Guevara & Irons (2011) Part I

8/3/2019 Guevara & Irons (2011) Part I

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Simulation of Slag Freeze Layer Formation:Part I. Experimental Study

FERNANDO J. GUEVARA and GORDON A. IRONS

This study was conducted to understand the freeze layer formation and heat transfer that isrequired to design cooling systems in pyrometallurgical operations in which a frozen slag layer isused to protect the furnace wall. Similar Grashof and Prandtl numbers for operating furnaces

were obtained in a 200-mm square cavity differentially heated on the sides containing anaqueous solution of calcium chloride. The solid front was tracked using a digital camera, andthe temperature field was measured with thermocouples. The flow velocity field was measuredusing the two-dimensional particle image velocimetry technique. Experiments were conductedover a range of superheat conditions, and the solidification front was planar ( i.e., neither cellularnor dendritic) because the system slowly approached steady state. The two-phase zone com-prised particles circulating slowly with the liquid in the bulk of the cavity; at the vertical walls,velocities were higher.

DOI: 10.1007/s11663-011-9524-3 The Minerals, Metals & Materials Society and ASM International 2011

I. INTRODUCTION

IN nonferrous pyrometallurgy, the concept of freez-ing a slag layer at the furnace walls to protect therefractory is well accepted.[1,2] Figure 1 shows a crosssection of a six-in-line electric smelting furnace (ESF) fornickel calcine smelting. The cooling system freezes a slaglayer that is shown in the detail on the right of thediagram. To date, the liquidus temperature of the slaghas been used as the boundary condition at the sidewallin the numerical simulations of fluid flow and heattransfer,[37] which is a major simplification of the

boundary condition because slag freezes over a temper-ature range that can be as large as 393 K (120 C), andsome slags may pass through a glass transition. The rightside of Figure 1 shows a counter-current circulation ofslag primarily caused by bubble-induced buoyancy and,to a lesser extent, by natural convection from the Jouleheating of the slag.[4] The hot slag then descends over therefractory surface, and the amount of heat that can beextracted as well as the thickness of the frozen slagdepend on the velocity and temperature profiles in thetwo-phase zone and boundary layers. This issue is crucialfor furnace integrity but has never been studied in detail.

Small-scale, low-temperature models have been madeof ESFs to analyze precipitation and solidification

effects. Banerjee and Irons[8] used mineral oil and anaqueous calcium chloride solution to simulate the slagand matte layers, respectively, in a 1/10-scale model ofthe Falconbridge ESF. In that study, precipitation in

the matte phase was simulated. In a later study, Shengand Irons[9] presented a mathematical model for heatand momentum transfer in the small-scale model. Thevertical temperature stratification measured in theinterface oil/solution agreed well with the numericalsimulations.

Robertson and Kang[10] experimentally studied theheat transfer and fluid flow in a slag-cleaning furnace,which works similar to an ESF. Eicosene wax was usedto simulate molten slag, and a heated aluminum cylinderwas used to represent the electrode. Eicosene wax has amelting point of 309 K (36 C) and a Prandtl (Pr)

number of 80, similar to fayalite slags (Pr % 50 to 150),but it does not allow for the formation of a two-phasezone.

The heat transfer at the cold solid wall is governed bythe size of the hydrodynamic and thermal boundarylayer, which depends strongly on the Pr number of thefluid.[11] Fayalite slags that are used in nonferrouspyrometallurgy typically have Prandtl numbers between50 and 150.[12,13] Solidification in natural convectionconditions has been studied in rectangular cavities thathave opposing side walls maintained at different tem-peratures, with all other surfaces being adiabatic; thisconfiguration is often referred to as a Bridgman cell.These studies have been performed for liquids with alow Prandtl number to understand the solidification ofmetallic alloys and other liquids such as water, withvarious Rayleigh numbers, Ra. Experimental data forfreezing water (Pr = 13 and Ra = 1.5 9 106),[14] leadtin solidification (Pr = 0.015 and Ra = 6.8 9 104),[15]

Gallium solidification (Pr = 0.02 and Ra = 6.0 9105),[16] and aqueous ammonium chloride solidification(Pr = 30 and Ra = 2.6 9 107)[17] are available in theliterature. As such, this study focuses on the phenomenaat the solidification front and is separate from thephenomena at the electrode in an ESF. In the second

FERNANDO J. GUEVARA, Senior Mechanical Engineer, iswith Fluor Canada Ltd., Vancouver, British Columbia V6E 4M7,Canada. Contact e-mail: [email protected] GORDONA. IRONS, Director, is with Steel Research Centre, McMasterUniversity, Hamilton, Ontario L8S 4L7, Canada.

