Forced Periodic Operation Research Articles

A Novel Data-Driven Approach to Analysis and Optimal Design of Forced Periodic Operation of Chemical Reactions

Forced periodic operation is a technique that periodically changes the manipulating variable of a chemical reaction system in order to exploit nonlinear dynamics to improve reactant conversion rate. However, the analysis and design of a periodically operated chemical process is a significant challenge. To resolve this problem, recently, Nonlinear Frequency Response (NFR) based methods have been proposed. However, because of the need to derive the NFR from a first principle model, existing NFR methods can only perform qualitative analysis to simple processes and are often difficult to be applied in engineering practice. The present study proposes a novel data driven approach to the analysis and optimal design of forced periodic operation of chemical reactions. From the data generated numerically using the first principle model or experimentally from experimental tests, the approach produces a data-driven NFR model that can readily be used for both quantitative study and optimal design of forced periodic operation of any complexities. This can fundamentally address the challenges faced by the existing NFR methods, and provides an effective approach that can potentially be applied in engineering practice. Simulation studies and experimental works are carried out on the application of the new method to an isothermal CSTR system and a laboratory-scale carbon dioxide absorption process, respectively. The results verify the effectiveness and advantage of the newly proposed data driven approach and demonstrate the potential of the new approach in engineering applications.

Forced periodic operation via pulsing and feed switching to address methane dehydroaromatization catalyst deactivation challenge

Methane Dehydroaromatization is a one-step catalytic reaction that converts methane into aromatics and hydrogen with zero CO2 emission from the reaction stoichiometry. This work represents an engineering approach to address the rapid catalyst deactivation challenge by implementing forced periodic operation with shorter times at the scale of minutes. A systematic study was carried out by investigating pulse and periodic feed switching on a typical Mo/HZSM-5 catalyst. A fixed operating temperature and space velocity was maintained in both reaction and regeneration, while the operational time and type of regenerating molecules using reductive (H2) and oxidative (CO2) were investigated. For both modes of regeneration, CO2 was not an efficient oxidative molecule. Pulse feeding with hydrogen improved the catalyst stability, but it minimized the rate of aromatics production. The best performance was observed for the periodic switch with H2. Even a very short regenerating during 1 min H2 after 15 min reaction, was sufficient to achieve stable operation in the tested range. The global benzene production doubled compared to the base case which is a significant improvement reached by optimizing the mode of regeneration and conditions. A mathematical model was constructed and successfully used to simulate and explain the repeated reaction-regeneration cycles for different conditions. The result here is aiming to shed light on the opportunity of the forced periodic operation as an engineering strategy to mitigate the catalyst deactivation challenge and develop a reliable and stable MDA reactor operation.

Forced Periodic Operation of Methanol Synthesis in an Isothermal Gradientless Reactor

AbstractMethanol synthesis from synthesis gas with heterogeneous Cu/ZnO/Al2O3 catalysts in an isothermal gradientless reactor is described. In a theoretical study, the potential of forced periodic operation (FPO) for improving reactor performance in terms of methanol production rate and methanol yield is explored. The approach is based on a detailed kinetic model and combines nonlinear frequency response (NFR) analysis with rigorous numerical multi‐objective optimization. Optimal steady‐state operation is compared with optimal forced periodic operation for a given benchmark problem with and without inert nitrogen in the feed. Further, the significant influence of the saturation capacity of the solid phase on the dynamic behavior in response to step changes and periodic input modulations is studied.

Corrigendum to “Analysis and experimental demonstration of forced periodic operation of an adiabatic stirred tank reactor: Simultaneous modulation of inlet concentration and total flow-rate” [Chem. Eng. J. 410 (2021) 128197

Corrigendum to “Analysis and experimental demonstration of forced periodic operation of an adiabatic stirred tank reactor: Simultaneous modulation of inlet concentration and total flow-rate” [Chem. Eng. J. 410 (2021) 128197

