## 2 Materials and Methods
### Stage 3--Development and Validation of the Dynamic HE Model  
The planning of efficient use of HEs under changing operating conditions (e.g., conditions resulting from fouling build-up with time) requires the application of adequate dynamic models. Stage 3 is based on the mathematical model proposed by [PERSON] et al. [6], of transient heat exchange with the influence of thermal resistance of fouling taken into account. According to the scheme shown in Figure 4, the three main steps of Stage 3 are:  
Figure 5: Fouling factor versus time during three years of operation of a selected HE. * Determination of a cell-based dynamic HE model based on the operator transmittances. By solving the equations of the mathematical model, relationships employing operator transmittances can be obtained between disturbances occurring at cell inlet and changes in temperature at the cell outlet. Operator transmittance \(G\)(s) is a widely used tool for describing a dynamic system. This step is extensively described in Section 2 of the authors' previous work [6]. * Implementation of the dynamic HE model. Starting from HEN block diagram in which the role of operator transmittances was visualized and using MATLAB/Simulink program package, a software module was developed to simulate the performance of HEN control. In order to make simulation possible, a database is needed for providing the values of relevant parameters in all the cells at steady state (from stage 2), of all the HEs in the HEN. This step is extensively described in Section 2.2 of the authors' previous work [6]. * Validation of the dynamic HE model using operational data of a real-life HEN coupled with a CDU. The values of simulated and real temperature at heat exchanger outlet in transient states of the exchanger were compared and found to be in close agreement. This step is in more detail described in Section 3 of the authors' previous work [6]. Stage 4--Identification and Evaluation of the Influence of Fouling on the Dynamic Behavior of PID-Controlled HEs and on the Control-Quality Indices  
The first step is to study the open-loop unit step responses simulated at different periods of fouling build-up for all controlled HEs. When parameter \(R_{f}\) is increased, the thermal inertia of the HE is changed, leading to changes in its dynamic behavior. Such changes can be detected by studying the open-loop unit step responses simulated at different stages of fouling build-up. A typical response of a HE system is illustrated in Figure 6a. In Figure 6b, open-loop responses of the HE model are plotted for a step upset +5% in the shell-side flowrate \(M_{s}\). As can be seen in these responses, fouling build-up on the exchanger's heat transfer surface induces changes in the values of gain \(K_{o}\), delay time \(t_{d}\) and time constant \(t_{1}\). The second step is modeling of a control-loop unit including the dynamic exchanger model together with PID control configurations in Matlab-Simulink (see Figure 7a). In the next two steps, assuming PID control of the exchanger unit, three gain coefficients (proportional \(K_{p}\), integral \(K_{i}\) and derivative \(K_{d}\)) are needed to determine its closed-loop characteristics. The values of gain coefficients can be determined by the Ziegler-Nichols method [18], which is commonly used in industry. Figure 6: A typical open-loop unit step response of a thermal system (**a**) and the step responses of a HE under fouling conditions (**b**). For specific dynamic characteristics similar to that shown in Figure 7b, description and evaluation of the control quality can be based on the quality indices including:  
* Overshoot \(M_{p}\)--percentage of the maximum deviation of step response \(y(t)\) from its steady-state value: \[M_{p}=((y_{max}-y_{\text{ss}})/y_{\text{ss}})\times 100\%,\] (1) where: \(y_{max}-y(t)\) at its maximum, \(y_{ss}-y(t)\) at steady state (\(y_{ss}\leq y_{max}\)). The value of overshoot is determined during control-system design and may be used as a measure of system stability; large overshoot values are not recommended.