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An improved full-spectrum correlated-k-distribution model for non-gray radiative heat transfer in combustion gas mixtures
International Communications in Heat and Mass Transfer 114 (2020) 104566
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An improved full-spectrum correlated-k-distribution model for non-gray radiative heat transfer in combustion gas mixtures Shu Zhenga, Ran Suib, Yu Yanga, Yujia Sunc, Huaichun Zhoud, Qiang Lua,
T
⁎
a
National Engineering Laboratory for Biomass Power Generation Equipment, North China Electric Power University, Beijing 102206, China Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA c School of Atmospheric Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China d School of Energy and Power Engineering, Northeast Electric Power University, Jilin 132012, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Radiative heat transfer Full-spectrum correlated-k-distribution (FSCK) Planar flames Efficiency improvement
To improve the efficiency of full-spectrum correlated-k-distribution (FSCK), a new method FSCK-RSM has been proposed based on response surface methodology (RSM) in this paper. The k-distributions of FSCK was used to fit the response surface model based on radial basis function and the radiative calculation efficiency was improved by avoiding multiple computations and interpolation in the FSCK-RSM. The thermal radiation heat transfer of five combustion gases (H2O, CO2, CO, C2H2 and C2H4) in a one-dimensional layer was investigated and the radiative sources calculated by the LBL, SNB, FSCK and FSCK-RSM methods were given at different distributions of temperature and gas concentration. The results showed that the needed amount of input data was reduced by 677 times using FSCK-RSM comparing to the FSCK and the maximum of the average normalized deviation for FSCK-RSM was 2.46% in the non-isothermal homogeneous medium. The model was finally used for radiation reabsorption calculations in planar C2H4/O2/N2/CO2 flames with full coupling to heat transfer and multi-species chemistry. The computational time using the FSCK-RSM was found to be at most half of that using the FSCK method. This FSCK-RSM model was an effective method for addressing the radiation problems that occur in combustion systems.
1. Introduction Due to the linear structure of emission and absorption spectra of most combustion gases, radiation heat transfer in combustion gases becomes extremely complicated. For example, the absorption spectra of CO2 and H2O in different versions of HITRAN contain 105 to 106 spectral lines over each entire spectrum [1]. Despite many studies have been performed in the treatment of non-gray gas radiation properties in the past two decades, it is still difficult to achieve good computational performance in precision and efficiency. Currently, the most accurate model for calculating radiative properties is the line-by-line (LBL) [2], which however, requires a large amount of calculation. For example, radiative transfer equation (RTE) has to be solved 4.95 × 105 times if the interval is 0.02 cm−1 in the spectral range of 100 cm−1-10,000 cm−1. For this reason, LBL is usually used for solving low dimensional problems or benchmarking solutions to check the accuracy of other models. The statistical narrow band models (SNB) [3,4], including the standard SNB, the exponential SNB [3], and the Malkmus SNB [4] are considered to be the most
⁎
Corresponding author. E-mail address: [email protected] (Q. Lu).
https://doi.org/10.1016/j.icheatmasstransfer.2020.104566
0735-1933/ © 2020 Elsevier Ltd. All rights reserved.
successful band models. In some complex and computational expensive cases, SNB models can be used as an alternative of LBL for benchmarking. However, it can only give the transmissivity of gas in a certain path and cannot be applied to all RTE solutions. Recently, k-distribution methods based on the reordering concept have been widely studied [5–8]. Using these methods, the RTE can be rewritten from the wavenumber space to the probability distribution space of the absorption coefficient. Furthermore, the wavenumber-dependent absorption coefficient is transformed into the k-distribution which monotonically increases with the cumulative distribution function. To this direction, the full-spectrum k-distribution (FSK) method proposed by Modest [5] has become one of the most promising k-distribution methods. Using highprecision Gauss quadrature method for spectral integration, FSK method can reduce the number of RTE calculations from more than 1 million to about ten without losing accuracy [9]. To improve the calculation efficiency and to guarantee the accuracy of radiation heat transfer calculation, an FSK look-up table with two species H2O and CO2 was proposed by Wang [10] such that the k-distributions can be obtained by applying an appropriate mixing model.
International Communications in Heat and Mass Transfer 114 (2020) 104566
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Fig. 1. Change of (a) R2 and (b) emax with the number of sampling states M and the shape parameter of Gaussian function c. The degree of polynomial n was fixed as 2.
Subsequently, Wang et al. constructed an FSK look-up table with three species H2O, CO2 and CO [11] and the k-distributions were assembled directly by summing the linear absorption coefficients. Moreover, since the k-distributions obtained by these FSK look-up tables need multi-step interpolations, Wang et al. further proposed an improved full-spectrum correlated-k-distribution (FSCK) look-up table by storing the correlated k-values in a new table [12]. The k-distributions can be retrieved directly from the new tabulated values, avoiding multiple computations and interpolation, thus saving considerable CPU time during the radiation calculation. Nevertheless, the radiation characteristics of gas mixtures still occupy a large amount of data space, whose size increases exponentially with the number of gas species in the FSCK model. In the past two decades, the response surface methodology (RSM) [13] has been widely used as a comprehensive statistical method to predict the relation between the input and output of complex systems, which is exactly how to calculate the radiation characteristics of gas mixtures from spectral line data of spectral database by FSCK. For instance, only 32 radiation characteristic parameters k*(Tref, Φ, gref) are needed for one state when using 32 points Gauss quadrature method with FSCK [14]. In a mathematical sense, the RSM is an interpolation technique that fits a multidimensional function with finite sample points in any function domain [15]. Subsequently, the optimal problem can be reconstructed and solved efficiently by using the selected response surface model [16]. To provide an interpolating surface through all sampled data points, some nonparametric metamodeling methods have been proposed with RSM, such as polynomial response surface
Fig. 2. Change of (a) R2 and (b) emax with the degree of polynomial n and the shape parameter of Gaussian function c. Number of sampling states M was fixed as 2000.
Fig. 3. Change of R2 and emax with the shape parameter of Gaussian function c.
2
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Table 1 The cases for the validation of FSCK-RSM. Case
Medium
1 2 3 4 5 6 7
H2O H2O H2O H2O H2O H2O H2O
+ + + + + + +
CO2 CO2 CO2 CO2 CO2 CO2 CO2
+ CO + C2H2 + C2H4 + CO + C2H2 + C2H4
+ CO + C2H2 + C2H4
T (K)
XH2O
XCO2
XCO
XC2H2
XC2H4
1000 1000 1000 1000 1000 500[1 + sin(πx/L)] 500[1 + sin(πx/L)]
0.3 0.2 0.15 0.1 0.5×(1-x/L)/L 0.5×(1-x/L)/L 0.1
0.3 0.2 0.1 0.1 0.5×(1-x/L)/L 0.5×(1-x/L)/L 0.1
– 0.02 – 0.05 – – 0.05
– – 0.1 0.05 – – 0.05
– – 0.3 0.5 – – 0.5
Fig. 4. Distributions of the radiative source and the normalized deviation for Case 1: the temperature 1000 K, XH2O = 0.3 and XCO2 = 0.3.
3
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Fig. 5. Distributions of the radiative source and the normalized deviation for Case 2: the temperature of 1000 K, XH2O = 0.2, XCO2 = 0.2 and XCO = 0.02.
by R-Squared (R2) and normalized maximal relative errors (emax). Since a(T, Tref, gref) is a function of gaussian point and temperature while k*(Tref, Φ, gref) is a function of gaussian point, temperature and state parameter, the data space of a-values is much less than that of k-values. Therefore, only the table data for k*(Tref, Φ, gref) was used to fit the response surface model and the a-values were kept unaltered. In Section 3, the radiative transfers of five gas species mixture in a one-dimensional layer are investigated by FSCK-RSM and the results are validated by LBL and SNB. Futhermore, the model has been used for radiation reabsorption calculations in a planar C2H4/O2/N2/CO2 flame with full coupling to the conservation equations for mass, momentum, energy, and species. Finally, the conclusions are summarized in Section 4.
