Globex Julmester 2017Lecture #3

05 July 2017

Modelling of dispersed, multicomponent, multiphase flows in resource industries

Section 4: Non-Newtonian fluids and rheometry(PART 1)

Agenda – Lecture #3

• Section 4: Examples of analyses conducted for Newtonian fluids– 4.1 Motivation & context– 4.2 Lecture 1 review: Non-Newtonian behavior– 4.3 Objectives: what do we wish to provide through

the study of this material?– 4.4 Different approaches / philosophies– 4.5 Useful measurement devices

• Pipeline (tube) viscometers• Rotational devices• Yield stress measurements

4.1 Context and motivation

• Many industrial mixtures cannot reasonably be assumed to exhibit Newtonian behavior

• The design, control and optimisation of such processes cannot be properly achieved if one does not have a realistic, quantitative description of the fluid’s non-Newtonian behavior

• Therefore, it is critical that one knows how high-quality rheometrymeasurements are conducted

• Even if one never conducts rheometry measurements themselves, they still must be able to assess the quality and suitability of measurements that someone else has made!

Non-Newtonian fluid behaviour

Visco-elasticViscous

Time-dependent Time-independent

Reversible Irreversible

Viscoplastic

Polymer solutions

Bread dough

“Silly putty”

Bingham model

HB modelNewtonian fluids

Power Law model

4.2. Lecture 1 review

Non-Newtonian fluid behaviourSh

ear s

tress

()

Rate of shear (du/dy)

Newtonian

Bingham

Pseudoplastic

Dilatant

Some time-independent rheology models

B

Or “shear-thinning”

Or “shear-thickening”p

4.2 Lecture 1 review

Example 4.1Consider the figure given below, which shows the behaviour of a sample of “red mud” (tailings sample from the alumina/bauxite industry in Australia). Would you characterize this as “TIME-DEPENDENT” or “TIME-INDEPENDENT” behaviour? Justify your answer.

Figure from “Non-Newtonian Flow in the Process Industries”, by RP Chhabra and JF Richardson (1999). Boston: Butterworth-Heinemann.

Example 4.2For the figure shown below, suggest which rheological model would best describe the rheogram (shear stress vs. shear rate curve) for the “Meat Extract”. Would the same model also fit the Carbopolrheogram? If not, which would most likely be a better fit?

Figure from “Non-Newtonian Flow in the Process Industries”, by RP Chhabra and JF Richardson (1999). Boston: Butterworth-Heinemann.

Example 4.3Which rheological model(s) would best fit the data shown in the figure, below? Explain / justify your answers.

Figure from “Non-Newtonian Flow in the Process Industries”, by RP Chhabra and JF Richardson (1999). Boston: Butterworth-Heinemann.

4.3. Objectives• To provide an engineering-based introduction to

rheology measurement techniques and data analysis• To review the principles of operation of the most useful

“rheometers”• To identify the most basic (but most critical) issues that

often arise in making rheology measurements

4. Non-Newtonian fluids and rheometry

4.4 Different approaches / philosophies• Use a constitutive rheological model• Use apparent viscosity• Use a rheogram• Use only yield stress

4. Non-Newtonian fluids and rheometry

4.5 Useful measurement devices• Pipeline (“tube”) viscometer• Rotational viscometers

– Parallel plate– Cone and plate– Concentric cylinder

• Vane shear measurements

4. Non-Newtonian fluids and rheometry

D = 25 mm pipe viscometer P

Flow meter

Pump

Heat exchangers

P

Slide courtesy of Saskatchewan Research Council Pipe Flow Technology Centre

Integrated equations for laminar pipe flow

V (m/s)

dP/d

z (P

a/m

)(1/s)

rz

(Pa)

?

Lecture #2 Review: Laminar, Newtonian flow

• Integrated equations give:– Velocity profile: uz (r)– Wall shear stress, w:

2w

z 2R ru 1

2 R

zrz

dudr

+ rz

w

rR

zA

Q u dA + QVA

w

8 VD

Newton’s Law of Viscosity

Shear stress decay law

Poiseuille’sEquation (3.8)

(3.9)

zrz rz

dudr

zrz B p

dudr

nz

rzduKdr

Newtonian fluid

Useful rheology models--- written here for pipe flow ---

Bingham fluid

Power Law fluid

Yield-Pseudoplastic fluid

or Ostwald de Waele model

or Herschel-Bulkley model

nz

rz HduKdr

Casson fluid 21 2 1 2 zrz c

dudr

(4.1)

(4.2)

(4.3)

(4.4)

(4.5)

Laminar, non-Newtonian pipeline flows• Integrated equations developed in the same way as for

the Newtonian, laminar flow case (see Lecture #2)

• Friction losses: use the integrated form of the selected rheology model

This applies to any rheological model…but let’s use the Bingham model as an example

zrz B p

dudr

(4.3)

