Analysis of the quench propagation along Nb$_{3}$Sn Rutherford cables with the THELMA code. Part I: geometric and thermal models

The paper describes the new lumped thermal model recently implemented in THELMA code for the coupled electromagnetic–thermal analysis of superconducting cables. A new geometrical model is also presented, which describes the Rutherford cables used for the accelerator magnets. A first validation of these models has been given by the analysis of the quench longitudinal propagation velocity in the Nb$_{3}$Sn prototype coil SMC3, built and tested in the frame of the EUCARD project for the development of high field magnets for LHC machine. This paper shows in detail the models, while their application to the quench propagation analysis is presented in a companion paper.