Engineers and other applied scientists are frequently faced with models of complex systems for which no rigorous mathematical solution can be calculated. To predict and calculate the behaviour of such systems, numerical approximations are frequently used, either based on measurements of real life systems or on the behaviour of simpler models. This is essential work for example for the process engineer implementing simulation, control and optimization of chemical processes for design and operational purposes. This fourth in a suite of five practical guides is an engineer's companion to using numerical methods for the solution of complex mathematical problems. It explains the theory behind current numerical methods and shows in a step-by-step fashion how to use them. The volume focuses on differential and differential-algebraic systems, providing numerous real-life industrial case studies to illustrate this complex topic. It describes the methods, innovative techniques and strategies that are all implemented in a freely available toolbox called BzzMath, which is developed and maintained by the authors and provides up-to-date software tools for all the methods described in the book. Numerous examples, sample codes, programs and applications are taken from a wide range of scientific and engineering fields, such as chemical engineering, electrical engineering, physics, medicine, and environmental science. As a result, engineers and scientists learn how to optimize processes even before entering the laboratory. With additional online material including the latest version of BzzMath Library, installation tutorial, all examples and sample codes used in the book and a host of further examples.

A cutting-edge guide to modelling complex systems with differential-algebraic equations, suitable for applied mathematicians, engineers and computational scientists.

Systems of differential-algebraic equations (DAEs) are used to model an incredible variety of dynamic phenomena. In the chemical process industry in particular, the numerical simulation of detailed DAE models has become a cornerstone of many core activities including, process development, economic optimization, control system design and safety analysis. In such applications, one is primarily interested in the behavior of the model solution with respect variations in the model inputs or uncertainties in the model itself. This thesis addresses two computational problems of general interest in this regard. In the first, we are interested in computing a guaranteed enclosure of all solutions of a given DAE model subject to a specified set of inputs. This analysis has natural applications in uncertainty quantification and process safety verification, and is used for many important tasks in process control. However, for nonlinear dynamic systems, this task is very difficult. Existing methods apply only to ordinary differential equation (ODE) models, and either provide very conservative enclosures or require excessive computational effort. Here, we present new methods for computing interval bounds on the solutions of ODEs and DAEs. For ODEs, the focus is on efficient methods for using physical information that is often available in applications to greatly reduce the conservatism of existing methods. These methods are then extended for the first time to the class of semi-explicit index-one DAEs. The latter portion of the thesis concerns the global solution of optimization problems constrained by DAEs. Such problems arise in optimal control of batch processes, determination of optimal start-up and shut-down procedures, and parameter estimation for dynamic models. In nearly all conceivable applications, there is significant economic and/or intellectual impetus to locate a globally optimal solution. Yet again, this problem has proven to be extremely difficult for nonlinear dynamic models. A small number of practical algorithms have been proposed, all of which are limited to ODE models and require significant computational effort. Here, we present improved lower-bounding procedures for ODE constrained problems and develop a complete deterministic algorithm for problems constrained by semi-explicit index-one DAEs for the first time.

Scientific Computing in Chemical Engineering gives the state of the art from the point of view of the numerical mathematicians as well as from the engineers. The application of modern methods in numerical mathematics on problems in chemical engineering, especially reactor modeling, process simulation, process optimization and the use of parallel computing is detailed.

The term differential-algebraic equation was coined to comprise differential equations with constraints (differential equations on manifolds) and singular implicit differential equations. Such problems arise in a variety of applications, e.g. constrained mechanical systems, fluid dynamics, chemical reaction kinetics, simulation of electrical networks, and control engineering. From a more theoretical viewpoint, the study of differential-algebraic problems gives insight into the behaviour of numerical methods for stiff ordinary differential equations. These lecture notes provide a self-contained and comprehensive treatment of the numerical solution of differential-algebraic systems using Runge-Kutta methods, and also extrapolation methods. Readers are expected to have a background in the numerical treatment of ordinary differential equations. The subject is treated in its various aspects ranging from the theory through the analysis to implementation and applications.