Toor-Stewart-Prober method

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 development of the Toor-Stewart-Prober method in Section 8.4 is based on the assumptions that the molar density and the matrix of Fick diffusion coefficients in the molar average reference velocity frame can be assumed constant along the diffusion path. Develop the theory anew in the mass average reference velocity frame that is, assuming and [D° ] can be considered constant. You will need to work with mass fluxes and mass diffusion fluxes 7). 

Repeat Example 8.6.1 (diffusion in a Stefan tube) using the Toor-Stewart-Prober method of determining the fluxes. Compare the profiles computed from the linearized equations to the profiles obtained with the exact method and the experimental data given in Example 2.2.1. 

Repeat Example 8.5.1 (evaporation into two inert gases) using the Toor-Stewart-Prober method of determining the fluxes. 

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). 

Numerical simulations of Sardesai s experiments are discussed by Webb and Sardesai (1981) and Webb (1982) (who used the Krishna-Standart (1976), Toor-Stewart-Prober (1964) and effective diffusivity methods to calculate the condensation rates), McNaught (1983a, b) (who used the equilibrium model of Silver, 1947), and Furno et al. (1986) (who used the turbulent diffusion models of Chapter 10 in addition to methods based on film theory). It is the results of the last named that are presented here. 

Repeat Example 8.7.1 using the linearized method of Toor-Stewart-Prober (1964) discussed in Section 8.4. 

In 1964 Toor and Stewart and Prober independently put forward a general approach to the solution of multicomponent diffusion problems. Their method, which was discussed in detail in Chapter 5, relies on the assumption of constancy of the Fick matrix [D] along the diffusion path. The so-called Tinearized theory of Toor, Stewart, and Prober is not limited to describing steady-state, one-dimensional diffusion in ideal gas mixtures (as we have already demonstrated in Chapter 5) however, for this particular situation Eq. 5.3.5, with [P] given by Eq. 4.2.2, simplifies to... 

We emphasize the difference between the Toor, Stewart, and Prober approach and the exact method considered above by using the subscript av thus, [Pavl- For practical purposes, this means that [Pavl has to be evaluated by employing suitable average mole fractions, av iri the definition of the (Eqs. 2.1.21 and 2.1.22). The arithmetic average mole fraction av = + F/sX normally is recommended for calculation of (Stewart... 

Both the exact solution and the Toor, Stewart, and Prober methods discussed above require an iterative approach to the calculation of the fluxes. In addition, the calculations are somewhat time consuming, especially when done by hand. It would be nice to have a method of calculating the fluxes that involved no iterations and yet was sufficiently accurate (when compared to these more rigorous methods) to be useful in engineering calculations. 

If Eqs. (5-200) and (5-201) are combined, the multicomponent diffusion coefficient may be assessed in terms of binary diffusion coefficients [see Eq. (5-214)]. For gases, the values Dy of this equation are approximately equal to the binary diffusivities for the ij pairs. The Stefan-Maxwell diffusion coefficients may be negative, and the method may be applied to liquids, even for electrolyte diffusion [Kraaijeveld, Wesselingh, and Kuiken, Ind. Eng. Chem. Res., 33, 750 (1994)]. Approximate solutions have been developed by linearization [Toor, H.L., AlChE J., 10,448 and 460 (1964) Stewart and Prober, Ind. Eng. Chem. Fundam., 3,224 (1964)]. Those differ in details but yield about the same accuracy. More recently, efficient algorithms for solving the equations exactly have been developed (see Taylor and Krishna, Krishnamurthy and Taylor [Chem. Eng. J., 25, 47 (1982)], and Taylor and Webb [Comput Chem. Eng., 5, 61 (1981)]. 

The solution of multicomponent diffusion problems is a little more complicated than the solution of binary diffusion problems because the differential equations governing the process are coupled. In the early 1960s a versatile and powerful method of solving multicomponent diffusion problems was developed independently by Toor (1964a) and by Stewart and Prober (1964). The method they proposed is described and illustrated in this chapter. 

Equations 5.1.6 represent a set of n - 1 coupled partial differential equations. Since the Fick matrix [ )] is a strong function of composition it is not always possible to obtain exact solutions to Eqs. 5.1.6 without recourse to numerical techniques. The basis of the method put forward by Toor and by Stewart and Prober is the assumption that c and [D] can be considered constant. (Actually, Toor worked with the generalized Fick s law formulation, whereas Stewart and Prober worked with the Maxwell-Stefan formulation. Toor et al. (1965) subsequently showed the two approaches to be equivalent.) With this assumption Eqs. 5.1.6 reduce to... 

The general method of solution that was proposed by Toor and by Stewart and Prober exploits the properties of the modal matrix [P] whose columns are the eigenvectors of [7)] (see Appendix A.4). The matrix product... 

The linearized theory of Toor (1964a) and of Stewart and Prober (1964) is probably the most important method of solving multicomponent diffusion problems. Very often, the method provides the only practical means of obtaining useful analytical solutions of multicomponent diffusion problems. Additional applications of the method are developed in Chapters 8-10 and still more can be found in the literature [see Cussler (1976), Krishna and Standart (1979) and Taylor (1982c) for sources]. 

We have developed the solution to the linearized equations as a special case of an exact solution in Section 8.3.5 in order to emphasize the close relationship that exists between the two methods. It should be noted, however, that this is not the way in which Toor or Stewart and Prober obtained their results. Indeed, these equations are not to be found in this form in the papers that first presented the linearized theory. Both Toor and Stewart and Prober obtained their results using the procedure described in Chapter 5 that is, by diagonalizing the matrix [D] and solving sets of uncoupled equations. The final result is... 

Toor (1964) and Stewart and Prober (1964) did not use the method presented above they used the method described in Chapter 5. For the multicomponent penetration model, the following expression for the matrix of mass transfer coefficients is obtained (cf. Section 8.4.2) ... 

A number of investigators used the wetted-wall column data of Modine to test multicomponent mass transfer models (Krishna, 1979, 1981 Furno et al., 1986 Bandrowski and Kubaczka, 1991). Krishna (1979b, 1981a) tested the Krishna-Standart (1976) multicomponent film model and also the linearized theory of Toor (1964) and Stewart and Prober (1964). Furno et al. (1986) used the same data to evaluate the turbulent eddy diffusion model of Chapter 10 (see Example 11.5.3) as well as the explicit methods of Section 8.5. Bandrowski and Kubaczka (1991) evaluated a more complicated method based on the development in Section 8.3.5. The results shown here are from Furno et al. (1986). 

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