Nucleation and Growth of Graphite in Eutectic Spheroidal Cast Iron: Modeling and Testing

Abstract

A new model of graphite growth during the continuous cooling of eutectic spheroidal cast iron is presented in this paper. The model considers the nucleation and growth of graphite from pouring to room temperature. The microstructural model of solidification accounts for the eutectic as divorced and graphite growth rate as a function of carbon gradient at the liquid in contact with the graphite. In the solid state, the microstructural model takes into account three stages for graphite growth, namely (1) from the end of solidification to the upper bound of intercritical stable eutectoid, (2) during the intercritical stable eutectoid, and (3) from the lower bound of intercritical stable eutectoid to room temperature. The micro- and macrostructural models are coupled using a sequential multiscale approach. Numerical results for graphite fraction and size distribution are compared with experimental results obtained from a cylindrical cup, in which the graphite volumetric fraction and size distribution were obtained using the Schwartz–Saltykov approach. The agreements between the experimental and numerical results for the fraction of graphite and the size distribution of spheroids reveal the importance of numerical models in the prediction of the main aspects of graphite in spheroidal cast iron.

Appendices

Appendix A: Phase Diagrams

The equilibrium carbon concentrations for eutectoid phase changes together with the lower and upper bounds of the intercriticals stable and metastable eutectoid of the Fe-C-Si and Fe-Fe₃C-Si systems are as follows:

• Fe-C-Si:

$$ C_{\text{C}}^{\gamma /g} = 1.60 \times 10^{ - 3} + 8.13 \times 10^{ - 5} C_{\text{Si}} - 6.46 \times 10^{ - 6} T + 5.47 \times 10^{ - 6} (C_{\text{Si}} )^{2} + 1.85 \times 10^{ - 8} T^{2} - 1.05 \times 10^{ - 4} C_{\text{Si}} T $$

$$ C_{\text{C}}^{\alpha /g} = - 9.53 \times 10^{ - 4} + 1.02 \times 10^{ - 2} C_{\text{Si}} + 1.55 \times 10^{ - 6} T + 9.59 \times 10^{ - 4} (C_{\text{Si}} )^{2} + 1 \times 10^{ - 10} T^{2} - 1.76 \times 10^{ - 5} C_{\text{Si}} T $$

$$ C_{\text{C}}^{\alpha /\gamma } = 1.92 \times 10^{ - 3} - 1.59 \times 10^{ - 2} C_{\text{Si}} - 4.12 \times 10^{ - 6} T - 2.62 \times 10^{ - 3} (C_{\text{Si}} )^{2} + 2.31 \times 10^{ - 9} T^{2} + 2.5 \times 10^{ - 5} C_{\text{Si}} T $$

$$ C_{\text{C}}^{\gamma /\alpha } = 1.29 \times 10^{ - 1} + 5.13 \times 10^{ - 3} C_{\text{Si}} - 2.56 \times 10^{ - 4} T + 2.9 \times 10^{ - 4} (C_{\text{Si}} )^{2} + 1.24 \times 10^{ - 7} T^{2} + 3.27 \times 10^{ - 4} C_{\text{Si}} T $$

$$ T_{{a_{T} }}^{\alpha } = - \left( {\sqrt {1.10 \times 10^{11} + 2.74 \times 10^{35} \left( {C_{\text{Si}} } \right)^{2} - 3.25 \times 10^{35} C_{\text{Si}} + 1.46 \times 10^{34} } + 1.74 \times 10^{23} C_{\text{Si}} - 1.02 \times 10^{23} } \right)/8.53 \times 10^{19} $$

$$ T_{{A_{1} }}^{\alpha } = - \left( {\sqrt {3.47 \times 10^{11} + 1.60 \times 10^{22} \left( {C_{\text{Si}} } \right)^{2} - 2.19 \times 10^{21} C_{\text{Si}} + 5.85 \times 10^{19} } + 7.4 \times 10^{16} C_{\text{Si}} - 9.85 \times 10^{15} } \right)/7.69 \times 10^{12} $$

• Fe-Fe₃C-Si:

$$ C_{C}^{\gamma /\theta } = - 7.34 \times 10^{ - 3} + 1.82 \times 10^{ - 1} C_{\text{Si}} + 1.7 \times 10^{ - 6} T + 2.92 \times 10^{ - 2} (C_{\text{Si}} )^{2} + 1.72 \times 10^{ - 8} T^{2} + 4.17 \times 10^{ - 5} C_{\text{Si}} T $$

$$ C_{C}^{\alpha /\theta } = 2.99 \times 10^{ - 3} + 1.37 \times 10^{ - 4} C_{\text{Si}} - 9.84 \times 10^{ - 6} T + 1.10 \times 10^{ - 5} (C_{\text{Si}} )^{2} + 8.23 \times 10^{ - 9} T^{2} - 1.89 \times 10^{ - 6} C_{\text{Si}} T $$

$$ T_{{a_{T} }}^{P} = - \left( {\sqrt {2.16 \times 10^{10} + 5.07 \times 10^{20} \left( {C_{\text{Si}} } \right)^{2} - 3.94 \times 10^{20} C_{\text{Si}} + 5.23 \times 10^{19} } + 3.08 \times 10^{15} C_{\text{Si}} - 2.82 \times 10^{15} } \right)/2.32 \times 10^{12} $$

$$ T_{{A_{1} }}^{P} = - \left( {\sqrt {4.05 \times 10^{11} + 6.72 \times 10^{21} \left( {C_{\text{Si}} } \right)^{2} - 7.3 \times 10^{20} C_{\text{Si}} + 7.5 \times 10^{19} } + 5.46 \times 10^{16} C_{\text{Si}} + 1.16 \times 10^{16} } \right)/2.4 \times 10^{13}, $$

where $C_{\text{Si}}$ is the Si content in austenite at different interfaces expressed in weight percentage and T is the temperature of the alloy in Celsius degrees.

