Accurate knowledge of the flow rate is essential for hydraulic systems, enabling the calculation of hydraulic power when combined with pressure measurements. These data are crucial for applications such as predictive maintenance. However, most flow rate sensors in fluid power systems operate invasively, disrupting the flow and producing inaccurate results, especially under transient conditions. Utilizing pressure transducers represents a non-invasive soft sensor approach since no physical flow rate sensor is used to determine the flow rate. Usually, this approach relies on the Hagen–Poiseuille (HP) law, which is limited to steady and incompressible flow. This paper introduces a novel soft sensor with an analytical model for transient, compressible pipe flow based on two pressure signals. The model is derived by solving fundamental fluid equations in the Laplace domain and converting them back to the time domain. Using the four-pole theorem, this model contains a relationship between the pressure difference and the flow rate. Several unsteady test cases are investigated and compared to a steady soft sensor based on the HP law, highlighting our soft sensor’s promising capability. It exhibits an overall error of less than $0.15$% for the investigated test cases in a distributed-parameter simulation, whereas the HP-based sensor shows errors in the double-digit range.

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.