Absorption coefficient regression ... - School of Engineering

Powder Technology 265 (2014) 76–82

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Absorption coefficient regression scheme for splitting radiative heatsources across phases in gas–particulate mixtures

Jian Cai a, Michael F. Modest b,⁎a School of Engineering, University of California, Merced, United Statesb Shaffer and George Professor of Engineering, University of California, Merced, 5200 North Lake Road, Merced, CA 95343, United States

⁎ Corresponding author.E-mail address: [email protected] (M.F. Modes

0032-5910/$ – see front matter © 2014 Elsevier B.V. All rihttp://dx.doi.org/10.1016/j.powtec.2014.01.026

a b s t r a c t

Available online 13 January 2014

Keywords:Radiative heat transferk-distribution methodsNon-gray particulate radiationGas-particulate radiationRegression

In a gas–particle mixture particles of different makes and sizes may have different local temperatures andare, therefore, modeled with separate energy equations. If the important radiative heat transfer contribu-tions are determined with sophisticated spectral models such as k-distributions, spectral information islost during the reordering process, making it difficult to separate the resulting radiative heat source acrossthe various energy equations. It is shown here how radiative sources can be determined accurately with thefull-spectrum k-distribution assembled from a narrowband database and combinedwith particle propertieson a narrowband level. Proper distribution of the radiative source across the various phases is achievedusing regression analysis. Numerical examples demonstrate the accuracy of the model.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

Thermal radiation is one of themajor heat transfermodes in fluidizedbed and pulverized coal combustors, due to strong particle radiation athigh temperature [1,2]. A high-fidelity radiation model is necessary toimprove numerical predictions of the overall heat transfer.

In gas–particle flows, polydisperse particles are grouped into phasesby their size and material because of distinguishing characteristics ofmomentum transfer [3]. Particles of each phase also have individual dy-namical responses to heat transfer, which requires a dedicated energyequation and each energy equation requires an individual radiativeheat source. On the other hand, the radiation field is described by asingle radiative transfer equation (RTE), which determines the totalradiative heat source. Therefore, an accurate scheme is needed to splitthe total heat source across phases in gas–particle mixtures.

Solving the RTE requires spectralmodels for radiatively participatinggases commonly occurring in coal combustion, such as CO2, CO andH2O.Their absorption coefficients have strong spectral dependency. Com-mon particles in coal combustion include soot, coal, char, ash and lime-stone [4]. Their radiative properties are not gray even for gray complexrefractive indices due to dependency on the size parameter [5].

While the line-by-line (LBL) spectral model [6] provides the mostaccurate results of radiative heat transfer, its large computationaldemands prevent its use in practical engineering applications. Recently,k-distribution methods have been developed for gases [7–9]. A k-distribution is a spectrally reordered absorption coefficient over anarrow-band or the full spectrum. Using k-distributions, radiative heat

t).

ghts reserved.

transfer can be solved with excellent accuracy, but at a small fractionof the effort compared to LBL calculations [9].

Although the total heat source can be accurately predicted usingk-distribution methods, the reordering process makes splitting thetotal heat source across phases challenging because of the loss of spec-tral information. For example as shown in Fig. 2, for a nongray solidgas–particle mixture, multiple solid absorption coefficient values maycorrespond to one total absorption coefficient value. After reordering,due to the loss of spectral information, the individual solid phase absorp-tion of radiation is indeterminate.

In this paper a regression scheme is introduced to extend the validityof the k-distribution scheme to multi-phase gas–particulate mixtureswith multiple emission temperatures, and to accurately distribute theresulting radiative source across the phases. The new method is testedwith several examples, demonstrating its validity and accuracy.

