Models for Sorption Isotherms for Foods- A Review

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Models for Sorption Isotherms for Foods: A Review

Santanu Basu,1 U. S. Shivhare,1 and A. S. Mujumdar2

1Department of Chemical Engineering & Technology, Panjab University, Chandigarh, India2Department of Mechanical Engineering, National University of Singapore, Singapore

This article presents basic concepts related to the thermody-namics of sorption of water and measurement of sorption isothermfor food materials. A comprehensive review of the widely used sorp-tion models is presented. Various statistical techniques used toascertain the effectiveness of a model to describe the sorption dataare discussed. It is anticipated that this article will provide useful

information to researchers pursuing work on sorption behavior offood materials as well as modeling of drying processes.

Keywords Heat of sorption; Isosteric heat; Model; Sorption iso-therm; Statistical analysis

INTRODUCTION

The state of water plays a crucial role in food preser-vation. The quality of preserved foods depends upon themoisture content, moisture migration, or moisture uptakeby the food material during storage. Extent of sorptionof water by or desorption from a food product dependson vapor pressure of water present in the food sample

and that in the surroundings. Moisture content at whichvapor pressure of water present in the food equals that ofthe surroundings is referred to as equilibrium moisture con-tent (EMC).[1] Relationship between EMC and corre-sponding relative humidity at constant temperature yieldsthe so-called moisture sorption isotherm. For a givenmaterial the EMC increases with relative humidity butdecreases with increase in temperature. The phenomenonwhere the EMC during the adsorption and that duringthe desorption process is different is called hysterisis.Water activity is commonly used to characterize the foodquality and is defined as

aw PP0ERH100 1

wherePis vapor pressure of water in the food material atany given temperature (Pa), P0 is the vapor pressure ofpure water at that temperature (Pa),awis the water activity

(dimensionless), and ERH is the equilibrium relativehumidity (%).

Five types of isotherms were described by Brunaueret al.[2] (Fig. 1). Type 1 is the well-known Langmuir iso-therm, obtained assuming monomolecular adsorption ofgas by the porous solids in a finite volume of voids. Type 2is the sigmoid isotherm obtained for soluble products,which exhibits asymptotic trend as water activityapproaches 1. Type 3, known as the Flory-Higgins iso-therm, accounts for a solvent or plasticizer such as glycerolabove the glass transition temperature. Type 4 isothermdescribes adsorption by a swellable hydrophilic solid untila maximum of hydration sites are reached. Type 5 is theBET[3] multilayer adsorption isotherm, observed foradsorption of water vapor on charcoal; it is related to types2 and 3 isotherms. The two isotherms most commonlyfound in food products are types 2 and 4.[4]

Water adsorption by foods is a process wherein watermolecules progressively and reversibly combine with the

food solids via chemisorption, physical adsorption, andmultilayer condensation.[5] An isotherm can typically bedivided into three regions as shown in Fig. 2. The waterin region A represents strongly bound water with enthalpyof vaporization considerably higher than that of purewater. The bound water includes structural water(H-bonded water) and monolayer water,[6] which is sorbedby hydrophilic and polar groups of the food components(polysaccharides, proteins, etc.). Bound water is unfreez-able and is not available for chemical reactions or as a plas-ticizer. In region B, water molecules bind less firmly than inthe first zone. The vaporization enthalpy is slightly higherthan that for pure water. This class of constituent water

can be looked upon as the continuous transition of thebound to the free water. Properties of water in regionC are close to those of the free water that is held invoids, capillaries, crevices, and loosely binds to the foodmaterials.[1,7]

Measurement and modeling of sorption isotherms offood materials has attracted numerous researchers becauseof their value in industrial practice. Comprehensive reviewson sorption behavior of foods have been published.[813]

Correspondence: U. S. Shivhare, Department of ChemicalEngineering & Technology, Panjab University, Chandigarh,160014 India; E-mail: [email protected]

Drying Technology, 24: 917930, 2006

Copyright# 2006 Taylor & Francis Group, LLC

ISSN: 0737-3937 print/1532-2300 online

DOI: 10.1080/07373930600775979

917


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More than 1000 scientific papers have been published

covering a very wide range of products. This article gives

a comprehensive review of the main theories of the sorp-

tion phenomenon in foods and different mathematical

models commonly used to describe the sorption behavior.

The article elaborately discusses various statistical analysis

tools that should be considered for drawing conclusions

from any set of experimental and predicted values. Notethat in drying it is the desorption isotherm that is of greater

interest than the adsorption isotherm. The word sorp-

tion covers both adsorption and desorption.

