Quenching is one of the most important heat-treatment steps for steel castings. It is intended to improve the final mechanical properties of steel castings by producing a hard martensitic or bainitic microstructure. However, the quenching of steel castings involves complex interactions among temperature change, solid-state phase transformation, and internal stress. If the quenching process is not properly designed, steel castings may exhibit large distortion, high residual stress, or even cracks. In my research, I developed a numerical simulation system especially for steel castings during quenching. The system can calculate the transient temperature field of steel castings, obtain the local cooling rate at 700°C, and predict the as-quenched microstructure and hardness distribution. The main objective is to provide a fast and reliable engineering tool for optimizing the quenching treatment of steel castings.
In this article, I present the heat-transfer model, the finite-difference solution technique, the Maynier–Casay type microstructure prediction model, the software development, and two numerical examples: a stepped steel castings specimen and a crane brake wheel made of cast steel. The results show that the numerical simulation can describe the cooling behavior of steel castings in different quenching media and can quantitatively predict the as-quenched microstructure and hardness. Thus, the proposed system is useful for reducing the cost and time in the development of quenching processes for steel castings.
1. Heat-Transfer Mathematical Model for Quenched Steel Castings
During quenching, heat is first transferred from the interior of steel castings to the surface by conduction, and then from the surface to the surrounding quenchant by convection and radiation. The transient temperature field inside steel castings is governed by the three-dimensional Fourier heat-conduction equation. The density of steel castings is treated as constant, while the thermal conductivity and specific heat are temperature-dependent. The heat-conduction equation used in my model is expressed as:
$$
\rho c_p \frac{\partial T}{\partial t}
=
\lambda \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) + S
$$
where \(T\) is temperature in °C, \(t\) is time in seconds, \(\rho\) is density in kg/m³, \(c_p\) is specific heat in J/(kg·°C), \(\lambda\) is thermal conductivity in W/(m·°C), and \(S\) is the internal heat-source term caused by the solid-state phase transformation of steel castings during quenching. The latent heat released by the transformation of supercooled austenite into martensite, bainite, and ferrite–pearlite is included in the source term \(S\).
For the quenching process of steel castings, the initial temperature distribution is usually uniform after austenitization:
$$
T(x,y,z,0) = T_0
$$
where \(T_0\) is the austenitizing temperature. For the boundary condition, the third kind of thermal boundary condition is most suitable for convection-dominated quenching of steel castings:
$$
-\lambda \left. \frac{\partial T}{\partial n} \right|_{s}
=
h \left( T_w – T_c \right)
$$
In this equation, \(h\) is the convective heat-transfer coefficient between the surface of steel castings and the quenching medium, \(T_w\) is the surface temperature of steel castings, and \(T_c\) is the temperature of the cooling medium. The heat-transfer coefficient \(h\) is not constant; it depends on the instantaneous surface temperature of steel castings and on the type of quenchant.
For the phase-transformation latent heat, I applied the equivalent heat-source method. At each time step, the latent heat generated by phase transformation is treated as an internal heat source:
$$
S = \frac{\Delta V}{\Delta t} \Delta H
$$
where \(\Delta V\) is the increment of the phase fraction, \(\Delta t\) is the time-step size, and \(\Delta H\) is the volumetric enthalpy change associated with the phase transformation. This method improves the accuracy of the temperature field prediction for steel castings because the cooling curve near the transformation temperature is no longer artificially accelerated.
