Mathematical Models Based on Energy Balance

The rate of energy input to the bath E = lie), which is actually contributes to the mixing process, is the sum of the rate of energy input due to buoyancy effect of rising bubble ( b = 11 b) and a fraction of the rate of total kinetic energy associated with the gas at the nozzle exit ( k = llek). Thus, 

Adopting the approach used by Bhavaraju et al. [69] in modeling bubble columns, Sano and Mori [70] andMurty et al. [71] assumed the rate of energy dissipation to be equal to the sum of the rate of energy dissipation associated with liquid circulation and that due to bubble sUp. 

The net energy dissipation due to bubble break up and from bubble coalescence is generally assumed to be negligible [72]. The amount of viscous drag at the walls depends on liquid viscosity. For low viscosity liquids such as molten metals and water, the energy dissipation due to viscous drag at the walls can be neglected. Thus at steady state  

The rate of energy loss due to liquid circulation in the bath (Ec) is calculated as the rate of energy associated with rising liquid in the plume ( lip) minus the rate of kinetic energy associated with liquid flowing downward outside of the plume ( Lop)- The rate of energy associated with the rising liquid in the plume can be 

This implies that (p = total volume of gas in the plume volume/volume of the plume. Finally, 

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