Fluid Mechanics, KU, 2007 Chap. 4: Flow of Particulates Dimensional analysis & experimentation Physical phenomena do not depend on the frame of reference of the observer - Fixed particles and moving fluids … 1. Introduction 2. Flow past a sphere Drag coefficient Drag coefficient Steady motion of a spherical particle in an infinite expanse of Newt. fluid (No interaction btw spheres under volume fraction < 0.1) = Uniform motion of infinite fluid past a stationary sphere. Sphere velocity & force on the sphere ? - Net force on the sphere: viscous force & gravity

Fluid Mechanics, KU, 2007 From dimensional analysis: two independent dimensionless group 5 variables (N=5) 3 dimensions (D=3)  Indep. groups, G=N-D=2 Definition of friction factor: F k : force by the fluid motion A: characteristic area f: friction factor k: characteristic kinetic energy per unit volume For flow around stagnant particles, (Derive above groups using methods 1 & 2)

Fluid Mechanics, KU, 2007 C D -Re data - Stokes region: Re < 1 (inertialess) - Intermediate region: 1 < Re < Newton region: 10 3 < Re < 2  10 5 (Ch.12)

Fluid Mechanics, KU, 2007 Settling (terminal) velocity Settling (terminal) velocity Motion of the sphere in viscous fluid Settling velocity ? using C D -Re data - Stokes regime: Re < 1 - Intermediate regime: 1 < Re < Newtonian region: 10 3 < Re < 2  10 5 Note: Different diameter dependence of the velocity ! For Re < 1

Fluid Mechanics, KU, 2007 Device for removing solids from fluid streams Gravity acting on the particles Effective in particles, D p >0.042mm No interaction btw particles (particle vol.% <10%) Friction-sphere viscometer Friction-sphere viscometer Measurement of the terminal velocity of a sphere Elapsed time of a sphere flowing between distance “L” Ex. 4-1) Ex. 4-2) Separation of particles Separation of particles “Gravity settling chamber” Viscosity under the Stokes regime

Fluid Mechanics, KU, 2007 Calculation of the max. particle size passing through a chamber, without being removed. From A  B (for no eddying motions) (v p ~ indep. of H) Ex. 4-3)

Fluid Mechanics, KU, 2007 Sizing a distillation column Sizing a distillation column Plate distillation column No entrainment: settling velocity of the droplet = upward gas stream Newton regime: Wall effects Wall effects Influence of the container walls In Stokes regime, D p /D c <  unbounded with error 0.5% Ex. 4-4)

Fluid Mechanics, KU, 2007 General definition of C D : 3. Other submerged objects A p : projection of the solid object on a plane normal to the flow direction Long cylinder case: 4. Beds of particles Porous media: Porous media: for filtering, textiles, packed bed, … Flow of a fluid through a porous solid bed (e.g., packed tubular reactors…) Unconsolidated spherical particles with same D p - Particle diameter, D p & void fraction (porosity),  - Definition of D p : Eq. (4.38) & Re < 1:

Fluid Mechanics, KU, 2007 f-Re relation f-Re relation Determination of  P (velocity without particles) simplify From dimensional analysis - No information about  D eff & v eff measurable ? Derivation ?

Fluid Mechanics, KU, 2007 Packed bed friction factor: Packed-bed Reynolds number: Ergun Eq.: Neglected in the inviscid Newton region, Re p  1,000 Neglected at Re p  10 (inertialess) Ex. 4-6)

Fluid Mechanics, KU, 2007 Special Case: Re p  0 Indep. of fluid Operating variable Material (fluid) variable Darcy’s law,

Fluid Mechanics, KU, 2007 Chaotic movement of solid particles in the gas (or liq.) stream - Mixing/ particle-particle interaction/ particle-wall contact - Useful for highly exothermic chemical reactions Fluidized bed Fluidized bed Net upward force on the bed of particles: Net gravitational and buoyant force: (Vol. of solid Particles)

Fluid Mechanics, KU, 2007 “Weightless” state: - Free to move about unhindered by gravity - Minimum superficial velocity, v f : Point of incipient fluidization: Upper limiting velocity at which a fluidized bed can be operated. Maximum superficial velocity without particle entrainment, v max : = settling velocity for a sphere Ex. 4-7)