ISSN 09670912, Steel in Translation, 2016, Vol. 46, No. 2, pp. 83–87. © Allerton Press, Inc., 2016. Original Russian Text © A.V. Fedosov, 2016, published in “Izvestiya VUZ. Chernaya Metallurgiya,” 2016, No. 2, pp. 82–87.
Simulation of Slab Formation in a ContinuousCasting Machine A. V. Fedosov Priazovsk State Technical University, Mariupol, Ukraine email: fedosov
[email protected] Received October 15, 2015
Abstract—A discrete analog of the differential heatconduction equation permits the use of nonuniform cal culation grids in the simulation of continuous casting. That allows the distribution of the temperature gradi ents in the model to be taken into account, with corresponding increase in the accuracy of the approximation and the results. A mathematical model is developed for the solidification and shrinkage of continuouscast slab in the mold. The adoption of a nonuniform grid permits the use of elements measuring 1–2 mm in the simulation. This model is used to study the distortion of the slab cross section at the mold walls. Calculation of the geometric profile permits refinement of the thermal and mechanical interaction of the solidifying shell and the mold walls and determination of the optimal mold taper so as to reduce the risk of surface and sub surface cracking in the slabs. Keywords: finitedifference method, differential equation, discrete analog, continuouscast slab, mold, taper, heat flux, heattransfer coefficient, shrinkage DOI: 10.3103/S0967091216020054
cretization of the model. Such grids cannot take account of the distribution of temperature gradients within the model. That leads to less accurate results. Accordingly, the development of finitedifference methods in which nonuniform calculation grids may be used will improve their effectiveness to match that of finiteelement methods, with significantly fewer demands on the computer. In the literature, however, little attention has been paid to nonuniform calcula tion grids in the finitegrid simulation of continuous casting. In the present work, we develop a finitedifference model of the solidification and shrinkage of free slab in the mold of a continuouscasting machine on the basis of a nonuniform calculation grid, so as to improve the speed and precision of thermal and geometric calcula tions. The nonsteady cooling of the continuouscast slab is simulated on the basis of the heatconduction equa tion [12, 13]
Although the mold takes up little of the strand length in the continuouscasting machine, processes in the mold have a significant influence on the quality of the cast slab. Most defects in the continuouscast slab appear and develop within the mold [1–3]. Accordingly, the processes in the mold require careful attention. With the development of computer technology, mathematical simulation is widely used in the study of metallurgical processes. Today, many different models of slab formation in the mold of a continuouscasting machine are available. The simplest onedimensional models are generally used to study slab cooling and solidification [4]. Twodimensional models permit calculation of the strength and shrinkage of the slab cross section [5–7]. In hydrodynamic problems, threedimensional models require three measure ments [8–10]. The numerical methods most com monly employed are the finitedifference method and the finiteelement model. The finiteelement model has numerous benefits: for example, it is universal; it permits the description of practically any object; and nonuniform calculation grids may be employed [8, 11, 12]. However, it makes considerable demands on the personal computers employed. The finitedifference method considerably reduces the demands on the computer. Since the continuous cast slab is rectangular in most cases, the finitediffer ence method may be used with high precision. Accordingly, it has been widely used in the simulation of continuous casting. However, a deficiency of most known finitedifference models is uniform spatial dis
∂T ρC = div ( λgradT ) + Qν, ∂τ
(1)
where T is the metal temperature, °C; C is the specific heat of the metal, J/kg °C; ρ is the metal density, kg/m3; λ is the thermal conductivity of the metal, W/m °C; τ is the time, s; Qν is the bulk heat source, W. The use of the finitedifference method calls for the replacement of the infinitesimal components in the differential equation by finite values, on the basis of spatial discretization and the assumption that the tem 83
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Fig. 1. Example of the discretization of a twodimensional space by means of uniform (a) and nonuniform (b) calculation grids.
perature varies linearly within the element. Then Eq. (1) is replaced by its discrete analog, from which a system of linear equations may be obtained. The model is usu ally developed into rectangular elements of equal vol ume. That considerably simplifies the construction of the grid. However, in view of the deficiencies of this approach already noted, we employ a nonuniform cal culation grid. The basic requirement on the grid is that the points must be at the intersection of lines parallel to the coor dinate axes; only the distance between the lines varies. With decrease in the line spacing, the approximation is potentially more precise. For the modeling of slab cooling with allowance for the local heattransfer coefficients, it is expedient to increase the precision at the shell of the slab—that is, at the boundary of the calculation grid. Taking account of the parabolic tem perature distribution in the calculation region, we pro pose the following equation –1
n L ⎛ 1⎞ , Δx i = ⎜ ⎟ i ⎝ k = 1 k⎠
∑
(2)
where Δxi is the linear dimension of element i, m; L is the total linear dimension of the model, m; n is the n 1 total number of elements; and is a normal k=1 k izing factor.