2. Theoretical analysis

Consider a gas–particlemixturewith particles grouped intoM differ-ent phases according to their material and/or sizes withm = 1, 2,…,Mand in particular letm = 0 be the gas phase. Particles of different sizespossess different radiative properties and may have different tempera-tures due to different reaction rates and responses to heat transfer.The spectral radiative transfer equation (RTE) of radiation intensity Iηatwavenumber η including absorption, scattering andmultiple temper-ature emission for a gas–particle mixture may be stated as [5]

dIηds

¼XMm¼0

κmηIbη Tmð Þ−XMm¼0

κmηIη−XMm¼0

σ smηIη þXMm¼0

σ smη14π

Z4πIη s′� �

Φmη s; s′; η� �

dΩ

#"

ð1Þwhere κmη are the absorption coefficient, Ibη the Planck function, σsmη

the scattering coefficient and Φmη the scattering phase function of

Fig. 1. Line-by-line scatter plot of gas phase absorption coefficient (κgη) vs total absorptioncoefficient (κη). The mixture has 10% CO2 and 90% N2 by volume. The particles are charwith a complex refractive index m = 2.2 − 1.12i and a uniform diameter of 400 μm.The volume fraction of particle is 0.001. The mixture has a temperature of 600 K.

Fig. 2. Line-by-line scatter plot of solid phase absorption coefficient (κsη) vs total absorp-tion coefficient (κη). The regression of solid absorption coefficient against the total absorp-tion coefficient is shown by red circles. Mixture conditions are the same as in Fig. 1.

77J. Cai, M.F. Modest / Powder Technology 265 (2014) 76–82

phase m. Both absorption and scattering coefficients are functions ofstate variable arrays ϕ and ϕ

s, i.e., ϕ contains the gas properties, such

as temperature, pressure and species concentration, and ϕs contains in-formation of local particle properties such as particle number density,radius and species.

We now apply reordering according to the total absorption coeffi-cient at reference state (κη0) over the full spectral range [5,10]:

Zη•½ �δ k−κ0

η

� �dη: ð2Þ

The superscript 0 indicates a reference state, i.e., κ0η ¼ κηðϕ0;ϕ0

sÞ in-

dicates the absorption coefficient at a reference state specified by statevariable arrays ϕ0 and ϕ0

s. Similar to the full spectrum k-distribution

method [5], after performing the integration indicated by Eq. (2), eachterm in Eq. (1) is transformed (reordered) as follows:

The path derivative (term on the left) becomes

Zη

dIηds

δ k−κ0η

� �dη ¼ d

ds

ZηIηδ k−κ0

η

� �dη ¼ d

dsIk ð3Þ

where

Ik ¼Z

ηIηδ k−κ0

η

� �dη: ð4Þ

The emission term (first on the right) becomes

Zη

XMm¼0

κmηIbη Tmð Þδ k−κ0η

� �dη

¼XMm¼0

ZηκmηIbη Tmð Þδ k−κ0

η

� �dη

¼ k�0 kð Þ þZ

ηIbη T0ð Þδ k−κ0

η

� �dηXMm¼1

km k; Tmð ÞZ

ηIbη Tmð Þδ k−κ0

η

� �dη

¼ k�0Ib T0ð Þ f FS T0;ϕ0;ϕ0

s; k

!XMm¼1

km k; Tmð ÞIb Tmð Þ f FS Tm;ϕ0;ϕ0

s; k

!# "

ð5Þ

where the gas absorption coefficients are assumed to be correlatedwith total absorption coefficients, i.e., there exists a map K0 [10]

κ0η ϕ;ϕs;η�¼ K0 ϕ;ϕ

s; κη ϕ0

;ϕ0s;η��

¼ K0 ϕ;ϕs; κ0

η

�����ð6Þ

which gives

k�0 ¼ K0 ϕ;ϕs; k;ϕ0

;ϕ0s

�:

