WAVE EFFECTS ON RADIATIVE TRANSFER IN ABSORBING AND EMITTING THIN-FILM MEDIA

This work answers two questions in microscale radiative heat transfer. The first question is how to take electromagnetic wave effects into consideration in formulating radiative transfer problems in a participating medium, and the second question is what the differences are between the wave solution of the problem and the solution of the equation of radiative transfer (ERT)? Based on the electric fluctuation theory and the solution of the electromagnetic wave equations, a wave formulation is presented for radiative transfer across a thin film sandwiched between two semi-infinite media. The solution includes absorption and emission in both the film and the surrounding media. Numerical results are obtained for the special case of thermal emission from a thin film with a large internal temperature gradient. A comparison between the wave and the ERT solutions demonstrates the interference and tunneling effects in the thin-film limit.     

Numerical investigation of coupled natural convection and radiation in a differentially heated cubic cavity filled with humid air. Effects of the cavity size

A numerical study of natural convection with surface and air/H₂O mixture radiation in a differentially heated cubic square cavity is presented. The coupled flow and heat transfers in the cavity are predicted by coupling a finite volume method with a spectral line weighted sum of gray gase model to describe gas radiative properties. The radiative transfer equation is solved by means of the discrete ordinate method. Simulations are performed at Ra?=?10⁶, considering different combinations of passive wall and/or gas radiation properties and different cavity length. It was found that in presence of a participative medium representative of building, cavity length has a strong influence on temperature and velocity fields which affect the global circulation and heat transfers in the cavity. For each steady-state solution, the convective and radiative contributions to the global heat transfer are discussed. More specifically, boundary layer thickness, thermal stratification parameter, and three-dimensional effects are compared to pure convective case results. The results suggest that radiative effects, often considered as negligible in view of the relatively low optical thickness, may not be neglected when trying to predict regime transitions.     

NATURAL CONVECTION AND RADIATION IN A SQUARE ENCLOSURE

Natural convection of a radiating fluid in a square enclosure is studied numerically. The coupled momentum, energy, and radiative transfer equations are solved by an iterative procedure. The solutions to the equation of radiative transfer are obtained by the discrete ordinates method using S4 and S8 quadratures. The method is based on control volume formulation and is fully compatible with the SIMPLER algorithm used to solve the momentum and energy equations. The effects of optical thickness and scattering on the flow and temperature fields and heat transfer rates are analyzed. The changes in the buoyant flow patterns and temperature distributions due to the presence of radiation in inclined or heat generating enclosures are also studied. Comparative results obtained by the P-I differential approximation are presented.     

Combined heat transfer of radiation and natural convection in a square cavity containing participating gases

This article deals with analyzing the effect of radiative heat transfer on natural convection heat transfer in a square cavity under normal room conditions. The governing equations of natural convection and radiative transfer are solved simultaneously to obtain the temperature, velocity and heat flux distributions inside the participating medium. The finite volume method has been adopted to solve the governing equations and the discrete ordinates method (DOM) is used to model the radiative transfer in absorbing-emitting media. The radiative–convective model is validated by comparison with test cases solutions from the literature. Then, the effects of Rayleigh number from 10² to 10⁶ and optical thickness in a broad range from 0 to 100 on temperature and velocity distributions and Nusselt numbers are investigated. The results show that even under normal room conditions with a low temperature difference, the radiation plays a significant role on temperature distribution and flow pattern in the cavity. Also, several interesting effects of radiation are observed such as a sweep behavior on the isotherms, streamlines and velocity distributions of the cavity along the optical thickness and a reverse behavior on maximum stream function and convective Nusselt number at different Rayleigh numbers.     

RADIATION-CONDUCTION INTERACTION IN A PLANAR,ANISOTROPICALLY SCATTERING MEDIUM WITH FLUX BOUNDARY

Steady, combined radiation and conduction heat transfer in an absorbing, emitting, and anisotropically scattering planar medium is investigated theoretically. The problem is considered with a constant net heat flux imposed at one boundary and a constant temperature at the other. Both specular and diffuse reflectivities are included. The influence of radiation heat transfer is obtained by solving the exact integral equations of radiative transfer with the nodal approximation technique. The technique reduces the integral equations into a discrete system of algebraic equations and permits obtaining exact numerical solutions for any scattering-phase function. Temperatures are obtained from the energy equation with an iterative procedure. The effects of scattering anisotropy as well as radiation parameters such as albedo and wall reflectivities on temperatures are analyzed.     