THERMODYNAMICS OF SORPTION

When water is removed from a food product, heat is

absorbed because water has to be removed against a water

activity gradient or against increasing osmotic pressure.

The isosteric heat of sorption, also called differential

enthalpy, is an indicator of the state of water held by the

solid material. Net isosteric heat of sorption (DHS) is the

difference of total heat of sorption (DHd

) in the food and

the heat of vaporization of water (DHvap) associated with

the sorption process and can be computed from experi-

mental data using Clausius-Clapeyron equation

dln awd1=T

M

DHSR

2

where the isosteric heat of sorption is associated with

sorbed molecules at a particular moisture content (M).

Integrating Eq. (2) using the associated boundary con-

ditions yields

Z aw2

aw1

d

ln aw

DHS

RZ

T2

T1

d 1

T

ln aw2

aw1

DHS

R

1

T1 1

T2

3

where aw1 and aw2 represent water activities at tempera-

tures T1 and T2, respectively, and R is the universal gas

law constant (8.314 kJ=molK)Several assumptions are made implicitly in applying the

Clausius-Clapeyron equation. First, the heat of vaporiza-

tion of pure water (DHvap) and the excess heat of sorption

DHS are assumed not to change with temperature. Sec-

ondly, the equation applies only when the moisture content

of the system remains constant with respect to time.[14]

From the isotherms determined at least at 10C apart, awat any other temperature should be predictable from

Eq. (3) if the assumptions are correct. Plotting the experi-

mental sorption isotherm in the form ln(aw) vs. 1=T for aspecific moisture content, DHS is determined from the

slopeDHs=R.An empirical relationship between the net isosteric heat

of sorption (DHS) and the moisture content (M, dry basis)

is expressed as[14]

FIG. 1. Five types of isotherms.[4]

FIG. 2. A typical sorption isotherm showing the phenomenon of

hysteresis.[1]

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DHS DH0expM=MC 4where DH0 is the net isosteric heat of sorption when moist-

ure content (M)0. The constant MC is a characteristicmoisture content (kg water per kg dry matter) of a food

material at which the net isosteric heat of sorption (DHS)

has been reduced by 63%. Due to the exponential decay,

the net isosteric heat of sorption becomes very small at highmoisture content; e.g., at M3 MC, DHS is less than 5%ofDH0.

The net heat of sorption of water Q (kJ per kg dry mat-

ter) from dryness (M0) to moisture content M isexpressed as:[14]

QZ M

0

DHSdM 5

SubstitutingDHS from Eq. (4) into Eq. (5) and integrating

yields

QDH0MC 1expbM=MCc 6

For very high moisture content (M! 1), Eq. (6) gives thetotal net isosteric heat in terms of heat of sorption of water

(QT).

QT DH0MC 7The net heat of sorption of water for a change of moisture

content from M1 to M2 (Q12) is expressed as

Q12 DH0MCbexpM1=MC expM2=MC c 8Equation (8) can be used to estimate the energy required for

drying a food material from moisture content M1 to M2.

In drying, M1> M2, and the net heat of desorption (Q12)is negative, because an extra amount of energy is required,

in addition to heat of vaporization of water (DHvap), to

remove the adsorbed water from the food product.

The relationship between the isosteric heat (DHd) and

differential entropy (DSd) of sorption is expressed as:[15]

lnawM DHd=RT DSd=R 9Applying Eq. (9) at different moisture contents, the

dependence of DHd and DSd on moisture may be

determined.[16,17]

The compensation theory proposes a linear relationship

between DHd and DSd[15,18]

DHd

TbDSd

a

10

The isokinetic temperature (Tb) and constanta are

computed using linear regression of Eq. (10). Parameter a

may be neglected due to its negligible contribution to the

enthalpy change. The compensation theory may be further

used to evaluate the effect of temperature on the sorption

behavior by applying Eq. (11).[16]

lnaw DHd=R1=T1=Tb 11

MEASUREMENT OF SORPTION ISOTHERMS

Many methods are available for determination of water

sorption isotherm. These methods may be classified into

three categories: (1) gravimetric, (2) manometric, and (3)

hygrometric. The gravimetric method involves the measure-

ment of mass changes that can be measured both continu-

ously and discontinuously in dynamic and static systems.

