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2014-04-05 1Scale up of Powder BlendingMarch 2014

1 of 50Tehran University of Medical

SciencesSchool of Pharmacy

Batch Size Increase in Dry Blending and Mixing

Chapter by:Albert W. Alexander and Fernando J. Muzzio




Introduction

• Dry particle blending is a critical step that has a direct impact on content uniformity

• There are currently no mathematical techniques to predict blending behavior of granular components without prior experimental work.

• Blending studies start with a small-scale, try-it-and-see approach.




A Typical Problem• A 5ft3 tumble blender filled to 50% of capacity and

run at 15 rpm for 15 min produces the desired mixture homogeneity. What conditions should be used to duplicate these results in a 25ft3 blender?

• The following questions might arise:– 1. What rotation rate should be used?– 2. Should filling level be the same?– 3. How long should the blender be operated?– 4. Are variations to the blender geometry between scales

acceptable?




Tumbling Blenders




Defining Mixedness• The final objective of any granular mixing process is

homogeneity.• No reliable techniques for on-line measuring of

composition have been developed yet.• Granular mixtures are usually quantified by removing

samples from the mixture. • Interrupting the blend cycle and repeated sampling may

change the state of the blend.• The mean value and sample variance are determined

and then often used in a mixing index.




Online Processing of Homogenicity

• Imaging (NIR)• Image Processing




Mixing in Tumbling Blenders

• Particle motions is in a thin, cascading layer at the surface, while the remainder of the material rotates with the vessel as a rigid body.

• Most tumbling blenders are symmetrical which impedes achieving a homogeneous mixture.

• The mixing rate often becomes limited by the amount of material that can cross from one side of the symmetry plane to the other




Mixing in Tumbling Blenders• Blending process takes place by three essentially

independent mechanisms: • Convection causes large groups of particles to move in

the direction of flow (orthogonal to the axis of rotation), the result of vessel rotation.

• Dispersion is the random motion of particles as a result of collisions or inter-particle motion, usually orthogonal to the direction of flow (parallel to the axis of rotation).

• Shear separates particles that have joined due to agglomeration or cohesion and requires high forces.




Mixing in Tumbling Blenders• Loading of multiple ingredients will have a dramatic

effect on mixing rate if dispersion is the critical blending mechanism.

• Care must be taken when loading a minor (~1%) component into the blender

• The order of constituent addition can also have significant effects on the degree of final homogeneity, especially if ordered mixing (bonding of one component to another) can occur within the blend




Mixing in Tumbling Blenders• Intershell flow is the slowest step in a V-blender,

because it is dispersive in nature, while intrashell flow is convective. Both processes can be described by similar mathematics, typically using an equation such as:

• Where σ2 is mixture variance, N is the number of revolutions, A is an unspecified constant, and k is the rate constant.

• The rate constants for convective mixing are orders of magnitude greater than for dispersive mixing.




Process Parameters

• One parameter consideration that arises is whether rotation rate should change with variations in size.

• When far from the critical speed of the blender, the rotation rate does not have strong effects on the mixing rate

• The number of revolutions was the most important parameter governing the mixing rate.




Process Parameters

• Given a geometrically similar blender and the same mixture composition, the fill level should also be kept constant with changes in scale.

• However, an increase in vessel size at the same fill level may correspond to a significant decrease in the relative volume of particles in the cascading layer compared to the bulk.

• A large decrease in mixing rate.




SCALE-UP APPROACHES

• Froude number Fr=Ω2R/g

Where Ω is the rotation rate, R is the vessel radius, and g is the acceleration from gravity is often suggested for tumbling blender scale-up.

• Matching Tangential SpeedA less commonly recommended scaling strategy is to

match the tangential speed (wall speed) of the blender; however, this hypothesis also remains untested

What is the dimension of Fr?




Back to our problem with a Fr approach

• We now look at our general problem of scaling the 5ft3 blender using Fr as the scaling approach:

• The requisites are: – Geometric Similarity– Keep total number of revolutions constant





• the 25ft3 blender must look like a photocopy enlargement of the 5ft3 blender. So the linear increase is 51/3, or 71%.

