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Monster Media 1996 #14
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Monster Media No. 14 (April 1996) (Monster Media, Inc.).ISO
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AGITATOR.TXT
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AGITATOR DESIGN METHODS
The routines used in the agitator design section encompass much of
the known theory of mixing. Different correlations and methods are
used for different types of agitators. All methods and procedures
used in this program are available in the literature through
technical articles or in standard texts on mixing.
POWER CORRELATIONS
The basic methods used for all agitators is based upon pump
technology. The power required by an impeller is proportional to
the pumping flowrate and the velocity head produced.
P = f(Q*H*density)
Power is also proportional to the the rotational speed to the third
power and the diameter of the impeller to the fifth power.
P = f(N^3*D^5*density)
These relationships are usually correlated in terms of power
numbers and flow numbers that are measured and are specific for
different types of agitators. The correlations are in general
based upon a tank that contains four baffles.
The Flow number Nq is a dimensionless group that is used to
quantify the flow characteristics of the agitator. It is
given by the following relationship.
Nq = Q/N*D^3
Values of experimentally derived flow numbers are presented in Uhl
and Gray and in OldShues Fluid Mixing Technology Table 8-3. and
also in the Chemineer articles referenced earlier.
The power number Np is correlated against the mixer Reynolds
number. These two dimensionless variables are defined as follows.
Np = (1.523 * 10^13 * P) / (N^3 * D^5 * density)
Nre = (10.754 * N * D^2 * density)/Viscosity
In order to determine the power requirements of a specific agitator
it is necessary to read the power number for that type of agitator
from the Reynold's number correlations. There are power number
correlations in the literature for a major types, such as Axial
flow, Radial Flow, propellers and other types. Once the power
number has been read from the graph the power requirement is
determined by multiplying the power number by the liquid density,
the RPM to the third power and the Agitator Diameter to the fifth
power.
The Power requirements for a given agitator are calculated
from the power number by the following equation.
P = Np * N^3 * D^5 * density / 1.523*10^13
The correlations for power numbers are given for different
agitators in several standard texts. The graphs given here were
published in Chemical Engineering Progress Sept 1950,
Page 470 and Vol 2 Oct 1963.
The power number of the agitators under turbulent conditions
reaches a constant value that is specific for the agitator. The
graphs were also developed for a certain number of blades and
standard blade widths. They are also based upon a standard tank
geometry with 4 baffles sized at T/12 width.
The program uses a curve fit of these graphs, and covers the range
from Newtonian flow at very low Nre to very high levels where the
power number is constant. Additional correlations are used to
adjust the power results for different number of blades, blade
width, and proximity of the agitator to the bottom and top of the
tank.
The specific correlations for the different agitators selected are
defined as follows for each type.
Flat and Disk Blade Turbines
This type of agitator (also known as a Rushton Type turbine)
generally consists of 6 blades that are horizontal to the shaft.
The Blade width is normally 0.2 of the turbine diameter. A disk
agitator has a center disk that the blades are attached. The disk
serves to direct the flow of the liquid (or gas) to the blades.
Disk agitators have a somewhat higher power number than a flat
blade turbine without a disk.
AGITATOR POWER
For agitator Reynold's number above 10,000 ie highly turbulent
flow, the power numbers reach a constant value.
The power number of the flat blade turbine is 4.0 for a BW of 1/5.
A disk blade turbine has a Np of 5.0 for this case.
For both the flat blade turbine:
The power Number is adjusted for blade width as follows:
Np = Np * BW/0.2
This relationship is based upon Nagata, where he shows that power
is roughly proportional to blade width, for W/D ration of 0.1 to
0.4 (Nagata 'Mixing Principles and Applications').
The power number is adjusted for number of blades by correction for
the Power number of the 6 bladed turbine by the number of blades to
the 0.8 power as follows:
Np = Np * Np(6blades) * (Number of Blades / 6 ) ^ 0.8
The power numbers at different Reynolds Numbers were curve fit from
the following graph.
