Excess properties and solid-liquid equilibria for aqueous solutions of sugars using a UNIQUAC model

ELSEVIER

Fluid Phase Equilibria,

96 ( 1994) 33-50

Excess properties and solid-liquid equilibria for aqueous solutions of sugars using a UNIQUAC model Marianne

CattC, Claude-Gilles

Laboratoire

Dussap,

de Ginnie Chimique Biologique,

Christian

Universitt

Achard,

Jean-Bernard

Gros

*

Blaise Pascal, 63177 AubiSre Cedex, France

(Received August 3, 1993; accepted in final form November 25, 1993)

Abstract Cat& M., Dussap, C.-G., Achard, C. and Gros, J.-B., 1994. Excess properties and solid-liquid aqueous solutions of sugars using a UNIQUAC model. Fluid Phase Equilibria, 96: 33-50.

equilibria

for

A modified UNIQUAC model was used to describe thermodynamic properties of binary water-carbohydrate mixtures. Interaction parameters are determined for glucose, fructose and sucrose. A new equation was developed to describe carbohydrate solubility in water from knowledge of dilution enthalpy rather than fusion enthalpy, and particularly when sugar crystallizes as a hydrated form. For all thermodynamic properties (water activity, osmotic coefficients, excess Gibbs energy, excess enthalpy, activity coefficients, boiling temperature, freezing temperature and solubility) the model agrees with experiments when reliable data are available. Good predictions are also obtained for water activity and osmotic coefficients in ternary systems (water-sucrose-glucose). Keywords: Theory; Computer simulation;

Excess functions; Solid-liquid

equilibria; Vapour-liquid

equilibria; Water;

Carbohydrates

1. Introduction Carbohydrates have long interested chemists because of their prominent and significant role in a variety of areas including biological and technological applications. The stabilities of these compounds and their positions in chemical equilibria are largely dictated by thermodynamic considerations. Despite their importance, relatively few thermodynamic data are available for sugars; that is the reason why the correlation and prediction of phase equilibrium data are of major interest.

* Corresponding

author.

0378-3812/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDIO378-3812(93)02458-Y

34

M. Cattk et al. 1 Fluid Phase Equilibria 96 (1994) 33-50

Much work has been done on developing thermodynamic models for this purpose, and reliable methods are now available: ASOG, NRTL, UNIQUAC, UNIFAC (Reid et al., 1986). Although these models were not developed to deal with carbohydrate-water systems, Achard et al. ( 1992) used a modified UNIFAC model (Larsen et al., 1987) to predict water activity, boiling and freezing temperatures in aqueous solutions containing mono- and/or disaccharide. Le Maguer (1992) used a UNIQUAC equation with revised structural parameters by the building block method to correlate excess Gibbs energy and excess enthalpy of five aqueous carbohydrate systems, and Abed et al. (1992) used a UNIFAC equation to describe solubility of two sugars in water. However, no method exists to describe in the same way all the thermodynamic properties of similar mixtures. The purpose of the work reported here was to use a modified UNIQUAC model (Larsen et al., 1987) to describe excess properties (water activity, excess Gibbs energy, excess enthalpy), vapour-liquid equilibria characterized by the boiling temperature of the mixture, and solid-liquid equilibria including the determination of freezing temperature and solubilities of the anhydrous or the hydrated form of the sugar. Aqueous mixtures of glucose, fructose and sucrose will be examined.

2. Model Water activity, osmotic coefficient and activity coefficients are directly related to the excess Gibbs energy function. The UNIQUAC model (Abrams and Prausnitz, 1975) offers a suitable temperature dependency to ensure correct calculations of the activity coefficient of water over a range exceeding 100 K. However, for excess enthalpy and solute (sugar) solubilities, more robust expressions must be used to obtain precise predictions of the successive temperature derivations. The model of Larsen et al. (1987) is used in this study. The molar excess Gibbs energy is calculated as the sum of a combinatorial and a residual contribution: gE/RT = gEC/RT + gER/RT

Conversely, the symmetric activity coefficient of component combinatorial and a residual part: lily, =lnyF

+lnrr

(1) i in the mixture is the sum of a

(2)

Larsen et al. (1987), with Kikic et al. (1980), left out the Staverman-Guggenheim correction and introduced a modified calculation of the volume fractions; the resulting expression of the combinatorial part of the molar excess Gibbs energy is gEC/RT = C Xi ln(oi/xi) i

which leads to In 7” = ln(w,/x,) + 1 -z I

(4)

