The and Mathematical Modelling of Gas Stirred Ladle SYstems

ISIJ International, Vol. 35 (1 995), No. 1, pp. 1-20

Review

The Physical and Mathematical Modelling of Gas Stirred Ladle

SYstems

Dipak MAZUMDARand Roderick l. L. GUTHRIE1)

Department of Materials and Metallurgical Engineering, Indian Institute of Technology, Kanpur, INDIA.1)McGill Metals Processing Centre, Departmentof Mining &Metallurgical Engineering, McGill University. Montreal, C~nada,H3A2A7.

(Received on July 5. 1994; accepted in final form on September16. l994)

Considerable efforts have been madeduring the past two decadesto investigate gas injection operationsin steelmaking ladles. Towards these, numerousphysical and mathematical model studies embodyingaqueousas well as full sca[e systems have been reported. Onthe basis of an extensive literature search, asummary,discussion and analysis of these are nowpresented. For the sake of convenience and clarity ofpresentation, studies have been categorised into three major groups: (1 ) physical mode]]ing studies, (2)

combined physica] and mathematical modelling studies and (3) mathematical modelling studies. In eachof these categories, a great numberof publications on various phenomena,such as gas-liquid interac-

tions, turbulent fluid flow, mixing, solid-Iiquid masstransfer, etc. have been reported. Accordingly, and asdiscussed in the text, considerable improvements have resulted in our understanding of the various gasinjection induced phenomenain ladle metallurgy operations. Coupled with these, extensive mathematicalmodelling studies have also lead to a reasonably accurate framework for carrying out engineering designand process calculations. Nonetheless, someobscu~ities and uncertainties still remain and these are pointedout, together with those areas where further work is needed.

KEYWORDS:overview; gas stirred ladles; fluid dynamics; heat transfer; masstransfer physical modelling;mathematical modelling.

1. Introduction

Since first envisioned and practised by Sir HenryBessemeralmost a century and half ago for his bottomblown steelmaking process, submergedgas injection into

melts contained in furnaces, Iadles and transfer vessels,

has becomecommonin to-day's metallurgical industries.

In steelmaking, gas injection is applied ona routine basis,

at various stages of melt refining, to enhance reaction

rates, eliminate thermal and/or composition gradients,

to remove particulates and so on. Similarly, parallel

examplescan be cited for the non-ferrous industries. Forinstance, submergedgas injection plays a vital role in the

copper and aluminium industries. The manydiverse

applications of gas stirring in liquid metal processingoperations have naturally resulted in a large numberofinvestigations covering widely varying conditions (for

example, see Refs. l)-5)). In addition to these, a sig-

nificant amount of literature has accumulated dealingwith the fundamental aspects of gas injection into liquids

(e.g., see Refs. 6) through 13)). Indeed, the volume ofliterature on the subject of "gas injection into liquids"is so huge that summarising every study would appearto be beyondthe scope of any single up to-date review.

Asummaryof research work on the important and spe-cific applications of gas stirring, such as those in ladle

1

metallurgy steelmaking operations, has been considered

as the appropriate subject of the current review. Con-sequently, the purpose of the present work has been tobring together the results of a large numberof investi-

gations in this area into a comprehensivedocumentandto present a critical assessmentof the subject as a whole.However,wherever appropriate in the text, brief remarksand references have beenmadeto other relevant studies.

Since ladle injection metallurgy is becomingsuch anintrinsic part of steel processing operations, the the-oretical and industrial aspects of these processes havebeen considered at somelength at a numberof sym-posial4 ~ 23) over the last two decades. Morerecently, theprinciples and practices underlying ladle steelmakingmetallurgy have been summarisedand discussed in ateference text24) by Fruehan. Furthermore, over the

years, several review articles,25~29) addressing specific

aspects of gas injection operations in ladles have also

beenpublished. The limited scope of these reviews25 ~29)

exclude a great numberof studies reported to-date onthe ladle refining of steei. In subsequent sections there-fore, Iaboratory, pilot scale, andmathematical studies of

gas stirred ladle systems are summarised. For the sakeof convenience, these research studies have been cate-gorised into three groups: (1) physical models, (2) com-bined physical and mathematical investigations and (3)

C 1995 ISIJ

ISIJ International, Vol. 35 (1 995), No. 1

Table l. Various investigators and their specific

contribution to gas stirred ladle systems.area of

Referencenumbers

in the text

Stopper-rod assemb[y

SI.

No.Investigator/Group of

investigators

Specific area ofcontribution

1 Asai and coworkers

2 Brimacombeand coworkers

34567

Cross and coworkersDebroy and coworkersFarouk and coworkersFruehanand coworkersGuthrie and coworkers

8 Irons and coworkers

9 Johansenand coworkers

10 Koria and coworkers

l I Lehner and coworkers

12 Mazumdarand coworkers

l3 Krishnamurthy andcoworkers

14 Oeters and coworkers

15 Salcudean and coworkers

16 Sanoand coworkers

17 Schwerdtfeger andcoworkers

18 Szekely and coworkers

19

20Taniguchi and coworkersThemelis and coworkers

21 Wright, Schwarzandcoworkers

Mixing andmasstransfer

Fluid dynamics,mixing andCFDCFDCFDCFDMasstransfer

Fluid dynamics,mixing, CFD,turbulencemodellingFluid dynamics,

CFDandturbulencemodellingFluid dynamicsand CFDFluid dynamicsand masstransfer

Fluid dynamicsand mixingMixing, masstransfer, CFDand turbulencemodellingFluid dynamicsand mixingMixing andmasstransfer

CFD

Fluid dynamicsand mixingFluid dynamics

Mixing, masstransfer, CFDand turbulencemodellingMasstransfer

Mixing andmasstransfer

CFDand masstransfer

49, 105

27, 28, 3033,ll4, I15

118-1 19

75, 109, 11012(~12364, 66, 6734, 46, 56, 59,

76, 79, 112,

126, 127, 130,

131

37, 38, 129

42, 44, 84

35, 36, 106

25, 43

56, 79-81, 98,

99, 104, I13,

126, 128, 130,

13154, 55, 93, 94

51, 52, 69

ll2, I14, 124,

1259, 10, 57

32, 39

47, 72, 73, 78,

89, 115-117

68, 96, 9726, 53

85, 100, 101

mathematical models. Asummaryof the various groupsof investigators, together with their specific areas ofcontribution, are outlined in Table I for ready reference.

2. Physical Modelling Studies

Figure I provides a schematic of gas injection into afilled ladle. There, as seen, Ar/N2 is injected through aporous plug, Iocated at the base of the ladle, into a bathof molten steel contained in a slightly tapered cylindrical

vessel. The injected gas, given its buoyancy, rises to the

free surface, thereby inducing a turbulent recirculatory

flow of liquid, well known for enhancing the rate ofchemical and thermal homogenisation, as well as ac-

Refractory[ined wat[

Fig. l. Schematic of gas purging in a filled ladle of steel.

celerating the absorption of harmful non-metallic inclu-

sions into an overlying slag phase. While typically, underindustrial conditions, only relatively low gas flow rates

are applied to achieve thermal and/or chemical homo-genisation, somewhatintense stirring conditions can also

be practiced for accelerating slag-metal reactions. Con-sequently, depending on the specific objectives of a ladle

refining operation, a wide range of gas flow rates maybe applied29) (0.001 to 0.015 Nm3(STP)/t•min).

As seen from Fig. I,the physical processes involved

in gas stirred ladle systems are numerous. They arecomplex owing to the multi-dimensional (2 or 3 di-

mensional), multiphase (gas-metal and slag) nature ofthe system. Furthermore, several distinct phenomenacanbe identified from Fig. I which include gas-liquid inter-

actions, the development of a plume-induced turbulentrecirculatory motion in the bulk liquid steel, masstrans-fer between slag and metal in the vicinity of the free

surface, and so on. High temperatures and the visual

opacity of liquid steel, as well as the large size of indus-trial ladles, makethese processing units rather cumber-

somefor direct experimental measurementsandobserva-tions. Physical modelling embodyingaqueousas well asliquid metal-gas systems have therefore beencarried out(water and steel haveequivalent kinematic viscosities) toinvestigate underlying process dynamics. Manyphysicalmodelling studies have been reported over the past twodecades on various aspects of gas stirred ladle systems.These are discussed in the present section under thefollowing three headings: (I) fluid dynamics (2) mixingand (3) two fluid masstransfer phenomena.

(a) Fluid DynamicsFigure 2 illustrates schematicaily the physical phe-

nomenaonewould observe during gas bubbling through

a tuyere30) at a moderate gas flow rate (0.5 Nm3/min't)

into an aqueous model of a gas stirred ladle system.There, the gas-liquid two phase region has beensubdivided into four physically distinct regions, primarybubble, free bubble, plume and spout respectively. Ofthese, the plume region, characterised by dispersedspherical cap bubbles in an air-water mixture, is thelargest, andoccupies mostof the bath depth. In contract,the spout typically occupies approximately 3to 4olo ofthe equilibrium bath depth (in aqueousandmolten steel

C 1995 !SIJ 2

ISIJ International, Vol. 35 (1995), No. 1

C ..1L

R i' *

Jl-

lll

,.~i~"I"' ' ~'~'!"a~'~'

'dP

' "51~'

~)

~j'~)

G)

* L

Fig. 2. Characteristics of the two phase air-water plumesduring gas injection through a tuyere30) in a cylindrical

vessel.

systems).It is nowgenerally accepted29-31) that during ladle

refining of steel, in the immediate vicinity of the nozzle,the input gas kinetic energy as well ~s the modeof gasinjection are important variables, while in the fully

developed region, these variables will have practically noinfiuence on the overall development of the gas-liquid

two phase region (i.e., bubble sizes, their spatial dis-

tributions and rise velocities are largely independent ofinlet operating conditions) since the "primary" as well

as the "free bubble" regions, occupy very little volumeof the reactor. Consequently, it is reasonable to generalisethat under typical ladle refining conditions, any largebubbles or gas envelopes forming at the nozzle/plug will

typically shatter a short way above the nozzle into anarray of smaller bubbles. Bubbles in the fully developedregion, with continuous coalescence and disintegration

phenomenaat work, will tend to establish a dynamicrange of sizes in the spherical cap regime, the equilibriumsize of which are determined by the thermophysicalproperties of the system and not by the inlet operatingvariables (gas injection device, orifice diameter etc.).

