Chapman-Enskog relation

Dij (here D12) is the diffusion coefficient of the pair i-j. It can be experimentally measured by Wicke-Callenbach type (isobaric) measurements (see Sections 9.2.4.3 and 9.4.2,3) or calculated with the help of the first order approximation Chapman-Enskog relation [1,4] which is written as... 

The Chapman-Enskog theory (Chapman and Cowling, 1970) is a model which is positioned between the two approaches, the empirical relation by Fuller et al. (Eq. 18-44) and the theoretically stringent Equation 18-43. This theory improves the absolute size of the expression by taking into account the individual sizes and interactions of the diffusing molecules. However, the numerical values obtained with the model by Fuller et al. (Eq. 18-44) are still better than both the Chapman-Enskog theory and Eq. 18-43. 

We shall organize the Chapman-Enskog reduction into four steps Step 1) Initial suggestion M w C Mi for the slow manifold Msiow C Mi is made. It is a manifold that has a one-to-one relation with the space M2. We can regard it as an imbedding of M2 in Mi. Step 2) The vector field v.f.) 1 is projected on M w (i.e., v.f. jf 1 denoting the vector field v. /.) 1 attached to a point of M, is projected on the tangent plane of M ow a at point). The projected vector field is denoted by the symbol... 

Muckenfuss, C., Stefan-Maxwell Relations for Multicomponent Diffusion and the Chapman Enskog Solution of the Boltzmann Equations, J. Chem. Phys., 59, 1747-1752 (1973). 

Maxwell suggested that the porous material itself be described as a supplementary dust species, consisting of very large molecules that are kept motionless by some unspecified external force. The Chapman-Enskog kinetic theory is then applied to the new pseudo-gas mixture, in which the interaction between the dust and gas molecules simulates the interaction between the solid matrix and the gas species. In addition, one is no longer faced with the problem of flux and composition variations across a pore and problems related to catalyst geometry. 

The kinematic viscosity of the fluid is related to the relaxation parameter through the relation v = c (t — l/2)i5(, which is obtained by means of a multiscale Chapman-Enskog analysis [5]. 

For show that ERB can use for describing the fluid s behavior, NS equations are derivate by process are named Chapman-Enskog s expansion or multi-scale analysis. It depends of Knudsen s number it was mentioned at the first part of this chapter it is the relation between the free mean trajectory and the characteristic length. 

We now foUow the Chapman-Enskog procedure. To lowest order the transport relations are those appropriate to local equilibrium. Since we have assumed the local velocity to vanish, the heat flux also vanishes in this approximation, and consequently the time derivative of the temperature vanishes. (Actually there is a contribution to the conservation law from the sources w, but this can be neglected at points far from a boundary.) Equation (101) then reduces to... 

Viscosity is a measure of fluid resisfance to mofion, and if relates fhe strain rate to applied shear stress. A functional dependence of gas viscosify on temperature at low density is given by Chapman-Enskog based on kinetic theory (Bird et al., 1960) using Lennard-Jones potentials. The theory has been also extended to multicomponent gas mixtures. For most common applications, however, a simplified semiempirical formula of Wilke (1950) is used ... 

The Chapman-Enskog kinetic theory actually gives general expressions for the transport properties in terms of the intermolecular potential energy which is related to the intermolecular force as expressed by (2.54) and (2.55). The molecular interaction is most frequently described by the empirical Lennard-Jones 12-6 potential. 

Thermal Diffusion. The existence of a concentration gradient in a gas mixture subject to a temperature gradient, thermal diffusion, was predicted by Enskog and by Chapman in the development of the kinetic theory of non-uniform gases. The phenomenon was demonstrated experimentally by Chapman and Dootson. The transport equation relates the separation, q, to the temperature gradient by the equation... 

This pressure tensor closure was derived by Gidaspow [22] in accordance with the Enskog theory presented by Chapman and Cowling [11], chap 16. That is, with the restitution coefficient e equal to one, the y factor substituted by go, and bp = Aad this relation corresponds to equation (16.34 — 2) in Chapman and Cowling [11]. 

The popular problems of kinetics theory is the derivation of hydrodynamic equations, in certain conditions, solution of f (r, v,t) transport equation is similar the form that can relate directly to continuous or hydrodynamic description. In certain conditions the transport process is like hydrodynamic limit. In 1911 David Hilbert was who ptropwsed the existence Boltzmann equations solutions (named normal solutions), and these are determinate by initial values of hydrodynamic variables it return to collision invariant (mass, momentum and kinetics energy), Sydney Chapman and David Enskog in 1917 were whose imroUed a systematic process for derivate the hydrodynamic equations (and their corrections of superior order) for these variables. 

When a gas mixture is subjected to a temperature gradient, diffusion occurs and a concentration gradient is established. Enskog (1911) and Chapman (1916) deduced the transport equation, which relates separation factor, 9, to the temperature gradient. 

A more accurate and rigorous treatment must consider the intermolecular forces of attraction and repulsion between molecules and also the different sizes of molecules A and B. Chapman and Enskog (H3) solved the Boltzmann equation, which does not utilize the mean free path X but uses a distribution function. To solve the equation, a relation between the attractive and repulsive forces between a given pair of molecules must be used. For a pair of nonpolar molecules a reasonable approximation to the forces is the Lennard-Jones function. 

On multiplying both sides of the first approximation equation of Chapman and Enskog, 75, by ipy, integrating with respect to dVy, summing over y, and rewriting with use of the well known relation dVydv = dy dVf, we obtain... 

The correction function x is related to the viral expansion of the equation of state and fitted to experimental data [55], Enskog ([37], p. 9) and Chapman and Cowling ([20], p. 299) gave an approximate relation for the value for x for a fairly rare uniform gas ... 

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