Second-Order Radiative Transfer Equation and Its Properties of Numerical Solution Using the Finite-Element Method

The original radiative transfer equation is a first-order integrodifferential equation, which can be taken as a convection-dominated equation. The presence of the convection term may cause nonphysical oscillation of solutions. This type of instability can occur in many numerical methods, including the finite-difference method and the finite-element method, if no special stability treatment is used. To overcome this problem, a second-order radiative transfer equation is derived, which is a diffusion-type equation similar to the heat conduction equation for an anisotropic medium. The consistency of the second-order radiative transfer equation with the original radiative transfer equation is demonstrated. The perturbation characteristics of error are analyzed and compared for both the first- and second-order equations. Good numerical properties are found for the second-order radiative transfer equation. To show the properties of the numerical solution, the standard Galerkin finite-element method is employed to solve the second-order radiative transfer equation. Four test problems are taken as examples to check the numerical properties of the second-order radiative transfer equation. The results show that the standard Galerkin finite-element solution of the second-order radiative transfer equation is numerically stable, efficient, and accurate.     

Combined Mode Conduction and Radiation Heat Transfer in a Porous Medium and Estimation of the Optical Properties of the Porous Matrix

This article deals with the analysis of combined mode conduction and radiation heat transfer in a porous medium, and simultaneous estimation of the optical properties of the porous matrix. Simultaneous solution of the gas- and solid-phase energy equations encompasses local thermal nonequilibrium, while the convective heat exchange term couples the gas- and the solid-phase energy equations. A localized uniform volumetric heat generation zone is the source of heat transfer in the porous matrix. With volumetric radiative information needed in the solid-phase energy equation computed using the discrete transfer method, the solid- and gas-phase energy equations are simultaneously solved using the finite difference method. For a given set of boundary conditions and operating parameters, the computed temperature distribution serves as the exact temperature profile necessary in the estimation of parameters. In the estimation of parameters using inverse analysis, the objective function is minimized using the genetic algorithm. Effects of measurement error, number of generations, population size, crossover probability, and mutation probability are studied in regard to the accuracy of results and the computational time required. Reasonably accurate estimations of extinction coefficient, scattering albedo, and emissivity of the porous matrix are obtained.     

Radiative heat transfer to flow of a combustible mixture in a vertical channel

The paper studies the flow of a combustible mixture in a vertical channel in the presence of radiative heat transfer as a model for biomass moving bed gasifiers operating in the temperature range 750–1500 K. The simplistic binary reaction A → B is assumed, and both the optically thick (high density gas) and the optically thin (low density gas) situations are considered for the radiative heat transfer. Analytical and numerical solutions are obtained and discussed quantitatively.     

SIMULATION OF RAPID THERMAL PROCESSING IN A DISTRIBUTED COMPUTING ENVIRONMENT

Rapid thermal processing (RTP) has become a key technology in the fabrication of advanced semiconductor devices. As wafers get larger and chip dimensions become smaller, the understanding of the highly coupled physics, such as radiative heat transfer, transient fluid flow, heat transfer, and chemical reactions through numerical modeling using high-performance computing, is the key to the design, optimization, and control of RTP reactors. In this study, an accurate and efficient simulation tool for RTP in a distributed computing environment is developed by implementing various new models and algorithms. Thegoverning equations for highly coupled and transient transport phenomena inRTP are solved by anunstructured finite volume method (FVM). Surface radiative heat transfer is the most dominant mode of heat transfer in RTP and it is modeled by the modified discrete transfer method (MDTM). The radiative properties on the patterned wafer are quite different from those on the bare silicon and they are predicted by the matrix method. To enhance thecomputationefficiency, anefficient parallelalgorithmis implemented in the solution procedure. Data communication among the processors is carried out by the Parallel Virtual Machine (PVM). To evaluate the present simulation tool, an actual commercial RTP reactor is investigated under various conditions. The accuracy of the present model is validated through the comparisons of wafer temperature profile between different models for steady state andtransient flow andheat transfer cases. To demonstrate the importance of the pattern effects in RTP systems, a transient case containing the patterned wafer is investigated. The temperature profile and its uniformity for the patterned wafer are found to be quite different from those for the unpatterned wafer. To examine the performance on parallel computation, the previous transient case is studied with different processor numbers. As the processor number increases, the computationtime is seen to reduce; however, the parallel performance is seen to degrade A larger solution iteration number and higher communication overhead are believed to be the major reasons for the degradation of the parallel performance. The present case studies indicate that the simulation tool developed in this study can be used to systematically investigate various effects in RTP systems because of its high accuracy and efficiency.     

