Kinetic polynomial coefficients

The non-linear theory of steady-steady (quasi-steady-state/pseudo-steady-state) kinetics of complex catalytic reactions is developed. It is illustrated in detail by the example of the single-route reversible catalytic reaction. The theoretical framework is based on the concept of the kinetic polynomial which has been proposed by authors in 1980-1990s and recent results of the algebraic theory, i.e. an approach of hypergeometric functions introduced by Gel fand, Kapranov and Zelevinsky (1994) and more developed recently by Sturnfels (2000) and Passare and Tsikh (2004). The concept of ensemble of equilibrium subsystems introduced in our earlier papers (see in detail Lazman and Yablonskii, 1991) was used as a physico-chemical and mathematical tool, which generalizes the well-known concept of equilibrium step . In each equilibrium subsystem, (n—1) steps are considered to be under equilibrium conditions and one step is limiting n is a number of steps of the complex reaction). It was shown that all solutions of these equilibrium subsystems define coefficients of the kinetic polynomial. 

Proposition 3. If p> 1 and property (35) is valid, then the cyclic characteristic C is contained in the coefficient Bi of kinetic polynomial with an exponent equal... 

Figure 8 A convergence domain (rhomboid) and coefficients of the kinetic polynomial (ovals). The ovals represent the coefficients b2 and b3 as parametric functions of parameter fj at different values of parameter Parameters f = 1.4, = 0.9 and = 0.4.

The obtained steady-state kinetic equations (46) are the kinetic model required for both studies of the process and calculations of chemical reactors. The parameters of eqns. (46) are determined on the basis of experimental data. It is this problem that is difficult. The fact is that, in the general case, eqns. (46) are fractions whose numerator and denominator are the polynomials with respect to the concentrations of observed substances (concentration polynomials). Coefficients of these polynomials can be cumbersome complexes of the initial model parameters. These complexes can also be related. 

Using the kinetic polynomial it is convenient to specify the regions of critical phenomena, e.g., regions of multiplicity of steady-states, self-oscillations. Also it is possible to determine correlations between critical parameters and to identify coefficients of the model. 

Eq. (4.179) contains (1) the kinetic apparent coefficient k+ (2) the potential term, or driving force related to the thermodynamics of the net reaction (3) the term of resistance, ie, the denominator, which reflects the complexity of reaction, both its multi-step character and its non-hnearity finally, (4) the non-linear term (N k,c)), a polynomial in concentrations and kinetic parameters, which is caused exclusively by nonlinear steps. In the case of a linear mechanism, this term vanishes. In classical kinetics of heterogeneous catalysis (LHHW equations), such term is absent. 

These equations look innocuous, but they are highly nonlinear equations whose solution is almost always obtainable only numerically. The nonlinear terms are in the rate r (Ca, T), which contains polynomials in Ca, especially the very nonlinear temperature dependence of the rate coefficient k(T). For first-order kinetics this is... 

Calculation of the coefficients dt for a given matrix is a very laborious process. We will give a method to calculate these coefficients proceeding directly from the complex reaction graph. Like a steady-state kinetic equation, a characteristic polynomial will be represented in the general (struc-turalized) form ... 

We believe that it is not necessary to consider the overall kinetic order of steps above three in mechanism (4). We have analyzed comprehensively [97, 102, 103] all the possible versions for mechanism (4) assuming that the stoichiometric coefficients n, m, p, and q can amount to 1 or 2, p + q < 3, and k 3 = 0. The principal results of this analysis are listed in Table 2. By using the method of general analysis and the Sturm and Descartes theorem concerning the number of positive roots in the algebraic polynomial (ref. 219, pp. 248 and 255), we could show that there exist four detailed mechanisms providing the possibility of obtaining three steady states with a non-zero catalytic reaction... 

When an accurate model of the reaction kinetics is not available (e.g., due to the lack of reliable data for identification), the previously developed approach may be ineffective and model-free strategies for the estimation of the effect of the heat released by the reaction, aq, must be adopted. In detail, the approach in [27] can be considered, where aq is estimated, together with the heat transfer coefficient, via a suitably designed nonlinear observer [24], Other model-free approaches can be adopted, e.g., those based on the adoption of universal interpolators (neural networks, polynomials) for the direct online estimation of the heat [16] and purely neural approaches [11], Approaches based on the combination of neural and model-based paradigms [2] or on tendency models [25] can be considered as well. 

