RESEARCH

Mathematical modelling of the complex heat exchange of a glass melt in a cylindricalinduction furnace

V. K. Schiff

S. I. Vavilov State Optical Institute, St. Petersburg, Russia~Submitted August 27, 1999!Opticheski� Zhurnal67, 11–16~September 2000!

This paper describes the radiative–conductive–convective heat exchange in a semitransparentglass melt with internal heating in terms of a mathematical model. The model includesthe Navier–Stokes equation, the radiant-energy-transport equation, and the heat-transport equation.The Navier–Stokes equation is solved in the variablesC, V, and the radiation-transportequation is solved by the discrete-ordinate method. The contributions of the different componentsof heat exchange to the complex process are compared. ©2000 The Optical Society ofAmerica.@S1070-9762~00!00209-8#

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INTRODUCTION

Glass-melting furnaces are an interesting objectthermophysical studies. A glass melt in an electrical glamelting furnace~EGMF! is an absorptive, radiative, viscouelectrical conductive liquid with internal heating. Therefowhen heat exchange in an EGMF is considered, it is nesary to take into account three heat-transport mechanisconductive, convective, and radiative. When melts of setransparent glasses in small furnaces with a geometricalof the order of 0.1–1 m are studied, the usual effectithermal-conductivity approximation introduces large error1

The study of heat exchange in an EGMF is complicatedthe presence of internal heating, which has a nonlinear cacter in an induction furnace. This paper discusses the plem of conductive–convective–radiative heat transport fogray, viscous, semitransparent liquid with an open mirplaced in the EGMF with gray walls.

THE MAIN EQUATIONS

The mathematical model of steady-state complex hexchange in an EGMF includes the following coupled eqtions: the Navier–Stokes equation, the radiant-enertransport equation, and the heat-transport equation. Assing that the motion of the glass mass in a cylindricinduction furnace is two-dimensional, we shall describeconductive heat transport in terms of the BoussinesOberbeck approximation in a cylindrical coordinate systin the variables of the current functionC and the vorticityV, coupled with the longitudinal and transverse componeof velocity vector (u,v) according to2

V5]u

]z2

]n

]r, u52

]C

r ]z, n5

]C

r ]r. ~1!

Let us write a system of equations of the complex hexchange in dimensionless form~as units we choose temperatureT0, internal radiusR of the EGMF, thermal conduc

787 J. Opt. Technol. 67 (9), September 2000 1070-9762/2000/0

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tivity l0 and absorption coefficientkabs0 of the glass melt,kinematic viscosityn0, and velocityu05n0 /R):

]

]z S ]V

]z2nV D1

]

]r S ]~rV!

r ]r2uV D2Gr

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]r50, ~2!

]2C

r ]z21

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r ]r D52V, ~3!

]

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Pr]z2nTD1

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r ]r S r ]T

Pr]r2uTrD1Sel2

¹Q8

Bo50, ~4!

cosu]I ~r ,z,u,w!

]z2

]@sinu sinwI ~r ,z,u,w!#

r ]w

1sinu cosw

r

]~rI ~r ,z,u,w!!

]r1kabsBuI ~r ,z,u,w!

5kabsBuI b~T!. ~5!

Equations~1!–~5! contain the following dimensionlesvariables: r 85r /R, z85z/R, v85v/u0 , u85u/u0 , V8

5VR/u0, andC85C/u0R, T85T/T0 . Qr 85Qr /4n2sT04 is

the radiative thermal flux vector, which is the first momentthe radiation intensityI (r ,z,u,w) at the point with coordi-natesr ,z in the directionv(u,w),

Qr5E1p

vI ~r ,z,u,w!sinududw, ~6!

