# Analytical Solution of the Non-Stationary Heat Conduction Problem in Thin-Walled Products during the Additive Manufacturing Process

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## Abstract

**:**

## 1. Introduction

## 2. Model and Methods Description

#### 2.1. Problem Statement

- The physical properties of the substrate and the filler material (specific heat capacity c, density ρ, thermal conductivity λ, thermal diffusivity a) are temperature-independent.
- The effect of convection of liquid metal is not considered.
- Heat flux distribution of the heat source q
_{h}is presented as a surface normally distributed heat source. - Heat transfer occurs according to Newton’s law.

_{h}(x, y) is the heat flux density.

_{h}(x, y). In the x0y plane, the power-density distribution is described by the Gaussian function:

_{h}—heat-source power, η—heat efficiency, R

_{H}—effective radius of the heat source, and β—tilt angle.

#### 2.2. Analytical Model of Non-Stationary Heat Transfer

^{−b(t−t′)}into Equation (5). It means only a decrease in the average temperature in the section, but does not consider the temperature unevenness along the wall thickness. Thus, heat transfer is equivalent to a volumetric heat sink, while the condition of the adiabatic boundary is still satisfied. Then Equation (5) takes the form:

_{1}to t

_{2}and make elementary transformations:

_{1}—the start time of the source action, t

_{2}—the time when the source ends its action, and t > t

_{2}≥ t

_{1}≥0.

_{2}) is the difference between t and t

_{2}by an infinitesimal value o(t).

_{1}, t

_{2}), it is necessary to calculate the integral in Equation (7) with the limits of integration t

_{1}and t

_{2}. For this, the integral in Equation (7) is represented as the difference of two integrals. Then the solution for a moving point source can be obtained using the substitution ${u}^{2}=1/\sqrt{t-{t}^{\prime}}$ and the known integral 1.3.3.20 [29]:

_{s}+ H

_{w}), and also from the side wall boundaries y = W/2, y = −W/2, where W is the wall width. As a result, we obtain a system of an infinite number of heat sources. A cylindrical wall (generally a closed wall) can be represented as a single wall by unwrapping the wall around one of its generatrices (Figure 1). Figure 2 and Figure 3 show the schematic of the reflection of sources along the x-axis for a single wall and a closed single wall, respectively. The red color denotes imaginary sources for which k = 1, and the blue color denotes that k = 1. The temperature field is calculated at an arbitrary point p.

_{w}—for the case of a closed wall. Summation over k and n considers the limited length; while for summation over j and p, over width and height, respectively.

_{1}to t

_{2}, repeating all the steps that have been done to obtain Equation (8) and take into account that the source can be distributed over the surface of the computational domain. The result is the equation:

_{1}and t

_{2}in such a way that t

_{1}= 0 always, and ${t}_{2}^{}=\{\begin{array}{ll}t-o(t),& ift\le \frac{{L}^{*}}{v};\\ \frac{{L}^{*}}{v},& ift\frac{{L}^{*}}{v};\end{array}$. In this case, t

_{1}and t

_{2}are not arguments to the dT function.

_{b}). Then the heating temperature ∆T

_{preh}(x, y, z, t, t

_{1}, t

_{2}) can be obtained as:

_{1}= 0.254829592, a

_{2}= −0.284496736, a

_{3}= 1.421413741, a

_{4}= −1.453152027, and a

_{5}= 1.061405429.

#### 2.3. Influence of the Substrate on the Temperature Field

_{w}—the wall volume.

_{s}′—truncated substrate volume, and V

_{w}(n)—wall volume on the nth passage.

_{n}and the total energy of all sinks E

_{n}, which is already known. The action time of the sinks is proportional to the distribution time of uneven temperature; that is, $t{s}_{n}~\frac{{R}_{n}{}^{2}}{4a}$, where R

_{n}is the characteristic size of the deposition wall after n passes. So, the power of each sink is:

_{n}is presented as the sum of the temperature fields as a result of the heat-source action and the heat-sink action at each pass.

_{n}after the deposition of the nth number of layers, taking into account the linearity of the thermal problem, is represented as a temperature field as a result of the heat-source action depositing the nth layer, in front of which are n−1 heat sources, and also the action of sinks. These heat sources and sinks have equal or different power and operate at equal or different intervals of time, depending on the deposition strategy or scheme. Then, the heating temperature can be calculated using the following equation:

_{pause}—the pause time between passes, index i = 0 corresponds to the last pass, and index i = n − 1 corresponds to the first pass.

## 3. Results and Discussions

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Scheme of introducing imaginary sources to consider the effect of the limited wall size for a single wall.

**Figure 3.**Scheme of introducing imaginary sources to consider the effect of the limited wall size for a closed wall.

**Figure 4.**Scheme of the introduction of sinks to consider the influence of the substrate in the AM process (the drains are shown for the current passage).

**Figure 5.**Comparison of calculated and experimental thermal cycles during deposition of the single wall of 20 layers (t

_{pause}= 0 s). Experimental results obtained by Peyre [31].

**Figure 6.**Calculated temperature field of the deposited wall in the longitudinal section during cladding of the 20th layer.

**Figure 7.**Comparison of calculated and experimental thermal cycles during deposition of the cylindrical wall of 10 layers (t

_{pause}= 33 s). Experimental results obtained by Xiong [32].

Process | Heat Source | Power (W) | Cladding Speed (mm·s^{−1}) | Heat Convection (W·K^{−1}·m^{−2}) | Heat Efficiency | Pause Time(s) |
---|---|---|---|---|---|---|

DLD | laser | 600 | 6 | 20 | 0.35 | 0 |

WAAM | arc | 2850 | 5 | 5.7 | 0.85 | 33 |

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**MDPI and ACS Style**

Mukin, D.; Valdaytseva, E.; Turichin, G. Analytical Solution of the Non-Stationary Heat Conduction Problem in Thin-Walled Products during the Additive Manufacturing Process. *Materials* **2021**, *14*, 4049.
https://doi.org/10.3390/ma14144049

**AMA Style**

Mukin D, Valdaytseva E, Turichin G. Analytical Solution of the Non-Stationary Heat Conduction Problem in Thin-Walled Products during the Additive Manufacturing Process. *Materials*. 2021; 14(14):4049.
https://doi.org/10.3390/ma14144049

**Chicago/Turabian Style**

Mukin, Dmitrii, Ekaterina Valdaytseva, and Gleb Turichin. 2021. "Analytical Solution of the Non-Stationary Heat Conduction Problem in Thin-Walled Products during the Additive Manufacturing Process" *Materials* 14, no. 14: 4049.
https://doi.org/10.3390/ma14144049