6. Moisture sorption isotherm models

Equations for fitting the moisture sorption isotherms are of special importance in many aspects of crop and food preservation by drying. These include the prediction of the drying times, shelf life of the dried product in a packaging material and the equilibrium conditions after mixing products with varying water activities [87]. Others are the analytical determination of control for undesirable chemical and enzymatic reactions [88] and control of moisture migration in multidomain foods [9]. Moisture sorption isotherm models, therefore, not only constitute an essential part of the overall theory of drying but also provide information directly useful in the accurate and optimum design of drying equipment [1]. They are needed in the evaluation of the thermodynamic functions related to moisture sorption in biological materials [89].

Several theoretical, semi-theoretical and empirical models have been proposed and used by investigators to fit the equilibrium moisture content data of food and agricultural products. Chirife and Iglesias [87] reviewed part of the isotherm equations and presented a discussion of 23 common models, while Van den Berg and Bruin [5] presented a more comprehensive list. Ngoddy-Bakker-Arkema [1] developed a generalized moisture sorption isotherm model for biological materials based primarily on the BET and capillary condensation theories and indirectly on Polanyi's potential theory. This model appears to possess very high versatility but needs to be modified to reduce the number of parameters and incorporate the temperature term. A thorough going and extensive testing of the model on various categories of food is also necessary to confirm its versatility and prove the generalized posture. Ferro Fontan et al. [2] and Chirife et al. [90] presented a new model, which Iglesias and Chirife [91] compared with the GAB model and reported to be an alternative. Chen [92] derived a new moisture sorption isotherm model from a reaction engineering approach. The Brunauer-Emmett-Teller (BET) [87] and Guggenheim-Anderson-de Boer (GAB) [56, 91, 93] models have been used for estimating the monolayer moisture content of agricultural and food products. Boquet et al. [94] noted that the Hailwood and Horrobin model has a remarkably good ability to fit the experimental data for most food types. A test of the model on moisture sorption data of native cassava and sorghum starches [95] showed that it has good predictive Moisture Sorption Isotherms and Isotherm Model Performance Evaluation for Food… DOI: http://dx.doi.org/10.5772/intechopen.87996

performance with R<sup>2</sup> ranging from 0.92 to 0.99. It, however, lacked the temperature term and was modified to incorporate the term. Other commonly used models include modified Henderson, modified Chung-Pfost, modified Halsey and modified Oswin and the GAB. The modified Henderson [96] and modified Chung-Pfost [97] models have been adopted as the standard equations by the American Society of Agricultural and Biological Engineers (ASABE) for describing the EMC-aw data for cereals and oil seeds [98]. The modified Halsey [85] has been reported as the best model for predicting the EMC-aw relationships of several tropical crops [99] and alongside with the modified Oswin [100] has been shown to describe the EMC-aw data of many seed satisfactorily [101, 102]. The Guggenheim-Anderson-de Boer (GAB) model has been recognized as the most satisfactory theoretical isotherm Equation [103–106] and has been recommended as the standard model for use in food laboratories in Europe [105] (1985) and the USA [107]. The GAB does not incorporate a temperature term; therefore, the determination of the effect of temperature on isotherms using the model usually involves the evaluation of up to six constants. Jayas and Mazza [108], however, developed a modified form of the GAB, which incorporates the temperature term. The MSI models considered in this study were selected from the above list and presented as follows:

1.Brunauer-Emmett-Teller (BET) model

$$M = \frac{M\_m \text{Ca}\_w}{(1 - a\_w)[1 + (\text{C} - \text{1})a\_w]} \tag{7}$$

2.Guggenheim-Anderson-de Boer (GAB) model

$$M = \frac{\text{CKM}\_m a\_w}{(1 - K a\_w)[1 - K a\_w + \text{CKa}\_w]} \tag{8}$$

3.Modified GAB model

be hemispherical in which the Kelvin equation is applied. The open-pore theory is

out, the force of attraction causes water-holding spaces to shrink (molecular shrinkage). This permanent shrinkage reduces the water-binding polar sites and water-holding capacity of the material; hence less amount of water is absorbed

isotherm using bulk moduli determined as a function of moisture content.

Equations for fitting the moisture sorption isotherms are of special importance in many aspects of crop and food preservation by drying. These include the prediction of the drying times, shelf life of the dried product in a packaging material and the equilibrium conditions after mixing products with varying water activities [87]. Others are the analytical determination of control for undesirable chemical and enzymatic reactions [88] and control of moisture migration in multidomain foods [9]. Moisture sorption isotherm models, therefore, not only constitute an essential part of the overall theory of drying but also provide information directly useful in the accurate and optimum design of drying equipment [1]. They are needed in the evaluation of the thermodynamic functions related to moisture sorption in biologi-

Several theoretical, semi-theoretical and empirical models have been proposed and used by investigators to fit the equilibrium moisture content data of food and agricultural products. Chirife and Iglesias [87] reviewed part of the isotherm equations and presented a discussion of 23 common models, while Van den Berg and Bruin [5] presented a more comprehensive list. Ngoddy-Bakker-Arkema [1] developed a generalized moisture sorption isotherm model for biological materials based primarily on the BET and capillary condensation theories and indirectly on Polanyi's potential theory. This model appears to possess very high versatility but needs to be modified to reduce the number of parameters and incorporate the temperature term. A thorough going and extensive testing of the model on various categories of food is also necessary to confirm its versatility and prove the generalized posture. Ferro Fontan et al. [2] and Chirife et al. [90] presented a new model, which Iglesias and Chirife [91] compared with the GAB model and reported to be an alternative. Chen [92] derived a new moisture sorption isotherm model from a reaction engineering approach. The Brunauer-Emmett-Teller (BET) [87] and Guggenheim-Anderson-de Boer (GAB) [56, 91, 93] models have been used for estimating the monolayer moisture content of agricultural and food products. Boquet et al. [94] noted that the Hailwood and Horrobin model has a remarkably good ability to fit the experimental data for most food types. A test of the model on moisture sorption data of native cassava and sorghum starches [95] showed that it has good predictive

