Journal of Food Engineering 49 (2001) 103±114

www.elsevier.com/locate/jfoodeng

Prediction of water activity of osmotic solutions

A.M. Sereno a,*, M.D. Hubinger b, J.F. Comesa~
na c, A. Correa c
a
Department of Chemical Engineering, University of Porto, Rua Dr. Roberto Frias s/n, 4200-465 Porto, Portugal
b
Faculty of Food Engineering, University of Campinas, 13.083-970 Campinas, S. Paulo, Brazil
c
Department of Chemical Engineering, University of Vigo, Apdo. 874, 36200 Vigo, Spain
Received 1 June 2000; accepted 1 December 2000

Abstract
Models to correlate and predict water activity in aqueous solutions of single and multiple solutes, including electrolytes, relevant
for osmotic processing of foods are reviewed. During the last decade a signi®cant number of theoretical thermodynamic models that
are applicable to these systems have been developed and published. Though their use is still limited, their performance is in general
very good, similar to the best traditional empirical equations. Their predictive character together with built-in capabilities to work at
di€erent temperatures and in some cases pressure suggests that an increased e€ort to their wide use should take place. It was found
that predictions of water activity in aqueous solutions may easily be made with average relative deviations of less than 2%; this value
is of the same order or in some cases less than the typical error of current instrumentation available to measure water activity. Ó 2001
Elsevier Science Ltd. All rights reserved.

Keywords: Osmotic dehydration; Vapour±liquid equilibria; Water activity; Sugars; Electrolytes; Aqueous solutions; Predictive models

1. Introduction Rios, & Raoult-Wack, 1998; Sabadini, Carvalho, So-

bral, & Hubinger, 1998), as well as for fruit products,
Osmotic treatments involve the contact of a material, when the correct choice of solutes and a controlled ratio
usual of vegetal or animal origin, with a concentrated of water removal and impregnation allow to enhance
aqueous solution and include namely osmotic dehydra- their natural ¯avour and colour retention (Raoult-Wack
tion and solute impregnation processes. These constitute et al., 1994; Guerrero, Alzamora, & Gerschenson, 1996;
simple food processing operations conducted at ambient Alzamora, 1997; Forni, Sormani, Scalise, & Torregiani,
or near ambient temperature, which achieve a signi®cant 1997; Spiess & Behsnilian, 1998).
degree of dewatering without phase change and allowing As a preliminary step to convective air drying,
some degree of product formulation and even restruc- several studies are found in the recent literature as well
turing. as its in¯uence on ®nal food properties (Lenart, 1996;
A special mention should be made about the use of Del Valle, Cuadros, & Aguilera, 1998; Sa, Figueiredo,
osmotic treatments in the preparation of intermediate & Sereno, 1999; Salvatori, Andres, Chiralt, & Fito,
moisture foods (IMF) and the so-called fourth genera- 1999; Lewicki & Lukaszuk, 2000; Krokida, Kiranou-
tion or minimally processed foods. Such processes have dis, Maroulis, & Marinos-Kouris, 2000). The combi-
been also referred to as dewatering and impregnation nation of osmotic dehydration to some less common
soaking (Raoult-Wack, Rios, Saurel, Giroux, & Guil- drying steps, such as microwave drying (Venkatacha-
bert, 1994) and are mainly being used as a pre-treatment lapathy & Raghavan, 1999) and freeze-drying (Ka-
introduced in any conventional fruit and vegetable rathanos, Anglea, & Karel, 1996) has also been
processing, in order to improve quality, reduce energy described.
costs or even formulate ®nal products. Applications Osmotic pre-treatments may be used before vacuum
have been reported for ®sh and certain meat products frying, by immersion in solutions of high osmotic pres-
(Collignan & Raoult-Wack, 1994; Bohuon, Collignan, sure (Spiess & Behsnilian, 1998), the e€ect on mass
transfer rates of high hydrostatic pressure (HPP) pro-
cessing, previous to osmotic treatment of vegetable tis-
*
Corresponding author. Fax: +351-22-508-1440. sues was also investigated (Rastogi & Niranjan, 1998;
E-mail address: sereno@fe.up.pt (A.M. Sereno). Rastogi, Angersbach, & Knorr, 2000).
0260-8774/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved.
PII: S 0 2 6 0 - 8 7 7 4 ( 0 0 ) 0 0 2 2 1 - 1
104 A.M. Sereno et al. / Journal of Food Engineering 49 (2001) 103±114

