Appl Phys B (2009) 96: 695–707 DOI 10.1007/s00340-009-3664-z

The interpretation of the LII signal in optically dense combusting sprays R. Ochoterena

Received: 12 January 2009 / Revised version: 8 June 2009 / Published online: 24 July 2009 © Springer-Verlag 2009

Abstract A numerical investigation was made of the generation and behaviour of the LII signal in optically dense combusting sprays at conditions similar to those in the combustion chamber of compression ignition engines and gas turbines. The influence of particle size, particle morphology and size distribution on the behaviour of the LII signal, and the scattering and absorption of light, and the consequences that different calibration procedures have on the accuracy of the results were studied. Results show that, as the particle size or aggregation increases, light extinction is not caused only by absorption but also by scattering, which contributes more than 10% to the total extinction of light. Particle shape effects are important, irrespective of particle size. The form, soot concentration gradients and optical thickness of the flame cause an uneven laser fluence across the measuring volume that affects the generation of the LII signal. In addition, the quotient between the transmitted and incoming laser pulses across the flame borders can be as small as a percentage of unity. The interpretation of the induced signal is further challenged by the loss of signal between the measuring volume and the detection arrangement, thus causing the detection of spectrally distorted and weaker signals with an erroneous profile of the local amount of carbonaceous particles. An appropriate calibration procedure must be followed to obtain results that are quantitatively representative. External calibration was found to be inappropriate for these systems R. Ochoterena () Department of Applied Mechanics, Chalmers University of Technology, 412 96 Göteborg, Sweden e-mail: [email protected] Fax: +46-0-31180976

since it can lead one to underestimate the local volume fraction for almost two orders of magnitude. Implementing an in situ calibration along a line can lead to underestimate or overestimate the local mean volume fraction by a factor of two. However, the use of an in situ calibration procedure using a laser sheet that propagates through the complete measuring volume can reduce the error in estimating the mean soot volume fraction to a 30%. The latter was found to be the most adequate among the studied calibration routines. PACS 87.64.Cc · 87.64.K- · 07.07.Df · 47.70.Pq

1 Introduction Laser Induced Incandescence (LII) has been used extensively as a diagnostic tool to determine the amount of carbonaceous particles in combusting sprays inside compression ignition engines and gas turbines [1]. The customary pressure and temperature inside the combustion chambers of these devices yield highly radiative flames [2] but with elevated local concentrations of carbonaceous particles that give aerosols with high optical density. Nonetheless, the optical thickness of these flames can negatively affect the quality and veracity of the results of LII measurements if certain aspects, such as the non-linearity of the LII signal or the calibration procedure used to adjust the detected signal against the local concentration of particles, are neglected or improperly assessed. The inherent optical thickness of these flames results in a high and progressive absorption of the excitation laser light along its path and a significant loss of signal between the measuring volume and the detection arrangement [3]. The former causes an uneven laser fluence across the flame, while the latter yields a signal whose detected amplitude is

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lower than the induced signal. The complex non-linearity of the LII signal against the laser fluence has been given special attention by several groups in extensive studies [4, 5]. Unfortunately, this complex behaviour is further accentuated by the uneven laser fluence across the flame and by detection difficulties attributable to signal trapping [6–8]. A calibration procedure is required to correlate the signal response to the local soot concentration, thus being able to obtain quantitative information. This calibration procedure can be done either externally or in situ. In external calibration procedures the response of the system is compared, for instance, against a controllable radiation source such as a carbon lamp [9] or an LII measurement done in a well-characterised burner [10, 11]. It is assumed thereafter that the calibration constant calculated from the external calibration procedure is universally valid. Even if this latter assumption appears to be acceptable, it may be adequate only if signal trapping is negligible and both systems, the reference and the studied systems, operate at the same ambient pressure and have similar flame temperature and gas composition. In in situ procedures, the average volume fraction is measured with the aid of a light extinction measurement, either along a line using a continuous light source and a pair of photo detectors [12], or employing a laser sheet across a plane and imaging the sampled incoming and transmitted laser beams onto a detector [13–15]. Although the former option is preferred by some workers because of its simplicity, it can lead to erroneous conclusions in the interpretation of the LII signal if the fluence across the beam is not homogeneous or if variations in the soot concentration across the measuring volume are large. Nevertheless, this error can be reduced if the planes for the light extinction and LII measurements are coincident, since the signal response against the local soot volume fraction is obtained across the entire measuring volume. The extensive use that LII has had in recent years as an almost standard technique for measuring carbonaceous particles in aerosols from different combustion appliances raises the necessity of highlighting certain procedures that are commonly followed and can diminish the quality of the measurements. Of special interest in combusting spray studies are: the effect on the generation and interpretation of the LII signal caused by an elevated optical density in high pressure environments, and the consequences that different calibration procedures have on the accuracy of the results. This paper presents a theoretical study of the behaviour of the LII signal in optically dense combusting sprays at conditions similar to those prevailing inside the combustion chamber of compression ignition engines and gas turbines. For this purpose, the effects on the detection of the LII signal caused by particle concentration and particle morphology, together with flame geometry and the wavelengths chosen for excitation and detection, were studied.

