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How to Scale-Up ScientificallyMichael Levin, Ph. D.

Metropolitan Computing CorporationProcess Analytical Instrumentation - Monitoring and Control

East Hanover, New Jersey, USA www.mcc-online.com

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Dimensional analysis is a method for producing dimensionless numbers that completely describe the

process. The analysis should be carried out before the measurements have been made, because

dimensionless numbers essentially condense the frame in which the measurements are performed and

evaluated. It can be applied even when the equations governing the process are not known.

Dimensional Analysis

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Dimensional AnalysisDimensional Analysis

Similarity:• geometrical• kinematic• dynamic

“For any two dynamically similar systems, all the dimensionless numbers necessary to describe the

process have the same numerical value”.

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Dimensional AnalysisDimensional Analysis

Np = P / (ρ n3 d5) Newton (power) Fr = n2 d / g Froude Re = d2 n ρ / η Reynolds

P - power consumption [ML2T-5]ρ - specific density of particles [M L-5]n - impeller speed [T-1]d - impeller diameter [L]g - gravitational constant [LT-2]η - dynamic viscosity [M L-1 T-1]

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Dimensional AnalysisDimensional Analysis

Scientific scale-up procedure:1. Describe the process using a complete

set of dimensionless numbers, and 2. Match these numbers at different

scales. This dimensionless space in which the

measurements are presented or measured will make the process

scale invariant.

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Dimensional AnalysisDimensional Analysis

Π-theorem (Buckingham)Every physical relationship between n dimensional variables and constants

ƒ(x0, x1, x2, … , xn) =0

can be reduced to a relationship

ƒ (Π0 ,Π1, … , Πm) = 0

between m = n - r mutually independent dimensionless groups,

where r = number of dimensional units,

i.e. fundamental units (rank of the dimensional matrix).

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Dimensional AnalysisDimensional Analysis

Relevance List

list of all variables thought to be crucial for the process being

analyzed

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Dimensional AnalysisDimensional Analysis

Dimensional Matrix

Rows: basic dimensions

Columns: quantities from the Relevance List

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Application to mixing-granulation process

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Typical Instrumentation Signals

s3

s4s5

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Endpoint Determination

• Target particle size mean• Target particle size distribution• Target granule viscosity• Target granule density

• Principle of equifinality

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Granulation End Pointand Product Properties

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LmhBowl height8

LT-2m / s2gGravitational constant7

L3m3VbBowl volume6

MkgmBinder amount5

T-1rev / snBlade angular velocity4

LmdBlade diameter3

M L-3kg / m3ρSpecific density2

ML2 T-3WattPPower consumption1

DimensionsUnitsSymbolQuantity

The Relevance List

Case Study I: Leuenberger (1979,1983) Bier HP, Leuenberger H, Sucker H. Determination of the uncritical quantity of granulating liquid by power measurements on planetary

mixers. Pharm Ind 4:375-380, 1979 Leuenberger H. Scale-up of granulation processes with reference to process monitoring. Acta Pharm Technol 29(4), 274-280, 1983

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0-200–3–100Time T

1130201–3Length L

00011001Mass M hgVbmPndρ

Residual MatrixCore matrixThe Dimensional Matrix

02003100-T113350103M + L

00011001M

hgVbmPndρ

Residual MatrixUnity matrixThe transformed Dimensional Matrix

Case Study I: Leuenberger (1979,1983)

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Ratio of Lengths= h / dh / (ρ0 * d1 * n0)Π4 =Froude Number= Fr-1g / (ρ0 * d1 * n2)Π3 =Fractional Particle Volume= (Vp / Vb)-1t / (ρ0 * d3 * n0)Π2 =

Specific Amount of LiquidVp ≡ Volume of particlesq = binder addition ratet = binder addition time

= q t / (Vp ρ)q / (ρ1 * d3 * n0)Π1 =Newton (Power) number= NpP / (ρ1 * d5 * n3)Π0 =

DefinitionExpressionΠ group

Conclusion: Π0 = ƒ (Π1)Assumption: groups Π2, Π3, Π4 are “essentially constant”Π0 = ƒ (Π1, Π2, Π3, Π4)

