1. 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

2. 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

3. Consistent distillation curves

4. 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

5. 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

6. 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 [2,3]
[1] Bruno et al., Energy & Fuels, 2008
[2] Govindaraju, Ihme, IJHMT, 2016
[3] Edwards, Internal Communication

7. Utilizing Experimental Uncertainty

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. Sensitivity analysis of surrogates

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. Thank you! Questions?