Sunday, 9 October 2011

Granulation: Endpoint Determination and Scale-Up

In addition to active ingredients, formulations are complex mixtures of diluents, binders, disintegrants, surface active agents, glidants, lubricants, colorants, coating substances, surfactants and many other raw materials that impart different properties to the final solid dosage product. Granulation is often required to improve the flow of powder mixtures and mechanical properties of tablets.  Granules are usually obtained by adding liquids (binder or solvent solutions).  Larger quantities of granulating liquid produce a narrower particle size range and coarser and harder granules, i.e. the proportion of fine granulate particles decreases.  The optimal quantity of liquid needed to get a given particle size should be known in order to keep a batch-to-batch variations to a minimum. Wet granulation is used to improve flow, compressibility, bio-availability, homogeneity, electrostatic properties, and stability of solid dosage forms.          
The particle size of the granulate is determined by the quantity and feeding rate of granulating liquid. Wet granulation is used to improve flow, compressibility, bio-availability, and homogeneity of low dose blends, electrostatic properties of powders, and stability of dosage forms.
This review is limited to a wet granulation process in low and high shear mixing devices where a low viscosity liquid (usually water) is added to a powder blend containing binder that was dry mixed with the rest of the formulation.  Due to time constraints, the following subjects will be excluded: melt granulation, fluid bed granulation, dry mixing, liquid binder addition, and high viscosity binders.
Let me just mention in passing that wet massing in a high-shear mixing is frequently compared to fluid bed mixing and to roller compaction technique, for example [6], and the results seem to be formulation dependent. 
For the comparison of high and low shear granulator performance, see, e.g. [115].  Compared to high shear granulation, low shear or fluid bed process requires less fluid binder, resulting in a shorter drying time, but also in a less cohesive material.
The generally known facts about types of mixers and measurement instruments available in the field will be briefly summarized in the following slides.  For an excellent review of the wet granulation process, equipment and variables see Holm [47].
Factors in Wet Granulation Densification include:
§       Agglomeration
§       Shearing and compressing action of the impeller
§       Mixing, granulation and wet massing
§       Possibility of overgranulation due to excessive wetting
§       Possibility of producing low porosity granules
§       Liquid bridges
§       Coalescence
§       Breakage of the bonds
Due to rapid densification and agglomeration that are caused by the shearing and compressing action of the impeller in a high-shear single pot system, mixing, granulation and wet massing can be done relatively quickly and efficiently.   The dangers lie in a possibility of overgranulation due to excessive wetting and producing low porosity granules thus affecting the mechanical properties of the tablets.
As the liquid bridges between the particles are formed, granules are subjected to coalescence alongside with some breakage of the bonds. 
Additional Factors in Wet Granulation:
§       Specific surface area
§       Moisture content
§       Liquid saturation
§       Intragranular porosity
§       Heating
§       Evaporation
§       Mean granule size
§       Apparent viscosity
It stands to reason that mean granule size is strongly dependent on the specific surface area of the excipients, as well as the moisture content and liquid saturation of the agglomerate.
During the wet massing stage, granules may increase in size to a certain degree while the intragranular porosity goes down.  However, some heating and evaporation may also take place leading to a subsequent decrease in the mean granule size, especially in small-scale mixers.
Load on the main impeller is indicative of granule apparent viscosity and wet mass consistency.
The following forces act on the particles:
§       acceleration (direct impact of the impeller),
§       centrifugal,
§       centripetal, and
§       friction
·       Binder addition rate controls  granule density
·       Impeller speed control granule size and granulation rate
·       End point controls the mix consistency and reproducibility
These statements are substantiated in [18].  Other factors that affect the granule quality include spray position and spray nozzle type, and, of course, the product composition.
Such variables as mixing time and bowl or product temperature are not independent factors in the process but rather are responses of the primary factors listed above.

Endpoint Determination

The formulator can define endpoint as a target particle size mean or distribution, or in terms of granulate viscosity or density.  It has been shown [26] that once you have reached the desired endpoint, the granule properties and the subsequent tablet properties are very similar regardless of the granulation processing factors, such as impeller or chopper speed or binder addition rate.  This can be called “the principle of equifinality”.
The ultimate goal of any measurement in a granulation process is to estimate viscosity and density of the granules, and, perhaps, to obtain an indication of the particle size mean and distribution.  One of the ways to obtain this information is by measuring load on the main impeller.

