Optimal control of diffusion-convection-reaction processes using reduced-order models

Two approaches for optimal control of diffusion-convection-reaction processes based on reduced-order models are presented. The approaches differ in the way spatial discretization is carried out to compute a reduced-order model suitable for controller design. In the first approach, the partial differential equation (PDE) that describes the process is first discretized in space and time using the finite difference method to derive a large number of recursive algebraic equations, which are written in the form of a discrete-time state-space model with sparse state, input and output matrices. Snapshots based on this high-dimensional state-space model are generated to calculate empirical eigenfunctions using proper orthogonal decomposition. The Galerkin projection with the computed empirical eigenfunctions as basis functions is then directly applied to the high-dimensional state-space model to derive a reduced-order model. In the second approach, a continuous-time finite-dimensional state-space model is constructed directly from the PDE through application of orthogonal collocation on finite elements in the spatial domain. The dimension of the derived state-space model can be further reduced using standard model reduction techniques. In both cases, optimal controllers are designed based on the low-order state-space models using discrete-time and continuous-time linear quadratic regulator (LQR) techniques. The effectiveness of the proposed methods are illustrated through applications to a diffusion-convection process and a diffusion-convection-reaction process.     

Rate‐based modelling of reactive absorption of acid gases in an aqueous methyldiethanolamine (MDEA) solution

In this work different tools for accurate prediction of acid gas absorption are used. At first, simulation of reactive absorption is carried out using the RATEFRAC module of Aspen Plus, which is tested against pilot plant data. The limitations and disadvantages of this module are presented. In order to present a more predictive approach a rate‐based model for the gas scrubbing process is developed. In this model the assumption of thermodynamic equilibrium is considered only at the gas–liquid interphase. Chemical equilibrium among the reacting species in the liquid phase is assumed just for the bulk phase. Mass transfer is modelled using mass transfer coefficients calculated from available correlations which are then improved using an enhancement factor to account for the chemical reactions. The validity of the suggested model is established by comparison of model results with published pilot plant data. The prediction results using the proposed model are improved by around 17% AAD for CO₂ and around 7.5% AAD for H₂S compared to simulation results using Aspen Plus.     

基于MPCA-MDPLS的间歇过程的故障诊断

针对间歇过程的故障诊断问题,提出了一种新的混合模型方法——MPCA-MDPLS.这种方法包括两个模型：多向主元分析(MPCA)模型和多向判别部分最小二乘(MDPLS)模型.这两个模型的建模数据不仅包括正常工况的数据,而且还包含了各种已知故障数据.因此,MPCA模型具有检测未知故障的能力.给出了MDPLS模型故障诊断限,对经MPCA模型检测不是未知故障的故障做进一步诊断.如果故障是未知的,可以采取其他的方法来分析新的故障,并按不同类别存入到数据库中.当多次出现这种故障之后（一般≥5次）,把新的故障数据加入到建模数据中,并重新建立MPCA-MDPLS模型.通过对实际工业链霉素发酵过程数据的分析,表明了提出的算法是可行的、有效的,并具有识别未知新故障的能力.     

Presumed joint probability density function model for turbulent combustion

A presumed joint probability density function (pdf) model of turbulent combustion is proposed in this paper. The turbulent fluctuations of reactant concentrations and temperature are described using a presumed joint pdf of three-dimensional Gaussian distribution based on first and second-order moments of reactant concentration and temperature. Mean reaction rates in both premixed and diffusion combustion are obtained by mean of integration under the presumed joint pdf. This model is applied to predict turbulent premixed combustion of sudden-expansion flow and turbulent jet diffusion methane/air flame. For turbulent premixed combustion, the predicted results of temperature distribution and maximum temperature using the proposed model agree better with the experiment than that using the conventional eddy-breakup (EBU)-Arrhenius model. For the turbulent jet diffusion methane/air flame, the predicted results of velocity, temperature and species concentrations using the proposed model, the Arrhenius, EBU-Arrhenius, and laminar flamelet models are compared with experiment data. Results obtained with the presumed pdf model and that obtained by the laminar flamelet model both agree well with experiments, while results using the other models have a significant difference. The presumed joint pdf model is used to predict the NO formation process, which also agrees well with the experiment data. A unified turbulent combustion model, in which both effects of turbulent diffusion and chemical dynamics are considered, is established for both premixed and diffusion combustion, especially for the process of NO formation.     

