A correlation between char emissivity and temperature

Fuel 256 (2019) 115889

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A correlation between char emissivity and temperature a,⁎

b

a

a

b

Martin Schiemann , Tim Gronarz , Philipp Graeser , Jeanette Gorewoda , Reinhold Kneer , Viktor Scherera a b

T

Department of Energy Plant Technology, Ruhr-University, Bochum, Germany Institute of Heat and Mass Transfer, RWTH Aachen University, Aachen, Germany

ARTICLE INFO

ABSTRACT

Keywords: Emissivity Coal char Heat transfer Coal combustion

The radiative behaviour of coal and char particles is an important input parameter for simulations of coal combustion and gasification processes, as those typically feature elevated reactor temperatures. For consideration of radiative heat transfer, accurate knowledge of particle emissivity is a pre-requisite. Combining theoretical considerations and experimental data from literature, a temperature dependent relation for char emissivity is provided.

1. Introduction Coal and biomass particles are utilized in many technical applications for heat and power generation, e.g. pulverized fuel [1,2] or circulating fluidized bed [3–5] power plants, blast furnaces [6,7] or rotary kilns for cement processing [8,9]. In these systems thermal radiation accounts for a significant portion or even the dominating part of heat transfer. The majority of papers utilizing char particle emissivity can be attributed to pulverized fuel combustion simulations. For this, often simple approaches have been chosen. Many authors use a fixed value for char particle emissivity εp. Gronarz et al. [10] carried out simulations of a coal fired boiler to identify the influence of char particle emissivity on radiative heat transfer. In this study, char emissivity of a Rhenish lignite was measured in the range εp = 0.3–0.5 and thus, the simulations exhibited a strong influence of εp on radiative heat transfer (details on the experiments follow in Section 2.2). Including the effect of chemical composition (carbon and ash content) on the particle emissivity, other authors use a formulation depending on conversion, thus p

= 0.6 + 0.4· Uc

(1)

[11–13], which results in an emissivity of unity for unreacted char (unburned carbon Uc = 1) and 0.6 for ash particles. Char emissivity not only is important for combustion calculations, but also for prior determination of reaction rates. These are typically determined through simplified laboratory experiments, where e.g. the energy balance for a burning particle is employed for calculation of the corresponding burning rate. Typical examples are drop tube or ⁎

entrained flow reactor experiments with optical particle temperature measurement [14–18] or the application of burnout models to conversion measurements in a similar reactor geometry [19]. Publications from this community typically refer to similar char emissivity values as mentioned above. In the following this work summarizes experimental data on char particle emissivity. The experimental findings are strengthened by a discussion on ash emissivity, which has been investigated experimentally in more detail than coal char emissivity. Based on these findings a model for the temperature-dependent spectrally averaged char emissivity is derived. The third section of the manuscript describes the concept of spectrally averaged emissivity calculation and includes some theoretical considerations of this parameter, which are based on Mie Theory. Finally, high-temperature data on char emissivity are analysed. Based on these results a universal expression for particle emissivity depending on conversion and temperature is presented, which covers the experimental data base to a reasonable degree and enhances the accuracy of radiative heat transfer calculations. 2. Literature review on particle emissivity experiments 2.1. Coal and char emissivity Experimental data on coal and char particle emissivity are available for a limited number of coals and chars in different temperature ranges. Baxter et al. [20] studied the emissivity of raw and devolatilized particles suspended on a NaCl window by emission FTIR spectroscopy. They varied the coal rank from lignite to bituminous coal and investigated particles in narrow size cuts with dp = 40 µm and

Corresponding author. E-mail address: [email protected] (M. Schiemann).

https://doi.org/10.1016/j.fuel.2019.115889 Received 31 May 2018; Received in revised form 10 July 2019; Accepted 24 July 2019 Available online 01 August 2019 0016-2361/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature

