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Development of an absorption coefficient calculation method potential for combustion and gasification simulations
International Journal of Heat and Mass Transfer 91 (2015) 1069–1077
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Development of an absorption coefficient calculation method potential for combustion and gasification simulations Linbo Yan a, Guangxi Yue a, Boshu He b,⇑ a b
Department of Thermal Engineering, Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Tsinghua University, Beijing 100084, China Institute of Combustion and Thermal System, School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing 100044, China
a r t i c l e
i n f o
Article history: Received 10 February 2015 Received in revised form 15 August 2015 Accepted 15 August 2015
Keywords: Absorption coefficient Total emissivity Line by line (LBL) Exponential wide band (EWB) model Efficient exponential wide band (E-EWB) model Computational fluid dynamics (CFD)
a b s t r a c t Efficient and accurate calculation of the absorption coefficient is very important to the modeling of combustion and high-temperature gasification where the thermal radiation plays an important role. It is therefore important to develop a radiative absorption coefficient calculation method which is accurate enough, universal but not time consuming. In this work, a computationally efficient exponential wide band (E-EWB) model based on the exponential wide band (EWB) model is put forward and programmed so that it can be integrated into the computational fluid dynamics (CFD) software. The E-EWB calculation results are then validated against the results obtained via the line by line (LBL) model, and the runtime of this model is also evaluated. Through this work, the E-EWB model is found effective and reliable. The maximum deviation between the results obtained by the E-EWB model proposed in this work and the LBL results is within 0.8%. Moreover, the calculation speed of this model can be about 350 times faster than that of the original EWB model and is fast enough to be applied to the combustion and gasification simulations. Thus, the E-EWB model is a good choice to do the future absorption calculations during the combustion and gasification CFD simulations. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Numerical simulations of the combustion and gasification processes are very important to the design and optimization of combustors, gasifiers and the corresponding operating parameters. For almost all the combustion processes and most of the gasification ones, the operation temperatures are usually high and the radiative energy occupies a great part of the total heat flux. It is therefore important to address the radiative heat transfer carefully for the numerical simulations of these cases. Besides the numerous radiation models which can affect the accuracy of the radiative heat transfer calculation, one of the material physical property parameters used in the radiative transfer equations (RTE), the absorption coefficient a, is also essential to the accurate radiant energy calculation [1]. In most of the recent computational fluid dynamic (CFD) packages, the weighted sum of gray gas model (WSGG) is used to calculate a. Although the WSGG model is very efficient and generates acceptable errors for the ranges of temperatures and pressures that cover almost all the traditional combustion and gasification processes, there are also a lot of inherent defections. First, it is easy for the WSGG model to consider ⇑ Corresponding author. Tel.: +86 10 5168 8542; fax: +86 10 5168 8404. E-mail address: [email protected] (B. He). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.08.047 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
two radiative species at one time, e.g., H2O and CO2, and the corresponding parameters of this model are calculated in terms of the concentration ratio of the two species [2]. If there are more radiative species, the calculation with the WSGG model will be very complex and even not feasible. However, for many gasification processes, the effect of other species like CO and CH4 on the total radiation properties cannot be avoided since they are of parallel amount of H2O and CO2 and even more. Second, the application of the WSGG model is limited to the volume of the homogeneous gas, whose characteristic length should be greater than 1 cm [3]. However, in order to accurately simulate the flame front and capture the denotation wave, the grid characteristic length should be less than 0.5 mm [4]. Third, the original WSGG model is only proper for a narrow range of H2O to CO2 concentration ratio [5]. For the simulation of the oxy-fuel combustion in which the concentration of CO2 is very high, the original WSGG model will lost its accuracy. Yin once proposed a refined WSGG model for the conventional air–fuel combustion and found remarkable difference between their model and the existing model in the CFD software if the particle radiation was not considered [6]. Kangwanpongpan once proposed an optimized WSGG model for the oxy-coal combustion and found that the flame temperature can be better predicted by their model [7]. Kangwanpongpan once proposed another WSGG model based on the new HITEMP-2010 spectral
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Nomenclature
Parameters total band absorptance [cm1] Ain polynomial coefficients b1–9 bin self-broadening coefficient of species i and band n C the speed of light [cm/s] C2 Planck second radiation constant [cm K] lower state energy of the transition [cm1] En g F T fractional function of blackbody radiation degeneracy number of the fundamental band gd parameter calculated with Eq. (11) gv,max g(N N0) line shape function [cm] Ibg black body intensity at wave number g [W m2 Sr1/ lm] IbDg;n mean block intensity of the nth band [W m2 Sr1/lm] L beam length [cm] Mw molecular weight [kg/kmol] n coefficient of temperature dependence of air broadened half width nin the fitting parameter of species i and band n Ni molecular concentration of species i [molecules/(cm3 atm)] P0 the reference pressure [1 atm] Pe,in dimensionless equivalent broadening pressure of species i band n Pi partial pressure of species i [atm] Pt total gas pressure [atm] Q(T) total internal partition function R the universal gas constant [8.3145 J/(K mol)] Sk line intensity [cm1/molecule cm2] T gas temperature [K] T0 reference temperature [K] xi mole fraction of species i X pressure path length [atm cm] Greek symbols a0 reference band intensity [cm/(g/m2)] ae effective absorption coefficient [m1] ain integrated band intensity [cm/(g/m2)] ap Planck mean absorption coefficient [(cm atm)1] ag spectral absorption coefficient [(cm atm)1]
database for the oxy-fuel combustion and validated their model with the line by line method [8]. Although many new WSGG parameters have been put forward by researchers so that this model can be applied to the oxy-fuel combustion calculation, the model is still mainly used in the cases where CO2 and H2O are the dominant absorptive gases. In addition to the WSGG model and its variants, there are also many other models can be used to calculate the absorption coefficient. These models can be generally classified as the line by line (LBL) model, the band models and the global models [9]. The band interval of the LBL model is generally 0.0002–0.02 cm1. Thus, the LBL model is the most accurate but very time consuming. It is hence usually used as the benchmark to validate the other radiation models. The band models usually refer to the narrow band model (NBM) and the wide band model (WBM). The band interval of the narrow band model is generally 5–50 cm1. In principle, the narrow band model can be as accurate as the line by line model. This model, however, is also very time consuming and is hard to be applied to the nonhomogeneous media. The band interval of the wide band model is generally 100–1000 cm1. It is much faster than the LBL model and the
ag,n b0 bin
cair cD cp cself cV dd
et g N0 gin gLm gU,m m0,d md sg,in sDg,m U(T)
vi w(T) x0 xin
absorption coefficient contributed by the nth band [(cm atm)1] reference mean line width to spacing ratio the mean line width to spacing ratio parameter air-broadened half width parameter [(cm atm)1] half width at half height of the Doppler feature [cm1] half width at half height of the Lorentzian feature [cm1] self-broadened half width [(cm atm)1] half width at half height of the Voigt feature [cm1] effects of the photon transition on the vibrational quantum number total emissivity wave number [cm1] wave number at the special line center [cm1] nth band width of species i [cm1] lower limits of block m [cm1] upper limits of block m [cm1] the lowest possible initial state the vibrational quantum number transmissivity of species i and band j the mth block transmissivity temperature dependent intermediate parameter for calculating bin mass path length of species [g/m2] temperature dependent intermediate parameter for calculating ai reference band width [cm1] the band width parameter [cm1]
Abbreviations CFD computational fluid dynamics E-EWB efficient EWB EWB exponential wide band LBL line by line LSF line shape function RTE radiative transfer equation WSGG weighted sum of gray gas model UDF user defined function
narrow band model with slight accuracy lost. The global models calculate the radiation properties over the whole spectral range [10]. One of the well-known global models is the WSGG model. Based on the WSGG model, a spectral line based WSGG (SLW) model has been improved by Denison and Webb [11]. Modest and Zhang have also developed a full spectrum k-distribution (FSK) model [12]. There are also some other global non-gray gas models such as the absorption distribution function (ADF) model [13] and the spectral-line moment-based (SLMB) model [14]. The SLW and the FSK of the global models are very promising because they have been extensively developed for gas radiation calculations in nonhomogeneous gas mixtures [15]. When using the FSK model for the nonhomogeneous media, gas absorption coefficients must be assumed to be scaled or correlated before the reordering process. But it is reported that for strong temperature and gas composition variations, the assumption will be challenged [16]. Most of these models can avoid the disadvantages of the WSGG model. However, most of them are either computationally uneconomical or not well and sufficiently developed, so they are not chosen to calculate the absorption coefficient for the radiative heat transfer
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in the CFD calculation. Among these advanced models, the exponential wide band (EWB) model is by far the most successful wide band models and is also very popular for its relatively simple correlation structure [9]. In this work, a computationally efficient scheme to calculate the absorption coefficient is used and programmed so that it can be integrated into the CFD software through the user defined function (UDF). The scheme is based on the EWB model and the calculation results are validated against the results calculated by the LBL and EWB models. Since some key reference EWB parameters were fitted against the emissivity data [17], some of the reference EWB parameters are recalculated in this work based on the LBL method and the HITEMP-2010 [18] spectroscopic database which contains much more hot lines than the HITRAN [19] database and is more accurate for the high temperature radiation calculation. Then, mathematical regression is done to simplify the EWB model correlations to generate the efficient EWB (E-EWB) model. The E-EWB model is accurate, time aving and can be generally used. It can be integrated into the commercial CFD packages to calculate the absorption coefficients for both the engineering and scientific research usage. It should be noted that the HITEMP database only contains five species up to now and CH4 have not been included yet, so the radiative properties of H2O, CO2 and CO are recalculated. Although only four species which usually the dominant ones for most of the combustion and gasification cases are considered in this work, the number of species in the calculation can be expanded arbitrarily. Since the E-EWB model in this work is based on the EWB model, it will share some weakness of the EWB model. For example, the EWB correlations are less accurate compared with the line by line method and the narrow band model. Moreover, when the EWB model is used for the nonhomogeneous media, the model should be modified. In fact, if the E-EWB model parameters are obtained by fitting the results from the LBL model with the HITEMP-2010 spectral database rather than the results from the original EWB model, the E-EWB model results can be as accurate as the LBL model results in the fitting temperature range. As has been mentioned, parameters for calculating the radiation properties of H2O, CO2 and CO have been updated except CH4. Thus, the accuracy of calculation for CH4 in the E-EWB model will be limited by the original EWB model. 2. LBL model calculation methodology For the engineering application of radiation heat transfer, the total emissivity, et, and the effective absorption coefficient, ae, are usually concerned [20]. It is believed that the LBL calculation results are the most accurate ones and can be treated as the benchmark data. So, in this section, the LBL calculations are implemented to generate the reference emissivity and absorption coefficient database. The effective absorption coefficient can be calculated from the total emissivity, so the absorption coefficient generated with the LBL method is, instead, the Planck mean coefficient [21], ap, which is also one of the most important mean absorption coefficients and is used to adjust the reference EWB parameters along with et in this work. Recently, there are two high resolution spectroscopic databases for the LBL calculation, the HITRAN and the HITEMP. The HITEMP database contains more hot lines and it is more accurate for the high temperature calculation, while the HITRAN database can only be accurate enough for the temperature lower than 750 K [22]. The HITRAN database contains 47 species while the HITEMP database only contains 5 including H2O, CO2, CO, NO and OH so far. Since the combustion and gasification always run at high temperatures, the HITEMP database is therefor used for the calculation of the radiative parameters. The Planck mean absorption coefficient is calculated with Eq. (1) [23].
