Gray Approximation Research Articles

A Radiative Transfer Module for Relativistic Magnetohydrodynamics in the PLUTO Code

Abstract We present a numerical implementation for the solution of the relativistic radiation hydrodynamics and magnetohydrodynamics equations, designed as an independent module within the freely available code PLUTO. The radiation transfer equations are solved under the gray approximation and imposing the M1 closure, which allows the radiation transport to be handled in both the free-streaming and diffusion limits. Equations are integrated following an implicit–explicit scheme, where radiation–matter interaction terms are integrated implicitly, whereas transport and all of the remaining source terms are solved explicitly by means of the same Godunov-type solvers included in PLUTO. Among these, we introduce a new Harten–Lax–van Leer–contact (HLLC) solver for optically thin radiation transport. The code is suitable for multidimensional computations in Cartesian, spherical, and cylindrical coordinates using either a single processor or parallel architectures. Adaptive grid computations are also made possible by means of the CHOMBO library. The algorithm performance is demonstrated through a series of numerical benchmarks by investigating various different configurations with a particular emphasis on the behavior of the solutions in the free-streaming and diffusion limits.

Modulated heat conduction in a two-layer dielectric system with dynamical interface thermal resistance

Heat conduction in a two-layer dielectric system excited with a laser beam of modulated intensity is studied in terms of a dynamical interface thermal resistance predicted by the phonon Boltzmann transport equation under the gray relaxation time approximation. This is done by using accurate expressions for both the modulated temperature and heat flux profiles, which describe both the diffusive and ballistic regimes of heat transport. It is shown that (i) for modulation frequencies much smaller than the phonon collision frequency f1 of the finite layer, the values of this dynamical resistance in the pure ballistic regime agree well with those of the diffuse mismatch model, while they differ by about 10% in the diffusive one. (ii) In the diffusive regime, the thermal resistance reaches a maximum at the characteristic modulation frequency fc≃(10/2π)(l1/L)2f1, where l1 and L are the phonon mean free path and thickness of the finite layer, respectively. This maximum thermal resistance is associated with the minimum of the modulated heat flux at the interface. The theoretical basis is used to establish a methodology to determine the dominant thermal relaxation time and phonon mean free path of the finite layer. The obtained results can thus be applied for describing the modulated heat conduction in dielectric thin films through the comparison of our theoretical model with experimental data measured by thermoreflectance or other relevant photothermal techniques.

Assessment of SLW-1 model in the presence of gray and non-gray particles

In this study, predictive accuracy of Gray Gas and SLW-1 approximations is benchmarked against Spectral Line-Based Weighted Sum of Grey Gases Model (SLW) in multidimensional enclosures involving gray/non-gray absorbing, emitting and scattering particles. Input data required for the radiation code and its validation are provided from two combustion tests previously carried out in a 300 kWt Atmospheric Bubbling Fluidized Bed Combustor (ABFBC) test rig burning low calorific value Turkish lignite with high volatile matter/fixed carbon (VM/FC) ratio in its own ash. Comparisons reveal that SLW-1 approximation leads to one order of magnitude higher accuracy in heat flux and source term predictions compared to that of Gray Gas approximation in the presence of gray particles while maintaining a similar computational efficiency. In the presence of non-gray particles, SLW-1 approximation is again found to improve the predictive accuracy compared to that of Gray Gas approximation. However, it is seen that errors in heat flux and source term predictions with SLW-1 approximation are noticeably higher in the presence of non-gray particles compared to those with gray particles.

An integrated MCDM approach considering demands-matching for reverse logistics

Reverse logistics (RL) has been regarded as a key driving force for remanufacturing. However, there are great uncertainties in terms of quality and quantity of used components for RL. There are also complexities in suppliers and operations. These make decision-making of RL very complex. In order to identify the best collection mode for used components, a demand-matching oriented Multiple Criteria Decision Making (MCDM) method is established. In this method, the damage level and remaining service life are firstly incorporated into the evaluation criteria of reuse modes, then a hybrid method (AHP-EW) that integrates Analytic Hierarchy Process (AHP) and Entropy Weight (EW) method is applied to derive criteria weights and the grey Multi-Attributive Border Approximation Area Comparison (MABAC) is adopted to rank the collection modes. Finally, sensitivity analysis is implemented to test the stability of the proposed method, and a demands-matching method is proposed to validate and evaluate the feasibility of the optimal alternative. The collection of used pressurizers is taken as case study to validate the applicability of the proposed model. The results showed the effectiveness of the proposed method in MCDM of RL.

