Model reduction of a direct spring-loaded pressure relief valve with upstream pipe

A mathematical model is developed of a spring-loaded pressure relief valve connected to a reservoir of compressible fluid via a single, straight pipe. The valve is modelled using Newtonian mechanics, under assumptions that the reservoir pressure is sufficient to ensure choked flow conditions. The usual assumptions of ideal gas theory lead to a system of first-order partial differential equations for the motion, energy and momentum of the fluid in the pipe. A reduced-order model is derived using a collocation method under the assumption that the dominant pipe dynamics corresponds to a standing quarter wave. The model comprises just five non-linear ordinary differential equations, representing the position and velocity of the valve body, the pressure in the tank and the velocity and pressure amplitudes of the pipe quarter wave. Through comparison with simulations of the full model using a Lax–Wendroff method, it is shown that the reduced model is quantitatively accurate and is able to predict the onset of an oscillatory valve-chatter instability. The basic trends of this instability are shown to be robust to the inclusion of pipe friction and convective effects.