The Full-Spectrum Correlated-k Distribution for Thermal Radiation From Molecular Gas-Particulate Mixtures

Transcription

1 Michael F. Modet Fellow ASME Hongmei Zhang Mem. ASME Dept. of Mechanical Engineering, Penn State Univerity, Univerity Park, PA The Full-Spectrum Correlated-k Ditribution for Thermal Radiation From Molecular Ga-Particulate Mixture A new Full-Spectrum Correlated-k Ditribution ha been developed, which provide an efficient mean for accurate radiative tranfer calculation in aborbing/emitting molecular gae. The Full-Spectrum Correlated-k Ditribution can be ued together with any deired olution method to olve the radiative tranfer equation for a mall number of pectral aborption coefficient, followed by numerical quadrature. t i hown that the Weighted-Sum-of-Gray-Gae model i effectively only a crude implementation of the Full-Spectrum Correlated-k Ditribution approach. Within the limit of the Full-Spectrum Correlated-k Ditribution model (i.e., an aborption coefficient obeying the o-called caling approximation ), the method i exact. Thi i demontrated by comparion with line-by-line calculation for a one-dimenional CO 2 -N 2 ga mixture a well a a twodimenional CO 2 -H 2 O-N 2 ga mixture with varying temperature and mole fraction field. DO: 0.5/ Keyword: Combution, Gaeou, Radiation ntroduction Radiative heat tranfer in gae ha important application from combution ytem to modeling atmopheric procee. The magnitude of radiative heat fluxe can have profound effect on combution performance and environmental impact. Radiative heat tranfer calculation in combution gae may be looely grouped into the following three method in order of decreaing complexity: line-by-line calculation; 2 band model; and 3 global model. Line-by-line calculation are the mot accurate to date, but they require vat amount of computer reource. Thi i undeirable even with the availability of powerful upercomputer, ince radiative calculation are only a mall part of a ophiticated fire/ combution code. Many tudie have been devoted to narrow and wide band model, uch a the Malkmu narrow band model, the correlated-k CK model and many other. The CK method i baed on the fact that inide a pectral band, which i ufficiently narrow to aume a contant Planck function, the precie knowledge of each line poition i not required for the computation 7. n thi paper the CK approach i extended to the whole pectrum by defining a Planck function weighted cumulative k-ditribution function. The mot common global method i the o-called Weighted- Sum-of-Gray-Gae model. The concept of the WSGG approach wa firt preented by Hottel and Sarofim 8 within the framework of the zonal method. The method could be applied to arbitrary geometrie with varying aborption coefficient, but wa limited to noncattering media confined within a black-walled encloure. Modet 9 ha hown that thi model may be generalized for ue with any arbitrary olution method. n thi method the nongray ga i replaced by a number of gray gae, for which the heat tranfer rate are calculated independently by olving the RTE with weighted emiive power for each of the gray gae. The total heat flux i then found by adding the fluxe of all gray Contributed by the Heat Tranfer Diviion for publication in the JOURNAL OF HEAT TRANSFER. Manucript received by the Heat Tranfer Diviion November 28, 200; reviion received June, 200. Aociate Editor: J. P. Gore. gae. The different aborption coefficient i and emiive power weight factor for each ga are found from total emiivity data. Denion and Webb 0 5 have improved on the WSGG model and have developed the Spectral-Line-Baed Weighted- Sum-of-Gray-Gae SLW model baed on detailed pectral line data. They alo extended the SLW model to noniothermal and nonhomogeneou media by introducing a cumulative ditribution function of the aborption coefficient, calculated over the whole pectrum and weighted by the Planck function. The aborption ditribution function ADF approach 6 8 i almot identical to the SLW model and differ from the SLW only in the calculation of the gray-ga weight. Thee weight are choen in uch a manner that emiion by an iothermal ga i rigorouly predicted for actual pectra. Thi method ha been further generalized 7 by introducing fictitiou gae ADFFG employing a joint ditribution function that eparate the into two or more fictitiou gae, and i deigned to be more uitable for the treatment of nonhomogeneou media. n thi paper, the Full-Spectrum Correlated-k Ditribution approach i developed baed on Weighted-Sum-of-Gray-Gae argument. Thi approach provide a moother and, therefore, more eaily integrated et of weight function than the ADF method. Through thee argument the Weighted-Sum-of-Gray-Gae model i hown to be imply a crude implementation of the FSCK Full-Spectrum Correlated-k Ditribution developed here. Therefore, it i clear that the WSGG method, like the correlated-k approach, i not limited to black-walled encloure without cattering 9, but can accommodate gray wall a well a gray cattering particle. Thi i alo hown through direct WSGG argument. Theoretical Formulation Conider an aborbing/emitting medium inide a black encloure. For implicity, we will firt aume here that the medium doe not catter, and that it conit primarily of molecular combution gae uch a H 2 O and CO 2 mixed in air with their thouand of pectral line, plu perhap ome non-cattering particle, uch a oot, all encloed by opaque black wall. However, 30 Õ Vol. 24, FEBRUARY 2002 Copyright 2002 by ASME Tranaction of the ASME