Benefits of Mixer instrumentation:

·       Machine Troubleshooting
Ø     detect worn-out gears and pulleys
Ø     identify mixing and binder irregularities
·       Formulation Fingerprints
Ø     batch record becomes a batch and mix ID
·       Batch Reproducibility
Ø     use end point to achieve consistency
·       Process Optimization
Ø     raw material evaluation
Ø     ideal end point determination
·       Use of experimental design to minimize the effort
·       Process Scale-Up
Ø     move the end-point value along the scale-up path

Mixer Measurements:

Conductivity of the damp mass [118] measures uniformity of liquid distribution and packing density.
Probe vibration analysis [120, 121] requires a specially constructed probe that includes a target plate attached to an accelerometer (in-process monitoring).  This measurement is based on the theory that increasing granule size results in the increase of the acceleration of agglomerates striking the probe target.  The method has a potential for granulation monitoring and end-point control.
Boots Diosna Probe measures densification and increase of size of granules (changes in momentum of granules moving with constant velocity due to a mass change of the granules) [53].  Did not gain popularity because of its invasive nature.
Laser beam diffraction in free-flowing systems could be used in-process by creating a product loop similar to those available for the fluid bed granulators and measuring the reflection of the beam and losses due to diffusion [48].
Current  in DC motors can be used as some indication of the load on the main impeller because torque T is proportional to current in some intervals [19] and therefore a current meter (ammeter) can be used for small scale direct current (DC) motors.
However, for alternating current (AC) motors (most often used in modern mixers), there may be no significant change in current as motor load varies up to 50% of full scale.  At larger loads, current draw may increase but this increase is not linearly related to load, and, consequently, current is completely ineffective as a measurement of load.
Voltage measurement has no relation to load.
Capacitance sensor [20, 21, 32, 33, 122] responds to moisture distribution and granule formation. It provided similar endpoints (based on the total voltage change) under varying rates of agitation and liquid addition. Capacitance sensor can be threaded into an existing thermocouple port for in-process monitoring.
Chopper Speed has no significant effect on the mean granule size [40], [41].
Impeller or Shaft Speed could be used as some indication of the work being done on the material [5].
Motor Slip [124, 125, 5] is the difference between rotational speed of an idle motor and motor under load.  Motor slip measurements, although relatively inexpensive, do not offer advantages over the power consumption measurements and did not gain popularity, probably because the slip is not linearly related to load [30] despite some claims to the contrary.
Impeller Tip Speed corresponds to shear rate and has been used as a scale-up parameter in fluid mixing [84, 97].  For processing of lactose granulations in Gral mixers, however, it was shown by Horsthuis et al. [49] that the same tip speed did not result in the same end point (in terms of particle size distribution).  These findings were contradicted by other studies in Fielder mixers indicating that, for a constant tip speed, successful scale-up is possible when liquid volume is proportional to the batch size and wet massing time is related to the ratio of impeller speeds [97].
Relative Swept Volume, that is, the volume swept by the impeller (and chopper) per unit time, divided by the mixer volume, has been suggested as a scale-up factor [107, 106, 112].  This parameter is related to work done on the material and was studied extensively at various blade angles [46]. 
Higher swept volume leads to higher temperature and denser granules.
However, it was shown by Horsthuis et al. [49] that the same relative swept volume did not result in the same end point (in terms of particle size distribution).
Product and jacket temperature is usually measured by thermocouples.  These response variables are controlled by a variety of factors, notably, the speed of the main impeller and the rate of binder addition.
