Wet Granulation - End-Point Determination and Scale-Up 2007

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Wet Granulation: End-Point Determination and

Scale-UpMichael Levin a

a Metropolitan Computing Corporation, East Hanover, New Jersey, U.S.A.

Online Publication Date: 02 October 2006

To cite this Section: Levin, Michael (2006) 'Wet Granulation: End-Point

Determination and Scale-Up', Encyclopedia of Pharmaceutical Technology, 1:1,

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Wet Granulation: End-Point Determinationand Scale-Up

Michael LevinMetropolitan Computing Corporation, East Hanover, New Jersey, U.S.A.

INTRODUCTION

Wet granulation is used mainly to improve flow andcompressibility of powders, and to prevent segregationof the blend components. Particle size of the granulateis affected by the quantity and feeding rate of thegranulating liquid.

Wet massing in a high shear mixing is frequentlycompared to the fluid bed mixing and to the roller

compaction technique,

[1]

and the results seem to be for-mulation dependent. Compared to high shear granu-lation, low shear or fluid bed process requires lessfluid binder, not only resulting in a shorter drying time,but also in a less cohesive material.[2–4]

For excellent classical review of the wet granulationprocess, equipment and variables, and measurementinstruments available in the field, see papers by Holmet al.[5–12] These papers have become a standard refer-ence for numerous subsequent publications.

Because of rapid densification and agglomeration thatare caused by the shearing and compressing action of theimpeller in a high shear single pot system, mixing, granu-

lation, and wet massing can be done relatively quicklyand efficiently. The dangers lie in the possibility of over-granulation because of excessive wetting and producinglow porosity granules, thus affecting the mechanicalproperties of the tablets. As the liquid bridges betweenthe particles are formed, granules are subjected tocoalescence alongside with some breakage of the bonds.

It stands to reason that mean granule size is stronglydependent 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, gran-ules may increase in size to a certain degree while theintragranular porosity goes down. However, some

heating and evaporation may also take place leadingto a subsequent decrease in the mean granule size,especially in small-scale mixers.

Load on the main impeller is indicative of granuleapparent viscosity and wet mass consistency. It canbe seen as an interplay of acceleration (direct impactof the impeller), centrifugal, centripetal, and frictionforces that act on the particles.

According to Cliff,[13,14] binder addition rate con-trols granule density, while impeller and chopper speed

control the granule size and granulation rate. The end-point controls the mix consistency and reproducibility.Other factors that affect the granule quality includespray position and spray nozzle type, and of course,the product composition. Such variables as mixingtime and bowl or product temperature are not inde-pendent factors in the process but rather are responsesof the primary factors listed above.

WHAT IS AN END-POINT?

End-point can be defined by the formulator as a targetparticle size mean or distribution. Alternatively, theend-point can be defined in rheological terms. It hasbeen shown[15] that once you have reached the desiredend-point, the granule properties and the subsequenttablet properties are very similar regardless of thegranulation processing factors, such as impeller orchopper speed or binder addition rate. I would call this‘‘the principle of equifinality.’’

The ultimate goal of any measurement in a granu-

lation process is to estimate the viscosity and densityof the granules, and, perhaps, to obtain an indicationof the particle size mean and distribution. One of theways to obtain this information is by measuring theload on the main impeller.

Mixer instrumentation, in general, has numerousbenefits. In addition to a possible end-point determi-nation, it can be used to troubleshoot the machine per-formance (for example, help detect worn-out gears andpulleys or identify mixing and binder irregularities).Instrumentation can serve as a tool for formulationfingerprinting, assure batch reproducibility, aid in rawmaterial evaluation, process optimization, and scale-up.

WHAT CAN BE MEASURED ON AMIXER-GRANULATOR?

Current

Current in direct current (DC) motors can be used assome indication of the load on the main impellerbecause impeller torque is proportional to current in

Encyclopedia of Pharmaceutical Technology DOI: 10.1081/E-EPT-120014089Copyright# 2007 by Informa Healthcare USA, Inc. All rights reserved.4078



some intervals[13] and therefore, a current meter(ammeter) can be used for small scale DC motors.However, for alternating current motors (most oftenused in modern mixers), there may be no significantchange in the current as motor load varies up to 50%of full scale. At larger loads, the current draw mayincrease but this increase is not linearly related to load,and, consequently, the current is completely ineffectiveas a measurement of load. Moreover, the current base-line may shift with time.

Voltage

Voltage measurement generally has no relation to load.

Capacitance

Capacitive sensor responds to the moisture distri-bution and granule formation.[16–19] It provided similar

end-points (based on the total voltage change) undervarying rates of agitation and liquid addition. Capaci-tive sensor can be threaded into an existing thermo-couple port for in-process monitoring.

Conductivity

Conductivity of the damp mass[20] makes it possible toquantify uniformity of liquid distribution and packingdensity during wet massing time.

Probe Vibration

Probe vibration analysis[21,22] requires a specially con-structed probe that includes a target plate attached toan accelerometer (for in-process monitoring). Thismeasurement is based on the theory that increasinggranule size results in the increase of the accelerationof agglomerates striking the probe target. The methodhas a potential for granulation monitoring and end-point control.

Boots Diosna Probe

This probe

[23]

measured the densification and increasein size of the granules (changes in momentum of gran-ules moving with constant velocity because of a masschange of the granules). The method did not gainpopularity because of its invasive nature.

Chopper Speed

Chopper speed has no significant effect on the meangranule size.[5,6]

Impeller or Motor Shaft Speed

Rate of impeller rotation could be used as some indi-cation of the work being done on the material. [24] Asthe motor or impeller power consumption is pro-portional to the product of torque and speed, the latteris an important factor in evaluating the correspondingload.

Motor Slip and Motor Load Analyzer

Motor slip is the difference between the rotationalspeed of an idle motor and the motor under load. [25,26]

Motor slip measurements, although relatively inex-pensive, do not offer advantages over the powerconsumption measurements. The method did notgain popularity, probably because the slip is notlinearly related to the load,[27] despite some claims tothe contrary.

Impeller Tip Speed

Impeller tip speed corresponds to the shear rate andhas been used as a scale-up parameter in fluid mix-ing.[28] For processing of lactose granulations in Gralmixers, however, it was shown by Horsthuis et al.[29]

that the same tip speed did not result in the same end-point [in terms of particle size distribution (PSD)].These findings were contradicted by other studies withFielder mixers indicating that for a constant tip speed,successful scale-up is possible when liquid volume isproportional to the batch size and wet massing timeis related to the ratio of impeller speeds.[30]

Relative Swept Volume

Relative swept volume, that is, the volume swept by theimpeller (and chopper) per unit time, divided by themixer volume, has been suggested as a scale-upfactor.[11,12,31] This parameter is related to work doneon the material and was studied extensively at variousblade angles.[32] Higher swept volume leads to highertemperature and denser granules. However, it wasshown by Horsthuis et al.[29] that the same relativeswept volume did not result in the same end-point (interms of PSD).