R1 a I dg ap ¼ R0 1 g bg I dg 0 bg
ð1Þ
where, ag and Ibg are the absorption coefficient and black body intensity at wave number g. The total emissivity can be calculated with Eq. (2) [24]. Based on the total emissivity, the effective absorption coefficient, ae , can then be calculated with Eq. (3) [25].
R
1 e I dg et ¼ R01 g bg ¼ I dg 0 bg
R1 0
Ibg ð1 eag X Þdg R1 I dg 0 bg
ð2Þ
1 lnð1 et Þ L
ae ¼
ð3Þ
where, X denotes the pressure path length and equals piL; pi is the partial pressure of species i and L is the beam length. The spectral absorption coefficient, ag , is the convolution of the line intensity and the line shape factor, and can be calculated with Eq. (4) [26].
ag ¼
X Sk gðg g0 ÞNi
ð4Þ
k
where, Sk denotes the line intensity and can be obtained from the spectroscopic database. It should be noted that the line intensity given in the database are based on the reference temperature of 296 K [27]. The values of line intensity at any temperature are calculated with Eq. (5). n is the number of the radiation line. Ni is the molecular concentration of species i and can be calculated based on the Loschmidits number. g(N N0) is the line shape function (LSF). N and N0 are the wave numbers at any position and at the special line center, respectively. LSF caused by Doppler broadening is calculated with Eq. (6); LSF caused by pressure broadening is calculated with Eq. (7); LSF caused by the combination of the two broadening effects is calculated with Eq. (8) [28].
C E C 2 gi T QðT 0 Þ exp T2 n 1 exp Si ðTÞ ¼ Si ðT 0 Þ Q ðTÞ exp C 2 En 1 exp C 2 gi T0
ð5Þ
T0
where, Q(T) is the total internal partition function which accounts for the distribution of various energy states of molecules in thermodynamic equilibrium [27]; C2 denotes the Planck second radiation constant; En denotes the lower state energy of the transition; T and T0 are the gas temperature and the reference temperature, respectively.
g D ðg g0 Þ ¼
1
1=2 ln 2
cD
p
"
exp ln 2
ðg g 0 Þ2
#
ð6Þ
c2D
where, cD denotes the half width at half height of the Doppler feature and can be calculated with Eq. (9).
g P ðg g0 Þ ¼
cP =p ðg g0 Þ2 þ c2P
ð7Þ
where, cp denotes the half width at half height of the Lorentzian feature and can be calculated with Eq. (10).
" # g V ðg g0 Þ cP g g0 2 exp 2:772 ¼ 1 g V;max cV 2cV 2 c 1 cP cP þ P 2 þ 0:016 1 cV 1 þ 4 gg0 cV cV 2cV 8 9 " > > # < = g g0 2:25 10 þ exp 0:4 2:25 > > 2cV : ; 10 þ g2cg0 V
ð8Þ
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where, g V;max can be calculated with Eq. (11); cV denotes the half width at half height of the Voigt feature and can be calculated with Eq. (12).
cD ¼
g0 2RT ln 2 c
1=2
ð15Þ
where, n is calculated with Eq. (16).
ð9Þ
Mw
where, c is the speed of the light; R is the universal gas constant; and Mw is the molecular weight.
n T cP ¼ ½cair ðP t Pi Þ þ cself Pi 0 T
ð10Þ
where, cair and n are the air-broadened half width parameter and the coefficient of temperature dependence of air broadened half width [24]. cself is the self-broadened half width. These two parameters, cair and cself, can be found in the spectroscopic database. Pt is the total gas pressure. Pi denotes the partial pressure of species i.
g V;max ¼
" !# 1 15 X ejn 3 3n2 6n 6 F ¼ 4 n þ þ 2 þ 3 T p j¼1 j j j j g
1 2 2cV 1:065 þ 0:447 ccP þ 0:058 ccP V
ð11Þ
V
n¼
C2g T
ð16Þ
The total band absorptance, Ain, and the transmissivity, sg;in , of species i and band n can be calculated with the three-region (linear region, square root region and logarithmic region) expressions which are not listed here [29]. To solve the three-region expressions, the following five parameters should be calculated firstly. They are the mass path length of species i, vi; the dimensionless equivalent broadening pressure of species i band n, Pe,in; the band width parameter xin; the integrated band intensity ain; and the mean line width to spacing ratio parameter bin. The five parameters can be calculated with Eqs. (17)–(21) [25].
vi ¼
xi PT M wi L RT
ð17Þ
where, xi is the mole fraction of species i. 2
2 1=2
2cV ¼ ð0:5346Þð2cP Þ þ ½0:2166ð2cP Þ þ ð2cD Þ
ð12Þ
The choice of the line shape functions can be made according to the value of cP/cV. When the value is less than 0.1, the Gaussian line shape is used. When the value is greater than 0.1 and less than 5, the Voigt profile is chosen. When the value is greater than 5, the Lorentzian profile can be used [27]. The mathematical models mentioned above are then programmed to calculate ap and et based on the HITEMP-2010 database. In order to accelerate the LBL calculation speed, parallel calculation scheme is used. The continuous wave number for the integration is divided into several parts but the spectral line database should not be divided. Then, the divided parts can be calculated separately so that the computer source can be fully utilized. Moreover, some parts can usually be neglected since the blackbody intensity is very small in the wave number ranges at given temperature according to the Wien displacement law [21]. Thus, the computation amount can be reduced and the efficiency can be elevated.
where, ag;n denotes the absorption coefficient contributed by the nth band; IbDg;n is the mean block intensity of the nth band.
X gL;m gU;m ð1 sDg;m Þ F F T T m
T T0
xin ðTÞ ¼ x0
ð19Þ
h
P i 1 exp Dd¼1 nd dd wðTÞ P i ain ðTÞ ¼ a0 h 1 exp Dd¼1 n0;d dd wðT 0 Þ
ð20Þ
where, dd describes the effects of the photon transition on the vibrational quantum number and can be found in the reference EWB parameter table [9]. a0 is the reference band intensity. w(T) in Eq. (20) is defined by Eq. (22).
T T0
0:5
UðTÞ UðT 0 Þ
ð21Þ
where, b0 is the reference mean line width to spacing ratio parameter. U(T) in Eq. (21) is defined by Eq. (23).
QD P1 wðTÞ ¼
d¼1
md ¼m0;d
QD P1
ðmd þg d þjdd j1Þ! nd md e ðg d 1Þ!md !
ð22Þ
ðmd þg d 1Þ! n m d d md ¼0 ðg d 1Þ!md ! e
where, md is the vibrational quantum number; gd denotes the degeneracy number of the fundamental band and they can be found in the reference EWB model table [9]. m0;d denotes the lowest possible initial state and is equal to zero when dd is nonnegative, otherwise it equals to the absolute value of dd .
QD P 1
ð14Þ
where, sDg;m is the mth block transmissivity, it is the products of all the band transmissivities, sg;im , that belong to this block; and gL;m and gU;m are the lower and upper limits of block m and they are determined according to the band center wave number and the calculated nth band width of species i, Dgin . Dgin can be calculated using the total band absorptance, Ain, and the band transmissivity. F gT is the fractional function of blackbody radiation and can be calculated with Eq. (15) [32].
0:5
where, x0 is the reference band width parameter.
d¼1
ð13Þ
ð18Þ
where, P0 denotes the reference pressure which is 1 atm; bin and nin are the self-broadening coefficient and the corresponding fitting parameter of species i and band n, respectively.
bin ðTÞ ¼ b0
The EWB model was developed by Edwards & Menard [29] and Edwards & Balakrishnan [30]. It is much faster than the LBL and NBM and the results are accurate enough, so it has a good potential to be imbedded into the CFD package. The calculations of ap and et with the band model can be cast into Eqs. (13) [31] and (14) [32].
et ¼
nin PT Pi þ ðbin 1Þ P0 P0
3. EWB model calculation methodology
R1 Z X IbDg;n a I dg 1 X 1 ap ¼ R0 1 g bg ¼ ag;n Ibg;n dg ¼ ain Ib n 0 Ib I dg n 0 bg
Pe;in ¼
UðTÞ ¼
d¼1
md ¼m0;d
QD P 1 d¼1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 n m
md ¼m0;d
ðmd þg d þjdd j1Þ!e ðg d 1Þ!md !
d d
ðmd þg d þjdd j1Þ! n m e dd ðg d 1Þ!md !
