Bulk Molecular Diffusion

Bulk (Molecular) Diffusion Bulk or molecular diffusion is the predominant mode of diffusion in large pores. For a gas, molecular diffusion occurs when r km- For liquids, molecular diffusion occurs when r is much greater than the radius of the diffusing molecule. In the molecular diffusion regime, collisions between molecules are much more frequent than collisions between molecules and the walls of the pore. Diffusion takes place through molecule-molecule interactions. 

The equation for the flux of component A is more complex for bulk diffusion than for Knudsen diffusion. In bulk diffusion, the flux of A in the z-direction is given by 

In this equation, Ni is the flux of component in the z-direction, Da u is the diffiisivity of 

A in the mixture, yA is the mole fraction of A, and iV is the number of compounds in the mixture. The first term on the right-hand side of Eqn. (9-19) is Ae diffusive flux of A and the second term is the flux of A due to bulk flow. The total flux Ni is a vector that is directed 

For a single reaction taking place in a catalyst particle at steady state, the molar fluxes are related through stoichiometry, i.e., 

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In Equation (9.6), x is the direction of flux, nt [mol m-3 s 1 ] is the total molar density, X [1] is the mole fraction, Nd [mol m-2 s 1] is the mole flux due to molecular diffusion, D k [m2 s 1] is the effective Knudsen diffusion coefficient, D [m2 s 1] is the effective bimolecular diffusion coefficient (D = Aye/r), e is the porosity of the electrode, r is the tortuosity of the electrode, and J is the total number of gas species. Here, a subscript denotes the index value to a specific specie. The first term on the right of Equation (9.6) accounts for Knudsen diffusion, and the following term accounts for multicomponent bulk molecular diffusion. Further, to account for the porous media, along with induced convection, the Dusty Gas Model is required (Mason and Malinauskas, 1983 Warren, 1969). This model modifies Equation (9.6) as ... 

Diffusion in macropores occurs mainly by the combined effects of bulk molecular diffusion (as in the free fluid) and Knudsen flow, with generally smaller contributions from other mechanisms such as surface diffusion and Poiseuille flow. Knudsen flow, which has the characteristics of a diffusive process, occurs because molecules striking the pore wall are instantaneously adsorbed and re-emitted in a random direction. The relative importance of bulk and Knudsen diffusion depends on the relative frequency of molecule-molecule and molecule-wall collisions, which in turn depends on the ratio of the mean free path to pore diameter. Thus Knudsen flow becomes dominant in small pores at low pressures, while in larger pores and at higher pressures diffusion occurs mainly by the molecular mechanism. Since the mechanism of diffusion may well be different at different pressures, one must be cautious about extrapolating from experimental diffusivity data, obtained at low pressures, to the high pressures commonly employed in industrial processes. 

Maxwellian diffusion (bulk molecular diffusion) in moderately large pores (macropores) or Knudsen diffusion in pores (micropores) which have a diameter smaller than the mean free path of the adsorbate molecules ... 

From the section Bulk (Molecular) Diffusion (Section 9.3.4.2) the flux of reactant A through the boundary layer, in the z-direction, is given by... 

For a given solid structure, or the pore size is fixed however, in many cases it is still possible to effect a change from bulk (molecular) diffusion to Knudsen diffusion or vice versa, by altering the total pressure of the system. 

We note that bulk (molecular) diffusion is well understood and its laws hold irrespective of the materials involved. Moreover, good general approximations may also be made regarding the pore diffusion of gases. In the consideration of adsorption and chemical reaction phenomena, matters will become highly specific to the nature of the substances involved. 

However, polymer molecules are very long and generally exist in a highly coiled state in the reaction mass with the functional groups situated at the chain ends. Therefore, in addition to the bulk molecular diffusion of P and P , the chain ends must diffuse close to each other (called segmental motion) before the chemical reaction can occur. This can be represented schematically as... 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

Macropore Diffusion. Transport in a macropore can occur by several different mechanisms, the most important of which ate bulk molecular... 

Although molecular diffusion itself is very slow, its effect is nearly always enhanced by turbulent eddies and convection currents. These provide almost perfect mixing in the bulk of each Hquid phase, but the effect is damped out in the vicinity of the interface. Thus the concentration profiles at each... 

Concentration and temperature differences are reduced by bulk flow or circulation in a vessel. Fluid regions of different composition or temperature are reduced in thickness by bulk motion in which velocity gradients exist. This process is called bulk diffusion or Taylor diffusion (Brodkey, in Uhl and Gray, op. cit., vol. 1, p. 48). The turbulent and molecular diffusion reduces the difference between these regions. In laminar flow, Taylor diffusion and molecular diffusion are the mechanisms of concentration- and temperature-difference reduction. 

