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A Closed-Form Solution for Temperature Profiles in Selective Laser Melting of

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A Closed-Form Solution for Temperature Profiles in Selective Laser
Melting of Metal Additive Manufacturing

Steven Y. Liang1,a, Jinqiang Ning1,b and Elham Mirkoohi1
1George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, 801 Ferst

Drive, Atlanta, GA 30332, USA.
Correspondence: asteven.liang@me.gatech.edu, bjinqiangning@gatech.edu

Keywords: Selective Laser Melting; Closed-form solution; Temperature prediction; Selective laser
melting; Part boundary; Latent heat; High computational efficiency.

Abstract. This paper presents a closed-form solution for the temperature prediction in selective laser
melting (SLM). This solution is developed for the three-dimensional temperature prediction with
consideration of heat input from a moving laser heat source, and heat loss from convection and
radiation on the part top boundary. The consideration of heat transfer boundary condition and latent
heat in the closed-form solution leads to an improvement on the understanding of thermal
development and prediction accuracy in SLM, and thus the usefulness of the analytical model in the
temperature prediction in real applications. A moving point heat source solution is used to calculate
the temperature rise due to the heat input. A heat sink solution is used to calculate the temperature
drop due to heat loss from convection and radiation on the part boundary. The heat sink solution is
modified from a heat source solution with equivalent power due to heat loss from convection and
radiation, and zero-moving velocity. The temperature solution is then constructed from the
superposition of the linear heat source solution and linear heat sink solution. Latent heat is considered
using a heat integration method. Ti-6Al-4V is chosen to test the presented model with the assumption
of isotropic and homogeneous material. The predicted molten pool dimensions are compared to the
documented values from the finite element method and experiments in the literature. The presented
model has improved prediction accuracy and significantly higher computational efficiency compared
to the finite element model.

Introduction
Selective laser melting (SLM) is a widely used metal additive manufacturing (AM) process, in

which high-density laser powder is used to fully melt and fuse metal powders to build parts in a
layer-by-layer manner. Defects such as distortion, crack and balling effect are frequently observed
due to the large thermal gradient caused by the repeatedly rapid heat and solidification in SLM [1-3].

Experimental temperature measurements are difficult and inconvenient due to the restricted
accessibility under extremely high temperatures [4-6]. Infrared (IR) camera can measure the
temperature profile only on the exposed surfaces [7]. The embedded thermocouple can measure the
temperature history only in the far field, typically inside the substrate [8]. The metallographic
technique is also employed for post-process measurement of molten pool geometry based on the
solidification microstructure, which requires extensive experimental work [9]. Therefore, numerical
models and analytical models are developed for convenient temperature prediction in SLM.

Numerical models have been developed based on the finite element method (FEM), in which the
temperature, residual stress, and distortion were investigated [9-12]. Although the numerical models
have made considerable progress in the prediction of the SLM process, the expensive computational
cost is still the major drawback.

Analytical models have demonstrated their high computational efficiency in the prediction of
manufacturing processes [13-14]. To overcome the aforementioned drawbacks, analytical models
were developed for temperature prediction in the AM processes using closed-form solutions [15].
Temperature models were developed using point moving heat source, line moving heat source,
semi-ellipsoidal moving heat source, and uniform moving heat source [16,17]. All solutions were
developed for a 3D semi-infinite body without considering heat loss at the part boundary from

Materials Science Forum Submitted: 2019-06-12
ISSN: 1662-9752, Vol. 982, pp 98-105 Accepted: 2019-06-27
© 2020 Trans Tech Publications Ltd, Switzerland Online: 2020-03-20

All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans
Tech Publications Ltd, www.scientific.net. (#537514887, Morris M. Bryan, China-06/03/20,22:53:15)

convection and radiation. The temperature distribution in the direct metal deposition was predicted
using the moving point heat source solution with the assumption of the homogeneous and isotropic
solid workpiece. Good agreements were observed based upon validation to experimental results [18].
The temperature model was further developed with consideration of build layers, latent heat and
temperature-sensitive material properties [19]. An in-process temperature model was developed to
predict molten pool evolution with consideration of laser absorption, latent heat, scanning strategy
and powder packing [20]. However, the heat loss due to the convection and radiation at part boundary
has not been considered in the developed models. FEM must be used with the analytical model to
properly impose the boundary conditions [21], which resulted in an unoptimized computational
efficiency. The neglection of the heat transfer boundary condition significantly reduced the
usefulness of other developed analytical models in real applications because of the temperature
overestimation. The neglection of the latent heat also leads to the overestimation because of the
neglection of energy required for phase transformation.

