Nonideal Mixing

Usually, the typology of batch reactors also includes the semi-batch gas-liquid reactors, in which a gaseous phase is fed continuously in order to provide one of the reactants. A typical example is given by the reactors used both in different oxidative industrial processes and in the active sludge processes for the treatment of wastewater. It is possible to distinguish between the bubble columns (Fig. 7.1(c)), in which the gas rises undisturbed in the liquid phase, and the bubble stirred reactor, in which a mechanical mixer is added. Finally, the slurry reactors can be considered, in which the liquid phase contains a finely dispersed solid phase as well, which can act as a reactant or as a heterogeneous catalyst these reactors assume in general the features of Fig. 7.1(d). 

As discussed in Sect. 2.1, physical and mathematical models of ideal chemical reactors are based on two very simplified fluid dynamic assumptions, namely perfect mixing (BR and CSTR) and perfect immiscibility (PFR). On the contrary, in real tank reactors the stirring system produces a complex motion field made out of vortices of different dimensions interacting with the reactor walls and the internal baffles, as schematically shown in Fig. 7.2(a). As a consequence, a complex field of composition and temperature is established inside the reactor. 

The main features of this behavior may be captured by a simple modeling approach based on a proper combination of ideal reactors. The simplest example is 

The modeling can be improved by increasing the number of ideal reactors so that a complex network composed of fictitious interacting compartments is created. This approach can be applied to describe a large number of industrial reactors, in which partial mixing phenomena take place [3, 10]. 

On the basis of the considered macroscopic flow pattern, the dominant circulation flows (/ c and Fc/2) subdivide the reactor into three parallel levels, where each level is then divided into Nc/3 equally sized compartments of equal volume Vc = Vr/Nc. Every compartment is modeled as a nonstationary ideal continuous stirred tank reactor, with a main inlet and outlet flow, which connects the given compartment with adjacent compartments on the same level, and secondary exchange flow rates accounting for the turbulent mixing with adjacent compartments laying on the upper and/or lower level (Fig. 7.3). 

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Barnes and co-workers have studied mixed-monolayer systems [278,281,283,284] and found some striking nonidealities. Mixed films of octadecanol and cholesterol, for example, show little evaporation resistance if only 10% cholesterol is present [278] apparently due to an uneven granular microstructure in films with cholesterol [284]. Another study of cellulose decanoate films showed no correlation between holes in the monolayer and permeation rate [285]. Polymerized surfactants make relatively poor water evaporation retarders when compared to octadecanol [286]. There are problems in obtaining reproducible values for r [287] due to impurities in the monolayer material or in the spreading solvent. 

Reactions in which a product remains in the him (as above) are complicated by the fact that the areas of reactant and product are not additive, that is, a nonideal mixed him is formed. Thus Gilby and Alexander [310], in some further studies of the oxidation of unsaturated acids on permanganate substrates, found that mixed hlms of unsaturated acid and dihydroxy acid (the immediate oxidation product) were indeed far from ideal. They were, however, able to ht their data for oleic and erucic acids fairly well by taking into account the separately determined departures from ideality in the mixed hlms. 

Since it has been shown that nonideal mixing occurs in the 2.5-15.0 dyn cm 1 range, the excess free energies of interaction were calculated for compressions of each pure component and their mixtures to each of these surface pressures. In addition, these surface pressures are below the ESPs and/or monolayer stability limits so that dynamic processes arising from reorganization, relaxation, or film loss do not contribute significantly to the work of compression. 

The nonideal mixing factor k accounts for conditions in the enclosure that are less than well mixed. It follows that... 

The nonideal mixing factor varies from 0.1 to 0.5 for most practical situations.7 For perfect mixing k = 1. 

Table 3-12 Nonideal Mixing Factor kfor Various Dilution Ventilation Conditions1...

Equations 3-9, 3-12, and 3-14 are used to compute the ventilation rates required. Table 3-12 lists values for k, the nonideal mixing factor used with these equations. 

Psat is the saturation vapor pressure of the liquid, k is the nonideal mixing factor,... 

The nonideal mixing factor k ranges between 0.1 and 0.5. Because no information is given about the ventilation, k is used as a parameter. Substituting into Equation 3-14, we obtain... 

