Water activity in concentrated sucrose solutions and its consequences for the availability of water in the film of syrup surrounding the sugar crystal* Wasseraktivität in konzentrierten Saccharoselösungen und die Auswirkungen auf das Vorhandensein von Wasser in dem den Kristall umgebenden Sirupfilm Maciej Starzak and Mohammed Mathlouthi
A thermodynamically rigorous chemical model of water activity in the sucrose-water system is proposed. The model covers practically the entire composition range including very high sugar concentrations. It employs the concept of the semi-ideal solution according to which all the departures from ideal behavior are attributed to chemical reactions occurring in the system, while physical interactions are neglected. The model accounts for sucrose hydration and clustering as well as water association. All the reactions are assumed to attain chemical equilibrium. Two different mechanisms of sucrose hydration (linear and bridging) are included. The concept of independent binding sites, originally proposed by Schönert (1986), has been extended to describe both the probability of sucrose/water and sucrose/ sucrose interactions. An equilibrium “mixture” model of liquid water, involving the formation of simple water clusters, has been incorporated in order to account for the water/water interactions. Experimental data (vapor pressure, boiling point elevation, equilibrium relative humidity, and heat of dilution) from selected literature sources have been used to validate the model and estimate its parameters. The proposed model can be used as a mathematical tool in solving various practical problems encountered in the sugar and food industry, such as the behavior of crystalline sugar during conditioning and storage, stickiness, re-crystallization of amorphous sugar, its stability in foods and pharmaceuticals. It is postulated that the driving force for migration of water from sugar crystals should be defined in terms of the water activity rather than the nominal content of water in solution. Diffusion processes occurring in the film of syrup surrounding the sugar crystal are briefly outlined. The key role of sucrose hydration and clustering in establishing the driving force of monomeric water in the syrup film is discussed in detail.
Ein thermodynamisches, ausschließlich chemisches Modell der Wasseraktivität im System Saccharose/Wasser wird vorgeschlagen. Das Modell deckt praktisch den gesamten Bereich von Zusammensetzungen – einschließlich sehr hoher Zuckerkonzentrationen – ab. Es nutzt das Konzept der semi-idealen Lösung, laut dem alle Abweichungen vom idealen Verhalten im System auftretenden chemischen Reaktionen zuzurechnen sind, während physikalische Interaktionen vernachlässigt werden. Das Modell erklärt Saccharose-Hydratisierung und -Clusterbildung ebenso wie Wasserbindungen. Bei all diesen Reaktionen wird davon ausgegangen, dass sich das chemische Gleichgewicht einstellt. Zwei verschiedene Mechanismen der Saccharose-Hydratisierung (lineare und Brückenbildung) sind enthalten. Das Konzept unabhängiger Bindungsplätze, ursprünglich vorgeschlagen von Schönert (1986) wird genutzt, um das Vorhandensein von sowohl Saccharose/Wasser- als auch Saccharose/Saccharose-Interaktionen zu beschreiben. Ein Gleichgewichts-Mischungs-Modell von flüssigem Wasser, einschließlich der Bildung einfacher WasserCluster, wurde integriert, um die Wasser/Wasser-Interaktionen zu erklären. Experimentelle Daten (Dampfdruck, Siedepunkterhöhung, Gleichgewichtsfeuchte, Lösungswärme) von ausgewählten Literaturquellen wurden genutzt, um das Modell zu bewerten und seine Parameter zu bestimmen. Das vorgeschlagene Modell kann als mathematisches Werkzeug zur Lösung verschiedener praktischer Probleme in der Zucker- und Lebensmittelindustrie genutzt werden, wie z.B. dem Verhalten kristallinen Zuckers bei der Konditionierung und Lagerung, Klebrigkeit, Rekristallisation amorphen Zuckers und dessen Stabilität in Lebensund Arzneimitteln. Es wird gefordert, dass die treibende Kraft für die Migration des Wassers von den Zuckerkristallen eher in Einheiten der Wasseraktivität als im nominalen Gehalt des Wassers in Lösung definiert werden sollte. Die im Sirupfilm um das Zuckerkristall herum stattfindenden Diffusionsprozesse werden kurz umrissen. Die Schlüsselrolle der Saccharose-Hydratisierung und -Clusterbildung bei der Bestimmung der treibenden Kraft des monomeren Wassers im Sirupfilm wird detailliert diskutiert.
1 Introduction
that is conditions typically encountered in the sugar industry. In this study, an activity model of the sucrose-water system derived from specific interactions between molecules of water and sucrose is presented. The purpose of this model is to reproduce the qualitative behavior of the system at high concentrations, retaining, however, the full predictability of already existing models for this system for dilute solutions. Although the proposed model is general enough to predict activity coefficients of both water and sucrose in solution, the presented analysis and discussion are focused mainly on the water activity.
In a number of unit operations of the sugar industry, such as crystallization, drying, sugar conditioning and storage, the migration of water and diffusion of sucrose within the region between the sugar crystal and the concentrated sucrose solution are process rate-controlling [1, 2]. Simultaneous mass and heat transfer in the film of syrup surrounding the sugar crystal depends on the corresponding concentration and temperature driving forces across the film. Rigorous evaluation of the mass transfer driving forces should be based on the thermodynamic activities of both water and sucrose rather than their nominal contents in solution. It is imperative, therefore, to develop a thermodynamically sound activity model of the sucrose-water system which would be valid for highly concentrated sugar solutions, Zuckerindustrie 127 (2002) Nr. 3, 175–185
* Paper presented at the VIII AvH Symposium, Reims, March 2001
175
2 Water activity in the sucrose/water system 2.1 Fitting experimental data Due to the common availability of sugar and ease of its purification, the system sucrose/water has been a subject of countless physicochemical studies over more than a century. Starzak and Peacock [3] gave a comprehensive review of vapor pressure, boiling point and relative humidity data that could be used to determine the water activity coefficient γw in aqueous solutions of sucrose. Since their publication in 1997, some new data have been generated as well as a few older studies have been re-discovered. Authors updated database contains currently 1536 experimental points. Although, more than 2/3 of this collection involves data for sucrose concentrations above 50% w/w, the highly concentrated solutions (> 90% w/w) constitute only about 5% of all the data, revealing, in addition, a relatively large statistical uncertainty. In general, the water activity coefficient is a function of the mixture composition, temperature and total pressure. However, up to moderate pressures, the pressure effect can be ignored. Since the available literature data include mainly isobaric vapor-liquid equilibrium (VLE) measurements, the corresponding γw values refer to temperatures varying along the boiling point curve. For these conditions, up to about 75% w/w sucrose, the experimental points show a trend characteristic for systems with negative deviations from the Raoult’s law. However, from 75% w/w onwards, where in fact the spread of experimental error is evidently wider, the negative trend flattens and, at about 95–96% w/w sucrose, a minimum value of γw can be identified. There is also a certain indication, albeit based on a very limited number of experimental data, that at this point the negative trend reverses towards positive deviations from the Raoult’s law, leading eventually to γw values well above one. The robust statistical data analysis presented in [3] confirmed these visual observations. As shown in Figure 1, the resulting empirical 4-suffix Margules equation clearly reveals these dramatic changes in γw behavior. It is also interesting to note that some previously published water activity correlations (see [4]), when extrapolated to extreme sucrose concentrations, show a similar pattern (Fig. 2a). In all these cases, however, the predicted behavior should be considered a physically meaningless mathematical artefact rather than reflection of any phenomenological argument. Whether the behavior demonstrated for VLE conditions may also be observed under isothermal conditions cannot be definitely concluded from the available experimental data. At low temperatures, very high concentrations of sucrose in solution are practically not attainable because of the solubility limit. Although maintaining a certain degree of liquid supersaturation was proven possible especially when studying VLE (as a thermodynamically metastable state), some authors question the reproducibility and accuracy of such measure-
1.8 exptl. data Margules eq.