�ð7Þ

The correlation assumption for a mixture of a gas with nongray par-ticles may be examined by plotting line-by-line gas absorption coeffi-cients against the total absorption coefficients as shown in Fig. 1, for atypical CO2–N2–char mixture with properties given in the figure cap-tion. The gas absorption coefficients cluster around a monotonic curve,suggesting the existence of themap K0 and the validity of correlation as-sumption. The correlation between gases at varying temperatures andconcentrations has been found to be good in early work on nonlumi-nous reacting flows [9]. Such correlation is not generally available forsolid absorption coefficients as shown in Fig. 2 for the same mixture.For a given total absorption coefficient, several different solid absorp-tion coefficients are possible. However, an apparent (or relevant

average) solid absorption coefficient may be determined from a regres-sion as

km k; Tð Þ ¼

ZηκmηIbη Tð Þδ k−κ0

η

� �dηZ

ηIbη Tð Þδ k−κ0

η

� �dη

: ð8Þ

This allows us to find a proper average value of κmη that occurs in thepresence of a total absorption coefficient κη0. The definition of the fullspectrum k-distribution for a multi-temperature mixture remains thesame as its definition for a single-temperature medium:

f FS�T;ϕ0

;ϕ0s; k�¼ 1

Ib Tð ÞZ

ηIbη Tð Þδ k−κ0

η

� �dη: ð9Þ

78 J. Cai, M.F. Modest / Powder Technology 265 (2014) 76–82

The absorption term (second on the right) becomes

Zη

XMm¼0

κmηIηδ k−κ0η

� �dη

¼Z

ηκ0ηIηδ k−κ0

η

� �dηþ

XMm¼1

ZηκmηIηδ k−κ0

η

� �dη

¼ k�0

ZηIηδ k−κ0

η

� �dηþ

XMm¼1

(ZηκmηIηδ k−κ0

η

� �dηZ

ηIηδ k−κ0

η

� �dη

)ZηIηδ k−κ0

η

� �dη

≈k�0Ik þXMm¼1

km

#Ik

"

ð10Þ

where the term in the curly braces is approximated as

ZηκmηIηδ k−κ0

η

� �dηZ

ηIηδ k−κ0

η

� �dη

≈

ZηκmηIbη Tmð Þδ k−κ0

η

� �dηZ

ηIbη Tmð Þδ k−κ0

η

� �dη

¼ km k; Tmð Þ: ð11Þ

This approximation is equivalent to assuming that Iη ∝ Ibη(Tm)over the spectrum. The proportionality factor may vary with spaceand direction. For example, if space and spectral dependence areseparable

Iη x; s;ηð Þ ¼ p x; sð ÞIbη Tmð Þ: ð12Þ

Eq. (11) is still exact when p is not a function of wavenumber.Eq. (12) is valid in an isothermal homogeneous mixture bounded bycold black walls, or for a medium with a scaled absorption coefficient.Deviations may arise near a wall with a different temperature, in thepresence of strong temperature gradients and if phase temperaturesare very different.

The treatment of scattering will follow Modest and Riazzi [9], inwhich they demonstrated that gray scattering coefficients (σsmη = σsm)and phase functions (Φmη = Φm) are valid approximations even for ex-tremely nongray scatterers. By dropping spectral dependence in scatter-ing properties and reordering according to Eq. (2), the scattering termsbecome

Zη

(XMm¼0

σ smηIη þXMm¼0

σ smη14π

Z4πIη s′� �

Φmη s; s′; η� �

dΩ

#" )Zδ k−κ0

η

� �dη

¼XMm¼0

σ smIk þXMm¼0

σ sm14π

Z4πΦm s; s′� �

Ik s′� �

dΩ

#:

" ð13Þ

Assuming gas phase scattering to be negligible, the summation overphases starts from m = 1.

Combining the above terms, the reordered RTE becomes

dIkds

¼ k�0 kð ÞIb T0ð Þ f FS�T0;ϕ

0;ϕ0

s; k�þXMm¼1

"km k; Tmð ÞIb Tmð Þ f FS

�Tm;ϕ

0;ϕ0

s; k�#

− k�0 kð Þ þXMm¼1

km�k; Tm

�#Ik−

XMm¼1

σ smIk þXMm¼1

σ sm14π

Z4πΦm s; s′� �

Ik s′� �

dΩ

#:

""