Numerical investigation of ballistic-diffusive heat transfer through a constriction with the Boltzmann transport equation

Thermal constriction resistance plays an important role in the study of thermal contact resistance. It also helps with better understanding and tuning the thermal properties of nano porous structures. For investigating the size effect on thermal constriction resistance, the phonon Boltzmann transport equation (BTE) can be adopted as it can cover the ballistic to diffusive regime. In this research, by incorporating phonon dispersion and polarization, the nongray phonon BTE is adopted to reveal details of thermal transport through a constriction. The numerical scheme is implemented over different ranges of the Knudsen numbers and the constriction ratios. The results of the gray, two- and five-band nongray BTE simulations for different constriction widths are compared. Under the gray assumption, it is found that with the same geometry a more diffusive case will result in smaller resistance. In the ballistic limit, with the same total contact area, the number of contact points does not affect the constriction resistance. In the diffusive limit, more contact points will lead to smaller resistance. For the nongray cases, the contribution of each band is analyzed. It is shown that in the ballistic limit, the gray approximation results reasonably agree with the nongray results. There is larger difference between the gray and the nongray results for a larger constriction width. While the conductance for larger constriction width increases for all bands, the percentage contribution of the conductance of each band can be increased or decreased according to its mean free path.

Effective interface thermal resistance and thermal conductivity of dielectric nanolayers

Analytical expressions for the effective interface thermal resistance (ITR) and thermal conductivity k of dielectric nanolayers are derived and analyzed, based on the analytical solution of the phonon Boltzmann transport equation under the gray relaxation time approximation. This is achieved by using accurate expressions for the temperature and one-dimensional heat flux propagating across nanolayers supporting a diffusive phonon scattering at their interfaces. It is shown that the effective ITR between two layers can be symmetric on their thermal properties, such that its asymptotic value in the ballistic regime is higher than that in the diffusive one. In the ballistic-diffusive regime, the effective ITR depends strongly on the ratio λ=L/l, between the layer thickness L and mean free path l of phonons. Our predictions for the effective ITR in the ballistic regime are in good agreement with those of the diffuse mismatch model, while they differ by about 16% in the diffusive regime. On the other hand, k increases with λ until reaching saturation for bulk layers and agrees rather well with previous predictions reported in the literature. The obtained results could be useful for analytically describing the heat transport in dielectric nanothin films and superlattices, in which the gray approximation is valid.

Comparison of approximate solutions to the phonon Boltzmann transport equation with the relaxation time approximation: Spherical harmonics expansions and the discrete ordinates method

The phonon Boltzmann transport equation is used to analyze model problems in one and two spatial dimensions, under transient and steady-state conditions. New, explicit solutions are obtained by using the P1 and P3 approximations, based on expansions in spherical harmonics, and are compared with solutions from the discrete ordinates method. For steady-state energy transfer, it is shown that analytic expressions derived using the P1 and P3 approximations agree quantitatively with the discrete ordinates method, in some cases for large Knudsen numbers, and always for Knudsen numbers less than unity. However, for time-dependent energy transfer, the PN solutions differ qualitatively from converged solutions obtained by the discrete ordinates method. Although they correctly capture the wave-like behavior of energy transfer at short times, the P1 and P3 approximations rely on one or two wave velocities, respectively, yielding abrupt, step-changes in temperature profiles that are absent when the angular dependence of the phonon velocities is captured more completely. It is shown that, with the gray approximation, the P1 approximation is formally equivalent to the so-called “hyperbolic heat equation.” Overall, these results support the use of the PN approximation to find solutions to the phonon Boltzmann transport equation for steady-state conditions. Such solutions can be useful in the design and analysis of devices that involve heat transfer at nanometer length scales, where continuum-scale approaches become inaccurate.

The direct collocation meshless method based on a second-order phonon Boltzmann equation for ballistic–diffusive phonon heat transport

A second-order phonon Boltzmann equation (SOPBE) is proposed based on the phonon Boltzmann equation (PBE) under gray relaxation-time approximation, and the direct collocation meshless (DCM) method is employed to solve the SOPBE. Several numerical tests for phonon transport over a broader range of Knudsen numbers under different boundary conditions are carried out. The results show that the SOPBE solved by the DCM method is applicable for different transport regimes. Moreover, it overcomes the numerical error “ray effects” of other numerical methods under the ballistic limit to a certain extent. When modeling phonon transport in materials with inhomogeneous acoustic property, the superiority of SOPBE will be more obvious compared with the PBE. The results demonstrate the capability of our methodology for ballistic–diffusive phonon heat transport.

