Solid Nitrogen (SN2) can provide a uniform and stable cryogenic environment for High Temperature Superconducting (HTS) systems such as bulk samples during their magnetization and/or characterization. In this paper, we are studying a SN2 cooling system consisting of a cryocooler Sumitomo CH-110 and an exchanger in a Liquid Nitrogen (LN2) bath. In order to design this cooling system, an analytical model based on a nodal method coupled to a formulation of the thermal capacity was realized. The model considers the thermal parameters variation as well as the phase change of the Nitrogen. In order to compare our results, we performed a 3D simulation on COMSOL Multiphysics. The performance of the cooling system was evaluated and we estimates that 50 L of LN2 can be cooled down to 20 K in 50 hours.