Studies in the control of tubular reactors—III stabilization by observer design

The stabilization of an unstable nonlinear distributed chemical reactor system is examined when concentration measurements are not possible. The linearized form of the finite dimensional approximate model developed in Parts I[1] and II[2] is used to show that the observability index is equal to two. Furthermore it is shown that the dynamical characteristics of the reactor are such that, by a proper design of the Luenberger observer, the concentration estimation at a given collocation point can be made independently of the estimation at other collocation points.For purposes of control a one-dimensional observer is designed to directly estimate the control variable. Simulations of the nonlinear model of the reactor show that the observer design is quite successful in the stabilization of an unstable steady state when only temperature measurements along the reactor are available.     

A STUDY OF NON-NEWTONIAN FLOW AND HEAT TRANSFER OVER A NON-ISOTHERMAL WEDGE USING THE HOMOTOPY ANALYSIS METHOD

The thermoconvective boundary layer flow of a generalized third-grade viscoelastic power-law non-Newtonian fluid over a porous wedge is studied theoretically. The free stream velocity, the surface temperature variations, and the injection velocity at the surface are assumed variables. A similarity transformation is applied to reduce the governing partial differential equations for mass, momentum, and energy conservation to dimensionless, nonlinear, coupled, ordinary differential equations. The homotopy analysis method (HAM) is employed to generate approximate analytical solutions for the transformed nonlinear equations under the prescribed boundary conditions. The HAM solutions, in comparison with numerical solutions (fourth-order Runge-Kutta shooting quadrature), admit excellent accuracy. The residual errors for dimensionless velocity and dimensionless temperature are also computed. The influence of the “power-law” index on flow characteristics is also studied. The mathematical model finds important applications in polymeric processing and biotechnological manufacture. HAM holds significant promise as an analytical tool for chemical engineering fluid dynamics researchers, providing a robust benchmark for conventional numerical methods.     

Dynamic recurrent radial basis function network model predictive control of unstable nonlinear processes

A multistep model predictive control (MPC) strategy based on dynamically recurrent radial basis function networks (RBFNs) is proposed for single-input single-output (SISO) control of uncertain nonlinear processes. The control system consists of two automatically configured RBFNs, a trained network representing the plant model and a network with on-line learning to function as controller. The automatic configuration and learning of the networks is carried out by using a hierarchically self-organizing learning algorithm. This control strategy is structurally simple and computationally efficient since a single output node of each RBFN is configured to provide multistep predictions for plant output and controller. The performance of the proposed RBFNMPC strategy is evaluated by applying to two unstable nonlinear chemical processes, a chemical reactor and a biochemical reactor, and also a stable polymerization reactor. Further, the results of the RBFNMPC is compared with similar RBFN model based control strategies and also with well tuned PID/PI controller. The results show the better performance of the proposed RBFNMPC for the control of open-loop unstable nonlinear processes that exhibit multiple steady-state behavior.     

Selectivity analysis in electrochemical reactors. II. Engineering models of a batch reactor with a complex reaction sequence

This paper presents a mathematical model of a batch stirred-tank electrochemical reactor where a required cathodic reduction reaction is coupled with a complex reaction sequence between the reactant and the key product. The set of coupled, non-linear differential equations is solved numerically and simple dimensionless quantities characterizing the cell performance and selectivity are derived. The experimental results presented in Part I of this paper are found to be in excellent agreement with the model. In the particular case where the homogeneous chemical reactions may be neglected in the cathodic diffusion boundary layer, a simplified analytical expression of the process selectivity is proposed. This quantifies the effects of the operating conditions by means of a single dimensionless criterion.     

Adaptive nonstationary operation of reactor systems for catalytic combustion

NO_x removal from automobile exhaust by selective catalytic reduction is considered as an example of a system with competing chemical reactions and selectivity strongly dependent on temperature. In thermally coupled monolithic reactors, temperature profiles can be successfully kept within the required range by periodic operation — switching between a hot stream containing reactants and an inert cooling stream. This work presents a methodology for determining optimum (i.e. minimum emissions) operating parameters of a system of monolithic reactors in situations where the inlet gas properties vary in time. An adaptive control strategy is employed, based on predicting the inlet properties by a feed-forward artificial neural network, and then using a set of heuristic rules combined with nonlinear optimization to select the most favorable reactor configuration. This adaptive control scheme leads to significant improvements (in terms of integral emissions of pollutants), compared with a situation where system configuration and operating parameters are fixed a priori.     

