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Liquid mixing equation
Q: I know there is an equation that can be used to determine the adequate mixing time of two liquids (viscosity of water). Can you give me the name of the equation? I saw it once a long time ago and, if I remember correctly, it was a person's name.
A: The only person's name I have ever heard associated with the time required to blend liquids is Betty Crocker. I have heard, and even referred myself, to the dimensionless blend time as the "Betty Crocker Number."
Dimensionless blend time is essentially the product of actual blend time and the rotational speed of the mixer, along with some empirical geometry factors, as shown in the attached equation for a pitched-blade turbine.
In the turbulent range (low viscosity, waterlike liquids), dimensionless blend time is a constant. For any given geometry or geometrically similar system, the blend time is inversely proportional to the rotational speed. In other words, the time required to achieve a given blend uniformity depends on a certain number of mixer revolutions.
To calculate the blend time, the dimensionless equation is rearranged to calculate time (as shown below).
The time is in minutes or seconds depending on whether the rotational speed of the mixer is in revolutions per minute or revolutions per second, respectively. For a typical pitched-blade turbine the dimensionless blend time for 99% uniformity is:
6.34 = Blend time x Rotational Speed x (Impeller Diameter/Tank Diameter)^2.3 x (Liquid Level/Tank Diameter)^-0.5
as shown in the attached dimensionless equation. For a straight-blade, radial flow turbine, the dimensionless value is 4.8. Although by the blend time value the straight-blade turbine takes less time, it takes more power and/or torque than the pitched-blade turbine for the same result. The dimensionless blend time value for a hydrofoil or propeller is 16.4 but the exponent on the impeller to tank diameter ratio is 1.7 instead of 2.3. Simple factors associated with an exponential decay function can be used to estimate times for 95% and 99.9% blend uniformity.
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Q = 1Nm3/hr
He= 0.99
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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.
