The Thermochemistry Library THERMOCHIMICA Markus H.A. Piro,
April 2014
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Outline Introduction Background Applications and capabilities
Example problem Numerical methods and algorithms Accessing software
Future plans Summary
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Introduction THERMOCHIMICA is an open-source software library
for computing thermodynamic equilibria with the primary purpose of
direct integration into multi-physics codes. The software is
written in Fortran and it can be called from a Fortran, C, or C++
Application Programming Interfaces (API) on a desktop workstation
or high performance computing environment. Software development
began during PhD at RMC*, it evolved during a Post- Doctoral
fellowship at ORNL and it is currently being maintained by M.H.A.
Piro. * M.H.A. Piro, Computation of Thermodynamic Equilibria
Pertinent to Nuclear Materials in Multi-Physics Codes, PhD Thesis,
Royal Military College of Canada (2011).
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Brief Background Conditions for thermodynamic equilibrium:
Gibbs Phase rule, Conservation of mass, and Gibbs energy of a
closed system at constant T & P is a global minimum (derived
from first and second laws of thermodynamics). Thermodynamic
equilibrium is assumed (i.e., time dependency is not considered).
The appropriateness of this assumption is problem specific. This is
generally a good assumption when temperature is high and time scale
is long. * M.H.A. Piro, Computation of Thermodynamic Equilibria
Pertinent to Nuclear Materials in Multi-Physics Codes, PhD Thesis,
Royal Military College of Canada (2011).
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Applications The software is intended to provide input to
material properties and boundary conditions for continuum mechanics
and phase field simulations. THERMOCHIMICA can be used for various
applications: Combustion Metallurgy Geochemistry Batteries Nuclear
materials
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Species mole fraction Chemical Potential Element Mass Database
Gibbs energy Moles of Phases Enthalpy Heat capacity Pressure
THERMOCHIMICA Temper- ature Input Output I/O
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Applications of Thermochimica to Nuclear Engineering
Applications Fuel performance and safety analysis: Fuel chemistry,
Fuel melting, Fission gas retention (predicting fission product
speciation), Iodine-induced stress corrosion cracking (I-SCC) /
Pellet-cladding interaction (PCI), and Zirconium hydriding.
Potential applications (more development needed): Aqueous
chemistry: CRUD formation, fuel storage, fuel transportation.
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Capabilities Parse ChemSage data-files as input. Data-files
containing a maximum of 48 chemical elements, 1500 chemical species
and 24 solution phases. Thermodynamic models: Pure condensed
phases, Ideal solution phases, Substitutional Kohler-Toop model
with regular polynomials, Substitutional Redlich-Kister-Muggiano
model with Legendre polynomials, and Compound energy formalism with
Legendre polynomials (up to 5 sublattices).
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Compound Energy Formalism UO 2 Fluorite Crystal Structure
M.H.A. Piro, PhD Thesis, Royal Military College, 2011. Reproduced
from D.R. Olander, U.S. Dept. of Commerce, 1976.
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Compound Energy Formalism UO 2 Fluorite Crystal Structure
M.H.A. Piro, PhD Thesis, Royal Military College, 2011. Reproduced
from D.R. Olander, U.S. Dept. of Commerce, 1976. Non-stoichiometric
UO 2x (U 3+, U 4+, U 5+, O 2- ) Modelled with three sublattices by
C. Gueneau et al, J. Nucl. Mater., 419 (2011) 145-167. This
treatment is being expanded to represent irradiated fuel by T.M.
Besmann et al, to be published.
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Example Nuclear Fuel Thermochemistry Engineering motivation:
Extend PWR fuel to very high burnup (i.e., ~ 100 GWd/t(U)).
Maximize performance and safety. Experiments are extremely
time-consuming (i.e., ~10 years in reactor + ~5 years in storage),
expensive and dangerous. Simulations may help guide/minimize
experiments. Description of problem: Fuel irradiated in European
PWR to 100 GWd/t(U). Oxidation and compositional measurements
performed at ITU (Germany). Numerical simulations predict fuel
behaviour (chemistry, isotopic evolution and heat transfer).
Coupled: AMP, Origen-S and Thermochimica. M.H.A. Piro, J. Banfield,
K.T. Clarno, S. Simunovic, T.M. Besmann, B.J. Lewis and W.T.
Thompson, J. Nucl. Mater., 441 (2013) 240-251.
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Example Nuclear Fuel Thermochemistry Cont Oxygen partial
pressure predictions with Thermochimica are in very good agreement
with experimental measurements. Most codes that account for fuel
chemistry assume fresh fuel. M.H.A. Piro, J. Banfield, K.T. Clarno,
S. Simunovic, T.M. Besmann, B.J. Lewis and W.T. Thompson, J. Nucl.
Mater., 441 (2013) 240-251.
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Example Nuclear Fuel Thermochemistry Cont O/M cannot be
measured directly. Experimentally inferred values for O/M were
derived by ICP-MS, EPMA and assumptions regarding phase equilibria.
M.H.A. Piro, J. Banfield, K.T. Clarno, S. Simunovic, T.M. Besmann,
B.J. Lewis and W.T. Thompson, J. Nucl. Mater., 441 (2013)
240-251.
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Example Nuclear Fuel Thermochemistry Cont SEM in high burnup
structure [~75 GWd/t(U)] Noble metal HCP white phase Figure kindly
provided by T. Wiss and V.V. Rondinella (ITU) M.H.A. Piro, J.
