Implementation of Models for reheating processesin industrial furnaces

Daniel Rene Kreuzer Andreas WernerVienna Universtiy of Technology , Institute of Energy systems and Thermodynamics

Getreidemarkt 9/302, 1060 Wien

Abstract

Components were developed for the modeling of in-dustrial furnaces in the iron and steel industry likepusher-type and walking beam furnaces. A cell modelon the basis of the dynamic pipe model of the Mod-elica Fluid Library was designed. The cell modelcomputes the heat transfer between furnace walls, fluegas and the processed steel goods. The radiative heattransfer is modeled by a 1-dimensional method basedon Hottel’s net radiation method. Furthermore, mod-els for furnace walls, slabs, hearth and the transport ofthe slabs were designed. The models are suitable foranalyzing operation modes and designing control con-ceptsKeywords: reheating; furnace; simulation; radiation

1 Introduction

Currently the manufacturing industry has focused onincreasing energy efficiency of their production pro-cesses. The effective and sustainable use of energyis becoming more important, due to the issues of cli-mate change and therefore the demand of reducingCO2 emissions. The iron and steel industry in partic-ular incorporates a multiplicity of reheating processesin the production chain of their commodities. For in-stance the reheating of slabs in pusher type or walkingbeam furnaces to set up the right temperature intervalbefore hot rolling as well as annealing of steel coils incontinuous or batch furnaces to trigger the micro struc-ture and therefore the properties of the steel products.The furnaces mentioned are commonly operated byusing natural gas and off gas from blast furnace, cokeoven and basic oxygen furnace. While in the past thedevelopment of reheating processes was concentratedon increasing the production output, recently the de-crease in energy consumption became more and moreimportant. Raising energy efficiency requires the op-timization or redesign of the reheating process steps,

Figure 1: Walking beam furnace

e.g. by implementing advanced control concepts orby analyzing and assessing different operation modes.These engineering tasks indicate the need for accuratedynamic furnace models. There exists a variety of fur-nace models as e.g. described in [5],[8], [4], [2]. Sev-eral models which are used by the industry are basedon experimental modeling (system identification) or socalled black box modeling. The disadvantage of suchmodels is the restricted validity to the considered sys-tems. The main intention of the presented work is thedesign of highly reusable models contained within alibrary which are able to simulate the unsteady heattransfer phenomena of reheating steel goods in indus-trial furnaces. Due to the focus on modeling the phys-ical phenomena and the reuse ability of the models theModelica language standard and the simulation envi-ronment Dymola were chosen for the modeling task.

2 Basic considerations

Figure 1 shows the layout of a typical reheating fur-nace which can be used for heating semi-finishedsteel products before hot rolling. Basically theywork as counter current heat exchangers, thereforethe transport direction of the processed goods is di-

Figure 2: Heat transfer mechanisms of reheating process

rected against the main flow of the flue gas. Reheat-ing furnaces usually consist of a convective, heatingand soaking zone. The heating and soaking zone arecommonly operated by gaseous fuel-type burners, andthe main heat transfer phenomenon is radiation. Dueto the high flue gas temperatures up to 1400◦C, fur-naces are lined with refractory. The walls have a typi-cal multiple layer configuration of different materials.During processing, the steel goods are positioned on ahearth, which has to be cooled for e.g. by evaporativecooling. The basic phenomena of reheating processeswhich have to be modeled are

• Heat transfer due to radiation between walls, fluegas, flame and processed goods

• Heat transfer due to convection between flue gasand walls as well as flue gas and processed goods

• Heat conduction in furnace walls, processedgoods and hearth

• Combustion of gases (burners)

• Transport of flue gas - Fluid Flow Models (FFM)

• Transport of processed goods - Solid Flow Mod-els (SFM)

Figure 2 depicts the appearing heat flow rates of re-heating slabs in a walking beam furnace. The furtherpresented models consider primarily the reheating of

slabs in walking beam or pusher-type furnaces. How-ever the models for heat transfer phenomena are gen-erally valid and may only need a minor adaptation foruse at other reheating processes. The major diversi-ties in modeling reheating processes are evident for theSFM and their impact on the heat transfer models.

