mathematical and physical modelling of fluid flow and heat transfer in electric smelting

Canadian Metallur`ical Quarterly\ Vol[ 26\ No[ 2Ð3\ pp[ 154Ð162\ 0887Þ 0887 Canadian Institute of Mining and Metallurgy[ Published by Elsevier Science Ltd\ Pergamon Printed in Great Britain[ All rights reserved

9997Ð3322:87 ,08[99¦9[99

PII] S9997Ð3322"86#99914Ð5

MATHEMATICAL AND PHYSICAL MODELLING OFFLUID FLOW AND HEAT TRANSFER IN ELECTRIC

SMELTING

Y[ Y[ SHENG� and G[ A[ IRONS$%

�Noranda Technology Centre\ Pointe Claire\ Quebec\ H8R 0G4 Canada^ $Department of MaterialsScience and Engineering\ McMaster University\ Hamilton\ Ontario\ L7S 3L6 Canada

"Received 08 Au`ust 0886#

Abstract*Heat and momentum transfer in a submerged electric smelting furnace were investigated in aphysical model\ using oil and an aqueous calcium chloride solution to simulate the slag and matte phases\respectively[ Gas evolution at the electrode was simulated by the injection of gas through the electrode inthe model[ A mathematical model for ~uid ~ow and heat transfer in the model was also developed[The measured temperature distributions near the oil:solution interface could only be reproduced in themathematical model by the imposition of a no!slip boundary condition at the interface[ This conditionimpedes the transfer of heat and momentum into the lower phase^ the implications for smelting arediscussed[ Þ 0887 Canadian Institute of Mining and Metallurgy[ Published by Elsevier Science Ltd[ Allrights reserved[

Re�sume�*On a e�tudie� avec un mode�le physique\ le transfert de chaleur et de movement dans un four a�fusion e�lectrique submerge�\ en utilisant de l|huile et une solution aqueuse de chlorure de calcium poursimuler les phases de laitier et de matte\ respectivement[ Dans le mode�le\ l|e�volution de gaz a� l|e�lectrodee�tait simule�e par l|injection de gaz a� travers l|e�lectrode[ On a aussi de�veloppe� un mode�le mathe�matique del|e�coulement de ~uide et de transfert de chaleur dans le mode�le "physique#[ Dans le mode�le mathe�matique\on pouvait reproduire les distributions de tempe�rature mesure�e pre�s de l|interface huile:solution seulementsi on imposait une condition limite de non!glissant a� l|interface[ Cette condition empe¼che le transfert dechaleur et de movement vers la phase infe�rieure^ on discute des implications pour le traitement en fonderie[Þ 0887 Canadian Institute of Mining and Metallurgy[ Published by Elsevier Science Ltd[ All rights reserved[

NOMENCLATURE

A area "m1#Cp Heat capacity "J:kg:K#` Gravitational acceleration "m:s1#H Electrode immersion in oil "m#HB Depth of the bath "m#Jf Flux term of fk Turbulent kinetic energy "m1:s1# or thermal conductivity

"W:m:K#L length scale "m#p Pressure "N:m1#q Heat ~ux "J:m1#

P Electrical power input "W#Qg gas ~ow rate "ml:s#Sf Source term for fT Temperature ">C#[x\ y\ z Cartesian coordinatesU\ V\ W Time averaged liquid velocity components "m:s#Up Rising velocity of bubbly plume "m:s#a Time averaged void fraction "*#r Liquid density "kg:m2#m Liquid viscosity "Pa s#e Dissipation of turbulent kinetic energy "m1:s2#t Shear force resulting from velocity gradient "N:m1#s Interfacial tension "N:m#

