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3 - AGITATION & MIXING
3.1 PURPOSE
∗ Agitation: Applies to those operations whose primary purpose is to
promote turbulence in a liquid
∗ Mixing: Operation in which two or more materials are intermingled to
attain a desired degree of uniformity
∗ Main purpose of mixing and agitation in water and wastewater treatment:
1. Uniform distribution of a chemical
Typically rapid mixing or flash mixing
Less than 10 s
Coagulation
Dispersion of oxygen in activated sludge
Food industry, fabrication, dyes
2. Promotion of aggregate particle formation by collisions
Typically slow mixing
Minutes to hours
Flocculation
∗ Other agitation and mixing objectives (rapid mixing):
1. Suspending solid particles
2. Dispersing immiscible liquids
3. Promoting heat transfer at a wall
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3.2 RAPID MIXING DEVICES
3.2.1 MECHANICAL AGITATORS
1. Turbine Impeller
Various types of turbine blades
Turbine blades can be vertical or pitched
Impeller diameter 30 to 50% of tank diameter or width
Mounted one impeller diameter above tank bottom
Range in speed of 10 to 150 rpm
Baffling minimizes vortexing and rotational flow
Radial flow
Figure 3-1: Types of Turbine Impellers
Figure 3-2: Flow Regime in a Turbine Impeller Tank
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2. Paddle Impeller
Typically 2 or 4 blades
Paddle blades can be vertical or pitched
Paddle width 1/6 to 1/10 of diameter
Paddle impeller diameter 50 to 80% of tank diameter or width
Mounted ½ of a paddle diameter above tank bottom
Paddle speed range 20 to 150 rpm
Baffling required to minimize vortexing and rotational flow
Radial flow
Figure 3-3: Types of Paddle Impellers
Figure 3-4: Flow Regime in a Paddle Impeller Tank
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3. Propeller Impeller
Typically 2 or 3 blades
Blades are pitched
Pitch Axial distance of Liquid Movement / Revolution
Propeller Diameter=
Usually pitch is 1.0 or 2.0
Maximum propeller diameter is 18 inches
Propeller speed 400 to 1750 rpm
Axial flow
Figure 3-5: Types of Propeller Impellers
Figure 3-6: Flow Regime in a Propeller Impeller Tank
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3.2.2 PNEUMATIC AGITATORS
Mixing action effected by rising air bubbles
Tanks and aeration devices similar to those used in aeration
Detention times and velocity gradients of same magnitude and range
as those used in mechanical mixing
Not affected by variations in influent flow rate
Relatively small hydraulic headlosses
Figure 3-7: Pneumatic Rapid Mixing
3.2.3 BAFFLE BASIN
Mixing depends on gravity and hydraulic turbulence
Very little short circuiting
Headloss usually varies from 0.3 to 0.9 m
Not suitable for wide variations in flow rates
Velocity gradient can not be varied
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Figure 3-8: Baffle Basin Rapid Mixing
3.2.5 HYDRAULIC JUMP
Mixing results from turbulent movement of liquid and high energy
losses
Used where sufficient head is available
Velocity gradient can not be varied
Figure 3-9: Hydraulic Jump for Rapid Mixing
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3.3 VELOCITY GRADIENT
∗ The intensity of agitation represents the relative motion of fluid particles
the velocity gradient
∗ The velocity gradient is averaged in some way to represent the entire body
of liquid affected by the mixing
dudy
L tL
t G= = =−/ 1
∗ The symbol G is often utilized to express this velocity gradient
∗ Need to quantify the amount of agitation. This is accomplished through
dimensional analysis
G ⋅ t = (Intensity)X(Duration)
G ⋅ t is dimensionless
∗ Consider a volume ∀ of fluid of viscosity µ upon which power P is imparted
to create a velocity gradient G
∗ Performing a dimensional analysis using these parameters will yield an
expression for the velocity gradient in terms of power, volume and fluid
properties
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P f G= ∀( , , )µ
[P] = FLt-1
[G] = t-1
[µ] = FtL-2
[∀] = L3
∗ Dimensional analysis yields
∗ From the Buckingham Pi theory there will be m - n = 1 dimensionless
groups
PG
G P
a b cµ
µ
∀ =
=∀
Π1
∗ The power imparted per unit volume is expressed as
P W∀
=
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3.4 FLOW VELOCITIES AND CIRCULATION 3.4.1 IMPELLER FLOW THEORY
∗ Consider a simple, vertical impeller blade with a diameter Da
∗ Impeller rotates at a speed of n rpm
∗ Velocity at the tip of the impeller can be expressed as follows
u Da2 = π n
n
