Modeling Hydrogen Transport and Precipitation of Hydrides in Zircalloy using Thermo-Chemistry Solver in BISON Theodore M. Besmann (Principal Investigator) Stewart L. Voit Markus H. A. Piro Srdjan Simunovic (presenting) April 30, 2014 2 Outline Project highlights Introduction to thermodynamics and T HERMOCHIMICA New implementation of T HERMOCHIMICA and reaction-diffusion equation for Zr hydride precipitation in BISON Examples 3 Highlights This project contributes to the NEAMS vision by: Developing an efficient thermochemical code, T HERMOCHIMICA, for coupling to depletion, transport and material models Utilizing thermochemical models to simulate chemical evolution in nuclear fuel and reactor components Providing information to various material and process models The tasks for FY13 were: Advancement of T HERMOCHIMICA utilizing fuel models/databases Integration of T HERMOCHIMICA with BISON Expansion of a thermochemical database for fuel-fission products The main accomplishments in FY13 were: Initial integration of T HERMOCHIMICA with BISON Demonstration of prediction of oxygen potential across oxide fuel radius at high burnup Initial demonstration of hydriding of clad using transport coupled with T HERMOCHIMICA 4 Highlights (cont.) FY14 Tasks Refine/improve coupling of T HERMOCHIMICA with Moose/Bison Couple T HERMOCHIMICA with Marmot Expand T HERMOCHIMICA capability for handling complex thermochemical models Simulate fuel-fission product behavior including transport 5 Thermochemistry Applications in Nuclear Fuel Example applications include: Zirconium hydriding, Irradiated fuel chemistry in-reactor, Irradiated fuel chemistry out-reactor, Fission gas retention, Iodine-induced stress corrosion cracking (I-SCC) / Pellet-cladding interaction (PCI) CRUD formation. SEM in high burnup structure UO 2 fuel Above: Figure kindly provided by T. Wiss and V.V. Rondinella (ITU) Top right: Markowitz, WAPD-TM-351, Bettis Atomic Power Lab., Noble metal HCP white phase. Sunburst formation (hydride blister) in fuel clad. LWR Clad Image 6 Thermodynamic Modeling Thermodynamic databases Define components, species, phases, conditions Define solution models, activities Develop thermodynamic model and database Define initial species, pressures, temperatures Calculate chemical equilibrium Thermodynamic software T HERMOCHIMICA Process model e.g. BISON 7 Thermodynamic Databases FACT (Facility on Analysis of Chemical Thermodynamic) SGTE (Scientific Group Thermodata Europe) 8 Thermodynamics Software with APIs FACTSAGE & ChemApp MTDATA NPL Windows only Temperature O/M Ratio Simulations with ChamApp in 2009 9 History of T HERMOCHIMICA T HERMOCHIMICA is an open-source thermochemistry library. History: Initiated during M.H.A. Piro thesis at RMC, continued during Post-Doc at ORNL, currently maintained by Piro, his students and S. Simunovic. The main source code is maintained on a repository managed by Piro. A copy in the Moose/Bison repository contains additional APIs. The solver has similar capabilities as commercial codes. Several algorithms have been advanced/developed to speed up the computations and make the solver more robust. M.H.A. Piro, Computation of Thermodynamic Equilibria Pertinent to Nuclear Materials in Multi- Physics Codes, PhD Thesis, Royal Military College of Canada (2011). M.H.A. Piro, S. Simunovic, T.M. Besmann, B.J. Lewis and W.T. Thompson, Comp. Mater. Sci., 67 (2013) 10 Thermochimica: Input / Output Input: Temperature, Hydrostatic pressure, Mass of each chemical element, and Thermodynamic data-file. Output: Which phases are stable and their quantities (e.g., does it rust? does it melt?), What is the composition of each phase (e.g., how much Cs/I is in gas and how much is solid?), Chemical potentials of every component in the system (useful for mass diffusion), and Heat capacity, enthalpy, entropy and Gibbs energy of system (useful for Phase Field simulations). 