Evidence also exists that similar phenomenaare at playin equivalent metallic systems.6 ~ Il)

The physical characteristics of the gas-liquid (viz.,

ail~water, mercur~hnitrogen, etc.) plumes of relevanceto ladle processing have been investigated extensive-ly.32~39) Gas volume fraction, bubble frequency andbubble rise velocities within the two phase region, haveall been measuredby numerousinvestigators in vesselsof widely varying geometries and gas flows for a numberof gas-liquid systems. Typically, electro-resistivity probeshave been applied to measure these parameters in thetwo-phase plumes. Thus, while Tackeet al. ,34) Koria andSingh35,36) and Shengand lrons37,38) applied a single

element probe, Brimacombeandcoworkers30- 33) as well

as Castello-Branco and Schwerdfeger39) applied acomputer aided, two element resistivity probe. Theirexperimental set-ups28~31,39) allowed simultaneousmeasurementsof bubble size and velocity in addition toroutine bubble frequency and gas volume fractiondistributions.

\E U 688 m/s dL A de=4.10 mm:h 2.5 Ua= 28.68 m/s v v d0=6.35 mmv a = 876 Ncm3/soa' hb= 400 mm:'

a'

L 1.S

~l]::'

GotZ Oo

100 200 300 400,,,

~ VerticaL position, mm

Fig. 3. Variation of bubble rise velocity at the plume centre-line as a function of axial height.33)

The results of these studies32~38) indicate that nor-malised radial profiles of gas volume fraction and bub-ble frequency are essentially Gaussian. Consequently, acomplete characterisation of the full spatial distribution

of gas and liquid within the plume (viz., plume geome-try and resultant gas voidage distributions) can beconveniently expressed using only two parameters, thesebeing the maximumaxial value of gas voidage (ocmax) andthe plume's half radius value (ra../2), in the followingdimensionless form:

r -v= .ioema' Fr0.30 (1)50L0.20

mdand

9 1/5 r pra ~

zmax/2 2

0'42 ' L0'20 Fr0'30mQ d .

(2)

As shown in several studies,30,33,34) the values of the

exponent pand y, in Eqs. (1) and (2), are functions ofthe gas/liquid density ratio. Thus, the dimensions of the

plumeand the gas volume fraction within themare alsoliquid/gas property dependent.

In Fig. 3, average bubble rise velocity at the axis ofthe plumeas a function of axial height33) is shown.There,except in the vicinity of the gas injection nozzle and thefree surface, measuredbubble rise velocities are seen tobe practically independent of axial height. Morerecentexperimental results by Shengand lrons38) confirm sim-ilar trends for bubble rise velocity. These experimentalobservations33,38) also appear to confirm that hydro-dynamicconditions at an orifice or nozzle, are not crit-

ical to bubble rise velocity over a significant portion ofthe axial height.

A still photographic technique was applied byKrishnamurthy et al.40) to macroscopically characterisethe geometry of two phase axisymmetric plumes in acylindrical vessel (ID =0.24 m) for a wide range of gasflow rates, bath depths and orifice diameters. Througha dimensional analysis of the experimental data,

Krishnamurthy et al. proposed an expression for the

extent of gas dispersion in terms of a plumecone angle,e. (expressed in degrees) in gas stirred systems, accordingto :

'= . ---e 0915 Fr0.12 L 02s4 d 0.441(3)

~180 D D

3 C1995 ISIJ

ISIJ International. Vol.

The specific objective of these workers40) was to derive

an expression for the recirculating rate of liquid usingEq. (3), so as to arrive at a quantitative expression for

recirculation or mixing time (see later).

As seen from Eqs. (1) and (3), a modified Froudenumber, Fr~, has been applied in a large numberofstudies32~40) as an important dimensionless numbertorepresent various parameters (gas voidage, plume coneangle, etc.) in the gas stirred system. The definition ofthe modified Froude numberembodiesa characteristic

velocity scale, which in practically all studies31~36,40)

was considered to be synonymouswith the free spacegas velocity through the orifice. However, under ladle

refining conditions (viz., Q~**-- 0.01 Nm3/min't), as has

been pointed out already, hydrodynamic conditions at

the orifice only marginally influence the physicalcharacteristics of the two phase region, so that the useof a modified Froude numberincorporating an orifice

velocity would appear to be an apparent contradiction

and an issue for debate. A more appropriate andphysically plausible velocity scale in the definition of amodified Froude numberis clearly warranted.

35 (1 995), No. 1

1.o

0.75

Eo,

co'* O.Sto

~o

0.25

o

X e olX• O l

e O

, o

X -

o

I

Subsequentto their studies on the physical characteris-

tics of the two phase plume zone,30-33) Sahajwalla etal.41) reported on their experimental investigation of the

spout region in a gas-water plume. Their experimentalresults indicated that the trend in the distribution of gasvolumefraction in the spout is opposite to those observedwithin the plume i.e., minimumgas voidages were foundat the axis. However, the bubble frequency distribution

remained nearly Gaussian. Bubble velocities within the

spout were found to decrease with increasing axial

distance. This contrasts the plume, where, shortly beyondthe nozzle, the axial bubble velocity remains constant(viz., Fig. 3).

Apart from the measurementsof various parameterswithin the rising two phase plume region, gas inducedliquid flows have also been measuredand reported byvarious investigators. To this end, widely different flowmeasuring devices and techniques have been applied.

These include an electromagnetic flow meter,42) a dragprobe,43) Laser Doppler Velocimetry,31.44) combinedelectro-resistivity probe and Laser Doppler Veloci-metry,37,38) video recordings, and so on. Measurementson individual phasevelocities within the two phasezone,as well as in the bulk liquid, have all been reported.

Hsiao et al.43) with their early but extensive ex-perimental study on fluid fiow in pilot scale (7T) andindustrial (60T) Iadles and corresponding gas stirred

viater model ladles, observed that the centreline liquid

velocity at the plume axis, at any given gas fiow rate,

was practically independent of the height of the ladle.

Theonly deviations from this trend were observed nearthe injection nozzle and near the free surface. Theirexperimental results are illustrated in Fig. 4. The morerecent study of Sheng and lrons38) confirms suchobservations. Under ladle refining conditions, one cantherefore anticipate nearly constant bubble and liquid

velocities in the major portion of the upwelling two phaseregion. Referring back to Figs. 3and 4, onemayfurther

note that near the free surf4ce, the rising liquid loses a

C 1995 ISIJ

3dm nmin

x 50

e 100

o 200

I 300

4

Fig. 4.

Fig. 5.

O 0.4 1.2 1.60.8

Centreline rise veLocity, m/sVariation of iiquid rise velocity at the plumecentreline

as a function of gas flow rates.43)

0.40

E0,30a'

vcou'

1' 0.20

ov

o> OIO

o

f""tttdf'

tt,,,~

tt",t

O 0.10 0.20

Radial distance , mCharacteristic liquid fiow pattern in an axisymmetric

gas stirred ladle.38)

significant portion of its vertical momentumin favourof an upwelling spout and is thereby forced to fiowradially outward from the plume's axis. Thesephenome-naaccordingly slow both liquid andbubble rise velocities.

Figure 5shows38)experimentally measuredfiow patternsin an axisymmetric gas stirred water model ladle. There,

a relatively high vertical velocity at the plume axis is

readily apparent. The rising fluid is seen to flow radially

outwards in the vicinity of the free surface and thencertically downwardsadjacent to the ladle side-walls.

This motion creates a recirculating flow pattern in thevessel with its toroidal "eye" Iocated up in the ladle anddisplaced towards the side wall. Thebottom portion ofthe ladle is relatively quiescent. Thecharacteristic natureof the flow recirculation in the bulk liquid, as has beenexperimentally confirmed by Anagbo and Brima-combe,31) is insensitive to the modeof gas injection, aswell as the specific gas flow rates applied.

Bubble and liquid rise velocities within the plumeregion were measuredsimultaneously for the first timein an aqueousgas bubble driven system by Shengand

ISIJ International, Vol.

lrons.3s,36) While their experimental results37.38) on gasvolume fraction, bubble rise velocity, etc. were similar

in trend to earlier studies, several newand interesting

findings emerged. Thus, Shengand lrons were, on thebasis of their experimental data, able to suggest that therelative velocity of bubbles in a bubbly plume can bereasonably well represented by the terminal rise velocity

of a single bubble (of the samesize). Their study,38) aswell as that by Johansen et al.,44) confirm that rising

bubbles within the plumecontribute significantly to the

production of turbulence and that this turbulence withinthe two phase plume region is somewhatanisotropic(e.g., turbulence within the plume is slightly skewedtothe vertical direction).

In contrast to the large numberof experimental studies

carried out on the air-water systems summarisedabove,31~44) physical model studies on the hydro-dynamicsof equivalent gas-metal-slag systemshavebeenrare. Tanakaand Guthrie45) investigated someaspectsof slag metal interactions under relatively high gas flowrates (e.g., corresponding to BOFsteelmaking) andsuggested that gas bubbling can lead to the generationof droplets torn from around the rim of the plume (see

also later). Subsequently, Mazumdaret al.46) measuredflow phenomena, turbulence characteristics, averagekinetic energy of fluid motion, etc. within a water modelladle (ID =0.30 mand L=0.21 m), in the presence of anupper buoyant phase using Laser Doppler Velocimetry.It wasobserved experimentally that at any gas flow rate,

the kinetic energy of the meanand fluctuating motions(i.e., the total specific kinetic energy of motion in thesystem) are considerably reduced in the presence of the

top buoyant phase. The deformation of the upperbuoyant phase together with its entrainment in the bulkphase liquid was found to be responsible for observedrates of energy dissipation andslowing of the lower phase

~rfo

ol!x

\l~:,

o:,

c

17,

c

vQ,c:

~v

,lu,

aoh

17

16

15

14

13

12

11

10

e Without oi[

o With woodenbtock on topof bu[k Liquid

x with oiL

e

o

e

o

9 e8 o

7 e65 o x4321O 72 3 4 5 6O 1

Specific energy input rate. W/kgx

10~3

Plot of total kinetic energy contained in recirculating

aqueousphase vs. specific potential energy input rate

under various upper slag phase conditions i.e., rigid,

fluid and no slag.46)

35 (1995), No. 1

liquid. This is illustrated in Fig. 6, in which the variationin total kinetic energy contained in the recirculating

aqueousphaseis plotted against the specific energy inputrate, for various upper slag phaseconditions (viz., rigid,

fluid and no-slag conditions are shown). From suchobservations, it is clear that under practical ladle refining

conditions, significant energy can be dissipated by anupper slag phase, and that this will be a function of slag

thickness, viscosity and density, etc. Further experi-

mental work is needed to adequately quantify suchphenomena.

(b) MixingThe intrinsic efficiencies of manychemical processing

operations carried out in present day steelmaking ladles

are intricately related to mixing phenomena.Mixingenhances chemical reactions by bringing reactantstogether and removing products from reaction sites. It

also influences the extent of thermal and particulate

inhomogeneities within the ladle. It is therefore desirable

to ascertain the extent of mixing, in order to evaluatethe process performance of argon or nitrogen stirred

ladles.

