Angle-dependent light scattering in materials with controlled diffuse solar optical properties

Light scattering plays a prominent role in a wide range of energy-efficient materials and solar applications. Some examples are materials for daylighting, diffusely reflecting sunscreens, foils for radiative cooling and nanocrystalline solar cells. Measurements of the angular profile of light scattering are very useful for obtaining a detailed characterization of the light scattering mechanisms. We review recent theoretical results on the forward and backward light scattering profiles. Forward scattering is of major importance for novel pigmented polymeric daylighting materials. Measurements of scattering profiles are in good agreement with Mie theory. Backscattering profiles from highly diffusely reflecting paints containing titanium oxide-based pigments have also been measured. It seems that scattering from the paint surface dominates at low pigment volume fractions. Results for paints with high pigment volume fractions are interpreted in terms of coherent backscattering effects from the pigment particles.     

Thermal Radiative Properties of Complex Media: Theoretical Prediction Versus Experimental Identification

In many engineering applications and natural phenomena, thermal radiation interacts with complex media composed of dispersed phases that may be of different type: solid/solid, solid/gas, or liquid/gas. Most of them are semitransparent media that emit, absorb, and scatter thermal radiation. Heat transfer by combined radiation with conduction or convection in such media is a problem of high practical importance, mostly in situations where radiation is a dominant mode. Improvement of thermal performance of such materials or of the manufacturing processes that involve these media requires the availability of efficient methods (i) for radiative transfer modeling, and (ii) to predict and to experimentally determine the thermophysical properties intended to feed the models. This paper is focused on radiative properties assessment. After a brief overview of the materials and properties of interest, the emphasis is put on methodology of property investigation combining both theoretical prediction and experimental identification. Examples related to different particulate media are presented, showing recent advances and needs for further investigation.     

Modeling the radiation of anisotropically scattering media by coupling Mie theory with finite volume method

A new mathematical model and code for radiative heat transfer of particulate media with anisotropic scattering for 2-D rectangular enclosure is developed. The model is based on the coupling of (i) finite volume method for the solution of radiative transfer equation with (ii) Mie equations for the evaluation of scattering phase function. It has not been done before to the authors’ best knowledge. The predictions were compared against the only found results, published 15 years ago. For those results the S-N discrete ordinates method for the solution of radiative transfer equation and the Legendre polynomials expansions for the evaluation of scattering phase function were used. The agreement between the results is very good. The advantages of new model and code are in their straight forward application to any given particles parameters without the need for previously designed analytical expression for scattering phase function. In addition, that analytical expression, with generated expansion coefficients, is restricted and can be used only for that particular case of particle parameters. The new model was applied to the solid particles of several various coals and of an ash and the series of 2-D predictions are performed. The effects of particle size parameter and of scattering albedo on radiative heat flux and on incident radiation were analyzed. It was found that the model developed is reliable and very accurate and thus suitable for extension towards: (i) 3-D geometries, (ii) mixtures of non-gray gases with particles as well as for (iii) incorporation in computational fluid dynamics codes.     

Simultaneous Estimation of Properties in a Combined Mode Conduction–Radiation Heat Transfer in a Porous Medium

Simultaneous estimation of thermophysical and optical properties such as the thermal conductivity, the scattering albedo, and the emissivity of a 1‐D planar porous matrix involving combined mode conduction and radiation heat transfer with heat generation is reported. Coupled energy equations for the gas and solid phase account for the nonlocal thermal equilibrium between the two phases. Performances of the genetic algorithm (GA) and the global search algorithm (GSA) in simultaneous estimation of three properties are analyzed. Both the GA and the GSA utilize a priori knowledge of the axial gas temperature distribution, and the magnitudes of the convective and the radiative heat fluxes at the outer surface of the porous matrix. With volumetric radiative information needed in the solid‐phase energy equation computed using the discrete transfer method, the two energy equations are simultaneously solved using the finite volume method. GSA provides better estimation, and computationally, it is much faster than the GA.     

Effect of light scattering on the performance of a direct absorption solar collector

Recently, a solar thermal collector often employs nanoparticle suspension to absorb the solar radiation directly by a working fluid as well as to enhance its thermal performance. The collector efficiency of a direct absorption solar collector (DASC) is very sensitive to optical properties of the working fluid, such as absorption and scattering coefficients. Most of the existing studies have neglected particle scattering by assuming that the size of nanoparticle suspension is much smaller than the wavelength of solar radiation (i.e., Rayleigh scattering is applicable). If the nanoparticle suspension is made of metal, however, the scattering cross-section of metallic nanoparticles could be comparable to their absorption cross-section depending on the particle size, especially when the localized surface plasmon (LSP) is excited. Therefore, for the DASC utilizing a plasmonic nanofluid supporting the LSP, light scattering from metallic particle suspension must be taken into account in the thermal analysis. The present study investigates the scattering effect on the thermal performance of the DASC employing plasmonic nanofluid as a working fluid. In the analysis, the Monte Carlo method is employed to numerically solve the radiative transfer equation considering the volume scattering inside the nanofluid. It is found that the light scattering can improve the collector performance if the scattering coefficient of nanofluid is carefully engineered depending on its value of the absorption coefficient.     

