Peclet longitudinal

However, the two mechanisms interact and molecular diffusion can reduce the effects of convective dispersion. This can be explained by the fact that with streamline flow in a tube molecular diffusion will tend to smooth out the concentration profile arising from the velocity distribution over the cross-section. Similarly radial dispersion can give rise to lower values of longitudinal dispersion than predicted by equation 4.39. As a result the curves of Peclet versus Reynolds number tend to pass through a maximum as shown in Figure 4.6. 

In this section, we will obtain the non-dimensional effective or upscaled equations using a two-scale expansion with respect to the transversal Peclet number Note that the transversal P let number is equal to the ratio between the characteristic transversal timescale and longitudinal timescale. Then we use Fredholm s alternative to obtain the effective equations. However, they do not follow immediately. Direct application of Fredholm s alternative gives hyperbolic equations which are not satisfactory for our model. To obtain a better approximation, we use the strategy from Rubinstein and Mauri (1986) and embed the hyperbolic equation to the next order equations. This approach leads to the effective equations containing Taylor s dispersion type terms. Since we are in the presence of chemical reactions, dispersion is not caused only by the important Peclet number, but also by the effects of the chemical reactions, entering through Damkohler number. 

It should be pointed out that for a low pressure gas the radial- and axial diffusion coefficients are about the same at low Reynolds numbers (Rediffusion effects may be important at velocities where the dispersion effects are controlled by molecular diffusion. For Re = 1 to 20, however, the axial diffusivity becomes about five times larger than the radial diffusivity [31]. Therefore, the radial diffusion flux could be neglected relative to the longitudinal flux. If these phenomena were also present in a high-pressure gas, it would be true that radial diffusion could be neglected. In dense- gas extraction, packed beds are operated at Re > 10, so it will be supposed that the Peclet number for axial dispersion only is important (Peax Per). 

This shows that the secohd term, i.e., the longitudinal diffusion term, is negligible compared to the first Iterm, i.e., the radial diffusion term, if the quantity Re Pr is large. The quantity RePr is termed the Peclet number, Pe. Generally, if the Peclet number is greater t ian 10, the effects of longitudinal diffusion are, indeed, negligible. 

Dispersion is most commonly modeled as a diffusive process. For flow in a packed column, dispersion is captured by the DV C term in the one-dimensional convection-diffusion equation. The longitudinal dispersion coefficient, D, is a function of the Peclet number, Pe = vR/Dj (where is the molecular... 

Taking into account the previous discussion, we restrict ourselves to the case of high Peclet numbers, for which the longitudinal molecular heat transfer may be neglected. The corresponding equations of the thermal boundary layer and the boundary conditions have the form... 

Statement of the problem. Now let us consider mass transfer from a solid wall to a liquid film at high Peclet numbers. Such a problem is of serious interest in dissolution, crystallization, corrosion, anodic dissolution of metals in some electrochemical processes, etc. In many practical cases, dissolution processes are rather rapid compared with diffusion. Therefore, we assume that the concentration on the plate surface is equal to the constant Cs and the incoming liquid is pure. As previously, we introduce dimensionless variables according to formulas (3.4.5). In this case, the convective mass transfer in the liquid film is described by Eq. (3.4.1), the boundary condition (3.4.2) imposed on the longitudinal variable x, and the following boundary conditions with respect to the transverse coordinate ... 

Convective mass and heat transfer to a plate in a longitudinal flow of a non-Newtonian fluid was considered in [443]. By solving the corresponding problem in the diffusion boundary layer approximation (at high Peclet numbers), we arrive at the following expression for the dimensionless diffusion flux ... 

Fig. 3-5. Empirical relationship between the dimensionless longitudinal dispersion coefficient and the Peclet number (Fried Combamous, 1971).

It is interesting to consider the dependence of Deff on the molecular diffusion coefficient. For small molecular diffusivity, D < UL, i.e. large Peclet number, the second term dominates in (2.51) and Deff U2L2/D ( DPe2). Thus weak molecular diffusion leads to large effective diffusivity. This somewhat counterintuitive result can be explained as follows. Longitudinal dispersion arises due to... 

Figure 7. Longitudinal dispersion (Dl) divided by the diffusion coefficient (Df) for tracers measured in column experiments as a function of the particle scale Peclet number (Npe). It is defined as the product of the average pore fluid velocity, u, and the grain diameter, d, divided by the free fluid diffusion coefficient, D/. The magnitude of the dispersion is independent of the pore fluid velocity (Vp) for very small Peclet numbers (or fluid velocities). Note that the effective diffusion coefficient in a porous media is smaller than the diffusion coefficient in a free fluid phase due to the tortuosity. The dispersion increases linearly with increasing flow velocity (increasing Peclet number). Modified from Appelo and Postma (1999).

In a circular bed, the effective dispersion tensor is anisotropic and is composed of the longitudinal and lateral dispersion coefficients D Jff and Djjf, respectively. The longitudinal dispersion coincides with the direction of the mean fluid flow with the lateral dispersion normal to this direction. At high Peclet numbers, the longitudinal dispersion is large in comparison with the lateral dispersion, since the component of the fluid velocity parallel to the mean flow direction has the largest gradients. The lateral dispersion D if is associated with the weaker lateral fluid motion, whence D fj. 

The Taylor-Aris result for the dispersion coefficient (Eq. 4.6.35) has been applied to the empirical correlation of measured and calculated longitudinal dispersion coefficients in flow through packed beds and porous media (see Eidsath et al. 1983). Typically, the velocity in the Peclet number of the Taylor-Aris formula is identified with the superficial velocity, and the capillary diameter with the hydraulic diameter for spherical particles. An alternative velocity suggested by the capillary model is the interstitial velocity, and an alternative length is the square root of the permeability. In an isotropic packing of particles is about one-tenth the particle diameter (Probstein Hicks... 

There is lack of agreement in the literature on the behavior of the longitudinal dispersion coefficient with Peclet number, other than it tends to increase monotonically with Pe. For large Peclet numbers the increase is generally found to have the power law behavior Pe , with 1 < k <2 (Plumb c Whitaker 1990, Adler 1992, Brenner Edwards 1993). 

Note that in the inequality of (5-33) that the quantity uR/Dm) can be considered a radial Peclet number defined analogously to the longitudinal quantity defined previously. It has been suggested that the right inequality is a little tight, and that iiR/Dm) > 50 is a more comfortable approximation [V. Ananthakrishnan, W.N. Gill and A.I. Barduhn, Amer. Inst. Chem. Eng. J., 11, 1063 (1965)). 

The second BC is due to Danckwerts and has been used for chemical reactor models. This leads, of course, to a split boundary value problem, which needs to be solved by an appropriate numerical technique. The resulting longitudinal profiles of solid moisture content and tanperature in a dryer for various Peclet numbers (Pe = u LIE) are presented in Figure 3.10. 

This makes the Peclet number inversely related to the longitudinal dispersivity (Knapp, 1989). 

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