Chemical Engineering August 2017 - 48

Impeller
TABLE 2: MEAN VELOCITY PROFILE SHEAR RATES



Narrow-blade hydrofoil 0.25 0.14 0.80 0.16 0.17 1.00
Pitched-blade turbine
Rushton turbine
0.35 0.07 0.65 0.12 0.53 3.11
0.60 0.16 0.12 0.04 5.50 24.4
  /HYDFL
at equal VTIP
/HYDFL
at equal 
1.00
1.91
9.55
Table 2 also shows the ratio of the shear rates when
the impellers operate at equal power input per mass.
Taking power numbers from Table 1, the ranking of the
impellers does not change. Therefore,
whether compared
at equal tip speed or power input the Rushton
turbine generates the highest shear rate followed by the
pitched-blade turbine then the hydrofoil. This ranking
can be tested against a process result that is dependent
on shear, namely the break-up of droplets to create a
liquid-liquid dispersion.
Process result
Mass transfer between two immiscible liquid phases, with
or without reaction, is an important process result. The interfacial
area available for mass transfer is proportional to
the volume fraction of the dispersed phase and inversely
proportional to the Sauter mean droplet size [8].
If the dispersion is agitated for a long period of time
(several hours), an " equilibrium droplet size " is achieved
that is stable in the mixing environment in which the
droplets are being formed. This means that there is an
equilibrium between the forces breaking-up the droplets
and the forces resisting break-up resulting from the interfacial
tension between the two liquids and the viscosity of
the dispersed phase liquid.
Figure 6 shows the Sauter mean droplet size plotted
versus the average power input per unit mass for Rushton,
two pitched-blade turbines, with blades angled at
45 and 60 deg, a hydrofoil and a high-shear disperser
impeller. The experiments were carried out with low viscosity
silicone oil as the dispersed phase and at a very
low concentration so that the effect of coalescence on
the droplet size can be ignored.
If the hydraulic efficiency and shear rate comparison
quantify the performance characteristics of the impellers,
when compared at the same power input per mass,
r/R, %
0%
5%
10%
15%
20%
25%
30%
35%
40%
0%
20%
40%
60%
- Hydrofoil
- Pitched-blade turbine
80%
100%
The standard deviation for this correlation is ±15%.
Where measurements have been made, typical values
of x are given in Table 1. Generally, the scale of
the trailing vortex for the Rushton and pitched-blade
turbines is equal to one-half of the projected height
of the blade at its tip. For hydrofoils the scale of the
trailing vortex is equal to the projected height of the
blade at its tip.
Equations (13) and (24) can be combined to show that
the ratio of the maximum energy dissipation rate to the
average power input per mass, K, is:
FIGURE 4. This graph shows a plot of the mean velocity profiles for pitchedblade
turbine and hydrofoil impellers
48
CHEMICAL ENGINEERING WWW.CHEMENGONLINE.COM
AUGUST 2017
the Rushton should create the smallest droplets
and the hydrofoil, the largest, with the pitchedblade
turbines falling somewhere between these
two. In fact, the hydrofoil creates smaller droplets
than the Rushton and two pitched-blade turbines
and the droplets created by the turbines are indistinguishable
experimentally. This result has also
been observed by Pacek and others [9].
There is another geometrical property of impellers
that determines how they create the " fluid dynamic effect "
that achieves this desired " process result. " This
is the trailing vortices that form at the tip of the impeller
blades.
Trailing vortex
As the impeller moves through the fluid, the pressure on
the leading face of the blade is higher than on the back.
The high- and low-pressure zones meet at the tip of the
blade and the fluid moves from the high- to low-pressure
region creating the trailing vortex. This phenomenon can
often be observed on airplane wings [10, 11].
In a stirred tank, the velocity and size of the vortices
can be measured using laser-Doppler or particle-image
velocimetry, then the kinetic energy and energy dissipation
rate (the local power input per mass) can be calculated.
The kinetic energy of the trailing vortex is often
non-dimensionalized by dividing by the impeller tip speed
squared. Grenville and others [12] have shown that for
impellers with blades:
kMAX
VTIP
2 = 0.104 Po12
(22)
Where VTIP = πN.D.
The standard deviation for this correlation is ±10%.
The maximum energy dissipation rate within the vortex
is given by the following equation [13]:
MAX = A
kMAX
32
l 0
(23)
l0 is a length scale related to the flow near the impeller
and it is a fraction of the impeller diameter. Substituting
Equation (22) into Equation (23) and setting l0 = D/x and
A = 1:
MAX =1.04 x Po34
N3
D2
(24)
v/VTIP, %
http://WWW.CHEMENGONLINE.COM
Chemical Engineering August 2017

Table of Contents for the Digital Edition of Chemical Engineering August 2017