m n T Qabs Tp Uc FTIR hvb; hvb MIR NIR pc SEM

Symbol/acronym name unit α absorptance [–] ε emissivity [–] λ wavelength [µm] бε Uncertainty in ε [–] c speed of light 3·108 ms−1 dp particle diameter [µm] h Plancks constant 6.63·10−34 m2 kg s−1 IBB blackbody intensity [W m−2 K−4] k imaginary part of m [–] kB Boltzmann constant 1,38·10−23 m2 kg s−2 K−1 dp = 115 µm. To prevent the particles from further thermochemical changes, the temperature of the experiments was varied around 455 K. The measured spectral emissivity in this temperature range was between εp(λ) = 0.55 and 1 with characteristic dips in the ranges λ = 3.3–5 and below 3 µm, the latter being more pronounced for the smaller particle size range. Bhattacharya and Wall [21] investigated the effects of coal rank, particle size, and conversion and devolatilization conditions. They produced chars at high temperature and heating rate and took samples thereof to investigate char emissivity at a temperature of 473 K in the wave length range 2–12 µm. Their results show that each of the mentioned factors affects the emissivity of coal and char particles at the given temperature. The reported coal spectra show emissivities significantly below unity at λ < 5 µm, and also at longer wavelengths emissivity never exceeds εp(λ) = 0.87. However, samples of partially burned coal showed an increase of emissivity in the short wavelength range. Increasing devolatilization led to increasingly grey radiation characteristics. According to their measurements preceding devolatilization leads to an increase in total (spectrally averaged) emissivity from εp ≈ 0.7 to εp ≈ 0.8. The emissivity was measured at 473 K, while the extent of devolatilization was controlled by varying the devolatilization temperature in the range 473–1273 K. Solomon et al. [22] performed measurements combining FTIR and emission/transmission spectroscopy for coal and char particle streaks in a heated wall reactor in a particle temperature range Tp = 670–1050 K. Their results are in agreement with Baxter et al., but they found some differences and stronger deviations from grey emissivity: Studying the spectral emissivity of coals of various rank, a clear trend to radiative intensities smaller than expected for an assumed εp = 0.9 was found for λ < 6.3 µm for all coals with less than 90% C. Increasing the temperature to 780 K caused a decrease in emissivity to εp(λ < 3 µm) < 0.2. Only in the range of λ = 6.3–10 µm grey radiation behaviour with εp ∼ 0.9 was observed. At Tp ∼ 1050 K a significant reduction of emissivity in the regions of dominating absorption bands of hydrocarbons being characteristic for coals or young char was observed. Finally a spectrally averaged εp = 0.73 at Tp = 990 K was reported, with uncertainties being Δεp = 0.07 and ΔTp = 30 K. RegoBarcena et al. [23] performed measurements with a 64-channel spectrometer, but only the signal at λ = 3.95 µm was used. The measurement was carried out in a coal-fired boiler, taking radiation signals from the luminous coal flame in front of the burners. Their measurements show a strong decrease from εp = 0.7 at 1400 K to εp ∼ 0.3 at 1600 K, which is significantly lower than the values reported at lower temperature mentioned above. Graeser and Schiemann [24–26] measured the temperature dependent emissivity of single burning pulverized coal char particles in-flight. The experimental setup consisted of an entrained flow reactor and a pyrometer/spectrometer setup, measuring particle temperature by means of two-colour pyrometry in the visible spectral range and using a fibre spectrometer (resolved emissivity at λ = 1.25–2.25 µm, NIR) and an InSb detector (integral emissivity at λ = 2.4–5.5 µm, MIR). Two coals, a high volatile bituminous (hvb) coal

complex refractive index [–] Real part of n [–] temperature [K] absorption efficiency [–] particle temperature [K] unreacted fraction of carbon [–] Fourier transformation infrared spectroscopy coal High-volatile bituminous coal Medium-Infrared Near-Infrared pulverized coal Scanning electron microscopy

and a lignite, were investigated in the temperature window 1800–2500 K (hvb) [24,25] and 2200–2500 K (lignite) [24], respectively. In the upper temperature range, for the hvb coal char (carbon content 90%) a spectrally averaged εp ≈ 0.4 was measured, while for the lignite (carbon content 64%) εp ≈ 0.35 was reported [24] with a steeper rising gradient from the NIR to the MIR compared to the hvb char. In the lower temperature range of 1800–2000 K, only measurements of the hvb char are available, with an average εp ≈ 0.5. A slight decrease of the spectrally averaged emissivity in the NIR and MIR with the ash content of the char increasing from 12.4 to 16.3%, corresponding to 49–62% conversion, has been reported. As the ash of the coal sample was rather silicon rich and contained only little iron [25], the observed reduction suits to the common knowledge on ash emissivity. Summarizing, available data in literature covers many aspects of pulverized coal combustion. Coals of different rank have been investigated with different techniques, at different conversion levels and at different temperature. Although several factors are identified to have (potential) influence on char emissivity, the current data base does not allow to derive a universal correlation for spectrally resolved char emissivity considering all known factors. However, the following sections provide deeper insight into the physical theory which can be used to calculate spectrally averaged char emissivity depending on temperature and to derive an empirical law for its value. 2.2. Ash emissivity Ash optical properties have always been of interest for combustion systems and pc boilers in particular for different reasons. A large number of studies investigated the properties of ash deposits on furnace walls [27–30] and clarified their influence on heat transfer [10,13,31]. Fly ash, which originates from completely burned coal particles, is an important factor not only for ash deposit formation, but also for radiative heat transfer, which has been investigated e.g. in [10,32,33]. However, this study focusses on the emissivity of char particles, which are larger than fly ash and contain a non-negligible fraction of carbon. Given the fact that this carbon content decreases during burnout the relative ash content increases simultaneously. It is supposed that ash is accumulating on the char surface to a certain extent during combustion [34–36]. The aspects of physical structure and chemical composition have been investigated in more detail for ash than for coal char, a fact which will help to assess the necessary assumptions for char emissivity in Section 3.2. Real coal ashes have been investigated as well as synthetic mixtures which represent characteristic ash constituents [37–43]. Global trends of emissivity have been observed for several factors. The total (spectrally averaged) emissivity decreases with increasing temperature [38,41,43]. Its calculation requires weighting the spectral emissivity ( , T ) with the Planck intensity IBB ( , T ) , 2

