Equation-oriented modeling

Sarma, P. V. L. N. and G. V. Reklaitis. Optimization of a Complex Chemical Process Using an Equation Oriented Model. Math Prog Study 20 113-160 (1982). 

Modeling Equation-oriented models and large scale solvers for... 

To construct equation-oriented models for an entire process, it becomes important to identify specifications that are consistent, avoiding overspecifying or underspecifying subsets of the equations. When convergence is not achieved, facilities are provided to examine the values of selected variables and the residuals of selected equations. This requires well-designed programs that can display subsets of variables and equation residuals. 

Figure 2. Molecular orientation model of bilayer membranes and schematic explanation of Kasha s molecular exciton theory (see equation (2)).

The older modular simulation mode, on the other hand, is more common in commerical applications. Here process equations are organized within their particular unit operation. Solution methods that apply to a particular unit operation solve the unit model and pass the resulting stream information to the next unit. Thus, the unit operation represents a procedure or module in the overall flowsheet calculation. These calculations continue from unit to unit, with recycle streams in the process updated and converged with new unit information. Consequently, the flow of information in the simulation systems is often analogous to the flow of material in the actual process. Unlike equation-oriented simulators, modular simulators solve smaller sets of equations, and the solution procedure can be tailored for the particular unit operation. However, because the equations are embedded within procedures, it becomes difficult to provide problem specifications where the information flow does not parallel that of the flowsheet. The earliest modular simulators (the sequential modular type) accommodated these specifications, as well as complex recycle loops, through inefficient iterative procedures. The more recent simultaneous modular simulators now have efficient convergence capabilities for handling multiple recycles and nonconventional problem specifications in a coordinated manner. 

The more common method could best be described as the Equation Oriented Method (EOM). A good example of this method s use is by Schobeiri et al. [6], who applied it to detailed gas turbine modeling. Individual models are divided into two main types of elements, flow elements and pressure elements. 

Industrial Applications of Plant-Wide Equation-Oriented Process Modeling—2010... 

Figure 2 Sequential modular (SM) and equation-oriented (EO) modeling difficulty versus complexity.

One approach to a solution of this problem was put forward by Hansen (14), who derived general equations which express the overall experimental film absorbance in terms of the external reflectance of the substrate. These relations contain within them expressions for the individual anisotropic extinction coefficients in each geometric orientation. Solution of these general equations for the anisotropic extinction coefficients allows for an unambiguous description of the dipole orientation distribution when combined with a defined orientation model. 

The modelling tools in current commercial simulators may roughly be classified into two groups depending on their approach block-oriented (or modular) and equation-oriented. 

When there are multiple recycles present, it is sometimes more effective to solve the model in a simultaneous (equation-oriented) mode rather than in a sequential modular mode. If the simulation problem allows simultaneous solution of the equation set, this can be attempted. If the process is known to contain many recycles, then the designer should anticipate convergence problems and should select a process simulation program that can be run in a simultaneous mode. 

Physical property estimation procedures are at the heart of the design procedures. The group-contribution section provides facilities for entering group contribution and equation oriented correlations. Models in the form of LISP code are entered for both types of estimation techniques. Additionally, groups and their contributions are specified for group contribution techniques. 

Another most important question in anomalous dielectric relaxation is the physical interpretation of the parameters a and v in the various relaxation formulas and what are the physical conditions that give rise to these parameters. Here we shall give a reasonably convincing derivation of the fractional Smoluckowski equation from the discrete orientation model of dielectric relaxation. In the continuum limit of the orientation sites, such an approach provides a justification for the fractional diffusion equation used in the explanation of the Cole-Cole equation. Moreover, the fundamental solution of that equation for the free rotator will, by appealing to self-similarity, provide some justification for the neglect of spatial derivatives of higher order than the second in the Kramers-Moyal expansion. In order to accomplish this, it is first necessary to explain the concept of the continuous-time random walk (CTRW). 

We shall now almost exclusively concentrate on the fractal time random walk excluding inertial effects and the discrete orientation model of dielectric relaxation. We shall demonstrate how in the diffusion limit this walk will yield a fractional generalization of the Debye-Frohlich model. Just as in the conventional Debye relaxation, a fractional generalization of the Debye-Frohlich model may be derived from a number of very different models of the relaxation process (compare the approach of Refs. 22, 23, 28 and 34—36). The advantage of using an approach based on a kinetic equation such as the fractional Fokker-Planck equation (FFPE) however is that such a method may easily be extended to include the effects of the inertia of the dipoles, external potentials, and so on. Moreover, the FFPE (by use of a theorem of operational calculus generalized to fractional exponents and continued fraction methods) clearly indicates how many existing results of the classical theory of the Brownian motion may be extended to include fractional dynamics. 

We shall now demonstrate how the CTRW in the diffusion limit may be used to justify the fractional diffusion equation. We consider an assembly of permanent dipoles constrained to rotate about a fixed axis (the dipole is specified by the angular coordinate unit circle with fixed angular spacing A. We note that A may not necessarily be fixed for example, if we have a Gaussian distribution of jumps, the standard deviation of A serves as a fixed quantity. A typical dipole may remain in a fixed orientation at a given site for an arbitrary long waiting time. It may then reorient to another discrete orientation site. This is the discrete orientation model. 

A second solution method is based on an equation-oriented approach, which represents each process unit with a set of equations that are either part of the simulator model library or provided by the user. This method frequently uses an ordered set of equations. Procedures have been developed to automatically determine this information after the simulation flow sheet topology and the complete set of equations have been defined for the process simulation. The equations for all of the process units are solved simultaneously. The equations describing every process in the simulation flow sheet are solved by standard numerical methods that are discussed later. The methods for simulation flow sheet decomposition and equation ordering are covered in the work of Cameron and Hangos and Westerberg et al. ... 

---