%To whom all correspondence should be addressed[

154

INTRODUCTION

In submerged arc smelting\ the material to be smelted is chargedto the top of the furnace[ As it descends\ it is smelted to producea slag and either a metallic or matte phase[ Current is carriedbetween the electrode tips in the liquid slag to generate the heatrequired for the process[ Heat losses are experienced throughthe bottom and sides of the furnace\ so\ in essence\ the furnaceis heated from the top and cooled from the bottom\ resultingin stable thermal strati_cation[ Consequently\ heat and momen!tum transfer to the underlying metallic phase may be poor\unless additional measures\ such as gas stirring\ are employed[

Gas stirring may be inherent to the process if gas is evolvedin the smelting reactions\ or may be injected through a lance[The buoyancy of the gas bubbles then works against the strati!_cation imposed by the thermal gradient ð0Ł[ In the presentwork\ measurements were made in a low!temperature\ small!scale model of Falconbridge|s submerged arc electric furnaceused for smelting nickel calcine[ A mathematical model for the~uid ~ow and heat transfer was also developed[ Attention isfocused on the boundary condition between the phases simu!lating the slag and matte[ Intuitively\ one would expect that theinterface between overlaying liquids would be free!moving[ Aswill be shown\ such is not the case^ a motionless boundarycondition results in better simulation of the results[ A simplemechanistic model is proposed to explain the phenomenon[

Y[ Y[ SHENG AND G[ A[ IRONS] MATHEMATICAL AND PHYSICAL MODELLING OF FLUID FLOW155

This _nding also explains some of the problems experienced infull!scale furnaces with respect to heat transfer to the lowerphase[

EXPERIMENTAL APPARATUS AND PROCEDURE

Apparatus

The Falconbridge furnace to smelt nickel calcine has beendescribed elsewhere ð1Ł[ It consists of a six!in!line submergedarc electric furnace[ A one!tenth scale model of half of thefurnace was constructed from polycarbonate sheets "Fig[ 0#[Mineral oil "Drakeol 23# and a 42 wt) aqueous calcium chlor!ide solution were selected to simulate the slag and matte phases\respectively[ Calcium chloride was chosen so that solid hydratescould be precipitated from the solution in an analogous mannerto ironÐnickel metallics which precipitate in the full!scalefurnace[ The model was heated by resistance heaters embeddedin steel electrodes\ and cooled by a water!cooled jacket on thebottom[ Details of the apparatus\ the similarity criteria andthe selection of the experimental materials have been reportedearlier ð0Ł[

Two types of gas injection were investigated]

0[ The evolution of carbon monoxide at the carbon electrodes

Fig[ 0[ Illustration of the physical model "a] plan view^ b] elevation#[The location of the computational cell is shown as the double!hatched

area\ whereas the measurement cell is single!hatched[

was simulated by the injection of gas through a ~ush!moun!ted hole in the underside of each electrode "14 mm below theoil surface#[

1[ Deeper injection of gas was simulated with lances positionedat various locations in the model at depths between 14 and037 mm in three di}erent con_gurations]

a] six lances located at the centre of each cell "Fig[ 0#\b] three lances at the centres of cells 1\ 2 and 5\ andc] a single lance located in the measurement cell[

Cartesian coordinates were chosen with the origin located atthe centre of the free surface of the oil phase and with the z!axisdownward into the liquid "Fig[ 0#[ Temperature measurementswere taken with four chromelÐalumel thermocouples "Type K\9[35 m long and 0[5 mm diameter# at the following locations]

T0 x � 9[9 y � 9[9 z � 03[4 mm "in the oil#T1 x � 099 y � 9[9 z � 32[4 mm "in the oil#T2 x � 199 y � 9[9 z � 092[99 mm "at the oil:solution

interface#T3 x � 299 y � 9[9 z � 037[9 mm "in the solution#

The temperature gradients at the centre of the bath wereobtained by measurements at the centre of the model at thefollowing depths]

