## 1 Introduction
conducted a comprehensive review of analytical models for accelerated incompressible Newtonian fluid flow in pipes. They examined different approaches with imposed pressure gradients and flow rates, focusing on their mathematical complexity and applicability to laminar and turbulent flows. Although existing models predict specific laminar flow conditions well, they struggle with turbulent flow scenarios [12]. In 2023, [PERSON] et al. developed an analytical solution for modeling laminar water hammer events in pipes, validated through numerical simulations and experimental measurements [13]. Later that year, [PERSON] et al. published new analytical models for wall shear stress in water hammer scenarios. By employing quasi-steady and transient hydraulic resistance assumptions, these models broadened the range of validity and simplified the mathematical descriptions. Explicit analytical expressions for wall shear stress during water hammer events were derived and numerically validated [14]. In two separate studies, [PERSON] et al. explored wave propagation models for water hammer events in pipes. The first paper developed a rheology-based model for viscoelastic pipes and validated it using experimental data [15], while the second study provided a wave propagation model in the Laplace domain applicable to a variety of boundary conditions in pipe scenarios [16]. This paper aims to provide the mathematical foundation for a volumetric flow rate soft sensor based on two pressure signals by deriving an equation that accurately describes the flow rate in a pipe while considering the system's transient behavior. The derived equation consists of the steady HP solution with an added term describing unsteady effects. This will be achieved by taking the Laplace transform (LT) of the fundamental fluid mechanics equations, solving them in the Laplace domain, and then deriving the inverse Laplace transform (ILT) of the equations in the Laplace domain to obtain a solution in the time domain. The Laplace transform has a significant advantage over the Fourier transform, as the Fourier domain is a subset of the Laplace domain; thus, the benefits of the Fourier transform are inherently included in the Laplace transform. Regarding the area of convergence, the Laplace transform exhibits a broader region of convergence, which is a half-plane compared to a unit circle for the Fourier transform. This extension results in the Laplace transform capturing both the system's dissipation and oscillatory behavior, making it applicable to arbitrary pressure signals. A one-step integration method is utilized to decrease the computational costs of the soft sensor. The analytical model of the soft sensor based on two pressure transducers exhibits three main benefits compared to currently available sensors and numerical solvers: high accuracy of transient compressible flow, real-time capability, and, regarding application on a real test rig, no requirement of an actual volumetric flow rate sensor but two non-invasive pressure transducers. Analysis of a system's high-frequency behavior allows the user to capture transients in the volumetric flow rate caused by fast changes in pressure. Valuable areas of application of this soft sensor include dynamic measurements of volumetric flow rates achieved by pumps and controlling hydraulic actuators through the correlation of the velocity of the piston to the volumetric flow rate times the cross-section. Additionally, the soft sensor allows for predictive maintenance of hydraulic systems by calculating the power from the pressure and the volumetric flow rate. Fast flow rate calculation allows a detailed simulative examination of a hydraulic system compared to the computationally demanding classical iterative solver. Section 2 introduces the fundamentals of fluid mechanics and the Laplace techniques. Afterward, the analytical model for the soft sensor is derived and investigated in Section 2.3 to Section 2.8. The implementation of the soft sensor for the actual application is presented in Section 3.