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How do you calculate the power number of an impeller?

Q: How do you go about calculating the power number (Np) of an impeller?

A:

The calculation is easy. Getting good data to use in the calculation can be hard. The power number is a dimensionless group, so you need to be sure that the units cancel.
 
For SI Metric: Np = P / (rho N^3 D^5)
     where P is power in Watts, rho is liquid density in kg/m^3, N is rotational speed in rev/s, D is impeller diameter in meters.
     simply power divided by density, rotational speed cubed, and diameter to the fifth power.
 
For common engineering units, you have a large conversion factor: Np = 1.524x10^13 P / (sp.gr. N^3 D^5)
     where power is in hp, sp.gr. is specific gravity with respect to water, N is rotational speed in rpm, and D is impeller diameter in inches. 
Be sure not to substitute the wrong units.
 
The real problem is getting a valid power reading.  In an industrial application, it is not the motor power (motors are never fully loaded, unless by poor design), it is the actual power (in watts), not voltage times amperage applied by the motor.  Rotational speed is the actual rotational speed of the mixer and impeller diameter needs to be measured accurately a 1% error in diameter results in about a 5% error in the power number.  In a laboratory measurement, instead of power, usually torque is measured and multiplied by rotational speed (in the correct units) to get power.
 
Probably more than you thought you needed to know, but for power number to be useful it needs to be accurate.  It then will tell you how much power is needed to rotate a specific impeller at a given speed in a certain liquid.  Power number will not tell you whether the mixing intensity is adequate to mix your product.

Editor's Note: Our expert, Dave Dickey of MixTech Inc., offered additional information to his answer. It appears below:

I may not have understood your question.  Although, I did give you the right answer the first time.  An impeller power number is not a calculated quantity based solely on geometry factors, such as blade width, blade shape, blade angle, number of blades, etc.  Impeller power numbers are essentially correlated experimental results.  Without measurements, only estimates can be made based on experiments with similar impeller types.

Because many measurements have been made for different impeller types, reasonable estimates can be made for other impellers sometimes.  General power number information is scattered through the literature.  Some of the data is better than others.  Measurement techniques are often biased because of friction, approximation, or other inaccuracies.

Sometimes with detailed geometric information, estimates can be made.  The more complicated the impeller shape, camber, curvature, etc., the more difficult and less accurate the estimate.

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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 )

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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