Nomenclature T temperature (K)

x mole fraction
a activity
ARD average relative deviation (%) Greeks
f fugacity c activity coecient
GE Gibbs free energy (J) tFH
i no. of non-hydrogen atoms in molecule i
gE molar Gibbs free energy (J mol 1 ) tki no. of non-hydrogen atoms in group k of component i
h molar heat of mixing (J mol 1 )
Subscripts
M molar mass …kg mol 1 †
i component i
P pressure (Pa)
W water
p partial pressure (Pa)
R gas constant Superscripts
RH relative humidity (%) ° reference state
RMS root mean square

The transfer of mass, water and solutes observed It is usually assumed that under normal working
during the contact of a solid material with an osmotic conditions of ambient temperature and atmospheric
solution is due to di€erences in chemical potential inside pressure, gas phases behave ideally and so the ratio of
and outside the material, usually expressed in terms of fugacities can be taken as the ratio of partial pressures
the corresponding activity coecients. As dehydration is (Reid, Prausnitz, & Poling, 1987):
the main objective of osmotic treatments, the activity of fW pW
water in both the material and the solution and its ˆ 0 ; …2†
fW0 pW
prediction is of major importance. This relevance is
0
particularly emphasised when mathematical models are where pW and pW are the vapour pressures of water in
used to describe the process. For cellular materials, as is the system and of pure liquid water at the same tem-
the case of most foods, water transfer takes place in the perature, respectively. Under this assumption, from
vicinity of the cell through the semi-permeable cell Eq. (1):
membrane (Le Maguer, 1988); again di€erence of water pW RH
activity in both sides governs its physical behaviour and aW ˆ 0
ˆ ; …3†
pW 100
mass transport.
The thermodynamic description of these osmotic so- where RH is the percent relative humidity of the air
lutions has been the object of intense research all along layer in equilibrium with the sample. The activity coef-
the last century, particularly those involving sugars and/ ®cient cW may be calculated directly from the partial
E
or salts. Excellent reviews of such e€orts have been molar excess Gibbs energy of water, gW (Prausnitz et al.,
written by Van den Berg and Bruin (1981) and Le Ma- 1986):
guer (1992) to mention only two papers. Most thermo- E
gW ˆ RT ln cW ; …4†
dynamic models used to describe vapour±liquid
E
equilibrium of osmotic solutions are based on relations and for the total molar excess Gibbs energy, g :
X
involving Gibbs free energy of the system. Of particular gE ˆ RT xi ln ci …5†
interest is the excess Gibbs energy (GE ) and for each of i
the components the partial molar excess Gibbs energy
for an ideal solution all the activity coecients,
(giE ), both allowing a convenient way to quantify the
ci …T ; P ; x†, are equal to one, corresponding to an excess
deviations from ideal behaviour.
Gibbs energy equal to zero.
From GE , a large number of physical parameters may
Several approaches have been used to calculate or
be calculated such as water and solute activities, partial
estimate excess Gibbs energy in liquid solutions, in-
equilibrium properties (solubility, relative volatility,
cluding empirical models based on solution composi-
etc.) and others (Le Maguer, 1989).
tion, the use of equations of state extended to the
The activity of water in aqueous solutions is de®ned
description of condensed phases and probably more
as (Prausnitz, Lichtenthaler, & Azevedo, 1986):
often, di€erent theories developed to describe solution
fW …T ; P ; x† structure and interactions among the chemical species.
aW …T ; P ; x† ˆ cW …T ; P ; x† xW ˆ ; …1† In Table 1 the major methodological groups used by
fW0 …T ; P 0 ; x0 †
published contributions to the prediction of water ac-
where aW is the water activity, xW is the mole fraction of tivity of solutions are indicated.
water, cW is the activity coecient for water, fW ; fW ° are Most of the models included in Table 1 are general in
the fugacities of water in the system and at reference themselves and may be used with food and related sys-
conditions, respectively. tems (Prausnitz et al., 1986; Le Maguer, 1992). The
 A.M. Sereno et al. / Journal of Food Engineering 49 (2001) 103±114 105