Temperature

Product and jacket temperature are usually measuredby thermocouples. These response variables are con-trolled by a variety of factors, notably, the speed of the main impeller and the rate of the binder addition.

Wet Granulation: End-Point Determination and Scale-Up 4079



Binder Addition Rate

There are conflicting reports on the preferred methodof adding the binder. For example, Holm[32] does notgenerally recommend adding dry binder to the mix(as commonly done in order to avoid preparation of a binder solution) because homogeneity of binder dis-tribution cannot be assured. Others recommend justthe opposite.[34–36]

Slow continuous addition of water (in case the water-soluble binder is dry mixed) or a binder solution to themix is a granulation method of choice.[5,6,10–12,37–44]

The granulating fluid should be added at a slow rate toavoid local overwetting.[33]

If the binder solution is added continuously, thenthe method of addition (pneumatic or binary nozzle,atomization by pressure nozzle) should be consideredin any end-point determination and scale-up.

An alternative to a continuous binder liquid additionmethod is to add binder liquid all at once[29] to assure the

ease of processing and reproducibility, reduce processingtime, and to avoid wet mass densification that may occurduring the liquid addition. This latter phenomenonmay obscure the scale-up effect of any parameter underinvestigation.

Power Consumption

One of the most popular and relatively inexpensivemeasurements is the power consumption of the mainmixer motor. It is measured by a watt transducer ora power cell utilizing Hall effect (a measurable trans-

versive voltage between the two radial sides of a cur-rent conductor in a magnetic field, an effectdiscovered by E.H. Hall in 1879).

Power consumption of the mixer motor for end-point determination and scale-up is widely used(Leuenberger[43] and subsequent work, Holm[5] andsubsequent work, Landin et al.[45–48], Faure et al.[49–53],andmany others[16,33,37–44,54,55]) because the measurementis economical, does not require extensive mixer modifica-tions, and is well correlated with the granule growth.

Power consumption correlates with the mean gran-ule size of a granulation,[8] although the correlation isnot always linear in the entire range. Intragranular

porosity also shows some correlation with power con-sumption.[56] Normalized work of granulation (powerprofile integrated over time) can accurately determineend-points and is correlated well with properties of the granulates.[57]

The main problem with the power consumptionmeasurements is that this variable reflects load on themotor rather than load on the impeller. It relates to theoverall mixer performance, depends on the motorefficiency, andcan changewith time regardless of theload.

Motor power consumption is a product of current,voltage, and the so-called power factor. In the rangeof interest, motor power consumption is generally pro-portional to load on the motor and, to some degree,can reflect the load on the impeller (Fig. 1).

However, up to 30% of the power consumption of amotor can be attributed to no-load losses because of windage (by cooling fan and air drag), friction in thebearings, and core losses that comprise hysteresis andeddy current losses in the motor magnetic circuit. Loadlosses include stator and rotor losses (resistance of materials used in the stator, rotor bars, magnetic steelcircuit) and stray load losses such as current losses inthe windings.[58]

Attempts to use a no-load (empty bowl, or dry mix)value as a baseline may be confounded by a possiblenonlinearity of friction losses with respect to theload.[59] As the load increases, so does the current drawof the motor. This results in heat generation thatfurther impacts the power consumption.[27] A simple

test might be to run an empty mixer for several hoursand see if there is any shift in the baseline. Also, asthe motor efficiency drops with age, the baseline mostdefinitely shifts over time.

Motor power consumption is non-linearly related tothe power transmitted to the shaft[60] and the degree of this non-linearity could only be ‘‘guestimated.’’

Impeller Torque

In a mixing process, changes in torque on the blades andpower consumption of the impeller occur as a result of

Power

CurrentVoltage

ldle FullLoad

Fig. 1 Voltage, current, and power consumption of a typicalmixer motor.

4080 Wet Granulation: End-Point Determination and Scale-Up



change in the cohesive force or the tensile strength of theagglomerates in the moistened powder bed.

Direct torque measurement requires installation of strain gages on the impeller shaft or on the couplingbetween the motor and impeller shaft (Fig. 2). As theshaft is rotating, a device called slip ring is used totransmit the signal to the stationary data acquisitionsystem.

Planetary mixer instrumentation for direct torquemeasurement does not substantially differ from thatof a high shear mixer. Engineering design should onlytake into account the planetary motion in addition toshaft rotation.[61]

Impeller torque is an excellent in-line PAT measure-ment of the load on the main impeller.[39,62]

Torque Rheometer

A torque rheometer is a device that provides an off-line

measurement of torque required to rotate the blades of the device and this torque can be used to assess rheolo-gical properties of the granulation. It has been exten-sively used for end-point determination.[45,63–65] Thetorque values thus obtained were termed as ‘‘measureof wet mass consistency.’’[50,51,66]

One of the main concerns is that using the torquevalue that the unit is reporting instead of the dynamicviscosity for calculation of Reynolds numbers rendersthe latter to become dimensional. Therefore, theReynolds number calculated from torque rheometerdata is referred to as ‘‘pseudo-Reynolds’’ dimensionalnumber. Because of the fact that torque was shown tobe proportional to a kinematic (rather than dynamic)

viscosity,[67] it can have a conditional use in the dimen-sional analysis of the process, as will be shown below.

Reaction Torque

By the third law of Newton, for every force there is acounterforce, collinear, equal and opposite in direc-tion. As the impeller shaft rotates, the motor tries torotate in the opposite direction, but it does not becauseit is bolted in place. The tensions in the stationarymotor base can be measured by a reaction torquetransducer.

Reaction torque is a less expensive alternative todirect impeller torque and is recommended for mixersthat have the motor and impeller shafts axially aligned(in this case, the reaction torque is equal to direct tor-que and is opposite in sign).

Other Possibilities

When the agglomeration process is progressing veryrapidly, neither power consumption nor torque onthe impeller may be sensitive enough to adequatelyreflect changes in the material. Some investigators feelthat other measurements, such as torque or force onthe impeller blades may be better suited to monitorsuch events.

There are other ideas floating around, for example,use of neural network to describe and predict the beha-vior of the wet granulation[68] or control the end-pointby rapid image processing system.[69]

A technique for measuring tensile strength of gran-ules, in addition to power consumption measurement,to facilitate optimal end-point determination was recentlydescribed by Betz, Burgin, and Leuenberger.[54]

Powder flow patterns in wet granulation can bestudied using positron emission particle tracking.[70]

Eventually, this and similar techniques can be used tovalidate various mathematical and statistical modelsof the process.

