(196d) Canonical Primal-Dual (CPD) Formulation of Process Models

This paper presents a canonical form for chemical process systems models for steady state. The approach is motivated by the fact that current mainstream technologies for process modeling at the level of process flowsheet simulation and similar "rigorous" fidelity models have seen little fundamental change in more than 20 years. The sequential modular architecture is still relied upon. The successor "equation oriented" architecture has afforded new capabilities, but has not realized its full promise. The skilled modeling specialist can accomplish results if the required time can be invested, but the capabilities available to the mainstream user/modeler have improved little. The drawbacks of the standard methods are well known and remain.

The CPD modeling approach is intended to provide a means to mitigate or avoid the technical issues that to date make modeling software development a demanding task, and model use problematic. The core issues to be resolved are (I) Numerical convergence of the inherently nonlinear models; (II) Adequacy/consistency/specification of complex formulations and positive diagnosis of misformulation; and (III) Flexibility to embody all relevant physical phenomena, extending to applications beyond the conventional.

The formulation employs a single, uniform basic submodel structure that instantiates to form all parts of the entire large scale process model. The complete system is a collected composite of these coupled subsystems, possibly in nested hierarchy. Such a uniform representation is possible because the basic submodel is derived directly from the fundamental physics of the phenomenology.

Fundamental conservations are identified according to the underlying symmetries of the physical behavior. In addition to strain, momentum, and energy, relevant species are identified along with allowable interconversions. Nonconserved species such as photons are also included as needed. Then all physical phenomena are expressed as variational problems subject to the conservation constraints.

Physical modes are partitioned into microscopic and macroscopic. The microscopic modes are treated with a local equilibrium (LE) approximation. This partition of the subsystem evolves the thermodynamic behavior and the macroscopic transport diffusivities. The macroscopic modes are governed by a convection/diffusion/conduction model. Significantly, all these macroscopic phenomena are captured by a single variational problem.

Formulation in terms of variational principles is a very important feature. It allows the convexity characteristics of the governing physical principles to be recognized and exploited. When representing process models simply as algebraic equations it is difficult or impossible to do this - perhaps this partly explains the numerical difficulties that frequent the modeling-via-equations paradigm.

The main characteristics of the CPD formulation are summarized:

1. The physical phenomena are formulated in a single basic subsystem that applies to all points in space.

2. The phenomena are represented by a pair of coupled macroscopic-microscopic variational principles with convexity properties that are established a priori.

3. The macroscopic variational principle represents the behavior in terms of a set of primal fields (generalized densities) and dual fields (potentials).

4. The macroscopic behavior is integrated over the geometry of macroscopic volumes to obtain low-dimensional characterizations of equipment regions. Simplified basis functions are used to approximate the primal and dual fields. These could be empirically determined, for instance from pressure drop, heat, and mass transfer correlations for turbulent flow. Alternatively, methods ranging from simple approximate calculations to detailed CFD might be employed. The LE material properties are averaged over space by associated simple quadratures.

5. All equipment subsystems inherit the convexity characteristics of the intrinsic physical phenomena. They are interconnected at physical interfaces by continuity of both the primal flows and dual field quantities. Geometric congruence of the interconnections is unambiguous.

6. Model feasibility can be determined by analyzing a global linear system. Misformulation diagnosis is obtained using standard linear algebraic analysis.

7. The convexity properties of the variational principles rigorously guarantee that the combined system can be solved via homotopy.

This paper will explain the basic subsystem model that provides the core of the canonical primal-dual formulation. It will also briefly outline the significant properties mentioned above and their consequences for modeling large scale nonlinear process systems.