Water activity coefficient γ
W
1.6
1.4
1.2
1
0.8
0.6
0.4 0
10
20
30
40 50 60 Sucrose, % w/w
70
80
90
100
Fig. 1: Water activity coefficient at VLE conditions (760 mm Hg) as a function of sucrose content: comparison of values predicted by the Margules equation [3] with experimental data 176
ments because of possible sugar crystallization during the experiment. As a matter of fact, the empirical Margules equation mentioned earlier [3] does predict the occurrence of γw minimum at isothermal conditions as well. This is, however, merely due to the simplified temperature dependence and the type of polynomial used for data regression, and by no means can be used as a firm experimental evidence. 2.2 Molecular interactions in aqueous solutions of sucrose Physicochemical properties of aqueous solutions of sucrose are heavily affected by the specific chemical nature of the two types of molecules originally involved. Hydrogen bonds formed by water molecules make pure liquid water a highly self-associating compound as well as a solvating (hydrating) agent. Although the long and obstinate dispute between the adherents of the mixture and continuum model of liquid water has not been resolved, according to many investigators [5–8], bulk water can be seen as an equilibrium mixture of small clusters of single water molecules (monomers, dimers, trimers, etc.). Such a physical picture seems to be supported by both spectral data [9–11] (esp. Raman spectra in the OH stretching vibration region) as well as recent results of molecular dynamics simulation [12, 13]. Similarly, the sucrose molecule, with its 8 hydroxyl groups, 3 hydrophillic oxygen atoms and 14 hydrogen atoms, can readily interact through hydrogen bonding with water as well as other molecules of sucrose. However, not all the eleven hydrogen-binding sites on a sucrose molecule exhibit the same affinity for hydrogen. In general, the fructosyl moiety of sucrose seems preponderant in establishing intermolecular H-bonds. It is well known that the most reactive hydroxyl groups on a sucrose molecule are Of(6)-H, Of(1)-H, and Og(6)-H [14, 15], whereas the three acetalic (non-hydroxyl) oxygens are engaged to a lesser extent as H-bond acceptors (often not considered as potential H-bond sites at all [2]). Furthermore, recent molecular dynamics simulation studies [16, 17], although involving only very dilute solutions of sucrose (about 3% w/w), have shown the presence of a stabilized water bridge between glucose and fructose residues of the sucrose molecule (Og(2)···H2O···H-Of(1)). This water molecule seems to have a relatively long lifetime and might play a key role in sucrose crystallization as it is the most strongly bound and the most difficult to remove. As such, it could be the last hydration water molecule to dissociate prior to sucrose incorporation in the crystal. It is very likely that the broad supersaturation range prevailing before nucleation occurs is caused by difficulties in expelling the water from the interresidue water bridge [16]. In dilute solutions the association of sucrose molecules is highly unlikely. However, with the increasing sugar concentration, especially when entering the region of liquid supersaturation, the probability of sucrose-sucrose collisions increases dramatically. Kelly and Mak [18] postulated a hypothetical sucrose hexamer as the simplest theoretically possible representation of the embryonic crystal of sucrose at the onset of crystallization. VanHook [19] estimated that at 90 °C more than 20% of the total sucrose is involved in the formation of sucrose hexamers. A recent study by van Drunen [20] using the technique of photon correlation spectroscopy, shows the existence of strong attractive forces between the sucrose molecules at low supersaturation as well as in slightly unsaturated solutions. Even below the saturation concentration, clusters of 2–3 molecules develop, apparently, much smaller than the critical cluster size of crystalline sucrose nucleus (approx. 80–100 sucrose units [2]). Once the solution contains simple sucrose hydrates and small clusters of both water and sucrose, the formation of larger heteromolecular aggregates such as partly hydrated polymers of sucrose becomes possible. These may result from interactions between monomers and the existing polymerized forms, as well as between clusters of different size and molecular composition. Culp [21] postulated a structural model of sucrose solutions at higher concentrations in which four of the eight hydroxyl groups in the sucrose molecule are tied up pairwise by hydrogen bonds, each pair through a single “internal” molecule of water situated in between. The remaining four hydroxyl groups are bonded individually by “external” water molecules. Most Zuckerindustrie 127 (2002) Nr. 3, 175–185
The chemical and the physical theories of solutions are extreme. More recent developments involve models compromising the two limiting cases by including both physical and 1.6 1 chemical forces. One of the most successful attempts is the Extended Real Associated So1.4 lution Model (ERAS) [24]. It has been ap0.8 plied extensively over the last decade to model relatively simple systems involving 1.2 highly associating substances (e.g. acetic acid, various alcohols). 0.6 Both the physical and the chemical ap1 proaches have been used in the past to describe the activity of water in sucrose solu0.4 tions. The physical models are exemplified 0.8 by studies employing the UNIQUAC method [25–28], ASOG method [29] and some varia0.2 tions of the basic UNIFAC method [30–33]. 0.6 Typically, the physical models give fairly Abed et al. [30], ASOG Correa et al. [29], UNIFAC good predictions of the water activity coeffiLeMaguer [27], UNIQUAC cient γw up to moderate sucrose concentra0.4 0 40 50 60 70 80 90 100 20 40 60 80 100 tions and fail for highly concentrated soluSucrose, % w/w Sucrose, % w/w tions. Their performance at VLE conditions is presented in Figure 2b. A more detailed Fig. 2a (left): A minimum of water activity coefficient on the boiling curve (760 mm Hg) as discussion of these results can be found in predicted by selected empirical equations [4]. Fig. 2b (right): Water activity coefficient (VLE, 760 mm Hg) as predicted by selected physThe first and, in fact, fairly complex chemiical models cal model of the sucrose-water system was proposed by Scatchard [34]. He considered a mechanism of sucrose hydration in which either hexa- or heptahydrate was formed in a single-step reaction. This of the “external” water molecules are shared by two hydroxyl groups has also been the only known model which takes into account the on adjacent sucrose molecules. These, however, do not adhere very effect of water association (formation of water dimers was assumed). tightly, so the solution resembles a “soup” of hydrated sucrose clusA natural