## Rochester Institute of Technology

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Thesis/Dissertation Collections

7-8-1996

CFD analysis of flow field in a mixing tank with and

without baffles

Carlos Rezend

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## Recommended Citation

CFD ANALYSIS OF FLOW FIELD IN A MIXING

TANK WITH AND WITHOUT BAFFLES

by

Carlos F. Rezende

A Thesis Submitted

InPartial Fulfillment

of the

Requirements for the Degree of

MASTER OF SCIENCE

m

Mechanical Engineering

Approved by:

Dr.

Ali

Ogut (Thesis Advisor

Prof.

_{ }

-Prof.

_{ }

## -Dr. AlanNye

Prof.

- - - -

Dr. P. Venkataraman

Prof.

_{ }

-Dr. C. Haines

DEPARTMENT OF MECHANICAL ENGINEERING

COLLEGE OF ENGINEERING

PERMISSION FROM AUTHOR REQUIRED:

I

Carlos Frois Rezende

prefer to be contacted each time a request for reproduction

of the thesis entitled

CFD Analysis of Flow Field in a Mixing Tank with and

## Without Baffles

is made. I can be reached at the following address:

577 Pinnacle Road

Rochester, NY 14623

USA

Phone: (716) 359-1408

Date: 07/0811996

Acknowledgment:

I

wouldlike

to expressmy

appreciationto,

Dr.

Ali

Ogut,

for his

evaluationandsuggestions with regardto

the thesis

organization.This

thesisis dedicated

tomy

parents and tomy

_{wife,}

Julie,

for

thelove,

Abstract:

A

computational techniquehas

been

## developed

tostudy

the

fluid

motionin

abaffled

and unbaffledmixing

tank.This

technique wasdevised for

a cylindricalmixing

tankusing

aComputational Fluid Dynamics

_{code,}

FIDAP.

Two

modelanalyses are presented

in

this thesis:Unbaffled

tanksteady

state analysis.Baffled

tank transientstate analysis.The

method presentedin

thisthesis

uses subroutinesincorporated into the CFD

programto

impose

the

boundary

conditions associatedwith each## baffle

in

the

tank.These

boundary

conditions are rebuiltfor

eachtimestep

to capturethe

effects ofthe relative motion

between

the

baffles

andthe

impeller.

In

addition,

the

use ofparameters

in

defining

the

coordinates ofthe_{geometry, the}

density

ofthe

_{mesh,}

and the

fluid

properties allow a straightforward automation ofthe

model.This

automation

considerably

reduces thetime

to create a new tankgeometry from

scratch.

## Furthermore,

the

automation permits rapid modeldevelopment

tobe

usedin selecting

optimaltanks and radialimpeller

combinations.Table

ofContents:

Chapter

1Introduction

ofMixing

Technology

andObjective

ofthisWork 11.1

Introduction

1 -21.2

Objective

ofthisWork

2-3Chapter 2

Background

andLiterature

SearchonFluid

Mixing

42.1

Background

42.1.1

Experimental.

52.1.2 NumericalComputation 6- 7

2.2 Typesof

Fluid

Mixing

7- 82.3

Mixing

Processes

82.4

Types

ofImpellers

for

Mixing

Tanks - 122.5

## Types

ofMixing

Tanks

13 - 152.6

Mixing

Tank Optimization 16-_{17}

Chapter 3

Basic Concepts

183.1 Dimensionless

Group

18 -203.2 Power

Consumption

20- 223.3 Baffle Design 23 - 24

3.4

Pumping

Capacity

24- 26Chapter 4

Theory

on FluidMixing

274.1

Theory

27-284.2

Coordinate System

28

4.3

Analysis

of aFluid

Elementin

anInertial Reference Frame

294.3. 1

Conservation

ofMass

29

- 304.3.2 Motionof a

Fluid

Element30-31

4.3.3 Fluid Rotation 31 -33

4.3.4 Accelerationof a

## Particle

33

4.3.5

Forces

Acting

onaFluid Particle

33

-344.3.6

Navier Stokes

Equation(N-S)

35

Chapter

SCFD Model

39### 5.1 FIDAP 39-42

5.2

## Application

toFluid

Mixing.

42-455.3

Problem

Considerations

- 495.4

Assumptions.

49 - 505.5

Parameters

515.5.1

Geometry

Parameters 51 - 525.5.2

Mesh Parameters

52-535.5.3

Fluid Properties

andTransient Parameters 545.6

Simulation

Time 55 - 56### 5.7 FI-GEN

(Geometry)

56- 575.7.1

## Points

57-585.7.2

Curve

58 -59

5.7.3 Mesh

Edges

59- 605.7.4

MeshFace

61-625.7.5

Mesh Solid

62- 645.7.6 Mesh

Generation

645.7.6.1 Two-Dimensional Mesh 65- 69

5.7.6.2

Three-Dimensional

Mesh 70-72

5.7.6.3

Element

Type - 745.8

Boundary

Conditions

75 - 765.8.1

FI-BC

Program Module5.8.2 FIPREP Program

Module Programs

81 - 885.9 Programs 88

5.9.1 Start.f Program 88

5.9.2 user.f

Program.

89-915.9.3 Baff4.f Program - 92

5.9.4 plotld.f Program 91

5.10 Model

Application

toa## Mixing

Problem.

93

Chapter 6

Results 94

6.1

Mixing

TankModel.

94-956.2

Steady

State Unbaffled

ModelSolution.

95 -99

6.3 Transient

State Baffled

ModelSolution.

100- 1026.3.1

Summary

ofFigures

6.7to6.30102

- 1156.3.2

Summary

ofFigures

6.31 to6.66.

116-135

6.3.3

Summary

ofFigures

6.67 to6.96.

136 - 153Chapter

7Conclusions

154Conclusions

154-156Bibliography

157-158Appendix

A(Unbaffled

Steady

State

Analysis)

159- 160Appendix B

(Transient Analysis

ofa## Baffled

Mixing Tank)

161 - 186Appendix C

(Description

onHowtoRuntheModel).

187- 201List

ofFigures:

Figure 2.1 Typical Radial Flow Pattern 11

Figure 2.2 Typical Axial Flow Pattern U

Figure 2.3 TypicalRadialImpellers 12

Figure 2.4 Typical Axial Impellers 12

Figure 2.5 Reference

Geometry

15Figure 2.6 Typical

Mixing

TankWithBaffles 15Figure3.1 Power Number Versus

## Mixing

ReynoldsNumber 20Figure 3.2 Relative Power VersusRelativeSpeed 22

Figure3.3 Power NumberComparisons for SeveralImpellerTypes

Figure 3.4 BaffleArrangements 24

Figure3.5

Velocity

ProfileattheBiade Edge 25Figure 3.6 Control Volume Aroundthe Impeller 25

Figure4.1 Differential Control Volume in Cylindrical Coordinates 28

Figure4.2 InfinitesimalElementofFluid Figure4.3 ComponentsofFluidMotion

Figure4.4 VortexFormationinanUnbaffled

Mixing Tank.