11 I/O Species mole fraction Chemical Potential Element Mass Database Gibbs energy Moles of Phases Enthalpy Heat capacity Pressure THERMOCHIMICA Temper- ature Input Output 12 Gibbs Energy Minimization in a Nutshell At thermodynamic equilibrium, the Gibbs energy of a closed system at constant temperature and pressure is at minimum. Satisfying conservation of mass, the first and second laws of thermodynamics, and Gibbs phase rule. The numerical procedure determines a unique combination of co- existing phases with a unique composition of constituents that yields a minimum in the integral Gibbs energy function. Different types of thermodynamic models are capable of describing different physical-chemical behavior. Time dependency is not considered. This is generally a good assumption when temperature is high and time scale is long. 13 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) M.H.A. Piro, S. Simunovic, T.M. Besmann, B.J. Lewis and W.T. Thompson, Comp. Mater. Sci., 67 (2013) M.H.A. Piro, T.M. Besmann, S. Simunovic, B.J. Lewis and W.T. Thompson, J. Nucl. Mater., 414 (2011) M.H.A. Piro and B. Sundman, to be published. 14 Numerical Methods Local Minimization Minimize the integral Gibbs energy of the system (i.e., 1 st and 2 nd law of thermodynamics): 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) 15 Numerical Methods Global Minimization At thermodynamic equilibrium, the following linear equality constraints must be satisfied for all stables phases: and the following non-linear inequality constraints must hold true for all other phases in the system (i.e., the system is not metastable. Equivalently, a global minimum has been reached and not a local minima): 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. M.H.A. Piro and B. Sundman, to be published. 16 Current Status / Future Plans of T HERMOCHIMICA Current capabilities: Ideal solution phases and pure condensed phases, Substitutional Kohler-Toop model with regular polynomials, Substitutional Redlich-Kister-Muggiano model with Legendre polynomials, Compound energy formalism with Legendre polynomials (up to 5 sublattices). Parse ChemSage data-file on input. Data-files can contain a maximum of 48 chemical elements, 1500 chemical species and 24 solution phases. Limitations and future work: Only ideal gases can be considered (not real gases), Aqueous phases cannot be considered (needed for CRUD/corrosion), FactSage (commercial software) is required to generate a database, More work is needed in software quality assurance, More work is needed in enhancing computational performance, and Expand user group. 17 Hydride Precipitation in Zircalloy: Motivation for Modeling As Zirconium is corroding, it is absorbing a fraction of the H released by the oxidation reaction H precipitates as hydrides when the solubility limit is exceeded Zr-base system phases: hcp, low-temperature phase, and bcc high-temperature phase have different H solubility and mobility Crack grows by increments that are proportional to the size of hydride region. Sub-critical crack growth mechanism (DHC). The large local volume mismatch associated with hydride formation makes it difficult to measure a true equilibrium value of TSS. 18 Hydrogen Transport and Hydride Formation Heat Diffusion ( T ) Hydrogen Transport ( H ) Hydride Precipitation ( ZrH x ) Use material model to calculate thermo-chemical equilibrium using Thermochimica Calculate source/sink, Q, for hydrogen in transport equation based on the material model data First order model Other physics (a.k.a. kernels) and material models can be now easily added Make sure