Particle Size Distribution of Soot from a Laminar/Diffusion Flame

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1 Aerosol and Air Quality Research, 7: 95 9, 7 Copyright Taiwan Association for Aerosol Research ISSN: print / 7-49 online doi: 9/aaqr Particle Size Distribution of Soot from a Laminar/Diffusion Flame Jian Wu, LinghongChen *, Jianwu Zhou, Xuecheng Wu, Xiang Gao, Gérard Gréhan, Kefa Cen State Key Laboratory of Clean Energy Utilization, Zhejiang University, Hangzhou 7, China UMR 664/CORIA, CNRS, BP 768 Saint Etienne du Rouvray, France ABSTRACT A practical method is proposed to determine the inflame soot particle size distribution via an explicit solution derived from time-resolved laser-induced incandescence (TiRe-LII. Three appropriate time intervals are selected from the TiRe- LII decay signal. The equivalent mean sizes as well as relative ratios of number densities for three classes of monodisperse particles are determined with the mono-exponential fit to each interval, which allows an explicit solution for the particle size distribution. Simulations show that inversed log-normal distributions from the explicit solution are coincident with the input parameters in terms of trend, and there exist critical time intervals where inversed results are most insensitive to the variation of interval length. The error of inversion is critically dependent on the geometric standard deviation but weakly dependent on the count median particle diameter. Influences of experimental conditions on the inversed error are additionally evaluated. The results show that flame temperature has a significant impact on the error of inversion. Thus, a database of the error as a function of flame temperature at a fixed aggregate size is established. The error database allowed the inversed results from experimental TiRe-LII signals to be readily corrected at various flame locations by interpolation. The corrected inversed log-normal distributions were consistent in trend with those determined from the established nonlinear regression method, and the modeling LII signals reproduced agree with the experimental data. The small deviation of the results potentially stemmed from the statistical noise contained in recorded LII signals. Keywords: Soot particle size; Laser-induced incandescence; Explicit solution; Error database. INTRODUCTION Numerous scientific studies have demonstrated that soot particles produced from incomplete combustion processes pose serious risks to human health and the environment (Pope and Dockery, 6; Kennedy, 7; Das et al., 5. The strong absorption from soot in the visible and near-infrared regions is related to positive radiative forcing (Jose, 6, which is believed to cause global warming. Furthermore, the deposition of soot can accelerate the melting of snow and ice due to additional surface absorption, and decrease the reflectivity of snow and ice, which also enhances global warming (Hansen and Nazarenko, 4. The absorption of soot is a million times stronger than that of CO, but the residence time in the Earth s atmosphere of soot is much shorter than that of CO (Kim et al.,. Thus, reducing soot emissions into the atmosphere is regarded as a near-term climate change mitigation approach (Shindell et al.,. * Corresponding author. Tel.: ( address: chenlh@zju.edu.cn Soot is formed through the high temperature condensation of hot gases produced from hydrocarbon fuels during combustion (Lin et al., 6, and is often emitted along with polycyclic aromatic hydrocarbons (PAHs (Zhang et al., 6. The physical and chemical properties of soot are very different from those of fuels (Veilleux et al., 9 and other kinds of particles that are generated simultaneously (Pu et al., 5; Zhan et al., 6. Most of the properties are closely related to particle size (Wahlin et al., ; Wu et al., 5; Fan et al., 6. Soot particles are usually polydisperse and particles of different sizes show distinctly different behaviors (Zhao et al.,. Therefore, quantitatively characterizing soot particle size distribution is necessary for better controlling soot emission. Time-resolved laserinduced incandescence (TiRe-LII is a powerful tool for the measurement of soot particles, including their size distribution (Schulz et al., 6; Michelsen et al., 5. In this technique, soot particles are heated by an incident high-power pulsed laser and subsequently cooled by evaporation, heat transfer, and thermal radiation. The particle cooling process exhibits size-dependent behavior and quantitative evaluation of the particle size distribution can be achieved by analyzing the LII decay signal (Vander Wal et al., 999; Snelling et al., b; Wu et al., 7.

