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ISIJ International, Vol, 33 (1993), No. 1, pp. 224-232
Physica l Model of Slag Foaming
Yuji OGAWA.Didier HUIN.1) Henri GAYE1)Naokl TOKUMITSUSteelmaking Process. Process Technology Research Laboratories. Technical DevelopmentBureau. Nippon Steel Corporation,Shintomi, Futtsu. Chiba-ken, 299-12 Japan. 1) Physical Chemistry Department. IRSiD, Voie Romaine. BP320.Mazieres-les-Metz. 57214 France.
(Received on May27. l992, accepted in final form on October 16. 7992)
Physical model of slag foaming wasdeveloped by using results of cold and hot model experiments. Thebubble size at the slag/metal interface, the void fraction of foam and the film life of a bubble at the topsurface of slag were calculated. The governing factors of slag foaming have been clarified with this model.It was confirmed that the bubble size at the slag/metal interface is determined basically by the staticbalance betweenthe buoyancyforce and the adhesive force to the slag/metal interface. It wasresulted thatthe slag/metal interfacial tension and the surface tension of metal also affect the foam height besides thesurface tension and the viscosity of slag because they change the bubble size.
KEYWORDS:slag foaming; physical model; numerical calculation: foam; bubble; slag; metal; interface;interfacial tension; contact angle; surface tension; viscosity.
1. Introduction
Slag foaming has been intensively investigated invarious steelmaking processes such as converter process,bath smelting, hot metal treatment and electric arcfurnace.
But quite a few models of slag foaming have beenproposed. Tatsukawa et al.1) proposed an empiricalequation which describes the change in foam height in
BOFprocess. This model is simple and effective to use.However, the knowledge of the average foam life is
necessary to predict the foam height. It is difficult toknowthe average foam life in general.
Ito and Fruehan2,3) introduced a foaming index as acharacteristic parameter of slag foaming. This param-eter can be obtained from the slag properties. The slagheight in the bath smelting and the electric arc furnace
processes can be predicted with this model.3) Thecoefficient of this model must be measuredwith eachslag because it is an entirely empirical model, Iackingphysical background. For example, the effect of bubblesize on the foaming index is ignored in this model.
In this study, the authors have tried to makeup aslag foaming model with clear physical background.Phenomenain three stages of slag foaming weredescribed based on the previous X-ray fluoroscopicobservations4) and two kinds of water model experi-
ments. With these physical models, the governingfactors of slag foaming were investigated.
2. Model
2.1. Concept of Slag FoamingModelThe slag foaming phenomenonis divided into the
following three principal stages, by X-ray fiuoroscopicobservations.4)
1) Formation ofCObubbles at the slag/metal interface
and their detachment,2) Rise of the bubbles in the slag layer and their
accumulation under the free surface (formation ofthe foam), and
3) Coalescenceof bubbles in the foamand the ruptureof bubble films at the top surface of the slag whichtends to destroy the foam.
The physical model should correspond to these stages.Figure I schematically shows the • concept of the
present model. The foam height is decided by thebalance of the gas evolution rate at the slag/metalinterface and the gas escape rate at the top surface ofthe slag. The rate equation for the change in foamheight is expressed as Eq. (1).
dH Qi* - Q-
'~t ..........(1)
dt Awhere His the foam height. Q. and Q are the gas.~t*"evolution rate and the gas escape rate respectively. A is
the cross section area of the vessel.
As for the estimation of the gas evolution rate,
data of previous experiments5,6) or somereaction ratemodels7) can be used. In this study, the works wereconcentrated to makeclear the governing factors of the
gas escape rate which determines the foam life.
Thegas escape rate is determined by the rupture rateof bubble films at the top surface of slag, the size andthe numberof the bubbles. The rate of bubble rupture
maybe governed by the bubble size, the film thicknessbetween bubbles and physical properties of the slag.
Moreover, the numberof bubbles and the film thickness
C 1993 ISIJ 224
~
e
ISIJ International, Vol. 33 (1993), No, 1
foamhei9ht~LQout
4~rate ot film rupture - film thickness
betweenbubblestdistributiOn of blucll~tetbj,P"I_1_,ne)sIfin fOam(void traetiOn)
+9as evolutiOn
(bubble size) Fig. l.
Concept of present foaming model of siag.
C+(O)--CO
in the foam are considered to be determined by the gas~volution rate, the bubble size and the slag properties.
