Predictability of Chemical Systems

Transcription

1 Predictability of Chemical Systems Habib N. Najm Sandia National Laboratories, Livermore, CA, USA Collaborators: M.T. Reagan & B.J. Debusschere O.M. Knio, A. Matta, & R.G. Ghanem O.P. Le Maitre Sandia National Labs, Livermore, CA, USA Johns Hopkins University, Baltimore, MD, USA Univ. d Evry Val d Essonne, Evry, France Acknowledgement: US Dept. of Energy, Basic Energy Sciences, Div. Chemical Sciences, Biosciences, and Geosciences Sandia National Labs, LDRD Defense Advanced Research Projects Agency (DARPA) Workshop on the Elements of Predictability, The Johns Hopkins University, November 13-14, HNN-SNL JHU Pred. 03-1

2 Preliminaries Scope Systems involving chemical reactions, in the gas, liquid, or solid phases Combustion, Atmospheric chemistry, Electrochemistry, Biochemistry, Catalysis,... Definition of predictability A system is predictable with respect to a given observable when we have a model which predicts that observable over the requisite range of operating conditions with known and acceptable accuracy and precision. Accuracy: agreement between mean model predictions and experimental measurements Precision: uncertainty in model predictions resulting from model and parametric uncertainties and numerical discretization errors HNN-SNL JHU Pred. 03-2

3 Uncertainty Quantification (UQ) in Chemically Reacting Flow Rational model validation with respect to experimental measurements requires estimates of ranges of error in each set of data Experimental error-bars are (usually) available How large are the error bars on the computational results? Sources of error/uncertainty in the prediction Model uncertainty Parametric uncertainty Numerical discretization errors Determining that numerical and experimental error bars do not overlap enables a decision on the efficacy of the model as distinct from the role of the parameters Focus on quantification of parameteric uncertainty (UQ) in chemically reacting flow computations HNN-SNL JHU Pred. 03-3

4 Computational Issues in Reacting Flow Modeling Reacting Flow Model Formulation Chemical & Transport Models Low Mach number or Compressible flow Model Spatial Discretization Large range of length scales & sharp moving fronts Adaptive Mesh Refinement (AMR) Turbulence High Order spatial discretization Time Integration Large range of time scales & Stiffness Implicit chemistry Diffusive stability implicit or stabilized explicit Diffusion constructions Convection accuracy, CFL, explicit convection Operator-Splitting splitting errors HNN-SNL JHU Pred. 03-4

5 Governing Dimensionless Low Mach Number Equations (ρv) t ρ t + (ρv) = 0 + (ρvv) = p + 1 Re [ µ[( v) + ( v) T ] 2 3 µ( v)u ] ρc p DT Dt (γ 1) dp o = γ dt + 1 ReP r (λ T ) ρ ReSc N c p,i V i T Da i=1 N h i w i i=1 (ρy i ) t + (ρvy i ) = 1 ReSc (ρy iv i ) + Da w i p o = ρt W i = 1,, N Low Mach No., no body forces, no radiation, mixture-averaged transport Neglect Soret and Dufour effects HNN-SNL JHU Pred. 03-5

6 Dominant Reaction Pathways, GRImech1.2, Atm. CH 4 +Air C 2 H 6 H, OH C 2 H 5 C 2 H 4 C 2 H 3 C 2 H 2 C 2 H H, M H, OH H, M OH O CH 3 + M OH, H, O H + M OH + M H O O OH OH, H CH 3 CH * 2 N 2, CO 2, H 2 O HO 2 O 2 O, O 2 O 2 CH 2 H, OH CH* M CH H 2 O O 2 O 2 CH 3 OH CH 3 O O OH* H M O CO 2 * M CH 2 OH O 2, M CH 2 O HCO H, OH, O, CH 3 H, OH, M, O 2, H 2 O CO OH CO 2 HNN-SNL JHU Pred. 03-6

7 General Approach for Forward Parametric UQ Define the physical model and associated parameters Determine which parameters are uncertain, and by how much Identify the set of important parameters Define a representation for uncertainty Interval Arithmetic, Fuzzy Sets, Stochastic,... Propagate the uncertainty through the model Non-intrusive, Intrusive Error estimation/propagation, Stochastic Sensitivity analysis Evaluate resulting uncertainty in model predictions HNN-SNL JHU Pred. 03-7

8 Sensitivity Analysis For an ODE system: dx = w(x, t; A) dt where X = {X i } is the solution vector, and A = {A k } is the vector of model parameters, the first order sensitivity coefficients, ζ ik = X i A k are integrated in time using the ODE system dζ ik dt = w i A k + j w i X j ζ jk obtained by differentiation of dx i /dt w.r.t. A k. The normalized sensitivity coefficients are given by: S ik = ln X i ln A k = A k X i ζ ik HNN-SNL JHU Pred. 03-8

