Numerical simulation of dissolved air flotation using a lattice Boltzmann method

In this paper, the behavior of a bubble and droplet rising in a system, namely, a dissolved air flotation system, is investigated under different conditions. A lattice Boltzmann model which is based on the Cahn-Hilliard equations for ternary flows is implemented. This model can handle high density and viscosity ratios, remove parasitic currents, and capture partial and total spreading conditions. Two classical problems, such as spreading of a liquid lens and the Rayleigh-Taylor instability are used to determine the accuracy of the model. As a practical application, three-component flow in a tank is studied and the dynamics of bubble and droplet under different conditions is investigated. We then concentrate on the dimensionless average velocity and locations of bubble and droplet at different density ratios, viscosity ratios, and diameter ratios. Also, total spreading and partial spreading conditions are compared. The numerical results are justifiable physically and show the ability of this model to simulate three-component flows.

Figure 1

A liquid lens at the equilibrium state for different surface tension coefficients. (a) Schematic of liquid lens, (b) initial condition of the liquid lens for each set of surface tension coefficients. Surface tensions coefficients $\left({\sigma }_{12},{\sigma }_{13},{\sigma }_{23}\right)$ are (c) $\left({10}^{-4},{10}^{-4},{10}^{-4}\right)$, (d) $\left(6\times {10}^{-5},6\times {10}^{-5},{10}^{-4}\right)$, (e) $\left(8\times {10}^{-5},1.4\times {10}^{-4},{10}^{-4}\right)$, (f) $\left({10}^{-4},\frac{4}{3}\times {10}^{-4},{10}^{-4}\right)$, (g) $\left({10}^{-4},3\times {10}^{-4},{10}^{-4}\right)$, (h) $\left(3\times {10}^{-4},{10}^{-4},{10}^{-4}\right)$, (i) $\left({10}^{-4},{10}^{-4},3\times {10}^{-4}\right)$.

Figure 2

$L^2$ norm error versus number of grids in the $y$ direction.

Figure 3

Phase-field contour at different times for the Rayleigh-Taylor instability.

Figure 4

Time evolution of (a) bubble front position and (b) spike tip position.

Figure 5

Grid independency.

Figure 6

Time evolution of purifying the red component from the green component using the DAF method.

Figure 7

Time evolution of the bubble rising in the total spreading state.

Figure 8

Vertical component of the velocity field at $t^{*}=4.72$.

Figure 9

Time evolution of the bubble rising in the total spreading state.

Figure 10

Time evolution of the bubble rising with different bubble sizes.

Figure 11

Time evolution of the rising bubble in the partial spreading state.

Figure 12

Evolution of (a) the green droplet location and (b) the blue bubble location at different density ratios in the partial spreading state.

Figure 13

Evolution of the green droplet dimensionless velocity at different density ratios in the partial spreading state.

Figure 14

Evolution of the green droplet location at different density ratios in the total spreading state.

Figure 15

Evolution of the green droplet dimensionless velocity at different density ratios in the total spreading state.

Figure 16

Evolution of (a) the green droplet location and (b) the blue bubble location at different diameter ratios.

Figure 17

Evolution of (a) the green droplet and (b) the blue bubble dimensionless velocity at different diameter ratios.

Figure 18

Evolution of (a) the green droplet and (b) the blue bubble location at different viscosity ratios.

Figure 19

Evolution of (a) the green droplet and (b) the blue bubble dimensionless velocity at different viscosity ratios.

Figure 20

Evolution of (a) the green droplet and (b) the blue bubble location at different surface tension coefficients.