research article extension of weighted sum of gray gas

Research ArticleExtension of Weighted Sum of Gray Gas Datato Mathematical Simulation of Radiative Heat Transferin a Boiler with Gas-Soot Media

Samira Gharehkhani1 Ali Nouri-Borujerdi2 Salim Newaz Kazi1 and Hooman Yarmand1

1 Department of Mechanical Engineering University of Malaya 50603 Kuala Lumpur Malaysia2 School of Mechanical Engineering Sharif University of Technology Tehran 11365-956 Iran

Correspondence should be addressed to Salim Newaz Kazi salimnewazyahoocom

Received 10 December 2013 Accepted 19 January 2014 Published 6 March 2014

Academic Editors V Bubnovich and M Yao

Copyright copy 2014 Samira Gharehkhani et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In this study an expression for soot absorption coefficient is introduced to extend the weighted-sum-of-gray gases data to thefurnace medium containing gas-soot mixture in a utility boiler 150 MWe Heat transfer and temperature distribution of walls andwithin the furnace space are predicted by zone method technique Analyses have been done considering both cases of presence andabsence of soot particles at 100 load To validate the proposed soot absorption coefficient the expression is coupledwith the Taylorand Fosterrsquos data as well as Trueloversquos data for CO

2-H2O mixture and the total emissivities are calculated and compared with the

Trueloversquos parameters for 3-term and 4-term gray gases plus two soot absorption coefficients In addition some experiments wereconducted at 100 and 75 loads to measure furnace exit gas temperature as well as the rate of steam production The predictedresults show good agreement with the measured data at the power plant site

1 Introduction

One of the most important modes of heat transfer in a boilerfurnace of a large power plant is radiation Thus determi-nation of the radiative properties of combustion products isvital to predict the temperature andheat flux distributionThemain products of combustion in an enclosure contain carbondioxide and water vapor [1] Some studies have been doneon modeling of gas mixture in furnaces Different modelsfor calculation of the radiative properties of real gases havebeen proposed by researchers such as statistical narrow band(SNB) the full-spectrum correlated-k (FSCK) distributionand the weighted-sum-of-gray gases (WSGG) [2ndash6] Amongthese models the last one is more reasonable in engineeringcalculations in view of the accuracy and computing timeThe WSGG model was developed by Hottel and Sarofim[1] Modest [7] stated that this model can be applied forany solution methods for the transport equation Based onWSGG concept Taylor and Foster [8] utilized a ldquothree gray

plus one clear gasrdquo model A three-term mixed gray gasmodel with third order polynomial for weighting factorswas employed by Smith et al [9] Soufiani and Djavdan[10] proposed a five-order polynomial for gas combustionwhere 119875

119908119875119888= 2 After that a lot of efforts were given to

model the radiative properties from semitransparent mediacontaining a mixture of nongray gases and soot [11 12] whichwas applied for analyzing the radiative heat transfer insidethe furnace [13] Among the different methods of modelingthe combusting environments such as inverse methodolo-gies hybrid method discrete-ordinates method and finitevolume method [14ndash17] the zone method is more practicaland many attempts on radiative heat transfer analysis wereconducted based on this method [18ndash20] Zone methodwas originally developed by Hottel and Cohen [21] for anabsorbing emitting nonscattering gray gas with constantabsorption coefficient Later Hottel and Sarofim [1] extendedit to deal with three-dimensional problems Also Larsen andHowell [22] presented amethod for calculations related to the

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 504601 9 pageshttpdxdoiorg1011552014504601

2 The Scientific World Journal

direct exchange areas in zonal analysis based on last-squaressmoothing Tucker [23] conducted a numerical integrationand suggested an exponential expression for exchange areaswhich covers a range of optical thickness from 0 to 18Lawson [24] proposed an improved method for smoothingapproximate exchange areas To achieve the total exchangeareas Noble [25] presented the explicit matrix relationsBatu and Selcuk [26] analyzed the radiative heat transferin the freeboard of a fluidized bed combustor by usingthe zone method Bordbar and Hyppanen [27] employedthe zone method for predicting temperature and heat fluxon the water walls of a steam boiler furnace RecentlyMechi and coworkers [19] proposed a radiative model toextend the zonal method to semitransparent inhomogeneouscomposed of nongray gas and soot Also Moghari et al[18] used the zone method to predict thermal radiationbehavior in the D-type water-cooled steam boiler furnaceCrnomarkovic et al [28] used the simple gray gas (SGG) andWSGG to model the radiative properties of the two-phasemixture composed of gas and particles inside the lignite firedfurnace

In this study a new expression for soot absorption coef-ficient has been presented depending on temperature whichcould be coupled with nonluminous flame data containingseveral gray gases and one clear gas The results are basedon the suggested soot absorption coefficient coupled withthe data generated by Taylor and Foster The validity ofthe calculated soot absorption coefficient is confirmed bycomparison with the obtained total emissivities and thecalculated values from Trueloversquos models In addition forreconfirmation of the results the soot expression is utilizedin the zone method to model the furnace of a utilityboiler 150MWe The temperature and heat flux distribu-tions are discussed for 2 cases (with and without sootparticles) at 100 load Furthermore the furnace exit gastemperature and amount of steam production by consid-ering the effect of soot for the loads of 100 75 arepresented and compared with the captured data from thesite

2 Mathematical Model

21 The Weighted Sum of Gray Gas The weighted-sum-of-gray gases (WSGG) is one of the accurate techniques formodeling the radiative behavior of combustion gases Thetotal emissivity of real gas can be represented mathematicallyby a mixture of119873 gray gases [1]

120576119892=

119873

sum

119899=1

119886119892119899

[1 minus exp (minus119870119892119899

(119875) 119871)] (1)

where 119870119892119899 119875 and 119871 represent the absorption coefficient for

the 119899th gray gas sum of the partial pressure of all radiatinggases in the mixture and effective path length respectivelyand 119886