T00 03[4 mmT01 32[4T02 092[9T03 037[9T04 at the solution:precipitate interface[

Procedure

It took two weeks for the model furnace to reach a steady!state condition^ at that time the temperature distribution in thefurnace before gas injection was measured[ During gas injec!tion\ the temperatures were monitored at thermocouples T0\ T1\T2 and T3[ The data rate ranged from 4 data per minute in the_rst half hour of gas injection to one in three minutes at theend of the gas injection[ Immediately after gas injection\ thetemperature distribution in the bath was measured once again[To re!establish steady!state thermal strati_cation\ the measure!ments were taken 24 to 39 h after each change in the electricpower level\ and 09 to 01 h between measurements at the samepower level[

EXPERIMENTAL RESULTS

Temperature pro_les before `as injection

Prior to gas injection\ the temperature distributions in boththe oil and the calcium chloride layers were strati_ed\ as shownin Fig[ 1[ The temperatures at the same depth were almostidentical\ being only 0 to 2>C lower at the sides than in themiddle[ The vertical temperature pro_les are illustrated in Fig[2 for three power levels[ These results are similar to the previousstudy\ in which it was shown that under stable strati_cationconditions\ heat is transported by conduction[ Consequently\the gradients are linear\ and inversely proportional to thethermal conductivity of the ~uid ð0Ł[

Y[ Y[ SHENG AND G[ A[ IRONS] MATHEMATICAL AND PHYSICAL MODELLING OF FLUID FLOW 156

Fig[ 1[ Strati_ed temperature pro_le prior to gas injection with an electrical power input of 599 W\temperatures in >C[

Temperature pro_les with `as injection

The temperatures in the oil and calcium chloride solutionduring gas injection through the electrodes are shown in Fig[ 3[Temperatures at T0 and T1 locations dropped as soon as gasinjection began\ while T2 rose[ In this particular case\ steady!state was achieved after approximately 4999 s[ The steady!statetemperatures are shown as a function of the gas ~ow rate inFig[ 4[ It is clear that the oil became homogenized at the higher~ow rates\ but the underlying calcium chloride solution wasmuch less a}ected[ The same picture can be seen across the bathin Fig[ 5^ the calcium chloride layer remained strati_ed[

The steady!state temperature distributions during gas injec!tion at the centre of the bath are shown in Fig[ 6^ the curves

Fig[ 2[ Vertical temperature distributions for three electrical powerlevels prior to gas injection[

Fig[ 3[ Temperature variations at four locations during gas injectionthrough the electrodes in which the power level was 899 W\ and the gas

~ow rate was 799 ml:s[

Fig[ 4[ E}ect of gas ~ow rate on the temperature pro_les after gasinjection through the electrodes while electrical power input was 599 W[


Fig[ 5[ Strati_ed temperature pro_le after gas injection through the electrodes for electrical power of 599 Wand gas ~ow rate of 599 ml:s[

are computational results from the mathematical model to bepresented in the following section[ The data show that thevertical temperature gradient is almost eliminated in the oillayer\ except near the oil:solution interface[ The number oflances and lance immersion depths above the interface did nothave a signi_cant impact on the shape of these temperaturepro_les\ unless the lances were close to or below the interface[Then\ strong mixing caused local emulsi_cation\ almost elimin!ating the interface[ Consequently\ thermal homogenizationexpanded well into the lower phase\ and produced temperaturepro_les similar to those in a single phase bath[

Fig[ 6[ Vertical temperature distributions after gas injection for threeelectrical power input levels and gas ~ow rate of 599 ml:s^ com!putational results from the mathematical model are shown as the curves[

MATHEMATICAL MODELLING

Mathematical formulation

The transport equations for steady!state momentum and heattransfer were solved in three dimensions[ The three!dimensionalconservation equation for a transportable variable\ f\ is expre!ssed in the following form]

1"rUf#1x

¦1"rVf#

1y¦

1"rWf#1z

¦1"Jf\yz#

1x

¦1"Jf\yz#

1y¦

1"Jf\xy#1z

� Sf "0#

where f stands for U\ V\ W and T[ Transport equations werealso written for turbulent kinetic energy\ k\ and the rate ofdissipation\ e\ in the kÐe model for turbulence[

