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Eurotherm Seminar N° 81 Reactive Heat Transfer in Porous Media, Ecole des Mines d’Albi, France June 4 – 6, 2007 ET81 – XXXX MODELLING OF A ROTARY KILN FOR THE PYROLYSIS OF ALUMINIUM F. Marias Laboratoire de Thermique Energétique et Procédés Ecole Nationale Supérieure d’Ingénieurs en Génie des Technologies Industrielles Universite de Pau et des Pays de l’Adour Rue Jules Ferry, BP 7511 64075 Pau Cedex [email protected] H. Roustan Alcan, centre de recherhce de Voreppe 725, rue Aristide Berges, BP 27 38341 Voreppe Cedex [email protected] A. Pichat Alcan, centre de recherhce de Voreppe 725, rue Aristide Berges, BP 27 38341 Voreppe Cedex [email protected] This paper deals with the mathematical modelling of a rotary kiln which is used for the recycling of aluminium waste. This model is mainly based on the coupling between:“a bed model” describing the processes occurring within the bed of aluminium waste flowing inside the kiln, “a kiln model” describing heat transfer within the kiln itself, and, “a gas model” describing processes occurring within the gaseous phase held inside the furnace. The “bed model” is mainly based on a plug flow of particles of aluminium inside the kiln. Mass balances as well as energy balances allow for the prediction of the fraction of the organic material within the particles of aluminium as well their temperature. Relevant equations for the “kiln model” include heat conduction and heat exchange with solid and gaseous material. The equations for the “gas model” are mainly based on fluid mechanics equations coupled with turbulence, radiation, and combustion. The software Fluent TM will be used in order to solve this last model. In this paper, some insights will be given on the description of the “bed model” and “the kiln model” and on the procedure used for the coupling of these models. Exchange variables as well as solving procedure will also be included. Numerical results will be compared to experimental ones, obtained from the pilot scale rotary kiln at Alcan research centre Keywords. Rotary kiln, Pyrolysis, Waste, CFD, Modelling 1. Introduction The goal of this study is to build a mathematical model able to predict the physico-chemical processes occurring when particles of aluminium coated with organic materials (polish in the case of cans, cardboard in the case of milk packaging …..) are introduced into a rotary kiln in order to be cleared from their organic content. Figure 1 sketches the experimental device under consideration. The goal of the study is to predict: The composition of the particles of aluminium leaving the furnace, Their temperature at kiln exit, The composition of the exhaust gas P elec P elec Particles Aluminium Alu. Air Gas Rotary kiln Electrical heaters Figure 1: Sketch of the device under consideration

MODELLING OF A ROTARY KILN FOR THE …eurotherm81.mines-albi.fr/papers_pdf/29-Marias.pdfMODELLING OF A ROTARY KILN FOR THE PYROLYSIS OF ... This paper deals with the mathematical modelling

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Page 1: MODELLING OF A ROTARY KILN FOR THE …eurotherm81.mines-albi.fr/papers_pdf/29-Marias.pdfMODELLING OF A ROTARY KILN FOR THE PYROLYSIS OF ... This paper deals with the mathematical modelling

Eurotherm Seminar N° 81 Reactive Heat Transfer in Porous Media, Ecole des Mines d’Albi, France June 4 – 6, 2007