Emerging Technology

Acoustic

Applicability of piezoelectric acoustic emission sensorsto end-point determination has been studied since thebeginning of this century.[71] The technique is verypromising, especially because it is non-invasive, sensi-tive, and relatively inexpensive. Granulation processsignatures obtained with acoustic transducer can beused to monitor changes in particle size, flow, andcompression properties.[72,73]Fig. 2 Schematic of a direct impeller torque transducer.




Near-infrared

Use of a refractive near-infrared (NIR) moisture sen-sor for end-point determination of wet granulationwas described by several authors.[74,75] There are tech-nological challenges associated with this approach, asthe sensor can only measure the amount of water atthe powder surface.

NIR monitoring of the granulation process wasattempted by researchers at many major pharmaceuti-cal corporations with a modest success. In particular,yet unpublished work by David Rudd of Glaxo-SmithKline in England should be mentioned as a partof the global effort in the field of Process AnalyticalTechnology.

Focused beam reflectance

measurement

Focused beam reflectance measurement (FBRM) is aparticle size determination technique based on a laserbeam focusing in the vicinity of a sapphire windowof a probe. The beam follows a circular path at speedsof up to 6 m/sec. When it intersects with the edge of aparticle passing by a window surface, an optical collec-tor records a backscatter signal. The time interval of the signal multiplied by the beam speed represents achord length between two points on the edge of a par-ticle. The chord length distribution (CLD) can berecalculated to represent either a number or volumeweighted PSD.

In many cases, where precision is more important

than accuracy, CLD measurements are adequate tomonitor dynamic changes in process parametersrelated to the particle size and shape, concentration,and rheology of fluid suspensions.

Several attempts were made to evaluate the useof FBRM particle size analyzer as a potential toolfor granulation end-point determination.[76] Dilworthet al.[77] have compared power consumption, FBRM,and acoustic signals in a study of a wet granulationprocess in Fielder PMA 200 mixer. It was found thatthese techniques were complimentary, with FBRMprobe capable to follow median granule size growtheven when the power consumption curve showed a

plateau.A major disadvantage of the FBRM method is that

the measured CLD does not directly represent a PSD.Conversion of CLD to PSD is not straightforward andrequires sophisticated mathematical software that isnot easy to validate. Moreover, CLD depends onoptical properties and shape of the particles, as wellas the focal point position. The total number of countsmeasured is a function both of solids concentrationand probe location.

END-POINT DETERMINATION

End-point detection in wet granulation has become amajor scientific and technological challenge.[78] Moni-toring granulation is most commonly achieved by col-lecting either power or torque signals, or both. In whatfollows, we will compare both methods.

Torque vs. Power

When we say, ‘‘power consumption,’’ we usually referto the main motor. It reflects the load on the motorbecause of useful work, as well as the power neededto run the motor itself (losses because of eddy currents,friction in couplings, etc.).

It is quite possible (and, indeed, quite pertinent) totalk about the power consumption of the impeller,which is, obviously, quantitatively less than the powerconsumption of the motor and relates directly to the

load on the impeller.

Power Torque Speed

Impeller power consumption can be calculated as aproduct of the direct torque, rotational impeller speed,and a coefficient (usually equal to 2p times a unitconversion factor, if required).

The power consumption of the mixer motor differsfrom that of the impeller by the variable amount of power draw imposed by various sources (mixercondition, transmission, gears, couplings, motor con-dition, etc.).

Compared to impeller torque, motor power con-sumption is easier to measure; wattmeters are inexpen-sive and can be installed with almost no downtime.However, motor power signal may not be sensitiveenough for specific products or processing conditions.Wear and tear of mixer and motor may cause powerfluctuations. Moreover, power baseline may shift withload.

Impeller torque, on the other hand, is closer towhere the action is, and is directly related to theload on the impeller. Torque is not affected by mixercondition.

Although the motor power consumption is strongly

correlated with the torque on the impeller,[39] it is lesssensitive to high frequency oscillations caused by directimpact of particles on the blades as evidenced by FastFourier Transform (FFT) technique.[16]

Power consumption or torque fluctuations are influ-enced by granule properties (PSD, shape index, andapparent density) and the granulation time. Fluctua-tion of torque/power consumption and intensity of spectrum obtained by FFT analysis can be used forend-point determination.[38]




It was observed that when the end-point region of a granulation is reached, the frequency distribution of a power consumption signal reaches a steady state. [79]

It should be repeated here that torque shows moresensitivity to high frequency oscillations.

Torque and Power Profiles

Fig. 3 illustrates the classical power and torqueprofiles that start with a dry mixing stage, rise steeplywith binder solution addition, level off into a plateau,and then exhibit overgranulation stage. The powerand torque signals have similar shape and are stronglycorrelated. The pattern shows a plateau region wherepower consumption or torque is relatively stable.

The peak of the derivative indicates the inflectionpoint of the signal. Based on the theory by Leuenberger(1979 and subsequent work), usable granulates can beobtained in the region that starts from the peak of the

signal derivative with respect to time and extends wellinto the plateau area.[44] Prior to the inflection point,a continuous binder solution addition may requirevariable quantities of liquid. After that point, the pro-cess is well defined and the amount of binder solutionrequired to reach a desired end-point may be more orless constant.

Torque or power consumption pattern of a mixeris a function of the viscosity of both the granulateand binder. With the increasing viscosity, the plateau isshortened and sometimes vanishes completely, therebyincreasing the need to stop the mixer at the exactend-point.

At low impeller speeds or high liquid addition rates,the classic S-shape of the power consumption curvemay become distorted with a steep rise leading intoovergranulation.[9]

The area under the torque-time curve is related tothe energy of mixing and can be used as an end-pointparameter. Area under power consumption curvedivided by the load gives the specific energy consumedby the granulation process. This quantity is well corre-lated with the relative swept volume.[11,12,32]

The consumed energy is completely converted intoheat of the wet mass,[7] so that the temperature riseduring mixing shows some correlation with relativeswept volume and Froude number[29] that relates theinertial stress to the gravitational force per unit areaacting on the material.

Fig. 4 represents a record of a typical granulationbatch done by an experienced operator on largeHobart mixer. You can see that the batch was stoppedon the downslope of the derivative.

On a Fig. 5 you can see another batch made by thesame operator. This time it is a power consumptiontrace, but again it extends beyond the peak of thederivative and the end-point thus can be deemed repro-

ducible.In the batch represented in Fig. 6, a novice operatortrainee has stopped the batch well before the peak of the derivative. This required a major adjustment of the tableting operation (force and speed) to producetablets in an acceptable range of material properties(hardness and friability).

In this batch (Fig. 7), the same novice operator hasstopped the granulation process, opened the lid, took asample, and decided to granulate for another 10 sec.You can see that there is no indication that the peakof the derivative was reached at the end-point.

