Diffuse-gray Surfaces Research Articles

A general ray tracing approach for solving thermal radiation in regular one-dimensional variable index media via the Monte Carlo method

In the heat transfer community, one-dimensional media refers to some media where the temperature and other thermophysical properties vary along one direction. Plane-parallel, concentric cylindrical, and concentric spherical geometries are among of regular one-dimensional media. In this paper, a general ray tracing approach is presented to track the curve paths of energy particles in regular one-dimensional variable index media. The domain of interest is limited by specular sidewalls and the medium is divided into slices with constant temperature and radiative properties. Then, the path of random energy particles emitted from boundary surfaces and volume elements are traced, element by element, until they absorb by diffuse-gray surfaces or the gray medium. After counting the number of absorbed particles by each boundary surface and volume element, the Monte Carlo method is performed to simulate thermal radiation heat transfer. The results are presented for three regular geometries, including plane-parallel, concentric cylindrical, and concentric spherical media, and the effects of radiative properties are investigated by some numerical experiments.

Mechanistic modeling and analysis of thermal radiation in conventional, microwave-assisted, and hybrid freeze drying for biopharmaceutical manufacturing

In freeze drying, thermal radiation has a significant effect on the drying process of vials located near the corner and edge of the trays, resulting in non-uniformity of the products. Understanding and being able to predict the impact of thermal radiation are therefore critical to accurate determination of the drying process endpoint given the variation in heat transfer of each vial. This article presents a new mechanistic model that describes complex thermal radiation during primary drying in conventional, microwave-assisted, and hybrid freeze drying. Modeling of thermal radiation employs the diffuse gray surface model and radiation network approach, which systematically and accurately incorporates simultaneous radiation exchange between every surface including the chamber wall and vials, allowing the framework to be seamlessly applied for analyzing various freeze-dryer designs. Model validation with data from the literature shows accurate prediction of the drying times for all vials, including inner, edge, and corner vials. The validated model is demonstrated for thermal radiation analysis and parametric studies to guide the design and optimization of freeze dryers.

Modeling and Analysis of Temperature Compensation for Multi-temperature Zone Sintering Furnace Temperature Sensing

The surface temperature of workpieces in a multi-temperature zone sintering furnace is an important parameter to characterize the performances of a multi-temperature zone system. Due to the practical structural properties of the sintering furnace, however, the conventional way of temperature measurement cannot detect the exact surface temperature of the workpieces directly, making it difficult to control the multi-temperature zone system performances precisely. To address such an issue, this paper proposes, for the first time to the best of our knowledge, a temperature compensation algorithm based soft-measurement technique to compensate for the temperature of the measuring points. Specifically, we analyze the heat-transfer mechanism within the electric heating surface and the workpiece surface and establish a mathematical model for it first, and then calculate the radiative heat transfer coefficient between the diffuse gray surface within sintering furnace using the discrete radiation heat transfer method. Finally, the heat transfer mechanism based soft measurement technique is proposed and applied to compensate for the temperature. Both simulations and experiments with a practical sintering furnace are conducted to verify the correctness and effectiveness of the proposed temperature compensation algorithm in different cases. Results show that the proposed algorithm could help maintain the temperature difference within a range limit of ±5°, which is much better as compared with the conventional temperature measurement methods.

Analysis of Heat Transfer Through Optically Participating Medium in a Concentric Spherical Enclosure: The Role of Dual-Phase-Lag Conduction and Radiation

This paper deals with the analysis of the effects of combined dual-phase-lag (DPL) heat conduction and radiation in a concentric spherical enclosure with diffuse-gray surfaces. The enclosed medium is optically participating, i.e., it is radiatively absorbing, emitting, and scattering. Lattice Boltzmann method (LBM) is used to solve the energy equation, and finite volume method (FVM) is used to compute the radiative information. To establish the accuracy of this approach, the combined energy equation is also solved with the finite difference method. Radial temperature profiles and energy contributions by conduction and radiation at various instances and prior to steady-state are elaborated for different kind of thermal perturbations Influence of numerous conductive and radiative parameters over the heat transport process have been investigated in detail. It is observed that higher contribution of radiation to the heat transport process can destroy the thermal wave in the medium completely. Sample results for pure non-Fourier heat conduction, pure radiation, and steady-state response of combined Fourier conduction and radiation in spherical geometry are compared with the results available in literature. In all the cases, comparison shows good agreement with the reported results.

