Eurotherm Seminar 110 – Computational Thermal Radiation in Participating Media - VI April 11-13, 2018, Cascais, Portugal
RANK TRANSMUTATION MAPPING TECHNIQUE FOR THE FSK AND SLW MODELS Frederic Andre*,§, Vladimir P. Solovjov**, Brent W. Webb** and Denis Lemonnier*** Centre d’Energétique et de Thermique de Lyon, INSA de Lyon – 9 rue de la Physique – 69621 Villeurbanne, France ** Brigham Young University, 435 CTB, Provo, UT 84602 USA ***ISAE-ENSMA, BP 40109, 86961 Futuroscope Chasseneuil Cedex, France § Corresponding author:
[email protected] *
ABSTRACT. The aim of the present paper is to describe a computationally efficient method to evaluate inverse full spectrum k-distribution functions which provides accurate solutions to the implicit equations encountered in full spectrum models of gas radiation, both in uniform and non-uniform situations. The method uses some recent mathematical tools, called Rank Transmutation Maps, which are applied within the frame of global spectral k-distribution methods. Model outputs are assessed against reference LBL calculations and illustrate the relevance of the proposed approach for radiative heat transfer applications.
INTRODUCTION The principle of k-distribution methods was introduced by Ambartzumian [1]. However, the approach only became suitable for radiative heat transfer calculations when Lacis and co-workers [2] extended the technique from uniform to non-uniform situations by proposing the assumption of correlated spectra. It must be noticed that, in modern textbooks where “correlation” refers to linear dependence and corresponds to the concept of scaling between spectra, this model of association between spectra in distinct states is called comonotonicity [3] and refers to a perfectly increasing dependence between gas spectra treated as random variables. Several mostly equivalent models based on full spectrum k-distributions were developed since the 90s such as SLW [4,5], FSK [6] or ADF [7]. A general formulation, which encompasses all these techniques, was proposed recently [8]. Full spectrum k-distribution methods are presently among the most widely used approximate gas radiation models for studying coupled heat transfer problems in engineering. In non-uniform situations, the efficiency of these methods is however strongly affected by the necessity to solve implicit equations at local states in the medium to define values of absorption coefficients through the correlated / comonotonicity assumption. This process can be time consuming in 3D geometries. The objective of the present paper is to describe a technique which addresses the inefficiencies associated with current methods for inverting the k-distribution function F k to find k F . RANK TRANSMUTATION MAPS IN FS k-DISTRIBUTION MODELS FS k-distribution formulation of total emissivities In full spectrum k-distribution methods, inside a gas in a thermophysical state Tg , Y , p symbolized by a vector that encloses all the quantities
required to define the state of the gas, viz. its temperature Tg , total pressure p and species concentrations Y, the total emissivity of a uniform path of length L is written as:
L, , Tg
1 exp L I T d 1 exp k L dF k , , Tg b, g Tg4 0 k 0
(1)
where F k , , Tg is the distribution function of absorption coefficients (ALBDF when formulated in terms of absorption cross-sections, as used in SLW models [4,5]). F represents the fraction of blackbody intensity emitted at temperature Tg over the intervals of wavenumbers such that the spectral absorption coefficient of the gas, , is lower than k. F can thus be interpreted as the probability to find a value of
lower than k over the full spectrum. This function takes values between 0 and 1 and is strictly increasing with respect to k. In many situations, gas spectra are a combination of transparent zones over which the gas neither absorbs nor emits radiation, and spectral intervals over which the gas molecules and the radiation field interact. The union of all intervals over which the gas spectrum takes strictly positive values will be hereafter written * : 0 . With this notation, Eq. (1) can be equivalently written as:
L, , Tg S , Tg 1 exp k L dF * k , , Tg
(2)
0
where S , Tg is the fraction of blackbody intensity emitted over * :
S , Tg
I T d Tg4 : b, g
(3)
*
and F * k , , Tg is the restriction of the k-distribution function to the same set of wavenumbers. As the distribution function F * is invertible, the integral at the RHS in Eq. (2) can be reformulated through a change of variable of integration:
1
L, , Tg S , Tg 1 exp k L dF k , , Tg S , Tg 1 exp k F * L dF * (4) *
0
0
Eq. (4) allows application of simple quadrature schemes to approximate the full spectrum emissivity as: 1
L, , Tg S , Tg 1 exp k F * L dF * S , Tg wi 1 exp k Fi L (5) 0
n
i 1
where wi , i 1,..., n and Fi , i 1,..., n are respectively the weights and nodes of a numerical quadrature at order n over the 0,1 interval. When such a quadrature is chosen, application of Eq. (5) allows calculating an approximation of the total emissivity of the gas. This mostly requires evaluating the function k F * at the quadrature nodes. Indeed, Eq. (5) includes the inverse of the distribution function F * k , , Tg as k F * F *1 F * F *1 F * k , , Tg .