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2 Methodology The effect of elevated optical density in the interpretation of the LII signal in combusting sprays was numerically investigated in two fields, first by studying the behaviour of the LII signal and the transmission of light through a virtual combusting spray and, second, by analysing the effects of using different calibration procedures on the accuracy of the results. The transmission of light through the flame was investigated by examining both the transmission of the laser pulse along the flame and the propagation of the induced signal from the measuring volume across the detection path. The influence on the scattering and absorption of light caused by particle size, particle morphology and particle size distribution was also studied. Multiple scattering effects were not included in this study. The behaviour of the LII signal in the combusting spray was investigated by studying the generation and transmission of the LII signal, the importance of the wavelength interval chosen for detection and the effect in the detected signal of variations in the optical path. Three calibration procedures used by different workers for LII measurements in these types of flames were analysed and compared. The group of procedures studied includes an external calibration procedure and two in situ procedures, where the average volume fraction was determined with the aid of an extinction measurement. 2.1 Model flame The behaviour of the LII signal and the transmission of light in a particle-laden combusting spray were numerically studied simulating a compression ignited, optically dense spray 1 ms after the start of combustion (ASOC) as a model flame. The virtual flame conserves a radial symmetry along its longitudinal axis, and has values of soot volume fraction, particle size and particle concentration found in the literature [12, 16] which intend to mimic a real flame. Variations among contiguous cells were intentionally included in the calculations to simulate inhomogeneities across the combusting spray. For this model flame it is assumed that diesel flames are governed by diffusion, thus containing either vaporised or pyrolysed fuel inside the reaction borders and small amounts of oxygen and nitrogen. It is also considered that the fuel molecules do not absorb and do not emit visible light, and that molecular diffusion governs the mixing of the fuel cloud [17]. The simulated spray basically consists of three joint geometric structures: a partially hollow column with a variable width, a cylinder and a half sphere. As a consequence, at the top of the spray, the volume fraction has a dip at its centre that fades gradually along the axis while the mean volume fraction increases. A cut at the middle coronal plane is shown in Fig. 1.

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Fig. 1 Mean particle size, Dp , particle concentration, N , and volume fraction, fv , in the middle sagittal cut of the simulated flame

2.2 Modelling of the LII signal in the flame The LII signal was simulated using a model that was developed for LII studies in combusting sprays at high gas pressures and temperatures [18]. For convenience to the reader, the main features and assumptions of the model are presented in the appendix. The simulations were made using the same experimental configuration described in that work, with a gas pressure of Pgas = 50 bar and an average flame temperature of Tf = 2500 K. The excitation laser light has a wavelength of λ = 1064 nm and a total fluence of 0.55 J/cm2 before impinging the flame. It is further assumed that the energy across the laser sheet is homogeneously distributed and that its cross section is constant across the flame. The laser light propagates through the flame in the left to right direction. The signal was virtually detected at three wavelength intervals using ideal interference filters centred at λ = 450 nm, λ = 550 nm and λ = 650 nm with an FWHM of 10 nm. The gate of the detectors was assumed to have an ideal square profile with a width of 50 ns. The spectral response of the detectors was assumed to be constant across the entire detection range. 2.3 Particle morphology and size distribution effects 2.3.1 Particle shape Diesel spray combustion occurs predominantly under fuel rich conditions where equivalence ratios near the nozzle around 10 have been measured during the injection period.