An important mixing operation involves bringing different molecular species together to obtain a chemical reaction. The components may be miscible liquids, immiscible liquids, solid particles and a liquid, a gas and a liquid, a gas and solid particles, or two gases. In some cases, temperature differences exist between an equipment surface and the bulk fluid, or between the suspended particles and the continuous phase fluid. The same mechanisms that enhance mass transfer by reducing the film thickness are used to promote heat transfer by increasing the temperature gradient in the film. These mechanisms are bulk flow, eddy diffusion, and molecular diffusion. The performance of equipment in which heat transfer occurs is expressed in terms of forced convective heat transfer coefficients. 

Thus, mixing in liquids is achieved by several mechanisms which gradually reduce the size or scale of the fluid elements and then redistribute them in the bulk. If there are initially differences in concentration of a soluble material, uniformity is gradually achieved, and molecular diffusion becomes progressively more important as the element... 

Kishinev ski/23 has developed a model for mass transfer across an interface in which molecular diffusion is assumed to play no part. In this, fresh material is continuously brought to the interface as a result of turbulence within the fluid and, after exposure to the second phase, the fluid element attains equilibrium with it and then becomes mixed again with the bulk of the phase. The model thus presupposes surface renewal without penetration by diffusion and therefore the effect of diffusivity should not be important. No reliable experimental results are available to test the theory adequately. 

In many applications of mass transfer the solute reacts with the medium as in the case, for example, of the absorption of carbon dioxide in an alkaline solution. The mass transfer rate then decreases in the direction of diffusion as a result of the reaction. Considering the unidirectional molecular diffusion of a component A through a distance Sy over area A. then, neglecting the effects of bulk flow, a material balance for an irreversible reaction of order n gives ... 

In a gas absorption process, the solute gas A diffuses into a solvent liquid with which it reacts. The mass transfer is one of steady state unidirectional molecular diffusion and the concentration of A is always sufficiently small for bulk flow to be negligible. Under these conditions the reaction is first order with respect to the solute A. 

Mass transfer from a single spherical drop to still air is controlled by molecular diffusion and. at low concentrations when bulk flow is negligible, the problem is analogous to that of heat transfer by conduction from a sphere, which is considered in Chapter 9, Section 9.3.4. Thus, for steady-state radial diffusion into a large expanse of stationary fluid in which the partial pressure falls off to zero over an infinite distance, the equation for mass transfer will take the same form as that for heat transfer (equation 9.26) ... 

From the penetration theory, the mass transfer rate per unit area N, is given in terms of the concentration difference AC, between the interface and the bulk fluid, the molecular diffusivity D and the age t of the. surface clement by ... 

The devolatilization of a component in an internal mixer can be described by a model based on the penetration theory [27,28]. The main characteristic of this model is the separation of the bulk of material into two parts A layer periodically wiped onto the wall of the mixing chamber, and a pool of material rotating in front of the rotor flights, as shown in Figure 29.15. This flow pattern results in a constant exposure time of the interface between the material and the vapor phase in the void space of the internal mixer. Devolatilization occurs according to two different mechanisms Molecular diffusion between the fluid elements in the surface layer of the wall film and the pool, and mass transport between the rubber phase and the vapor phase due to evaporation of the volatile component. As the diffusion rate of a liquid or a gas in a polymeric matrix is rather low, the main contribution to devolatilization is based on the mass transport between the surface layer of the polymeric material and the vapor phase. 

The modeling of mass transport from the bulk fluid to the interface in capillary flow typically applies an empirical mass transfer coefficient approach. The mass transfer coefficient is defined in terms of the flux and driving force J = kc(cbuik-c). For non-reactive steady state laminar flow in a square conduit with constant molecular diffusion D, the mass balance in the fluid takes the form... 

The use of excess inert electrolyte so as to reduce differences in transport properties of the solution at the electrode surface and in the bulk. In such a solution, the ionic diffusivity of the reacting ion, for example, Cu2 + or Fe(CN)g, should be employed in the interpretation of results, and not the molecular diffusivities of the compounds, for example, CuS04 or K3Fe(CN)6. 

Bulk or forced flow of the Hagan-Poiseuille type does not in general contribute significantly to the mass transport process in porous catalysts. For fast reactions where there is a change in the number of moles on reaction, significant pressure differentials can arise between the interior and the exterior of the catalyst pellets. This phenomenon occurs because there is insufficient driving force for effective mass transfer by forced flow. Molecular diffusion occurs much more rapidly than forced flow in most porous catalysts. 

The only instances in which external mass transfer processes can influence observed conversion rates are those in which the intrinsic rate of the chemical reaction is so rapid that an appreciable concentration gradient is established between the external surface of the catalyst and the bulk fluid. The rate at which mass transfer to the external catalyst surface takes place is greater than the rate of molecular diffusion for a given concentration or partial pressure driving force, since turbulent mixing or eddy diffusion processes will supplement ordinary molecular diffusion. Consequently, for porous catalysts one... 

The Schmidt and Prandtl numbers must be evaluated in order to be able to determine concentration and temperature differences between the bulk fluid and the external surface of the catalyst. The Schmidt number for naphthalene in the mixture may be evaluated using the ordinary molecular diffusivity employed earlier, the viscosity of the mixture, and the fluid density. 

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