Sel8 5SelR

2

rCpn0T0is the specific density of the Joule hea

ing, and

I b8(T)5I b(T)

4sn2T04 5

T84

4pis the total radiation intensity o

an absolute blackbody.The primes on the dimensionless variables are omitte

Eqs.~1!–~5! and are not needed in what follows. The equtions also use the following dimensionless criterional coplexes:N5l0kabs0/4n2sT0

3 is the radiative–conductive pa

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rameter, Bu5kabs0R is the Bouguer radial parameter, B5aR/l0 is the Biot parameter, Bo5rCpn0/4n2sT0

3R is theBoltzmann parameter, and Pr5rCpn0 /l0 is the Prandtlnumber. Heren is the refractive index of the medium,s isthe Stefan–Boltzmann constant,r is the specific density othe melt, andCp is the specific heat of the melt. The last terin energy Eq.~4! ~the divergence of the radiative thermflux! can be obtained by integrating transfer Eq.~5! over theangular variables:

2¹Qr52kabsS T4~r !2E4p

I ~r ,z,u,w!sinududw D .

~7!

INTERNAL HEATING

The heat source in an induction EGMF is an electromnetic wave; therefore, to estimate the energy source in htransport Eq.~4!, the electromagnetic field equations needbe solved. Because of the difference in the time it takesthermal and electromagnetic processes to evolve~1–10 and1026 sec, respectively!, the electric and thermal problemcan be solved separately in high-frequency devices, witstep correction of the interconnected quantities. Therefthe specific density of the Joule heating can be averagedoscillation periodt of the electromagnetic waves:

Sel~r ,z!51

t E0

t

E~r ,z,t !H~r ,z,t !dt, ~8!

where E(r ,z,t) is a complex electric field vector, anH(r ,z,t) is the complex- conjugate magnetic field vector.

For simplicity, we assume that the electromagnetic fiis quasi-steady-state and harmonic, neglect displacementrents in the glass melt, and assume that the length ofinductor and the height of the melt level are significangreater than the transverse linear dimensions of the apptus. The infinite length of the inductor makes it possibleneglect edge effects. The magnetic field in this case willuniform in the gap between the inductor and the melt awill equal zero outside the inductor. Since the tangencomponent of the magnetic field has no discontinuity atinterface of different media, the magnetic fieldHg in the gapequals the fieldHs at the surface of the inductor winding anthe side surface of the heated melt; i.e.,Hs5Hg . By usingthe total current law and recalling that the external magnfield of the inductor equals zero, we express the magnfield at the surface of the inductor winding in terms of tcurrentI IN in the inductor as

Hs5ÃI IN

A2

2, ~9!

whereà is the number of turns in the inductor, anda is thelength of the inductor. For the assumptions made above,tangential component of the magnetic field is describedthe Bessel equation

]2Hz

]r 21

]Hz

r ]r5 iH z , ~10!

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whose analytical solution is expressed in Thompsofunctions,3

Hz5Hs

ber~m!1 jbei~m!

ber~m2!1 jber~m2!, ~11!

where parameteri 5 j 2p f gm0m, andm5A2r /d is the coor-dinate normalized tod51/Apm0mg, which is the penetra-tion depth of the magnetic field in the glass melt,m0 is thepermeability of free space,m is the permeability of the me-dium (m51), g is the electrical conductivity of the melt, andf is the frequency of the electromagnetic field. On the sidsurface of the heated melt,r 5R, we havem5m2.

BOUNDARY CONDITIONS

Boundary conditions for the equation of motion

We shall assume that the conditions of adhesiou(r ,0)50 and v(R,z)50, and of nonflowv(r ,0)50 andu(R,z)50, are satisfied for the melt on the walls of thinduction furnace. The boundary conditions will have thfollowing forms on the axis of symmetry (r 50) and at thefree surface (z5H), respectively:u(0,z)50, dv(0,z)/dr50 and v(r ,H)50, du(r ,H)/dz50. Going to the trans-formed variables on the symmetry axis and at the free sface, C and V take the following forms;C(0,z)50,V(0,z)50, C(r ,H)50, V(r ,H)50, C(r ,0)50, C(R,z)50.2 The variableV has no exact expression at the soliwall and the bottom and is found by approximation. Wtherefore use the second-order approximation proposedBuds.2 We write the expression for the vorticity at theboundary of the melt~the bottom! as

V~r ,0!520.5V~r ,Dz!23C~r ,Dz!