119892119894is weighting factors [5 9 10 12] of various commonly

used correlations for the mixture of combustion productswhich have been reported by Taylor and Foster [8] Smith etal [9] and Soufiani and Djavdan [10]

In fact WSGG is an appropriate tool which could beapplied in the modeling of media containing CO

2 H2O

and soot and in this subject some approaches have beendeveloped to consider the effect of soot particles [12 29]

Based on the suggestion of Truelove [12] for the gas-sootmixture the two absorption coefficients (gas mixture andsoot) are contributed in the calculations The expressions foremissivity of the combustion product-soot mixture (120576

119898) can

be presented by

120576119898=

119873

sum

119899=1

119886119899(119879) [1 minus exp minus119870

119892119899(119875) 119871 minus 119870

119904119862119904119871] (2)

where 119862119904is the soot concentration

In order to determine the soot absorption coefficient arelationship from the wavelength dependence of 119870

119904120582derived

from experimental investigations is [30] as follows

119870119904120582

= 119886120582minus119887 (3)

where 119886 = 271 times 103 and 119887 = 1090

By integrating 119870119904120582

over wavelength we have

119870119904=

1

1205901198794int

infin

0

119870119904120582119890119887120582119889120582 =

1

1205901198794int

infin

0

21205871198621119886120582minus119887119889120582

1205825 (119890(1198622120582119879) minus 1) (4)

Introducing 119911 = 1198622119879120582 into above equation the soot

absorption coefficient becomes

119870119904=2120587119886119862

1(1198791198622)4+119887

1205901198794int

infin

0

1199113+119887

(119890119911 minus 1)

119889119911

=2120587119886119862

1(1198791198622)4+119887

1205901198794Γ (4 + 119887) 120585 (4 + 119887)

(5)

where1198621=3742times 10Wsdot120583m4m2 and119862

2= 14388times 104 120583msdotK

are the first and second Planck function constants respec-tively 120590 = 5669 times 108Wmminus2Kminus4 is Stephane-Boltzmanconstant Γ(119911) is Gamma function and 120585(119911) is Rieman zetafunction The above temperature dependence 119870

119904as deter-

mined from (5) is expressed by following the simple polyno-mial equation

119870119904= minus4337 + 06691119879 + 2 times 10

minus51198792 (6)

where 119879 is the temperature of the radiation source in Kelvin

22 Zonal Method In zone method the enclosure is sub-divided into surfaces and volumes zones which could beassumed isothermal [31]Then by using the gas flow and com-bustion pattern themass flow rate fromto each volume zonegenerated heat by combustion and convection coefficients areobtained A steady state energy balance is considered for eachzone and then a set of simultaneous equations based on thetemperatures and heat fluxes are produced By solving theseequations the temperature and heat flux distributions areobtained

The Scientific World Journal 3

dAj

dVigi

120579j

sj

r

Figure 1 The schematic of volume and surface zone

Calculation of the Direct Exchange Areas and Total ExchangeAreas For finding the radiative heat transfer between twozones the first step is to calculate the direct exchange areas(DEA) and then the total exchange areas (TEA) Thereare three types of DEAs surface-surface volume-surfaceand volume-volume For instant the volume-surface directexchange area as illustrated in Figure 1 can be determined asfollows

119892119894119904119895= int

119881119894

int

119860119895

119870119905cos 120579119895exp (minus119870

119905119903119894119895)

1205871199032

119894119895

119889119881119894119889119860119895 (7)

The DEAs obey the reciprocity definitions where 119904119894119904119895=

119904119895119904119894and 119892

119894119892119895= 119892119895119892119894 Direct numerical integration can be

applied to calculate the respective areasFor the gray gas the total flux between two zones 119894 and

119895 must be proportional to 120590(1198794

119894minus 1198794

119895) and the proportion-

ality constant called the total exchange area is indicatedby 119878119878 119866119878 119866119866 [1] All of these terms are calculated by usingthe methods that have been reported by Hottel and Sarofim[1] and Modest [7]

Direct Flux Areas The radiant energy between any two zonesis proportional to the a-weighted summation of the totalexchange areas for each gas For example the net flux betweenzones 119894 and 119895 is given by [1]

119894119895=

119873

sum

119899=1

[119886119892119899

(119879119894)] (119866119894119878119895)119899119864119892119894

minus

119873

sum

119899=1

[119886119904119899(119879119895)] (119866119894119878119895)119899 times 119864

119904119895

equiv997888997888997888rarr119866119894119878119895119864119892119894minuslarr997888997888997888119866119894119878119895119864119904119895

(8)

where 997888997888997888rarr119866119894119878119895and larr997888997888997888

119866119894119878119895are replacing the terms in the brackets

These are called directed-flux areas [1 29] Similarly expres-sion for surface-surface transfer is

119876119894119895=997888997888rarr119878119894119878119895119864119904119894minuslarr997888997888119878119894119878119895119864119904119895 (9)

And for gas-gas transfer it is expressed by

119876119894119895=997888997888997888rarr119866119894119866119895119864119892119894minuslarr997888997888997888119866119894119866119895119864119892119895 (10)

Total Energy Balance For a volume zone 119894 the total energybalance can be stated by

119897

sum

119895=1

larr997888997888997888119866119894119866119895119864119892119895

+

119898

sum

119895=1

larr997888997888997888119866119894119878119895119864119904119895

minus 4

119873

sum

119899=1

[119886119892119899

(119879119892)119870119892119899119881119894119864119892119894] minus (conv)

119894

+ (119866net + 119886)

119894+ (enth)

119894= 0

(11)

where 119897 and 119898 are the number of volume and surface zonesrespectively119873 is the number of gases in themodel (conv)119894 isthe convection heat transfer to all surfaces in contact with thevolume zone and (enth)119894 is the total sensible heat presentedby

enth = 1198981198941015840rarr119894(119862119875119879)1198941015840 minus 119898119905119894(119862119875119879)119894

