Fluid Mechanics for Chemical Engineering

Mathieu Mory, Université de Pau et des pays de l'Adour, France.

Preface xiii

PART I. ELEMENTS IN FLUID MECHANICS 1

Chapter 1. Local Equations of Fluid Mechanics 3

1.1. Forces, stress tensor, and pressure 4

1.2. Navier–Stokes equations in Cartesian coordinates 6

1.3. The plane Poiseuille flow 10

1.4. Navier–Stokes equations in cylindrical coordinates: Poiseuille flow in a circular cylindrical pipe 13

1.5. Plane Couette flow 17

1.6. The boundary layer concept 19

1.7. Solutions of Navier–Stokes equations where a gravity field is present, hydrostatic pressure 22

1.8. Buoyancy force 25

1.9. Some conclusions on the solutions of Navier–Stokes equations 26

Chapter 2. Global Theorems of Fluid Mechanics 29

2.1. Euler equations in an intrinsic coordinate system 30

2.2. Bernoulli’s theorem 31

2.3. Pressure variation in a direction normal to a streamline 33

2.4. Momentum theorem 36

2.5. Evaluating friction for a steady-state flow in a straight pipe 38

2.6. Pressure drop in a sudden expansion (Borda calculation) 40

2.7. Using the momentum theorem in the presence of gravity 43

2.8. Kinetic energy balance and dissipation 43

2.9. Application exercises 47

Exercise 2.I: Force exerted on a bend 47

Exercise 2.II: Emptying a tank 48

Exercise 2.III: Pressure drop in a sudden expansion and heating 48

Exercise 2.IV: Streaming flow on an inclined plane 49

Exercise 2.V: Impact of a jet on a sloping plate 50

Exercise 2.VI: Operation of a hydro-ejector 51

Exercise 2.VII: Bypass flow 53

Chapter 3. Dimensional Analysis 55

3.1. Principle of dimensional analysis, Vaschy–Buckingham theorem 56

3.2. Dimensional study of Navier–Stokes equations 61

3.3. Similarity theory 63

3.4. An application example: fall velocity of a spherical particle in a viscous fluid at rest 65

3.5. Application exercises 69

Exercise 3.I: Time of residence and chemical reaction in a stirred reactor 69

Exercise 3.II: Boundary layer on an oscillating plate 69

Exercise 3.III: Head capacity curve of a centrifugal pump 70

Chapter 4. Steady-State Hydraulic Circuits 73

4.1. Operating point of a hydraulic circuit 73

4.2. Steady-state flows in straight pipes: regular head loss 78

4.3. Turbulence in a pipe and velocity profile of the flow 81

4.4. Singular head losses 83

4.5. Notions on cavitation 87

4.6. Application exercises 88

Exercise 4.I: Regular head loss measurement and flow rate in a pipe 88

Exercise 4.II: Head loss and cavitation in a hydraulic circuit 89

Exercise 4.III: Ventilation of a road tunnel 91

Exercise 4.IV: Sizing a network of heating pipes 92

Exercise 4.V: Head, flow rate, and output of a hydroelectric power plant 93

4.7. Bibliography 93

Chapter 5. Pumps 95

5.1. Centrifugal pumps 96

5.2. Classification of turbo pumps and axial pumps 105

5.3. Positive displacement pumps 106

Chapter 6. Transient Flows in Hydraulic Circuits: Water Hammers 111

6.1. Sound propagation in a rigid pipe 111

6.2. Over-pressures associated with a water hammer: characteristic time of a hydraulic circuit 115

6.3. Linear elasticity of a solid body: sound propagation in an elastic pipe 118

6.4. Water hammer prevention devices 120

Exercise 121

Chapter 7. Notions of Rheometry 123

7.1. Rheology 123

7.2. Strain, strain rate, solids and fluids 126

7.3. A rheology experiment: behavior of a material subjected to shear 129

7.4. The circular cylindrical rheometer (or Couette rheometer) 132

7.5. Application exercises 136

Exercise 7.I: Rheometry and flow of a Bingham fluid in a pipe 136

Exercise 7.II: Cone/plate rheometer 137

PART II. MIXING AND CHEMICAL REACTIONS 139

Chapter 8. Large Scales in Turbulence: Turbulent Diffusion – Dispersion 141

8.1. Introduction 141

8.2. Concept of average in the turbulent sense, steady turbulence, and homogeneous turbulence 142

8.3. Average velocity and RMS turbulent velocity 145

8.4. Length scale of turbulence: integral scale 146

8.5. Turbulent flux of a scalar quantity: averaged diffusion equation 151

8.6. Modeling turbulent fluxes using the mixing length model 153

8.7. Turbulent dispersion 157

8.8. The k-ε model 159

8.9. Appendix: solution of a diffusion equation in cylindrical coordinates 163

8.10. Application exercises 165

Exercise 8.I: Dispersion of fluid streaks introduced into a pipe by a network of capillary tubes 165