Eq. (5) can be written in the same form as full spectrum k-distribution methods viz.:
L, , Tg ai , Tg 1 exp k Fi L n
(6)
i 1
if we simply set the weights in Eq. (6) to the following values:
ai , Tg S , Tg wi The weight corresponding to the clear gas is in this case:
(7)
a0 , Tg 1 ai , Tg 1 S , Tg wi 1 S , Tg n
n
i 1
i 1
(8)
Notice that formulas for the extension of these weights to absorptivity calculations can be found in Ref. [8] Section 3. We will focus in the next section on approximations of k F * and will restrict our analysis to uniform cases, viz. to direct applications of Eq. (5). However, one can notice that providing a method to estimate k F * indirectly solves the problem of association of spectra in non-uniform cases through the correlated / comonotonicity assumption. This is because the corresponding implicit equation [9], in which exponent “ref ” refers to some arbitrarily chosen reference state of the gas and where the reference blackbody source temperature is chosen equal to that of the gas in state :
F * k , , Tg F * k ref , ref , Tg
(9)
can be equivalently rewritten as: k k F * , , Tg k F * k ref , ref , Tg
(10)
which involves function k F * evaluated at the same thermophysical state of the gas and blackbody source temperature Tg as used in Eq. (5) to calculate the emissivity of the gas in the same thermophysical condition. Principle of Rank Transmutation Maps in FS k-distribution methods The main idea behind the concept of Rank Transmutation Map was described in Ref. [10]. This mathematical tool was already applied within the frame of gas radiation models for the development of the l-distribution approach in Ref. [11]. It consists in the following. Let F * k be the distribution function of a spectral absorption coefficient assumed to take values inside a bounded interval kmin , kmax , where kmin 0 , and F0* k be any approximate model for F * k with the only requirement that F0* k is a strictly increasing function from kmin , kmax to 0,1 . In this case,
F0* k has an inverse F0*1 : 0,1 kmin , kmax . Then, the Rank Transmutation Map GR X which
relates functions F * k and F0* k is defined as:
GR X F * F0*1 X , X 0,1
(11)
Application of the Rank Transmutation Map to F0* k provides the following equality:
GR F0* k F * F0*1 F0* k F * k , k kmin ; kmax
(12)
which establishes a one-to-one relationship between the two distribution functions F0* k and F * k . Eq. (12) can be reversed to yield: k F * F0*1 GR1 F *
(13)
which is exactly the type of relationship we are seeking for, as can be noticed in Eq. (5). Accordingly, if ones chooses F0* and F0*1 in such a way that these two functions are analytical, then it
suffices to tabulate the inverse of function GR , which maps the interval 0,1 into itself by definition, to obtain estimates of k F using Eq. (13). This approach has the potential advantages of a more
manageable scale in k (which varies by as much as ten orders of magnitude), and more efficient and more accurate interpolation. The method to derive this function together with several possible choices for function F0* k , that will be from now on called the germ distribution function because its choice fixes all the other parameters of the model, are detailed in the next section. APPLICATION Method of construction of Rank Transmutation Maps The first step in the building of the Rank Transmutation Map is the choice of an interval kmin ; kmax assumed to be shared by the two distributions functions F0* k and F * k . In the present work, kmax was fixed to the maximum value of the spectral absorption coefficient found inside the LBL dataset and kmin was arbitrarily set (but this lower limit is reasonable considering usual FS models [4-7]) to 105 cm-1.atm-1. All values of kmin were then assumed to belong to the transparency region of the absorption spectrum. The choice of kmin thus fixes
