# Numerical Modeling for the Prediction of Microstructure and Mechanical Properties of Quenched Automotive Steel Pieces

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## Abstract

**:**

## 1. Introduction

## 2. Description of the Industrial Quenching Process

## 3. Numerical Model

#### 3.1. Thermal Model

- A first one, where the pieces are transported from the outlet of the furnace to the quenching bath, being in contact with the surrounding air and the transportation tray.
- A second one, with the pieces submerged in the fluid.

- Piece length $L$, diameter $D,$ and thickness $e$, and on the following quenching parameters:
- Bulk quenching liquid temperature ${T}_{b}$
- Velocity $V$ of the quenching liquid upstream of the piece
- Thermophysical properties of the quenching fluid: liquid and vapor densities, viscosities, conductivities, and specific heats ${\rho}_{l}$, ${\rho}_{v}$, ${\mu}_{l}$, ${\mu}_{v}$, ${k}_{l}$, ${k}_{v}$, ${C}_{p,l}$, ${C}_{p,v}$, saturation temperature ${T}_{sat}$, vapor surface tension ${\sigma}_{st}$, and latent heat of vaporization ${i}_{lv}$.

#### 3.1.1. Heat Flux in Surrounding Air

#### 3.1.2. Film Boiling

#### 3.1.3. Transition Boiling

#### 3.1.4. Fully Developed Boiling

#### 3.1.5. Partial Boiling

#### 3.1.6. Single-Phase Heat Flux

#### 3.2. Validation of the Thermal Model

#### 3.3. Metallurgical Model

#### 3.4. Mechanical Model

#### 3.5. Numerical Implementation

## 4. Results

#### 4.1. Direct Quenching Prediction

## 5. Conclusions

- -
- A prediction tool has been developed for the design of the industrial quenching process.
- -
- The computational cost-accuracy relationship has been optimized with respect to the models proposed in the literature for this type of process.
- -
- The appearance of unwanted microstructures can be predicted by the model with good accuracy.
- -
- The proposed model allows us to find out the mechanical properties of the product based on the composition of the steel used and the process parameters. Innumerable plant tests required for the adjustment of process parameters can be avoided by using this tool.
- -
- The proposed model can be used to redesign and optimize the quenching process, for example, to reduce energy consumption, e.g., by adjusting the homogenization temperature, minimizing operating times, or even eliminating the homogenization step (direct quenching), if process times and parameters are adjusted to avoid the appearance of ferrite.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${C}_{p}$ | specific heat at constant pressure |

$D$ | averaged piece diameter |

$E$ | Young modulus |

$Gr$ | Grashof number $\frac{g{\rho}^{2}\beta {L}^{3}\left({T}_{w}-{T}_{ref}\right)}{{\mu}^{2}}$ |

$L$ | piece length |

${L}_{c}\text{}$ | characteristic length $\sqrt{\frac{{\sigma}_{st}}{g\left({\rho}_{l}-{\rho}_{v}\right)}}$ |

${M}_{s}$ | onset temperature for martensitic transformation |

$Pr$ | air Prandtl number $\frac{{C}_{p}\mu}{k}$ |

$P{r}_{l}$ | liquid Prandtl number |

$Q$ | heat flux per unit of volume generated by metallurgical transformations. |

$R{a}_{L}$ | air Rayleigh number, $G{r}_{L}Pr$, based on piece length $L$ |

$R{e}_{l}$ | liquid Reynolds number $\frac{\rho eV}{\mu}$ |

$T$ | temperature |

${T}_{b}$ | liquid bulk temperature |

${X}_{i}$ | proportion of the metallurgical phase $i$ |

$R$ | hardening law for multiphasic materials |

$V$ | fluid velocity |

${V}_{flot}$ | characteristic velocity induced by buoyancy $\sqrt{\frac{{\rho}_{l}-{\rho}_{v}}{{\rho}_{l}}gL}$ |

