Simultaneous diffusion

Comparison of Eq. (184) with Eq. (183) shows the effect of size distribution for the case of fast chemical reaction with simultaneous diffusion. This serves to emphasize the error that may arise when one applies uniform-drop-size assumptions to drop populations. Quantitatively the error is small, because 1 — is small in comparison with the second term in the brackets [i.e., kL (kD)112). Consequently, Eq. (184) and Eq. (183) actually give about the same result. In general, the total average mass-transfer rate in the disperser has been evaluated in this model as a function of the following parameters ... 

Danckwkrts. P. V. Trans. Faraday Soc. 46 (1950) 300. Absorption by simultaneous diffusion and chemical reaction. 

Mathematical approaches used to describe micelle-facilitated dissolution include film equilibrium and reaction plane models. The film equilibrium model assumes simultaneous diffusive transport of the drug and micelle in equilibrium within a common stagnant film at the surface of the solid as shown in Figure 7. The reaction plane approach has also been applied to micelle-facilitated dissolution and has the advantage of including a convective component in the transport analysis. While both models adequately predict micelle-facilitated dissolution, the scientific community perceives the film equilibrium model to be more mathematically tractable, so this model has found greater use. 

The analysis of simultaneous diffusion and chemical reaction in porous catalysts in terms of effective diffusivities is readily extended to geometries other than a sphere. Consider a flat plate of porous catalyst in contact with a reactant on one side, but sealed with an impermeable material along the edges and on the side opposite the reactant. If we assume simple power law kinetics, a reaction in which there is no change in the number of moles on reaction, and an isothermal flat plate, a simple material balance on a differential thickness of the plate leads to the following differential equation... 

Soled catalyzed roocUono. These involve simultaneous diffusion, adsorption and surface reaction. See Chapter <5. 

These TPD techniques reflect the kinetics (not thermodynamics) of adsorption, and are quite useful for determining trends across series of catalysts, but are often not suitable for the derivation of quantitative information on surface kinetics or energetics, in particular on ill-defined real catalysts. Besides averaging the results from desorption from different sites, TPD detection is also complicated in porous catalysts by simultaneous diffusion and readsorption processes [58],... 

Since interaction phenomena due to simultaneous diffusion of several components play an important role, the Maxwell-Stefan theory has been selected to describe the mass transfer processes. The general form of the flux expressions can be represented by (Taylor and Krishna, 1993)... 

If there is simultaneous diffusion of more than one component in the crystal, the flux of A in direction X depends on the individual diffusivities of all diffusing components (Darken, 1948), and the individual diffusivity coefficient in equations 4.87 and 4.88 is replaced by interdiffusion coefficient D i.e., for the simultaneous diffusion of two ions A and B,... 

At one extreme diffusivity may be so low that chemical reaction takes place only at suface active sites. In that case p is equal to the fraction of active sites on the surface of the catalyst. Such a polymer-supported phase transfer catalyst would have extremely low activity. At the other extreme when diffusion is much faster than chemical reaction p = 1. In that case the observed reaction rate equals the intrinsic reaction rate. Between the extremes a combination of intraparticle diffusion rates and intrinsic rates controls the observed reaction rates as shown in Fig. 2, which profiles the reactant concentration as a function of distance from the center of a spherical catalyst particle located at the right axis, When both diffusion and intrinsic reactivity control overall reaction rates, there is a gradient of reactant concentration from CAS at the surface, to a lower concentration at the center of the particle. The reactant is consumed as it diffuses into the particle. With diffusional limitations the active sites nearest the surface have the highest turnover numbers. The overall process of simultaneous diffusion and chemical reaction in a spherical particle has been described mathematically for the cases of ion exchange catalysis,63 65) and catalysis by enzymes immobilized in gels 66-67). Many experimental parameters influence the balance between intraparticle diffusional and intrinsic reactivity control of reaction rates with polymer-supported phase transfer catalysts, as shown in Fig. 1. 

Since the membrane is also permeable to the solvent, the solvent simultaneously diffuses into the bag, diluting the colloid. Ample air space must be present in the bag at the beginning, otherwise it will rupture owing to the pressure developed by the solvent imbibed. There is also a danger that the porosity of the membrane will increase if the bag is stretched as a result of internal pressure buildup. Cellophane tubing is most commonly used as the membrane material. It is sold in rolls for this purpose and may be cut to length and tied at the ends to make the required bags. 

