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

David Mikaelian, Benoît Haut* and
Benoit Scheid

* Corresponding author: Email:
bhaut@ulb.ac.be

Laboratory Transfers,
Interfaces and Processes (TIPs), CP 165/67,

Université Libre de
Bruxelles, Av. F. D. Roosevelt 50, 1050 Brussels, Belgium

1.     Introduction

Gas-liquid
two-phase flow patterns in microfluidic devices were investigated in circular
and square microchannels by Kawahara et al. (2002) and Cubaud and Ho (2004).
The mass transfer between gas bubbles in a microchannel and the surrounding
liquid was studied in capillaries and in square microchannels by Kashid et al.
(2011), Sun and Cubaud (2011) and Cubaud et al. (2012).

In
this work, the dynamics of spherical bubbles in square and circular
microchannels and the mass transfer between these bubbles and the surrounding
liquid are investigated in the bubbly flow regime. The bubbly flow corresponds
to discrete spherical bubbles, with diameters smaller than the channel
hydraulic diameter, moving in a continuous liquid phase. Two boundary
conditions are considered on the bubble-liquid interface: a stress-free
condition (referred hereafter as a stress-free interface) or a no slip
condition (referred hereafter as a rigid interface).

2.     Problem
statement

Gas-liquid
bubbly flows in a square and in a circular microchannel of length L_c and hydraulic diameter d_h are considered. For each type of microchannel, a
model segment of length L containing, at its center, a
single spherical bubble of diameter d is studied (see Fig. 1). Taking
benefit of the symmetries, only a quarter of the square microchannel is
analyzed and the analysis of the flow and the mass transport in the circular
microchannel is reduced to a two-dimensional axisymmetric problem.

Figure 1: (a) Model segment
of the square microchannel and notations (b) Model segment of the circular microchannel
and notations.

In a
laboratory reference frame (x
̃,y ̃,z ̃) (see
Fig. 1), the considered bubble moves along the microchannel at a
velocity V_B in
the positive x ̃
direction in a liquid moving in the same direction.  The volumetric flow rates
of the liquid and the gas are written Q_L and Q_G, respectively, and the
total superficial velocity is defined as, J_A = (Q_L + Q_G)/A_Σ with A_Σ the area of the
cross-section of the microchannel. In a reference frame (x,y,z) (see
Fig. 1) attached to the center of the bubble, the walls of the microchannel
move at a velocity V_B in
the positive x direction
and the liquid enters the considered segment of the microchannel at an average
velocity V_B - J_A  in
the positive x direction.

As the density
and the viscosity of the liquid are much higher than those of the gas, a
one-sided approach is used where only the liquid flow is considered. The liquid
flow and the mass transfer around the bubble inside the square or the circular
microchannel are analyzed by solving, in (x,y,z),
the continuity, the Navier-Stokes and the mass transport equations in
stationary conditions.

The boundary
conditions for the liquid flow and the mass transport are presented in Table 1.
Periodic boundary conditions are used between the IN and the OUT planes in
order to reproduce a chain of bubbles that can be generated in real
microchannels.

Table 1: Boundary conditions
for the liquid flow and the mass transport.

Three
non-dimensional control parameters are used: d/d_hRe_JA=(ρJ_Ad_h)/μ and Sc=μ/ρD, with ρ the density of the
liquid [kg/m³], μ
the dynamic viscosity of the liquid [Pa s] and D
the diffusion coefficient of the dissolved gas in the liquid [m²/s].
The ranges covered by these three non-dimensional control parameters are 0.15 ≤ d/w ≤ 0.75,
5.74 ≤ Re_JA ≤ 28.7 and 150
≤ Sc ≤ 550. These ranges ensure covering realistic values of
the parameters encountered for usual gas and liquid combinations. Two non-dimensional
variables are defined for the post-processing: V_B/J_A and
the Sherwood number Sh=(k_ld)/D, with k_l the mass transfer
coefficient across the bubble interface.

3.    
Numerical procedure

Based on the
computational domain of Fig. 1, a grid is generated for each value of d/d_h using
the software Gambit 2.4. Refined zones for the mesh are used around the bubble
and next to the walls to ensure that the diffusion boundary layers are
correctly captured. The grid independence is checked for each simulation.                                         

For each
generated grid, the system of equations defining the problem is solved in
stationary conditions in the liquid phase with the boundary conditions of Table
1 for both cases of stress-free and rigid interfaces. The three-dimensional or the
two-dimensional axisymmetric version with a double precision of the solver
Ansys Fluent 14.5 are used in the case of the square microchannel or the
circular microchannel, respectively.

