Multicomponent model equations

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

Although the pure component LRC coefficients, as applied in the multicomponent model of equation (4), generally gave an excellent fit of the breakthrough loadings, there was one series of runs (the low concentration C02/air/5A system) for which the pure component LRC s did not provide a satisfactory fit of the data. For these cases, the A3 and Xq constants for the pure component CO2/5A LRC correlation were modified to provide a better fit of the breakthrough loadings. 

Powers MF, Vickery DJ, Arehole A, Taylor R. A nonequilibrium-stage model of multicomponent separation processes—V. Computational methods for solving the model equations. Computers Chem Eng 1988 12 1229-1241. 

Although the multicomponent Langmuir equations account qualitatively for competitive adsorption of the mixture components, few real systems conform quantitatively to this simple model. For example, in real systems the separation factor is generally concentration dependent, and azeotrope formation (a = 1.0) and selectivity reversal (a varying from less than 1.0 to more than 1.0 over the composition range) are relatively common. Such behavior may limit the product purity attainable in a particular adsorption separation. It is sometimes possible to avoid such problems by introducing an additional component into the system which will modify the equilibrium behavior and eliminate the selectivity reversal. 

Deeper insight into the consequences of counterion condensation is gained by an effective monomer-monomer and counterion-counterion potential, respectively. The idea is to reduce the multicomponent system (macromolecules + counterions) to effective one-component systems (macromolecules or counterions, respectively). We define the simplified model in such a way that the effective potential between the counterions or monomers, respectively, of the new system yields exactly the same correlation function (gcc, gmm) as found in the multicomponent case at the same density. Starting from the correlation function gcc -respectively gmm-of the multi-component model we calculate an effective direct correlation function cefy via the one-component Ornstein-Zernike equation. An effective potential is then obtained from the RLWC closures of the one- and multicomponent models [24]. For low and moderate densities the effective potential is well approximated by... 

The model equations for multicomponent distillation under constant boilup rate (Robinson, 1969) are ... 

In addition to the above-mentioned problem, numerical difficulties may arise. The system (model equations) describing the multicomponent off-cut recycle operation needs to be reinitialised at the end of each main-cut and off-cut to accommodate the next off-cut to the reboiler. To optimise these initial conditions (new mixed reboiler charge and its composition) it is essential to obtain the objective function gradients with respect to these initial conditions. 

A multicomponent model is often used to describe spherical micelles or globular proteins in solution. In this case the ions are treated as charged hard spheres immersed in a solvent of dielectric constant uy r. In this way the micellar solution is depicted as an electrolyte where the ions grossly differ in size and charge. The solvent averaged potential in this case is given by (2aab = aa+crb) the equation,... 

Effectively the parameter m for the width of the distribution function in the ordinary multicomponent LF equation is replaced by a product of two parameters p representing the intrinsic affinity, nx the ion specific non ideality. The ion specific non ideality can be due to residual heterogeneity or other non ideality effects typical for the ion studied. On the expense of one additional parameter (nx) for each adsorbing component this model is far more flexible for multicomponent adsorption on heterogeneous surfaces than the fully coupled models. For nx = 1 for all X the NICA equation reduces to Eq. (89). The NICA model has been used successfully for proton and metal ion binding to humic acids [116-118], but it is not yet applied to heterogeneous metal oxides. 

Algorithm 13.1 Procedure for Solving Equilibrium Model Equations with Multicomponent Tray... 

In this chapter we have considered models of multicomponent condensation. In particular, we have considered various approaches to calculating the rates of mass and energy transfers in the vapor and condensate, respectively. Methods of solving the model equations have also been discussed. 

Powers, M. F., Vickery, D. J., Arehole, A., and Taylor, R., A Nonequilibrium Stage Model of Multicomponent Separation Processes—V. Computational Methods for Solving the Model Equations, Comput. Chem. Eng., 12, 1229-1241 (1988). 

Not all the terms in these equations have the same importance in determining the flow solution in chemical reactors. The only body force considered in most reactor models, gj (per unit mass), is gravitation which is the same for all chemical species, g. The model equations for momentum and energy can then be simplified. In the momentum equation Pc c = f cS = PS-In the energy equation Xlc=i(jc Sc) = Sc=i jc S = 0- Furthermore, in most multicomponent flows, the energy or heat flux contributions from the interdiffusion processes are in general believed to be small and omitted in most applications, ft-cV jc 0 (e.g., [148], p. 816 [89], p. 198 [11], p. 566). 

The design of a complete set of governing equations for the description of reactive flows requires that the combined fluxes are treated in a convenient way. In principle, several combined flux definitions are available. However, since the mass fluxes with respect to the mass average velocity are preferred when the equation of motion is included in the problem formulation, we apply the species mass balance equations to a (/-component gas system with q — independent mass fractions Wg and an equal number of independent diffusion fluxes js. However, any of the formulations derived for the multicomponent mass diffusion flux can be substituted into the species mass balance (1.39), hence a closure selection optimization is required considering the specified restrictions for each constitutive model and the computational efforts needed to solve the resulting set of model equations for the particular problem in question. 