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(T) =

3. Emissivity model

( , T ) IBB ( , T ) d

1 2 1

As literature reveals the dependency of char and ash emissivity on temperature is evident. The temperature of burning char particles can easily exceed 1500 K in the near-burner region of a furnace. Toporov et al. [52] reported particle temperatures up to 1800 K in a 100 kWth laboratory swirl burner under oxyfuel conditions. Similar values were reported by Butler et al. [53] in a 80 MWth utility boiler. Rego-Barcena et al. [23] measured particle temperatures up to 1600 K in an industrial system. In laboratory experiments, char particle temperatures in the range from 2000 to 3150 K are reported [15,24,54,55]. Comparing the experimental data on char particle emissivity (Section 2.1) to the initially summarized constant values of εp = 0.7–0.9 or the conversion dependent expression yielding εp = 0.6–1 for ash and unreacted char, it appears that these values are too high.

. IBB ( , T ) d

(2)

At shorter wavelengths (λ < 6 µm) the spectral emissivity typically decreases for minerals [27,43]. In combination with spectral averaging according to Eq. (2) this causes the observed reduction in total emissivity as the peak of the Planck intensity is shifted to shorter wavelengths with increasing temperature (e.g. from λmax = 5.96 µm (T = 500 K) to λmax = 1.49 µm (T = 2000 K) as described by Wien’s displacement law. An influencing factor for spectral and total emissivity is the chemical composition of the ash. Fe2O3 is sometimes called a “colouring agent” causing high emissivity in ashes [41,44]. Another component known to increase the emissivity is carbon [40]. A physical factor is particle size, which has been studied experimentally [40,41,43] and theoretically by consideration of Mie theory [27,41,45], showing that increasing particle size causes increasing emissivity. The latter investigations showed that with decreasing particle size the effect of chemical properties, which basically affect the complex index of refraction, becomes less important for single particles. Experimental results, which demonstrate both effects, are summarized in Figs. 1 and 2. Two exemplary results obtained from the test rig described in [43] are plotted. In this test rig, powdery samples are filled in a crucible and heated. At defined temperatures, the thermal spectrum of the powder layer is recorded and compared to a black body spectrum at the same temperature. A South African coal ash has been investigated at three different temperatures (Fig. 1). The spectral emissivity clearly shows the decreasing trend to shorter wavelengths, with ε = 0.2–0.3 below 2 µm. The total emissivity of three different, typical constituents (Fe2O3, SiO2 and CaCO3 which completely decomposes to CaO at T > 1173 K [43]) of coal ash is also depicted for two different particle size distributions (Fig. 2). It is clearly visible, that layers of particles with dp < 32 µm have a significantly lower emissivity than those of the size fraction 125 µm < dp < 160 µm. This is evident for all materials investigated. This short summary on findings on ash layers adds three facts from experimental investigations of ash deposits to the further consideration: i) The emissivity is typically low in the NIR range with a decrease to the lowest values in the wavelength range around 2 µm, which is the most important spectral range for burning char particles. ii) The spectrally averaged emissivity for different ash constituents and for real coal ashes decreases with increasing temperature. iii) Finally the emissivity decreases with particle size for a given material at constant temperature. Combining these facts a trend for the effect of ash enrichment on (partially) converted char particles can be deduced. During burnout, ash layers on the char surface are typically not homogeneous. In fact, the ash layers formed are of irregular structure [47] constituting from microscopic ash “grains”, as mentioned by Niu et al. [48]. Furthermore parts of the ash are supposed to diffuse back into the char matrix [47–49], which makes the exact determination of ash structure and composition on a partially reacted particles’ surface difficult. This means that potential ash deposits on the surface of char particles will contribute to the total emissivity in the same way as smaller particles contribute to the total emissivity of deposits, where smaller grains have been identified to reduce the total emissivity. Fly ash particles have been characterized to be in the size of 10 µm [32,50,51] with irregular surface structure, indicating that coalescing grains are responsible for fly ash particle formation. Fly ash is responsible for scattering of thermal radiation [32,33], which might be influenced by the particles structure, but its active contribution to the radiative heat flux due to emission of thermal radiation is typically small, therefore this topic is not investigated in further detail here.

