Computation Of Mathematical Models For Complex Industrial Processes

Yu-Chu Tian, Tonghua Zhang, Hongmei Yao, Moses O Tadé

• Publisher: WSPC
• Year: 2014
• eBook ISBN: 9789814616577
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

Designed for undergraduate and postgraduate students, academic researchers and industrial practitioners, this book provides comprehensive case studies on numerical computing of industrial processes and step-by-step procedures for conducting industrial computing. It assumes minimal knowledge in numerical computing and computer programming, making it easy to read, understand and follow. Topics discussed include fundamentals of industrial computing, finite difference methods, the Wavelet-Collocation Method, the Wavelet-Galerkin Method, High Resolution Methods, and comparative studies of various methods. These are discussed using examples of carefully selected models from real processes of industrial significance. The step-by-step procedures in all these case studies can be easily applied to other industrial processes without a need for major changes. Thus, they provide readers with useful frameworks for the applications of engineering computing in fundamental research problems and practical development scenarios.

Contents:

• Introduction
• Fundamentals of Process Modelling and Model Computation
• Finite Difference Methods for Ordinary Differential Equation Models
• Finite Difference Methods for Partial Differential Equation Models
• Wavelets-Based Methods
• High Resolution Methods
• Comparative Studies of Numerical Methods for SMB Chromatographic Processes
• Conclusion

Readership: Students, academics and practitioners in the field of chemical engineering, numerical analysis and computational mathematics.

Chapter 1

Introduction

This book focuses on computation of mathematical models for complex industrial processes. As the computation serves a specific purpose in a real application, it is tightly related to process modelling, from which the mathematical models are developed for the purpose. For an industrial process, different types of mathematical models may be required for different applications of the models. For example, models for real-time process control may be different from those for process optimization. Therefore, process modelling will also be discussed in this book for better understanding of model computation methods and technologies.

While computation of mathematical models is a very broad topic, this book will investigate computational problems from the practical application perspective. Nevertheless, special attention will also be paid to fundamental concepts and theory of mathematical computation in problem solving for real industrial processes. This is based on our understanding that fundamental concepts and theory evolve much more slowly than technologies and applications and form the foundation from which new technologies and applications can be developed.

In the development of the content of this book, step-by-step procedures will be provided for practical industrial process modelling and model computation. Comprehensive case studies will also be demonstrated through carefully selected and experimentally verified complex industrial process examples. These procedures and case study examples form some useful problem solving frameworks, which the readers may follow in real industrial process applications.

1.1 Background

What is computation of mathematical models?

In this book, computation of mathematical models refers to a set of techniques with which the mathematical models formulated for a specific problem can be solved through arithmetic and logical operations. This is a traditional understanding of the computation of mathematical models and gives the focus of the computational problem onto the development and theoretical justification of various computing methods. Many such traditional numerical analysis books have been published, which were written in this way with detailed theoretical derivations.

The computation of mathematical models can also be understood as a process of deriving a solution with a computer or multiple computers from a given set of mathematical models. With this understanding, the focus of the investigations would be on problem solving, i.e., on the choice and application of numerical methods for deriving a satisfactory solution. Theoretical derivations and rigorous analysis of the computation may be largely ignored or not fully discussed. For a specific computational problem of mathematical models, techniques and algorithms could be recommended for this type of models from previous experience of successful applications. To a large extent, examples of such successful applications form a foundation of useful frameworks with which many practical problems could be solved straightaway without the need to carefully consider related mathematical theory.

Computation of mathematical models is discussed in this book in a different way from the above two typical understandings in both intend and content. While intending to emphasis more on solving practical problems with appropriate frameworks and techniques which are well verified and validated through successful applications, we still discuss necessary mathematical background and theoretical analysis. The content of the book is designed so that we could cover the three major types of effective methods and algorithms for computation of mathematical models of complex industrial processes: finite-difference methods, wavelet-based methods and high-resolution methods.

Computation of mathematical models serves a specific purpose in a real application. This book addresses more on model computation for online optimization and real-time control. Therefore, computational methods which we believe are not quite suitable for this purpose will not be investigated in this book. One of such methods is the widely used and well developed finite element technique for process design and optimization. Another type of such methods is those for fluid dynamics analysis, which is a broad topic of research and development in process systems engineering.