thermo-chemical equilibrium is calculated once per integration point because it is expensive 19 Thermodynamic Model for Zr-H System (FactSage File) Calculations were in a close agreement with the commercial thermochemical equilibrium solver FactSage 20 Implementation of Material Model for Zr Hydride Formation void MaterialHZrH::computeQpProperties(){ FORTRAN_CALL(Thermochimica::ssinitiatezrhd)(); // read model data file, read only once HinZrHppm = _hhydride_old[_qp]; // H in ZrH, immobile, ppm Hppm = _transported[_qp]; // H in solid solution, mobile, ppm HTotalppm = Hppm + HinZrHppm; FORTRAN_CALL(Thermochimica::settemperaturepressure)(&Temperature, &Pressure); // set T, p iElement=40; FORTRAN_CALL(Thermochimica::setelementmass)(&iElement, &dBMol); // Zr iElement=1; FORTRAN_CALL(Thermochimica::setelementmass)(&iElement, &dAMol); // H FORTRAN_CALL(Thermochimica::thermochimica)(); // calculate thermochemical equilibrium iH=2; // get second element in the model, H char s1[] = "ZRH2_DELTA,ZRH2_EPSILON"; // Hydride phases to search for in eq. solution FORTRAN_CALL(Thermochimica::ssfindsolnspecies)(s1, iPhNameL, iH, dMolSum); // mol of H in s1 dZrHRatio = dMolSum / dAMol; // fraction of hydrides in total moles of H _hhydride[_qp] = HTotalppm * dZrHRatio; // H in hydrides in ppm, new value _dhhydride[_qp] = _hhydride[_qp] - _hhydride_old[_qp]; // difference in hydrides for source rate } Partial source code. 21 Source/Sink for Zr Hydride Formation template InputParameters validParams () { InputParameters params = validParams (); params.addParam ("scaling", 1.0, "The scaling factor for the source."); return params; } HZrHSource::HZrHSource(const std::string & name, InputParameters parameters) : Kernel(name, parameters), _scaling(getParam ("scaling")), _dhhydride(getMaterialProperty ("dhhydride")) {} Real HZrHSource::computeQpResidual() { Real value; value = _dhhydride[_qp] / _dt; value *= _scaling; return _test[_i][_qp]*value; } Entire source code! 22 Examples One element problem Multiple-elements 23 Example Model Parameters Thermal Model = 6551 kg/m 3 k = 16 W/m-K c = 330 J/kg-K T=573K (573K-673K for larger problems) Hydrogen Diffusion Coefficient Kearns, J of Nuclear Materials, 43, 1972 D 0 = 7.9e-7 m 2 /sec Q = -4.49e+4 J/mol Hydrogen Heat of Transport Sawatzky, J of Nuclear Materials, 2, 1960 Q* = 2.51e+4 J/mol Thermo-Chemical Equilibrium Model FactSage 24 Problem Data H content based on 50 MWd(kgU) -1, 400 wppm P.Bouffioux (EDF R&D) B.Cheng (EPRI) 25 Problem Data H content dependence on radiation A. McMinn et al., "The Terminal Solubility of Hydrogen in Zirconium Alloys", 1998, ASTM-STP- 1354, pp H solubility B. F. Kammenzind et al., Hydrogen Pickup and Redistribution in Alpha-Annealed Zircaloy-4, WAPDT-3047, Bettis Atomic Power Laboratory, 1995. 26 Single Element Boundary Condition T=573K Initial Condition H ss = 400 wppm H ZrHx = 0 wppm End result H ss = 53.5 wppm H ZrHx =346.5 wppm 27 Element Strip, 20 FE Boundary Conditions Left T=573K, Right T=673K Initial Condition H ss =400 wppm T=0, H ss T=0, H ZrHx T=10000, H ss T10000, H ZrHx 28 Animation of H in Solid Solution 29 Animation of H in ZrH x-2 30 Animation of H in Solid Solution Double FEM Discretization 31 Animation of H in ZrH x-2 Double FEM Discretization 32 Circular Strip Simulation Boundary Conditions Outer T=573K, Inner T=673K H ss H ZrHx Initial Condition H ss =400 wppm 33 Summary New implementation of T HERMOCHIMICA in Bison enables transport and reaction simulations Can provide material data to support other physics models The approach is also applicable to the systems with larger number of diffusing species as long as the thermodynamic calculations are done only as needed. Thermodynamic model gives results similar to the ones reported in the literature New models are being developed for the fuel chemistry Fuel chemistry models will be implemented with the transport models using the same approach 34 Questions 35 Hydriding in Zircalloy 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. 