It was discovered that the tracer used in problem 19-9 was subject to decay, and the measured concentrations could reflect both nonideal mixing and tracer decay. Therefore, a study of the... 

The TIS and DPF models, introduced in Chapter 19 to describe the residence time distribution (RTD) for nonideal flow, can be adapted as reactor models, once the single parameters of the models, N and Pe, (or DL), respectively, are known. As such, these are macromixing models and are unable to account for nonideal mixing behavior at the microscopic level. For example, the TIS model is based on the assumption that complete backmixing occurs within each tank. If this is not the case, as, perhaps, in a polymerization reaction that produces a viscous product, the model is incomplete. 

Of the possible types of measurements, heats of micellar mixing obtained from the mixing of pure surfactant solutions are perhaps of the greatest interest. Also of interest is the titration (dilution) of mixed micellar solutions to obtain mixed erne s. While calorimetric measurements have been applied in studies of pure surfactants (6,7) and their interaction with polymers ( ), to our knowledge, applications of calorimetry to problems of nonideal mixed micellization have not been previously reported in the literature. 

Among the purposes of this paper is to report the results of calorimetric measurements of the heats of micellar mixing in some nonideal surfactant systems. Here, attention is focused on interactions of alkyl ethoxylate nonionics with alkyl sulfate and alkyl ethoxylate sulfate surfactants. The use of calorimetry as an alternative technique for the determination of the cmc s of mixed surfactant systems is also demonstrated. Besides providing a direct measurement of the effect of the surfactant structure on the heats of micellar mixing, calorimetric results can also be compared with nonideal mixing theory. This allows the appropriateness of the regular solution approximation used in models of mixed micellization to be assessed. 

The derivation of a pseudo-phase separation model for treating nonideal mixed micellization is given in detail in reference 3. This leads to the generalized result... 

Consideration of the thermodynamics of nonideal mixing provides a way to determine the appropriate form for the activity coefficients and establish a relationship between the measured enthalpies of mixing and the regular solution approximation. For example, the excess free energy of mixing for a binary mixture can be written as... 

Titration results for the mixed erne s of the SDS/CgE4 and C12E2S/C8E4 systems as a function of their relative mole fraction in solution are shown in Figures 2 and 3, respectively. Here, the experimentally determined points are compared with calculated results from the nonideal mixed micelle model (solid line) and the ideal mixed micelle model (dashed line). Good agreement with the nonideal model is seen in each case. 

Figure 2. Cmc s of mixtures of SDS and CgE4 in distilled water (at 25°C). The plotted points are experimental data, the solid line is the result for the nonideal mixed micelle model with B = -3.3, and the dashed line is the result for ideal mixing.

The finding that the assumptions of the regular solution approximation do not hold for the mixed micellar systems investigated here suggests a re-examination of how the thermodynamics of mixing enter the nonideal mixed micelle model. 

Calorimetric measurements can be used to obtain heats of mixing between different surfactant components in nonideal mixed micelles and assess the effects of surfactant structure on the thermodynamics of mixed micellization. Calorimetry can also be successfully applied in measuring the erne s of nonideal mixed surfactant systems. The results of such measurements show that alkyl ethoxylate sulfate surfactants exhibit smaller deviations from ideality and interact significantly less strongly with alkyl ethoxylate nonionics than alkyl sulfates. 

The mixed cmc behavior of these (and many other) mixed surfactant systems can be adequately described by a nonideal mixed micelle model based on the psuedo-phase separation approach and a regular solution approximation with a single net interaction parameter B. However, the heats of micellar mixing measured by calorimetry show that the assumptions of the regular solution approximation do not hold for the systems investigated in this paper. This suggests that in these cases the net interaction parameter in the nonideal mixed micelle model should be interpreted as an excess free energy parameter. 

Phase behavior of lipid mixtures is a much more difficult problem, due to nonideal mixing of lipid components. Ideal mixing implies like and unlike lipids have the same intermolecular interactions, while nonideal mixing results from differential interactions between lipid types. If the difference is too great, the two components will phase separate. While phase separation and lateral domain formation have been observed in many experiments, we lack a molecular-level physical description of the interactions between specific lipids that cause the macroscopic behavior. The chemical potential of a lipid determines phase separation, as phase coexistence implies the chemical potential of each type of lipid is equal in all phases of the system [3,4],... 

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