[17], Kriging model [18], multivariate adaptive regression splines [19] and radial basis function (RBF) [20]. RBF model has been proved to be the most effective multidimensional approximation method, with its performance being independent on the dimensionality of the problem [21], and is thus suitable for dealing with high dimensional problems. In this paper, an FSCK-RSM method is proposed to improve the calculation efficiency without affecting the computation accuracy for radiative transfer in inhomogeneous gas mixtures (H2O, CO2, CO, C2H2 and C2H4). The principles of the FSCK-RSM are derived in Section 2: a certain number of sampling states are selected to calculate the k-distributions of FSCK model by latin hypercube sampling method and the response surface model based on RBF is fitted. The fitting is evaluated 4
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Fig. 6. Distributions of the radiative source and the normalized deviation for Case 3: the temperature of 1000 K, XH2O = 0.15, XCO2 = 0.1, XC2H2 = 0.1 and XC2H4 = 0.2.
where ϕ is the state vector containing all the parameters that affect the spectral absorption coefficient, κη the spectral absorption coefficient calculated from the spectral database, and δ the Dirac-delta function. The full spectrum Planck-function weighted cumulative k-distribution is described as:
2. Models 2.1. The full-spectrum correlated-k model (FSCK) The correlated k distribution model in narrow band has been extended to the full spectrum. Since the blackbody radiation intensity changes with wave number in the full spectrum, a Planck-function weighted k-distribution was introduced [22]:
f (T , ϕ, k ) =
1 Ib (T )
∫0
∞
Ibη (T ) δ (k − κ η (ϕ, η))dη
g (T , ϕ, k ) =
∫0
k
f (T , ϕ, k ′) dk ′
(2)
Due to the smoothness and monotony of the cumulative function, the integration of the reordered wavenumber can be obtained by a given integration scheme.
(1) 5
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Fig. 7. Distributions of the radiative source and the normalized deviation for Case 4: the temperature of 1000 K, XH2O = 0.1, XCO2 = 0.1, XCO = 0.05, XC2H2 = 0.05 and XC2H4 = 0.5.
Based on the assumption that the absorption coefficient distributions of different gases are not correlated and the product of transmissivities model [23], Modest and Riazzi [6] proposed a new narrowband mixing model for calculating the full spectrum k-distributions of multi-gas mixtures. In this paper, mixing is performed at the narrow band (NB) level by using this method. The narrow-band parameters of H2O, CO2, CO, C2H2 and C2H4 were provided by Soufiani [24] and Qi [25]. The ten-point Gauss–Lobatto quadrature was adopted in this study.
To apply the FSK in a nonhomogeneous mixture, radiative properties need to be conducted in a unified reference g-space gref: g (T, ϕref, k) = g(T, ϕ, k∗). Then the radiative transfer equation (RTE) in FSCK based on gref-space can be expressed as [22]:
dIg ds
= k ∗ (Tref , ϕ, gref ) [a (T , Tref , gref ) Ib (T ) − Ig ]
where
1
I = ∫0 Ig dg = ∑ wi Ig (gi ) ,
(Tref, ϕref, k).
(3)
N
a(T, Tref, gref)
=
dg(T, ϕref, k)/dg
i=1
6
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Fig. 8. Distributions of the radiative source and the normalized deviation for Case 5: the temperature of 1000 K, XH2O = XCO2 = 0.5×(1-x/L)/L. M
2.2. Response surface model based on RBF
∑ Pi (x (j) )⋅λj = 0
f (x) =
M
f (x) is the predicted absorption coefficient, x = {x1, x2, …, xl} where the vector of variables (reference temperature, local temperature and gas concentration), l the number of variables, ||x-x(j)|| the Euclidean distance between an arbitrary state x and a sampling state x(j), M the number of sampling states, λ = {λ1, λ2, …, λM} the regression coefficient vector of radial basis function, b = {b1, b2, …, bN} the regression coefficient of the additional polynomials, N =
N
∑ λj φ (‖x − x (j) ‖) + ∑ bi Pi (x) j=1
i=1
(5)
j=1
The weighted sum of a set of functions was used to fit the response surface in the RBF method. The values of these functions only depend on the Euclidian distance between the independent variable and the sampling state φ(x,x(j)) ≡ φ(||x-x(j)||). As long as the sample points are different, these functions are orthogonal. To improve the fitting characteristics of the near-linear region, some additional polynomials and constraint conditions were adopted for the FSCK-RSM in the RBF [26]:
1 n!
n
∏ (l + i) the number i=1
of mutually orthogonal polynomial terms, and n is the degree of polynomial. Eqs. (4) and (5) can be rewritten in matrix form as:
(4)
7
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Fig. 9. Distributions of the radiative source and the normalized deviation of Case 6: the temperature of 500[1 + sin(πx/L)] K, XH2O = XCO2 = 0.5×(1-x/L)/L.
⎡ ΦT P ⎤ ⎡ Λ ⎤ = ⎡ F ⎤ ⎣P 0 ⎦ ⎣ B ⎦ ⎣ 0 ⎦
λ = {λ1, λ2, …, λM} and b = {b1, b2, …, bN} can be obtained by solving Eq. (6). The response surface fitting effect needs to be evaluated with both global average and worst-case properties. A good response surface fitting should have not only satisfying average fitting, but also acceptable fitting under the worst conditions, which should not drastically deviate from the actual response. Otherwise, the model accuracy will fluctuate in response to the actual input. f (x) : R-Squared (R2) was used to assess the quality of the predicted
(6)
11 1M ⎡ φ (r ) ⋯ φ (r ) ⎤ Φ=⎢ ⋮ ⋱ ⋮ ⎥, rij = ||x(i)-x(j)||, ⎢ ⎥ M 1 MM ⎣ φ (r ) ⋯ φ (r ) ⎦ (1) ) ⋯ P (x (1) ) P ( x 1 N ⎡ ⎤ ⎥,Λ = [λ1 ⋯ λM]T, B = [ b1 ⋯ bN ]T , ⋮ ⋱ ⋮ P=⎢ ⎢ ⎥ ( M ) ( M ) ⎣ P1 (x ) ⋯ PN (x ) ⎦ F = [ f1 ⋯ fm ]T and fj is the real response absorption coefficient corresponding to each sampling state. To get the global characteristics of the actual response problem via a minimum number of sampling states, the Latin hypercubes design method [27] was applied. The Gaussian function φ(r) = exp.(−cr2) (c is the shape parameter) was used in the RBF. Subsequently, the
where
m
f (xi)]2 ∑ [f (xi) − R2 = 1 −
i=1 m
∑ [f (xi) − f ]2 i=1
(7)
where f(xi) is the real response absorption coefficient corresponding to 8
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Fig. 10. Distributions of the radiative source and the normalized deviation for Case 7: the temperature of 500[1 + sin(πx/L)] K, XH2O = 0.1, XCO2 = 0.1, XCO = 0.05, XC2H2 = 0.05 and XC2H4 = 0.5.
uu0 = su0
f (xi) the results predicted by the Eq. (6), f the the ith testing state, average value of f(xi) at m testing states. A closer-to-unity value of R2 indicates a higher accuracy of approximation. The normalized maximal relative error (emax) was used to assess the worst case:
T
0 b
Yu downstream upstream
reaction sheet
emax = max ⎜⎛ ⎝
Tu Y =0 0 b
m⋅|f (xi) − f (xi)| ⎞ ⎟ m ∑i = 1 f (xi) ⎠
(8)
A value of emax closer to 0 indicates that a higher accuracy of approximation is achieved at the worst state.
x
0D
3. Results and discussion
Fig. 11. Schematic showing the premixed flame structure at the transport and reaction-sheet level [29].
Seven variables were choosen to establish the response surface model: five gas species concentrations (XCO2, XH2O, XCO, XC2H2 and 9
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Fig. 12. Laminar flame speeds and the reabsorption effect on the flame speed of C2H4-(0.21O2/0.74N2/0.05CO2) flames at various equivalence ratios 0.7–1.0 by different radiation models.
