V. Le Dez

Discrete ordinates solution of radiative transfer across a slab with variable refractive index

The radiative transfer equation (RTE) is derived in both conservative and non conservative forms for a plane slab made of an absorbing-emitting material with a continuous transverse variation of refractive index. The RTE is set in a form which displays an angular redistribution term analogous to what appears in curvilinear media with uniform index. Numerical solutions are provided by means of the discrete ordinates method. Results are given for a slab bounded by diffuse surfaces and submitted on each exterior side to a black body radiation at prescribed temperature. Comparisons are made with previously published predictions either on a similar problem but with a different solution technique (ray-tracing) or on a different model (composite medium with diffuse sublayers).

Temperature field inside an absorbing–emitting semi-transparent slab at radiative equilibrium with variable spatial refractive index

Abstract The temperature field inside an absorbing–emitting slab of semi-transparent grey medium at radiative equilibrium has been determined with the help of a curved ray-tracing technique when the spatial variation of the refractive index in the medium is assumed to be linear. The integration of the radiative transfer equation has been carried out on the trajectories on which radiation propagates inside the medium, leading to the absorbed radiative energy at an internal point. For a linear refractive index, existence of totally reflected internal trajectories producing mirage effects have to be taken into account in the resolution of the radiative problem. Results obtained for different optical depths with low and strong gradients of refractive index display significant differences from the case of a constant refractive index.

Thermal emission of a semi-transparent slab with variable spatial refractive index

Abstract The directional emission intensities emerging from a parallel plane slab or semi-transparent media in which is imposed a variable spatial refractive index, have been determined by a curved ray-tracing method determining the trajectories of radiation propagation and integrating the radiative transfer equation on each of these trajectories. Cases of linear and sinusoidal refractive indexes are examined, for both isothermal and non-isothermal situations, which display significant differences from the case of a constant refractive index.

Application of the discrete ordinates method to combined conductive and radiative heat transfer in a two-dimensional complex geometry

Abstract This paper describes a new approach for determining the radiative intensity and temperature fields in a semi-transparent medium for coupled radiative-conductive heat transfer in two-dimensional enclosures. The boundary surfaces are uniformly gray with prescribed emissivities and temperatures. The medium is radiatively absorbing-emitting-scattering and gray. The method is a modification of the discrete ordinates method based on the incorporation of directional ray propagation relations within the cells. The algorithm is applicable to enclosures of arbitrary geometry and does not generate numerical oscillations and negative intensities which can appear in the traditional technique. This is made possible by solving the radiative transfer equation exactly along a set of discretized directions. The method can handle triangular grids of any type, structured or unstructured, and is thus compatible with the finite element technique—which is used for the conduction part of the present coupled problem. A summary of the basic equations is given, followed by a brief assessment of the method for pure radiation. Cases of combined conduction-radiation are then presented and the results are compared with those obtained by other researchers. It is shown that the method has no limitation with respect to geometry and is accurate over a wide range of optical thicknesses.

Thermal emission of a two-dimensional rectangular cavity with spatial affine refractive index

Abstract The directional emission intensities emerging from a two-dimensional rectangular cavity of semi-transparent media in which is imposed a spatial variable refractive index, have been determined by a curved ray-tracing method determining the trajectories of radiation propagation and integrating the radiative transfer equation on each of these trajectories. Cases of affine refractive indexes with respect to the space variables are examined, for both isothermal and nonisothermal situations, which display significant differences from the case of a constant refractive index.

Radiative flux field inside an absorbing–emitting semi-transparent slab with variable spatial refractive index at radiative conductive coupling

Abstract The temperature and radiative flux fields inside an absorbing–emitting slab of semi-transparent grey medium at radiative conductive coupling has been determined with the help of a curved ray-tracing technique when the spatial variation of the refractive index in the medium is assumed to be linear. The exact expression of the radiative flux field in such a situation is derived by using previous expressions of the spatial and directional intensity field inside the medium. Results obtained for different optical depths and thermal conductivities with low and strong gradients of refractive index display significant differences from the case of a constant refractive index.

Radiative heat transfer in three-dimensional enclosures of complex geometry by using the discrete-ordinates method

Abstract This paper presents a new algorithm based on the discrete-ordinates method for the calculation of radiative transfer in three-dimensional, gray participating media of complex geometry. The key feature of the new technique is the utilization of general characteristic equations which replace the classical spatial differencing schemes. These equations result from the formal integration of the radiative transfer equation followed by an averaging procedure on the surfaces. This feature enables the use of tetrahedral grids which can be fitted to any geometrical enclosure. Furthermore, the new algorithm does not generate the numerical oscillations and negative intensities that can be encountered with the traditional method. The method uses the techniques of recognition of neighboring cells. It is directly compatible with the solution of the conservation equations by finite element techniques. This paper is a sequel to a previous paper recently published by the same authors on two-dimensional cases. An assessment of the method is given by comparing the results for simple cases with those obtained by other authors. Application is then made to more complicated cases. The method is proved to be general and accurate.

Influence of a spatial variation of the thermo-optical constants on the radiative transfer inside an absorbing–emitting semi-transparent sphere

Abstract The temperature field inside an absorbing–emitting nonscattering sphere of a semi-transparent grey medium at radiative equilibrium with an internal heat generation density has been determined with the help of a curved ray-tracing technique when the spatial variation of the refractive index (and possibly of the absorption coefficient) in the medium is spatially variable. The integration of the radiative transfer equation has been carried out on the trajectories on which radiation propagates inside the medium, leading to the absorbed radiative energy at an internal point. For a variable refractive index, existence of totally reflected internal trajectories with one or two total reflection points producing mirage effects and inducing a local heating in confinement areas have to be taken into account in the resolution of the radiative problem. Results obtained for weak and strong gradients of refractive index display significant qualitative and quantitative differences from the case of a constant refractive index.

Thermal emission of axisymmetric bodies of semi-transparent materials

Abstract The spectral directional emission intensities of axisymmetric, semi-transparent materials have been determined by a ray-tracing method using differential geometry and the geometrical approximation. As an example, the emission mapping of a semi-transparent paraboloid is examined, both for isothermal and non-isothermal cases, which display unusual emission properties.

Conductive–radiative coupling in an absorbing–emitting axisymmetric medium

Abstract A modeling of simultaneous radiation–conduction in axisymmetric semi-transparent motionless bodies (hereafter labeled stm) is proposed. The stm is an emitting–absorbing medium with gray spectral radiative properties. Its interfaces are smooth with specular reflection; they are assumed to be opaque or transparent. The two-dimensional modeling is based on a double-grid representation: (i) a physical one corresponding to curvilinear coordinates adapted to the stm body shape, in which the radiative source is calculated by means of a ray-tracing technique and the formal solution of the radiative transfer equation (RTE), and (ii) an abstract one with an orthogonal grid for which the enthalpy equation is solved by a finite differences scheme. The nonlinear discretized equations are solved iteratively by passing from one to the other representation. Results are presented for the cases of a finite length cylinder and a truncated paraboloid both fitted with prescribed temperatures at the ending black cross sections, the lateral surface being opaque or transparent, and subjected to an imposed uniform temperature. In the case of the cylinder, results are compared with those obtained by a discrete ordinates method in curvilinear orthogonal coordinates.