This reaction is rather slow in the absence of the enzyme carbonic anhydrase, which is usually the case with fermentation broths, although this enzyme exists in the red blood cells. Thus, any increase of for CO, desorption from fermentation broths due to simultaneous diffusion of HCO3 seems negligible. 

As mentioned in Section 15.3.3, most CO2 in blood exists as HCO3". In evaluating the CO2 desorption performance of blood oxygenators, we must always consider the simultaneous diffusion of HCOg", the rate of which is greater than that of physically dissolved COg. Experimental data on the rates of CO2 desorption... 

Simultaneous diffusion and dipole—dipole (Forster) energy transfer can be described by eqn. (66) with /(r) given by eqn. (62), i.e. 

Figure 10-10. Representation of the chemical potential of A during the heterogeneous solid state reaction A+B = AB. a) Diffusion control, b) interface control at b2, c) rate control by rearrangement (relaxation) of A in B in zone A (B), d) simultaneous diffusion and interface control (bj).

Figure 9.8 Isolated-boundary (Type-B) self-diffusion associated with a stationary grain boundary, (a) Grain boundary of width 6 extending downward from the free surface at y = 0. The surface feeds tracer atoms into the grain boundary and maintains the diffusant concentration at the grain boundary s intersection with the surface at the value cB(y = 0, t) = 1. Diffusant penetrates the boundary along y and simultaneously diffuses transversely into the grain interiors along x. (b) Diffusant distribution as a function of scaled transverse distance, xi, from the boundary at scaled depth, yx, from the surface. Penetration distance in grains is assumed large relative to 5.

Nielsen, P. H. and Villadsen, J. (1984) An analysis of the multiplicity pattern of models for simultaneous diffusion, chemical reaction and adsorption. Chem. Engng Sci. 40, 571-587. 

As was shown in a paper by Frank-Kamenetskii and the author [9], simultaneous diffusion and heat exchange between the fresh mixture and the hot reaction products under the simplest assumptions leads to the same relation between the concentration and temperature as their direct mixing. Since in a propagating flame the entire range of temperatures from the initial one to the combustion temperature of the given mixture is simultaneously realized, it is obvious that the bulk of the substance reacts precisely in the temperature region in which the chemical reaction rate is close to a maximum. 

The catalyst packing of the reactor consists of an iron oxide Fe20s, promoted with potassium carbonate K COo, and chromium oxide Cr O-s,. The catalyst pellets are extrudates of a cylindrical shape. Since at steady state the problem of simultaneous diffusion and reaction are independent of the particle shape, an equivalent slab geometry is used for the catalyst pellet, with a characteristic length making the surface to volume ratio of the slab equal to that of the original shape of the pellet. 

Liu, R, Higuchi, W., Ghanem, A., Bergstrom, T., and Good, W. Assessing the influence of ethanol on simultaneous diffusion and metabolism of P-estradiol in hairless mouse skin for the asymmetric situation in vitro. Int. J. Pharm. 78 123-136, 1992. 

As an example of simultaneous flows, we will consider simultaneous diffusion and heat transfer. This is illustrated in Fig. 3. 

Fig. 1.8. Schematic representation of the growth process of the ApBq layer between elementary substances A and B due to the simultaneous diffusion of both components.

Fig. 4.5. Schematic diagram to illustrate the growth process of the ArBs layer in the ApBq-AiBn reaction couple in the case of simultaneous diffusion of both components.

The proposed mechanism391 of growth of the ZrAl2 and Zr02 layers appears to be more likely than an alternative one372 involving the simultaneous diffusion of oxygen and aluminium atoms. Indeed, the self-... 

The remarkable capability of oil-water multilaminates to separate permeants in the nonsteady state can be best demonstrated by studying the asymptotic solutions of the simultaneous diffusion equations (.2,3). An alternating series of n oil and n-1 water laminates (Figure 1) separate a well-stirred, infinite aqueous source compartment of solute concentration C and an aqueous receptor compartment of zero solute concentration.0 Within the ith membrane phase, the solute concentration, obeys Fick s second law,... 

Simulations were carried out for the case of simultaneous diffusion and uptake of oxygen in a viscous fermentation broth. 

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