Equimolal diffusion

The combined diffusivity is, of course, defined as the ratio of the molar flux to the concentration gradient, irrespective of the mechanism of transport. The above equation was derived by separate groups working independently (8-10). It is important to recognize that the molar fluxes (Ni) are defined with respect to a fixed catalyst pellet rather than to a plane of no net transport. Only when there is equimolar counterdiffusion, do the two types of flux definitions become equivalent. For a more detailed discussion of this point, the interested readers should consult Bird, Stewart, and Lightfoot (11). When there is equimolal counterdiffusion NB = —NA and... 

For diffusion in dilute non-reacting liquid systems which are not flowing (v = 0) and for equimolal counterdiffusion in ideal non-reacting gases at constant temperature and pressure (w = 0), Eqs. (50) and (51) simplify to ... 

Figure 13.44. Factors in Eqs. (13.239) and (13.240) for HTUs of liquid and vapor films and slopes m and m" of the combining Eqs. (13.235) and (13.236) [Bolles and Fair, Inst. Chem. Eng. Symp. Ser. 56(2), 3.3/3.S, (1979)]. (a) Definitions of slopes m and m" in Eqs. (13.235) and (13.236) for combining liquid and gas film HTUs / = 1 for equimolal counter diffusion / = (jtB)mean for diffusion through a stagnant film, (b) Factor (j> of the liquid phase Eq. (13.239). (c) Factor C of the liquid phase, Eq. (13.239). (d) Factor ip of the gas phase, Eq. (13.240), for metal pall rings.

Unsteady state diffusion in monodisperse porous solids using a Wicke-Kallenbach cell have shown that non-equimolal diffusion fluxes can induce total pressure gradients which require a non-isobaric model to interpret the data. The values obtained from this analysis are then suitable for use in predicting effectiveness factors. There is evidence that adsorption of the non-tracer component can have a considerable influence on the diffusional flux of the tracer and hence on the estimation of the effective diffusion coefficient. For the simple porous structures used in these tests, it is shown that a consistent definition of the effective diffusion coefficient can be obtained which applies to both the steady and unsteady state and so can be used as a basis of examining the more complex bimodal pore size distributions found in many catalysts. 

This is because of the effect of the total pressure gradient which develops which must be accounted for in the diffusion model. Clearly, although this may not play an important role in carefully designed experiments for determining diffusion coefficients using this technique, it has a considerable bearing on the use of such information where non-equimolal fluxes can arise, as with chemical reaction (6). 

The use of the effective diffusion coefficients in situations where a pressure gradient arises from non-equimolal fluxes, such as when chemical reactions occur, should then be based on the non-isobaric equations. Although this means that the models to be used are more complex, the parameters will be consistent. Where the pore size distribution is not monodisperse, the additional structural parameters which influence the effective diffusion coefficient will make the problem even more complex and requires further study. 

For the reaction A B, reaction and diffusion (at steady state) in a pore would require equimolal counterdiffusion that is, Ng = — Then a = 0, and the effective diffusivity is... 

This example illustrates the following point. The variation of D with depends on the importance of bulk diffusion. At the extreme where the Knudsen mechanism controls, the composition has no effect on D. When bulk diffusion is significant, the effect is a function of a. For equimolal counterdiffusion, a = 0 and yJ has no influence on D. In our example, where a = 0.741, and at 10 atm pressure, D increased only from 0.044 to 0.050 cm /sec as y increased from 0.5 to 0.8. 

W. Wakao and J. M. Smith [Chem. Eng. ScL, 17, 825 (1962)] thoroughly analyzed the diffusion data of Rothfeld. These data were obtained in an apparatus of the type shown in Fig. 11-1. For the butane-helium system this means that A He/ c is 3.80. Diffusion is far from equimolal, suggesting that Eqs. (11-26) and (11-27) for D values are not exact. For this particular case Eq. (11-2) should be used. In most reaction systems the counterdififusion of reactants and products is much closer to equimolal, so that Eqs. (11-26) and (11-27) are better approximations. 

Estimate the volume of bubble-free slurry required to obtain a conversion of 30% for a hydrogen feed rate of 100 ft /min (at 60°F and 1 atm). By a light-transmission technique, Calderbank measured gas-liquid interfacial areas of 0.94 to 2.09 cm /cm for bubble sizes likely to be encountered in this system. Suppose for this illustration Mg = 1.0 cm-/cm of bubble-free slurry. The Henry s law constant for hydrogen in toluene at 50°C is 9.4 (g mole/cm )/(g moles/cm ), and its diffusivity is 1.1 x 10 " cm /sec. The density and viscosity of toluene at 50°C are 0.85 g/cm and 0.45 centi-poises, respectively. Equimolal feed rates of ethylene and hydrogen will be used. 

Steady-State Equimolal Counterdiffusion and Unimolal Unidirectional Diffusion... 

Equation (7.1-16) reduces to two special cases of molecalar difiiision which are customarily considered. In equimolal counterdiffusion, component A diffuses through component B, which is diffusing at the same molal rate as A relative to stetionaiy coordinates, but in die opposite direction. This process is often approximated in the distillation of a binary system. In unimolol unidirectional diffusion, only one molecalar species—component A—diffuses through component B, which is motionless relative to stationary coordinates. This type of transfer is approximated frequently in the operations of gas absorption, liquid-liquid extraction, and adsorption. 

There are several types of situations covered by Eq, (21.16). The simplest case is zero convective flow and equimolal counterdiffusion of A and B, as occurs in the diffusive mixing of two gases. This is also the case for the diffusion of A and B in the vapor phase for distillations that have constant molal overflow. The second common case is the diffusion of only one component of the mixture, where the convective flow is caused by the diffusion of that component. Examples include evaporation of a liquid with diffusion of the vapor from the interface into a gas stream and condensation of a vapor in the presence of a noncondensable gas. Many examples of gas absorption also involve diffusion of only one component, which creates a convective flow toward the interface. These two types of mass transfer in gases are treated in the following sections for the simple case of steady-state mass transfer through a stagnant gas layer or film of known thickness. The effects of transient diffusion and laminar or turbulent flow are taken up later. 

EQUIMOLAL DIFFUSION. For equimolal diffusion in gases, the net volumetric and molar flows are zero, and Eq, (21.16) or Eq. (21.17) can be used with the convective term set to zero, which makes them equivalent to Eq. (21.6). Using a... 

Concentration gradients for equimolal and unicomponent diffusion ( ) components A and B diffusing at same molal rates in opposite directions (b) component A diffusing, component B stationary with respect to interface. 

Example 21.1. (a) For the diffusion of solute A through a layer of gas to an absorbing liquid, with = 0.20 and y, - = 0.10, calculate the transfer rate for one-way diffusion compared to that for equimolal diffusion. (6) What is the value oiy halfway through the layer for one-way diffusion ... 

In this case the transfer rate with one-way diffusion is about 18 percent greater than that with equimolal diffusion. 

The significance of k is brought out by combining Eq. (21.31) with Eq. (21.20) for steady-state equimolal diffusion in a stagnant film. This gives... 

In the absence of a fiilly developed kinetic theoiy for liquids, the relationships lor molecular diflusion are usually assumed to parallel those for gases, although difliisivities ate often mote substantially dqiendeni on concentration of the diffusing components. In the case of equimolal courtterdifliision, die expression analogous to Eq. (7.1-20) is... 

Equimolal Counterdiffusion. This leads to Fick s law. If Na and NB are the number of moles of A and B, respectively, diffusing per unit time through a cross section B,... 

Diffusion of Equimolal A through (x>UDterdiffusion nondiffusing B Units of coefficient... 

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