Thestudy of mixing phenomenahasjustifiably receivedconsiderable attention over the years. 'Most often, the

concept of amixing time, T~, hasbeenapplied to representthe state of agitation (and hence, an inde~ of processefficiency) in the reactor vessels. Efforts have beenmadeto quantify, experimentally, mixing time as a functionof operating variables (e.g.. L, D, Q, etc.), using appro-priately scaled aqueous models. Towards these, dif-

ferent experimental techniques have been applied, al-

though methodsbasedon pHandelectrical conductivity

measurementshave been relatively more popular. Inequivalent high temperature systems, radio active tracerdispersion tests have been carried out to estimate the

rates of metal mixing.Nakanishi et al.47) were the first to plot experimental

data on mixing time against global rates of specific ener-

gy dissipation for an RHvacuumdegassing unit, aninduction stirred ASEA-SKFsystem, and for watermodelsof argon stirred ladles. Theseauthors47) providedthe following correlation betweenmixing time andenergydissipation rates according to (SI units):

V

,~

o

~~V:,v,

cE .~

,:E

Fig. 6.

5

Fig. 7.

1000

500

100

50

10

xl X o q) Tmc ~~O.4

oo

o

o 50 Targon stirred ladte

e 50 TASEASXF e

l 200 T R-HX 65 kg water modet

1 2 5 10 20 50 100 200 5003Specific energy input rate (Em), Wott/kg xlO~

The experimental work of Nakanishi et al.4?) show-ing the functional relationship betweenperfect mixingtimes and specific energy input rates for a range ofmetal processing operations.

C 1995 ISIJ

ISIJ International, Vol. 35 (1995). No. 1

T~= 12.68 x 103e~o.40 ..........(4)

In a later study, Nakanishi et al.,48) while performingmixing studies on bottom blown reactors for steelmak-ing, found that the preceding relationship (viz., Eq. (4))

performed remarkably well for all single tuyereexperiments. Their "universal correlation" is showninFig. 7. This, as well as Eq. (4), appear to indicate thatthe vessels' shape and size and modeof energy input,etc. have no influence on mixing. Such claims werehowever, refuted in later years by several investiga-tors49~57) and it is nowgenerally accepted the metal-lurgical vessel's geometry, modeof energy input, etc.

influence mixing phenomenaconsiderably.Since that early study of Nakanishi et al., a large

numberof experimental studies on mixing phenomena,of relevance to gas stirred ladle systems, have beenreported in the literature. In these, widely varying gasflow rates, vessel geometries and nozzle configurations

were applied and their influence on mixing investigated.

Mixing times were measuredvia different experimentaltechniques and to this end, various criteria were applied.

As a consequence, a variety of functional relationshipsbetween mixing time, specific energy input rate andoperating variables (e.g., L. R, Q) have been proposed.

To illustrate these, experimental conditions, togetherwith the various proposed mixing time correlationsreported in the literature, have been surnmarised andpresented in Table 2. There, as seen, dissimilar ob-~ervations have been reported. Furthermore, proposedmixing time correlations differ between investigators.Several reasons can be attributed for such diverseobservations and these are now addressed. Thus

,referring back to Table 2, it is readily seen that someofthe vessel's aspect ratios as well as the specific gas flowrates applied, are very different to those expected undertypical ladle refining operations. Asalready pointed out,specific flow rates in industrial gas stirred ladles, having

aspect ratios (L/D) in the range of 0.5 and 2.0, typical-ly lie in the range of 0.0015 to 0.0lNm3/min't.29)Furthermore, in someexperiments and analyses, thekinetic energy of the incoming gases has been includedin the expression for specific energy input rate to thesystems,54,5s,57,58) while others have ignored such kinet-ic energy contributions. While consideration of inputkinetic energy makesthe energy balance more realistic

from a conceptual standpoint, it is instructive to notehere that for the gas flow rates and orifice dimensionsnormally applied in actual practice, the contribution ofgas kinetic energy in the overall energy balance is

rendered insignificant for all practical purposes (typicallyless than 5o/, or so). Onthe basis of such, the kinetic

energy of the incoming gas has little relevance under ladlemetallurgy steelmaking operations and therefore, thepotential energy afforded by the rising gas bubbles canbe conveniently regarded as the dominant modeof ener-gy supply to the system, as far as current ladle refiningoperations are considered. Apart from these considera-tions, widely varying expressions have been applied toestimate potential energy input rates to gas stirred

systems. Someof the expressions commonlyapplied to-gether with corresponding estimates of potential energyinput rate, e~, for a typical situation are shownin Table3. These expressions, as one would note here wereformulated considering the work doneby the gas bubbles(viz., the buoyancywork) during their rise through themelt. The contributions of the expansion work due to

pressure and temperature near the nozzle/tuyere and thetransfer of kinetic energy from the gas to the liquid nearthe tuyere were ignored as these are not of muchrelevanceto ladle gas injection operations. Contrary to expectation,different values of e~ results from these formulae for anunique operating conditions, as illustrated in Table 3.

There, while the expressions adopted by MazumdarandGuthrie,56) Sinha and McNallan58) and Krishnamurthyet al.55) provide practically similar estimates, those of

Table 2. Asummaryof experimental configurations and mixing time correlations reported by various investigators.

Investigators Exp. tech. MlxmgcrrtenaVessel dimensions and

massof fluid

Specific gas fiow rates(m3/min/T) Mixing time correlation

Nakanishi et al.47)

Asai et al.49)

pH Undefined

Electrical 990/.

conductivity

Sinha &McNallan58) pH

Themelis and Stapurewicz53) Photocell

Mietz and Oeters52) Electrical

conductivity &colorometr y

Mazumdarand Guthries6) Electrical

conductivityKrishnamurthy et al.5s) Electrical

conductivity

97.7010

950/o

950/0

950/0

Bulk99.90/0

L=0.465 m,D=0.42 m; 64kgD=0.405, 0.2 &0.10;

L/D=0.5 to 1;

52kg (max) &0.4kg(min)

L=0.48 m,D=0.45m; 76 kgL=0.67 to 1.0m,D=0.66m;310kg(max)

L=1.0 m, D=0.63 m;311kg

L=0.5 to 1.1 m,D=1.12m; IOOOkgL=0,1 to 0.45m,D=0.48m;81 kg(max)

0.015 to 0.06

Numerous;say, 0.019to 0.90 with ref. to

D=0.405 m&L=0.40 m0.02 to 0.4

0.13 to 0.96

0.038 to O.29

O.012 to 0.06

0.1 1to 2.67

r =800e 04

T 2748- o. 33L~ IRl .36

Tm=6928~0.89

Tm 164e~ o. 39LO' 39m

1,~ CIQ~"CI and n arefunctions of traceraddition and monitoringlocations

T~=37e~0.33L~ IR1'66

1,~=CIQ~" C and n arefunctions of flow regimeand liquid depth

C 1995 ISIJ 6


Staphrewicz andThemelis53) andNakanishi et al.47) Iead

to markedly different values. A flawed derivation,47) anerror in the use of "In" vs. "log",53) are the reasons for

the discrepancy amongestimates shown in Table 3.

These, together with the reasons already enumerated, aretherefore likely to introduce somevariability in the final

results, as is reflected in Table 2.

As illustrated in numerousexperimental investigations(viz., Table 2), operating conditions (particularly gas flowrates and vessel geometry) in gas stirred systems havealso considerable bearing on mixing phenomena.As anexample of this, in Figs. 8(a) through 8(c), functionalrelationships between experimentally measuredmixingtimes and energy input rates (or gas flow rate) reportedby three different groups of investigators49'55,s6) areillustrated. There, over a wide range of operatingconditions, two distinct kinds of dependencehave beensuggested between 8~ and mixing times by the three

Table 3. Various expressions for rate ofpotential energy in-

put to the gas stirred systemand the correspondingestimates in an aqueoussystem for a typical ex-perimental situation. (L= 1.0m, D=1.0m, Q=l.10~4Nm3/s, TL=298K)

Value of total

InvestigatorsExpressions for total specific

specific energyenergy input rate

input rate (W/t)

Nakamshietal 47)0.0285 . QTL L(log~l +W 148

[L (cm); Q(Nl/min); W(t)]

742.QTLln(1+ LW 10.34

14.54

35 (1995), No. 1

Themelis andStapurewicz53)

[for water model only; Q(m3/s)]

Mazumdarand pLgQLGuthries6)PL7rR2L

[Q (m2/s)]

Sinha andMcnallan58)

854• QTLIog I+

PLqL

W P*

Krishnamurthyet al.55)

[Q (m3/s)]

In(1+

4QPaTL PL9L298.21~D2L P,,

[Q (Nm'/s)]

2.5

1.13

l.3

l .29

200 r ac e0,68

405cx 400Hu,

• IooQ,

.~ lj ?cc e-0.32

~ 50o,,:

)( o2:

10(a)

0.5 1 5 Io 50 10C

Specific ener y input rate kg/ms3

investigators. As a possible reason for the shift in thefunctional relationship between 8~ and T~ in Fig. 8(b),

Krishnamurthy et a/. attributed the phenomenato theswirling motion of plumeat relatively high gas flow rates(e.g., Iarge energy input rate). Contrary to this, Asai andcoworkers49) suggested that the shift in the functionalrelationship in Fig. 8(a), might be the result of a flowtransition from laminar to a fully turbulent situation.

Although suchargumentshavenot beenoften adequatelysupported with direct experimental evidence, these two~xperimental studies,49'55) pertaining to widely different

operating conditions, nevertheless, appear to indicatethat different hydrodynamic phenomenaare at playunder different operating regimes. These, as alreadypointed out, are likely to induce variations in observa-tions and as a consequence, will tend to further com-plicate comparisons amongvarious studies.

Several investigators51,52,56,58,59) have reported thatthe state of mixing in a gas stirred bath under a givenset of operating conditions is a function of tracer additionandmonitoring point locations. Suchaphenomenonwasobserved at relatively low specific gas flow rates52,56,59)

(e.g., in the range applied to typical ladle gas injectionoperations), irrespective of the criteria applied to assessthe degree of mixing. As shown in Fig. 9, at variouslocations in the vessel, different functional relationships

were observed to govern the mixing phenomena.There,

as seen, the probe location as well as the tracer additionpoint in combination, appear to have a significant

influence on measuredmixing times.

