MEDIAN (D50) CUT POINTS

In spray discharge, for any set of cone diameter, feed slurry compositions,
flow properties, volumetric flow rates, and pressure conditions, some particles’ size (mass) is 100% discarded from the apex.

For every size and design of cone operating at a given pressure with
feed slurry of a given viscosity, density, and solids distribution, there is
a certain size (mass) of particle that shows no preference for either top or
bottom discharge. As a result, 50% of this particular size exits through
the vortex and 50% exits through the apex. This particle size is termed
the median cut, the median-size particle, or more frequently in drilling
operations, the D50 cut point.

The median cut, or D50 cut point, does not mean that all larger
particles exit at the apex and smaller particles exit at the vortex. The D50
cut point of a solids-separation device is defined as that particle size
at which one half of the weight of specific-size particles go to the
underflow (discard) and one half of the weight go to the overflow
(returned to the active system). For example, a D30 cut point references
a particle size that is concentrated 30% in the underflow and 70% in the
overflow.

As stated earlier, the cut point is related to the inside diameter of the
hydrocyclone. For example, a 12-inch cone has a D50 cut point for lowgravity solids in water of approximately 60 to 80 microns; a 6-inch cone, around 30 to 60 microns, and a 4-inch cone, around 15 to 20 microns (Table 1.). However, the cut point will vary with the size and amount of solids in the feed, as well as fluid viscosity.

table1.Hydrocyclone Size Versus D50 Cut Point (CP)

Cone Diameter (in.)	D50 CP in Water	D50 CP in Drilling Fluid
2	8-10	15
4	15-20	35-70
6	30-35	70-100
12	60-70	200+

For comparative purposes, consider a 50-micron-equivalent drilledsolid
diameter. Relatively speaking, the percentage of discharge is as follows:

. A 6-inch cone discharges 80% at underflow.
. A 4-inch cone discharges 95% at underflow.
. A 3-inch cone discharges 97% at underflow.

Now consider a 10-micron-equivalent drilled-solid diameter:

. A 6-inch cone discharges 7% at underflow.
. A 4-inch cone discharges 11% at underflow.
. A 3-inch cone discharges 17% at underflow.

If a graph of particle size versus percentage of particles recovered to underflow is plotted, the portion of the curve near the D50, or 50%, recovery point (median cut point) is very steep when separations are efficient.

Stokes’ Law

Stokes’ law defines the relationship between parameters that control the
settling velocity of particles in viscous liquids, not only in settling pits but
also in equipment such as hydrocyclones and decanter centrifuges.

The force of gravity and the viscosity of the suspending fluid (drilling
mud) control separation in a settling pit. A large, heavy particle settles
faster than a small, light particle. The settling process can be increased by reducing the viscosity of the suspending fluid, increasing the gravitational
forces on the particles, or increasing the effective particle size with flocculation or coagulation.

Hydrocyclones and centrifuges increase settling rates by application
of increased centrifugal force, equivalent to higher gravity force. Stokes’
law for the settling of spherical particles in a viscous liquid applies:

Vs=CgD²e(ρs-ρl)⁄µ

. Vs = Settling or terminal velocity, in ft/sec
. C = Units constant, 2.15 107
. g = Acceleration (gravity or apparatus), in ft/sec2
. DE = Particle equivalent diameter, microns
. s = Specific gravity of solids (cutting, barite, etc.)
. L = Specific gravity of liquid phase
. µ = Viscosity of media, centipoise

Particles of differing densities and varying sizes can have the same settling rates. That is, there exists an equivalent diameter for every 2.65-SG drilled solid, be it limestone, sand, or shale, that cannot be separated by gravimetric methods from barite particles of corresponding equivalent
diameter. Presently, it is not possible to separate desirable barite particles
from undesirable drilled-solid particles that settle at the same rate.

The recognized rule of thumb is: A barite particle (SG 4.25) will settle
at the same rate as a drilled-solids particle (SG 2.65) having 1½ times the
barite particle’s diameter. This rule of thumb may be verified applying
Stokes’ law.