∗ A liquid particle leaving the impeller tip has both radial (Vr2l) and tangential
velocity (Vu2l) components
∗ Assume that the tangential velocity is some fraction k of the impeller tip
velocity due to slip between impeller and fluid
V k Dul
a2 = π
Figure 3-10: Velocity Vectors at Tip of Turbine Impeller Blade
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∗ Impeller receives flow axially from above and below and discharges
radially; the volumetric flow rate q through the impeller is represented by:
q V Ar
lp= 2
∗ Ap is taken to be the area swept out by the tips of the impeller blades
(projected area of impeller) and is a function of impeller circumference πDa
and blade width W
A Dp a= π W
∗ From geometry
( )
( )
tan
tan
β
β
22
2 2
2 2 2
l rl
ul
rl
ul l
Vu V
V u V
=−
= − 2
β
∗ Substituting in relationships for u2 and Vu2
l, the radial velocity is found to be
( )
( )
V D n k D n
V D n k
rl
a al
rl
al
2 2
2 21
= −
= −
π π
π β
tan
tan
∗ The flow rate through the impeller then becomes
( )q D nW ka
l= −π β2 221 tan
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∗ However, this assumes a constant velocity profile across the blade which is
not the case under real flow conditions
∗ Introduce a parameter K to account for ‘non-ideal’ effects
( )q K D nW ka
l= −π β2 221 tan
Figure 3-11: Velocity Profile of Flow from a Straight Blade Turbine
3.4.2 FLOW NUMBER
∗ For geometrically similar impellers the blade width (W) must be
proportional to the impeller diameter (Da)
∗ Parameters K, k and β2
l can be assumed to remain constant. Hence,
q nDa∝ 3
∗ Ratio of these two quantities is called the Flow Number (NQ) 81.301 3 - 11 Environmental Engineering Unit Operations
3 - AGITATION & MIXING
N qnDQ
a
= 3
∗ Flow Number is expected to be constant for each type of impeller
Standard flat blade in baffled vessel NQ = 1.3
Marine propeller (square pitch) NQ = 0.5
Four-blade 45o turbine (W/Da = 1/6) NQ = 0.87
∗ The flow rate (q) expression developed in the previous section accounts for
radial flow leaving the tip of the impeller
∗ High velocity stream of liquid leaving the tip of the impeller entrains some
of slower moving bulk liquid: slows down jet, but increases the total flow
rate
∗ Tracer tests for turbine impellers give indication of this entrainment
phenomenon and have yielded a relationship which also takes tank diameter
Dt into account
q nDDDT a
t
a=
0 92 3.
∗ Estimated total flow for flat blade turbines
2 4< <DD
t
a
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3.5 POWER CONSUMPTION
∗ For turbulent flow power represents the product of the kinetic energy
imparted to the fluid and the total fluid flow
P energyvolume
volumetime
energytime
=
=
3.5.1 MECHANICAL AGITATORS
∗ Power required to generate turbulent flow in a reactor using impeller
agitation
( )P Vg
nD Nl
ca Q=
ρ( )22
3
2
∗ Velocity V2
l is slightly smaller than the tip velocity u2. If the ratio of V2l/u2
is denoted by α (similar to k) and V2l = απnDa (similar to V2u
l), then the
power requirement becomes
( )P nD Ng
nD
P n Dg
N
a Qc
a
a
cQ
=
=
3 2
3 5 2 2
2
2
ρ α π
ρ α π
∗ This relationship can be rearranged to obtain a dimensionless form
Pgn D
N Nc
aQ P3 5
2 2
2ρα π
= =
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∗ The left hand side of the equation is called the Power Number NP
∗ For a standard 6-bladed turbine impeller with a Flow Number of 1.3 and a
velocity ratio of 0.9, Np = 5.2
∗ However, α and NQ are not readily measured, so must establish a correlation
between P or NP and other easily measured variables defining the system
∗ Factors that can be expected to affect power consumption include
Viscosity of fluid (µ)
Density of fluid (ρ)
Rotational speed (n)
Gravity (g and gc if engineering units are used)
Diameter of tank (Dt)
Height of tank (H)
Diameter of impeller blade (Da)
Width of impeller blade (W)
Distance of impeller from tank bottom (E)
Width of baffles (J)
Length of impeller blade (L)
∗ The linear dimensions can all be converted to dimensionless ratios by
dividing each one by say impeller blade diameter
ttaatt
a
DHS
DJS
DWS
DLS
DES
DDS ====== 654321
∗ S1, S2, S3, S4, S5 and S6 are dimensionless shape factors particular to
impeller
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∗ Can utilize dimensional analysis to establish relationship with other
variables
( )P f n D g ga c= , , , , ,µ ρ
[n] = t-1
[Da] = L
[gc] = LMF-1t-2
[µ] = FLT-2 or ML-1t-1
[ρ] = ML-3 or Ft2L-4
[g] = Lt-2
∗ Dimensional analysis yields the following dimensionless groups (VERIFY!)