Development of a Dynamic Gray-Box Model of a Fermentation Process for Spore Production

Fermentation processes are difficult to describe using purely mechanistic relations as the underlying biochemical phenomena are complex and often not fully understood. In order to cope with this challenge, we developed an approach to augment standard dynamic model equations by data-based components that are fitted to data using machine learning techniques, which results in dynamic gray-box models. This methodology is applied here to the batch fermentation process of the sporulating bacterium Bacillus subtilis, using experimental data from a lab-scale fermenter. The key step in developing the model is the estimation of a training set for the machine learning submodels. The quality of the resulting model is analyzed, and the predictions are compared with real data.     

ROBUST NONLINEAR PREDICTIVE CONTROL USING A DISTURBANCE ESTIMATOR

A robust nonlinear predictive control strategy using a disturbance estimator is presented. The disturbance estimator is comprised of two parts: one is the disturbance model parameter adaptation and the other is future disturbance prediction. A linear discrete model is proposed as a disturbance model which is formulated by using process inputs and available process measurements. The recursive least square (RLS) method with exponential forgetting is used to determine the uncertain disturbance model parameters and for the future disturbance prediction, future disturbances projected by the future process inputs are used. Two illustrative examples: a jacketed CSTR as a SISO system: an adiabatic CSTR as a MIMO system, and experimental results of the distillation column control are presented. The results indicate that a substantial improvement in nonlinear predictive control performance is possible using the disturbance estimator.     

A USM turbulence-chemistry model for simulating NOx formation in turbulent combustion

A unified second-order moment (USM) turbulence-chemistry model for simulating NO_x formation in turbulent combustion is proposed. All the correlations, including the correlation of the reaction-rate coefficient fluctuation with the concentration fluctuation, are closed by the transport equations in the same form. This model abandons the series expansion approximation of the exponential term or the approximation of using a product of two single-variable PDFs instead of a joint PDF. The proposed model is used to simulate methane-air jet diffusion combustion and NO_x formation. The combustion prediction results are compared with those using the EBU-Arrhenius model and other two versions of the second-order moment model. The NO_x prediction results are compared with those using the pure presumed PDF model. Validation of predictions using the experimental data given by the Sandia National Laboratory, USA indicates that the proposed model gives better results than other models, and it is much economical than other refined models.     

An application of a unified relaxation model to the aging of polystyrene below its glass temperature

Torsional creep measurements of physical aging in high-molecular-weight glassy polystyrene are presented. The data are analyzed using a relaxation model which has been successful in quantitatively describing a wide range of relaxation phenomena. The model predicts the time-functional dependences of the creep compliance and, upon physical aging, a possible systematic change of the functional dependence as creep is shifted to longer time scales. These features are established by experimental measurements. Good quantitative fits are found using a restricted application of the model in which all but one of the model parameters are held constant with annealing. Relations of the present study to other polymer relaxation phenomena are discussed.     

A structural model for the catalytic polymerization of ethylene using chromium catalysts. Part I: Description and solution

A mathematical model for the polymerization of ethylene using silica-supported chromium catalysts is presented. The fundamentals for the physical and chemical representation of the phenomenon are detailed. The emerging mathematical problem is attached and solved using a multiple moving-boundary numerical scheme. Predictions are made using the model for the cases of low and high activity catalysts tested by experiments, and the results used to evaluate the model. The main feature of the model is its structural nature: catalyst characterization and morphology data are used to create a physical scheme that does not require the assumptions concerning particle geometry, usually made in models. The initial catalyst fragmentation process is modeled using a pore-structure scheme based upon porosimetry data. The whole fragmentation sequence is treated as a set of steps that follow the actual rupture process of the initial solid into fragments with irregular shapes. Parameters are evaluated from data for the initial support, and time evolution is followed using a realistic interpretation of the pore filling process. A novel application of moving-boundary solution techniques permits a relatively simple mathematical scheme to handle the transport and reaction processes that take place in the polymerizing particle. Only the results obtained for the experimental laboratory conditions are analyzed. In subsequent parts, the case of thermal effects typical of industrial conditions will be discussed. The results obtained are in excellent agreement with the experimental data.     

