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Influence of scattering and probe-volume heterogeneity on soot measurements using optical pyrometry
Combustion and Flame 143 (2005) 1–10 www.elsevier.com/locate/combustflame
Influence of scattering and probe-volume heterogeneity on soot measurements using optical pyrometry J.J. Murphy ∗ , C.R. Shaddix Combustion Research Facility, Sandia National Laboratories, P.O. Box 969, Livermore, CA 94550, USA Received 2 December 2003; received in revised form 15 September 2004; accepted 26 September 2004 Available online 22 October 2004
Abstract The equations describing radiative emission from soot are reexamined in light of recent measurements showing that scattering by aggregated soot particles is not negligible in the near-infrared region. When measuring soot temperature and concentration using two-color optical pyrometry, emission is usually best approximated by Iλ (s) = [1 − exp(−ka,λ s)]Ib,λ , using an absorption coefficient, ka,λ , determined while taking into account the effects of scattering. This equation gives good results in most practical measurement environments, where scattering-induced contributions to the measured light intensity are approximately balanced by scattering losses. It is also shown that probe-volume heterogeneity biases temperature measurements made with two-color pyrometry toward the higher temperatures in the probe volume. Soot concentration measurements are also biased, reflecting the concentration of hot soot in the probe volume, rather than the mean concentration of hot and cold soot. Methods to minimize these effects are suggested. 2005 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Two-color pyrometry; Soot; Optical properties
1. Introduction Optical pyrometry has been used extensively to measure soot temperature in environments ranging from laboratory flames and shock tubes to internal combustion engines and large pool fires. Most often, two-color pyrometry is used, in order to simplify the calibration procedure and to make the measurement insensitive to variations in the optical throughput of the light collection system. * Corresponding author. Current address: The Aerospace
Corporation, P.O. Box 92957, Los Angeles, CA 90009, USA. E-mail addresses: [email protected] (J.J. Murphy), [email protected] (C.R. Shaddix).
In addition to temperature, soot concentration can also be measured using two-color pyrometry. This measurement, however, requires absolute calibration of the detection system, consistent optical throughput, and knowledge of the emissive properties of the soot that is being probed. As a consequence, simultaneous measurements of soot concentration and temperature have often relied upon the combination of two-color pyrometry and optical extinction (usually using a visible-wavelength laser source). Soot concentration can be calculated from optical extinction without difficulty, although knowledge of the soot optical properties is still required. The combined measurement is usually called the “three-line,” or “threewavelength,” soot diagnostic, and is easy to implement using two fiber-optic probes that face each other
0010-2180/$ – see front matter 2005 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustflame.2004.09.003
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Nomenclature d dp fv Iλ Iλ,0 Ib,λ ka,λ ke,λ ks,λ Ka
sight-tube diameter (see Fig. 1) [mm] soot-particle diameter [nm] soot volume fraction optical intensity [W m−2 nm−1 ] incident light intensity [W m−2 nm−1 ] blackbody emission intensity [W m−2 nm−1 ] absorption coefficient [mm−1 ] extinction coefficient [mm−1 ] scattering coefficient [mm−1 ] dimensionless absorption coefficient = ka,λ λ/fv
across the probe volume. Sivathanu, Gore, and coworkers have used this technique extensively in buoyant, turbulent diffusion flames and pool fires [1–6]. The effect of optical property uncertainty on the accuracy of the derived temperature and volume fraction from two-color pyrometry and laser extinction measurements has long been recognized [7–9]. Furthermore, when interpreting laser extinction signals, it is necessary to apply the correct spectral extinction coefficient so that light scattering as well as absorption is taken into account [10–12]. The need for accurate values of soot extinction coefficients has motivated an extensive series of measurements by Choi, Mulholland, and co-workers [13–16], and by Faeth and co-workers [12,17–20]. In contrast, soot pyrometry signals have always been interpreted with the assumption that the soot particles are in the smallparticle, or Rayleigh, limit of Mie theory, such that light scattering is negligible at the detected wavelengths. With this assumption, the extinction and absorption coefficients are equal, and simple relations are readily derived to express the optical emission intensity and to deduce the temperature from two-color pyrometry. The justification for applying the Rayleigh limit assumption in the interpretation of soot pyrometry signals has been loosely based on the small size of soot primary particles (dp ≈ 10–50 nm) in comparison to the long wavelengths of light (> 700 nm) generally used for pyrometry measurements. However, for aggregated soot particles such as those that exist in most regions of sooty flames (particularly in pool fires [21], older diesel engines, and other strongly sooting environments), the contribution of soot scattering signals is largely dependent on the aggregate size distribution. The importance of soot aggregate scattering has been recognized at visible wavelengths, where scattering albedos (ratios of scattering to total extinction) of 20–30% have often been measured for
Ke p s sc sh x γ λ
dimensionless extinction coefficient = ke,λ λ/fv phase function probe-volume length [mm] path length through “cold” soot [mm] path length through “hot” soot [mm] dimensionless optical size parameter = π dp /λ Henyey–Greenstein phase function parameter (see Eq. (8)) wavelength [nm]
postflame soot [12,18]. On the other hand, the contribution of scattering at longer wavelengths has historically been discounted, on the basis of the strong dependence of Rayleigh light-scattering intensity on the optical size parameter x = π dp /λ, where λ is the wavelength of the light. However, recent measurements [20,22] indicate that soot scattering continues to be significant at near-infrared wavelengths for aggregated soot, because of increases in the soot index of refraction at these wavelengths. For example, Zhu et al. [22] measured soot-scattering albedos that ranged from 20 to 24% at 856 nm for soot emitted from laminar acetylene– and ethylene–air flames. In a study of soot emitted from a number of buoyant turbulent diffusion flames, Krishnan et al. [20] derived scattering albedos that ranged from 0.15 to 0.35 over 800–1000 nm for six different fuels. In light of these developments, it is appropriate to revisit the derivation of the equations for optical soot pyrometry without resorting to Rayleigh limit simplifications. We also quantitatively evaluate the effects of another important complication of soot pyrometry, the presence of probe-volume heterogeneity.
2. General relations A schematic of the idealized system to be analyzed is shown in Fig. 1. The emission measurement is made across a probe volume bounded by two cylindrical “sight tubes” of diameter d whose open ends face one another across the interrogated medium. Rather than having an actual emission detector placed within the optical probe, a lens-coupled fiber optic cable is typically used to displace the detector from the sampling environment, but this is irrelevant to the analysis performed here. This sampling geometry has been extensively used to perform two-color soot pyrometry in pool fires and turbulent flames. For emission mea-
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on the local soot temperature, which can in turn depend on z. The phase function p(θ , φ ) specifies how much light is scattered from the (θ , φ ) direction into the direction (θ = 0, φ = 0) along the measurement axis, and is normalized such that 1 4π
2ππ
p(θ , φ ) sin θ dθ dφ = 1.