Manometric methods involve sensitive manometers to

measure vapor pressure of water in equilibrium with a food

material of given moisture content. Hygrometric methods

measure the equilibrium relative humidity of air in contact

with a food material at given moisture content. Dew point

hygrometers detect the condensation of cooling water

vapor. Electronic hygrometers measure the change of

conductance or capacitance of hygrosensors.[19]

The most common technique, for which a recommended

procedure has been defined in the European project COST

90, uses thermostatted jars filled at the bottom with super-

saturated salt solutions to maintain the desired air relative

humidity.

[20]

However, this method encounters some prob-lem at high humidity ranges due to (a) excessive equili-

bration times and (b) its inability to produce and control

high relative humidities.

Baucour and Daudin[20] developed a rapid but accurate

method to measure moisture sorption isotherms of solid

foods in the water activity range 0.91. This method avoids

the drawbacks of saturated salt solution method by blow-

ing calibrated air along thin slices of material at a high

velocity to impose an intensive water vapor exchange

between air and samples.

MATHEMATICAL MODELS OF ISOTHERMS

Numerous attempts have been made to describe thesorption isotherms mathematically. While some models

have been derived theoretically based on thermodynamic

concepts, others are an extended or modified form of these

models. Some of the widely used mathematical models are

presented in the following paragraphs.

LANGMUIR EQUATION

On the basis of monomolecular layers with identical,

independent sorption sites, Langmuir[21] proposed the fol-

lowing physical adsorption model:

aw

1

M 1

M0

1

CM0 12

whereM0is monolayer sorbate constant andCis a constant.

The relation described by Eq. (12) yields the type I isotherm.

BRUNAUER-EMMETT-TELLER (BET) EQUATION

The BET isotherm equation[3] is one of the most widely

used models and gives good fit for a variety of foods over

the region 0.05< aw< 0.45.[22] It provides an estimate of

MODELS FOR SORPTION ISOTHERMS: A REVIEW 919


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monolayer value of moisture adsorbed on the surface. The

BET equation is expressed as

M

M0 Caw1aw1 C1aw 13

oraw

1aw M 1

M0CC

1

M0Caw 14

where M is the equilibrium moisture content (kg water=kg dry matter), M0 is the monolayer moisture content on

the internal surface (kg water=kg dry matter), and C is adimensionless parameter related to heat of sorption of

monolayer region.

The theory behind the development of the BET equation

has been questioned due to the assumptions that (a) the

rate of condensation on the first layer is equal to the rate

of evaporation from the second layer, (b) binding energy

of all of the adsorbate on the first layer is same, and (c)

binding energy of the other layers is equal to that of pureadsorbate. The assumptions of uniform adsorbent surface

and absence of lateral interactions between adsorbed

molecules are incorrect in view of the heterogeneous food

surface interactions.[9]

MODIFIED OSWIN EQUATION

The Oswin[23] equation is a mathematical series expan-

sion for a sigmoid-shaped curve and is represented as

M K aw1aw

N15

Here,Kand Nare constants.The parameter K was found to be a linear function of

temperature.[24] The Oswin equation may be modified as

M ABT aw1aw

N16

or

aw MABT

C1

" #117

withC1=N.

MODIFIED HALSEY EQUATION

Halsey[25] developed an equation to describe conden-

sation of multilayers, assuming that the potential energy

of a molecule varies inversely as the Cth power of its

distance from the surface. The equation is

awexp AMC

RT

18

Because the use of the term RTdoes not eliminate the tem-

perature dependence of constants A and C, Iglesias and

Chirife[26] simplified it to the form

awexpA0MC 19where A 0 is a new constant.

Iglesias and Chirife[27]

analyzed the parameter A of theHalsey equation and found that it could be related to tem-

perature by an empirical exponential function. A new

modified Halsey equation was therefore proposed:

awexp expABTMC

20

where A, B, and Care constants.

Iglesias and Chirife[28] reported that the Halsey equation

described 220 experimental sorption isotherms of 69 differ-

ent foods in the range of 0.1 < aw< 0.8.

MODIFIED HENDERSON EQUATION

Henderson[29] presented an equation of the form

aw1expbATMCc 21Thompson et al.[30] modified the Henderson equation by

adding another constant to the temperature term. The

modified Henderson equation becomes

aw1expbATBMCc 22where A, B, and Care constants.

Hendersons equation has been applied to many

foods,[3134] but compared to the Halsey equation, its appli-

cability has been rather limited.