3.1. Numerical assessment of Planck mean emissivity For ash deposits and particles being suspended in the gas phase (fly ash and coal) the calculation of optical properties is possible based on Mie’s theory (see e.g. [56]). A detailed explanation of this theory and its application on coal, char and fly ash particles is given in [57], therefore only a brief description of the methodology is given in the following including the most necessary equations: According to Kirchhoff’s law the emissivity of a particle equals its absorptance,

( )=

(3)

( )

The determination of the absorptance directly leads to the absorption efficiency Qabs , as = Qabs Cp , where Cp is the particle concentration per cross section of the control volume (unity for a single particle):

Qabs (T , m) =

Qabs (T , m , x )· IBB ( , T ) d IBB ( , T ) d

(4)

The absorption efficiency is a function of the scattering parameter or Mie parameter x = dp 1, which links the absorption efficiency to the wavelength of radiation, and the complex refractive index m. The calculation of Qabs (T , m , x ) requires significantly more math work, which shall not be repeated in this publication, instead the reader is referred to chapter 2 in [57] or the work of Modest [56]. Theoretical studies based on Mie-theory show similar trends as being observed in experiments. Greff-rath et al. [41] calculated the Planck mean emissivity of particles by Mie theory for particles at different temperatures and particle diameters and compared the influence of different complex indices of refraction. In the calculation procedure, the spectrally resolved emissivity was averaged over wavelength and weighted by Planck’s law. A Sauter mean diameter was applied to avoid

Fig. 1. Spectral emissivity of a South African coal ash [46], measured at three different temperatures with the test rig described in [43]. 3

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depending on particle temperature and is shown in Fig. 3. The corresponding complex refractive indices m, which were used to calculate emissivity, are listed in Table 1. The values M1-M4 were derived by Goodwin and Mitchner [44] (GM) from four different synthetic coal slags measured at room temperature and resemble an approximation for a wavelength-independent m. Further wavelength-dependent approximations are derived from this (GM) and other data (GM combined) on spectrally resolved measurements of m. The influences of carbon (C in ash) and Fe2O3 (Iron) was considered based on specific data. The results of the calculations show, that even for particles with a diameter of 200 µm, only complex refractive indices from the Goodwin and Mitchner data set ([44], ash combined, ash GM and ash GM/Carbon in ash in Fig. 3/Table 1) cause an emissivity above 0.9 for all temperatures, which is far away from the predictions given from emissivity measurements. For all other complex refractive indices a clear decrease in total emissivity is visible; at 2200 K the result varies between 0.2 and 0.55. Reducing the particle diameter to 20 µm causes a more distinct trend. All models, except the Ash models with combined complex indices of refraction based on carbon in ash and Goodwin/Mitchner values, show an emissivity dropping from εp = 0.3–0.4 below 0.2 at 2200 K. Only one data point, resulting from the complex index of refraction for Ash GM with C in ash, reports an emissivity above 1 at lowest temperature. This seems unreasonable, but can be a result of the application of Mie theory for particle radiation interaction. A comment on this is given in Appendix B. With another step down to a particle diameter of 2 µm, the total emissivity at low temperature is in the range between εp = 0.05 and 0.7 at 700 K. The three curves based on calculations using complex refractive indices for ash are highest again. Again, the general trend of decreasing total emissivity with increasing temperature is clearly visible. The same trend from highest to lowest

Fig. 2. Exemplary total emissivities of three typical constituents of coal ashes (0 < dp < 32 µm, 125 µm < dp < 160 µm), data extracted from [43].

averaging over particle size distribution. In a numerical experiment by Gronarz et al. [58], the absorptivity (and thus the emissivity) of particles during burnout was modelled with a shell model consisting of a layer of ash surrounding a core of coal based on Mie theory for layered particles. Therein, Planck mean properties obtained in a similar way to [49] were investigated. It is important to note that data for the complex index of refraction for particles undergoing burnout is not available. In literature, several refractive indexes are available for char particles and ash, which forms on the char surface with increasing burnout. Based on these literature values, the spectrally averaged emissivity of particles with diameters of 2, 20 and 200 µm has been calculated