9 Error Propagation Let the standard deviation in each ln A k be σ k, then the total contribution of each reaction k to the variance in the mole fraction of species i is ( ) 2 zki 2 X i = A k σk 2 A k Variances are additive, and the total variance in the mole fraction of each species i, is the sum of the N reaction contributions: N [σ(x i )] 2 = Issues: Linear... no higher order information or coupling between parameters Intrusive... involves recoding k=1 Concerns about overprediction of output uncertainties z 2 ki HNN-SNL JHU Pred. 03-9

10 Spectral Stochastic UQ Formulation Model uncertain parameters as random variables A stochastic process u(x, t, θ) can be described by : a Polynomial Chaos (PC) expansion in terms of Hermite polynomials Ψ k, their associated Gaussian basis ξ(θ), and spectral mode strengths u k (x, t) P u(x, t, θ) = u k (x, t)ψ k (ξ(θ)) u k (x, t)ψ k (ξ(θ)) k=0 Literature: Wiener : 1938 : Homog. Chaos span of Hermite pol. functionals of a Gaussian process k=0 Cameron & Martin : 1947 : L 2 Convergence for any L 2 stochastic process Ghanem & Spanos : 1991 : Application to UQ in Stochastic Finite Element Method Le Maître et al. : 2001,2002 : Application to Fluid Flow Xiu & Karniadakis : 2002 : Conv. rate for Gaussian/non-Gaussian processes Debusschere et al. : 2003 : Application to electrochemistry in microfluid flow Reagan et al. : 2003 : Application to reacting thermofluid flow J. Comp. Phys. 2001,2002; Phys. Fluids 2003; Comb. Flame 2003 HNN-SNL JHU Pred

11 Intrusive Spectral Stochastic UQ Formulation: ODE Example du Sample ODE with parameter λ: dt = λu, u(0) = u 0 Let λ be uncertain : Represent it as a stochastic quantity Introduce a new dimension ξ, where ξ is a Normal random variable Use P -th order Polynomial Chaos (PC) expansions: λ = P λ k Ψ k (ξ), u = k=0 P u k Ψ k (ξ), k=0 The Ψ k s are orthogonal < Ψ i Ψ j >=< Ψ 2 i > δ ij (λ k known, u k (t) unknown) Substitute PC expansions in the ODE, and apply Galerkin projection: du i dt = λuψ i P P Ψ 2 i λu i = λ p u q C pqi, i = 0,, P p=0 q=0 where the C pqi = Ψ p Ψ q i are known coefficients. HNN-SNL JHU Pred

12 Pseudo-Spectral Implementation Spectral Product : w = uv w = u v w i = uv i, i = 0,, P Psuedo-spectral higher-order polynomial terms : w = λu 2 v w = λ (u (u v)) Division : w = u v vw i = u i, i = 0,, P solve linear equation system for w i Arbitrary functions u = f(x) where u = u(x, u), and u i are algebraically found from x i, u i, i = 0,, P : P (xb ) j P u k (x b ) u k (x a ) = C ijk ( u) i dx j j=0 (x a ) j i=0 HNN-SNL JHU Pred

13 Pseudo-Spectral UQ Formulation: low M 2D Reacting Flow ρv q t T q t + v T q = ρ q t + ρv q = 0 + ρvv q = p q + 1 Re µ[( v) + ( v) T ] 2 3 µ( v)u (γ 1) γρc p dp o + 1 dt q ReP r (λ T ) ρc p q 1 N c p,i V i T Da ReSc c i=1 p q q 1 ρc p N h i w i i=1 q ρy i q t + ρvy i q = 1 ReSc ρy iv i q + Da w i q i = 1,, N Time Integration: Operator-Split reaction-diffusion integration of (P + 1)(N + 1) species and energy/density equations Stochastic Projection Method integration of (P + 1) momentum equations HNN-SNL JHU Pred

14 0D Intrusive H 2 -Air Ignition : Uncertainty in [H 2 O 2 ] 1.5e 10 1e 22 all modes 1.0e 10 c H2 O 2 σ 2 H 2 O 2 5e 23 reaction 5.0e 11 enthalpy 0.0e t (s) H 2 -Air (Yetter et al., 1991), 9 species, 19 reactions 3 rd -order PC, 2 uncertain parameters 0e t (s) Rxn. 11 pre-exponential: A 11 ]A m 11/3, 3 A m 11[ Lognormal distribution H enthalpy: σ hh = 1% Normal distribution Fast rise in the mean [H 2 O 2 ], little amplification in its uncertainty HNN-SNL JHU Pred

15 0D Intrusive H 2 -Air Ignition : Uncertainty in [HO 2 ] Very fast rate of growth of the mean [HO 2 ] Followed by a similarly fast rise in the standard deviation Much larger uncertainty than H 2 O 2 COV of HO 2 is about 40% persists near equilibrium Amplification of enthalpy uncertainty 5e 10 4e 10 3e 10 c HO2 2e 10 1e 10 0e t (s) HNN-SNL JHU Pred

