## 4 Discussion  
Using Simulink software, a validated multi-cell dynamic model of a shell-and-tube HE was applied in simulating the operation of PID-controlled HEs (see Figures 8 and 9). A control-theory based approach was proposed for the identification and evaluation of the influence of fouling on the dynamic behavior of the HEN and on the quality of its control (Figure 4). The dynamic model was applied to a case study on the HEs and HENs operated in the crude distillation unit under fouling conditions. In case no. 1, control loops no. 1 and 2 include all the HEs operated in the HEN and interactions between the HEs are significant (Figure 2). The simulated step responses prove that as fouling was building up, the quality indices of network control remained nearly unchanged even if the tuning of PI controllers was not adjusted (see Figure 16 and Table 5). In case no. 2, control loop no. 3 includes a set of HEs that interact with other exchangers present in the HEN (Figure 3). From the qualitative and quantitative estimates presented above, it can be inferred that the effect of fouling on HEN operation is more pronounced than that observed in case no. 1. Although most indices of control quality remain unchanged as the fouling increases, the settling time becomes longer. Periodic adjustments of PID controller tuning are required in the consecutive  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \multirow{2}{*}{\ \begin{tabular}{c} **Closed-Loop** \\ **Responses for** **Case No. 2** \\ \end{tabular}  } & \multirow{2}{*}{
\begin{tabular}{c} **\(R_{\text{\emph{\emph{\emph{\emph{\emph{\emph{\emph{\emph{\emph{\emph{\emph{ \emph{ \emph{ \emphemph { \{ \emph  
stages of fouling build-up because the value of parameter \(t_{s}\) is significantly increased (see Figure 17 and Table 6). In previous publications [6; 21], the present authors evaluated dynamic characteristics of four different sets of heat exchangers operated in simple control loops according to the scheme shown in Figure 7a, that is, without meaningful interactions with other exchangers in the HEN. The performance of those control loops have now been simulated and their closed-loop step responses are presented in Appendix A, Figures A1-A4, while the corresponding values of control quality indices are shown in Table A1. In each of the studied PID-controlled HEs, it was found that increased fouling led to the deterioration of all the indices of control quality so that periodic adjustments of PID-controller tuning appeared necessary. It can be mentioned that the dynamic model of shell-and-tube heat exchangers developed by the present authors found application in the work done by [PERSON] et al. [22], who performed the dynamic analysis of fouling build-up in the HEs designed according to TEMA standards. The same author team also tested several tuning strategies for the PID-controlled HEs under fouling conditions [23] and arrived at results that appear to be consistent with those of the present authors. Overall, according to the results of the mentioned case studies, the higher the number of heat exchangers in the PID control loop and the more interactions occur between heat exchangers in the network, the weaker the influence of fouling on the control quality indices (see Table 7). This observation may be attributed to underestimated values of \(R_{f}\) (calculated according to TEMA standards) and/or to the compensation of the negative impacts of fouling on the heat transfer in the HEs. Such a compensation is possible only in the network where significant interactions occur between the HEs (that is, if antecedent exchangers are operated on the process streams--Figure 18). As previously observed by the present authors [17; 24], the larger the number of interacting exchangers, the better the compensation of the detrimental effects of fouling. Fouling on the heat transfer surface of a HE operated in the HEN brings about a change in the exchanger capacity as well as changes in the outlet temperatures of process streams.