### SOLUTION OF RADIATIVE TRANSFER EQUATION I

#### A PARABOLIC FORMULATION OF THE DISCRETES ORDINATES METHOD FOR THE TREATMENT OF COMPLEX GEOMETRIES

R. Koch*+, W. Krebs*, S. Wittig*, R. Viskanta**
*Lehrstuhl und Institut fü Thermische Strömungsmaschinen
Universität Karlsruhe, Karlsruhe, Germany
**School of Mechanical Engineering
Purdue University, West Lafayette, U.S.A.

Among the various methods proposed for numerically solving the radiative transfer equation, the discrete ordinates method is presently judged as one of the most promising. The usual procedure in the method is to solve one first order differential equation for each of the discrete directions. The numerical treatment of these first order differential equations is well known for its convenient programming and its small computer memory requirements. However, this approach possesses some major shortcomings. The most serious one is due to the nature of the differential equations (first order, hyperbolic type). The method is difficult to implement in advanced finite volume or finite element codes for combusting flows which are designed to handle complex geometries.

In order to overcome this disadvantage, an alternate methodology based on the even parity formulation of the discrete ordinates equations is proposed. This approach leads to a set of second order differential equations as governing transfer equations. The equations are of the parabolic type and their structure is formally similar to the differential equation describing a diffusion process. Hence, this formulation of the discrete ordinates equations is compatible with the numerical structures employed by computer codes for combusting flows.

The parabolic formulation of the discretes ordinates method has been implemented in a computer program for three-dimensional combusting flows developed at the University of Karlsruhe. The code is based on a finite volume formulation and can handle complex geometries by using body-fitted, non-orthogonal grids.

The major objective of the paper is to demonstrate the capabilities of the method.As basic evaluation, sample calculations of benchmark solutions are presented. The test cases are defined on a Cartesian coordinates system. The results reveal that the achievable accuracy of the method is comparable to the conventional discrete ordinates method.

With respect to the application of curvilinear, bodyfitted grids, special emphasis is placed on the various effects encountered when a non-orthogonal grid is used. As a typical example, the rotation of the grid with respect to the principal coordinates system has been studied. It was found to have an unexpected effect on the radiative flux distribution. A close examination reveals that this effect is related to the directional biasing inherent to the angular quadrature schemes.

Finally, ray effects which are well known to be present in the conventional discrete ordinates method were studied. It was found that they are also present within the parabolic formulation of the discretes ordinates method and that they may be suppressed by applying higher order quadrature schemes.

+ Presently visiting Assistant Professor at Purdue University

#### RADIATIVE-CONDUCTIVE HEAT TRANSFER IN AXISYMMETRIC SEMI-TRANSPARENT SHELLS USING THE DISCRETE ORDINATES METHOD

Rodolphe VAILLON, Michel LALLEMAND and Denis LEMONNIER
Laboratoire d'Etudes Thermiques (URA CNRS 1403)
ENSMA, Site du Futuroscope, BP 109
86960 Futuroscope, France

Since pioneering works of the 60s (1-2), the Discrete Ordinate Method (DOM) has become of a widespread use in the heat transfer community during the last decade. However, it has so far mainly been applied to cartesian geometries and its extension to curvilinear systems is still limited to cylindrical or spherical coordinates (3-4-5). In particular, we are not aware of any previous works where the DOM is implemented for general orthogonal curvilinear coordinates.

One of the major difficulties of the DOM in non cartesian geometries is the so-called angular redistribution which appears in the RTE when expressed in a given direction and then projected over the coordinate axes. Extra directional terms arise from the fact that the orientation of the local system varies with the position in the spatial frame. As a consequence, a same direction is seen under carious angles at different locations. Furthermore, for non cartesian geometries, the integration of the RTE over a control volume may lead to incorrect results since the space derivative of a constant intensity will in general not give zero (6). Thus, it requires a specific task to formulate the discrete ordinate method in curvilinear coordinates.

A general expression for the pathlength derivative of radiation intensity is given for spatial directional orthogonal curvilinear coordinates system. The main steps involved in the DOM are described in the case of the configuration described in Fig. 1. As an application the DOM is used for solving the coupled radiative-conductive heat transfer problem in an absorbing-emitting (but non scattering) gray medium confined between two axisymmetric shells whose generatrices are either ellipses or paraboles. The chosen thermal boundaries conditions are temperature or fluxes assessments and the radiative properties of the shells can be black or diffusely reflectives. The problem is solved for a set of curvilinear coordinates (s,n) moving along the inner wall.