In contrast to these observations,51'52,56,58,59) Krish-namurthy and coworkers54.55) on the basis of their

experimental measurements, suggested that a single,

characteristic mixing time can be ascribed to any reactorvessel for a given set of operating conditions, provided

a degree of homogeneity up to 99.5 o/o is applied as theacceptable mark. As already pointed out and illustrated

in Table 2, the state of agitation in the experimentalsystem of Krishnamurthy et al. was far too intense thanthose to be expected in the experimental systems of Mietzand Oeters,52) Mazumdarand Guthrie56) etc. Therefore,it is not unlikely that at large values of specific gas flowrates applied in a relatively small vessel, the state ofmixing in mostparts of the reactor wasrelatively uniformand muchdifferent from those encountered at low flowrates in relatively larger vessels, particularly when themeasuringprobe is placed near the baseof a vessel (which

100

t,,

a,

E 50

:bcx~ 20

10

H(m]o 0.10A 0.15a 0.20e 0.25

,Al,

HClm

)0.300.35O.400.45

e

e

dn = 6.66 mmD =0.48m

(b)

200

Q, 100Ecbc

::50

40

O

rmGcQ0.48

fmeCQO34~

(C)

100 1.67 3.33 5.0 e.675 10 20 40 70 100 2003Specific energy input rate

,kg/ms Specific energy input rate, kg/m5 Ges fLOWrate,

Fig. 8. Functional relationship betweenmixing time and specific energy input rate/gas flow rate reproduced fromthe work of various investigators. (a) Asai et al.,49) (b) Krishnamurthy et al.s5) and (c) MazumdarandGuthrie. 56)

m3/s

16.67

7 C 1995 ISIJ


50

40

' 30

Ea)c

'x~_ 20~;

10

o

tmix = 59•Q O06

r__1L'

~ Ii- I~ ~T--~l\

tmix :: 280 QO381\ '

\\\\\\

mlx '= 631 QO54\e'\t '\ \\

\ e, \\+\\ c'

+\\\~e.+

\ + ~e++__ + ~~1~

+ *+

tl ~~2 1 Position I +

3 + position 21 eb Position 3

O 500 1000 1500

Gas flow rate, Ncm3ls

Fig. 9. Theexperimental work of Mietz and Oeters52) show-ing the influence of monitoring point location on the

measuredmixing times for central nozzle position andtracer addition in the dead zone.

is often adeadregion). 51,5 2, s6, 58,59) Further investigationis needed to resolve such anamolies in experimentalobservations. :

Attempts have been madein the past to characterisemixing phenomenain gas stirred systems in terms ofeither bulk convection or eddy diffusion dominatedphenomena.It was first suggested by Nakanishi andcoworkers47) that mixing in metallurgical reactors is

caused primarily by eddy diffusion rather than by bulkcirculation of the melt. In a later study, Asai et a/.49)

expressed a similar view. Subsequently, however,MazumdarandGuthrie56) through a combinedthebreti-

cal and experimental investigation on mixing demon-strated that such classifications do not always hold andthat mixing in gas stirred ladle systems is the result ofthe combined actions of bulk recirculation and eddydiffusion phenomena,in practically equal measure.

Despite the diverse nature of the experimental ob-servations discussed so far, similar empirical correla-tions have beensuggested by numerousinvestigators forestimating mixing times as a function of operatingvariables at relatively low specific gas flow rates in

cylindrical vessels (0.5 ~L/D~2.0) agitated by acentrallyrising bubble plume. Thus, Sanoand Mori57) found thatthe following relationship describes mixing phenomenareasonably accurately:

T~ Cle~033R133L~0.67 ..........(5)

Similarly, Asai et al.49) suggested a very similar ex-pression for axisymmetric gas bubble driven systemsviz.,

T~=C28~0.33R1'36L~1 ..........(6)

In a later study, Mazumdarand Guthrie,56) found asimilar functional relationship between operating vari-

ables and mixing times, as those suggested by Eqs. (5)

and (6):

C 1995 ISIJ 8

T~=C38~0.33R1'66L-1.

..........(7)

As seen, these expressions illustrate the fact that mixingtimes are inversely related to the height of the liquidwithin the vessel, that they increase with vessel radiusand decrease according to the third powerof the specific

energy input rate. Amorerecent study by Neifer et al.60)

also appears to indicate that a functional relationshipequivalent to these equations provides reasonableestimates of mixing times in laboratory, as well asindustrial size, Iadles. The values of the fitted constantsC1' C2 and C3 in Eqs. (5) through (7), are functions ofthe degrees and definitions of mixing time l' dapp re

.

Mixing studies in an aqueousmodel (di~. =0.66 m) ofa gas stirred ladle system was reported by Stapurewiczand Themelis.53) Theeffect of two different gas injectiondevices (viz., tuyere andporous plug) anda large numberof gas fiow rates, in the range of 19 to 68 Iit/min (e.g.,

approximately 0.005 Nm3/min't), on mixing wasinvesti-

gated. Basedon their experimental findings, Stapurewiczand Themelis53) confirmed that mixing conditions arerelatively insensitive to the gas injection device applied(e.g., porous plug, tuyere, etc.).

The experimental studies considered above were es-sentially carried out in water models in which the effect

of an upper buoyant phase on mixing phenomenawasneither simulated nor investigated. As one would note,the presence of a slag rather than no slag is moretypicalof industrial argon stirring operations. Thus. Haida andcoworkers,6 1) investigated the role of an upper slag layer(e.g., oil) on bulk liquid mixing with the aid of a watermodel. Theseauthors found that mixing times measuredwith a similated slag tended to be considerably differentto those for equivalent "no slag" situations. Such ob-servations61) can be readily rationalised, since the hy-drodynamic state of the vessel, at any gas flow rate, is

knownto be different in the presence of an overlyingsecond phase46) Iiquid. As discussed already, the upperslag phase dissipates a part of the input energy rate andtherefore, mixing times in ladles, in the presence of theoverlying second phase liquid will be somewhatlongerthan those to be expected under an equivalent no slagsituation.

Several investigators have also reported on the indus-trial scale measurementsof mixing phenomenain gasstirred ladles and torpedo cars. It has been suggest-ed57,60,62,63) that correlations deduced from carefullydesigned water model investigations can be extrapolatedto industrial scale operations. However, despite the bulkof theoretical anc experimental work on the subject ofgas injection into ladles, comprehensive statement~ onthe scale up of model trial results are not availableto-date. Although someinvestigators49'57,60,62,63) haveattempted to address "scale-up" in the past, divergentcriteria have been advocated. Throughconsideration ofmeasuredmixing time data in industrial vessels as well

as in water model ladles of various sizes and gas fiowrates, it has been shown57,62,63) that mixing timesregardless of the configurations of the gas stirred ladle

systemapplied, can be adequately expressed as a functionof mixing energy, 8~ (e.g., specific potential energy input


10 e

5 eo

'~A v JLv~A o e

_ 2 AVOAO.~ v' e~~

o~oA l~~ 1= oo

:~ 0.5 PLant scaLe water modeLz Ga5 bUbbung e o

Induction stirrin90.2 vpward A A

Downward v v0.11

2 5 Io 20 50 Ioo 200 500 Iooo2/3

~v,

wattlton'm2

Fig. 10. The experimental work of Ogawaand Onoue63) il-

lustrating the adequacyof e[MLlpL] ~ 0.66 as the scale

up criterion for mixing time estimation.

rate) and a scale factor (MI Ipl) ~0.66. This, after Ogawaand Onoue,63) is shownin Fig. 10. This criterion, it is

instructive to note, produces a considerably different

functional relationship between 8~ and the operatingvariables than those embodiedin Eqs. (5) through (7).

In contrast to this, Asai andcoworkers49) suggested that

water modelexperiments are to be carried out in similar

vessel geom~try in order to establish a functional

relationship amonge~, Iiquid depth and mixing times.

Furthermore, multiplying the results by 1.9 [=(pF./p~)0.33], the mixing times in the corresponding liquid

steel system can be estimated. Such arguments werehowever, not supported with any experimental measure-ments. Onthe other hand, Neifer et al.60) suggested that

the following correlation (in SI units) e.g,,

Pg.B~T~=71.84~Qg

Pg." o.38D2.0L066 .........(8)( -

(pg,"Ipg,B being the ratio of gas density at normal to bathtemperature) provides excellent estimates of mixing times(950/0) in various gas stirred ladle systems therebyessentially indicating that Eq. (8) can be applied to

estimate mixing times in water models as well as full

scale steel systems.

(c) TwoFluid MassTransfer

Chemical reactions between injected gas and bulkliquid, as well as between two immiscible liquids (e.g.,

slag and metal) in gas stirred systems, are the subject ofconsiderable interest and importance to metallurgical

engineers. Thus, numerous experimental studies havebeen carried out and reported to investigate slag-met-

al-gas interactions in gas stirred ladle systems, 'em-bodying widely varying experimental conditions (viz.,

vessel geometry, gas fiow rate, gas injection devices, etc.).

Manyof these studies,48,64,6s) while of considerable

fundamental importance to steelmaking, are not of di-

rect relevance to ladle metallurgy operations since the

experimental conditions (gas fiow rates, nozzle configura-tions, etc.) applied do not exactly correspond to ladle

metallurgy operations. Consequently, an extensive dis-

cussion on these has been avoided.

Chemical reactions between gas bubbles and a bulkliquid has been commonlysimulated53,62,66) using ap-propriate reacting aqueoussystems. Often C02(g) and

9

35 (1995), No. 1

NaOH(1)have been applied to simulate gas and metalphases respectively. Typically, gas injection devices (e.g.,

porous plugs, tuyeres, Iance) were found to have largeeffects on the absorption rates in aqueoussystems. Forexample, at any given flow rate, porous plugs enhance

gas absorption rates significantly53) in comparison toinjection with a tuyere. Aporous plug produces relatively

more, uniformly distributed, smaller size bubbles, Ieading

to an incre~se in the effective interfacial area and thereby,

an increased gas-liquid mass transfer rate. However,owing to the non-wetting behaviour of liquid steel onrefractory pozzles, which lead to the production of muchlarger "blisters" of gas, the situation may be quitedifferent in practice.

In a slightly modified experimental situation, KimandFruehan67) as well as Taniguchi et al.68) investigated the

absorption of gas from the environment to the bath via

the eye of the plume. In these studies, while inert gas

wasbubbled through an aqueoussolution of NaOH,aC02 atmosphere was maintained as the ambient en-vironment above the bath. It wasobserved67) that withincrease in gas flow rates, gas absorption through the

plume's "eye" increased significantly. Furthermore, anupper "slag" phasewasfound to be helpful in minimising

gas absorption in comparison to an equivalent situation

without any protective slag covering.

Watermodelstudies have also been reported on masstransfer phenomenabetween slag and metal in argonstirred ladles. In these, water andorganic fluids (oil,66,67)

benzene,68) cyclohexane,69) etc.) were used as simulatingfluids to represent the metal and slag phases respectively.

Typically, the intensity of slag~netal reactions havebeenstudied by monitoring the transport of a tracer, whichhas an equilibrium partition ratio betweenwater and the

upper phase liquid. In such systems, owing to thedifficulties in estimating the effective slag-metal inter-

facial area accurately under gas stirring conditions, avolumetric masstransfer coefficient (=K~A; K~=aque-ous phase masstransfer coefficient) has often been ap-plied to quantify the experimental results.

