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PROBLEM1
THE FEASIBILITY OF CASTINGSTEEL IN A CONTINUOUSHEET
1. INTRODUCTION
The general concern of this problem was to investigate the feasibility of
casting sheet steel by direct solidification of mol ten metal on a large,
cooled, rotating drum or wheel. If such a procedure were possible, it would
lead to considerable savings, notably on capital and energy costs associated
wi th rolling steel slabs or billets down to sheet metal thickness. The t ypi+
cal design parameters that were envisaged by the clients (BHP Melbourne
Research Laboratories) were to cast sheet steel between 1 and 10 mm thick and
~1
at peripheral drum speeds of about 1 m sec The Mathematics~in~Industry
Study Group (MISG) was specifically asked to investigate the heat transfer and
fluid mechanics in the immediate region where the molten metal was poured onto
the wheel. We broadened this goal to consider heat transfer in the metal
sheet up to the end of the solidification process, but did not consider s ubs e-
quent cooling from solidification down to a temperature where coiling or r In-
Lsrnng rolling could be applied.
Substantial preliminary work was undertaken before the MISG. In par-t t cu-
lar, a computerised literature search was made based on the key words "con~
tinuous", "casting" and "steel sheet or strip". The search unearthed more
than 50 references (abstracts and some articles passed on to the BHP r-epr es en-
tative), although it was immediately clear that only a few of these were us e-
ful. It was apparent that several groups in the U.S.A. planned to spend of
the order of U.S. 10 million on investigating the problem, but they did not
appear to have pilot plants operating yet and their technical research find-
ings were generally not published. One of the references (Swanson et al.,
1984) was both technical and directed towards the same goals as the present
2study. However, as explained later, we disagree wi th them on many aspects of
the work.
The other major preliminary preparation was to have extensi ve discussions
with Drs Day, Dunlop and Foley at the CSIRO Division of Applied Physics. This
group have a lot of experience with the melt spinning of glassy metals: that
is, the production of very thin ribbons of metal with amorphous structure by
squirti ng mol ten metal directly onto a rapidly spinning small wheel. In par-
ticular, Dr Foley was of great assistance and gave a background talk on melt
spinning at the MISG. The Applied Physics group had succeeded in slowing down
their melt spinning process to such an extent that crystalline metal was
formed, but only coarse samples of metal strips had been so obtained.
Clearly, much more experimental work is required.
We would like to frame the rest of this report under the headings of
general fluid mechanics considerations, (three) simple heat transfer calcula-
tions to give the puddle length, fluid mechanics stability considerations, and
con cl usions.
2. GENERALFLUIDMECHANICSCONSIDERATIONS
Careful geometric design should minimise the importance of fluid motion
in the process. Basically, the molten metal should be poured onto the wheel
in as smooth a fashion as possible, rather as shown in Figure 1(a) below. The
thickness and mean speed of the feed stream of molten metal should be adjusted
to approximate the design thickness of the steel sheet and the speed of the
wheel's surface. Nozzles could be designed to minimise fluid effects: for
example, a short nozzle would mean that fully developed plane Poiseuille flow
would not exist at the exit of the nozzle. This would pr-esunabl y ease the
transi ti on to free surface shear flow of the mol ten steel layer on the wheel.
Attention to these precepts would mean that fluid motions above the s ot i di r r-
3cation front passing upwards through the metal layer could be neglected in a
preliminary calculation to obtain the molten puddle length.
On the other hand, we believe it would be disastrous to squirt molten
metal normal to the wheel fran sane distance as shown in Figure 1(b). This
procedure would involve the risk of metal splashes, liquid metal going the
wrong way (against the direction of the wheel), and transient effects such as
waves and ripples on the surface of the molten metal sheet.
/
SOLIDIFIED
STEEL SHEET
h
Figure l(a). The correct way: molten
steel poured smoothly onto the rotating
wheel.
Figure l(b). The wrong way: molten
metal squirted onto the rotating
wheel.
It is clear that solidification would need to be completed before the
metal sheet was removed fran the wheel and before the sheet achieved such an
angle to the horizontal that surface instabilities had a chance to grow and
become significant. Accordingly, we believe that the most important calcula-
tion is to obtain the length of the molten puddle on the wheel (neglecting
fluid motions above the solidification front), and this problem is attacked in
3 different ways in the next section. It is believed that curvature of the
wheel's surface and surface tension do not affect the approximate heat
transfer cal oul at t ons , although these effects should be considered in a full
4feasibility study. Some possible sources of fluid instabilities are described
in Secti on 4.
