4. Numerical solution

Figure 1 illustrates the important geometrical features of the octagonal-shaped billet. Considering that the flow of heat takes place only in the radial direction toward the center of the octagon, and certainly toward the center of its circumscribed circle, it appears that due to symmetry, only a small trigonal part becomes the important region (the shaded area in Figure 1) for the solution of the DE of heat transfer. Actually, when uniform cooling takes place around the solidifying billet, then symmetrical conditions prevail in the relevant heat transfer. Consequently, heat flow takes place only in the radial direction. The strongly implicit scheme as presented by Patankar [23] was deployed for the solution of the DE of heat transfer. Figure 3 illustrates the control-volume selection for a small number of nodal points.

The five-point formulation for the discretization equations was applied. Special attention was paid in the boundary condition formulation of the discretization equations, and a more detailed analysis on this is presented in Appendix I. One may realize that once the boundary conditions are properly written, then a finer grid will ultimately permit the convergence to the engineering solution. In Figure 3, only the lower trigonal part is necessary to be examined. The vertical side, MZ, is the boundary from which the main heat flow (cooling) takes place; the horizontal side OZ and the diagonal side OM, with an inclination of π/8, or 22.5°, are considered to be adiabatic to heat flow due to the aforementioned symmetry reasons. This study is part of a series of published works with respect to the numerical solution of the heat transfer equation in 2D (square and rectangular) and 3D domains [24, 27–29]. Although the core of the existing program remained intact, due to the nature of the present trigonal domain, the part of the program related to the

Figure 3. Presentation of control volumes and nodal points for a coarse grid.

A Numerical Solution Model for the Heat Transfer in Octagonal Billets DOI: http://dx.doi.org/10.5772/intechopen.84305

boundary conditions had to be developed from scratch. The octagonal billet with R = 162.84 mm can be discretized with a fine grid of 200 ˜ 200 nodal points. In this way, the space intervals (δx, δy) were about δx = R cos(π/8)/199 = 0.756 mm and δy = R sin(π/8)/199 = 0.313 mm, respectively. Actually, keeping the ratio, δy/ δx = 0.313/0.756 = 0.414 = tan(π/8), exhibited a relatively good convergence; the time interval was Δt = 1 sec.

#### 5. Running the computer program

The Gauss-Seidel algorithm was applied for the iterative solution of the matrix equations in the case of the 42CrMo4 grade. Over-relaxation was used for the fastest possible convergence, and the over-relaxation parameter used was ω = 1.870, which has exhibited good behavior for this type of studies [30]. The applied maximum error (tolerance) for the computed temperature at each nodal point was 5 ˜ <sup>10</sup>°<sup>3</sup> . In addition to this, the Strongly Implicit Procedure (SIP) [31] was deployed for the iterative solution of the matrix equations in the case of the S355 J2 grade. In this case, however, the same tolerance (5 ˜ <sup>10</sup>°<sup>3</sup> ) was used for the maximum residual value and not the temperature difference at each nodal point. The C++ language was selected for the development of the program, and the deployed compiler was the Intel Parallel Studio XE 2015 Composer Edition for C++; the open source, crossplatform Code::Blocks, version 17.12 software was used as an integrated development environment for program edition and compiler invoking. The software was run in a DELL Alienware laptop with the Intel i7-6700HQ CPU (8 cores) at 2.6 GHz, 16GB RAM, running under a 64-bit Windows 10 Professional O/S.

### 6. Results and discussion

In order to validate the new developed program, a solution against the aforementioned theoretical one had to be sought. For this reason, an octagonal billet initially at 500°C, cooled afterwards by an hypothetical fluid at 30°C and with a heat transfer coefficient h = 200 W/m2 /K, was simulated. The values for the properties of density, thermal conductivity, and heat capacity were the same with the ones used in the theoretical solution. In total, 38 sets of data were randomly selected at various positions inside the cylindrical billet and were compared with similar results deduced from the numerical solution described here in order to verify its validity. Figure 4 presents the compared results of these two sets of computed data. An analysis of variance (ANOVA) was performed for these two sets of results, and the following statistical results were derived: residual standard error, 3.133 on 36 degrees of freedom (DF); multiple-R squared, 0.9957; and F-statistic, 8333 on 1 and <sup>35</sup>DF, with a p-value <2.2 ˜ <sup>10</sup>°16. Consequently, a standard error of 3.1°C was found to represent the mean statistical error between the results obtained from the theoretical and current numerical solutions, respectively.

