3.1. Analysis of Micromixing Intensity Distribution within Ribbed TCR
The intensity of micromixing within the ribbed TCR is closely related to the turbulent kinetic energy dissipation rate (
) within the reactor. The distribution of the energy dissipation rate within the reactor can be transformed into the distribution of the micromixing time using Equation (13), which can be can be obtained through CFD simulations. In this study, the response surface methodology (RSM) model was used to estimate the turbulent kinetic energy and energy dissipation rate in the TCR. For example, for a rotor with a rib spacing of 8 mm, rib width of 3 mm, and rib height of 1 mm, the energy dissipation rate distribution inside the reactor varies with Re
z and Ta, as shown in
Figure 4 and
Figure 5, respectively.
By combining
Figure 4 and
Figure 5, it is evident that the distribution of the micromixing intensity within the ribbed TCR is non-uniform. From cylinders to vortex core, the vertical plane between adjacent ribs can be divided into three regions based on the intensity: (1) the strong micromixing region near the wall; (2) the moderate micromixing region at the inflow and outflow regions of the Taylor vortex; and (3) the weak micromixing region in the vortex core region. In contrast, the conventional TCR has the strongest energy dissipation rate and micromixing intensity in any flow region between vortices, which is the instability flow region. However, the presence of ribs in the ribbed TCR stabilizes the flow field and eliminates the high energy dissipation region between the Taylor vortex pairs between the ribs [
15,
28]. As a result, the strong micromixing region within the annular gap is mostly distributed near the wall, with the highest energy dissipation rate at the edge of the ribs, indicating the role of the ribs in enhancing the micromixing performance of the TCR. The low micromixing region due to the lower velocity gradient of the vortex core region is consistent with the conventional TCR. Therefore, the process of micromixing enhancement within the ribbed TCR can be achieved by increasing the high micromixing region at the wall surface and decreasing the low micromixing region at the vortex core.
According to
Figure 4, as Re
z increases, the flow pattern changes gradually. The strong micromixing region near the wall remains mostly unchanged, while the weak micromixing region slightly decreases. However, in the
Figure 4f–h, the strong micromixing region near the wall gradually decreases, while the weak micromixing region also reduces due to the dominance of the mainstream flow and loss of turbulence. This trend suggests that the role of axial flow in enhancing micromixing performance is not significant enough and may even become harmful after crossing over the double vortex pattern.
In
Figure 5, at Re
z = 9.11, the distribution of energy dissipation rate within the TCR is shown to vary with increasing rotor speed. Increasing the rotor speed leads to a significant increase in the strong micromixing region near the wall and a compression of the weak micromixing region. This suggests that increasing the Taylor number is a more effective means of enhancing micromixing for ribbed TCRs.
3.4. Calculation of Micromixing Time
In a reactor, rapid reactions typically involve two time scales, namely the characteristic time of the reaction,
tR, and the micromixing time,
tm. The relative relationship between these two time scales also determines the rate-controlled step in the reactor. For instance, precipitation reactions often require a micromixing time that is smaller than the characteristic time of the reaction (
tm <
tR). To ensure this requirement is met, reactor design often needs to enhance micromixing performance. In this regard, the characteristic time of the reaction is usually determined by the reaction conditions, while the micromixing time is typically determined by fluid dynamics. Currently, several micromixing models are available to estimate the mixing time, among which the Fournier model is a simple and accurate calculation widely used to estimate mixing time in flow reactors. According to this model, in our experiments, the injected acid was split into multiple agglomerates that were continuously invaded by the ambient medium containing iodide and iodate ions. The acid agglomerates continued to agglomerate with the surrounding ambient medium, and
R1 and
R2 reactions occurred continuously, with the agglomeration time assumed to be equal to the micromixing time. Yang et al. [
21] simplified the process of solving
tm by using the agglomeration model to obtain the segregation index
XS.
In summary, the micromixing time is a critical parameter in reactor design for rapid reactions. The micromixing time should be less than the characteristic time of the reaction to ensure that the rate-controlled step is not the micromixing process. The micromixing time is usually determined by fluid dynamics, and several micromixing models were developed to estimate this parameter. The Fournier model is a simple and accurate calculation widely used to estimate mixing time in flow reactors. In experiments, the agglomeration model is used to solve the micromixing time by the segregation index XS. Overall, improving micromixing performance is essential in reactor design to ensure efficient and effective rapid reactions.
