Kinetic equation multicomponent

First let us give some generalization of the kinetic equations. So far we have considered only a one-component system now we will deal with a multicomponent system. In this case it is necessary to indicate the species of a particle by an index, a,b,c,. Moreover, however, we will consider the kinetic equations in a more simple form taking into account only for Boltzmannlike contributions to the collision terms, which means, we neglect, for example, the terms coming from the retardation. Then we may obtain... 

An example of multicomponent system that can be dealt with by using kinetics equations of the Smoluchowski type is provided by a step growth alternating copolymerization of two bifunctional monomers [16]. This system requires fit-tie more laborious, but quite straightforward algebra. 

The investigation of the copolymerization dynamics for multicomponent systems in contrast to binary ones becomes a rather complicated problem since the set of the kinetic equations describing the drift of the monomer feed composition with conversion in the latter case has no analytical solution. As for the numerical solutions in the case of the copolymerization of more than three monomers one can speak only about a few particular results [7,8] based on the simplified equations. A simple constructive algorithm [9] was proposed based on the methods of the theory of graphs, free of the above mentioned shortcomings. 

Micke, A., and Bulow, M., Application of Volterra integral equations to the modelling of the sorption kinetics of multicomponent mixtures in porous media Fundamentals and elimination of apparatus effects. Gas Sep. Purif.,4(3), 158-170(1990). 

The preceding kinetic equation does not take the spontaneous formation of the deactivator during polymerization into account and therefore the actual kinetic law appears to be more complex. ATRP is a multicomponent initiating system and the structure and the concentration of all the components affect the polymerization rate and the properties of the resultant polymers. 

Menon and Landau" point out a number of complications associated with modeling alloy deposition. First, the values for the kinetics parameters of the alloy species are likely to be different from those measured for the pure components. Furthermore, these parameters may vary with the composition of the alloy. Determination of such parameters can be done experimentally from measured alloy composition in well-controlled deposition experiments. A second issue has to do with the difficulty in determining the activities of the alloy components in the solid phase. The activity, which affects the electrode kinetics equation (82), is unity for single-component deposition however, in multicomponent alloy deposition it varies with the nature of the deposited alloy." ... 

Copolymer composition can be predicted for copolymerizations with two or more components, such as those employing acrylonitrile plus a neutral monomer and an ionic dye receptor. These equations are derived by assuming that the component reactions involve only the terminal monomer unit of the chain radical. The theory of multicomponent polymerization kinetics has been treated (35,36). 

Table 5.4-3 summarizes the design equations and analytical relations between concentration, C/(, and batch time, t, or residence time, t, for a homogeneous reaction A —> products with simple reaction kinetics (Van Santen etal., 1999). Balance equations for multicomponent homogeneous systems for any reaction network and for gas-liquid and gas-liquid-solid systems are presented in Tables 5.4-7 and 5.4.8 at the end of Section 5.4.3. 

Cases with more complex multicomponent kinetics will require similar balance equations for all the components of interest. 

Kinetic analysis usually employs concentration as the independent variable in equations that express the relationships between the parameter being measured and initial concentrations of the components. Such is the case with simultaneous determinations based on the use of the classical least-squares method but not for nonlinear multicomponent analyses. However, the problem is simplified if the measured parameter is used as the independent variable also, this method resolves for the concentration of the components of interest being measured as a function of a measurable quantity. This model, which can be used to fit data that are far from linear, has been used for the resolution of mixtures of protocatechuic... 

Multicomponent diffusion in the films is described by the Maxwell-Stefan equations, which can be derived from the kinetic theory of gases (89). The Maxwell-Stefan equations connect diffusion fluxes of the components with the gradients of their chemical potential. With some modification these equations take a generalized form in which they can be used for the description of real gases and liquids (57) ... 

Equations (10) are generally valid for both liquid and gas phases if reactions take place there. They represent nothing but a differential mass balance for the film region with the account of the source term due to the reaction. To link this balance to the process variables like component concentrations, some additional relationships - often called constitutive relations (see Ref. [16]) - are necessary. For the component fluxes Ni, these constitutive relations result from the multicomponent diffusion description (Eqs. (1), (2)) for the source terms, from the reaction kinetics description. The latter strongly depends on the specific reaction mechanism [27]. The reaction rate expressions lli usually represent nonlinear dependencies on the mixture composition and temperature of the corresponding phase. 

Rieckmann and Keil (1997) introduced a model of a 3D network of interconnected cylindrical pores with predefined distribution of pore radii and connectivity and with a volume fraction of pores equal to the porosity. The pore size distribution can be estimated from experimental characteristics obtained, e.g., from nitrogen sorption or mercury porosimetry measurements. Local heterogeneities, e.g., spatial variation in the mean pore size, or the non-uniform distribution of catalytic active centers may be taken into account in pore-network models. In each individual pore of a cylindrical or general shape, the spatially ID reaction-transport model is formulated, and the continuity equations are formulated at the nodes (i.e., connections of cylindrical capillaries) of the pore space. The transport in each individual pore is governed by the Max-well-Stefan multicomponent diffusion and convection model. Any common type of reaction kinetics taking place at the pore wall can be implemented. 

Equations 3.15, 3.17 and 3.19 provide the flux relationships in the limiting regimes. There remains the problem of finding the flux relationships in intermediate situations, where the pore size is comparable to the mean free path and the mixture is a multicomponent one. At present, no quantitative kinetic theory exists for flow in the transition region where the dimensions of A and dt are comparable. Therefore different simplified models have been developed. 

In the late 1800s, the development of the kinetic theory of gases led to a method for calculating multicomponent gas diffusion (e.g., the flux of each species in a mixture). The methods were developed simultaneously by Stefan and Maxwell. The problem is to determine the diffusion coefficient Dim. The Stefan-Maxwell equations are simpler in principle since they employ binary diffusivities ... 

The equations for conservation of mass, momentum, and energy for a one-component continuum are well known and are derived in standard treatises on fluid mechanics [l]-[3]. On the other hand, the conservation equations for reacting, multicomponent gas mixtures are generally obtained as the equations of change for the summational invariants arising in the solution of the Boltzmann equation (see Appendix D and [4] and [5]), One of several exceptions to the last statement is the analysis of von Karman [6], whose results are quoted in [7] and are extended in a more recent publication [8] to a point where the equivalence of the continuum-theory and kinetic-theory results becomes apparent [9]. This appendix is based on material in [8]. 

The appropriate generalization of equation (33) to arbitrary geometries in multicomponent mixtures can be shown from kinetic theory [5] or from a reasonable continuum treatment to be... 

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