3 Fluid Flow

The general concept of the reheating process mod-els is based on a homogeneous cell model which in-corporates fluid flow and heat transfer. The dynamicpipe flow model of theModelica.Fluid.Libraryis used for modeling the 1-dimensional fluid flowof flue gases in the furnaces. A comprehen-sive description of solving the fluid transport equa-tions with theModelica.Fluid.Library is given in[3]. Properties of the fluids are determined by theModelica.Media.Library , e.g. internal energy,specific enthalpy, density, thermal conductivity, dy-namic viscosity. The heat transfer between the gases,the walls and the slabs is realized by designing a newheat transfer model which replaces the standard mod-els.

Figure 3: Cell model based on Modelica fluid library

Figure 4: Net radiation heat flow of surface i in a cellcontaining the emitting gas

4 Heat Transfer

4.1 Radiative heat transfer

The radiative heat transfer plays a key role in mod-eling high temperature industrial furnaces. The dif-ferent mathematical radiation models can be classifiedby their dimensionality. in literature 0-dimensional, 1-dimensional and 3-dimensional models are discribed.3D-models achieve the highest accuracy but requirehigh efforts in modeling and computing time, thosekind of models are usually implemented in commer-cial CFD-packages. 0D-models are based on thestirred-tank reactor and give only a rough estimationfor the radiative heat transfer of the regarded systems.1D-models are in between the two aforementionedtypes and seem to be a proper trade-off in accuracyand modeling effort. For modeling the radiative heattransfer a 1D-model derived by [6] was selected. Ithas been developed from the net radiation method ofHottel, which is described in [7], [1]. Scholand [6]depicts the equations needed for calculating radiativeheat exchange between radiating surfaces in an enclo-sure comprising isothermal radiative gases.

Figure 5: Discretization of a combustion chamber

4.1.1 Governing Equations

The following section states the equations according[6] used for the radiative heat transfer model. Thegrey and diffuse radiating surfaceAi , shown in Figure4, receives the radiative heat flowHi composed of theheat flow emitted by the different grey surfaces and thegas. The grey surface i again emits with its tempera-tureεiEbi and reflectsHiρi . The net-radiation heat flowof the surface i, which exchanges heat with n differentsurfaces of the enclosure, yields

qi = Wi −Hi (1)

at which the outgoing radiative heat flow is given by

Wi = εi ·Ebi +ρi ·Hi (2)

εi is the emissivity of the grey wall andEbi = σ ·T4i is

the hemispherical total emissive power of a black bodywhereasσ is the Stefan-Boltzmann constant andTi thesurface temperature. The incoming radiative heat flowis given by

Hi = εgi ·σT4g +

1Ai

n

∑j=1

A jWjFji τ ji (3)

and equation 1 results in

qi = εi ·Ebi − εi · εgi ·σT4g − εi ·

1Ai

n

∑j=1

A jWjFji τ ji (4)

εgi is the emissivity of the gas andTg is the gastemperature.Wj is the outgoing heat flow of the jth-surface element,Ai andA j are the associated surfaceareas,Fji is the view configuration factor between sur-face j and surface i andτ ji is the transmittance of thegas. The discretization of a furnace is done by con-necting several of the above described cells. The heattransfer due to radiation between those cells is real-ized by a diathermic wall approach depicted in figure5. The diathermic wall presents the interface between

two cells, e.g. the relation between celln and the celln−1 yields:

Hi,n = Wi,n−1 (5)

Hi,n−1 = Wi,n (6)

