LIISim: a modular signal processing toolbox for laser-induced incandescence measurements

Abstract

Evaluation of measurement data for laser-induced incandescence (LII) is a complex process, which involves many processing steps starting with import of data in various formats from the oscilloscope, signal processing for converting the raw signals to calibrated signals, application of models for spectroscopy/heat transfer and finally visualization, comparison, and extraction of data. We developed a software tool for the LII community that helps to evaluate, exchange, and compare measurement data among research groups and facilitate the application of this technique by providing powerful tools for signal processing, data analysis, and visualization of experimental results. A common file format for experimental data and settings simplifies inter-laboratory comparisons. It can be further used to establish a public measurement database for standardized flames or other soot/synthetic nanoparticle sources. The open-source concept and public access to the software development should encourage other scientists to validate and further improve the implemented algorithms and thus contribute to the project. In this paper, we present the structure of the LIISim software including the materials database concept, signal-processing algorithms, and the implemented models for spectroscopy and heat transfer. With two application cases, we show the operation of the software how data can be analyzed and evaluated.

Appendices

Appendix A

1.1 Levenberg–Marquardt algorithm

For a given data vector y of the length m and for n parameters a and a vector of standard deviations ${\sigma _i}$, the residuals for a given model ${y_{{\text{mod}}}}\left( {{x_i},{\mathbf{a}}} \right)$ (spectroscopic or heat-transfer model) can be described as

$${f_i}\left( {\mathbf{a}} \right)={\text{~}}\frac{{{y_i} - {y_{{\text{mod}}}}\left( {{x_i},{\mathbf{a}}} \right)}}{{{{{\upsigma}}_i}}}\quad i=1,2, \ldots ,m,$$

(11)

and

$${\mathbf{F}}\left( {\mathbf{a}} \right)={\text{~}}\left[ {\begin{array}{*{20}{c}} {{f_1}({\mathbf{a}})} \\ \vdots \\ {{f_m}({\mathbf{a}})} \end{array}} \right] \in {{\mathbb{R}}^m}.$$

(12)

The goal is now to minimize the nonlinear least-squares problem for the parameters a

$${\text{arg}}\mathop {\hbox{min} }\limits_{{\mathbf{a}}} f\left( {\mathbf{a}} \right)$$

(13)

with

$$f\left( {\mathbf{a}} \right)={\text{~}}\mathop \sum \limits_{{i=1}}^{m} {f_i}{\left( {\mathbf{a}} \right)^2}.$$

(14)

The gradient of $f\left( {\mathbf{a}} \right)$ can be written in matrix notation as

$$\nabla f\left( {\mathbf{a}} \right)=2{\mathbf{J}}{\left( {\mathbf{a}} \right)^\text{T}}{\mathbf{F}}\left( {\mathbf{a}} \right) \in {{\mathbb{R}}^m},$$

(15)

where J(a) is the Jacobian

$${\mathbf{J}}({\mathbf{a}})=~\left[ {\begin{array}{*{20}{c}} {\frac{{\partial {f_i}}}{{\partial {a_1}}}}& \cdots &{\frac{{\partial {f_1}}}{{\partial {a_n}}}} \\ \vdots & \ddots & \vdots \\ {\frac{{\partial {f_m}}}{{\partial {a_1}}}}& \cdots &{\frac{{\partial {f_m}}}{{\partial {a_n}}}} \end{array}} \right] \in {{\mathbb{R}}^{m \times n}},$$

(16)

and D half of the Hessian matrix:

$${\nabla ^2}f\left( {\mathbf{a}} \right) \approx 2{\mathbf{J}}{\left( {\mathbf{a}} \right)^\text{T}}{\mathbf{J}}\left( {\mathbf{a}} \right)=2{\mathbf{D}}.$$

(17)

For each iteration, the gradient of the parameters a can be found by solving [51]:

$$\left( {{\mathbf{J}}{{\left( {\mathbf{a}} \right)}^\text{T}}{\mathbf{J}}\left( {\mathbf{a}} \right)+{{\uplambda}}{\mathbf{I}}} \right){{\Delta}}{\mathbf{a}}={\text{~}} - {\mathbf{J}}{\left( {{{\mathbf{a}}^{k}}} \right)^\text{T}}{\mathbf{F}}\left( {{{\mathbf{a}}^k}} \right),$$