the spectral interval * as defined in the previous section. Coefficient S , Tg
can be calculated
directly from its definition as the fraction of blackbody intensity emitted over the set of wavenumbers * viz. Eq. (3). The next step consists of the choice of a germ model F0* k . In this study, we have considered the germs provided in Table 1. Two of them are related to single spectral lines with Lorentz or Doppler profiles. The method to derive these analytical solutions for single lines can be found in Refs. [12,13]. Notice that their inverse (third column) are analytical, strictly increasing from 0,1 to kmin , kmax and thus comply with the requirements of the preceding section. They are plotted in Figure 1 for a mixture of 10 % CO2 – 90% N2 at 800 K and atmospheric pressure. The actual k-distribution function restricted to the interval * is also depicted for comparison purpose. Corresponding GR functions are shown in Figure 2. Table 1: Germ models and their inverse inside a bounded interval kmin ; kmax k F0* F0*1
F0* k
Germ model Log-uniform
1
ln kmax k ln kmax kmin
Single Lorentz line
1
kmax k 1 kmax kmin 1
Single Doppler line
1
ln kmax k ln kmax kmin
Power exponential (here = 2)
ln kmax k 1 ln kmax kmin
kmax kmin kmax
1 F0
kmax
1 kmax kmin 11 F0
2
2 kmax exp ln kmax kmin 1 F0
1 kmax exp ln kmax kmin 1 F0
The log-uniform case corresponds to the power exponential (PE) with = 1 ; single Doppler line is the PE with = 1/2.
The final stage is to tabulate function GR1 with respect to the variable X between 0 and 1. This requires several steps: 1/ the interval 0,1 is uniformly discretized into N steps to provide a regularly spaced sequence
X i , i 0, N , X 0 0, X N 1 ;
2/ for each index i, k F0* X i is calculated using the formulas of the last column in Table 1; 3/ the value of function GR X i is then evaluated directly from its definition, equivalent to Eq. (11): 1
GR X i I b, Tg d I b , Tg d : * : * and k X i
(14)
4/ the corresponding table of values of GR X i is reversed and projected, using linear interpolations, over the set X i , i 0, N . This provides a vector which contains the values of function GR1 X at the nodes X i , i 0, N . This choice for the building of the table is not arbitrary. It is used to allow a simple search of index for the interpolation of GR1 X at some given value X inside 0,1 . Indeed, for any X, the index j such that
X X j ; X j 1 is, with this choice of tabulation, simply given as j Int X N where Int Y is the function that returns the integer value which is the closest to (and lower than) Y. The method allows finding efficiently the index for interpolation without any iterative searching schemes known to be one of the main sources of computational cost in FS k-distribution methods.
Figure 1. Comparison of the true positive k-distribution F * and germ models F0 (see Table 1) – 10% CO2 – 90% N2 at 1 atm and 800 K. k is the pressure based absorption coefficient, in cm-1.atm-1.
Once the table is built, it can be stored for later use together with coefficients S , Tg
(already
calculated at the beginning of the process) and kmax . All these coefficients, when combined with the formulas provided in Table 1, are then sufficient to apply the method viz. use Eqs. (5,10) for the calculation of total emissivities, as discussed in the next section.
Figure 2. Examples of Rank Transmutation Maps GR – 10% CO2 – 90% N2 at 1 atm and 800 K. Inverse GR function are obtained by a symmetry with respect to the diagonal Y = X.