The solution process starts with assuming a value of
tm and substituting it into the system of equations to calculate
XS, and then comparing the calculated model value with the experimental value and continuously adjusting the value of
tm until the relative deviation between the model value
XS,mod and the experimental value
XS,exp is less than 10
−3. In this way, the correlation plot between the micromixing time calculated by the agglomeration model
and the experimentally obtained segregation index
XS is obtained and shown in
Figure 16. The correlation coefficient R
2 = 0.9998 in fitting
tm and
XS; Equation (15):
In this study, the range of
XS was obtained from 3.1 × 10
−4 to 1.1 × 10
−1, and the corresponding range of
tm was obtained from 2.0 × 10
−5 to 8.0 × 10
−3 s. The lower limit of the micromixing time is small, indicating that the ribbed TCR is suitable for many fast reactions. Additionally, a wider range of
tm values can greatly enhance the applicability of the ribbed TCR.
Table 5 summarizes the range of
tm values for various rotating reactors or mixers, which corresponds to the speed range. It is evident that the ribbed TCR can obtain a lower micromixing time at a lower rotational speed (energy input) compared to other rotating reactors or mixers, which further proves the advantage of the ribbed TCRs used in this study in terms of micromixing performance.
Meanwhile, rib is an adjustable parameter that greatly broadens the range of the micromixing performance of the TCR, and the rib structure can be precisely designed for different applicable scenarios and reaction types to achieve the best target product yields and conversions.
In order to measure the micromixing time
tm of the reactor by the Villermaux–Dushman reaction system, it is also a common means to predict the micromixing time of the reactor by the engulfing diffusion model. According to Equation (14),
tm is related to the kinematic viscosity of the fluid as well as the local energy dissipation rate. In Baldyga and Bourne’s [
5] theory, the micromixing time scale at the molecular level equals to the Kolmogorov time scale,
where
tK is the Kolmogorov time scale, s,
v is the kinematic viscosity, m
2/s, and
is the local turbulent kinetic energy dissipation rate, m
2/s
3.
In order to obtain the local energy dissipation rate, it is necessary to predict a volume-averaged energy dissipation rate as a reference value. The volume-averaged energy dissipation rate within the ribbed TCR can be divided into two parts. One part is the energy dissipation caused by the rotor rotation
and the other part is the energy dissipation used to overcome the pressure drop in the system
. The average dissipation rate is equal to the sum of the two components:
Rotation-induced energy dissipation
is equal to the specific power input applied to the fluid due to rotor rotation [
32]: The
where
P is the effective power applied by the rotor to the fluid, W;
is the density of the fluid, kg/m
3;
VR is the effective volume of the reactor, m
3;
U is the operating voltage driving the motor, V;
IW is the current when the rotor is loading, A; and
IE is the current when the rotor is idling, A.
The energy dissipation to overcome the pressure drop
can be expressed as [
33]:
where
Q is the volume flow rate through the reactor, m
3/s; and Δ
P is the pressure drop flowing through the reactor, Pa.
Through measuring the pressure drop of the TCRs under the range of conditions used in this study and calculating
, it was determined that the order of magnitude of
is approximately 1/1000 of that of
. The primary source of energy dissipation within the ribbed TCR is the effective energy input from rotor rotation, i.e.,
. To verify the accuracy of the volume-average dissipation rate obtained by CFD, the average energy dissipation rate was experimentally measured for a rotor with a rib pitch of 8 mm in the Ta range of 1000–6000. The experimental and CFD volume-average dissipation rates were compared, as shown in
Figure 17, with an error of less than 10%. This indicates that the energy dissipation in the TCR can be predicted accurately. Therefore, we correlated the experimentally obtained micromixing time
tm with the volume-average dissipation rate
obtained by CFD under the same conditions using Equation (20), resulting in a correlation coefficient R
2 = 0.951:
where A = 9.84 × 10
−4 m/s
0.5,
tm on
are fitted as shown in
Figure 18.
According to Fournier [
34], a factor
ϕ is introduced as a constant value, which characterizes the relationship between the local energy dissipation rate and the volume-averaged energy dissipation rate,
, and Equation (20) can be transformed into
Equation (21) has the same form as the defining equation of
tK in Kolmogorov’s turbulence theory, where
tm is proportional to (
ν/
ε)
0.5. According to Baldyga and Bourne’s proposed [
27] phagocytic diffusion model,
K is equal to (12/ln2), and brought into Equation (22), we obtain
ϕ is 300.