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THE METHOD OF LINESSOLUTION OF THE DISCRETEORDINATES METHOD FORRADIATIVE HEAT TRANSFER INENCLOSURESNevin Selcuk, Gorkem KirbasPublished online: 29 Oct 2010.

To cite this article: Nevin Selcuk, Gorkem Kirbas (2000) THE METHOD OFLINES SOLUTION OF THE DISCRETE ORDINATES METHOD FOR RADIATIVE HEATTRANSFER IN ENCLOSURES, Numerical Heat Transfer, Part B: Fundamentals: AnInternational Journal of Computation and Methodology, 37:3, 379-392, DOI:10.1080/104077900275459

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Numerical Heat Transfer, Part B, 37:379 –392, 2000Copyright Q 2000 Taylor & Francis1040–7790 rrrrr 00 $12.00 H .00

THE METHOD OF LINES SOLUTION OF THEDISCRETE ORDINATES METHOD FOR RADIATIVEHEAT TRANSFER IN ENCLOSURES

Nevin SelcË uk and Gorkem Kõ rbasËÈDepartment of Chemical Engineering, Middle East Technical University,Ankara 06531, Turkey

( )A radiation code based on the method of lines MOL solution of the discrete ordinates( )method DOM for transient three-dimensional radiative heat transfer in rectangular

( )enclosures for use in conjunction with a computational fluid dynamics CFD code based onthe same approach was developed. Assessment of the predictive accuracy of the code bybenchmarking its steady-state solutions against exact solutions on one- and three-dimen-sional test problems shows that the MOL solution of the DOM provides accurate andcomputationally efficient solutions for radiative heat fluxes and source terms and can beused with confidence in conjunction with CFD codes for transient problems.

1. INTRODUCTION

Recent emphasis on the prediction of transient behavior of turbulent, react-

ing, and radiating flows in most industrial combustion chambers such as gas turbine

combustors, rocket engines, etc., has led to the investigation of accurate and

computationally efficient solution methods of the time-dependent conservation

equations for mass, momentum, energy, and species. Thermal radiation is thepredominant mode of heat transfer in such high-temperature systems. This means

( )that the radiative transfer equation RTE must be solved in conjunction with the( )time-dependent partial differential equations PDEs governing the other transport

processes and that the solution methods must be compatible. In an attempt to

produce a radiation code compatible with the existing standard computational fluid( ) w xdynamics CFD codes, Fiterman et al. 1 proposed a discrete ordinates with time

( )stepping DOTS algorithm and verified its predictive accuracy against exact and

numerical solutions on multidimensional problems with black inner walls contain-

ing cold media.

Standard CFD codes are based on the simplest and most practical approach

in use at present, i.e., the solution of the Reynolds-Averaged Navier-Stoke s( )RANS equations along with a turbulence model. On the other hand, the moststraightforward approach to the solution of turbulent flows is direct numerical

( )simulation DNS in which the governing equations are discretized and solved

numerically for the time development of detailed, unsteady structures in a turbu-

lent flow field without using any turbulence models. However, a lot of grid points

as well as time steps are needed for DNS and hence the computational effort is

Received 1 August 1999; accepted 24 September 1999.

Address correspondence to Prof. Nevin SelcË uk, Department of Chemical Engineering, Middle

East Technical University, Ankara 06531, Turkey. E-mail: selcuk@metu.edu.tr

379

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N. SELCË UK AND G. KIRBASË380

NOMENCLATURE

d slope of the axial temperature curve t optical thicknessy2E emissive power, W m f azimuthal angle

y2 y 1I radiant intensity, W m sr V direction vector of intensityy1k absorption coefficient, ma

zz direction cosine SubscriptsL half of the side length of the

rectangular enclosure , m b black-body

m discrete direction bw black wall

M total number of directions burner burner wall

N order of approximation; number e exit

of grids g gasy2q radiative heat flux, W m i inlet

r position vector max maximum

s distance, m o reference valuey3S radiative energy source term, W m side side wall

