Diffusion in cylindrical coordinates

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... 

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

A similar simplification for radial diffusion in cylindrical coordinates is valid at large Schmidt numbers ... 

For one-dimensional diffusion in Cartesian coordinates For radial diffusion in cylindrical coordinates For radial diffusion in spherical 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. 

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,... 

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 ... 

There are certain practical diffusion problems, which can be treated most appropriately in cylindrical or in spherical coordinates. In many cases, choosing the natural coordinate system allows for the coordinates to be separated, and one is left with the simpler problem of dealing with one-dimensional diffusion along the radial coordinate. Basically, the only technical complication which arises as compared to the one-dimensional diffusion in Cartesian coordinates treated so far, concerns the approximation of the spatial derivative of the concentration involved by the diffusion equation. 

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... 

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

Consider the diffusion of solute A from the surface of a cylinder of radius R into a homogeneous tissue (Figure 3.4b). For example, the cylinder might represent the external surface of a capillary that contains a high concentration of a drug. The concentration within the tissue, in the region r > R, can be determined by solving Equation 3-31 in cylindrical coordinates ... 

As in the planar membrane, the concentration of drug in the cylinder wall can be found by solving the diffusion equation, now written in cylindrical coordinates ... 

Solution of the mass transfer equation in cylindrical coordinates is not as simple as the procedure described above in rectangular coordinates. Radial diffusion and first-order irreversible chemical kinetics in long cylinders produce the following linear second-order ordinary differential equation with variable coefficients ... 

In the absence of convective mass transfer and chemical reaction, calculate the steady-state liquid-phase mass transfer coefficient that accounts for curvature in the interfacial region for cylindrical liquid-solid interfaces. An example is cylindrical pellets that dissolve and diffuse into a quiescent liquid that surrounds each solid pellet. The appropriate starting point is provided by equation (B) in Table 18.2-2 on page 559 in Bird et al. (1960). For one-dimensional diffusion radially outward, the mass transfer equation in cylindrical coordinates reduces to... 

One last example should suffice to illustrate the way the point method can be used. If we take the diffusion equation in cylindrical coordinates and add a homogeneous first-order chemical reaction. 

Impedance may also be studied in the case of forced diffusion. The most important example of such a technique is a rotating disk electrode (RDE). In a RDE conditions a steady state is obtained and the observed current is time independent, leading to the Levich equation [17]. The general diffusion-convection equation written in cylindrical coordinates y, r, and q> is [17]... 

As the mean firee path of the atoms (0.043 cm at 200 Pa) is less than the diameter of the reactor (5 cm), the atom concentration is given by the diffusion equation written in cylindrical coordinates that describes the variation of the concentration Co of an oxygen atom versus time for a fixed point in the cylinder (r. X) ... 

Other hydrodynamic cases for corrosion studies can be found elsewhere [10]. On the other hand, for a continuous metal removal from solution in electrowinning, rotating cylinders and disks are used as cathodes (Figme 7.6). The classical rotating-disk cathode is known as Weber s disk [25-26] having a diameter of 2a. The corresponding differential equation in cylindrical coordinates is free of the diffusivity and it is given by... 

The condition that the electrochemical processes occur largely in the region of the meniscus is only met if the thin film of electrolyte is absent or if the thickness is very small. The simple-pore model [64] is an example of the first case. The meniscus is assumed to form at the intersection of micropores with macropores. While the micropores are filled with electrolyte up to the intersection, the macropores are filled with gas. The meniscus may be treated as flat in a first approximation. The walls of micropores are the seat of the electrode reaction. The simple-pore model was suggested [64] as applying to non-wetted systems like the Teflon-bonded platinum black electrodes. The limiting current due to the diffusion of species into a micropore was derived [64] as the steady-state solution of the two-dimensional diffusion equation in cylindrical coordinates. The summation of the currents of the individual pores leads to ... 

The convective-diffusion Equation 7.8 in the case of two-dimensional system in a steady state [(dddt) = 0] in cylindrical coordinates is as follows ... 

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