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We perform direct numerical simulations of a gas bubble dissolving in a surrounding liquid. The bubble volume is reduced due to dissolution of the gas, with the numerical implementation of an immersed boundary method, coupling the gas diffusion and the Navier–Stokes equations. The methods are validated against planar and spherical geometries’ analytical moving boundary problems, including the classic Epstein–Plesset problem. Considering a bubble rising in a quiescent liquid, we show that the mass transfer coefficient $$k_L$$ can be described by the classic Levich formula $$k_L = (2/\sqrt {{\rm \pi} })\sqrt {\mathscr {D}_l\,U(t)/d(t)}$$ , with $d(t)$ and $U(t)$ the time-varying bubble size and rise velocity, and $$\mathscr {D}_l$$ the gas diffusivity in the liquid. Next, we investigate the dissolution and gas transfer of a bubble in homogeneous and isotropic turbulence flow, extending Farsoiya et al. ( J. Fluid Mech. , vol. 920, 2021, A34). We show that with a bubble size initially within the turbulent inertial subrange, the mass transfer coefficient in turbulence $$k_L$$ is controlled by the smallest scales of the flow, the Kolmogorov $$\eta$$ and Batchelor $$\eta _B$$ microscales, and is independent of the bubble size. This leads to the non-dimensional transfer rate $${Sh}=k_L L^\star /\mathscr {D}_l$$ scaling as $${Sh}/{Sc}^{1/2} \propto {Re}^{3/4}$$ , where $${Re}$$ is the macroscale Reynolds number $${Re} = u_{rms}L^\star /\nu _l$$ , with $$u_{rms}$$ the velocity fluctuations, $L^*$ the integral length scale, $$\nu _l$$ the liquid viscosity, and $${Sc}=\nu _l/\mathscr {D}_l$$ the Schmidt number. This scaling can be expressed in terms of the turbulence dissipation rate $$\epsilon$$ as $${k_L}\propto {Sc}^{-1/2} (\epsilon \nu _l)^{1/4}$$ , in agreement with the model proposed by Lamont & Scott ( AIChE J. , vol. 16, issue 4, 1970, pp. 513–519) and corresponding to the high $Re$ regime from Theofanous et al. ( Intl J. Heat Mass Transfer , vol. 19, issue 6, 1976, pp. 613–624).
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Bubble-mediated transfer of dilute gas in turbulence
Bubble-mediated gas exchange in turbulent flow is critical in bubble column chemical reactors as well as for ocean–atmosphere gas exchange related to air entrained by breaking waves. Understanding the transfer rate from a single bubble in turbulence at large Péclet numbers (defined as the ratio between the rate of advection and diffusion of gas) is important as it can be used for improving models on a larger scale. We characterize the mass transfer of dilute gases from a single bubble in a homogeneous isotropic turbulent flow in the limit of negligible bubble volume variations. We show that the mass transfer occurs within a thin diffusive boundary layer at the bubble–liquid interface, whose thickness decreases with an increase in turbulent Péclet number, $$\widetilde {{Pe}}$$ . We propose a suitable time scale $$\theta$$ for Higbie ( Trans. AIChE , vol. 31, 1935, pp. 365–389) penetration theory, $$\theta = d_0/\tilde {u}$$ , based on $$d_0$$ the bubble diameter and $$\tilde {u}$$ a characteristic turbulent velocity, here $$\tilde {u}=\sqrt {3}\,u_{{rms}}$$ , where $$u_{{rms}}$$ is the large-scale turbulence fluctuations. This leads to a non-dimensional transfer rate $${Sh} = 2(3)^{1/4}\sqrt {\widetilde {{Pe}}/{\rm \pi} }$$ from the bubble in the isotropic turbulent flow. The theoretical prediction is verified by direct numerical simulations of mass transfer of dilute gas from a bubble in homogeneous and isotropic turbulence, and very good agreement is observed as long as the thin boundary layer is properly resolved.
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- Award ID(s):
- 1844932
- PAR ID:
- 10308645
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 920
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript
- Journal Article:
- https://doi.org/10.1017/jfm.2021.447