Pervaporation performance analysis and prediction—using a hybrid solution-diffusion and pore-flow model

A hybrid mathematic model for pervaporation is proposed which incorporates the concepts of solution-diffusion model and pore model. The model allows performance prediction as well as the establishment of the internal concentration and pressure profiles within the membrane. The model parameters specific to the particular membrane and mixture system are determined using liquid sorption and pervaporation experimental data. The model is experimentally examined using ethanol–water mixtures and poly(dimethyl siloxane)–poly(vinyldiene fluoride) (PDMS–PVDF) composite membranes. The characteristics of flux and separation factor predicted using the model are in fair agreement with the experimental data under various feed concentrations and downstream pressures for different membrane arrangements, including single-layer, reverse single-layer and double-layer PDMS–PVDF composite membranes. Internal profiles of pressure, concentration and component mole fraction can be established using the model. Concentration polarization phenomena for ethanol and water are located at membrane interfaces and vapor–liquid interfaces, respectively. Performances of several different membrane designs are compared using the model.     

Diagnostics for Time Series Analysis

Test statistics are proposed to determine the goodness of fit of a time series model. The test statistics are based on a sequence of random variables that are independent and standard normal if the model is correct. The paper shows how to compute this sequence of random variables efficiently using a combination of Markov chain Monte Carlo and importance sampling. The power of the statistics to detect outliers and level shifts is studied for an autoregressive model. The methodology is illustrated using both simulated and real data.     

A DYNAMIC MODEL WITH ON-LINE IDENTIFICATION FOR ROTARY SUGAR DRYING PROCESSES

A dynamic modelling methodology, which combines on-line variable estimation and parameter identification with physical laws to form an adaptive model for rotary sugar drying processes, is developed in this paper. In contrast to the conventional rate-based models using empirical transfer coefficients, the heat and mass transfer rates are estimated by using on-line measurements in the new model. Furthermore, a set of improved sectional solid transport equations with localized parameters is developed in this work to replace the global correlation for the computation of solid retention time. Since a number of key model variables and parameters are identified on-line using measurement data, the model is able to closely track the dynamic behaviour of rotary drying processes within a broad range of operational conditions. This adaptive model is validated against experimental data obtained from a pilot-scale rotary sugar dryer. The proposed modelling methodology can be easily incorporated into nonlinear model based control schemes to form a unified modelling and control framework.     

A MILP model for N-dimensional allocation

This paper presents a Mixed Integer Linear Programming (MILP) model for the solution of N-dimensional allocation problems. The applicability of the model is presented and demonstrated through some illustrative examples with different numbers of dimensions. Several problems, previously presented in the literature, are solved using the proposed model, such as, one-dimensional scheduling problems, two-dimensional cutting problems, as well as plant layout problems and three-dimensional packing problems. Additionally, some problems in four dimensions are presented and solved using the considered model. The presented model is applicable to a wide variety of allocation problems as it offers a general framework for modelling allocation problems with any given number of continuous or discrete dimensions. The presented problems are formulated as MILP problems where the first four dimensions usually are continuous spatial and time dimensions. Additional dimensions are often of a discrete nature.     

A DYNAMIC MODEL WITH ON-LINE IDENTIFICATION FOR ROTARY SUGAR DRYING PROCESSES

A dynamic modelling methodology, which combines on-line variable estimation and parameter identification with physical laws to form an adaptive model for rotary sugar drying processes, is developed in this paper. In contrast to the conventional rate-based models using empirical transfer coefficients, the heat and mass transfer rates are estimated by using on-line measurements in the new model. Furthermore, a set of improved sectional solid transport equations with localized parameters is developed in this work to replace the global correlation for the computation of solid retention time. Since a number of key model variables and parameters are identified on-line using measurement data, the model is able to closely track the dynamic behaviour of rotary drying processes within a broad range of operational conditions. This adaptive model is validated against experimental data obtained from a pilot-scale rotary sugar dryer. The proposed modelling methodology can be easily incorporated into nonlinear model based control schemes to form a unified modelling and control framework.     

