Hyperbolic model

Rules. Eliminate temperature terms in the denominator. (Terms with negative exponents in the power law model are considered to belong to the denominator, in the hyperbolic model. Author.)... 

The preferred kinetic model for the metathesis of acyclic alkenes is a Langmuir type model, with a rate-determining reaction between two adsorbed (complexed) molecules. For the metathesis of cycloalkenes, the kinetic model of Calderon as depicted in Fig. 4 agrees well with the experimental results. A scheme involving carbene complexes (Fig. 5) is less likely, which is consistent with the conclusion drawn from mechanistic considerations (Section III). However, Calderon s model might also fit the experimental data in the case of acyclic alkenes. If, for instance, the concentration of the dialkene complex is independent of the concentration of free alkene, the reaction will be first order with respect to the alkene. This has in fact been observed (Section IV.C.2) but, within certain limits, a first-order relationship can also be obtained from many hyperbolic models. Moreover, it seems unreasonable to assume that one single kinetic model could represent the experimental results of all systems under consideration. Clearly, further experimental work is needed to arrive at more definite conclusions. Especially, it is necessary to investigate whether conclusions derived for a particular system are valid for all catalyst systems. 

Figure 5. Dependence of rate of dissolution of 5pM Y-FeOOH in pH 4.0, 0.01M NaCl on concentration of a) tartaric acid, and b) salicylic acid. Fitted parameters obtained for rectangular hyperbolic model are given. Light source mercury arc lamp with 365nm band-pass filtering.

The hyperbolic model types have very commonly been used in the analysis of kinetic data, as discussed in Section I. Such applications are sometimes justified on the theoretical bases already alluded to, or simply because models of the form of Eq. (2) empirically describe the existing reaction-rate data. Considerably more complex models are quite possible under the Hougen-Watson formalism, however. For example, Rogers, Lih, and Hougen (Rl) have proposed the competitive-noncompetitive model... 

The amount of uncertainty in parameter estimates obtained for the hyperbolic models is particularly large. It has been pointed out, for example, that parameter estimates obtained for hyperbolic models are usually highly correlated and of low precision (B16). Also, the number of parameters contained in such models can be too great for the range of the experimental data (W3). Quantitative measures of the precision of parameter estimates are thus particularly important for the hyperbolic models. (Cl). 

The covariance (off-diagonal) terms of Eq. (53) are also needed to calculate approximate intervals for parameters in the hyperbolic models. For a model such as... 

When any hyperbolic model is written in terms of fractional conversions instead of partial pressures, two groupings of terms inherently arise within the denominator These two groupings will be called the intrinsic parameters Cx and C2. For example, when data are taken for the olefinic dehydration of a pure alcohol feed to a reactor, Eq. (80) becomes... 

In the following discussion, we shall again separate the terms of a hyperbolic model and identify two parameters Cl and C2. As before, each of these two parameters will be a collection of terms, one of which is multiplied by conversion and one not multiplied by conversion. In previous formulations, however, we have oriented the discussion toward a familiar type of experimental design in kinetics conversion versus space-time data at several pressure levels. Consequently, the parameters Cx and C2 were defined to exploit this data feature. Another type of design that is becoming more common is a factorial design in the feed-component partial pressures. 

Balakotaiah and Chakraborty introduce a four-mode hyperbolic model but with non-linear reactions, in three-dimensional geometry and with much bigger Damkohler number (Balakotaiah, 2004 Chakraborty and Balakotaiah, 2005). The effective model cannot be directly compared with our system (12)-(13). Nevertheless, in Section 3.1 we derive a four-mode hyperbolic model, analogous to the models from Balakotaiah (2004) and Chakraborty and Balakotaiah (2005). We show that it is formally equivalent to our model at the order This shows the relationship between the upscaled models... 

Nevertheless, there is the possibility of obtaining hyperbolic models, at same order of precision, We note that such models where... 

The theory for classifying linear, second-order, partial-differential equations is well established. Understanding the classification is quite important to understanding solution algorithms and where boundary conditions must be applied. Partial differential equations are generally classified as one of three forms elliptic, parabolic, or hyperbolic. Model equations for each type are usually stated as... 