Flow visualisation studies reported so far45,46,66~69)

on bubble stirred oil-water systems indicate that the

oil-water interface (viz., slag-metal interface) canundergo considerable disturbances and therefore, pro-duce a host of complex physical phenomena,such asextensive deformation of the interface, formation of oil

ligaments, droplet generation andentrainment, etc. Thus,at relatively low gas flow rates, the interface remaihsrelatively imperturbed but with increasing flow rates, oil

ligaments and droplets tend to form at the water-oilinterface (viz.. Fig. 11) Ieading to a significant increasein the oil-water interfacial area. At still higher specific

flow rates, the entire oil layer breaks downinto numerousdroplets leading to the formation of an oil~vateremulsion. Experimentally measured66)volumetric masstransfer coefficients (=K~A) as a function of gas flow

rates are illustrated in Fig. 12. There, corresponding 'to

these three operating regimes, different functional

relationships are seen to govern interphase massexchangerates. Furthermore66,67) the flow rates at the transition

points in Fig. 12, are a function of the volume as well

C 1995 ISIJ


~1~;:F~;,:~,1

~•~~~~LUME

DIRECTION

WATER

':~c'~'~ ~~~Zi~~~~~'~)~~V'•~/;1~1

f ~~, l/ t

OIL LleAMENT_f FB

INVERSION

Fig. Il. Schematic ofthe interactions betweenthe plumeandthe upper and lower phase liquids.68)

35 (1995). No. 1

\fQE

;~

4.5

4,0

3.5

3.0

2.5

2,0

1,5

1.o

05

oo

O60kwAcc: a '

kwAccQ1.43

kwAacCL2.51

Fig. 12.

-1 1 2 3OIn (Q), L/min

Theexperimental work of Kimet al.69) showing the

variation of K~Aas a function of gas flow rate for

central tuyere injection,

well as interfacial tension (a) in the aqueous and hightemperature systems are likely to be someof the factorsfor such observations.

Entrainment as well as emulsification phenomenabetweenan upper buoyant phaseanda bulk lower phaseduring submerged gas injection operation has beeninvestigated recently by Lin and Guthrie71) using lowtemperature oil/water and oil/mercury analogues. Theseauthors demonstrated experimentally that these dropletentrainment phenomenadependprimarily on the densitydifference (Ap) betweenthe two liquid phases. Thus, for

10wdensity differences, entrainment of the upper phaseinto the lower phasewasfound to be pronouncedwhilefor large density differentials, such entrainment wasfound to be less significant than the dispersion of thebulk lower phase into the upper phase.

E

U)

0.1

0.05

0,01

dc =75 cmhM= 3.7 ~3.8 cmh5L=1.6 cm

*Ql o

Region I : Region II

Q

JIIJIi,t

*,F

Region 111

1 5 10 50 100

Q, cm3/s

Fig. 13. Variation in masstransfer coefficient as a function of

gas fiow rates in a high temperature gasslagmetalsystem

.71)

as the thermophysical properties of the upper phase fluid

and therefore, would vary from one physical systemto another. The experimental studies of Kim andcoworkers,66,67) as well as of Mietz and Oeters69) havealso indicated that an off-centred nozzle position hinders

masstransfer rates vis a vis acentral nozzle position. Thistrend, as onewould note here, is contrary to observations

madeon mixing times, which are found to be accelerated(i.e., reduced) by an eccentric nozzle position in thebath.52,59)

Qualitatively, similar trends of results between masstransfer coefficients and gas flow rates have beenobserv-ed by Hirasawa et al.70) in pilot/laboratory scale high

temperature systems. This is shownin Fig. 13. However,close examination and a comparison between Figs. 12

and 13, particularly for region II, indicate that the func-tional relationship between K and Qare different for

the two systems. Dissimilar density differential (Ap) as

C 1995 ISIJ 10

3. Combined Physical and Mathematical ModellingStudies

The operating conditions (viz., high temperature andvisual opacity of liquid metal, massive size of industrial

reactors, etc.) prevalent in steel plants pose serious

problems for any direct or elaborate experimentalinvestigation. Consequently, mathematical models, in

conjunction with appropriately scaled down physicalmodels, have been a reasonable alternative towardseffective process analysis. To date, a large numberofcombinedtheoretical andexperimental investigations onwidely varying aspects of argon stirred ladles have beencarried out andreported on. In these, considerable efforts

have been madeto develop a reliable and predictive

mathematical frame-work which can be convenientlyextrapolated to investigate relevant phenomenain

industrial systems. These are summarisedbelow undertwo headings: (1) hydrodynamicsand (2) heat and masstransfer phenomena.

(a) HydrodynamicsSzekely and coworkers72,73) were the first to attempt

numerical simulation of turbulent fiow phenomena(embodying the Navier Stokes equations in conjunctionwith the k-Wtwo equation model of turbulence74)) in

the bulk liquid outside the plume region. A physically

morerealistic approach wasthen adopted by Debroy etal. ,7 5) whorecognised the importance of plumebuoyancyrather than shearing and proposed a computationalschemein which the gas liquid mixture contained withinthe two phase region was represented by a homogene-ous fluid of somewhatreduced density (viz., p*i*-oegPg+(1-ceg)pL; ceg being the gas volume fraction). In

essence, an additional term (=pLgoeg) was incorporatedin the axial direction momentumbalance equation to

model the buoyancy force generated by the differencesin density between the bulk single phase and plume twophase regions. This procedure,75) requires a priori

specification of the field distribution of gas volumefractions in the plume within the calculation domain,thereby allowing computation of hydrodynamicvariables

and turbulence parameters within the entire system. The"quasi single phase calculation procedure", also known


Table 4. Expressions of gas volume fractions in quasi single phase calculation procedure as applied by various investigators.

InvestigatorGeometryof the two

phase plumeExpressions for gas volume fraction, ee Remarks

Debroyet a/.75)

Szekely et al.77)

Sahai and Guthrie76)

Mazumdarand Guthrie79)

Balaji and Mazumdar80)

Castillejos et al.1 14)

Empirically determinedand specified a priori in

the calculation procedure

ec:=

e,;=

121r

Q

o

r(u + Us)dr

Q-Icr~ Usoe(1 -

oc)

~c27~rUzdr

L

cc=

Q'Up

7rr*~L

r*. is the radius of an equivalent volumecylindrical shapedplume.

L

oc=

Q'Up

7rr"2~L

Q-7~r~.U*ec(1

-cc)

oc:=

p

~auU 2lcrdr

ecmax:=

rmax/2

'r r 24oc

expL~O' 7

oemaxetc' are correlated to modified

Froude No', density ratio, nozzle

dimension etc'

Both slip and no-slip models wereapplied.

U. estimated by solving a separateordinary differential equation

Noslip model: constant centreline

velocity applied as a boundarycondition

Noslip model

Slippage assumed:UF=4.5Ql!3L1/4R ~ 113

Gasvolume fraction within the plumeestimated on the basis of empirically

determined correlations

as the "single phase variable density formulation", hasbeenpopular amongmathematical modellers. Numerousinvestigations,7 5~ 8l) embodyingsuch an approach, havebeenreported in the literature, Nonetheless, in estimating

o(g, different concepts and procedures have been applied

by the various investigators. These, as illustrated in Table4, have resulted in different versions of the quasi-single

phase calculation procedure,In almost all the mathematical modelling studies

reported to-date,75 ~ 81) major approximations havebeenapplied (e.g., flat free surface, conical plumeshape, etc.),

to effectively modelthe fiow phenomenain the gas stirred

bath. Parallel to the numerical computations, experi-

mental measurementson liquid velocity fields via video77,7 8) wererecording76,79) andLaser Doppler Velocimetry

also carried out. Comparisonsbetween such measure-ments and corresponding predictions have generally

indicated that a quasi single phasecalculation procedure,in conjunction with the standard coefficient k-8 modelof turbulence,82) provides estimates of flow parametersin the bulk of the liquid, that are in reasonable agreementwith those measured.This is shownin Fig. 14. In contrast

to this, the correspondence between predicted (via the

standard coefficient k-8 model of turbulence) andexperimentally measured turbulence parameters (e.g.,

r.m.s, of the fiuctuating velocity components, Reynoldsstress components, etc.) was found to be less sat-

isfactory. On the basis of their detailed LDVmea-surements, Grevet and coworkers77) pointed out that

Fig. 14.

e +0,i

, +0.1

oh/Z=0.8 - 0.1

,e +0.l

o, 0.6 -0.1

,

,+0.:

o0,4 -o.:

oe

,+0.1

o,

o. 2 -0,l

1o 0.8 06 0.4 0.2

+0 2+0.1

+0.1

+0.1

-0.1

+0.1

-0.1

o

Ito

E:h

uoa,

>

r/R

• Ex~erimental Predictedh/z = Dimensionless bath depth

Comparisonof experimentally measuredand theo-

retically predicted vertical velocity component76)at

different depths in a water model ladle.

the standard coefficient k-8 turbulence model was in-

adequate to realistically simulate turbulence phenomenain such gas stirred systems. In a later and more recent

study, Mazumdarand coworkers81) expressed similar

11 C 1995 ISIJ


apprehensions. However, despite the uncertainties

associated with predicted turbulence quantities, com-puted results, together with experimental measurementsshownin Fig. 14, indicate that for reasonably accuratemodelling of flow phenomena, precise modelling ofturbulence is to someextent secondary. These observa-tions76,77,79) therefore, appear to suggest that bulk flowsin gas stirred ladle systems are largely dominated byinertial phenomena.

Parallel to their numerical model study,76) 'a

macroscopic model was proposed by Sahai andGuthrie83) to predict plume rise velocity and averagespeed of liquid recirculation in gas stirred ladles as afunetion of operating variables. Appreciating the

relevance of hydrodynamic coupling between widelydispersed, Iarge rising bubbles andentrained liquid withinthe upwelling plume, Sahai and Guthrie derived fromthe first principles (via an energy balance in which, the

rate of energy supplied by the rising bubbles wasequatedwith the turbulence energy dissipation losses within the

system) an expression for estimating the average rise

velocity of the bubble plume, Up, according to (in SIunits) :

Up~i 4.5 (Q0.33L0.25)/Ro 33.......

...(9)

in which, Qis the gas flow rate corrected to the meanliquid temperature and height, L is the depth of liquid

and R is the radius of the vessel. In their analyticaltreatment,76) energy dissipation via bubble slippage phe-

nomenawas entirely ignored. Similarly, through con-sideration of their ownand others43) experimental data,

Sahai andGuthrie provided the following correlation for

estimating the average speed of bath recirculation, U, as

a function of operating variables (in SI units), viz.,

~=

JoLJoR(u2+v2)o52lrerdrdz

o86 ' Q0'33r0'25D-0'58' L l\foR

21Lrdrdz

.(10)

Subsequently, Mazumdarand Guthrie79) modified the

plume equation83) to model flows in ladles stirred with

a partially submergedvertical lance. To this end, analternative form of the plume equation was proposedaccording to (in SI units):

(pQ)0.33L0.25.........