Thus, it seems that the monitoring torque or power

can fingerprint not only the product, but the processand the operators as well.

A number of publications relate to the practicalexperience of operators on the production floor.[33,80–82]

200

150

100

50

50 100 150 2000

0.5

1

2

2.5

3

3.5

1.5

00

T o r q u e ,

N - m

P o w e r ,

k W

Time, sec

Start of

binder addition

Torque

Power

Torque slope

Power slope

Fig. 3 Impeller torque and motorpower consumption for a small highshear mixer (Fielder PMA 10).




End-Point Optimization

Agglomeration of particles in wet granulation havebeen studied extensively.[24,83] The optimal end-pointcan be thought of as the factor affecting a number of agglomerate properties (Fig. 8).

With so many variables involved in a granulationprocess, it is no wonder that more and more research-ers throw in a number of factors together in an attemptto arrive at an optimum response.[34,84–92]

The final goal of any granulation process is a soliddosage form, such as tablets. Therefore, when optimiz-ing a granulation process, the list of factors affectingtablet properties may include both the granulation

end-point and the tableting processing parameters,such as compression force or tablet press speed.

In one of the most interesting works based on theexperimental design approach, an attempt was madeto find a statistical relationship between the majorfactors affecting both granulation and compaction,namely, granulation end-point, press speed (dwell time),and compression force.[93] The resulting equationallowed optimization of such standard response param-eters as tablet hardness, friability, and disintegrationtime. This study has also investigated the possibility of adjusting the tableting parameters in order to accountfor an inherent variability of a wet granulation process.

Multivariate optimization of wet granulation mayinclude hardness, disintegration, and ejection asresponse variables.[94] Compressibility property of

800

600

400

200

0 50 100 150 200

T o r q u e , l b - i n c h

Time, sec

TorqueTorque slope

Start of

binder addition

Batch No. JJ0202A

Fig. 4 A torque profile in a typical production batch.

8.0

6.0

4.0

2.0

50 100 1500

P o w e r ,

k W

Time, sec

PowerPower_slope

Start ofbinder addition

Batch No. JJ0202C

Fig. 5 Another batch by the same operator (powerconsump-tion profile).

6.0

4.0

2.0

0 50 100 150 200 250 300

P o w e r , k W

Time, sec

PowerPower slope

Batch No. JJ0214A

Start ofbinder addition

Fig. 6 A batch by a novice operator (power consumptionprofile).

Fig. 7 Another batch by inexperienced operator (torqueprofile).




granulations is extremely sensitive to various proces-sing parameters of wet granulation.[95]

Recently, the experimental design procedure was

applied to low shear wet granulation[96]

with a factorialdesign used to evaluate the influence of such factors asbinder strength and agitator speed.

End-Point Reproducibility

As will be shown in the following section, for everyblend and a fixed set of values for processing factors(such as mixer geometry, blade speed, powder volume,amount, and method of addition of granulatingliquid), a wet granulation process state (end-point) iscompletely characterized by rheological properties of

the wet mass (density, viscosity), which are, in turn, afunction of particle size, shape, and other properties.The process can be quantified with the help of dimen-sionless Newton Power Number N p that will assume acertain numerical value for every state (condition) of the granulate. Under fixed processing conditions,N p will be proportional to Net Power ConsumptionDP for any end-point (defined, in part, by wet massdensity). Thus, in order to reproduce an end-point, itis sometimes sufficient to monitor power of theimpeller (or the motor) and stop when a predefinednet level of the signal is reached. If, however, any of the processing variables or the rheological definition

of the end-point has changed, a more sophisticatedapproach is required, as described below.

END-POINT SCALE-UP

Scale-Up Attempts

Numerous studies were undertaken in an attemptto determine empirically (and, lately, with a solid

theoretical foundation) useful scale-up parameters of the wet granulation process.[50,91,97]

In a seminal and elegant work published in 1993,Horsthuis et al. from Organon in The Netherlandshave studied granulation process in Gral mixers of 10, 75, and 300 L size.[29] Comparing relative swept vol-ume, blade tip speed and Froude numbers with respectto end-point determination (as expressed by the timeafter which there is no detectable change in particlesize), they have concluded that only constant Froudenumbers result in a comparable end-point.

In another attempt to determine good scale-upparameters, the University of Maryland group underthe direction of Dr. Larry Augsburger[30] has appliedthe ideas of Leuenberger and Horsthuis to show that,for a specific material, end-point can be expressed interms of wet massing time. For a constant ratio of abinder volume to a batch size, this factor was foundto be inversely proportional to impeller speed whenthe impeller tip speed was held constant for all batches.

However, this result was not corroborated by otherstudies or other materials.Yet another example of semiempirical scale-up

effort[97] was based on the fact that normalized powerprofiles are very similar and allow for direct compari-son of different size granulators, at least for the equip-ment and materials used in the study. Normalizedpower curve rose at a relatively constant rate in theregion where the ratio of water to dry mass is 0.1–0.2(‘‘slope plateau’’). Despite a rapid increase in the slopeof the power curve, the desired end-point was stilldetectable at a moment when the slope of the powerconsumption signal exceeded the plateau level by a fac-

tor of 5 (empirical observation). Using this approach,an acceptable end-point (target particle size of 135mm) was first established on a 10-L Fielder andthen scaled to 65-L Fielder and 250-L Diosna.

Dimensional Analysis

A rational approach to scale-up using dimensionalanalysis has been in use in chemical engineering forquite some time. This approach, based on the use of process similarities between different scales, was beingapplied to pharmaceutical granulation since the early

work of Leuenberger in 1979.[43]Dimensional analysis is a method for producing

dimensionless numbers that completely describe theprocess. The analysis should be carried out before themeasurements are made because dimensionless num-bers essentially condense the frame in which the mea-surements are performed and evaluated. The methodcan be applied even when the equations governingthe process are not known. Dimensional analyticalprocedure was first proposed by Rayleigh in 1915.[98]

End Point Level

Mean Particle Size

Disintegration

Friability

HardnessOptimum

Fig. 8 Wet granulation end-point as a factor in tabletingoptimization.




Principle of Similitude

Imagine that you have successfully scaled up from a10-L batch to 300-L batch. What exactly happened?You may say: ‘‘I got lucky.’’ Apart from luck, therehad to be similarity in the processing and the end-pointconditions of the wet mass of the two batches.

According to the modeling theory, two processesmay be considered similar if there is a geometrical,kinematic, and dynamic similarity.[99]

Two systems are called geometrically similar if theyhave the same ratio of characteristic linear dimensions.For example, two cylindrical mixing vessels are geome-trically similar if they have the same ratio of height todiameter.

