Matrix mass transfer coefficients

In general, all elements of the mass transfer matrix depend on the process variables, and in particular on the vapor phase composition. The mass transfer mechanisms in membranes can be rather complicated. However, for the conceptual analysis of the considered membrane process, it is not advantageous to go into the details of mass transport. Therefore, in the following the effective binary mass transfer coefficients k,j are assumed to be constants. 

In analogous manner, residue curve maps of the reactive membrane separation process can be predicted. First, a diagonal [/e]-matrix is considered with xcc = 5 and xbb = 1 - that is, the undesired byproduct C permeates preferentially through the membrane, while A and B are assumed to have the same mass transfer coefficients. Figure 4.28(a) illustrates the effect of the membrane at nonreactive conditions. The trajectories move from pure C to pure A, while in nonreactive distillation (Fig. 4.27(a)) they move from pure B to pure A. Thus, by application of a C-selective membrane, the C vertex becomes an unstable node, while the B vertex becomes a saddle point This is due to the fact that the membrane changes the effective volatilities (i.e., the products xn a/a) of the reaction system such that xcc a. ca > xbbO-ba-... 

As demonstrated by means of residue curve analysis, selective mass transfer through a membrane has a significant effect on the location of the singular points of a batch reactive separation process. The singular points are shifted, and thereby the topology of the residue curve maps can change dramatically. Depending on the structure of the matrix of effective membrane mass transfer coefficients, the attainable product compositions are shifted to a desired or to an undesired direction. 

Here, k/, is called the matrix of high flux mass transfer coefficients, and is defined by... 

The [I] is the identity matrix. When N, -> 0, the matrix of correction factors [Mf] reduces to the identity matrix, and the matrix [k ] becomes k0], which is defined as the matrix of zero flux mass transfer coefficients... 

Step 1 Calculate the matrix of zero flux mass transfer coefficients from the inversion of the matrix [B0], which is obtained from Eqs. (6.70), (6.71), and (6.77)... 

Step 1 From Eqs. (6.89) to (6.92), estimate the matrix of zero flux mass transfer coefficients for both the liquid and vapor phases at the interface [kx ] and [ky ]. Since the interface compositions are needed, first assume a value for xn. 

The extension of ideal phase analysis of the Maxwell-Stefan equations to nonideal liquid mixtures requires the sufficiently accurate estimation of composition-dependent mutual diffusion coefficients and the matrix of thermodynamic factors. However, experimental data on mutual diffusion coefficients are rare, and prediction methods are satisfactory only for certain types of liquid mixtures. The thermodynamic factor may be calculated from activity coefficient models such as NRTL or UNIQUAC, which have adjustable parameters estimated from experimental phase equilibrium data. The group contribution method of UNIFAC may also be helpful, as it has a readily available parameter table consisting of mam7 species. If, however, reliable data are not available, then the averaged values of the generalized Maxwell-Stefan diffusion coefficients and the matrix of thermodynamic factors are calculated at some mean composition between x0i and xzi. Hence, the matrix of zero flux mass transfer coefficients [k ] is estimated by... 

It was shown I8 that the binding process is the rate-limiting step, with an adsorption rate constant of kd = 4.8 x 104 dm mol s. The calculated mass transfer coefficient for the diffusion into the pores of the support contributes 11% to the overall adsorption process. The value of ka is an order of magnitude lower than that reported in Table 2. The binding properties of the polyclonal immunoadsorbents used in these two studies may differ because of the different methods employed for protein immobilization. Another possible explanation may be an underestimation of the contribution for the diffusion rate-limiting step as the polyclonal anti-HSA antibody was attached to a silica matrix of large pores [18]. 

The transformed mass transfer coefficients may be predicted as follows from the results of Section 3.2 and the ith eigenvalue of the matrix A introduced above ... 

DA TA The matrix of multicomponent volumetric mass transfer coefficients [ AT ] is given in Example 5.6.1. as... 

The development for multicomponent mixtures is best carried out by using n — l dimensional matrix notation. We, therefore, define a matrix of finite flux mass transfer coefficients [fc ] by... 

With the finite flux mass transfer coefficients matrix defined by Eq. 7.1.9 we have... 

The first step is to estimate the molar fluxes. This can be done as described above and elsewhere (Section 8.5). The mass transfer coefficients are calculated and the values of the discrepancy functions evaluated. To reestimate the molar fluxes we must evaluate the Jacobian matrix [J]. The elements of this matrix are obtained by differentiating the above equations with respect to the independent variables. These derivatives may be approximated by... 

An alternative method is summarized in Algorithm 8.5 based on the fact that we really only need to determine the total flux in order to completely determine the matrix of mass transfer coefficients and, therefore, the molar fluxes N-. We start from Eq. 7.2.22 and substitute for (7) using Eqs. 8.4.23 and 8.4.24 to get... 

In order to evaluate the discrepancy functions F -F we must compute the matrix of finite flux mass transfer coefficients. The procedure is illustrated below. 

The eigenvalues of the matrix of low flux mass transfer coefficients are computed from Eq. 8.4.26 as... 

In their original development of the linearized theory Toor (1964) and Stewart and Prober (1964) proposed that correlations of the type given by Eqs. 8.8.5 and 8.8.7 could be generalized by replacing the Fick diffusivity D by the charactersitic diffusion coefficients of the multicomponent system that is, by the eigenvalues of the Fick matrix [ >]. The mass transfer coefficient calculated from such a substitution would be a characteristic mass transfer coefficient an eigenvalue of [/c]. For example, the Gilliland-Sherwood correlation (Eq. 8.8.5) would be modified as follows ... 

This approach is, in fact, equivalent to replacing the binary diffusivity D by the matrix of multicomponent diffusion coefficients [D] and the binary mass transfer coefficient with the... 

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