A unified heat transfer model in a pressurized volumetric solar receivers

In the present work, we developed an overall mathematical model adequately describing the main heat transfer processes in a pressurized volumetric receiver. The key components, a windowed cavity, incorporating with the irradiated surface of the absorber, were theoretically modeled as a closed diffuse-gray surfaces system. Accordingly, a boundary condition for the absorber concerning its porous structure surface was developed using net radiation method (NRM) under local thermal non-equilibrium (LTNE) condition. The same method is also applied to the back cavity. Then a modified P1 approximation with collimated irradiation was introduced to incorporate the radiation transfer penetrating in the absorber. The major characteristic of the heat transfer behavior combining radiation, thermal conduction, and convection in the windowed cavity, absorber and the back cavity, are detailedly presented. Also, the key design parameters, such as those relating to pore structure (φ and dp), the volumetric heat transfer coefficient hv, the emissivity ε for window and absorber, and their thickness La and Lg were systematically analyzed. Optimization design can be carried out for both of the solar thermal system and the receiver itself in the future work based on our model.

A hybrid inverse method for the thermal design of radiative heating systems

This work applies the inverse analysis for the thermal design of a three-dimensional radiative enclosure formed with diffuse-gray surfaces. The objective is to determine the powers and locations of the heaters to attain prescribed uniform temperature and radiative heat flux on the design surface. The solution is based on a hybrid solution that couples two different approaches to solve the inverse problem, optimization and regularization. The determination of the locations of the heaters is treated via optimization with the generalized extremal optimization (GEO), whereas the evaluation of the required heat inputs in the heaters, a problem that embodies the Fredholm equation of first kind, is achieved through the truncated singular value decomposition (TSVD) regularization of the ill-conditioned system of equations. The results illustrate that the hybrid solution is a more flexible and accurate approach than the explicit solution based on regularization, and less time consuming than the implicit solution based solely on optimization.

Geometric optimization of radiative enclosures using PSO algorithm

A Shape optimization of a two-dimensional radiative enclosure with diffuse-gray surfaces is studied by the particle swarm algorithm. The goal of the design problem is to achieve an optimized geometry to satisfy the desired temperature and heat flux distributions over a part of radiative enclosures, namely the design surface. The radiative transfer equation is solved by the discrete transfer method. In order to optimize the geometry of enclosure, the positions of the control points of the B-spline curves which represent the geometry of the enclosure are optimized. Some numerical examples are presented to demonstrate the effects of mesh refinements and shape parameters on the resultant geometry and the convergence rate.

Entropy flow and generation in radiative transfer between surfaces

Entropy of radiation has been used to derive the laws of blackbody radiation and determine the maximum efficiency of solar energy conversion. Along with the advancement in thermophotovoltaic technologies and nanoscale heat radiation, there is an urgent need to determine the entropy flow and generation in radiative transfer between nonideal surfaces when multiple reflections are significant. This paper investigates entropy flow and generation when incoherent multiple reflections are included, without considering the effects of interference and photon tunneling. The concept of partial equilibrium is applied to interpret the monochromatic radiation temperature of thermal radiation, T λ ( λ, Ω), which is dependent on both wavelength λ and direction Ω. The entropy flux and generation can thus be evaluated for nonideal surfaces. It is shown that several approximate expressions found in the literature can result in significant errors in entropy analysis even for diffuse-gray surfaces. The present study advances the thermodynamics of nonequilibrium thermal radiation and will have a significant impact on the future development of thermophotovoltaic and other radiative energy conversion devices.

Mathematical Modeling of Heat and Mass Transfer during Continuous Infrared Micronization

An attempt was made to present a logical development of a mathematical model describing temperature and moisture changes in the infrared (IR) processed product in a trough-type IR micronizer. The model was based on a radiation exchange principle in an enclosure composed of diffuse-gray surfaces. Validation of the model was achieved based on experiments conducted with yellow peas tempered to different initial moisture contents. The experiments were carried out on a pilot-scale gas-fired IR micronizer with a vibrating trough. The simulation results were in reasonable agreement with the experimental values.

