Гидрогеология Загрязнений
и их Транспорт в
Окружающей Среде
Yoram Eckstein, Ph.D.
Fulbright Professor 2013/2014
Tomsk Polytechnic University
Tomsk, Russian Federation
Fall Semester 2013
Basic Concepts
Mass Balance and the Control Volume
Physical Transport of Chemicals
 Advective Transport
 Fickian Transport
• Turbulent Diffusion
• Dispersion
• Molecular Diffusion
The Advection-Dispersion-Reaction Equation
Mass Balance
and the Control Volume
Min
S
Mout
M – mass of a chemical
S – storage
RS – reaction product
RS = Generation - Consumption
Accumulation = ΔS = Min - Mout ± RS
S M in M out RS



t
t
t
t
An open system mass balance in
a control volume
(контрольный объем)
Control volume
(контрольный объем)
Control volume
Control volume
An open system mass balance
0 = ΔS = Min - Mout ± RS
An open system mass balance
Question: what is the magnitude of internal
sinks of butanol? ΔS = Min - Mout ± RS
Answer
Input + Generation = Output + Accumulation + Consumption
RS = Generation – Consumption = 0 – x = - x
ΔS = Accumulation = 0
Min= 20kg/d
Mout= Qout.10-4kg/m3 = 3.104m3/d .10-4kg/m3 = 3kg/d
RS = Min – Mout = 20kg/d - 3kg/d = 17kg/d
Transport phenomena in the
natural environment
Advection
transport mechanisms of a substance or a
conserved property with a moving fluid. The
fluid motion in advection is described
mathematically as a vector field, and the
material transported is typically described as
a scalar concentration of substance, which
is contained in the fluid.
Transport phenomena in the
natural environment
Advection
An example of advection is the
transport of pollutants or silt in a river:
the motion of the water carries these
impurities downstream.
Advection
Advective transport velocity
𝜹
𝜹
𝜹
𝒗 × 𝛁 = 𝒖𝒙
+ 𝒖𝒚
+ 𝒖𝒛
𝜹𝒙
𝜹𝒚
𝜹𝒛
where the velocity vector v has components
u, ω and w in the x, y and z directions
respectively and  is the divergence
operator.
Quantification of advective
transport
Assuming one-dimensional
steady-state transport velocity v
we can define flux density (or
mass flux):
J = C∙v
Quantification of advective
transport
Assuming one-dimensional
transient transport velocity v we
can define flux density (or mass
flux):
C
C
 v
t
x
X
Another commonly
advected property is
heat, and here the
fluid may be water,
air, or any other
heat-containing fluid
material. Any
substance, or
conserved property
(such as heat) can be
advected, in a similar
way, in any fluid.
Advection is important for the formation of
orographic cloud and the precipitation of water
from clouds, as part of the hydrological cycle.
Advective & convective heat
transport in the ocean
Tectonic cycle
Turbulent diffusion
The fluid flow in the natural environment such as
the atmosphere, world oceans, lakes and rivers
occurs as random complex turbulent movements.
Superimposed on a mean flow circulation are
eddy-like motions of varying intensities and
temporal and spatial scales. These eddy-like
motions are three dimensional, but the horizontal
eddies are much larger than those of the vertical
eddies. A direct consequence of the turbulent
diffusion processes is the transport and dispersion
of chemical and biological species.
Turbulent diffusion
of smoke
Turbulent diffusion
of cloud vapors
Vincent Van Gogh
Turbidities at the feet of a continental
slope
Paleoseismic turbidities
Molecular diffusion
Turbulent diffusion occurs within
fluids/gases moving at certain
velocity
Molecular diffusion, driven solely
by chemical concentration
gradients, occurs within static (or
quasi-static) fluids
Molecular diffusion
Molecular diffusion
Molecular diffusion
Mass transfer equations:
Fick’s Law
C
J 
x
C
J  D
x
C
J  D
x
J– The Mass Flux – the movement of mass
from one point to another in a given
time. The flux is what we are measuring
when studying diffusion (течение, поток)
Units: the units of moles/(time ∙ area).
Note: area has units of length2.
Example: mol/(h ∙ ft2), mol/(s ∙ m2).
C
J  D
x
D– Diffusivity – is the constant that describes how
fast or slow an object diffuses.
Units: the units of area/time. Example: ft2/h, or cm2/s
D is proportional to the velocity of the diffusing particles,
which depends on the temperature, viscosity of the fluid
and the size of the particles according to the StokesEinstein relation. In dilute aqueous solutions the diffusion
coefficients of most ions are similar and have values that
at room temperature are in the range of 0.6∙10-9 to 2∙10-9
m2/s.
C
J  D
x
C – Concentration – is the amount of mass in a
given volume. The symbol ΔC refers to the change
in concentration from when the object had not
diffused at all, to the final concentration when the
object was done diffusing.
Units: amount of substance/volume.
Note: volume is the representation of size in three
dimensions. Therefore it has the units of length3.
Example: mol/cm3, mol/L
Fick’s
st
1
Law of
Diffusion
J   D C
3
g/cm /s
Fick’s
st
1
Law: example
Think of the last time that you washed the dishes. You
placed your first greasy plate into the water, and the
dishwater got a thin film of oil on the top of it, didn’t it?
Data: the sink is 18 cm deep (Δx = 18 cm)
D = 7 ∙ 10-7cm2/s
the concentration of oil on the plate Cplate= 0.1 mol/cm3;
the concentration of oil on the top of the sink Csurface= 0
mol/cm3;
Find the flux, J, of oil droplets through the water to the top
surface.
The answer:
J = -D ∙ΔC/Δ x
J = -(7 ∙ 10-7 cm2/s) ∙ (0 - 0.1 mol/cm3)/(18 cm)
J = 4 ∙ 10-4 mol/(cm2s)
Fick’s
nd
2
Law of Diffusion
C   C 
 D 
t x  x 
3
g/cm /sec
nd
2
Fick’s
Law of
Advection-Diffusion
C   C    Cv 
 D 
t x  x 
x
3
g/cm /sec
nd
2
Fick’s
Law of
Advection-Diffusion
with sources or sinks
C   C    Cv 
 D 
R
t x  x 
x
3
g/cm /sec
nd
2
Fick’s
Law of AdvectionDiffusion with sources or sinks
in 3-dimensions