2 thi approach i alo valid for gray cattering and/or gray wall, which will be hown later. For uch a ituation the radiative equation of tranfer RTE i given by 9 d d b, () where i the pectral intenity varying along a path, i wavenumber, b the Planck function, and i the pectral aborption coefficient, which beide wavenumber depend on local temperature T, preure p and mole fraction x. Thi i the tarting point to make line-by-line calculation. The formal olution to Eq. i 9 bw e 0 d 0 b Te d d, (2) where the ubcript w denote a value at the wall. ntegrated over the entire pectrum thi become 9 where 0 d bw T w,0 0 b T, d, (3) T, b T b Te 0 d d (4) i the total aborptivity for a ga column for irradiation from a blackbody ource at temperature T. We will now aume that the commonly ued caling approximation applie, i.e., that pectral and patial dependence of the aborption coefficient are eparable,,t,p,x k ut,p,x. (5) Thi approximation i ued, for example, in the popular Curti- Godon approximation to calculate narrow band aborptivitie for nonhomogeneou path 20, and in correlated-k model applied to nonhomogeneou atmophere 3,2. While a good approximation for oot, it accuracy i more limited for molecular gae: if we aume pectral line of Lorentz hape we get 9 S j b j,t,p,x j j 2 2, (6) b j where j i the pectral poition of the center of the j th line, b j i the line half width at half height, and S j i the line intenity. n order for Eq. 5 to hold, we need all b j to be contant uually fairly true if total preure i contant and all S j to have the ame patial dependence the S j depend only on temperature and denity of the aborbing pecie. Equation 5 ha been hown to give very accurate reult in atmopheric application, even for trong variation in total preure 2,2, but due to hot line with trongly different temperature dependence of the line intenitie may become le accurate in the preence of field with extreme temperature variation 4,6,22. Moreover, the line intenitie are directly proportional to the partial preure of the aborbing ga. Thu, in ga mixture with locally varying mole fraction ratio, Eq. 5 i alo certain to be violated. Sticking Eq. 5 into Eq. 2 and 4 yield bw e k X0, b e 0 k X, k ud, (7) where T, b T b e 0 k X, d, (8) ud. (9) X, n the k-ditribution method it i recognized that over a mall pectral interval over which b may be aumed contant the aborption coefficient attain the ame value many time. f the medium i homogeneou i.e., () only, and even in a nonhomogeneou medium if Eq. 5 applie, the aborption coefficient may be reordered into a monotonically increaing function without lo of accuracy. For a narrow pectral interval we then write for the narrow band tranmiivity e k X e d0 kx f e kdk0 kgx dg, (0) where the cumulative k-ditribution gk0 k f kdk () i an equivalent, nondimenional wavenumber. n Eq. 0 the integration can be witched from to k, ince for each there i only a ingle value of k but many different for each k. A imilar argument can be applied to the entire pectrum. Defining a fractional Planck function a it, b T b d, (2) 0 it i obviou that i(t,0)0 and i(t,), i.e., the fractional Planck function i monotonically increaing with 0i. Equation 8 can then be rewritten a e T, T, 0 k ix, di. (3) We note there i only one value of k for each value of i but many value of i for a ingle value of k. Thu, we may reorder Eq. 3 in the ame way a Eq. 0, leading to e 0 k ix e di0 kx f e T,kdk0 kt,gx dg. (4) Here, g i no longer an equivalent wavenumber, but an equivalent fractional Planck function. Note that, ince i i a function of temperature, o i k, i.e. kk(t,g). Since we would like to be able to ue arbitrary method for the olution of the radiative tranfer problem, employing the implified kg relation of Eq. 4 rather than k, the temperature dependence of k(t,g) i rather inconvenient. Following Modet 9 we deire a form, which, upon ubtitution into Eq. 3, can be hown to reduce back to the original RTE, Eq.. We will, therefore, carry out one more reordering tep to modify Eq. 4 to e 0 kt,gx at,g ref e dg0 kt ref,g ref X dg ref, (5) where k(t ref,g ref ) i the kg ditribution evaluated at a reference temperature T ref, i.e., we have moved the temperature dependence from the exponent in k to a bae function a. How thi i done i bet undertood by looking at Fig., which how two typical k-ditribution for T ref the reference temperature and ome other temperature T. Both function have identical value for kk min at g0 the minimum aborption coefficient acro the pectrum, uually zero, and kk max at g the maximum aborption co- Journal of Heat Tranfer FEBRUARY 2002, Vol. 24 Õ 3

3 What remain to be done i to olve the gray medium RTE, Eq. 9, by any arbitrary olution method for a mall number of gray aborption coefficient k(g) ince k i a mooth, monotonic function in g, followed by numerical quadrature over g. n the correlated-k approach thi i generally done by Gauian quadrature, ince thi give a high degree of accuracy with relatively few RTE evaluation. The mot primitive and leat accurate quadrature cheme would be the ue of the trapezoidal rule; i.e., 0 n g dg i g kg i g i ; n i g i, (20) which would imply Fig. The weight function a, obtained from k-ditribution at different temperature efficient acro the pectrum; for each value of k the correponding value for g i imply hifted. Now etting k(t,g) k(t ref,g ref ), and differentiating lead to dg kt ref,g ref /g ref kt,g/g dg ref f T,kg f T