An Extended Two-Dimensional Mathematical Model of Vertical Ring Furnaces

S. PETER, A. CHARETTE, R.T. BUI, A. TOMSETT, and V. POTOCNIK

An extended two-dimensional (2-D*) mathematical model of vertical anode baking furnaces has been developed. The work was motivated by the fact that a previous 2-D model was unable to predict the nonuniform baking in the transverse direction, i.e., perpendicular to the longitudinal axis of the furnace. The modeling strategy based on dividing each section in four zones (underlid, pit, underpit, head wall and fire shaft zones) and introducing two symmetry planes in the exterior pits is explained. The basic heat-transfer relations used are also detailed. Selected results shown include draught and oxygen concentration profiles in the flue, gas and anode temperature distributions and fuel con- sumption in the back fire ramp. Simulation and experimental results are compared.

I. INTRODUCTION

RING furnaces are widely used in the aluminum indus- try for the baking of carbon electrodes. They can be of the horizontal or vertical type. The authors have acquired pre- viously extensive experience in the mathematical modeling of the horizontal furnace, t~-4~ Furthermore, in 1990, a first model of the vertical furnace was elaborated5 si The geo- metrical arrangement of this type of furnace is significantly different from that of the horizontal type. Figure t describes a typical fire train consisting of three fire ramps, three cool- ing and four preheat sections. The gas flow is from right to left, while the fire group equipment is moved one section ahead in the same direction at each fire cycle. Figure 2 gives the details of a section illustrating four different zones: the head wall and fire shaft, the underlid, the pit, and the tmderpit zones. The fuel combustion takes place in the fire shafts. The hot gases leave the fire shaRs, mix with infiltrated air, and exchange heat with the upper part of the furnace (lid) before they ftow down in more than a hundred separate vertical ducts along the brick holes which are uni- formly distributed among the pits. In the pits, the gases exchange heat with the brick wails enclosing the coke cov- ered anodes before reuniting in the underpit region. As they pass the pillars in the underpit region, they exchange heat with the solids before entering the next fire shaft down- stream. While the electrodes are heated, volatiles (mainly tar, methane, and hydrogen) escape from the solid mass and enter the low pressure flue duet where they burn.

The 1990 model was two-dimensional (2-D) (y and z in Figure 2). Assumption was made that the solid temperatures and the gas flow were uniform across a section of the fur- nace (along the transverse plane normal to the overall gas flow), and as a result, the model was unable to predict non- uniform baking in that direction. In this work, the model is improved by taking into account the variations in solid tern-

S. PETER, Research Professional, and A. CttARETTE and R.T. BUI. Proti~ssors of Engineering, are with the Department of Applied Sciences, University. of Quebec at Chieoutimi, Chicoutimi, PQ, Canada G7H 2BI. A. TOMSETT, Senior Research Scientist, Cornalco Research Center, Carbon Products. M.R.U., Thomastm,,-n 3074, Australia. V. POTOCNIK, Consultant, is with the Department of Reduction. Alean International Ltd., Jonquirre, PQ, Canada G7S 4K8.

Manuscript submitted April 12, 1995.

peratures across the furnace considering three subcontrol volumes in the pit zone (extended 2-D + model). The effect of uneven gas flow distribution in these subcontrol volumes on anode temperatures is also determined.

The same authors have published recently a three-di- mensional (3-D) model of the vertical ring furnace, t6~ This model is very useful for the detailed description of a given section (the fourth preheat section was analyzed in the ar- ticle); however, it cannot predict the behavior of the com- plete fire train. Input data to the 3-D model have to be taken from other sources, e.g., experimental data or running the 2-D* model. Therefore, the 2-D' model presented in this article is an upgraded general predictive model, whereas the 3-D model is used as a complement for added information in specific sections.