ð23Þ
With the above correlations, ap and et can then be calculated from the EWB model. In order to calculate the two parameters more accurately, some of the reference band intensities are recalculated based on the LBL calculation results in this work. Some literature once gave different reference band intensity values, but the values were not accurate enough compared with the LBL results obtained
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from the HITEMP database [33]. In addition, the reference parameters were only fitted by emissivity. When they are used to calculate the Planck mean absorption coefficient which is also one very important radiation parameter, substantial errors will be caused. In this work, the reference rotational band intensity of H2O and some reference band intensities which once were constant at different temperatures are adjusted to fit ap, as well as et in the temperature range of 500–2000 K. For H2O, the reference band intensities for the bands centered at 140 cm1 and 1600 cm1 are recalculated. For CO2, the reference band intensities for the band centered at 667 cm1 and the band with upper head wave number limit of 2410 cm1 are recalculated. For CO, the reference band intensity for the band centered at 2143 cm1 is recalculated. Since the integrated band intensity is only the function of temperature as can be seen from Eq. (21), the recalculated reference band intensities are considered as the function of gas temperature and the recalculating polynomial correlations proposed in this work are shown by Eq. (24). The corresponding polynomial coefficients are listed in Table 1. Since CH4 has not been contained in the HITEMP database, the reference band values of CH4 are not
adjusted. During our calculation, it is found that the accuracy of
ap and et cannot be met simultaneously for CO using the EWB model, so only the total emissivity of CO is refined.
a0;k ¼ b0 þ b1 T þ b2 T 2 þ b3 T 3 þ b4 T 4 þ b5 T 5 þ b6 T 6 þ b7 T 7 þ b8 T 8 þ b9 T 9
ð24Þ
4. E-EWB model calculation methodology The calculation speed of EWB model is much faster than those of the LBL and NBM. It is, however, still too computationally intensive to be applied to the CFD calculation. In order to further improve the calculation speed of EWB model, mathematical regressions are needed according to the characteristic of this model and the E-EWB model is proposed in this work. In the original EWB model calculation, the time consuming parts are the calculation of ain ðTÞ and bin ðTÞ as defined by Eqs. (20)–(23). It is assumed in the EWB model that ain ðTÞ, bin ðTÞ and xin ðTÞ are solely functions of temperature and the effect of pressure is accounted for through
Table 1 Polynomial coefficients for the modified reference band intensities.
a0;H2 O;140 a0;H2 O;1600 a0;CO2 ;667 a0;CO2 ;2410 a0;CO;2143
b0
b1
b2
b3
b4
b5
b6
b7
b8
b9
28119.6 594.3822 1300.236 14692.34 2.62492
199.5666 4.39809 9.50528 71.3231 0.086348
0.53632 0.013095 0.030733 0.16175 0.000162
0.0007868 2.02E05 5.7E05 0.000205 1.317E07
6.72E07 1.769E08 6.82E08 1.544E07 4.4E11
3.303E10 8.89E12 5.3E11 6.86E11 2.37E15
8.6E14 2.38E15 2.71E14 1.66E14 5.36E18
9.26E18 2.62E19 8.7E18 1.7E18 9.9E22
– – 1.6E21 – –
– – 1.3E25 – –
Table 2 Polynomial coefficients for ain ðTÞ.
aH2 O;3760 aH2 O;5350 aH2 O;7250 aCO2 ;960 aCO2 ;1060 aCO2 ;3660 aCO2 ;5200 aCO;4260 aCH4 ;4220 aCH4 ;5861
b0
b1
b2
b3
b4
b5
b6
b7
24.90321 3.104651 2.495858 0.069745 0.070363 4.162493 0.072587 0.134803 3.049165 0.459851
6.5E05 0.00055 2.7E05 0.0005 0.0005 0.00089 4.2E05 5.227E05 0.001 0.00025
4.75E08 6.09E07 3.03E07 1.24E06 1.25E06 8.28E07 7.55E08 1.97E07 1.99E06 4.75E07
1.718E10 7.648E10 8.77E10 1.25E09 1.25E09 2.395E09 1.17E11 3.39E10 5.8E10 1.4E10
2.12E13 9.68E13 1.102E12 7.304E13 7.318E13 2.8E12 7.33E15 2.7E13 7.6E14 1.1E14
1.141E16 5.106E16 6.27E16 2.53E16 2.54E16 1.442E15 7.894E18 1.206E16 1.25E16 2.23E17
3.1E20 1.4E19 1.74E19 4.83E20 4.87E20 3.7E19 2.7E21 2.9E20 4.1E20 1E20
3.3613E24 1.4812E23 1.929E23 3.91E24 3.97E24 3.968E23 3.396E25 2.92E24 4.71E24 1.38E24
Table 3 The polynomial coefficients for bin ðTÞ.
bH2 O;140 bH2 O;1600 bH2 O;3760 bH2 O;5350 bH2 O;7250 bCO2 ;667 bCO2 ;960 bCO2 ;1060 bCO2 ;2410 bCO2 ;3660 bCO2 ;5200 bCO;2413 bCO;4260 bCH4 ;1310 bCH4 ;3020 bCH4 ;4220 bCH4 ;5861
b0
b1
b2
b3
b4
b5
b6
b7
0.143246 0.087998 0.224847 0.077023 0.113097 3.83E02 2.60E02 7.93E02 1.65E01 8.74E02 2.40E01 0.0796287 0.1805136 0.081379 0.066498 0.340122 0.683294
0.00033 0.00021 0.00051 0.00018 0.00027 3.51E05 1.34E05 2.47E05 7.43E05 3.36E05 1.88E04 0.000223 0.000522 0.00023 0.00019 0.001 0.00215
5.63E07 4.53E07 1.01E06 3.94E07 5.34E07 2.09E07 1.06E07 3.54E07 6.19E07 3.8E07 8.91E07 4.67E07 1.12E06 7.25E07 5.68E07 3.01E06 6.5E06
6.1E10 4.7E10 9.9E10 4E10 5.3E10 1.78E10 1.16E10 2.82E10 6.63E10 4.36E10 1.95E09 5.21E10 1.23E09 7.2E10 5.7E10 2.8E09 6.1E09
4.222E13 3.452E13 6.943E13 2.98E13 3.879E13 1.35E14 1.04E13 3.6E13 3.37E13 4.31E13 1.59E12 3.58E13 8.38E13 8.13E13 5.88E13 2.93E12 6.23E12
1.77E16 1.49E16 2.83E16 1.26E16 1.62E16 2.3E17 5.8E17 2E16 2.3E16 2.7E16 1E15 1.5E16 3.5E16 1.7E16 9E17 4.7E17 4.61E16
4.152E20 3.564E20 6.426E20 2.916E20 3.749E20 7.7E21 1.46E20 5.01E20 6.4E20 7.08E20 2.78E19 3.49E20 8.06E20 2.59E20 8.44E21 6.6E20 3.2E19
4.1E24 3.6E24 6.2E24 2.9E24 3.7E24 9.4E25 1.5E24 5.2E24 6.9E24 7.5E24 3E23 3.5E24 8E24 2E24 2.8E25 9.28E24 3.99E23
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the equivalent broadening pressure, which makes it possible that these temperature dependent parameters and their combinations can be fitted by polynomials with temperature as the argument. Since the calculations of ain ðTÞ and bin ðTÞ are very time consuming, mathematical regressions are implemented for these parameters and the corresponding fitting formula for H2O, CO2, CH4 and CO is shown in Eq. (26). The corresponding polynomial coefficients are listed in Tables 2 and 3. Besides the time consuming calculations of the band parameters, the calculation of the black body fractional function shown by Eq. (16) will also appear time consuming when the calculation runtime is within 1 ms. This is because the solution of Eq. (16) needs at least 50 loops in the program. Thus, in order to further improve the calculation efficient, the table look-up scheme is substituted for the calculation of
Planck Mean Absorption Coefficient of H2O, αp (cm·atm)-1
0.20 0.18 0.16 0.14
Eq. (16) in this work and it is found that this scheme can further reduce the calculation time substantially. Firstly, a black body fractional function table is built with Eq. (16), then the table can be used for any cases and the value needed during the calculation can be obtained by interpolation.
ain ðTÞ ¼ b0 þ b1 T þ b2 T 2 þ b3 T 3 þ b4 T 4 þ b5 T 5 þ b6 T 6 þ b7 T 7 ð25Þ The integrated band intensities of the other bands of these species that are not listed in the table are set as their corresponding reference values. If the values are modified in this work, the modified values should be chosen. If not, the reference values listed in literature can be used [9]. 0.40
LBL EWBM E-EWBM
0.12 0.10 0.08 0.06 0.04 0.02
Planck Mean Absorption Coefficient of CO 2, αp (cm·atm)-1
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0.30 0.25 0.20 0.15 0.10 0.05 0.00
0.00 400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature (K)
Total Emissivity of CO2, εt
Total Emissivity of H2O, εt
LBL EWBM E-EWBM
0.23 0.22 0.21 0.20 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12
400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature (K)
(b) Comparisons of εt of H2O 0.75 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30
LBL EWBM E-EWBM
400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature (K) (b) Comparisons of εt of CO2
0.26
LBL EWBM E-EWBM
400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature (K)
(c) Comparisons of αe of H2O Fig. 1. Comparisons of ap, et and ae of H2O calculated with LBL, EWB and E-EWB models.