As digital computers are used for solving the computational problems, the computation we are investigating is actually numerical computation. Numerical computation by digital computers is naturally conducted by methods of approximation. This means that approximation methods are critical to the computation of mathematical models. To justify the approximation methods, mathematical justification needs to be provided for a number of issues, such as accuracy and precision of the approximation methods, computational complexity, computational stability, and the speed of convergence. For a specific computational problem of an industrial process model, an important task is to choose a right approximation method for the numerical computation in order to get the solution within an acceptable period of time under the constraints of the available computing resources.

Depending on the scale and complexity of an industrial process model, numerical computation of the mathematical behaves with different performance in computing resource requirements and execution time. In some scenarios, it can be completed very quickly without demanding excessive computing resources. In many other cases, it may be computationally expensive with a high demand of computing resources and significant time consumption. With the rapid development of modern computer technologies, many computational tasks of industrial process models can be easily handled with existing numerical methods. Such computing tasks would not be comprehensively discussed in this book.

Without discussing easy tasks in process model computing, this book will address computational problems for complex industrial processes, which require significant computing effort and thus demand innovative numerical methods. The complex processes we will cover in this book include galvanizing processes, biological fermentation processes, crystallization processes, chemical reaction processes and simulated moving bed chromatographic separation processes. These processes are either typical unit operation processes in process industries or an integration of several unit operation processes. They are also widely deployed in real process industries, and are thus industrially significant.

We emphasize again that the computation of a mathematical model serves a specific purpose for an application. In this sense, numerical methods for computation of mathematical models should be chosen to meet the requirements for the purpose. This means that from a mathematical model of an industrial process, different computational methods may be required for different application purposes. This highlights the need of careful investigations into the application requirements and the performance of the chosen computational methods.

In order to fulfill the application requirements, process modelling needs to be investigated before the computational problems of the mathematical models are tackled. We consider process modelling as an integrated and significant step in the whole procedure of the computation of mathematical models for complex industrial processes. Therefore, it will be discussed in this book for each of the selected industrial process examples.

Process modelling itself is a broad topic of research and development in process systems engineering. A complete discussion of process modelling is beyond the scope of this book. In general, a process model can be established from one of the following three methods:

(1) from the first principal and theoretical development;

(2) from empirical analysis and experimental studies; and

(3) from a combination of the above two methods.

In our observation, the majority of process models for real industrial applications are established through the third modelling method, i.e., through a combined application of theoretical development and empirical analysis. This is also the method we will adopt in this book for establishment of mathematical models of complex industrial processes. Techniques and approaches we will use for process modelling will be embedded into the discussions of our process modelling.

The results derived from numerical computation of mathematical models need to be carefully validated and verified before they are applied to real systems. This is because of two main reasons. The first reason is the approximation nature of numerical computing, which works in discrete-time domain to approximate system dynamics in continuous-time domain. Obviously, a discrete-time representation with finite number of time instances may not well approximate a continuous-time problem with infinite number of time instances. The second reason is the uncertainties in the mathematical models of the industrial process under investigation. Industrial processes in chemical industries are known to show significant uncertainties, which cannot be well captured in process modelling. A detailed discussion of this topic will not be carried out in this book. Interested readers are referred to process modelling books for comprehensive discussions on this topic.

1.2 Motivation

Why should we study computation of mathematical models for complex industrial processes?

Numerical computation of mathematical models is an integrated part of process optimization and real-time control in modern process industries. Optimization and control of industrial processes rely on various optimization strategies and control actions in real-time. These strategies an actions are typically designed from mathematical models of the processes. Therefore, numerical computation of mathematical models of the processes becomes an integrate part of process optimization and real-time control for many processes in modern process industries.

Most processes in modern process industries can be well optimized and controlled by using simple strategies such as proportional-integral-derivative (PID) algorithm. For these processes, the process dynamics are well understood and the implementation of numerical methods for process model computation is mature. Solutions to the process models can be obtained numerically with an acceptable consumption of computing resources and execution time. Thus, there is no need to design advanced and sophisticated computational methods for computation of the mathematical models. Therefore, this book does not address computational problems of such processes.

