Incompressible Liquid Mixtures

In this and the next section we consider liquid-liquid phase separation in liquid mixtures terminating in either an upper or a lower critical solution point. Since the pressure does not affect concentration fluctuations we neglect in first approximation the contribution of the pressure to the independent scaling fields, hi and 

Equation 10.62 yields for the mole fraction x of an incompressible liquid mixture  

Except for a trivial factor x, eq 10.64 has exactly the same form as the expression of eq 10.42 for the density of a one-component fluid. Hence, in analogy with eq 10.52, we can immediately conclude that the mole fractions x and x of the solute along the two sides of the phase boundary will vary with temperature as  

In eq 10.65b Af and ficr now refer to the critical behaviour of the isomorphic specific heat capacity which for liquid mixtures is the isobaric specific heat capacity. 

Just as eq 10.52b for Ap j, we see that the eoexistence diameter Ax for the mole fraction of a liquid mixture depends on the asymptotic behaviour of the specific heat capacity and on two asymmetry coefficients, namely eff= - Xcai/(l-Xcai) and h = b -xJ 2)lxJ -b2Sc). 

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The thermodynamic behavior of fluids near critical points is drastically different from the critical behavior implied by classical equations of state. This difference is caused by long-range fluctuations of the order parameter associated with the critical phase transition. In one-component fluids near the vapor-liquid critical point the order parameter may be identified with the density or in incompressible liquid mixtures near the consolute point with the concentration. To account for the effects of the critical fluctuations in practice, a crossover theory has been developed to bridge the gap between nonclassical critical behavior asymptotically close to the critical point and classical behavior further away from the critical point. We shall demonstrate how this theory can be used to incorporate the effects of critical fluctuations into classical cubic equations of state like the van der Waals equation. Furthermore, we shall show how the crossover theory can be applied to represent the thermodynamic properties of one-component fluids as well as phase-equilibria properties of liquid mixtures including closed solubility loops. We shall also consider crossover critical phenomena in complex fluids, such as solutions of electrolytes and polymer solutions. When the structure of a complex fluid is characterized by a nanoscopic or mesoscopic length scale which is comparable to the size of the critical fluctuations, a specific sharp and even nonmonotonic crossover from classical behavior to asymptotic critical behavior is observed. In polymer solutions the crossover temperature corresponds to a state where the correlation length is equal to the radius of gyration of the polymer molecules. A... 

The theory of crossover critical phenomena has been extended to binary mixtures. This extension is based on a principle of isomorphism of critical phenomena which states that the thermodynamic behavior of fluid mixtures is similar to that of one-component fluids provided that the mixtures are kept at a constant value of a hidden field variable C [80-82]. For mixtures with a simple phase diagram in which the critical points of the two components are connected by a continuous critical locus, this hidden field C may be taken as a function of the difference of the critical potentials of the two components [83-85]. Based on this principle crossover equations have been proposed for the thermodynamic properties of a variety of fluid mixtures near the vapor-liquid critical locus [68,69,79,86-89]. A systematic procedure for extending the application to fluids with more complex phase diagrams has been developed by Anisimov et al. [90-92]. This procedure also incorporates crossover between the one-component vapor-liquid critical limit and the liquid-liquid critical limit of incompressible liquid mixtures [90, 91, 93]. 

The structure of a simple mixture is dominated by the repulsive forces between the molecules [15]. Any model of a liquid mixture and, a fortiori of a polymer solution, should therefore take proper account of the configurational entropy of the mixture [16-18]. In the standard lattice model of a polymer solution, it is assumed that polymers live on a regular lattice of n sites with coordination number q. If there are n2 polymer chains, each occupying r consecutive sites, then the remaining m single sites are occupied by the solvent. The total volume of the incompressible solution is n = m + m2. In the case r = 1, the combinatorial contribution of two kinds of molecules to the partition function is... 

Another example of phase coexistence that is described by a single scalar one-component order parameter is provided by incompressible binary mixtures. One considers a dense liquid of two components, A and B the composition of... 

Most chemical engineers relate the term incompressible flow to incompressible fluid systems. For non-reactive ideal liquid mixtures operated at nearly constant temperatures, the incompressible flow limit is obviously a reasonable approximation in practice. 

