Diffusion convective radial

As time goes on, the action of radial diffusion continues to inhibit axial dispersion by diffusion and convection and makes the mixed zone more uniform, as shown at time tj. Finally, at still larger times a quasi equilibrium is established. Here, convection, radial diffusion, and axial diffusion all contribute to the dispersion, with the net effect appearing as if the fluid were in plug flow, whereas in fact the velocity is radially distributed. With a further increase in time, the effect is only to increase the length of the mixed zone. 

Just as pure axial diffusion is in a sense an opposite limit to pure convection, so also is convective axial diffusion an opposite limit to convective radial diffusion (Taylor dispersion). The convective axial diffusion limit, as with the Taylor dispersion limit, characterizes the convection at the mean flow speed U, whereas true convection is at the actual local speed. The criterion for the axial convection term to be of the same order as the axial diffusion term is... 

What important dimensionless number(s) appear in the dimensionless partial differential mass transfer equation for laminar flow through a blood capillary when the important rate processes are axial convection, radial diffusion, and nth-order irreversible chemical reaction ... 

In this case study we assiune a constant even flow and concentration at the inlet and only consider steady state. However, the concentration will have a radial variation downstream due to the reaction and the radial difference in residence time. Assuming an even inflow and a minor effect of gravity in the radial and tangential directions, the convective radial transport terms can be removed. Axial viscous transport, diffusion, and conduction can also be neglected because the convective axial transport is much larger. After a sufficiently long period of time, the transient terms become very small, and we obtain a model with steady axial convection, radial diffusion, and conduction with a reaction soiuce term. 

This section derives a simple version of the convective diffusion equation, applicable to tubular reactors with a one-dimensional velocity profile V (r). The starting point is Equation (1.4) applied to the differential volume element shown in Figure 8.9. The volume element is located at point (r, z) and is in the shape of a ring. Note that 0-dependence is ignored so that the results will not be applicable to systems with significant natural convection. Also, convection due to is neglected. Component A is transported by radial and axial diffusion and by axial convection. The diffusive flux is governed by Pick s law. 

With the carrier stream unsegmented by air bubbles, dispersion results from two processes, convective transport and diffusional transport. The former leads to the formation of a parabolic velocity profile in the direction of the flow. In the latter, radial diffusion is most significant which provides for mixing in directions perpendicular to the flow. The extent of dispersion is characterized by the dispersion coefficient/). 

The simplest and most commonly used convection apparatus consists of a disc electrode rotating with a constant angular velocity u [1-5]. The disc sucks the solution toward its surface, much in the way a propeller would as the solution approaches the disc, it is swept away radially and tangentially (see Fig. 14.1). The transport of the reacting species to the disc occurs both by convection and diffusion. Though the mathematics are complicated, the rate of transport can be calculated exactly for an infinite disc. A particularly nice feature of this setup is the fact that the transport is uniform so that the surface concentration of any reacting species is constant over the surface of the electrode. 

As noted earlier, air-velocity profiles during inhalation and exhalation are approximately uniform and partially developed or fully developed, depending on the airway generation, tidal volume, and respiration rate. Similarly, the concentration profiles of the pollutant in the airway lumen may be approximated by uniform partially developed or fully developed concentration profiles in rigid cylindrical tubes. In each airway, the simultaneous action of convection, axial diffusion, and radial diffusion determines a differential mass-balance equation. The gas-concentration profiles are obtained from this equation with appropriate boundary conditions. The flux or transfer rate of the gas to the mucus boundary and axially down the airway can be calculated from these concentration gradients. In a simpler approach, fixed velocity and concentration profiles are assumed, and separate mass balances can be written directly for convection, axial diffusion, and radial diffusion. The latter technique was applied by McJilton et al. 

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... 

Although convection, axial diffusion, and radial diffusion actually occur simultaneously, a multistep procedure was adopted in the finite-difference calculation. For each 5-cm increment in tidal volume and for each time increment At, the differential mass-balance equations were solved for convection, axial difihision, and radial diffusion in that order. This method may slightly underestimate the dosage for weakly soluble gases, because the concentration gradient in the airway may be decreased. 

When a tube or pipe is long enough and the fluid is not very viscous, then the dispersion or tanks-in-series model can be used to represent the flow in these vessels. For a viscous fluid, one has laminar flow with its characteristic parabolic velocity profile. Also, because of the high viscosity there is but slight radial diffusion between faster and slower fluid elements. In the extreme we have the pure convection model. This assumes that each element of fluid slides past its neighbor with no interaction by molecular diffusion. Thus the spread in residence times is caused only by velocity variations. This flow is shown in Fig. 15.1. This chapter deals with this model. 

