Mean residence time, T
rotation rate (rd/time), L-1
The rotation rate of the inner cylinder is an important variable of operation in VFRs. Changing this speed, it is possible to span a series of different flow patterns, ranging from an almost plug-flow to an almost perfectly mixed continuous stirred tank reactor (CSTR) (16). This operational flexibility may be useful when the equipment is employed in multipurpose bioprocess units.
VFRs may sustain suspended particles in the gap between cylinders even for low axial flow rates (i.e., high residence times); conventional fluidized bed reactors would need a recycle in this situation. Another interesting aspect for further exploration is the use of varying cross-sections, for instance, in conical units when the medium viscosity changes significantly along the reactor.
Figure 2 shows start-up results of a VFR running the tailor-made hydrolysis of cheese whey proteins (17-21), using Alcalase® (Novo Nordisk, Bagsvaerd, Denmark) immobilized on glyoxyl-agarose. As shown in Fig. 1, the VFR was a jacketed glass vessel (working volume 2.46 x 10-4 m3), with a polypropylene inner cylinder, radius ratio n = 0.48, and aspect ratio, r = 11.9 (see Table 1 for the definition of these quantities).
Characteristic Dimensionless Quantities in Taylor-Couette-Poiseuille Flow
Ree Rotational Reynolds number ( oRd
Re0,c Critical rotational Reynolds number for the onset of Taylor vortices
Ta Taylor number
oR d among several other definitions
Tac Critical Taylor number for the onset of vortex flow
Reax axial Reynolds number / U d
Geometry n radius ratio (R1/Ro) r Aspect ratio (L/d)
Rotation rates and reaction temperature (50°C) were controlled by a T&S (Sao Carlos, Brazil) computer interface unit, with accuracy of 0.1 s-1 and 1°C, respectively. Bed porosity was 98%. The biocatalyst was agarose gel, loaded with 7.8 x 103 UBAEE/kggel (1 UBAEE is the amount of Alcalase that hydrolyzes 1.0 Mmol of benzoil arginine ethyl ester (BAEE)/min at pH 8.0 and 25°C). The enzyme was covalently bonded to porous particles of agarose-glyoxyl gel 6% (weight basis; Hispanagar, Spain) through multipoint links (18). The substrate was 50.0 kg/m3-sweet cheese whey (proteins weight fractions: B-lactoglobulin 56%; a-lactoalbumin 20%; bovine serum albumin [BSA] 11%, immunoglobulins and other fractions 13%), provided by Cooperativa de Laticinios (Sao Carlos, Brazil). Proteolytic activities of the biocatalyst were always checked before and after each reaction assay. Enzyme losses from the biocatalyst were negligible in all assays, proving the stability of the multipoint attachments. Figure 3 schematically depicts the experimental apparatus used to perform VFR cheese whey enzymatic hydrolysis assays.
All assays had Ree = 3500 and Reax= 1.10. Concentrated cheese whey (50.0 kg/ m3), with 0.25 M sodium tetraborate and 0.1 M NaOH was fed to the reactor from a thermostatic bath (Neslab) at 50°C. The pH was adjusted to 9.5 and 2.20 x 10-4 m3 of this solution was used in the jacketed reactor. The inner cylinder rotation was 20.9 s-1. Biocatalyst beads were maintained in 0.25Msodium tetraborate/1.0M NaOH buffer solution at pH 9.5, with 17% volume particles/volume buffer, under stirring. At the beginning of the assay, the gel was added to the reactor (up to 2% v/v). The feed of gel (17% v/v) started immediately, to replace the loss at the VFR outlet, which was recycled to the reactor (see Note 3). The mass of gel in the VFR was measured at the end of the assay, and the steady-state regime was confirmed. Samples were analyzed through HPLC, according to the procedure previously described (17). Five markers were used to define the molecular weight distribution (MWD) intervals of the products: BSA (67,000 Da), P-lactoglobulin (18,000 Da), insulin (5000 Da), angiotensin II (Asp-Arg-Val-Tyr-Ile-His-Pro-Phe) (MW 1046.7 Da), and leucine enkephalin (Tyr-Gly-Gly-Phe-Leu) (555.6 Da).
The next subheadings summarize some useful equations for the design of a VFR. It should be emphasized once again that this is by no means a comprehensive review, but a personal choice of some correlations, among several others, available in the literature.
3. Stability Criteria: The Onset of Taylor Vortices
Taylor flow patterns may be classified in laminar (immediately after the onset of the Couette flow instability), wavy (with distinct numbers of azimuthal waves), and turbulent vortices, for increasing rotation rates of the inner cylinder. When operating with biocatalyst particles, the conditions for minimum fluidization generally impose the operation in the region of turbulent vortices, for biocatalysts slightly denser than the medium.
In practice, the bioreactor designer must calculate the critical rotation rate, either for closed (batch reactors) or for open systems (continuous reactors, in the Taylor-Couette-Poiseuille [TCP] regime). Different authors use different dimen-sionless numbers to describe Taylor flow, depending on the formalism employed in the solution of the Navier-Stokes equations. Table 1 shows one possible set of dimensionless quantities that describe TCP flow. Two options are given for the dimensionless number that takes into account the rotation of the inner cylinder: the tangential Reynolds number (Ree), and one of the definitions of the Taylor number (Ta); any of them may be used without any loss of generality.
The critical rotational Reynolds number for a closed VFR may be predicted by Esser and Grossman (22) equation:
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