~Dz!2, z50. ~12!

Boundary conditions for the radiation-transport equation

The boundary conditions at the outer boundary detemine the radiation entering the region under considerationconsists of the radiation of the external sources and of tradiation reflected by the ambient medium. The outer bounary can be represented as consisting of two sections:

a completely nontransparent section~the melt–floor andmelt–side-wall boundaries!

I ~r ,z,v!5«I b~Ta!1r

p Ekv8.0

~kv8!I ~r ,z,v8!dv8,

~13!kv,0, r 51, z50

and a transparent section with an optically smooth surfa~the upper melt–air boundary!,

I ~r ,z,v!5I ~r ,z,v8!, kv8~w,q8!.0,~14!

kv~w,q!,0, q8.q0 , z5H

R

I ~r ,z,v!5«I b~T!1r

p Ekv8.0

~kv8!I ~r ,z,v8!dv8,

~15!

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where « and r are the emissivity and reflectance of thboundary surface,Ta is the temperature of the ambient mdium, q0 is the angle of total internal reflection, determinfrom sinq051/n; and k is the unit vector normal to theboundary surface.

Boundary conditions for the energy equation

The boundary conditions for the energy equation aretermined by the energy balance at the edge of the region.assumed that there is convective heat exchange with thebient medium at the external boundary of the region aradiative heat exchange at the transparent boundary; i.e

2l]T

]r1

Bu

NcE

kv.0~kv!I ~r ,z,u,w!dV

5Bu

NcE

kv,0~kv!I ~r ,z,u,w!dv

1Bi~T2Ta!1~12c!s«~T42Ta4!, ~16!

wherea is the heat-output coefficient, which allows for thwater cooling of the furnace walls and the thermal lossethe wall and bottom of the furnace. Coefficientc in Eq. ~6!takes the valuec50 for the open surface of the mirror of thmelt and c51 for the nontransparent glass–vessel-wboundary.

NUMERICAL SOLUTION

It is very hard to solve the system of Eqs.~2!–~5! ana-lytically. It has therefore become traditional to use numerimethods for such problems, in particular the method of findifferences. For Navier–Stokes Eqs.~2! and ~3! and energyEq. ~4!, the control-volume method is used to construct coservative finite-difference equations. The radiation intensdepends upon the temperature field, the frequency, andspatial and angular variables. In developing numerical meods for solving the transport equation, the greatest effortsapplied to the preliminary total or partial integration ovfrequency and angles. This paper uses the approximationgray medium, which represents total integration over thediation spectrum; i.e., it assumes that the radiation characistics of the medium are independent of the wavelengththe radiation. To simplify the angular dependence, this pauses a method of discrete ordinates, first proposedChandrasekhar.4,5 Three steps can be distinguished in tdiscrete-ordinate method. In the first step, the transport etion is angularly discretized. In the second step, for ediscrete direction, defined byu and w, an integro–interpolation method is used to construct a finite-differenapproximation of the transport equation on the given spanetwork. In the third step, additional linear equationsintroduced that couple the intensities at the centers ofcontrol volumes with the corresponding intensities atfaces of the control volumes. After angular discretizatio

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transport Eq.~5! and the corresponding boundary valuesthe intensity, Eqs.~13!–~15!, for the angular directionl, mtake the following form:

m l

]I l ,m

]z2

1

r

]~h l ,mI l ,m!

]w1

j l ,m

r

]~rI l ,m!

]r

52BukabsI l ,m1BukabsI b , ~17!

