## 3 The Analytical Description of the Temperature Field during Multi-Pass Surfacing
\begin{array}{l}F_{H}(t^{\prime\prime})=\frac{1}{t^{\prime\prime}+t_{0}}\exp \bigg{(}-\frac{\xi_{k}z^{2}+\eta z^{2}}{4a(t^{\prime\prime}+t_{0})}^{2}-\frac{ v^{2}t^{\prime\prime}}{4a}\bigg{)}\\ \bigg{\{}\bigg{(}1-\frac{z^{2}+2 at^{\prime\prime}}{z_{0}^{2}}\bigg{)}\,\text{ erf}\bigg{(}\frac{z+z_{0}}{2(at^{\prime\prime})^{0.5}}-\Phi(z)\text{erf}\bigg{(}\Phi(z) \frac{z-z_{0}}{2(at^{\prime\prime})^{0.5}}\bigg{)}\bigg{)}+\\ \frac{4 at^{\prime\prime}}{z_{0}^{2}}\bigg{(}\frac{z+z_{0}}{(4\pi at^{\prime \prime})^{0.5}}\exp\bigg{(}-\frac{(z-z_{0})^{2}}{4 at^{\prime\prime}}\bigg{)}- \frac{z-z_{0}}{(4\pi at^{\prime\prime})^{0.5}}\exp\bigg{(}-\frac{(z+z_{0})^{2} }{4 at^{\prime\prime}}\bigg{)}\bigg{)}\bigg{\}}\end{array} , \tag{10}\]  
\[A_{C}=\frac{3\dot{q}}{8C_{p}\rho\pi az_{0}}, \tag{11}\]  
Figure 6: Scheme for describing the temperature field during multi-pass surfacing. \[ \begin{array}{l}F_{\rm C}(t^{\prime})=\frac{1}{t+t_{0}-t^{\prime}}\exp(-\frac{ \left(x-\tau b^{\prime}-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}}{4a(t+t_{0}-t ^{\prime})})\\ \left\{\left(1-\frac{z^{2}+2a(t-t^{\prime})}{z_{0}^{2}}\right)\left(erf\left( \frac{z+z_{0}}{2(a(t-t^{\prime}))^{0.5}}\right)-\Phi(z)erf\left(\Phi(z)\frac{ z-z_{0}}{2(a(t-t^{\prime}))^{0.5}}\right)\right)+\right.\\ \frac{4a(t-t^{\prime})}{z_{0}^{2}}\left(\frac{z+z_{0}}{4a(t-t^{\prime}))^{0.5} }\exp(-\frac{(z-z_{0})^{2}}{4a(t-t^{\prime})})-\frac{z-z_{0}}{4a(t-t^{\prime} ))^{0.5}}\exp(-\frac{(z+z_{0})^{2}}{4a(t-t^{\prime})})\right)\right\}\\ \xi_{k}=x-v\left(t-(k-1)\left(\frac{l_{k}}{v}+t_{pk}\right)\right)-x_{0k}, \end{array}  \tag{13}\]  
\[\Phi(z)=\left\{ \begin{array}{rl}-1&for\;\;z\in<0,z_{0i}>\\ 1&for\;\;z\in(z_{0i},\infty)\end{array} \right., \tag{14}\]  
\[\dot{q}=\eta UI-\dot{q}_{\
u}, \tag{15}\]  
\[t_{c}=\frac{\sum\limits_{j}l_{j}}{v}, \tag{16}\]  
where functions \(F_{1}(y,z)\)-\(F_{4}(y,z)\) and \(G_{1}(y,z)\)-\(G_{4}(y,z)\) are solutions of integrals using Gauss-Legendre quadrature described in the appendix, \(a\) denotes thermal diffusivity (m\({}^{2}\)/s), \(C_{p}\)--specific heat (J/kg K), \(\rho\)--density (kg/m\({}^{3}\)), \(z_{0}\) (m) denotes the depth of heat source deposition, and quantity \(t_{0}\) (s) characterizes the surface heat source distribution thus satisfying the relation (cf. Figure 3):  
\[r_{B}^{2}=4 at_{0}, \tag{17}\]  
where the \(r_{B}\) denotes the averaged radius of heat source with Gaussian distribution [63], \(U\) (V), \(I\) (A) and \(\eta\) voltage, current and arc efficiency respectively, \(v\) (m/s)--velocity of heat source (welding speed), \(t_{bj}\) (s) and \(t_{qj}\) (s)--denote starting and finishing time of applying \(j\)-th weld, defined as:  
\[t_{bj}=(j-1)\left(\frac{l_{j}}{v}+t_{pj}\right), \tag{18}\]  
\[t_{ej}=(j-1)t_{pj}+j\frac{l_{j}}{v}, \tag{19}\]  
\(l_{j}\) (m)--the length of \(j\)-th weld with coordinates \(x_{0j}\) and \(x_{bj}\):  
\[l_{j}=x_{kj}-x_{0j}, \tag{20}\]  
\(t_{pj}\) auxiliary time preceding the application of \(j\)-th weld. The total amount of heat \(q_{v}\) contained in the material of a melted electrode is expressed by the relationship [64]:  
\[q_{v}=\Delta q_{solid}+\Delta q_{f}+\Delta q_{liquid}, \tag{21}\]  
where \(\Delta q_{solid}\)--heat necessary to heat up the electrode from the initial temperature to the melting temperature, \(\Delta q_{f}\)--heat used for melting the electrode (heat of fusion), \(\Delta q_{liquid}\)--heat used for heating up melted material to the temperature \(T_{L}\), in which the drop of metal falls on the surface of the welded material. The value of the initial temperature of the electrode \(T_{e}\) from the welding head is set at 100 \({}^{\circ}\)C. In correspondence to the above:  
\[\dot{q}_{v}=\dot{m}\left(C_{p}(T_{L}-T_{e})+L\right), \tag{22}\]  
where  
\[\dot{m}=\rho_{e}\frac{\pi d^{2}}{4}v_{e}, \tag{23}\]  
\(v_{e}\) (m/s)--velocity of passing electrode wire with diameter \(d\) (m) and density \(\rho_{e}\) (kg/m\({}^{3}\)).