Transcript Pharmaceutical Processing and Manufacturing

Special Topics - Modules in
Pharmaceutical Engineering
ChE 702
Liquid Mixing
Fundamentals
Piero M. Armenante
2008©
Instructional Objectives of
This Section
 By the end of this section you will be
able to:
Identify the geometric, physical and
dynamic variables of importance for the
analysis of mixing in a stirred tank
Assess the relative importance of those
variables
Quantify the power dissipation, pumping
effects, and blend time in a mixing vessel
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Summary




Basic Rheology
Power Dissipation
Impeller Pumping Effects
Blend Time in Stirred Tanks
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Basic Rheology
Basic Rheological Concepts
 Consider a fluid contained between
two plates separated by a distance y.
 One plate is set in motion parallel to
the other, with velocity vx.
 For many fluids it has been found
experimentally that the force applied
to the plate is directly proportional to
vx and inversely proportional to y.
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Basic Rheological Concepts
vx=v
y
x
vx=0
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Newton’s Law of Viscosity
 Mathematically:
Force F
vx  0
v x
  constant
 
Area
A
y
y
i.e.:
 yx
dv x
 
     yx
dy
 This constitutes Newton’s Law of
Viscosity.
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Newton’s Law of Viscosity
Definitions
 Shear Stress:  yx
 Shear rate:
 yx
dv

dy
 (Dynamic) Viscosity: 

 Kinematic Viscosity:  

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Newtonian Fluids
 Newtonian fluids are fluids having
constant viscosity.

 xy
Increasing
Viscosity
 xy
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Dynamic Viscosities of
Various Fluids
Fluid
Gases
Viscosity
(centipoise, cP)
~ 0.001
Organics
<1
Water
~1
Kerosene
~10
Lubricants
~100
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Dynamic Viscosities of
Various Fluids
Fluid
Glycerol
Viscosity
(centipoise, cP)
~1000
Corn Syrup
~10,000
Molasses
~100,000
Molten Polyethylene
~1,000,000
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Focus of This Section
Only the mixing behavior
of Newtonian fluids, and,
more specifically, liquids,
will be examined in this
section.
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Schematic of a Stirred Tank
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Important Variables in the
Analysis of Mixing Phenomena
 The variables of importance in the
analysis of mixing phenomena in
stirred tanks can be classified as:
geometric variables
physical variables
dynamic variables
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Geometric Variables
Geometric variables include the
geometric characteristics of:
 tank (shape, sizes)
 shaft
 liquid height
 baffles (shape, size, position)
 impellers (type, dimensions, position)
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Geometric Variables: Tank,
Shaft, and Liquid Height
 Tank shape (e.g., cylindrical)
 Tank bottom shape (e.g., dish, flat)
 Internal diameter, T
 Internal height, HT
 Shaft diameter
 Shaft length
 Liquid height, H (or Z)
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Geometric Variables: Baffles
 Number of baffles, nB
 Shape (e.g., rectangular)
 Baffle width, B
 Baffle height (e.g., full, half)
 Baffle thickness
 Gap between baffles and tank wall
 Gap between baffles and tank bottom
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Geometric Variables:
Impellers
 Number of impellers, n
 Impeller type (e.g., disc turbine)
 Diameter, D
 Blade angle
 [Pitch, p]
 Blade width (height), w
 [Blade width projected across the
vertical axis, wb]
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Geometric Variables:
Impellers (continued)
 Clearance off the tank bottom measured
from the midpoint, C
 [Clearance off the tank bottom measured
from the impeller bottom, Cb]
 Spacing between impellers, S
 Disc diameter (disc turbines)
 Blade thickness
 Hub diameter
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Physical Variables
 Liquid density,  or L
 Liquid “rheology” (e.g., newtonian,
non-netwonian, shear-thinning, etc.)
and corresponding parameters (e.g.,
power law exponent)
 Dynamic viscosity, 
 [Kinematic viscosity,  (= /)]
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Dynamic Variables
 Impeller rotational (agitation) speed,
N
 Impeller angular velocity, 
 Impeller tip speed, vtip
 Torque, 
 Power dissipation (consumption), P
 Impeller pumping flow, Q
 Gravitational acceleration, g
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Relationship Between N, 
and Vtip
 The agitation speed, N, must be expressed
in revolutions per unit time such as:
 revolutions per minute (rpm)
 revolutions per second (rps)
 The tip speed, vtip, is not independent of N
but it is related to N as follows (with  in
rad/s, N is in rps, D in m, vtip in m/s):
vtip
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D
   R  2 N    N D
2
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Power Dissipation in Low
Viscosity Liquids in Stirred
Tanks
Instructional Objectives of
This Section
 By the end of this section you will be
able to:
Calculate Re, Fr in stirred tanks
Distinguish agitation regimes
Calculate the power dissipated by an
impeller from available power numbers
Calculate the power dissipation as a
function of operating variables
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Turbulence and Mixing
 Turbulent flows are associated with
rapid, apparently random
fluctuations of all three components
of the local velocity vector with time
 To this day turbulence is still a
relatively poorly understood
phenomenon
 Many mixing phenomena are
associated with turbulence
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Velocity Fluctuations in
Turbulent Flow
v(t)
v
t
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Turbulent Flow
 In a turbulent flow, “pulsations”
consisting of disorderly displacement
of fluid bodies (eddies), are
superimposed on an average flow.
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Isotropic Turbulence
 In isotropic turbulence all the
fluctuation components are equal,
and there is no correlation between
the fluctuations in different directions
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Energy Cascade in Isotropic
Turbulent Flow
 During the process of energy
transfer and ultimate decay in a
turbulent system the largest eddies
receive fresh kinetic energy from an
outside source (e.g., an impeller) and
pass it on to smaller eddies that are
produced as a result of the instability
of the primary eddies
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Energy Cascade in Isotropic
Turbulent Flow
 During this process smaller and
smaller eddies are generated
 One can conceptually introduce an
eddy Reynolds Number:
Re eddy 
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 eddy veddy

inertial forces

viscous forces
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Energy Cascade in Isotropic
Turbulent Flow
 As long as Reeddy>>1 no viscous
dissipation will occur, and the kinetic
energy will simply be transferred to
smaller and smaller eddies
 However, at Reeddy~1 viscous forces
will begin to dominate
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Energy Cascade in Isotropic
Turbulent Flow
 For Reeddy<<1, the eddy will not
break up and the eddy kinetic energy
will be transformed into heat by the
viscous forces (energy dissipation).
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Energy Cascade in Isotropic
Turbulent Flow
 Such a transition occurs at the
Kolmogoroff’s length scale, equal to:
 3 
k     
 
1
4
where  is the power dissipated per unit
mass and  is the kinematic viscosity.
 k is the size of the smallest eddy in
the turbulent fluid
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Energy Cascade: Summary
Big whorls have little whorls
That feed on their velocity,
And little whorls have lesser whorls
And so on to viscosity.
Lewis F. Richardson
(1881-1953)
The poem summarizes Richardson's 1920 paper ‘The Supply of
Energy from and to Atmospheric Eddies‘.
(A play on Jonathan Swift's "Great fleas have little fleas upon their backs to bite 'em,
And little fleas have lesser fleas, and so ad infinitum." (1733))
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Energy Cascade: Summary
Big whirls have little whirls,
That feed on their velocity,
Little whirls have smaller whirls,
And so on to viscosity.
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Power Dissipation
 The power dissipated (or consumed)
by the impeller, P, is one of the most
important variables to describe the
performance of an impeller in a tank
 P is a function of all the geometric and
physical variables of the system
 Dimensional analysis can be used to
establish a relationship between P and the
independent variables
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Experimental Determination
of Power Consumption
 The power dissipated by various
impellers under different conditions
has been experimentally obtained by
many investigators
 Power data are available in the
literature (as non-dimensional Power
Numbers)
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Experimental Determination
of Power Consumption
 It is relatively easy to determine the
cumulative overall power drawn by a
mixing system (including motor, drives,
seals, impellers, etc.)
 It is much more difficult to determine the
power dissipated by the impeller alone
 The power dissipated by the impeller in
the fluid is the only important power
dissipation parameter for the mixing
process
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Experimental Determination
of Power Consumption
 The total power dissipation in a
system is given by:
Ptotal  Pmotor  Pgearbox  Pseal  Pimpeller
 If one needs to know Pimpeller, Ptotal
and all other power dissipation
sources must be known under the
dynamic conditions in which the
impeller operates
 This can be quite difficult
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Experimental Determination
of Power Consumption
 A number of methods have been
used to measure the power
dissipated by impellers including:
electric measurements
dynamometers (coupled to the
motor or the tank)
strain gages and torquemeters
calorimetric measurements
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Example of Strain Gage System for
Power Measurement
Motor
Controller
Strain Gage Conditioner
Probe
Slip Ring
Tachometer
Interface
Strain Gages
Vessel
Computer
Armenante and Chang, Ind. Eng. Chem. Res., 1998.
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Power Dissipation
 For the case in which a number of
geometric variables have been
defined (e.g., tank shape, tank
bottom, impeller type, baffle position,
etc.) the dependence between P and
the other variables can be written as:
P  f (N, D,T , H,C, B,w, n, nB , g, , ,
impeller type )
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Power Dissipation
 Using dimensional analysis
(Buckingham pi theorem) the previous
equation can be rewritten in nondimensional terms, as:
P
NP  Po  Ne 
N 3 D 5
  ND 2 N 2 D T H C w B

