Size Determination of Micron-Nano Scale Carbon Particulates Based on Time-resolved Laser-induced Incandescence

Size Determination of Micron-Nano Scale Carbon Particulates Based on Time-resolved Laser-induced Incandescence

CHEN Linghong¹, ZUO Lei¹, JIANG Yiqi¹, Jiao Li², GREHAN Gérard³, CEN Kefa¹

¹State Key Laboratory of Clean Energy Utilization, Zhejiang University, Hangzhou 310027, China

²Hangzhou Environmental Monitoring Center, Hangzhou 310007, China

³LESP, UMR 6614/CORIA, CNRS/Universit´eet INSA de ROUEN, 76800 Saint Etienne du Rouvray, France

ABSTRACT
Micron-nano scale carbon particulates resulting from the incomplete combustion of fossil fuel are highly related to the combustion efficiency, atmospheric environment and human health, etc. Time-resolved laser-induced incandescence (LII) was proved to be a potential useful tool to measure carbon particulate concentration and primary particulate size in combustion, particulate synthesis, and environmental applications. LII involves heating particulates with a high power pulsed laser and measuring the radiative emission. Using LII technique for quantitative determination of particulate concentration and size requires a detail understanding of the heat- and mass-transfer mechanisms in the process. Two mass- and energy-balance models were established to simulate different sizes of laser-heated carbon particulates during the laser-heated and subsequent cooling process. A uniform internal temperature was assumed in model N while internal temperature gradient was considered in model M. The simulated LII signal profiles using two models were compared for different particulates sizes. The results showed that models N and M could fit together when the particulate diameter was in the range between 1.0 and 2.5μm. The magnitude of LII curve using model M was significantly lower than using model N when the particulate diameter was less than 0.1μm. This could be explained that the tradition heat transfer mechanism was not fit in this size region. The results showed that the internal temperature gradient could not be neglected when the particulate diameter was larger than 5.0μm. In addition, experiments of pointwise (0D) and imaging (2D) LII applications on a methane flame were set up. The measured temporally LII signals were compared to theoretical calculations. The results indicated that the carbon particulate size was of 1-2nm in flame. Simultaneously extinction measurement showed that the carbon particulate concentration was lower than 3ppm along the flame height.

KEYWORDS
Micron-nano scale; Carbon particulate; Laser-induced incandescence; Concentration; Particulate-size
NOMENCLATURE

D particulate diameter, cm;

D₀ initial particulate diameter, cm;

D_m mean particulate diameter, cm;

r radial coordinate, cm;

t time, s;

M particulate mass, g;

T₀ initial flame temperature, K;

T particulate temperature, K;

T^* temperature in K at which carbon vapor pressure p = p^* (p = 1 atm T^* = 3915K);

Q_abs absorption efficiency factor;

q(t) laser temporal power density, W/cm²;

F laser fluence, J/cm²;

k_a thermal conductive of air, = 5.83×10^-5(T/273)^0.82 W/(cm·K);

l mean free path, =2.355×10^-8·T cm for air at 1 atm;

G geometry-dependent heat transfer factor, = 53.33;

ΔH_v heat of sublimation of carbon, = 7.68×10⁵ J/mol;

W_s molecular weight of solid carbon, = 12 g/mol;

W_v molecular weight of vapor taken at 36 g/mol;

P vapor pressure of carbon;

ρ_s solid carbon density, = 2.26 g/cm³;

C_s specific heat of carbon, = 2.41 J/(g·K);

k_s thermal conductivity of graphitic carbon, = 1.5 W(cm·K);

α_s thermal diffusivity of graphitic carbon, = 0.275 cm2/s;

R gas constant, = 8.31 J/(K·mol)

σ Stefan-Boltzman constant, =5.67×10^-12 W/(cm²·K⁴);

c₁ Planck First radiation constant, = 3.7419×10^-16 W·m²;

c₂ Planck Second radiation constant, = 1.4388×10^-2 m·K;

λ wavelength of incident light, μm;