2 96 Wu et al., Aerosol and Air Quality Research, 7: 95 9, 7 Recovering the particle size distribution from the TiRe- LII signal is a challenging task because this is an ill-posed problem that is difficult to solve (Daun et al., 7. Thus, implicit methods are routinely employed to solve the problem based on the assumption of a log-normal distribution for particle size through a best-fit comparison of the LII decay signal to the simulated database from modeling prescription. Lehre et al. ( used direct non-linear regression to solve the size distribution by minimizing the deviation between the entire experimental signal and the simulated LII signals modeled by varying the input of the distribution function. To avoid the time-consuming non-linear regression in the best-fit comparison, Dankers and Leipertz (4 proposed a simple engineering approach to determine the particle size distribution through an exponential fit to the LII decay signal in two different time intervals after a laser pulse by using multivariate minimization. With a temporal temperature measurement of the laser-heated soot, an alternative approach was presented by Liu et al. (6b, where the multivariate minimization problem was transformed into an easier-to-solve univariate minimization problem. According to the above methods, many worthwhile contributions have been made both in the gases of combustion processes and in their exhausts (Kock et al., 6; Hadef et al.,. However, in the least-squares fit procedure used in most of the references cited above, various experimental conditions, such as aggregate size and flame temperature, lead to a great effort in the establishment of different databases and repeated comparisons with simulated LII signals. Thus, further improvements are still needed for recovering the soot particle size distribution in practical applications. An important aspect in the evaluation of the size distribution from a TiRe-LII signal is that the smaller particles cool faster than the larger ones and their contribution to the overall LII signal decreases over time, with the remaining signal increasingly dominated by the larger particles. This causes an increase in signal decay with time. Dankers and Leipertz (4 exploited the decay rates derived from exponential fits to two time intervals selected from the LII signal to represent two different individual particle size classes. An improved strategy called two-exponential reverse fitting (TERF was proposed by Cenker et al. (5a to determine the monodisperse equivalent mean sizes for small and large particle groups and evaluate the relative ratios of particle number densities for both groups. The proposed strategy allows the particle size distribution to be roughly characterized without a distribution function. Based on this strategy, an alternative improved method may be feasible for deriving the particle size distribution directly, where three appropriate time intervals are selected from the LII decay signal induced by polydisperse particles to infer the information of three particle size groups. It is possible to resolve the generally used log-normal distribution explicitly via the particle information. Unlike with conventional implicit methods, the explicit solution can be used in practical applications and the cumbersome minimization may be simplified. A practical method with an explicit solution derived from the TiRe-LII signal is proposed in the present study to determine the soot particle size distribution. The effects of the selected time interval length (TIL and the experimental conditions, including the flame temperature, aggregate size, refractive index function, thermal accommodation coefficient, and noise from the detection system, on the inversed results are quantitatively evaluated. An error database of the inversion is established, and based on this database, the method is tested on experimental LII signals acquired in a C H 4 /air laminar diffusion flame at various heights above the burner (HAB. The particle size distributions are determined and compared to the data obtained using the traditional non-linear regression strategy. THEORY LII Signal Evolution The model used to describe the heat and mass transfer process of laser-heated soot particles is based on that proposed by Melton (984 and further developed by Snelling et al. (a and Liu et al. (6a. According to the Rayleigh- Debye-Gans theory for fractal aggregates (RDG-FA (Koylu and Faeth, 99, the absorption and radiation of aggregates can be expressed as the sum of the same processes for individual primary particles. Considering the aggregation tendency of soot particles in flames, the energy equation can be written as: d T p dpe( m H v dm Npdpscs Np q( t Np 6 dt Mv dt q q c rad where the term at the left side is the change in the internal energy of an aggregate particle. The terms on the right side are the rate of laser energy absorption by the aggregate, the soot sublimation, the rate of heat loss by conduction from the aggregate to the surrounding gas, and thermal radiation, respectively. The parameter N p represents the number of primary soot particles within the aggregate, d p is the primary soot particle diameter, ρ s and c s denote the soot particle density and specific heat, which are set to the values used by Liu et al. (6a, and T p and t are the laser-heated soot particle temperature and time, respectively. The variable E(m in the first term on the right-hand side represents the refractive index function, and λ and q(t are the excitation wavelength and pulsed laser temporal intensity, respectively. Since the influence of aggregation on the soot sublimation is unknown, the total area of the primary particles is used in the sublimation term, as done by Bladh et al. (8. An ensemble of sublimed carbon species (C C 7 is taken into account. The heat of vaporization of soot, H v, and molecular weight of soot vapor, M v are set to the temperaturedependent value proposed by Smallwood et al. (. The detailed expression for the rate of soot particle mass reduction, dm/dt, can be found in the study of Snelling et al. (a. For heat conduction, the approach proposed by Filippov and Rosner ( based on the Fuchs model is chosen. This model was found to be the most accurate to define the (

