A Modified Exponential Wide Band Model for Gas Emissivity

Jan 23, 2018 - (24) Thereby, an emissivity database for the main radiant species and their mixtures are generated first by the LBL model, as the basis...
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Article Cite This: Energy Fuels 2018, 32, 1634−1643

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A Modified Exponential Wide Band Model for Gas Emissivity Prediction in Pressurized Combustion and Gasification Processes Linbo Yan,*,† Yang Cao,† Xuezheng Li,† and Boshu He*,†,‡ †

Institute of Combustion and Thermal System, School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing 100044, China ‡ Department of Mechanical and Electrical Engineering, Haibin College of Beijing Jiaotong University, Huanghua 061199, Hebei Province, China ABSTRACT: Accurate and fast prediction of the gas emissivity in a wide range of pressure and temperature is very essential for the accurate and efficient simulation of the combustion and gasification processes. In this work, the total emissivities of the main radiant species including H2O, CO2, CO, and their mixtures are first generated with the line-by-line (LBL) model and the HITEMP-2010 spectroscopic database at different total pressures varying from 0.1 atm to 30 atm and at different temperatures varying from 300 K to 2500 K. Then, the exponential wide band model (EWBM) is modified according to the LBL results and simplified using the polynomial fitting method and the table look-up scheme. Finally, a modified EWBM (M-EWBM) is presented. The performances of the EWBM, the M-EWBM, and the LBL model are comprehensively compared, and it is found that the M-EWBM is more accurate than the EWBM and can be as efficient as the weighted-sum-of-gray-gas model, making it very promising for the total gas emissivity prediction in the pressurized combustion and gasification simulations. with a spectral interval of 0.0002−0.02 cm−1. It has been, until now, the most accurate but the most computationally intensive model.10 Because of the formidable computation task, the LBL prediction is only used as a benchmark for the prediction of the other spectral models. The NBM11 generally works with a spectral (band) interval of 5−50 cm−1 and it can be as accurate as the LBL model. However, this model also requires vast amounts of computer resources. The WBM12 generally works with a spectral (band) interval of 100−1000 cm−1. It is much faster than the LBL model and the NBM with only modest accuracy penalty. Unlike the LBL model or the band models, the global models calculate the radiation properties over the entire spectral range. One typical global model is the weighted-sum-of-gray-gas model (WSGGM),1 which frequently appears in engineering applications and generates acceptable errors for temperatures ranging from 600 K to 2400 K at 1 atm.13 Besides the WSGGM, the global models also include the spectral-line-based WSGG (SLW) model proposed by Denison and Webb14 and the full-spectrum k-distribution (FSK) model proposed by Modest and Zhang.15 In addition to the gray-gas global models, there are also some other non-gray-gas global models, such as the absorption distribution function (ADF) model16 and the spectral-line momentbased (SLMB) model.17 All the approximate spectral models suffer from at least one of the following issues: less efficiency, less accuracy, or less generality. Computation efficiency is of primary importance for the numerical simulations, because radiation calculations are usually only a small part of a sophisticated combustion or gasification code, and the spectral properties can be calculated million times in one iteration. This explains why only the WSGGM is preferred in the

1. INTRODUCTION Thermal radiation dominates the heat-transfer processes in combustion or high-temperature gasification.1,2 To give accurate prediction of the radiative heat source/sink distribution in a concerned space, much effort has been done and many models, including the discrete transfer radiation model,3 the P−N radiation model,4 the Rosseland radiation model,5 the surface-to-surface radiation model,5 and the discrete ordinates radiation model,6 have been developed by researchers worldwide to solve the radiative transfer equation (RTE), d Ir ( r ⃗ , s ⃗ ) + (αr + σs)Ir( r ⃗ , s ⃗) ds 4π σT4 σ = ar nr 2 r + s I( r ⃗ , s ⃗′)Θ( s ⃗ · s ⃗′) dΩ′ 4π 0 π

∫

(1)

where Ir is the radiation intensity, r ⃗ is the position vector, s ⃗ is the direction vector, s denotes the beam path length, ar is the absorption coefficient, σs is the scattering coefficient, nr is the refractive index, σr is the Stefan−Boltzmann constant, s′⃗ denotes the scattering direction vector. Θ is the phase function, and Ω′ denotes the solid angle. Besides the radiation models, eq 1 indicates that the two apparent spectral properties of the participating medium, ar and σs, are also of great importance for the tenable prediction of radiative heat transfer.7 The scattering coefficient can sometimes be ignored without great accuracy penalty.8 However, the absorption coefficient usually determines the accuracy of the RTE predictions and should be carefully addressed. Until now, many approximate spectral models, including the line-by-line (LBL) model, the narrow band model (NBM), the wide band model (WBM) and the global models have been developed and can be used to predict the absorption coefficients of the participating medium.9 The LBL model generally works © 2018 American Chemical Society

