## 3 The Analytical Description of the Temperature Field during Multi-Pass Surfacing
Figure 2: Geometry of the weld: \(h_{w}\)—height of the weld, \(w_{w}\)—width of the weld, \(d_{p}\)—depth of filling. In the temperature field modeling during multi-pass surfacing, it is necessary to take into account temperature increments, caused by overlaying consecutive welding sequences and self-cooling of areas previously heated. Since during surfacing weld overlaps (30% to 60% of the cross-sectional area of the deposit) are used in order to obtain a smooth weld surface, in the numerical model the distances between the axes of the welds must be adequately smaller than the width of the weld. For the proper mapping process, auxiliary time connected with the idle movement of an electrode (welding head) or other activity resulting from the needs of technological processing (e.g., correction or change of technological parameters) should also be considered. Then, the temperature field during the application of \(k\)-th weld is determined by:  
\[T(x,y,z,t)-T_{0}=\sum_{j=1}^{k-1}\Delta T_{j}^{C}+\Delta T_{k}^{H}, \tag{2}\]  
where \(\Delta T_{j}^{C}\) denotes the increment of temperature caused by the already applied (cooling) \(j\)-th weld, while \(\Delta T_{j}^{H}\) the increment of temperature during application of \(k\)-th weld. Whereas the temperature field after application of all welds is described by:  
\[T(x,y,z,t)-T_{0}=\sum_{j=1}^{k}\Delta T_{j}^{C}. \tag{3}\]  
On the basis of solution for single-pass surfacing [60] the temperature increment at point \(P(x,y,z)\) during multi-pass surfacing (Figure 6) can be described by the following equations:  
Figure 4: Parabolic change of the heat source in relation to depth. Figure 5: Calculated heat-affected zone after single-pass surfacing: dashed line—fusion line obtained experimentally [62]. For time \(t\leq t_{c}\), where \(t_{c}\) means the total time of welds application:  
\[ \begin{array}{l}T(x,y,z,t)-T_{0}=\\ =\sum\limits_{j=1}^{k-1}\left\{A_{w}\int\limits_{t_{ij}}^{t_{ij}}\{\quad H_{C}(t ^{\prime})\left(G_{2}(y,z)+G_{3}(y,z)-G_{4}(y,z)-G_{1}(y,z)\right)\}dt^{\prime} \right.+\left.A_{C}\int\limits_{t_{bj}}^{t_{ej}}F_{C}(t^{\prime})dt^{\prime} \right\}+\\ +A_{w}\int\limits_{t_{bl}}^{t}\{\quad H_{H}(t^{\prime\prime})\left(F_{2}(y,z)+ F_{3}(y,z)-F_{4}(y,z)-F_{1}(y,z)\right)\}dt^{\prime\prime}+A_{k}\int\limits_{t_{ lk}}^{t}F_{H}(t^{\prime\prime})dt^{\prime\prime}\end{array}  \tag{4}\]  
for time \(t>t_{c}\):  
\[ \begin{array}{l}T(x,y,z,t)-T_{0}=\\ =\sum\limits_{j=1}^{k}\left\{A_{w}\int\limits_{t_{bj}}^{t_{ej}}\{\quad H_{C}(t ^{\prime})\left(G_{2}(y,z)+G_{3}(y,z)-G_{4}(y,z)-G_{1}(y,z)\right)\}dt^{\prime} \right.+\left.A_{C}\int\limits_{t_{bj}}^{t_{ej}}F_{C}(t^{\prime})dt^{\prime} \right\}\end{array} , \tag{5}\]  
where  
\[A_{w}=\frac{\dot{q}_{v}}{8C_{p}\rho\sqrt{\pi a}}, \tag{6}\]  
\[H_{H}(t^{\prime\prime})=\frac{1}{\sqrt{t^{\prime\prime}}}\bigg{(}\text{erf} \bigg{(}\frac{\Delta l-2(\xi+vt^{\prime\prime})}{4\sqrt{at^{\prime\prime}}} \bigg{)}+\text{erf}\bigg{(}\frac{\Delta l+2(\xi+vt^{\prime\prime})}{4\sqrt{at ^{\prime\prime}}}\bigg{)}\bigg{)}, \tag{7}\]  
\[H_{C}\big{(}t^{\prime}\big{)}=\frac{1}{\sqrt{(t-t^{\prime})}}\bigg{(}\text{erf }\bigg{(}\frac{\Delta l-2(x-vt^{\prime}-x_{0})}{4\sqrt{a(t-t^{\prime})}} \bigg{)}+\text{erf}\bigg{(}\frac{\Delta l+2(x-vt^{\prime}-x_{0})}{4\sqrt{a(t- t^{\prime})}}\bigg{)}\bigg{)}, \tag{8}\]  
\[A_{k}=\frac{3\dot{q}}{8C_{p}\rho\pi az_{0}}\exp\bigg{(}-\frac{v\xi_{k}}{2a}- \frac{v^{2}t_{0}}{4a}\bigg{)}, \tag{9}\]  
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