ref,kg ref dg ref at,g ref dg ref, (6) where we have arbitrarily et a(t ref,g ref ) at the reference condition. For implicity of notation we will, from now on, drop the ubcript ref from g ref, a well a the argument T ref from k(t ref,g ref ), which then i imply written a k(g). Subtituting Eq. 5 into Eq. 3, and noting that we obtain at,ge 0 kgx, kg X dg at,ge 0 kgx kgudg, at w,ge kx0, dg bw0 b 0 0 at,ge kgx, kgudgd. (7) f we now introduce a new pectral intenity g (), we get 0 0 g dg at w,g bw e kx0, at,g b 0 e kx, kuddg. (8) Comparing with Eq. 7 we find that g atifie the general RTE, Eq., ubject to the ame boundary condition, but for a pectrally-integrated, gray cae with the Planck function replaced by a weighted value, a b : d g d kgua b g, 0g. (9) 0 n i at,ge kgx dg A i Te k i X, A i g i adg, (2) which i commonly known a the Weighted-Sum-of-Gray- Gae WSGG method. Therefore, the Weighted-Sum-of-Gray- Gae method, while a very powerful method in itelf, i effectively only a crude implementation of the Full-Spectrum Correlated-k approach preented here. Neverthele, the WSGG method a preented in Eq. 2 wa a coniderable improvement over the previou tate-of-the-art, which allowed only cumberome and inaccurate application to inhomogeneou media 9,3,4. Alternative Development. Equation 9 can alo be derived directly from the pectral RTE, alo including gray cattering and gray boundarie. Under the caling approximation, the pectral RTE i then given by 9 d d k u b k u 4 ŝŝ,ŝd, 4 (22) where i the cattering coefficient and (ŝ,ŝ) i the cattering phae function. n it mot general form, Eq. 22 i ubject to the boundary condition 9 at a wall, w w bw w nˆ ŝd, nˆ ŝ0 (23) where w i the pectral intenity leaving the encloure wall, due to diffue gray emiion and/or diffue gray reflection, w i the emiivity of the wall, and ŝ and nˆ are unit vector for direction and the urface normal pointing out of the wall, repectively. A reditributed RTE i obtained by multiplying Eq. 22 by (kk )/f(t ref,k), where (kk ) i the Dirac-delta function and f T ref,k b T ref kk kk di b 0 d0 (24) i the Planck-function-weighted k-ditribution at the reference temperature, a already given in Eq. 4. Note that integrating the Dirac-delta function acro a ingle occurrence of kk yield kk d kk d dk dk d dk. (25) ntegrating the o-multiplied Eq. 22 acro the pectrum, and auming gray cattering propertie yield the deired form 32 Õ Vol. 24, FEBRUARY 2002 Tranaction of the ASME

4 d g d kua bku g 4 g ŝŝ,ŝd, 4 (26) where g 0 kk d f T ref,k, (27a) a b kk b d f T ref,k f T,k/ f T ref,k, 0 (27b) ubject to the boundary condition at a wall, g wg w a g b T w w nˆ ŝ0 g nˆ ŝd. (28) Thi reordering proce require that any factor accompanying radiative intenity with the exception of k itelf mut be independent of wavenumber; i.e., like any global model the FSCK method i limited to gray urface and/or gray cattering propertie. Evaluation of Weight Function a. Note that expreing the tranmiivity in term of the correlated-k ditribution f (T,k) econd formulation in Eq. 4 alo atifie the RTE, replacing the a b in Eq. 9 by f (T,k) b, in which cae the reulting intenity ha to be integrated over k-pace, i.e., 0 k dk 7. The advantage of the preent formulation beide demontrating the equivalence between k-ditribution and the WSGG model i the fact that the weight function a(t,g) i much moother and better behaved than the k-ditribution, and thu require fewer quadrature point i.e., gray-ga evaluation for the accurate determination of full pectrum reult uch a heat flux. t remain to determine the k(t,g) ditribution for a given ga mixture followed by tranformation to k(t ref,g ref ), i.e., the evaluation of the a(t,g ref ). Thi can be done in a number of way, the two mot extreme one being calculation from i total emiivity tranmiivity data, and ii from line-by-line data uch a the HTRAN 23,24 or HTEMP 25 databae. n thi paper, we will limit our conideration to high-reolution databae. HTRAN92 23 ha been ued uccefully in meteorological application, but i known to be inaccurate for combution cenario ince many hot line are miing in that databae. HT- RAN96 24 ha remedied thi problem to ome extent and may now be ued with confidence for up to 600K, although many hot line are till miing. Very recently, HTEMP 25 ha become available for carbon dioxide and water vapor and hould be accurate for up to 000 K. However, it ha many time the number of pectral line than HTRAN96, requiring ubtantially more computer time, and i limited to carbon dioxide and water vapor mixture. n either cae k(t,g) i determined from Eq. 4 or 24 for a fixed reference condition. Recall that the temperature dependence in k(t,g) originate from the fractional Planck function i, not from the aborption coefficient, which i evaluated at the reference condition which remain fixed. The tranformation function a(t,g) i bet determined by ratioing the lope of the preferably lightly moothened g ditribution function for the actual and the reference temperature for the ame k, ince a may have dicontinuitie if f (T,k)/f(T ref,k) i employed ince the f may have ingularitie, albeit at identical k-value. Once the correlated-k ditribution and the weight function have been determined, the temperature and additional preure dependence given by the function u(t,p,x ) in Eq. 5 mut be potulated and/or determined in ome optimal way. There are everal different way to obtain k(t,g) from line-byline data a decribed in everal paper 2,3,2,26. We prefer