and T denote gas-phase equilibrium pressure of the sorbing species and the absolute temperature, respectively. Z is the compressibility coefficient (Z = 1 for an ideal gas and Z≠1 for a real gas) and vn and vg denote the partial molar volume of the sorbing species in the sorption phase. For an ideal gas, Eq.(6) is simplified to

ln

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From isosteres, or data sets (T, X, ) for a given system, available over the entire sorbate concentration region, the sorption equilibrium is comprehensively and fully described. With this experimental data is possible to evaluate the isosteric heat of sorption using Eq.(7).

The accuracy of fit was evaluated by calculating the root mean square percent error (RMS %).

% 100 ∗1

8

Where, and are experimental and computed activity, respectively, and n is the observation number.

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RESULTS AND DISCUSSION

In Figure 3, the experimental and theoretical water sorption isotherms for morel (Morchela Sculenta) are compared. The experimental data were taken from Mulet et al. (1999), and the theoretical data were predicted using the theory of the temperature invariance of the characteristic curve of sorption (from Eq. (5)) and the isotherm at 5°C. The equilibrium moisture content increased with increased water activity at different temperatures (15, 25 and 35°C). The model allows the prediction and description of the behavior of the isotherms at 15, 25 and 35°C with RSM %< 4.48. The discrepancy between the data and the PT model is probably within the experimental error given the difficulty in measuring water isotherms for morel. However, a good accuracy between experimental and computed isotherms is observed at both low and high moisture.

Figure 3. Experimental sorption isotherms for morel (Morchella Sculenta) at 15, 25 and 35°C and curves estimated using the PT model based on the 5°C experimental isotherm.

Points denote the experimental data and lines denote the prediction using Eq. (5).

Figure 4 compares the experimental and theoretical (from Eq. (5)) water sorption isotherms in lemon juice. The experimental data were taken from Martinelli et al. (2007), and the theoretical data were predicted using the proposed model (from Eq. (5)) and the isotherm at 20°C. Figure 4 also shows an increase in equilibrium moisture content with increasing water activity at different

0

0,3

0,6

0 0,2 0,4 0,6 0,8 1

X

aw

5°C

0

0,3

0,6

0 0,2 0,4 0,6 0,8 1

X

aw

15°C

0

0,3

0,6

0 0,2 0,4 0,6 0,8 1

X

aw

25°C

0

0,3

0,6

0 0,2 0,4 0,6 0,8 1

X

aw

35°C

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temperatures (20, 30, 40 and 50°C). The model allows the prediction and description of the behavior of the isotherms at 30, 40 and 50°C with RSM %< 0.38. Excellent accuracy between experimental and computed isotherms is observed at both low and high moisture.

Figure 4. Experimental sorption isotherms for lemon juice at 20, 30, 40 and 50°C and curves estimated using the PT model based on the 20°C experimental isotherm. Points

denote the experimental data and lines denote the prediction using Eq. (5).

The experimental and theoretical water sorption isotherms for acid casein from buffalo skim milk, biceps femoris muscle of pork and Spray-Dried yogurt powder are compared in Fig. (5), (6) and (7), respectively. The experimental data of acid casein from buffalo skim milk, biceps femoris muscle of pork and Spray-Dried yogurt powder were taken from Sawhney et al. (2010), Clemente et al.(2009) and Koç et al. (2010), respectively.

0

0,3

0,6

0,9

0 0,2 0,4 0,6 0,8 1

X

aw

20°C

0

0,3

0,6

0,9

0 0,2 0,4 0,6 0,8 1

Xaw

30°C

0

0,3

0,6

0,9

0 0,2 0,4 0,6 0,8 1

X

aw

40°C

0

0,3

0,6

0,9

0 0,2 0,4 0,6 0,8 1

X

aw

50°C

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Figure 5. Experimental sorption isotherms for acid casein from buffalo skim milk at 25, 35 and 45°C and curves estimated using the PT model based on the 25°C experimental isotherm. Points denote the experimental data and lines denote the prediction using Eq.

(5).

Figure 6. Experimental sorption isotherms for biceps femoris muscle of pork at 25, 30, 35 and 40°C and curves estimated using the PT model based on the 25°C experimental

0

0,2

0,4

0,6

0 0,2 0,4 0,6 0,8 1

X

aw

25°C

0

0,2

0,4

0,6

0 0,2 0,4 0,6 0,8 1

X

aw

35°C

0

0,2

0,4

0,6

0 0,2 0,4 0,6 0,8 1

X

aw

45°C

0

0,3

0,6

0,9

0 0,2 0,4 0,6 0,8 1

X

25°C

0

0,3

0,6

0,9

0 0,2 0,4 0,6 0,8 1

X

30°C

0

0,3

0,6

0,9

0 0,2 0,4 0,6 0,8 1

X

35°C

0

0,3

0,6

0,9

0 0,2 0,4 0,6 0,8 1

X

40°C

isotherm. Points denote the experimental data and lines denote the prediction using Eq. (5).

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Figure 7. Experimental sorption isotherms for Spray-Dried yogurt powder at 10, 25 and 40°C and curves estimated using the PT model based on the 10°C

experimental isotherm. Points denote the experimental data and lines denote the prediction using Eq. (5).