II. E X T E N D E D M O D E L (2-D +)

The global extended model predicts the temperature dis- tributions in the gas and solids and is assisted by several submodels: gas mass flow, gas composition (linked to the packing coke and volatile release and combustion submo- dels), and transient conduction submodels. The control vol- ume encompasses the complete fire train. As shown in Section I, each section is divided into four zones, each of which is given a specific simulation strategy in the overall model and is connected to its neighbors by common bound- ary conditions. The solid behavior is governed by transient conduction, while the gas flow is considered to adjust itself rapidly to the changing conditions in the solid. The basic heat balance for the gas can be cast in a general discretized form as follows:

b , -b , . ,+b ,~ [1j The elementary volume to which this equation applies is different from zone to zone, and the equation will be adapted to the specific cases. Qi - Q,+T is the rate at which the thermal energy changes in the elementary volume (= thg, hg , - th~,,,h.~.+,),.Q~f is the extensive enthalpy rate of the infiltrated air, Qcr is the combined convective and ra- diative heat-transfer rate, and Qg~, regroups energy per unit time coming from the combustion of the fuel, the volatiles, and the packing coke.

METALLURGICAL AND MATERIALS TRANSACTIONS a VOLUME 27B. APRIL 199(~-2~7

E l EXHAUST

FIRST PREHEAT PREHEAT

Fig. l---Furnace fire train arrangement.

i +1

8

PREHEAT

RRE RAMPS ~ GAS DIRECTION

FRESH AIR i INTAKE

7 6 5 4 3 2 1

PREHEAT FIRE FIRE FIRE F I R S T COVERED NATURAL COVERED COOUNG COOLING COOUNG

oov?-,,,

'.'." ":':'~.-;- ':r~:. i~.'-~.:.

PACK NG ~, 2?+ ~:i ~;,';; ~;~, '~,i;;". M A T E R I A ~ t - ' :

LA~ERS ~ ~:i ~iii: ~ i~ "

I \

T Z

~ ~LL/~- HEADWALL

~'~r; "- FIRESHAFT

I, ~ FOUNDATION

Fig. 2---Side view of a section illustrating the four zones: head wall and fire shaft (I), undertid (2), pit (3), and underpit (4).

The pressure drop along the flue is very important since it dictates the air infiltration distribution. This information can be obtained by performing a momentum balance on the gas:

P,+', = P, _ k}_,in~ 2 (p ,+ , _ P,) p-

- 2 k l hz~ I - - ~ - ~ - P, [2] }

- ~ k, - ~ g (Z~+, - Z , ) P

where kj = 0.7/A~ (,-I~ being the equivalent passage area), k, = JLJ2D,A~ or being the friction factor and L, and D, the equivalent length and equivalent diameter of the duct, respectively), rh~ is the mass flow rate, Z is the elevation, and the bar sign denotes an average value. Also,

~h~., = &~, + &~.~ [31

with

nzi,l- = k3 V p A p [4]

AP being the pressure difference between the flue and the ambient air and ks a constant dependent on the size of the openings available for infiltration.

A. Underlid Zone

This zone receives the upflowing gas from the fire shafts and redistributes it in the small vertical ducts surrounding the pits. It is considered to be a unique well-mixed zone; i.e., the outflowing stream temperature equals the effective gas temperature T~ of that zone. Convective and radiative heat transfer occurs between the gas and both the inner surface of the lid and the upper surface of the pit (Q~z and Qs.,-,2, respectively, for radiative transfer). There is also a net radiative transfer {)2,-,t between the same two surfaces. A radiative heat balance written for the gas gives

0 , = [ a s , - r,,) + a s , ( r ; - r r tsl

Under the assumptions that the gas is gray and that surfaces S, and $2 form a speckled surface of total area S , the total exchange areas GSI and GS,_ are given bym

G S , = C~r [61

where

and

GS2 = (1 - C.J eg&Sr/R [71

C = S , /ST

R = e~ + [C,~:. + (1 - C,)~] (1 - ec)

The gas emissivity e~ is calculated by taking into account water vapor and carbon dioxide. In the underlid zone, ~)~=, comes from the combustion of packing coke only. It is as- sumed that all the volatiles burn in the pit and the under'pit zones and that the injected fuel bums in the fire shafts.

