Modelling of dispersed, multicomponent, multiphase flows in resource industries Section 4: Non-Newtonian fluids and rheometry (PART 1)

Transcription

1 Modelling of dispersed, multicomponent, multiphase flows in resource industries Section 4: Non-Newtonian fluids and rheometry (PART 1) Globex Julmester 2017 Lecture #3 05 July 2017

2 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

3 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 rheometry measurements 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!

4 4.2. Lecture 1 review Non-Newtonian fluid behaviour Time-dependent Time-independent Reversible Irreversible Viscous Newtonian fluids Power Law model Viscoplastic Bingham model HB model Visco-elastic Polymer solutions Bread dough Silly putty

5 4.2 Lecture 1 review Non-Newtonian fluid behaviour Some time-independent rheology models B Bingham Shear stress () Pseudoplastic p Or shear-thinning Dilatant Newtonian Or shear-thickening Rate of shear (du/dy)

6 Example 4.1 Consider 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.

7 Example 4.2 For 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 Carbopol rheogram? 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.

8 Example 4.3 Which 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.

9 4. Non-Newtonian fluids and rheometry 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

10 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

11 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

12 D = 25 mm pipe viscometer P Heat exchangers P Flow meter Slide courtesy of Saskatchewan Research Council Pipe Flow Technology Centre Pump

13 Integrated equations for laminar pipe flow rz (Pa)? dp/dz (Pa/m) (1/s) V (m/s)

14 Lecture #2 Review: Laminar, Newtonian flow Integrated equations give: Velocity profile: u z (r) Wall shear stress, w : du dr z rz + Newton s Law of Viscosity rz w r R Shear stress decay law R r u 1 2 R 2 w z 2 Poiseuille s Equation (3.8) Q u da + V Q A A z w 8 V D (3.9)

15 Useful rheology models --- written here for pipe flow --- Newtonian fluid Power Law fluid or Ostwald de Waele model Bingham fluid Yield-Pseudoplastic fluid or Herschel-Bulkley model duz rz rz dr n duz rz K dr duz rz B p dr du rz c dr du rz H K dr Casson fluid z (4.1) (4.2) (4.3) z (4.4) n (4.5)

16 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 duz rz B p dr (4.3)

17 Bingham fluid pipe flow behaviour y/d Shear Stress Decay Law: y/d τ τ rz w r R w rz u z w

18 The integrated equation for the Bingham fluid model (laminar flow) duz rz B p dr + rz w r R Integrate not so easy this time 8V 4 1 D 3 3 w 4 1 ; p B w (4.6) This is called the Buckingham equation!

19 20 18 Wall Shear Stress, w (Pa) V Bulk Velocity, V (m/s) Pipeline flow of a Bingham fluid (laminar flow) Bingham Fluid Model Parameters: Pipe Diameter, D (m) Wall Roughness, k (mm) Density, (kg/m 3 ) 1200 Yield Stress, y (Pa) 15.0 Plastic Viscosity, p (mpa.s) 10.0

20 Example 4.4 Flow through a horizontal, 50 mm (diameter) pipeline is driven by a constantspeed positive displacement pump, such that the pressure gradient is always 1.58 kpa/m. If a Newtonian fluid ( = 65 mpas; = 1100 kg/m 3 ) 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.

21 The integrated equations for laminar pipe flow Newtonian (3.9) Bingham (4.6) Power Law (4.7) 8V w D 8V w ; D 3 3 p 1 4n n w 8V D 3n 1 K B w Casson (4.8) Hershel-Bulkley (4.9) 8V ; D w 12 4 c 8V 2 1 D K a b c ; a 1 ; b 2 ; c 3 n n n 1n 2 w a b c ; H w w

22 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

23 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/m 3 V (m/s) -(dp/dz) f (Pa/m)

24 Ex. 4.5: Solution Step 1: Plot the data w Wall shear stress (Pa) V/D (1/s) Models we might try: (i) Pseudoplastic; (ii) Bingham

25 Ex. 4.5: Solution Try the pseudoplastic model: Step 2: rewrite Eqn (4.7) as 3n 1 8V K 4n D w n n or n 8V w K D We now use a power law curve fit (regression) to obtain values of K and n

26 Ex. 4.5: Solution Using a power law regression curve: Wall shear stress (Pa) V/D (1/s) n 8V w K D n = K =

27 Ex. 4.5: Solution Since: 3n 1 8V K 4n D w n n and n 8V w K D n n 3n 1 K K 4n n Therefore: n = K = Pa.s n

28 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 V t, the laminar-to-turbulent transition velocity. You will have to find an expression from the literature that allows you to predict V t. Hint: you cannot use the Newtonian version of the Reynolds number for the prediction of V t! Please a copy of your spreadsheet to rseans7955@yahoo.com no later than the assignment submission date and time.

29 20 18 Wall Shear Stress, w (Pa) V Bulk Velocity, V (m/s) Pipeline flow of a Bingham fluid (laminar flow) Bingham Fluid Model Parameters: Pipe Diameter, D (m) Wall Roughness, k (mm) Density, (kg/m 3 ) 1200 Yield Stress, y (Pa) 15.0 Plastic Viscosity, p (mpa.s) 10.0