Transcript Smith-D
Thermodynamics and chemical transport through a deforming porous medium Exploration Geodynamics Lecture David Smith, Glen Peters
The
University
of
Newcastle Callaghan, Australia
Chemical reaction
aA
bB
...
m M
nN
...
G r
G r
0
RT
ln
a m M a n N
...
a a A a b B
...
G r
0
G
0
products
G
0
reac
tan
ts
At thermodynamic equilibrium
G r
0 and so
G r
0
RT
ln
K
Where K = equilibrium constant
Structural Engineering
Principle of Minimum Potential Energy Principle of Minimum Complementary Potential Energy (Castigliano’s Theorem)
Deformation of a Truss
Consilience
(Edward Wilson) Physics (mechanics) traditionally been separate from chemistry.
But they are not really separate.
Two forces: gravity and electromagnetic.
Electrical and chemical forces (actually merge one into another) Inorganic Chemistry: Shriver and Atkins
Thermodynamics is required to understand: Solid mechanics ( e.g. fully coupled thermoelasticity ) Material science Interfacial phenomena Geochemistry
OUTLINE
Brief history of thermodynamics What is thermodynamics? Concept of thermodynamic potentials Coupling between (ir)reversible processes Briefly discuss 2 applications Darcy’s law and why water flows through soil Thermodynamics of dissolution processes Getting the governing differential equations right: Transport through a deforming porous media
Rumford 1782 Carnot 1820’s Joule Kelvin Clausius 1840’s -60’s Helmholtz 1882 Caratheodory Gibbs 1880’s-90’s Truesdell Coleman Noll 1909 Katchalsky& Curran Broecker& Oversby 1950’s-60’s 1960’s-70’s Onsager Slater Prigogine 1930’s Bowen 1980 1939 Mitchell Collins Houlsby 1970’s,80’s,90’s 1950’s
What is Thermodynamics?
One approach to thermodynamics is through the atomic theory of matter (statistical mechanics) -1 gram MW of a substance contains 6.023 x 10 23 atoms or molecules 602,300,000,000,000,000,000,000
To completely define 1 litre of water, the position and velocity of every nuclei and every electron in the litre of water would have to be specified 6.6 x 10 26 co-ordinates However, the litre of water can be characterized by the temperature, pressure and strength of the electro magnetic field surrounding the water 3 co-ordinates 6.6 x 10 26 Statistical averaging 3
Ludwig Boltzmann
came up with a way of getting a statistical measure of the likelihood of a particular configurations of nuclei and electrons
S
k
ln
W
The Second Law of Thermodynamics
d
S universe
0 d
S system
d
S surroundin gs
0 d
S surroundin gs
d
q T
This equation is crucial because it allows us to concentrate on the system alone d
S system
d
q T
0 Clausius inequality Corollary: At equilibrium, S is maximised
Time’s Arrow
(Arthur Eddington) The Diffusion Equation: 2
T
x
2
T
t
2
T
x
2
T
Leads to counter-intuitive solutions
The Wave Equation 2
h
x
2
c
2
h
t
2 2
h
x
2
c
2
h
t
2 The wave equation is unchanged if time is reversed
The First Law of Thermodynamics
U
q
w
U
Change in internal energy
q
w
Heat flow across the system boundary Work done on the system Definition: Internal energy of the system (U) is the sum of the total potential and the kinetic energies of the atoms in the system
Thermodynamic Potentials
For adiabatic systems, the amount of work required to change the internal energy of the system is independent of how the work is performed The system is dependent on its initial and final states but independent of how it got there Hence the internal energy is a state function (or potential) [A state function (or potential) has a path independent integral between two points in the same state space]
‘p
v’
is also a state function (or potential) p d (
p
)
p
d
v
d
p
( (
p
p
) 0 )
f
d (
p
) (
p
)
f
(
p
) 0 v Addition (or subtraction) of two potentials gives another potential
U
p
H
(
enthalpy
)
‘TS’ is another potential, and this may be subtracted from U
U
TS
A
(Helmholtz free energy) ‘TS’ may be subtracted and ‘pv’ may be added to U
U
TS
pv
G
(Gibbs free energy)
Legendre Transformations
d
H
d
A
dU
p
d
v
d
U
dp
T
d
S
S
d
T
d
G
d
U
d
U
d
q
T
d
S
d
w
S
d
T
T
d
S
p
d
v
p
d
v
d
p
(S,v indep. variables) And substituting dU shows d
H
T
d
S
v
d
p
d
A
d
G
S
d
T
S
d
T
p
d
v
v
d
p
(S,p indep. variables) (T,v indep. variables) (T,p indep. variables)
Irreversible Processes
d
S system
dq T
d
S system
dq T
0
dS irreversib le
For many slow processes of interest to engineers
T .
S irreversib le
ν i
G
x i v i
G
x i
= thermodynamic force
Well known thermodynamic fluxes:
i
k ij
G
x j i i f i
D ij
R ij
a
x j
V
x j h i
c ij T
T
x j
(Darcy’s Law) (Fick’s Law) (Ohm’s Law) (Fourier’s Law)
Rates of Entropy Production
2 .
S irreversib le
k ij T
G
x j
(Darcy’s Law) 2
D ij T
a
x j
(Fick’s Law) 2
R ij T
v
x j
(Ohm’s Law) 2
c ij T
2
T
x j
(Fourier’s Law)
Lord Kelvin postulated existence of a ‘dissipation potential’ (D)
D
f
(
ij
) A potential implies the general reciprocal relationships between thermodynamics forces and fluxes;
ij
lm
lm ij
These are known as the Onsager reciprocal relationship (Onsager (1931)) Coupled flows (and the Onsager relationships) are important in two phase materials e.g. clays (Mitchell, 1991).
Zeigler (1983) assumed the existence of a dissipation potential in solid mechanics
T
.
S irreversib le
D
ij
ij implying dD ij d
ij
ij
D
ij
Performing a Legendre transform on D d
D
d
ij
ij dD dD
ij
d ij implying ij
D
ij
If D is a homogeneous function of degree one, then D 0
ij
D
ij
yield condition flow rule
Couplings of Irreversible Processes Flow J
Fluid Thermodynamic Force (Gradient of Potential) Hydraulic head
Hydraulic conduction
Darcy’s law Temperature
Thermo osmosis
Electrical
Electro osmosis
Chemical concentratio
osmosis
Heat
Isothermal heat transfer Thermal conduction
Fourier’s law
Peltier effect
Dufour effect
Current Ion
Streaming current Streaming current Thermoelectrici ty
Seeback effect
Electric conduction
Ohm’s law
Diffusion potential and membrane potential Thermal diffusion of electrolyte
Soret effect
Electrophores is Diffusion
Fick’s law
F l o w J
Hydraulic head Thermodynamic Force (Gradient of Potential) Temperature Electrical Chemical concentratio n Stress Fluid
Hydraulic conduction
Darcy’s law
Thermo-osmosis Density changes Electro-osmosis Chemical osmosis Density change consolidation
Heat Current Ion
Isothermal heat transfer Thermal conduction
Fourier’s law
Streaming current Streaming current
Peltier effect Dofour effect
Thermoelectricity
Seebeck effect
Electric conduction
Ohm’s law
Thermal diffusion of electrolyte
Soret effect
Electrophoresis Diffusion potential and membrane potential Diffusion
Fick’s law
Fully coupled thermoelastcity Phase change Piezoelectricity Dissolution/ precipitation
Strain
consolidation (change in effective stress) fracture Thermal expansion Density changes Piezo- electricity Dissolution and precipitate Consolidation (double-layer contraction) Elasticity Viscoelasticity Plasticity Viscous flow Consolidation
Couplings through constitutive equations Permeability =
k
f
(
ij
, ,
T
,
c i
) Resistance =
r
f
(
ij
,
T
,
c i
) Diffusion coeff. =
D
f
( ,
ij
,
T
,
c i
) Young’s modulus =
E
f
(
ij
,
T
,
c i
,
t
) Viscosity =
f
(
ij
,
T
,
c i
) Yield surface =
f
f
(
ij
,
ij
,
T
,
c i
, 0 )
EXAMPLE 1 Darcy’s law and why water flows through soil
Darcy’s law - flow of water through soil
v x
k x
h w
x
h w
total head
u
w
z
G w
w
h w
u
z
w
where,
G w
= Gibbs free energy of an incompressible pore fluid per unit volume
G w
w
w
u
0
v du
0
z
w dz
RT
ln
w
0
w
SdT
Standard state Pressure contribution Position contribution entropy component thermal component
Why water flows through soil?
v i
k ij
G
w
x j G
w
0
w
p
z
os
thermal
.
S irrev ersib le
k ij T
G
*
w
x j
2
EXAMPLE 2: Dissolution and precipitation
Dissolution and precipitation
A s
m M
nN
...
G r
G r
0
RT
ln
a m M a a n N A s
...
, assume
a A s
1
G r
0
RT
ln
K so IAP
K so IAP
K so
supersaturated undersaturated
EXAMPLE: reactive transport (i.e. transport with precipitation)
Soil Physical Chemistry 1999: Ed Sparks
Ion Activity Product varies from soil to soil: Solid not pure Amorphous or crystalline (size of crystals important) Surface is charged (leads to concept of intrinsic and apparent IAP i.e. concentrations at surface of solid are critical, not those in the bulk solution).