The source term for the vertical momentum equation is com!posed of two terms\ the _rst is the buoyancy due to the naturalconvection of the oil\ and the second is due to buoyancy of thebubbles]

Sw\z � `rb"T−Tref#¦rmag "1#

The mixture model was used for the slag!gas two!phase zonein which the void fraction\ a\ was speci_ed[ The mixture density\rm\ is given by]

rm � "0−a#rs¦ar` "2#

Based on previous work by the authors on gas injection ð3\4\ 5Ł\ the void fraction was estimated to be 9[91 in a region9[90 m around the electrode[ The void fraction was imposed onthe nodes adjacent to the electrode[

Geometry of the computational cell

To save computational time\ only a portion of the modelfurnace was chosen as the computational cell[ The location andvarious boundaries of the computational cell are shown in Fig[0\ and in more detail in Fig[ 7[ It was assumed that Planes I\ II


Fig[ 7[ Perspective view of the computational cell indicating the dimensions and the boundaries of the cell[

and III were all symmetrical planes for both heat and momen!tum transfer[

Boundary conditions of the mathematical model

The following boundary conditions were used]

0[ The side wall was held constant at room temperature\ 14>C\and slip was not permitted at the wall[

1[ Planes I\ II and III were assumed to be symmetrical planesfor both the velocity and temperature _elds[

2[ The curved bottom surface was approximated by a hori!zontal section and two inclined sections[ The inner surfaceswere maintained at 14>C in the model[ Again\ slip was notpermitted at the wall[

3[ The top oil surface was treated as a moving solid wall formomentum ð6\ 7Ł[ For heat transfer\ experimentally mea!sured surface temperatures were used as boundary con!ditions to simplify the computation process[

4[ At the electrode surface\ a constant heat ~ux was assigned\based on the electrical energy input[

5[ At the boundary between the oil and calcium chloride solu!tion\ a no!slip condition was adopted\ and heat ~uxes weremade equal across the boundary[

METHOD OF SOLUTION

A commercial code\ PHOENICSTTM "CHAM\ London\UK#\ was used to carry out the numerical calculation ð8Ł[ A05×01×06 grid system was used[ The grid sizes were not evenlydistributed^ in the vicinity of the electrode and the free surface\the grids were smaller than those close to side wall and bottomwall[ The calculations were performed on a work station in amatter of 09 min for each condition[

Computational results

Without às injection To illustrate the results\ attention will befocused on Plane II "see Fig[ 7#[ Natural convection at the

electrode surface and the cooler walls was the only source ofmomentum[ As Fig[ 8a shows\ the velocities are rather weak^they were at most 2 mm:s at the electrode surface[ A horizontal~ow radiates from the electrode\ but is mainly restricted to ashallow zone close to the free surface[ Such a weak ~ow providessome lateral mixing\ but is not strong enough to move thecooler\ denser liquid deeper than the electrode[ This accountsfor the strati_ed temperature distribution in the companiondiagram\ Fig[ 8b\ and the strati_cation observed in Fig[ 1[ Amore quantitative comparison with the measured temperaturepro_les is shown in Fig[ 09[ The computations account quitenicely for the deviations from straight lines placed through thedata in Fig[ 2[

With às injection[ When gas was injected\ ~ow around theelectrode was greatly enhanced by the buoyancy of the bubbles\as shown in Fig[ 00a[ The ~ow was no longer limited to theshallow zone near the free surface[ However\ due to the immo!bility of the oil:solution interface\ the ~ow had little e}ect belowthe interface^ the lower phase remained virtually stagnant[ Thetemperature distribution was also quite di}erent\ as shown inFig[ 00b[ The bulk of the oil phase was very well homogenized\but on both sides of the interface\ large temperature gradientsexisted[ The calculated vertical temperature pro_les agreedquite well with the experimental measurements as shown inFig[ 6[ To demonstrate the enormous e}ect of the boundarycondition at the oil:solution interface\ a free!moving interfacewas employed for the calculations shown in Fig[ 01[ The homo!genized zone penetrates much deeper into the matte phasebecause heat was carried by conduction and by convection[ Itis clear that this boundary condition grossly overestimates thetransport across the interface[