Therefore, if the bubble size, the distribution of bubblesin the foam and the rupture rate of bubble films at the
top surface of the slag are known, the foam height canbe predicted.
Mathematical modeling in each stage of slag foaming
wasconstructed as follows:l) a model to calculate the size of bubbles evolved at
the slag/metal interface,
2) a model to estimate the distribution of bubbles in
the foam, and3) a model to calculate the rate of film rupture at the
top surface of the slag.
2.2. Size of COBubbles at the Slag/Metal Interface
2.2.1. Theoretical Basis
Recently, Terashima et al.8) calculated the configura-tion of CObubbles at the slag metal interface. Thetheoreticai basis of the calculation in this study is
almost the sameas that of their calculation. Also in
this study, the size of CObubbles at the slag/metalinterface was assumedto be determined by the static
balance of the three interfacial tensions and the static
pressure.The calculations of the configuration of a bubble at
the slag/metal interface are based on the resolution
of Laplace's equation of capillarity. This equation
expresses the difference of pressure across the interface
between two phases I and 2.
AP=yl2(1/R +1/R ) ..........(2)
where yl2 is the interfacial tension between the twophases and R1 and R2 are the two principal radii of
curvature of the interface.
At equilibrium this pressure difference is balanced bythe difference of static pressure, which is a function ofthe position on the interface, the difference between the
densities of the two phases and the gravity. If the
descending vertical axis is called Oz, this balance gives
at any point of the interface:
AP=yl2(1/RI + l/R2)
= (pl - p2)9(z-zo) +yl2(11RiO + 1/R20) -"""(3)
where, pl and p2 are the densities of the two phases, gis the gravity acceleration and Ois a point of reference
where the principal radii of curvature are known.
225
Ys
eG eS
C7lvl YSM
@MFlg' 2. Schematic drawing of the joining point between three
phases.
At the joining point of the three phases, if theinterfacial tensions are considered as force vector (Fig.
2), the angles formed by the three phases are expressedby the relationship:
cos((9G) =(- y~-y~ +y~M)/2ysyM
............,,(4)
cos((9M) =(- y~+y~- y~M)/2ySMyM
...........(5)
cos((9s) = (y~ -y~
- y~M)/2ysMys""""""" . . .
(6)
where yM, ys and ysM are the interfacial tensions ofgas/slag, gas/metal and slag/metal respectively. Theangle Os is commonlyreferred to as the contact angle
between slag and metal.
2.2.2. Equations for the Three Interfaces
Gas/Slag Interface
The gas bubble is assumedto be axisymetric as Fig.
3(a). At the apex Oof the bubble, the two radii of
curvature equal to the samevalue called b.
A point M(x, z) maybe associated with an angle ep
between the normal at the interface and the vertical
axis. If the curvilinear abscissa on the interface is noteds, the two principal radii of curvature of the interface
are such that:
l/R1=ec/as and 1/R2=sinc/x ........,.(7)
Thus, the gas/slag interface is described by the system:
ac/as = -sin c/x+ ((pG- Ps)9/ys)z+ 2/b.....
(8a)
ex/as =cos c az/as =sin c ..............(8b)
with the initial conditions at the apex O: x=z=s=c=O.
Gas/Metal InterfaceIt is possible to show that the curvature of this
C 1993 ISIJ
ISIJ International, Vol. 33 (1993), No. 1
o x
z
(a)
Fig.
M
s~~gglaoJr~
z' z
cc
9G_ssls_ag
Os
-- -----C- ---- -
meral
(b)
3. Schematic drawing of the interface between two phases.(a) gaslslag interface (b) gas/metal interface (c)
(c)
slag/metal interface
c*1
a
x
s"
interface has an opposite sign compared to that ofgas/slag interface as shownin Fig. 3(b). Let O'(O, h) bethe position (unknown) of the bottom of the bubbleand b' the value of the two radii of curvature at this
point. The generic equation is written in the referenceaxis (O'x. O'z) where z' =z-h.
The shape of the gas/metal interface is described bythe following system:
ac'/es' = - sin ep'/x +((pG- pM)g/yM)z' +2/b' ,...,..(9a)
ax/as cos ep',
az'las' - ... ...(9b)= sin c'
. .
with the initial conditions at the reference point O':
x =z' =s' = c' =O.