T temperature , K w wall

w quadrature weight

x, y, z coordinate axes in Cartesian Superscriptsgeometry

z 9 dimensionless axial distance q leaving

u polar angle y arrivingm , h , j direction cosines for any geometry ; dimensionless

s Stefan-Boltzmann constant m discrete direction

enormous. Use of efficient numerical methods can decrease the computation time

considerably. This can only be achieved in two-ways: the first way is to increase the

order of the spatial discretization method, resulting in high accuracy with less grid

points; and the second way is to use not only a highly accurate but also a stable( )numerical algorithm for the time integration. The method of lines MOL , which

meets the latter requirement, is an alternative approach for time-dependent

problems that does not involve the discretization of all variables. The MOL

consists of converting the system of partial differential equations into an ordinary( )differential equation ODE initial-value problem by discretizing the spatial deriva-

tives together with the boundary conditions via Taylor series and integrating the

resulting ODEs using a sophisticated ODE solver which takes the burden of timediscretization and chooses the time steps in a way that maintains the accuracy and

stability of the evolving solution. The most important advantage of the MOL

approach is that it has not only the simplicity of the explicit methods but also the( )superiority stability advantage of the implicit ones unless a poor numerical

method for the solution of ODEs is employed. In recent years, studies carried out(on a novel CFD code , MOLS4MEE MOL Solution for Momentum and Energy

)Equations , based on the DNS technique , demonstrated that conservation equa-

tions for mass, momentum, and energy for transient problems can be solved

w xsimultaneously with ease by the MOL approach 2 ] 8 .

With the objective of producing a radiation code compatible with

w x ( )MOLS4MEE 7, 8 , use has been made of 1 the discrete ordinates method( )DOM , an accurate and efficient technique for multidimensional steady-state

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METHOD OF LINES 381

w x ( ) w xproblems 9 ] 13 , and 2 the approach suggested by Yucel 14 , in which discreteÈordinates equations are represented in transient forms by using a false-transientsapproach. Recently, the predictive accuracy of the MOL solution of the DOM was

investigated by comparing its steady-state predictions with the analytical solution of

w xthe DOM reported earlier in the literature by SelcË uk and Kayakol 11 on a 1-D

w xslab containing a gray absorbing- emitting medium with uniform temperature 15 .

Encouraging agreement between the two has led to the validation of MOL solution( )of the DOM and the finite-volume method FVM , originally proposed by Raithby

w x w xand Chui 16 , against the exact solution of the RTE 17 on the same test problem.

Comparisons for the same angular and spatial subdivisions show that the MOL

solution of the DOM is superior to the FVM, particularly in optically thin media

w x15 .

In the present article , a three-dimensional radiation code based on the MOL

solution of the DOM was developed and subjected to a more challenging test case ,a box-shaped enclosure problem with black walls bounding a gray absorbing- emit-

ting medium with steep temperature gradients, typical of operating furnaces and

combustors. The method is evaluated from the viewpoints of predictive accuracy

and computational economy by comparing its steady-state predictions with the

w xexact numerical solutions available for the same test problem 18 . The validation

of the method, the DOM, and the FVM against the exact solution of the RTE on aone-dimensional test problem is also presented for the sake of integrity.

2. DESCRIPTION OF THE METHOD

The physical situation to be considered is that of a three-dimensional rectan-gular enclosure containing an absorbing- emitting, nonscattering, radiatively gray

medium whose absorption coefficient is the same at all points and of known

magnitude . Values of black-body emissive power are assumed to be available at all

points within the enclosed medium and at all points on the interior bounding

surfaces of the enclosure , which are assumed to be black.

( )2.1. Radiative Transfer Equation RTE

For the problem under consideration, the RTE takes the following form:

dI( ) ( ) ( ) ( ) ( ) ( ) ( )s V ? = I r , V s yk r I r , V q k r I r 1a a bds

( ) ( )where I r, V is the radiation intensity at position r in the direction V . k r is thea( ) ( 4( ) )absorption coefficient of the medium and I r ’ s T r r p is the black-bodyb

radiation intensity at the temperature of the medium. The expression on the

left-hand side represents the change of the intensity in the specified direction V .

The first term on the right-hand side is attenuation through absorption , and the

second term represents augmentation due to emission.

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N. SELCË UK AND G. KIRBASË382

As the surface bounding the medium is black at a specified temperature , Eq.( )1 is subject to the boundary condition

s T 4g

( ) ( ) ( )I r s I r s 2w b w p

( )where I r is the black-body intensity at the walls.b w

Once the radiation intensities are evaluated by using the RTE together withits boundary condition, quantities of interest such as radiative flux and energy

source term distributions can be readily evaluated.