 f 
,
, , , , , , n, nB , impeller type 
g D D D D T
 

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Power Number, NP (also
referred to as Po or Ne)
 The impeller Power Number, Np
(also called Po, or the Newton
number, Ne) is a non-dimensional
variable defined as:
P
Np  Po  Ne 
3 5
N D
 If English units are used then:
P gc
Np  Po  Ne 
N 3D5
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Power and Power Number
 The power consumed by an
impeller and the Power Number are
related to each other via the
equation:
P  NP N 3D5
where Np is a function of the impeller
type and the geometric and dynamic
characteristic of the system
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Impeller Reynolds Number
 The impeller Reynolds number, Re,
defined as:
2
ND
Re 

is a product of the non-dimensional
analysis.
 Compare this Re with the Reynolds
number for a pipe:
vD
Re 
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pipe

46
Impeller Reynolds Number
 As usual, a physical interpretation can
be associated with the impeller
Reynolds number, Re. Accordingly:
Inertial forces
Re 
Viscous forces
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Froude Number
 Another non-dimensional number
arising from the non-dimensional
analysis is the Froude number, Fr,
defined as:
DN
Fr 
g
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Froude Number
 It can be shown that the Froude
number has the following physical
interpretation:
Inertial forces
Fr 
Gravitatio nal forces
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Power Equation
 The power equation can be rewritten
P
as:
NP 
N D
3
5

T H C w B


f  Re, Fr , , , , , , n, nB , impeller type 
D D D D T


 i.e.:
 Re, Fr , geometric ratios,
P
NP 
 f 
3 5
N D
 impeller type
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


50
Geometrical Similarity
 Two systems are geometrically
similar if all corresponding
dimensional ratios are the same in
both systems
1.5 H
H
1.5 Cb
Cb
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D
1.5 D
T
1.5 T
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Geometrical Similarity
 For geometrically similar systems:
T
T2 T1
 constant 

D
D2 D1
H
H 2 H1
 constant 

D
D2 D1
C
C2 C1
 constant 

D
D2 D1
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Power Equation for
Geometrically Similar Systems
 For geometrically similar (including
same type of impeller) stirred tanks
and impeller all geometric ratios
are the same
 Hence, NP does not change with
scale between tanks:
P
NP 
 f Re, Fr 
3 5
N D
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Power Equation for Baffled,
Geometrically Similar Systems
 When baffles are present, no vortex
occurs, i.e., the gravitational forces
become unimportant, and the Power
Number becomes independent of
Fr:
P
NP 
 f Re 
3 5
N D
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Typical Power Curve for
Impellers in Baffled Tanks
10
La
in
m
ar
Power Number, Po
100
al
n
o
i
t
i
s
ran
Turbulent
T
1
1
10
100
1000
10000
100000
Reynolds Number, Re
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Power Curve:
Laminar Flow Regime
 For Re<10 the flow in a baffled tank
is laminar
 Theoretical and experimental evidence
shows that:
 i.e.:
P
1

NP 


3 5
N D
Re  N D 2
P   N 2 D3
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Power Curve:
Laminar Flow Regime
 In the laminar flow region the
power dissipated by an impeller is
given by:
P  k"  N D
2
3
where k” is a proportionality constant
that depends on:
type of impeller
geometry ratios for the system
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Power Dissipation in the
Laminar Flow Regime
 In the laminar regime power
dissipation is:
independent of the density of the liquid
directly proportional to the viscosity
strongly affected by the agitation speed
(PN2)
strongly affected by the impeller diameter
(PD3)
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Power Curve:
Transitional Flow Regime
 For ~10<Re<~10,000 the flow regime
cannot be well characterized as either
fully laminar or fully turbulent
 Depending on the type of impeller NP
may decrease with Re or decrease and
then increase with Re before entering
the turbulent flow regime
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Power Curve:
Turbulent Flow Regime
 At high Reynolds numbers
(Re>10,000) the flow in a baffled
tank is turbulent
 Theoretical and experimental evidence
shows that NP is independent of Re:
P
NP 
 constant
3 5
N D
 i.e.:
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P   N 3 D5
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Power Curve:
Turbulent Flow Regime
 In the turbulent flow region the
power dissipated by an impeller is
given by:
3 5
3
P  k '  N D  NPT  N D
5
where k’ is a proportionality constant
equal to NPT, the asymptotic value of
NP that depends on:
type of impeller
geometry ratios for the system
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Power Dissipation in the
Turbulent Flow Regime
 In turbulent regime, power
dissipation is:
independent of viscosity
directly proportional to the density of the
liquid
very strongly affected by the agitation
speed (PN3)
extremely sensitive to the impeller
diameter (PD5)
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Sensitivity of Power
Dissipation
 In the turbulent regime P is very
sensitive to N and D
 Examples:
a 10% increase in agitation speed, N
increases the power dissipated by 33%
a 20% increase in N increases P by 73%
a 10% increase in impeller diameter, D
increases the power dissipated by 61%
a 20% increase in D, increases P by
148%
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Sensitivity of Power
Dissipation
 Because of the sensitivity of the power
dissipation to impeller diameter and
agitation speed small adjustments
to the impeller size or agitation
speed can rectify situations in which
an existing motor is underpowered
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Power Number Curves for
Various Impellers
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After Bates et al., Ind. Eng. Chem. Proc. Des. Devel. 1963
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Power Number Curves for 45° Pitched-Blade
Turbines (4-Blades) and HE-3 Impeller
After K. Myers and R. J. Wilkens, Personal Communication
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Equation for Power Number
Curves
 An equation for Power Number as a
function of Re has been proposed:
A
Re
NP 
 BC
Re
1000  Re
where A, B, and C are coefficients that
depend on the type of impeller.
After John Smith, Unpublished Data
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Coefficients in Equation for
Power Number Curves
Impeller Type
A
B
C
Rushton Turbine
67
3.2
1.8
45 Pitched-Blade
Turbine (4-blades)
60 Pitched-Blade
Turbine (4-blades)
49
1.5
0.3
50
4.0
1.0
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Turbulent Power Number
 Most low viscosity systems and
industrial stirred tanks operate in the
turbulent regime where NP is constant
 A simple and meaningful way to compare
the power performance of various
agitators is to compare their turbulent
Power Numbers, NPT
 The term “Power Number” is often used to
mean “Turbulent Power Number”
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Turbulent Power Numbers
 Turbulent Power Numbers have been
obtained experimentally for many
impellers
 Typically, NPT is measured for a
“standard” configuration of the agitation
system (H=T, D/T=1/3, C=D)
 Data also exist for other non-standard
systems (e.g., NP as a function of C/D)
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Turbulent Power Numbers
for Various Impellers
Impeller Type
NPT
Flat-Blade Turbine (24 Blades)
9.8
Flat-Blade Turbine (12 Blades)
8.5
Gate
5.5
Disc Turbine (Rushton Type)
5.0
Smith Turbine (Concave-Blade
Turbine)
3.2
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Turbulent Power Numbers
for Various Impellers
Impeller Type
Flat-Blade Turbine
(6 Blades, w/D=1/5)
Flat-Blade Turbine
(6 Blades, w/D=1/8)
Curve-Blade Turbine
(6 Blades, w/D=1/8)
45 Pitched-Blade Turbine
Prochem
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NPT
4.0
2.6
2.6
1.3-1.7
1.0
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Turbulent Power Numbers
for Various Impellers
Impeller Type
NPT
Glass-Lined Impeller
(Pfaudler Type)
MIG Impeller
0.75
Marine Propeller
0.35
Lightnin A310
0.30
Chemineer HE-3
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0.65
0.26-0.30
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Effect of D/T Ratio on Power
Number for Disc Turbines
6
5
Ne
4
3
2
1
0
0.2
0.25
0.3
0.35
0.4
D/T
Armenante and Chang, Ind. Eng. Chem. Res., 1998.
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Effect of D/T Ratio on Power Number for
Pitched-Blade Turbines
6-PBT; C
6-PBT; C
6-PBT; C
6-PBT; C
3
/D=0.24
/D=0.10
/T=0.25
/T=0.05
4-PBT; C
4-PBT; C
4-PBT; C
4-PBT; C
/D=0.84
/D=0.54
/T=0.28
/T=0.19
Ne
2
1
0
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
D/T
Armenante et al., Ind. Eng. Chem. Res., 1999.
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Effect of Impeller Clearance on
Power Number for Disc Turbines
6
5
Ne
4
3
D/T=0.352; H/T=1
D/T=0.264; H/T=1
D/T=0.352; H/T=2
D/T=0.264; H/T=2
Regression Curve
2
1
0
0
1
2
3
4
5
6
Cb1/D
Armenante and Chang, Ind. Eng. Chem. Res., 1998.
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Effect of Impeller Clearance on
Power Number for Disc Turbines
 A correlation between the Power
Number and the impeller clearance off
the impeller bottom, Cb1, is:
Cb1 

