Diffusion steady-state analytical approximations

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. 

The rotating ring—disc electrode (RRDE) is probably the most well-known and widely used double electrode. It was invented by Frumkin and Nekrasov [26] in 1959. The ring is concentric with the disc with an insulating gap between them. An approximate solution for the steady-state collection efficiency N0 was derived by Ivanov and Levich [27]. An exact analytical solution, making the assumption that radial diffusion can be neglected with respect to radial convection, was obtained by Albery and Bruckenstein [28, 29]. We follow a similar, but simplified, argument below. 

For hydrodynamic electrodes, in order to solve the convective-diffusion equation analytically for the steady-state limiting current, it is necessary to use a first-order approximation of the convection function(s) (such as the Leveque approximation for the channel). These approximate expressions for the steady-state mass transport limited currents were introduced in Section 4 (see Table 5). 

Equation was derived without approximations. It is noteworthy that these solutions do not couple tensorial components of different orders and that they confirm that rotational diffusion and cis—>trans thermal isomerization are isotropic processes that do not favor any spatial direction. In Section 3.4, I discuss, through the example of azobenzene, how Equation 3.11 can be used to study reorientation processes during cis—>trans thermal isomerization after the end of irradiation. The next subsection gives analytical expressions at the early-time evolution and steady-state of photo-orientation, for the full quantification of coupled photo-orientation and photoisomerization in A<- B photoisomerizable systems where B is unknown. 

It should be emphasized that Oh and Cavendish assumed that the reactions only occur on the surface of the channel wall. This assumption is less realistic for a layer of washcoat (typically y-alumina) dispersed with catalyst applied on to the wall surface. Ramanathan, Balakotaiah, and West showed that the diffusion in the washcoat has a profound influence on the light-off behavior of a monolith converter. They derived an analytical light-off criterion based on a onedimensional two-phase model with position-dependent heat and mass transfer coefficients. The derivation of this criterion is based on the two key assumptions a positive exponential approximation (i.e., the Frank-Kameneskii approximation) and negligible reactant consumption in the fluid phase. The light-off is defined as the occurrence of multiple steady states with the attainment of the ignited steady state. Here, we discuss only the results of their analysis, without going into the details of their derivation. 

Assuming a uniform accessibility of the tip surface, e.g., a uniform concentration of electroactive species, an analytical approximation of the tip feedback current can be derived (see Chapter 5). For convenience, we repeat the main equations here. Such a model represents a thin layer cell (TLC) with a diffusion-limiting current expressed by Eq. (8). The approximate equation for a quasi-reversible steady-state voltammogram is as follows (11) ... 

Concentration distribution is described by the steady-state equation (3.1.1) and boundary conditions (3.1.2) and (3.1.5), where = Y is the distance from plate surface. In the diffusion boundary layer approximation, the exact analytical... 

Analytical solutions for cases of temperature-dependent thermal conductivity are available [22, 23]. In cases where the solid s thermophysical properties vary significantly with temperature, or when phase changes (solid-liquid or solid-vapor) occur, approximate analytical, integral, or numerical solutions are oftentimes used to estimate the material thermal response. In the context of the present discussion, the most common and useful approximation is to utilize transient onedimensional semi-infinite solutions in which the beam impingement time is set equal to the dwell time of the moving solid beneath the beam. The consequences of this approximation have been addressed for the case of a top hat beam, p 1 = K = 0 material without phase change [29] and the ratios of maximum temperatures predicted by the steady-state 2D analysis. Transient ID analyses have also been determined. Specifically, at Pe > 1, the diffusion in the x direction is negligible compared to advection, and the ID analysis yields predictions of Umax to within 10 percent of those associated with the 2D analysis. 

Within the effective chemical isolation thickness of a cap, as defined by Eq. 4 under transient conditions or Eq. 5 under steady-state conditions, the chemical migration processes are limited to advection and diffusion (that is, no significant bioturbation or erosion). The dynamics of the chemical migration behavior within this layer can be estimated by the advection-diffusion equation. Traditionally, the cap is often approximated as semi-infinite and the transient behavior estimated using an analytical solution of the advection-diffusion equation [1]. The approach can be extended to reactive contaminants using the solution of van Genuchten [2]... 

The transient and steady-state voltammetric responses of GNEs have been analyzed by simulation, theory, and experiment (1, 2). An approximate analytical expression for the diffusion-limited steady-state current at a GNE, t,i , is given by equation (6.3.11.3) ... 

Building on the results of Ref. [61], the Unwin group proposed a family of equilibrium perturbation-based approaches to study lateral surface diffusion [62]. The theory based on numerical solution of a time-dependent diffusion problem was developed to describe a triple potential step transient experiments [62a], chronoamperometric experiments, and steady-state approach curves [62b]. An approximate analytical expression for the approach curves produced by the combination of bulk diffusion in the tip/substrate gap with heterogeneous ET and lateral charge transport on the substrate surface is also available [63],... 

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