Exercise 8.II: Grid turbulence and k-ε modeling 167

Chapter 9. Hydrodynamics and Residence Time Distribution – Stirring 171

9.1. Turbulence and residence time distribution 172

9.2. Stirring 178

9.3. Appendix: interfaces and the notion of surface tension 185

Chapter 10. Micromixing and Macromixing 193

10.1. Introduction 193

10.2. Characterization of the mixture: segregation index 195

10.3. The dynamics of mixing 198

10.4. Homogenization of a scalar field by molecular diffusion: micromixing 201

10.5. Diffusion and chemical reactions 202

10.6. Macromixing, micromixing, and chemical reactions 204

10.7. Experimental demonstration of the micromixing process 205

Chapter 11. Small Scales in Turbulence 209

11.1. Notion of signal processing, expansion of a time signal into Fourier series 210

11.2. Turbulent energy spectrum 213

11.3. Kolmogorov’s theory 214

11.4. The Kolmogorov scale 218

11.5. Application to macromixing, micromixing and chemical reaction 221

11.6. Application exercises 222

Exercise 11.I: Mixing in a continuous stirred tank reactor 222

Exercise 11.II: Mixing and combustion 223

Exercise 11.III: Laminar and turbulent diffusion flames 225

Chapter 12. Micromixing Models 229

12.1. Introduction 229

12.2. CD model 233

12.3. Model of interaction by exchange with the mean 245

12.4. Conclusion 250

12.5. Application exercise 251

Exercise 12.I: Implementation of the IEM model for a slow or fast chemical reaction 251

PART III. MECHANICAL SEPARATION 253

Chapter 13. Physical Description of a Particulate Medium Dispersed Within a Fluid 255

13.1. Introduction 255

13.2. Solid particles 257

13.3 Fluid particles 270

13.4. Mass balance of a mechanical separation process 273

Chapter 14. Flows in Porous Media 277

14.1. Consolidated porous media; non-consolidated porous media, and geometrical characterization 278

14.2. Darcy’s law 280

14.3. Examples of application of Darcy’s law 282

14.4. Modeling Darcy’s law through an analogy with the flow inside a network of capillary tubes 289

14.5. Modeling permeability, Kozeny-Carman formula 291

14.6. Ergun’s relation 293

14.7. Draining by pressing 293

14.8. The reverse osmosis process 298

14.9. Energetics of membrane separation 301

14.10. Application exercises 301

Exercise: Study of a seawater desalination process 301

Chapter 15. Particles Within the Gravity Field 305

15.1. Settling of a rigid particle in a fluid at rest 306

15.2. Settling of a set of solid particles in a fluid at rest 309

15.3. Settling or rising of a fluid particle in a fluid at rest 312

15.4. Particles being held in suspension by Brownian motion 315

15.5. Particles being held in suspension by turbulence 319

15.6. Fluidized beds 321

15.7. Application exercises 329

Exercise 15.I: Distribution of particles in suspension and grain size sorting resulting from settling 329

Exercise 15.II: Fluidization of a bimodal distribution of particles 330

Chapter 16. Movement of a Solid Particle in a Fluid Flow 331

16.1. Notations and hypotheses 332

16.2. The Basset, Boussinesq, Oseen, and Tchen equation 333

16.3. Movement of a particle subjected to gravity in a fluid at rest 336

16.4. Movement of a particle in a steady, unidirectional shear flow 339

16.5. Lift force applied to a particle by a unidirectional flow 341

16.6. Centrifugation of a particle in a rotating flow 350

16.7. Applications to the transport of a particle in a turbulent flow or in a laminar flow 355

Chapter 17. Centrifugal Separation 359

17.1 Rotating flows, circulation, and velocity curl 360

17.2. Some examples of rotating flows 364

17.3. The principle of centrifugal separation 377

17.4. Centrifuge decanters 381

17.5. Centrifugal separators 385

17.6. Centrifugal filtration 388

17.7. Hydrocyclones 391

17.8. Energetics of centrifugal separation 396

17.9. Application exercise 397

Exercise 17.I: Grain size sorting in a hydrocyclone 397

Chapter 18. Notions on Granular Materials 401

18.1. Static friction: Coulomb’s law of friction 402

18.2. Non-cohesive granular materials: Angle of repose, angle of internal friction 403

18.3. Microscopic approach to a granular material 405

18.4. Macroscopic modeling of the equilibrium of a granular material in a silo 407

18.5. Flow of a granular material: example of an hourglass 413

Physical Properties of Common Fluids 417

Index 419