Diffusion equation spherical, steady-state

Equation (14) also shows that for microorganisms with radii that are less than a few microns with a typical diffusion layer thickness > 10 pm, radial diffusion should predominate over linear diffusion [46], Under steady-state conditions, the area integrated cellular flux (mols-1), Q, for a small, spherical cell of surface = 4tt q, is given by ... 

Write the equations for steady-state diffusion across a flat membrane, across the wall of a tube, and out of a spherical shell. 

This expression is the sum of a transient tenu and a steady-state tenu, where r is the radius of the sphere. At short times after the application of the potential step, the transient tenu dominates over the steady-state tenu, and the electrode is analogous to a plane, as the depletion layer is thin compared with the disc radius, and the current varies widi time according to the Cottrell equation. At long times, the transient cunent will decrease to a negligible value, the depletion layer is comparable to the electrode radius, spherical difhision controls the transport of reactant, and the cunent density reaches a steady-state value. At times intenuediate to the limiting conditions of Cottrell behaviour or diffusion control, both transient and steady-state tenus need to be considered and thus the fiill expression must be used. Flowever, many experiments involving microelectrodes are designed such that one of the simpler cunent expressions is valid. 

The estimation of the diffusional flux to a clean surface of a single spherical bubble moving with a constant velocity relative to a liquid medium requires the solution of the equation for convective diffusion for the component that dissolves in the continuous phase. For steady-state incompressible axisym-metric flow, the equation for convective diffusion in spherical coordinates is approximated by... 

Mass transfer from a single spherical drop to still air is controlled by molecular diffusion and. at low concentrations when bulk flow is negligible, the problem is analogous to that of heat transfer by conduction from a sphere, which is considered in Chapter 9, Section 9.3.4. Thus, for steady-state radial diffusion into a large expanse of stationary fluid in which the partial pressure falls off to zero over an infinite distance, the equation for mass transfer will take the same form as that for heat transfer (equation 9.26) ... 

Neal and Nader [260] considered diffusion in homogeneous isotropic medium composed of randomly placed impermeable spherical particles. They solved steady-state diffusion problems in a unit cell consisting of a spherical particle placed in a concentric shell and the exterior of the unit cell modeled as a homogeneous media characterized by one parameter, the porosity. By equating the fluxes in the unit cell and at the exterior and applying the definition of porosity, they obtained... 

The mathematical formulations of the diffusion problems for a micropippette and metal microdisk electrodes are quite similar when the CT process is governed by essentially spherical diffusion in the outer solution. The voltammograms in this case follow the well-known equation of the reversible steady-state wave [Eq. (2)]. However, the peakshaped, non-steady-state voltammograms are obtained when the overall CT rate is controlled by linear diffusion inside the pipette (Fig. 4) [3]. 

If the electric field E is applied to a system of colloidal particles in a closed cuvette where no streaming of the liquid can occur, the particles will move with velocity v. This phenomenon is termed electrophoresis. The force acting on a spherical colloidal particle with radius r in the electric field E is 4jrerE02 (for simplicity, the potential in the diffuse electric layer is identified with the electrokinetic potential). The resistance of the medium is given by the Stokes equation (2.6.2) and equals 6jtr]r. At a steady state of motion these two forces are equal and, to a first approximation, the electrophoretic mobility v/E is... 

Diffusion of electroactive species to the surface of conventional disk (macro-) electrodes is mainly planar. When the electrode diameter is decreased the edge effects of hemi-spherical diffusion become significant. In 1964 Lingane derived the corrective term bearing in mind the edge effects for the Cotrell equation [129, 130], confirmed later on analytically and by numerical calculation [131,132], In the case of ultramicroelectrodes this term becomes dominant, which makes steady-state current proportional to the electrode radius [133-135], Since capacitive and other diffusion-unrelated currents are proportional to the square of electrode radius, the signal-to-noise ratio is increased as the electrode radius is decreased. 

For ease of solution, it is assumed that the spherical shape of the pellet is maintained throughout reaction and that the densities of the solid product and solid reactant are equal. Adopting the pseudo-steady state hypothesis implies that the intrinsic chemical reaction rate is very much greater than diffusional processes in the product layer and consequently the reaction is confined to a gradually receding interface between reactant core and product ash. Under these circumstances, the problem can be formulated in terms of pseudo-steady state diffusion through the product layer. The conservation equation for this zone will simply reflect that (in the pseudo-steady state) there will be no net change in diffusive flux so... 

Steady-state diffusion in three dimensions with spherical symmetry (i.e., the concentration is a function of r only) is described by an ordinary differential equation (which is Equation 3-28 simplified for spherical symmetry, cf. Equation 3-66b later) ... 

Figure 3-3c displays the steady-state concentration profile for a spherical shell. One application of this solution is for a spinel crystal inside a magma chamber, where the spinel contains a large melt inclusion at its core. The diffusion profile in the spinel (which is a spherical shell) in equilibrium with the melt inclusion and the outside melt reservoir would follow Equation 3-3 Ig. 

The second term represents a correction for spherical diffusion. This result is approximate and assumes that there is a steady state in the reaction layer. Numerical solution using the expanding-plane model leads to the approximate equation... 

As in Fig. 11.13, the loop can be represented by an array of point sources each of length R0. Using again the spherical-sink approximation of Fig. 11.126 and recalling that d Rl Ro, the quasi-steady-state solution of the diffusion equation in spherical coordinates for a point source at the origin shows that the vacancy diffusion field around each point source must be of the form... 

The steady state diffusion equation in spherical polar coordinates relates concentration c to only the radius r because of spherical symmetry ... 

Mass transfer from the water phase to the microdroplet under stationary conditions quickly reaches the steady-state because of spherical diffusion. When the extraction of X is diffusion-limited in the water phase. Equation (3) is obtained using the diffusion coefficient of X in the water phase (Z w) [18]. 

For a simple electron transfer [see (1), (2)], it is possible to solve the diffusion equation analytically at steady state, as described for a microdisc by (91) and for a spherical electrode by (99). 

For planar or spherical electrodes, where the mass transport is a diffusion function in one dimension, it is possible to solve the diffusion equation as a function of time. In Section 3 the principles of how the cyclic voltammetric peak current could be calculated for a simple electron transfer reaction were presented. It is also possible to solve the material balance equations for the spherical electrode at steady state for a few first-order mechanisms (Alden and Compton, 1997a). In order to tackle second-order kinetics, more complex mechanisms, solve time-dependent equations or model other geometries with... 

Using these equations for heat conduction in the porous network emd the pseudo-steady state emalysis described for pore diffusion, the time to dry a spherical green body with pore heat conduction as the rate determining step is given by... 

The rate of heterogeneous condensation depends on the exchange of matter and heat between a particle and the continuous phase. The extreme cases of a particle much larger or much smaller than the mean free path of the suspending gas are easy to analyze. In the continuum range (dp ip), diffusion theory can be used to calculate the transport rate. For a single sphere in an infinite medium, the steady-state equation of diffusion in spherical coordinates takes the form... 

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