Slag/Metal Interface
For this interface the joining point Cis the referencepoint as shownin Fig. 3(c). The shape of the slag/metalinterface can be computedby resolution of the system:
ac"/as" = - smc"/x+((pM Ps)9/ysM)z
+ (2ys/b-2yM/b' +(pG-pM)gh)/vsM ""(10a)
ex/as" =cosc",
az/as" =sin ep".....
(10b)
with the initial conditions at the point C:
x=xc; z zc, s" Oc" cc+6, Ir
Equation (IO) takes into account the actual equilibrium
curvature of the slag/metal interface at the point C,resulting from the pressure difference across this inter-
face.
For each one of the three interfaces, the positions ofsuccessive points spaced at fixed distance ~s apart, arecalculated step by step. In order to compute the
displacement (dx, dz) corresponding to the relative
position of two successive points, the following third
order Taylor'series expansion is made:
dx=(ax/es)8s+ (a2x/ds2)5s2/2 + (a3x/as3)8s3/6 ...(1 Ia)
dz= (azlas)~s + (a2z/as2)~s2/2 + (a3z/as3)8s3/6....
(1 Ib)
the various partial derivatives being obtained by dif-
ferentiation of Eq. (8), (9) or (lO).
C 1993 ISIJ 226
2.2.3. ComputingProcedureThe calculations were performed as follows:(1) First the value of the radius bof curvature at
the apex of the gas/.slag interface is fixed, and thetheoretical gas/slag contour satisfying Eq. (8) is com-puted.
(2) For each one of these points C, there is only oneconfiguration of the gas/metal interface that satisfies the
contact angle condition expressed in Eq. (4). Thepropervalue of b' is obtained by successive approximations.
(3) Starting from each point C, the slag/metalinterface is computedaccording to Eq. (lO). The con-figuration of the slag/metal interface depends on theposition of C. The proper position with which theslag/metal interface far from the bubble becomesflat is
numerically determined by successive approximations.Whenb is increased, there appears a value beyondwhich no position of C Ieading to a flat slag/metalinterface can be found: no bubble with such a large
value of the top radius of curvature can be in a stableposition at the flat slag/metal interface.
If the properties of slag and metal are given, the
maximumvolume of a bubble which exists stably atthe flat slag/metal interface can be obtained. It wasassumedthat this volume is that of a bubble just beforedetachment from the slag/metal interface. The size of abubble evolved at the slag/metal interface wasobtainedfrom this maximumvolume, assuming that the bubbleafter detachment is spherical.
2.3. Bubble Distribution in the Slag LayerThe X-ray observations of slag foaming showedthe
.4)emergenceof two superposed layers in the slag.
(1) in lower layer, the bubbles seemed to movefreely and the void fraction is low,
(2) in upper layer, they formed a dense packed bed(foam) where the void fraction is muchhigher.
This behavior is typical two-phase flows. Thus it seemspossible to analyze the foaming phenomenonwith the
help of two-phase flow correlations.Wallis9) proposed a theoretical model to describe
ISIJ International. Vol.
two-phase flows in a vertical pipe. He assumed fiat
velocity profiles in the cross section of the pipe. Secondassumption is that the rise velocity W* in an infinite
liquid mediumfollows the modified Stokes law:
W*= (pL - PG)9R213,4L.....
..........(12)
where pL and pGare the densities of liquid and gas, /lLis the viscosity of liquid and R is the radius of thebubbles.
From the relationship between the actual and thesuperficial velocities of gas phase, two values of voidfraction (oci and oc2) which correspond to two layers aretheoretically obtained.
o(1=(1+fr4J/W*)/2 and oc2=1-0(1 ......(13)
whereJ is the suD*erficial velocity of gas phase (ratio gasflow rate to cross section area). Wallis proposed adiagram as shown in Fig. 4. A simple reading of this
diagram gives directly the two values of void fraction.
Thus, for given properties of gas and slag and afixed bubble size this model allows to estimate the voidfraction in the two parts of the flow, that is, the
dispersed gas layer (10wer part) and the foam layer
(upper part). In the present study, the ability of Wallismodel to describe the void fraction in foamwasverified.
2.4. Rupture of Bubble Films at the TopSurface of Slag
As mentioned before, if the film life of single bubbleat the top surface of the foam is estimated, the gasescape rate from the foamcan be evaluated as functionof the numberand the size of the bubbles. However, it
is difficult to calculate directly the rate of film ruptureof a bubble at the top of the foam. Therefore, in the
present model, it was assumedthat the film life of abubble at the top of the foam is the sameas that at the
top surface of slag without foam.