the following method, which i imple, quick, adaptive and particularly well-uited for full-pectrum calculation: the pectrum 0 i ubdivided into N equal ubinterval, and N equally paced pectral location. Similarly, the k-range i ubdivided into J range becaue of the large order-of-magnitude range of the aborption coefficient, the k-range i ubdivided equally on a log 0 k-bai. A et of temperature T i can be conidered imultaneouly. A can i now made over the N pectral location, the local value of k i calculated, and the j th k-bin for the i th temperature T i i incremented by the correponding fractional Planck function if k j kk j, i.e., b, with a reolution fine enough that b (T i ) i contant acro. At the end of the can all bin value are multiplied by /T 4 i, after which they reflect the calculated value for f (T,k j )k j and the cumulative k-ditribution for each temperature follow from j gt,k j j f T,k j k j gt,k j f T,k j k j. (29) Note that the number of k-bin a well a temperature bin can be made arbitrarily large without any appreciable increae in computation time. While that may reult in empty bin, the g(t,k) ditribution would imply remain contant for adjacent j-value. After each can the number of N i doubled, reulting in N additional knot point, and N additional k-value are calculated and placed in the f (T,k j ) bin, until uch time when the g j no longer change beyond ome criterion. Note alo that, in the limit of k 0, 0, it follow that f (T,k) wherever k i maximum or minimum, reulting in mall dicontinuitie for a(t,k) at thee point. Thu, the moothne of f and a are trongly affected by the numerical implementation. By making a tranformation from f (k)dk to the weight function a(t,g)dg it wa hoped to obtain a moothened function for eaier quadrature imilar to the tranformation from f (k)dk to dg, with k(g) being a much moother function than f (k). Thi i demontrated for two extreme temperature in Fig. 2a, howing f (T cold,k), f (T hot,k), and a(t cold,k), uing the hot ga temperature a the reference tate. The correponding k(t,g), together with a(t cold,g) are hown in Fig. 2b. While the weight function a(t,g) i not a mooth a the k(t,g) function, a i coniderably better behaved than the f (T,k): high frequency ocillation are reduced from approximately 25 percent of maximum to about 5 percent a dicued earlier, maxima of f and, therefore, a, depend on the numerical implementation; value given are for our preent calculation. Low frequency ocillation are alo much le evere. A the temperature move cloer to the reference value, a become progreively moother hovering around an average value of a. Therefore, accurate numerical quadrature of Eq. 20 become relatively eay. Efficient quadrature can be further improved by moothing the weight function through where 0 agkgdg0 0 g g agkgdgdg āgkgdg, (30) ā agdg (3) g g ince k(g) i eentially contant acro a mall interval g. n thi expreion (k) i any function that depend on g through the function k(t,g), uch a g. The moothened weight function ā i alo indicated in Fig. 2b along with typical quadrature point ued in later example. Alo indicated are typical tep value k i for the WSGG trapezoidal integration. Scaling of Aborption Coefficient. The FSCK method i exact a compared to LBL calculation that ue the ame caled Journal of Heat Tranfer FEBRUARY 2002, Vol. 24 Õ 33

5 Planck Mean Temperature. P T 4 ref V V P T 4 dv (33d) Emiion Weighted Temperature. T ref VT4 P T,p,x T 4 dv A TT 4 /da V 4 P T,p,x T 4 dv A T 4 /da (33e) Fig. 2 a Comparion of k-ditribution at different temperature and the weight function a, b Planck function weighted cumulative k-ditribution g, and the weight function a aborption coefficient. Error arie only from the fact that actual ga mixture do not obey the caling approximation. Therefore, optimal caling methodology i extremely important for the accuracy of the FSCK method, although thi remain omewhat of a black art. While the FSCK method i, in principle, valid for arbitrary ga mixture, we will limit ourelve here to ytem with contant total preure, which reduce the determination of a caled aborption coefficient ditribution from a line-by-line databae to two tep. Firt, a reference condition mut be choen. Auming contant total preure throughout, it appear natural to take a volume average a the reference mole fraction ditribution, or x ref V Vx dv. (32) Chooing an optimal reference temperature i le obviou; different poibilitie are lited and dicued a follow: Maximum Temperature in Sytem. T ref T hot Minimum Temperature in Sytem. T ref T cold Volume Averaged Temperature. (33a) (33b) T ref V VTdV (33c) Since at the reference tate the aborption coefficient i et to coincide with that of the databae, an intermediate temperature may be expected to do better than chooing the maximum Eq. 33a or minimum temperature Eq. 33b a the reference temperature, uch a a patially averaged temperature Eq. 33c. However, traight volume averaging neglect local variation in mole fraction, a well a the fact that emiion from hot region often dominate the radiative field. Therefore, uing a Planck mean temperature baed on overall emitted energy or an emiion-weighted temperature can be expected to give better reult. The performance of different reference temperature will be teted later within thi paper. Once a reference tate ha been etablihed, an appropriate patial variation function u(t,p,x ) mut be found. f we aume contant total preure throughout the ytem and neglect