Table 1
Di€erent approaches to calculate GE and activity coecients

Empirical equations based on solution composition

 Simple equations, often including semi-theoretical basis, which often produce reasonable predictions of liquid±vapour and liquid±liquid
equilibria; served in some cases as the basis for more complex models: Margules equation, the Wohl expansion, Van Laar equation, Redlich±Kister
equation, Scatchard±Hamer equation
Use of equations of state (EOS)
 Hu, Liu, Soane, and Prausnitz (1991): use double lattice (Freed & Flory-Huggins) to calculate Helmholtz energy of mixing, used to describe
equilibrium in non-polar or slightly polar non-ideal systems
 Wu, Lin, Zhu, and Mei (1969): use modi®ed version of Pitzer (1973) virial EOS with Fowler and Guggenheim (1939) instead of the more
common Debye±Huckel contribution of long-range electrostatic forces
 Simonin (1999): uses McMillan±Mayer theory, assuming molecules as hard bodies interacting through van der Waals short range forces; to allow
for the presence of polar molecules (alcohols) and weak electrolytes, and a volume exclusion contribution using Carnahan±Starling hard sphere
EOS
Solution theories
 Van der Waals±van Laar theory: valid only for nearly ideal systems
 Regular solution theory (Hildebrand, 1929; Scatchard, Hamer, & Wood, 1938): systems without interactions among polar molecules (athermal),
e.g. hydrocarbons
 Debye and Huckel (1923) introduced a successful description of long-range polar and ionic forces contribution
 Lattice theory due mostly to Guggenheim (1935, 1952) and Eyring and Marchi (1963)
 Flory (1941, 1942) and Huggins (1942) extension of lattice theory to polymer solutions
 Local-composition models, namely Wilson (1964); NRTL model of Renon and Prausnitz (1968); UNIQUAC model of Abrams and Prausnitz
(1975)

diculties usually arise from the fact that as most food a technique for aW prediction for complex systems
systems are not well characterised both from a chemical without requiring parameters for each speci®c com-
and structural point of view, and consequently ther- pound, but instead characteristic interaction parameters
mophysical property data required to use (and ®t) the for the chemical groups that constitute each molecule.
models to such systems are not available. One way to The e€ect in the solution of each of such chemical group
overcome these diculties in practical applications is to is assumed to be the same, irrespective of the molecule
look for correlations between these thermodynamic where it is contained.
properties and some easy method to measure physical The ASOG method, based on the Wilson model
properties. This approach produced a series of empirical (Wilson, 1964), has been used for this purpose by Correa
and semi-empirical models which provide in many cases et al. (1994), who obtained good aW predictions for
excellent results and constitute relevant contributions of aqueous solutions of both sugar/polyol and sugar/urea.
widespread use in the food industry (Table 2). Kawaguchi et al. (1981, 1982) have combined the
All the models mentioned in Tables 1 and 2 refer to method with an hydration model of ionic species origi-
solutions, more speci®cally aqueous solutions. A parallel nated in aqueous solutions of electrolytes; later Correa
e€ort has been dedicated to the correlation and predic- et al. (1997) have rede®ned the groups in solution, ob-
tion of sorption isotherms in solid foods; a large number taining improved predictions for several binary and
of equations and models used to describe and predict multicomponent salt solutions.
water activity and sorption isotherms of food systems Attempts to predict water activity using UNIFAC
containing solid phases (Van den Berg & Bruin, 1981; have also been extensively reported. Choudhury and Le
Iglesias & Chirife, 1982). Such models were not included Maguer (1986) used this method to predict aW in glucose
here, as only aqueous solutions constituted the scope of solutions and Achard et al. (1992) described its use to
this work. estimate activity coecients in aqueous systems con-
An alternative approach to the use of empirical and taining sugars and polyols; Catte et al. (1995) used the
semi-empirical equations consists in the application of same group typology as suggested by Correa et al.
group contribution methods, namely ASOG (analytical (1994) to characterise sugars, and successfully estimated
solutions of groups, Derr & Deal, 1969; Kojima & several thermodynamic properties. Christensen et al.
Tochigi, 1979) and UNIFAC (UNIQUAC functional (1983) and Kikic et al. (1991) applied the same model to
group activity coecients, Fredenslund, Jones, & predict equilibria in salt solutions. Very interesting ex-
Prausnitz, 1975; Larsen, Rasmussen, & Fredenslund, tensions to cover organic acids (Velezmoro & Meirelles,
1987). These two proposals probably constitute the only 1998), polyethylene glycol and other polyols (Ninni et
general accessible predictive technique for water activity al., 1999a, 2000), aminoacids (Ninni et al., 1999b) have
of osmotic solutions. Their main advantage is to provide been developed by Meirelles and co-workers.
106 A.M. Sereno et al. / Journal of Food Engineering 49 (2001) 103±114