Figure4.5 Stress Tensoron aFluid

## Element.

34Figure4.6 CentripetalandCoriolisAcceleration 37

Figure 5.1 Two Time Steps forthe

Sliding

Mesh Techniquesusedin Fluent Software 43Figure5.2 Comparisonofthe

Geometry

Figure 5.3 RadialPaddle Impellers 47

Figure5.4 CoordinatesoftheTank

Geometry.

Figure 5.5 HorizontalandVerticalCurves 59

Figure 5.6 MeshEdgeAppliedtoLinesandCircles 60 Figure 5.7 Examplesof

Surfaces,

MeshLoops,

andMeshFacesFigure 5.8 InitialStageofthe MeshSolidProjection 63

Figure 5.9 FinalStageoftheMeshSolidProjection

Figure5.10 Difference in Shape Between"Map"and

"Pave" Mesh

Figure 5.11 Bottom SurfaceoftheTank

## Geometry

Figure5.12 MeshFaceofthe

Top

SurfaceoftheTank 68 Figure5.13 ImpellerMesh FaceFigure 5.14 Walland

Top

Surface,

MeshFaceFigure5.15

Boundaiy Group

Elements FortheTankMeshSolid Definition 71 Figure5.16 The CompleteThree-DimensionalMeshfortheTankGeometry

72 Figure5.17 Two-DimensionalandThree-DimensionalElements 74Figure5.18

Boundary

Conditions fortheVelocity

oftheBottom Plate inaRotating

FrameofReference 78

Figure 5.19

Mixing

Tank inaRotating

FrameofReference 83Figure5.20 Baffle Motion AroundtheImpeller 92

Figure6.1

Velocity

Vector(FrontPlane)

inanUnbaffled Tank 97Figure 6.2

Velocity

Vector (at Different LevelsoftheTank)

inanUnbaffled Tank 97 Figure 6.3Velocity

Vector(ImpellerPlane)

inanUnbaffled Tank 98 Figure6.4## Velocity

Vector0?laneCutting Surface)

inanUnbaffled Tank 98 Figure 6.5 Pressure Distribution inanUnbaffled Tank 99Figure6.6 ShearDistributioninanUnbaffled Tank 99

Figure6.7

Velocity

Vector (FrontPlane)

inanBaffledTank,

TimeStep

1 104 Figure 6.8Velocity

Vector (at Different LevelsoftheTank)

inaBaffledTank,

TimeStep

1.

104Figure 6.9

Velocity

Vector(ImpellerPlane)

inaBaffledTank,

TimeStep

1 105 Figure 6.10Velocity

Vector (PlaneCutting

Surface)

inaBaffledTank,TimeStep

1 105Figure6.11 Pressure Distribution inabaffled

Tank,

TimeStep

1 106 Figure 6.12 Shear Distribution inaBaffledTank,

TimeStep

1 106 Figure 6.13## Velocity

Vector (FrontPlane)

inanBaffledTank,

TimeStep

2 107 Figure 6. 14Velocity

Vector (at Different LevelsoftheTank)

inaBaffledTank,

TimeStep

2 107 Figure 6.15Velocity

Vector (ImpellerPlane)

inaBaffledTank,

TimeStep

2 108 Figure 6.16Velocity

Vector (PlaneCutting

Surface)

inaBaffledTank,

TimeStep

2 108 Figure 6.17 PressureDistribution inabaffledTank,

TimeStep

2 109 Figure 6.18 Shear Distribution inaBaffledTank,

TimeStep

2 109 Figure 6.19Velocity

Vector (FrontPlane)

inanBaffledTank,

TimeStep

3 110 Figure 6.20Velocity

Vector (at Different LevelsoftheTank)

inaBaffledTank,

TimeStep

3 110 Figure 6.21Velocity

Vector(ImpellerPlane)

inaBaffledTank,

TimeStep

3 Ill Figure 6.22## Velocity

Vector (PlaneCutting Surface)

inaBaffledTank,

TimeStep

3.

Ill Figure 6.23 Pressure Distribution inabaffledTank,

TimeStep

3 112 Figure 6.24 Shear Distribution inaBaffledTank,

TimeStep

3 112 Figure 6.25Velocity

Vector OFrontPlane)

inanBaffledTank,

TimeStep

4 113 [image:10.562.119.495.43.698.2]Figure6.63

Velocity

VectorGmpellerPlane)

inaBaffledTank,

TimeStep

10 134 Figure 6.64Velocity

Vector (PlaneCutting Surface)

inaBaffledTank,

TimeStep

10.

134Figure 6.65 PressureDistribution inabaffled

Tank,

TimeStep

10. 135Figure 6.66 Shear Distribution inaBaffled

Tank,

TimeStep

10 135 Figure 6.67## Velocity

Vector(FrontPlane)

inaBaffledTank,

TimeStep

1i

139Figure 6.68

Velocity

Vector (at DifferentLevelsoftheTank)

inaBaffledTank,

TimeStep

11.

139 Figure 6.69Velocity

Vector (ImpellerPlane)

inaBaffledTank,

TimeStep

11 140Figure 6.70

Velocity

Vector 0?laneCutting

Surface)

inaBaffledTank,

TimeStep

11 140 Figure6.71 PressureDistribution inabaffledTank,

TimeStep

11 141 Figure 6.72 Shear Distribution inaBaffledTank,

TimeStep

1 1 141 Figure 6.73Velocity

Vector (FrontPlane)

inaBaffledTank,

TimeStep

i2

142 Figure 6.74Velocity

Vector (at Different LevelsoftheTank)

inaBaffledTank,

TimeStep

12.

142 Figure 6.75Velocity

Vector(Impeller## Plane)

inaBaffledTank,

TimeStep

12.

143 Figure 6.76Velocity

Vector (PlaneCutting

Surfece)

inaBaffledTank,

TimeStep

12 143 Figure 6.77 Pressure Distribution inabaffledTank,

TimeStep

12.

144 Figure6.78 Shear Distribution inaBaffledTank,

TimeStep

12 144 Figure 6.79Velocity

Vector O^rontPlane)

inaBaffledTank,

TimeStep

13 145 Figure6.80Velocity

Vector (at Different LevelsoftheTank)

inaBaffledTank,

TimeStep

13 145 Figure 6.81Velocity

Vector(ImpellerPlane)

inaBaffledTank,

TimeStep

13 146 Figure 6.82Velocity

Vector OPlaneCutting Surface)

inaBaffledTank,

TimeStep

13.

146 Figure 6.83 Pressure DistributioninabaffledTank,

TimeStep

13 147Figure 6.84 Shear Distribution inaBaffled

Tank,

TimeStep

13 147 Figure 6.85## Velocity

Vector (FrontPlane)

inaBaffledTank,

TimeStep

14 148Figure 6.86

Velocity

Vector (at Different LevelsoftheTank)

inaBaffledTank,

TimeStep

14 148 Figure6.87Velocity

Vector (ImpellerPlane)

inaBaffledTank,

TimeStep

14 149 Figure 6.88Velocity

Vector 0laneCutting

Surfece)

inaBaffledTank,

TimeStep

14.