preure effect on the line half width b j generally a good approximation for ytem with roughly contant total preure, then pectral line become temperature and preure dependent through only the line intenitie S j, which, for a linear aborption coefficient, i linearly proportional to the partial preure of the aborbing ga, time a function of temperature only. Since radiative heat fluxe from a layer are governed by emiion rate attenuated by elf aborption, the caling function for a ga mixture with only one participating ga u(t,x) i found here from the implicit relation 0 b T em exp T,xL m d 0 b T em expk ut,xl m d, (34) where k (T ref,x ref ) and L m i the mean beam length of the volume under conideration. Note that there are two temperature involved in Eq. 34, an emiion temperature T em and the reference temperature T ref. Uing an emiion temperature different from T ref may give better reult, but will involve a larger amount of precalculation and interpolation. For implicity, one may conider the ue of the reference temperature alo a the emiion temperature. For optically thin ituation, Eq. 34 enure that the caling produce the correct Planck-mean aborption coefficient at all location weighted by T em. For optically thick ituation, Eq. 34 enure that the caling produce the correct heat flux ecaping a layer with a thickne of L m. For ga mixture with more than one participating ga pecie, we ue here an aumed hape of N ut,x n x n u n T, (35) where x n u n (T) i the caling function for the n th ga pecie and i evaluated, independently for each pecie, uing Eq. 34. Thi implification ha the diadvantage that it neglect line overlap between pecie in the function of optimal caling parameter only, not in the heat tranfer calculation. t ha the advantage that the volumetric caling function can be databaed independently for each pecie. 34 Õ Vol. 24, FEBRUARY 2002 Tranaction of the ASME

6 Fig. 3 Local radiative flux in an iothermal N 2 CO 2 mixture TÄ500 K, pä bar, x CO2 Ä0., LÄ cm and LÄ m bounded by cold, black wall Sample Calculation The validity of the preent model, it application to non-black wall and cattering media, and it limitation due to the caling approximation will be hown through a number of relatively imple one-dimenional example in which CO 2 -N 2 mixture confined between two infinite parallel wall are conidered. Alo, a two-dimenional practical combution problem will be tudied to tet the model, with more than one participating ga coexiting in a cylindrical axiymmetric combution chamber. The P approximation i employed in the following example, ince it i a popular method with reaonable level of effort and accuracy. Since the FSCK method i a pectral model that can be ued with any RTE olution method, comparion of the LBL benchmark with FSCK demontrate the accuracy of the FSCK method a long a the ame RTE olver i employed in both cae. Any other olution method than the P approximation would alo be acceptable. The HTEMP a well a HTRAN96 databae are ued in the following calculation to validate the new approach, and method to determine optimally caled aborption coefficient will be dicued. One-Dimenional Slab. Firt an iothermal medium confined between two parallel, cold and black plate i conidered. Since the medium i homogeneou, the k-ditribution at only one temperature i needed, i.e., at the temperature of the medium, o that a in thi cae. The medium i a nitrogen-carbon dioxide mixture at 500 K, bar total preure, with a 0 percent mole fraction of CO 2. Uing the HTEMP databae for the evaluation of aborption coefficient, benchmark line-by-line reult are compared in Fig. 3 with the Full-Spectrum Correlated-k Ditribution FSCK method for two lab width demontrating that the FSCK method i indeed exact for homogeneou media. Uing 0 Gauian quadrature point, the FSCK reult eentially coincide with the LBL reult for which approximately 600,000 quadrature point were needed. Uing only 6 quadrature point how light dicrepancie for optically thick cae (L m). Similar to LBL calculation, more accurate reult can be obtained by uing more quadrature point. Within numerical accuracy, the wall heat flux predicted by the FSCK i exact for the homogeneou and iothermal cae. Taine et al. 22 have hown that the caling approximation may produce ubtantial error when radiation emitted in a hot region travel through a cold layer, ince i k-ditribution alway ort aborption coefficient according to magnitude auming that thi produce conitent wavenumber orting, while ii in Fig. 4 Radiative flux exiting from the cold column of a two-column CO 2 -nitrogen mixture at different temperature T hot Ä2000 K, l hot Ä50 cm; T cold Ä300 K, l cold variable; uniform pä bar, x CO2 Ä0., cold and black wall uing HTEMP and HTRAN96 databae; the relative error hown are for HTEMP reult trongly non-iothermal media thi ordering conitency i violated by hot line, which have large aborption coefficient at high temperature, while being eentially negligible at low temperature. We will conider two type of non-iothermal media. Firt, we will look at the extreme cae of an iothermal hot layer adjacent to an iothermal cold layer. Thi extreme tet will allow u to find out optimum way to determine accurately caled aborption coefficient ditribution from the HTEMP databae. Figure 4 how the radiative heat flux arriving at the cold black wall ofan 2 -CO 2 mixture with a tep in temperature. Preure and CO 2 mole fraction are contant throughout at bar and 0 