It is possible to combine the two equations in one hyperbolic second-order PDE. This has the property of finite wave speed, both boundary conditions at the entrance are easily calculable, and it accounts for some of the phenomena of unmixing. This is not the place to treat this model in detail and, indeed, it is still finding fruitful applications.5 Another method for a hyperbolic model is to be found in [173]. 

As our first application, we consider the classical Taylor-Aris problem (Aris, 1956 Taylor, 1953) that illustrates dispersion due to transverse velocity gradients and molecular diffusion in laminar flow tubular reactors. In the traditional reaction engineering literature, dispersion effects are described by the axial dispersion model with Danckwerts boundary conditions (Froment and Bischoff, 1990 Levenspiel, 1999 Wen and Fan, 1975). Here, we show that the inconsistencies associated with the traditional parabolic form of the dispersion model can be removed by expressing the averaged model in a hyperbolic form. We also analyze the hyperbolic model and show that it has a much larger range of validity than the standard parabolic model. 

This result was first derived by Aris (1956) using the method of moments. While the resulting model now includes both the effects (axial molecular diffusion and dispersion caused by transerverse velocity gradients and molecular diffusion) it has the same deficiency as the Taylor model, i.e. converting a hyperbolic model into a parabolic equation. 

Now, the averaged hyperbolic model, Eq. (52), defines a characteristic initial-value problem (Cauchy problem). To complete the model, we need to specify Cm only along the characteristic curves x = 0 and f — 0. Thus, the initial and boundary conditions for the averaged model are obtained by taking the mixing-cup averages of Eqs. (31) and (32) ... 

When the assumption Per 6.93 is not valid, it is better to leave the averaged model in the more general hyperbolic form given by Eqs. (45) or (46) with boundary and initial conditions given by Eqs. (53)-(54). The important point to be made is the wave forms of the averaged model [either Eqs. (46) or (52)] have much larger domain of validity than the parabolic form as shown below. (Remark The hyperbolic model is also much easier to solve numerically compared to the parabolic model.)... 

Before closing this section, we compare the hyperbolic model derived here with the wave model of Westerterp et al. (1995). For the classical Taylor problem (without axial dispersion), the model of Westerterp et al. may be written in the present notation as... 

We now present the solution of the hyperbolic model defined by Eqs. (52) and (53)-(54) and compare the solution to that of the classical parabolic model with Danckwerts boundary conditions. We use the axial length and convective time scales to non-dimensionalize the variables and write the hyperbolic model in the following form ... 

The exit concentration Cm(z — 1, t) for the case of a unit impulse (Delta function) input (f(z) — 0, git) — S(i)) is known as the dispersion (or RTD) curve. For the hyperbolic model, this can be found either by Laplace transformation or from the general solution of the model [see Balakotaiah and Chang (2003), for a general analytical solution of Eqs. (57)—(59)]. It is easily seen that the Laplace transform of the dispersion curve is given by... 

We now compare the solution of the hyperbolic model with that of the parabolic model used widely in the literature to describe dispersion in tubular reactors. The parabolic model with Danckwerts boundary conditions (in dimensionless form) is given by... 

Fig. 6. Dispersion curves predicted by the hyperbolic model [Eq. (57)] for various values of the effective local Peclet number, jj.

We also present solutions of the full hyperbolic model... 

This shows again that when X is small (or equivalently, Per 1), we can combine the small axial dispersion term with the mixed derivative term and simplify the general hyperbolic model [Eq. (72)] to the simpler model [Eq. (57)]. However, for X values of order unity or larger, this cannot be justified. The inverse transform of Eq. (73) can be found by integrating around the branch points but we will not pursue it here. Instead, we show in Figs. 7 and 8 the numerically determined dispersion curves for r — 0.1,1 and various values of X. As can be expected, the qualitative behavior of the full hyperbolic model [Eq. (72)] is similar to that of the simpler case of X — 0. Only for X 1, the peak value changes and shifts to lower times. 

Hyperbolic Model for Describing Dispersion Effects in Turbulent flow in a Tube... 

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