1.1. Dynamic Process Intensification and Forced Periodic Operation Analysis
The idea of intensifying a process by switching from a steady-state (SS) to a forced periodic (FP) operation was first introduced by Douglas and Rippin in 1966 [
1], and it was further expanded by Douglas [
2], Horn and Lin in 1967 [
3], and Bailey and Horn in 1971 [
4]. Though in the beginning, the investigations of FP operations were limited to theoretical studies, in the next 25 years, there were at least 29 different experimental dynamic process intensification (PI) investigations of industrially important processes [
5]. The initial investigations studied the FP operations with the modulation of one input variable: the inlet flow rate or the chemical composition (reactant concentration). The studies focused on enhancing the reaction rate, selectivity, and product distribution [
5]. However, except for the reverse flow reactors, no remarkable improvements, i.e., process enhancements, were achieved by applying PI concepts in the time-domain. This was the case until the emergence of sophisticated process system engineering (PSE) techniques and methods that were mostly used for PI in multi-functional units [
6].
The surge of the computer industry at the end of the 20th century and the use of advanced numerical solvers and new algorithms had a great impact in the area of chemical engineering and led to new PI concepts in the field of process synthesis and design [
7]. Different dynamic single-objective optimization (SOO) techniques were available for research on a plant-level [
8] and the optimization of processes on a unit-level (e.g., reactor operation) [
9,
10,
11,
12,
13,
14].
For optimizing periodic operations in chemical engineering, a variety of different numerical methods have been introduced [
11,
14,
15,
16,
17,
18,
19,
20,
21,
22]. Most methods, in the beginning, were single-objective optimization algorithms that computed the time-average performance of a process [
15,
16]. Dynamic optimization methods, which were very suitable for the optimization of periodic processes, allowed users to find an optimal solution around a predefined SS point and rely on local optimization algorithms [
11]. In cases where there were many different possibilities regarding the modulated inputs and no fast and accurate way of calculating time-average performance, the dynamic local SOO proved to be time-consuming, numerically expensive, and inaccurate (far from the global optimum), leading to only marginal process improvements [
14]. As Özgülşen et al. pointed out, the most challenging difficulty in determining optimal parameter values in forced periodic operations was locating the global maximum [
15]. The use of global optimization methods would remove this difficulty, and further advances would eliminate the need for initialization strategies to solve nonlinear programming problems [
7].
Recently, Rangaiah et al. [
23] published a review and tutorial of a five-step procedure used to create, solve, and select the optimal results of chemical engineering problems using multi-objective optimization (MOO). One of the most widely used MOO algorithms for the optimization of engineering problems is the non-dominated sorting hybrid algorithm [
24]. In the last five years, there has been a significant usage of MOO in chemical engineering [
25,
26,
27,
28,
29,
30,
31,
32,
33]. The advantage of MOO over SOO is that it allows for a multi-criteria analysis for choosing the best solution [
26]. Additionally, MOO does not depend on user-supplied initialization procedures. However, not many examples can be found where global dynamic SOO or MOO were used for the optimization of FP operation with modulation of several input variables [
11,
14].
As Baldea and Edgar pointed out [
34], although dynamic PI (FP operation) can significantly enhance process performance, several hurdles still need to be overcome:
A systematic framework for the identification of possible dynamically intensified processes is required.
Various mathematical and numerical challenges while optimizing and computing the cyclic processes need to be addressed.
New techniques that encompass the previous points should be part of intuitive software tools that can be used both in academia and industry.
The idea of this research was to address the beforementioned challenges using a completely new approach based on an analytical PSE tool for predicting the time-average behavior of periodic processes—the Nonlinear Frequency Response (NFR) method [
35]. Recently, this method was upgraded by developing a new PSE software application, leading to the so-called computer-aided Nonlinear Frequency Response (cNFR) method [
36].
1.2. The NFR Method for Evaluating the Potential of Process Intensification through Forced Periodic Operations
The Nonlinear Frequency Response (NFR) method is an analytical PSE tool based on the concept of higher-order frequency response functions (FRFs) [
37]. The NFR method allows users to transfer a dynamic mathematical model of a weakly nonlinear system from the time domain to the frequency domain. In the frequency domain, the system is mathematically represented with a set of the first, second, and higher-order FRFs [
37], which are functions of frequency. Though an infinite sequence of FRFs is needed for full representation of the system, it was shown that good approximations can be achieved by using only the first and second-order FRFs [
38].