Manuscript submitted May 14, 2009.

METALLURGICAL AND MATERIALS TRANSACTIONS B


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part of this article, a mathematical model will bepresented for the solidification phenomena that can beused in furnace simulation models.

II. PHYSICAL MODEL AND EXPERIMENTALPROCEDURE

A. Dimensional Analysis

The purpose of this section is to show that the small-scale model is reasonably similar in many respects to afull-scale ESF. The physical properties and conditionsused for the calculations are representative of a Cu-Nismelting furnace, but of course, wide ranges are presentin the operation of various furnaces. The point is thatbest efforts were made to make the model as similar asphysically possible to real operations so that themathematical model in the second portion of this articleis verified against a physical system that operates inreasonably similar thermal and flow regimes.

The dimensionless groups that are most relevant insolidification under natural convection are the Prandtl,Grashof (Gr), and Stefan (St) numbers, which aredefined in the Nomenclature section. For convenience,the Rayleigh number (Ra), the product of the Pr and Gr,is also provided. Physical properties for typical Cu-Nislag and aqueous calcium chloride are presented inTable I.[12,13,18,19] An aqueous solution of calcium chlo-ride was selected as the slag-simulating phase because itsPr is 50, which close to that of the selected slag (47), asshown in Table II. However, other nonferrous smeltingslags may have Prandtl numbers up to 150. To calculatethe Grashof number, a characteristic velocity isrequired; this outcome was taken as the bulk flowvelocity. For the ESF, the characteristic velocities fromthe numerical simulations of Sheng and Irons[9] wereused, whereas the characteristic velocity for the modelwas taken from the present measurements. Table IIshows that the Grashof numbers are an order ofmagnitude smaller in the model because of the depen-dency of Gr on L3; nevertheless, both the ESF and themodel operate in the transition between laminar andturbulent natural convection. The Stefan number is thedimensionless ratio of the liquid superheat above theliquidus temperature and the latent heat. For the ESF,

the superheat term was taken as the difference between atypical bulk slag temperature, 1583 K (1310 C)[20] anda liquidus temperature of 1410 K (1137 C) for anonferrous smelting slag.[21] For the model, mostpresent experiments were carried out with the hot wallat 313 K (40 C)only 2 C above the liquidus. Usingthis temperature difference in the Stefan number makesit much smaller than in the ESF, as shown in Table II.This usage was deliberate so that a larger two-phaseregion could be investigated. Therefore, the modelsystem provides a real system to verify the mathematicalmodel in the second portion of the study, particularly

Table I. Physical Properties for Slag* and CaCl2-H2OSolution

[12,13,18,19]

Slag CaCl2-H2O

Density, kg/m3 3200 1600Thermal expansion, 1/K 2.70 9 104 5.96 9 104

Heat capacity, J/kg K 1250 2010Viscosity, Pa s 0.3 0.015Thermal conductivity, W/m K 8.0 0.565Thermal diffusivity, m2/s 2.0 9 106 1.8 9 107

Latent heat, J/kg 6.45 9 105 2.1 9 105

Liquidus temperature, K (C) 1410 (1137) 311 (38)Solidus temperature, K (C) 1363 (1090) 303 (30)

*Properties at 1583 K (1310 C) and a composition of, in wt pct,55 pct FeO, 35 pct SiO2, 7 pct CaO, 2 pct Cu2O, 0.5 pct MgO, and0.5 pct MnO.

Table II. Characteristic Parameters and DimensionlessNumbers

ESF Model ESF/Model

Superheat aboveliquidus, K (C)

410 (137) 275 (2)

Vertical Length, m 1.2 0.2 6.0Velocity, m/s 0.05* 0.001 50.0Prandtl number, Pr 47 50 0.9Reynolds number, Re 53 8 6.7Grashof number, Gr 1.1 9 108 1.2 9 107 9.6Rayleigh number, Ra 5.4 9 109 4.3 9 108 12.6Stefan number, St 0.34 0.019 17.5

*From Sheng and Irons.[9]From experimental results.

