ABE 527
Computer Models in Environmental and Natural Resources Engineering
Spring 2004
Rabi H. Mohtar
Transport Phenomena
We will discuss three transport (mass transport) in water:
1)
Advection: transport by the current of water
2)
Dispersion: due to mixing within the water body
3)
Transport of sediment particles
Dissolved chemicals move water
Adsorbed chemicals move with the soil particles. These may undergo sedimentation,
deposition, resuspension. These will retard the substance movement relative to water.
Dispersion has 3 contributing processes:
1)
Molecular diffusion: random walk of molecules induced by concentration
gradient Fick’s law. It is very slow: 10 days for 1 mg/L to move through 10
cm water column from a concentration of 10 mg/L
2)
Turbulent diffusion: Eddy or mixing of fine particles caused by microscale
turbulence. It is advection at a small scale. More than molecular diffusion by
several order of magnitude.
3)
Dispersion: Velocity gradient induced. It is the mechanism of transport in
lakes and estuaries.
Advection:
J=ūAC=QC
J=flux of substances
ū = avg velocity
A = area
Q = flow rate
v.c = mass of substance
volume * concentration
  vc    Qa Ca  Qb Cb  t
  vc 
 Q a Ca  Q b C b
t
v=AX
C
QC

t
AX
C
1   QC  -ūC


t
A X
X
ABE 527
Computer Models in Environmental and Natural Resources Engineering
Spring 2004
Rabi H. Mohtar
Diffusion/Dispersion
dc
Fm=-D
dx
Fm=mass flux per unit area
D = molecular diffusion coefficient
dc
 concentration gradient
dx
Fick’s second law:
We may write Fick’s first law as a different equation.
C
J = -DA
X
c
C
V
 DA
t
x
c
c
 DA
t
x A x
c
c
 DA
t
x x
c
 2C
lim
D 2
t  o t
x
This is useful if the concentration is varying with time
Advective dispersion equation:
Based on Fick’s law and continuity equation
c
c

c
 ui

Ei
R
t
x i x i x i
Rate of
change of
mass in
C.V.
Advection rate of
change
Diffusion rate of change
Degradation
reaction
This will be coupled with open channel flow equations.
Analytical solutions available to the advection dispersion equation.
E values are reported for various locations and conditions.
Importance of advection compared to dispersion
Pe  uL
E
L = segment length (L)
E=10-5 Molecular diffusion
U = velocity LT-1
10-7-10-5 Compacted sediment
E = dispersion coefficient L2T-1
102 – 107 River in estuaries
Pe>>1 advection predominates
ABE 527
Computer Models in Environmental and Natural Resources Engineering
Spring 2004
Rabi H. Mohtar
Pe <<1 dispersion predominates
Sediment Transport
Cp
 KpM
C
Cp = particulate chemical concentration μgL-1
C = dissolved chemical concentration μgL-1
Kp = sediment/water partition coefficient Lkg-1
M = suspended solids concentration, kgL-1
KpM
Cp 
CT
1+K p M
1
CT
1+K p M
CT = total concentration
 suspended loads = flow rate * concentration
 sedimentation:
 g 
2
W=8.64 
  s   w  ds
 18 
w = particle fall velocity ft/s
ps = density of sediment particle 2-2.7 g/cm3
pw = density of water 1 g/cm3
g = 981 cm/s2 gravitational constant
ds = sediment particle diameter
μ=viscosity
W
W
ks 
 ku
H
H
Ks = net sedimentation rate constant T-1
W = mean particle fall velocity L/T
Ku = scour/resuspension rate constant, T-1
C=
Next week assignment:
Present the problem you will be solving. Example: physical problem, with the
boundary conditions for same
The watershed and the conditions for other
Scenario under which input were selected