Among the promising directions in the development of continuous casting of steel is casting of steel in horizontal continuous casting machines [i, 2].A very important part of this machine is the part for delivery of the metal from the metal receiver to the water-cooled mold (Fig. I). A feature of a horizontal continuous casting machine distinguishing it from other designs of continuous casting machines is direct connection through a fitting of the refractory metal line and the watercooled mold which, on the one hand~ prevents the possibility of secondary oxidation of the metal supplied to the mold while, on the other, it makes more severe the service conditions of th~ face portion of the metal line in contact with the mold material. Obviously, the portion of the metal line (feeder) located in contact with the mold or in a butt joint with the mold is made of a more wear-resistance refractory than the whole metal line [i, 3].In the horizontal method of continuous casting of steel, the refractory feeder fulfills the function of the first zone of the mold and solidification of the skin of the billet starts on the surface of the refractory of this portion, which in combination with the interrupted character of withdrawal of the billet from the mold and, consequently, the cyclic character of the skin of the billet on the refractory, has a significant influence on the life of the refractory and the quality of the billet and determines the reliability and effectiveness of operation of the machine [I, 4].To select the optimum design parameters of the refractory feeder-mold assembly and also to obtain information on the conditions of solidification of the billet skin with the selected design of this assembly it: was necessary to develop an appropriate mathematical model of the process. The model is the theoretical basis for planning of experimental tests for the purpose of optimication of the design and technical decisions for the investigated assembly of the continuous casting machine.The unique service conditions of the refractory feeder for the horizontal continuous casting machine determines the complex character of the temperature field in the portion of the refractory in contact with the copper water-cooled mold. On one side the refractory is in contact with molten steel, and on another with the surface of the mold, the temperature of which does not exceed 500~ and on the third with the molten steel and the cyclically freezing skin of the billet. The design features of the continuous casting machine make difficult measurements in service of the temperature field in the refractory.The withdrawal process includes movement and stoppage (Fig. 2a). As the result in each withdrawal cycle solidification of the metal, the formation of the skin of the billet both on the wall of the mold and in some portion of the refractory feeder, occurs. To a known degree the length of this portion determines the quality of the billet cast. Figure 2b shows the process of formation of the skin of the billet during the time of a single cycle....