Table 2
Models to calculate water activity in aqueous solutions relevant to food systems

1. Empirical equations
 Simple early formulas applicable to the candy industry: Grover and Nicol (1940), Money and Born (1951) and Dunning, Evans, and Taylor
(1951)
 Zdanovskii (1936) and Stokes and Robinson (1966) proposed independently, equivalent models, valid, respectively, for electrolytes and non-
electrolytes, whose merits were discussed in detail by Chen, Sangster, Teng, and Lenzi (1973)
 Several authors attempted to correlate water activities with freezing point depression, with moderate success: Ferro-Fontan and Chirife (1981),
Lerici, Piva, and Rosa (1983) and Chen (1987)
 Lin, Zhu, Mei, and Yang (1996) proposed a simpli®ed formula with two parameters per solute and one additional per binary interaction;
Comesa~ na, Correa, and Sereno (2000) extended to sugars and salts
 Roa and Tapia (1998) proposed a very simple yet reasonably successful formula with a single parameter per solute in mixed multicomponent
solutions
2. Semi-empirical equations
 A ®rst group includes empirical approximation of deviations from Raoult's law, Caurie (1985) and the very successful model by Chen (1989,
1990), which unfortunately involves three empirical parameters per solute
 Based on the identity between equilibrium relative humidity and thermo-dynamic activity, Norrish (1966), developed one of the most successful
equations applicable to non-ionic solutes; extensions were made by Chuang and Toledo (1976) and Chirife, Ferro-Fontan, and Benmergui (1980)
calculated an improved set of parameters which has been generally adopted since then
 Based on the Gibbs±Duhem equation for multicomponent aqueous solutions, Ross (1975) proposed a simple and successful mixing rule to
predict water activity of multicomponent solutions; extensions made by Ferro-Fontan, Chirife, and Boquet (1981) and Ruegg and Blanc (1981)
3. Local composition models
 WILSON model has been used in some models combined with other contributions and is the basis of ASOG group contribution method to
predict thermodynamic equilibria. Sorrentino, Voilley, and Richon (1986), used it to predict activity coecients of aroma compounds and
Kawaguchi, Kanai, Kajiwara, and Arai (1981, 1982), of electrolytes; Correa, Comesa~ na, and Sereno (1994) applied to sugar/polyol/urea; later
Correa, Comesa~ na, Correa, and Sereno (1997) applied to electrolyte solutions, introducing modi®cations to Kawaguchi's approach
 UNIQUAC model was used by Le Maguer (1981), Saravacos and Marino-Kouris (1990) and Sander, Fredenslund, and Rasmussen (1986),
combined with Debye±Huckel contribution and applied to electrolyte systems. Le Maguer (1992) proposed a building block concept to calculate
individual molecular parameters. This model is the basis of the well-known UNIFAC group contribution method; Choudhury and Le Maguer
(1986) used it with glucose solutions; Gabas and Laguerie (1992) with sugar solutions; Achard, Gros, and Dussap (1992) with sugars and polyols;
Catte, Dussap, and Gros (1995) and Peres and Macedo (1997) introduced some improvements to describe sugars; Christensen, Sander,
Fredenslund, and Rasmussen (1983) and Kikic, Fermeglia, and Rasmussen (1991) applied to electrolytes. Velezmoro and Meirelles (1998) applied
the concept to systems of organic acids; Ninni, Camargo, and Meirelles (1999a, 2000), to polyethylene glycol and other systems with polyols;
Ninni, Camargo, and Meirelles (1999b), to aminoacids

