Counterdiffusion. equimolar

This situation is sometimes referred to as equimolar counterdiffusion (or mass transfer). 

In an isobaric closed system like the diffusion cell or Loschmidt tube (see Chapter 5), any 

In this case, there is no bulk-motion contribution to the flux, and the flux is related linearly to the concentration difference driving force. Special k -type mass-transfer coefficients are defined specifically for equimolar counterdiffusion as follows  

Equations (2-12) and (2-13) are always valid for equimolar counterdiffusion, regardless of whether the solutions are dilute or concentrated. It is easily shown that 

Example 2.3 Mass-Transfer Coefficient in a Packed-Bed Distillation Column 

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, 

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The creatmenc of the boundary conditions given here ts a generali2a-tion to multicomponent mixtures of a result originally obtained for a binary mixture by Kramers and Kistecnaker (25].These authors also obtained results equivalent to the binary special case of our equations (4.21) and (4.25), and integrated their equations to calculate the p.ressure drop which accompanies equimolar counterdiffusion in a capillary. Their results, and the important accompanying experimental measurements, will be discussed in Chapter 6 ... 

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 ... 

Rate Equations with Concentration-Independent Mass Transfer Coefficients. Except for equimolar counterdiffusion, the mass transfer coefficients appHcable to the various situations apparently depend on concentration through thej/g and factors. Instead of the classical rate equations 4 and 5, containing variable mass transfer coefficients, the rate of mass transfer can be expressed in terms of the constant coefficients for equimolar counterdiffusion using the relationships... 

Equation 55 is a tigoious expression for the number of overall transfer units for equimolar counterdiffusion, in distillation columns, for instance. 

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... 

Equimolar counterdiffusion takes place in catalyst pores when a reaction with a stoichiometry of the form A - B occurs under steady-state conditions. 

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 ... 

When chemical reaction occurs, a must be defined in terms of molar fluxes. For the case of equimolar counterdiffusion, a = 1, and Eq. (4.81) reduces to (4.80). 

Based on such analyses, which of course do imply a film model in which the resistance to mass transfer is supposed to be confined to a film of finite thickness (see Volume 1, Chapter 10), it is possible to estimate the effect which mass transport external to the solid surface has on the overall reaction rate. For equimolar counterdiffusion of a component A in the gas phase, the rate of transfer of A from the bulk gas to the interface can be expressed as ... 

The right-hand side of equation 3.60 contains the mass transfer coefficient hD which is used if the driving force is expressed in terms of gas concentrations. Because of the stoichiometric demands imposed by chemical reaction, equimolar counterdiffusion of components may not necessarily occur and the effects of bulk flow must be taken into account (see Volume 1, Chapter 10). 

It is now essential to write the mass and energy balance equations for the two dimensions z and r. For the sake of completeness, we will include the effect of longitudinal dispersion and heat conduction and deduce the material and energy balances for one component in an elementary annulus of radius Sr and length Sz. We assume that equimolar counterdiffusion occurs and by reference to Fig. 3.25 write down, in turn, the components of mass which are entering the element in unit time longitudinally and radially ... 

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 ... 

For equimolar counterdiffusion, the film factor reduces to one. Combination of (7.A15) and ... 

When 8a 0 that represents equimolar counterdiffusion, thenyfA = 1. 

Equimolar counterdiffusion (iV, = —Ng) can often be assumed and further simplification is thus possible to give ... 

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