Appendix B: Surface of Spheroids in Contact with Ferrite and Austenite

For the jth spheroid, the surfaces of graphite spheroids in contact with ferrite and austenite are given as fraction:

$$ \begin{aligned} A_{{\alpha /g_{j} }} & = \frac{{\sum\nolimits_{i = 1}^{{n_{\text{f}} }} {\left( {R_{{\alpha_{i} }} } \right)^{2} } }}{{4\left( {R_{{g_{j} }} } \right)^{2} }} \\ A_{{\gamma /g_{j} }} & = 1 - A_{{\alpha /g_{j} }} \\ \end{aligned}, $$

(B1)

where $n_{\text{f}}$ is the number of ferrite grains nucleated on each graphite spheroid (see Table B-IV) and $R_{{\alpha_{i} }}$ is the radius of ferrite grains nucleated on the spheroid. $A_{{\alpha /g_{j} }}$ could be computed with higher precision by means of a surface integral and assuming that each ferrite grain is located on the spheroid; however, the differences with the results of Eq. [B1] are negligible.

Table B-IV Thermo-physical Properties and Material Parameters of Cast Iron

Appendix C: Volume Fraction of Graphite and Austenite

With the radius increment of graphite spheroid in a time integration interval $\Delta t$, $t + \Delta t$, the radius of a graphite spheroid corresponding to nucleation event j at time $t + \Delta t$ (${}^{t + \Delta t}R_{{g_{j} }}$) is

$$ {}^{t + \Delta t}R_{{g_{j} }} = {}^{t}R_{{g_{j} }} + \Delta R_{{g_{j} }}. $$

The value of $\Delta R_{{g_{j} }}$ is obtained from the differential Eqs. [6], [7], [9], or [10], depending on the temperature of the alloy and on the characteristics of the transformations.

With the values of radius of graphite spheroids, the graphite volume fraction is obtained from

$$ f_{g} = \frac{4}{3}\pi \sum\limits_{j = 1}^{k} {N_{{g_{j} }}^{V} } \left( {R_{{g_{j} }} } \right)^{3}, $$

where k is the number of events of nucleation of graphite spheroids and $N_{{g_{j} }}^{V}$ the number of graphite spheroids per unit volume associated with the j event of nucleation.

As the austenite fraction is transformed into graphite, ferrite, and/or pearlite, its value should be computed again as

$$ f_{\gamma } = \left( {1 - f_{g} - f_{\alpha } - f_{\text{P}} } \right), $$

where $f_{\alpha }$ and $f_{\text{P}}$ are the volume fractions of ferrite and pearlite, respectively.

Details of the microstructure models from which $f_{\alpha }$ and $f_{P}$ are calculated are given by Carazo.[12]

Appendix D: Carbon Quantity in Austenite

The value of $C_{\text{C}}^{\gamma }$ per unit volume of RVE is computed as

$$ {}^{t + \Delta t}C_{C}^{\gamma } = \frac{{{}^{t}C_{C}^{\gamma } \rho_{\gamma } \left( {1 - {}^{t}U_{g} - {}^{t}U_{\alpha } - {}^{t}U_{P} } \right) + C_{g} \rho_{g} \left( {{}^{t}U_{g} - {}^{t + \Delta t}U_{g} } \right) + C_{{\alpha_{C} }} \rho_{\alpha } \left( {{}^{t}U_{\alpha } - {}^{t + \Delta t}U_{\alpha } } \right) + C_{{P_{C} }} \rho_{P} \left( {{}^{t}U_{P} - {}^{t + \Delta t}U_{P} } \right)}}{{\rho_{\gamma } \left( {1 - {}^{t + \Delta t}U_{g} - {}^{t + \Delta t}U_{\alpha } - {}^{t + \Delta t}U_{P} } \right)}}, $$

(D1)

where $C_{{X_{C} }}$, $\rho_{X},$ and ${}^{t}U_{X}$ are carbon concentrations in weight percentage, density, and carbon quantity in a micro-constituent X, respectively. X may be austenite, graphite, ferrite, or pearlite. The derivation of Eq. [D1] may be seen in Carazo.[12]

Appendix E: Thermo-physical Properties and Material Parameters used in the Numerical Simulations

Tables B-IV and E-V show the values of coefficients and thermo-physical properties of the alloy and sand used in the numerical simulation. The initial temperature of alloy is the same as the maximum value recorded in the experiments: 1478 K (1205 °C). The initial temperature for the cylindrical cup is the environmental temperature at the moment of conducting the experiments: 293 K (20 °C).

Table E-V Thermo-physical Properties of Sand

The values of specimen–mold conductance coefficient, specimen–environment and mold–environment convection heat transfer coefficients, and specimen thermocouple conductance coefficient are shown in Tables E-VI, E-VII, and E-VIII, respectively.

Table E-VI Specimen–Mold Conductance Coefficient

Table E-VII Specimen–Environment and Mold–Environment Convection Heat Transfer Coefficients

Table E-VIII Specimen–Thermocouple Conductance Coefficient