Nongray inverse boundary design problems in enclosures at radiative equilibrium

In this paper, an inverse analysis is used to find an appropriate heat flux distribution over the heater surface of radiant enclosures, filled with nongray media at radiative equilibrium from the knowledge of desired (prespecified) temperature and heat flux distributions over the given design surface. Regular and irregular 2D enclosures filled with nongray combustive gas products are considered. Radiation is considered the dominant mode of heat transfer and the medium temperature is obtained from the energy equation. To evaluate the nongray behavior of the participating gases properly, the spectral‐line weighted‐sum‐of‐gray‐gases (SLW) model with updated correlations is used. The dependence of absorption coefficients and the weights of the SLW model on the temperature of the medium makes the inverse problem nonlinear and difficult to handle. Here, the inverse problem is formulated as an optimization problem and the Levenberg‐Marquardt method has been used to solve it. The finite volume method is exploited for the discretization of the energy equation and the spatial discretization of the radiative transfer equation (RTE). The discrete ordinates method (TN quadrature) is used for the angular discretization of RTE. Five test cases, including homogeneous and inhomogeneous media, are investigated to prove the ability of the present methodology for achieving the desired conditions.     

Inverse Boundary Design Problem of Turbulent Forced Convection Between Parallel Plates With Surface Radiation Exchange

The inverse methodology is employed to estimate the unknown heat flux distribution over the heater surface of a channel formed by two parallel plates with forced convection and surface radiation exchange, from the knowledge of the desired temperature and heat flux distributions over a given design surface. The energy and radiative transfer equations are solved by the finite-volume method and the net radiation method, respectively. The conjugate gradient method is used for minimization of an objective function, which is expressed by the sum of square residuals between estimated and desired heat fluxes over the design surface. The performance and accuracy of the present method for solving inverse problems are evaluated by some numerical experiments.     

Inverse boundary design of square enclosures with natural convection

An optimization technique is applied to design of heat transfer systems in which the natural convection is important. The inverse methodology is employed to estimate the unknown strengths of heaters on the heater surface of a square cavity with free convection from the knowledge of the desired temperature and heat flux distributions over a given design surface. The direct and the sensitivity problems are solved by finite volume method. The conjugate gradient method is used for minimization of an objective function, which is expressed by the sum of square residuals between estimated and desired heat fluxes over the design surface. The performance and accuracy of the present method for solving inverse convection heat transfer problems is evaluated by comparing the results with a benchmark problem and a numerical experiment.     

Non-gray radiative convective conductive modeling of a double glass window with a cavity filled with a mixture of absorbing gases

Coupled radiation and natural convection heat transfer occurs in vertical enclosures with walls at different temperatures filled with gas media. In glass window thermal insulation applications in hot climates, infrared absorbing gases appear as an alternative to improve their thermal performance. The thermal modeling of glass windows filled with non-gray absorbing gases is somewhat difficult due to the spectral variation of the absorption coefficients of the gases and the phenomena of natural convection. In this work, the cumulative wavenumber (CW) model is used to treat the spectral properties of mixtures of absorbing gases and the radiative transport equation is solved using CW model and the discrete ordinates method. Due to the range of temperature variation, the mixture of gases is considered as homogeneous. The absorption coefficients were obtained from the database HITRAN. First, the natural convection in a cavity with high aspect ratio is modeled using a CFD code and the local and global Nusselt numbers are computed and compared with available empirical correlations. Also, the flow pattern for different Rayleigh numbers is analyzed. Then, the heat transfer in the gas domain is approximated by a radiative conductive model with specified heat flux at boundaries which is equivalent to convective transport at the walls surroundings. The energy equation in its two-dimensional form is solved by the finite volume technique. Three types of gas mixtures, highly absorbing, medium and transparent are investigated, to determinate their effectiveness in reducing heat gain by the gas ambient. Reflective glasses are also considered. The numerical method to solve radiative heat transport equation in gray and non-gray participant media was validated previously. The temperatures distributions in the gas and the glass domain are computed and the thermal performance of the gas mixtures is evaluated and discussed. Also, comparison with pure radiative conductive model is shown.     