VOF(Volume Of Fluid) methodlo) was adopted tothe calculation of film rupture of the bubble. In this
method, the free surface is treated by introducing afunction F. The value of F in a cell equals to thefractional volume of the cell occupied by fluid. Bycalculating F, the new free surface orientation in eachcell is determined. In the cell with Fvalue betweenzeroand unity, the curvature of the free surfaces are
0.15
J/W=O.1Q
0.00
oe( I -a)2
(1-a)J/W*
33 (1993), No. 1
obtained from F values of neighboring cells. Thepressure and velocity field are calculated with use of thesurface tension of liquid and this curvature.
The fiow of slag around a bubble at the top surface
of the slag was calculated with the VOFmethod. Thebubble was fixed to be spherical and the size wasranged from 2to 6cmin diameter, becauseof difficulty
in convergence for smaller bubbles. The initial positionof slag surface was taken as I.5 cmabove the top of thebubble. The average decreasing rate in film thickness
from 5 to Ommat the apex of the bubble was calcu-lated. Although the effect of the bubble deformation is
ignored and the bubble size is large, this calculation will
be available to evaluate the effects of slag properties
and the bubble size on the rate of film rupture.
3. Water Model Experiment
Two kinds of water model experiments with gasbubbling as shown in Fig. 5 were performed. Experi-
ment I is to verify the ability of the Wallis model todescribe the foaming phenomenon.Experiment 11 is to
know the difference of film life of a bubble at the topof foam from that at the top surface of the slag withoutfoam.
In experiment I, an initial height Ho of liquid
(water +saccharose) was put in a vertical cylindrical
tube (about Im in height, inner diameter of 37mm).Nitrogen gas wasinjected at the bottom through a glassfilter. In order to reduce the bubble coalescence at the
outset of the glass filter, 2"/o ethanol was added inwater;1 1) the size of the bubbles wasmuchlarger (about
5mmin diameter) without ethanol. Increasing the gasflow rate, two superposed layers appeared. The voidfractions in the two layers were evaluated by the heightof the layers H1 and H2'
In experiment II, argon gas wasinjected into aqueoussolution containing I o/o gelatin. The diameter of the
cylinder was 60mmand the four kinds of glass fiters
which have different size of pores were used. The
O a I O.2 0.4 0.6 0.8 a2 1.o
void traction Oe
Fig. 4. Wallrs diagram 8)
227
".
OOOO OOO._;2 oO15~~
OOOoooOooOOOOooOOOOoO OOo OOOOaO OOO OOo OQoo
-=~ flow meter
tH1 fOamlayer
H2 dispersedgas layer
glass fi,ter
tgas In]ection
Fig. 5. Schematic representation of the water model.
C 1993 ISIJ
ISIJ International, Vol. 33 (1993). No. 1
foaming phenomenonwas recorded by using a highspeedcamera(400 frames/sec). The foamheight and thebubble size was measuredwith these images. After thegelatin solution was foamed, gas supply was stoppedand the change in the height of solution wasmeasured.
4. Results and Discussion
4.1. Bubble Size at the Slag/Metal Interface
From the calculation of the bubble at the slag/metalinterface, it was found that the critical size of bubbles
wasdependent on all of three interfacial tensions.
Figure 6 shows the effect of the contact angle onthe bubble size by changing the slag-metal interfacial
tension. The critical diameter of bubbles increasedalmost linearly with the increase in the contact angle.
The relationship between the critical bubble diametermeasured in the X-ray fiuoroscopic observations andthe contact angle is also shownin the samefigure in the
case of two sulphur concentration. The contact angledata used were those obtained by Crambet al,12) based
on the measurementsof Gayeet a!.13) The measuredbubble size agreed well with the calculated one with this
model.Crambet al. showed that the increase in oxygen
content and the decrease in sulphur content lower the
contact angle between slag and metal. Mukai et al.14)
indicated that the contact angle decreases sharply with
increasing the iron oxide content in slag. Therefore, it is
concluded that the decrease in the contact angle bythese surface-active elements causes the decrease in thebubble size.