Fuel concentration within the spray decreases with increasing the radial and axial position at any given time but maintaining an equivalence ratio above 2 [19]. Particle size distributions measured from laboratory flames burning also under fuel rich conditions, with equivalence ratios higher than or equal to 2, have shown to be bimodal having one mode centred at large sizes (10–20 nm), which represents soot particles, and the other mode centred at smaller sizes, which has been attributed to denote soot precursors [20]. Even though the contribution of soot precursors to the generation of the LII signal is much smaller than of larger particles [20], soot precursors contribute to the scattering and absorption of light. It has been suggested that, contrary to single and non-aggregated particles that can be considered spherical, the form of soot precursors can deviate from this spherical shape [21]. Calculations of the scattering and absorption cross sections for non-spherical particles that are small compared to the wavelength, using as an example the optical properties of carbon [22], show that a needle-shaped particle has a scattering or absorption cross section almost twofold larger than that of an equivalent sphere, while a disk-shaped particle has cross sections nearly three times larger than a sphere with the same volume. These shapes may be extreme and uncommon cases for the study of soot particles in flames; nonetheless, these observations show that particles substantially smaller than the wavelength, such as soot precursors, have optical properties that are morphology dependent. On the other hand, even if the optical properties of soot precursors are not well established, absorption of light in the near UV region has been

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attributed to these particles [21]. The absorption of light in this range can be relevant when detecting LII signals at short wavelengths. 2.3.2 Size distribution and aggregation The effect on the absorption and scattering of light, and behaviour of the LII signal, of particle aggregation and size distribution was investigated using a distribution composed by a number of independent spherical particles and a cluster of aggregated particles. The cluster is constituted by a lognormal number distribution of equally in size particles with a constant diameter Dpa = 30 nm while the independent particles have a normal size distribution with mean diameter sp Dμ . The total mass of the distribution, Mtot , for the studied cases, was kept constant and equal to hundred times the mass of the cluster’s primary particles. Five different mean diameters of independent partisp cles (Dμ ) were studied (30, 24, 18, 14 and 11 nm), being the volume of each of the independent particles a fraction of the volume of the cluster’s primary particles (1, 1/2, 1/5, 1/10, 1/20). The relation between the amount of independent particles and size of the cluster, ω, was varied such as the total mass of the distribution, Mtot , was divided between the mass of the cluster of agglomerated particles, Magg , and the mass of the distribution of independent single particles, Msp , as shown in (1) and (2).

The absorption and scattering of light by the aggregate was treated according to the model suggested by Dobbins et al. [25]. Each of the aggregates is conformed by a lognormal number distribution of monodisperse spherical particles of size Dpa = 30 nm. The scattering and absorption cross sections are calculated according to (7) and (8), respectively, where E(m) and F (m) are functions of the complex refractive index m, Rg is the radius of gyration, xp = πDpa /λ and k = 2π/λ. The fractal dimension and the fractal factor are Df = 1.9 and Kf = 5.8 in the order already mentioned. The mean number of particles per cluster is N¯ agg .  σs =

p(Nagg ) · 

σa =

p(Nagg ) ·

2 x 6 F (m) 8πNagg   p g k 2 R¯ g2 · dNagg 2 3k

4πNagg xp3 E(m)

 ¯ 1/Df Nagg Kf     4 2 ¯ 2 −Df /2 k Rg g k 2 R¯ g2 = 1 + 3Df   2 m −1 E(m) = Im 2 m −2 and

Msp = ω · Mtot

(2)

2 m − 1 2 F (m) = 2 m + 2

(3) (4)

For the independent single particles the scattering, σs , and absorption, σa , cross sections are calculated by solving the Mie equations using a complex refractive index, m = 1.57 + 0.56i [23] and computed according to (5) and (6), respectively. These calculations use a standard deviation σ = 3 nm [24] and a wavelength λ = 500 nm. The diameter of the independent particles is D sp .      σs = p D sp · σs D sp · dD sp (5)  σa =

    p D sp · σa D sp · dD sp

(6)

(8)

Rg = Dpa

(1)

    π a −1 ¯ D Nagg = Magg · 6 p     π sp −1 Nsp = Msp · Dμ 6

· dNagg

where

Magg = (1 − ω) · Mtot

The mean number of particles aggregated in the cluster, N¯ agg , and the number of independent single particles, Nsp , are calculated according (3) and (4).

k2

(7)

(9) (10) (11)

(12)