NP  4.93  3.44exp   5.38

D 


 C '1 w b 
 4.93  3.44exp  5.38


 D 2D 

Armenante and Chang, Ind. Eng. Chem. Res., 1998.
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ChE702
77
Effect of Cb on Power Number
for Disc Turbines
6
Po
5
4
D/T
0.217
0.261
0.348
3
2
0
0.05
0.1
0.15
0.2
0.25
Cb /T
Armenante and Uehara Nagamine, Chem. Eng. Sci., 1998.
Piero M. Armenante
ChE702
78
Effect of Cb on Power Number
for Flat-Blade Turbines
3
D/T
0.217
0.261
0.348
2.8
Po
2.6
2.4
2.2
2
0
0.05
0.1
0.15
0.2
0.25
Cb /T
Armenante and Uehara Nagamine, Chem. Eng. Sci., 1998.
Piero M. Armenante
ChE702
79
Effect of Impeller Clearance on Power
Number for Pitched-Blade Turbines
3
6-PBT; D/T=0.264; H/T=1
6-PBT; D/T=0.264; H/T=2
4-PBT; D/T=0.343; H/T=1
4-PBT; D/T=0.507; H/T=1
Ne
2
1
0
0
1
2
3
4
5
6
Cb1/D
Armenante et al., Ind. Eng. Chem. Res., 1999.
Piero M. Armenante
ChE702
80
Effect of Cb on Power Number
for Pitched-Blade Turbines
3
Po
D/T
0.217
0.261
0.348
2
1
0
0.05
0.1
0.15
0.2
0.25
Cb /T
Armenante and Uehara Nagamine, Chem. Eng. Sci., 1998.
Piero M. Armenante
ChE702
81
Effect of Cb on Power Number
for HE-3 Impellers
D/T=0.348; T=0.292 m
D/T=0.391; T=0.292 m
D/T=0.304; T=0.584 m
D/T=0.348; T=0.584 m
D/T=0.391; T=0.584 m
0.5
0.45
Po
0.4
0.35
0.3
0.25
0
0.05
0.1
0.15
0.2
0.25
Cb /T
Armenante and Uehara Nagamine, Chem. Eng. Sci., 1998.
Piero M. Armenante
ChE702
82
Power Dissipation in
Multiple Impeller Systems
Piero M. Armenante
ChE702
83
Power Dissipation in
Multiple Impeller Systems
 If the H/T ratio is larger than 1.2-1.5
multiple impellers are typically used
 The Power Number and the power
drawn by two impellers mounted on
the same shaft and spaced by a
distance S is not usually twice that of
the individual impeller
 For large S, NP double  2 NP single
Piero M. Armenante
ChE702
84
Power Dissipation in Multiple
Disc Turbine Systems
3
H
2
H
1
S
D
Piero M. Armenante
Cb2
Cb1
2
S23
S13
1
S12
D
T
T
Double DT
System
Triple DT
System
ChE702
Cb2
Cb1
85
Power Dissipation in Double
Disc Turbine Systems
10
Ne
8
6
4
D/T=0.264
H/T=1
S/D=1.5
2
0
0
0.2
0.4
0.6
0.8
1
1.2
Cb1/D
Armenante and Chang, Ind. Eng. Chem. Res., 1998.
Piero M. Armenante
ChE702
86
Power Dissipation in Double
Disc Turbine Systems
10
Ne
8
6
4
D/T=0.264
H/T=1
Cb1/D=1
2
0
0
0.5
1
1.5
2
S/D
Armenante and Chang, Ind. Eng. Chem. Res., 1998.
Piero M. Armenante
ChE702
87
Power Dissipation in Double
Disc Turbine Systems
10
Ne
8
6
S/D=3
H/T=2
Cb1/D=1
4
Impeller Location
1
Total
2
Single Std.
2
0
0.1
0.2
0.3
0.4
D/T
Armenante and Chang, Ind. Eng. Chem. Res., 1998.
Piero M. Armenante
ChE702
88
2
air entrainment
1.8
1.6
S/D
0.667
1
1.333
1.5
1.667
1.4
D/T=0.264
H/T=1
1.2
4
S/D
Ne 2/Ne 1 = P2 /P1
Ne tot /Ne Single Std. = Ptot /PSingle Std.
Power Dissipation in Double
Disc Turbine Systems
1
0
(a)
0.5
1
1.5
2
2.5
Cb1/D
D/T=0.264
H/T=1
3
2
0.667
1
1.33
1.5
1.667
1
air entrainment
0
(b)
0
0.5
1
1.5
2
2.5
Cb1/D
Armenante and Chang, Ind. Eng. Chem. Res., 1998.
Piero M. Armenante
ChE702
89
2
air entrainment
1.8
D/T=0.264
H/T=1
1.6
Cb1 /D
1.4
0.167
0.333
0.5
0.667
1
1.2
4
Cb1 /D
1
0
0.5
(a)
1
1.5
2
Ne 2/Ne 1 = P2 /P1
Ne tot /Ne Single Std. = Ptot /PSingle Std.
Power Dissipation in Double
Disc Turbine Systems
2.5
S/D
0.167
0.333
0.5
0.667
1
3
2
1
D/T=0.264
H/T=1
air entrainment
0
(b)
0
0.5
1
1.5
2
2.5
S/D
Armenante and Chang, Ind. Eng. Chem. Res., 1998.
Piero M. Armenante
ChE702
90
Power Dissipation in Triple
Disc Turbine Systems
16
14
12
D/T=0.264
H/T=2
S13/D=5
S12/D=2.5
Ne
10
8
Impeller Location
1
Total
2
Single Std.
3
4
2
0
0
(a)
0.5
1
Cb1/D
1.5
2
Ne tot /Ne Single Std. = Ptot /PSingle Std.
6
(b)
4
3
2
S12 /D
0.67
1
1.5
D/T=0.264
H/T=2
S13/D=5
2.5
4
1
0
0.5
1
1.5
2
Cb1/D
Armenante and Chang, Ind. Eng. Chem. Res., 1998.
Piero M. Armenante
ChE702
91
Power Dissipation in Triple
Disc Turbine Systems
70
Cb1/D=0.33
Pi /P tot (%)
60
50
40
30
20
10
0
0
(a)
2
3
4
5
S12/D
D/T=0.264
H/T=2
S13/D=5
70
60
Pi /P tot (%)
1
Cb1/D=0.5
50
40
30
20
10
0
0
1
2
3
4
5
S12/D
(b)
70
Cb1/D=1
Pi /P tot (%)
60
50
40
30
20
Impeller Location
1
2
10
3
0
(c)
0
1
2
3
4
5
S12/D
Armenante and Chang, Ind. Eng. Chem. Res., 1998.
Piero M. Armenante
ChE702
92
Power Dissipation in Double
Pitched-Blade Turbine Systems
5
Impeller Location
1
Total
2
Single Std.
4
Ne
3
2
D/T=0.264
H/T=1
S/D=1.5
1
5
0
0
0.2
0.6
0.8
1
4
1.2
Cb1/D
3
Ne
(a)
0.4
Impeller Location
1
Total
2
Single Std.
2
D/T=0.264
H/T=1
Cb1/D=1
1
0
0
(b)
0.5
1
1.5
2
S/D
Armenante and Chang, Ind. Eng. Chem. Res., 1998.
Piero M. Armenante
ChE702
93
2.2
S/D
2
0.667
1
1.5
1.8
1.6
1.4
2
D/T=0.264
H/T=1
1.2
air entrainment
D/T=0.264
H/T=1
1
0
0.5
(a)
1
1.5
2
Ne 2/Ne 1 = P2 /P1
Ne tot /Ne Single Std. = Ptot /PSingle Std.
Power Dissipation in Double
Pitched-Blade Turbine Systems
2.5
Cb1/D
1
S/D
0.667
1
1.5
air entrainment
0
(b)
0
0.5
1
1.5
2
2.5
Cb1/D
Armenante and Chang, Ind. Eng. Chem. Res., 1998.
Piero M. Armenante
ChE702
94
2.2
air entrainment
2
D/T=0.264
H/T=1
1.8
1.6
1.4
Cb1 /D
0.333
0.5
1.2
0.667
1
2
1
0
0.5
(a)
1
1.5
2
Ne 2/Ne 1 = P2 /P1
Ne tot /Ne Single Std. = Ptot /PSingle Std.
Power Dissipation in Double
Pitched-Blade Turbine Systems
2.5
S/D
Cb1 /D
0.333
0.5
0.667
1
1
D/T=0.264
H/T=1
0
(b)
0
0.5
1
1.5
2
2.5
S/D
Armenante and Chang, Ind. Eng. Chem. Res., 1998.
Piero M. Armenante
ChE702
95
Power Curves for Impellers in
Baffled and Unbaffled Tanks
Power Number, Po
100
Baffled Tank
10
Unbaffled Tank
1
1
10
100
1000
10000
100000
Reynolds Number, Re
Piero M. Armenante
ChE702
96
Power Curves for Impellers in
Baffled and Unbaffled Tanks
 NP vs. Re plots for baffled systems
show that NP reaches an asymptotic
value at high Reynolds Number
 NP vs. Re plots for unbaffled
systems show that NP keeps
decreasing with Re even at high
Reynolds Numbers
Piero M. Armenante
ChE702
97
Power and Torque
 The power drawn by an impeller, P,
and the torque, , required by the
same impeller rotating at N are related
to each other by the following
equation:
P     2 N 
 Remark: the same power dissipation
can be achieved using a higher torque
and smaller agitation speed or vice
versa
Piero M. Armenante
ChE702
98
Power Dissipation and
Operating Cost of Mixing
 The power dissipated by the impeller,
P, is just the energy consumed by the
impeller per unit time, typically as
electric energy
 Hence, the operating cost of the
mixing operation are proportional to
P:
Operating Cost  P
Piero M. Armenante
ChE702
99
Torque and Capital Cost
 The capital cost of a mixing
operation is significantly dominated by
the cost of the gear box
 The cost of the gear box is directly
related to the its torque rating,
typically through an power law:
Capital Cost   0.8
Piero M. Armenante
ChE702
100
Important Mixing Operating
and Scale-up Parameters
 Traditionally, mixing processes have
been scaled up and operated by
maintaining constant one the
following parameters:
Power per unit liquid volume in the
tank, P/V, or per unit liquid mass, P/V
Torque per unit liquid volume in the
tank, /V, or per unit liquid mass, /V
Piero M. Armenante
ChE702
101
Power per Unit Volume
 The power dissipated by the impeller
per unit liquid volume in the tank:
PV
is one of the most important
mixing parameters used in scale up
of mixing processes
 The units for P/V are W/L, kW/m3 or
hp/1000 gal
Piero M. Armenante
ChE702
102
Power per Unit Mass
 The power dissipated by the impeller
per unit liquid mass in the tank, :
P