extension of the Scatchard model was the model develters of varying sizes. According to Culp [21], such a structure affects oped by Stokes and Robinson [35] in which rather unrealistic singlethe ability of sucrose to diffuse. When the number of water molestep mechanism of sucrose hydration was replaced by a sequence of cules is just sufficient to bind together all of the pairs of hydroxyl stepwise equilibria. The maximum number of binding sites available groups, sucrose loses its mobility and diffusion is suppressed. Any on a single molecule of sucrose was assumed to be equal to 11, the excess water breaks the bonding pattern and permits the diffusion of total number of oxygen atoms. For the sake of simplicity, all hydrasucrose. tion equilibrium constants were assumed to be identical. As a result In consequence, the aqueous solution of sucrose may contain a of that last assumption, the model predicts unrealistically that nonhywhole variety of true chemical species differing in size, shape and drated (free) sucrose is the most abundant sucrose-containing species molecular content. In general, the presence and actual quantities of in solution while the largest hydrate (with 11 molecules of water) is these species in solution depend on the nominal concentration of suthe scarcest species. Moreover, this general trend is predicted for crose, temperature and pressure. both dilute and concentrated solutions (see Fig. 3). 2.3 Models of water activity Schönert [36, 37] postulated the concept of independent binding sites. According to this concept, all hydrogen bonding sites show the In ideal (non-interacting) systems, activities of the species making same affinity to hydrogen. However, the probability of hydration/deup the solution are represented by their mole fractions. Therefore, in hydration of a given sucrose molecule depends on the number of the symmetric convention, the corresponding activity coefficients sites already occupied by water molecules. The Schönert model preare all equal to one. On the other hand, systems involving interacting dicts the distribution of hydrates exhibiting a characteristic maximolecules are thermodynamically nonideal. Their nonideal character mum. The location of this maximum depends on the nominal sucrose is caused by both physical and chemical interactions between molecontent. For example, as shown in Figure 3, sucrose pentahydrate is cules, in other words by weak and strong forces, respectively. the peak sucrose component at 40% w/w, whereas at 90% w/w Roughly speaking, weak forces between molecules are those that mono- and dihydrate predominate. This seems to be in line with the have energy less than RT (~2.5 kJ/mol at room temperature), while opinion that the dehydration of sucrose monohydrate is the last stage strong forces are those whose energies are considerably larger than which precedes the appearance of sucrose seeds and their developRT [22]. In general, both types of forces contribute towards the obment in the process of crystallization. served non-ideality of solution. In certain limiting cases, however, Finally, VanHook [19] proposed a simple model which accounts for one type predominates over the other. In the so-called “physical” sucrose-sucrose interactions. The model allows for the formation of models of activity, only weak forces between molecules are considsucrose hexahydrate as well as nonhydrated hexameric clusters of ered. As a result, no new stable chemical species are formed in solusucrose. Although VanHook’s model exhibits some key features of tion, so that the true species are the same as the apparent species. chemical models, it is not detailed enough to be studied in the rigorMany modern physical models of activity originate from the Gugous terms of chemical equilibrium. A semi-empirical version of this genheim quasi-crystalline lattice theory of liquids which accounts for model enabling γw prediction at high sucrose contents was discussed potential energy of the weak London dispersion forces as well as the in [38]. relative sizes of molecules and their shapes [23]. Contrarily, in the Figure 4 compares the water activity coefficient γw (at isobaric VLE “chemical” models of activity, forces between molecules are asconditions) predicted by the above four chemical models and the sumed to be strong. They result in the formation of chemical bonds empirical Margules equation. Clearly, none of the discussed models and constitution of new chemical species (hydrogen bonds, charge was able to reproduce the characteristic minimum occurring at about transfer complexes, etc.). 1.2
1.8
Zuckerindustrie 127 (2002) Nr. 3, 175–185
A
Water activity coefficient γ
Water activity coefficient γA
Bates [55] Crapiste & Lozano [56] Kadlec et al. [57, 58] Lyle [59]
177
Fig. 3: Distribution of sucrose-containing species at 25 °C for three different nominal concentrations of sucrose – a comparison of the Stokes-Robinson model [35] with the Schönert model [36, 37]
95% w/w sucrose. Only the van Hook model in its modified version [38], produces roughly a similar pattern. However, from the quantitative point of view, the predictions are rather poor. The shallow minimum produced by the Scatchard model is caused by the water dimerization effect rather than sucrose clustering since the latter was not considered. The other models fail completely in the region of high sucrose concentration. Despite its partial failure, VanHook’s model reveals the key role of the sucrose association mechanism on the water activity behavior in highly concentrated solutions. The incorporation of this mechanism remarkably improves the model performance beyond 80% w/w su-
crose. The water activity coefficient resulting from the VanHook model shows a minimum at about 85% w/w sucrose and then starts to increase, rapidly exceeding the value of one (Fig. 4). The predicted effect has a simple physical interpretation. Sucrose clustering reduces the number of free sucrose molecules in the solution and at the same time lowers the total number of molecules in the system. As a result, at a sufficiently high degree of sucrose clustering (high sucrose content), the mole fraction of free water becomes higher than the nominal mole fraction of water resulting from the original quantity used to make up the solution. This is, in fact, what makes the observed water activity coefficient greater than one. One must bear in mind, however, that VanHook’s model gives a very simplistic picture of the chemical transformations taking place in the real solution. Thus, the incorporation of more realistic mechanisms of water association, sucrose hydration and sucrose clustering should result in a highly predictable chemical model of the sucrosewater system valid in the entire range of sucrose concentrations. An attempt at developing such a model will be presented below.