Fig. 1Cross section of a six-carbon electrode ESF, with a schematic cooling system and its details.



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for the two-phase region. The model and full-scalesystems are similar enough that use of the model for realfurnaces can be done with some confidence.

B. Physical Model

The aqueous solution of calcium chloride wasprepared using solid calcium chloride dihydrate, ACS-CaCl2-2H2O (74.0 to 78.0 pct CaCl2) from Alfa AesarCompany (Ward Hill, MA). In this system, several

precipitate phases are possible (Figure 2a)). Figure 2(b)shows a detail of the working concentration range. A53 pct solution was selected because the temperature isconvenient and because limited solute depletion wasavailable, as will be discussed later. The peritectictemperature and composition are 302.9 K (29.9 C)and 49.8 pct CaCl2, respectively.

An acrylic glass cavity of 200 9 200 9 100 mm(internal dimensions) was constructed to study thefreeze layer formation under natural convection condi-tions between two walls held at different temperatures(Figure 3). Polystyrene thermal insulation (25-mmthick) was used to insulate the top, front, back, andsidewalls. Under the bottom, 50 mm of insulation wasused. Two bath controllers (immersion circulator, model12112-01; Cole Parmer, Chicago IL), working indepen-dently were used to maintain constant temperature oneach side of the cavitycold on the left and hot on theright. Aluminum plates (2-mm thick) on each side wereused to minimize the heat-transfer resistance. Distilledwater was used as the circulation fluid, on the primarycircuit, at a rate of 9 L/min. A cooling secondary circuitusing an open loop with tap water was used to extractthe heat in the cooling side of the primary circuit.Thermocouples type T (Copper-Constantan) were fixedin the middle plane of the aluminum plates to assure thedesired working temperature. A thermocouple type T,

2.5-mm diameter and 300-mm long, was used tomeasure temperature inside the cavity in the CaCl2-H2O solution. The gain and offset for all thermocoupleswere adjusted using the freezing and boiling tempera-tures of water, according to the International PracticalTemperature Scales.[22] In the top wall of the cavityperforations, every 20 mm was used to introduce thethermocouple. The first perforation was made 10 mmfrom the aluminum walls on both sides.

A charge-coupled device camera (FlowMaster 4

model; LaVision Ypsilanti, MI positioned 2 m in frontof the cavity) was used to record the advance of the solidprofile over time. Pictures were captured with a resolu-tion of 2040 9 1536 pixels. Software was used to correctthe pictures because of curvature of the lensless thanhalf millimeter. Velocity measurements were performedusing a two-dimensional particle image velocimetry(PIV) system from LaVision with data acquisition andvisualization software DaVis 6.2.[23] To obtain goodresolution in the velocity field, the cavity was dividedinto quadrants. Pictures were acquired using a single-frame/single-exposure technique, with an exposure timeof 10 ls and 0.25 seconds between pictures. A total of

30 frames were used to calculate an average for eachflow field measured. Root mean square (RMS) velocityfluctuations were less than 10 pct with respect to theabsolute velocity value. A double-pulsed Nd:YAG laserwas used to illuminate the cavity. The laser wasmounted over the cavity as shown in Figure 2. Crystalsproduced during the solidification process were used asseed material to track the velocity field. The particleswere observed to be 300- to 500-lm diameter; therefore,they moved with virtually the same velocity as theliquid. The postprocessing was performed using a cross-correlation algorithm over two consecutive images.[24]

For each velocity field calculation, multipasses withdecreasingly smaller sizes were used, with an initial and

Fig. 2Phase diagram for CaCl2-H2O system.



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final interrogation window size of 32 9 32 and 16 9 16pixels, respectively. An overlap final window of 50 pctwas used. No mask technique was used in the solidphase (i.e., velocity vectors computed inside the solidfront must be considered as unfiltered). For the pictureresolution of 2040 9 1536 pixels and the minimuminterrogation area of 16 9 16 pixels greater than 25 pctof the cavity (100 9 100 mm), the interrogation windowis 0.78 9 1 mm. The pictures were acquired at intervalsof 0.25 seconds, so for the maximum velocity of 2 mm/s,the maximum displacement of a particle is about0.5 mm.

