Bubbly flow and gas–liquid mass transfer in square and circular microchannels for stress-free and rigid interfaces: CFD analysis

Abstract

In this paper, the dynamics of bubbles and the mass transfer between bubbles and the surrounding liquid in square and circular microchannels are investigated, in the bubbly flow regime. For this purpose, a computational fluid dynamics analysis is used to carry out numerical simulations of the liquid flow and the mass transport around a spherical bubble in a square or a circular microchannel, for a stress-free or a rigid gas–liquid interface. The corresponding results are consolidated into correlations to calculate the bubble velocity and the interfacial rate of mass transfer as functions of the control parameters of the system. For each considered case, the flow structure, the concentration field around the bubble and the local interfacial rate of mass transfer are presented and shown to be intricately related.

Appendices

Appendix 1: Grid

1.1 Square microchannel

The geometry of the square microchannel is shown in Fig. 2. The principal volume is constituted of 5 subvolumes: Vol1, Vol2, Vol3, Vol4 and a zone Vol5 around the bubble. This principal volume is surrounded by smaller volumes forming a layer. These small volumes are referred to as $V_{\text {layer}}$ hereafter. The meshing parameters of the edges presented in Fig. 2 are provided in Table 4. When an edge mentioned in Table 4 is meshed, the same mesh is applied to the other edges presented in Fig. 2, which are parallel to it and of the same length. Numerical values of the meshing parameters presented in Table 4 are provided in Table 5. For the edge cr, the number of intervals is calculated such that the ratio of the size of the last interval to the size of the first interval does not exceed 5.

Table 4 Meshing parameters of the edges presented in Fig. 2

Table 5 Numerical values of the parameters used for creating and meshing the edges presented in Table 4

Once the edges are meshed, Vol1, Vol4 and $V_{\text {layer}}$ are meshed using Hex(ahedral) elements of type Map. It enables having the same mesh on the planes IN and OUT, which is necessary for using pseudo-periodic boundary conditions. Vol5 is meshed using Hex/Wedge elements and Vol2 and Vol3 are meshed using Tet(rahedral) elements. The Tet(rahedral) elements of Vol2 and Vol3 are combined in order to form polyhedra. It enables reducing the number of elements, improving the quality of the mesh and fastening the convergence.

$V_{\text {layer}}$ and Vol5 are used in order to accurately capture the diffusion boundary layers at the walls of the microchannel and at the bubble surface, as explained in Sect. 3.1.

1.2 Circular microchannel

The two-dimensional geometry used for the circular microchannel is shown in Fig. 3. The principal surface is divided in 5 subsurfaces: S1, S2, S3, S4 and a surface S5 around the bubble. Smaller surfaces forming a layer are present at the top of the principal surface. These small surfaces are referred to as $S_{\text {layer}}$ hereafter. $S_{\text {layer}}$ and S5 are used in order to accurately capture the diffusion boundary layers at the walls of the microchannel and at the bubble surface. The edges have the same names as for the square microchannel and are created and meshed in the same way as described in “Square microchannel” in Appendix 1. Once the edges are meshed, S1, S4, S5 and $S_{\text {layer}}$ are meshed using Quad(rilateral) elements and S2 and S3 using Tri(angular) elements.

Appendix 2: Numerical results

The values of the dimensionless control parameters and the postprocessed parameters are presented in Tables 6, 7, 8 and 9 for all the numerical simulations of the flow and the mass transport around the bubble with a stress-free or a rigid interface in the square or the circular microchannel.

Table 6 Values of the dimensionless control parameters and the postprocessed parameters for all the numerical simulations of the flow and the mass transport around the bubble with a stress-free interface in the square microchannel, with $\hbox {Re}=\hbox {Re}_{J_A}\frac{V_B}{J_A}\frac{d}{d_h}$

Table 7 Values of the dimensionless control parameters and the postprocessed parameters for all the numerical simulations of the flow and the mass transport around the bubble with a rigid interface in the square microchannel

Table 8 Values of the dimensionless control parameters and the postprocessed parameters for all the numerical simulations of the flow and the mass transport around the bubble with a stress-free interface in the circular microchannel

Table 9 Values of the dimensionless control parameters and the postprocessed parameters for all the numerical simulations of the flow and the mass transport around the bubble with a rigid interface in the circular microchannel