The quasi-steady-state approximation QSSA

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 QSSA has close connections with slow manifold techniques in that it depends on the existence of time-scale separation in the variables. Specifically, it involves finding those species which react on a very short time-scale so that the system can be assumed to be in equilibrium with respect to their motion. The application of the QSSA to mechanism reduction implies that the concentrations of fast intermediate species can be expressed algebraically in terms of other species, since it is assumed that their rates of change can be decoupled from the differential equations and the righthand sides set to zero. The application of the QSSA to a subset of the original species converts equation (4.1) into the following system of differential-algebraic equations  

The difference in the concentrations of the QSSA species calculated from differential equation (4.1), and algebraic equation (4.90), while the concentrations of the non-steady-state species are the true concentrations calculated from equation (4.1), is the instantaneous error of the quasisteady-state approximation. The instantaneous error induced by the application of the QSSA to a single species Ac/ [158] can be used to identify the possible steady-state species and is calculated by the following expression  

Since the local lifetime of a species is equal to the reciprocal of the diagonal Jacobian element for that species, 

An important feature of QSSA species is that their concentrations are completely determined by the concentrations of other species through equation (4.90). If the concentrations of the QSSA species calculated in equation (4.1) are perturbed slightly, this perturbation must vanish within a short time. This property of QSSA species has been noted previously, (see, for example, Klonowski [159], p. 83) who stated that the fast 

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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. 

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. 

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. 

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. 

There are many ways one can try to reduce the computational burden. Ideally, one would find numerical methods which are guaranteed to retain accuracy while speeding the calculations, and it would be best if the procedure were completely automatic i.e. it did not rely on the user to provide any special information to the numerical routine. Unfortunately, often one is driven to make physical approximations in order to make it feasible to reach a solution. Common approximations of this type are the quasi-steady-state approximation (QSSA), the use of reduced chemical kinetic models, and interpolation between tabulated solutions of the differential equations (Chen, 1988 Peters and Rogg, 1993 Pope, 1997 Tonse et al., 1999). All of these methods were used effectively in the 20th century for particular cases, but all of these approximated-chemistry methods share a serious problem it is hard to know how much error is... 

Assumption 5. In addition to the steady state—the quasi-steady state approximation (QSSA) applies locally to ions and olefins. 

The quasi-steady-state approximation (QSSA) is commonly made for the moments of living polymer chains since, for most practical situations, an equilibrium is achieved instantaneously between chain initiation and chain transfer, fc,C [M] = ( cpR + daclMo- This equilibrium results from the fast dynamics of the initiation and transfer reactions compared to that of the overall polymerization rate. In this case, an even simpler system of equations is obtained than the one listed in Table 2.6. 

Another useful and widely used approach is called the quasi steady-state approximation (QSSA). In this case we hypothesize the existence of at least one (or more) intermediates involved in the reaction mechanism whose concentration in the reacting mixture is very low and can be considered as quasi constant. 

An overview of the methods used previously in mechanism reduction is presented in Tomlin et al. (1997). The present work uses a combination of existing methods to produce a carbon monoxide-hydrogen oxidation scheme with fewer reactions and species variables, but which accurately reproduces the dynamics of the full scheme. Local concentration sensitivity analysis was used to identify necessary species from the full scheme, and a principle component analysis of the rate sensitivity matrix employed to identify redundant reactions. This was followed by application of the quasi-steady state approximation (QSSA) for the fast intermediate species, based on species lifetimes and quasi-steady state errors, and finally, the use of intrinsic low dimensional manifold (ILDM) methods to calculate the mechanisms underlying dimension and to verify the choice of QSSA species. The origin of the full mechanism and its relevance to existing experimental data is described first, followed by descriptions of the reduction methods used. The errors introduced by the reduction and approximation methods are also discussed. Finally, conclusions are drawn about the results, and suggestions made as to how further reductions in computer run times can be achieved. 

A robust and realistic approximation is the quasi-steady-state approximation (QSSA) [6], in which it is assumed that the rate at which R population changes is very slow compared to the other processes (decomposition, initiation, propagation, and termination) that is, it is assumed that... 

One can make the quasi-steady state approximation (QSSA) for radicals (P and P). This assumes that radical reactions are fast compared with other reactions and so can be considered to be always at steady state thus the left-hand sides of Equations 16.65 and 16.66 may be set to zero. Solution of Equation 16.66 for P and substitution of Equation 16.65 into the result gives the concentration of live chains ... 

The quasi-steady-state approximation (QSSA) is also called the Bodenstein principle, after one of its first users (Bodenstein 1913). As a first step, species are selected that will be called quasi-steady-state (or QSS) species. The QSS-species are usually highly reactive and low-concentration intermediates, like radicals. The production rates of these species are set to zero in the kinetic system of ODEs. The corresponding right-hand sides form a system of algebraic equations. These... 