Two geometrically similar systems are called kine-matically similar if they have the same ratio of veloci-ties between the corresponding system points. Twokinematically similar systems are dynamically similarwhen they have the same ratio of forces between

the corresponding points. Dynamic similitude for wetgranulation would imply that the wet mass flowpatterns in the bowl are similar.

The gist of dimensionless analysis is as follows: Forany two dynamically similar systems, all the dimen-sionless numbers necessary to describe the processhave the same numerical value.[100] Once a process isexpressed in terms of dimensionless variables, we aremagically transferred to a world where there is nospace and no time. Therefore, there is no scale and,consequently, there are no scale-up problems. The pro-cess is characterized solely by numerical values of thedimensionless variables (numbers). In other words,

dimensionless representation of the process is scale-invariant.

Lack of geometrical similarity often is the mainobstacle in applying the dimensional analysis to solv-ing the scale-up problems. It was shown, for example,that Collette Gral 10, 75, and 300 are not geometricallysimilar.[29] In such cases, a proper correction to theresulting equations is required.

Dimensionless Numbers

Dimensionless numbers most commonly used to

describe the wet granulation process are Newton,Froude, and Reynolds:

N p ¼ DP =ðr n3 d5Þ NewtonðpowerÞ

Fr ¼ n2 d= g Froude

Re ¼ d2 n r=Z Reynolds

(for list of symbols, notation, and dimensions, seeAppendix).

Newton (power) number, which relates the dragforce acting on a unit area of the impeller and the iner-tial stress, represents a measure of power requirementto overcome friction in fluid flow in a stirred reactor.In mixer-granulation applications, this number canbe calculated from the power consumption of theimpeller or estimated from the power consumption of the motor.

Froude Number[101] has been described for powderblending and was suggested as a criterion for dynamicsimilarity and a scale-up parameter in wet granulation.[29]

The mechanics of the phenomenon was described asinterplay of the centrifugal force (pushing the parti-cles against the mixer wall) and the centripetal forceproduced by the wall, creating a ‘‘compaction zone.’’

Reynolds numbers relate the inertial force to theviscous force.[102] They are frequently used to describemixing processes and viscous flow, especially in chemi-cal engineering.[103]

We have seen that there exists a sort of ‘‘principle of

equifinality’’ that states: ‘‘An end-point is an end-pointis and end-point, no matter how it was obtained.’’Different processing pathways can lead to differentend-points, each with its own set of granulation pro-perties. However, once an end-point is reached, it ischaracterized by certain numerical values of thedimensionless variables describing the process, andthese values will be independent of scale.

At the same end-point, no matter how defined, therheological and dimensional properties of the granulesare similar. As we will see from the examples describedbelow, that means that the density and dynamic vis-cosity of the wet mass are constant, and the only vari-

ables that are left are the process variables, namelybatch mass, impeller diameter and speed, and thegeometry of the vessel.

Comparison of Attainable Froude Numbers

Horsthuis et al.[29] showed that an end-point could bereproduced and scaled up in Gral mixers by keepingthe Froude numbers constant. For the same end-point,in dynamically similar mixers (same geometrical ratios,and same flow patterns), all dimensionless numbersdescribing the system should have the same numerical

value, but Froude numbers for any mixer are easiest tocompute.

Each mixer has a range of attainable Froude num-bers, and an end-point transfer between mixers can onlybe achieved when such ranges overlap. Fig. 9 repre-sents such a range for Collette Gral mixers. It can beseen that Gral 10 and Gral 150 have no overlap of Froude number ranges, and therefore, a direct scale-up is not possible (in addition, Gral mixers are notexactly similar geometrically, as was stated elsewhere).






matrix (you will see examples in the case studiesbelow).

The rows of the matrix consist of the basic dimen-sions, while the columns represent the physical quan-tities from the relevance list. The most important

physical properties and process-related parameters, aswell as the ‘‘target’’ variable (that is, the one we wouldlike to predict on the basis of other variables) areplaced in one of the columns of the residual matrix.

The core matrix is then linearly transformed into amatrix of unity where the main diagonal consists onlyof ones and the remaining elements are all zero. Thedimensionless numbers are then created as a ratio of the residual matrix and the core matrix with the expo-nents indicated in the residual matrix. This rather sim-ple process will be illustrated below in the examples.

Case Study I: Leuenberger (1979,1983)

This example is based on the groundbreaking studiesconducted by Leuenberger at the University of Baseland Sandoz AG.[43,105–108]

The Relevance List in Table 1 reflects certainassumptions used to simplify the model, namely, thatthere are short-range interactions only and no viscosityfactor (and therefore, no Reynolds number).

Why do we have to consider the gravitational con-stant? Well, imagine the same process to be done onthe moon—would you expect any difference?

One target variable (Power consumption) and sevenprocess variables/constants thus represent the numbern ¼ 8 of the P-theorem. The number of basic dimen-sions r ¼ 3 (M, L, and T). According to the theorem,the process can be reduced to the relationship betweenm ¼ n r ¼ 8 3 ¼ 5 mutually independentdimensionless groups.

The Dimensional Matrix in Table 2 was constructedas described above, with the rows listing the basicdimensions and the columns indicating the physicalquantities from the relevance list.

Transformation of the dimensional matrix (Table 3)into a unity matrix is straightforward. To transform-3in L-row/r-column into zero, one linear transform-ation is required. The subsequent multiplication of the T-row by 1 transfers the 1 of then-column to þ1.

The five dimensionless groups are formed from thefive columns of the residual matrix by dividing eachelement of the residual matrix by the column headers

of the unity matrix, with the exponents indicated inthe residual matrix.The residual matrix contains five columns; therefore

five dimensionless P groups (numbers) will be formed(Table 4).

The end result of the dimensional analysis is anexpression of the form

P0 ¼ f ðP1;P2;P3;P4Þ

Assuming that the groups P2, P3, P4 are ‘‘essentiallyconstant,’’ the P-space can be reduced to a simplerelationship P0 ¼ f (P1), that is, the value of Newtonnumber N p at any point in the process is a function of the specific amount of granulating liquid.

Up to this point, all the considerations were rathertheoretical. From the theory of modeling, we knowthat the above dimensional groups are functionallyrelated. The form of this functional relationship f ,however, can be established only through experiments.

Leuenberger and his group have empirically estab-lished that the characteristic (that is, relative to thebatch size) amount of binder liquid required toreach a desired end-point (as expressed by the absolutevalue of N p and, by proxy, in terms of net power

Table 1 The relevance list used by Leuenberger (1983)

Quantity Symbol Units Dimensions

Power consumption P Watt M L2 T3

Specific density r kg/m3 M L3

Blade diameter d m L

Blade velocity n rev/sec T1

Binder amount s kg MBowl volume V b m3 L3

Gravitationalconstant

g m/sec2 L T2

Bowl height H m L

Table 2 The dimensional matrix for case study I

Core matrix Residual matrix

q d n P s V b g H

Mass (M) 1 0 0 1 1 0 0 0

Length (L) 3 1 0 2 0 3 1 1

Time (T) 0 0 1 3 0 0 2 0




consumption DP ) is ‘‘scale-up invariable,’’ that is,independent of the batch size (Fig. 11), thus specifyingthe functional dependence f and establishing rationalbasis for granulation scale-up.

