SOLID PARTICLES IN A CARRYING LIQUID
INTRODUCTION: Forces acting on solid particles submerged in a liquid have their origin either in a particle-liquid interaction or in a particle-particle interaction. Particles moving in a conduit may also interact with a conduit boundary. The forces acting on a single particle in a dilute suspension are the body forces. The particle-liquid body’s forces are the buoyancy force drag force and lift force. When a solid particle is transported in the turbulent flow of a carrying liquid the turbulent diffusive force from carrier eddies is an additional particle-liquid force. Forces acting on solid particles due to particle-particle interaction are transmitted as the antiparticle stress via the particle contacts. Columbic stresses occur in a granular body occupied by particles in continuous contact. When a granular body is sheared and antiparticle contacts are only sporadic, Bagnold stresses are transmitted through the granular body.
Gravitational and buoyancy force: The body force due to gravitational acceleration is determined from the solid particle volume and density. The gravitational force on a spherical solid particle of diameter is
According Archimedes law, a solid particle immersed in a liquid obeys a buoyancy effect, which reduces its weight in the carrying medium. The submerged weight of the solid particle is a result of gravitational and buoyancy effects on the solid particle immersed in the liquid.
Drag force: When the surrounding liquid moves relative to a solid particle, an additional force is exerted from the liquid onto the submerged particle. The drag force, FD, acts in the direction of the relative velocity vr = vf - vsbetween the liquid and the solid particle. The magnitude of the drag force is expressed in terms of the drag coefficient CD. This comes from dimensional analysis of the function
FD = fn(ρf, µf, d, vr)
Lift force: The lift force, FL, on a single solid particle is a product of simultaneous slip (given by relative velocity vr = vf - vs) and particle rotation. The force (sometimes called the Magnus force) acts in a direction normal to both the relative velocity vr and the particle rotation vector. A particle rotation combined with a slip results in a lower hydrodynamic pressure in flow above the particle than in that below the particle. Lift force is due to this pressure gradient.
The lift force is most active near a pipeline wall where the velocity gradient is high. However, the lift forces due to particle spin play a minor role in the majority of mixture flow regimes compared to the Bagnold and Columbic forces.
Note: Lift force on a rotating solid body.
(a) Lift force on a rotating cylinder,
(b) The Saffman force, i.e. lift force due to shear and slip
Turbulent diffusive force: Solid particles are also subject to additional liquid-solids interactions when they are transported in a turbulent stream of the carrying liquid. An intensive exchange of momentum and random velocity fluctuations in all directions are characteristic of the turbulent flow of the carrying liquid in a pipeline. Scales of turbulence are attributed to properties of the turbulent eddies developed within the turbulent stream. According to Brandt’s picture of turbulence, the length of the turbulent eddy is given as the distance over which the lump of liquid transports its momentum without losing its identity, i.e. before the lump is mixed with liquid in a new location. This distance is called the mixing length and since it is supposed to represent a mean free path of a pulse of liquid within a structure of turbulent flow it is considered a length scale of turbulence. A turbulent eddy is responsible for the transfer of momentum and mass in a liquid flow. The instantaneous velocity of liquid at any point in the flowing liquid and in arbitrary direction (x, y or z) is given by v = v v' where v is the time-averaged velocity and v' is the instantaneous fluctuation velocity. The turbulent fluctuating component v' of the liquid velocity v is associated with a turbulent eddy.
It is well known that turbulent eddies are responsible for solid particle suspension. The intensity of liquid turbulence is a measure of the ability of a carrying liquid to suspend the particles. The size of the turbulent eddy and the size of the solid particle are also important to the effectiveness of a suspension mechanism. The characteristic size of turbulent eddies is assumed to depend on the pipeline diameter.