Page 260 - Process Modelling and Simulation With Finite Element Methods

Coupling Variables Revisited 247

multiphysics is not an afterthought for dealing with awkward situations. It
permits treating discrete systems on an even footing with continuum systems
characterized not only by pde constraints, but by integral constraints as well.
Essentially, coupling variables permit nonlocal and discrete modelling.
In sections 7.3 (scalar), 7.4 (projection) and 7.5 (extrusion) we revisit
coupling variables to explore FEMLAB treatment of inverse problems, line
integrals, and integralhtegro-differential equations, respectively.
Scalar Coupling Variables

Undoubtedly, scalar coupling variables are the conceptually easiest to grasp. In
chapter 4, scalar coupling variables were used to link up a recycle stream in a
flowsheet for a heterogeneous chemical reactor - the output of the reactor,
suitably scaled, re-enters with the feed stream. An abstract 0-D element in a
second geometry was created for the purpose of modeling a buffer tank that
achieved the algebraic relationship between the recycle stream and the reactor
outlet. Very simply, a scalar coupling is a single value passed to the destination
domain, subdomain, boundary, or edge, where it is used anywhere in the
description of the domain FEM residuals. The scalar coupling variable is
created by an integration on the source domain. Since in our example, sources
were 0-D (endpoints or the single element construct), the integrations were
trivially the same as the integrand. Furthermore, that buffer tank model was
artificial since the recycle relations could have been more readily incorporated in
a weak boundary constraint without recourse to the second domain. So we have
yet to see an example of scalar coupling variables where the source integration
was non-trivial and the coupling itself essential. In the next subsection, we
tackle an inverse problem where coupling is essential and intricate. An inverse
problem has the connotation that there is an associated forward problem that is
well-posed, but that the inverse problem is ill-posed. Our selected inverse
problem is a tomographic inversion for electrical capacitance tomography.

Electrical Capacitance Tomography
Process tomography has matured as an engineering science in the past decade.
One of the most common configurations is electrical capacitance tomography,
frequently used for imaging processes with multiphase flows in cylindrical
pipelines. Sensing of multiphase pipeline flows with information about the
distributed flow of dispersed phases can be crucial to tight control of chemical
and processing unit operations. Non-invasive and non-intrusive measurements
of two-phase flow are notoriously difficult to obtain. The difficulty is often
exacerbated by the highly time-varying flows some times encountered in gas-
liquid flows in the oil and gas production industry. Accurate measurements of
transients in the flow and instantaneous phase distributions cannot be achieved.