AIAA-97-2563RADIATION IN A THREE-DIMENSIONAL GAS TURBINE COMBUSTOR
Seung Wook Back* and Man Young Kirn*Department of Aerospace Engineering, Korea Advanced Institute of Science and Technology, 373-1
Kusong-dong, Yusong-ku, Taejon 305-701, Korea
ABSTRACT
The finite-volume method for radiation in a three-dimensional non-orthogonal geometry is combinedhere with geometric relations to be applied to aproblem of gas turbine combustion chamber. Anabsorbing, emitting and anisotropically scatteringmedium is assumed to exist inside. The scatteringphase function is modeled by a Legendrepolynomial series. After a benchmark solution forthree-dimensional rectangular combustor isobtained to validate the present formulation, aproblem in three-dimensional non-orthogonal gasturbine combustor is investigated by changing suchparameters as scattering albedo, scattering phasefunction and extinction coefficient. Forwardscattering is found to produce higher radiative heatflux at hot and cold wall than backward scattering,and the optical thickness is also shown to play animportant role in the problem.
NOMENCLATURE
D™ = drectional weights^x,e ,ez = unit vectorsgq = co variant metric tensor
I = radiation intensity, W/(m2 • srjM = number of total radiation directionni = unit normal vector at i surfaceP(cos *P) = Legendre polynomial of order ;'q^ = radiative heat flux at wall, W/m2
r = position vectorSnr = nonradiative volumetric heat source,
oxides of nitrogen, and sulfur dioxide, that areharmful to both human beings and the environment.Environmental regulations on these species inindustrial and propulsive gas turbine enginesystems have become more stringent due toconcerns about acid rain and depletion ofstratospheric ozone.1'2 In order to successfullypredict these pollutants, thermal radiation effect hasto be taken into account in gas turbine modeling.This is, however, a formidable problem, since theradiative transfer equation (RTE) has to be solved ingas turbine geometry.3 Therefore, a very reliablemethod for solving the RTE in a complex orcurvilinear geometry is highly demanded.
Among several radiation models devised tomeet various engineering problems,4 a specific onehas to be chosen for one's own concern, hi theproblem of gas turbine combustion chamber someaspects are needed to be taken into account such assolution accuracy, easiness, applicability tomultidimensional problem5 as wall as efficiency incomputational time. Among others the zonal6 andMonte-Carlo methods7 seem to be inappropriate formodeling multidimensional enclosures because oflong computational time, complexity of involvedequations, especially incompatibility with otherfinite differenced equations to solve momentumand energy equations. From this point of view,spherical harmonics PN approximation and thediscrete-ordinates method (DOM) are ratherpreferred.
While Menguc and Viskanta8 applied thespherical harmonics P3 approximation to study athree-dimensional rectangular enclosure radiationcontaining inhomogeneous, anisotropically scatter-ing media, Song and Viskanta8 used Pl approxi-mation to investigate the turbulence/radiationinteraction in two-dimensional combustion system.Since Carlson and Lathrop9 developed the DOM tosolve the neutron transport equation, Fiveland10
and Truelove11 applied it to pure radiative heattransfer problem in multidimensional furnacechamber with remarkable accuracy. Fiveland,12
Jamaluddin and Smith,13 and Truelove14 also usedthe DOM to obtain three-dimensional radiative heattransfer solutions. By using S4 approximation, acoupled conductive and radiative heat transfer inrectangular enclosure has been solved by Kim andBaek,15 whereas a problem of diffusion flamebehavior over a combustible solid was examined byKim et al.16 with the Arrhenius type of second-ordersingle-step chemical reaction. However, thesemethods have not yet been applied to curvilineargeometries such as gas turbine combustion chamber.
