Micromixing Performance in a Taylor–Couette Reactor with Ribbed Rotors

Abstract

The Taylor–Couette reactor (TCR) is becoming an increasingly significant topic in chemical industry. This study investigates the micromixing performance of a ribbed TCR with axial flow in the Villermaux–Dushman reaction system. The local micromixing mechanism of the ribbed TCR was analyzed, and the volume-averaged energy dissipation rate was calculated using CFD. The effects of operating parameters and rib structural parameters on micromixing performance were investigated. The results show that the introduction of ribs eliminates the high shear region between the vortex pairs, resulting in the strong micromixing region being situated on the inner and outer cylinder wall surfaces and the ribbed surface region. Smaller rib spacing, larger rib width, and rib height can strengthen micromixing and result in a smaller segregation index. Micromixing times of ribbed TCRs were calculated using the incorporation model, t_m, in the range of 2.0 × 10⁻⁵ to 8.0 × 10⁻³. The results show that ribbed TCRs require a lower energy consumption to achieve a lower t_m than other rotating reactors. A correlation equation between t_m and five parameters was developed, with a correlation coefficient of 0.951. The accuracy of the volume-averaged energy dissipation rate obtained via CFD was verified through experimental analysis. The correlation between the micromixing time and the volume-averaged energy dissipation rate was established in a form that satisfies Kolmogorov’s turbulence theory for t_m. To convert the volume-averaged energy dissipation rate into a local energy dissipation rate, a factor ϕ was introduced and solved using the engulfing diffusion model. This study provides insights into the design and optimization of ribbed TCRs.

1. Introduction

Mixing plays a vital role in the process industry and is considered as one of the fundamental operations. In continuous flow reactors, chemical reactions occur at a specific reaction rate, given reactant concentration and temperature. If the mixing rate fails to match the reaction rate, the reaction will be delayed, resulting in a decrease in conversion and selectivity. Consequently, the performance of the reactor’s mixing significantly influences the process’s feasibility and economics. To achieve a well-mixed state, the reactor should attain three primary objectives, including uniform concentration distribution, defined segregation scales X_S (such as particle size, droplet size, and streak thickness), and the mixing time or rate that satisfies the prerequisites. Therefore, understanding and describing mixing are crucial for designing continuous flow reactors [1].

It is widely accepted that mixing can be divided into three scales, namely macroscopic mixing, mesoscopic mixing, and microscopic mixing [2,3]. Baldyga [4] introduced the Batchelor scale as the measure of micromixing and defined macromixing as diffusion that occurs above the Kolmogorov scale. The mesoscopic mixing scale lies between the Batchelor scale and the Kolmogorov scale [5]. Although the term mesoscopic mixing is not commonly used, it is often studied in conjunction with microscopic mixing [6]. In this paper, the term microscopic mixing encompasses both mesoscopic and microscopic mixing scales.

Micromixing performance is closely related to the flow structure within the reactor. Taylor–Couette flow (TCF) is a special flow phenomenon, which can offer a superior micromixing performance and have significant potential for fast reaction applications. Basing on the TCF, a device named the Taylor–Couette reactor (TCR) is utilized for fast reaction. The flow structure of TCRs is characterized by a continuous sequence of Taylor vortices in the annular gap. TCRs offer several advantages [7,8,9,10,11] over conventional continuous flow reactors, including homogeneous mixing, high specific surface area, mild and uniform shear distribution, controllable residence time distribution, and suitability for mixing and reacting non-Newtonian fluids.

Micromixing studies on the TCR are limited, but it was shown that the vortex motion of the Taylor vortex can significantly improve micromixing efficiency. Liu [12] and colleagues investigated this by studying the micromixing performance of TCR under different flow patterns using the Villermaux–Dushman reaction system. They found that the rotor speed had little impact on micromixing efficiency under laminar Couette flow, but significantly improved micromixing efficiency under laminar Taylor vortex flow and weakly turbulent wave Taylor vortex flow. Banaga [13] and colleagues studied the effect of feeding methods on micromixing performance in a rotating bar reactor (RBR), which has a similar structure to the TCR but uses a higher rotor speed. They found that tangential feed resulted in better micromixing performance than radial feed, and increasing rotational speed and flow rate reduced micromixing time.

