Settling

Settling is the process by which particulates settle to the bottom of a liquid and form a sediment. ]Particles that experience a force, either due to gravity or due to centrifugal motion will tend to move in a uniform manner in the direction exerted by that force. For gravity settling, this means that the particles will tend to fall to the bottom of the vessel, forming a slurry at the vessel base.

Settling is an important operation in many applications, such as mining, wastewater treatment, biological science and particle mechanics.

Physics

Outline

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 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 settling velocity, fall velocity or terminal velocity of a the particle.

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.

Stoke's drag

For dilute suspensions, Stokes' Law predicts the settling velocity of small spheres in fluid, either air or water. Stokes' Law finds many applications in the natural sciences, and is given by:

w={\frac {2(\rho _{p}-\rho _{f})gr^{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 Stoke's law is found to hold within 1% for for $Re\leq 1$ , wihtin 3% for $Re\leq 0.5$ and within 9% $Re\leq 1.0$ ^[1]. With increasing Reynolds numbers, stokes law begins to break down due to fluid inertia, requiring the use of empirical solutions to calculate drag forces.

Newtonian drag

Defining a drag coefficient, $C_{d}$ , 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.

$C_{d}={\frac {A}{{\frac {1}{2}}\rho _{f}U^{2}}}$

In the Stoke's regime, for a spherical particle this value is not constant, however in the Newtonian drag regime $1000\leq Re_{p}\geq 2\times 10^{5}$ 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, indicating that the fluid viscosity plays a decreased role in the drag on the particle ^{[citation needed]}.

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

$w=1.75\left({\frac {(\rho _{p}-\rho _{f})gr}{\rho _{f}}}\right)^{-{\frac {1}{2}}}$

Transitional drag

In the intermediate region between Stoke's 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.

Non-ideal settling

Stoke's, transitional and newtonian settling describe the behaviour of a single spherical particle in an infinite fluid, however this model has limitations in practical applications. Typical practical applications of settling are for the settling of a polydisperse distribution of particle sizes, with non-spherical particles, possibly of differing densities, where the particles are dispersed in the fluid, but quickly come into contact upon settling to a bed at the base of the vessel.

As such these drag formulae can only be applied in limited situations. Semi-analytic or empirical solutions must be used to perform meaningful settling calculations.

Applications

Settleable solids are the particulates that settle out of a still fluid. Settleable solids can be quantified for a suspension using an Imhoff tank or cone.

Settling tanks are used for separating solids and/or oil from another liquid. In food processing, the vegetable is crushed and placed inside of a settling tank with water. The oil floats the top of the water then is collected. In water and waste water treatment a flocculant is often added prior to settling to form larger particles that settle out quickly in a settling tank leaving the water with a lower turbidity.

Winemaking

In winemaking, the French term for this process is débourbage. This step usually occurs in white wine production before the start of fermentation.^[2]

References

^ Martin Rhodes. Introduction to Particle Technology.
^ J. Robinson (ed) "The Oxford Companion to Wine" Third Edition pg 223 Oxford University Press 2006 ISBN 0198609906

External links

[Rhodes-1] Martin Rhodes. Introduction to Particle Technology.

[Oxford_pg_223-2] J. Robinson (ed) "The Oxford Companion to Wine" Third Edition pg 223 Oxford University Press 2006 ISBN 0198609906

[1]

[2]