Transient Radiative Transfer Research Articles

Transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate

In this work, we applied the discrete ordinates method (DOM) with a first-order spatial scheme and adapted a modified DOM (MDOM) to solve transient radiative transfer in a refractive, absorbing and scattering slab suddenly exposed to a diffuse strong irradiation at one of its boundaries. The other boundary is diffusely reflecting. From the comparison of the results obtained by the first-order DOM, the MDOM and the Monte Carlo method, it can be seen that the results obtained by the three methods are in excellent agreement as the time is long enough. Besides, the solutions of optically thin and moderate cases obtained by the first-order DOM include some early transmitted radiation due to numerical diffusion and those obtained by the MDOM do not show numerical diffusion in the beginning of a transient process. The reason is that the MDOM solves exactly the reduced incident intensity which dominates radiative transfer in the beginning of a transient process. The time-resolved hemispherical reflectance and transmittance of the slab are obtained for various linearly varying refractive indices, optical thicknesses, scattering albedos and substrate reflectivities. Effects of those parameters are investigated.

Spectral Collocation Method for Transient Combined Radiation and Conduction in an Anisotropic Scattering Slab With Graded Index

The spectral collocation method for transient combined radiation and conduction heat transfer in a planar participating medium with spatially variable refractive index is introduced and formulated. The angular dependence of the problem is discretized by discrete ordinates method and the space dependence is expressed by Chebyshev polynomial and discretized by spectral collocation method. Due to the exponential convergence of spectral methods, very high accuracy can be obtained even using a small resolution for present problem. Numerical results in one-dimensional planar slab by Chebyshev collocation spectral-discrete ordinates method (SP-DOM) are compared with those available data in references. Effects of various parameters such as the variable thermal conductivity, the scattering albedo, the emissivity of boundary, the conduction-radiation parameter, the optical thickness, and the graded index are studied for absorbing, emitting, and anisotropic scattering medium. The SP-DOM has been found to successfully and efficiently deal with transient combined radiation and conduction heat transfer problem in graded index medium.

About the solution of the even parity formulation of the transient radiative heat transfer equations

This paper studies the existence and uniqueness questions, in classical spaces, for a certain system of differential equations. From the physical point of view, the interest of this system lies in that it becomes, for particular choices of its coefficients, the even parity formulation of the S 2 approximation of the transient radiative heat transfer equations in the one-dimensional slab. A priori lower and upper bounds of the solution are obtained as well.

About the solution of the even parity formulation of the transient radiative heat transfer equations

This paper studies the existence and uniqueness questions, in classical spaces, for a certain system of differential equations. From the physical point of view, the interest of this system lies in that it becomes, for particular choices of its coefficients, the even parity formulation of the S 2 approximation of the transient radiative heat transfer equations in the one-dimensional slab. A priori lower and upper bounds of the solution are obtained as well.

Control of Transient Coupled Radiative-conductive Heat Transfer Equation

Control of the combined heat transfer process is presently under consideration for better glass manufacture with less defects. Solution of transient combined radiative and conductive heat transfer in a medium requires the solution of radiative transfer equation, which usually takes hours to converge using latest finite volume codes. Indeed transient combined radiative and conductive heat transfer in a medium is governed by nonlinear partial integro-differential equations. Simple models are necessary for real time control purposes. First this paper present a reduced model. It is obtained by a polynomial approximation for the combined heat transfer and a discretization of PDEs by finite difference method. Then a PI controller of the approximate plant is designed. Convergence is confirmed by simulation results.

