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.

Precise flow measurement is crucial in fluid power systems. Especially in combination with pressure, hydraulic power can be particularly beneficial for predictive maintenance and control applications. However, conventional flow sensors in fluid power systems are often invasive, thus disrupting the flow and yielding unreliable measurements, especially under transient conditions. A common alternative is to estimate the flow rate using pressure differentials along a pipe and the Hagen–Poiseuille law, which is limited to steady, laminar, and incompressible flows. This study advances a previously introduced analytical soft sensor, demonstrating its ability to accurately determine the transient pipe flow beyond laminar conditions, without requiring a dedicated flow rate sensor. This method provides a robust and computationally efficient solution for real-world hydraulic systems by applying two pressure transducers. A key contribution of this work is the investigation of signal filtering, revealing that even a simple first-order low-pass filter with a 100 Hz cutoff frequency significantly improves accuracy, which is demonstrated for pulsation frequencies of 5, 10, and 15 Hz, where the filtered results closely match experimental data from a test rig. These findings underscore the soft sensor’s potential as a reliable alternative to traditional flow sensors, offering high accuracy with minimal computational overhead for a wide range of flow conditions.

Accurate flow measurement is critical for hydraulic systems because it represents a crucial parameter in the control of fluid power systems and enables the calculation of hydraulic power when combined with pressure data, which is valuable for applications such as predictive maintenance. Existing flow sensors in fluid power systems typically operate invasively, disturbing the flow and providing inaccurate results, especially under transient conditions. A conventional method involves calculating the flow rate using the pressure difference along a pipe via the Hagen–Poiseuille law, which is limited to steady, laminar, incompressible flow. This paper presents a novel soft sensor with an analytical model for transient pipe flow based on two pressure signals, thus eliminating the need for an actual volumetric flow sensor. The soft sensor was derived in previous research and validated with a distributed parameter simulation. This work uses a constructed test rig to validate the soft sensor with real-world experiments. The results highlight the potential of the soft sensor to accurately and computationally efficiently measure transient pipe volumetric flow based on two pressure signals.

Precise flow measurement is pivotal in hydraulic systems, serving as an essential parameter for fluid power control. When combined with pressure data, it also facilitates calculating hydraulic power, proving invaluable for applications like predictive maintenance. Conventional flow sensors in fluid power systems are typically invasive, often disrupting the flow and delivering unreliable measurements, particularly during transient conditions. One common approach relies on determining the flow rate by measuring the pressure difference along a pipe and applying the Hagen-Poiseuille law, constrained to steady, laminar, and incompressible flows. This study presents the capability of a previously introduced analytical soft sensor to determine transient pipe flow for non-laminar flow rates. This method does not require a flow rate sensor and utilizes two pressure transducers to determine the pipe flow. Furthermore, the relevance of filtering is investigated, and the robustness of the soft sensor is demonstrated by the outcome that even basic filtering results in a good match between the soft sensor and test rig. The findings underscore the soft sensor’s capability to deliver accurate and computationally efficient measurements of transient volumetric flow based solely on pressure data for laminar and non-laminar flow with simple filtering of real-world signals.

Accurate flow measurement is critical for hydraulic systems because it represents a crucial parameter in the control of fluid power systems and enables calculating hydraulic power when combined with pressure data, which is valuable for applications such as predictive maintenance. Existing flow sensors in fluid power systems typically operate invasively, disturbing the flow and providing inaccurate results, especially under transient conditions. A conventional method involves calculating the flow rate using the pressure difference along a pipe via the Hagen-Poiseuille law, which is limited to steady, laminar, incompressible flow. This paper presents a novel soft sensor with an analytical model for transient pipe flow based on two pressure signals, thus eliminating the need for an actual volumetric flow sensor. The soft sensor was derived in previous research and validated with a distributed parameter simulation. This work uses a constructed test rig to validate the soft sensor with real-world experiments. The results highlight the potential of the soft sensor to accurately and computationally efficiently measure transient pipe volumetric flow based on 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.