Radial diffusion

Consider a long circular cylinder in which a solute diffuses radially. The concentration is a function of radial position (r) and time (t). In the case of constant diffusion coefficient, the diffusion equation is... 

Another typical example of inhomogeneity in rheometry is the oxidation of a polymer in a rotational rheometer in which a disk-shaped sample is held between metal fixtures. The oxygen enters the sample through the free surface (at the outer diameter) and diffuses radially inwards. The result is a radial gradient in properties which changes with time. If the reaction with oxygen results in... 

The processes of convection, axial diffusion, radial diffusion, and chemical reaction in the liquid and tissue layers all occur simultaneously. A rigorous approach requires solution of several simultaneous differential equations. To avoid this complexity in preliminary models, the transfer... 

A slight variant on the plume model of Park et al (42) is shown in Fig. 9. Coal particles injected at the bottom of the bed are convected upward and diffuse radially as a consequence of bubble motion. Volatiles will be released at a decaying rate for several seconds, depending upon particle size and bed temperature. The radius r of the cross-section over which the volatiles are released is determined by the solid diffusivity VQ. To a first approximation... 

First calculations of beryllium transport in PISCES have also been carried out. The axial profile of Bell emission can be well reproduced. The simulated amount of re-deposition of sputtered beryllium is relatively small (about 2%). The main reason for this is the radial transport (caused by diffusion, radial electrical field and friction) leading to a loss of beryllium, which cannot contribute to a re-deposition. However, additional calculations and a detailed comparison with experimental findings is necessary. The influence of the beryllium deposition at the graphite samples on the chemical erosion is one of the open questions still to be addressed and is a matter of particular interest for ITER. 

In the bubble column the velocity profile of recirculating liquid is shown in Fig. 27, where the momentum of the mixed gas and liquid phases diffuses radially, controlled by the turbulent kinematic viscosity Pf When I/l = 0 (essentially no liquid feed), there is still an intense recirculation flow inside the column. If a tracer solution is introduced at a given cross section of the column, the solution diffuses radially with the radial diffusion coefficient Er and axially with the axial diffusion coefficient E. At the same time the tracer solution is transported axially Iby the recirculating liquid flow. Thus, the tracer material disperses axially by virtue of both the axial diffusivity and the combined effect of radial diffusion and the radial velocity profile. 

The capillary phenomena in filter paper have been the subject of numerous investigations. It is possible to tell when the sensitivity limit of an indicator paper is being approached simply from the appearance of the reactions taking place. A drop of 0.001 N HCl placed upon congo paper will, of course, diffuse radially. The center of this region shows the red alkaline color of the indicator. This is surrounded by a circle showing the acid reaction, which is followed in turn by a water circle. Other indicator papers such as dimethyl yellow, azolitmin, litmus papers, etc., behave in similar fashion. 

Category 2 this comprises electroactive zones of micro size, which are separated by sufficiently large inert blocking material. This electrode as a whole behaves as a collection of isolated microelectrodes, each of which exhibits convergent diffusion (radial and axial). 

The third variation that exploits unsliced gels is based on the following principle the gel is, after isoelectric focusing, incubated with antiserum and the non-precipitated proteins are eluted. Further processing, e.g., staining and destaining is conventional. Methods that use slicing can be also modified in at least three different ways the individual sections can be tested by double gel diffusion, radial diffusion or immunoelectrophoresis. 

We also can use these microscopic models to describe diffusion radially inward, accounting for the changing pore shape and the presence of void or solid as a function of any position (r, 0, mass transfer is solved rigorously in a pellet at the level depicted in Figure 7.1D to give an equation of the form... 

Hence, high rates of mass transfer are obtained when Sc(0) is very small. Tangential convection parallel to the interface is responsible for sweeping away the mobile component as it diffuses radially into the boundary layer. The thin boundary layer simplification proceeds as follows ... 

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... 

Given the laminar nature of the flow, mixing of separate vapor streams is dominated by diffusion. The mixing will be complete and uniform, when the vapors diffuse radially to the point of filling the entire flow diameter, which occurs in characteristic time Xd ... 

On the other hand, the drug released from the posterior surface of the CL needs to diffuse radially over the long distance of the lens radius to reach tear-lake and then absorb into conjunctiva. 

The remaining equations in Table 7.6 can be derived in a similar fashion. Note that no partial derivatives with respect to appear in the polymer-phase balances on the functional groups ([A], [11] and [Z]) because it was assumed that these groups cannot diffuse. Radial concentration gradients in end-groups are observed when the equations are solved, because the local reaction rates ... 

The Biemann concentrator consisted of a heated glass jacket surrounding a sintered glass tube. The eluent from the chromatograph passed directly through the sintered glass tube and the helium diffused radially through the... 

The most striking change is in the A-term that accounts for multipath dispersion (Equation [3.33] in the van Deemter theory) and decreases rapidly with decreasing i.d. This effect was interpreted (Karlsson 1988) as reflecting a decrease in the extent of the flow rate variation over the column cross-section as the i.d. decreases, together with shorter time for solute molecules to diffuse radially through the flow velocity profile (parabolic in the case of a nonpacked column) together these phenomena will decrease the effect of flow anisotropy that is reflected in... 

The third mechanism that can result in the separation of large particles from dissolved polymer is transient dispersion. If, during flow through a capillary, a species with a low diffusion coefficient does not have sufficient time to diffuse radially, it will emerge from the capillary in a peak located well before that of a fast-diffusing species. The location and shape of the eluting peak are determined by the size (diffusion coefficient) of the eluting species. This mechanism has been used to determine the diffusion coefficients of monodisperse biopolymers [14,15]. 

Homogeneity of cell layer stuck to fibers oxygen diffuses radially though the layers and is consumed because of cell respiration and other metabolic reactions, with a zero-order reaction. 

For the case of diffusion radially through a cylinder wall of inner radius r, and outer and length L,... 

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