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

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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.

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Acknowledgements

We gratefully thank Stanislav Musikhin (University of Duisburg-Essen, Germany) for testing the software and giving helpful feedback. We acknowledge funding through the German Research Foundation via SCHU1369/14 and SCHU1369/20.

Author information

Correspondence to Raphael Mansmann.

Electronic supplementary material

Appendices

Appendix A

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

Kock model

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

Gas mixture properties (Kock-Nitrogen-100%)

Composition:

Gas Variable Symbol Fraction
Nitrogen_Kock x \(x\) 1.0

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

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)

Liu model

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 \(~f\left( T \right)={a_0}+{a_1}{\text{exp}}({a_2}+{a_3}T+{a_4}{T^2}+{a_5}{T^3}+{a_6}{T^4}+{a_7}{T^5})\)

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

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

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)

Melton model

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 \({C_{p,{\text{mol}}}}={c_{s,\text{Melton}}}{M_{\text{p}}}\)

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

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 \(\gamma (1800\;{\text{K}})=1.3=\frac{{{C_{\text{p}}}}}{{{C_{\text{p}}} - R}} \Rightarrow {C_{\text{p}}}=\frac{{1.3}}{{0.3}}R\)

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.

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Mansmann, R., Terheiden, T., Schmidt, P. et al. LIISim: a modular signal processing toolbox for laser-induced incandescence measurements. Appl. Phys. B 124, 69 (2018). https://doi.org/10.1007/s00340-018-6934-9

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