Particle (s) motion

OBJECTIVES

Solid-Liquid Flow Conveying of solid by a fluid in a pipe can involve a wide range of flow conditions and phase distributions, depending on the density, viscosity, and velocity of the fluid and the density, size, shape and concentration of the solid particle.

Acting Forces on Particle All particles immersed in a fluid are subject to a buoyancy force, B A gravitational force G is always acting on the particle due to its mass Whenever relative motion exists between a particle and a surrounding fluid, the fluid will exert a drag upon the particle creating a drag force, F Flow direction B G F In a flowing viscous fluid, the drag force is made up of two components: A pressure drag force, Fp A shear stress drag force, Fs B Flow direction

F = Fs + Fp = 2Dv + Dv = 3Dv For creeping flow (flow at very low velocities, v relative to the sphere), the total drag force F on the particle with a diameter D in a fluid of viscosity  is given by Stoke’s law: F = Fs + Fp = 2Dv + Dv = 3Dv B creeping flow

Reynolds Number, Re To characterize the fluid flows on particle Dimensionless group Definition: : density of the fluid : viscosity of the fluid v : the velocity of the fluid relative to the particle D : the diameter of the particle

Drag Force on a Spherical Particle The most satisfactory way of representing the relation between drag force and velocity involves the use of two dimensionless groups: Reynolds number, Re Drag coefficient, CD Stokes Law R’ is the force per unit projected area of particle in a plane perpendicular to the direction of motion

show that Drag coefficient, CD has the following formula for stokes law (laminar region) Turbulent Flows: Turbulent Turbulence may arise either from an increased fluid velocity or from artificial roughening of the forward face of the immersed body (or particle)

Drag Coefficients for Different Regions of Re Region (a) (10-4 < Re < 0.3) : Stoke’s Law Region (b) (0.3 < Re < 500) : Intermediate region Region (c) (500 < Re < 2 x 105) : Newton’s Law Region (d) (Re > 2 x 105): Boundary layer changes from laminar to turbulent

Total Drag Force on Particle for Different Regions of Re Region (a) (10-4 < Re < 0.3) : Stoke's Law Region (b) (0.2 < Re < 500) : Intermediate region Region (c) (500 < Re < 2 x 105) : Newton’s Law Region (c) (Re > 2 x 105): Boundary layer changes from laminar to turbulent

Terminal Falling Velocity B If a spherical particle is allowed to settle in a fluid under gravity, its velocity will increase until the accelerating force is exactly balanced by the resistance force In general, the forces of buoyancy, drag and gravity act on the particle: gravity – buoyancy – drag = acceleration force In balance condition:

Assumptions That settling is not affected by the presence of other particles in the fluid (or “Free Settling”) That walls of the containing vessel do not exert an appreciable retarding effect

Non-spherical Particles The effect of the shape of non-spherical particles on their drag coefficient is described by their sphericity, the ratio of the surface area of particle and the surface area of sphere of equal volume) If the fluid is moving relative to some surface other than that of the particle, there will be a superimposed velocity distribution and the drag on the particle may be altered. The drag force is then determined by the difference in the velocities of the fluid and the particle at the axis

Cases

x = D = Diameter