POSTER SESSION II

METHOD OF RADIATIVE COEFFICIENTS AND ITS APPLICATION IN THERMAL CALCULATIONS OF PRACTICAL INDUSTRIAL CASES

Considering heat exchange equipment in general, heat transfer is usually a combined process consisting of conduction, convection and radiation. In Calculation procedures there is a need to determine a certain strategy taking into consideration a prevalent component of heat transfer, the influence of other components, the degree of simplification of a mathematical model being created, available data from experiment, the ratio of operation etc.

The application of the method of radiative coefficients has proved itself to be an efficient and useful tool for the evaluation of the radiative component of combined heat transfer in various industrial cases.

The dependence between the temperature of surfaces and the heat fluxes in an enclosure with isothermal gas is defined by a set of non-linear algebraic equations. By their modification, a direct expression of the radiative heat flux at the surface has been obtained through radiative coefficients and source and sink temperatures. Radiative coefficients can be determined analytically for some fundamental systems - infinite parallel plates, part of plate and enclosing area, infinitely long cylinders and concentric spheres. These systems are represented by an enclosure consisting of two surfaces with given emissivities and temperatures filled with gas of a known constant temperature. Radiative coefficients are functions of configuration factors, emissivities of surfaces and emissivity of gas for determined mean beam length dependent on the system in question. The application of this method is very effective especially in the case of simple isothermal enclosures consisting of two surfaces with given emissivities and temperatures, filled with gas of a known temperature.

Using the method of radiative coefficients, quite new and original methods for the evaluation of the radiative component in the case of combined heat transfer have been developed. The creation of these methods was initiated by attempts to solve industrial cases.

The mathematical model for the evaluation of the radiative component in the case of bare tube banks/bundles considers individual tubes surrounded by gaseous layers. The actual surrounding volume of these layers was substituted by an equivalent cylindrical surrounding volume for the simplification of geometry to be able to apply the fundamental system with infinitely long cylinders. The main result of the calculation is the evaluation of heat flux density at the tube outer surface which is represented by the inner cylinder of the substituting system. Consequently the radiative heat transfer coefficient is evaluated.

As an application the thermal calculation of the heat exchanger primary reformer - type TANDEM can be mentioned. The radiation shares on the total heat flux with 18 to 30% dependent on the number of baffles in this special shell -and- tube heat exchanger.

The mathematical model based on the method of radiative coefficients can be incorporated into any program used to predict the thermal performance of a tubular heat exchanger and/or of a convective section with bare tubes in a furnace. A computer program based on the above method has been incorporated into HTRI software.

A method for comparatively simple calculation of combined heat transfer in the case of finned tubes (convection + radiation + conduction in fins) is compatible with the above method for calculation of heat transfer in the case of bare tubes. The application of the method of radiative coefficients is a common point of the two methods. The resulting value of heat flux density is given by the product of this value in the case of bare tubes, correction factor for fin thickness and above all the heat flux multiplicator which expresses the fins influence from the radiation point of view. This factor is given by the ratio of the radiative heat flux on the tube surface in the case with the presence of fins and that on the bare tube surface. The values of both heat fluxes can be obtained by solving equations of the heat balance for surface zones (using a simple application of the zone method) considering conduction in fins. In the case of calculating a convective section in the furnace, considering cooling the flue gas from about 400^oC to 250^oC the share of the radiative heat transfer is around 6%.

A model for the calculation of the radiative heat flux from a radiation chamber to the shield tubes of a furnace and the evaluation of the radiative heat transfer coefficient as a boundary condition in calculating temperature distribution using the method of finite elements can be considered as another applications of the method of radiative coefficients.

Relatively simple procedures based on this method have been developed with the purpose of being convenient for practising engineers. These procedures are based on some simplifications enabling the evaluation of the radiative component in combined heat transfer. Computer programs enable us to explore the sensitivity of selected parameters as a result of e.g. the variation in geometrical parameters and operation conditions.

