Schematic representation of  the  Single Roll Continuous Casting of Steel Strips is shown in Figure 1. The liquid metal at a particular temperature Tl it held in the res-ervoir where the liquid metal level is always kept cons-tant. The  metal continuously  enters  the annular space between the rotating chilled drut  and  the nozzle block through a graphite (refractory)  nozzle. As  soon as the

molten metal comes in contact  with  the  drum a skin of solid metal is formed which continues to grow as long as the liquid is in contact with this solidifies skin. Water at an input temperature TC extracts heat from the copper drum between the angles h and P2 . The drum rotates at an angular velocity W is anticlockwise directio

Mehrotra, S P

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MATHEMATICAL MODELLING OF CONTINUOUS CASTING OF STEEL STRIPS BY SINGLE ROLLPROCSSDr S.P. MehrotraDepartment of Metallurgical EngineeringIndian Institute of Technology, Kanpur- 208 016PROCESS DESCRIPTIONSchematic representation of the Single Roll Continuous Casting of SteelStrips is shown in Figure 1. The liquid metal at a particular temperature Tl itheld in the reservoir where the liquid metal level is always kept constant. ThEmetal continuously enters the annular space between the rotating chilled drutand the nozzle block through a graphite (refractory) nozzle. As soon as themolten metal comes in contact with the drum a skin of solid metal is formewhich continues to grow as long as the liquid is in contact with this solidifiesskin. Water at an input temperature TC extracts heat from the copper druibetween the angles h and P2 . The drum rotates at an angular velocity W isanticlockwise direction. Skin formation begins at angle h and completes aP2 (see Figure 1). The liquid metal near the interface has a temperaturTp which is close to its freezing point. At the interface the followintransformation reaction takes place:LatentheatLiquid -------> SolidMATHEMATICAL MODELLINGMacroscopic ModelA macroscopic model is described below which is based on the rate of growth osolidifihation front in the angular direction as a function of enthalpy. TiModel predicts speed of rotation of the drum and the final strip thickness.The model is based on the following assumptionsi. The process is assumed to be at steady state.ii. Heat flow occurs primarily in the angular and radial directions.iii. Liquid metal at temperature Ti is contained in the reservoir and it hasconstant height. Inside the pool, the liquid is perfectly mixed' and hensat uniform temperature. This is designated by TP which is the temperatutof liquid adjacent to the liquid-solid interface.1iv. A particular overall heat transfer coefficient is assumed for the processand the equations solved for the above predictions. The heat transfercoefficient used here is between the liquid - solid interface to water.v. The skin formation starts at angle h and the final strip thickness tf isobtained at angle .The model consists of the following equations:( i) Angular relationship between the dimensional parameters.(ii) Continuity equation for the description of process along with boundaryconditions.k 1 d dT dT[ - - ( r - )] - W - - O^SCps r dr dr da(For meaning of symbols please refer to the list of symbols at the end ofthis report.)