Semitransparent Graded Index Research Articles

Analysis of transient radiative transfer in two-dimensional scattering graded index medium with diffuse energy pulse irradiation

The natural element method (NEM) is extended to solve transient radiative transfer in two-dimensional semitransparent graded index media with energy pulse irradiation diffusely incident on the left boundary. The surfaces of the medium are diffusely reflecting and semitransparent. By using a fully implicit backward differencing scheme to discretize the transient term in the radiative transfer equation and employing the least-squares (LS) weighted residuals approach to discretize the radiative transfer equation, a LSNEM approach is devised. Firstly, LSNEM solutions to transient radiative transfer are validated by comparison with results reported in the literature. Afterward, LSNEM is employed to investigate two-dimensional transient radiative transfer in graded index medium with a refractive index discontinuity at the surfaces. Effects of the optical thickness, scattering phase function and scattering albedo on transmittance and reflectance signals are investigated. Finally, the transient radiative transfer process of the medium subject to square diffuse pulse irradiation train are investigated, and the transient behavior of the radiative energy transport inside the medium with two complex graded index distributions is analyzed.

The DRESOR method for radiative heat transfer in semitransparent graded index cylindrical medium

During a numerical analysis of radiative transfer in some cylindrical optical thermal analysis and thermal design, applying a cylindrical coordinate system would be much more convenient and precise than that using a Cartesian coordinate system. In this paper, the DRESOR method under a cylindrical coordinate system is proposed to address radiative transfer in a semitransparent graded index cylindrical medium. The dimensionless incident radiation and net radiative heat flux are obtained using the DRESOR method. The accuracy and validity of the proposed method is verified by comparison with other techniques. The effects of isotropic scattering albedo and graded index on radiative transfer are also considered. Additionally, the high directional radiative intensity information is obtained to show the performance of the DRESOR method. It shows that the DRESOR method is an effective technique to address the radiative transfer problem in the graded index cylindrical medium with complex surface temperature characteristics.

Least-squares natural element method for radiative heat transfer in graded index medium with semitransparent surfaces

The least-squares natural element method (LSNEM) is employed for solving radiative heat transfer problem in two-dimensional semitransparent graded index medium with semitransparent and diffusely reflecting surfaces. LSNEM is an extension of natural element method (NEM). Unlike the NEM based on Galerkin discretization, the least-squares weighted residuals approach is employed to spatially discretize the radiative heat transfer equations in LSNEM. Radiative heat transfer problem in rectangular enclosure filled with graded index medium having opaque surfaces is examined to verify the LSNEM. Afterwards, radiative heat transfer in the semicircular enclosure with an inner circle filled with graded index medium is studied. Two kinds of refractive index distributions are considered. The studies show that the refractive index distributions and optical boundary conditions of the surfaces greatly influence the radiative transfer. Effects of the extinction coefficient are investigated on the temperature distributions and radiative heat fluxes.

Numerical Simulation of Radiation Heat Transfer in a One-Dimensional Graded Index Sphere

The rays will propagate along a curved path determined by the Fermat principle in medium with inhomogeneous refractive index distribution. To avoid the complicated computation of ray trajectories, a finite element method is extended to solve the radiative transfer problem in a one-dimensional absorbing-emitting semitransparent spherical graded index medium. A problem of radiative transfer inside a semitransparent spherical graded index medium is taken as an example to verify the method. The predicted temperature distributions are determined by the proposed method, and are compared with the results available in references. The results show that finite element method can predict the radiative heat transfer in one-dimensional absorbing-emitting semitransparent spherical graded index medium accurately.

Upwind Finite Element Method for Solving Radiative Heat Transfer in Graded Index Media

Radiative transfer equation (RTE) in Cartesian coordinates can be considered as a special kind of convective-diffusive equation with strong convection characteristics. For this convection dominated problem, standard finite element solutions often suffer from spurious oscillations. To avoid this problem, the upwind finite element methods based on streamline upwind (SU) and streamline upwind Petrov-Galerkin (SUPG) schemes are developed to solve multidimensional radiative heat transfer in semitransparent uniform and graded index media. Comparison between these two upwind schemes on the solution of RTE is carried out. The SUPG scheme is demonstrated to be better than the SU scheme as far as solution accuracy is concerned and have good accuracy in solution of radiative heat transfer in semitransparent graded index media.

