# A Numerical Investigation of Dimensionless Numbers Characterizing Meltpool Morphology of the Laser Powder Bed Fusion Process

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations of the LPBF Process

#### 2.1. Thermo-Fluidic Model of the LPBF Process

**∇**, are shown in bold to distinguish them from other scalar quantities.

#### 2.2. Non-Dimensional Formulation of the Governing Equations

#### Weak Formulation

#### 2.3. Computational Implementation

## 3. Experimental and Numerical Validation

## 4. Empirical Analysis of the Energy Absorbed by the Meltpool

#### 4.1. Process Variables, Material Properties, and Output Variables

#### 4.2. Parametrization in Terms of the Dimensionless Quantities

## 5. Results

#### 5.1. Influence of Péclet Number on Advection Transport in the Meltpool

#### 5.2. Influence of Marangoni Number on the Meltpool Aspect Ratio

#### 5.3. Influence of Stefan Number on Meltpool Volume

#### 5.4. Influence of the Heat Absorbed on the Solidification Cooling Rates

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic depicting the part-building process in Laser Powder Bed Fusion (LPBF). Laser irradiation on the powdered material causes localized melting and fusion of the metal powder on top of the partially built part. The localized melting results in a small pool of liquified metal referred to as the meltpool. Shown in the inset figure are the state of the powder under the laser—with the newly solidified region and a meltpool with convective flow of the liquified metal, and this region comprises the computational domain ($\Omega $) of the numerical model presented in this work.

**Figure 2.**Schematic of the 3D finite element (FE) computational domain indicating the laser scan path and the relevant boundary conditions. Shown also is the underlying adaptive mesh that evolves with the location of the heat source. Representative dimensionless temperature profile and location of the meltpool obtained from the FE simulation of SS316 alloy AM are shown. The numerical parameters and material properties used in this simulation are given in Section 2.3 and in the Supplementary Information.

**Figure 3.**Dependence of cooling rates obtained from experiments and the FE model on the energy density, $\frac{P}{{\mathit{\nu}}_{p}}$. The average cooling rate from the FE model was estimated using the relation: $\dot{T}=G{\mathit{\nu}}_{p}=|\mathbf{\nabla}T|{\mathit{\nu}}_{p}$. Laser power (W) and scan speed (mm/s) combinations used for this study were $(P,{\mathit{\nu}}_{p})=(90,575),(90,675)$.

**Figure 4.**Validation of the FE model results by comparing with corresponding values reported in the literature. (

**a**) Variation in point temperature with time for the case P = 200 W. (

**b**) Variation in point temperature with time for the case P = 100 W. (

**c**) Variation in maximum pool velocity with time for the case P = 100 W [58].

**Figure 5.**Measure of total advection measured as $\mathit{Pe}{\mathit{\nu}}_{\mathit{max}}$ vs. surface tension based advection $\mathit{Ma}\widehat{\mathit{U}}={a}_{0}\mathit{Ma}+{a}_{1}\mathit{MaE}+{a}_{2}\mathit{MaPe}$ on a log–log scale for (

**a**) AlSi10Mg, (

**b**) SS316, (

**c**) Ti6Al4V alloys. Corresponding plots comparing IN718 and AZ91D alloys, and a comparison of all the five alloys considered in this work can be found in Figures S1 and S2 of the Supplementary Information, respectively. The advection measure corresponds to the degree of fluid flow inside the meltpool. Each point in these plots represent a single simulation result for the relevant quantities plotted, and is obtained from the FEM framework.

**Figure 6.**Correlation of the aspect ratio with $\mathit{Ma}\widehat{\mathit{U}}={a}_{0}\mathit{Ma}+{a}_{1}\mathit{MaE}+{a}_{2}\mathit{MaPe}$, plotted on a log–log scale, for (

**a**) Ti6Al4V alloy, and for (

**b**) three alloys (Ti6Al4V, SS316 and AlSi10Mg) shown in a single plot to demonstrate clustering. A combined plot demonstrating this clustering for all the five alloys (Ti6Al4V, SS316, AlSi10Mg, IN718 and AZ91D) considered in this work can be found in Figure S3 of the Supplementary Information. Each point in these plots represent a single simulation result for the relevant quantities plotted, and is obtained from the FEM framework.