This work presents an analytical model for the temperature prediction in SLM with consideration
of heat loss at the top boundary due to convection and radiation. The heat input from laser power is
calculated using the moving point heat source solution. The heat source solution is modified for the
consideration of the heat loss due to the convection and radiation with equivalent power loss and zero
velocity. The final solution is constructed from the superposition of the heat source solution and heat
sink solutions. The presented model is employed to predict the temperature distribution in the SLM of
Ti-6Al-4V under various process conditions. Molten pool dimensions are determined from the
predicted temperatures by comparing them to the material melting temperature as illustrated in Fig. 1.
The molten pool dimensions are validated to the experimental values in the literature, which were
measured based on the solidification microstructure. In addition, the molten pool dimensions from the
presented model are compared to that adopted from the FEM model regarding prediction accuracy
and computational efficiency.

Fig. 1. Schematic view of molten pool geometry. P, W, L, D denote the moving laser power, molten
pool width, molten pool length, and molten pool depth respectively. The red arrow denotes laser heat

source input. The blue arrows denote heat loss on the top surface due to convection and radiation

Material and Methodology
In this work, a closed-form solution is developed for the temperature prediction in SLM with

consideration of convection and radiation at the part boundary. Temperature increase due to the laser
power input is calculated using the moving point heat source solution as

the following.

𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) =
𝑃𝑃𝑃𝑃

4𝜋𝜋𝜋𝜋𝜋𝜋(𝑇𝑇𝑚𝑚 − 𝑇𝑇0)
exp �

−𝑉𝑉(𝜋𝜋 + 𝑥𝑥)
2𝜅𝜅

�
(1)

where 𝑃𝑃 is laser power, 𝑃𝑃 is laser absorption, 𝑉𝑉 is scanning speed, 𝜋𝜋 is conductivity, 𝜅𝜅 is diffusivity
(𝜅𝜅 = 𝑘𝑘

𝜌𝜌𝑠𝑠
, 𝜌𝜌 is materials density, c is specific heat), 𝜋𝜋 is the distance from the laser heat source location

Materials Science Forum Vol. 982 99

(𝜋𝜋 = �𝑥𝑥2 + 𝑦𝑦2 + 𝑧𝑧2, x, y, z are the distance from the laser heat source along three mutually
perpendicular axes), 𝑇𝑇0 is the room temperature, and 𝑇𝑇𝑚𝑚 is the material melting temperature.

Temperature decrease due to the heat loss from convection and radiation is calculated using the
heat sink solution, which is derived by modifying the heat source solution with equivalent power loss
and zero moving speed. The non-moving part boundary is mathematically discretized into many
sections, and each section is now named heat sink. The equivalent power loss due to convection and
radiation at each heat sink are expressed as the following.
𝑄𝑄𝑠𝑠𝑠𝑠𝑐𝑐𝑐𝑐 = 𝐴𝐴ℎ(𝑇𝑇 − 𝑇𝑇0)

(2)
𝑄𝑄𝑠𝑠𝑟𝑟𝑟𝑟 = Aεσ(𝑇𝑇4 − 𝑇𝑇04)

(3)
where ℎ is heat convection coefficient, ε is emissivity, σ is Stefan-Boltzmann constant, A is the area
of each heat sink, T is the temperature of each heat sink that can be estimated using the point moving
heat source solution.

The heat sink solution at the part boundary is expressed as the following.

𝜃𝜃𝑠𝑠𝑠𝑠𝑐𝑐𝑘𝑘(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) = �
𝐴𝐴𝑠𝑠

4𝜋𝜋𝜅𝜅𝜋𝜋𝑠𝑠(𝑇𝑇𝑚𝑚 − 𝑇𝑇0)
[ℎ(𝑇𝑇𝑠𝑠 − 𝑇𝑇0) + εσ(𝑇𝑇𝑠𝑠

4 − 𝑇𝑇04) ]
𝑐𝑐

𝑠𝑠=1

(4)
where 𝑖𝑖 denotes the index of the heat sink, 𝑛𝑛 denotes the total number of heat sinks, 𝜋𝜋𝑠𝑠 is the distance
from the heat sink location.