the R2 and emax, the best accuracy was achieved when n = 2 and c = 7. It is hence concluded based on the above discussion that the choice of M = 2000 and n = 2 as parameter fitting response surface provides not only an accurate model, but also significantly reduces the computational complexity. The corresponding optimal c should be around 7 when M = 2000 and n = 2, as manifested in Fig. 2. To get a more accurate value c, further intensive numerical calculations were carried out near c = 7, as shown in Fig. 3. When c = 7.2, R2 reached its the maximum value, and thus c = 7.2 was selected as the RBF parameter in the response surface model. To study the gas mixtures with the five given gas species, the table data for k*(Tref, Φ, gref) calculated by the original FSCK model contain: 21 points for the reference temperature (300–2300 K, every 100 K), 21 points for the local temperature (300–2300 K, every 100 K), 5 points for the 5 gas species volumetric concentration (1 × 10−4-1.0, every 0.25), and 10 points Gauss–Lobatto quadrature. A total of 13,781,250 data storage was required in the FSCK. However, with the new FSCK-RSM
XC2H4), reference temperature Tref and local temperature Tloc. Tref and Tloc varied in the temperature range of 300–2300 K. XCO2, XH2O and XC2H4 varied from 0 to 1, while XCO and XC2H2 0 to 0.5. To simplify the problem, the gas species concentrations at the reference state were fixed as XCO2 = 0.1, XH2O = 0.1, XCO = 0.05, XC2H2 = 0.05 and XC2H4 = 0.5. The Latin hypercubes design method was used to generate sampling state following normalization of the parameter value range. Invalid sampling states with sum of concentrations of the five gas species greater than 1 were discarded until the number of effective sampling states reached the required number. To determine the appropriate number of sampling states M, the shape parameter of Gaussian function c and the degree of polynomial n, numerical calculations were first carried out over a large range. Since the maximal absorption coefficient appears at the last gaussian quadf (x) at this node were used to evaluate the rature node, R2 and emax of optimal values in the Eq. (4). As shown in Fig. 1, R2 and emax varied with different numbers of sampling states M and the shape parameter of Gaussian function c when the degree of polynomial n was fixed as 2. It is evident that the response surface fitting performance did not rise monotonously with the increase of sampling state number M. Even after the number of sampling state reaches 3500 or higher, obvious overfitting phenomenon appears, and the fitting error increases rapidly. As shown in Fig. 1 (a), R2 decreased with the c with M = 100, while it first increased and then decreased with the c within the range of sampling state numbers M = 200–5000. Since a close-to-unity R2 indicates a higher accuracy, the best R2 is 0.987 was obtained with M = 2000 and c = 7. In Fig. 1 (b), emax increased with the c when M = 100–200, while it first decreased and then increased with the c when M was in the range of 500–5000. As higher accuracy is achieved when the value of emax is closer to 0, the best obtained emax was 1.29 with M = 2000 and c = 7. As shown in Fig. 2, R2 and emax varied with the different degree of polynomial n and the shape parameter of Gaussian function c, when the number of sampling states M was fixed as 2000. It was observed that for high shape parameter of Gaussian function c the response surface fitting performance gets better with the increase of the degree of polynomial n. While with low c values, the lower-order polynomials had better performance. For all investiaged degrees of polynomial n, R2 and emax changed non-monotonically with c. The error first decreased with rising c until a critical value at ~4–7, and then increased monotonically. The results of R2 and emax in Fig. 2 indicated that the optimal c values were 5, 6 and 7 for all degrees of polynomial. Combining the results of both
model, as M was selected as 2000 and N =
1 n!
n
∏ (l + i) = 36 in the i=1
response surface model, only 20,360 data storage was required. The amount of input data has been reduced by 677 times. Moreover, the calculation efficiency is further improved as multiple computations and interpolation are avoided in FSCK-RSM. To validate the accuracy of FSCK-RSM, the thermal radiation heat transfer in a one-dimensional gas layer was investigated and different mixtures of absorbing-emitting gases (XCO2, XH2O, XCO, XC2H2 and XC2H4) were studied. The thickness of the layer was 0.08 m. The total gas pressure of all the cases was fixed as 1 atm, the temperature of the two walls 300 K and the emissivity of both walls 1.0. The RTE was solved by discrete ordinates method (DOM) [28]. The resolution of absorption coefficients is set to 0.01 cm−1 for the LBL model while 0.1 cm−1 for FSCK model. The numerical calculations were carried out at the same spatial and angular discretizations for all the cases. The distributions of gas temperature and species mole fractions for the seven test cases are summarized in Table 1. For non-isothermal medium, the reference temperature was calculated according to the radiation intensity weighted average temperature [7]:
10
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heat source by FSCK or FSCK-RSM comparing to that by LBL, which is ∇ ⋅ qLBL − ∇ ⋅ qFSCK used as the benchmark, is defined as: error(%) = × 100 . ∇⋅q LBL,max
As shown in Fig. 4, the average normalized deviation of FSCK-RSM (1.66%) was only slightly higher than that of FSCK (1.45%). In addition, the maximal normalized deviation of FSCK-RSM was 2.11%, significantly lower than that of FSCK (3.22%). CO was added in the gas mixture of Case 2. As shown in Fig. 5, the FSCK and FSCK-RSM predictions agreed very well with the LBL prediction, while the radiative source calculated by SNB was less than the other models. The maximal normalized deviation for FSCK-RSM was 1.35%, a little higher than that of FSCK, 0.87%. The average normalized deviation of FSCK-RSM was 0.69%, which was also higher than that of FSCK (0.18%). It indicates that as a penalty for the much lower computational cost, FSCK-RSM only scarified accuracy by ~0.5% comparing to FSCK in this case. In Case 3, CO was removed while C2H2 and C2H4 was added in the gas mixture. As shown in Fig. 6, the FSCK-RSM results agreed very well with the LBL results while the FSCK predictions were closer to those by the SNB. It is further noted that despite a somewhat higher error at the two boundary points (3.56%), the FSCK-RSM predictions had excellent average normalized deviation (0.64%), significantly lower than those of FSCK (2.60%). Hence, for the investigated condition of Case 3, FSCKRSM had not only much lower computational cost, but also much better accuracy than the FSCK. Five gas species (H2O, CO2, CO, C2H2 and C2H4) were considered in the gas mixtures of Case 4. The gas concentrations data were taken from the centerline of a laminar ethylene/air diffusion flame in [25]. As shown in Fig. 7, the FSCK-RSM predicted radiative source agrees well with the SNB results, while the LBL prediction was higher than the other models. Similar to Case 3, FSCK-RSM had a high normalized deviation (4.52%) at the two boundary points, while its average normalized deviations (1.97%) were much lower than those of FSCK (3.37%). 3.2. Isothermal inhomogeneous medium (Case 5) In Case 5, an isothermal inhomogeneous medium filled with H2O and CO2 was studied. The mole fractions of the of H2O and CO2 gas species were set as parabolic distributions. The distributions of the radiative sources calculated by LBL, FSCK and FSCK-RSM are shown in Fig. 8. FSCK-RSM had similar accuracy (note the positive/negative magnitudes) near the two walls while worse accuracy in the central area than FSCK. The average normalized deviation of FSCK-RSM predictions was 1.87%, which was higher than those of FSCK (1.29%) by ~0.6%. Similar to Case 2, FSCK-RSM did not scarify the computational accuracy much comparing to FSCK.
Fig. 13. Distributions of temperature (a) and volumetric radiative heat loss (b) of C2H4-(0.21O2/0.74N2/0.05CO2) flame at ϕ = 1.0.
3.3. Non-isothermal inhomogeneous medium (Case 6)
N
∑ ∑ vi Xav, sp kp (Ti ; sp) Ti4
∑ Xav,sp kp (Tref ; sp) Tref4 = sp
i = 1 sp
Furthermore, a non-isothermal inhomogeneous medium filled with H2O and CO2 was investigated in Case 6. The temperature profile of the gas mixture was set as a sine function 500[1 + sin(πx/L)]. The mole fractions of the gas species were the same as Case 5. The distributions of the radiative source and the normalized deviation calculated by LBL, FSCK and FSCK-RSM are compared in Fig. 9. FSCK-RSM resulted in very close results to LBL near the two walls, better than FSCK. The difference between the FSCK-RSM and LBL predictions in the central area was similar to that between FSCK and LBL. Furthermore, the average normalized deviation of FSCK-RSM (1.12%) was also similar to that of FSCK (1.22%).
N
∑ vi i=1
(9)
where sp is different gas species, vi the volume of the ith discrete element and kp(T; sp) the Plank mean absorption coefficient. 3.1. Isothermal homogeneous medium (Cases 1, 2, 3 and 4) In Case 1, the 1-D parallel-plane layer was filled with H2O and CO2, and was considered as an isothermal homogeneous medium. The distributions of the radiative source calculated by LBL, SNB, FSCK and FSCK-RSM are shown in Fig. 4. It can be seen that the FSCK and FSCKRSM predictions were in excellent agreement with the SNB prediction, while the radiative source calculated by the LBL was slightly less than other models. Since the radiative heat source was very small in the whole domain, the normalized deviation of the calculated radiative
3.4. Non-isothermal homogeneous medium (Case 7) In the last model validation case, Case 7, a non-isothermal homogeneous medium filled with H2O, CO2, CO, C2H2 and C2H4 was studied. 11
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flame speed predicted by SNB was lower than those predicted by FSCK and FSCK-RSM.
The temperature of the gas mixture was the same as that of Case 6, the mole fractions of the gas species were the same as those in Case 4 (see Table 1). The distributions of the radiative source and the normalized deviation calculated by the LBL, FSCK and FSCK-RSM are shown in Fig. 10. The maximal normalized deviation of FSCK-RSM (3.82%) was less than that of FSCK (4.54%), and the average normalized deviation of FSCK-RSM (2.46%) was also lower than that of FSCK (3.48%). This indicated that FSCK-RSM had better accuracy comparing to FSCK in non-isothermal homogeneous cases. The computation time for Case 7 with 180 discrete directions in Fig. 9 was 9.7 s using FSCK-RSM while the computation time for FSCK was 19.3 s. Consequently, FSCK-RSM showed not only better accuracy, but also better computational efficiency than FSCK in the non-isothermal homogeneous case. The much higher computational efficiency of FSCK-RSM was due to two reasons: first, as discussed previously the amount of input data was reduced by 677 times using FSCK-RSM; second, the absorption coefficient was directly calculated by Eq. (4) and no interpolation was needed.
