s4 Pharmaceutical Technology SCALING UP MANUFACTURING 2005 www.pharmtech.com
rational approach to scale-up has
been used in the physical sciences
for quite some time. This approach,
called dimensional analysis, is
a proven method of developing functional
relationships that describe any
given process in a dimensionless form
to facilitate modeling and scale-up or
scale-down. This short article reviews
and demonstrates the dimensional
analysis method as applied to pharmaceutical
processes and includes references
to works that have become classics
in the field of granulation scale-up.
What is dimensional analysis?
Dimensional analysis is a method for
producing dimensionless numbers
and deriving functional relationships
among them that completely characterize
the process. The analysis can be
applied even when the equations governing
the process are not known.
Imagine a dimensionless space in
which you have no mass, no length,
and no time. It is obvious that in such
a space, there are no scale-up problems
because there is no scale.
Dimensional analytical procedure
was first systematically applied to fluid
How to Scale Up
Scientifically
Michael Levin
A
The author reviews
dimensional analysis as a
straight-forward approach
to scale up.
COMSTOCK
Michael Levin is president
and CEO of Metropolitan
Computing Corporation,
tel. 973.887.7800,
fax 973.887.8447,
michael.levin@mcc-online.com.
DIMENSIONAL ANALYSIS
s6 Pharmaceutical Technology SCALING UP MANUFACTURING 2005 www.pharmtech.com
flow 90 years ago by Lord Rayleigh (1),
on the basis of the principle of similitude
referred to by Newton in one of
his early works.
Dimensionless numbers
Physical quantities such as force or
speed have the basic dimensional qualities
of length (L), mass (M), time (T),
and so forth. For example, speed has
dimensions of L/T, and force has a dimensional
composition of ML/T2
.
Unlike the regular physical quantities,
dimensionless numbers have
no dimensions. Such numbers are
frequently used to describe the ratios
of various physical quantities.
Most relevant to the forthcoming
discussion are Newton (Ne, power),
Froude (Fr), and Reynolds (Re)
numbers, which, respectively, are expressed
as
Ne ϭ P/(␳n3
d5
)
Fr ϭ n2
d/g
Re ϭ d2
n␳ր␩
in which P is the power consumption
(ML2
/T3
), ␳ is the specific density of
particles (M/L3
),n is the impeller speed
(TϪ1
), d is the impeller diameter (L), g
is the gravitational constant (L/T2
),
and ␩ is the dynamic viscosity (M/LT).
The Newton (power) number,
which relates the drag force acting on
a unit area of the impeller and the inertial
stress, is a measure of the power
required to overcome friction in fluid
flow in a stirred reactor. In mixergranulation
applications, this number
can be calculated from the power consumption
of the impeller.
The Froude number,first introduced
to quantify the resistance of ships (2),
has been described for powder blending
(3) and was suggested as a criterion
for dynamic similarity as well as a scaleup
parameter in wet granulation (4).
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,
thereby creating a“compaction zone.”
The Reynolds number, which relates
the inertial force to the viscous
force, is frequently used to describe
mixing processes (5), 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 to
906 L, which is a scale-up factor of
1:71 (see, for example, Reference 6).
Theory of models
Scale-up, then, is simple: Express the
process using a complete set of dimensionless
numbers and try to match
them at various scales. This dimensionless
space in which the measurements
are presented or measured will
make the process scale invariant.
According to the modeling theory,
two processes are considered similar
if there is a geometrical, kinematic,
and dynamic similarity (7). Two systems
are called geometrically similar if
they have the same ratio of characteristic
linear dimensions. For example,
two cylindrical mixing vessels are geometrically
similar if they have the same
ration of height to diameter. Two geometrically
similar systems are called
kinematically similar if they have the
same ratio of velocities between corresponding
points. Two kinematically
DIMENSIONAL ANALYSIS
Pharmaceutical Technology SCALING UP MANUFACTURING 2005 s7
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 (8).
Lack of geometrical similarity is
often the main obstacle when applying
dimensional analysis to solve scaleup
problems. It has been shown, for
example, that Collette Gral 10, 75, and
300 are not geometrically similar (4).
In such cases, a proper correction to
the resulting equations is required.
Buckingham’s ⌸ theorem
The fundamental theorem of the dimensional
analysis, the ⌸ theorem,
states: Every physical relationship between
n dimensional variables and constants
can be reduced to a relationship
between m = n Ϫ r mutually independent
dimensionless groups,in which r is
the number of dimensions; that is,fundamental
dimensional units (rank of
the dimensional matrix). The theorem
is universally ascribed to Buckingham,
who introduced the term and popularized
it in the United States (9),although
it was proven earlier by others.