Optimization of Methanol Synthesis under Forced Periodic Operation

Traditionally, methanol is produced in large amounts from synthesis gas with heterogeneous Cu/ZnO/Al2O3 catalysts under steady state conditions. In this paper, the potential of alternative forced periodic operation modes is studied using numerical optimization. The focus is a well-mixed isothermal reactor with two periodic inputs, namely, CO concentration in the feed and total feed flow rate. Exploiting a detailed kinetic model which also describes the dynamics of the catalyst, a sequential NLP optimization approach is applied to compare optimal steady state solutions with optimal periodic regimes. Periodic solutions are calculated using dynamic optimization with a periodicity constraint. The NLP optimization is embedded in a multi-objective optimization framework to optimize the process with respect to two objective functions and generate the corresponding Pareto fronts. The first objective is the methanol outlet flow rate. The second objective is the methanol yield based on the total carbon in the feed. Additional constraints arising from the complex methanol reaction and the practical limitations are introduced step by step. The results show that significant improvements for both objective functions are possible through periodic forcing of the two inputs considered here.

Analysis and experimental demonstration of forced periodic operation of an adiabatic stirred tank reactor: Simultaneous modulation of inlet concentration and total flow-rate

It is well known, that forced periodic operation possesses the potential for process improvements. Nevertheless, only a small number of applications is reported, due to complex realization, limited predictability and high inertia of larger units.Nonlinear frequency response (NFR) analysis has proven to predict efficiently time-averaged performance of reactor effluent streams originating from forced periodic changes of one or several input(s).Focus of this paper was an experimental demonstration of forced periodic operation applied to the hydrolysis of acetic anhydride carried out in an adiabatic CSTR. Theoretical results provided a guideline for experiments exploiting simultaneous sinusoidal modulations of the anhydride inlet concentration and the total volumetric flow-rate. Influences of the forcing parameters (amplitudes and the phase difference) were also studied. Confirming the predictions of NFR analysis a significantly higher time-averaged product yields were experimentally achieved compared to conventional steady-state operation with simultaneous modulation of two inputs using an optimized phase shift.

Introduction to Chemical Reactor Analysis

Nonlinear frequency response (NFR) method was used to evaluate the forced periodic operations and their potential for improving the system performance compared to steady-state operations. The NFR method was used to evaluate the time-average performances of forced periodically operated systems subjected to simultaneous periodic modulation of two-inputs of general waveforms. As an example, a forced periodically operated adiabatic CSTR with simple nth order reaction A → νPP and square-wave modulation of two pairs of inputs were analyzed in detail. The theory is illustrated analyzing the hydrolysis of acetic acid anhydride performed in a laboratory scale adiabatic CSTR. The results showed that significant improvements could be achieved by simultaneous modulation of both pairs of inputs. Comparison with the results of numerical integration showed that the approximation using only the asymmetrical second order FRFs and the first three harmonics of the inputs provide very reliable predictions of the DC component.

Evaluation of Electrochemical Process Improvement Using the Computer-Aided Nonlinear Frequency Response Method: Oxygen Reduction Reaction in Alkaline Media.

The intensification of an electrochemical process by forced periodic operation was studied for the first time using the computer-aided Nonlinear Frequency Response method. This method enabled the automatic generation of frequency response functions and the DC components (Faradaic rectification) of the cost (overpotential) and benefit (current density) indicators. The case study, oxygen reduction reaction, was investigated both experimentally and theoretically. The results of the cost–benefit indicator analysis show that forced periodic change of electrode potential can be superior when compared to the steady-state regime for specific operational parameters. When the electrode rotation rate is changed periodically, the process will always deteriorate as the dynamic operation will inevitably lead to the thickening of the diffusion layer. This phenomenon is explained both from a mathematical and a physical point of view.

