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Modeling of particle radiative properties in coal combustion depending on burnout

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Abstract

In the present study, absorption and scattering efficiencies as well as the scattering phase function of a cloud of coal particles are described as function of the particle combustion progress. Mie theory for coated particles is applied as mathematical model. The scattering and absorption properties are determined by several parameters: size distribution, spectral distribution of incident radiation and spectral index of refraction of the particles. A study to determine the influence of each parameter is performed, finding that the largest effect is due to the refractive index, followed by the effect of size distribution. The influence of the incident radiation profile is negligible. As a part of this study, the possibility of applying a constant index of refraction is investigated. Finally, scattering and absorption efficiencies as well as the phase function are presented as a function of burnout with the presented model and the results are discussed.

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Abbreviations

a :

Curvature parameter

E :

Distribution of black body radiation, normalized

T :

Temperature (K)

B :

Burnout

D :

Particle size (μm)

s :

Spread factor in particle size distribution

ΔD :

Local particle interval(μm)

I :

Radiative intensity (W/m2)

k :

Imaginary part of index of refraction (m)

m :

Complex index of refraction

n :

Real part of the index of refraction (m)

P(D):

Particle size distrib. (propability density formulation)

Q :

Efficiency

x :

Size parameter \(x =\pi {D}/{\lambda }\)

λ :

Wavelength (μm)

Δλ :

Local wavelength interval (μm)

Φ :

Scattering phase function

θ :

Azimuthal angle (rad)

φ :

Polar angle (rad)

ω :

Solid angle (sr)

Δω :

Solid angle interval (sr)

Δθ :

Interval in θ (rad)

T:

At a certain temperature

a:

Ash

abs:

Absorption

sca:

Scattering

c:

Coal

Δω :

In a solid angle interval

Δφ :

In a φ interval

Δθ :

In a θ interval

inner:

Indicating inner diameter of combusting particle

λ :

Spectral/wavelength dependent

max:

Maximal value

Mie:

According to Mie theory

min:

Minimal value

m:

Mean

–:

Averaged or integrated value

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Acknowledgments

This work has been financed by the German Research Foundation (DFG) within the framework of the SFB/Transregio 129 “Oxyflame”. The authors also thank Mischka Laemmerhold for his contributions to this work.

Author information

Authors and Affiliations

  1. Institute of Heat and Mass Transfer, RWTH Aachen University, Augustinerbach 6, 52056, Aachen, Germany

    Tim Gronarz, Martin Habermehl & Reinhold Kneer

Authors
  1. Tim Gronarz
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  2. Martin Habermehl
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  3. Reinhold Kneer
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Corresponding author

Correspondence to Tim Gronarz.

Appendix

Appendix

1.1 Calculation of a representative index of refraction

The calculation of an effective representative constant refractive index is possible to some degree. However, these refractive indices need to be handled with care and have to be identified for each case specifically. A different weighting by the incident radiation or particle size distribution results in a different representative refractive index. Therefore, this representative refractive index is not a pure material property anymore but a parameter that depends on other variables (such as size distribution, incident radiation profile) as well.

By minimizing the differences between scattering phase function calculated with a spectrally resolved and a constant refractive index, a representative constant refractive index was found. The best fit yielded a value of m = 1.52 + i0.0005 for the case of ash. Note that the absorptive index (the imaginary part, k) is much smaller than the average value applied by several authors. The scattering efficiency with this representative value was then determined to \(Q_{{\rm sca}} = 2.275\) compared to a—in terms of the method presented here—correct value of \(Q_{{\rm sca,\lambda }} = 2.217\). The absorption efficiency yielded \(Q_{{\rm abs}} = 0.0316\), compared to a “correct” value \(Q_{{\rm abs,\lambda }} = 0.0567\).

The efficiencies are matched well. Nevertheless, the need for such representative values is questionable, if spectral values for m λ are available. Additionally, it is not possible to calculate a representative value m, if no spectral m λ exists. Finally, series of experiments need to be performed to gain more detailed information on radiative properties for a broad range of coals and ashes.

1.2 Weighting functions applied in Eq. 6

Weighting by blackbody radiation (or intensity profile):

$$E_{T,\Delta \lambda } = \int _{\Delta \lambda }E(T,\lambda ) \, d \lambda$$
(8)

Weighting by particle size distribution

$$P_{\Delta D} = \int _{\Delta D}P(D) \, dD$$
(9)

Note that the integral of both functions, P and E T are equal to 1, when integrated from 0 to ∞.

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Gronarz, T., Habermehl, M. & Kneer, R. Modeling of particle radiative properties in coal combustion depending on burnout. Heat Mass Transfer 53, 1225–1235 (2017). https://doi.org/10.1007/s00231-016-1896-0

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