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

Analytical methods for vacuum simulations in high energy accelerators for future machines based on the LHC performance