percent, repectively. The hot layer i at T2000 K and ha a fixed width of 50 cm, while the cold layer i at 300 K, and i of varying width. The LBL reult obtained from both the HTEMP and the HT- RAN96 databae are compared with variou caling approximation. Note that heat fluxe predicted from the HTEMP databae are more than double no cold layer to five-fold thick cold layer of thoe predicted from the HTRAN96 databae: while the HT- RAN96 databae can be ued with confidence up to about 600 K 27, for temperature beyond that level it appear to be miing many hot line, which are etimated in the HTEMP databae. Since FSCK require quadrature over a ingle monotonically increaing function and need about 0 quadrature point, while LBL calculation require about million quadrature point, the FSCK method will greatly peed up the calculation. n thi example, baed on a well-etablihed aborption coefficient databae for both FSCK and LBL, the FSCK calculation required le than 0.05 econd 0 quadrature point on an SGO200 ingle proceor R0000 at 50 MHz, while the LBL calculation required 25 minute, or a factor of approximately 00,000:. Journal of Heat Tranfer FEBRUARY 2002, Vol. 24 Õ 35

7 Fig. 6 Geometry of the cylindrical combutor Fig. 5 Same a Fig. 4, but for medium bounded by gray wall a well a for gray cattering media From Fig. 4 it can be een that uing two temperature caling T hot a T em in Eq. 34, plu a Planck mean reference temperature give the bet reult with a maximum error of only 8 percent. n thi problem, the emiion weighted temperature i very cloe to the hot temperature of 2000 K and, uing it for both T em and T ref, give a maximum error of 9 percent. Uing the patially averaged temperature and the cold temperature 300 K a T ref and T em give maximum error of 3 percent and 9 percent, repectively. Uing the Planck mean temperature for both T ref and T em produce a maximum error of 25 percent for thi extreme temperature cae becaue, with increaing cold layer, the Planck-mean reference temperature move cloer to the cold temperature, greatly overpredicting emiion from the hot layer. Although producing large error in thi extreme example, we feel that the Planck mean temperature, together with the emiion-weighted temperature, are the bet choice for reference temperature. A will be hown later, Planck mean and emiion-weighted temperature are actually very cloe in more realitic combution ytem. Denion and Webb have already hown that the WSGG method i applicable to gray boundarie. To demontrate that the FSCK method and, therefore, alo the WSGG method i equally valid not only for media bounded by gray wall, but alo for gray cattering media, heat fluxe through the mixture of the previou example were alo calculated for the cae of gray wall 0.5, the addition of a gray cattering medium cattering coefficient /(l hot l cold ), and the combination of both. The choice of here i arbitrary, and i choen to give an optical thickne of unity, where one would expect cattering to have the larget effect. Repreentative calculation uing line-by-line calculation together with the caled aborption coefficient confirmed that the FSCK method produce exact reult for the caled aborption coefficient, even in the preence of non-black wall and gray cattering. n all cae uing the emiion-weighted temperature a the reference value gave again the mot accurate reult. npection of Fig. 5 how that with the preence of a nonblack wall the heat flux to the wall i reduced and the maximum relative error remain approximately the ame. The influence of cattering and combined effect are alo hown in Fig. 5. Qualitatively, the trend remain the ame, with maximum error at l cold 0 cm of 9.5 percent and 0.4 percent, repectively. Again, HTEMP reult a compared to HTRAN96 reult are higher by a factor of 2 no cold layer to about 6 thick cold layer. The previou example with a tep change in temperature were deigned to be a wort-cae cenario, i.e., to undertand the limit of the caling approximation, and a a tool to find method to determine optimum caling parameter for a ga mixture. Onedimenional nitrogen-carbon dioxide mixture with moothly varying temperature and/or mole fraction profile, uch a one may expect to occur in actual combution application, make LBL and FSCK reult virtually coincide 28. Two-Dimenional Ga Mixture. The model will now be teted further by applying it to a practical combution problem. A mixture of combution product i.e., CO 2 and H 2 O a well a fuel i.e., CH 4 in a cylindrical axiymmetric geometry i tudied, a hown in Fig. 6. A mall nozzle at the center of the combutor introduce methane at high peed and ambient air enter the combutor coaxially at a lower peed. Fuel and air mix and are allowed to react uing a imple eddy diipation reaction model. The liner wall i aumed black and inulated, and it temperature i equal to the local ga temperature. Since fuel i injected from the inlet together with cold air, temperature near the inlet are relatively low 300 K. The combution reaction produce a bell-haped flame heet with high downtream outlet temperature. Temperature level inide the chamber range from 300 K to around 700 K, a hown in Fig. 7a. The mole fraction ditri- Fig. 7 Temperature and mole fraction ditribution in a twodimenional cylindrical combution chamber, a temperature ditribution; b mole fraction ditribution of CO 2 and H 2 O; and c mole fraction ditribution of CH 4 ga mixture with methane and without methane are both conidered. 36 Õ Vol. 24, FEBRUARY 2002 Tranaction of the ASME