The above stated equations presume a grey gas entirelysurrounded by grey walls. For a grey gas the depen-dency of emissivity, absorptivity and transmittance isgiven byαg+τg = 1, αg = εg andρg = 0, whereas fora grey bodyτ = 0, α + ρ = 1 andε = α . A real gaslike flue gas from a combustion reaction is composedof several species which have different properties. Im-portant radiative gases likeCO2 andH2O are selectiveemitters which means their emissivity , absorptivityand transmittance strongly depend on the wave lengthinterval. Due to this fact the assumption of a singlegrey gas surrounded by grey walls has to be abandonedwhereas the assumption of grey body radiation of thewalls remains valid. Therefore the equation 4 yields:

qi = εi ·Ebi − εi ·

∫ ∞

λ=0ελgi ·σT4

g dλ (7)

−εi ·1Ai

∫ ∞

λ=0

n

∑j=1

A jWjFji τ ji dλ (8)

The clear-gray gas model approach according to [9] isused for the calculation of the emissivity, absorptiv-ity and transmittance of the gas mixture. The emit-ting species are considered as grey gases and the otherspecies are concentrated as non emitting clear gas. Theemissivity of a gas mixture withH2O andCO2 as emit-ting gases is hence:

εG =3

∑i=1

ai(1−e−kgi·pg·SG) (9)

wherekgi is the absorption coefficient of the differentconstituents,pg equals the sum of the partial pressureof pH2O andpCO2, SG is the radiation beam length andai are weighting factors which are given by a linearapproach

ai = b0i +b1i ·TG (10)

and the conditional equation

3

∑i=1

ai = 1 (11)

The absorptivity is calculated with a similar approachto equation 9 and 10, only the weighting factors are de-termined with the wall temperatures instead of the gastemperature. For solving the equations of the radiativeheat transfer the calculated radiative gas properties areapplied to equation 4.

Figure 6: Cell model for heat transfer

4.2 Convective heat transfer

The convective heat transfer can simply be added tothe net-radiation equation of the surface i

qi = Wi −Hi + qconv (12)

qconv= α · (Tg−Ti) (13)

The heat transfer coefficients for the single surfacesare determined with the relation of forced convectionat a single plate according to [9]

4.3 Thermal conduction

The transient heat conduction problem in walls, slabsand hearth is modeled by Fourier’s 1-dimensional par-tial differential equation

∂T∂ t

= a·∂ 2T∂x2 +

Qe

ρcp(14)

4.4 Implementation of heat transfer models

4.4.1 Radiation and convection models

The net-radiation method and the convective heattransfer equations presented in section 4.1 and4.2 were integrated into a new designed modelRadiationAndConvectionHeatTransfer . Themodel calculates all the data necessary for the heattransfer mechanisms, e.g. the view configurationfactors of the involved bodies, the radiative propertiesof the gases, the Nußelt-Numbers, the convective heattransfer coefficients etc. Those data in conjunctionwith the temperatures lead finally to the net-radiation

Figure 9: Code for computation ofQb_dot_radconv for a single cell

Figure 7: HeatPort_Radiation Connector

Figure 8: GeometryPort_Radiation Connector

and convective heat flows. A bottom-cell and atop-cell are developed for the discretization of thefurnaces to take the different geometric configurationsinto account. For the energy exchange between fluid,walls, slabs, hearth and the neighbouring cells a newtype of connectors HeatPort_Radiation was devel-oped. It is not working like real physical connectordue to the fact that it may be connected to diathermicwalls. A diathermic wall usually represents theinterface between two discretized cells. Unlike a realwall a diathermic wall has to enable a heat flow inboth directions at the same time. Therefore anotherflow variable Q_flow_res and the correspondingpotential variableT_g which is equal to the gastemperature of the cell were introduced. The radiativeheat transfer depends on the temperature differencesof the involved bodies and it depends on the position,attitude, propagation and emissivity of the emittingsurfaces. That fact leads to a further none physicalconnector GeometryPort_Radiation which transmitsthe necessary data from the involved bodies to theheat transfer model and vice versa. The generatedconnectors are combined to composite connectors.Figure 6 shows the heat transfer model of a top-celland the connections to walls, slabs and the hearth.Two further connectors are present one which sends asignal with needed data for heat conduction betweenslabs and hearth and the other one which realizes aheat flow due to radiation between a top and a bottomcell via a diathermic wall.The connection between the heat transfer model andthe fluid flow model is made via the energy balance

