SESSION 5
DISCRETE ORDINATES METHOD
Chairman: B. Webb
ASSESSMENT OF DISCRETE ORDINATES METHOD FOR RADIATIVE TRANSFER IN CYLINDRICAL FURNACES
Nevin Selçuk* and Nuray Kayakol** *Department of Chemical Engineering Middle East Technical University Ankara 06531, Türkiye **Glass Research Center Sisecam, Istanbul, Türkiye
The solution accuracy of S_{n} or Discrete Ordinates Method (DOM) was assessed by
comparing its predictions with exact solutions reported earlier on a cylindrical
enclosure problem with steep temperature gradients. The enclosure has interior black
walls and an absorbingemitting medium of constant properties. Predictive accuracy
was investigated for two angular quadrature schemes: 1) the S_{n} discrete ordinates
approximation (S_{2} and S_{4}) and 2) the modified piecewise constant angular (PCA)
approximation. Comparative testing shows that all predictions agree well with the
exact solutions and that S_{n} approximations produce slightly better accuracy than the
modified PCA approximation in radiative heat flux densities. The sensitivity of the
predictions of the DOM to radiative properties was also examined by performing
calculations for various absorption coefficients in the range 0.2 to 2.0. Comparative
testing shows that the discrepancies between the predictions increase with the
coefficient. The sensitivity analysis performed illustrates that testing a radiation model
by predicting wall heat fluxes from measured gas and wall temperatures and estimated
radiative properties and comparing the predictions with the measurements may be
misleading. Hence, the accuracy of a radiation model should be tested in isolation from
radiative property estimation.
Oscillations in the predicted dimensionless flux densities are present for low values of
gas absorption coefficient but get smaller as the coefficient increases.
RADIATIVE EQUILIBRIUM OF AN HYDROGENHELIUM MIXTURE WITHIN AN ELLIPSOIDAL SHELL USING THE DISCRETE ORDINATES METHOD AND A SPECIFIC FREQUENCY INTEGRATION NIETHOD
R. VAILLON*, M. LALLEMAND*, D. LEMONNIER* and C. STEHLE** * LET, UMR CNRS 6608, ENSMA, 86960 Futurosco e FRANCE ** DARC, UPR CNRS 176, Observatoire de Paris, 92195 Meudon Cedex, FRANCE
ABSTRACT The radiative equilibrium of an hydrogenhelium ellipsoidal gas layer confined between
an internal black surface at prescribed temperature and an external one either transparent or
black is investigated. The temperature field is calculated by considering the coupling between the
chemistry and the optical properties of the reactive medium in the range [3000  30000] K. The
theoretical determination of the hydrogenhelium mixture absorption properties is performed under the
assumption of local chemical and thermodynamical equilibria in the working temperature range, at
a pressure of one atmosphere and a given initial molar fractions of the mixture. Absorption
crosssections calculations are presented for the following processes: boundfree and freefree electronic
transitions of the negative hydrogen ion, boundfree, freefree and boundbound transitions of the
hydrogen atom. The determination of the lines shapes is carried out by means of the Model Microfield
Method. Since total radiative quantities have to be calculated, a specific frequency integration method
suitable for the hydrogenhelium absorption coefficient spectrum is presented. It is based on the
constitution of a spectral data bank that makes total radiative quantities integration fast and accurate.
Finally, the proposed method is used to solve the radiative equilibrium of such a system. The spectral
directional intensity and incident energy fields are determined by using the Discrete Ordinates Method
extended to orthogonal curvilinear coordinates.
ANALYSIS OF RADIATIVE HEAT TRANSFER IN ENCLOSURES OF COMPLEX GEOMETRY USING THE DISCRETE ORDINATES METHOD
A. CHARETTE*, M. SAKAMI* and V. LE DEZ** * Groupe de Recherche en Ingénierie des Procédés et Systèmes, Département des Sciences Appliquées, Université du Québec à Chicoutimi, 555, boulevard de 1'université, Chicoutimi, G7H 2B 1, Québec, Canada ** Laboratoire d'Etudes Thermiques, E.N.S.M.A (URA 1403) B.P 109 86960 Futuroscope, France
This paper presents a new discrete ordinates algorithm for the numerical treatment of radiative
participating media in both two and threedimensional enclosures. The algorithm is based on the
utilization of general characteristic relations in lieu of the traditional differencing schemes for
the spatial marching procedure. It is ideally suited for the treatment of complex geometries, the
grid being formed from triangles (2D) or tetrahedra (3D). The method uses the techniques for
recognition of neighboring cells and is exempt of any numerical oscillation. The paper summarizes
the technique by presenting a unified derivation for 2D and 3D cases.