.(1 1)Up~;4.5R0.33

in which pis the fractional depth of lance submergence(e.g., O~p~1). It is important to note here that in the

absence of any differential model solution, the siinple

algebraic relationships embodiedin Eqs. (9) through (1 1),

can provide a reasonable description of the state ofmotion/agitation under a given set of operating con-ditions.

Experimental studies on ladle flows in conjunction with

more elaborate mathematical models have also beenreported.60,84,85) For instance, Johansenand Boysan84)

used a combined Lagrangian-Eulerian two phasecomputational approach for bubble stirred ladles.

C 1995 ISIJ 12

35 (1 995). No. 1

Instead of the plume dimension and gas volume frac-

tion being specified a priori, these authors set up acomputational procedure to predict the distributions of

gas volume fraction, Iiquid velocity, and fluid turbulencein an axisymmetric gas bubble driven system. In this, anordinary differential equation describing bubble motionwas solved numerically in addition to the liquid phasecontinuity and equations of motions (see also later).

Basedon their experimental findings,44) a modified formof the standard coefficient k-8 model was applied, in

which, for the first time, the effect of turbulencegeneration by bubbles within the plumeregion wastakeninto account via an additional source tenu. It wasdemonstrated that predicted flows and isotropicturbulence fluctuations within the plume and elsewherein the bath, comparewell against ~quivalent experimentalmeasurements.

Schwarzet al. 85) Iater reported a combinedtheoretical

and experimental investigation of flow phenomenain alaboratory scale, nitrogen stirred, I kg carbon saturatedbath of iron at 1400'C. A PHOENICS86)based codeincorporating a two phase computational approachembodyingthe standard coefficient k-e turbulence modelwasapplied to predict flow phenomenaand turbulencequantities in the reactor as a function of nitrogen flowrates. Parallel to these, attempts were also madeto

measureplumevelocities by carrying out experiments onthe isothermal dissolution of steel rods within the twophase plume region of the iron bath. Significant

differences between experimental plume velocities andthose predicted via their two phasemodelwere reported.

In view of such discrepancies, modelimprovements, such

as morerealistic descriptions of interphase friction (e.g.,

drag coefficient,87) etc.) and turbulence phenomenawereadvocated by those authors.

(b) Heat and MassTransfer PhenomenaIn addition to the studies discussed in the preceding

section, many combined physical and mathematicalmodelling studies have also been carried out during thelast two decades to investigate numerousheat and masstransfer phenomenaof relevance to ladle metallurgysteelmaking opeirations. These, for example, includestudies on such phenomenaas, mixing, slag-metalreactions, melting anddisso~ution of solids, and so forth.

These studies are summarised in the two sub-sections

below.

(i) MixingExperimental aspects of liquid mixing phenomenain

gas stirred ladle systems have been elaborated upon in

a previous section. So far as mathematical modelling ofmixing times is concerned, different approacheshavebeenadopted by researchers. Agoodsummaryof the variousprocedures available for calculation has recently beenpresented by Mietz and Oeters.88) Froma conceptualstand point, two different approacheshave beenapplied

and these, for example include, modelling via numericalsolutions of the species conservation equation56, 59, 89 ~ 92)

and the circulation time model.53,5 7,93,94) In conjunctionwith mathematical modelling, experimental measure-


ments of mixing times in water models and industrial

argon stirred ladles were also carried out, some ofwhich haveagain, already beendiscussed. Suchmeasure-ments were often applied to validate model predic-tions.53,56,s7,59)

In the presence of a two dimensional velocity field, the

massconservation of an inert tracer i (e.g., mi) can beexpressed in terms of, say a cylindrical polar co-ordinate

system, via the following convectioniturbulent diffusion

equation:

a I aa-(pmi) + (pumi) +r ar

(prvmi)azat

=eaz (T aa~~i )++ ,( ami

...(12)Tr ..ar

Theeddy diffusivity, DE(=Tlp), and the eddy kinematicviscosity, VE (=pElp), are conventionally taken to benumerically equal. Fromthe view point of engineering

calculations, the assumption of equality (viz., Sct =DE/vE) ~i 1) has proven to be reasonably adequate for a large

variety of turbulent flows. It is therefore apparent that

provided the fiow parameters (u and v) and turbulence

quantities (uE) are known with reasonable certainty,

mixing times [viz., mi(z, r, t) fields] canbe fairly accurately

predicted since physically realistic andaccurate boundaryconditions can be applied to Eq. (12). This has been apopular approachs6,59,89-92) that has been applied in

numerous mathematical modelling studies in which

numerically predict mixing phenomenain aqueous aswell as industrial gas stirred ladle systems are predicted

ab initio.

In a muchearlier study, Szekely et al. 89) reported acombined theoretical and experimental investigation ofliquid mixing phenomenain industrial scale argon stirred

ladles. Mixing rates in three different size industrial ladles

(viz., 7, 40 and 60 t) weremeasuredfor various operating

conditions by monitoring a transient local concentration

of a radio active tracer addition. In conjunction withtheir experimental work, they predicted mixing times

through solution of Eq. (12) via a two dimensional el-

lipitic, turbulent flow model.72,73) Despite somesim-

plifying assumptions,72,73) experimental me~surementsand numerical predictions were found to be in rea-

sonable agreement. This is illustrated in Fig. 15.

Similar observations on water models of argon stirred

ladle havealso beenreported by Ying et al. ,90) Salcudean

and coworkers,91) Mazumdarand Guthrie56) and morerecently by Joo and Guthrie.59) Computational studies

of mixing phenomenain industrial size and water modelladles confirm that gas flow rate and vessel geometryaffect mixing considerably56,s9,89-91) and that mixing

rates vary considerably from one location to another in

such gas stirred vessels.56,59)

In a subsequent investigation,92) El-Kaddah andSzekely improved their earlier hydrodynamicmodel72, 73)

to predict the turbulent fiow fields in 6and40 t industrial

ladles. Embodyingthe predicted flow parameters andturbulence quantities in the species conservation equation(viz., Eq. (12)) and invoking some thermodynamicequilibrium relationships, desulphurisation kinetics in the

35 (1995), No. 1

eo*\o

3

2

1

o

,

,

,et

e ExperimentaLPredicted

_'• __ '__~__-1-

13

O 4 12 168Time, min

Fig. 15. Comparisonbetween numerically predicted and ex-

perimentally measuredmixing times in a7t gas stirred

ladle.89)

industrial argon stirred ladle were numerically predicted

and reasonably good agreement with experimental

measurementsdemonstrated.

Mixing phenomenain gas stirred ladles designed fof

single and dual plug bubbling operations was reported

recently by Joo and Guthrie.59) Their59) computationalresults and experimental observation confirmed that as

the bubblers are movedoff-centre, angular momentaincrease, reducing mixing times considerably. It wasfound that a mid-radius placement of a porous plug

represents an optimumlocation for single plug bubbling,

while diametrically opposed, mid radius placement of

bubblers were found to be best for dual plug bubbling.These56,s9,89 - 92) numerical and experimental modellingexercises indicate that mixing phenomenain gas stirred

ladle systemscanbe reasonably accurately predicted fromfirst principles.

Analternative to the differential modelling approach

to investigate mixing phenomenain gas stirred ladle

systems was first applied by Sano and Mori.57) Thecalculation procedure, commonlyknownas the "circula-

tion time model", is essentially basedon the concept that

the circulation time is proportional to the mixing time.

Throughan energy balance (e.g., at steady state, the rate

of potential energy supplied by the rising bubbles is

balanced by turbulence energy and interphase frictional

dissipation losses), SanoandMori derived an expression

for estimating circulation times in terms of operating

variables. Furthermore, by assuming "mixing times" to

be equivalent to three times the circulation times (e.g.,

equivalent to a degree of mixing equal to IOO~5o/.),

mixing times in gas stirred cylindrical vessels containing

molten iron were predicted mathematically and com-pared with experimental measurements.The calculated

results were however, found to agree only roughly with

measurements.In a later study, the sameapproach wasalso adopted by Stapurewicz and Themelis53) to in-

vestigate mixing phenomenatheoretically.

Asimilar approach57) wasadopted by Krishnamurthy

et al. 93) to theoretically investigate mixing phenomenain gas stirred baths. Onthe basis of manyexperimentaldata54,55) derived from water model studies, Krishna-

murthy et al.93) suggested that circulation number

C 1995 ISIJ


(=circulation time/mixing time), in contrast to the earlier

supposition of Sanoand Mori, is not a constant andcanrather assumea value between2and 12(for any givendegree of homogeneity). In a subsequent study,Krishnamurthy94) applied a "tank in series model" to

compute the circulation numberand hence, the mixingtime. Model predictions were assessed against experi-

mental measurements54,55) and reasonable agreementclaimed. However, for high temperature industrialsystems,47) agreementwasfound to be less satisfactory.

The models of mixing time adopted by Sano andMori,57) Stapurewicz and Themelis,53) as well as byKrishnamurthy and coworkers,93'94) are somewhatsimplistic. Although, through tuning of some keyparameters in the modelequations, the "circulation timemodels" can be madeto describe a particular set ofexperimental conditions, nevertheless, such empiricalmodels, in general will have only limited utility, sincethese cannot be readily extrapolated with muchconfidence to other experimental configurations. Theconcepts applied in the formulation of the modelequations are fundamentally weak since these do notincorporate the physics of the mixing process (e.g.,

that macro-mixing process via bulk convection, turbu-lent diffusion and molecular diffusion phenomena)ade-quately. Thus, Mietz and Oeters88) suggested that therecirculating flow, turbulent diffusion from the tracerenriched cloud and the massexchangebetweenthe deadzone and the remaining volume of liquid in tanks areparameters that have to be considered in such "tank in

senes" or "recrrculation time" models.

(ii) Solid-Liquid Interactions

Thermal and mass interactions between solids andfluids are acharacteristic of manyprocesses (viz., alloyingaddition, powder injection and so on) that are carried

out in refining ladles. Overthe years, several experimentalstudies9 s- I oI )havebeenre ported andnewempirical heatand masstransfer correlations have been developed toestimate melting an!d/or dissolution rates in gas stirred

ladles. These investigations,95~98) have in general, ap-peared to indicate that in turbulent metal processingsystems, given that the intensity of turbulence is oftenappreciable, classical correlations for translatory fiow areinsufficient to connect the hydrodynamic transportphenomenato process rates. Thus, in order to estimateheat and mass transfer rates in gas stirred systemseffectively, functional relationships embodying the

combined influence of flow and turbulence have been

proposed.Szekely et al.89) investigated the dissolution rates of

graphite rods dipped into industrial size gas stirred ladles.