Albedo of an irradiated plane-parallel atmosphere with finite optical depth

Abstract We analytically derive albedo for a plane-parallel atmosphere with finite optical depth, irradiated by an external source, under the local thermodynamic equilibrium approximation. Albedo is expressed as a function of the photon destruction probability ε and optical depth τ, with several parameters such as dilution factors of the external source. In the particular case of the infinite optical depth, albedo A is expressed as $A=[1 + (1-W_J/W_H)\sqrt{3\varepsilon }/3]/(1+\sqrt{3\varepsilon })$, where WJ and WH are the dilution factors for the mean intensity and Eddington flux, respectively. An example of a model atmosphere is also presented under a gray approximation.

Numerical simulations of microgravity ethylene/air laminar boundary layer diffusion flames

Microgravity ethylene/air laminar boundary layer diffusion flames were studied numerically. Two oxidizer velocities of 250 and 300 mm/s and three fuel injection velocities of 3, 4, and 5 mm/s were considered. A detailed gas-phase reaction mechanism, which includes aromatic chemistry up to four rings, was used. Soot kinetics was modeled by using a pyrene-based model including the mechanisms of nucleation, heterogeneous surface growth and oxidation following the hydrogen-abstraction acetylene-addition (HACA) mechanism, polycyclic aromatic hydrocarbon (PAH) surface condensation and soot particle coagulation. Radiative heat transfer from CO, CO2, H2O and soot was calculated using the discrete ordinate method (DOM) coupled to a wide-band correlated-k model. Model predictions are in quantitative agreement with the available experimental data. Model results show that H and OH radicals, responsible for the dehydrogenation of sites in the HACA process, and pyrene, responsible for soot nucleation and PAH condensation, are located in a thin region that follows the stand-off distance. Soot is produced in this region and, then, is transported inside the boundary layer by convection and thermophoresis. The combustion efficiency is significantly lower than 1 and is reduced as the flow residence time increasing, confirming that these sooting micro-gravity diffusion flames are characterized by radiative quenching at the flame trailing edge. In particular, this quenching phenomenon explains the increase in flame length with the oxidizer velocity observed in previous experimental studies. The effects of using approximate radiative-property models, namely the optically-thin approximation and gray approximations for soot and combustion gases, were assessed. It was found that the re-absorption and the spectral dependence of combustion gases and soot must be taken into account to predict accurately temperature, soot volume fraction, flame geometry and flame quenching.

A Demands-Matching Multi-Criteria Decision-Making Method for Reverse Logistics

A demand matching oriented Multi-Criteria Decision-Making method is presented to identify the best collection mode for used components. In this method, the damage condition and remaining service life are incorporated into the evaluation criteria of reuse mode, then a hybrid method (AHP-EW) integrating Analytic Hierarchy Process (AHP) and Entropy Weight (EW) is used to derive the criteria weights and the grey Multi-Attributive Border Approximation Area Comparison (MABAC) is adopted to rank the collection modes. Finally, a sensitivity analysis is used to test the stability of the method and a demands-matching method is proposed to validate the feasibility of the optimal alternative. The method is validated using the collection of used pressurizers as case study. The results of which show the effectiveness of the proposed method.

A limit problem for three-dimensional ideal compressible radiation magneto-hydrodynamics

General radiation magnetic hydrodynamics models include two main parts that are coupled: one part is the macroscopic magnetic fluid part, which is governed by the ideal compressible magnetohydrodynamic (MHD) equations with additional radiation terms; another part is the radiation field, which is described by a transfer equation. It is well known that in radiation hydrodynamics without a magnetic field there are two physical approximations: one is the so-called P1 approximation and the other is the so-called gray approximation. Starting out with a general radiation MHD model one can derive the so-called MHD-P1 approximation model. In this paper, we study the non-relativistic type limit for this MHD-P1 approximation model since the speed of light is much larger than the speed of the macroscopic fluid. This way we achieve a rigorous derivation of a widely used macroscopic model in radiation magnetohydrodynamics.