Effective Absorption Coefficient for CO2, αe (m-1)
Effective Absorption Coefficient of H2O, αe (m-1)
0.70
400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature (K)
(a) Comparisons of αp of CO2
(a) Comparisons of αp of H2O 0.52 0.50 0.48 0.46 0.44 0.42 0.40 0.38 0.36 0.34 0.32 0.30 0.28
LBL EWBM E-EWBM
0.35
0.24 0.22
LBL EWBM E-EWBM
0.20 0.18 0.16 0.14 0.12
400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature (K)
(c) Comparisons of αe of CO2 Fig. 2. Comparisons of ap, et and ae of CO2 calculated with LBL, EWB and E-EWB models.
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0.48
Total Emissivity of CH4, εt
0.44 0.40
0.32 0.28 0.24 0.20
LBL EWBM E-EWBM
Total Emissivity for CO, εt
0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02
400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature (K) (a) Comparisons of εt of CO
Effective Emissivity of CO, αe (m-1)
0.11 LBL EWBM E-EWBM
0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02
400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature (K)
(b) Comparison of αe of CO Fig. 3. Comparisons of et and ae of CO calculated with LBL, EWB and E-EWB models.
400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature (K)
(a) Comparisons of εt of CH4 0.48 0.44 0.40
EWBM E-EWBM
0.36 0.32 0.28 0.24 0.20 0.16
0.11
EWBM E-EWBM
0.36
0.16
Effective Absorption Coefficient for CH4, αe (m-1)
The calculated ap, et and ae at one atmospheric pressure for H2O and CO2 from the LBL, EWB and E-EWB models are shown and compared in Figs. 1 and 2. As aforementioned, the accuracy of ap and et of CO cannot be satisfied simultaneously in the whole temperature range of 500–2000 K, so only the accuracy of et of CO is considered in this work. This is because in the CFD calculation, et is usually used. For the case where ap is also required, maybe another set of the polynomial coefficients that are accurate for ap can be chosen. The calculated et and ae at one atmospheric pressure for CO from the LBL, EWB and E-EWB models are shown and compared in Fig. 3. Since the spectral data of CH4 have not been included in the HITEMP-2010 database, the reference band intensities for CH4 are not adjusted. The calculated et and ae at one atmospheric pressure for CH4 from the EWB and E-EWB models are shown and compared in Fig. 4. From Fig. 1, it can be seen that the EWB model with the original band parameters can cause substantial errors when calculating ap of H2O when the temperature is lower than 1000 K. When the temperature is lower than 700 K, et and ae calculated from the original EWB model will deviate from the LBL results obviously. The values of ap, et and ae calculated from the E-EWB model are almost the same with the LBL results in the investigated temperature range. The maximum relative errors between the E-EWB results and the LBL results in the temperature range are 0.78%, 0.15% and 0.22%, respectively for ap, et and ae. From Fig. 2, it can be seen that the EWB model with the original reference band parameters can cause substantial errors when calculating ap of CO2 in the whole investigated temperature range.
400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature (K)
(b) Comparisons of αe of CH4 Fig. 4. Comparisons of et and ae of CH4 calculated with EWB and E-EWB models.
When the temperature is lower than 1400 K, et and ae calculated from the original EWB model will deviate from the LBL results obviously. The values of ap, et and ae calculated from the E-EWB model are almost the same with the LBL results in the investigated temperature range. The maximum relative errors between the E-EWB results and the LBL results in the investigated temperature range are 0.80%, 0.54% and 0.38%, respectively for ap, et and ae. From Fig. 3, it can be seen that et and ae of CO calculated with the EWB model deviate much from those calculated from the LBL method with the HITEMP spectral database in the whole investigated temperature range. The E-EWB results, however, fit very well with the LBL results. The maximum errors between the E-EWB results and the LBL results in the temperature range are 0.32% and 0.33%, respectively for et and ae. From Fig. 4, it can be seen that the E-EWB results fit very well with the EWB results. This is because the parameters for CH4 radiation property are fitted with the original EWB model. The spectral lines for H2O, CO2 and CO have been added in the HITEMP-2010 database but these for CH4 have not been included. Hence, only the parameters, ain and bin, for H2O, CO2 and CO are recalculated and these parameters for CH4 are still the same as the original EWB model. In the future when the spectral data of CH4 is included in the HITEMP database, the accuracy of et and ae of CH4 can be further improved. The CPU times for the calculation of et for the species of H2O, CO2, CO and CH4 with the E-EWB and EWB models are compared in Table 4. For the comparison, the radiation temperature is set as 1000 K and the pressure is 1 atm. The testing CPU type is Intel (R) Core(TM) i7-4810MQ and its CPU Clock Speed is 2.80 GHz.
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Table 4 Calculation time duration comparisons between the EWB and E-EWB models.
EWB model (ms) E-EWB model (ms)
H2O
CO2
CO
CH4
18.57 0.053
18.30 0.049
18.32 0.052
18.35 0.049
From Table 4, it can be seen that the calculation speed of E-EWB model is about 350 times faster than the EWB model, which makes the E-EWB model possible to be implemented in the CFD calculation.
the oxy-fuel combustions and gasifications which contain multiply absorptive species like CH4, CO, H2O and CO2. However, it is noted that for some species such as CO, the accuracies of ap and et cannot be satisfied simultaneously in a wide range of temperature by only adjusting the fundamental band intensity. Although in the CFD calculation, only et is concerned, ap is also one important radiation parameter and the EWB model still needs being improved. Conflict of interest None declared.
5. Conclusions The objective of this investigation is to set up an efficient, accurate and universal method to calculate the radiative absorption coefficient during the CFD simulations. After being compared with the recent models, the EWB model is found having this potential, and the E-EWB method is proposed based on the EWB model in this work. First, the LBL calculations are implemented with the HITEMP-2010 database to generate a series of reference values including the total emissivity and the Planck mean absorption coefficient. Then, some of the original EWB fundamental band intensities are recalculated to fit the values generated by the LBL method. Finally, the E-EWB model is put forward with mathematical regression based on the values from the modified EWB model. The novelty and enhancement of this work mainly include two folds. First, the parameters of the EWB model are updated in the E-EWB model with the LBL method and the HITEMP-2010 spectral database. This new data base has been extended to include the temperature up to 4000 K and the pressure range up to 80 bar. Second, in addition to the polynomial fitting procedure to reduce the time consumption of the wide band model, the table look-up procedure of the black body fractional function is also proposed in this work. With the two simplifications, the E-EWB model is finally found about 350 times faster than the EWB model, which makes the E-EWB model can be definitely used in the large CFD calculations. With the work done, the following conclusions can be drawn: (1) When the temperature is lower than 700 K, et and ae of H2O calculated from the original EWB model will deviate from the LBL results obviously. When the temperature is lower than 1400 K, et and ae of CO2 calculated from the original EWB model will deviate from the LBL results obviously. et and ae of CO calculated with EWB model deviate much from LBL results in the whole investigated temperature range. (2) The E-EWB model proposed in this work is more accurate than the original EWB model when calculating the total emissivity, the effective absorption coefficient and the Planck mean absorption coefficient. For H2O, the maximum relative errors between the E-EWB results and the LBL results in the temperature range are 0.78%, 0.15% and 0.22%, respectively, for ap, et and ae. For CO2, the maximum relative errors between the E-EWB results and the LBL results in the temperature range are 0.80%, 0.54% and 0.38%, respectively, for ap, et and ae. For CO, the maximum relative errors between the E-EWB results and the LBL results in the temperature range are 0.32% and 0.33%, respectively, for et and ae. (3) The calculation speed of the E-EWB model is about 350 times faster than the original EWB model, which makes it possible to be imbedded into the CFD package. (4) Unlike the WSGG model, the E-EWB model can contain more than two species and it is not restricted by the species concentration. Thus, the E-EWB model can be used to model
Acknowledgments The authors gratefully acknowledge financial supports from the National Natural Science Foundation of China (NSFC, 51176009) and the China Postdoctoral Science Foundation (2015M570096) for this work. References [1] Y.K. Hu, J.Y. Yan, Numerical simulation of radiation intensity of oxy-coal combustion with flue gas recirculation, Int. J. Greenhouse Gas Control 17 (2013) 473–480. [2] H.C. Hottel, A.F. Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967. [3] N. Lallermant, A. Sayre, R. Weber, Evaluation of emissivity correlations for H2O–CO2–N2/air mixtures and coupling with solution methods of the radiative transfer equation, Prog. Energy Combust. Sci. 22 (1996) 543–574. [4] A.A.F. Peters, R. Weber, Modeling of swirling natural gas and pulverized coal flames with emphasis on nitrogen oxides, IFRF Document no. F 36/y/21D, 1993. [5] G. Krishnamoorthy, M. Sami, S. Orsino, A. Perera, M. Shahnam, E.D. Huckaby, Radiation modelling in oxy-fuel combustion scenarios, Int. J. Comput. Fluid D 24 (2010) 69–82. [6] C.G. Yin, Refined weighted sum of gray gases model for air–fuel combustion and its impacts, Energy Fuel 27 (2013) 6287–6294. [7] T. Kangwanpongpan, R.C. Silva, H.J. Krautz, Prediction of oxy-coal combustion through an optimized weighted sum of gray gases model, Energy 41 (2012) 244–251. [8] T. Kangwanpongpan, F.H.R. Franca, R.C. Silva, S.S. Paulo, H.J. Krautz, New correlations for the weighted-sum-of-gray-gases model in oxy-fuel conditions based on HITEMP 2010 database, Int. J. Heat Mass Transfer 55 (2012) (2010) 7419–7433. [9] M.F. Modest, Radiative Heat Transfer, 2nd ed., Academic Press, California, 2003. [10] C.Y. Ye, High-resolution radiative gas model based on the K-distribution method (Master’s thesis), Beijing University of Aeronautics and Astronautics, Beijing, 2013 (in Chinese). [11] M.K. Denison, B.W. Webb, The weighted sum of gray gases model for arbitrary solution methods in radiative transfer, ASME J. Heat Transfer 113 (1991) 650– 656. [12] M.F. Modest, H. Zhang, The full-spectrum correlated-k distribution for thermal radiation from molecular gas-particulate mixtures, ASME J. Heat Transfer 124 (2002) 30–38. [13] L. Pierrot, R. Philippe, S. Anouar, J. Taine, A fictitious-gas-based absorption distribution function global model for radiative transfer in hot gases, J. Quant. Spectrosc. Radiat. Transfer 62 (1999) 609–642. [14] F. André, R. Vaillon, The spectral-line moment-based (SLMB) modeling of the wide band and global black body-weighted transmission function and cumulative distribution function of the absorption coefficient in uniform gaseous media, J. Quant. Spectrosc. Radiat. Transfer 109 (2008) 2401–2416. [15] H.Q. Chu, F.S. Liu, J.L. Consalvi, Relationship between the spectral line based weighted sum of gray gas model and the full spectrum k-distribution model, J. Quant. Spectrosc. Radiat. Transfer 143 (2014) 111–120. [16] J. Cai, M.F. Modest, Improved full-spectrum k-distribution implementation for inhomogeneous media using a narrow-band database, J. Quant. Spectrosc. Radiat. Transfer 141 (2014) 65–72. [17] A.T. Modak, Exponential wide band parameters for the pure rotational band of water vapor, Radial Transfer 21 (1979) 131–142. [18] L.S. Rothman, I.E. Gordon, R.J. Barber, H. Dothe, R.R. Gamache, A. Goldman, V.I. Perevalov, S.A. Tashkun, J. Tennyson, HITEMP, the high-temperature molecular spectroscopic database, J. Quant. Spectrosc. Radiat. 111 (2010) 2139–2150. [19] L.S. Rothman, I.E. Gordon, Y. Babikov, A. Barbe, D.C. Benner, P.F. Bernath, M. Birk, L. Bizzocchi, V. Boudon, L.R. Brown, A. Campargue, K. Chance, E.A. Cohen, L.H. Coudert, V.M. Devi, B.J. Drouin, A. Fayt, J.M. Flaud, R.R. Gamache, J.J. Harrison, J.M. Hartmann, C. Hill, J.T. Hodges, D. Jacquemart, A. Jolly, J. Lamouroux, R.J. Le Roy, G. Li, D.A. Long, O.M. Lyulin, C.J. Mackie, S.T. Massie, S. Mikhailenko, H.S.P. Müller, O.V. Naumenko, A.V. Nikitin, J. Orphal, V. Perevalov, A. Perrin, E.R. Polovtseva, C. Richard, M.A.H. Smith, E. Starikova, K.
