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1 Preface... xiii PART I. ELEMENTS IN FLUID MECHANICS... 1 Chapter 1. Local Equations of Fluid Mechanics Forces, stress tensor, and pressure Navier Stokes equations in Cartesian coordinates The plane Poiseuille flow Navier Stokes equations in cylindrical coordinates: Poiseuille flow in a circular cylindrical pipe Plane Couette flow The boundary layer concept Solutions of Navier Stokes equations where a gravity field is present, hydrostatic pressure Buoyancy force Some conclusions on the solutions of Navier Stokes equations Chapter 2. Global Theorems of Fluid Mechanics Euler equations in an intrinsic coordinate system Bernoulli s theorem Pressure variation in a direction normal to a streamline Momentum theorem Evaluating friction for a steady-state flow in a straight pipe Pressure drop in a sudden expansion (Borda calculation) Using the momentum theorem in the presence of gravity Kinetic energy balance and dissipation Application exercises Exercise 2.I: Force exerted on a bend... 47

2 vi Fluid Mechanics for Chemical Engineering Exercise 2.II: Emptying a tank Exercise 2.III: Pressure drop in a sudden expansion and heating Exercise 2.IV: Streaming flow on an inclined plane Exercise 2.V: Impact of a jet on a sloping plate Exercise 2.VI: Operation of a hydro-ejector Exercise 2.VII: Bypass flow Chapter 3. Dimensional Analysis Principle of dimensional analysis, Vaschy Buckingham theorem Example the oscillating pendulum Dimensional study of Navier Stokes equations Similarity theory An application example: fall velocity of a spherical particle in a viscous fluid at rest Application of the Vaschy Buckingham theorem Forces exerted on the ball The hydrodynamic force opposing the particle s movement relative to the fluid Fall velocity for a small Reynolds number Fall velocity for a large Reynolds number Application exercises Exercise 3.I: Time of residence and chemical reaction in a stirred reactor Exercise 3.II: Boundary layer on an oscillating plate Exercise 3.III: Head capacity curve of a centrifugal pump Chapter 4. Steady-State Hydraulic Circuits Operating point of a hydraulic circuit Steady-state flows in straight pipes: regular head loss Turbulence in a pipe and velocity profile of the flow Singular head losses Notions on cavitation Application exercises Exercise 4.I: Regular head loss measurement and flow rate in a pipe Exercise 4.II: Head loss and cavitation in a hydraulic circuit Exercise 4.III: Ventilation of a road tunnel Exercise 4.IV: Sizing a network of heating pipes Exercise 4.V: Head, flow rate, and output of a hydroelectric power plant Bibliography... 93

3 vii Chapter 5. Pumps Centrifugal pumps Operating principle Similarity laws and head/capacity curves Implementation of a centrifugal pump Classification of turbo pumps and axial pumps Positive displacement pumps Chapter 6. Transient Flows in Hydraulic Circuits: Water Hammers Sound propagation in a rigid pipe Over-pressures associated with a water hammer: characteristic time of a hydraulic circuit Linear elasticity of a solid body: sound propagation in an elastic pipe Water hammer prevention devices Exercise Chapter 7. Notions of Rheometry Rheology Strain, strain rate, solids and fluids A rheology experiment: behavior of a material subjected to shear The circular cylindrical rheometer (or Couette rheometer) Application exercises Exercise 7.I: Rheometry and flow of a Bingham fluid in a pipe Exercise 7.II: Cone/plate rheometer PART II. MIXING AND CHEMICAL REACTIONS Chapter 8. Large Scales in Turbulence: Turbulent Diffusion Dispersion Introduction Concept of average in the turbulent sense, steady turbulence, and homogeneous turbulence Average velocity and RMS turbulent velocity Length scale of turbulence: integral scale Turbulent flux of a scalar quantity: averaged diffusion equation Modeling turbulent fluxes using the mixing length model Turbulent dispersion The k-ε model

