Equation with diffusion, approximate

Approximate Solutions of Chemical Separation Equations with Diffusion... 

The reaction of Si02 with SiC [1229] approximately obeyed the zero-order rate equation with E = 548—405 kJ mole 1 between 1543 and 1703 K. The proposed mechanism involved volatilized SiO and CO and the rate-limiting step was identified as product desorption from the SiC surface. The interaction of U02 + SiC above 1650 K [1230] obeyed the contracting area rate equation [eqn. (7), n = 2] with E = 525 and 350 kJ mole 1 for the evolution of CO and SiO, respectively. Kinetic control is identified as gas phase diffusion from the reaction site but E values were largely determined by equilibrium thermodynamics rather than by diffusion coefficients. 

In filtration, the particle-collector interaction is taken as the sum of the London-van der Waals and double layer interactions, i.e. the Deijagin-Landau-Verwey-Overbeek (DLVO) theory. In most cases, the London-van der Waals force is attractive. The double layer interaction, on the other hand, may be repulsive or attractive depending on whether the surface of the particle and the collector bear like or opposite charges. The range and distance dependence is also different. The DLVO theory was later extended with contributions from the Born repulsion, hydration (structural) forces, hydrophobic interactions and steric hindrance originating from adsorbed macromolecules or polymers. Because no analytical solutions exist for the full convective diffusion equation, a number of approximations were devised (e.g., Smoluchowski-Levich approximation, and the surface force boundary layer approximation) to solve the equations in an approximate way, using analytical methods. 

A large number of analytical solutions of these equations appear in the literature. Mostly, however, they deal only with first order reactions. All others require solution by numerical or other approximate means. In this book, solutions of two examples are carried along analytically part way in P7.02.06 and P7.02.07. Section 7.4 considers flow through an external film, while Section 7.5 deals with diffusion and reaction in catalyst pores under steady state conditions. 

Figure 17. Comparison of Ju versus t plots predicted by different submodels for a system with one type of site (adsorbing and internalising) linear isotherm with dSS approximation (O) applying equation (43) with A)i = 5.2 x 10 6 m Langmuirian isotherm with dSS approximation (continuous line) applying equation (46) Langmuirian isotherm with semi-infinite diffusion (dotted line) by numerically solving integral equation (7)). Other parameters c(, = 5x 10-4 mol m-3, Dm = 8 x 10 10 m2 s-1, Kn = 2 x 10-5 m, k — 5 x 10 4 s 1, ro = 1.8 x 10 6 m, r0 + <5M = 10 5 m, KM — 2.88x 10 3mol m 3, Tmax = 1.5 x 10-8 mol m-2...

Finally, as described in Box 4.1 of Chapter 4, an exact numerical solution of the diffusion equation (based on Fick s second law with an added sink term that falls off as r-6) was calculated by Butler and Pilling (1979). These authors showed that, even for high values of Ro ( 60 A), large errors are made when using the Forster equation for diffusion coefficients > 10 s cm2 s 1. Equation (9.34) proposed by Gosele et al. provides an excellent approximation. 

The light fluxes are now linear functions of the depth coordinate z as it is predicted also by Fick s first law for steady-state diffusion without sink. For weak absorption, the equations for Td and Ro of the Kubelka-Munk formalism are also directly equivalent to the results of the diffusion approximation. Comparing Eqs. (8.22) and (8.23) with Eqs. (8.11), (8.12), and (8.14) under diffuse irradiation or under //o = 2/3, the Kubelka-Munk coefficients can be expressed by<31 34)... 

The Gaussian expressions are not expected to be valid descriptions of turbulent diffusion close to the surface because of spatial inhomogeneities in the mean wind and the turbulence. To deal with diffusion in layers near the surface, recourse is generally had to the atmospheric diffusion equation, in which, as we have noted, the key problem is proper specification of the spatial dependence of the mean velocity and eddy difiusivities. Under steady-state conditions, turbulent diffusion in the direction of the mean wind is usually neglected (the slender-plume approximation), and if the wind direction coincides with the x axis, then = 0. Thus, it is necessary to specify only the lateral (Kyy) and vertical coefficients. It is generally assumed that horizontal homogeneity exists so that u, Kyy, and Ka are independent of y. Hence, Eq. (2.19) becomes... 

As noted earlier, air-velocity profiles during inhalation and exhalation are approximately uniform and partially developed or fully developed, depending on the airway generation, tidal volume, and respiration rate. Similarly, the concentration profiles of the pollutant in the airway lumen may be approximated by uniform partially developed or fully developed concentration profiles in rigid cylindrical tubes. In each airway, the simultaneous action of convection, axial diffusion, and radial diffusion determines a differential mass-balance equation. The gas-concentration profiles are obtained from this equation with appropriate boundary conditions. The flux or transfer rate of the gas to the mucus boundary and axially down the airway can be calculated from these concentration gradients. In a simpler approach, fixed velocity and concentration profiles are assumed, and separate mass balances can be written directly for convection, axial diffusion, and radial diffusion. The latter technique was applied by McJilton et al. 