ð14Þ

Defining the cumulative k-distribution as

g k; T0;ϕ0

;ϕ0s

!¼Z k

0f FS T0

;ϕ0;ϕ0

s; k′!dk′

ð15Þ

and

Ig g kð Þ; r; sð Þ ¼ Ik k; r; sð Þ= f FS T0;ϕ0

;ϕ0s; k��

ð16Þ

where T0 is the reference temperature and T0 the gas phase tempera-ture. The new RTE for Ig becomes

dIgds

¼ k�0 gð Þa T0; T0; g

� �Ib T0ð Þ þ

XMm¼1

km g; Tmð Þa Tm; T0; g

� �Ib Tmð Þ

−"k�0 gð Þ þ

XMm¼1

km g; Tmð Þ#Ig−

XMm¼1

σ smIg

þXMm¼1

σ sm14π

Z4πΦm s; s′� �

Ig s′� �

dΩ

#"ð17Þ

where the nongray stretching function a is defined as

a Tm; T0; g

� �¼ f FS

�Tm;

ϕ�0;ϕ0

s; k gð Þ

�f FS T0;ϕ0

;ϕ0s; k gð Þ

�:

� ð18Þ

To split the heat source across phases, one may use the same as-sumption as in the derivation of the absorption term in the RTE forsolid phases as given by Eq. (11). The heat source for solid phase m(m ≥ 1) then becomes

Sm ¼ −∇ � qm ¼ −Z

η∇ � qmηdη ¼

Z4π

ZηκmηIηdηdΩ−4π

ZηκmηIbηdη

¼Z

4π

ZηκmηIη

Z ∞

k¼0δ k−κ0

η

� �dkdηdΩ−4πκPmIb Tmð Þ

¼Z

4π

Z ∞

k¼0

ZηκmηIηδ k−κ0

η

� �dηdkdΩ−4πκPmIb Tmð Þ

≈Z

4π

Z ∞

k¼0km k; Tmð Þ

ZηIηδ k−κ0

η

� �dηdkdΩ−4πκPmIb Tmð Þ

¼Z ∞

k¼0km k; Tmð Þ

Z4πIkdΩdk−4πκPmIb Tmð Þ

¼Z ∞

k¼0km k; Tmð ÞGkdk−4πκPmIb Tmð Þ

¼Z 1

g¼0km k gð Þ; Tmð ÞGgdg−4πκPmIb Tmð Þ:

ð19Þ

The gas phase heat source is, assuming the gas to be correlated,

S0 ¼ −∇ � q0 ¼ −Z

η∇ � q0ηdη ¼

Z4π

Zηκ0ηIηdηdΩ−4π

Zηκ0ηIbηdη

¼Z

4π

Zηκ0ηIη

Z ∞

k¼0δ k−κ0

η

� �dkdηdΩ−4πκP0Ib T0ð Þ

¼Z

4π

Z ∞

k¼0

Zηκ0ηIηδ k−κ0

η

� �dηdkdΩ−4πκP0Ib T0ð Þ

¼Z

4π

Z ∞

k¼0k�0

ZηIηδ k−κ0

η

� �dηdkdΩ−4πκP0Ib T0ð Þ

¼Z ∞

k¼0k�0 kð Þ

Z4πIkdΩdk−4πκP0Ib T0ð Þ ¼

Z ∞

k¼0k�0 kð ÞGkdk−4πκP0Ib T0ð Þ

¼Z 1

g¼0k�0 k gð Þð ÞGgdg−4πκP0Ib T0ð Þ:

ð20Þ

The total heat source is obtained by integrating Eq. (17) over all solidangles and all g, and is conserved because

XMm¼0

Sm ¼Z 1

0

"k�0 k gð Þð Þ þ

XMm¼1

km k gð Þ; Tmð Þ#Ggdg−

XMm¼0

4πκPmIb Tmð Þ

¼Z 1

0

"k�0 gð Þ þ

XMm¼1

km gð Þ#Ggdg−4π

Z 1

g¼0k�0 gð Þa0 gð ÞIb T0ð Þdg

−4πXMm¼1

Z 1

0km gð Þam gð ÞIb Tmð Þdg ¼

Z 1

0

("k�0 gð Þ þ

XMm¼1

km gð Þ#Gg

−4π

"k�0 gð Þa0 gð ÞIb T0ð Þ þ

XMm¼1

km gð Þam gð ÞIb Tmð Þ#)