To estimate the mass transfer coefficient theoretically,

the correlation proposed by Lavender et al.102) wasapplied e.g.,

Sh 20+072 Re05Re~•25Sc0.33...........(13)

Acomparison between observed and predicted rates ofdissolution89) (via Eq. (13)) is shownin Fig. 16. There,despite the simplicity of the turbulent flow model89) andthe difficulty in obtaining reproducible measurements

10

Ig5v

r~'

o.2xc'

EOJ:: 1

o

Predicted

e ExperimentaL

,e e

Fig. 16.

10 20 50 100 200 5003 -1O, dmnmin

Comparisonof experimentally measuredand numer-ically predicted masstransfer coefficient as a functionof gas flow rates in the vicinity of the free surface in

an industrial ladle.89)

under industrial conditions, the agreement betweentheory and experiment appears to be very satisfactory.

Acomprehensive theoretical and experimental inves-tigation on melting phenomenain turbulent recircula-

tory gas bubble driven aqueous and high temperaturesystems has been reported by Taniguchi and cowork-ers.96,97) Theseauthors measuredthe melting rates (e.g.,

Nu) of ice spheres using a photographic technique andcomparedthese with estimates derived from three dif-

ferent heat transfer correlations reported in the litera-

ture. Such comparisons96) demonstrated explicitly thatthe classrcal "Ranz Marshall" correlation (viz., Nu=S+0.6 •

(Re)o.s(pr)0.33) underpredicts heat transfer ratesin gas bubble driven systems by about 40 to 50 olo. Onthe other hand, the heat transfer correlation proposedby Whitakerl03) (e.g., a modified version of the Ran2hMarshall correlation and applicable to turbulent flowsituation) viz.,

(Nu 2) (04 Reo5+006 Reo66)pro.4 'xb 0.25

'l o.(14)

wasfound to produce better estimates andwastherefore

moreappropriate and adequate for investigating meltingphenomenain both aqueousand high temperature gasstirred baths.

In a later study, Szekely, Grevet and El-Kaddah,95)

went on to propose a correlation (applicable in the rangeof values for Re(between 100and2OOO)and Tu(~: O. 15))

for estimating melting rates of ice cylinders in gas bubbledriven aqueoussystem according to:

Nu=0.338(ReTu)0.8pro.33 .......(15)

In order to estimate various non-dimensional numbersand hence the corresponding melting rates, a photo-graphic technique in conjunction with a turbulent flowmodel77) wasapplied.

An similar approach,95) was adopted by Mazumdarand coworkers,98,99) for estimating mass transfercoefficients for the dissolution of benzoic acid c. ylinders

in agas stirred aqueousbath. Onthe basis of experimental

measurementsand numerical computations. Mazumdaret al.98) demonstrated that masstransfer coefficients for

vertically submergedcylinders in awater modelladle can

C 1995 ISIJ 14


be adequately expressed via a similar correlation of the

type expressed via Eq. (13), e.g.,

Sh O73 (Reio")o 2s(ReT)0.32Sc0.33.

..(16)

in which the nominal and turbulent Reynolds numbers(Re andReT)were the local flow variables anddefined

io','

as (dp(u2 +v2)o.5/v) and (dpa/v) respectively. In thosestudies,89,99) the meanand fluctuating velocity scales

embodied in the definition of nominal and turbulence

Reynolds numberwere estimated theoretically from the~1~~~~k-8 modelof turbulence (e.g., a= 0.66 k). In a separate

study, Mazumdaret al.104) extrapolated the aboveequation and demonstrated that isothermal dissolution

rates of steel cylinders in a carbon saturated iron melt

can be predicted with reasonable certainty via Eq. (16).

Theseinvestigations,98,99, I 04) together with manyothers,

are therefore indicative of the fact that appropriatelydesigned water model investigations can be reasonableand effective alternatives for studying various high

temperature phenomehain ladle refining operations.

Solid and particulate dissolution rates in laboratoryscale, gas stirred iron-carbon melts were reported byWright and coworkers.loo,lol) Mass transfer rates for

the isothermal dissolution of steel rods were measuredin two different sized (1 and 25 kg) bathsloo) for a wide

range of gas fiow rates. It wasconfirmed experimentallythat gas injection enhances the rate of dissolution

considerably. In their subsequent study,lol) recarburisa-tion phenomenain a gas stirred bath were investigated

via multi-particulate dissolution experiments. To com-pare measuredandpredicted particle masstransfer rates,

Wright and coworkerslol) applied the following cor-relation, derived from Kolmogorofrs theory of local

isotropy, e.g.,

ed~ )0.25Sco33Sh=2+0.4 ...........(17)

It was demonstrated that Eq. (17) can adequatelysimulate multi particle dissolution phenomena(e.g.,

recarburisation etc.) in high temperature gas stirred

system.Asai and coworkersl05) have compiled and discussed

someof the reported masstransfer correlations from the

viewpoint of their applicability to ladle refining

operations. Oneof these correlations, viz.,

Sh=0.079 • Re0.7Sc0.356......

.....(1 8)

wassubsequently applied by Korial06) to estimate liquid

velocity (e.g., Re) from measureddissolution rates (e.g.,

Sh) of solid oxalic acid compacts. The fluid velocities

estimated from the measured dissolution rates werecompared with corresponding theoretical estimates

obtained from simplified macroscopic relationships76)

and found to be in excellent agreement. This appears to

contradict the findings of many other investiga-tions,95 ~ 99) since the object Reynolds numberalone hasbeen shownto be inadequate to simulate observed heat

andmasstransfer rates in gas stirred ladles. Nevertheless,in view of the relatively small size of the vessel (L =O. 15mand D=O. 15 m)106) applied in conjunction with low gas

35 (1995), No. 1

flow rates,106) it is not unlikely that the intensity ofturbulence in that system was small and therefore, acorrelation (essentially valid under laminar flowconditions) embodying only object Reynolds numberproduced reasonable estimates of dissolution rates. It

is also useful to note that the intensity of turbulencewithin a gas stirred ladle is strongly spatially depen-dent.76,77,79,83) Furthermore, fiows across the ladle's

base and regions deep within the ladle are amongthe

slowest. Consequently, the regions in which dissolutionis measuredmust greatly affect the conclusions drawn.

The subsurface motions and trajectories of solid

additions to gas stirred ladle systems were investigat-

ed both theoretically and experimentally by Mazumdarand Guthrie.107) Their studyi07) suggested that underindustrial conditions, buoyant additions, irretpective

of their size, have practically no chance to undergoprolonged subsurface motion but rather almost in-

stantaneously resurface. Similarly they found that heavieradditions would always sink to the bottom of the vessel.

In contrast to these, neutrally buoyant additions (e.g.,

somegrades of ferromanganese), have the potential to

undergo prolonged subsurface motion, and therefore

melt within the bulk metal bath. Onthe basis of their

physical and mathematical modelling studies, it wassuggested that buoyant additions such as Al, FeSi,

etc. would, in conjunction with variable amounts of

carry-over slag, exhibit poor and erratic recovery rates,

if added to ladles during argon bubbling operations. Inthis context, these authors highlighted the role of the

C.A.S (~omposition Adjustment through ~ealed argonbubbling) procedure. I08)

4. Mathematical Modelling Studies

Extensive numerical modelling of various aspects ofladle refining operations e.g., hydrodynamics,80,81,84,

l09- 115,1 18,ll9,124- 126) heat and mass transfer,60,120-

124) turbulence phenomenal16,1 17, 127~ 129) etc., has beencarried out and reported during the last two decades. Asalready mentioned, hydrodynamic modelling of gasstirred ladle systemshas beenessentially carried out usingthree separate approaches, namely, (i) the quasi single

phase or the single phase variable density formulationprocedures,80,81,109-114) (ii) the Lagrangian-Eulerian

two phase approach60,84,i26) and (iii) the Eulerian twophasell6~125) models. Of these, in terms of computa-tional complexities, the quasi single phase procedure is

by far the simplest. In contrast, the Eulerian two phasemodelentails major computational efforts and these aresomewhatcomplex. Thus, except for the studies reported

by Lai and Salcudean,124,i25) practically all the twophase simulations of gas stirred ladle systemsll5~123)

embodyingthe Eulerian approach have beencarried outusing the commercial PHOENICS86)computer code.

Coupling the hydrodynamic models with appropriate

statements of mass and thermal energy conservation,

numerical computations of thermal and material mixing

were also performed. Furthermore, in some of the

computational studies, different versions of turbulence

viscosity (ranging from an algebraic to two equation

15 C 1995 ISIJ


turbulence models) were applied. Computational studies

concerned with the mathematical modelling of mo-mentum,heat and mass transfer phenomena,and notaddressed so far, are discussed below.

Table 5 presents the structure of the three hydro-dynamicmodelling techniques illustrating their commonelements and distinguishing features. As seen, the liquid

phasecontinuity andmomentumconservations, togetherwith the k-e modelandthe systemof associated boundaryconditions represent a commoncore. The probleminvolves body force terms induced by gas injection (as

well as phase volume fractions for the two phase for-

mulations) and therefore, further inputs are neededfor closure. To this end, in the two fluid model,115~125)

a set of additional partial differential equations governingthe conservation of massandmomentumof the gas phaseare coupled to the core model, while in the Lagrangiantwo phase approach,60,84,126) and ordinary differential

equation describing the ttajectory of single rising gasbubbles is applied. In contrast to these, a set of auxilliary

relationships (in the form of algebraic equations) togeth-

er with the dimensions of the two phase plume havebeen applied in the quasi single phase calculation pro-cedure.109-114) Consequently, the distribution of gasvoidages, plume shapes, etc., which are integral parts ofthe solution of the two phase calculation procedures arespecified, a priori, in the single phase model.

The liquid phase massand momentumconservationequations in the core mathematical model, as a.pplied to

Table 5. General structure of the mathematical models asapplied to the hydrodynamic simulation of gasstirred ladle systems.

CommonelementsModel category

(core model)Distingurshing elements

Eulerian two phase Liquid phasemodelsl 16~ 12s) continuity and

momentumbalanceequations

Lagrangian two phase +model60,84, 126) Theturbulence model

Quasi single phase +models8o, 81, I o9- 114} Appropriate

boundary conditions

Gasphase continuity

and momentumconservation equations

Bubble trajectory

equation

Plumemodel; Drift

flux model andempirical plumedimension

Table

these calculation procedures are very similar apart fromthe body force term embodiedin the momentumbalanceequations. These forces are derived from different

standpoints andas shownin Table 6, havewidely varyingexpressions. Furthermore, descriptions of body force

terms, summarised in Table 6, show that all threemodelling procedures require a numberof parameterssuch as bubble size, plumedimensions, drag coefficients,

etc., that need to be specified "a priori". Similarly, in

Table 7, various drag coefficient-Reynolds numberrelationships that have been embodiedin the two phasemodelstudies are presented. Expressions for body forces

anddrag coefficient shownin Tables 6and7indicate thatthe accuracy of numerical predictions should, for eachcategory of model, be sensitive to an appropriate set ofchoices of values for the various empirical parameters.