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Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Development of an absorption coefficient calculation method potential for combustion and gasification simulations Linbo Yan a, Guangxi Yue a, Boshu He b,⇑ a b
Department of Thermal Engineering, Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Tsinghua University, Beijing 100084, China Institute of Combustion and Thermal System, School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing 100044, China
a r t i c l e
i n f o
Article history: Received 10 February 2015 Received in revised form 15 August 2015 Accepted 15 August 2015
Keywords: Absorption coefficient Total emissivity Line by line (LBL) Exponential wide band (EWB) model Efficient exponential wide band (E-EWB) model Computational fluid dynamics (CFD)
a b s t r a c t Efficient and accurate calculation of the absorption coefficient is very important to the modeling of combustion and high-temperature gasification where the thermal radiation plays an important role. It is therefore important to develop a radiative absorption coefficient calculation method which is accurate enough, universal but not time consuming. In this work, a computationally efficient exponential wide band (E-EWB) model based on the exponential wide band (EWB) model is put forward and programmed so that it can be integrated into the computational fluid dynamics (CFD) software. The E-EWB calculation results are then validated against the results obtained via the line by line (LBL) model, and the runtime of this model is also evaluated. Through this work, the E-EWB model is found effective and reliable. The maximum deviation between the results obtained by the E-EWB model proposed in this work and the LBL results is within 0.8%. Moreover, the calculation speed of this model can be about 350 times faster than that of the original EWB model and is fast enough to be applied to the combustion and gasification simulations. Thus, the E-EWB model is a good choice to do the future absorption calculations during the combustion and gasification CFD simulations. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Numerical simulations of the combustion and gasification processes are very important to the design and optimization of combustors, gasifiers and the corresponding operating parameters. For almost all the combustion processes and most of the gasification ones, the operation temperatures are usually high and the radiative energy occupies a great part of the total heat flux. It is therefore important to address the radiative heat transfer carefully for the numerical simulations of these cases. Besides the numerous radiation models which can affect the accuracy of the radiative heat transfer calculation, one of the material physical property parameters used in the radiative transfer equations (RTE), the absorption coefficient a, is also essential to the accurate radiant energy calculation [1]. In most of the recent computational fluid dynamic (CFD) packages, the weighted sum of gray gas model (WSGG) is used to calculate a. Although the WSGG model is very efficient and generates acceptable errors for the ranges of temperatures and pressures that cover almost all the traditional combustion and gasification processes, there are also a lot of inherent defections. First, it is easy for the WSGG model to consider ⇑ Corresponding author. Tel.: +86 10 5168 8542; fax: +86 10 5168 8404. E-mail address: [email protected] (B. He). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.08.047 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
two radiative species at one time, e.g., H2O and CO2, and the corresponding parameters of this model are calculated in terms of the concentration ratio of the two species [2]. If there are more radiative species, the calculation with the WSGG model will be very complex and even not feasible. However, for many gasification processes, the effect of other species like CO and CH4 on the total radiation properties cannot be avoided since they are of parallel amount of H2O and CO2 and even more. Second, the application of the WSGG model is limited to the volume of the homogeneous gas, whose characteristic length should be greater than 1 cm [3]. However, in order to accurately simulate the flame front and capture the denotation wave, the grid characteristic length should be less than 0.5 mm [4]. Third, the original WSGG model is only proper for a narrow range of H2O to CO2 concentration ratio [5]. For the simulation of the oxy-fuel combustion in which the concentration of CO2 is very high, the original WSGG model will lost its accuracy. Yin once proposed a refined WSGG model for the conventional air–fuel combustion and found remarkable difference between their model and the existing model in the CFD software if the particle radiation was not considered [6]. Kangwanpongpan once proposed an optimized WSGG model for the oxy-coal combustion and found that the flame temperature can be better predicted by their model [7]. Kangwanpongpan once proposed another WSGG model based on the new HITEMP-2010 spectral
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Nomenclature
Parameters total band absorptance [cm1] Ain polynomial coefficients b1–9 bin self-broadening coefficient of species i and band n C the speed of light [cm/s] C2 Planck second radiation constant [cm K] lower state energy of the transition [cm1] En g F T fractional function of blackbody radiation degeneracy number of the fundamental band gd parameter calculated with Eq. (11) gv,max g(N N0) line shape function [cm] Ibg black body intensity at wave number g [W m2 Sr1/ lm] IbDg;n mean block intensity of the nth band [W m2 Sr1/lm] L beam length [cm] Mw molecular weight [kg/kmol] n coefficient of temperature dependence of air broadened half width nin the fitting parameter of species i and band n Ni molecular concentration of species i [molecules/(cm3 atm)] P0 the reference pressure [1 atm] Pe,in dimensionless equivalent broadening pressure of species i band n Pi partial pressure of species i [atm] Pt total gas pressure [atm] Q(T) total internal partition function R the universal gas constant [8.3145 J/(K mol)] Sk line intensity [cm1/molecule cm2] T gas temperature [K] T0 reference temperature [K] xi mole fraction of species i X pressure path length [atm cm] Greek symbols a0 reference band intensity [cm/(g/m2)] ae effective absorption coefficient [m1] ain integrated band intensity [cm/(g/m2)] ap Planck mean absorption coefficient [(cm atm)1] ag spectral absorption coefficient [(cm atm)1]
database for the oxy-fuel combustion and validated their model with the line by line method [8]. Although many new WSGG parameters have been put forward by researchers so that this model can be applied to the oxy-fuel combustion calculation, the model is still mainly used in the cases where CO2 and H2O are the dominant absorptive gases. In addition to the WSGG model and its variants, there are also many other models can be used to calculate the absorption coefficient. These models can be generally classified as the line by line (LBL) model, the band models and the global models [9]. The band interval of the LBL model is generally 0.0002–0.02 cm1. Thus, the LBL model is the most accurate but very time consuming. It is hence usually used as the benchmark to validate the other radiation models. The band models usually refer to the narrow band model (NBM) and the wide band model (WBM). The band interval of the narrow band model is generally 5–50 cm1. In principle, the narrow band model can be as accurate as the line by line model. This model, however, is also very time consuming and is hard to be applied to the nonhomogeneous media. The band interval of the wide band model is generally 100–1000 cm1. It is much faster than the LBL model and the
ag,n b0 bin
cair cD cp cself cV dd
et g N0 gin gLm gU,m m0,d md sg,in sDg,m U(T)
vi w(T) x0 xin
absorption coefficient contributed by the nth band [(cm atm)1] reference mean line width to spacing ratio the mean line width to spacing ratio parameter air-broadened half width parameter [(cm atm)1] half width at half height of the Doppler feature [cm1] half width at half height of the Lorentzian feature [cm1] self-broadened half width [(cm atm)1] half width at half height of the Voigt feature [cm1] effects of the photon transition on the vibrational quantum number total emissivity wave number [cm1] wave number at the special line center [cm1] nth band width of species i [cm1] lower limits of block m [cm1] upper limits of block m [cm1] the lowest possible initial state the vibrational quantum number transmissivity of species i and band j the mth block transmissivity temperature dependent intermediate parameter for calculating bin mass path length of species [g/m2] temperature dependent intermediate parameter for calculating ai reference band width [cm1] the band width parameter [cm1]
Abbreviations CFD computational fluid dynamics E-EWB efficient EWB EWB exponential wide band LBL line by line LSF line shape function RTE radiative transfer equation WSGG weighted sum of gray gas model UDF user defined function
narrow band model with slight accuracy lost. The global models calculate the radiation properties over the whole spectral range [10]. One of the well-known global models is the WSGG model. Based on the WSGG model, a spectral line based WSGG (SLW) model has been improved by Denison and Webb [11]. Modest and Zhang have also developed a full spectrum k-distribution (FSK) model [12]. There are also some other global non-gray gas models such as the absorption distribution function (ADF) model [13] and the spectral-line moment-based (SLMB) model [14]. The SLW and the FSK of the global models are very promising because they have been extensively developed for gas radiation calculations in nonhomogeneous gas mixtures [15]. When using the FSK model for the nonhomogeneous media, gas absorption coefficients must be assumed to be scaled or correlated before the reordering process. But it is reported that for strong temperature and gas composition variations, the assumption will be challenged [16]. Most of these models can avoid the disadvantages of the WSGG model. However, most of them are either computationally uneconomical or not well and sufficiently developed, so they are not chosen to calculate the absorption coefficient for the radiative heat transfer
