Modeling mass transfer and reaction of dilute solutes in a ternary phase system by the lattice Boltzmann method

In this work, we propose a general approach for modeling mass transfer and reaction of dilute solute(s) in incompressible three-phase flows by introducing a collision operator in lattice Boltzmann (LB) method. An LB equation was used to simulate the solute dynamics among three different fluids, in which the newly expanded collision operator was used to depict the interface behavior of dilute solute(s). The multiscale analysis showed that the presented model can recover the macroscopic transport equations derived from the Maxwell-Stefan equation for dilute solutes in three-phase systems. Compared with the analytical equation of state of solute and dynamic behavior, these results are proven to constitute a generalized framework to simulate solute distributions in three-phase flows, including compound soluble in one phase, compound adsorbed on single-interface, compound in two phases, and solute soluble in three phases. Moreover, numerical simulations of benchmark cases, such as phase decomposition, multilayered planar interfaces, and liquid lens, were performed to test the stability and efficiency of the model. Finally, the multiphase mass transfer and reaction in Janus droplet transport in a straight microchannel were well reproduced.

Figure 1

Program chart of LBM scheme to model the mass transfer of the dilute species in multiphase flows. Bulk phase equations are described by a color-gradient LBM model in Sec. 2a and the solute equations are given in Sec. 2b.

Figure 2

The temporal evolution of the one-phase soluteble compound in Case 1 during phase segragation, where the dilute compound only stays in the Red phase. The inset shows the coresponding bulk phase flow field by a RGB image as the color defination of the fluids (The black, dark gray, and light gray represent Blue phase, Red phase, and Green phase, respectively). The coordinates of $x, y$ axes are repreneted by the number of lattices. The time step (t.s. in short) is (a) 100 t.s., (b) 500 t.s., (c) 1 000 t.s., (d)10 000 t.s., (e) 50 000 t.s., (f) 100 000 t.s.

Figure 3

The temporal evolution of adosrbed coumpound at the interface R-B and R-G in Case 2 during the phase segragation. The inset shows the coresponding bulk phase flow field by a RGB image as the color defination of the fluids (The black, dark gray, and light gray, represent Blue phase, Red phase, and Green phase, respectively). The coordinates of $x, y$ axes are repreneted by the number of lattices. The time step (t.s. in short) is (a) 100 t.s., (b) 500 t.s., (c) 1 000 t.s., (d)10 000 t.s., (e) 50 000 t.s., (f) 100 000 t.s.

Figure 4

The temporal evolution of the two-phase soluble compound in Case 3 during phase segragation, where the compound is more soluble in Red phase than that in Green phase. The inset shows the coresponding bulk phase flow field by a RGB image as the color defination of the fluids (The black, dark gray, and light gray represent Blue phase, Red phase, and Green phase, respectively). The coordinates of $x, y$ axes are repreneted by the number of lattices. The time step (t.s. in short) is (a) 100 t.s., (b) 500 t.s., (c) 1 000 t.s., (d)10 000 t.s., (e) 50 000 t.s., (f) 100 000 t.s.

Figure 5

The temporal evolution of the three-phase soluteble compound in Case 1 during phase segragation, where the equilibrium concentration is $C_{s,R}>C_{s,G}>C_{s,B}$. The inset shows the coresponding bulk phase flow field by a RGB image as the color defination of the fluids (The black, dark gray, and light gray represent Blue phase, Red phase, and Green phase, respectively). The coordinates of $x, y$ axes are repreneted by the number of lattices. The time step (t.s. in short) is (a) 100 t.s., (b) 500 t.s., (c) 1 000 t.s., (d)10 000 t.s., (e) 50 000 t.s., (f) 100 000 t.s.

Figure 6

The effect of ${\lambda }_{s,BG},{\lambda }_{s,RB}$ on the Henry coefficient of the three-phase soluble compound condition.

Figure 7

The evolution of solute mass transfer across the interface compared with the penetrate theory from Eq. (48); (a) ${\lambda }_{s,RG}=1$; (b) ${\lambda }_{s,RG}=0$. The dotted lines show the numerical results at each time step (t.s.) and the solid lines show the corresponding theoretically results.

Figure 8

The pseudocolor images representing the mass transfer of the two-phase soluble compound in the equilibrium shapes of liquid lens. The solute is initialized in Red phase as the initial condition. The black solid line shows the contour of the concentration at [0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1] respectively and the upper inset in the first picture is the stabilized condition of bulk phase (The black, dark gray, and light gray represent Blue phase, Red phase, and Green phase, respectively). The coordinates of $x, y$ axes are repreneted by the number of lattices. The time step (t.s. in short) is (a) 100 t.s., (b) 500 t.s., (c) 1 000 t.s., (d)10 000 t.s., (e) 50 000 t.s., (f) 100 000 t.s.

Figure 9

The pseudocolor images representing the mass transfer of the three-phase soluble compound in the equilibrium shapes of the lens. The solute is initialized in Red phase. The black solid line shows the contour of the concentration at [0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1], respectively, and the upper inset in the first picture is the stabilized condition of bulk phase (The black, dark gray, and light gray represent Blue phase, Red phase, and Green phase, respectively). The coordinates of $x, y$ axes are repreneted by the number of lattices. The time step (t.s. in short) is (a) 100 t.s., (b) 500 t.s., (c) 1 000 t.s., (d)10 000 t.s., (e) 50 000 t.s., (f) 100 000 t.s.

Figure 10

Schematic of the initial condition of the dispersed phase Red (dark gray) and Green (light gray), the reactive dilute species A and B.

Figure 11

Pseudocolor images of concentration profile of each solute compound at instant times during the mass transfer and reaction in the Janus droplet flowing through microchannels. An average velocity $U_{\mathrm{aver}}=0.003\mathrm{l}.\mathrm{u}./\mathrm{t}.\mathrm{s}$. is toward the right side of $x$ direction. The black solid line shows the contour of the concentration at [0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1], respectively. The time step (kt.s. in short) is shown in the gragh. The reaction rate parameter (a) $k_r=0$, (b) $k_r={10}^{-2}$, (c) $k_r={10}^{-3}$.