Rapid Multi-Objective Optimization of Periodically Operated Processes Based on the Computer-Aided Nonlinear Frequency Response Method

The dynamic optimization of promising forced periodic processes has always been limited by time-consuming and expensive numerical calculations. The Nonlinear Frequency Response (NFR) method removes these limitations by providing excellent estimates of any process performance criteria of interest. Recently, the NFR method evolved to the computer-aided NFR method (cNFR) through a user-friendly software application for the automatic derivation of the functions necessary to estimate process improvement. By combining the cNFR method with standard multi-objective optimization (MOO) techniques, we developed a unique cNFR–MOO methodology for the optimization of periodic operations in the frequency domain. Since the objective functions are defined with entirely algebraic expressions, the dynamic optimization of forced periodic operations is extraordinarily fast. All optimization parameters, i.e., the steady-state point and the forcing parameters (frequency, amplitudes, and phase difference), are determined rapidly in one step. This gives the ability to find an optimal periodic operation around a sub-optimal steady-state point. The cNFR–MOO methodology was applied to two examples and is shown as an efficient and powerful tool for finding the best forced periodic operation. In both examples, the cNFR–MOO methodology gave conditions that could greatly enhance a process that is normally operated in a steady state.

Nonlinear frequency response analysis of forced periodic operations with simultaneous modulation of two general waveform inputs with applications on adiabatic CSTR with square-wave modulations

Nonlinear frequency response (NFR) method was used to evaluate the forced periodic operations and their potential for improving the system performance compared to steady-state operations. The NFR method was used to evaluate the time-average performances of forced periodically operated systems subjected to simultaneous periodic modulation of two-inputs of general waveforms. As an example, a forced periodically operated adiabatic CSTR with simple nth order reaction A → νPP and square-wave modulation of two pairs of inputs were analyzed in detail. The theory is illustrated analyzing the hydrolysis of acetic acid anhydride performed in a laboratory scale adiabatic CSTR. The results showed that significant improvements could be achieved by simultaneous modulation of both pairs of inputs. Comparison with the results of numerical integration showed that the approximation using only the asymmetrical second order FRFs and the first three harmonics of the inputs provide very reliable predictions of the DC component.

Optimization of forced periodic operations in milli-scale fixed bed reactor for Fischer-Tropsch synthesis

Abstract One-dimensional pseudo-homogenous dynamic reactor model, incorporating detailed Fischer-Tropsch kinetics, was applied in a theoretical analysis of forced periodic operations. A milli-scale fixed-bed reactor was analyzed, using design and operation parameters, obtained previously in a steady-state optimization. Dynamic optimization and NLP methods were utilized to obtain optimal values of amplitude(s), frequency and phase shift(s) of sine-wave variation of inputs, around the corresponding optimal steady-state values, which maximize the productivity of C5+ hydrocarbons. Inlet variables that were modulated are: coolant temperature, reactants molar ratio, mass flow rate and pressure. In addition to the single input forcing, simultaneous modulations of multiple inputs were also considered, with combinations of the listed inlet variables. Among the single input cases, periodic variation of the coolant temperature resulted in the highest relative improvement of C5+ productivity by 30%. Multiple inputs forcing showed additional potential for improvement, resulting in relative C5+ productivity increase of 52% for synchronized modulation of the coolant temperature, reactants molar ratio and mass flow rate. However, the increase in C5+ productivity is accompanied with relative increase in methane selectivity of 22–33% (relative to the steady-state value). The results suggest that, in the case of multiple input variations with high amplitudes, modulation of the inlet reactants molar ratio mainly contributes to the increase of CO conversion (e.g. reaction rate), the coolant temperature forcing slightly increases selectivity towards the desirable higher hydrocarbons (C5+), while the variation of the inlet mass flow rate enables better reaction temperature control and prevents a thermal runway.

Nonlinear frequency response analysis of forced periodic operation of non-isothermal CSTR with simultaneous modulation of inlet concentration and inlet temperature

The nonlinear frequency response (NFR) method is applied for evaluation of possible improvement through simultaneous periodic modulation of two inputs of a non-isothermal continuously stirred tank reactor (CSTR) in which homogeneous nth order reaction A→product(s) takes place. The two modulated inputs are the concentration of the reactant in the feed steam and the temperature of the feed stream. The cross asymmetrical second order FRF which correlates the outlet concentration with both modulated inputs is derived and analyzed. The optimal phase difference which should be used in order to maximize the conversion is determined. The method is tested on three numerical examples of non-isothermal CSTRs: (a) one which is oscillatory stable with strong resonant behavior, (b) one which is oscillatory stable with weak resonant behavior and (c) one which is nonoscillatory stable. Good agreement between the results of the approximate NFR method and the results of “exact” numerical integration is obtained except for the reactor with strong resonance for forcing frequencies which are close to the resonant frequency and for the reactor with weak resonant behavior for forcing frequency equal to the resonant one in case of high forcing amplitudes.