It starts with a relevance list
Dimensional analysis starts with a relevance
list, which is a list of all variables
thought to be crucial for the
process being analyzed. To set up a relevance
list for a 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. All entries in the list can be
further subdivided as geometric, physical,
or operational. Each relevance list
should include only one target (i.e.,
dependent “response”) variable.
Pitfalls of dimensional analysis often
relate to selecting the reference list,target
variable, or measurement errors
(e.g., friction losses of the same order
of magnitude as the power consumption
of a motor). The larger the scaleup
factor, the more precise the measurements
of the small scale must be (8).
Dimensional matrix
Dimensional analysis can be simplified
by arranging all relevant variables
from the relevance list into a matrix,
with a subsequent transformation
yielding the required dimensionless
numbers. The dimensional matrix
consists of a square core matrix and a
residual matrix. The rows of the matrix
consist of the basic dimensions,
whereas the columns represent the
physical quantities from the relevance
list. The most important physical
properties and process-related parameters,
as well as the target variable (i.e.,
the one we would like to predict on
the basis of other variables) are placed
in one of the columns of the residual
matrix.
The core matrix is linearly transformed
into a matrix of unity in which
the main diagonal consists only of
ones and the remaining elements are
all zero. The dimensionless numbers
are then created as a ratio of columns
in the residual matrix and the core
matrix, with the exponents indicated
in the residual matrix. The following
examples illustrate this rather simple
process.
s8 Pharmaceutical Technology SCALING UP MANUFACTURING 2005 www.pharmtech.com
Case study I
Hans Leuenberger
et al. proposed a
theory of granulation
end point determination
and
scale up using dimensional
analysis,
starting with the
relevance list in
Table I (7, 10, 11).
The list reflects certain
assumptions
that are used to simplify the model;
namely, that there are short range interactions
only and no viscosity factor
(and therefore, no Reynolds number).
Why is the gravitational constant
included? Well, imagine the same
process to be done on the moon.
Would you expect any difference?
One target variable (power consumption)
and seven process variables
or constants thus represent the number
n ϭ 8 of the ⌸ theorem, and there
are three basic dimensions (r); namely,
M, L, and T. According to the theorem,
the process can be reduced to a
relationship between m ϭ n Ϫ r ϭ 8
Ϫ 3 ϭ 5 mutually independent dimensionless
groups.
To find these groups, or numbers,
form the dimensional matrix shown in
Figure 1. One transformation changes
the Ϫ3 in the L-row, ␳-column to 0.
Subsequent multiplication of the Trow
by Ϫ1 transfers the Ϫ1 of the ncolumn
to ϩ1 (see Figure 2). Five dimensionless
groups are formed from
the five columns of the residual matrix
by dividing each element of the residual
matrix by the column headers of
the unity matrix, with the exponents
shown in the residual matrix.
The residual matrix contains five
columns, therefore five dimensionless
⌸ groups (numbers) are formed (see
Table II).
The end result of the dimensional
analysis is an expression of the form
⌸0 = f(⌸1, ⌸2, ⌸3, ⌸4)
Assuming that the groups ⌸2, ⌸3,
and ⌸4 are “essentially” constant, the
⌸-space can be reduced to a simple reDIMENSIONAL
ANALYSIS
Table I: Variables used in dimensional analysis.
Quantity Symbol Units Dimensions
Power consumption P Watt ML2
/T 3
Specific density ␳ kg/m3
M/L3
Blade diameter d m L
Blade speed n rev/s TϪ1
Binder amount m kg M
Bowl volume Vb m3
L3
Gravitational constant g m/s2
L/T 2
Bowl height h m L
Core matrix Residual matrix
␳ d n P m Vb g h
1 0 0 1 1 0 0 0
–3 1 0 2 0 3 1 1
0 0 –1 –3 0 0 –2 0
Mass M
Length L
Time T
Figure 1: Initial dimensional matrix for
case study I.
Unity matrix Residual matrix
␳ d n P m Vb g h
1 0 0 1 1 0 0 0
0 1 0 5 3 3 1 1
0 0 1 3 0 0 2 0
M
3M ϩ L
ϪT
Figure 2: Initial dimensional matrix after
one linear transformation.
s10 Pharmaceutical Technology SCALING UP MANUFACTURING 2005 www.pharmtech.com
DIMENSIONAL ANALYSIS
lationship ⌸0 = f(⌸1); that is, the value
of Newton number Ne at any point in
the process is a function of the specific
amount of granulating liquid.