Finally, the temperature solution is constructed from the superposition of heat source solution and
heat sink solutions as expressed in the following.
𝜃𝜃(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) = 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 − 𝜃𝜃𝑠𝑠𝑠𝑠𝑐𝑐𝑘𝑘 =

𝑃𝑃𝑃𝑃

4𝜋𝜋𝜋𝜋𝜋𝜋(𝑇𝑇𝑚𝑚 − 𝑇𝑇0)
exp �
−𝑉𝑉(𝜋𝜋 + 𝑥𝑥)
2𝜅𝜅

� − �
𝐴𝐴𝑠𝑠

4𝜋𝜋𝜅𝜅𝜋𝜋𝑠𝑠(𝑇𝑇𝑚𝑚 − 𝑇𝑇0)
[ℎ(𝑇𝑇𝑠𝑠 − 𝑇𝑇0) + εσ(𝑇𝑇𝑠𝑠
4 − 𝑇𝑇04) ]
𝑐𝑐
𝑠𝑠=1

(5)
where 𝜃𝜃 is the dimensionless temperature that can be expressed as

𝜃𝜃 =
𝑇𝑇 − 𝑇𝑇0
𝑇𝑇𝑚𝑚 − 𝑇𝑇0

(6)
The latent heat is considered in the presented model using an integration method as expressed in

the following.

�
𝑇𝑇 = 𝑇𝑇𝑆𝑆 (𝑇𝑇𝑆𝑆 < 𝑇𝑇 < 𝑇𝑇𝐿𝐿)

𝑇𝑇 = 𝑇𝑇 −
𝐿𝐿𝑓𝑓
𝑐𝑐

(𝑇𝑇 > 𝑇𝑇𝐿𝐿)

(7)
where 𝑇𝑇𝑆𝑆 is solidus temperature, 𝑇𝑇𝐿𝐿 is liquidus temperature, 𝐿𝐿𝑓𝑓 is the latent heat, 𝑐𝑐 is specific heat.

Results and Discussion
To investigate the prediction accuracy and computational efficiency of the presented model, the

temperature distribution in the SLM of Ti-6Al-4V was predicted under various process conditions as
given in Table 1. The build body was assumed to be isotropic and homogeneous with boundary
condition imposed on the top surface. The materials properties of Ti-6Al-4V and heat transfer
coefficients are given in Table 2.

100 Advanced Materials and Processing Technologies II

Table 1. Process conditions in the selective laser melting of Ti-6Al-4V [9]
Test 𝑃𝑃 [𝑊𝑊] 𝑉𝑉[mm/s]

1 20 200
2 40 200
3 60 200
4 80 200

Table 2. Material properties of Ti-6Al-4V [9]

Name Value Unit

Density (𝜌𝜌) 4428 [Kg/m3]

Heat capacity (𝐶𝐶𝑝𝑝) 580 [J/kg-K]

Bulk thermal conductivity (𝜋𝜋𝑡𝑡) 7.2 [W/m-K]

Melting temperature (𝑇𝑇𝑚𝑚) 1655 [oC]

Room temperature (𝑇𝑇0) 20 [oC]

Solidus temperature (𝑇𝑇𝑠𝑠) 1605 [oC]

Liquidus temperature (𝑇𝑇𝑙𝑙) 1655 [oC]

Absorption (𝑃𝑃) 0.77 [1]

Latent heat (𝐻𝐻𝑓𝑓) 365000 [J/Kg]

Heat Convection coefficient (ℎ) 24 [W/m2-K]

Stefan–Boltzmann constant (𝜎𝜎) 5.67 10-8 [W/m2-K4]

Radiation emissivity (𝜀𝜀) 0.9 [1]

The temperatures distribution was predicted near laser heat source location under various process
conditions as illustrated in Fig. 2, in which the temperature profiles were plotted with a top view
while laser scanned along x-direction at x = 0.4 mm, y = 0.25 mm. The larger the laser power, the
larger the heat affected zone, and vice versa. This observation is consistent with the instinctive trend.

Materials Science Forum Vol. 982 101

Fig. 2. Top view temperature distribution in SLM of Ti-6Al-4V alloy under (a) test 1, (b) test 2, (c)
test 3, (d) test 4 process conditions

The temperature distribution was also plotted along the molten width direction and molten pool
depth direction under the test 1 process condition as illustrated in Fig. 3. The maximum temperature
in the SLM under the test 1 process condition was about 4000 oC. The maximum temperature was
higher than the material melting temperature due to the highly concentrated energy from the point
heat source, which indicated the existence of material evaporation. The material evaporation was
confirmed from the experimental observation in the literature, in which a laser with a spot radius of
26 μm was employed [9]. The smaller the laser spot radius, the higher the concentrated energy.
Symmetric temperature distributions were observed along the width direction (y-direction) as shown
in Fig. 3a. Small regions of constant temperatures were observed due to the consideration of latent
heat, in which the phase transformation took place instead of temperature rise with continuous heat
input.

