Settling

Physics

For settling particles that are considered individually, i.e. dilute particle solutions, there are two main forces enacting upon any particle. The primary force is an applied force, such as gravity, and a drag force that is due to the motion of the particle through the fluid. The applied force is usually not affected by the particle's velocity, whereas the drag force is a function of the particle velocity.

For a particle at rest no drag force will be exhibited, which causes the particle to accelerate due to the applied force. When the particle accelerates, the drag force acts in the direction opposite to the particle's motion, retarding further acceleration, in the absence of other forces drag directly opposes the applied force. As the particle increases in velocity eventually the drag force and the applied force will approximately equate, causing no further change in the particle's velocity. This velocity is known as the terminal velocity, settling velocity or fall velocity of the particle. This is readily measurable by examining the rate of fall of individual particles.

The terminal velocity of the particle is affected by many parameters, i.e. anything that will alter the particle's drag. Hence the terminal velocity is most notably dependent upon grain size, the shape (roundness and sphericity) and density of the grains, as well as to the viscosity and density of the fluid.

Single particle drag

Stokes' drag

For dilute suspensions, Stokes' law predicts the settling velocity of small spheres in fluid, either air or water. This originates due to the strength of viscous forces at the surface of the particle providing the majority of the retarding force. Stokes' law finds many applications in the natural sciences, and is given by:

${\displaystyle w=\frac{2({\rho }_p-{\rho }_f)g{r}^2}{9\mu }}$

where w is the settling velocity, ρ is density (the subscripts p and f indicate particle and fluid respectively), g is the acceleration due to gravity, r is the radius of the particle and μ is the dynamic viscosity of the fluid.

Stokes' law applies when the Reynolds number, Re, of the particle is less than 0.1. Experimentally Stokes' law is found to hold within 1% for ${\displaystyle Re\le 0.1}$, within 3% for ${\displaystyle Re\le 0.5}$ and within 9% ${\displaystyle Re\le 1.0}$. With increasing Reynolds numbers, Stokes law begins to break down due to the increasing importance of fluid inertia, requiring the use of empirical solutions to calculate drag forces.

Newtonian drag

Defining a drag coefficient, ${\displaystyle C_d}$, as the ratio of the force experienced by the particle divided by the impact pressure of the fluid, a coefficient that can be considered as the transfer of available fluid force into drag is established. In this region the inertia of the impacting fluid is responsible for the majority of force transfer to the particle.

${\displaystyle C_d=\frac{F_d}{\frac{1}{2}{\rho }_f{U}^{2}A}}$

For a spherical particle in the Stokes regime this value is not constant, however in the Newtonian drag regime the drag on a sphere can be approximated by a constant, 0.44. This constant value implies that the efficiency of transfer of energy from the fluid to the particle is not a function of fluid velocity.

As such the terminal velocity of a particle in a Newtonian regime can again be obtained by equating the drag force to the applied force, resulting in the following expression

${\displaystyle w=2.46{\left(\frac{({\rho }_p-{\rho }_f)gr}{{\rho }_f}\right)}^{\frac{1}{2}}.}$

Transitional drag

In the intermediate region between Stokes drag and Newtonian drag, there exists a transitional regime, where the analytical solution to the problem of a falling sphere becomes problematic. To solve this, empirical expressions are used to calculate drag in this region. One such empirical equation is that of Schiller and Naumann, and may be valid for ${\displaystyle 0.2\le Re\le 1000}$:

${\displaystyle \frac{F}{{\rho }_f{U}^{2}A}=\frac{12}{\mathrm{R}\mathrm{e}}\left(1+0.15{\mathrm{R}\mathrm{e}}^{0.687}\right).}$

Hindered settling

Stokes, transitional and Newtonian settling describe the behaviour of a single spherical particle in an infinite fluid, known as free settling. However this model has limitations in practical application. Alternate considerations, such as the interaction of particles in the fluid, or the interaction of the particles with the container walls can modify the settling behaviour. Settling that has these forces in appreciable magnitude is known as hindered settling. Subsequently, semi-analytic or empirical solutions may be used to perform meaningful hindered settling calculations.