Received: November 29, 2017 Revised: January 14, 2018 Published: January 23, 2018 1634

DOI: 10.1021/acs.energyfuels.7b03747 Energy Fuels 2018, 32, 1634−1643

Article

Energy & Fuels commercial computational fluid dynamics (CFD) software, although it is not the most accurate one and lacks generality. Actually, the WSGGM suffers some other intrinsic deficiencies.18,19 First, the accuracy of the original WSGGM is limited, especially when used to predict some special combustion processes, such as oxy-fuel combustion. Although some researchers have tried to improve the WSGGM so that this model can be tenable for some specific cases,20−22 the generality of the modified WSGGM is challenged. Second, it only addresses problems that occur at 1 atm. Actually, until now, very little work has been done for cases of nonatmospheric pressures.23 Since the WBM is much faster than the LBL model and NBM with only modest accuracy penalty, it is believed to be a good candidate for the fast and accurate prediction of the total gas emissivity and absorptivity for different species and their mixtures in a wide range of pressure and temperature after modification and simplification. The novelty and contribution of this work is the development of a modified exponential WBM (M-EWBM), which has better accuracy than the original EWBM and can be as efficient as the WSGGM. Therefore, the M-EWBM proposed in this work can be readily embedded in the commercial or opensource CFD software through the user-defined functions to predict the total emissivities of the participating medium for combustion and gasification simulations under different conditions.

exp Q (T0) Sk(T ) = Sk(T0) × × Q (T ) exp

εt =

∫0 εηIbη dη ∞

∫0 Ibη dη

∑ Skg(η − η0)Ni k

× 1 − exp

−C 2ηi

( ) T

−C 2ηi

( ) T0

η0(p) = η0 + Ptδ

(5)

where δ denotes the shift of transition due to pressure and Pt is the total pressure. The line shape function due to Doppler broadening33 is calculated with ⎡ ⎛ 1 ⎞⎛ ln 2 ⎞1/2 (η − η0)2 ⎤ ⎥ ⎟ gD(η − η0) = ⎜⎜ ⎟⎟⎜ exp⎢ −ln 2 ⎢⎣ γD2 ⎥⎦ ⎝ γD ⎠⎝ π ⎠

(6)

where γD denotes the half width at half height of the Doppler feature and is calculated by η ⎛ 2RT ln 2 ⎞ γD = 0 ⎜ ⎟ c ⎝ Mw ⎠

1/2

(7)

where c is the speed of light, R is the universal gas constant, and Mw is the molecular weight. The line shape function due to pressure broadening33 is calculated by using the expression gP(η − η0) =

γP/π (η − η0)2 + γP 2

(8)

where γp denotes the half width at half height of the Lorentzian feature and is calculated by using the expression ⎛ T ⎞n γP = [γair(Pt − Pi) + γself Pi ]⎜ 0 ⎟ ⎝T ⎠

(9)

where γair and γself are the air-broadened half width and the selfbroadened half width, respectively; n is the coefficient of temperature dependence of the air-broadened half width, and Pi denotes the partial pressure of species i. The line shape function due to the combination of the two broadening effects33 is calculated by using the expression gV (η − η0) gV ,max

∞

⎡ ⎛ ⎛ η − η ⎞2 ⎤ γ ⎞ 0 ⎥ ⎟⎟ = ⎜⎜1 − P ⎟⎟ exp⎢− 2.772⎜⎜ ⎢ ⎥ γV ⎠ 2 γ ⎝ ⎝ V ⎠ ⎦ ⎣ ⎛ γ ⎞2 1 + ⎜⎜ P ⎟⎟ ⎝ γV ⎠ 1 + 4 η − η0 2γ

(2)

2

( )

where εη, αη, and Ibη are the emissivity, absorptivity, and blackbody intensity at wavenumber η. X denotes the pressure path length. The spectra absorptivity (αη) is the convolution of the line intensity and the line shape factor31 and is calculated using the expression αη =

−C 2En T0

1 − exp

(4)

∫0 Ibη[1 − exp( − αηX )] dη ∫0 Ibη dη

)

where Q(T) is the total internal partition function,32 C2 is the Planck second radiation constant, and En is the lower state energy of the transition. T0 and T are the reference temperature and actual gas temperature, respectively. Note that the line location may experience a shift due to total pressure, and is calculated25 as

∞

=

−C 2En T

( )

2. THE LBL EMISSIVITY DATABASE 2.1. The LBL Methodology. As is commonly believed, the LBL prediction is often used to benchmark the accuracy of the other spectral models.24 Thereby, an emissivity database for the main radiant species and their mixtures are generated first by the LBL model, as the basis of this work. When the LBL model is used, the spectra line data, including the line position and intensity, the air- and self- broadened half-width, the coefficient of temperature dependence of the air broadened half-width, the lower state of energy, and the shift of transition due to pressure should be known. Recently, there have been three spectra databases that provide this information: HITRAN, HITEMP, and CSCD. Thereinto, the HITRAN database can only be accurate enough for temperatures lower than 750 K,25,26 and the CSCD database is only available for CO2,27 so these two databases are not suitable for this work. The HITEMP-2010 database28 has much more hot lines than HITRAN and contains more species than CSCD, so this spectra database is chosen in this work. With the total emissivity known, the absorptivity can then be calculated by using a simple equation.29 The LBL model calculates the total emissivity (εt)30 as ∞