The vorticity-transport equation has convective and diffusive terms in both the axial and radial directions. For the stagnation flow these terms contribute in such a way that the... 

As with the Navier-Stokes equations, consider the behavior of the leading coefficients of the radial and axial diffusion terms. Presume that the radial diffusion terms are order one, that is, comparable to the convective terms,... 

A second assumption made is that radial diffusion is unimportant relative to radial convection. This problem has been investigated by Smyrl and Newman [22] who show that inclusion of the radial diffusion term only increases the limiting current by 0.12%. It can thus be neglected in practice. 

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. 

Analogously to rotating electrodes, we take p = 1 for the upstream of two electrodes (generator) and p = 3 for the downstream (detector). Since, in electrochemical experiments, radial diffusion will be much less than axial convection, we can say that... 

Table 23.1 reproduces data affected by breakdown of linear diffusion for the one-electron reduction of Cp2Rh at a Pt disk electrode (r = 2 mm). At sweep rates above about 0.1 V/s, the current function is essentially constant, consistent with the simple one-electron reaction of Equation 23.8. The increase of over 10% in X that is observed at lower scan rates arises from the breakdown of linear diffusion, rather than from additional reactions coupled to Equation 23.8. The radial diffusion contribution is less than 3% [5], with convection accounting for most of the additional mass transport. 

The use of the Coanda effect is based on the desire to have a second passive momentum to speed up mixing in addition to diffusion [55, 163], The second momentum is based on so-called transverse dispersion produced by passive structures, which is in analogy with the Taylor convective radial dispersion ( Taylor dispersion ) (see Figure 1.180 and [163] for further details). It was further desired to have a flat ( in-plane ) structure and not a 3-D structure, since only the first type can be easily integrated into a pTAS system, typically also being flat A further design criterion was to have a micro mixer with improved dispersion and velocity profiles. 

Thus in turbulent flow, the dispersion coefficient is independent of the diffusion coefficient, but in laminar flow, the dispersion coefficient depends inversely on the diffusion coefficient. This counterintuitive inverse dependence, the result of axial convection coupled with radial diffusion, is the foundation of the Goulay equation describing peak spreading in chromatography. We now return from this dispersion tangent back to diffusion and in particular, to mass transfer. 

The plug-flow model indicates that the fluid velocity profile is plug shaped, that is, is uniform at all radial positions, fact which normally involves turbulent flow conditions, such that the fluid constituents are well-mixed [99], Additionally, it is considered that the fixed-bed adsorption reactor is packed randomly with adsorbent particles that are fresh or have just been regenerated [103], Moreover, in this adsorption separation process, a rate process and a thermodynamic equilibrium take place, where individual parts of the system react so fast that for practical purposes local equilibrium can be assumed [99], Clearly, the adsorption process is supposed to be very fast relative to the convection and diffusion effects consequently, local equilibrium will exist close to the adsorbent beads [2,103], Further assumptions are that no chemical reactions takes place in the column and that only mass transfer by convection is important. 

Figure 1 shows the schematic of a tubular reactor, of radius a and length L, where a — a/Lis the aspect ratio. Clearly, ifa>S> 1, or a <3C 1, a physical length scale separation exists in the reactor. This length scale separation could also be interpreted in terms of time scales. For example, a 1 implies that the time scale for radial diffusion is much smaller than that of either convection and axial diffusion, and concentration gradients in the transverse direction are small compared to that in the axial direction. 

Regardless of the source of the vorticity, it remains confined near the body surface for Re >> I because the time scale for radial diffusion is limited by convection around the body... 

Sample dispersion is altered when the flowing sample merges with a confluent stream [27]. At the vicinity of the confluence site, convective mass transport is strongly altered by the sudden change in concentrations caused by the convergence of the confluent and sample carrier streams. The laminar flow regime tends to be maintained therefore, interactions between the sample carrier stream and the confluent stream are mainly dictated by radial diffusion. 

Temperature influences both molecular diffusion and viscosity, and hence the distribution of velocities of the different fluid lines (see also Fig. 3.1). Consequently, both convective and diffusive mass transport are, in principle, affected. An increase in temperature (and the concomitant decrease in viscosity) promotes radial mixing, thus reducing sample broadening and increasing the recorded peak height. 

The reactant molecules transfer from the entrance of the reactor to the neighbourhood of the catalyst pellets. This transfer takes place by convection and/or diffusion. When axial diffusion is negligible and radial diffusion is instantaneous, we get the simplest description for the bulk phase, that is one-dimensional piug/flbw. 

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