ET81 – XXXX

MODELLING OF A ROTARY KILN FOR THE PYROLYSIS OF ALUMINIUM F. Marias Laboratoire de Thermique Energétique et Procédés Ecole Nationale Supérieure d’Ingénieurs en Génie des Technologies Industrielles Universite de Pau et des Pays de l’Adour Rue Jules Ferry, BP 7511 64075 Pau Cedex [email protected] H. Roustan Alcan, centre de recherhce de Voreppe 725, rue Aristide Berges, BP 27 38341 Voreppe Cedex [email protected] A. Pichat Alcan, centre de recherhce de Voreppe 725, rue Aristide Berges, BP 27 38341 Voreppe Cedex [email protected] This paper deals with the mathematical modelling of a rotary kiln which is used for the recycling of aluminium waste. This model is mainly based on the coupling between:“a bed model” describing the processes occurring within the bed of aluminium waste flowing inside the kiln, “a kiln model” describing heat transfer within the kiln itself, and, “a gas model” describing processes occurring within the gaseous phase held inside the furnace. The “bed model” is mainly based on a plug flow of particles of aluminium inside the kiln. Mass balances as well as energy balances allow for the prediction of the fraction of the organic material within the particles of aluminium as well their temperature. Relevant equations for the “kiln model” include heat conduction and heat exchange with solid and gaseous material. The equations for the “gas model” are mainly based on fluid mechanics equations coupled with turbulence, radiation, and combustion. The software FluentTM will be used in order to solve this last model. In this paper, some insights will be given on the description of the “bed model” and “the kiln model” and on the procedure used for the coupling of these models. Exchange variables as well as solving procedure will also be included. Numerical results will be compared to experimental ones, obtained from the pilot scale rotary kiln at Alcan research centre Keywords. Rotary kiln, Pyrolysis, Waste, CFD, Modelling

1. Introduction

The goal of this study is to build a mathematical model able to predict the physico-chemical processes occurring when particles of aluminium coated with organic materials (polish in the case of cans, cardboard in the case of milk packaging …..) are introduced into a rotary kiln in order to be cleared from their organic content. Figure 1 sketches the experimental device under consideration. The goal of the study is to predict:

• The composition of the particles of aluminium leaving the furnace, • Their temperature at kiln exit, • The composition of the exhaust gas

PelecPelec

ParticlesAluminium

Alu.

AirGas

Rotary kiln

Electrical heaters

Figure 1: Sketch of the device under consideration

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Eurotherm Seminar N° 81 Reactive Heat Transfer in Porous Medi, Albi, France June 4 – 6, 2007 ET81-XXXX

Before entering the section devoted to the formulation of the mathematical model, it is necessary to check the relevant processes occurring within the furnace. These phenomena can be described by tracking a particle’s path from its supply to the kiln to its exit (Boateng et Barr (1996), Chen & al. (1993), Chen et Lee (1994), Heydenrych & al. (2002), Leger & al. (1993, a, b), Li & al. (1999 a, b), Martins & al. (2001), Marias (2003), Patisson & al: (2000), Sudah & al. (2002)):

• Transport of the particle under the rotation and inclination of the kiln, • Heating of the particle (radiation, heat transfer inside and outside of the particle…), • Decomposition of organic material (pyrolysis), • Diffusion of volatiles outside of the particle, • Potential oxidation of aluminium.

From the gaseous side of the process, the relevant phenomena are turbulent flow with chemical reaction and radiation (Jackway & al. (1996), Leger & al. (1993 a,b), Mastorakos & al. (1999)):

• Turbulence, • Chemical oxidation of the products of pyrolysis • Interaction between turbulence and combustion • Heat transfer (convection and radiation).

Finally, conduction inside the kiln, heat exchange between the kiln and the bed of solids as well as heat exchange between the kiln and the gaseous material are important phenomena. 2. Mathematical modelling

This section is devoted to the description of the model that has been used to translate the physical and chemical

processes occurring within the kiln into mathematical formalism 2.1. The kiln

The kiln plays a major role in the total amount of heat transferred within the furnace. Indeed, it receives the thermal power released by the heating furnaces, and releases it towards the bed of solid particles of aluminium to be reprocessed. That is why, for our modelling purposes, the measure of its temperature is required. The kiln is supposed to be a cylinder of stainless steel of inner diameter D, of length L and of thickness e. It is fully three dimensional. It undergoes several heat fluxes, and their total amount results in its thermal equilibrium. These specific heat fluxes are (Figure2):

bedϕ

elecϕ

Pelec

extϕ

lnki,gasϕ

Figure 2 Sketch of specific heat fluxes to the kiln

• extϕ : Heat exchange with surrounding air (maybe external heating), • ln,kigasϕ : Heat exchange with the gas held within the furnace, • bedϕ : Heat exchange with the bed of particles to be reprocessed.

