Diffusion equimolar counter

In a mixture of two gases A and B provided that the total pressure, and hence the total molar concentration is constant, d[A]/dx and d[B]/dx are equal and opposite and A and B tend to diffuse in opposite directions. In equimolar counter-diffusion, the two components diffuse at equal and opposite rates =- ba- Equation (108) can then be integrated 

---

Condition (iv). For equimolar counter diffusion in a binary mixture, the... 

In dilute systems the logarithmic-mean insoluble-gas and nonvolatile-hquid concentrations approach unity, and Eq. (5-261) reduces to the dilute-system formula. For equimolar counter diffusion (e.g., binary distillation), these log-mean factors should be omitted. See Eq. (5-189). 

For small values of [A], this reduces to eqn. (110) which was derived from equimolar counter-diffusion and thus, at low concentration, the effect of bulk flow can be neglected. 

It should be noted that D g is a property of the mixture of A and B, and is defined with reference to the mixture and not to the fixed coordinates. Except in the case of equimolar counter-diffusion of A and B, the diffusion of A would result in the movement of the mixture as a whole. However, in the usual case where the concentration of A is small, the value of ) g is practically equal to the value defined with reference to the fixed coordinates. 

It is easy to define the number of transfer units for equimolar counter-diffusion (distillation) and for dilute systems, when the mole fraction of the solute is so small that the... 

Equimolar Counter-Diffusion. The diffusion of component A in a mixture to and from the interface is balanced by an equal and opposite molar flow of component B, so that there is no net molar flow, thus ... 

The essential difference between distillation and absorption is that in the former, the vapour has to be introduced in each stage by partial vapourisation of the liquid, which is therefore at its boiling point, whereas in absorption the liquid is well below its boiling point. In distillation, there is diffusion of molecules in both directions, so that for an ideal system, equimolar counter-diffusion exists. In absorption, gas... 

Mass transfer in catalysis proceeds under non-equilibrium conditions with at least two molecular species (the reactant and product molecules) involved [4, 5], Under steady state conditions, the flux of the product molecules out of the catalyst particle is stoi-chiometrically equivalent (but in the opposite direction) to the flux of the entering reactant species. The process of diffusion of two different molecular species with concentration gradients opposed to each other is called counter diffusion, and if the stoichiometry is 1 1 we have equimolar counter diffusion. The situation is then similar to that considered in the case of self-or tracer diffusion, the only difference being that now two different molecular species are involved. Tracer diffusion may be considered, therefore, as equimolar counter diffusion of two identical species. 

This method provides the exact solutions for ideal systems at constant temperature and pressure. It is successful in describing diffusion flow in (i) nearly ideal mixtures, (ii) equimolar counter diffusion where the total flux is zero (Nt = 0), (iii) diffusion of one component through a mixture of n — 1 inert components, and (iv) pseudo-binary case and the diffusion of two very similar components in a third. 

By using Fick s and Fourier s laws in one-dimensional transport in a slab catalyst pellet (Figure 9.1) with, equimolar counter-diffusion under mechanical equilibrium, Eqs. (9.14) and (9.15) become... 

Example 9.3 Effectiveness factor for first-order irreversible reaction-diffusion system Consider a first-order reaction occurring on the pore walls of a catalyst with equimolar counter diffusion. Assume that isothermal conditions are maintained, and a catalyst with simple slab geometry is used (Figure 9.1). If the -coordinate is oriented from the centerline to the surface, the steady-state reaction diffusion equation for reaction A — B between reactant A and product B is... 

The effective transition diffusivity is calculated from the Bosanquet equation assuming equimolar counter diffusion, which is what happens with isomerization reactions ... 

Equimolar counter diffusion appears in the distillation of binary mixtures. In a distillation column the liquid falls downwards, and the vapour flows upwards, Fig. 1.43. As the liquid flowing down the column is colder than the vapour flowing upwards, chiefly the component with the higher boiling point, the so called least volatile component condenses, whilst the vapour from the boiling liquid mainly consists of the components with the lower boiling points, the more volatile components. The molar enthalpy of vaporization is, according to Trouton s rule, approximately constant for all components. If a certain amount of the least volatile component condenses out from the vapour, then the same number of moles of the more volatile substance will be evaporated out of the liquid. At the phase boundary between liquid and vapour we have cAwA = —cBwB. The reference velocity u is zero because cu = cAwA + cBwB. The molar flux transported to the phase boundary from (1.158) and (1.160) is... 