With reference to the various boundary conditionsapplied to the hydrodynamic models, considerableidealisations have been applied, particularly in thevicinity of the liquid's free surface and the location ofthe gas injection nozzle (for two phase calculationprocedures). In all three categories of model to date, free

surfaces have always beenassumedto be essentially flat.

As such, the presence of a spout, as well as waves, havealtogether been ignored in computational procedures.Similarly, for the two phase models, free stream gasvelocity, regardless of the modeof gas injection (viz.,

porous plug/tuyere/lance), been commonlyapplied as

one of the relevant boundary conditions to the gasphase/bubble trajectory equations. All these aspects areof concern, particularly whenfull scale predictions areto be made.

A recent analysis by the present authors summarisedelsewherel30) indicates that it is within the plume regionand predominantly in the vicinity of the free surface andthe gas injection plug, that the three modelling proceduresproduce somewhatdifferent estimates of fiow velocity,

gas volumefraction, etc. However,since the plumebarelyoccupies 2 to 3"/o of the reactor volume under ladle

refining conditions, any variations in computed rise

velocities within the plume (particularly in the vicinity

of the free surface andplug/nozzle), are refiected by onlymarginal variations within the bulk of the liquid.

Consequently, predicted fiows, particularly outside the

two phaseplumeregion for all three categories of models,correspond reasonably well with experimental measure-

6. Characteristic expression of the body force in the liquid phase momentumconservation equations as applied in thethree different calculation procedures.79,84, 123)

Modelling procedureExpression for volumetric body in the liquid phase

momentumequationsEmpirical parameters in the expression of

body force

Quasi single phase79) PL90e; C, =Q- It:ravUsOe(1 -

OC)

p

~auU 27Trdr

Slip velocity, U*(f(db)) and the plumedimension, r~~

and Up=4.5Q0.33L025R025

Lagrangian two phase84)

Eulerian two phasel23)

Q ;=:f:~d~

' 3CDPRe(Vh v)dt~=

NAVCf(Vb-v)' C=f(Re d C " ')

' f ' b' D'

Bubble size db and the, drag coefficient CD

Bubble size db and the drag coefficient CD

C 1995 ISIJ 16


Table 7. Drag coefficien~Reynolds numberrelationships as applied in various two phase hydrodynamic models of bubblestirred ladle.

Investigators Drag coefficientReynolds numberrelationships Remarks

Johansenand Boysan84)

Turkoglu and Faroukl20) C -D~24

Red

CD;:

0.622

+0'235)1Eo

(1 +0.15 - Re~.6B1)+0.42

(1+ Re~

16)42 500

Valid for 500~Red~5OOO:Eo=Eotvosnumber

Rd is the dispersed phase volume fraction.

and

Lai and Salcudeanl24)

Ilegbusi and Szekelyll6)

Neifer et al.60)

CD=:24

Red

CD=2.66 (1 -Rd)2

CD=48/Red

(1 +0,15 - Re~.687) +0.42

(1+ Re~'16

42 500

CD=al +a2Re~+a3Rend

Red=2RUpl,t

FromRef. 87)

al, a2 ' ' ' etc. are function of Red.

\E~Vo)

~~

0.6

0.5

0.4u'

\E 03

g 0.2

a'

>~ 0.1

.~

O

-0.1

1.O

0.75

0.50

0.25

o

-0.25

- 0.2

o

Fig. 17.

o

\\

\.\

\\

\\

\

1L

\

1

Ca)2phase Euterian approach

-.- 2phase Lagrangian (lpproach

1---- auasi single phase approach\ e Experimenta[~\\\

z/H = 0,83.~\\

\\

=~~\,

\\\

\\,

\

o

\

.2 0,4

Dimension[ess

O.e

radia[

0.8

distance1.O

(b)

2phase Euterian approach

---- Quasi singLe pha5e approoche ExperimentaL

z = 300 mm+1L

~ ~1~-

e ~

0.05 0.20 0.2S0.150,10

Radia[ position, mComparison of experimentally measured and nu-merically predicted velocity distribution in a gasstirred aqueousbath illustrating the relative effective-

ness of three different hydrodynamic modelling pro-cedures. 131)

ments, as illustrated in Fig. 17.

As pointed out already, procedures adopted to mod-el turbulence phenomenaare also likely to introduce

someuncertainties in the predicted results. To assessthis, the role of bulk effective viscosity modelsi27,128)

and differential models of turbulence74,82) have beeninvestigated in numerouscomputational studies. Themuchearlier work of Mazumdarand Guthrie79) as well

as of Grevet and coworkers77) indicates that bulk fiowsin gas stirred ladles are largely dominated by inertial

rather than turbulence viscous forces. The more recentstudy by Shengand lronsl29) goes onto showthat flowswithin the two phasezone itself are in fact quite sensitive

to the choice of turbulence modelsused in the calculationprocedure. Similarly, Salcudeanand coworkers91) notedthat a bulk effective viscosity formula simulates the

mixing process in a water model ladle better than thek-e turbulence model. A detailed analysis of such di-

vergent views expressed by various investigators hasbeen presented by Mazumdarand coworkers81) in arecent publication. There, it was demonstrated that

assumptions of bubble slippage (or no slippage), theapplication of wall functions with a bulk effective

viscosity formula (or no wall functions at all), etc. canall have a significant effect on computed results andtherefore, conclusions drawn from mathematical modelstudies are a function of these considerations. Thecomputational results of Mazumdaret al. 81) indicatedthat as far as predictions of flow phenomenaareconcerned, the k-8 model, together with bubble slippage,is adequate for all practical purposes and providesreasonably accurate estimates. Nevertheless, on a local

basis and particularly within the two phaseplume, somedifferences betweenmeasurementsandpredictions mightexist, as the recent work of Shengand lrons39) appearsto indicate. In contrast, considerable differences betweenvarious predicted and experimentally measuredturbu-lence quantities (e.g., intensity of turbulence, turbulencekinetic energy, etc.) have been reported.81) As shownin

Table 8, predictions andmeasurementsof various quanti-ties are different by considerable limits. Similar observa-tions have beenmadeby other investigators.77,129)

Despite the inadequacies of the k-e turbulence model,the large numberof studies reported in the literature and

17 C 1995 ISIJ

Table


8. Numerically predicted (via standard coefficient k8 turbulence model) bulk average values of various turbulencequantities in a gas stirred bath (L =0,21 m, D=0,15m) as functions of gas flow rates and their comparison with

corresponding experimental measurements.81)

Gasflow rate

x 105, m3ls

Turbulence kinetic energyx 104, m2/s2

Turbulence intensity Turbulence energy dissipation rate

x 104, m2/s3

Experimental Predicted Experimental Predicted Experimental Predicted

l .66

3.33

5.0

0.8242.0

2.39

7.3

8.8

14.6

0.218O.2540.236

O.51

0.520.52

6.23

12.5

18.7

3.95

5.9

ll.7

discussed already, indicate that realistic predictions ofheat and masstransfer rates can be derived via the k-gmodel embodied in a turbulent flow cornputationalprocedure. Towards these, it is important to recognisethat heat and mass transfer rates do not dependsignificantly on the predicted turbulence kinetic energyor turbulence intensity (e.g., for example, from Eq. (15),

it is seen that Kocu0.32). A simple calculation indicates

that uncertainties in predicted turbulence intensity values

by a factor of 2(e.g.. Table 8), are only likely to introducecorresponding uncertainties in predicted mass transfer

coefficient values, K, of about 11o/•. Similarly, if it is

assu~nedthat the specific energy input rate is proportion-al to the rate of turbulence energy dissipation losses

(the constant being the efficiency of turbulence gen-eration/dissipation), ohe would then readily derive fromthe macroscopic mixing time correlations, that errors in

the predicted 8 values to the extent shownin Table 8,

would induce muchless error in predicted mixing times.

However, errors of such magnitude in predicted results,

as one might anticipate, are well within the commonrange of uncertainties inherently associated withcorresponding experimental techniques.

It is worth noting that no realistic attempts have yet

beenmadeto model flows in the presence of an overly-

ing slag phase.131) Also, most computational studies

havebeenreported for essentially isothermal conditions.

Free convection phenomenahave been considered in

only a handful of cases60,91,1 14.122) together with atten-

dant thermal stratifications and also a study on gas-liquid heat transfer.120) Furthermore, although com-puted results were presented to illustrate the influence

of operating conditions on these and related aspects,

most often such results have not been validated

against experimental measurements.Nevertheless, thesestudies60,91,114,122) appear to suggest that temperaturedistribution in gas stirred ladles can be described

reasonably via the standard convective heat transfer

equation having a: form essentially similar to Eq. (12).

5. Concluding Remarks

Manytheoret.ical andexperimental investigations havebeen carried out on the subject of gas injection in ladle

metallurgy steelmaking operations. Symmetrically andasymmetrically placed lances/tuyeres/porous plugs havebeenused as gas injection devices and studies have beenreported on both gas-liquid, as well as gas-liquid-slag,

systems for wide range of vessel geometries and gas fiow

C 1995 ISiJ 18

rates. Theseindicate that under ladle refining conditions,

apart from the vicinity of the nozzle, well dispersedspherical caps bubbles can be expected in the two phaseregion. Over most of the two phase zone, nozzleconfigurations have little influence on bubble and liquid

rise velocities, gas volume fraction distributions, etc. andare therefore not critical to the overall flow recircula-

tion produced. The rising gas liquid plume induces arecirculatory motion of liquid within the vessel, whichtypically has its "eye" Iocated in the upper quadrant ofthe ladle anddisplaced towards the side wall. Undersuchconditions, the bottom of the vessel can be expected tobe relatively quiescent. Furthermore, the intensity ofliquid motions and liquid mixing in the vessel arerelatively sluggish in the presence of an upper secondphase liquid in comparison to an equivalent no slag

situation.

Theinfluence of operating variables (viz., gas injection

rate, vessel geometry, Iocation of gas injection nozzle,etc.) on bath hydrodynamics and associated transport

processes has beenextensively studied and the influence

these variables exert under practical ladle refining

conditions are nowknownwith reasonable certainty.

Similarly, extensive mathem!atical modelling of fluid fiow

and the associated transport phenomena(viz., mixing,slag-metal reactions, solid liquid masstransfer, etc.) andthe validation of model predictions against laboratory

as well as plant scale experimental data indicate that areasonable mathematical frame work now exists toeffectively carry out design and process analysis cal-

culations in refining ladles.

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M. J. McNallanandT. B. King: Metall. Trans., 13B(1982), 165.

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