$d$ | grain size |

$e$ | averaged piece thickness |

$g$ | gravity |

$h$ | heat transfer coefficient |

${h}_{l}$ | single phase (liquid) heat transfer coefficient |

${i}_{lv}^{\text{'}}$ | modified latent heat |

${i}_{lv}$ | latent heat of evaporation |

$k$ | thermal conductivity |

$q$ | heat flux per unit of surface |

$t$ | time |

Greek symbols | |

$\alpha $ | solid thermal dilatation coefficient |

$\mathsf{\Delta}H$ | enthalpy of solid phase change |

$\u03f5$ | emissivity |

$\epsilon $ | strain tensor |

$\dot{\lambda}$ | plasticity multiplier |

$\mu $ | viscosity |

$\nu $ | Poisson coefficient |

$\rho $ | density |

$\sigma $ | stress tensor |

${\sigma}_{SB}$ | Stefan-Boltzmann constant |

${\sigma}_{st}$ | liquid-vapor surface tension |

${\sigma}_{eq}$ | Von Mises stress |

${\sigma}_{y}$ | yield stress |

${\sigma}^{D}$ | deviatoric stress tensor |

$\tau $ | incubation time for a micro-constituent |

Subscripts | |

$CHF$ | critical heat flux |

$LDF$ | Leidenfrost point |

$FDB$ | fully developed boiling |

$ONB$ | onset of the nucleate boiling |

$c$ | convection |

$b$ | bainite |

$f$ | ferrite |

$\gamma $ | austenite |

$i$ | metallurgical phase index |

$l$ | liquid phase |

$m$ | martensite |

$p$ | pearlite |

$rad$ | radiation |

$sat$ | saturation |

$v$ | vapor phase |

$w$ | wall |

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**Figure 1.**(

**a**): Manufactured axle spindles: Aspect of the piece after quenching (left spindle) and a final manufactured spindle (right spindle). (

**b**) Images of the manipulation of the spindles after the homogenizing furnace. (

**c**) Submersion in the quenching bath. Images provided by CIE GALFOR S.A.

**Figure 2.**$q$ vs. ${T}_{w}$ given by the thermal model when applied to the analyzed spindle. (

**a**): heat flux during air transportation. (

**b**): heat flux inside the quenching bath for $V=0.34$ m/s and ${T}_{b}$ = 35 °C.

**Figure 3.**Heat flux dependency on different bath agitation velocities $V$ for ${T}_{b}={35}^{\circ}\mathrm{C}$.

**Figure 4.**(

**a**): Picture of the experimental Inconel 600 standard probe tested (

**left**) and the experimental quenching container with agitation (

**right**). (

**b**) Computational mesh and temperature field in the probe for Test #5 at t = 5 s. (

**c**): Evolution of temperature T vs. time t (solid lines) and cooling rate dT/dt vs. T (dashed lines) at the center of the standard experimental probe for Test #5.

**Figure 5.**(

**a**): Sketch of the spindle geometry and part of the supporting device. (

**b**): Computational mesh.

**Figure 7.**JMAK model resolution algorithm (referenced in the metallurgical model resolution algorithm described in Figure 6).

**Figure 8.**(

**a**): Final bainite content along the symmetry plane that divides the piece and the supporting device. (

**b**): Detail of the upper part of the spindle.

**Figure 9.**Micrographs (500×) corresponding to the marked points 1 to 5 and 0 taken from probes of a spindle subjected to the described industrial quenching process. Ruler in micrographs indicated 50 $\mathsf{\mu}\mathrm{m}$.

**Figure 10.**(

**a**): Bainite content along lines A (red) and B (blue) of Figure 6. Solid lines correspond to the support side. (

**b**): Numerical deviations (in mm) predicted by the model after the quenching treatment.

**Figure 11.**Prediction of final content of (

**a**) bainite, (

**b**) martensite and (

**c**) ferrite for direct quenching.

**Figure 12.**Micrographs (500×) corresponding to the marked points 1 to 5 and 0 taken from probes of a spindle subjected to the alternative direct quenching process.