The main source of error in the velocity measurementcame from the resolution of individual measurements ofthe displacement distance. For the pixilation described,each pixel corresponds to approximately 50 lm, so that,for a displacement of 0.5 mm, the resolution is only10 pct of the displacement. Accordingly, 30 readingswere taken for each measurement, as described previ-ously, and it is reasonable to estimate that the RMSvelocity fluctuations were also of the order of 10 pct.Vertical velocities were hard to resolve near the vertical

aluminum plate; horizontal velocities were measuredmore easily near the bottom of the cavities and were ingood agreement with those computed under naturalconvection.

C. Experimental Procedure

The experiments were carried out as follows:

(a) First, the solution was melted by setting 323 K(50 C) on both sides of the cavity. The cavity wasfilled to the top. This took almost 24 hours, and was

accelerated by bubbling air at 1 to 2 L/min throughthe bath to increase heat transfer.

(b) Once all solution was fully liquid, the hot side wasshifted down to 313 K (40 C) and the cold side to308 K (35 C). This temperature gradient producedsome natural convection.

(c) After 2 hours, the experiment was started by addingice cubes in the cold side bath controller to reach thedesired temperature. This procedure achieved acooling rate of 2 C/min, so that the cold wall couldreach the set point in about 5 minutes.

The following different hot and cold wall temperatureswere investigated: (1) 321 K to 301 K (48 C to 28 C),(2) 318 K to 298 K (45 C to 25 C), and (c) 313 K to293 K (40 C to 20 C).

III. RESULTS

A. Freeze Layer Profile

Pictures of the solid profile growth were recordedevery 30 minutes. Figure 4 shows an example of a frozenlayer formed after 15 hours. The distinct interface is thesolidus. For the experimental condition of TC = 298 K(25 C) (left side of the cavity) and TH = 318 K (45 C)(right side), the freeze layer formed at the upper and atthe lower part of the cavity was approximately 5 mmand 40 mm, respectively. The difference in thickness wascaused by the greater heat transport in the upper part ofthe cavity than in the lower part. Another importantfeature is the growth of the solid across the bottom,which suggests that the adiabatic boundary condition

Fig. 3Experimental setup to study a freezing layer formation under natural convection in a cavity using 53 wt pct calcium chloride aqueoussolution.



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comparison. Figure 8(b) shows such a plot, yielding adimensionless experimental stratification slope of 0.53for TC = 293 K (20 C) and TH = 313 K (40 C). Adimensionless stratification slope of 0.63 results for298 K and 318 K (25 C and 45 C) cold and hot walltemperatures, respectively, reported in Reference 27.These values are in good agreement with experimentalvalues reported by other authors under similar turbulentnatural convection conditions.[26] All experimental stud-ies reporting this thermal stratification have beendeveloped for air cavity filled Pr = 0.7.[2830] In thepresent experiments, the temperature working condi-tions result in a large mushy zone inside the cavity.

C. Velocity Field

The following results correspond to the test condi-tions of TH = 313 K (40 C) and TC = 293 K (20 C).Figures 9(a) and (b) show the velocity field distributionwith the solid profile in the background for the twoquadrants near the solid front. Vectors are plottedevery three columns or rows for clarity. The counter-clockwise circulation of the flow is clear, descendingover the cold face. The flow is slow 2 mm/s at most.The hot face vertical velocities have similar maximumvalues, although the profile is different, which will be

discussed in a later section. The center of the cavity ismore quiescent. Figures 10(a) and (b) show the hori-zontal velocity profiles at x/L = 0.5 at the top andbottom part of the cavity. Maximum horizontalvelocities are around 0.4 to 0.5 mm/s. It is clear thatthe horizontal components of the circulation arepredominantly at the top and bottom of the cavity.Figures 11(a) and (b) present the vertical velocities athorizontal planes at y/H = 0.4 and y/H = 0.75,respectively. The velocities are zero in the solid layer,which is thicker nearer the bottom. A comparison ofthese profiles shows that the boundary layer flow isaccelerating as it descends, reaching a maximum ofapproximately 2 mm/s.

