Adaptational parameter

Additionally to and a third adjustable parameter a was introduced. For a-values between 14 and 15, a form very similar to the Lennard-Jones [12-6] potential can be obtained. The Buckingham type of potential has the disadvantage that it becomes attractive for very short interatomic distances. A Morse potential may also be used to model van der Waals interactions in a PEF, assuming that an adapted parameter set is available. 

The adaptive parameters in the model were estimated by nonlinear and multiresponse regression, performed using the Fortran subroutine BURENL23 based... 

In order to obtain the best lit of the forty-six experimental data sets considered, we adopted thirteen adaptive parameters. The regression was performed using as experimental responses the CO conversion and the CH4, C2H4, C2H6,... 

For the range of industrially relevant conditions, the developed model could accurately predict both the observed CO conversion and the products distribution up to n = 49, in terms of total hydrocarbons, n-paraffins, and a-olefins. In particular, using thirteen adaptive parameters, the model is able to describe the typical deviations of the product distribution from the ASF model, i.e., the methane high selectivity, the low selectivity to C2 species, and the change of the slope of the ASF plot with growing carbon number. Accordingly, the present model can be applied to identify optimized process conditions that are suitable to grant the desired conversion with the requested products distribution. 

Laboratory data collected over honeycomb catalyst samples of various lengths and under a variety of experimental conditions were described satisfactorily by the model on a purely predictive basis. Indeed, the effective diffusivities of NO and NH3 were estimated from the pore size distribution measurements and the intrinsic rate parameters were obtained from independent kinetic data collected over the same catalyst ground to very fine particles [27], so that the model did not include any adaptive parameters. 

Canonical correlation analysis was used to relate small subsets of physicochemical parameters to the MDS space. Small subsets were necessary because in canonical correlation analysis, the number of stimuli should be greater than the number of dimensions and physicochemical parameters combined. The analysis revealed that a linear combination of two ADAPT parameters in Table 3 (number of oxygen atoms and chemical environment of substructure (7)) in addition to a concentration variable accounted for 63% of the arrangement of the pyrazine odor space. 

The rate expressions Rj — Rj(T,ck,6m x) typically contain functional dependencies on reaction conditions (temperature, gas-phase and surface concentrations of reactants and products) as well as on adaptive parameters x (i.e., selected pre-exponential factors k0j, activation energies Ej, inhibition constants K, effective storage capacities i//ec and adsorption capacities T03 1 and Q). Such rate parameters are estimated by multiresponse non-linear regression according to the integral method of kinetic analysis based on classical least-squares principles (Froment and Bischoff, 1979). The objective function to be minimized in the weighted least squares method is... 

Here rref is a suitable reference temperature (e.g. the mean temperature in the experimental field) and Aj and Bj the modified adaptive parameters obviously related to the original parameters kQj and Ej. Replacing Eq. (36) with Eq. (37) can... 

Barrer (1984) suggested a further refinement of the dual-mode mobility model, including diffusive movements from the Henry s law mode to the Langmuir mode and the reverse then four kinds of diffusion steps are basically possible. Barrer derived the flux expression based on the gradients of concentration for each kind of diffusion step. This leads to rather complicated equations, of which Sada (1987, 1988) proved that they describe the experimental results still better than the original dual-mode model. This, however, is not surprising, since two extra adaptable parameters are introduced. 

Krupp has defined Zq = 4x 10 cm as a measure for the atomic distance. For Figure 66(b), was calculated for different adhesion mechanisms using this value of Zq, although it does not represent a true atomic distance. Rather, it is an approximate or adaptive parameter the accurate value still needs to be determined. 

For convenience, two symmetry-adapted parameters were introduced... 

DeNOx - Scope of the model analysis was to evaluate on a quantitative basis the effective dependence of the intrinsic activity of the monoliths on the thermal sintering, and separate it from the contributions of inter-phase mass transfer and the effect of morphological modifications on intra-porous diffusion. When excess ammonia is present, as in the case of the experiments herein analyzed, then the Ealy-Rideal kinetic expression which is contained in the model of the SCR reactor reduces to a first order dependence on NO concentration under such conditions, an unique adaptive parameter, kc, accounts for the DeNOx intrinsic activity. Estimation of kc for the three calcined catalysts was obtained by fitting the model to each set of experiments. Input data included the operating conditions, the geometrical... 

FIGURE 65.6 Enlarged part of Figure 67.5 showing the influence of the values given to the adaptation parameter s. 

Let us indicate with v(a, x) an opportune function with P, adaptive parameters a, and with V (a., x) and v"(a, x) the first and second derivatives, respectively, obtained analytically with respect to the variable x. By introducing the function v and its derivatives into the differential eqpiation and in the boundary conditions, we obtain... 

Also from a general viewpoint, it is opportune that the function v(a, x) that approximates y and contains P adaptive parameters a has the following features. 

The adaptive parameters a must be easy to obtain once the support points Xi y are assigned. 

If the Galerkin method is adopted, the first derivative of v with respect to all the adaptive parameters must also be easy to evaluate. 

The most obvious selection, which is also adopted here, is a (P - l)-degree polynomial with P being the number of unknown adaptive parameter a. 

Which Variables Should Be Selected as Adaptive Parameters 1231... 

With the knowledge of the kinetic parameters for a given reaction, which are relative easily accessible by a test or even found in the relevant literature, and the membrane performance the optimum ratio Q/m of membrane area to mass of the reaction mixture can be determined for that reaction, with the initial ratio of the educts as an adaptable parameter. 

Fig. 9.48 Distribution P m) of m quantum jumps per second measured over a period of 150 s (black bars) and 600 s (open bars). The curves are fits of Poisson distributions with two adapted parameters [1232]...

Even if some mechanistic background allows a deeper interpretation of the parameters so that they have more biological significance, with such an approach gray boxes still remain the approach is still, more or less, an optical illusion (Rods and Kossen, 1978), since the final subsystems are always black boxes. In conclusion, the level of modeling to be considered as adequate depends on the principle of analogy, in which the model parameters are adapted to experiments ( adaptational parameters, according to Hofmann, 1975). 

Having properly described the pure component systems, the adaptable parameters a , and g (T) remain to be determined. It was already showed by Van der Waalsl2) that such can be very well done from accurate experimental data liquid-vapour critical points in small-molecule mixtures or 1 iquid-1 iquid critical points in polymer systems showing partial miscibility. 

In the work reported here the coil and the fire box were simulated simultaneously by means of an optimized computer package in which the design of the radiant section of the furnace is an extension and refinement of Hottel s zone method (3 ). In this paper the approach is applied to the simulation of an industrial ethane cracking furnace. The only adaptable parameter left in the simulation model is a burner design factor, namely the fraction of the heat generated in the burner that is transferred to the burner cup. The parameter is determined by matching the exit conver s ion. 

The results of the simulation are given in Table II and Fig. 4. As mentioned already the only adaptable parameter in the model is the split factory This factor was varied untill the simulated exit conversion matched the experimental. The agreement of the calculated product distribution with the industrial results simply reflects the reliability of tie kinetic scheme of Table I. The value of 0.076 fory is considered to be realistic. From this value a burner cup temperature of 1230 C was calculated in agreement with industrial values. In further support of this y-value the effluent gas temperature was found to be 820°C, as compared with 830 C for the plant. Also, the flue gas temperatures are close to those observed in such a furnace. Notice that the flue gas temperatures are not too different between the four horizontal volume zojes. The heat flux is seen to tend rapidly from 3Kcal/m sec to a constant value of about 14Kcal/m sec, again a plausible value. 

---