Studies explored the new structure of the TCR to enhance micromixing. Chen [14] developed a miniaturized annular rotating flow mixer (MARFM) based on TCF and conducted experiments to assess its micromixing performance and the impact of operational parameters. Li [15] used CFD techniques to analyze local micromixing times within a TCR with a lobbed rotor and found that micromixing time was not uniformly distributed, with shorter times for intro-vortices than inter-vortices. This was attributed to the flow field between adjacent annular vortices, which generates strong turbulence and dissipation of kinetic energy. Smaller inlet sizes were found to result in poorer micromixing performance in Chen’s study, while Li’s study identified the strongest turbulence generation and kinetic energy dissipation in the region between two adjacent annular vortices. However, to the best of our knowledge, micromixing performance in ribbed TCRs was not investigated.

In this study, TCRs with ribbed rotors are used to intensify micromixing performance. The local energy dissipation rate and its distribution in ribbed TCRs were analyzed using CFD. The micromixing performance of ribbed TCRs was also examined using the Villermaux–Dushman reaction system, with operational parameters such as axial Reynolds number (Re_z) and Taylor number (Ta), and structural parameters such as rib spacing (L), rib width (d), and rib edge thickness (h) being evaluated. The micromixing time was correlated with each parameter using the partial least squares regression method, and the influence of each variable was evaluated using variable importance projection. Additionally, the segregation index X_S obtained experimentally was compared to the volume-averaged energy dissipation rate obtained by CFD. Finally, the experimental micromixing times were estimated using an incorporation model and correlated with X_S.

2. Experiments

2.1. Geometrical Structures of Ribbed TCRs

This study used a self-made 3D-printed Taylor–Couette reactor (TCR), which had a detailed structure and dimensional definition shown in Figure 1. The ribbed rotor’s structural parameters are listed in

Table 1. Structural parameters of the ribbed TCR.

Structural Parameters	Range
Inner cylinder radius (Ri), mm	12.5
Outer cylinder radius (Ro), mm	16.5
Annular gap width (e), mm	4
Rotor length (l), mm	200
Rib spacing (L), mm	8–12
Rib width (d), mm	1–3
Rib thickness (h), mm	1–12
Number of regions (N)	10–50

. This TCR had an annulus width of 4 mm, which was a reasonable size for 3D printing and ensured a feasible liquid flux. This size also avoided the negative effects of too small a flow rate and unstable pump delivery when lower axial flow rates were studied. The miniaturization of the TCR was in line with the concept of a process-enhanced flow reactor device, which significantly improved the device’s mixing performance while maintaining the reactant flux. The TCR used a tangential feed system that enhanced micromixing performance [13,14]. The fluid entered the reactor’s annular gap from the bottom, generating a series of flow structures that exited the reactor from above. Figure 2 shows the experimental setup and tangential feed mode schematically.

2.2. Villermaux–Dushman Reaction System

The Villermaux–Dushman reaction system is a parallel competitive reaction system proposed by Fourier et al. [16] that is widely used to study the micromixing performance of continuous flow reactors [17,18,19,20]. The system consists of three chemical reactions as follows:

$${\mathrm{H}}^{+}+{\mathrm{H}}_2{\mathrm{BO}}_3^{-}\stackrel{k_1}{\to }{\mathrm{H}}_3{\mathrm{BO}}_3$$

$$6{\mathrm{H}}^{+}+5{\mathrm{I}}^{-}+{\mathrm{IO}}_3^{-}\stackrel{k_2}{\to }3{\mathrm{I}}_2+3{\mathrm{H}}_2\mathrm{O}$$

$${\mathrm{I}}^{-}+{\mathrm{I}}_2\underset{k_4}{\overset{k_3}{\rightleftharpoons }}{\mathrm{I}}_3^{-}.$$

The complex calculation of the reaction system was simplified by Yang et al. [21]. Only the kinetics of the second reaction R₂ are necessary in the calculation of the micromixing time.

$$R_2=k_2{\left[{\mathrm{I}}^{-}\right]}^2\left[{\mathrm{IO}}_3^{-}\right]{\left[{\mathrm{H}}^{+}\right]}^2$$

(1)

The kinetic constant k₂ depends on the ionic strength of the reaction medium:

$$\lg (k_2)=9.28105-3.664\sqrt{I_t},I_t<0.166\mathrm{mol}\cdot {\mathrm{L}}^{-1}$$

(2)

$$\lg (k_2)=8.383-1.5112\sqrt{I_t}+0.23689I_t,I_t>0.166\mathrm{mol}\cdot {\mathrm{L}}^{-1}$$

(3)

${\mathit{I}}_{\mathit{t}}$ is the ionic strength of the reaction medium, defined as

$$I_t=\frac{1}{2}{\displaystyle \sum C_i{Z}_i^2}$$

(4)

where C_i is the molar concentration of ion i, mol/L; $Z_i$ is the valence of the ion.

