Signal Processing for High-Frequency Flow Rate Determination: An Analytical Soft Sensor Using Two Pressure Signals

Accurate knowledge of flow rates is essential for hydraulic systems, as it allows for calculating hydraulic power when combined with pressure measurements. This data is helpful in applications such as predictive maintenance. However, most flow rate sensors in fluid power systems operate invasively, disrupting the flow and producing inaccurate results, particularly for transient flow conditions. A common approach is calculating the flow rate using the pressure difference along a pipeline based on the Hagen-Poiseuille law. However, this method is limited to laminar, steady, incompressible flow. This paper presents a novel soft sensor with an analytical model for transient, compressible pipe flow based on two pressure signals, thus no actual volumetric flow rate sensor is required. The model is derived by solving the fundamental fluid equations in the Laplace domain and converting the solution back to the time domain to obtain a physical representation. The relationship between the pressure difference signal and flow rate is derived using the four-pole theorem. The resulting analytical solution, including a convolution integral with a weighting function, shows high accuracy for transient and steady flows. This new equation enables the development of a soft sensor capable of non-invasive flow rate measurements for various pipe flow conditions.