Crnomarković, N. Dj., et al.: Influence of the Gray Gases Number in the Weighted THERMAL SCIENCE, Year 2016, Vol. 20, Suppl. 1, pp.

Transcription

1 Crnomarkovć, N. Dj., et al.: Influence of the Gray Gases Number n the Weghted S197 INFLUENCE OF THE GRAY GASES NUMBER IN THE WEIGHTED SUM OF GRAY GASES MODEL ON THE RADIATIVE HEAT EXCHANGE CALCULATION INSIDE PULVERIZED COAL-FIRED FURNACES by, Nenad Dj. CRNOMARKOVI] *, Srdjan V. BELOŠEVI], Ivan D. TOMANOVI], and Aleksandar R. MILI]EVI] Laboratory for Thermal Engneerng and Energy, Vnca Insttute of Nuclear Scences, Unversty of Belgrade, Belgrade, Serba Orgnal scentfc paper DOI: /TSCI C The nfluence of the gray gases number n the weghted sum n the gray gases model on the calculaton of the radatve heat transfer s dscussed n the paper. A computer code whch solved the set of equatons of the mathematcal model descrbng the reactve two-phase turbulent flow wth radatve heat exchange and wth thermal equlbrum between phases nsde the pulverzed coal-fred furnace was used. Gas-phase radatve propertes were determned by the smple gray gas model and two combnatons of the weghted sum of the gray gases models: one gray gas plus a clear gas and two gray gases plus a clear gas. Investgaton was carred out for two values of the total extncton coeffcent of the dspersed phase, for the clean furnace walls and furnace walls covered by an ash layer depost, and for three levels of the approxmaton accuracy of the weghtng coeffcents. The nfluence of the number of gray gases was analyzed through the relatve dfferences of the wall fluxes, wall temperatures, medum temperatures, and heat transfer rate through all furnace walls. The nvestgaton showed that there were condtons of the numercal nvestgatons for whch the relatve dfferences of the varables descrbng the radatve heat exchange decrease wth the ncrease n the number of gray gases. The results of ths nvestgaton show that f the weghted sum of the gray gases model s used, the complexty of the computer code and calculaton tme can be reduced by optmzng the gray gases number. Key words: furnace, pulverzed coal, smple gray gas model, relatve dfference, numercal smulaton, weghted sum of gray gases model Introducton In numercal nvestgatons of processes nsde pulverzed coal fred furnaces, the smple gray gas model (SGM) or weghted sum of gray gases model (WSGM) s used as the gas-phase radatve property model (). When the SGM s used, a real gas s replaced by a gray gas. In the WSGM, a real gas s replaced by a mxture consstng of several gray gases plus a clear gas. The WSGM s used n numercal nvestgatons of conventonal combuston wth ar [1-3] and oxy-fuel combuston [4-6]. In the WSGM based on total radatve propertes, the number of gray gases s usually not bgger than three. The emssvty of a real gas s represented by summng the terms, each of whch s a product of the emssvty of a gray gas column and temperature-dependent weghtng coeffcent. The nfluence of the number of gray gases on the emssvty accuracy * Correspondng author; e-mal: ncrn@vnca.rs

2 S198 Crnomarkovć, N. Dj., et al.: Influence of the Gray Gases Number n the Weghted has been well documented [7-9]. The ncrease n the number of gray gases decreases the dfferences between the emssvty determned by the WSGM and the emssvty of a real gas. The nfluence of the number of gray gases n the WSGM on the calculaton of the radatve heat transfer were descrbed for the WSGM based on the spectral radatve propertes. Coelho [10] used the spectral lne based WSGM wth 3 and 20 gray gases to evaluate the performances of two radaton models for the calculaton of wall fluxes and radatve heat sources nsde a rectangular 3-D enclosure wth fxed temperature and concentraton dstrbuton. The exact values were obtaned by the ray tracng method and statstcal narrow band model [11]. Wth some exceptons, an ncrease n the number of gray gases from 3 to 20 decreased the average and maxmal relatve errors. Park and Km [12] used the regroupng technque to nvestgate the possblty of reducng the number of gray gases of the narrow band based WSGM. The exact values of the wall fluxes and average ntensty of radaton nsde the cubc 3-D enclosure wth fxed temperature and concentraton spatal dstrbuton were found usng the statstcal narrow band model. Nne hundred gray gases were regrouped nto 3, 5, 7, 10, 15, and 20 gray gases. The average and maxmal relatve errors generally decreased wth the ncrease n the number of regrouped gray gases, and when the number of gray gases was bgger than fve, the