2.4 Transmission of light through the combusting spray The ratio between the incoming Ii and transmitted It intensities of light through the flame along a differentially thin section, dϕ, which has a local particle concentration of Nl and an extinction optical cross section of σx was calculated with the Beer–Lambert law for inhomogeneous media, which is shown in (13).    +r Ii = Nl (ϕ) · σx (ϕ) · dϕ ln It −r

(13)

The extinction cross section per particle, σx , is the sum of the absorption cross section, σa , and the scattering cross section, σs , as shown in (14), ζ is the quotient between the scattering and absorption cross sections (σs /σa ). σx = σa [1 + ζ ]

(14)

The interpretation of the LII signal in optically dense combusting sprays

3 Results and discussion 3.1 Behaviour of the LII model at engine-like conditions The normalised LII signal, spectrally and time integrated as a function of the laser fluence, is plotted in Fig. 2 for various gas pressures and in Fig. 3 for diverse particle sizes. The former figure was prepared for a particle diameter of Dn = 16 nm and the latter for a gas pressure of Pgas = 50 bar. For both cases a detection wavelength of λ = 650 with a FWHM of 10 nm and a detection gate of 50 ns were assumed. Figures 2 and 3 show that, as expected, the amplitude of the integrated LII signal increases as the laser fluence grows until it reaches a maximum, whereafter the integrated signal decreases monotonically as the laser fluence is increased. It

Fig. 2 Integrated LII signal as a function of the relative laser fluence and gas pressure (Dn = 16 nm)

Fig. 3 Integrated LII signal as a function of the relative laser fluence and particle size (Pgas = 50 bar)

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can also be noted that the integrated signal is larger for lower gas pressures and for larger particles. The highly non-linear behaviour of the LII signal to the laser fluence can be observed in both figures. 3.2 Particle morphology and size distribution effects The absorption and scattering cross sections, the quotient between the scattering and absorption cross sections, ζ = σs /σa , together with the integrated LII signal are plotted against ω in Figs. 4, 5, 6 and 7, respectively. As mentioned above, ω is the relation between the total mass of the independent particles and total mass of the cluster. For ω = 0 the total mass of the distribution is represented solely by aggregated particles, and for ω = 1 by independent, nonaggregated particles. The total mass among distributions re-

Fig. 4 Absorption cross section for different size distributions

Fig. 5 Scattering cross section for different size distributions

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Fig. 6 Scattering to absorption ratio for different size distributions

Fig. 7 Integrated LII signal for different size distributions

mains constant regardless of the mean size of the indepensp dent particles (Dμ ) or the value of ω. Figure 4 shows that the absorption cross section of the distributions remains almost constant regardless of the value of ω, the size of the independent particles or the size of the cluster. This is explained by the fact that the absorption cross section is mainly a function of the total volume of the particles which, for these distributions, was kept constant for all cases. The slight increase in the absorption cross section, noticed as ω increases, occurs because the absorption cross section of the independent particles is calculated by solving the Mie equations, while the absorption cross section of the aggregate is calculated using an approximation for particles that are small relative to the wavelength in which edge effects and the interaction between neighbour particles are not considered. The use of this approximation, in the case of

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single particles of diameter equal to 30 nm, results in underestimating the absorption cross section by almost 3%. Figure 5 shows that an increase in the size of the cluster, which consequently reduces the amount of independent particles, yields an increase in the scattering cross section of the distribution. For a given value of ω, a reduction in the size of the independent particles and increasing in quantity, leads to smaller scattering cross sections. These differences are more notorious for variations among larger particles than for variations among smaller ones since the scattering cross section is a function of the square of the volume of the particle. Even if the scattering cross section is usually very small compared to the absorption cross section for particles that are substantially smaller than the wavelength, this may not be true for small particles with capricious forms, clusters of aggregated particles or as the size of the particles becomes comparable to the wavelength. This can be noticed in Fig. 6 where, as presumed, extinction is dominated by absorption rather than scattering for small and single particles. However, as the particle size or aggregation increases, scattering becomes comparable to absorption, indicating that extinction is caused by both scattering and absorption in the case of larger aggregates. Figure 7 plots the integrated LII signal for different size distributions. The signal is detected at λ = 650 nm with an exposure time of 50 ns. In this plot it is noticed that, for independent and non-aggregated particles (ω = 1), the integrated LII signal is reduced as the mean particle size of the distribution decreases even if the total mass of the distribution remains unaltered. A pair of distributions with the same volume fractions and different mean particle size, one composed of a set of larger particles than the other, will result in integrated LII signals of different amplitude. In addition, an escalate in aggregation (ω → 0) results in integrated LII signals of higher amplitude due to a reduction in the surface to volume ratio of the set of particles. Although there are clear differences among the studied distributions, it is difficult to discern which of them best predicts the behaviour of the LII signal as well as the scattering and absorption of light from carbonaceous particles inside combusting sprays shortly after the start of combustion, particularly if the morphology and internal structure of the carbonaceous particles inside these flames remains unknown. Notwithstanding the existence of numerous studies of the morphology of carbonaceous particles indicating the feasibility of treating them as fractal aggregates [26–32], these studies were carried out either in particles sampled from flames with long residence times or particles sampled from the exhaust gases of diesel engines. Unfortunately, these structures do not necessarily resemble the structure of carbonaceous particles encountered in combusting diesel sprays shortly after the start of combustion (∼1 ms ASOC), mostly because of the very short formation and residence times in these flames.