Figure 7 shows the effect of surface tension of theslag on the bubble size whenthe other parameters werefixed. The bubble size decreased with decreasing thesurface tension of slag. Figure 8 shows the effect ofsurface tension of metal on the bubble size. The bubblesize decreased with the increase in the surface tension
EEa'
~,DEa~i
odanB~lo
,~-
87654321o
7~ = 1.6 (N/m)
T*~= 1.3 (N/m)
p.= 2.5 (glcnlP)
Fig. 7.
0.4 0.5 0.6
surface tension of slag (N/m)
The effect of the surface tension of slag on the critical
size of adhesive bubble to the slag/metal interface.
EE~*O~OEcc
1,,D
!:Q==Lcco
:S:
o
12
10
8
6
4
2
O
Is = 0.5(N/m)TM= 1.6(N/m)ps = 2.5(glem3)
e'F-exp. 1
(S:0.40/0)
Present work\ex
p.2(S:0.050/0)
20 60 8040COntact angle (deg.)
The effect of the contact angle betweenmetal on the critical bubble size.
1oo i 20
EEO~,g11
:J:I
-8
:E
*O
o
87654321o
7s = 0.5 (Nlm)
lr.*= 1.3 (N/m)
p.= 2.5 (glcnf)
Fig. 6. slag and Fig. 8.
1.41.O I .61.2
surface tension of metal (Nlm)
Theeffect of the surface tension of slag on the critical
size of adhesive bubble to the slag/metal interface.
mm2
1
O
(a)
l,
mm8
6
4
2
1 mm O
(b)
2 4 6 8 mm
Fig. 9.
Successive calculated contours of a bubblegrowing at the slag/metal interface for twovalues of the contact angle ((a) es=20degrees, (b) Os=100 degrees).
Dotted lines and continuous lines show the
intermediate contours and the maximumbubble size.
C1993 ISIJ 228
ISIJ International, Vol. 33 (1993), No. 1
15
~~a 10EE~1
5
JL
e
JL
A
JL
t
JL
~,L
~LA
\RbubbLc
=0.18 mmA AAe A,
eA
t, t
O, 16 mm
0.18 mm
O, 16 mJ11
e uL= l0-3 kg/m/S-water + ethanol
'r water + ethanoluL = 2. 10 3kg/m/s
+ saccharose
A
A A
e
e
J';
O
Fig. lO.
0.5 1.O
VOid traetiOn oc (-)
The relationship between the void fraction and the superficial velocity of gas obtained in the water modelexperiment, for two values of viscosity.
of metal beyond 1200dyn/cm. It is considered that
these effects are due to the change in contact angle bychanging the surface tension of slag or metal.
Bubble contours during their growth are lllustrated
in Fig. 9 for two extreme values of contact angle. Inthe case of low contact angle, the bubble was almostspherical and the gas/metal contact area was small.
On the other hand, when the contact angle is large,
the bubble is compressed near the bottom and the
gas/metal contact area is relatively large. The shape ofbubble agreed with that observed in the experirnent
with X-ray fiuoroscopy.4) The buoyancy force calcu-
lated by the volume of the bubble and the density ofslag and metal agreed well with the sumof the vertical
componentof slag/metal interfacial tensions around the
joining boundary of the three interfaces. Therefore, the
decrease in the gas/metal contact area will cause the
decrease in the force which should be balanced with the
buoyancy forces.
4.2. Void Fraction in FoamIn the water modelexperiment, it wasverified at first
that the value of W*compatible with the values of ocl
and c(2 remained reasonably constant, which showsthatthe bubble size is independent of the gas flow rate.
Therefore, the bubble size was estimated by assumingthat Wallis model is true.
Measurementswere madewith two kinds of liquid
which have different viscosities. The results are pre-sented in the Wallis diagram (Fig. lO). For comparisonwith Wallis theory, the curves corresponding to fixed
values of bubble radius (0.16 and O.18mm)have beendrawn in the samefigure. Thecorrelation wasrelatively
Table l. The relation between the pore size of filters andthe observed bubble size.
Filter No. 1 2 3 4
Pore size (pm) 5l O 2(~30 40-50 lOC~120Bubble diameter (mm) O. l0.3 0.3-0.5 0.6-1 .O 0.7-1 .5
good in spite of the important infiuence of the bubblesize. Somecoalescence occurred during the rise ofbubbles in the column and their size dependedon the
gas flow rate; this explains the scattered results for high
gas fiow rate.