3. HEAT TRANSFER CALCULATIONS FOR THE PUDDLE LENGTH
A useful heat transfer calculation has been given by Kuiken (1977). His
mode.l considers a moving belt or wall maintained at a constant temperature
below the freezing temperature emerging into a semi-infini te fluid at uniform
temperature above freezing. Solidification occurs adjacent to the moving sur-
face, and Kuiken has analysed the temperature and velocity fields using the
boundary layer equations for fluid motion and by looking for similarity solu-
tions. This enables a calculation to be made of the heat transfer through the
moving wall, and thereby provides an input to the calculation of Swanson et
al. (1984). Swanson et al. considered heat transfer from a solidifying steel
layer to a conducting metal wheel or drum, and arrived at numerical results
for mol ten puddle length and wall temperature for various values of the peri-
pheral wheel speed and wheel diameter and at a fixed small sheet metal thick-
ness. Unfortunately, Swanson et al , did not give information for a range of
values of the sheet metal thickness. What we desire is an expression for the
molten puddle length for general values of sheet thickness, wheel speed, and
temperature drop from the mol ten metal to the deep core temperature in the
wheel. We attempt to provide this expression by three different arguments in
this section.
Model 1
The first model comes from the theory of heat transfer in solid rollers.
Suppose that the heat to be extracted comes only from latent heat liberated as
the liquid steel sheet solidifies. The latent heat extraction required per
unit time per unit width of the sheet is
5where Ps is the steel density, L the specific latent heat per unit mass, V the
peripheral speed and h the sheet thickness. Fran strip rolling theory (Yuen,
1985), the heat transferred per unit time per unit width to the roller is
Q = ~8k (VA/a )~(T -T )c y1f c c s c
where 8 ~ k Ip I (k Ip +k Ip ), k is the thermal conducti vi ty, P = VA/a thes s c c s s
Peclet number, a the thermal diffusivity, A the molten puddle length, Ts the
solidification temperature, Tc the deep core temperature in the drum, and sub-
scripts sand c refer to the sheet steel and roller (copper) properties
respectively. It is assuned that the wheel or drum is sufficiently thick that
Yuen's res ul ts for sol id rolls ar e appl i cabl e and we ha ve negl ected radi ati ve
heat transfer fran the top surface of the steel sheet.
After algebraic manipulation, we obtain the expression
A = [
la la 121f c s
--+ --4 k kc s
By substi tuting the approxi mate numeri cal values a =10-4 m2s-1c
k =20 wm-10C-1,
s
°c and T =150c
k =400 W -10C-1 ( -6 2-1cm, appropriate for copper), as·4x10 ms,
Ps=7.6x10
3 kgm-3, L=2.7x105 Jkg-1, V=1 ms-1, h-0.01 m, Ts·11100
°c, we find the estimate for A of 3.8 metres.
Sensible heat effects can also be incorporated in a crude way. If the
temperature profile in the solidified sheet is considered to be linear (as is
also assuned in model 2 below), the heat extraction due to sensible heat is
Qs = ~P VhC (T -T )s s s w
where Cs is the specific heat of the sheet steel and Tw the wall temperature
gi ven by (Yuen, 1985)
Tw Tc+(Ts-Tc)8.
If Qc is now equated to the sun of QL and Qs' the following expression for A
is obtained
6A = ~r~+ ICLS]2p2 CL + ic (T -T )(1-rl)]2 Vh2(T -T t2
~ kS a s s c s c •
With C
S
=600 Jkg-10C-1 and the same data used above, A is approximately 17
metres. Note that these estimates for A would be decreased by allowing for
radiati ve heat transfer at the top of the steel sheet.
Model 2
The previous model gave an estimate of the mol ten puddle length based on
the amount of heat that could be transferred away through the underlying
material. The second model looks at heat transfer processes in the solidified
steel sheet, and balances the latent heat released at the solidification front
against the heat which can be conducted through the solidified steel to the
underl yi ng materi al. Agai n, we negl ect fl ui d moti on above the solidif i cati on
front; indeed, we assune that the overlying mol ten steel is all at a uniform
temperature just above the solidification temperature of the metal. If it is
further assuned that the temperature profile in the solidified sheet is linear
and if we balance the rate of latent heat production against heat flux through
the solid sheet, we have the equation
in which Ps' L, ks' Ts and Tw are as before, and 0 is the distance of the
solidification front above the wall. The above equation is a first order
ordinary differential equation for 0 and it can be integrated to give 0 as a
function of t , If t is identified with AlV, the length of the puddle when the
thickness is h is found to be
If we now substi tute the previous numerical values and T -T *8000C, we get thes w
estimate for 1.0 of approximately 6 metres.