(18)

which can be transformed using the Cholesky decomposition to the form ${\mathbf{L}}{{\mathbf{L}}^{\text{T}}}{\mathbf{x}}={\mathbf{b}}~$ with

$$\begin{gathered} {\mathbf{L}}{{\mathbf{L}}^{\text{T}}}=\left( {{\mathbf{D}}+{{\uplambda}}{\mathbf{I}}} \right) \hfill \\ {\mathbf{x}}={\Delta\mathbf{a}} \hfill \\ {\mathbf{b}}={\text{~}} - {\mathbf{J}}{\left( {{{\mathbf{a}}^k}} \right)^{\text{T}}}{\mathbf{F}}\left( {{{\mathbf{a}}^k}} \right). \hfill \\ \end{gathered}$$

(19)

Now x can be found by forward ${\mathbf{Ly}}={\mathbf{b}}$ and backward substitution ${{\mathbf{L}}^{\text{T}}}{\mathbf{x}}={\varvec{y}},$ which gives the new parameter approximation for the next iteration k:

$${{\mathbf{a}}^{{k}+1}}={{\mathbf{a}}^{k}}+\Delta {\mathbf{a}}.$$

(20)

In LIISim, the parameters ${{\mathbf{a}}^{k}}$ are visualized for the temperature fit in “AnalysisTools Temperature Fit” and for the heat-transfer modeling in the FitCreator module.

Appendix B

Implemented heat-transfer models from literature [22]. The heat transfer rates are defined in the HeatTransferModel child classes in the “calculations/models/” folder of the source code.

The following materials and gas mixture properties are calculated for all models according:

Name	Variable	Symbol (original)	Symbol (LIISim)	Equation	Unit
Specific heat capacity of the particle	c_p_kg	${c_{\text{s}}}$	${c_{\text{p}}}$	${c_{\text{p}}}={C_{{\text{p}},{\text{mol}}}}/{M_{\text{p}}}$	J kg−1 K−1
Thermal velocity of gas molecules	c_tg	${c_{{\text{tg}}}}({T_{\text{g}}})$	${c_{{\text{tg}}}}({T_{\text{g}}})$	${c_{{\text{tg}}}}=~{\left( {\frac{{8~{k_{\text{B}}}{N_{\text{A}}}{T_{\text{g}}}}}{{\pi {M_{mix}}}}} \right)^{\frac{1}{2}}}$	m s−1
Molar heat capacity of gas mixture	C_p_mol	–	${C_{{\text{p,mix}}}}$	${C_{p,{\text{mix}}}}=\mathop \sum \limits_{i}^{n} {x_i}{C_{{\text{p}},{\text{g}},i}}$	J mol−1 K−1
Heat capacity ratio	gamma	$\gamma ({T_{\text{g}}})$	$\gamma ({T_{\text{g}}})$	$\gamma \left( {{T_{\text{g}}}} \right)=\frac{{{C_{{\text{p}},{\text{mix}}}}}}{{{C_{{\text{p}},{\text{mix}}}} - R}}$	–
Molar mass of gas mixture	molar_mass	–	${M_{{\text{mix}}}}$	${M_{{\text{mix}}}}=\mathop \sum \limits_{i}^{n} {x_i}{M_{{\text{g}},i}}$	kg mol−1