λ_em emission wavelength, nm;

1. INTRDUCTION

Black carbon is one of the most important pollutants emitting from a variety of combustion processes and is found throughout the Earth system. Black carbon has a unique and important role in the Earth’s climate system because it absorbs solar radiation, influences cloud processes, and alters the melting of snow and ice cover [¹]. In addition, carbon particulates will carry plenty of toxicants (such as heavy metal and organic matter) because of its high absorption. These particulates with diameters from 10nm to 100nm can be breathed into alveolus, which can harm breath system of human [²,³]. Besides, carbon particulates from the incomplete combustion will reduce combustion efficiency significantly [⁴]. Thus, developing carbon particulates measurement techniques is important to the understanding of physical and chemical mechanisms of carbon formation and the validation of carbon formation models, which ultimately helps to increase combustion efficiency and decrease pollutant emission.

Laser-induced incandescence (LII) has proven to be a useful tool for carbon particulate concentration and primary particulate size measurements in combustion, particulate synthesis, and environmental applications. LII involves heating particulates with a high power pulsed laser and measuring the radiative emission. The magnitude of the LII signal depends on the particulate volume fraction, and the decay rate of the LII signal is mainly governed by the specific surface of the particulates, which in turn depends on primary size [⁵]. Using such measurements for quantitative determinations of particulate volume fraction and size requires a detailed understanding of experimental parameters and physical mechanisms that control the LII signal.

One approach to developing such a description of LII signals generation involves modeling the processes that control signal production. A first mathematical model of the laser-induced incandescence phenomenon, based on the energy- and mass-balance equations to predict temporal response of the particulate to pulsed laser heating, has been developed by Eckbreth and Melton [⁶,⁷]. Subsequently considerable effort has been devoted to developing models capable of predicting LII signals over a range of fluences [⁸]. Some models have been optimized to fit LII signal decay curves for inferring particulate size distributions [⁹,¹⁰] and aggregate size distributions [¹¹], whereas others have been used primarily to understand the influence of factors such as experimental parameters [¹²,¹³,¹⁴], particulate characteristics [¹⁵,¹⁶], and particulate phase changes during particulate heating and cooling [¹⁷]. As the evolution of LII emission is often a combination of the size-dependent LII decay of the different particulate size classes, it is critical to detection small particulates in the presence of large ones. Bladh and Bengtsson [¹⁸] investigated the impact of Gaussian and lognormal size distributions of different width on temporal LII-signal behavior. The result is a significant bias towards larger particulate sizes because of the higher influence of the larger particulates on the LII signal. Allouis et al. [¹⁹] extended Melton model for different kinds of large carbonaceous particulate assuming a temperature profile inside. The theoretical calculations showed a good agreement compared to the temporally resolved LII emissions in a diesel oil spray flames, but the accuracy of the model needed to be drastically improved [²⁰].

Our motivation in this paper is to demonstrate an appropriate model applying to size measurement of micron-nano scale carbon particulates. Firstly, we will establish the differential equations for the spatial- and time-dependence of the temperature and diameter of a laser-heated spherical particulate with different classes of particulate sizes. Then a detailed analysis of the particulate-size that determines the uncertainty of LII technique will be present. Finally, the carbon particulate concentration and size distribution in a methane flame will be given through the fitting of theoretical calculations of the model.

