A time-delay model for molecular gas flow into vacuum

Flow of molecular gas into a complex vacuum system is investigated by a lumped parameter model to estimate the time evolution of gas pressure $p_g$, which for the first time takes into account the realistic effect of time-delay arising due to multiple reasons such as valve response, pumping and transport of gas, conductance of the pipe network, etc. The net effect of all such delays taken together into a single (constant) delay term gives rise to a scalar delay differential equation (DDE). Analytical solutions of a linear DDE in presence of external forcing due to the (pulsed) injection of gas and outgassing from a stainless steel (SS) wall are then derived using the Laplace transform method, where the transcendental characteristic equation is approximated by means of rational transfer functions such as diagonal Pad{\'e} approximation. A good agreement is obtained with the numerical results and it is shown from these solutions that a reasonably good match with experimental results is obtained only in the presence of a nonzero time-delay. An attempt is also made to simplify the DDE into an ordinary differential equation (ODE) by using a low-order Taylor series expansion of the time-delay term. A singular perturbation method is used to solve the resultant initial value problem (IVP) type second-order ODE. It is found that unlike the Laplace transform method, the ODE closely approximates the DDE only if the delay is small. Emergence of stable periodic oscillations in $p_g$, as a generic supercritical Hopf bifurcation, if time-delay exceeds a critical value is then established by numerical and Poincar{\'e}-Lindstedt method.