Banfield, K.T. Clarno, S. Simunovic, T.M. Besmann, B.J. Lewis and
W.T. Thompson, J. Nucl. Mater., 441 (2013) 240-251.
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Numerical Methods From a mathematical point of view, this is a
numerical optimization problem of a non-convex function with linear
and non-linear equality and inequality constraints. Also, the
active set of constraints change throughout the iteration process.
The overall objective is to minimize the integral Gibbs energy of
the system subject to the mass balance constraints and Gibbs Phase
Rule. Numerical methods employed by THERMOCHIMICA are described in
the literature (1-4). 1. M.H.A. Piro and S. Simunovic, CALPHAD, 39
(2012) 104-110. 2. M.H.A. Piro, S. Simunovic, T.M. Besmann, B.J.
Lewis and W.T. Thompson, Comp. Mater. Sci., 67 (2013) 266-272. 3.
M.H.A. Piro, T.M. Besmann, S. Simunovic, B.J. Lewis and W.T.
Thompson, J. Nucl. Mater., 414 (2011) 399-407. 4. M.H.A. Piro and
B. Sundman, to be published.
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Numerical Methods Overview Initialization Local Minimization
Global Minimization Leveling & Post-Leveling algorithms Gibbs
energy minimization algorithm Modified branch and bound
algorithm
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Numerical Methods Initialization (Leveling) A procedure is
required to initiate the non-linear solver. The Leveling algorithm
of Eriksson & Thompson is first used. The premise is to
temporarily drop the non-linear terms(i.e., mixing) from the
chemical potentials, converting this to a linear minimization
problem. G. Eriksson and W.T. Thompson, CALPHAD, 13(4) 1989
389-400. M.H.A. Piro and S. Simunovic, CALPHAD, 39 (2012) 104-110.
Initialization Local Minimization Global Minimization One can then
compute the chemical potentials of the system components directly.
An iterative process is required to determine a unique assemblage
of stable phases (i.e., species are treated as pure phases).
00
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Numerical Methods Initialization (Post-Leveling) The
Post-Leveling algorithm of Piro and Simunovic is then used to
improve upon the estimates from Leveling. The premise is to include
the ideal mixing terms of only the dominant species, which are
treated numerically as phases. M.H.A. Piro and S. Simunovic,
CALPHAD, 39 (2012) 104-110. Initialization Local Minimization
Global Minimization 0 Performance is enhanced with this algorithm.
0 ~300%! ~40%
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Numerical Methods Local Minimization Minimize the integral
Gibbs energy of the system (i.e., 1 st and 2 nd law of
thermodynamics). Minimize a system of Lagrangian multipliers:
Subject to the following linear equality constraints (i.e.,
conservation of mass): and inequality constraints (i.e.,
non-negative mass and Gibbs Phase Rule): M.H.A. Piro, S. Simunovic,
T.M. Besmann, B.J. Lewis and W.T. Thompson, Comp. Mater. Sci., 67
(2013) 266-272. Initialization Local Minimization Global
Minimization
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Numerical Methods Global Minimization At equilibrium, the
following must be satisfied for all stables phases: M.H.A. Piro and
B. Sundman, to be published. M.H.A. Piro, T.M. Besmann, S.
Simunovic, B.J. Lewis and W.T. Thompson, J. Nucl. Mater., 414
(2011) 399-407. Initialization Local Minimization Global
Minimization and the following must be satisfied for meta-stable
phases:
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Numerical Methods Global Minimization A modified branch and
bound approach has been adopted for the non-linear inequality
constraints. This is tested when a local minima has been reached.
Minimize the following function (for meta-stable phases): M.H.A.
Piro and B. Sundman, to be published. Initialization Local
Minimization Global Minimization Which is subject to the following
linear equality and inequality constraints: By exploiting the fact
that the variables (i.e., x) are linearly constrained (i.e.,
bounded), the domain is decomposed into multiple sub-domains (i.e.,
branches) to search for a global minimum.
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Numerical Methods Updating the Phase Assemblage Throughout all
of the foregoing processes, provisions must be made to allow for
the predicted assemblage of stable phases to change. This is the
most challenging component of the entire programming:
Singularities, Cyclical sets of constraints, Inefficiencies
resulting from poor choices, Initialization Local Minimization
Global Minimization M.H.A. Piro and S. Simunovic, CALPHAD, 39
(2012) 104-110. The Euclidean Norm Method of Piro & Simunovic
greatly accelerates the process. Compute the Euclidean norm in
multi- dimensional space between the composition of the phase to be
added to the system relative to all other existing phases. Very
simple and inexpensive. ~30% ~120%
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Accessing the code The software is maintained on a Subversion
(SVN) repository and can be accessed online:
https://sites.google.com/site/thermochimica/
https://sites.google.com/site/thermochimica/ Prerequisites: BLAS /
LAPACK linear algebra libraries Fortran compiler (gfortran/Intel)
Operating system: Intended for Linux/Mac OS-X
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Current and Future plans (Piro) A database conversion tool is
under development to convert between various established formats
(i.e., TDB, DAT). A thermodynamic model optimization tool is being
developed to facilitate model development.
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Summary THERMOCHIMICA is an open-source thermodynamic
equilibrium solver for integration into multi-physics codes to
provide material properties and boundary conditions. THERMOCHIMICA
can be used for a multitude of applications, including combustion,
metallurgy, geochemistry, nuclear materials and batteries. Please
feel free to contact me ([email protected]) should you have any
[email protected]