These extensive and successful applications of the two some of the compounds found in food-related systems
major group contribution methods to predict water ac- are too complex and sometimes new groups have to be
tivity of osmotic solutions constitute an example, pos- identi®ed and characterised. This happened with sug-
sibly one of the earliest, of the use of advanced ars. Sugars in aqueous solutions, both monosaccha-
thermodynamic models to describe the behaviour of rides, like glucose and fructose, and disaccharides, like
food-related systems. sucrose, adopt a cyclic structure made of pyranose and
furanose unit rings (Fig. 1). To successfully describe
such compounds by the ASOG method, Correa et al.
2. How group contribution methods work (1994) de®ned three new groups named GR, FR and
CPOH standing for `glucose ring', `fructose ring' and
The main principle underlying this technique is the `cyclic-polyalcohol' (OH groups linked to consecutive
assumption that all chemical functional groups in solu- carbon atoms in a cyclic sugar structure). For polyhy-
tion interact with other speci®c groups in a similar way, dric alcohols (e.g. glycerol, mannitol and sorbitol) a
independent of the types of groups in presence and of new group POH referring to OH groups linked to
the speci®c molecules where they are contained. To use consecutive carbon atoms in a linear chain was also
the method the ®rst step consists in `breaking' the adopted. A ®fth `group' de®ned was UREA for the
structure of the molecule in all its basic groups, for urea molecule, as it could not be adequately repre-
which interaction and individual geometric parameters sented by a combination of available ASOG groups
are available. Since then, the method is applied as if the (Table 3).
solution was made of known groups instead of unknown The method uses two speci®c individual group pa-
molecules. rameters tFH
i and tki , and four interaction parameters for
Since their development the list of identi®ed and each binary pair of groups. Table 4 lists the values of
well-characterised groups is increasing. Unfortunately these parameters for some relevant compounds.
 A.M. Sereno et al. / Journal of Food Engineering 49 (2001) 103±114 107

theoretical understanding of these systems has been

minimal. Several reviews have been published during the
last 20 years, focusing particular lines of research; a
special mention about the work of Zemaitis, Clark,
Rafal, and Scrivner (1986) where much of the published
previous information including data are collected and
that of Renon (1986). Loehe and Donohue (1997) give a
list of such reviews together with a general broad view of
recent advances.
Before 1985, most of the works have been conducted
along the principles introduced by Debye and Huckel
(1923) theories and their modi®cation by Pitzer (1973),
Bromley (1973) and Sandler (1977). In terms of new
theoretical research on electrolyte systems, modern de-
velopments follow, according to Loehe and Donohue
Fig. 1. Ring structures for sugars in aqueous solution. (1997), four major lines: (i) extensions to Debye±Huckel
theories, which assume the system to be made of
3. Solutions with electrolytes charged ions in a continuous dielectric medium, and
employ di€erent forms of the Poisson±Boltzman equa-
Although some of the general models included in tions to calculate potential energies in the system; (ii)
Tables 1 and 2 have proved to be able to describe perturbation theories that use a series expansion of the
aqueous electrolyte solutions, this group of systems has Helmholtz free energy about a given reference state; (iii)
deserved, from the early stages, a special separate at- integral equation theories that include solutions of the
tention. E€ective empirical and semi-empirical equa- Ornstein±Zernicke equation, namely using the spherical
tions were derived, but their contribution to the approximation concept and (iv) ¯uctuation solution or