The iodine monomers generated by the reaction R₂ may further react with ${\mathrm{I}}^{-}$ to form ${\mathrm{I}}_3^{-}$, which is the reversible reaction R₃. The equilibrium constant of the reaction k_B is a function of temperature and is defined as

$$\lg k_3=555T+7.355-2.575\lg T.$$

(5)

Reaction R₁ is a quasi-instantaneous neutralization reaction and reaction R₂ is a fast oxidation reaction, but slower than reaction R₁. Under ideal micromixing conditions, H⁺ is completely consumed by the most rapid reaction R₁. Actually, due to incomplete micromixing, H⁺ is simultaneously consumed by reaction R₂, and the ${\mathrm{I}}_2$ generated further transformed to ${\mathrm{I}}_3^{-}$ by reaction R₃, which accelerate the competition with reaction R₁. Therefore, ${\mathrm{I}}_3^{-}$ is the key substance to examine the micromixing efficiency, and the higher amount of ${\mathrm{I}}_3^{-}$ indicates that the micromixing process of the reactor deviates from the ideal one. The micromixing intensity can be defined by the segregation index X_S.

$$X_S=\frac{Y}{Y_{ST}}$$

(6)

where Y is the selectivity of ${\mathrm{H}}^{+}$ in a real reactor environment (the ratio of H⁺ consumed by reaction R₂ to the total injection); and Y_ST is the selectivity of H⁺ based on stoichiometric ratios, which is the ideal Y for the fully segregated case.

$$Y=\frac{2(n_{{\mathrm{I}}_2}+n_{{\mathrm{I}}_3^{-}})}{n_{{\mathrm{H}}_0^{+}}}$$

(7)

$$Y_{\mathrm{ST}}=\frac{6{\left[{\mathrm{IO}}_3^{-}\right]}_0}{6{\left[{\mathrm{IO}}_3^{-}\right]}_0+{\left[{\mathrm{H}}_2{\mathrm{BO}}_3^{-}\right]}_0}$$

(8)

Bringing Equations (8) and (7) into Equation (6) yields

$$X_S=\frac{Y}{Y_{ST}}=\frac{Q_A+Q_B}{Q_B}\cdot \frac{\left[{\mathrm{I}}_2\right]+\left[{\mathrm{I}}_3^{-}\right]}{{\left[{\mathrm{H}}^{+}\right]}_0}\cdot \frac{6{\left[{\mathrm{IO}}_3^{-}\right]}_0+{\left[{\mathrm{H}}_2{\mathrm{BO}}_3^{-}\right]}_0}{3{\left[{\mathrm{IO}}_3^{-}\right]}_0}$$

(9)

where Q_A and Q_B represent the flow rates of buffer and acid solutions, respectively.

The value of X_S is in the range of 0 to 1 according to the Equation (9) (0 < X_S < 1). When the device is in a complete micromixing state, X_S = 0. If it is in a total segregation, X_S = 1. The smaller the X_S is, the better the micromixing performance is. According to Lambert’s law, the concentration of ${\mathrm{I}}_3^{-}$ can be calculated by measuring the absorbance at 353 nm by UV spectrophotometer (Ling Guang 759s, China).

The concentration of iodine monomers can be obtained from a material balance as follows:

$$\left[{\mathrm{I}}^{-}\right]={\left[{\mathrm{I}}^{-}\right]}_0-\frac{5}{3}\left(\left[{\mathrm{I}}_2\right]+\left[{\mathrm{I}}_3^{-}\right]\right)-\left[{\mathrm{I}}_3^{-}\right].$$

(10)

The equilibrium constant for the reaction R₃ is

$$k_B=\frac{\left[{\mathrm{I}}_3^{-}\right]}{\left[{\mathrm{I}}_2\right]\left[{\mathrm{I}}^{-}\right]}.$$

(11)

Bringing Equation (11) into (10) yields:

$$-\frac{5}{3}{\left[{\mathrm{I}}_2\right]}^2+\left({\left[{\mathrm{I}}^{-}\right]}_0-\frac{8}{3}\left[{\mathrm{I}}_3^{-}\right]\right)\left[{\mathrm{I}}_2\right]-\frac{\left[{\mathrm{I}}_3^{-}\right]}{k_B}=0.$$

(12)

Once the concentration of ${\mathrm{I}}_3^{-}$ is known, the concentration of iodine monomers can be found by solving the equation.