(iii) Development of solidified shell.dT dt-k - - hMS(Tp -TF) = FS - W [ H(Tp) - HS(TF)Jdr da( Equation at the interface )(iv) Heat balance on the liquid pool( v) Solidification and removal of latent heat.(vi) Analytical solution of the governing equation based on the assumption thatthe solidified shell thickness is small enough to make the thermalconductivity large.The following final equation gives the strip thickness)* Rd tf 1 * hMS(TP - TF )(°< -Alt - ) (q-1) (1 - exp[-( - )( - (t + * * * * )J)tf Rd q H (T P - H (TF )*where HSW (TF -TW )qHMS(TP - TF), ZDimensionless thickness t - t/tf* TX TSDimensionless temperature T =iTL - TSwhere subscript i may be P, F, 1 or W refering to liquid metal pool adjacent todrum, freezing, liquid metal in the resevoir, and water, respectively.Dimensionless heat transfer coefficient h* _jwhere the subscript j may be MS or SW.2 IT hjcps FS W tfDimensionless Entholpy at temperature T*, H*(T*)H(T)cpS (TL - TS)Dimensionless heat capacity , cp = cp/cpsDimensionless latent heat, L* _LcPS (TL - Ts)Dimensionless angle 0 = 0 /27Twhere Omay refer to h, k or oCAll equations are expressed in the form of dimensionless variables. Theequations are solved analytically for various sets of input conditions whichinclude :i. Superheat of liquidii. Approximated freezing temperature for the alloyiii. R.P.M..of the rolliv. Final strip thicknessv. Heat transfer coefficientsSince at the present moment the heat transfer coefficient between the waterspray and copper drum surface is not known the model equations have been solvedfor a wide range-;of heat trnasfer coefficients. The other data used in the modelare given below.Heat capacity of liquid ( steel ) : 866.0 W kg-l oC lHeat capacity of solid ( steel ) 690.0 W kg-1 O C-1o3Latent Heat evolved 272142. 0 W kg-1Melt-Interface ht.tr.coefficient : 10000.0 W M-2 °C lSteel thermal conductivity 16.0 W M-1 ° C-lCopper thermal conductivity 380.0 W m-1 o C-1The solution of model equations results in selecting that heat transfercoefficient which would give the assumed final strip thickness at P2 andconfirms the R.P.M. which made it possible. A few typical results obtained usingthis model are presented below.Figure 2 represents the operating conditions required to produce a strip offinal thickness of 0.001 m. Figure 3 is a similar representation of a strip offinal thickness of 0.005 m. A typical computer output showing these results inthe tabular form is shown in Table I. I "Similar results have also been obtained for the following operating conditionsi. Strip thickness : 0.002, 0.003, 0.01 mii. Different superheats : 1650 & 1700 °CMicroscopic Model Based on Enthalpy BalanceThe macroscopic model described above presents only a simplified description ofthe process. To present a more realistic situation a microscopic model based onoverall enthalpy balance is developed.To develop such a model the various process components are discretised intoseveral small elements as shown in Figure 4. It can be seen that the solidifiedstrip is divided into 100 cylindrical elements of equal width along its lengthand 10 equal elements at the exit