Diffusion equimolar counterdiffusion

Equimolar Counterdiffusion in Binary Cases. If the flux of A is balanced by an equal flux of B in the opposite direction (frequently encountered in binary distillation columns), there is no net flow through the film and like is directly given by Fick s law. In an ideal gas, where the diffusivity can be shown to be independent of concentration, integration of Fick s law leads to a linear concentration profile through the film and to the following expression where (P/RT)y is substituted for... 

Multicomponent Diffusion. In multicomponent systems, the binary diffusion coefficient has to be replaced by an effective or mean diffusivity Although its rigorous computation from the binary coefficients is difficult, it may be estimated by one of several methods (27—29). Any degree of counterdiffusion, including the two special cases "equimolar counterdiffusion" and "no counterdiffusion" treated above, may arise in multicomponent gas absorption. The influence of bulk flow of material through the films is corrected for by the film factor concept (28). It is based on a slightly different form of equation 13 ... 

Equimolar Counterdiffusion. Just as unidirectional diffusion through stagnant films represents the situation in an ideally simple gas absorption process, equimolar counterdiffusion prevails as another special case in ideal distillation columns. In this case, the total molar flows and are constant, and the mass balance is given by equation 35. As shown eadier, noj/g factors have to be included in the derivation and the height of the packing is... 

Xm are not. For unimolecular diffusion through stagnant gas = 1), and reduce to T and X and and reduce to and equation 64 then becomes equation 34. For equimolar counterdiffusion = 0, and the variables reduce tojy, x, G, and F, respectively, and equation 64 becomes equation 35. Using the film factor concept and rate equation 28, the tower height may be computed by... 

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... 

In this equation the entire exterior surface of the catalyst is assumed to be uniformly accessible. Because equimolar counterdiffusion takes place for stoichiometry of the form of equation 12.4.18, there is no net molar transport normal to the surface. Hence there is no convective transport contribution to equation 12.4.21. Let us now consider two limiting conditions for steady-state operation. First, suppose that the intrinsic reaction as modified by intraparticle diffusion effects is extremely rapid. In this case PA ES will approach zero, and equation 12.4.21 indicates that the observed rate per unit mass of catalyst becomes... 

First, consider the gradient of cA. Since A is consumed by reaction inside the particle, there is a spontaneous tendency for A to move from the bulk gas (cAg) to the interior of the particle, first by mass transfer to the exterior surface (cAj) across a supposed film, and then by some mode of diffusion (Section 8.5.3) through the pore structure of the particle. If the surface reaction is irreversible, all A that enters the particle is reacted within the particle and none leaves the particle as A instead, there is a counterdiffusion of product (for simplicity, we normally assume equimolar counterdiffusion). The concentration, cA,at any point is the gas-phase concentration at that point, and not the surface concentration. 

For convenience, let the flux of A within the ash layer be expressed by Fick s law for equimolar counterdiffusion, though other forms of this diffusion equation will give the same result. Then, noting that both (2a dCJdr are positive, we have... 

In the specific instance of A diffusing through a catalytic pore and reacting at the end of the pore to form gaseous component B, equimolar counterdiffusion can be assumed, and an effective transition region diffusivity, D a is independent of concentration and can be calculated from the Knudsen and binary diffusion coefficients ... 

This simplified diffnsivify is sometimes used for diffusion in porous cafalysfs even when equimolar counterdiffusion is nol occurring. This greafly simplifies fhe equations. When no reactions are occurring, the diffusivity is a function of concentration (in terms of mole fraction, xa) ... 

Combiue your iuformatiou to estimate the trausitioual diffusivity for uitrogeu,. You can use au average couceutratiou for uitrogeu iu your calculatiou. How does it compare to the two diffusivities you calculated iudividually How does it compare to the transitional diffusivity if there were equimolar counterdiffusion—that is reaction at the end of the pore ... 

For ideal gases the effective binary diffusion coefficient can be calculated from molecular properties (see Appendix A). The film thickness, 5, is determined by hydrodynamics. Correlations are given in the literature which allow the calculation of the transfer coefficient in the case of equimolar counterdiffusion, kf, rather than the film thickness, 5 ... 

The driving force for the transport is provided by a concentration gradient as the reactant moves further towards the center of the pellet its concentration is decreased by reaction. The resistance to the transport mainly originates from collisions of the molecules, either with each other or with the pore walls. The latter dominate when the mean free path of the molecules is larger than the pore diameter. Usually both type of collisions are totally random, which amounts to saying that the transport mechanism is of the diffusion type. Hence the rate of transport, expressed as a molar flux in mol mp2 s-1, in the case of equimolar counterdiffusion can be written as ... 