V
is an alternative to the use of P/V (since the
only difference is the presence of )
  is also widely used for scale-up
 The units for  are m2/s3
Piero M. Armenante
ChE702
103
Power per Unit Volume
 Substituting for P and V gives:
P NP  N D

2
V  / 4T H
3
5
P NP  3 2  D   D 

N D    
V  /4
T   H 

Piero M. Armenante

ChE702
2
104
Power per Unit Volume at
Different Scales
 The ratio of P/V at two different
scales is:
P / V large scale
P / V small scale
Piero M. Armenante
 NP  3 2
N D

  / 4

 NP  3 2
N D

  / 4


ChE702


2
 D 
  
 H  large scale
2
 D 
  
 H  small scale
D
 
T 
D
 
T 
105
Power per Unit Volume at
Different Scales
 For geometrically similar systems:
D / T  large scale D / H  large scale NP  large scale


D / T small scale D / H small scale NP small scale
1
 and the P/V ratio becomes:
P / V  large scale N D  large scale
 3 2
P / V small scale N D small scale
3
Piero M. Armenante
ChE702
2
106
Scale-up Based on Constant
Power per Unit Volume (P/V)
 If P/V is kept constant during
scale-up of geometrically similar
systems:
P / V  large scale
P / V small scale
N
Piero M. Armenante
3
D2


N
 1
N
large scale

3
3

D 
D2
 N 3 D2
ChE702
large scale
2
small scale

small scale
107
Tip Speed and Torque per
Unit Volume
 For geometrically similar systems (for
which D  T) in fully turbulent
regimes, or for the same system at
different agitation speeds, if the
torque per unit volume, /V, is kept
constant, then:

3
5
3
P
NP N D
NP N D



3
V 2 N V
N T
N  D3
Piero M. Armenante
ChE702
5
108
Tip Speed and Torque per
Unit Volume
 Simplifying:

2
 vtip 
N D
2
2
2
2






N
D

N
D


v
tip
3


V
ND
  

2
 v tip
V
3
5
i.e., keeping constant the tip speed is
equivalent to keeping /V constant,
provided that the geometry of the systems
is similar and the flow is fully turbulent
Piero M. Armenante
ChE702
109
Typical P/V Values for
Common Mixing Processes
Power
Level
P/V, kW/m3
(hp/1000 gal)
Low
0.2-0.6
(1-3)
Moderate
0.6-1
(3-5)
High
1-4
(5-20)
Very High
>4
(>20)
Piero M. Armenante
Applications
Low  blending, light solid
suspension
Solid suspension, liquid
dispersion
Heavy solid suspension,
emulsification, gas
dispersion
Very intense gas
dispersion; mixing pastes,
dough
ChE702
110
Typical Tip Speed and P/V for
Various Mixing Equipment
Equipment
Turbines
Close Clearance
Impellers
High Shear
Dispersers
In Line Mixers
Vtip, m/s
(ft/s)
3-4
(10-12)
0.2-4
(0.6-12)
6-27
(20-80)
20-40
(60-120)
P/V, kW/m3
(hp/1000 gal)
0.2-0.6
(1-3)
4-?
(20-?)
10-14
(50-70)
20-40
(100-200)
After Arthur Etchells, Unpublished Data
Piero M. Armenante
ChE702
111
Additional Power Sources in
Stirred Tanks
 In the vast majority of cases
mechanical power input in stirred
tanks is provided by impellers
 Additional mechanical power
sources can also be present, and
their contribution should be
incorporated in power calculations
Piero M. Armenante
ChE702
112
Additional Power Sources in
Stirred Tanks
 Mechanical power can be supplied
to stirred tanks via three primary
different sources, i.e.:
mechanical agitation (e.g., impellers)
power delivered by the expansion of a
compressed gas (e.g., gas dispersers,
diffusers)
power delivered by the kinetic energy of
a liquid (e.g., jets)
Piero M. Armenante
ChE702
113
Additional Power Sources in
Stirred Tanks
 Important Remark: some mechanical
power sources, e.g., sparging a gas,
typically reduces the mechanical
power input by the impeller (e.g., a
gassed impeller)
Piero M. Armenante
ChE702
114
Power Input by Gas
Sparging
 The mechanical power input
contribution of a gas sparged inside a
liquid is:
Pexpanding gas  Qgas L g H
Piero M. Armenante
ChE702
115
Power Input of a Liquid Jet
 The mechanical power input
contribution of a liquid jet injected
inside a liquid is:
Pliquid jet
Piero M. Armenante
1
 L Q jet v 2jet
2
ChE702
116
Total Mechanical Power Input
 The total mechanical power input
to a liquid in a stirred vessel is:
PTotal
mechanical
 Pimpeller  Pexpanding
gas
 Pliquid
jet
 In the presence of a sparged gas:
Pgassed impeller  Pungassed impeller
Piero M. Armenante
ChE702
117
Impeller Pumping Effects
Instructional Objectives of
This Section
 By the end of this section you will be
able to:
Distinguish the flow patterns generated
by different impellers under different
operating conditions
Calculate the impeller discharge flow
from available flow numbers
Piero M. Armenante
ChE702
119
Impeller Pumping Action
 Both radial and axial impellers exert a
pumping action within the tank
 The mixer can then be regarded as a
caseless pump
 Different types of impellers produce
different pumping actions resulting in
the establishment of fluid flow
circulation patterns inside the tank
Piero M. Armenante
ChE702
120
Vortices Generated by
Impeller Blades
 Both radial and axial impellers
produce strong vortices behind them
 These vortices are primarily
responsible for a number of
mixing phenomena, including
bubble and droplet breakup, rapid
mixing of homogeneous fluids, and
power dissipation
Piero M. Armenante
ChE702
121
Vortices Generated by a Disc
Turbine
Blade
Disc
Ulbrecht and Patterson, Mixing of Liquids by Mechanical Agitation, 1985.
Piero M. Armenante
ChE702
122
Vortices Generated by a Disc
Turbine
 A balanced vortex
pair develops
behind a Rushton
turbine blade,
conveying away
turbulent energy
Source: John Smith, Mixing XX
Piero M. Armenante
ChE702
123
Vortices Generated by a
Pitched Blade Turbine
Ulbrecht and Patterson, Mixing of Liquids by Mechanical Agitation, 1985.
Piero M. Armenante
ChE702
124
Vortices Generated by a
Pitched Blade Turbine
 The single line
vortices from
pitched blade or
hydrofoil impellers
are less intense
that those
generated by the
flat blade of a
Rushton turbine
Source: John Smith, Mixing XX
Piero M. Armenante
ChE702
125
Flow Pattern for Axial
Impellers in Baffled Tanks
 Axial impellers tend to pump
downward or upward, depending
on the direction of rotation
 Downward pumping impellers
produce an axial (or angled) main
flow that:
impinges on the tank bottom first
moves upwards near the tank wall
converges radially inwards, and then
returns to the impeller to feed it
Piero M. Armenante
ChE702
126
Flow Pattern for a Typical
Axial Impeller
Piero M. Armenante
ChE702
127
Flow Pattern for Radial
Impellers in Baffled Tanks
 Radial impellers pump the liquid
radially, forming a radial jet
 If C/T is sufficiently high, as the
liquid jet impinges on the tank wall it
splits upwards and downwards
 Both upward and downward flows
move vertically first, converge
radially inwards, and then return to
the impeller to feed it (“doubleeight” flow pattern)
Piero M. Armenante
ChE702
128
Flow Pattern for a Typical
Radial Impeller (high C/T)
Piero M. Armenante
ChE702
129
Flow Pattern for Radial
Impellers in Baffled Tanks
 If C/T is low, the liquid jet impinging on
the tank wall only forms an upward flow
that first moves vertically near the wall,
then converges radially inwards, and
returns to the impeller to feed it (“singleeight” flow pattern)
 In the “single-eight” regime the lower
circulation patter is suppressed
because of the proximity with the tank
bottom
Piero M. Armenante
ChE702
130
Flow Pattern for a Typical
Radial Impeller (low C/T)
Piero M. Armenante
ChE702
131
Impeller Clearance and Flow
Pattern Change with Disc Turbines
 For disk turbines a flow transition
from “double-eight” to “singleeight” regimes occurs when the C/T
ratio drops below a specific value:
For C/T >0.2  “double-eight” flow
pattern
For 0.16<C/T<0.2  either flow pattern
can exist
For C/T <0.16  “single-eight” flow
pattern
Armenante et al., Can. J. Chem. Eng., 1998.
Piero M. Armenante
ChE702
132
Impeller Clearance and Flow Pattern
Change with Flat-Blade Turbines
 Also for flat-blade turbines the flow
pattern changes from “double-eight”
to “single-eight” regimes as C/T
varies.
For C/T >0.25  “double-eight” flow
pattern
For 0.20<C/T<0.25  either flow pattern
can exist
For C/T <0.20  “single-eight” flow
pattern
Armenante et al., Can. J. Chem. Eng., 1998.
Piero M. Armenante
ChE702
133
Velocity Flow Field in a
Stirred Tank
Akiti and Armenante, MIXING XVII, 1999.
Piero M. Armenante
ChE702
134
Experimental Velocity
Measurement
 Local velocity measurements inside
a stirred tank are generally difficult
 Techniques include:
laser-Doppler velocimetry (LDV)
hot-wire anemometry
whole flow visualization
Piero M. Armenante
ChE702
135
Laser-Doppler Velocimetry
(LDV) System
Motor
Laser
Mixing Vessel
Color Separator
Box
Control
Volume
Beam Expander and
Transmitting Lens
y
Photodetectors
and Frequency Shifters
Transmitting Optical
Train
x
z
Traversing
Apparatus
Multicolor
Receivers
Piero M. Armenante
ChE702
Computer and Data
Acquisition System
136
Impeller Discharge Flow
 The pumping action of an impeller
results in a discharge flow rate out
of the impeller region, Qout, balanced
by an incoming flow toward the
impeller (inflow rate= Qin). Since mass
is conserved it must be that:
Qin  Qout  Q
where Q is the discharge flow rate
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Impeller Discharge Flow
Cylindrical envelope to determine
flow out of impeller region
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Impeller Discharge Flow
 The impeller discharge flow rate, Q,
can be obtained by summing up the
outflow contributions from all the
surfaces of the cylinder enveloping the
impeller:
Q  Qout axial
lower face
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 Qout axial
upper face
ChE702
 Qout
radial
139
Impeller Discharge Flow
Q
r D / 2