2. Finite-Difference Solution of the Temperature Field
I used the finite-difference method to solve the heat-conduction equation for steel castings. The three-dimensional domain of steel castings is discretized into regular hexahedral cells. For a typical interior cell \(i\), heat is exchanged with six neighboring cells. The energy balance can be written as:
$$
\rho_i c_{p,i} (\Delta x)^3 \left( T_i^{t+\Delta t} – T_i^t \right)
=
\sum_{j=1}^{6} A_{ij} \left( T_j^t – T_i^t \right) \Delta t
+
S_i (\Delta x)^3 \Delta t
$$
where \(T_i^t\) is the temperature of cell \(i\) at time \(t\), \(T_i^{t+\Delta t}\) is the temperature at the next time level, \(\Delta x\) is the cell size, and \(A_{ij}\) is the equivalent conductance between cell \(i\) and neighbor cell \(j\). For internal faces, the equivalent conductance is:
$$
A_{ij} = \frac{2 \lambda_i \lambda_j}{\lambda_i + \lambda_j} \Delta x
$$
For boundary cells of steel castings, the convective heat exchange with the quenchant is added. If one face of cell \(i\) is exposed to the quenching medium, the boundary conductance is:
$$
B_i = h_i (\Delta x)^2
$$
Therefore, the explicit update formula for a boundary cell of steel castings is:
$$
T_i^{t+\Delta t}
=
T_i^t
+
\frac{\Delta t}{\rho_i c_{p,i} (\Delta x)^3}
\left[
\sum_{j=1}^{6} A_{ij} \left( T_j^t – T_i^t \right)
+
\sum_{b} h_b (\Delta x)^2 \left( T_c – T_i^t \right)
+
S_i (\Delta x)^3
\right]
$$
Because the explicit finite-difference scheme is conditionally stable, the time step must satisfy the following stability condition:
$$
\Delta t
\le
\frac{\rho_i c_{p,i} (\Delta x)^3}{
\sum_j A_{ij} + \sum_b h_b (\Delta x)^2
}
$$
For a uniform cube with six internal neighbors and constant thermal conductivity, this condition reduces to:
$$
\Delta t
\le
\frac{\rho c_p (\Delta x)^2}{6 \lambda}
$$
In my program, the temperature-dependent thermal properties of steel castings are updated at each time level according to the current temperature. For the cast steel ZG45, the thermal conductivity and specific heat were fitted by the following functions:
$$
\lambda(T) = 0.0123 T + 16.697
$$
$$
c_p(T) = -5 \times 10^{-5} T^2 + 0.2259 T + 453.04
$$
These fitted equations describe the measured thermal conductivity and specific heat of steel castings in the temperature range from room temperature to 900°C. The heat-transfer coefficient curves of the quenching media were also expressed as functions of the surface temperature of steel castings. In particular, I studied water, L-AN22 oil, and 10% NaCl aqueous solution. Their cooling characteristics are quite different, which strongly influences the final microstructure of steel castings.
3. Microstructure and Hardness Prediction Model
The final microstructure of quenched steel castings depends on the continuous-cooling transformation behavior of austenite. I used the Maynier–Casay model, which was originally developed from statistical regression of many continuous-cooling-transformation diagrams. This model is suitable for steel castings with typical low-alloy compositions. The critical cooling velocities are calculated from the chemical composition and austenitizing conditions of steel castings:
$$
\log_{10} V_i
=
K
–
A (\%C)
+
B (\%Si)
+
D (\%Mn)
+
E (\%Ni)
+
F (\%Cr)
+
G (\%Mo)
$$
where \(V_i\) is a critical cooling velocity in °C/h, and \(K\), \(A\), \(B\), \(D\), \(E\), \(F\), and \(G\) are regression coefficients. The austenitizing parameter \(P_a\) is used to consider the austenitizing temperature and time:
$$
P_a
=
\frac{1}{R}
\left(
\frac{1}{T_m}
–
\frac{1}{T}
\right)
\ln t
$$
where \(T\) is the austenitizing temperature, \(t\) is the austenitizing time, \(R\) is the gas constant, and \(T_m\) is a reference temperature. The coefficients for the Casay model are summarized in the following table.
| Parameter | V1 | V190 | V150 | V110 | V10 | V20 | V210 | V250 | V290 | V2 |
|---|---|---|---|---|---|---|---|---|---|---|
| K | 9.81 | 8.76 | 8.20 | 9.80 | 8.56 | 10.55 | 9.06 | 8.04 | 8.40 | 8.56 |
| A | 4.62 | 4.04 | 3.00 | 3.90 | 1.50 | 4.80 | 4.11 | 3.40 | 2.80 | 1.50 |
| B | 0.78 | 0.86 | 0.79 | -0.54Mn+2.45√Mn | 1.84 | 0.80 | 0.90 | 1.15 | 1.51 | 1.84 |
| D | 0.41 | 0.36 | 0.57 | 0.46 | 0.70 | 0.72 | 0.60 | 0.96 | 1.03 | 0.78 |
| E | 0.80 | 0.58 | 0.67 | 0.50 | 1.24 | 1.07 | 1.00 | 1.00 | 1.10 | 1.24 |
| F | 0.66 | 0.97 | 0.94 | 1.16 | 1.46 | 1.58 | 2.00 | 2.00 | 2.31 | 2√Mo |
| G | 0.0018 | 0.0010 | 0.0012 | 0.0020 | 0.0020 | 0.0026 | 0.0013 | 0.0070 | 0.0014 | 0.0020 |
In this table, \(V_1\) is the minimum cooling velocity required for 100% martensite in steel castings, \(V_1^{90}\) is the cooling velocity producing 90% martensite and 10% bainite, and \(V_2\) is the maximum cooling velocity producing 100% ferrite–pearlite. The meanings of the other critical cooling velocities are similar and are applied exactly in my prediction program.