An inverse radiation boundary design problem for an enclosure filled with an emitting,absorbing, and scattering media

This work is an inverse radiative design problem in which the objective is to determine the spatial distribution of heat source strengths which produces a desired temperature and heat flux distribution on the design surface. The furnace whose walls are diffuse-grey is assumed to be filled with an absorbing, emitting, and scattering medium. The function to be minimized is the sum of squares of the differences between the desired and calculated radiative heat fluxes at the design surface. Radiative heat flux calculations are accomplished by means of the Modified Discrete Transfer Method MDTM using the correction factors suggested by Coelho and Carvalho [P.J. Coelho, M.G. Carvalho, Conservative formulation of the discrete transfer method, ASME J. Heat Transfer, 119 (1997) 118–128.] and Cumber [P.S. Cumber, Improvements to the discrete transfer method of calculating radiative heat transfer, Int. J. Heat Mass Transfer, 38 (12) (1995) 2251–2258.]. For inverse design calculations the Conjugate Gradient Method CGM is employed, in which the sensitivity coefficients are defined and used as needed by the algorithm. Our investigation shows that the presented algorithm is able to estimate heater strengths accurately.     

Inverse radiation–conduction design problem in a participating concentric cylindrical medium

Inverse conduction–radiation problem for design analysis in a two-dimensional concentric cylindrical absorbing, emitting and isotropically scattering medium has been solved, when the desired boundary conditions are available on the design surface. The finite-volume method was adopted to deal with energy conservation equation including conduction and radiation. The radiative transfer equation was also taken into consideration in direct problem, whereas the Levenberg–Marquardt method was used to solve a set of equations in inverse problem, which are expressed by errors between estimated and desired total heat fluxes on the design surface. The automatic differentiation as well as the Broyden combined update was utilized to reduce computational time in calculating the sensitivity matrix. The results have shown that the desired total heat flux distribution on design surface could be successfully estimated with less computational time using the present inverse procedure developed here.     

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.     

Application of a Particle Swarm Algorithm for Parameter Retrieval in a Transient Conduction-Radiation Problem

This article addresses the application the particle swarm optimization (PSO) algorithm as an optimization tool for retrieval of parameters in a combined mode 1-D transient conduction-radiation heat transfer problem. In the chosen problem, the participating medium is absorbing, emitting, and scattering. The boundaries are taken to be diffuse gray. In both direct and inverse methods, the energy equation is solved using the lattice Boltzmann method (LBM) and the finite volume method (FVM) is used to compute the radiative information. In the inverse method, the objective function is minimized using the PSO algorithm. The objective function considered in the inverse formulation is an error function evaluated with the exact and inverse temperature fields for the simultaneous retrieval of the extinction coefficient and the scattering albedo. The inverse analysis constituted the effect of measurement errors on solution efficacies. In addition, the effect of important PSO parameters such as swarm size, inertia factor and constriction factor on the parameter retrieval is considered. For the chosen problem, it is found that the PSO with 20 discrete particles and 50 iterations is adequate for accurate parameter retrieval. The PSO has been found to provide a better accuracy than the genetic algorithm.     

Reconstruction of Temperature Distribution in the Combustion Region of a Non-Gray Medium

Two-dimensional temperature distribution in the combustion region of a radiant furnace with non-gray gaseous products is reconstructed in an inverse manner. The discrete ordinate method is used to solve the radiative transfer equation, and the non-gray behavior of gaseous medium is modeled by the spectral line weighted sum of gray gas model. Conjugate gradient method is employed for the inverse analysis. The domain of interest is divided into two regions: the unknown- and known-temperature zones. The temperature distribution in the unknown-temperature zone, near the flame, is recovered in an inverse manner by the measurement of heat fluxes over the opposite walls in the direction of flame diffusion. Effects of the size of unknown-temperature zone, variable concentrations, and measurement errors in temperature reconstruction are discussed.     

Inverse Geometry Design of Radiating Enclosure Filled with Participating Media Using Meshless Method

A new inverse geometry design methodology is presented in this work for designing a two-dimensional radiating enclosure filled with participating media to meet the pre-specified radiative heat flux distribution on a designed boundary wall. Akima cubic interpolation is employed to approximate the shape of the unknown design surface and transform the continuous geometry shape design to the discrete points' position design. To avoid the tedious remeshing of the variable computational domain in the inverse geometry design processes, the direct collocation meshless method is adopted to solve the radiative transfer problem in the enclosure. The geometry shape of the design surface is optimized using the conjugate gradient method, and the zeroth order regularization method is chosen to stabilize the inverse solutions. A test example is taken to verify the new method presented in this work. The inverse design results show that pre-specified design requirement on the boundary wall can be successfully obtained using the new methodology.     

