## 3 Case Study--Results
### Dynamic analysis of the HEN  
#### 3.3.1 Study the Open-loop Step Responses in Case No. 1  
For the different periods of HEN operation during which fouling was building up, simulations have been carried out in Simulink. According to the obtained results, when the thermal resistance of fouling is increased, the thermal inertia of every HE is changed leading to changes in the dynamic behavior of the interacting \(A\), \(B\), \(C\), \(D\) branches shown in Figure 2. Such changes can be detected by studying the open-loop step responses of the end temperatures (after parallel branches \(T_{ABend}\), \(T_{CDend}\) \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**HE** & \multicolumn{2}{c}{**Fouling Factor**} & \multicolumn{2}{c}{**Total Heat**} \\
**no.** & \multicolumn{2}{c}{\(R_{f}\times\) 10\({}^{-3}\) (m\({}^{2}\)K/W)} & \multicolumn{2}{c}{**Transfer**} \\  & \multicolumn{2}{c}{**after Period of Operation:**} & \multicolumn{2}{c}{**Coefficient**} \\  & **1** & **2** & **3** & **for Clean HE** \\  & **year** & **years** & **years** & **_U_ (W/m\({}^{2}\)K)** \\ \hline E3-11 AB & 1.869 & 2.672 & 3.611 & 429 \\ E3-12 AB & 1.486 & 2.478 & 3.437 & 434 \\ E3-13 AB & 0.957 & 1.522 & 2.589 & 237 \\ E3-14 AB & 1.563 & 2.587 & 3.523 & 443 \\ E3-15 AB & 0.982 & 1.207 & 1.694 & 294 \\ E3-16 ABC & 1.234 & 1.894 & 2.896 & 295 \\ E3-17 ABCD & 1.623 & 2.543 & 3.431 & 810 \\ E3-18 AB & 0.323 & 0.623 & 1.196 & 745 \\ E3-21 ABCD & 1.587 & 2.452 & 3.257 & 894 \\ E3-22 AB & 0.128 & 0.273 & 0.532 & 327 \\ E3-23 AB & 2.077 & 4.448 & 6.075 & 383 \\ E3-24 AB & 0.444 & 0.699 & 1.259 & 623 \\ E3-25 AB & 0.677 & 0.823 & 1.647 & 603 \\ E3-26 ABC & 0.279 & 1.116 & 1.628 & 410 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of the \(R_{f}\) and heat transfer coefficient for the studied HEN in case no. 2. Figure 9: Scheme of the HEN with PID-control loop 3 implemented in Simulink/MATLAB. simulated at the different stages of fouling build-up. The features of a typical response of a heat exchanger system (network branch) are illustrated in Figure 6a. In Figure 10a,b, simulated open-loop responses of the studied HEN models are plotted for a step upset +1\({}^{\circ}\)C in the tube-side inlet temperatures of the branches \(A\) (exchanger E1-11 AB) and \(B\) (exchanger E1-21). The open-loop responses of the branches to a step upset +1\({}^{\circ}\)C in the shell-side inlet temperatures of the exchangers (E1-13, 14, 27 and E2-14, 15 AB, 25 AB, 26, 27), are presented in Figure 11a,b. Next, Figure 12a,b shows the open-loop responses of the branches to +10% step change in the shell-side flowrate of the HEs (+8.61 kg/s in E1-14, +3.75 kg/s in E1-27, +1.22 kg/s in E2-11, +2.55 kg/s in E2-12 AB, +3.46 kg/s in E2-23 and +1.59 kg/s in E2-24). Finally, Figure 13a,b depicts the open-loop responses of the HEN models to +10% and -10% step change in the tube-side flowrates of branches \(A\), \(C\) (+6.11 kg/s) and \(B\), \(D\) (\(-6.11\) kg/s), respectively. Figure 11: Open-loop responses of the HEN models ((a)—the end temperature \(T_{ABend}\) and (b)—the end temperature \(T_{CDend}\), after parallel branches) to +1\({}^{\circ}\)C step change in the shell-side inlet temperature of selected exchangers. Figure 10: Open-loop responses of the HEN models ((a)—the end temperature \(T_{ABend}\) and (b)—the end temperature \(T_{CDend}\), after parallel branches) to +1 \({}^{\circ}\)C step change in the tube-side inlet temperature of the branches \(A\) and \(B\). It can be seen in the open-loop responses that in each of the studied branches, variations induced by fouling build-up on the exchangers' heat transfer surfaces are visible in the values of gain \(K_{0}\), delay time \(t_{d}\) and time constant \(t_{1}\). In practical terms, the changes in the delay time in the most open-loop step responses are insignificant but the increased/decreased time constants and reduced/increased gain values may impair the quality of PID control considerably.