36 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). 37 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). 38 As Fab 0 MWd/t thermal shock 1 MWd/t 10 MWd/t 100 MWd/t 10 3 MWd/t 10 4 MWd/t 10 5 MWd/t 10 6 MWd/t Adapted from Christensen, J.A. Trans. Am. Nucl. Soc., 15 (1972) 214. High burnup structure MOX Fuel Shows Extensive Phase and Microstructure Changes Burnup Rondinella, Wiss, Mat. Today, 13 (2010) 39 Isotopic Modeling of Nuclear Fuel Fresh Fuel Irradiated Fuel UO 2 Isotopes 235 U, 238 U and 16 O Isotopes 235 U, 238 U and 16 O + >2000 isotopes r/R Power ORIGEN-S: Depletion Decay Transmutation Neutron flux depression Rim effect 40 Adapted from D.R. Olander, Fundamental Aspects of Nuclear Reactor Fuel Elements, U.S. Dept. of Commerce (1976). Thermodynamic Modelling of Nuclear Fuel Fresh Fuel Irradiated Fuel UO 2 Elements U and O Gas (Xe, Kr, Cs, ) Elements U, O, Pu, Xe, Zr, Mo, Ru, Nd, Ce, Cs, Sr, Ba, Pd, La, Tc, Pr, Y, Rh, Kr, Sm, Rb, Te, Np, I, Pm, Nb, Ag, Se, Eu, Cd, Sn, Gd, Br, Am, Sb, Hg, In, Cu, Cm, Tb, Ge, Dy, As, Ho, Er, Th, Zn, Pa, Ga, Pb, Ra, Ac, Bk, Cf, Bi, Po, Rn, Tl, Es, Fr, At Complex oxide inclusions Fluorite oxide Noble metal inclusions (i.e., white phase) Various other phases Fluorite oxide 41 Outline Project Highlights Introduction to Thermochimica New implementation of Thermochimica and reaction-diffusion equation in BISON Examples 42 Outline Project Highlights Introduction to Thermochimica New implementation of Thermochimica and reaction-diffusion equation in BISON Examples 43 Applications to Nuclear Engineering Applications Fuel performance and safety analysis: Fuel chemistry (akin to previous slides), 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. 44 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) M.H.A. Piro and S. Simunovic, CALPHAD, 39 (2012) 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 45 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) Initialization Local Minimization Global Minimization 0 Performance is enhanced with this algorithm. 0 ~300%! ~40 % 46 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) Initialization Local Minimization Global Minimization 47 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) Initialization Local Minimization Global Minimization and the following must be satisfied for meta-stable phases: 48 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. 49 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) 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% 50 Thermodynamic Models in a Nutshell UO 2 UO 3 UO Regular substitutional model Multi-sublattice model M.H.A. Piro, PhD Thesis, Royal Military College, UO 2x is treated as a mechanical mixture of UO, UO 2 and UO 3 molecules. UO 2x is modelled with 3 sublattices, ionic charge and defects are considered. More realistic (added in FY13) Simpler (added in FY12) 51 Thermodynamic Model Predictions U-O Phase Diagram O 2(g) Partial Pressure Predictions C. Gueneau, N. Dupin, B. Sundman, C. Martial, J.-C. Dumas, S. Gosse, S. Chatain, F. DeBruycker, D. Manara and R.J.M. Konings, J. Nucl. Mater., 419 (2011) 52 Applications of Thermochimica in Nuclear Fuel Simulations Simulation below coupled AMP/Origen-S/Thermochimica. UO 2 fuel; 3.5% enriched; 100 GWd/t(U) burnup. 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) Simulation predictions can be improved with more sophisticated thermodynamic models developed at ORNL. Since this publication, Thermochimica is now able to handle these models.