Settling

Dimensionless force versus Reynolds number for spherical particles

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:

${\displaystyle w={\frac {2(\rho _{p}-\rho _{f})gr^{2}}{9\mu }}}$

Deviation from the Stokes' Model from increased fluid drag as a particle increases in size.

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 ${\displaystyle Re\leq 0.1}$, within 3% for ${\displaystyle Re\leq 0.5}$ and within 9% ${\displaystyle Re\leq 1.0}$.[2] 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.

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

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

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 ${\displaystyle 0.2\leq Re\leq 1000}$:[3]

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

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.

Settleable solids are the particulates that settle out of a still fluid. Settleable solids can be quantified for a suspension using an Imhoff cone. The standard Imhoff cone of transparent glass or plastic holds one liter of liquid and has calibrated markings to measure the volume of solids accumulated in the bottom of the conical container after settling for one hour. A standardized Imhoff cone procedure is commonly used to measure suspended solids in wastewater or stormwater runoff. The simplicity of the method makes it popular for estimating water quality. To numerically gauge the stability of suspended solids and predict agglomeration and sedimentation events, zeta potential is commonly analyzed. This parameter indicates the electrostatic repulsion between solid particles and can be used to predict whether aggregation and settling will occur over time.

The water sample to be measured should be representative of the total stream. Samples are best collected from the discharge falling from a pipe or over a weir, because samples skimmed from the top of a flowing channel may fail to capture larger, high-density solids moving along the bottom of the channel. The sampling bucket is vigorously stirred to uniformly re-suspend all collected solids immediately before pouring the volume required to fill the cone. The filled cone is immediately placed in a stationary holding rack to allow quiescent settling. The rack should be located away from heating sources, including direct sunlight, which might cause currents within the cone from thermal density changes of the liquid contents. After 45 minutes of settling, the cone is partially rotated about its axis of symmetry just enough to dislodge any settled material adhering to the side of the cone. Accumulated sediment is observed and measured fifteen minutes later, after one hour of total settling time.[5]