2.1 Kock model

2.1.1 Materials properties (Soot_Kock)

Name	Variable	Type	Symbol (original)	Symbol (LIISim)	a 0	a 1	a 2	a 3	a 4	a 5	a 6	a 7	a 8	Unit	Comment
Accommodation coefficient	alpha_T_eff	const	${\alpha _{\text{T}}}$	${\alpha _{\text{T}}}$	0.23	–	–	–	–	–	–	–	–	–
Accommodation coefficient	theta_e	const	${\alpha _{\text{M}}}$	${\theta _{\text{e}}}$	1.0	–	–	–	–	–	–	–	–	–
Molar heat capacity	C_p_mol	poly2a	–	${C_{p,{\text{mol}}}}$	22.5566	0.0013	–	–	–	− 1.8195 × 106	–	–	–	J mol−1 K−1	Calculated from given ${c_{\text{s}}}$
Total emissivity	eps	const	$\varepsilon$	$\varepsilon$	1.0	–	–	–	–	–	–	–	–	–
Molar mass	molar_mass	const	${W_{\text{s}}}$	${M_{\text{p}}}$	0.012011	–	–	–	–	–	–	–	–	kg mol−1
Molar mass of vapor	molar_mass_v	const	${W_{\text{v}}}$	${M_{\text{v}}}$	0.036033	–	–	–	–	–	–	–	–	kg mol−1
Density	rho_p	const	${\rho _{\text{s}}}$	${\rho _{\text{p}}}$	1860	–	–	–	–	–	–	–	–	kg m− 3
Enthalpy of evaporation	H_v	const	${{\Delta}}{H_{\text{v}}}$	${{\Delta}}{H_{\text{v}}}$	790776.6	–	–	–	–	–	–	–	–	J mol− 1
Reference pressure	p_v_ref	const	${p_{{\text{ref}}}}$	${p_{\text{v}}}^{{\text{*}}}$	61.5	–	–	–	–	–	–	–	–	Pa	Clausius–Clapeyron
Reference temperature	T_v_ref	const	${T_{{\text{ref}}}}$	${T_{\text{v}}}^{{\text{*}}}$	3000	–	–	–	–	–	–	–	–	K	Clausius–Clapeyron

1. ^a $f\left( T \right)={a_0}+{a_1}T+{a_2}{T^2}+{a_3}{T^3}+{a_4}{T^{ - 1}}+{a_5}{T^{ - 2}}$

2.1.2 Gas mixture properties (Kock-Nitrogen-100%)

Composition:

Gas	Variable	Symbol	Fraction
Nitrogen_Kock	x	$x$	1.0

2.1.3 Gas properties (Nitrogen_Kock)

Name	Variable	Type	Symbol (original)	Symbol (LIISim)	a 0	a 1	a 2	a 3	a 4	a 5	Unit
Molar mass	molar_mass	const	${M_{\text{g}}}$	${M_{\text{g}}}$	0.028014	–	–	–	–	–	kg mol−1
Molar heat capacitya	C_p_mol	poly2b	${C_{{\text{mp}},{\text{g}}}}$	${C_{p,{\text{g}}}}$	28.58	0.00377	–	–	–	− 50,000	J mol− 1 K− 1

1. ^aFor nitrogen from [52]
2. ^b $f\left( T \right)={a_0}+{a_1}T+{a_2}{T^2}+{a_3}{T^3}+{a_4}{T^{ - 1}}+{a_5}{T^{ - 2}}$

2.2 Heat-transfer model (HTM_KockSoot)

Evaporation:

$${\dot {Q}_{{\text{evap}}}}= - \frac{{{{\Delta}}{H_{\text{v}}}}}{{{M_{\text{v}}}}}~{\dot {u}_{{\text{evap}}}},$$

(21)

$${\dot {u}_{{\text{evap}}}}= - {{{\theta}}_{\text{e}}}\frac{1}{4}{{\uppi}}~d_{{\text{p}}}^{2}~{c_{{\text{tv}}}}{\rho _{\text{v}}},$$

(22)

$${c_{{\text{tv}}}}=~{\left( {\frac{{8~{k_{\text{B}}}{N_{\text{A}}}{T_{\text{p}}}}}{{\pi {M_{\text{v}}}}}} \right)^{\frac{1}{2}}},$$

(23)

$${\rho _{\text{v}}}=\frac{{{p_{\text{v}}}~~{M_{\text{v}}}}}{{R~{T_{\text{p}}}}},$$

(24)

$${p_{\text{v}}}={p_{\text{v}}}^{*}~{\text{exp}}\left( { - \frac{{{{\Delta}}{H_{\text{v}}}~}}{R}~\left( {\frac{1}{{{T_{\text{p}}}}} - \frac{1}{{{T_{\text{v}}}^{*}}}} \right)} \right).$$

(25)

