Diffusion Cylindrical coordinates

Equation (8.12) is a form of the convective dijfusion equation. More general forms can be found in any good textbook on transport phenomena, but Equation (8.12) is sufficient for many practical situations. It assumes constant diffusivity and constant density. It is written in cylindrical coordinates since we are primarily concerned with reactors that have circular cross sections, but Section 8.4 gives a rectangular-coordinate version applicable to flow between flat plates. 

Unsteady-State Heat Conduction and Diffusion in Spherical and Cylindrical Coordinates... 

Db R) Radial dispersion coefficient, general dispersion model in cylindrical coordinates Molecular diffusivity Exit age distribution function, defined in Section I... 

The diffusive flux rates would be treated similarly. The area of the control volume changing with radius is the reason the mass transport equation in cylindrical coordinates, given below - with similar assumptions as equation (2.18) - looks somewhat different than in Cartesian coordinates. 

Any flow with a nonuniform velocity profile will, when spatial mean velocity and concentration are taken, result in dispersion of the chemical. For laminar flow, the well-described velocity profile means that we can describe dispersion analytically for some flows. Beginning with the diffusion equation in cylindrical coordinates (laminar flow typically occurs in small tubes) ... 

The solution of the diffusion equation for the quasi-steady state in cylindrical coordinates shows that each dislocation line source will have a vacancy concentration diffusion field around it of the form... 

The general solution of the diffusion equation in cylindrical coordinates is a = ai lnr+ a2, and using the boundary conditions above to determine the constants ai and <12,... 

Using the solution of the Laplace equation for diffusion in cylindrical coordinates given by Eq. 5.10, fitting it to the boundary conditions given by Eq. 16.80, and employing Eq. 13.3 for the flux, the total diffusion current of atoms (per unit pipe length) passing radially from R,n to f out is... 

Construct a cylindrical cell of radius Rc centered on a single pore as illustrated in Fig. 16.13 and solve the diffusion problem within it using cylindrical coordinates and the same basic method employed to obtain... 

Solution. Starting with the diffusion equation in cylindrical coordinates (see Eq. 5.8) and using the scaling parameter to change variables, the diffusion equation in 77-space becomes... 

The diffusion equations in cylindrical coordinates for the redox species O and R, with no chemical reactions occurring in solution are given below ... 

Assuming constant diffusion coefficient, the equation describing the radial diffusion in cylindrical coordinates may be written ... 

If the geometry of the system is cylindrical, it is convenient to switch to cylindrical coordinates x along the cylinder, r the radial distance from the axis and 6 the angle. In most cases, concentration is independent of the angle and the diffusion equation is then... 

Consider Fig. 12.1, depicting the UMDE in a cylindrical coordinate system. The electrode of radius a is flush with an infinite insulating plane. The pde that governs diffusion around the UMDE is then... 

Here, the four major mapping functions for the disk electrode are presented, as well as the form that the diffusion equation for the disk electrode takes in the mapped spaces. We assume that the cylindrical coordinates, time and concentrations have all been normalised by the disk radius as in (12.14). 

Let us consider a shallow fluidized bed combustor with multiple coal feeders which are used to reduce the lateral concentration gradient of coal (11). For simplicity, let us assume that the bed can be divided into N similar cylinders of radius R, each with a single feed point in the center. The assumption allows us to use the symmetrical properties of a cylindrical coordinate system and thus greatly reduce the difficulty of computation. The model proposed is based on the two phase theory of fluidization. Both diffusion and reaction resistances in combustion are considered, and the particle size distribution of coal is taken into account also. The assumptions of the model are (a) The bed consists of two phases, namely, the bubble and emulsion phases. The voidage of emulsion phase remains constant and is equal to that at incipient fluidization, and the flow of gas through the bed in excess of minimum fluidization passes through the bed in the form of bubbles (12). (b) The emulsion phase is well mixed in the axial... 

Figure 16 Projection of Brownian trajectories (in cylindrical coordinates) for rodlike molecules for different values of the dimensionless diffusivity perpendicular to the rod axis ()). The initial orientation of the molecule is along the z direction (Gupta and Khakhar [65]).

A uniformly accessible electrode is an electrode where, at the interface, the flux and the concentration of a species produced or consumed on the electrode are independent of the coordinates that define the electrode surface. The mass flux at the interface is obtained by solving the material balance equation. If migration can be neglected, the material balance equation for dilute electrol5 c solutions is reduced to the convective-diffusion equation. For an axisymmetric electrode, the concentration derivatives with respect to the angular coordinate 9 are equal to zero, and the convective-diffusion equation can be expressed in cylindrical coordinates as... 

There are three possibilities corresponding to the dimension of the distribution. The first is a ID concentration distribution (d = 1), in which the diffusing species spreads evenly in the z directions from an initial line pulse at z = 0 on the xz plane. In this case, the variable r in (6-37) is the Cartesian variable z. The second case is a circularly symmetric distribution for c (d = 2), which evolves by diffusion on a plane from an initial compact planar pulse. In this case, r in (6 37) is the radial component of a polar (or cylindrical) coordinate system that lies in the diffusion plane. The third case is a spherically symmetric distribution corresponding to d = 3, which evolves at long times from a compact 3D pulse that diffuses outward into the frill 3D space. In this case, r is the radial variable of a spherical coordinate system. To obtain the long-time form of the distribution we must solve (6-37), but subject to the integral constraint that the total amount of the diffusing species is constant, independent of time ... 

The considerable complexity of SECM theory is due to the combination of a cylindrical diffusion to the ultramicroelectrode (UME) tip and a thin-layer-type diffusion space. The time-dependent diffusion problem for a simple quasireversible reaction in cylindrical coordinates is as follows (2,3) ... 

The experiment is carried out under diffusion control. Theoretical concentration profiles are calculated by solving Fick s second law of diffusion in the steady state with boundary conditions appropriate to the solution domain and to the substrate, taking into account its geometry and the type of reaction occurring on it. Assumption is made that the redox species are stable and not involved in a homogeneous reaction in solution. Two geometries known to produce steady-state concentration profiles have been considered (72,77) the hemisphere and the microdisk. The former only requires a radial dimension, and the diffusion equation can be solved analytically. The latter, on the other hand, necessitates cylindrical coordinates and the solution becomes much more complex. With the latter a closed form analytical ex-... 

It follows from the previous discussion and the results of Section 3.1 that the diffusion equation and the boundary conditions have the following form in the cylindrical coordinates ... 

The convective diffusion equation in cylindrical coordinates has the form... 

Pore-water profiles were used together with solid-phase dissolution rates in diagenetic models to determine first-order anoxic precipitation rate constants for both Mn and Fe. A two-dimensional cylindrical coordinate model was employed to account for the effects of biogenic irrigation of burrows on pore-water Mn " distributions. Two-dimensional diffusion can result in a decrease in Mn " with depth that would be interpreted as evidence for precipitation and cause overestimation of precipitation rates in a one-dimensional model. 

Cylindrical coordinate along symmetry axis in stability analysis of liquid jet Fraction of column cross-sectional area available to solute Thermal diffusivity Amplification factor in jet instability Particle diameter to minimum separation between particles, Eq. (9.3.9)... 

The solvent is assumed to be in solid body rotation at an angular speed (o, and the solute is assumed to move circumferentially with the solvent. A single solute is considered, that is, a binary mixture, and a cylindrical coordinate system rotating with the angular speed (o is adopted. The solute concentration is then a function only of the time t and radial distance r from the rotation axis. The continuity (diffusion) equation (Eq. 3.3.15) can therefore be written... 

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