However, there are many complex industrial processes which behaves with complex behaviours or a large number of degrees of freedom. Nonlinear dynamics, multiplicity which shows co-existence of several steady states, bifurcation and a huge number of coupled model equations are just a few of many examples. To capture the dynamic features of such complex systems, effective methods are required in order to efficiently derive numerical solutions to the process models effectively and efficiently.

Also, modern process industries tend to increasingly integrate multiple processes into a single unit or a tightly coupled production line for improved system performance and/or reduced energy consumption. For example, integration of chemical reaction and distillation separation into a single unit leads to the design of compact reactive distillation columns. As another example, integration of multiple chemical reactors, each of which may exhibits complex spatiotemporal concentration patterns and other dynamics, leads to tighter system coupling, more delay variables, and more types of symmetry, inducing more complicated bifurcation and coexistence of multiple steady states. A further example is a continuous galvanizing production line, which will be discussed later in Chapter 4, is an integration of multiple and tightly coupled processes. Integration of multiple processes in modern process industries results in large-scale and complex process models, which require significant computing effort for numerical computation.

Investigations into efficient computation of mathematical models for complex industrial processes also greatly enhance our capability to tackle many problems that would otherwise be considered to be too complicated to handle. Typically, such problems may not be easily solved analytically and thus require effective and efficient numerical methods for model computation. Numerical computation makes it possible to deal with complex process dynamics and large-scale model problems with affordable computing resources and execution time.

Furthermore, a good understanding of model computation theory underlying various numerical methods enables appropriate use of software tools to solve computational problems for complex industrial processes. Nowadays, there are many commercial and open source software packages available for numerical computation of mathematical models. On the commercial side, typical examples are MATLAB, Maple, and Aspen Plus. Among open source software packages is SciLab. Knowledge of theoretic developments of computing methods is crucial for deriving useful numerical results and interpreting the results.

Finally, many computational problems of mathematical models cannot be directly solved using existing software packages. In this case, a good understanding of computation of mathematical models will facilitate design and development of computer programs to solve the problems. There are many occasions that we have to write our own programs for model computational problems under investigation.

With all the above mentioned reasons, we are well motivated to study computation of mathematical models for complex industrial processes. We are now ready to describe several specific and major problems we are going to address in computation of mathematical models for complex industrial processes. They include process modelling, model approximation, algorithm design and setup, and interpretation and verification of results. When discussing these problems, we will also identify some challenges around the problems.

It is worth mentioning that there are also many other issues in the broad area of process modelling and model computation. However, these issues are not the focus of this book dedicated to model computation for complex industrial processes.

1.3 Process Modelling

Process modelling is the first major problem to be addressed in this book for computation of mathematical models for a specific application. Depending how and where the process models to be established will be used, methods for process modelling may differ significantly. For example, in comparison with online process monitoring, real-time control typically requires much faster computation of process models and model-based control actions. In this sense, it is crucial to build process models with sufficient details of the process dynamics yet simple enough for efficient computation. Furthermore, a decision has to be made on whether the process models should be established through theoretical derivation or experimental investigations.

In modeling complex industrial processes, a significant challenge is to understand the process dynamics. This has not been an easy task. One reason is that some industrially important process variables, which are ideal control variables, are not measurable in real time with existing sensing technologies. An example is industrial crystallization processes, in which the quality of crystals as represented by crystal size and its distributio...

Table of contents

• Cover
• Halftitle
• Advances in Process Systems Engineering
• Title
• Copyright
• Preface
• Contents
• List of Figures
• List of Tables
• 1. Introduction
• 2. Fundamentals of Process Modelling and Model Computation
• 3. Finite Difference Methods for Ordinary Differential Equation Models
• 4. Finite Difference Methods for Partial Differential Equation Models
• 5. Wavelets-Based Methods
• 6. High Resolution Methods
• 7. Comparative Studies of Numerical Methods for SMB Chromatographic Processes
• 8. Conclusion