(2)
0 0
Fig. 1. Schematic of the probe geometry.
surements through walls bounding a process (such as across an I.C. engine cylinder), the probe ends may be thought of as defining the locations of the walls. The measurement takes place in a cylindrical probe volume of length s aligned with the measurement axis. The system is symmetric about the measurement axis. The spatial variable z is used to represent distance along the measurement axis. φ indicates azimuthal angle and θ indicates polar angle measured from the z-axis. The use of θ , rather than φ, to represent polar angle is consistent with the notation used by Brewster [23]. Scattering effects on soot emission measurements are seen to be dual-natured: emitted light originally directed forward along the measurement axis that is scattered results in a loss of detected light intensity, and is herein termed “out-scattering”; light from the surroundings originally directed across the measurement axis that is scattered in the forward direction of the measurement axis results in a gain in detected light intensity, and is herein termed “inscattering.” The radiative transfer equation of an absorbing, emitting, and scattering medium can be found in a variety of textbooks. Here, we use the expression given in Chapter 7 of Brewster [23]: dIλ (z) dz = − ke,λ Iλ (z) + ka,λ Ib,λ extinction
emission
2ππ
+
ks,λ Iλ (z, θ , φ )p(θ , φ ) sin θ dθ dφ . 4π 0 0 in-scattering
(1) Iλ (z, θ, φ) represents the light intensity coming from the (θ, φ) direction at location z along the measurement axis. Iλ (z) is shorthand for Iλ (z, 0, 0). Ib,λ is Planck’s blackbody radiation function, and depends
The coefficients ka,λ and ks,λ are the dimensional absorption and scattering coefficients, respectively, and ke,λ = ka,λ + ks,λ is the dimensional extinction coefficient. These coefficients may vary spatially, and are thus functions of z. By Kirchoff’s Law, the emission coefficient is equal to the absorption coefficient. We note that for laser-based extinction measurements, where there is absorption and out-scattering but negligible in-scattering and no emission, Eq. (1) becomes dIλ (z) = −ke,λ Iλ (z). (3) dz This equation, with the incident light intensity given by Iλ,0 , integrates to Iλ (s) = Iλ,0 e−ke,λ s .
(4)
Accurate implementation of Eq. (4) for media with substantial forward scattering (such as soot) is complicated by the likelihood that some fraction of the forward-scattered light will reach the extinction detector. This effect was recently analyzed [24] and shown to depend on the optical depth of the probed medium, as well as the geometry of the detection system. In practice, it is important to approximately match the angle of acceptance of the measurement system with that used to generate the nondimensional extinction coefficient used to calibrate the laser extinction.
3. Out-scattering effects For emission measurements, such as those used in two-color pyrometry, there are two distinct complications that need to be considered when light scattering is nonnegligible compared to absorption. The first complication concerns out-scattering from the probe volume, which has the consequence that the extinction coefficient in Eq. (1) is not identical to the absorption coefficient. In this case, for negligible inscattering, Eq. (1) integrates to Iλ (s) =
ka,λ 1 − e−ke,λ s Ib,λ . ke,λ
(5)
The three-line diagnostic literature has used the classical derivation of Eq. (5), where all scattering is
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Fig. 2. Comparison of soot emission equations for the case of negligible in-scattering. The unit ppm corresponds to a soot volume fraction of 10−6 .
neglected, to arrive at Iλ (s) = 1 − e−ka,λ s Ib,λ .
(6)
More recently, extensive measurements have been performed of the dimensionless soot extinction coefficient, Ke ≡ ke,λ λ/fv , in postflame soot. It was suggested that the use of these values would improve the comparison between soot volume fraction measurements based on extinction and soot volume fraction measurements based on emission (presumably by using the measured Ke for interpreting both measurements). In this case, it is tacitly assumed that it is permissible to use Eq. (6) with ka,λ = ke,λ : Iλ (s) = 1 − e−ke,λ s Ib,λ . (7) This equation has been used in applying the technique of modulation absorption/emission for soot concentration and temperature, as introduced by Dyer and Flower [7] and more fully implemented by Jenkins and Hanson [9]. The consequences of utilizing Eq. (7), in comparison to the traditionally used Eq. (6) and the correct expression for out-scattering but negligible inscattering, Eq. (5), are shown in Fig. 2. The calculations shown in this figure assume 850 nm emission, a Ke of 8, and a ka,λ /ke,λ of 0.7, which is appropriate for the postflame soot where the measurements of Ke have typically been made. The use of Eq. (7) with an experimentally determined ke,λ overestimates soot emission to a larger extent than the use of Eq. (6), where the emission term itself is accurate but extinction due to out-scattering is ignored. The emissivity difference between Eq. (7) and Eq. (5) is equal to ke,λ /ka,λ and is therefore independent of the sootlayer thickness. In contrast, the emissivity difference between Eq. (6) and Eq. (5) involves the terms e−ke,λ s and e−ka,λ s and thus increases with increasing sootlayer thickness. For reference, the peak soot-layer thicknesses for the 11-mm-diameter nonsmoking and incipient-smoking ethylene flames originally studied
Fig. 3. Apparent soot-layer thickness as a function of actual thickness when applying two-color pyrometry for the case of negligible in-scattering and soot optical properties as given in the main text.
by Santoro and co-workers vary from 30 to 35 ppmmm [25,26]. In large JP-8 pool fires, mean soot-layer thicknesses of 50 ppm-mm have been probed, and instantaneous soot-layer thicknesses have exceeded 100 ppm-mm [27,28]. In diesel engine studies, peak soot-layer thicknesses during the power stroke can easily reach 120 to 480 ppm-mm, corresponding to a “KL factor” of one to four [29,30]. In turbulent flame geometries such as pool fires and engines, the probed soot-layer thickness is largely a function of the sample volume length. When applied to two-color pyrometry, the differences in emissivity evident in Fig. 2 lead to differences in the derived soot concentration and soot temperature. Because the soot temperature is dependent on the ratio of emissivities at the two wavelengths, the use of Eq. (7) yields the identical soot temperature to that using Eq. (5). However, the use of Eq. (6) results in slightly elevated derived soot temperatures. Assuming 1600 K soot and the values of Ke and ka,λ /ke,λ listed above, the use of Eq. (6) yields an approximately linear increase in apparent soot temperature from 1600 K for negligible soot-layer thickness to 1616 K for a soot-layer thickness of 100 ppmmm. More substantial effects manifest themselves for the apparent soot-layer thickness, as shown in Fig. 3. Here, the use of Eq. (6) is shown to underestimate soot concentrations by 25% for a soot-layer thickness of 100 ppm-mm, and the use of Eq. (7) results in an even greater underestimation of the actual soot concentration.
4. In-scattering effects In general, the in-scattering term in Eq. (1) is not negligible and couples the emission measurement to the entire emitting flowfield (whether it be a flame, pool fire, or engine cylinder, for example). This aspect
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Fig. 4. Schematics of the two in-scattering cases considered. Shown are typical values of the incident intensity Iλ (z, θ, φ), the phase function p(θ, φ), and the scattered intensity Iλ (z, θ, φ)p(θ, φ), which is the in-scattering integrand in Eq. (1).
of the soot emission measurement adds an element of uncertainty that cannot be readily taken into account, particularly on an instantaneous basis. However, some bounding cases of in-scattering contribution can be described and analyzed, to aide in quantifying the magnitude of this effect. For simplicity, we consider the case in which the medium surrounding the probe volume is optically thick in all directions and has the same temperature as the soot in the probe volume. Referring back to Fig. 1 we see that the extent to which the environment couples into the measurement will depend on how large the sight tubes are compared to the length of the measurement volume. When this aspect ratio, d/s, is small, radiation from most of the surrounding environment is able to reach the probe volume. Conversely, when the aspect ratio is large, the effective region of radiant interaction with the probe volume approximates a vertical, onedimensional slab. The geometries of these two cases are shown in Fig. 4. This figure also shows schematically the values of the incident intensity Iλ (z, θ, φ), the phase function p(θ, φ), and the in-scattering integrand, which is composed of the product of these two terms. Since our idealized geometry is axisymmetric, the terms inside the in-scattering integral do not depend on the azimuthal angle φ. Note that we are assuming that the probe volume itself is not optically thick (ke,λ s < 1). For the first case, where d/s → 0, the medium surrounding the probe contributes blackbody radiation to the probe volume from all angles except along the measurement axis. Thus, Iλ (θ, φ) ≈ Ib,λ can be taken out of the in-scattering integral, and the in-scattering term becomes equal to ks,λ Ib,λ . Using ke,λ = ka,λ + ks,λ , Eq. (1) is solved to yield Eq. (7).