A Finite-Element Model for the Thermal Radiative Properties of Graded Index Fiber Coated with Thin Absorbing Film

A finite-element model in combination with the wave optical approach is developed based on the radiative transfer equation for graded index medium in cylindrical coordinate system to predict the total hemispherical thermal radiative properties of semitransparent graded index fiber coated with thin absorbing film. The film is made of a strong absorbing medium with thickness less than or on the order of the wavelength of peak magnitude of thermal radiation. Radiative absorptance of the fiber-film system is directly obtained by solving the radiation deposited in the system. Radiative transfer in the fiber is solved by a least squares finite-element method, while radiative transfer in the thin film is treated through wave optics, and the film is formulated as a special kind of semitransparent boundary condition for the fiber medium. The results obtained by the finite-element model for uniform index fiber are in good agreement with the results in the literature obtained through the ray tracing model. The effects of fiber refractive index distribution on predicted thermal radiative properties are investigated. For the fiber with or without film, the variation of refractive index distribution has a substantial influence on the effective emittance.

Chebyshev collocation spectral approach for combined radiation and conduction heat transfer in one-dimensional semitransparent medium with graded index

The Chebyshev collocation spectral method for discrete ordinates equation is presented to solve combined radiation and conduction heat transfer problem in semitransparent graded index media. The angular dependence of the problem is discretized by discrete ordinates method, and the space dependence is expressed by Chebyshev polynomial and discretized by collocation spectral method. The exponential convergence characteristic of the spectral methods is studied. The comparisons between the present results and those available in references indicate that, the Chebyshev collocation spectral-discrete ordinates method (SP-DOM so called) for combined radiation and conduction heat transfer in one-dimensional semitransparent medium with graded index is accurate and efficient.

Finite element method for modeling radiative transfer in semitransparent graded index cylindrical medium

Both Galerkin finite element method (GFEM) and least squares finite element method (LSFEM) are developed and their performances are compared for solving the radiative transfer equation of graded index medium in cylindrical coordinate system (RTEGC). The angular redistribution term of the RTEGC is discretized by finite difference approach and after angular discretization the RTEGC is formulated into a discrete-ordinates form, which is then discretized based on Galerkin or least squares finite element approach. To overcome the RTEGC-led numerical singularity at the origin of cylindrical coordinate system, a pole condition is proposed as a special mathematical boundary condition. Compared with the GFEM, the LSFEM has very good numerical properties and can effectively mitigate the nonphysical oscillation appeared in the GFEM solutions. Various problems of both axisymmetry and nonaxisymmetry, and with medium of uniform refractive index distribution or graded refractive index distribution are tested. The results show that both the finite element approaches have good accuracy to predict the radiative heat transfer in semitransparent graded index cylindrical medium, while the LSFEM has better numerical stability.

Thermal emission characteristics of a graded index semitransparent medium

This paper develops a numerical model for thermal radiative transfer in a two-dimensional semitransparent graded index medium. A piecewise continuous refractive index model, the linear refractive index bar model, is presented. This model is established based on three hypotheses, and has a higher precision than the bar model used previously. This paper also studies the thermal emission from a two-dimensional graded index medium, which is scattering or non-scattering. We find that it can present an obvious pattern of directional distribution at times. The refractive index distribution and absorption coefficient are the two main influential factors. This finding differs from the common belief that thermal sources, such as the incandescent filament of a light bulb, emit a quasi-isotropic light. The finding also suggests that there maybe other important applications of artificial GRIN materials.

Solution of radiative heat transfer in graded index media by least square spectral element method

Least square spectral element method based on discrete-ordinates equation is extended to solve multidimensional radiative heat transfer problems in semitransparent graded index media. Chebyshev polynomial is employed as expansion set for the spectral element discretization. Five various test problems were taken as examples to verify the least square spectral element formulation for solving radiative heat transfer in semitransparent graded index media. The predicted distributions of temperature and radiative heat flux are determined by the least square spectral element method and compared with data in the references. The results show that the least square spectral element method has good accuracy for solving multidimensional radiative heat transfer problems in semitransparent graded index media.

Discontinuous finite element method for radiative heat transfer in semitransparent graded index medium

To avoid the complicated and time-consuming computation of curved ray trajectories, a discontinuous finite element method based on discrete ordinate equation is extended to solve the radiative transfer problem in a multi-dimensional semitransparent graded index medium. Two cases of radiative heat transfer in two-dimensional rectangular gray semitransparent graded index medium enclosed by opaque boundary are examined to verify this discontinuous finite element method. Special layered and radial graded index distributions are considered. The predicted dimensionless net radiative heat fluxes and dimensionless temperature distributions are determined by the discontinuous finite element method and compared with the results obtained by the curved Monte Carlo method in references. The results show that the discontinuous finite element method has a good accuracy in solving the multi-dimensional radiative transfer problem in a semitransparent graded index medium.