The most popular measurements are direct and reaction torque, and the power consumption of the main motor. In the following slides we will examine the benefits and relative disadvantages of the torque and power measurements.
Power  Consumption of the main mixer motor is a relatively inexpensive measurement.  It is done by a watt transducer or a power cell utilizing Hall effect.
Power  consumption of the impeller motor for endpoint determination and scale-up is widely used [69, 71, 72, 70, 66, 122, 131, 9, 20, 42, 43, 56, 100, 113, 119, 133] because the measurement is inexpensive, it does not require extensive mixer modifications and is well correlated with granule growth [100].  Mean granule size of a granulation does not vary linearly with the absolute value of the power consumption of the motor but intragranular porosity does show some correlation with power consumption [100].
Power is a product of
·       Current
·       Voltage
·       Power factor
Power is proportional to load and reflects system performance. The main problem with power consumption measurements is that this variable reflects the overall mixer performance and mixer motor efficiency, as well as the load on the main impeller. 
Up to 30% of the power consumption of a motor can be attributed to no-load losses due to windage (by cooling fan and air drag), friction in bearings, and core losses that comprise hysteresis and eddy current losses in the motor magnetic circuit.  Load losses include stator and rotor losses (resistance of materials used in the stator, rotor bars, magnetic steel circuit) and stray load losses (current losses in the windings) [39].
Attempts to use a no-load (empty bowl, or dry mix) values as a baseline may be confounded by a possible nonlinearity of friction losses with respect to load [25]. As the load increases, so does the current draw of the motor.  This results in heat generation that further impacts the power consumption [30].  A simple test might be to run an empty mixer for several hours and see the shift in the baseline.  Also, as the motor efficiency drops, the baseline most definitely shifts with time.
Motor power consumption is non-linearly related to the power that is transmitted to the shaft [47] and the degree of this non-linearity could only be “guestimated”.      
In the mixing process, changes in torque on the blades and power consumption occur as a result of change in the cohesive force or the tensile strength of the agglomerates in the moistened powder bed.
Direct Impeller Torque measurements require installation of strain gages on the impeller shaft or on the coupling between the motor and impeller shaft.  Since the shaft is rotating, a device called slip ring is used to transmit the signal to the stationary data acquisition system.
Impeller torque is an excellent in-line measure of the load on the main impeller [12, 20, 21, 35, 127, 100] and was shown to be more sensitive to high frequency oscillations than power consumption [20, 56].
Torque Rheometer is an off-line technique of measuring rheological properties of the granulation.  It has been extensively used for endpoint determination [38, 37, 63, 65, 87, 102, 114]
Reaction torque is a less expensive alternative and is recommended for mixers that have the motor and impeller shafts axially aligned (in this case it is equal to direct torque and opposite in sign). By the third law of Newton, for every force there is a counter-force, collinear, equal and opposite in direction. As the impeller shaft rotates, the motor tries to rotate in the opposite direction, but it does not because it is bolted in place.  The reaction torque transducer can measure the tensions in the stationary motor base.
Planetary mixer instrumentation for direct torque measurement does not substantially differ from that of a high shear mixer.  Engineering design should only take into account the planetary motion in addition to shaft rotation.
Other possibilities:
When the agglomeration process is progressing very rapidly, neither power consumption nor torque on the impeller may be sensitive enough to adequately reflect changes in the material.  Some investigators feel that other measurements, such as torque or force on the impeller blades may be better suited to monitor such events.