Conduction:

$${\dot {Q}_{{\text{cond,fm}}}}=\frac{{{\alpha _{{\text{T~}}}}\pi ~d_{{{\text{p~}}}}^{2}{p_{{\text{g~}}}}{c_{{\text{tg}}}}}}{8}~\left( {\frac{{\gamma +1}}{{\gamma - 1}}} \right)\left( {\frac{{{T_{\text{p}}}}}{{{T_{\text{g}}}}} - 1} \right),$$

(26)

$${c_{{\text{tg}}}}=~{\left( {\frac{{8~{k_{\text{B}}}{N_{\text{A}}}{T_{\text{g}}}}}{{\pi {M_{{\text{mix}}}}}}} \right)^{\frac{1}{2}}}.$$

(27)

Radiation:

$${\dot {Q}_{{\text{rad}}}}=\pi ~d_{{\text{p}}}^{2}~\varepsilon ~\sigma (T_{{\text{p}}}^{4} - T_{{\text{g}}}^{4})~.$$

(28)

2.3 Liu model

2.3.1 Materials properties (Soot_Liu)

Name	Variable	Type	Symbol (original)	Symbol (LIISim)	a 0	a 1	a 2	a 3	a 4	a 5	a 6	Unit	Comment
Accommodation coefficient	alpha_T_eff	const	${\alpha _{\text{T}}}$	${\alpha _{\text{T}}}$	0.37	–	–	–	–	–	–	–
Accommodation coefficient	theta_e	const	${\alpha _{\text{M}}}$	${\theta _{\text{e}}}$	0.77	–	–	–	–	–	–	–
Molar heat capacity	C_p_mol	polya		${C_{p,{\text{mol}}}}$	3.54288	3.55694 × 10− 2	− 2.55018 × 10− 5	9.83713 × 10− 9	− 2.10385 × 10− 12	2.35752 × 10− 16	− 1.07879 × 10− 20	J mol− 1 K−1	Valid from 1200 to 5500 K; calculated from given ${c_{\text{s}}}$
Total emissivity	eps	const	$\varepsilon$	$\varepsilon$	0.4	–	–	–	–	–	–	–
Molar mass	molar_mass	const	${W_{\text{v}}}$	${M_{\text{p}}}$	0.012011	–	–	–	–	–	–	kg mol−1
Molar mass of vapor	molar_mass_v	const	${W_1}$	${M_{\text{v}}}$	17.179 × 10−3	6.8654 × 10− 7	2.9962 × 10− 9	− 8.5954 × 10− 13	1.0486 × 10− 16	–	–	kg mol−1
Density	rho_p	const	${\rho _{\text{s}}}$	${\rho _{\text{p}}}$	1860	–	–	–	–	–	–	kg m−3
Enthalpy of evaporation	H_v	polya	${{\Delta}}{h_{\text{v}}}$	${{\Delta}}{H_{\text{v}}}$	2.05398 × 105	7.366 × 102	− 0.40713	1.1992 × 10− 4	− 1.7946 × 10− 8	1.0717 × 10− 12	–	J mol−1
Vapor pressure	p_v	exppolyb	${p_v}$	${p_{\text{v}}}$	–	101,325 (unit conversion)	− 122.96	9.0558 × 10− 2	− 2.7637 × 10− 5	4.1754 × 10− 9	− 2.4875 × 10− 13	Pa	Original unit: [atm] from fits to data

1. ^a $f\left( T \right)={a_0}+{a_1}T+{a_2}{T^2}+{a_3}{T^3}+{a_4}{T^4}+{a_5}{T^5}+{a_6}{T^6}+{a_7}{T^7}+{a_8}{T^8}$
2. ^b $~f\left( T \right)={a_0}+{a_1}{\text{exp}}({a_2}+{a_3}T+{a_4}{T^2}+{a_5}{T^3}+{a_6}{T^4}+{a_7}{T^5})$

2.3.2 Gas mixture properties (Liu_Flame)

Composition:

Gas	Variable	Symbol	Fraction
FlameAir_Liu	x	$x$	1.0

Properties (manually set for composition):

