Fusion of Low Carbon Steel Scrap in the Middle Carbon Steel Melt
Taljenje niskougljičnog čeličnog otpatka u talini srednje ugljičnog čelika
V. Grozdanič,1 Metallurgical faculty Sisak, Croatia
Prejem rokopisa - received: 1996-10-01; sprejem za objavo - accepted for publication: 1996-11-04
A quasi three-dimensional mathematical model of fusion of cylindrical steel scrap In converter melt was developed. The model vvas solved using the implicit alternating direction method. The obtained algorithm vvas programmed in ASCII FORTRAN for the computer SPERRY 1100/72. In the model temperature dependent thermophysical properties of material vvere incorporated. That gives to the model a nonlinearity. On the basis of the model it vvas concluded that the addition of 1% of Iow carbon steel scrap decreases the temperature of a middle carbon steel melt for cca 20°C. This is in good agreement vvith experimental data from literature. The mathematical model was tested for one-dimensional exact solution using the Bessel functions. A good agreement was found.
Key vvords: fusion, steel scrap, mathematical model
Razvijen je i istražen kvazitrodimenzijski matematički model taljenja valjkastog čeličnog otpatka u talini kod konverterskog procesa. Matematički model riješen je implicitnom metodom promjenljivog smjera. Dobiveni algoritam programiran je u programskom jeziku ASCII FORTRAN za računalo SPERRY 1100/72. U matematički model inkorporirana su temperaturno ovisna toplofizička svojstva materijala, što modelu daje nelinearnost. Na temelju matematičkog modela zaključeno je da 1 % niskougljičnog čeličnog otpatka snizi temperaturu srednje ugljične celične taline za cca 20°C, što se dobro slaže s eksperimentalnim podacima iz literature. Matematički model testiran je na jednodimenzionalnom egzaktnom rješenju pomoču Besselovih funkcija. Konstatirano je njihovo medusobno dobro slaganje.
Ključne riječi: taljenje, čelični otpadak, matematički model
1 Introduction
From the increase of the share of metalic scrap in the charge, an increased economy of steel manufacturing in the converter processes is expected. The share of steel scrap should approach that in the openhearth process. Steel scrap, usually as low carbon steel refuse, is a very economical means of melt cooling. Iron ore is only better means of cooling. Data in ref.l show that 1% of steel scrap decreases the bath temperature for 12 to 15°C, vvhile 1% iron ore decreases the temperature for 30 to 40°C. Hovvever, production expirience shows that steel scrap is better than iron ore. For example, by using steel the quantity of metal ejected from the converter is dimin-ished, the resistance of refractory lining is increased and a better utilization of excess heat in the bath obtained. The regime of scrap fusion depends in significant degree on scrap size, and affects the bath temperature, the slag forming processes, as well as the oxidation of carbon and the metal desulfurization. For instance, small sized scrap melts faster and cools quickly the bath. This decreases the rate of slag forming, carbon oxidation, desulfurization, and lower the quantity of blovvn oxygen. Very large pieces of scrap don't melt completly during the process-•ng in the oxygen converter. A mathematical model of low carbon steel scrap melting in the carbon steel melt was developed and tested vvith the aim to determine the optimal scrap size. Since the bath temperature is above
Dr. Sc. Vladimir GROZDANIČ Univcrsily of Zagreb
Metallurgical facully Sisak. Aleja narodnih heroja 3. 40<X> Sisak. Crnalia
1550°C the so-called diffusion melting is not considered. In figure 1 the investigated system is illustrated. It con-sists of a volume element of melt in which cylindrical steel scrap is immersed.
2 Mathematical model
For the start of the melting of a cylindrical scrap piece in a volume element of melt in figure 1 the Fourier's partial differential equation of heat conduction
has the form2:
dT dt
= a
V
čPT J_3T čPT dr2 + r dr + dz1
(D
Since for the horizontal axis of the system r — 0 the equation (1) is modified according to L'Hospital's in the form:
Figure 1: Volume element of melt with cylindrieal steel scrap
3T 52T 32T
(2)
The basic assumption for the validity of the differen-tial equations (1) and (2) is that scrap piece is immersed is physicaly realistic since the difference in density of steel scrap (7860 kg/m3) and melt (7507 kg/m3) is very small. Considering the system in figure 1, it can be con-cluded that mathematical model is quasi three-dimen-sional. In the time t = 0 the temperature of the melt is Tl, and that of the scrap piece is Ts. The initial temperature at the steel scrap/melt boundary interface is obtained by solving the Fourier's differential equation for heat flow through the contact area of two semifinite medias3:
Ti = T. + -
tl-t,
k„, a.
