Sensitivity to Experimental Uncertainty in Surrogate Descriptions Pavan B. Govindaraju Matthias Ihme Stanford University US National Combustion Meeting‘17 April 25, 2017 University of Maryland Special thanks to Tim Edwards, AFRL CRECK Modeling Group in Politecnico Di Milano Sponsors : FAA and ASCENT Thanks to NERSC for computing resources

Motivation Gas turbine and IC engines use liquid hydrocarbon fuels - mixtures of many compounds Surrogate approach for fuels - palette represents entire fuel Two different notions : experimental and computational Further division of concepts : physical and chemical surrogates Consistent descriptions of various surrogates is required Experimental uncertainty is not incorporated in surrogate compositions Objective : Develop rigorous computational methods for construction of surrogate Consistent inclusion of both physical and chemical properties Focus on distillation curves (volatility) Incorporate experimental uncertainty into surrogate descriptions Sensitivity analysis of surrogate compositions to uncertainty

Consistent distillation curves

Physical properties in surrogates Notion of physical surrogate [1] - designed to reproduce Density, Molecular weight, H/C ratio (computable) Volatility : historically done by `matching’ distillation curve Computational distillation curves : two approaches Flash Fractional one equilbrium interface infinite equilbrium interfaces [1] Violi et al., CST, 2002

Distillation Curve - Algorithm Flash Fractional For each T Calculate K for each species and note boiling Note mass fraction boiled compared to initial mass Need to solve

Matching experimental and computational curves Experimentally obtained curves D86, Advanced Distillation Curve (ADC) [1] These are, at best, bounded between flash and fractional limits [2] as transient effects still exist and emulate the effect of finite equilibrium interfaces D2887 standard circumvents this - exactly found to match computational fractional distillation limits (because injection and oven temperature are now identical) Provides consistent description between computational and experimental volatility and hence, physical properties POSF 10325 [2,3] [1] Bruno et al., Energy & Fuels, 2008 [2] Govindaraju, Ihme, IJHMT, 2016 [3] Edwards, Internal Communication

Utilizing Experimental Uncertainty

Necessity to use experimental uncertainty Computational approaches [1,2] to surrogate construction rely on an optimization procedure and assume exact target properties Visualizing the optimization procedure: Physical properties like molecular weight and H/C ratio are linear properties and create a `polytope’ for the feasible region in composition space Ignition delay time (IDT) can be experimentally indistinguishable within this feasible region. Enforced MW and H/C constraint IDT error ~ 15% marked in brown JP-8 Surrogate from Vasu et al. [3] JP-8 Surrogate from Violi et al. [4] [1] C. Mueller et al., Energy & Fuels, 2012 [2] Sarathy and others, Fuel, 2015 [3] Vasu et al., CnF, 2007 [4] Violi et al., CST, 2002

Utilizing experimental uncertainty Combustion property targets (CPTs) are constraints with error thresholds Linear constraints => Convex feasible region => Easy optimization Non-linear constraints - approximate feasible region using convex hull Target Threshold Composition Example : Distillation Curve Error

Utilizing experimental uncertainty - II Feasible region is now a `high-dimensional polytope’ Not amenable to use as description for surrogates The objective function can be cleverly designed to reflect more information about the feasible region Most of all, let’s keep the problem convex Experimental errors : reported as error bars Can we design `error bars’ for surrogate composition? Geometrically, these represent (hyper)cuboids in composition space Objective : `best-fit’ cuboids for feasible regions of surrogates Solution to convex optimization problem(s) Finding `best-fit’ objects : diamond cutting problem

Utilizing uncertainty : `best-fit’ (hyper)cuboids Polytopes can be represented using The outer bounding box is just maximum/minimum of each coordinate Solved using multiple convex optimization problems Inner bounding box is also the result of a convex optimization problem

Utilizing uncertainty : Cuboids results CPTs utilized with 15% uncertainty bounds Molecular Weight H/C Ratio Distillation curve error Ignition Delay Time at DCN conditions (22 bar and 833 K) Test case : 4-compound surrogate [1] Experimental composition lies inside computational bounds [1] Dooley et al., CnF, 2012