(

⎛ γ ⎞⎛ γ ⎞ + 0.016⎜⎜1 − P ⎟⎟⎜⎜ P ⎟⎟ γV ⎠⎝ γV ⎠ ⎝

V

⎧ ⎪ ⎛ η − η ⎞2.25⎤ ⎪ ⎡⎢ 0 ⎟⎟ ⎥ ⎨ + exp −0.4⎜⎜ ⎪ ⎢⎣ ⎝ 2γV ⎠ ⎥⎦ ⎪ ⎩

(3)

10

−

where Ni is the molecular concentration of species i and g(η−η0) is the line shape function. η0 and η are the wave numbers at the special line center and at any position, respectively. Sk denotes the line intensity at the actual temperature and is calculated by

10 +

⎫ ⎪ ⎪ ⎬ 2.25 ⎪ ⎪ ⎭

( ) η − η0 2γV

(10)

where gV,max is calculated with 1635

DOI: 10.1021/acs.energyfuels.7b03747 Energy Fuels 2018, 32, 1634−1643

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Energy & Fuels g V,max =

1 ⎡ 2γV ⎢1.065 + 0.447 ⎣

( ) + 0.058( ) ⎥⎦ γP

γP

γV

γV

(αin), and the mean line width to spacing ratio parameter (βin). The five parameters29 are determined by eqs 16−22:

2⎤

(11)

χi =

where γV denotes the half width at half height of the Voigt feature and is calculated by using the expression 2γV = (0.5346)(2γP) + [0.2166(2γP)2 + (2γD)2 ]1/2

∏ τi ,η = 1 − ∏ exp( − αi ,ηXi) i

i

(12)

Pe,in

⎡ e − jξ ⎛ 3ξ 2 6ξ 6 ⎞⎤ ⎜ξ 3 + + 2 + 3 ⎟⎥ j ⎝ j j j ⎠⎥⎦ j=1 ⎣

(18)

where ω0 is the reference bandwidth parameter; ⎡⎣1 − exp( −∑dD= 1 ξdδd)⎤⎦ψ (T ) αin(T ) = α0 ⎡⎣1 − exp( −∑dD= 1 ξ0, dδd)⎤⎦ψ (T0)

(19)

where δd describes the effects of the photon transition on the vibrational quantum number,9 α0 is the reference band intensity, and ψ(T) is defined by

(13)

ψ (T ) =

(νd + gd + |δd| − 1) !

∞

D

∏d = 1 ∑ν = ν d

(gd − 1) ! νd !

0, d

D

∞

(νd + gd − 1) !

d

(gd − 1) ! νd !

∏d = 1 ∑ ν = 0

exp( − ξdνd)

exp( − ξdνd)

(20)

where νd is the vibrational quantum number, gd denotes the degeneracy of the fundamental band,9 and ν0,d denotes the lowest possible initial state and is equal to zero when δd is non-negative (otherwise, it is equal to the absolute value of δd); ⎛ T ⎞0.5 Φ(T ) βin(T ) = β0⎜ ⎟ ⎝ T0 ⎠ Φ(T0)

(21)

where β0 denotes the reference mean line width to spacing ratio, and Φ(T) is defined by

Φ(T ) =

⎧ D ⎨∏d = 1 ∑∞ νd = ν0, d ⎩ D

∞

∏d = 1 ∑ν = ν d

0, d

(νd + gd + |δd| − 1) ! exp(−ξdνd) (gd − 1) ! νd ! (νd + gd + |δd| − 1) ! (gd − 1) ! νd !

⎫2 ⎬ ⎭

exp( − ξdνd) (22)

(14)

With the exception of ξ, which is a function of temperature, variables νd, gd, and δd are all nondimensional. Thereby, ψ(T) and Φ(T) are functions of the temperature. 3.2. Modification of EWBM. It is very intractable to find a proper way to modify the EWBM so that it can give better predictions in a wide range of temperatures and pressures for different species and their mixtures. Much pre-exploratory work has been done, and, finally, a modification scheme is found to work well. Note that the EWBM uses the equivalent broadening pressure to allow for the effect of pressure on the total gas emissivity, as expressed by eq 17, where the self-broadening parameter bin and the fitting parameter nin are adjustable. Unlike the LBL model, the effect of pressure on the band position is not considered in the EWBM. However, this work shows that, if all the band center positions maintain the original values, the improvement of the EWBM performance can be greatly limited, especially at high pressures. Also note

where τΔη,m is the transmissivity of the mth block, and it is the product of all the band transmissivities (τg,in) that belong to this block; ηL,m and ηU,m are the lower and upper limits of block m, respectively, and they are determined by the total band absorbance (Ain) and the band transmissivity. F(η/T) is the fractional function of blackbody radiation35 and can be calculated by using the expression ∞