reducton of the average and maxmal relatve errors became unmportant. Results presented by Coelho [10] and Park and Km [12] showed that the dfference n radatve heat transfer calculatons n 3-D enclosures wth fxed temperature and concentraton dstrbutons decreased wth ncreases n the number of gray gases. The objectve of ths nvestgaton was to fnd the nfluence of the number of gray gases n the WSGM based on the total radatve propertes on the calculaton of the radatve heat exchange nsde pulverzed coal-fred furnaces. For a selected pulverzed coal-fred furnace, an n-house computer code whch solved a set of equatons of the reactve two-phase turbulent flow wth radatve heat exchange and wth thermal equlbrum between phases was used. The radatve propertes of the gas-phase furnace medum were determned by the SGM and two WSGM: a combnaton of one gray gas plus a clear gas (WSGM1) and a combnaton of two gray gases plus a clear gas (WSGM2). The nfluence of the number of gray gases n the WSGM was found through analyses of the relatve dfferences of the wall fluxes, wall temperatures, medum (. e. flame) temperatures, and heat transfer rates through all furnace walls. The selected varables drectly descrbe or affect the radatve heat exchange. The nvestgatons were started under numercal condtons used prevously, such as clean furnace walls wth constant wall temperature [13]. After that, the effects of the ash depost layer on the furnace walls wth varous levels of weghtng coeffcent approxmaton accuraces (WCAC) were ncluded. Ths nvestgaton showed that the optmzaton of the number of gray gases was possble not only for the narrow lne and spectral lne based WSGM but also for the WSGM based on total radatve propertes. The mathematcal model and computer code, developed for tangentally-fred furnaces, were appled for the furnace of a utlty boler of 210 MWe. The furnace was parallelepped-shaped wth a hopper at the bottom and a contracton at the upper part. The furnace was 40.0 m hgh, 16.5 m wde, and 14.5 m deep. The furnace geometry wth mass flow rates of coal and ar as well as coal propertes has been descrbed earler [14, 15]. In the followng text, the method and results of the nvestgaton and the conclusons are descrbed. Method of nvestgaton Ths nvestgaton was based on the analyses of results obtaned from numercal smulatons. The descrpton of the method of nvestgaton ncludes descrptons of the mathematcal model and radatve propertes.

3 Crnomarkovć, N. Dj., et al.: Influence of the Gray Gases Number n the Weghted S199 Mathematcal model The mathematcal model of the process nsde the furnace descrbed the reactve two-phase turbulent flow wth radatve heat exchange and wth thermal equlbrum between phases. Only the man characterstcs are descrbed here, because the mathematcal model was descrbed n [13]. The gas phase s descrbed by the tme averaged dfferental equatons of the conservatons of momentum, enthalpy, gas-phase concentratons, and turbulent knetc energy and ts rate of dsspaton n an Euleran reference frame. The general form of the gas-phase equaton s: Φ ( ρg UΦ) = ΓΦ + SΦ + SΦ,p (1) x x x where ρ [kgm 3 ] s the gas-phase densty, U [ms 1 ] the component of the gas-phase velocty vector, Φ the general gas-phase varable, Γ Φ [kgm 1 s 1 ] the transport coeffcent for Φ, S Φ the source term, and S Φ,p the source term due to the partcles presence. The set of equatons s closed by the k-ε turbulence model. The equaton for the pressure feld s obtaned from the contnuty and momentum equatons. The pressure feld s solved by the SIMPLE algorthm. The dspersed phase s descrbed by the dfferental equatons of moton and change of mass and energy n a Lagrangan reference frame. The moton of the partcles s tracked along trajectores wth a constant number of partcles. The partcle velocty vector s the sum of the convectve and dffuson components. Heterogeneous reactons of coal combuston are modeled n the knetc-dffuson regme. The combuston model s descrbed n detal n [16]. The radatve heat exchange s determned by Hottels zonal method of radaton, based on dvson of the furnace volume nto volume zones and furnace walls nto surface zones. For every par of zones, drect exchange areas (DEA) and total exchange areas (TEA) are calculated. The wall heat flux of the surface zone s s the dfference between the energy absorbed and lost: Nvz Nsz SσT A σt GmSσTg + m Sn sn s s s m= 1 n= 1 w, s = As, = 1,, sz q N (2) where G m S [m 2 ] s the volume-surface TEA, S n S [m 2 ] the surface-surface TEA, N vz the number of volume