Torque vs. Power

Power consumption is easier to measure since wattmeters are inexpensive and can be installed with almost no downtime.
·       Power may not be sensitive enough for specific products or processing conditions.
·       Wear and tear of mixer and motor may cause power fluctuations.
·       Power baseline may shift with load.
·       Torque is sensing higher frequency events due to material impact on the impeller.
Although the motor power consumption is strongly correlated with the torque on the impeller [56], it is less sensitive to high frequency oscillations caused by direct impact of particles on the impeller as evidenced by FFT technique [20].
Impeller power consumption (as opposed to motor power consumption) is directly proportional to torque multiplied by the rotational speed of the impeller. The power consumption of the mixer motor differs from that of the impeller by the variable amount of power draw imposed by various sources (mixer condition, transmission, gears, couplings, motor condition, etc.).

These are the classical power and torque profiles that start with a dry mixing stage, rise steeply with binder solution addition, level off into a plateau, and then exhibit overgranulation stage. The power and torque signals have similar shape and are strongly correlated. The pattern shows a plateau region where power consumption or torque is relatively stable.
Based on the theory by Leuenberger [9, 69-72], useable granulates can be obtained in the region that starts from the peak of the signal derivative with respect to time and extends well into the plateau area. The peak of the derivative indicates the inflection point of the signal.  Prior to this point, a continuous binder solution addition may require variable quantities of liquid.  After that point, the process is well defined and the amount of binder solution required to reach a desired endpoint may be more or less constant.
It seems that monitoring torque or power can fingerprint not only the product, but the process and the operators as well. A number of publications relate to practical experience of operators on the production floor ([126], [133],[78], [95] ).
Power Consumption or Torque –  Frequency Analysis
Power consumption or torque fluctuations are influenced by granule properties (particle size distribution, shape index, apparent density) and the granulation time. Fluctuation of torque / power consumption and intensity of spectrum obtained by FFT analysis can be used for end point determination [131, 123].
Another interesting fact was reported recently by Terashita et al. [123] who observed that when the end point region of a granulation is reached, the frequency distribution of a signal reaches a steady state.
It should be repeated here that torque shows more susceptibility to high frequency oscillations.
Torque or power consumption pattern of a mixer is a function of the viscosity of the granulate and binder. With the increasing viscosity, the plateau is shortened and sometimes vanishes completely thereby increasing the need to stop the mixer at the exact end point.  At low impeller speeds or high liquid addition rates, the classic S-shape of the power consumption curve may become distorted with a steep rise leading into overgranulation [43]. 
The area under the torque-time curve is related to the energy of mixing and can be used as an endpoint parameter [38]. Area under power consumption curve divided by the load gives the specific energy consumed by the granulation process.  This quantity is well correlated with the relative swept volume [46, 107, 106].  The consumed energy is completely converted into heat of the wet mass [42], so that the temperature rise during mixing shows some correlation with relative swept volume and Froude number [49] that relates the inertial stress to the gravitational force per unit area acting on the material.

Method of Binder Addition

There are conflicting reports on the preferred method of adding the binder.  For example, Holm [47] does not generally recommend adding dry binder to the mix in order to avoid preparation of a binder solution because a homogeneity of binder distribution can not be assured.   Others recommend just the opposite [133, 75, 61].
Slow continuous addition of water (in case the water-soluble binder is dry mixed) or a binder solution to the mix is a granulation method of choice [40, 41, 42, 45, 43, 51, 58, 75, 106, 107, 135, 136, 66 and many others].  The granulating fluid should be added at a slow rate to avoid local overwetting [133].
If the binder solution is added continuously, then the method of addition (pneumatic or binary nozzle, atomization by pressure nozzle) should be considered in any endpoint determination and scale-up.
An alternative to a continuous binder addition method is to add binder all at once [49] to assure ease of processing and reproducibility, reduce processing time and to avoid wet mass densification that may occur during the liquid addition.  This latter phenomenon may obscure the scale-up effect of any parameter under investigation.

End Point Optimization

Mixing and agglomeration of particles in wet granulation have been studied extensively ([60], [3]).  The optimal endpoint can be thought of as the factor affecting a number of granule properties.
With so many variables involved in a granulation process, it is no wonder that more and more researchers throw in a number of factors together in an attempt to arrive at an optimum response.  Examples of the experimental design references that include processing factors are: [128, 74, 97, 1, 132, 130, 129, 83, 82].
The final goal of any granulation process is a solid dosage form, such as tablets.  Therefore, when optimizing a granulation process, it stands to reason to include, alongside the end-point factor, the tableting processing parameters, such as compression force or tablet press speed. 
Searching through literature I could find but one reference [1] that applied this idea using the tablet hardness, friability and disintegration as response variables. 
This study has also investigated the possibility of adjusting the tableting parameters in order to account for an inherent variability of a wet granulation process.
Granulation and mixing scale-up was specifically addressed in numerous recent publications [29, 17, 49, 83, 84, 97, 130].
A rational approach to scale-up using dimensional analysis has been in use in chemical engineering for quite some time.  This approach, based on the use of process similarities between different scales, is being applied to pharmaceutical granulation since the early work of Hans Leuenberger in 1982 [69, 72
Scale-up - Example
In this seminal and elegant work published in 1993, Horsthuis and his colleagues from Organon in The Netherlands have studied granulation process in Gral mixers of 10, 75, and 300 liter size. 
• relative swept volume
• blade tip speed
• Froude number
with respect to end point determination (as expressed by the time after which there is no detectable change in particle size), they have concluded that only constant Froude numbers results in a comparable end point.
Scale-up - Example
In this work, the University of Maryland group under the direction of Dr. Larry Augsburger has applied the ideas of Leuenberger and Horsthuis to show that, for a specific material, end point can be expressed in terms of wet massing time. 
In a scale-up study, for a constant ratio of a binder volume to a batch size, this factor was found to be inversely proportional to impeller speed when the impeller tip speed was held constant for all batches. However, this result was not corroborated by other studies or other materials.

