Chemical Engineering August 2021 - 31

per mass or constant bulk-fluid velocity
allow the engineer to size an
agitator without necessarily having
a good understanding of the role
mixing has on the outcome of their
process [15]. Most mixing processes
are promoting a change in the contents
of the vessel and the success
of the process will be determined, to
some extent, by the rate at which this
change occurs. If the rate-limiting
process can be identified, then the
goal of scaleup should be to match
the rate of mixing to the rate-limiting
step, although care must be taken to
determine whether the rate-limiting
step changes with scale.
Blend time
Blending rate is defined as " the rate
that concentration differences are reduced
by large-scale circulation and
convective flow down to a selected
level of variation everywhere in the
whole vessel [16]. " The blend time
is inversely proportional to this rate.
The rate of blending is proportional
to the rotational speed of the impeller
and, in the turbulent regime, the
product of blend time and impeller
speed is a constant. This value represents
the number of revolutions
that the impeller must make to reach
the desired level of variation or homogeneity.
It is dependent on the
impeller type and its size.
There have been many studies
on blending using a variety of
experimental methods, including
the following:
1. Adding acids and bases and
recording the time for color change
with a pH indicator [17, 18, 19].
2. Adding tracer of hot fluid and
measuring temperature change [19,
20, 21].
3. Adding a concentrated brine
tracer and measuring conductivity
changes [19, 22, 23, 24, 25].
Landau and Procházka [19] compared
all three methods and concluded
that they are mutually consistent,
so it is possible to reliably
compare results measured with
these techniques.
Figure 2a shows a typical conductivity
trace for a blend-time experiment.
In this experiment, the
conductivity is recorded for about
25 s to establish
the
initial conditions
and
then
the tracer
is added,
shown by
the green
vertical line.
There is a
short lag as
the packet
of tracer is
carried to
the probe
and then
there is a
spike in the
conductivity
reading.
The packet
is carried
away and
the measured
conductivity
falls
as the salt
within the
packet is diluted
by the
surrounding
liquid. Eventually,
the
conductivity
reaches
a steady
state and
the vessel
contents are
homogeneous.
The
4
33.999966
37.66629
3.5
3
2.5
2
1.5
1
0.5
Tracer Injection
Time tinital
Blend Time 훉c
20 30 40 50 60 70 80 90
Time, s
(a) Concentration vs. Time
10
1
0.1
0.01
0.001
0.001
20
40
(b) Variance vs. Time
FIGURE 2. The plot of conductivity versus time (a) shows the time taken to reach 95%
homogeneity (the vertical red line). Alternatively, normalized variance versus time (b)
is plotted when systems are using multiple conductivity probes, resulting in a measure
of homogeneity for the whole vessel based on the individual probe responses.
The slope of the plot represents the blending rate
time taken to reach 95% homogeneity
(conductivity
fluctuations
within ±5% of the final
value) is shown by the vertical red
line. An alternative method for extracting
the blend time from the
measurement is to plot the log of the
variance in the conductivity measurement
versus time, as shown in
Figure 2b. This is especially useful
when conductivity is measured with
several probes, since the variance
gives a measure of homogeneity for
the whole vessel based on the individual
probe responses [24]. The
slope of the curve represents the
mixing rate.
Results are usually expressed by
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plotting the dimensionless blend
time, which is the product of the
measured blend time and the impeller
speed, Nθ. This represents
the number of revolutions that the
impeller must make in order to
achieve the desired level of variation
or homogeneity. In the turbulent
regime, the value of the dimensionless
blend time is constant, and it
depends on the type and size of
the impeller. Figure 3 shows a plot
of dimensionless blend time versus
Reynolds number (Re) for two diameters
of pitched-blade turbines
measured in four vessels ranging
from 1 to 9 ft in diameter. There are
two regimes identified in Figure 3. In
the turbulent regime, Nθ is constant
29
60
80
Time, s
100 120 140
Blending rate, k = 0.103/s
y=18.302e-0.103x
R2=0.751
Finish Time ≈ 88.7s
Variance (All time)
Variance (Blending period)
95% Homogeneous, Var. = (1-0.95)2
1st. Order
Concentration
Change, ∆C
100 110 120
Circulation
Time
Normalized concentration
Initial concentration
Final concentration
Start time
Finish time
±5%
Concentration
Fluctuations <±5% (95%
Homogeneous), tFinal
Normalized Concentration [1-C]2
Normalized Concentration
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