The Quasi-Steady-State Approximation

We hinted in Sect. 2.3.6 that the timescale separation present in most kinetic systems can be exploited in terms of model reduction. The next sections will therefore cover the use of timescale analysis for the reduction of the number of 

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This procedure constitutes an application of the steady-state approximation [also called the quasi-steady-state approximation, the Bodenstein approximation, or the stationary-state hypothesis]. It is a powerful method for the simplification of complicated rate equations, but because it is an approximation, it is not always valid. Sometimes the inapplicability of the steady-state approximation is easily detected for example, Eq. (3-143) predicts simple first-order behavior, and significant deviation from this behavior is evidence that the approximation cannot be applied. In more complex systems the validity of the steady-state approximation may be difficult to assess. Because it is an approximation in wide use, much critical attention has been directed to the steady-state hypothesis. 

Develop a suitable rate expression using the Michaelis-Menten rate equation and the quasi-steady-state approximations for the intermediate complexes formed. 

Frequently function R can be written as a single term having the simple form of equation 1. For Instance, with the aid of the long chain approximation (LCA) and the quasi-steady state approximation ((JSSA), the rate of monomer conversion, I.e., the rate of polymerization, for many chain-addition polymerizations can be written as... 

The assumption made is called the quasi-steady-state approximation (QSSA). It is valid here mainly because of the great difference in densities between the reacting species (gaseous A and solid B). For liquid-isolid systems, this simplification cannot be made. 

Using the quasi-steady-state approximation and the conservation of total enzyme Et = E) + [ZsS], the concentration of the complex is given as... 

As for the quasi (pseudo)-steady-state case, the basic assumption in deriving kinetic equations is the well-known Bodenshtein hypothesis according to which the rates of formation and consumption of intermediates are equal. In fact. Chapman was first who proposed this hypothesis (see in more detail in the book by Yablonskii et al., 1991). The approach based on this idea, the Quasi-Steady-State Approximation (QSSA), is a common method for eliminating intermediates from the kinetic models of complex catalytic reactions and corresponding transformation of these models. As well known, in the literature on chemical problems, another name of this approach, the Pseudo-Steady-State Approximation (PSSA) is used. However, the term "Quasi-Steady-State Approximation" is more popular. According to the Internet, the number of references on the QSSA is more than 70,000 in comparison with about 22,000, number of references on PSSA. 

The quasi-steady-state approximation then allows the concentration time derivatives to be set equal to zero,... 

This ratio provides the criterion for the applicability of the quasi-steady-state approximation for the concentration as shown in Table V. 

The quasi steady state approximation is a powerful method of transforming systems of very stiff differential equations into non-stiff problems. It is the most important, although somewhat contradictive technique in chemical kinetics. Before a general discussion we present an example where the approximation certainly applies. 

The following treatment applies to the case where the solids are stationary in a shallow packed bed, so that they can be considered to be in well-mixed conditions, and that the solute initially saturates the solid, as in the case of vegetable oil in crushed seeds. For the quasi--steady-state approximation, Brunner [51] derived a practical equation ... 

The governing equation is therefore identical with that for the irrotational flow of an ideal fluid through a circular aperture in a plane wall. The stream lines and equipotential surfaces in this rotationally symmetric flow turn out to be given by oblate spheroidal coordinates. Since, from Eq. (157), the rate of deposition of filter cake depends upon the pressure gradient at the surface, the governing equation and boundary conditions are of precisely the same form as in the quasi-steady-state approximation... 

At bg -> oo, reactions involving the participation of the greatest number of gas molecules (k = raax) become predominant. When choosing a new time scale t = we can go to the quasi-steady-state approximation at s - 0... 

Consequently, starting from some sufficiently low bs, the quasi-steady-state approximation can be applied after a certain period of time ("boundary layer )... 

At e = SI V -> 0, the Tikhonov theorem is applicable, hence starting from sufficiently small , we can use the quasi-steady-state approximation. 

Of great importance is the fact that the quasi-steady-state approximation is the solution asymptote of the initial system at e -> 0, but it is applied at finite e. To establish a starting value from which this approximation can be used with the prescribed accuracy is a rather difficult problem in each particular case. 

Fig. 15.6(c)]. At the center of each element there is a node. The nodes of adjacent elements are interconnected hy links. Thus, the total flow field is represented by a network of nodes and links. The fluid flows out of each node through the links and into the adjacent nodes of the network. The local gap separation determines the resistance to flow between nodes. Making the quasi-steady state approximation, a mass (or volume) flow rate balance can be made about each node (as done earlier for one-dimensional flow), to give the following set of algebraic equations... 

If the quasi steady state approximation is imposed for all h, then (alternative )... 