The equilibrium-diffusion limit for radiation hydrodynamics

The equilibrium-diffusion approximation (EDA) is used to describe certain radiation-hydrodynamic (RH) environments. When this is done the RH equations reduce to a simplified set of equations. The EDA can be derived by asymptotically analyzing the full set of RH equations in the equilibrium-diffusion limit. We derive the EDA this way and show that it and the associated set of simplified equations are both first-order accurate with transport corrections occurring at second order. Having established the EDA’s first-order accuracy we then analyze the grey nonequilibrium-diffusion approximation and the grey Eddington approximation and show that they both preserve this first-order accuracy. Further, these approximations preserve the EDA’s first-order accuracy when made in either the comoving-frame (CMF) or the lab-frame (LF). While analyzing the Eddington approximation, we found that the CMF and LF radiation-source equations are equivalent when neglecting O(β2) terms and compared in the LF. Of course, the radiation pressures are not equivalent. It is expected that simplified physical models and numerical discretizations of the RH equations that do not preserve this first-order accuracy will not retain the correct equilibrium-diffusion solutions. As a practical example, we show that nonequilibrium-diffusion radiative-shock solutions devolve to equilibrium-diffusion solutions when the asymptotic parameter is small.

The relative influence of H2O and CO2 on the primitive surface conditions and evolution of rocky planets

AbstractHow the volatile content influences the primordial surface conditions of terrestrial planets and, thus, their future geodynamic evolution is an important question to answer. We simulate the secular convective cooling of a 1‐D magma ocean (MO) in interaction with its outgassed atmosphere. The heat transfer in the atmosphere is computed either using the grey approximation or using a k‐correlated method. We vary the initial CO2 and H2O contents (respectively from 0.1 × 10−2 to 14 × 10−2 wt % and from 0.03 to 1.4 times the Earth Ocean current mass) and the solar distance—from 0.63 to 1.30 AU. A first rapid cooling stage, where efficient MO cooling and degassing take place, producing the atmosphere, is followed by a second quasi steady state where the heat flux balance is dominated by the solar flux. The end of the rapid cooling stage (ERCS) is reached when the mantle heat flux becomes negligible compared to the absorbed solar flux. The resulting surface conditions at ERCS, including water ocean's formation, strongly depend both on the initial volatile content and solar distance D. For D > DC, the “critical distance,” the volatile content controls water condensation and a new scaling law is derived for the water condensation limit. Although today's Venus is located beyond DC due to its high albedo, its high CO2/H2O ratio prevents any water ocean formation. Depending on the formation time of its cloud cover and resulting albedo, only 0.3 Earth ocean mass might be sufficient to form a water ocean on early Venus.

Modeling quasi-ballistic transient thermal transport with spatially sinusoidal heating: A McKelvey-Shockley flux approach

Ballistic phonon effects, arising on length scales comparable to the mean-free-path, result in non-diffusive heat flow and alter the thermal properties of materials. Simple theoretical models that accurately capture non-diffusive transport physics are valuable for experimental analysis, technology design, and providing physical insight. In this work, we utilize and extend the McKelvey-Shockley (McK-S) flux method, a simple and accurate framework, to investigate ballistic effects in transient phonon transport submitted to a spatially sinusoidal heating profile, simulating a transient thermal grating. We begin by extending a previous McK-S formulation to include inelastic scattering, then obtain an analytical solution in the single phonon energy case (gray approximation), and after show how this approach can readily support a full phonon dispersion and mean-free-path distribution. The results agree with experimental data and compare very well to solutions of the phonon Boltzmann transport equation in the diffusive and weakly quasi-ballistic transport regimes. We discuss the role of ballistic and non-equilibrium physics, and show that inelastic scattering is key to retrieving the heat equation solution in the diffusive limit. Overall the McK-S flux method, which takes the form of a diffusion-like equation, proves to be a simple and accurate framework that is applicable from the ballistic to diffusive transport regime.

Benchmarking grey particle approximations against nongrey particle radiation in circulating fluidized bed combustors

ABSTRACTInvestigation of the effect of grey/nongrey particle property models on radiative heat fluxes and source terms is performed in the dilute zone of the lignite-fired 150 kW Middle East Technical University circulating fluidized bed combustor test rig. Predictive accuracy and computational economy of several grey particle models, geometric optics approximation (GOA) with average particle reflectivity (GOA2), GOA with Fresnel solution for particle reflectivity (GOA3), and Planck mean particle properties from spectral Mie solution are tested by benchmarking their predictions against spectrally banded solution of radiative transfer equation (RTE). Comparisons reveal that all grey models lead to accurate and CPU efficient radiative heat flux predictions. On the other hand, only GOA3 and Planck mean properties are in favorable agreement with the benchmark solution for both incident fluxes and source terms. These findings indicate that grey particle approximation with GOA3 is a more practical choice in solution of RTE as it eliminates the need for spectral calculations.