Results are given in terms of temperature mappings and fluxes profiles for various values of the optical thickness and of the Planck number. Fig. 1: the problem under consideration and its associated curvilinear coordinates system

1. CHANDRASEKHAR S. -Radiative Transfer. Dover Ed., 1960
2. CARLSON B.G and LATHROP K.D. -Transport Theory The Method of Discrete Ordinates. Computing Method in Reactor Physics, Gordon and Breach 1968
3. YÜCEL A. and WILLIAMS M.L. - Azimuthal dependent Radiative Transfer in Cylindrical Geometry. ASME HTD 72, pp.29-37, 1987
4. BOUGUERRA E.H. and LEMONNIER D. - Prediction of coupled conductive-radiative heat transfer in cylindrical and annular enclosures by SN methods. Heat Transfer in Semitransparent Media, Ed. Europeennes de Thermique, pp.165-173, 1992
5. JENDOUBI S., LEE H.S and KIM T.K Discrete Ordinates solutions for radiatively participating media in a cylindrical enclosure. J. Thermophys. Heat Transfer, Vol.7, pp.213-219, 1993.
6. JONES P.D and BAYAZITOGLU Y. Coordinates systems for radiative transfer equation in curvilinear media. J. Quant. Spect.Rad.Transfer, Vol.48 (4), pp.427-440, 1992
7. VAILLON R., LALLEMAND M. and LEMONNIER D. Radiative-conductive heat transfer in curvilinear coordinates by the discrete ordinates method., EUROTHERM 36, LET ENSMA, Sept. 1994.

#### APPLICATION OF THE SECOND ORDER DISCRETE ORDINATES METHOD TO A RADIATION PROBLEM IN A SQUARE GEOMETRY

Kyeong-Beom CHEONG, Tae-Ho SONG
Department of Mechanical Engineering,
Korea Advanced Institute of Science and Technology,
Kusong-dong 373-1, Yusong-ku, Taejon, KOREA

Radiative heat transfer in a two dimensional square enclosure containing gray absorbing/emitting and nonscattering media was investigated to explore validity of the second order discrete ordinates method which had been reformulated from the conventional discrete ordinates (SN) method. Discretization equations of governing equation and boundary condition were obtained using a Taylor series expansion method (TE) and using an exponential scheme with cubic interpolation method (EXP3) and the results were compared with those from the conventional SN method and the zonal computations (in the zonal computations, the direct exchange areas were integrated numerically using the GAUSS-Legendre quadrature). Two sample problems were taken; in the first, the medium has known heat source and the walls are cold (gas emission problem), and in the second, the medium has no heat source and one of the walls is hot while the others are cold (boundary emission problem); all the walls are black and diffuse in both problems. The two problems were solved to obtain the medium temperature and the wall heat flux distributions using S2 and S6 methods with varying optical depth. When the optical depth is as small as 0.1, S2 results of the second order and the conventional discrete ordinates method deviate significantly from the zonal one in both problems while the S6 results of any discrete ordinates method are in fair agreement with the zonal results. When the optical depth is large (10 here), the heat flux obtained from S6/EXP3 is closer to the zonal result than that of conventional S6 near the corner in both problems.When the optical depth is unity, an intermediate behavior is observed. On the whole, the second order SN results show very good agreement with those of conventional SN method and when the number of discrete ordinates is not too small, they also agree well with the zonal computations. When the optical depth is small in the boundary emission problem, the second order SN method shows wall heat flux greater than the blackbody emissive power near the corner (TE is worse than EXP3), which calls for further improvement.

#### CALCULATION OF THE RADIATION FLUX DIVERGENCE NEAR THE REGION OF LOCAL HEAT RELEASE BY QUADROMOMENT METHOD

Sergey T. SURZHIKOV
Institute for problems in Mechanics
Russian Academy of Sciences, Moscow, Russia

Accurate and general solutions of the radiation heat transfer equation in two-dimensional geometry are required for several low temperature plasma applications. To study various radiation transfer calculation methods in two-dimensional cylinder-shaped geometry the stimulating problem is the one of mathematical simulation of radiation processes in the continuous optical discharge, which is used in laser plasma generators and laser sustained rocket engines. As a rule, continuous optical discharge is generated at pressures around the atmospheric level, therefore, the model of local thermodynamic equilibrium may be used as the basis for a radiant heat exchange problem. A typical dimension of the continuous optical discharge is ~1 cm, the temperature in its central part is ~12000-20000 K.

The radiation transfer equation has been modeled by using method of quadromoments for two dimensional cylindrical geometry. The scattering of heat radiation is disregarded.

The original for method of quadromoments was first given by Özýþýk, Menning and Halg1 (the half-range method for a spherically symmetric geometry), Sherman2 (the half-moment method for a plane-parallel geometry), Mengüç and Iyer3 (the multiple spherical harmonics approximations for plane-parallel and two-dimensional cylindrical geometries).

At present formulation, the simplest zero-order moments are used to provide a solution of the radiation heat transfer equation for two-dimensional cylindrical volume with high temperature inhomogeneities.

