Computation of the gradient and sensitivity coefficients in sum of squares minimization problems with differential equation models

In parameter estimation problems where the system model consists of differential equations, methods for minimizing a sum of squares of residuals objective function require derivatives of the residuals with respect to the parameters being estimated (sensitivity coefficients) or the gradient of the objective function (depending on the numerical optimization method). This paper considers two methods for generating such derivatives: (1) the adjoint equation — gradient formula; and (2) complimentary sensitivity coefficient differential equations. Particular attention is given to the consistency between the method used to solve the model equations and the proper formulation of the additional equations required by the two methods. Two example problems illustrate computational experience using a modified quasi-Newton method with the adjoint method used to generate gradients and applying a modified Gauss-Newton approach with the sensitivity coefficient equations to calculate both the Gauss-Newton matrix and the objective function gradient. Results indicate the superiority of the sensitivity coefficient approach. When comparing the computational effort required by the two methods and the results from the simple examples, it appears that the use of complimentary sensitivity coefficient equations is much more efficient than using only the gradient of the sum of squares function.     

Optimization and sensitivity analysis for multiresponse parameter estimation in systems of ordinary differential equations

Methodology for the simultaneous solution of ordinary differential equations (ODEs) and associated parametric sensitivity equations using the Decoupled Direct Method (DDM) is presented with respect to its applicability to multiresponse parameter estimation for systems described by nonlinear ordinary differential equations. The DDM is extended to provide second order sensitivity coefficients and incorporated in multiresponse parameter estimation algorithms utilizing a modified Newton scheme as well as a hybrid Newton/Gauss-Newton optimization algorithm. Significant improvements in performance are observed with use of both the second order sensitivities and hybrid optimization method. In this work, our extension of the DDM to evaluate second order sensitivities and development of new hybrid estimation techniques provide ways to minimize the well-known drawbacks normally associated with second-order optimization methods and expand the possibility of realizing their benefits, particularly for multiresponse parameter estimation in systems of ODEs.     

Zur kinetik von planetenkugelmühlen

Least-squares techniques used to adjust experimental data, such as flow rates and concentrations (or size distributions, …) in a process flow-sheet, produce variables estimates which are consistent with respect to mass conservation laws. The sensitivity of this processing method is studied by calculation of the estimate variances. The method used consists of linearizing the mass balance equations and studying the effect of small disturbances created on the true values. The technique is applied to the generalized least-squares method as well as to the two-step least-squares method and in each case separate formulae are given for flow rates and concentrations. To illustrate the influence of various conditions on the mass balance results reliability, the equations are applied to a single node separator. It appears that very different amounts of information, with respect to the flow rates calculation, may be contained in stream concentrations. It is stated that the component which concentrates in the low flow rate product stream greatly contributes to flow rate value reliability. Application of this sensitivity analysis technique to a cement clinker grinding circuit, with its dedusting system, is used to demonstrate the usefulness of the method to evaluate mass balance reliability and to determine the experimental procedures and the data required to correctly balance industrial processes.     

The reaction engineering of electrochemical cells—I. Axial packed bed electrodes

It is shown that the graphical method of solving the coupled equations for an endothermic reaction in a plug-flow, non-isothermal reactor can be extended to electrochemical reactions in three-dimensional electrodes. The method of solution is described and as a first example, it has been applied to a simple reversible reaction in an axial packed bed electrode. The sensitivity to various parameters has been examined. The method is ideally suited to computer-aided design procedures.     

Sensitivity analysis of systems of differential and algebraic equations

Formulae are derived for parametric sensitivity analysis of mathematical mo dels consisting of sets of differential and algebraic equations. Such equations often arise in dynamic modeling of equilibrium stage processes, and in solution of partial differential equations via the numerical method of lines. These formulae can be used to efficiently produce the model sensitivity coefficients, simultaneously with the solution of the model.     

An ESDIRK method with sensitivity analysis capabilities

A new algorithm for numerical sensitivity analysis of ordinary differential equations (ODEs) is presented. The underlying ODE solver belongs to the Runge–Kutta family. The algorithm calculates sensitivities with respect to problem parameters and initial conditions, exploiting the special structure of the sensitivity equations. A key feature is the reuse of information already computed for the state integration, hereby minimizing the extra effort required for sensitivity integration. Through case studies the new algorithm is compared to an extrapolation method and to the more established BDF based approaches. Several advantages of the new approach are demonstrated, especially when frequent discontinuities are present, which renders the new algorithm particularly suitable for dynamic optimization purposes.     