3 Wu et al., Aerosol and Air Quality Research, 7: 95 9, 7 97 heat conduction process of spherical nanoparticles in a wide range of the Knudsen (Kn number (Liu et al., 6a. It is a two-layer model where the space surrounding the particle is divided into inner and outer regions separated by a limiting sphere, as detailed by Liu et al. (6a. In the approach, the goal is to find the distance between the soot particle and the limiting sphere, δ, and the temperature of the limiting sphere, T δ. Inside the sphere, the heat conduction is given by: where f is the Eucken correction, which is given by (9γ 5/4. The mean free path follows the expression reported by McCoy and Cha (974, i.e.,: kg WgTg g ( fpg R The parameters Λ and Λ are calculated as: / (8 d p RT q T T ( * T eff g δ c * p δ Tδ Wg δ Reff ( δ Reff (9 where α T, p g, R, and W g are the thermal accommodation coefficient of soot, ambient gas pressure, universal gas constant, and average molecular weight of gas, respectively. The equivalent spherical diameter of the aggregate d eff is defined as (Snelling et al., 4: deff dp, Np / Dh Np deff d, p Np kh where the scaling parameters k h and D h are calculated with a thermal-accommodation-coefficient-dependent function (Liu et al., 6c. The average value of the specific heat ratio, γ *, is used in Eq. (, which can be evaluated as: Tp * d TP Tδ Tδ T (4 where the temperature-dependent expression of the specific heat ratio γ is used (Liu et al., 6a. In the region outside the limiting sphere, heat conduction is expressed as: T δ c eff g Tg q 4 R k dt (5 where R eff is the radius of the equivalent sphere, T g is the surrounding gas temperature, and k g is the heat conduction coefficient of the surrounding gas, with a temperaturedependent value chosen from the report by Liu et al. (6a. The distance is given as: R R 5 5 eff δ 5 5/ eff The mean free path inside the limiting sphere λ δ is related to the mean free path of the gas λ g, according to kg ( Tδ T δ ( Tδ f( Tg δ = g kg ( Tg T g ( Tg f ( Tδ ( (6 (7 Integration of the Planck function over the full wavelength range yields the thermal radiation from the aggregate (Michelsen,, i.e.,: 5 99 dpnp( kbtp E( m q rad ( hhc ( where k B, h, and c are respectively the Boltzmann s constant, Planck s constant, and speed of light. By solving Eqs. ( (, the primary particle temperature evolution can be obtained. For the given polydisperse soot particles and the detection wavelength λ det, the cumulative TiRe-LII signal in the detection volume J λdet can be written as: det dmax J ( t S (, t d f( d dd ( dmin det p p p where S is the temporal LII signal induced by a soot particle with size of d p, f(d p is a function used to describe the particle size distribution, and d min and d max are the smallest and largest particles in the detection volume, respectively. Determination of Particle Size Distribution Although inflame soot size distributions are sometimes bimodal (Stirn et al., 9, a numerical analysis performed by Johnsson et al. ( showed that the TiRe-LII signal was insensitive to the smaller-size mode. The following widely used assumption for the inflame soot particle size distribution is adopted here: f( d exp (ln d ln d p cmd p d (ln pln g g ( This function is defined by the count median particle diameter and the geometric standard deviation σ g. The particle size distribution can be determined as long as and σ g are available. A typical decay signal produced from the polydisperse particle ensemble with log-normal distribution is presented in Fig.. The TiRe-LII signal excited from a polydisperse particle ensemble is a superposition of the signals produced from

4 98 Wu et al., Aerosol and Air Quality Research, 7: 95 9, 7 J nm (t.. S (t=a exp(-t/ Simulated LII signal Exponential fit to the first interval Exponential fit to the second interval Exponential fit to the third interval Additional signal for the first interval Additional signal for the second interval S (t=a exp(-t/ S (t=a exp(-t/ t (ns Fig.. Simulated LII decay signal for polydisperse soot particles with a log-normal distribution ( = nm, σ g =.4. The signal intensity is normalized to the peak value. Three time intervals are selected from the decay signal curve. An exponential function is fitted to each interval and extrapolated back to time t = ns (dashed line. The additional signal for the first interval represents the difference between the LII signal within the first interval and the back-extrapolated curve fitted to the second interval. The additional signal for the second interval represents the difference between the LII signal within the second interval and the back-extrapolated curve fitted to the third interval. each unique size group. As validated by Cenker et al. (5a, a shortened time interval selected from the whole signal induced by polydisperse particles can be fitted with an exponential function (Fig., which provides a signal decay time, τ, corresponding to a specific monodisperse equivalent mean particle size, d p. The probability of d p in the detection volume can be approximately expressed by: F f ( d d ( p p where d p is the particle size interval. If two different intervals are chosen from the decay signal, two kinds of particle size are determined. With an assumption of a uniform size interval, the ratio of particle number densities for the two classes in the detection volume is: A Cnd p (5 where C is a constant associated with the detection system. Thus, the ratio of particle number densities for the two classes, d p and d p, can be written as: A d p A d p n (6 n Combining Eq. (4 with Eq. (6 yields: p A dp p A d p f ( d f( d (7 n F f ( d p n F f( d p (4 where n is the particle number density, and subscripts and represent the two kinds of particle size, respectively. As shown in Fig., the fitting curves are extrapolated back into the time domain before the selected time interval to represent the theoretical cooling of the monodisperse equivalent mean particles. The peak value of each backextrapolated curve equals the pre-exponential factor A, which is linearly proportional to the soot volume fraction in the detection volume as the particles are heated to the same temperature by a laser with a uniform profile, i.e.,: Substituting Eq. ( into Eq. (7, it can be found that except for parameters and σ g, all parameters (A, A, d p, d p can be determined by fitting to the experimental LII signal (see App. A.. Since two parameters ( and σ g have to be determined, at least two equations such as Eq. (7 should be built. In this regard, the information of the three classes of particles must be extracted from the TiRe-LII signal. Here, three time intervals are selected from the whole LII decay signal (see Fig.. Each interval is fitted with a mono-exponential function. In terms of the LII signal in Fig., the values of τ, τ, and τ are linearly related to the three classes of particles with monodisperse equivalent mean sizes of d p, d p, and d p, respectively. Similar to Eq. (7, an equation can also be established for d p and d p. Thus,

