## 1 Introduction  
Heat Exchanger Network (HEN) fouling is a chronical problem encountered in many process industries. The operation of a Heat Exchanger (HE) may be affected by fouling which builds up on the heat transfer surface. For example, fouling of HENs in the oil refining industry results in increased energy consumption (burning extra fuel to compensate for reduced heat recovery), reduced plant throughput when the exchangers are cleaned, and induced costs of cleaning interventions [1; 2]. In recent years, various approaches to the mitigation of fouling effects in industrial HEs and HENs have been reported in the literature [3; 4]. Fouling leads to the reduction of steady-state heat recovery [5], but also to changes in the transient states of HEs [6] and inefficient control of HEs [7; 8] that may have an adverse effect on the overall performance of the HEN [9; 10]. In the literature, publications devoted to the effect of fouling on the dynamic behavior of HEs and the role of fouling in control issues are rare and limited in scope [11; 12]. For a more regular introduction please refer to the authors' previous work [6], where the relevant research field has been reviewed and key publications have been cited. Figure 1 shows an example of dynamic characteristics of a HE (output signal \(y\) as a function of time \(t\), in response to a step change in an input signal that occurred at \(t=0\)) and indicates possible  
deformation of the characteristics induced by the deterioration of control quality. The dynamic characteristics are described by the following parameters: gain \(K_{0}\), delay time \(t_{d}\) and time constant \(t_{1}\). The gain is a coefficient determining the change of the output signal with respect to the change in the input control signal; the larger the gain value, the stronger output signal's response to the input signal. The delay time defines the waiting period between the step change in the input signal (in Figure 1, \(t=0\)) and the change of the output signal (that is, delay time describes the speed of response on given control). The time constant is a measure of the capacity of the process and determines how long after the cessation of the delay does the output signal reach 63.2% of its final value. When fouling builds up on the heat transfer surface, the dynamic characteristics of the HEs operated in the network may be changed. In a previous work [6], the present authors studied the influence of fouling deposition in the individual HE units on their dynamic characteristics and on the quality of their control, with the aim of describing fouling effects quantitatively. Moreover, four examples of control loops with single HE and without significant interactions from the other HEs were investigated. In Appendix A, as a complement to the mentioned closed-loop characteristics, the corresponding values of control-quality indices are shown in Table A1. In the present paper, using the validated dynamic model, the behavior of specific crude preheat trains (branches of HENs interacting with each other--Figures 2 and 3) together with their PID-control loops is simulated and the influence of fouling build-up on the specific indices of the quality of operation is investigated. Figures 2 and 3 show two considered examples of real-life PID-controlled HENs coupled with a Crude Distillation Unit (CDU). The graphical form of Figures 2 and 3 was developed by the present authors on the basis of schemes and process data made available by the owner of two different CDUs that operated in a Polish oil refinery. The conclusions of the previous publication [6] seemed to suggest that fouling build-up in heat exchangers usually leads to the significant deterioration of control-quality indices of PID-based control loops. However, from the results of the present research in which more complex PID-controlled loops in large HENs have been investigated, different conclusions can be drawn.