dC
   D(C )  V  C  R
dt
dC
   D(C ) 
dt
3
g/cm /sec
Diffusion through
porous materials
D
D' 
f
where f is the formation factor
Diffusion through
porous materials
f = f(φ,τ)
where:
φ is the porosity (fraction)
τ is the tortuosity (length)
(извилистость, кривизна, уклончивость)
Tortuosity
τ = l/m
Fick’s 2nd Law of Diffusion through
saturated porous medium
 ( C ) J

R
t
x
and
dC
J   D
  vC
dx
Fick’s 2nd Law for Diffusion
and Advection through
Saturated Porous Medium
 ( C )  
C    vC 
  D  
R
t
x 
x 
x
Mechanical Dispersion
Mechanical Dispersion
Mechanical Dispersion
Mechanical Dispersion
Dl = alv’
and
Dt = atv’
where:
Dl & Dt are longitudinal and transversal
coefficiants of dispersion, respectively
al & at are longitudinal and transversal
dispersivity, respectively
v’ is seepage velocity
The Mechanism of
Mechanical Dispersion
Fick’s Law for
Mechanical Dispersion
C
C
C
D
 v'
t
x
x
2
l
2
Peclet Number
2rv ' 2rq
Pe 

D D
molecular
diffusion
1 > Pe > 1
mechanical
dispersion
where:
v’ is the mean fluid velocity in m s-1
r is the characteristic grain diameter in m
q is the flow rate through a unit cross-section
Reynolds Number
In fluid mechanics and aerodynamics, the Reynolds
number is a measure of the ratio of inertial forces
(vsρ) to viscous forces (μ/L) and, consequently, it
quantifies the relative importance of these two types
of forces for given flow conditions. It is also used to
identify and predict different flow regimes, such as
laminar or turbulent flow. Laminar flow occurs at
low Reynolds numbers, where viscous forces are
dominant, and is characterized by smooth, constant
fluid motion, while turbulent flow, on the other hand,
occurs at high Reynolds numbers and is dominated
by inertial forces, which tend to produce random
eddies, vortices and other flow fluctuations.
Reynolds Number
v '
2
p
v ' r v ' r
r
Re 



v '
s


r
d
s
2
where:
v’ is the mean fluid velocity in m s-1
r is the characteristic grain diameter in m
μ is the (absolute) dynamic fluid viscosity in N s m-2 or Pa·s
υ is the kinematic fluid viscosity, defined as υ = μ/ρ, in m2 s-1
ρ is the density of the fluid in kg m-3
Prandtl Number
 viscous diffusion rate c 
Pr  

 thermal diffusion rate k
p
where:
ν : kinematic viscosity, ν = μ / ρ, (SI units : m2/s)
α : thermal diffusivity, α = k / (ρcp), (SI units : m2/s)
μ : viscosity, (SI units : Pa s)
k : thermal conductivity, (SI units : W/(m K) )
cp : specific heat, (SI units : J/(kg K) )
ρ : density, (SI units : kg/m3 )
Schmidt Number
Schmidt number is a dimensionless number
defined as the ratio of momentum diffusivity
(viscosity) and mass diffusivity, and is used to
characterize fluid flows in which there are
simultaneous momentum and mass diffusion
convection processes.
It is expressed as the ratio of the shear
(cдвижение силы) component for diffusivity
viscosity/density to the diffusivity for mass
transfer D.
Schmidt Number


Sc  
D D
where:
ν is the kinematic viscosity
D is the mass diffusivity.
μ is the Viscosity
ρ is the Density
The heat transfer analog of the Schmidt
number is the Prandtl number.
Reynolds, Peclet,
Prandtl, Schmidt
In fluid dynamics, the Péclet number is a
dimensionless number relating the rate of
advection of a flow to its rate of diffusion,
often thermal diffusion. It is equivalent to the
product of the Reynolds number with the
Prandtl number in the case of thermal
diffusion, and the product of the Reynolds
number with the Schmidt number in the case
of mass dispersion.
Reynolds, Peclet,
Prandtl, Schmidt
For thermal diffusion, the Péclet number is defined as:
PeL = Lv/α = ReL∙ Pr
For mass diffusion, it is defined as:
PeL = Lv/D = ReL∙ Sc
where:
L = characteristic length
α = thermal diffusivity = k/(ρCp)
D = mass diffusivity
v = velocity
ρ = density
Cp = heat capacity