CHUNG-PFOST EQUATION

Chung and Pfost[35,36] developed a model based on the

assumption that the change in free energy for sorption is

related to the moisture content

awexp ART

expBM

23

Later, in order to obtain a better fit, Pfost et al.[37] added a

new parameter to the temperature term and combined the

R parameter in constant term. The modified Chung-Pfost

equation is expressed as:

awexp ATCexpBM

24

FERRO FONTAN EQUATION

The Ferro Fontan equation[38] is represented as

ln c

aw

aMr 25



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where c is the parameter that accounts for the structure of

sorbed water, a is a constant, and r is a constant that

involves net isosteric heat with moisture content adjustable

parameters. Iglesias and Chirife[39] compiled sorption data

for 156 food items and documented that Ferro-Fontan

equation is an accurate tool for the mathematical descrip-

tion of food isotherms. The Ferro-Fontan equation accu-

rately represents the sorption isotherm in the range ofwater activity 0.10.9 with only 24% error in the pre-

dicted moisture content.

GAB EQUATION

van den Berg[40] refined the Langmuir and BET theories

and proposed a new equation with three parameters having

physical meanings

M

M0 CKaw1 Kaw1 KawCKaw 26

M

M0 C1Kaw

1 KawCKaw Kaw

1 Kaw 27

whereC is a dimensionless GAB parameter related to heat

of sorption of monolayer region and Kis a dimensionless

GAB parameter related to heat of sorption of multilayer

region.

Equation (27) is divided into two additive terms, the first

one describes the classical mono-molecular layer

expression in Langmuirs adsorption isotherms and the

second term describes the multilayer adsorption corre-

sponding to Raoults law.[40]

ParametersKand Ccan be expressed by Arrhenius type

equations:

CC0exp DHcRT

28

K K0exp DHkRT

29

where DHc DHmDHq; DHk DHl DHq (kJ mol1);DHl is the heat of condensation of pure water

(kJ mol1); DHm is the total heat of sorption of mono layer

(kJ mol1); DHq is the total heat of sorption of multilayer

covering the monolayer (kJmol1); and C0, K0 are theconstants of entropic character.

The GAB model underestimates the water content values

at high water activities (aw> 0.93). The discrepancy under-lines two facts: (a) this type of model is unsuitable for high

humidity range, and (b) the saturated salt solution method

does not afford sufficient information to get a complete sorp-

tion curve. The GAB model was refined for higher water

activities by Timmerman and Chirife;[41] and Viollaz and

Rovedo.[42] They modified the GAB model by inserting a

new parameter to add a third sorption stage without changing

the values of the usual parameters defined in the GAB model.

ANALYSIS OF GAB ISOTHERM

Equation (27) can be rewritten in one-parameter form:

w 11 X

1

1 1 CX 30

where w

M=M0 and X

Kaw are new coordinates.

Equation (30) can be described for five conditions: (1) forX0, (2) for w1 for the so-called mono-molecularlayer, (3) for XK (i.e., for a

w 1), (4) at the point of

inflection wX, and (5) potential singular points. Thecharacteristic properties of the isotherm at these points give

the limits on the parameters Kand C.[43]

1. For X 0; Eq. (30) gives the value w0 identically.The shape of the sigmoid sorption isotherm may be

obtained by differentiating Eq. (30) corresponding to

the condition dw

dawKdw

dX 0:

dw

dX 1

1 X 2 1

C

1 1 C X 2 31

Substituting X0 into Eq. (31), we obtain dw=dX C,so the mentioned condition leads to the final form

KC 0. It infers that both parameters K and C haveto be of the same sign.

2. The existence of a mono-molecular layer for w1 leadsto the quadratic equation X2m1C 2Xm10where Xm is the value of variable X in the mono-

molecular state. The equation has the following solution

for 0< X< 1:

Xm 1

1 ffiffiffiffiCp for C0: 32In combination with condition (1) it means that both C

and Khave to be non-negative.

3. Substituting XK into Eq. (31) and rearranging, thefollowing equation is obtained:

w KC1 K1 K1 C forX K 33

that has to be greater than 0. This equation shows that

both parameters Kand Chave to be greater than zero.

Moreover, the following condition

1 K1 K1 C > 0

has to be fulfilled. This inequality has two solutions:

a=0< K; C>K1K

; i:e:;C> 0 34a

b=K> 1; 0< C 1.[43]

PELEG MODEL

Peleg[44] developed a two-parameter model to describe

the sorption curves

Mt Mi tk1k2t 39

where M(t)

moisture content after time t (% dry basis),

Miinitial moisture content (% dry basis) and k1, k2 areconstants. According to this model, equilibrium moisture

contentMEwhen t! 1 is given by

ME Mi 1k2

40

Similarly, the instantaneous sorption ratedM(t)=dtis givenby

dMtdt

k1k1k22 41

and the initial rate (at t

0) by 1=k1.