**Figure 7.**Correlation of the meltpool volume $\left({l}_{m}{w}_{m}{d}_{m}\right)$ with $\frac{\mathit{Ste}\widehat{\mathit{U}}}{{\mathit{T}}_{\mathit{c}}}={a}_{0}\frac{\mathit{Ste}}{\mathit{Tc}}+{a}_{1}\frac{\mathit{SteE}}{\mathit{Tc}}+{a}_{2}\frac{\mathit{StePe}}{\mathit{Tc}}$, plotted on a log–log scale, for (

**a**) AlSi10Mg, (

**b**) SS316, and (

**c**) Ti6Al4V alloys. Corresponding plots comparing IN718 and AZ91D alloys, and a comparison of all the five alloys considered in this work can be found in Figure S4 and S5 of the Supplementary Information, respectively. Each point in these plots represent a single simulation result for the relevant quantities plotted, and is obtained from the FEM framework.

**Figure 8.**Dimensionless temperature gradient $\left(G\right)$ with the $\widehat{\mathit{U}}$ for different alloys. Plots corresponding to IN718 and AZ91D alloy material can be found in Figure S6 of the Supplementary Information. Each point in these plots represent a single simulation result for the relevant quantities plotted, and is obtained from the FEM framework.

**Figure 9.**(

**a**) Variation of the dimensional cooling rate, $G{\mathit{\nu}}_{p}$, with $\widehat{\mathit{U}}={a}_{0}+{a}_{a}\mathit{E}+{a}_{2}\mathit{Pe}$, plotted on a log–log scale. Here, $\widehat{\mathit{U}}$ is changed by changing $\mathit{E}$, but keeping $\mathit{Pe}$ fixed for SS316 alloy, (

**b**) Variation of dimensional cooling rate, $G{\mathit{\nu}}_{p}$, with $\widehat{\mathit{U}}={a}_{0}+{a}_{a}\mathit{E}+{a}_{2}\mathit{Pe}$, plotted on a log–log scale. Here, $\widehat{\mathit{U}}$ is changed by changing $\mathit{Pe}$, but keeping $\mathit{E}$ fixed for SS316 alloy.

Parameter | Expression | Physical Interpretation |
---|---|---|

$\tilde{l}$ | $\frac{{l}_{p}}{{l}_{p}}$ | Dimensionless powder layer thickness |

$\tilde{r}$ | $\frac{{r}_{s}}{l}$ | Dimensionless laser spot radius |

$\tilde{t}$ | $\frac{t{\mathit{\nu}}_{p}}{{l}_{p}}$ | Dimensionless time |

$\tilde{T}$ | $\frac{T-{T}_{\infty}}{{T}_{l}-{T}_{\infty}}$ | Dimensionless temperature |

$\tilde{\mathit{\nu}}$ | $\frac{\mathit{\nu}}{{\mathit{\nu}}_{p}}$ | Dimensionless velocity |

$\tilde{\mathit{p}}$ | $\frac{\begin{array}{c}\hfill \mathit{p}\end{array}}{\rho {\mathit{\nu}}_{p}^{2}}$ | Dimensionless pressure |

$\tilde{\mathbf{\nabla}}$ | $\frac{1}{{l}_{p}}\mathbf{\nabla}$ | Dimensionless gradient operator |

Parameter | Expression | Physical Interpretation |
---|---|---|

Prandtl $\left(\mathit{Pr}\right)$ | $\frac{\nu}{\alpha}$ | Ratio of momentum to thermal diffusivity |

Grashof $\left(\mathit{Gr}\right)$ | $\frac{g{l}^{3}\beta ({T}_{l}-{T}_{\infty})}{{\nu}^{2}}$ | Ratio of buoyancy force to viscous force |

Darcy $\left(\mathit{Da}\right)$ | $\frac{\kappa}{{d}_{\varphi}^{2}}$ | Ratio of permeability to the cross-sectional area |

Marangoni $\left(\mathit{Ma}\right)$ | $\frac{d\gamma}{dT}\frac{{l}_{p}\Delta T}{\mu \alpha}$ | Ratio of advection (surface tension) to diffusive transport |

Péclet$\left(\mathit{Pe}\right)$ | $\frac{{l}_{p}{\mathit{\nu}}_{p}}{\alpha}$ | Ratio of advection transport to diffusive transport |

Stefan $\left(\mathit{Ste}\right)$ | $\frac{c({T}_{l}-{T}_{s})}{L}$ | Ratio of sensible heat to latent heat |

Power $\left(\mathit{Q}\right)$ | $\frac{P}{\rho c({T}_{l}-{T}_{\infty}){\mathit{\nu}}_{p}{l}_{p}^{2}}$ | Dimensionless power with velocity dependence |

