Prediction of the reaction yield in a X-micromixer given the mixing degree and the kinetic constant

An adequate comprehension of the flow regimes and mixing process inside microreactors is a crucial factor to obtain high reaction yields. In the present paper, we carry out jointly simulations and experiments to characterize the flow regimes and reaction occurring in a X-microreactor up to Reynolds number, Re, based on the inlet bulk velocity and mixing channel hydraulic diameter, equal to 600. For $\text{Re}>375$, an unsteady periodic regime is found, characterized by the presence of a central vortical flow structure that periodically collapses leading to symmetric vorticity shedding in both outlet channels. In the meanwhile, two counterrotating vortices form in the confluence region of the inlet streams, come closer, and merge creating again the central vortex. Compared to the steady engulfment regime, previously described in the literature and characterized by a single steady central vortex, a higher mixing degree is found in the unsteady regime. However, despite the increased mixing, the reaction yield remains similar, as the enhanced mixing is counterbalanced by the reduced residence time of the reactants. Based on the obtained results we developed a model to predict the reaction yield, given the mixing degree and the nominal kinetic constant as input data. The proposed law successfully predicts the reaction yield in all the flow regimes, including the unsteady engulfment one regime, and for the Damköhler numbers in the range ${10}^{-1}<\text{Da}<{10}^3$ (chemistry slower and faster than flow convection).

Figure 1

(a) Geometry of the X-microreactor and (b) experimental device.

Figure 2

Isosurfaces of the ${\lambda }_2$ vortex-indicator (left panel) and contours of the normalized methylene blue concentration in the outlet channel (right panel). Red dashed lines denote the vortex cores. Simulation at $\text{Re}=450$ and $\left[\mathrm{HCl}\right]=2.19$ mol/l. Considered times: $\mathrm{t}/\tau =0$, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875 (from top to bottom).

Figure 3

Comparison between (a) numerical reaction-rate field and (b) experimental visualization of the contact area between AsA solution and water (b) at $\text{Re}=450$ and $\left[\mathrm{HCl}\right]=2.19$ mol/l. Considered times: $\mathrm{t}/\tau =0$, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875 (from top to bottom). For the video of numerical reaction-rate field, see Supplemental Material [48]. For the video of experimental visualization of the contact area between AsA solution and water, see Supplemental Material [48].

Figure 4

Isosurfaces of the ${\lambda }_2$ vortex-indicator (left panel) and contours of the normalized methylene blue concentration in the outlet channel (right panel). Red dashed lines denote the vortex cores. Simulation at $\text{Re}=600$ and $\left[\mathrm{HCl}\right]=2.19$ mol/l. Considered times: $\mathrm{t}/\tau =0$, 0.167, 0.333, 0.5, 0.667, 0.833 (from top to bottom).

Figure 5

Comparison between (a) numerical reaction-rate field and (b) experimental visualization of the contact area between AsA solution and water (right panel) at $\text{Re}=600$ and $\left[\mathrm{HCl}\right]=2.19$ mol/l. Considered times: $\mathrm{t}/\tau =0$, 0.167, 0.333, 0.5, 0.667, 0.833 (from top to bottom).

Figure 6

(a) Numerical time behaviors of the degree of mixing and (b) comparison between experimental and numerical time behaviors of the normalized ${\mathrm{MB}}^{+}$ concentrations, $C_{{\mathrm{MB}}^{+}}/C_{{\mathrm{MB}}^{+},0}$. The results are evaluated at $Y=25$ cross-section for $\text{Re}=450$ (left panel) and $\text{Re}=600$ (right panel).

Figure 7

Mixing degree numerically evaluated at $Y=25$ as a function of the Reynolds number. The results are visually distinguished by color, considering those that fall within a mixing degree range of $\pm 4–7\%$.

Figure 8

The reaction yield numerically evaluated at $Y=25$, plotted as a function of Reynolds number. The computational results are color-coded based on the mixing degree levels defined in Fig. 7 and Table 2.

Figure 9

Reaction yield numerically evaluated at $Y=25$ cross-section as a function of Damköhler number. Considered cases: (a) $\left[\mathrm{HCl}\right]=5.36$ mol/l, (b) $\left[\mathrm{HCl}\right]=2.19$ mol/l, (c) $\left[\mathrm{HCl}\right]=1.00$ mol/l, and (d) $\left[\mathrm{HCl}\right]=0.10$ mol/l. Markers represent numerical results. The solid and dotted lines scale as ${\text{Da}}^{0.3}$, while the dashed one the scaling in Eq. (17).The results are colored according to the mixing degree levels defined in Fig. 7 and in Table 2.

Figure 10

Reaction yield numerically evaluated at $Y=25$ cross-section as a function of Damköhler number. Considered cases: (a) $\left[\mathrm{HCl}\right]=5.36$ mol/l, (b) $\left[\mathrm{HCl}\right]=2.19$ mol/l, (c) $\left[\mathrm{HCl}\right]=1.00$ mol/l, and (d) $\left[\mathrm{HCl}\right]=0.10$ mol/l. Markers represent numerical results and lines represent Eq. (18). The results are colored according to the mixing degree levels defined in Fig. 7 and in Table 2.

Figure 11

(a) Parameter $\alpha$ as a function of the mixing degree. Considered cases: (a) $\left[\mathrm{HCl}\right]=5.36$ mol/l, (b) $\left[\mathrm{HCl}\right]=2.19$ mol/l, (c) $\left[\mathrm{HCl}\right]=1.00$ mol/l, and (d) $\left[\mathrm{HCl}\right]=0.10$ mol/l. The computational results are colored according to the mixing degree levels defined in Fig. 7 and in Table 2.

Figure 12

Coefficient $m/m_{\infty }$ as a function of the kinetic constant $k_{r,0}$.