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Can you predict the vortex depth?

Q: I am looking for an equation that will predict the vortex depth in an unbaffled tank with a centrally mounted mixer. Process is a solid liquid dispersion where powder must be drawn from top of the tank and incorporated into the mix.

A:

The simple answer: There is no such equation.
 
The more complicated answer is plenty of small-scale data.  All of the data is for a liquid-level-to-tank-diameter-ratio of one (Z/T = 1). In the turbulent range, Reynolds numbers greater than 10,000, the ratio of vortex depth over impeller diameter (X/D) is a constant with respect to Froude number (X/D / Fr = constant).  The data includes results for 6-blade disk turbines, 4-blade straight-blade turbines, 4-blade pitched-blade turbines at impeller to tank diameter ratios (D/T) of D/T = 0.29, 0.34, 0.40, 0.57, and for 3-blade hydrofoil impellers at D/T = 0.37, 0.42, 0.47, 0.53.  Thus, liquid level, viscosity, impeller type, impeller to tank diameter ratio, and rotational speed all enter into the effect on vortex depth.

The best answer is a geometric similarity, small-scale test with equal Froude number scale-up.

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