Shrinkage theory: This states that while agricultural and food product is drying

Capillary condensation and swelling fatigue theory: In this theory proposed by Ngoddy-Bakker-Arkema [86], the sorption hysteresis is considered linked with condensation and evaporation in irregular voids (capillary condensation) and influence of adsorbed water molecules on such physical properties of agricultural and food products as strength, elasticity, rigidity, swelling and evolution of heat (swelling fatigue). The above combination was simulated by adopting the Cohan theory of capillary condensation with modifications and combining it with the ink bottle theory in the first approximation. The theory presented expressions for calculating the desorption isotherms of biomaterials from corresponding adsorption

illustrated in Figure 15.

Sorption in 2020s

cal materials [89].

160

during the adsorption process.

6. Moisture sorption isotherm models

$$M = \frac{AB\left(^{C}\_{T}\right)a\_{w}}{\left(1 - Ba\_{w}\right)\left[1 - Ba\_{w} + \frac{C}{T}Ba\_{w}\right]}\tag{9}$$

4.Hailwood-Horrobin model

$$\mathcal{M} = \left(\frac{\mathcal{A}}{a\_w} + \mathcal{B} - \mathcal{C}a\_w\right)^{-1} \tag{10}$$

5.Modified Hailwood-Horrobin model

$$\mathcal{M} = \left( T \left( \frac{A}{a\_w} + B \right) - \frac{C}{T^n} a\_w \right)^{-1} \tag{11}$$

6.Modified Chung-Pfost model

$$M = \frac{-1}{C} L n \left[ -\frac{(T+B)}{A} L n a\_w \right] \tag{12}$$

7.Modified Halsey model

$$M = \left[\frac{-Lna\_w}{\exp\left(A + BT\right)}\right]^{-1/c} \tag{13}$$

8.Modified Henderson model

$$M = \left[\frac{-Ln(\mathbf{1} - a\_w)}{A(T + B)}\right]^{\text{l/c}} \tag{14}$$

variation, Chen [92] used both coefficient of determination and mean relative percent error and Sun [110] and Sun [111] employed the residual sum of squares, standard error of estimate and mean relative percent error in comparing moisture sorption isotherm models for food. Other combinations of parameters that have been used include standard error of estimate and mean relative percent error [120], coefficient of determination and residual sum of squares [18] and standard error of estimate, mean relative percent deviation and residual plots [15, 41, 65, 101, 102].

Moisture Sorption Isotherms and Isotherm Model Performance Evaluation for Food…

DOI: http://dx.doi.org/10.5772/intechopen.87996

A model is considered acceptable for predictive purpose, if the residuals are uniformly scattered around the horizontal value of zero showing no systematic tendency towards a clear pattern [41, 45, 64, 65]. A model is considered better than another if it has lower standard error of estimate and mean relative percent deviation and higher fraction explained variation and coefficient of

Menkov [37] reported that of five moisture sorption isotherm models fitted to

the experimental data on the EMC of vetch seeds, the modified Oswin model proved the best for describing the adsorption and desorption branches. Aviara et al. [41] and Oyelade [121] reported that the modified Oswin model gave the best fit to the EMC of soya bean and lafun, respectively. Santalla and Mascheroni [39] in a similar study on the EMC of sunflower seeds and kernels reported that the GAB model gave the best fit to the experimental data. Other crops whose moisture sorption isotherms have recently been studied include quinoa grains [40], crushed

The procedure followed in evaluating a moisture sorption isotherm model depends on the nature of the model. For the selected models (Eqs. (7)–(16)), the

a. BET model: the BET model (Eq. (7)) can be linearized thorough algebraic

C � 1 MmC

MmC

<sup>M</sup> <sup>¼</sup> Aaw<sup>2</sup> <sup>þ</sup> Baw <sup>þ</sup> <sup>C</sup> (18)

<sup>M</sup> against aw at each temperature and fitting a

<sup>M</sup>ð Þ <sup>1</sup>�aw against aw within the water activity range of 0.01–0.5 at each

MmC, and from these, the values of Mm and C can be obtained and used as the starting values in nonlinear regression. The nonlinear regression analysis procedure minimizes the sum of deviation in the evaluation of a model using a series of iterative steps. The procedure could require that initial parameter estimates be

b. GAB model: the GAB model (Eq. (8)) can be transformed to a quadratic form

polynomial of the second order to the plots. This will yield the following functions

aw (17)

and intercept on the y-

<sup>¼</sup> <sup>1</sup> MmC <sup>þ</sup>

chillies [59], amaranth grains [122] and black gram nuggets [44].

aw Mð Þ 1 � aw

temperature yields a straight line with the slope as <sup>C</sup>�<sup>1</sup>

by algebraic manipulation to yield Eq. (18):

Eq. (18) can be solved by plotting aw

aw

7.1 Model parameter evaluation procedures

manipulations to yield Eq. (17):

procedures are as follows:

A plot of aw

chosen close to the true values.

axis as <sup>1</sup>

from Eq. (8):

163

determination.