35

Act. Cat&! et al. 1 Fluid Phase Equilibria 96 (1994) 33-50

where the volume fraction oi of component

i is given by

mi = Xi*:‘3 C XjYj213 I j ri is the molecular volume parameter of component

(5)

i, calculated as the sum of the group volumes

Rk: Yi = C v~‘R, k

VP)is the number of groups of type k in molecule i. Group parameters Rk are obtained from the van der Waals group volumes given by Bondi ( 1968). The residual contribution of the molar excess Gibbs energy is gER/RT = -C

i

(7)

qixi ln(C Ojrji‘, \j

/

(8) qi is the molecular surface area parameter for component i calculated as for ri from group area parameters Qk obtained from van der Waals group surface areas given by Bondi (1968):

8i is the surface area fraction for component tli = qiXi

I

C

i

i in the mixture given by

(10)

qjXj

The Boltzmann factors Zij are obtained from the temperature-dependent aij as zu =

interaction parameters

exp( -aij/T)

(11)

To account for excess enthalpies and excess heat capacities, Larsen et al. ( 1987) proposed describing the temperature dependency of the interaction parameters by a three-coefficient expression: aij(T) =aij,l +au,z(T-

T”) +a,3[Tln(P/T)

+ T-

To]

where T“ is an arbitrary reference temperature, set at 298.15 K. This model is used throughout this work to describe excess properties, vapour-liquid and solid-liquid equilibria of mixtures of water and simple carbohydrate.

(12)

equilibria

3. Excess properties and partial molar properties Subscripts 1 and 2 will refer respectively to solvent (water) and solute (sugar). As already mentioned, the UNIQUAC models, which are systematically proposed in flowsheeting programs for equilibrium calculations, allow the calculation of the molar excess Gibbs

36

M. Cattk et al. 1 Fluid Phase Equilibria 96 (1994) 33-50

energy gE and the activity coefficients yi with respect to the so-called symmetrical convention, i.e. with respect to a hypothetical reference state taken to be pure liquid for all species at the temperature and pressure of the system. For carbohydrates, however, the experimental data are generally available in the asymmetrical conventions in either the molal scale or the mole fraction scale. The total Gibbs energy of the mixture is used as a starting point to express the partial ideal Gibbs energies gfd, the partial excess Gibbs energies g” and the activity coefficients yi. The working equations are summarized in Tables 1 and 2 where all the properties have been established as a function of the symmetrical activity coefficients. y? denotes the symmetrical activity coefficient of the infinitely diluted solute. The reference state in the asymmetrical convention using the molal scale is the hypothetical ideal solution with a molality of rni for the solute; for practical use, the reference molality rn;is generally chosen as rn$= 1 mol kg-‘. The quantity ml is 55.5084 mol kg-’ for aqueous solutions. Also, knowledge of activity coefficients affords the water activity a, and the osmotic coefficient @ of the solutions studied, given by a, = ylxl = S% = Y? exp( -xz/xl)

(13)

The excess enthalpies are calculated by the general relation -hEIT

(15)

= $(gE/T)

Assuming that the extension of the UNIQUAC model proposed by Larsen et al. (1987) gives a sufficiently correct temperature dependency for derivation, the excess enthalpy hE with respect to Table 1 Conversion

equations for partial ideal Gibbs energy (J mol-*)

Symmetrical convention. g’d=gy+RTlnx 1 gip=g$+

Mole fraction scale. Reference state: pure liquid for all the species at T and P of the system.

1

RTlnx,

Asymmetrical convention. Mole fraction scale. Reference state: infinite dilution for the solute, pure liquid for the solvent, at T and P of the system. gi,d'=

gy+RTlnx,

g~*=g~+RTlny,“x, g; =g! + RT In y: Asymmetrical convention. Molal scale. Reference state: hypothetical ideal solution of molality rn; for the solute, pure liquid for the solvent, at T and P of the system. g’,“” = gy - RTmJm, g;“” = g: + RT.ln(yFm$/m,) g$” = g! + RT ln(r~m~/m,)

+ RT ln(m,/m;)

M. CattL et al. 1 Fluid Phase Equilibria 96 (1994) 33-50

31

Table 2 Conversion equations for activity coefficients, partial excess Gibbs energies and excess Gibbs energy (J mol-’ unless stated otherwise), partial Gibbs energies and Gibbs energy (J mol-’ unless stated otherwise) Symmetrical

convention.