Transient Radiation and Conduction Heat Transfer in Glass Sheets by the Thin Layer Approximation

This paper is devoted to the simulation of 3D transient radiation and conduction heat transfer occurring inside thin glass sheets undergoing high temperature processing. The glass is considered as an absorbing, emitting, and nonscattering medium. The zonal method is used to establish the governing radiation transfer model. Direct exchange areas are calculated by the flux planes approximation. The thin layer approximation (TLA) is then introduced for increasing CPU efficiency. Three different numerical integration schemes made possible by the TLA are presented. Comparisons are made, with calculations performed using the finite volume method (FVM). The transient coupled energy equation is solved by a full implicit control volume method using the incomplete Cholesky conjugate gradient method. The heat transfer analysis of a glass sheet residing inside a hot rectangular enclosure is studied. Results obtained by the zonal method, with or without the TLA, are in close agreement with those obtained by the FVM. CPU requirements for radiative heat transfer analysis of the zonal method with TLA are, depending on the numerical integration scheme used, between 8 and 23 times smaller than those of the zonal method without TLA. The difference between the results of the different models never exceeds 4%. The zonal method with the TLA offered significant improvements in CPU time when compared with the original zonal method with similar or acceptable accuracy.

On Galerkin Technique for Transient Radiative Heat Transfer in Finite Thin Media

The transient radiative heat transfer problem in an absorbing and isotropically scattering plane-parallel medium is proposed. The medium is considered to be nonemitting and the boundaries are nonreflecting and nonrefracting, exposed to an external incident flux. The transient problem is transformed into a stationary-like one. Then, Galerkin technique is extended to obtain the analytical solution for the transient radiative heat transfer problem. The transient reflectivity and transmissivity of the medium are calculated for various values of optical thickness and scattering albedo at different times. The results are in fair agreement with those available in the literature using Pomraning-Eddington approximation.

Three-Dimensional Transient Radiative Transfer Modeling Using Discontinuous Spectral Element Method

Three-Dimensional Transient Radiative Transfer Modeling Using Discontinuous Spectral Element Method

Transient radiation and conduction heat transfer inside a plane-parallel participating gray medium with boundaries having different reflecting characteristics

A hybrid ray-tracing method is developed for the solution to the radiative transfer in a plane-parallel participating medium having one specular surface and another diffuse surface. By this method, radiative transfer coefficients (RTCs) for specular–diffuse (S–D) surfaces are deduced. The medium surfaces are considered to be semitransparent. The effects of convection–radiation parameter, conduction–radiation parameter and refractive index on transient coupled heat transfer are investigated. Results show that the temperature curves of the medium having S–D surfaces is higher than those of the medium having S–S surfaces (two specular surfaces); the total heat flux at steady state for the S–D surfaces is lower than that for the S–S surfaces.

Monte Carlo simulation of transient radiative transfer in a medium with a variable refractive index

In this work, a Monte Carlo method is developed to simulate transient radiative transfer in a refractive planar medium exposed to a collimated pulse irradiation. The time of flight in closed form is derived for a medium with a linearly varying refractive index. The time-resolved reflectance and transmittance of the slab are obtained by tracing photon bundles and calculating the time of flight. There is a very satisfying correspondence between the present results and the discrete ordinates solutions. The magnitude of numerical uncertainty decreases with the increase of bundles. To simulate transient radiative transfer in an optically thick medium, we need a large number of bundles. The reflectance decreases with the increase of the positive gradient of the refractive index considered. The appearance instant of the transmittance peak postpones as the refractive index increases. Influence of the optical thickness, the scattering albedo and the anisotropically scattering coefficient on the time-resolved reflectance and transmittance is also investigated.

Differential approximations for transient radiative transfer in refractive planar media with pulse irradiation

Transient radiative transfer in an anisotropically scattering refractive planar medium with pulse irradiation is solved by various approximation methods, such as P−1, P−1 parabolic, P 1/3 and two-flux. The time-resolved transmittance and reflectance are calculated for various radiative parameters, and are compared with those obtained by the discrete ordinate method (DOM). Among the approximation methods considered, the P 1/3 approximation is the better one, because its results are in overall good agreement with those obtained by the more rigorous DOM, except the transmittance around the peak for neither thin nor very thick slabs. It is found that the curved paths of radiation and the internal reflection of the back scattered radiation enhance the effect of scattering.