The net radiative heat flux on a wall is calculated as

q y ( )q s q y q 3ne t

where q q and qy are leaving and arriving wall heat fluxes, respectively. For anabsorbing- emitting gray medium bounded by black walls, q q and qy can be

written as

q 4 ( )q s p I s s T 4bw w

y ( ) ( )q s z z I r , V d V 5H2 p

where zz represents the direction cosines.

The radiative energy source term distribution for problems where prescribed

temperature distributions are available is expressed as

( ) ( )S s k 4p I y I r , V d V 6Ha b( )4 p

( )2.2. Discrete Ordinates Method DOM

The discrete ordinates representation of RTE for an absorbing-emitting gray( )medium in a rectangular coordinate system see Figure 1 takes the following form:

­ I m ­ I m ­ I mm ( )m q h q j s yk I q k I 7m m m a a b­ x ­ y ­ z

Figure 1. Coordinate system.

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METHOD OF LINES 383

m w ( )x ( )where I ’ I x , y, z ; m , h , j is the radiation intensity at position x , y, z inm m m( )the discrete direction V m s cos u , h s sin u sin f , j s sin u cos f , m de-m m m m

notes the discrete direction, m s 1,2, . . . , M , and M is the total number of

directions used in the approximation. The resulting set of differential equations form ( )I is solved together with the boundary condition, Eq. 2 , for each discrete

direction.

Once the intensity distribution is determined, the incident radiative heat flux( ) ( )and source term described by Eqs. 3 ] 6 can be obtained from the following

equations:

q y m ( )q s q y q s p I y w z z I 8pnet bw mm

m ( )S s k 4 p I y w I 9pa b m p( )m

( ) ( )As can be noted from Eqs. 7 ] 9 , the accuracy of the discrete ordinatesmethod depends on the choice of the angular quadrature scheme , the order of

approximation, and the spatial differencing scheme adopted for the solution.

( )2.3. Method of Lines MOL Solution of DOM

The solution of discrete ordinates equations with the MOL approach iscarried out by the adoption of a false transients approach which involves addition

of the time derivative of intensity into the discrete ordinate equation for each

w x ( )direction 14 . Applying this approach to Eq. 7 and rearranging yields

­ I m ­ I m ­ I m ­ I mm ( )s y m y h y j y k I q k I 10m m m a a b­ t ­ x ­ y ­ z

The PDE system for all discrete directions is transformed into an ODE

initial-value problem by using the method of lines approach. The transformation is

carried out by using any discretization technique , such as Taylor series, for thespatial derivative. Starting from an initial condition for radiation intensities in all

discrete directions, the resulting ODE system is integrated until steady state by

using a powerful ODE solver. The ODE solver takes the burden of time discretiza-

tion and chooses the time steps in a way that maintains the accuracy and stability

of the evolving solution. Any initial condition can be chosen to start the integra-

tion, as its effect on the steady-state solution decays to insignificance.Once the steady-state intensities at all grid points are available , the net

radiative heat fluxes on enclosure boundaries and radiative energy source terms at( ) ( )interior grid points can be evaluated by using Eqs. 8 ] 9 .

3. PERFORMANCE OF MOL SOLUTION OF DOM

The MOL solution of the DOM was evaluated on two test cases for which

analytical and exact numerical solutions are available , the first being a one-dimen-

w xsional slab containing an absorbing- emitting medium at uniform temperature 17 ,

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N. SELCË UK AND G. KIRBASË384

and the second being the three-dimensional rectangular enclosure problem of

w xSelcË uk 18 with steep temperature gradients.As the MOL is only a transient solution technique used for the solution of

discrete ordinates equations, basic parameters affecting the accuracy of the DOM,

i.e., order of approximation, angular and spatial discretization schemes, also

determine the accuracy of the MOL solution. For implementation of the DOM in

this study, the S angular quadrature scheme proposed by Carlson and LathropN

w x19 was selected. The choice was based on an assessment study carried out by

w xSelcË uk and Kayakol 9 for the same three-dimensional test problem. Therefore ,( )only the effect of order of approximation N s 2,4,6,8 and spatial discretization

and its order on the solution accuracy were investigated in this study.