Figure 12 shows the vertical and horizontal velocityprofiles in greater detail at different horizontal andvertical planes close to the solidification front. Higherup, such as at y = 120 or 140 mm, vertical velocities areabout 1.0 mm/s, which are comparable in magnitudewith the small eddies outside the boundary layer. Thenthe fluid is accelerated to 2.0 mm/s at y = 70 mm.Below that point, the fluid is deflected by the curvedsolidus profile, and vertical velocities decrease again. At

y = 50 mm, vertical velocity is approximately 1 mm/swith a wider profile.

Fig. 6Solid profile for two experimental conditions. Solidliquid interface is showed at the initial time (30 min) and after steady-state conditionis reached (15 h).

Fig. 7Experimental profile of solidified layer under different oper-ating conditions.



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The right side of Figure 12 shows that the horizontalvelocities are one order of magnitude smaller than thevertical velocities. The fluid is accelerated now in thepositive x direction. After the fluid has turned atx/L = 0.25 (x = 50 mm), the horizontal velocity isaround 0.15 mm/s and reaches 0.28 mm/s at x/L = 0.5(x = 100 mm).

IV. DISCUSSION

This dimensionless analysis was used to ensure thatthe experimental conditions were representative of theconditions that would prevail at the freeze layer of anESF. The picture that emerges from the experimental

results is that it takes up to 15 hours to reach steady-state conditions. This finding results in a slow solidi-fication rate and in the planar solidification regime.The temperature range was selected so that a largetwo-phase region would be present, and Figure 8shows that it does indeed extend through most of thecavity. In this region, 300- to 500-lm calcium chlorideparticles are moving with the natural convection flow.These particles are convenient seeds for the PIVmeasurement of velocity. They are moving with veloc-ities that are at most 2 mm/s but, in the center of thecavity, are almost motionless. It is a simple matter toshow that for such small particles the diffusion-controlled rates of heat and mass transfer are fast (ofthe order of a few seconds) enough that the thermal

Fig. 8(a) Experimental isothermal for TC = 293 K (20 C) and TH = 313 K (40 C) and (b) dimensionless temperature stratification at themiddle horizontal plane of the cavity.

DISTANCE FROM THE COLD WALL, mm

DIST

ANCEFROMT

HEBOTTOM,mm

25 50(a) (b)

75 100

100

75

50

25

0 DISTANCE FROM THE COLD WALL, mm

DISTANCEFROMT

HEBOTTOM,mm

0 25 50 75 100

200

175

150

125

100

Fig. 9Velocity field distribution after 15 h of solidification tune at (a) the bottom and (b) the top quarter of the cavity fi = 2 mm/s.



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and chemical equilibrium is closely approached at eachlocation in the cavity. See Appendix A for thesecalculations. Therefore, the lever rule can be applied ateach point according to the temperatures observed inFigure 8. As demonstrated in Section IIIA, precipita-tion of CaCl2-4H2O does not significantly alter thebulk solution concentration.

Solidification at the cold wall in full-scale, nonferroussmelting furnaces is also expected to be slow. Thesefurnaces are large (e g., the Xstrata furnace is 10 m 9 30m, with 1 m of slag depth). They operate with a 373 K to473 K (100 C to 200 C) superheat above the liqui-dus,[20] and the freeze layers are up to 1-m thick. Heattransfer through the wall is mainly controlled byconduction through the freeze layer and refractory, soit is slow. Because of the large mass and insulation, thetime constants for furnace response are of the order of aday.[31] Therefore, fluctuations in the thickness of thefreeze layer caused by the normal operational fluctua-tions should be slow, and the solidus front should movein a planar manner (i.e., neither cellular nor dendritic).As explained in Section IIA, the Stefan number issmaller in the real furnace, so the two-phase layerthickness will be comparatively small.

The group of Blanpain and Wollants has conductedseveral studies of the freezing of industrial slags.[32,33]

They dipped cold fingers into laboratory melts andexamined the freeze layers chemically and microscopi-cally. The cooling was most intense at the surface of thefinger, so a glassy phase was frozen, and farther from thesurface, the morphology changed to equiaxed crystalsand then to columnar crystals. For long submergencetime (~1 hour), an amorphous freeze layer was formedin contact with bath. The layer composition was close tothe bath composition. Small crystals were observed onthe surface when the finger was removed, but they weredifficult to recover when the finger was removed fromthe melt.[32] The cooling rate at the sample surface forthe long submergence experiments was slow, so thatconstitutional undercooling (according to Eq. [1]) wasnot possible, just as in the present experiments. Thisfinding explains why the morphologies for the longersubmergence times are similar to the present morphol-ogies (i.e., planar in the solidification sense with smallcrystals). These findings suggest that these morphologieswould be found in the nonferrous smelting furnacesbecause they have slow thermal responses and thatsolidification would occur in the planar regime.