Equation (39) can be easily transformed to a linearrelationship

t

Mt Mi k1k2t 42

Peleg[45] also developed a four-parameter model

M k1an1w k2an2w 43

wherek1,k2,n1, andn2 are constants (n1< 1) andn2> 1).The model expressed by Eq. (43) has no monolayer incor-

porated in it. However, plots ofaw=M1awvs.awusedto determine the monolayer value with the BET model are

still expected to be practically linear over a water activity

range of up to about 0.4. This is because at this range of

aw Eq. (43) can be approximated by

Mffik1an1w n1 < 1 44and

aw

M1aw aw

k1an1w1aw 45

The plot ofaw= an1w1aw

vs. aw(k11) is linear in this

water activity range and for a range ofn1values. The inter-

cept is positive when n1 is greater than 0.3.

aw

M1aw aw=k1

an1w1 k1=k2an1n2w 1aw 46

the added term 1 k1=k2an2n1w in the denominatortends to lower the magnitude of aw=M1aw as themagnitude ofaw increases.

TIMMERMANN GAB MODEL

Compared to the classical BET isotherm, the GAB

isotherm contains a third constant, K, which measures the

differences between the standard chemical potential of the

molecules in the second sorption stage and in the pure

liquid. Some researchers[46,47] noted that at very high water

activities some systems showed a sorption stage larger than

the predicted by the GAB model. Timmermann

[41,47]

sug-gested that the second sorption stage introduced by the

GAB model may be limited to a certain number of layers

and that, thereafter, a third stage becomes available

for the sorbate molecules, which has true liquid-like pro-

perties, as postulated by the original BET model. The

so-called third sorption stage isotherm developed by

Timmermann[47] is

M

M0 CKawHH

0

1Kaw1 CH1Kaw 47

where H and H0 are functions containing a fourth dimen-sionless parameter h.

H 11KK

Kawh1aw 48

H01H1H

1Kaw1aw

h 1haw 49

Forh! 1,H;H0!1 and with this result Eq. (47) trans-forms into a GAB equation.



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VIOLLAZ GAB MODEL

Equation (27) was modified by Viollaz and Rovedo [42] in

an empirical way by adding a term

M

M0 CKaw1Kaw1 C1Kaw

CKK2a2w

1Kaw1aw

50

K2 is an additional dimensionless parameter.The second term of the RHS allows the necessary flexi-

bility to obtain a good fitting for high awvalues. This term

has a very low weight for low values ofaw, so the values of

M0,C, andKare not substantially affected by the addition

of this new term. It can be observed that if the value ofK2is equal to zero, the GAB isotherm is obtained. The predic-

tive capacity of Eq. (50) was evaluated using the sorption

data for starch and gluten by Viollaz and Rovedo.[42]

LEWICKI MODEL

Lewicki[48] developed a model on the basis of Raoults

law. The model assumed that water present in food occursin two states, as free water with properties of the bulk

water and as water of hydration. The Lewicki model of

food sorption isotherm is

M A 1aw

1 b1

51

where A and b are constants. Equation (51) described the

water sorption isotherms of 38 products and 31 model

mechanical mixtures.[48]

WATER ACTIVITY OF SOLUTION/MIXTURES

For water solution, Raoults law shows that the water

activity in ideal solution is equal to mole fraction of water

(xw); that is, awxw.For nonideal solutions, the activity coefficient c corrects

for deviation from nonideality.

awxwc 52Norrish[49] proposed a semiempirical correction to Raoults

law for the nonideal solution

awxwkx2s 53where xs is mole fraction of solute and k is an empirical

constant.

Ross

[50]

extended the Norrish equation for multi compo-nent systems by using the Gibbs-Duhem equation with the

assumption that the activity coefficient of individual com-

ponents with water are equal to their activity coefficients

in binary mixtures in water.

The Ross equation is represented as:

lnaw Xkj1

lna0wj 54

Here,awjis the water activity of the binary mixture with the

jcomponent.

The Ross equation has been applied by several research-

ers, e.g., Chirife et al.,[51] Herman et al.,[52] and Roman

et al.[34] The water activities of non-solute parts were also

evaluated with the Henderson equation based on literature

data for starch, protein, and fiber by Roman et al. [34]

ARTIFICIAL NEURAL NETWORK MODELING

Artificial neural network (ANN) modeling is a method

to describe relations between independent and dependent

variables when the explicit form of mapping is not known.

In recent years, the concept of ANN has gained wide

acceptance in food engineering for predictive modeling.