zones, N sz the number of surface zones, T gw [K] the temperature of the volume zone g m, T s [K] the temperature of the volume zone s, s the emssvty of the surface zone s, and A s [m 2 ] the surface area of the zone s. The sum of the frst two terms n the numerator of eq. (2) s the heat transfer rate of the ganed energy and the thrd term s the heat transfer rate of the energy loss because of the emsson of radaton from the surface zone s. The heat transfer rate of the zone s s determned by multplyng the wall flux by the surface area: Q s = qw,, s A s and the heat transfer rate through all furnace walls s found by summng the heat transfer rates of all surface zones whch represent the wall: Q fur =ΣQ s. When the gas-phase radatve propertes are modeled by the WSGM, the wall heat flux of the surface zone s s determned from the relaton: Nvz Nsz GmSσTg + m SnSσTs A n s s σt s m= 1 n= 1 qw, s =, = 1,, N sz (3) A s

4 S200 Crnomarkovć, N. Dj., et al.: Influence of the Gray Gases Number n the Weghted where GmS [m 2 ] s the volume-surface drected flux area (DFA), and SS n [m 2 ] the surface-surface DFA, [9]. The flow feld was solved on a fne numercal grd composed of N cv control volumes. Dscretzaton and lnearzaton of the gas-phase equatons are acheved usng the method of the control volumes and hybrd dfference scheme. The stablty of the teratve procedure s provded by the under-relaxaton method [17]. The thermophyscal propertes of the gas phase are determned from the equaton of the state, tabulated values, and emprcal relatons. The method of calculaton of the ash depost temperatures s as descrbed prevously [18]. The furnace walls consst of two layers: the metal wall layer (4.0 mm thck) and the ash depost layer (l ash = 0.3 mm). The dependence of the effectve thermal conductvty of the ash depost layer on temperature was adopted from reference [19] (the sample whose slca rato was 62), whereas the thermal conductvty of the metal wall layer was that of carbon steel [20, 21]. For the clean furnace walls (l ash = 0.0 mm), the wall temperature was constant (T w = K). The wall emssvty was 0.8. Radatve propertes The absorpton coeffcents of the gray gases for the WSGM were determned accordng to the procedure descrbed n [7]. The absorpton coeffcents for the SGM and for the WSGM1 were the same as descrbed prevously [22]: K a,g = /m for the SGM and K a,gg,1 = /m for the WSGM1. The absorpton coeffcents n the WSGM2 were K a,gg,1 = /m and K a,gg,2 = /m. The weghtng coeffcents a(t) were determned by fttng the expresson for the gas-phase emssvty: N g= a,gg, m j = 0 { aj( T) 1 exp( K jl ) } gg (4) where L m [m] s the mean beam length, and g the emssvty of the real gas, determned by Leckner s model [23]. The weghtng coeffcents were determnd n the range of K. The weghtng coeffcents were determned n the form of polynomals: PD = 0 f( θ ) = bθ (5) where b desgnates the polynomal coeffcent (PC), and θ [K] s the scaled temperature, θ = T/1000. Each weghtng coeffcent was represented by the polynomals of the thrd, fourth, and ffth degrees. The PC were determned by the least-squares method [24] and ther values are gven n tabs Table 1. The PC for the thrd-degree polynomal PC WSGM1 WSGM2 a 1 a 0 a 1 a 2 b b b b

5 Crnomarkovć, N. Dj., et al.: Influence of the Gray Gases Number n the Weghted S201 Table 2. The PC for the fourth-degree polynomal PC WSGM1 WSGM2 a 1 a 0 a 1 a 2 b b b b b Table 3. The PC for the ffth-degree polynomal PC WSGM1 WSGM2 a 1 a 0 a 1 a 2 b b b b b b The values of the weghtng coeffcents were set by means of weghts [25] n the range of K, whch was the expected temperature nterval of the coarse numercal grd. The WCAC were determned by the root-mean-square error: N = 1 χ = dp [ at ( ) f( T) ] N dp 2 (6) where N dp s the number of Table 4. Values of χ dscrete ponts. Values of χ WSGM1 WSGM2 are gven n tab. 4. PD For the same furnace a 1 a 0 a 1 a 2 and for the condtons of thermal equlbrum between phases, t was already found that the results of numercal nvestgatons agreed wth the expermental data for the medum total extncton coeffcent between 0.2 and 2.0 1/m and for the scatterng albedo between 0.0 and 0.5 [13]. As the gas-phase absorpton coeffcent was found to be K a,g = /m, t was concluded that the values of K t,m were determned by the radatve propertes of the dspersed phase. In ths nvestgaton, the values of the total extncton coeffcents of the dspersed phase were as follows: K t,p = 0.5 1/m (K a,p = /m, K s,p = /m) and K t,p = 2.0 1/m (K a,p = 1.0 1/m, K s,p = 1.0 1/m).

