UDK 669.18:519.85 ISSN 1580-2949 Izvirni znanstveni članek MTAEC9, 38(6)303(2004) THE TEMPERATURE DISTRIBUTION IN THE STRAND DURING SECONDARY COOLING OFTHE CONTINUOUSLY CAST BILLET PORAZDELITEV TEMPERATURE NA ŽILI MED SEKUNDARNIM OHLAJANJEM KONTINUIRNO LITEGA JEKLA Vladimir Grozdani}, Anto Markoti} University of Zagreb, Metallurgical Faculty, Aleja narodnih heroja 3, 44103 Sisak, Croatia Prejem rokopisa – received: 2004-04-13; sprejem za objavo - accepted for publication: 2004-11-29 On the basis of mathematical model a new equation for temperature distribution in the continuous cast strand in the secondary zone has been obtained. From this equation the time of cooling of the strand surface to a selected temperature can be deduced, witch is in good agreement with experimental values from the literature. Key words: continuous casting of steel, temperature distribution, secondary cooling Na osnovi matematičnega modela je bila razvita nova enačba za porazdelitev temperature v zoni sekundarnega ohlajanja. Iz enačbe je mogoče izračunati čas hlajenja, ko se površina gredice ohladi na izbrano temperaturo, ki se dobro ujema z empiričnimi podatki iz literature. Ključne besede: kontinuirno litje jekla, sekundarno ohlajanje, porazdelitev temperature 1 INTRODUCTION Primary, secondary and, according to some authors, tertiary cooling (on air) are distinguished during the continuous casting of steel. The primary cooling occurs in the mould. The secondary cooling is from the bottom of the mould to the level where the section is partially solidified. In this section the cooling is accelerated by spraying the strand with water. Of utmost importance for the secondary cooling is to achieve a surface temperature of the strand above the A3 point in Fe-C equilibrium diagram. Namely, if the surface temperature is below the A3 point, the strain is increased because of y -> a transformation and that increases the danger of surface cracking. In this paper a new equation for the temperature distribution in the strand of continuous casting is derived. Based on this equation, it is possible to deduce the surface temperature of the strand and the time for witch a determined temperature of the section is achieved. 2 MATHEMATICAL MODEL The equation for the temperature distribution in the strand of continuous cast billet is obtained on the base of the Fourier's partial differential equation of heat conduction 1: with the initial condition: T(x,0) = T0 (2) and boundary conditions: \T(x,t)\ < M (3) MATERIALI IN TEHNOLOGIJE 38 (2004) 6 -k(T) = h(Tw-Tok) (4) Č ox A=0 where T0 - initial temperature in the strand Tw - surface temperature of the strand Tok - ambient temperature a - temperature conductivity k - thermal conductivity h - convective heat transfer coefficient. The boundary condition (3) means that the temperature is bound to x and t, while M is a positive real constant. Equation of heat conduction (1) is solved using the Laplace transforms 2: Lš T(x,t)} = 6(x,s)= J ečst T(x, t)dt (5) 0 The equation of heat conduction can be written: dT(x,t) _ d2T(x,t) ,,, Then the transforms of the both side of the equation is determined as: rldrr, J ? -stdT(x,t)J ,. } _stdT(x,t)J L\—T(x,t)}=\e —ČČd t= lim \e —ČČd t = \dt y '} J dt ßČ-0 dt lim \e-stT(x, t)I+ s |estT(x, t)dt \ = M ° J0 J =sje-stT(x,t)dt-T(x,0) (7) 0 Consequently: 303 V. GROZDANI], A. MARKOTI]: THE TEMPERATURE DISTRIBUTION IN THE STRAND fdT(x,t)] = _ = sQ_ \ dt 0 (8) Using Leibnitz rule for derivation inside the integral we obtain: J J dt d . do = — su(x s ) — — dx dx dt dx By analogy we obtain than: L — T(x,t) Šdt d2 e dx (9) (10) The partial differential equation is written as ordinary differential equation (linear differential equation of second order with constant coefficients): s0(x, s)-T(x,0) = a d2 fl(x, s) dx2 respectively: d2 O 1 sO 1 dxa a The general solution of the equation (12) is: 0(x, s) = c 1 e Tx s / a + c2 exVs / a + T (11) (12) (13) Tw = k \n ok 0 1+ ha 2Jt k V TT (21) The time necessary to obtain the surface temperature of the strand is obtained from equation (21): tI 0 = TOk 2 4h 2 a T 1 w Tw T okJ (22) If thermophysical properties depend on temperature, equation (22) is written as: np(T0)cp(T0)k(T0)(T0-Tw) tI x=0- 4h2 T w-ToJ (23) To obtain a clear conception on the cooling time of the continuous strand in the zone of secondary cooling the following example, published by IRSID 3 is quoted. It is necessary to define the time when the surface temperature of the water cooled strand is not below a certain level if the observed part of the strand is in liquid state. The casting temperature is of 1580 °C. For the steel with 0,5 % C the liquidus temperature is of 1482 °C, and thermophysical properties are temperature dependent 4. The examined