Fig. 3. Single particle emissivity predicted by Mie calculations in accordance to [58]. Refractive indices are listed in Table 1. 4

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Table 1 Complex refractive indices used for the calculations presented in Fig. 3. m = n + ik

M1 M2 M3 M4 Ash combined

Ash GM

Reference

n

k

1.5 1.5 1.5 1.5 1.5 1.5 − 0.35(λ − 6) 0.8 + 0.5(λ − 8) 2.3 + 0.5(λ − 11) 1.8 1.5 1.5 − 0.35(λ − 6) 0.8 + 0.5(λ − 8) 2.3 + 0.5(λ − 11) 1.8

5·10−5i 1·10−4i 2·10−4i 3·10−4i −1.5·10‐3 + 7.5·10‐3λ 0.0119 + 2.33·10‐3λ −3.39 + 0.488λ Ref. [44]

λ < 2.6 2.6 < λ < 7 7<λ <9 λ>9

[59] reports this data as a combination of refractive indices reported in [44,60,61] derived from different coal ashes

10‐3 ‐ 1.86·10‐3λ −2.45·10‐4 + 6.3·10‐4λ 7·10‐4 −9.02·10 − 3+ 2.43·10‐3λ −3.46 + 0.496λ 1 8.7 − 0.7λ 0.3 0.01 −3.2·10−3 + 6.6·10−3 λ −1.18 + 0.243 λ 1 8.7 − 0.7 λ 0.3 1e−3 − 1.86e−3λ −2.45e−4 + 6.3e−4λ 7e−4 −9.02e−3 + 2.43e−3λ −3.46 + 0.496λ 1 8.7 − 0.7λ 0.3

λ < 0.5 0.5 < λ < 1.5 1.5 < λ < 4 4<λ <7 7<λ <9 9 < λ < 11 11 < λ < 12 λ > 12 λ<2 2<λ <5 5<λ <9 9 < λ < 11 11 < λ < 12 λ > 12 λ < 0.5 0.5 < λ < 1.5 1.5 < λ < 4 4<λ <7 7<λ <9 9 < λ < 11 11 < λ < 12 λ > 12

[59] reports this as parametrised function solely based on data from [44]

Ash GM/C in Ash

1.5

Ash GM/Iron

1.51.5 − 0.35 (λ − 6)0.8 + 0.5 (λ − 8)2.3 − 0.5 (λ − 11) 1.8

Neubronner BK Neubronner SK

m = m(λ)

λ<6 6 < λ <8 8 < λ < 11 11 < λ < 12 λ > 12 λ<6 6 < λ <8 8 < λ < 11 11 < λ < 12 λ > 12

λ<6 6<λ<8 8 < λ < 11 11 < λ < 12 λ > 12

[44], derived for coal slags at λ ≤ 4 µm, m(λ > 4 µm) = 1.5 + 1·10−2i

l

values for different complex refractive indices remains, but for these small particles even the highest total emissivity is below 0.4 at 2200 K curves based on calculations using complex refractive indices for ash are highest again. Again, the general trend of decreasing total emissivity with increasing temperature is clearly visible. The same trend from highest to lowest values for different complex refractive indices remains, but for these small particles even the highest total emissivity is below 0.4 at 2200 K. As the results from Mie-calculations show, the total emissivity in the temperature range of burning char particles is lower than unity, and even εchar = 0.6, as presumed in Eq. (1), is above the predicted values considering most complex refractive indices available in literature, when a char particle temperature above 1500 K and a char diameter below 200 µm are considered. In Section 2.2, the influence of particle size on emissivity of ash particle layers has been described, where the size of the radiating surface is much larger than that of the single particles. Clear evidence is given that layers of smaller particles have a lower total emissivity than those formed by larger particles. The surface of (partially reacted) char particles in pulverized fuel environments resembles a coarse structure with macroscopic pores typically. Images of such particles, taken by scanning electron microscopy (SEM), are manifold in literature. Two examples are given in Fig. 4. It has to be mentioned that different coals may exhibit different types of porosity, e.g. cenospheres, sponge-like structures, or surface coverage with ash particles [64]. All these char structures have in common, that the characteristic size of the porous structure is in the size range 1–10 µm, with particles being in the range of 10–100 µm. Comparing this to the findings for ash particles and layers thereof discussed above, the influence of the surface structure on char emissivity becomes evident.

values from [62] using data from [61] (λ < 5 µm) and [44] (λ > 5 µm)

data from [44], only samples with Fe2O3 content, parameters derived in [62]