3 Model development 3.1 Activity coefficients in associated/solvated systems The associated/solvated mixtures are examples of thermodynamically nonideal systems. In particular, in the “ideal” associated/solvated solutions [39], where all the effects of nonideal behavior are attributed exclusively to chemical interactions between molecules, the activities are given by equilibrium mole fractions of the individual species formed in solution as a result of molecular association/solvation. However, a special precaution is needed when evaluating the activity coefficients in these solutions. The thermodynamically consistent activity coefficient can be derived by considering the chemical potential on both the macro- and microscopic level. The macroscopic chemical potential of component i in an associated/solvated solution is:
µ i = µΘ i + RT ln( γ i xi )
(1)
µ i(1) = µΘ i(1) + RT ln xi(1)
(2)
where µΘ i is the chemical potential of pure component i in a state of association determined by temperature and pressure only, γi is the macroscopic (observed) activity coefficient, and xi is the nominal (macroscopic) mole fraction of i in solution. On the other hand, the microscopic chemical potential of the monomer molecules of component i in solution is: µΘ i(1)
Fig. 4: Water activity coefficient in the region of concentrated solutions (at 25 °C) as predicted from the Margules equation [3], and the models of Scatchard [34], Stokes and Robinson [35], Schönert [36, 37] and VanHook [19, 38] 178
where is the chemical potential of pure monomeric (non-associated) form of component i, and xi(1) is the true (microscopic) equilibrium mole fraction of monomers of i in solution. Note that ignoring physical interactions makes the activity coefficient of monomer equal to one. Prigogine and Defay [53, 54] showed for mixtures with associating/solvating molecules that at the conditions of chemical equiZuckerindustrie 127 (2002) Nr. 3, 175–185
librium the two chemical potentials, µi and µi(1), must be equal: µi = µi(1) (3) Hence, by comparing eqs.(1) and (2): Θ µΘ i – µ i(1) = RT ln
xi(1)
(4)
γ i xi
If xi → 1 , i.e. when solution approaches pure i in a certain state of self-association, the true mole fraction of monomers xi(1) tends to a * certain limit xi(1) , the mole fraction of monomers in the pure fluid i (fixed at a given temperature and pressure), while the macroscopic activity coefficient γi tends to unity. Written for these conditions, eq.(4) yields: Θ * µΘ i − µ i(1) = RT ln xi(1)
(5)
Combining then eq.(4) with (5) results in the effective formula for the calculation of the macroscopic activity coefficient γi:
γi =
xi(1) (6)
* xi(1) xi
In fact, this is exactly how γi was calculated in the original Scatchard’s paper [34], although his arguments were more intuitive than thermodynamically rigorous. Eq.(6) shows clearly that in order to evaluate γi for the sucrose-water system, the problem of chemical equilibrium has to be solved independently for two different systems (at the same p-T conditions): 1. the given solution (sucrose in water), in order to determine xi(1) * 2. pure solvent (liquid water), in order to determine xi(1) Note that eq.(6) predicts a water activity coefficient of 1 for pure liquid water as required, regardless of temperature and pressure. 3.2 Molecular interactions In the proposed chemical model of activity in the sucrose-water system, three different types of molecular interactions are assumed to occur. These are: water association, sucrose hydration and sucrose clustering (association). Hydrogen bonding is the reaction mechanism common to all these processes. The following system of instantaneous reversible reactions was chosen to model the resulting chemical equilibrium: KAi Wi + W ← → Wi+1 ,
i =1 , 2 , 3 , 4
KH,ij SiWj–1 + W ← → SiWj, KCi Si + S ← → Si+1 ,
i =1, 2 ,..., m; j = 1, 2,..., ni
i = 1, 2,..., m – 1
(7) (8)
(10)
where n is the maximum available number of hydrogen-bond sites on a single molecule of sucrose, and α is the average number of HZuckerindustrie 127 (2002) Nr. 3, 175–185
3.3 Chemical equilibrium The state of chemical equilibrium can be mathematically identified with the criterion that at equilibrium in a closed system at constant T and p, the Gibbs free energy G is a minimum [43]. Consequently, the resulting so-called mass action law is given in terms of activities rather than concentrations. However, since the proposed chemical model ignores all physical interactions, activities of all true species involved (symbols in brackets used below) can be replaced by the corresponding mole fractions. 3.3.1
Water association equilibrium
K Ai =
x [Wi+1 ] = W,i+1 , i = 1, 2, 3, 4 [Wi ][W ] xWi xW
(11)
The equilibrium constants of water association KAi at 25 °C together with the corresponding reaction enthalpies ∆HAi are given in Table 1. The water association model was derived from the laser-Raman spectra of liquid water in the OH stretching vibration region [10, 40] enhanced by the most recent results of molecular dynamics simulation [12, 13].
(9)
where m is the maximum admissible number of sucrose molecules in a single aggregate containing sucrose, and ni is the maximum number of hydrogen-bond sites on a single molecule of i-th aggregate (composed of i molecules of sucrose). A few additional assumptions must be made in order to evaluate the maximum association and hydration numbers required by the model. Firstly, following the conceptual scenario of water association derived by Luu et al. [10, 40] from the analysis of Raman spectra of liquid water at various temperatures, pentamer was assumed to be the largest possible water cluster. The determination of ni, the maximum hydration number for a given aggregate, is more problematic. This number seems to depend on the number of sucrose units present in the aggregate. Basically, a single sucrose/sucrose association utilizes only one hydrogen-bond site which is located on one of the two sucrose units involved. However, with the growing size of a sucrose/water aggregate, one can expect the appearance of steric effects making some Hbond sites inaccessible for further molecular aggregation [41]. In order to make a simple provision for the steric effects, the following self-explanatory formula for ni is proposed: ni = n · i – α (i – 1)
bonds utilized (directly involved or made inaccessible because of steric effects) per one sucrose/sucrose attachment. No distinction is made between hydrates possessing the same number of attached water molecules but exhibiting different isomeric configurations. Generally, the numbers n, m and α can be considered adjustable model parameters. For the purpose of this study, however, some rational estimates have been made. In particular, the maximum hydration number for a single sucrose molecule n = 8 is taken, as this is the number of hydroxyl oxygen in a single molecule of sucrose. The maximum sucrose clustering number is chosen to be m = 6 , as proposed by Kelly and Mak [18] for a sucrose embryo (produced prior to the formation of a crystal nuclei). The average number of H-binding sites engaged to produce a single sucrose-sucrose association α = 2 is assumed. It is worth noting that the reaction network given by Equations (7–9) is not the only theoretically possible scheme which yields the required variety of clusters and hydrates. One can easily imagine different modes of association and hydration resulting in the same products. A proof demonstrating that more general reaction schemes can always be derived from the scheme adopted above is presented in [42].
