  The Future Circular Collider (FCC), currently in the design phase, will
address many outstanding questions in particle physics. The technology to
succeed in this 100 km circumference collider goes beyond present limits.
Ultra-high vacuum conditions in the beam pipe is one essential requirement to
provide a smooth operation. Different physics phenomena as photon-, ion- and
electron- induced desorption and thermal outgassing of the chamber walls
challenge this requirement. This paper presents an analytical model and a
computer code PyVASCO that supports the design of a stable vacuum system by
providing an overview of all the gas dynamics happening inside the beam pipes.
A mass balance equation system describes the density distribution of the four
dominating gas species $\text{H}_2, \text{CH}_4$, $\text{CO}$ and
$\text{CO}_2$. An appropriate solving algorithm is discussed in detail and a
validation of the model including a comparison of the output to the readings of
LHC gauges is presented. This enables the evaluation of different designs for
the FCC.
