## 1 Introduction
In the computational procedure for the analysis of temperature field in multi-pass TIG (tungsten inert gas) welding of steel pipe, [PERSON] and [PERSON] [36] applied a volumetric Goldak heat source. The results of numerical simulations were compared to experimental data from the thermocouples. [PERSON] et al. [37] studied several aspects of evaluating the geometric parameters of Goldak's double-ellipsoidal heat source using an experimental case of GMAW welding of two overlapping (between 40% and 80%) beads. In the paper by [PERSON] et al. [38], the 3D transient temperature during multi-pass overlapping was computed using TBW (temper bead technique). Numerical simulation was performed using FEA software. Also [PERSON] et al. [39] used a 3D FE model for analysis of temperature field and stresses during a multi-pass welding of thick structural steel S355J2 under high restraint conditions confirming the experimentally calculated residual stresses. [PERSON] and [PERSON] [40] reported the results of analysis of the microstructure and mechanical properties after multi-pass SAW. Some works refer only to experimental research results on multi-pass welding. [PERSON] et al. [41] conducted an experimental analysis of the temperature field in the multi-pass welding of steel plates. [PERSON] et al. [42] examined the welding heat cycles and microhardness in addition to conducting metallographic analysis of the tempered area in repeatedly welded steel samples. [PERSON] et al. [43] measured the heat cycles related to each welding pass. In most cases of the temperature field analytical solution thus far, the energy of an electric arc was used as the heat source. For large surface deformations of a weld pool during GMAW, [PERSON] and [PERSON] [44] proposed the model based on a bimodal distribution of arc heat flux. [PERSON] and [PERSON] [45] proposed taking into account the area of melted metal in a fillet weld by a moving bivariate Gaussian distributed heat source. Analogically, [PERSON] and [PERSON] [46] described the temperature field in a welding model using the GTA method taking into consideration filler wire. In the cited works, as well as in other solutions that concern the determination of the temperature field during welding, the impact of heat transferred to the surfaced object from the molten electrode material is not taken into account. The necessity of taking the heat of molten metal into consideration in temperature field solutions in arc weld surfacing (rebuilding) processes is therefore advisable. An analytical solution of a heat conduction equation allows rapid assessment of the temperature field and its dependence on parameters such as the source speed and its power. Therefore, this method is very popular in the description of the temperature field not only in welding processes, but others, e.g., laser heat treatment. In addition, it is difficult to find work presenting an analytical description of the temperature field during multi-pass welding induced by different point models of a heat source and the actual technological parameters of the process. In the work, an analytical model of temperature field during multi-pass GMAW surfacing of items satisfying the infinite-model conditions of a body is presented. A bimodal heat source model has been used in the description of the temperature field. The model takes into account both the heat of an electric arc directly acting on the object and the heat of the molten electrode material. The adoption of a bimodal heat source will allow the reconstruction of the irregular shape of the fusion line, which is often found in welding practice and impossible to obtain using a single-distributed heat source. The influences of heat caused by previously padded beads as well as the weld overlap effect have been taken into account in the algorithm. This approach brings the model closer to the real phenomena occurring in the welding process.