Model diffusion/convection flow

Meyers, J.J., Liapis, A.I. Network modeling of the convective flow and diffusion of molecules adsorbing in monoliths and in porous particles packed in a chromatographic column,... 

The F(t) data were transformed to give the corresponding intensity function, X(t), curves shown in Figure 5. Comparison of the curves clearly shows a difference between the two reactors. The two impeller reactor exhibits well-mixed behavior whereas the one impeller reactor exhibits behavior indicative of a distributed system. This notion can be supported by examining a sisiple, transient, diffusion/convection flow model ... 

Bakd et al. theoretically analyzed simultaneous gas flow and diffusion in Weibel s symmetric model. Th applied a time-varying flow with simultaneous longitudinal diffusion and concluded that convective mixing is much less important than mixing induced by molecular diffusion. 

Other Springer model derivatives include those of Ge and Yi, ° van Bussel et al., Wohr and co-workers, and Hertwig et al. Here, the models described above are slightly modified. The model of Hertwig et al. includes both diffusive and convective transport in the membrane. It also uses a simplified two-phase flow model and shows 3-D distributions... 

There have been various models that try to incorporate both diffusive flow and convective flow in one type of membrane and using one governing transport equation. They are based somewhat on... 

When fluid is pumped through a cell such as that shown in Fig. 12, transport of dissolved molecules from the cell inlet to the IRE by convection and diffusion is an important issue. The ATR method probes only the volume just above the IRE, which is well within the stagnant boundary layer where diffusion prevails. Figure 13 shows this situation schematically for a diffusion model and a convection-diffusion model (65). The former model assumes that a stagnant boundary layer exists above the IRE, within which mass transport occurs solely by diffusion and that there are no concentration gradients in the convection flow. A more realistic model of the flow-through cell accounts for both convection and diffusion. As a consequence of the relatively narrow gap between the cell walls, the convection leads to a laminar flow profile and consequently to concentration gradients between the cell walls. 

The study of rotating disk electrode behavior provides a unique opportunity to develop a model that predicts the effect of diffusion and convection on the current. This is one of the few convective systems that have simple hydrodynamic equations that may be combined with the diffusion model developed herein to produce meaningful results. The effect of diffusion is modeled exactly as it has been done previously. The effect of convection is treated by integrating an approximate velocity equation to determine the extent of convective flow during a given At interval. Matter, then, is simply transferred from volume element to volume element in accord with this result to simulate convection. The whole process repeated results in a steady-state concentration profile and a steady-state representation of the current (the Levich equation). 

The Peclet numbers are useful for estimating the relative contributions of convection and diffusion to mass and heat transfer. If Pe is large (>10), convection dominates, and a plug-flow model may be appropriate for simple reactor computations. When Pe is small (<<1), diffusion dominates, and the system behaves like a well-stirred reactor. Thus, Pe may be used to estimate whether downstream impurities can diffuse into the deposition zone. 

Most of the models assume that neutral-species transport can be represented with either a well-mixed model or a plug flow model. The major drawback to these assumptions is that important inelastic rate processes such as molecular dissociation are usually localized in space in the reactor and are often fast compared with rates of diffusion or convection. As a result, the spatial variation of fluid flow in the reactor must be accounted for. This variation introduces a major complication in the model, because the solution of the nonisothermal Navier-Stokes equations in multidimensional geometries is expensive and difficult. 

Needless to say, the assumption of plug flow is not always appropriate. In plug flow we assume that the convective flow, i. e., the flow at velocity qjAt = v that is caused by a compressor or pump, is dominating any other transport mode. In practice this is not always so and dispersion of mass and heat, driven by concentration and temperature gradients are sometimes significant enough to need to be included in the model. We will discuss such a model in detail, not only because of its importance, but also because the techniques used to handle the ensuing boundary value differential equations are similar to those used for other diffusion-reaction problems such as catalyst pellets, as well as for counter-current processes. 

In laminar flows through porous media, the pressure is proportional to velocity and C2 can be taken as zero. Ignoring convective acceleration and diffusion, the porous media model can be changed into Darcy s Law ... 

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. 

Although the diffusion layer model is the most commonly used, various alterations have been proposed. The current views of the diffusion layer model are based on the so-called effective diffusion boundary layer, the structure of which is heavily dependent on the hydrodynamic conditions, fn this context, Levich [102] developed the convection-diffusion theory and showed that the transfer of the solid to the solution is controlled by a combination of liquid flow and diffusion. In other words, both diffusion and convection contribute to the transfer of drug from the solid surface into the bulk solution, ft should be emphasized that this observation applies even under moderate conditions of stirring. 

Returning to the Nissan and Hansen model, they use a finite difference numerical analysis model to determine both the temperature profile of a sheet material and the subsequent water removal as it passes over the cylinder. Their experimental results match well with the predicted values. However, their experiments were limited to a cylinder surface temperature of 93.3°C. Accordingly, the maximum vapor pressure of the evaporated water is less than one atmosphere. The diffusion model advanced by Hartley and Richards is in close agreement with experimental work of Dreshfield(12l. However, the boundary conditions are still relatively uncertain since the convective flow region outside the sheet is relatively unknown. Later work by this author studies this convective flow. 

The first approach is the discretization of the convection and the diffusion operators of the PDEs, which gives rise to a large (or very large) system of effective low-dimensional models. The order of these low-dimensional models depend on the minimum mesh size (or discretization interval) required to avoid spurious solutions. For example, the minimum number of mesh points (Nxyz) necessary to perform a direct numerical simulation (DNS) of convective-diffusion equation for non-reacting turbulent flow is given by (Baldyga and Bourne, 1999)... 

Once the descriptive model has been realized, we need to make the mathematical model of the process, which can be used to identify the mean pore radius of the membrane pores and the associated tortuosity. Before starting with the establishment of the model, we consider that the elementary processes allowdng the gas flow through the membrane are a combination of Knudsen diffusion with convective flow. If we only take into account the linear part of the curve of the pressure increase with time then we can write ... 

A good agreement is generally obtained between the models based on transport equations and the SDE for mass and heat molecular transport. However, as explained above, the SDE can only be applied when convective flow does not take place. This restrictive condition limits the application of SDE to the transport in a porous solid medium where there is no convective flow by a concentration gradient. The starting point for the transformation of a molecular transport equation into a SDE system is Eq. (4.108). Indeed, we can consider the absence of convective flow in a non-steady state one-directional transport, together with a diffusion coefficient depending on the concentration of the transported property... 

In another example, Richard et al. (2002) simulated the transport of water in a two-dimensional mantle convection model. They found that mantle flow, not diffusion, was the primary control on water distribution, which led to a homogeneous distribution of water in the mantle. If this is the case, the transition zone may contain less water than could be dissolved into the nominally anhydrous phases present there. Because of the low solubility of water in lower-mantle nominally anhydrous phases (Bolfan-Casanova et al., 2000), Richard et al. proposed that there might be a water-rich fluid phase in the lower mantle. They did not, however, consider the possibility of water-induced partial melting, leading to a melt rather than a fluid. 

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