(3)
On the contact steel scrap/melt area a continuous heat flow occurs with boundary condition of the fourth kind:
Km i ~~ Ks -n
dn dn
(4)
In developing the model it was asummed that thermal properties of low carbon steel scrap (0,2% C) and middle carbon steel melt (0,6% C) are temperature dependent4.
3 Implicit alternating direction method
The differential heat flow equations (1) and (2) with the corresponding initial and boundary conditions were numerically solved using the implicit alternating direction method5 and dividing the time interval into two steps.
In the first half of the time interval the equation is solved implicitly for the z and explicitly for the r direction. The procedure is reversed in the second half of time interval.
Consequently, for the differential equation (1) and first half of time interval At/2 we obtain:
T11 _Tn	i "T*
J 2rAr z a,jn At/2	7
Whereas for the second At/2 we obtain
pn+I _'pn+l
- 4- i.
+ i-H-1 i.j-1 + 32 ^ = J_ hA L, 2rAr
r
ai„i.n At/2
(6)
The numerical solution of the differential equation (2) of heat flow for first At/2 is:
npn _nrn
— + 3; T
(Ar)2
and for second At/2
. -LlLzZIi
z ,1_a,,„ At/2
Tn+1 "-pn+1	, --pn+1 hp*
i,2 Ai.l , rr*	1 i.l i.
(Ar)2
+ d2rT',=-
ai.j.n At/2
(8)
The solution of equations (1) and (2) for the net point (i, j) in the melt and scrap piece as well as for net points on their boundary surface are given in Appendix 2.
Generally it holds: bivi+civi	= di
a2Vl+b2V2+C2V3	= d2
a3V3+b3V3+C3V4	= d3
aivi-i+bivj-t-civi+i
= di (9)
aN-lVN-2+bN-lVN-l+CN-lVN = dN-1 aNVN-l+bNVN =	dN
where v is the unknown temperature, and N is a real number. On the base of the presented algorithm of fusion a computer program was written in ASCII FORTRAN and solved on SPERRY 1100/72 computer.
4	Discusion
The simulation of fusion of a low carbon steel scrap in the carbon steel melt is carried out by space steps Az = Ar = 1 cm and the time step At = 30 s till tmax = 630 s. The initial melt temperatures of 1700°C for the melt, 25°C for the scrap piece and 883°C for the boundary surface were assumed. On the basis of succesive temperatures prints out for particular net points the fusion time of 540 s was obtained for a low carbon steel cylinder of size <|) 50 x 100 mm. The weight of the scrap piece was 4,3% of total vveight of the melt. Thus, it can be con-cluded that 1 % of steel scrap decrease the temperature of steel melt for 20 to 22°C, a value in good agreement with published experimental data1. Three-dimensional mathematical model of fusion of scrap piece was tested through the one-dimensional exact solution of the fusion of the low carbon steel cylinder and a good agreement was established. The derivation of the exact solution using Bessel functions is given in Appendix 3.
5	Conclusions
A quasi three-dimensional mathematical model of fusion of low carbon steel scrap piece in carbon steel melt in oxygen converter was developed. The model was checked in the base of experience data. The simulation of the fusion is carried out on cylindrical piece of diameter of 50 mm and length of 100 mm. The fusion time of 540 s was calculated. Also it has been established that the ad-dition of 1% of steel scrap decrease the melt temperature for cca 20°C, a value in good agreement with experimen-tal data, and also with the exact one-dimensional solution of the equations, which are the model base.
