## 2 Materials and Methods
### Derivation of a Solution for a Fixed Dissipation Number for a Compressible Fluid  
The previous chapter demonstrated that the poles of the weighting function depend on the dissipation number, making it impossible to derive a singular solution through an inverse Laplace transform (ILT) of the weighting function. This chapter derives the ILT for a fixed dissipation number of \(Dn=0.0011\). This value is selected based on the pipe characteristics used in the test bench later in this project. Specifically, \(Dn=0.0011\) corresponds to a pipe length of \(L=1.5\,\mathrm{m}\), a diameter of \(D=14\,\mathrm{mm}\), and standard fluid properties of HLP46, including a density of \(\rho=860\,\mathrm{kg/m^{3}}\), a kinematic viscosity of \(\
u=46\times 10^{-6}\,\mathrm{m^{2}/s}\), and a bulk modulus of \(K=14,000\,\mathrm{bar}\). The general formula for calculating residues, provided in Equation (4), is now applied to determine the residue of the double pole at \(\zeta_{0}=0\). \[ \begin{split}\mathrm{Res}(W^{*}_{\rm inc}(\zeta))|_{\zeta_{0}=0 }&=\lim_{\zeta\to\zeta_{0}}\frac{1}{(m-1)!}\frac{\mathrm{d}^{m-1 }}{\mathrm{d}\zeta^{m-1}}[W^{*}_{\rm inc}(\zeta)(\zeta-\zeta_{0})^{m}]\\ &\stackrel{{ m=2}}{{=}}\lim_{\xi\to\zeta_{0}}\frac{ \mathrm{d}}{\mathrm{d}\zeta}[W^{*}_{\rm inc}(\xi)(\zeta-0)^{2}]\\ &=\lim_{\xi\to\zeta_{0}}\frac{\mathrm{d}}{\mathrm{d}\zeta}[W^{*} _{1}(\zeta)\;\zeta^{2}]\\ &=\lim_{\xi\to\zeta_{0}}\frac{\mathrm{d}}{\mathrm{d}\zeta}\left[ \frac{8\mathrm{I}_{2}(\sqrt{\zeta})}{\zeta^{2}\mathrm{I}_{0}(\sqrt{\zeta})} \right)\zeta^{2}\right]\\ &=\lim_{\xi\to\zeta_{0}}\frac{\mathrm{d}}{\mathrm{d}\zeta}\left[ \frac{8\mathrm{I}_{2}(\sqrt{\zeta})}{\mathrm{I}_{0}(\sqrt{\zeta})}\right]\\ &=1\end{split}  \tag{36}\]  
The equation for the residues of simple, complex conjugated poles is given in Equation (5). \[\left.\mathrm{Res}(W^{*}_{1}(\zeta))\right|_{\zeta_{k}}=\frac{\mathrm{numerator }[W^{*}_{1}(\zeta_{k})]\;\mathrm{e}^{\zeta_{k}t_{n}}}{\frac{\partial}{ \partial\zeta}\left(\mathrm{denominator}[W^{*}_{1}(\zeta_{k})]\right)} \tag{37}\]  
The poles of the compressible component of the weighting function can be identified by solving \(\sinh[x]=0\) along with Equation (32). This leads to the following expression:  
\[1+\frac{\mathrm{I}_{0}(\sqrt{\zeta_{k}})\zeta_{k}^{2}Dn^{2}}{k^{2}\mathrm{I}_ {2}(\sqrt{\zeta_{k}})\pi^{2}}=0,\quad\text{for }k\in\mathbb{N}. \tag{38}\]  
Rearranging for \(\zeta_{k}\), we obtain:  
\[\frac{\mathrm{I}_{0}(\sqrt{\zeta_{k}})\zeta_{k}^{2}}{\mathrm{I}_{2}(\sqrt{ \zeta_{k}})}=\left(\frac{k\pi}{Dn}\right)^{2}-1. \tag{39}\]  
Assuming \(Dn=0.0011\), the poles are located at:  
1. \(\zeta_{1}=-38.30\pm 2818.21i\)
2. \(\zeta_{2}=-53.95\pm 5658.55i\)
3. \(\zeta_{3}=-65.96\pm 8502.53i\)\[ \begin{split} W_{1,\text{comp}}(t_{\text{n}})=\\ -(0.0000180+0.00270i)\text{e}^{(-38.30-2818.21i)\,t_{\text{n}}}+( -0.000018+0.00270i)\text{e}^{(-38.30+2818.21i)\,t_{\text{n}}}\\ +(0.0000066+0.00140i)\text{e}^{(-53.95-5658.55i)\,t_{\text{n}}}+( 0.0000066-0.00140i)\text{e}^{(-53.95+5658.55i)\,t_{\text{n}}}\\ -(0.000036+0.00093i)\text{e}^{(-65.96-8502.53i)\,t_{\text{n}}}+( -0.0000036+0.00093i)\text{e}^{(-65.96+8502.53i)\,t_{\text{n}}}\\ +(0.0000023+0.00070i)\text{e}^{(-76.08-11348.4i)\,t_{\text{n}}}+( 0.000023-0.00070i)\text{e}^{(-76.08+11348.4i)\,t_{\text{n}}}\\ -(0.000016+0.00056i)\text{e}^{(-85.00-14195.4i)\,t_{\text{n}}}+( -0.0000016+0.00056i)\text{e}^{(-85.00+14195.4i)\,t_{\text{n}}}.\end{split}  \tag{40}\]  
The method is based on a sum of poles with increasing imaginary parts. Thus, the equation can be expanded to include the system behavior corresponding to any desired frequency. The five poles considered were sufficient for the model to accurately capture the fluid flow behavior in all examined test cases. The curve-fitted ILT of the incompressible fluid is independent by \(Dn\) and is presented in Equation (41):  
\[