Table 3
New ASOG interaction groups

POH Polyalcohol OH groups linked to consecutive carbon atoms in a linear chain

FR Fructose ring Furanose cyclic structure of fructose in water
GR Glucose ring Pyranose cyclic structure of glucose in water
CPOH Cyclic polyalcohol OH groups linked to consecutive carbon atoms in a cyclic sugar structure
UREA Urea Urea molecule

Table 4
Values of tFH
i and tki for some compounds considered

Compound tFH
i Groups: tki

Glucose 12 GR:6 CPOH:4 CH2:1 OH:1

Fructose 12 FR:5 CPOH:3 CH2:2 OH:2
Sucrose 23 GR:6 FR:5 CPOH:5 CH2:3
Glycerol 6 POH:3 CH2:2.8 OH:3 O:1
Urea 4 UREA:4
Water 1 H2O:1.6

Partial list of ASOG group pair parameters, including the new groups

H2 O GR FR O CPOH CH2 OH POH Urea

H2 O ± )1.6667 )0.8312 )20.26 0.2826 0.5045 1.4318 )3.714 )0.0067 m
± 0.7717 1.5856 )1.2186 2.170 )2382. )280.2 )2.418 )2.186 n
GR 0.7435 ± 0.8393 1.8119 2.014 0.3248 0.1652 )0.9060 1.0312 m
1.3432 ± )1.2930 )6.468 0.3987 3.473 1.5168 7.833 )4.198 n
FR )0.0377 )14.561 ± 1.2042 0.6475 0.8477 0.7318 )1.6816 )2.126 m
)0.8764 )1.5918 ± 0.2919 1.5562 1.6635 0.9593 2.309 )4.814 n
108 A.M. Sereno et al. / Journal of Food Engineering 49 (2001) 103±114

Table 5
Equations for electrolyte solutions

1. Hydration models: ion±solvent interaction is modelled as producing solvated aggregates characterised through the use of solvating numbers and
solvating energy; this approach has been applied for many years
 Zdanovskii (1936) and Stokes and Robinson (1966): based on a linear relation existing between the molalities of isopiestic systems of single or
mixed solutes
 Robinson and Bower (1965): use speci®c expressions to describe hydration and association of ions in solution
 Pitzer (1973): is a virial development of GE in terms of molalities of the solutions; is very widely used
 Meissner and Tester (1972): use a relation of mean ionic activity coecient and ionic force of the solution through a single parameter per salt
 Bromley (1973): improved previous model including extended Debye±Huckel equation
 Chirife et al. (1980): proposed a relation between solutions of mixed solutes and single solutes of same ttal ionic strength
 Heyrovska (1989): considers the association degree and hydration number as parameters for direct correlation of equilibrium data
 Schoenert (1990a,b) and subsequent work: used expressions of GE of hydrated and associated ions, modifying Robinson and Stokes hydration
model
2. Local composition models: model ion±solvent interaction on a local basis, trying then to generalise to the whole system
 Chen, Britt, Boston, and Evans (1982) and Chen and Evans (1986): use NRTL + Pitzer±Debye±Huckel; good for low and moderate
concentrations of strong electrolytes
 Liu, Wimby, and Gren (1989) and Liu and Gren (1991): use WILSON + Debye±Huckel contribution
 Haghtalab and Vera (1991): two parameter non-random model to calculate GE
 Cardoso and O'Connel (1987): combined UNIQUAC+Debye±Huckel contribution
 Kawaguchi et al. (1981, 1982) and Correa et al. (1997): use WILSON based ASOG group contribution coupled with hydration model for cations
3. Aqueous mixed electrolytes: concentrates in developing mixing rules that can be used with single solute models
 Clegg and Pitzer (1992) and Lu and Maurer (1993): combine Debye±Huckel with UNIQUAC for mean ionic activity and osmotic coecient,
proposing a mixing rule for mixed systems
4. Equation of state models: used mostly to predict the behaviour at high temperature and pressures
 Pitzer and Tanger (1989), Anderko and Pitzer (1993,) and Economou, Peters, and Arons (1995): equations of state able to predict phase
behaviour of electrolyte systems at high temperature
 Ikonomou and Donohue (1986) and Jin and Donohue (1988a,b): introduced the associated-perturbed-anisotropic-chain theory to describe
aqueous halides between 200°C and 500°C
 Wu et al. (1969): combine Pitzer virial equation with Fowler and Guggenheim (1939), long-range contribution