(12)

where 1198981198941015840rarr119894

is the mass flow rate of gas entering the zone 119894from a neighboring zone 1198941015840 and 119898

119905119894is representing the total

mass flow rate of gas leaving the zone 119894 Also (119866net + 119886)119894 is

heat released due to combustion plus the heat content in thecombustion air so this term can be expressed [27]

119866net + 119886 =

119866[119862Vnet + 119877119904 (1 +

119883

100) 120588119900

119886(119867119886(119879119886))]

(13)

On the other hand for a surface zone 119894 the total energybalance could be represented by

119898

sum

119895=1

larr997888997888119878119894119878119895119864119904119895+

119897

sum

119895=1

997888997888997888rarr119866119895119878119894119864119892119895

minus 119860119894120576119894119864119904119894+ 119860119894119902119894conv =

119894 (14)

where 119894is heat transfer rate to water walls

Finally the energy balance for the total number of volumeand surface zones generates a series of nonlinear algebraicequations These equations should be solved by the iterativetechniques in order to achieve the temperature distribution inzones In this study the surfaces have been assumed gray andthe combustion is complete in the zones in front of burners


Drum

Reheater

Economizer

Superheater

Burner

Furnace

Figure 2 Overview of boiler

Table 1 Plant operation at 100 load and fuel characteristic

Boiler load (MWe) 150Fuel lower heating value (KjKg) 50000Fuel flow rate (Kghr) 30597Fuel temperature ∘C 30Excess air ratio 5Ambient temperature ∘C 28Fuel chemical composition (Vol)

CH4 907C2H6 62C3H8 21C4H10 1

3 Experimental Facility

The experimental data were obtained from the furnace ofa 150MWe utility boiler Schematic of boiler is shown inFigure 2 The dimensions of the boiler furnace are 92m times

92m times 23m which is equipped with 9 natural gas firedburners in three rows of three They are located at the leftside wall of the furnace chamberThe operating conditions ofthe boiler and fuel characteristics are mentioned in Table 1The experiments were conducted to measure the furnace exitgas temperature by a thermocouple with reasonable accuracy(005 of reading) located adequately far from the last raw ofburners at the furnace outlet at 100 and 75 loads

4 Results and Discussion

The soot absorption coefficient suggested in (6) can becoupled with available models for nonluminous flam withgray gases and one clear gas to obtain the total emissivitiesof gas-soot mixture and to be applied in the zone method

Validation of Presented Soot Absorption Coefficient with Tru-eloversquos Model The total emissivities of gas-soot mixture are

0

01

02

03

04

05

06

07

08

09

1

800 1200 1600 2000 2400

Tota

l em

issiv

ity

Temperature (K)

Present model coupled with four-term Trueloversquos modelFour-gray gas plus two-soot Trueloversquos model (benchmark)Present model coupled with three-term Trueloversquos modelThree-gray gas plus two-soot Trueloversquos model (benchmark)Present model coupled with Taylor model

L = 01m

L = 1m

L = 10m Pc = 01 atmPwPc = 2Cs = 00001 kgm3

Figure 3 Total emissivity based on present calculations andTrueloversquos models for gas combustion 119875

119908119875119888= 2

obtained by coupling the calculated 119870119904by using (6) and

Taylorrsquos data The evaluated total emissivities at the differenttemperatures with soot concentration of 00001 Kgm3 forgas combustion are well compared against benchmark datawhich are ldquothree-gray gas plus two-sootrdquo and ldquofour-gray gasplus two-sootrdquo models suggested by Truelove (Figure 3) Alsoto show the suitability of using the present expression in othermodels the suggested 119870

119904is coupled with Trueloversquos models

without soot which are three-term (two-gray plus one clear)and four-term (three-gray plus one clear) gas models and theresults are presented in Figure 3 as well

Table 2 presents the discrepancies between the computedtotal emissivities by using the coupled models and bench-marks

Figure 4 demonstrates the comparison between the cal-culated gas-soot mixture total emissivity and benchmarkdata The total emissivities are plotted versus path lengthon a logarithmic axis with two different soot concentrations(00001 Kgm3 and 0005Kgm3) for gas at 800 1600 and2400∘K

It is seen that the obtained results coincide with thebenchmarks deviations are acceptable specially in compari-son of two cases of models (i) present model coupled withTaylorrsquos data and (ii) present model coupled with 3-termTrueloversquos data with the Trueloversquos model (three-gray gas plus2-soot) The errors are not greater than 48 percent at 119871 gt

01m for the first case and 2 percent at all path lengths forsecond case for small soot concentration (00001 Kgm3) at1600∘K

Application in the Zone Method The proposed soot absorp-tion coefficient coupled with Taylorrsquos data is used for radiativeheat transfer analysis inside the boiler furnaceThe emissivity


001

01

1

001 01 1 10

Tota

l em

issiv

ity

Path length (m)

Pc = 01 atmPwPc = 2Cs = 00001 kgm3

Cs = 0005 kgm3

T = 800K

(a)

001

01

1

001 01 1 10

Tota

l em

issiv

ity

Path length (m)

Cs = 00001 kgm3

Cs = 0005 kgm3

Pc = 01 atmPwPc = 2

T = 1600K

(b)

Four-gray gas plus two soot Trueloversquos model (benchmark)Present model coupled with Taylorrsquos modelPresent model coupled with four-term Trueloversquos modelree-gray gas plus two-soot Trueloversquos model (benchmark)Present model coupled with three-term Trueloversquos model

001

01

1

001 01 1 10

Tota

l em

issiv

ity

Path length (m)

Cs = 00001 kgm3

Cs = 0005 kgm3

Pc = 01 atmPwPc = 2

T = 2400K

(c)