DISCUSSION

Bath temperature distribution

Without às injection[ In the oil layer\ the horizontally strati!_ed distribution of the temperature was a result of the natural


Fig[ 8[ Results of computations for no gas injection and 599 W power input[ a] Velocity vectors\ and b]isotherms in >C[ The dashed line indicates the slag:matte interface[

convection originating in the hotter region near the electrode[Since the ~uid at lower levels was cooler and heavier\ the ~owwas limited to a shallow region near the free surface[ The hori!zontal velocities close to the electrode were of the order of9[90 m:s\ as calculated with the mathematical model[ The ~uxof heat due to this convection is]

qcv � UrCpDT "3#

To compare this ~ux with the ~ux due to conduction]

qCD � kDTL

"4#

one may take the ratio of these two ~uxes which is a PecletNumber for heat transfer[ The convective ~ux is about 3 to 4orders of magnitude higher than conduction in the horizontaldirection[ Thus\ heat transferred by convection in the shallowregion near the free surface dominates heat transfer\ virtuallyeliminating any horizontal temperature gradient[ In contrast\heat is mainly transferred by conduction in the vertical direc!tion[ This _nding provides a good explanation for the exper!imental observation that the solid calcium chloride precipitatesin the physical model always assumed a ~at surface ð0Ł[

Effects of `as injection

Gas injection completely changed the temperature contoursin the oil layer even for shallow injection situations[ Momentumsupplied by the bubbles created a signi_cantly stronger ~ow inthe oil layer than without gas injection[ However\ due to theinterface between the two liquids\ momentum and heat transferbetween the two phases was much di}erent than if there wereonly one liquid in the bath[ As indicated by the temperaturegradients shown in Fig[ 6\ mixing was always limited to theupper oil layer for gas injections with lance immersion abovethe liquid:liquid interface[ In the vicinity of the interface\ thetemperature dropped sharply\ in a similar manner to a solidwall[ This implies that the interface was the major resistance toheat transfer in the vertical direction[ A fundamental under!standing of the reasons behind this _nding are required^ this isaddressed in the following section[

Mobility of the interface between two liquid layers

Experimental data\ combined with the mathematical model\provide strong evidence that the oil:solution interface was


Fig[ 09[ Comparison of the computed vertical temperature pro_le withthe experimental measurements at various electrical power input levels

without gas injection[

motionless during gas injection[ However\ this behaviour isunexpected[ There are two hypotheses to explain the immobilebehaviour of the interface[

The _rst hypothesis may be found in the literature related toimmobilization of interfaces ð09\ 00Ł[ In that work\ force bal!ances are made between surface shear forces and those due tosurface tension gradients[ These concepts have been successfullyapplied to immobilization of bubble interfaces ð01Ł\ but therehas been much less work on large planar interfaces pertinent tothe present work[ Mansell et al[ ð02\ 03Ł investigated the inter!face between a ~uid ~owing in a channel and another immiscible~uid inside a square cavity below the channel[ They used thefollowing equation as a boundary condition]

t−xy−t−

xy �1so

1y"5#

where t¦xy is the shear from the upper\ moving layer\ whereas t−

xy

is resisting shear in the lower liquid\ and so is the surface tension[This equation may be used to provide an explanation for theimmobile behaviour of the interface[ At the side wall of thevessel\ the interface is restrained\ so that the horizontal andvertical velocities are zero[ At the centre line a similar boundarycondition exists due to symmetry\ although in reality the pointof ~ow separation may oscillate[ The interface may be pos!tulated to move only when the shear stress above the interfaceovercomes the interfacial tension holding the interface rigid\i[e[\ the shear stress times the interfacial area must exceed resist!ing force due to interfacial tension[ From the mathematicalmodel\ the average shear stress in the upper liquid at the inter!face is 9[994 to 9[90 N:m1[ The resulting total shear force alongthe interface of length 9[4 m and unit width is approximately9[992 to 9[994 N[ The interfacial tension between oil and cal!