Case Study I: Leuenberger (1979,1983)

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Case Study I: Leuenberger (1979,1983)

Constant rate of binder addition proportional to the batch size

Bin

der a

mou

nt

Batch Size

Adapted from Bier, Leuenberger and Sucker (1979)

S3

Power number vs. Granulating Liquid

0

5

10

15

20

25

7 12 17 22 27

Specific Amount of Granulating Liquid

Np

S4

S5

S3

S4

S5

5 different planetary mixers (Dominici, Glen, Molteni); batch sizes from 3.75 kg up to 60 kg.

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LmhBowl height7

LT-2m / s2gGravitational constant6

M L-1 T-1Pa * sηDynamic viscosity5

T-1rev / snBlade speed4

LmdBlade diameter3

M L-3kg / m3ρSpecific density2

ML2T-3WattPPower consumption1

DimensionsUnitsSymbolQuantity

The Relevance List

Case Study II: Landin et al. (1996)Landin M, York P, Cliff MJ, Rowe RC, Wigmore AJ. Scale-up of a pharmaceutical granulation in fixed bowl

mixer-granulators. Int J Pharm 133:127-131, 1996

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The Dimensional Matrix

The transformed Dimensional Matrix

Case Study II: Landin et al. (1996)

0-2-1–3–100Time T11-1201–3Length L 0011001Mass M

hgηPndρ

Residual MatrixCore matrix

0213100-T11250103M + L 0011001M

hgηPndρ

Residual MatrixUnity matrix

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Case Study II: Landin et al. (1996)

Π0 = ƒ (Π1, Π2, Π3,)

or

Ne = ƒ (Re, Fr, h/d).

Ratio of Lengths= h / dh / (ρ0 * d1 * n0)Π3 =

Froude Number= Fr-1g / (ρ0 * d1 * n2)Π2 =

Reynolds number= Re-1η / (ρ1 * d2 * n1)Π1 =

Newton (Power) number= NpP / (ρ1 * d5 * n3)Π0 =

DefinitionExpressionΠ group

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Power Number Relationships

0.1

1

10

100

100 1000 10000 100000Re * Fr * h / D

Np

PMA 25

PMA 100

PMA 600

Np = 7.96 x 102 (Re * Fr * h / d)-0.732

Case Study II: Landin et al. (1996) • Fielder PMA 25, 100 and 600 Liter

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Concerns:

• Geometric Similarity

• Interpretation of data from

Mixer Torque RheometerKinematic viscosity vs. dynamic viscosity

ΨRe = “pseudo Reynolds number” = “wet mass consistency number”

Case Study II: Landin et al. (1996)

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• Planetary MixersHobart AE240, dual bowl 5L and 8.5L

• Np = k (ΨRe * Fr * h/d)-s

• r2 > 0.92Assumptions:

Drive speed ~ blade speed

h/d ~ Vm / Vb (fill ratio) ~ m / (ρ R3) fill ratio for Hobart bowl

ΨRe = “pseudo Reynolds number” = “wet mass consistency number”

Case Study III: Faure et al. (1998) Faure A, Grimsey IM, Rowe RC, York P, Cliff MJ. A methodology for the optimization of wet granulation in a model

planetary mixer. Pharm Dev Tech 3(3):413-422, 1998

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• Planetary MixersCollette MP20, MP90, and MPH 200Bowl sizes (L): 5, 20, 45, 90, 200

• Np = k (ΨRe * Fr * h/d)-s

• r2 > 0.95Assumptions:

Drive speed ~ blade speed

h/D ~ Vm / Vb (fill ratio) ~ m / (ρ R3)

ΨRe = “pseudo Reynolds number” = “wet mass consistency number”

Case Study IV: Landin et al. (1999)Landin M, York P, Cliff MJ, Rowe RC. Scaleup of a pharmaceutical granulation in planetary mixers.

