One question frequently asked about mixing scale-up is whether to use equal tip speed or equal power per volume. While one of these criteria may guide successful scale-up, you may need to factor in some additional limitations or qualifications to get proper results. Scale-up using these concepts most often involves equipment with geometric similarity, i.e., length dimensions in the large-scale mixing equipment are in the same proportion as those in the small-scale equipment. Because geometric similarity sets all the dimensions and impeller features in the large-scale equipment, the only remaining variable for scale-up is the rotational speed.

When mixer speed is the only variable in scale-up, you can reduce the calculation of the large-scale speed to an expression starting with the successful small-scale speed times the inverse scale ratio (small-scale length/large-scale length) raised to an exponent:

*N _{Large} = N_{Small} (T_{Small}/T_{Large})^{n} = N_{Small} (D_{Small}/D_{Large})^{n}*

where *N* is the rotational speed, typically expressed in revolutions per minute, *T* is the tank diameter, and *D* is the impeller diameter. (You can use either the tank or impeller ratio because they are the same with geometric similarity.) The exponent, *n*, provides a convenient means for adjusting the magnitude of the speed change from the small scale to the large scale. To calculate a large-scale speed for equal tip speed, the exponent is one, i.e., *n =* 1. Whatever the successful small-scale speed is, you must reduce the large-scale speed by the ratio of the small-scale to large-scale length dimensions. For instance, if the effective small-scale speed is 250 rpm and the large-scale length dimensions are five times the small-scale dimensions, you must set the large-scale speed at one-fifth the small-scale speed or 50 rpm.

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Using equal power per volume for geometric scale-up usually runs into an additional limitation for turbulent mixing conditions. With geometric similarity in turbulent mixing, power is proportional to speed cubed and the impeller diameter to the fifth power. With geometric similarity, volume is proportional to the tank diameter cubed or, alternatively, the impeller diameter cubed. For geometric similarity, the tank diameter and impeller diameter scale ratio are the same. If power is defined by speed cubed and impeller diameter to the fifth power and volume is proportional to impeller diameter cubed, then power per volume must be proportional to speed cubed and impeller diameter squared. So, re-arranging the power-per-volume relationships to calculate the large-scale speed from the small-scale speed raises the inverse scale ratio to an exponent of two thirds, i.e., *n = ⅔*. An exponent of two-thirds reduces the large-scale speed by less of a factor than for equal tip speed. Again, using the successful small-scale speed of 250 rpm and a five-to-one length increase as an example, the large-scale speed for equal power per volume is 85.5 rpm. For turbulent conditions, where power is proportional to speed cubed, the large-scale power for equal power per volume will be 4.9 times the power for equal tip speed. This difference in power becomes even greater with larger scale changes and may be impractical for some.

### Other Factors

Beyond the obvious differences in the speed and power changes between equal tip speed and equal power per volume, the fluid dynamic reasons for choosing one or the other set of criteria differ. With geometric similarity and turbulent conditions, the flow pattern in a stirred tank is a constant. In other words, the local velocity magnitude at any point in the tank is proportional to the impeller tip speed. Equal tip speed for turbulent mixing and geometric similarity will result in similar local speeds and relative velocities throughout a stirred tank. The velocity of the impeller tip relative to the surrounding fluid may define important velocity gradients, which can affect certain types of dispersion. Drop size in two-phase liquid/liquid systems and agglomerate breakup size in solid/liquid systems may be closely related to impeller tip speed in scale-up. Power per volume, which also is power per mass, can be related to turbulence factors, such as micro-scale length and time or energy dissipation. These power-per-volume effects may influence certain types of chemical reactions and product distributions.

Although some process generalizations may point in favor of tip speed or power per volume, you should determine scale-up behavior in small-scale tests for best scale-up results. Varying the impeller size as well as the speed in these tests often may help better differentiate between scale-up by tip speed or power per volume.

### Non-Geometric Scale-Up

Geometric similarity, while reducing the number of variables, isn’t essential for successful scale-up. One of the most common geometry changes is the impeller-to-tank-diameter ratio. By simple logic, a small impeller operating at a high speed should provide similar results to a large impeller running as a low speed. What “similar results” means depends on the process. Because impeller pumping capacity and power input don’t have the same functionality with respect to impeller diameter and rotational speed, the tip speeds or power requirements likely will differ depending on the impeller-to-tank-diameter ratio. Small impellers tend to operate at higher tip speeds and power inputs than large impellers.

Sometimes geometric similarity isn’t practical or even advisable. Unfortunately, non-geometric scale-up is a more difficult process. It may involve several different combinations of constant or changing mixing parameters. A step-by-step scale-up process may begin with a geometric similarity scale-up to the large-scale tank diameter. Once some conditions have been established in the large scale, you can adjust liquid level to alter the volume. Then, you can make further changes to impeller diameter or type with assumptions about equal tip speed, equal power per volume, or other factors (such as torque per volume, mixing intensity, surface motion, blend time, and heat or mass transfer rates) being kept constant or changed. The combination of factors best suited for successful non-geometric scale-up will depend on the particular aim of the mixing.

Perhaps the most difficult scale-up occurs when viscosity is a significant factor in mixer performance. High viscosity almost always makes mixing tougher. However, the effect of viscosity on mixing isn’t measured just by the magnitude of the viscosity. Instead, the impeller Reynolds number, *Re*, typically is used. It includes the effect of impeller size, rotational speed, fluid density and apparent viscosity. The *Re* is the best way to judge whether fluid motion is turbulent, transitional or laminar. Because the impeller diameter appears as a squared factor in the *Re* numerator with viscosity in the denominator, scale-up from a small mixer to a large mixer increases the *Re* and decreases the effect of viscosity magnitude. This rise in *Re* with size means that viscosity will have less impact in the large-scale mixer — and that mixing may get easier as the scale of the process goes up.

### Go Beyond Simple Rules

The real problem with mixing scale-up is that the simple rules like tip speed and power per volume are only part of the answer. Other factors may help or hurt the results of using the simple rules. You can successfully use scale-up to design a mixer for a process — if you understand the process needs and keep the essential features the same with scale-up. Take advantage of the many studies of mixing scale-up reported in books and technical literature. Scale-up requires some process knowledge and background information.

DAVID S. DICKEY is a senior consultant at MixTech, Inc., Coppell, Texas. E-mail him at [email protected].