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For particles sampled from the exhaust gases of diesel engines, it is particularly complicated to discern among the processes that occurred during the short period in which the particles were inside the flame (<2 ms) and the processes that happened during the relatively long time inside the cylinder before the exhaust valve opens (∼50 ms) or in the exhaust pipe before sampling. On the other hand, even if it can be suspected that aggregates together with a number of single particles exist simultaneously in the flame, yet extensive research is required before a solid conclusion can be made with respect to the morphology of carbonaceous particles inside combusting diesel sprays. Furthermore, it has been shown that particles with short residence times sampled in laboratory flames are in early stages of development [33, 34] in which coagulation rather than aggregation is dominant [35]; in addition, since particles inside combusting diesel sprays are predominately small (D sp < 30 nm) [16], this study considers particles as spherical and non-aggregated (ω = 1) for the investigation of the behaviour of the LII signal across the modelled combusting diesel spray. 3.3 Transmission of light and behaviour of the LII signal in the combusting spray Figure 8 displays: (a) the relative laser fluence across the flame and (b) the induced signal and the detected LII signal at the spectral intervals centred at (c) λ = 450 nm and (d) λ = 650 nm. The induced signal is also plotted in Fig. 9 together with the local soot volume fraction and the detected LII signal captured at λ = 450 nm, λ = 550 nm and λ = 650 nm at four vertical positions. Inspecting Fig. 8(a) reveals that the excitation laser pulse is strongly absorbed as it travels through the flame. Furthermore, the relative gradient of the laser fluence between the entering and the exiting sides of the flame is almost in the order of two decades where the flame is optically thickest. On the other hand, close to the top of the flame, where the total optical path is shorter and the concentration of particles is moderate, the relative laser fluence has a decreasing, yet moderate, gradient between the flame borders. Figures 8(b) and 9 give the impression that the induced (LII) signal is indeed less affected by the optical thickness of the flame than the laser fluence itself. This is caused by the non-linear behaviour of the LII signal as a function of the laser fluence, as seen in Figs. 2 and 3, where, departing from the standard laser fluence before impinging on the flame, the amplitude of the LII signal increases as the fluence is reduced. This behaviour continues until a maximum is reached at relatively low fluences (∼0.2 J/cm2 ), depending on particle size and gas pressure. Nonetheless, if the laser fluence is further reduced, the amplitude of the LII signal falls abruptly.

Fig. 8 (a) Relative laser fluence across the flame (Plaser ), (b) Induced and calibrated LII signal and (c) LII signal detected at λ = 450 nm and (d) LII signal detected at λ = 650 nm. The laser light enters into the flame from the left hand side

The overall consequences of the uneven laser fluence across the measuring volume, the optical thickness of the flame and the non-linear behaviour of the LII signal are observed in the detected signals in Figs. 8(c), (d) and 9. As expected, the detected signals are much weaker than the originally induced signal, especially where the optical path is longest and when the signal is detected at a shorter wavelength. At first glance, it might be surprising to note that the detected signal is slightly stronger at the location where the laser light exits the flame than at the spray centre. Notwithstanding, it must be taken into account that the signal is trapped between the measuring volume and the detectors, and the amplitude of the detected signal is therefore lower in places where the total signal extinction is stronger.