A one-dimensional (l-D) submodel of transient heat con-

298--VOLUME 27B, APRIL 1996 METALLURGICAL AND MATERIALS TRANSACTIONS B

~ * ' * ~ ~ ~ E lementary /

i

B>..----~

:l:: !1!ili ? .. . . .

[: .",..

I. ,.:. I.. .~.

�9 I~:.

I

.F '~ i - i

I

I Q-o

Fig. 3--Cross section the pit zone.

m g a ~

8

�9 , . . .~

I I

mgg2.~

B2

I I I I

Q=O Q=O

J.y

of the furnace showing the modeling approach in

duction through the cover bricks is linked to the underlid zone model to determine the thermal losses and the external cover temperature. The internal radiation heat transfer is part of the heat balance on the lid and is obtained by

Q~. . , = o- s 2 s , I T : - r : ) [8]

with

$2S, = S,S: = Cs (1 - Cs)(1 - g~)&e:Sr/R [9]

B. Pit Zone

The pit zone is subdivided in three subzones, namely, middle zone B and external pit zones B1 (including side- wall) and B2 (including central wail) (Figure 3). The mid- dle zone B is bounded by two adiabatic planes which pass through the middle of the external pits. Rigorously, sym- metry does not exist at these planes: they should be some- what displaced and their positions found by trial and error. However, this choice considerably simplifies the problem since the control volume of zone B is henceforth reduced to only one equivalent pit. Moreover, the validity of the position of these adiabatic planes is confirmed by the de- tailed calculations performed with the 3-D model: ~j as men-

tioned later in Section lIl. On the other hand, perfect symmetry exists at the middle of the central wall, which explains the presence of the third adiabatic plane delimi- taring zone B2. Finally, zone B1 is allowed to exchange heat with the surroundings. Such an arrangement makes possible a more realistic treatment of the exterior pits which, experimentally, are at lower temperatures than the middle ones. It is assumed that there is no temperature var- iation throughout one section in the x direction and that solid temperatures vary only along y and z. Allowance is made also for nonuniformity of mass flow along y:

J;z~.,~ = ,h~, .... + rh,~.,,, + r ~ : .... [10]

= fB~ rn..,,, + fu the.,,, + A2 nits,,,

The fractionfcan be obtained either from the experimental measurements of flow rates or by solving the 3-D motion equations under the lid.t"]

Equation [1] is solved for each subdivision in the z di- rection. In this case, i.e., in the pit zone, infiltration is as- sumed absent (O~,,r = 0), the heat generation comes from the combustion of the volatiles (0,o, = Q,.), and the con- vection and radiation terms depend on the, given subzones as follows: for zones B I and B2, Qr = Q,, + Q,,, where 0* is the heat flow on the brick wall enclosing the coke covered anodes and 0~,, is the heat flow on the external or central wall; and for zone B, Q,r = 0~ (no heat loss)�9 The radiation heat-transfer component is obtained by using the following form of the heat-transfer coefficient:

/,, = o-C (V. ~ ,- r:,,.'- + r , : d + r,,:) [ t l ]

where

1 C =

1 1 - - + - - - I c. e~

e,. and eg being the wall and gas emissivities, respectively. For the central and sidewalls in zones B2 and B1, re-

spectively, the number of material layers, the material prop- erties, and the initial temperature distribution are taken the same in both walls. The sidewall is assumed to lose heat to ambient air by natural convection and radiation�9 The en- erg3" contribution of the volatiles is calculated from the amount of volatiles combusted. The volatile evolution is obtained from Tremblay and Charette:tg]

aX,. = Xo__2, [exp ( -Eo /RT) ] (1 - X~)", [12] d T a

where X~ is the amount of tar, hydrogen, or methane evolved at a given temperature divided by the total amount of that constituent evolved at the end of the pyrolysis, R o is the pre-exponential factor, Eo is the activation energy, n is the order of reaction, and a is the heating rate. The kinetic parameters n, Eo, and Ko are given in Table I. The mass flow rate of volatiles can thus be obtained fbr all the op- erational conditions in the furnace. As tbr the combustion of these gases, any volatile that evolves when the adjacent gas temperature is below the ignition temperature (425 ~ 575 ~ and 630 ~ for tar, hydrogen, and methane, re- spectivelyO:~l) is considered lost. For higher adjacent gas temperatures, hydrogen and methane are assumed to bum instantly and reaction kinetics are used only for tar.

