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

Abstract

A model is proposed to describe the dissolution of a chain of spherical pure gas bubbles into a nonvolatile liquid, along square and circular microchannels. The gas–liquid interface is considered stress-free or rigid. This model enables predicting, in each considered case, the evolution along the microchannel of the bubble diameter, the bubble velocity, the separation distance between two successive bubbles, the liquid fraction, the pressure in both the liquid and the gas phases, the concentration of the dissolved gas in the liquid phase and the mass transfer coefficient between the bubble and the liquid phase. The influence on the gas dissolution of the interfacial boundary condition, the microchannel type, the operating conditions and the physicochemical properties of the liquid and gas is analyzed. Existing experimental data for a nearly square microchannel are convincingly reproduced using the model. The validity and the different applications of the model are also discussed.

Appendix: Supplementary materials for the model construction

1.1 Separation distance between two successive bubbles along the microchannel

For $\Delta x \rightarrow 0$, it can be considered that a bubble moves from x to $x+\Delta x$ at a velocity $V_B(x)$. The time for this bubble to travel the distance $\Delta x$ is equal to $\Delta x/V_B(x)$. During this time, the preceding bubble moves on a distance $\Delta x - \ell (x+\Delta x)+\ell (x)$ at a velocity $V_B(x- \ell (x))$:

$$\begin{aligned} \frac{\Delta x}{V_B(x)}=\frac{\Delta x- \ell (x+\Delta x)+\ell (x)}{V_B(x- \ell (x))}, \end{aligned}$$

(17)

or, after rearrangement,

$$\begin{aligned} \frac{\ell (x+\Delta x)-\ell (x)}{\Delta x}=\frac{V_B(x)-V_B(x- \ell (x))}{V_B(x)}. \end{aligned}$$

(18)

For $\Delta x \rightarrow 0$, Eq. 18 leads to:

$$\begin{aligned} \ell '(x)=\frac{V_B(x)-V_B(x-\ell (x))}{V_B(x)}, \end{aligned}$$

(19)

where the prime denotes the derivative with respect to x. As $\ell (x)\ll x$, except of course in the vicinity of the inlet, $V_B(x)-V_B(x-\ell (x))$ is approximated by the first order of its Taylor expansion, i.e., $V_B'(x) \ell (x)$. Equation 19 then simplifies into Eq. 11.

1.2 Concentration of the dissolved gas in the liquid phase along the microchannel

For a control segment of the microchannel of length $\Delta x$ (see Fig. 6), the following mass balance can be written for the transferred species between the gas phase and the liquid phase, using the ideal gas law:

$$\begin{aligned} Q_L C(x)+ Q_G(x)\frac{p_G(x)}{R T}=Q_L C(x+\Delta x) + Q_G(x+\Delta x)\frac{p_G(x+\Delta x)}{R T}. \end{aligned}$$

(20)

Fig. 6

Control segment of the microchannel of length $\Delta x$

Q _L is taken independent of x thanks to the fact that the gas density is much lower than the liquid density, such that the amount of gas dissolved in the liquid does not significantly affect its flow rate.

Using Eq. 9, Eq. 20 can be rearranged as

$$\begin{aligned} C(x)+\frac{1-\alpha _L(x)}{\alpha _L(x)}\frac{p_G(x)}{R T}=C(x+\Delta x)+\frac{1-\alpha _L(x+\Delta x)}{\alpha _L(x+\Delta x)}\frac{p_G(x+\Delta x)}{R T}. \end{aligned}$$

(21)

For $\Delta x \rightarrow 0$, Eq. 21 becomes

$$\begin{aligned} C'(x)=-\frac{1}{RT} \left( \frac{1-\alpha _L(x)}{\alpha _L(x)} p_G(x) \right) ', \end{aligned}$$

(22)

or equivalently recasts into Eq. 15.

1.3 Bubble diameter along the microchannel

If a bubble is at the coordinate x at time t and at the coordinate $x+\Delta x$ at time $t+\Delta t$ and if $\Delta t \rightarrow 0$ is considered, $\Delta t$ can be replaced by $\Delta x/V_B(x)$ and the following mass balance can be written for the transferred species between the gas phase and the liquid phase, using the ideal gas law:

$$\begin{aligned} \frac{p_G(x+\Delta x)}{R T}\frac{\pi d^3(x+\Delta x)}{6}-\frac{p_G(x)}{R T}\frac{\pi d^3(x)}{6} = -\frac{\Delta x}{V_B(x)} \pi d(x)^2 k_l(x) [C_{\rm sat}(x)-C(x)], \end{aligned}$$

(23)

For $\Delta x \rightarrow 0$, Eq. 23 becomes

$$\begin{aligned} \frac{1}{6RT}\left( p_G(x)\,d^3(x)\right) ' = -\frac{d^2(x)k_l(x)}{V_B(x)} [ C_{\rm sat}(x) - C(x)] , \end{aligned}$$

(24)

or equivalently recasts into Eq. 16.