Appendix 1
Abbreviations used: a - temperature conductivity
ai, bi, ci,di - coefficients adjoining to unknowns in
tridiagonal system of algebric equations cp - specific heat at constant pressure k - thermal conductivity n - vertical direction r - space coordinate t - time
T - temperature
vi - unknown in system of simultaneous algebric
equations z - space coordinate
Appendix 2
Constant whivh appear in tridiagonal coefficients aAt
pi =■
2(Ar)2 aAt
P2 " 4rjAr P3 = Pt - P2 P4 = pi + P2
At(kA + kB
ps
P6 = '
2c(Ar)2 At(kA + kB 4crjAr
kA , kB C =-H--
aA aB
qi =
=
q3 =
q4 =

q6 =
aAt 2(Az)2 kAAt c(At)2 kBAt c(At)2
At (kA + kB)
2c(Az)2 kAAt
c(Ar)2 kBAt c(Ar)2
Tridiagonal coefficients
1- Point (i,j) in the melt or scrap piece
-	first At/2:
ai = Ci = -q[ bi = 1 + 2q,
di = p3Tni,j_i + (l-2pi)T\j + p4Tni,j+i
-	second At/2:
aj = "P3 bj = 1 + 2p i cj = -p4 ^
dj = qiT*n j + (l-2qi)T*ij + qiT*i+i j
(10)
2.	Point (i.j) on the boundary surface parallel to r axis separating the materials A (left) and B (right)
-	first At/2: ai = -qi
bi = 1 + q2 + q3 Ci = -q3
di = (p5-p6)Tni,j-i + (l-2p5)Tni,j + (p5+p6)Tni,j+i (12)
-	second At/2: aj = -(P5-P6)
bj = l+2p5 Cj = -(ps+pe)
dj = q2T j-ij + (l-q2-q3)T i.j + q3T i+ij	(13)
3.	Point (i,j) on the boundary surface parallel z axis separating the materials A (down) and B (up)
-	first At/2: ai = Q = -q4 bi = l+2q4
di = (q5-q6)Tni,j-i + (l-2p5)Tni,j + (q6+p6)Tni,j+i
-	second At/2: aj = pe - q5 bj = l+2p5 Cj = -(qe;Pe)
dj = q4T*i-ij + (l-2q4)T*ij + q4T*i+i,j
4.	Point (i,l) out of the boundary surface
-	first At/2: ai = Ci = -qi bi = l+2qi
di = (l-4pi)T*i,i +4piTni,2
-	second At/2: bj = l+4pi cj = 4pi
* * * dj = qiT i-i,i + (l-2qi)T i,i + qiT i+i,i
(14)
(15)
(16)
(17)
5. Point (i,l) on the boundary surface which separates the materials A (left) and B (right)
-	first At/2: ai = -q2
bi = 2q4 + 1 ci = -q3
di = (l-4p5)Tni,i +4p5Tni,2
-	second At/2: bj = 4p5 + 1 cj = -4p5#
dj = q2T*i-i,i + (l-2q4)T\i + q3T*i+l,l
(18)
(19)
Appendix 3
One-dimensional mathematical model of fusion of steel cylinder consists of the solution of a differential equation of heat conduction vvith adequate initial and boundary conditions
5T
,52T
(11)
- 1 3Tv
at
0<r<l, t>0 (20)
T(r,0) = To T(l,t) = TL I T(r,t) | < M
vvhere M is a positive real number.
It is convenient to consider insted of the equation (20) the equation
9T 32T J_ 3T at " 3r2 + r dr
and then to repleace t by at.
Applying the Laplace transform6, we find
(21)
d2e 1 dO
T^ + T - s6 = _T<> dr r dr
(22)
9(1,s) = Tl/s, 0(r,s) is connected.
The general solution of this equation is given in terms of Bessel functions as
9(r,s) = c,J0 (ir Vš) + c2Y0 (ir Vš) + -
(23)
Since Yo(iWs) is unbounded as r —> 0, we must choose ca = 0. Than
T
9(r,s) = c,J(1 (ir Všj + —
From the boundary conditions we find 9(1,s) = c,J0	= ^
c, =
tl-t„ sJ„ (i Vš)
(24)
(25)
(26) (27)
T„	J« (ir Vs)
Thus 8(r,s)=-f+ (Tl-T„) r s	sJ0(iVs)
After complex inversion this equation acquires the form
1 r^" estJ0 (ir Vš) T(r,t) = T„ + (Tl-T„) — - J
2n\ i„sJ0 (t Vs) ds
and the final solution is obtained as
T(r,t) = TL - 2(Tl-T0) ^ r f
. (K)
(29)
where A.2, ..., Xn, ... are positive zeros of equation Jo(^-n) = 0, vvhich are given in Table 1.
Table t: Zeros of equation Jo(A.n) = O7.
n	An
1	2,40482 55577
2	5,52007 81103
3	8,65372 79129
4	11,79153 44391
5	14,93091 77086
6	18,07106 39679
7	21,21163 66299
8	24,35247 15308
9	27,49347 91320
10	30,63460 64684
6 References
1	M. Ja. Madžibožski, Osnovi termodinaraiki i kinetiki staleplaviteljnih processov, Viša škola, Kiev, 1986
2	F. Oeters, Metallurgie der Stahlerstellung. Springer-Verlag, Berlin, 1989
3	V. Grozdanic. Metalurgija, 30, 1991, 1/2, 47-50
4	V. Grozdanic, Livarstvo, 5, 1990, 1, 3-11
5	J. Douglas, H. H. Rachford, Trans. Amer. Malh. Soc., 82, 1956, 421
6	M. R. Spiegel. Laplace Transforms, McGraw-Hill, New York, 1965
1 M. Abramovvitz, I. A. Stegun (eds.), Handbook of Mathematical Functions vvith Formulas, Graphs and Mathematical Tables, National Bu-reau of Standards. VVashington, 1964