5 Wu et al., Aerosol and Air Quality Research, 7: 95 9, 7 99 and σ g can be inversed as: dp nd p dp nd p ln dp d p ln ln ln d p d p ln ln dp n dp d p n dp ln dcmd = dp nd p dp nd p ln ln ln ln dp n dp d p n d p (8 d p (ln dp d p ln d cmd ln d p ln g (9 nd p ln nd p where the ratio of particle number densities n /n, n /n can be obtained through Eq. (6. Additional information about the derivation of Eqs. (8 and (9 can be found in Appendix A. Because the intervals selected from the decay signal to evaluate d p, d p, and d p contain overlapping distributions, the ratios of particle number densities determined do not exactly equal the values calculated from Eq. (. Thus, there exist errors in the inversed results of and σ g. It should be pointed out that additional signals of the first and second intervals (Fig., defined by Cenker et al. (5a as the difference between the LII signal and the backextrapolated curves, contain somewhat narrower distributions compared to the original ones since the contribution of the large ensemble is eliminated. However, the additional signals are not used to evaluate the particle sizes of d p or d p because inversions based on the additional signals yield results that are not as good as those obtained using the proposed method. EXPERIMENT An experiment on a laminar diffusion ethylene/air Santoro flame was performed to validate the applicability of the proposed method. The flame was stabilized on a stainless steel burner consisting of two concentric tubes with inner diameters of. and mm, as described by Santoro et al. (987. The ethylene and air flow rates were controlled by mass flow meters to be maintained at. and 4.78 standard liters per minute (SLPM, resulting in a visible stabilized flame height above the burner of 88 mm. A schematic illustration of the LII experimental setup is shown in Fig.. The soot particles generated in the flame were heated by a Nd:YAG laser (Quantal, Brilliant B with a pulse width of 7 ns; the repetition rate was Hz. To strictly avoid the fluorescence induced by PAHs, a fundamental wavelength of 64 nm was selected in the experiment. After the laser output, an attenuator was applied to adjust the laser energy. Subsequently, a stainless steel diaphragm was used to select the central portion of the ~8 mm laser beam, the diameter of which was measured by a beam profiler (Duma Optronics, BeamOn. In combination with a telescope system consisting of two sphere lens with focus lengths of 5 and mm, respectively, the diameter of the selected portion of the laser beam was reduced to nearly mm. From observations by the beam profiler placed behind the telescope system, the standard deviation of the spatial intensity from the pixel average was around 5%, suggesting that a laser beam with nearly homogeneous intensity distribution was obtained, as shown in Fig.. The laser fluence was set to 8 J cm, which was monitored by a power meter. Although the calculated laser fluence was actually lower than that in the flame due to attenuation of the flame, the theoretical analysis demonstrated that the deviation in particle diameter caused by the difference can be neglected. Modeling showed that the peak temperature of soot under such laser fluence was about K. This is far lower than 7 K, a typical level above which considerable sublimation occurs (Cenker et al., 5b. In order to obtain sufficiently strong LII signals with good signal-to-noise ratios, the probe zones at various heights in the flame were located at a -mm distance from the Fig.. Experimental setup.

6 Wu et al., Aerosol and Air Quality Research, 7: 95 9, 7 (a. HAB=5 mm Fig.. Spatial intensity profile of the laser beam. burner axis. The LII signals were imaged onto a slit, resulting in a cylindrical zone to be measured in the flame with a diameter of mm and a width of mm. Thus, the volume of the detection zone was about.785 mm. A photomultiplier (PMT, Hamamatsu H7- connected to a digital 6- GHz oscilloscope (Agilent Technologies, DSO864B was applied to record the LII signals. To reduce the flame luminosity additional to the LII signals and prevent the scattering of laser light, a narrow band-pass filter with a center wavelength of 45 nm (full width at half maximum = nm was employed in front of the PMT. LII signals were detected at several heights between and 7 mm above the burner in steps of mm. A typical single-shot LII signal recorded at HAB = 5 mm is displayed in Fig. 4(a. In order to improve the signal-to-noise ratio, at least 5 single shots were recorded for each height to give an average result (see Fig. 4(b. The baseline signal before the laser pulse, representing the line-of-sight integrated flame luminosity, was subtracted at each location to suppress the influence of the background signal. Note that this treatment also resulted in a subtraction of the signal from the LII sampled region, which may lead to small errors in the evaluation of decay time. For the exponential fit, the time zero was shifted to the peak of the LII signal. RESULTS AND DISCUSSION Theory Validation Simulations were conducted to validate the proposed method. The refractive index function E(m and the thermal accommodation coefficient α T were chosen as and.5, respectively (Cenker et al., 5a. The flame temperature T g, also regarded as the initial soot temperature under the assumption of thermal equilibrium between the particles and the surrounding gas, was taken to be 6 K in the model, which is a typical temperature level for the flame (Schulz et al., 6. The other input parameters, including Normalized LII signal (b Normalized LII signal t (ns HAB= mm HAB=4 mm HAB=5 mm HAB=6 mm HAB=7 mm t (ns Fig. 4. (a Single-shot LII signal at HAB = 5 mm and (b averaged LII signals at various flame heights. the laser fluence, excitation wavelength, and detection wavelength, were consistent with those employed in the experiments. For simplification of the calculation, the aggregate size N p employed in Eq. ( was set to. A set of LII signals was generated by varying the particle size distribution as an input for modeling. The count median particle diameter varied from to 6 nm with an interval of nm and the geometric standard deviation σ g varied from. to.7 with an interval of.. Thus, a total of 5 cases were considered. In each case, the heat and mass transfer equations were solved for 5 primary soot particle diameters in the range of 5 nm with uniform intervals. Although the heat loss due to sublimation and thermal radiation is of minor importance at such a low laser fluence, these effects were still included in the calculations. Solutions for all isolated particles were then taken into the numerical integration of Eq. ( in combination with a simple trapezoidal algorithm to generate an LII signal. Subsequently, three time intervals were selected from the simulated LII signal and then employed to perform the inversion with the proposed method. Due to the complex and long procedures required for the treatment of the fits with various interval lengths, the three time intervals were set to a uniform length. As shown in Fig., each 4-ns interval was arbitrarily selected. The three intervals are consecutive and there is no delay between them. Using exponential fitting, the particle sizes were determined based