Sample calculations The following section provides results of comparisons between LBL reference calculations (using a Riemann sum for the spectral integration – integrals are calculated over all wavenumbers, including transparency regions) and Eq. (5) with absorption coefficients k F obtained by application of the method detailed in the paper. High resolution (10-2 cm-1) gas spectra were computed with CDSD-4000 [14] for CO2. Details about these LBL calculations can be found in Ref. [15]. A n = 64 points Gauss-Legendre quadrature shifted to the 0,1 interval was used to generate the nodes and weights in Eq. (5). The size of the table (Format 2(E14.8)) including both the direct and inverse mapping functions for a single state of the gas and reference blackbody source temperature is 30 kBytes. This corresponds to a total size of 23.5 MB per value of total pressure. In Ref. [16], the same set of states required 0.5 MB for the same configuration (CO2 at a single total pressure). Total computational time to generate 106 values of k, and thus solve 106 implicit equations, is 0.12 seconds on a single core of a Xenon E5620 at 2.4 GHz using 1,000 values of X together with the PE germ. This corresponds to 3.7 seconds for generating 8 values of gray gases absorption coefficients over a 100x100x100 grid for CO2-N2 mixtures (which requires a two-dimensional interpolation scheme with respect to Tg and p and thus evaluation of absorption coefficients at 4 nodes for each quadrature point). Results of comparisons of the output of the present method are provided in Table 2 for various choices of germ distribution functions. In these calculations, all other quantities (minimum value of the absorption coefficient, parameter S and number of values of X here equal to N=1,000) are the same. The best results are obtained by application of the PE germ. However, both SDL and LU provide in this case very acceptable results. The use of the SLL germ cannot be recommended. Table 3 provides the same results as for Table 2 with a number N of values of X equal to 100. In this case, PE clearly outperforms all the other germs. This can be explained by the plot depicted in Figure 2. Indeed, from this figure, the choice of the PE germ provides the mapping functions with the smallest overall rate of change with respect to X. Accordingly, using a linear interpolation scheme yields smaller interpolation errors with this germ, even with coarse grids, than with the others. The optimal germ has the highest agreement with the actual k-distribution function, and provides the mapping function that is the nearest to the straight line Y GR X X corresponding to the exact germ. As may be seen in Fig. 2, the Power Exponential germ most nearly fits this optimum.
Table 2: Mean absolute relative errors on total emissivities, in %, for 100 values of path lengths logarithmically scaled between 0.01 and 1,000 cm.atm – p = 1 atm ; 10 % CO2, 90 % N2. LU: Loguniform; SLL: Single Lorentz Line; SDL Single Doppler Line; PE: Power Exponential. N=1,000. Gas temperature (K) 300 1,300 2,300 3,300
LU
SLL
SDL
PE
0.1 0.2 0.5 0.3
2.0 2.1 1.3 0.8
0.2 0.3 0.7 0.5
0.1 0.0 0.2 0.1
Table 3: Mean absolute relative errors on total emissivities, in %, for 100 values of path lengths logarithmically scaled between 0.01 cm.atm and 1,000 cm.atm – p = 1 atm ; 10 % CO2, 90 % N2. LU: Log-uniform; SLL: Single Lorentz Line; SDL Single Doppler Line; PE: Power Exponential. N=100. Gas temperature (K) 300 1,300 2,300 3,300
LU
SLL
SDL
PE
4.2 2.9 5.2 3.9
15.6 7.8 4.4 3.1
6.7 5.2 8.9 7.0
1.5 0.9 1.9 0.8
Finally, Figure 3 depicts direct comparisons of total emissivities calculated LBL and with Eq. (5) at fixed weights wi , i 1,.., n (n 64) together with the inversion method described in the present work to calculate k Fi , i 1,.., n . Only results for the Power Exponential germ are shown. Maximum errors are observed at low values of emissivities but remain lower than 0.5 % for temperature between 300 K and 3,300 K.
Figure 3. Total emissivities calculated LBL (top) and absolute relative errors obtained with the approximation developed in this work (bottom) – Power Exponential germ, N=1,000.