r 0
vz

out z   w / 2
r D / 2

r 0

vz
out z w / 2
2 r dr
z w / 2

z  w / 2
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2 r dr
ChE702
vr
out r D / 2
 D dz
140
Flow Number (or Pumping
Number) NQ
 In order to make the impeller
discharge flow rate non-dimensional
one can define the Flow Number, or
Pumping Number, NQ:
Q
NQ 
N D3
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Turbulent Flow Numbers
 The Flow Number is to the discharge
flow rate what the Power Number is to
power
 Turbulent Flow Numbers, NQT (or
simply NQ) have been obtained
experimentally for many impellers
 Typically, NQ is measured for a
“standard” configuration of the
agitation system (H=T, D/T=1/3,
C=D)
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Flow Numbers for Various
Impellers in Baffled Tanks
Impeller
NQ
Disc Turbine
0.7-0.85
45 Pitched-Blade Turbine (4 blades)
0.7-0.8
45 Pitched-Blade Turbine (6 blades)
0.9
Marine Propeller
0.4-0.55
Lightnin A310
0.55-0.7
Chemineer HE-3
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0.48
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143
Flow Numbers for 45° PitchedBlade Turbines (4 Blades)
After K. Myers and R. J. Wilkens, Personal Communication
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Flow Numbers for HE-3
Impellers
After K. Myers and R. J. Wilkens, Personal Communication
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Relationship Between Power
and Flow
 In a number of industrial cases it may
be advantageous to use impellers that
produce significant circulation within
the tank, but consume little power.
 To determine the optimal impeller
design and operation the following
ratio:
Q P
should be maximized.
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Relationship Between Power
and Flow
 For a fixed impeller geometry it is:


NQ  N D
NQ 1
Q
1


 
2
3
5
P NP   N D
NP  N D 

3

 i.e.:
Q  2 NQ 1


 2
P
 NP vtip
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NQ/NP for Various Impellers
in Baffled Tanks
Impeller
NQ/NP
Disc Turbine
0.14-0.17
45 Pitched-Blade Turbine (4 blades)
0.4-0.6
45 Pitched-Blade Turbine (6 blades)
0.5-0.7
Marine Propeller
1.1-1.6
Lightnin A310
1.8-2.3
Chemineer HE-3
1.6-1.9
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Variation of Flow and Power
Dissipation
 Most impellers have flow numbers in
the relatively narrow range of 0.4-0.9
(typically 0.5-0.8), i.e., their ability
to pump is of the same order of
magnitude
 The same impellers have power
numbers ranging between 0.25 and 6,
a much wider range
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Variation of Flow and Power
Dissipation
 Impellers with blades oriented parallel to
the shaft produce radial flow, and have
high power dissipation rates although
their pumping action is significant
 As a consequence, their NQ/NP ratios is
low
 Radial impellers generate significant
turbulence and produce high shear
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Variation of Flow and Power
Dissipation
 Impellers with blades forming a
(small) angle with the plane of
rotation produce axial flow, and
have relatively low power
dissipation rates although their
pumping action is also significant
 As a consequence, their NQ/NP ratios
will be high
 Axial impellers generate less
turbulence and shear
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Optimization Strategies to
Maximize Pumping Efficiency
 To maximize pumping efficiency
(i.e., maximize the Q/P ratio):
choose impellers with high NQ/NP ratios
if capital cost must be minimized select
impellers with the same vtip (=ND) but
lower D, since this decreases N and
hence the torque  (=P/2N) [recall that
the cost of the gear box is proportional
to the torque]
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Optimization Strategies to
Maximize Pumping Efficiency
 If a specific flow rate Q must be
achieved then, by rearranging it is:
3
1
3
 P  1
 P 
4
3



Q  NQ 

N
D
4
Q

5

N

N
P  N
P 


5
 To lower P at constant Q one can
lower N while increasing D. This
approach decreases the operating cost
( P).
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Optimization Strategies to
Maximize Pumping Efficiency
 If a specific power input P must be
maintained then, by rearranging it is:
5
3
Q 
Q  1
4
3
 N  NP 
 4
P  NP 

N  D
N
 Q
 Q
3
 To increase Q at constant P one can
lower N while increasing D. This
approach increases the capital cost
(proportional to the torque =P/2N).
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Optimization Strategies to
Maximize Pumping Efficiency
 The preceding analysis is valid if NP
and NQ are constant. This is correct if
the flow is fully turbulent.
 Changing the D/T ratio usually has
little influence on NP and NQ provided
that it is not too small or large (0.25<
D/T <0.7) [Too large a D/T ratio
chokes the recirculation flow].
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Circulation Time
 One can define the circulation time,
tcirc, as:
t circ
V

Q
 tcirc is a measure of how long it takes
the impeller to pump the same
volume of liquid as that contained in
the tank (V=Qtcirc)
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Circulation Time
 The circulation time, tcirc, is directly
related to how long it takes:
 a small, neutrally buoyant tracer particle
to pass consecutively through the
same region (e.g., the impeller region)
 a tracer to produce two consecutive
concentration peaks in the region
where the detector is
 The blend time is typically a multiple
of the circulation time
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Blend Time in Stirred Tanks
Instructional Objectives of
This Section
 By the end of this section you will be
able to:
Describe the concepts of blend time and
degree of uniformity and how the can be
determined in the lab
Calculate the blend time for any desired
degree of uniformity in a mixing tank
Determine the blend time as a function of
geometry and operating parameters
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Blend Time
(Mixing Time)
 If a miscible tracer is added to a
homogenous liquid in an agitated tank the
local concentration (measured with a
detector) fluctuates with time
 The amplitude of the concentration
fluctuations will decrease with time
 Eventually the tracer concentration will
become completely uniform in the tank
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Blend Time
 Blend Time (also referred to as
“Mixing Time”) is the time it takes
the tracer-liquid system to reach a
desired (and pre-defined) level of
uniformity
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Blend Time Facts
 Blend time and the achievement of a
homogeneous state can be critical in
some operations (e.g., fast chemical
reactions)
 In any real mixing tank, blend time
is never zero
 Homogeneous phases do not mix
instantaneously!
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Experimental Determination
of Blend Time
 Detection of tracer can be
accomplished with a variety of
techniques including:
acid-base indicators (e.g., pH meters)
ion-specific electrodes
electric conductivity meters
thermometers
refractometers (for refractive index)
light adsorption meters
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Experimental Determination
of Blend Time
 A tracer is typically added to the tank
(typically at the surface)
 The concentration of the tracer is
determined at one or more locations
in the tank as a function of time
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Experimental Determination
of Blend Time
Tracer
Sensor
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Concentration Fluctuations at
Sensor and Experimental Blend
Time
C
CFinal
t
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Concentration Fluctuations at
Sensor and Experimental Blend
Time
C
C90%
CFinal
t
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t90%
167
Concentration Fluctuations at
Sensor and Experimental Blend
Time
C
C95%
CFinal
t
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t95%
168
Equations for the
Determination of Blend Time
 Here, two approaches/equations for the
determination of the blend time will be
presented:
Approach 1: Fasano, Bakker, and
Penney’s approach
Approach 2: Grenville’s approach
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Equations for the
Determination of Blend Time
 Approach 1
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Blend Time and
Non-Uniformity
 The level of non-uniformity (or
unmixedeness) X is defined as:
CFinal  C t 
X (t ) 
CFinal  Co
where Co and CFinal are the initial and
final tracer concentrations in the liquid
 Before the tracer addition (t=0) C=Co
and X=1; for t, C=CFinal and X=0
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Non-Uniformity vs. Time
X(t)=(CFinal - C(t))/CFinal - Co)
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
1
2
3
4
5
6
7
8
9
10
t (s)
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Non-Uniformity vs. Time
X(t)=(CFinal - C(t))/CFinal - Co)
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
1
2
3
4
5
6
7
8
9
10
t (s)
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Non-Uniformity vs. Time
X(t)=(CFinal - C(t))/CFinal - Co)
1
0.8
exp(-kt)
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-exp(-kt)
-0.8
-1
0
1
2
3
4
5
6
7
8
9
10
t (s)
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Non-Uniformity vs. Time
p(t)
X(t)=(CFinal - C(t))/CFinal - Co)
1
0.8
exp(-kt)
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-exp(-kt)
-0.8
-1
0
1
2
3
4
5
6
7
8
9
10
t (s)
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Mixing Rate Constant, k
 For “long” enough times the value of
X(t) oscillates while decaying
exponentially
 These damped oscillations are
enveloped between an upper and
lower decaying exponential curves
(X=e-kt and X=-e-kt)
 The parameter k is called the mixing
rate constant (in min-1)
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Mixing Rate Constant, k
 The greater the k value is:
the faster the oscillations will die out
the faster blending will be
the shorter the mixing time will be
 The extent of the damping effect
will depend on the geometric (e.g., D,
T, H) as well as dynamic (e.g., N)
variables
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Non-Uniformity Peaks
 The absolute values of the height of
the oscillation peaks, p(t), in the Xt curve will determine whether a
required level of homogeneity has
been achieved
 The values of p(t) can be found from:
p(t )  max X t   ek t
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Non-Uniformity Peaks
 p(t) determines the level of nonhomogeneity (non-uniformity)
 One can arbitrarily decide when
sufficient uniformity has been
achieved by selecting a small enough
p(t) value (e.g., 0.05, implying that
the largest fluctuation is 5% of the
final X value)
 For t p(t) 0
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Degree of Uniformity, U
 It is convenient to introduce the
Degree of Uniformity, U, defined as:
U t   1  p(t )
where U is just the complement of p
(for example, if p=0.05, U=95%,
implying that the liquid is 95%
homogeneous).
 Then:
k t
U t   1  e
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Blend Time and Degree of
Uniformity
 ln1  U 
tU 
k
 This equation relates the blend time,
tU, required to achieve a desired level
of U, to U and k. For example, the
time required to achieve 99%
homogeneity is:
 ln1  0.99 4.6
t99 

