# Development of a CA-FVM Model with Weakened Mesh Anisotropy and Application to Fe–C Alloy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Description

#### 2.1. Nucleation Model

_{max}is the maximum nucleation density, m

^{−1}, and ΔT

_{n}and ΔT

_{σ}are the average and the standard deviation of the nucleation undercooling, K, respectively. The increment of the nucleation density, δn, induced by the change of the melt undercooling δ(ΔT) is calculated as follows:

_{n}of a liquid cell at the domain bottom is determined as follows:

_{n}, and simultaneously ΔT is higher than ΔT

_{n}, nucleation will occur. The preferential growth orientation is randomly chosen within −45° to 45°.

#### 2.2. CA Model

**n**is the norm of the solidification interface,

**V**is the norm growth velocity of the interface, m/s, ${C}_{0}$, ${C}_{\mathrm{l}}^{*}$, and ${C}_{\mathrm{s}}^{*}$ are the initial content, equilibrium liquid, and solid concentrations at the interface, wt%, respectively, k

_{n}_{0}is the equilibrium redistribution coefficient of the solute, m

_{l}is the slope of the liquidus line in the equilibrium Fe–C phase diagram, K·wt%

^{−1}, Γ is the Gibbs–Thomson coefficient of Fe–C alloy, K·m, $\phi $ and θ are angles of the interface norm and the preferential growth orientation with respect to the x axis, °, respectively, ε is the anisotropy parameter, D

_{l}and D

_{s}are diffusion coefficients of the solute in liquid and solid phases, m

^{2}·s

^{−1}, respectively, T

_{l}is the equilibrium liquidus temperature of Fe–C alloy, K, and ∂

_{x}f

_{s}, ∂

_{y}f

_{s}, ∂

_{xx}f

_{s}, ∂

_{yy}f

_{s}, and ∂

_{xy}f

_{s}are derivatives of solid fraction f

_{s}determined with the bilinear interpolation method [13,14].

**V**is determined as follows [7]:

_{n}_{x}and V

_{y}are growth velocities along x and y axes, m·s

^{−1}, and W and E represent left and right neighbors of the present interface cell P, respectively.

_{dia}is updated as follows [9,35,36]:

#### 2.3. Transport Models

^{−3}, λ is the thermal conductivity, W·m

^{−1}·K

^{−1}, c

_{p}is the specific heat capacity, J·kg

^{−1}·K

^{−1}, and L is the solidification latent heat, J·kg

^{−1}.

_{n}

_{,max}is the maximum growth velocity of the interface, m·s

^{−1}, and D

_{max}is the maximum solute diffusion coefficient, m

^{2}·s

^{−1}. Physical properties of Fe–0.82C alloy are listed in Table 1 [37,38].

## 3. Model Evaluation and Application

#### 3.1. Free Growth of Equiaxed Dendrite

^{−1}) predicted by the LGK model. Therefore, before the dendritic growth reaches the steady state at ΔT = 5 K, it will be influenced by boundary conditions. Beltran-Sanchez and Stefanescu [7] proposed that the steady state can be determined as the concentration at the boundary contrary to the dendritic tip reaches 1.01 times the initial value. So, the domain size is adjusted according to the melt undercooling in the present work. Steady states at melt undercoolings of 5 K (n = 701) and 8 K (n = 301) are properly determined according to Beltran-Sanchez and Stefanescu [7], as shown in Figure 2. Figure 3 shows steady tip growth velocities of equiaxed dendrites predicted by CA-FVM model and the comparison with analytical results. Steady tip growth velocities agree with analytical results. Meanwhile, compared with our previous model based on Neumann rule [37,38,41], the accuracy is much improved. However, because the present approach is deterministic and depends on the mesh layout, the mesh size greatly influences the dendritic radius.

#### 3.2. Interface Type and Growth Consistence

#### 3.3. Segregation among Equiaxed Dendrites

_{l}according to the preset cooling rate. Figure 7 shows the solute distribution around equiaxed dendrites at the solid fraction of 0.5, as the cooling rate (CR) varies from 0.5 K·s

^{−1}to 50 K·s

^{−1}. As CR is 0.5 K·s

^{−1}, the nuclei develop as cellular structures. With the improvement of CR, the equiaxed dendritic structure gradually forms (as shown in Figure 7b). As CR increases to 50 K·s

^{−1}, primary arms become thinner, with developed secondary arms.

