Sedimentation and Sedimentary Rocks
Stokes Settling Velocities
The settling of particles through fluids is an important geologic process in
many geologic systems. Crystals settle in magmas, particles settle in flowing
streams or through the static ocean. We can better understand the controls
on this process using an equation called Stokes' Law, named after its inventor
in 1851, George Gabriel Stokes. The motion of a settling particle is controlled
mainly by gravity and by viscous bouyant forces resisting its downward movement.
For Stokes' Law to apply, you must consider a spherical grain of radius
r and
density d1 falling through a fluid of
density d2 and viscosity µ.
The velocity of fall V is then given by Stokes Law
V=2/9(gr2)(d1-d2)/µ
where
- V = velocity of fall (cm/sec)
- g = acceeleration of gravity (cm/sec2)
- r = "equivalent" radius
of particle
- d1 = density of particle (g/cm3)
- d2 = density of medium (g/cm3)
- µ = viscosity of medium
(g/cm-sec)
As you can see from this equation, the settling velocity of a mineral fragment
sinking in water is essentially a function of its weight or size. If the settling
velocity (v) is much less that the velocity of stream flow, the particle will
travel far downstream before it reaches the river's bed. Coarse grains for which
v is large, travel shorter distances before settling to the floor of the river.
Moreover, a grain of sediment will be suspended in a moving river if the settling
velocity is less than the velocity of turbulent upward moving currents found
in a flowing stream.
According to the theory behind Stokes' Law, when a sphere settles in
a fluid of constant temperature and viscosity it is acted on by three forces,
1) The force of gravity (acting downward); 2) The buoyant force of the liquid
(acting upward); and 3) The drag force (friction between the particle and the
fluid), which appears whenever there is relative motion between the particle
and the fluid. The buoyant force
depends very
much on the diameter of the particle and is the basis for Stokes' Law.
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1. Begin by viewing this Excel
tutorial (QuickTime) or here (HTML) on calculating formulas (Note: If you don't have QuickTime Player you
can download it for free here).
Next, calculate
the settling velocities of six different grains of quartz in water. Assume
that the density of quartz is 2.7 g/cm3 (grams per cubic
centimeter) and that the density of liquid water is 1.0 g/cm3. Gravitational
acceleration (g) is 980 cm/s2 (centimeters per second squared).
Viscosity has dimensions of mass, length and time and can be expressed in terms
of g/(cm sec). Assume for this exercise that the viscosity of water is 1 x
10-2 g/(cm
sec) (Hint: you can enter this in Excel as either 0.01 or 1E-02) and that the
grains are spherical. (Tabular flakes of mica or clays will settle
at slower
velocities than spheres.) Note: Set up your calculations
in Excel by creating separate columns for each component of the equations
and submit your calculations with your answers.
| |
Radius (cm) |
| Grain 1 |
0.0002 (clay) |
| Grain 2 |
0.002 (silt) |
| Grain 3 |
0.02 (fine sand) |
| Grain 4 |
0.2 (coarse sand) |
| Grain 5 |
2.0 (granule) |
| Grain 6 |
12 (pebble) |
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2.
Typical turbulent streams have upward velocities
of about 200 cm/s. Based upon your calculations above, are any of the
particles going to settle in a turbulent stream? If yes, which ones? If no,
why not?
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3.
It
is important to note that Stokes' Law is technically valid only for fine sand
to clay sized
particles
(less than
0.2
mm in diameter).
In fact,
experiments have shown that
particles larger than about 0.2 mm settle somewhat slower than predicted by
Stokes' Law. (However, the effect is so small that Stokes' Law is still
a good
estimate.) Can you think of a reason why larger grains might settle more
slowly than predicted by Stokes' Law?
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4.
Now we will apply Stokes' Law to a geological situation. Imagine that a stream
with non-turbulent laminar flow receives a sediment mixture consisting
of grains of quartz (SiO2),
magnetite (Fe3O4) and
gold (Au). Since these particles are of different densities (densities are given
in the table below) their settling velocities will vary. Note: Before you do any calculations make a prediction about the relationship between the density of a particle and its settling velocity. Remember from your calculations in question 1that settling velocity is also a function of size. Thus, if a sediment mixture containing particles of different densities enters a stream, the particles of different densities that settle at the same rate will be different sizes.
Assuming
spherical shapes for all particles and the same physical parameters as in question
#1 (i.e. viscosity, gravity, fluid density), what is the diameter of magnetite
and gold particles that settle at the same velocity as a 0.2 cm diameter
grain of quartz?
Hint: Since settling velocities are the same, we can solve
for the unknown size parameter by setting the two equations equal to each
other, cancelling out common values (such as gravity, viscosity, etc.), and
solving for the unknown value (i.e. For mineral x, r2(x) *
(d1-d2x) = r2(qtz) * (d1-d2qtz), or r = the square root of ((d1-d2qtz)*r2qtz/(d1-d2x)). You can either do this in Excel or simply calculating by hand, but be sure to show all calculations. (Online
calculators are available at numerous sites such as http://www.math.com/students/tools.html)
| Mineral |
Density (g/cm3) |
Diameter (cm) |
| Quartz |
2.7 |
0.2 |
| Magnetite |
5.1 |
|
| Gold |
19.3 |
|
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5.
Grain Sorting: Sorting refers to whether the particles in a mixture are relatively
uniform in size or are many different sizes. A well-sorted sediment consists of grains
of uniform size and a poorly-sorted sediment contains particles of many sizes.
In the figure below the examples are described, from left to right, as poorly
sorted, moderately poorly sorted, moderately well sorted and well sorted. For
the sediment accumulation described in question #4 which sorting category best
describes the sediment mixture composed of grains that settled at the same rate as quartz?
Submit your answers to the Assignments page in Blackboard by the due date
announced in class.