∑ ⎢⎢

(17)

⎛ T ⎞0.5 ωin(T ) = ω0⎜ ⎟ ⎝ T0 ⎠

3. DEVELOPMENT OF M-EWBM 3.1. The EWBM Methodology. The EWBM that was developed by Edwards and Menard12 and Edwards and Balakrishnan34 calculates the total emissivity using the expression

⎛ η ⎞ 15 F⎜ ⎟ = 4 ⎝T ⎠ π

⎡P ⎤nin Pi T =⎢ + (bin − 1)⎥ P0 ⎣ P0 ⎦

where P0 is the reference pressure (P0 = 101 325 Pa), and bin and nin are the self-broadening parameter and the corresponding fitting parameter of species i and band n, respectively;

where τi,η denotes the transmissivity of species i at wavenumber η. 2.2. The Emissivity Database. Note that the effect of pressure on the total gas emissivity is permitted by the LBL model in twofold. One method involves the effect on the spectra line position, as shown by eq 4, and the other involves the line shape, as shown by eq 5. The above correlations are programmed and the total emissivities of H2O, CO2, CO, and their mixtures are calculated in the temperature range of 300−2500 K with an interval of 100 K and in a pressure range of 0.1−30 atm, including 0.1, 0.25, 0.5, 1, 5, 10, 20, and 30 atm. For the gas mixtures, the H2O/CO2 molar ratios of 1/1, 1/8, and the H2O/CO2/CO mole fractions of 0.1/0.1/0.3 are selected. The first two mixtures refer to two typical oxy-fired combustion cases: one with the wet flue gas recycling and the other with the dry flue gas recycling.30 The last mixture refers to the typical steam gasification case.

⎡ ⎛ ηL, m ⎞ ⎛ ηU, m ⎞⎤ ⎟ − F⎜ ⎟⎥ εt = ∑ (1 − τΔη , m)⎢F ⎜ ⎝ ⎠ ⎝ T ⎠⎦ ⎣ T m

(16)

where xi is the molar fraction of species i;

The choice of the line shape functions can be made according to the value of γP/γV. When the value is 5, the line shape function due to pressure broadening (Lorentzian profile) is used.33 For H2O, CO2, and CO, the spectral lines whose intensities are greater than 1 × 10−45 cm−1/(mol cm−2) are chosen for integration. The wavenumber interval during the calculation is chosen as 0.01 cm−1 for all the species. For mixtures, the emissivity can be calculated with εη = 1 −

⎛ xiPTM wi ⎞ ⎜ ⎟L ⎝ RT ⎠

(15)

where ξ denotes C2η/T. To calculate τg,in, the following five parameters should be calculated first: the mass path length of species i (χi), the equivalent broadening pressure of species i band (nPe,in), the bandwidth parameter (ωin), the integrated band intensity 1636

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Energy & Fuels that the time-consuming parameters αin(T) and βin(T) are only functions of temperature, which makes it possible to use simple polynomials to simplify the calculation. In addition, the fractional function of blackbody radiation is also computationally intensive, so direct calculation should be avoided to further improve the EWBM efficiency. Given the above consideration, the detailed

EWBM modification scheme can be summarized, using the following steps: (1) The first step is to modify the band positions of the main medium components, so that the following fitting can give better results at high pressures. For H2O, the bands centered at 140, 3760, and 5350 cm−1 are changed to 180, 3300, and 4350 cm−1,

Table 1. Values of bCO2,750 Versus Temperature (T) and Pressure (P) bCO2,750 T (K)

P = 0.1 atm

P = 0.25 atm

P = 0.5 atm

P = 0.75 atm

P = 1 atm

P = 5 atm

P = 10 atm

P = 20 atm

P = 30 atm

300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500

0.21 0.07 0.03 0.03 0.03 0.03 0.05 0.09 0.20 0.86 6.43 2.77 1.92 2.57 2.27 1.11 0.27 0.09 0.04 0.06 0.76 0.03 0.02

0.09 0.03 0.005 0.003 0.002 0.002 0.003 0.003 0.006 0.019 0.15 1.37 2.09 0.79 0.53 0.33 0.10 0.08 0.19 0.08 0.17 0.02 0.02

0.05 0.0061 0.0013 0.0005 0.0003 0.0003 0.0003 0.0005 0.0013 0.0069 0.0609 0.19 1.25 0.42 0.32 0.27 0.10 0.09 0.11 0.05 0.10 0.02 0.02

