Thermodynamics andchemical transport through a deforming porous medium

Exploration Geodynamics Lecture

David Smith, Glen Peters

The University of NewcastleCallaghan, Australia

At thermodynamic equilibrium and soChemical reactionWhere K = equilibrium constant

Principle of Minimum Potential EnergyStructural Engineering

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 intoanother)

Inorganic Chemistry: Shriver and Atkins

Thermodynamics is required to understand:Solid mechanics (e.g. fully coupled thermoelasticity)Material scienceGeochemistryInterfacial phenomena

OUTLINEWhat is thermodynamics? Concept of thermodynamic potentialsBriefly discuss 2 applicationsDarcys law and why water flows through soilBrief history of thermodynamics Thermodynamics of dissolution processesGetting the governing differential equations right: Transport through a deforming porous mediaCoupling between (ir)reversible processes

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 specified6.6 x 1026 co-ordinatesHowever, the litre of water can be characterized by the temperature, pressure and strength of the electro-magnetic field surrounding the water 3 co-ordinates

Ludwig Boltzmann came up with a way of getting a statistical measure of the likelihood of a particular configurations of nuclei and electrons

The Second Law of ThermodynamicsCorollary: At equilibrium, S is maximised

Times Arrow (Arthur Eddington)

The First Law of ThermodynamicsDefinition: Internal energy of the system (U) is the sum of the total potential and the kinetic energies of the atoms in the system

Thermodynamic PotentialsFor adiabatic systems, the amount of work required to change the internal energy of the system is independent of how the work is performedThe system is dependent on its initial and final states but independent of how it got thereHence 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)Addition (or subtraction) of two potentials gives another potential

TS is another potential, and this may be subtracted from UTS may be subtracted and pv may be added to U

And substituting dU shows(S,p indep. variables)(T,v indep. variables)(T,p indep. variables) (S,v indep. variables)Legendre Transformations

Irreversible Processes

Well known thermodynamic fluxes:(Darcys Law)(Ficks Law)(Ohms Law)(Fouriers Law)

Rates of Entropy Production(Darcys Law)(Ficks Law)(Ohms Law)(Fouriers Law)

Lord Kelvin postulated existence of a dissipation potential (D)A potential implies the general reciprocal relationships between thermodynamics forces and fluxes;

Performing a Legendre transform on D

Couplings of Irreversible Processes

Thermodynamic Force (Gradient of Potential)

Flow J

Hydraulic

head

Temperature

Electrical

Chemical concentration

Stress

Fluid

Hydraulic conduction

Darcys law

Thermo-osmosis

Density changes

Electro-osmosis

Chemical osmosis

Density change

consolidation

Heat

Isothermal heat transfer

Thermal conduction

Fouriers law

Peltier effect

Dofour effect

Fully coupled thermoelastcity

Phase change

Current

Streaming current

Thermoelectricity

Seebeck effect

Electric conduction

Ohms law

Diffusion potential and membrane potential

Piezoelectricity

Ion

Streaming current

Thermal diffusion of electrolyte

Soret effect

Electrophoresis

Diffusion

Ficks law

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

EXAMPLE 1

Darcys law and why water flows through soil

Darcys law - flow of water through soil

EXAMPLE 2: Dissolution and precipitation

Dissolution and precipitation

EXAMPLE: reactive transport (i.e. transport with precipitation)

Soil Physical Chemistry 1999: Ed Sparks

Ion Activity Product varies from soil to soil:

Nucleation (Stumm and Morgan 1996)

Change in solubility product with pressureConsider reaction:

Change in solubility product with pressureStandard partial molar compressibilityFor the reaction we have

Change in solubility product with pressure (Langmuir 1997)Change in pressure of 180 bar increases solubility by 50%.

Soil Physical Chemistry 1999: Ed SparksEffect of shear stress

The Advection-Dispersion Equation:

Boundary Conditions

Solute Mass FluxFlux = Advection + Diffusion

Advection and Mechanical DispersionThese processes cause perturbations of soluteconcentration and pore water velocity, hence,

The Advection-Dispersion Equation

Boundary ConditionsAt 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 exampleConsider 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=c0vPorous Mediumc=ceInlet BoundaryOutlet Boundary

The Inlet Boundary ConditionThis boundary condition ensures mass is conserved.

Solute concentration is not continuous at boundary (solute concentration is continuous at the microscopic scale).

The outlet boundaryHowever, this does not agree with experiment.

Experiment agrees with the semi-infinite model evaluated at the point x=L.

Why?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 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.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.

Conclusions The great task that lies ahead of the engineering and the applied sciences this century, is consilience between the different disciplines.