dg

¼ −Z 1

0∇ � qgdg ¼ −∇ � q

ð21Þ

79J. Cai, M.F. Modest / Powder Technology 265 (2014) 76–82

2.1. Regression

To calculate the full-spectrum k-distribution of a gas–particle mix-ture, we use databased narrowband k-distributions [11] of the combus-tion gases and mix them based on the Modest–Riazzi model [9] toobtain the narrowband k-distribution of the gasmixture. Over each nar-rowband, the solid absorption coefficient of each phasemay be approx-imated by a constant value equal to its average absorption coefficientover the narrowband. Then the narrowband k-distribution of the gas–particle mixture is obtained by adding the narrowband solid absorptioncoefficients of all solid phases to the narrowband gas k-distribution overeach narrowband [9]. The full-spectrum k-distribution of the gas–particlemixture is then compiled fromthenarrowband k-distributions of the gas–particle mixture.

The regression of solid absorption coefficient in Eq. (8) is calculatedwith narrowband k-distributions of themixture. Suppose that the spec-trum is partitioned into NNB narrowbands and the particle absorptioncoefficient of phase m is approximated by a constant value km,n overthe nth narrowband. Over this narrowband, the cumulative k-distribution of the gas–particle mixture is gn,i for a sequence of increas-ing kis (i = 1, 2,…, Ni, where Ni is the total number of k values). The re-

gression km is expressed as a weighted average of km,n from allnarrowbands with the weights proportional to the narrowband k-distributions and the narrowband integral Planck function, i.e.,

km kið Þ ¼

ZηIbηκmηδ ki−κη

� �dηZ

ηIbηδ ki−κη

� �dη

≈

Z kiþkiþ1ð Þ=2kiþki−1ð Þ=2

ZηIbηκmηδ k−κη

� �dηdk

Z kiþkiþ1ð Þ=2kiþki−1ð Þ=2

ZηIbηδ k−κη

� �dηdk:

ð22Þ

The numerator may be expressed as

Z kiþkiþ1ð Þ=2kiþki−1ð Þ=2

ZηIbηκmηδ k−κη

� �dηdk ¼

Zη

Z kiþkiþ1ð Þ=2kiþki−1ð Þ=2

Ibηκmηδ k−κη

� �dkdη

¼XNNB

n¼1

Z ηnþΔηn

ηn

Z kiþkiþ1ð Þ=2kiþki−1ð Þ=2

Ibηnκmηnδ k−κη

� �dkdη

¼XNNB

n¼1

Ibηnκmηn

Z ηnþΔηn

ηn

Z kiþkiþ1ð Þ=2kiþki−1ð Þ=2

δ k−κη

� �dkdη

¼XNNB

n¼1

IbηnΔηκm;n

Z kiþkiþ1ð Þ=2kiþki−1ð Þ=2

1Δη

Z ηnþΔηn

ηn

δ k−κη

� �dηdk

¼XNNB

n¼1

Ibnκm;n

Z kiþkiþ1ð Þ=2kiþki−1ð Þ=2

1Δη

Z ηnþΔηn

ηn

δ k−κη

� �dηdk

¼XNNB

n¼1

Ibnκm;n

Z kiþkiþ1ð Þ=2kiþki−1ð Þ=2

f NB k;nð Þdk ¼XNNB

n¼1

Ibnκm;nΔgi;n

ð23Þ

where κm,n is the constant solid phase absorption coefficient of phasem

over narrowband n, Ibn ¼ ∫ηnþΔηnηn

Ibηdη is the fraction of Planck function

over narrowband n, fNB(k;n) is the narrowband k-distribution at k over

narrowband n,Δgi;n ¼ ∫ kiþkiþ1ð Þ=2kiþki−1ð Þ=2 f NB k;nð Þdk is the change of narrow-

band k-distribution around ki over narrowband n. Similarly, thedenominator reads,