Mole fraction scale. Reference state: pure liquid for all the species at T and P of the

system. gy=RTlny, gf=RTlny, gE = RT(x, In y, +x2 In y2) g, =g?+RTlny,x, g, = g!: + RT In y2x2 g = x,g? + x2& + RT(x, In ylxl +x2 In y2x2) Asymmetrical convention. Mole fraction scale. Reference. state: infinite dilution for the solute, pure liquid for the solvent, at T and P of the system. Y:=Yl

gf* = RT In y,

g?* = RT ln(y,ly?) gE* = RT[x, ln yl +x2

g:

=g,

d

=g2

Wr2M’ll

g*=g

Asymmetrical convention. Molal scale. Reference state: hypothetical ideal solution of molality m; for the solute, pure liquid for the solvent, at T and P of the system. . YY=ylxI exp(x2/xl)

g:” = RTln y,x, + RT: gz”” = RT My,x, Iy?) gEm = Ram1 In ylxl + m2 + m2 m(y2xl I~91

gf

(J kg-’ of solvent)

=g,

ts=g2 gm

=

m,g’: + m,g: + RT[m, In yl xl + m2 In y2x21

(J kg-’ of solvent)

38

M. Catt& et al. 1 Fluid Phase Equilibria 96 (1994) 33-50

Table 3 Calculation

of the excess enthalpies (J mol-’ unless stated otherwise)

Symmetrical convention. Mole fraction scale. Reference state: pure liquid for all the species at T and P of the system. UNIQUAC modified by Larsen et al. (1987): i

hE=R i

qixi

0, exp[ -a,j(T)/~l[ali,l

- ai,,zp + +,,(T

-

01

j=’

i=I i

flj exp(

-aij(T)/T)

j=l 1 i Asymmetrical convention. Mole fraction scale. Reference state: infinite dilution for the solute, pure liquid for the solvent, at T and P of the system.

Asymmetrical convention. Molal scale. Reference state: hypothetical ideal solution of molality rn; for the solute, pure liquid for the solvent, at T and P of the system. hEm = (m, + m,)hE + RT2m2 ET(ln y?) h Em= (m, + m,)h E*

(J kg-’ of solvent)

(J kg-’ of solvent)

pure liquid reference state is calculated (Table 3). The expressions for excess enthalpies in the asymmetrical conventions are also given in Table 3. These conversion equations (Tables l-3) allow the modified UNIQUAC model (Larsen et al., 1987) to be used on the experimental data for activity coefficients, molar excess Gibbs energy and excess enthalpy given in the asymmetrical conventions (molal and mole fraction scales).

4. Vapour-liquid equilibria As sugars are not present in the vapour phase, the treatment of vapour-liquid equilibria concerns the determination of boiling temperature and vapour pressure of the mixture only. The basic relationship used to determine these properties is P=CPi I

(16)

Considering the vapour phase as ideal, and the Poynting factor equal to 1, Eq. ( 16) is simplified to P = C YiXiPQ

(17)

i

where P$’ is the vapour pressure of pure component written as p=p,=y,x,py

i.

For the systems studied, this equation is (18)

Py is calculated through the international DIPPR correlation (1984). Knowing T, P is easily determined. Knowing P, the boiling temperature is determined iteratively.

39

M. CattC et al. 1 Fluid Phase Equilibria 96 (1994) 33-50

5. Solid-liquid equilibria The treatment solubility.

of solid-liquid

equilibria involves the determination

of freezing point and

5.1. Freezing point Assuming the freezing point depression is suRiciently small that changes in pure liquid water and pure ice heat capacities can be neglected, the relationship between the freezing point depression of an aqueous solution and its water activity is given by (Ferro Fontan and Chirife, 198 1) In a, = In y1x1 (19) where Ah,, is the enthalpy of fusion at freezing point T,,,, (273.15 K) of pure water: Ah,,,, =hyL-hyS=6002

Jmol-’

(20)

AC,, is the difference in the heat capacities of liquid water and ice at Tm, assumed to be independent of temperature in the range Tm, - TmmiX: AC,,, = Cz: - CFy = 38.03 J mol-’ K-’ Equation (19) yields iteratively the freezing point T,,,,iX of carbohydrate-water known composition.

(21) mixtures of

5.2. Solubility The available phase diagrams of carbohydrate-water systems show for some sugars (e.g. glucose and fructose) the existence of a hydrated and an anhydrous solid form with a transition occurring at a precise temperature. Knowing this transition temperature T,, two different equations are used to determine the carbohydrate solubility according to whether the temperature is lower or higher than Tt . Anhydrous solid form If T is higher than T,, or if no hydrated form exists, the carbohydrate crystallizes in an anhydrous form. Solubility is usually determined, as for freezing point depression, using fusion enthalpy. However, very few reliable data on enthalpies of fusion of carbohydrates are available (Table 9). Expressions in Tables l-3 afford a different equation to determine carbohydrate solubility in water at T, using dilution enthalpy:

(22)

40

M. Cattt! et al. / Fluid Phase Equilibria 96 (1994) 33-50

where A&(T,&

is the dilution enthalpy at melting point T,, of carbohydrate:

Ah+ = h,” - h;S

(23)

AC,, is the difference in the carbohydrate heat capacities at infinite dilution in water and in the pure solid state at 298.15 K. AC,, is assumed to be independent of temperature: AC,, = C; - C;;

(24)

As dilution enthalpies of carbohydrates is rewritten as

in water are generally given at P = 298.15 K, Eq. (22)

(25) A similar equation is obtained when AC,, is considered as a linear function of temperature, ACM,(T) = AA + AB(T - To), where AA = AC@,(P) + ABT?