7 Wu et al., Aerosol and Air Quality Research, 7: 95 9, 7 on the relation between the particle size and signal decay rate, and the number densities ratios based on Eq. (6. Finally, the count median particle diameter and geometric standard deviation were calculated in conjunction with Eqs. (8 and (9. To distinguish the inversed results from the original values of, and σ g,, used as the input for generating the simulated LII signals, the former are denoted as, and σ g,. The inversed results are respectively shown in Figs. 5(a and 5(b. It can be seen that the inversed results exhibit similar trends but with a certain deviation compared to those of the initial input parameters. For the count median particle diameter, the inversed results are mostly smaller than the original inputs, whereas the inversed geometric standard deviations are mostly larger. Generally, relatively remarkable deviations are introduced for the log-normal distributions with smaller and σ g. However, for all cases, the inversed results increase monotonically with the original inputs for modeling. The explicit solution shows some potential for determining particle size distribution even though the monodisperse equivalent mean particle sizes d p, d p, and d p evaluated actually represent overlapping size distributions. The additional signals (Fig. are also employed to infer the distribution parameters in combination with Eqs. (8 (a, (nm g =. g =. g =. g =.4 g =.5 g =.6 g =.7, =, , (nm (b = nm. = nm d. cmd = 4 nm. g, = = 5 nm g, = 6 nm g, g, Fig. 5. (a Inversed results of count median particle diameter and (b geometric standard deviation at various log-normal distributions based on TIL = 4 ns. and (9. Most of the inversions are far smaller than the original values (Fig. 6. Note that with a large count median particle diameter ( = 5, 6 nm, σ g, remains nearly constant. The small dynamic range of inversions over the entire range of σ g results in a low sensitivity according to the analysis reported by Cenker et al. (5b. The nonmonotonicity of the results leads to complicated inversion in practice. Since the particle size distribution is a continuous function, contributions of particles with various sizes to the total signal continuously change with time. Thus, the three groups of d p, d p, d p evaluated from exponential fits critically depend on the length set for each time interval, and in turn, it is anticipated that the length significantly influences the inversion. In order to evaluate this effect quantitatively, the inversions were conducted by setting the time interval to a length varying from 5 to 65 ns in.5- ns (the time step used for the simulation of the LII signal increments. Calculations indicate that invalid results are obtained when the TIL is set to be smaller than ~4 ns because of a negative radicand in Eq. (9. Thus, the results presented in Fig. 7 are inversed at TIL values larger than 5 ns. For a given log-normal distribution, with increasing TIL, the results in Fig. 7 demonstrate that, exhibits a trend opposite to that for σ g,. According to Fig. in the work by Cenker et al. (5a, a possible explanation is that log-normal distributions defined by different combinations of and σ g can lead to an almost identical LII signal when a larger comes along with an appropriately smaller σ g. Inversions of larger or σ g are less sensitive to time interval variation. For instance, the curve calculated for =, σ g =. changes more rapidly than those for =, σ g =.4 and = 4, σ g =.. Fig. 7 indicates that there exists a critical TIL for each log-normal distribution where the inversions are insensitive to the variation of the interval length. The critical length can be taken as the optimal choice for the interval to deduce the particle size distribution. The critical TILs for inversions of and σ g are not always identical. In g, = nm = nm =4 nm =5 nm =6 nm g, Fig. 6. Inversions of geometric standard deviation with additional signals.

8 Wu et al., Aerosol and Air Quality Research, 7: 95 9, 7 (a, (nm (b g, =4 nm, g =. =4 nm, g =.4 =4 nm, g =.6 = nm, g =. = nm, g =.4 = nm, g = TIL (ns TIL (ns Fig. 7. Effects of time interval length on the inversed results; (a count median particle diameter and (b geometric standard deviation. The time step for markers on the curves is larger than.5 ns because some data were selected to be represented by the markers for a clear description. particular, there is a discernable difference at a large σ g. For instance, when is fixed at nm, the critical TILs are both.5 ns for, and σ g, with σ g =., but 6.5 ns for, and 5.5 ns for σ g, with σ g =.6. However, due to very slight changes for the inversed results at larger and σ g (Fig. 7, the critical TIL can be set to that from σ g. The error introduced by this treatment is far less than %. As shown in Fig. 8, the critical TIL increases with and σ g. Note that the TIL for the log-normal distribution of = 4 nm and σ g =.7 is not accurate because the calculation result is larger than 65 ns. The inversed results with critical intervals are replotted in Fig. 9. The presented inversed results for the cases where σ g =.7 with = 4, 5, and 6 nm are derived at an interval length of 65 ns because the critical TILs are beyond the calculated range of 5 65 ns. However, the inversions remain nearly constant with increasing interval at large geometric standard deviation (see Fig. 7, and thus the additional errors for the inversed results can be neglected. Compared to the results in Fig. 5, the inversions are less scattered at various and σ g. Except for the cases where the initial geometric standard deviation is set to.6 or.7, the inversions, including, and σ g,, are tightly distributed around an identical value at each and σ g. The relative errors in the inversed results with respect to the original values were also calculated; the results are shown in Fig.. Except at σ g =.6 or.7, the error is critically dependent onσ g but weakly dependent on, and declines with σ g. Differences in error are less than.5% for and less than.5% for σ g at a fixed count median particle diameter, Critical TIL (ns = nm =4 nm Fig. 8. Critical time interval length of inversions under various lognormal distributions. (a, (nm (b g, g =. g =. g =. g =.4 g =.5 g =.6 g =.7 g, =, , (nm = nm. = nm d. cmd = 4 nm. g, = = 5 nm g, = 6 nm g, Fig. 9. (a Inversed results of count median particle diameter and (b geometric standard deviation at various log-normal distributions based on the critical TIL.

