Page 77 - Process Modelling and Simulation With Finite Element Methods

64 Process Modelling and Simulation with Finite Element Methods

expressible as a PDE system, but traditionally thought too difficult to solve given
the complexity of the fundamental physical chemistry. These simpler
methodologies are still preferable for quick estimates desired for preliminary
design calculations, but may be insufficient for detailed design, retrofit, or
process analysis and optimization purposes. For fundamental science, these
methods are still migrating from chemical engineers to biotechnologist or
material scientists in the first approach to multidisciplinary work. Nevertheless,
computational fluid dynamics (CFD) has forever changed the paradigm for what
is considered the state-of-the-art in transport modeling. Phenomenological
methods may still have a niche, and a particular important one in interpreting
distributed system models, yet CFD has a unique role for visualization and
quantification of transport phenomena.
FEMLAB is not a “commercial CFD code”, but it will do some CFD. There
are several general purpose CFD packages available, with their own advantages
in supporting certain applications. By CFD, most process engineers would
envisage support for many turbulence and combustion models. FEMLAB,
however, has a different niche in the area of multiphysics. In addition to the
traditional transport phenomena that CFD treats, FEMLAB includes application
modes for electrodynamics, magnetodynamics, and structural mechanics,
permitting simultaneous treatment of these and transport phenomena. But its
greatest strengths are actually least trumpeted - first, the ease of “user defined
programming”, which is the ability to implement the user’s own model or
parametric variation of coefficients, boundary conditions, initial conditions and
to link to simultaneous physics, even on other domains; second, that it is built on
MATLAB so that all the programming functionality needed to set up greater
complexity of models or simulations is available, treating FEMLAB as a
convenient suite of subroutines for high-level finite element programming and
analysis. In the last chapter, we saw some of the power of user defined
programming and analysis. In this chapter, we introduce FEMLAB’s core
strength of finite element modeling of higher dimensional PDE systems. The
greater functionality of multiphysics, extended physics, and treatment on non-
PDE constraints will be left for later chapters.

Partial Differential Equations

PDEs are classified according to their order, boundary condition type, and
degree of linearity (yes, no or quasi). Amazingly, most PDEs encountered in
science and engineering are second order, i.e. the highest derivative term is a
second partial derivative. Is this a coincidence? Lip service is usually paid at
this point to variational principles underlying most of physics. Yet, recently
Frieden [2] has demonstrated that all known laws of physics can be derived from
the principle of minimum Fisher information, which naturally introduces a
second order operator of a field quantity as the highest order term in the