DEVELOPMENT OF A MATHEMATICAL MODEL FOR HIGH–INTENSITY INFRARED PROCESSING (MICRONIZATION) OF PEAS

A mathematical model of temperature and moisture changes for granular material exposed to infrared (IR) heatprocessing (micronization) was developed. The model was based on a radiation exchange principle in an enclosure composedof diffusegray surfaces. The validation of the model was performed for experiments conducted with peas tempered to threeinitial moisture contents of 20%, 25%, and 30% w.b. (wet basis). The simulation results were in good agreement with theexperimental values. Further simulation experiments quantified the effect of such micronization variables as the temperatureof the IR emitter, emissivity, configuration factor, temperature of the conveying trough, temperature of the surroundings, andemissivity of the processed material.

FINITE-ELEMENT MODELING OF COUPLED RADIATIVE AND DIFFUSIVE HEAT TRANSFER IN NONPARTICIPATING MEDIA INCLUDING SYMMETRY AND PERIODICITY CONDITIONS

Thermal radiation exchanges in enclosures constituted of diffuse gray surfaces separated by a nonparticipating medium become a leading heat transfer mode as soon as high temperatures are involved. From the numerical point of view, radiation brings a strong nonlinearity that is difficult to handle due to couplings among all the surfaces. In this article, we present a finite-element implementation of coupled radiative and diffusive heat transfer in enclosures including symmetry or periodicity conditions. The formulation proposed enables us to define different solution strategies for the nonlinear finite element equations. Three-dimensional application examples show the efficiency of the proposed approach.

Approximate Solution of a Class of Radiative Heat Transfer Problems

An approximate solution is presented for a class of radiative heat transfer problems within enclosures having black or diffuse-gray surfaces based on a modified Taylor series expansion method; such radiative transfer problems are generally represented by integral equations. The approach avoids use of any boundary/initial conditions associated with the original Taylor series expansion method and leads to an approximate solution in a simple closed form to the radiant integral equations, which can be computed straightforwardly on a modern personal computer using symbolic computing codes such as Maple. The method can be effectively and efficiently applied to deal with enclosures involving more than one or two surfaces, for which direct numerical integration may be subject to instability, or require an excessive amount of computation. The computed numerical results for representative thermal problems are in excellent agreement with those obtained by other numerical approaches. [S0022-1481(00)00203-6]

Effect of internal radiative heat transfer on the convection in CZ oxide melt

A numerical study was carried out to investigate the effect of internal radiative heat transfer on the convection in the Czochralski oxide melt. Results were obtained for a wide range of the absorption coefficients of the melt. The flow and temperature fields showed a remarkable change with the variance in the absorption coefficient. In this study, we adopted two different kinds of boundary conditions for the radiative heat transfer at the melt surface, i.e., an opaque and a semitransparent diffuse gray surface, and for each case, the influence of Marangoni convection was investigated. The optical characteristic of the melt surface and the Marangoni effect greatly affected the convection and temperature field in the melt. When the surface was assumed to be semitransparent, in the range of a small absorption coefficient, the smaller the absorption coefficient was, the lesser the Marangoni effect became. This was caused by the disappearance of the temperature gradient at the melt surface, and consequently the flow became as weak as in the case without the Marangoni effect. These numerical results give a clue for discussion on the effect of internal radiative heat transfer in the melt of the Czochralski system.

FINITE-ELEMENT FORMULATIONS FOR ACCURATE CALCULATION OF RADIANT HEAT TRANSFER IN DIFFUSE-GRAY ENCLOSURES

Accurate formulations based on finite-element methods are developed for calculating radiant heat transfer among diffuse-gray surfaces in an enclosure. The method of Gebhart is compared with solutions of the rigorous integral equations using these formulations. A Swartz-Wendroff approximation applied to the finite-element discretizations yields higher-order-accurate analogs of view factors that depend on geometry only and can be readily employed for more accurate solution of general radiant heat transfer problems. The accuracy and efficiency of these approaches are presented in the context of a model problem representative of a high-temperature crystal growth system.