9 Wu et al., Aerosol and Air Quality Research, 7: 95 9, 7 Error of inversion (% = nm = nm =4 nm =5 nm =6 nm Fig.. Errors of inversion as a function of geometric standard deviation. Solid and open symbols stand for the errors in σ g and, respectively. suggesting that the errors for all cases can be acquired by evaluation at a specific with various σ g. The inversed results can thus be readily corrected based on the error database built from the experimental conditions. For instance, we just need to model the errors at = nm with σ g varying from. to.6; then, the inversed results from the experimental LII signals can be corrected based on these errors. Error of Inversion under Various Experimental Conditions It is anticipated that variations of the LII signal have an impact on the inversion since the inversion of particle size distribution directly relies on the signal. Through systematic analysis of the experiment, parameters such as aggregate size and flame temperature as the input for modeling vary with location. Preliminary numerical simulations based on different aggregate sizes and flame temperatures also show tightly distributed results similar to those in Fig. 9. But the relative errors with respect to the initial distribution parameters are different. By systematically changing an individual input parameter for generating LII signals and based on findings of the critical time interval, the inversed errors in and σ g can be quantitatively evaluated. As addressed by Daun et al. (7, the statistical noise introduced by the detection system also affects inversed results. Hence, it is necessary to consider the detection noise in the establishment of the error database. Similar to the primary soot particles, aggregates formed in flames often have a kind of distribution in size (Bladh et al.,. For simplification, the aggregate size is assumed as monodisperse. The assumption is somewhat reasonable since in the free-molecular regime at high ambient temperature, the effect of aggregate size polydispersity is relatively unimportant on the signal decay compared to the effect corresponding to the primary particle diameter polydispersity (Liu et al., 6c. The LII decay signal allows the extraction of information on primary particle diameter as long as an estimate of the mean aggregate size is available. Fig. (a g shows that more notable deviations are produced by the proposed method in the presence of aggregation. The distribution width derived from the experimental LII signal is corrected to be somewhat smaller compared to that for the case without aggregation. Note that the shielding effect from aggregation on the calculation of the count median particle diameter has been taken into account. When N p >, the relative errors remain nearly constant at σ g =. or.4 but slightly increase at σ g =.6 with increasing aggregate size. A plausible explanation for this is that the relative contribution of the conductive cooling reduces in the overall heat exchange with increasing aggregate size; the assumption of mono-exponential approximation may thus lose its validity. The evaluation of flame temperature impacts the particle sizing significantly (Will et al., 998. In the atmospheric diffusion flame considered in the present study, the flame temperature falls in a range of 5 7 K for the current detection zones according to measurements in a previous work (Chen et al., 7. Since the variation of aggregate size slightly affects the inversed errors (see Fig. (a, the input for modeling is chosen as N p =. The calculation results in Fig. (b demonstrate a similar trend in error for different flame temperatures, and show that a higher flame temperature (e.g., 7 K leads to a larger error in the inversion. The maximum difference in relative error for different flame temperatures at a given distribution reaches 5% in and % in σ g. Since the probe volumes of LII are located at a -mm distance from the flame axis, the soot maturity is subjected to the variations of height in the present diffusion flame with a converging flame front geometry. In other words, measurements at HAB = mm were performed with the soot in the center flame region, whereas for those at HAB = 7 mm, the probe volume was on the flame edge. The different locations may lead to differences in the refractive index function (E(m and thermal accommodation coefficient (α T. Thus, three sets of typical values are set to E(m and α T. The corresponding errors of inversion are presented in Figs. (c and (d, respectively, where opposite trends can be found; i.e., the error of inversion increases with refractive index function but decreases with thermal accommodation coefficient. In the experiments, the TiRe-LII signals were captured by the PMT. The statistical noise from the detection system, regarded as the intensity recorded without laser heating, was evaluated as 5% of the peak intensity measured at HAB = 5 mm (see Fig. 4(a. Based on a previous study (Chen et al., 7, a random function was applied to generate the noise. The phantom single-shot LII signal with added noise is plotted in Fig. (e. The effect of PMT shot noise is usually mitigated by averaging a large set of single-shot measurements. Thus, the signal averaged over shots is also presented in Fig. (e. From Fig. (f, the curves for inversed error oscillate with a few shots averaged, and become steady as the number of averaged shots increases to several hundreds; their values nearly equal those deduced from the signals without noise. As a corollary, to obtain a relatively accurate result, it is essential to record and average hundreds of shots to improve the signal-to-noise ratio.

