Sub-structured 1-D radial models

The purpose of simplified 1-D radial models is to study the heat transfer through the HSS shell during HT, taking into account compositional variations. Therefore, the 1-D radial model was sub-structured, consisting of multi-layers with individual material properties. Two types of 1-D radial model are considered, having 3 and 11 homogenized layers, representing the roll structure with uniform and high compositional gradients, respectively. In Figure 23 the 1-D FE model is shown;

where the centrally cut radial rod includes the multi-layered HSS shell, intermix zone and core.

The number of layers depended upon the desired degree of accuracy and the actual chemical variability through the shell (Figure 24). The shown 11-layer model has 9 sub-domains in HSS shell, intermediate layer (intermix) and core domain; whereas in contrast the 3-layer model has the homogenized shell and the same intermix and core. The 3-layer model represents improved roll structure and material. The predicted material properties for this 1-D radial 3-layer model were also used for the 3-layer axisymmetric (see Section 3.5.2.2) and 3-layer full-scale roll models (see Section 3.5.2.3).

Figure 24. Radial sub-structured models of the roll

Each particular layer (i.e. domain) is considered isotropic and homogeneous along the roll.

The core material was assumed to be uniform, neglecting common cast segregations. Full nonlinear thermal transient analysis was done on the sub-structured models (11 and 3 layers), representing the compositional variation of the roll in the radial direction. The 11-layer model (i.e.

9 layers of HSS shell, intermix and core) represented the conventional as-cast material, while the 3-layer model (improved case) consisted only of a homogenized HSS shell, intermix and core. A

simplified 1-D approach was compared with axisymmetric models, which took advantage of the axial symmetry to reduce computational time.

The material properties of the layers were predicted based on corresponding actual chemical compositions, which were measured as a function of the radius, using optical emission spectrometer (OES). The carbon and sulfur contents were determined by combustion, using Leco combustometric methods in accordance with ASTM standard E1019-03. All other elemental contents were determined by glow discharge-optical emission spectrometry (GD-OES) in general accordance with ISO 14707:2000E First Edition (2000-18-15). In Figure 25 actual radial cut-off sample from the HSS shell is shown, where GD-OES spark spots are visible on the surface, representing measured locations.

Figure 25. Radial cut-off HSS sample with sparks on surface from GD-OES measurements

Figure 26 shows a curve-fitting of the results, measured by GD-OES, i.e. chemical distribution (weight %, vertical axis) in conventional as-cast material as a function of radial depth (distance from the surface of the roll, horizontal axis).

Figure 26. Measured compositional gradient along the HSS shell, intermix and core

The nonlinear material properties are predicted for each individual layer, using measured radial compositional gradient (Figure 26) and thermodynamic analysis (see Sections 3.4 and 3.5.2.1). The diffusion coefficient is determined by thermal conductivity, heat capacity and

density, which are temperature and domain dependent in the composite model. For example, based on OES analysis, these properties were predicted, using JMatPro. Figures 27, 28, 29 show density, thermal conductivity and heat capacity, respectively, for sub-structured 11-layer radial model (Figure 24). The “Pl1” to “Pl9” are abbreviations of corresponding OES sparks (“burn spots”, Figure 25) made from surface to the core, which represent individual domains in HSS shell.

Figure 27. Predicted density of the layers vs. temperature

Figure 29. Predicted specific heat of the layers vs. temperature

A scattering of the heat capacity values at high temperature range (Figure 29) can be attributed to both the phase transformation reactions (e.g., dissolution) and existing database (or/and numerical approximations) of the thermodynamic software.

Three scenarios of boundary conditions (BC) were investigated. Using the 1-D models built with 3- and 11-layers, the radial heat transfer was studied. Theoretical austenitizing temperatures and corresponding initial conditions of the HT were applied on the surface of the 3- and 11-layer FE models. Two theoretical cases were considered: Tγ2 [Ti2] (Case I) > Tγ1 [Ti1] (Case 2), where Tγ and Ti are the austenitizing and initial temperatures, respectively (Figure 30). In the third scenario (Case 3, Figure 30), actual point-wise pyrometer readings acquired from the surface of the spinning roll were used as the tabulated temperature input for the developed semi-1-D models (possible temperature fluctuations during cooling due to the roll rotation are neglected). The time required to heat through the shell was determined, assuming adiabatic conditions (i.e. no heat

dissipation). The thermal gradients during heating, isothermal hold and quenching were determined.

Figure 30. Considered surface boundary condition scenarios

In order to investigate the effects of meshing on the solutions, 2-3 different meshing techniques (e.g. MultiZone, Hex Dominant or Sweep) were used individually for different domains to consider mesh quality, solution efficiency (rapid meshing and solution) and approximation (convergence and stability). The meshes in the high gradient regions had to be refined even more, using the “Sizing Method”, but keeping in mind the factors like: micro-scale, iterative solution, nonlinear material properties. Because the further refinement at the domain’s boundaries raises the cumulative numerical error (i.e. global truncation error, precision, and round-off) exponentially. Therefore, a compromise had to be made. The hex-dominant meshing or other types of higher-order elements (hexahedra-dominant and/or quadrilateral-dominant mesh) were found to be more robust, reducing computational time. Speaking of the mesh quality controls, different

found in Meshing User’s Guide [99]. Below in Figure 31 an example of mesh quality control is given for 1-D model during transient thermal analysis. The highlighted regions with red-green and light blue represent detected locations, giving raise in accumulated error as a function of time.

These shown locations are right at the boundaries between the HSS shell and the intermix and fusion line, which was explained by the different element size used and introduced contact elements between domains after importing the CAD model. In spite of that, a small error value was estimated and was neglected.

(a)

(b)

Figure 31. Example of mesh quality control in 11-layer radial model: (a) original mesh, (b) highlighted error estimates at contacts due to different mesh sizes