I l ,m5«wI bw1~12«w! (l 8,m8

wl 8,m8j l 8m8I l 8,m8 ,

~18!j l 8m8.0, j l ,m,0, r 51,

I l ,m5«wI bw1~12«w! (l 8,m8

wl 8,m8um l 8uI l 8,m8 ,

~19!m l 8,0, m l.0, z50,

I l ,m5I l 8,m8 , j l ,m5j l 8,m8 , r 50, ~20!

I l ,m5«wI bw1~12«w! (l 8m8

wl 8,m8m l 8I l 8,m8 ,

~21!m l 8.0, m l,0, z5H.

After completing the transformation, the transport eqution depends on the spatial variables, and the method of fidifferences, generally speaking, can be applied to it.

The divergence of the radiative thermal flux in the eergy equation is expressed by Eq.~6!, in which the spatialdensity of the incident radiation is written out in explicform. Separating out the power dependence on temperain Eq. ~7! for the divergence of the radiation flux enterininto energy Eq.~4! and into the boundary conditions makespossible to linearize it using the formula

T j4'T j 21

4 14Tj 213 ~Tj2Tj 21!. ~22!

Here subscriptsj and j 21 denote the correspondinsubsequent approximations.

TABLE I. Parameters of the medium and characteristics of the inteheating.

Internal radius of EGMF, m 0.053Height of melt level, m 0.1Energy of induction heating, kW 10.5Density of melt, kg/m3 2500Electrical conductivity of melt, S/m 35Specific heat of melt, J/kg K 1256Frequency of electromagnetic field, MHz 5.28Thermal conductivity of melt, W/m K 1.5Absorption coefficient of melt, m21 0.34Refractive index of melt 1.5Dynamic viscosity of melt, Pa sec 30Effective heat-output coefficient, W/m2 183Temperature unit 1000Ambient temperature, K 300Radiative–conductive parameter 0.001Biot parameter 6.47Boltzmann parameter 1.43103

Prandtl number 2.53104

Radial Bouguer parameter 0.018

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The algorithm for solving the interconnected systemEqs.~2!–~5! includes independent sequential iterational pcesses for solving the radiative-energy-transport equatthe equation for the transport of quantity of motion, andenergy equation. In this case, the internal iterational pcesses are carried out until a given accuracy is reached.

FIG. 1. Radiative–conductive heat exchange in an EGMF. Radial distrtions of heat fluxes~1—conductive and2—radiative! for N50.001 alongthe line z5H/2 and temperatures for various values of parameterN: N50.001~3! andN→` ~4).

FIG. 2. Conductive–convective heat exchange in an EGMF.~a! radial dis-tributions of convective~1! and conductive~2! heat fluxes and temperature~3! along the linez5H/2; ~b! isolines of the current functionsQ152331024 and Q252131024 W/m2; ~c! isotherms forT151900 K andT2

52200 K.

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main part of the calculations is done on spatial and angnetworks in which the number of nodes equals 20320 and80, respectively. In the case of the angular network, tcorresponds toSg—the Carlson quadrature.

RESULTS OF THE CALCULATIONS

Keeping the parameters of the medium~see Table I! andthe characteristics of the internal heating fixed, the followifour problems of heat-and-mass exchange were calculaconductive, radiative–conductive, convective–conductiand radiative–conductive–convective. The results of theculations are presented in the form of the distributions oftemperature and of the conductive, radiative, and convecheat fluxes at the half height of the EGMF and the isolinesthe temperature field and the current function~Figs. 1–3!.