The advantage of studying the heat transfer in an octagonal billet marks the importance of such an experiment: the potential to test some results against theoretical ones gives the confidence about attaining the proper numerical solution; in fact, the octagonal geometry approximates the circular cross-section better than a square one for this comparison to be accomplished.

Table 1 presents the chemical analysis for the two selected grades under examination: 42CrMo4 and S355 J2. In the bloom caster of Stomana, in Pernik, the billet size of 250 ˜ 300 mm2 is normally used for the production of special steels. The equivalent billet to this size in octagonal shape has a circumradius of

#### Figure 4.

Comparison of temperature values between the theoretical (for a cylindrical billet) and numerical (for an octagonal billet) solutions.


#### Table 1.

Chemical analysis for grades 42CrMo4 (1) and S355 J2 (2).

R = 162.84 mm, as aforementioned. An interesting idea is that putting the octagonal billet in practice, there is a potential to increase productivity by using more cooling water, an advantage that has been tested in practice [21] for normal rebar grades. Consequently, in this study a surplus of mold water by 12% for both of the two grades 42CrMo4 and S355 J2 was used compared to the normal water-cooling process applied in Stomana for these grades, respectively. Furthermore, a surplus of water in the air-mist region by 3% for the high-carbon grade 42CrMo4 and 4% for the peritectic grade S355 J2 was applied, respectively.

Figure 5 illustrates the shell thickness (curve 3) increase as an octagonal billet of grade 42CrMo4 travels down Stomana's continuous casting machine. One may notice that the solidification completes at about 32.4 m from the liquid-steel meniscus in the mold.

At the same time, the centerline temperature (curve 2) drops appreciably at that point revealing the complete solidification at that point, as well; the surface temperature is also presented by curve 1. The casting speed was 0.70 m/min, and the superheat (SPH) was 30°C in the computation. The SPH is actually the excess temperature above liquidus temperature; the liquidus temperature is exclusively a function of the chemical composition of a steel grade.

Figure 6 depicts the temperature distribution in the domain of interest of a 42CrMo4-grade octagonal billet at a position of 19.2 m from the meniscus. Carbon A Numerical Solution Model for the Heat Transfer in Octagonal Billets DOI: http://dx.doi.org/10.5772/intechopen.84305

Figure 5.

Surface (1) and centerline (2) temperature values in an octagonal billet of the 42CrMo4 grade; thickness evolution (3) as the billet solidifies downwards the continuous caster.

#### Figure 6.

Temperature distribution in the selected part of a 42CrMo4-grade octagonal billet. Solid fraction values (fs) are presented in the mushy zone.

steels normally solidify gradually from liquid to solid phase. The degree of solidification is generally described by a parameter that is called solid fraction (fs) and represents the percentage of the solid phase in the mixture of solid and liquid phases. When fs = 0, then we have 100% liquid phase, and the steel temperature at this point is the liquidus temperature, Tl. When fs = 1, then we have 100% solid phase, and the steel temperature at that point is the solidus temperature,Ts. Although the liquidus temperature is always a function of the chemical composition of a steel grade, this is not the case with the solidus temperature. The solidus

temperature is also affected by the local cooling rate at solidification (CR), which is expressed as

$$\mathbf{C}\_{R} = \frac{T\_{P} - T\_{P}^{\mathrm{0}}}{\Delta t} \tag{11}$$

Eq. (11) shows that the temperature difference between a temperature TP and the previous one TP <sup>0</sup> at point P inside a solidifying billet within a time interval Δt, divided by this time interval, defines that local cooling rate. Consequently, the solidus temperature may be given by a formula of the type:

$$T\_{\mathcal{S}} = \mathcal{G}\_1(chemical\ analysis, \mathcal{C}\_{\mathcal{R}}) \tag{12}$$

On the other hand, the solid fraction may be considered a function of the local cooling rate and temperature:

$$f\_{\mathcal{S}} = \mathcal{G}\_{\mathcal{Z}}(T, \mathcal{C}\_{\mathcal{R}}) \tag{13}$$

Therefore, during solidification of carbon steels, there is always a range between solidus and liquidus temperatures in which the solid fraction is in the range from 0 to 1; this physical range inside a solidifying steel body is called mushy zone, and the whole phenomenon is related to micro-segregation. The simple micro-segregation model [32] has been adopted in this work, and the analysis has been described in similar previous studies [24, 29, 33]. It appears that the larger the carbon content in a steel grade, the larger the mushy zone gets, and central porosity becomes inevitable in the final stages of solidification.

