Matrix Sylvester expansion formula

The method of successive substitution can be a very effective way of computing the from Eqs. 8.3.24 when the mole fractions at both ends of the diffusion path y-g and y g, are known. In practice, we start from an initial guess of the fluxes and compute the rate factor matrix [< >]. The correction factor matrix [a] may be calculated from an application of Sylvester s expansion formula (Eq. A.5.20)... 

The approach of Toor, Stewart, and Prober and that of Krishna and Standart are equivalent for the two limiting cases discussed in Section 8.3.1 ideal gas mixtures in which the binary D-j sltq equal or very dilute mixtures. In our experience the two approaches almost always give virtually identical results [although Young and Stewart (1986) might disagree]. We, therefore, recommend the use of [ vl [7 av] in view of the ease of computation and because the results are sufficiently accurate. In the Toor-Stewart-Prober approach we need to evaluate fractional powers of matrices. This calculation can be done using Sylvester s expansion formula or the modal matrix transformation approach but either way is rather more involved than the direct use of binary k-j in the calculation of [ ] (or equivalent matrix). 

SOLUTION For the purposes of this calculation we shall assume the drop to be noncirculating. Thus, the matrix of multicomponent mass transfer coefficients [A] may be computed from Eqs. 9.4.18 and 9.4.19 with the help of Sylvester s expansion formula or the modal transformation. At both the long and short contact time limits, however, we may calculate the ratios of mass transfer coefficients ky2/kii and A 21/A 22 without evaluating the series expansions needed in Eq. 9.4.19. 

Equation A.6.1 arises when the Maxwell-Stefan equations are solved for the case of steady-state, one-dimensional mass transfer, as discussed in Chapter 8. The matrices [A ] and [O] are as defined in Chapter 8, is the molar density of the mixture and a scalar, and (Ax) is a column matrix of mole fraction differences. All matrices in Eq. A.6.1 are of order n — 1 where n is the number of components in the mixture. For the purposes of this discussion we shall assume that the matrices [/ ] and [O] have already been calculated. The matrix function [0][exp[] - [7]] denoted by [2], can be computed using Sylvester s expansion formula (see, however, below) so the immediate problem is the calculation of the column matrix (7) from... 

Acrivos and Amundson (SO, 51) applied matrix algebra to the unsteady-state behavior of stagewise operations in chemical processes. Instead of using a characteristic vector expansion, they emphasize the use of the Sylvester-Lagrange-Buchheim formula (52). Even though this formula is equivalent to the characteristic vector expansion, it is more difficult to manipulate and is not easily related to physical concepts such as straight line reaction paths. 

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