3.1. One-Dimensional Slab with an Absorbing-Emitting Medium atUniform Temperature

The physical situation is two infinitely large , parallel , cold, and black plates( )bounding an absorbing-emitting nonscattering gray medium with uniform absorp-

tion coefficient at uniform temperature.

The predictive accuracy of the method for varying optical thicknesses be-

tween 0.1 and 3.0 is examined by comparing its steady-state solutions with analyti-cal solutions of the DOM and predictions of the FVM reported earlier in the

w xliterature 15 on the same test problem, for which an exact solution exists.

Comparisons are based on the same angular and spatial subdivisions for all

methods. For polar angle range , 0 ( u ( p r 2, S to S angular quadrature2 8

schemes in the MOL solution of the DOM correspond to 1 to 4 control angles in

the FVM, respectively, so that the number of solid angles in the FVM is equal tothe number of discrete directions in the MOL solution of the DOM.

w xBased on the findings of the previous study 15 , the incident wall heat fluxes(of the MOL solution of the DOM calculated by using LSODE Livermore Solver

) w xfor ODEs 20 as quality ODE solver and a three-point upwind differencing( )scheme DSS014 as spatial discretization technique will only be displayed in what

follows.Incident radiative fluxes calculated for all methods were normalized by

dividing them by s T 4. Comparisons are presented in terms of absolute percentageg

error in dimensionless incident radiative flux, defined as the ratio of the absolute

value of the difference between the exact solution and the predicted values to the

exact solution, multiplied by 100. Figure 2 shows variation of absolute percentageerror with number of solid angles for 61 grids and the DSS014 approximation for

three different optical thicknesses. As can be seen from the figure , error decreases

with the increasing number of solid angles for all methods and for all optical

thicknesses. It can also be noted that the MOL provides better predictive accuracy

than the FVM and that the largest discrepancy between the MOL and FVM( )solutions appears in the optically thin case t s 0.1 , where the error of the MOL

is two orders of magnitude smaller than that of the FVM for four solid angles,

which is the finest angular subdivision employed in this study.

Variation of absolute percentage error in the dimensionless incident flux

predicted by the MOL, DOM, and FVM with spatial discretization for four solid

angles and using DSS014 for various optical thicknesses is illustrated in Figure 3. It

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METHOD OF LINES 385

Figure 2. Variation of solution error of MOL, DOM, and FVM with number of solid angles for( )different optical thicknesses DSS014 with 61 grids .

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N. SELCË UK AND G. KIRBASË386

Figure 3. Variation of solution error of MOL, DOM, and FVM with spatial discretization for( )various optical thicknesses S and DSS014 .8

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METHOD OF LINES 387

can be seen that accuracies of both the MOL solution of the DOM and FVM

increase with number of grids utilized and that only slight improvement is obtainedafter a certain number of grids for both methods. Predictions of the DOM are not

expected to be a function of spatial discretization, as it provides analytical solution.

CPU time comparison between the methods is not possible , as the MOL

provides a transient solution and only its steady-state solution is compared with

those of the FVM and DOM.

3.2. Three-Dimensional Rectangular Enclosure Problem Containingan Absorbing-Emitting Medium with Steep Temperature Gradients

The accuracy of the MOL solution of the DOM for three-dimensionalradiative heat transfer has been assessed by applying this radiation model to the

prediction of distributions of radiative flux and energy source term of a highly

demanding rectangular enclosure problem with a steep temperature gradient and a

nonscattering medium of small optical thickness and by comparing its predictions

w xwith exact solutions produced previously 18 . The rectangular enclosure under

consideration has interior black walls and an absorbing-emitting medium of con-stant properties. The problem was based on data taken from a large-scale experi-

mental furnace with steep temperature gradients, typical of operating furnaces.

Table 1 shows the complete dimensionless data used as input data for both the

exact solution and the MOL solution of the DOM. The temperature of the gases at( ) ( )any point within the enclosure were calculated by using Eqs. 11 and 20 together

( ) ( ) w xwith either of Eqs. 26 and 29 of 18 , depending on the position of the pointunder consideration. A detailed description of the data can be found elsewhere

w x18 .