Let us assume that two containers, each containing a different gas are linked by a thin pipe between them, Fig. 1.44. Equimolar counter diffusion will also take place in this case, if the pressure and temperature of both the gases are the same and obey the thermal equation of state for ideal gases. 

The value of a varies with the system under consideration. For example, in equimolar counter diffusion, Na and Nb are of the same magnitude, but in opposite direction. As a result, a is equal to 1 and hence, Eq. (2) reduces to Eq. (1), where is equal to Convective mass transfer coefficients are used in the design of mass transfer equipment. However, in most cases, these coefficients are extracted from empirical correlations that are determined from experimental data. The theories, which are often used to describe the mechanism of convective mass transfer, are the film theory, the penetration theory, and the surface renewal theory. 

Dependent upon the mode of mass diffusion, i.e., diffusion through a nondiffusing layer, counter-diffusion, or equimolar counter-diffusion, the relationship between the mass transfer coefficient and the diffusivity may take different forms. For example, under the condition of equimolar counter diffusion for a binary system, the molar flux in x-direction can be expressed in terms of the diffusivity as ... 

Equimolar Counter-Diffusion vs. Diffusion through Stagnant Fluid... 

Case I If we use a capillary tube to connect two gas reservoirs having the same total pressure but different amounts of gases A and B (see Fig. 6.2-1), we will obtain equimolar counter-diffusion of A and B (i.e. N z - -Ng-. In this way, the mole-average velocity v = 0 and the total pressure in both reservoirs remains constant. For N z = -Ngz (125) gives... 

For the same driving force (concentration gradient), J is the same for both cases, but different. Since i-y is always less than one, we see that equimolar counter-diffusion (126) is slower than diffusion through a stagnant fluid (127). This can be qualitatively understood as follows. Suppose that to get to class, you need to walk down a corridor that s crowded with other students. If everyone else was standing almost still (stagnant fluid), it would be easier to walk around them than if everyone is walking toward you (counter diffusion). 

All of the flux expressions above apply locally at every point in the fluid. For 1-D steady equimolar counter-diffusion (from a reservoir having to a second reservoir having c 2) (126) integrates to... 

We will first consider equimolar counter diffusion (EMCD). 

Equimolar Counter Diffusion. In equimolar counter diffusion (EMCD). for every mole of A that diflfuse.s in a given direction, one mole of B diffuses in the opposite direction. For example, consider a species A that is diffusing at steady state from the bulk fluid to a catalyst surface, where it isomerizes to form B. Species B then diffuses back into the bulk (see Figure 11-1). For every mole of A that diffuses to the surface. 1 mol of the isomer B diffu.ses away from the surface. The fluxes of A and B are equal in magnitude and flow counter to each other. Stated mathematically,... 

In which N k is the molar flux of component k in the pore. For binar> systems components A and B) with equimolar counter diffusion, Equa tion 7.6 reduces to... 

Equimolar counter diffusion, EMD, in which the number of moles of A diffusing in one direction equals the number of moles of B diffusing in the opposite direction that is, = —N, and (16-2) reduces to... 

It will be recalled from transport phenomena, we know that it is most useful to define a mass transfer coefficient to describe only the diffusive transport and not the total diffusive plus convective. The coefficients are identical only for the special case of equimolar counter-diffusion and this is the value of the coefficient k/, which is actually correlated in handbooks. 

Let us first consider diffusion in an idealized single cylindrical pore. Pick s law for a binary system with equimolar counter diffusing occurring is ... 

The restriction to equimolar counter diffusion should be removed. 

If the diffusivity does not change too much with composition (actually, an average diffusivity is computed corresponding to the average composition between initial and final state), the transient form of Pick s law for equimolar counter diffusion and constant molar density is... 

When (5a = 0 that represents equimolar counter diffusion, then yfA = 1 ... 

---