0.0247 0.0037 0.0007 0.0003 0.0002 0.0002 0.0002 0.0003 0.0008 0.005 0.0399 0.12 0.33 0.39 0.23 0.20 0.16 0.15 0.08 0.04 0.02 0.02 0.02

0.02 0.0027 0.0005 0.0002 0.00005 0.00004 0.00006 0.00014 0.00054 0.00401 0.02981 0.302 0.041 0.154 0.182 0.155 0.051 0.109 0.057 0.005 0.052 0.003 0.002

0.00842 0.00096 0.00015 0.00005 0.00004 0.00005 0.00009 0.00027 0.00126 0.009 0.092 0.111 0.078 0.024 0.029 0.015 0.015 0.01 0.027 0.008 0.003 0.004 0.002

0.006 0.0007 0.0002 0.00004 0.00003 0.00005 0.0001 0.00035 0.00175 0.013 0.062 0.081 0.022 0.017 0.015 0.039 0.011 0.004 0.005 0.004 0.005 0.004 0.002

0.0025 0.0004 0.00006 0.00003 0.00003 0.00004 0.00011 0.00044 0.00263 0.0326 0.0346 0.0162 0.0091 0.0186 0.0116 0.0031 0.0019 0.0008 0.0011 0.0013 0.0017 0.0027 0.0002

0.00163 0.0002 0.00005 0.00003 0.00003 0.00004 0.00011 0.0005 0.00337 0.019 0.028 0.02 0.009 0.016 0.008 0.008 0.005 0.002 0.003 0.003 0.004 0.002 0.002

Table 2. Fitting Parameters of CO (nCO,2143 and nCO,4260) Versus Pressure (P) and Temperature (T) nCO,2143 and nCO,4260 T (K)

P = 0.1 atm

P = 0.25 atm

P = 0.5 atm

P = 0.75 atm

P = 1 atm

P = 5 atm

P = 10 atm

P = 20 atm

P = 30 atm

300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500

−0.2 0.35 0.28 0.32 0.35 0.4 0.43 0.46 0.49 0.53 0.56 0.59 0.59 0.61 0.61 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.7

−0.9 1.5 1.3 1.3 1.5 1.6 1.8 1.9 2.1 2.2 2.3 2.4 2.5 2.5 2.5 2.6 2.7 2.7 2.7 2.7 2.8 2.8 2.8

0.7 −0.8 −0.9 −0.95 −1.0 −1.1 −1.2 −1.3 −1.4 −1.5 −1.6 −1.7 −1.8 −1.8 −1.8 −1.8 −1.8 −1.8 −1.9 −1.9 −1.9 −1.9 −1.9

0.35 −0.4 −0.45 −0.45 −0.5 −0.55 −0.6 −0.65 −0.7 −0.75 −0.8 −0.85 −0.85 −0.85 −0.9 −0.92 −0.94 −0.95 −0.95 −0.95 −0.95 −0.95 −0.95

0.2 −0.3 −0.33 −0.33 −0.35 −0.39 −0.43 −0.48 −0.52 −0.56 −0.58 −0.6 −0.62 −0.63 −0.64 −0.66 −0.67 −0.68 −0.69 −0.7 −0.71 −0.72 −0.73

1.2 0.35 0.25 0.24 0.24 0.24 0.24 0.24 0.23 0.23 0.22 0.22 0.21 0.21 0.2 0.19 0.18 0.18 0.17 0.17 0.16 0.16 0.15

1.0 0.57 0.35 0.31 0.31 0.31 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29

0.8 0.8 0.52 0.38 0.35 0.33 0.31 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3

0.8 0.8 0.8 0.45 0.39 0.36 0.34 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32

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Energy & Fuels respectively. For CO2, the bands centered at 667 and 2413 cm−1 are changed to 750 and 2013 cm−1, respectively. Since the pressure-relating parameters must be adjusted to fit the LBL results, the modified band positions are assumed not to change against pressure, to simplify the modification process. (2) The second step is to fit the pressure parameters of CO2 and CO against the LBL results to generate good predictions at different temperatures and different total pressures. The selfbroadening parameter of CO2 for the band centered at 750 cm−1, bCO2,750, and the fitting parameters of CO for the bands centered at 2143 and 4260 cm−1 (nCO,2143 and nCO,4260, respectively) are modified to fit the discrete LBL results in the temperature and pressure ranges concerned in this work. The corresponding modified values of bCO2,750 are listed in Table 1, and those of nCO,2143 and nCO,4260 are listed in 2. The values at other positions can be obtained via interpolation. The self-broadening parameters of CO for the modified bands are set at a constant value of 3. (3) The third step is to modify the pressure parameters of H2O, so that that the model can give good prediction for the mixed medium components. The fitting parameters of H2O for the bands centered at 180 and 1600 cm−1 (nH2O,180 and nH2O,1600, respectively) are modified to fit the discrete LBL results at different temperatures, pressures and H2O mole fractions concerned in this work. The corresponding modified values are given in Tables 3 and 4, respectively. The values at other positions can be obtained via interpolation. The corresponding self-broadening parameters for the modified bands are set at a constant value of 3. (4) Finally, the fifth-order polynomial fitting scheme (a0 + a1T + a2T2 + a3T3 + a4T4 + a5T5) is used to simplify the calculation of the time-consuming parameters, αin(T) and βin(T). The coefficients of the fifth-order polynomial for αin(T) and βin(T) of the medium components are listed in Tables 5−7. Since the fifth order