Nonlinear frequency response analysis of forced periodic operation of non-isothermal CSTR using single input modulations. Part II: Modulation of inlet temperature or temperature of the cooling/heating fluid

Periodic operation of a non-isothermal CSTR subject to a single input modulation is analyzed considering a nth order reaction and using the nonlinear frequency response (NFR) method, introduced in our previous publications. The method is based on deriving the asymmetrical second order frequency response function (FRF) and analyzing its sign. In Part I of this paper, the periodic modulations of inlet concentration or flow-rate of the reaction stream were investigated. In this Part II, periodic operations are analyzed for modulations of the temperature of the inlet reaction stream or the temperature of the heating/cooling fluid. Conditions that need to be fulfilled in order to achieve process improvement through periodic operation are identified. The method is applied for the same numerical example taken from literature which was used in Part I. The results obtained by the NFR method are compared with the results of numerical simulations. Good agreement is obtained, except for forcing frequencies close to the resonant frequency. In these cases, even the sign of the DC component was not predicted correctly.

Nonlinear frequency response analysis of forced periodic operation of non-isothermal CSTR using single input modulations. Part I: Modulation of inlet concentration or flow-rate

Periodic operations of a non-isothermal CSTR with n-th order reaction, subject to a single input modulation, is analysed using the nonlinear frequency response (NFR) method, introduced in our previous publications. The method is based on deriving the asymmetrical second order frequency response function (FRF) and analysing its sign. In Part I of this paper, periodic operation with modulation of the inlet concentration or flow-rate of the reaction stream is analysed. As a result, conditions regarding the reaction order, process parameters and frequency of the input modulation are identified that need to be fulfilled in order to achieve process improvement through the periodic operation compared to conventional steady state operation. The method is applied for a numerical example from literature and the results obtained by the NFR method are compared with the results of numerical simulation. Good agreement is obtained, except for imposed forcing frequencies close to the resonant frequency and high forcing amplitudes.

Evaluation of periodic processes with two modulated inputs based on nonlinear frequency response analysis. Case study: CSTR with modulation of the inlet concentration and flow-rate

In our previous work, a new, fast and easy, nonlinear frequency response method for analysing potential improvements of reactor performance by forced periodic operation was presented. This method, which is based on Volterra series, generalized Fourier transform and the concept of higher-order frequency response functions (FRFs), gives an approximate value of the average process performance directly, without numerical simulation of the complete process. It was shown that the asymmetrical second order frequency response function, (G2(ω,−ω)) corresponds to the dominant term of the non-periodic (DC) component of the periodic steady-state response and determines the average performance of the periodic process. Thus, in order to evaluate the potential of a periodic reactor operation, it is enough to derive and analyse this function.In this work this method is extended to evaluating periodic operations with forced oscillations of two modulated inputs. In this case the nonlinear system has to be defined by three sets of frequency response functions, two of them correlating the output to each of the inputs and one set of cross-FRFs. The general methodology for this case is developed. It is further used to analyse the time-average performance of an isothermal continuous stirred tank reactor (CSTR) with forced periodic modulation of the inlet concentration and flow-rate, for a simple nth order homogeneous reaction. The analysis is performed for cases when the inlet concentration and flow-rate are modulated simultaneously. The optimal choice of the phase shift between the two inputs is discussed in detail.

Periodic control of a reverse osmosis desalination process

This paper addresses the forced periodic operation of a tubular reverse osmosis process for improved performance. The investigation is carried out through simulation of a previously validated model for the RO process. The feed pressure and feed flow rate are transformed into periodic behavior in the form of sinusoidal functions. A nonlinear model predictive control algorithm is utilized to regulate the amplitude and period of the sinusoidal functions that formulate the input signal. The control system managed to generate the cyclic inputs necessary to enhance the closed-loop performance in terms of higher permeate production and lower salt concentration. The proposed control algorithm can attain its objective with and without defining a set point for the controlled outputs. In the latter case, the process is driven to the best achievable performance. Similar successful results in terms of improved production and quality of water were also obtained in the presence of modeling errors and external disturbances.