8 Fig. 8 Two-dimenional cylindrical combution chamber with a ga mixture containing CO 2 and H 2 O: a LBL calculation for the radiative heat ource "q WÕcm 3 ; b relative error of FSCK reult, "q LBL À "q FSCK Õ "q LBL,max. Fig. 9 Two-dimenional cylindrical combution chamber with a ga mixture containing CO 2,H 2 O and CH 4 : a LBL calculation for the radiative heat ource "q WÕcm 3 ; b relative error of FSCK reult, "q LBL À "q FSCK Õ "q LBL,max. bution of the combution product baically follow the pattern of the temperature change, a hown in Fig. 7b, with x H2 O 2x CO2 everywhere uing a imple global reaction for methane. The fuel, on the other hand, ha large mole fraction near the inlet where it ha not yet been conumed, and i barely preent beyond where combution ha taken place, a hown in Fig. 7c. LBL calculation were carried out a a benchmark. Since three emitting and aborbing gae coexit in thi chamber in thi cae, methane, carbon dioxide and water vapor, the FSCK approach need now to be applied to a ga mixture of more than one participating ga. n the following calculation, the reference and emiion temperature in Eq. 34 are taken a the Planck mean temperature. Chooing the emiion weighted temperature from Eq. 33e intead, yield eentially the ame reult, ince both temperature are very cloe to each other. We will firt conider the cae of CO 2 and H 2 O being the only radiatively participating gae, with the temperature ditribution hown in Fig. 7a and mole fraction ditribution hown in Fig. 7b. Although the mole fraction of the combution product vary throughout the volume, the mole fraction ratio of CO 2 to H 2 Oi 0.5 everywhere. The radiative heat ource q determined from LBL calculation for thi cae i hown in Fig. 8a and the relative error of the FSCK method with repect to the LBL benchmark, defined a errorpercent) q LBL q FSCK 00 (36) q LBL,max i hown in Fig. 8b. t can be een that the maximum error are around 4 percent acro the harp gradient jut outide the flame heet, and 3 percent in the hot downtream ection. Thu, one may conclude that the FSCK method predict heat tranfer rate very well in ituation where gae have contant ratio of mole fraction. n thi problem, the CPU time required for LBL calculation i about 60 h, while that for the FSCK method i 5 ec. Next we conider the ame mixture of CO 2 and H 2 O, but will alo include the radiative participation of CH 4. Since methane, a the fuel, ha large mole fraction only near the inlet a hown in Fig. 7c, the gae in the mixture no longer have the ame mole fraction ratio throughout the combution chamber. Again, LBL calculation are carried out a a benchmark and are hown in Fig. 9a. The ditribution of the radiative heat ource differ from the previou problem only in the inlet region becaue of the preence of CH 4. Shown in Fig. 9b, the maximum error of the FSCK approach now increae to 50 percent in the inlet region with it mole fraction dicontinuity, although the error remain well below 0 percent throughout mot of the combution chamber. t can be een that non-contant ratio of mole fraction of participating gae have a big effect on the accuracy of the method, ince at one location ga a may be prominent, and ga b at another, cauing evere breakdown of aborption coefficient caling. To overcome thi problem we have alo developed a multi-cale FSCK method 29. Summary and Concluion A Full-Spectrum Correlated-k Ditribution FSCK ha been developed, which within it limitation gray wall, gray cattering, pectral aborption coefficient obeying the caling approximation allow very efficient exact evaluation of radiative fluxe for arbitrary molecular ga mixture, uing any deired RTE olver. Nongray urface and/or nongray cattering would require a multi-band approach rather than full-pectrum. t ha been hown that the popular Weighted-Sum-of-Gray-Gae WSGG method i imply a crude implementation of the FSCK method; therefore, it i implied that the WSGG method can alo be applied to gray encloure a well a gray cattering media. Limitation of the caling approximation have alo been invetigated and procedure to find optimally caled ditribution have been dicued. Comparion of reult uing the HTRAN96 and HTEMP databae how that, beyond 000 K, HTEMP radiative fluxe are everal time larger than thoe from HTRAN, thu indicating the application limit of HTRAN96. Acknowledgment The author gratefully acknowledge the financial upport of the National Science Foundation under the contract CTS Temperature and mole fraction profile for the axiymmetric combution problem in Fig. 7 were provided by Mr. G. Li. Nomenclature A weight function for WSGG method Journal of Heat Tranfer FEBRUARY 2002, Vol. 24 Õ 37