Recently, the finite-volume method (FVM) forradiation was derived by Chui and Raithby17 andChai et al.18 It has successfully been applied to aproblem of two-dimensional curvilinear cavity. Kimand Baek19 also applied it for analyzing combinedconduction, convection and radiation in a graduallyexpanding channel. Unlike the DOM, the FVM canbe applied to body-fitted coordinates, since theconservation constraint resulting from integratingthe RTE over a control volume and a control angleis satisfied, In this method, the inflow and outflowof radiant energy across control volume faces arebalanced with attenuation and augmentation ofradiant energy within a control volume and acontrol angle. Total solid angle, kn steradians isdiscretized into a finite number of discrete solid(control) angle in any convenient manner,depending on the problem dealt with. Thereby,Chai et al.20 performed accurate prediction ofcollimated incidence by matching its control angleto the collimated beam direction, which could notbe described by the DOM if the coUimated beamdirection differs from ordinate direction set by theDOM.
The objective of this work is to extend theFVM for radiation to solve a problem of three-dimensional gas turbine combustion chamber, ofwhich shape is curvilinear. A non-orthogonal three-dimensional discretization formulation will bederived in the following by using the step schemefor spatial differencing. The scattering phasefunction is modeled by a Legendre polynomialseries as in Kim and Lee.21 After a benchmarksolution for radiation is validated by comparisonwith that in rectangular combustor,5 the RTE issolved in a three-dimensional gas turbinecombustor. Then, its thermal characteristics areinvestigated by changing such parameters asscattering albedo, scattering phase function, and theoptical thickness of the system.
ANALYSIS
Radiative Transfer Equation
A geometrical schematic, in which the RTE forintensity is analyzed, is shown in Fig. 1. Theradiation intensity for gray medium at any position,r , along a path, s through an absorbing, emittingand scattering medium is given by22
where, /?„ = Ka + as is the extinction coefficientand G)O = crJPs is the scattering albedo.<t>(s' — > s\ is the scattering phase function forradiation from incoming direction s' to scattereddirection s . This equation/ if temperature of themedium, 7fc(r) and boundary conditions forintensity are given, provides a distribution of theradiation intensity in medium. The boundarycondition for a diffusely emitting and reflectingwall can be denoted by
" AA; D™ = fi0 (-Im +Srm)pAI/AQm (3a)
w ww
(2)where, ew is the wall emissivity and subscript wdenotes the location of the wall, while nw is theunit normal vector. The above equation illustratesthat the leaving intensity is a summation of emittedand reflected intensities.
The Finite-Volume Method for Radiation
To derive the discretization equation, Eq. (1) isintegrated over a control volume, AV, and acontrol angle, AQm , as shown in Fig. 2. Byassuming that magnitude of intensity is constant,but its direction varies within control volume andcontrol angle given,18 following finite volumeformulation can be obtained
Fig. 1 Geometrical schematic for three-dimensionalgas turbine combustor.
where
,!»+ ff"- ,ff"
An1" = I dQ.m = I IJlK- if- iff"-
(3b)
s = sin 0cos ̂ ex + sin #sin ̂ ey + cos 0ez (3c)
«£ = «*; e* + «y; ?y + Mz/i e2 (3d)
S;=(l-o»0)l t+-?2-L . J-'O-'^-dn' (3e)
(3f)
AA, and AV represent the surface area and controlvolume, respectively, while ni is the outward unitnormal vector at the control volume faces as shownin Fig. 2. This equation indicates that a net outgoingradiant energy across control volume faces must bebalanced by a net generation of radiant energywithin control volume and control angle, hi thisFVM, the directional weights,19 D" should becarefully evaluated, for it determines an inflow oroutflow of radiant energy across the control volumeface, depending on their sign
Next, required is a scheme which relates acontrol volume face intensity to a nodal one.Among many others such as diamond scheme,positive scheme of Fiveland10 variable weightscheme of Jamaluddin and Smith13 and exponentialtype scheme of Carlson and Lathrop,9 Chui andRaithby,17 and Chai et al.,20 here is adopted a stepscheme in which a downstream face intensity is setequal to the upstream nodal value.19-20 It is not onlysimple and convenient, but also ensures positiveintensity while not considering complex geometricand directional information. According to thisscheme, typical relations are as follows
(D:,o)-I*inax(-D*,0) (4a)
) (4b)Similar ones can be also obtained for other faceintensities. This simplicity and convenience,however, may degenerate the solution accuracy. Byusing this scheme, Eq. (3a) can be recast into thefollowing general discretization equation;
In Eq. (5b), subscript 7 represents E, W, N, S,T and B while f does e, w, n, s, t and b,respectively.