2.3. Materials

The micromixing performance of the ribbed TCR is investigated in this paper using the Villermaux–Dushman reaction system. This reaction system contains two configurations of solutions: a buffer solution A containing boric acid (H₃BO₄), sodium hydroxide (NaOH), potassium iodide (KI), and potassium iodate (KIO₃), and a solution B of aqueous sulfuric acid (H₂SO₄).

Table 2. Reagent specifications and producer.

Reagents	Molecular Formula	Specification
Boric acid	${\mathrm{H}}_3{\mathrm{BO}}_4$	Analytically pure
Sodium hydroxide	$\mathrm{NaOH}$	Analytically pure
Potassium iodide	$\mathrm{KI}$	Analytically pure
Potassium iodate	${\mathrm{KIO}}_3$	Analytically pure
Concentrated sulfuric acid	${\mathrm{H}}_2{\mathrm{SO}}_4$	98%

presents the specifications of the reagents used in this study.

For the Villermaux–Dushman reaction system, the ratio of the concentration between the two solutions affects the experimental results of the micromixing performance. It is usually necessary to ensure that the generation of ${\mathrm{I}}_3^{-}$ is only due to poor micromixing performance in the reactor. Therefore, in this paper, we chose concentration configuration 1b for buffer A, recommended by Commenge et al. [22]. The detailed concentration configuration is shown in

Table 3. Initial concentrations of reagents in buffer solution A.

Reagents	Initial Concentration (mol/L)
${\mathrm{H}}_3{\mathrm{BO}}_4$	0.009
$\mathrm{NaOH}$	0.009
$\mathrm{KI}$	0.03
${\mathrm{KIO}}_3$	0.006

. The choice of acid type and concentration of acid solution B is also an important influencing factor to improve the accuracy of the experiment. The acid selected in this study was sulfuric acid, and the concentration of the acid was chosen to be 0.03 mol/L based on the upper measurement limit of the UV spectrophotometer [23]. Both solutions A and B were prepared by dissolving the reagents in deionized water and were ready to use to avoid spoilage. The flow ratio $R$ of buffer A to acid solution B was chosen as 1.

Buffer solution A and acid solution B were fed into the reactor at a fixed flow ratio (R = 1) from two tangential inlets at the bottom of the TCR, respectively, using two high-performance chromatography pumps (Flash 300, Laballiance, USA). The two solutions are mixed and reacted in the annular gap of the reactor. After the solution is feeding for about four times the residence time (depends on the volume flow rate), the system can be considered as steady state and three samples’ absorbance of ${\mathrm{I}}_3^{-}$ can be measured at the outlet by a UV. After an additional time interval equal to the residence time, three samples were taken again for measurement. All the micromixing performance experiments in this paper were carried out at 20 ± 1 °C. The range of operating conditions for the experiments is shown in

Table 4. Operation conditions in this work.

Operating Conditions	Range
Volume flow rate ratio of buffer A to sulfuric acid solution B	1
Total volume flow rate (mL/min)	10–500
Rotation speed (rpm)	0–1000

.

2.4. Numerical Simulation

The simulations were conducted using ANSYS Fluent 2020R1. The working fluid selected has the same properties as water (density ρ = 998.2 kg/m at room temperature and viscosity µ = 10⁻³ Pa-s). The governing equations for continuity and momentum for incompressible, constant viscosity within the annular gap of the TCR reactor are:

Continuity equation:

$$\frac{\partial u_i}{\partial x_i}=0.$$

(13)

Momentum equation:

$$\frac{\partial u_i}{\partial t}+\frac{\partial u_i{u}_j}{\partial x_j}=-\frac{1}{\rho }\frac{\partial P_i}{\partial x_i}+\frac{\partial }{\partial x_j}\left(v\frac{\partial u_i}{\partial x_j}\right).$$

(14)

The instantaneous velocity u_i can be converted into an average velocity plus the fluctuating quantity of

$$u_i=\overline{u_i}+u_i^{’}.$$

(15)

The mean momentum equation or Reynolds equation can be obtained by bringing Equation (5) into (4) as follows:

$$\frac{\partial \overline{u_i}}{\partial t}+\frac{\partial \overline{u_i{u}_j}}{\partial x_j}=-\frac{1}{\rho }\frac{\partial \overline{P_i}}{\partial \overline{x_i}}+\frac{1}{\rho }\frac{\partial }{\partial x_j}\left(\mu \frac{\partial \overline{u_i}}{\partial x_j}+{\tau }_{ij}\right)$$

(16)