secondary cooling part of the Selecting c2 = 0 so that 0(x,s) is bound for xČ>oo, we obtain: 0(x,s) = c1e-xs / a + T (14) The Laplace transform of the boundary condition (4) is: -k dfl(x, s) dx = -kVs/ac1e xvs / a=-(Tw-Tok) (15) h ( Tw-T ok) ksyfs/a The final solution in Laplace area is: 0(x, s) = - h kss / :(T w-Tok)e xys / a ¦ 0 (16) (17) s/a The crossing from Laplace area to real area is obtained with: LČ/Vas 2 exs/a! = 2 — expŠ - x- | -xerfc 2 4atn 24atj (18) and: T(x,t) = - k i at I x 2 J— exp 4at H 2at ( Tw-T ok)+T 0 (19) The equation (19) represents the temperature distribution in the strand in the secondary cooling zone. If T(x,t) = T(0,t) = Tw (20) the surface temperature of strand is: Č30 20 *0 *v*o ysoö mo 4ShO T0/°C 4SQO 4StO Figure 1: Dependence cooling time for the surface temperature of 1193 °C versus strand initial temperature in zone I Slika 1: Čas ohlajanja do temperature površine 1193 °C v odvisnosti od začetne temperature gredice v zoni I Figure 2: Dependence cooling time for the surface temperature of 1164 °C versus strand initial temperature in zone II Slika 2: Čas ohlajanja do temperature površine 1164 °C v odvisnosti od začetne temperature gredice v zoni II s x x 304 MATERIALI IN TEHNOLOGIJE 38 (2004) 6 V. GROZDANI], A. MARKOTI]: THE TEMPERATURE DISTRIBUTION IN THE STRAND Figure 3: Dependence cooling time for the surface temperature of 1064 °C versus strand initial temperature in zone III Slika 3: Čas ohlajanja do temperature površine 1064 °C v odvisnosti od začetne temperature gredice v zoni III Figure 4: Dependence cooling time for the surface temperature of 1036 °C versus strand initial temperature in zone IV Slika 4: Čas ohlajanja do temperature površine 1036 °C v odvisnosti od začetne temperature gredice v zoni IV continuous – cast strand consists of four zones 3. In the first zone, with the strand length, L = 1 m, the surface temperature of the strand is Tw = 1193 °C, and the convective coefficient of heat transfer for water is of h = 618 W/m2 K. Figure 1 shows the time – temperature dependence in zone I, based on the results obtained from equation (23), Figure 2 shows the time – temperature dependence in zone II (L = 2 m, Tw = 1164 °C, h = 500 W/m2K), Figure 3 shows the time – temperature dependence in zone III (L = 3 m, Tw = 1064 °C, h = 471 W/m2K) and Figure 4 shows the time – temperature dependence in zone IV (L = 4 m, Tw = 1036 °C, h = 368 MATERIALI IN TEHNOLOGIJE 38 (2004) 6 Figure 5: Dependence cooling time for the surface temperature of 1000 °C versus strand initial temperature for a casting machine with standard cooling Slika 5: Čas ohlajanja do temperature površine 1000 °C v odvisnosti od začetne temperature gredice za livno napravo s standardnim ohlajanjem W/m2 K). For a casting machine with standard cooling, the average exchange coefficient of h = 418 W/m2 K and surface temperature of the strand of Tw = 1000 °C, the time – temperature dependence is presented in Figure 5. The dependence cooling time versus strand initial temperature is moderately hyperbolic. The simulation is performed for the real continuous casting machine in Maizieres-les-Metz in France. From the data in Figure 4 and equation (19) we determined that at the time of 234 s, the surface temperature is of 1036 °C and for casting billet of size 100 mm × 100 mm the thickness of the solid skin is of 25 mm and the thickness of mushy zone is of 7 mm. Equation (23) has the following limitations for a metallurgical height of 10 m, it is possible to withdraw billets of 100 mm × 100 mm at maximum speed of 50 mm/s. 3 CONCLUSION A mathematical model of the cooling of the continuous cast strand in the secondary zone was developed. The Fourier's partial differential equation of heat conduction with physically realistic assumptions is solved using Laplace transforms. A new equation for the temperature distribution in the continuous cast strand in the zone of secondary cooling was obtained. The equation can be used to deduce the critical time when the transformation are ? ? a can generate internal stresses and also cracks. 305 V. GROZDANI], A. MARKOTI]: THE TEMPERATURE DISTRIBUTION IN THE STRAND 4 LITERATURE 1 E. R. G. Eckert, R. M. Drake, Analysis of heat and mass transfer, McGraw-Hill Kogakuska, Tokyo, 1972 2 M. R. Spiegel, Laplace transforms, McGraw-Hill, New York, 1965 3 R. Alberny, Heat transfer and solidification in continuous casting, in: Commission of the European Communites, Casting and solidifi- cation of steel, Luxembourg, 1977, Vol. 1, IPC Science and Technology Press Ltd., Guildford, 1977, 278–338 R. D. Pehlke, A. Jeyarajan, H. Wada, Summary of thermal properties for casting alloys and mold materials, University of Michigan, Ann Arbor, 1982 306 MATERIALI IN TEHNOLOGIJE 38 (2004) 6