4 viii Fluid Mechanics for Chemical Engineering 8.9. Appendix: solution of a diffusion equation in cylindrical coordinates Application exercises Exercise 8.I: Dispersion of fluid streaks introduced into a pipe by a network of capillary tubes Exercise 8.II: Grid turbulence and k-ε modeling Chapter 9. Hydrodynamics and Residence Time Distribution Stirring Turbulence and residence time distribution Notion of residence time distribution Modeling RTD via a turbulent diffusion approach: cases of a tubular reactor with axial dispersion and of a CSTR Stirring Mechanical characterization of a stirrer Stirring and mixing time Emulsions and foams Appendix: interfaces and the notion of surface tension Interface between two non-miscible fluids and surface tension Equilibrium in the contact line between three phases, Jurin s law Chapter 10. Micromixing and Macromixing Introduction Characterization of the mixture: segregation index The dynamics of mixing Homogenization of a scalar field by molecular diffusion: micromixing Diffusion and chemical reactions Macromixing, micromixing, and chemical reactions Experimental demonstration of the micromixing process Chapter 11. Small Scales in Turbulence Notion of signal processing, expansion of a time signal into Fourier series Turbulent energy spectrum Kolmogorov s theory The Kolmogorov scale Application to macromixing, micromixing and chemical reaction

5 ix Application exercises Exercise 11.I: Mixing in a continuous stirred tank reactor Exercise 11.II: Mixing and combustion Exercise 11.III: Laminar and turbulent diffusion flames Chapter 12. Micromixing Models Introduction CD model Principle CD model in a closed reactor without reaction CD model in an open reactor without reaction CD model in the presence of a chemical reaction Model of interaction by exchange with the mean Principle IEM model without a chemical reaction IEM model with a chemical reaction Conclusion Application exercise Exercise 12.I: Implementation of the IEM model for a slow or fast chemical reaction PART III. MECHANICAL SEPARATION Chapter 13. Physical Description of a Particulate Medium Dispersed Within a Fluid Introduction Solid particles Geometrical characterization of a particle Grain size distribution in a granular medium Determination of a solid s density using a pycnometer Concentrations Formation of clusters, coagulation, and flocculation Fluid particles Mass balance of a mechanical separation process Chapter 14. Flows in Porous Media Consolidated porous media; non-consolidated porous media, and geometrical characterization Darcy s law Examples of application of Darcy s law Laboratory permeameters

6 x Fluid Mechanics for Chemical Engineering Membrane resistance to filtration Dead-end filtration and cross-flow filtration Modeling Darcy s law through an analogy with the flow inside a network of capillary tubes Modeling permeability, Kozeny-Carman formula Ergun s relation Draining by pressing Draining the liquid Mechanical equilibrium of forces applied on the solid skeleton and on the liquid Force transmission in the structure Characteristic time of draining by pressing The reverse osmosis process Energetics of membrane separation Application exercises Exercise: Study of a seawater desalination process Chapter 15. Particles Within the Gravity Field Settling of a rigid particle in a fluid at rest Settling of a set of solid particles in a fluid at rest Settling or rising of a fluid particle in a fluid at rest Particles being held in suspension by Brownian motion Particles being held in suspension by turbulence Fluidized beds Flow regimes Mechanical equilibrium in a fluidized bed Fluid flow in a fluidized bed Application exercises Exercise 15.I: Distribution of particles in suspension and grain size sorting resulting from settling Exercise 15.II: Fluidization of a bimodal distribution of particles Chapter 16. Movement of a Solid Particle in a Fluid Flow Notations and hypotheses The Basset, Boussinesq, Oseen, and Tchen equation Movement of a particle subjected to gravity in a fluid at rest Movement of a particle in a steady, unidirectional shear flow Lift force applied to a particle by a unidirectional flow Lift force exerted on a particle in a fluid flow in an infinite medium Lift force exerted on a particle in the vicinity of a wall Centrifugation of a particle in a rotating flow

7 xi Applications to the transport of a particle in a turbulent flow or in a laminar flow Application to laminar flows Application to turbulent flows Chapter 17. Centrifugal Separation Rotating flows, circulation, and velocity curl Some examples of rotating flows Solid-body rotation in a rotating tank Vortex flow Flow in a hydrocyclone The principle of centrifugal separation Centrifuge decanters Discontinuous centrifuge decanters Continuous centrifuge decanters Centrifugal separators Centrifugal filtration Hydrocyclones Separation by a hydrocyclone of particles that are denser than the fluid Separation by a hydrocyclone of particles less dense than the fluid Energetics of centrifugal separation Application exercise Exercise 17.I: Grain size sorting in a hydrocyclone Chapter 18. Notions on Granular Materials Static friction: Coulomb s law of friction Non-cohesive granular materials: Angle of repose, angle of internal friction Microscopic approach to a granular material Macroscopic modeling of the equilibrium of a granular material in a silo Flow of a granular material: example of an hourglass Physical Properties of Common Fluids Index

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