The whole domain is discretized intoIN.,, x Ny x Nz I control volumes. Note that ni,
AA,. and AV have to be calculated carefullyaccording to grid skewness, as will be shown in thefollowing section. Total solid angle, 4 n steradiansis divided into (NexN^\ = M directions with
equal A0= 0m+ -0m~ = jr/Ng and A<* = </>m+ -(j>m~= 27T/N,,, where 6 is polar angle and ^ isazimuthal angle, ranging from 0 to n and from 0 to2it, respectively. Boundary condition in Eq. (2) canbe discretized as
Z r' um'w ™ (6)
The iterative solution is terminated when thefollowing convergence is attained
(a) control volume
x'
(b) control angle
Fig. 2 Schematics of control volume and controlangle: (a) control volume, (b) control angle.
max I™ -J (7)where J™ is the previous iteration value of I™ .
Geometric Relations
In order to close the general discretization equation(5), the volume of control volume, AV, surfacearea, AAf, and outward unit normal vector, nt,have to be provided. This geometric relations,which transform the Cartesian coordinate into ageneral body-fitted coordinate, can be obtained inthe same way as in the field of computational fluiddynamics.3'23
The unit normal vector and surface area foreast control volume face can be evaluated as
where
and
g22 = xl+yl+zl (8c)g*=x2r+V2r+*1r <8d)
ga=x^r+y«yr+znzr <8e)is the covariant metric tensor23 and
A^= A 77 = A/ = 1 • Similarly, the unit normal vectorand surface area for north control volume is
where
For the top control volume they are
(9c)(9d)
(lOa)
(lOb)where
Similarly, other unit normal vectors, nw, ns andK f t, and surface areas, &AW, AAS and AAj, can beeasily obtained. The volume of a control volume,which is the Jacobian of the transformation,23 can bedenoted by
The directional weights of Eq.calculated analytically as follows
+ -0m-)-0.25(sin20m+ -
(11)(3b) can be
+0.5nM(sin2 0m+ -sin2 0m-)(<T+ -0-) (12)This would complete the formulation of the finitevolume radiation method in three-dimensionalbody-fitted coordinate system. It is noted that three-dimensional formulations derived here are similarto that of cylindrical formulations24 ifcorresponding geometric relations are provided.
Supplementary Equations
For many engineering applications, there existsnonradiative volumetric heat source, Snr, in themedium. At steady state the divergence of theradiative heat flux, V-^R , which is the net loss ofradiant energy from a control volume,22 has to beequal to this volumetric heat source through thefollowing radiation energy balance equation;5-14
J /o= 2m
Fig. 3 (a) Schematic of the three-dimensionalrectangular combustion chamber; (b) Spatial gridsystem in x-y plane.
nr i 0 \ ^^0/1 P \ / l_ . \ ' / I
The temperature field, T (r), obtained from theabove equation, is then inserted into Eq. (3e),thereby, the intensity field could be calculated. Itmust be noted that in the case of radiativeequilibrium, Sm =0 in the above radiation energybalance equation.
Once the intensity field is obtained, the wallradiative heat flux can be estimated as follows
M
where «„ means ez for the case of axial heat flux.
RESULTS AND DISCUSSIONS
Three-Dimensional Rectangular Combustor
hi order to validate present predictions using thenon-orthogonal FVM, a three-dimensionalrectangular combustor is chosen as studied byMenguc and Viskanta,5 Fiveland,12 Jamaluddin andSmith,13 Truelove14 and Chai et al.20 The modelcombustor shown in Fig. 3(a) contains radiativelyparticipating medium with J30=OJ5m~1 andSnr =5kW/m3 . The hot wall conditions at z = 0are Tw = 1200 K and ew = 0.85, while the cold wallconditions at z = 4m are TW=400K andsw — 0.7. The other wall conditions are atTw = 900 K and £w = 0.7 . After Eq. (5) for intensityis solved with boundary condition (6), the radiationenergy balance equation (13) is unraveled tocalculate the temperature field. The spatial andangular grid systems are (N^ xNy x N z j =
(10x10x20) and (Ne^N^) = (6x8), (10x12)and (14x16), which correspond to the S6, S8
and S14 DOM in the number of radiative directions.In the present computation, a non-orthogonal meshshown in Fig. 3(b) was used to check the effect ofgrid skewness and a uniform mesh wasincorporated in the z — direction.