Consequently, the same equation that clearly overestimates emission intensity for soot when only considering scattering out of the probe volume (with negligible in-scattering) becomes the appropriate equation to use when an unhindered in-scattering contribution is made from an optically thick, homogeneous medium. Measurements made within a large pool fire, for example, would generally fit this description, as long as the probe is small enough not to significantly shield the measurement volume. For the case of large aspect ratio and arbitrary phase function, there is no closed-form solution immediately available; Eq. (1) must be integrated numerically. If, however, we assume that the soot is a strong forward scatterer, strong enough that the phase function approximates a Dirac delta function, Eq. (6) will hold. Although the presumption of forward-only scattering seems severe, aggregated soot particles are in fact strong forward scatterers. Scattering measurements of soot from ethylene–air [31] and acetylene– air [18] flames indicate that at an angle of 10◦ from forward, the phase function falls to less than 20% of its value at an angle of 0◦ . At an angle of 180◦ , the measured phase function is less than 1% of its 0◦ value. To further examine the large aspect ratio case, we used a one-dimensional discrete-ordinates program [32] to integrate Eq. (1). For convenience, the phase function of the soot is modeled using the Henyey–Greenstein equation [33]: p(θ) =
1−γ2 . (1 + γ 2 − 2γ cos θ)3/2
(8)
This function uses the parameter γ to characterize the scattering. Possible values range from γ = 1,
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Fig. 5. Result of numerical integration of Eq. (1) compared to Eq. (6), assuming a slab geometry (d/s → ∞).
a forward-scattering delta function, to γ = −1 for perfect reflection; γ = 0 describes isotropic scattering. Here, we assume γ = 0.8, consistent with the scattering properties of aggregated soot particles. The discrete-ordinates calculation used 32 computational polar angles, resulting in numerical errors of approximately 10−7 . A calculation was performed where γ was set equal to 0.9999 for comparison with the analytical solution, given by Eq. (6), where γ = 1.0. The analytical and numerical results were found to differ by less than 5 × 10−6 . Results of the numerical integration are shown in Fig. 5. We see that for the case of forward-scattering soot (γ = 0.8), the effective emissivity is close to that predicted by Eq. (6). If the soot is assumed to be an isotropic scatterer (γ = 0), there is more in-scattering from the surrounding medium. The results, however, still differ by less than 10% from Eq. (6).
5. Effects of probe-volume heterogeneity Two-color pyrometry and three-line measurements of soot concentrations and temperatures are often performed in combustion systems in which heterogeneity in soot concentration and soot temperature are expected over the length scale of the optical probe volume. All of the expressions (such as those presented above) that have been derived and used to interpret these measurements are based on the assumption of a homogeneous probe volume. For extinction measurements of soot volume fraction, the occurrence of heterogeneity does not pose any difficulties, because of the linearity of the relation between the logarithm of the extinction ratio and the product of the extinction coefficient and path length (see Eq. (4)). For emission measurements, Eqs. (5)– (7) show a nonlinearity that makes them susceptible to heterogeneous effects. This quality has often been invoked in the literature as the presumed cause of low values of soot concentrations deduced from two-color
Fig. 6. Schematic of the layered system, resulting in a step profile in temperature, used to quantify the effects of probe-volume heterogeneity.
pyrometry in heterogeneous environments [2–6,27]. Some [2,34] have also noted that a heterogeneous probe volume will yield temperature measurements that are weighted high. Some quantitative calculations regarding the effect of probe-volume heterogeneity on both soot volume fraction and temperature as deduced from emission measurements in diesel engines were previously reported by Matsui et al. [35] and by Yan and Borman [29], as later summarized by Zhao and Ladommatos [36]. To demonstrate the influence of a heterogeneous probe volume, we compute the result when using Eq. (6) over an idealized probe volume consisting of two uniform regions of soot. The geometry is shown in Fig. 6. The regions are assumed to have identical soot concentrations, but different temperatures; there is a “hot” region of 1800 K soot and a “cold” region of 1200 K soot, resulting in a step profile in temperature across the probe volume. Apparent soot temperature and soot-layer thickness (fv s, in ppmmm) are calculated as functions of the thickness of hot soot layer in the probe volume, fv sh . The calculations are performed at a constant total soot-layer thickness, fv (sc + sh ), of 100 ppm-mm. We assume the same optical constants as in the previous section, Ke = 8.0 and Ka = 5.6. The calculations are performed at two sets of two-color pyrometry wavelengths: 850 and 1000, and 750 and 900 nm. Figs. 7 and 8 show the results of the calculations. The apparent temperature is indeed biased toward the temperature of the hot soot, validating the presumption referenced earlier in this section. The apparent soot-layer thickness approximates the thickness of the hot soot region except when there is very little hot soot in the probe volume. The results are also dependent on whether the hot region or the cold region is adjacent to the detector. The cold soot acts as a shield when adjacent to the detector, absorbing and scattering some of the emission from the hot soot. The effect of the choice of pyrometry wavelengths is minor, with
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Fig. 7. Apparent soot temperature calculated for the system shown in Fig. 6. The calculations are performed at wavelengths of 750 and 900 nm, and at longer wavelengths of 850 and 1000 nm. The results depend on whether the detector is placed adjacent to the hot soot layer (the “hot side”) or the cold soot layer.
Fig. 8. Apparent soot-layer thickness calculated for the system shown in Fig. 6.
only a slight reduction in the bias with the use of longer wavelengths. We also examine the more realistic case of a linear temperature profile across the probe volume. To accomplish this, we integrate Eq. (1) numerically, with ks,λ set to zero, using a Runge–Kutta–Verner fifthorder and sixth-order method. For these calculations, the average temperature of the profile is kept constant at 1500 K. The total difference in temperatures across the probe volume is varied from zero (a uniform profile) to 600 K. We use the same optical constants and constant soot-layer thickness as in the previous section. The calculations are performed at one set of wavelengths, 850 and 1000 nm. Results of these calculations are shown in Figs. 9 and 10 and are compared to the results when assuming a step profile of soot temperature. The apparent temperature and soot-layer thickness are approximately constant for temperature differences less than 100 K. For larger temperature differences, the difference between the apparent values and actual mean values is larger for the step profile than for the linear profile. The results for the linear profile, however, show a
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Fig. 9. Comparison of the apparent soot temperature calculated for a step temperature profile (see Fig. 6) and a linear temperature profile based on emission measurements at 850 and 1000 nm. The average temperature of the profile was kept constant at 1500 K, and the soot-layer thickness was 100 ppm-mm. The x-axis indicates the total difference in temperature across the soot layer.
Fig. 10. Comparison of the apparent soot-layer thickness calculated for a step temperature profile and a linear temperature profile.
larger dependence on whether the detector is on the hot side or the cold side of the probe volume.
