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