9.Modified Oswin model

$$\mathbf{M} = (\mathbf{A} + BT) \left[ \frac{a\_w}{\mathbf{1} - a\_w} \right]^{\mathbf{1}\_{\odot}} \tag{15}$$

10.Ngoddy-Bakker-Arkema model

$$\begin{split} M &= \frac{\rho \varepsilon}{\eta} \left\{ 3.2^{\eta} \left[ \left( \left( \frac{1.75}{Ln \left( \frac{P\_o + P\_m}{P + P\_m} \right)} \right)^{\circ\_2} + \frac{\sigma V}{R\_g T L n \left( \frac{P\_o + P\_m}{P + P\_m} \right)} \right)^{\eta} \right] \\ &- \left( \left( \frac{1.75}{Ln \left( \frac{P\_o + P\_m}{P\_m} \right)} \right)^{\circ\_2} + \frac{\sigma V}{R\_g T L n \left( \frac{P\_o + P\_m}{P\_m} \right)} \right)^{\eta} \right] \end{split} \tag{16}$$

where M is the moisture content, (db); Mm is monolayer moisture content, (db); aw is water activity; T is absolute temperature, (K); A, B, C and k are constants; η is primary characteristic parameter of pore structure; ε is secondary characteristic parameter of pore structure; σ is surface tension of sorbate in bulk liquid form, (N/m); Rg is universal gas constant; V is molal volume of sorbate in its bulk liquid condition, (m<sup>3</sup> /mol); Pm is vapor pressure corresponding to monolayer, (N/m<sup>2</sup> ); Po is saturated vapor pressure, (N/m<sup>2</sup> ); and P is vapor pressure at the condition under which the study is carried out, (N/m<sup>2</sup> ).

### 7. Isotherm model predictive performance evaluation

Sun and Byrne [109], Sun [110] and Sun [111] evaluated the predictive performance of the moisture sorption isotherm models that have been reported for fitting the EMC and ERH data of rapeseed, rice, other grains and oilseeds and selected the models that gave the best fits.

Coefficient of terms in the moisture sorption isotherm equations is usually determined using nonlinear regression procedure, and the predictive performance of an equation on sorption data is evaluated using such goodness of fit parameters as standard error of estimate (estimate of the residual mean square), residual sum of square, coefficient of determination, mean relative percent error, fraction explained variation and residual plots. Several investigators used these parameters to evaluate the fitting ability of EMC-aw equations. For instance, Ajibola [35–37], Ajibola and Adams [34], Ajibola [112], Gevaudan et al. [24], Talib et al. [8], Pezzutti and Crapiste [113], Tsami et al. [17] and Ajibola et al. [64] used the standard error of estimate, and Young [48] and Jayas et al. [114] used the residual sum of squares to compare the fitting ability of different models. Boquet et al. [94], Chirife et al. [90], Weisser [31], Saravacos et al. [12], Pollio et al. [115], Iglesias and Chirife [91] and Khalloufi et al. [10] used the mean relative percent deviation (MRE), while Shepherd and Bhardwaj [33], Demertzis et al. [116], Diamante and Munro [117] and Sopade et al. [118] employed coefficient of determination in evaluating the fitting ability of several models. Pappas and Rao [119] used the fraction explained

Moisture Sorption Isotherms and Isotherm Model Performance Evaluation for Food… DOI: http://dx.doi.org/10.5772/intechopen.87996

variation, Chen [92] used both coefficient of determination and mean relative percent error and Sun [110] and Sun [111] employed the residual sum of squares, standard error of estimate and mean relative percent error in comparing moisture sorption isotherm models for food. Other combinations of parameters that have been used include standard error of estimate and mean relative percent error [120], coefficient of determination and residual sum of squares [18] and standard error of estimate, mean relative percent deviation and residual plots [15, 41, 65, 101, 102]. A model is considered acceptable for predictive purpose, if the residuals are uniformly scattered around the horizontal value of zero showing no systematic tendency towards a clear pattern [41, 45, 64, 65]. A model is considered better than another if it has lower standard error of estimate and mean relative percent deviation and higher fraction explained variation and coefficient of determination.

Menkov [37] reported that of five moisture sorption isotherm models fitted to the experimental data on the EMC of vetch seeds, the modified Oswin model proved the best for describing the adsorption and desorption branches. Aviara et al. [41] and Oyelade [121] reported that the modified Oswin model gave the best fit to the EMC of soya bean and lafun, respectively. Santalla and Mascheroni [39] in a similar study on the EMC of sunflower seeds and kernels reported that the GAB model gave the best fit to the experimental data. Other crops whose moisture sorption isotherms have recently been studied include quinoa grains [40], crushed chillies [59], amaranth grains [122] and black gram nuggets [44].

