Approximate solutions equations with diffusion

Approximate Solutions of Chemical Separation Equations 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). 

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. 

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. 

After the electrode reaction starts at a potential close to E°, the concentrations of both O and R in a thin layer of solution next to the electrode become different from those in the bulk, cQ and cR. This layer is known as the diffusion layer. Beyond the diffusion layer, the solution is maintained uniform by natural or forced convection. When the reaction continues, the diffusion layer s thickness, /, increases with time until it reaches a steady-state value. This behaviour is also known as the relaxation process and accounts for many features of a voltammogram. Besides the electrode potential, equations (A.3) and (A.4) show that the electrode current output is proportional to the concentration gradient dcourfa /dx or dcRrface/dx. If the concentration distribution in the diffusion layer is almost linear, which is true under a steady state, these gradients can be qualitatively approximated by equation (A.5). 

These equations, for the case of solid diffusion-controlled kinetics, are solved by arithmetic methods resulting in some analytical approximate expressions. One common and useful solution is the so-called Nernst-Plank approximation. This equation holds for the case of complete conversion of the solid phase to A-form. The complete conversion of solid phase to A-form, i.e. the complete saturation of the solid phase with the A ion, requires an excess of liquid volume, and thus w 1. Consequently, in practice, the restriction of complete conversion is equivalent to the infinite solution volume condition. The solution of the diffusion equation is... 

Equation (11) is also applicable as a good, or reasonably good, approximation to a number of techniques classified as d.c. voltammetry , in which the response to a perturbation is measured after a fixed time interval, tm. The diffusion layer thickness, 5/, will be a function of D, and tm and the nature of this function has to be deduced from the rigorous solution of the diffusion problem in combination with the appropriate initial and boundary conditions [21—23]. The best known example is d.c. polarography [11], where the d.c. current is measured at the dropping mercury electrode at a fixed time, tm, after the birth of a new drop as a function of the applied d.c. potential. The expressions for 5 pertaining to this and some other techniques are given in Table 1. 

Methods to solve the diffusion equation for specific boundary and initial conditions are presented in Chapter 5. Many analytic solutions exist for the special case that D is uniform. This is generally not the case for interdiffusivity D (Eq. 3.25). If D does not vary rapidly with composition, it can be replaced by successive approximations of a uniform diffusivity and results in a linearization of the diffusion equation. The... 

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 ... 

Thus, in this paper we have obtained an exact solution of the diffusion equation for one-dimensional motion of an incompressible fluid, and determined the effective diffusion coefficient. We have constructed an approximate theory of turbulent diffusion as a cascade process of motion interaction on different scales. We have obtained an expression for the turbulent diffusion coefficient with the correct transformation properties under time reversal. 

In Eq. (4.39), the upper sign refers to solution soluble product and the lower one to amalgam formation. When species R is amalgamated inside the electrode, the applicability of this analytical equation is limited by Koutecky approximation, which considers semi-infinite diffusion inside the electrode, neglecting its finite size and simplifying the calculations. Due to the limitations of this approximation, the analytical and numerical results coincide only for < 1 with a relative difference < 1.7% [20]. For higher values of c,. a numerical solution obtained with the condition (3cR/3r)r=0 = 0 should be used. 

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. 

Predicting fast and slow rates of sorption and desorption in natural solids is a subject of much research and debate. Often times fast sorption and desorption are approximated by assuming equilibrium portioning between the solid and the pore water, and slow sorption and desorption are approximated with a diffusion equation. Such models are often referred to as dual-mode models and several different variants are possible [35-39]. Other times two diffusion equations were used to approximate fast and slow rates of sorption and desorption [31,36]. For example, foraVOCWerth and Reinhard [31] used the pore diffusion model to predict fast desorption, and a separate diffusion equation to fit slow desorption. Fast and slow rates of sorption and desorption have also been modeled using one or more distributions of diffusion rates (i.e., a superposition of solutions from many diffusion equations, each with a different diffusion coefficient) [40-42]. 

Fig. 9.11. Case diagram for the approximate solutions of equation (9.99). Different approximations hold for different values of the Laplace variable, s. and y where y(= XJX.) compares the distance a photogenerated electron diffuses through solution on a particle before participating in a loss reaction with the distance over which the light is absorbed.

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]. 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

There are a number of possible approaches to the calculation of influences of finite-rate chemistry on diffusion flames. Known rates of elementary reaction steps may be employed in the full set of conservation equations, with solutions sought by numerical integration (for example, [171]). Complexities of diffusion-flame problems cause this approach to be difficult to pursue and motivate searches for simplifications of the chemical kinetics [172]. Numerical integrations that have been performed mainly employ one-step (first in [107]) or two-step [173] approximations to the kinetics. Appropriate one-step approximations are realistic for limited purposes over restricted ranges of conditions. However, there are important aspects of flame structure (for example, soot-concentration profiles) that cannot be described by one-step, overall, kinetic schemes, and one of the major currently outstanding diffusion-flame problems is to develop better simplified kinetic models for hydrocarbon diffusion flames that are capable of predicting results such as observed correlations [172] for concentration profiles of nonequilibrium species. 

The heat-loss term is analogous to that appearing in equation (B-52) and is assumed to be nonnegative and to vanish at the initial temperature (t = 0). The heat-loss distribution, k (t), is a critical factor in the structure of the nonadiabatic flame. It is found that if k (t) is of order unity or larger over a range of of order unity, then flame-structure solutions do not exist with the present formulation therefore, at first let us assume that k t) is of order p everywhere. In this case, upstream from the reaction zone in equation (10) there is a convective-diffusive balance with negligible loss in the first approximation, and the approximate solution... 

The basic BD algorithm developed by Ermak and McCam-mon (64) provides an approximate solution to the Langevin equation in the highly damped diffusive limit by using the positions of a solute particle at time t, together with the forces acting on them, to estimate the particle s new position at time, t + At. The translational behavior of the particle is described by ... 

Dispersion Models Based on Inert Pollutants. Atmospheric spreading of inert gaseous contaminant that is not absorbed at the ground has been described by the various Gaussian plume formulas. Many of the equations for concentration estimates originated with the work of Sutton (3). Subsequent applications of the formulas for point and line sources state the Gaussian plume as an assumption, but it has been rigorously shown to be an approximate solution to the transport equation with a constant diffusion coefficient and with certain boundary conditions (4). These restrictive conditions occur only for certain special situations in the atmosphere thus, these approximate solutions must be applied carefully. 

In the case of dilute solutions, the translational diffusivities are related by Dj o = 110 2, hence assuming the translational diffusion to be isotropic ( >j o= l o) with effective diffusivity >, = (2Dj g+ oy2 [13], gives reasonably accurate results (cf. Section 6.3.1). With this approximation the diffusion equation becomes... 

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