Evaluation of the potential of periodically operated reactors based on the second order frequency response function

A new, fast and easy method for analysing the potential for improving reactor performance by replacing steady state by forced periodic operation is presented. The method is based on Volterra series, generalized Fourier transform and the concept of higher-order frequency response functions (FRFs). The second order frequency response function, which corresponds to the dominant term of the non-periodic (DC) component, G 2( ω, − ω), is mainly responsible for the average performance of the periodically operated processes. Based on that, in order to evaluate the potential of periodic reactor operation, it is enough to derive and analyze G 2( ω, − ω). The sign of this function defines the sign of the DC component and reveals whether a performance improvement by cycling is possible compared to optimal steady state process. The method is used to analyze the periodic performance of a continuous stirred tank reactor (CSTR), a plug flow tubular reactor (PFTR) and a dispersive flow tubular reactor (DFTR), after introducing periodic changes of the input concentrations. A homogeneous, n-th order reaction is studied under isothermal conditions.

A critical appraisal of the [formula omitted]-criterion through continuation/optimization

Periodic forcing may improve bioreactor performances with respect to stationary operating conditions. An analytical procedure based on the so-called π -criterion was used by Parulekar [(1998). Analysis of forced periodic operations of continuous bioprocesses—single input variations. Chemical Engineering Science, 53(14), 2481–2502] to assess whether periodic forcing is beneficial, and, when this is the case, the criterion also gives an indication of the optimal forcing frequency. Such procedure is exact only in the limit of infinitesimal amplitude of the periodic forcing. In this work the applicability of the π -criterion to nonlinear models is investigated. The analysis is carried out by comparing the analytical predictions of the criterion with numerical predictions obtained with a continuation/optimization algorithm. As an example, two different reaction schemes are considered. The continuation/optimization procedure gives information on optimal forcing frequency and also on optimal forcing amplitude.

Unsteady-state operation of trickle-bed reactors

An experimental and theoretical study of forced unsteady-state operation of trickle-bed reactors in comparison to the steady-state operation is the subject of this paper. It is well known that changes in the control variables influence the regime and the performance of trickle-bed reactors. In this study as one example for a forced periodic operation of a trickle-bed reactor an unsteady-state technique was used in which the catalyst bed is contacted periodically with different liquid flow rates. The unsteady-state operation was considered as square-waves cycling liquid flow rate at the reactor inlet. The hydrogenation of alpha-methylstyrene to cumene: C 6 H 5 (CH 3 )=CH 2 ( L ) + H 2 ( G ) → C 6 H 5 CH(CH 3 ) 2 ( L ) over a Palladium-catalyst (0.7% Pd/ γ -Al 2 O 3 ) was selected as a model reaction. The experimental results of the alpha-methylstyrene hydrogenation in a laboratory-scale trickle-bed reactor showed, that the reactor performance can be significantly improved by feed liquid flow modulation. The simulation studies demonstrate that the liquid flow variation has a strong influence on the liquid hold-up oscillation and on the catalyst wetting efficiency. Consequently the time average alpha-methylstyrene conversion will be increased, because the mass transfer resistance between the phases (gas–liquid, gas–solid and liquid–solid) affect the overall reaction rate and consequently the conversion will be improved.

Understanding performance improvement of forced periodic operation of stirred tank reactors

It is well-known that the performance of chemical reaction processes can be improved by periodic forcing. The π-criterion has been introduced and applied extensively to quantitatively assess the potential of periodic forcing in a particular case, though it is difficult to understand the physical reasons for the performance improvement. A theoretical analysis in the frequency domain and a study of typical model reaction systems in stirred tanks including parallel, consecutive and reversible reactions is presented here. It is shown that the performance improvement can clearly be related to resonance phenomena displayed by the linearized dynamics in conjunction with a nonlinear variation of the process gain in the region of forcing. The theoretical findings are illustrated by a number of examples.