4.3 Effect of Reynolds number on Xs
Figure 6 (a) shows the effect of the variation of Reynolds number onXs under the condition of 1.0 mL sulfuric acid solution injected
within 120 s. With the increase of Reynolds number, Xs decreases
in both the CTC and LTC. When Reynolds number is greater than
25128 (corresponding to 600 rpm),
the decrease in Xs becomes small. At a low rotational speed,
i.e., a small Reynolds number, Xs presents a very high value, and
the difference between the CTC and LTC is very small, which can be
attributed to the excessive turbulence generated by the lobed inner
cylinder being still small. Although the geometry modification can
enhance the micromixing to some extent, flow pattern has not become
fully turbulent for both the CTC and LTC. The degree of the occurrence
of the micromixing may still rely on the molecule-scaled diffusion. The
reactant fluid elements that contributes to the micromixing still hold a
relatively large size compared with the molecular diffusion length
scale. In such case, the micromixing may not be sufficient. With the
increase of Reynolds number, turbulence intensity is gradually built up
and the flow in the reactor develops to the turbulent state, and the
micromixing improves evidenced by drop in Xs . Although the
chemical reaction occurs at molecular level, the intensified turbulence
can provide the environment for reactant fluid elements to break into
much smaller size eddies with the surface area for the mass transfer
being increased. As a result, mixing diffusion improves and the
micromixing rate can be accelerated. Finally, as Reynolds number exceeds
25128, it was observed that Xs levels off, reaching a minimum of
about 0.15 and 0.08 for the CTC and LTC, respectively.
We cautiously mention here that the difference of Xs between the
CTC and LTC becomes remarkable with the flow in the TC reactor to be
judged to be fully turbulent. The LTC shows a much better micromixing
than the CTC. This may be explained by the facts: Firstly, with the
rotation of the inner cylinder, gap size of the LTC varies periodically
so that the formed Taylor vortices change and the vortices are deformed.
Consequently, this type of perturbation due to the deformation Taylor
vortices will induce the generation of small turbulent eddies down to
the scales beneficial to the micromixing. Secondly, Liu et al. ,
(2020) have compared the turbulent flows generated by the CTL and LTC
and shown that the impinging jet region existing between the two
toroidal counter-rotating Taylor vortices induces a stronger outward
shear gradient in the LTC than that in CTL when the same rotational
speed was taken. Thus, it can be claimed that the reactant micro
elements entrapped by the turbulent eddies generated by the impinging
jet flow shear in the LTC can have a shorter entrainment time than the
CTC.
In order to quantitatively describe how Xs changes with the
Reynolds number, the following relation is proposed, given by
\(Xs=C\text{Re}^{b}\) (18)
By taking the logarithmic transformation of both sides, a liner
relationship is obtained. Using this regression fitting, it was found
that well fitted relation for the CTC is \(lnXs=-0.451lnRe+2.963\)with R2 =0.968 and the same fitted relation for
the LTC is\(\ lnXs=-0.635lnRe+4.319\) withR2 =0.986, respectively. As the slops bfor both relations show negative values, the smaller value of bindicates Xs to be more sensitive to turbulent eddies.
As the turbulent intensity can be used to determine the micromixing
efficiency as suggested by Qin et al. , (2017), the turbulent
intensity measured on the surface of the inner cylinder for both the CTC
and LTC based on CFD simulation is shown in Figure 7. For three
representative rotational speeds, 100, 600 and 1000 rpm, the
corresponding Reynolds numbers are
4188, 25128 and 41880, respectively. It can be seen clearly from the
figure that the turbulent intensity is enhanced with the increase of
Reynolds number for both the CTC and LTC but the enhancement for the LTC
is significantly larger than that in the CTC. Also, the highest
turbulent intensity appears at regions of three concaved arcs,
corresponding to the smallest gap regions in the LTC. We postulate that
the best micromixing may happen in these regions. To demonstrate this,
the correlation between the turbulence intensity and 1/Xs is
proposed.
\(R_{\text{IXs}}=I\frac{1}{X_{s}}\) (19)
where <I > and
<Xs > are the volume average turbulence
intensity and Xs in the reactor. Figure 8 shows such correlation,
clearly indicating that the micromixing can be improved through the
modification of the inner cylinder configuration of the TC reactor.