7This model ignores thermal and flow conditions in the mol ten metal zone
above the solidification front, and it forces a linear temperature profile in
the solidified sheet. The validity of the linear temperature profile can be
tested by using results of Carslaw and Jaeger (1980, pp.286-289) for the
advancement of a solidification front into a stationary liquid. Carslaw and
Jaeger's r-eeul ts srow that the puddle length depends on the ratio of sensible
to latent heat requirements according to the expression
where
'Y4: (T -T )C /Ls w s
If we use the above data and Cs=600 Jkg-
10C-1, we find that 'Y-0.9 which indi-
cates that it is unwise to assune a linear temperature profile in the solidi-
fied sheet. It also implies that '\0 is a crude underestimate of the molten
puddle length.
Model 3
It is clear that the above two models need an improved calculation for
the temperature profile in the SOlidified steel sheet. A model, which looks at
the coupled heat transfer in the sheet and the underlying material is now
presented.
Again, we assune that the molten steel above the solidification front is
at a constant temperature and that all the latent heat liberated at the front
is conducted through the metal sheet. Moreover, assune there is perfect t ner+
mal contact between the solidified sheet and the underlying wheel. In 00-
ordinates moving wi th the speed of the wheel's surface, the temperature equa-
ti ons to be solved are
sheet, 0<y<6( t), aT a
2T
at - as 2'ay
wheel I -"'<y<OI aT a2Tat - ac ay2"
8subject to the conditions
metal interface:
y-6(t), T-Ts'
y=O, T=Tw'
solidification front:
cor e eondi ti on: y+-'" , T+Tc '
This problem has the similari ty solution
sheet metal: T-Tw+(Ts-Tw)tj>U;),
wheel: TaTc+(Tw-Tc)~(n),
where tj>and ~ satisfy
_I
l;=y(Cl
S
t ) ~;
_I
n=y (Cl t) ~c
d2cp ~l;dCP_ 0,-+
dl;2 dl;
d2~
I~- O.-+
dn2
~ d n
I
The solidification front is given by I;=A, that is /j(t )-A(Cl
S
t)~; and the
conditions there are
I;=A, cp(A)=l, ~AR=CP'(A)
L
R = (T -T )e
s w s
At the interface between the sheet metal and the wheel, the conditions are
found to be
1;-0, n=O, 4>(0)=0, ~(O)-l, Y-r4>'(O) 1/1' (0)
where
I
(kspses)~
I
(k p e )~
c c c
As n+-"', we also have ~(n)+O.
Y and T -
The solutions to the above equations are
erf (1;/2)
erf(A/2) ,
1/1( n) = 1 + erf ( n/2) •
9Then, the interfacial conditions may be used to eliminate Tw in favour of A
and other par amet ers; thi s gi ves (after simpl if Lcat i on)
2
~/n A eA /4 [Y+erf(A/2)] = Cs(Ts-Tc)/L.
This transcendental equation has to be solved for A given the parameters Y,
Cs' Ts' Tc and L. Now, in this model, the molten metal will have disappeared
1
when h=A(ets t)~ or, equi valentl y, when
Vh2
A • -2-'
A et
S
Again, this estimate for the puddle length is proportional to Vh2, but it is
now required to estimate A.
To this end, we note the previous numerical values and the additional
approximate values Cs=600 Jkg-
10C-1, Cc=400 Jkg-
10C-1, Ts=1400 °c, Tc.150 °c,
-62-1 -1
Ks=5Xl0 ms, Vel ms , h- 0.01 m. The constants Y and Cs(Ts-Tc)/L are
found to be approximately 0.25 and 2.7 respecti vely, and the root of the t r-an-
scendental equation is about A-1.6. We find that A is about 7.7 metres and Tw
is about 460 °c.