Experiments with five different planetary mixerswith batch sizes ranging from 3.75 kg to 60 kg showedthat, if the binder is mixed in as a dry powder and thenliquid is added at a constant rate proportional to thebatch size, the ratio of the granulation liquid quantityto a batch size is constant. This was shown for non-

viscous binders.The ratio of quantity of granulating liquid to batchsize at the inflection point of power vs. time curve isconstant irrespective of batch size and type of machine.Moreover, for a constant rate of low viscosity binderaddition proportional to the batch size, the rate of change (slope or time derivative) of torque or powerconsumption curve is linearly related to the batch sizefor a wide spectrum of high shear and planetary mix-ers. In other words, the process end-point, as determ-ined in a certain region of the curve, is a practicallyproven scale-up parameter for moving the productfrom laboratory to production mixers of different sizes

and manufacturers.As we have indicated before, for any desired end-

point, the power consumption will be proportional tothe Newton power number, at a constant mixer speed.

The Leuenberger’s ideas relating to the use of powerconsumption for wet granulation end-point determi-nation were tested and implemented by numerousresearchers.[9,33,37,41,43]

In 2001, Holm, Schaefer, and Larsen[78] haveapplied the Leuenberger method to study various pro-cessing factors and their effect on the correlationbetween power consumption and granule growth. Theyhave found that such a correlation did indeed exist butwas dependent, as expected, on the impeller design, theimpeller speed, and the type of binder. The conclusionwas that it was possible to control the liquid additionby the level detection method whereby the liquidaddition is stopped at a predetermined level of power

consumption. An alternative approach involves aninflection point (peak of the signal derivative withrespect to time).

Different vessel and blade geometry will contributeto the differences in absolute values of the signals.However, the signal profile of a given granulate com-position in a high shear mixer is very similar to theone obtained in a planetary mixer.

For accuracy, in power number N p calculations, thepower of the load on the impeller rather than the mixermotor should be used. Before attempting to use dimen-sional analysis, one has to measure/estimate powerlosses for empty bowl or dry stage mixing. Unlike

power consumption of the impeller (based on torquemeasurements), the baseline for motor power con-sumption does not stay constant and changes signifi-cantly with load on the impeller, mixer condition, ormotor efficiency. This may present inherent difficultiesin using power meters instead of torque. Torque, of course, is directly proportional to power drawn bythe impeller (the power number can be determined

Table 4 Dimensionless P groupsa

P group Expression Definition

P0 P /(r1 d5 n3) ¼ N p Newton (Power) number

P1 s/(r1 d3 n0) q/(rV p) Specific amount of liquidV p volume of particlesq ¼ binder addition ratet ¼ binder addition time

P2 V b/(r0 d3 n0) (V p/V b)1 Fractional particle volume

P3 g /(r0 d1 n2) ¼ Fr1 Froude Number

P4 H /(r0 d1 n0) ¼ H /d Ratio of lengthsaFormed from the matrix in Table 3.

Table 3 The transformed dimensional matrix for case study I

Unity matrix Residual matrix

q d n P s V b g H

M 1 0 0 1 1 0 0 0

3M þ L 0 1 0 5 3 3 1 1

T 0 0 1 3 0 0 2 0




from the torque and speed measurements) and has arelatively constant baseline.

Case Study II: Landin et al. (1996)

Scale-up in fixed bowl mixer-granulators has beenstudied by Rowe and Cliff’s group[46] using the clas-sical dimensionless numbers of Newton (power),Reynolds, and Froude to predict the end-point ingeometrically similar high-shear Fielder PMA 25,100, and 600 L machines.

The relevance list (Table 5) included power con-sumption of the impeller (as a response) and six factorquantities: impeller diameter, impeller speed, vesselheight, specific density and dynamic viscosity of the

wet mass, and the gravitational constant.Note that dynamic viscosity has replaced the binder

amount and bowl volume of the Leuenberger’s rel-evance list, thus making it applicable to viscous bindersand allowing long-range particle interactions respon-sible for friction.

The dimensional matrix for Case Study II (Table 6)is different from Table 2: the columns for mass [M]and bowl volume [L3] are replaced by a viscosity

[M L1 T1] column. Evidently, it was assumed thatthe mass and volume could be adequately representedin the relevance list by the density and powder heightin a semicylindrical vessel of a known diameter.

The residual matrix (Table 7) contains fourcolumns; therefore four dimensionless P groups(numbers) will be formed, in accordance with theP-theorem (7 variables 3 dimensions ¼ 4 dimen-sionless groups).

Table 8 lists the resulting groups; they correspond toNewton power, Reynolds, and Froude numbers, andthe ratio of characteristic lengths.

Under the assumed condition of dynamic similarity,from the dimensional analysis theory, it follows thatP0 ¼ f (P1, P2, P3,), and, therefore, N p ¼ f (Re,Fr, H /d).

When corrections for gross vortexing, geometricdissimilarities, and powder bed height variation weremade, data from all mixers (Fielder PMA 25, 100and 600 L) correlated to the extent that allows predic-

tions of the optimum end-point conditions. The linearregression of Newton number (power) on the productof adjusted Reynolds number, Froude number, andthe Geometric number (in log/log domain) yields(Fig. 12) an equation of the form:

log10 N p ¼ a log10ðRe Fr H =dÞ þ b

where b ¼ 796 and a ¼ 0.732.Theoretically, in such a representation of the granu-

lation process, a slope a ¼ 1 would signify a truelaminar flow whereby a slope significantly less than

1 or approaching 0 would indicate turbulence. Thus,one would expect planetary mixers to have a slope clo-ser to 1 compared to that of high shear granulators.However, the results described here and in subsequentstudies do not show a clear difference betweenthe slopes of regression for planetary and high-shearmixers.

However, the correlation coefficient of 0.7854 forthe final curve fitting effort indicates the presence of many unexplained outlier points. One of the possibleconcerns was an inherent error in measuring the heightof the powder bed from the wet mass density.

In a subsequent communication[47] it was shown

that, in order to maintain geometric similarity, it isimportant to keep the batch size proportional to thebowl shape.