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in the CFD calculation. Among these advanced models, the exponential wide band (EWB) model is by far the most successful wide band models and is also very popular for its relatively simple correlation structure [9]. In this work, a computationally efficient scheme to calculate the absorption coefficient is used and programmed so that it can be integrated into the CFD software through the user defined function (UDF). The scheme is based on the EWB model and the calculation results are validated against the results calculated by the LBL and EWB models. Since some key reference EWB parameters were fitted against the emissivity data [17], some of the reference EWB parameters are recalculated in this work based on the LBL method and the HITEMP-2010 [18] spectroscopic database which contains much more hot lines than the HITRAN [19] database and is more accurate for the high temperature radiation calculation. Then, mathematical regression is done to simplify the EWB model correlations to generate the efficient EWB (E-EWB) model. The E-EWB model is accurate, time aving and can be generally used. It can be integrated into the commercial CFD packages to calculate the absorption coefficients for both the engineering and scientific research usage. It should be noted that the HITEMP database only contains five species up to now and CH4 have not been included yet, so the radiative properties of H2O, CO2 and CO are recalculated. Although only four species which usually the dominant ones for most of the combustion and gasification cases are considered in this work, the number of species in the calculation can be expanded arbitrarily. Since the E-EWB model in this work is based on the EWB model, it will share some weakness of the EWB model. For example, the EWB correlations are less accurate compared with the line by line method and the narrow band model. Moreover, when the EWB model is used for the nonhomogeneous media, the model should be modified. In fact, if the E-EWB model parameters are obtained by fitting the results from the LBL model with the HITEMP-2010 spectral database rather than the results from the original EWB model, the E-EWB model results can be as accurate as the LBL model results in the fitting temperature range. As has been mentioned, parameters for calculating the radiation properties of H2O, CO2 and CO have been updated except CH4. Thus, the accuracy of calculation for CH4 in the E-EWB model will be limited by the original EWB model. 2. LBL model calculation methodology For the engineering application of radiation heat transfer, the total emissivity, et, and the effective absorption coefficient, ae, are usually concerned [20]. It is believed that the LBL calculation results are the most accurate ones and can be treated as the benchmark data. So, in this section, the LBL calculations are implemented to generate the reference emissivity and absorption coefficient database. The effective absorption coefficient can be calculated from the total emissivity, so the absorption coefficient generated with the LBL method is, instead, the Planck mean coefficient [21], ap, which is also one of the most important mean absorption coefficients and is used to adjust the reference EWB parameters along with et in this work. Recently, there are two high resolution spectroscopic databases for the LBL calculation, the HITRAN and the HITEMP. The HITEMP database contains more hot lines and it is more accurate for the high temperature calculation, while the HITRAN database can only be accurate enough for the temperature lower than 750 K [22]. The HITRAN database contains 47 species while the HITEMP database only contains 5 including H2O, CO2, CO, NO and OH so far. Since the combustion and gasification always run at high temperatures, the HITEMP database is therefor used for the calculation of the radiative parameters. The Planck mean absorption coefficient is calculated with Eq. (1) [23].
R1 a I dg ap ¼ R0 1 g bg I dg 0 bg
ð1Þ
where, ag and Ibg are the absorption coefficient and black body intensity at wave number g. The total emissivity can be calculated with Eq. (2) [24]. Based on the total emissivity, the effective absorption coefficient, ae , can then be calculated with Eq. (3) [25].
R
1 e I dg et ¼ R01 g bg ¼ I dg 0 bg
R1 0
Ibg ð1 eag X Þdg R1 I dg 0 bg
ð2Þ
1 lnð1 et Þ L
ae ¼
ð3Þ
where, X denotes the pressure path length and equals piL; pi is the partial pressure of species i and L is the beam length. The spectral absorption coefficient, ag , is the convolution of the line intensity and the line shape factor, and can be calculated with Eq. (4) [26].
ag ¼
X Sk gðg g0 ÞNi
ð4Þ
k
where, Sk denotes the line intensity and can be obtained from the spectroscopic database. It should be noted that the line intensity given in the database are based on the reference temperature of 296 K [27]. The values of line intensity at any temperature are calculated with Eq. (5). n is the number of the radiation line. Ni is the molecular concentration of species i and can be calculated based on the Loschmidits number. g(N N0) is the line shape function (LSF). N and N0 are the wave numbers at any position and at the special line center, respectively. LSF caused by Doppler broadening is calculated with Eq. (6); LSF caused by pressure broadening is calculated with Eq. (7); LSF caused by the combination of the two broadening effects is calculated with Eq. (8) [28].
C E C 2 gi T QðT 0 Þ exp T2 n 1 exp Si ðTÞ ¼ Si ðT 0 Þ Q ðTÞ exp C 2 En 1 exp C 2 gi T0
ð5Þ
T0
where, Q(T) is the total internal partition function which accounts for the distribution of various energy states of molecules in thermodynamic equilibrium [27]; C2 denotes the Planck second radiation constant; En denotes the lower state energy of the transition; T and T0 are the gas temperature and the reference temperature, respectively.
g D ðg g0 Þ ¼
1
1=2 ln 2
cD
p
"
exp ln 2
ðg g 0 Þ2
#
ð6Þ
c2D
where, cD denotes the half width at half height of the Doppler feature and can be calculated with Eq. (9).
g P ðg g0 Þ ¼
cP =p ðg g0 Þ2 þ c2P
ð7Þ
where, cp denotes the half width at half height of the Lorentzian feature and can be calculated with Eq. (10).
" # g V ðg g0 Þ cP g g0 2 exp 2:772 ¼ 1 g V;max cV 2cV 2 c 1 cP cP þ P 2 þ 0:016 1 cV 1 þ 4 gg0 cV cV 2cV 8 9 " > > # < = g g0 2:25 10 þ exp 0:4 2:25 > > 2cV : ; 10 þ g2cg0 V
ð8Þ
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where, g V;max can be calculated with Eq. (11); cV denotes the half width at half height of the Voigt feature and can be calculated with Eq. (12).
cD ¼
g0 2RT ln 2 c
1=2
ð15Þ
where, n is calculated with Eq. (16).
ð9Þ
Mw
where, c is the speed of the light; R is the universal gas constant; and Mw is the molecular weight.
n T cP ¼ ½cair ðP t Pi Þ þ cself Pi 0 T
ð10Þ
where, cair and n are the air-broadened half width parameter and the coefficient of temperature dependence of air broadened half width [24]. cself is the self-broadened half width. These two parameters, cair and cself, can be found in the spectroscopic database. Pt is the total gas pressure. Pi denotes the partial pressure of species i.
g V;max ¼
" !# 1 15 X ejn 3 3n2 6n 6 F ¼ 4 n þ þ 2 þ 3 T p j¼1 j j j j g
1 2 2cV 1:065 þ 0:447 ccP þ 0:058 ccP V
ð11Þ
V
n¼
C2g T
ð16Þ
The total band absorptance, Ain, and the transmissivity, sg;in , of species i and band n can be calculated with the three-region (linear region, square root region and logarithmic region) expressions which are not listed here [29]. To solve the three-region expressions, the following five parameters should be calculated firstly. They are the mass path length of species i, vi; the dimensionless equivalent broadening pressure of species i band n, Pe,in; the band width parameter xin; the integrated band intensity ain; and the mean line width to spacing ratio parameter bin. The five parameters can be calculated with Eqs. (17)–(21) [25].
vi ¼
xi PT M wi L RT
ð17Þ
where, xi is the mole fraction of species i. 2
2 1=2
2cV ¼ ð0:5346Þð2cP Þ þ ½0:2166ð2cP Þ þ ð2cD Þ
ð12Þ
The choice of the line shape functions can be made according to the value of cP/cV. When the value is less than 0.1, the Gaussian line shape is used. When the value is greater than 0.1 and less than 5, the Voigt profile is chosen. When the value is greater than 5, the Lorentzian profile can be used [27]. The mathematical models mentioned above are then programmed to calculate ap and et based on the HITEMP-2010 database. In order to accelerate the LBL calculation speed, parallel calculation scheme is used. The continuous wave number for the integration is divided into several parts but the spectral line database should not be divided. Then, the divided parts can be calculated separately so that the computer source can be fully utilized. Moreover, some parts can usually be neglected since the blackbody intensity is very small in the wave number ranges at given temperature according to the Wien displacement law [21]. Thus, the computation amount can be reduced and the efficiency can be elevated.
where, ag;n denotes the absorption coefficient contributed by the nth band; IbDg;n is the mean block intensity of the nth band.
X gL;m gU;m ð1 sDg;m Þ F F T T m
T T0
xin ðTÞ ¼ x0
ð19Þ
h
P i 1 exp Dd¼1 nd dd wðTÞ P i ain ðTÞ ¼ a0 h 1 exp Dd¼1 n0;d dd wðT 0 Þ
ð20Þ
where, dd describes the effects of the photon transition on the vibrational quantum number and can be found in the reference EWB parameter table [9]. a0 is the reference band intensity. w(T) in Eq. (20) is defined by Eq. (22).
