Bubble Dynamics in a Multi-Componenet Solution

Consider the dynamics of a bubble in a multicomponent solution under the condition that its lift can be neglected. At % = 0, the system of equations (22.1)-(22.6) can be solved by the same approximate method that used in Section 22.2 for a binary solution. For each component dissolved in liquid, select its own thin diffusion layer with the thickness 5 R, around the bubble. In each layer, we seek for the distribution of concentration in the form 

Plug ug = 0 into (22.1) and integrate this equation over r from R up to R + (5 . We then obtain Eq. (22.27) for the layer thickness Si, which is accurate to the order ofdi/R  

In new variables, the system of equations describing (in the currently considered approximation) the diffusion growth of a bubble in a multicomponent solution will become  

In order to solve this system of equations, it is necessary to set the initial density values PiQo of gas components in the bubble. These values cannot be known in advance. The calculations above show that for any initial composition of gas in the bubble, the componential composition of gas is established very quickly and practically does not change afterwards. The initial composition of the solution is conventionally specified by mass percentages Ili of components. Mole fractions are expressed in terms of mass percentages as follows  

Denote by yiQ the initial mole fractions of components in the bubble (Y iYio = 1). It is obvious that 

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