The applicability of the multicomponent mass diffusion models to chemical reactor engineering is assessed in the following section. Emphasis is placed on the first principles in the derivation of the governing flux equations, the physical interpretations of the terms in the resulting models, the consistency with Pick s first law for binary systems, the relationships between the molar and mass based fluxes, and the consistent use of these multicomponent models describing non-ideal gas and liquid systems. 

The equilibrium ratio [Equation (12.1)] involves physical equilibrium between phases. The system involved may be binary or multicomponent and ideal or nonideal, according to the terminology used in solution thermodynamics. Methods for correlating or predicting equilibria are based on an application of thermodynamics to each phase and to the solutions of the components within each phase. The techniques are described in standard reference works such as Walas and Reid et al. For multicomponent systems, the usual approach is to model the equilibria for each of the binary pairs and then combine the binary models in special ways to obtain the multicomponent model. 

The modeling of activity coefficients in multicomponent systems and the application of the general model [Equation (12.21)] are discussed in detail by Walas and Reid et al. ... 

A multicomponent model of adsorption can be reduced to three main components first, H2S, second, VOC with average properties of found species, and third, H2O. Moreover, H2O adsorption can be described as quasi-equilibrium due to the high concenbation of H2O and the long operation time of the carbon bed. Rectangular type isotherms are used to model the equilibrium of VOC and H2S. The amount of H2S adsorbed depends on available space left after VOC adsorption and it can be described bj equation ... 

Tip 13 (related to Tip 12) Copolymerization, copolymer composition, composition drift, azeotropy, semibatch reactor, and copolymer composition control. Most batch copolymerizations exhibit considerable drift in monomer composition because of different reactivities (reactivity ratios) of the two monomers (same ideas apply to ter-polymerizations and multicomponent cases). This leads to copolymers with broad chemical composition distribution. The magnirnde of the composition drift can be appreciated by the vertical distance between two items on the plot of the instantaneous copolymer composition (ICC) or Mayo-Lewis (model) equation item 1, the ICC curve (ICC or mole fraction of Mj incorporated in the copolymer chains, F, vs mole fraction of unreacted Mi,/j) and item 2, the 45° line in the plot of versus/j. 

Sun SS, Chumlea WC, Heimsfield SB et al. 2003. Development of bioelectrical impedance analysis prediction equations for body composition with the use of a multicomponent model for use in epidemiological surveys. Am Clin Nutr 77,331-340. 

Write out the complete equation representing g / RT for a ternary mixture modeled by the multicomponent Margules equation (5.6.22). Also write out the complete expression for lny3. 

The effects of capillary condensation were included in the network model, by calculating the critical radius below which capillary condensation occurs based on the vapor composition in each pore using the multicomponent Kelvin Equation (23.2). Then the pore radius was compared with the calculated critical radius to determine whether the pore is liquid- or vapor-filled. As a significant fraction of pores become filled with capillary condensate, regions of vapor-filled pores may become locked off from the vapor at the network surface by condensate clusters. A Hoshen and Kopelman [30] algorithm is used to identify vapor-filled pores connected to the network surface, in which diffusion and reaction continue to take place after other parts of the network filled with liquid. It was assumed that, due to the low hydrogen solubility in the liquid, most of the reaction takes place in the gas-filled pores. The diffusion/reaction simulation is repeated, including only vapor-filled pores connected to the network surface by a pathway of other vapor-filled pores. 

The model equations (11.6-2) to (11.6-4) are validated with the experimental data of binary and ternary systems of ethane, propane and n-butane onto activated carbon. All the necessary equilibrium and kinetic parameters are obtained from the single component fitting as done in the last section. In this sense the multicomponent model is the predicting tool, and it has been shown in Do (1997) that this multicomponent heterogeneous model is a good predictive model. It is capable of predicting well simultaneous adsorption, simultaneous desorption and displacement. Readers are referred to a review paper by Do (1997) for further details. 

This chapter summarizes the thermodynamics of multicomponent polymer systems, with special emphasis on polymer blends and mixtures. After a brief introduction of the relevant thermodynamic principles - laws of thermodynamics, definitions, and interrelations of thermodynamic variables and potentials - selected theories of liquid and polymer mixtures are provided Specifically, both lattice theories (such as the Hory-Huggins model. Equation of State theories, and the gas-lattice models) and ojf-lattice theories (such as the strong interaction model, heat of mixing approaches, and solubility parameter models) are discussed and compared. Model parameters are also tabulated for the each theory for common or representative polymer blends. In the second half of this chapter, the thermodynamics of phase separation are discussed, and experimental methods - for determining phase diagrams or for quantifying the theoretical model parameters - are mentioned. 

In the calculation of the predicted response curves the axial dispersion coefficient and the external mass transfer coefficient were estimated from standard correlations and the effective pore diffusivily was determined from batch uptake rate measurements with the same adsorbent particles. The model equations were solved by orthogonal collocation and the computation time required for the collocation solution ( 20 s) was shown to be substantially shorter than the time required to obtain solutions of comparable accuracy by various other standard numerical methods. It is evident that the fit of the experimental breakthrough curves is good. Since all parameters were determined independently this provides good evidence that the model is essentially correct and demonstrates the feasibility of modeling the behavior of fairly complex multicomponent dynamic systems. 

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