149 Figure6.89 Pressure DistributioninabaffledTank,

TimeStep

14.

150 Figure6.90 Shear Distribution inaBaffledTank,

TimeStep

14 150Figure 6.91

Velocity

Vector O^rontPlane)

inaBaffledTank,

TimeStep

15 151 Figure 6.92Velocity

Vector(atDifferent LevelsoftheTank)

inaBaffledTank,

TimeStep

15.

151 Figure 6.93Velocity

Vector(Impeller## Plane)

inaBaffledTank,

TimeStep

15 152 Figure 6.94Velocity

Vector 0PlaneCutting Surface)

inaBaffledTank,

TimeStep

15 152 Figure 6.95 Pressure Distribution inabaffledTank,

TimeStep

15.

153 Figure6.96 Shear DistributioninaBaffledTank,

TimeStep

15 153AppendixC:

Figured input.readfile Parameters 190

Figure C.2 FIDAP User interfece

Startup

Screen 191Figure C.3 READFTLECommand 192

Figure C.4 CONFIGURE Command 194

Figure C.5 CREATECommand 194

Figure C.6 RUN

Command.

197Figure C.7 FDPOSTCommand 201

List

of## Tables:

Table 2.

1

Mixing

Processes

8

Table 5.1

Tank

Coordinates

52

Table 5.2

MeshDensity

53

Table 5.3

Fluid Properties

andTransient Parameters

54

List

ofSymbols:

Symbol

Definition

on angularacceleration

>

V

blade

velocity

vectorh

elementlength

scale-*

m rotation of a

fluid

particle>

vorticity

Q

volumeflow

rateT

total torqueFs

surfaceforce

P

fluid

density

r circulation

T|

efficiency

of amixing

tank3

pitch anglea yaw angle

It absolute

## viscosity

T

deviatoric

stress?

specificenergy dissipation

ratea surfacetension

angular

velocity

s

Kronecker

delta

Oij

stresstensorEijk

alternating

symbola acceleration

in

thenon-rotating

frame

A

areaa'

Ao

accelerationofthemoving

systemCp

specificheat

at constantpressured

lever

armlength

D

diameter

oftheimpeller

F

force

fi

body

forces

per unit massg gravitational acceleration

N

rotational speed oftheimpeller(rad/sec)

Nfr

Froude

NumberNP

Power NumberNQ

Flow NumberNRe

mixingReynolds

NumberNw

Weber NumberP

pressureP power

Qax

axial_{pumping capacity}

Qr

radial## pumping

_{capacity}

r position vector

in

thenon-rotating frame

r'

position vector

in

the_{rotating}

frame

r,e,z cylindricalcoordinates

T

temperatureUi velocityvector

V

velocity in

the_{non-rotating}

frame

V

volumetankV

velocity

in

the_{rotating}

frame

Ve

angular_{velocity}

vr

radial_{velocity}

vz

axial_{velocity}

Chapter

1

Introduction

ofMixing Technology

andObjective

ofthis

Work

1.1

Introduction:

The

development

ofmixing

processtechnology

is

alarge

area ofcurrentresearchespecially in the field

ofchemical engineering.The

number of opportunitiesfor

## mixing

technology

are almost unlimiteddue

tothe

immense variety

ofapplicationsin the

chemical,pharmaceutical,

fermentation

process,bio-reactor,

paper,

mining,water

treatment

wastetreatment,

andfood industries.

Fluid mixing has

an enormousimpact in

almostevery

production process andevery industrial

andconsumerproduct.For

instance,

the

beer

industry

utilizesthe

mixing

process to ensureintimate

contact and good## mixing

ofthe stirred phases.This

is

essential to controlthe

fermentation

andthe

quality

ofthe

product.Another

specific example ofthe

fermentation

processis

the

commercialization ofmixing

tanksfor the

production of penicillin.The mixing

processis

usedfor

baby

food

wherethe

formula

orfood is

mixedin

abatch

process.Mixing

is

also usedin

the

fabrication

of materialsthrough

chemical process andin

the

petroleumindustries.

Another

applicationis in the

film

## industry

wheredyes

and chemicalsextendedto themolecular

level

whereit

is

the result ofmoleculardiffusion.

Most

of

mixing

processes onthe

molecular scale takes placein laboratories.

Many

scientists

may

usemixing

processes toharvest

cells whichmay

provideinsights

for the

development

of new medications orformulas.

Fluid

mixersplay

animportant

rolefor

theenvironment as well.For

_{example,}

Flue

Gas

Desulfurization

(FGD)

process usesfluid

mixers to absorb the sulfurdioxide

whichhelps

tocontrol

the

air pollution(acid

rain).1.2

## Objective

ofthis

Work:

The

objectives ofthis thesis

are:1.

To

show a completetransient state analysis ofamixing

tankwhere anuniformradial

flow

patternis

achievedthroughout

the entire content ofthe

tank andwhere

the

velocity

vectorfiled

remains constant with time atevery

pointin

space.

The steady

state analysisfor

the unbaffledtankis

also presented.2.

To

present allthe

requirementsfor constructing

amixing

tank model and toshow

the

steps needed to run amixing

tank analysis(more

emphasis onthe

transient solution).

Computer

analysis of## mixing

problemsis steadily growing

among

the engineering

community.Since

experimental analysisis

timeconsuming

for

equipmentsetup

anddata

analysis.Furthermore,

the

equipmentcomputer analysis.

This

work should provideindustries

with a moreeconomical,

effective,

andfaster

toolin

designing

mixing

tanks.3.

To

inform

the

reader withthe different

aspects ofthisexciting

field.

Mixing

technology

deserves

more attentiondue

to theeconomic significance of## mixing

Chapter

2

Background

andLiterature

Search

onFluid

Mixing

The

purpose ofthis

chapteris

tointroduce

the principle offluid mixing

and thetechniques

usedin

thepasttoanalyzetheflow field in

amixing

tank.This

chapteris

divided

into

twoparts asfollows:

1.

The

background

section(2.1

to2.1.2)

discusses

some ofthe

experimental andcomputational techniques utilized

in

the

past to analyzethe

flow field

of amixing

tank.2.

The

## literature

section(2.2

to2.6)

introduces the

fundamentals

offluid

mixing.and

discusses

the

different

types of_{mixture,}

impellers,

tanks,

mixing

processes, andtankoptimization.

2.1

Background:

Most

ofthe mixing

tanks analysis arebased

on experimental results and empiricalformulations. Although

experimental results give a greatdeal

ofinformation,

it

is

often time

consuming

and expensive.Another

technique ofanalyzing the

flow

field

in

amixing

tankis

## through CFD

(Computational Fluid

Dynamics).

CFD

is

2.1.1 Experimental:

A

paper writtenby

Weetman

[12]

shows thedesign

of amixing impeller

through

experimentaltechniquesand

CFD

analysis.The

experimental technique consistedof a

dual

channelLaser Doppler

Velocimeter

(LDV)

which measuresthe

fluid

velocities.