CONCLUSION A simple and accurate method was described to evaluate solutions to the implicit equations encountered in Full Spectrum k-distribution models. It was assessed against reference LBL calculations for the approximation of total emissivities of CO2-N2 mixtures at high temperature. The technique opens new possibilities for building and handling FS k-distribution databases and does not require computationally expensive schemes to search and interpolate model parameters stored in look-up tables. This is the main advantage of the present approach, based on Rank Transmutation Map, compared to other existing methods. Its extension to mixtures of radiating species is scheduled as future work. KEYWORDS gas radiation, Rank Transmutation Map, full spectrum k-distribution, SLW REFERENCES 1. V. Ambartzumian, “The effect of the absorption lines on the radiative equilibrium of the outer layers of the stars”, Publ. Obs. Astron. Univ. Leningrad, vol. 6, pp. 7-18, 1934. 2. A. A. Lacis, W. C. Wang, J. E. Hansen, “Correlated k-distribution method for radiative transfer in climate models: Application to effect of cirrus cloud on climate”, NASA Goddard Space Flight Center 4th NASA Weather and Climate Program Sci. Rev., pp. 309-314, 1979. 3. R. B. Nelsen, An introduction to copulas, Second Edition, Springer series in statistics, 2006. 4. M. K. Denison, B. W. Webb, “A spectral line based weighted-sum-of-gray-gases model for arbitrary RTE solvers”, ASME J. Heat Transfer, vol. 115, pp. 1004-1012, 1993. 5. V. P. Solovjov, F. André, D. Lemonnier, B. W. Webb, “The rank correlated SLW model of gas radiation in non-uniform media”, J. Quant, Spectrosc. Radiat. Transfer, vol. 197, pp. 26-44, 2017. 6. M. F. Modest, Radiative Heat Transfer, Third Edition, Academic Press, 2013. 7. Ph. Rivière, A. Soufiani, M. Y. Perrin, H. Riad, A. Gleizes, “Air mixture radiative property modeling in the temperature range 10,000-40,000 K”, J. Quant, Spectrosc. Radiat. Transfer, vol. 56, pp. 29-45, 1996. 8. F. André, V. P. Solovjov, B. W. Webb, D. Lemonnier, “Comonotonic global spectral models of gas radiation in non-uniform media based on arbitrary probability measures”, Appl. Math. Modelling, vol. 50, pp. 741-754, 2017. 9. V.P. Solovjov, B.W. Webb, F. André, “Radiative Properties of Gases”, in Handbook of Thermal Science and Engineering, Springer, pp. 1-74, 2017. 10. W. T. Shaw, I. R. C. Buckley, “The alchemy of probabilistic distributions: beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map”, in First IMA Computational Finance Conference, March 23 2007. 11. F. André, “The l-distribution method for modeling non-gray absorption in uniform and non-uniform gaseous media”, J. Quant, Spectrosc. Radiat. Transfer, vol. 179, pp. 19-32, 2016. 12. S. J. Young, Band model theory of radiation transport, The Aerospace Press, 2013. 13. F. André, R. Vaillon, “Generalization of the k-moment method using the maximum entropy principle: Application to the NBKM and full spectrum SLMB gas radiation models”, J. Quant, Spectrosc. Radiat. Transfer, vol. 113, pp. 1508-1520, 2012. 14. S. A. Tashkun, V. I. Perevalov, “CDSD-4000, High-resolution, high temperature carbon dioxide spectroscopic databank”, J. Quant, Spectrosc. Radiat. Transfer, vol. 103, pp. 1-42, 1993. 15. F. André, L. Hou, M. Roger, R. Vaillon, “The multispectral gas radiation modeling: a new theoretical framework based on a multidimensional approach to k-distribution methods”, J. Quant, Spectrosc. Radiat. Transfer, vol. 147, pp. 178-195, 2014. 16. J. T. Pearson, B. W. Webb, V. P. Solovjov, J. Ma, “Efficient representation of the absorption line blackbody distribution function for H2O, CO2 and CO at variable temperature, mole fraction, and total pressure”, J. Quant, Spectrosc. Radiat. Transfer, vol. 138, pp. 82-96, 2014.