Figure 10: Surface configuration of top cell

for a cell stated by [3] for a single flow segment i.

der(Us[i])=Hb_flows[i]+Ib_flows[n]+Qb_flows[i]

Figure 9 shows the code for the computation ofvector Q_dot_radconv for n different flow seg-ments. An accurate implementation of heat transferallows only the discretization of one flow segment.The vector Qb_flows is set equal to the vectorQ_dot_radconv which is determined from the heatflow vectorsq and q_res of the participating m+7surfaces. The number of surfaces is determined fromthe number of slabs m which are present during thesimulation, the number of furnace walls, the hearthsurface and the surface of the gap between slabs andhearth. For all real wallsq_res is equal to 0 andq isthe net heat flow due to radiation and convection. Thenet heat flowq of the slabs is set to 0 if the slab isnot present in the considered cell. The vectorA_agg

aggregates the surface areas of all participating bodies.

4.4.2 Wall, hearth and slab models

The wall hearth and slab models are based on theModelica.Thermal.HeatTransfer.Components,which provide a solution for the 1-dimensionalheat conduction equation. Figure 11 showsthe model of a layered wall which is dis-cretized with the elementsHeatCapacitor andThermalConductor. The models have connectorsto the heat transfer model which represents theconnection to the radiative and convective trans-

Figure 11: Layered slab and wall model

fer equations and a connectorHeatPort of theModelica.Thermal.HeatTransfer.Components

which serves to set up the boundary conditions. Theinterface between the connectors of the cell and theModelica.Thermal.HeatTransfer.Components

is realized by a special surface element. This elementrepresents the emitting surface and is the first dis-cretization of the wall and it is based on the equationsof a theHeatCapacitor. The slabs and the hearthare modeled in the same way as the layered wall.The slab model possesses two surface models for theheat transfer between a top cell and a bottom cell,further it has connections to the hearth for the heattransfer due to conduction and radiation as well asto the solid flow model which determines the slabposition. The hearth model is split into a model for atop cell and a bottom cell and presents an equivalentsystem for the real configuration, because the exactgeometry configuration can hardly be modeled due tothe characteristics of a 1-dimensional analysis. TheHearthBottom model has a connector for the celland one for a boundary condition which is usuallya temperature condition. TheHearthTop modelpossesses connectors for the heat conduction with theslabs, the radiative and convective heat transfer withthe cell, the radiative heat transfer with the slabs anda signal connector which transmits needed data forcomputing the phenomena mentioned before. Figure13 shows the parameter needed for a layered wall.The number of layers, the number of discretizationper layer, the thickness of the single layers andthe material properties of each layer as well as thestart temperatures of the different layers have to besupplied.

5 Combustion

The release of energy in reheating furnaces is typicallyrealized by the combustion of gaseous fuels. The fun-damental combustion calculation is utilized to deter-

Figure 12: Hearth model for bottom and top cell

Figure 13: Parameter window for layered wall

mine the required mass flow of air, the flue gas amountand the flue gas composition. The energy input of thefuels is considered by the lower heating valueHu. Thedesigned models neglect the dynamics of the combus-tion reactions. ThePartialLumpedVolume serves asbase class for the combustion model.The energy bal-ance for the combustion volume is given by

∂ (ρuV)

∂ t= mAir ·hAir + mFuel(hFuel +Hu) (15)