Figure 4 Total emissivity of the coupled models and benchmarks for different path lengths and soot concentrations

i

kj

Furnace exit

Rare wall

Right wall

Left wall

Front wall

Figure 5 Schematic of the simplified model of the furnace


0

20

40

60

80

100

120

140

160

180

200

220

0 1 2 3 4 5Height index (j)

Hea

t flux

(kW

m2)

(a)

0

20

40

60

80

100

120

140

160

180

200

220

0 1 2 3 4Height index (j)

Hea

t flux

(kW

m2)

(b)

Include sootExclude soot

0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(c)


0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(d)

Figure 6 Heat flux distribution with and without soot particles along the (a) front wall (119894 = 1 119895) (b) front wall (119894 = 2 119895) (c) right wall(119896 = 1 119895) and (d) left wall (119896 = 1 119895)

of tube wall is considered 08 The zones of furnace areobtained by dividing the height (119895 direction) into five equalsections the length (119894 direction) into 2 equal sections and thewidth (119896 direction) into 2 equal sections thus the furnace hasbeen divided into 46 surface and 18 volume zones as shownin Figure 5

The results are based on the effects of existence andabsence of soot particles Tables 3 and 4 show the temperaturedistribution on height direction of front wall at 100 load forcase 1 (excluding soot particles) and case 2 (including sootparticles) It is noticeable that in the results of case 2 the 119870

119904

value is fixed at an average of zones temperature which isobtained from results by excluding soot particlesThe amountof excess air is 5 and soot concentration is 000005Kgm3

The heat flux distribution along walls (front wall rightside wall and left side wall) for both cases with soot andwithout soot is shown in Figure 6 It is observed that existenceof soot raised considerably the heat flux on the wall Similarresult was reported in the literature [19 32ndash34] In fact theexistence of soot enhances the radiative intensity because ofcontinuum radiation in the visible and infrared regions of thewavelength spectrum [35] and in this situation the radiative


Table 2 Comparison of the calculated total emissivities using the coupledmodels with the benchmarks 119879 = 1600∘K and119862

119904= 00001Kgm3

Present model coupled byDiscrepancies () with ldquo3-gas plus 2-sootrdquo

Trueloversquos modelDiscrepancies () with ldquo4-gas plus 2-sootrdquo

Trueloversquos model [12]119871 = 01m 119871 = 1m 119871 = 10m 119871 = 01m 119871 = 1m 119871 = 10m

Taylorrsquos data minus485 minus0679 525 minus671 minus1088 minus2463-term Trueloversquos data minus0768 minus183 249 minus270 minus1192 minus5024-term Trueloversquos data minus514 minus361 minus398 minus699 minus1351 minus364

Table 3 Temperature distribution in furnace with and without sooteffect 119894 = 1 and 119895 = 1 2 5

Height index 119895 Temperature KCase 1 (without soot)

Temperature KCase 2 (with soot)

1 17960 171912 18034 172403 17818 169954 16432 153295 15875 14732




1 17213 162842 17287 163353 17194 162204 16767 15717

transfer is conducted towards the wall and as a result of thistemperature of the medium has reduced

To confirm the obtained results and validate the appliedmathematical model Tables 5 and 6 present a comparisonbetween the calculated data and measured practical dataAvailable results of measurements are furnace exit gas tem-perature and rate of steam production at 100 and 75 loads

The table shows that there is a good agreement betweenthe present results and the experimental data

5 Conclusions

A new soot absorption coefficient proposed in this paperhas been assessed though coupling with WSGG parameterssuggested by Taylor It has been utilized for modeling theradiation heat transfer in a utility boiler 150MWe The totalemissivities are calculated and compared with the Trueloversquosparameters for 3-term and 4-term gray gases plus two-sootabsorption coefficients In addition some experiments wereconducted at 100 and 75 loads to measure furnace exit gastemperature as well as the rate of steam production and thefollowing results are obtained

(1) The soot absorption coefficient model is compatiblewith WSGG models containing gray gases and oneclear gas

Table 5 Comparison of the calculated furnace exit gas temperaturewith the measured data

Load () Gas temperature (K) Discrepancy ()Experimental data Present data

100 1605 14732 minus8275 1465 13969 minus464

Table 6 Comparison of the calculated steam generation rates withmeasured data

Load () 100 75Fuel flow rate (Tonh) 305 229Measured in the site (Tonh) 503 375Calculated data (Tonh) 4925 40158Discrepancy () minus208 minus69

(2) The existence of soot particles leads to a decrease ingas temperature and an increase in wall heat flux

(3) The exhaust gas temperature and steam productioncould be estimated with reasonable accuracy at dif-ferent loads

Nomenclature

119860 Surface area (m2)119886 Weighting factor119862 Mass concentration (Kgm3)1198621 First Planck function constant (W

120583m4m2)1198622 Second Planck function constant (120583m

K)119862119875 Specific heat capacity at constant

pressure (kJkg K)119864 Black body emissive power (Wm2)119892119894119892119895 119866119894119866119895997888997888997888rarr119866119894119866119895 Direct exchange area total exchangearea and flux exchange area for volume119894 to volume 119895 (m2)

119892119894119904119895 119866119894119904119895997888997888997888rarr119866119894119904119895 Direct exchange area total exchange

area and flux exchange area for volume119894 to surface 119895 (m2)

119870 Extinction coefficient of the medium(mminus1)

119870119904 Soot absorption coefficient (m2Kg)

119871 Effective path length (m)119873 Number of gases


119875 Partial pressure of gas (atm) Heat transfer rate (W)119877 Distance between two zones (m)119877119904 Stoichiometric airfuel volume ratio

119904119894119904119895 119878119894119878119895997888997888rarr119878119894119878119895 Direct exchange area total exchangearea and flux exchange area for surface 119894to surface 119895 (m2)

