Now suppose we want to duplicate our process results in an 84.0 in.-dia. tank with a 3,000 gal capacity using four-blade, pitched-blade turbines. As we will see while doing the calculations, the tank geometry and impeller type both change in the scale-up process. Such changes are common. In this particular case, a combination of experience, literature research and experimentation leads us to believe that the tip speed should be held constant when we scale-up and that torque-per-volume may represent mixing intensity.

As a first step in our scale-up calculation, we will use geometrical similarity to do a scale-up from our 11.5 in.-dia. pilot tank to an 84-in.-dia. process tank. Geometric similarity means that our scale-up length ratio of 84/11.5 = 7.3 will multiply all of the length dimensions. Thus, the 5.0 in.-dia. pilot turbine will scale-up to a 36.5 in.-dia turbine in the large-scale tank and the liquid level will be 7.3 × 11.12 = 81.2 inches. However, this liquid level provides a volume of only 1,948 gal. We will need to adjust the liquid level later to get our desired results.

To maintain the same tip speed in the large-scale mixer as in the pilot unit, we must adjust rotational speed because the impeller diameters differ and impeller diameter times rotational speed gives tip speed. So, for equal tip speed, the operating speed for the large-scale mixer is:

300 (11.5/84) = 41.1 rpm

At 41.1 rpm, we can do the same calculations we did for the pilot-scale results.

As shown in the table, geometric scale-up with constant tip speed increased the Reynolds number, power and torque. However, power per volume decreased and torque per volume remained constant. For this first step in our scale-up process, both tip speed and torque per volume remained constant, which satisfies our process scale-up objectives.

The next step
At this point we will change to the four-blade turbines required in our large-scale process. The four-blade turbines have a power number of 1.37 instead of 1.19; so we can round down the turbine diameter to 36.0 in. and round up the speed to 42.0 rpm. Such adjustments seem both reasonable and practical. However, we should recalculate our mixing results to be sure.

With this turbine change, tip speed essentially stays constant and our torque per volume rises by a little more than 20%. This increased torque per volume represents a possible boost in apparent mixing intensity, which could be judged to be beneficial, because the larger tank will blend more slowly than the small pilot tank.

Adjust the tank volume
Now we need to increase tank volume to the required 3,000 gal. for the full-scale process. We can achieve this by raising the liquid level to 125 in. from the geometric value of 81.2 in. We must check the effects of the larger volume on the mixing variables.

Because the volume changed but not the mixer, only the power per volume and torque per volume changed. The torque per volume is now only 80% of our original pilot and geometric-scale-up values, which might translate into too little mixing and thus uneven results.

Several possible remedies exist for the undesired reduction in torque per volume. Suppose that with the volume increase we had held both the tip speed and torque per volume constant. For such a scenario, both the turbine diameter, D, and rotational speed, N, must change as the volume increases. To keep tip speed constant, we can use the relationship:

vTIP ∝ND

Then, per Eq. 1, for constant tip speed:

N₂D₂ = N₁D₁

For torque per volume, the following relationship based on turbulent conditions applies:

t/V ∝ N²D⁵/V

For constant torque per volume:

N₂²D₂⁵/V₂ = N₁²D₁⁵/V₁

Factoring tip speed out of the constant torque per volume relationship gives:

(N₂² D₂²) D₂³/V₂ = (N₁² D₁²) D₁³/V₁

which for constant tip speed can be reduced to:

D₂³/V₂ = D₁³/V₁  

Rearranging Eq. 7 and solving for the new turbine diameter gives:

D₂ = D₁(V₂/V₁)^1/3 = 36(3000/1948)^1/3 = 41.6 in.

Using the new turbine diameter, a speed can be selected to satisfy the constant tip-speed condition:

N₂ = N₁(D₁/D₂) = 42 (36.0/41.6) = 36.3 rpm

According to the algebra, this larger turbine and lower speed should keep both tip speed and torque per volume constant; this can be checked by recalculating mixing values.

By similar methods other combinations of variables, such as tip speed and power per volume, could be held constant. Also, impeller type can be changed and mixing variables held constant by including the power number in the expressions for power or torque. Many different adjustments to mixer characteristics can be handled by similar expressions and manipulations. Usually only a couple of mixing variables can be held constant while the others change. Sometimes this will lead to impractical solutions such as extreme impeller diameters or rotational speeds. If viscosity, through Reynolds number, affects power number, the exact mathematical relationships do not hold for power and torque. In those cases estimate approximate values and adjust calculated mixing values if necessary.