Finally, the kiln is subject to conduction. In polar coordinates both temperature and heat fluxes depend on r, θ and z (r being held in the range [D/2 D/2+e]). Because the load is not evenly distributed over the circumference of the kiln, its temperature is very dependant upon θ . However, in the frame of this study, in order to reduce the complexity of the system, this dependence has been lumped, and the choice has been made to compute an average temperature over the circumference of the kiln. Moreover, because of the measure of the thickness of the kiln and of the order of magnitude of the thermal exchange coefficient, the low value of Biot’s number has led us to consider the kiln as being a thin material. Indeed, it is assumed that, for Biot’s number less than 0.1 this assumption is valid. Thus, the temperature of the kiln is supposed to be one dimensional. Because of the low velocity of the particles of solid inside the furnace, heat transfer between the moving bed and the kiln is expected to be mainly conductive and radiative. Following the work of Patisson & al (2000), the heat flux is expressed as follows:

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Eurotherm Seminar N° 81 Reactive Heat Transfer in Porous Medi, Albi, France June 4 – 6, 2007 ET81-XXXX

( )πα

ϕ bedbedbedbed zTzThz )()()( kiln−= (1)

where hbed=25 W.m-2.K-1 is the heat exchange coefficient between the kiln and the bed of solids, bedT is the local temperature of the bed, and bedα is the filling angle of the bed inside the kiln. Hence, bedϕ denotes heat transfer from the kiln to the bed An energy balance on an annulus of kiln of length dz leads to the following equation

bedgasexterzTe ϕϕϕλ +−=∂

∂kiln,2

kiln2

kiln (2)

The solving of this equation requires: the temperature of the bed Tbed and he heat flux from the gas to the kiln ln,kigasϕ . 2.2. Particles of aluminium waste

The particles are considered to be a basis of inert material (aluminium) covered with one organic material sharing a same surface S0 (Figure 3). In order to simplify the mathematical formulation of the problem the following assumptions have been considered

• The surface of the particle is not modified as it passes through the furnace 0SS = • Inert material does not undergo any transformation, thus, its thickness remains constant along

the kiln. 0,inertinert ee = • The organic material coating the particle to be processed is gradually consumed along the

furnace. Its thickness is variable ( orge ).

0S

inerte0,orge

Figure 3 Sketch of the particles to be processed As the waste is processed inside the furnace, it undergoes several phenomena. The first one is heating up with

simultaneous (or not) drying. However, because the particles used in the experimental section are dry at the input of the furnace, drying is not taken into account in our model. Then, when an ignition temperature is reached, the organic material begins to be subjected to chemical reaction. Because of the low oxidant concentration inside the deep bed, it is assumed that this chemical reaction to is mainly pyrolysis. Under a sufficient temperature, the organic coating is gradually converted into lighter volatiles chemical species (hydrogen, methane, carbon monoxide…) and tars. Then, if the length of the furnace is sufficient, all the organic coating has been released once the particle leaves the kiln. These pyrolysis phenomena are very complex and can lead to a wide range of chemical species and tars. However, in the frame of this study, it is assumed that the pyrolysis can be sufficiently described by a global reaction. This global conversion leads to the formation of a set of chemical species. The composition of the products of pyrolysis is reached by atomic and energy balances. This means that all of the atoms present in the organic coating are present in the produced gas, and that the energy released by the combustion of this gas is the same that the one released by combustion of the organic coating itself (Marias, 1999). Moreover, it is assumed that the kinetics of the reaction follows the kinetics (Bockhorn & al. 1996)

( )na

RTE

Adtd αα

−⎟⎠

⎞⎜⎝

⎛ −= 1exp (3)

∞−

−=

,0,

0,

orgorg

orgorg

mmmm

α (4)

increases from 0 to 1 within the furnace. Thus, given the composition of the organic part of the waste, and given the kinetic parameters of its pyrolysis

reaction, one is able to compute the mass flow rate and composition of the gas leaving the particle at a given temperature.