−mFlueGashFlueGas+∑Q

For the adiabatic combustion∑Q = 0. For in-stance,∑Q can be used to define heat transfer dueto flame radiation but the geometric configurationand the propagation of the flame has to be knownas well as the flame emissivity. The volume modelhas connections for the combustion air, the gaseousfuel and the flue gas. It is assumed that the en-tire volume is filled with homogeneously distributedflue gas. The flue gas in the volume model iscompletely combusted without any residues of com-bustibles. Three new ideal gas mixtures were definedwith the Modelica.Media.Library . The mixtureFuelGaseous which consists of the speciesH2, N2,CO2, CO, H2S, CH4, C2H2, C3H8, C4H10,C3H6, C2H2

and C4H8, FlueGas which consists ofO2, Ar, N2,CO2, H2O, SO2 andAir which consists ofO2, Ar, N2,CO2, H2O. The combustion is realized by computingthe rate of change dependent on time of the single flue

Figure 14: Volume for combustion of gases

gas masses∂mO2

∂ t ,∂mN2

∂ t , ∂mAr∂ t ,

∂mCO2∂ t ,

∂mH2O

∂ t ,∂mSO2

∂ t ac-cording to the fundamental combustion equations anda predefined excess air.

6 Solid Flow Model

The solid flow model consists of 3 different mod-els shown in figure 15. One model represents theheating goods, in this case slabs and therefore theSlabDistributer. It possesses the same connectorsas the slabs and aggregates all slabs, which are presentduring simulation. The number of slabs m has to bespecified before starting a simulation and determineshow many objects "slab" are present. Every singleobject slab has to be connected to each heat transfercell, which is done via theSlabDistributer. TheSlabFeedmodel is the most important element for thesolid flow because it computes the position of the sin-gle slab models, refers the slab properties to the slabmodels and triggers the slab feed movement. Com-puting the slab positions and the heat transfer betweenslabs and the other involved bodies led to the intro-duction of a reference coordinate system. Thereforeall models which represent geometric objects, e.g. thecells, walls slabs etc., get an origin referenced to theintroduced coordinate system. The propagation of themodels is specified into the positive direction of thex(1)-, y(2)- and z(3)-axis. The fluid flow is definedpositive in the positive x-direction and the solid flowis directed against the positive x-axis. TheSlabFeedhas a connection to theSlabDistributer and trans-mits slab position and properties. A further connec-tion is needed via boolean input to a pulse generator.At every pulse the slabs are moved one step forwardand a new slab is fed into the furnace or cell. Theslab transportation occurs step wise which means theyare allocated to a certain discrete position at each timestep of the simulation. That means that the time of slabmovement is equal to 0. Due to the nature of a pusher

Figure 15: Solid Flow Models

type or a walking beam furnace all slabs are moved si-multaneously through the cells as long as they are inthe furnace. If they leave the boundaries of the furnacethe slabs are brought to an end position. If more thenm slabs are processed during one simulation the posi-tion of the completed slabs is reset to a starting posi-tion, new property values and new input temperaturesare set. TheSolidFlowSystem model is defined asan inner system wide component which provides im-portant data for most of the models, e.g. maximumnumber of slabs in simulation, slab properties, generalfurnace data etc..

7 Testing the heat transfer models

Figure 16: Configuration of reheating furnace

The heat transfer models are tested by modelingthree zones of a reheating furnace depicted in figure16. The regarded furnace incorporates two heatingzones, a convective zone and the soaking zone. Thereheating furnace is operated by natural gas and theenergy input is evenly distributed over the two heatingzones. The combustion air is preheated in a recupera-tor, which is not considered in the model.

7.1 Experimental setup

For the experimental setup the two heating zones andthe convective zone are modeled. The soaking zoneis neglected because it serves only to ensure that the

Figure 17: Discretization of furnace

core of the slab reaches the target temperature. Fig-ure 17 shows the discretization of the furnace zones.Each zone is discretized by two cell models, a top cellmodel and a bottom cell model. The heating zone1 is represented by cell2T and cell2B, heating zone2 by cell1T and cell1B and the convective zone bycell3T andcell3B. The postfix ’B’ and ’T’ distinguishbetween bottom and top cells. The heating zones areequipped with 12 burners, which are modeled by theburner elements. One burner model represents 6 burn-ers and is connected to either a top or a bottom cell.The cells of the convective zone (cell3T and cell3B)are not connected to burner elements. The top cellsand bottom cells are connected via the fluid connec-tors and the diathermic walls among each other. Topand bottom cells are not connected via fluid connec-tors, there exists no fluid exchange between the topand the bottom cells. There is only an exchange ofheat due to radiation between the top and the bottomcells. Furthermore the cells are connected to the wallmodels, the slab models and the hearth models. Figure18 and 19 show the burner elements and the connectedcell model. Figure 20 shows the whole configurationin Dymola.

