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