Keeping a dimensionless group constant often seems a convenient way to scale up mixing of liquids, powders or combinations of phases. However, holding a dimensional variable, such as tip speed or power per volume, constant more often yields a successful scale-up. Indeed, maintaining a dimensionless group constant frequently leads to incorrect or impractical scale-up results.
So, let's look at three common dimensionless groups — Reynolds number, power number and Froude number — to understand why their use in mixing scale-up causes problems.
Most engineers know about Reynolds number from pipe flow. The value of the Reynolds number in a pipe helps to indicate the conditions where flow transitions between laminar and turbulent regimes. The Reynolds numbers also is used in correlations for pressure drop. The impeller Reynolds number in mixing does about the same thing. The value can describe a transition from laminar to turbulent mixing — but that transition occurs over several orders of magnitude of the Reynolds number. In the transition range, the motion near the impeller blade may be turbulent while the conditions near the wall may be more laminar. The impeller Reynolds number also correlates impeller power, blend time and heat transfer.
However, just as for pipe flow, the Reynolds number for mixing isn't kept constant as vessel diameter increases. In pipe flow, linear velocity often is held constant as pipe diameter increases, causing the Reynolds number to increase. Similarly in mixing, the impeller tip speed, which can be considered a velocity, may be held constant for scale-up. At constant tip speed, the impeller Reynolds number increases as equipment size becomes larger. Keeping Reynolds number constant for scale-up of either pipe flow or impeller mixing will result in lower velocities in the larger diameter pipe or vessel. This reduces flow and the effectiveness of either the pipe or mixer.
A higher Reynolds number in a larger vessel offers an interesting advantage — it has the equivalent effect of operating with a lower viscosity fluid. If the Reynolds number in the laminar or transition range increases because of tip speed scale-up, then the flow pattern will be the same as if the viscosity had been less in the small-scale test. An effective flow pattern always is important, especially in the laminar and transition range.
Also called the Newton number, the power number is almost unique to mixing. It is analogous to a friction factor for pipe flow and exists at the basic level of pump impeller and boat propeller design. The power number can be correlated to Reynolds number. However, it effectively becomes a constant independent of Reynolds number for turbulent mixing applications. Thus, keeping power number constant for turbulent mixing is a default and does nothing useful to describe how to do scale-up.
The power number for a mixing impeller primarily depends upon the impeller geometry. The impeller type, number of blades, blade angle, width and shape, etc., define the value of the turbulent power number. Impeller design also substantially influences power number behavior as a function of Reynolds number in the transition and laminar range. Because power number correlates to Reynolds number, keeping power number constant in the transition or laminar range is equivalent to keeping Reynolds number constant, with all of the same problems described in the previous section.
The power number tells nothing about mixing intensity, other than how much power is required to rotate an impeller at a specified diameter and speed. It doesn't indicate whether the mixer will move all of the fluid or not. The power required to rotate a specific impeller at a given speed is about the same whether the mixer is in a small tank, a big tank or a lake. However, the mixer will generate drastically different mixing intensity depending upon the quantity of material. About the only scale-up use for power number is to decide the size of a motor needed for an otherwise effectively scaled mixer. Power number in conjunction with motor size and impeller speed also can determine how large an impeller can be for a given motor size. In any case, all of the power used to rotate the impeller will be converted to heat.
This number was developed to describe wave shape and height for ship design. For impeller mixing, it usually is defined as rotational speed squared times impeller diameter divided by the acceleration of gravity. The Froude number is the ratio of inertial forces to gravitational forces. In mixing, it can be used to describe vortex depth or wave height on the liquid surface. Unfortunately, the Froude number often has been misapplied in mixing. While it may seem to roughly correlate experimental data, you can be certain the experimenter never tried changing the acceleration of gravity to see if the functionality was correct.
Scaling up by equal Froude number will result in increased power per volume on the larger scale, which may lead to very intense mixing, which usually is impractical. The only reason scale-up using the Froude number may work is that it's extremely conservative and over-mixing rarely is as much of a problem as under-mixing.
Scaling up using approaches such as constant tip speed or power per volume will give a lower Froude number at the large scale. The net effect is that the vortex depth or wave height will decrease as the vessel diameter increases. A small-scale experiment may show a deep vortex or a splashing surface but larger equipment will have a shallower vortex and a quieter surface. The surface will look different with scale-up.
SUCCEED AT SCALE-UP
The best approach for most mixing scale-ups is to keep constant an important variable, not a dimensionless group. While dimensionless groups provide a way of correlating flow behavior on different scales, holding a dimensionless group constant usually gives poor or ambiguous mixing results. Dimensionless groups typically represent a ratio of two forces or a modified ratio of other effects. The forces or effects describe mechanisms that may contribute to mixing. So, dimensionless groups may assist in understanding which mechanisms become more important with scale-up. If a mechanism's change positively impacts the process, scale-up may be easy and a bit conservative. If the effect decreases with scale-up and is important to the process, problems may develop unless some degree of overdesign in the appropriate direction is provided. Testing the effects of mixing over a range of variables is essential to good scale-up because the best approach may not be an obvious one.
DAVID S. DICKEY is senior consultant for MixTech, Inc., Coppell, Texas. E-mail him at email@example.com.