point of the strip from the liquid pool. TheCapper drum is similarly divided into 100 equal elements along the length and 50elements along the width at exit point.The model is based on the following assumptionsi. An initial thickness ti is obtained as soon as the liquid strikes the drumat Pi. The final thickness tf is attained at . Between P1 and thethickness increases linearely.,V Lfii. Heat transfer coefficient between copper drum and water is specified forthe particular spray nozzle geometry and the water flow rate.iii. Constant high temperature thermal conductivity value prevails in thesolid.iv. Freezing temperature of the liquid alloy is assumed to be constant.v. There is perfect thermal contact between Copper drum and strip and there isno slip between the two during the rotation of the drum. IThe model involves enthalpy balance along the boundaries of the systeacomprising of elements which results in 10 algebraic equations.The solution of model equation results in predicting the required speed ofrotation of the drum for a strip of specified thickness. It also predicts thecomplete temperature profile in :(i) the strip, while it is growing(ii) the Copper drumAs in previous case the computations are being carried out for various assume(values of heat transfer coefficient between the water spray and the inne:surface of Copper drum. The results are being predicted for(i) strip thicknesses of 0.003, 0.005, 0.010 m(ii) Liquid metal temperature ranging between 1550 and 1650 °C(iii)Heat transfer coefficients (Water-inner surface of Cu drum) varying betwee1000 - 100000 W m 2 °C 1Microscopic Model Based on Mass, Momentum and Heat BalanceAs a next step to the microscopic model based on enthalpy balance, we havundertaken the development of a model based on microscopic balance of massmomentum and heat balance.The model involves a set of partial differential equations of Second orderThese equations are valid for the three regions:(1) solidifying strip, (2) solid/liquid interface and (3) liquid metal poccontained between the nozzle block and the drum.An important aspect of the model is that the velocity field in the annular spacebetween the nozzle block and drum is predicted by solving :(1) Conservation of momentum equation(ii) Conservation of mass equation(iii)Conservation of energy equation(Navier-Stokes Equation)iFor the process under observation these equations reduce to a two dimensionalform. Further it is assumed that there is no radial gradient for pressure - thisreduces the momentum balance to 1 - dimension.By solving the momentum balance equation, the velocity field in the liquidregion can be mapped and the convective heat transfer quantified.The boundary conditions to solve the model equation involve :(i) An overall heat transfer coefficient from the solidified strip to thecooling water in the spray.(ii) At the refractory-liquid interface heat flow outwards is taken as zero,because the refractory is assumed to be a perfect insulator.