## 2 Materials and Methods
### Analytical Model for Signal Processing of Pressure Transducer Signals  
The process of the derivation of an equation that can be used in a soft sensor to calculate the volumetric flow rate in dependency on the pressure difference over the length of a pipe is presented in Figure 2. First, the Navier-Stokes equations and the equation of conservation of mass are simplified by using assumptions, yielding a set of partial differential equations that describe the transient flow of fluid in a pipe. The assumptions are listed in Section 2.3. In the second step, the time dependencies in the differential equations are eliminated by taking the Laplace transform. Afterward, the differential equations are solved in the Laplace domain to obtain a Laplace domain solution for the volumetric flow rate. These two steps can be found in detail in a previously published work by [PERSON] et al. [31]. The resulting equation in the Laplace domain is shown in Section 2.3. Lastly, as shown in Sections 2.4-2.8, the inverse Laplace transform of the solution is derived to obtain a solution for the volumetric flow rate in the time domain. Section 3 treats the resulting time domain solution. Generally, the Navier-Stokes equations govern fluid flow in three-dimensional space. However, solving the full Navier-Stokes equations is unnecessary for this case of liquid flow in a rigid pipe. A simplified version of the equations can be used to obtain a solution. The following assumptions are applied to simplify the Navier-Stokes equations:  
1. The flow is laminar (Reynolds number \(Re\leq 2300\)), meaning the fluid layers do not mix. Consequently, the pressure remains constant across the pipe's cross-section, and the pressure gradient in the radial direction is negligible. Figure 2: Flowchart of the derivation of the soft sensor equations. Both the flow and pipe geometry are axisymmetric, resulting in no gradient in the angular direction, i.e., \(\left(\frac{\partial}{\partial\varphi}=0\right)\). 3. The axial flow velocity \(v_{\mathrm{x}}\) is much smaller than the speed of sound \(a\), which governs the fluid's pressure wave propagation speed. Therefore, \(v_{\mathrm{x}}\ll a\), and the Mach number \(Ma=v_{\mathrm{x}}/a\ll 1\), so supersonic effects can be ignored. 4. The pipe radius is much smaller than the pipe length (\(R\ll L\)), meaning no pressure reflections occur at the pipe walls. 5. The significant viscous effects in the motion equations are limited to those involving the radial distribution of axial velocity [5]. 6. Gravitational forces are negligible as the pipe is horizontal, keeping the forces constant over its length. 7. Changes in fluid density due to vertical positioning are negligible because the pipe's diameter is small. 8. Heat transfer is ignored since the focus is on liquids, excluding gases [32]. Using these assumptions, the Navier-Stokes equations can be simplified, and the general equation for the volumetric flow rate in the Laplace domain was derived in prior works [5,33]. The solution for the volumetric flow rate at the outlet of the pipe (\(Q_{2}^{*}\)) can be expressed as follows:  
\[Q_{2}^{*}=\frac{1}{Z_{\mathrm{c}}\sinh(\gamma L)}p_{1}^{*}-\frac{\coth(\gamma L )}{Z_{\mathrm{c}}}p_{2}^{*}=W_{1}^{*}(\zeta)\ p_{1}^{*}-W_{2}^{*}(\zeta)\ p_{2}^{*}. \tag{7}\]  
Inserting the definitions of the impedance \(Z_{\mathrm{c}}\) and the propagation operator \(\gamma\) into the equation and multiplying both sides by \(R_{\mathrm{H}}\) yields Equation (8).