Krishna-Standart method

Run 7 of the simulations of Modine s experiments is particularly sensitive to the mass transfer model used. This experiment was used as the basis for the flux calculation in Example 10.4.1. For this experiment none of the models does well at predicting the total amount of acetone transfer. The Krishna-Standart method, the linearized theory and the effective diffusivity model predict the wrong direction of mass transfer while the experimental data show that there is net vaporization of acetone, the models predict net condensation. This erroneous prediction comes about in part because of the extremely small magnitude of the acetone flux relative to the total flux. 

Revise the analysis of Example 11.5.3 and show how a method based on the film models of Chapter 8 could be used to compute the rates of mass transfer. Then use the Krishna-Standart method (of Sections 8.3 and 8.8.3) and compute the molar fluxes. Binary pair mass transfer coefficients may be estimated using the Chilton-Colburn analogy. 

Diffusional interaction methods have also been applied successfully to packed columns. Gorak (204) found that the Krishna-Standart model (205) is relatively simple and sufficiently accurate to predict multicomponent composition profiles. Gorak s own variation of the diffusional interaction method was also reported to predict experimental data well, while use of HETP was reported to give poor data predictions. 

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). 

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. 

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). 

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]. 

SOLUTION We shall use the method of Krishna and Standart described in Section 8.3 to compute the molar fluxes. Algorithms 8.3.1 or 8.3.2 may be used to determine the molar fluxes. Convergence is very rapid in this example. No more than two iterations are needed. Only the final results of the relevant computations are summarized below. 

Repeat Example 8.8.1 (ternary distillation in a wetted wall column) using the method of calculating the fluxes developed in Section 8.3 but following the suggestion of Krishna and Standart as described in Section 8.8.3 for estimating the binary mass transfer coefficients. 

Krishna, R. and Standart, G. L., A Multicomponent Film Model Incorporating an Exact Matrix Method of Solution to the Maxwell-Stefan Equations, AIChE J, 22, 383-389 (1976a). 

In recent years, considerable progress has been made to improve further upon this method for use with multicomponent mixtures. Detailed discussions of these methods may be found in Stephan [25], Hewitt et al. [193], and Webb [194]. The procedure of Sardesai et al. [205], which outlines the work of Krishna and Standart [206], is briefly described below. 

R. Krishna and G. L. Standart, A Multicomponent Film Model Incorporating a General Matrix Method of Solution to Maxwell-Stephan Equations, AIChE I, 22, pp. 383-389,1976. 

To relate the multicomponent mass-transfer rates to binary mass-transfer coefficients, the method of Krishna and Standart [71] can be used. By this method, the diffusional fluxes for the liquid phase are calculated from... 

In plate column distillation the basic performance concept is that of plate efficiency, usually the Murphree type, although the Hausen and Standart variations have occasionally been recommended and used. The mass transfer process enters the plate efficiency concept through the relation between the flux, and hence mass transfer coefficients, and the point efficiency which, in turn, is related to the plate efficiency by flow behaviour on and between the plates. Since in multicomponent mixtures the mass transfer coefficients of the components are different, the same must apply to the point efficiencies and also to the plate efficiencies. Again, the method of Krishna and Standart can be used to predict plate efficiencies of individual components and it has been successfully tested for ternary mixtures. However, it has not as yet been accepted in general distillation practice. 

Expanding the Maxwell-Stefan equations to systems with more conponents is straightforward. Matrix solution methods for these multicon onent systems were originally developed by Krishna and Standart (1976). The generalized form for diffusion in a film of thickness 6 is. 

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