Rational lumping

First, a priori criteria are needed for a rational lumping basic mechanism and/or model simplification (see section 3). 

A relatively simple mathematical model composed of 21 or 23 transcendental and rational equations numbered (7.25) to (7.47) was presented to describe the steady-state behavior of type IV FCC units. The model lumps the reactants and products into only three groups. It accounts for the two-phase nature of the reactor and of the regenerator using hydrodynamics principles. It also takes into account the complex interaction between the... 

In an efficient digital simulation, lumped loss factors of the form Gk (0)) are approximated by a rational frequency response Gk(c,mT). In general, the coefficients of the optimal rational loss filter are obtained by minimizing I Ylk (go) - Gk d r ) I with respect to the filter coefficients or the poles and zeros of the filter. To avoid introducing frequency-dependent delay, the loss filter should be a zero-phase, finite-impluse-response (FIR) filter [Rabiner and Gold, 1975], Restriction to zero phase requires the impulse response g.k(n) to be finite in length (i.e., an FIR filter) and it must be symmetric about time zero, i.e., ) k(-n) = gk(n). In most implementations, the zero-phase FIR filter can be converted into a causal, linear phase filter by reducing an adjacent delay line by half of the impulse-response duration. 

Note that we have replaced here the rational potential E by the poten tial E, measured versus some reference electrode. The difference between them is a constant, which we can lump into the equilibrium constant. 

For small bodies, Po 1, obviously, whereas for large lumps [11], Po A/r. These properties suggest the rational approximation... 

The DancofF correction in the rational approximation is thus equivalent to a Active reduction of the surface area of the lumps. This is consistent with the alternative interpretation of the Dancofif correction as the fractional reduction of the flux into an absorber lump due to the shadowing by the adjacent ones. These considerations suggest that a somewhat better approximation than Equation (7.11) will be obtained by simply introducing this surface correction into the average chord length, i.e., by equating... 

The rational basis of any reaction lumping is the concentra-tion-reactiontime-diagram of the respective reaction as it is derived from experimental results /I7/ as e.g. in Fig. 6 for the butene oxidation. 

In general the contour maps of the laplacian of the electron density nicely show the links between the regions of the electron density depletion with the regions of its concentration this was rationalized recently by Eskandari and coworkers in terms of the lump-hole interactions concept [55, 56]. 

These relations, sometimes called the Nernst-Planck equations (Bard and Faulkner, 2000), could be written down directly as a definition for Z),-. If this were done, then the restriction to dilute solutions in Eq. 6.1-3 and the implicit neglect of a reference velocity in the first line of Eq. 6.1-5 would be hidden in the final flux equation, lumped into the experimental coefficient D,. I find the derivation a sensible, reassuring rationalization, even though I know that it is arbitrary. 

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