Table 4. Fitting Parameters of H2O (nH2O,180 and nH2O,1600) against Temperature (T) and Pressure (P) at H2O Mole Fractions of 1/2 and 1/9 nH2O,180 and nH2O,1600 H2O mole fraction = 1/2

H2O mole fraction = 1/9

T (K)

P= 1 atm

P= 10 atm

P = 30 atm

P= 1 atm

P= 10 atm

P= 30 atm

300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500

−0.52 −0.8 −0.25 0.15 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

0.15 0.2 0.46 0.65 0.67 0.63 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55

0.085 0.18 0.41 0.59 0.69 0.56 0.51 0.46 0.41 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35

−2.2 −4.4 −5.7 −7 −8.5 −10 −10 −10 −10 −10 −10 −10 −10 −10 −10 −10 −10 −10 −10 −10 −10 −10 −10

0.33 0.19 0.19 0.25 0.25 0.2 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11

0.26 0.18 0.25 0.31 0.3 0.26 0.22 0.18 0.17 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

of temperature can be a very big value, the significant digits are kept to ensure that the calculation is accurate. To further improve the

Table 3. Fitting Parameters of H2O (nH2O,180 and nH2O,1600) against Temperature (T) and Pressure (P) at a H2O Mole Fraction of 1 nH2O,180 and nH2O,1600 T (K)

P = 0.1 atm

P = 0.25 atm

P = 0.5 atm

P = 0.75 atm

P = 1 atm

P = 5 atm

P = 10 atm

P = 20 atm

P = 30 atm

300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500

0.1 0.07 0.05 0.01 −0.01 −0.05 −0.15 −0.2 −0.3 −0.45 −0.8 −2 −5 −5 −5 −5 −5 −5 −5 −5 −5 −5 −5

−0.7 −0.6 −0.6 −0.7 −0.5 −0.4 −0.3 −0.2 −0.3 −0.5 −0.7 −1 −2 −2.5 −3 −4 −5 −5 −5 −5 −5 −5 −5

1.5 1.3 1.3 1.1 0.8 0.55 0.3 0.2 0.05 0.05 −0.05 −0.05 −0.05 −0.05 −0.05 −0.05 −0.05 −0.05 −0.05 −0.05 −0.05 −0.05 −0.05

1 0.9 0.85 0.75 0.6 0.4 0.25 0.15 0.05 −0.1 −0.1 −0.15 −0.2 −0.2 −0.2 −0.2 −0.2 −0.2 −0.2 −0.2 −0.2 −0.2 −0.2

0.9 0.72 0.65 0.55 0.4 0.28 0.15 0.05 −0.05 −0.1 −0.12 −0.2 −0.2 −0.2 −0.2 −0.2 −0.2 −0.2 −0.2 −0.2 −0.2 −0.2 −0.2

0.75 0.63 0.61 0.56 0.49 0.42 0.35 0.3 0.27 0.27 0.3 0.35 0.4 0.8 1.5 4 4 4 4 4 4 4 4

0.68 0.58 0.58 0.51 0.45 0.4 0.35 0.31 0.28 0.32 0.38 0.5 0.8 4 4 4 4 4 4 4 4 4 4

0.6 0.5 0.5 0.46 0.41 0.36 0.31 0.29 0.29 0.29 0.35 0.65 1.5 4 4 4 4 4 4 4 4 4 4

0.56 0.48 0.47 0.43 0.38 0.33 0.29 0.26 0.26 0.29 0.36 0.65 1.5 4 4 4 4 4 4 4 4 4 4

1638

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Energy & Fuels Table 5. Polynomial Coefficients for αin(T) and βin(T) of H2O α H2O,180

α H2O,3300

α H2O,4350

a0 a1 a2 a3 a4

−3.87977423154383 × 10 1.34730020931322 × 100 2.67163949708715 × 10−3 −1.26504030833507 × 10−6 1.89631354117659 × 10−10

2.49037869295819 × 10 −8.99238158678600 × 10−5 1.64829094074161 × 10−7 −5.28205247334168 × 10−11 6.62991026264585 × 10−15

α H2O,7250

βH2O,180

βH2O,1600

a0 a1 a2 a3 a4

2.50218262400667 × 100 2.97766456031507 × 10−5 −1.57345799906255 × 10−7 1.84823321136002 × 10−10 −3.35286970212567 × 10−14