Pharm Dev Tech 4(2):145-150, 1999

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• Collette Gral Mixers(8, 25, 75 and 600 Liter)

• no geometric similitude: significant “distortion factor”

• no dynamic similitude due to different wall adhesion, lid interference -> PTFE lining

Np = k (ΨRe * Fr * m / (ρ R3))-s

r2 > 0.93

Case Study V: Faure et al. (1999)Faure A, Grimsey IM, Rowe RC, York P, Cliff MJ. Applicability of a scale-up methodology for wet

granulation processes in Collette Gral high shear mixer-granulators, Eur J Pharm Sci, 8(2):85-93, 1999

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LmlBlade length8

LmhPowder bed height7

LT-2m / s2gGravitational constant6

M L-1 T-1Pa * sηDynamic viscosity5

T-1rev / snBlade speed4

LmrBlade radius3

M L-3kg / m3ρSpecific density2

ML2T-3WattPPower consumption1

DimensionsUnitsSymbolQuantity

The Relevance List

Case Study VI: Hutin et al. (2004)Hutin S, Chamayou A, Avan JL, Paillard B, Baron M, Couarraze G, Bougaret J.

Analysis of a Kneading Process to Evaluate Drug Substance–Cyclodextrin Complexation. Pharm Tech, October 112-123, 2004

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• Aoustin kneader (2.5 and 5 Liter)

• Np = k (ΨRe * Fr * h/r * r/l))-s

r2 > 0.99

Case Study VI: Hutin et al. (2004)

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• Dimensional Analysis provides a powerful scientific technique for scale-up

• This technique is proven by a century of engineering applications

• Rational scale-up should replace empirical approach in pharmaceutical applications

Finale

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• Bier HP, Leuenberger H, Sucker H. Determination of the uncritical quantity of granulating liquid by power measurements on planetary mixers. Pharm Ind 4:375-380, 1979• Buckingham E. On physically similar systems; Illustrations of the use of dimensional equations. Phys Rev NY 4:345-376, 1914• Faure A, Grimsey IM, Rowe RC, York P, Cliff MJ. A methodology for the optimization of wet granulation in a model planetary mixer. Pharm Dev Tech 3(3):413-422, 1998• Faure A, Grimsey IM, Rowe RC, York P, Cliff MJ. Applicability of a scale-up methodology for wet granulation processes in Collette Gral high shear mixer-granulators, Eur J Pharm Sci, 8(2):85-93, 1999• Horsthuis GJB, van Laarhoven JAH, van Rooij RCBM, Vromans H. Studies on upscaling parameters of the Gral high shear granulation process. Int J Pharm 92:143, 1993• Hutin S, Chamayou A, Avan JL, Paillard B, Baron M, Couarraze G, Bougaret J. Analysis of a Kneading Process to Evaluate Drug Substance–Cyclodextrin Complexation. Pharm Tech, October 112-123, 2004• Landin M, York P, Cliff MJ, Rowe RC, Wigmore AJ. Scale-up of a pharmaceutical granulation in fixed bowl mixer-granulators. Int J Pharm 133:127-131, 1996• Landin M, York P, Cliff MJ, Rowe RC. Scaleup of a pharmaceutical granulation in planetary mixers. Pharm Dev Tech 4(2):145-150, 1999•Leuenberger H. Scale-up of granulation processes with reference to process monitoring. Acta Pharm Technol29(4), 274-280, 1983• Levin M. (ed.). Pharmaceutical Process Scale-Up. Marcel Dekker, New York, 2002. • Merrifield CW. The experiments recently proposed on the resistance of ships. Trans Inst Naval Arch (London) 11:80-93, 1870• Rayleigh, Lord. The principle of similitude. Nature 95 (2368, March 18):66-68, 1915• Reynolds O. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinusous, and of the law of resistance in parallel channels. Philos Trans R Soc London 174:935-982, 1883• Zlokarnik M. Dimensional Analysis and Scale-Up in Chemical Engineering. Springer-Verlag, 1991• Zlokarnik M. Problems in the application of dimensional analysis and scale-up of mixing operations. ChemEng Sci 53(17):3023-3030, 1998

References