## VENECIA: Venecia software package for thermal hydraulic analysis of forced flow cooled superconducting equipment and their primary cryogenic subsystems

# Numerical Solution

For computations, the analysis equations (1)-(3), (8), (9) were re-written in the non-conservative form using pressure P, enthalpy H and velocity V as flow variables. The Gruneisen parameter *f* and isentropic speed of sound *c* were introduced in the equations [1]. Using the basic thermodynamic identity

the equations (1)-(3) are transformed to the form

where

Equations (8) and (9) are re-written as

So, the time derivatives for different helium parameters are separated and assigned in explicit form.

All equations related to *channels* and *conductors* are represented via finite differences with respect to the space variable x using the following approximation for the first and second order derivatives in a mesh node "*i*"

where *h* is a step of uniform spatial discretization. A special attention is paid to approximation of derivatives in the boundary nodes. In **VENECIA**, this approximation is significantly improved and allows provide stable calculation for very fast process at the boundaries (explosion of pipe etc.). As a result of such discretization procedure, the initial system of three partial differential equations for the *P-H-V*channel parameters is transformed into *3N* ordinary differential equations for the parameters in the mesh nodes with respect to time.

The number of steps (nodes) on the space variable x is arbitrary and individual for each *channel*. For the*channels* coupled by heat and mass transfer in the longitudinal direction the number of nodes should be identical.

where t is the time step for integration.

The numerical method for solving equation (14) is based on a semi-explicit spliting-up method for parabolic partial differential equations [3].