2.1 Reaction kinetics
The reaction system can be described by three sub-reactions, which are expressed as
\(H_{2}BO_{3}^{-}+H^{+}{}H_{3}\text{BO}_{3}\ (quasi-instantaneous)\)(1)
\(5I^{-}+\text{IO}_{3}^{-}+6H^{+}{3I}_{2}+{3H}_{2}O\ (\text{fast})\)(2)
\({I_{2}+I^{-}}_{\overset{\leftarrow}{k_{3}^{{}^{\prime}}}}I_{3}^{-}\) (3)
where H+ stands for the hydrogen ion of the sulfuric acid (H2SO4). Such a system can be regarded as having a competition between the first neutralisation reaction and the second Dushman reaction that produces the by-product of iodine, leading to the occurrence of the third reaction. The concentration of triiodide (\(I_{3}^{-}\)) in the third reaction can be measured by a UV spectrophotometer. Reaction (1) is quasi-instantaneous with a second-order rate constant k1 at about 1011 Lmol-1s-1(Unadkat et al. , 2013), while Reaction (2) is very fast and has the same order of magnitude as the micromixing process as indicated by Fournier et al. (1996). The rate constantk2 depends on the ionic strength I(Guichardon et al. , 2000) and will change during the micromixing process, which was found to be well approximated by the following empirical relationships:
\(\lg k_{2}=9.28-3.66\sqrt{I_{i}},\ \ I_{i}<\ 0.16\ {mol\bullet L}^{-1}\)(4)
\(\lg k_{2}=8.38-1.51\sqrt{I_{i}}+0.23I_{i},\ \ I_{i}>0.16\ {mol\bullet L}^{-1}\)(5)
The rate laws of Reactions (1) and (2) can be expressed as
\(r_{1}=k_{1}c_{H_{2}\text{BO}_{3}^{-}}c_{H^{+}}\) (6)
\(r_{2}=k_{2}c_{I^{-}}^{2}c_{\text{IO}_{3}^{-}}c_{H^{+}}^{2}\) (7)
The equilibrium constant K3 of Reaction (3) is given by
\(K_{3}=\ \frac{k_{3}}{k_{3}^{{}^{\prime}}}=\ \frac{c_{I_{3}^{-}}}{c_{I_{2}}c_{I^{-}}}\)(8)
For a given reaction, the value of K3 is only dependent on temperature. For \(I_{3}^{-}\) formation,K3 is given by Palmer et al. (1984) and is expressed as follows:
\(\lg K_{3}=\ \frac{555}{T}+7.355-2.575\log_{10}T\) (9)
Due to these three reactions occurring in one system, there is a material balance on each component. On the basis of yield of iodide ion (i.e.,\(\ I^{-}\)), its material balance can be written as
\(c_{I^{-}}=\ \frac{c_{I_{0}^{-}}V_{1}}{V_{0}+\ V_{1}}\ -\ \frac{5}{3}\left(c_{I_{3}^{-}}+\ c_{I_{2}}\right)-\ c_{I_{3}^{-}}\)(10)
where \(c_{I_{0}^{-}}\ \)stands for the initial concentration of\(I^{-}\). V0 represents the volume of H2SO4 solution, whileV1 is the volume of the mixture solution of reactants. H2SO4, serving as the limiting agent, is additionally injected to the system to trigger the parallel competing reaction between Reactions (1) and (2).
Combining Equations (8) and (9) with (10), the concentration of iodine (I2) can be calculated from Equation (11).
\(-\frac{5}{3}c_{I_{2}}^{2}+\left(\frac{c_{I_{0}^{-}}V_{1}}{V_{0}+\ V_{1}}-\frac{8}{3}c_{I_{3}^{-}}\right)c_{I_{2}}-\frac{c_{I_{3}^{-}}}{K_{3}}=0\)(11)
The main by-product of the Villermaux iodide-iodate reaction system is I2. With the presence of excessive\(\ I^{-}\), I2 will further react with \(I^{-}\) to generate\(I_{3}^{-}\) until an equilibrium is reached. Since \(I_{3}^{-}\) will have absorption peaks at the wavelength of 288 nm and 353 nm in the spectrum, the concentration of \(I_{3}^{-}\) can be measured by a UV spectrophotometer. However, \(I^{-}\) also presents the absorption peak at around 288 nm in the spectrum. In order to reduce interference from the other components, the use of 353 nm as an indicator is preferable for the concentration measurement of\(\text{\ I}_{3}^{-}\). According to Beer-Lambert’s law, the absorption A of a component across a quartz cell with a thickness Ψ is linearly dependent on its concentration c and molar extinction coefficient e , i.e.
