Low-dimensional models

By adopting the no-pair approximation, a natural and straightforward extension of the nonrelativistic open-shell CC theory emerges. The multireference valence-universal Fock-space coupled-cluster approach is employed [25], which defines and calculates an effective Hamiltonian in a low-dimensional model (or P) space, with eigenvalues approximating some desirable eigenvalues of the physical Hamiltonian. The effective Hamiltonian has the form [26]... 

Using the methods presented in Chapter 2, the above formulation can be used to derive a state-space realization of the slow dynamics of the type in Equation (2.48). The resulting low-dimensional model should subsequently form the basis for formulating and solving the control problems associated with the slow time scale, i.e., stabilization, output tracking, and disturbance rejection at the process level. 

We proposed a method for deriving nonlinear low-dimensional models for the dynamics in each time scale. Subsequently, we proposed a hierarchical controller design framework that takes advantage of the time-scale multiplicity, and relies on a multi-tiered structure of coordinated decentralized and supervisory controllers in order to address distributed and process-level control objectives. 

The non-stiff, low-dimensional model (6.63) is ideally suited for control purposes. We used it to design an input-output linearizing controller with integral action that manipulates QH and enforces a first-order response in the Tr dynamics, namely... 

Product state analysis offers a flexible way to obtain detailed state resolved information on simple surface reactions and to explore how their dynamics differ from the behaviour observed for H2 desorption [7]. In this chapter, we will discuss some simple surface reactions for which detailed product state distributions are available. We will concentrate on N2 formation in systems where the product desorbs back into the gas phase promptly carrying information about the dynamics of reaction. Different experimental techniques are discussed, emphasising those which give fully quantum state resolved translational energy distributions. The use of detailed balance to relate recombinative desorption measurements to the reverse, dissociation process is outlined and the influence of the surface temperature on the product state distributions discussed. Simple low dimensional models which provide a reference point for discussing the product energy disposal are described and then results for some surface reactions which form N2 are discussed in detail, emphasising differences with the behaviour of H2. 

At the other extreme, it may be argued that the traditional low-dimensional models of reactors (such as the CSTR, PFR, etc.) should be abandoned in favor of the detailed models of these systems and numerical solution of the full convection-diffusion reaction (CDR) equations using computational fluid dynamics (CFD). While this approach is certainly feasible (at least for singlephase systems) due to the recent availability of computational power and tools, it may be computationally prohibitive, especially for multi-phase systems with complex chemistry. It is also not practical when design, control and optimization of the reactor or the process is of main interest. The two main drawbacks/criticisms of this approach are (i) It leads to discrete models of very high dimension that are difficult to incorporate into design and control schemes. 

In all of the above cases, a strong non-linear coupling exists between reaction and transport at micro- and mesoscales, and the reactor performance at the macroscale. As a result, the physics at small scales influences the reactor and hence the process performance significantly. As stated in the introduction, such small-scale effects could be quantified by numerically solving the full CDR equation from the macro down to the microscale. However, the solution of the CDR equation from the reactor (macro) scale down to the local diffusional (micro) scale using CFD is prohibitive in terms of numerical effort, and impractical for the purpose of reactor control and optimization. Our focus here is how to obtain accurate low-dimensional models of these multi-scale systems in terms of average (and measurable) variables. 

The first approach is the discretization of the convection and the diffusion operators of the PDEs, which gives rise to a large (or very large) system of effective low-dimensional models. The order of these low-dimensional models depend on the minimum mesh size (or discretization interval) required to avoid spurious solutions. For example, the minimum number of mesh points (Nxyz) necessary to perform a direct numerical simulation (DNS) of convective-diffusion equation for non-reacting turbulent flow is given by (Baldyga and Bourne, 1999)... 

Using the above dimensionless parameters and variables, Eqs. (188)—(198) are written in dimensionless form and spatially averaged over transverse dimensions to obtain the low-dimensional model for non-isothermal homogeneous tubular reactors, which is given to order p by Eqs. (130) (134) with r((c)) being replaced by r((c), (0f and... 

Low-dimensional models for loop, recycle, and tank reactors could similarly be derived starting from the coupled mass and thermal balances. Here, we present the reduced models and refer to a previous publication (Chakraborty and Balakotaiah, 2004) for the derivation of these models. 

The low-dimensional model for a non-isothermal homogeneous tank reactor with premixed feed is given by... 

For the special case of a simple reaction A — B, the low-dimensional model for a CSTR with premixed feed consists of three differential equations and two algebraic equations. When the mass and thermal micromixing effects are ignored (r) = t]H — 0), cm — (c), (Of) — dfm — (ds), and we get the classical pseudohomogeneous CSTR model... 

Here, we extend the low-dimensional models derived for the case of a single reaction to the case of multiple homogeneous reactions represented by... 

The low-dimensional model for non-isothermal wall-catalyzed reaction in a tubular reactor is given by Eqs. (283)-(285) and... 

The accuracy of low-dimensional models derived using the L S method has been tested for isothermal tubular reactors for specific kinetics by comparing the solution of the full CDR equation [Eq. (117)] with that of the averaged models (Chakraborty and Balakotaiah, 2002a). For example, for the case of a single second order reaction, the two-mode model predicts the exit conversion to three decimal accuracy when for (j>2(— pDa) 1, and the maximum error is below 6% for 4>2 20, where 2(= pDd) is the local Damkohler number of the reaction. Such accuracy tests have also been performed for competitive-consecutive reaction schemes and the truncated two-mode models have been found to be very accurate within their region of convergence (discussed below). 

This regularized form [Eqs. (370) and (371)] has a much larger region of validity than the original low-dimensional model and gives qualitatively correct results for any p>0. As discussed in Section III, we can combine the above two equations to get a single hyperbolic regularized equation for Cm. 

Preference mapping can be accomplished with projection techniques such as multidimensional scaling and cluster analysis, but the following discussion focuses on principal components analysis (PCA) [69] because of the interpretability of the results. A PCA represents a multivariate data table, e.g., N rows ( molecules ) and K columns ( properties ), as a projection onto a low-dimensional table so that the original information is condensed into usually 2-5 dimensions. The principal components scores are calculated by forming linear combinations of the original variables (i.e., properties ). These are the coordinates of the objects ( molecules ) in the new low-dimensional model plane (or hyperplane) and reveal groups of similar... 

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