Z kiþkiþ1ð Þ=2kiþki−1ð Þ=2

ZηIbηδ k−κη

� �dηdk ¼

XNNB

n¼1

IbnΔgi;n ð24Þ

Combining Eqs. (23) and (24), the regression may be evaluated as

km kið Þ ¼ km;i ¼XNNB

n¼1Ibnκm;nΔgi;nXNNB

n¼1IbnΔgi;n:

ð25Þ

This model is implemented based on the narrowband k-distributiondatabase for common radiatively participating gases in combustion

[11,12]. The gas spectra between 200 cm−1 and 15,000 cm−1 are divid-ed into 248 narrowbands. The solid phase absorption coefficient overeach narrowband is approximated by its average value across each nar-rowband. After adding the narrowband constant solid phase absorptioncoefficients to themixture narrowband k-distributions, the regression isevaluated using Eq. (25) while the narrowband k-distribution of themixture is compiled into its full-spectrum counterpart [11].

A sample regression is included in Fig. 2, in which 256 k values areused, with the corresponding km marked by circles. Note that forsmall k values, gas absorption is negligible, and the km takes thesame value as k. For large k values (k N 3 cm−1), the total absorptioncoefficient is dominated by gas absorption at the line centers, and kmapproaches the solid absorption coefficient at the correspondingwavenumber.

3. Sample calculations

First consider the radiation in a homogeneous gas–particle fieldbounded by two parallel black cold walls. The gas phase has 10% CO2

and 90% N2. The particles are char with a complex refractive indexm = 2.2 − 1.12i [13] and a uniform diameter of 400 μm. The volumefraction of the particles is 0.001. The mixture slab has a temperatureof 600 K and a thickness of 7 cm. The gray scattering coefficient isevaluated at the wavenumber where the Planck function peaks andscattering is assumed to be isotropic. The gas absorption coefficientis calculated from databased narrowband k-distributions and thesolid absorption coefficient is approximated as constant across eachnarrowband [9].

Eq. (17) is solved with the P1 approximation using 64 quadraturepoints for g. The results of total, gas and solid phase radiative heatsources are compared with line-by-line calculations in Fig. 3. The k-distribution method is capable of predicting all of the heat sourcesaccurately (to within 1%).

To demonstrate the accuracy of the regressionmethod in inhomoge-neous media, one-dimensional slabs, bounded by two parallel blackcold walls are considered. The slabs have two homogenous gas–particlemixture layers with equal thickness. For all five inhomogeneous testcases, the left layer has a solid temperature of 500 K, a gas temperatureof 600 K and a CO2 mole fraction of 5% diluted by O2 and N2, while theright layer has a solid temperature of 1300 K, a gas temperature of1200 K and a CO2 mole fraction of 10% diluted by O2 and N2. CO2 is as-sumed to be the only radiatively participating gas species, whichreduces errors from k-distributions since only temperature levels (asopposed to concentration) contribute to uncorrelation. The inaccuracyhereby primarily comes from the approximation in Eq. (11). In bothlayers the particles are char with a complex refractive indexm = 2.2 − 1.12i. The particle volume fractions and sizes are listed inTable 1. These configurations approximate both unburned and burninggas–particle environments in typical pulverized and both fixed and cir-culating fluidized-bed coal combustion conditions. For large particlevolume fractions, the test layer thicknesses are reduced due to largeoptical thicknesses.

For all five inhomogeneous test cases, the reference states are evalu-atedwith themethod proposed byModest [10] using only gas phase ab-sorption coefficients, i.e., the reference state does not have particles. Allresults suggest that the proposed regression is not sensitive to the par-ticle volume fractions in the choice of reference state. The correlation as-sumption for the total absorption coefficients does not pose additionalsevere constraints upon the correlation assumption proposed for thegas phase previously [10]. This is largely due to the fact that solidphase absorption coefficients have much smaller spectral variationsthan gases. As a result, the gas phase absorption coefficient is wellcorrelated with the total absorption coefficients, e.g. Fig. 1. Theuncorrelatedness in the solid phase absorption coefficient is approxi-mately accounted for by the proposed regression model.

Table 1Test configurations for inhomogeneous media: layer thickness (l), particle diameter(d) and particle volume fraction (fv).