Even if it seemsrelatively simplistic, the Wallis modelpredicts the emergenceof two flow configurations in the
samecolumn: dispersed gas layer and foam layer. It wasconfirmed that Wallis model is applicable to evaluatethe void fraction of the foam as functions of the gasevolution rate, the bubble size and the slag properties.
4.3. Film Life of a Bubble at the Top Surface of Slag
4.3.1. Water Model ExperimentIn the experiment with gelatin solution, the average
bubble size increased with the increase in the pore size
as shown in Table l. The average bubble size wasalmost constant with the change in the gas flow rate
from 100 to IOOOcm3/min.Figure 11 shows the effect of the average bubble
size on the gas escape rate. The gas escape rate wascalculated by the foam decline rate after gas supply
was stopped. The gas escape rate increased with the
increase in the bubble size. In other words, the foam life
increases with decreasing the bubble size. The result
229 C 1993 ISIJ
ISIJ International, Vol. 33 (1993), No. 1
40o
=~3oo
E~o~co* 20Q
oa,5o~Ioococao'
o
)(
o o.5 1average bubble diameter (mm)
Fig, Il, The effect of the average bubble size on the gasescape rate from the foam.
e,
\~10.5~
o~E I0.4~-oo P*= 2.5 (g/cnf)
~~ 11* = IO (P)o,Dc IO 3 d = 5 (cm)2~ 0.4 0.5 0.6
surface tension of slag (Nlm)
Fig. 13. The effect of the surface tension of slag on the
decreasing rate of thickness of bubble film.
(i) d=2 (cm) (ii) d=3 (cm)
Fig. 12. Calculated configuration of the top surface of slag
around a bubble just before the film rupture.
agreed with the result in the case of molten slag.4)
Gas escape rate for a single bubble (Imm in
diameter) at the top surface of the liquid without foamwasestimated in a following method.
(1) The time from beginning of the surface de-
formation by the effect of the bubble rise to the
momentof the rupturc of the bubble film wasmeasuredby using the high speed TVcamera.
(2) From the result of the foaming experiment, the
void fraction was calculated by the foam height in the
case that the average bubble diameter was Imm.(3) Thenumberofbubbles at the top surface of the
foam were evaluated by the void fraction.
(4) Thegas escape rate wasestimated by the time offilm rupture of a single bubble and the number ofbubbles at the top surface of the foamcalculated in (3).
TheX mark In Fig. I I represents the result of estima-tion. The estimated value agreed with that obtained bythe foam decline rate. Fromthis result, it is consideredthat the gas escape rate from the foamcan be predicted
from the rate of film rupture of a single bubble at the
top surface of liquid without foam.
4.3.2. Numerical Calculation
Figure 12 shows the configuration when the filrn
thickness becomeszero. In the case of 2cmin diameterof the bubble, the level of slag surface was about the
half height of the bubble. Onthe other hand, the level
of the slag surface wasnear the bottom of the bubble in
C 1993 ISIJ
co.\Eo*J
20
J',S
5EN:*O 15
~
~lo
P.= 2.5 (g/cnf)
n.= 10 (P)
T* = 0.5 (N/m)
230
32 4 5 6bubble diameter (cm)
Fig. 14. The effect of the bubble diameter on the decreasingrate of the thickness of bubble film.
the case of 3cmof the bubble. Thecurvature of the slag
surface at the same level of the surface increases withthe decrease in the diameter of the bubble. Therefore, it
is considered that the decrease in film thickness is
accelerated by the pressure difference between at the
apex and near the bottom of the slag surface due to this
curvature.The effects of surface tension on the rate of the
decrease in film thickness is shown in Fig. 13. Theincrease in the surface tension of slag enlarged thedecreasing rate of film thickness. The smaller thesurface tension of slag becomes, the smaller is the
pressure difference mentioned above. This mayresults
in the suppression of the flow in the film.
Figure 14 presents the effect of the bubble diameter.
The decreasing rate of film thickness increased with thedecrease in the bubble size below 5cmin diameter. This
maybe because of the increase in pressure difference
due to the increase in curvature of the slag surface.
However, there is a ininimum point in this rate between
5and 6cmin diameter of the bubble. The decreasingrate of film thickness maybe accelerated also by the
hydrostatic pressure. Therefore, it is regarded that theinteraction between the pressure due to the curvatureof the surface and the hydrostatic pressure causes the
[SIJ International, Vol. 33 (1993), No. 1
'O*~SO~'O'O
O:~OEE~:.hO'D
~SO'O,eO8U
10
9
8
p . = 2.5 (glcnf)
T. = 0.45 (N/m)
d= 5 (cm)
Fig. 15.