(iii)At the solid/liquid interface the exact location of the solid boundary isdetermined by interpolation and the latent heat evolved at this reactionzone accounted for.Solution of Model EquationFinite difference scheme is employed to solve the partial differentialequations. After substituting the finite differences for the partialderivatives, a set of linear Ist order simultaneous equations are obtained.These are solved using IMPROVED GAUSSIAN ALGORITHM. The solution gives thetemperature at various points.After knowing the temperature distribution, the phase boundaries are located byrunning a subprogram, and the solidified strip thickness development from thestart to the final value calculated., I",64The model can predict the following :i. Strip thickness as a function of speed of rotation of the drum.ii. Strip thickness as a function of liquid superheat.iii. Strip thickness as function of heat transfer coefficient.iv. Limiting conditions for the process.LIST OF SYMBOLSTemperatures:T Variable temperature, KTi Temperature of liquid in reservoir, KTP Temperasture of liquid in pooladjacent to the solid/liquid interface, KTF Freezing temperature of liquid, KTL Liquidus temperature, KTSTcSolidus temperature, KTemperature of water, KInner surface temperature of the drum, KGeometrical Dimensionsti Initial Strip thickness at angle h, mtf Final strip thickness at angle k) mt Variable strip thickness between h and h, m0C Variable angle, radiansRd Radius of drum, mAngle at which solidification starts , radiansAngle at which strip attains final thickness , radiansr Variable radius, m; Rd < r < Rd+th Height of liquid in reservoir, mx Entry radius for liquid at nozzle, my Exit radius for liquid at nozzle, mW Angular velocity of drum, radian/secThermal QuantitiescpS Heat capacity of solid, J kgloC 1cpl Heat capacity of liquid, J kg1oC 1-1H[ ] Enthalpy at the temperature mentioned, J kg-1L Latent heat of liquid-solid transformation, J kgF"^T8liquid metal and interface, W m-2 oC 1k Thermal conductivity of solid, W m-2 o C-1Note: Starred quantities refer to the dimensionless forms defined in the text.9TABLE 1TABLE 1A : A typical computer output showing the resultsof macroscopic heat balance modelSTRIP THICKNESS - 0.001 METRESLIQUIDTEMPDEG.COVERALL HT.TR.COEF.W/SQ.M.CR.P.M.POOLTEMP.DEG.CANGLE P2RADIANS1600.0 780.00 3.00 1533.9 2.1901600.0 820.00 3.00 1533.9 2.1471600.0 860.00 3.00 1533.9 2.1081600.0 900.00 3.00 1533.9 2.0741600,0 940.00 3.00 1533.9 2.0431600.0 980.00 3.00 1533.9 2.0161600.0 1020.00 3.00 1533.9 1.9901600.0 1060.00 3.00 1533.9 1.9661600.0 1100.00 3.00 1533.9 1.9451600.0 1140.00 3.00 1533.9 1.9251600.0 1180.00 3.00 1533.9 1.9071600.0 1220.00 3.00 1533.9 1.8911600.0 1260.00 3.00 1533.9 1.8751600.0 1300.00 3.00 1533.9 1.8611600.0 1340.00 3.00 1533.9 1.8471600.0 1380.00 3.00 1533.9 1.8341600.0 1420.00 3.00 1533.9 1.8221600.0 1460.00 3.00 1533.9 1.8111600.0 1500.00 3.00 1533.9 1.8011600.0 1540.00 3.00 1533.9 1.7901600.0 1020.00 4.00 1538.5 2.2071600.0 1060.00 4.00 1538.5 2.1731600.0 1100.00 4.00 1538.5 2.1421600.0 1140.00 4.00 1538.5 2.1141600.0 1180.00 4.00 1538.5 2.0881600.0 1220.00 4.00 1538.5 2.0641600.0 1260.00 4.00 1538.5 2.042-1600.0 1300.00 4.00 1538.5 2.0211600.0 1340.00 4.00 1538.5 2.0011600.0 1380.00 4.00 1538.5 1.9831600.0 1420.00 4.00 1538.5 1.9661600.0 1460.00 4.00 1538.5 1.9501600.0 1500.00 4.00 1538.5 1.9351600.0 1540.00 4.00 1538.5 1.9211600.0 1300.00 5.00 1542.4 2.1891600.0 1340.00 