M E T A L L U R G I C A L A N D M A T E R I A L S T R A N S A C T I O N S B V O L U M E 2 7 B , A P R I l . 1 9 9 6 - - 2 9 9

]'able !. Devolatilization Kinetic Parameters (~

E,, K,, Constituent n (K J/tool) (h- ~)

TAR 0.7 4.167 In (a) -~ 35.0 exp (0.273E,, - 6.417)

H, 1.1 5.882 In (a) + 57.65 exp (0.233Ec,- 10.39)

CH~ 0.8 10.0 In (a) + 76.0 exp (0.225Eo - 11.325)

Table II. Assumed Packing Coke Consumption Distribution

Section Position Percentage

Second cooling 10.8 First cooling 10.8 Back fire 9.4 Middle fire 52.5 Front fire 16.5

Total 100.0

The conduction submodel, which calculates the temper- ature distribution in the brick walls, packing coke, and an- odes, uses a 2-D explicit scheme with 80 nodes in each half-pit and a time interval of 160 seconds. The boundary conditions are of the second type (Neumann) and are de- rived from the gas temperature profile and the temperature distribution at the surface of the brick walls obtained from the previous time interval. The thermophysical properties of the bricks, packing coke, and anodes (density, heat ca- pacity, and conductivity) are considered to vary with tem- perature.

C. Unde~Tait Zone

The different gas streams coming from the pit zone merge into a cavity reinforced by pillars, on their way to the fire shafts of the next section. The treatment given to this zone is similar to that of the underlid zone. It is taken as a unique well-mixed zone with an effective gas temper- ature Tg. Heat transfer occurs by convection and radiation to the underpit surface (SO and foundation floor (S~), the radiation components still being found by expressions [5] through [9]. No infiltration occurs in that zone (Q~,r = 0). The value of Q,~, comes from the partial combustion of the volatiles, part of which are entrained from the pit zone. An additional term, Qp~, is introduced as a heat sink in the heat balance expression [1] to take into account the heat accu- mulated in the pillars sustaining the pits:

To,, '~a' - Tp,!' [13] Q~ = rn~176 At

where mp~ is the total mass of the pillars and Tp~ is assumed to be the average of the underpit surface and foundation floor temperatures at a given time t. This latter temperature, as well as the heat losses in the foundations, is obtained via a I-D transient heat conduction submodel applied to the foundations.

D. Head wall and b'ire shql't Zone

Since the temperature distribution in the head wall is nol of critical importance, only its global influence on the fur- nace behavior is considered. Therefbre, a unique average temperature is assigned to the head wall and fire shaft, and this temperature is taken equal to the average solid tem- p.erature of the two adjacent sections. In this zone, the tema Q,., in Eq. [1] stands for the heat losses through the corre- sponding area of the foundations and through the central and sidewalls.

The energy accumulated in the head wall also constitutes a heat sink, and it is given by

A = , h , , , c ; , , , . r , . , , , ' . ' , - - r, , , , ' [14] At

where nr~,,, is the total mass of the head wall and fire shaft. No infiltration is considered in this zone (.Q~,,- = 0) and

Q,~~ comes solely from the combustion of the fuel fed to the furnace.

E. Packing Coke Combustion

Packing coke combustion is known to be responsible for a coke depletion of 40 kg per ton of baked anodes. How- ever, its distribution along the fire train depends on the coke temperature, the gas oxygen content, and the resistance to air infiltration; therefore, it cannot be predicted easily. The combustion profile, rather, is chosen by trial and error, and it is given in Table II. Packing coke is assumed to be pure carbon associated with a certain amount of nonreacting ash. By assuming only complete reactions, the concentration distribution in the flue due to the combustion of packing coke, volatiles, and fuel can be easily calculated.