\(A_{353}=\ e_{353}c_{I_{3}^{-}}\Psi\). (12)
For a particular product and a quartz cell, the molar extinction coefficient e and thickness Ψ are fixed values. Thus, the absorption is linearly dependents on the concentration of\(\text{\ I}_{3}^{-}\). The calibration curve of \(I_{3}^{-}\) was firstly prepared before experimental work was performed as shown in Figure 1. The calibration curve with R2 equal to 0.987 highlights that the Beer- Lambert’s law is valid within the range of \(I_{3}^{-}\) concentration chosen in the present study. For the micromixing experiment test, once the absorption of triiodide ion is obtained from the UV spectrophotometer, the concentration of\(I_{3}^{-}\) can be calculated from the calibration curve.
Definition of segregation index
The segregation index (Xs ) is defined as the relative amount of H+ consumed by Reaction (2). For a perfect micromixing condition, all H+ would need to be evenly distributed in the system and then immediately consumed by borate ion (i.e.,\(\ H_{2}BO_{3}^{-}\)) without the appearance of Reaction (2). On the contrary, for poor micromixing, \(H_{2}BO_{3}^{-}\) and\(\ I^{-}\), iodate ion (i.e.,\(\text{\ IO}_{3}^{-}\)) would compete with H+ simultaneously. According to stoichiometry, the yield of I2 from Reaction (2) for a total segregation is expressed as,
\(Y_{\text{ST}}=\ \frac{6n_{\text{IO}_{3,0}^{-}}}{6n_{\text{IO}_{3,0}^{-}}+\ n_{H_{2}\text{BO}_{3,0}^{-}}}\)(13)
In practice, I2 exists in two parts. One is generated in Reaction (2), and the other one is consumed by Reaction (3). Thus, the yield of I2 should be calculated through the ratio of total mole of both I2 and \(\text{IO}_{3}^{-}\) to the initial mole of H+, as defined by
\(Y=\ \frac{2(n_{I_{2}}+\ n_{I_{3}^{-}})}{n_{H_{0}^{+}}}\) (14)
Therefore, Xs can be written as
\(X_{S}=\ \frac{Y}{Y_{\text{ST}}}=\ \frac{n_{I_{2}}+\ n_{I_{3}^{-}}}{n_{H_{0}^{+}}}(2\ +\ \frac{n_{H_{2}\text{BO}_{3,0}^{-}}}{3n_{\text{IO}_{3,0}^{-}}})\)(15)
Equation (15) can be converted in terms of the concentration given by Equation (16).
\(X_{S}=\ \frac{(c_{I_{2}}+\ c_{I_{3}^{-})V_{1}}}{n_{H_{0}^{+}}}(2\ +\ \frac{c_{H_{2}\text{BO}_{3,0}^{-}}}{3c_{\text{IO}_{3,0}^{-}}})\)(16)
By definition, the value of XS varies between 0 and 1 with a lower value indicating a better micromixing performance.
\(X_{s}=0\) Perfect micromixing\(X_{s}=1\) Total segregation\(0<X_{s}<1\) Partial segregation
CFD modelling
In order to get a better understanding of the mixing conditions in the TC reactor, the flow fields of two different inner cylinders were simulated using commercial CFD code, FLUENT 17.0. Based on the structures of the two types of inner cylinder shown in Figure 2, the geometry was created by ANSYS ICEM. Then, the computational domain was divided into two zones, connected by the predefined interface. The total meshes have around 1,100,000 cells, with each direction of 16×147 ×480 (radial × circumferential × axial). Our previous work (Liu et al. , 2020) presents the details for such flow field simulation, where RNG k- ε turbulent model was adopted. The boundary conditions were set as velocity inlet and pressure outlet with no slip wall. The discretized equations were realized by the SIMPLEC algorithm.
Experimental work