Dave Dickey Forum Moderator 287 Posts
Re: During scale-up, which dimensionless number should match?10 November 2011 at 1:29pm
The first thing to understand about mixing scale-up for any type of process is that keeping a dimensionless number constant is rarely effective. The typical requirement for scale-up is the same process result. An equal dimensionless number may be obtained at several conditions and is often a range of conditions, rather than a single best result. In all cases, scale-up is best done if the large granulator is geometrically similar to the small granulator. Geometric similarity means that the shape of the blades is the same and the diameter of the blades is in the same proportion as the diameters of the two different size granulators. Thus, if the diameter of the granulator bowl increases by 50%, then the diameter of the mixer blade should increase by 50%, and the diameter of the chopper blade should increase by 50%.
First, equal Froude number makes sense only if gravity has a demonstrated effect on the process. Froude number is the ratio of inertial to gravitational forces. Since gravity is rarely changed in mixing processes, the effect to be match for scale-up needs to be a characteristic that would obviously change with gravity. For granulation, the relative lift height of the moving particles might be affected by gravity, but is not likely to be related to dry mixing. Actually, scale-up of dry mixing is most often accomplished at equal tip speed and geometric similarity. However, the time required for blending will be increased as the speed of the mixer blade decreases. For equal tip speed and a 50% increase in diameter, the rotational speed will be two thirds the small-scale speed and the time required for blending will need to be increased by 50%. The reason that the rotational speed is not held constant to keep the blend time the same is that the increased tip speed may break dry particles and increase the temperature more rapidly. The same speed with a 50% increase in diameter will cause the powder temperature to increase 30% faster in the larger granulator.
Second, keeping the Newton (power) number constant will happen as long as the geometry of the blade is similar in both scales. One of the benefits of many dimensionless groups is that the value of the number is the same independent of size. Thus with geometric similarity, the blade in a large granulator will have the same power number as the blade in a small granulator, regardless of diameter or speed. The only use for the power number is to predict the power input for a given speed and blade diameter. Keeping power input constant may not hold much advantage for binder addition. Again, equal tip speed for binder addition is probably a good start for scale-up of the blade speed. However, the rate of addition of the binder probably needs to be reduced in proportion to the surface to volume ratio of the granulator to avoid over-wetting the particles on the surface of the material. The area of the surface of the powder is proportional to the bowl diameter squared and the volume of the powder is proportional to the diameter cubed. So the rate of addition for a 50% larger diameter granulator should be adjusted so that the 3.375 times the quantity of liquid goes in over a period about 80% longer than in the small granulator.
Third, equal Reynolds almost never works for any type of mixing, even for granulation. The Reynolds number is the ratio of inertial to viscous forces. What is the viscosity of the granulation? Can it be measured? Even for kneading, equal tip speed is usually the best scale-up approach, otherwise, the potentially lower speed associated with the assumption of constant viscosity may not cause sufficient motion during kneading to get move the entire batch. Dead spots are not good.
In fact, for most stages of a granulation process, equal tip speed should be a good first choice. In some cases, a large difference in size between the small and large granulators may obtain the same results at less than equal tip speed. The possibility even exists that better results will be obtained in the large granulator. Equal results are far more important than equal dimensionless groups. The customer is rarely interested in having equal dimensionless groups, they want good product.
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