Since diffusion along the interface is negligible compared with that normal to the interface, the convective diffusion equation can be approximated by... 

Though there is fluid flow in the bulk of the electrolyte, it is found that there is a layer adjacent to the electrode in which the electrolyte is stationary, or stagnant. Thus the electron acceptors travel by convection from the bulk up to the stagnant layer and then cross the remaining boundary layer by diffusion. This transport by a convection-with-diffusion mechanism has not been taken into account so far. The equations for the time and space variation of concentration [i.e., Eq. (7.178)], for the transition time [Eq. (7.190)], and for the time variation of potential [Eq. (7.192)] have been derived for convection-free conditions, and they break down when convection becomes significant. The first approximation theory given above, therefore, deviates from experiment if the constant current is applied sufficiently long (times on the order of seconds) for convection to be important. 

Comparison with (X.2.16) shows the following drastic differences. The dominant term of (X.2.16) is absent and therefore no equation for the macroscopic part of X can be extracted. In other words, on the macroscopic scale the system does not evolve in one direction rather than the other. The remaining evolution of P is merely the net outcome of the fluctuations. Accordingly the time scale of the change is a factor slower than in the preceding case, compare (X.2.14). Since P is not sharply peaked the coefficients a(x) cannot be expanded around some central value but they remain as nonlinear functions in the equation. The first line of (1.4) contains the main terms and is called the diffusion approximation... 

Fig. 4.15. A comparison of experimental delayed kinetics of an increase of tunnelling luminescence intensity after sudden change of their mobility (temperature increase from 175 to 180 K) in KC1 with theory [86], 1 - hopping kinetics for A = 2n> obtained by means of equation (4.4.1), 2 - experimental curve, 3 - results of continuous diffusion approximation...

The kinetic equations for diffusion-controlled defect accumulation are presented in the rest of the Table 7.6. Equations (7)—(11) are approximations of the kinetics of accumulation that are not substantiated theoretically in any way but give curves with saturation, which qualitatively resemble the form of equations (1) to (5). 

In order to develop an intuition for the theory of flames it is helpful to be able to obtain analytical solutions to the flame equations. With such solutions, it is possible to show trends in the behavior of flame velocity and the profiles when activation energy, flame temperature, diffusion coefficients, or other parameters are varied. This is possible if one simplifies the kinetics so that an exact solution of the equation is obtained or if an approximate solution to the complete equations is determined. In recent years Boys and Corner (B4), Adams (Al), Wilde (W5), von K rman and Penner (V3), Spalding (S4), Hirschfelder (H2), de Sendagorta (Dl), and Rosen (Rl) have developed methods for approximating the solution to a single reaction flame. The approximations are usually based on the simplification of the set of two equations [(4) and (5)] into one equation by setting all of the diffusion coefficients equal to X/cpp. In this model, Xi becomes a linear function of temperature (the constant enthalpy case), and the following equation is obtained ... 

Ehrlich (El) uses the Crank-Nicholson (C16) finite-difference procedure for the integration of the diffusion equation, with a three-point approximation of the space derivatives on either side of the moving... 

Wilson (79, 80) pointed out that A is not the dimensionless thickness of the diffusion boundary layer scaled with D/Vg, as originally suggested by Burton et al. (74), except in the limit at which the velocity field in the layer is dominated by the bulk flow, that is, X >> 1. In this case, the analysis reduces to the one first presented by Levich (81), and the integral in equation 25 is approximated as follows ... 

From general considerations, the use of equation (2) seems to be more reasonable. However, we used both approximations (1) and (2) to understand how the choice of boundary conditions affected the D value. The data in Fig. 2 (these data were sufficiently typical) and in Table 1 show that boundary conditions had a weak effect on the diffusion coefficients. Therefore, results obtained with the approximation (2) will be discussed below. It may be noted, however, that the approximation... 

Hence, under the quasi-steady approximation, the movement of the species is dictated by a macroscopic convection-diffusion-reaction equation with an instantaneous adsorption/desorption source term. A notable consequence of the three-scale approach is the double-averaging representation for the partition coefficient A which is defined as... 

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 above reasoning shows that the stretched exponential function (4.14), or Weibull function as it is known, may be considered as an approximate solution of the diffusion equation with a variable diffusion coefficient due to the presence of particle interactions. Of course, it can be used to model release results even when no interaction is present (since this is just a limiting case of particles that are weakly interacting). 

When the concentration boundary layer is sufficiently thin the mass transport problem can be solved under the approximation that the solution velocity within the concentration boundary layer varies linearly with distance away from the surface. This is called the L6v que approximation (8, 9] and is satisfactory under conditions where convection is efficient compared with diffusion. More accurate treatments of mass transfer taking account of the full velocity profile can be obtained numerically [10, 11] but the Ldveque approximation has been shown to be valid for most practical electrodes and solution velocities. Using the L vSque approximation, the local value of the concentration boundary layer thickness, 8k, (determined by equating the calculated flux to the flux that would be obtained according to a Nernstian diffusion layer approximation that is with a linear variation of concentration across the boundary layer) is given by equation (10.6) [12]. 

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