Page 121 - Process Modelling and Simulation With Finite Element Methods

108 Process Modelling and Simulation with Finite Element Methods

Of course either definition is a Byzantine notion, so let’s make it concrete by
examples. Are fluid dynamics multiphysics? Yes, but only on the technicality that
pressure is an independent variable which has different units to velocity. Is it
multiphysics in FEMLAB? No, because there is a single Navier-Stokes application
mode. Is heat transfer multiphysics? No, there is only one independent variable -
temperature, and only one FEMLAB application mode. What about thermofluids?
Yes, as velocity, temperature, and combustion conversion are three independent
variables with different units, and there are three transport equations coupled.
Many typical research areas in chemical engineering are multiphysics:
physicochemical hydrodynamics, magneto-hydrodynamics, electrokinetic flow,
multiphase flow, double diffusion, and separations.
FEMLAB deals with specific common multiphysics applications by creating
application modes that are a full description of single field models, but can be
readily coupled to other application modes. The user provides the coupling by
specifying PDE terms and boundary and initial conditions symbolically.
FEMLAB does the “bookkeeping” to make sure that application modes have
different specifications for the field variables and derived quantities that are
commonly computed for a typical application. For two coupled application modes,
FEMLAB assembles the FEM description (through the sparse matrices K,N,L,M)
for each mode, including the user specified coupling terms. If the pre-made
application modes do not cover the user’s coupled system, then the user can adapt
as many coefficient form, general form, or weak form systems as necessary to
describe their dynamics.
A few examples will illustrate multiphysics modelling to much better effect.
Chapter 8 of the book of Ramirez [l] has a wealth of multiphysics PDE models
with chemical engineering applications. They are computed on simple domains
with finite difference methods coded with full detail in MATLAB. We shall adapt
several such examples to FEMLAB models here. But first we will attempt some
simple buoyant convection problems.

3.2 Buoyant Convection
Coupling momentum transport and heat transport is a well studied area of
transport phenomena. The governing equations are
au
-+ u . VU = - -Vp + VV~U + -T w
1
at P P
v*u=o (3.1)
JT
_+U.VT = KV~T
dt