4. Conclusions In this paper, an improved FSCK model, FSCK-RSM was developed in order to improve the computational efficiency for non-gray radiative heat transfer in combustion gas mixtures. Radiation transfer in a onedimensional layer filled with gas mixtures of H2O, CO2, CO, C2H2 and C2H4 with various configurations of gas temperatures and species concentrations were studied. The FSCK-RSM predictions exhibited good agreement with the LBL benchmarks that were generated using the same spectral database. The maximal average relative difference of the radiative source predicted by FSCK-RSM was 2.46% for a non-isothermal homogeneous medium, lower than that of the original FSCK model. FSCK-RSM showed a better computational efficiency than FSCK by reducing the amount of input data by around 677 times, and by avoiding interpolations in the RBF. FSCK-RSM was successfully tested for the predictions of radiation reabsorption in planar C2H4/O2/N2/CO2 flames. The results indicated that the computational time was saved by ~50% using FSCK-RSM comparing to FSCK without affecting the accuracy in predicting the laminar flame speeds.
3.5. Planar C2H4/O2/N2/CO2 flames Finally, the energy and chemical species conservation equations of realistic flames were taken into consideration. Premixed steady C2H4(0.21O2/0.74N2/0.05CO2) planar flames were solved by the PREMIX package [30] of CHEMKIN. The GRI Mech 3.0 chemistry was used [31]. The pressure was fixed as 1 atm and equivalence ratio varied in the range of 0.7–1.0. For the propagation of the standard premixed flame shown in Fig. 11, the upstream mixture approaches the flame with velocity uu = su0 and temperature Tu, and leaves the flame with velocity ub0 and temperature Tb0. As the mixture approaches the flame, it is gradually heated up by the heat conducted forward from the chemical heat release region, resulting in a continuously increasing temperature profile until Tb0 is reached. The profile is not linear due to the presence of convective transport. The continuous heating of the mixture will eventually lead to its ignition and subsequent reaction. To study and to compare the radiation effects, five radiation models were applied: the adiabatic model (ADI) in which radiation is ignored, the optically thin model (OTM) where only the radiation emission of H2O, CO2 and CO was considered [32], the SNB, the FSCK and the FSCK-RSM. As shown in Fig. 12, the laminar flame speed increased when the radiation reabsorption (SNB, FSCK and FSCK-RSM) was considered for every equivalence ratio. The reason was that radiation from the downstream hot products was reabsorbed by the upstream cold mixture. Moreover, the reabsorption effect increased with droping of equivalence ratio as a result of the increased CO2 mole fraction. It can be seen that the laminar flame speeds predicted by FSCK and FSCKRSM are very close to each other but both larger than that by SNB. Characteristically, the relative increment of the laminar flame speed was 4.06% for FSCK-RSM and 3.84% for FSCK at ϕ = 1. Therefore, the FSCK-RSM exhibited a similar accuracy with FSCK in predicting the laminar flame speed. However, the computation time was only 67.5 s using FSCK-RSM while the computation time of FSCK was 136.3 s. To further understand how the radiation model affects the laminar flame speed, distributions of gas temperatures and volumetric radiative heat losses of a C2H4-(0.21O2/0.74N2/0.05CO2) mixture (ϕ = 1.0) are shown in Fig. 13. It is shown in Fig. 12(a) that the gas temperatures decreased in the downstream when considering the radiation (see the ADI curves and others). The temperatures calculated by SNB, FSCK and FSCK-RSM were higher than that calculated by OTM due to the radiation reabsorption. In the upstream, however, the radiative heat loss was negative as some radiation emitted at the high temperature reaction zone is reabsorbed by the upstream lower temperature region. Furthermore, the radiation reabsorption energy in the upstream predicted by SNB was less than those by FSCK and FSCK-RSM. Consequently, the increment of temperature in the upstream calculated by SNB was less than those calculated by FSCK and FSCK-RSM. This is why the laminar
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This research was supported by the National Key Research Development Program of China (No.2017YFB0601900), the National Natural Science Foundation of China (No. 51976057, 51922040 and 51827808), the China Scholarship Council and the Fundamental Research Funds for the Central Universities (No. 2017ZZD005). RS acknowledges the Swiss National Science Foundation postdoctoral fellowship (Award Number 178619). References [1] H. Chu, S. Zheng, W. Liang, H. Zhou, Effects of radiation reabsorption of C1-C6 hydrocarbon flames at normal and elevated pressures, Fuel. 266 (2020) 117061. [2] J. Arnold, E. Whiting, G. Lyle, Line by line calculation of spectra from diatomic molecules and atoms assuming a voigt line profile, J. Quant. Spectrosc. Radiat. Transf. 9 (6) (1969) 775–798. [3] R. Goody, A statistical model for water-vapour absorption, Q. J. R. Meteorol. Soc. 78 (338) (1952) 638–640. [4] W. Malkmus, Random Lorentz band model with exponential-tailed S− 1 line-intensity distribution function, JOSA 57 (3) (1967) 323–329. [5] M.F. Modest, Narrow-band and full-spectrum k-distributions for radiative heat transfer—correlated-k vs. scaling approximation, J. Quant. Spectrosc. Radiat. Transf. 76 (1) (2003) 69–83. [6] M.F. Modest, R.J. Riazzi, Assembly of full-spectrum k-distributions from a narrowband database; effects of mixing gases, gases and nongray absorbing particles, and mixtures with nongray scatterers in nongray enclosures, J. Quant. Spectrosc. Radiat. Transf. 90 (2) (2005) 169–189. [7] Y. Du, Z. Peng, Y. Ding, A high-accurate and universal method to characterize the relative wavelength response (RWR) in wavelength modulation spectroscopy (WMS), Opt. Express. 28 (2020) 3482–3494. [8] Y. Sun, S. Zheng, B. Jiang, J. Tang, F. Liu, One-dimensional P1 method for gas radiation heat transfer in spherical geometry, Int. J. Heat Mass Transf. 145 (2019) 118777. [9] C. Wang, M.F. Modest, B. He, Full-spectrum correlated-k-distribution look-up table for use with radiative Monte Carlo solvers, Int. J. Heat Mass Transf. 131 (2019) 167–175. [10] A. Wang, High-accuracy, compact database of narrow-band k-distributions for water vapor and carbon dioxide, ICHMT DIGITAL LIBRARY ONLINE, Begel House Inc., 2004. [11] C. Wang, W. Ge, M.F. Modest, B. He, A full-spectrum k-distribution look-up table for radiative transfer in nonhomogeneous gaseous media, J. Quant. Spectrosc. Radiat. Transf. 168 (2016) 46–56. [12] C. Wang, B. He, M.F. Modest, T. Ren, Efficient full-spectrum correlated-k-
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[23] J. Taine, A. Soufiani, Gas IR radiative properties: from spectroscopic data to approximate models, Advances in Heat Transfer, Elsevier, 1999, pp. 295–414. [24] P. Rivière, A. Soufiani, Updated band model parameters for H2O, CO2, CH4 and CO radiation at high temperature, Int. J. Heat Mass Transf. 55 (13–14) (2012) 3349–3358. [25] C. Qi, S. Zheng, H. Zhou, Calculations of thermal radiation transfer of C2H2 and C2H4 together with H2O, CO2, and CO in a one-dimensional enclosure using LBL and SNB models, J. Quant. Spectrosc. Radiat. Transf. 197 (2017) 45–50. [26] L. Zhou, G. Yan, J. Ou, Response surface method based on radial basis functions for modeling large-scale structures in model updating, Computer-Aided Civil Infrastruct. Eng. 28 (3) (2013) 210–226. [27] E.R. Van Dam, B. Husslage, D. Den Hertog, H. Melissen, Maximin Latin hypercube designs in two dimensions, Oper. Res. 55 (1) (2007) 158–169. [28] I. Ayranci, N. Selçuk, MOL solution of DOM for transient radiative transfer in 3-D scattering media, J. Quant. Spectrosc. Radiat. Transf. 84 (4) (2004) 409–422. [29] C.K. Law, Combustion Physics, Cambridge university press, 2010. [30] R.J. Kee, J.F. Grcar, M.D. Smooke, J.A. Miller, E. Meeks, PREMIX: A Fortran Program for Modeling Steady Laminar One-Dimensional Premixed Flames, Sandia National Laboratories Report, 1985 SAND85-s8249. [31] http://www.me.berkeley.edu/gri_mech/. [32] Y. Ju, H. Guo, F. Liu, K. Maruta, Effects of the Lewis number and radiative heat loss on the bifurcation and extinction of CH 4/O 2-N 2-He flames, J. Fluid Mech. 379 (1999) 165–190.