Kirkhood±Bu€ theories, accounting for composition In spite of this elegant way of expressing temperature
¯uctuations in an open system. dependence, some empirical equations have been pro-
From an applied engineering point of view recent posed (e.g. Scott & Bernard, 1983; Kitic, Jardim, Fav-
results on electrolyte systems may be divided into three etto, Resnik, & Chirife, 1986), often involving a
major groups (Loehe & Donohue, 1997): (i) local com- considerable number system speci®c parameters.
position and hydration models, (ii) empirical and semi- Those models which involve local composition based
empirical equations of state for electrolyte systems and concepts (Wilson, NRTL, UNIQUAC) as well as
(iii) equations for mixed electrolytes and mixed solvents. equations of state have explicit dependence on temper-
These are summarised in Table 5. ature (and in some cases pressure), making this depen-
dence automatic.

4. Temperature e€ect on aW
5. Comparing model predictions with experimental data
Dependence of activity coecient on temperature
may be expressed in terms of the excess partial molar As all the required information to apply the models
enthalpy or the partial molar excess heat of mixing as included in Tables 1, 2 and 5 is detailed in the original
(Prausnitz et al., 1986): papers and some of the indicated reviews, they will not
  be reproduced here; the reader is invited to look at those
d ln ci hEi papers for such information.
ˆ : …6†
dT P RT 2 Several comparative studies have been presented in
After integration between two di€erent temperatures, a the past, most of them involving only a small group of
form of the well-known Clausius±Clapeyron equation is models. It is apparent from these studies that for many
obtained: systems of growing interest for the food industry, there
  is still a very important lack of reliable experimental
…aW †T 1 hEW 1 1
ln ˆ ; …7† data on thermophysical properties. Experimental tech-
… aW † T 2 R T1 T2
niques to measure them are delicate and very time
when hEW is constant along the path T1 to T2 . consuming.
 A.M. Sereno et al. / Journal of Food Engineering 49 (2001) 103±114 109

Table 6
Prediction of water activity in glucose±water solutions

Reference Model/Equation ARD (%) RMS Experimental data

Norrish (1966) and Chirife et al. Norrish (1966) 0.63 0.89 Stokes and Robinson
(1980) (parameters) (1966)
Chen (1989) Chen (1989) 0.65 0.95 Stokes and Robinson
(1966)

Table 7
Prediction of water activity in sucrose±water solutions

Reference Model/Equation ARD (%) RMS Experimental data

Norrish (1966) and Chirife et al. Norrish (1966) 2.8 3.9 Robinson and Stokes
(1980) (parameters) (1961)
Chen (1989) Chen (1989) 2.8 3.9 Robinson and Stokes
(1961)
Lerici et al. (1983) Lerici et al. (1983) 12.7 14.5 Lerici et al. (1983)

Fig. 2. Water activity of aqueous glucose solution: experimental data Fig. 3. Water activity of aqueous sucrose solution: experimental data
(symbols) and predictive models (lines). (symbols) and predictive models (lines).