Inverse radiation analysis of simultaneous estimation of temperature field and radiative properties in a two-dimensional participating medium

An inverse radiation analysis is presented for simultaneous estimation of temperature field and radiative properties including absorption and scattering coefficients in a two-dimensional rectangular, absorbing, emitting and scattering gray medium from the knowledge of the exit radiative energy received by charge-coupled device (CCD) cameras at boundary surfaces. The backward Monte Carlo method was introduced to describe the radiative heat transfer for its efficiency. The inverse problem is formulated as an optimization problem and solved by the least-square QR decomposition (LSQR) method. The effects of measurement errors, optical thickness and search step length on the accuracy of the estimation were investigated and the results show that the temperature field and radiative properties can be reconstructed accurately for the exact and noisy data.     

Optimization of Heat Fluxes on the Heater and the Design Surfaces of a Radiating-Conducting Medium

In a radiating-conducting planar medium with a boundary as the heater surface using an inverse analysis, this work deals with the design methodologies to understand the inherent relationship between heater surface temperature/flux, design surface temperature/flux, and medium properties. The heat flux on the heater surface is chosen as the fitness function. Subsequently, to achieve maximum and minimum design surface heat fluxes, an optimization was done to evaluate the zone of operation of the heater. In addition, the effect of medium properties on the temperature-flux relationships on both surfaces has been studied. The distance between the two surfaces is also considered a parameter. The medium properties, the distance between the surfaces, and the heater surface temperature have been found to have a great impact on the design surface heat flux. The inverse mixed boundary problem has been solved using the lattice Boltzmann method (LBM), the finite-volume method (FVM), and the genetic algorithm (GA). Results of the present study provide a guideline towards the efficient design of a heater in which conduction and radiation are the dominant modes of heat transfer.     

Assessment of gas radiative property models in the presence of nongray particles

In this study, a radiation code based on the method of lines solution of the discrete ordinates method for the prediction of radiative heat transfer in nongray gaseous media is developed by incorporation of two different spectral gas radiative property models, banded spectral line-based weighted sum of gray gases (banded SLW) and gray wide band (GWB) approximation in the presence of nongray absorbing–emitting–scattering particles. The aim is to introduce an accurate and CPU efficient spectral gas radiation model, which is compatible with spectral fuel/ash particle property models. Input data required for the radiation code and its validation are provided from two combustion tests previously performed in a 300 kWt atmospheric bubbling fluidized bed combustor test rig burning low calorific value Turkish lignite with high volatile matter/fixed carbon (VM/FC) ratio in its own ash. The agreement between wall heat fluxes and source term predictions obtained by global and banded SLW models reveal that global SLW model can be converted to an accurate wide band gas model (banded SLW) which can directly be coupled with spectral particle radiation. Furthermore, assessment of GWB approximation by benchmarking its predictions against banded SLW model shows that GWB gives reasonable agreement with a higher CPU efficiency when the particle absorption coefficient is at least in the same order of magnitude with the gas absorption coefficient.     

Statistical modeling of thermal radiation transfer in buoyant turbulent diffusion flames

A statistical (Monte Carlo) method for radiative heat transfer has been incorporated in CFD modeling of buoyant turbulent diffusion flames in stagnant air and in a cross-wind. The model and the computational tool have been developed and applied to simulate both burner flames with controlled fuel supply rate and in self-sustained pool fires with burning rates coupled with flame radiation. The gas–soot mixture was treated either as gray (using the effective absorption coefficient derived from total emissivity data or the Planck mean absorption coefficient) or as non-gray (using the weighed sum of gray gases model). The comparison of predicted radiative heat fluxes indicates applicability of the gray media assumption in modeling of thermal radiation in case of high soot content. The effect of turbulence-radiation interaction is approximately taken into account in calculation of radiation emission, which is corrected to allow for temperature self-correlation and absorption-temperature correlation. In modeling buoyant propane flames in still air above 0.3 m diameter burner, extensive comparison is presented of the predictions with the measurements of gas species concentrations, temperature, velocity and their turbulent fluctuations, and radiative heat fluxes obtained in flames with different heat release rates. Similar to previously published experimental data, the predicted burning rate of flames above the acetone pools exposed to flame radiation increases with the pool diameter and approaches a constant level for large pool sizes. The magnitude of predicted burning rates is shown to be in agreement with the reported measurements. Augmentation of burning rate of the pool fire in a cross-wind because of increased net radiative heat flux received by the fuel surface and non-monotonic dependence of burning rate on cross-wind velocity, subject to the pool diameter, is predicted. The statistical treatment of thermal radiation transfer has been found to be robust and computationally efficient.     

