The Laminar Regime (Best known as Stokes Law; for Re < 2)
The Intermediate Regime (Best known as the Transition Regime Law, for 2 <Re <500)
The Turbulent Regime (Best known as Newton's Law, for 500 < Re < 200,000)
From this point on, it's some very simple algebra. You can solve equation 5 for us, and then substitute an expression in for CD for each of the three flow regimes (laminar, turbulent and intermediate). You can work through the details, but the working expressions are summarized for you in the sidebar discussion on this page. Remember that to apply these equations you need to know the flow regime, and so you need to make some assumptions when applying any one of these expressions. We will get to this in a moment. One point we can make is that the expressions can be further developed to give us an idea of the maximum size particles that will settle out in the first two regimes of settling. If we take Stokes law for example, the maximum size particle whose velocity follows Stokes' law can be found by substituting /¿Re/dpP for the settling velocity into the first sidebar equation, and then setting Re = 2 (the limiting Reynolds number value for the flow regime). This then gives us the following useful expression:
The minimum size particles that do not follow Stokes' law occurs at Re ~ 10"4. The settling velocity in this lower bound regime is less than that computed by the Stokes' Law expression, and generally an empirical correction factor is applied to account for particle slippage. This correction factor, which is applied by dividing the value into the Stokes' law calculated us value is: K= 1 + A A./d, where X is defined as the mean free path of a fluid molecule, and constant A varies between 1.4 and 20 (as a point of reference, A = 1.5 for air). But this is a correction factor we will likely never have to consider in a conventional water treatment assignment. A more convenient set of expressions for settling velocity can be derived by expressing the three settling regime equations in terms of dimensionless groups. We won't get tangled in the derivations, although they are reasonably straightforward, but rather just list these expressions. The relationships are based on the dimensionless Archimedes number, defined as:
Note that the Archimedes number is a dimesnionless group that describes the physical properties of the heterogeneous system. It can be related to the Reynolds number (and hence the settling velocity, us) for each settling regime as follows: For the Stake's settling regime:
Note that the upper limiting or critical value of the Archimedes number for this range occurs at Re = 2, and hence Arcr [ = 18 X 2 = 36. This means that the laminar settling regime corresponds to Ar < 36.
For the intermediate settling regime, where 2 < Re < 500, we have the following expression:
For the critical value Re = 500, the limiting value of Ar for the intermediate settling regime is Arcr 2 = 83,000. In other words, the intermediate settling regime corresponds to 36 < Ar < 83,000.
For the Newton's law (turbulent settling regime) region, where Ar > 83,000, the expression of interest is:
The usefulness of these relationships lies in the recognition that by evaluating the Archimedes number, we can establish the theoretical settling range for the particles we are trying to separate out of a wastewater stream. This very often gives us a starting point for evaluating the settling characteristics of suspended solids for dilute systems. Note that from the definition of the Reynolds number, we can readily determine the settling velocity of the particles from the application of the above expressions (us = ¿tRe/dpP). The following is an interpolation formula that can be applied over all three settling regimes:
For low values of Ar, the second term in the denominator may be neglected, and equation 11 simplifies to equation 8; at high Ar values, we may neglect the first term in the denominator and the expression simplifies to equation 10, which corresponds to the Newton's law range.
The settling velocity of a nonspherical particle is less than that of a spherical one. A good approximation can be made by multiplying the settling velocity, us, of spherical particles by a correction factor, , called the sphericity factor. The sphericity, or shape factor is defined as the area of a sphere divided by the area of the nonspherical particle having the same volume:
The factor < 1 must be determined experimentally for particles of interest. Typical values are ij; = 0.77 for particles of rounded shape; ij; = 0.66 for particles of angular shape; = 0.43 for particles of a flaky geometry. The above analysis applies only to the free settling velocities of single particles and does not account for particle-particle interactions. Hence, the application of these formulas only applies to very dilute systems. At high particle concentrations, mutual interference in the motion of particles exists, and the rate of settling is considerably less than that computed by the given expressions. In the latter case, the particle is settling through a suspension of particles in a fluid, rather than through a simple fluid medium.
The above provides us with a theoretical staring basis for particle settling. Let's now take a closer look at some of the standard hardware.
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