Bingham fluid pipe flow behaviour

Rr

ττ

w

rz Shear Stress Decay Law:

y/D

uz

y/D

rzw

w

The integrated equation for the Bingham fluid model (laminar flow)

4w B

p w

8V 4 11 ;D 3 3

This is called the Buckingham equation!

zrz B p

dudr

rz

w

rR

+

Integrate…not so easy this time…

(4.6)

Pipeline flow of a Bingham fluid (laminar flow)

0

2

4

6

8

10

12

14

16

18

20

0.0 0.5 1.0 1.5

Wal

l She

ar S

tres

s, w

(Pa)

Bulk Velocity, V (m/s)

Bingham Fluid Model Parameters:Pipe Diameter, D (m) 0.100Wall Roughness, k (mm) 0.045Density, (kg/m3) 1200Yield Stress, y (Pa) 15.0Plastic Viscosity, p (mPa.s) 10.0

V

Example 4.4Flow through a horizontal, 50 mm (diameter) pipeline is driven by a constant-speed positive displacement pump, such that the pressure gradient is always 1.58 kPa/m. If a Newtonian fluid ( = 65 mPas; = 1100 kg/m3) is pumped through the line, what will the operating velocity be?

What will the operating velocity be if a homogeneous mixture exhibiting Bingham fluid properties (p = 65 mPas; y = 10Pa) is pumped through the line? Assume the flow is laminar.

The integrated equations for laminar pipe flow

Newtonian w8VD

4w B

p w

8V 4 11 ;D 3 3

Bingham

1n

w8V 4nD 3n 1 K

Power Law

Casson1 2 4 cw

w

8V 16 4 11 ;D 7 3 21

Hershel-Bulkley 1 n 2

a b cw

H

w

8V 2 14 1 1 1 ;D K a b c

1 1 1; a 1 ; b 2 ; c 3n n n

(4.9)

(4.8)

(4.6)

(4.7)

(3.9)

Interpretation of data

• Data regression (more work, more accurate)– Example 4.5

• Trial-and-error (less work, less accurate)– We will demonstrate this later in the course

Example 4.5

A mixture of wood fibre and water (“pulp”) was tested in a 25 mm tube viscometer at 50°C.

Select the appropriate rheology model and then determine the best-fit values of the model parameters.

Notes:(i) The mixture density is 1105 kg/m3

V (m/s) -(dP/dz)f

(Pa/m)

0.25 770

0.65 1360

1.00 1830

1.30 2110

1.85 2640

2.20 2910

0

2

4

6

8

10

12

14

16

18

20

0 100 200 300 400 500 600 700 800

Wall she

ar stress (P

a)

8V/D (1/s)

Ex. 4.5: SolutionStep 1: Plot the data

w

Models we might try: (i) Pseudoplastic; (ii) Bingham

Ex. 4.5: SolutionTry the pseudoplastic model:

We now use a power law curve fit (regression) to obtain values of K´ and n´

or

n n

w3n 1 8VK

4n D

n

w8VKD

Step 2: rewrite Eqn (4.7) as

Ex. 4.5: SolutionUsing a power law regression curve:

n´ = 0.620

n

w8VKD

K´ = 0.3135

0

2

4

6

8

10

12

14

16

18

20

0 100 200 300 400 500 600 700 800

Wall she

ar stress (P

a)

8V/D (1/s)

Ex. 4.5: SolutionSince:

andn n

w3n 1 8VK

4n D

n

w8VKD

n n

n3n 1K K4n

Therefore:

n = 0.620

K = 0.287 Pa.sn

Assignment #2 – due 1:00pm, Mon 10 July(Total = 30 marks)Prepare an Excel spreadsheet to calculate the bulk velocity of a homogeneous Bingham slurry, in laminar pipe flow, as a function of wall shear stress.

The spreadsheet should be designed such that the following inputs can be easily specified by the user: pipe diameter (D); slurry density (m); Bingham plastic viscosity (p) and Bingham yield stress (B).

The spreadsheet should provide to the user: a graph of wall shear stress (w) on the y-axis against average velocity (V) on the x-axis.

IMPORTANT NOTE: The maximum value of the average velocity shown on your graph should be Vt, the laminar-to-turbulent transition velocity. You will have to find an expression from the literature that allows you to predict Vt.

Hint: you cannot use the Newtonian version of the Reynolds number for the prediction of Vt!

Please email a copy of your spreadsheet to rseans7955@yahoo.comno later than the assignment submission date and time.

Pipeline flow of a Bingham fluid (laminar flow)

0

2

4

6

8

10

12

14

16

18

20

0.0 0.5 1.0 1.5

Wal

l She

ar S

tres

s, w

(Pa)

Bulk Velocity, V (m/s)

Bingham Fluid Model Parameters:Pipe Diameter, D (m) 0.100Wall Roughness, k (mm) 0.045Density, (kg/m3) 1200Yield Stress, y (Pa) 15.0Plastic Viscosity, p (mPa.s) 10.0

V