In Fig. 4 the radiative heat flux distribution isshown at both hot and cold walls and comparedwith other works. Irrespective of refinement ofangular grid system from (6x8) to (14x16), thesolution based on the FVM is in considerably goodagreement with those by the exact zone and S6
discrete-ordinates methods. Figure 5 compares thetemperature profiles at three axial locations. Present
• Menguc and Viskanta (1985)• Truelove (1988)4 Jamaluddin and Smith (1988)
0 0.5 1.0 15
y = l m a n d z = 4 m (COLD WALL)
• 4 • 4 | | 4 •. 4 -•
present FVM——— (6X8) •--- (10X12) ----- (14X16)
0 0.5 1.0 1.5
-
-
2.
-
-
2.
Fig. 4 Comparison of radiative heat flux at hot andcold walls for absorbing, emitting medium.
11UU
1050
1000
950C
1000
950
900C
950
900
850
800
y = 1 m and z= 0.4 m
• a • • e -
• Menguc and Viskanta (1985)• Truelove (1988)+ Jamaluddin and Smith (1988)e ChaietaLCI994c)
.0 05 1.0 15 2
y — 1 m and z = 2 m
«--y-T» i| u f a • • • • —— • .
present FVM
• i i i.0 05 1.0
y = 1 m and z =
Z) - - - - - (14x16)
15 2
3.6m
e "•̂ jfc-.. w w *•— — - s
-
OJO 05 1.0 15 2J)
non-orthogonal finite-volume solutions are in goodagreement with that obtained by zone method aswell as orthogonal finite-volume solutions by Chaiet al.20 It must be noted that Chai et al.20 adopted amore denser spatial and angular grid system, i.e.(N, x Ny x Nz ) x (Ng x N,) = (25 x 25 x 25) x (4 x 20) .
For (Ne xN^l = (6x 8) angular grid system, thesolution takes about 3 min on a HP 712 workstationuntil convergence criterion equation (7) is satisfied,whereas it takes about 18 min with \Ng xN)(] =
(14x16)
Thtee-Dimensional Gas Turbine Combustor
In this section, an applicability of the present non-orthogonal FVM to three-dimensional gas turbinecombustion chamber is sought. The results arepresented by changing parameters such asscattering albedo, scattering phase function andoptical thickness. The test combustor constructedhere is similar to Menguc and Viskanta combustor
Table 1 Physical conditions of the three-dimensionalnon-orthogonal gas turbine combustor.
Dimension (nondimensionalized by y0 )y e = 1.423, yf t=3.78, x0=xe=2, z0=7.6
Wall temperatures and wall emissivitieswall(l) - waU(4) : Tw = 900 K, ew = 0.7wall(5): TW=1200K, £v
wall(6): TW=400K, £W
Properties of the mediumSBr=5.0kW/m3
Fig. 5 Comparison of temperature profiles at three Fi& 6 Grid syste™ for the three-dimensional non-axial locations for absorbing emitting medium. orthogonal gas turbine combustor.
Fig. 7 Effect of angular discretization on heat fluxprofiles at hot and cold walls for the three-dimensional gas turbine combustor.
65
I60
55
HOTWALL(5)
0.0 0.5 1.0x
1.5 2.0
au
1
I2'cp
20o
COLD WALL(6),F2
/^/,iso tropic
/
B2
0 0.5 1.0 1.5 2.*
Fig. 8 Effect of scattering phase function on heatflux profiles at hot and cold walls for the three-dimensional gas turbine combustor.
used above. Table 1 describes basic physicalconditions for the combustion chamber as depictedin Fig. 1. If any condition is changed from them, itwould be mentioned where necessary. In the table itmust be noted that there exists a volumetricheatsource Sm. A spatial grid system of thecombustor is shown in Fig. 6, in which total numberof grids is (N., x Ny x Nzj = (10 x 10 x 20) .
