agitation mixing

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

81.301 3 - 1 Environmental Engineering Unit Operations


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



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



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



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



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



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



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∀

=



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



∗ 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



∗ 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


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



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

= =



∗ 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



∗ 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)



∗ 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



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



( )

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



∗ 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



∗ 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



∗ 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



∗ 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



∗ 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


agitation mixing

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