## 3 Case Study--Results
### Dynamic analysis of the HEN
#### 3.3.2 Study the Open-loop Step Responses in Case No. 2  
In case no. 2, the changes in the dynamic behavior of the E3-18 AB HE unit operated in HEN (see Figure 3) can be detected by studying the open-loop step responses of the outlet temperature \(T_{3}\) simulated at the different stages of fouling build-up.  
Figure 15a shows the open-loop responses of the studied outlet temperature \(T_{3}\) to +1% step change in the shell-side flowrate (+0.47 kg/s) of the HE. Next, Figure 15b depicts the open-loop responses of the studied outlet temperature \(T_{3}\) to \(-1\)% step change in the tube-side flowrate (\(-0.73\) kg/s) of the HE. In Figure 15c, simulated open-loop responses of the studied outlet temperature \(T_{3}\) are plotted for a step upset +1\({}^{\circ}\)C in the shell-side inlet temperature of the HE. Finally, the open-loop responses of the studied outlet temperature \(T_{3}\) to a step upset +1\({}^{\circ}\)C in the tube-side inlet temperature of the exchanger E3-18 AB, are presented in Figure 15d.  
Figure 14: Open-loop step responses under fouling conditions of the control variable \(CV_{1}\) (**a**) and \(CV_{2}\) (**b**) on a 10% increase in the inlet mass flow \(M_{At}\) and \(M_{Ct}\).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**HEN** & **Control** & **Loop 1** & **Control** & **Loop 2** & **PID** & **PID 1** & **PID 2** \\
**Operating** & \(\mathbf{K_{0}}\) & \(\mathbf{t_{1}}\) & \(\mathbf{K_{0}}\) & \(\mathbf{t_{1}}\) & **Parameters** & **PID 1** & **PID 2** \\
**Condition** & (\({}^{\circ}\)Cs/kg) & (**s**) & (\({}^{\circ}\)Cs/kg) & (s) & & **Parameters** & **PID 1** & **PID 2** \\ \hline \(R_{f}=0\) & \multirow{2}{*}{\(-1.52\)} & \multirow{2}{*}{\(110\)} & \multirow{2}{*}{\(-2.13\)} & \multirow{2}{*}{96} & \(K_{p}\) & \(-0.8790\) & \(-0.6261\) \\ (clean) & & & & & \(K_{i}\) & \(-0.0080\) & \(-0.0065\) \\  & & & & & \(K_{d}\) & 0 & 0 \\ \hline \(R_{f}\) after & \multirow{2}{*}{\(-1.63\)} & \multirow{2}{*}{99} & \multirow{2}{*}{\(-2.21\)} & \multirow{2}{*}{89} & \(K_{p}\) & \(-0.8175\) & \(-0.6042\) \\
1 year & & & & & \(K_{i}\) & \(-0.0082\) & \(-0.0068\) \\  & & & & & \(K_{d}\) & 0 & 0 \\ \hline \(R_{f}\) after & \multirow{2}{*}{\(-1.66\)} & \multirow{2}{*}{97} & \multirow{2}{*}{\(-2.24\)} & \multirow{2}{*}{87} & \(K_{p}\) & \(-0.8036\) & \(-0.5954\) \\
2 years & & & & & \(K_{i}\) & \(-0.0083\) & \(-0.0069\) \\  & & & & & \(K_{d}\) & 0 & 0 \\ \hline \(R_{f}\) after & \multirow{2}{*}{\(-1.77\)} & \multirow{2}{*}{88} & \multirow{2}{*}{\(-2.33\)} & \multirow{2}{*}{79} & \(K_{p}\) & \(-0.7522\) & \(-0.5723\) \\
3 years & & & & & \(K_{i}\) & \(-0.0086\) & \(-0.0072\) \\ \hline \end{tabular}
\end{table}
Table 3: Values of PID controller parameters obtained using the Skogestad method [20] in case no. 1.  
Because of the fouling build-up on the exchangers' heat transfer surfaces, the visible changes in the delay time \(t_{d}\), time constant \(t_{1}\) and gain \(K_{0}\) values in the studied open-loop step responses, may impair the quality of PID control considerably. In order to prevent that from happening, it is also advisable to investigate the components of the tuning PID controller 3 (\(K_{p}\), \(K_{i}\), \(K_{d}\)) and to check the resulting transient responses.  
PID controller for loop 3 (see Figure 3) was tuned according to the Ziegler-Nichols method [18]. The values of the parameters of the open-loop characteristics (shown in Figure 15a) for each of the selected operation periods and the resulting values of the tuning parameters for PID controller 3 are presented in Table 4.  
Figure 15: Open-loop step responses under fouling conditions of the outlet temperature \(T_{3}\) of E3-18 AB exchanger: (**a**) on a +1% step change in the shell-side flowrate, (**b**) on a \(-\)1% step change in the tube-side flowrate, (**c**) on a +1 \({}^{\circ}\)C step change in the shell-side inlet temperature and (**d**) on a +1 \({}^{\circ}\)C step change in the tube-side inlet temperature.