Juan Carlos Morales*, Vigain Harutunian*, Masahito Oguma**, and John R. Howell*
*Department of Mechanical Engineering, The University of Texas at Austin
**IHI Research Institute, Inc., Yokohama, Japan

Standard analysis of radiative transfer in enclosures with participating media defines a geometry and a specified temperature or energy flux distribution on each surface. The temperature or temperature distribution in the participating medium within the enclosure may also be given. Unknown temperatures and fluxes are then computed. Here, an inverse design procedure is described for radiating enclosures containing an isothermal participating medium. In inverse design, both a temperature and heat flux distribution are given for one enclosure surface (the “design” surface), as would be the case for heat treating furnaces and applications where conditions are required on a particular surface to meet process needs. On one surface of the enclosure, no conditions are imposed; these are then determined from the conditions required on the design surface. The inverse problem requires solving a Fredholm equation of the first kind, which is notoriously ill-conditioned. Description and examples are given of one useful technique.

Solutions are shown for the necessary emissive power distribution on an enclosure surface that will provide a specified temperature and heat flux distribution on another enclosure surface. Effects of the temperature and absorption coefficient of the medium within the enclosure are shown, as are the effects of enclosure aspect ratio and surface properties.

The methods for ascertaining the accuracy and approach to oscillatory solutions inherent in this type of problem are discussed, and the behavior of the singular values is shown to be a useful tool for reducing the numerical labor involved in the solution.

Some useful solutions are presented that could not be generated by the usual radiative analysis techniques without multiple iterative solutions of the radiative energy equations.

It is argued that inverse solutions present a very useful tool for the thermal analyst, particularly for problems where radiative transfer is a dominant energy transfer mode.


Kazuhiko Kudo, Akiyoshi Kuroda, Amr Eid, Takahiko Saito
Department of Mechanical Engineering, Faculty of Engineering, Hokkaido University
N-13, W-8, Kita-ku, Sapporo 060, JAPAN
Masahito Oguma
Advanced Technology Department 2, Research Institute, Isikawajima-Harima Heavy Industries Co., Ltd.
1, Shin-Nakahara-cho, Isogo-ku, Yokohama 235, JAPAN

A method is developed to solve inverse radiative load problems. The method estimates the heat load distribution and temperature profile within gas region from the values of wall temperature and heat flux distributions, while the shape of the analytical domain and the profile of the optical property values within the system are given. The method can treat arbitrarily shaped multi-dimensional systems with mass flow. In the method, the system is divided into many gas and wall elements, and the radiative heat exchange between each element is calculated by using the MonteCarlo method. By using the results and the mass-flow distributions, the energy equations for each elements are expressed by a matrix form. When the values of the heat-load distribution within the gas region and temperature profile along the wall are given, a set of the matrix equations can easily be solved to obtain the profiles of the gas temperature and wall-heat flux. This is called as forward problem. To obtain the heat-load distribution from the profiles of the temperature and heat flux along the wall: inverse radiative load problem, the matrix which relates the gas temperature to the wall-heat flux should be converted into the inverse form. Due to the ill- posedness of inverse problems, the matrix inversion is usually difficult by using ordinary methods and/or obtained inverse matrix gives very fluctuating results. In the present study, the inverse matrix is obtained by singular value decomposition method.

To check the validity of the method, a forward problem is solved by giving an arbitrarily distributed heat load within a square gray-gas region of 10mx 10m surrounded by gray walls with a temperature of 300K to obtain the corresponding wall-heat-flux distribution. Then, the inverse problem is solved by using the present method by giving the resulting wall-heat flux as the input, and the heat-load distribution within the gas region is estimated. The heat-load distribution obtained from the inverse problem fits very well with those given to the forward problem, which shows the validity of the present method for solving radiative heat-load problems.

By using the method, the radiative heat-load distribution within gas region satisfying the constant heat-flux condition along the wall is also obtained when the condition number of the matrix is reduced to an appropriate lower value by setting some singular values to be 0. The wall-heat flux obtained by using forward problem by giving the resulting heat-load distribution is almost constant with only 1% fluctuations.

The present method can be used not only to estimate the heat-load distribution within furnaces, but also be applied to the problems to decide an optimum burner arrangement within furnaces satisfying prescribed heat-flux profiles.


M. Sakami and M. Lallemand
Laboratoire d'Etudes Thermiques (URA 1403 CNRS)
ENSMA 86960 Futuroscope, France

Spectral absorption and temperature profiles in a cross section of a premixted propane-air flame have been carried out by inversion of the radiative transfer equation (RTE) from simultaneous directional transmission and emission measurements.