2. METHODOLOGY
2.1 Model description
For a carbon particulate, the mass balance can be written as follows [6]:
₍₁₎
where P is the vapor pressure of carbon, which is given by the Clapeyron equation:
₍₂₎
However, the form of energy balance equation between a spherical particulate and its surroundings varies from different classes of particulate sizes. This depends on the form of heat flux due to Kundsen number [²¹]: continuum region (K_n < 0.1), transition region (0.1 < K_n < 2.0), slip region (2.0 < K_n < 10), free molecular region (K_n > 10). The Kundsen number represents the ratio between mean free path of heat-transfer medium and particulate diameter. Basing on different classes of particulate diameter, heat flux between the particulate and surrounds and inside the particulate varies and ultimately affects the temperature profile appears inside the particulate. The nano-scale carbon particulates may be assured to have a uniform internal temperature since the laser, which is assumed to have a triangular pulse of 10ns FWHM (Full Width Half Maximum), is much longer than the characteristic time for internal gradient dissipation of 10^-3ns for a diameter D = 10nm [7]. In contrast, a particulate with a diameter of 10μm has a characteristic diffusion time, t_D, of about 24ns. This time rises with the second power of the particulate diameter, as predicted by the boundary layer approximation [²²]. Therefore, a spatial- and time-dependent temperature profiles appears inside the micron-scale particulates. Here we gave the description of two different models due the different shapes of temperature profiles inside the particulates.
(1)Model N
The particulates are assumed to have a uniform internal temperature in the process of heating by laser absorption and cooling by conduction to surrounding atmosphere, sublimation of carbon clusters, and radiative emission. The energy balance is expressed as a function of time, temperature, mass, and primary particulate diameter:
(3)
The terms on the right of the equation are the laser energy absorption per second, the energy expended in sublimation of the carbon, the rate of heat transfer to the surrounding atmosphere (taken to be air at temperature T₀) and the rate of energy loss by black-body radiation respectively. The thermal conductivity must be corrected considering the particulates are smaller compared to the mean free path.
(2)Model M
An internal temperature gradient is considered inside the particulate in this model. The energy balance is given by the heat diffusion equation in spherical coordinates:
(4)
The initial condition is

at t = 0, D = D₀ and T = T₀∀r

The boundary conditions are
(5)


2.2 Absorption efficiency factor
Equation (3) and (5) involved the calculation of Q_abs, which represents the absorption efficiency factor and is a function of the complex refractive index (m = n - ik). Chang and Charalmpopoulos gave the values of n and k of carbon particulate [²³]:
(6)

(7)
Given the complex indices of refraction for the medium and the particulate, Q_abs can be computed from the Mie equations. A FORTRAN program was used to compute Q_abs. Instead of using the Rayleigh approximation to calculate the absorption efficiency factor as in the previous works [8], we adopt the full Lorenz-Mie theory when the particulate size is beyond the Rayleigh size limit.
2.3 Prediction of LII signals
The radiation emitted by one particulate follows the Planck law and the LII signal can be expressed as follows at any time during the laser pulse heating and cooling process [²⁴]:
(8)
During these processes the LII signal rises very fast, becomes limited by the energy going into sublimation and then falls with a time constant appropriate thermal conduction to the medium. The magnitude of the signal depends on the particulate volume fraction and the signal decay rate depends on the primary particulate size.

3. EXPEIMENTAL PROCEDURE

Experiments on a methane flame of pointwise (0D) and imaging (2D) LII applications have been set up to illustrate the accuracy of the present models. Figure 1 shows schematics of the apparatus arrangements. A Nd: YAG laser generating 10ns pulses at 1064nm with a repetition rate of 10Hz is used for LII. A 1064nm laser is used rather than at 532nm because: (i) At longer excitation wavelengths, the generation of electronically excited C₂ fragments is less pronounced [²⁵], (ii) laser-induced fluorescence (LIF) of polycyclic aromatic hydrocarbons (PAH) is induced with excitation in the visible and the UV [²⁶], which is not easily separated with detection filters from LII signal. The experiments proceed under an incident laser pulse fluence of 0.1J/cm².

Figure 1a presents the two-dimensional measurement of the LII from the carbon particulates in a methane flame. To form the laser sheet, two lenses, cylindrical (f = -75mm) and spherical (f = +300m), are installed at the laser output. The laser sheet is 30mm in width and 1mm in thickness. An ICCD camera is located perpendicularly to the laser sheet, allowing the best view of 2D LII images. The camera is synchronized with the laser pulse through the camera control unit. The LII signal has a long temporal decrease and therefore a 30ns delay after laser pulse permits to eliminate light scattered by carbon particulates [25].A 100ns gate width is chosen to have enough LII signal [²⁷]. Furthermore, LII signal is detected at 600nm to achieve a good signal-to-noise ratio and avoid signal disturbances [5].