1.20834413771582 × 10−1 −1.68173067295022 × 10−4 1.37749114005749 × 10−7 −5.35070116763474 × 10−11 7.77978695598821 × 10−15

7.17425645933008 × 10−2 −8.39254822472201 × 10−5 1.19579583015509 × 10−7 −1.78041460192485 × 10−11 4.46284709442596 × 10−15

βH2O,3300

βH2O,4350

βH2O,7250

a0 a1 a2 a3 a4

1.87103019554815 × 10−1 −2.35855403871490 × 10−4 2.86269408261456 × 10−7 −4.14370576029613 × 10−11 1.24924522464568 × 10−14

6.18222648221342 × 10−2 −6.98110337488596 × 10−5 9.41559212526602 × 10−8 −3.52797202797194 × 10−12 3.33003952569168 × 10−15

9.15806648637394 × 10−2 −1.13909565386302 × 10−4 1.30629147975084 × 10−7 −9.70315636562658 × 10−12 5.60018864938755 × 10−15

2

3.10893882003329 × 100 −6.86716710938835 × 10−4 1.20133856617577 × 10−6 −3.06096050986908 × 10−10 3.24441217313088 × 10−14

1

Table 6. Polynomial Coefficients for αin(T) and βin(T) of CO2 a0 a1 a2 a3 a4

α CO2,960

α CO2,1060

α CO2,3660

−7.73302891617955 × 10−4 −6.57483064930208 × 10−5 2.50974426028457 × 10−7 −1.19611396752242 × 10−10 1.88719704526339 × 10−14

−4.35840898690673 × 10−4 −6.80138154792047 × 10−5 2.55316705214438 × 10−7 −1.21339578321500 × 10−10 1.91151116871940 × 10−14

4.20660441547744 × 100 −1.40654528857139 × 10−3 2.71499045391433 × 10−6 −8.12770475570744 × 10−10 9.70305235065027 × 10−14

βCO2,750

βCO2,960

α CO2,5200 −2

−2

2.49100072810502 × 10−2 3.22947907165267 × 10−5 2.97692970454435 × 10−8 2.48645754321955 × 10−10 −1.56046337339242 × 10−14

a0 a1 a2 a3 a4

7.05051676929481 × 10 −3.23490828688234 × 10−5 6.24778815927289 × 10−8 −9.12333127081494 × 10−12 1.24010879788926 × 10−15

4.43787740794668 × 10 6.45557396709520 × 10−6 2.50014610548365 × 10−7 1.68563172652417 × 10−10 −1.10815601891116 × 10−14

βCO2,1060

βCO2,2013

βCO2,3660

a0 a1 a2 a3 a4

7.37176889952210 × 10−2 9.55576554846003 × 10−5 8.81306814403593 × 10−8 7.35833233763436 × 10−10 −4.61789119855474 × 10−14

1.80052189931346 × 10−1 3.85120470592315 × 10−5 5.37000867235980 × 10−7 9.99515821833905 × 10−10 −7.11405553510824 × 10−14

9.25981420844631 × 10−2 5.41990748623299 × 10−5 1.72456835206498 × 10−7 9.14720724228731 × 10−10 −6.72079801713665 × 10−14 βCO2,5200

2.79080794258378 × 10−1 1.61619756764947 × 10−4 2.95143896454453 × 10−7 3.63121790337807 × 10−9 −2.80698233879009 × 10−13

a0 a1 a2 a3 a4

Table 7. Polynomial Coefficients for αin(T) and βin(T) of CO α CO,4260 a0 a1 a2 a3 a4 a5

βCO,2413 −1

1.44692785978783 × 10 −2.30877149500734 × 10−5 2.18486760093608 × 10−8 1.98584173009685 × 10−11 −1.16166207933901 × 10−14 1.79360050069376 × 10−18

βCO,4260 −2

6.98469244851239 × 10 −1.39192120545355 × 10−4 2.10920083203089 × 10−7 −1.36246539057178 × 10−10 4.26136252509244 × 10−14 −5.18639128478939 × 10−18

computation efficiency, the table look-up scheme is used to simplify the calculation of the blackbody fractional function. With this scheme, the blackbody intensity is first calculated against the discrete wavenumber and temperature to create a three-dimensional

1.56946216767210 × 10−1 −3.23821485097401 × 10−4 5.14392394238096 × 10−7 −3.31065482318339 × 10−10 1.02864079094055 × 10−13 −1.24504684230083 × 10−17

table, which can then be referenced by the computation code to avoid the direct calculation of the blackbody fractional function. 3.3. The Performance of M-EWBM. After modification, the performance of the M-EWBM is assessed by comparing its 1639

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Figure 1. Total emissivities of H2O calculated by LBL, M-EWBM, and EWBM.

Figure 2. Total emissivities of CO2 calculated by LBL, M-EWBM, and EWBM.

