Page 142 - Process Modelling and Simulation With Finite Element Methods

Multiphysics 129

Now try the initial condition c(t0)=nli,. With the time dependent solver, set
output times to 0:0.001:0.1, then animate the solution. You should be able to
watch the initial condition pass completely out of the reactor and converge to the
steady state solution found by the two previous methods. This should be a clear
signal that the time dependent solver may be an essential tool in attacking non-
convergence. Even in problems that have no inherent unsteady time scale, pseudo-
time dependent solution may be essential to finding a converged stationary
solution. If so, then we can be fairly certain that the steady state so found is stable,
since it is attractive.

Exercise 3.1
This chapter is entitled “Multiphysics.” The problem statement is definitely for
two physics modes (heat and mass transport with reaction), yet due to
Amundson’s technique, the problem could be simplified to “single physics” for
a
-- - 1. Try implementing the calculation with the ChemEng Module modes
3
convection and conduction (cc) and convection and diffusion (cd) with the same
parameters, but as written in equations (3.7), (3.8). Take the initial condition to
be uniform temperature n=l. Solve for the steady state after a long time, or use
the steady solver. Compare you results with the Amundson technique solution
given here.

3.4 Heterogeneous Reaction in a Porous Catalyst Pellet

It would be an injustice not to draw on the FEMLAB Model Library for an
example of multiphysics. Although the chemical engineering curriculum does
not contain many examples of multiphysics partial differential equations, the
same cannot be said for the chemical engineering model library of FEMLAB. In
keeping with the complementary of this text with the FEMLAB manual set,
however, we must treat any problem that we extract differently. This section is
inspired by the heterogeneous reaction modeling in a porous catalyst pellet,
treated in [9]. The model uses the incompressible Navier-Stokes application
mode and couples the results to the convection and diffusion mode through the
multiphysics facility of FEMLAB in two-dimensions (c.f eqn (3.7)), adding the
additional reaction term in the pellet subdomain, but without convection
(reaction-diffusion model). This clearly counts as multiphysics by either
definition, since there are two different PDEs with independent variables u,v,p,
and c having different units. The twist that we add to the model is to decouple
the multiphysics.
Because the reaction is taken to be isothermal and constant density, there is no
back action coupling the concentration field into the Navier-Stokes equations. The
mass transport requires knowledge of the velocity field to compute convection, but