One-Parameter Models

In many practical situations, one is mainly interested in C t), which for most kinetics (power law, Langmuir-Hinshelwood, bimolecular reactions, etc.) can be accurately estimated or exactly calculated from the following simple expression 

Here Cq = C t) Cf and the subscript q signifies that depends on the adjustable parameter q. Both z and 7 can be determined from information on 

The point here is that Eq.(22) is all that is needed for an estimate of C t), which requires characterization of only the most refractory portion of the feed. A much improved estimate needs a data point on C t) at an intermediate conversion, Eq.(23). 

Note that the predicted values for at each time are in the vector Fj. That is, 

7 7 -1-7 7 1 and the correlation coefficient matrix is R where the elements of R are obtained from the elements of A  

To illustrate the derivation of a confidence interval, let s first determine the parameter value for a one parameter model. That is, for an isothermal, constant 

take the natural logarithm of each side of equation (9.283) to obtain 

use the following measured values of to find f, then find k using least squares. 

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Because non-adiabatic collisions induce transitions between rotational levels, these levels do not participate in the relaxation process independently as in (1.11), but are correlated with each other. The degree of correlation is determined by the kernel of Eq. (1.3). A one-parameter model for such a kernel adopted in Eq. (1.6) meets the requirement formulated in (1.2). Mathematically it is suitable to solve integral equation (1.2) in a general way. The form of the kernel in Eq. (1.6) was first proposed by Keilson and Storer to describe the relaxation of the translational velocity [10]. Later it was employed in a number of other problems [24, 25], including the one under discussion [26, 27]. 

A one-parameter model, termed the bubbling-bed model, is described by Kunii and Levenspiel (1991, pp. 144-149,156-159). The one parameter is the size of bubbles. This model endeavors to account for different bubble velocities and the different flow patterns of fluid and solid that result. Compared with the two-region model, the Kunii-Levenspiel (KL) model introduces two additional regions. The model establishes expressions for the distribution of the fluidized bed and of the solid particles in the various regions. These, together with expressions for coefficients for the exchange of gas between pairs of regions, form the hydrodynamic + mass transfer basis for a reactor model. 

Even though the Einstein and Debye models are not exact, these simple one-parameter models illustrate the properties of crystals and should give reliable estimates of the volume dependence of the vibrational entropy [15]. The entropy is given by the characteristic vibrational frequency and is thus related to some kind of mean interatomic distance or simpler, the volume of a compound. If the unit cell volume is expanded, the average interatomic distance becomes larger and the... 

The Gaussian is a one parameter model, the variance. The RTD curves are bell shaped. At smaller values of the variance (or the larger values of n of the Erlang), the Erlang and Gaussian RTD curves approach each other, and the... 

Audisio studied the microtacticity of the 1,4-rrans-polypentadiene [—CH2—CH=CH—CH(CH3)—] in connection with that of the poly-(methyltetramethylene) [—CH2—CH2—CH2—CHfCHs)—] obtained from the preceding compound by reduction (109, 110), and succeeded in evaluating the distribution of the triads mm, mr, and rr. He has proposed an interpretation according to a one-parameter model based on enantiomorphic catalyst sites (111) (see Table 4, column 4, 3/ = 1). 

The tanks-in-series model is a flexible one-parameter model which amounts to characterising a system in terms of the general transfer function of the equation... 

Dispersion and Tanks in Series Model. The first attempts at modeling naturally tried the simple one-parameter models however, observed conversion well below mixed flow cannot be accounted for by these models so this approach has been dropped by most workers. 

Models vary in complexity. One-parameter models seem adequate to represent packed beds or tubular vessels. On the other hand models involving up to six parameters have been proposed to represent fluidized beds. 

The tanks-in-series and dispersion models are one-parameter models. In general, the number of parameters in a combined model is... 