To calibrate the LII signal, an extinction measurement through the flame is made. A semiconductor laser at 532nm coupled with a photodiode with a 532nm filter is applied to the extinction measurement. According to the Beer-Lambert law, the carbon volume fraction is calculated by the ratio between the initial intensity I₀ and decayed intensity I transmitted through the flame [²⁸]. Given by the LII distribution at the detected height, the temporal ratio on the optical depth provides the absolute carbon volume fraction along the width in flame.

Figure 1b shows the simultaneous pointwise measurement of the LII and light absorption along the flame height. A homogeneous laser light sheet (3mm×3mm) is formed by an aperture. The signal is collected by a PMT with a narrow pass bond filter at 600nm and recorded on an oscilloscope.

Figure 1 Experimental apparatus for carbon particulate measurement in a methane flame with LII

4. SIMULATION RESULTS AND DISCUSSION

4.1 Calculation of Q_abs
Figure 2 demonstrates the effects of particulate size on the particulates’ emissivity based on theoretical calculations. With increasing particulate diameter the absorption efficiency factor increases, then decreases slowly and finally approaches a constant when the diameter goes forward beyond 3μm. It is as if that Q_abs becomes more size-dependent effect for smaller particulates (D < 0.5μm) and less size-dependent effect for larger particulates (D > 1μm). Compared with curves of different wavelengths of incident light, the effect of wavelength seems to be less pronounced when the wavelength is chosen at the infrared range.

FIGURE 2 Calculated absorption efficiency factor Q_abs for spheres carbon particulates with primary diameter of 0-5μm. The wavelength of incident light was in a wide regime from the ultraviolet range to the infrared range (λ = 355, 532 and 1064nm). The refractive index was assumed to be independent of wavelength.

4.2 Comparisons between Model N and M

In order to elaborate which model would be suitable to use in an accurate classes of particulates, comparisons between Model N and M were performed under the same laser temporal profile, wavelength and fluences. It is a well-known fact that LII signal first increases with the laser fluence and then reaches a plateau. With increasing laser fluence carbon sublimation during the laser pulse increases and evaporative cooling and mass loss cause the maximum LII signal to remain constant [12]. In our case, we will consider laser intensities causing excessive sublimation, i.e. laser fluence F = 3.0J/cm² would be sufficient for different classes of particulates to approach carbon sublimation temperature.



FIGURE 3 Comparisons of LII signal emit from one carbon particulate between model N and M with different initial diameters. Calculations were perform for a fluence of 3.0J/cm², at λ = 1064nm and FWHM = 10ns; T₀ = 2000K, λ_em = 600nm.

To better understand the differences between model N and M, it is useful to compare the results calculated from the two model types having the same particulate size. Combined effects on the LII signal profiles of the different particulates are represented in Figure 3, it seems that signal curves of the two model types could fit together in a restrict condition, i.e. D = 1.0-2.5μm demonstrated in Figure 3b and Figure 3c. In Figure 3a, the signal of Model N first rises, approaches the peak and then falls with a time constant appropriate thermal conduction to the medium, compared to lower curve of Model M. The reason why the signal calculated in Model M are weird is that the surface heat flux of particulate in Model M is calculated by Fourier’s law, which cannot be applied in nano-scale heat transfer. The thermal conductivity of carbon of nano-scale is significantly lower than the values in the conventional scale [21], which results in rapid rise and decay rate of LII signals emit from nano-scale carbon particulates. In contrast, the signal data from Model N is much lower than Model M in Figure 3d. This is because particulate in Model N is assumed to have a uniform temperature since the laser pulse, but in fact the internal temperature gradient could not be neglected of particulate of micron-scale. As a consequence Model N calculates the mean temperature of a heated-particulate, which is definitely lower than the surface temperature. These behaviors affect directly the time evolution of the LII signal, which strongly depends on particulate surface temperature.