Utilizing uncertainty : Cuboids summary Palette size independent approach 4 and 5-compound palettes tested Consistent surrogates : experimental compositions all lie within (inner) bounds [1] Violi et al., CST, 2002 [2] Dooley et al., CnF, 2012 [3] Vasu et al., CnF, 2007

Utilizing uncertainty : Other geometric objects `Best-fit’ ellipsoids also solutions of convex optimization problems Ellipsoids described using by center and symmetric matrix in general Two current approaches: Axis-Aligned : Only tolerances needed to describe Smaller packing fractions in general Maximum-Volume : Allows for rotation Needs rotation matrix to completely describe 2D Projections for Dooley et al. surrogate

Utilizing uncertainty : Ellipsoids summary Similar agreement on consistency of experimental surrogates Transformation matrix Q1/2 : eigenvalues gives semi-axes lengths MV ellipsoid has better agreement and higher packing ratios Not as straightforward to check new candidate [1] Violi et al., CST, 2002 [2] Dooley et al., CnF, 2012 [3] Vasu et al., CnF, 2007

Sensitivity analysis of surrogates

Sensitivity analysis of surrogates Previous computational approaches [1,2] use scalar objective function formed using `weighting factors’ for each CPT. Common technique in optimization of vector functions - scalarization Pareto-optimal : No other feasible point better in every component of the objective function Any positive weighting factor gives a pareto-optimal surrogate Weighting factors thus only help explore pareto-optimal surface Directly related to dual optimization problem and sensitivity coefficients [1] C. Mueller et al., Energy & Fuels, 2012 [2] Sarathy and others, Fuel, 2015

Sensitivity analysis of surrogates Objective function of regression-based approaches [1,2] Fi - matching deficit for each CPT This is identical to objective of dual problem for current approach Optimal weighting factors are the minimum solution to the dual problem. Other factors only give a lower bound on optimal Unless strong duality holds, even the best weighting factors don’t give optimal Strong duality holds for current convex approach Weighting factors can now be defined using thresholds and objective functions Equally obtainable by solving the dual problem. [1] C. Mueller et al., Energy & Fuels, 2012 [2] Sarathy and others, Fuel, 2015

Sensitivity analysis of surrogates - II Sensitivity coefficients (or weighting factors/dual optimals) depend on the objective function Should not be used to decide importance of particular CPT - easier if optimal could be evaluated as function of thresholds Multi-parametric optimization approach solves the issue [1] Violi et al., CST, 2002 [2] Dooley et al., CnF, 2012 [3] Vasu et al., CnF, 2007

Sensitivity analysis : Multi-parametric optimization Theory well-developed for linear objective functions [1] H/C ratio is no longer linear - product of εMW and x - both variables Lower and upper bounds derived enforcing MW constraint as example Computations performed using MPT toolbox [2] with Cantera [1] Pistikopoulous et al., Wiley, 2007 [2] Kvasnica et al., 2004

Multi-parametric optimization : Results 4-compound Jet-A surrogate [1] 5-compound JP-8 surrogate [2] Lower and upper bound as function of the error threshold Piecewise linear function agrees with theoretical closed-form solutions [1] Dooley et al., CnF, 2012 [2] Vasu et al., CnF, 2007

Summary Framework to incorporate experimental uncertainty into surrogate description Consistent inclusion of distillation curve Previous work, at best, provided lower and upper bounds D2887 standard aligns with fractional distillation Convex optimization approach for including uncertainty Constraint-based approach Error bounds using hypercuboids and hyperellipsoids Consistent experimental surrogate compositions Sensitivity analysis Regression-based approaches = scalarization Under strong duality, optimal weights = dual solutions = sensitivity coefficients Multi-parametric optimization Motivated by calculation of sensitivity coefficient Lower and upper mole fraction bounds for a linear (MW) constraint

Thank you! Questions?