4.4 Characterisation of micromixing time
In order to compare the micromixing efficiency of the adopted TC reactor especially the lobed inner cylinder with conventional stirred tank, the micromixing times for different reactors are evaluated. Many models have been proposed to estimate the micromixing time. Amon these models, the IEM model (Costa and Trevissoi, 1972), the EDD model (Baldyga and Bourne, 1984), the E-model (Baldyga and Bourne, 1989), and the incorporation model (Villermauxet al. , 1994) are representatives. However, the incorporation model has been widely used and recognized as illustrated in Figure 9. This model assumes that the limited agent, acid occupying Environment 2, is divided into several aggregates, which then are progressively invaded by surrounding solution from Environment 1. Consequently, the volume of acid aggregates gradually grows due to the incorporation, based on\(\text{\ V}_{2}=V_{20}g(t)\). The characteristic incorporation time is assumed to be equivalent to the micromixing time. Fournier et al. (1990) proposed a dilution-reaction equation in the reaction Environment 2, which is found to be suitable for the description of the present employed TC reactor system, given by
\(\frac{\text{dc}_{j}}{\text{dt}}=\left(c_{j10}-c_{j}\right)\frac{1}{g}\frac{\text{dg}}{\text{dt}}+r_{j}\)(20)
where cj is the reactant concentration, and species j denotes\(\text{\ H}_{2}BO_{3}^{-}\), H+, \(I^{-}\),\(\text{\ IO}_{3}^{-}\), I2, and \(I_{3}^{-}\). cj10 is the concentration of surrounding solution (i.e., the initial concentration of species j in Environment 1).rj is the net production rate of speciesj , and g denotes the mass exchange rate between reactant fluid particle and its surrounding solution. A large value ofdg/dt indicates a fast dilution, indicating a good mixing performance between the feeding acid and its surrounding solution. The empirical equation of the growing law for acid aggregates can be expressed as an exponential function of micromixing time,tm , which reads
\(g\left(t\right)=exp(\frac{t}{t_{m}})\) (21)
Thus, Equation (20) can be converted into the following form,
\(\frac{dc_{j}}{\text{dt}}=\frac{c_{j10}-c_{j}}{t_{m}}+r_{j}\)(22)
From Equation (22), the mass balance equation of individual species can be obtained. In total, there are six transport equations to be solved. In order to reduce computational cost, the W-Z transformation was adopted to reduce the number of solutions and the simplification yields
\(\frac{\text{dW}}{\text{dt}}=-\frac{c_{H_{2}\text{BO}_{3}^{-},\ \ 10}+W}{t_{m}}-{6r}_{2}\)(23)
\(\frac{\text{dY}}{\text{dt}}=\frac{c_{I^{-},\ \ 10}-Y}{t_{m}}-8r_{2}\)(24)
\(\frac{\text{dZ}}{\text{dt}}=\frac{c_{I^{-},\ \ 10}-Z}{t_{m}}-5r_{2}\)(25)
\(\frac{dc_{\text{IO}_{3}^{-}}}{\text{dt}}=\frac{c_{\text{IO}_{3}^{-},\ \ 10}-c_{\text{IO}_{3}^{-}}}{t_{m}}-r_{2}\)(26)
where\(\ W=c_{H^{+}}-c_{H_{2}\text{BO}_{3}^{-}}\),\(\ Y=c_{I^{-}}-c_{I_{2}}\), and \(Z=c_{I^{-}}+c_{I_{3}^{-}}\). Equations (23)-(26) can be solved numerically by iteration, where the initial conditions are given by\(\ W=c_{H^{+},\ \ 0}\),\(\ Y=0\),\(\ Z=0\), and\(\text{\ c}_{\text{IO}_{3}^{-}}=0\). The iteration ends as H+ concentration approaches 0. Acid concentration reaches its highest value at the inlet, then, it disperses within a very limited range and is consumed quickly. Thus, H+concentration is assumed to be at its initial value, \(c_{H^{+},\ \ 0}\)during the iteration before it is complete consumed. The forth-order Runge-Kutta method was adopted in the present study to calculatetm . Firstly, a series value oftm is assumed. Following the Runge-Kutta iteration, Equations (23)-(26) are solved, and the concentration of individual species are obtained. Subsequently, a set of segregation indexXs can be calculated based on Equation (16). Figure 10 depicts the obtained relation of Xsagainst the micromixing time tm(\(Xs=37991\text{\ t}_{m}\)) based on fitting the calculated values ofXs . This relation can be used to evaluate the micromixing time in TC reactor based on the value of Xs obtained from the experimental results, which are also shown in Figure 10. Figure 11 shows the relationship between the Reynolds number and micromixing time in the CTC and LTC. For better description, the contour of turbulent intensities in the circumferential direction for both the CTC and LTC is also shown in Figure 10, where the intensified regions by geometry modification can be seen clearly. By using power law fitting, the micromixing time for both the CTC and LTC can be approximated by\(\text{\ t}_{m}=0.0025\text{Re}^{-0.664}\)and\(\text{\ t}_{m}=0.0006\text{Re}^{-0.456}\), respectively.