A steady
bubble velocity V_B
implies that the x-component of the force exerted by the liquid on the bubble
surface (F_x) should
be equal to zero. It provides a relation between J_A and V_B. Here, J_A is imposed (through Re_JA) and
V_B is
then evaluated such that F_x vanishes.
From the results of the numerical simulation of the mass transport around the
bubble, the mass transfer coefficient k_l is
calculated by dividing the mass flow rate across the bubble surface by the
bubble surface area and by the driving concentration difference C_sat - C_bulk.
The length of the computational domain L
is adjusted to a value, depending on d,
J_A and D, such that, when L is increased by 200
μm, the relative change in k_l is
lower than 1%.

4.     Results

Based
on the numerical results, the following correlations are proposed:

Table 2: Correlations with .

The
liquid flow around the bubble is analyzed using a dimensionless velocity
defined as the bubble
velocity V_B minus the liquid velocity in the
laboratory reference frame (x
̃,y ̃,z ̃)
divided by J_A. As it can be seen in Fig. 2, a
recirculation can be present between two successive bubbles depending on
d/d_h and the boundary condition at
the bubble-liquid interface.

Figure
2: Pathlines
(colored by the dimensionless velocity magnitude) around bubbles of various
d/d_h with a stress-free or a rigid interface in the symmetry
plane z = 0 of a square microchannel (a) and contours of the
dimensionless velocity magnitude in the plane x = 0 (b)
for Re_JA = 28.7.

The
mass transport around the bubbles is analyzed using a dimensionless
concentration C^* defined as . The comparison of Figs. 2 and 3
highlights the
influence of the liquid flow on the mass transport around the bubbles.

Figure
3: C^* field around bubbles of various d/d_h with a stress-free or a rigid interface in the symmetry
plane z = 0 of a square microchannel for Re_JA = 28.7 and Sc = 551.3.

The
mass transfer rate between a bubble and the surrounding liquid is characterized
locally on the bubble surface by defining a local Sherwood number as

,

where
  is the normal vector to the
bubble surface pointing inward the liquid and C
is the concentration of the dissolved gas in the liquid phase.

Figure
4: Sh_loc on the bubble surface for bubbles of various d/d_h with a stress-free or a rigid interface, Re_JA = 28.7 and Sc = 551.3, in the case of a square
microchannel.

As
it can be seen in Fig. 4, the highest mass transfer rate is observed a bit in
front of the side of the bubble surface. Furthermore, when d/d_h increases, an azimuthal
asymmetry of the local mass transfer rate on the bubble surface is appearing.

The
velocity, C^* and Sh_loc fields around bubbles with a
stress-free or a rigid interface in a circular microchannel appears to be quite
similar to the one obtained in a square microchannel. The main difference
between the two types of microchannels is the presence, in the square microchannel,
of an azimuthal asymmetry of the velocity field in the vicinity of the bubble
surface (see for example Fig. 2 d/d_h = 0.75) which leads to an azimuthal asymmetry of the mass
transfer rate on the bubble surface (see Fig. 4).

References

T.
Cubaud and C.M. Ho, Transport of bubbles in square microchannels, Physics of
fluids, 16, 4575 (2004).

T.
Cubaud, M. Sauzade and R. Sun, CO₂ dissolution in water using long
serpentine microchannels, Biomicrofluidics, 6, 022002 (2012).

M.
N. Kashid, A. Renken and L. Kiwi-Minsker, Gas-liquid and liquid-liquid mass
transfer in microstructured reactors, Chemical Engineering Science, 66(17),
3876-3897 (2011).

A.
Kawahara, P. Y. Chung and M. Kawaji, Investigation of two-phase flow pattern,
void fraction and pressure drop in a microchannel, International Journal of
Multiphase Flow, 28(9), 1411-1435 (2002).

R.
Sun and T. Cubaud, Dissolution of carbon dioxide bubbles and microfluidic
multiphase flows, Lab on a Chip, 11(17), 2924-2928 (2011).