The LDV

measures twovelocity

components simultaneously, and thedata

acquisitionis

sent to a computer whichdisplays the information

received.The

second tool usedby

Weetman in the development

of a transitionalflow

## mixing impeller is

a program calledFLUENT,

createdby

Fluent

Inc..

The

velocity

field

calculatedby

the

CFD

program(FLUENT)

agrees well withthe

experimental results.

Besides

the

LDV

experimental technique mentioned_{above,}

many

other methodsexists

for measuring the velocity

ofthe

particles.Some

ofthese

methods are"transit-time

or correlative techniques"(Laser Transit

Velocimetry,

andParticle

Tracking Velocimetry [12,

page145]

).

## According

tothe

literature,

the

first

method

does

not allowthe

determination

ofthevelocity

sign.The

second andthe

third methods use a

fast

camera todetermine the

velocity

field.

Unfortunately

these

techniques require a greatdeal

ofmixing

expertise to analyzethe

results.Another

draw back in

determining

the velocity

field in

a## mixing

tankthrough

2.1.2 Numerical Computation:

In

thepast,

CFD

software usedtobe

limited

to single-phaseflow

analysisdue

tothe

lack

ofknowledge

ofbubble

dynamics,

turbulentflows,

and computationallimitation [12].

Today,

CFD

softwareis

capable ofsolving

two-phaseflow

andmore complex

geometry

problems.This

is important because

most ofthemixing

processes

in industries

involve

more than onemixing

phase.As

a_{result,}

numerical computation of

## mixing

tanksarebecoming

more realistic.The

numerical computation of theflow field in

stirred tankshas

receivedsignificant attention since

the

beginning

ofthe

1980's,

starting

withHarvey

(1980)

and

Harvey

andGreaves (1982).

In

(1981)

Issa

andGosman

calculatedthe

flow

in

a gassedmixing

tank equippedwith aRushton

turbineimpeller.

However,

the

results of

their

analysis were never verifiedbecause

no experimentaldata

wasavailable.

Using

conservation of mass and momentum_{equations,}

## Looney

(1985)

presented a model

for

turbulentflow

ofsolid/liquid suspensionin

amixing

tank.Numerical

computational analysis was also performedfor

variousimpeller

typesfor both

single-phase and two-phases.This

analysis wasfirst

calculatedby

Pericleous

andPatel

(1987),

but the

results could notbe

verified withexperimental

data.

Mann

(1988)

modeledthe

flow

createdby

adisc-turbine

by

applying the

continuity

equationto a'network

of zones'.This

'network

ofzones'

more complete model was

## developed

by

Trdgardh (1988).

Where

the momentumexchange

between

the

gasphase andthe

liquid

phase wasincorporated.

Trdgardh

alsomodeled

local

masstransferfor both

phases.2.2

Types

ofFluid

Mixing

Many

different

ways substances(gas

orliquid)

canbe

mixed.One

exampleis

the

dispersion

ofgases and chemicalsin

_{soils,}

rivers,

and oceansin

our environment.This

type of mixturewiththe

environmenttakesplace onthe molecularlevel,

andit

givesrise

to an environmental pollution.Radiation

is

another example ofhow

small particles can

be

mixed with air.It

can occur## naturally

through cosmicradiation, or

it

canbe

createdartificially

by

humans.

In

both

_{cases,}

the

mixtureoccurs at microscopic

levels

by

emitting energy in

theform

of electromagneticwaves.

Radiation

may

consist of threedifferent

particles of_{energy,}which

is

known

asalpha,beta,

and gamma particles.Mixing

canbe

achievedby

using

some conventional agitation systems orincreasingly

as a continuous operation system.In the

latest

_{case,}

mixing

is

accomplished

by

implementing

motionless mixers or static mixers.For

example,

static mixers are often encountered on pipes to split

the

## fluid into

streams whichare

then

combined and split again.This

is

done

in

such away that mixing

effectsoccur

by

entanglement ofthe fluid layers [16].

The

classical agitationsystem,

rotating

system(impeller).

Depending

onthe

rotational speed of theimpeller,

mixing

can take place atlow

orhigh mixing Reynolds

number.For

low

angularspeeds of

the

impeller,

the

flow

is

morelikely

tobe in the laminar

region.In

thiscase,

## mixing

takes placedue

tothe

continuous change ofthe

relative positions ofthe

particles.On the

otherhand,

for high

angular speedthe flow

is

morelikely

tobe turbulent;

andmixing

resultsfrom flow

instabilities

anddisordered flow

pattern.

The

quantitativedistinction between laminar

and turbulentflow in

amixing

tankis clearly

statedin Chapter 3 (Basic Concepts

section).2.3

Mixing

Processes:

## According

toOldshue

andHerbst

[16],

the requirementfor mixing may

vary

depending

onhow

the substances arebeing

mixed.This

information

is

important,

because it

givesinsight

onhow

to select an optimalmixing

apparatus.Table

2. 1

provides a general classification of

mixing

applications asthey

apply

to physicaland chemicalprocesses.

Table

2.1

Mixing

processesPhysical

Processing

Application Class

## Chemical

Processing

Suspension

Dispersion

Emulsification

Blending

Pumping

Liquid-Solid

Liquid-Gas

Immiscible

Liquids

Miscible

Liquids

Fluid Motion

Dissolving

Absorption

Extraction

Reactions

2.4

Types

ofImpellers for

Mixing

Tanks:

There

arebasically

two types ofimpellers (axial

and_{radial),}and

they

arecharacterized

by

## the direction

ofthefluid

motion generatedby

their

blades. Axial

flow

impellers have

a principaldirection

ofdischarge

which coincides withthe

axis of

impeller

rotation[16].

In

_{contrast,}radial

flow

impellers

have

a principaldirection

ofdischarge

normaltothe

axis ofrotation[16].

However,

both

types ofimpellers

(radial

andaxial) may

alsodischarge flow in

oppositedirections.

For

example, an axial

flow impeller discharges flow radially

atlow

off-bottomclearances and

## large D/T (impeller diameter/tank

diameter)

ratios.The flow

pattern produced

by

radial and axialimpellers

canbe

seen onFigures 2. 1

and2.2

[10].

Often,

radialmixing impellers

canbe

distinguished from

axialimpellers

by

simply observing

thedifference in

the geometry.However,

the clearance andthe

ratio of

impeller diameter

versusthe

tankdiameter may have

significant effectsin

the type of

the

impeller.

Typical

radial and axialmixing

## impellers

are shown onFigures

2.3

and2.4

[16].

Impellers

are also classifiedaccording

tothe mixing

flow

regime(laminar

orturbulent).