Table 1 shows the RMS% of the water sorption isotherms for 21 foods at different temperatures. The highest root mean square percent error is 6.65% for tea Leaf Juice at 40 °C. The other foods have RMS% values of less than 6.65%, which is in good agreement (Lopes-Filho et al., 2002). A good correspondence between experimental and calculated isotherms is observed at low and high activities between 0.1 and 0.9. Also, Table 1 shows the highest RMS% of the isosteric heat for Prickly Pear Fruit (Lahsasni et al, 2004) is 5%, the highest value for the evaluated foods at a wide range of moisture levels. The reported RMS% value is far less than 5% for the eleven foods. These results show the accuracy and the effectiveness of the approach based on the Polanyi theory (Polanyi 1914; 1916). Additionally, Table 1 compares the root mean square for the GAB and our model, where the GAB model shows a higher RMS% than our model for temperatures different and several foods.

0

0,2

0,4

0,6

0,8

0 0,2 0,4 0,6 0,8 1

X

aw

10°C

0

0,1

0,2

0,3

0,4

0 0,2 0,4 0,6 0,8

X

aw

25°C

0

0,2

0,4

0,6

0,8

0 0,2 0,4 0,6 0,8

X

aw

40°C

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Table 1. Estimated RMS% of water sorption isotherms and isosteric heats for different foods types.

Food Sorption properties RMS% RMS% (GAB model)

Biceps femoris of pork (Clemente et al, 2009)

Isosteric heat (J/g) 2.04 -

Isotherm(°C) 25 30 35 40

0.00 9.62 2.16 8.37 3.53 9.11 5.62 10.62

Semimembranosus muscle of pork (Clemente et al, 2009)


Isotherm (°C) 25 30 35 40

0.00 8.36 1.02 7.54 2.98 7.32 3.71 9.34

Lemonene-MA (Borgognoni et al., 2008)

Isotherm (°C) 25 30 35

0.00 2.47 0.22 2.42 0.54 2.01

Lemonene-CH (Borgognoni et al., 2008)


0.00 3.28 0.00 11.93 0.82 10.1

Beef (Delgado and Sun, 2002a)

Isosteric heat (J/g) 1.94 - Isotherm (°C) 10 20 25 35 49

0.00 5.6 6.00 15.3 5.07 9.1 5.52 12.2 4.59 11.9

Chicken (Delgado and Sun, 2002b)

Isosteric heat (J/g) 3.17 - Isotherm (°C) 4 8 20 30

0.00 11.6 5.97 10.31 4.31 7.31 5.66 10.27

Goat (Singh, 2006)



0.00 6.5 3.68 7.93

5.20 6.99

Morel (Mulet, 2002)

Isosteric heat (J/g) 1.80 - Isotherm (°C) 5 15

0.00 2.38 3.13 5.15

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25 35

4.48 7.66 0.94 4.78

Lemon Peel (García-Peréz et al., 2008)

Isosteric heat (J/g) 0.89 - Isotherm (°C) 20 30 40 50

0.00 3.82 1.08 4.20 0.66 2.38

0.90

3.65

Starch Potatoes (Al-Muhtaseb et al., 2004)

Isosteric heat (J/g) 1.24 - Isotherm (°C) 30 45 60

0.00 8.3 4.90 8.8 6.08 15.1

Grapes (Kaymak-Ertekin and Gedik, 2004)


0.00 2.4 4.90 4.9 6.08 14.3

Reticulata C (Jamali, 2006)


0.00 - 4.10 - 4.59 -

Prickly Pear Fruit (Lahsasni et al., 2004)


0.00 0.30 3.65 3.82 4.45 4.62

Powdered Milk (Abdenouri et al., 2010)


0.00 3.32 0.52 3.10 0.36 3.20

Chitosan (Rosa et al., 2009)

Isotherm (°C) 20 30 40 50 60

0.00 5.84 0.36 6.58 0.42 7.37 2.70 9.52 1.42 4.32

Acid Casein (Sawhney et al., 2011)


0.00 5.24 1.91 1.93 0.29 5.78

Lemon Juice Isotherm (°C) 20 0.00 -

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(Martinelli et al., 2007)

30 40 50

0.00 - 0.17 - 0.38 -

Lemon Juice- Maltodextrin (Martinelli et al., 2007)

Isotherm (°C) 20 30 40 50

0.00 - 0.92 - 0.16 - 0.84 -

Tea Leaf Juice (Sinija & Mishra, 2008)

Isotherm (°C) 20 30 40 50

0.00 - 2.09 - 6.65 - 6.61 -

Tomato Pulp (Goula et al., 2007)

Isotherm (°C) 20 30 40 50

0.00 - 0.19 - 1.32 - 2.21 -

Spray-Dried Yogurt Powder (Koç et al., 2010)


0.00 - 0.10 - 0.59 -

CONCLUSION

The sorption isotherm can be estimated using the Polanyi theory with less experimental time required compared to the other method and with good accuracy compared to the experimental data in the literature. This novel method is an easy tool and good agreement is found between the food sorption isotherms predictions made using this method and the experimental data. Additionally, the isosteric heat computed using the theoretical data was verified using the values obtained from experimental data and compared with theoretical data obtained from GAB. In principle, the Polanyi postulate may be constructed using one experimentally measured isotherm. No other adsorption model allows an experimentally measured isotherm to be extrapolated to other temperatures.

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A New Model for Predicting Sorption Isotherm of Water in Foods

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