Simultaneous Diffusion and Chemical Reaction

There are numerous practical situations in which both diffusion and chemical reaction may be occurring, for instance, during low-pressure chemical vapor 

This example involves the derivation of the differential equation and boundary conditions for a process step that is integral to microelectronics processing. 

Diffusion is the mass transfer mechanism in the region between any two wafers. 

Gap between two wafers (inter-wafer region) is 28 long. 

Surface reaction dominates over homogeneous reaction. 

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

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... 

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. 

Christman, P. G. Analysis of simultaneous diffusion and chemical reaction in the calcium oxide/sulfur dioxide system, Ph. D. Dissertation (1983). 

A stepwise change in pH can be applied outside a protein-containing membrane applied on top of an ISFET, at the interface between the membrane and the electrolyte, using a flow-through system [ 10]. This pH step will lead to simultaneous diffusion and chemical reaction of protons and hydroxyl ions in the membrane. A theoretical description of these phenomena, elaborated in the next subsection, leads to the conclusion that the diffusion of protons in the membrane is delayed by a factor that depends linearly on the protein concentration. Consequently, the time needed to reach the end point in the obtained titration curve also depends linearly on the protein concentration. The effect of both the incubation time and the protein concentration will be simulated and experimentally verified. 

The titration of an acid or base can be carried out by choosing a cathodic or anodic current, respectively. The ions produced cause a local change in the pH. which can easily be measured by the pH-sensitive ISFET. located in the direct vicinity of the actuator electrode [15]. This change in pH will lead to simultaneous diffusion and chemical reaction of protons and hydroxyl ions in the membrane. These diffusion processes will be delayed as a result of protein dissociation reactions of immobilized protein molecules 110[. The description of the diffusion and the effect of the concentration of immobilized protein on the... 

The overall symmetry of the system can be used to show that some coefficients in the L or R matrices are zero. If, for example, the force Xp is a vector quantity but the flow Ja is a scalar flow, the coefficient Lap must be a vector quantity. This is, however, impossible in an isotropic homogeneous system in the absence of external forces. Thus a scalar force cannot induce a vector flow and Lap = 0. An example is that of a mixture in which there are chemical reactions. According to the above, the chemical affinity, a scalar force, cannot induce a flow of matter Jj in any particular direction thus simultaneous diffusion and chemical reaction cannot be coupled. 

The aim is to introduce basic concepts and to establish the general mathematical background. Therefore attention is restricted here to those conditions of simultaneous diffusion and chemical reaction which can be regarded as limiting or asymptotic cases. Significant simplifications may arise, for example, when the ch mical reaction is extremely fast or extremely slow, as compared to diffusion phenomena. 

The mass-transfer coefficient is proportional to. However, since we have shown that in Eq. (7.3-13), Jp is proportional to (f ab then k[ oc. Hence, the film theory is not correct. The great advantage of the film theory is its simplicity where it can be used in complex situations such as simultaneous diffusion and chemical reaction. 

This example covers simultaneous diffusion and chemical reaction in a tubular reactor [15]. 

Examples are provided from heat transfer mass transfer simultaneous diffusion and convection simultaneous diffusion and chemical reaction simultaneous diffusion, convection, and chemical reaction and viscous flow. 

The third chapter addresses linear second-order ordinary differential equations. A brief discourse, it reviews elementary differential equations, and the chapter serves as an important basis to the solution techniques of partial differential equations discussed in Chapter 6. An applications section is also included with ten worked-out examples covering heat transfer, fluid flow, and simultaneous diffusion and chemical reaction. In addition, the residue theorem as an alternative method for Laplace transform inversion is introduced. 

Chapter 7 is dedicated entirely to worked-out examples taken from the chemical engineering research literature. This chapter relies on the mathematics of the previous six chapters to solve problems in heat transfer mass transfer simultaneous diffusion and convection simultaneous diffusion and chemical reaction simultaneous diffusion, convection, and chemical reaction and viscous flow. 

Equation 7.52 describes simultaneous diffusion and chemical reaction in the liquid film. The concentration profile of component i, cu(z), can in principle be solved by Equation 7.52 and the flux, iVy, is obtained from the derivative dcu/dz in Equation 7.36. Equation 7.52 has the boundary conditions... 

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