#### 7.1 Model parameter evaluation procedures

The procedure followed in evaluating a moisture sorption isotherm model depends on the nature of the model. For the selected models (Eqs. (7)–(16)), the procedures are as follows:

a. BET model: the BET model (Eq. (7)) can be linearized thorough algebraic manipulations to yield Eq. (17):

$$\frac{a\_w}{M(1-a\_w)} = \frac{1}{M\_m \mathcal{C}} + \left(\frac{\mathcal{C}-1}{M\_m \mathcal{C}}\right) a\_w \tag{17}$$

A plot of aw <sup>M</sup>ð Þ <sup>1</sup>�aw against aw within the water activity range of 0.01–0.5 at each temperature yields a straight line with the slope as <sup>C</sup>�<sup>1</sup> MmC and intercept on the yaxis as <sup>1</sup> MmC, and from these, the values of Mm and C can be obtained and used as the starting values in nonlinear regression. The nonlinear regression analysis procedure minimizes the sum of deviation in the evaluation of a model using a series of iterative steps. The procedure could require that initial parameter estimates be chosen close to the true values.

b. GAB model: the GAB model (Eq. (8)) can be transformed to a quadratic form by algebraic manipulation to yield Eq. (18):

$$\frac{d\_w}{M} = A a\_w \,^2 + B a\_w + C \,\tag{18}$$

Eq. (18) can be solved by plotting aw <sup>M</sup> against aw at each temperature and fitting a polynomial of the second order to the plots. This will yield the following functions from Eq. (8):

8.Modified Henderson model

Sorption in 2020s

9.Modified Oswin model

condition, (m<sup>3</sup>

162

10.Ngoddy-Bakker-Arkema model

Po is saturated vapor pressure, (N/m<sup>2</sup>

models that gave the best fits.

under which the study is carried out, (N/m<sup>2</sup>

<sup>M</sup> <sup>¼</sup> ρε η

8 ><

>:

0

B@

0 @

� <sup>1</sup>:<sup>75</sup> Ln PoþPm Pm � �

7. Isotherm model predictive performance evaluation

<sup>M</sup> <sup>¼</sup> �Lnð Þ <sup>1</sup> � aw A Tð Þ þ B � �1=<sup>C</sup>

<sup>M</sup> <sup>¼</sup> ð Þ <sup>A</sup> <sup>þ</sup> BT aw

Ln PoþPm PþPm � �

1=2

where M is the moisture content, (db); Mm is monolayer moisture content, (db); aw is water activity; T is absolute temperature, (K); A, B, C and k are constants; η is primary characteristic parameter of pore structure; ε is secondary characteristic parameter of pore structure; σ is surface tension of sorbate in bulk liquid form, (N/m); Rg is universal gas constant; V is molal volume of sorbate in its bulk liquid

1 A

<sup>3</sup>:2<sup>η</sup> <sup>1</sup>:<sup>75</sup>

0 @

0

B@

6 4

1 � aw � �1=<sup>C</sup>

> 1 A

<sup>þ</sup> <sup>σ</sup><sup>V</sup> RgTLn PoþPm Pm � �

/mol); Pm is vapor pressure corresponding to monolayer, (N/m<sup>2</sup>

).

Sun and Byrne [109], Sun [110] and Sun [111] evaluated the predictive performance of the moisture sorption isotherm models that have been reported for fitting the EMC and ERH data of rapeseed, rice, other grains and oilseeds and selected the

Coefficient of terms in the moisture sorption isotherm equations is usually determined using nonlinear regression procedure, and the predictive performance of an equation on sorption data is evaluated using such goodness of fit parameters as standard error of estimate (estimate of the residual mean square), residual sum of square, coefficient of determination, mean relative percent error, fraction explained variation and residual plots. Several investigators used these parameters to evaluate the fitting ability of EMC-aw equations. For instance, Ajibola [35–37], Ajibola and Adams [34], Ajibola [112], Gevaudan et al. [24], Talib et al. [8], Pezzutti and Crapiste [113], Tsami et al. [17] and Ajibola et al. [64] used the standard error of estimate, and Young [48] and Jayas et al. [114] used the residual sum of squares to compare the fitting ability of different models. Boquet et al. [94], Chirife et al. [90], Weisser [31], Saravacos et al. [12], Pollio et al. [115], Iglesias and Chirife [91] and Khalloufi et al. [10] used the mean relative percent deviation (MRE), while Shepherd and Bhardwaj [33], Demertzis et al. [116], Diamante and Munro [117] and Sopade et al. [118] employed coefficient of determination in evaluating the fitting ability of several models. Pappas and Rao [119] used the fraction explained

1 =2

2 <sup>η</sup>

<sup>þ</sup> <sup>σ</sup><sup>V</sup> RgTLn PoþPm PþPm � �

1

η3 7 5

9 >=

>;

CA

); and P is vapor pressure at the condition

1

CA

(14)

(15)

(16)

);

Sorption in 2020s

$$A = \frac{k}{M\_m} \left(\frac{1}{C} - 1\right), B = \frac{1}{M\_m} \left(1 - \frac{2}{C}\right), C = \frac{1}{M\_m C k} \tag{19}$$

The values of Mm, C and k obtained at each temperature are then used as the initial values of the parameters in the nonlinear regression procedure of Eq. (8) to evaluate the model.

c. Modified GAB model: the modified GAB model (Eq. (9)) like the original GAB model can be transformed to a quadratic form by algebraic manipulation to yield Eq. (20):

$$\frac{d\_w}{M} = Xa\_w^{-2} + Ya\_w + Z \tag{20}$$

Linearizing Eq. (25) by logarithmic transformation is carried out as follows:

�A

A

A plot of Ln½ � �Ln að Þ <sup>w</sup> against M at each temperature yields a straight line with

With the expression for the slope, further algebraic manipulation is carried out as follows in order to solve for the temperature-related parameters of the model:

A plot of T against exp ð Þ �b yields a straight line with A as slope and intercept

In the nonlinear regression procedure, the avC as C and A and B are used as the

aw <sup>¼</sup> exp � exp ð Þ <sup>A</sup> <sup>þ</sup> BT <sup>M</sup>�<sup>C</sup> � � (30)