∑
The elements are numbered from the edge of the grid to the thermal center of the cross section.
In most cases, the finitedifference method does not include the precise derivation of formulas from Eq. (1). Instead, simplified heatbalance equations are formulated on the basis of fluxes parallel to the coor dinate axes that travel from point to point. We calcu late the mean value of the heat fluxes through the boundaries of the control volume in the direction of the coordinate axes. The control volumes are shaded in Fig. 1. The boundaries of the control volume are deter mined by the intersection of lines through the mid points between the coordinates of the grid points. When using a uniform calculation grid, the geometric dimensions of the control volumes and grid elements are the same, while the thermal center of the control volume is its geometric center (Fig. 1a). That signifi cantly simplifies the formulation of the discrete analog for the differential heatconduction equation. When using a nonuniform calculation grid, there are two possible configurations of the control volume and the grid points. (1) When the grid point is at the intersection of the diagonals in the control volume and is associated with the thermal center. In this case, it is impossible to obtain a point at the grid boundary. The remedy here is to introduce an additional layer of zero thickness around the grid and to create points at the open boundaries of the control volumes (Fig. 1a). The new volumes do not disrupt the physical description, since they are characterized by zero specific heat and ther mal conductivity along the boundaries and infinitely larger thermal conductivity from the boundary into STEEL IN TRANSLATION
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the grid. However, the introduction of additional points increases the size of the system of equations. (2) When the boundary of the control volume is midway between the grid points, which accordingly are not at the geometric center of the control volume (Fig. 1b). This formulation looks simpler, and the points at the boundary may be obtained without the need for any additional measures. However, the dis placement of the grid points relative to the geometric center of the control volumes complicates the deriva tion of relations between the temperature field and the heatflux field. It is also difficult to take account of the specific heat and the other temperaturedependent properties of the metal, which may affect the accuracy. In the present work, we outline a twodimensional model of the slab cross section. A quarter of the slab cross section may be considered, in view of the heat transfer conditions in the mold. In that case, half the crosssectional width and height may be measured along the symmetry axes X and Y. In the calculations, we must also take account of the layer thickness of the cross section, which depends on the casting condi tions. The thickness runs along the Z direction, corre sponding to the longitudinal axis of the continuous casting machine, along which the plane coordinate system of the cross section moves. For these condi tions, we develop a discrete analog of Eq. (1) in which nonuniform calculation grids with displacement of the points relative to the center of the control volume may be used k+1
k
T i, j – T i, j ρC ei, j V ( i, j ) Δτ = ( Qx ( i + 1, j ) + Qx ( i – 1, j ) + Qy ( i, j + 1 ) + Qy ( i, j – 1 ) ),
(3)
here Ce(i, j) is the effective specific heat of the material k
in the control volume at point (i, j), J/kg; T i, j and
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To calculate the heat fluxes along the axes, we use the following equations k
k
λ ( i – 1, j ) + λ ( i, j )⎞ ⎛ T i, j – T i – 1, j⎞ Qx ( i – 1, j ) = ⎛ Sy ( j ) , ⎝ ⎠ ⎝ x(i) – x(i – 1) ⎠ 2 k
(6)
k
λ ( i + 1, j ) + λ ( i, j )⎞ ⎛ T i + 1, j – T i, j⎞ Qx ( i + 1, j ) = ⎛ Sy ( j ) , (7) ⎝ ⎠ ⎝ x(i + 1) – x(i) ⎠ 2 where λ is the thermal conductivity at the correspond ing point, W/m K; Sy(j) is the area of the control vol ume’s boundary parallel to the Y and Z axes y(j + 1) + y(j – 1) k Sy ( j ) = Z . 2
(8)
The heat flux Qy in the direction of the Y axis and the area Sx(i) in the direction of the X and Z axes are calculated analogously to Eqs. (6)–(8). For points at the outer boundary, the external heat fluxes are determined on the basis of boundary condi tions of the third kind. The heattransfer coefficient is calculated for each external point, on the basis of the conditions of contact heat transfer and radiant heat transfer through the gas gap [14, 15]. In calculating the contact heat transfer, we take account of the thickness of the slagforming layer in the gap between the mold wall and the solidifying shell. The thickness of the slag forming layer initially is assumed to be 1–3 mm. Then, over the height of the mold, the thickness of the flux layer varies as a function of the shrinkage of the slab cross section and the taper of the mold walls. The shrinkage of the slab cross section is determined on the basis of the change in linear dimensions of the elements along the orthogonal directions X and Y k+1
l i, j
( α tk + 1 + α tk ) k + 1 k k i, j i, j ( T i, j – T i, j )⎞ , = l i, j ⎛ 1 + ⎝ ⎠ 2
(9)
k+1
