Nonlinear Frequency Response Method Research Articles

Process Intensification in Photocatalytic Decomposition of Formic Acid over a TiO2 Catalyst by Forced Periodic Modulation of Concentration, Temperature, Flowrate and Light Intensity

The effect of forced periodic modulation of several input parameters on the rate of photocatalytic decomposition of formic acid over a TiO2 thin film catalyst has been investigated in a continuously stirred tank reactor. The kinetic model was adopted based on the literature and it includes acid adsorption, desorption steps, the formation of photocatalytic active sites and decomposition of the adsorbed species over the active titania sites. A reactor model was developed that describes mass balances of reactive species. The analysis of the reactor was performed with a computer-aided nonlinear frequency response method. Initially, the effect of amplitude and frequency of four input parameters (flowrate, acid concentration, temperature and light intensity) were studied. All single inputs provided only a minor improvement, which did not exceed 4%. However, a modulation of two input parameters, inlet flowrate and the acid molar fraction, considerably improved the acid conversion from 80 to 96%. This is equivalent to a factor of two increase in residence time at steady-state operation at the same temperature and acid concentration.

Forced periodic operations of a chemical reactor for methanol synthesis – The search for the best scenario based on Nonlinear Frequency Response Method. Part II Simultaneous modulation of two inputs

The analysis of the potential to improve performance of a methanol synthesis reactor through forced periodical operations by Nonlinear Frequency Response method is presented. The methanol synthesis in an isothermal and isobaric lab-scale CSTR is considered. First, the analysis was performed for single input modulations (in Part I), which showed that significant improvements can’t be achieved. Here, the study is extended to analysis of simultaneous modulations of two inputs. All possible input combinations (6 cases) are analysed and the optimal forcing parameters, maximizing the time-average methanol production, were determined. For all combinations the improvement is possible, but for some cases it is not significant. The highest improvement is predicted for simultaneous modulation of the inlet partial pressure of CO and the inlet volumetric flow rate. This case, for which it is possible to achieve up to 33.51 % of methanol production, is analysed it detail and optimized using multi-objective optimization.

Forced periodic operations of a chemical reactor for methanol synthesis – The search for the best scenario based on Nonlinear Frequency Response Method. Part I Single input modulations

In this two-part paper a comprehensive study of the potential to improve performance criteria of a methanol synthesis reactor through forced periodical operations is presented. The study uses the Nonlinear Frequency Responsemethod, a powerful analytical and approximate tool which gives an answer whether and under which conditions certain periodic operation would lead to improvement of process performance. To demonstratethe method, isothermal and isobaric methanol synthesis in a lab-scale CSTR is considered. In Part I, the analysis is performed for single input modulations. Partial pressures of each reactantin the feed stream and the total inlet volumetric flow-rate are considered as possible modulated inputs. The results show that modulations of single inputs essentially do not provide potential for significant improvements. In Part II, the study will be extended to analysis of periodic operations with simultaneous modulations of two inputs and conditions offering significant performance enhancements will be identified.

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.

Computer-Aided Nonlinear Frequency Response Method for Investigating the Dynamics of Chemical Engineering Systems

The Nonlinear Frequency Response (NFR) method is a useful Process Systems Engineering tool for developing experimental techniques and periodic processes that exploit the system nonlinearity. The basic and most time-consuming step of the NFR method is the derivation of frequency response functions (FRFs). The computer-aided Nonlinear Frequency Response (cNFR) method, presented in this work, uses a software application for automatic derivation of the FRFs, thus making the NFR analysis much simpler, even for systems with complex dynamics. The cNFR application uses an Excel user-friendly interface for defining the model equations and variables, and MATLAB code which performs analytical derivations. As a result, the cNFR application generates MATLAB files containing the derived FRFs in a symbolic and algebraic vector form. In this paper, the software is explained in detail and illustrated through: (1) analysis of periodic operation of an isothermal continuous stirred-tank reactor with a simple reaction mechanism, and (2) experimental identification of electrochemical oxygen reduction reaction.

Forced periodic reactor operation: Analysis of process and forcing parameters exploiting the nonlinear frequency response method

Special Issue: 10. ProcessNet‐Jahrestagung und 34. DECHEMA‐Jahrestagung der Biotechnologen 2020: Processes for Future

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.

Nonlinear Frequency Response Method for Evaluating Forced Periodic Operations of Chemical Reactors

AbstractNonlinear Frequency Response (NFR) method is a relatively new method which can be used for fast evaluation of possible process improvements through periodic operations. The method is analytical and approximate. Its main task is to give an answer whether periodic modulation of one or more process inputs can result in improved process performance. The method is explained in brief and an overview of the existing applications is given. The review covers simple reactions performed in isothermal and non‐isothermal stirred tank reactors exposed to different modulated inputs (inlet concentration, flow‐rate, inlet temperature, temperature of the heating/cooling medium). Processes with two simultaneously modulated inputs and different shapes of the periodic input are also considered. The results of the NFR method are compared with the results of numerical simulation and a critical evaluation of the method is given.