• Also the fill level must remain the same. • To maintain the same Froude number, since R has

increased by 71%, the rpm (Ω) must be reduced by a factor of (1.71)-1/2 = 0.76, corresponding to 11.5 rpm

• The speed closest to 11.5 rpm would be selected.





• If the initial blend time were 15 minutes at 15 rpm, the total revolutions of 225 must be maintained with the 25ft3 scale. Assuming 11.5 rpm were selected, this would amount to a 19.5-minute blend time.




NEW APPROACH TO THE SCALE-UP PROBLEM IN TUMBLING BLENDERS

• We begin by proposing a set of variables that may control the process.

• The driving force for flow in tumbling blenders is the acceleration from gravity.

• Vessel size is obviously a critical parameter, as is the rotation rate, which defines the energy input to the system.

• These variables define the system parameters but do not cover the mixture response.





• In case of Newtonian fluids:

Driving Forces (pressure gradients, gravity, shear)

Fluid Response (velocity gradients)

Viscosity

• For granular mixtures:

Driving Forces (gravity, vessel size )

Particle Size and Interactions

Fluid Response (velocity gradients, blending rate)





• For granular mixtures we will define particle size and particle velocity as our “performance variables.”

• Particle size plays a large role in determining mixing (or segregation) rates because dispersion distance is expected to vary inversely with particle size.





• Although previous studies have indicated that rotation rate (and, hence, probably particle velocities) does not affect mixing rate, these experiments were done in very small blenders. It is conceivable that at larger scales, these variables could become important.

• We can now address the development of non-dimensional scaling criteria.




Applying Rayleigh’s Method• Variables that are believed to govern particle dynamics

in tumbling blenders are listed:

• Using these variables and the Rayleigh method, we derive the following equation:

Lord Rayleigh’s sarcastic comment with which he began his short essay on “The Principle of Similitude”

“I have often been impressed by the scanty attention paid even by original workers in physics to the great principle of similitude. It happens not infrequently that results in the form of ‘laws’ are put forward as novelties on the basis of elaborate experiments, which might have been predicted a priori after a few minutes’ consideration.”




Dimensional Homogenization

• Applying the rule of dimensional homogeneity and making c and e the unrestricted constants leads to:




Correlating Particle Velocities to Vessel Rotation Rate and Radius

• In order to determine particle velocities, an empirical approach is taken.

• A digital video camera was used to record the positions of individual particles on the flowing surface in clear acrylic, rotating cylinders of 6.3, 9.5, 14.5, and 24.8cm diameter filled to 50% of capacity.

• nearly monodisperse 1.6mm glass beads dyed for visualization was used.





• The displacement of particles from one frame to the next was converted into velocities.

• To calculate velocity, only the motion down the flowing layer was used, and all cross-stream (i.e., dispersive) motion was ignored.




• By differentiating the polynomial fit, we obtain an estimate of the downstream acceleration:

• Over the initial upper third of the flowing layer, the acceleration profiles for all cylinders are nearly identical.





• maximum accelerations are nearly equal, implying that tangential velocity may be proportional to maximum acceleration.

• Maximum accelerations were determined for all experiments; the results are plotted against the tangential velocity





• An Approximate Linear fit:

• where TV is the tangential velocity (=2πRΩ) and α = 17 sec-1, is seen relating acceleration and tangential velocity for all cylinders and rotation rates.





• The distance to reach zero acceleration, l, by itself, has little meaning but the parameter l/r, where r is the cylinder radius, that has a quantitative effect on the velocity profile and maximum velocities.

• When all values of l/r were compiled, a strong correlation to rotation rate was noted.




Because l/r determines the shape of the velocity profile, experiments run at the same rotation rate should show qualitatively similar velocity profiles, regardless of cylinder size.




Developing a Model

• The simplest possible model for particle velocity relates velocity and distance when acceleration is constant,

• Where V0 is the initial downstream velocity and x is the downstream coordinate.




