## 3 Implementation of the Soft Sensor in the Time Domain  
In this chapter, the solution of the equation for volumetric flow rate in the time domain is evaluated. This involves a detailed examination of the convolution integral obtained as a solution, which will be simplified in this section. To begin, we revisit Equation (12). For clarity, this equation is restated as Equation (47). To determine the volumetric flow rate in the time domain, we need to revert this equation into that domain:  
\[Q_{2}^{*}R_{\text{H}}=W_{1}^{*}\zeta p_{1}^{*}-W_{1}^{*}\zeta p_{2}^{*}. \tag{47}\]  
By applying the principle that the multiplication of two functions in the Laplace domain corresponds to the convolution of those same functions in the time domain, along with the fact that multiplying a function by the Laplace variable equates to taking the time derivative, we obtain:  
\[Q_{2}(t_{\text{n}})R_{\text{H}}=\bigg{[}\int_{0}^{t_{\text{n}}}W_{1}(t_{\text{ n}}-\tau)\frac{\partial p_{1}(t_{\text{n}}-\tau)}{\partial t_{\text{n}}}d\tau- \int_{0}^{t_{\text{n}}}W_{2}(\tau)\frac{\partial p_{2}(t_{\text{n}}-\tau)}{ \partial t_{\text{n}}}d\tau\bigg{]}. \tag{48}\]  
Figure 6: Position of the poles of \(W_{1}^{*}(\zeta)\) for various \(Dn\). Since the first term of both weighting functions is a constant, we can separate \(W_{1}\) and \(W_{2}\), placing the constant term in a separate integral:  
\[ \begin{split} Q_{2}(t_{\mathrm{n}})R_{\mathrm{H}}=& \bigg{[}\int_{0}^{t_{\mathrm{n}}}1\ \frac{\partial p_{1}(\tau)}{\partial t_{\mathrm{n}}}d\tau+\int_{0}^{t_{ \mathrm{n}}}(W_{1}(t_{\mathrm{n}}-\tau)-1)\frac{\partial p_{1}(\tau)}{ \partial t_{\mathrm{n}}}d\tau\bigg{]}\\ &-\bigg{[}\int_{0}^{t_{\mathrm{n}}}1\ \frac{\partial p_{2}(\tau)}{ \partial t_{\mathrm{n}}}d\tau+\int_{0}^{t_{\mathrm{n}}}(W_{2}(t_{\mathrm{n}}- \tau)-1)\frac{\partial p_{2}(\tau)}{\partial t_{\mathrm{n}}}d\tau\bigg{]}. \end{split}  \tag{49}\]  
It is practical to reorganize the equations into constant and dynamic components. The constant integral can be evaluated as the pressure difference given by \(p_{1}(t_{\mathrm{n}})-p_{2}(t_{\mathrm{n}})=\Delta p\). \[ \begin{split} Q_{2}(t_{\mathrm{n}})R_{\mathrm{H}}=& \bigg{[}\int_{0}^{t_{\mathrm{n}}}1\ \frac{\partial p_{1}(\tau)}{\partial t_{\mathrm{n}}}d\tau-\int_{0}^{t_{ \mathrm{n}}}1\ \frac{\partial p_{2}(\tau)}{\partial t_{\mathrm{n}}}d\tau\bigg{]}\\ &+\bigg{[}\int_{0}^{t_{\mathrm{n}}}(W_{1}(t_{\mathrm{n}}-\tau)-1) \frac{\partial p_{1}(\tau)}{\partial t_{\mathrm{n}}}d\tau-\int_{0}^{t_{\mathrm{ n}}}(W_{2}(\tau)-1)\frac{\partial p_{2}(\tau)}{\partial t_{\mathrm{n}}}d\tau \bigg{]}\\ =&(p_{1}(t_{\mathrm{n}})-p_{2}(t_{\mathrm{n}}))\\ &+\bigg{[}\int_{0}^{t_{\mathrm{n}}}(W_{1}(t_{\mathrm{n}}-\tau)-1) \frac{\partial p_{1}(\tau)}{\partial t_{\mathrm{n}}}d\tau-\int_{0}^{t_{\mathrm{ n}}}(W_{2}(\tau)-1)\frac{\partial p_{2}(\tau)}{\partial t_{\mathrm{n}}}d\tau \bigg{]}\\ =&\Delta p+\bigg{[}\int_{0}^{t_{\mathrm{n}}}(W_{1}(t _{\mathrm{n}}-\tau)-1)\frac{\partial p_{1}(\tau)}{\partial t_{\mathrm{n}}}d \tau-\int_{0}^{t_{\mathrm{n}}}(W_{2}(\tau)-1)\frac{\partial p_{2}(\tau)}{ \partial t_{\mathrm{n}}}d\tau\bigg{]}\end{split}  \tag{50}\]  
The resulting equation now contains a stationary term that resembles the Hagen-Poiseuille law \((QR_{H}=\Delta p)\) alongside dynamic terms for each pressure function.