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Fig. 9 Original, (I0 ), induced, (Ilii ) and detected signals, (Id ), using three wavelengths at four vertical positions. λ1 = 450 nm, λ2 = 550 nm and λ3 = 650 nm

Figure 9(a) shows that, at the top of the flame where the total optical thickness is low, the detected signals have a much lower amplitude than the induced signal, although the shape is similar. Further, the amplitude of the induced and detected signals is faintly higher at the outgoing than at the incoming border due to the non-linear behaviour of the LII signal. However, as the optical density or the total optical path increases, as occurs downstream of the flame, the induced and detected signals are highly asymmetrical and do not resemble the local concentration of particles, as can be observed in Figs. 9(b)–(d). The plots in Fig. 9 show that the fluence of the laser light, used to generate the LII signal, is inhomogeneous across the measuring volume and the quotient between the transmitted

and incoming laser pulses across the flame borders can be much smaller than unity. Although these factors represent serious difficulties in estimating the local volume fraction, yet it is possible to mitigate the negative effects of the uneven laser fluence across the measuring volume [3] by estimating the local fluence and compensating the loss of signal between the measuring volume and the detection arrangement [18]. Detecting of the LII signal at two wavelength intervals, or resolved in time at two wavelength intervals, in order to estimate the particle cooling after being heated by the laser light have gained popularity as procedures for determining the size of carbonaceous particles in aerosols [36–38]. However, results of these techniques can be seriously affected by

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Fig. 10 Quotient between the detected signals at different wavelengths for four vertical positions. λ1 = 450 nm, λ2 = 550 nm and λ3 = 650 nm

the optical density of the flame [6, 7], since the transmission of shorter wavelengths is lower than the transmission of longer wavelengths [39]. Figure 10 shows plots of the quotient among detected signals captured at different wavelengths for different flame heights. These plots show that the quotient between a given pair of signals detected at different wavelengths depends strongly on the optical thickness of the flame. Figure 11 shows the averaged LII signal detected at λ = 650 nm as a function of the optical path length, ϕd , and the mean soot volume fraction of the flame, fv . The figure was prepared for particles with a mean size of Dn = 16 nm. It is observed that, as the optical path length increases, the amplitude of the detected signal decreases. Nevertheless, for

relatively high concentrations of particles, an increase in the average concentration of particles does not necessarily lead to stronger detected signals but, on the contrary, to the detection of weaker signals due to signal attenuation. For instance, for a given constant optical path length, ϕd = 8 mm, the amplitude of the detected signal increases together with the mean volume fraction until an inflexion point is reached at a mean volume fraction, fv  3 × 10−5 . Thereafter, an increment in the mean volume fraction yields a reduction in the amplitude of the detected signal. Furthermore, there are silhouettes where the amplitude of the detected signal remains constant for certain combinations of the optical path and the mean volume fraction.

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Fig. 11 Detected LII signal as function of the optical path length, ϕd , and mean soot volume fraction of the aerosol, fv . The figure was prepared for a mean particle size of Dn = 16 nm

In the given scenario that aggregation would have occurred resulting in an aerosol with the same volume fraction but containing clusters of N¯ agg = 100 aggregated particles of diameter Dpa = 30 nm, as with ω = 0 for the above studied distributions, the total absorption would have remained unchanged. However, the extinction of light for both the laser light and the induced signal would have been increased in nearly 10% due to the light scattering. The overall consequences would be LII signals with lower amplitude and less qualitative resemblance to the local soot concentration. 3.4 Influence of the calibration procedure on the accuracy of the results The influence of the calibration procedure on the accuracy of the results of LII measurements in optically dense combusting sprays was studied for one external and two in situ procedures. 3.4.1 External calibration If the calibration procedure is carried out externally using a well-characterised burner at ambient conditions to adjust the signal amplitude to the local soot concentration of the combusting spray in a high pressure environment, the calibration procedure may be affected by, among others, three factors: ambient gas pressure, mean particle size and optical density. The difference in the ambient pressure at which the reference burner and the combusting spray operate, the latter enclosed in a chamber in which the gas pressure is elevated, will result in overestimating the local volume fraction in the combusting spray. This is caused by the reduction of the integrated LII signal as the ambient pressure is raised, as shown in Fig. 2, where, for instance, a sevenfold rise in the ambient pressure leads to a reduction of approximately 20%