16 Experience with Instabilities When integrating a chemical system, e.g. ignition, regions of explosive mode growth (positive eigenvalues) can lead to instabilities. Instability manifested in the fast growth of higher order modes, and fast drift of the solution towards unphysical values Typically occurs when the standard deviation increases significantly, becoming a sizeable fraction of the mean. Consider a model problem du = u(u + 10)(1 u) dt Attractors at u = 10, u = 1, and a repulsive fixed point at u = 0. Let the initial condition u(t = 0) = U be stochastic, U = P k=0 U kψ k. Integrate the reformulated chaos system for the time evolution of u k, k = 0,..., P HNN-SNL JHU Pred

17 Model Problem: Consequence of Initial PDF tail zero crossing 6.0 U 0 =0.2,0.3; U 1 = U 0 =0.2,0.3; U 1 = PDF(U) u U time HNN-SNL JHU Pred

18 Non-intrusive Spectral Projection (NISP) UQ Formulation Construct spectral stochastic descriptions of uncertain parameters λ Sample parameter space and compute Monte-Carlo (MC) realizations of the deterministic model u i (t), i = 1,..., N Project MC statistics on the spectral mode strengths u k (t) u k = uψ k Ψ 2 k = 1 Ψ 2 k uψ k (ξ)ρ(ξ)dξ, k = 0,..., P Evaluate integral numerically Sacrifices efficiency for reduced complexity and improved stability May require excessively large number of samples to converge Retains spectral sensitivity information Allows designing UQ wrappers around legacy code HNN-SNL JHU Pred

19 Sampling Issues in NISP UQ Need to minimize the number of samples required for evaluating spectral mode strengths Collocation techniques (DEMM, SRSM) Minimize errors at sample points High efficiency : number of samples number of unknowns Galerkin projection (NISP) Minimize RMS error Less efficient but potentially more robust to nonlinearities Projection is a Quadrature operation - samples are quadrature points Latin Hypercube Sampling Gauss-Hermite Quadrature Sparse Quadrature / Cubature HNN-SNL JHU Pred

20 NISP UQ Application: Premixed H 2 -O 2 Chemistry at Super-Critical Water Oxidation (SCWO) Conditions Allow uncertainties in reaction rate constants and thermodynamic properties, per published experimental data Wrap NISP processing around a deterministic reacting flow code Using 8-step simplified SCWO Hydrogen mechanism (McRae) Reaction A n E a /R UF 1. OH + H H 2 O 1.620E H 2 + OH H 2 O + H 1.024E H + O 2 HO E HO 2 + HO 2 H 2 O 2 + O E H 2 O 2 + OH H 2 O + HO E H 2 O 2 + H HO 2 + H E H 2 O 2 OH + OH E OH + HO 2 H 2 O + O E Species µ 0 2σ H OH H 2 O H 2 O HO HNN-SNL JHU Pred

21 0D H 2 -O 2 Isothermal Isobaric SCWO Ignition 2.0e e 12 +σ 6.00e 13 All 1st order c OH (mol/cm 3 ) 1.0e e 13 σ 0.0e t (s) σ (OH) 4.00e e 13 Rxns 7,5 Rxns 8,7,5 Rxn e t (s) Mean and standard deviation predictions validated against published data Initial fast growth in uncertainty followed by a slower approach to a steady-state with large OH uncertainty Reactions 7 & 8 have dominant roles in the OH uncertainty HNN-SNL JHU Pred

22 Convergence with Spectral Order 6.00e 25 3rd order σ 2 (OH) 4.00e e 25 2nd order Nord=3 Nord=2 1st order 0.00e t (s) HNN-SNL JHU Pred

23 1D H 2 -O 2 SCWO Flame NISP UQ/Chemkin-Premix σ Rxns 1,6 Rxns 5,8,1,6 Y OH 0.10 σ σ (OH) Rxns 8,1, All 1st order x (cm) x (cm) 1D freely propagating H 2 -O 2 flame at SCWO conditions Fast growth in OH uncertainty in the primary reaction zone Steady level of uncertainty and mean of OH in the post-flame region Uncertainty in pre-exponential of Rxn.5 (H 2 O 2 +OH=H 2 O+HO 2 ) has largest contribution to uncertainty in the predicted OH field HNN-SNL JHU Pred