Dimensional Analysis

Dimensionless analysis is a method for producing dimensionless numbers and it can be applied when the equations governing the process are not known. Dimensional analytical procedure was first proposed by Lord Rayleigh in 1915 [93].
p-theorem (or Buckingham theorem) [72, 139, 138, 11] states:
Every physical relationship between n dimensional variables and constants can be reduced to a relationship between m=n-r mutually independent dimensionless groups, where r = number of dimensional units, i.e. fundamental units (rank of the dimensional matrix).
Imaging that you have successfully scaled up from a 10 liter batch to 300 liter batch.  What exactly happened?  You may say: “I got lucky”.  Apart from luck, there had to be similarity in the processing of the two batches
According to the modeling theory, two processes may be considered similar if there is a geometrical, kinematic and dynamic similarity [580, 72]. 
Two systems are geometrically similar if they have the same ratio of linear dimensions.  Two geometrically similar systems are kinematically similar if they have the same ratio of velocities between corresponding points.  Two kinematically similar systems are dynamically similar when they have the same ratio of forces between corresponding points.
For any two dynamically similar systems, all the dimensionless numbers necessary to describe the process have the same numerical value [139].
It was shown, for example, that Collette Gral 10, 75 and 300 are not geometrically similar [49].
Dimensionless representation of the process is scale-invariant and thus can be easily scaled up. For notation, formulas and units, see the one of the appendices.
Dimensionless Reynolds numbers that relate the inertial force to the viscous force are frequently used to describe mixing processes [99], especially in chemical engineering. For example, for problems of water-air mixing in vessels equipped with turbine stirrers where scale-up can range from 2.5 l to 906 l (scale-up factor of 1:71) – see, e.g. Zlokarnik [139].
Froude Numbers [31] has been described for powder blending [84] and was suggested as a criterion for dynamic similarity and a scale-up parameter in wet granulation [49].  The mechanics of the phenomenon was described as an interplay of the centrifugal force (pushing the particles against the mixer wall) and the centripetal force produced by the wall, creating a “compaction zone”.
We have seen that there exists sort of a “principle of equifinality” that states: “An endpoint is an endpoint is and endpoint, no matter how it was obtained”.  The rheological and dimensional properties of the granules are similar [26].  That means that the density and dynamic viscosity are constant, and the only two variables that are left are impeller diameter and speed.
It is therefore seem appropriate to characterize and compare different mixers by the range of the Froude numbers they can produce.  A matching range of the Froude numbers would indicate the possibility of a scale-up even for the mixers that are not geometrically similar [49].
We have attempted to compute Froude numbers for mixers of different popular brands and the results are presented on the following slides. Such representation is akin to tablet press characterization using dwell time ranges. It allows some measure of comparison between otherwise incomparable devices.  I realize, of course, that this measure should not be used in absolute terms.  Rather, its use and relative usefulness will be evident during scale-up and technology transfer between various stages of product development.

Collette-Gral Mixers.  Looking at the chart, we can notice, that both the minimum and the maximum Froude numbers tend to decrease with mixer scale.  This essentially is a restatement of the fact that laboratory scale mixers tend to produce higher shear and intensity of agglomeration having relatively more powerful motors.
As was shown by Horsthuis et al. [49], it is possible to match dynamic conditions of Gral 10 and 75, or Gral 75 and 300, but not between Gral 10 and 300.

The Fielder PMA mixers exhibit essentially the same pattern of Froude number distribution as the Grals.  Again, we notice a tendency of larger mixers to have smaller Froude numbers - both extremes and ranges.
It is then understandable why scaling up the process quantified on a 10 liter mixer is sometimes so difficult to apply to PMA 600 - it is virtually impossible to match Froude numbers for a comparable dynamic conditions.
The impeller speed is the governing factor in Froude representation because it is squared.  The geometrical dimensions of the blades are of a secondary significance.

The popular Diosna mixers show a distribution pattern of Froude numbers similar to those of other brands.
It can be seen that one may expect to scale-up easily from P10 mixer to P100 but not to P600.
A word of caution: in addition to matching Froude numbers, certain corrections may be needed to account for geometric dissimilarity of vessels in the machines of different size.