Usually,

for laminar

flow,

thediameter

ofthe

impeller

approaches tothe diameter

ofthe

mixing

tankwhichbrings

the

fluid

motiontothe

entire contentof

the

tank.This

is due

tothe

poorfluid

mixing

contentin the

laminar

## regime,

since

the

momentum ofthe

fluid

is

fairly

weakwhen compared toturbulent

flow.

impellers

for laminar flow

tend tobe relatively

small.Typical

laminar

impellers

used

in mixing

applications arehelical

ribbons,

_{screws,}

helical ribbon

screws,anchor

impellers,

sigma,

MIG,

disks,

paste_{rollers,}

high

_{shear,}and gate

impellers.

Turbulent mixing

flow does

not need## impellers that

approachthe

tankdiameter,

since

it has

enough momentumfor

the particles todiffuse

during

the mixing

process.

The impeller for

turbulentflow

areusually

one-fourth to one-halfofthe

tank

diameter.

According

to'Herbst

andOldshue'

[16]

impellers in

the turbulentmixing

regimecan

be

classifiedbased

upon whether theimpeller

is

aflow,

_{pressure,}or a shear

impeller.

Impellers

withlarge

pumping capacity

areflow

## impellers,

impellers

that

work againstpressure gradient are pressure

impellers,

andimpellers

that produce,i

Figure

2.1

(Typical

Radial

Flow

Pattern)

/'

*

Y

\

\Unfortunately,

no universal standard methodfor

identifying

animpeller

typeexists.

Figures 2.3

and2.4

show the most common type ofmixing

impellers for

radialand axial

flow.

R100

R510

R400

Figure

2.3

(Typical Radial

Impellers)

Radial Impellers

Type:

## Applications:

(R

100)

Rushton-Type

generally

usedfor

gas-liquiddispersion

(R 5

10)

Disk

Style Flat Blade Turbine

usedfor high

intensity

agitationandhigh

power

inputs

(R

400)

Anchor

Impeller

usedfor

high

viscosity

blending

A310<

A100

A200

A400

Figure 2.4

(Typical Axial

Impellers)

Axial

Impellers

Type:

(A

100)

Marine-Type Propeller

(A

200)

## Pitched-Blade Turbine

Applications:

used

for relatively

small scaleblending

operations

2.5

Types

ofMixing

Tanks:

There

aremany different

types ofmixing tanks

availablein the

markettoday

depending

on specific application.However,

somegeometry

ofmixing

tanks arenot available on

the

market,

andthey

require customdesign.

The

most commontype of mixers are

the

cylindricaltype,

andit

willbe

the

## focus

of this work.Mixers usually

consist of an electrical_{motor, shaft,}

baffles,

and one or moreimpellers. A

typical arrangement of abaffled mixing

tankis

shownin

Figure 2.6.

The

employmentofbaffles

in mixing

tanksis widely

usedtodisturb

or redirectthe

flow. The

most commontype ofbaffle is

wallbaffle,

whichis nothing

more thana

long

rectangularplate placed paralleltothe

cylinder wall.There

are also othertypesof

baffles,

i.e.,

bottom

baffles,

disk

baffles

placed onthe impeller

_{shaft,}and

surface

baffles

which## float

onthe fluid

surface.The

baffles

studiedhere

arethe

walltype.

The

following

are some examples of mixersfound in the literature:

The first

class ofmixers arethe

Laboratory

Mixers.

These

are small unitsusually

less

than 1/8

hp,

which canbe

mounted on aring

stand.The

size ofthe

impellers

are suitable

for

laboratory

beakers

and smallvessels.The

second class of mixers arethe

## Portable

Mixers.

The

sizetypically

rangesfrom 1/4

to3

hp,

andthey

are equipped with aclamping

device

which canbe

mounted on

the

rim of a vessel.These

arevery

versatilemixers,

because

they

The

third class ofmixers are theTop-Entering

Mixers.

These

type of mixers areused on permanent

supports,

which canbe

placed either above the tank ormounted on

the

top

ofthetank.The

size variesfrom

5

to250

hp depending

onthe

mixing

requirements andtheshaftis

mountedvertically

withone ormore## impeller.

This

particular mixeris

the one studiedhere.

The

fourth

class ofmixers arethe

Side-Entering

Mixers.

These

types ofmixersare mounted on a

flange

on the side ofatank.They

arenormally located

on thebottom

ofthe

tank,

andthe

shaftis horizontal

relative to the tank.A

single axialimpeller is

usedin

this type of application.The

sizeusually

rangefrom

1

to75

hp.

The fifth

class of mixers are theBottom-Entering

Mixers.

They

are mountedpermanently

on the## bottom

of thetank,

and a shaftsealing device is

alwaysrequired

for

this type of mixer.This

seal preventsany

possiblefluid leakage

through the

bottom

ofthe

tank.The

last

class ofmixersis

the Line

Mixer.

This

type of mixermay be

aStatic

orMotionless

Mixers.

It

is

usedto mixthe

fluid

during

its

passageto alarge

storagetank.

Mixing

is

achievedby

splitting the

_{stream,}

## reversing

stream rotation and### STATIC LIQUID LEVEL

\

N

T

-*

1 _{'T}

'

fT/3-1

1

C\

T/12 O-T/3 Z-T C-T/3-D B-T/12

## Figure

2.5

(Reference

Geometry)

Figure

2.6

(Typical

Mixing

Tank

With

Baffles)

2.6

Mixing

Tank

Optimization:

Optimizing

a3

-Dimensional mixeris

not atrivialtask,

andusually

requires alot

of computer powerto produce optimalresults.

The

optimization processbecomes

even more

difficult

whenthe role ofbaffles is

analyzed.The

model createdhere

will give an optimal tank

## geometry

orimpeller geometry

by

trial and erroranalysis.

This

is

accomplishedby

inputting

different

parametersin

the

model(see

chapter

5

for

moredetails)

andanalyzing

theflow field for

thedesire

result.Perhaps,

afterthis

_{work,}the next

step

towardsanalyzing

abaffled mixing

tankis

to create an actual optimization process without trial and error analysis.

For

example,

the

user would providethe

variables and the constraints to the model(i.e.,

type offlow,

flow

_{rate,}and

## mixing

properties)

and the computer wouldcalculate

automatically the

required size of_{geometry,}

impeller,

andthe number ofbaffles.

There

are a number offluid mixing

variables anddesign

parameters whichmay

affect

the

optimization process.However,

before

designing

an optimal stirredtank,

it is important

toquantify

the performance ofoneimpeller

and tankdesign

withrespectto another.

The

objective ofoptimizing

amixing

tankis

to achieve arapid mass

transfer,

rapidblending

time,

and an uniform suspension.## Therefore,

the

mechanical elements and the parameters of a mixerplay

animportant

rolein

designs,

the

referencegeometry

shownin Figure 2.5

[16]

is

used as abasis for

comparison.

For

most ofthe

industrial

_{applications, the}goal of

any

optimization processis

tofind

a

balance

between

cost and efficiency.The

cost of a productis

usually

directly

proportional to

its

efficiency.Therefore,

whenever a productis

optimizedusually

atrade exists

between efficiency

and cost.In

this

_{case,}

it

is up

tothe

designer

to makethe

## decision

whether the productjustifies

ahigher investment.

