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.