Diffusion-reaction theory

This study employed conventional diffusion-reaction theory, showing that with diffusion-limited reactions the internal effectiveness factor of a heterogeneous catalyst is inversely related to the Thiele modulus. Using a standard definition of the Thiele modulus [100], the observed reaction rate of an immobilized-enzyme reaction will vary with the square root of the immobilized-enzyme concentration in a diffusion-limited system. In this case, a plot of the reaction rate versus the enzyme loading in the catalyst formulation will be nonlinear. 

Thiele modulus. This can be interpreted by current diffusion-reaction theory-The reaction occurs mainly inside the pores of zeolites at lower Thiele modulus. Thus, the pore entrance sites are not important and the effects of pore entrance deactivation on the product selectivity and effectiveness are not obvious. With the increase of Thiele modulus, the reaction is gradually diffusion-controled, the pore entrance sites become important, and the effects of pore entrance deactivation become distinct. 

Like the gas holdup the gas-liquid interfacial area, a, represents an important quantity. If the reaction takes place in the fast reaction regime of diffusion-reaction theory, the interfacial area is the main design criterion. Gas holdup and interfacial area are related by... 

To apply the diffusion-reaction theory to reactive dyeing, we use the fact that the overall behavior is generally similar to that of regime 2-3, The following equation based on the penetration theory (Danckwerts, 1970) can be derived for regime 2-3 ... 

Wardlaw, C.W A commentary on Turing s diffusion-reaction theory of morphogenesis. New Phytol. 52(1), 40-47 (1953). http //www.jstor.org/stable/2429242 Wardlaw, C.W The chemical concept of organization in plants. New Phytol. 54(3), 302-310 (1955). http //www.jstor.org/stable/2429313... 

In addition to the Burke and Schumann model (34) and the Displacement Distance theory, a comprehensive laminar diffusion flame theory can be written using the equations of conservation of species, energy, and momentum, including diffusion, heat transfer, and chemical reaction. 

According to the transition state theory, the pre-exponential factor A is related to the frequency at which the reactants arrange into an adequate configuration for reaction to occur. For an homolytic bond scission, A is the vibrational frequency of the reacting bond along the reaction coordinates, which is of the order of 1013 to 1014 s 1. In reaction theory, this frequency is diffusion dependent, and therefore, should be inversely proportional to the medium viscosity. Also, since the applied stress deforms the valence geometry and changes the force constants, it is expected... 

This article presents a brief account of theory and practical aspects of rotating hemispherical electrodes. The fluid flow around the RHSE, mass transfer correlations, potential profile, and electrochemical application to the investigations of diffusivity, reaction rate constants, intermediate reaction products, passivity, and AC techniques are reviewed in the following sections. 

According to the diffusion layer theory, for which the transport process is rate-limiting, kT kR, so that k = kT. According to the interfacial barrier theory, for which the surface reaction is rate-limiting, kR kT, so that it, = R-... 

If finite chemical reaction times are put into the columnar diffusion flame theory (76), burning rates are predicted to be linearly proportional to pressure at low pressure and independent of pressure (plateau burning) at high pressure. Based on this model, von Elbe et al. (97) proposed the simple equation ... 

Figure 12. Effect of activation energy of surface reaction on burning rate behavior of granular diffusion flame theory with collapsed A/PA flame zone...

Normal-burning propellants such as PBAA-AP and PB(CT)-AP are in qualitative agreement with the behavior predicted by the granular diffusion flame theory when the NH3/HCIO4 reaction is considered distended. Quantitative comparison between theory and experiment is not... 

Thus, we find a remarkable lack of sensitivity for the calculated burning rates of an adiabatic droplet-burning process in which the reactions go to completion. This observed lack of sensitivity to reaction rates may well be related to the known successes (11,12, 22) of simplified diffusion-flame theories in theoretical predictions of droplet burning rates. 

Nevertheless, the kinetic modelling of spurs is by far the most complex problem to which diffusion-limited chemical reaction theory has been applied. The radiation chemistry of water is of especial importance to both radiotherapy and nuclear engineering. 

Another important test of the accuracy of the superposition approximation is the diffusion-controlled A + B — 0 reaction. For the first time it was computer-simulated by Toussaint and Wilczek [27]. They confirmed existence of new asymptotic reaction laws but did not test different approximations used in the diffusion-controlled theories. Their findings were used in [28] to discuss divergence in the linear and the superposition approximations. Since analytical calculations [28] were performed for other sets of parameters as used in [27], their comparison was only qualitative. It was Schnorer et al. [29] who first performed detailed study of the applicability of the superposition approximation. 

A wide range of condensed matter properties including viscosity, ionic conductivity and mass transport belong to the class of thermally activated processes and are treated in terms of diffusion. Its theory seems to be quite well developed now [1-5] and was applied successfully to the study of radiation defects [6-8], dilute alloys and processes in highly defective solids [9-11]. Mobile particles or defects in solids inavoidably interact and thus participate in a series of diffusion-controlled reactions [12-18]. Three basic bimolecular reactions in solids and liquids are dissimilar particle (defect) recombination (annihilation), A + B —> 0 energy transfer from donors A to unsaturable sinks B, A + B —> B and exciton annihilation, A + A —> 0. 

Information relating to the diffusion of metal-bearing compounds in catalytic materials at reaction conditions has been obtained indirectly through classic diffusion and reaction theory. Shah and Paraskos (1975) calculated effective diffusitivities of 7 x 10-8 and 3 x 10-8 cm2/sec for V and Ni compounds in reduced Kuwait crude at 760°F. These low values may be indicative of a small-pore HDS catalyst. In contrast, Sato et al. (1971) report that the effective diffusivity of vanadium compounds was one-tenth that of the nickel compounds on the basis of metal deposition profiles in aged catalysts. This large difference may be influenced by relative adsorption strengths not explicitly considered in their analysis. 

The location of a metal sulfide deposit in a catalyst pellet is dependent on the relative rates of reaction and diffusion. The theory describing diffusion and reaction in catalysts was first developed by Thiele (1939) and extended by many others, including Wheeler (1955), Weisz (1962) and Satterfield (1970), and has been discussed in Section IV. Sato et al. (1971) and Tamm et al. (1981) have discussed metal deposit profiles in the context of the Thiele analysis. 

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