Invariant for one-dimensional heat conduction in dielectrics and metals

We theoretically and experimentally demonstrate that the one-dimensional heat conduction in dielectrics and metals is ruled by the invariant , where T is the temperature and z an arbitrary position within the heated material of length L. This is achieved using the integral expressions predicted by the Boltzmann transport equation, under the gray relaxation time approximation, for the steady-state temperature and heat flux, and measuring the temperature at three equidistant positions in rods of Si, Cu, and Fe-C excited with temperatures much smaller than their corresponding Debye ones. The obtained temperature invariant for symmetrical positions could be applied to describe the heating of materials supporting one-dimensional heat conduction.

Effect of multiphase radiation on coal combustion in a pulverized coal jet flame

The accurate modeling of coal combustion requires detailed radiative heat transfer models for both gaseous combustion products and solid coal particles. A multiphase Monte Carlo ray tracing (MCRT) radiation solver is developed in this work to simulate a laboratory-scale pulverized coal flame. The MCRT solver considers radiative interactions between coal particles and three major combustion products (CO2, H2O, and CO). A line-by-line spectral database for the gas phase and a size-dependent nongray correlation for the solid phase are employed to account for the nongray effects. The flame structure is significantly altered by considering nongray radiation and the lift-off height of the flame increases by approximately 35%, compared to the simulation without radiation. Radiation is also found to affect the evolution of coal particles considerably as it takes over as the dominant mode of heat transfer for medium-to-large coal particles downstream of the flame. To investigate the respective effects of spectral models for the gas and solid phases, a Planck-mean-based gray gas model and a size-independent gray particle model are applied in a frozen-field analysis of a steady-state snapshot of the flame. The gray gas approximation considerably underestimates the radiative source terms for both the gas phase and the solid phase. The gray coal approximation also leads to under-prediction of the particle emission and absorption. However, the level of under-prediction is not as significant as that resulting from the employment of the gray gas model. Finally, the effect of the spectral property of ash on radiation is also investigated and found to be insignificant for the present target flame.

Second-order discretization in space and time for radiation-hydrodynamics

Second-order accurate discretizations for radiation-hydrodynamics are currently an area of interest in the high energy density laboratory physics and astrophysics communities. Second-order methods used to solve the hydrodynamics equations and second-order methods used to solve the radiation transport equation often differ fundamentally, making it difficult to combine them in a second-order manner. Here, we present an implicit–explicit (IMEX) method for solving the 1D equations of radiation-hydrodynamics that is second-order accurate in space and time. Our radiation-hydrodynamics model consists of the 1D Euler equations coupled with a gray radiation S2 approximation. Our RH method combines the MUSCL-Hancock method for solving the Euler equations with the TR/BDF2 time integration scheme and the linear-discontinuous Galerkin finite-element spatial discretization scheme for the S2 radiation equations. The MUSCL-Hancock method is commonly used for hydrodynamic calculations and the linear-discontinuous Galerkin scheme is the standard for the Sn equations of radiative transfer. While somewhat similar, these schemes vary fundamentally with respect to the treatment of spatial slopes. We address the challenges inherent to coupling these different numerical methods and demonstrate how these challenges can be overcome. Using the method of manufactured solutions, we show that the method is second-order accurate in space and time for both the equilibrium diffusion and streaming limit, and we show that the method is capable of computing radiative shock solutions accurately by comparing our results with semi-analytic solutions.

Thermal transport at the nanoscale: A Fourier's law vs. phonon Boltzmann equation study

Steady-state thermal transport in nanostructures with dimensions comparable to the phonon mean-free-path is examined. Both the case of contacts at different temperatures with no internal heat generation and contacts at the same temperature with internal heat generation are considered. Fourier's law results are compared to finite volume method solutions of the phonon Boltzmann equation in the gray approximation. When the boundary conditions are properly specified, results obtained using Fourier's law without modifying the bulk thermal conductivity are in essentially exact quantitative agreement with the phonon Boltzmann equation in the ballistic and diffusive limits. The errors between these two limits are examined in this paper. For the four cases examined, the error in the apparent thermal conductivity as deduced from a correct application of Fourier's law is less than 6%. We also find that the Fourier's law results presented here are nearly identical to those obtained from a widely used ballistic-diffusive approach but analytically much simpler. Although limited to steady-state conditions with spatial variations in one dimension and to a gray model of phonon transport, the results show that Fourier's law can be used for linear transport from the diffusive to the ballistic limit. The results also contribute to an understanding of how heat transport at the nanoscale can be understood in terms of the conceptual framework that has been established for electron transport at the nanoscale.