Diffusion with a second order

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 ... 

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. 

Recent studies on the electron-transfer between a number of flavin analogs with equine cytochrome c shows that flavin semiquinone reduction of cytochrome c reflects a diffusion-controlled process with a second-order rate of 6x 10 sec" ... 

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

The convective terms were solved using a second order TVD scheme in space, and a first order explicit Euler scheme in time. The TVD scheme applied was constructed by combining the central difference scheme and the classical upwind scheme by adopting the smoothness monitor of van Leer [193] and the monotonic centered limiter [194]. The diffusive terms were discretized with a second order central difference scheme. The time-splitting scheme employed is of first order. 

Methylnicotinamide radicals (1-MNA ), produced pulse radiolytically, react with Ps-NiR concomitantly with the appearance of an absorption increase at 554 nm, a characteristic wavelength for heme-c reduction [Fig. 18(a)] (105, 106). At 670 nm, where heme-t/i reduction can be independently monitored, no absorption changes were observed in the fast time domain [Fig. 18(h)], indicating that no parallel bimolecular reduction of the heme-di center by 1-MNA is taking place. Therefore it was concluded that only heme-c is reduced directly by 1-MNA and in an essentially diffusion controlled process, with a second-order... 

Having considered cases in which lattice or volume diffusion does not explain the data, it should be noted that lattice diffusion seems to dominate many natural systems, in argon and other noble gases (cf Farley 2002, this volume). Lattice diffusion follows a second order diffusion mechanism (McDougall and Harrison 1999) with an Arrhenius relationship given by the equation ... 

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... 

Errors in advection may completely overshadow diffusion. The amplification of random errors with each succeeding step causes numerical instability (or distortion). Higher-order differencing techniques are used to avoid this instability, but they may result in sharp gradients, which may cause negative concentrations to appear in the computations. Many of the numerical instability (distortion) problems can be overcome with a second-moment scheme (9) which advects the moments of the distributions instead of the pollutants alone. Six numerical techniques were investigated (10), including the second-moment scheme three were found that limited numerical distortion the second-moment, the cubic spline, and the chapeau function. 

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. 

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. 

The second class of materials, which we will consider herein are carbons with a highly ordered porosity prepared by a template technique [15-18]. The pores are characterized by a well-defined size determined by the wall thickness of the silica substrate used as substrate for carbon infiltration. They can be also interconnected, that is very useful for the charge diffusion in the electrodes. Figure 1 presents the general principle of the carbon preparation by a template technique, where the silica matrix can be, for example, MCM-48 or SBA-15. 

The cyclohexadienyl radicals decay by second-order kinetics, as proven by the absorption decay, with almost diffusion-controlled rate (2k = 2.8 x 109 M 1 s 1). The cyclohexyl radicals 3 and 4 decay both in pseudo-first-order bimolecular reaction with the 1,4-cyclohexadiene to give the cyclohexadienyl radical 5 and cyclohexene (or its hydroxy derivative) (equation 15) and in a second order bimolecular reaction of two radicals. The cyclohexene (or its hydroxy derivative) can be formed also in a reaction of radical 3 or... 

The Hatta criterion compares the rates of the mass transfer (diffusion) process and that of the chemical reaction. In gas-liquid reactions, a further complication arises because the chemical reaction can lead to an increase of the rate of mass transfer. Intuition provides an explanation for this. Some of the reaction will proceed within the liquid boundary layer, and consequently some hydrogen will be consumed already within the boundary layer. As a result, the molar transfer rate JH with reaction will be higher than that without reaction. One can now feel the impact of the rate of reaction not only on the transfer rate but also, as a second-order effect, on the enhancement of the transfer rate. In the case of a slow reaction (see case 2 in Fig. 45.2), the enhancement is negligible. For a faster reaction, however, a large part of the conversion occurs in the boundary layer, and this results in an overall increase of mass transfer (cases 3 and 4 in Fig. 45.2). 

The general equation of convective diffusion in liquids, equation (15), is a second-order, partial differential equation with variable coefficients. Its solution yields the spatial distribution of c, as a function of time, namely its transient behaviour. On an analytical level, solution of equation (15) into the transient c(t) is possible only for a number of relatively simple systems with well-defined geometry and flow properties. The problem is greatly simplified if the concentration function Cj(x,y,z) is essentially independent of time t, i.e. in the steady-state. Then equation (15) reduces to ... 

---