Dynamics of Gas Bubbles in a Multi‐Component Liquid

Abstract

If a bubble of radius R 0 was initially placed into a supersaturated (at the given pressure and temperature) mixture, then, due to the concentration difference Dr ¼ r Ly À r Lw between the gas dissolved in the liquid far away from the bubble and the appropriate equilibrium value at the bubble surface, there arises a directional diffusion flux of the dissolved substance toward the surface. At the interface, the transition of substance from the liquid to the gaseous state takes place. The result is the increase in the bubble volume. The growth of the bubble, in its turn, results in the increase of its lift velocity, as well as the increase of convective diffusion flux. The statement of the problem and the basic dynamic equations for a bubble in a solution were described in Section 6.8.Consider a multicomponent solution. The distribution of concentrations of the dissolved substances is described by the equations of convective diffusion qr iL qt þ u r qr iL qr þ u y 1 r qr iL qy ¼ D iL 1 r 2 q qr r 2 qr iL qr ; ð22:1Þwhere u r and u y are the radial and tangential velocity components of the flow that goes around the bubble, r and y are spherical coordinates, r iL are the mass concentrations of dissolved components, D iL is the coefficient of binary diffusion. The mass balance equation for the gas in the bubble is d dtwhere r iG is the mass concentration of i-th gas component in the bubble. The dynamic balance equation for the bubble, known as Rayleigh's equation, looks likeHere r L is the liquid phase density; R -the radius of the bubble; n L -kinematic viscosity of the liquid; S -the coefficient of surface tension, p G -gas pressure inside the bubble; p y -gas pressure far away from the bubble.