The technique of lowering the order of the system of multigroup equation in the quadromoment form is described. The problem, in essence, is this: integration of radiation transfer equation in a selective formulation or in group approximation (including statistical simulation at sections) for the purpose of determination of the total energy emission rate stipulated by radiation processes provides for multiple solutions in individual sectors of the spectrum with subsequent adding up of the results. This is the main reason for the fact that the labor intensity of solution of selective radiation transfer usually supersedes the labour intensity of solution of the associated mechanical problems. Therefore methods of effective reduction of dimensionalities of the system of selective equations are often used in radiation gas dynamics. The common idea of these methods is that the full system of selective equations is not solved at each stage of solution of gas dynamics equations, but is solved periodically (as the need arises). The main result of solution of this full system of equations is determination of effective coefficients in the transfer equation integrated through the spectrum. It is this last equation that is solved at each step jointly with gas dynamics equations until the given replacement of the system by just one equation becomes too rough. The advantage of the such formulation of the quadromoment method is that unlike methods of averaging by the full solid angle (for example, PN-approximations of Spherical Harmonic method), where the average integral absorption coefficient may have gaps of the second type, the given case presents a smooth continuous functions.

Numerical solutions of the equations for the two-dimensional cylindrical geometry are obtained using a software code and the results are compared with those available by P1-approximation of Spherical Harmonic method. These calculations were made for two temperature distributions: with temperature in the hot area centre T1=10000K and T1=18000K. Different calculation grids, absorption coefficients k and coefficients defining the value of artificial calculation diffusion were used. It has been established that the zero approximation of the quadromoment method with =1-10 in a wide range of optical thicknesses allows to obtain results close to the results based on the P1 approximation of the spherical harmonics. If =0.1 is entered, it leads to incorrect distribution of heat radiative flux divergence in the peripheral area of the volume. Relative error is getting higher as we move further from the central high-temperature area.

A question is discussed as to whether the method of quadromoments can describes the heat radiative flux divergence in low temperature plasma near the regions of local heat release.

1. Özýþýk, M. N., Menning, J., and Halg, W., Half-range moment method for solution of the transport equation in a spherically symmetric geometry, JQSRT, Vol.15, p. 1101-1106, 1975.
2. Sherman, M.P., Moment methods in radiative transfer problem, JQSRT, Vol. 7, No. 1, pp. 89-109,1967.
3. Mengüç, M.P., Iyer, R.K., Modeling of radiative transfer using multiple spherical harmonics approximations, JQSRT, Vol. 39, No.6, pp.445-461, 1988.

#### RADIATIVE HEAT TRANSFER OF ARBITRARY 3-D PARTICIPATING MEDIA AND SURFACES WITH NON-PARTICIPATINGMEDIA BY A GENERALIZED NUMERICAL METHOD REM2

Shigenao MARUYAMA and Toshio AIHARA
Institute of Fluid Science, Tohoku University, Sendai, Japan

Radiative heat transfer of absorbing, emitting and scattering media and of diffuse and specular surfaces containing non-participating media is analyzed by a generalized numerical method: Radiation Element Method by Ray Emission Model, REM2. Arbitrary thermal conditions can be specified for each radiation element. A generalized radiative heat transfer analysis can be achieved without recognizing participating media and surface elements by introducing the ray emission model and extinction view factors.

We consider a system composed of N radiation elements. N elements are divided into M participating media or absorbing and diffuse reflecting surfaces and L non-participating media or perfect specular surfaces. The radiation elements are numerically modeled by arbitrary triangles, quadrilaterals, tetrahedrons, wedges and hexahedrons generated by a general purpose pre- and post-processor package for the finite element method.

Discretized radiation rays are emitted from each participating radiation element according to a ray mission model. The ray tracing is performed for all N radiation elements; however, only M participating elements are considered for view factors and radiation transfer. In the present analysis, the elements are classified into participating and non-participating radiation elements and the memory needed for the calculation is minimized.

The present method was applied to a two-dimensional participating square and a three-dimensional participating cube covered with black isothermal walls. The results are compared with a semi-analytical solution and the results obtained by a zonal method, respectively. The present method shows good agreement with existing solutions, using only a small number of ray emission and radiation elements. Comparison of the dimensionless temperature distributions of the cubic medium show some deviation between solutions for large and small ray emission numbers. However, the distribution in each case is almost identical except for the very small region near corners.

A spherical participating medium with uniform heat generation contained in a square isothermal and adiabatic walls is analyzed. The temperature in the participating sphere shows spherical distributions for the case of small and medium optical thickness in spite of the non-spherical boundary conditions of the outer wall when the heat generation is uniform in the spherical medium.

As an example of an arbitrary configuration, a torus plasma in a large helical device for the research of a fusion reactor was analyzed, in which plasma is approximated as a gray participating medium and a specular and diffuse surface is assumed for the vacuum chamber.

The dimensionless temperature distribution of the model plasma and surface heat flux of the vacuum chamber are demonstrated. The dimensionless temperature distribution of the model plasma and the dimensionless surface heat flux are shown in the Figure below. The analysis model is consist of 2430 elements. The figure represents the total model of the helical device comprised of 24300 elements and a part of the wall and plasma are removed to show the distributions. The temperature distribution of the plasma has a maximum value at the center, and a minimum in the region near the end major axis. The heat flux on the wall is negative due to the definition. The distribution shows a maximum flux on the bottom of trapezoidal grooves and has a minimum value at the corner of the edge of the grooves. Temperature distributions of modeled plasma in large helical device used for study of fusion reactor and the wall heat flux on vacuum chamber. 