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Eurotherm Seminar N° 81 Reactive Heat Transfer in Porous Medi, Albi, France June 4 – 6, 2007 ET81-XXXX

2.3. The bed of particles

The previous paragraph has given insights into the description of one particle. Nevertheless, in the kiln, particles are fed continuously. Thus, they are in the shape of a “bed of solids” inside the furnace. The next paragraph is devoted to the description of this bed.

Several studies have focused on the modelling of the motion of solid particles inside a rotary kiln. (Boateng et Barr (1996), Boateng et Barr (1997), Henein & al. (1983), Heydenrych & al., (2002), Li & al., (2002 a,b)). Most detailed models describe the bed of solids as a pseudo tri dimensional media. In the framework of this study, and in order to simplify the mathematical description of the system, and in accordance with several researchers (Chen et Lee (1994), Patisson & al. (2000)) the bed is supposed to be in plug flow inside the kiln. Thus the state variables of the model depend solely upon the axial coordinate. Finally, it is supposed that the bed of solids proceeds through the kiln (under its motion of rotation and its slope) at a constant speed. This is computed as the ratio of the length of the kiln to the mean residence time of a particle inside the furnace: τ/Lubed = . This residence time is computed according to Li & al (2002 b)

As of now, N will stand for the lineic concentration of particles within the kiln. If it is assumed that these particles are neither subject to attrition nor to agglomeration, the concentration is constant inside the furnace and equals its value at the furnace input. Indeed, at the feed of the kiln, the number of particles must match the total mass flow rate to be processed. Thus, if wQ and bedu respectively denote this flow rate and the velocity of the bed, one can write:

bedpartpat

w

ueSQ

NN0,00,

0 ρ== (5)

Given the value of α (4), one is able to compute the specific mass flow rate of volatiles released from the bed and supplied to the gaseous phase

( )npart

aorgorg

litpyro RT

EASe

lNm αρ −⎟

⎟⎠

⎞⎜⎜⎝

⎛ −−= 1exp00,

''& (6)

If bedS stands for the cross sectional area of the bed, the mass balance over a slice of thickness dz of the kiln leads to:

( ) ''1 pyrobedpartbed

bed mldz

dSu &=−

ρε (7)

and the energy balance leads to

( ) ( )( ) Dhhmldz

hdSu bedbedgaspyroorgrvolpyrobed

partpartbedbed πϕϕ

ρε −+Δ+=− ,,

''1 & (8)

where bedgas,ϕ stands for the heat flux released by the gas to the bed, volh the enthalpy to weight of the volatiles leaving the bed and pyroorgr h ,Δ the enthalpy to weight of the pyrolysis reaction.. Further details about the computation of

bedgas,ϕ will be given later. The enthalpy of a single particle is computed as the sum of the enthalpy of the inert material it holds plus the

enthalpy of the volatiles resulting from complete pyrolysis of the organic content of the waste (as it has been quoted, there is equivalence between these energy as the composition of the volatiles is computed on this basis).