After the local cooling velocity \(v\) of steel castings is obtained from the temperature-field simulation, it is compared with the calculated critical cooling velocities. The phase fractions are obtained by linear interpolation between the two neighboring critical cooling velocities:
$$
W_M
=
W_{M,L}
+
(W_{M,H} – W_{M,L})
\frac{v – v_L}{v_H – v_L}
$$
$$
W_B
=
W_{B,L}
+
(W_{B,H} – W_{B,L})
\frac{v – v_L}{v_H – v_L}
$$
$$
W_{F+P}
=
W_{F+P,L}
+
(W_{F+P,H} – W_{F+P,L})
\frac{v – v_L}{v_H – v_L}
$$
where \(v_L\) and \(v_H\) are the bounding critical cooling velocities, and \(W\) represents the phase fraction. The hardness of the single-phase constituents is also calculated from the Maynier–Casay formulas. For the martensite hardness of quenched steel castings, I used:
$$
HV_M = 121.156 + 902.6 (\%C) + 26.68 \log_{10} v
$$
For the bainite hardness:
$$
HV_B
=
323
+
185 (\%C)
+
330 (\%Si)
+
153 (\%Mn)
+
65 (\%Ni)
+
144 (\%Cr)
+
191 (\%Mo)
+
\left[
89
+
53 (\%C)
+
55 (\%Si)
+
22 (\%Mn)
+
10 (\%Ni)
+
20 (\%Cr)
+
33 (\%Mo)
\right]
\log_{10} v
$$
For the ferrite–pearlite hardness, I used the modified Casay expression:
$$
HV_{F+P}
=
-437
+
3300 (\%C)
–
5343 (\%C)^2
+
\left[
1329 (\%C)
–
744 (\%C)^2
+
4 (\%Ni)
+
15 (\%Cr)
+
135.4
\right]
\log_{10} v
$$
The total hardness of quenched steel castings is then calculated by the weighted-average rule:
$$
HV
=
W_M HV_M
+
W_B HV_B
+
W_{F+P} HV_{F+P}
$$
One important contribution of my work is the direct calculation of the 700°C cooling rate for every cell of steel castings. During the finite-difference temperature simulation, I record the time at which the temperature crosses 700°C. The cooling velocity is then calculated as:
$$
v(i,j,k)
=
\frac{T(i,j,k,t) – T(i,j,k,t+1)}{\Delta t}
$$
where the temperature of cell \((i,j,k)\) at time \(t\) is above 700°C and at time \(t+1\) is below 700°C. This technique avoids the manual extraction of cooling curves and allows the complete three-dimensional hardness map of steel castings to be generated directly.
4. Development of the Simulation and Prediction System
I developed a dedicated software system for the quenching simulation of steel castings using Visual Studio 2005. The system consists of three main functional modules: temperature-field calculation, microstructure prediction, and hardness prediction. The temperature-field module reads a finite-difference mesh of steel castings, accepts the material thermophysical properties, the heat-transfer coefficients of the cooling medium, and the quenching process parameters. It then performs the transient heat-conduction calculation and saves the temperature history and the 700°C cooling velocity file.
The microstructure-prediction module reads the chemical composition of steel castings, the austenitizing temperature, and the austenitizing time. It calculates all critical cooling velocities using the Maynier–Casay model and then compares them with the cooling velocity file. The predicted martensite, bainite, and ferrite–pearlite fractions are saved for visualization. The hardness-prediction module calculates the single-phase hardness values and the final weighted hardness distribution. The user-friendly interface allows engineers to evaluate different quenching processes for steel castings without repeated experiments.