METHODS FOR MODELLING PROCESS FURNACES

Furnaces are inherent equipment in many processes in both the chemical and petrochemical industry. In a number of this processes, particularly in those for bulk chemical production, a process furnace forms the core of the chemical/petrochemical plant. It is quite usual that cost of the furnace system ranges between 10 and 30 % of the total investment and, as for the operating cost, fuel burned in the furnace can represent up to 90 % of the energy bill. It is therefore evident that furnace design methods play an important role. An experimentally verified mathematical model which enables us to simulate the behavior of various arrangements of furnace system belongs to a list of designer's powerful tools. The application of the model in assisting effective operation is of even greater importance.

The basic requirements and aspects concerning the methods for modelling have to be specified in connection with the choice of methods (the purpose, use and the field of application of a mathematical model, the complexity of furnace geometry, the input data available for simulation, etc.). Two different approaches are presented:

(i) the first one deals with a detailed and complex mathematical model for a specific purpose;
(ii) a simple mathematical model convenient for rapid calculation of tubular fired heaters is the subject of the second one.

The mathematical model of a radiation chamber in steam reforming based on the zone method (i) is a result of long term research. It enables us to simulate thermal chemical processes both in the combustion space and inside the reaction tubes. This model was verified by measurements in a process plant.

A complete mathematical model capable of simulating the conditions in a reaction furnace is formed by connecting the combustion space model with that of reactions inside the tubes during its development the model has passed through several stages due to its continuous confrontation with the plant data and measured values obtained in the specially designed testing program on an industrial size steam reformer furnace. The permanent comparison of calculated and measured values has made it possible to introduce a few simplifying assumptions and some improvements in the practical application of the zone method. A two dimensional section of radiation chamber cross-section can be considered for modelling. This section is limited by two tube rows which are considered as so called "specled walls". An approximative way for the fast evaluation of direct exchange areas has been developed based on analytical relations and approved comparing the resulting values with those obtained using the Gaussian method for numerical integration. the real gas (flue gas) is substituted by three gray and one diathermic component. A universal set of absorption coefficients independent on temperature and gas composition was used which is advantageous for repeated calculations (e.g. different regime of furnace operation ) because direct exchange areas can be calculated only once. As the calculations in the early developmental stage of a model considering plug flow of flue gas led to an unrealistic maximum of the tube outer surface temperature ( compared with the measured one) a suitable level of back mixing, based on flue gas flow visualization, has been introduced into the model.

By means of the model the influence of selected parameters on main characteristic quantities in steam reforming was investigated, a method for steam reforming design was developed (and applied in practise) and a knowledge base for fuzzy expert system was created. The widest possible application of the verified mathematical model is obviously in the sphere of furnace operation in technological processes. The model enables technologists and operators to simulate expected operation conditions.

The simple mathematical model of furnaces (ii) based on Hottel's One-Gas-Zone Method is extended a verified. The mathematical model is constructed in such a way as to enable the calculation to be applied to any tubular fired heater. This implies that the furnace with an arbitrary geometry and consisting of both radiation chamber and convective sections can be calculated. The computer program based on this model basis involves two major alternatives for calculation. The first one deals with complete calculation of process fluid heating inside tubes by considering the pressure drop. In the second one the model of heat transfer inside the tubes is replaced by given values of the heat transfer coefficient and input/output temperature of the medium.

Special attention is given to a model for the calculation of heat flux from a radiation chamber to the shield tubes. Heat flux at the shield tubes is evaluated using the application of the method of radiative coefficients for an enclosure where the analytical solution is known. Assuming the radiative heat transfer from the radiation chamber flue gas to the shield tubes is approximated by that of an equivalent gaseous volume belonging to the shield tubes and surrounded by two surfaces- the absorbing one (shield tubes transferred to the "specled wall" ) and the radiatively adiabatic one, substitution by a simple fundamental system is feasible and the radiative heat flux density at the tube surface can be evaluated.

The radiative component of heat transfer is also considered in the calculation of convective sections.

Compared with the model based on the zone method a simple mathematical model for relatively rapid calculations for process furnaces provides designers and operators with a useful tool in those cases where distribution of temperatures, heat fluxes and other quantities are not required. An algorithm for a furnace with arbitrary geometry consisting of radiation chamber and convective sections has been created. Special emphasis was laid on simplicity and universality.

