Radiative Heat Transfer Equation Research Articles

Experimental and Numerical Study of the Flow Field and Temperature Field for a Large-Scale Radiant Syngas Cooler

Experimental and numerical studies on the gas-particle flow field and temperature field in the industrial-scale radiant syngas cooler (RSC) have been carried out. The bench-scale cold model experiment was presented for measuring the gas flow field in the RSC. The accuracy and performance of four turbulent flow models were evaluated according to the comparison of predicted results with experimental results. The ash particle trajectories were predicted by the stochastic Lagrangian model and the interaction between the gas and particle phase was also considered by two-way coupling method. A discrete ordinate model (DOM) was used for solving the radiative heat transfer equation when the radiative properties were calculated by weighted-sum-of-gray-gases model (WSGGM). The Ranz−Marshall correlation for the Nusselt number was used to account for convection heat-transfer between the gas phase and the particle phase. The ash particle radiative heat transfer was also considered. The physical properties of gas mixtu...

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.

Nonequilibrium Calculation of High-Temperature Radiating H2-He Flowfield

Chemical kinetics of high-temperature hydrogen-helium gas mixture behind a shock wave is numerically investigated by integrating rate equations for species concentration in time. State-to-state transition rates are used to determine quasi-steady-state rate coefficients for atomic hydrogen ionization. The electron concentrations in front of the shock wave are deduced by solving the radiative heat transfer equation of molecular hydrogen. The computed incubation time of avalanche ionization is compared with the experimental data and those appeared in past studies. It is found that the precursor photoionization and the associative ionization of the molecular hydrogen are important to determine the ionization time behind the shock wave. The present chemical kinetic models are found to reproduce the shock tube experimental data of the ionization time reasonably well.

Numerical modelling and simulation of pulverized solid‐fuel combustion in swirl burners

AbstractA finite‐volume numerical model for computer simulation of pulverized solid‐fuel combustion in furnaces with axisymmetric‐geometry swirl burner is presented. The simulation model is based on the k − ε single phase turbulence model, considering the presence of the dispersed solid phase via additional source terms in the gas phase equations. The dispersed phase is treated by the particle source in cell (PSIC) method. Solid fuel particle devolatilization, homogenous and heterogeneous chemical reaction processes are modelled via a global combustion model. The radiative heat transfer equation is also resolved using the finite volume method. The numerical simulation code is validated by comparing computational and experimental results of pulverized coal in an experimental furnace equipped with a swirl burner. It is shown that the developed numerical code can successfully predict the flow field and flame structure including swirl effects and can therefore be used for the design and optimization of pulverized solid‐fuel swirl burners.

The Existence and Behavior of Solutions for Nonlocal Boundary Problems

The purpose of this work is to investigate the uniqueness and existence of local solutions for the boundary value problem of a quasilinear parabolic equation. The result is obtained via the abstract theory of maximal regularity. Applications are given to some model problems in nonstationary radiative heat transfer and reaction-diffusion equation with nonlocal boundary flux conditions.

Existence Theory for Radiative Flows

We consider the coupling of radiative heat transfer equations and the energy equation for the temperature Tof a compressible fluid occupying a bounded convex region D with smooth boundary. Using the technique of upper and lower sequences associated with integro-parabolic equations, we establish the existence and uniqueness of a solution T, 0 ≤ Λ− ≤ T(x,t) ≤ Λ+ < ∞ with corresponding radiative intensity I(x,Ω,ν,t) where the total incident radiation satifies ∫S2 I(x,Ω,ν,t)dΩ = Sg B(ν,t)+Sb B(ν,Tb), and where Sb and Sg are positivity preserving linear operator, Tb is the external temperature of the boundary, and B is Planck's function. We also establish certain energy estimates for T.

Validation of a Rapid Thermal Processing model in steady-state

Rapid Thermal Processing (RTP) is widely used in advanced semiconductor manufacturing. The present work deals with the heat transfer from infrared lamps to the silicon wafer in a commercial RTP equipment. Both numerical and experimental approaches are considered. For numerical purposes, the RTP system is modelled in two (2D) and three dimensions (3D). Calculations are performed in steady-state. The computational fluid dynamics method (CFD) is used for solving the mass and heat conservation equations. The radiative heat transfer equation is solved with the Monte Carlo method. In order to validate these models, measurements of the wafer temperature are realized for five electric power values supplied to the infrared lamps. The experimental wafer temperature profiles are in good agreement with the numerically calculated ones. Moreover, a confrontation between the experimental temperature of the infrared lamp filaments evaluated from the Ohm law and the one used in the numerical calculations shows a good agreement with the 3D model. The slight difference observed with the 2D model is explained. So the numerical simulations are fully validated. Two relations are established in order to predict the power which has to be applied to infrared lamps to obtain the required wafer temperature.

