*2.1. Analytical Modeling*

There are various engineering applications, such as turning, grinding, welding, and 3D printing in which the computation of the temperature field in the solid is modeled as a problem of heat conduction involving a moving heat source. The objective of this section is to present the mathematical formulation and the method of solution of heat conduction by considering the moving heat source, which indeed is the case in metal additive manufacturing [32].

In this study, the basic premise is that the powder is situated in the desirable location relative to the melt pool. In other words, there is no moment or mass transfer consideration in this work, and only the heat transfer is considered. Although the effect of time difference between the two consecutive irradiations on temperature profile is not considered in this work, it is worth noting that considering the existence of the time difference between two consecutive irradiations may cause an increase in predicted temperature during the metal AM processes. This is because the predicted temperature at time *t* + Δ*t* will be the materials-response-coupled superposition considering the temperature sensitivity of thermal properties at time *t* and *t* + Δ*t*.

By considering a line heat source of constant strength *g<sup>c</sup> <sup>l</sup>* (W/m) located at the *x*-axis and oriented parallel to the *z*-axis, the source releases its energy continuously over time as it moves with a constant velocity of *v* in the positive *x*-direction. The medium is initially at room temperature. It is assumed (*∂T*/*∂z*) = 0 everywhere in the medium. Hence, the differential equation of heat conduction in the *x, y* coordinates in now taken as

$$\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{1}{k} \mathbf{g}(x, y, t) = \frac{1}{a} \frac{\partial T}{\partial t} \tag{1}$$

where *<sup>T</sup>* ≡ *<sup>T</sup>*(*x*, *<sup>y</sup>*, *<sup>t</sup>*). *<sup>k</sup>* is thermal conductivity, and *<sup>α</sup>* is the thermal diffusivity. The line heat source *<sup>g</sup><sup>c</sup> l* (W/m) is related to the equivalent volumetric source *g*(*x*, *y*, *t*) (W/m3) by the delta function notation as

$$\mathbf{g}(\mathbf{x}, y, t) = \, \mathbf{g}\_l^c \, \delta(y) \delta(\mathbf{x} - vt) \tag{2}$$

In order to consider the moving heat source, it is assumed that the coordinate system transfers from the *x*, *y* fixed coordinate system to *ζ*, *y* coordinate moving with the line heat source by using the transformation

$$\mathcal{J} = \mathfrak{x} - vt\tag{3}$$

Using the abovementioned transformation, the heat conduction equation for the moving coordinate system can be written as

$$\frac{\partial^2 T}{\partial \zeta^2} + \frac{\partial^2 T}{\partial y^2} + \frac{1}{k} g\_l^x \delta(\zeta) \delta(y) = \frac{1}{a} (\frac{\partial T}{\partial t} - v \frac{\partial T}{\partial \zeta}) \tag{4}$$

Equation (4) can be solved by the assumption of the quasi-stationary condition [32]. Using the separation of variables, the closed form solution of the temperature field can be obtained as

$$T = \frac{P}{4\pi KR} \exp\frac{-v(R-x)}{2\alpha} + T\_0 \tag{5}$$

where *P* is the laser power, *K* is thermal conductivity, *v* is scan speed (laser velocity). *α* is thermal diffusivity which can be calculated as *<sup>K</sup> <sup>ρ</sup><sup>c</sup>* . In which *ρ* is material density and *c* is material heat capacity. *R* = *x*<sup>2</sup> + *y*<sup>2</sup> is the radial distance from the heat source. *T*<sup>0</sup> is the initial temperature. The material is considered homogeneous and isotropic. As the laser moves along the surface it deposited some energy. Figure 1 depicts the heat transfer in metal AM. The heat loss from the surface by radiation and convection are not considered in this study.

**Figure 1.** Heat transfer during laser-based metal additive manufacturing.

It is worth noting that the process parameters such as laser power, scanning speed, powder size, powder distribution, etc. have influence on material properties in the metal AM processes since it may change the predicted temperature [33]. As a result, the material properties are assumed to be temperature dependent as shown in Table 1 [34].