The implication of distinguishing between fast and slow variables is that a short time after the perturbation, the values of the fast variables are determined by the values of the slow ones. Appropriate algebraic expressions to determine the values of the fast variables as functions of the values of the slow ones can therefore be developed. This is the starting point of model reduction methods based on timescale analysis. One such method was introduced in Sect. 2.3 where the quasi-steady-state approximation (QSSA) was demonstrated for the reduction in the number of variables of a simple example. In this case, the system timescales were directly associated with chemical species. We shah see in the later discussion that this need not always be the case. 

For the stiff case ki = 100, the dynamic concentration profiles are very different (Figure 4.9). Except for a very short initial period where ca increases rapidly from ca-o = 0, it appears that ca remains proportional to ca. Such an observation leads to the quasi-steady state approximation (QSSA), which states that the concentration of a very active species such as A is in dynamic equilibrium between generation and destruction i.e.. 

Assuming that the quasi-steady-state approximation (QSSA) is valid, the concentration of the intermediate species I can be found from Eq. (5.3.3) ... 

The kinetic model of styrene auto-initiation proposed by Hui and Hameilec [27] was used as a starting point for this work. The Mayo initiation mechanism was assumed (Figure 7.2) but the acid reaction was of course omitted. After invoking the quasi-steady-state assumption (QSSA) to approximate the reactive dimer concentration, Hui and Hameilec used different simplifying assumptions to derive initiation rate equations that are second and third order in monomer concentration. 

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. 

Conversely, /react (I lbind) = when the ligand-receptor interaction proceeds and is related to the standard quasi-steady-states approximation (QSSA). The probability of the occurrence of products in L-R reactions lies between these limits. Similarly, in the case where specific interactions do not take place. 

Furthermore, mathematical procedures can be applied to the detailed mechanism or the skeletal mechanism which reduces the mechanism even more. These mathematical procedures do not exclude species, but rather the species concentrations are calculated by the use of simpler and less time-consuming algebraic equations or they are tabulated as functions of a few preselected progress variables. The part of the mechanism that is left for detailed calculations is substantially smaller than the original mechanism. These methods often make use of the wide range of time scales and are thus called time scale separation methods. The most common methods are those of (i) Intrinsic Low Dimensional Manifolds (ILDM), (ii) Computational Singular Perturbation CSF), and (iii) level of importance (LOl) analysis, in which one employs the Quasy Steady State Assumption (QSSA) or a partial equilibrium approximation (e.g. rate-controlled constraints equilibria, RCCE) to treat the steady state or equilibrated species. 

A time scale separation method makes use of the fact that the physical and chemical time scales have only a limited range of overlap. The time scales of some of the more rapid chemical processes can thus be decoupled and be described in approximate ways by the Quasi Steady State Assumption (QSSA) or partial equilibrium approximations for the selected species. This reduces the species list to only the species left in the set of differential equations. Also, eliminating the fastest time scales in the system solves the numerical stiffness problem that these time scales introduce. Numerical stiffness arises when the iteration over the differential equations need very small steps as some of the terms lead to rapid variations of the solution, typically terms involving the fastest time scales. 

To be able to predict the result of a specific degradation process, possibilities for modelling are of interest. An easily applicable model for the evolution of the MWD was developed by Tzoganakis et al. [1] using a quasi-steady state approximation (QSSA) technique. It )delds the evolution of the moments of the MWD as function of peroxide type, concentration and efficiency. If one considers the moments of the MWD as... 

Despite deviations from the ideal kinetics at early and late conversion, rationalized by existing models, quasi-steady state approximation (QSSA) proved to be an appropriate approximation for analyzing the reactions and obtaining the associated rate constants. Several reaction features such as the dependence of and initiator decay rates on l/[/]o and respectively, and the first-order decay of monomer during the majority of the conversion allowed verification of the ideal kinetics applicability. 

We discussed in Chapter 5 that, under quasi-steady-state approximation (QSSA), the net production of all intermediate species in a reaction mass can be taken as zero after a small induction time. The polymer in radical polymerization is formed only after polymer radicals are generated. At any given time, therefore, we have two distributions one for polymer radicals (P , n= 1, 2, 3) and one for... 

Then, employing the in vitro conditions, the enzyme can always be saturated with the substrate, so that the quasi-steady-state (or equilibrium) approximation (QSSA) may apply to the intermediate formed complex in Eq. (1.4). It implies imposing on Eq. (1.12) the mathematical constrain (Segel, 1975, 1988, 1989) ... 

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