( Graphs and Specific Correlations are deleted from Demo Text)
POWER NUMBER CORRELATIONS
RADIAL FLOW TURBINES
AGITATOR FLOW
The total circulation flow that is generated from a turbine is
comprised of the flow from the blades of the turbine and an induced
flow. The flow from the turbine blades is called the discharge
flow from the turbine. The discharge flow entrains fluid to
produce the total circulation flow.
It is the discharge flow numbers that have been measured and
presented in the standard references (see Oldshues Fluid
Mixing Technology Table 8-3 )
The discharge flow from a radial turbine is given by
Q = Nq * N * D^3
Nq is a constant known as the pumping of discharge flow
number, it is dependent on the type of impeller, system
geometry and the Reynolds number.
Under turbulent flow conditions, the radial flow discharge number
is calculated by the following equation.
Nq = 6.0 * B * BW * (D/T)^0.3
WHERE B = (NUMBER BLADES / 6)^ 0.7
D = IMPELLER ID inches
T = tank diameter inches
The Nq number for a radial flow turbine is generally about 0.8, the
total induced flow rate is generally about 3 times higher, giving
a flow number of around 2.75, For axial flow turbines the Nq is
also about 0.77 but the induced flow is less giving a total flow
Number of around 1.73. (Oldshue Mixing Tech p 174)
The flow profile produced by a radial flow impeller is horizontal
toward the vessel walls, and it splits with somewhat more than 50
percent flowing upward. The flow pattern is determined by the
vessel geometry and depth of the agitator.
In order to be consistent with the bulk of the literature, the
radial discharge flow numbers are used in determining the
circulation rates and for correlations of blend time.
The vertical flow velocity from a radial turbine in the tank is
calculated from the circulation flow generated from the equation.
This flow is divided by 2 to account for the split flow at the
walls with a radial turbine, and by the cross sectional area of the
tank to get the vertical velocity.
Flow = Nq * D^3 * N
SHEAR
Radial flow turbines are used for applications where shear is
require to achieve the process result. Shear rate is calculated
for the flat blade and disk turbines, but is not done for the axial
blade or other types. The shear rate produced by these other
agitators is insignificant. The flat blade turbines force the gas
or fluid to be dispersed horizontally into the high velocity
regions associated with the edge of the turbine blade. The axial
flow turbines pump the fluid vertically though the inpeller and
have very low shear rates. Shear rate calculation is important to
judge the effectiveness of gas dispersion and other applications
such as dispersing a second immisible liquid phase, such as acid or
caustic, where the size of the droplets formed can affect reaction
rates.
The correlation used for maximum shear were taken from
Bowen's article in Chemical Engineering June 9 1986.
The correlation is based upon methods for predicting the velocity
profiles for the turbines. The maximum velocity at the centerline
of a radial flow turbine is given by the equation. This
correlation is based upon measurements of the resultant flow
generated by a turbine from the data of Cooper and Wolf.(Canadian
Journal Echem. Eng. Vol 46, 1968 p 94).
Vmax = 4.9 * N * D * (D/T)^0.3
This velocity is about 50 % greater than the radial discharge
velocity. If the velocity profile is determined across the blade
width of the impeller, the shear rate (dv/dz) is given by the slope
of the curve. Shear rate is given in units of reciprocal seconds.
The maximum shear rate is found on the sides of the profile where
the slope is steepest. The maximum velocity decreases rapidly over
the blade width, moving away from the centerline of the blade, and
levels off beyond the blade.
The average shear rate across the velocity profile is found by
taking the difference between the centerline velocity and a point
where the velocity of the fluid has leveled out, and dividing this
by the distance between the measurements. The leveling off point
is assumed to be a one blade width past the centerline.
The velocity at one blade width from the centerline is about 15% of
the centerline velocity. Consequently, the velocity difference is
0.85 * the maximum velocity.
(dv/dz)ave = 0.85 * Vmax / W
(dv/dz)ave = (0.85 * 4.9) * ND(D/T)^0.3/(W/D) * D
where W = Blade width
This equation shows that for geometrically similar impellers at the
same D/T ration the average shear varies only with speed, and is
independent of diameter. The maximum shear is 2.3 times the
average shear. This relationship can also be shown in terms of the
agitator RPM and Diameter to Tank ratio and Blade Width as follows.