For an easier understanding of the NFR method,
Figure 1 shows a schematic representation of a chemical reactor with forced, sinusoidal modulation of an input
around its SS value
. For a stable system, after going through a transient, any output from the reactor will eventually reach a quasi-(periodic)-steady-state
. Most processes in chemical engineering, including chemical reactors, are weakly nonlinear (all nonlinear terms in their model equations are continuous and differentiable, i.e., can be represented in the Taylor series form). The NFR method can only be applied to stable, weakly nonlinear systems that can be represented with convergent Volterra series. Another limitation is that the analyzed systems must not have multiple steady states. For such systems, the quasi-steady- state of the response is obtained as a sum of its SS value, the first harmonic (of the same frequency as the input), a theoretically indefinite number of higher harmonics, and a nonperiodic term (the so-called DC component), as shown in
Figure 1. The DC component represents the difference between the time-average of the output in the FP regime and its SS value and is the measure of PI owing to the FP operation.
The NFR method based on the concept of higher-order frequency response functions (FRFs) [
39] enables the fast and easy estimation of the DC component. For the sinusoidal modulation of a single input
, with amplitude
and frequency
, the DC component can be approximately evaluated by using only the asymmetrical second-order FRF
relating the output
and input
:
The sign of , which is the same as the sign of , answers whether the FP operation would result in process improvement or not, while its value gives a very good estimate of the extent of the improvement.
When two input signals (
and
) are modulated, with amplitudes
and
and a phase angle
between them, then:
the DC component can be approximately calculated as a sum of contributions of the inputs
and
, separately, and the contribution of their cross-effect [
40]:
where
is the cross asymmetrical second-order FRF. From Equation (4), it follows that the simultaneous modulation of two inputs has a higher PI potential in FP operations than single-input modulations (Equation (1)), as the sign of the cross-term can be easily adjusted by correctly selecting
[
40]. Due to this, by modulating the system with two inputs with an optimal phase difference, improvements become possible even in cases when modulating one or both inputs, separately, leads to process deterioration [
41].
The NFR method is a PSE tool for the fast and easy prediction of the approximate time-average values of the outputs of interest (as shown in Equations (1) and (4)). Even though the NFR method gives an approximate value of the DC component, it has been shown that good estimates can be achieved by using only the second-order asymmetrical FRFs (Equations (1) and (4)) [
35,
39,
40,
41,
42,
43,
44,
45,
46,
47,
48,
49]. Thus far, the NFR method has been used for the dynamic PI analysis of isothermal [
39,
40,
42] and nonisothermal [
41,
43,
44,
45,
46,
47] reactors with sinusoidal [
39,
40,
41,
42,
43,
45,
46,
49] or other periodic shapes [
44,
50] of the modulated inputs.
It is important to note that different performance criteria of interest (e.g., conversion, yield, selectivity, and productivity) of FP operation can be directly calculated from the time-average values of the outputs obtained using the NFR method. Using Equations (1) and (4), they can be expressed in the algebraic form as functions of the input modulating parameters (frequency, amplitudes, and phase shift).
One of the most important steps of applying the NFR method is the derivation of the needed FRFs. Though the procedure for the analytical derivation of FRFs is well-established and has been used and presented in many publications [
35,
39,
40,
41,
43,
44,
45,
46,
47,
48,
49,
50,
51,
52,
53,
54], it is rather complex, time-consuming, and requires certain mathematical skills in the complex domain. As such, the NFR method can be unappealing to beginners, especially when applied to more complex systems.
The recently developed computer-aided Nonlinear Frequency Response (cNFR) method [
36] is an upgrade of the NFR method using a software application for the automatic derivation of the FRFs of interest. It has an easy-to-use interface where the user can define the dynamic model equations of the system and automatically generate MATLAB files for the desired FRFs and DC components [
36]. The generated MATLAB files contain codes with algebraic expressions of the derived FRFs and DC components in the vector form, and they enable very fast simulation of the key system performance criteria and rapid rigorous global and local optimization. No time-consuming and numerically expensive methods are required for solving the systems of differential equations. In this way, the cNFR method provides answers to three key questions about the potential of the investigated FP operation:
Which input or combination of inputs should be periodically modulated?
What are the optimal forcing parameters of the modulated inputs: frequency, amplitudes, or phase shift ( and , or , respectively)?
What is the expected process improvement if we switch from SS to FP operation (a maximal increase in the performance criteria of interest)?
By giving answers to these questions in a fast and easy manner, the cNFR method overcomes most challenges listed by Baldea and Edgar. As is shown in this paper, the cNFR method has the potential to become a very strong software PSE tool for analyzing the dynamic PI potential.
The cNRF software tool was explained in detail by Živković et al. [
36]. Thus far, it was used for the analysis of the FP operation of an isothermal continuous stirred tank reactor (CSTR) with the simultaneous modulation of feed concentration and flow-rate [
24], for experimental identification [
55], and for the evaluation of the FP operation of an electrochemical process [
41].
In this work, a new methodology for the rigorous multi-objective optimization (MOO) of forced periodic (FP) operations is presented. It is based on the cNFR method and cost–benefit indicator analysis.