This model proved itself to be very useful in a new method for effective furnaces integration into processes based on Pinch Technology.

MONITOR SYSTEM FOR RADIATION TRANSFER

Monitor System for Radiation Transfer (MSRT) is an interactive computer system intended for creating radiative models.

Thee concept radiative model includes the following three parts:

• optical model (spectral, group and total absorption and emission coefficients);
• model for radiation heat transfer.

1. Base of Models.
   This part of the MSRT consists of the program modules which are realize the mathematical models of radiative processes in gases, plasma, dispersive media:
   • The calculation methods for modeling of elementary radiative processes: probabilities and cross-section at the bound-bound, bound-free and free-free transitions; oscillator strengths, Franck-Condon factors; modeling of scattering processes by theory Mie;
   • The calculation methods for modeling of complex radiative processes in gases and plasma: the group and spectral models of low temperature plasma optical properties: absorption coefficients, half-spherical emission capability, Plank and Rosseland coefficients.
   • The calculation methods for problem of radiative transfer in selective-emitting, selective-absorbing and selective-scattering media.
2. Data Base
   This part of the MSRT contains the following data:
   • probability, cross-section, rate constant;
   • spectroscopic properties of the particles (atoms, molecules, ions).
3. Software
   This part of the MSRT consists of computer implementation of the models, service procedures, program modules for data manipulation.

The different radiative models are presented: spectral absorption coefficients of low-temperature plasma with atomic lines structure, "smeared rotational line"absorption coefficients for diatomic molecules, absorption coefficients of diatomic molecules with rotational line structure.

For example, the figures show spectral absorption coefficients of air plasma without and with atomic lines structure.

THE INTERACTION OF RADIATION WITH CONVECTION IN SUBSONIC LAVAL NOZZLES OF LASER'S PLASMA ACCELERATORS

The laser-heated plasma accelerator (LHPA) operation is based on the phenomenon of stationary existence of a low-temperature plasma (continuous optical discharge, COD) in a focused laser beam (as a rule, CO₂-continuous wave laser)^1-3. The major parameters of LHPA are^1-5: temperature region - 300-20000 K, pressure - 1 atm, inlet velocity - 1-100 m/s, nozzle length- 20-30 cm, height of the plane laval nozzle critical section - 1-2 cm.

• the laser radiation power, the laser radiation wave number, the quality and geometry of the beam, and time characteristics of laser radiation;
• type of gas and working pressure;
• the velocity and the spatial distribution of gas flow in the vicinity of COD;
• the structural peculiarities of the gas dynamic flow, first of all, the channel shape and the manner of gas flow supply and heating.

The evolution of temperature and gas dynamical characteristics distributions in the laser radiation field in a Laval nozzle is described by a system that comprises the two-dimensional equations of conservation of energy, the continuity and Navier-Stokes equations, the equations of selective thermal radiation transfer (in the form of a multigroup P1-approximation of the spherical harmonics method), and laser radiation transfer (in the geometrical optics approximation)⁵.

The implicit-explicit numerical method is developed for solving the self-consistent equations system in a curvilinear calculated grid. Because the pressure variation was small, only the temperature dependence of the thermodynamic, optical, and transport properties of the gas was taken into account. The main iteration process was organized using the equations of conservation of energy and the equations of transfer of selective thermal and laser radiations. At this stage, we were in fact solving the truly nonstationary problem of radiative-conductive heat exchange. On reaching the convergence of the iteration process at each time step, the iteration solution of the system of gas dynamical equations was started, after which all of the indicated iteration processes were repeated until a full convergence of the unknown functions at a time step was reached.

The results of numerical modelling of subsonic air flow in various Laval nozzles are discussed. The parameters of LHPA operating under conditions of radiative combustion of COD are found by calculations, and conditions are established under which stationary eddy flow structures may be expected to occur because of a large gas mass flowing around the COD combustion region. The calculation results make it possible to point out one of the probable mechanisms of COD stability loss in a gas flow, namely, the mechanism associated with the occurrence of intense vortex motion behind a discharge. This suggests sum means of improving the efficiency of operation of power devices of the indicated class.