W,*x2) = +

Ah,,(P) R

+ AA - ABP R

AA - ABF R

MT/T,,)

AB (T - T,,) - UGVm2)I + 2~

Importantly, the heat of fusion of carbohydrate can be calculated with knowledge dilution enthalpy by way of the equations listed in Table 3: A&,(T,,)

= Ahd2(L2) + BT:, $

{Wr?(Tm2)1)

(26) of the

(27)

Ahrn,(T,,) = Ahd2(T? + (AA - ABT9(Tm2 - T’?

+AB -yIT:, - V'?'l

+ Wi,

&

PW'(Tm2)1~

From a knowledge solely of the following thermodynamic properties of the studied sugar (dilution enthalpy, melting point, infinite dilution heat capacity and specific heat capacity), its solubility in water can be determined iteratively with Eq, (26). This equation can be used for other solutes besides carbohydrates. When a hydrated form exists this equation can be used for T lower than T,, though the’results do not represent the stable solid-liquid equilibrium, but a metastable anhydrous form. Hydrated solid form If T is lower than T,, the carbohydrate crystallizes in a hydrated form (e.g. glucose. 1H20, fructose.2H20). To determine solubility in this case, an equation based on the same basic principles as Eq. (22) has been established.

41

M. Catte et al. 1 Fluid Phase Equilibria 96 (1994) 33-50

Subscripts 1, 2 and 3 refer respectively to water, sugar and hydrated sugar, and $, represents the number of water molecules in hydrated form. The equilibrium can be represented as follows: SugarL + Q, Hz oL t-) (Sugar?z,H,O) L ++ (Sugar*nhH*O)s The basic equation for solid-liquid

equilibrium is (29)

gS(T) =&V) Two major assumptions are made: (1) The solid phase is composed of pure hydrated sugar only:

(30)

gS(T) = gW) (2) Hydrated sugar concentration anhydrous sugar. Then: g%)

=&(T)

+

in the liquid phase is very low compared with that of the (31)

nhd-(T)

=gz”(T) + RT ln(!&%) +

nhk?“(T)

+

RT

(32)

~n(wdl

Equation (29) can be written as nh

Ink

x1 > +

ln($d

=

As a/aT(gi /T) = -hi/T’,

j&i

k?(T)

-

t??(T)

-

(33)

nhgyL(T)]

Eq. (33) becomes, after integration,

---

+

gP(Tm,)

-

&Wim,)

-

nhdL(Tm3)

RTmJ where AC,,

(34)

and CE are calculated, if not available in the literature, by

AC,, = (Cp”,+ n,.,C;)

- C;;

(35)

cos = cos + n h cos P3 P2 PI

(36)

Also, at T = Tm3, Eq. (33) leads to

gt%n,) - dVm3) - nhdL(Tm3) RG3 In this way, Eq. (34) becomes

1[ +ln

1

~2*Cn3)--nh+

1

1

(37)

42

M. Cattt et al. 1 Fluid Phase Equilibria 96 (1994) 33-50

(39) Knowing some thermodynamic properties of the sugar studied (infinite dilution enthalpy and heat capacity), of its hydrated form (pure solid enthalpy and heat capacity, melting point and n,,) and of water (pure liquid enthalpy and heat capacity), the solubility of this sugar in water can be determined iteratively with Eq. (38) at T below Tt. It must be noted that solubility equations are valid for binary systems only, which is not the case for the other thermodynamic properties studied in this work.

6. Estimation of parameters The modified UNIQUAC method (Larsen et al., 1987) requires knowledge of the molecular volume rk and surface area qk parameters, and ajj,k coefficients which describe the temperature dependency of interaction parameters. The decomposition of sugar molecules into groups and the rk and qi, parameters are given in Table 4. Interaction parameters were estimated from experimental data (water activity, osmotic coefficient, activity coefficient, excess Gibbs energy, excess enthalpy, boiling temperature, freezing point, solubility of anhydrous sugar) on binary water-carbohydrate systems. The data base is given in the Appendix. For each sugar, the parameters are estimated simultaneously from all data by minimizing, by a Gauss Newton method, the following objective function: F = C (Pi Kxp -Pi KAc> *

where subscripts exp and talc refer respectively to experimental and calculated total number of data points and pi is a weighting factor corresponding to property i. The aji,k coefficients are given in Table 5; numbers of data points absolute deviation (MADev) and average percentage error (APErr) per property are given in Table 6.