In general, Model N suits the calculation of nano-scale particulates while Model M is appropriate to use in micron-scale particulates. Both Model types are adapted in the calculation of particulate size of 1.0-2.5μm. As fine (D < 2.5μm) particulates is of special interest of our study, Model N is capable to use in the simulation on the measurement of carbon particulates of PM_2.5 with LII technology.
4.3 Effect of carbon particulate size on LII signals
As we are seizing a way to measure carbon particulate size with LII technique, it is critical to discuss the effect of particulate size on LII signals. Figure 4 shows the dependence of the temperature and normalized LII signal on time of different particulate sizes (D = 0.1, 1.0 and 2.5μm). Figure 4a indicates that small carbon particulates approaches higher temperature-addition rate than large ones. With increasing particulate size of nano-scale the final temperature increases, and then drops down tardily when particulate size moves forward in the range of micron-scale. This is because when the particulate is sufficiently large, the energy is better distributed and cooling goes on rapidly because of the higher exchange surface, which limits the particulate temperature goes higher. As shown in Figure 4b, it is as if LII signal decay rates keep the same at the beginning and then vary from different particulate sizes. When particulate temperature is sufficient to enable the incandescence phenomenon which occurs around 4000K, sublimation of carbon particulate contributes significantly to signal decay rate. When temperature drops down under 4000K, signal decay rate depends primarily on conductive cooling rate. Therefore, the larger the particulates, the lower their signal decay rates.

FIGURE 4 Calculated values of (a) the temperature and (b) the normalized LII signal using Model N on dependence of time for different particulate sizes. Of the normalized LII signal time t=0 was fixed to correspond to the peak position of the signal for each size of particulate. Calculations were performed for a fluence of 3.0J/cm², at λ = 1064nm and FWHM = 10ns; T₀ = 2000K, λ_em = 600nm.

5. EXPERIMENTAL MEASUREMENTS

5.1 2D LII signals
Figure 5 shows a relationship between the instantaneous photograph of a methane flame and its corresponding two-dimensional LII image. Figure 5a reveals two different structures of the flame in height, i.e. a stable triangle shape of laminar flame below 80mm compared with relative streaky turbulent flame above 80mm, which is in coincidence with the image of ICCD, present in Figure 5b. As it is well known that the LII signal intensity is proportional to the carbon volume fraction [7], the intensity of LII signals recorded in the 2D ICCD image indicates the spatial distribution of carbon particulates. A very high LII intensity around the outer edge of the flame implies a very high carbon concentration. In contrast with low intensity within the flame, it seems that carbon particulates generate at the burning region. The luminosity is highlighted around the surface of the turbulent flame, which suggests amount of carbon particulates aggregating at the top of flame in the consequence of flame movement and propagation.

FIGURE 5 Direct photograph (exposure time: 1/50s) of a methane flame (a) and its corresponding two-dimensional distribution of LII signals (b). The flame was generated by a gaseous burner with methane at 1L/min. The instantaneous photograph of the methane flame shows its structure is 130mm in height and 15mm in width.

5.2 Particulate concentration measurement

The 2D distribution of LII signals from the ICCD image at different heights is calibrated by the simultaneous measurement of optical extinction, as shown in Figure 6. It seems the absolute carbon particulate concentration grows with increasing measurement height in the flame, limiting in 3ppm. The trend of the particulate distribution varies from different heights, as a typical bimodal distribution moving towards a unimodal distribution. However measurements from light absorption may cause the underestimation of the carbon volume fraction. This is because in optical extinction measurement the absorption of a carbon aggregate is lower than the sum of the total absorptions of its constituent primary particulates, and thus leads to a result of lower carbon volume fraction.

Figure 6 Carbon volume fraction profiles calibrated by the simultaneous measurement of light absorption in the radial direction at different heights.