径向流动反应器流体力学特性研究

本文提出了表征径向流动反应器内流体流动规律的数学模型.该模型为一特殊的二阶微分方程的边值问题.对于Ⅱ型流动:Au′u″+Bu′u+Cu~2=0对于Z型流动:Au′u″+Bu′u+Cu′+Eu~2+F(1-u)~2=0边界条件为y=0,u=1;y=1,u=0式中模型参数A、B、C、E、F决定于径向反应器分布流道结构尺寸、穿孔阻力系数和径向床层特性等因素.为简化Ⅱ型径向反应器边值问题的求解过程,本文推荐一种解析求解方法.该模型应用于径向氨氧化炉操作工况和径向氨合成塔设计工况的模拟,揭示了径向流动反应器内的流体力学特性.     

A numerical investigation of aerosol dynamics in a wall-less reactor

This paper describes a numerical investigation of aerosol formation during silane decomposition in a wall-less reactor. The wall-less reactor is amenable to numerical investigation because the homogeneous chemical reactions leading to the formation of solid particles are isolated from heterogeneous effects, such as occur at the walls of a laminar flow aerosol reactor. The flow/heat transfer and gas-phase chemical kinetics are simulated utilizing separate one-way coupled models. The aerosol dynamics model is based on a simplified sectional model originally developed by Okuyama et al. This model is modified to allow for the simulation of particle growth via condensation. Simulations have been performed which indicate that particle growth via condensation may be an important process. Additionally, the effects of total reactor pressure, temperature and inlet silane concentration on the dynamics of the aerosol population have been investigated. Conditions which result in the formation of larger and more numerous particles have been identified.     

On Danckwerts’ Boundary Conditions for the Plug-Flow with Dispersion/Reaction Model

Accurate predictions of oxygen utilization with position in plug-flow like (aka serpentine) reactors employed for aerobic activated-sludge treatment of municipal and industrial wastewaters are desired. Three non-ideal reactor models—completely-mixed flow reactors (CMFRs) in series, segregated flow, and plug-flow with dispersion (PFD)—and the ideal plug-flow model were investigated considering typical reaction kinetics and a typical reactor configuration to perform such predictions. Significant differences arose between predictions from the PFD model relative to those from the CMFR in series and segregated flow models. These differences led to re-examination of the PFD model and the inlet/outlet boundary conditions postulated more than six decades ago by P. V. Danckwerts. Major criticisms are found in the literature addressing the PFD model arising from the concentration “jump” at the inlet boundary necessitated by specification of conservation of reactant flux at the expense of continuity of target reactant concentration. A fresh theoretical examination of the dispersive/reactive processes within a closed reactor with particular attention paid to the inlet and outlet boundary planes led to alternative statements for the inlet and outlet boundary conditions. Revisions proposed herein to Danckwerts’ boundary conditions for the PFD/reaction model preserve both continuity of reactant concentration and conservation of reactant flux at both inlet and outlet boundaries. Hypothetical computations based on pseudo-first-order kinetics for a reaction carried out within a serpentine reactor of typical configuration illustrate the differences between concentration profiles through the reactor predicted using the proposed and Danckwerts’ boundary conditions.     

乙炔前加氢催化剂活性分析方法研究

乙炔加氢催化剂活性对反应器入口CO浓度变化非常敏感,量化CO浓度对催化剂活性的影响程度并预测催化剂活性与反应器入口温度的对应关系十分必要。通过反余切三角函数建立催化剂活性随CO浓度变化的模型,定量描述CO浓度对催化剂活性的影响,并结合反应器入口温度、入口原料气总质量、入口与出口各物质的含量等实际生产数据,优化动力学参数和失活模型参数,定性和定量描述催化剂活性变化后反应器入口温度的调整方法,利用此模型结合相关数据考察反应器入口温度对催化剂活性的影响。     

Sequential experimental design for estimation and analysis of thermal parameters in a fixed bed

The analysis of the effective radial thermal conductivity and the film heat transfer coefficient were carried out in a fixed bed. The temperature profiles were described by two‐dimensional pseudo‐homogeneous model. The thermal parameters were estimated using a sequential experimental design technique. The minimum volume criterion was used to design the next point for temperature measurement in the bed. The utilization of T = T₀ (constant) as the boundary condition at the bed inlet resulted in an axial variation of thermal parameters, which was the factor responsible for the inadequacy of the model in fitting experimental data of different bed heights simultaneously. Using T = T(r) as the boundary condition makes the thermal parameters independent of the axial position and the model statiscally adequate to describe the axial and radial temperature profiles throughout the bed.     