10 4 Wu et al., Aerosol and Air Quality Research, 7: 95 9, 7 (a Error of inversion (% g -error at g =. g -error at g =.4 g -error at g =.6 -error at g =. -error at g =.4 -error at g = N p (b Error of inversion (% g -error at T g =5 K -error at T g =5 K g -error at T g =6 K -error at T g =6 K g -error at T g =7 K -error at T g =7 K (c Error of inversion (% g -error at E(m=.5 -error at E(m=.5 g -error at E(m= -error at E(m= g -error at E(m=5 -error at E(m= (d Error of inversion (% - - g -error at g -error at T = T = g -error at T =.5 -error at T =.5 g -error at T = -error at T = (e J nm (t... g Single shot LII -shot averaged LII t (ns (f Error of inversion (% g g -error at g =. g -error at g =.4 g -error at g = Number of averaged shots Fig.. Relative error of the inversion as a function of aggregate size (a, flame temperature (b, refractive index function (c, thermal accommodation coefficient (d, number of averaged shots added with noise (f. The curves in (e represent simulated signals added with noise. The count median particle diameter keeps constant as nm. Determination of Soot Particle Size Distributions Using Experimental LII Signals To validate the proposed method experimentally, the captured LII signals shown in Fig. 4(b were employed to perform the inversion in conjunction with Eqs. (8 and (9, with the inversed results corrected based on Fig.. The flame temperature at each measurement position is given in our previous work (Chen et al., 7. Based on a previous study (Cenker et al., 5a, the aggregate sizes observed using transmission electron microscopy (TEM with sampling at different heights in the outer layer of a given flame were taken into account. The inversions from Eqs. (8 and (9 take into account the real aggregate sizes and are corrected based on the error database established with N p = ; the difference of the inversed error due to variation of aggregate size is negligible, i.e., less than ~% (see Fig. (a. The refractive index function E(m and thermal accommodation coefficient α T are still subject to large

11 Wu et al., Aerosol and Air Quality Research, 7: 95 9, 7 5 uncertainties and unknown at different locations of the flame (Michelsen et al., 7. Thus, the typical values of for E(m and.5 for α T were fixed throughout the entire flame. As aforementioned, inversions should be carried out based on the critical time interval. Thus, it is necessary to find the critical time interval first. The inversed geometric standard deviations as a function of the TIL are plotted in Fig.. The calculations show an opposite tendency for the inversion of the count median particle diameter, identical to the numerical results displayed in Fig. 7. To reduce the influence of experimental statistical noise on the exponential fit, the calculations started at an interval length of ns. The increment was set to ns, a value equal to the sampling interval of the experimental data. According to the modeling in Fig. 7, the critical time interval can be obtained where the inversion is most insensitive to variation of the TIL. Based on the critical time intervals, the log-normal distributions inversed from the experimental LII signals were obtained; they are summarized in Table. The error database shown in Fig. (b allows correction of the inversed results combining with the flame temperature by interpolation. For instance, distributions directly inversed from the experimental LII signal at HAB = 5 mm are, =. nm, σ g, =.55. From Fig. (b, errors of the inversed geometric standard deviation at the corresponding flame temperature, 585 K, can be calculated by interpolation. Then, the relative errors for, =. nm, σ g, =.55 can be derived by minimizing the deviation in comparison with the calculations, which are evaluated as 7 and 6, respectively. Hence, the inversed results are corrected using these errors; the final results are, = 4.8 nm and g, HAB= mm HAB=4 mm HAB=5 mm HAB=6 mm HAB=7 mm TIL (ns Fig.. Inversed results as a function of time interval length. σ g, =.444. Similarly, particle size distributions at other flame heights also can be corrected, as shown in Table. To assess the quality of the obtained results, particle size distributions were additionally determined using the established non-linear regression method proposed by Lehre et al. (. The method is conducted globally by minimizing the least-squares estimator: N exp exp mod mod Jt ( i, dcmd, g Jt ( i, dcmd, g = ( i i where J is the LII signal, superscripts exp and mod indicate experimental and modeling parameters, respectively, and the standard deviation σ i is simply set to one. The flame temperatures and aggregate sizes taken as input parameters for prediction are shown in Table. The other parameters are identical to those used in the proposed method. Particle size distributions determined using non-linear regression method and the proposed method are plotted in Fig. (a. As can be seen, there exist disagreements in the distributions obtained using these two methods. The particle sizes derived using the proposed method are generally larger than those obtained using the non-linear regression method. Comparisons show that the proposed method overpredicts the count median particle diameter but underestimates the geometric standard deviation. Nevertheless, overall satisfactory agreements still exist between the experimental signals and the modeling ones reproduced from the corrected inversed log-normal distributions (Fig. (b. From Table and Fig. (a, in contrast with the inversed results with correction, the results without correction are closer to those derived using the non-linear regression method. Noise contained in the signal may be responsible for the corrected inversed results that small particles are omitted, since the signals of small particles within a polydisperse ensemble are relatively weak and the information extracted from such particles is more likely to be affected by noise. The sensitivity to noise in Fig. (f also indicates that signals with stronger noise allow inversions closer to the true values. Thus, a plausible explanation for the larger discrepancy that resulted from correction is that the inversion is sensitive to the signal noise, and the real noise of the experimental signal is still remarkable, leading the inversion to be closer to the real one, as shown in Fig. (f. Although the results in Fig. (f suggest that the number of LII signals averaged is sufficient to deduce reliable results in the experiment, the noise intensity presented in Fig. (e is a mean value roughly evaluated from the Table. Particle size distributions inversed from experimental LII signals. HAB (mm T g (K (Chen et N p (Cenker et Inversion without correction Inversion with correction al., 7 al., 5 (nm σ g (nm σ g

