Diffusion cylindrical geometry

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

A moving front is usually observed in swelling glassy polymers. A diffusion-controlled front will advance with the square root of time, and a case II front will advance linearly with time. Deviations from this simple time dependence of the fronts may be seen in non-slab geometries due to the decrease in the area of the fronts as they advance toward the center [135,140], Similarly, the values of the transport exponents described above for sheets will be slightly different for spherical and cylindrical geometries [141],... 

Kempter50 studied the thermal decomposition of 88% dense NbC cylinders from 2273 to 3473 K in 1 atm of He. Data at 3273 K will be used to test our diffusion-coupled vaporization mass loss model. We transposed the cylindrical geometry into an equivalent slab by dividing the volume by the average vaporizing area. One face of the cylinder was not included in the calculation because it rested on a NbC pedestal in the furnace. Table 3.13. summarizes analytical X-ray data for average C/Nb compositions. 

FIGURE 17.2 Dissolution pro Jes of ne (open circles) and coarse ( lied circles) hydrocortisone (Lu et al., 1993). Simulated curves were drawn using spherical geometry without (a) and with (b) a time-dependent diffusion-layer thickness and cylindrical geometry without (c) and with (d) a time-dependent diffusion-layer thickness. Error bars represent 95% 6I=(= 3). (Reprinted from Lu, A. T. K., Frisella, M. E., and Johnson, K. C. (1993pharm. Res., 10 1308-1314. Copyright 1993. With permission from Kluwer Academic Plenum Publishers.)... 

Both of these methods may be applied to nonplanar electrodes if the results are obtained at electrolysis times sufficiently short that the diffusion layer remains thin in comparison to the radius of curvature of the nonplanar electrode surface. For example, the spherical hanging-mercurcy-drop electrode provides chronoamperometric data that deviate less than 1-2% from the linear-diffusion Cottrell equation out to times of about 1 s. With solid wire electrodes of cylindrical geometry, similar conclusions apply, but at short times surface roughness effects yields a real surface area that is larger than the geometric area. 

Although the emphasis of this section will be on the most recent mechanistic approaches, the work of Fu et al. [70] published in 1976 should be mentioned since it deals with the fundamental release problem of a drug homogeneously distributed in a cylinder. In reality, Fu et al. [70] solved Fick s second law equation assuming constant cylindrical geometry and no interaction between drug molecules. These characteristics imply a constant diffusion coefficient in all three dimensions throughout the release process. Their basic result in the form of an analytical solution is... 

In this section we present expressions for the mass transfer coefficients for diffusion in spherical and cylindrical geometries. The results presented here are useful in the modeling of mass transfer in, for example, gas bubbles in a liquid, liquid droplets in a gas, or gas jets in a liquid as shown in Figure 9.6. 

For radial, unsteady diffusion in a cylindrical geometry the matrix [F] is given by expf-y Fo f[n ]]... 

Total electrode impedance consists of the contributions of the electrolyte, the electrode solution interface, and the electrochemical reactions taking place on the electrode. First, we consider the case of an ideally polarizable electrode, followed by semi-infinite diffusion in linear, spherical, and cylindrical geometry and, finally a finite-length diffusion. 

The DE (3-95) is identical in form to the familiar heat equation for radial conduction of heat in a circular cylindrical geometry. Thus we see that the evolution in time of the steady Poiseuille velocity profile is completely analogous to the conduction of heat starting with an initial parabolic temperature profile -(1 - r2)/4. In our problem, the final steady velocity profile is established by diffusion of momentum from the wall of the tube so that the initial profile for w eventually evolves to the asymptotic value uT - 0 as 1 oo. The characteristic time scale for any diffusion process (whether it is molecular diffusion, heat conduction, or the present process) is (f y cli fl iisivity ), where tc is the characteristic distance over which diffusion occurs. In the present process, tc = R and the kinematic viscosity v plays the role of the diffusivity so that... 

Theoretical characteristics for the EQ mechanism, in the feedback mode under chronoamperometric and steady-state conditions, were obtained initially by solving numerically the diffusion equations in the axisymmetric cylindrical geometry defining the tip and substrate, in the SECM configuration (Fig. 1). The results were later generalized and simplified equations presented for the collection efficiency and the relationship between the substrate and tip responses (TG/SC mode) under steady-state conditions (5). 

FIG. 2 Principles of SECMID using H+ as a model adsorbate. Schematic of the transport processes in the tip/substrate domain for a reversible adsorption/desorption process at the substrate following the application of a potential step to the tip UME where the reduction of H+ is diffusion-controlled. The coordinate system and notation for the axisymmetric cylindrical geometry is also shown. Note that the diagram is not to scale as the tip/substrate separation is typically <0.01 rs. 

Integral transforms can be used to solve ordinary differential equations by converting them to algebraic equations. In what follows, the convolution properties of the different transforms have been listed, followed by the methods of integral transform used to solve (a) one-dimensional diffusion equations in the infinite and semi-infinite domains and (b) Laplace equations in the cylindrical geometries. 

Metal deposition occurs with sharp gradients within a catalyst pellet, usually concentrated on the outside of catalyst pellets forming a U-shaped distribution. Sato et at [3] related this metal deposition with simultaneous diffusion and reaction, and suggested a value of 8 for the Thiele modulus in a slab geometry. Tamm [4] suggested that this distribution can be characterized by a theta factor defined in a cylindrical geometry as... 

Tannock (1968) used the diffusion equation for a cylindrical geometry to calculate the radius of viable tissue around a capillary on the basis of... 

Calculations of the spin-echo intensity are complicated by the fact that surface relaxation may play a significant role. A general formalism for calculating PFG spin-echo attenuation for restricted diffusion in isolated pores has recently been proposed that allows for wall relaxation effects. Expressions have been obtained for the cases of diffusion within a sphere, and for planar and cylindrical geometries.These show that diffraction effects are still apparent even when surface relaxation is rapid. Also, the locations of the minima in the spin-echo intensities are not particularly affected by varying the surface relaxation parameter, Analysis of PFG spin-... 

In cylindrical geometry, the surface through which diffusion occurs is not constant. The external surface of the silica layer is larger than the inner surface and both change with time. As long as AJm. (thus, e) remains small (<10%), this difference and the gas evolution can be neglected to a first approximation, and the kinetic constant can be calculated by applying the classical parabolic law for diffusion across a planar interface ... 

Fig. 2.11 The cylindrical diffusion (a) geometry of the electrode and (b) schematic presentation of a protrusion growing inside the diffusion layer of the macroelectrode under the conditions for this type of diffusion (Reprinted from Ref. [34] with permission from Elsevier)...

An important limiting case of interest in this monograph is the purely diffusive case (no convection and chemical reaction) in the radial direction (radius = ro) of a cylindrical geometry, which occurs when a fluid within a small tube is rapidly cooled. Thus, Eq. (2.2.9) reduces to... 

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