Kalman filter approach for uncertainty quantification in time-resolved laser-induced incandescence

Paul J. Hadwin; Timothy A. Sipkens; Kevin A. Thomson; Fengshan Liu; Kyle J. Daun

doi:10.1364/JOSAA.35.000386

Journal of the Optical Society of America A
Vol. 35,
Issue 3,
pp. 386-396
(2018)
•https://doi.org/10.1364/JOSAA.35.000386

Kalman filter approach for uncertainty quantification in time-resolved laser-induced incandescence

Paul J. Hadwin, Timothy A. Sipkens, Kevin A. Thomson, Fengshan Liu, and Kyle J. Daun

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Paul J. Hadwin, Timothy A. Sipkens, Kevin A. Thomson, Fengshan Liu, and Kyle J. Daun, "Kalman filter approach for uncertainty quantification in time-resolved laser-induced incandescence," J. Opt. Soc. Am. A 35, 386-396 (2018)

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Abstract

Time-resolved laser-induced incandescence (TiRe-LII) data can be used to infer spatially and temporally resolved volume fractions and primary particle size distributions of soot-laden aerosols, but these estimates are corrupted by measurement noise as well as uncertainties in the spectroscopic and heat transfer submodels used to interpret the data. Estimates of the temperature, concentration, and size distribution of soot primary particles within a sample aerosol are typically made by nonlinear regression of modeled spectral incandescence decay, or effective temperature decay, to experimental data. In this work, we employ nonstationary Bayesian estimation techniques to infer aerosol properties from simulated and experimental LII signals, specifically the extended Kalman filter and Schmidt–Kalman filter. These techniques exploit the time-varying nature of both the measurements and the models, and they reveal how uncertainty in the estimates computed from TiRe-LII data evolves over time. Both techniques perform better when compared with standard deterministic estimates; however, we demonstrate that the Schmidt–Kalman filter produces more realistic uncertainty estimates.

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Notation	Value	Unit	Notes
$ρ_{s} (T)$	1.9	$g / {cm}^{3}$	Liu model [31]
$c_{s} (T)$	$- 9.7768 \times 10^{- 4} + 2.7943 \times 10^{- 4} T + 1.4554 \times 10^{- 5} T^{2} - 3.4432 \times 10^{- 8} T^{3}$	J/gK	Taken from Liu model [31]
$α_{M}$	0.77	—	[38]
$m_{v} (T)$	$17.179 + 6.8654 \times 10^{- 4} T + 2.9962 \times 10^{- 6} T^{2} - 8.5954 \times 10^{- 10} T^{3} + 1.0486 \times 10^{- 13} T^{4}$	g/mol	Liu model [31]: Fits to data from [34]
$Δ H_{v} (T)$	$2.05398 \times 10^{5} + 7.3660 \times 10^{2} T - 0.40713 T^{2} + 1.1992 \times 10^{- 4} T^{3} - 1.7946 \times 10^{- 8} T^{4} + 1.0717 \times 10^{- 12} T^{5}$	J/mol	Liu model [31]: Fits to data from [34]
$p_{v} (T)$	Clausius–Clapeyron, Eq. (9)	atm	$p_{ref} = 1 atm$ , $T_{ref} = 3915 K$
$α_{T}$	0.37	—	Liu model [31]
$p_{g}$	1	atm	—
$T_{g}$	1800	K	—
$F_{0}$	1	$mJ / {mm}^{2}$	—
$E (m_{λ_{lsr}})$	0.4	—	Liu model [31]

Parameter	Mean	Standard Deviation	Unit	Variable Type
$T_{eff}$	3550	880	K	State
$f_{v}$	2.5	0.7	ppm	State
$d_{p, g}$	30	8	nm	State
$σ_{g}$	1.25	0.3	—	State
$ρ_{s}$	1.8	0.2	$g / {cm}^{3}$	Nuisance
$c_{s}$	2.1	0.2	J/gK	Nuisance
$T_{g}$	1800	300	K	Nuisance
$E (m_{λ})$	0.36	0.03	—	Nuisance
$Δ H_{v}$	7.9078	0.8	$10^{5} J / mol$	Nuisance
$m_{v}$	36	4	g/mol	Nuisance
$α_{M}$	0.75	0.08	—	Nuisance

$E (m_{λ_{1}})$	$E (m_{λ_{2}})$	$f_{v}$	$d_{p, g}$	$σ_{g}$
0.356	0.376	3 ppm	15 nm	1.15

Notation	Value	Unit	Notes
$ρ_{s} (T)$	1.9	$g / {cm}^{3}$	Liu model [31]
$c_{s} (T)$	$- 9.7768 \times 10^{- 4} + 2.7943 \times 10^{- 4} T + 1.4554 \times 10^{- 5} T^{2} - 3.4432 \times 10^{- 8} T^{3}$	J/gK	Taken from Liu model [31]
$α_{M}$	0.77	—	[38]
$m_{v} (T)$	$17.179 + 6.8654 \times 10^{- 4} T + 2.9962 \times 10^{- 6} T^{2} - 8.5954 \times 10^{- 10} T^{3} + 1.0486 \times 10^{- 13} T^{4}$	g/mol	Liu model [31]: Fits to data from [34]
$Δ H_{v} (T)$	$2.05398 \times 10^{5} + 7.3660 \times 10^{2} T - 0.40713 T^{2} + 1.1992 \times 10^{- 4} T^{3} - 1.7946 \times 10^{- 8} T^{4} + 1.0717 \times 10^{- 12} T^{5}$	J/mol	Liu model [31]: Fits to data from [34]
$p_{v} (T)$	Clausius–Clapeyron, Eq. (9)	atm	$p_{ref} = 1 atm$ , $T_{ref} = 3915 K$
$α_{T}$	0.37	—	Liu model [31]
$p_{g}$	1	atm	—
$T_{g}$	1800	K	—
$F_{0}$	1	$mJ / {mm}^{2}$	—
$E (m_{λ_{lsr}})$	0.4	—	Liu model [31]

Parameter	Mean	Standard Deviation	Unit	Variable Type
$T_{eff}$	3550	880	K	State
$f_{v}$	2.5	0.7	ppm	State
$d_{p, g}$	30	8	nm	State
$σ_{g}$	1.25	0.3	—	State
$ρ_{s}$	1.8	0.2	$g / {cm}^{3}$	Nuisance
$c_{s}$	2.1	0.2	J/gK	Nuisance
$T_{g}$	1800	300	K	Nuisance
$E (m_{λ})$	0.36	0.03	—	Nuisance
$Δ H_{v}$	7.9078	0.8	$10^{5} J / mol$	Nuisance
$m_{v}$	36	4	g/mol	Nuisance
$α_{M}$	0.75	0.08	—	Nuisance

Abstract

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Equations (30)

Journal of the Optical Society of America A