Download Spectral Uncertainty Analysis of Combustion Reaction Systems

Spectral Uncertainty Analysis of Combustion Reaction Systems
using Sparse Adaptive Polynomial Chaos Expansions
Daesang Kim([email protected]) Jie Han Fabrizio Bisetti Aamir Farooq Omar Knio
SRI Center for Uncertainty Quantification in Computational Science and Engineering, KAUST, KSA
Clean Combustion Research Center, KAUST, KSA
1
G(ξ) =
10
Reaction ID
a
0.4
0.2
0
0
326
126
125
Reaction ID
10
(b)
Figure 7: Sensitivity indices of the 22 reactions on (a) [E−]max and (b) t50%,E-.
Only reactions 326 and 125 have good sensitivities for further analyses.
5
1
A/A0
0.6
A = 1.5 A
5
0
0.5
125
0.8
A = 0.5 A0
15
10
126
(a)
A = A0
a0
326
Reaction ID
Ea = Ea0
E = 1.1 E
0.2
0
20
Ea = 0.9 Ea0
0.4
20
Sens. index, t50%,E−
20
0
1.5
0.9
0.95
1
Ea/Ea0
1.05
1.1
A125/A125,0=1.0
17
2
Figure 4: A surrogate model (solid lines) for the maximum slope using
the PC expansion is constructed for A(0.5A0 − 1.5A0), Ea(0.9Ea,0 − 1.1Ea,0),
T (1100K − 1500K), and [H2]0(0.5% − 5%). The plots compare the surrogate
model (solid lines) with SENKIN computation results (crosses) for different A
and Ea values.
(2)
ck Ψk (ξ)
15
x 10
40
1.5
[E−]max
∞
X
5
Figure 3: Sensitivity indices for ±50% uncertainties of the 20 reactions on the
maximum slope using the PC expansion. The index of H+O2 ↔ OH+O is 0.99
and the sum of the other indices is only 0.01.
Max slope
where pf and Ω are respectively the probability density of, and the support
of ξ. G can be represented by a polynomial chaos (PC) expansion
1
0.6
1
0.5
1.1
A125/A125,0=0.5
A125/A125,0=1.5
30
t50%,E−
0
Adaptive pseudo spectral analysis
Ω
Sens. index, [E−]max
0.5
15
• ξ = (ξ1, ..., ξα) is a vector of independent and identically distributed (iid)
random variables, G(ξ) = (G1(ξ), ..., Gβ (ξ)) is a vector of observables, and
{Ψk } is a set of orthogonal polynomials with respect to an inner product
Z
< f, g >=
f (ξ)g(ξ)pf (ξ)dξ
(1)
0.8
Max slope
We applied a sparse adaptive pseudo-spectral method to two model reaction
systems, H2 oxidation and methane ignition in shock tubes, for the investigation of parametric uncertainty in the reaction rates. The non-intrusive algorithm creates polynomial chaos expansions of the reaction systems with
orthogonal polynomial terms, and the expansion coefficients are determined
through a weighted inner product which requires the reaction model realizations at quadrature points. Fast and efficient performance is acheived by
adopting sparse adaptive quadrature grids based on the notions of arbitrary
admissible Smolyak multi-index sets. The method provides sensitivities of
quantities of interest on uncertain reaction rate parameters. Also provided
were efficient surrogate polynomial models, which enabled us to avoid intensive computations of simulation models to perform least squares and Markov
Chain Monte Carlo to demonstrate inferences of uncertain rates from measured data.
Sens. Index
Abstract
20
10
k=0
0
0.5
10
where ck =< G, Ψk > / < Ψk , Ψk >.
Ea/Ea,0
• Numerical integration of ck =< G, Ψk > / < Ψk , Ψk > requires multidimensional quadrature grids.
8
Max. slope
• In this study, ξ is uniformly distributed, Ψk (ξ) are multidimensional Legendre polynomials on Ω = [−1, +1]α, the PC expansion is truncated to have a
finite number of terms.
6
1.0
A326/A326,0
0
0.5
1.5
1.0
A326/A326,0
1.5
Figure 8: A polynomial surrogate model has been constructed for T (2250 K
- 2750 K), [CH4]0 (0.2% - 0.75%), [O2] (1% - 3%), and the 22 reaction rate
constants. The surrogate model (solid lines) are compared with TChem simulations (crosses).
1.0
4
2
17
1100
1300
1500
0.9
0.5
1
A/A0
1.5
3
ξy
Level y
T(K)
1.5
1100K
1500K
0
2
[E−]max(m−3)
4
25
t50%,E−(µs)
1
x 10
1
20
15
10
pdf
0
ξx
1
ξy
Level y
0
2
3
Level x
4
0
0
ξx
1
( b ) Classical Smolyak sparse quadrature whose multi-index set (left)
satifies kx + ky <= 5. The number of the sparse quadrature points is
considerably reduced (right).
1
4
ξy
2
3
Level x
0
1
k1/k1,0
1.05
1.1
• TChem, a software library for numerical simulations of complex reaction
systems, simulates CH4 oxidation in shock tubes, and the reaction mechanism of GRIMECH 3.0 is used for neutral species while 11 ion species and
67 ionization reactions are added to the reaction mechanism.