Figure 7 indicates the effect of angulardiscretization on heat flux at hot wall (5) and coldwall (6). This test is conducted since north andsouth bounding walls are curvilinear. For simplegeometries like a rectangular combustor as shownin Fig. 3(a), each control angle is always devised tolie tangent to the bounding wall. However, forcurvilinear geometries like one in Fig. 6, thispractice is impractical or sometimes impossible. Inthe present work, the incoming and outgoingradiant energy at the wall is determined by the signof the directional weights, D ™ . The maximumdifference in the results between fine angular gridsystems, \Ne x N^ j = (14 x 16) and coarse system,
(Ng x N#) = (6x8) is less than 0.6%. Hereafter, thecoarse angular system is adopted since only a slightincrease in accuracy with grid refinement requires asignificant increase in computation time.
Effects of isotropic scattering and scatteringphase function on heat flux profiles at hot and coldwalls are illustrated for co0 = 0.7 in Fig. 8.Mathematically, for an isotropic scattering case withalbedo, «»0 and specified heat generation, Snr inthe medium, the RTE becomes independent ofscattering albedo, which can be easily derived bysubstituting radiation energy balance equation (13)into the RTE (1) with 0> = 1.«
The scattering phase function, whichdescribes the manner in which radiant energy isscattered, is here approximated by a finite series ofLegendre polynomials as follows
;0(s ', s) = O(cos Y) = ]T CtP} (cos Y) (15)
j=0
where *¥ is scattering angle between incidentdirection, s', and scattered direction, s , and C^are the expansion coefficients dependent on the sizeand refractive index of the particle. Forward (F2,F3) and backward (Bl, B2) scattering phasefunctions as given by Kim and Lee21 are consideredhere. Asymmetry factors for the phase function F2,F3, Bl and B2 are 0.670, 0.40, -0.188 and -0.40,respectively. Radiative heat flux at both walls isobserved to be larger for forward scattering case.
Fig. 9 Effect of extinction coefficient on heat fluxprofiles at hot and cold walls for the three-dimensional gas turbine combustor.
900K
400K
=0.1
900K
400K
(b) #,=5.0
Fig. 10 Temperature contour for two differentextinction coefficient at mid-plane : (a) J30 = 0.1, (b)/?0=5.€.
Since the scattering type dearly affects wallradiative heat flux, a more accurate scattering datawould help simulate thermal radiation forpulverized coal, char, fly-ash, soot, liquid dropletand other particles in the combustion chamber.
Figure 9 illustrates the effect of extinctioncoefficient, /?„ on wall heat flux profiles. Physically,the decrease in 00 means an increase in the meanfree path, which is considered to be a characteristicdistance penetrated by radiation. As J30 decreases,the local temperature of radiating medium becomeshigher through the far-reaching effects of radiationas shown in Fig. 10. Therefore, the wall radiativeheat flux is also seen to be higher for the case withsmaller optical thickness.
CONCLUSIONS
Thermal radiation in a shape of gas turbinecombustor is solved by using three-dimensionalnon-orthogonal FVM put together with geometricrelations. A final form of general discretizationequation is coordinate-independent when the unitnormal vector, surface area and volume of thecontrol volume are provided according to thespecific problem dealt with. Anisotropic scatteringis considered by using finite series of Legendrepolynomials. Our formulation is validated byapplying it to a three-dimensional rectangularcombustor containing radiatively participatingmedium. Then, the three-dimensional non-orthogonal finite-volume formulation is applied to aproblem of gas turbine combustion chamber. Theresults show that (1) radiative heat flux at both hotand cold walls is higher for forward scatteringcompared with that for backward scattering; (2) asthe optical thickness decreases, the radiative heatflux at both hot and cold walls increases, since themedium heats up faster.
ACKNOWLEDGMENTS
The financial assistance by the Objective ResearchFund of the Korea Science and EngineeringFoundation (KOSEF 95-0200-05-01-3) is gratefullyacknowledged.
REFERENCES
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