(40) values, N is the thermodynamic used (n), mean and per system

(41)

(42) The thermodynamic data needed for determination of some properties (solubility, freezing point) are presented in Tables 7 and 8. Some of these, not available in the literature, are estimated simultaneously with interaction parameters. The thermodynamic data needed for solubility of the hydrated form are estimated separately using only data on hydrated form solubility, with aji,k coefficients fixed. It is noteworthy that the melting point of hydrated

M. Cattt? et al. 1 Fluid Phase Equilibria 96 (1994) 33-50 Table 4 Decomposition

Glucose Fructose Sucrose Water

Table 5 Interaction

of carbohydrates

1 2 3

43

into groups and structural parameters CH

C

OH

CHO

4 2 5

0 1 1

5 5 8

1 1 3

rk

qk

8.1528 8.1529 14.5496 0.92

8.102 8.186 14.310 1.4

parameters (first row gives ajj,,, second row ai,,* and third row a&

Hz0 H,G

0 0 0

Glucose

Fructose

Sucrose

26.2775 - 1.4567 - 2.5222

17.4626 - 1.7294 - 2.7504

92.6880 -0.5538 0.5935

Glucose

- 5.6142 1.7631 -0.5151

0 0 0

0 0 0

0 0 0

Fructose

0.8591 2.0314 - 0.4329

0 0 0

0 0 0

0 0 0

Sucrose

- 69.6757 0.5589 -0.7496

0 0 0

0 0 0

0 0 0

carbohydrate and the transition temperature between the hydrated aqueous solution represent two different thermodynamic data.

and anhydrous

forms in

7. Results Figure 1 shows the results obtained for water activity in water-fructose and water-sucrose mixtures. The calculated values are in close agreement with experimental data up to the solubility limit. The results obtained for excess properties gEm and hEm are shown respectively in Figs. 2 and 3. As for water activity, the model results are in close agreement with experiment, but no concentrated data are available for fructose and sucrose solutions. The prediction of boiling temperatures of water-sucrose mixtures is presented in Fig. 4. Interestingly, the concentrated data of Lees and Jackson ( 1973) are not used for the determination of interaction parameters: the boiling temperature is well represented over the whole concentration range.

44

IU. Cattk et al. / Fluid Phase Equilibria 96 (1994) 33-50

Table 6 Identification characteristics: number of data points (first row), mean absolute deviation (second row) and average percentage error (third row) per system and per property Water-glucose

(-)

4

Q,C-1

rf

C-1

Water-fructose

Water-sucrose

22 7.04 x 10-4 7.35 x lo-??

16 4.79 x 10-s 0.556%

17 1.49 x 10-s 0.156%

27 5.83 x 1O-3 0.550%

a

116 1.30 x 10-z 1.147%

Tb W)

23 8.87 x 1O-3 0.721% 46 8.80 x 10-3 0.558%

a

11 5.68 x 10-2 1.51 x lo-2%

1.67 x lo-* 4.47 x lo-3%

G W)

42 6.15 x 1O-3 2.28 x 10-3%

46 5.75 x 10-2 2.15 x 10-2%

44 4.56 x lo-* 1.72 x 1O-2o/,

gEm (J mol-‘)

25 2.54 x lo-* 1.170%

21 4.92 x lo-’ 0.324%

21 4.67 x lo-* 1.378%

hem (J mol-‘)

20 3.56 x 1O-3 0.271%

30 1.09 x 10-2 0.605%

21 1.66 x 10-2 0.496%

8 1.48 x 1O-3 0.192%

34 6.99 x 1O-4 8.63 x lo-*%

19 6.45 x 1O-4 8.90 x lo?/,

Solubility (weight fraction)

28 8.69 x lo-* 2.29 x lo-*%

J

a No reliable data available.

Table 7 Thermodynamic

data used for carbohydrates

Glucose Fructose Sucrose a b c d e

419.1 377.15 461.15

cos

(J mol-‘)

(J’&ol-’ K-‘)

( Jp&ol-’ K-r)

AB d (J mol-’ K-*)

10820 e 10080 ’ 5850 b

347 352 649.4

219 233 427.8

-1.1140 0 1.4230

utd2

Lide (1991). Hinz (1986). Jasra and Ahluwalia (1982). Estimated parameters. Goldberg and Tewari (1989).