THE EFFECT OF ASH PARTICLES' COMPLEX REFRACTIVE INDEX ON RADIATIVE HEAT TRANSFER WITHIN A BOILER FURNACE

A three-dimensional mathematical model for predicting turbulent flow with combustion and radiative heat transfer within a furnace was developed. The model consists of two linked sections: (1) the transport equations which are nonlinear partial differential equations solved by a finite difference scheme, and (2) the radiative heat transfer which was analyzed by the zone method. The Monte Carlo method was used to evaluate total radiative interchange in the system between the zones. An existing rigorous theory of interaction between infrared waves and solid particles developed for several simple particle geometries was used. The resulting set of equations, called the Mie equations, enable the calculation of absorption and scattering coefficients, for given particle shape, particle size, complex refractive index, mass concentration, density of particle materials, and wavelength of the incident radiation. With this model the series of calculations with varying gas radiative properties models, loads, types of fuel, excess air, burner tilt angles, and several particle parameters can be performed. The effect of these variables on the gas temperature and heat flux distribution within the furnace can be studied.

The mathematical model was previously validated against experimental data collected on two large furnaces by comparison of the measured and predicted gas temperature and heat flux distribution. The purpose of this paper is to study only the influence of the ash particles' complex refractive index on the radiative heat transfer within the furnace of a pulverized-coal-fired boiler. The method of approach is mathematical modelling and simulation. Three various complex refractive indices (for three particle materials: soot, alumina and simulated ash) were considered. The effects of these complex refractive indices on: 1. Heat absorbed by water wall and secondary superheater from combustion chamber gases (radiation, convection and total); 2. Combustion chamber exit temperature and 3. Heat flux distribution on furnace walls were presented.

Among the others, the following conclusions are drawn. Soot and alumina were compared for 25 m mean particle diameter, and it was found that complex refractive index of the soot gives a higher heat transfer to the sink surfaces and a lower gas temperature at the furnace exit than the alumina complex refractive index. The comparison of complex refractive index for simulated ash and for soot, for 5 m mean particle diameter, shows that the soot complex refractive index gives a higher heat flux to the water walls and lower gas temperature at furnace exit than the complex refractive index for simulated ash. The real value of the complex refractive index for ash material in pulverized-coal-fired boiler would be somewhere between the values given for soot and alumina. The mathematical model given here is believed to be widely applicable for the analyses of furnace combustion chambers.

THERMAL RADIATION IN A PASSIVE CONTAINMENT COOLING SYSTEM BY NATURAL AIR CONVECTION

The Karlsruhe Research Center and the University Karlsruhe have proposed a new containment concept for future pressurized water reactors. This containment should ensure that it remains intact even in severe core meltdown accidents and the decay heat can be removed in a passive way by natural air convection and by thermal radiation.

To determine the coolability limit of such a passive containment cooling system, experimental investigations are performed in the PASCO (acronym for passive containment cooling) test facility of the Karlsruhe Research Center. By these experiments different effects, e.g. the effect of the heated wall temperature and the wall emissivity, on heat transfer are studied. Moreover, a data base is provided for validating advanced multi-dimensional computer codes.

In addition to the experimental work, numerical calculations are performed by using the three-dimensional computer code FLUTAN to simulate flow conditions and heat transfer behavior in the PASCO test channel, an one-sided heated rectangular flow channel. Even though the FLUTAN code was successfully applied to many complex thermalhydraulic problems, it did not contain radiative heat transfer. To extend the application of the FLUTAN code to the PASCO problem, a radiation model is developed with the following main features:

• Radiatively non-participating fluids, grey and diffuse wall surfaces.
• Net radiation method for enclosures.
• Analytical methods for view factor calculation.
• Macro-element method for improving numerical efficiency.

THERMAL SIMULATION OF A GLASS PANEL SUBJECTED TO A FIRE RADIATIVE AND CONDUCTIVE MODELIZATION OF THE SEMI-TRANSPARENT MEDIUM

This paper presents a study of the modelization of the thermal behavior of glass used for windows in buildings subjected to the typical solicitation of an accidental fire. The manufacturers of this glass are concerned about predicting its behavior in the kind of difficult conditions which generally lead to its mechanical destruction, given that glass breaks when subjected to large, rapid temperature variations. Safety standards require that sheet glass resist for a certain length of time, to enable the evacuation of people inside a burning building.