In the discussion that follows, all physical quantities are expressed in dimen-

sionless forms, which are obtained by dividing them by the shortest dimension of

the enclosure or by the maximum emissive power of the gas, depending on the

quantity.In implementation of the MOL solution of the DOM to the three-dimen-

sional enclosure problem, S for the angular quadrature scheme , DSS014 for theN

order of spatial discretization were utilized. As for time integration, two quality

ODE solvers, LSODE and RKF45, were evaluated from the viewpoints of numeri-

Table 1. Dimensionless data fed to the exact solutionsa

Ä Ä ÄDimensions of the furnace L s 1, L s 1, L s 6x y z

Optical thickness t s 1 r 6o

Ä Ä( ) ( )Wall black-body intensities I s 0.0020; I s 0.0574;bw side bw burner

Ä( )I s 0.0167bw end

Ä Ä ÄGas temperatures T s 0.1775, T s 0.6222, T s 1i e maxXPosition of the peak z s 0.8max

Slope of gas temperature d s y0.220e

aReference values used to make the experimental data dimensionless:

L s 0.48 m, T s 1,673 K, E s 4.4419 = 105 W my2 , I s 1.4139 = 10 5o o o o

W my2 sry1 .

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N. SELCË UK AND G. KIRBASË388

cal accuracy and computing time. It was found that both result in the same

predictive accuracy, with LSODE requiring 30% more CPU time than RKF45. This

was considered to be due to the implicit nature of LSODE as opposed to the

explicit character of RKF45. This assessment led to the choice of RKF45 for the

three-dimensional enclosure problem.

The accuracy and computational economy of the MOL solution of the DOM( )with respect to order of approximation S to S and spatial discretization2 8

(4 = 4 = 24 and 12 = 12 = 72 control volumes in the x , y, and z directions,)respectively were investigated by comparing its steady-state predictions with exact

point values of radiation variables and illustrated in condensed form in Table 2. As

can be seen from the table , S is too inaccurate to use , and percentage errors do2

not improve significantly with the order of approximation beyond S for both4

spatial discretizations. With regard to the effect of spatial discretization, errors

decrease as mesh size is reduced for all orders except S , while CPU time increases2

considerably. However, the improvement in accuracy is not significant enough to

compensate for the orders-of-magnitude increase in CPU time. Hence , from the

viewpoints of accuracy and CPU time , the use of S and 4 = 4 = 24 control4

volumes was found sufficient for the problem under consideration.

It can also be noted from the same table that the use of finer mesh yields

larger discrepancies between the predicted heat fluxes and source terms and the

exact solutions for the coarsest angular discretization, S . This implies that angular2

and spatial discretizations are intimately linked and optimization has to be achieved

for reasonable accuracy.

The CPU times reported in Table 2 correspond to IBM Risc r 6000 model

590.

Figure 4 illustrates the comparison between the exact values of dimensionless

heat fluxes to the side wall and those predicted by the MOL solution of the DOM( )for S and 4 = 4 = 24 control volumes. Points on the lines x s 1, y s 0.25 , z andÄ Ä Ä4

( )x s 1, y s 0.75, z represent points near the center of the face and near theÄ Ä Ä

Table 2. Comparison between the exact values and the MOL solution of the DOM predictions of

dimensionless heat flux and source term distributions for medium grid points using DSS014

Dimensionless Dimensionless

flux densities source terms

Average Average

absolute Maximum absolute Maximum

Number of Order of percentage percentage percentage percentage CPU

control volumes approximation error error error error time, s

4 = 4 = 24 S 8.58 y16.95 1.67 y4.92 592

S 5.03 y16.73 2.36 y10.63 1404

S 5.04 y15.49 3.06 y12.60 3316

S 5.37 y16.03 2.19 y8.44 5378

12 = 12 = 72 S 19.81 y42.12 2.53 12.41 7,2962

S 3.52 y8.69 0.58 3.03 11,1454

S 3.79 y8.73 0.52 y2.15 22,4176

S 3.67 y9.94 0.69 2.20 28,9588

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METHOD OF LINES 389

Figure 4. Comparison between the exact values and the MOL solution of DOM predictions of

dimensionless flux densities to the side wall.

corner of the face , respectively. As can be seen from the figure , the heat fluxes

predicted by the MOL solution of the DOM are found to be in good agreement

with the exact solutions over the whole length of the enclosure.