The

computer modelcreated

in this

analysisfor

themixing

tank allowstheuserto selectdifferent

variablessuch as

geometry,

fluid

_{medium,}shaft

_{speed,}

boundary

_{conditions,}and other

Chapter 3

Basic Concepts

In

fluid

mixing

technology

the

dimensionless

groups are perhapsthe

mostimportant

prerequisitefor

thestudy

ofmixing

tanks.Therefore,

it is

theintent

ofthis chaptertopresent

these basic

concepts.3.1

## Dimensionless

Group:

When analyzing the flow field in

amixing

tank,

dimensional

analysisbecomes

important in correlating scale-up

parameters.Dimensional

analysis reducesthe

number of

independent

variables whichsimplify

some ofthe

dimensionless

groups.

Dimensionless

groups are a convenientway

ofcorrelating engineering

data

and scientificinformation [10].

There

aremany different

dimensionless

equations

that

## describe the

properties oftheflow field in

amixing

tank,

and someof

these

equationsaredescribed

here [16].

The

mixing

Reynolds

numberis

defined

as the ratio ofinertia forces

(Fi)

_{to}

viscous

forces

(Fv)

acting

atthe

end oftheimpeller blade.

The mixing

Reynolds

Equation

(3.1)

defines

themixing

Reynolds

number.Figure

3.1

[16]

shows therelationship between

power number(see

equation3.4)

andmixing

## Reynolds

number.

Looking

atthisgraph,

the curvetends towardthe

turbulentregime asthe

shaft speed

(N)

increases.

This

trendis easily

explained since themixing

Reynolds

numberis

directly

proportionalto the shaft_{speed,}and

the

powernumberis

inversely

proportionalto thecubeofthe shaft speed.F,

ND2p

N-=t=^r

<31>

Where;

Fi

=inertia force

D

=impeller diameter

Fv

=viscous

force

_{p}=

density

offluid

orsolid\x=_{absolute}

fluid

viscosity

The Froude

number(NFr)

is defined

astheratio ofinertia

## force

(FO

to gravitationalforce

(Fg),

andit is

givenby

thefollowing

equation:Fi

N2D

Nrr-ft-(3-2)

Another Dimensionless

equationis

theWeber

number(Nwe),

whichis

defined

asthe

ratio ofinertia force

to surfacetensionforce (Fa).

F,

N2D3p

N^=vr~v~

(3-3)

Where,

a=surfacetension

Equations

(3.4)

and(3.5)

define

the

power number(NP)

andthe

flow

number(NQ).

Power

Number

is

defined

asthe

ratio ofappliedforce

to masstimes

acceleration,

Nz

Pg

NzD5p

Q

Nq

"ND>

(3.4)

(3.5)

Where;

P

=power

Q

=impeller pumping

capacity

### TURBULENT

to'

,0-REYNOLOS NUMBER

## Figure

3.1

(Power

Number

Versus

Mixing

Reynolds

Number)

3.2

Power

Consumption:

It is

important

toquantify

powerconsumption,

_{since}

this

information

allows

engineers to

design

more efficientmixing

tanks and_{electrical motors.}

This

information

is

important,

because

the

powerinput

of arotating mixing

impeller

depends

onits

geometry,

diameter

(D),

_{and speed}

(N).

Power

consumption also## helps

todetermine

theoperating

costs of the process.The

amount of powerconsumed

from

amixing

impeller

is

_{the}

_{result of}

_{the}

pumping capacity

and thePower

consumptionis

directly

proportional to thedensity,

to theimpeller

speedcubed, and to the

impeller

diameter

raised to thefifth

power.The

log

plotin

Figure 3.2

[16]

showsthe relationship between the

relative power versus therelative

impeller

speedfor different impeller diameters.

## Power

consumption canbe

measuredby determining

the torqueonthe

shaft.One

way

ofmeasuring

torqueis

by

using

adynamometer

motor todrive

the shaft.A

torqueindicator is

thenconnected to

the

dynamometer

motor to measure the amount oftorque producedby

the shaft.After getting the

_{reading, the}power consumption can

be

calculatedby

equation(3.6)

[10].

Experimental

measurements can alsobe done

through

electrical methods.

For

_{example,}a wattmeter can measure the power

input from

the

drive

motor.Power

=Torque

x

Rate

ofAngular Displacement

or(

d

\

P

=6600x\

Fxxco

V 12

J

(3.6)

## Where;

P

=impeller

horse

_{power}

_{oo}=

impeller

speed, radians/second

F

=Force in Pounds Force

d

=Lever

Arm

Length

in inches

The

amount of powerdrawn

by

the impeller

canbe

calculatedby

integrating

the

specific

energy dissipation

(<j>)

over_{the tank volume,}as

defined

by

equation(3.7).

Another approach

is

by integrating

the

total stress tensor(r)

timesthe

impeller

velocity

overthe impeller

area.## This integral

is

givenby

equation(3.8),

where xblade

velocity

vector[page

25,

1 1]. The

power numberfor

varioustypes of radialimpellers is

shownin Figure

3.3 [14].

P

=\dV

(3.7)

\v

rdA

1000

o

5 UJ cc

### RELATIVE SPEED.N

(3.8)

RELATIVE OIAMETER

## Figure

3.2

(Relative Power Versus Relative

Speed)

3.3

Baffle

Design:

As

statedpreviously,

baffles

are parallel plates attached to the tank wall.The

purpose of

baffles

is

tostop

theswirling

andvortexing

formation

so efficientmixing

canbe

obtained.Without

baffles

very

little

realmixing

takes place.The

reason

is because

unbaffledtankshave

very

little

vertical currents and novelocity

gradients.

According

to theliterature

## found,

asliquid

viscosity

increases,

theneedfor

baffles

toreducevortexing decreases, [page

17,

10].

In

other_{words,}unbaffled

tanksare

generally

usedfor

sticky

materials(high

viscosity)

orif

perfectcleaning

ofatank

is

required.For

_{example,}unbaffledtanksproduce acentral vortexwhich

"facilitate

wetting

offine

particles"

[page

59,

17]

when pigments are addedto themixing

solution(i.e.;

food

andpainting

industries).

There

aremany different

types ofbaffles

depending

onthe## mixing

application.If

high

viscosity

liquids

are_{mixed,}then

the

width ofthe

baffle

shouldbe

reduced(usually

1/20

oftankdiameter).

For

_{this situation,}

baffles

are more efficient whenthey

are placedaway from the

wall or at an angle as shownin Figure 3.4-c.

If

moderate

viscosity

liquids

are_{mixed,}the

baffles

shouldbe

placedaway from

thewall, as shown

in Figure 3.4-b.

In

_{this case,}

liquid

canflow

along the

wallavoiding

any

stagnant areas toform

## behind

the

baffles.

If

the viscosity

is

relatively

low,

thenthebaffles

shouldbe

placednear thewalls which causes moreresistance to

the fluid.

Thus,

mixing

takes placedue

toliquid

swirl anddrag.