Ln að Þ¼� <sup>w</sup> exp ð Þ <sup>A</sup> <sup>þ</sup> BT <sup>M</sup>�<sup>C</sup> (31) �Ln að Þ¼ <sup>w</sup> exp ð Þ <sup>A</sup> <sup>þ</sup> BT <sup>M</sup>�<sup>C</sup> (32)

Ln½ �¼ �Ln að Þ <sup>w</sup> ð Þ� A þ BT CLnM (33)

aw <sup>¼</sup> <sup>1</sup> � exp �A Tð Þ <sup>þ</sup> <sup>B</sup> <sup>M</sup><sup>C</sup> � � (34)

g. Modified Halsey model: the modified Halsey model (Eq. (13)) can be

A plot of Ln½ � �Ln að Þ <sup>w</sup> against LnM at each temperature yields a straight line with slope as –C and intercept on the y-axis as A + BT. Using the intercept on y-axis for different temperature plots of the above, the values of the intercepts are then plotted against temperature to yield another straight line with slope as B and intercept on y-axis as A. In the nonlinear regression analysis, the avC as C and A and B values are used as the starting values in parameter estimates for the model.

h. Modified Henderson model: the modified Henderson model (Eq. (14)) is

Eq. (34) is linearized by logarithmic transformation as follows:

transformed by algebraic manipulations to yield Eq. (30):

Linearizing Eq. (30) by logarithmic transformation yields

ð Þ T þ B � �

> ð Þ TþB h i.

ð Þ <sup>T</sup> <sup>þ</sup> <sup>B</sup> exp ð Þ �CM (26)

ð Þ <sup>T</sup> <sup>þ</sup> <sup>B</sup> exp ð Þ �CM (27)

� CM (28)

exp ð Þ <sup>b</sup> <sup>¼</sup> Aexpð Þ �<sup>b</sup> (29)

Ln að Þ¼ <sup>w</sup>

�Ln að Þ¼ <sup>w</sup>

slope as -C and intercept on the y-axis as Ln <sup>A</sup>

DOI: http://dx.doi.org/10.5772/intechopen.87996

From the above, T ¼ Aexpð Þ� �b B.

initial parameter estimates in the equation.

transformed to yield Eq. (34):

exp ð Þ¼ <sup>b</sup> <sup>A</sup>

on the y-axis as -B.

So

165

Ln½ �¼ �Ln að Þ <sup>w</sup> Ln <sup>A</sup>

Moisture Sorption Isotherms and Isotherm Model Performance Evaluation for Food…

ð Þ <sup>T</sup> <sup>þ</sup> <sup>B</sup> , implying that <sup>T</sup> <sup>þ</sup> <sup>B</sup> <sup>¼</sup> <sup>A</sup>

Plotting aw <sup>M</sup> against aw and fitting a polynomial of the second order to the plot yield the following functions from Eq. (9):

$$X = \frac{B}{A} \left( \frac{T}{C} - 1 \right), Y = \frac{1}{A} \left( 1 - \frac{2T}{C} \right), Z = \frac{1}{ABC} \tag{21}$$

The average values of A, B and C are obtained and used as initial parameter estimates in the nonlinear regression analysis to evaluate the model.

d. Hailwood-Horrobin model: The Hailwood-Horrobin model (Eq. (10)) is mathematically similar to the GAB and can after algebraic manipulations be represented in the form

$$\frac{d\_w}{M} = \text{Ca}\_w\text{ }^2 + \text{Ba}\_w + A\tag{22}$$

Plotting aw <sup>M</sup> against aw and fitting a polynomial of the second order to the plot at each temperature yield the values of C, B and A for use as initial parameter estimates in the nonlinear regression procedure for the model evaluation.

e. Modified Hailwood-Horrobin model: this model (Eq. 11) also has mathematical similarity with the GAB. It can be transformed algebraically to yield Eq. (23):

$$\frac{d\_w}{M} = \lambda a\_w^{-2} + \mu a\_w + \rho \tag{23}$$

Plotting aw <sup>M</sup> against aw and fitting a polynomial of the second order to the plot yield the following functions from Eq. (11):

$$
\lambda = \frac{C}{T^n}, \mu = BT \text{ and } \rho = TA \tag{24}
$$

The average values of A, B and C are obtained and used as initial parameter estimates in the nonlinear regression analysis to evaluate the model.

f. Modified Chung-Pfost model: the modified Chung-Pfost model (Eq. (12)) is transformed by algebraic manipulations to yield Eq. (25):

$$\mathfrak{a}\_w = \exp\left[\left(\frac{-A}{(T+B)}\right)\exp\left(-\text{CM}\right)\right] \tag{25}$$

Moisture Sorption Isotherms and Isotherm Model Performance Evaluation for Food… DOI: http://dx.doi.org/10.5772/intechopen.87996

Linearizing Eq. (25) by logarithmic transformation is carried out as follows:

$$Ln(a\_w) = \frac{-A}{(T+B)} \exp\left(-CM\right) \tag{26}$$

$$-Ln(a\_w) = \frac{A}{(T+B)} \exp\left(-\text{CM}\right) \tag{27}$$

$$Ln[-Ln(a\_w)] = Ln\left[\frac{A}{(T+B)}\right] - CM\tag{28}$$

A plot of Ln½ � �Ln að Þ <sup>w</sup> against M at each temperature yields a straight line with slope as -C and intercept on the y-axis as Ln <sup>A</sup> ð Þ TþB h i.

With the expression for the slope, further algebraic manipulation is carried out as follows in order to solve for the temperature-related parameters of the model:

$$\exp\left(b\right) = \frac{A}{\left(T + B\right)}, \text{ implying that } T + B = \frac{A}{\exp\left(b\right)} = A \exp\left(-b\right) \tag{29}$$

From the above, T ¼ Aexpð Þ� �b B.