Figure 5 shows the comparison between dimensionless source term distribu-

tions predicted by the MOL solution of the DOM and exact values for grid points( ) ( ) ( )x s 0.25 , y s 0.25 , z , x s 0.75 , y s 0.25 , z , and x s 0.75 , y s 0.75 , z , whichÄ Ä Ä Ä Ä Ä Ä Ä Ärepresent the points at the center of the row of control volumes nearest to the

furnace axis, points nearer to the side wall , and near the corner of the enclosure ,

respectively. It can be seen that the trend of the distributions predicted by the

MOL solution of the DOM is the same as that of the exact distributions and that

the trends of the source term profiles comply with those of temperature profiles,

displaying steep gradients near the furnace axis and nearly uniform distributions

close to the side walls. As can also be seen from the figure , source terms predicted

by the MOL solution of the DOM are in perfect agreement with the exact

solutions.

To provide a global check on the accuracy of the predictions of the MOL

solution of the DOM, the total rate of removal of radiative energy through walls

and the total rate of generation of radiative energy within the enclosed medium

w xwere evaluated and compared with the exact values 18 . Percentage errors in

generated and removed radiative energy were found to be y1.66 and 1.34,

respectively, implying that the solution produces consistent results, although slightly

different from the exact values.

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N. SELCË UK AND G. KIRBASË390

Figure 5. Comparison between the exact values and the MOL solution of DOM predictions of

dimensionless radiative energy source terms.

4. CONCLUSIONS

Predictive accuracy of a radiation code based on the MOL solution of the

DOM for transient three-dimensional radiative heat transfer in rectangular enclo-

sures was tested by first applying it to the prediction of radiative heat transfer

between two infinitely large , cold, black plates that enclose a gray absorbing- emit-

ting medium with uniform temperature and benchmarking its steady-state solutionstogether with analytical solution of the DOM and solutions of the FVM against

exact solutions of the RTE on the same problem. Comparative testing was carried

out by utilizing the same number of angular and spatial subdivisions for all

methods. Comparisons show that the MOL technique is significantly superior to

the FVM when the medium is optically thin.

The code was then assessed from the viewpoints of predictive accuracy andcomputational economy by applying it to a three-dimensional rectangular enclo-

sure problem with black walls enclosing an absorbing-emitting gray medium with

steep temperature gradients typical of operating furnaces and comparing its

steady-state solutions against exact solutions available for the same test problem.

Effects of order of approximation and spatial discretization on the predictive

accuracy of the MOL solution of the DOM were investigated. Comparisons showthat the MOL solution of the DOM with S approximation for order 4 = 4 = 244

control volumes for spatial discretization, and DSS014 for its order provides

accurate and computationally efficient solution for radiative heat fluxes and source

terms for the three-dimensional problem under consideration. Therefore , the

method can be recommended for use in conjunction with CFD codes for transient

problems.

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METHOD OF LINES 391

REFERENCES

1. A. Fiterman, R. Ben-Zvi, and A. Kribus, DOTS: Pseudo-Time Stepping Solution of the

Discrete Ordinate Equations, Numer. Heat Transfer B , vol. 35, pp. 163 ] 183, 1999.

2. O. Oymak and N. SelcË uk, MOL vs FDM Solutions of an Unsteady Viscous Flow

Problem, Proc. 8th Int. Conf . on Numerical Methods in Laminar and Turbulent Flows,

VIII, Swansea, U.K., vol. 1, pp. 153 ] 162, 1993.

3. O. Oymak and N. SelcË uk, Method of Lines Solution of Time-Dependent Two-Dimen-

sional Navier-Stokes Equations, Int. J. Numer. Meth. Fluids, vol. 23, pp. 455 ] 466, 1996.

4. O. Oymak and N. SelcË uk, Transient Simulation of Internal Separated Flows Using an

Intelligent Higher-Order Spatial Discretization Scheme, Int. J. Numer. Meth. Fluids,

vol. 24, pp. 759 ] 769, 1997.

5. O. Oymak, The Method of Lines Solution of Time-Dependent 2D Navier-Stokes

Equations for Incompressible Separated Internal Flows, Ph.D. thesis, Middle East

Technical University, Ankara, Turkey, 1997.

6. O. Oymak and N. SelcË uk, The Method of Lines Solution of Time-Dependent 2D

Navier-Stokes Equations Coupled with Energy Equation, Proc. Int. Symp. on Advancesin Computational Heat Transfer, CË esË me, Turkey, pp. 256 ] 264, Begell House , New York,

1998.

