Azimuthally dependent radiative transfer in a slab with variable refractive index

Applying Fourier series expansion and a discrete ordinates method, we obtain accurate solutions of azimuthally dependent radiative transfer in an anisotropically scattering slab with variable refractive index and oblique irradiation. It is found that in a slab with a positive gradient of refractive index the optical distance traveled by incident rays decreases, and thus the value of transmittance increases. Besides, the increase of gradient of refractive index enhances the above effect. Influence of the optical thickness, scattering albedo and phase function of the medium, the incident angle of irradiation and the spatial variation of refractive index on the angular distribution of bidirectional reflectance is also investigated.     

Discrete ordinate solutions associated with the finite Legendre transform for radiative transfer in a slab with sinusoidal refractive index

In this work, we develop an alternative discrete ordinate approximation for radiative transfer in a refractive slab. The present method treats the angular derivative term of the radiative transfer equation for a planar medium with varying refractive index (VRI) by using a finite Legendre transform which gives a simple expression of the angular derivative term. Thus, the solution procedure does not march along direction, and so is not restricted to a monotonic variation of refractive index. We apply this method to study radiative heat transfer in a cold slab with anisotropic scattering, diffuse boundaries and sinusoidal VRI. We also solve the problems by the discrete curved ray tracing (DCRT). The hemispherical reflectance and transmittance of slabs with irradiation from the upper surroundings obtained by the present method and those obtained by the DCRT are in excellent agreement. For a slab of a sinusoidal refractive index with the minimum at the center plane, the gradient of refractive index causes the internal reflection of a part of irradiation, which reduces the transmittance of the slab. Other effects of the VRI, the optical thickness, the scattering albedo, the anisotropically scattering coefficient and the boundary reflection are also investigated.     

Transient Radiative Transfer in a Graded Index Medium with Specularly Reflecting Surfaces

A modified Monte Carlo (MC) method has been developed for solving transient radiative transfer in a one-dimensional scattering medium with a graded refractive index. The accuracy and computational efficiency of the algorithm are validated initially. With the introduction of time shift and superposition principle into the MC model, the computational efficiency is greatly improved. We make a comparative analysis of the time-resolved incident radiation (and radiative heat flux) distributions in the media with diffuse and specular reflection boundaries. Results show that the temporal and spatial radiative signals of the medium with specular reflection boundaries greatly differ from those having diffuse reflection boundaries.     

Solution of integral equations of intensity moments for radiative transfer in an anisotropically scattering medium with a linear refractive index

In this work, we derive the integral equations of radiative transfer in terms of intensity moments for radiative transfer in an anisotropically scattering slab with a spatially varying refractive index (VRI). The integral equations are solved by the Nyström method. We apply this method to study radiative heat transfer in a cold slab with higher-degree anisotropic scattering and linearly VRI. The slab lays on an opaque substratum. The refractive index may have a jump at the interface between the surroundings and the slab, while the interface between the slab and the substratum is assumed to be non-reflecting. To exemplify the application of the integral formulation, we consider the case with irradiation from external source in the surroundings and the case with an emitting substratum. We also solve the problems by the Monte Carlo method (MCM). The hemispherical reflectance and transmittance of the slabs obtained by solving integral equations are in excellent agreement with those obtained by the MCM. A positive gradient of refractive index (n′) enhances forward radiative transfer, and so the dimensionless radiative heat flux increases with the increase of n′ for the cases with irradiation from the surroundings. Effects of the optical thickness, the scattering albedo and the scattering phase function are also investigated.     

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.     

Development and comparison of the DTM,the DOM and the FVM formulations for the short-pulse laser transport through a participating medium

The present article deals with the analysis of transient radiative transfer caused by a short-pulse laser irradiation on a participating medium. A general formulation of the governing transient radiative transfer equation applicable to a 3-D Cartesian enclosure has been presented. To solve the transient radiative transfer equation, formulations have been presented for the three commonly used methods in the study of radiative heat transfer, viz., the discrete transfer method, the discrete ordinate method and the finite volume method. To show the uniformity in the formulations in the three methods, the intensity directions and the angular quadrature schemes for computing the incident radiation and heat flux have been taken the same. To validate the formulations and to compare the performance of the three methods, effect of a square short-pulse laser having pulse-width of the order of a femtosecond on transmittance and reflectance signals in case of an absorbing and scattering planar layer has been studied. Effects of the medium properties such as the extinction coefficient, the scattering albedo and the anisotropy factor and the laser properties such as the pulse-width and the angle of incidence on the transmittance and the reflectance signals have been compared. In all the cases, results of the three methods were found to compare very well with each other. Computationally, the discrete ordinate method was found to be the most efficient.     