Up to this point, all considerations
were rather theoretical. From the theory
of modeling, we know that the
above dimensional groups are functionally
related. The form of this functional
relationship f, however, can be
established only through experiments.
Leuenberger and his
group empirically established
that the amount
of binder required to
reach a desired endpoint
(as expressed by
the absolute value of
Ne) is essentially proportional
to the batch
size (see Figure 3), thus
specifying the functional
dependence f and establishing
a rational basis
for granulation scale-up.
According to Leuenberger,
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 functionality
was shown for nonviscous binders.
The ratio of quantity of granulating
liquid to batch size at the inflection
point S3 of power versus time
curve 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 the 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 various
sizes and manufacturers.
Different vessel and blade geometries
will contribute to differences in
Batch size
Binderamount
S3
Figure 3: Binder amount versus batch
size. Adapted from Reference 11.
Table II: Dimensionless ⌸ groups for case study I.
⌸ group Expression Definition
⌸0 P/(␳1
d 5
n3
) ϭ Ne
Newton (power)
number
⌸1
q /(␳1
d 3
n0
) ϭ
qt /(Vp␳)
Specific amount
of liquid Vp is
volume of particles,
q is the binder
addition rate, and t
is the binder
addition time
⌸2
t/(␳0
d3
n0
) ϭ
(Vp /Vb)Ϫ1
Fractional particle
volume
⌸3 g/(␳0
d1
n2
) ϭ FrϪ1
Froude number
⌸4 h/(␳0
d1
n0
) ϭ h/d Ratio of lengths
Pharmaceutical Technology SCALING UP MANUFACTURING 2005 s11
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.
To calculate Ne, the power of the
load on the impeller rather than the
mixer motor should be used. Before
attempting to use dimensional analysis,
one must measure or estimate
power losses for empty-bowl mixing.
This baseline, however, does not stay
constant; it changes significantly with
load, mixer condition, or motor efficiency,
which may present inherent
difficulties in using power meters instead
of torque. Torque, of course, is
directly proportional to the power
drawn by the impeller so that the
power number can be calculated from
the torque and speed measurements.
Case study II
Scale-up in fixed-bowl mixer granulators
has been studied using the classical
dimensionless numbers of Newton
(power), Reynolds, and Froude to
predict end-points in geometrically
similar, high-shear Fielder PMA 25-,
100-, and 600-L machines (12).
The relevance list included power
consumption of the impeller (as a response)
and six factor quantities: specific
density, impeller diameter, impeller
speed, gravitational constant,
vessel height (all with units and dimensions
shown in Table I), and viscosity
of the wet mass (␩, units of Pas,
dimensions M/LT). Dynamic
viscosity has replaced the binder
amount and bowl volume of Leuenberger’s
relevance list, thus making it
applicable to viscous binders.
The dimensional matrix and the
matrix after the already familiar linear
transformation are shown in Figures
4 and 5, respectively. The residual matrix
contains four columns, therefore
four dimensionless ⌸ groups (numbers)
are formed in accordance with
the ⌸-theorem (see Table IV). Under
the assumed condition of dynamic
similarity, therefore, Ne = f(Re, Fr, h/d).
When corrections for gross vortexing,
geometric dissimilarities, and
powder-bed height variation are made,
data correlations from all mixers allow
predictions of optimum end-point
conditions. The linear regression of
the Newton number (power) on the
adjusted Reynolds number (in log/log
domain) yields Ne ϭ 7.96 ϫ 102
(Re
Fr h/d)Ϫ0.732
(see Figure 6).
The 0.7854 correlation coefficient
for the final curve-fitting effort, however,
indicates the presence of many
unexplained outlier points. To mainCore
matrix Residual matrix
␳ d n P g h
1 0 0 1 1 0 0
–3 1 0 2 –1 1 1
0 0 –1 –3 –1 –2 0
Mass M
Length L
Time T
␩
Figure 4: Dimensional matrix for case
study II.