C”

b

c

nh

1 2 -

344.30 293.87 -

Tt e (K)

Md,

323.15 298.15 -

20900 = 26594 d -

(J mol-‘)

M. Cattk et al. 1 Fluid Phase Equilibria 96 (1994) 33-50 Table 8 Thermodynamic

45

data used for water a

Ah,,,

COL PI (J mol-’ K-‘)

(J mol-‘) 6002

273.15

75.38 (298.15 K) 76.15 (273.15 K)

cos Pl

(J mol-’ K-l) 38.12 (273.15 K)

a DIPPR Tables (1984).

Table 9 Comparison

between calculated and experimental

fusion enthalpies

Sugar

Calculated fusion enthalpy (J mol-‘)

Literature fusion enthalpy (J mol-‘)

Reference

Glucose

31397

Fructose Sucrose

26030 56946

32429 31400 32429 41076 19169

Raemy and Schweizer (1983) DIPPR Tables (1984) Raemy and Schweizer (1983) Raemy and Schweizer (1983) Kohlrausch (1962)

0.61 0

I SUGM

KIGHT

FRMTION

8

SUGClR UEIGHT FRRCTION

1.6

Fig. 1. Experimental and calculated water activity of water-sugar mixtures at 298.15 K: 0, fructose (Ruegg and Blanc, 1981); A, sucrose (Ruegg and Blanc, 1981). Arrows indicate solubility limits. Fig. 2. Experimental and calculated excess Gibbs energy of water-sugar mixtures at 298.15 K: 0, fructose (Barone et al., 1986); A, sucrose (Barone et al., 1981); 0, glucose (Barone et al., 1981).

M. Cattt! et al. /Fluid Phase Equilibria 96 (1994) 33-50

0

u

SUGfiR UEIGHT FRACTION

5

3701 0

I SUCROSE WEIGHT FRACTION

Fig. 3. Experimental and calculated excess enthalpy of water-sugar mixtures at 298.15 K: 0, glucose (Barone et al., 1981); A, sucrose (Barone et al., 1981); 0, fructose (Barone et al., 1981). Fig. 4. Experimental and calculated boiling temperature of water-sucrose mixtures: 0, Bryselbout and Fabry (1984); 0, Pancoast and Junk (1980); n , International Critical Tables (1926); 0, Lees and Jackson (1973).

Figure 5 shows the accuracy of the model in describing fusion temperatures up to the eutectic point. The solubility results for glucose and fructose are presented in Fig. 6. Below the transition temperature the model is also able to represent the metastable anhydrous form of glucose. Likewise the model is able to describe the solid-liquid equilibrium of water-fructose mixtures below the eutectic point. The fusion enthalpies of carbohydrates were calculated with Eq. (28) and interaction parameters of Table 5. The results are presented in Table 9 and compared with literature data. Table 10 Comparison between calculated and experimental osmotic coefficients for water-sucrose-glucose Sucrose molality

Glucose molality

Experimental value a

Calculated value

Absolute deviation

2.8039

0.5542 1.0996 24489 1.8737 2.9503 3.4947

1.271 1.237 1.165 1.197 1.144 1.120

1.281 1.243 1.164 1.198 1.142 1.119

0.010 0.006 0.001 0.001 0.002 0.001

2.3504 1.2166 1.7391 0.8286 0.3648

a Lilley and Sutton (1991). b IExperimental - Calculated valuei. c IExperimental - Calculated value1 x 1oo Experimental value

b

systems at 298.15 K Relative deviation (%) c 0.787 0.485 0.086 0.083 0.175 0.089

M. Cattt et al. 1 Fluid Phase Equilibria 96 (1994) 33-50 421

3,

,L

DO

SUGAR

HEIGHTFRRCTION

7

I SUGClR WEIGHT FMCTION

2401 0

Fig. 5. Experimental and calculated freezing temperature of water-sugar mixtures: 0, fructose (Young et al., 1952); A, sucrose (International Critical Tables, 1926). The arrow indicates the eutectic point. Fig. 6. Experimental and calculated carbohydrate solubility in water and freezing temperature: and Stephen, 1963, Young et al., 1952); 0, glucose (Stephen and Stephen, 1963).