F. Overall Solution Procedure

The model thus proposed is dynamic and extends from the first preheat section to the last covered cooling section. Users are asked to define all relevant operational and geo- metrical parameters and the total simulation time, the latter being equal to the duration of the number of permutations (fire changes) they want to calculate. The user also has to define initial conditions (mass flow, gas, and solid temper- atures at time t = 0 for each section in the fire train). The overall solution procedure is shown in Figure 4. At time t = 0, the initial solid temperature distribution is known. Assuming these temperatures to be constant over a time increment At, the program calculates the corresponding gas temperatures (Eq. [ 1]) which are applied during At seconds to the solid to reach new temperatures. The procedure it- erates over time until it reaches the fire cycle time. A new (cold) section is then added to the fire train and the last cooling section is dropped. A new permutation starts and the program runs until the total number of permutations is covered or steady state (5 ~ temperature difference of the gas temperature between two fire changes) is reached. The total number of permutations necessary to achieve steady state depends on the assumed initial temperature distribu- tion. If the initial distribution is adequately approximated, the total number of permutations will be small, usually <20.

300~VOLUME 27B. APRIL 1996 .'VIETAL[.URGICAL AND MATERIALS TRANSACTIONS B

RAMETERS, MAXNP. FC. ETC... /

t [DETERMINATIoNQFINITIALSOLII~TEMPI:RATUREI I oTNP ::

IO'ETERM~NiTION OF {}AS MASS[ I FLOWAND cx~gosmoN I

DETERMINATICN OF BURNED I

t ~)ErERMINA"P~N OF GAS TE,~EF~'rURE l

J ' IDErERMIN*TION OF HEAT LOSSES THROUGH[ [ COVER,SIDEWALL AND FOUNOATION �9

t DETERMINATION OF 2D TEMPEF~TIJRE I

DISTRIBUTION IN BRICK, COKE AND ANOOEJ

PP4NT RESULTS: TEMPERATURES, / NSUMPTIONS,HEAT BALANCE ETC../

VARIABLE DESCRIgTION- : TOTAL NUMBER C,F

PERMUTATIONS : FIRE CYCLE

OURAT;ON : PEqMUTATION NUMBER

: TIME INCREMENT

ADD NEW CX~.D [ SECTION AND [ OROP t.Asr I

copt, e~o BECT~N I

F i g . 4 - - T h e s o l u t i o n p r o c e d u r e .

IlL RESULTS

The model has been applied to an anode baking furnace at COMALCO Aluminium Limited. The fire train is pre- sented in Figure 1 and the fire cycle time is 36 hours. It may be noted that the natural cooling (NC) section is not considered in the simulation. Each section comprises five pits and the anodes are stacked up in three layers of six for a total of 90 anodes per section. The results shown in Fig- ures 5 through 7 are obtained with a uniform mass flow distribution in the pits 0c~ = 4/6 andft~ = ft,2 = 1/6), and the comparisons given with the results of the previous 2-D model relate to the pits in the middle section B. In the 2- D model, heat losses were also calculated at the external wall, but they were distributed evenly among all the pits in a given section. The effect of nonuniform mass flow dis- tribution is also dealt with in Figure 8 and Table III.

Figure 5 gives the measured and calculated draught pro- files along the furnace. The pressure drop increases as the gas passes through each section from NC (section 1) to first preheat (section 10). The predictions of the 2-D + model agree with the experimental data and with the 2-D model predictions in the cooling and firing sections. There is a discrepancy in pressure drop of 5 Pa in the preheat sections, between the experimental data and the results of both 2-D and 2-D § models.

The gas temperature predicted by the 2-D ~ model is com- pared with the experimental and 2-D model results in Fig-

o ~ ~~

,02~ ,;

.8~ L~___o,,~.,:=, I "i

- 1 0 0 I t I ~ I t I = I ~ I . I I I I I I 0 1 2 3 4 5 6 7 8 9 10

SECTION POSITION

Fig. 5---Measured and calculated draught profiles.

