The problem of heat transport during sub-cooling below lambda point in long, narrow channels containing superconductors immersed in liquid helium is discussed. In order to describe the propagation of phase transformation front the travelling plane wave description with suitable change of variables has been adopted. Nonlinear temperature profile corresponding to the conductivity function in He II has been assumed. The heat diffusion equation is solved in the volume of He I with the assumption that the main contribution to the specific heat is due to helium and the main contribution to longitudinal conduction is due to superconductor. For one dimensional problem with the temperatures clamped at both extremities of the channel the closed form analytical solutions for temperature profile are obtained. Channels with discontinuity in diameter are also analysed. Experimental validation of the model is based on 54 m long superconducting lines applied in the Large Hadron Collider (LHC). (C) 2008 Elsevier Ltd. All rights reserved.