Properties The diffusion coeflicient of helium in air (or air in helium) at normal atmospheric conditions is Dgg = 7.2 x 10 m% (Table 14-2). The molar masses of air and helium are 29 and 4 kg/kmol, respectively (Table A-1). Analysis This is a typical equimolar counterdiffusion process since the problem involves two large reservoirs of ideal gas mixtures connected to each other by a channel, and the concentrations of species in each reservoir (the pipeline and the atmosphere) remain constant. 

The rate processes of diffusion and catalytic reaction in simple square stochastic pore networks have also been subject to analysis. The usual second-order diffusion and reaction equation within individual pore segments (as in Fig. 2) is combined with a balance for each node in the network, to yield a square matrix of individual node concentrations. Inversion of this 2A matrix gives (subject to the limitation of equimolar counterdiffusion) the concentration profiles throughout the entire network [14]. Figure 8 shows an illustrative result for a 20 X 20 network at an intermediate value of the Thiele modulus. The same approach has been applied to diffusion (without reaction) in a Wicke-Kallenbach configuration. As a result of large and small pores being randomly juxtaposed inside a network, there is a 2-D distribution of the frequency of pore fluxes with pore diameter. 

Equimolar Counterdiffusion. In equimolar coimterdrffusion (EMCD), for every mole of A that diffuses 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 smface, 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 smface, 1 mol of the isomer B diffuses away from iihe surface. The fluxes of A and B are equal in magnitude and flow counter to each other. Stated mathematically. 

If we had assumed equimolar counterdiffusion the molar fluxes would be equal to our first estimate of the diffusion fluxes... 

Equations 15.9 and 15.10 are empirical with respect to the dehnition of the mass transfer coefficients, but the form of the equations is based on molecular diffusion theory. Applying the theory to a multi-component mixture where each component has a distinct diffusivity is impractically complex and must rely on diffusivity data for all the components in the mixture. To derive usable equations from the diffusion theory, certain simplifying assumptions must be made. The basis for the derivation of Equations 15.9 and 15.10 is to assume that mass transfer takes place either as equimolar counterdiffusion or as unimolar diffusion under dilute conditions. 

In equimolar counterdiffusion or unimolar dilute diffusion, V and L are assumed constant, allowing Equations 15.22 and 15.23 to be written as... 

Calculate molar fluxes for steady-state gaseous diffusion of A through stagnant B, and for equimolar counterdiffusion. 

Consider the following numerical example. A binary gaseous mixture of components A and B at a pressure of 1 bar and temperature of 300 K undergoes steady-state equimolar counterdiffusion along a 1-mm-thick diffusion path. At one end of the path... 

The two situations noted in Chapter 1, equimolar counterdiffusion and diffusion of A through stagnant B, occur so frequently that special mass-transfer coefficients are usually defined for them. These are defined by equations of the form... 

A packed-bed distillation column is used to adiabatically separate a mixture of methanol and water at a total pressure of 1 atm. Methanol—the more volatile of the two components—diffuses from the liquid phase toward the vapor phase, while water diffuses in the opposite direction. Assuming that the molar latent heat of vaporization is similar for the two components, this process is usually modeled as one of equimolar counterdiffusion. At a point in the column, the mass-transfer coefficient is estimated as 1.62 x 10-5 kmol/m2-s-kPa. The gas-phase methanol mole fraction at the interface is 0.707, while at the bulk of the gas it is 0.656. Estimate the methanol flux at that point. 

Equimolar counterdiffusion can be assumed in this case (as will be shown in a later chapter, this is the basis of the McCabe-Thiele method of analysis of distillation columns). Methanol diffuses from the interface towards the bulk of the gas phase therefore, yM = 0.707 and yA1 = 0.656. Since they are not limited to dilute solutions,... 

When we deal with situations which do not involve either diffusion of only one substance or equimolar counterdiffusion, or if mass-transfer rates are large, F-type coefficients should be used. The general approach is the same, although the resulting expressions are more cumbersome than those developed above. Thus, in a situation like that shown in Figures 3.3 to 3.5, the mass-transfer flux is... 

When reaction occurs on the pore walls simultaneously with diffusion, the process is not a strictly consecutive one, and both aspects must be considered together. Comprehensive discussions are available in Satterfield [40] and in Aris [74], We first consider the simplest case of a first-order reaction, equimolar counterdiffusion, and isothermal conditions—generalizations will be discussed later. Also, the simplest geometry of a slab of catalyst will be used. When the z-coordinate is oriented from the center line to the surface, the steady-state diffusion equation is... 

On the other hand, calculation of diffusion in distillation columns tends to be easier if the molar average reference velocity v gf moi is used. In distillation, constant molal overflow is often valid or close to valid fSection 4.2T The resulting equimolar counterdiffusion results in = -Ng, and there is no convection in the reference frame with v,.pf mni = 0. If we choose the reference velocity as the molar average velocity, then Eq. ri5-16ei becomes... 

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