Another concern is the interpretation of data frommixer torque rheometer that was used to assess the vis-cosity of wet granulation. The torque values obtainedfrom the rheometer were labeled ‘‘wet mass consist-ency’’ and were used instead of dynamic viscosity tocalculate Reynolds numbers. It was shown[45,66] thatsuch torque values are proportional to kinematic

20

16

12

N p

Granulating liquid quantity

(% of batch size)

8

7 12 17 22 27

Fig. 11 Newton power number as a function of the charac-teristic liquid quantity. (Adapted from Ref.[43].)

Table 5 Relevance list for case study II (Landin et al., 1996)


Power consumption P Watt M L2 T3


Blade diameter d m L

Blade speed n rev/sec T1

Dynamic viscosity Z Pa sec M L1 T1


g m/sec2 L T2

Bowl height H m L




viscosity n ¼ Z/r rather than dynamic viscosity Z

required to compute Reynolds numbers. The degreeof proportionality between n and Z was found to beformulation dependent.

Consequently, it was prudent to acknowledge thatthe above regression equation is not dimensionlessbecause for all practical purposes, the Reynolds num-ber Re was replaced by CRe, what the authors calleda ‘‘pseudo Reynolds number’’ with the dimensions[L3 T]. This predicament did not deter a plethora of other studies in the same line of reasoning to be pub-

lished in recent years. Note that this pseudo Reynoldsnumber has a physical meaning: it is a reciprocal of volumetric flow rate.

Case Study III: Faure et al. (1998)

The same approach was applied to planetary HobartAE240 mixer with two interchangeable bowls, 5 and8.5 L.[49] Assuming the absence of chemical reactionand heat transfer, the following relevance list for thewet granulation process was suggested (Table 9):

One difference from Table 5 of the previous study is

the use of net power DP that was defined as motorpower consumption under load minus the dry blendingbaseline level.

An assumption was made that a motor drive speedis proportional to the impeller blade speed. Anotherconsideration was that the ratio of characteristiclengths h/d is proportional to (and, therefore, can bereplaced by) a fill ratio V m/V b, which was, in turn,shown to be proportional to (and therefore, could bereplaced in the final equation by) the quantity m/(r d3). This is a preferred method of representinga fill ratio because the wet mass m is easier to measurethan the height of the granulation bed in the bowl.

Dimensional analysis and application of the Buck-ingham theorem lead to four dimensionless quantitiesthat adequately describe the process: Ne, CRe, Fr,and h/d. As before, a relationship of the form

N p ¼ 10bðCRe Fr rR3b=mÞa;or

log10 N p ¼ a log10ðCRe Fe rR3b=mÞ þ b

was postulated and the constants a and b (slope andintercept in a log–log domain) were found empirically

(b ¼ 2.46 and a ¼ -0.872) with a good correlation(>0.92) between the observed and predicted numbers(Fig. 13). Radius of the bowl Rb cubed was used torepresent the bowl volume V b. The graph indicatesa collection of end-points produced with differentmixers and different processing factors.

It was noted that the above equation could be inter-preted to indicate that

DP Z d2 V m=V b

the net power consumption of the impeller variesdirectly with the fill ratio, wet mass viscosity, and the

surface swept by the blades (proportional to d2).Wet masses produced at the same end-point

(regardless of bowl and batch size, impeller speed,and moisture content) have been consistently shownto result in the same final dry granule size distribution,bulk density, flow, and mechanical strength.

Case Study IV: Landin et al. (1999)

Following the methodology developed in the previousCase Study using the same assumptions, this study wasalso performed on planetary mixers Collette MP20,

Table 6 The dimensional matrix for case study II

Core matrix Residual matrix

q d n P g g H

Mass (M) 1 0 0 1 1 0 0

Length (L) 3 1 0 2 1 1 1

Time (T) 0 0 1 3 1 2 0

Table 7 The transformed dimensional matrix for case study II

Unity matrix Residual matrix

q d n P g g H

M 1 0 0 1 1 0 0

3M þ L 0 1 0 5 2 1 1

T 0 0 1 3 1 2 0




MP90, and MPH 200.[48] The relevance list and dimen-sional matrix were the same as before, and torquemeasurements from torque rheometer were again usedto represent kinematic viscosity (instead of dynamicviscosity) in Reynold numbers.

Fig. 14 represents the resulting regression line

log10 N p ¼ a log10ðCRe Fr rR3b=mÞ þ b

for the combined results from three mixers with bowlsizes 20, 90, and 200 L showed a pretty good fit to data(r 2 > 0.95). The values for the slope and interceptwere found to be: a ¼ 0.68, b ¼ 1280. Data fromtwo other mixers with bowl sizes 5 and 40 L produceslines that were significantly different from the first setof mixers. The authors explained this difference byan assumption of ‘‘different flow patterns’’ in thetwo groups of mixers.

Case Study V: Faure et al. (1999)

This study was done on Collette Gral Mixers (8, 25, 75,and 600 L) and followed the accepted—and by now,standard—methodology developed earlier.[52] Theproblem with the scale-up in the Gral mixers was thelack of geometric similitude: there was significant‘‘distortion factor’’ between the bowl geometries atdifferent scales. In addition, the researchers had to takeinto account the lack of dynamic similitude because of

different wall adhesion and lid interference that waspartially relieved by using a Polytetrafluoroethylene(PTFE) lining.

The end result of the dimensional analysis andexperimental work was, again, a regression equation(Fig. 15) of the form

log10 N p ¼ a log10ðCRe Fr rR3b=mÞ þ b

The regression coefficient was r 2 > 0.88 using the datafrom the 8, 25 and 75-L bowls with PTFE lining, andthe 600-L bowl that did not require the lining. Theslope was found to be a ¼ 0.926, and the interceptb ¼ 3.758.

Case Study VI: Hutin et al. (2004)

In this study, the foregoing methodology of dimen-sional analysis was applied to a kneading process of drug-cyclodextrin complexation.[109] Aoustin kneaderwith dual Z blades was instrumented for torque mea-surements and multiple runs were made at two scales(2.5 and 5L).

The relevance list for this study (Table 10) differsfrom those discussed previously by addition of bladelength as one of the crucial factors affecting theprocess.

Introduction of the blade length, after the properoperations with the dimensional matrix, createsanother dimensionless quantity, namely, d/l, so thatthe resulting regression equation has the form of

N p ¼ bðCRe Fr h=d d=lÞa

Experiments showed that the model fits data remark-ably well (r 2 > 0.99).

Unfortunately, the Pharmaceutical Technology journal does not grant permissions to reproduceindividual graphs; therefore, an interested reader isreferred to the source article to see the regression linesfrom this study.

Table 8 Dimensionless P groupsa

P group Expression Definition

P0 P /(r1 d5 n3) ¼ N p Newton (Power) number

P1 Z/(r1 d2 n1) ¼ Re1 Reynolds number

P2 g /(r0 d1 n2) ¼ Fr1 Froude number

P3 H /(r0 d1 n0) ¼ H /d Geometric number(ratio of characteristic lengths)

aFormed from the matrix in Table 7.

