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• Correction
• Wetting angle
• Particle sizes

Soil Physics 2010
Correction

Principle, part 1:
An electrical pulse propagating along a wire reflects
back from the end of the wire:

Knowing the speed of propagation (around c), we can
figure out the distance to the end – hence “Cable
Tester”
Soil Physics 2010                Animation courtesy of Dr. Dan Russell, Kettering University
Time Domain Reflectometry
Principle, part 2:
An electrical pulse propagating along a wire has its
velocity changed according to the dielectric permittivity
of the surrounding medium:
c
v
A wire running
through water
er                                  – even if insulated –
will transmit a
signal more slowly!

The dielectric permittivity er (sometimes called the dielectric
constant, which it isn’t!) is expressed relative to the
permittivy of a vacuum (1 by definition), so it is unitless.
Soil Physics 2010                       Animation courtesy of Dr. Dan Russell, Kettering University
TDR setup                                                           The coaxial cable
is shielded from
Cable Tester          1) A pulse is sent                          soil’s dielectric
through the cable
to the probe
2) The pulse goes
down the bare
5) The returned                                           wires, surrounded
pulse shows the                                           by soil
effect of this delay
+
-
4) Both ways through the
needle, the pulse is slowed by               3) The pulse reflects off
the dielectric of the soil                   the ends of the needles.
c
v
er
Soil Physics 2010                          Animation courtesy of Dr. Dan Russell, Kettering University
TDR in practice

travel time
in needle

Montmorillonite
trace q
Needles have length L           2L   c a 4
v       b 11
t   e c 22
d 30     r

Soil Physics 2010
Wetting angles

But air’s wetting angle
is around 150°

Here, water’s wetting
angle is around 30°         a < 90° : “wetting phase”
a > 90 ° : “non-wetting phase”

The wetting angle is defined as that angle
passing through the fluid being described.

Soil Physics 2010
Young’s equation
relates the energies of the 3 interfaces
 SL   LG cos a   SG
subscripts: S solid, L liquid, G gas

The contact point is pulled equally
each way along a (flat) solid surface

a

Soil Physics 2010
Back to the capillary tube

 SL   LG cos a   SG

a

Soil Physics 2010
Capillary equation – final version

2 cos a
h
r w  r a  g r
a

Soil Physics 2010
Particle sizes

Which is bigger?

Soil Physics 2010
How to decide which is bigger?

Volume?
Surface area?
Projected area?
Longest transect?
Largest inscribed sphere?
Smallest circumscribed sphere?
Largest circle inscribed in projection?
Smallest circle circumscribing projection?
Soil Physics 2010
…?
Likewise for soil particles

Volume?
Surface area?
Projected area?
Longest transect?
Largest inscribed sphere?
Smallest circumscribed sphere?
Largest circle inscribed in projection?
Smallest circle circumscribing projection?
…?
Soil Physics 2010
Equivalent sphere
All methods attempt to
relate each real soil
particle to a sphere that
in some sense is the
same (“equivalent”) size
Equivalent by:
Volume?
Surface area?
Projected area?
Longest transect?
Largest inscribed sphere?
Smallest circumscribed sphere?
Largest circle inscribed in projection?
Soil Physics 2010   Smallest circle circumscribing projection?
Measuring particle size: first Disperse

Soil particles aggregate.
Do we want to know about the primary
particles, or the secondary particles?
If primary, how do we disperse (dis-
aggregate) the secondary particles?
Why not measure both?
How are the two distributions related?

Soil Physics 2010
Measuring soil particle sizes: Sieving

Sieving:
• Related to smallest circle circumscribing projection
• Nested sieves                • Discrete sizes
• Labor-intensive              • \$
• Time-dependence              • Errors each way
• Mass-dependence              • Slower with more mass
• Energy-dependence            • Jumping
• Size- and shape-dependence • 50 mm smallest
• Rounder is better
Soil Physics 2010
Measuring soil particle sizes: Sedimentation
Gravitational Sedimentation
Stokes Settling
Imagine a sphere sinking
through a viscous fluid –
say, a silt grain in water.
At terminal velocity,
Force up = Force down
Newton’s 1st law:
Objects at rest tend to stay at rest
→
An object moving at a constant speed is acted upon
by forces (if any) equal in magnitude: Forces
slowing it, and forces accelerating it.
Soil Physics 2010
Measuring soil particle sizes: Sedimentation

At terminal velocity,
Force up = Force down
(Newton’s 1st law)

Force down:
Force = Mass * acceleration
= (rs-rw)(4/3 p r3) * g

(Newton’s 2nd law)

Soil Physics 2010
Measuring soil particle sizes: Sedimentation

At terminal velocity,
Force up = Force down
(Newton’s 1st law)

Force up (viscous drag):
=6prhv

viscosity

(Stokes said so)

Soil Physics 2010
Measuring soil particle sizes: Sedimentation

At terminal velocity,
Force up = Force down
(Newton’s 1st law)

6prhv= (rs-rw)(4/3 p r3) * g
Solve for v:
4p r s  r w r g 2r s  r w r g
3               2
v                   
18prh               9h
Soil Physics 2010
Measuring soil particle sizes: Sedimentation
2r s  r w r g
2
v
9h
Particles ≥ r will fall at least   Dh
Dh within a known time t.

Sampling at a known depth
and time, you know the size
of the biggest particle in

Soil Physics 2010
Measuring soil particle sizes: Sedimentation

2r s  r w r g  2
v
Assumptions:               9h
Particles are smooth spheres
Particles fall slowly (laminar flow)
All particles have the same density
Dilute: particles don’t affect each other
Fluid is otherwise at rest
Terminal velocity is reached instantly
Soil Physics 2010

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