τ_ij is the stress term and can be written as

$${\tau }_{ij}=\mu \left(\frac{\partial \overline{u_i}}{\partial x_j}+\frac{\partial \overline{u_j}}{\partial x_i}\right)-\rho u_i{u}_j.$$

(17)

The Reynolds stress term $\left(-\rho u_i{u}_j\right)$ is the last term of the equation $\left(-\rho u_i{u}_j\right)$ of Equation (7) that can be modeled by various turbulence models. One such model is the Reynolds stress model (RSM), which is particularly accurate for cyclonic flows. The RSM model is proven to be suitable for calculations within the TCR [24,25,26]. Thus, in this work, RSM was used to model the turbulence in ribbed TCRs. Details of the simulation and grid independence test are shown in the Supplementary Materials.

In this study, three-dimensional steady-state simulations were utilized to solve the continuity and momentum governing equations. The working fluid chosen possesses the same properties as water (density ρ = 998.2 kg/m³ at room temperature and viscosity µ = 10⁻³ Pa·s), enabling the application of the Navier–Stokes equations for incompressible fluids with constant viscosity within the annular gap of the TCR reactor.

According to the engulfment deformation diffusion model proposed by Baldyga and Bourne [27], the micromixing time can be estimated from the turbulent kinetic energy dissipation rate:

$$t_m=\frac{12}{\ln 2}\sqrt{\frac{\nu }{\epsilon }}$$

(18)

where $t_m$ is the micromixing time, s; $v$ is the kinematic viscosity of the fluid, m²/s; and $\epsilon$ is the turbulent kinetic energy dissipation rate, m²/s³.

The local micromixing time is closely related to the local energy dissipation rate, and the increase in energy dissipation rate can strengthen the micromixing. The local energy dissipation rate can be obtained from the local flow field within the TCR annular gap region, and the relationship between the local turbulent energy dissipation rate and the velocity strain rate tensor is:

$$\epsilon =\frac{1}{2}v{\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)}^2.$$

(19)

The local turbulent kinetic energy and energy dissipation rate were estimated using the Reynolds stress model (RSM) in CFD, and solving the volume-averaged energy dissipation rate $\overline{\epsilon }$ within the TCR. $\overline{\epsilon }$ can be used to estimate the global micromixing time of the TCRs, and Jiang’s results demonstrate the accuracy of volume-averaged energy dissipation rates in predicting reactor micromixing times.

The same meshing method is used for all computational cases in this study. An unstructured mesh with a better fit to the geometric model was chosen for the main body of the reactor fluid domain, with a mesh size of 0.04 cm, and a refined boundary layer was generated near the wall to use a finer mesh to delineate the structure. Depending on the number of fins on the rotor surface, the maximum grid can be up to about 10.5 million. The fine mesh division and the larger number of total grids can ensure the accuracy and stability of the calculation. The grid division model is shown in Figure 3. The grid independence tests are necessary to be carried out, which are shown in the Supplementary Materials.

3. Results and Discussion

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 ($\epsilon$) 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.

3.2. Effect of Operating Parameters on X_S

3.2.1. Effect of Axial Reynolds Number

The axial Reynolds number Re_z or the volumetric flow rate Q are crucial operating parameters for the ribbed TCR. In this part, the effect of ${\mathrm{Re}}_{\mathrm{z}}$ is investigated at 200 rpm (Ta = 1041.46) and 600 rpm (Ta = 3124.38) for rotors with rib spacing of 8 mm and 12 mm, respectively. Figure 6 shows the plot of the effect of Re_z on the micromixing performance. In general, the micromixing performance decreases as $\mathrm{Rez}$ increases. Such a result can also be obtained from the trend of the volume-averaged energy dissipation rate $\overline{\epsilon }$ with Re_z for different conditions obtained by CFD in Figure 7; $\overline{\epsilon }$ is inversely proportional to the axial Reynolds number, and according to Equation (14), a decrease in the energy dissipation rate increases the micromixing time, which corresponds to a decrease in the micromixing performance.

This results in a decrease in the micromixing performance with an increase in axial flow. The ribs in the ribbed TCR cause additional turbulence and mixing, so the role of axial flow in enhancing micromixing performance is not as significant as in the conventional TCR [14]. It is because a larger Re_z makes a lower intensity of vortices, and the flow characteristics are closer to a laminar flow. The energy dissipation is reduced. The micromixing performance of the ribbed TCR has a low dependence on the axial Reynolds number, which is different to other rotating reactors [17] or passive mixers [28]. Micromixing performance of the ribbed TCR cannot be enhanced by increasing the flow rate. Increasing the Taylor number, which enhances the turbulence and mixing caused by the ribs, is a more effective way to improve micromixing performance in ribbed TCRs.