100

10

0.1

1

100 1000 10000 100000

N

p

Re.Fr.H/d

PMA 25

PMA 100

PMA 600

Fig. 12 Case study II. Regression lines of the Newton powernumber on the product of Reynolds number, Froude Num-ber, and the length ratio for three different Fielder mixers.(Reprinted from Ref.[46], copyright 1996, with permission fromElsevier.)






and then recalculate the target DP with the changedvalues of speed n or wet mass density r.

Wet mass viscosity Z can be calculated from NetImpeller power DP , blade radius d, and impeller speedn, using the following equations:

DP ¼ 2p Dt n

Z ¼ j Dt=ðn d3Þ

where Dt is the net torque required to move wet mass,n is the speed of the impeller, d is the blade radius or

diameter, and j is mixer specific ‘‘viscosity factor’’

relating torque and dynamic viscosity (note: the corre-lation coefficient j can be established empirically bymixing a material with a known dynamic viscosity,e.g., water). Alternatively, you can use impeller torquet as a measure of kinematic viscosity and use it toobtain a non-dimensionless ‘‘pseudo-Reynolds’’ num-ber, based on the so-called ‘‘mix consistency’’ measure,that is, the end-point torque, as described in the casestudies.

Fill Ratio h/d can be calculated from a powderweight, granulating liquid density (1000 kg/m3 forwater), rate of liquid addition, time interval for liquid

addition, and bowl volume V b. The calculations areperformed using the idea that the fill ratio h/d (wetmass height to blade diameter) is proportional toV /V b, and wet mass volume V can be computed as

V ¼ m=r

Fig. 14 Regression graph of Case StudyIV. The Reynolds Number Re was, infact, a dimensional ‘‘pseudo Reynolds’’number CRe. Data from ColletteMP20, MP90, and MPH200 planetarymixers. (Reproduced from Ref.[48].)

Table 10 Relevance list for Hutin et al. (2004)


Powerconsumption

P Watt M L2 T3


Blade radius d M L

Blade speed n rev/sec T1

Dynamic viscosity Z Pa sec M L1 T1


g m/sec2 L T2

Powder bed height h M L

Blade length l M L

L o g 1 0

N p

Log10 (ψ Re·Fr·ρRb3 /m)

2.5

2.0

1.5

600L75L + PTFE insert

25L + PTFE insert8L + PTFE insert600LOther bowls

1.0

0.5

1.4 2.0 2.6 3.2

Fig. 15 Regression graph of Case Study V. Data fromCollette Gral 8, 25, 75, and 600 liter mixer-granulators. (Rep-rinted from Ref.[52], copyright 1999, with permission fromElsevier.)




where m is the mass (weight) of the wet mass and r isthe wet mass density.

Now, the weight of the wet mass is computed as theweight of powder plus the weight of added granulatingliquid. The latter, of course, is calculated from therate and duration of the liquid addition and the liquiddensity.

Finally, following the examples discussed in thecase studies, you can combine the results obtained atdifferent end-points of the test batch or from differentbatches or mixer scales (assuming geometrical simi-larity).

Given wet mass density r, wet mass viscosity Z, fillratio h/d m V b/r, setup speed n, and bladeradius or diameter d, you can calculate the Reynoldsnumber Re (or the ‘‘pseudo-Reynolds’’ number) andthe Froude number Fr. Then you can estimate theslope ‘‘a’’ and intercept ‘‘b’’ of the regression equation

N p ¼ b ðRe Fr h=dÞa

or

logN p ¼ log b þ a logðRe Fr h=dÞ

And, inversely, once the regression line is established,you can calculate Newton power number N p (whichis the target quantity for scale-up) and net power DP (which can be observed in real time as a true indicatorof the target end-point) for any point on the line.

ACKNOWLEDGMENT

Selected excerpts and figures from M. Levin, ‘‘Granu-lation: End-Point Theory, Instrumentation, and Scale-Up, Education Anytime, CD-ROM Short Course,AAPS 1999’’ are reprinted with permission.

APPENDIX

List of Symbols and Dimensions

a, b Slope and intercept of a

regression equationd Impeller (blade) diameter or

radius (m); dimensional units[L]

g Gravitational constant (m/sec2); dimensional units [L T2]

h Height of granulation bed inthe bowl (m); dimensional units[L]

H Bowl height (m); dimensionalunits [L]

l Blade length (m); dimensionalunits [L]

n Impeller speed (rev/sec);dimensional units [T1]

P Power required by the impeller

or motor (W ¼ J/sec); dimen-sional units [M L2 T5]Rb Radius of the bowl (m); dimen-

sional units [L]q Binder liquid addition rates Amount of granulating liquid

added per unit time (kg);dimensional units [M]

t Binder addition time (sec);dimensional units [T]

V p Particle volume (m3); dimen-sional units [L3]

V m Wet mass volume (m3); dimen-

sional units [L3

]V b Bowl volume (m3); dimensional

units [L3]w Wet mass; dimensional units

[M]r Specific density of particles

(kg/m3); dimensional units [ML3]

n ¼ Z/r Kinematic viscosity (m2/sec);dimensional units [L2 T1]

Z Dynamic viscosity (Pa sec);dimensional units [M L1 T1]

t Torque (N-m); dimensional

units [M L2 T

2]. End-pointtorque values were describedas ‘‘wet mass consistency’’numbers. Note: torque has thesame dimensions as work orenergy

j ¼ Z n d3/Dt

Dimensionless ‘‘viscosity fac-tor’’ relating net torque Dtand dynamic viscosity Z

Fr ¼ n2 d/ g Froude number. It relates theinertial stress to the gravi-tational force per unit area act-ing on the material. It is a ratio

of the centrifugal force to thegravitational force

N p ¼ P /(r n3 d5)

Newton (power) number. Itrelates the drag force actingon a unit area of the impellerand the inertial stress

Re ¼ d2

n r/ZReynolds number. It relates theinertial force to the viscousforce




CRe ¼ d2 n r/t

‘‘Pseudo Reynolds number’’(m3/s); dimensional units [L3

T]. Note: this variable physi-cally is a reciprocal of volumeflow rate

Ga ¼ Re2/Fr Galileo number

ARTICLES OF FURTHER INTEREST

Electrical Power Systems for PharmaceuticalEquipment, p. 1482.

Fluid Bed Processes for Forming FunctionalParticles, p. 1773.

Fractal Geometry in Pharmaceutical andBiological Applications, p. 1791.

Roller Compaction Technology for the

Pharmaceutical Industry, p. 3159.Scale-Up and Post Approval Changes (SUPAC),

p. 3188.Tablet Press Instrumentation, p. 3684.

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