Design equations with axial dispersion

Table 10 J Design Equations for Adiabatic Fixed Beds with Axial Dispersion...

It has been shown in Section 10.3 that the solution of the plug-flow model leads directly to that of the one-dimensional model with axial dispersion. These results can be applied to reactions affected by diffusion. Use of the results leads to the design equations for the fixed-beds with axial dispersion given in Table 10.3. Note that Co is the solution for the plug-flow model and that Eq. (E) in Table 10.3 is used for kb in Eq. (B), whereas Eq. (G) is in Eq. (F) (see Problem 10.9 for the case of = 0). 

In most adsorption processes the adsorbent is contacted with fluid in a packed bed. An understanding of the dynamic behavior of such systems is therefore needed for rational process design and optimization. What is required is a mathematical model which allows the effluent concentration to be predicted for any defined change in the feed concentration or flow rate to the bed. The flow pattern can generally be represented adequately by the axial dispersed plug-flow model, according to which a mass balance for an element of the column yields, for the basic differential equation governing llie dynamic behavior,... 

This model is very often used to design and optimize chromatographic systems (e.g., Guiochon et al, 2006). Compared with the ideal model (Equation 6.39) a term describing axial dispersion (Equation 6.22) is included in the mass balance of the mobile phase ... 

For turbulent flow in pipes the velocity profile can be calculated from the empirical power law design formula (1.360). Similar balance equations with purely molecular diffusivities can be used for a fully developed laminar flow in tubular reactors. The velocity profile is then parabolic, so the Hagen Poiseuille law (1.359) might suffice. It is important to note that the difference between the cross section averaged ID axial dispersion model equations (discussed in the previous section) and the simplified 2D model equations (presented above) is that the latter is valid locally at each point within the reactor, whereas the averaged one simply gives a cross sectional average description of the axial composition and temperature profiles. 

The simplest packed bed design arises with a single dilute adsorbate in a carrier fluid when it can be assumed that the process is isothermal, that there is plug flow, and that there are no mass transfer resistances. In such a situation, instantaneous equilibrium exists at all points in the system. Without the axial dispersion term and taking the velocity outside the partial differential term for the convective flow, equation (6.19) is simplifled to ... 

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