Shear(max) = 9.7 * N/60 * (D/T)^0.3 / (W/D)
One implication of these equations is that the shear rate can be
increased and the power reduced by reducing the width of the blade
below the industry standard of 0.2. The flow is also reduced
however by reducing the blade width and care must be taken to
maintain pump circulation. Blade widths down to 0.1 times the
impeller diameter can be used to save power on applications where
a high shear rate is critical to the process.
PROXIMITY FACTORS
The location of the agitator in the tank affects the power
consumption and circulation rates. If a radial flow turbine is
located close to the bottom of a tank the power consumption is
reduced because the circulation flow to the Bottom impeller eye is
restricted by the tank geometry.
An axial flow impeller will have an increase in power consumption
when the impeller is closer to the bottom because the resistence to
flow increases the head generated by the impeller for the same flow
rate.
Graphs for the proximity factors for radial and axial flow
impellers are presented in OldShues text Figures 3-20,through 3-22.
This program uses similar relationships, with different
correlations for each type and location of agitator. The power
requirement without the proximity factor is also generated by the
program. For multiple impeller designs the average proximity
factor for the impellers is used.
AXIAL FLOW IMPELLERS
POWER NUMBER
The power number of Axial turbines are also correlated against the
mixing Reynold's numbers. These correlations were curve fit for
the 6 blade 45 degree pitch turbines with a W/D of 1/8, as was done
for the Radial flow turbine. The specific correlations used are as
follows as a function of Nre. The program adjusts for the actual
number of blades on the turbine by the same equation described
earlier for flat blade turbines.
In the viscous region at low Reynold's Numbers the power number is
approximately:
Np := (50.0 / Nre)* Sqrt(Wd/0.125);
In the turbulent range the power number becomes constant and given
by:
Np := 1.4* ( Wd/0.125);
Note Wd = (W/D) above
FLOW
The correlation used for the flow number Nq of a pitched blade
turbine as a function of the Reynold's number is based upon the
Chemineer Article reference earlier. The correlation is based upon
4 blades at a blade width of 0.2. The value for the Nq is adjusted
by the program for the actual number of blades by the correlation
given in Nagata on page 138.
Nq = Nq * (Wd/0.2)
Nq = Nq * ((Number Blades)/4)^0.7
BLEND TIME
The Correlations used to predict Blend Time for axial flow
agitators are based upon the correlation presented by Fenic and
Fondy (1966 AIChE Atl. City). The correlation gives
blend times as a function of Nre. Data taken from Uhl and
Gray Vol 1 p 219 were used to modify this correlation for
different impellers. The minimum blend time presented by the
program will be at least 5 batch turnovers of the vessel by the
liquid pumping rate.
The correlations for blend time are only approximate and can be
used for comparisons. They are no substitute for pilot
plant data for critical applications.
RETREAT BLADE AGITATORS
Retreat blade agitators are similar to flat blade turbinesand
propellers except that the blades curve backward. This type of
agitator is popular with Pfaudler for use in glass lined reactors.
The correlations are based upon retreat blade agitators with the
use of two finger baffles.
If Nre <= 10 then
Np := 50.0 / Nre;
If (Nre > 10 ) and (Nre <= 100.0 ) then
Np := 21.832446 * Pow(Nre,-0.66690987);
If (Nre > 100 ) and (Nre <= 400.0) then
Np := 2.818063 * Pow(Nre,-0.20327403);
If (Nre > 400 ) and (Nre <= 10000. ) then
Np := 1.0399062 * Pow(Nre,-0.0559205);
If Nre > 10000. then
Np:= 0.55 ;
FLOW
The correlations for flow were taken from Nagata Page 138
Nq := 0.29 ; { with two baffles }
CORRECTIONS FOR ONE BAFFLE
If (Nre > 400) and (Baffle = 1.0)
Np := 0.77 * Np; { for one baffle}
Nq := 0.23;
PROPELLER AGITATORS
POWER
The following correlations were used for the power number for
propellers with a pitch of 1.5
In the viscous range the power number is given by
Np = 45 / Nre
In the turbulent range the Power Number becomes constant.