Numerical method for reducing stress level in GaAs crystals

We have developed a numerical method based on the finite element technique for calculating heat transfer in crystal growth furnaces. Our model is global, as it entirely relies upon external control parameters, such as the pulling rate and the power input, while internal heat exchange is calculated on the basis of geometrical and material properties of all constituents of the puller. Radiative exchanges within the furnace are calculated with the assumptions of an axisymmetric geometry and of diffuse-gray surfaces. The viewed and hidden parts of the bodies are taken into account. The influence of the meniscus at the tri-junction is also investigated. It is generally agreed that the presence of structural defects in the grown crystal is related to thermoelastic stresses generated during growth. A finite element prediction of these thermal stresses is coupled to the global heat transfer calculation. The aim of this paper is to show how the thermal stress level in the crystal can be decreased by modifying the thermal exchanges inside the furnace. We study the evolution of the temperature and stress fields during growth and research the optimal pulling rate to obtain a flat solid-liquid interface. Moreover we point out how the temperature gradients and the stress field in the crystal could be affected using a heat shield inside the crucible. Different configurations are studied. Performing quasi-steady simulations at different growth stages, we show that our global numerical model provides a powerful tool for the computer-aided design of crystal growth furnaces.

Heat Transfer Analysis in an Automated Venetian Blind Window System

The heat transfer through an automated venetian blind window system is theoretically studied. The assumptions for the solution of the problem were: Heat transfer is unidimensional, heat exchanges are at steady-state for the whole system, thermal exchange by convection over the blind is negligible, materials used are diffuse-gray surfaces, and radiative exchanges with the room walls are considered as if all the surfaces are blackbodies at indoor air temperature. The solution of the governing equations (system of nonlinear equations) is carried out by using an iterative numerical method. Summarized results are presented and compared to the case of a regular window in a typical Canadian climate during winter and summer seasons and with different louver angles. It is found that by using an automated blind system, the predicted auxiliary load can be reduced by up to 36 percent and 47 percent for winter and summer, respectively.

Radiative exchange between gray fins using a coupled integral equation formulation

This paper presents an analytic methodology for determining the temperature and radiosity distributions in a rectangular, gray fin array through a primitive formulation that explicitly couples the two distributions. The primitive formulation, used in conjunction with the Green's function method, produces a set of coupled nonlinear Fredholm integral equations for temperature and radiosity that must be solved simultaneously. Accurate numerical results are produced rapidly and efficiently using the trapezoidal method and a standard iterative scheme. Initially, an integrodifferential equation is derived from the one-dimensional steady-state energy balance. This mixed mode equation displays the coupling between the temperature and radiosity. In deriving the energy balance, a temperature-dependent thermal conductivity is included in terms of a Taylor series expansion. The Green's function method is used to convert the integro-differential equation describing the conservation of energy into an equivalent integral equation. The second integral equation is developed from the balance of radiant energy for a diffuse-gray surface. The coupled integral equations are solved simultaneously by a simple numerical integration scheme. A parametric study considering the effects of the conduction-radiation number, emissivity, spacings, and thermal conductivity is presented.

Numerical calculation of the global heat transfer in a Czochralski furnace

Abstract A quantitative prediction of the thermal field and of the location of the interface in the Czochralski process requires a precise knowledge of the heat transfer taking place in the entire furnace. The problem is highly complex, since it involves radiation, heat conduction, heating elements and convection-diffusion in the melt. These features are taken into account in our model while outer control parameters - such as power dissipated in the heater and convective losses on the enclosure - are used as input instead of arbitrary boundary conditions. Each constituent such as the heater, the crucible, the melt and the crystal is covered by a finite element mesh. The radiative exchanges within each enclosure are calculated with the assumption of the diffuse-gray surfaces and with the use of a viewed and hidden part algorithm together with a Galerkin discretization. The model allows one to investigate the sensitivity of the temperature field with respect to the geometry of the furnace, the presence of a heat shield and the material properties of the constituents.

Calculation of emittance of a disperse system

The emittance of a disperse system is calculated under the assumption that the particles forming the system have a diffuse gray surface. The calculated results are compared with experiment.

Three-dimensional radiative heat-transfer problem with shading

The problem of radiative heat transfer between diffuse gray surfaces bounding a closed volume of arbitrary configuration is discussed.