Unsteady flow rate in transient, incompressible pipe flow

Abstract

Knowledge regarding current flow rates through pipes and other components is crucial for most hydraulic systems. The hydraulic power can be computed in combination with the pressure, which may be used in many applications, such as predictive maintenance. Most flow rate sensors used in the field of fluid power operate invasively. Therefore, the measurement process itself alters the flow rate. Furthermore, most sensors operate accurately for stationary flow but produce inaccurate measurements for transient flow. A well‐known method of determining the flow rate is to measure the pressure difference between two points along a pipeline and calculate the flow rate based on the law of Hagen and Poiseuille. However, since the relation mentioned above only applies to laminar, steady, and incompressible flow, its usefulness for transient flows is limited. This paper derives a system equation based on the fundamental laws of fluid mechanics, which describe transient, incompressible pipe flow. As a result, the fundamental equations are solved in the Laplace domain and subsequently transformed back into the time domain. The four‐pole theorem relates the pressure difference and the volumetric flow rate. The analytical solution consists of a convolution integral containing a weighting function and the pressure difference. Compared to a simulation, the novel equation displays high accuracy for transient and stationary incompressible pipe flow. This equation paves the way for a soft sensor, which allows the noninvasive measurement of arbitrary volumetric flow rates within pipes.