Inverse design involving combined radiative and turbulent convective heat transfer

This work considers an inverse boundary design problem which involves radiation and convection heat transfer. The objective is finding the heat flux distribution required on heaters located on the top and side walls of a two-dimensional enclosure that satisfies both the temperature and heat flux distributions prescribed on the design surface of the enclosure. A turbulent air flow is generated by a fan located inside the chamber. The problem is described by a system of non-linear, ill-conditioned equations, which is solved by an iterative procedure. The solution are obtained by regularizing the system of equations by means of the TSVD method.     

Analysis of conduction-radiation problem in absorbing, emitting and anisotropically scattering media using the collapsed dimension method

Combined conduction-radiation problem is solved using the collapsed dimension method. One-dimensional gray planar absorbing, emitting and anisotropically scattering medium is considered. Non-dimensional medium temperature and heat flux distributions are found for various values of boundary temperature, optical thickness, boundary emissivity and conduction-radiation parameter. Effects of scattering albedo and anisotropy factor are also discussed. For comparison, problems considered are also analyzed with the discrete transfer method and the exact method. Collapsed dimension method is found to give excellent comparison for various radiative parameters considered.     

Modeling the 3-D radiation of anisotropically scattering media by two different numerical methods

An original model and code for 3-D radiation of anisotropically scattering gray media is developed where radiative transfer equation (RTE) is solved by finite volume method (FVM) and scattering phase function (SPF) is defined by Mie Equations (ME). To the authors’ best knowledge this methodology was not developed before. Missing the benchmark, another new 3-D model and code, which solve the same problems, based on a combination of zone method (ZM) and Monte Carlo method (MC), as a solution of RTE, is developed. Here SPF is also calculated by Mie Equations. The conception ZM + MC is numerically expensive and is used and recommended only as a benchmark. The 3-D rectangular enclosure and the spherical geometry of particles are considered. The both models are applied: (i) to an isotropic and to four anisotropic scattering cases previously used in literature for 2-D cases and (ii) to solid particles of several various coals and of a fly ash. The agreement between the predictions obtained by these two different numerical methods for coals and ash is very good. The effects of scattering albedo and of wall reflectivity on the radiative heat flux are presented. It was found that the developed 3-D model, where FVM was coupled with ME, is reliable and accurate. The methodology is also suitable for extension towards: (i) mixture of non-gray gases with particles and (ii) incorporation in computational fluid dynamics.     

Inverse boundary design of a radiant furnace with diffuse-spectral design surface

An optimization technique is applied to inverse design of radiative furnaces with diffuse-spectral surfaces. The variation of emissivity with respect to the wavelength is approximated by considering a set of spectral bands with constant emissivities and then the radiative transfer equation is solved by the net radiation method for each band. The conjugate gradient method is used for estimation of temperatures over reflector and heater surfaces. The sensitivity problem is approximated by differentiation of the radiative transfer equation with respect to the unknown variables. The performance of the present method is evaluated by comparing the results with the results obtained by considering a diffuse-gray design surface.     

Transient thermal analysis of semitransparent composite layer with an opaque boundary

Transient coupled radiative and conductive heat transfer in a two-layer, absorbing, emitting, and isotropically scattering non-gray slab is investigated by the ray tracing method in combination with Hottel's zonal method. One outer boundary is opaque, and another is semitransparent. The radiative energy transfer process in a semitransparent composite is divided into two sub-processes, one of which considers scattering, the other does not. The radiative transfer coefficients of the composite are deduced under specular and diffuse reflection and combined specular and diffuse reflection, respectively. The radiative heat source term is calculated by the radiative transfer coefficients. Temperature and heat flux are obtained by using the full implicit control-volume method in combination with the spectral band model. The method presented here needs only to disperse the space position, instead of the solid angle. A comparison with previous results shows that the results are more accurate.     

Inverse estimation of boundary conditions on radiant enclosures by temperature measurement on a solid object

In this paper, we present an inverse analysis to estimate the thermal boundary conditions over a two-dimensional radiant enclosure from the knowledge of the measured temperatures for some points on a solid object within the enclosure. The conduction heat transfer in the solid object and the radiative heat transfer between the surface elements of the enclosure are formulated by the finite volume method and the net radiative method, respectively. The resultant set of nonlinear equations is solved by the Newton's method. The inverse problem for estimation of boundary conditions over the radiant enclosure is solved by the conjugate gradient method.