Multi-component models

Zielinski, J.M. and Hanley, B.F. A.I.Ch.E.Jl. 45 (1999) 1. Practical friction-based approach lo modeling multi component diffusion. 

Modeling multi-component mass transfer in porous media is complicated when the mean-free-path length of the molecules is of the order of magnitude of the pore diameter. The difficulties posed by this case may be circumvented by a method... 

In this book we considered mass transfer and elemental migration between the atmosphere, hydrosphere, soils, rocks, biosphere and humans in earth s surface environment on the basis of earth system sciences. In Chaps. 2, 3, and 4, fundamental theories (thermodynamics, kinetics, coupling model such as dissolution kinetics-fluid flow modeling, etc.) of mass transfer mechanisms (dissolution, precipitation, diffusion, fluid flow) in water-rock interaction of elements in chemical weathering, formation of hydrothermal ore deposits, hydrothermal alteration, formation of ground water quality, seawater chemistry. However, more complicated geochemical models (multi-components, multi-phases coupled reaction-fluid flow-diffusion model) and phenomenon (autocatalysis, chemical oscillation, etc.) are not considered. 

HGSystem offers the most rigorous treatments of HF source-term and dispersion analysis a ailable for a public domain code. It provides modeling capabilities to other chemical species with complex thermodynamic behavior. It treats aerosols and multi-component mixtures, spillage of a liquid non-reactive compound from a pressurized vessel, efficient simulations of time-dependent... 

Crystallization-based separation of multi-component mixtures has widespread application. The technique consists of sequences of heating, cooling, evaporation, dilution, diluent addition and solid-liquid separation. Berry and Ng (1996, 1997), Cisternas and Rudd (1993), Dye and Ng (1995), Ng (1991) and Oyander etal. (1997) proposed various schemes based on the phase diagram. Cisternas (1999) presented an alternate network flow model for synthesizing crystallization-based separations for multi-component systems. The construction... 

Q. Han, J. D. Hunt. Numerical modelling of the growth of a cellular/dendritic array in multi-component alloys. Mater Sci Eng A 255 192, 1997. 

The framework for constructing such multi-component equilibrium models is the Gibbs phase rule. This rule is valid for a system that has reached equilibrium and it states that... 

Because of the relatively small number of experiments done on commercial-scale equipment before submission, and the often very narrow factor ranges (Hi/Lo might differ by only 5-10%), if conditions are not truly under control, high-level models (multi-variate regressions, principal components analysis, etc.) will pick up spurious signals due to noise and unrecognized drift. For example, Fig. 4.43 summarizes the yields achieved for... 

An alternate approach is to utilize the chromatogram heights as representative of individual concentrations of molecular size. From the kinetic modeling viewpoint, this leads to treating the polymerization as a well-characterized, multi-component reaction system. 

The set of possible dependent properties and independent predictor variables, i.e. the number of possible applications of predictive modelling, is virtually boundless. A major application is in analytical chemistry, specifically the development and application of quantitative predictive calibration models, e.g. for the simultaneous determination of the concentrations of various analytes in a multi-component mixture where one may choose from a large arsenal of spectroscopic methods (e.g. UV, IR, NIR, XRF, NMR). The emerging field of process analysis,... 

Here va and va are the stoichiometric coefficients for the reaction. The formulation is easily extended to treat a set of coupled chemical reactions. Reactive MPC dynamics again consists of free streaming and collisions, which take place at discrete times x. We partition the system into cells in order to carry out the reactive multiparticle collisions. The partition of the multicomponent system into collision cells is shown schematically in Fig. 7. In each cell, independently of the other cells, reactive and nonreactive collisions occur at times x. The nonreactive collisions can be carried out as described earlier for multi-component systems. The reactive collisions occur by birth-death stochastic rules. Such rules can be constructed to conserve mass, momentum, and energy. This is especially useful for coupling reactions to fluid flow. The reactive collision model can also be applied to far-from-equilibrium situations, where certain species are held fixed by constraints. In this case conservation laws... 