Left Right

Case l (cm) d (μm) fv l (cm) d (μm) fv

A 5 200 0.001 5 100 2.5 × 10−4

B 5 2 1 × 10−4 5 0.4 4 × 10−6

C 5 20 0.001 5 8 1 × 10−4

D 0.5 600 0.6 0.5 400 0.05E 0.5 600 0.3 0.5 400 0.1

80 J. Cai, M.F. Modest / Powder Technology 265 (2014) 76–82

Eq. (17) is solved with the P1 approximation using 64 quadraturepoints for g. The results of total, gas and solid phase radiative heatsources are compared with line-by-line calculations in Figs. 4–6. Thek-distribution method is capable of predicting major trends of totaland phase-split heat source terms (Fig. 4). In particular, the gas phaseheat source is predicted accurately (mostly within 5%) in spite of thefact that it is at least one order less in magnitude than the total heatsource. This is because gas radiation primarily comes from strongbands, and the proposed regression scheme successfully picks thesolid absorption coefficients at the corresponding wavenumbers(e.g. large k values in Fig. 2). Accurate predictions of the gas phase radi-ative heat source terms are important for gas phase combustion becauseof much less gas phase volumetric specific heat. For the solid phase heatsource (Fig. 4), the largest deviation occurs near the interface betweenthe two layers. Eq. (11) is greatly challenged by the sharp temperaturegradient at the interface. In the left (cold) layer, the spectral incident ra-diation deviates significantly from the emission spectrum at the localtemperature because of the emission from the hot layer. Away fromthe interface, such deviation is reduced and accuracy improves. Howev-er, this effect is hardly observed in the hot layer because emission fromthe cold layer is negligible. In the hot layer, the error is less than 1%,which is due to the temperature difference between gas and solidphases. Such deviation becomes smaller in more realistic situations(Figs. 5 and 6).

For pulverized coal cases (Fig. 5), the left layer has a larger particlevolume fraction and thus a larger optical thickness. Strong absorptioninside of this layer by both gas and solid phases causes the radiativeheat sources to decay to zero within a few centimeters, e.g., −∇ ⋅ q isclose to zero near x = 2 cm (3 cm away from the temperature jump).The gas phase radiative heat source is at least an order of magnitudesmaller than the solid phase radiative heat source, and therefore, thesolid radiative heat source is nearly equal to the total radiative heatsource. Even so, the regressionmodel successfully recovers the gas radi-ative heat source.

The proposed model is more successful at higher solid volume frac-tions as found in fluidized beds, as shown in Fig. 6. Because of the

x (cm)

-∇q

(M

W/m

3 )

0 1 2 3 4 5 6 7

-0.1

-0.08

-0.06

-0.04

-0.02

0

Total, LBLGas, LBLSolid, LBLTotal, regression-FSKGas, regression-FSKSolid, regression-FSK

Fig. 3. Heat source calculation of a homogeneous slab between line-by-line (solid lines)and full-spectrum k-distribution with regression (dots). Line colors are red (total heatsource), black (gas phase heat source) and blue (solid phase heat source).

increased solid absorption coefficients, the gas radiative heat source istwo orders of magnitude less than that of the solid, causing the totaland solid radiative heat sources to overlap. Again, even in these extremecases, the proposed regression model is capable of predicting gas radia-tive heat sources accurately. Note that inside the left layer, due to largesolid absorption coefficients, the radiation effect decays within a fewmillimeters from the temperature jump (e.g. at x = 0.2 cm). Therefore,the test case layer thickness appropriate for fluidized bed configurationsis reduced to 0.5 cm from 5 cm for the pulverized coal cases. Neverthe-less, the accuracy of the proposedmodel is improved in a dense particleflow as indicated by the overlap of regression model predictions andline-by-line predictions. This is because a larger optical thickness causesthe local radiation intensity to be closer to the local Planck function, andimproves the accuracy of approximation in Eq. (11).