Damköhler number (Da ), defined as the ratio of the chemical reaction timescale (reaction rate) to the mixing timescale (mixing rate), is also used to characterize the impact of hydrodynamics in the TC reactor on chemical reaction. Here, we use the obtained relations for the micromixing time to estimate\(\ \text{Da}_{1}\) and\(\ \text{Da}_{2}\) for Reactions (1) and (2), respectively. The chemical reaction time for Reactions (1) and (2) is given by
\(t_{r_{1}}=\frac{min(c_{H_{2}\text{BO}_{3}^{-},0};c_{H^{+},0})}{r_{1}}\)(27)
\(t_{r_{1}}=\frac{min(\frac{3}{5}c_{I^{-},0};{3c}_{\text{IO}_{3}^{-},0};\frac{1}{2}c_{H^{+},0})}{r_{2}}\)(28)
Thus, Da1 and Da2 can be estimated using Equations (29) and (30).
\(\text{Da}_{1}=\frac{t_{m}}{t_{r_{1}}}=t_{m}k_{1}c_{H^{+},0}\) (29)
\(\text{Da}_{2}=\frac{t_{m}}{t_{r_{2}}}=t_{m}k_{2}c_{I^{-},0}^{2}c_{H^{+},0}^{2}\)(30)
The estimatedDa1 =4.2×105-2.8×106is much great than 1, indicating that Reaction (1) is an instantaneous reaction.Da2 =3.2×10-3-2×10-2has the order of 10-2, which is small than 1. Both results indicate that the iodide-iodate reaction system used for evaluation of the micromixing performance in the TC reactor to be suitable.
Compared with the conventional stirred tank reactor, in which the micromixing time is the order of 20 ms (Fournier et al. , 1996), the order of micromixing time in the TC reactor is evaluated to be 10-5 s based on the above discussion. It thus can be claimed that the TC reactor can have a better micromixing performance than the traditional stirred tank reactor as far as those fast chemical processes controlled by the mixing are concerned. The use of the lobed inner cylinder configuration in the TC reactor can further shorten the micromixing time due to the local turbulence intensification.
Conclusions
The micromixing performance in a TC reactor with two different inner cylinder geometries has been evaluated based on the parallel competing iodide-iodate reaction system to characterise the impact of the inner cylinder configuration variations on the micromixing process that will significantly affect the hydrodynamic environment of the particle synthesis. Segregation index Xs was employed as an indicator to characterise the micromixing efficiency. In order to assess the effects of various factors, the sample collection time has been carefully determined to ensure the reliable UV results. The acid concentration was also carefully chosen to avoid over-loading, while the injection of acid was controlled to keep the feeding as slow as possible in order to eliminate the impact on the macromixing in the TC reactor. The conclusions reached for the present study can be summarised as follows:
(1) The segregation experimental results have indicated that the value of Xs decreases with the increase of Reynolds number for both inner cylinder configurations but the LTC exhibits a better micromixing performance than the CTC as Xs for the LTC is smaller than the CTC.
(2) CFD simulation results have revealed that the turbulence shear generated in the jet regions in vicinity of the inner cylinder of the LTC is stronger than that of the CTC, which subsequently enhances the local micromixing, which can be characterised by the enhancement of the turbulence intensity in these regions close to the inner cylinder surface. This clearly indicates that the modification of the inner cylinder configuration (here, the use of a lobed cross-sectional profile) may improve the micromixing action significantly.
(3) Predictions made by employing the incorporation model show that the micromixing time is estimated to be of the order of 10-5 s for the TC reactor, much smaller than that of the traditional stirred tank reactor according to the open literature. In addition, the LTC shows a shorter micromixing time than the CTC.
Nomenclature