6. Practical considerations The sensitivity of the soot emission measurement to probe-volume heterogeneity favors, when practical, the use of shorter probe volumes, where soot temperatures may be expected to be more uniform. The use of short probe volumes, however, typically results in more probe-induced perturbation of the flowfield and greater uncertainty in the effective probe-volume length. The effects of scattering on emission measurements cannot be easily controlled in most experimental configurations, but the relative magnitude of these effects should be evaluated and understood in reporting uncertainties associated with emission-based measurements. Also, the appropriate relation for soot emission, with balanced or imbalanced in-scattering and out-scattering effects, should be used, depend-
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Fig. 11. Comparison of the apparent soot temperature calculated for a step temperature profile (left) and a linear temperature profile (right), as described in Fig. 9. The traditional two-color pyrometry result is compared with the results from one-color pyrometry with a correct mean soot-layer thickness.
ing on the geometry of the measurement. The use of significantly longer wavelengths for performing soot pyrometry may be beneficial in reducing the magnitude of soot scattering relative to absorption, which would reduce uncertainty in the measurement. Unfortunately, performing pyrometry at longer wavelengths reduces the temperature sensitivity of the measurement and makes it more prone to contamination by background radiation. For measurement of soot concentration, the bias to low soot-layer thickness in heterogeneous environments and the sensitivity to scattering effects makes two-color pyrometry inferior to extinction measurements of soot concentration. However, if scattering effects are well understood, a combination of extinction and two-color pyrometry measurements will yield information on both the mean soot concentration (from extinction) and the amount of “hot” soot in the probe volume (from two-color pyrometry). For measurements of soot temperature, the modulated absorption/emission (MAE) technique [7,9] offers the distinct advantage over traditional twocolor pyrometry of not requiring an assumed ratio of wavelength-dependent optical properties of the soot. However, as shown here, the MAE approach is only accurate when soot particles have not experienced significant agglomeration and therefore exhibit minimal light scattering, or when the experimental configuration is such that there is unhindered in-scattering to the probe volume from an optically thick, homogeneous medium. Neither of these conditions is generally applicable. On the other hand, the general concept of combining extinction and emission measurements is promising for reducing the distorting effects of a heterogeneous probe volume on the derived soot temperature. Because the extinction measurement accurately measures the mean soot concentration in heterogeneous probe volumes (within the accuracy of knowledge of Ke ), this information can be used to improve the interpretation of the emission signals.
For example, Fig. 11 shows the derived soot temperature for hot-side detection of two-color pyrometry at 850 and 1000 nm, in comparison to onecolor pyrometry at each of these wavelengths combined with accurate knowledge of the mean sootlayer thickness (as provided by an extinction measurement). As is evident in this figure, the one-color approach reduces the error in the derived temperature by approximately 40% at 850 nm and 50% at 1000 nm. Calculations demonstrate that an even larger reduction in the derived temperature error occurs for cold-side detection. Of course, the downside of relying on one-color pyrometry is the necessity of performing an absolute intensity calibration of the detector and for maintaining consistent optical throughput throughout the measurement and calibration procedure. Finally, the dependence of the emission measurement on the location of hot and cold soot within the probe volume suggests that there is significant value in performing emission measurements on both ends of a double-ended sampling configuration. In this way, the relative magnitude of probe-volume heterogeneity may be diagnosed and, in principle, one could combine the output of the double-ended, two-color emission measurements in such a way as to fit a presumed profile of soot temperature across the probe volume (e.g., yielding a mean temperature as well as the slope of a presumed linear temperature profile).
7. Conclusions Recent measurements indicate that scattering makes a significant contribution to radiation extinction by soot, even in the near-infrared region. Thus, contrary to traditional Rayleigh limit approximations, absorption and extinction coefficients cannot be used interchangeably in interpreting soot emission measurements of soot volume fraction and temperature.
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Scattering of radiation out of the measurement probe volume may be explicitly taken into account, but inscattering is a confounding factor that cannot generally be quantified, leading to uncertainties in the soot volume fraction and temperature derived from twocolor pyrometry. Idealized limiting conditions allow one to bracket the net emissive strength between two simple expressions, one that assumes in-scattering is negligible and another that assumes in-scattering is unhindered by the probe and is from an optically thick, homogeneous medium. Numerical integration of the radiative transfer equation indicates that in-scattering approximately balances out-scattering when the effective region of interaction with the probe volume resembles a onedimensional slab. In this case, the classical emission expression Iλ (s) = [1−exp(−ka,λ s)]Ib,λ can be used to estimate emission from the probe volume. This result is expected to hold true in many practical environments. Another complication in soot pyrometry concerns the effects of a heterogeneous probe volume. The simulations presented here demonstrate that both the emissive soot volume fraction and the derived temperature are strongly weighted toward the properties of the “hot” soot in a bimodal probe volume. A soot population with a more evenly distributed temperature yields a weaker but still significant bias in the resultant measurements. Measures to alleviate the deleterious effects of probe-volume heterogeneity are suggested, including the use of single-color pyrometry in combination with an extinction-based measurement of soot concentration.
Acknowledgments The authors thank Prof. Adel Sarofim of the University of Utah and Drs. George Mulholland and Anthony Hamins of the National Institute of Standards and Technology for stimulating discussions regarding some of these issues. Dr. Mark Musculus and Dr. William Houf of Sandia provided useful suggestions for improving the manuscript. Funding for this work was provided by the Laboratory Directed Research and Development program of Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94-AL85000.
References [1] Y.R. Sivathanu, G.M. Faeth, Combust. Flame 81 (2) (1990) 133–149.
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[2] Y.R. Sivathanu, J.P. Gore, J. Dolinar, Combust. Sci. Technol. 76 (1–3) (1991) 45–66. [3] Y.R. Sivathanu, J.P. Gore, J. Heat Transfer 114 (3) (1992) 659–665. [4] M. Klassen, Y.R. Sivathanu, J.P. Gore, Combust. Flame 90 (1) (1992) 34–44. [5] M. Klassen, J.P. Gore, Y.R. Sivathanu, A. Hamins, T. Kashiwagi, Proc. Combust. Inst. 24 (1992) 1713– 1719. [6] M. Klassen, J.P. Gore, Combust. Flame 93 (3) (1993) 270–278. [7] T.M. Dyer, W.L. Flower, in: D.C. Siegla, G.W. Smith (Eds.), Particulate Carbon: Formation During Combustion, Plenum, New York, 1981, pp. 363–389. [8] S. di Stasio, P. Massoli, Meas. Sci. Technol. 5 (12) (1994) 1453–1465. [9] T.P. Jenkins, R.K. Hanson, Combust. Flame 126 (3) (2001) 1669–1679. [10] W.L. Grosshandler, Combust. Flame 78 (3–4) (1989) 257–262. [11] M.Y. Choi, A. Hamins, G.W. Mulholland, T. Kashiwagi, Combust. Flame 99 (1) (1994) 174–186. [12] Ü.Ö. Köylü, G.M. Faeth, J. Heat Transfer 116 (1) (1994) 152–159. [13] M.Y. Choi, G.W. Mulholland, A. Hamins, T. Kashiwagi, Combust. Flame 102 (1–2) (1995) 161–169. [14] G.W. Mulholland, M.Y. Choi, Proc. Combust. Inst. 27 (1998) 1515–1522. [15] Z.-Q. Zhou, T.U. Ahmed, M.Y. Choi, Exp. Therm. Fluid Sci. 18 (1) (1998) 27–32. [16] J. Zhu, M.Y. Choi, G.W. Mulholland, L.A. Gritzo, Int. J. Heat Mass Transfer 43 (18) (2000) 3299–3303. [17] Ü.Ö. Köylü, G.M. Faeth, J. Heat Transfer 118 (2) (1996) 415–421. [18] J.-S. Wu, S.S. Krishnan, G.M. Faeth, J. Heat Transfer 119 (2) (1997) 230–237. [19] S.S. Krishnan, K.-C. Lin, G.M. Faeth, J. Heat Transfer 122 (3) (2000) 517–524. [20] S.S. Krishnan, K.-C. Lin, G.M. Faeth, J. Heat Transfer 123 (2) (2001) 331–339. [21] J.M. Williams, L.A. Gritzo, Proc. Combust. Inst. 27 (1998) 2707–2714. [22] J. Zhu, M.Y. Choi, G.W. Mulholland, L.A. Gritzo, Proc. Combust. Inst. 28 (2000) 439–446. [23] M.Q. Brewster, Thermal Radiative Transfer and Properties, Wiley, New York, NY, 1992. [24] L. Wind, W.W. Szymanski, Meas. Sci. Technol. 13 (3) (2002) 270–275. [25] R.J. Santoro, H.G. Semerjian, R.A. Dobbins, Combust. Flame 51 (1983) 203–218. [26] C.R. Shaddix, K.C. Smyth, Combust. Flame 107 (1996) 418–452. [27] L.A. Gritzo, Y.R. Sivathanu, W. Gill, Combust. Sci. Technol. 139 (1–6) (1998) 113–136. [28] J.J. Murphy, C.R. Shaddix, Combust. Sci. Technol. (2005), in press. [29] J. Yan, G.L. Borman, SAE Trans.: J. Engines 97 (6) (1989) 1623–1644, paper 881315. [30] M.P. Musculus, J.E. Dec, D.R. Tree, SAE Trans. 111 (2002), paper 2002-01-0889.
