Phase separation, mathematical description

Remember that the hump which causes the instability with respect to phase separation arises from an unfavorable AH considerations of configurational entropy alone favor mixing. Since AS is multiplied by T in the evaluation of AGj, we anticipate that as the temperature increases, curves like that shown in Fig. 8.2b will gradually smooth out and eventually pass over to the form shown in Fig. 8.2a. The temperature at which the wiggles in the curve finally vanish will be a critical temperature for this particular phase separation. We shall presently turn to the Flory-Huggins theory for some mathematical descriptions of this critical point. The following example reminds us of a similar problem encountered elsewhere in physical chemistry. 

The mathematical model comprises a set of partial differential equations of convective diffusion and heat conduction as well as the Navier-Stokes equations written for each phase separately. For the description of reactive separation processes (e.g. reactive absorption, reactive distillation), the reaction terms are introduced either as source terms in the convective diffusion and heat conduction equations or in the boundary condition at the channel wall, depending on whether the reaction is homogeneous or heterogeneous. The solution yields local concentration and temperature fields, which are used for calculation of the concentration and temperature profiles along the column. 

Now we will use the ideal solution model to develop a mathematical description of vapor-liquid equilibrium in a multicomponent solution. We will make the assumption that we have a system that is separated into a coexisting vapor and liquid phase. The vapor phase will be assumed to behave like an ideal gas, while the liquid phase will be assumed to behave as an ideal solution. 

Mathematical Description of Phase Separation. The thermodynamic state of a system of two or more components with limited miscibility can be described in terms of the free energy of mixing.40 At constant pressure and temperature, three different states can be distinguished ... 

The position of the T,-divide that separates soluble from insoluble (hydrophobically associated) states in the phase diagram depends on seven variables on the six intensive variables of temperature, chemical potential, electrochemical potential, mechanical force, pressure, and electromagnetic radiation, and on polymer volume fraction or concentration. Therefore, diverse protein-catalyzed energy conversions by the consilient mechanism result from designs that control the location of the Tfdivide in this seven-dimensional phase transitional space. Complete mathematical description has yet to be written for representation of the T,-divide in seven-dimensional phase transitional space, but it may prove to be more relevant to... 

The second approach is based on the different activation energies of separate particles. However, if at the adsorption on the inhomogeneity surface this approach is physically grounded, then under consideration of processes in the liquid phase the assumption about activation energy dependence upon time looks unconvincingly. In spite of this fact, the mathematical description of problem practically does not differ from the previous one [10-12]. 

In the following part of this section, we provide simple mathematical descriptions of a few common features of two-phase/two-region countercurrent devices, specifically some general considerations on equations of change, operating lines and multicomponent separation capability. Sections 8.1.2, 8.1.3, 8.1.4, 8.1.5 and 8.1.6 cover two-phase systems of gas-Uquid absorption, distillation, solvent extraction, melt crystallization and adsorption/SMB. Sections 8.1.7, 8.1.8 and 8.1.9 consider the countercurrent membrane processes of dialysis (and electrodialysis), liquid membrane separation and gas permeation. Tbe subsequent sections cover very briefly the processes in gas centrifuge and thermal diffusion. 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

Equations of state relate the phase properties to one another and are an essential part of the full, quantitative description of phase transition phenomena. They are expressions that find their ultimate justification in experimental validation rather than in mathematical rigor. Multiparameter equations of state continue to be developed with parameters tuned for particular applications. This type of applied research has been essential to effective design of many reaction and separation processes. 

Basic physical phenomena occurring during a chromatographic separation are described in Chapter 2. A quantitative description is possible using suitable mathematical models, which are typically based on material, energy, and momentum balances, in addition to equations that quantify the thermodynamic equilibria of the distribution of the solutes between the different phases. A good model has to be as... 

Many of these software systems work (particularly for Drylab ) by using mathematical algorithms to predict separations for a number of other conditions after a few experimental runs have been performed. Typical predictions are made for changes made to the mobile phase conditions, temperature, isocratic or gradient separations, or changes to the column conditions (e.g. column dimensions, particle size and flow rate). The references listed in Table 2.3 provide more specific descriptions of each type of software system. 

However, in spite of the known advantages and applications of liquid membrane separation processes in hollow-fiber contactors, there are scarce examples of industrial application. The industrial application of a new technology requires a reliable mathematical model and parameters that serve for design, cost estimation, and optimization purposes allowing to accurate process scale-up. " The mathematical modeling of liquid membrane separation processes in HFC is divided into two steps (1) the description of the diffusive mass transport rate and (2) the development of the solute mass balances to the flowing phases. 

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