Purely diffusive flow

In general, for each value of 2, there must be a different value of the eigenvalue T. Figure 5.11 illustrates this relationship for the radially inward flow. The two limiting situations (i.e., purely diffusive flow and purely convective flow) can be seen in the functional form of T ( 2). At high 2, the value of T may be fit from the solutions asT = —0.062(— 2)0 911. At low values, T = -1.9. 

For purely diffusive flows, sharp (fresh versus saline water resistivity) discontinuities always smear in time. The dynamics of such flows are very important in log interpretation. For this class of problems, the speed of the fresh-to-saline water interface slows appreciably once the mudcake establishes itself at the borehole walls, as we have demonstrated in Chapter 17. This is especially true in the case of radial flows, where geometric spreading significantly slows the front. For such problems, the speed of the underlying flow U can be neglected after some time, when diffusion predominates. The problem is shown in Figure 21-2. 

This gives a model for Eq. (4,305b), but not a model for force While force gives the flow force caused by the material, it is normal to represent this fact so that gives the pure diffusion resistance force that is not caused by the material. This requires treating independently from the material or porosity. For (f) = 1 or E = °°, where 0 EQ- (4.306) gives... 

As an example, consider a simple reaction of the type (6.2) taking place under pure diffusion control. At all times the electrode potential, according to the Nemst equation, is determined by the reactant concentrations at the electrode surface. It was shown in Section 11.2.3 that periodic changes in the surface concentrations which can be described by Eq. (11.19) are produced by ac flow. We shall assume that the amplitude of these changes is small (i.e., that Ac electrode polarization. With this substitution and using Eq. (11.19), we obtain... 

Schutz s correlation for free convection at a sphere, Eq. (25) in Table VII, takes pure diffusion into account by means of the constant term Sh = 2. According to his measurements using local spot electrodes, the flow here is not laminar but already in transition to turbulence. 

Let us consider the transport of one component i in a liquid solution. Any disequilibration in the solution is assumed to be due to macroscopic motion of the liquid (i.e. flow) and to gradients in the concentration c,. Temperature gradients are assumed to be negligible. The transport of the solute i is then governed by two different modes of transport, namely, molecular diffusion through the solvent medium, and drag by the moving liquid. The combination of these two types of transport processes is usually denoted as the convective diffusion of the solute in the liquid [25] or diffusion-advection mass transport [48,49], The relative contribution of advection to total transport is characterised by the nondimensional Peclet number [32,48,49], while the relative increase in transport over pure diffusion due to advection is Sh - 1, where Sh is the nondimensional Sherwood number [28,32,33,49,50]. 

As described above, spatial transport in an Eulerian PDF code is simulated by random jumps of notional particles between grid cells. Even in the simplest case of one-dimensional purely convective flow with equal-sized grids, so-called numerical diffusion will be present. In order to show that this is the case, we can use the analysis presented in Mobus et al. (2001), simplified to one-dimensional flow in the domain [0, L (Mobus et al. 1999). Let X(rnAt) denote the random location of a notional particle at time step m. Since the location of the particle is discrete, we can denote it by a random integer i X(mAt) = iAx, where the grid spacing is related to the number of grid cells (M) by Ax = L/M. For purely convective flow, the time step is related to the mean velocity (U) by16... 

Diffusion is due to the random motion of particles (atoms, ions, molecules). The random motion is excited by thermal energy. In the case of pure diffusion, there is no bulk flow, only the redistribution of the components. Nonetheless, exchange of components may result in a shift of the mass center if a heavier particle such as Fe exchanges with a lighter particle such as it may result in a... 

The homogeneous catalysis method is suitable to measure rate constants over a very wide range, up to the diffusion limit. The lower limit is determined by interferences, such as convection, which occur at very slow scan rates. It is our experience that, unless special precautions are taken, scan rates below lOOmV/s result in significant deviations from a purely diffusion-controlled voltammetric wave. For small values of rate constants (down to 10 s ), other potentiostatic techniques are best suited, such as chronoamperometry at a rotating disk electrode UV dip probe and stopped-flow UV-vis techniques. ... 

Once this flux equality condition (7.173) is formulated, one simply works out transport as a Pure transport problem and equates it to 1 InF times the current density across the interface since n faradays per mole are required for the transported material to be electronated. If the transport process consists of pure diffusion (i.e., there is no contribution from either migration or hydrodynamic flow), then the flux is given by Fick s first law (see Section 4.2.2), i.e.,... 

The flow modulation technique, in general, appears therefore very well suited for this specific purpose of quantitative diffusivity measurement. However, it also reveals any deviation from a purely diffusion controlled kinetics more clearly than do steady state measurements when, for example, a slow series process is concealed in an apparent diffusion plateau. 

Figure 10.3 is a plot of c/c0 versus time, measured at x = HI2 for monodisperse particles having a diameter of 1 xm when H - 2 cm. Note that there is essentially no change in concentration until sometime after 105 s, and then a fairly rapid decrease in concentration takes place. It can be concluded that except for very small particles or very small tubes, pure diffusion will have a small to negligible influence on the concentration changes in an aerosol flowing through a tube. 

Equation (3.113) contains the convection flow of the total energy and energy changes due to the diffusion flows. If j is the pure heat conduction without a flow of internal energy due to diffusion of the substance, the total energy conservation given in Eq. (3.99) becomes... 

Example 7.4 Modified Graetz problem with coupled heat and mass flows The Graetz problem originally addressed heat transfer to a pure fluid without the axial conduction with various boundary conditions. However, later the Graetz problem was transformed to describe various heat and mass transfer problems, where mostly heat and mass flows are uncoupled. In drying processes, however, some researchers have considered the thermal diffusion flow of moisture caused by a temperature gradient. 

When we particularize this relationship for the mass transport of the humidity into a porous medium (w = 0, because there is no microscopic displacement), we can observe the superposition of the thermo-diffusion and of the diffusive filtration (where p is the humidity flowing by filtration) over the pure diffusion process ... 

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