Gas permeation and separation in asymmetric hollow fiber membrane permeators: Mathematical modeling,sensitivity analysis and optimization

Mathematical modeling is useful for analysis of process design and performance and is widely used for membrane separation and other important technologies in the energy sector. This study presents the results of our investigations on the mathematical modeling and optimization of hollow fiber membrane permeators specifically used for air separation as well as natural gas purification. The governing equations and mathematical models are developed based on the consideration of ideal and non-ideal conditions often involved in the separation of gas mixtures using membrane permeators. The influence and consequences of adoption of two distinct numerical methods for solving governing equations are investigated in details. The results obtained by using the models as well as the effect of numerical method type are examined and compared to the experimental data. The findings highlight the important role of the solution method on the validity and accuracy of the models. Moreover, the effect of variations in the operating conditions and physical geometries of the membrane are investigated through comprehensive sensitivity analysis. Accordingly, a set of optimal input parameters is determined using an appropriate statistical method. The findings provide useful information for the design and development of high performance membrane permeators and processes particularly in the case of binary gas mixtures for energy applications.     

Sensitivity analysis of a batch polymerization reactor

The sensitivity of model output variables for a batch polymerization reactor to uncertainties in the kinetic parameters and initial conditions is studied. Differential equations that describe the time variation of sensitivity coefficients for the batch reactor are derived. Numerical integration of the sensitivity equations reveals that the system output responses are very sensitive to parameter variations especially when the polymerization exhibits an autoacceleration of the reaction rate.     

A METHOD FOR SEPERATING VELOCITY FLUCTUATION AND TEMPERATURE FLUCTUATION IN THE COURSE OF MEASURING TURBULENT PARAMETERS

A double hot-wire method is adopted to separate the mixed signal of temperature fluctuation and velocity fluctuation appearing in high temperature turbulences. The relationships among the various coefficients in the sensitivity equation for measuring turbulent fluctuations with constant temperature hot wire anemometer are deduced. Once these equations are being related to the temperature fluctuation sensitivity equation of a constant current hot wire anemometer, the temperature fluctuation signal may be taken out from the mixed signal, then the turbulent velocity fluctuation signal and other main turbulent parameters may be obtained.     

Hydrodynamics of Reduced Pressure Fluidization Employing Particles with Variable Density

A series of experiments was carried out to study the hydrodynamics of a fluidized bed operating at reduced pressure and employing particles with variable moisture content; that is, of variable density. In these experiments, four different types of particles were used and fluidization characteristics very similar to the ones encountered in atmospheric pressure operations were observed. A novel method to measure the minimum fluidization velocity of particles with varying density was proposed. The experimental results demonstrated that the minimum fluidization velocity increased with decreased operating pressure, increased operating temperature, and increased particle moisture content. However, the bed voidage under minimum fluidization conditions showed very little sensitivity to variations in operating pressure. Two equations were developed to predict the minimum fluidization velocity and the results were compared with the experimental data as well as with the predictions of equations available in the technical literature.     

Dynamic sensitivity analysis of chemical reaction systems: A variational method

An approach to the sensitivity analysis of systems governed by ordinary differential equations is presented. It is based on a variational approach and is developed here in the context of chemically reacting systems, although in principal it is more general. The linear analysis generates sensitivity indexes which are a measure of the sensitivity of an objective function to the jth reaction of a mechanism. Both instantaneous and time-averaged sensitivities can be obtained. It also gives the sensitivity to initial concentrations of reactants, or to concentration perturbations at arbitrary times. The use of an objective function permits one to obtain the sensitivity of several species simultaneously in a single analysis, in contrast to other methods where the sensitivity of one species at a time must be determined. In this work the objective functions used are norms involving an arbitrary number of dependent variables. A ranking of the relative importance of each reaction in governing the concentrations of species appearing in the objective function is obtained. Since the sensitivity indexes are time dependent, the analysis may be said to be a dynamic sensitivity analysis. The sensitivity indexes given by this method are related to other standard sensitivity indexes.     

Analysis of a CFSTR with two consecutive reactions in transient states

The sensitivity of a chemical reactor is defined by the matrix which relates the steady state gain of output variables to unit step changes in input variables. This definition proves applicable to reactors with arbitrary residence time distributions. The sensitivity calculation is illustrated by treating a specific example. Three ordinary differential equations which govern (he transient behavior of a CFSTR with two consecutive reactions are numerically solved to demonstrate the parametric sensitivity and temperature runaway phenomena for some parameters such as the feed temperature, the feed concentration, the heat transfer coefficient and the wall temperature. The estimation of the regions of parametric sensitivity is briefly discussed.     

非线性分布参数系统状态估计的最佳测量位置

研究了分布参数系统状态估计中特有的最佳测量位置问题．建立了基于后集中方法的分布参数系统的非线性状态估计器,包括状态估计偏微分方程和微分灵敏度矩阵偏微分方程,并用适当的数值计算方法实现状态估计器的求解;以一个最小化的空间域上积分函数表达最佳测量位置的目标函数,并相应地用非线性约束优化方法求解系统具有一个或多个测量时的最佳测量位置．还以壁冷式单管固定床反应器为例,讨论了各种因素对最佳测量位置的影响及其灵敏度,并得出了一些有普遍意义的结论．     

Axial Dispersion Model for Estimation of Ozone Self-Decomposition

A new method using the axial dispersion model for estimation of ozone self-decomposition kinetics in a semibatch bubble column reactor was developed. The reaction rate coefficients for literature equations of ozone decomposition and the gas phase dispersion coefficient were estimated and compared with literature data. The reaction order in the pH range 7–10 with respect to ozone 1.12 and 0.51 the hydroxyl ion were obtained, which is in good agreement with literature. The model parameters were determined by parameter estimation using a nonlinear optimization method. A sensitivity analysis was conducted to obtain information on reliability and identifiability of the estimated parameters.     