RDFI determination of anisotropic and scattering dependent radiative conductivity tensors in porous media: Application to rod bundles

A radiative conductivity model is developed for porous media with a solid opaque phase and a transparent fluid phase. In a first step, an effective semi transparent medium occupying the volume of the real fluid phase is characterized, assuming the validity of the Beer’s laws. For example, rod bundles in squared or triangular configurations can be directly characterized by effective strongly anisotropic extinction, absorption and scattering coefficients, optical index and phase function, which depends on both the incident and scattering unit vectors, by generalizing the Radiative Distribution Function Identification method of Tancrez and Taine [M. Tancrez, J. Taine, Direct identification of absorption and scattering coefficients and phase function of a porous medium by a Monte Carlo technique, Int. J. Heat Mass Transfer 47 (2004) 373–383]. The validity and accuracy of the associated Beer’s laws are discussed in this case. In a second step, at the limit of an optically thick porous medium, an original model based on a perturbation method of the Radiative Transfer Equation directly leads to the determination, under an accurate validity criterion, of a radiative conductivity tensor for the fluid phase. Examples of results are given in the case of rod bundles versus porosity, specific area and local wall absorptivity.     

Modeling of the radiative contribution to heat transfer in porous media composed of spheres or cylinders

A theoretical model for evaluating the radiative conductivity tensor of a porous media is developed in this paper. The porous media is composed of a transparent fluid and opaque particles with characteristic lengths longer than the radiation wavelength. The main features of the proposed approach are (i) take into account the interaction between conduction and radiation heat transfers, (ii) allow the modeling of the radiative transfer in anisotropy system such as an assembly of cylinders, and (iii) have an easy numerical implementation into the energy equations of the porous media. In order to study the accuracy of the approach, the paper evaluates the model for porous media composed of spheres or cylinders. The predictions of the model agree well with experimental data and with results obtained from finite element simulations. The numerical results also show that the radiative conductivity can be strongly influence by the effect of temperature distribution across the particle surface and by the effect of the multiple scattering of radiation in the porous media.     

DRESOR法对含有边界的一维各向异性散射介质辐射传递方程的求解

通过DRESOR(d istribu tions of ratios of energy scattered or reflected)法求解一维黑体边界发射、介质各向异性散射的辐射传递问题,与理论解、辐射元法、蒙特卡洛方法和有限体积法的计算结果进行比较,验证了该方法计算的准确性;同时,对DRESOR法给出的高方向分辨率的辐射强度进行了分析。计算了带有漫反射边界介质具有吸收、发射、各向异性散射特性的辐射传递问题,对不同工况下的辐射强度和辐射热流进行了比较分析。     

Inverse radiation problem of temperature field in three-dimensional rectangular enclosure containing inhomogeneous,anisotropically scattering media

An inverse radiation analysis is presented for determining the three-dimensional temperature field in an inhomogeneous, absorbing, emitting and anisotropically scattering media of known radiative properties from the knowledge of the exit radiative energy received by charge-coupled device (CCD) cameras at boundary surfaces. The forward Monte Carlo method was employed to describe the radiative energy propagation. The inverse problem was formulated as an ill-posed matrix equation and solved by least square QR decomposition (LSQR) method. The measured data were simulated by adding random errors to the exact solution of the direct problem. The effects of measurement errors, combinations of CCD cameras, concentration distributions of particles, and coefficient fluctuating errors on the accuracy of the inverse problem were investigated. The results show that the three-dimensional temperature field can be estimated accurately, even for the noisy data.     

Experimental characterization of thermal radiation properties of dispersed media

As dispersed materials generally are semi-transparent media which absorb, emit and scatter thermal radiation, the predictive modeling of thermal processes involving such kind of materials requires the knowledge of a number of radiative properties to feed the models. These properties cannot be directly measured but are identified from a set of experimental data of radiative flux collected from a sample submitted to appropriate experimental conditions. This paper focuses on identification methodology for thermal radiation properties of dispersed media such as fibers, foams, pigmented coatings, ceramics. After a brief introductive overview of the subject, the parameter identification methodology and two experimental facilities used for radiative properties determination are firstly described. As the identification process involves a solution model for the Radiative Transfer Equation inside the sample, some attention is then paid to the development of RTE solution models well matched to this specific purpose. Two examples of application are described before concluding on the advantages, limitations and remaining difficulties connected to this new and promising metrology of thermal radiative properties of dispersed media.     