Figure 1 also shows that the highest total emissivity appears at a temperature of ∼600 K, and increasing the total pressure can gradually increase the total emissivity of H2O over the entire pressure range investigated. Figure 2 shows that the total emissivities of CO2 obtained by M-EWBM predictions agree very well with the LBL results over the entire temperature and pressure ranges. The maximum relative error between the M-EWBM prediction and the LBL prediction at different total pressures and temperatures is 8.67%. The EWBM can give good predictions at temperatures of >1000 K, while big deviations between the EWBM predictions and the LBL results are detected at lower temperatures. It can also be detected that the highest total emissivity appears at ∼1000 K, based on the results of the LBL model, and increasing the total pressure has a very slight effect on the total emissivity of CO2. Figure 3 shows that the total emissivities of CO calculated by M-EWBM predictions agree very well with the LBL results in the temperature and pressure ranges investigated. The maximum

predictions with the LBL results and the results generated by the original EWBM. The comparisons of the total emissivities of H2O, CO2, and CO calculated with the three models against temperature and pressure are depicted in Figures 1−3. The comparisons of the total emissivities of H2O/CO2 mixtures against temperature, pressure, and H2O/CO2 molar ratio are depicted in Figure 4. The comparisons of the total emissivities of H2O/CO2/ CO mixture against temperature, pressure are depicted in Figure 5. All the total emissivities in the comparisons are calculated for the unit pressure path length.36 Figure 1 shows that the total emissivities of H2O predicted by M-EWBM agree very well with the LBL results and are more accurate than the EWBM. The maximum relative error between the M-EWBM prediction and the LBL prediction is 3.81% when the total pressure is higher than 0.25 atm at different temperatures. The EWBM can give pretty good results at relatively higher pressures, but big deviations between the EWBM predictions and the LBL results are detected when the total pressure is 10 atm. Figure 5 shows that both the EWBM and M-EWBM can give reasonable predictions for the H2O/CO2/CO mixture. The maximum relative error between the M-EWBM prediction and the LBL prediction at different total pressures and temperatures

relative error between the M-EWBM prediction and the LBL prediction at different total pressures is 2.01% when the temperature is >500 K. However, the EWBM results deviate greatly from the LBL results, especially at higher temperatures in the entire pressure range. The highest total emissivity appears at ∼800 K, and increasing the total pressure can increase the total emissivity of CO. When the pressure is 0.5 atm, the EWBM has a tendency to overpredict the total emissivity. It can also be detected from Figures 1−3 that the effect of pressure on the total emissivities is larger at lower temperatures. The total emissivity is observed to be more sensitive to the temperature than to the total pressure. For H2O and CO2, when the total pressure is >10 atm, further increases in pressure lead to 1641

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Figure 5. Total emissivity of the H2O/CO2/CO mixture calculated by LBL, M-EWBM, and EWBM.

commercial or open-source computational fluid dynamics (CFD) software. (3) The gas total emissivities are more sensitive to the temperature than to the total pressure. The effect of pressure on the total emissivities is greater at lower temperatures. (4) The total emissivity increases as the total pressure increases, and the change becomes insignificant when the pressure is >10 atm.

is 3.97%. The M-EWBM is more accurate than the EWBM in comparison. When the total pressure is >10 atm, increasing the pressure has only a minor effect on the total emissivity of the mixture. From all the above comparisons, it can be concluded that the M-EWBM is more accurate than the EWBM and can give satisfying predictions for the total emissivities of both the individual gases and their mixtures in a wide range of temperature and pressure. In addition to the accuracy, the time intensity is also very essential. To test the computation efficiency of the M-EWBM proposed in this work, the time consumptions of the LBL model, as well as the EWBM, M-EWBM, and WSGGM, are compared using a personal computer with an Intel Core i7−4810MQ CPU. The LBL model requires more than 10 h to generate one emissivity datum. The CPU time of the EWBM, M-EWBM, and WSGGM are 18.45, 0.052, and 0.0095 ms, respectively. It can be seen that the M-EWBM has similar efficiency with the WSGGM, making it ready to be embedded in the commercial or opensource CFD software such as the WSGGM through the userdefined function. For example, when embedding the M-EWBM model to the commercial CFD software Fluent, the DEFINE_EXECUTE_ON_LOADING macro can be used to generate the table for the blackbody fractional function, and the values are stored in a two-dimensional (2D) array. Then, the 2D array is directly used in the subsequent calculation, so that the computation time consumption is further reduced.

■

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (Linbo Yan). *E-mail: [email protected] (Boshu He). ORCID

Linbo Yan: 0000-0001-7996-7908 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors gratefully acknowledge financial support from the National Natural Science Foundation of China (NSFC, No. 51706012) and the Talent Research Start-up Fund of Beijing Jiaotong University (No. M17RC00030) for this work.

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REFERENCES

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