Powrex mixers, distributed by Glatt, show a better distribution of Froude numbers than Grals or Fielders. 
Despite the fact that mixers with 100 liter capacity and above have only one shaft speed available, the ranges of the laboratory scale mixers seem to be wide enough to create a possibility of a match.
Once again, formulators should be encouraged to experiment with low speeds in an attempt to simulate dynamic conditions that exist in production mixers.

When a number of selected mixers from the preceding charts are placed on the same chart for comparison, an interesting conclusion can be made:  One may expect to scale up successfully from Gral 10 to Fielder PMA 65 but not to Gral 300, while PMA 65 can be matched with Diosna P250 but not with P600.  Powrex VG-10, on the other hand, covers the whole range rather nicely.
The rationale for using Froude numbers to compare different mixers may be found in the fact that at any desired endpoint (as defined by identical particle size mean and distribution), viscosity and density of the wet mass are similar in any mixer regardless of its brand and model, and the Newton Power Number Ne will depend solely on the diameter d and speed n of the impeller.  In this case any assumed relationship between Re and Fr can be reduced to Ga = Re 2 /Fr.

End Point Scale-Up using Dimensional Analysis

To set up a relevance list for mixing-granulation process, one needs to compile a complete set of all dimensional relevant and mutually independent variables and constants that affect the process.  The word “complete” is crucial here.  All entries in the list can be subdivided into geometric, physical and operational.  Each relevance list should include only one target (dependent “response”) variable.
Many pitfalls of dimensional analysis are associated with the selection of the reference list, target variable, or measurement errors (e.g. when friction losses are of the same order of magnitude as the power consumption of the motor). The larger the scale-up factor, the more precise the measurements of the smaller scale have to be [139]
In  Power number calculations, the power of the load (blades), not the idle motor should be taken into account!  Before attempting to use dimensional analysis, one has to measure / estimate power losses for empty bowl mixing. However, this baseline does not stay constant and changes significantly with load.  This may present inherent difficulties in using power meters instead of torque.  Torque, of course, is directly proportional to power drawn by the impeller so that the power number can be calculated from the torque measurements
Based on certain simplifying assumptions, Hans Leuenberger [72] suggested the following relevance list for the wet granulation process (in its final dimensionless form):
Ne  =  P / (r n3 d5)
Newton (power) number, which relates the drag force acting on a unit area of the impeller and the inertial stress.
m t / (V r)
Specific amount of granulating liquid       
V / C
Fraction of volume loaded with particles 
Fr   =  n2 d / g
Froude number, which relates the centrifugal and gravitational energy.       
d / l
Geometric number (ratio of characteristic lengths)
One of the assumptions implied that viscosity is not a system property (i.e. there is only a short range particle interaction), thus effectively excluding Reynolds number from the list.
It was then postulated that the target variable p0 is a function of the other four dimensionless numbers, i.e.  p0 = f(p1, p2, p3, p4), and p0 = f(p1) when  p2, p3, p4  ”were essentially kept constant” [72].  This was shown for planetary mixers (Dominici, Glen, Molteni) ranging from 5.75 to 60 kg.
According to Leuenberger’s school, the correct amount of granulating liquid per batch is a scale-up invariable, provided that the binder is mixed in as a dry powder and then water is added at a constant rate.  This was shown for non-viscous binders.
The ratio of quantity of granulating liquid to batch size at the inflection point S3 is constant irrespective of batch size and type of machine.
Moreover, for a constant rate of low viscosity binder addition proportional to the batch size, the rate of change (slope or time derivative) of torque or power consumption curve is linearly related to the batch size for a wide spectrum of high shear and planetary mixers.  In other words, the process end point, as determined in a certain region of the curve, is a practically proven scale-up parameter for moving the product from laboratory to production mixers of different sizes and manufacturers.
Different vessel and blade geometry will contribute to differences in absolute values of the signals but the signal profile of a given granulate composition in a high shear mixer is very similar to one obtained in a planetary mixer.
Another approach
Scale-up in fixed bowl mixer-granulators has been studied by Ray Rowe and Mike Cliff’s group using the classical dimensionless numbers of Newton (Power), Reynolds and Froude to predict end-point in geometrically similar high-shear Fielder PMA25, 100, and 600 liter machines [64].
The relevance list included power consumption of the impeller (as a response) and six factor quantities: impeller diameter, impeller speed, vessel height, specific density and dynamic viscosity of the wet mass, and the gravitational constant.
Note: Why do we have to include the gravitational constant?  Well, imagine the same process to be done on the moon - would you expect any difference?
In a subsequent communication [62] it was stated that, in order to maintain the geometric similarity between mixers, it is important to keep the batch size in proportion to the overall shape of the mixer and especially its bowl height.
The problem with using torque values from mixer torque rheometer for the viscosity of wet granulation is that it is proportional to kinematic viscosity
n  =  h / r rather than dynamic viscosity h required to calculate the Reynolds numbers.
Most recently, the same approach was applied to planetary Hobart AE240 mixer with two interchangeable bowls [29, 28].  Assuming the absence of chemical reaction and heat transfer, the following relevance list for the wet granulation process was suggested:
Dimensional quantity
net impeller power consumption (motor power consumption minus the dry blending baseline level)
impeller diameter (or radius)
impeller speed
height of granulation bed in the bowl
granulation bulk or specific density
granulation dynamic viscosity
Dimensional analysis and application of the Buckingham theorem indicates that there are 4 dimensionless quantities that adequately describe the process: Ne, Re, Fr, and h/d (the latter corresponds to Leuenberger’s dimensionless group p2). 
Again, a relationship of the form Ne = k [Re * Fr * (h/d)]-r  was postulated and the constants k and r were found empirically with a good correlation (>0.92) between the observed and predicted numbers.
The proposed quantification of mixer performance using the dimensionless Froude numbers may be utilized in a variety of scale-up paths. As a possible variation on the dimensional analysis techniques, one may want to investigate the usefulness of the Galileo dimensionless number Ga = Re 2 /Fr  for identification of an endpoint.

