## 2 Materials and Methods
### Principle of Continuous Fibre Integration
#### 2.1.1 Influencing Factors on Roving Integration  
An analytical consideration of the existing heat flows is used to derive the influencing variables. For simplicity, only one-dimensional heat flows were considered in this paper. The heat transferred during the roving integration can be described by the heat flows involved, as defined in Equation (1). \[\dot{\mathrm{Q}}_{\mathrm{FI}}=\dot{\mathrm{Q}}_{\mathrm{HM3}}+\dot{\mathrm{Q}}_ {\mathrm{FN}} \tag{1}\]  
The additional and parallel heat source at the bottom of the fibre integration unit forms the first heat flow summand \(\dot{\mathrm{Q}}_{\mathrm{HM3}}\). This constellation is a special case of free convection with internal flow within two parallel horizontal plates [23; 24]. The additional heat source forms the hotter upper plate, and the powder bed surface forms the colder lower plate. According to [23], no flow is generated since the air is stably layered between the upper plate (heat source) and the lower plate (powder bed surface). As described in [23; 24], heat is transferred mainly by conduction and radiation. The heat flow transferred to the powder bed surface by the additional heat source is given by Equation (2) below. All quantities are shown in Table 1.  
\[\dot{\mathrm{Q}}_{\mathrm{HM3}}=\ \lambda_{\mathrm{L}}\cdot\mathrm{A}_{ \mathrm{HM3}}\cdot\frac{\mathrm{T}_{\mathrm{HM3}}-\mathrm{T}_{\mathrm{O}}}{ \mathrm{h}_{\mathrm{HMG}}}+\frac{\sigma}{\varepsilon_{\mathrm{HM3}}+\frac{1} {\varepsilon_{\mathrm{F}}}-1}\cdot\mathrm{A}_{\mathrm{HM3}}\cdot(\mathrm{T}_ {\mathrm{HM3}}^{4}-\mathrm{T}_{\mathrm{O}}^{4}) \tag{2}\]  
The first summand in Equation (2) represents the heat conduction law applied to the LS machine. The second summand represents the Stefan-Boltzmann law with an additional radiation exchange between the additional heat source and the powder bed surface. The second summand in Equation (1) represents the heat flow generated by the heated fibre nozzle. This heat flow is mainly responsible for the targeted transfer of heat energy and, thus, for HAZ formation. The heat flux transferred through the fibre nozzle is defined in Equation (3). \[\dot{\mathrm{Q}}_{\mathrm{FN}}=\lambda_{\mathrm{L}}\cdot\mathrm{A}_{\mathrm{ D}}\cdot\frac{\mathrm{T}_{\mathrm{D}}-\mathrm{T}_{\mathrm{O}}}{\mathrm{h}_{ \mathrm{D}}}+\frac{\sigma}{\frac{1}{\varepsilon_{\mathrm{D}}}+\frac{1}{ \varepsilon_{\mathrm{F}}}-1}\cdot\pi\cdot\left(\mathrm{T}_{\mathrm{D},\mathrm{ O}}^{2}-\mathrm{r}_{\mathrm{D},i}^{2}\right)\cdot(\mathrm{T}_{\mathrm{D}}^{4}- \mathrm{T}_{\mathrm{O}}^{4}) \tag{3}\]  
Figure 1: 3D view of the fibre integration unit (without powder bed) (**a**) and a schematic of the process zone (**b**) [22]. The first summand in Equation (3) represents the heat conduction law between the ring-shaped fibre nozzle with inner diameter \(\mathrm{d}_{\mathrm{D,i}}\) (guidance and transmission for the roving inside the fibre nozzle), outer diameter \(\mathrm{d}_{\mathrm{D,o}}\), nozzle curvature \(\kappa\) and the powder bed surface. The second summand represents the Stefan-Boltzmann law with an additional radiation exchange between the front surface of the fibre nozzle and the powder bed surface. In \(\mathrm{Q}_{\mathrm{FN}}\), the time period \(\mathrm{t}_{\mathrm{FN}}\) of the heat transfer (i.e., the amount of heat energy transferred in a given time period) is also implied. The time period of the heat transfer can be influenced by the feed rate \(\mathrm{v}_{\mathrm{D}}\) of the fibre nozzle. Possible convection flows between the fibre nozzle and the additional heat source, as well as heat losses at the edges of the additional heat source to the surroundings, such as losses due to the fact of convection \(\mathrm{Q}_{\mathrm{K,B}}\) and radiation \(\mathrm{Q}_{\mathrm{S,B}}\), were assumed but not considered in this analysis.