Table
1 5 10 50 100slag viscosity (P)
The effect of the viscosity of slag onrate of the thickness of bubble film.
he decreasing
2. The effects ofthe properties ofslag and metal onthe bubble size at the slag/metal interface, the
void fraction of foam, the rupture rate of bubblefilm and the foam height.
Foamheight up ure ra eot
bubble fllm, Void fractionBubble s~zej of foam
Slag viscos~y A increase v vSurface tension Aof slag l
decrease A A ASlag-metal Ainterfacia] tension ,
decrease A A +Surface tension Aof metal l
increase v A v
minimumpoint. Themaximumpoint in bubble lifetime
was recognized also in the measurementwith p]astisols
by Harrell et al.Is) According to the measurementwith
aqueoussolution of soap,16) the lifetime increased withdecreasing the bubble size. In this case, because the
bubble is quite stable, the lifetime may be governedmainly by the stability of the liquid film, not the
mechanism above. In the case of the actual slag
foaming caused by slag/metal reaction, the bubble size
is less than 2mm.4)Therefore, the rate of film ruptureof bubbles in the slag foam will increase with the
decrease in the bubble size. But the increasing rate is
not so much, while the volume of a bubble is propor-tional to the third power of the diameter. Thus, total
gas escape rate will increase with the increase ofthe bubble size as observed in the water and X-rayexperiments.
Figure 15 shows the effect of the slag viscosity. Therate of the decrease in film thickness decreased withincreasing the slag viscosity.
6. The Effects of Properties of Slag and Meta] on the
FoamHeight
The governing factors of the bubble size evolved at
the slag/metal interface, the void fraction in the foamand the rate of fiim rupture of the bubbles at the topsurface of the foam wasmadeclear with the three mod-els of each foaming process. Therefore, the qualitative
evaluation of the effects of the physical properties of
s]ag and metal on the foam height could be made.Table 2showsthe effects of the properties of slag and
metal on the foam height. The effect of each property is
considered as follows:
Viscosity of SlagAccording to Wallis diagram, the void fraction of the
foam decreases with the increase in the slag viscosity,
then the film between two bubbles in the foam maybecome thick. The downward flow in the film ofthe bubble at the top surface of the foam may besuppressed by the increase in the slag viscosity. Thesetwo effects will cause the decrease in the rate of film
rupture of the bubbels at the top of the foam and thenthe increase in the foam height.
Surf~ce Tension of SlagAccording to Fig. 6, the bubble size increases with
increasing the surface tension of slag. The film betweenbubbles becomesthin, because the void fraction in the
foam increases with the increase in the bubble size. Thedownwardflow in the film of the bubble at the top ofthe foam will be promoted because of the increase in
the pressure difference due to the curvature of the slag
surface around the bubble. Therefore, the rate of film
rupture of the top bubbles will increases with the surfacetension of slag and the foam height will decrease.
Slag/Metal Interfacial TensionAccording to Fig. 8, the bubble size increases with
the slag/metal interfacial tension. It will also cause theincrease in the rate of film rupture of the top bubblesand then the decrease in the foam height.
Surface Tension of MetalAccording to Fig. 7, the increase in the surface
tension of metal decreases the bubble size. Therefore,the foam height increases with the surface tension ofmetal.
The effects of viscosity and surface tension of slag onthe foam height obtained in this study agree with thoseof most of the previous works. It has been clarified thatthe slag/metal interfacial tension and the surface tensionof metal also affect the foam height because they
maychange the bubble size evolved at the slag/metalinterface.
7. Conclusion
Physical model of slag foaming was derived byusing results of cold and hot model experiments. Thegoverning factors of slag foaming have been clarified
more in detail with this model. The effects of thephysical properties of slag and metal on the foamheight have been madeclear. It was confirmed thatthe bubble size evolved at
,
the slag/metal interface is
determined basically by the static balance between the
buoyancy force and the adhesive force to the slag/metalinterface, and the slag/metal interfacial tension and thesurface tension of metal also affect the foam heightbesides the surface tension and the viscosity of slag
through the change in the bubble size.
231 C 1993 ISIJ
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