12 6 Wu et al., Aerosol and Air Quality Research, 7: 95 9, 7 f (d p f (d p f (d p f (d p f (d p HAB= mm, T g =64 K =6.8 nm g =.9 =. nm g =.4 HAB=4 mm, T g =6 K =5.8 nm g =.468 =.5 nm g =.5 HAB=5 mm, T g =585 K =4.8 nm g =.444 =9. nm g =.5 HAB=6 mm, T g =555 K =8.8 nm g =.44 =4. nm g =.5 HAB=7 mm, T g =55 K =4. nm g =.69 =. nm g = d p (nm J 45nm (t J 45nm (t J 45nm (t J 45nm (t J 45nm (t Experimental LII signal Phantom LII signal (present method t (ns Fig.. (a Particle size distributions determined from experimental LII signals. Open circles represent the inversed results obtained with the proposed method and solid circles represent the results derived using the non-linear regression method. (b Comparison of simulated signal to the experimental data. The simulations were performed with the log-normal distributions derived using the proposed method. experimental signal without laser heating, which is different from that presented in Fig. 4(a in terms of noise intensity and pattern. In order to examine the accuracy of the proposed method, the noise of the experimental signal was completely removed by replacing the experimental signal with the numerical one generated from the distribution determined using the non-linear regression method. According to Daun et al. (7, the non-linear regression method proposed by Lehre et al. ( is quite resilient to measurement noise and is recommended to get unbiased estimates. It is reasonable to assume that the results obtained using the non-linear regression method are the true ones. Combined with the LII signal numerically reproduced from = 9. nm, σ g =.5 at HAB = 5 mm, the inversed results as a function of TIL (Fig. 4 indicate that, = 6.7 nm and σ g, =.68 at the critical interval. Since the flame temperature is T g = 585 K, interpolation with the error database in Fig. (b shows that the errors for and σ g are 5 and 58, respectively. The correction leads to, = 8. nm and σ g, =.549, which are much closer to = 9. nm, σ g =.5. It is believed that the small additional deviation is produced from the error database (Fig. (b since it is built based on N p =. The analysis above suggests that further improvement of the signal-to-noise ratio in the experimental signal is essential to obtain an accurate result with the proposed method. Nevertheless, as an explicit solution, the present approach

13 Wu et al., Aerosol and Air Quality Research, 7: 95 9, 7 7 g, g, TIL (ns, , (nm Fig. 4. Inversed results as a function of time interval length based on the numerical signal from = 9. nm, σ g =.5 at HAB = 5 mm. is more efficient in computation. Although the inversed results deviate from the exact distributions, it is easy to correct the inversions because the errors of the inversed results only depend on the geometric standard deviation. Although some errors still exist in the corrected inversions, the obtained particle size distributions as a function of flame height demonstrate a trend identical to that for the results obtained using the non-linear regression method (Fig. (a. Daun et al. (7 utilized the Cramér-von Mises (CVM goodnessof-fit parameter to quantify the accuracy of the recovered distribution, which was defined as the area contained between the recovered and exact particle size cumulative distribution functions. Compared to the distribution determined using the non-linear regression method, the CVM goodness-of-fit parameter of the corrected inversed results was evaluated as.74 at HAB = 5 mm, a value comparable to those calculated using most traditional recovery methods summarized by Daun et al. (7. The error database established based on various flame temperatures readily allows the results at different flame locations to be corrected by interpolation. This is quite different from the traditional methods, most of which need an LII signal database numerically calculated for each flame temperature and repeated comparisons. CONCLUSION An explicit solution was proposed for determining the lognormal distribution of inflame soot particle size based on a TiRe-LII signal processing method. Three time intervals are selected from the LII decay signal and each interval is applied to fit with a mono-exponential function. Monodisperse equivalent mean particle sizes and corresponding number density ratios for three size groups can be determined. Thereby, the count median diameter and geometric standard deviation σ g are solved in conjunction with the log-normal distribution function. Although each size group contains a distribution and the groups possibly overlap, inversed results from the simulated LII signals demonstrate a coincidence in trend with the input log-normal distributions for modeling. The influence of time interval length on the calculation was investigated. Opposite trends for inversed count median diameter and geometric standard deviation were found. There existed a critical time interval where the inversion was most insensitive to the variation of the interval length. The critical time interval for inversion of was almost identical to that of σ g, and based on the critical time intervals, errors of the inversed results strongly depended on σ g but weakly depended on, and declined with σ g. Quantitative evaluations demonstrated that slightly larger errors occur for inversions performed with aggregations, and remarkable larger ones occur under high flame temperature. Accordingly, an error database as a function of flame temperature but at a fixed aggregate size could be built and based on which the inversions can be easily corrected by interpolation. The proposed method was applied to the experimental TiRe-LII signals acquired at various locations in an atmospheric laminar diffusion flame. After being corrected, the inversed log-normal distributions exhibited a trend similar to that for results obtained using the established non-linear regression method. Modeling LII signals reproduced from the corrected inversed log-normal distributions also gave satisfactory agreement with the experimental data. The small deviation of the results potentially stemmed from the statistical noise contained in the recorded LII signals. It suggested that further improvement of the signal-to-noise ratio for the experimental signal is essential for obtaining an accurate result with the proposed method. Nevertheless, the evaluation showed that the accuracy of the results derived using the proposed method is comparable to that of those obtained using most conventional methods. As an explicit solution, just one error database built based on various flame temperatures is sufficient to deduce the particle size distributions at different flame locations, whereas traditional methods, such as full non-linear regression, require various databases to be established and repeated comparisons to the simulated LII signals, which may preclude online analysis. The present approach is more efficient in computation under different flame temperatures, making it more applicable for practical measurements. ACKNOWLEDGMENTS This work was support by the Natural Science Foundation of China (No. 5644, the public project of Ministry of Environmental Protection (498-4, the National Basic Research Program of China (5CB55, the project of Hangzhou Technology (6A6, the Program of Introducing Talents of Discipline to University (B86, and the Innovative Research Groups of the National Natural Science Foundation of China (565. APPENDIX A Derivation of Eqs. (8 and (9 This appendix presents the detailed derivation of Eqs. (8 and (9. The definitions of the parameters involved in the following equations are consistent with those used in the theory section.