A plot of T against exp ð Þ �b yields a straight line with A as slope and intercept on the y-axis as -B.

In the nonlinear regression procedure, the avC as C and A and B are used as the initial parameter estimates in the equation.

g. Modified Halsey model: the modified Halsey model (Eq. (13)) can be transformed by algebraic manipulations to yield Eq. (30):

$$\mathfrak{a}\_w = \exp\left[-\exp\left(A + BT\right)\mathcal{M}^{-C}\right] \tag{30}$$

Linearizing Eq. (30) by logarithmic transformation yields

$$Ln(a\_w) = -\exp\left(A + BT\right)M^{-C} \tag{31}$$

$$-Ln(a\_w) = \exp\left(A + BT\right)M^{-C} \tag{32}$$

So

<sup>A</sup> <sup>¼</sup> <sup>k</sup> Mm

yield the following functions from Eq. (9):

represented in the form

<sup>X</sup> <sup>¼</sup> <sup>B</sup> A

evaluate the model.

Sorption in 2020s

Plotting aw

Plotting aw

Plotting aw

164

yield Eq. (23):

yield the following functions from Eq. (11):

to yield Eq. (20):

1 <sup>C</sup> � <sup>1</sup> 

aw

T <sup>C</sup> � <sup>1</sup> 

estimates in the nonlinear regression analysis to evaluate the model.

aw

, B <sup>¼</sup> <sup>1</sup> Mm

The values of Mm, C and k obtained at each temperature are then used as the initial values of the parameters in the nonlinear regression procedure of Eq. (8) to

c. Modified GAB model: the modified GAB model (Eq. (9)) like the original GAB model can be transformed to a quadratic form by algebraic manipulation

, Y <sup>¼</sup> <sup>1</sup>

The average values of A, B and C are obtained and used as initial parameter

d. Hailwood-Horrobin model: The Hailwood-Horrobin model (Eq. (10)) is mathematically similar to the GAB and can after algebraic manipulations be

each temperature yield the values of C, B and A for use as initial parameter esti-

mates in the nonlinear regression procedure for the model evaluation.

aw

<sup>λ</sup> <sup>¼</sup> <sup>C</sup>

estimates in the nonlinear regression analysis to evaluate the model.

transformed by algebraic manipulations to yield Eq. (25):

aw <sup>¼</sup> exp �<sup>A</sup>

e. Modified Hailwood-Horrobin model: this model (Eq. 11) also has

<sup>M</sup> against aw and fitting a polynomial of the second order to the plot

<sup>A</sup> <sup>1</sup> � <sup>2</sup><sup>T</sup> C 

<sup>M</sup> against aw and fitting a polynomial of the second order to the plot at

mathematical similarity with the GAB. It can be transformed algebraically to

<sup>M</sup> against aw and fitting a polynomial of the second order to the plot

The average values of A, B and C are obtained and used as initial parameter

ð Þ T þ B 

f. Modified Chung-Pfost model: the modified Chung-Pfost model (Eq. (12)) is

<sup>1</sup> � <sup>2</sup> C 

, C <sup>¼</sup> <sup>1</sup>

<sup>M</sup> <sup>¼</sup> Xaw<sup>2</sup> <sup>þ</sup> Yaw <sup>þ</sup> <sup>Z</sup> (20)

, Z <sup>¼</sup> <sup>1</sup>

<sup>M</sup> <sup>¼</sup> <sup>C</sup>aw<sup>2</sup> <sup>þ</sup> Baw <sup>þ</sup> <sup>A</sup> (22)

<sup>M</sup> <sup>¼</sup> <sup>λ</sup>aw<sup>2</sup> <sup>þ</sup> <sup>μ</sup>aw <sup>þ</sup> <sup>φ</sup> (23)

<sup>T</sup><sup>n</sup> , <sup>μ</sup> <sup>¼</sup> BT and <sup>φ</sup> <sup>¼</sup> TA (24)

exp ð Þ �CM

(25)

MmCk (19)

ABC (21)

$$Ln[-Ln(a\_w)] = (A+BT) - CLnM\tag{33}$$

A plot of Ln½ � �Ln að Þ <sup>w</sup> against LnM at each temperature yields a straight line with slope as –C and intercept on the y-axis as A + BT. Using the intercept on y-axis for different temperature plots of the above, the values of the intercepts are then plotted against temperature to yield another straight line with slope as B and intercept on y-axis as A. In the nonlinear regression analysis, the avC as C and A and B values are used as the starting values in parameter estimates for the model.

h. Modified Henderson model: the modified Henderson model (Eq. (14)) is transformed to yield Eq. (34):

$$a\_w = 1 - \exp\left[-A(T+B)M^C\right] \tag{34}$$

Eq. (34) is linearized by logarithmic transformation as follows:

$$\mathbf{1} - a\_w = \exp\left[-A(T+B)\mathbf{M}^C\right] \tag{35}$$

7.2. Moisture sorption isotherm model predictive indicators

Moisture Sorption Isotherms and Isotherm Model Performance Evaluation for Food…

predicted values of the EMC) against the measured values.

SE ¼

MRE <sup>¼</sup> <sup>100</sup> N

RSS ¼

of squares due to the model and SST is the total sum of squares.

from that of adsorption leading to moisture sorption hysteresis.