With the model we have constructed it is possible to follow, over a period of time the thermal state of a glass sheet subjected to a solicitation which is representative of a fire situation. Radiation data are processed with the P1 method, transfer being taken as unidirectional, given the thickness of the glass. Conduction is represented by the heat equation, radiative and conductive transfers being coupled. A detailed representation is given of the boundaries of the mediums, radiative as well as conductive. The aim of this study is to bring out not only the influence of the thickness of the glass, but also its thermo-optic characterization, using a spectrum of nongray medium, in relation to the traditional opaque medium hypothesis. There is also to be an analysis of the influence of the optic properties of the surfaces, which are different for the two sides of the physical boundaries: those on the outside, facing the external conditions, and those on the inside of the glass. The results are given in the form of diagrams showing sensitivity to these hypotheses.

IDENTIFICATION OF SIZE AND STRUCTURE OF SOOT AGGLOMERATES AT THE EXHAUST OF DIESEL ENGINES

Feasibility of multi-wavelength laser diagnostic to characterize the soot agglomerates structure at diesel engine exhaust is invstigated. Normalized scattered intensities and dissymmetry ratios are obtained numerically for different soot agglomerates. Primary soot diameters considered are between 20 and 50 nm, and the number of primary spheres varies from 25 to 125. The results are obtained for incident radiation wavelength spectrum of 200 to 500 nm. From the preliminary analyses it is found that if the primary sphere diameter is estimated a priori from TEM pictures (or other means), measurement of spectral variation of dissymetry ratios can be used in characterizing the morphology of the agglomerates accurately. It is concluded that a more general study, which includes multiple fractal dimensions for mono-or poly - dispersed agglomerates can easily be performed by extending the approach presented here.

PRELIMINARY STUDY ON AIR-FUEL MIXING AND COMBUSTION OF A DIVIDED-CHAMBER DIESEL ENGINE SYSTEM

In recent years, numerous experimental studies were carried out in optically accessible closed vessel, model combustion chamber and rapid compression machines in order to understand the Diesel combustion process. The advantage of using these devices is to study the influence of a single parameter on the engine internal process and to have large optical access for non-intrusive optical measurement techniques. However, it is difficult to operate simultaneously at high ambient temperature and pressure and with air swirl typical of real engines. Therefore, optically accessible engines are used to overcome these intrinsic limitations, even though severe constructive modifications are need. In particular, measurement techniques such as laser Doppler velocimetry (LDV), light extinction, Mie scattering, 2-D particle image velocimetry (PIV), laser induced incandescence (LII) and laser induced fluorescence (LIF) techniques require special design since the orthogonal optical accesses could influence negatively the engine behavior.

In the present paper, an alternative way to investigate the combustion process with optical techniques which allows a realistic design is proposed. It consists of a divided-chamber with longitudinal and lateral optical accesses, connected to a flat piston engine that supplies air swirl and typical of both direct injection and prechamber diesel engines. This device allows to apply the optical techniques mentioned above and the direct cinematography and flame spectroscopy. Moreover, the internal processes can be reproduced using the new KIWA-3 developed by A. Amsden at Los Alamos National Laboratory, a computer program with block-structured mesh for complex geometry.

The present preliminary study aims at analyzing experimentally and numerically the in-cylinder processes focusing the attention on two main aspects of Diesel combustion: I) air- fuel mixture preparation and ii) spatial distribution of soot temperature and concentration. High-speed direct cinematography at 8,000 frame/s was used to visualize the spray distortion and to follow the entire Diesel combustion process. The appearance of flame luminosity was used to indicate the spatial region of the Diesel jet where the combustion starts and the temporal evolution of the combustion. The ignition start downstream the spray in the region where the fuel is well atomized and the flame expands rapidly filling up the chamber volume. This trend is confirmed by the computations implemented with a new droplet breakup model and turbulent mixing-controlled combustion model.