Analysis of transient radiative transfer in semitransparent graded index medium

The formulation of transient radiative transfer in semitransparent graded index medium is outlined. The discontinuous finite element method is extended to solve the transient radiative transfer equation within semitransparent graded index medium both in the time- and the frequency-domain. Four numerical examples, including one- and two-dimensional geometries exposed to collimated pulse irradiation, are studied. The results show that the discontinuous finite element method is efficient, accurate and stable, and can be used for solving the transient radiative transfer problems in semitransparent graded index medium. Due to ray-trajectory bend resulting from the gradient of refractive index, the transient radiative transfer in semitransparent graded index medium exhibits distinctive behavior from that in uniform index medium. Therefore, the effect of graded index in the medium, when exists, has to be taken into consideration when the time-resolved radiation signals are used to infer the medium's optical and radiative property distributions.

Least-squares finite element method for radiation heat transfer in graded index medium

To avoid the complicated and time-consuming computation of curved ray trajectories, a least-squares finite element method based on discrete ordinate equation is extended to solve the radiative transfer problem in a multi-dimensional semitransparent graded index medium. Four cases of radiative heat transfer are examined to verify this least-squares finite element method. Linear and nonlinear graded index are considered. The predicted dimensionless net radiative heat fluxes are determined by the least-squares finite element method and compared with the results obtained by other methods. The results show that the least-squares finite element method is stable and has a good accuracy in solving the multi-dimensional radiative transfer problem in a semitransparent graded index medium, while the Galerkin finite element method sometimes suffers from nonphysical oscillations.

Finite Volume Method for Radiation Heat Transfer in Graded Index Medium

Because a ray goes along a curved path determined by the Fermat principle, curved ray tracing is very difficult and complex in a graded index medium. To avoid the complicated and time-consuming computation of curved ray trajectories, a finite volume formulation is developed to solve the radiative-transfer problem in a multidimensional absorbing-emitting-scattering semitransparent graded index medium. Three test problems in radiative transfer are taken as examples to verify this finite volume formulation. The predicted temperature distributions and apparent directional emissivity are determined by the proposed method and compared with data in the references. The results show that the finite volume formulation presented in this paper has a good accuracy in solving the radiative transfer problem in a multidimensional absorbing-emitting-scattering semitransparent graded index medium. Nomenclature A = surface area of cell, m 2 b = asymmetry factor of scattering D = direction cosine integrated over control solid angle Erms = rms error of apparent directional emissivity, defined in Eq. (27) ei = unit outward normal vector at the ith surface of cell volume H = height of medium, m I = radiative intensity, W/m 3 sr Ib = blackbody radiative intensity, W/m 3 sr

Meshless method for radiation heat transfer in graded index medium

Because ray goes along a curved path determined by the Fermat principle, curved ray tracing is very difficult and complex in graded index media. To avoid the difficult and complex computation of curved ray trajectories, a meshless local Petrov–Galerkin approach based on discrete-ordinate equations is developed to solve the radiative transfer problem in multi-dimensional absorbing–emitting–scattering semitransparent graded index media. A moving least square approximation is used to construct the shape function. Two particular test problems in radiative transfer are taken as examples to verify this meshless approach. The predicted temperature distributions and the dimensionless radiative heat fluxes are determined by the proposed method and compared with the other benchmark approximate solutions. The results show that the meshless local Petrov–Galerkin approach based on discrete-ordinate equations has a good accuracy in solving the radiative transfer problems in absorbing–emitting–scattering semitransparent graded index media.

Finite element method for radiation heat transfer in multi-dimensional graded index medium

In graded index medium, ray goes along a curved path determined by Fermat principle, and curved ray-tracing is very difficult and complex. To avoid the complicated and time-consuming computation of curved ray trajectories, a finite element method based on discrete ordinate equation is developed to solve the radiative transfer problem in a multi-dimensional semitransparent graded index medium. Two particular test problems of radiative transfer are taken as examples to verify this finite element method. The predicted dimensionless net radiative heat fluxes are determined by the proposed method and compared with the results obtained by finite volume method. The results show that the finite element method presented in this paper has a good accuracy in solving the multi-dimensional radiative transfer problem in semitransparent graded index medium.