.

where Y is the measured EMC value, Y<sup>0</sup> is the EMC value predicted by the model, N is the number of data points, df is the degree of freedom, SSM is the sum

Moisture sorption phenomena govern several technological processes (drying, storage, mixing and packaging to mention a few) involving agricultural and food products. Moisture sorption isotherms of these products are generally of the type II, sigmoidal in shape and temperature dependent. The isotherms can be determined using the static or dynamic gravimetric, vapor pressure manometric, hygrometric and inverse gas chromatographic methods. Desorption isotherm path could differ

Commonly used moisture sorption isotherm models include the BET, GAB, modified GAB, Hailwood-Horrobin, modified Hailwood-Horrobin, modified Chung-Pfost, modified Halsey, modified Henderson and modified Oswin models. Ngoddy-Bakker-Arkema model which was proposed as a generalized model was considered. While some of the models can be evaluated by fitting polynomial functions of the second order to them and applying nonlinear regression procedure, others can be solved thorough linearization by logarithmic transformation and nonlinear regression. For the Ngoddy-Bakker-Arkema model, the initial parameter

of fit is determined using the following indices:

DOI: http://dx.doi.org/10.5772/intechopen.87996

b. Standard error of estimate given as

c. Mean relative percent deviation given as

d. Fraction explained variation given as

e. Residual sum of squares (RSS) given as

f. Coefficient of determination, R<sup>2</sup>

8. Conclusions

167

After values of model constants have been determined using the nonlinear regression analysis, the suitability of a model for predictive purpose or its goodness

a. Residual plots: these are plots of residuals (difference between measured and

<sup>P</sup> <sup>Y</sup> � <sup>Y</sup><sup>0</sup> ð Þ<sup>2</sup> df " #1=<sup>2</sup>

> X Y � Y<sup>0</sup> Y

� � � �

SST (46)

<sup>N</sup> (47)

� � � �

FEV <sup>¼</sup> SSM

<sup>P</sup> <sup>Y</sup> � <sup>Y</sup><sup>0</sup> ð Þ<sup>2</sup>

(44)

(45)

$$Ln(1 - a\_w) = -A(T + B)M^C\tag{36}$$

$$-Ln(\mathbf{1} - \mathbf{a}\_w) = A(T + B)\mathbf{M}^C\tag{37}$$

$$Ln[-Ln(1-a\_w)] = Ln[A(B+T)] + CLnM\tag{38}$$

A plot of Ln½ � �Lnð Þ 1 � aw against LnM at each temperature yields a straight line with slope a1 = C and intercept on the y-axis b1 = Ln A B ½ � ð Þ þ T . To solve for the temperature-related parameters, intercept on the y-axis is used.

$$\text{Therefore, } \exp\left(b\_1\right) = A(T+B) = AT + AB. \tag{39}$$

A plot of exp ð Þ b<sup>1</sup> against T yields a straight line with slope a2 as A and intercept on y-axis b2 as AB. In the nonlinear regression procedure, avC and A and B are used as initial parameter estimates for the model.

i. Modified Oswin model: the modified Oswin model (Eq. (15)) can be manipulated algebraically to yield Eq. (40):

$$a\_w = \frac{1}{\left[\frac{(A+BT)}{M}\right]^C + 1} \tag{40}$$

$$\frac{\mathbf{1}}{a\_w} = \left[\frac{(A+BT)}{M}\right]^C + \mathbf{1} \text{ and } \frac{\mathbf{1}}{a\_w} - \mathbf{1} = \left[\frac{(A+BT)}{M}\right]^C \tag{41}$$

Linearizing Eq. (41) by logarithmic transformation yields

$$\text{CLn}(A+BT) - \text{CLnM} = \text{Ln}\left(\frac{\mathbf{1} - a\_w}{a\_w}\right) \tag{42}$$

A plot of Ln <sup>1</sup>�aw aw � � against LnM at each temperature yields a straight line with slope as –C and intercept on the y-axis as CLn Að Þ þ BT .

The expression for intercept on the y-axis is solved further to evaluate the temperature-related parameters of the model and yield Eq. (43),

$$\exp\left(^{b}\!\!/\_{\ell}\right) = \left(\mathcal{A} + BT\right) \tag{43}$$

A plot of exp <sup>b</sup>ð Þ =<sup>c</sup> against T yields a straight line with slope as A and intercept on the y-axis as B. In the nonlinear regression procedure, avC as C and A and B are used as the initial parameter estimates in the model evaluation.

j. Ngoddy-Bakker-Arkema model: the Ngoddy-Bakker-Arkema model, which has been postulated to be a generalized model, has the following parameters (unknowns): σ, V, Pm, ρ, ε and η.

Evaluating the model requires a lot of care. The starting values of parameters for application in the nonlinear regression procedure can be obtained as follows:

σ, V, ρ and Po can be obtained at different temperatures from the steam table P can be calculated using the expression P = aw, ε can be taken as having a typical value of 0.1 though its value can be less, Pm is the monolayer value of P and η can be assumed to lie between �1 and þ1 in the form of �1 ≤ η ≤ þ1 with 0.1 as a typical starting value.

