## 3 The Analytical Description of the Temperature Field during Multi-Pass Surfacing  
The bimodal model of the heat source in description of the temperature field during surfacing by welding is applied, finding justification in the way of transmitting heat generated by an electric arc to the surfaced object. In this model, only one heat source is physically present--an electric arc. It is assumed that part of the heat generated by the electric arc is transferred to the surfaced object by the direct impact of an electric arc. In contrast, part of the heat generated by the electric arc is consumed to heat up the electrode's material, its melting and further heating up, until the melted material of an electrode in the form of droplets is detached under the influence of electromagnetic forces and transferred to the developing weld under their own weight. In the analytical solutions, the influence of phase transformation heat (both solid and liquid) is generally ignored, thermo-physical properties of the material are assumed to be independent of temperature, and only the conduction effect is taken into consideration (the radiation phenomenon and the gravity field are omitted), also Joule heat (for electric arc welding) is omitted [6; 7; 8; 9; 10; 11; 12; 53; 54; 55; 56; 57; 58]. Such assumptions were also adopted in this work. The point of departure in the description of the temperature field in a homogeneous and isotropic body is a basic differential equation of heat conduction based on the law of conservation of energy [53]. On the basis of the solution to this equation for a semi-infinite body with a moving heat source presented by [PERSON] [54] and [PERSON] [55], the author of this article describes the temperature field  
Figure 1: Characteristic structural areas of a welded joint, depending on the temperature and the share of the carbon in the steel. caused by an electric arc during single-pass surfacing, assuming a volumetric model of the heat source [59]. In the paper, it is assumed that the volume of electrode's melted material is equal to the volume of the weld reinforcement and the amount of heat stored in the melted electrode's material equals the quantity of heat transferred to the weld. Furthermore, it is assumed that the wire speed determines the speed of its melting. Taking into account the direct impact of an electric arc, as well as the heat transferred by the applied metal, the author determines analytically the temperature field during single-pass surfacing [60]:  
\[T(x,y,z,t)-T_{0}=T_{w}(x,y,z,t)+T_{a}(x,y,z,t), \tag{1}\]  
where \(T_{w}(x,y,z,t)\) and \(T_{a}(x,y,z,t)\) are temperature fields caused by heat of the weld reinforcement (used to melt the electrode) and direct influence of an electric arc respectively, while \(T_{0}\) denotes initial temperature. For the weld a volumetric model of the heat source is assumed, the lower limit of which is the shape of welded surface, which in turn is assumed in the solution as flat or parabolic (Figure 2). The face of the weld is determined mainly by surface tension forces. Based on experimental research, [PERSON] et al. [61] proposed the parabolic shape of the face of the weld. Therefore, in this work the face of the weld is assumed to be parabolic. For the description of the temperature field caused by an electric arc a volumetric heat source of Gaussian surface distribution is assumed (Figure 3) and parabolic change in relation to depth (Figure 4) [59]. Calculations of the temperature field are performed for a single-pass surfaced S235 steel plate. The accuracy of solution is verified comparing the calculated fusion line to that obtained experimentally by other authors [61]. In the calculations, the same technological parameters as in the experiment were assumed. In Figure 5, the calculated heat-affected zone and fusion line obtained experimentally [62] are presented. The calculated fusion line coincides with the line obtained in experiment. Figure 3: Gaussian surface distribution of the heat source.