0.0 0.1 0.2 0.3 0.4 0.50

20

40

60

80

100

TC = 20oC

TH = 40oC

Experimental Conditions

DISTAN

CEFROMT

HEBOTTOM,mm

U - VELOCITY, mm/s

(a) (b)

-0.5 -0.4 -0.3 -0.2 -0.1 0.0100

120

140

160

180

200

TC = 20oC

TH = 40oC


DISTAN

CEFROMT

HEBOTTOM,mm

U - VELOCITY, mm/s

Fig. 10Horizontal velocity profile (a) x/L = 0.5 at the bottom and (b) x/L = 0.5 at the top sides.

0 20 40 60 80 100-2.5

(a) (b)

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0


TC = 20oC

TH = 40oC

y/H = 0.40

V-VELOCITY,

mm/s


0 20 40 60 80 100-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

TC = 20oC

TH = 40oC

y/H = 0.75


V-VELOCITY,mm/s


Fig. 11Vertical velocity profile at (a) y/H = 0.40 (at 80 mm from the bottom) and (b) y/H = 0.75 (at y = 150 mm from the bottom).



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Fig. 12Vertical (left) and horizontal velocity (right) profile component extracted from PIV measurements after 15 h. Vertical distances mea-sured from the bottom. Horizontal distances measured from the cold wall. TC = 293 K (20 C) and TH = 313 K (40 C).



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V. CONCLUSIONS

1. ESFs used in nonferrous smelting operate with rela-tively low superheat over the liquidus temperature ofthe slag, (373 K to 473 K (100 C to 200 C)) that iscomparable with the difference in temperaturebetween the solidus and liquidus of the slag. ThePrandtl number for these slags is typically 40 to 150.Dimensional analysis reveals that the Grashof num-ber is of the order of 108 in the transition between

laminar and turbulent natural convection. The Stefannumber in these systems is of the order of 0.3,meaning that the superheat corresponds to 0.3 of thelatent heat of fusion.

2. A low-temperature experimental technique wasdeveloped to measure the freeze layer thickness,temperatures, and velocities in a Bridgman cell dur-ing freezing under natural convection conditions.

3. A 53 pct aqueous solution of calcium chloride wasselected as the low-temperature analog for the slagbecause it has a similar Prandtl number (50) andconvenient liquidus (311 K (38 C)) and solidus(303 K (30 C)) temperatures could be obtained witha peritectic precipitation reaction. The Grashof

number was an order of magnitude smaller in themodel but still in the transition regime betweenlaminar and turbulent convection. The superheat wasvaried in the experiments but made much smaller toenlarge the two-phase region, corresponding toStefan numbers of the order of 0.02.

4. For the conditions employed, the freezing was soslow that a planar (in solidification terminology)front was observed. It was thinner at the top of thecavity because of hot liquid that encountered it. Mostof the cavity was occupied by a mixture of 300- to500-lm particles and solution that circulatedtogether.

5. This study of freeze layer formation seems to be thefirst in a square cavity under natural convectionconditions for high Prandtl number fluids.

6. Temperature stratification was observed undersolidification conditions. The dimensionless experi-mental stratification slope, dh/dY, has a value of 0.53for TC = 293 K (20 C) and TH = 313 K (40 C).Also, a dimensionless stratification slope of 0.63results for 298 K and 318 K (25 C and 45 C) coldand hot wall temperatures, respectively.

7. Velocity measurements show that the motion is onlyclose to the walls and solid front. The main coreremains virtually motionless. The vertical velocitymeasured close to the solid profile showed a maxi-

mum value of 2 mm/s. Horizontal velocities are oneorder of magnitude smaller (i.e., close to 0.3 mm/s).

8. These results may be useful as benchmarks to testnumerical codes for solidification under naturalconvection for high Prandtl number fluids.