119879 Temperature (K)119881 Volume (m3)119883 Percentage excess air level

Greek Symbols

120572 Gas absorptivity120576 Gas emissivity120576119898 Emissivity of gas-soot mixture

120579 Angle between the beam joining thecenter points of two zones and thenormal to one of the two zones (Rad)

Γ(119911) Gamma function120585(119911) Rieman zeta function120590 Stephane Boltzman constant (Wm2 K4)120588 Density (Kgm3)

Subscripts

119886 Air119892 Gas119894 119895 Surface or volume zone119897 Number of volume119898 Number of surface119899 119899th gray gas119904 Surface or soot

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge High Impact ResearchGrant UMCHIRMOHEENG45 and UMRG GrantRP012D-13AET University of Malaya Malaysia for supportto conduct this research work The authors are also gratefulto the persons in charge in power plant site for their technicalsupport during the experiments

References

[1] H CHottel andA F SarofimRadiative TransferMcGraw-Hill1967

[2] M N Borjini K Guedri and R Saıd ldquoModeling of radiativeheat transfer in 3D complex boiler with non-gray sooting

mediardquo Journal of Quantitative Spectroscopy and RadiativeTransfer vol 105 no 2 pp 167ndash179 2007

[3] V Goutiere F Liu and A Charette ldquoAn assessment of real-gas modelling in 2D enclosuresrdquo Journal of Quantitative Spec-troscopy and Radiative Transfer vol 64 no 3 pp 299ndash326 1999

[4] P J Coelho ldquoNumerical simulation of radiative heat transferfrom non-gray gases in three-dimensional enclosuresrdquo Journalof Quantitative Spectroscopy and Radiative Transfer vol 74 no3 pp 307ndash328 2002

[5] I H Farag ldquoNon-luminous gas radiation approximate emissiv-ity modelsrdquo in Proceedings of the 7th International Heat TransferConference vol 2 pp 487ndash492 Miinchen Germany 1982

[6] H Q Chu F S Liu and H C Zhou ldquoCalculations of gas radi-ation heat transfer in a two-dimensional rectangular enclosureusing the line-by-line approach and the statistical narrow-bandcorrelated-k modelrdquo International Journal of Thermal Sciencesvol 59 pp 66ndash74 2012

[7] M F Modest Radiative Heat Transfer Academic Press 2003[8] P B Taylor and P J Foster ldquoThe total emissivities of luminous

and non-luminous flamesrdquo International Journal of Heat andMass Transfer vol 17 no 12 pp 1591ndash1605 1974

[9] T F Smith Z F Shen and J N Friedman ldquoEvaluation ofcoefficients for the weighted sum of gray gases modelrdquo Journalof Heat Transfer-Transactions of the ASME vol V 104 no 4 pp602ndash608 1982

[10] A Soufiani and E Djavdan ldquoA comparison between weightedsum of gray gases and statistical narrow- band radiationmodelsfor combustion applicationsrdquo Combustion and Flame vol 97no 2 pp 240ndash250 1994

[11] A T Modak ldquoRadiation from products of combustionrdquo FireSafety Journal vol 1 no 6 pp 339ndash361 1979

[12] J S Truelove A Mixed Grey Gas Model for Flame RadiationThermodynamics Division AERE 1976

[13] R Yadav A Kushari A K Verma and V Eswaran ldquoWeightedsum of gray gas modeling for nongray radiation in combustingenvironment using the hybrid solution methodologyrdquo Numeri-cal Heat Transfer Part B vol 64 no 2 pp 174ndash197 2013

[14] S Payan S M H Sarvari and A Behzadmehr ldquoInverseestimation of temperature profile in a non-gray medium withsoot particles between two parallel platesrdquo Numerical HeatTransfer Part A vol 63 no 1 pp 31ndash54 2013

[15] F C Lockwood and N G Shah ldquoA new radiation solutionmethod for incorporation in general combustion predictionproceduresrdquo Symposium (International) on Combustion vol 18no 1 pp 1405ndash1414 1981

[16] A S Jamaluddin and P J Smith ldquoPredicting radiative-transferin rectangular enclosures using the discrete ordinates methodrdquoCombustion Science and Technology vol 59 no 4ndash6 pp 321ndash340 1988

[17] P J Coelho ldquoFundamentals of a newmethod for the solution ofthe radiative transfer equationrdquo International Journal ofThermalSciences vol 44 no 9 pp 809ndash821 2005

[18] M Moghari S Hosseini H Shokouhmand H Sharifi and SIzadpanah ldquoA numerical study on thermal behavior of a D-typewater-cooled steamboilerrdquoAppliedThermal Engineering vol 37pp 360ndash372 2012

[19] R Mechi H Farhat K Guedri K Halouani and R SaidldquoExtension of the zonal method to inhomogeneous non-greysemi-transparent mediumrdquo Energy vol 35 no 1 pp 1ndash15 2010

[20] H Ebrahimi A Zamaniyan J S SMohammadzadeh andAAKhalili ldquoZonal modeling of radiative heat transfer in industrial


furnaces using simplified model for exchange area calculationrdquoApplied Mathematical Modelling vol 37 no 16-17 pp 8004ndash8015 2013

[21] H C Hottel and E S Cohen ldquoRadiant heat exchange ina gas-filled enclosure allowance for nonuniformity of gastemperaturerdquo AIChE Journal vol 4 no 1 pp 3ndash14 1958

[22] M E Larsen and J R Howell ldquoLeast-squares smoothingof direct-exchange areas in zonal analysisrdquo Journal of HeatTransfer vol 18 pp 239ndash242 1986

[23] R J Tucker ldquoDirect exchange areas for calculating radiationtransfer in rectangular furnacesrdquo Journal of Heat Transfer vol108 p 707 1986