3.2.2. Effect of Taylor Number

The Taylor number (Ta), which characterizes the rotation, is an important parameter in all rotating devices and can lead to an unstable flow within the TCR if it is too high. Figure 8 illustrates the effect of the Taylor number on X_S for different rib spacings ranging from 8 to 12 mm. The variation in the volume-averaged energy dissipation rate with the Taylor number obtained by CFD under the same conditions is shown in Figure 9. It is worth noting that the effect of the Taylor number on X_S is the same for all rib spacings. Both figures indicate that increasing Ta can significantly improve the micromixing performance. There are four main reasons for this improvement. Firstly, an increase in Ta enhances the fluid shear force due to the rotation of the rotor, which is beneficial to fluid dispersion and leads to a larger fluid contact area [29]. Secondly, an increase in rotational speed splits the fluid into finer fluid elements, reducing the local concentration of injected hydrogen ions [13]. Thirdly, an increase in rotational speed enhances the fluid circumferential flow velocity, which increases the degree of fluid collision and the energy dissipation rate. Finally, an increased Ta intensifies the process of dispersion–agglomeration–redispersion, leading to a faster and stronger fluid surface renewal. Taken together, increasing Ta can substantially improve the micromixing performance of the ribbed TCR. Furthermore, it is worth noting that while decreasing the rib spacing can also improve micromixing performance (as shown in Figure 8 and Figure 9), the effect is smaller than that of increasing the Taylor number. When the Taylor number is greater than 2500, the effect of rib spacing on X_S gradually decreases, and the X_S eventually level off at around 10⁻² for different rib spacings.

3.3. Effect of Rib Spacing

3.3.1. Effect of Rib Spacing

Figure 10 and Figure 11 further illustrate the effect of rib spacing on micromixing performance. As the rib spacing increases, X_S increases and $\overline{\epsilon }$ decreases, indicating a decrease in micromixing performance. This is because a larger rib spacing means fewer Taylor vortices are generated in the annular gap, leading to weaker micromixing performance. In contrast, smaller rib spacing results in more Taylor vortices and stronger micromixing performance. Additionally, smaller rib spacing also increases the contact area between the rotor and the fluid, leading to more energy input and better micromixing performance. However, it is important to note that rib spacing can also affect the flow pattern transition within ribbed TCRs, and this should be taken into consideration when designing such reactors. Overall, while rib spacing can slightly enhance micromixing performance, increasing the Taylor number remains the most effective way to improve micromixing in ribbed TCRs.

3.3.2. Effect of Rib Width

The rib width is also an important parameter that affects both the macroscopic and micromixing performance of ribbed TCRs. Figure 12 shows that increasing the dimensionless rib width at a fixed annular gap width (d/e) improves the micromixing performance of the reactor. This is because a wider rib increases the contact area between the rotor and the fluid, leading to more energy input and frequent collisions that enhance micromixing. Additionally, the outflow area between the rib and the fluid has the highest shear force and energy dissipation rate, so increasing the rib width can increase the high-energy dissipation region in the TCR and improve micromixing performance. Volume-average energy dissipation rate for different ribs width in Figure 13 has also verified this.

Based on these findings, a rib width of 3 mm is chosen in this paper, with a distance of only 1 mm between the rib edge and the outer cylinder. This design ensures maximum micromixing performance while limiting axial diffusion to the greatest extent possible. Overall, the rib width is an important factor in optimizing the performance of ribbed TCRs, and careful consideration should be given to this parameter in reactor design.

3.3.3. Effect of Rib near Wall Thickness

The effect of rib thickness on micromixing performance in ribbed TCRs is also an important factor to consider in reactor design. Figure 14 shows the linear decrease in X_S and linear increase in the volume-averaged energy dissipation rate with increasing rib thickness, indicating that larger rib thicknesses are beneficial for enhanced micromixing. This is due to the increase in rotor surface area and energy input as well as the compression of the annular gap space, resulting in a shorter diffusion path and higher flow rate in the annular gap region.

Figure 15 also shows the variation in volume-averaged energy dissipation rate with rib thickness for five different rib spacings obtained by CFD, indicating that rib thickness is a more influential structural parameter than rib spacing and rib width. This is because rib thickness has a more pronounced effect on the rotor surface area and fluid compression.