6 S202 Crnomarkovć, N. Dj., et al.: Influence of the Gray Gases Number n the Weghted The nfluence of the number of gray gases was found by comparng the relatve dfferences of the numercal smulaton results when WSGM1 (on one sde) and SGM or WSGM2 (on the other sde) were used. The relatve dfferences of the varables were determned as the rato of the absolute dfference and the value of the varable obtaned when the WSGM1 was used: η ηwsgm1 δ = (7) ηwsgm1 where the symbol η stands for the medum temperature T m, wall heat flux q w, and wall temperature T w (only for l ash = 0.3 mm), and the subscrpt means SGM or WSGM2. The average values of the relatve dfference were determned on the bass of the relatve dfference of all control volumes or surface zones: δ 1 N δ, N = 1 = (8) where N stands for N cv (for T m ) or N sz (for q w and T w ). The results also nclude the relatve dfferences of Q fur. Results and dscusson Table 5. Average and maxmal values of the relatve dfferences [%], K t,p = 0.5 1/m and l ash = 0.0 mm The coarse numercal grd was composed of volume zones of cubc shape wth edge dmensons of 1.0 m. Volume zones composed the coarse numercal grd: The total numbers of volume and surface zones were N vz = 7956 and N sz = 2712, respectvely. The DEA of the close zones were determned usng correlatons [9]. TEA were calculated usng Hottel and Sarofm s method of orgnal emtters of radaton [7]. Leavng fluxes were calculated usng the method of LU decomposton [26]. TEA and DEA were corrected usng the generalzed Lawson s smoothng method [27]. The DFA were determned as defned n [9]. The fne numercal grd was obtaned by dvdng each volume zone nto a number of control volumes. The grd-ndependent soluton was acheved for the numercal grd contanng control volumes. The grdndependent soluton and agreement of the numercal results wth the expermental data have been prevously descrbed [13]. All results were obtaned after 4000 teratons. The results are presented n tabs The average values of the relatve dfferences decrease wth the ncrease n the number of gray gases for both values of the total extncton coeffcent of the dspersed phase. The Average Maxmal q w T m q w T m SGM WSGM Table 6. Average and maxmal values of the relatve dfferences [%], K t,p = 2.0 1/m and l ash = 0.0 mm Average Maxmal q w T m q w T m SGM WSGM Table 7. Relatve dfferences of the heat transfer rates, l ash = 0.0 mm K t,p [m 1 ] Q fur [MW] δ SGM [%] δ WSGM2 [%]

7 Crnomarkovć, N. Dj., et al.: Influence of the Gray Gases Number n the Weghted S203 Table 8. Average and maxmal values of the relatve dfferences [%], K t,p = 2.0 1/m and l ash = 0.3 mm Average Maxmal q w T m T w q w T m T w SGM WSGM Table 9. Average and maxmal values of the relatve dfferences [%] wth the weghtng coeffcent approxmated by the polynomal of the fourth degree, K t,p = 2.0 1/m and l ash = 0.3 mm Average Maxmal q w T m T w q w T m T w SGM WSGM Table 10. Average and maxmal values of the relatve dfferences [%] wth the weghtng coeffcent approxmated by the polynomal of the ffth degree, K t,p = 2.0 1/m and l ash = 0.3 mm Average Maxmal q w T m T w q w T m T w SGM WSGM Table 11. Relatve dfferences of the heat transfer rates for the polynomals of the thrd, fourth, and ffth degrees, K t,p = 2.0 1/m and l ash = 0.3 mm PD Q fur [MW] δ SGM [%] δ WSGM2 [%] maxmal values of T m (for K t,p = 0.5 m 1 ) and q w (for K t,p = 2.0 m 1 ) ncrease wth the ncrease n the number of gray gases for dfferent values of K t,p. For K t,p = 2.0 m 1, the values and relatve dfferences of Q fur, gven n tab. 7, decrease wth the ncrease n the