Fig. 3. Temperature distribution (a) along width direction (y-direction) and (b) depth direction
(z-direction)

The molten pool dimensions are determined from the predicted temperatures by comparing to the
material melting temperature. The obtained molten pool dimensions are compared to FEM values and
to the experimental values in the literature [9]. Experimental measurements of the molten pool depth
and width were conducted based on the solidification microstructure. The molten pool volume was
calculated as the following.

𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 =
𝜋𝜋𝜋𝜋𝑊𝑊𝐿𝐿

(8)

102 Advanced Materials and Processing Technologies II

where 𝜋𝜋, 𝑊𝑊, 𝐿𝐿 are the molten pool depth, molten pool width and molten pool length
respectively.

The molten pool dimensions of molten pool width, depth, length, and volume under four different
process conditions were shown in Fig. 4. The larger the laser power, the larger the molten pool
dimensions, and vice versa. The predicted molten pool depth and length using the presented model
demonstrated an improved prediction accuracy, compared to the available FEM results in the
literature [9]. The prediction using the presented model was implemented using a MATLAB program
on a personal computer running at 2.8 GHz. The average computational time was recorded as 12 s.
For comparison, FEM usually needs hours of time depending on the part size and mesh resolution for
a comparable prediction accuracy [22,23].

In addition, the balling effect was investigated based on the molten pool length to width ratio. The
molten pool length to width ratio from the prediction was smaller than the critical value 𝜋𝜋 [24],
which indicated that no concentrated balling effect existed. This finding confirmed the experimental
observation in the literature [9].

Fig. 4. Comparison between experimental measurements (black color), documented results from the
finite element model (red color) and predicted results using the presented model (blue color). Plots

(a-d) represents the molten pool width, depth, length, and volume respectively under various process
conditions

Conclusion
This paper presents a closed-form solution for the temperature prediction in SLM with

consideration of heat loss at the part top boundary, and latent heat. The lack of consideration of the
heat transfer boundary condition in previously developed analytical models reduced their prediction
accuracy. This temperature solution is constructed from the superposition of the moving point heat
source solution and heat sink solutions. The latter is derived by modifying the former with equivalent
power due to heat loss from convection and radiation at the part boundary and zero moving velocity.
Ti-6Al-4V was chosen to investigate the prediction accuracy and computational efficiency of the
presented temperature model under various process conditions. Molten pool dimensions obtained

Materials Science Forum Vol. 982 103

from the predicted temperatures were compared to the FEM results and experimental measurements
in the literature. Improved prediction accuracy and considerably higher computational efficiency
were observed with the presented model, compared to the documented FEM results. In the future, the
developed closed-form solution can be used to impose boundary condition on the lateral boundary
and geometrical complex boundary.

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three-dimensional finite element analysis of the temperature field during laser melting of metal
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[24] Kruth, J. P., Levy, G., Klocke, F., & Childs, T. H. C. (2007). Consolidation phenomena in laser
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https://doi.org/10.1016/j.cirp.2007.10.004

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• Type your report in word or pdf format: at least 6 pages excluding the cover page (title, name, affiliation, etc.)

• Prepare the report using ASME conference paper template (can be downloaded from ASME website)

• The report should include the followings: – Title, abstract, and your name – Literature review (It is suggested that least 10 related papers are reviewed to describe the state-of-the-art on the subject). – Summary of the work in the reference paper you selected, such as PDFs, BSc, ICs, solution method, solutions, etc. – Your proposed work and its objectives.

• Describe what’s new or the difference in your work as compared to the existing papers (they can be different assumptions, BC/IC conditions, physics, or solution methods).

• Use 1-2 schematic drawings to illustrate the physical problem if variations are made. – Results and discussions

• Use a tool, such as Excel or any other software, to plot the typical solution from the reference paper your selected.

• Investigate the effects of key parameters on the solutions using existing solutions • Compare your new solution with existing ones.

• Discuss the pros and cons of different solutions in the context of heat transfer analysis. – Conclusions – List of References Cited.

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