Modeling laser-induced incandescence of soot: a new approach based on the use of inverse techniques

Abstract

Two LII models derived from the literature have been tested to simulate signals provided in a recently published extensive set of experimental data collected in a non-smoking laminar diffusion flame of ethylene. The first model classically accounts for particle heating by absorption and cooling by radiation, sublimation and conduction. The second one also integrates an alternative absorption term that accounts for saturation of the linear, single-photon and multi-photon absorption leading to C₂-photodesorption at high fluence, a heating flux attributable to oxidation and a cooling process based on thermionic emission. Obtained results illustrate that both models fail to reproduce the LII signals experimentally monitored on a wide range of fluences (up to ~1 J cm⁻²) regardless of the value implemented for the main parameters involved in the energy- and mass-balance equations. We therefore originally proposed a new modeling approach based on the use of inverse techniques to gain information about the specific terms that should be integrated into the calculation. The inverse procedure allows inferring the temporal evolution of the soot diameter as well as the temporal and fluence dependence of additional energy rates that have to be considered to fulfill the particle energy and mass balances while providing a complete fit with experimental data. Conclusions issued from the present work indicate that modeling soot LII using only absorption, radiation, conduction and sublimation rates (as these fluxes are generally expressed and computed in the literature) is inadequate to correctly simulate the soot heating and cooling mechanisms over a wide range of fluences. The inverse modeling procedure also gave insights concerning the relevance of integrating photolytic mechanisms such as multi-photon absorption and carbon cluster photodesorption as previously proposed by Michelsen. Additional calculations performed using a new model formulation integrating such processes finally led to predictions merging on a single curve with experimental data. Additional works should be undertaken, however, to complete this first-approach analysis especially to address the large uncertainties existing in the input parameters and equations accounting for photolytic processes that are likely to significantly impact soot LII.

Appendices

Appendix 1: Algorithm of the inverse calculation procedure

The algorithm describing the iterative process implemented to solve the inverse problem can be summarized as follows:

Step 1: Start with an initial guess for $\left( {D_{k = 0} = \left[ {D_{1} , D_{2} , \ldots ,D_{i} , \ldots , D_{{N_{t} }} } \right]^{T} } \right)$ and with a (N _t  × N _t ) positive-definite symmetric matrix H _κ=0. Usually H _κ=0 is taken as the identity matrix I. For D _k=0, the solution obtained using one of the models implemented in Sect. 2 is considered.

Step 2: Solve the direct problem described by Eq. 29 to obtain the time-dependent temperature and compute the LII signal based on Eq. 28.

Step 3: Compute the gradient of the cost function using Eq. 38, the direction of descent based on Eq. 33 and the sensitivity coefficients for each component of the unknown vectors $\left( {D_{k} = \left[ {D_{1} , D_{2} , \ldots ,D_{i} , \ldots , D_{{N_{t} }} } \right]^{T} } \right)$ using the formulation proposed in Eq. 40.

Step 4: Find the optimal step size η _κ by solving the iterative inexact line search problem based on the Armijo rule given by Eq. 41.

Step 5: Calculate the new estimation for the diameter D _κ+1 using Eq. 31.

Step 6: Test the new estimation for optimality (stopping criterion). If D _κ+1 is optimal, terminate the iteration process. Otherwise, go to next step.

Step 7: Update H _κ+1 based on the BFGS formula (Eq. 37).

Step 8: Set the new iteration number κ = κ + 1 and go to step 2.