Second-order-accurate discrete ordinates solutions of transient radiative transfer in a scattering slab with variable refractive index

The discrete ordinates method (DOM) with a second-order upwind interpolation scheme is applied to solve transient radiative transfer in a graded index slab suddenly exposed to a diffuse strong irradiation at one of its boundaries. The planar medium is absorbing and anisotropically scattering. From the comparison of the results obtained by the first-order DOM, the second-order DOM, the modified DOM and the Monte Carlo method, it can be seen that the numerical diffusion in the transient solutions obtained by the second-order DOM is less than that in the solutions obtained by the first-order DOM, but the numerical diffusion is still noticeable, especially for optically thin and moderate cases. By contrast, for optically thick cases the numerical diffusion due to the finite difference of the advection term of the transient radiative transfer equation is minor. In general, it is still necessary to adopt a DOM with a higher order scheme to capture the wave front of transient radiative transfer accurately. Besides, the influence of numerical diffusion is a little less noticeable for the case with a larger gradient of refractive index, and the distribution of direction-integrated intensity around the irradiation boundary decreases and that around the other boundary increases with the increase of the anisotropically scattering coefficient.     

Participating media exposed to collimated short-pulse irradiation – A Laguerre–Galerkin solution

A new method is developed for the solution of radiative transfer in a one-dimensional absorbing and isotropically scattering medium with short-pulse irradiation on one of its boundaries. The time-dependent radiative intensity is expanded in a series of Laguerre polynomials with time as the argument. Moments of the radiative transfer equation, as well as of the boundary conditions, then yield a set of coupled time-independent radiative transfer problems. This set, in turn, is reduced to a set of algebraic equations by the application of the Galerkin method. The transient transmittance and reflectance of the medium are evaluated for various values of the optical thickness, scattering albedo and pulse duration. It is demonstrated that the Laguerre–Galerkin method is not only easier to implement and more efficient but also yields more accurate results compared to the direct application of the Galerkin method. The results are in very good agreement with those available in the literature.     

Transient Prediction of Radiation Response in a 3-D Scattering-Absorbing Medium Subjected to a Collimated Short Square Pulse Train

Transient radiative transfer characteristics in a three-dimensional scattering-absorbing medium subjected to collimated irradiation of a short pulse train were investigated. A basic problem in which a cube is exposed to collimated irradiation of a single unit step square pulse was solved via the transient discrete-ordinates method, and Duhamel's superposition theorem was used to construct the responses of various pulse trains. The effects of optical thickness, scattering albedo, pulse width, and pulse train interval between two successive pulses on the temporal profiles of divergence of radiative heat flux, reflectance, and transmittance were scrutinized.     

Integral equation solutions based on exact ray paths for radiative transfer in a participating medium with formulated refractive index

The exact analytical path length of radiation traveling in a slab with formulated variable refractive index is derived. Based on the analytical path lengths, the integral equations in terms of intensity moments for radiative transfer in a participating slab with one of the family of spatially varying refractive indices are developed. We solve the integral equations for radiative transfer in a slab at radiative equilibrium or for radiative transfer in an isothermal slab. The boundaries are assumed to be black for the slab at radiative equilibrium and the index jumps at both boundaries for the isothermal slab are considered. For comparison purpose, we also solve the radiative equilibrium problems by the discrete ordinates method (DOM). The nondimensional emissive power and nondimensional radiative heat flux obtained by solving integral equations show an excellent agreement with those obtained by the DOM. For the slab at radiative equilibrium and with positive gradient of refractive index, the jump of the emissive power at bottom boundary decreases with the increase of optical thickness for the cases with slightly varying refractive index, but the trend may not hold for the cases with significantly varying refractive index. For the non-scattering slab with positive gradient of refractive index and fixed refractive indices at the boundaries, the directional emittances at both boundaries for the case with linear refractive index are smaller than those for the case with a refractive index of slope-increasing profile. Effects of the scattering albedo and the scattering phase function coefficient are investigated too.     

