Page 88 - Process Modelling and Simulation With Finite Element Methods

Partial Differential Equations and the Finite Element Method 15

4000 002 004 006 008 01 012 014 016 018 02
Time
Figure 2.5 Temperature at position x=OS
Exercise 2.2

Solve for the same plots as Figure 2.4 and 2.5 with constant diffusivity f(T)=l.
How do the profiles differ?

2.1.3 The wave equation: A hyperbolic PDE

The 1-D wave equation (2.3) has also been studied to death. Nor does it
particularly turn up in chemical engineering applications. The obvious place is
the study of sound waves, which receives little attention in the chemical
engineering curriculum. So the major reason for including it here is
completeness. Does this mean that the wave equation is unimportant in the
chemical and process industries? Probably not. For instance, reactors are
known to exhibit chemical waves, waves on interfaces in condensers, swirl
atomizers, and distillation columns effect mass transfer, and acoustics, power
ultrasound, and sonochemistry are receiving much attention on the research
front. It just so happens that chemical engineers are taught little about waves,
and thus it is difficult to find classical textbook analyses of chemical engineering
unit operations in which waves play any role.
In this subsection, we will attempt to make the demonstration of wave
dynamics slightly more interesting by the use of periodic boundary conditions
and animation. Start up FEMLAB and enter the Model Navigator: