Download 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  ...
 a m a n ... 
0
Gr  Gr  RT ln M N 
b 
 a a aB
 A ... 
0
Gr0   G 0products   Greac
tants
At thermodynamic equilibrium
Gr  0
Gr0   RT ln K
Where
K = equilibrium constant
and so
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
Carnot
1782
1820’s
Joule
Kelvin
Clausius
1840’s
-60’s
Caratheodory
Onsager
Gibbs
1880’s-90’s
1909
Helmholtz
1930’s
1882
Slater
1939
Truesdell
Coleman
Noll
Katchalsky&
Curran
Broecker&
Oversby
Bowen
Mitchell
Collins
Houlsby
1950’s-60’s
1960’s-70’s
1980
1970’s,80’s,90’s
Prigogine
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 1023
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 1026 co-ordinates
However, the litre of water can be characterized by
the temperature, pressure and strength of the electromagnetic field surrounding the water
3 co-ordinates
6.6 x 1026
3
Statistical averaging
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 lnW
The Second Law of Thermodynamics
dSuniverse  0
dSsystem  dSsurroundin gs  0
dSsurroundin gs
 dq

T
This equation is crucial because it allows us to
concentrate on the system alone
dq
dSsystem 
0
T
Clausius inequality
Corollary: At equilibrium, S is maximised
Time’s Arrow (Arthur Eddington)
The Diffusion Equation:
 2T T
 2 
x
t
If t

 2T
T


2

x
Leads to counter-intuitive solutions
The Wave Equation
 2h
 2h
c 2
2
x
t
If t

 h
 h
c 2
2
x
t
2
2
The wave equation is unchanged if time is reversed
The First Law of Thermodynamics
U  q  w
U 
Change in internal energy
q  Heat flow across the system
boundary
w  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]
‘pv’ is also a state function (or potential)
d( p )  pd  vdp
p

( p )f
( p )0
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
dH  dU  pdv  dp
dA  dU  TdS  SdT
dG  dU  TdS  SdT  pdv  dp
d U  d q  dw
(S,v indep. variables)
TdS  pdv
And substituting dU shows
dH  TdS  vdp
dA  SdT  pdv
(S,p indep. variables)
dG  SdT  vdp
(T,p indep. variables)
(T,v indep. variables)
Irreversible Processes
dq
dSsystem 
0
T
dq
dSsystem 
 dSirreversible
T
For many slow processes of interest to engineers
.
T S irreversible
where

G
 νi
xi
vi
= thermodynamic flux
G
x i
= thermodynamic force
Well known thermodynamic fluxes:
G
 i   kij
x j
(Darcy’s Law)
a
f i   Dij
x j
(Fick’s Law)
V
ii   Rij
x j
(Ohm’s Law)
cij T
hi  
T x j
(Fourier’s Law)
Rates of Entropy Production
.
Sirreversible
kij  G 

 
T  x j 
Dij  a


T  x j
2
(Darcy’s Law)




2
(Fick’s Law)




2
cij  T 

 2
T  x j 
2
Rij  v


T  x j
(Ohm’s Law)
(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 Sirreversible  D   ij ij
dD  ijdij
implying
D
 ij 
 ij
Performing a Legendre transform on D
dD  d ij ij   dD
dD   ij d ij

implying
D
 ij 
 ij
If D is a homogeneous function of degree one, then
D 0


D
 ij  
 ij
yield condition
flow rule

Couplings of Irreversible Processes
Thermodynamic Force (Gradient of Potential)
Flow J
Fluid
Hydraulic
head
Hydraulic
conduction
Temperature
Electrical
Thermoosmosis
Electroosmosis
Isothermal
heat transfer
Thermal
conduction
Peltier
effect
Thermoelectrici
ty
Electric
conduction
Diffusion
potential and
membrane
potential
Thermal
diffusion of
electrolyte
Electrophores
is
Diffusion
Darcy’s law
Heat
Current
Ion
Streaming
current
Streaming
current
Fourier’s law
Seeback effect Ohm’s law
Soret effect
Chemical
concentratio
n
Chemical
osmosis
Dufour
effect
Fick’s law
Thermodynamic Force (Gradient of Potential)
F l o w J Hydraulic
Temperature
Electrical
head
Chemical
Stress
concentratio
n
Fluid
Hydraulic
conduction
Darcy’s law
Thermo-osmosis
Density changes
Electro-osmosis Chemical
osmosis
Density change
consolidation
Heat
Isothermal heat
transfer
Thermal
conduction
Fourier’s law
Peltier effect
Dofour effect
Fully coupled
thermoelastcity
Phase change
Current
Streaming
current
Thermoelectricity
Seebeck effect
Electric
conduction
Ohm’s law
Piezoelectricity
Ion
Streaming
current
Thermal diffusion Electrophoresis
of electrolyte
Soret effect
Diffusion
potential and
membrane
potential
Diffusion
Fick’s law
Strain
consolidation
(change in
effective stress)
fracture
Thermal
expansion
Density changes
Dissolution and
precipitate
Consolidation
(double-layer
contraction)
Elasticity
Viscoelasticity
Plasticity
Viscous flow
Consolidation
Piezoelectricity
Dissolution/
precipitation
Couplings through constitutive equations
Permeability = k  f ( ij ,  , T , ci )
Resistance =
r  f ( ij , T , ci )
Diffusion coeff. = D  f ( ,  ij , T , ci )
Young’s modulus = E  f ( ij , T , ci , t )
Viscosity =   f (ij , T , ci )
Yield surface = f  f ( ij , ij , T , ci ,  0 )
EXAMPLE 1
Darcy’s law and why water flows
through soil
Darcy’s law - flow of water through soil
hw
v x  k x
x
hw 
total head

u
w
 z
Gw   w hw  u  z w
where,
Gw = Gibbs free energy of an incompressible pore
fluid per unit volume
For water,
Gw  w

Gw   w   w

u
, the chemical potential is
z
 vdu  
0
0
wdz 
Standard Pressure
Position
state contribution contribution
If
v  f u 
RT ln
w
0
w
 SdT
entropy
thermal
component component
Hubbert potential (1940)
Why water flows through soil?
Gw
vi  kij
x j

0
Gw
 w
  p   z  os  thermal
.
Sirrev ersible
k ij  G w
 
T  x j
*




2
EXAMPLE 2:
Dissolution and precipitation
Dissolution and precipitation
As  m M  nN  ...
 m n 
 aM a N ... 
0
Gr  Gr  RT ln

 aA

s


Gr0   RT ln K so
IAP Kso
supersaturated
IAP Kso
undersaturated
, assume a As  1
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).
(zF o ) mn  m n  (zF o ) mn
 aM a N ... 
app
int
RT
RT
K so  K so e

e
 aA

s


Stress induced change in IAP
(IAP strong function of temperature)
Nucleation (Stumm and Morgan 1996)
G j  Gbulk  Gsurface
 4 r 3 
kT ln s  4 r 2
G j  
 3V 


1
 IAP  

s  

K
so


Dissolution (Sparkes 1999)
  interfacial energy
  atoms in formulae unit
G j  Gbulk  Gsurface  Gdislocation
Change in solubility product with pressure
Consider reaction:
As  mM  nN  ...
0
Gr0   G 0products   Greac
tants
d (Gr0 )  Vr0dP  Sr0dT
  (G 0 ) 
r   V 0

r
 P 
r
0
Gr0   RT ln K so
  (ln K 0 ) 
Vr0
so

 
RT
 P 
r
Change in solubility product with pressure
Standard partial molar compressibility
 V 
Ci0  

 P T
For the reaction we have
 V
Ci0  
 P


T
  2 (ln K 0 ) 
 (Vr0 )
so

 
2
RT P
 P

r
K p 
Vr0 ( P  1) Ci0 ( P  1) 2
so
 
ln 

0
RT
2 RT
 K so 
r
Change in solubility product with pressure
(Langmuir 1997)
Pressure generally increases the solubility of minerals
Considering the reaction
SrSO4  Sr2  SO42
Vr0  50.43cm3 / m ol
Cr0  1.514 103 cm3 / m olbar
Change in pressure of 180 bar increases
solubility by 50%.
Effect of shear stress
d (Gr0 )  Vr0dP  Sr0dT  ij0dsij
  (G 0 ) 
r    0

ij

s

ij 
r
0
Gr0   RT ln K so
0

0 



(ln
K
)
ij
so  

 s 0 
RT
ij


r
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
c
f   nD
e
e x
Advection and Mechanical Dispersion
These processes cause perturbations of solute
concentration and pore water velocity, hence,
c  c  c'
v  v  v'
cv cc' vv'





















cv c'v'
Mechanical dispersion
Mean Advection or
Plug Flow
This represents a cross-correlation
between concentration and velocity
fluctuations.
Fickian under “ideal” conditions
c
cv  nD
md x
The Advection-Dispersion Equation
The solute mass flux is

c

c
f  nvc  nDe  nD
md x
x
And leads to the standard ADE
c  nD  c  v c
t
x 2 x
2
where D = De+Dmd = De+av
Boundary Conditions
 (nc)  f
t
x
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.
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.
c=c0
Porous Medium
c=ce
v
Inlet Boundary
Outlet Boundary
The Inlet Boundary Condition
Solute mass flux must be continuous,
c
q0c0  q0c  nD
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
0
x
However, this does not agree with experiment.
Experiment agrees with the semi-infinite model
evaluated at the point x=L.
Why?
Consider solute advection
c
c
 v
t
x
Solution
 c0
c
0
for x  vt
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.