7. N. SelcË uk and O. Oymak, A Novel Code for the Prediction of Transient Flow Field in a

Gas Turbine Combustor Simulator, Proc. NATO r RTA r AVT Symp. on Gas TurbineEngine Combustion, Emissions and Alternative Fuels, Lisbon, Portugal, NATO r RTO

Meeting Proceedings 14, 11 r 1 ] 10, 1999.

8. T. Tarhan, Numerical Simulation of Flow through Inlet Flues of Heat Recovery Steam

Generators, M. S. thesis, Middle East Technical University, Ankara, Turkey, 1999.

9. N. SelcË uk, and N. Kayakol, Evaluation of Angular Quadrature and Spatial Differencing

Schemes for Discrete Ordinates Method in Rectangular Furnaces, Proc. 31st Natl. HeatTransfer Conf., ASME HDT-Vol. 325 , Houston, Texas, USA, vol. 3, pp. 151 ] 158, 1996.

10. N. SelcË uk and N. Kayakol, Evaluation of Discrete Transfer and Discrete Ordinates

Methods for Radiative Heat Transfer in Rectangular Furnaces, Invited Lecture, Proc.

Int. Symp. on Glass Problems, ICG, vol. 1, pp. 255 ] 276, 1996.

11. N. SelcË uk and N. Kayakol, Evaluation of Discrete Ordinates Method for Radiative

Transfer in Rectangular Furnaces, Int. J. Heat Mass Transfer, vol. 40, pp. 213 ] 222, 1997.

12. N. SelcË uk and N. Kayakol, Assessment of Discrete Ordinates Method for Radiative

Transfer in Cylindrical Furnaces, Proc. 1st Int. Symp. on Radiative Heat Transfer,

ICHMT, KusË adasõ , Turkiye, pp. 221 ] 236, Begell House , New York, 1998.È13. N. Kayakol and N. SelcË uk, Effect of Angular Quadrature Schemes on the Predictive

Accuracy of Discrete Ordinates Method for Radiative Transfer, CHMT 99 , Proc. Int.Conf . on Computational Heat and Mass Transfer, Gazimagusa, Northern Cyprus, pp.Ï347 ] 357, Eastern Mediterranean University Printing House , Gazimagusa, NorthernÏCyprus, 1999.

14. A. Yucel, Solution of the Discrete Ordinates Equations for a Radiatively ParticipatingÈMedium by the Method of Lines, in Advances in Computer Methods for Partial Differen-

tial Equations VII , pp. 838 ] 844, 1992.

15. N. SelcË uk, G. Kõ rbasË , and T. Tarhan, Evaluation of Method-of-Lines Solution of

Discrete Ordinates Method and Finite Volume Method in a Planar Medium, CHMT 99 ,

Proc. Int. Conf . on Computational Heat and Mass Transfer, Gazimagusa, NorthernÏCyprus, pp. 358 ] 364, Eastern Mediterranean University Printing House , Gazimagusa,ÏNorthern Cyprus, 1999.

16. G. D. Raithby and E. H. Chui, A Finite-Volume Method for Predicting a Radiant Heat

Transfer in Enclosures with Participating Media, J. Heat Transfer, vol. 112, pp. 415 ] 423,

1990.

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

0:05

23

Nov

embe

r 20

14

N. SELCË UK AND G. KIRBASË392

17. J. S. Truelove , An Evaluation of the Discrete Ordinates Approximation for Radiative

Transfer in an Absorbing, Emitting, and Scattering Planar Medium, HTFS Rep. No.

R8478 , 1976.

18. N. SelcË uk, Exact Solutions for Radiative Heat Transfer in Box-Shaped Furnaces, J. HeatTransfer, vol. 107, pp. 648 ] 655, 1985.

19. B. G. Carlson and K. D. Lathrop, Transport Theory } The Method of Discrete Ordi-( )nates, in H. Greenspan, C. N. Kelber, and D. Okrent eds. , Computing Methods in

Reactor Physics, pp. 165 ] 266, Gordon & Breach, New York, 1968.

20. A. C. Hindmarsh, LSODE and LSODI, Two New Initial Value Ordinary Differential

Equations Solvers, SIGNUM Newslett., vol. 15, no. 4, pp. 10 ] 11, 1980.

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

0:05

23

Nov

embe

r 20

14