• Some simplifying assumptions:1. Particles emerge into the flowing layer with zero

initial downstream velocity V0= 02. Peak acceleration is proportional to the tangential

velocity (TV), 3. Particles accelerate over the distance l.4. Acceleration (a) is not constant over the distance l,

but the rate of change in acceleration scales appropriately with the value of l (i.e., a = amaxƒ(x/l), where x is the distance down the cascade).

Developing a Model




Developing a Model

• Using these assumptions a new relation for particle velocity would be:

• Which relates particle velocities to the rotation rate and the radius, can be used as the basis for scaling particle velocities with changes in cylinder diameter and rotation rate.




Returning to Dimensional Analysis

• This equation can be used to complete the dimensional analysis discussed earlier.

• Applying dimensional homogeneity and solving leads to:




Testing the Model

• To test the scaling criteria suggested by equation we will look at velocity profiles between 10 and 30 rpm. Figure shows the scaled velocity profiles (i.e., all data are divided by using RΩ2/3(g/d)1/6, and the distance down the cascade is divided by the cylinder diameter




• Un-scaled Data




Returning to our example

• scaling from a 5-ft3 blender to a 25-ft 3 blender, again the relative change in length is 71%. This time, to scale surface velocities using this approach, the blending speed (Ω) must be reduced by a factor of (1.71)-3/2 = 0.45, corresponding to 6.7 rpm (assuming the particle diameter, d,remains constant).

• Again, the total number of revolutions would remain constant at 225, for a blend time of 33.6 min.




TESTING VELOCITY SCALING CRITERIA

• Experimental work has not validated the preceding scaling procedure with respect to scale-up of blending processes.

• This model should not be favored over other approaches currently in use, though it may provide additional insight.





• recent work has indicated that particle velocities may be critical for determining segregation dynamics in double-cone blenders and V-blenders.

• Segregation occurs within the blender as particles begin to flow in regular, defined patterns that differ according to their particle size.





• In a 1.9-quart-capacity V-blender at fixed filling (50%), incrementally changing rotation rate induced a transition between two segregation patterns

• At the lower rotation rate, the “small-out” pattern forms• At a slightly higher rotation rate, the “stripes” pattern forms




Segregation Patterns





• To validate both the particle velocity hypothesis and our scaling criteria, similar experiments were run in a number of different-capacity V-blenders.





• All the vessels are constructed from clear plexiglass, enabling visual identification of segregation patterns.

• For these experiments, a binary mixture of sieved fractions of 150 to 250-(nominally 200) and 710- to 840- (nominally 775-) glass beads was used.

• A symmetrical initial condition (top-to-bottom loading) is implemented.

• The blender is run at constant rotation rate; a segregation pattern was assumed to be stable when it did not discernibly change for 100 revolutions.





• The transition speeds (rotation rates) were determined for the change from the “small-out” pattern to “stripes” at 50% filling for all the blenders listed

• As discussed, the most common methods for scaling tumbling blenders have used one of two parameters, either the Froude number (Fr) or the tangential speed of the blender




RECOMMENDATIONS AND CONCLUSIONS

• The analysis of particle velocities provides a good first step toward the rigorous development of scaling criteria for granular flow, but it is far from conclusive.

• It is important in granular systems to first determine the dynamic variable that governs the process at hand before determining scaling rules

• A systematic, generalized approach for the scale-up of granular mixing devices is still far from attainable. Clearly, more research is required both to test current hypotheses and to generate new approaches to the problem.




some simple guidelines that can help through the scale-up

Make sure that changes in scale have not changed the dominant mixing mechanism in the blender (i.e., convective to dispersive)

Number of revolutions is a key parameter, but rotation rates are largely unimportant.

When performing scale-up tests, be sure to take enough samples, be wary of how you interpret your samples.

One simple way to increase mixing rate is to decrease the fill level. It also reduces the probability that dead zones will form.

Addition of asymmetry into the vessel, either by design or baffles, can have a tremendous impact on mixing rate.

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