Diffusion steady-state radial

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) ... 

Therefore, the steady-state radially-symmetric diffusion equation becomes... 

Equation (1-77) is one of the most fundamental relations in the analysis of mass transfer phenomena. It was derived by integration assuming that all the molar fluxes were constant, independent of position. Integration under conditions where the fluxes are not constant is also possible. Consider, for example, steady-state radial diffusion from the surface of a solid sphere into a fluid. Equation (1-70) can be applied, but the fluxes are now a function of position owing to the geometry. Most practical problems which deal with such matters, however, are concerned with diffusion under turbulent conditions, and the transfer coefficients which are then used are based upon a flux expressed in terms of some arbitrarily chosen area, such as the surface of the sphere. These matters are discussed in detail in Chapter 2. 

Most theories of droplet combustion assume a spherical, symmetrical droplet surrounded by a spherical flame, for which the radii of the droplet and the flame are denoted by and respectively. The flame is supported by the fuel diffusing from the droplet surface and the oxidant from the outside. The heat produced in the combustion zone ensures evaporation of the droplet and consequently the fuel supply. Other assumptions that further restrict the model include (/) the rate of chemical reaction is much higher than the rate of diffusion and hence the reaction is completed in a flame front of infinitesimal thickness (2) the droplet is made up of pure Hquid fuel (J) the composition of the ambient atmosphere far away from the droplet is constant and does not depend on the combustion process (4) combustion occurs under steady-state conditions (5) the surface temperature of the droplet is close or equal to the boiling point of the Hquid and (6) the effects of radiation, thermodiffusion, and radial pressure changes are negligible. 

Solute flux within a pore can be modeled as the sum of hindered convection and hindered diffusion [Deen, AIChE33,1409 (1987)]. Diffusive transport is seen in dialysis and system start-up but is negligible for commercially practical operation. The steady-state solute convective flux in the pore is J, = KJc = where c is the radially... 

Convective diffusion to a growing sphere. In the polarographic method (see Section 5.5) a dropping mercury electrode is most often used. Transport to this electrode has the character of convective diffusion, which, however, does not proceed under steady-state conditions. Convection results from growth of the electrode, producing radial motion of the solution towards the electrode surface. It will be assumed that the thickness of the diffusion layer formed around the spherical surface is much smaller than the radius of the sphere (the drop is approximated as an ideal spherical surface). The spherical surface can then be replaced by a planar surface... 

Figure 3.98 Comparison of a reversible conventional cyclic voltammogram (linear diffusion) and reversible steady-state voltammogram obtained at a single microelectrode disc where mass transport is solely by radial diffusion. Current axis not drawn to scale. From A.M. Bond and H.A.O. Hill, Metal Inns in Biological Systems, 27 (1991) 431. Reprinted by courtesy of Marcel...

Dispersion in packed tubes with wall effects was part of the CFD study by Magnico (2003), for N — 5.96 and N — 7.8, so the author was able to focus on mass transfer mechanisms near the tube wall. After establishing a steady-state flow, a Lagrangian approach was used in which particles were followed along the trajectories, with molecular diffusion suppressed, to single out the connection between flow and radial mass transport. The results showed the ratio of longitudinal to transverse dispersion coefficients to be smaller than in the literature, which may have been connected to the wall effects. The flow structure near the wall was probed by the tracer technique, and it was observed that there was a boundary layer near the wall of width about Jp/4 (at Ret — 7) in which there was no radial velocity component, so that mass transfer across the layer... 

We consider steady-state, one-dimensional laminar flow (q ) through a cylindrical vessel of constant cross-section, with no axial or radial diffusion, and no entry-length effect, as illustrated in the central portion of Figure 2.5. The length of the vessel is L and its radius is R. The parabolic velocity profile u(r) is given by equation 2.5-1, and the mean velocity u by equation 2.5-2 ... 

In the common case of cylindrical vessels with radial symmetry, the coordinates are the radius of the vessel and the axial position. Major pertinent physical properties are thermal conductivity and mass diffusivity or dispersivity. Certain approximations for simplifying the PDEs may be justifiable. When the steady state is of primary interest, time is ruled out. In the axial direction, transfer by conduction and diffusion may be negligible in comparison with that by bulk flow. In tubes of only a few centimeters in diameter, radial variations may be small. Such a reactor may consist of an assembly of tubes surrounded by a heat transfer fluid in a shell. Conditions then will change only axially (and with time if unsteady). The dispersion model of Section P5.8 is of this type. 

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 ... 

In the model, the internal structure of the root is described as three concentric cylinders corresponding to the central stele, the cortex and the wall layers. Diffu-sivities and respiration rates differ in the different tissues. The model allows for the axial diffusion of O2 through the cortical gas spaces, radial diffusion into the root tissues, and simultaneous consumption in respiration and loss to the soil. A steady state is assumed, in which the flux of O2 across the root base equals the net consumption in root respiration and loss to the soil. This is realistic because root elongation is in general slow compared with gas transport. The basic equation is... 

In problems such as the drying of droplets or diffusion through films around spherical catalyst pellets, it is more convenient to use Eqs. (40b) and (49) in spherical coordinates. Then for steady state diffusion in the radial direction alone, one has in the absence of chemical reactions... 

The rotating ring—disc electrode (RRDE) is probably the most well-known and widely used double electrode. It was invented by Frumkin and Nekrasov [26] in 1959. The ring is concentric with the disc with an insulating gap between them. An approximate solution for the steady-state collection efficiency N0 was derived by Ivanov and Levich [27]. An exact analytical solution, making the assumption that radial diffusion can be neglected with respect to radial convection, was obtained by Albery and Bruckenstein [28, 29]. We follow a similar, but simplified, argument below. 

Find an expression for the steady-state concentration profile during the radial diffusion of a diffusant through a cylindrical shell of thickness, AR, and inner radius, i ln, in which the diffusivity is a function of radius D(r). The boundary conditions are c(r = i ln) = cln and c(r = Rln + Ail) = cout. 

Fig. 1.19. The radial pair correlation function of the steady-state overlayer generated by the A + B -> 0 annihilation reaction, with no particle diffusion. Averaged over five simulations.

Here, r0 is the radius of the hemispherical electrode A = Anr for a sphere and A = 27t/q for a hemisphere. The first term on the right-hand side of (7.18) is the Cottrell term (7.17) and the second is the correction for radial diffusion to the microelectrode. With time, the first term becomes negligible compared to the second. The time te required for the current to reach the steady-state value depends on the desired accuracy (e%) and on the diameter of the electrode d = 2ro (in Am). It can be estimated by making the first term in (7.18) negligible against the second term, according to the formula... 

Several different membrane materials have been used, namely Teflon, polyethylene, and silicon rubber among others. It is possible to obtain some degree of selectivity by choosing the material of this membrane according to the conditions of the application. The diffusion through such a structure is more complicated. For radial geometry, the steady-state current is given as... 

A maximum reactor temperature of 500 K is used in this study. This maximum temperature occurs at the exit of the adiabatic reactor under steady-state conditions. Plug flow is assumed with no radial gradients in concentrations or temperatures and no axial diffusion or conduction. 

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