[24] D A Lawson ldquoAn improved method for smoothing approxi-mate exchange areasrdquo International Journal of Heat and MassTransfer vol 38 no 16 pp 3109ndash3110 1995

[25] J J Noble ldquoThe zone method explicit matrix relations fortotal exchange areasrdquo International Journal of Heat and MassTransfer vol 18 no 2 pp 261ndash269 1975

[26] A Batu and N Selcuk ldquoModeling of radiative heat transferin the freeboard of a fluidized bed combustor using thezone method of analysisrdquo Turkish Journal of Engineering andEnvironmental Sciences vol 26 no 1 pp 49ndash58 2002

[27] M H Bordbar and T Hyppanen ldquoModeling of radiation heattransfer in a boiler furnacerdquo Advanced Studies in TheoreticalPhysics vol 1 no 12 pp 571ndash584 2007

[28] N CrnomarkovicM Sijercic S Belosevic D Tucakovic and TZivanovic ldquoNumerical investigation of processes in the lignite-fired furnace when simple gray gas and weighted sum of graygases models are usedrdquo International Journal of Heat and MassTransfer vol 56 no 1 pp 197ndash205 2013

[29] T R Johnson and J M Beer ldquoRadiative heat transfer infurnaces further development of the zone method of analysisrdquoSymposium (International) on Combustion vol 14 no 1 pp639ndash649 1973

[30] J M Rhine and R J Tucker Modelling of Gas-Fired Furnacesand Boilers and Other Industrial Heating Processes British Gas1991

[31] R Siegel and J R Howell Thermal Radiation Heat TransferTaylor amp Francis 2002

[32] F Liu H A Becker and Y Bindar ldquoA comparative study ofradiative heat transfer modelling in gas-fired furnaces using thesimple grey gas and the weightedsum-of-grey-gases modelsrdquoInternational Journal of Heat and Mass Transfer vol 41 no 22pp 3357ndash3371 1998

[33] NW Bressloff ldquoThe influence of soot loading on weighted sumof grey gases solutions to the radiative transfer equation acrossmixtures of gases and sootrdquo International Journal of Heat andMass Transfer vol 42 no 18 pp 3469ndash3480 1999

[34] Y-L Hwang and J R Howell ldquoLocal furnace data andmodelingcomparison for a 600-MWe coal-fired utility boilerrdquo Journal ofEnergy Resources Technology vol 124 no 1 pp 56ndash66 2002

[35] T L Farias M G Carvalho and U O Koylu ldquoRadiative heattransfer in soot-containing combustion systems with aggrega-tionrdquo International Journal of Heat and Mass Transfer vol 41no 17 pp 2581ndash2587 1998

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



direct exchange areas in zonal analysis based on last-squaressmoothing Tucker [23] conducted a numerical integrationand suggested an exponential expression for exchange areaswhich covers a range of optical thickness from 0 to 18Lawson [24] proposed an improved method for smoothingapproximate exchange areas To achieve the total exchangeareas Noble [25] presented the explicit matrix relationsBatu and Selcuk [26] analyzed the radiative heat transferin the freeboard of a fluidized bed combustor by usingthe zone method Bordbar and Hyppanen [27] employedthe zone method for predicting temperature and heat fluxon the water walls of a steam boiler furnace RecentlyMechi and coworkers [19] proposed a radiative model toextend the zonal method to semitransparent inhomogeneouscomposed of nongray gas and soot Also Moghari et al[18] used the zone method to predict thermal radiationbehavior in the D-type water-cooled steam boiler furnaceCrnomarkovic et al [28] used the simple gray gas (SGG) andWSGG to model the radiative properties of the two-phasemixture composed of gas and particles inside the lignite firedfurnace

In this study a new expression for soot absorption coef-ficient has been presented depending on temperature whichcould be coupled with nonluminous flame data containingseveral gray gases and one clear gas The results are basedon the suggested soot absorption coefficient coupled withthe data generated by Taylor and Foster The validity ofthe calculated soot absorption coefficient is confirmed bycomparison with the obtained total emissivities and thecalculated values from Trueloversquos models In addition forreconfirmation of the results the soot expression is utilizedin the zone method to model the furnace of a utilityboiler 150MWe The temperature and heat flux distribu-tions are discussed for 2 cases (with and without sootparticles) at 100 load Furthermore the furnace exit gastemperature and amount of steam production by consid-ering the effect of soot for the loads of 100 75 arepresented and compared with the captured data from thesite

2 Mathematical Model

21 The Weighted Sum of Gray Gas The weighted-sum-of-gray gases (WSGG) is one of the accurate techniques formodeling the radiative behavior of combustion gases Thetotal emissivity of real gas can be represented mathematicallyby a mixture of119873 gray gases [1]

120576119892=

119873

sum

119899=1

119886119892119899

[1 minus exp (minus119870119892119899

(119875) 119871)] (1)

where 119870119892119899 119875 and 119871 represent the absorption coefficient for

the 119899th gray gas sum of the partial pressure of all radiatinggases in the mixture and effective path length respectivelyand 119886

119892119894is weighting factors [5 9 10 12] of various commonly

used correlations for the mixture of combustion productswhich have been reported by Taylor and Foster [8] Smith etal [9] and Soufiani and Djavdan [10]

In fact WSGG is an appropriate tool which could beapplied in the modeling of media containing CO

2 H2O

and soot and in this subject some approaches have beendeveloped to consider the effect of soot particles [12 29]

Based on the suggestion of Truelove [12] for the gas-sootmixture the two absorption coefficients (gas mixture andsoot) are contributed in the calculations The expressions foremissivity of the combustion product-soot mixture (120576

119898) can

be presented by

120576119898=

119873

sum

119899=1

119886119899(119879) [1 minus exp minus119870

119892119899(119875) 119871 minus 119870

119904119862119904119871] (2)

where 119862119904is the soot concentration

In order to determine the soot absorption coefficient arelationship from the wavelength dependence of 119870