( ) volorgrefpartinertpinertpart hYTTcYh +−= , (9) The enthalpy of the volatiles is computed using the enthalpy of formation as reference and ideal mixing

assumption:

⎟⎟⎟

⎜⎜⎜

⎛+= ∫∑

=

part

ref

volT

T

kpkf

N

k

volkvol dTTchzh )(,

0,

1

(10)

2.4. Gas held within the furnace

Several researchers have shown that Computational Fluid Dynamics is well suited for the description of processes

occurring within the gaseous phase of a rotary kiln (Jackway & al. (1996), Leger & al. (1993 a,b), Marias (2003); Mastorakos & al. (1999)). Thus, in the frame of this study, the commercial software FluentTM is used for this specific topic. The governing equations of this software will not be developed here. The interested reader should refer to the Fluent User’s guide. The following models have been used:

• Turbulence : ε−k model • Chemistry of the system : chemical equilibrium computed by minimisation of Gibb’s free energy • Interaction turbulence-combustion : density probability function (pdf) • Radiation : discrete ordinates model

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Dealing with radiation, the discrete ordinates model allows for the computation of the radiation intensity at every location of the gaseous phase:

( ) ( ) ( ) '4

0

''4

2 .),(4

),(),(. ΩΦ+=++∇ ∫ dsssrITansrIassrI ss

π

πσ

πσσ

rrrrrrrrr (11)

From knowledge of the radiation intensity I, in every direction (it depends on the discretisation) one is able to compute the incident radiation on a wall:

'4

0

)',( Ω= ∫ dsrIG ww

πrr

(12)

As it will been explained in the next paragraph, the knowledge of the incident radiation on the walls of the domain will lead to the determination of bedgas,ϕ and ln,kigasϕ .

In the CFD package, the simulation is fully three dimensional. The domain under consideration is the gas hold within the kiln. This domain is limited by a cylindrical shaped wall (the kiln) and a surface representative of the free surface of the bed. Thus, in order to run the CFD software, the following is required

• Equation of the free surface of the bed (required for the construction of the geometrical domain and of its mesh)

• Profile of mass flow rate and composition along this free surface (boundary condition of the domain) • Profile of turbulence intensity and dissipation along this free surface (boundary condition of the domain) • Profile of temperature along this free surface (boundary condition of the domain) • Profile of temperature along the kiln (boundary condition of the domain) Among the information yield after convergence, the following data is known • Incident radiation on the kiln surface • Incident radiation on the free surface of the bed • Convective heat flux to the kiln surface • Convective heat flux to the free surface of the bed

3. Solving

This section is devoted to the description of the solving procedure used to complete the computation. As it has been

explained earlier, two sub-models (kiln+bed of solids, gaseous phase) allow for specific computation once specific boundary conditions have been yield to each of these sub-models. Figure 4 illustrates the way the system has been cut, and what information should be exchanged from one sub-model to another. In the first part of this section, insights are given on the computation of the exchange variables. Then, in a second part, the general solving procedure is presented.

Kiln (1D model)

Bed (1D model)

Gas (3D model)

Surface informationMass flow rate of volatilesTemperature

Heat flux

Temperature

Heat fluxes

Original system Separated system

Figure 4 : Sketch of the system under consideration and of the exchanged variables

3.1. The kiln

The first difficulty to tackle in the overall solving procedure is linked to the nature of the free surface of the bed. Indeed, as the organic material is released from the bed of solids, its cross section area diminishes. Because this free surface is the limit of the domain taken into account by the CFD package, the same surface should be shared by the two sub-models. However, this means that a new meshing of the computational domain should be performed as iterations proceed. This would make the automation of the process really difficult. Thus it has been assumed that this surface had the constant shape of a flat plane.

The information brought by the “solid system” sub model is one dimensional. However, in the CFD package two dimensional profiles are required for the limits of the domain (kiln and free surface of the bed). Regarding temperatures, this is not such a problem. The one dimensional profile is extended over the width of the concerned surface. This is a

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Eurotherm Seminar N° 81 Reactive Heat Transfer in Porous Medi, Albi, France June 4 – 6, 2007 ET81-XXXX

little bit different for the specific mass flow rate of volatiles. Indeed, very nearer form the kiln wall, the thickness of the bed is quasi nil. Because there are no particles at this particular location, there is no reason for volatile material to be released from this point. On the other hand, in the middle of the bed (from the transversal point of view), the thickness of the bed is maximal. One can expect that the yield of volatiles will be maximal at this location. To tackle this drawback of the model, we have chosen to weight the specific mass flow rate of volatiles by the depth of the bed.