5.3 Particulate-size measurement

For point measurements the entire LII decay curve can be used to obtain primary particulate-size information from fits of numerical simulation to the experimental decay curve. Figure 7 demonstrates experimental LII signal with the best fitting simulation curve in flame height. A mean particulate diameter represents the size distribution, which cannot be evaluated since normalized LII signals can biased towards particulate sizes. It seems the mean carbon particulate size remains at a region of 1-2nm in the flame and gets larger in height. The results accord with the conclusion reported in the literature that during the early stage of carbon formation carbon particulates are very big molecules with diameters less than 2nm [²⁹].





Figure 7 Experimental LII signal and its respective fitting simulation curve in flame height. The LII signals were detected at 600nm for a fluence of 0.1J/cm².

6. CONCLUDION

Two mass- and energy-balance models were established to simulate different sizes of laser-heated carbon particulates during the laser-heated and subsequent cooling process. A uniform internal temperature was assumed in model N while internal temperature gradient was considered in model M. The simulated LII signal profiles using two models were compared for different particulates sizes. The results showed that models N and M could fit together when the particulate diameter was in the range between 1.0 and 2.5μm. The magnitude of LII curve using model M was significantly lower than using model N when the particulate diameter was less than 0.1μm. This could be explained that the tradition heat transfer mechanism was not fit in this size region. The results showed that the internal temperature gradient could not be neglected when the particulate diameter was larger than 5.0μm. In addition, experiments of pointwise (0D) and imaging (2D) LII applications on a methane flame were set up. The measured temporally LII signals were compared to theoretical calculations. The results indicated that the carbon particulate size was of 1-2nm in flame. Simultaneously extinction measurement showed that the carbon particulate concentration was lower than 3ppm along the flame height.

ACKNOWLEDGEMENT

This work is supported by the National Basic Research Program of China (grants 2009CB219802), the National Science Foundation (51206144), the Natural Science Foundation of Zhejiang province (LY12E06003), the Program of Introducing Talents of Discipline to University (B08026).

REFERENCES

1[] Bond T C, Doherty S J, Fahey D W, et al. Bounding the role of black carbon in the climate system: a scientific assessment[J]. Journal of Geophysical Research: Atmospheres, 2013.

2[] Dumka U C, Moorthy K K, Kumar R, et al. Characteristics of aerosol black carbon mass concentration over a high altitude location in the Central Himalayas from multi-year measurements[J]. Atmospheric Research, 2010, 96(4): 510-521.

3[] Lou L, Luo L, Wang W, et al. Impact of black carbon originated from fly ash and soot on the toxicity of pentachlorophenol in sediment[J]. Journal of hazardous materials, 2011, 190(1): 474-479.

4[] Al-Omari S A B, Kawajiri K, Yonesawa T. Soot processes in a methane-fueled furnace and their impact on radiation heat transfer to furnace walls[J]. International journal of heat and mass transfer, 2001, 44(13): 2567-2581.

5[] Schulz C, Kock B F, Hofmann M, et al. Laser-induced incandescence: recent trends and current questions[J]. Applied Physics B, 2006, 83(3): 333-354.

6[] Eckbreth A C. Effects of laser‐modulated particulate incandescence on Raman scattering diagnostics[J]. Journal of Applied Physics, 1977, 48(11): 4473-4479.

7[] Melton L A. Soot diagnostics based on laser heating[J]. Appl. Opt, 1984, 23(12): 2201-2208.

8[] Michelsen H A, Liu F, Kock B F, et al. Modeling laser-induced incandescence of soot: a summary and comparison of LII models[J]. Applied physics B, 2007, 87(3): 503-521.

9[] Lehre T, Jungfleisch B, Suntz R, et al. Size distributions of nanoscaled particulates and gas temperatures from time-resolved laser-induced-incandescence measurements[J]. Applied optics, 2003, 42(12): 2021-2030.

10[] Krüger V, Wahl C, Hadef R, et al. Comparison of laser-induced incandescence method with scanning mobility particulate sizer technique: the influence of probe sampling and laser heating on soot particulate size distribution[J]. Measurement Science and Technology, 2005, 16(7): 1477.