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[31] Ü.Ö. Köylü, Combust. Flame 109 (3) (1996) 488– 500. [32] K. Stamnes, S.-C. Tsay, W. Wiscombe, K. Jayaweera, Appl. Opt. 27 (12) (1988) 2502–2509. [33] L.C. Henyey, J.L. Greenstein, Astrophys. J. 93 (1) (1941) 70–83.
[34] A. Coppalle, D. Joyeux, Combust. Flame 96 (3) (1994) 275–285. [35] Y. Matsui, T. Kamimoto, S. Matsuoka, SAE Trans. 97 (3) (1981) 3043–3055, paper 800970. [36] H. Zhao, N. Ladommatos, Prog. Energy Combust. Sci. 24 (1998) 221–255.
Influence of scattering and probe-volume heterogeneity on soot measurements using optical pyrometry J.J. Murphy ∗ , C.R. Shaddix Combustion Research Facility, Sandia National Laboratories, P.O. Box 969, Livermore, CA 94550, USA Received 2 December 2003; received in revised form 15 September 2004; accepted 26 September 2004 Available online 22 October 2004
Abstract The equations describing radiative emission from soot are reexamined in light of recent measurements showing that scattering by aggregated soot particles is not negligible in the near-infrared region. When measuring soot temperature and concentration using two-color optical pyrometry, emission is usually best approximated by Iλ (s) = [1 − exp(−ka,λ s)]Ib,λ , using an absorption coefficient, ka,λ , determined while taking into account the effects of scattering. This equation gives good results in most practical measurement environments, where scattering-induced contributions to the measured light intensity are approximately balanced by scattering losses. It is also shown that probe-volume heterogeneity biases temperature measurements made with two-color pyrometry toward the higher temperatures in the probe volume. Soot concentration measurements are also biased, reflecting the concentration of hot soot in the probe volume, rather than the mean concentration of hot and cold soot. Methods to minimize these effects are suggested. 2005 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Two-color pyrometry; Soot; Optical properties
1. Introduction Optical pyrometry has been used extensively to measure soot temperature in environments ranging from laboratory flames and shock tubes to internal combustion engines and large pool fires. Most often, two-color pyrometry is used, in order to simplify the calibration procedure and to make the measurement insensitive to variations in the optical throughput of the light collection system. * Corresponding author. Current address: The Aerospace
Corporation, P.O. Box 92957, Los Angeles, CA 90009, USA. E-mail addresses: [email protected] (J.J. Murphy), [email protected] (C.R. Shaddix).
In addition to temperature, soot concentration can also be measured using two-color pyrometry. This measurement, however, requires absolute calibration of the detection system, consistent optical throughput, and knowledge of the emissive properties of the soot that is being probed. As a consequence, simultaneous measurements of soot concentration and temperature have often relied upon the combination of two-color pyrometry and optical extinction (usually using a visible-wavelength laser source). Soot concentration can be calculated from optical extinction without difficulty, although knowledge of the soot optical properties is still required. The combined measurement is usually called the “three-line,” or “threewavelength,” soot diagnostic, and is easy to implement using two fiber-optic probes that face each other
0010-2180/$ – see front matter 2005 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustflame.2004.09.003
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Nomenclature d dp fv Iλ Iλ,0 Ib,λ ka,λ ke,λ ks,λ Ka
sight-tube diameter (see Fig. 1) [mm] soot-particle diameter [nm] soot volume fraction optical intensity [W m−2 nm−1 ] incident light intensity [W m−2 nm−1 ] blackbody emission intensity [W m−2 nm−1 ] absorption coefficient [mm−1 ] extinction coefficient [mm−1 ] scattering coefficient [mm−1 ] dimensionless absorption coefficient = ka,λ λ/fv
across the probe volume. Sivathanu, Gore, and coworkers have used this technique extensively in buoyant, turbulent diffusion flames and pool fires [1–6]. The effect of optical property uncertainty on the accuracy of the derived temperature and volume fraction from two-color pyrometry and laser extinction measurements has long been recognized [7–9]. Furthermore, when interpreting laser extinction signals, it is necessary to apply the correct spectral extinction coefficient so that light scattering as well as absorption is taken into account [10–12]. The need for accurate values of soot extinction coefficients has motivated an extensive series of measurements by Choi, Mulholland, and co-workers [13–16], and by Faeth and co-workers [12,17–20]. In contrast, soot pyrometry signals have always been interpreted with the assumption that the soot particles are in the smallparticle, or Rayleigh, limit of Mie theory, such that light scattering is negligible at the detected wavelengths. With this assumption, the extinction and absorption coefficients are equal, and simple relations are readily derived to express the optical emission intensity and to deduce the temperature from two-color pyrometry. The justification for applying the Rayleigh limit assumption in the interpretation of soot pyrometry signals has been loosely based on the small size of soot primary particles (dp ≈ 10–50 nm) in comparison to the long wavelengths of light (> 700 nm) generally used for pyrometry measurements. However, for aggregated soot particles such as those that exist in most regions of sooty flames (particularly in pool fires [21], older diesel engines, and other strongly sooting environments), the contribution of soot scattering signals is largely dependent on the aggregate size distribution. The importance of soot aggregate scattering has been recognized at visible wavelengths, where scattering albedos (ratios of scattering to total extinction) of 20–30% have often been measured for
Ke p s sc sh x γ λ
dimensionless extinction coefficient = ke,λ λ/fv phase function probe-volume length [mm] path length through “cold” soot [mm] path length through “hot” soot [mm] dimensionless optical size parameter = π dp /λ Henyey–Greenstein phase function parameter (see Eq. (8)) wavelength [nm]
postflame soot [12,18]. On the other hand, the contribution of scattering at longer wavelengths has historically been discounted, on the basis of the strong dependence of Rayleigh light-scattering intensity on the optical size parameter x = π dp /λ, where λ is the wavelength of the light. However, recent measurements [20,22] indicate that soot scattering continues to be significant at near-infrared wavelengths for aggregated soot, because of increases in the soot index of refraction at these wavelengths. For example, Zhu et al. [22] measured soot-scattering albedos that ranged from 20 to 24% at 856 nm for soot emitted from laminar acetylene– and ethylene–air flames. In a study of soot emitted from a number of buoyant turbulent diffusion flames, Krishnan et al. [20] derived scattering albedos that ranged from 0.15 to 0.35 over 800–1000 nm for six different fuels. In light of these developments, it is appropriate to revisit the derivation of the equations for optical soot pyrometry without resorting to Rayleigh limit simplifications. We also quantitatively evaluate the effects of another important complication of soot pyrometry, the presence of probe-volume heterogeneity.
