Application of the method of lines to the analysis of single fluid-solid reactions in porous solids

A mathematical model for the isothermal reaction between a fluid and a porous solid, in pelletgrain form, is studied. First, an analytical derivation of the time for total conversion is presented. Then the differential equations, given as a coupled pair of time-dependent and time-independent equations, are altered to a fully time-dependent form, without loss of accuracy, but with much greater amenability to an efficient numerical solution method. The solution is by the method of lines, whereby finite difference discretization in the spatial variable yields a stiff system of ordinary differential equations (ODEs), and the ODE initial value problem is solved with a modern general-purpose ODE solver. A further alteration to the equations overcomes the difficulty caused by the discontinuous ODEs for the solid variable. Results are given for two grain geometries and for a wide range of reaction moduli.     

Dynamic optimization of dissipative PDE systems using nonlinear order reduction

This article presents computationally efficient methods for the solution of dynamic constraint optimization problems arising in the context of spatially distributed processes governed by highly dissipative nonlinear partial differential equations (PDEs). The methods are based on spatial discretization using the method of weighted residuals with spatially global basis functions (i.e., functions that cover the entire domain of definition of the process and satisfy the boundary conditions). More specifically, we perform spatial discretization of the optimization problems using the method of weighted residuals with analytical or empirical (obtained via Karhunen-Loève expansion) eigenfunctions as basis functions, and combination of the method of weighted residuals with approximate inertial manifolds. The proposed methods account for the fact that the dominant dynamics of highly dissipative PDE systems are low dimensional in nature and lead to approximate optimization problems that are of significantly lower order compared to the ones obtained from spatial discretization using finite-difference and finite-element techniques, and thus, they can be solved with significantly smaller computational demand. The resulting dynamic nonlinear programs include equality constraints that constitute a low-order system of coupled ordinary differential equations and algebraic equations, which can then be solved with combination of standard temporal discretization and nonlinear programming techniques. We employ backward finite differences (implicit Euler) to perform temporal discretization and solve the nonlinear programs resulting from temporal and spatial discretization using reduced gradient techniques (MINOS). We use two representative examples of dissipative PDEs, a diffusion-reaction process with constant and spatially varying coefficients, and the Kuramoto-Sivashinsky equation, a model that describes incipient instabilities in a variety of physical and chemical systems, to demonstrate the implementation and evaluate the effectiveness of the proposed optimization algorithms.     

Estimating the parameters of chemical kinetics equations from the partial information about their solution

The inverse problem of identifying the parameters of sets of ordinary differential equations using experimental measurements of three functions that correspond to some components in the vector solution of a set is considered. A private case that is important for applications of chemical and biochemical kinetics when reduced equations linearly depend on the combinations of initial unknown parameters has been studied. An analysis and the numerical results are presented for two types of sets of chemical kinetics equations, such as the Lotka–Volterra model that describes the coexistence of a predator and a prey and the chemical kinetics equations that model enzyme catalysts reactions, including the Michaelis–Menten equations. The search for unknown parameters is confined to the problem of minimizing a quadratic function. In this case, the reduced differential equations of systems are used instead of their vector solutions, which are unknown in most cases. The cases of both stable and unstable search for unknown parameters are analyzed.     

Enclosing all solutions of two-point boundary value problems for ODEs

The two-point boundary value problem (TPBVP) occurs in a wide variety of problems in engineering and science, including the modeling of chemical reactions, heat transfer, and diffusion, and the solution of optimal control problems. A TPBVP may have no solution, a single solution, or multiple solutions. A new strategy is presented for reliably locating all solutions of a TPBVP. The method determines narrow enclosures of all solutions that occur within a specified search interval. Key features of the method are the use of a new solver for parametric ODEs, which is used to produce guaranteed bounds on the solutions of nonlinear dynamic systems with interval-valued parameters and initial states, and the use of a constraint propagation strategy on the Taylor models used to represent the solutions of the dynamic system. Numerical experiments demonstrate the use and computational efficiency of the method.     

碳钢在管流体系中的流动腐蚀动力学模型

针对管流体系 ,根据动量、能量和质量守恒原理 ,应用壁函数、k -ε湍流运动模型确立了管流体系中的计算流体力学模型和质量传递模型 ;结合必要的实验分析碳钢流动腐蚀过程中的主要控制因素及其影响程度 ,建立了碳钢在流动的 3.5 %NaCl溶液中的流动腐蚀动力学模型 .同时 ,采用数值计算方法计算了流动腐蚀速度 ,并与实验结果进行了验证 .揭示出腐蚀电化学因素在碳钢流动腐蚀过程中起着主导的作用.     