The cpu time cost of k-distribution evaluations involving one partic-ipating gas species and one solid phase is 0.59 s per cell, which involvestwo k-distribution compilations from the narrowband database, one forthe local temperature and the other for the reference temperature. Theone-time global evaluation of the k-distributions for the reference statesis negligible when the number of mesh cells is large. Most cpu time isused to interpolate between gas narrowband k-distributions, and as-semble narrowband to full-spectrum k-distributions. The solid phaseproperty and regression calculations (Eq. (25)) add less than 1%overhead.

x (cm)

-∇q

(M

W/m

3 )

0 2 4 6 8 10

-2

-1.5

-1

-0.5

0

0.5

Total, LBLGas, LBLSolid, LBLTotal, regression-FSKGas, regression-FSKSolid, regression-FSK

Fig. 4. Heat source calculation of Case A between line-by-line (solid lines) and full-spectrum k-distribution with regression (dots). Line colors are red (total heat source),black (gas phase heat source) and blue (solid phase heat source).

x (cm)

-∇q

(M

W/m

3 )

-∇q

(M

W/m

3 )

0 2 4 6 8 10

0

5

10

15

20

25

-0.3

-0.2

-0.1

0

0.1

0.2

0.3Total, LBLGas, LBLSolid, LBLTotal, regression-FSKGas, regression-FSKSolid, regression-FSK

x (cm)

-∇q

(M

W/m

3 )

-∇q

(M

W/m

3 )

0 2 4 6 8 10

-5

0

5

10

15

20

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Total, LBLGas, LBLSolid, LBLTotal, regression-FSKGas, regression-FSKSolid, regression-FSK

Fig. 5.Heat source calculation of Cases B (top) and C (bottom) between line-by-line (solidlines) and full-spectrum k-distribution with regression (dots). Line colors are red (totalheat source), black (gas phase heat source) and blue (solid phase heat source).

x (cm)

-∇q

(M

W/m

3 )

-∇q

(M

W/m

3 )

0 0.2 0.4 0.6 0.8 1

-50

0

50

100

-0.4

-0.2

0

0.2

0.4Total, LBLGas, LBLSolid, LBLTotal, regression-FSKGas, regression-FSKSolid, regression-FSK

x (cm)

-∇q

(M

W/m

3 )

-∇q

(M

W/m

3 )

0 0.2 0.4 0.6 0.8 1

0

50

100

150

200

250

-0.4

-0.2

0

0.2

0.4Total, LBLGas, LBLSolid, LBLTotal, regression-FSKGas, regression-FSKSolid, regression-FSK

Fig. 6.Heat source calculation of Cases D (top) and E (bottom) between line-by-line (solidlines) and full-spectrum k-distribution with regression (dots). Line colors are red (totalheat source), black (gas phase heat source) and blue (solid phase heat source).

81J. Cai, M.F. Modest / Powder Technology 265 (2014) 76–82

Although the test cases involve only one solid phase, it should benoted that all theoretical derivations are presented for an arbitrarynumber of solid phases. For example, all derivations assume multiplesolid phases as indicated in the subscriptm. The regression calculationsare with respect to the total narrowband k-distributions after all thesolid phase narrowband absorption coefficients have been added. Thescenarios for multiple solid phases are the same as for a single solidphase,making extensions tomore than one solid phase straightforward.Its accuracy is determined only by the correlation assumption andEq. (11), e.g., the departure of local incident radiation from blackbodyemission.

4. Conclusion

In a gas–particle mixture, particles of different makes and sizes mayhave different local temperatures and, therefore, are modeled withseparate energy equations. To model the radiative heat transfer insuch a mixture the full-spectrum k-distribution model was extendedto the case of multi-phase mixtures with individual temperature for

each phase. Full spectrum k-distributions were assembled from a nar-rowbanddatabase for gases, by assuming particles to be gray across nar-rowbands, and by using a regression to determine the “effective”absorption coefficients for particles.

It was shownhowproperly distributed radiative source terms can bedetermined for the individual energy equations, by using a similar re-gression analysis. The accuracy of the method was demonstrated withseveral examples.

Acknowledgment

This research is sponsored by NETL under Grant number DE-FE0003801.

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