5.00 1542.4 2.1641600.0 1380.00 5.00 1542.4 2.1401600.0 1420.00 5.00 1542.4 2.1171600.0 1460.00 5.00 1542.4 2.0971600.0 1500.00 5.00 1542.4 2.0761600.0 1540.00 5.00 1542.4 2.0581600.0 1540.00 6.00 1545.9 2.200^0TABLE 1 CONTINUED ...TABLE 1B : A typical computer output showing the resultsof macroscopic heat balance modelSTRIP THICKNESS - 0.001 METRESLIQUID OVERALL HT.TEMP TR.COEF. R.P.M.DEG.C W/SQ.M.CPOOL ANGLE P2TEMP.DEG.C RADIANS----------------1650.0 580.00 2.00 1536.6 2.1971650.0 620.00 2.00 1536.6 2.1341650.0 660.00 2.00 1536.6 2.081k1650.0 700.00 2.00 1536.6 2.0351650.0 740.00 2.00 1536.6 1.9951650.0 780.00 2.00 1536.6 1.9611650.0 820.00 2.00 1536.6 1.9311650.0 860.00 2.00 1536.6 1.9041650.0 900.00 2.00 1536.6 1.8801650.0 940.00 2.00 1536.6 1.8581650.0 980.00 2.00 1536.6 1.8381650.0 1020.00 2.00 1536.6 1.8201650.0 1060.00 2.00 1536.6 1.8041650.0 1100.00 2.00 1536.6 1.7891650.0 1140.00 2.00 1536.6 1.7751650.0 1180.00 2.00 1536.6 1.7631650.0 1220.00 2.00 1536.6 1.7511650.0 1260.00 2.00 1536.6 1.7401650.0 1300.00 2.00 1536.6 1.7301650.0 1340.00 2.00 1536.6 1.7201650.0 1380.00 2.00 1536.6 1.7121650.0 1420.00 2.00 1536.6 1.7031650.0 1460.00 2.00 1536.6 1.6961650.0 1500.00 2.00 1536.6 1.6881650.0 1540.00 2.00 1536.6 1.6821650.0 860.00 3.00 1545.0 2.2091650.0 900.00 3.00 1545.0 2.1651650.0 940.00 3.00 1545.0 2.1261650.0 980.00 3.00 1545.0 2.0911650.0 1020.00 3.00 1545.0 2.0601650.0 1060.00 3.00 1545.0 2.0311650.0 1100.00 3.00 1545.0 2.0051650.0 1140.00 3.00 1545.0 1.9821650.0 1180.00 3.00 1545.0 1.9601650.0 1220.00 3.00 1545.0 1.9401650.0 1260.00 3.00 1545.0 1.9211650.0 1300.00 3.00 1545.0 1.9041650.0 1340.00 3.00 1545.0 1.8881650.0 1380.00 3.00 1545.0 1.8731650.0 1420.00 3.00 1545.0 1.8591650.0 1460.00 3.00 1545.0 1.8461650.0 1500.00 3.00 1545.0 1.8331650.0 1540.00 3.00 1545.0 1.8221650.0 1140.00 4.00 1552.3 2.214TABLE 1B CONTINUED ...STRIP THICKNESS - 0.001 METRES------------------------------------------------LIQUID OVERALL HT. POOL ANGLETEMP TR.COEF. R.P.M. TEMP.DEG.C W /SQ.M.C DEG.C RADIANS------------------------------------------------1650.0 1180 . 00 4.00 1552 . 3 2.1811650.0 1220 . 00 4.00 1552.3 2.1511650 . 0 1260 . 00 4.00 1552.3 2.1231650.0 1300.00 4 . 00 1552.3 2.0971650 . 0 1340.00 4.00 1552.3 2.0731650.0 1380.00 4.00 1552 . 3 2.0511650.0 1420 . 00 4.00 1552.3 2.0311650 . 0 1460.00 4 . 00 1552.3 2.0111650 . 0 1500 . 00 4.00 1552.3 1.9931650.0 1540 . 00 4.00 1552 . 3 1.9761650.0 1460.00 5.00 1558.6 2.1911650.0 1500 . 00 5.00 1558.6 2.1661650 . 0 1540.00 5.00* 1558.6 2.143-----------------------------------------------TABLE 1 CONTINUED ...TABLE 1C : A typical computer output showing the resultsof macroscopic heat balance modelSTRIP THICKNESS - 0.001 METRESLIQUIDTEMPDEG.COVERALL HT.TR.COEF.W/SQ.M.CR.P.M.POOLTEMP.DEG.CANGLE P2RADIANS1700.0 340.00 1.00 1531.1 2.1401700.0 380.00 1.00 1531.1 2.0381700.0 420.00 1.00 1531.1 1.9621700.0 460.00 1.00 1531.1 1.9041700.0 500.00 1.00 1531.1 1.8571700.0 540.00 1.00 1531.1 1.8191700.0 580.00 1.00 1531.1 1.7881700.0 620.00 1.00 1531.1 1.7611700.0 660.00 1.00 1531.1 1.7381700.0 700.00 1.00 1531.1 1.7191700.0 740.00 1.00 1531.1 1.7021700.0 780.00 1.00 1531.1 1.6861700.0 820.00 1.00 1531.1 1.6731700.0 860.00 1.00 1531.1 