Iterative and direct Chebyshev collocation spectral methods for one-dimensional radiative heat transfer

In this paper, the Chebyshev collocation spectral method for one-dimensional radiative heat transfer equation with participating media is presented; and sequentially the iterative and direct solvers are developed. Implementation of the new method shows its flexibility to complex problems: highly anisotropic and space-dependent scattering. The new solvers can provide exponential convergence in space and can capture large oscillations. Numerical results verified the high accuracy of the new method, and its competitive ability compared with other newly appeared methods.

Radiative Gas Dynamic Model of Hydrogen Laser-Supported Plasma Generator

Radiative gas dynamics of a hydrogen laser-supported plasma generator and radiative heating of the internal surface of the plasma generator are studied numerically. For this purpose, a radiative gas dynamic model of the laser-supported wave in the cylindrical channel of the laser-supported plasma generator is created. The model is based on the Navier-Stokes equations, energy conservation equation for heat conducting gas in chemical equilibrium, and the radiation heat transfer equation using the multigroup spectral approximation. The numerical study is performed for the typical parameters of the laser-supported plasma generator: input velocities u 0 = 10-200 m/s, pressure p = 1 atm, and CW CO 2 laser radiation power P L = 100-200 kW.

Comparison of the Finite-Volume and Discrete-Ordinate Methods and Diffusion Approximation for the Radiative Heat Transfer Equation

Comparison of the Finite-Volume and Discrete-Ordinate Methods and Diffusion Approximation for the Radiative Heat Transfer Equation

A two-scale method for radiative heat transfer in non-grey absorbing and emitting media

Multiscale radiative heat transfer (RHT) problems are formulated and methods to approximate their numerical solutions are developed. We focus on RHT problems in participating media with heterogeneous optical properties leading to both optically thick and thin regimes within the same media spectrum. By introducing a diffusive scale and using an asymptotic expansion in the RHT equations we formulate the simplified P N approximations. The optical spectrum is decomposed in wavelength bands and the RHT equations are solved for bands with low absorption while the simplified P N equations are solved for bands with high absorption. The hybrid models solve the multiscale RHT more accurately than the simplified P N approximations and with a computational costs less than using the full RHT solver. Accuracy and effectiveness of the proposed models are demonstrated on three-dimensional RHT problems arising in combustion systems.

Radiative and Conductive Heat Exchanges in High-Temperature Glass Melt with the Finite-Volume Method Approach. Influence of Several Spatial Differencing Schemes on RTE Solution

ABSTRACT This article is devoted to heat transfer involving radiation and conduction. Considering a nongray, purely absorbing medium, the radiative heat transfer equation (RTE) and the energy balance equation are both solved with the finite-volume method (FVM). The energy equation is coupled to the RTE through the radiative source term. Several differencing scheme efficiencies are discussed for the case of strong opacities. Validation of the model with various benchmarks simulating high-temperature glass melting, including one- and two-dimensional geometries, is achieved.

Effects of Boundary Conditions on Thermal Response of a Cellulose Acetate Layer Using Hottel’s Zonal Method

Energy can transfer internally by radiation in addition to conduction in translucent polymers. Since radiant propagation is very rapid, it can provide energy within the layer more quickly than diffusion by heat conduction. Thus, the transient thermal response of a layer for combined radiative and conduction may be extremely different from that of conduction alone. In this paper, the behavior of a heat conducting, absorbing, and emitting layer of Cellulose Acetate layer is investigated during the transient interval when both conductive and radiative heat transfer are considered. Specifically, the effects of boundary conditions on the response of the layer are considered here. These boundary conditions include both conductive boundary conditions, such as convection coefficient and convective fluid temperature, and radiation boundary conditions, like radiation surrounding temperature and specular reflectivity. To this end, the governing differential equations including the equation of radiative heat transfer within the material coupled to the transient energy equation with radiative terms are presented. The solution procedure is based on nodal analysis and Hottel’s zonal method extended by the ray tracing method. The transient energy equation including the radiative internal energy source is solved using a time marching finite difference procedure with variable space and time increments.