11[] Liu F, Smallwood G J, Snelling D R. Effects of primary particulate diameter and aggregate size distribution on the temperature of soot particulates heated by pulsed lasers[J]. Journal of Quantitative Spectroscopy and Radiative Transfer, 2005, 93(1): 301-312.

12[] Michelsen H A, Witze P O, Kayes D, et al. Time-resolved laser-induced incandescence of soot: the influence of experimental factors and microphysical mechanisms[J]. Applied optics, 2003, 42(27): 5577-5590.

13[] Charwath M, Suntz R, Bockhorn H. Influence of the temporal response of the detection system on time-resolved laser-induced incandescence signal evolutions[J]. Applied Physics B, 2006, 83(3): 435-442.

14[] Wang F, Yan J, Ma Z Y, et al. Simulation on soot concentration measurement with laser induced incandescence[C]. Zhongguo Dianji Gongcheng Xuebao (Proceedings of the Chinese Society of Electrical Engineering). 2006, 26(7): 6-11

15[] Therssen E, Bouvier Y, Schoemaecker-Moreau C, et al. Determination of the ratio of soot refractive index function E (m) at the two wavelengths 532 and 1064 nm by laser induced incandescence[J]. Applied Physics B, 2007, 89(2-3): 417-427.

16[] Liu F, Smallwood G J. The effect of particulate aggregation on the absorption and emission properties of mono-and polydisperse soot aggregates[J]. Applied Physics B, 2011, 104(2): 343-355.

17[] Michelsen H A. Understanding and predicting the temporal response of laser-induced incandescence from carbonaceous particulates[J]. The Journal of chemical physics, 2003, 118: 7012.

18[] Bladh H, Bengtsson P E. Characteristics of laser-induced incandescence from soot in studies of a time-dependent heat-and mass-transfer model[J]. Applied Physics B, 2004, 78(2): 241-248.

19[] Allouis C, Rosano F, Beretta F, et al. A possible radiative model for micronic carbonaceous particulate sizing based on time-resolved laser-induced incandescence[J]. Measurement Science and Technology, 2002, 13(3): 401-410.

20[] Allouis C, Beretta F, D’Alessio A. Sizing soot and micronic carbonaceous particulate in spray flames based on time resolved LII[J]. Experimental thermal and fluid science, 2003, 27(4): 455-463.

21[] Liu J. Micro/nano scale heat transfer[M]. Science Press, 2001: 179.

22[] Bird R B, Stewart W E, Lightfoot E N. Transport phenomena[M]. Wiley, 2006: 357.

23[] Chang H, Charalampopoulos T T. Determination of the wavelength dependence of refractive indices of flame soot[J]. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 1990, 430(1880): 577-591.

24[] Snelling D R, Smallwood G J, Liu F, et al. A calibration-independent laser-induced incandescence technique for soot measurement by detecting absolute light intensity[J]. Applied optics, 2005, 44(31): 6773-6785.

25[] Vander Wal R L, Weiland K J. Laser-induced incandescence: development and characterization towards a measurement of soot-volume fraction[J]. Applied Physics B, 1994, 59(4): 445-452.

26[] Moreau C S, Therssen E, Mercier X, et al. Two-color laser-induced incandescence and cavity ring-down spectroscopy for sensitive and quantitative imaging of soot and PAHs in flames[J]. Applied Physics B, 2004, 78(3-4): 485-492.

27[] Hebert D, Coppalle A, Talbaut M. 2D soot concentration and burning rate of a vertical PMMA slab using Laser-Induced Incandescence[J]. Proceedings of the Combustion Institute, 34(2013): 2575-2582.

28[] Hofmann M, Bessler W G, Schulz C, et al. Laser-induced incandescence for soot diagnostics at high pressures[J]. Applied optics, 2003, 42(12): 2052-2062.

29[] Haynes B S, Wagner H G. Soot formation[J]. Progress in Energy and Combustion Science, 1981, 7(4): 229-273.