Np = 0.5
The power number is adjusted for different blade pitch by the
ratio
Np = Np * (pitch/1.5)^ 1.5
FLOW
The discharge flow number is calculated as follows as a
function of pitch.
Nq = 0.55 * pitch
ANCHOR AGITATORS
The correlations for the anchor agitator are for a pure U anchor
with no cross bars. The cross bars should be estimated separately
pitched turbine to estimate the effect on power of the cross bars.
POWER
The power number correlations are curve fit from basic data present
in Oldshue text on Fiqure 3-15 Page 61. The correlation is based
upon a Height to diameter ratio of 1 for the agitator. The
diameter of the impeller is selected by the program at 0.95 * the
tank diameter. This can be over ridden by the user from the
command line. Anchors are usually only used for high viscosity
applications to maximize the heat transfer coefficient at the
walls. For blending applications in high viscosity service the
selection of a spiral agitator is generally far superior.
In the viscous range the power number is approximately.
NP = 280 / Nre;
In the turbulent range the power number becomes constant at
Np = 0.4;
BLEND TIME
Blend time for anchor agitators was taken from Nagata's correlation
Fiq 4.24
Time = (5/3) * 60000 / N
( Nagata used 3 turnovers I prefer to use 5)
The values for circulation rate and velocity were backed out of
Nagata blend time correlation assuming 5 batch turnovers for
uniformity, instead of 3. I am not aware of published data on Nq
values for Anchor agitators.
The shear rate of the agitator was calculated by the following
equation.
DvDs := N * Pi * D / ((T - D) * 60);
HELIX AGITATORS
Helix agitators are describe very throughly in Nagata's work. The
correlations used are as follows for double helix agitators. The
Power requirements for a single helix with a center screw is
approximately the same but somewhat lower.
The correlations assume that the Diameter of the Helix is equal to
0.95 the tank diameter. The blade pitch is assumed equal to the
diameter of the impeller. The Blade width is assumed equal to 0.1
* impeller diameter.
POWER
The following correlation is used for the power number as a
function of the Reynold's Number at low Nre.
Np := ( Nre * 245.663404)^ -0.8524365114);
The program adjusts for different screw pitch, diameter and blade
widths by the following relationship.
Np = 78.0 / (Nre * A * B)
Where A = SQRT ( screw pitch)
B = SQRT((T - D)/D))
BLEND TIME
The blend time is calculated by the relationship
Time =(5/3) * 500 * 0.1 * 60 * ScrewPitch / N
based upon 5 batch turnovers instead of 3.
The circulation rate is given by
Qcirc := (5 * A/time)*60; {for Neutonian fluids}
Vel := Qcirc/Csa;
The maximum shear rate is given by
DvDs := rpm * Pi * D / ((T - D) * 60);
based upon 5 batch turnovers instead of 3.
The maximum shear rate is given by the same equation used above for
Anchor Agitators.
AGITATOR TORQUE
The agitator torque is calculated in Inch Pounds and used in the
shaft sizing programs. The following equation is used.
TORQUE := Bhp * 6.3025E4 / Rpm ;
Where Bhp is sum of all impellers
MISCELLANEOUS RELATIONSHIPS
The Prandtl number is calculated by
Npr := Mfcp * Mfvis * 2.42/mftc;
MfCp = Mix Phase Heat Capacity
MfVis = Mix Phase Viscosity
MfTc = Mix Phase Thermal Conductivity
The Impeller tip speed is calculated by
TS = (D/12) * Rpm * Pi
The relationship called mixing intensity is used by Chemineer to
correlate blending relationship. It is simply the
vertical velocity of the tank contents divided by 6. The
velocity is the impeller pumping rate divided by the CSA of the
tank.
MI = Vel/6.0 ; { Mixing intensity }