[63], values for lignite (BK) and hard coal (SK) ashes, wavelengthdependent, approximations in [59]

3.2. Derivation of a generalized emissivity model The experimental findings reported in Section 2.1 and the theoretical analysis of emissivity boundaries in the previous section motivate for a temperature dependent description of spectrally averaged char emissivity. Many numerical studies rely on such data and also experimentalists need this parameter to quantify their results. Based on the experimental data being available in the temperature range above 2000 K from this group [10,24–26], a general expression for the spectrally averaged emissivity of pulverized coal char particles in dependence of particle temperature can be derived. The data sets in the corresponding references are based on the measurement of temperature, diameter and thermal (NIR and MIR) radiation of single burning char particles. Statistically significant numbers of particles were recorded for each set of experimental parameters (coal rank, atmosphere, residence time/conversion) reducing the impact of particle to particle variations and data scattering. A complete list of data being available in the high temperature regime is listed in Appendix A. In a first step, the experimental data need to be extrapolated, as the test rig has already been used in afore mentioned publications delivers the emissivity spectrally resolved in the NIR range (1.25–2.25 µm) and integral in the MIR range (2.4–5.5 µm). Before providing detailed information on the extrapolation scheme, a short calculation of the fraction of thermal radiation which has been covered experimentally in the given spectral range and its significance for the spectrally averaged emissivity is presented. The fraction of measured thermal radiation can be calculated by integration of Planck’s law

IBB ( , T ) =

2 hc 2 5

1 e hc /

kb T

1

(5)

for Δλ = 1.25–5.5 µm and comparing it to the intensity of the entire 5

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Fig. 4. SEM images of chars: (top left) extracted from pulverized coal drop tube experiments at 1300 °C: 53–75 µm chars produced which resemble industrial pulverized coal char [65], (top right) 90–125 µm chars produced in a 30% O2/70 CO2 atmosphere [66]. Bottom right: 160–200 µm lignite char produced in a 25% O2/56% CO2/19% H2O atmosphere ([24], image unpublished). The porous surface structure is clearly visible. Top left: Adopted with permission from [65]. Copyright 2009 American Chemical Society. Top right: Reprinted from [66] Copyright 2016, with permission from Elsevier.

black body spectrum. The result is plotted in Fig. 5. As the diagram shows, the fraction of thermal radiation being detected by the two detectors is in the range of 66–82% for temperatures between 1500 and 2500 K. Additionally the peak intensity max = 2, 9 mmK/ T falls into the detectors’ sensitivity range according to Wien’s displacement law. To the authors’ best knowledge high temperature emissivity data is not available in the missing wave length bands (λ < 1.25 µm and λ > 1.25 µm). Therefore, the effect of different extrapolation schemes to close the gaps to higher and lower wavelengths was tested based on several assumptions. The extrapolation range has been limited to λ = 0–10 µm because 97% of the complete thermal black body radiation fall into this interval at 1500 K with increasing tendency for higher temperature. Elements of the procedure are depicted in Fig. 6. In a) the spectrally resolved NIR and integral MIR measurements of a high volatile bituminous (hvb) coal (Tp,avg = 2423 K, dp,avg = 168 µm, [24]) are shown. While the NIR spectrum shows a certain trend, the MIR data represent an integral value, which is indicated by the shaded area covering λ = 2.4–5.5 µm. The NIR trend is approximated by a linear relation up to 2.25 µm (black dashed line) in (b). For the MIR data the integral value of ε (shaded area) is kept constant, but the slope is calculated to match εNIR(2.25 µm) resulting from the linear fit in the NIR. Note that the slope of the resulting linear functions for NIR and MIR is not necessarily equal as only εNIR(2.25 µm) is considered to calculate

the slope of εMIR(λ), the apparent agreement in the figure is a result of the data used. The small green line shows the interpolation between NIR and MIR. Three different cases are used to determine the influence of the extrapolation to shorter and longer wavelengths (c–e). The highest results when it is assumed that the spectral emissivity shows an increase to unity for λ approaching the range limits λ = 10 and 0 µm (c). It has to be noted that the increase to longer wavelength is in agreement with general expectations, which are known from other materials (e.g. the ash example in Fig. 1), but reaching ε = 1 means a

Fig. 5. Black body radiation intensity fraction I for Δλ = 1.25–5.5 µm compared to full spectrum (Itotal). 6

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Fig. 6. Extrapolation strategies to calculate the global emissivity from NIR and MIR emissivity data. a) and b) are equal for all approaches, but c)–e) are applied alternatively. Exemplary data taken from a hvb coal with Tp,avg = 2423 K, dp,avg = 168 µm, [24]. Dashed lines: linear fit for λ = 1.25–2.25 µm, extrapolation below 1.25 and above 5.5 µm.