1.6611700.0 900.00 1.00 1531.1 1.6501700.0 940.00 1.00 1531.1 1.6411700.0 980.00 1.00 1531.1 1.6321700.0 1020.00 1.00 1531.1 1.6241700.0 1060.00 1.00 1531.1 1.6161700.0 1100.00 1.00 1531.1 1.6101700.0 1140.00 1.00 1531.1 1.6031700.0 1180.00 1.00 1531.1 1.5971700.0 1220.00 1.00 1531.1 1.5921700.0 1260.00 1.00 1531.1 1.5871700.0 1300.00 1.00 1531.1 1.5821700.0 1340.00 1.00 1531.1 1.5781700.0 1380.00 1.00 1531.1 1.5741700.0 1420.00 1.00 1531.1 1.5701700.0 1460.00 1.00 1531.1 1.5671700.0 1500.00 1.00 1531.1 1.5631700.0 1540.00 1.00 1531.1 1.5601700.0 660.00 2.00 1544.6 2.1721700.0 700.00 2.00 1544.6 2.1141700.0 740.00 2.00 1544.6 2.0651700.0 780.00 2.00 1544.6 2.0221700.0 820.00 2.00 1544.6 1.9851700.0 860.00 2.00 1544.6 1.9531700.0 900.00 2.00 1544.6 1.9241700.0 940.00 2.00 1544.6 1.8981700.0 980.00 2.00 1544.6 1.8751700.0 1020.00 2.00 1544.6 1.8541700.0 1060.00 2.00 1544.6 1.835TABLE 1C CONTINUED,.STRIP THICKNESS - 0.001 METRESLIQUIDTEMPDEG.COVERALL HT .TR.COEF.W / SQ.M.CR.P.M.POOLTEMP.DEG.CANGLERADIANS1700.0 1100.00 2.00 1544.6 1.8181700.0 1140.00 2.00 1544.6 1.8021700.0 1180.00 2.00 1544.6 1.7881700.0 1220.00 2.00 1544.6 1.7751700.0 1260.00 2.00 1544. 6 1.7621700.0 1300 .00 2.00 1544.6 1.7511700.0 1340. 00 2.00 1544.6 1.7401700.0 1380.00 2.00 1544.6 1.7301700.0 1420.00 2.00 1544.6 1.7211700.0 1460 . 00 2.00 1544 .6 1.7121700.0 1500. 00 2.00 1544.6 1.7041700.0 1540.00 2.00 1544.6 1.6971700.0 980.00 3.00 1556.1 2.1831700.0 1020 .00 3.00 1556.1 2.1441700.0 1060.00 3.00 1556.1 2.1081700.0 1100 .00 3.00 1556.1 2.0761700.0 1140. 00 3.00 1556.1 2.0471700.0 1180 . 00 3.00 1556.1 2.0201700.0 1220 . 00 3.00 1556.1 1.9961700.0 1260 . 00 3.00 1556.1 1.9741700.0 1300 . 00 3.00 1556.1 1.9531700.0 1340.00 3.00 1556.1 1.9341700.0 1380 . 00 3.00 1556.1 1.9171700.0 1420. 00 3.00 1556.1 1.9001700.0 1460 . 00 3.00 1556.1 1.8851700.0 1500 . 00 3.00 1556.1 1.871 11700.0 1540. 00 3.00 1555 . 1 1.8571700.0 1300 . 00 4.00 1566.1 2.1891700.0 1340.00 4.00 1566.1 2.1591700.0 1380 . 00 4.00 1566.1 2.1321700.0 1420. 00 4.00 1566.1 2.1061700.0 1460 .00 4.00 1566.1 2.0831700.0 1500 . 00 4.00 1566.1 2.0611700.0 1540. 00 4.00 1566.1 2.040FIGURE 1 Schematic representation of one roll stripcaster2.3wCaVaD`C02.22.1c 2.0U'0 1.90aa1.8E1,70UL-01.6w 1.50000CqxqxStrip Thickness : 0.001 mliquid Metal Temp. : 1600 °CRPMo 3.0x 4.0o 5.000x 0x 0x0xqx0xqx0xqxq0x00xq0x0rn1.4500 700 900 1100 1300 1500Heat Transfer Coefficient, W m ?K 1FIGURE 2 Operating conditions required to produce astrip of thickness 0.001 metresitU)C2.302.20OC2.1C00 2.0U1.90WaaaE0U0I*--1.81.71.600CxxxX0 x0 x X000ra-Strip Thickness : 0.005 mLiquid Metal Temp . : 1600 °CRPM0 0.6x 0.90 1.20 W,0 0 0 Fai a0 0 0XFw- 0x0< 1.41 1 1 1 1 1 1 1 1500 700 900 1100 1300 1500Heat Transfer Coefficient, Wm ^KFIGURE 3 Operating conditions required to produce astrip of thickness 0.005 metres71 -7GFIGURE 4 Schematic representation of microscopic modelbased on enthalpy balancergr 1,6

Mathematical Modelling of Contrinuous Casting of Steel Strips by Single Roll Process

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Mathematical Modelling of Contrinuous Casting of Steel Strips by Single Roll Process

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