Name	Variable	Type	Symbol (original)	Symbol (LIISim)	a 0	a 1	a 2	a 3	a 4	Unit
Heat capacity ratioa	gamma_eqn	polyb	$\gamma$	$\gamma$	1.4221163416	− 1.8636002383 × 10−4	8.0783894569 × 10−8	− 1.6425082302 × 10−11	1.2750021975 × 10−15	–

1. ^aFor flame mixture from [53]
2. ^b $f\left( T \right)={a_0}+{a_1}T+{a_2}{T^2}+{a_3}{T^3}+{a_4}{T^4}+{a_5}{T^5}+{a_6}{T^6}+{a_7}{T^7}+{a_8}{T^8}$

2.3.3 Gas properties (FlameAir_Liu)

Name	Variable	Type	Symbol (original)	Symbol (LIISim)	a 0	Unit
Molar mass	molar_mass	const	–	${M}_{\text{g}}$	0.02874	kg mol− 1

2.4 Heat-transfer model (HTM_Liu)

Evaporation:

$${\dot {Q}_{{\text{evap}}}}= - \frac{{~\Delta {H_{\text{v}}}}}{{{M_{\text{v}}}}}{\dot {u}_{{\text{evap}}}},$$

(29)

$${\dot {u}_{{\text{evap}}}}= - \frac{{\pi d_{{\text{p}}}^{2}{M_{\text{v}}}{\theta _{\text{e}}}{p_{\text{v}}}}}{{R{T_{\text{p}}}}}{\left( {\frac{{R{T_{\text{p}}}}}{{2\pi {M_{\text{v}}}}}} \right)^K},$$

(30)

with $K=0.5.$

Conduction:

$${\dot {Q}_{{\text{cond}}}}=\frac{{\pi d_{{\text{p}}}^{2}{\alpha _{\text{T}}}{p_0}}}{{2{T_{\text{g}}}}}~\sqrt {\frac{{R{T_{\text{g}}}}}{{2\pi {M_{{\text{mix}}}}}}} \left( {\frac{{{\gamma ^*}+1}}{{{\gamma ^*} - 1}}} \right)\left( {{T_{\text{p}}} - {T_{\text{g}}}} \right),$$

(31)

$$\frac{1}{{{\gamma ^*} - 1}}=\frac{1}{{T - {T_0}}}\mathop \int \limits_{{{T_0}}}^{T} \frac{1}{{\gamma (T^{\prime}) - 1}}{\text{d}}T^{\prime}.$$

(32)

This heat-transfer model uses polynomial fitting coefficients for calculation of $\gamma \left( T \right)$. These are provided through the “gamma_eqn” property of the LIISim implementation in the GasMixture database. If $~~\gamma \left( T \right)$ is not defined, the heat capacity of the gas mixture ${C_{p,{\text{mix}}}}(T)$ is used to calculate $\gamma \left( T \right)$ according to:

$${{\upgamma}}\left( T \right)=\frac{{{C_{{\text{p}},{\text{mix}}}}(T)}}{{{C_{{\text{p}},{\text{mix}}}}(T) - R}}.$$

(33)

Radiation:

$${\dot {Q}_{{\text{rad}}}}=\frac{{199{{{\uppi}}^3}d_{{\text{p}}}^{3}{{\left( {{k_{\text{B}}}T} \right)}^5}\varepsilon }}{{h{{\left( {hc} \right)}^3}}}.$$

(34)