Appropriate expressions for the fluxes of each of the reactions in the system must be determined. Typically, biochemical reactions proceed through multiple-step catalytic mechanisms, as described in Chapter 4, and simulations are based on the quasi-steady state approximations for the fluxes through enzyme-catalyzed reactions. (See Section 3.1.3.2 and Chapter 4 for treatments on the kinetics of enzyme catalyzed reactions.)... 

Fig. 5.11. Time evolution of the vibrational (v = 0 — 5) populations of the H2 molecule ground state (XHere, it is assumed that all the molecules are initially populated in the v = 0 level. The initial v = 0 level population is normalized to be unity. The broken and continuous lines indicate the populations calculated with and without the quasi-steady-state approximation, respectively. Here, the electron temperature is 2 eV, and the electron density is 1 x 1020 m 3. The upper scale indicates the distance corresponding to the time when the kinetic energy of the H2 molecules corresponds to a temperature of 300° C...

SE Interval II begins at the cessation of nucleation, or in hght of the nucleation theory just reviewed, when the particle number becomes relatively constant. Most theories developed for this interval assume a constant particle number and use the quasi-steady-state approximation (QSSA) for average number of radicals per particle. The kinetics and mechanisms of Interval II have been some of the most studied aspects of macroemulsion polymerization. SE Interval II ends when the monomer droplets disappear and the monomer concentration in the particles begins to decrease. 

It should be noted that the time dependence of the resin viscosity, t/, appears only through Equation 10. This is because of the quasi-steady-state approximation implicit in the present analysis. Equation 8 reduces to an expression similar to that derived by Aung (4) when k 0... 

The application of the quasi-steady-state approximation and the site balance (assuming A is the mart) gives the following expression for the reaction rate ... 

Quasi-steady-state periodic regime (T Tj. The input variable varies rather slowly compared to the dynamics of the system, and the system follows the input variable almost exactly. The time-averaged performance of the reactor is calculated applying the quasi-steady-state approximation to the state of the system and averaging out the resulting performances at any time. 

The reduction techniques which take advantage of this separation in scale are described below. They include the quasi-steady-state approximation (QSSA), the computational singular perturbation method (CSP), the slow manifold approach (intrinsic low-dimensional manifold, ILDM), repro-modelling and lumping in systems with time-scale separation. They are different in their approach but are all based on the assumption that there are certain modes in the equations which work on a much faster scale than others and, therefore, may be decoupled. We first describe the methods used to identify the range of time-scales present in a system of odes. 

The application of the quasi-steady-state approximation is a well established technique introduced at the start of this century. The importance of the early applications led to the analytical solution of non-linear reaction systems which, without the aid of computer technology, could not otherwise be solved at that time [149-154]. Since the advent of computers and advanced software for the solution of stiff systems of equations there have been suggestions that the QSSA is an obsolete technique. Even if such an argument was valid, an understanding of the basis and applicability of the QSSA would still be needed, as emphasized by Come [155] since the QSSA has been used to elucidate most reaction mechanisms and to... 

The error of the quasi steady-state approximation in spatially distributed systems has recently been studied by Yannacopoulos ef al. [160]. It has been shown qualitatively that QSSA errors, which might decay quickly in homogeneous systems, can readily propagate in reactive flow systems so that the careful selection of QSSA species is very important. A quantitative analysis of QSSA errors has not yet been carried out for spatially distributed systems but would be a useful development. 

T. Turdnyi and J. Toth, Comments to an Article of Frank-Kamenetskii on the Quasi-Steady-State Approximation, Acta Chim. Hung. 129 (1992) 903-914. 

T. Turanyi, A.S. Tomlin and M.J. Pilling, On the Error of the Quasi-Steady-State Approximation, J. Phys. Chem. 97 (1993) 163-172. 

Even if the quasi-steady-state approximation is valid, all of the forms of the reaction rate equation (5.3) given above involve the concentrations of at least some of the intermediate species. If we are to be concerned with the intermediate species, and particularly when we wish to employ the methods of non-linear mathematics (or even direct numerical computation) then we must find ways of including such quantities. The appropriate rate equations for the concentration of H, O and OH are readily constructed... 

The quasi-steady-state approximation works by replacing the differential equations for the rates of change of the intermediate species by algebraic conditions obtained by setting d[H]/dt = 0 etc. (see Section 4.8.5). In some cases, the resulting equations can be solved and manipulated algebraically allowing substitution into the overall rate equation to obtain a form that only involves explicitly the concentrations of the reactants and (perhaps) products. Such rate equations can then be compared with the empirical rate equations determined from experiment to test the validity of the assumed mechanism and to obtain quantitative values for the rate coefficients involved. 

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