Solution of the Multicomponent Diffusion Equations

The analysis of turbulent eddy transport in binary systems given above is generalized here for multicomponent systems. The constitutive relation for j y in multicomponent mixtures taking account of the molecular diffusion and turbulent eddy contributions, is given by the matrix generalization of Eq. 10.3.1 

Let us proceed to integrate the differential Eq. 10.4.3. As in our analysis of binary mass 

The solution to the matrix differential Eq. 10.4.6 can be found using the method of successive substitution (Appendix B.2). Here we follow closely the treatment by Taylor (1981b) (see, also Krishna, 1982). The solution to Eq. 10.4.6 can be written as 

Let us now consider the evaluation of the matrizant [ll ( )]. The matrix [ (y )] is given by Eq. 10.4.5 and it is easy to see that the inverse matrix [ (y )] exhibits a very simple dependence on the position coordinate y ([Sc] is assumed to be constant in our model) only the diagonal elements of [ (y )] are position dependent. Furthermore, the position dependence is the same for all the principal diagonal elements because turb/ is not species dependent. The matrices [Sc] [[Sc] + Sc Jb( turb/ )[ ]], [ (y )] and [/l(y )] all commute with each other and with /[/l(y )1 dy. All this means is that the integrations required by Eq. B.2.16 can be carried out by parts to give 

Taking the upper limit of this integral to be, the reduced distance from the wall at which the bulk compositions (ft,) are attained, we define a matrix of rate factors [O] by 

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Despite these differences both solutions of the multicomponent diffusion equations will give identical results if... 

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