In case of sorption isotherm prediction, Myhara et al., [53]

Myhara and Sablani,[54] and Kaminski and Tomczak[55]

used ANN modeling for different food materials. Myhara

et al.[53] demonstrated that when chemical composition

data is combined with physical data through an ANN

approach, significant improvements in the prediction ofwater sorption behavior can be achieved.

COMPARISON OF BET AND GAB MODELS

The BET and the GAB isotherms are closely related as

they are derived from the same statistical model. The

GAB model represents a refinement over the BET model

and shares with it two original BET constants (M0, the

monolayer capacity, and C, the energy constant) and owes

its versatility due to the introduction of a third constant

(K). The regression of an experimental sorption data by

each of these two isotherms will give two sets of values of

M0 and C. The same type of differences between both sets

of values ofM0 and Chave always been observed:

M0B< M0G; CB> CG 55

where subscript B and G represent BET and GAB,

respectively.

The M0 value given by BET isotherms is always lower

than the monolayer value derived from GAB and, reversi-

bly, the BET value of energy constant is always higher than

the GAB value. It has been demonstrated that there exist

mathematical and physical reasons for the inequalities set

by Eq. (55).[56,57]

The BET equation may be rewritten as

M CBawM0B1aw1 CB1aw 56

To obtain the two characteristic constants, the BET equa-

tion is linearized by the following function

FBET aw1awM 1

CBM0B CB1

CBM0Baw 57



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The so-calledF(BET) vs.awplot usually gives a linear part

at low activities (0.05< aw< 0.30.5), after which alwaysan upward curvature is observed. This deviation shows that

at higher activities, less gas or vapor is sorbed than that

indicated by the BET equation using the values of the con-

stants corresponding to the low activity range.

The GAB equation is rewritten as

M CGKawM0G1Kaw1 CG1Kaw 58

and

CGK CGG 59Here,M0Gis the GAB monolayer capacity andCG(G)is the

analogue of this formulation to the BET energy constant

CB. The constant (K) is a measure of the difference of free

enthalpy of the sorbate molecules in the pure liquid and

the second sorption stage, the layer above the monolayer.

It is always found that K< 1. Finally, with K1, theGAB isotherm reduces to the original BET equation

(M0BM0G; CBCG(G)).To determine the three constants of the GAB equation,

it is linearized like the BET equation and the function

F(GAB) is introduced:

FGAB aw1Kaw M

1

CGKM0G CG1

CGM0Gaw 60

F(GAB) vs. aw should be linear if the correct K value is

used. From the two linear regression coefficients, M0GandCGare obtained.

Equation (57) can be rewritten in a modified form by

incorporatingF(GAB)

FBET aw1awM1Kaw

1aw FGAB 61

Substituting the value of F(GAB) from Eq. (60) into

Eq. (61) a new relationship represented by F(BET) isobtained, in terms of the three constants of the GAB

isotherm:

FBET 1CGKM0G

CG1CGM0G

aw

1Kaw1aw

1CGGM0G

CGG121KCGGM0G

aw

1KCGG1K

CGGM0G aw

1aw 62

The second expression for F(BET) shows that, if K< 1,F(BET) will not be linear in aw but will present a hyper-bolic behavior.

FBET ABaw Caw1aw

AC BCaw C1aw 63

where

A 1CGGM0G

64a

BCGG121KCGGM0G

64b

C1KCGG1KCGGM0G

64c

QUANTITATIVE EXPRESSION OF INEQUALITIES

The experimental values ofF(BET) are adjusted by the

linear polynomial

PBET a0a1awi 65By minimizing the sum over the n experimental points

Xni1 FBETi a0a1awi

2 minimum 66Here, awistands for aw at the point i. The coefficients a0and a1are given by the solutions of the system of normal equations

associated to the extreme condition (Eqs. (67a) and (67b).

The least square estimates of a0 and a1, a00 and a

01 are

given directly by the BET relations by which the BET con-

stants are usually computed.

BET:a00 1

CBM0B67a

a01

CB1CBM0B

67b

and hereafter

M0B 1a00a01

68a

CBa00a01a00

68b

It should be noted that the energy constantCB is inversely

proportional to the intercept a00 of the linear regressionpolynomialP(BET) of Eq. (68b) and thereforeCBis highly

sensitive to the value ofa 00, which is usually very low.The second expression ofF(BET) may also be adjusted

by the same linear polynomial (Eq. (65)), but now using an

analytical formulation as F(BET) is known as a functionofaw and not by a set of numerical data. The calculation

implies the adjustment of a function of a known functional

dependence of a higher degree than one to a straight line.