Improvement of state profile accuracy in nonlinear dynamic optimization with the quasi‐sequential approach

Quasi‐sequential methods are efficient and flexible strategies for the solution of dynamic optimization problems. At the heart of these strategies lies the time discretization and approximation of dynamic systems for nonlinear optimization problems. To address this question, we employ a time derivative analysis within the quasi‐sequential approach and derive a finite element placement strategy. In addition, methods for direct error prediction are applied to this approach and extended with a proposed time derivative analysis. According to the information for current time derivatives, subintervals are introduced that improve accuracy of state profiles. Since this is only done in the simulation layer, the nonlinear programing solver need not be restarted. An efficient gradient computation is also derived for these subintervals; the resulting enhanced accuracy accelerates convergence performance and increases the robustness of the solution to initialization. A beer fermentation process case study is presented to demonstrate the effectiveness of the proposed approach. © 2011 American Institute of Chemical Engineers AIChE J, 2011     

PRISM: Piecewise Reusable Implementation of Solution Mapping. An Economical Strategy for Chemical Kinetics

In a chemical kinetics calculation, a solution-mapping procedure is applied to parametrize the solution of the initial-value ordinary differential equation system as a set of algebraic polynomial equations. To increase the accuracy, the parametrization is done piecewise, dividing the multidimensional chemical composition space into hypercubes and constructing polynomials for each hypercube. A differential equation solver is used to provide the solution at selected points throughout a hypercube, and from these solutions the polynomial coefficients are determined. Factorial design methods are used to reduce the required number of computed points. The polynomial coefficients for each hypercube are stored in a data structure for subsequent reuse, since over the duration of a flame simulation it is likely that a particular set of concentrations and temperature will occur repeatedly at different times and positions. The method is applied to H₂–air combustion using an 8-species reaction set. After N₂ is added as an inert species and enthalpy is considered, this results in a 10-dimensional chemical composition space. To add the capability of using a variable time-step, time-step is added as an additional dimension, making an 11-dimensional space. Reactive fluid dynamical simulations of a 1-D laminar premixed flame and a 2-D turbulent non-premixed jet are performed. The results are compared to identical control runs which use an ordinary differential equation solver to calculate the chemical kinetic rate equations. The resulting accuracy is very good, and a factor of 10 increase in computational efficiency is attained.     

Numerische Methoden zur Simulation verfahrenstechnischer Prozesse

Numerical Methods for Simulation of Chemical Engineering Processes. Essential fundamentals and the current state of the art in simulating the dynamic and the steady state behaviour of chemical engineering processes are discussed. It is shown that discretization of the spatial derivatives in the balance equations leads to a system of so-called DAE (differential algebraic equations), consisting of ordinary differential equations in time and algebraic equations. The paper discusses necessary steps to solve the DAE and mentions approved standard software for these steps as well as for the solution of the DAE as a whole.     

Optimal moving and fixed grids for the solution of discretized population balances in batch and continuous systems: droplet breakage

The numerical solution of droplet population balance equations (PBEs) by discretization is known to suffer from inherent finite domain errors (FDE). Tow approaches that minimize the total FDE during the solution of discrete droplet PBEs using an approximate optimal moving (for batch) and fixed (for continuous systems) grids are introduced. The optimal grids are found based on the minimization of the total FDE, where analytical expressions are derived for the latter. It is found that the optimal moving grid is very effective for tracking out steeply moving population density with a reasonable number of size intervals. This moving grid exploits all the advantages of its fixed counterpart by preserving any two pre-chosen integral properties of the evolving population. The moving pivot technique of Kumar and Ramkrishna (Chem. Eng. Sci. 51 (1996b) 1333) is extended for unsteady-state continuous flow systems, where it is shown that the equations of the pivots are reduced to that of the batch system for sufficiently fine discretization. It is also shown that for a sufficiently fine grid, the differential equations of the pivots could be decoupled from that of the discrete number density allowing a sequential solution in time. An optimal fixed grid is also developed for continuous systems based on minimizing the time-averaged total FDE. The two grids are tested using several cases, where analytical solutions are available, for batch and continuous droplet breakage in stirred vessels. Significant improvements are achieved in predicting the number densities, zero and first moments of the population.     

Graphite foams infiltrated with phase change materials as alternative materials for space and terrestrial thermal energy storage applications

In this work, a numerical study is proposed to investigate and predict the thermal performance of graphite foams infiltrated with phase change materials, PCMs, for space and terrestrial energy storage systems. The numerical model is based on a volume averaging technique while a finite volume method has been used to discretize the heat diffusion equation. A line-by-line solver based on tri-diagonal matrix algorithm has been used to iteratively solve the algebraic discretization equations. Because of the high thermal conductivity of graphite foams, the PCM-foam system thermal performance has been improved significantly. For space applications, the average value of the output power of the new energy storage system has been increased by more than eight times. While for terrestrial applications, the average output power using carbon foam of porosity 97% is about five times greater than that for using pure PCM.     