The computations confirm also the flame emissivity measurements carried out by multicolor pyrometry at the same crank angle. Spectral emissivity measurements, in the range 300 to 700 nm, were used to investigate more quantitatively the combustion phenomenon. In fact the chemical species such as OH and CH radicals as well as the soot temperature and concentration were estimated in the different locations of chamber volume.

SURFACE RADIATION EFFECTS ON NATURAL CONVECTION COOLING OF A DISCRETE HEAT SOURCE IN AN OPEN-TOP CAVITY

The heat transfer performance of an electronic heat generating device mounted on one of the vertical conductive walls of a two-dimensional open-top cavity is modelled numerically. Natural convection in the fluid, radiation heat exchange between the surfaces of the cavity, and condition in the vertical walls are considered. The results are obtained for air in a cavity of 100 mm height, 20 mm width and with 2 mm thick vertical walls. The imposed heat flux on the heating element is varied leading to a variation of the heat flux based Rayleigh number from 1.148x10⁴ to 8.03x10⁴. Walls with different values of thermal emissivity are considered in order to explore the effects of surface radiation. Initial results were obtained neglecting radiation. However, comparison of the numerical results with the experimental data highlighted the importance of this mode of heat transfer. This led to the inclusion of a radiative heat transfer model in the computations. Noticeable changes in the thermal and flow fields in the cavity are observed when the radiation is included. A reversed flow in the cavity with the formation of a recirculation zone in the proximity of the cold vertical wall is observed for the conjugate radiation-convection case. This is attributed to a rise in the temperature of the cold wall due to the radiation heat exchange with the heating device. This behavior can not be predicted if only natural convection is considered. Hence, it was concluded that fully conjugate analysis, i.e. natural convection, radiation and conduction, must be considered in the numerical modelling of heat transfer from cavities with discrete heat sources for accurate prediction of heat transfer data and flow and thermal fields.

EFFECT OF AN EVOLVING AEROSOL CLOUD ON AN UPWARD RADIATIVE HEAT TRANSFER

The present paper deals with radiative heat transfer through a dense aerosol cloud.

We explain here the theoretical basis of a numerical model. In this model we solve the radiative heat transfer equation which coefficients are calculated with respect to the aerosol optical properties (extinction and scattering coefficients). These optical properties are determined using the Mie or Rayleigh theory, depending on the aerosol size.

This model can be used in different cases, here we applied it to the later phase of an hypothetical severe Pressurized Water Reactor (PWR) accident, in a Molten Core Concrete Interaction (MCCI) framework. We then show that the numerous aerosols generated during the concrete ablation by the corium melt have a significant effect on the radiative transfer in the reactor containment.

Indeed, we present an example of calculation using given aerosol size and concentration distributions which allow us to determine extinction and scattering coefficient profiles. Our instance does not exactly model any experiment, however the physical conditions are very close to the test L4 from the Advanced Containment Experiment (ACE) program.

We also provide a calculation with constant values (with respect to the position) for the extinction and scattering coefficients. This last study will then allow us to simplify the radiative transfer equation in our model and therefore shows the effect of such a hypothesis.

RADIATIVE AND CONVECTIVE HEAT TRANSFER IN CIRCULAR SECTORS

The radiative and convective heat transfer in circular sectors is numerically investigated in this paper. The flow is assumed to be a steady and fully developed incompressible flow and the entrance effects are considered. The momentum method is applied to simulate the radiation of the fluid, which is nonscattering absorptive gas. The thermal boundary condition is a constant temperature on the wall. The control volume method is used to discrete the governing equations for flow, heat transfer, and radiation. The results show that the participation of radiation leads to the enhancement of heat transfer, but changes the behavior of the variation of the local Nusselt number along the axial direction. For pure forced convection, the local Nusselt number decreases monotonically from a large value at the entrance to an asymptotic value. However, for combined radiative and convective heat transfer, the local Nusselt number decreases until it reaches a minimum value at certain downstream locations and then increases again.

NUMERICAL SIMULATION FOR RADIANT GAS FLOWS IN COMPLEX STRUCTURES

A new hybrid method is described for numerically solving radiative-convective heat transfer in complex structures. The simulated thermal installations can be combustion chambers or furnaces in which the obstructing walls are set.