Baffles at the wall for low viscosity liquids

-a-Baffles set _{awoy}
from wall for moderate

viscosity liquids

Baffles set _{away}

from wall at an

angle for high viscosity liquids

## -C-Figure

3.4

(Baffle

Arrangements)

3.4

Pumping

Capacity:

One

ofthe

mostimportant

components ofamixing

tankis

theimpeller,

because it

identifies the

effectiveness of a stirred tank.The

mechanicalefficiency

of amixing

tank canbe

calculatedbased

on the power consumption and thepumping

capacity.

The efficiency

of amixing

systemis

givenby

equation(3.9),

andthe

flow

rate through theimpeller in the

radialdirection

is

givenby

equation(3.10)

[16].

See Figure

3.5

tocalculatethe radialvelocity from

equation(3.10).

+w

TJ=

N

(3.9)

Q

=_{nD]Vrd.}

r"Z

(3.10)

-w2

## Where;

NQ

=Flow

Number

NP

=Power Number

Where;

D

=Impeller

Diameter

Vr

=Radial

Velocity

Shaft

Impeller Blade

Figure 3.5

(Velocity

Profile

attheBlade

Edge)

Where;

Vr

=Vcosa

cosp,

Ve

=Vsina

cosp,Vz

=VsinP,

V

=(Vr2_{+}Ve2

_{+}Vz2)0"5

Vz

=Axial

Velocity,

Ve

=_{Angular}

Velocity,

Vr

=Radial

Velocity

a=

Yaw

angle,

P

=Pitch

_{angle}

To

calculatethe pumping

_{capacity,}one must

## first

construct a control volumearoundthe

impeller

(the

control volumeis

shownin Figure 3.6).

Then,

a programcan

be

developed

to calculatethe

radial and axialflow

rates.These

programs andthe control volume around the

impeller

are presentedin

Kellem's

thesis.It

is

important

to reemphasizethatimpellers

are characterizedby

radial and axialflow

rates.

Depending

onthe

impeller

diameter

and tankdiameter

ratio(D/T),

radialimpellers

canproduceboth

axialflow

rates and radialflow

rates.tcnpsaarKaia

I*

Figure

3.6

## (Control

Volume

Around

the

Impeller)

The

axialpumping capacity

of theimpeller

canbe

calculatedby

the

following

equation

[11]:

Qm

=4nr\vz^dr

(3.11)

0

The

radialpumping capacity

of theimpeller

canbe

calculateby

thefollowing

equation

[11]:

Qr

=1*rt

s\vr^dz

J_{r,avg}

(3.12)

Chapter 4

Theory

onFluid

Mixing

This

chapter presents thebasic

equationsof motion ofafluid

particle anddefines

the principles of conservation of _{mass,} momentum and

## energy

equations.The

equations of motion are shown

in

aninertial

referenceframe

and non-inertialreference

frame.

The

purpose ofthis chapteris

to present thedynamics

and theforces

acting

on afluid

particlefor

anon-inertial coordinate system(i.e.,

such asthe

fluid

particlesmoving in

amixing

tank).4.1 Theory:

The

following

sectionswilldescribe

thekinematics

and the momentum equationsofthe

fluid

motionin

a cylindricalmixing

tank.The

equations andformulations

shown

here

areonly

validfor

## incompressible

flow. A

closed-form solution(exact

solution)

of athreedimensional mixing

tankproblem wouldbe

impossible.

Such

asolution wouldrequire

the

use ofdynamic

equationsin

describing

themotion of afluid

particle.The

closed-form solutionfor

aNewtonian

fluid

would require thedevelopment

ofdifferential

equationsfor

conservation of mass andNewton's

second

law

of motion(Navier-Stokes Equation).

One

must evaluatethe Navier

be

carried outin

cylindricalform

by

applying

conservation of mass to adifferential

controlvolumeusing

allthreecoordinatesin

space(r, 0, Z),

and atimevariant

for

the transient solution.If

constant## viscosity

andincompressible

conditions are not _{assumed, the} problem

becomes

even moredifficult.

A CFD

program such as

FIDAP greatly

simplifies these tasks ofsolving

3

-Dimensionalflow

problems.Further

details

ofCFD

programsarepresentedin

chapter5.

4.2

Coordinate System:

Cartesian

frame

of referencedoes

not provide aparticularly

usefulframe

ofreference

for

describing

the equations of motionbecause

flows do

not## generally

occur

in

a straightline. This is especially

truefor

themixing

tankpertainedto thisthesis.

Since

thefluid is

confinedin

a cylindricaltank,

the coordinates willbe

presented

in

cylindricalform

exceptfor

theboundary

conditions of the tankgeometry.

The

cylindrical coordinate system applied to adifferential

control4.3

Analysis

of aFluid

Element

in

anInertial

Reference Frame:

For

aninertial

frame

of reference theN-S

equation canbe

used todescribe

the

motion of a viscous

fluid.

## However,

if

theflow

involves

arotating

boundary (i.e.,

tankwalls and

baffles)

whichmovesthroughthe

fluid,

then the acceleration ofthenon-inertial coordinate system must

be

included in

the equations ofmotion.Since

a

mixing

tank probleminvolves

a non-inertial coordinate_{system, the}additional

terms of the acceleration are presented

in

section4.4.

For

its

_{appreciation,}

it

is

worth

mentioning

the

basic

equationsfor the

conservation ofmass and## Newton's

second

law

of motion.Therefore,

theNavier-Stokes

equation and the equationsthat

constitutetheN-S

are also presentedhere in

aninertial

referenceframe.

Let's

first

discuss

thebasic

equations that govern the motion of a viscousfluid,

before

developing

theN-S

equation.It

shouldbe

reemphasizedthat thegoverning

equations ofmotionpresented

here

areonly

validfor laminar flow

ofNewtonian

fluid. Non-Newtonian

fluids

present more complex_{relationships,}and

they

are notpresented

in

this## thesis. The

following

equation appliestoNewtonian

fluids:

ra-(4.1)

Where;

x=_{shear}

_{stress}

_{and}

du/dy

=_{shear}

rate

(rate

ofdeformation)

4.3. 1

Conservation

ofMass

:The differential

conservation of mass equationis

obtainedfrom

aTaylor

seriesFigure 4.1.

This

differential

equationfor

conservation of massis

givenby

equation

(4.2)

[7].

I^_+I_^)+(^)+^=0

(4.2)

r

dr

r60

62

dt

v 'This

equation can alsobe

writtenin

vectornotation as:->

dp

V.pV+-

=_{0}

(4.3)

Where:

*

a

-i

e

~a

'

dr

erdO

dZ

V

=erVr+eeV6+ekVz

Note:

er,eg

,andet are unit vectorsinther,6

,andZ

directions

For steady

state orincompressible flow

(p

=constant)

the time## derivative

ofequation

(4.3)

reducesto:r

dr

r39

dZ

v 'For

incompressible flow

thefollowing

vectornotation mustbe

satisfied:V./?