Conclusions based on the simple heat transfer calculations
At our level of approximation, the most that can be said of all these
modaLs is that they gi ve estimates for the puddle length of about 10 metres at
the design parameters of 0.01 m thickness and 1 m/sec speed. The results of
model 3 should be more reliable than those of models 1 and 2. The resul ts are
considerably more pessimistic than those of Swans on et al. who predicted a
puddle length of about 0.1 m for a sheet of thickness 0.005 m east at 1 m/sec,
whereas model 3 would gi ve a puddle length about 20 times longer. Moreover,
their results did not show the linear dependence with V that we have cat cu-
lated and they did not mention the h dependence of their results.
10
4. REMARKS ON THE STABILITY OF THE HDLTEN STEEL LAYER
Clearly, it is necessary that the molten steel layer should solidify
before surface instabilities have time to grow and become significant. The
most obvious instability that needs to be considered is the surface I ns t abi Lr-
ity in fluid flowing down an inclined plane. This problem was analysed in the
low Reynolds number, low wavenumber limit by Benjamin and Yih in the 1950s;
they found that instabilities would occur (in the linear theory) when the Rey-
nolds number R-2Uhl3v (U is the surface veloci ty) was greater than (5/6) cot e
where e is the angle of the layer to the horizontal. However, it was pointed
out that the growth rates of these instabilities would be small, and the large
Reynolds nunber behaviour of the neutral stability curve was unknown.
A second sort of stability was also discussed. This was related to
unpublished work and examined the linear stability of profiles sketched in
Figure 2 below. In the inviscid theory, this profile was stable to small ct e-
turbances for low Reynolds nunbers , unstable at moderate Reynolds nunbers, and
stable again at large Reynolds numbers.
h
y
FLUID SPEED
Figure 2. Typical velocity profile for the
molten steel on the spinning wheel.
11
We therefore thought that many possible sorts of instability could mam -
fest themselves, even without the complicating effects of solidification and
surface tension. It was pointless trying to estimate growth rates for the
possible instabilities without being guided by experiments as to which form of
instability was the most important to look for. Therefore, we make the finn
recommendation that an experimental program should be set up to look at poasf-
ble instabilities on ei ther a sloping surface or a rotating drum. In such a
program, it would not be necessary to have the fluid solidifying, rather it
would be important to get the nozzle geometry and the Reynolds mmber of the
flow correct.
5. CONCLUSIONS
Our concl us i ons ar e as f 011ows:
1. We feel that careful nozzle design should minimise fluid mechanical
effects in the solidification process, and that it should then be poset+
ble to model the casting process by neglecting fluid motions and heat
conduction in the molten layer overlying the solidification front.
2. Based on our estimates for the molten puddle length, it would seem to be
impossible to continuously cast steel sheet at the design parameters of
0.01 m thickness and at a speed of about 1 m/sec on any reasonable size
drum (say of diameter about 2 metres). Moreover, our estimates for the
molten puddle length are much greater than those of Swanson et al.
3. However, all of our estimates for the puddle length are proportional to
2Vh ; hence, it may be possible to continously cast thin strip of typical
thi ckness 0.001 m.
4. An experimental program should be initiated to investigate possible
instabilities in sheets of fluid flowing down inclined planes. Solidifi-
12
cation should not be considered here, rather the Reynolds numbers should
be made to correspond to those anticipated in the continous casting pro-
cess.
5. Further analytical and numerical work is justified if instabilities do
not appear to be troublesane and if thinner sheets and slower casting
speeds V are acceptable. This work could include (a) the importance of
fluid effects in the transition fran the nozzle flow to the free surface
shear flow on the drum, (b) temperature-dependent properties, (c) heat
and mass transfer in the molten steel layer overlying the solidification
front, and (d) surface tension (progressively more important for thinner
sheets) •
REFERENCES
Carslaw, H.S. & Jaeger, J.C. Conduction of heat in solids (Oxford University
Press, 1980) •
Kuiken, H.K. "Solidification of a liquid on a moving sheet", Int. J. Heat &
Mass Transfer 20 (1977),309-314.
Swanson , B.W., Wllliams, R.S. & Draper, R. "Heat transfer in melt spinning
cooling belts and drums", ~. Res. Soc. Symp. Proc. (Eds B.H. Kear and
W.C. Giessen) 20 (1984),113-117.
Yuen, W.Y.D. "On the heat transfer of a moving composite strip compressed by
two rotating cylinders", ASMEJ. Heat Transfer 107 (1985),541-548.


The feasibility of casting steel in a continuous sheet

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