2. General relations A schematic of the idealized system to be analyzed is shown in Fig. 1. The emission measurement is made across a probe volume bounded by two cylindrical “sight tubes” of diameter d whose open ends face one another across the interrogated medium. Rather than having an actual emission detector placed within the optical probe, a lens-coupled fiber optic cable is typically used to displace the detector from the sampling environment, but this is irrelevant to the analysis performed here. This sampling geometry has been extensively used to perform two-color soot pyrometry in pool fires and turbulent flames. For emission mea-
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on the local soot temperature, which can in turn depend on z. The phase function p(θ , φ ) specifies how much light is scattered from the (θ , φ ) direction into the direction (θ = 0, φ = 0) along the measurement axis, and is normalized such that 1 4π
2ππ
p(θ , φ ) sin θ dθ dφ = 1.
(2)
0 0
Fig. 1. Schematic of the probe geometry.
surements through walls bounding a process (such as across an I.C. engine cylinder), the probe ends may be thought of as defining the locations of the walls. The measurement takes place in a cylindrical probe volume of length s aligned with the measurement axis. The system is symmetric about the measurement axis. The spatial variable z is used to represent distance along the measurement axis. φ indicates azimuthal angle and θ indicates polar angle measured from the z-axis. The use of θ , rather than φ, to represent polar angle is consistent with the notation used by Brewster [23]. Scattering effects on soot emission measurements are seen to be dual-natured: emitted light originally directed forward along the measurement axis that is scattered results in a loss of detected light intensity, and is herein termed “out-scattering”; light from the surroundings originally directed across the measurement axis that is scattered in the forward direction of the measurement axis results in a gain in detected light intensity, and is herein termed “inscattering.” The radiative transfer equation of an absorbing, emitting, and scattering medium can be found in a variety of textbooks. Here, we use the expression given in Chapter 7 of Brewster [23]: dIλ (z) dz = − ke,λ Iλ (z) + ka,λ Ib,λ extinction
emission
2ππ
+
ks,λ Iλ (z, θ , φ )p(θ , φ ) sin θ dθ dφ . 4π 0 0 in-scattering
(1) Iλ (z, θ, φ) represents the light intensity coming from the (θ, φ) direction at location z along the measurement axis. Iλ (z) is shorthand for Iλ (z, 0, 0). Ib,λ is Planck’s blackbody radiation function, and depends
The coefficients ka,λ and ks,λ are the dimensional absorption and scattering coefficients, respectively, and ke,λ = ka,λ + ks,λ is the dimensional extinction coefficient. These coefficients may vary spatially, and are thus functions of z. By Kirchoff’s Law, the emission coefficient is equal to the absorption coefficient. We note that for laser-based extinction measurements, where there is absorption and out-scattering but negligible in-scattering and no emission, Eq. (1) becomes dIλ (z) = −ke,λ Iλ (z). (3) dz This equation, with the incident light intensity given by Iλ,0 , integrates to Iλ (s) = Iλ,0 e−ke,λ s .
(4)
Accurate implementation of Eq. (4) for media with substantial forward scattering (such as soot) is complicated by the likelihood that some fraction of the forward-scattered light will reach the extinction detector. This effect was recently analyzed [24] and shown to depend on the optical depth of the probed medium, as well as the geometry of the detection system. In practice, it is important to approximately match the angle of acceptance of the measurement system with that used to generate the nondimensional extinction coefficient used to calibrate the laser extinction.
3. Out-scattering effects For emission measurements, such as those used in two-color pyrometry, there are two distinct complications that need to be considered when light scattering is nonnegligible compared to absorption. The first complication concerns out-scattering from the probe volume, which has the consequence that the extinction coefficient in Eq. (1) is not identical to the absorption coefficient. In this case, for negligible inscattering, Eq. (1) integrates to Iλ (s) =
ka,λ 1 − e−ke,λ s Ib,λ . ke,λ
(5)
The three-line diagnostic literature has used the classical derivation of Eq. (5), where all scattering is
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Fig. 2. Comparison of soot emission equations for the case of negligible in-scattering. The unit ppm corresponds to a soot volume fraction of 10−6 .
neglected, to arrive at Iλ (s) = 1 − e−ka,λ s Ib,λ .
(6)
More recently, extensive measurements have been performed of the dimensionless soot extinction coefficient, Ke ≡ ke,λ λ/fv , in postflame soot. It was suggested that the use of these values would improve the comparison between soot volume fraction measurements based on extinction and soot volume fraction measurements based on emission (presumably by using the measured Ke for interpreting both measurements). In this case, it is tacitly assumed that it is permissible to use Eq. (6) with ka,λ = ke,λ : Iλ (s) = 1 − e−ke,λ s Ib,λ . (7) This equation has been used in applying the technique of modulation absorption/emission for soot concentration and temperature, as introduced by Dyer and Flower [7] and more fully implemented by Jenkins and Hanson [9]. The consequences of utilizing Eq. (7), in comparison to the traditionally used Eq. (6) and the correct expression for out-scattering but negligible inscattering, Eq. (5), are shown in Fig. 2. The calculations shown in this figure assume 850 nm emission, a Ke of 8, and a ka,λ /ke,λ of 0.7, which is appropriate for the postflame soot where the measurements of Ke have typically been made. The use of Eq. (7) with an experimentally determined ke,λ overestimates soot emission to a larger extent than the use of Eq. (6), where the emission term itself is accurate but extinction due to out-scattering is ignored. The emissivity difference between Eq. (7) and Eq. (5) is equal to ke,λ /ka,λ and is therefore independent of the sootlayer thickness. In contrast, the emissivity difference between Eq. (6) and Eq. (5) involves the terms e−ke,λ s and e−ka,λ s and thus increases with increasing sootlayer thickness. For reference, the peak soot-layer thicknesses for the 11-mm-diameter nonsmoking and incipient-smoking ethylene flames originally studied
Fig. 3. Apparent soot-layer thickness as a function of actual thickness when applying two-color pyrometry for the case of negligible in-scattering and soot optical properties as given in the main text.
by Santoro and co-workers vary from 30 to 35 ppmmm [25,26]. In large JP-8 pool fires, mean soot-layer thicknesses of 50 ppm-mm have been probed, and instantaneous soot-layer thicknesses have exceeded 100 ppm-mm [27,28]. In diesel engine studies, peak soot-layer thicknesses during the power stroke can easily reach 120 to 480 ppm-mm, corresponding to a “KL factor” of one to four [29,30]. In turbulent flame geometries such as pool fires and engines, the probed soot-layer thickness is largely a function of the sample volume length. When applied to two-color pyrometry, the differences in emissivity evident in Fig. 2 lead to differences in the derived soot concentration and soot temperature. Because the soot temperature is dependent on the ratio of emissivities at the two wavelengths, the use of Eq. (7) yields the identical soot temperature to that using Eq. (5). However, the use of Eq. (6) results in slightly elevated derived soot temperatures. Assuming 1600 K soot and the values of Ke and ka,λ /ke,λ listed above, the use of Eq. (6) yields an approximately linear increase in apparent soot temperature from 1600 K for negligible soot-layer thickness to 1616 K for a soot-layer thickness of 100 ppmmm. More substantial effects manifest themselves for the apparent soot-layer thickness, as shown in Fig. 3. Here, the use of Eq. (6) is shown to underestimate soot concentrations by 25% for a soot-layer thickness of 100 ppm-mm, and the use of Eq. (7) results in an even greater underestimation of the actual soot concentration.