The central element of the experimental setup is a Fourier Transform Infrared Spectrometer working at 4cm-1 resolution. In a given optical line of sight of prescribed optical extension it receives selected angular radiation coming up either from an external source crossing through the flame or from the emission of the flame itself. A black body at 1200K allowed the intenity calibration. The propane-air flame is produced by an axisymmetric burner in which the gases are mixted in a tranquilizer and homogeneizer chamber. Its richness is fixed close to one.

In a given cross section, perpendicular to the symmetry axis of the flame (of space dependent absorption coefficient kn(r), the outgoing spectral directional intensity Ln(p) in the line of sight to, characterized by the positionning parameter p, can be written as

where R is a working radius, u=p (see Fig.1), Lno(r) the Planck function and An(u) is the transmission which is related to the absorption profile by the relationship:

Fig.1-Geometry in a flame cross section

Both absorption coefficient and the temperature profiles settled up in the medium may be retrieved as solutions of the coupled Volterra integral equations (1) and (2) when the projections sets An(pi) and Ln(pi) are known from measurements, with i=1...20.

For the resolution of these ill-posed inverse radiative problems several methods of inversion have been used: the Abel inversion with data preconditionning, the Regularised-Adjoint-Conjugate-Gradient method, the Mollification method and the Fourier-Bessel inversion method. They were associated with special filtering and symmetrization techniques.

As a result in the lower part of the flame the absorption profiles of propane and CO2 have been reconstructed at 2980 cm-1 and 2280 cm-1, respectively, in a satisfactionning way and the CO2 profile may be recovered at the top of the flame. Similarly, the temperature profiles have been retrieved and the results tested with thermo-couple measurements.


Loretta Bonfanti*, Leonardo Castellano**, Sauro Pasini*, Nice Pintus*, Christine Mounaim-Rousselle***
*ENEL-CRT, Pisa, Italy, **MATEC, Milano, Italy,
***present address: L.C.S.R. CNRS, Orleans, France

Over the last few years, increasing attention has been paid to the problem of reducing pollutant emissions from the combustion processes of fossil fuels. To meet the requirements concerning Nox emissions, techniques based on a fuel/air staging were utilized. It is not difficult to imagine that the effectiveness of these techniques is very dependent on boiler configuration (boiler type, excess oxygen level, burner zone heat release rate, rate of fuel air mixing, atomization quality and others) and, at the same time, these new combustion configurations can have an impact on the thermal performance of the boiler itself.

Strong efforts have been dedicated to improve the knowledge on the methods for clean combustion, but many topics in the description of the physical behavior of a full-scale plant have not yet had a satisfactory solution. The present paper deals with one of the most important problems encountered in evaluating the performance of advanced low-pollutant combustion systems: the measurement of the concentration of combustion products that exist in form of solid particles (soot) in the combustion chamber. Experimental tests were conducted on an oil-fired 150 Mwe power station (Livorno unit#1). The basic idea of the study was to develop and verify the compatibility of different techniques for measuring the extinction coefficient and volume fraction of soot in furnace. The main conclusion is that a rough theoretical analysis of the data collected by two optical methods and the information given by the application of a Lidar apparatus provide almost similar values. More detailed information available from chemical analysis, which is not yet completed, could prove to b crucial to corroborate or disprove this statement.


Neven Duic, Zeljko Bogdan, Daniel Rolph Schneider, Naim Afgan*
Faculty of Mechanical Engineering and Naval Architecture
University of Zagreb
*Instituto Superior Tecnico
Technical University of Lisbon

A common failure in the boiler furnace is tube rupture. It may cause large financial losses. An early diagnostics of tube leakage would help in preventing the failure by planning maintenence. For this reason it is of the importance to have the detection system which will give an early warning for the eventual development of the tube rupture. The detection system should be based on the reading of the selected variables which are dependent on the water leakage mass content in the flue gases. In the vicinity of the tube leakage the concentration of water vapor is increased and heat flux density distribution on the furnace wall changed. The degree of change is measure of the incident.

The aim of this paper is to investigate the sensitivity of the radiation heat flux changes due to the water leakage into the furnace. The zone method based mathematical model was developed in order to enable that analyses. The three-dimensional mathematical model describes steady-state behavior of the steam generator furnace using PISO numerical algorithm for solving conservation equations. Nine conservation equations for each control volume, including continuity, momentum, enthalpy and mass transport of fuel, oxygen, carbon dioxide and water vapor are used for determination of the temperature, velocity and concentration distributions. The combustion model is described by overall reaction rate. Solution of each conservation equation is obtained as a quasi-linear system with the additive correction multigrid technique. The Monte Carlo method is used to calculate radiation heat transfer between elementary gas and surface zones. Radiation properties are obtained by three grey gases model. The flue gas mixture radiation properties are primarily influenced by water vapor and carbon dioxide content. Boiler furnace behavior simulation results of the tube leakage conditions are represented graphically and compared with the normal operating conditions. It was shown that the radiation heat flux is sufficiently sensitive to detect the water leakage into the furnace.

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