24 Higher-Order Terms Rxns 1,6 All modes Rxns 5,8,1,6 σ (OH) Rxns 8,1,6 All 1st order x (cm) HNN-SNL JHU Pred

25 1D H 2 -O 2 SCWO Flame NISP UQ/Chemkin-Premix Y H2 O σ σ (H 2 O 2 ) All 1st order Rxns 5,8,7,6 Rxns 8,7,6 Rxns 7, Rxn 6 σ x (cm) x (cm) Very large uncertainty in H 2 O 2 prediction, COV=σ/µ=100%!! Dominant source is again Rxn.5 (H 2 O 2 +OH=H 2 O+HO 2 ) Not a robust model for predicting H 2 O 2 under SCWO conditions Results highlight the utility of additional experimental measurements of A 5 HNN-SNL JHU Pred

26 Higher-Order Terms σ (H 2 O 2 ) All modes All 1st order Rxns 5,8,1,6 Rxns 8,1,6 Rxns 1,6 Rxn x (cm) HNN-SNL JHU Pred

27 CPU-time Savings with Intrusive Spectral Strategy 0D H 2 -O 2 SCWO ignition NISP standard deviation tends to that from the intrusive construction NISP comes to within 50% of the intrusive value after 6000 Latin- Hypercube realizations 6000 sample runs 48 CPU hrs 1 intrusive run 2 hrs max σ(h 2 O) MC cycles HNN-SNL JHU Pred

28 Discussion State of predictability awareness Useful definitions of predictability, examples Obstacles in the way of predictability What can realistically be done to mitigate these obstacles Benefits of improved predictability, potential sci/tech achievements Elements of a predictability aware meta-model HNN-SNL JHU Pred

29 Predictability Awareness in the Modeling of Chemical Systems Awareness of uncertainty in chemical models, thermodynamic properties, & kinetic rate constants Significant focus on prediction accuracy; only a few studies on precision/uncertainty. Pan et al. Atmospheric Chemistry J. Geophys. Res.-Atmos., Phenix et al. H 2 -O 2 ignition Combustion & Flame, DEMM, Polynomial Chaos, sampling, collocation Isukapalli et al. Environmental & Bio. systems Risk Analysis, Stoch. Resp. Surf. Meth., Polynomial Chaos, Collocation, Regression Turanyi et al. CH 4 -Air flame Phys. Chem. Chem. Phys., Sensitivity analysis, Error propagation Debusschere et al. Electrochemistry & Biodetection Phys. Fluids, Polynomial Chaos, Galerkin, intrusive Reagan et al. H 2 -O 2 flame Combustion & Flame, Polynomial Chaos, Latin Hypercube Sampling, non-intrusive HNN-SNL JHU Pred

30 Useful Definitions of Predictability in Chemical Systems A system is predictable with respect to a given observable when we have a model which predicts that observable over the requisite range of operating conditions with known and acceptable accuracy and precision. Accuracy: agreement between mean model predictions and experimental measurements Precision: uncertainty in model predictions resulting from model and parametric uncertainties and numerical discretization errors Some systems are simply not predictable w.r.t. a given observable, independent of the model, e.g. predictions of local weather patterns. HNN-SNL JHU Pred

31 Obstacles in the way of Predictability of Chemical Systems Identification of chemical kinetic network/mechanism structure Robust model inference under noise and uncertainty Measurement/Computation of chemical rate constants and thermo properties Estimation of uncertainty in chemical/thermo parameters Availability of uncertainties in published/online chemical mechanisms Strong nonlinearities, sensitivity to IC s, high amplification of uncertainties Need for accurate reduced-order/subgrid turbulent reacting flow models CPU intensive reacting flow computations sampling limitations Large size and complexity of chemical models Large ranges of length and time scales (turbulence, thin flames, stiffness) Multiphysics, Heterogeneous Multiscale Models HNN-SNL JHU Pred

32 What can realistically be done to mitigate these Obstacles More experimental/ab-initio data on chemical model structure, chemical rate constants, and thermodynamic properties Development of techniques for robust model inference under noise and uncertainty Development of techniques for identifying important parameters w.r.t. specific observables in complex reacting flow models Development of accurate/efficient formulations and numerical techniques for UQ in higb-dimensional models with strong non-linearities and/or bifurcations Public dissemination of uncertainties in thermochemical model parameters, along with the typically published mean values Development of accurate reduced order models for turbulent combustion and other multiscale reactive systems Development of efficient direct numerical simulation constructions for turbulent reacting flow in laboratory-scale settings HNN-SNL JHU Pred

33 Benefits of improved predictability, new S & T Enhanced understanding of physical, chemical, and biological systems More efficient engineering development of chemically reacting systems... engines, chemical processing plants, etc. HNN-SNL JHU Pred

34 Elements of a predictability-aware model A predictive model has to predict requisite observables with acceptable accuracy and precision. To do this, it is necessary for such meta-models to include accurate models of the physical and chemical phenomena at hand data on nominal values and uncertainties of model parameters means of error/uncertainty estimation in predicted results means of identification of selective parametric contributions to uncertainties in model predictions HNN-SNL JHU Pred