−1
−1
4
0.95
• Fuel mixtures consists of CH4 (0.2% - 0.75%), O2 (1% - 3%), and Ar as
a diluent. Temperature changes over a range from 2250 K to 2750 K, but
pressure is fixed at 1 atm because pressure has very little effect on ionization reactions.
2
1
0.9
Ion chemistry of methane oxidation
3
1
0.01
Figure 5: Synthetic data of the maximum slope are generated with 10% Gaussian error (95% confidence) at 20 temperatures uniformly distributed over the
range from 1100 K to 1500 K (red points on the top left). On the top right
are shown 10,000 sample points from an adaptive Markov chain Monte Carlo
(H. Haario et. al., 2001) for a posterior distribution of A and Ea from the uniform prior on the A - Ea domain and Gaussian likelihood. The rate constant
k is computed by the Arrhenius equation at 1100 K and 1500 K and the 95%
confidence intervals of k are ±4.7% and ±4.4% respectively (bottom right).
2
1
0.005
σ2ε
3
−1
−1
0
ξx
2200
2400
2600
T(K)
2800
1.5
( a ) Full tensor product of 1D Gauss-Kronrod-Patterson quadrature of
level 4. Multi-index notation (left) and grid points (right) of the quadrature.
1
4
1
2800
1
• A reaction rate constant is determined by the Arrhenius equation, k =
AT b exp(−Ea/RT ). In this study, only the pre-exponential factor is an uncertain parameter and there are 22 uncertain parameters.
( c ) An example of an adaptive sparse quadrature whose multi-index
set (left) is admissible.
1.0
0.5
0.5
1
A326/A326,0
1.5
Figure 9: Synthetic data of [E−]max (top left) and t50%,E- (top right) are produced at 20 different temperatures and the synthetic data have 10% Gaussian random errors (95% confidence). An adaptive Markov chain Monte Carlo
(MCMC) random simulation has been applied to a posterior of A326 and A125
from the uniform prior on the domain and Gaussian Likelihood function (bottom). The red curve surrounds the 95% confidence region of the posterior.
17
2
x 10
80
70
1.5
t50%,E−(µs)
4
2400
2600
T(K)
A125/A125,0
2
3
Level x
[E−]max(m )
1
−1
−1
−3
1
Level y
0.5
2200
1
0.5
60
50
40
30
20
10
0
2
3
4
5
[CH ]
4
6
7
0
8
2
3
4
−3
x 10
5
[CH ]
4
6
7
8
−3
x 10
1.5
Table 1: Ionization reactions
ID
Reaction
Uncertainty
326 CH + O ↔ HCO+ + E −
±50%
331 H3O+ + E − ↔ OH + H + H
±25%
332 H3O+ + E − ↔ H2 + OH
±25%
350
OH − + H ↔ H2O + E −
2.0±
349
OH − + O ↔ HO2 + E −
±50%
126
CH + H2 ↔ H + CH2
±50%
93 OH + CH2 ↔ CH + H2O
±25
125
CH + O2 ↔ O + HCO
±50
...
H2 oxidation in shock tubes
• Goal: to determine the rate constants k of H+O2 ↔ OH+O. The rate constant k has two parameters, pre-exponential factor A and activation energy
Ea and is evaluated by the Arrhenius equation k(T ) = A exp(−Ea/RT ).
• The maximum slope from the graph of the measured H2O determines a
value of the reaction rate constant k through a relation between the maximum slope and k provided by a simulation model.
1.2k
10
10
0
Acknowledgement
0.05
0
0
x 10
6
[E−] (m−3)
0.1
Max. slope
[H2O] (%)
0.8k0
4
[E−]max
5
1
time (ms)
2
1.5
15
8
0.15
1
15
16
x 10
k0
0.5
0.5
Figure 10: Synthetic data of [E−]max (top left) and t50%,E- (top right) are produced at 20 different values of [CH4]0 by adding 10% Gaussian random errors
(95% confidence). The same MCMC simulation has been applied (bottom)
and the 95% confidence region surrounded by the red curve is smaller than
that in Figure 9.
16
0.2
1.0
2
0.5
Daesang Kim, Fabrizio Bisetti, and Omar Knio are members of the KAUST
SRI Center for Uncertainty Quantification in Compuational Science and Engineering.
5
t50%,E−
1.0
k/k0
1.5
0
0
20
40
time (µs)
Figure 2: Time histories of H2O concentrations (left) from three SENKIN simulations at the nomial value k0 from Hong et al (2011), and 0.8k0 and 1.2k0. The
observable is the maximum slope of each curve and the simulation results at
different k values have different maximum slopes. On the right is shown the
relation between the maximum slope and k.
60
0
0
20
40
60
time(µs)
80
100
Figure 6: A time history of [E−] from a TChem simulation of CH4 oxidation
(left). The observables are the maximum concentration [E−]max of E− and the
time t50%,E − between [E−]max and the half of it. TChem simulation realizations
at all quadrature points for adaptive sparse pseudo spectral projection. (right)
Future Work
• MCMC simulations with more parameters of uncertain reaction rate constants and various error sources.
• Bayesian experimental design study on the methane oxidation.
• Shock tube experiments and uncertainty quantification of the reaction rate
constants via the experiment data.