The Solution of Transient Radiative Transfer With Collimated Incident Serial Pulse in a Plane-Parallel Medium by the DRESOR Method

A time-dependent distribution of ratios of energy scattered by the medium or reflected by the boundary surfaces (DRESOR) method was proposed to solve the transient radiative transfer in a one-dimensional slab. This slab is filled with an absorbing, scattering, and nonemitting medium and exposed to a collimated, incident serial pulse with different pulse shapes and pulse widths. The time-dependent DRESOR values, representing the temporal response of an instantaneous, incident pulse with unit energy and the same incident direction as that for the serial pulse, were proposed and calculated by the Monte Carlo method. The temporal radiative intensity inside the medium with high directional resolution can be obtained from the time-dependent DRESOR values. The transient incident radiation results obtained by the DRESOR method were compared to those obtained with the Monte Carlo method, and good agreements were achieved. Influences of the pulse shape and width, reflectivity of the boundary, scattering albedo, optical thickness, and anisotropic scattering on the transient radiative transfer, especially the temporal response along different directions, were investigated.

Ray-Tracing Method for Radiative Heat Transfer in a Two-Layer Anisotropic Scattering Medium

Transient radiation heat transfer in a two-layer anisotropic scattering medium is investigated. Each layer of the medium can have different thermal and radiative properties. The two surfaces are assumed to be semitransparent or opaque, and the interface between the two layers is assumed to be semitransparent. Specular reflection is considered at the surfaces and the interface. A general criterion for total reflection at the surfaces and the interface is proposed to overcome the integral singularity problem above the critical angle. A fully implicit control-volume method is used to solve the transient energy equation of coupled radiation and conduction, and a ray-tracing/nodal-analyzing method is applied in the solution of the radiative transfer. Effects of emissivities of the surfaces, extinction coefficients, and optical boundary conditions on transient radiative heat transfer are examined. Results show that for the medium layer having a larger refractive index, its average temperature level may be higher, mainly due to the contribution of total reflection at the interface facing the layer, even if its absorption coefficient is relatively small compared to the medium layer having a larger absorption coefficient and a smaller refractive index.

A frequency-based numerical approach for unsteady radiative transfer in participating media

The one-dimensional transient radiative transfer problem in the Cartesian coordinate system – an absorbing and scattering medium illuminated by a short laser pulse – is solved by the use of a discrete ordinates–finite volume method. Previous works have shown that the original numerical approach, based in the space–time domain, induces transmitted flux emerging earlier than the minimal time required by the radiation to leave the medium. Therefore, a frequency-based numerical method is formulated, implemented, and validated in this paper. Results for transmittances are accurate, without physically unrealistic behaviors at early time periods. However, the frequency-dependent approach is computationally expensive; it requires approximately five times more computational time than its temporal counterpart. Ongoing research is devoted to the optimization of these CPU requirements.

Transient radiative heat transfer in an inhomogeneous participating medium with Fresnel’s surfaces

This paper studies the radiative heat transfer within an inhomogeneous and isotropically scattering medium with reflecting Fresnel’s surfaces. Thermal radiation transfers in a curve inside a medium with an inhomogeneous distribution of a refractive index. The inhomogenous medium is divided into n homogenous isothermal sub-layers and in each sub-layer the radiation transfers in a straight line. By adopting a multilayer radiative transfer model and using a ray-tracing/nodal-analyzing method, a radiative transfer model is built for the inhomogenous participating medium. In the multilayer model, a criterion for refraction / total reflection at the interfaces between neighboring sub-layers is introduced, avoiding the integral singularity and reflection at physically inexistent interfaces (only the total reflection is considered). Transient thermal behavior is examined when the parameters of the radiative properties such as refractive indexes, extinction coefficients, and single-scattering albedoes vary continually along the thickness direction.

