Determining the Settling Velocity

First, a Little More Theory

Whenever you have have a particle moving through a horizontally-flowing fluid, confined within a certain space, a number of forces come into play. Gravity, the type of fluid involved, how smoothly the fluid is flowing, the size of the particles, the dimensions of the piping or chamber, the bouyancy force, and a few other negligible forces. All in all, there are quite a few things to look into.

First, there was a man named Stokes, who did a great deal of work in these areas. To find out more about him, go to your library or text and read about him. Basically, his one important contribution was a formula for calculating Drag Force in completely laminar environments, with perfectly spherical bodies. And for situations with basically spherical particles and mostly laminar flow, those equations work. The equation, when coupled with one other and trying to solve for Drag Coefficient, works out to this.

Drag Coefficient = 24 / Reynold's Number

However, we (usually) need to go a bit beyond that. I have tried to make this simple, I hope, and so I broke it down into three more subsections, each dealing with a certain type of settling.

Free Settling

The total amount of force exerted on a particle can be broken down into four categories.

Force due to Acceleration = Gravity Force - Buoyancy Force - Drag Force

I won't go into the derivation of these formula; I figure if you want to look for them, that's a project you can endeavor on yourself. The important information is this. For a particle, there are two stages when it falls. The acceleration portion and then the portion of constant velocity, also known as the terminal velocity or free settling velocity

The following is a diagram of the correlation between Reynold's Number and Drag Coefficient for Rigid Spherical Bodies. Any other type of particle has it's own special charts, but once more, I must forget those, due to space and time and scope considerations.

Free Settling, with Wall Effect

When the diameter of the particle becomes fairly noticeable with respect to the diameter of the container, the the particles tend to get forced away from the wall through something known, appropriately, as wall effect.

To compensate for this, you need only know whether or not the flow is laminar, as well as the diameter of the particle and the container. Below are two fudge factors, or correction factors, that you can multiply your previously calculated terminal velocities by in order to allow for wall effect. The most important ratio in this case is what I am terming DR, which is equal to the following.

DR = Particle Diameter / Container Diameter

For laminar flow, the correction factor is

k = 1 / (1 + 2.1 * DR )

And for turbulent flow, the correction factor is slightly altered.

k = ( 1 - DR² ) / ( square root of [ 1 + DR * DR * DR * DR ] )

Hindered Settling

Hindered settling is called hindered settling for a reason -- the added number of particles in an enclosed area creates a slower-moving mixture than would normally be expected.

In this case, everything revolves around epsilon (e), which is the volume fraction of the slurry mixture occupied by the liquid.

What comes from that is that yet another dimensionless variable, Psi, was created for the sole purpose of adding more confusion to this mess.

Psi = 1 / (10 raised to the 1.82*(1-epsilon) power)

Now, using this wonderful variable, we can calculate the effective viscosity of the mixture due to the hinderance of other particles.

effective viscosity = viscosity / Psi

And now the density of the fluid phase (rhom) is altered, which is now calculated as

rhom = epsilon * rho + (1 - epsilon)*rhop

Now we get to the important point. Calculating, as before, the terminal velocity, for laminar flow. If it does not fall into laminar flow, then other equations well beyond the scope of what I am doing here are needed.

Vt = g*Dp*Dp*(rhop - rho)*epsilon*epsilon*Psi / (18*viscosity)

(Here is a sketched formula for Stokes' Law. The notation is Vs for settling velocity instead of terminal velocity if you want to compare with the equation as typed by Chris Patillo that has other factors.)

To figure out whether the flow is laminar, use the following equation. If the value is below one, you are okay; the flow is laminar. If it is above one, then you need to look elsewhere.

Reynolds Number = Diameter of particle * Vt * rhom / (effective viscosity * epsilon)

How to measure settling velocity experimentally

You can measure the height of the clear liquid interface as it changes over time. After that, you can plot that. You get a plot something like this.

The average settling velocity for a particular plot at any given time is then equivalent to

settling velocity = (height at time 1 - original height) / (time required to reach current height)

Quick Conclusions

The fastest settling particles are huge, heavy, spherical molecules. The slowest settling particles, which sometimes cannot be settled accurately or properly, are tiny, light, irregularly shaped molecules. And for anything in between, here is a general guide as to what characteristics increase the rate of thickening.