**Figure 13.**(

**a**): Ferrite content along the depth of the piece: experimental values (blue dots) and numerical prediction (blue line). HV hardness: experimental measurements (red dots) and numerical (red line). (

**b**): Numerical deviations (in mm) predicted by the model after the direct quenching process conditions.

**Table 1.**Experimental piece superficial temperatures T

_{w}at different times during the transportation to the quenching bath.

Test | T_{w}_{1} [K] | T_{w}_{2} [K] | T_{w}_{3} [K] | T_{w}_{4} [K] |
---|---|---|---|---|

1 | 1082 | 1073 | 1063 | 1062 |

2 | 1093 | 1093 | 1078 | 1068 |

**Table 2.**Deviations (averaged and maximum temperature deviations, maximum cooling rate, and ${T}_{CHF}$ deviations) between experimental measurements and numerical results of the thermal problem for the cylindrical probe.

Test | V (m/s) | T_{b} (°C) | Aver. (%) | Max (%) | Max. Cooling Rate (%) | T_{CHF} (%) |
---|---|---|---|---|---|---|

#1 | 0.34 | 35 | 4.0 | 11.2 | 5.2 | 9.3 |

#2 | 0.5 | 35 | 3.9 | 12.4 | 1.8 | 6.2 |

#3 | 0.75 | 35 | 4.6 | 16 | 8.6 | 3.4 |

#4 | 0.5 | 20 | 4.8 | 15.6 | 6.9 | 4.2 |

#5 | 0.5 | 50 | 7.5 | 13 | 1.6 | 1.9 |

Mesh | Nº of elements | ∆t[s] | ε_{1}[%] | ε_{2}[%] |

1# | 264664 | 0.025 | 4.29 | 0.71 |

2# | 391530 | 0.025 | 1.38 | 0.93 |

3# | 607746 | 0.025 | −− | −− |

Mesh | Nº of elements | ∆t[s] | ε_{1}[%] | ε_{2}[%] |

2# | 391530 | 0.1 | 1.4 | 1.36 |

2# | 391530 | 0.025 | 0.13 | 0.13 |

2# | 391530 | 0.01 | −− | −− |

**Table 4.**Comparison of the final bainite proportion at points 0 and 0S predicted by the numerical model and obtained from the metallurgical analysis of the treated spindles.

Point | X_{b} Num. | X_{b} Exp. | ε_{b} [%] |
---|---|---|---|

0 | 0.18 | 0.15 | 20 |

0S | 0 | 0 | 0 |

**Table 5.**Comparison of the final ferrite proportion at points 1–5 and 0 (Figure 9) given by the metallurgical analysis and the numerical model for the spindles subjected to direct quenching process.

Point | X_{f} Exp. | X_{f} Num. | ε_{f} [%] | HV exp. |
---|---|---|---|---|

1 | 0.36 | 0.39 | 8.3 | 216 |

2 | 0.37 | 0.40 | 8.1 | 218 |

3 | 0.35 | 0.41 | 17.1 | 236 |

4 | 0.32 | 0.34 | 6.25 | 244 |

5 | 0.15 | 0.20 | 33 | 262 |

0 | 0 | 0 | 0 | 280 |

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## Share and Cite

**MDPI and ACS Style**

Coroas, C.; Viéitez, I.; Martín, E.; Román, M.
Numerical Modeling for the Prediction of Microstructure and Mechanical Properties of Quenched Automotive Steel Pieces. *Materials* **2023**, *16*, 4111.
https://doi.org/10.3390/ma16114111

**AMA Style**

Coroas C, Viéitez I, Martín E, Román M.
Numerical Modeling for the Prediction of Microstructure and Mechanical Properties of Quenched Automotive Steel Pieces. *Materials*. 2023; 16(11):4111.
https://doi.org/10.3390/ma16114111

**Chicago/Turabian Style**

Coroas, Carlos, Iván Viéitez, Elena Martín, and Manuel Román.
2023. "Numerical Modeling for the Prediction of Microstructure and Mechanical Properties of Quenched Automotive Steel Pieces" *Materials* 16, no. 11: 4111.
https://doi.org/10.3390/ma16114111