Chemical Engineering August 2013 - 35

B2
B1
D = the impeller dia., in. or m or mm
T = the tank dia., in. or m or mm
n = the exponent on the scale ratio,
unitless
Z2
Z1
D1
C1
T1
T2
FIGURE 2. With geometric similarity, linear dimensions are in the same proportion
in both scales. For instance, the ratio of impeller dia. to tank dia. will be the same in
both scales.The relative size of all of the large-scale dimensions will be in the same
proportion to the small-scale dimensions
made to ensure the desired outcome.
Other examples of problem definition
arise when considering solids
suspension. What aspect is the most
important for the specific application:
Keeping solids from settling on the bottom?
Or distributing solids uniformly?
Or dissolving solids in the liquid? Or
reacting the solids with something in
the liquid? Each may be governed by
a different scaleup requirement. The
box on p. 35 shows sn off-bottom solids
suspension, which is one type of suspension
that can be observed visually.
Similarly, gas dispersion may be
limited by several factors, including:
* Mass transfer between the bubbles
and the liquid
* Concentration uniformity in liquid
* A reaction taking place in the liquid
* Stoichiometric depletion of a component
in the sparged gas
Correct identification of the primary
or a key process element is essential.
2. Small failures. One major advantage
of conducting small-scale tests
before designing full-scale applications
is that mistakes or problems can
be observed using a smaller quantity
of material. For instance, a reaction or
formulation failure on the small scale
is less likely to create significant hazard
or excessive quantities of waste.
For investigators, one essential aspect
of those small-scale failures is
noting the conditions that lead to the
failure, and establishing the reason
for failure - whether the problem is
formulation or mixing related - to
avoid repeating those conditions.
Insufficient mixing is more likely to
cause a problem than excessive mixing,
but both extremes need to be understood.
Often knowing the lowest
intensity of mixing that is necessary
for success will result in the most economical
mixer scaleup.
3. Geometric similarity. Geometric
similarity means that all of the length
dimensions in the different scale tanks
are in the same relative proportions
to one another (Figure 2). Geometric
similarity is not essential for all
small-scale mixing tests, but it usually
helps. Geometric similarity alone
may be a sufficient reason for conducting
some small-scale tests. Unusual
tank geometry, impeller type or fluid
properties may be the primary reason
for small-scale testing. Geometry will
dictate what flow patterns are created
and whether they will effectively control
the entire tank.
Scaleup with geometric similarity
means that the only remaining variable
to be chosen is the rotational
speed of the large-scale mixer. Because
geometric scaleup from one size to another
means that every length dimension
of the small scale is in the same
proportion to the corresponding length
dimension in the large-scale unit, any
convenient length ratio can be used to
calculate a large-scale mixer speed.
The most common length ratios to use
for this approach are the impeller dia.
or the tank dia.
W1
D2
C2
W2
Because all calculations are ratios, the
units for each variable should be the
same for the different sizes.
During scaleup with geometric similarity,
the ratio of impeller diameter
(from small scale to large scale) is the
same as the ratio of the tank diameter
(from small scale to large scale)
so the calculation can be carried out
either way.
The exponent, n, on the scale ratio
determines how much the impeller
speed changes from small to large
scale. For any positive value of n and
any practical scaleup criterion, the
large-scale rotational speed will be less
than that of the small-scale mixer.
Some values for the exponent n
have a physically significant meaning
for turbulent mixing. For instance, an
exponent value of n = 1 means that the
rotational speed is reduced in proportion
to the linear dimension increase.
This speed change means that the
impeller tip speed, πND, will be held
constant for scaleup. The constant π
cancels out of the ratio of tip speeds,
so ND is held constant on scaleup.
If n = 2/3, the power per volume or
power per mass is held constant with
scaleup. The exponent n can be developed
by simple algebra, using Equation
2:
(2)
where:
P = impeller power, hp, W or kW
ρ = fluid density, lb/gal or kg/m3 or
g/cm3
(1)
where:
N = the rotational speed of the mixer,
rpm or rps
N = rotational speed, rpm or rps
D = impeller dia., in. or m or mm
Equation 2 is only a proportionality
not an equality, so appropriate
constants and conversion factors are
needed to calculate actual values.
The constant of proportionality for
power involves the impeller power
number, which is a characteristic of
the impeller geometry. Conversion factors
are necessary to ensure consistent
units for power, speed, and diameter.
If the same fluid is used in both scales,
the fluid density will be a constant for
scaleup, and will cancel out of the ratio
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