Overall, optimizing rib thickness is an important factor in enhancing micromixing performance in ribbed TCRs, and careful consideration should be given to this parameter in reactor design.

3.4. Calculation of Micromixing Time

In a reactor, rapid reactions typically involve two time scales, namely the characteristic time of the reaction, t_R, and the micromixing time, t_m. 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 (t_m < t_R). 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 t_m by using the agglomeration model to obtain the segregation index X_S.

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 X_S. 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 t_m and substituting it into the system of equations to calculate X_S, and then comparing the calculated model value with the experimental value and continuously adjusting the value of t_m until the relative deviation between the model value X_S,mod and the experimental value X_S,exp is less than 10⁻³. In this way, the correlation plot between the micromixing time calculated by the agglomeration model $t_m$ and the experimentally obtained segregation index X_S is obtained and shown in Figure 16. The correlation coefficient R² = 0.9998 in fitting t_m and X_S; Equation (15):

$$t_m=0.077X_S^{1.02}.$$

(20)

In this study, the range of X_S was obtained from 3.1 × 10⁻⁴ to 1.1 × 10⁻¹, and the corresponding range of t_m 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 t_m values can greatly enhance the applicability of the ribbed TCR.

Table 5. Micromixing time ranges for different rotating reactors and mixers.

Authors	Reactors or Mixers	Rotational Speed (rpm)	Micromixing Time (s)
Martínez [30]	Rotor-stator spinning disk reactor	100–2000	1.1 × 10−4–8.8 × 10−3
Wang [17]	Rotor-stator spinning disc extractor	600–1400	10−5–10−4
Banaga [13]	Rotating bar reactor	500–2500	5.0 × 10−6–1.1 × 10−5
Chu [31]	Rotating packed bed reactor	200–1600	3.3 × 10−3–4.6 × 10−3
Liu [12]	Taylor–Couette reactor	0–300	2.0 × 10−3–4.0
Chen [14]	Annular rotating flow mixer	0–5400	2.7 × 10−5–4.1 × 10−3
This work	Ribbed Taylor–Couette reactor	100–1000	2.0 × 10−5–8.0 × 10−3

summarizes the range of t_m 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 t_m 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), t_m 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,

$$t_K=\sqrt{\frac{\nu }{\epsilon }}$$

(21)

where t_K is the Kolmogorov time scale, s, v is the kinematic viscosity, m²/s, and $\epsilon$ is the local turbulent kinetic energy dissipation rate, m²/s³.

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 ${\epsilon }_r$ and the other part is the energy dissipation used to overcome the pressure drop in the system ${\epsilon }_p$. The average dissipation rate is equal to the sum of the two components:

$$\overline{\epsilon }={\epsilon }_p+{\epsilon }_r.$$

(22)

Rotation-induced energy dissipation ${\epsilon }_r$ is equal to the specific power input applied to the fluid due to rotor rotation [32]: The

$${\epsilon }_r=\frac{P}{\rho V_R}=\frac{U\left(I_W-I_E\right)}{\rho V_R}$$

(23)

where P is the effective power applied by the rotor to the fluid, W; $\rho$ is the density of the fluid, kg/m³; V_R is the effective volume of the reactor, m³; U is the operating voltage driving the motor, V; I_W is the current when the rotor is loading, A; and I_E is the current when the rotor is idling, A.

The energy dissipation to overcome the pressure drop ${\epsilon }_p$ can be expressed as [33]:

$${\epsilon }_p=\frac{Q\Delta P}{\rho V_R}$$

(24)

where Q is the volume flow rate through the reactor, m³/s; and ΔP 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 ${\epsilon }_p$, it was determined that the order of magnitude of ${\epsilon }_p$ is approximately 1/1000 of that of ${\epsilon }_r$. The primary source of energy dissipation within the ribbed TCR is the effective energy input from rotor rotation, i.e., $\overline{\epsilon }\approx {\epsilon }_r$. 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 t_m with the volume-average dissipation rate $\overline{\epsilon }$ obtained by CFD under the same conditions using Equation (20), resulting in a correlation coefficient R² = 0.951:

$$t_m=A{\left(\overline{\epsilon }\right)}^{-0.5}$$

(25)

where A = 9.84 × 10⁻⁴ m/s^0.5, t_m on $\overline{\epsilon }$ 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, $\epsilon =\varphi \overline{\epsilon }$, and Equation (20) can be transformed into

$$t_m=A\frac{{\varphi }^{0.5}}{v^{0.5}}{\left(\frac{\nu }{\epsilon }\right)}^{0.5}=K{\left(\frac{\nu }{\epsilon }\right)}^{0.5}.$$

(26)

Equation (21) has the same form as the defining equation of t_K in Kolmogorov’s turbulence theory, where t_m 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.