To use the energy balances, we will need to relate the energy to more easily measurable properties, such as temperature and pressure (and in later chapters, when we consider mixtures, to composition as well). The interrelationships between energy, temperature, pressure, and composition can be complicated, and we will develop this in stages. In this chapter and in Chapters 4,5, and 6 we will consider only pure fluids, so composition is not a variable. Then, in Chapters 8 to 15, mixtures will be considered. Also, here and in Chapters 4 and 5 we will consider only the simple ideal gas and incompressible liquids and solids for which the equations relating the energy, temperature, and pressure are simple, or fluids for which charts and tables interrelating these properties are available. Then, in Chapter 6, we will discuss how such tables and charts are prepared. 

The ideal solution assumptions are applicable to mixtures of isomers or homologs that have relatively close boiling points. Here, only limited extrapolations of pure-component fugacities into hypothetical regions are required. A frequently used extrapolation technique is as follows. Whether hypothetical or not, (4-71), which assumes incompressible liquid, is utilized to determine vfi.. Substitution of (4-71) into (4-85) gives... 

In pervaporation and vapor-permeation processes the partial vapor pressures of the components at the feed side are fixed by the nature of the components, composition, and temperature of the feed, whereas the total pressure is of no influence, as long as the liquid mixture can be regarded as incompressible. 

For a practically useful tubular reactor model, the reacting (polymerizing) mixture is considered homogeneous and only axial dispersion is considered. At each specific position along the tube, perfect radial mixing and a uniform velocity profile are assumed. The tube, therefore, can be modeled as a one-dimensional tubular reactor. Also, instantaneous fluid dynamics are assumed because of the incompressibility of the liquid mixture (hence the calculation of the velocity profile is simplified). 

When viewed at this description, the correct representation of the motion is the mean local motion due to the bubble growth, superposed with the mean global motion of the macroelement dv. We illustrate the theory by examining the flow of spherical bubbles dispersed in an incompressible liquid. In such a case, the volumetric evolution of the mixture as a whole (compressibility) is due only to spherical growth of the gas, i.e, the mean local motion induces spherical... 

Since in this case different gas species move independent of each other for both type 1 as well as Knudsen transport, (34.12) can be used for single gases as weU as mixtures. Because the gas viscous flow permeance (34.10) contains a pressure-dependent term p ), it is not possible to use it in expressions as simple as (34.12). However, for incompressible liquids, an expression similar to (34.12) can also be written for a meso/macroporous... 

The stationary phase matrices used in classic column chromatography are spongy materials whose compress-ibihty hmits flow of the mobile phase. High-pressure liquid chromatography (HPLC) employs incompressible silica or alumina microbeads as the stationary phase and pressures of up to a few thousand psi. Incompressible matrices permit both high flow rates and enhanced resolution. HPLC can resolve complex mixtures of Upids or peptides whose properties differ only slightly. Reversed-phase HPLC exploits a hydrophobic stationary phase of... 

Mass balance on the mixture, at a cross section in the slug zone (both liquid and gas are assumed incompressible) ... 

The DPS is a measure of the pore saturation and is defined as the ratio of volume of the suspending liquid to pore volume (see Figure 4.3). The mechanical compressibility of a collection of particulates, X, is defined as (AV/Vq)/AP, where AV is the volume change resulting from an incremental pressure change AP, and Vq is the initial volume. As more liquid is added to the particles, the mixture becomes essentially incompressible, and X approaches zero. 

Note there is a difference between the Darcy s law for a gas (a compressible fluid) and that for a liquid (an incompressible fluid). Thus, before doing an injection test, it is important to know whether the injected acid gas mixture is in the gas phase or in the liquid phase. 

Figure 11.3-3 shows the vapor-liquid and liquid-liquid equilibrium behavior computed for the system of methanol and n-hexane at various temperatures. Note that two liquid phases coexist in equilibrium to temperatures of about 43°C. Since liquids are relatively incompressible, the species liquid-phase fugacities are almost independent of pressure (see Illustrations 7.4-8 and 7.4-9), so that the liquid-liquid behavior is essentially independent of pressure, unless the pressure is very high, or low enough for the mixture to vaporize (this possibility will be considered shortly). The vapor-liquid equilibrium curves for this system at various pressures are also shown in the figure. Note that since the fugacity of a species in a vapor-phase mixture is directly proportional to pressure, the VLE curves are a function of pressure, even though the LLE curves are not. Also, since the methanol-hexane mixture is quite nonideal, and the pure component vapor pressures are similar in value, this system exhibits azeotropic behavior. 

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