ENGG 199 Reacting Flows Spring 2006 Lecture 2b Blending of Viscous, Non-Newtonian Fluids ·...

Copyright © 2000, A.W. Etchells, R.K.Grenville & R.D. LaRocheAll rights reserved.

ENGG 199 Reacting FlowsSpring 2006

Lecture 2bBlending of Viscous,

Non-Newtonian Fluids

ENGG 199 Lecture 2b Slide 2Copyright © 2000, A.W. Etchells, R.K.Grenville & R.D. LaRoche. All rights reserved.

Re-Cap

In turbulent regime, viscosity does not influence:Power input.

Flow.

Blend time.

In transitional regime, viscosity influences:Flow.

Blend time.

In laminar regime, viscosity influences:Power input.

Flow.

Blend time (for turbines - not for helical ribbons).

Need to know viscosity.


Newtonian Fluids

Newtonian Fluids:Ratio of shear stress to shear rate is constant.

Constant of proportionality is the fluid s viscosity.

Viscosity will be uniform throughout stirred tank:Shear rates are high in the impeller zone.

Low elsewhere.


Non-Newtonian Fluids

Non-Newtonian Fluids:Ratio of shear stress to shear rate is not constant.

Non-linear relationship.

Measure shear stress over range of shear rates in Viscometer.

Regression of data - Power Law:

Apparent viscosity at given shear rate:

nK

)1(nn

A KK


Yield Stress Fluids

More extreme non-Newtonian behavior.Fluid will not move until a certain level of shear stress is exerted.

No motion -No Shear Rate.

Shear stress at which motion starts is called the Yield Stress .

Two models for data:

n

YA

n

Y

YAY

KK


Design Problems

Power consumption by impellers in laminar regime:

Dimensionless blend time in the transitional regime:

Motion of Yield Stress fluids:

P

N

Y


Power Consumption in the Laminar Regime

Work of Metzner et al. (Late 50 s - Early 60 s).

Measure power consumption in the laminar regime with Newtonian fluids. Get values of KP for impellers tested.

Measure power consumption in non-Newtonian fluids. Value of KP allows apparent viscosity to be estimated.

32DN

PKP

32DNK

P

P

A


Power Consumption in the Laminar Regime

Having measured the power law constants, K and n:

Found that shear rate is proportional to impeller speed.

For turbines:

For helical ribbons:

1

1

nA

K

NkS

1410 Sk

D

ckS 14434


Power Calculation

In the laminar regime:

Check for Newtonian case. What happens when n = 1?

3)1()1(

)1(

32

)(

DNKkKP

NkK

DNKP

nn

SP

n

SA

AP


Blending in the Transitional Regime

Blending rate for vessel is determined by local blending rate in region of baffle.

What is the shear rate and apparent viscosity here?

Several papers published using Metzner s method:Does not work!

Gives shear rate in impeller region - not wall / baffle.

Does not account for processes in bulk of vessel.

Metzner s method only valid in laminar regime.

Need to estimate shear rate for:Wall / baffle region.

Transitional regime.


Force Balance

Impeller exerts force on fluid causing it to move.

Wall and baffles exert equal and opposite force.

From Transport Phenomena:

Assume that the shear rate at the wall does not vary with position:

ABaff

WS

ARpSr

vRTq dd

W

W r

v


Force Balance

Substituting equations:

Making assumptions about the fluid velocity at the baffles and integrating:

Having measured K and n:

ABaff

SW ARpSRTq dd

2

3)(0638.0

622.11

vT

TqW

)1(

1

n

WW

nW

W KK


Results for Pitched Blade Turbine - RMS


Correlation - Pitched Blade Turbine


Correlation - All Impellers


Blend Time Calculation

Use wall viscosity to calculate ReW and FoW.

Plot:

Compare with Newtonian data.

Excellent agreement.

Implications for impeller selection?

WW

W

FoRePo

ReN

/1 versus

versus

3/1


Impeller Selection

In the transitional regime:

For Newtonian fluids, no effect of impeller type.

For non-Newtonian fluids, high torque impeller (high Po) will give shorter blend time for equal power input.

3/2

3/23/2

3/1

1

186

TD

T

FoRePo

W

W

W


Why?

High torque per unit volume gives high shear stress (and rate) at wall:

Results in low apparent viscosity at the wall.

n

n

n

A

nn

W

W

T

Tq

T

Tq

K

TTq

T

Tq

)1(

3

)1(

1

3

13

3

/


Yield Stress Fluids

Important class of fluids for industry.

Fluid must experience certain shear stress before it will move:

Typical yield stress is 10 - 50 Pa.

At low shear rates, apparent viscosity is very high.

n

YA

n

Y

YAY

KK


Shear Stress - Shear Rate Models

These models are simply fits of data to a model.

No physical basis.

A yield stress fluid may be considered a shear-thinning fluid with very low n.

A shear-thinning fluid may be considered a yield stress fluid with a low yield stress.

Need to design mixers that work and yield stress model allows us to do this.


Observations

A cavern of moving fluid is created around the impeller.

A the boundary, the shear stress generated by the impeller is equal to the fluid s yield stress.

As impeller speed increases, cavern size increases.

Until cavern reaches wall:

Once cavern reaches wall, cavern grows axially.

CC xDH


Hirata & Aoshima, ChERD, 1996.


Cavern Diameter

Prediction of Cavern Diameter:

Minimum speed when cavern reaches vessel wall:Y

Y

YC

DNRe

RePoD

D

22

2

332.1

2

32

32.1 DPoD

TN

then

TD

When

YC

C


Growth of Cavern

Once cavern reaches vessel wall, it grows axially:

Need to compare power and speed of single impeller design versus multiple impellers (intersecting caverns).

Value of x dependent on impeller type:Rushton turbine: x = 0.40

Paddle: x = 0.45

Pitched blade turbine: x = 0.55

Propeller: x = 0.75

C

C

N

Nx

T

H


Shear-thinning vs. Yield stress Correlations

For yield stress fluids, when cavern reaches wall:

For shear-thinning fluids:

33

52

223

T

Tq

T

DNPo

DNPo

D

T

Y

Y

3T

TqW

ENGG 199 Reacting Flows Spring 2006 Lecture 2b Blending of Viscous, Non-Newtonian Fluids ·...

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Transcript of ENGG 199 Reacting Flows Spring 2006 Lecture 2b Blending of Viscous, Non-Newtonian Fluids ·...