Generalized radiative transfer equation for porous medium upscaling: Application to the radiative Fourier law

Many porous media cannot be homogenized as Beerian semi-transparent media. Effective extinction, absorption and scattering coefficients can indeed have no physical meaning for small or intermediate optical thicknesses. A generalized radiative transfer equation (GRTE), directly based on the extinction cumulative distribution function, the absorption and scattering cumulative probabilities and the scattering phase function is established for this optical thickness range. It can be solved by a statistical Monte Carlo approach. For a phase of a porous medium that is optically thick at local scale, the GRTE degenerates into a classical Beerian RTE. In these conditions, a radiative conductivity tensor is directly obtained, by a perturbation method, and expressed with the radiative coefficients of this RTE and temperature. As illustrations, exhaustive radiative conductivity results are given for a set of overlapping transparent spheres within an opaque phase and for opaque rod bundles.     

Bulk and surface light scattering from transparent silica aerogel

Elastic light scattering has been used to study structural properties of different transparent aerogels, which may be used as filling materials in super-windows. With a goniometer having an angular resolution better than 0.6° and a He-Ne laser as the light sourcewe investigated the angular distribution of scattered intensity from transparent silica aerogels and one xerogel. The densities ranged between 0.11 and 0.60 g cm^-3. An exponential correlation function for the density fluctuations of a random porous medium has been utilised to analyse the large-angle scattering, which is dominated by bulk scattering, for different polarisation of the incident light. The determination of correlation lengths in the nanometre range was possible, because the absolute scattering intensities were determined. For relative angular dependence measurements, this range would have been accessible only to small angle X-ray scattering (SAXS). The resulting mean pore sizes between 8 nm and 50 nm and specific surface areas between 500 and 700 m²/g agree well with nitrogen-porosimetry data from the literature.The data compare quite well with correlation lengths calculated from specular transmittance data from an ordinary spectrophotometer. This method, which is not sensitive to the angular distribution of superposed forward scattering with large correlation lengths, has also been applied to a series of base-catalysed TMOS aerogels with different catalystconcentrations.The forward scattering peak of the signal may be attributed to correlation lengths in the micrometre range. Experimental results for aerogel surfaces with evaporated aluminium indicate that this might be due to the surface properties. A quantitative analysis, however, is not possible yet.     

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.     

Closure of non-equilibrium volume-averaged energy equations in high-conductivity porous media

A formulation for closure of the volume-averaged energy equations under local thermal non-equilibrium conditions has been proposed, providing a general method for obtaining the relevant effective properties for high-conductivity porous materials. The method requires numerical solution of the flow field, as well as closure problems, over a representative volume of porous media. This study considers an array of highly conductive circular cylinders for which all relevant effective properties are obtained. Comparisons are made to results in literature for low Reynolds numbers, while results at higher Reynolds numbers are presented to provide insight into the effects of inertia on the effective properties.     

Transient thermal investigation of a fully wet porous convective–radiative rough cylindrical pin fin

The microelectromechanical systems technologies frequently produce rough surfaces, and the repercussion of roughness on the thermal performance is more prominent in structures of smaller dimensions. In this regard, the present article intends to examine the unsteady thermal behavior of a fully wet, porous, and rough micropin-fin structure under convective–radiative conditions. Here, a pin fin of a cylindrical profile has been chosen. The problem is modeled by incorporating the roughness parameters in the perimeter and cross-sectional area of the pin fin. Further, the study of the porous structure has been carried out by implementing the Darcy model. The resulting partial differential equation is nonlinear and of the second order which has been solved by employing the finite difference method. The impact of the roughness parameter, wet porous parameter, dimensionless time, convective parameter, base radius-to-length ratio, radiative parameter, thermal conductivity parameter, power index, and ambient temperature on the thermal performance and efficiency of rough micropin-fin structures has been established graphically. According to the findings, for $0.15\%$ rise in roughness, the rough micropin fin has $12\%$ more thermal drop rate and $13\%$ less efficiency than the smooth one. Further, the work is beneficial in the field of microelectronics, especially in the design of micropin-fin structures.     