Numerical investigation on optical and heat transfer characteristics of a rapid thermal annealing system for LCD manufacturing

The objectives of present study are to propose a method to evaluate the quality of polycrystalline silicon film by using the thin film optics analysis, and to investigate heat transfer characteristics in a rapid thermal annealing system for liquid crystal display (LCD) manufacturing. The transmittance of polycrystalline silicon film is calculated by the characteristic transmission matrix method, and predicted results are compared with the experimental data for two different samples. The transient one-dimensional conduction and radiation heat transfer equations are additionally solved to predict the surface temperature of thin films. The two-flux method is employed to evaluate radiation heat transfer, and the ray-tracing method is utilized to take into account of the wave interference effect. As the film thickness increases, the peak transmittance value increases and the wavelength where the peak appears becomes longer due to wave interference. These characteristics can be used for in situ and practical estimation of the extent of crystallization of the silicon film during the process. From the thermal analysis, it is shown that the selective heating in the multilayer film structure acts as an important mechanism during the annealing of silicon film deposited on the glass.

Partial moment entropy approximation to radiative heat transfer

We extend the half moment entropy closure for the radiative heat transfer equations presented in Dubroca and Klar [B. Dubroca, A. Klar, Half moment closure for radiative transfer equations, J. Comput. Phys. 180 (2002) 584–596] and Turpault et al. [R. Turpault, M. Frank, B. Dubroca, A. Klar, Multigroup half space moment approximations to the radiative heat transfer equations, J. Comput. Phys. 198 (2004) 363–371] to multi-D. To that end, we consider a partial moment system with general partitions of the unit sphere closed by an entropy minimization principle. We give physical and mathematical reasons for this choice of model and study its properties. Several numerical examples in different physical regimes are presented.

Formula omitted] solutions of radiative heat transfer in a slab with reflective boundaries

The spherical harmonics method is used to obtain solution for the radiative heat transfer equation for a slab with reflective boundaries. An absorbing, emitting, non-isothermal, gray medium is considered with linearly anisotropic scattering. Under the condition of the thermal equilibrium, the slab boundaries are subjected to specular and diffuse reflection. The analytical form of solutions is obtained for both conservative and non-conservative cases. The accuracy of the method was verified by benchmark comparisons against the solutions of an earlier work performed by the normal-mode expansion technique. The present predictions of heat flux were found to be in good agreement with the benchmark data.

Transient radiative and conductive heat transfer in non-gray semitransparent two-dimensional media with mixed boundary conditions

This paper is devoted to transient heat transfer involving radiation and conduction. Considering a non-gray purely absorbing media, the radiative heat transfer equation (RTE) is solved iteratively with the Discrete Ordinates Method (DOM) using an exponential differencing scheme. The energy balance equation is used to compute temperature at each time step with the Crank–Nicholson technique. Energy equation is coupled to the RTE through the radiative source term. Both equations are discretized with finite differencing schemes. The energy conservation leads to the sparse system of linear equations A× T=B which is solved with a bi-conjugate stabilized gradient technique (BCSG). Validation of the model with different test cases is achieved and application to transient heating of glass is also studied.

Thermal analysis for transient radiative cooling of a conducting semitransparent layer of ceramic in high-temperature applications

A method is developed for obtaining transient temperature distribution in a cooling semitransparent layer of ceramic. The layer is emitting, absorbing, isotropically scattering and heat conducting with a refractive index ranging from 1 to 2. The solution involves solving simultaneously the energy equation and the integral equation for the radiative flux gradient. The energy equation is solved using an implicit finite volume scheme and the integral equation of radiative heat transfer is solved using the singularity technique and Gaussian integration. The effects of scattering are investigated. It is shown that scattering has a significant effect on the transient temperature distribution and the transient mean temperature of the layer.

Estimation of Microwave Radiation Intensity from a Multilayered Cloud Model

A direct analysis for the estimation of brightness temperatures in realistic, multilayered cloud hydrometeor profiles over a range of microwave frequencies (6.6-183 GHz) is presented. The sensitivity of microwave radiative transfer through the atmosphere with respect to its constituents is investigated with a one-dimensional numerical model. The atmosphere is modeled as nongray, absorbing, emitting, and anisotropically scattering with nonisothermal, vertically stratified layers. The governing time-independent radiative heat transfer equation through the atmosphere is solved using a finite volume method. Results of homogeneous and inhomogeneous media, in the presence and absence of scattering constituents, are compared to those published earlier. In the absence of scattering constituents, the differences in the brightness temperature values between the present study and with those published earlier never exceeded 2 K, and the differences that are caused in the presence of scattering constituents are discussed.