The use of more recent techniques, like group con- ARD ˆ average relative deviation
tribution methods or equations of state, has not yet been
1Xn
…aWi †calc …aWi †exp
the object of extensive comparative studies, justifying  100; …8†
thus this analysis targetting the case of osmotic solu- n 1 1 …awi †exp
tions. For such a purpose, using some of the more
successful models, predictions of water activity for two RMS ˆ root mean square
v
!2
sugar solutions (glucose and sucrose, Tables 6 and 7 and u n
u1 X …aWi † …a † …9†
Figs. 2 and 3), sodium chloride solution (Table 8, Fig. 4) t calc W i exp
 100 :
and two mixed solute solutions (sucrose±sodium chlo- n 1 1 …aWi †exp
ride and glucose±sodium chloride, Tables 9 and 10 and
Figs. 5 and 6) were made and compared. These systems
are believed to be the most common solutions for os- 6. Conclusions
motic treatment processes.
The meaning of the statistical parameters ARD and Analysis of the comparative performance of di€erent
RMS in the tables is as follows: predictive models for water activity has revealed that
110 A.M. Sereno et al. / Journal of Food Engineering 49 (2001) 103±114

Table 8
Prediction of water activity in sodium chloride±water solutions

Reference Model/Equation ARD (%) RMS Experimental data

Pitzer (1973) Pitzer (1973) 0.14 0.25 Robinson and Stokes (1965)
Correa et al. (1997) ASOG 1.67 2.33 Robinson and Stokes (1965)
Chen (1989) Chen (1989) 2.8 3.9 Robinson and Stokes (1965)
Roa and Tapia (1998) Roa and Tapia (1998) 9.35 10.7 Robinson and Stokes (1965)

good estimations may be obtained, not only from em-

pirical equations with parameters ®tted to experimental
data but also from theoretical models derived from ex-
pressions of the excess Gibbs free energy and from
equations of state.
It is somehow surprising how good the predictions
made from group contribution models are. These truly
predictive models use group interaction parameters not
necessarily calculated from data involving the compo-
nents of interest. Due to these capabilities, further e€ort
and testing should be devoted to this group of tech-
niques.
For practical applications of osmotic treatments, the
most widely used equations for prediction of water ac-
tivity in binary are the ones of Norrish (1966) and Ross
(1975) (see also Lopez-Malo, Palou, Welti, Corte, &
Argaiz, 1994; Guerrero et al., 1996; Alzamora, 1997;
Nieto, Salvatori, Castro, & Alzamora, 1998). Parame-
ters calculated by Chirife et al., 1980 should be preferred
for use with Norrish equation. For solutions containing Fig. 4. Water activity of aqueous sodium chloride solution: experi-
electrolytes, Pitzer (1973) and Chen (1989, 1990) equa- mental data (symbols) and predictive models (lines).

Table 9
Prediction of water activity in sucrose±sodium chloride±water solutions

Reference Model/Equation ARD (%) RMS Experimental data

Teng and Seow (1981) Ross (1975) 1.09 1.13 Robinson, Stokes, and Marsh (1970)a
Chen (1990) Chen (1989) and Ross (1975) 1.25 1.41 Robinson et al. (1970)
Comesa~na et al. (2000) Lin et al. (1996) 1.78 1.92 Robinson et al. (1970)a
Correa et al. (1997) ASOG 2.48 3.54 Robinson et al. (1970)a
Teng and Seow (1981) Chirife et al. (1980) 2.62 3.1 Robinson et al. (1970)a
Teng and Seow (1981) Zdanovskii±Stokes±Robinson 2.99 3.3 Robinson et al. (1970)a
Roa and Tapia (1998) Roa and Tapia (1998) 7.15 9.52 Robinson et al. (1970)a
Chuang and Toledo (1976) Modi®ed Norrish/Ross (1975) 14.2 16.4 Chuang and Toledo (1976)
a
As reported by Teng and Seow (1981).

Table 10
Prediction of water activity in glucose±NaCl solutions

Reference Model/Equation ARD (%) RMS Experimental data

Chen (1990) Chen (1990)/Ross 1.83 2.66 Comesa~

na et al. (2000)
Comesa~na et al. (2000) Lin et al. (1996) 1.61 2.33 Comesa~
na et al. (2000)
This work ASOG 8.4 9.3 Comesa~
na et al. (2000)
Roa and Tapia (1998) Roa and Tapia (1998) 8.9 9.7 Comesa~
na et al. (2000)
 A.M. Sereno et al. / Journal of Food Engineering 49 (2001) 103±114 111

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