cium chloride is 9[90 to 9[94 N:m ð04Ł[ The restraining force dueto interfacial tension over unit width is therefore 9[90 to 9[94 N^consequently\ the interface remains motionless[ According tothese calculations\ the shear stress could be increased tenfoldbefore the critical condition is reached[

The other hypothesis is that the interface is immobilized bythe presence of a solid phase at the interface[ It would havebeen prohibitively expensive to use reagent grade calcium chlor!ide for these experiments[ The material used contained smallamounts of iron which would precipitate as iron chloride[ Giventhe weak shear stresses discussed in the previous paragraph\ itmay be surmised that the solids impedes the motion of theinterface[

Implications for smeltin` operations

Some preliminary computations have been carried out forthe full!scale furnace conditions[ It was found that the velocitiesare also small\ and that the furnace is highly strati_ed withoutgas stirring[ Such a highly strati_ed temperature distribution inthese processes is undesirable for a number of reasons[ Mostnoticeably\ it lowers the power e.ciency of the furnace since alarge amount of heat is lost at the free surface and to the sidewalls\ instead of being used to heat the liquid in the bottom ofthe furnace[ The high heat ~ux towards the side wall also createshot spots on the side walls[ This latter problem has promptedfurnace designers to install copper cooling blocks in the wallsadjacent to the slag ð05Ł[ These coolers are designed to freeze alayer of slag to the refractory to prevent erosion[

An immobilized slag:matte interface is also undesirablebecause it prevents heat from being convected into the mattephase[ In the case of the Falconbridge furnace\ this contributesto the precipitation of ironÐnickel metallics on the hearth whichreduces the e}ective furnace volume and creates problems intapping[

It must be recognized that in the present work the stirring isvery weak\ characteristic of submerged arc smelting[ In con!verters or ladles\ the gas ~ow rates are generally much greater\and the gas rises through the interface[ Therefore\ the present_ndings regarding the immobility of the interface do not applyin such situations[

CONCLUSIONS

Measurements carried out in a physical model of submergedarc electric furnace revealed the temperature distributions inboth the upper oil and the lower calcium chloride solutionphases under di}erent gas injection conditions[ The interfacebetween the phases behaved as an immobile surface\ and the~ow generated in the upper oil layer was greatly damped\ sothat little momentum was transferred to the lower phase[ Conse!quently\ in the vicinity of the interface\ signi_cant temperaturegradients persisted\ despite homogenization of the bulk of theoil layer[ A mechanistic model of the mobility of the interfacewas developed which demonstrated that a motionless interfacewas reasonable\ and accounted for the observed temperaturegradient in the oil phase near the interface[


Fig[ 00[ Results of computations with 599 ml:s of gas injection through the electrode\ and 599 W powerinput[ a] Velocity vectors\ and b] isotherms in >C[ The dashed line indicates the slag:matte interface[

Fig[ 01[ Same as Fig[ 6\ except that a free interface boundary condition between the slag and the matte wasused to show that it overestimates heat transfer through the interface[


Acknowledèments*The authors are very grateful to Falconbridge\Limited\ for _nancial support of this project^ in particular\ Mr I[ A[Cameron and Dr D[ Tisdale have been very helpful[ The assistance ofMr O[ Kelly during the experiments and discussions with Dr P[ E[Wood are also gratefully acknowledged[

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mathematical and physical modelling of fluid flow and heat transfer in electric smelting

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