CO2 Conversion Enhancement in a Periodically Operated Sabatier Reactor: Nonlinear Frequency Response Analysis and Simulation‐based Study

AbstractConversion of CO2 into synthetic CH4 via thermocatalytic hydrogenation (the Sabatier reaction), has recently gained increasing interest as a possible route for CO2 utilization and energy storage pathway. Herein, we analyze the possibility of increasing the CO2 conversion through periodic operation of the reactor. The analysis is performed by using the Nonlinear Frequency Response (NFR) method, a recently developed analytical technique, suitable for fast evaluation of periodic reactor operations. The NFR analysis predicts a significant conversion gain (up to 50 %) for certain frequencies of the feed flow rate modulation. This prediction is validated by numerical simulations with a reaction rate expression obtained by CO2 conversion experiments using a Ni/Al2O3 catalysts. Both the NFR analysis and numerical simulations predict that it is possible to obtain 70 % CO2 conversion at 500 K, 5 bar, and average space velocity of 7600 h−1 by a periodic modulation of the feed flow rate, as compared to the corresponding steady state CO2 conversion of 43 %.

Periodic Operation with Modulation of Inlet Concentration and Flow Rate Part II: Adiabatic Continuous Stirred‐Tank Reactor

AbstractThe nonlinear frequency response (NFR) method, which is an analytical, fast, and easy method for evaluating the performance of forced periodically operated chemical reactors, was used to investigate possible improvements to a nonisothermal continuous stirred‐tank reactor (CSTR) when inlet concentration and/or flow rate are periodically modulated. The product yield was used for evaluating the performance improvements. Part I of this paper considered the general nonisothermal case. In this part, the results are applied to an adiabatic CSTR. A laboratory‐scale adiabatic CSTR was analyzed by applying kinetic parameters for the hydrolysis of acetic anhydride. It is shown that improvement can be obtained for simultaneous modulation of the two inputs with appropriately chosen forcing parameters.

Periodic Operation with Modulation of Inlet Concentration and Flow Rate. Part I: Nonisothermal Continuous Stirred‐Tank Reactor

AbstractThe nonlinear frequency response (NFR) method, which is an analytical, fast, and easy method for evaluating the performance of forced periodically operated chemical reactors, was used to investigate possible improvements to a nonisothermal continuous stirred tank reactor (CSTR) when inlet concentration and/or flow rate is periodically modulated. The product yield corresponding to periodic operation is defined, expressions for its estimation, based on the NFR method, are derived, and it is used to evaluate the performance improvements due to periodic operation. Part I considers the general nonisothermal case. In Part II, these results are applied to an adiabatic CSTR and used to evaluate possible improvements for the case of the hydrolysis reaction of acetic anhydride.

Evaluation of Performance of Periodically Operated Reactors for Single Input Modulations of General Waveforms

AbstractThe nonlinear frequency response (NFR) method is used for evaluating the time‐average performance of a chemical reactor subjected to single input modulations of general waveforms, by using Fourier series for representing the input and Volterra series for representing the output. Both the input and the output are approximated by finite sums. The obtained results are applied for the case of a square‐wave input modulation. As a case study, the improvement of an isothermal continuous stirred‐tank reactor with simple reaction mechanism with modulation of the inlet reactant concentration is used and the results are tested on a numerical example.

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 measurements of gas adsorption equilibrium and kinetics: New apparatus and experimental verification

A new apparatus for experimental validation of the Nonlinear Frequency Response (NFR) method for investigation of gas adsorption is presented. The apparatus has been designed as a batch adsorber with sinusoidal modulation of the volume, with an amplitude that can be varied. Fast and accurate measurements of pressure, gas temperature and particle temperature responses are employed. The measurement routine and data analysis are demonstrated by CO2 adsorption on zeolite 5A particles of two different diameters. The responses for input amplitudes of 3% and 7% of the volume change are measured. The concept of blank FRFs is introduced as a new way to incorporate the results of blank experiments for elimination of spurious effects. The first- and second-order FRFs which relate the adsorbed quantity to the pressure (F1(ω) and F2(ω,ω)) are calculated based on the measured pressure responses. The first-order FRF which relates the particle temperature to the pressure (H1(ω)) is calculated from the particle temperature responses. The characteristics of F1(ω), F2(ω,ω) and H1(ω) for spherical beads of 1.7mm agree with the corresponding theoretical ones for nonisothermal macropore diffusion model. The equilibrium and kinetic parameters are estimated from these functions. The results of this study confirm the applicability of NFR method for characterization of gas adsorption and show significance of the extension of analysis to the nonlinear range.

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.

A Study of Applicability of Nonlinear Frequency Response Method for Investigation of Gas Adsorption Based on Numerical Experiments

A study of the applicability of the Nonlinear Frequency Response (NFR) approach for investigation of equilibrium and kinetics of gas adsorption, based on numerical simulations, is presented. Pressure responses of a batch adsorber are simulated, using literature data, for a sinusoidal volume change with different input amplitudes, up to 10%, around several steady-state points. The procedure for application of the NFR approach is demonstrated step-by-step, using the simulated pressure responses. As a result, quasi-experimental first- and second-order FRFs are estimated and used for estimation of the equilibrium and kinetic parameters of the adsorption system. The estimated parameters agree well with the original ones used for simulations. Influence of noise on the accuracy of the estimated FRFs and model parameters is also analyzed. The results of this study show that the application of the NFR approach for investigation of equilibrium and kinetics of gas adsorption in a batch system is feasible. Furthermore, valuable information regarding the design of the NFR equipment and experiments is gained.

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.