7

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clear overestimation for most surfaces. Case d) calculates a lower , as linear interpolation is used for wavelengths shorter than 1.25 and longer than 5.5 µm. The extrapolation for λ > 5.5 µm was varied between ε = 0.6 and 1 at λ = 10 µm, but the total influence was below Δε = 0.02, so ε(10 µm) = 0.8 was used as target value. For λ < 1.25 µm the linear trend from the NIR data was kept down to λ = 0 µm. As third case, the assumption of decreasing emissivity for smaller and larger λ has been tested (e). Emissivity decreasing to zero was assumed for λ = 0.625 and 10 µm, which results in the lowest calculated emissivity using Eq. (2). The resulting global emissivity from the example is = 0.33, 0.35 and 0.42 for cases c), d) and e), respectively. A clear trend was observed for minimum and maximum global emissivity values to occur for cases c) and e), the maximum deviation was Δεmin/max = 0.09, but from 16 samples which are listed in = 0.06 or less. Summarizing the Appendix A 14 have a deviation of extrapolation effect is recognizable, but below the experimental scatter being reported in [24–26]. In the following the linear fit (d) is chosen as this represents the median value. Based on data from this group [10,24–26,67], for char particle temperatures in the range 2000–2500 K, and experimental data on emissivity presented by Solomon et al. [22], Bhattacharya and Wall [21] and Baxter et al. [20] in the temperature range below 800 K a generalized correlation between char particle emissivity and temperature seems obvious. Fig. 7 shows available data points and a fitted trend line. Data has been grouped into non-reacting particle data for T < 800 K and burning particle data in the high temperature regime. The non-reacting particle data contains both char data from previously pyrolyzed samples [21,22] as well as raw coal data [20], while burning particles have consistently been produced by injection of raw coal particles into the burner environment described in the corresponding literature. The involved reader will note that earlier versions of this diagram [24,25] contained data from Rego-Barcena et al. [23], which covers the temperature range 1400–1600 K. However, these measurements were performed in a narrow radiation band at 3.95 µm, such that an extrapolation to calculate a total emissivity seems less reasonable. The clear trend to decreasing total emissivity, which has been reported in previous work [25], is visible from the experimental data, and motivates fitting of a temperature-dependent emissivity correlation, which results in

(Tp) =

1.86 10 4Tp + 0.8.

3.3. Discussion The correlation between char total emissivity and temperature needs to be discussed from several points of view. The application of a linear fit is questionable when extrapolation to larger temperatures is needed, e.g. for pressurized oxy-fuel combustion [68]. The trend to low emissivity values at higher temperature is not surprising. As the weighting function for total emissivity calculations (Eq. (2)) shifts the importance of spectral fractions to shorter wavelengths, the typical trend of increasing emissivity with wavelength causes this behaviour. The fact that surface properties, which are represented by roughness or pore size for coal chars, have been identified as one of the influencing factors for surface emissivity experimentally [41,43], with smaller structures causing reduced emissivity, and sintering as a factor which causes an emissivity increase [41,69,70], supports the idea of relatively low total emissivity values for porous char particles. Considering the typical total emissivity of coal ash in dependence on temperature (ash components in Fig. 1 (right), other examples in [31,41]) and the theoretical predictions for temperature dependent total emissivity summarized in Section 3.1, the exponential expression might be more feasible. Calculating the difference between εlin and εexp, the largest deviation is 0.03 at 1380 K. This difference appears negligible when the experimental uncertainties, which are in the range of 0.15 , are taken into account. The generalized expression in Fig. 7 does not include effects of coal rank, burnout level, porosity or chemical composition of the ash. While chemical composition is known to affect the emissivity of surfaces like ash and slag deposits, for coal char no clear trend is reported in literature. First experimental results comparing coal with natural mineral content and demineralized samples with the same origin show an almost negligible effect of coal minerals on emissivity in the early char combustion phase, at least for the investigated sample [26]. The current data base does not provide strong evidence for the influence of coal rank. Data for the intermediate temperature range seems to be lacking in Fig. 7. Experimental data in this range would be a final support for the given emissivity relation, however the general data base in combination with theoretical considerations and comparison to ash emissivity behaviour is a significant basis for the current state.

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4. Conclusions

Using a linear function has no theoretical background in this case, the data set would allow for an exponential interpolation 4 ( (Tp) = 0.837 e 3.54 10 Tp ), but the clear difference in the regression 2 2 = 0.772 ) is = 0.834 shows better agreement than Rexp coefficient (Rlin the main reason to keep the linear regression.