in the amplitude of the induced signal for the same volume fraction at moderate fluences. Differences in the particle size distributions between the reference flame and the combusting spray will result in signals of different amplitude even if the local soot volume fractions of both systems are equivalent. As shown in Fig. 7, a two fold increase in the mean particle size, but with equal volume fractions, yields an increase of nearly 30% in the amplitude of the integrated induced signal. Furthermore, if the optical density of the studied flame is higher than that of the reference flame, signal trapping and laser attenuation can lead to a severe underestimation of the volume fraction in the combusting spray. For the combusting spray used in this study, neglecting the optical density of the system would lead to underestimating the mean volume fraction for almost two orders of magnitude in the centre of the spray, as can be seen in Fig. 9(c). Although the differences in ambient pressure and mean particle size between the reference burner and the combusting spray lead to important errors in the correct assessment of the local volume fraction, the errors caused by these factors are small when compared to the consequences of not considering the high optical density of the combusting spray. 3.4.2 In situ calibration An in situ calibration procedure in which light extinction is measured along a line using a continuous light source and a pair of photo detectors has the advantage that the absolute mean volume fraction is known in the path of the measured line. This calibration procedure is advantageous, since it is relatively simple and the average volume fraction can be known regardless of signal trapping and with high repetition rates. Unfortunately, if the optical density or mean particle size varies substantially across the measuring plane, the use

The interpretation of the LII signal in optically dense combusting sprays

of a single calibration constant across the flame might lead to errors in the interpretation of the induced signal. If, for the modelled flame presented above, a calibration constant is calculated 20 mm downstream of the top of the flame and the same constant is then used 10 mm above the calibration location, its local mean volume fraction would be overestimated by a factor of two. An alternative to the former in situ calibration procedure employs a laser sheet that propagates through the measuring volume in which the incoming and transmitted laser planes are imaged onto a detector. Even if this procedure is slightly more complicated than the former one, since it requires at least one two-dimensional detector, the error that the former calibration procedure incurs is reduced. The error is reduced, among horizontal lines at least, because the planes for the extinction and LII measurements are coincident, and the signal response against the local soot volume fraction is therefore known across the entire measuring volume. Due to theoretical and experimental uncertainties an error of 30% is expected in the soot volume fraction measurements even if this calibration procedure is implemented [15]. Regardless of the usefulness of the light extinction measurements for calibration and compensation purposes, these measurements are seldom performed because, among other obstacles, of the limited optical access in the combustion chambers of turbomachinery and diesel engines. Nevertheless, making LII measurements in these types of devices without following an in situ calibration procedure may lead to results with little or no resemblance to the local volume fraction of carbonaceous particles in the flame.

4 Conclusions The effect of elevated optical density in the generation and interpretation of the LII signal in combusting sprays at elevated gas pressures was numerically investigated in a model flame. The influence of particle size, morphology and size distribution on the behaviour of the LII signal, scattering and absorption of light was studied. Results show that, as particle size or aggregation increase, the amplitude of the LII and scattering signals increase and light extinction is caused by absorption and scattering together. For severe aggregation cases, scattering is responsible for more than 10% of the total extinction. Furthermore, even if the shape of small particles is often considered unimportant, there are shape effects on the scattering and absorption cross sections that may become relevant in flames burning under very rich conditions where bimodal distributions of soot particles and soot precursors have been detected. The fluence of the laser light, used to generate the LII signal was found to be inordinately inhomogeneous across

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the measuring volume in optically dense systems. In addition, the quotient between the transmitted and incoming laser pulses across the flame borders can be as small as a percentage of unity. The loss of signal is shown to be important for the detection of the LII signal at two wavelength intervals where the quotient between signals can be more a function of the optical thickness than of the characteristics of the LII signal. Three calibration procedures were discussed. External calibration, which can lead to underestimate the local volume fraction for almost two orders of magnitude, was found to be inappropriate for these systems. Doing an in situ calibration along a line can lead to underestimate or overestimate the local mean volume fraction by a factor of two. On the other hand, the use of an in situ calibration procedure using a laser sheet that propagates through the complete measuring volume can reduce the error in estimating the mean soot volume fraction to a 30%. Failing of doing a correct calibration procedure results in an underestimation of magnitude and an erroneous spatial distribution of the local amount of carbonaceous particles. Acknowledgements The author acknowledges the financial support of CERC (Combustion Engine Research Centre) at Chalmers and its member companies. Discussions with A. Magnusson, M. Andersson and S. Andersson are appreciated.

