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What are the power requirements for turbulent mixing?

Q: We have an agitator with the following designation:

Speed: 31/46 rpm
Absorbed power:1.5/5.4 kW
Impeller type: paddle (and blades angle is reversed in tip area)
Shaft dia: 85 mm
Shaft length: 3300 mm
Impeller dia: 1600 mm
Impeller num.: 2
Space in: 650 mm
Motor rating: 12/3.7 kW
Motor speed: 1470/980 rpm

Materials specification: density (kg/m³)/viscosity (cp)
1- liquid 1166/1
2- powder 600/-
3- liquid 794/0.32
4- liquid 971/0.36
5- liquid 1408/9.1

This agitator is installed on a drum with 2400 mm diameter and 3754 mm height and the agitator is top entry.

We want to change the process media with a liquid that its density is 1200 kg/m³ and the viscosity is 200 cp.

Is the change in density and viscosity (as above mentioned) allowed or is this agitator (with no change in motor rating, gear box, shaft diameter and blade length) capable to mix the new media without any failure?
Could you please to send us the required formula to obtain the shaft power, power number and etc.?

A: The process description is sufficient to see that the density change from whatever the average is for the current formulation that the change to a 1200 kg/m³ fluid will have little effect. Power requirements for turbulent mixing are proportional to the average density of the fluid.

The process is fully turbulent, especially for the low viscosity fluids in the current formulation. A viscosity increase to 200 cp. will have no observable effect on the power requirements. The viscosity change may have an effect on the powder addition, but that cannot be determined from the information supplied.

The described change in the density and viscosity will have no effect on the motor rating, gear box, shaft diameter or blade length.

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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)

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N_Q= (Vk_mix)/(ND^3 )

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

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