number of gray gases. Therefore, that value of the total extncton coeffcent of the dspersed phase was chosen for the rest of the nvestgaton. The average and maxmal relatve dfferences for the furnace walls covered by the ash depost layer, gven n tab. 8, decrease wth the ncrease n the number of gray gases, except for T w, for whch a small ncrease n the maxmal value s obtaned. In tabs. 5-8, the weghtng coeffcents are approxmated by the thrddegree polynomal. Tables 9-11 show the average and maxmal values of the relatve dfferences of the selected varables for two levels of the WCAC and the relatve dfferences of Q fur for three levels of the WCAC. The average values of the relatve dfferences of the selected varables and the relatve dfferences of Q fur decrease wth the ncrease n the number of gray gases for every level of the WCAC, but the character of dependence on the approxmaton accuracy s changeable. An ncrease n the maxmal values of the relatve dfferences was obtaned only for T w and for the polynomal of the ffth degree. The results of ths nvestgaton show that there are condtons of numercal nvestgatons for whch the relatve dfferences of the varables descrbng the radatve heat exchange nsde the pulverzed coal fred furnace decrease wth the ncrease n the number of gray gases n the WSGM. For example, both K t,p = 0.5 1/m and K t,p = 2.0 1/m provde agreement wth expermental data and ether can be used for numercal nvestgatons [13]. The decrease n the average values of the relatve dfferences of the selected varables and heat transfer rates through all furnace walls wth the ncrease n the number of gray gases s obtaned only for K t,p = 2.0 m 1. For that value of K t,p, a decrease n the relatve dfferences of the selected varables and Q fur s obtaned when the effects of the exstence of the ash depost layer on the furnace walls are ncluded wth three levels of the WCAC. A small ncrease n the maxmal values of the relatve dfference of T w s of small mportance, because the average

8 S204 Crnomarkovć, N. Dj., et al.: Influence of the Gray Gases Number n the Weghted values are obtaned on the bass of the varables n all control volumes (or surface zones), whereas the maxmal values of the relatve dfferences represent the rato of varables n one control volume (or surface zone). The decrease n the dfferences s acheved at the expense of the computer code complexty and computatonal effcency. For example, n the case of the computer code used n ths nvestgaton, three approxmatng polynomals for WSGM2 are needed nstead of one (for WSGM1) and also a larger number of the data fles for the TEA storage s needed for WSGM2 (77 fles per gray gas are needed). The ncrease n the computatonal tme from 60 (for WSGM1) to 84 hours (for WSGM2), usng the Intel Core 3 processor, s also mportant. Ths analyss ndcates the exstence of the optmal number of gray gases n the WSGM, for whch the ncrease n the computer code complexty and computatonal tme s not justfed by the mprovement of the results. Conclusons The nfluence of the number of gray gases n the WSGM based on the total radatve propertes on the calculaton of the radatve heat exchange nsde the pulverzed coal fred furnace was dscussed n the paper. A computer code whch solved the set of equatons of the mathematcal model descrbng the reactve two-phase flow wth radatve heat exchange nsde the pulverzed coal fred furnace was used. Gas-phase radatve propertes were determned by the SGM and two combnatons of the WSGM: a combnaton of one gray gas plus a clear gas and a combnaton of two gray gases plus a clear gas. The