## 3 Case Study--Results
### Closed-loop Control Analysis  
Using the dynamic HE model outlined in Section 2.3, the entire HENs together with control configurations were modelled employing Simulink software; the block diagram of the HEN model is presented in Figures 8 and 9.  
For case no. 1, the simulations of transient responses were carried out and their results shown in Figure 16a,b (for control loops 1 and 2) demonstrate that fouling build-up induces insignificant changes in \(CV\) rise time \(t_{r}\) and settling time \(t_{s}\). This can be seen as an indication that in the studied case, no adjustments of \(K_{p}\) and \(K_{i}\) values are needed and the indices of control quality would not be adversely affected by fouling of heat-exchanger surfaces--see Table 5.  
Figure 16: Closed-loop step responses (case no. 1) under fouling conditions of the control variable (**a**) \(CV_{1}\) and (**b**) \(CV_{2}\) to setpoints step change.  
For case no. 2 (HEN with control loop 3), the closed-loop step responses under fouling conditions were simulated with three different sets of the PID parameters:  
1. Assuming constant values of the base PID parameters obtained using Ziegler-Nichols method [18] for clean HEN--Figure 17a
2. With the adjusted PID parameters, for the consecutive periods of fouling build-up, in accordance with the data shown in Table 4--Figure 17b
3. With the optimal PID parameters obtained using Signal Constraint toolbox in SIMULINK [21] under fouling conditions--Figure 17c  
In case no. 2, control loop 3 comprises heat exchangers E3-18 AB whose operation is affected by the interactions with the remaining exchangers in the studied HEN (see Figure 3). Qualitative evaluation of the obtained dynamic closed-loop characteristics (Figure 17a-c) can be complemented by the values of quality indices--Table 6.  
In this case, judging from the information presented in Figure 17 and Table 6, the effect of fouling that builds up during HEN operation is more pronounced than that observed in case no. \(1\). At unchanged PID-controller settings, settling time \(t_{\text{s}}\) is increased from 124.6 s for the clean exchanger to 173.2 s for the exchanger fouled after 3 years of HEN operation. While some quality indices including peak time \(t_{p}\), delay time \(t_{d}\) and rise time \(t_{r}\), are changed, overshoot \(M_{p}\) remains nearly constant (Figure 17a). If adjustments of PID-controller settings were applied in reaction to a fouling build-up, then the resulting dynamic characteristics and control-quality indices would not deteriorate (Figure 17b and Table 6). Using the Ziegler-Nichols method [18] or the Signal Constraint toolbox in Simulink [21], optimal controller settings can be determined to eliminate excessively large values of overshoot \(M_{p}\) (Figure 17c). In this context, a more advanced approach recently introduced by [PERSON] [12] in cooperation with [PERSON] and [PERSON] can be mentioned. In their work, robust Model Predictive Control - MPC with integral action is used for optimizing the control performance when the operation of heat exchangers has been affected by fouling that induces changes of the exchangers' parameters.