The relevant information yielded by the CFD package is relative to the heat fluxes to the bed of solids and to the kiln. In paragraph 2.4 it has been shown that these heat fluxes were the sum of a convective and a radiative term. The CFD package can directly yields the convective specific heat flux. Dealing with the radiative term, one needs to transfer the information brought through the incident radiation on a wall into appropriate heat fluxes. Hence, these two dimensional heat fluxes have to be computed according to

( ) ( )bedbedbed

bedfluentbedgas GT −

−= 4

, 422

σε

εϕ (13)

( ) ( )ln4

lnln

lnln, 4

22 kikiki

kifluentkigas GT −

−= σ

εε

ϕ (14)

where bedε and lnkiε stand respectively for the emissivities of the bed and the These heat fluxes being fully three dimensional, it is necessary to sum up these fluxes over the filling angle in order

to make them “one dimensional” (as is required by the bed model):

∫=bed

dfluentbedgasbedgas

α

θϕϕ0

,, (15)

∫−

=bed

dfluentkigaskigas

απ

θϕϕ2

0

ln,ln, (16)

3.1. General solving procedure

Because of the coupling through boundary conditions and exchange variables, the only way to solve the overall problem is iterative. An iteration is composed of two successive calls to MatlabTM, then one call to FluentTM then one call to MatlabTM. The convergence criteria are based on a non evolution of the exchange variables. This means that for any of the exchange variables ( pyrom& , lnkiT , bedT , bedgas,ϕ , ln,kigasϕ ) the convergence is validated once:

φεφ

φφ<

−∑=

−nodesDN

iki

ki

ki

NodesDN

1

1

1

1

1 (16)

Generally, less than 50 global iterations are required for convergence to be reached. This is not that much. However, it can be tiresome to ensure the solving sequence by “hand”. The limiting step being the introduction of the new boundary profiles into the CFD package. That is why we have constructed a tool (using JavaTM language) to automate the solving procedure.

3. Results

In this section, an application of our modelling strategy is presented. It is devoted to the processing of aluminium

wastes in a rotary kiln. The centre de recherche Alcan de Voreppe (CRV) owns a rotary kiln at laboratory scale. Unfortunately, for reasons of confidentiality, information about the characteristic length and diameter of the kiln cannot be provided. The kiln is heated externally, by two electrical heaters (an appropriate function for extϕ has been used for our modelling purposes). The particles that are fed to the kiln are can leads.

In order to check for the validity of the model, a “cold experiment” is firstly performed. This means that the nominal mass flow-rate of air is fed to the kiln which is kept free of particles to process. Two electrical powers are supplied to the electrical heaters in order to reach 250°C in the middle of the first heater and 500°C at the centre of the second one. Figure 5 illustrates the comparison between the experimental profile of temperature on the axis of the furnace and the numerical one for this particular experiment. As it can be seen, there is a very good agreement between them.

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0 LL/20

200

400

600

(°C)

z

ExperimentModelTaxis

Figure 5: Temperature in the axis of the kiln.

Comparison between numerical prediction and experimental results

In a second experiment, the kiln is fed with the nominal mass flow rates of waste and air. Electrical powers sent to the heaters are controlled in order to keep the temperature at their centre at the desired value (250°C, 570°C). Figure 6 shows the numerical results obtained after convergence in terms of profiles of temperature (kiln and bed) and specific heat fluxes. It reveals that the temperature of the kiln is very closed from the temperature of the bed. Also, the high negative value of bedgas,ϕ at the end of the kiln indicates high heat transfer between the hot particles leaving the furnace and the cold air entering it.