Mixed convection heat transfer in inclined rectangular ducts with radiation effects

A numerical study was carried out to investigate the radiation effect on the characteristics of the mixed convection fluid flow and heat transfer in inclined ducts. The three-dimensional Navier–Stokes equations and energy equation are solved simultaneously with the vorticity–velocity method. The integro-differential radiative transfer equation was solved by the discrete ordinates method. The effects of the thermal buoyancy and the radiative transfer on the distributions of the bulk fluid temperature, the friction factor and the Nusselt number are emphasized in detail. Results indicate that radiation effects have a considerable impact on the heat transfer and tend to reduce the thermal buoyancy effects. In addition, the development of the bulk fluid temperature is enhanced by the radiation effects.     

Modeling of a Solar Water Collector with Water‐Based Nanofluid Using Nanoparticles

The pressure‐velocity form of the Navier–Stokes equations, energy equation, and concentration equation are used to represent the mass, momentum, energy, and concentration conservations of the nanofluid medium in the solar collector. The governing equations and corresponding boundary conditions are converted to dimensionless form and solved numerically by the finite element method. The physical domain is discretized by triangular mesh elements with six nodes. The working fluid is water‐based nanofluid with two nanoparticles, namely, silver (Ag) and copper oxide (CuO). The study includes computations for different values of buoyancy ratio (Nr) and Schmidt number (Sc). Flow, heat, and mass transfer characteristics are presented in the forms of streamlines, isotherms, and iso‐concentrations. In addition, results for the average radiative, convective heat and mass transfer, mean temperature and concentration of nanofluid, mid‐height horizontal‐vertical velocities, and subdomain average velocity field are offered and discussed for the above‐mentioned parametric conditions. Results show that the effects of Nr and Sc on the convective‐radiative heat and mass transfer phenomenon inside the collector are significant for all values of Nr and Sc studied. Comparison and validation with the standard experimental/numerical data is given in brief. The variation of the obtained result is presented as 34% with the result of experimental data. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(3): 270–287, 2014; Published online 30 September 2013 in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21080     

Radiative and conductive transfer for a real gas in a cylindrical enclosure with gray walls

The purpose of this study is to examine the interaction of radiative and conductive transfer for a radiatively participating real gas stagnant in a cylindrical enclosure with gray diffuse walls. Consideration of reflecting boundaries represents an extension of previous black wall studies. Examination of radiative transfer was made by the zone method with gas radiative properties furnished by the weighted sum of gray gases model. Directed flux areas are expressed as the weighted sum of gray gas total exchange areas which are evaluated using the matrix formulation method from direct exchange areas. Axial and radial gas temperatures are examined along with wall heat flux or temperature for respective cases of either specified wall temperatures or heat fluxes. Emphasis is placed on examining results to show the effects of wall emittance and duct diameter. Results for heat generation within the gas are also presented.     

Numerical simulation of LEC growth of InP crystal with an axial magnetic field

A set of numerical analyses for momentum, heat transfer and thermal stress for a 3 in. diameter liquid encapsulant Czochralski growth of single-crystal InP with an axial magnetic field is carried out by using the finite element method. Firstly, convective and conductive heat transfers, radiative heat transfer between diffuse surfaces and the Navier–Stokes equations for both melt and encapsulant and electric current stream function equations for the melt and crystal are all combined and solved simultaneously. The thermal stress distribution in the crystal is calculated by using the obtained temperature field. The effects of some process parameters on flow, heat transfer, growing interface shape and thermal stress are investigated.     

