Non-linear lumping

we define the necessary and sufficient conditions for exact nonlinear lumping. If we start with the system of equations (4.1) then the new lumped variables are defined by an fl-dimensional non-linear transformation c = h(c) and the new n-dimensional equation system becomes 

The conditions for exact lumping can be expressed in different ways. If we define the Jacobian of the transformation h(c) as Z h,c(c) = dh/dc, then 

Comparison with the linear case becomes more apparent if we redefine the system using a linear partial differential operator A, which, using indices notation, is given by 

By comparison with previous work on linear lumping, it is clear that finding the non-linear lumping function h depends on finding canonical forms for the operator A and/or on finding the invariant manifolds of the original system. This parallels the linearized case where we searched for canonical forms for the Jacobian J and its invariant subspaces. 

---

Approximate non-linear lumping in systems with time-scale separation... 

Application of approximate non-linear lumping to the hydrogen oxidation example... 

G. Li, A.S. Tomlin, H. Rabitz and J. Toth, A General-Analysis of Approximate Non-Linear Lumping in Chemical-Kinetics, 1 Unconstrained Lumping, J. Chem. Phys. 101 (1994) 1172-1187. 

Calculated reaction rates can be in the spatially ID model corrected using the generalized effectiveness factor (rf) approach for non-linear rate laws. The effect of internal diffusion limitations on the apparent reaction rate Reff is then lumped into the parameter evaluated in dependence on Dc>r, 8 and Rj (cf. Aris, 1975 Froment and Bischoff, 1979, 1990 Leclerc and Schweich, 1993). 

Typically, a non-linear system dynamic model is made up of individual lumped models of the components which at a minimum conserve mass and energy across the given component, but may also have a momentum equation if pressure drops must also be analyzed. For most dynamic problems of interest in hybrid studies, however, the momentum equation may be taken as quasi-steady (unless the solver requires the dynamic form to perform the numerical solution). Higher fidelity individual models or reduced order models (ROMs) can also be used, where the connection to the system model would be made at each subcomponent boundary. Since dynamic systems modeling is not as common as steady-state modeling, some discussion of modeling approaches will be given. There are two primary methods used to provide solutions for the pressure-flow dynamics of a system model. 

An important problem in catalysis is to predict diffusion and reaction rates in porous catalysts when the reaction rate can depend on concentration in a non-linear way.6 The heterogeneous system is modeled as a solid material with pores through which the reactants and products diffuse. We assume for diffusion that all the microscopic details of the porous medium are lumped together into the effective diffusion coefficient De for reactant. 

These authors [32, 33] have considered an alternative classification based on the nature of the variables involved in the model. They classify models by grouping them into opposite pairs deterministic vs. probabilistic, linear vs. non-linear, steady vs. non-steady state, lumped vs. distributed parameters models. In a lumped parameters model, variations of some variable (usually a spatial one) are ignored and its value is assumed to be uniform throughout the entire system. On the other hand, distributed parameters models take into account detailed variations of variables throughout the system. In the kinetic description of a chemical system, lumping concerns chemical constituents and has been widely used (see Sects. 2.4 and 2.5). 

Antos, D., Kaczmarslci, K., Wojciech, P., Seidel-Morgenstem, A. Concentration dependence of lumped mass transfer coefficients - Linear versus non-linear chromatography and isocra-tic versus gradient operation, J. Chromatogr. A, 2003, 1006, 61-76. 

We circumvent this problem in 2fa-like methods by lumping the non-linear terms into the local V and determining its symmetry. Now we can use an expression like Eq. (20) to simplify matrix elements over atom-centered basis functions when there are many symmetry-related atoms. For a given symmetry-adapted basis function centered on atom type C we expand the basis function into terms centered on the N symmetry-equivalent atoms,... 

The model was completed with the adsorption-oligomerization equilibria for formaldehyde, and the lumped kinetic parameters were estimated with non-linear regression. Generally, the model gave a very good agreement with experimental data obtained from aldolization of propion- and butyraldehyde. 

The presence of adsorbate interactions make this approach fail totally, as the isotherms are non-linear. The Newton-Raphson algorithm is a practical method of calculation for surface coverages of a 13 component FG isotherm. We make the further assumption that for species within a given lump, the adsorbate interaction energies are identical. In addition, we need to specify only the interaction energy between dissimilar pairs. Thus we have recourse to a Monte- Carlo Procedure(15)... 

Nevertheless, in the seventies and eighties complete new instruments came to our disposal in the forth of new computer technology, without leading very much to new process technology This is the more remarkable since in reality processes have a very high non-linear character, e g. heterogeneously catalysed processes. Partly, since linear partial processes are lumped together non-linearly and partly since the partial processes are non-linear. 

Eq. 6.2.6 was solved analytically to obtain the operation curve of the reactor (X vs t). Lumped kinetic parameters were determined by non-linear regression of experimental data using the numerical method of Newton-Raphson with first-order Taylor series expansion. Lumped parameters were smooth functions of temperature all parameters were adequately fitted to second order polynomials except for D that required a fourth order polynomial. The model can be used for reactor temperature optimization and can be extended to prolonged sequential batch operation provided that a sound model for enzyme inactivation is validated (Illanes et al. 2005b). 

The seismic analysis of the core is performed with the two-dimensional special purpose computer codes CRUNCH-2D and MCOCO, which account for the non-linearities in the structural design. Both CRUNCH-2D and MCOCO are based on the use of lumped masses and inertia concepts. A core element, therefore, is created as a rigid body while the element flexibilities are input as discrete springs and dampers at the corners of the element. CRUNCH-2D models a horizontal layer of the core and the core barrel structures (Figure 3.7-7). The model is one element deep and can represent a section of the core at any elevation, MCOCO models a strip of columns in a vertical plane along a core diameter and includes column support posts and core barrel structures (Figure 3.7-8). The strip has a width equal to the width of a permanent reflector block. Both models extend out to the reactor vessel,... 

---