Discontinuous Spectral Element Approach for Solving Transient Radiative Transfer Equation

A discontinuous spectral element method is presented to solve transient radiative heat transfer in multidimensional semitransparent media. The transient term is discretized by the Crank-Nicolson scheme, and the spatial discretization is conducted using the discontinuous spectral element approach. The salient features of the discontinuous spectral element method are that it possesses high-order accuracy, has properties such as local conservation, and its solutions are allowed to be discontinuous across interelement boundaries. The inherent discontinuity acceptable property of the discontinuous spectral element method allows efficient and accurate capturing of the sharp wave front of the transient radiative transfer process. A special treatment for the problem with diffuse irradiation at the boundary is presented to mitigate the ray effect encountered in solving the transient radiative transfer equation. The performance of the discontinuous spectral element method is verified by three test cases. The predicted results by the discontinuous spectral element method agree well with reported benchmark solutions in the references.

Laguerre Runge--Kutta--Fehlberg Method for Simulating Laser Pulse Propagation in Biological Tissue

An efficient algorithm for solving the transient radiative transfer equation for laser pulse propagation in biological tissue is presented. A Laguerre expansion is used to represent the time dependency of the incident short pulse. The Runge-Kutta-Fehlberg method is used to solve the intensity. The discrete ordinates method is used to discretize with respect to azimuthal and zenith angles. This method offers the advantages of representing the intensity with a high accuracy using only a few Laguerre polynomials, and straightforward extension to inhomogeneous media. Also, this formulation can be easily extended for solving the 2-D and 3-D transient radiative transfer equations.

Time shift and superposition method for solving transient radiative transfer equation

Due to the long-time transient response of pulse irradiation, the computational time required for solving transient radiative transfer (TRT) is often very long, especially for the case in which the boundary is subjected to continuous pulse train and the geometry is complicated. In addition, sometimes, before actual experiments are carried out, a suitable pulse shape or type often needs to be selected by numerical simulation and comparison. Because the numerical solution of TRT needs to be repeated many times, the selection processes is very time-consuming. In this paper, by considering that the TRT equation and its initial and boundary conditions are linear, a time shift and superposition method is developed for solving TRT equation in non-emitting media, in which only the transient response of a short square pulse needs to be solved, and the solution of TRT under any pulse shape can be constructed by time shift and then superposition using the basis solution of the short square pulse. Three numerical examples are studied to illustrate the peformance of the superposition method in solving TRT problems. The results show that the superposition is effective, accurate and very suitable for solving TRT in the medium subjected to a series of pulse train.

An Insight Into The Modeling of Short-Pulse Laser Transport Through A Participating Medium

Thermal radiation is a time-dependent process in the analysis of a short-pulse laser irradiation. The numerical modeling of the radiative phenomenon considers a finite number of intensity directions over a spherical space. Thus, in time-dependent radiative transfer problems, radiative intensities take different times to reach a particular location in the medium, and accordingly, collectively they remain available for durations different than the laser pulse width. The present work proposes that the above aspect be taken into account in the analysis of transient radiative transfer problems. To validate this concept, transport of a short square-pulse laser in a plane-parallel participating medium is studied. It is found that the magnitude of the transmittance signals differ for higher values of the extinction coefficient and also that the signals remain available for a longer duration. Reflectance signals are found to be less affected.

Numerical predictions of short-pulsed laser transport in absorbing and scattering media. I—A time-based approach with flux limiters

A one-dimensional transient radiative transfer problem in the Cartesian coordinate system involving an absorbing and scattering medium illuminated by a short laser pulse has computationally been solved by use of a finite volume method (FVM). Previous works have shown that first order spatial interpolation schemes cannot represent the physics of the problem adequately as transmitted fluxes emerge before the minimal physical time required to leave the medium. In this paper, the Van Leer and Superbee flux limiters, combined with the second order Lax–Wendroff scheme, are used in an attempt to prevent this. Results presented in this work show that, despite significant improvement, flux limiters fail to completely eliminate the physically unrealistic behaviour. Therefore, a numerical approach into which temporal variables are transformed into frequency-dependent variables is presented in the companion paper to this work.