Boundary layer model for char combustion and gasification

A non-steady boundary layer model is developed for numerical simulation of combustion and gasification of a single shrinking char particle. The model considers mass and energy conservation coupled with heterogeneous char reactions producing CO and homogeneous oxidation of CO to CO₂ in the boundary layer surrounding the char particle. Mass conservation includes accumulation, molecular diffusion, Stefan flow and generation by chemical reaction. Energy conservation includes radiation transfer at the particle surface and heat accumulation within the particle. Simulation results predict experimentally measured conversion and temperature profiles of a burning Spherocarb particle in a laminar flow reactor. Effects of bulk oxygen concentration and particle size on the combustion process are addressed. Predicted particle temperature is significantly affected by boundary layer combustion of CO to CO₂. With increasing particle size, char gasification to char combustion ratio increases, resulting in decreasing particle temperature and increasing peak boundary layer temperature.     

Stability of non-isothermal flow in channels—II: Temperature dependent power-law fluids without heat generation

The simplified lubrication type model used in earlier work on temperature dependent Newtonian fluid flow in a cooled channel is here extended to take account of shear dependent viscosity and is applied not only to plane channel flow but also to radial flow between parallel circular discs. Two additional dimensionless parameters are thereby introduced: m, a power-law index, and R, a geometrical variable based on the inlet and exit radii of the discs.The results are unsurprising: symmetrical flow curves of pressure P vs flow rate V are found to be multi-valued if B, the dimensionless inlet temperature, is large enough; the minimum value of B for which this occurs decreases as the fluid becomes more pseudoplastic, but is only weakly dependent upon the disc geometry, R, until the inner radius is a very small fraction of the outer radius. Linearized stability analysis for the flow is carried out in the same way as before: this shows that all symmetrical flow regimes for which dP/dV ? 0 are unstable to small perturbations and so in practice steady flows showing less symmetry than the boundary conditions are to be expected.Entrance and exit pressure losses are shown to lead to greater stability.     

Dynamic analysis and control of a tubular autothermal reactor at an unstable state

The dynamic behavior of an autothermal reactor with internal countercurrent heat exchange is represented by a relatively simple mathematical approximat which retains essentially all of the steady state and dynamic features of real reactors. The system of partial differential equations is discretized in space by the method of orthogonal collocation and the convergence of the eigenvalues of the linearized model is found to be extremely difficult to achieve in the unstable region near blow-off.Different model reduction techniques are investigated and compared. Modal control with state or output feedback and a single manipulated input is used an attempt to stabilize the system. A technique based on a low-order collocation model provides good control action, whereas most of the conventional, approximate methods fail to stabilize the unstable model of the reactor. The features of the stable, closed-loop system are confirmed through simulated reactor transient tests.     

绝热固定床反应器的研究——(Ⅰ)反应器的着火

本文建议了一种确立绝热固定床反应器着火温度的方法.无因次温度θ和无因次浓度(?)组成的两相平面可以用来表示单个催化剂的着火线和熄火线.同时也可以在该平面上画出表示反应进程的轨迹——反应器操作线.此操作线为一直线,其斜率始终为—45°,当它触及着火线即表明反应器的着火.本文推荐一个无需动力学的准确测定但可确立使反应器着火的临界进口温度的简单方法.可以假定反应为拟零级而仅引起极有限的误差,从而使实验简化.     

不均匀流动对径向氨合成塔性能的影响

通过导出考虑床层流速分布的径向床二维非均相模型,考察了不均匀流动对三段径向冷激式氨合成塔性能的影响,模拟计算表明了不均匀流动会对径向合成塔性能造成一定的影响,在各种不同的床层进口温度,人口氨浓和活性系数下,流动的不均匀度所造成的影响是不同的,提高进口温度、降低入口氨浓和提高活性系数有利于降低流动不均匀度的影响。流动不均匀对三段径向合成塔净值和生产能力的影响是有限的,在计算条件下,均匀流动和不均匀度     

Two observer-based nonlinear control approaches for temperature control of a class of continuous stirred tank reactors

In this paper, two nonlinear observer based controllers for temperature control of a continuous stirred tank reactor in which a special class of parallel exothermic reactions take place are proposed. A reduced order nonlinear observer is constructed to estimate the concentration in the reactor. The observer is coupled with two nonlinear controllers, designed based on two well-known techniques, namely input-output linearization and backstepping for controlling the reactor temperature. For dampening the effect of observer error dynamics, a compensating term is used in each control law. The asymptotical stability of the closed-loop system is shown by the Lyapunov's stability theorem. The effectiveness of the proposed controllers has been demonstrated through computer simulations.     