0

200

400

600

0 L/2 Lz

lnkiT

bedT

T (°C)

-7000

-5000

-3000

-100001000

3000)(W.m -2ϕ

ln,kigasϕbedgas,ϕ

bedϕexterϕ

elecϕ

0 L/2 Lz Figure 6 : Results of the model for the “hot experiment”. Left: profiles of temperature along the bed of particles and

along the kiln. Right: specific heat fluxes Figures represents the fields of mass fraction of O2 inside the furnace. Transversal slices are located at the kiln

input, (from the particle’s point of view) at the centre of the first electrical heater, at the centre of the second electrical heater and at the output of the kiln. It yields information on the location where volatile oxidation takes place. Indeed, it is clearly located between the two heaters, where there is a sharp decrease in the oxygen mass fraction.

Dealing with gas analysis at the exit (from the gas side) of the furnace, experimental results have revealed: - ∈2% COvol [10 -14]

- ∈2% Ovol [7 – 11] the average values computed by the model give 10% for O2 and 10% for CO2 which corresponds perfectly

Figure 7: Predicted profile of mass fraction of O2 (-) inside the furnace

3. Conclusion A general mathematical model for the processing (pyrolysis and subsequent combustion of organic material coating aluminium) of waste inside a rotary kiln has been presented in this paper. This model is decomposed in two sub-models.

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In the first one, governing equations for the transport of a bed of particle submitted to pyrolysis are solved. It takes into account heat transfer with the kiln, with gas held within the kiln and with the surroundings. It allows for prediction of the bed and kiln temperatures as well as for the prediction of the composition of the particle to be processed (in terms of organic content). The second of these sub-models is a CFD package. Indeed, it allows for the prediction of combustion and radiation occurring within the gas held inside the furnace. This combustion is due to the release of volatile from the bed and leads to subsequent radiation towards the free surface of the bed. These two data (specific mass flow rate of volatiles, incident radiation) constitute variables that make the system fully coupled. Hence, we have presented in this paper a method that makes this coupling feasible, from a computer point of view. The whole set of exchange variables has been detailed as well as the general solving procedure. Moreover, a tool that allows for automation of this solving procedure is presented. Also, numerical convergence and relaxation have been defined.

This model has been applied to a laboratory scaled rotary kiln processing particles of aluminium coated with organic material. The model has been validated on a cold experiment and by an analysis of the exhaust gas leaving the furnace. It yields a lot of information for the understanding of the phenomena occurring within the device, making it a good tool for the optimisation or scale up of the overall process

4. References Boateng, A.A., Barr, P.V. “A thermal model for the rotary kiln including heat transfer within the bed”, International

Journal of Heat and Mass Transfer, 2131-2147, 39, 1996. Boateng, A.A., Barr, P.V. “Granular flow behaviour in the transverse plane of a partially filled rotating cylinder”,

Journal of Fluid Mechanics, 233-249, 330, 1997. Bockhorn, H., A. Hornung, U. Hornung, S. Teepe, and J. Weichmann, "Investigation of the kinetics of thermal

degradation of commodity plastics.", Combusion. Scence. Technology, 116-117, 129 (1996). Chen, K.S., Tu, J.T., Chang, Y.R., “Simulation of steady-state heat and mass transfer in a rotary kiln incinerator”,

Hazardous Waste and Hazardous Materials, 397-411, 10, 1993 Chen, Y.Y., Lee, D.J., “A steady State Model of a Rotary Kiln Incinerator”, Hazardous Waste and Hazardous

Materials, 541-559, 11, 1994 Ames, W.F., “Numerical Methods for Partial Differential Equations. Third Edition”, Academic Pressin Computer

Science and Scientific computing. 1992 Henein, H., Brimacombe, J.K, Watkinson, A.P., “Experimental study of transverse bed motion in rotary kilns”,

Metallurgical Transactions, B; 191-205, 14B, 1983. Heydenrych, M.D., Greef, P., Heesink, A.B., Versteeg, G.F., “Mass transfer in rolling rotary kilns: a novel approach”,

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