The denominator in equation (116) is a polynomial containing the partial pressures having positive coefficients kfj. From the structure of the kinetic equation, the maximum power of in the denominator will be equal to the power in the numerator. Thus, the qualitative description of the reaction rate in terms of any partial pressure (aU other partial pressures and the temperature being held constant) assumes the form ... 

Chapman and Cowling [12] have shown that in kinetic theory the transport coefficients can be expressed in terms of the Sonine polynomial expansion coefficients which are complicated combinations of the bracket integrals. In the given solutions these integrals are written as linear combinations of a set of these collision integrals. See also Hirschfelder et al [39], sect 7-4. 

Hempenstall et al.333 reported a calculation method that can be performed by simple computers, whereby the degradation curve is represented by a polynomial equation in order to easily obtain a rate constant kT at a temperature T. Using Eq. (2.90), kT can be represented by Eq. (2.91) in the case of first-order degradation kinetics. Inserting the coefficients a0, ah. . . , an, [which are obtained by fitting the drug concentration versus time data to Eq. (2.90)]... 

In deriving the equation system, it is commonly assumed with respect to the collision integrals that the atoms or molecules are at rest before the collision events. Furthermore, each collision integral is additionally expanded with respect to the mass ratio nig/M, and only the leading term with regard to m. /M of each collision integral has been taken into account in each coefficient of the Legendre polynomial expansion of the kinetic equation. 

Under these conditions system (9.1) still admits a unique steady state, but linear stability analysis shows that the latter is always stable (Goldbeter Dupont, 1990) this rules out the occurrence of sustained oscillations around a nonequilibrium unstable steady state. This result holds with previous studies of two-variable systems governed by polynomial kinetics these studies indicated that a nonlinearity higher than quadratic is needed for limit cycle oscillations in such systems (Tyson, 1973 Nicolis Prigogine, 1977). Thus, in system (9.1), it is essential for the development of Ca oscillations that the kinetics of pumping or activation be at least of the Michaelian type. Experimental data in fact indicate that these processes are characterized by positive cooperativity associated with values of the respective Hill coefficients well above unity, thus favouring the occurrence of oscillatory behaviour. 

How should you process the coefficients of the polynomial, which are determined from the regression analysis, to calcnlate the temperature-dependent kinetic and equilibrium constants ... 

According to several different real situations different definitions have been given of the random event that a polynomial differential equation is kinetic (Toth, 1981b, pp. 44-8). The results can be summarised as follows. If one selects a polynomial differential equation with fixed coefficients and the random selection only concerns the exponents than the probability of getting a kinetic differential equation is 1. If the exponents are fixed and the coefficients are randomly chosen then the probability of getting a kinetic differential equation is 0. Finally, as a consequence of the statements above, if both the coefficients and the exponents are randomly selected then the probability of getting a kinetic differential equation is again 0. 

Figure 5. Mean proton kinetic energy, ( k), as a function of pressure P (25 P 400 MPa at r = 268K (left axis, solid circles). The right axis shows the pressure dependence of the water selfdiffusion coefficient at P = 268 K (right axis, open triangles). Self-diffusion coefficient data are taken from Ref. [39]. Solid line is a polynomial fit of the self-diffusion coefficient data over the pressure range 20 P < 400 MPa.

If / / is to be kinetic and Hamiltonian then c,e,f,C,E, E O, A=-b, B=-a, D=-d must hold. This, however, excludes that / / be conservative too, except the trivial - zero right hand side - case. This is so, because conservativity would imply the existence of positive numbers r and s such that /4/ holds for all nonnegative numbers x and y. Consequently, all the coefficients of the polynomial on the left hand side of / / are equal to zero. Again, a short investigation of the coefficients shows, that a=...=E=0 thus the only conservative kinetic Hamiltonian system is /6/. 

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