nvestgaton was carred out for two values of K t,p : for clean furnace walls and furnace walls covered by an ash layer depost wth three levels of WCAC. The nfluence of the number of gray gases was analyzed through the average and maxmal values of the wall fluxes, wall temperatures, medum temperatures, and heat transfer rates through all furnace walls. The nvestgaton showed that there were condtons of numercal nvestgaton for whch the relatve dfferences of the varables descrbng the radatve heat exchange (ncludng the Q fur ) decreased wth the ncrease n the number of gray gases n the WSGM. The results of ths nvestgaton show that f the WSGM s used as the gas-phase, the complexty of the computer code and calculaton tme can be reduced by optmzng the number of gray gases. Acknowledgment Ths work s a result of the project Increase n energy and ecology effcency of processes n pulverzed coal-fred furnace and optmzaton of utlty steam boler ar preheater by usng n-house developed software tools, supported by the Mnstry of Educaton, Scence and Technologcal Development of the Republc of Serba (project No. TR-33018). Nomenclature A surface area, [m 2 ] a weghtng coeffcent, [ ] b polynomal coeffcent, [ ] G S j volume-surface total exchange area, [m 2 ] GS j volume-surface drected flux area, [m 2 ] K a absorpton coeffcent, [m 1 ] K s scatterng coeffcent, [m 1 ] K t total extncton coeffcent, [m 1 ] k turbulent knetc energy, [m 2 s 3 ] L m mean beam length, [m] thckness of the ash depost layer, [m] l ash N cv number of control volumes, [ ] N dp number of dscrete ponts, [ ] N gg number of gray gases n the WSGM, [ ] N vz number of volume zones, [ ] N sz number of surface zones, [ ] Q heat transfer rate, [W] q heat flux, [Wm 2 ] S S j surface-surface total exchange area, [m 2 ] SS j surface-surface drected flux area, [m 2 ] S Φ source term for the gas-phase varable Φ T temperature, [K]

9 Crnomarkovć, N. Dj., et al.: Influence of the Gray Gases Number n the Weghted S205 U tme-averaged velocty component, [ms 1 ] x co-ordnate n general ndex notaton, [m] Greek symbols Γ transport coeffcent, [kgm 1 s 1 ] δ relatve dfference, [%] total emssvty, [ ] ε rate of dsspaton of the turbulent knetc energy, [m 2 s 3 ] θ scaled temperature, [K] ρ gas-phase densty, [kgm 3 ] σ Stefan-Boltzmann constant, [Wm 2 K 4 ] Φ gas-phase varable χ root-mean-square error, [ ] Subscrpts and superscrpts g gas phase gg gray gas g volume zone fur furnace m medum p s w Φ partcle, dspersed phase surface zone wall related to general varable Φ Abbrevatons DEA DFA PC PD SGM TEA WCAC drect exchange area drected flux area polynomal coeffcent polynomal degree radatve property model smple gray gas model total exchange area weghtng coeffcent approxmaton accuracy WSGM weghted sum of the gray gases model WSGM1 WSGM, combnaton of one gray gas plus a clear gas WSGM2 WSGM, combnaton of two gray gases plus a clear gas References [1] Flkosk, R. V., et al., Optmsaton of Pulverzed Coal Combuston by Means of CFD/CTA Modellng, Thermal Scence, 10 (2006), 3, pp [2] Constenla, I., et al., Numercal Study of a 350 MWe Tangentally Fred Pulverzed Coal Furnace of the As Pontes Power Plant, Fuel Processng Technology, 116 (2013), Dec., pp [3] Schuhbauer, C., et al., Coupled Smulaton of a Tangentally Hard Coal Fred 700 C Boler, Fuel, 122 (2014), Apr., pp [4] Yn, C., Nongray-Gas Effects n Modelng of Large-Scale Oxy-Fuel Combuston Processes, Energy & Fuels, 26 (2012), 6, pp [5] Nakod, P., et al., A Comparatve Evaluaton of Gray and Non-Gray Radaton Modelng Strateges n Oxy-Coal Combuston Smulatons, Appled