SLA (Second-law analysis) of transient radiative transfer processes

This paper concerns a SLA (second-law analysis) of transient radiative heat transfer in an absorbing, emitting and scattering medium. Based on Planck’s definition of radiative entropy, transient radiative entropy transfer equation and local radiative entropy generation in semitransparent media with uniform refractive index are derived. Transient radiative exergy transfer equation and local radiative exergy destruction are also derived based on Candau’s definition of radiative exergy. The analytical results are consistent with the Gouy–Stodola theorem of classical thermodynamics. As an application concerning transient radiative transfer, exergy destruction of diffuse pulse radiation in a semitransparent slab is studied. The transient radiative transfer equation is solved using the discontinuous finite element based discrete ordinates equation. Transient radiative exergy destruction is calculated by a post-processing procedure.     

Transient coupled heat transfer in multilayer composite with one specular boundary coated

By the ray tracing?node method, the transient coupled radiative and conductive heat transfer in absorbing, scattering multilayer composite is investigated with one surface of the composite being opaque and specular, and the others being semitransparent and specular. The effect of Fresnel’s reflective law and Snell’s refractive law on coupled heat transfer are analyzed. By using ray tracing method in combination with Hottel and Sarofim’s zonal method and spectral band model, the radiative intensity transfer model have been put forward. The difficulty for integration to solve radiative transfer coefficients (RTCs) is overcame by arranging critical angles according to their magnitudes. The RTCs are used to calculated radiative heat source term, and the transient energy equation is discretized by control volume method. The study shows that, for intensive scattering medium, if the refractive indexes are arranged decreasingly from the inner part of the composite to both side directions respectively, then, the total reflection phenomenon in the composite is advantageous for the scattered energy to be absorbed by the layer with the biggest refractive index, so at transient beginning a maximum temperature peak may appear in the layer with the biggest refractive index.     

Radiative heat transfer in a participating medium with specular–diffuse surfaces

The results obtained by ray-tracing method can be regarded as benchmarks for its good accuracy. However, up to now, this method can be only used to solve radiative transfer within medium confined between two specular surfaces or two diffuse surfaces. This article proposes a hybrid ray-tracing method to solve the radiative transfer inside a plane-parallel absorbing–emitting–scattering medium with one specular surface and another diffuse surface (S–D surfaces). By the hybrid ray-tracing method, radiative transfer coefficients (RTCs) for S–D surfaces are deduced. Both surfaces of the medium under consideration are considered to be semitransparent or opaque. This paper examines the effects of scattering albedo, opaque surface emissivity and anisotropically scattering on steady-state heat flux and transient temperature fields. From the results it is found that the effects of anisotropic scattering is more for a bigger optical thickness medium; and keeping other optical parameters unchanged, anisotropic scattering affects transient temperature distributions so much in a small refractive index medium.     

Radiative heat transfer in semitransparent solidifying slab considering space–time dependent refractive index

Radiative heat transfer in semitransparent phase-change media is of great interest in many engineering fields. Its essence is the transient coupled heat transfer of radiation and conduction along with liquid–solid phase change. The difficulty is to solve radiative heat transfer with the consideration of time–space dependent radiative properties. Especially when the refractive index is considered to vary with space and time in phase change, the problem becomes more complicated. This paper investigates the problem of the variable radiative properties with space and time during phase change in semitransparent media. The phase-change medium is assumed to have solid, mushy and liquid zones, and the solid/mushy and liquid/mushy interfaces are considered to be semitransparent and diffuse reflecting. In different zones, there are different physical property parameters. Phase interfaces are always moving in phase change, while the interfaces of control volumes are fixed. Therefore, the interfaces of control volumes and phase interfaces are not always coincided, which will bring errors into the simulation of radiative transfer in phase-change media. However, the errors can be reduced by dividing the medium into enough sub-layers. As long as the number of sub-layers is big enough, the errors can be limited in a very small range. Then using the multilayer radiative transfer model, we can solve the radiative transfer problem in the semitransparent phase-change medium. Considering time–space dependent refractive index, this paper analyzes coupled radiative and conductive heat transfer in semitransparent solidifying media. The results show that the effects of variable refractive index with time and space on transient coupled heat transfer are significant and could not be neglected inside the semitransparent phase-change medium under some conditions.     

Combined conduction and radiation heat transfer with variable thermal conductivity and variable refractive index

This article deals with the solution of conduction–radiation heat transfer problem involving variable thermal conductivity and variable refractive index. The discrete transfer method has been used for the determination of radiative information for the energy equation that has been solved using the lattice Boltzmann method. Radiatively, medium is absorbing, emitting and scattering. To validate the formulation, transient conduction and radiation heat transfer in a planar participating medium has been considered. For constant thermal conductivity and constant and variable refractive indices, results have been compared with those available in the literature. Effects of conduction–radiation parameter and scattering albedo on temperature have been studied for variable thermal conductivity and constant and/or variable refractive index. Lattice Boltzmann method and the discrete transfer method have been found to successfully deal with the complexities introduced due to variable thermal conductivity and variable refractive index.     

