Diffusion with a second order reaction

Example 3.2.1. Series Solutions for Diffusion with a Second Order Reaction... 

Consider diffusion with a second order reaction in a rectangular pellet. [18] The dimensionless concentration is governed by ... 

The numerical method of lines described in the previous example can be used for nonlinear elliptic partial differential equations, also. For example, consider the following nonlinear boundary value problem (diffusion with a second order reaction) ... 

Examples 3.2.1 and 3.2.3, diffusion with a second order reaction is solved here again. Consider diffusion with a second order reaction in a rectangular pellet (Rice and Do, 1995). The dimensionless concentration is governed by ... 

Consider diffusion in a slab with a second order reaction. The governing equation in dimensionless form is (see example 5.1.5)... 

Thus a zero-order reaction appears to be 1/2 order and a second-order reaction appears to be 3/2 order when dealing with a fast reaction taking place in porous catalyst pellets. First-order reactions do not appear to undergo a shift in reaction order in going from high to low effectiveness factors. These statements presume that the combined diffusivity lies in the Knudsen range, so that this parameter is pressure independent. 

Gradient diffusion was assumed in the species-mass-conservation model of Shir and Shieh. Integration was carried out in the space between the ground and the mixing height with zero fluxes assumed at each boundary. A first-order decay of sulfur dioxide was the only chemical reaction, and it was suggested that this reaction is important only under low wind speed. Finite-difference numerical solutions for sulfur dioxide in the St. Louis, Missouri, area were obtained with a second-order central finite-difference scheme for horizontal terms and the Crank-Nicolson technique for the vertical-diffusion terms. The three-dimensional grid had 16,800 points on a 30 x 40 x 14 mesh. 

According to Eqs. (18)-(20), a generalized isothermal model of reaction and diffusion in a catalyst slab with the second order reaction can be solved to obtain the solution series ... 

Figure 9 shows the approximate solutions of dimensionless potential and concentration with different terms for a second order reaction in a porous slab electrode, and shows the comparisons between the approximate and numerical solutions. The potential and concentration profiles are obtained by using the coupled equation model with diffusion. 

The modelling of kinetics at modified electrodes has received much attention over the last 10 years [1-11], mainly due to the interest in the potential uses of chemically modified electrodes in analytical applications. The first treatment published by Andrieux et al. [5] was closely followed by a complimentary treatment by Albery and Hillman [1, 2]. Both deal with the simplest basic case, that is, the coupled effects of diffusion and reaction for a second-order reaction between a species freely diffusing in the bulk solution and a redox mediator species trapped within the film at the modified electrode surface. The results obtained by the two treatments are essentially identical, although the two approaches are slightly different. 

Exercise 9.9.4. Show that the distribution function of residence times for laminar flow in a tubular reactor has the form 2z /Zp, where tp is the time of passage of any fluid annulus and the minimum time of passage. Diffusion and entrance effects may be neglected. Hence show that the fractional conversion to be expected in a second order reaction with velocity constant k is 2B[1 + j lnu5/(5 + 1)] where B = akt n and a is the initial concentration of both reactants. (C.U.)... 

At high Peclet numbers, for an nth-order surface reaction withn=l/2, 1,2, Eq. (5.1.5) was tested in the entire range of the parameter ks by comparing its root with the results of numerical solution of appropriate integral equations for the surface concentration (derived in the diffusion boundary layer approximation) in the case of a translational Stokes flow past a sphere, a circular cylinder, a drop, or a bubble [166, 171, 364], The comparison results for a second-order surface reaction (n = 2) are shown in Figure 5.1 (for n = 1/2 and n = 1, the accuracy of Eq. (5.1.5) is higher than for n = 2). Curve 1 (solid line) corresponds to a second-order reaction (n = 2). One can see that, the maximum inaccuracy is observed for 0.5 < fcs/Shoo < 5.0 and does not exceed 6% for a solid sphere (curve 2), 8% for a circular cylinder (curve 3), and 12% for a spherical bubble (curve 4). 

How docM the FEMLAB result compare with the solution to Example 14-2 Repeat (a) for a second-order reaction with k = 0.5 dm-Vraol min. Repeat (a) but a,ssume laminar (low and consider radial gradient. in concentration. Use for both the radial and axial diffusion coefficients. Plot the axial and radial profiles. Compare your results with pan (a). 

The catalyzed reaction is extremely rapid with a second-order rate constant on the order of 10 M s at ambient temperature and neutral pH. This is well within the range of diffusion-controlled reactions in aqueous solutions under these conditions. Reaction (1) is generally regarded to proceed... 

In order to visualize the reaction-diffusion process of a second-order reaction in a T-shaped micromixer, Baroud et cd. used the reaction between Ca and CaGreen, a fluorescent tracer for calcium. The experimental measurements were compared with the 2D numerical simulation of the reaction-diffusion equations and showed good agreement between theory and experiment. From this study, it is possible to extract... 

Time-resolved optical absorption spectroscopy experiments have shown that arenesul-fonyl radicals decay with clean second-order kinetics14 the values of 2 k,/a h where s2 is the extinction coefficient at the monitoring wavelength, increased linearly with decreasing viscosity of the solvent, further indicating that reaction 16 is clearly a diffusion-controlled process. 

A soluble gas is absorbed into a liquid with which it undergoes a second-order irreversible reaction. The process reaches a steady-state with the surface concentration of reacting material remaining constant at (.2ij and the depth of penetration of the reactant being small compared with the depth of liquid which can be regarded as infinite in extent. Derive the basic differential equation for the process and from this derive an expression for the concentration and mass transfer rate (moles per unit area and unit time) as a function of depth below the surface. Assume that mass transfer is by molecular diffusion. 

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