Nonlinear mechanical response of high density polyethylene. Part II: Uniaxial constitutive modeling

Based on the experimental data presented in Part I, two uniaxial constitutive models are constructed. The first, a nonlinear viscoelastic (NVE) model, is formulated using the mechanical analogy consisting of one independent spring and six Kelvin elements in series. Creep data are used to determine the model parameters. The second model, a viscoplastic (VP) formulation, is developed using the viscoplastic theory proposed by Bodner to characterize the uniaxial viscoplastic behavior of metals. Inelastic strain rate is introduced into the state variable in addition to inelastic work to depict the strong rate dependent behavior of HDPE. Experimental data from constant strain rate tests are employed to construct the material functions of the model. Limitations in the application of each model are discussed in conjunction with possibilities for future work.     

Experimental validation of CFD models for fluidized beds: Influence of particle stress models, gas phase compressibility and air inflow models

This work compares numerical simulations of fluid dynamics in fluidized beds using different closure models and air feed system models. The numerical results are compared to experiments by means of power spectral density distributions of fluctuating pressure signals and bubble statistics obtained from capacitance probe measurements. Two different particle rheology models are tested in combination with two different values of the maximum particle volume fraction. The first particle model predicts the particle pressure by an exponential power law and assumes a constant particle viscosity (CPV), and the second model predicts the stresses using the kinetic theory of granular flow (KTGF). Furthermore, two model approaches for the air inflow are evaluated. The first inflow model includes the coupling between the air-feed system and the fluidized bed in the simulation, and the second model assumes a constant mass flow of gas into the fluidized bed. Finally, the influence of the compressibility of the gas phase on the numerical predictions is investigated. The numerical simulations are made using the CFX-4.4 commercial flow solver.The simulations show that the KTGF model gives a more evenly distributed bubble flow profile over the bed cross-section, while the CPV model gives a more parabolic bubble flow profile, with a higher bubble flow in the central part of the bed. This work shows that the KTGF model results are in significantly better agreement with the experiments. It is furthermore shown that the modelling of the air-feed system is crucial to for predicting the overall bed dynamic behaviour.     

Measurement of particle size and shape by FBRM and in situ microscopy

In this work a model is defined allowing for a rapid calculation of chord length distributions as well as the prediction of in situ microscopy data. Both calculations are done using the same underlying algorithm. The model assumes convex polyhedral particles that are defined by their vertices only, connected by straight lines, but imposes no further restrictions on particle geometry. Due to its speed, the model can easily be used for the prediction of experimental data from in situ monitoring tools based on whole particle populations, also with non-constant shape. The model has been verified using in situ microscopy to characterize a population of disc shaped particles.The applications of the model are focused on crystallization processes, but are not limited to these. Several relations between data measured by in situ instruments and the underlying multidimensional particle size distribution have been derived. The model is used extensively in a method that is presented allowing for the calculation of bidimensional growth rates from Focused Beam Reflectance Measurement or in situ microscopy measurements.     

MASS TRANSFER WITH CHEMICAL REACTION IN A FROTH BED REACTOR

A model to predict conversions in a froth bed reactor has been developed. The model is then compared against the available experimental data on the oxidation of sodium sulfide in a foam bed contactor. The predictions using the present model are also compared against those based on the model of a foam bed reactor developed earlier. The predictions using the present model agree fairly well with the experimental data and, in some cases, are even in better quantitative agreement than the previous single stage model of a foam bed reactor. The case of significant surface resistance due to surfactant has also been analyzed theoretically, obtaining analytical solutions for the concentration profile and fractional gas absorption in a liquid froth shell.     

Kinetic parameter estimation in hydrocracking using hybrid particle swarm optimization

This paper provides a continuous lumping model for hydrocracking using molecular weight and true boiling point as the basis. Isomerization and cracking are the two typical global reactions which occur on acid sites of a bifunctional acid-metal hydrocracking catalyst. The latter reaction is considered to model the hydrocracking process. It is assumed that the rate coefficient of the cracking reactions depend on the size of the feed hydrocarbons. The kinetic parameters involved in the continuous lumping model are estimated by using a hybrid particle swarm optimization method. The proposed kinetic methodology is validated with the experimental data cited in the literature.