Design under uncertainty using parallel multiperiod dynamic optimization

A technique for optimizing dynamic systems under uncertainty using a parallel programming implementation is developed in this article. A multiple‐shooting discretization scheme is applied, whereby each shooting interval is solved using an error‐controlled differential equation solver. In addition, the uncertain parameter space is discretized, resulting in a multiperiod optimization formulation. Each shooting interval and period (scenario) realization is completely independent, thus a major focus of this article is on demonstrating potential computational performance improvement when the embedded dynamic model solution of the multiperiod algorithm is implemented in parallel. We assess our parallel multiperiod and multiple‐shooting‐based dynamic optimization algorithm on two case studies involving integrated plant and control system design, where the objective is to simultaneously determine the size of the process equipment and the control system tuning parameters that minimize cost, subject to uncertainty in the disturbance inputs. © 2014 American Institute of Chemical Engineers AIChE J, 60: 3151–3168, 2014     

Electrochemical model of a lithium-ion battery implemented into an automotive battery management system

This paper presents the development of an electrochemical model that can be implemented into automotive battery management systems (BMSs). Compared with empirical models, the electrochemical model features more accurate state estimates over a broader and longer use of the battery. In this work, model implementation schemes are devised to make the electrochemical model uncomplicated enough to be embedded into the BMS. A nonlinear system of partial differential equations in the model is discretized into a linearized system of algebraic equations (AEs). A solver selected to evaluate the resulting system of AEs is modified for its application to the BMS. As the BMS is preoccupied by its existing tasks, the reformulated equations and optimized solver are reorganized such that the limited computational resources of the BMS are appropriately exploited. The electrochemical model is consequently implemented into the BMS, predicting battery behaviors in 1 s intervals while occupying a 14 kB RAM.     

A solution method for solving coupled nonlinear boundary-value problems

A Strum-Liouville integral transform technique is novelly applied to solve system of coupled nonlinear boundary-value problems approximately. The systems of differential equations consist of a linear differential operator and a nonlinear function of the dependent variables. To illustrate the potential of this technique we consider an example which comes from the modeling of diffusion and nonlinear chemical reaction systems in chemical engineering. The approximate solutions obtained by our technique agree surprising well with the numerically exact solutions obtained by the orthogonal collocation technique. To improve the approximation an iteration scheme in transform space is also defined.

Scope—Today, mathematical modeling of physical phenomena often produces (single or coupled) nonlinear differential equations. The true physical situation can, in many cases, be more closely described if the differential equations are allowed to be nonlinear. However, nonlinear differential equations are generally too difficult to be solved analytically apart from a few “tricks” or substitutions which apply only to a handful of equations [1]. An alternative approach is to look for a method which will reduce the problem, via analytical techniques, to a point where a “simple” computer program can solve the rest of the problem. The method introduced in this paper belongs to this class of solution techniques.

The method, which in this paper is applied to solving coupled nonlinear boundary-value problems, is a generalization of an idea in a paper by Do and Bailey [2] who apply it to a single nonlinear differential equation of boundary-value type. The equations, to which the technique is applied, arise from Fick's law diffusion into a porous solid and nonlinear reaction within the solid.

The solution method employs a Strum-Loiuville integral transform and to account for the nonlinear part an approximation is introduced. An iteration scheme is defined to improved the accuracy of the solution. The system of coupled nonlinear differential equations is reduced to a system of coupled nonlinear algebraic equations which is solved using a Newton-Raphson process. Finally, the solution is expressed as an infinite series, which is summed using a computer.

In response to papers by Do and Bailey [3] and Do and Weiland [4], Jerri [5] has tried to put this method on a more mathematical footing, and he shows that this method is a special case of a more general technique he has devised. Jerri uses the idea of Fourier transforms and convolution products to justify his method. The results for the example he considered are good, but he did not state how many iterations he required to obtain the solutions reported.

Conclusions and Significance—This paper has presented a very powerful method of solving boundary-value problems with linear operators and a nonlinear function of the dependent variable. The method works well for a single equation or coupled equations and can handle any kind of nonlinear function. We have shown through extensive numerical calculation the accuracy of this solution method, where the accuracy is measured in terms of a ratio of norms. In most cases an error of 4% can be achieved with just one iteration (Tables 2 and 3). Even though the present method has been applied to problems which have arisen from the modeling of chemical engineering problems, it would also be applicable to differential equations arising in other areas, provided they are of the same form.     

