The possibility of molten salt freezing in piping systems is a key concern for the solar energy industry and a safety issue in the new generation of molten-salt reactors worthy of careful consideration. This paper tackles the problem of coolant solidification in complex pipe networks by developing a transient thermohydraulic model and applying it to the 'Direct Reactor Auxiliary Cooling System' (DRACS), the passive safety system proposed for the generation-IV molten salt reactors. The results indicate that DRACS, as currently envisioned, is prone to failure due to freezing in the air/molten-salt heat exchanger. The occurrence of this scenario is related to an unstable behaviour mode of DRACS in which newly formed solid-salt deposit on the pipe walls acts to decrease the flow-rate in the secondary loop, facilitating additional solid-salt deposition. Conservative criteria are suggested to facilitate the preliminary assessments of DRACS designs. Such study is, to the knowledge of the authors, first in its type and it serves to illustrate the possible safety concerns in molten salt reactors, which are otherwise considered very safe in the literature. On a broader prospective, the analytical tools developed in this study can be used to model the freezing of molten salts in complex piping systems where standard, finite element approaches are computationally too expensive.