Thermal Engneerng, 54 (2013), 2, pp [6] Hu, Y., et al., Numercal Investgaton of Heat Transfer Characterstcs n Utlty Bolers of Oxy-Fuel Combuston, Appled Energy, 130 (2014), Oct., pp [7] Hottel, H. C., Sarofm, A. F., Radatve Transfer, McGraw-Hll, New York, USA, 1967 [8] Taylor, P. B., Foster, P. J., The Total Emssvtes of Lumnous and Non-Lumnous Flames, Internatonal Journal of Heat and Mass Transfer, 17 (1974), 12, pp [9] Rhne, J. M., Tucker, R. J., Modellng of Gas-Fred Furnaces and Bolers, McGraw-Hll, New York, USA, 1991 [10] Coelho, P. J., Numercal Smulaton of Radatve Heat Transfer from Non-Gray Gases n Three- -Dmensonal Enclosures, Journal of Quanttatve Spectroscopy & Radatve Transfer, 74 (2002), 3, pp [11] Lu, F., Numercal Solutons of Three-Dmensonal Non-Grey Gas Radatve Transfer Usng the Statstcal Narrow-Band Model, Journal of Heat Transfer, 121 (1999), 1, pp [12] Park, W. H., Km, T. K., Numercal Soluton of Radatve Transfer wthn a Cubc Enclosure Flled wth Nongray Gases Usng the WSGGM, Journal of Mechancal Scence and Technology, 22 (2008), 7, pp [13] Crnomarkovc, N., et al., Radatve Heat Exchange nsde the Pulverzed Lgnte Fred Furnace for the Gray Radatve Propertes wth Thermal Equlbrum between Phases, Internatonal Journal of Thermal Scences, 85 (2014), Nov., pp [14] Crnomarkovć, N., et al., Influence of Applcaton of Hottel s Zonal Model and Sx-Flux Model of Thermal Radaton on Numercal Smulatons Results of Pulverzed Coal Fred Furnace, Thermal Scence, 16 (2012), 1, pp [15] Crnomarkovc, N., et al., Influence of Forward Scatterng on Predcton of Temperature and Radaton Felds Insde the Pulverzed Coal Furnace, Energy, 45 (2012), 1, pp

10 S206 Crnomarkovć, N. Dj., et al.: Influence of the Gray Gases Number n the Weghted [16] Belosevc, S., et al., Numercal Predcton of Pulverzed Coal Flame n Utlty Boler Furnaces, Energy & Fuels, 23 (2009), 11, pp [17] Sjer~}, M., Mathematcal Modelng of the Complex Turbulent Transport Processes (n Serban), Yugoslav Socety of Thermal Engneers and VINCA Insttute of Nuclear Scences, Belgrade, 1998 [18] Crnomarkovć, N., et al., Numercal Determnaton of the Impact of the Ash Depost on the Furnace Walls to the Radatve Heat Exchange nsde the Pulverzed Coal Fred Furnace, Proceedngs, Power Plants 2014, Zlatbor, Serba, 2014, pp [19] Boow, J., Goard, P. R. C., Fresde Deposts and Ther Effect on Heat Transfer n a Pulverzed-Fuel- Fred Boler. Part III: The Influence of the Physcal Characterstcs of the Depost on ts Radant Emttance and Effectve Thermal Conductance, Journal of the Insttute of Fuel, 42 (1969), pp [20] ***, Combuston, Fossl Power (Ed. J. G. Snger), Combuston Engneerng Inc., Wndzor, Conn., USA, 1991 [21] Kaye, G. W. C., Laby, T. H., Tables of Physcal and Chemcal Constants, Longman, London, 1995 [22] Crnomarkovć, N., et al., Numercal Investgaton of Processes n the Lgnte-Fred Furnace when Smple Gray Gas and Weghted Sum of Gray Gases Models are Used, Internatonal Journal of Heat and Mass Transfer, 56 (2013), 1-2, pp [23] Modest, M. F., Radatve Heat Transfer, Academc Press, New York, USA, 2013 [24] Froberg, C. E., Introducton to Numercal Analyss, Addson-Wesley Publshng Company, London, 1969 [25] Sastry, S. S., Introductory Methods of Numercal Analyss, PHI Learnng Prvate Lmted, New Delh, Inda, 2013 [26] Press, W. H., et al., Numercal Recpes, Cambrdge Unversty Press, New York, USA, 1986 [27] Mech, R., et al., Extenson of the Zonal Method to Inhomogeneous Non-Grey Sem-Transparent Medum, Energy, 35 (2010), 1, pp Paper submtted: June 3, 2015 Paper revsed: December 2, 2015 Paper accepted: December 4, 2015