Numerical investigation of combustion with non-gray thermal radiation and soot formation effect in a liquid rocket engine

A numerical analysis was carried out in order to investigate the combustion and heat transfer characteristics in a liquid rocket engine in terms of non-gray thermal radiation and soot formation. Governing gas and droplet phase equations with PSIC model, turbulent combustion model with liquid kerosene fuel, soot formation, and non-gray thermal radiative equations are introduced. A radiation model was implemented in a compressible flow solver in order to investigate the effects of thermal radiation. The finite-volume method (FVM) was employed to solve the radiative transfer equation, and the weighted-sum-of-gray-gases model (WSGGM) was applied to model the radiation effect by a mixture of non-gray gases and gray soot particulates. After confirming the two-phase combustion behavior with soot distribution, the effects of the O/F ratio, wall temperature, and wall emissivity on the wall heat flux were investigated. It was found that the effects of soot formation and radiation are significant; as the O/F ratio increases, the wall temperature decreases. In addition, as the wall emissivity increases, the radiative heat flux on the wall increases.     

Effect of absorption of solar radiation in glass-cover(s) on heat transfer coefficients in upward heat flow in single and double glazed flat-plate collectors

Absorption of solar radiation in the glass cover(s) of a flat plate solar collector increases the temperature of cover(s) and hence changes the values of convective and radiative heat transfer coefficients. The governing equations for the case of single as well as double glazed collector have been solved for inner and outer surface temperatures of glass cover(s) with/without including the effect of absorption of solar radiation in the glass cover(s), with appropriate boundary conditions. The effects of absorption of solar radiation on inner and outer surface temperatures and consequently on convective and radiative heat transfer coefficients have been studied over a wide range of the independent variables. The values of glass cover temperatures obtained from numerical solutions of heat balance equations with and without including the effect of absorption of solar radiation in the glass cover(s) are compared. For a single glazed collector the increase in glass cover temperature due to absorption of solar radiation could be as high as 6°. The increase in temperatures of first and second glass covers of a double glazed collector could be as high as 14° and 11°, respectively. The effect on the convective heat transfer coefficient between the absorber plate and the first glass cover is substantial. The difference in the values of the convective heat transfer coefficients between the absorber plate and the first glass cover (h_cp1) of a double glazed collector for the two cases: (i) including the effect of absorption and (ii) neglecting the effect of absorption in glass cover, could be as high as 49%. Correlations for computing the temperatures of inner and outer surfaces of the glass cover(s) of single and double glazed flat plate collectors are developed. The relations developed enable incorporation of the effect of absorption of solar radiation in glass cover(s) in the relations for inner and outer surface temperatures in a simple manner. By making use of the relations developed for inner and outer surface temperatures of glass cover(s) the convective and radiative heat transfer coefficients can be calculated so close to those obtained by making use of surface temperatures of glass cover(s) obtained by numerical solutions of heat balance equations that numerical solutions of heat balance equations are not required.     

RADIATIVE HEAT TRANSFER IN FINITE CYLINDRICAL HOMOGENEOUS AND NONHOMOGENEOUS SCATTERING MEDIA EXPOSED TO COLLIMATED RADIATION

Radiative heat transfer is studied in a finite axisymmetrical cylindrical enclosure exposed to collimated radiation. The integral equations for radiative transfer are solved by the YIX method and the quadrature method for comparison. Integrated intensity and radiative heat flux are presented in homogeneous and nonhomogeneous scattering media exposed to both uniform and Gaussian distributions of normal collimated incident radiation. The effects of aspect ratio, different incident radiation, and anisotropic scattering phase function as well as nonhomogeneous property distribution are discussed. Ray effects appear in the YIX solution for the case of a nonhomogeneous step change in the extinction coefficient. In order to eliminate the ray effect, an adaptive angular quadrature scheme is described and applied.     