Table 1: Water association equilibrium parameters i
Reaction step
1 2 3 4
H2O + H2O = (H2O)2 (H2O)2 + H2O = (H2O)3 (H2O)3 + H2O = (H2O)4 (H2O)4 + H2O = (H2O)5
3.3.2 KH,ij =
Equilibrium constant at 25 °C, K°Ai
Enthalpy of reaction ∆HAi in J/mol
320.1 1,882.2 61.38 5.93
–22,200 –48,100 –40,200 –31,000
Sucrose hydration equilibrium [SiWj ] [SiWj–1 ][W ]
=
xij xi,j–1 xW
, i = 1, 2,..., m; j = 1, 2,..., ni (12)
For the sake of simplicity, following Schönert’s concept of independent binding sites [36, 37], it is assumed that all the eleven hydrogen-binding sites on a sucrose molecule exhibit the same affinity for hydrogen. However, ordinary (linear) hydration and hydration involving a water bridge are treated differently by assigning to each a different equilibrium constant and enthalpy of hydration. In addition, it was assumed that: 179
– each molecule of sucrose may possess only one interresidual water bridge; – two H-binding sites are needed to establish a single water bridge; only one of them is provided by the sucrose molecule, the other one comes from water; – bridging-type hydration has absolute priority; consequently, the formation of sucrose monohydrate always occurs via the bridging mechanism; similarly, a sucrose dimer, when exposed to hydration, binds two bridging waters first (each associated with a different molecule of sucrose) and all subsequent hydrations are ordinary (linear). Based on these assumptions, the following calculation procedure has been developed to determine the hydration equilibrium constants for sucrose/water aggregates. First of all, it was accepted after Schönert [36, 37] that each equilibrium constant can be expressed as the product of two terms: KH,ij = pij · KH
(13)
where pij is a temperature independent term being the ratio of hydration and dehydration frequency factors for a given sucrose-water aggregate, and KH is a temperature dependent constant satisfying the van’t Hoff equation. Since the model is to predict VLE in the entire range of sucrose concentrations and consequently, a large temperature variation is expected (from 0 °C to 170 °C), the molar enthalpy of hydration ∆HH has to be considered a function of temperature. Assuming a constant heat-capacity increase attending hydration, ∆CpH, leads to the following integrated form of the van’t Hoff equation: ∆H H° – ∆CpHTo 1 1 ∆CpH T ° KH = KH exp ( – )+ ⋅ ln R To T R To
[total bridging sites in SiWj–1 ] – [occupied bridging sites in SiWj–1 ] [occupied bridging sites in SiWj ]
Sucrose clustering equilibrium
KCi =
x [Si+1 ] = i+1,0 , i = 1, 2,..., m – 1 [Si ][S] xi,0 xS
where xS ≡ x1,0 is the mole fraction of free sucrose (monomer). The sucrose clustering equilibrium constant KCi can be viewed also as a product of two factors: KCi = qi · KC
(19)
where qi is a temperature independent term representing the ratio of association and dissociation frequency factors for a given cluster, while KC depends on temperature only. Since the enthalpy of sucrose clustering, ∆HC, is assumed to be constant, the integrated van’t Hoff equation takes the form:
∆H 1 1 KC = KC° exp C – R To T
(20)
By analogy to hydration, it is assumed that the association rate constant is proportional to the number of vacant hydrogen-bond sites on the cluster Si, whereas the dissociation rate constant is proportional to the number of the existing sucrose-sucrose hydrogen bonds in the cluster Si+1. Consequently, the frequency ratio qi is:
qi =
[total H -binding sites in Si ] – [sucrose-sucrose H -bonds in Si ] n ni – α(i –1)) = i = α ⋅i [sucrose-sucrose H -bonds in Si +1 ] α⋅i
(14)
=
3.4 Equilibrium composition of solution The method of calculation of the equilibrium composition in the assumed multi-reaction system is based on the equilibrium relationships (11), (12) and (18), in addition to the nominal material balance equations for water and sucrose which will be derived below. Eventually, this mathematical description results in a system of two algebraic equations that can be solved for xW and xS, the true equilibrium mole fractions of free water and sucrose, respectively, in solution. The nominal total number of moles of water (i.e. the amount of water ° which is needed to make up the solution), N W , is equal to the number of moles present in all multimeric forms of water (including monomer) plus the number of moles attached to sucrose: m
5
∑
° Nw =(
i xWi +
i=1
i – ( j – 1) j
ni
∑ ∑ j x )N ij
t
(22)
i=1 j=1
where Nt is the total number of moles in the system at equilibrium. In turn, the total nominal number of moles of sucrose, NS° , is equal to the number of moles of sucrose present in all different kinds of hydrates and clusters (including free sucrose):
(15)
m
ni
∑ ∑ i x )N
NS° = (
Note that eq.(15) applies only if the number of water molecules in the aggregate does not exceed that of sucrose molecules, that is for i = 1,..., m and j = 1,..., i. In the case of ordinary (linear) hydration, pij is calculated according to: pij =
(18)
(21)
° and ° refer to the reference temperature T = 25 °C. where KH ∆HH o ° were assigned to ordinary and bridging hyDifferent values of KH ° ° dration (KHo and KHb , respectively). Furthermore, since the bridging hydration always engages two hydrogen bonds, it was assumed that the enthalpy of bridging hydration was twice as high as that for ordinary hydration, i.e. ∆HHb(T) = 2 ∆HHo(T). The way in which the frequency ratio pij is evaluated depends on the mechanism of hydration considered. In the case of bridging hydration, it is assumed that the rate constant (probability) of hydration reaction is proportional to the number of vacant bridging sites on SiWj–1 aggregate, whereas the rate constant (probability) of the reverse reaction is proportional to the number of already occupied bridging sites on SiWj aggregate. Since the hydration equilibrium constant KH,ij is the ratio of these two rate constants, the term pHij is defined as follows:
pij =
3.3.3
ij
t
(23)
i=1 j=0
[total H -binding sites in SiWj–1 ] – [sucrose-sucrose H -bonds in SiWj–1 ] – [total bridging sites in SiWj–1 ] – [ordinary sucrose-water H -bonds in SiWj–1 ] [ordinary sucrose-water H-bonds in SiWj–1 ] (16) The combination of Equations (22) and (23) yields:
which leads to: pij =
ni – i – [( j – 1) – i] j–i
5
(17)
Eq.(17) applies, in turn, to situations when all the bridging sites are occupied, i.e. if the number of water molecules in the aggregate exceeds that of sucrose (i = 1,..., m; j = i+1,..., ni).
xW = xS
∑
m
i xWi +
i =1
m
ni
∑∑ j x
i =1 j =1 ni
∑∑i x
ij
(24)
ij
i =1 j = 0
where xW and xS denote the nominal (apparent) mole fractions of water and sucrose, respectively, in solution. In addition, mole frac180
Zuckerindustrie 127 (2002) Nr. 3, 175–185
tions of all the species involved are subject to the obvious constraint: m
5
∑
xWi +
i=1
ni
∑∑ x
ij
=1
(25)
i=1 j=0
It is worth noting that the system of equilibrium relationships, given by eqs.(11), (12) and (18), can be solved for all the unknown equilibrium concentrations xWi and xij, solely in terms of xW and xS. However, due to eqs.(24) and (25), the latter can be found as well. The prediction of the equilibrium composition must therefore begin with finding expressions for the mole fractions of associated species. Because of its algebraic complexity, the corresponding analytical solution is not given here, however, its complete form is presented in [42].
reported for O-H···O bonds in the liquid phase [46]. However, the hydration enthalpy was found to depend strongly on temperature. For example, its predicted value at 150 °C is about –25 kJ/mol. The predicted hydration heat capacity change ∆CpH is unrealistically high and, at this stage of the model development, should be considered merely an adjustable parameter. The enthalpy of sucrose clustering, ∆HC, is within the expected range of values (–13.4 kJ/mol). Figure 5a shows a comparison of the predicted and experimental values of the water activity coefficient γW at VLE conditions. Despite the evident presence of some outliers, the data fit is acceptable. One has to remember that the experimental data at high sucrose concen-
3.4 Equilibrium composition of pure solvent
1.5
Predicted γW
According to eq.(6), the composition of pure liquid water at equilibrium is required for the prediction of the water activity coefficient in solution. If the pressure effect can be ignored, this composition depends on temperature only. The mole fraction of water monomer in pure liquid water can be determined by solving a polynomial resulting from eq.(11). More details can be found in [42].