T T0
0:5
UðTÞ UðT 0 Þ
ð21Þ
where, b0 is the reference mean line width to spacing ratio parameter. U(T) in Eq. (21) is defined by Eq. (23).
QD P1 wðTÞ ¼
d¼1
md ¼m0;d
QD P1
ðmd þg d þjdd j1Þ! nd md e ðg d 1Þ!md !
ð22Þ
ðmd þg d 1Þ! n m d d md ¼0 ðg d 1Þ!md ! e
where, md is the vibrational quantum number; gd denotes the degeneracy number of the fundamental band and they can be found in the reference EWB model table [9]. m0;d denotes the lowest possible initial state and is equal to zero when dd is nonnegative, otherwise it equals to the absolute value of dd .
QD P 1
ð14Þ
where, sDg;m is the mth block transmissivity, it is the products of all the band transmissivities, sg;im , that belong to this block; and gL;m and gU;m are the lower and upper limits of block m and they are determined according to the band center wave number and the calculated nth band width of species i, Dgin . Dgin can be calculated using the total band absorptance, Ain, and the band transmissivity. F gT is the fractional function of blackbody radiation and can be calculated with Eq. (15) [32].
0:5
where, x0 is the reference band width parameter.
d¼1
ð13Þ
ð18Þ
where, P0 denotes the reference pressure which is 1 atm; bin and nin are the self-broadening coefficient and the corresponding fitting parameter of species i and band n, respectively.
bin ðTÞ ¼ b0
The EWB model was developed by Edwards & Menard [29] and Edwards & Balakrishnan [30]. It is much faster than the LBL and NBM and the results are accurate enough, so it has a good potential to be imbedded into the CFD package. The calculations of ap and et with the band model can be cast into Eqs. (13) [31] and (14) [32].
et ¼
nin PT Pi þ ðbin 1Þ P0 P0
3. EWB model calculation methodology
R1 Z X IbDg;n a I dg 1 X 1 ap ¼ R0 1 g bg ¼ ag;n Ibg;n dg ¼ ain Ib n 0 Ib I dg n 0 bg
Pe;in ¼
UðTÞ ¼
d¼1
md ¼m0;d
QD P 1 d¼1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 n m
md ¼m0;d
ðmd þg d þjdd j1Þ!e ðg d 1Þ!md !
d d
ðmd þg d þjdd j1Þ! n m e dd ðg d 1Þ!md !
ð23Þ
With the above correlations, ap and et can then be calculated from the EWB model. In order to calculate the two parameters more accurately, some of the reference band intensities are recalculated based on the LBL calculation results in this work. Some literature once gave different reference band intensity values, but the values were not accurate enough compared with the LBL results obtained
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from the HITEMP database [33]. In addition, the reference parameters were only fitted by emissivity. When they are used to calculate the Planck mean absorption coefficient which is also one very important radiation parameter, substantial errors will be caused. In this work, the reference rotational band intensity of H2O and some reference band intensities which once were constant at different temperatures are adjusted to fit ap, as well as et in the temperature range of 500–2000 K. For H2O, the reference band intensities for the bands centered at 140 cm1 and 1600 cm1 are recalculated. For CO2, the reference band intensities for the band centered at 667 cm1 and the band with upper head wave number limit of 2410 cm1 are recalculated. For CO, the reference band intensity for the band centered at 2143 cm1 is recalculated. Since the integrated band intensity is only the function of temperature as can be seen from Eq. (21), the recalculated reference band intensities are considered as the function of gas temperature and the recalculating polynomial correlations proposed in this work are shown by Eq. (24). The corresponding polynomial coefficients are listed in Table 1. Since CH4 has not been contained in the HITEMP database, the reference band values of CH4 are not
adjusted. During our calculation, it is found that the accuracy of
ap and et cannot be met simultaneously for CO using the EWB model, so only the total emissivity of CO is refined.
a0;k ¼ b0 þ b1 T þ b2 T 2 þ b3 T 3 þ b4 T 4 þ b5 T 5 þ b6 T 6 þ b7 T 7 þ b8 T 8 þ b9 T 9
ð24Þ
4. E-EWB model calculation methodology The calculation speed of EWB model is much faster than those of the LBL and NBM. It is, however, still too computationally intensive to be applied to the CFD calculation. In order to further improve the calculation speed of EWB model, mathematical regressions are needed according to the characteristic of this model and the E-EWB model is proposed in this work. In the original EWB model calculation, the time consuming parts are the calculation of ain ðTÞ and bin ðTÞ as defined by Eqs. (20)–(23). It is assumed in the EWB model that ain ðTÞ, bin ðTÞ and xin ðTÞ are solely functions of temperature and the effect of pressure is accounted for through
Table 1 Polynomial coefficients for the modified reference band intensities.
a0;H2 O;140 a0;H2 O;1600 a0;CO2 ;667 a0;CO2 ;2410 a0;CO;2143
b0
b1
b2
b3
b4
b5
b6
b7
b8
b9
28119.6 594.3822 1300.236 14692.34 2.62492
199.5666 4.39809 9.50528 71.3231 0.086348
0.53632 0.013095 0.030733 0.16175 0.000162
0.0007868 2.02E05 5.7E05 0.000205 1.317E07
6.72E07 1.769E08 6.82E08 1.544E07 4.4E11
3.303E10 8.89E12 5.3E11 6.86E11 2.37E15
8.6E14 2.38E15 2.71E14 1.66E14 5.36E18
9.26E18 2.62E19 8.7E18 1.7E18 9.9E22
– – 1.6E21 – –
– – 1.3E25 – –
Table 2 Polynomial coefficients for ain ðTÞ.
aH2 O;3760 aH2 O;5350 aH2 O;7250 aCO2 ;960 aCO2 ;1060 aCO2 ;3660 aCO2 ;5200 aCO;4260 aCH4 ;4220 aCH4 ;5861
b0
b1
b2
b3
b4
b5
b6
b7
24.90321 3.104651 2.495858 0.069745 0.070363 4.162493 0.072587 0.134803 3.049165 0.459851
6.5E05 0.00055 2.7E05 0.0005 0.0005 0.00089 4.2E05 5.227E05 0.001 0.00025
4.75E08 6.09E07 3.03E07 1.24E06 1.25E06 8.28E07 7.55E08 1.97E07 1.99E06 4.75E07
1.718E10 7.648E10 8.77E10 1.25E09 1.25E09 2.395E09 1.17E11 3.39E10 5.8E10 1.4E10
2.12E13 9.68E13 1.102E12 7.304E13 7.318E13 2.8E12 7.33E15 2.7E13 7.6E14 1.1E14
1.141E16 5.106E16 6.27E16 2.53E16 2.54E16 1.442E15 7.894E18 1.206E16 1.25E16 2.23E17
3.1E20 1.4E19 1.74E19 4.83E20 4.87E20 3.7E19 2.7E21 2.9E20 4.1E20 1E20
3.3613E24 1.4812E23 1.929E23 3.91E24 3.97E24 3.968E23 3.396E25 2.92E24 4.71E24 1.38E24
Table 3 The polynomial coefficients for bin ðTÞ.
bH2 O;140 bH2 O;1600 bH2 O;3760 bH2 O;5350 bH2 O;7250 bCO2 ;667 bCO2 ;960 bCO2 ;1060 bCO2 ;2410 bCO2 ;3660 bCO2 ;5200 bCO;2413 bCO;4260 bCH4 ;1310 bCH4 ;3020 bCH4 ;4220 bCH4 ;5861
b0
b1
b2
b3
b4
b5
b6
b7
0.143246 0.087998 0.224847 0.077023 0.113097 3.83E02 2.60E02 7.93E02 1.65E01 8.74E02 2.40E01 0.0796287 0.1805136 0.081379 0.066498 0.340122 0.683294
0.00033 0.00021 0.00051 0.00018 0.00027 3.51E05 1.34E05 2.47E05 7.43E05 3.36E05 1.88E04 0.000223 0.000522 0.00023 0.00019 0.001 0.00215
5.63E07 4.53E07 1.01E06 3.94E07 5.34E07 2.09E07 1.06E07 3.54E07 6.19E07 3.8E07 8.91E07 4.67E07 1.12E06 7.25E07 5.68E07 3.01E06 6.5E06
6.1E10 4.7E10 9.9E10 4E10 5.3E10 1.78E10 1.16E10 2.82E10 6.63E10 4.36E10 1.95E09 5.21E10 1.23E09 7.2E10 5.7E10 2.8E09 6.1E09
4.222E13 3.452E13 6.943E13 2.98E13 3.879E13 1.35E14 1.04E13 3.6E13 3.37E13 4.31E13 1.59E12 3.58E13 8.38E13 8.13E13 5.88E13 2.93E12 6.23E12
1.77E16 1.49E16 2.83E16 1.26E16 1.62E16 2.3E17 5.8E17 2E16 2.3E16 2.7E16 1E15 1.5E16 3.5E16 1.7E16 9E17 4.7E17 4.61E16
4.152E20 3.564E20 6.426E20 2.916E20 3.749E20 7.7E21 1.46E20 5.01E20 6.4E20 7.08E20 2.78E19 3.49E20 8.06E20 2.59E20 8.44E21 6.6E20 3.2E19
4.1E24 3.6E24 6.2E24 2.9E24 3.7E24 9.4E25 1.5E24 5.2E24 6.9E24 7.5E24 3E23 3.5E24 8E24 2E24 2.8E25 9.28E24 3.99E23
L. Yan et al. / International Journal of Heat and Mass Transfer 91 (2015) 1069–1077
the equivalent broadening pressure, which makes it possible that these temperature dependent parameters and their combinations can be fitted by polynomials with temperature as the argument. Since the calculations of ain ðTÞ and bin ðTÞ are very time consuming, mathematical regressions are implemented for these parameters and the corresponding fitting formula for H2O, CO2, CH4 and CO is shown in Eq. (26). The corresponding polynomial coefficients are listed in Tables 2 and 3. Besides the time consuming calculations of the band parameters, the calculation of the black body fractional function shown by Eq. (16) will also appear time consuming when the calculation runtime is within 1 ms. This is because the solution of Eq. (16) needs at least 50 loops in the program. Thus, in order to further improve the calculation efficient, the table look-up scheme is substituted for the calculation of
Planck Mean Absorption Coefficient of H2O, αp (cm·atm)-1
0.20 0.18 0.16 0.14
Eq. (16) in this work and it is found that this scheme can further reduce the calculation time substantially. Firstly, a black body fractional function table is built with Eq. (16), then the table can be used for any cases and the value needed during the calculation can be obtained by interpolation.