=0(4.5)

4.3.2

Motion

of aFluid

Element:

After

describing

the conservation of massequation,

let's

now consider aninfinitesimal

elementoffixed

mass(m)

in

aflow field

as shownin Figure 4.2

[15].

In

a_{mixer,}

this

element experiencesmany different

motions(as

shownin Figure

4.3

[7]),

such astranslation,

rotation, angulardeformation,

andlinear

deformation.

Obviously,

as the shaft starts to spinthe

element undergoes alinear

displacement

Figure

4.3,

the

elementmay

experience angular and## linear deformation.

In

thelatest

case,

the element changesin

shapebut

not the orientation.For

angulardeformation,

thedeformation involves

adistortion

of the element and the planeis

no

longer

perpendiculartothe

originalplane.element viewed along z axis

Figure 4.2

(Infinitesimal

Element

ofFluid)

^

\ \

\ J

truuUtna ROUtKXl

3 ()

'

r

I i

1 L j

tLnguUrdefornutioci Lineir fitCnution

Figure 43

(Components

ofFluid

Motion)

4.3.3 Fluid Rotation:

The

rotation of afluid

_{particle,}

m,

is

## defined

by

equation(4.6)

whichis "the

average of

any

twomutually

perpendicularline

elements ofthe particle"[7,

page211].

1

m=_{-VxV}

2

Rotation is

causedby

shear stress onthe surface of_{the particle,}and the presence of

viscous

flow.

If there

is

no viscousforces,

thefluid is

assumed tobe irrotational.

Another way

ofmeasuring

rotationis

by

calculating

aquantity

called_{vorticity,}which

is defined

tobe

twicethe

rotation.In

index

notation thevorticity

is

defined

by

equation

(4.7)

[7],

andin

cylindricalform the vorticity

is

givenby

equation(4.8)

[7].

C

=2m=WxV(4.7)

VxF

=_{e.}

r*"r w^

(dVr

dvA

:(\drVaIdV^

i_i.+.;

{r

d6

dz

dz

dr

+k

\r

dr

rdO

J

(4.8)

In

a## baffled mixing

tank,

one ofthe primary functions

ofthebaffles

is

to prevent thevortex

formation.

This

means that the pressurealong

thefree

surface shouldbe

constant.

Usually

a vortexforms

for

unbaffledmixing

tankscontaining

low

viscosity

liquid(s).

Mixing

efficiency

is

oftenlower for

unbaffledmixing

tanks,

and the vortexspeed

increases

withimpeller

speed.Figure

4.4

shows vortexformation for

anunbaffled

## mixing

tank[10].

Rotational

motion of afluid

particlein

amixing

tank can alsobe

demonstrated

in

terms ofcirculation.

Circulation,

T,

is

defined

asthe

line integral

ofthe

tangentialvelocity

ofany

closed pathin

aflow

field.

The

circulation equation(4.9)

[7

&

15]

is

a

direct

statement ofStokes

theoremwhich states thatthe circulation around a closedcontour

is

the

sum ofthe

vorticity

enclosed withinit

[7].

r=fv.dS

=J2c9zdA=\\yxVjdA

(4.9)

^jkS^J

Figure

4.4

(Vortex Formation in

anUnbaffled

## Mixing

Tank)

4.3.4

Acceleration

of aParticle:

Considering

thefluid

as a continuum_{element, the}acceleration ofa

fluid

particlefor

aninertial

reference system canbe derived from Newton's

secondlaw.

The

total acceleration of a particle

is

the sum ofthe convective acceleration plus thelocal

acceleration;

andit is

givenin

vector notationby

equation(4.

10)

[7].

DV

dV

ap=-=(V.V)V+

p

Dt

dt

(4.10)

4.3.5

Forces

Acting

on aFluid Particle:

The

nextstep

before

proceeding

to theNavier

Stokes

equationis the

description

ofthe

## forces

acting

onafluid

particle.In

afluid

elementtheforces

acting

on afluid

element can

be

classified asbody

forces

and surfaceforces.

The

body

forces

in

amixing

tank aregenerally

representedby

gravitational effects.Gravitational

forces

occur whenthere

is

adifference

in

density

between

liquid

and solid(assuming

thedensity

ofthe solidis

greaterthan the

density

oftheliquid),

andit

tends to move solid particles toward the

## bottom.

Surface forces

are subdividedinto

two parts, normalforces

andtangential(shear)

forces.

The

normalforces for

normal

forces

is

the viscous andinertial forces due

todrag

which tend to move particlesin

thesamedirection

astheliquid [19]. An

exampleof shearforces is

thefrictional

effectbetween

the

surfaces ofcolliding

particles.The

surfaceforces

onan element of mass

dm

and volumedV=drd$dz is

representedby

the stresscomponents.

The

normal and shear stresses are representedby

equations(4. 1 1).

In

afinite

element codethe

normal stresses act onthe

surface of an element andthe

shear stresses act onthe

sides.In

rectangular_{coordinates,}

the

stresstensor of a## fluid

elementis

shownin Figure

4.5

[TJ.

o-

=-p+2ft-dVr

dr

r

ee=-P+

2ju

\r

dO

r_{j}

<r=-P+

2M-dV.

dz

*re=Vr

ISYjl

dr\

rJ

+

ldVr

Bz

dVt

s +1

dvz

r

dO

\

\dz r

dd

)

(

dVr

dVr

+

dz

dr J

(4.11)

4.3.6

Navier

Stokes

Equation (N-S):

The N-S

equation canbe derived

from

thedifferential

equations ofNewton's

second

law

of motion andfrom

conservation of mass.After

developing

the

differential

equationfor

conservation of mass section4.3.1,

the nextstep

is

toapply

thedifferential

equation ofNewton's

secondlaw

of motion.The final form

ofthe

N-S

equationis

givenby

equations(4.

12).

This

equation## forms

the

basis for

the

analysisin this

thesis,

because

it

is only

validfor laminar

flows

ofNewtonian

fluids.

If

theabsoluteviscosity

(n)

is

_{zero, then}

the

fluid

equationof motion(N-S)

reducesto

Euler 's

equation.r-component

P

fDVr

V2^

Dt r

J

dh

dP

(

=

-YTr-~d~r~^

Vr

2dvA

V2

V -

-V

r r2 r2

de)

0-component:

(dv9

vrv0)

Y{dhdf\

fDt

rJ

Vd9

dO)

+ _{M}

Va

2dVr

0 r2

'

r2

dO

V2Va-^r

+(4.12)

z-component:

DV,

dh

dP

Dt =

-r

dz

dz

+vv2v,

4.4

Analysis

of aFluid Element in

aNon-inertial Reference

## Frame:

Since

this thesis

involves

rotating

elements,

the

equationsofmotion andthe

forces

acting

on afluid

particlefor

anon-inertial referenceframe

willbe discuss

in this

section.

In

section4.3

the equations of motionfor

a control volume withrectilinear accelerationwere presented.

The

goalhere is

to extendthose

equationstranslationandrectilinear acceleration.

The

momentum