0.5
4 Estimation of model parameters The proposed model of chemical equilibrium in the sucrose/water system involves two kinds of adjustable parameters: those which may change in a discrete way only (n, m, and α) and those which vary continuously (equilibrium constants and reaction enthalpies). For the purpose of this study, the set of discrete parameters was fixed arbitrarily (n = 8, m = 6 and α = 2). In turn, the set of continuous parameters was optimized numerically using a combination of the conventional simplex and gradient method of nonlinear regression. ° , ° The optimized parameters were: KHo and ∆CpH (ordinary ∆HHo ° hydration), KHb (bridging hydration), KC° and ∆HC (sucrose clustering). The least-squares performance index included selected water activity coefficient data (at VLE conditions ranging from 0 to 170 °C) [3] as well as values of the partial molar excess enthalpy of water E at 25 °C. The enthalpy data were generated from a truncated HW form of the McMillan-Mayer expansion obtained by regressing experimental heat of dilution data [44], as reported by Barone et al. [45]: =
hxx m•2
+ hxxx m•3
= 134.6m•2
–
7.05m•3
0.5
E HW =
Em 4.184(Hasym – m•
∂m•
in cal/kgH2O (26)
1 0
2 0
3 0
4 0
5 0
Gucker et al. [44]
6 0 0
10
) (27)
1000 / MW
E from the proposed model, the temperature In order to predict HW dependence of the water activity coefficient was used (as predicted by the model):
∂ ln γ W (28) ∂T Because of the tedious algebra of the model equations, the temperature derivative in eq.(28) was evaluated numerically rather than analytically. The results of parameter estimation have been summarized in Table ° value of about –34 kJ/mol (refer2. It must be noted that the ∆HHo ring to a single step of ordinary sucrose hydration at 25 °C) does not compare well with the typical energy values of 15–25 kJ/mol-H bond E HW = – RT 2
Zuckerindustrie 127 (2002) Nr. 3, 175–185
1.5
0
Em where Hasym is the molal excess enthalpy of solution in asymmetric convention, and m• is the molality of solution. The required experiE mental values of the partial molar excess enthalpy of water HW have been obtained from the available experimental data as follows (note that for the solvent, partial molar excess enthalpy is identical for both symmetrical and asymmetrical convention):
Em ∂Hasym
1
Experimental γW
10
Partial molar excess enthalpy of water, J/mol
Em Hasym
1
20 30 40 D.S., % w/w sucrose
50
Fig. 5a: Results of data regression for the water activity coefficient Fig. 5b: Results of data regression for the partial molar excess enthalpy of water
Table 2: Estimated parameters of the model Reaction
Equilibrium parameter at 25°C
Enthalpy of reaction in J/mol
Ordinary (linear) sucrose hydration
K°Ho
95.60
– 33,780 78.2
Bridging sucrose hydration Sucrose clustering
∆H°Ho ∆CpH *
K°Hb K°C
6,784 9.34 × 10–5
– ∆HC
– – 13,400
* Heat-capacity change attending sucrose hydration, J/(mol×K)
181
1.2
Water activity coefficient
1.1 1 0.9 0.8 proposed model
0.7
Margules equation [3] 0.6 0.5
0
20
40 60 D.S., % w/w sucrose
80
100
Fig. 6: Water activity coefficient as a function of the nominal sucrose content (VLE at 760 mm Hg) – model predictions compared with the empirical curve 10
100
9
90
7 6 5 4 3 2 1 0 0
S=5 1
S=2 150˚C
0.8
S=1
0.6
100˚C 0.4
10˚C
0.2
0 0
50˚C
0.2
0.6 0.4 Mole fraction sucrose
0.8
1
Fig. 8: Isotherms of the water activity coefficient – model predictions
Figure 8 shows a simulation of the isothermal γW for temperatures ranging from 10 to 150 °C. Up to about 100 °C, none of the isotherms exhibit a minimum. Apparently, the observed minimum on the VLE γW curve is due to the sucrose clustering effect being less exothermic and, as such, more resistant to the disintegrating role of temperature than the hydration effect. Furthermore, the intercepts of γW isotherms with the saturation line S = 1 show that the major portion of the isotherms is located within the region of supersaturation of the solution with sucrose. It is worth remembering that practically no experimental data are available in this region. Therefore, the predictions for the supersaturation region should be considered a modelbased extrapolation beyond VLE conditions. Note that the VLE conditions typically correspond either to nonsaturated solutions or, for highly concentrated solutions, to a slight supersaturation in the close proximity of the saturation line. At the present stage of model development, in particular with the numerical values of the equilibrium parameters given in Table 2, the model predicts a relatively low degree of sucrose association. Even for very concentrated sugar solutions, it is not higher than a few percent, typically in the form of sucrose dimers and trimers. This is apparently sufficient to explain the behavior of the water activity coefficient in solution, as well as to reconcile the heat of dilution data, but may not be enough to describe the onset of sucrose crystallization. Therefore, further investigation into the sucrose activity coefficient under conditions of crystallization is necessary for the model refinement.
5 Availability of water in the film of syrup surrounding the sugar crystal
80
Studies on the cluster composition of saturated water vapor, both experimental [47] and theoretical [48, 49], show that at p-T conditions normally encountered in the sugar industry, the vapor is composed almost exclusively of water monomers. This implies that water clusters are practically non-volatile and the process of water migration during sugar drying and conditioning is driven by the difference of concentrations of water monomer between the liquid and gas phase. In other words, the driving force used to calculate the rate of water migration should be defined in terms of the water activity rather than the nominal water content.
Margules eq. [3]
70 60 50 40 30 20 10
20 40 60 D.S., % w/w sucrose
80
0 80
5.1 Forms of water in sugar drying and conditioning 85 90 95 D.S., % w/w sucrose
100
Fig. 7: Boiling point elevation (at 760 mm Hg) as a function of the nominal sucrose content – model predictions compared with the empirical curve 182
Dashed lines represent the conditions of a constant supersaturation ratio S
Proposed model
Margules eq. [3]
Boiling point elevation [ ˚C ]
Boiling point elevation [ ˚C ]
Proposed model 8
1.2
Water activity coefficient
trations show large discrepancies between various literature sources. This is caused by the fact that the accurate determination of the equilibrium composition is difficult due to possible crystallisation effects during the experiment (note that for highly concentrated solutions the γW curve becomes extremely sensitive to the composition). A similar comparison for the partial molar excess enthalpy of water E shown in Figure 5b, is also satisfactory, although, since the exHW perimental data are much more reliable in this case, the obtained deviations of 10–15% may be considered too large. It is also likely that the global minimum of the performance index used in parameter estimation has not been found yet, and a further refinement of the results may still be possible. Figure 6 shows the results of γW simulations for the conditions of isobaric VLE. The results were compared with the empirical Margules equation [3]. Although the two curves do not match perfectly, the general trends seem to be properly reproduced. This proves that it is the implementation of the sucrose clustering mechanism in the thermodynamic model of the sucrose-water system which makes the model predict well in the region of high sucrose concentration. The model was also used to predict the boiling point elevation (BPE) as a function of the nominal mixture composition. The results of the calculation are presented in Figure 7 and have been compared with predictions based on the empirical γW correlation. As was expected, the largest deviations were observed in the same range where the γW simulation slightly failed, i.e. at 50–70% w/w. However, above 80% w/w the two predictions are surprisingly consistent.
