## 1 Introduction  
Cryogenic propellants are rocket propellants stored at very low temperatures. Typically, they are stored in a liquid state at their boiling point. Liquid oxygen (LOx, normal boiling point of 90.2 K), liquid hydrogen (LH2, normal boiling point of 20.3 K), and liquid methane (LCH4, normal boiling point of 111.5 K) are the most commonly used cryogenic propellants [1, 2]. Cryogenic solid propellants have also been proposed, but presently, no application exists [3]. On the contrary, propellants such as LOx and LH2 are widely used in space applications due to their excellent performance. High-performance liquid rocket engines can be realized by feeding the engine with high-pressure cryogenic propellants using turbopumps rotating at high shaft speeds. These turbopumps are usually driven by a turbine that is fed by hot gases coming either from a dedicated combustor or from the thrust chamber cooling system. The shaft connecting the turbine to the pump is supported by bearings, which operate at high rotational speeds. The bearings dissipate a large amount of power, thus requiring adequate cooling in order to avoid damage or failure. Due to the low temperatures, conventional lubricants cannot be used, and the only available coolant is the cryogenic propellant itself. Therefore, during engine operation, a small fraction of cryogenic propellant mass flow is redirected towards the bearings using dedicated channels, also called secondary flow systems. Given that the bearing housing is adjacent to the turbine, a -seal system must be incorporated to restrain the leakage of high-pressure cryogenic fluid toward the turbine. Shaft seals are among the most critical components of a liquid rocket engine turbopump, especially in the oxidizer turbopump, where direct contact between the oxidizer and the hot fuel-rich gases flowing through the turbine must be avoided in all conditions. However, a small leakage of cryogenic fuel toward the fuel-rich turbine gases can be accepted in fuel turbopumps; therefore, the seal system can be simplified in this latter case. In oxidizer turbopumps, a seal package usually provides separation through mechanical barriers (seals, rotating slingers) and by feeding an inert gas such as helium (He) at high pressure between the two fluids. The seal package, therefore, consists of at least three elements: the oxidizer seal, the He purge seal, and the turbine gas seal. The He purge seal is installed between the oxidizer seal and the turbine gas seal and supplies He as a barrier gas. The low temperature of the cryogenic propellants, in combination with the high temperature of the hot turbine gases and the He purge, results in severe temperature gradients through the seal system and across the seal assembly. These thermal gradients determine the thermal deformation of the components, which must be considered when evaluating the seal's clearance. Several factors must be considered during turbopump design to achieve the required system reliability. One of the main tasks is to accurately predict the pressure and temperature distribution in the secondary flow systems in order to correctly evaluate mass flows through seals and bearings. Modelling the secondary flow paths of a turbopump implies that complex phenomena must be considered. Such a network involves flow through narrow passages between rotating and stationary surfaces, phase changes, fluid mixing, and heat transfer; therefore, two-phase multi-component flows must be modelled. As can also be seen in simpler cases [4; 5; 6; 7], the development of appropriate methods is a really challenging task. Flow network solvers have proven to be the most appropriate tools for the preliminary design and analysis of components and systems involving two-phase heat transfer phenomena. This method simplifies the complex three-dimensional problem by relying on 1D modelling, represented by a series of interconnected elements, which make up the equivalent flow network (nodes linked by branches).