9 a weight function for FSCK method b line half-width, cm f k-ditribution function, cm g cumulative k-ditribution radiative intenity, W/m 2 r i fractional Planck function k aborption coefficient variable, cm k pectral aborption coefficient at reference tate, cm l geometric length, m L m mean beam length, m p preure, bar q radiative heat flux, W/m 2 S line intenity, cm 2, ditance along path, m T temperature, K u patial dependence function for aborption coefficient V Volume, m 3 X weighted path length, m x, x mole fraction, mole fraction vector Greek Symbol aborptivity emiivity wavenumber, cm cattering phae function aborption coefficient, cm olid angle, r cattering coefficient, cm tranmiivity Subcript 0 reference condition b blackbody emiion em emiion i gray ga in WSGG j line or bin P Planck mean w wall pectral Reference Goody, R., Wet, R., Chen, L., and Crip, D., 989, The Correlated-k Method for Radiation Calculation in Nonhomogeneou Atmophere, J. Quant. Spectroc. Radiat. Tranf., 42, No. 6, pp Laci, A. A., and Oina, V., 99, A Decription of the Correlated-k Ditribution Method for Modeling Nongray Gaeou Aborption, Thermal Emiion, and Multiple Scattering in Vertically nhomogeneou Atmophere, Journal of Geophyical Reearch, 96, No. D5, pp Fu, Q., and Liou, K. N., 992, On the Correlated-k Ditribution Method for Radiative Tranfer in Nonhomogeneou Atmophere, J. Atmo. Sci., 49, No. 22, pp Rivière, Ph., Soufiani, A., and Taine, J., 992, Correlated-k and Fictitiou Ga Method for H 2 O Near 2.7 m, J. Quant. Spectroc. Radiat. Tranf., 48, pp Rivière, Ph., Scutaru, D., Soufiani, A., and Taine, J., 994, A New CK Data Bae Suitable From 300 K to 2500 K for Spectrally Correlated Radiative Tranfer in CO 2 -H 2 O-Tranparent Ga Mixture, in Proceeding of the 0th nternational Heat Tranfer Conference, ed., G. F., Hewitt, Taylor & Franci, London. 6 Rivière, Ph., Soufiani, A., and Taine, J., 995, Correlated-k and Fictitiou Ga Model for H 2 O nfrared Radiation in the Voigt Regime, J. Quant. Spectroc. Radiat. Tranf., 53, pp Soufiani, A., and Taine, J., 997, High Temperature Ga Radiative Property Parameter of Statitical Narrow-Band Model for H 2 O, CO 2 and CO, and Correlated-k Model for H 2 O and CO 2, nt. J. Heat Ma Tranf., 40, No. 4, pp Hottel, H. C., and Sarofim, A. F., 967, Radiative Tranfer, McGraw-Hill, New York. 9 Modet, M. F., 99, The Weighted-Sum-of-Gray-Gae Model for Arbitrary Solution Method in Radiative Tranfer, ASME J. Heat Tranfer, 3, No. 3, pp Denion, M. K., and Webb, B. W., 993, An Aborption-Line Blackbody Ditribution Function for Efficient Calculation of Total Ga Radiative Tranfer, J. Quant. Spectroc. Radiat. Tranf., 50, pp Denion, M. K., and Webb, B. W., 993, A Spectral Line Baed Weighted- Sum-of-Gray-Gae Model for Arbitrary RTE Solver, ASME J. Heat Tranfer, 5, pp Denion, M. K., and Webb, B. W., 994, k-ditribution and Weighted-Sumof-Gray Gae: A Hybrid Model, in Tenth nternational Heat Tranfer Conference, Taylor & Franci, London, pp Denion, M. K., and Webb, B. W., 995, The Spectral-Line-Baed Weighted- Sum-of-Gray-Gae Model in Noniothermal Nonhomogeneou Media, ASME J. Heat Tranfer, 7, pp Denion, M. K., and Webb, B. W., 995, Development and Application of an Aborption Line Black-Body Ditribution Function for CO 2, nt. J. Heat Ma Tranf., 38, pp Denion, M. K., and Webb, B. W., 995, The Spectral-Line Weighted-Sumof-Gray-Gae Model for H 2 O/CO 2 Mixture, ASME J. Heat Tranfer, 7, pp Rivière, Ph., Soufiani, A., Perrin, Y., Riad, H., and Gleize, A., 996, Air Mixture Radiative Property Modelling in the Temperature Range K, J. Quant. Spectroc. Radiat. Tranf., 56, pp Pierrot, L., Rivière, Ph, Soufiani, A., and Taine, J., 999, A Fictitiou-Ga- Baed Aborption Ditribution Function Global Model for Radiative Tranfer in Hot Gae, J. Quant. Spectroc. Radiat. Tranf., 62, pp Pierrot, L., Soufiani, A., and Taine, J., 999, Accuracy of Narrow-Band and Global Model for Radiative Tranfer in H 2 O, CO 2, and H 2 O-CO 2 Mixture at High Temperature, J. Quant. Spectroc. Radiat. Tranf., 62, pp Modet, M. F., 993, Radiative Heat Tranfer, McGraw-Hill, New York. 20 Goody, R. M., and Yung, Y. L., 989, Atmopheric Radiation Theoretical Bai, 2nd ed., Oxford Univerity Pre, New York. 2 Goody, R., Wet, R., Chen, L., and Crip, D., 989, The Correlated-k Method for Radiation Calculation in Nonhomogeneou Atmophere, J. Quant. Spectroc. Radiat. Tranf., 42, pp Taine, J., and Soufiani, A., 999, Ga R Radiative Propertie: From Spectrocopic Data to Approximate Model, in Advance in Heat Tranfer, 33, Academic Pre, New York, pp Rothman, L. S., Gamache, R. R., and Tipping, R. H. et al., 992, The HT- RAN Molecular Databae: Edition of 99 and 992, J. Quant. Spectroc. Radiat. Tranf., 48, No. 5/6, pp Rothman, L. S., Rinland, C. P., Goldman, A., Maie, S. T., Edward, D. P., Flaud, J. M., Perrin, A., Camy-Peyret, C., Dana, V., Mandin, J. Y., Schroeder, J., McCann, A., Gamache, R. R., Watton, R. B., Yohino, K., Chance, K. V., Juck, K. W., Brown, L. R., Nemtchinov, V., and Varanai, P., 998, The HTRAN Molecular Spectrocopic Databae and HAWKS HTRAN Atmopheric Worktation: 996 Edition, J. Quant. Spectroc. Radiat. Tranf., 60, pp Rothman, L. S., Camy-Peyret, C., Flaud, J.-M., Gamache, R. R., Goldman, A., Goorvitch, D., Hawkin, R. L., Schroeder, J., Selby, J. E. A., and Watton, R. B., 2000, HTEMP, the High-Temperature Molecular Spectrocopic Databae, J. Quant. Spectroc. Radiat. Tranf., to appear. 26 Arking, A., and Groman, K., 972, The nfluence of Line Shape and Band Structure on Temperature in Planetary Atmophere, J. Atmo. Sci., 29, pp Modet, M. F., and Bharadwaj, S., 200, High-Reoultion, High-Temperature Tranmiivity Meaurement and Correlation for Carbon Dioxide-Nitrogen Mixture, in Proceeding of the CHMT 3rd nternational Sympoium on Radiative Tranfer, Antalya, Turkey. 28 Modet, M. F., and Zhang, H., 2000, The Full-Spectrum Correlated-k Ditribution and t Relationhip to the Weighted-Sum-of-Gray-Gae Method, in Proceeding of the 2000 MECE, HTD-366-, Orlando, FL, ASME, New York, pp Modet, M. F., and Zhang, H., 2002, A Multi-Level Full-Spectrum Correlated-k Ditribution for Radiative Heat Tranfer in nhomogeneou Ga Mixture, J. Quant. Spectroc. Radiat. Tranf., in print. 38 Õ Vol. 24, FEBRUARY 2002 Tranaction of the ASME