Problems with Algebraic Models

The following parameter estimation problems were formulated from research papers available in the literature and are left as exercises. 

---

However, all of these studies determine only approximate or parameterized optimal control profiles. Also, they do not consider the effect of approximation error in discretizing the ODEs to algebraic equations. In this section we therefore explore the potential of simultaneous methods for larger and more complex process optimization problems with ODE models. Given the characteristics of the simultaneous approach, it becomes important to consider the following topics ... 

We would like to highlight that in the mathematical programming approach with algebraic modeling systems, the declaration of models is expressed with an index notation, and not matrix notation, as seen in Table 10.4, where both notations are shown for an LP problem. 

The above parameter estimation problem can now be solved with any estimation method for algebraic models. Again, our preference is to use the Gauss-Newton method as described in Chapter 4. 

Based onEq. (3-51), the time response y(t) should be strictly overdamped. However, this is not necessarily the case if the zero is positive (or xz < 0). We can show with algebra how various ranges of K and X may lead to different zeros (—l/xz) and time responses. However, we will not do that. (We ll use MATLAB to take a closer look in the Review Problems, though.) The key, once again, is to appreciate the principle of superposition with linear models. Thus we should get a rough idea of the time response simply based on the form in (3-50). 

Dynamic simulations are also possible, and these require solving differential equations, sometimes with algebraic constraints. If some parts of the process change extremely quickly when there is a disturbance, that part of the process may be modeled in the steady state for the disturbance at any instant. Such situations are called stiff, and the methods for them are discussed in Numerical Solution of Ordinary Differential Equations as Initial-Value Problems. It must be realized, though, that a dynamic calculation can also be time-consuming and sometimes the allowable units are lumped-parameter models that are simplifications of the equations used for the steady-state analysis. Thus, as always, the assumptions need to be examined critically before accepting the computer results. 

One may therefore wish to know what are the potential functions V(r) that correspond to a given algebraic model. The general answer to this question is provided by the solution of the inverse Schrodinger problem Since one knows the spectrum of the algebraic model, one finds the potential that reproduces the spectrum.1 A simple approach consists in expanding the potential V(r) into a set of functions with unknown coefficients, say... 

Remark 1 If no approximation is introduced in the PFR model, then the mathematical model will consist of both algebraic and differential equations with their related boundary conditions (Horn and Tsai, 1967 Jackson, 1968). If in addition local mixing effects are considered, then binary variables need to be introduced (Ravimohan, 1971), and as a result the mathematical model will be a mixed-integer optimization problem with both algebraic and differential equations. Note, however, that there do not exist at present algorithmic procedures for solving this class of problems. 

A common alternative is to synthesize approximate state functions by linear combination of algebraic forms that resemble hydrogenic wave functions. Another strategy is to solve one-particle problems on assuming model potentials parametrically related to molecular size. This approach, known as free-electron simulation, is widely used in solid-state and semiconductor physics. It is the quantum-mechanical extension of the classic (1900) Drude model that pictures a metal as a regular array of cations, immersed in a sea of electrons. Another way to deal with problems of chemical interaction is to describe them as quantum effects, presumably too subtle for the ininitiated to ponder. Two prime examples are, the so-called dispersion interaction that explains van der Waals attraction, and Born repulsion, assumed to occur in ionic crystals. Most chemists are in fact sufficiently intimidated by such claims to consider the problem solved, although not understood. 

Algebraic Stress Models (ASM) Accounts for anisotropy Combines generality of approach with the economy of the k-s model Good performance for isothermal and buoyant thin shear layers Restricted to flows where convection and diffusion terms are negligible Performs as poorly as k-e in some flows due to problems with s equation Not widely validated... 

---