Radiation measure $\left(\frac{{\mathit{t}}_{\mathit{s}}}{\mathit{Bo}}\right)$ | $\frac{\sigma {({T}_{l}-{T}_{\infty})}^{3}}{\rho c{\mathit{\nu}}_{p}}$ | Measure of radiation contribution to the heat transfer |

Biot $\left(\mathit{Bi}\right)$ | $\frac{h{l}_{p}}{k}$ | Ratio of resistance to diffusion and convection heat transport |

**Table 3.**First attempt of the regression analysis to estimate the coefficients, ${a}_{i}$, using the linear least-squares approach. Asterisk (${}^{*}$) indicates the statistical significance of the coefficient using a t-test with a 95% confidence interval. Other statistics: ${R}^{2}=0.65$, Adjusted ${R}^{2}=0.61$, F-statistic = $20.12$, P(F) = $0.0$. Condition number = $8.11\times {10}^{7}$.

Parameter | Intercept | Q | ${\mathit{Pe}}^{-1}$ | $\frac{\mathit{Tc}}{\mathit{Ste}}$ | $\frac{\mathit{Bi}}{\mathit{Pe}}$ | $\frac{{\mathit{t}}_{\mathit{s}}}{\mathit{Bo}}$ |
---|---|---|---|---|---|---|

${a}_{i}$ | $1.45{\phantom{\rule{3.33333pt}{0ex}}}^{*}$ | $0.0053{\phantom{\rule{3.33333pt}{0ex}}}^{*}$ | $-0.1719{\phantom{\rule{3.33333pt}{0ex}}}^{*}$ | $-0.7076{\phantom{\rule{3.33333pt}{0ex}}}^{*}$ | $300.3$ | $-\mathrm{18,200}{\phantom{\rule{3.33333pt}{0ex}}}^{*}$ |

**Table 4.**Second attempt of the regression analysis to estimate the coefficients, ${a}_{i}$, using the linear least-squares approach. Asterisk (${}^{*}$) indicates the statistical significance of the coefficient using a t-test with a 95% confidence interval. Other statistics: ${R}^{2}=0.763$, Adjusted ${R}^{2}=0.750$, F-statistic = $55.95$, P(F) = $0.0$. Condition number = $1.12\times {10}^{3}$.

Parameter | Intercept | E | $\mathit{Pe}$ | $\frac{\mathit{Tc}}{\mathit{Ste}}$ |
---|---|---|---|---|

${a}_{i}$ | $0.6938{\phantom{\rule{3.33333pt}{0ex}}}^{*}$ | $0.0087{\phantom{\rule{3.33333pt}{0ex}}}^{*}$ | $-0.1677{\phantom{\rule{3.33333pt}{0ex}}}^{*}$ | $0.1521$ |

**Table 5.**Third attempt of the regression analysis to estimate the coefficients, ${a}_{i}$, using the linear least-squares approach. Asterisk (${}^{*}$) indicates the statistical significance of the coefficient using a t-test with a 95% confidence interval. Other statistics: ${R}^{2}=0.746$, Adjusted ${R}^{2}=0.737$, F-statistic = $83.61$, P(F) = $0.0$. Condition number = 422.

Parameter | Intercept | E | $\mathit{Pe}$ |
---|---|---|---|

${a}_{i}$ | $0.8146{\phantom{\rule{3.33333pt}{0ex}}}^{*}$ | $0.0082{\phantom{\rule{3.33333pt}{0ex}}}^{*}$ | $-0.1654{\phantom{\rule{3.33333pt}{0ex}}}^{*}$ |

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**MDPI and ACS Style**

Bhagat, K.; Rudraraju, S.
A Numerical Investigation of Dimensionless Numbers Characterizing Meltpool Morphology of the Laser Powder Bed Fusion Process. *Materials* **2023**, *16*, 94.
https://doi.org/10.3390/ma16010094

**AMA Style**

Bhagat K, Rudraraju S.
A Numerical Investigation of Dimensionless Numbers Characterizing Meltpool Morphology of the Laser Powder Bed Fusion Process. *Materials*. 2023; 16(1):94.
https://doi.org/10.3390/ma16010094

**Chicago/Turabian Style**

Bhagat, Kunal, and Shiva Rudraraju.
2023. "A Numerical Investigation of Dimensionless Numbers Characterizing Meltpool Morphology of the Laser Powder Bed Fusion Process" *Materials* 16, no. 1: 94.
https://doi.org/10.3390/ma16010094