List of Symbols

vessel capacity (m3) – dimensional units [L3]
impeller diameter (m) – dimensional units [L]
the gravitational constant (m / s2) – dimensional units [LT-2]
height of granulation bed in the bowl
characteristic length (or height) of the vessel (m) – dimensional units [L]
amount of granulating liquid added per unit time (kg) – dimensional units [M]
impeller speed  (revolutions / s) – dimensional units [T-1]
power required by the impeller (W = J / s) – dimensional units [ML2T-5], equal to motor power consumption assuming no losses due to eddy currents, friction in couplings, etc.
total mixing time (s) – dimensional units [T]
particle volume (m3) – dimensional units [L3]
specific density of particles (kg / m3) – dimensional units [M L-5]
n  =  h/ r
kinematic viscosity (m2 / s) = – dimensional units [L2T]
dynamic viscosity (Pa*s) = – dimensional units  [M L-1 T-1]
Fr   =  n2 d / g
Froude number.  It relates the inertial stress to the gravitational force per unit area acting on the material.  It is a ratio of the centrifugal force to the gravitational force.
Ne  =  P / (r n3 d5)
Newton (power) number.  It relates the drag force acting on a unit area of the impeller and the inertial stress.
Re  =  d2 n r / h
Reynolds number.  It relates the inertial force to the viscous force.
Ga = Re2 / Fr
Galileo number

10 l
[22], [109], [110]


[112], [71], [72]
Gral 10
[326], [49]
Gral 75
[56], [69], [49]
Gral 300
[133], [49]
Gral 400


[112], [52], [69] , [71]
[61], [8]
[5], [74], [75], [110]
[137], [95]
[19], [95]
[112], [71], [121], [24]
PMA 10
[97], [137]
PMA 25
[23], [46], [100], [80], [57], [42], [110]
PMA 65
[62], [52], [97], [137]
PMA 100
[18], [19], [23]
PMA 150
PMA 250
PMA 300
PMA 600
Fukae Powtec
[65], [131]
Key 5 liter
Littleford Lodige
[112], [69], [71], [72], [17]
[49], [33]
Morton M4E
[125], [124]
[82], [83], [115]
[12], [83], [26]
Roto J
[128], [129], [130]
Roto P
Planetary mixers
[9], [50], [71], [66], [119], [120]
[12], [52], [118], [29], [34], [35], [50], [28]