**Table 1.** Material properties of Ti-6Al-4V.

During the metal AM process such as SLM and DMD, the melting, solidification and solid-state phase transformation take place. This is considered using modified heat capacity.

$$\mathbf{C}\_P^m = \mathbf{C}\_P(T) + L\_f \frac{\partial f}{\partial T} \tag{6}$$

In which *Cp*(*T*) is temperature dependent specific heat, *Lf* is latent heat of fusion, and *f* is liquid fraction which can be calculated from

$$f = \begin{cases} 0, T < T\_s\\ \frac{T - T\_s}{T\_L - T\_s}, \ T\_s < T < T\_L\\ 1, \quad T > T\_L \end{cases} \tag{7}$$

where, *Ts* is solidus temperature and *TL* is liquidus temperature.

The process parameters such as laser power and scan speed are defined to start the calculation of the vertical distribution of temperature during the laser-based metal AM. At first, it is assumed the powder is at room temperature. As the laser moves along the *x*-axis, it deposits the energy on the powder and causes the powder to melt, as the laser passes the affected region, the melt pool starts to solidify. As it creates the first layer, the temperature profile is calculated for that layer. Next, it starts the second layer with the dwell time of zero. It is possible that the first layer has not had enough time to cool down to the room temperature when the second layer is starting to build. As a result, it affects the heat transfer during the metal AM processes. Considering the layer addition also has a substantial influence on thermal stress and residual stress predictions. The fellow chart of considering the build layers is illustrated in Figure 2.

**Figure 2.** Fellow chart of considering build layers.

#### *2.2. Numerical Modeling*

For further validation of this work, finite element analysis is used. The temperature profile is modeled using a moving heat source analysis. The user defined functions (UDF) code is written in ANSYS Fluent software using Equation (8) in order to run a FEA on a 2D geometry, as shown in Figure 3. The build part material is Ti-6Al-4V. The heat loss from the surface due to conduction and radiation is considered. The material properties are assumed to be temperature dependent as shown in Figure 4.

**Figure 3.** Representation of the mesh and numerical model.

The geometry of the build part is a rectangle shape of 30 × 10 mm. The quadratic element with the mesh size of 0.5 mm is chosen for all the simulations, as shown in Figure 3.

**Figure 4.** Material properties as a function of temperature, (**a**) specific heat, (**b**) thermal conductivity.

The laser power distribution on the laser beam focus plane is described by the Gaussian equation [32], as

$$q(x,y) = D \frac{P}{\pi r^2} e^{(\frac{-B(x-vt)^2}{r^2})} \tag{8}$$

where, *P* is the total laser power input, *r* is laser spot radius, *v* is scanning speed *B* is gaussian shape factor, and *D* is a numerical parameter used to fit the experimental data. It accounts for the absorptivity of the material, the heat lost to the metal powder before it falls into the melt pool and the angle of the surface with the laser beam. The values of the process parameters are listed in Table 2. The melting temperature of Ti-6Al-4V is in the range of 1538–1649 ◦C. In this study, 1620 is selected as the melting temperature of Ti-6Al-4V. The two-dimensional heat transfer in a rectangular surface could be described by

$$
\rho \mathbb{C} \left( \frac{\partial T}{\partial t} + v \frac{\partial T}{\partial x} \right) = \nabla (k \nabla T) + \mathbb{S} \tag{9}
$$

where *ρ* is material density, *C* is specific heat, *k* is thermal conductivity, and *S* is the heat sink.

The boundary condition on the laser heating surface is defined as

$$k\frac{\partial T}{\partial y} = q(\mathbf{x}, y) - h(T - T\_0) - \sigma \varepsilon (T^4 - T\_0^4) \tag{10}$$

where *q*(*x*,*y*) is laser power input, *h* is the heat transfer coefficient, *σ* is the thermal radiation coefficient, *ε* is the material emissivity, *T*<sup>0</sup> is the ambient temperature. The initial condition could be as


**Table 2.** Material parameters used for numerical modeling.
