Equimolal counterdiffusion

NTU (Number of Transfer Units) The NTU required for a given separation is closely related to the number of theoretical stages or plates required to cariy out the same separation in a stagewise or plate-type apparatus. For equimolal counterdiffusion, such as in a binary distillatiou, the number of overall gas-phase transfer units Nqg required for changing the composition of the vapor stream from yi to yo is... 

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... 

There are several factors that may be invoked to explain the discrepancy between predicted and measured results, but the discrepancy highlights the necessity for good pilot plant scale data to properly design these types of reactors. Obviously, the reaction does not involve simple first-order kinetics or equimolal counterdiffusion. The fact that the catalyst activity varies significantly with time on-stream and some carbon deposition is observed indicates that perhaps the coke residues within the catalyst may have effects like those to be discussed in Section 12.3.3. Consult the original article for further discussion of the nonisothermal catalyst pellet problem. 

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 ... 

Equimolal counterdiffusion between the phases, as in distillation with McCabe-Thiele approximations. 

Pick s law for steady-state equimolal counterdiffusion is then expression by Eq. (44)... 

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. 

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. 

In this section we indicate the predictions of the straight cylindrical pore model for isothermal reactions that are zero- or second-order in the gas phase concentration of reactant. Equimolal counterdiffusion is assumed (6 = 0). For a second-order reaction, a material balance on a differential element of pore length leads to the differential equation... 

Consider the packed distillation tower shown in Figure 16-1. Only binary distillation with constant molal overflow (CMO) will be considered. Let A be the more volatile conponent and B the less volatile component. In addition to making L/V constant and satisfying the energy balances, CMO automatically requires equimolal counterdiffusion, = -Ng. Thus, CMO sinplifies the mass balances, eliminates the need to solve the energy balances, and simplifies the mass transfer equations. We will also assume perfect plug flow of the liquid and vapor. This means that there is no eddy mixing to reduce the separation. 

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,... 

Equimolar counterdiffusion. Starting with the general equation (6.2-14), we can obtain for equimolal counterdiffusion where = —Ng, an equation similar to Eq. (6.1-11) for gases at steady state. 

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