K app so
K
int
so e
(
zF
o
)
m
n RT
a m M a a n N A s
...
e
(
zF
o
)
m
n RT
Stress induced change in IAP (IAP strong function of temperature)
Nucleation (Stumm and Morgan 1996)
G j
G bulk
G surface
G j
4
r
3 3
V
kT
ln
s
4
r
2
s
IAP K so
1 interfacial energy atoms in formulae unit Dissolution (Sparkes 1999)
G j
G bulk
G surface
G dislocatio n
Change in solubility product with pressure Consider reaction:
A s
mM
nN
...
G r
0
G
0
products
G
0
reac
tan
ts d
(
G r
0 )
V r
0
dP
S r
0
dT
(
G r
0
P
)
r
V r
0
G r
0
RT
ln
K
0
so
(ln
K
0
so
)
P
r
V r
0
RT
Change in solubility product with pressure Standard partial molar compressibility
C i
0
V
P
T
For the reaction we have
C i
0
V
P
T
2 (ln
P K
0
so
) 2
r
(
RT V r
0
P
) ln
K p so K
0
so
r
V r
0 (
P
1 )
RT
C i
0 (
P
1 ) 2 2
RT
Change in solubility product with pressure (Langmuir 1997) Pressure generally increases the solubility of minerals Considering the reaction
SrSO
4
Sr
2
SO
2 4
C r
0
V r
0 50 .
43 1 .
514 10 3
cm cm
3 3 / /
m ol m ol bar
Change in pressure of 180 bar increases solubility by 50%.
Effect of shear stress
d
(
G r
0 )
V r
0
dP
S r
0
dT
0
ij ds ij
(
G r
0
s ij
)
r
0
ij
G r
0
RT
ln
K
0
so
(ln
s K
0
ij
0
so
)
r
0
ij RT
Soil Physical Chemistry 1999: Ed Sparks
The Advection-Dispersion Equation: Boundary Conditions
Solute Mass Flux
Flux = Advection + Diffusion Two flows: 1) A mean flow (advection) 2) Perturbation about the mean (Mechanical Dispersion) Flow due to chemical potential gradients
f e
nD e
c x
Advection and Mechanical Dispersion
These processes cause perturbations of solute concentration and pore water velocity, hence,
c v c v
' '
cv
c
c
'
v
v
'
c v
c
'
v
' Mechanical dispersion Mean Advection or Plug Flow This represents a cross-correlation between concentration and velocity fluctuations.
Fickian under “ideal” conditions
c v
nD md
c
x
The Advection-Dispersion Equation The solute mass flux is
f
nvc
nD e
c x
nD md
c x
And leads to the standard ADE
t c
2
c nD
x
2
v
c x
where
D = D
e
+D
md
= D
e
+av
Boundary Conditions
(
nc
)
t
x f
At a boundary mass conservation requires the flux,
f
, to be continuous, that is,
f
(left of boundary
,t)
=
f
(right of boundary
,t) f
(0-,
t
)=
f
(0+,
t
) This holds for all times.
c=c
0
v
A simple example
Consider a porous medium between an upstream reservoir with a concentration of
c=c0
and a downstream reservoir that allows solute to drip freely from the porous medium.
Porous Medium
c=c
e
Inlet Boundary Outlet Boundary
The Inlet Boundary Condition
Solute mass flux must be continuous,
q
0
c
0
q
0
c
nD
c
x
This boundary condition ensures mass is conserved.
Solute concentration is not continuous at boundary
(solute concentration is continuous at the microscopic scale).
The outlet boundary
Both solute mass and solute mass flux must be continuous leading to the b/c
c
x
0
However, this does not agree with experiment.
Experiment agrees with the semi-infinite model evaluated at the point
x=L
.
Why?
Consider solute advection Solution
c
t
v
c
x c
c
0 for
x
vt
0 for
x
vt
Solute advection is only affected by upstream boundary conditions. However, the ADE requires downstream boundary conditions
(ADE is a parabolic equation).
Mechanical dispersion is inherently an advective process and so should be described by a hyperbolic equation (i.e. the ADE is incorrect).
Conclusions
Thermodynamics is a `keystone theory’ in modern physics, underpinning theories in all the applied sciences and engineering.
In some disciplines, the relation between thermodynamics and their discipline has become obscured by the continual telling and retelling by successive generations.
Conclusions
In much of engineering, thermodynamics is usually not taught in a systematic way, and first principles behind theories are skimmed over. This hampers fundamental research.
The contaminant transport equation requires some understanding of the underlying assumptions in order to use it properly.
The transport of chemicals through a deforming porous media requires the derivation of a suitable transport equation from first principles.
Conclusions
The great task that lies ahead of the engineering and the applied sciences this century, is consilience between the different disciplines.