Method of Taylor and Smith

The method of Taylor and Smith (1982) is a generalization of the method of Burghardt and Krupiczka for Stefan diffusion. We use the determinacy condition (Eq. 7.2.10) to eliminate the nth flux from the Maxwell-Stefan relations (Eq. 2.1.16) and combine the first n-1 equations in matrix form as 

It can be shown by inverting the bootstrap matrix [j8] using the Sherman-Morrison 

That is, the matrices [A] and [B][ ] are equal and not simply equivalent. Thus, the explicit method of Krishna could equally well be written in the form 

The generalization of the method of Burghardt and Krupiczka is based on the assumption that the matrix [A] divided by the mole fraction weighted sum of the 

The only difference between Eq. 8.5.6 obtained by Krishna and Eq. 8.5.18 obtained by Burghardt and Krupiczka (1975) and its generalization by Taylor and Smith (1982) is the inclusion of the scalar correction factor 3. For equimolar countertransfer (N = 0  

---

The great advantage of the methods described in this section over those described earlier is, of course, rapidity in computation. This gain in computational simplicity is, however, at the expense of theoretical rigor. It is, therefore, important to establish the accuracy of the methods described above using the exact method of Section 8.3 as a basis for comparison. The extensive numerical computations made by Smith and Taylor (1983) showed that the explicit method of Taylor and Smith ranked second overall among seven approximate methods tested (the linearized method of Section 8.4 was best). For some determinacy... 

Extensions of the method of Taylor and Smith (1982) are described by Kubaczka and Bandrowski (1990) and by Taylor (1991). 

Show that the explicit method of Taylor and Smith (1982) (discussed in Section 8.5) is an exact solution of the Maxwell-Stefan equations if all the binary diffusion coefficients are equal. Solutions are given by Burghardt (1984) and Taylor (1984). 

We thus have evidence that the differences between bond energies vary in simple hydro-carbons and in substituted hydrocarbons but so far this leaves us in the dark as to the variations of the individual values. Definite indications—if only of an approximate nature—of wide variations in bond energies were first postulated by Ogg and Polanyi from the variations in the rates of reaction between organic halides and sodium vapour observed by Hartel and Polanyi.More recently H. S. Taylor and Smith derived similar conclusions from the marked variations in the rate of reaction of methyl radicals with hydrocarbons. A fall in the bond energy of C—H was quite recently confirmed and quantitatively fixed by more direct methods. D. P. Stevenson has given the values as D CHs—H) = loi and D CjH5—H) = 96 while independently and by different methods Anderson, Kistiakowsky and van Arstdalen obtain D(CH3—H) = 102 and D(C H5—H) = 98 kcal. 

The flow patterns for single phase Newtonian and non-Newtonian fluids in tanks agitated by class 1 impellers have been reported in the literature by, amongst others, Metzner and Taylor [1960], Norwood and Metzner [1960], Godleski and Smith [1962] and Wichterle and Wein [1981]. The experimental methods used have included the introduction of tracer liquids, neutrally buoyant particles or hydrogen bubbles and measurement of local velocities by... 

Systems with highly nonideal VLE suffer from requiring very good initial profiles The sneaking-up technique can be used by first solving the column with a simple approximation of the VLE and then slowly introducing the nonideal VLE. This is described by Brierley and Smith (106) and is also the thermodynamic homotopy of Vickery and Taylor (81). As stated in Secs. 4.2.9 and 4.2.12, this can occur in the global Newton methods. The inside-out methods avoid these problems in their use of simple VLE models. 

These three formulae work for (almost) the whole field of values that undergo diffusional changes, up to the boundary lines. There is a problem area, as mentioned above, at / = 0, where the above approximation cannot be used, due to the singularity. This has been addressed by Crank and Furzeland [40] and again by Gavaghan [44]. The method they used is also described in detail by Smith [265]. It is the following. Expand (dC/dR) at some small R, using Maclaurin s expansion (a special case of Taylor s expansion) ... 

One especially good use for the Taylor-Smith/Burghardt-Krupiczka method is to generate initial estimates of the fluxes for use with the Krishna-Standart or Toor-Stewart-Prober methods. It is a very rare problem that requires more than two or three iterations if Eq. 8.5.26 is used to generate initial estimates of the fluxes (Step 3 in Algorithm 8.2) (Krishnamurthy and Taylor, 1982). 

The effective diffusivity formula of Stewart (Eq. 6.1.8) is by far the best of this class of methods. This should not come as a surprise since this method is capable of correctly identifying the various interaction phenomena possible in multicomponent systems. Indeed, for equimolar countertransfer, this effective diffusivity method is equivalent to the linearized theory and to both explicit methods discussed above. In fact, for some systems Stewart s effective diffusivity method is superior to Krishna s explicit method (Smith and Taylor, 1983). However, since the explicit methods are actually simpler to use than Stewart s effective diffusivity method (all methods require the same basic data) and, in general... 

A comparison of the film models that ignore diffusional interaction effects (the effective diffusivity methods) with the film models that take multicomponent interaction effects into account (Krishna-Standart (1976), Toor-Stewart-Prober (1964), Krishna, (1979b, c) and Taylor-Smith, 1982). 

---