In the discrete procedure,F(BET) given by Eq. (63) is putinto condition (Eq. (66)), which becomes



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Xni1

A C B Cawi C=1 awi f a0 a1awig2

minimum 69

This expression can be solved analytically for a0 and

a1 using the least squares technique. Coefficients a0 and

a1 become functions of the constants A, B, and C ofEq. (63) on one side, and of regression sums over the values

of the independent variable awion the other.

The final expressions are the following:

a0 A C Cd=d0 70a

a1 B C Cd1=d 70b

where a0 and a1 are the minimum squares estimates in

terms ofF(BET). The functions d, d0, and d1 contain only

the regression sums of aw over the employed regression

interval with the following signs: d0=d< 0, d1=d< 0, andd0=dd1=d< 0 for aw< 1.

[56,57]

The corresponding relations where Gaussian bracketshave been used are

d0 1=1 awa2w aw=1 awaw 71a

d1 aw=1 awn 1=1 awaw 71b

d na2w aw2 71c

wheren is the number of data included in the regression.

As Eqs. (65) and (66) remain valid, the BET constants

are now given by

M0B 1

a00 a01

1= A B Cd0 d1=d2 f g 72a

and

CBa

00 a

01

a00

ABCd0 d1=d2 f g= ACd0=d1 f g 72b

Finally, by Eqs. (71a)(71c), A, B, C fM0G;CG;Kand an algebraically explicit expression for M0B and CBin terms of the three GAB constants are obtained:

M0B M0G

121K=CGG

Rm

73a

CBCGG 121KCGG

RC 73b

where the functions Rm and RCare given by

Rm11K CGG 1K=CGG 21K

d0 d1 =d2 74a

RCRm= 11KCGG 1K d0=d1

74b

These functions are always greater than unity.[56] There-

fore, Eqs. (73a) and (73b) reproduce the inequalities

(Eq. (55)) ifK


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MRD1n

Xni1

Mj MCaljM

78

Plotting of the residuals (MMCal) against the inde-pendent variable is also used as a measure of the distri-

bution of errors. If the model is correct, then the

residuals should be only random independent errors with

a zero mean, constant variance and arranged in a normal

distribution. If the residual plots indicate a clear pattern,

the model should not be accepted.

In general, low values of the correlation coefficient, high

values of RSS, SE, and MRD, and clear patterns in the

residual plots mean that the model is not able to explain

the variation in the experimental data. It is also evident

that a single statistical parameter cannot be used to select

the best model and the model must always be assessed

based on multiple statistical criteria.[61]

TABLE 1

Recent works on sorption isotherms of food materials

Food product Temperature (C) aw Best fitted model Reference

Berries 4, 13, 27 0.110.87 GAB 63

Mushroom 4, 13, 27 0.120.75 GAB 63

3070 0.200.80 Chung-Pfost 64

Cocoa beans 25, 30, 35 0.080.94 BET (aw< 0.5)Harkins-Jura (aw> 0.5)

65

Starch, potato 3060 0.110.83 GAB, Halsey 15

3060 0.050.95 GAB, Ferro-Fontan, Peleg 66

Pumpkin seed flour 10, 25, 40 0.110.85 GAB 67

Cured beef and pork 10, 20, 25, 35, 49 0.100.94 Peleg, GAB 68

Chicken meat 430 0.250.94 Ferro-Fontan, GAB 69

Hazelnuts 25 0.110.90 GAB 70

Rough rice 0, 5, 10, 15, 20, 25, 30, 35 0.2590.935 Modified Chung-Pfost,

modified Henderson

71

Yogurt powder 35 0.40.99 Modified Chung-Pfost 72

Gelatine gel 20 0.750.98 Ferro-Fontan 20

Crystalline lactose powder 12, 20, 30, 40 0.110.98 Timmermann-GAB, GAB 73

TABLE 2

Data on goodness of fit of selected models for sardine sheet at selected temperatures (25, 40, and 50 C) and water activity(0.110.84)

Coefficients of linear regression Coefficients of nonlinear regression

Isotherm R2 S.E. R2 S.E.