Figure 18: Model of burner elements

Figure 19: Model of connected cell and wall models

Figure 20: Model of the furnace zones in Dymola

7.2 Simulation

For the simulation of the modeled furnace configura-tion the number of slab objects was set to m=25. Thatimplies a maximum number of 25 slabs can be simul-taneously processed by the furnace. Each slab is dis-cretized into 5 layers subdividing the thickness of theslab and two surface elements. It is necessary to setstart positions and temperatures for the slabs, which isbasically done via the solid flow model. The walls arebuilt of different layers of refractory, which are com-mon for those type of furnace, each layer is equal toa discretization of the heat conduction model. Theoutside connectors of the walls and further the wallmodels need proper temperature start values. The cellshave a fixed position according to a reference coordi-nate system and values for the propagation in x-, y-

and z-direction. The position of the cell origin is im-portant for the calculation of the view factors. Thefurnace has a length of 26.5 m in x-direction. The heatconduction between slabs and the hearth had to be pre-vented because of high values of the cell dimensions(8m to 10.5m) in x-direction compared to the width ofthe slabs(≈ 1.5m). The heat conduction between theslabs and the hearth of each cell is computed with themean temperature of all slabs which are present in acell. The energy input via the burners is set constantfor the whole simulation time at 68.75MW, the heat-ing value of the natural gas is set to 48.9kJ/kg, the sin-gle constituents of the gas remain constant during thesimulation and the temperature of the preheated com-bustion air is set to 350◦C. The emissivity of all wallsand the hearth is set to 0.9 and of the slabs to 0.8. Theslab movement starts after 500s and after the durationof 553s the slabs are moved one step forward as wellas a new slab is fed into the furnace. The value ofthe forward step is determined by the x-dimension ofthe incoming slab. 15000sof the furnace operation aresimulated by using the integration algorithm RadauIIafor stiff systems. The integration time for the simula-tion of 15000s lasted 1711.39s.

7.3 Results

Figure 21 shows the temperature distribution over thesimulation time of two different slabs. Figure 22shows the discrete position of the slab origin oversimulation time of the slab 19, furthermore it depictsthe position during a whole pass of the slab elementthrough the furnace. The dimensions of slab 19 at thattime areL(x) = 1.397m, H(y) = 0.218m andD(z) =12.12m. At 1053s the slab 19 is fed into the furnace.The processing of the slab ends at 11560swhich givesa time of residence of about 10507s. At the end of pro-cessing slab 19 has a surface temperature of 1351◦Cat the top and a temperature of 1335◦C in the middlelayer. The surface temperature at the bottom shows tobe similar compared to the top temperature. The tem-perature rise of the slabs show also the transport fromone zone into another by a change in the temperatureslope. After the heating process slab 19 is fed to an endposition where new properties and dimensions can bedefined, e.g. a new input temperature. Afterwards theslab is fed to a predefined position in front of the fur-nace where it remains until processing is started again.Figure 24 shows the temperature distribution of the gasand the top wall of the top cells. It can be seen thatthe temperature is strongly influenced by the incom-ing and outgoing slabs. During the slab movement a

high amount of energy contained in the outgoing slabis leaving the system and another much lower amountof energy connected to the incoming slab is enteringthe system. The resulting difference in enthalpy leadsto a temperature drop at each slab movement. Fig-ure 25 shows the emissivity and the absorptivity of gasfor the irradiating top wall oaf cell1T. The emissivitydepends on the gas temperature but it is independentfrom the wall temperature. As expected it decreaseswith increasing temperature. The absorptivity of thegas depends on the temperature of the irradiating wall,as a result it has not the same values as the emissivityof the gas.