4. In-scattering effects In general, the in-scattering term in Eq. (1) is not negligible and couples the emission measurement to the entire emitting flowfield (whether it be a flame, pool fire, or engine cylinder, for example). This aspect
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Fig. 4. Schematics of the two in-scattering cases considered. Shown are typical values of the incident intensity Iλ (z, θ, φ), the phase function p(θ, φ), and the scattered intensity Iλ (z, θ, φ)p(θ, φ), which is the in-scattering integrand in Eq. (1).
of the soot emission measurement adds an element of uncertainty that cannot be readily taken into account, particularly on an instantaneous basis. However, some bounding cases of in-scattering contribution can be described and analyzed, to aide in quantifying the magnitude of this effect. For simplicity, we consider the case in which the medium surrounding the probe volume is optically thick in all directions and has the same temperature as the soot in the probe volume. Referring back to Fig. 1 we see that the extent to which the environment couples into the measurement will depend on how large the sight tubes are compared to the length of the measurement volume. When this aspect ratio, d/s, is small, radiation from most of the surrounding environment is able to reach the probe volume. Conversely, when the aspect ratio is large, the effective region of radiant interaction with the probe volume approximates a vertical, onedimensional slab. The geometries of these two cases are shown in Fig. 4. This figure also shows schematically the values of the incident intensity Iλ (z, θ, φ), the phase function p(θ, φ), and the in-scattering integrand, which is composed of the product of these two terms. Since our idealized geometry is axisymmetric, the terms inside the in-scattering integral do not depend on the azimuthal angle φ. Note that we are assuming that the probe volume itself is not optically thick (ke,λ s < 1). For the first case, where d/s → 0, the medium surrounding the probe contributes blackbody radiation to the probe volume from all angles except along the measurement axis. Thus, Iλ (θ, φ) ≈ Ib,λ can be taken out of the in-scattering integral, and the in-scattering term becomes equal to ks,λ Ib,λ . Using ke,λ = ka,λ + ks,λ , Eq. (1) is solved to yield Eq. (7).
Consequently, the same equation that clearly overestimates emission intensity for soot when only considering scattering out of the probe volume (with negligible in-scattering) becomes the appropriate equation to use when an unhindered in-scattering contribution is made from an optically thick, homogeneous medium. Measurements made within a large pool fire, for example, would generally fit this description, as long as the probe is small enough not to significantly shield the measurement volume. For the case of large aspect ratio and arbitrary phase function, there is no closed-form solution immediately available; Eq. (1) must be integrated numerically. If, however, we assume that the soot is a strong forward scatterer, strong enough that the phase function approximates a Dirac delta function, Eq. (6) will hold. Although the presumption of forward-only scattering seems severe, aggregated soot particles are in fact strong forward scatterers. Scattering measurements of soot from ethylene–air [31] and acetylene– air [18] flames indicate that at an angle of 10◦ from forward, the phase function falls to less than 20% of its value at an angle of 0◦ . At an angle of 180◦ , the measured phase function is less than 1% of its 0◦ value. To further examine the large aspect ratio case, we used a one-dimensional discrete-ordinates program [32] to integrate Eq. (1). For convenience, the phase function of the soot is modeled using the Henyey–Greenstein equation [33]: p(θ) =
1−γ2 . (1 + γ 2 − 2γ cos θ)3/2
(8)
This function uses the parameter γ to characterize the scattering. Possible values range from γ = 1,
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Fig. 5. Result of numerical integration of Eq. (1) compared to Eq. (6), assuming a slab geometry (d/s → ∞).
a forward-scattering delta function, to γ = −1 for perfect reflection; γ = 0 describes isotropic scattering. Here, we assume γ = 0.8, consistent with the scattering properties of aggregated soot particles. The discrete-ordinates calculation used 32 computational polar angles, resulting in numerical errors of approximately 10−7 . A calculation was performed where γ was set equal to 0.9999 for comparison with the analytical solution, given by Eq. (6), where γ = 1.0. The analytical and numerical results were found to differ by less than 5 × 10−6 . Results of the numerical integration are shown in Fig. 5. We see that for the case of forward-scattering soot (γ = 0.8), the effective emissivity is close to that predicted by Eq. (6). If the soot is assumed to be an isotropic scatterer (γ = 0), there is more in-scattering from the surrounding medium. The results, however, still differ by less than 10% from Eq. (6).
5. Effects of probe-volume heterogeneity Two-color pyrometry and three-line measurements of soot concentrations and temperatures are often performed in combustion systems in which heterogeneity in soot concentration and soot temperature are expected over the length scale of the optical probe volume. All of the expressions (such as those presented above) that have been derived and used to interpret these measurements are based on the assumption of a homogeneous probe volume. For extinction measurements of soot volume fraction, the occurrence of heterogeneity does not pose any difficulties, because of the linearity of the relation between the logarithm of the extinction ratio and the product of the extinction coefficient and path length (see Eq. (4)). For emission measurements, Eqs. (5)– (7) show a nonlinearity that makes them susceptible to heterogeneous effects. This quality has often been invoked in the literature as the presumed cause of low values of soot concentrations deduced from two-color
Fig. 6. Schematic of the layered system, resulting in a step profile in temperature, used to quantify the effects of probe-volume heterogeneity.
pyrometry in heterogeneous environments [2–6,27]. Some [2,34] have also noted that a heterogeneous probe volume will yield temperature measurements that are weighted high. Some quantitative calculations regarding the effect of probe-volume heterogeneity on both soot volume fraction and temperature as deduced from emission measurements in diesel engines were previously reported by Matsui et al. [35] and by Yan and Borman [29], as later summarized by Zhao and Ladommatos [36]. To demonstrate the influence of a heterogeneous probe volume, we compute the result when using Eq. (6) over an idealized probe volume consisting of two uniform regions of soot. The geometry is shown in Fig. 6. The regions are assumed to have identical soot concentrations, but different temperatures; there is a “hot” region of 1800 K soot and a “cold” region of 1200 K soot, resulting in a step profile in temperature across the probe volume. Apparent soot temperature and soot-layer thickness (fv s, in ppmmm) are calculated as functions of the thickness of hot soot layer in the probe volume, fv sh . The calculations are performed at a constant total soot-layer thickness, fv (sc + sh ), of 100 ppm-mm. We assume the same optical constants as in the previous section, Ke = 8.0 and Ka = 5.6. The calculations are performed at two sets of two-color pyrometry wavelengths: 850 and 1000, and 750 and 900 nm. Figs. 7 and 8 show the results of the calculations. The apparent temperature is indeed biased toward the temperature of the hot soot, validating the presumption referenced earlier in this section. The apparent soot-layer thickness approximates the thickness of the hot soot region except when there is very little hot soot in the probe volume. The results are also dependent on whether the hot region or the cold region is adjacent to the detector. The cold soot acts as a shield when adjacent to the detector, absorbing and scattering some of the emission from the hot soot. The effect of the choice of pyrometry wavelengths is minor, with
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Fig. 7. Apparent soot temperature calculated for the system shown in Fig. 6. The calculations are performed at wavelengths of 750 and 900 nm, and at longer wavelengths of 850 and 1000 nm. The results depend on whether the detector is placed adjacent to the hot soot layer (the “hot side”) or the cold soot layer.