BET 0.9390.978 0.6480.968 0.9830.996 0.0040.009

Oswin 0.9630.984 0.1010.139 0.9860.997 0.0040.009

Smith 0.9540.974 0.0120.022 0.9540.974 0.0120.021

Hasley 0.9590.994 0.0650.164 0.9700.994 0.0200.042

Henderson 0.9310.974 0.1520.237 0.9480.978 0.360.054

Chung and Pfost 0.8850.906 0.250.299 0.9310.953 0.060.067

Iglesias and Chirife 0.9900.993 0.0060.008 0.990.993 0.0060.008

GAB 0.9890.996 0.0040.007

Peleg 0.9360.968 0.0140.026

Modified Oswin 0.8130.995 0.0050.029

Modified Henderson 0.9480.978 0.0360.054

Modified Chung-Pfost 0.9310.953 0.0600.067



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AN EXAMPLE

Published sorption data[62] (water activity range 0.11

0.84) of freeze-dried sardine fish at 25, 40, and 50Cwas selected for comparison of the different sorption iso-

therm models. Both linear and nonlinear approaches were

used to estimate the model coefficients. Standard error

and R2 values of different isotherm models for both linear

and nonlinear forms are given in Table 2. The R2 values

of the nonlinear models were always higher than that of

the linear equations. Similarly, standard error values were

also lower in nonlinear cases compared to the linearized

form, which suggests that nonlinear forms should be used

for estimation of model parameters. After choosing few

sorption models with higher R2 and lower SE values, their

residual plots were checked. Residual plot patterns of dif-

ferent models are presented in Table 3. Among all the

models, the GAB and Oswin models were close because

of their high correlation coefficient values and low stan-

dard error values. Both these models gave a scattered

residual plot. Although the Oswin model gave higher R2

(0.9860.997) and lower SE (0.0040.009) values, it does

not include temperature parameter. It may therefore be

inferred that GAB isotherm describes the sorption data

best over the experimental range of temperature and

water activity among all the isotherm models considered.

The monolayer moisture content (M0) and dimensionless

GAB parameters related to heat of sorption for the

monolayer and multilayer region (C and K) determined

from GAB equation are reported in Table 4. Figure 3

represents the sorption isotherm for sardine fish based

on GAB model and Fig. 4 shows a typical residual plot

of sorption data at 40C.

CONCLUSION

Theoretical aspects of sorption of water and commonlyused mathematical models to describe sorption isotherms

are discussed. Statistical criteria for selection of a model

have also been presented. Both the GAB and Oswin equa-

tions presented high R2 and low standard error values and

scattered residual pattern for freeze-dried sardine fish. But

because of lack of a temperature term in the Oswin equa-

tion, the GAB model is recommended for fitting the sorp-

tion data. From this discrimination approach based on

statistical measures, it is possible to select the best model

equation for describing experimental sorption data.

NOMENCLATUREaw Water activity

A, A0, B, r, k1, k2, n1, n2 ConstantsC Dimensionless parameter

related to heat of sorption

of monolayer region

df Degrees of freedom

ERH Equilibrium relative

humidity

TABLE 3

Pattern of residual plots of selected models

Model Residual plot pattern

BET Patterned

Peleg Patterned

Oswin Scattered

Hasley PatternedGAB Scattered

Modified Henderson Scattered

Modified Chung-Pfost Patterned

Iglesias and Chirife Patterned

TABLE 4

Calculated GAB parameters at selected temperatures

Temperature

(C)

Monolayer moisture

content

(kg water=kgdry matter) C K

25 2.0699 1.1172 0.0768

40 0.6834 1.0697 0.1857

50 7.3495 1.2121 0.1857

FIG. 3. Sorption isotherm for sardine fish at 25, 40, and 50C (GABisotherm).

FIG. 4. Residual plot of sorption data for sardine fish at 40C (GABisotherm).



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Hand H0 Functions containing a

fourth dimensionless

parameter h

K Dimensionless GAB

parameter related to heat of

sorption in multilayer

region

K2 Dimensionless parameter

M, M1, M2 Moisture content

MC Characteristic moisture

content

MCal Estimated value

MRD Mean relative deviation

M0 Monolayer moisture con-

tent on the internal surface

M0B(B), M0G(G), CB(B), CG(G) Constants

P Water vapor pressure in the

food material at any given

temperature

P0 Vapor pressure of purewater at that temperature

Q Net heat of sorption of

water

Q12 Net heat of desorption

R Universal gas constant

RSS Residual sum of squares

SE Standard error of the

estimate

T Absolute temperature

w M=M0X Kaw

Greek Letters

a, a0 Constants

c Parameter which accounts

for the structure of sorbed

water

DHd Total heat of sorption in

the food

DHS Net isosteric heat of sorp-

tion

DHvap Heat of vaporization of

water

d; d0; d1; a0; a1; a

0; a

1 Functions

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Documents

Models for Sorption Isotherms for Foods- A Review