0, fructose (Stephen

This Table reveals a large discrepancy between the different published values and this justifies our preference for dilution enthalpy for solubility calculations. However, when different authors give approximately the same value, as for glucose, the calculation agrees with the literature results. The melting temperature and decomposition temperature of sucrose are so close that a precise determination of fusion enthalpy is difficult. This could explain the wide differences observed in published values. Our proposed value should be considered as a purely predictive value without possible experimental comparison. Table 11 Comparison

between calculated and experimental

water activities for water-sucrose-glucose

Sucrose molality

Glucose molality

Experimental value a

Calculated value

Absolute deviation

2.8039 2.3504 1.2166 1.7391 0.8286 0.3648

0.5542 1.0996 2.4489 1.8737 2.9503 3.4947

0.9260 0.9260 0.9260 0.9251 0.9251 0.9251

0.9255 0.9257 0.9261 0.9250 0.9252 0.9252

5x 3x 1x 1x 1x 1x

a Lilley and Sutton (1991). b [Experimental - Calculated valuej. c [Experimental - Calculated value1 x 1oo Experimental value

10-4 10-4 10-4 10-4 10-4 10-4

b

systems at 298.15 K Relative deviation (%) c 0.054 0.032 0.011 0.011 0.011 0.011

48

M. CattC et al. 1 Fluid Phase Equilibria 96 (1994) 33-50

The interaction parameters determined from binary data only are used to predict osmotic coefficients and water activities of water-sucrose-glucose systems. Sucrose-glucose interaction parameters were set equal to zero. The results are presented in Tables 10 and 11. The model is able to predict simple thermodynamic properties of ternary systems. 8. Conclusions Water-sugar interaction parameters were determined using the modified UNIQUAC model. This model affords, with high accuracy, thermodynamic properties such as water activity, osmotic coefficient, excess Gibbs energy, excess enthalpy, boiling and freezing temperature for glucose, fructose and sucrose binary aqueous systems. Good predictions of water activity and osmotic coefficients are obtained for ternary systems too. A new equation accurately describes carbohydrate solubility in water over a temperature range including crystallization of anhydrous and hydrated sugar. 9. List of symbols aij a0.k 4 CR g

h mi nh

Pi P pi 4i

Qi

I;; Ri T Tb Tf T T; xi

interaction parameter (K) interaction parameter temperature dependency coefficients water activity heat capacity (J mol-’ K-‘) Gibbs energy (J mol-’ unless stated otherwise) enthalpy (J mol-’ unless stated otherwise) molality (mol kg-’ of solvent) number of water molecules in hydrate sugar form weighting factor of property i pressure (atm) partial pressure of component i (atm) surface area parameter of molecule i surface area parameter of group i volume parameter of molecule i gas constant (J mol-’ K-i) volume parameter of group i temperature (K) boiling temperature (K) freezing temperature (K) melting temperature of component i (K) transition temperature between hydrated and anhydrous carbohydrate mole fraction of component i

9.1. Greek letters YL AA, AB

activity coefficient of component i AC,, temperature dependency coefficients

(K)

M. Cattk et al. 1 Fluid Phase Equilibria 96 (1994) 33-50

44 44 Ah, Ah,,

e, Vp

zij @i wi

49

difference in infinite dilution and pure solid heat capacity (J mol-’ K-‘) difference in pure liquid and pure solid heat capacity (J mol-’ K-‘) dilution enthalpy of component i (J mol-‘) fusion enthalpy of component i (J mall’) surface area fraction of molecule j number of group of type k in molecule i Boltzmann factor osmotic coefficient volume fraction of molecule i

9.2. Subscripts mix 1 2 3

mixture water anhydrous sugar hydrated sugar form

9.3. Superscripts

C E id L m ; S 0 * co

combinatorial part of UNIQUAC model excess property ideal solution property liquid phase asymmetrical convention with molality as composition variable reference residual part of UNIQUAC model solid phase pure component property asymmetrical convention with mole fraction as composition variable infinite dilution property

10. References Abed, Y., Gabas, N., Delia, M.L. and Bounahmidi, T., 1992. Measurement of liquid-solid phase equilibrium in ternary systems of water-sucrose-glucose and water-sucrose-fructose, and predictions with UNIFAC. Fluid Phase Equilibria, 73: 175-184. Abrams, D.S. and Prausnitz, J.M., 1975. Statistical thermodynamics of liquid mixtures: a new expression of the excess Gibbs energy of partly or completely miscible systems. AIChE. J., 21: 116-128. Achard, C., Gros, J.B. and Dussap, C.G., 1992. Prediction de l’activite de l’eau, des temperatures d’tbullition et de congelation de solutions aqueuses de sucres par un modele UNIFAC. Ind. Agric. Aliment., 109: 93- 101. Bondi, A., 1968. Physical Properties of Molecular Crystals, Liquids, and Glasses. Wiley, New York. DIPPR Tables of Physical and Thermodynamic Properties of Pure Compounds, 1984. AIChE, New York. Goldberg, R.N. and Tewari, Y.B., 1989. Thermodynamic and transport properties of carbohydrates and their monophosphates: the pentoses and hexoses. J. Phys. Chem. Ref. Data, 18: 809-880. Hinz H.J., 1986. Thermodynamic Data for Biochemistry and Biotechnology. Springer-Verlag, Berlin/Heidelberg.

