Chemical Engineering August 2021 - 32

TABLE 2: VALUES OF 훂 AND 훃 REPORTED BY VARIOUS AUTHORS
Rushton
Pitched-blade Propeller
Authors
Prochaka and
Landau
Shiue and Wong
Grenville
Hydrofoil
Year T/D 훂 훃 훂 훃 훂 훃 훂 훃
1961 3-10 1.56 2.57 2.31 2.20 2.35 2.05 -
1984 2-3 8.57 2.40 -
-
-
Strand and Hensel 2018 2-5 5.19 2.30 5.76 2.21 -
1000
-
-
-
3.88 2.24
1992 2-3 5.20 2.00 5.20 2.00 5.20 2.00 5.20 2.00
8.12 1.93
100
T=0.3 m
T=0.6 m
T=1.8 m
T=2.7 m
T=0.3 m
T=0.6 m
T=1.8 m
T=2.7 m
Grenville, s θ
θ
Strand and
Hensel, s
TABLE 3. BLEND TIMES PREDICTED BY
CORRELATIONS OF GRENVILLE [24] AND
STRAND & HENSEL [25]
Rushton Pitchedblade
Power,
hp
19.84
15.81
21.94
6.35
25.73
35.02
1.18
39.85
57.61
case, the impeller diameter is 36 in.
and it operates at 100 rpm in a vessel
that is 108 in. in diameter (T/D
= 3). These results with the impeller
power draw are summarized in
Table 3.
While the blend-time predictions
10
Transitional
Turbulent
1
100
1000
10000
100000
Reynolds Number
FIGURE 3. Dimensionless blend time (Nθ) is plotted against Reynolds number for pitched-blade turbines,
1/3 and 1/2 of the vessel diameter (T), measured at four vessel scales
and the value of the constant depends
on the impeller type, through
its power number, and the ratio of
impeller to vessel diameters. At a
certain Re value, the blending regime
becomes transitional and Nθ
is inversely proportional to Reynolds
number - which means that it is
proportional to the fluid's viscosity.
The value of Reynolds number
at the boundary (ReB) between the
regimes is dependent on impeller
type and can be estimated from
Equation (19) [24]:
(19)
The reason for this is the difference
in power input by different impellers
when compared at the same
Reynolds number. For example, the
power input by a Rushton turbine (Po
= 5.0) will be approximately 15 times
higher than the power input by a hydrofoil
(Po = 0.3) at the same speed
and diameter, and, therefore, Reynolds
number. Khang and Levenspiel
measured blend times for Rushton
turbines and marine propellers and
reported the same result [23].
30
Most authors agree that, in
the turbulent regime (where
Po1/3Re > 6,400), the best correlation
of their data is in the form given
in Equation (20):
(20)
The power number accounts for
the effect of impeller type and the
vessel to impeller diameter ratio,
T/D, accounts for the impeller size.
ζ represents the " selected level of
variation " or degree of homogeneity
required. For example, for 99% homogeneity,
or a variation of ±1%, ζ =
99. Values of α and ß found by various
researchers are summarized in
Table 2. The values of α vary, but the
values of ß generally fall within ±15%
of 2.00.
To test the effect of these differences,
the blend times predicted
by the Grenville [24] and Strand
and Hensel [25] correlations have
been calculated for Rushton (Po =
5.0) and pitched-blade turbines (Po
= 1.5) and a narrow-blade hydrofoil
(Po = 0.3), since they used the same
experimental technique. In each
1000000 10000000
are in the same " ballpark, " generally,
the Strand and Hensel correlation
predicts a blend time that
is approximately 50% longer than
Grenville's. Both used changes in
conductivity to measure the blend
time, and the difference in prediction
is most likely due to the locations
chosen for the conductivity
probes. Strand and Hensel placed
their probes much closer to the liquid
surface than Grenville in a region
where the mixing rate was slower.
Even though there are differences
in predictions made by the two correlations,
the overall trends in terms
of impeller type, Po, and size, T/D,
are the same. Also, the blend time is
the order of tens of seconds, which
is very short compared to other process
timescales, such as filling or
emptying the vessel.
The degree of homogeneity measured
in most experimental work
is 95%. The reason for this is that
as the degree of homogeneity becomes
larger, the concentration
fluctuations being detected are so
small that they are lost in the background
signal noise of the measurement
equipment. The 95% blend
time can be converted into a different
degree of homogeneity because
the approach to uniformity
is exponential, since blending is a
first-order process. The relationship
between θ95 and θζ is shown
in Equation (21):
(21)
So, for example, the blend time
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