( )
Pgn D
fnD n D
g
N f Fr
c
a
a a
P
3 5
2 2
ρρ
µ=
=
,
Re,
∗ If the dimensionless shape factors are included
( )N f Fr S S SP = Re, , , , ,...1 2 3
∗ With the non-dimensional form of the functional relationship, can now
correlate data obtained experimentally in the lab
∗ It should be noted that the curves illustrated in the following figures do not
directly include the effect of the Froude Number (Fr)
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∗ The Froude Number becomes important only when there is significant wave
motion. i.e. at high Re and unbaffled tanks
∗ These conditions are normally avoided and are represented by dashed curves
∗ If dashed curves must be used, then
( )( )N N Fr
ma
b
P Pm
Figure=
=− log Re
∗ a and b are coefficients which would normally be given
a = 1.0 b = 40.0
Figure 3-12: NP versus Re for Six-Blade Turbines
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a = 1.7 b = 18.0
Figure 3-13: NP versus Re for Three-Blade Propeller
∗ At low Reynolds Number (laminar flow Re < 10), baffled and unbaffled
tanks give the same result
∗ log NP versus log Re gives a straight line with a slope of -1, now say
log log log Re
Re
N K
NK
P L
PL
= −
=
∗ KL is a function of impeller type and is constant for each type
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( )
c
aL
a
L
a
c
gDnKP
nDK
DnPg
µ
µρρ
32
253
=
=
∗ Note density does not play a role here
∗ At high Reynolds Number (turbulent flow Re > 10 000), NP is essentially
constant and NP = KT
Pg
n DK
PK n D
g
c
aT
T a
c
3 5
3 5
ρ
ρ
=
=
Table 3-1: Values of Constants KL and KT for Baffled Tanks Having Four
Baffles at Tank Wall with Width Equal to 10% of the Tank Diameter
3.5.2 PNEUMATIC AGITATORS
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∗ Power dissipated by rising bubbles released from a depth below the water
surface at a particular air flow rate
∗ The work done in compressing a volume of air at atmospheric pressure to a
particular compression pressure and volume, under isothermal conditions
can be used to calculate the work of expansion of the bubbles as they rise
∗ From ideal gas law, for a isothermal process
( )
+
=
∀=∀
∀∀
=∀=
∀=∀
∫∫
4.104.10log31.2100.1
ln
25 hQ
mNxP
pppdppdW
pp
a
a
caa
c
aa
ccaa
h = Depth to diffuser
Qa = Air flow rate at operating temperature and pressure (m3/s)
∗ Important design considerations include:
Bubbles should be uniform size and uniformly distributed in the
volume
Laterals are spaced 1 to 1.5 m apart
Diffuser openings: 1.5 mm - spaced 7.5 to 15 cm apart
3.5.3 BAFFLED BASINS
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∗ The hydraulic head or liquid flow rate required to impart the desired power
in a gravity driven system can be determined using
P ghL= ρ Q
∗ Headloss is a function of velocity head along the flow path
Changes in direction create eddies and hence, mixing and headloss
The head loss at each 180o bend is approximately
h to VgL = ( . )2 35
2
2
ρ = Density (kg/m3)
g = Gravity (m/s2)
Q = Liquid Flow rate (m3/s)
h = Hydraulic head (m)
∗ Velocity in the system should be within 0.15 to 0.45 m/s to optimize mixing
∗ Headloss through the system should range between 0.15 and 0.6 m 3.5.4 HYDRAULIC JUMPS
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∗ Momentum principle is most appropriate to describe this phenomenon
( )QEgP ∆=ρ
( )
1
11
21
1
2 11821
)(
gyVFr
Fryy
dtVmdF
=
−+=
=∑v
v
+−
+=∆
gVy
gVyE
22
22
2
21
1
∗ Froude Number is evaluated at the upstream face (y1)
∗ For the purpose of mixing, a good jump should have an upstream to
downstream depth ratio of y2/y1 > 2.38
∗ If 2 < Fr1 < 1, then get a series of undulations with little energy dissipated
∗ If Fr1 = 1, then have critical flow
∗ If Fr1 > 2, then have appreciable energy loss good mixing conditions 3.5.5 SPIRAL FLOW TANKS
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∗ A vortex action in the flow moving tangentially inward at center
∗ High inlet velocity required
∗ Headloss allows for determination of power dissipation
P ghL= ρ Q
Q
∗ Less headloss than in gravity mixing
3.5.6 PIPE MIXING
∗ Again, power dissipation is a function of headloss
P ghL= ρ
∗ Most significant headloss occurs in long pipe lines
∗ Headloss can be calculated using closed conduit and open channel equations
Darcy-Weisbach
Mannings
Hazen-Williams
Chezy
∗ Recall: Slope of the EGL (S) = hL/L
EXAMPLE 3.1: RAPID MIXING BY MECHANICAL AGITATION
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∗ A square rapid mixing basin, with a depth of water equal to 1.25 times the
width, is to be designed for a flow of 7570 m3/d. The velocity gradient is to
be 790 m/s/m, the detention time is 40 s, the operating temperature is 10oC
and the turbine shaft speed is 100 rpm. Determine:
a) The basin dimensions
b) The power required
c) The impeller diameter if a vane-disc impeller with 6 flat blades is
employed and the tank has 4 vertical baffles. The impeller diameter is
to be 30% to 50% of the tank width
d) The impeller diameter if no vertical baffles are used
e) The air required if pneumatic mixing is employed and the diffusers are
0.15 m above the tank bottom
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