Temperature distributions in an absorbing-emitting-scattering semitransparent slab with variable spatial refractive index

A Monte Carlo curved ray-tracing method is used to analyze the radiative heat transfer in one-dimensional absorbing-emitting-scattering semitransparent slab with variable spatial refractive index. A problem of radiative equilibrium with linear variable spatial refractive index is taken as an example in this paper. The predicted temperature distributions are determined by the proposed method and compared with the data in references. The results show that influences of refractive index gradient are important and the influences increase with the refractive index gradient, the temperature distribution approaches to the one obtained for a constant refractive index when the slab optical thickness is far greater than 1.0, and the effect of the scattering phase function is similar to that in the medium with constant refractive index.     

Conduction and radiation in a rectangular isotropic scattering medium with black surfaces by the RTNAM

The ray tracing-node analyzing method (RTNAM) has been successfully developed to solve 1-D coupled heat transfer in isotropic and anisotropic scattering media in the past, and in this paper it is further extended to solving the 2-D coupled heat transfer in a rectangular isotropic scattering medium. Using the control-volume method, the partial transient energy equation is discretized in implicit scheme. The effect of radiation on heat transfer is considered as a radiative source term (RST) in the discretized energy equation, and in combination with spectral band model, the RST is calculated using the radiative transfer coefficients (RTCs), which are deduced by the ray tracing method. The Patankar’s linearization method is used to linearize the RST and the opaque boundary condition, and the linearized equations are solved by the ADI method. Before solving the RTCs for isotropic scattering media, the RTCs without considering scattering must be solved at first. And then, the RTCs without considering scattering are normalized according to their integrality relationships. In addition, the correctness of the results obtained by the RTNAM is validated, and effects of scattering albedo and refractive index on transient temperature distribution are investigated.     

Integral equation method applied to radiative transfer in a 2-D absorbing–scattering refractive medium

In this paper, an investigation of radiative transfer in a rectangular medium with one-dimensional or two-dimensional graded index is presented. The integral equations of intensity moments are derived and then the cases of a cold medium exposed to diffuse irradiation at the left boundary are solved by the Nyström method. The results obtained by solving integral equations are in excellent agreement with those obtained by the Monte Carlo method and the discrete ordinates method. For the case with an increasing refractive index in the direction to the right boundary, the distribution of refractive index enhances the radiation in the direction to the right, and so the half-range flux toward the right boundary increases with the increase of the gradient of refractive index. Besides, the half-range flux toward the right boundary decreases as the position considered approaches the top or bottom boundary.     

偏振度对均匀折射率介质内矢量辐射传输过程中辐射熵产的影响

在半透明均匀折射率介质内矢量辐射传输过程中辐射熵传递方程及其数值模拟方法的基础上,研究了偏振度对矢量辐射传输过程中辐射熵产的影响。均匀折射率介质内辐射光束的起偏和改偏通过相距阵实现。计算结果表明:由介质内吸收发射过程的不可逆性产生的光谱辐射熵产数随着偏振度增加而减小,而由介质散射过程的不可逆性产生的光谱辐射熵产数随着偏振度增加而增加;偏振度对介质内的光谱辐射熵强度的影响很大,若不考虑偏振,光谱辐射熵强度的相对误差最大可达到18.04%;在整个系统中,光谱辐射熵产数满足热力学第二定律。     

Chebyshev collocation spectral method for one-dimensional radiative heat transfer in graded index media

Chebyshev spectral collocation method based on discrete ordinates equation is developed to solve radiative transfer problems in a one-dimensional absorbing, emitting and scattering semitransparent slab with spatially variable refractive index. For radiative transfer equation, the angular domain is discretized by discrete ordinates method, and the spatial domain is discretized by Chebyshev collocation spectral method. Due to the exponential convergence of spectral methods, a very high accuracy can be obtained even using few nodes for present problems. Numerical results by the Chebyshev collocation spectral-discrete ordinates method (SP-DOM) are compared with those available data in references. Effects of refractive index gradient on radiative intensity are studied for space dependent scattering media. The results show that SP-DOM has a good accuracy and efficiency for solving radiative heat transfer problems in even spatially varying absorbing, emitting, scattering, and graded index media.