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Jasra, R.V. and Ahluwalia, J.C., 1982. Enthalpies of solution, partial molal heat capacities and apparent molal volumes of sugars and polyols in water. J. Solution Chem., 11: 325338. Kikic, I., Fermeglia, M. and Rasmussen, P., 1980. On the combinatorial part of the UNIFAC and UNIQUAC models. Can. J. Chem. Eng., 58: 253-258. Kohlrausch, F. 1962. Praktische Physik, Band 2. B. G. Teubner Verlagsgesellschaft, Stuttgart. Larsen, B.L., Rasmussen, P. and Fredenslund, A., 1987. A modified group-contribution model for prediction of phase equilibria and heats of mixing. Ind. Eng. Chem. Res., 26: 2274-2286. Le Maguer, M., 1992. Physical Chemistry of Foods. In: Schwartzberg, H.G. and Hartel, R.W. (Eds.), Thermodynamics and Vapour-Liquid Equilibria. Marcel Dekker, New York, p. 1. Lees, R. and Jackson, E.B., 1973. Sugar Confectionery and Chocolate Manufacture. Specialized Publication Limited, Surbiton, UK. Lide, D.R., 1991. Handbook of Chemistry and Physics, 72nd edn. The Chemical Rubber Company, Boca Raton, FL, USA. Lilley, T.H. and Sutton, R.L., 1991. The prediction of water activities in multicomponent systems. Adv. Exp. Med. Biol., 302: 291-304. Raemy, A. and Schweizer, T.F., 1983. Thermal behaviour of carbohydrates studied by heat flow calorimetry. J. Therm. Anal., 28: 95-108. Reid, R.C., Prausnitz, J.M. and Poling, B.E., 1986. The Properties of Gases and Liquids, 4th edn. McGraw-Hill, New York.

11. Appendix: Data base Barone, G., Cacace, P., Castronuovo, G. and Elia, V., 1981. Excess enthalpies of aqueous solutions of mono- and oligosaccharides at 25°C. Carbohydr. Res., 91: 101-111. Barone, G., Castronuovo, G., Elia, V., Celotto, D., Santorelli, G. and Savino, V., 1986. Excess Gibbs free energy of aqueous solutions of carbohydrates and polyhydric alcohols at 25°C. Calorimetric Therm. Anal., 17: 464-468. Bonner, O.D. and Breazeale, W.H., 1965. Osmotic and activity coefficients of some nonelectrolytes. J. Chem. Eng. Data, 10: 325-327. Bryselbout, P. and Fabry, Y., 1984. Guide technologique de la confiserie industrielle, Tome 1. Societe d’Edition et de Promotion Agro-alimentaires, Industrielles et Commerciales, Paris. Chuang, L. and Toledo, R.T., 1976. Predicting the water activity of multicomponent systems from water sorption isotherms of individual components. J. Food Sci., 41: 923-927. Ferro Fontan, C. and Chirife, J., 1981. The evaluation of water activity in aqueous solutions from freezing point depression. J. Food Technol., 16: 21-30. Jnternational Critical Tables, 1926. Mc-Graw-Hill, New York. Lerici, C.R., Piva, M. and Dalla Rosa, M., 1983. Water activity and freezing point depression of aqueous solutions and liquid foods. J. Food Sci., 48: 1667- 1669. Miyajima, K., Sawada, M. and Nakagaki, M., 1983. Studies on aqueous solutions of saccharides. I. Activity coefficients of monosaccharides in aqueous solutions at 25°C. Bull. Chem. Sot. Jpn., 56: 1620-1623. Pancoast, H.M. and Junk, W.R., 1980. Handbook of sugars, 2nd edn. AVI Publishing Company Inc., Westport, CT. Robinson, R.A. and Stokes, R.H., 1961. Activity coefficients in aqueous solutions of sucrose, mannitol and their mixtures at 25°C. J. Phys. Chem., 65: 1954-1958. Ruegg, M. and Blanc, B., 1981. The water activity of honey and related sugar solutions. Lebensm. Wiss. Technol., 14: l-6. Stephen, H. and Stephen, T., 1963. Solubilities of Inorganic and Organic Compounds. Pergamon Press, London. Stokes, R.H. and Robinson, R.A., 1966. Interactions in aqueous nonelectrolyte solutions. I. Solute-solvent equilibria. J. Phys. Chem., 70: 2126-2130. Weast, R.C., 1973. Handbook of Chemistry and Physics, 53rd edn. The Chemical Rubber Company, Cleveland, USA. Young, F.E., Jones, F.T. and Lewis, H.J., 1952. d-Fructose-water phase diagram. J. Phys. Chem., 56: 1093-1096.