Appendix 2: Nomenclature

B λ1	Empirically determined saturation coefficient for linear absorption	(J cm−2)
B λn	Empirically determined saturation coefficient for multi-photon absorption	(J cm−2)
c	Speed of light	(cm s−1)
c s	Specific heat of solid carbon (graphite)	(J g−1 K−1)
C p	Molar heat capacity for ambient gases at constant pressure	(J mol−1 K−1)
C v	Molar heat capacity for ambient gases at constant volume	(J mol−1 K−1)
C CO p	Molar heat capacity of CO	(J mol−1 K−1)
C abs	Absorption cross section for soot particles in the Rayleigh regime	(cm2)
D	Primary particle diameter	(cm)
D eff	Total effective diffusion constant	(cm2 s−1)
d	Direction of descent	(–)
E(m)	Dimensionless refractive index function	(–)
F	Laser fluence	(J cm−2)
f	Dimensionless Eucken correction to the thermal conductivity of polyatomic gas	(–)
f 1	Empirical scaling factor for linear absorption	(–)
G	Geometry-dependent heat transfer factor	(–)
H	Approximation of the Hessian matrix	(–)
h	Planck constant	(J s)
I	Identity matrix	(–)
i	Time step index	(–)
k B	Boltzmann constant	(J K−1)
k p	Boltzmann constant in effective pressure units	(bar cm3 K−1)
k ox	Overall rate constant for oxidation	(s−1 cm−2)
k λn	Rate constant for removal of C 2 by photodesorption	(s−1)
Kn	Knudsen number	(–)
L c	Characteristic length	(cm)
M	Particle mass	(g)
m e	Mass of an electron	(J s2 cm−2)
n	Estimated number of 532-nm photons absorbed to photodesorb C 2	(–)
n g	Number density of gas molecules	(cm−3)
N A	Avogadro constant	(mol−1)
N ss	Density of carbon atoms on the surface of the particle	(cm−2)
N t	Total number of discrete time steps	(–)
p 0	Ambient pressure	(bar)
p v	Average partial pressure of sublimed carbon species	(bar)
P ref	Reference pressure	(bar)
$P_{\text{eq}}^{{C_{j} }}$	Thermal equilibrium partial pressure of C j	(bar)
$P_{\text{sat}}^{{C_{j} }}$	Saturation partial pressure of C j	(bar)
$P_{\text{surf}}^{{C_{j} }}$	Partial pressure of C j at the particle surface	(bar)
P λn	Effective pressure calculated from the rate of photodesorption of C 2	(bar)
q	Probe laser irradiance	(W cm−2)
q exp	Experimental temporal profile of the laser	(Arbitrary unit)
R	Universal gas constant	(J mol−1 K−1)
R m	Universal gas constant in effective mass units	(g cm2 mol−1 K−1 s−2)
R p	Universal gas constant in effective pressure units	(bar cm3 mol−1 K−1)
SLII	Calculated LII signal	(Arbitrary unit)
SLII*	Measured LII signal	(Arbitrary unit)
t	Time	(s)
t f	Final measurement time	(s)
t l	Laser pulse duration	(s)
T	Particle temperature	(K)
T g	Ambient gas temperature	(K)
$T_{\text{ref}}^{{C_{j} }}$	Reference temperature for C j	(K)
W 1	Average molecular weight of atomic carbon	(g mol−1)
W a	Average molecular weight of air	(g mol−1)
W v	Average molecular weight of sublimed carbon species	(g mol−1)
W j	Molecular weight of C j (j × 12.011)	(g mol−1)
${\mathcal{J}}$	Objective (or cost) function	(Arbitrary unit)
${\mathcal{R}}_{\lambda \det }$	Spectral characteristics of the detection system	(Arbitrary unit)
${\mathcal{X}}$	Sensitivity coefficient	(Arbitrary unit)
α j	Mass accommodation coefficient of vaporized species C j	(–)
α M	Species-independent mass accommodation coefficient of vaporized carbon clusters	(–)
α T	Thermal accommodation coefficient of ambient gases with the surface	(–)
β	Scaling factor for emissivity	(–)
γ	Heat capacity ratio for the ambient gas surrounding the particles	(–)
ΔH j	Enthalpy of formation of carbon vapor species C j	(J mol−1)
ΔH v	Average enthalpy of formation of sublimed carbon species C j	(J mol−1)
ΔH ox	Enthalpy of reaction for 2C + O2 → 2CO	(J mol−1)
ΔH λn	Enthalpy required to photodesorb C 2	(J mol−1)
Δλ det	Spectral range of the detection system	(cm)
ɛ λ	Emissivity at wavelength λ	(–)
ϵ	Real number used to calculate the sensitivity coefficient	(–)
η	Search step size	(–)
κ	Iteration index	(–)
κ a	Thermal conductivity of the surrounding gases	(W cm−1 K−1)
λ	Wavelength	(cm)
λ l	Laser wavelength	(cm)
λ g	Mean free path of gas molecules	(cm)
ξ	Dispersion exponent	(–)
ρ s	Density of graphite	(g cm−3)
σ	Average molecular cross section for ambient gases	(cm2)
$\bar{\sigma }$	Average molecular cross section for sublimed species	(cm2)
σ err	Standard deviation of the measurement errors	(Arbitrary unit)
σ j	Molecular cross section for sublimed species C j	(cm2)
σ λn	Empirically determined multi-photon absorption cross section for photodesorption of C 2	(Cm2n−1 J1−n)
ϕ	Work function	(J)
ϕ abs	Rate of energy gained by laser absorption	(W)
ϕ cnd	Rate of energy lost by conduction to the surrounding gases	(W)
ϕ int	Rate of change of the internal energy of the particle	(W)
ϕ loss	Additional rate of energy lost by the particle (inferred from the inverse model)	(W)
ϕ ox	Rate energy gained by oxidation of the particle	(W)
ϕ rad	Rate of energy lost by radiative emission	(W)
ϕ source	Additional rate of energy gained by the particle (inferred from the inverse model)	(W)
ϕ sub	Energy loss rate due to the sublimation mechanism	(W)
ϕ th	Energy loss rate by thermionic emission	(W)
ω	Tolerance used in the Armijo rule	(–)
∇	Gradient	(–)
∇2	Hessian matrix	(–)