Upscaling of superfluid helium flow in porous media

Although the Large Hadron Collider, the particle accelerator at CERN Geneva, did not reach full power, studies have begun for extending its performances. For that matter an increase of the luminosity of the beam by a factor of ten is desired. Achieving such luminosity implies the use of more powerful superconducting magnets experiencing higher heat load. Since the thermal resistance due to the electrical insulations of the superconducting cables is the main thermal resistance for cooling, new types of insulation, based on porous ceramic materials, are developed. In this context, studies of heat transfer in porous media saturated with superfluid helium are of great interest. We present the upscaling of the generalized two-fluid model describing the flow of superfluid helium in the Landau regime using the method of volume averaging. We show that a macro-scale model can be developed for some limiting cases. In particular, we obtain an upscaled version of the superfluid equations in the Landau regime proving the existence of a permeability concept and that allows us to study the relationship between this permeability and the classical intrinsic permeability. This puts the permeability concept previously introduced heuristically by some authors in this context on a much firmer basis. The results are validated against direct numerical simulations of the superfluid helium flow at the pore-scale as well as with comparisons with experimental data.