distribution look-up table, J. Quant. Spectrosc. Radiat. Transf. 219 (2018) 108–116. [13] R.H. Myers, Montgomery DC Response Surface Methodology: Process and Product Optimization Using Designed Experiments, John and Sons, Inc., New York, NY, 1995. [14] S. Mazumder, M.F. Modest, Application of the full spectrum correlated-k distribution approach to modeling non-gray radiation in combustion gases, Combust. Flame 129 (4) (2002) 416–438. [15] A.A. Mullur, A. Messac, Extended radial basis functions: more flexible and effective metamodeling, AIAA J. 43 (6) (2005) 1306–1315. [16] S. Ho, S. Yang, G. Ni, H.-c.C. Wong, A response surface methodology based on improved compactly supported radial basis function and its application to rapid optimizations of electromagnetic devices, IEEE Trans. Magnet. 41 (6) (2005) 2111–2117. [17] G.E. Box, K.B. Wilson, On the experimental attainment of optimum conditions, J. R. Stat. Soc. Ser. B Methodol. 13 (1) (1951) 1–38. [18] T.W. Simpson, T.M. Mauery, J.J. Korte, F. Mistree, Kriging models for global approximation in simulation-based multidisciplinary design optimization, AIAA J. 39 (12) (2001) 2233–2241. [19] J.H. Friedman, Multivariate adaptive regression splines, Ann. Stat. 19 (1) (1991) 1–67. [20] M.F. Hussain, R.R. Barton, S.B. Joshi, Metamodeling: radial basis functions, versus polynomials, Eur. J. Oper. Res. 138 (1) (2002) 142–154. [21] M.D. Buhmann, Radial basis functions, Acta Num. 9 (2000) 1–38. [22] M.F. Modest, Radiative Heat Transfer, Academic press, 2013.
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Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
An improved full-spectrum correlated-k-distribution model for non-gray radiative heat transfer in combustion gas mixtures Shu Zhenga, Ran Suib, Yu Yanga, Yujia Sunc, Huaichun Zhoud, Qiang Lua,
T
⁎
a
National Engineering Laboratory for Biomass Power Generation Equipment, North China Electric Power University, Beijing 102206, China Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA c School of Atmospheric Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China d School of Energy and Power Engineering, Northeast Electric Power University, Jilin 132012, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Radiative heat transfer Full-spectrum correlated-k-distribution (FSCK) Planar flames Efficiency improvement
To improve the efficiency of full-spectrum correlated-k-distribution (FSCK), a new method FSCK-RSM has been proposed based on response surface methodology (RSM) in this paper. The k-distributions of FSCK was used to fit the response surface model based on radial basis function and the radiative calculation efficiency was improved by avoiding multiple computations and interpolation in the FSCK-RSM. The thermal radiation heat transfer of five combustion gases (H2O, CO2, CO, C2H2 and C2H4) in a one-dimensional layer was investigated and the radiative sources calculated by the LBL, SNB, FSCK and FSCK-RSM methods were given at different distributions of temperature and gas concentration. The results showed that the needed amount of input data was reduced by 677 times using FSCK-RSM comparing to the FSCK and the maximum of the average normalized deviation for FSCK-RSM was 2.46% in the non-isothermal homogeneous medium. The model was finally used for radiation reabsorption calculations in planar C2H4/O2/N2/CO2 flames with full coupling to heat transfer and multi-species chemistry. The computational time using the FSCK-RSM was found to be at most half of that using the FSCK method. This FSCK-RSM model was an effective method for addressing the radiation problems that occur in combustion systems.
1. Introduction Due to the linear structure of emission and absorption spectra of most combustion gases, radiation heat transfer in combustion gases becomes extremely complicated. For example, the absorption spectra of CO2 and H2O in different versions of HITRAN contain 105 to 106 spectral lines over each entire spectrum [1]. Despite many studies have been performed in the treatment of non-gray gas radiation properties in the past two decades, it is still difficult to achieve good computational performance in precision and efficiency. Currently, the most accurate model for calculating radiative properties is the line-by-line (LBL) [2], which however, requires a large amount of calculation. For example, radiative transfer equation (RTE) has to be solved 4.95 × 105 times if the interval is 0.02 cm−1 in the spectral range of 100 cm−1-10,000 cm−1. For this reason, LBL is usually used for solving low dimensional problems or benchmarking solutions to check the accuracy of other models. The statistical narrow band models (SNB) [3,4], including the standard SNB, the exponential SNB [3], and the Malkmus SNB [4] are considered to be the most
⁎
Corresponding author. E-mail address: [email protected] (Q. Lu).
https://doi.org/10.1016/j.icheatmasstransfer.2020.104566
0735-1933/ © 2020 Elsevier Ltd. All rights reserved.
successful band models. In some complex and computational expensive cases, SNB models can be used as an alternative of LBL for benchmarking. However, it can only give the transmissivity of gas in a certain path and cannot be applied to all RTE solutions. Recently, k-distribution methods based on the reordering concept have been widely studied [5–8]. Using these methods, the RTE can be rewritten from the wavenumber space to the probability distribution space of the absorption coefficient. Furthermore, the wavenumber-dependent absorption coefficient is transformed into the k-distribution which monotonically increases with the cumulative distribution function. To this direction, the full-spectrum k-distribution (FSK) method proposed by Modest [5] has become one of the most promising k-distribution methods. Using highprecision Gauss quadrature method for spectral integration, FSK method can reduce the number of RTE calculations from more than 1 million to about ten without losing accuracy [9]. To improve the calculation efficiency and to guarantee the accuracy of radiation heat transfer calculation, an FSK look-up table with two species H2O and CO2 was proposed by Wang [10] such that the k-distributions can be obtained by applying an appropriate mixing model.
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Fig. 1. Change of (a) R2 and (b) emax with the number of sampling states M and the shape parameter of Gaussian function c. The degree of polynomial n was fixed as 2.
Subsequently, Wang et al. constructed an FSK look-up table with three species H2O, CO2 and CO [11] and the k-distributions were assembled directly by summing the linear absorption coefficients. Moreover, since the k-distributions obtained by these FSK look-up tables need multi-step interpolations, Wang et al. further proposed an improved full-spectrum correlated-k-distribution (FSCK) look-up table by storing the correlated k-values in a new table [12]. The k-distributions can be retrieved directly from the new tabulated values, avoiding multiple computations and interpolation, thus saving considerable CPU time during the radiation calculation. Nevertheless, the radiation characteristics of gas mixtures still occupy a large amount of data space, whose size increases exponentially with the number of gas species in the FSCK model. In the past two decades, the response surface methodology (RSM) [13] has been widely used as a comprehensive statistical method to predict the relation between the input and output of complex systems, which is exactly how to calculate the radiation characteristics of gas mixtures from spectral line data of spectral database by FSCK. For instance, only 32 radiation characteristic parameters k*(Tref, Φ, gref) are needed for one state when using 32 points Gauss quadrature method with FSCK [14]. In a mathematical sense, the RSM is an interpolation technique that fits a multidimensional function with finite sample points in any function domain [15]. Subsequently, the optimal problem can be reconstructed and solved efficiently by using the selected response surface model [16]. To provide an interpolating surface through all sampled data points, some nonparametric metamodeling methods have been proposed with RSM, such as polynomial response surface
Fig. 2. Change of (a) R2 and (b) emax with the degree of polynomial n and the shape parameter of Gaussian function c. Number of sampling states M was fixed as 2000.
Fig. 3. Change of R2 and emax with the shape parameter of Gaussian function c.
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Table 1 The cases for the validation of FSCK-RSM. Case
Medium
1 2 3 4 5 6 7
H2O H2O H2O H2O H2O H2O H2O
+ + + + + + +
CO2 CO2 CO2 CO2 CO2 CO2 CO2
+ CO + C2H2 + C2H4 + CO + C2H2 + C2H4
+ CO + C2H2 + C2H4
T (K)
XH2O
XCO2
XCO
XC2H2
XC2H4
1000 1000 1000 1000 1000 500[1 + sin(πx/L)] 500[1 + sin(πx/L)]
0.3 0.2 0.15 0.1 0.5×(1-x/L)/L 0.5×(1-x/L)/L 0.1
0.3 0.2 0.1 0.1 0.5×(1-x/L)/L 0.5×(1-x/L)/L 0.1
– 0.02 – 0.05 – – 0.05
– – 0.1 0.05 – – 0.05
– – 0.3 0.5 – – 0.5
Fig. 4. Distributions of the radiative source and the normalized deviation for Case 1: the temperature 1000 K, XH2O = 0.3 and XCO2 = 0.3.
