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Are power requirements for multiple impellers additive?

Q: With multiple impellers on a single shaft, are the effects on total power additive? For example, if two impellers are of the same type, would the power be determined as P = Np*rho*N^3*(D1^5+D2^5)? Also, what is a "rule of thumb" for empirically calculated power requirements vs. actual power input that a VFD might show?

A: In most cases, the power requirements for multiple impellers are additive, unless they are located less than half an impeller diameter apart, in which case, the power requirements are less than additive.  Your formula for multiple impellers is correct as long as the impellers are geometrically similar (same type, same width-to-diameter ratio, etc.), so that the power numbers are equal.  The best calculation is done by considering each impeller separately and then adding the results together.
 
The "rule of thumb" for calculated and measured power is that the results are only as good as the individual values.  Calculated power must be based on accurate fluid density (within 1%), viscosity (within 10%) effects if any, power numbers (within 5%), impeller diameters (within 1%), and rotational speeds (within 1%), for the results to be within about 10%.  Measured power with a VFD depends on the electronics.  Power measurement must be based on electronic power (including motor efficiency), not just amperage, and definitely not clamp-on amperage readings.  Good calculations and a good VFD should give equal results within 10% to 15%.  No inherent reason for the results to be different, other than inaccuracies.
 
If your differences are large, look for errors.

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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 )

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

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