119904120582derived

from experimental investigations is [30] as follows

119870119904120582

= 119886120582minus119887 (3)

where 119886 = 271 times 103 and 119887 = 1090

By integrating 119870119904120582

over wavelength we have

119870119904=

1

1205901198794int

infin

0

119870119904120582119890119887120582119889120582 =

1

1205901198794int

infin

0

21205871198621119886120582minus119887119889120582

1205825 (119890(1198622120582119879) minus 1) (4)

Introducing 119911 = 1198622119879120582 into above equation the soot

absorption coefficient becomes

119870119904=2120587119886119862

1(1198791198622)4+119887

1205901198794int

infin

0

1199113+119887

(119890119911 minus 1)

119889119911

=2120587119886119862

1(1198791198622)4+119887

1205901198794Γ (4 + 119887) 120585 (4 + 119887)

(5)

where1198621=3742times 10Wsdot120583m4m2 and119862

2= 14388times 104 120583msdotK

are the first and second Planck function constants respec-tively 120590 = 5669 times 108Wmminus2Kminus4 is Stephane-Boltzmanconstant Γ(119911) is Gamma function and 120585(119911) is Rieman zetafunction The above temperature dependence 119870

119904as deter-

mined from (5) is expressed by following the simple polyno-mial equation

119870119904= minus4337 + 06691119879 + 2 times 10

minus51198792 (6)

where 119879 is the temperature of the radiation source in Kelvin

22 Zonal Method In zone method the enclosure is sub-divided into surfaces and volumes zones which could beassumed isothermal [31]Then by using the gas flow and com-bustion pattern themass flow rate fromto each volume zonegenerated heat by combustion and convection coefficients areobtained A steady state energy balance is considered for eachzone and then a set of simultaneous equations based on thetemperatures and heat fluxes are produced By solving theseequations the temperature and heat flux distributions areobtained


dAj

dVigi

120579j

sj

r



119892119894119904119895= int

119881119894

int

119860119895

119870119905cos 120579119895exp (minus119870

119905119903119894119895)

1205871199032

119894119895

119889119881119894119889119860119895 (7)


119904119895119904119894and 119892




119894minus 1198794




119894119895=

119873

sum

119899=1

[119886119892119899

(119879119894)] (119866119894119878119895)119899119864119892119894

minus

119873

sum

119899=1

[119886119904119899(119879119895)] (119866119894119878119895)119899 times 119864

119904119895


(8)




119876119894119895=997888997888rarr119878119894119878119895119864119904119894minuslarr997888997888119878119894119878119895119864119904119895 (9)


119876119894119895=997888997888997888rarr119866119894119866119895119864119892119894minuslarr997888997888997888119866119894119866119895119864119892119895 (10)


119897

sum

119895=1

larr997888997888997888119866119894119866119895119864119892119895

+

119898

sum

119895=1

larr997888997888997888119866119894119878119895119864119904119895

minus 4

119873

sum

119899=1

[119886119892119899

(119879119892)119870119892119899119881119894119864119892119894] minus (conv)

119894

+ (119866net + 119886)

119894+ (enth)

119894= 0

(11)


enth = 1198981198941015840rarr119894(119862119875119879)1198941015840 minus 119898119905119894(119862119875119879)119894

(12)

where 1198981198941015840rarr119894





119866net + 119886 =

119866[119862Vnet + 119877119904 (1 +

119883

100) 120588119900

119886(119867119886(119879119886))]

(13)


119898

sum

119895=1

larr997888997888119878119894119878119895119864119904119895+

119897

sum

119895=1

997888997888997888rarr119866119895119878119894119864119892119895

minus 119860119894120576119894119864119904119894+ 119860119894119902119894conv =

119894 (14)




Drum

Reheater

Economizer

Superheater

Burner

Furnace




CH4 907C2H6 62C3H8 21C4H10 1







0

01

02

03

04

05

06

07

08

09

1

800 1200 1600 2000 2400

Tota

l em

issiv

ity

Temperature (K)


L = 01m

L = 1m



119908119875119888= 2











001

01

1

001 01 1 10

Tota

l em

issiv

ity

Path length (m)


Cs = 0005 kgm3

T = 800K

(a)

001

01

1

001 01 1 10

Tota

l em

issiv

ity

Path length (m)

Cs = 00001 kgm3

Cs = 0005 kgm3

Pc = 01 atmPwPc = 2

T = 1600K

(b)


001

01

1

001 01 1 10

Tota

l em

issiv

ity

Path length (m)

Cs = 00001 kgm3

Cs = 0005 kgm3

Pc = 01 atmPwPc = 2

T = 2400K

(c)


i

kj

Furnace exit

Rare wall

Right wall

Left wall

Front wall



0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(a)

0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(b)


0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(c)


0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(d)




119904





119904= 00001Kgm3








1 17960 171912 18034 172403 17818 169954 16432 153295 15875 14732




1 17213 162842 17287 163353 17194 162204 16767 15717




5 Conclusions





100 1605 14732 minus8275 1465 13969 minus464





Nomenclature














Greek Symbols




Subscripts




Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










dAj

dVigi

120579j

sj

r



119892119894119904119895= int

119881119894

int

119860119895

119870119905cos 120579119895exp (minus119870

119905119903119894119895)

1205871199032

119894119895

119889119881119894119889119860119895 (7)


119904119895119904119894and 119892




119894minus 1198794




119894119895=

119873

sum

119899=1

[119886119892119899

(119879119894)] (119866119894119878119895)119899119864119892119894

minus

119873

sum

119899=1

[119886119904119899(119879119895)] (119866119894119878119895)119899 times 119864

119904119895


(8)




119876119894119895=997888997888rarr119878119894119878119895119864119904119894minuslarr997888997888119878119894119878119895119864119904119895 (9)


