Accounting for Axial Mixing

Adiabatic Reactors. Like isothermal reactors, adiabatic reactors with a flat velocity profile will have no radial gradients in temperature or composition. There are axial gradients, and the axial dispersion model, including its extension to temperature in Section 9.4, can account for axial mixing. As a practical matter, it is difficult to build a small adiabatic reactor. Wall temperatures must be controlled to simulate the adiabatic temperature profile in the reactor, and guard heaters may be needed at the inlet and outlet to avoid losses by radiation. Even so, it is hkely that uncertainties in the temperature profile will mask the relatively small effects of axial dispersion. 

Accounting for Axial Mixing Differential-type column extractors are subject to axial (longitudinal) mixing, also c ed axial dispersion... 

Axial Mixing See Accounting for Axial Mixing under Liquid-Liquid Extraction Equipment. Many approaches nave been developed. Becker recommends the concept of the height of a dispersion unit (HDU) to correct the height of a transfer unit for axial mixing in a static contactor [Becker, Chem. Eng. Technol., 26(1), pp. 35-41 (2003) Chem. Ing. Tech., 74, pp. 59-66 (2002) and Becker and Seibert, Chem. Ing. Tech., 72, pp. 359-364 (2000)] ... 

Since the liquid flow rates are generally rather low it may be necessary to account for axial mixing in the liquid phase. This is done in terms of axial effective diffusion. The axial effective diffusivity for the liquid phase is given by Bdxkes [40] ... 

In an ideal PFR, fluid elements do not mix in the axial direction (i.e. flow direction). However, in an actual tubular reactor, some amount of axial mixing of fluid elements may occur due to a number of reasons (such as vortex formation at tube inlet). A mathematical model called axial dispersion model was proposed by P. V. Danckwarts to account for axial mixing of fluid elemenfs in the tubular (plug flow) reactor. 

Accounting for axial mixing, the steady-state continuity equation for a component A may be written... 

Reviews of calculation methods to account for axial mixing [41] (simplified procedures have been devised [47, 78]) and of axial-mixing data [26] are available. 

In a completely different interpretation Zefirov (242) proposed a new concept of frontier-orbital mixing (243) to explain how conformational and electronic effects in monosubstituted cyclohexanes are transmitted to remote 8-carbon atoms (Scheme 36). The orbitals at C(l) and C(4) in 112 are considered to be equatorial (242). A perturbation at C(l) (H is replaced by X) produces an electron-density shift from H(4) toward C(4) (242), which is associated with an upheld shift of the latter s signal. Although this approach appears to be quite crude and does not account for axial substituents, it deserves fiirther attention. 

At first sight, this simple model appears to have the capability of accounting only for axial mixing effects. It will be shown, however, that this approach can compensate not only for problems caused by axial mixing, but also for those caused by radial mixing- and other nonflat velocity profiles These fluctuations in concentration can result from different flow velocities and pathways and from moleeular and turbulent diffusion. 

The axial dispersion terms may be required to account for the mixing phenomena created by a non-ideal flow. However, the ideal plug flow model is often appropriate for packed bed reactors because the axial mixing is negligible compared to the convective flux for many processes. 

By changing D x, one may vary the reactor performance from plug flo v [Dax/(wl) = 0 or Bo = oo] to a CSTR [Dax/(w I) = oo or Bo = 0]. At first sight, this simple model appears to account only for axial mixing effects. However, this approach not only compensates for problems caused by axial mixing but also for those related to radial mixing and non-uniform velocity profiles (Aris, 1956), as shown in Section 4.10.6.3 for laminar flow in tubular reactors. 

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. 

We have discussed the tanks-in-series model in the sense that the composition in the system was constant over a cross-section. Recently Deans and Lapidus (D12) devised a three-dimensional array of stirred tanks, called a finite-stage model, that was able to take radial as well as axial mixing into account. Because of the symmetry, only a two-dimensional array is needed if the stirred tanks are chosen of different sizes across the radius and are properly weighted. By a geometrical argument. Deans and Lapidus arrived at the following equation for the (i, j) tank ... 

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