Effective diffusivity methods

The oldest, simplest, and still widely used methods, pioneered by Hougen and Watson (1947) and by Wilke (1950), employ the concept of an effective diffusion coefficient. The effective diffusivity concept was discussed in detail in Chapter 6 here we show how the effective diffusivity can be used to calculate mass transfer rates. 

The starting point for this analysis is the one-dimensional form of Eq. 6.1.1 

eff be assumed constant at some suitably averaged composition then the composition profiles are easily obtained as 

With the composition derivatives obtained from Eq. 8.6.2 we may define an effective mass transfer coefficient by 

The molar fluxes are obtained from the appropriate bootstrap relation as discussed in Chapter 1. 

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Solution of Multicomponent Diffusion Problems Effective Diffusivity Methods... 

There is a second class of effective diffusivity method in which is defined with... 

The flux of hydrogen (component 1) is not too different from the flux estimated using the linearized equations in Example 5.3.1. However, the effective diffusivity method predicts a very small flux of nitrogen (component 2), a result quite different from the predictions of the linearized theory. This, of course, is because the effective diffusivity method ignores the contribution due to the driving forces of the other components. We will investigate the consequences of this prediction in Example 6.4.1. ... 

Let us now proceed to see if the Wilke effective diffusivity method is able to model the diffusional process in the two bulb diffusion cell experiment of Duncan and Toor (1962). For convenience, we repeat the following information from Example 5.4.1. The two bulbs in the apparatus built by Duncan and Toor had volumes of 77.99 and 78.63 cm3, respectively. The capillary tube joining them was 85.9 mm long and 2.08 mm in diameter. The entire device... 

Figure 6.1 shows the concentration time history in the diffusion cell for the experiment of Duncan and Toor that was described in detail in Example 5.4.1. The mole fraction of hydrogen predicted by the effective diffusivity model compares well with the experimental data of Duncan (1960). However, the effective diffusivity model suggests that the mole fraction of nitrogen should remain almost constant at approximately 0.5. This is in stark contrast to the experimental data (Fig. 6.1). The results obtained with the effective diffusivity method for nitrogen are completely different from those obtained with the linearized theory. Additional comparisons between the data of Duncan and Toor and the predictions of both the linearized equations and the effective diffusivity models are shown in the triangular diagram in Figure 6.2.

These results are sufficiently different that we ought to be able to determine the better model by comparing the results to the experimental data. The complete concentration time history is shown in Figure 6.3 for this experiment. Note that the mole fraction of methane predicted by the effective diffusivity method is in reasonable agreement with the data (although the results from the linearized equations are better). However, the effective diffusivity method predicts almost no change in the mole fraction of argon a result that is in marked contrast to both the experimental data and to the predictions of the linearized equations. 

Additional data of Arnold and Toor are compared to the predictions of the linearized equations and of the effective diffusivity models in the triangular diagram in Figure 6.4. Clearly, the agreement with the data is very bad indeed. Thus, we have our second demonstration of the inability of the effective diffusivity method to model systems that exhibit strong diffusional interactions. ... 

Figure 6.4. Comparison between Loschmidt tube experiments of Arnold and Toor (1967) and the composition trajectories predicted by the linearized theory and effective diffusivity methods.

Algorithm 8.7 Algorithm Based on Repeated Substitution for Calculation of Mass Transfer Rates from an Effective Diffusivity Method... 

In Example 2.1.1 we described the experiments of Carty and Schrodt (1975) who evaporated a binary liquid mixture of acetone(l) and methanol(2) in a Stefan tube. Air(3) was used as the carrier gas. Using an effective diffusivity method calculate the composition profiles. 

If the nonzero fluxes have the same sign (i.e., they are all in the same direction), then effective diffusivity methods are more likely to give reasonable results. This is nearly always the case in condensation and absorption processes and this goes some way at least to explaining why effective diffusivity methods usually give good estimates of the total amount condensed and the total heat load even if the individual condensation rates are not so well predicted. Webb et al. (1981) discussed in detail the conditions that must apply for an effective diffusivity method to be a useful model in multicomponent condensation. 

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... 

In fact, through use of matrix models of mass transfer in multicomponent systems (as opposed to effective diffusivity methods) it is possible to develop methods for estimating point and tray efficiencies in multicomponent systems that, when combined with an equilibrium stage model, overcome some of the limitations of conventional design methods. The purpose of this chapter is to develop these methods. We look briefly at ways of solving the set of equations that model an entire distillation column and close with a review of experimental and simulation studies that have been carried out with a view to testing multicomponent efficiency models. 

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). 

Interaction effects are most important in systems containing species whose molecular size and nature differ widely, as is the case in Examples 2 and 4. Temperature, composition, and flux profiles for Example 4 are shown in Figures 15.16-15.18. There are significant differences between the matrix methods and the effective diffusivity methods. Without hydrogen in the mixture (Example 3) all of the models give very similar results. This result... 

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. 

We see from these figures that the mass transfer models that take diffusional interactions into account are quite a lot better than the effective diffusivity model, which underpredicts the rate of condensation of 2-propanol in every case. However, the effective diffusivity methods give good predictions of the overall temperature drops (Fig. 15.19) although there is little to distinguish any of the models here on this basis. 

Show that the effective diffusivity method leads to composition profiles in ternary systems that are straight lines when plotted on triangular diagrams. 

Repeat Exercise 5.5 using an effective diffusivity method. 

Repeat Example 8.3.2 (diffusional distillation) using an effective diffusivity method for determining the fluxes and composition profiles. Compare your results to those given in Example 8.3.2. 

Repeat Example 8.4.1 (dehydrogenation of ethanol) using an effective diffusivity method of determining the fluxes. 

Simplify the definition of the effective diffusivity given by Eq. 6.1.7 for the special case when two molar fluxes are zero, = N2 0. What is the relationship between the effective diffusivity method and the method of Section 8.5.2 for this special case. 

Repeat Example 8.7.1 using the effective diffusivity methods you derived in Exercise 6.1. Compare the results with those obtained in Example 8.7.1 and in Exercise 8.43. 

Frey, D. D., Prediction of Liquid Phase Mass Transfer Coefficients in Multicomponent Ion Exchange Comparison of Matrix, Film-Model, and Effective Diffusivity Methods, Chem. Eng. Commun., 41, 273-293 (1986). 

Correction factor for high fluxes in pseudobinary (effective diffusivity) methods [-]... 

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