5. Application to a Stepped Steel Castings Specimen
To validate the system, I first simulated a stepped specimen made of ZG45 cast steel. The chemical composition of this typical steel castings material is summarized in the following table.
| Element | C | Si | Mn | S | P | Cr | Ni | Mo | Cu | V |
|---|---|---|---|---|---|---|---|---|---|---|
| Content (%) | ≤0.50 | ≤0.60 | ≤0.90 | ≤0.04 | ≤0.04 | ≤0.35 | ≤0.30 | ≤0.20 | ≤0.30 | ≤0.05 |
The stepped specimen was austenitized at 860°C and then quenched in three different media: 20°C water, 20°C L-AN22 oil, and 25°C 10% NaCl aqueous solution. The temperature-dependent thermal properties of ZG45 steel castings are listed below.
| Temperature (°C) | 30 | 105 | 185 | 270 | 350 | 430 | 515 | 595 | 675 | 760 | 840 | 900 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Thermal conductivity (W/(m·°C)) | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 27.7 |
| Temperature (°C) | 30 | 100 | 200 | 300 | 400 | 450 | 500 | 550 | 700 | 750 | 850 | 900 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Specific heat (J/(kg·°C)) | 455.6 | 476 | 500 | 518 | 535.2 | 543.4 | 552.4 | 560 | 583.5 | 590.8 | 607.2 | 614.2 |
The simulated temperature fields of the stepped steel castings clearly show that the thin sections cool much faster than the thick sections. The surface temperature drops rapidly during water quenching, with local cooling rates on the surface exceeding 100°C/s. Oil quenching produces a much milder cooling history. The cooling capacity of the 10% NaCl solution is between water and oil above 200°C, but below 200°C its cooling rate becomes lower than that of the oil. This behavior is consistent with the heat-transfer coefficient curves used in the simulation.
After the temperature-field calculation, I applied the Maynier–Casay model to predict the microstructure and hardness at five central points of the stepped steel castings. The predicted phase fractions and hardness values are summarized below.
| Quenching medium | Point | Martensite (%) | Bainite (%) | Ferrite–pearlite (%) | Hardness (HV) |
|---|---|---|---|---|---|
| 20°C water | 1 | 41.8 | 58.2 | 0.0 | 434.7 |
| 2 | 55.2 | 44.8 | 0.0 | 483.9 | |
| 3 | 96.1 | 3.9 | 0.0 | 629.3 | |
| 4 | 100.0 | 0.0 | 0.0 | 642.9 | |
| 5 | 100.0 | 0.0 | 0.0 | 642.9 | |
| 20°C L-AN22 oil | 1 | 24.3 | 75.7 | 0.0 | 374.9 |
| 2 | 29.6 | 70.4 | 0.0 | 393.2 | |
| 3 | 51.7 | 48.3 | 0.0 | 471.5 | |
| 4 | 92.0 | 8.0 | 0.0 | 614.7 | |
| 5 | 100.0 | 0.0 | 0.0 | 642.9 | |
| 25°C 10% NaCl | 1 | 34.0 | 66.0 | 0.0 | 409.2 |
| 2 | 44.2 | 55.8 | 0.0 | 444.8 | |
| 3 | 76.5 | 23.5 | 0.0 | 559.5 | |
| 4 | 100.0 | 0.0 | 0.0 | 642.9 | |
| 5 | 100.0 | 0.0 | 0.0 | 642.9 |
The simulation results show that water quenching provides the strongest overall cooling ability for steel castings, producing the largest amount of martensite. Oil quenching produces the smallest amount of martensite and the largest amount of bainite. The NaCl solution gives intermediate results. These observations agree with the known cooling characteristics of the three media. Therefore, the proposed numerical simulation system is reliable for evaluating the quench severity and the resulting microstructure of steel castings.
6. Application to a Crane Brake Wheel
After the stepped specimen validation, I applied the system to an actual crane brake wheel made of ZG45 steel castings. The brake wheel has an outer diameter of 630 mm and a width of 260 mm. The finite-difference mesh was generated with a uniform grid size of 3 mm, giving approximately seven million cells. I compared three quenching processes. In the first process, the brake wheel was austenitized at 860°C and then directly quenched in 20°C water. In the second process, it was quenched in 20°C oil. In the third process, it was first quenched in water until the surface temperature reached about 200°C and then transferred to oil for final cooling.
