The history effect on bubble growth and dissolution. Part 2. Experiments and simulations of a spherical bubble attached to a horizontal flat plate
Summary (2 min read)
1. Introduction
- Mass transfer processes involving bubbles have gained a renewed interest over the last few years due to their relevance in modern microfluidic applications connected to topics such as carbon sequestration (Sun & Cubaud 2011; Volk et al. 2015).
- It has been shown that any recent history of growth and/or dissolution (triggered by past changes in ambient pressure) experienced by a particular bubble may leave, at least for some time, a non-negligible imprint on the current state of the concentration profile surrounding such a bubble.
- It is worth mentioning that history effects are common to problems in which diffusion plays a central role, such as the viscous drag around a body or, closer to the present mass transfer problem, the heat transfer around a spheroid (Michaelides 2003).
- With the purpose of maintaining the standard nomenclature, R0 will be used throughout this paper.
- Section 3 presents the general mathematical formulation of the problem and sheds light on the importance of the different physical effects involved in the experiments.
2. Experimental characterisation of the history effect
- The authors have carried out experiments to support their theoretical and numerical analyses by subjecting single bubbles to well-controlled, step-like pressure jumps that super- or undersaturate the liquid alternatively.
- It becomes apparent through the differences in the responses to successive identical pressure–time histories.
- Once the measurement tank is filled with the carbonated water, the following experimental procedure is followed: (i) The pressure is lowered below the saturation value until a bubble nucleates at the pit and grows up to the desired size, Ri. (ii) The tank pressure is set again to the saturation value.
- In all cases the growth rate during the second stage lies above that of the first one, although both curves eventually converge at longer times, when the memory of the previous dissolution stage damps out.
- It is interesting to compare this behaviour with that found when the bubble is forced to first dissolve and then to grow .
3. Numerical analysis: problem formulation
- Which involves a non-stationary boundary and that must be coupled with the equations of motion for the liquid assuming axisymmetry around the vertical axis.the authors.
- The initial concentration of dissolved gas is assumed to be uniform throughout the liquid and equal to C∞, equal to the gas concentration in the far field.
- This knowledge will now be used in the next section when implementing the equations into a numerical model.
- The vorticity field is then allowed to independently evolve through the vorticity transport equation, advancing with time step 1τv.
5. Simulation results and discussion
- The simulation predictions for the bubble size history are compared with the experiments in figure 11.
- The history effect on the growth rate is evident: the concentration contours in (b) are noticeably closer together than those in (a).
- The results of their simulations can also be used to validate the hypothesis made by Enríquez et al. (2014) about the existence of a ‘dead zone’ near the contact point where mass transfer is almost zero.
- In figure 14 the authors plot the local mass flux distribution along the bubble surface for different time instants during the first growth cycle of experiment 2.
- In dissolution, a low velocity recirculation region surrounding the bubble is observed.
6. Conclusions
- The authors have experimentally and numerically explored the influence of the past history of the ambient pressure experienced by a bubble on its instantaneous rate of mass transfer – the so-called history effect.
- This effect is caused by a history-induced preexisting concentration boundary layer of dissolved gas that surrounds the bubble at the beginning of a given growth or dissolution stage.
- The authors would naively expect that such a situation would lead to a monotonic dissolution, since the liquid is undersaturated during the whole process.
- By performing order of magnitude analyses, the authors show that their experiments belong to a regime dominated by mass and viscous diffusion.
- Thus, the momentum equation can be decoupled from the mass transfer problem.
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Frequently Asked Questions (10)
Q2. Why have bubbles gained a renewed interest in microfluidic applications?
Mass transfer processes involving bubbles have gained a renewed interest over the last few years due to their relevance in modern microfluidic applications connected to topics such as carbon sequestration (Sun & Cubaud 2011; Volk et al. 2015).
Q3. What is the molar flow rate of a spherical gas bubble?
The total gas pressure inside the bubble, Pg, considering liquid–gas surface tension γlg, but neglecting inertial and viscous effects inside the gas phase, is given byPg = P∞ + 2γlg/R. (3.4)The mass transfer problem is closed with Fick’s first law, which sets the molar flow rate of gas across the bubble surface S to beṅ=D ∫S ∇C · n̂ dS, (3.5)where dS is an infinitesimal area element of the bubble surface, and n̂ is the outwardpointing unit normal from the bubble surface.
Q4. What is the effect of advection on the bubble?
It can be concluded that, although the instantaneous rate of mass transfer may only be slightly affected by advection, its effect accumulates over time and becomes important to describe the evolution of the bubble when subjected to successive expansion–compression cycles.
Q5. How can the authors bypass the limitation of the model?
The authors may bypass this limitation by modelling the effect of stratification essentially through just an effective increase (decrease) of mass transfer towards (from) the bubble.
Q6. how much gas does the assumption of a perfectly spherical bubble yield?
the assumption of perfectly spherical bubble at all time yields a relative error of less than 3 % as compared to the actual gas volume of the spherical cap and the pit.
Q7. How is the vorticity field allowed to evolve?
The vorticity field is then allowed to independently evolve through the vorticity transport equation, advancing with time step 1τv.
Q8. What is the effect of convection on the concentration boundary layer near the bubble?
despite the changes that convection induces in the velocity field, its effect on the concentration boundary layer near the bubble is minute, as is revealed by the comparison between figures 15(c) and 16(c).
Q9. What is the effect that contributes to the diffusion-driven dynamics of a bubble?
Another effect that contributes to the diffusion-driven dynamics of a bubble is the so-called history effect, discussed in Part 1 and more recently in Chu & Prosperetti (2016b).
Q10. What is the a priori unknown corresponding apparent velocity field?
Let us define the a priori unknown corresponding (dimensionless) apparent velocity field as urel(η, ξ, τ )= urel,η êη + urel,ξ êξ .