A production mixing unit is usually not geometrically similar to the mixer used for process development. Such differences can make scale-up from the laboratory or pilot plant challenging. A solution to these problems is to systematically calculate and evaluate mixing characteristics for each geometry change.

Geometric similarity is often used in mixing scale-up because it greatly simplifies design calculations. Geometric similarity means that a single ratio between small scale and large scale applies to every length dimension (see figure). With geometric similarity, all of the length dimensions in the large-scale equipment are set by the corresponding dimensions in the small-scale equipment. The only remaining variable for scale-up to large-scale mixing is the rotational speed — one or more mixing characteristics, such as tip speed, can be duplicated by the appropriate selection of a large-scale mixer speed.

The two most popular and effective geometric scale-up methods are equal tip speed and equal power per volume. Equal tip speed results when the small-scale mixer speed is multiplied by the inverse geometric ratio of the impeller diameters to get the large-scale mixer speed:

N₂ = N₁(D₁/D₂)

Equal power per volume involves a similar calculation, except the geometry ratio is raised to the two-thirds power:

N₂ = N₁(D₁/D₂)^(2/3)

This expression for power per volume only applies strictly for turbulent conditions, where the power number is constant, but is approximately correct for transition-flow mixing.

Another important mixing characteristic is torque per volume, which often represents mixing intensity, in terms of fluid velocities. While torque is just power divided by speed, torque per volume is similar to momentum transfer and is closely related to the effective motion created by the mixer. For geometric similarity and turbulent conditions, torque per volume reduces to the same scale-up formula as equal tip speed (Eq. 1). Most of the practical scale-up rules for geometric similarity, including equal solids suspension, fall in the speed range between equal tip speed and equal power per volume. Other criteria, such as equal blend time or surface motion are difficult to scale-up because of the rapidly increasing mixing intensity for large-scale tanks.

The sidebar gives formulas for calculating mixing variables in conventional U.S. engineering units. Derivations and details on the various equations for geometric scale-up and mixer evaluation appear in many references such as widely available handbooks [Refs. 1 and 2].

Five steps 
The recommended approach for doing non-geometric scale-up is as follows:

 1. Calculate the laboratory- or pilot-plant-scale values for the mixing variables listed in the sidebar.
 2. Do a geometric-similarity scale-up to the large-scale tank diameter. Use a criterion such as tip speed to set the large-scale speed. Calculate the mixing variables for the new, large-scale conditions.
 3. As needed, change the impeller diameter at the large scale. Calculate the mixing variables and, if required, modify speed to keep an important variable such as tip speed or torque per volume constant.
 4. Switch impeller type, if desired or necessary, and recalculate mixing variables using the appropriate power number for the new impeller. Calculate the mixing variables and make decisions or adjustments as required.
 5. Alter the liquid level, if necessary, to match volume specifications at the large scale. Calculate the mixing variables again for the new liquid level. If undesirable changes occur, adjust mixer speed or number of impellers.

The key to success is making simple step-by-step changes in the design and calculating important variables each time. In this way, you can evaluate the cause and effects of changes and make corrections when necessary.

Applying the approach
Suppose for example, we did a series of experiments with a 5.0 gal. batch in an 11.5 in.-dia. pilot reactor, which had two 5-in.-diameter, three-blade, pitched-blade turbines, and found that we could get good results with the mixer operating at 300 rpm. The material we were mixing had a final viscosity of 85 cP and a specific gravity of 1.05. Rearranging the volume formula in the sidebar and plugging in values, we can find the liquid level for 5.0 gal. in a 11.5 in.-diameter cylindrical tank:

H = 4 (231)(5)/p(11.5)² = 11.12 in.

The 11.12-in. liquid level assumes that the tank bottom is flat and the tank internals (impellers, shaft, baffles, etc.) do not take up any space. In the real world, the tank may have a dished bottom and the internals typically will occupy 2% to 5% of the volume for this type of reactor. While corrections can be made for bottom shape and tank internals, for simplicity we will ignore these corrections here.

With this information, we can calculate the operating conditions that exist in our pilot-scale experiment (see table). At these conditions, the Reynolds number of 995 would suggest that the mixer is operating in the transition regime, which applies for 1 < N_Re < 20,000. For a pitched-blade turbine the power number, which remains constant in the turbulent range, will begin to increase in the transition and viscous regime. For simplicity, let’s assume that the power number does not begin to increase until the Reynolds number drops below 900. Had the Reynolds number been less than 900, we would have to make an appropriate correction to the power number and use that corrected number to calculate power and torque characteristics of the mixer. This additional step to correct power number should be done for each subsequent step in the scale-up process if the Reynolds number suggests that such a correction is necessary. (Power number corrections differ with type of impeller.)