Conductive heat exchange

The results of the calculation of the conductive hetransport are presented as a temperature distribution~curve4,Fig. 1!. The temperature distribution is characterized byincrease in the central part of the EGMF (DT'400 K in theinterval 0,r ,0.04 m! and by a sharp falloff at the boundarwith the side surface (DT'1000 K in the interval 0.04,r,0.054 m. This behavior of the temperature field is detmined by the character of the internal heat release for ind

-

FIG. 3. Radiative–conductive–convective heat exchange in an EGMF~a!radial distributions of convective~1!, conductive~2!, and radiative~3! heatfluxes and temperatures~4! along the linez5H/2; ~b! isolines of the currentfunctions Q152231024 and Q252531025 W/m2; ~c! isotherms T1

51700 K,T252200 K.

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tive heating~the main part of the heat is released in tneighborhood of the side wall1!, by the low thermal conductivity of the melt (l51.5 W/mK!, and by the intense heaexchange at the boundary due to the water cooling ofmetal walls of the EGMF~the effective heat-output coefficient, taking into account water cooling, is estimated in Re1,3,6 asa5183 W/m2K for an inductive furnace!.

Conductive–radiative heat exchange

Including a radiative mechanism of heat transport drothe temperature by about 100–200 K~curve 3, Fig. 1! withno appreciable change in the character of the temperafield distribution. This is explained by the low absorptiocoefficient of the medium (kabs0'0.34 m21).

Conductive–convective heat exchange

Allowing for convective heat transport changes the chacter of the temperature-field distribution in the central pof the EGMF. The convective heat flux plays the dominarole here and is directed toward the symmetry axis ofEGMF ~curve 1, Fig. 2a!. The current function in Fig. 2bdemonstrates a one-cell character of the motion in the inval 0.04,r ,0.054 m. This corresponds to a toroidal motiof the melt in the volume of the EGMF. A stagnant zoneobserved in the neighborhood of the side wall (r .0.045 m!.The main heat-transport mechanism in the stagnant zonconvective, which is characterized by a large temperatufield gradient in the boundary region~Fig. 2c!. In realEGMFs, the absence of convection in the near-boundarygions corresponds to the formation of an accretion layer

Radiative–conductive–convective heat exchange

Taking the three heat-transport mechanisms into accosimultaneously for the given criterional parameters~a low

791 J. Opt. Technol. 67 (9), September 2000

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value of parameterN and high values of parameters Bo anPr! results in a temperature distribution different from tone considered above~curve 3, Fig. 3a! and changes thedirection in which the convective flow moves, with the fomation of a second, oppositely directed vortex~Fig. 3b!. Thiscan be explained by the nonadditivity of the contributionthe separate components of the total thermal-energy fluthe heat exchange and by its continuous redistributiontween them.

CONCLUSION

Complex heat exchange in a cylindrical region in tpresence of internal heating and intense heat exchangethe ambient medium at the boundaries has been studieding as an example the heat exchange in an induction glmelting furnace. It is important to take all the heat-transpmechanisms into account when considering the melting pcess of semitransparent glasses in small EGMFs. The ltemperature gradient observed at the boundary of the mea water-cooled metal furnace must in practice promoteformation of an accretion layer.

1V. K. Schiff, A. N. Zamyatin, and A. A. Zhilin, ‘‘Numerical simulation ofthermal convection of a glass melt in a cylindrical induction furnaceGlastechn. Ber. Glass Sci. Technol.69, No. 12, 1~1996!.

2B. M. Berkovski� and V. K. Polezhaev,Computer Experiment in Convection ~Izd. Minsk. Gos. Univ., Minsk, 1988!.

3A. E. Slukhotski�, V. S. Nemkovet al., Apparatus for Induction Heating~Energoizdat, Leningrad, 1981!.

4L. P. Bass, A. M. Voloshchenko, and T. A. Germogenova,Discrete-Ordinate Methods in Radiation-Transport Problems~Moscow, 1986!.

5A. S. Jamaluddin and P. J. Smith, ‘‘Predicting radiative transfer in asymmetric cylindrical enclosures using the discrete ordinates methoCombust. Sci. Techn.62, 173 ~1988!.

6Yu. B. Petrov and D. G. Ratnikov,Cold Crucibles~Metallurgiya, Mos-cow, 1972!.

791V. K. Schiff