In this paper, Hottel zone method has been greatly improved. The development include three parts. First, based on the analysis of uniform sampling theorem and thermal radiation, the Uniform Deterministic Discrete Method (UDDM) is presented for radiative heat transfer in complex structures with participating media. By uniform sampling versus random sampling in radiation, this new technique is superior to Monte Carlo in accuracy, convergence, speed and computer cost. In present paper it is applied to evaluate the direct exchange areas of zone method even using fine grid. Second, for energy matrix equations, the Principal Variable Correction Method is described together with its physical principle analysis. By using this new approach instead of Newton-Raphson or Broyden iteration, the convergent solution can be achieved more fast and easily. Finally, considering the combination of convective heat exchange, a two-level grid is used, one for zone method, the other for the Navier-Stokes finite- difference equation of gas fluid flow.

MODELING OF A SOLAR RECEIVER DEDICATED TO LPG CRACKING FOR ETHYLENE PRODUCTION

A large solar central receiver plant for the cracking of liquid petroleum gas (LPG) to produce 20.000 tone/yr ethylene has been designed. The solar power required for this receiver is more than 30 MW at design point. This receiver has an hexagonal cross section with 8 m each side and 8 m height. Two hundred forty cracking tubes, 7.3 cm in diameter each, are distributed in parallel to the receiver walls. The solar energy is introduced through the top of this receiver.

The model of this receiver calculates the mode in which the concentrated radiation entering the receiver aperture is redistributed through radiative heat transfer between the various refracting and absorbing surfaces, and then calculates the heat fluxes on the reactor walls. These fluxes heat up the reacting gases and supply the enthalpy for the endothermic cracking reactions.

In a series of similar cases for radiation heat transfer calculations we used the zonal method where the internal surface of the receiver and external surface of the reactor tubes are divided into surface elements considered isothermal and equi-energetic. The dimensionality of the problem depends the number of zones considered. the computational time associated with formulating net radiative exchange in such a type of enclosure varies roughly with the cube of the number of zones into which the surface is divided. Therefore to treat, by zonal method, a receiver with hundreds reactor tubes, where each tube is considered a separately is an impossible task even if we were using the most powerful computer. And for reactor modelling, this accuracy can be superfluous.

In this case can be used successfully the classical concept of equivalent gray plane introduced by Hottel and Sarofim. According to this concept a row of tubes mounted parallel to a refractory wall which received the radiative heat from exterior can be replaced in calculations by an equivalent grey plane having that emissivity which is a full measure of the effects of tube spacing and of back wall.

Following this concept we considered the receiver with six walls having the equivalent emissivity as in Hottel and Serafim's model. Each of these equivalent walls are divided into a number of zones. In order to decrease the number of zones to a reasonable dimensionality of the problem, we assumed that a few neighboring tubes belong to the same zone. The height of this equivalent wall is divided into a few parallel regions, therefore each zone characterizes equal portion of the length of a group of few tubes.

The flux calculated on each zone following our previous model (see for example: A. Segal and M. Levy: Solar Chemical Heat Pipe in Closed Loop Operation: Mathematical Model and Experiments, Solar Energy, Vol 51, No.5, pp. 367-376) gives the possibility for evaluation of the flux on each segment from the length of each reactor tube. The fluxes on the reactor tube are used as an input to the endothermic reactions of LPG cracking for ethylene production. The interface between the radiative part of this model (into receiver enclosure) and its chemical part (inside the tube reactor) is done through the wall temperatures on each reactor segment. The chemical part generates a temperature profile on the reactor walls which is used as improved boundary conditions for the radiative part and a new heat flux calculation is generated. The calculation converges to a unique steady state solution and generates the final receiver wall temperatures, net heat flux profile, reactor wall temperature profile and various other parameters.

This model has been helpful in designing this receiver for better understanding of its properties. The computer program based on this model was run for a variety of flow conditions, feed composition, the catalyst activity, and pressure. Using this model made it possible to establish optimal conditions for operating the receiver and to establishment of the optimal configuration of the solar field of heliostats required for this application.