Slide 1

							                      Outline

                    • 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
   your sample.

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