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Calculating The Bending Moment on a Mixer Shaft

Q: We have one agitator with the following configuration: A single shaft with two mounted impellers -- one flat-blade impeller (inclined at 45 degrees to shaft) and 06 blades are present in an impeller assembly, and one gas-foil impeller (Parabolic contour), 06 blades are present in an impeller assembly. The power consumption is split in 70:30 with 70% for gas-foil blade. The agitator works in a gassed condition -- air bubbles are dispersed in liquid/solid stream, and for decontamination, the same agitator is run in 5% caustic solution. The question -- How do I calculate the force, which can cause a bending moment in the shaft, at gas-foil impeller and flat-blade impeller? And how can the resulting bending moment be calculated?

A:

To calculate bending moment [M] on a mixer shaft, you need to compute the bending moment at each impeller location.  The impeller location or shaft length [L] to each impeller is measured from the bottom support bearing of the mixer drive. 
You describe a 70:30 power split between the two impellers, make that split based on the motor horsepower [Hp]. You want to account for the maximum load on each impeller.  In addition to the shaft length [L] and power split [Hp], you need to know the rotational speed [N] of the mixer and the impeller diameter [D] of each impeller, then apply the following formula for each impeller and sum the results:
 
M (in-lbs) = 19000 Hp (fraction motor horsepower) × hydraulic factor × L (inches) / N (rpm) / D (inches)
 
The hydraulic factor should be 3.0 for the gas dispersion impeller, because of fluctuating loads caused by the gas.  The upper impeller should also have a hydraulic factor of 2.0 because of the gas or 3.0 if it must operate at the liquid level during filling or emptying of the tank.

The bending moment calculation will not complete a shaft analysis.  The shaft torque must be calculated and then tensile and shear stresses can be estimated, which will reflect the strength of the mixer shaft.  A natural frequency calculation is also appropriate to avoid undamped vibrations.

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What is the best way to approach gas-gas mixing?
I want to mix two low viscous gas streams in laminar regime. I am planning to mix them by passing through a pipe filled with spherical pebbles or some structured packing. I want to calculate the length required for the mixing expected. My required process stream details are as follows:

Q = 1Nm³/hr
He= 0.99
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Temp= 25 degrees C

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We would like to have a general idea of pumping number for our reactors. Can you help us with blend-time calculations?
To characterize one of the reactors at plant scale, we use the discoloration method. The parameters for the mixing time are calculated based according to the following formula:

t_mix= K/(aN(D/T)^b (T/Z)^0.5 )
N = impeller speed
D = diameter stirrer
T = diameter tank
Z = liquid height
The divisor is known as Kmix

Because we don't really know the uniformity (U) reached with this method we don't replace K with

K = -ln (1-U)

But get the best fit for K, a and b by means of the least square method. It is known that the pumping number can be determined by

N_Q= (Vk_mix)/(ND^3 )

We would like to have a general idea of pumping number for our reactors. So if we would like to deviate the pumping number from the above method would it be correct that for T=Z (so at a fixed volume based on the reactor geometry) we use the following formula.

N_Q= (VaN(D/T)^b)/(ND^3 )

Working in a turbulent regime, this result in a constant pumping number related to the tank geometry. Would the above approach be correct to compare pumping and mixing time capabilities of different reactor set ups? I think that not working at a fixed uniformity results in a gap in the above approach.

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