2.5 Melton model

2.5.1 Materials properties (Soot_Melton(workshop))

Name	Variable	Type	Symbol (original)	Symbol (LIISim)	a0	Unit	Comment
Accommodation coefficient	alpha_T_eff	const	${\alpha _{\text{T}}}$	${\alpha _{\text{T}}}$	0.3	–
Accommodation coefficient	theta_e	const	${\alpha _{\text{M}}}$	${\theta _{\text{e}}}$	1.0	–
Molar heat capacity	C_p_mol	const	–	${C_{p,{\text{mol}}}}$	22.8	J mol−1 K−1	acalculated from given ${c_{\text{s}}}$
Molar mass	molar_mass	const	${W_{\text{s}}}$	${M_{\text{p}}}$	0.012	kg mol− 1
Molar mass of vapor	molar_mass_v	const	${W_{\text{v}}}$	${M_{\text{v}}}$	0.036	kg mol− 1
Density	rho_p	const	${\rho _{\text{s}}}$	${\rho _{\text{p}}}$	2260	kg m− 3
Enthalpy of evaporation	H_v	const	${{\Delta}}{H_{\text{v}}}$	${{\Delta}}{H_{\text{v}}}$	7.78 × 105	J mol− 1
Reference pressure	p_v_ref	const	${p_{{\text{ref}}}}$	${p_{\text{v}}}^{{\text{*}}}$	100,000	Pa	Clausius–Clapeyron
Reference temperature	T_v_ref	const	${T_{{\text{ref}}}}$	${T_{\text{v}}}^{{\text{*}}}$	3915	K	Clausius–Clapeyron

1. ^a ${C_{p,{\text{mol}}}}={c_{s,\text{Melton}}}{M_{\text{p}}}$

2.5.2 Gas mixture properties (Melton-Nitrogen-100%)

Composition:

Gas	Variable	Symbol	Fraction
Nitrogen_Melton	x	$x$	1.0

Properties (manually set for composition)

Name	Variable	Type	Symbol (original)	Symbol (LIISim)	a 0	a 1	Unit	Comment
Thermal conductivity	therm_cond	const	${\kappa _{\text{a}}}$	${\kappa _{\text{a}}}$	0.1068	–	W/m/K	Original unit W/cm/K
Mean free path	L	polya	$L$	$L$	–	2.355 × 10−10	m	Original unit: cm

1. ^a $f\left( T \right)={a_0}+{a_1}T+{a_2}{T^2}+{a_3}{T^3}+{a_4}{T^4}+{a_5}{T^5}+{a_6}{T^6}+{a_7}{T^7}+{a_8}{T^8}$

2.5.3 Gas properties (Nitrogen_Melton)

Name	Variable	Type	Symbol (original)	Symbol (LIISim)	a0	Unit	Comment
Molar heat capacity	C_p_mol	const	$-$	${C_{{\text{p,g}}}}$	36.0295	J mol−1 K−1	a calculated from given$\gamma (1800\;{\text{K}})=1.3$

1. ^a $\gamma (1800\;{\text{K}})=1.3=\frac{{{C_{\text{p}}}}}{{{C_{\text{p}}} - R}} \Rightarrow {C_{\text{p}}}=\frac{{1.3}}{{0.3}}R$

2.6 Heat-transfer model (HTM_Melton)

Evaporation:

$${\dot {Q}_{{\text{evap}}}}= - ~\frac{{\Delta {H_{\text{v}}}}}{{{M_{\text{p}}}}}~{\dot {u}_{{\text{evap}}}}.$$

(35)

This model uses molar mass of solid species in Eq. (35).

$${\dot {u}_{{\text{evap}}}}= - \frac{{{{\uppi}}~d_{{\text{p}}}^{2}{M_{\text{v}}}{{{\uptheta}}_{\text{e}}}{p_{\text{v}}}~}}{{R{T_{\text{p}}}}}{\left( {\frac{{R{T_{\text{p}}}}}{{2{M_{\text{v}}}}}} \right)^{0.5}},$$

(36)

$${p_{\text{v}}}={p_{\text{v}}}^{*}~{\text{exp}}\left( { - \frac{{\Delta {H_{\text{v}}}}}{R}~\left( {\frac{1}{{{T_{\text{p}}}}} - \frac{1}{{{T_{\text{v}}}^{*}}}} \right)} \right).$$

(37)

Conduction:

$${\dot {Q}_{{\text{cond}}}}=\frac{{2{\kappa _{\text{a}}}\pi d_{{\text{p}}}^{2}}}{{{d_{\text{p}}}+G~L\left( {{T_{\text{g}}}} \right)}}~\left( {{T_{\text{p}}} - {T_{\text{g}}}} \right),$$

(38)

$$G=~\frac{{8f}}{{{\alpha _{\text{T}}}\left( {\gamma +1} \right)}},$$

(39)

$$f=~\frac{{9\gamma - 5}}{4}.$$

(40)

Radiation:

Not included in this model.