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Fig. 5. Distributions of the radiative source and the normalized deviation for Case 2: the temperature of 1000 K, XH2O = 0.2, XCO2 = 0.2 and XCO = 0.02.
by R-Squared (R2) and normalized maximal relative errors (emax). Since a(T, Tref, gref) is a function of gaussian point and temperature while k*(Tref, Φ, gref) is a function of gaussian point, temperature and state parameter, the data space of a-values is much less than that of k-values. Therefore, only the table data for k*(Tref, Φ, gref) was used to fit the response surface model and the a-values were kept unaltered. In Section 3, the radiative transfers of five gas species mixture in a one-dimensional layer are investigated by FSCK-RSM and the results are validated by LBL and SNB. Futhermore, the model has been used for radiation reabsorption calculations in a planar C2H4/O2/N2/CO2 flame with full coupling to the conservation equations for mass, momentum, energy, and species. Finally, the conclusions are summarized in Section 4.
[17], Kriging model [18], multivariate adaptive regression splines [19] and radial basis function (RBF) [20]. RBF model has been proved to be the most effective multidimensional approximation method, with its performance being independent on the dimensionality of the problem [21], and is thus suitable for dealing with high dimensional problems. In this paper, an FSCK-RSM method is proposed to improve the calculation efficiency without affecting the computation accuracy for radiative transfer in inhomogeneous gas mixtures (H2O, CO2, CO, C2H2 and C2H4). The principles of the FSCK-RSM are derived in Section 2: a certain number of sampling states are selected to calculate the k-distributions of FSCK model by latin hypercube sampling method and the response surface model based on RBF is fitted. The fitting is evaluated 4
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Fig. 6. Distributions of the radiative source and the normalized deviation for Case 3: the temperature of 1000 K, XH2O = 0.15, XCO2 = 0.1, XC2H2 = 0.1 and XC2H4 = 0.2.
where ϕ is the state vector containing all the parameters that affect the spectral absorption coefficient, κη the spectral absorption coefficient calculated from the spectral database, and δ the Dirac-delta function. The full spectrum Planck-function weighted cumulative k-distribution is described as:
2. Models 2.1. The full-spectrum correlated-k model (FSCK) The correlated k distribution model in narrow band has been extended to the full spectrum. Since the blackbody radiation intensity changes with wave number in the full spectrum, a Planck-function weighted k-distribution was introduced [22]:
f (T , ϕ, k ) =
1 Ib (T )
∫0
∞
Ibη (T ) δ (k − κ η (ϕ, η))dη
g (T , ϕ, k ) =
∫0
k
f (T , ϕ, k ′) dk ′
(2)
Due to the smoothness and monotony of the cumulative function, the integration of the reordered wavenumber can be obtained by a given integration scheme.
(1) 5
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Fig. 7. Distributions of the radiative source and the normalized deviation for Case 4: the temperature of 1000 K, XH2O = 0.1, XCO2 = 0.1, XCO = 0.05, XC2H2 = 0.05 and XC2H4 = 0.5.
Based on the assumption that the absorption coefficient distributions of different gases are not correlated and the product of transmissivities model [23], Modest and Riazzi [6] proposed a new narrowband mixing model for calculating the full spectrum k-distributions of multi-gas mixtures. In this paper, mixing is performed at the narrow band (NB) level by using this method. The narrow-band parameters of H2O, CO2, CO, C2H2 and C2H4 were provided by Soufiani [24] and Qi [25]. The ten-point Gauss–Lobatto quadrature was adopted in this study.
To apply the FSK in a nonhomogeneous mixture, radiative properties need to be conducted in a unified reference g-space gref: g (T, ϕref, k) = g(T, ϕ, k∗). Then the radiative transfer equation (RTE) in FSCK based on gref-space can be expressed as [22]:
dIg ds
= k ∗ (Tref , ϕ, gref ) [a (T , Tref , gref ) Ib (T ) − Ig ]
where
1
I = ∫0 Ig dg = ∑ wi Ig (gi ) ,
(Tref, ϕref, k).
(3)
N
a(T, Tref, gref)
=
dg(T, ϕref, k)/dg
i=1
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Fig. 8. Distributions of the radiative source and the normalized deviation for Case 5: the temperature of 1000 K, XH2O = XCO2 = 0.5×(1-x/L)/L. M
2.2. Response surface model based on RBF
∑ Pi (x (j) )⋅λj = 0
f (x) =
M
f (x) is the predicted absorption coefficient, x = {x1, x2, …, xl} where the vector of variables (reference temperature, local temperature and gas concentration), l the number of variables, ||x-x(j)|| the Euclidean distance between an arbitrary state x and a sampling state x(j), M the number of sampling states, λ = {λ1, λ2, …, λM} the regression coefficient vector of radial basis function, b = {b1, b2, …, bN} the regression coefficient of the additional polynomials, N =
N
∑ λj φ (‖x − x (j) ‖) + ∑ bi Pi (x) j=1
i=1
(5)
j=1
The weighted sum of a set of functions was used to fit the response surface in the RBF method. The values of these functions only depend on the Euclidian distance between the independent variable and the sampling state φ(x,x(j)) ≡ φ(||x-x(j)||). As long as the sample points are different, these functions are orthogonal. To improve the fitting characteristics of the near-linear region, some additional polynomials and constraint conditions were adopted for the FSCK-RSM in the RBF [26]:
1 n!
n
∏ (l + i) the number i=1
of mutually orthogonal polynomial terms, and n is the degree of polynomial. Eqs. (4) and (5) can be rewritten in matrix form as:
(4)
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Fig. 9. Distributions of the radiative source and the normalized deviation of Case 6: the temperature of 500[1 + sin(πx/L)] K, XH2O = XCO2 = 0.5×(1-x/L)/L.
⎡ ΦT P ⎤ ⎡ Λ ⎤ = ⎡ F ⎤ ⎣P 0 ⎦ ⎣ B ⎦ ⎣ 0 ⎦
λ = {λ1, λ2, …, λM} and b = {b1, b2, …, bN} can be obtained by solving Eq. (6). The response surface fitting effect needs to be evaluated with both global average and worst-case properties. A good response surface fitting should have not only satisfying average fitting, but also acceptable fitting under the worst conditions, which should not drastically deviate from the actual response. Otherwise, the model accuracy will fluctuate in response to the actual input. f (x) : R-Squared (R2) was used to assess the quality of the predicted
(6)
11 1M ⎡ φ (r ) ⋯ φ (r ) ⎤ Φ=⎢ ⋮ ⋱ ⋮ ⎥, rij = ||x(i)-x(j)||, ⎢ ⎥ M 1 MM ⎣ φ (r ) ⋯ φ (r ) ⎦ (1) ) ⋯ P (x (1) ) P ( x 1 N ⎡ ⎤ ⎥,Λ = [λ1 ⋯ λM]T, B = [ b1 ⋯ bN ]T , ⋮ ⋱ ⋮ P=⎢ ⎢ ⎥ ( M ) ( M ) ⎣ P1 (x ) ⋯ PN (x ) ⎦ F = [ f1 ⋯ fm ]T and fj is the real response absorption coefficient corresponding to each sampling state. To get the global characteristics of the actual response problem via a minimum number of sampling states, the Latin hypercubes design method [27] was applied. The Gaussian function φ(r) = exp.(−cr2) (c is the shape parameter) was used in the RBF. Subsequently, the
where
m
f (xi)]2 ∑ [f (xi) − R2 = 1 −
i=1 m
∑ [f (xi) − f ]2 i=1
(7)
where f(xi) is the real response absorption coefficient corresponding to 8
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Fig. 10. Distributions of the radiative source and the normalized deviation for Case 7: the temperature of 500[1 + sin(πx/L)] K, XH2O = 0.1, XCO2 = 0.1, XCO = 0.05, XC2H2 = 0.05 and XC2H4 = 0.5.
uu0 = su0
f (xi) the results predicted by the Eq. (6), f the the ith testing state, average value of f(xi) at m testing states. A closer-to-unity value of R2 indicates a higher accuracy of approximation. The normalized maximal relative error (emax) was used to assess the worst case:
T
0 b
Yu downstream upstream
reaction sheet
emax = max ⎜⎛ ⎝
Tu Y =0 0 b
m⋅|f (xi) − f (xi)| ⎞ ⎟ m ∑i = 1 f (xi) ⎠
(8)
A value of emax closer to 0 indicates that a higher accuracy of approximation is achieved at the worst state.
x
0D
3. Results and discussion
Fig. 11. Schematic showing the premixed flame structure at the transport and reaction-sheet level [29].