The local cooling rate distribution for the case of a 42CrMo4-grade at the caster position presented in Figure 6 is illustrated in Figure 7.

A short analysis showed that for the data presented in Figure 7, the CR (cooling rate in °C/s) has the following statistics: average value μ = 0.106, standard deviation σ = 0.128, minimum = 0.023, and maximum = 0.494. More than 99% of the CR population is within μ + 3\*σ = 0.490.

Figure 8 depicts the surface (1) and centerline (2) temperatures in an octagonal billet as the billet solidifies downstream the Stomana billet caster; the shell thickness (3) is presented as a function of the distance from the meniscus of liquid steel in the mold. It is worth mentioning that a S355 J2-billet cast at a speed of 0.85 m/min solidifies even faster than a similar 42CrMo4-grade billet at a lower speed.

Figure 7. Cooling rate distribution in the selected part of a 42CrMo4-grade octagonal billet.

A Numerical Solution Model for the Heat Transfer in Octagonal Billets DOI: http://dx.doi.org/10.5772/intechopen.84305

#### Figure 8.

Surface (1) and centerline (2) temperature values in an octagonal billet of the S355 J2 grade; thickness evolution (3) as the billet solidifies downwards the continuous caster.

Figure 9.

Temperature distribution in the selected part of a S355 J2-grade octagonal billet. In addition to the solid fraction values (fs) that are presented in the mushy zone, the percentages of δ- and γ- phases are shown, respectively.

Figure 9 illustrates the temperature distribution in the critical geometrical region of a S355 J2-grade octagonal billet at a distance of 18.4 m from the meniscus, cast at a speed of 0.85 m/min and a SPH = 30°C.

Peritectic grades have the tendency to crystallize in a mixture of delta (δ) and gamma (γ) phases of iron in an intermediate temperature range, before continuing to a 100% γ-phase solidification. For this reason, the percentages of δ- and γ-phases are also presented in Figure 9 at the selected solid fractions.

All these calculations come also from the simple micro-segregation model [32] that is adopted in the developed program. Figure 10 depicts the local cooling rates at the same position from the meniscus as for the aforementioned temperature distribution presented in Figure 9.

For the cooling rate distribution (CR in °C/s) presented in Figure 10, a short statistical analysis derived the following results: average value μ = 0.138, standard deviation σ = 0.156, minimum = 0.027, and maximum = 0.490. Almost 99% of the population is within μ + 3\*σ = 0.606.

The minimum average number of iterations required for convergence by the Strongly Implicit Procedure (SIP) was attained at the value of 0.95 for the iteration

Figure 10. Local cooling rate distribution in the selected part of a S355 J2-grade octagonal billet.

Figure 11. Effect of the iteration parameter on the convergence behavior of the strongly implicit procedure.

### A Numerical Solution Model for the Heat Transfer in Octagonal Billets DOI: http://dx.doi.org/10.5772/intechopen.84305

parameter α as explained by Stone [31] in order to speed up the convergence process. Actually, this value (α = 0.95) was used in order to solve the derived system of matrix equations in the case of the S355 J2 grade. Figure 11 illustrates the effect of this iteration parameter on the ratio of the required average number of iterations to the minimum number of iterations required at α equal to 0.95. In general, the convergence behavior improved as the selected iteration parameter increased. It is worth noting that convergence succeeded within an extensive range of the iteration parameter although above the limiting value of α = 0.95, the solution procedure started to diverge. For verification purposes, two sets of temperature results were generated by the Gauss-Seidel and Strongly Implicit Procedure, respectively, taken after a long simulation time (at the same time instant t = 2460 sec, equivalent to a distance of 34.85 m from meniscus). Using R [34], a Pearson's product–moment correlation test gave a coefficient of 0.9999344 for these two sets of results with a t-Student value equal to 17,455, 39,998 degrees of freedom and a p-value <sup>&</sup>lt; 2.2 ˜ <sup>10</sup>°16.
