First, matching dimensionless numbers rarely results in successful scale-up.  Successful scale-up is more often accomplished by matching equal process results or characteristics.

This scale-up question appears to represent a situation where a small-scale, high-shear granulator has been successfully applied to a new process.  Now, the objective is to get the same results in a larger granulator.  In most cases, successful results are more likely if the scale-up is done in geometrically similar equipment.  In the instance of this process, if the bowl diameter of the large granulator is twice the diameter of the small-scale granulator, then the impeller diameter and blade width should also be twice the small-scale dimensions.  In addition the shape of the bowl should be similar, the blade angle the same, and other features, like nozzle location, equivalent.  For convenience of discussion, assume that the scale ratio between the small and large scale is a factor of two.  A different factor would apply for different dimensional changes.

The Froude number expresses the ratio of inertial forces to gravitational forces.  For rotating mixing equipment the Froude number is usually expressed by rotational speed squared times impeller diameter divided by the acceleration of gravity.  This combination of variables is dimensionless, with both numerator and denominator having the units of length divided by time squared.  Unless this process is taking place on another planet, the acceleration of gravity is a constant.  To scale-up and keep Froude number constant, if the impeller diameter of the large scale is two times the impeller diameter of the small scale, then the large-scale rotational speed must be one over the square root of two times (0.707 times) the small-scale rotational speed.  As with all practical scale-up, the large-scale rotational speed is less than the small-scale rotational speed.

The Reynolds number expresses the ratio of inertial forces to viscous forces, which defines whether fluid motion is turbulent, laminar or transitional.  However, it does not tell whether sufficient mixing takes place or what the turbulence intensity is.  If physical properties are the same, as in the same materials in both scales, then Reynolds number can be held constant with scale-up, if the impeller diameter squared times the rotational speed is held constant.  Thus, the large-scale speed would be one-fourth (one divided by two [the scale ratio] squared) the small scale speed for a two-to-one scale ratio.  Obviously, whether equal Froude number or equal Reynolds number are used for scale-up, the large-scale rotational speed can be very different, 0.707 or 0.25 times the small-scale rotational speed.  Additionally, neither of these choices are likely to be a good choice for the process described.

Holding power number constant is a bit of a misinterpretation for scale-up.  The power number is a ratio of imposed forces (power) to inertial forces and essentially a function of impeller geometry.  With geometric similarity, the small-scale and large-scale impeller power number will be equal for turbulent flow conditions, as defined by a large Reynolds number.  A constant power number only means that if either or both the impeller diameter or rotational speed change, then the power requirement will change in proportion to the variables raised to the appropriate exponent.  However, if the relationship between speed and impeller diameter in the power number (rotational speed cubed times impeller diameter to the fifth power) is held constant the power will remain constant in the two scales.  Since the bowl diameter has doubled, the volume has increased by the diameter cubed or eight times the volume.  Equal power means that the power per volume in the large scale has been reduced to one-eighth of the power per volume in the small-scale.  For equal rotational speed cubed times impeller diameter to the fifth, the large scale speed needs to be 0.315 times the small-scale rotational speed.

None of the equal dimensionless groups addresses the objectives of the process.  Assuming that the small-scale tests have been run to establish the appropriate operating conditions for the three stages in the process, dry blending, liquid addition, and paste mixing, then the scale-up objective should be to duplicate those process objectives in the large scale.  The most likely scale-up requirement for mixing is equal material motion, or equal velocity.  Equal velocity should assure that all parts of the material that were moving in the small scale will be moving equally well in the large scale and no stagnant areas will develop.  For equal velocity in the material, the rotational speed in the large scale should be set for equal tip speed (peripheral speed of the impeller).  Equal tip speed means that the product of the rotational speed and the impeller diameter is held constant.  If the diameter is increased by a factor of two, then the rotational speed should be reduced by a factor of two, or the large-scale speed should be half of the successful small-scale speed for each step in the process.  The same scale-up criterion should apply to all stages, but to meet the process conditions, other operating conditions may change.

To accomplish the initial dry blending of the ingredients, not only was a rotational speed chosen and tested, but also a period of time necessary to obtain uniformity.  Blending typically depends on the actual number of revolutions of the impeller, effectively independent of the rotational speed.  Thus if the large-scale rotational speed is half the small-scale rotational speed for equal tip speed, then the time required for equal blending (the same number of impeller rotations) will need to be twice as long in the large-scale as it was in the small scale.

For the liquid addition, with two times the linear dimensions in the large scale, the volume of the dry material in the large scale is eight times (two cubed) the volume in the small scale.  To maintain the same formulation, the liquid addition must be eight times the small-scale volume for the large scale process.  Depending on the controlling conditions for liquid addition, the rate of addition will need to be adjusted.  If surface wetting is important, then with the same spray nozzle type and location, the spray will contact four times (two squared) the surface in the large scale.  To keep the rate of surface wetting constant, eight times the liquid must be applied over four times the surface, or the rate of addition should be set to take two times as long in the large scale as it was added in the small scale.  If blending is critical, then the rate of addition should match the time required for blending, which at equal tip speed, is two times as long.  So the rate of addition for either surface wetting or blending should be set for twice as long in the large scale if the scale ratio is two.  If the scale ratio is different, as in 1.5 times, then the time for addition is 1.5 times as long, and the speed change is a reduction to two thirds the rotational speed.

The final blending or kneading of the resultant paste, may depend on either the required uniformity or the rate of hydration..  For the equal blending criterion, as with the initial powder blending, the time should be twice as long in the large scale. If the kneading time is required for particle hydration, then the time for the large scale process could be equal to the time for the small-scale process.

The scale-up for a granulation process is most often a function of velocity and impeller type speed, so the speed adjustment for scale up may be the same factor applied to each of the small-scale operating conditions.  Depending on the small-scale operating conditions, the speeds may be proportionally different in the large scale.  To compensate for the speed change between sizes, the time for each step in the operation or rate of liquid addition must be changed according to the process requirements.  In all cases, a better understanding of the specific process objectives or requirements may change the speed adjustment or the time requirements.  Heat addition may be a factor, since the amount of power required for mixing is directly related to the rate of heat addition.  Power is heat.  However, the heat rate is being applied to a larger volume and the rate of temperature rise will be related the power per volume.  Equal tip speed scale-up will result in a lower power per volume and slower temperature rise in the large scale than in the small scale.  However, heat loss may be greater in the small scale.

Process requirements are the essential aspect of scale-up, not dimensionless groups.

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