Mechanical dispersion coefficient

Two approximations are introduced, for simplicity, to perform the integration of Eq. (4-9) for the Taylor mechanism dispersion coefficient Ezr-First, the term included in the second integral of Eq. (4-9) is eliminated from the integral by taking an effective mean value Vn, (= dvt), with d being a mean of a. Second, the local liquid holdup is eliminated from the multiple integral by using the mean liquid holdup Cl. with a correction factor/of order unity (/al). 

As in the case of turbulent diffusion, the chemical flux is often expressed by Fick s first law, as shown in Eqs. [1-3] and [1-4], but in this case D is called a mechanical dispersion coefficient. Dispersion also occurs at much larger scales than that of soil particles for example, groundwater may detour around regions of relatively less permeable soil that are many cubic meters in volume. At this scale, the process is called macrodispersion. 

Equation [3-17] does not hold at very low seepage velocities because mechanical dispersion no longer dominates Fickian mass transport. When the mechanical dispersion coefficient becomes less than the effective molecular diffusion coefficient, the longer travel times associated with lower velocities do not result in further decreases in Fickian mass transport. 

The mechanical dispersion coefficient D in the longitudinal direction can be approximated by Eq. [3-15] ... 

In a sand having a median grain size of 1 mm and porosity of 0.25, how high must specific discharge be to make the mechanical dispersion coefficient equal to the effective molecular diffusion coefficient ... 

It turned out that the mechanical dispersion may be formally described by the same Pick s laws if the diffusion coefficient is replaced with mechanical dispersion coefficient This is a proportionality coefficient between value of a deflection of component i migration rate from the average seepage velocity on the one hand, and the component concentration gradient between mixed waters, on the other. In a case of unidimensional and bidimensional fluxes this correlation by analogy with the first Pick s law has the following format... 

Experiments showed that mechanical dispersion coefficients are directly proportionate with filtration (flow) average seepage velocity, i.e.,... 

Values of mechanical dispersion coefficient and dynamic dispersivity coefficient (hydrodispersion constants) are usually considered in spatial coordinates, distinguishing these values along and across flow lines. The lengthwise mechanical dispersion, i.e., in the direction of flow, is... 

Its value D is determined as summary value of effective diffusion coefficient and mechanical dispersion coefficient ... 

The axial dispersion coefficient [cf. Eq. (16-51)] lumps together all mechanisms leading to axial mixing in packed beds. Thus, the axial dispersion coefficient must account not only for moleciilar diffusion and convec tive mixing but also for nonuniformities in the fluid velocity across the packed bed. As such, the axial dispersion coefficient is best determined experimentally for each specific contac tor. 

Comparison of Models Only scattered and inconclusive results have been obtained by calculation of the relative performances of the different models as converiers. Both the RTD and the dispersion coefficient require tracer tests for their accurate determination, so neither method can be said to be easier to apply The exception is when one of the cited correlations of Peclet numbers in terms of other groups can be used, although they are rough. The tanks-in-series model, however, provides a mechanism that is readily visualized and is therefore popular. 

Atmospheric stability and mechanical turbulence (important near to the ground) are used to derive the vertical and horizontal dispersion coefficients. Table 45.2 shows Pasquill s stability categories used to derive the coefficients by reference to standard graphs. 

Dispersion in packed tubes with wall effects was part of the CFD study by Magnico (2003), for N — 5.96 and N — 7.8, so the author was able to focus on mass transfer mechanisms near the tube wall. After establishing a steady-state flow, a Lagrangian approach was used in which particles were followed along the trajectories, with molecular diffusion suppressed, to single out the connection between flow and radial mass transport. The results showed the ratio of longitudinal to transverse dispersion coefficients to be smaller than in the literature, which may have been connected to the wall effects. The flow structure near the wall was probed by the tracer technique, and it was observed that there was a boundary layer near the wall of width about Jp/4 (at Ret — 7) in which there was no radial velocity component, so that mass transfer across the layer... 

The moments of the solutions thus obtained are then related to the individual mass transport diffusion mechanisms, dispersion mechanisms and the capacity of the adsorbent. The equation that results from this process is the model widely referred to as the three resistance model. It is written specifically for a gas phase driving force. Haynes and Sarma included axial diffusion, hence they were solving the equivalent of Eq. (9.10) with an axial diffusion term. Their results cast in the consistent nomenclature of Ruthven first for the actual coefficient responsible for sorption kinetics as ... 

If the radial diffusion or radial eddy transport mechanisms considered above are insufficient to smear out any radial concentration differences, then the simple dispersed plug-flow model becomes inadequate to describe the system. It is then necessary to develop a mathematical model for simultaneous radial and axial dispersion incorporating both radial and axial dispersion coefficients. This is especially important for fixed bed catalytic reactors and packed beds generally (see Volume 2, Chapter 4). 

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