k
k
 Then:
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Blend Times to Achieve
Various U’s
 It is possible to establish a
relationship (independent of k)
between a two blend times to
achieve two different degrees of
uniformity (e.g., U’ and U):
tU
 ln1  U  / k
 ln1  U 


tU '  ln1  U ' / k  ln1  U '
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Blend Times to Achieve
Various U’s
 Example: tU for a specific U and that
for 99% (t99):
tU
 ln1  U 
 ln1  U 


t99  ln1  0.99
4.6
 Example, it takes twice as long to
blend to U=99.99% that to blend to
99% t
 ln1  0.9999 
99 .99
t 99
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
 ln1  0.99 
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2
183
Blend Times to Achieve
Various U’s
U
tU/t99
U
tU/t99
90%
0.5
99.99%
2
95%
0.65
99.999%
2.5
99%
1
99.9999%
3
99.9%
1.5
99.99999%
3.5
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Mixing Rate Constant
 In order to calculate tU one needs to
determine the mixing rate constant, k
 As usual, dimensional analysis is
used:
k  f (N, D,T , H,C, B,w, n, nB , g,  , ,
impeller type )
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185
Mixing Rate Constant
 Then:
 ND 2 N 2D T H C w B


,
, , , , , , n, nB , 
k
 f 
g D D D D T

N
 impeller type



 For baffled, fully turbulent systems Re
and Fr have no effect
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Mixing Rate Constant
 Although all geometric variables could
play a role only a few are important.
 The most important geometric
variables affecting k/N are T, H, D,
and the impeller type. Hence:
k
T H

 f  , ,impeller type 
N
D D

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Mixing Rate Constant
 The equation for k/N is then:
b
k
D  T 
 a   
N
T   H 
0.5
where the parameters a and b depend
on the type of impeller used
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Parameters to Calculate the
Mixing Rate Constant
Impeller
a
b
Disc Turbine
1.06
2.17
Flat-Blade Turbine
(4 blades)
45 Pitched-Blade
Turbine (4 blades)
Marine Propeller
1.01
2.30
0.641
2.19
0.274
1.73
Chemineer HE3
0.272
1.67
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Blend Time Equation for
Fixed Geometries
 For a fixed set of geometric variables
i.e.:
same impeller
same D/T ratio
same H/T ratio
k/N =constant
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Blend Time Equation for
Fixed Geometries
 If k/N is constant:
 ln1  U   ln1  U  / N
tU 

k
k /N
i.e.:
tU N  constant
(for same impeller, same D/T ratio,
same H/T ratio), irrespective of
scale
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191
Blend Time Equations for
H/T=1, T/D=3
Impeller
Equation
Disc Turbine
t99 N=47
Flat-Blade Turbine
(4 blades)
45 Pitched-Blade Turbine
(4 blades)
Marine Propeller
t99 N=57
Chemineer HE3
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t99 N=79.7
t99 N=112.4
t99 N=106
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192
Blend Time and Impeller Speed
 The higher the agitation speed is the
shorter the blend time will be
 For geometrically similar systems this
equation:
1
tU 
N
does not change with scale
 Geometrically similar small and large
vessels have the same blend time only if
the agitation speed N is the same at both
scales
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193
Blend Time Equation for
Fixed Geometries
 From:
tU N  constant
it follows that if mixing time is to
remain unchanged during scale-up
the agitation speed N must remain
constant provided geometric
similarity is maintained
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Effect of Other Factors on
Blend Time
 The procedure outlined before can
be used to obtain the blend time tU for
the case in which:
the flow is turbulent
the viscosities of the added liquid
and the liquid in the tank are equal
the densities of the added liquid
and the liquid in the tank are equal
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Effect of Other Factors on
Blend Time
 Corrective factors can be applied to
tU to account for:
different flow regimes
viscosity differences between the
two liquids
density differences between the
two liquids
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196
Effect of Other Factors on
Blend Time
 The corrective factors can be applies as
follows:
tU Re;  *;   
where:
fRe f * f  tU Re  ;  *  1;   0
 f Re = corrective factor for the effect of Re
 f* = corrective factor to account for the effect of
viscosity differences
 f  = corrective factor to account for the effect of
density differences
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Effect of Other Factors on
Blend Time
 tU (Re; *=1; =0) is the
“standard” tU (i.e., under fully
turbulent conditions, with an added
fluid having the same viscosity an
density as the liquid in the tank)
calculated as outline before;
 tU (Re; *; ) is the mixing time
calculated to account for the effect of
Re, viscosity and density differences.
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Blend Time at Different
Reynolds Numbers
 fRe is the correction factor to account for Re
effects when Re is below 10,000 and the
fluid is not fully turbulent. Remark: fRe =1
for Re>10,000.
 Once tU has been calculated for Re >10,000
it is possible to obtain tU at other Reynolds
Number using the diagram obtained by
Norwood and Metzner (1964).
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Blend Time at Different
Reynolds Numbers
fRe=tU/tU (turbulent)
1000
100
10
1
0.1
10
100
1000
10000
100000
1E+006
Re
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Effect of Viscosity Ratio on
Blend Time
 f* is the correction factor to account
for viscosity effect, when the viscosity
of the added fluid is greater than that
of the liquid in the tank.
 In order to calculate f* the viscosity
ratio:
added liquid
* 
liquid in the tank
must be determined first.
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Effect of Viscosity Ratio on
Blend Time
Fasano et al., Chem. Eng., 1994
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Effect of Density Difference
on Blend Time
 f  is the correction factor to account
for the effect of differences in
densities between the added liquid an
the liquid in the tank.
 If the density difference is zero
(“standard” case) f =1.
 In order to calculate f  the
Richardson Number, Ri, must be
calculated first.
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Effect of Density Difference
on Blend Time
 To account for the effect of density
differences the Richardson Number is
introduced:
g H
Ri 
2
2
L N D
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Effect of Density Difference
on Blend Time
Fasano et al., Chem. Eng., 1994
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Blend Time and Geometric
Similarity
 In fully turbulent, geometrically similar
systems the equation below still holds:
tU N  constant
 This implies that blend time experiments
can be conducted in small scale
equipment to determine the above
constant, and that this equation can be used
for scale-up purposes
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Procedure to Calculate
Blend Time
 The procedure to calculate tU is then:
set the desired value of U
fix D, T, H and the impeller type
set N
calculate k
calculate the “standard” blend time
correct this value to account for Re,
viscosity and density effects
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Scale-up and Blend Time
 Scale-up based on blend time is
extremely costly to achieve since the
power consumption would increase
enormously. For constant N:
P V  large scale N D large scale
 3 2
P V  small scale N D  small scale
3
2
 D large scale 