11

ure 6. In the first three preheat sections, the 2-D" model prediction agrees better with the experimental results. The higher temperature in the fourth preheat section is due to the volatile combustion in that section. The only contribu- tion of energy to gas in the initial period of the first firing section is from the front fire fuel and from a small per- centage (16.5 pct) of packing coke burning. The target gas temperature is set at 1350 ~ in the back fire section and at 1250 ~ in the front and middle fire ramps. The set of data marked A, where the ta~et is fixed at 1250 ~ throughout the fire ramps, leads to lower gas and anode temperatures, and it is given only as a reference.

The gas temperature in the second covered cooling sec- tion (CC2), as predicted by the 2-D* model, is lower than the experimentally measured values. This could be ex- plained by the fact that the NC section tends to limit the gas flow in CC2. Because of a chimney effect (which is not included in the present model), an important outflow of gas is induced in the central holes of the brick walls of NC while inflow from outside air occurs mainly in the lateral holes. This effect cannot be counterbalanced by the nega- tive pressure responsible for the flow downstream in the fire train, so that only a small portion of the air entering NC reaches CC2, thus producing high experimental tem- peratures in this latter section.tt u

Figure 6 also gives the 2-D" predicted temperature profile of anodes located in any one of the middle pits. As shown in the figure, the anode temperature increases from preheat section to the last firing section (section 4). The maximum temperature is reached at the end of the last firing. In the cooling section, the anodes give off heat to the gas stream. From the profile in the fourth preheat section, it was deter- mined that the average rate of anode heating is 7. I ~ This confirms the value of 7.5 ~ used in the volatile evolution submodel.

As shown in Table III, with the 2-D § model, one can determine the average anode finishing temperature (highest temperature in a given portion) in the inside and outside half-pits and the middle four pits of the pit zone. When compared with the experimental dam, the results calculated with uniform mass flow are much lower in the inside and outside half-pits and higher in the middle four pits. The 2- D calculated results are logically lower than the measured values, since these pits are assigned irrelevant heat losses by the nonextended model. The discrepancies between ex- perimental and calculated 2-D + results can be explained by

M E T A L L U R G I C A L A N D M A T E R I A L S T R _ A N S A C ' I ' I O N S B V O L U M E 2 7 B , A P R I L 1 9 9 6 - - 3 0 1

1 4 0 0

C . ) 1 2 o o O

~ ' . 1 0 0 0 uJ r r

8 0 0

6 0 0

~t~ 4 0 0

2 0 0 I -

= : Experimental (1987) A - - - - A Experimental (1987)

m Experimental (1990) O . . . . -~ Model (2D) ~" - - - ' -O Model (2D +)

0

�9 ~ 1 7 6 ,,:.,.:7. ....

....-:.'.:

y "/ / /

/ ' ..,dl. ~....:,,/" ,.~/ ~ . . . ~ ~ Top Anodes

t Calculated

0 0 PH1 3 6 PH2 7 2 PH3 108 PH4 1 4 4 F1 1 8 0 F2 2 1 6 F3 2 5 2 CCl

TIME / (hours) Fig. 6--Gas and anode temperature profiles along the furnace: PH: preheat section; F: fire section; and CC: covered cooling section.

k

2 8 8 cc2 3 2 4

25

o•~ 20

10

,,=i, >- X 5 0

m

0 0

* i I J I i I i I l I i I i I i I I I l I

\ . . . .

tP---'----e Experimental (1986) A - - - - A Experimental (1987) (3, . . . . -(9 Model (2D) 0"---'-'0' Model (2D +)

t I t I I I I I I ! I I i I = I , I ~ ! 1 2 3 4 5 6 7 8 9 10

i

m

m

m

w

w

N

m

11

S E C T I O N P O S I T I O N Fig. 7---Oxygen content along the fire train.