Diffusion-type models are two-parameter models, involving kt or Ds and La, while BDST models are one-parameter models, involving only 0, as gmax is an experimentally derived parameter. The determination of La requires the whole experimental equilibrium curve, and in case of sigmoidal or other non-Langmiur or Freundlich-type isotherms, these models are unusable. From this point of view, BDST models are more easily applied in adsorption operations, at least as a first approximation. 

When DJuL is found to be large and the tracer response curve is skewed, as in Fig. 2.23b, but without a significant delay, a continuous stirred-tanks in series model (Section 2.3.2), may be found to be more appropriate. The tracer response curve will then resemble one of those in Fig. 2.8 or Fig. 2.9. The variance a2 of such a curve with a mean of tc is related to the number of tanks / by the expression a2 = t2/i (which can be shown for example by the Laplace transform method 7 from the equations set out in Section 2.3.2). Calculations of the mean and variance of an experimental curve can be used to determine either a dispersion coefficient Dl or a number of tanks i. Thus each of the models can be described as a one parameter model , the parameter being DL in the one case and i in the other. It should be noted that the value of i calculated in this way will not necessarily be integral but this can be accommodated in the more mathematically general form of the tanks-in-series model as described by Nauman and Buffham 7 . 

I now come to a very important observation. We have seen that interaction between the environments (or between the particles themselves if this is assumed equivalent to segregation decay) could be represented by different one parameter models, e.g. 

The activation barriers AE for dissociation and recombination belong to the same realm of relative energies as AQAB. For this reason, we shall not discuss here purely numerical calculations of AE. Remarkably, many authors tried to conceptualize their computational results in terms of simple analytic models, which have no direct relation to the computations. For example, the effective medium theory (EMT) is a band-structure model with a complex and elaborated formalism including many parameters (154). Nevertheless, while reviewing the numerical EMT applications to surface reactions, Norskov and Stoltze (155) discussed the calculated trends in the activation energies for AB dissociation in terms of a one-parameter model (unfortunately, no details were provided) projecting A b to vary as NJ, 10 - Nd), where Nd is the d band occupancy [cf. Eqs. (21a)—(21c) of the BOC-MP theory]. 

In this case we have a one-parameter model with two well-known limiting cases. One corresponds to a = 7r/4, for which 4>Apt/p = AAp"p, and the system decomposes into independent singlet pairs (Fig. 10) the other limiting case corresponds to a = 0 (our model reduces to the two-dimensional AKLT model in this case, the spins at each site forming a quintet). 

Our results suggest that the spin correlation functions decay exponentially with a correlation length 1 for an arbitrary parameter a. We also assume that the decay of the correlation function is of the exponential type for the 14 parameter model as well, i.e., for any choice of site spinor I>A/u/p. This assumption is supported in special cases 1) the partition of the system into one-dimensional chains with exactly known exponentially decaying correlation functions 2) the two-dimensional AKLT model, for which the exponential character of the decay of the correlation function has been rigorously proved [32], Further evidence of the stated assumption lies in the numerical results obtained for various values of the parameter in the one-parameter model. 

Figure 3. Chain stereostructures described by one-parameter model...

The simplest probability model is the one-parameter model as used by Coleman (4) and Newman (10) based on the preferred probability , such that after adding a monomer to the growing chain, the chain end is in + state. The typical stereostructures that can be described by this model are shown in Figure 3. 

For the one-parameter model, crystalline weight fraction, X, at any temperature is given by ... 

Equations 11-12,11-19, and 11-20 form the basis for calculating the percent crystallinity as a function of temperature for the one-parameter model. Calculations of crystalline weight fraction based on the above equations gave values at each temperature which were considerably higher than those found experimentally. 

The degree of crystallinity is calculated in the same manner as for the one-parameter model. 

The ability of this one parameter model to represent our experimental data is shown in Figure 3. This is a plot of the contact radii (p = D/2) of the gold clusters as measured by STM as a function of their free space volumes. The solid line in this figure is the prediction of the theoretical model, which is... 

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