Diffusion equation species

Thus far we have considered systems where stirring ensured homogeneity witliin tire medium. If molecular diffusion is tire only mechanism for mixing tire chemical species tlien one must adopt a local description where time-dependent concentrations, c r,f), are defined at each point r in space and tire evolution of tliese local concentrations is given by a reaction-diffusion equation... 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

Time-dependent diffusion equations, appropriate to the axisymmetrical cylindrical geometry of the SECM can be written for the species of interest in each phase. 

Biofilms adhere to surfaces, hence in nearly all systems of interest, whether a medical device or geological media, transport of mass from bulk fluid to the biofilm-fluid interface is impacted by the velocity field [24, 25]. Coupling of the velocity field to mass transport is a fundamental aspect of mass conservation [2]. The concentration of a species c(r,t) satisfies the advection diffusion equation... 

Starting from an initial state where half the system has species A and the other half B, a reaction front will develop as the autocatalyst B consumes the fuel A in the reaction. The front will move with velocity c. The reaction-diffusion equation can be solved in a moving frame, z = x — ct, to determine the front profile and front speed,... 

The first and second integrals have their coordinate systems centered on the catalytic C and noncatalytic N spheres, respectively. The local nonequilibrium average microscopic density field for species a is pa(r) = [Y = 5(r - ( )) The solution of the diffusion equation can be used to estimate this nonequilibrium density, and thus the velocity of the nanodimer can be computed. The simple model yields results in qualitative accord with the MPC dynamics simulations and shows how the nonequilibrium density field produced by reaction, in combination with the different interactions of the B particles with the noncatalytic sphere, leads to directed motion [117],... 

Since turbulent fluctuations not only occur in the velocity (and pressure) field but also in species concentrations and temperature, the convection diffusion equations for heat and species transport under turbulent-flow conditions also comprise cross-correlation terms, obtained by properly averaging products of... 

Eggels and Somers (1995) used an LB scheme for simulating species transport in a cavity flow. Such an LB scheme, however, is more memory intensive than a FV formulation of the convective-diffusion equation, as in the LB discretization typically 18 single-precision concentrations (associated with the 18 velocity directions in the usual lattice) need to be stored, while in the FV just 2 or 3 (double-precision) variables are needed. Scalar species transport therefore can better be simulated with an FV solver. 

For problems involving gradients in chemical species, the convection-diffusion equations for the species are also solved, usually for N— 1 species with the Nth species obtained by forcing the mass fractions to sum to unity. Turbulence can be described by a turbulent diffusivity and a turbulent Schmidt number, Sct, analogous to the heat transfer case. 

Diffusion is quantified by measuring the concentration of the diffusing species at different distances from the release point after a given time has elapsed at a precise temperature. Raw experimental data thus consists of concentration and distance values. The degree of diffusion is represented by a diffusion coefficient, which is extracted from the concentration-distance results by solution of one of two diffusion equations. For one-dimensional diffusion, along x, they are Fick s first law of diffusion ... 

This is known as Fick s second law of diffusion or more commonly as the diffusion equation. In these equations, J is called the flux of the diffusing species, with units of [amount of substance (atoms or equivalent units) m2 s-1], c is the concentration of the diffusing species, with units of [amount of substance (atoms or equivalent units) m-3] at position x (m) after time t (s) D is the diffusion coefficient, units (m2 s 1). 

Consider the generation of a species at a spherical electrode. In polar coordinates the diffusion equation is ... 

In the particular case dealt with now (fully labile complexation), due to the linearity of a combined diffusion equation for DmCm + DmlL ml, the flux in equation (65) can still be seen as the sum of the independent diffusional fluxes of metal and complex, each contribution depending on the difference between the surface and bulk concentration value of each species. But equation (66) warns against using just a rescaling factor for the total metal or for the free metal alone. In general, if the diffusion is coupled with some nonlinear process, the resulting flux is not proportional to bulk-to-surface differences, and this complicates the use of mass transfer coefficients (see ref. [II] or Chapter 3 in this volume). 

As implied in the previous section, the Russian investigators Zeldovich, Frank-Kamenetskii, and Semenov derived an expression for the laminar flame speed by an important extension of the very simplified Mallard-Le Chatelier approach. Their basic equation included diffusion of species as well as heat. Since their initial insight was that flame propagation was fundamentally a thermal mechanism, they were not concerned with the diffusion of radicals and its effect on the reaction rate. They were concerned with the energy transported by the diffusion of species. 

For the sake of simplicity, a common diffusion coefficient D is assumed for all species. The mass transport of the O form is described by the common diffusion equation (1.2). 

A mathematically simple case, that occurs frequently in solvent extraction systems, in which the extracting reagent exhibits very low water solubility and is strongly adsorbed at the liquid interface, is illustrated. Even here, the interpretation of experimental extraction kinetic data occurring in a mixed extraction regime usually requires detailed information on the boundary conditions of the diffusion equations (i.e., on the rate at which the chemical species appear and disappear at the interface). 

In a silicate melt or aqueous solution, a component may be present in several species. The species may interconvert and diffuse simultaneously. For example, the H2O component in silicate melt can be present as at least two species, molecular H2O (referred to as H20m) and hydroxyl groups (referred to as OH) (Stolper, 1982a). The diffusion of such a multispecies component is referred to as multispecies diffusion (Zhang et al., 1991a,b). Starting from Equation 3-5d, the one-dimensional diffusion equation for this multispecies component can be written as... 

There are two methods to write the diffusion equation for a multispecies component. One is to write the diffusion equation for the conserved component, and then relate the species concentrations by the reaction(s). Using one-dimensional H2O diffusion as an example, the diffusion equation is Equation 3-22a ... 

The second way to write the diffusion equations for a multispecies component is to write the diffusion-reaction equation for each species. Starting from Equation 3-5b, the diffusion-reaction equation for each species is... 

Assuming that the above reaction is an elementary reaction, then a onedimensional diffusion equation for such a species may be written as (the second method)... 

The total concentration (w) of a multispecies component is independent of species interconversions of the t)q)e of Reaction 3-81, but is affected by the diffusion flux of individual species. Because each species may have a distinct dif-fusivity, the diffusion equation for w may be written as... 

Assuming H20m is the diffusing species, the diffusion equation for R = 0 ... 

Integration of the stationary electro-diffusion equations in one dimension. The integration of the stationary Nernst-Planck equations (4.1.1) with the LEN condition (4.1.3), in one dimension, for a medium with N constant for an arbitrary number of charged species of arbitrary valencies was first carried out by Schlogl [5]. A detailed account of Schlogl s procedure may be found in [6]. In this section we adopt a somewhat different, simpler integration procedure. 

The behaviour of the simpler autocatalytic models in each of these three class A geometries seems to be qualitatively very similar, so we will concentrate mainly on the infinite slab, j = 0. For the single step process in eqn (9.3) the two reaction-diffusion equations, for the two species concentrations, have the form... 

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