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Finding the Hydraulic Load

Q: How do I find the hydraulic load to calculate the shaft deflection of a vessel agitator? A mixing agitator incorporates a six-blade turbine impeller (90 degree - flat) that spins at 650 rpm to mix 200L of a water-like fluid in a 260L fermenter. The tank has a 31" ID. What is the calculation used to determine the hydraulic load at this high rate of mixing? The impeller pulls a vortex to the bottom of the vessel, so it is not fully submerged in the fluid at all times. The next step would be to then calculate the shaft deflection that is experienced with this hydraulic load.

A: With the additional information and drawings provided, the lateral hydraulic force at the impeller is estimated to be 11 lbs.  That load is 3.0 times the load estimated for a fully submerged impeller.  The 1-hp motor is loaded to about 3/4 horsepower, without baffles in the tank.  The lack of baffles in the tank reduces the power input for the described conditions, but also greatly reduces the effectiveness of the agitator for both bulk mixing and gas dispersion.  The shaft deflection is estimated to be 0.042 inches.  With assumptions of a bearing in the seal housing and an estimated impeller weight of 13 lbs, the shaft natural frequency will be near 730 rpm, which could cause severe vibration with the mixer operating at 650 rpm.  Without details of the bearing supports and impeller weight, the critical speed is only an estimate.
 
For a data sheet the will provide more information about the mixer design and performance, click here.

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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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