Page 164 - Process Modelling and Simulation With Finite Element Methods

Extended Multiphysics 151

Simply, the mass transfer flux and surface reaction constraints are not time
evolution equations. FEMLAB naturally gives these equations in the weak
form a ddt accumulation term in the time dependent solver. Without the
time evolution terms, these algebraic constraints make the system rather
stiff. The additional coupling variables also subtract from the "sparsity" of
the system, thereby making the sparse matrix solvers strain harder to converge.
So don't be surprised if the solver steps in extended multiphysics problems
take longer.

Time=O.l, concentration of U and V) Time=O 1, concentration of US and VS

0'5 ; 1'5 ; 7'5 4 3'5 ; 4'5 ; 01 " " " " "
03A 0 05 1 15 2 25 3 35 4 45 5
z coordinate z coordinate
Figure 4.8 Bulk concentrations decay SIOwlY. Figure 4.9 US is still depleted throughout the
reactor.

The results are shown in Figures 4.8 and 4.9. One would typically expect that in
a nonlinear reactive-diffusion problem at low Peclet number (about OS), the
steady state would be well on its way at t=0.1. But the influence of the imperfect
mixing in the buffer tank is apparent. The bulk and surface concentrations are
slowly developing towards their steady state profiles, but the de-charging of the
capacitor is a slow process, which will hamper the development of the steady
state. If further system time lags were modelled, say the piping and control
systems, then it is likely that this buffering resistance to developing the steady
state from start up would be exacerbated.

Exercise 4.1
In 53.3, we added an oscillatory disturbance to the steady state solution of
a tubular reactor and observed the transient response. Store the steady
state solution to this heterogeneous reactor with buffer tank process, then create
m-file functions for the initial conditions, adding an oscillatory disturbance
to U only. How many m-file functions do you need to specify the steady
state condition with oscillation? Comment about the transient response of the
system.