302--VOLUME 27B, APRIL 1996 METALLURGICAL AND MATERIALS TRANSACTIONS B

0 . 0 3 0 ' I ' I ' I ' I ' I ' I ' I ' I ' I '

t,R

1 ~ 0 . 0 2 5 ..hr

C• 0 . 0 2 0

~ 0 . 0 1 5

=1 03 Z o.oi o 0

,.,,.I I ,LI 0 . 0 0 5

I.I,. 0 . 0 0 0

0

e. - - - - -e Experimental (1987)

O - ' - - - ~ Model (2D +) Uniform Flow

�9 ........ �9 Model (2D +) Non-Uniform Flow

, I ,, I , I , I , I I I I I , I , I I 4 8 1 2 1 6 2 0 2 4 2 8 3 2 3 6

TIME / (hours) Fig. 8--Back fire fuel consumption.

Table IIL Measured and Calculated Anode Finishing Temperatures

Calculated Calculated Calculated Measured Calculated 2-I3 + (~ (2-D § (2-D')

Position Layer* (~ 2-D (~ (Uniform Flow) (Nonuniform Flow) (Corrected)

Inside half-pit 1 1160 - - 1094 1126 1156 2 1086 - - 966 1012 1033 3 1019 -- 899 971 999

Average 1088 - - 986 1036 1063

Middle four pits 1 1171 1145 1207 1185 1185 2 1101 1043 1100 1054 1054 3 1031 994 1085 1026 1026

Average 1101 1061 1130 1088 1088

Outside half-pit l 1132 - - 993 1030 1108 2 1043 - - 851 918 986 3 985 - - 828 898 962

Average 1053 - - 891 949 1019

*1: top; 2: middle; and 3: bottom.

40

the fact that experimental measurements have revealed that the gas flow distribution among the pits is not uniform, a higher portion o f the flow being directed toward the inside and outside pits. It was therefore decided to run the model with the newly obtained distribution; i.e., fB = 2 .6 /6 , f~ = 1.9/6, and f s: = 1.5/6. This distribution, found in the first preheat section, was assumed to apply to all sections o f the fire train. The results obtained with this nonuniform mass flow are presented in the fourth temperature column of Ta- ble III. As seen, these results are now closer to the meas-

ured ones. The anode temperatures have been increased in the lateral (inside and outside) half-pits and decreased in the middle pits. However, significant differences still occur in the outside half-pit, and this is addressed subsequently.

As mentioned previously, the adiabatic plane cutting the internal and external pits should be moved away from the center. Several attempts to find exactly where the adiabatic line should be located produced no satisfactory results. This was done by changing the dimension of the subcontrol vol- umes. An iterative type o f solution might produce desirable

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 27B, APRIL 1996--303

results, but this will introduce another convergence criterion which means longer CPU time. An alternative simpler so- lution is 1o consider an average of the anode temperature for the whole external pits, that is, anodes in one whole inside pit and one whole outside pit, by combining those half-external pits with the other half-pit belonging to the middle subcontrol volume (volume B). This is reported in the last temperature column of Table lII, and it is seen that the calculated temperatures of the anodes in the external pits are now much closer to the experimental ones, thus validating the simple correction procedure.

Obviously, the nonuniform gas mass flow distribution will slightly modify the results already presented, namely, the draft profiles and the gas and anode temperature distri- butions (Figures 5 and 6). Because of space limitation, these corrected profiles will not be presented. Other results obtained include front, middle, and back fire ramp fuel con- sumption, external sidewall and lid temperatures, gas mass flow, and oxygen profiles, as well as the amount of unbur- ned volatiles at the furnace exhaust. Some of these are pre- sented subsequently.