The process of diffusive water migration between moist sugar and drying air is accompanied by simultaneous heat transfer. This makes driving forces for mass and heat transfer strongly coupled. However, the key factor complicating the whole process of water removal is the intricate macroscopic nature of the liquid phase region adjacent Zuckerindustrie 127 (2002) Nr. 3, 175–185
to the sugar crystal. The morphology of this region is variable and depends on the conditions of sugar crystallization and more importantly sugar drying. Many authors (see [50], for example) differentiate between three forms of water (strictly speaking, sugar solutions) associated with sugar crystals. Internal water is the type of water trapped within the crystal structure (lattice). It forms during the crystallization process in evaporating crystallizers. There is no evidence of the migration of this water to the crystal surface, so it can be released only by sugar dissolution or grinding. Free water refers to a thin film of unsaturated (relatively dilute) sugar solution carried by each crystal when the sugar is discharged from the centrifugals. Free water can be easily and rapidly removed using conventional methods of drying. This period of drying is referred to as constant-rate drying, since the rate of water removal is controlled only by the rate of heat transfer to the evaporating surface, which furnishes the latent heat of water evaporation. And finally, bound water, also known as migratable water, is represented by a film of concentrated sugar solution (typically, a supersaturated syrup) on the surface of the crystals, trapped by an external low-permeability crust of amorphous sugar. The latter is formed during initial drying of the sugar, most likely on the outer surface of the syrup film, when the rate of water evaporation is too high to allow the sucrose to diffuse and crystallize on the crystal surface. Because of its significant retarding effect on the rate of drying, the formation of the amorphous sugar layer corresponds to the so-called falling-rate part of the sugar drying curve. In sugar production and storage it is the bound water which represents the biggest potential problem. It is the greatest cause of caking, and must be removed by sugar conditioning. 5.2 Migration of water through the syrup film and the layer of amorphous sugar Due to the tight coupling between mass and heat transfer and the complex characteristics of moist sugar, the overall water transfer resistance during the falling-rate period of drying is a complex function of the following individual water transfer resistances: 1. through the film of sugar syrup; 2. across the layer of amorphous sugar; 3. through the film of vapor on the gas-phase side. This situation is shown schematically in Figure 9. In addition, because the activity of water depends on the sucrose content in the syrup film, the mass transfer rate for water is affected by the rate of sucrose diffusion and crystallization. Within the porous structure of the amorphous sugar layer, the situation becomes even more complicated, since it is not absolutely clear whether the ‘pores’ of this layer are still filled with liquid syrup or occupied by already formed water vapor. According to Meadows [51, 52], water evaporates before passing through the layer of amorphous sugar. If this is the case, then the rate of water migration is controlled by diffusion processes occurring in the syrup film.
Sugar crystal
Crystal – Film – Interface
Bound syrup film
Film – Layer – Interface
Amorphous sugar layer
Drying air
Water monomer Inherent moisture
Sucrose
Diffusion
Evaporation
Crystallization
Water monomer
Diffusion
Diffusion
The presented picture of physical phenomena governing the removal of water from moist sugar clearly demonstrates the key role of water activity, or in other words, the concentration of water monomer, in evaluating the rate of sugar drying or conditioning. Since the detailed modeling of sugar drying is beyond the scope of this study, the study will focus here only on the relationship between the availability of monomeric (microscopically unbound) water and the nominal sugar content in the syrup film. This relationship can be deduced from the proposed model of water activity in the sucrose/water system. Chemical reactions discussed in the previous sections are believed to occur within the film of syrup surrounding the sugar crystal during drying or conditioning. The rate processes involved in drying/conditioning, such as counter-current diffusion of water and sucrose, should be considered in conjunction with the accompanying reaction equilibrium. As long as viscosity of the syrup is not dramatically high and allows these reactions to occur rapidly, the system can attain chemical equilibrium in a relatively short time. Since in practice the reactions are much faster than other rate processes involved, it can be assumed that they are instantaneous and chemical equilibria establish immediately. For the purpose of illustration and to gain a better understanding of the critical role that sucrose hydration and clustering play in the syrup film, an extremely simplified reaction scheme will be consider in which water association is completely ignored, and sucrose hydration and clustering are represented, respectively, by the following single multimolecular reaction steps:
K’H S + nW ← → SWn
K’H =
K’C mS ← → Sm
K’C =
[SWn ] [S ][W ]n [Sm ] [S ]m
(29)
(30)
The role of hydration is fairly simple. According to eq.(29), by binding more and more molecules of water, sucrose makes less monomers of water available in solution. This is obviously manifested as a lowered water activity. Sucrose clustering has the opposite effect and seems to be more complicated. First of all, it reduces the total number of molecules in the system. This increases the mole fraction of monomeric water despite the fact that the actual absolute number of these molecules in the system has not changed. Consequently, the water activity also increases. Although this sounds like only a purely calculational result, a simple physical interpretation can be proposed as well. In the statistical sense, an increased mole fraction of the monomeric water means a higher probability of its molecules to interact with the neighboring molecules, and this is, in fact, how the activity is generally understood. The other effect of sucrose clustering on the water activity is by left-shifting the equilibrium of hydration. With the progressing sucrose clustering, the number of free sucrose molecules decreases. Since less sucrose is now available for hydration, the hydration equilibrium is shifted towards a lower concentration of sucrose hydrates and simultaneously free molecules of water are released to solution. This can be easily deduced from eqs.(29) and (30). It is also worth noting that, in the qualitative sense, clustering of sucrose is independent of the process temperature and mainly depends on the degree of saturation of the syrup film. According to the proposed model, sucrose clustering takes place at both low and high temperatures. As predicted by the van’t Hoff equation, at low temperatures, the value of the clustering equilibrium constant is high (note that, because of the hydrogen bonding nature of clustering, the enthalpy of clustering reaction is negative). This allows for a sufficient content of sucrose clusters, even though at these temperatures solubility of sucrose is low. On the other hand, at high temperatures the clustering equilibrium constant is smaller, so maintaining the same level of clusters requires a higher content of dissolved sucrose. This is possible, of course, due to the higher solubility of sucrose at elevated temperatures.
