Eddy diffusion mathematical models

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). 

The further development of mathematical representations of estuarine processes should proceed simultaneously with investigations of both specific sedimentary processes and regional sedimentary systems. For the model proposed here some of the specific processes that deserve attention in the future include the processes that control the rate of formation of marine mud at the base of the surficial layer of agglomerates and the relationship between the eddy-diffusion coefficient for sand transport and fluctuations in the water velocity. The study of specific processes tell us little about the long-term manifestations of these processes. For this there is the need to develop comprehensive descriptions of estuarine sedimentary systems and to begin to contrast and compare sediment budgets in different coastal areas. 

The mathematical description considered in Section 10.3.3 was used as a modeling basis for the specially developed completely rate-based simulator [80]. This tool consists of several blocks including model libraries for physical properties, mass and heat transfer, reaction kinetics and equilibrium as well as specific hybrid solver and thermodynamic package. It also contains different hydrodynamic models (e.g., completely mixed liquid - completely mixed vapor, completely mixed liquid - vapor plug flow, mixed pool model, eddy diffusion model [80]) and a model library of hydrodynamic correlations for the mass-transfer coefficients, interfacial area, pressure drop, holdup, weeping and entrainment that cover a number of different column internals and flow conditions. 

Dispersion Model An impulse input to a stream flowing through a vessel may spread axially because of a combination of molecular diffusion and eddy currents that together are called dispersion. Mathematically, the process can be represented by Fick s equation with a dispersion coefficient replacing the diffusion coefficient. The dispersion coefficient is associated with a linear dimension L and a linear velocity in the Peclet number, Pe = uL/D. In plug flow, = 0 and Pe oq and in a CSTR, oa and Pe = 0. 

Mathematically, dispersion can be treated in the same manner as molecular diffusion, but the physical background is different dispersion is caused not only by molecular diffusion but also by turbulence effects. In flow systems, turbulent eddies are formed and they contribute to backmixing. Therefore, the operative concept of dispersion, the dispersion coefficient, consists principally of two contributions, that is, the one caused by molecular diffusion and the second one originating from turbulent eddies. Below we shall derive the RTD functions for the most simple dispersion model, namely, the axial dispersion model. 

---