8 Conclusion

Components for modeling dynamic heat transfer in in-dustrial furnaces like pusher-type or walking beam-type have been developed. They can be used to modelsingle zones or entire furnace systems. Those physi-cal models would be particularly suitable for designingadvanced control concepts for the operation of the fur-naces. The simulation of the test configuration showssatisfactory results. The most important future task isthe validation of the heat transfer models by modelingan existing reheating furnace and comparing the sim-ulation with measurement results.

9 Symbols

Symbol Physical value

Ai(m2) surface area of wall iai(−) weighting factor gas radiationb0i(−) coefficient of weighting

factor polynomialb1i(1/K) coefficient of weighting

factor polynomialcp(J/kgK) specific heat capacityEb(W/m2) hemispherical total emissive

power of a black bodyFi j (−) view configuration factor

from surface i to jh(J/kg) specific enthalpyHi(W/m2) incoming radiative

heat flow of surface i

Symbol Physical value

Hu(J/kg) lower heating valuekgi(1/m·bar) absorption coefficient

of species im(kg/s) mass flowQe(W) power sourceqi(W/m2) net-radiation heat flow

of surface iqconv(W/m2) convective heat flow

of surface iSG(m) radiation beam lengthTi(K) temperature of wall iTg(K) gas temperatureu(J/kg) specific internal energyV(m3) volumeWi(W/m2) outgoing radiative heat flow

of surface iαi(−) absorptivity of surface iαg(−) absorptivity of gasεg(−) emissivity of gasεi(−) emissivity of surface iλ (m) beam lengthρi(−) reflectivity of surface iσ(W/m2K4) Stefan-Boltzmann constantτ(−) transmittance of gas

References

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[2] D IETZ, U. Einsatz mathematischer Modelezur Simulation industrieller Feuerräume unterbesonderer Berücksichtigung des Strahlungsaus-tausches. PhD thesis, Universität Stuttgart, Insti-tut für Verfahrenstechnik und Dampfkesselwesen,1991.

[3] FRANKE, R., CASELLA , F., SIELEMANN , M.,PROELSS, K., OTTER, M., AND WETTER, M.Standardization of thermo-fluid modeling in mod-elica.fluid. In 7th Modelica Conference, Como,Italy, Sep. 20-22,(2009).

[4] HACKL , F. Eine betriebsorientierte Methodezur Berechnung der instationären tehermischenVorgänge in Wärmeöfen. PhD thesis, Montanisi-tische Hochschule Leoben, 1974.

[5] K IM , M. Y. A heat transfer model for the anal-ysis of transient heating of the slab in a direct-fired wlking beam type reheating furnace.Inter-national Journal of Mass and Heat Transfer 50(2007), 3740–3748.

[6] SCHOLAND, E. Ein einfaches mathematischesModell zur Berechnung des Strahlungswärmeaus-tausches in Brennkammern, vol. Fortschr.-Ber.VDI-Z Reihe 6 Nr.111. VDI-Verlag GmbH-Düsseldorf, 1982.

[7] SIEGEL, R., AND HOWELL, J. R. Thermal Ra-diation Heat Transfer. Hemisphere PublishingCorporationher Publishing CorporationTaylor &Francis, 1992.

[8] STRACKE, H. Mathematisches Modell fürbrennstoffbeheizte Industrieöfen unter beson-derer Berücksichtigung der Gasstrahlung,vol. Forschungsberichte des Landes Nordreihen-Westfalen Nr. 2821/Fachgruppe MaschinenbauVerfahrenstechnik. Westdeutscher Verlag, 1979.

[9] V EREIN DEUTSCHER INGENIEURE, V.-G. V.U. C., Ed.VDI-Wärmeatlas 10. Auflage. Springer-Verlag, 2006.

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