Figure 7 compares the oxygen content along the fire train calculated from the 2-D* model with the experimental data and the calculated values from the 2-D model. The exper- imental data of 1986 are closer to the 2-D + model predic- tions. The oxygen content predicted by the 2-D + model is the lowest in sections 6 (front fire) and 7 (fourth preheat) due to volatile combustion at those locations. The curve relative to the 2-D ~ model is almost exactly the same for both uniform and nonuniform (not presented) gas mass flow distributions. The fuel consumption in the back fire ramp, however, is more affected by the nonuniform gas mass flow distribution. As shown in Figure 8, the fuel consumption for the uneven distribution is higher and nearly coincides with the experimental results. This can be explained as fol- lows. With the uniform distribution, the hottest part of the pit zone in the cooling sections, namely, the middle pits, receives a considerable fraction of gas flow, resulting in a higher average temperature of the gas than in the case with nonuniform gas distribution. The decrease in temperature of the gas in the cooling sections due to nonuniform gas distribution calls for an increased fuel consumption in the back fire to bring the gas temperature to 1350 ~

Detailed YZ anode temperature profiles obtained with the 3-D model in the fourth preheat section can be found in Reference 6. It can be seen from the temperature profiles displayed that the calculated adiabatic lines in the internal and external pits are located almost exactly in the middle. A closer analysis of the numerical results in other pits (not given in Reference 6) indicates that in some cases the ad- iabatic lines are displayed somewhat toward the inside of the pits but very slightly (one out of four anode cells).

The CPU time required to run the 2-D § model is 1 hour

and I0 minutes to 1 hour and 35 minutes, depending on the case studied on a SGI 4D/340 workstation, in compar- ison to 12 hours on the same computer for the 3-D model for only one section. This stresses the usefulness of the 2- D" model as a predictive tool, whereas the recently pub- lished 3-D model is dedicated to detailed investigations in specific sections.

Finally, it should be mentioned that numerous applica- tions have been performed with the 2-D + model both at COMALCO and ALCAN. It has been used to predict the furnace behavior for changes in fire cycle time, number of sections, position of the draught control section, firing se- quence, number of pits, etc.

IV. CONCLUSIONS

The extended 2-D* model allows the user to predict the baking nonuniformity in a plane perpendicular to the lon- gitudinal axis of a vertical anode furnace. The incorporation of a symmetry plane in the middle of the two external pits proves to be a simple way of accounting for the reduced heating in those pits. The technique is also flexible enough to include nonuniform flow effects. The improved model gives good agreement with the experimental data.

A C K N O W L E D G M E N T S

This work would not have been possible without the help of the COMALCO and ALCAN research centres and the plant personnel in Australia and Canada, respectively. The authors acknowledge their invaluable contributions.

REFERENCES

1. A. Charette, R.T. Bui, and T. Bourgeois: IEEE Trans. Ind. Appl., 1984, vol. IA-20 (4), pp. 902-07.

2. R.T. Bui, A. Charette, and T. Bourgeois: Metall. Trans. B, 1984, vol. 15B, pp. 487-92.

3. R.T. Bui, E. Dernedde, A. Charette, and T. Bourgeois: TMS-AIME (Light Met.), 1984, pp. 1033-40.

4. R.T. Bui, A. Charette, and T. Bourgeois: Can. d. Chem. Eng., 1987, vol. 65, pp. 96-I01.

5. T. Bourgeois, R.T. Bui, A. Charette, B.A. Sadler, and A.D. Tomsett: TMS-AIME (Light Met.), 1990, pp. 547-52.

6. ILT. Bui, S. Peter, A. Charette, A.D. Tomsett, and V. Potocnik: TMS- AIME (Light Met.), 1995, pp. 663-71.

7. H.C. Hottel and A.F. Sarofim: Radiative Transfer, McGraw-Hill Book Co., New York, NY, 1967, pp. 318-19.

8. R.T. Bui, S. Peter, A. Charette, A.D. Tomsett, and V. Potocnik: TMS- AIME (lAght Met.), 1992, pp. 731-38.

9. F. Trembtay and A. Charette: Can. d. Chem. Eng., 1988, vol. 66, pp. 86-96.

t0. E. Dernedde, A. Charette, T. Bourgeois, and L. Castonguay: TMS- AIME (Light Met.), 1986, pp. 589-92.

11. M.E. de Fernandez, J. Marletto, and H. Martirena: TMS-,dlME (Light Met.), 1983, pp. 805-17.

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