#### 3.4. Constrained Growth of Columnar Dendrite

_{l}, and cools the domain down with the average heat flux according to the mold cooling condition of SWRH82B steel billet [37], accordingly simulating the unidirectional solidification of Fe–0.82C alloy. Initial and boundary conditions for the solute diffusion are the same as those mentioned above. Figure 9 shows the comparison between the predicted columnar dendritic morphology and the experimental observation. Some columnar dendrites win from the initial competition, and are with strong primary branches and well developed secondary, even higher-order arms. Moreover, columnar dendrites with contrary growth directions alternately restrict each other, which is similar to the experimental observation. Additionally, average primary and secondary dendrite arm spacings (λ

_{1}and λ

_{2}) in regions A and B are concerned. The predicted λ

_{1}and λ

_{2}are 182.8 μm and 69.8 μm, while the corresponding experimental measurements are 142.2 μm and 101.5 μm, respectively. On the one hand, dendritic arms become ripened with repeated temperature cycles, rising up and going down in secondary and radiation cooling zones during the continuous casting of SWRH82B steel billet. On the other hand, λ

_{1}is determined before being adjusted by tertiary arms in the simulation, while secondary arms are promoted by the higher heat flux and the finer mesh. Additionally, the lack of one-dimensional space also contributes to the numerical deviation.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Basic concept of decentered square algorithm (DCSA): (

**a**) nucleation; (

**b**) capture of first nearest neighboring cells; and (

**c**) capture of second nearest neighboring cells.

**Figure 2.**Average tip growth velocities: (

**a**) ΔT = 5 K and (

**b**) ΔT = 8 K. CA-FVM: cellular automaton-finite volume method.

**Figure 3.**Comparison between predicted steady tip growth velocities and Lipton, Glicksman, and Kurz (LGK)’s results [40].

**Figure 5.**Equiaxed dendritic morphology at ΔT = 9 K and t = 0.6 s, with preferential growth orientations of: (

**a**) 0°; (

**b**) 15°; (

**c**) 30°; and (

**d**) 45°.

**Figure 7.**Equiaxed dendritic morphology of Fe–0.82C alloy at f

_{s}= 0.5 when cooling rates are (

**a**) 0.5 K·s

^{−1}; (

**b**) 10 K·s

^{−1}; and (

**c**) 50 K·s

^{−1}.

**Figure 8.**Average solute concentration in liquid phase during the solidification of multi-equiaxed dendrites. CR: cooling rate.

**Figure 9.**Columnar dendritic morphology of SWRH82B steel billet: (

**a**) experimental observation [37] (Reproduced with permission from Weiling Wang, Sen Luo, Miaoyong Zhu, Metallurgical and Materials Transactions A; published by Springer, 2015); and (

**b**) present prediction.

Physical Property | Symbol | Unit | Value |
---|---|---|---|

Melt temperature of pure iron | T_{m} | K | 1809.15 |

Liquidus line slope of Fe–C alloy | m_{l} | K·wt%^{−1} | −78.0 |

Thermal conductivity | λ | W·m^{−1}·K^{−1} | 33.0 |

Density of solid phase | ρ_{s} | kg·m^{−3} | 7400 |

Density of liquid phase | ρ_{l} | kg·m^{−3} | 7020 |

Specific heat capacity of solid phase | c_{p,s} | J·kg^{−1}·K^{−1} | 648 |

Specific heat capacity of liquid phase | c_{p,l} | J·kg^{−1}·K^{−1} | 824 |

Specific heat capacity at mushy state | c_{p,m} | J·kg^{−1}·K^{−1} | 770 |

Solidification latent heat | L | J·kg^{−1} | 27,200 |

Diffusion coefficient in solid phase | D_{s} | cm^{2}·s^{−1} | 0.0761exp(−16,185.2/T) |

Diffusion coefficient in liquid phase | D_{l} | cm^{2}·s^{−1} | 0.0767exp(−12,749.6/T) |

Partition coefficient | k_{0} | -- | 0.34 |

Anisotropy parameter | ε | -- | 0.04 |

Gibbs–Thomson coefficient | Γ | K·m | 1.9 × 10^{−7} |

Maximum nucleus density | n_{max} | m^{−1} | 16,736 |

Standard deviation of nucleation undercooling | ΔT_{σ} | K | 0.1 |

Average nucleation undercooling | ΔT_{n} | K | 1.0 |

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**MDPI and ACS Style**

Wang, W.; Luo, S.; Zhu, M.
Development of a CA-FVM Model with Weakened Mesh Anisotropy and Application to Fe–C Alloy. *Crystals* **2016**, *6*, 147.
https://doi.org/10.3390/cryst6110147

**AMA Style**

Wang W, Luo S, Zhu M.
Development of a CA-FVM Model with Weakened Mesh Anisotropy and Application to Fe–C Alloy. *Crystals*. 2016; 6(11):147.
https://doi.org/10.3390/cryst6110147

**Chicago/Turabian Style**

Wang, Weiling, Sen Luo, and Miaoyong Zhu.
2016. "Development of a CA-FVM Model with Weakened Mesh Anisotropy and Application to Fe–C Alloy" *Crystals* 6, no. 11: 147.
https://doi.org/10.3390/cryst6110147