Although the condensation of phenol with formaldehyde has been known for more than 100 years, it is only recently that the reaction could be studied in detail. Recent developments in analytical instrumentation like GC, GPC, HPLC, IR spectroscopy and NMR spectroscopy have made it possible for the intermediates involved in such reactions to be characterized and determined (1.-6). In addition, high speed computers can now be used to simulate the complicated multi-component, multi-path kinetic schemes involved in phenol-formaldehyde reactions (6-27) and optimization routines can be used in conjunction with computer-based models for phenol-formaldehyde reactions to estimate, from experimental data, reaction rates for the various processes involved. The combined use of precise analytical data and of computer-based techniques to analyze such data has been very fruitful. 

For other cases, such as La3+ where more detail is required about the nature of the species present in solution, titration data can be computer fit to more complicated multi-equilibrium models containing Mx 1 v( OR)v forms whose stoichiometry is suggested by information gained from independent spectroscopic or kinetic techniques. One must be mindful of the pitfalls of simply fitting the potentiometric data to complex multi-component models for which there is no independent evidence for the various species. Without some evidence for the species put into the fit, the procedure simply becomes an uncritical mathematical exercise of adding and removing various real and proposed components until the goodness of fit is satisfactory. 

The regular solution model can be extended to multi-component systems, in which case the excess Gibbs energy of mixing is expressed as... 

Finally, it should be noted that the model can be extended to multi-component systems [6], Furthermore, the Gibbs energy of the pair exchange reaction may depend on the relative proportions of the different pairs. In this case tab is a polynomial function of the pair fractions XAA and XBB [4],... 

The UNIQUAC model was successfully used to correlate the experimental liquid-liquid equilibria data. As it can be seen from figure 1, the predicted tie lines (dashed lines) are in good agreement with the experimental data (solid lines). In other words, the UNIQUAC equations adequately fit the experimental data for this multi-component system. 

This chapter is concerned with the design and improvement of chemically-active ship bottom paints known as antifouling paints. The aims have been to illustrate the challenges involved in working with such multi-component, functional products and to show which scientific and engineering tools are available. The research in this field includes both purely empirical formulation and test methods and advanced tools including mathematical modelling of paint behaviour. 

Recent developments and prospects of these methods have been discussed in a chapter by Schneider et al. (2001). It was underlined that these methods are widely applied for the characterization of crystalline materials (phase identification, quantitative analysis, determination of structure imperfections, crystal structure determination and analysis of 3D microstructural properties). Phase identification was traditionally based on a comparison of observed data with interplanar spacings and relative intensities (d and T) listed for crystalline materials. More recent search-match procedures, based on digitized patterns, and Powder Diffraction File (International Centre for Diffraction Data, USA.) containing powder data for hundreds of thousands substances may result in a fast efficient qualitative analysis. The determination of the amounts of different phases present in a multi-component sample (quantitative analysis) is based on the so-called Rietveld method. Procedures for pattern indexing, structure solution and refinement of structure model are based on the same method. 

The most common model for describing adsorption equilibrium in multi-component systems is the Ideal Adsorbed Solution (IAS) model, which was originally developed by Radke and Prausnitz [94]. This model relies on the assumption that the adsorbed phase forms an ideal solution and hence the name IAS model has been adopted. The following is a summary of the main equations and assumptions of this model (Eqs. 22-29). 

An equilibrium-flash calculation (using the same equations as in case A above) is made at each point in time to find the vapor and liquid flow rates and properties immediately after the pressure letdown valve (the variables with the primes F , F l, y], x j,.. . shown in Fig. 3.8). These two streams are then fed into the vapor and liquid phases. The equations describing the two phases will be similar to Eqs. (3.40) to (3.42) and (3.44) to (3.46) with the addition of (1) a multi-component vapor-liquid equilibrium equation to calculate Pi and (2) NC — 1 component continuity equations for each phase. Controller equations relating 1 to Fi and P to F complete the model. 

---