Fig. 8. Apparent soot-layer thickness calculated for the system shown in Fig. 6.
only a slight reduction in the bias with the use of longer wavelengths. We also examine the more realistic case of a linear temperature profile across the probe volume. To accomplish this, we integrate Eq. (1) numerically, with ks,λ set to zero, using a Runge–Kutta–Verner fifthorder and sixth-order method. For these calculations, the average temperature of the profile is kept constant at 1500 K. The total difference in temperatures across the probe volume is varied from zero (a uniform profile) to 600 K. We use the same optical constants and constant soot-layer thickness as in the previous section. The calculations are performed at one set of wavelengths, 850 and 1000 nm. Results of these calculations are shown in Figs. 9 and 10 and are compared to the results when assuming a step profile of soot temperature. The apparent temperature and soot-layer thickness are approximately constant for temperature differences less than 100 K. For larger temperature differences, the difference between the apparent values and actual mean values is larger for the step profile than for the linear profile. The results for the linear profile, however, show a
7
Fig. 9. Comparison of the apparent soot temperature calculated for a step temperature profile (see Fig. 6) and a linear temperature profile based on emission measurements at 850 and 1000 nm. The average temperature of the profile was kept constant at 1500 K, and the soot-layer thickness was 100 ppm-mm. The x-axis indicates the total difference in temperature across the soot layer.
Fig. 10. Comparison of the apparent soot-layer thickness calculated for a step temperature profile and a linear temperature profile.
larger dependence on whether the detector is on the hot side or the cold side of the probe volume.
6. Practical considerations The sensitivity of the soot emission measurement to probe-volume heterogeneity favors, when practical, the use of shorter probe volumes, where soot temperatures may be expected to be more uniform. The use of short probe volumes, however, typically results in more probe-induced perturbation of the flowfield and greater uncertainty in the effective probe-volume length. The effects of scattering on emission measurements cannot be easily controlled in most experimental configurations, but the relative magnitude of these effects should be evaluated and understood in reporting uncertainties associated with emission-based measurements. Also, the appropriate relation for soot emission, with balanced or imbalanced in-scattering and out-scattering effects, should be used, depend-
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Fig. 11. Comparison of the apparent soot temperature calculated for a step temperature profile (left) and a linear temperature profile (right), as described in Fig. 9. The traditional two-color pyrometry result is compared with the results from one-color pyrometry with a correct mean soot-layer thickness.
ing on the geometry of the measurement. The use of significantly longer wavelengths for performing soot pyrometry may be beneficial in reducing the magnitude of soot scattering relative to absorption, which would reduce uncertainty in the measurement. Unfortunately, performing pyrometry at longer wavelengths reduces the temperature sensitivity of the measurement and makes it more prone to contamination by background radiation. For measurement of soot concentration, the bias to low soot-layer thickness in heterogeneous environments and the sensitivity to scattering effects makes two-color pyrometry inferior to extinction measurements of soot concentration. However, if scattering effects are well understood, a combination of extinction and two-color pyrometry measurements will yield information on both the mean soot concentration (from extinction) and the amount of “hot” soot in the probe volume (from two-color pyrometry). For measurements of soot temperature, the modulated absorption/emission (MAE) technique [7,9] offers the distinct advantage over traditional twocolor pyrometry of not requiring an assumed ratio of wavelength-dependent optical properties of the soot. However, as shown here, the MAE approach is only accurate when soot particles have not experienced significant agglomeration and therefore exhibit minimal light scattering, or when the experimental configuration is such that there is unhindered in-scattering to the probe volume from an optically thick, homogeneous medium. Neither of these conditions is generally applicable. On the other hand, the general concept of combining extinction and emission measurements is promising for reducing the distorting effects of a heterogeneous probe volume on the derived soot temperature. Because the extinction measurement accurately measures the mean soot concentration in heterogeneous probe volumes (within the accuracy of knowledge of Ke ), this information can be used to improve the interpretation of the emission signals.
For example, Fig. 11 shows the derived soot temperature for hot-side detection of two-color pyrometry at 850 and 1000 nm, in comparison to onecolor pyrometry at each of these wavelengths combined with accurate knowledge of the mean sootlayer thickness (as provided by an extinction measurement). As is evident in this figure, the one-color approach reduces the error in the derived temperature by approximately 40% at 850 nm and 50% at 1000 nm. Calculations demonstrate that an even larger reduction in the derived temperature error occurs for cold-side detection. Of course, the downside of relying on one-color pyrometry is the necessity of performing an absolute intensity calibration of the detector and for maintaining consistent optical throughput throughout the measurement and calibration procedure. Finally, the dependence of the emission measurement on the location of hot and cold soot within the probe volume suggests that there is significant value in performing emission measurements on both ends of a double-ended sampling configuration. In this way, the relative magnitude of probe-volume heterogeneity may be diagnosed and, in principle, one could combine the output of the double-ended, two-color emission measurements in such a way as to fit a presumed profile of soot temperature across the probe volume (e.g., yielding a mean temperature as well as the slope of a presumed linear temperature profile).
7. Conclusions Recent measurements indicate that scattering makes a significant contribution to radiation extinction by soot, even in the near-infrared region. Thus, contrary to traditional Rayleigh limit approximations, absorption and extinction coefficients cannot be used interchangeably in interpreting soot emission measurements of soot volume fraction and temperature.
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Scattering of radiation out of the measurement probe volume may be explicitly taken into account, but inscattering is a confounding factor that cannot generally be quantified, leading to uncertainties in the soot volume fraction and temperature derived from twocolor pyrometry. Idealized limiting conditions allow one to bracket the net emissive strength between two simple expressions, one that assumes in-scattering is negligible and another that assumes in-scattering is unhindered by the probe and is from an optically thick, homogeneous medium. Numerical integration of the radiative transfer equation indicates that in-scattering approximately balances out-scattering when the effective region of interaction with the probe volume resembles a onedimensional slab. In this case, the classical emission expression Iλ (s) = [1−exp(−ka,λ s)]Ib,λ can be used to estimate emission from the probe volume. This result is expected to hold true in many practical environments. Another complication in soot pyrometry concerns the effects of a heterogeneous probe volume. The simulations presented here demonstrate that both the emissive soot volume fraction and the derived temperature are strongly weighted toward the properties of the “hot” soot in a bimodal probe volume. A soot population with a more evenly distributed temperature yields a weaker but still significant bias in the resultant measurements. Measures to alleviate the deleterious effects of probe-volume heterogeneity are suggested, including the use of single-color pyrometry in combination with an extinction-based measurement of soot concentration.
Acknowledgments The authors thank Prof. Adel Sarofim of the University of Utah and Drs. George Mulholland and Anthony Hamins of the National Institute of Standards and Technology for stimulating discussions regarding some of these issues. Dr. Mark Musculus and Dr. William Houf of Sandia provided useful suggestions for improving the manuscript. Funding for this work was provided by the Laboratory Directed Research and Development program of Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94-AL85000.
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