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Generally speaking, when r > 15 end effects are not significant. Because most of the bioreactors comply with this restriction, the equations transcribed here do not take into account the aspect ratio of the apparatus.

The superposition of an axial flow to this system, necessary for a VFR in continuous operation, induces a stabilization of the Couette flow, delaying the appearance of vortices. The equation of Recktenwald et al. (23) estimates the critical Ta number for a continuous VFR:

Parameters at may be found in reference (23), as a function of n. Notice the different definition of the Taylor number (here named Ta') adopted by the authors.

It should be stressed that the effect of the axial flow on the onset of the instability is negligible for typical values of the superficial velocity (Uax) in enzymatic reactors (i.e., for mean residence times around 10 to 30 min).

When the rotation rate of the inner cylinder increases, turbulent Taylor vortices arise. This is an interesting example of a complex system: the flow in this regime is locally random, but there is still a macroscopic degree of order, since the vortex structure remains present up to very high rotations. Di Prima and Swinney's (24) review reports that the full outcome of turbulence occurs above Ree/Ree,c «25 (for an apparatus with n = 0.875).

### 4. Particle Fluidization

In a heterogeneous VFR, with particles composed by the usual matrices for enzyme immobilization (for instance, agarose, chitosan, silica), dragged by the vortices in a typical medium (not too viscous), the minimum rotation for uniform fluidization (see Note 4) will generally be high enough to ensure turbulent vortices. Resende (20) has fitted an empirical correlation for the minimum fluidization conditions, when bed porosity is in the range 0.92-0.98:

° d dp for 0.48 < n < 0.66 and 7.1 < Ree,p < 75.6. Ar is the Archimedes number:

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