Page 29 - Process Modelling and Simulation With Finite Element Methods

16 Process Modelling and Simulation with Finite Element Methods

We can lay those at the user’s door morally, but since FEMLAB is not “idiot-
proof ’, we are free to specify “badly conditioned or inconsistent” models.
(Politely, wrong.) FEMLAB may never generate an error message at all. With
just about every novice user who has sought my advice, I have shown them
where they have specified an inconsistent boundary condition like 0=1 in
General PDE mode. Yet, in many cases, FEMLAB generates output that is not
superficially wrong, but certainly not satisfactory in the case of modeling,
conceptual, or syntactical errors. At this point, the “tough love” approach is all
that can be advised - there is no substitute for experience. This book
encapsulates many of my experiences. I haven’t tried to sugar-coat my chapters
so that all models are magically perfect. Think of this as a cookbook that shows
both good recipes and bad ones, but each labeled and the latter coming with a
health warning. For instance, in Chapter Seven, I tried four attempts at modeling
the population balance equations before the last came good. So you will learn
from my mistakes that I own up to as well as from my triumphs. For better or
worse, every modeling attempt I made during the six months of writing this book
has been included. I will pat myself on the back for persistence, because in the
end they all worked, but at many points I had my doubts and frustrations. I am
pleased not to have cherry picked the models. Of course I have not shown every
single computation nor “what if’ line that I pursued in each model.

0.3 Chapter Synopsis

Chapter One treats the basics of numerical analysis with FEMLAB. No doubt
many of the example models are artificial in that if you were handed the
modeling problems in Chapter One, FEMLAB would not be the obvious choice
of computational platform. The topics of root finding, numerical integration by
marching, numerical integration of ordinary differential equations, and linear
system analysis are universal to numerical analysis. They form the basis of my
previous lecture courses in FORTRAN programming and chemical engineering
problem solving with Mathernatica. For pedagological purposes, Chapter One
provides a firm basis for understanding what FEMLAB does. The common
applications in chemical engineering that are treated as examples, flash
distillation, tubular reactor design, diffusive-reactive systems, and heat
conduction in solids, are understandable to the non-chemical engineer as well.
Perhaps the single most important modeling feature introduced here, however, is
the use of a conceptual 0-dimensional model. Consisting of a single element, the
0-D construct introduces a variable which is a scalar for which an ODE in time
or an algebraic equation can be specified. This construct is important for
describing equations or systems of equations that are mixed (partia1)differential-
algebraic, and is utilized with the extended multiphysics feature of FEMLAB in
the more complicated models presented later.