ain ðTÞ ¼ b0 þ b1 T þ b2 T 2 þ b3 T 3 þ b4 T 4 þ b5 T 5 þ b6 T 6 þ b7 T 7 ð25Þ The integrated band intensities of the other bands of these species that are not listed in the table are set as their corresponding reference values. If the values are modified in this work, the modified values should be chosen. If not, the reference values listed in literature can be used [9]. 0.40
LBL EWBM E-EWBM
0.12 0.10 0.08 0.06 0.04 0.02
Planck Mean Absorption Coefficient of CO 2, αp (cm·atm)-1
1074
0.30 0.25 0.20 0.15 0.10 0.05 0.00
0.00 400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature (K)
Total Emissivity of CO2, εt
Total Emissivity of H2O, εt
LBL EWBM E-EWBM
0.23 0.22 0.21 0.20 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12
400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature (K)
(b) Comparisons of εt of H2O 0.75 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30
LBL EWBM E-EWBM
400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature (K) (b) Comparisons of εt of CO2
0.26
LBL EWBM E-EWBM
400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature (K)
(c) Comparisons of αe of H2O Fig. 1. Comparisons of ap, et and ae of H2O calculated with LBL, EWB and E-EWB models.
Effective Absorption Coefficient for CO2, αe (m-1)
Effective Absorption Coefficient of H2O, αe (m-1)
0.70
400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature (K)
(a) Comparisons of αp of CO2
(a) Comparisons of αp of H2O 0.52 0.50 0.48 0.46 0.44 0.42 0.40 0.38 0.36 0.34 0.32 0.30 0.28
LBL EWBM E-EWBM
0.35
0.24 0.22
LBL EWBM E-EWBM
0.20 0.18 0.16 0.14 0.12
400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature (K)
(c) Comparisons of αe of CO2 Fig. 2. Comparisons of ap, et and ae of CO2 calculated with LBL, EWB and E-EWB models.
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0.48
Total Emissivity of CH4, εt
0.44 0.40
0.32 0.28 0.24 0.20
LBL EWBM E-EWBM
Total Emissivity for CO, εt
0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02
400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature (K) (a) Comparisons of εt of CO
Effective Emissivity of CO, αe (m-1)
0.11 LBL EWBM E-EWBM
0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02
400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature (K)
(b) Comparison of αe of CO Fig. 3. Comparisons of et and ae of CO calculated with LBL, EWB and E-EWB models.
400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature (K)
(a) Comparisons of εt of CH4 0.48 0.44 0.40
EWBM E-EWBM
0.36 0.32 0.28 0.24 0.20 0.16
0.11
EWBM E-EWBM
0.36
0.16
Effective Absorption Coefficient for CH4, αe (m-1)
The calculated ap, et and ae at one atmospheric pressure for H2O and CO2 from the LBL, EWB and E-EWB models are shown and compared in Figs. 1 and 2. As aforementioned, the accuracy of ap and et of CO cannot be satisfied simultaneously in the whole temperature range of 500–2000 K, so only the accuracy of et of CO is considered in this work. This is because in the CFD calculation, et is usually used. For the case where ap is also required, maybe another set of the polynomial coefficients that are accurate for ap can be chosen. The calculated et and ae at one atmospheric pressure for CO from the LBL, EWB and E-EWB models are shown and compared in Fig. 3. Since the spectral data of CH4 have not been included in the HITEMP-2010 database, the reference band intensities for CH4 are not adjusted. The calculated et and ae at one atmospheric pressure for CH4 from the EWB and E-EWB models are shown and compared in Fig. 4. From Fig. 1, it can be seen that the EWB model with the original band parameters can cause substantial errors when calculating ap of H2O when the temperature is lower than 1000 K. When the temperature is lower than 700 K, et and ae calculated from the original EWB model will deviate from the LBL results obviously. The values of ap, et and ae calculated from the E-EWB model are almost the same with the LBL results in the investigated temperature range. The maximum relative errors between the E-EWB results and the LBL results in the temperature range are 0.78%, 0.15% and 0.22%, respectively for ap, et and ae. From Fig. 2, it can be seen that the EWB model with the original reference band parameters can cause substantial errors when calculating ap of CO2 in the whole investigated temperature range.
400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature (K)
(b) Comparisons of αe of CH4 Fig. 4. Comparisons of et and ae of CH4 calculated with EWB and E-EWB models.
When the temperature is lower than 1400 K, et and ae calculated from the original EWB model will deviate from the LBL results obviously. The values of ap, et and ae calculated from the E-EWB model are almost the same with the LBL results in the investigated temperature range. The maximum relative errors between the E-EWB results and the LBL results in the investigated temperature range are 0.80%, 0.54% and 0.38%, respectively for ap, et and ae. From Fig. 3, it can be seen that et and ae of CO calculated with the EWB model deviate much from those calculated from the LBL method with the HITEMP spectral database in the whole investigated temperature range. The E-EWB results, however, fit very well with the LBL results. The maximum errors between the E-EWB results and the LBL results in the temperature range are 0.32% and 0.33%, respectively for et and ae. From Fig. 4, it can be seen that the E-EWB results fit very well with the EWB results. This is because the parameters for CH4 radiation property are fitted with the original EWB model. The spectral lines for H2O, CO2 and CO have been added in the HITEMP-2010 database but these for CH4 have not been included. Hence, only the parameters, ain and bin, for H2O, CO2 and CO are recalculated and these parameters for CH4 are still the same as the original EWB model. In the future when the spectral data of CH4 is included in the HITEMP database, the accuracy of et and ae of CH4 can be further improved. The CPU times for the calculation of et for the species of H2O, CO2, CO and CH4 with the E-EWB and EWB models are compared in Table 4. For the comparison, the radiation temperature is set as 1000 K and the pressure is 1 atm. The testing CPU type is Intel (R) Core(TM) i7-4810MQ and its CPU Clock Speed is 2.80 GHz.
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Table 4 Calculation time duration comparisons between the EWB and E-EWB models.
EWB model (ms) E-EWB model (ms)
H2O
CO2
CO
CH4
18.57 0.053
18.30 0.049
18.32 0.052
18.35 0.049
From Table 4, it can be seen that the calculation speed of E-EWB model is about 350 times faster than the EWB model, which makes the E-EWB model possible to be implemented in the CFD calculation.
the oxy-fuel combustions and gasifications which contain multiply absorptive species like CH4, CO, H2O and CO2. However, it is noted that for some species such as CO, the accuracies of ap and et cannot be satisfied simultaneously in a wide range of temperature by only adjusting the fundamental band intensity. Although in the CFD calculation, only et is concerned, ap is also one important radiation parameter and the EWB model still needs being improved. Conflict of interest None declared.
5. Conclusions The objective of this investigation is to set up an efficient, accurate and universal method to calculate the radiative absorption coefficient during the CFD simulations. After being compared with the recent models, the EWB model is found having this potential, and the E-EWB method is proposed based on the EWB model in this work. First, the LBL calculations are implemented with the HITEMP-2010 database to generate a series of reference values including the total emissivity and the Planck mean absorption coefficient. Then, some of the original EWB fundamental band intensities are recalculated to fit the values generated by the LBL method. Finally, the E-EWB model is put forward with mathematical regression based on the values from the modified EWB model. The novelty and enhancement of this work mainly include two folds. First, the parameters of the EWB model are updated in the E-EWB model with the LBL method and the HITEMP-2010 spectral database. This new data base has been extended to include the temperature up to 4000 K and the pressure range up to 80 bar. Second, in addition to the polynomial fitting procedure to reduce the time consumption of the wide band model, the table look-up procedure of the black body fractional function is also proposed in this work. With the two simplifications, the E-EWB model is finally found about 350 times faster than the EWB model, which makes the E-EWB model can be definitely used in the large CFD calculations. With the work done, the following conclusions can be drawn: (1) When the temperature is lower than 700 K, et and ae of H2O calculated from the original EWB model will deviate from the LBL results obviously. When the temperature is lower than 1400 K, et and ae of CO2 calculated from the original EWB model will deviate from the LBL results obviously. et and ae of CO calculated with EWB model deviate much from LBL results in the whole investigated temperature range. (2) The E-EWB model proposed in this work is more accurate than the original EWB model when calculating the total emissivity, the effective absorption coefficient and the Planck mean absorption coefficient. For H2O, the maximum relative errors between the E-EWB results and the LBL results in the temperature range are 0.78%, 0.15% and 0.22%, respectively, for ap, et and ae. For CO2, the maximum relative errors between the E-EWB results and the LBL results in the temperature range are 0.80%, 0.54% and 0.38%, respectively, for ap, et and ae. For CO, the maximum relative errors between the E-EWB results and the LBL results in the temperature range are 0.32% and 0.33%, respectively, for et and ae. (3) The calculation speed of the E-EWB model is about 350 times faster than the original EWB model, which makes it possible to be imbedded into the CFD package. (4) Unlike the WSGG model, the E-EWB model can contain more than two species and it is not restricted by the species concentration. Thus, the E-EWB model can be used to model
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