3.5. Micromixing Time Correlation

Based on the above analysis, it can be concluded that the operating parameters (Re_z and Ta) and rib structural parameters (L/e, d/e, h/L) have varying degrees of influence on the micromixing performance of ribbed TCRs. To study the influence of these different parameters on micromixing time, partial least squares regression (PLS) was used for fitting to obtain the empirical correlation equation presented in this paper. The effect of each influencing factor on micromixing performance was analyzed. Equation (23) represents the empirical correlation between t_m and the operational and structural parameters studied in this research, using 141 data points and correlation coefficients R² = 0.951:

$$t_m=7.58(\mathrm{s})\cdot {\mathrm{Re}}_z^{0.20}{\mathrm{Ta}}^{-1.22}{\left(\frac{L}{e}\right)}^{0.27}{\left(\frac{d}{e}\right)}^{-0.13}{\left(\frac{h}{L}\right)}^{-0.049}.$$

(27)

Figure 19 presents the coefficient plots obtained through PLS fitting and the projection of variables importance (VIP). Upon considering the values of the coefficients and the VIP values of the variables together, it can be observed that among the five parameters that affect micromixing performance, the Taylor number, Ta, which characterizes rotor speed, has the most significant effect and is much more important than the others. This conclusion is consistent with the result that the energy dissipation rate mainly depends on the energy dissipation induced by rotor rotation. Since micromixing cannot be enhanced by increasing the flow rate, increasing rotational speed becomes the dominant way to boost micromixing. The three parameters characterizing the rib structure are less influential than Ta, but still play important roles in affecting micromixing performance. The increase in the strong micromixing region cannot be ignored. Figure 20 shows the comparison between the experimental t_m and those predicted by the empirical correlation. A total of 93% of the predicted data points lie within ±20% error bars, indicating good agreement between the predicted and experimental values.

4. Conclusions

In this study, we conducted experiments on the Villermaux–Dushman reaction system to investigate, for the first time, the micromixing performance of a ribbed TCR with axial flow. We analyzed the local micromixing mechanism of the ribbed TCR, and calculated the volume-averaged energy dissipation rate using CFD. We investigated the effects of operating parameters (Re_z, Ta) and rib structural parameters (L/e, d/e, h/L) on the micromixing performance. Our main findings are as follows:

(1) The distribution of strong and weak micromixing regions in the ribbed TCR is depicted in this study. The introduction of ribs eliminates the high shear region between the vortex pairs, resulting in the strong micromixing region being situated on the inner and outer cylinder wall surfaces and the ribbed surface region, whereas the vortex core region is the weak micromixing region due to its lower velocity gradient. To enhance micromixing, it is necessary to increase the strong micromixing region and decrease the weak micromixing region. By comparing variations in local micromixing at different rotational speeds and flow rates, it is observed that increasing the rotational speed can significantly strengthen the strong micromixing region on the wall surface, while increasing the flow rate has a lesser impact on it.

(2) All three structural parameters of ribs significantly affect the micromixing performance of ribbed TCRs. Smaller rib spacing, larger rib width, and rib thickness can strengthen micromixing and result in a smaller segregation index. The improvement in micromixing resulting from structural parameters can be attributed to the increased contact area between the ribs and fluid, which effectively enlarges the strong micromixing region at the wall and reduces the vortex size to decrease the weak micromixing region at the core of the vortex.

(3) Micromixing times of ribbed TCRs were calculated using the incorporation model, t_m, in the range of 2.0 × 10⁻⁵ to 8.0 × 10⁻³. The results show that ribbed TCRs require a lower energy consumption to achieve a lower micromixing time than other rotating reactors. This feature enhances the controllability of micromixing times due to the ribbed structure and makes ribbed TCRs applicable in a wider range of scenarios. Finally, a correlation between t_m and five parameters was developed, with a correlation coefficient of 0.951.

(4) The accuracy of the volume-averaged energy dissipation rate obtained via CFD was verified through experimental analysis. Additionally, the correlation between the micromixing time and the volume-averaged energy dissipation rate was established in a form that satisfies Kolmogorov’s turbulence theory for tm, i.e., t_m is proportional to (v/ε)^0.5. To convert the volume-averaged energy dissipation rate into a local energy dissipation rate, a factor ϕ was introduced and solved using the engulfing diffusion model. The value of ϕ was determined to be 300.