The transient temperature-field results show that the surface and thin sections of the brake wheel cool quickly, while the thick central section cools slowly. During direct water quenching, the surface temperature of the brake wheel drops below 200°C after about 30 s, while the center still remains above 600°C. This large temperature gradient can produce high thermal stress and may cause cracking of steel castings. Oil quenching produces a much more uniform temperature distribution, but the cooling rate is too low for the surface to obtain a fully martensitic hard layer. The water-oil quenching process combines the advantages of both media: the early water-cooling stage ensures a high surface cooling rate, while the later oil-cooling stage reduces the cooling rate in the temperature range where martensite transformation occurs, thereby reducing the distortion risk of steel castings.
The predicted microstructure and hardness distributions for the six characteristic points of the brake wheel are presented in the following table.
| Quenching process | Point | Martensite (%) | Bainite (%) | Ferrite–pearlite (%) | Hardness (HV) |
|---|---|---|---|---|---|
| Direct water quenching | A | 100.0 | 0.0 | 0.0 | 642.9 |
| B | 68.4 | 31.6 | 0.0 | 530.9 | |
| C | 100.0 | 0.0 | 0.0 | 642.9 | |
| D | 71.4 | 28.7 | 0.0 | 541.3 | |
| E | 53.8 | 36.2 | 0.0 | 479.1 | |
| F | 75.8 | 24.2 | 0.0 | 557.2 | |
| Direct oil quenching | A | 96.1 | 3.1 | 0.0 | 629.0 |
| B | 40.4 | 49.6 | 0.0 | 431.2 | |
| C | 73.8 | 26.2 | 0.0 | 547.3 | |
| D | 42.0 | 38.1 | 0.0 | 437.1 | |
| E | 34.6 | 65.4 | 0.0 | 411.0 | |
| F | 46.6 | 53.4 | 0.0 | 453.5 | |
| Water quenching followed by oil quenching | A | 100.0 | 0.0 | 0.0 | 642.9 |
| B | 50.0 | 50.0 | 0.0 | 465.6 | |
| C | 100.0 | 0.0 | 0.0 | 642.9 | |
| D | 52.0 | 48.0 | 0.0 | 472.7 | |
| E | 49.4 | 50.6 | 0.0 | 463.5 | |
| F | 57.0 | 43.0 | 0.0 | 490.3 |
From these results, I can conclude that direct water quenching gives the highest surface hardness of steel castings, but the center contains only a small amount of bainite, which means that the toughness of the brake wheel may be insufficient. Direct oil quenching provides more bainite and better toughness, but the surface martensite content is lower, so the wear resistance is reduced. The water-oil quenching process gives a fully martensitic surface layer and a bainite-rich core. Therefore, this process offers the best combination of wear resistance and toughness for the crane brake wheel made of steel castings.
7. Conclusion
In this research, I developed a complete numerical simulation and prediction system for the quenching process of steel castings. The main conclusions are as follows.
First, the three-dimensional heat-conduction equation with phase-transformation latent heat can accurately describe the transient temperature field of steel castings during quenching. The finite-difference method and the stability criterion provide a robust basis for the numerical implementation.
Second, the Maynier–Casay model is an effective method for predicting the microstructure and hardness of steel castings. By using the local 700°C cooling rate obtained from the temperature simulation, the phase fractions and hardness of every cell of steel castings can be predicted quickly and quantitatively.
Third, the system was successfully applied to a stepped steel castings specimen and to a crane brake wheel. The simulation results not only reveal the cooling behavior of steel castings in water, oil, and NaCl solution but also provide useful guidance for selecting the optimal quenching process. For the crane brake wheel, the water-oil quenching process is recommended because it combines high surface hardness with good core toughness.
In the future, I plan to extend the system by coupling the stress and strain fields, by adding a more complete database of heat-transfer coefficients, and by considering the initial as-cast microstructure of steel castings. These improvements will make the numerical simulation of quenching even more accurate and practical for the production of high-quality steel castings.