Char particle emissivity has been discussed with focus on the influence of char particle temperature. Available experimental data on the emissivity of single char particles provides a basis for a generalized expression which describes the temperature dependency of char

Fig. 7. Global char emissivity plotted against (average) particle temperature for T < 800 K [20–22] and T ≥ 2000 K from authors of this work [10,24–26,67]. 8

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M. Schiemann, et al.

emissivity. Mie theory as method of choice has been used to calculate the total (spectrally averaged) emissivity of particles in specific sizes. The results show, that total emissivity decreases with decreasing particle size and increasing temperature. This prediction from theory has been confirmed experimentally for ash materials, and the conclusion is drawn that surface structure needs to be considered when particle surface emissivity is of interest. Based on literature data a simple linear correlation between particle

temperature and emissivity has been derived. It represents the total emissivity of char in the temperature range from 300 to 2500 K, thus it covers the most relevant temperature range for technical applications. Acknowledgement Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Projektnummer 215035359 – TRR 129.

Appendix A. Emissivity experimental data See Table 2

Table 2 Complete list of references containing char emissivity data, carbon and ash content at measurement, average particle temperature and diameter, resulting average emissivity according to extrapolation assumptions.

#

Ref.

rank

C [%wf]

Ash [% wf]

Tp,avg [K]

dp,avg [µm]

εlin

εmin

εunity

[24] [67]

lignite hvb hvb

[26]

hvb*

64.2 90.1 77.8 85.7 86.8 88.4 79.5 80.3 83.9 #

33.9 8.9 16.3 8.2 7.0 7.8 7.8 8.4 13.1

[25]

hvb

[10]

lignite

81.1 82.2 82.3 83.7 85.1 #

16.3 14.7 12.4 13.9 13.1

2444 2423 2269 2250 2169 2207 2091 2048 2056 2107 2047 2022 2017 2058 2050 2036 2183

164 168 145 196 165 189 139 154 136 135 140 158 153 160 163 160 152

0.28 0.35 0.23 0.27 0.52 0.45 0.45 0.27 0.37 0.40 0.37 0.38 0.41 0.48 0.43 0.44 0.32

0.27 0.33 0.21 0.27 0.50 0.44 0.44 0.26 0.36 0.40 0.36 0.37 0.39 0.46 0.41 0.43 0.32

0.35 0.42 0.28 0.33 0.56 0.50 0.49 0.31 0.40 0.43 0.40 0.41 0.44 0.51 0.46 0.47 0.37

No solid samples for composition analyses; All hvb measurements are from one coal sample. * Sample leached to reduce mineral content.

Appendix B. Emissivity in the context of particles with dimensions close to wavelength In the context of the present investigation, a problem seems to arise, since absorption efficiency factors and thereby the absorptance can be above unity in some cases. This would, according to Kirchhoff’s law, lead to an emissivity larger than one, which is counter-intuitive but possible for the same reasons that the absorptance is larger than unity. Bohren and Huffman [71] commented on that problem. They stated that Kirchhoff’s definition of a black body includes the condition that a black body absorbs all radiation that is incident on it. This concept is derived from geometrical optics which is not valid for dimensions comparable or smaller than the wavelength, a restriction well known to Kirchhoff. For particles with a diameter in the size of the wavelength, this simply means that Kirchhoff's law does in principle not apply. However, the case of particles with an absorption efficiency factor greater than one is very rare and the application of Kirchhoff’s law for particles in coal combustion is well accepted in the available coal combustion literature. Therefore Kirchhoff’s law is applied in the present study for small particles. Another assumption made by Kirchhoff is that the body under consideration needs to be opaque, such that all radiation incident on it can be absorbed during its path through the body. This condition is related strongly to the conditions stated in the previous paragraph and is not necessarily met for small particles, radiation can enter the body on one side and exit it again on the opposite side. As a result, the effective absorptivity is smaller than it would be for a larger particle. This effect is accounted for the emissivities calculated from Mie theory. Hanssen and Campbell [72] derived an emissive power distribution for very small particles that are emitting radiation in high excitation states which is based on the energy of the particle instead of the temperature. Their formulation has similarities to Planck’s law, although serious distortions can be found in the visible and ultraviolet region of the spectrum as well as the low energy (long wavelength) region. In traditional textbooks on thermal radiation, e.g. the ones by Siegel and Howell [73] and Modest [56] no comment on this problem can be found, in the textbook by Michchenko, Travis and Lacis [74] the authors do not address the problem as well. Only in the book by Bohren and Huffman [71] the problem is addressed briefly.

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