Chemical Engineering August 2021 - 30

input, by the impeller to that volume
of liquid. Therefore, torque can be
used as a method for quantifying
agitation intensity.
Connolly and Winter developed a
design procedure that aimed to duplicate
velocities at critical points in
mixing vessels or create kinematic
similarity [12]. This can be achieved
by applying equal torque per volume.
Hicks and others [13] proposed a
scale of agitation, ranging from 1 to
10, where one unit on the scale corresponded
to a bulk fluid velocity of 6
ft/min. The bulk fluid velocity, UBF, is
simply the flow generated by the impeller
divided by the cross-sectional
area of the vessel. So, for a cylindrical
vessel, UBF is determined by
Equation (5):
(5)
The impeller tip speed is given by
Equation (6):
(6)
(12)
Equations (5) and (6) can be combined
to give Equation (7) below:
(7)
The torque acting on the impeller
shaft can be calculated from
Equation (8):
(8)
The torque per volume in a vessel
where the liquid depth is equal to the
vessel diameter (H = T) is given in
Equation (9):
(9)
One potential drawback of this approach
is that the torque and torque
per unit volume are proportional to
the density of the fluid, so the torque
per volume will be higher if the fluid is
denser in order to achieve the same
bulk fluid velocity. This is not an issue
if the approach is being applied to
28
Equations (7) and (12) can then be
combined into Equation (13):
(13)
So, the bulk fluid velocity is proportional
to the square root of the torque
per mass, but the actual relationship
also depends on the type of impeller,
through its power and flow numbers,
and its diameter relative to the vessel
diameter, or D/T ratio.
Once the bulk fluid velocity has
been estimated, the internal circulation
time of the fluid will be proportional
to the path taken by the packets
of fluid divided by the velocity of
the packets. The length of the path
will be proportional to a length scale
of the vessel, the diameter, as shown
in Equation (14):
(14)
This shows that the consequence
of scaling up with constant torque
scaleup, since the density of the
fluid will be the same at both scales.
However, it will be a problem if torque
per unit volume or bulk fluid velocity
are being used as the basis for design.
Hicks and others [13] proposed
using the " equivalent volume, " which
is the product of the fluid's specific
gravity and volume. In that case,
Equation (9) becomes Equation (10),
where S is the fluid specific gravity
and ρ = S × 1,000.
(10)
Alternatively, the agitation intensity
could be expressed in terms of
torque per mass or specific energy
input, as shown in Equation (11):
(11)
Equations (6) and (11) can be combined
into Equation (12):
per (equivalent) volume or mass is
that the circulation time increases in
proportion to the scaleup factor.
An alternative way to examine this
relationship is to define the circulation
time in terms of the operating
volume and pumping capacity of the
impeller, as shown in Equation (15):
(15)
Equation (16) below shows Equation
(15) in terms of the impeller
tip speed:
(16)
This approach also shows that,
when scaling up at constant tip
speed, the circulation time increases
in proportion to the scale factor.
The larger the impeller diameter, the
shorter the circulation time because
a larger-diameter impeller acts as a
more efficient pump [14]. Scaling up
at constant torque per volume and
tip speed (with geometrical similarity)
will result in the reduction of power
input per unit mass. The rotational
speed will reduce as the impeller diameter
increases, so the power per
volume will decrease in proportion
to the reduction in speed or increase
in scale.
Equation (15) shows that if equal
circulation time is required on scaleup
(with geometrical similarity), the agitator
must operate at the same rotational
speed at the large and small
scales. The power drawn by an impeller
is given in Equation (17):
(17)
The power input per mass of fluid
is given in Equation (18):
(18)
As a result, scaling up at constant
impeller speed and geometry leads
to an infeasible increase in power
input and power per mass, so this
approach is rarely taken.
Scaleup rules, such as constant
tip speed, torque per volume, power
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