Analytical Description of the Static and Dynamic Behaviour of a Pressure Relief Valve

The paper presents an explicitly straightforward formulation of the stationary and dynamic behaviour of a pressure relief valve (PRV). This makes it possible to consider the static, dynamic and robustness properties of a PRV during the analysis or design process. A PRV can be understood as a self-regulating, cross-domain system. The governing equations are well known and widely used in literature. Usually, these include: a geometrical description of the flow area and the pressure surface, a flow equation, the pressure build-up equations, a spring-like counterforce, a flow force, a term for viscous friction and the inertia force. Together they form a system of ordinary non-linear differential equations of third order. So far, these equations had to be solved numerically in order to analyse or adapt the static or dynamic properties of a particular PRV. In this paper, direct analytical solutions for stationary and dynamic cases are derived. This results in an explicit equation for the respective p-Q characteristic curve. In addition, a simple criterion for the stability of a PRV was found. As it turns out, the minimum requirement for viscous damping is directly anti-proportional to the gradient of the p-Q characteristic curve. It is empirically known that decreasing the gradient of the p-Q curve makes the system more susceptible to oscillations. However, this has not yet been shown mathematically elegant. The method presented here calculates the static p-Q curve, the stability and natural frequencies of a PRV in a simple procedure using only elementary mathematics — no numerical scheme is required. Thus, the new method offers four main advantages. First, it is several orders of magnitude faster because it is not necessary to solve the differential equation system numerically. Secondly, the user does not require any special knowledge or advanced calculation tools — a simple spreadsheet program is sufficient. This eliminates licensing and training costs. Third, sensitivity and robustness analyses can be carried out easily because the dependencies are explicitly known. Last but not least, the understanding of a PRV is improved by knowing directly which parameters have what influence. The new method is tested and verified by comparison with conventional non-linear numerical simulations.