T i, j are the temperatures at point (i, j) at times k and k + 1, respectively, K; Qx and Qy are the heat fluxes along the X and Y axes, respectively, W; V(i, j) is the local volume at point (i, j), m3; Δτ is the time step, s. In Ce(i, j), besides the direct dependence on the temperature, the heat of phase transition is taken into account [12]. The local volume is calculated from the formula ( y(j + 1) + y(j – 1) ) ( x(i + 1) + x(i – 1) ) k V ( i, j ) = Z , 4
(4)
where x(i), y(i) are the coordinates of the grid points rel ative to the X and Y axes, respectively, m; Z k is the layer thickness of the cross section in the finitedifference model, m. The layer thickness is determined in accordance with the casting speed v (m/s) k
Z = ν/Δτ. STEEL IN TRANSLATION
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k
k+1
where l i, j and l i, j are the linear dimensions of the ele ment at the present time and after the next time step, respectively, m; αt is the linearexpansion coefficient of the steel, which depends on the temperature, °C–1. Note that Eq. (9) describes the free shrinkage of grid elements. Assuming continuity of the material in the model, we may calculate the shape of the slab cross section due to free shrinkage of the steel on solidifica tion and cooling. On the basis of the model, we may derive the thermal field and the form of the slab cross section (Fig. 2). Predicting the slab profile, we may assess the inter action of the solidifying shell with the mold walls and role of wall taper in the formation of the continuous cast slab. The shrinkage of the steel shell over the broad face is significantly greater than that over the narrow face. That distorts the slab profile at its corners, as seen in Fig. 2. The nonuniform calculation grid permits
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FEDOSOV (a) X, mm 70
70
0
140
210
280
(b)
1
350 (c)
420
490
560
630
700 770 Y, mm
(d)
2
3 5
4
Fig. 2. Thermal and geometric profiles of the slab cross section at a lower point of the mold: (a) general view of calculation grid; (b) insufficient taper; (c) excess taper; (d) optimal taper; (1) mold; (2) liquid steel; (3) solid shell; (4) flux layer; (5) gas gap.
detailed assessment of the slab distortion. (The elements at the corners of the slab in Fig. 2 measure 1.5–2.0 mm.) In Figs. 2b–2d, we consider some possible interac tions of the steel shell with the mold walls. Analysis of the results permits the following conclusions. The physical properties of the cast steel and the heattransfer conditions over the broad face of the mold largely determine the shrinkage of the steel shell over the slab’s broad face. Local decrease in the heat transfer coefficient at the corners of the slab due to the contact conditions and the gas gap has practically no influence on the overall shrinkage. Sharp changes in the heat flux at the corners of the slab increase the thermal stress in the steel shell, with consequent increase in the likelihood of cracking. The ferrostatic pressure on the steel shell is in the opposite direction to the forces associated with shrink age. If the ferrostatic pressure is dominant, bulging of the slab will be observed, and tensile strain will lead to crack formation in the steel shell. In that case, the mold walls shape the slab and absorb the ferrostatic pressure. Depending on the relationship between the taper of the mold walls and the shrinkage of the steel shell, three different interactions of the slab and mold are possible. (1) Insufficient taper of the narrow mold walls (Fig. 2b). In that case, bulging of the narrow faces of the slab is accompanied by cracking at the corners of the slab. The sharp decrease in heat flux due to the appearance of a gas gap aggravates the situation. (2) Excess taper (Fig. 2c). In that case, the narrow mold walls act on the steel shell formed at the broad face of the slab, pushing it toward the slab axis. The forces applied to the two sides of the shell may result in
its deformation and change the contact conditions at the broad mold walls. In addition, the frictional forces at the lower corners of the mold increase, with conse quent increase in the wear of the working mold walls and loss of surface quality of the slab. (3) Optimal taper of the narrow mold walls (Fig. 2d). In that case, the problems in the first two cases are minimized and the surface quality of the slab is as good as possible. Optimal taper calls for maxi mum compensation of the shrinkage of the steel shell. That equalizes the heat flux at the contact perimeter of the slab and the mold walls; improves the thermal stress in the shell; and minimizes its deformation and the frictional forces between the mold wall and the steel shell. CONCLUSIONS A new discrete analog of the nonsteady differential heatconduction equation permits the use of nonuni form calculation grids in the finitedifference simula tion of continuous casting. That allows the distribu tion of the temperature gradients in the slab cross sec tion to be taken into account, with significant increase in the accuracy of the results. A mathematical model is developed for the solidification and shrinkage of con tinuouscast slab in the mold of the casting machine, permitting the prediction of the slab’s thermal and geometric profiles. Calculation of the geometric pro file permits refinement of the thermal and mechanical interaction of the solidifying shell and the mold walls and determination of the best mold taper so as to reduce the risk of surface and subsurface cracking in the slab. STEEL IN TRANSLATION
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Translated by Bernard Gilbert
