Delayed oscillator model of pressure relief valves with outlet piping

Abstract

The system of a vessel and a direct spring operated pressure relief valve is extended by a downstream pipe. The presented mathematical model consisting of ordinary and partial differential equations can be transformed to a system of differential difference equations by means of the travelling wave solutions of the partial differential equations. This model contains a time delay originated in the pipe length and the speed of sound in the pipe. The linear stability analysis presents stability charts in a parameter plane characterising the environment where the valve is built in. The boundary lines and the corresponding vibration frequencies of the self-excited vibrations are determined as compact closed form expressions. Several analogies are identified with other physical systems modelled by delayed oscillators. Both the stability charts and the vibration frequencies are checked with numerical simulations of the original nonlinear system consisting of coupled ordinary and partial differential equations. The connection between the travelling wave solution and the standard quarter standing wave approximation is explained. A qualitatively relevant phenomenon is identified, namely the appearance of the quasi-periodic vibration at certain parameter combinations.