SIMULATION OF STEADY DISTILLATION PROCESS ACCOMPANIED WITH MULTIPLE CHEMICAL EQUILIBRIUM REACTIONS BASING ON TRANSFORMED VARIABLES

The equations describing the steady state operation of distillation column with simultaneous equilibrium chemical reactions have been transformed into a new set or MESH equations by introducing transrormed composition, flow rate and enthalpy variables. The new set of equations are similar to equations describing traditional non-reactive distillation, and it is general when one or several reactions occur and inert components are present.

An algorithm combining relaxation method and modified Newton-Raphson method is proposed to solve the new equations, the relaxation method is used to estimate initial values, while the modified Newton-Raphson method is used to set solution. The reactive distillation processes of MTBE synthesis with n-butane and the separation of meta- and para-xylene are simulated as numerical examples     

Scaling and sensitivity analysis of a reverse flow reactor

Scaling analysis is presented as a systematic procedure to analyze and understand the operation of a complex process such as the autothermal reverse flow reactor (RFR). The reactor is complex from an operational point of view due to its hybrid and periodic nature. An adequate model of the RFR involves highly nonlinear equations. Using simple mathematical operations, these model equations are non-dimensionalized, scaled to order 1 and used to determine the contributions of the controlling physical phenomena taking place in it. The scale factors lead to several analytical expressions useful for suggesting efficient operational strategies for the RFR. Based on a specified error tolerance, we also illustrate how model approximation can be carried out and justified. The sensitivity of important operational parameters that determine sustainability (i.e., maximum temperature and overall conversion) to variables such as reactor length, switching time and mass transfer rate are also analyzed for the pseudo-steady-state condition. The results obtained prove that prudent ways of operating an RFR can be determined through scaling and sensitivity analysis.     

Sensitivity analysis of low-speed melt spinning of isotactic polypropylene

Linearized sensitivity analysis of fiber spinning has been studied using a two-phase constitutive model that includes the effects of crystallization. The analysis focuses on sensitivity predictions for the low-speed melt spinning of isotactic polypropylene and comparison to the experimental results of Young and Denn [1989, Disturbance propagation in melt spinning. Chemical Engineering Science 44, 1807-1818]. Modifications in an earlier-developed two-phase model enable comparisons of two different constitutive equations for the melt phase, namely the Giesekus and extended pom-pom models. Comparisons with sensitivity data for low-speed spinning conditions demonstrate that the incorporation of crystallization effects leads to improved predictions of the magnitude and trends of the perturbation frequency dependence for both constitutive equations. At low spin speeds where flow-enhanced crystallization effects are negligible, the sensitivity is predicted to decrease with increasing cooling and this trend is also shown to be consistent with increased crystallinity.     

Simulation of transient gas flow using the orthogonal collocation method

Proper evaluation of the dynamics of the transmission system is the key element in the design and operation of natural gas pipelines. Basic equations describing the transient flow of gas in pipes are derived from the Euler equations. The orthogonal collocation technique is employed as the mathematical method for the numerical solution of the governing equations. This method leads to a set of non-linear ordinary differential equations which can be solved by the Runge–Kutta–Fehlberg method. The performance of the proposed method is tested using two practical examples. The predicted results clearly demonstrate that the proposed method can successfully simulate the isothermal and non-isothermal unsteady flow in gas transmission systems.     

强制浓度振荡下的反应动力学参数特征函数估计法

研究了特征函数法(Karhunen-Lo(?)ve方法)在降低常微分方程组维数方面的可能性,并以CO氧化反应系统为例,说明该法的应用步骤.模拟结果证实了此方法的有效性.     

伴有多反应的精馏过程数学模拟

<正>在伴有多个反应的分离过程数学模拟中,反应量的存在使模型变量增多,非线性增强.文献[1～3]在普通精馏的基础上,提出了不同的算法来改进收敛性.他们对各种反应类型,均以动力学计算反应量,迭代变量多,算法也较复杂.Doherty针对平衡反应过程,提出了变换组成变量的概念计算反应相平衡.本文引入此概念,并对流率、焓等物理量进行了相应的变换,使变换后的反应精馏过程数学模型在形式上与普通精馏过程的模型完全一致.并以修正的Newton-Raphson法对对二甲苯和邻二甲苯的反应精馏过程进行了计算.