Page 76 - Process Modelling and Simulation With Finite Element Methods

Chapter 2

PARTIAL DIFFERENTIAL EQUATIONS AND THE
FINITE ELEMENT METHOD

W.B.J. ZIMMERMAN and B.N. HEWAKANDAMBY
Department of Chemical and Process Engineering, University of Shefield,
Newcastle Street, Sheffield Sl 3JD United Kingdom
E-mail: w.zimmerman @ shejac. uk

Partial differential equations (PDEs) arise naturally in science and engineering from
complex balance equations. Commonplace PDEs are derived from conservation laws for
transport of mass, momentum, species and energy. Because these conservation laws are
integral equations over the domain, the PDEs that arise from the continuum hypothesis
have a structure that is readily represented by the finite element method as an
approximation. In this chapter, the three different classes of differential equations that
arise in spatial-temporal systems - elliptic, parabolic, and hyperbolic - are defined and
representative cases are treated by FEMLAB computations. An overview of the finite
element method is given, but greater depth of detail will await later chapters where the
applications particularly exploit features of finite element methods that intrinsically
permit elegant and accurate computation.

2.1 Introduction
Partial differential equations are usually found in science and engineering
applications as the local, infinitesimal constraint imposed by conservation laws
that are typically expressed as integral equations. The whole class of transport
phenomena due to conservation of mass, momentum, species and energy lead to
PDEs in the continuum approximation. Chemical engineers are well acquainted
with shell balances in transport phenomena studies for heat, mass and
momentum transfer.
In contrast to the previous chapter, where 0-D and 1-D spatial systems were
treated by FEMLAB with example applications in chemical engineering, the
chemical engineering curriculum is not overflowing with 2-D and 3-D example
computations of the solutions to PDEs. A rare example is found in [I]. In fact,
historically, many of the common chemical engineering models and design
formula are simplifications of higher spatial dimension dynamics that are treated
phenomologically. Resistance coefficients in fluid dynamics, mass transfer and
heat transfer coefficients, Thiele moduli in heterogeneous catalysis, McCabe-
Thiele diagrams for distillation column design, and many more common
techniques are convenient semi-empiricisms that mask an underlying transport or
non-equilibrium thermodynamics higher spatial dimension system, possibly

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