Enhancement of Heat Transfer with Porous/Solid Insert for Laminar Flow of a Participating Gas in a 3-D Square Duct

In recent years, porous or solid insert has been used in a duct for enhancing heat transfer in high temperature thermal equipment, where both convective and radiative heat transfer play a major role. In the present work, the study of heat transfer enhancement is carried out for flow through a square duct with a porous or a solid insert. Most of the analyses are carried out for a porous insert. The hydrodynamically developing flow field is solved using the Navier–Stokes equation and the Darcy–Brinkman model is considered for solving the flow in the porous region. The radiative heat transfer is included in the analysis by coupling the radiative transfer equation to the energy equation. The fluid considered is CO₂ with temperature dependent thermophysical properties. Both the fluid and the porous medium are considered as gray participating medium. The increase in heat transfer is analyzed by comparing the bulk mean temperature, Nusselt number, and radiative heat flux for different porous size and orientation, Reyonlds number, and Darcy number.     

Mixed convection heat transfer in horizontal rectangular ducts with radiation effects

The study of mixed convection heat transfer in horizontal ducts with radiation effects has been numerically examined in detail. This work is primarily focused on the interaction of the thermal radiation with mixed convection for a gray fluid in rectangular horizontal ducts. The vorticity–velocity method is employed to solve the three-dimensional Navier–Stokes equations and energy equation simultaneously. The integro-differential radiative transfer equation was solved by the discrete ordinates method. The attention of the results is focused on the effects of thermal buoyancy and radiative transfer on the development of temperature, the friction factor and the Nusselt number. Results reveal that radiation effects have a considerable impact on the heat transfer and would reduce the thermal buoyancy effects. Besides, the development of temperature is accelerated by the radiation effects.     

Simultaneous estimation of four parameters in a combined-mode heat transfer in a 2D porous matrix with heat generation

This article reports a study on simultaneous estimation of four parameters for combined-mode conduction and radiation heat transfer in a 2D rectangular porous matrix with a localized volumetric heat generation source. Air flows at uniform velocity through the conducting and radiating porous matrix. In the heat generation zone, and its downstream, the gas temperature is higher than that of the solid, and in the upstream the reverse situation occurs. This temperature difference between gas and the solid results in heat transfer by convection between the two phases, and the analysis thus requires consideration of separate energy equations for the two phases. The solid being involved radiatively, the volumetric radiative source term, in the form of the divergence of radiative heat flux, appears only in the solid-phase energy equation. The two equations are coupled through the convective heat transfer term. Four parameters—scattering albedo, emissivity, solid conductivity, and heat transfer coefficient—are simultaneously estimated based on the solid and gas temperature distributions, and convective and radiative heat fluxes at the outer surface of the porous matrix. In both direct and inverse approaches, the energy equations are solved using the finite volume method. For a test case, determining the genetic algorithm is much more time-consuming than the global search algorithm; in other cases, parameter estimations are done using the global search algorithm. Parameters are found to be estimated accurately.     

Enhancement of heat transfer: Combined convection and radiation in the entrance region of circular ducts with porous inserts

Using porous ceramic inserts in high temperature equipment has been proven to be an effective means to enhance combined convective–radiative heat transfer. The porous ceramic insert was referred to as a convection-to-radiation converter (CRC) by previous investigators. We consider a novel application of CRC cores in a partial by pass flow system for heat transfer enhancement. Both hydrodynamically and thermally developing laminar flow is considered in the entrance region of a circular pipe with a porous insert located at the center. The momentum and Darcy–Brinkman equations are applied to the flow field in the annular gas layer and central porous layer respectively. The energy equation is coupled with the radiative transfer equation by the radiation source term. The radiative transfer is simulated by the newly developed integral equations [X.L. Chen, W. Sutton, Radiative transfer in finite cylindrical media using transformed integral equations, J. Quant. Spectrosc. Radiat. Transfer 77 (3) (2003) 233–271; W. Sutton, X.L. Chen, A general integration method for radiative transfer in 3-D non-homogeneous cylindrical media with anisotropic scattering, J. Quant. Spectrosc. Radiat. Transfer 84 (2004) 65–103] to avoid singularity problem and give high accuracy. The working fluid and porous medium are both considered as participating media. Finally, this highly non-linear system of equations is solved by a mixed iteration method. The results are compared between the cases with and without the porous insert. The porous insert enhances both convective and radiative transfer by about 35% and 105% respectively at the most. The effects of important parameters on this enhancement are discussed in detail.