According to the discrete ordinates method (DOM), the radiative transfer equation (RTE) may be
written for each discrete direction in cell i as:
(1)
where and defines a quadrature of M discrete directions to which
are associated the weights w_{m}. D^{i} stands for the cell domain (area of a triangle or volume of a
tetrahedron), b_{k}^{i} for the boundaries (length of a segment or area of a triangle). I_{m}^{i} and I_{km}^{i} are
averaged intensities in the cell domain and cell boundary, respectively.
To close the set of equations, lateral intensities I_{km}^{i} are needed. These are found by performing
the integration of the nonconservative form of the RTE over an optical path t = s_{f}  s_{i} in a given
cell. The procedure leads to the following exponential scheme:
(2)
where:
(3)
I_{m}^{si} is the incoming intensity at a given point on a cell and I_{m}^{sf} is the outgoing intensity of the
ray after it travelled through the cell.
The new method consists in integrating the exponential scheme over the complete domain of
a cell. Fig. 1 illustrates the technique used for 2D and 3D cases.
In the twodimensional case (Fig. la), it can be shown that when one segment receives radiation
from the two other segments, the average intensity on segment 1 may be expressed as:
(4)
Fig. la. Case where one face of a triangle
(2D cell) receives radiation from the two other faces.
Fig. lb. Case where one face of a tetrahedron
receives radiation from the three other faces.
where being the maximum optical thickness associated with the impact on
the plane normal to ABC and containing AB.
Similar equations can be obtained for the 3D case. For the special situation where one face
receives radiation from the three other faces (tetrahedron in Fig. lb), one obtains:
(5)
where is the maximum optical thickness associated with the
impact on subface IDC.
Other situations (different radiation directions relative to the orientation of the cell) can be
encountered. However the procedure is similar.
The new algorithm has been tested for a number of geometries and conditions, and the results
compare favorably with those obtained by other authors. We report here a case that was not
published earlier and shows the flexibility of our method. The chosen enclosure, depicted in
Fig.2, is delimited by a parallelepiped and an hemisphere. The medium is emittingabsorbing and
its temperature is 1000 K. The walls are black at 500 K. Solutions were obtained using 10220
tetrahedra arranged in an increasingly unstructured way as the hemisphere is approached, with 72
triangles on each face of the parallelepiped. The predicted S_{4} solution of the radiative heat flux in
the z direction along a line at x = 1.17m and y = 0 for different values of the absorption coefficient
is shown in Fig. 3. The fluxes are compared to those obtained in the absence of the hemisphere.
For high absorption coefficients, the effect of the hemisphere obstruction becomes less significant
and the flux distribution tends to be comparable to that obtained with the parallelepiped alone. For
high values of z (far from the obstruction) the fluxes are identical to those of the parallelepiped
(for all values of k). Conversely, when z is low, the effect of the obstruction is clearly perceptible.
Fig. 2. Complex enclosure bounded by a cubical cavity and an hemispherical obstruction.
Fig. 3. Net wall heat fiux along a vertical line of the parallelepiped /
hemisphere cavity defined by x = 1.17m, y = 0; effect of the presence
of the hemisphere and of the absorption coefficient (w = 0, e = 1.0)
A new DOM algorithm for radiative participating media has been developed. This technique
differs from the classical one by replacing the traditional differencing schemes with characteristic
equations issued from the formal integration of the RTE. A general formulation allows it to handle
either 2D or 3D situations. Furthermore, since the grid is formed from triangular or tetrahedral cells
that can be arranged in any unstructured manner, the algorithm can be easily adapted to arbitrary
geometries. The method is unconditionally stable and does not generate any numerical oscillation.
In this paper, after assessing the accuracy of the new algorithm against results taken from the
literature for cases of regular geometry, we prove its flexibility by applying it to two non
conventional enclosures: an infinite half elliptical ring (2D) and a parallelepiped containing an
hemispherical obstacle (3D).