Appendix A: LII heat transfer model The energy transfer in the LII process, where soot particles increase their internal energy when heated by the laser pulse and transfer this energy to the environment, is described in (15). ˙ abs − Q ˙ rad − Q ˙ vap − Q ˙ cond = dUint Q dt

(15)

˙ abs , by a single soot particle that The energy absorbed, Q is small compared to the wavelength, using a constant complex refractive index m, can be calculated according to (16), where the absorption is proportional to the intensity of the incident radiation, Ii , the cube of the particle diameter, Dp , and inversely proportional to λ, the wavelength of the incident light [22, 40]. ˙ abs = Q

π 2 Dp3 λ



 m2 − 1 · Im 2 · Ii (t) m +2

(16)

The energy radiated, Q˙ rad , by a single soot particle which is small compared to the wavelength can be calculated according to (17), where h, c, kb and Tp are Planck’s constant, speed of light in vacuum, Boltzman’s constant and the temperature of the particle, respectively [41–43].

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˙ radλ −λ Q i ∞  2   λ∞ m − 1 2πhc2 2 3 = π Dp · Im 2 · · m +2 λ6 λi

shown in (23), being mp and mg the masses of the soot particle and gas molecule, respectively.

1 e

hc kb Tp λ

−1

· dλ (17)

The energy loss by evaporation, Q˙ vap , is calculated according (18) [44]. Ru is the universal gas constant. Wv (Tp ) is the molecular weight of soot vapour in [kg/mole], Hv (Tp ) is the enthalpy of vaporisation in [J/mole·K], and Pv (Tp ) is the vapour pressure in [Pa]. These are functions of temperature and are approximated by polynomial functions [45].

Hv (Tp ) ˙ vap = β · · πDp2 · Pv (Tp ) · Q Wv (Tp )

Wv (Tp ) 2π · Ru · Tp

(18)

˙ cond , is estimated The energy transferred by conduction, Q as in (19). ˙ cond = ω · βE · EECT Q

(19)

Based on the ergodic collision theory (ECT) and by assuming that soot particles are considerably larger than the colliding molecules but comparable to the mean free path1 Λ, the average energy transfer per collision between a soot particle and a surrounding gas molecule, E, is determined by (20) and (21), where Tg is the mean temperature of the surrounding gas [46–48]. EECT =

3 · kb (Tp − Tg ) 2

(20)

The collision efficiency factor, βE , which is a function of the internal energy of the carbonaceous ensemble and the internal energy of the bathing gas, relates the average energy transferred to the medium per collision, E, and the ECT value, EECT . The collision efficiency factor is determined as in (21) [49, 50]. βE = E/EECT

(21)

The collision frequency, ω, estimated by a hard sphere approximation, is calculated according to (22). 1    8kb Tg 2 1 2 · π · (Dp + D¯ g ) · ω = γg · 4 πμm

(22)

where D¯ g is the mean diameter of the colliding gas molecules, γg is the number density of the medium molecules and μm is the reduced mass of the relative translation, as 1 Λ = [ √kb Tg ]. π 2Pg Dg2

μm =

mp · mg (mp + mg )

(23)

The energy change per particle through the process, dUint /dt, can be expanded as in (24). dTp dUint = mp · Cs (Tp ) · dt dt

(24)

The heat capacity of crystalline graphite, Cs (Tp ) in [kJ/kg·K], which is the sum of all possible forms of internal energy storage, is approximated by a polynomial function Cs (Tp ) = −9.68 × 10−15 · T 4 + 1.35 × 10−10 · T 3 − 6.98 × 10−7 · T 2 + 1.72 × 10−3 · T + 1.456 [51]. If particle aggregation is assumed, the collision diameter of the aggregate is considered to be proportional to the radius of gyration of the aggregate [52] and is used in (22) instead the physical diameter of the independent and nonaggregated particle, Dp . The absorption and radiation of the aggregate are calculated as the sum of the absorption or radiation of the single spherical particles in the aggregate. The total surface of the aggregate, important for calculating the energy loss by evaporation, Q˙ vap , is assumed to be the total surface of the particles constituting the aggregate due to of uncertainties regarding structural changes of the aggregate when being heated by the laser pulse [53].

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