二氯乙烷裂解管式反应器二维模拟

建立了二氯乙烷在管式反应器中进行气相热裂解的二维模型 ,模型考虑了二氯乙烷热解生成氯乙烯的主反应和生成焦前体的副反应以及气体密度变化对裂解反应的影响 .模拟计算表明 ,二氯乙烷和氯乙烯的浓度沿径向分布平坦 ;但是管内近壁面处由于存在边界层 ,始终存在着明显的径向温差 ;近管壁处始终是裂解的高速率区 ,副反应也主要发生在管壁区 .表明确定最优的炉管管径时必须考虑提高裂解速率与降低结焦速率之间的平衡 .在距进口量纲 1管程 0 .3左右的管壁处裂解速率达到最高 ;副反应速率的最大点位于出口管壁处 .与工业数据比较后发现 ,炉管出口的转化率、选择性、出口压力和温度等数据与模型预测值一致 ,表明模型具有较高的可信度     

Mass transfer and biochemical reaction in enzyme membrane reactor systems—II: Expanded analysis for single enzyme systems: effects of enzyme intermediates,denaturation and elution

A detailed study of the dynamics of a packed-bed reactor containing immobilized enzyme particles is presented. The analysis consists of (i) transient state behavior; (ii) models for interphase and interfacial mass transfer between fluid and solid phases and intraphase mass transfer for the solid phase; (iii) detailed reaction rate model for the Bodenstein intermediates; (iv) mass balances for substrates, Bodenstein intermediates, unoccupied enzyme active sites, and products; and (v) models for enzyme denaturation and elution. The general reactor model consists of a set of nonlinear, coupled, partial differential equations. Numerical solutions of the system equations were obtained, using the discrete-space, continuous-time method of lines and realistic parameter values. A generalized map of the range of validity of the Steady-State Hypothesis was established under conditions where multiple mass transfer gradients were present within the reactor.A detailed analysis of the computational errors was performed. It was conclusively shown that the computer simulation solutions obtained in the analyses were not disguised to any significant degree as a result of employing finite difference approximations to the spatial derivatives.It was shown that the level of “error” involved in invoking the Steady-State Hypothesis depends on the relative magnitude of the kinetic parameters and also on the level of “disturbance” at the reactor inlet (i.e. per cent change in substrate inlet concentration). The “error”, however, did appear to be strikingly insensitive to the magnitude of the resistances to mass transfer, as characterized by the Modified Sherwood Number. It was concluded that, given any complete set of kinetic parameters, a transient, heterogeneous, isothermal reactor model based on the Steady-State Hypothesis may be used for predicting time-varying concentration profiles for minor (i.e., less than 5 per cent change in substrate inlet concentration) “disturbances” at the reactor inlet. The corresponding “errors” would be at an acceptable level (i.e., less than 2 per cent in the concentration and less than 10 per cent in the time lag) under these conditions.Further, various mechanisms for enzyme denaturation and elution were incorporated in the general reactor model. Numerical solutions of the resulting system of partial differential equations were obtained, using hypothetical parameter values. Through extensive simulation research, it was shown that the loss in activity of immobilized enzyme reactors cannot be uniquely ascribed to any one particular set of mechanistic deactivation modes.     

Guaranteed cost distributed fuzzy observer‐based control for a class of nonlinear spatially distributed processes

The guaranteed cost distributed fuzzy (GCDF) observer‐based control design is proposed for a class of nonlinear spatially distributed processes described by first‐order hyperbolic partial differential equations (PDEs). Initially, a T–S fuzzy hyperbolic PDE model is proposed to accurately represent the nonlinear PDE system. Then, based on the fuzzy PDE model, the GCDF observer‐based control design is developed in terms of a set of space‐dependent linear matrix inequalities. In the proposed control scheme, a distributed fuzzy observer is used to estimate the state of the PDE system. The designed fuzzy controller can not only ensure the exponential stability of the closed‐loop PDE system but also provide an upper bound of quadratic cost function. Moreover, a suboptimal fuzzy control design is addressed in the sense of minimizing an upper bound of the cost function. The finite difference method in space and the existing linear matrix inequality optimization techniques are used to approximately solve the suboptimal control design problem. Finally, the proposed design method is applied to the control of a nonisothermal plug‐flow reactor. © 2013 American Institute of Chemical Engineers AIChE J, 59: 2366–2378, 2013     

尿素低压吸收系统的稳定与优化操作

介绍了影响尿素低压吸收系统稳定操作的主要因素;论述了低调水进口温度控制对系统的影响以及稳定控制低调水进口温度的措施;实施了逐步降低低调水进口温度、稳定缓冲罐液位、去除缓冲罐氮气管线、增加蒸汽盘管等项技术改造措施。结果表明,改造后低调水进口温度由61℃降低到56℃,甲铵液浓度相应得到提高。