Gear's procedure for the simultaneous solution of differential and algebraic equations with application to unsteady state distillation problems

The dynamic equations modeling a sieve plate at unsteady state are developed. Gear's procedure for the simultaneous solution of systems of stiff differential and algebraic equations is presented and demonstrated for the solution of unsteady state distillation problems. It is shown that the basic stage model can be modified by the addition of one variable and one equation such that Gear's procedures are readily applied. The proposed model and solution procedure is contrasted to recently published procedures. Numerical results are given for the solution of a problem involving an extractive distillation column at unsteady state.     

Objects for MWR

A computational framework has been developed for step-by-step implementation of global spectral projection methods used for solving boundary-value problems and for subsequent analysis of solutions produced using the numerical techniques of this framework. A set of matlab-based functions corresponding to each step in a Galerkin discretization procedure has been developed with emphasis on simplifying the implementation of discretization methods for nonlinear, distributed-parameter system models in up to three-dimensional physical domains. A key feature of this computational approach is that a set of object classes were developed to facilitate implementation of the weighted residual methods (MWR) in an effort to make the connection between the solution procedures and modeling equations as clear as possible. The utility of the computational procedures is demonstrated through applications to two-dimensional reaction-diffusion and fluid flow problems, and a three-dimensional heat transfer model relevant to semiconductor manufacturing.     

A multilevel unsymmetric matrix ordering algorithm for parallel process simulation

Computer simulation of complex chemical processes is increasingly being used in the design optimization and control of chemical facilities. Industrial-scale modeling involves the solution of large systems of algebraic differential equations. This is very computationally intensive with a large part of the computing time attributed to the repeated solution of large, sparse. unsymmetric systems of linear equations. One way of speeding up the simulation is to solve the linear systems efficiently in parallel by reordering the unsymmetric matrices into a bordered block-diagonal (BBD) form. In this paper a multilevel ordering algorithm is presented. A multilevel technique, which provides a global view, is combined with a Kernighan–Lin algorithm to form an effective unsymmetric matrix ordering algorithm MONET (Matrix Ordering for minimal NET-cut). Numerical results confirm that this algorithm gives ordering of better quality than existing algorithms.     

Numerical simulation of interphase mass transfer with the level set approach

A level set approach is applied for simulating the interphase mass transfer of single drops in immiscible liquid with resistance in both phases. The control volume formulation with the SIMPLEC (semi-implicit method for pressure-linked equations consistent) algorithm incorporated is used to solve the governing equations of incompressible two-phase flow with deformable free interface on a staggered Eulerian grid. The solution of convective diffusion equation for interphase mass transfer is decoupled with the momentum equations. Different spatial discretization schemes including the fifth-order WENO (weighted essentially nonoscillatory), second-order ENO (essentially nonoscillatory) and power-law schemes, are tested for the solution of mass transfer to or from single drops. The conjugate cases with different equilibrium distribution coefficients are simulated successfully with the transformation of concentrations, molecular diffusivities, mass transfer time and velocities. The predicted drop concentration, overall mass transfer coefficient and flow structure are compared with the reported experimental data of a typical extraction system, i.e., n-butanol-succinic acid-water, and good agreement is observed.     

SMB色谱分离过程计算机仿真

针对模拟移动床(SMB)综合速率模型，采用有限元法和正交配点法分别对柱向和颗粒径向模型进行离散化，利用MatLab ODE求解器对SMB过程进行数值求解。通过仿真，验证了该方法的可行性。     

Recursive state estimation techniques for nonlinear differential algebraic systems

Kalman filter and its variants have been used for state estimation of systems described by ordinary differential equation (ODE) models. While state and parameter estimation of ODE systems has been studied extensively, differential algebraic equation (DAE) systems have received much less attention. However, most realistic chemical engineering processes are modelled as DAE systems and hence state and parameter estimation of DAE systems is a significant problem. Becerra et al. (2001) proposed an extension of the extended kalman filter (EKF) for estimating the states of a system described by nonlinear differential-algebraic equations (DAE). One limitation of this approach is that it only utilizes measurements of the differential states, and is therefore not applicable to processes in which algebraic states are measured. In this paper, we address the state estimation of constrained nonlinear DAE systems. The novel aspects of this work are: (i) development of a modified EKF approach that can utilize measurements of both algebraic and differential states, (ii) development of a recursive approach for the inclusion of constraints, and (iii) development of approaches that utilize unscented sampling in state and parameter estimation of nonlinear DAE systems; this has not been attempted before. The utility of these estimators is demonstrated using electrochemical and reactive distillation processes.