Scaling anisotropic scattering in radiative transfer in three-dimensional nonhomogeneous media

Radiative heat transfer in three-dimensional nonhomogeneous participating medium was investigated by using REM² method. The anisotropic scattering phase function was dealt with the scaling technique based on delta function approximation. The three-dimensional scaled isotropic results were compared with the published anisotropic scattering computations. A good agreement between the scaled isotropic approaches and the anisotropic solutions was found. The effects of scattering albedo, forward fraction of phase function, and wall emissivity were discussed. It was found that, with the increase of the scattering albedo, the radiative heat flux increases for forward scattering media, but decreases for backward scattering media. The radiative heat flux is increased with the increases of forward fraction of phase function and wall emissivity. The emissive power at the center of a cubical nonhomogeneous medium in radiative equilibrium with gray diffuse walls equals to the averaged blackbody emissive power of the six walls.     

NUMERICAL MODEL FOR RADIATIVE HEAT TRANSFER ANALYSIS IN ARBITRARILY SHAPED AXISYMMETRIC ENCLOSURES WITH GASEOUS MEDIA

A numerical model is presented for evaluating thermal radiative transport in irregularly shaped axisymmetric enclosures containing a homogeneous, isotropically scattering medium. Based on the discrete exchange factor (DEF) method, exchange factors between arbitrarily oriented differential surface/volume ring elements are calculated using a simple approach. The present method is capable of addressing blockage effects produced by inner/outer obstructing bodies. The results obtained via the current method are found to be in excellent agreement with existing solutions to several cylindrical media benchmark problems. The solutions to several rocket-nozzle and plug-chamber geometries are presented for a host of geometric conditions and optical thicknesses.     

Simultaneous estimation of parameters in analyzing porous medium combustion—assessment of seven optimization tools

Simultaneous estimation of the gas velocity, scattering albedo, and downstream pore diameter in a two-layer planar porous matrix involving combined mode conduction, convection, and radiation heat transfer with combustion is reported. Non-local thermal equilibrium between the gas and the solid phase is accounted by separate energy equations for the two phases. Performances of the genetic algorithm, genetic algorithm parallel, simulated annealing, multiple starting point algorithm parallel, pattern search algorithm, pattern search algorithm parallel, and global search algorithm in the simultaneous estimation of three parameters are analyzed. All the algorithms utilize a priori knowledge of the axial gas temperature distribution, and the magnitudes of the convective and the radiative heat fluxes at the outer surface of the porous matrix. With volumetric radiative information needed in the solid phase energy equation computed using the discrete transfer method, the two energy equations are simultaneously solved using the finite volume method. Pattern search algorithm provides better estimation, and computationally it is also the fastest.     

多孔介质特性参数对燃烧温度及压力影响的数值模拟研究

考查了两段式多孔介质内预混气燃烧的温度与压力分布情况。建立了甲烷／空气预混气体在多孔介质内燃烧的二维数学模型,运用FLUENT软件求解瞬态控制方程的方法计算出燃烧稳定后多孔介质内的温度、与压力分布,并考查了不同当量比、多孔介质辐射衰减系数和导热系数对温度和压力分布的影响。结果表明,甲烷／空气预混气体在多孔介质中燃烧,当量比越大温度峰值越高,压力梯度越大;小孔介质辐射衰减系数的改变对温度分布和压力分布没有明显的影响,而大孔介质辐射衰减系数对温度分布和压力分布有较大的影响;增加多孔介质的导热系数,会使固相与气相温度均有所升高,燃烧区域压力降低。     

Radiative transfer within non Beerian porous media with semitransparent and opaque phases in non equilibrium: Application to reflooding of a nuclear reactor

A local radiative transfer model is developed for strongly anisotropic porous media with an opaque phase and a mixture of two semitransparent phases. At the optically thick limit, the homogenized phase associated with the opaque interfaces is characterized by generalized extinction and scattering coefficients at equilibrium, a phase function and an effective refraction index, by following the model of Taine et al. [1] for non Beerian media. The radiative transfer model is based on a Radiative Transfer Equation (RTE) with three source terms, which are associated with the temperature fields of the opaque interfaces and the two semitransparent phases. This RTE has been solved by a perturbation technique, which allows radiative interfacial fluxes and radiative powers per unit volume, that are exchanged between phases, to be computed at local scale. The main contributions are obtained at zeroth order perturbation. Corrective contributions at first order perturbation are also determined: Radiative fluxes and powers are then expressed from coupled Fourier’s laws, which are characterized by radiative conductivity tensors associated with each phase.Illustrative results are given for the radiative modeling of reflooding of a damaged nuclear reactor core. Pragmatic conclusions on the cooling efficiency by steam and water droplets are finally given.