Seven variables were choosen to establish the response surface model: five gas species concentrations (XCO2, XH2O, XCO, XC2H2 and 9
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Fig. 12. Laminar flame speeds and the reabsorption effect on the flame speed of C2H4-(0.21O2/0.74N2/0.05CO2) flames at various equivalence ratios 0.7–1.0 by different radiation models.
the R2 and emax, the best accuracy was achieved when n = 2 and c = 7. It is hence concluded based on the above discussion that the choice of M = 2000 and n = 2 as parameter fitting response surface provides not only an accurate model, but also significantly reduces the computational complexity. The corresponding optimal c should be around 7 when M = 2000 and n = 2, as manifested in Fig. 2. To get a more accurate value c, further intensive numerical calculations were carried out near c = 7, as shown in Fig. 3. When c = 7.2, R2 reached its the maximum value, and thus c = 7.2 was selected as the RBF parameter in the response surface model. To study the gas mixtures with the five given gas species, the table data for k*(Tref, Φ, gref) calculated by the original FSCK model contain: 21 points for the reference temperature (300–2300 K, every 100 K), 21 points for the local temperature (300–2300 K, every 100 K), 5 points for the 5 gas species volumetric concentration (1 × 10−4-1.0, every 0.25), and 10 points Gauss–Lobatto quadrature. A total of 13,781,250 data storage was required in the FSCK. However, with the new FSCK-RSM
XC2H4), reference temperature Tref and local temperature Tloc. Tref and Tloc varied in the temperature range of 300–2300 K. XCO2, XH2O and XC2H4 varied from 0 to 1, while XCO and XC2H2 0 to 0.5. To simplify the problem, the gas species concentrations at the reference state were fixed as XCO2 = 0.1, XH2O = 0.1, XCO = 0.05, XC2H2 = 0.05 and XC2H4 = 0.5. The Latin hypercubes design method was used to generate sampling state following normalization of the parameter value range. Invalid sampling states with sum of concentrations of the five gas species greater than 1 were discarded until the number of effective sampling states reached the required number. To determine the appropriate number of sampling states M, the shape parameter of Gaussian function c and the degree of polynomial n, numerical calculations were first carried out over a large range. Since the maximal absorption coefficient appears at the last gaussian quadf (x) at this node were used to evaluate the rature node, R2 and emax of optimal values in the Eq. (4). As shown in Fig. 1, R2 and emax varied with different numbers of sampling states M and the shape parameter of Gaussian function c when the degree of polynomial n was fixed as 2. It is evident that the response surface fitting performance did not rise monotonously with the increase of sampling state number M. Even after the number of sampling state reaches 3500 or higher, obvious overfitting phenomenon appears, and the fitting error increases rapidly. As shown in Fig. 1 (a), R2 decreased with the c with M = 100, while it first increased and then decreased with the c within the range of sampling state numbers M = 200–5000. Since a close-to-unity R2 indicates a higher accuracy, the best R2 is 0.987 was obtained with M = 2000 and c = 7. In Fig. 1 (b), emax increased with the c when M = 100–200, while it first decreased and then increased with the c when M was in the range of 500–5000. As higher accuracy is achieved when the value of emax is closer to 0, the best obtained emax was 1.29 with M = 2000 and c = 7. As shown in Fig. 2, R2 and emax varied with the different degree of polynomial n and the shape parameter of Gaussian function c, when the number of sampling states M was fixed as 2000. It was observed that for high shape parameter of Gaussian function c the response surface fitting performance gets better with the increase of the degree of polynomial n. While with low c values, the lower-order polynomials had better performance. For all investiaged degrees of polynomial n, R2 and emax changed non-monotonically with c. The error first decreased with rising c until a critical value at ~4–7, and then increased monotonically. The results of R2 and emax in Fig. 2 indicated that the optimal c values were 5, 6 and 7 for all degrees of polynomial. Combining the results of both
model, as M was selected as 2000 and N =
1 n!
n
∏ (l + i) = 36 in the i=1
response surface model, only 20,360 data storage was required. The amount of input data has been reduced by 677 times. Moreover, the calculation efficiency is further improved as multiple computations and interpolation are avoided in FSCK-RSM. To validate the accuracy of FSCK-RSM, the thermal radiation heat transfer in a one-dimensional gas layer was investigated and different mixtures of absorbing-emitting gases (XCO2, XH2O, XCO, XC2H2 and XC2H4) were studied. The thickness of the layer was 0.08 m. The total gas pressure of all the cases was fixed as 1 atm, the temperature of the two walls 300 K and the emissivity of both walls 1.0. The RTE was solved by discrete ordinates method (DOM) [28]. The resolution of absorption coefficients is set to 0.01 cm−1 for the LBL model while 0.1 cm−1 for FSCK model. The numerical calculations were carried out at the same spatial and angular discretizations for all the cases. The distributions of gas temperature and species mole fractions for the seven test cases are summarized in Table 1. For non-isothermal medium, the reference temperature was calculated according to the radiation intensity weighted average temperature [7]:
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heat source by FSCK or FSCK-RSM comparing to that by LBL, which is ∇ ⋅ qLBL − ∇ ⋅ qFSCK used as the benchmark, is defined as: error(%) = × 100 . ∇⋅q LBL,max
As shown in Fig. 4, the average normalized deviation of FSCK-RSM (1.66%) was only slightly higher than that of FSCK (1.45%). In addition, the maximal normalized deviation of FSCK-RSM was 2.11%, significantly lower than that of FSCK (3.22%). CO was added in the gas mixture of Case 2. As shown in Fig. 5, the FSCK and FSCK-RSM predictions agreed very well with the LBL prediction, while the radiative source calculated by SNB was less than the other models. The maximal normalized deviation for FSCK-RSM was 1.35%, a little higher than that of FSCK, 0.87%. The average normalized deviation of FSCK-RSM was 0.69%, which was also higher than that of FSCK (0.18%). It indicates that as a penalty for the much lower computational cost, FSCK-RSM only scarified accuracy by ~0.5% comparing to FSCK in this case. In Case 3, CO was removed while C2H2 and C2H4 was added in the gas mixture. As shown in Fig. 6, the FSCK-RSM results agreed very well with the LBL results while the FSCK predictions were closer to those by the SNB. It is further noted that despite a somewhat higher error at the two boundary points (3.56%), the FSCK-RSM predictions had excellent average normalized deviation (0.64%), significantly lower than those of FSCK (2.60%). Hence, for the investigated condition of Case 3, FSCKRSM had not only much lower computational cost, but also much better accuracy than the FSCK. Five gas species (H2O, CO2, CO, C2H2 and C2H4) were considered in the gas mixtures of Case 4. The gas concentrations data were taken from the centerline of a laminar ethylene/air diffusion flame in [25]. As shown in Fig. 7, the FSCK-RSM predicted radiative source agrees well with the SNB results, while the LBL prediction was higher than the other models. Similar to Case 3, FSCK-RSM had a high normalized deviation (4.52%) at the two boundary points, while its average normalized deviations (1.97%) were much lower than those of FSCK (3.37%). 3.2. Isothermal inhomogeneous medium (Case 5) In Case 5, an isothermal inhomogeneous medium filled with H2O and CO2 was studied. The mole fractions of the of H2O and CO2 gas species were set as parabolic distributions. The distributions of the radiative sources calculated by LBL, FSCK and FSCK-RSM are shown in Fig. 8. FSCK-RSM had similar accuracy (note the positive/negative magnitudes) near the two walls while worse accuracy in the central area than FSCK. The average normalized deviation of FSCK-RSM predictions was 1.87%, which was higher than those of FSCK (1.29%) by ~0.6%. Similar to Case 2, FSCK-RSM did not scarify the computational accuracy much comparing to FSCK.
Fig. 13. Distributions of temperature (a) and volumetric radiative heat loss (b) of C2H4-(0.21O2/0.74N2/0.05CO2) flame at ϕ = 1.0.
3.3. Non-isothermal inhomogeneous medium (Case 6)
N
∑ ∑ vi Xav, sp kp (Ti ; sp) Ti4
∑ Xav,sp kp (Tref ; sp) Tref4 = sp
i = 1 sp
Furthermore, a non-isothermal inhomogeneous medium filled with H2O and CO2 was investigated in Case 6. The temperature profile of the gas mixture was set as a sine function 500[1 + sin(πx/L)]. The mole fractions of the gas species were the same as Case 5. The distributions of the radiative source and the normalized deviation calculated by LBL, FSCK and FSCK-RSM are compared in Fig. 9. FSCK-RSM resulted in very close results to LBL near the two walls, better than FSCK. The difference between the FSCK-RSM and LBL predictions in the central area was similar to that between FSCK and LBL. Furthermore, the average normalized deviation of FSCK-RSM (1.12%) was also similar to that of FSCK (1.22%).
