In this work, we propose a new mathematical model to describe non-isothermal steady flows of a fluid with temperature-dependent viscosity in a 3D pipeline network. Our approach is based on the renouncement of the averaging procedure for the velocity and temperature fields and the use of the conjugation conditions that express the mass and energy balance for interior joints of the network. We give weak formulation of the problem. The main result is an existence theorem (in the class of weak solutions) for small data.