Unity matrix Residual matrix
␳ d n P g h
1 0 0 1 1 0 0
0 1 0 5 2 1 1
0 0 1 3 1 2 0
M
3M + L
–T
␩
Figure 5: Matrix for case study II after
linear transformation.
s12 Pharmaceutical Technology SCALING UP MANUFACTURING 2005 www.pharmtech.com
DIMENSIONAL ANALYSIS
tain 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. A special consideration
is required when using torque values
from mixer torque rheometers for
the viscosity of wet granulation, because
these values are proportional to
kinematic viscosity ␯ ϭ ␩/␳ rather
than dynamic viscosity ␩ required to
calculate the Reynolds numbers.
Case study III
Fraure et al. applied this same approach
to a planetary HobartAE240 mixer with
two interchangeable bowls (13). The
relevance list, assuming the absence of
chemical reaction and heat transfer,included
power consumption, specific
density,blade diameter,blade speed,dynamic
viscosity,gravitational constant,
and wet powder bed height (h, units of
m, dimensions of L).As a measure
of power consumption, net
impeller power consumption ⌬P
(motor power consumption
minus the dry blending baseline
level was used. Dimensional
analysis and application of the
Buckingham theorem indicated
that Ne, Re, Fr, and h/d (proportional
to the fill ratio) adequately
described the process.A relationship
Ne ϭ k[ReFr (h/d)]Ϫr
was postulated
and the constants k and r were
found empirically with a good correlation
(Ͼ0.92) between the observed and
predicted numbers.
Once the process similarity is established
or the appropriate corrections are
made,scale-up problems can be greatly
reduced or even completely eliminated,
because the same equation predicts the
power number of the granulation
process regardless of the batch size.
References
1. Lord Rayleigh,“The Principle of Similitude,”Nature
95,66–68 (18 March 1915).
2. C.W. Merrifield “The Experiments Recently
Proposed on the Resistance of
Ships,”Trans. Inst. Naval Arch. (London)
11, 80–93 (1870).
3. J.Y.Oldshue,“Mixing Processes,”in ScaleUp
of Chemical Processes: Conversion from
Laboratory-Scale Tests to Successful Commercial-Size
Design, A. Bisio and R.L.
Kabel, Eds. (Wiley, New York, NY, 1985).
100
10
1
0.1
100 1000 10000 100000
PMA 25
PMA 100
PMA 600
Re* Fr* h/d
Me
Figure 6: Power number versus dimensionless
processing factors. Adapted from Reference (12).
Table IV: Dimensionless ⌸ groups for Case study II.
⌸ group Expression Definition
⌸0 P/(␳1
d 5
n3
) ϭ Ne Newton (power) number
⌸1 ␩/(␳1
d 2
n1
) ϭ ReϪ1
Reynolds number
⌸2 g/(␳0
d1
n2
) ϭ FrϪ1
Froude number
⌸3 h/(␳0
d1
n0
) ϭ h/d Ratio of lengths
4. G.J.B. Horsthuis et al., “Studies on Upscaling
Parameters of the Gral HighShear
Granulation Process,” Int. J.
Pharm. 92 (143), (1993).
5. O. Reynolds,“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).
6. M. Zlokarnik, “Problems in the Application
of Dimensional Analysis and
Scale-Up of Mixing Operations,” Chem.
Eng. Sci. 53 (17), 3023–3030 (1998).
7. H. Leuenberger, “Scale-Up of Granulation
Processes with Reference to Process
Monitoring,” Acta Pharm. Technol. 29
(4), 274–280 (1983).
8. M. Zlokarnik, Dimensional Analysis and
Scale-Up in Chemical Engineering
(Springer-Verlag, 1991).
9. E. Buckingham, “On Physically Similar
Systems; Illustrations of the Use of Dimensional
Equations,” Phys. Rev. NY 4,
345–376 (1914).
10. H. Leuenberger, H.P. Bier, and H.B.
Sucker, “Theory of the GranulatingLiquid
Requirement in the Conventional
Granulation Process,” Pharm. Technol.
6, 61–68 (1979).
11. H.P. Bier, H. Leuenberger, and H. Sucker,
“Determination of the Uncritical Quantity
of Granulating Liquid by Power
Measurements on Planetary Mixers,”
Pharm. Ind. 4, 375–380 (1979).
12. M. Landin et al., “Scale-Up of a Pharmaceutical
Granulation in Fixed Bowl
Mixer–Granulators,”Int. J. Pharm. 133,
127–131 (1996).
13. A. Faure et al., “A Methodology for the
Optimization of Wet Granulation in a
Model Planetary Mixer,” Pharm. Dev.
Technol. 3 (3), 413–422 (1998). PT