Fig. 9: Conceptual model of water migration in the vicinity of a sugar crystal during the falling-rate period of sugar drying Zuckerindustrie 127 (2002) Nr. 3, 175–185
183
6 Conclusions An attempt has been made to model the water activity in highly concentrated aqueous solutions of sucrose using the chemical approach and the concept of semi-ideal solution. The model accounts for sucrose hydration and clustering as well as water association. Two different mechanisms of sucrose hydration have been adopted: ordinary linear hydration when a molecule of water interacts with a single Hbinding site (hydroxyl oxygen), and bridging-water hydration when the water molecule constitutes a bridge between glucose and fructose moieties. Model predictions in terms of the water activity coefficient, isothermal excess enthalpy of water, and boiling point elevation are satisfactory in the entire range of sucrose concentrations. Some minor modifications must be made, however, in order to improve the model performance between 50 and 70% w/w sucrose. The estimated values of the reaction enthalpies are in line with the known estimates of hydrogen bond energies, although further refinement is still necessary and possible, since the effect of the fixed model parameters, such as m, n and α, has not been studied. Including more data, especially water activity data from the supersaturation region, heat of dilution data at temperatures higher than 25 °C as well as sucrose activity coefficient data, should make the results of parameter estimation and finally the model itself more reliable. Due to its rigorous thermodynamic nature, the new model has a potential to predict the sucrose activity coefficient as well and may help provide a better understanding of the sucrose crystallization process. Water activity plays the critical role in evaluating the rate of water migration in the process of sugar drying and conditioning, especially when the rate is controlled by diffusion phenomena taking place within the film of concentrated syrup surrounding the sugar crystal. Due to molecular solvation and association effects occurring in the sucrose/water system, the driving force for water transfer across this film should be expressed in terms of water activity rather than the nominal concentration of water in solution. According to the presented chemical model of water activity, however, the latter is equivalent to the true mole fraction of water monomer. Therefore, the future attempts at modeling and controlling the process of sugar drying/ conditioning should account for the distribution of monomer water in the syrup film.
Nomenclature E partial molar excess enthalpy of water, J/mol HW K equilibrium constant/parameter M molecular mass, g/mol m maximum admissible number of sucrose molecules in a single sucrose cluster molality of solution, mol sucrose/kg water m• n maximum available number of hydrogen-bond sites on a single molecule of sucrose ni maximum available number of hydrogen-bond sites on a sucrose cluster containing i molecules of sucrose N number of moles P pressure in Pa pij frequency ratios for sucrose hydration equilibria qi frequency ratios for sucrose clustering equilibria R universal gas constant, 8.314 J/mol S supersaturation T absolute temperature in K x mole fraction α average number of H-bonds utilized per one sucrose-sucrose attachment ∆CpH heat-capacity increase attending sucrose hydration, J/(mol · K) ∆H enthalpy of reaction, J/mol γW water activity coefficient (observed) µ chemical potential, J/mol
184
Subscripts and superscripts A C H S W o t *
water association sucrose clustering sucrose hydration sucrose water at 25°C total pure water
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L’activité de l’eau en solution concentrée de saccharose et ses conséquences sur la disponibilité de l’eau dans le film de sirop entourant le cristal de sucre (Résumé) Un modèle chimique, thermodynamiquement rigoureux a été proposé pour le système binaire sacccharose-eau. Le modèle couvre pratiquement tout le domaine des concentrations y compris les plus élevées. Il utilise le concept de la solution semi-idéale suivant lequel tous les écarts par rapport à l’idéalité sont attribués à des réactions chimiques se produisant dans le système, alors que les interactions physiques sont négligées. Le modèle rend compte de l’hydratation du saccharose, des amas (clusters) de molécules de sucre, de même que des associations eau-eau. Toutes les réactions sont supposées atteindre l’équilibre. Deux mécanismes différents d’hydratation (linéaire et pontée) sont introduits. Le concept des sites indépendants de liaison, initialement proposé par Schönert (1986) a été étendu pour décrire aussi bien la probabilité des liaisons sucre-eau que celle entre sucre et sucre. Un modèle de «mélange» d’équilibre pour l’eau liquide comportant le formation de clusters simples de molécules d’eau a été inclus pour rendre compte des interactions eau-eau. Les données expérimentales (pression de vapeur, élévation du point d’ébullition, humidité relative d’équilibre et chaleur de dilution) tirées de sources bibliographiques ont été utilisées pour valider le modèle et pour estimer ses paramètres. Le modèle proposé peut être utilisé comme un outil mathématique pour résoudre différents problèmes pratiques rencontrés dans l’industrie sucrière et les industries alimentaires, comme par exemple le comportement du sucre cristallisé au conditionnement et au stockage, le collage, la recristallisation du sucre amorphe et sa stabilité dans les produits alimentaires et pharmaceutiques. On postule que la force motrice pour la migration de l’eau depuis les cristaux de sucre doit être définie en termes d’activité de l’eau plutôt qu’en termes de teneur en eau de la solution. Les processus de diffusion dans le film de sirop autour du cristal de sucre sont brièvement soulignés. Les rôles clefs de l’hydratation du sucre et de la formation de «clusters» dans l’établissement de la force motrice qu’est la formation d’eau monomérique dans le sirop, sont discutés en détail.
Zuckerindustrie 127 (2002) Nr. 3, 175–185
La actividad del agua en soluciones de sacarosa concentradas y sus efectos sobre el agua disponible en la capa de jarabe alrededor del cristal (Resumen) Se presenta un modelo termodinámico y exclusivamente químico de la actividad del agua en el sistema sacarosa–agua. El modelo cubre prácticamente todo el sector de composición, incluso concentraciones de azúcar sumamente altas. Además utiliza el concepto de la solución semiideal, según el cual todas las divergencias del comportamiento ideal son atribuidas a las reacciones químicas que ocurren en el sistema mientras que se descuidan las interacciones físicas. El modelo explica tanto la hidratación de sacarosa y la formación de clústers de sacarosa así como también la asociación de agua. Todas estas reacciones se basan en la adaptación del equilibrio químico. Dos distintos mecanismos de la hidratación de sacarosa, linear y formación de puentes, están incluidos. Para describir la presencia de interacciones de sacarosa–agua y de sacarosa–sacarosa se aprovechó el concepto de lugares de enlace independientes, originalmente propuesto por Schönert (1986). Para explicar las interacciones de agua–agua se integró el modelo del equilibrio de mezclas de agua líquida, incluso la formación de clústers de agua simples. Para la evaluación del modelo y la determinación de sus parámetros se aprovecharon datos experimentales de fuentes selectas de la literatura (presión de vapor, elevación del punto de ebullisión, humedad relativa de equilibrio y calor de disolución. El modelo propuesto puede ser empleado como utillaje matemático para la solución de diferentes problemas prácticos en la industria azucarera y la industria de la alimentación, tales como el comportamiento de azúcar cristalino en acondicionamiento y almacenamiento, viscosidad, recristalización de azúcar amorfo y su estabilidad en víveres y medicamentos. Se exige definir como fuerza motriz para la migración del agua de cristales de azúcar más bien las unidades de la actividad de agua que el contenido nominal del agua en solución. Además se describen brevemente los procesos de difusión que se llevan a cabo en la capa de jarabe alrededor del cristal de azúcar. Finalmente se discute detalladamente el papel clave de la hidratación de sacarosa y la formación de clústers de sacarosa para la determinación de la fuerza motriz de agua monómero en la capa de jarabe.
Authors’ addresses: Prof. M. Starzak, School of Chemical Engineering, University of Natal, Durban 4041, South Africa, e-mail:
[email protected]; Prof. Mohammed Mathlouthi, Laboratoire de Chimie Physique Industrielle, Faculté des Sciences, Université de Reims Champagne-Ardenne, 51687 Reims Cédex, France.
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