Moisture Sorption Isotherms and Isotherm Model Performance Evaluation for Food… DOI: http://dx.doi.org/10.5772/intechopen.87996

### 7.2. Moisture sorption isotherm model predictive indicators

After values of model constants have been determined using the nonlinear regression analysis, the suitability of a model for predictive purpose or its goodness of fit is determined using the following indices:


<sup>1</sup> � aw <sup>¼</sup> exp �A Tð Þ <sup>þ</sup> <sup>B</sup> <sup>M</sup><sup>C</sup> � � (35) Lnð Þ¼� <sup>1</sup> � aw A Tð Þ <sup>þ</sup> <sup>B</sup> <sup>M</sup><sup>C</sup> (36) �Lnð Þ¼ <sup>1</sup> � aw A Tð Þ <sup>þ</sup> <sup>B</sup> MC (37)

Ln½�Lnð Þ 1 � aw � ¼ Ln A B ½ �þ ð Þ þ T CLnM (38)

Therefore, exp ð Þ¼ b<sup>1</sup> A Tð Þ¼ þ B AT þ AB: (39)

þ 1

� <sup>1</sup> <sup>¼</sup> ð Þ <sup>A</sup> <sup>þ</sup> BT M � �<sup>C</sup>

> aw � �

exp <sup>b</sup>ð Þ¼ =<sup>c</sup> ð Þ A þ BT (43)

against LnM at each temperature yields a straight line with

(40)

(41)

(42)

A plot of Ln½ � �Lnð Þ 1 � aw against LnM at each temperature yields a straight line with slope a1 = C and intercept on the y-axis b1 = Ln A B ½ � ð Þ þ T . To solve for the

A plot of exp ð Þ b<sup>1</sup> against T yields a straight line with slope a2 as A and intercept on y-axis b2 as AB. In the nonlinear regression procedure, avC and A and B are used

i. Modified Oswin model: the modified Oswin model (Eq. (15)) can be

aw <sup>¼</sup> <sup>1</sup> ð Þ AþBT M h i<sup>C</sup>

<sup>þ</sup> 1 and <sup>1</sup>

CLn Að Þ� <sup>þ</sup> BT CLnM <sup>¼</sup> Ln <sup>1</sup> � aw

The expression for intercept on the y-axis is solved further to evaluate the

A plot of exp <sup>b</sup>ð Þ =<sup>c</sup> against T yields a straight line with slope as A and intercept on the y-axis as B. In the nonlinear regression procedure, avC as C and A and B are

j. Ngoddy-Bakker-Arkema model: the Ngoddy-Bakker-Arkema model, which has been postulated to be a generalized model, has the following parameters

Evaluating the model requires a lot of care. The starting values of parameters for

σ, V, ρ and Po can be obtained at different temperatures from the steam table P can be calculated using the expression P = aw, ε can be taken as having a typical value of 0.1 though its value can be less, Pm is the monolayer value of P and η can be assumed to lie between �1 and þ1 in the form of �1 ≤ η ≤ þ1 with 0.1 as a typical

application in the nonlinear regression procedure can be obtained as follows:

aw

temperature-related parameters, intercept on the y-axis is used.

as initial parameter estimates for the model.

1 aw

aw � �

A plot of Ln <sup>1</sup>�aw

Sorption in 2020s

starting value.

166

manipulated algebraically to yield Eq. (40):

<sup>¼</sup> ð Þ <sup>A</sup> <sup>þ</sup> BT M � �<sup>C</sup>

slope as –C and intercept on the y-axis as CLn Að Þ þ BT .

Linearizing Eq. (41) by logarithmic transformation yields

temperature-related parameters of the model and yield Eq. (43),

used as the initial parameter estimates in the model evaluation.

(unknowns): σ, V, Pm, ρ, ε and η.

$$SE = \left[\frac{\sum \left(Y - Y'\right)^2}{df}\right]^{1/2} \tag{44}$$

c. Mean relative percent deviation given as

$$\text{MRE} = \frac{\mathbf{100}}{N} \sum \left| \frac{Y - Y'}{Y} \right| \tag{45}$$

d. Fraction explained variation given as

$$FEV = \frac{\text{SSM}}{\text{SST}} \tag{46}$$

e. Residual sum of squares (RSS) given as

$$\text{RSS} = \frac{\sum \left( Y - Y' \right)^2}{N} \tag{47}$$

f. Coefficient of determination, R<sup>2</sup> .

where Y is the measured EMC value, Y<sup>0</sup> is the EMC value predicted by the model, N is the number of data points, df is the degree of freedom, SSM is the sum of squares due to the model and SST is the total sum of squares.

### 8. Conclusions

Moisture sorption phenomena govern several technological processes (drying, storage, mixing and packaging to mention a few) involving agricultural and food products. Moisture sorption isotherms of these products are generally of the type II, sigmoidal in shape and temperature dependent. The isotherms can be determined using the static or dynamic gravimetric, vapor pressure manometric, hygrometric and inverse gas chromatographic methods. Desorption isotherm path could differ from that of adsorption leading to moisture sorption hysteresis.

Commonly used moisture sorption isotherm models include the BET, GAB, modified GAB, Hailwood-Horrobin, modified Hailwood-Horrobin, modified Chung-Pfost, modified Halsey, modified Henderson and modified Oswin models. Ngoddy-Bakker-Arkema model which was proposed as a generalized model was considered. While some of the models can be evaluated by fitting polynomial functions of the second order to them and applying nonlinear regression procedure, others can be solved thorough linearization by logarithmic transformation and nonlinear regression. For the Ngoddy-Bakker-Arkema model, the initial parameter

estimates for use in nonlinear regression have to be obtained from the steam table. A model is considered acceptable for predictive purpose, if the residuals are uniformly scattered around the horizontal value of zero showing no systematic tendency towards a clear pattern. Model goodness of fit is determined using standard error of estimate, mean relative percent deviation, fraction explained variation, coefficient of determination and residual sum of squares.

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DOI: http://dx.doi.org/10.5772/intechopen.87996

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