N
∑ vi i=1
(9)
where sp is different gas species, vi the volume of the ith discrete element and kp(T; sp) the Plank mean absorption coefficient. 3.1. Isothermal homogeneous medium (Cases 1, 2, 3 and 4) In Case 1, the 1-D parallel-plane layer was filled with H2O and CO2, and was considered as an isothermal homogeneous medium. The distributions of the radiative source calculated by LBL, SNB, FSCK and FSCK-RSM are shown in Fig. 4. It can be seen that the FSCK and FSCKRSM predictions were in excellent agreement with the SNB prediction, while the radiative source calculated by the LBL was slightly less than other models. Since the radiative heat source was very small in the whole domain, the normalized deviation of the calculated radiative
3.4. Non-isothermal homogeneous medium (Case 7) In the last model validation case, Case 7, a non-isothermal homogeneous medium filled with H2O, CO2, CO, C2H2 and C2H4 was studied. 11
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flame speed predicted by SNB was lower than those predicted by FSCK and FSCK-RSM.
The temperature of the gas mixture was the same as that of Case 6, the mole fractions of the gas species were the same as those in Case 4 (see Table 1). The distributions of the radiative source and the normalized deviation calculated by the LBL, FSCK and FSCK-RSM are shown in Fig. 10. The maximal normalized deviation of FSCK-RSM (3.82%) was less than that of FSCK (4.54%), and the average normalized deviation of FSCK-RSM (2.46%) was also lower than that of FSCK (3.48%). This indicated that FSCK-RSM had better accuracy comparing to FSCK in non-isothermal homogeneous cases. The computation time for Case 7 with 180 discrete directions in Fig. 9 was 9.7 s using FSCK-RSM while the computation time for FSCK was 19.3 s. Consequently, FSCK-RSM showed not only better accuracy, but also better computational efficiency than FSCK in the non-isothermal homogeneous case. The much higher computational efficiency of FSCK-RSM was due to two reasons: first, as discussed previously the amount of input data was reduced by 677 times using FSCK-RSM; second, the absorption coefficient was directly calculated by Eq. (4) and no interpolation was needed.
4. Conclusions In this paper, an improved FSCK model, FSCK-RSM was developed in order to improve the computational efficiency for non-gray radiative heat transfer in combustion gas mixtures. Radiation transfer in a onedimensional layer filled with gas mixtures of H2O, CO2, CO, C2H2 and C2H4 with various configurations of gas temperatures and species concentrations were studied. The FSCK-RSM predictions exhibited good agreement with the LBL benchmarks that were generated using the same spectral database. The maximal average relative difference of the radiative source predicted by FSCK-RSM was 2.46% for a non-isothermal homogeneous medium, lower than that of the original FSCK model. FSCK-RSM showed a better computational efficiency than FSCK by reducing the amount of input data by around 677 times, and by avoiding interpolations in the RBF. FSCK-RSM was successfully tested for the predictions of radiation reabsorption in planar C2H4/O2/N2/CO2 flames. The results indicated that the computational time was saved by ~50% using FSCK-RSM comparing to FSCK without affecting the accuracy in predicting the laminar flame speeds.
3.5. Planar C2H4/O2/N2/CO2 flames Finally, the energy and chemical species conservation equations of realistic flames were taken into consideration. Premixed steady C2H4(0.21O2/0.74N2/0.05CO2) planar flames were solved by the PREMIX package [30] of CHEMKIN. The GRI Mech 3.0 chemistry was used [31]. The pressure was fixed as 1 atm and equivalence ratio varied in the range of 0.7–1.0. For the propagation of the standard premixed flame shown in Fig. 11, the upstream mixture approaches the flame with velocity uu = su0 and temperature Tu, and leaves the flame with velocity ub0 and temperature Tb0. As the mixture approaches the flame, it is gradually heated up by the heat conducted forward from the chemical heat release region, resulting in a continuously increasing temperature profile until Tb0 is reached. The profile is not linear due to the presence of convective transport. The continuous heating of the mixture will eventually lead to its ignition and subsequent reaction. To study and to compare the radiation effects, five radiation models were applied: the adiabatic model (ADI) in which radiation is ignored, the optically thin model (OTM) where only the radiation emission of H2O, CO2 and CO was considered [32], the SNB, the FSCK and the FSCK-RSM. As shown in Fig. 12, the laminar flame speed increased when the radiation reabsorption (SNB, FSCK and FSCK-RSM) was considered for every equivalence ratio. The reason was that radiation from the downstream hot products was reabsorbed by the upstream cold mixture. Moreover, the reabsorption effect increased with droping of equivalence ratio as a result of the increased CO2 mole fraction. It can be seen that the laminar flame speeds predicted by FSCK and FSCKRSM are very close to each other but both larger than that by SNB. Characteristically, the relative increment of the laminar flame speed was 4.06% for FSCK-RSM and 3.84% for FSCK at ϕ = 1. Therefore, the FSCK-RSM exhibited a similar accuracy with FSCK in predicting the laminar flame speed. However, the computation time was only 67.5 s using FSCK-RSM while the computation time of FSCK was 136.3 s. To further understand how the radiation model affects the laminar flame speed, distributions of gas temperatures and volumetric radiative heat losses of a C2H4-(0.21O2/0.74N2/0.05CO2) mixture (ϕ = 1.0) are shown in Fig. 13. It is shown in Fig. 12(a) that the gas temperatures decreased in the downstream when considering the radiation (see the ADI curves and others). The temperatures calculated by SNB, FSCK and FSCK-RSM were higher than that calculated by OTM due to the radiation reabsorption. In the upstream, however, the radiative heat loss was negative as some radiation emitted at the high temperature reaction zone is reabsorbed by the upstream lower temperature region. Furthermore, the radiation reabsorption energy in the upstream predicted by SNB was less than those by FSCK and FSCK-RSM. Consequently, the increment of temperature in the upstream calculated by SNB was less than those calculated by FSCK and FSCK-RSM. This is why the laminar
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This research was supported by the National Key Research Development Program of China (No.2017YFB0601900), the National Natural Science Foundation of China (No. 51976057, 51922040 and 51827808), the China Scholarship Council and the Fundamental Research Funds for the Central Universities (No. 2017ZZD005). RS acknowledges the Swiss National Science Foundation postdoctoral fellowship (Award Number 178619). References [1] H. Chu, S. Zheng, W. Liang, H. Zhou, Effects of radiation reabsorption of C1-C6 hydrocarbon flames at normal and elevated pressures, Fuel. 266 (2020) 117061. [2] J. Arnold, E. Whiting, G. Lyle, Line by line calculation of spectra from diatomic molecules and atoms assuming a voigt line profile, J. Quant. Spectrosc. Radiat. Transf. 9 (6) (1969) 775–798. [3] R. Goody, A statistical model for water-vapour absorption, Q. J. R. Meteorol. Soc. 78 (338) (1952) 638–640. [4] W. Malkmus, Random Lorentz band model with exponential-tailed S− 1 line-intensity distribution function, JOSA 57 (3) (1967) 323–329. [5] M.F. Modest, Narrow-band and full-spectrum k-distributions for radiative heat transfer—correlated-k vs. scaling approximation, J. Quant. Spectrosc. Radiat. Transf. 76 (1) (2003) 69–83. [6] M.F. Modest, R.J. Riazzi, Assembly of full-spectrum k-distributions from a narrowband database; effects of mixing gases, gases and nongray absorbing particles, and mixtures with nongray scatterers in nongray enclosures, J. Quant. Spectrosc. Radiat. Transf. 90 (2) (2005) 169–189. [7] Y. Du, Z. Peng, Y. Ding, A high-accurate and universal method to characterize the relative wavelength response (RWR) in wavelength modulation spectroscopy (WMS), Opt. Express. 28 (2020) 3482–3494. [8] Y. Sun, S. Zheng, B. Jiang, J. Tang, F. Liu, One-dimensional P1 method for gas radiation heat transfer in spherical geometry, Int. J. Heat Mass Transf. 145 (2019) 118777. [9] C. Wang, M.F. Modest, B. He, Full-spectrum correlated-k-distribution look-up table for use with radiative Monte Carlo solvers, Int. J. Heat Mass Transf. 131 (2019) 167–175. [10] A. Wang, High-accuracy, compact database of narrow-band k-distributions for water vapor and carbon dioxide, ICHMT DIGITAL LIBRARY ONLINE, Begel House Inc., 2004. [11] C. Wang, W. Ge, M.F. Modest, B. He, A full-spectrum k-distribution look-up table for radiative transfer in nonhomogeneous gaseous media, J. Quant. Spectrosc. Radiat. Transf. 168 (2016) 46–56. [12] C. Wang, B. He, M.F. Modest, T. Ren, Efficient full-spectrum correlated-k-
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