Diffusion convective axial

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. 

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... 

Using the above criterion and arguments similar to those to define the Taylor solution range, but with the constants chosen to agree with the Taylor solution, we find that for convective axial diffusion to be the dominant mode the inequality... 

This is termed the Taylor-Aris dispersion coefficient, and is simply the sum of the axial molecular diffusion coefficient and the Taylor radial dispersion coefficient. As can be seen, at large Peclet numbers D ffD increases as the square of the Peclet number (the Taylor dispersion limit), and at small Peclet numbers D ifD approaches 1 (the convective axial diffusion limit). 

Transport in the fluid phase inside the packed bed takes place through convection, axial diffusion and flow to or from the zeoHte crystals. A mass balance for a small volume element of the bed results in the following equation for the concentration Cz in the gas phase... 

The net transport of component A in the +2 direction in the centrifuge is equal to the sum of the convective transport and the axial diffusive transport. At the steady state the net transport of component A toward the product withdrawal point must be equal to the rate at which component A is being withdrawn from the top of the centrifuge. Thus, the transport of component is given by equation 72 ... 

The first term may be considered as the contribution of the internal circulation or convective flow to the stage length, the second term as the contribution of the axial diffusion to the stage length. The stage separation factor is given by... 

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. 

Surface Polarization in TFF The simplified model of polarization shown in Fig. 20-47 is used as a basis for analyzing more complex systems. Consider a single component with no reaction in a thin, two-dimensional boundary layer near the membrane surface. Axial diffusion is negligible along the membrane surface compared to convection. 

The development of the equations for the dynamic dispersion model starts by considering an element of tube length AZ, with a cross-sectional area of Ac, a superficial flow velocity of v and an axial dispersion coefficient, or diffusivity D. Convective and diffusive flows of component A enter and leave the element, as shown by the solid and dashed arrows respectively, in Fig. 4.12. 

One conclusion from these results is that the axial diffusion model begins to fail as Pe, - small, when an open boundary condition is used at the outlet. The case Pe, - small means increasing backmixing, or that the diffusive flux becomes increasingly significant compared with the convective flux. For an open boundary condition, it is also questionable whether the actual response C(e) can be identified with E(B). Furthermore, regardless of the boundary conditions chosen, it is difficult to envisage that cA... 

Convection + Axial Dispersion -t- Mass Transfer Resistance + Particle Diffusion + Kinetics of Adsorption... 

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,... 

The axial diffusion term can often be neglected, because the convective velocity can be much higher than the axial diffusion flow. The boundary conditions are as follows ... 

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. 

Using a (radius of the pipe) and L (length of the pipe) as the characteristic lengths in the transverse and axial directions, respectively, CR as the reference concentration, and Dm R as the reference molecular diffusivity, we obtain four time scales in the system associated with convection (ic), local/transverse diffusion (tD), axial diffusion tz), and reaction (tR),... 

On shell (ligMidl side. Mass balances of species A and B contains axial convection terms that are assumed to be uncoupled from the mass transport, and radial and axial diffusion terms ... 

Axial heat conductivity and axial diffusion in the fluid phase are neglected because of the usually large convective transport. 

Gondim, R.R (1997) Transient Internal Forced Convection with Axial Diffusion Solution by Integral Transforms, D.Sc. Thesis, COPPE / UFRJ (in Portuguese). 

Gondim, R.R., Cotta, R.M., Santos, C.A.C., and Mat, M. (2003) Internal Transient Forced Convection with Axial Diffusion Comparison of Solutions Via Integral Transforms, ICHMT International Symposium on Transient Convective Heat And Mass Transfer in Single and Two-Phase Flows, Cesme, Tmkey, August 17 - 22. 

Figure 11-2 Forced axial convection vdfii diffusion to surface.

The permeability of hardwoods (such as birch treated here) is 10 to 10 times higher in the axial direction than in the radial and tangential directions [14]. Therefore the axial flow was included in the conservation equations (Equation 2, 3 and 4). A discussion on the anisotropy of wood, the inclusion of axial convective terms and the omission of axial diffusive terms are presented in [7], In the conservation of gas species and of liquid phase the axial flow was estimated by Equation 7 and 8 assuming, for simplicity, a linear pressure gradient in the axial direction of the sample. For the energy equation (Equation 4) two cases were studied, q H) and an axial heat loss according to ... 

Diffusion with a convection and simultaneous first order reaction in a rectangular plate can be simulated using the program described above by using minor modifications. Consider the composition profile in a packed tube reactor undergoing isothermal linear kinetics with axial diffusion. The governing equation is... 

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