119876119894119895=997888997888997888rarr119866119894119866119895119864119892119894minuslarr997888997888997888119866119894119866119895119864119892119895 (10)


119897

sum

119895=1

larr997888997888997888119866119894119866119895119864119892119895

+

119898

sum

119895=1

larr997888997888997888119866119894119878119895119864119904119895

minus 4

119873

sum

119899=1

[119886119892119899

(119879119892)119870119892119899119881119894119864119892119894] minus (conv)

119894

+ (119866net + 119886)

119894+ (enth)

119894= 0

(11)


enth = 1198981198941015840rarr119894(119862119875119879)1198941015840 minus 119898119905119894(119862119875119879)119894

(12)

where 1198981198941015840rarr119894





119866net + 119886 =

119866[119862Vnet + 119877119904 (1 +

119883

100) 120588119900

119886(119867119886(119879119886))]

(13)


119898

sum

119895=1

larr997888997888119878119894119878119895119864119904119895+

119897

sum

119895=1

997888997888997888rarr119866119895119878119894119864119892119895

minus 119860119894120576119894119864119904119894+ 119860119894119902119894conv =

119894 (14)




Drum

Reheater

Economizer

Superheater

Burner

Furnace




CH4 907C2H6 62C3H8 21C4H10 1







0

01

02

03

04

05

06

07

08

09

1

800 1200 1600 2000 2400

Tota

l em

issiv

ity

Temperature (K)


L = 01m

L = 1m



119908119875119888= 2











001

01

1

001 01 1 10

Tota

l em

issiv

ity

Path length (m)


Cs = 0005 kgm3

T = 800K

(a)

001

01

1

001 01 1 10

Tota

l em

issiv

ity

Path length (m)

Cs = 00001 kgm3

Cs = 0005 kgm3

Pc = 01 atmPwPc = 2

T = 1600K

(b)


001

01

1

001 01 1 10

Tota

l em

issiv

ity

Path length (m)

Cs = 00001 kgm3

Cs = 0005 kgm3

Pc = 01 atmPwPc = 2

T = 2400K

(c)


i

kj

Furnace exit

Rare wall

Right wall

Left wall

Front wall



0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(a)

0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(b)


0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(c)


0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(d)




119904





119904= 00001Kgm3








1 17960 171912 18034 172403 17818 169954 16432 153295 15875 14732




1 17213 162842 17287 163353 17194 162204 16767 15717




5 Conclusions





100 1605 14732 minus8275 1465 13969 minus464





Nomenclature














Greek Symbols




Subscripts




Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Drum

Reheater

Economizer

Superheater

Burner

Furnace




CH4 907C2H6 62C3H8 21C4H10 1







0

01

02

03

04

05

06

07

08

09

1

800 1200 1600 2000 2400

Tota

l em

issiv

ity

Temperature (K)


L = 01m

L = 1m



119908119875119888= 2











001

01

1

001 01 1 10

Tota

l em

issiv

ity

Path length (m)


Cs = 0005 kgm3

T = 800K

(a)

001

01

1

001 01 1 10

Tota

l em

issiv

ity

Path length (m)

Cs = 00001 kgm3

Cs = 0005 kgm3

Pc = 01 atmPwPc = 2

T = 1600K

(b)


001

01

1

001 01 1 10

Tota

l em

issiv

ity

Path length (m)

Cs = 00001 kgm3

Cs = 0005 kgm3

Pc = 01 atmPwPc = 2

T = 2400K

(c)


i

kj

Furnace exit

Rare wall

Right wall

Left wall

Front wall



0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(a)

0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(b)


0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(c)


0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(d)




119904





119904= 00001Kgm3








1 17960 171912 18034 172403 17818 169954 16432 153295 15875 14732




1 17213 162842 17287 163353 17194 162204 16767 15717




5 Conclusions





100 1605 14732 minus8275 1465 13969 minus464





Nomenclature














Greek Symbols




Subscripts




Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










001

01

1

001 01 1 10

Tota

l em

issiv

ity

Path length (m)


Cs = 0005 kgm3

T = 800K

(a)

001

01

1

001 01 1 10

Tota

l em

issiv

ity

Path length (m)

Cs = 00001 kgm3

Cs = 0005 kgm3

Pc = 01 atmPwPc = 2

T = 1600K

(b)


001

01

1

001 01 1 10

Tota

l em

issiv

ity

Path length (m)

Cs = 00001 kgm3

Cs = 0005 kgm3

Pc = 01 atmPwPc = 2

T = 2400K

(c)


i

kj

Furnace exit

Rare wall

Right wall

Left wall

Front wall



0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(a)

0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(b)


0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(c)


0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(d)




119904





119904= 00001Kgm3








1 17960 171912 18034 172403 17818 169954 16432 153295 15875 14732




1 17213 162842 17287 163353 17194 162204 16767 15717




5 Conclusions





100 1605 14732 minus8275 1465 13969 minus464





Nomenclature














Greek Symbols




Subscripts




Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(a)

0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(b)


0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(c)


0

20

40

60

80

100

120

140

160

180

200

220


Hea

t flux

(kW

m2)

(d)




119904





119904= 00001Kgm3








1 17960 171912 18034 172403 17818 169954 16432 153295 15875 14732




1 17213 162842 17287 163353 17194 162204 16767 15717




5 Conclusions





100 1605 14732 minus8275 1465 13969 minus464





Nomenclature














Greek Symbols




Subscripts




Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119904= 00001Kgm3








1 17960 171912 18034 172403 17818 169954 16432 153295 15875 14732




1 17213 162842 17287 163353 17194 162204 16767 15717




5 Conclusions





100 1605 14732 minus8275 1465 13969 minus464





Nomenclature














Greek Symbols




Subscripts




Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Greek Symbols




Subscripts




Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article extension of weighted sum of gray gas

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