QSSA species

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 ... 

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... 

Assuming a QSSA species is perturbed from a fixed point by an amount Ac, and using the relationship... 

If we consider a quasi-stationary system, then the fixed point moves with a speed dc Vdr. The Jacobian elements for the QSSA species will be large and negative as discussed in Section 4.8.2, and so the initial approach will be fast. However, since the fixed point is moving in time, the perturbed solution will never exactly reach it, but will remain at a distance Ac The 1 2... 

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. 

Inner iteration. In some cases explicit expressions can be produced for most QSSA species, but for some other species the QSSA equations are still implicit coupled non-linear expressions. These equations can be solved separately and the concentration of QSSA species calculated by an iteration cycle. This so-called inner iteration method has proved to be a successful technique for this purpose. See Chapter 6 in [163] for an example of its application in methane and ethylene flames. 

The application of the QSSA to the above scheme leads to a simple set of differential and algebraic equations describing the system and to an algebraic relationship between the QSSA and the non-QSSA species. As above we choose O and OH as the QSSA species (HO2 is no longer in the scheme it is considered a stable product) and set d[0]/dr = d[OH]/dr = 0. This results in the following algebraic equations ... 

Substituting for the QSSA species in the original odes, the set of differential equations for this system becomes... 

There have been several empirical observations or conclusions based on the investigation of small model reaction systems that showed that the rates of consuming reactions of QSSA species are unusually high, that the concentrations, and the net rates of reaction of QSSA species are unusually low, that the induction period is usually short, and that most QSSA species are radicals, These observations are simple consequences of the physical pictures presented above and the error formulas derived from them. 

For k2 fei the rate of decomposition of B is much greater than, its rate of formation, w hich is a necessary condition for B to be a QSSA species. If component B is defined to be a QSSA species, we write... 

If the rate of production statement developed in step 1 contains the concentration of reaction intermediates that are QSSA species, their concentrations can be found in terms of reactant and product concentrations by writing a necessary number of algebraic statements. These algebraic statements come from applying the QSSA to reaction intermediates. In some cases, either by assumption or because of kinetic msight, an elementary reaction is treated as being at equilibrium. 

A system composed of Us species leads to differential equations, and the exact solution is found by solving Equations 5.48. The QSSA is applied to k QSSA species, where k < n. This leads to the following set of algebraic equations for the QSSA species... 

Similarly, for Scheme II, when component B is defined to be a QSSA Species,... 

If component B were a radical or atom, it could be treated as a QSSA species provided kz/ki were large enough because large kz/ki leads to small errors in the predicted concentration of product. In this simple example the net rate of production of the intermediate never reached zero over the values of kit examined, demonstrating it is not necessary for the net rate of production of QSSA species to be zero. At sufficiently large values of k-z/ki the intermediate B can be considered a QSSA species because the error in the actual and approximate. concentration of the, QSSA reache.s. an. acceptable level. 

The QSSA is a useful tool in reaction analysis. Material balances for hatch and plug-flow reactors are ordinary differential equations. By applying Equation 5.81 to the components that are QSSA species, their material balances become algebraic equations. These algebraic equations can be used to simplify the reaction expressions and reduce the number of equations that must be solved simultaneously. In addition, appropriate use of the QSSA can eliminate the need to know several difficult-to-measure rate constants. The required information can be reduced to ratios of certain rate constants, which can be more easily estimated from data. In the next section we show how the QSSA is used to develop a rate expression for the production of a component from a statement of the elementary reactions, and illustrate the kinetic model simplification that results from the QSSA model reduction.. 

Species 2, 3 and 5 are QSSA species. Apply the QSSA to each of these species... 

Species 3 and 4 are QSSA species. Applying the QSSA to these reaction inteonediates gives... 

Assuming the radicals CH3, C2H5 and H are QSSA species, develop an expression for the rate of ethylene formation. Verify that this approximation is valid. 

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. 

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