D

 small scale 
2
 This implies that P/V increases with the
square of the scale-up factor
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Scale-up and Blend Time
 If P/V is kept constant during scaleup:
3
2
P / V  large scale
P / V small scale
N
3
D2

N large scale
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
N D 
 1
N D 
 N D 
3
large scale
2
small scale
3
large scale
2
small scale
 Dsmall scale 

 Nsmall scale 
D

 large scale 
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3
209
Scale-up and Blend Time
 Recall that:
1
tU 
N
 If P/V is kept constant during scale-up of
geometrically similar systems:
tU large scale
 Dlarge scale 

 tU small scale 
D

 small scale 
2
3
the blend time increases with the (linear)
scale factor raised to the 2/3 power
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Blend Time and Other Time
Scales
 It is always important to make sure
that blend time is much shorter than
the other time scales that may be
important to the process
 If blend time is longer than other
critical time scales (e.g., reaction
time) mixing could become the
limiting step, often inadvertently
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Blend Time in Small Tanks
and Large Tanks
 Blend time is typically short in small
laboratory tanks, but much longer
in larger tanks
 Processes that are not affected by
blend time at small scales (since
mixing is fast) could be limited by
poor mixing at larger scales since
blending the tank’s contents typically
takes much longer
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Blend Time and Circulation
Time
 An empirical mixing rule of thumb
states that:
Blend Time  4  t circ
V
 4
Q
where the proportionality constant
(4) is often reported to be between 3
and 5 or even outside this range
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Blend Time and Circulation
Time
 From the definition of circulation time
(tcirc=V/Q) and Flow Number, NQ:
Q
NQ 
3
ND

Q  NQ N D
3
it follows that:
t circ
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V
V
 
Q NQ N D3
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214
Blend Time and Circulation
Time
 The relationship between tU and tcirc
can be obtained recalling that:
tU N  constant  

 Then:
tU

t circ
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N

V
NQ N D 3
ChE702
 NQ D 3
V
215
Blend Time and Circulation
Time
 Finally:
 NQ D
tU
4
D

   NQ    
2
t circ  / 4T H 
T 
3
2
D
 
H 
 For a given system, and for a preassigned level of uniformity, U, all the
factors on the right-hand side are
fixed. Hence tU/tcirc is constant
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Blend Time and Circulation
Time
 Example: Disk turbine in a standard
mixing system
 NQ=0.8
 D/T=D/H=1/3
 If U=99% (=47)  t99=5.3tcirc
 If U=95% (=0.6547=30.5)
 t95=3.5tcirc
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Blend Time and Circulation
Time
Impeller
t95/tcirc
Disc Turbine
3.0 – 3.7
45 Pitched-Blade Turbine
(4 blades)
Marine Propeller
5.1 – 5.9
Chemineer HE3
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4.1 – 5.7
4.7
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218
Blend Time and Circulation
Time
Impeller
t99/tcirc
4.6 – 5.6
Disc Turbine
45 Pitched-Blade Turbine
(4 blades)
Marine Propeller
Chemineer HE3
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7.9 – 9
6.3 – 8.8
7.2
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219
Blend Time and Circulation
Time
 The previous results confirm that the
blend time is typically a multiple
of the circulation time
 tU/tcircis typically in the range 3-6 for
t95/tcirc and 4-9 for t99/tcirc
 These results validate the empirical
mixing rule of thumb stating that
“Blend Time”  4  “Circulation Time”
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Equations for the
Determination of Blend Time
 Approach 2
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Blend Time Equation in
Turbulent Regime: Approach 2
 For mixing in the turbulent regime
(Re>~10,000) Grenville (1992)
found:
1 .5
0 .5
5.20  T   H 
t 95 N 
  
1/ 3 
Po  D   D 




0.33 < D/T < 0.50
C/T = 0.33
0.50 < H/T  1
Po = Impeller Power Number
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Blend Times to Achieve
Various U’s
 Recalling that it is possible to calculate
tU’ knowing tU:
tU '  ln1  U ' / k  ln1  U '


tU
 ln1  U  / k
 ln1  U 
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Blend Times to Achieve
Various U’s
 If U=95% is the reference degree of
homogeneity:
tU
 ln1  U 
 ln1  U 


t 95  ln1  0.95 
2.996
 For example:
t 99  ln1  0.99 

 1.537
t 95  ln1  0.95 
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Blend Time Equation in
Turbulent Regime: Approach 2
 For mixing in turbulent regime
(Re>~10,000) the Grenville equation
becomes:
 1.74 ln1  U   T   H 
tU N 
   
1/ 3
Po
D D 
1 .5




0 .5
0.33 < D/T < 0.50
C/T = 0.33
0.50 < H/T  1
Po = Impeller Power Number
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Blend Time for Turbulent Regime
with H/T=1, T/D=3: Approach 2
Impeller
Equation
Disc Turbine
t99 N=51
Flat-Blade Turbine
(4 blades)
45 Pitched-Blade Turbine
(4 blades)
Marine Propeller
t99 N=62
t99 N=127
Chemineer HE3
t99 N=132
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t99 N=81
226
Impeller Efficiency in Turbulent
Regime: Approach 2
 From:
 1.74 ln1  U   T   H 
tU N 
   
1/ 3
Po
D D 
3 5
3
5
and:
P
Po  N D
Po N D



2
3
 V   / 4T H
T
1 .5
it is:
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1
3
1
3
 1 T 
tU      T
  D
ChE702
0 .5
2
3
227
Impeller Efficiency in Turbulent
Regime: Approach 2
 From the previous equation for turbulent
regime, it is:
 all impellers of the same diameter are
equally energy efficient (i.e., achieve the
same tU at the same power input/mass)
 shorter tU are achieved with larger
impellers at the same power input/mass
 blend time is independent of fluid
properties
 when scaling at constant power
input/mass and similar geometry blend
time increases with the scale factor
raised to 2/3
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Blend Time Equation in Transitional
Regime: Approach 2
 For mixing in the transitional regime
(~200<Re<~10,000) Grenville
(1992) found:
2
33,500  T 
t 95 N 
 
2/3
Po  Re  D 




0.33 < D/T < 0.50
C/T = 0.33
0.50 < H/T  1
Po = Impeller Power Number
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Blend Time Equation in Transitional
Regime: Approach 2
 For mixing in transitional regime
(~200<Re>~10,000) the Grenville
equation becomes:
2
 11,181ln1  U   T 
tU N 
 
2/3
Po  Re  D 




0.33 < D/T < 0.50
C/T = 0.33
0.50 < H/T  1
Po = Impeller Power Number
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Impeller Efficiency in Transitional
Regime: Approach 2
 From:
 11,181ln1  U   T 
tU N 
 
2/3
Po  Re  D 
and:
it is:
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2
P
Po  N D
Po N D



2
3
 V   / 4T H
T
3
 1
tU   
 
2
3
5
2
3
   T 
   T
   D 
ChE702
3

5
2
3
231
Impeller Efficiency in Transitional
Regime: Approach 2
 From the previous equation for the
transitional regime, it is:
 all impellers of the same diameter are equally
energy efficient (i.e., achieve the same tU at the
same power input/mass)
 shorter tU are achieved with larger impellers at
the same power input/mass
 blend time is proportional to viscosity and
inversely proportional to density
 when scaling at constant power input/mass
and similar geometry blend time decreases with
the scale factor raised to 2/3 (however do not
forget that that Re increases with scale, and larger
system may no longer be in the transitional
regime)
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Conclusions: Power, Flow and
Blend Times in Mixing Tanks




Under turbulent conditions, the power dissipated by
an impeller depends on:
 agitation speed (PN3)
 impeller size (PD5)
 type of impeller (PNP)
 density of the fluid (Pρ)
Axial impellers and radial impellers generate
different circulation patterns
In general, axial impellers generate more flow per unit
of power dissipated than radial impellers
Blend time is inversely proportional to the impeller
agitation speed and is not generally significantly
affected by scale
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