9. Blainpain et al. froze industrial slags on cold fingersin their laboratory. The morphologies of their frozenslags that cooled the slowest had similar morpholo-gies to the present experiments. Industrial furnaceshave slow cooling rates because of their largemass and insulation, suggesting that the planar

solidification morphology will develop in the freezelayers in these furnaces.

ACKNOWLEDGMENTS

The financial support of the NSERC is gratefullyacknowledged. The technical assistance of Mr. O.Kelly and G. Bishop of McMaster University is alsogratefully acknowledged.

APPENDIX A

Time for Thermal Equilibration

Because the calcium chloride particles are relativelysmall, each one can be considered to have a uniformtemperature throughout, and the rate of heating can bedetermined as follows using Newtons law of cooling,and an appropriate heat-transfer coefficient:

qCpVdTS

dt hATS TL

which can be integrated for spherical particles to yieldthe following:

TS TLToS TL

exp6ht

qCPD

exp

t

t

The thermal time constant t* is expressed as follows:

t qCPD

6h

These particles are so small that they virtually movewith the liquid, so that the heat is transferred bydiffusion through the liquid (i.e., the Nusselt numberapproaches its limiting value of 2) and is expressed asfollows:

Nu hD

k 2

Substituting into the previous equation, and usingappropriate values for 500 lm calcium chloride particlesin SI units yields the following:

t qCPD

2

12k

18301250500x1032

120:

565 0:08 s

Therefore, the particles thermally equilibrate quicklywith any new fluid they encounter.

Time for Chemical Equilibration

A slightly different approach is taken for this calcu-lation. It is assumed that the bulk solution contains53 pct CaCl2, and that the particles are precipitating



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with a composition at the interface of 51 pct CaCl2corresponding to the liquidus composition. The time fora 10 pct increase in volume (3 pct increase in diameter iscalculated). The mass flux can be equated as follows tothe rate of increase in particle volume:

kACBL CSL mV

dV

dt

Because the particles are almost motionless withrespect to the fluid, the Sherwood number approachesits limiting value for radial diffusion as follows:

Sh kD

DL 2

After some algebraic manipulation and integration,the diameter of the particle can be related to the originaldiameter as follows:

D2 D2o 8DLC

BL C

SLt

mV

The liquid diffusivity is estimated to be 108 m2/s, themolecular volume of CaCl2.4H2O is 10

4 mole/m3, and

the 2 pct concentration difference is 276 mole/m3

. Underthese conditions, it takes 7.4 seconds for the particles toincrease 10 pct in volume. Therefore, in the centralregions of the cavity, where the particles are almostmotionless, local chemical and thermal equilibrium isalmost reached.

NOMENCLATURE

A area (m2)Cp specific heat at constant pressure (J/kg K)C

bulk liquid composition (pct)DL diffusivity in the liquid (m

2/s)GL temperature gradient in the liquid (K)Gr Grashof number, gbL3 DT/m2 (adim)h global convection heat transfer coefficient

(W/m2K)H cavity height (m)k thermal conductivity (W/mK)k equilibrium partition ratio between liquid and

solidL cavity width (m)L latent heat of solidification (J/kg)mL liquidus slope (K/wt pct)mV mass of the differential volume (kg)Nu Nusselt Number, hD/k

Pr Prandtl number, Pr = m/aR solidification interface growth rateRa Rayleigh number, Ra = Pr GrRe Reynolds number, Re = vx/mSt Stefan Number, Cp (Tb TL)/LSh Sherwood Number, kD/DLt time (s)t* thermal time constant (s)T temperature (K)Tb bulk temperature (K)TC temperature of the cold wall (K)

TH temperature of the hot wall (K)TL liquidus temperature (K)TS solidus temperature (K)TSP superheat, TSP = Tb TS (K)u, v horizontal and vertical velocity components

(m/s)V volume (m3)x horizontal position (m)X dimensionless horizontal position, X = x/L

y vertical position (m)

Y dimensionless vertical position, Y = y/Ha thermal diffusivity, a = k q1 Cp1 (m2/s)b fuid thermal expansion coefficient (l/K)DT temperature difference (K)DS stratification slope, Dh/DY (1/m)l viscosity (Pa s)m kinematic viscosity (m2/s)h dimensionless temperature, h = (T TS)/

(TH TS)q density (kg/m3)

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