The Quasi-Steady-State Assumption

Clearly, /t0(f) is the total number and is the total volume of the agglomerates in the charge. The latter is approximately independent of time in view of the quasi-steady-state assumption in Eq. (46). Kapur (K3) has shown that an asymptotic similarity solution to Eq. (47) exists in the following form ... 

In addition to the quasi-steady state assumption, the other assumptions required to arrive at equation (1) are 1. the aerosol itself does not coagulate 2. there is a fully developed concentration gradient around each aerosol particle and 3. the concentration of unattached radon progeny atoms is much greater than the concentration of aerosol particles (in order that concentration gradients of radon progeny atoms may exist). This last assumption is usually not valid since the radon progeny concentration is usually much less than the aerosol concentration. 

Very often, the quasi steady-state assumption for the hydrogen liquid phase concentration is proposed (Eq. (30)) ... 

Segel, L. A. Slemrod, M. The quasi-steady-state assumption - a case-study in perturbation. SIAM Rev 1989, 31 446-477. 

In this note, an equation is established for the rata cl reversible adsorption on the basis of the quasi steady state assumption over too entire thickness of too interaction -force boundary layer. When an interaction force boundary layer occurs fcT), we find that the epproxi-... 

By splitting the quasi-steady-state assumption of diffusion of particles under the action of the interaction force field into two parts, kinetic equations which account for accumulation at both the primary and secondary minimum are formulated. Conditions are established under which, after a short transient, reversible adsorption or coagulation can be treated by neglecting accumulation at the secondary minimum. The effect of tangential velocity of particles on the rate of reversible adsorption is analyzed and a criterion established when the effect... 

The last assumption is referred to as the quasi-steady-state assumption. The fraction of the bed which is poisoned is a function of time only and not of bed length, reactor space time, or the concentration of the reactant A external to the pellets. At any given time the bed activity will be constant, and only one concentration of the poison precursor species S will exist in the bed. Such a situation will be more likely to occur when deactivation rates are low compared to reaction rates. Under this condition S will spread evenly throughout the bed. Within particles, however, concentration gradients of S may still exist depending on the poisoning mechanism and the pore and pellet properties. 

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. 

This method integrally employs the quasi-steady state assumptions to relate the concentrations of H, OH and O in the overall radical pool, and can be applied to either fuel-rich or fuel-lean flames. Concentrations of HO2 were also calculated using the quasi-steady state condition, but because these were mostly much smaller than the other radical concentrations they were considered in the same manner as OH and O in the simpler method. Both methods lead to similar results for the low temperature, fuel-rich flames considered at present, indicating that the reverse reactions other than (—i) and (—iii) are relatively unimportant over most of these reaction zones. Three internally consistent sets of rate coefficients on which the more refined treatments may be based are given... 

Another interpretation is related to the dynamic behavior of a closed chemical system and the quasi-steady-state assumption. It is common in the kinetics of reaction systems to assume that the concentrations of intermediates reach a quasi-steady state The rate of production and the... 

The first 1 species (fly with 7 = 1,...,/) are assumed to be the intermediates, whereas the remaining T = A -1 species (fly, 7 = / -t- 1,..., / -I- T) are terminal species. If intermediates are present in very low concentrations, and their high rate of production is balanced by a high rate of consumption, the quasi-steady-state assumption allows dropping the accumulation term from the mass balance of an intermediate. 

An overall mechanism (Horiuti and Nakamura, 1967 Horiuti, 1973 Temkin 1973, 1979) is one that obeys the quasi-steady-state assumption It must consist of steps combined in specific proportions, such that the net transformation involves only terminal species. 

Although there is significant net production of some terminal species, and significant net consumption of others, within the quasi-steady-state assumption the concentrations of intermediates are low and the rate of production of each intermediate by some steps is approximately balanced... 

The intermediate complexes are in steady-state with respect to the rate of the overall reaction. This is sometimes called the quasi-steady state assumption, since not all molecular concentrations are required to be in steady-state. In many cases, intermediate complexes are treated as being in equilibrium with respect to the rate of the overall reaction. This is referred to as the quasiequilibrium assumption, which is gena-ally more restrictive. 

This need not be true in vivo where the concentrations of reactants and their enzymes in some cases are nearly comparable. Under these conditions, the nominal concentration of substrate could be significantly greater than the level of unbound substrate, and the reaction rate calculated with nominal concentrations inserted into the rate law clearly would overestimate the rate observed in vivo (Wright et al., 1992 Shiraishii and Savageau, 1993). This condition does not alter the basic chemical kinetic equations that describe the mechanism, but it does mean that the quasi-steady state assumption (e.g., see Peller and Alberty, 1959 Segel and Slemrod, 1989) may be inappropriate when reaction rates change with time in vivo. 

Consider a cooled circular tube in zero gravity (Fig. 14.20). Making the quasi-steady-state assumption used before, an energy balance yields... 

The quasi-steady-state assumption (QSSA) and the reaction equilibrium assumption allow us to generate reaction-rate expressions that capture the details of the reaction chemistry with a minimum... 

We see in the next section that this case is well described also by making the quasi-steady-state assumption on species B. 

The poisoning rate with respect to the main reaction rate is low, so the quasi-steady state assumption is valid. 

LH rate expressions often rely on the quasi steady state assumption that stipulates that one elementaiy step is rate determining whereas the other steps are at equilibrium. The assumption is debatable in me wide temperature range encountered in the monolith, because the rate determiniim step may be different according to tlie temperature level. It would thus be useful to know whether the rate expressions obtained at moderate temperature are still reliable at the higli temperature and low reactant concentrations observed after light-off has occurred. However, care must be taken of the possible transfer limitations, and of flie possible catalytically induced homogeneous reactions. 

According to the time dependence of P02, the quasi steady state assumption holds or not. Wlien Pp2 is constant and qox = Koxf(P02), (9) is... 

The deactivation model allows calculation of the rate of deactivation given the temperature and the activity of the catalyst (via site concentrations). Integration of the rate permits computation of the variation in activity with time. The model is incorporated into a reformer code by making the quasi-steady state assumption the rate of deactivation is slow so that the species and temperature profiles in the reformer are determined by the existing activity profile of the catalyst. 

To eliminate the concentrations of the propagating radicals from Equation 6.4, the quasi-steady-state assumption... 

In the partially filled case, there is a liquid-gas interface within the channel. By considering the quasi-steady-state assumption, the pressure at the inlet and at the outlet is related by... 

The total rate of polymer radical formation is given by (Rjnit + Ptr)- However, the net formation of polymeric radicals is Rinit, since transfer events both consume and create a polymeric radical species. With a continuous source of new radicals in the system, an equilibrium is achieved instantaneously between radical generation and consumption, such that Rinit = Rterm- This equilibrium, shown to be true for almost all FRP conditions [6], results from the fast dynamics of radical reactions compared to that of the overall polymerization system. Often referred to as radical stationarity or the quasi-steady-state assumption (QSSA), it leads to the well-known analytical expression... 

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. 

Once tbe "fast" and the "slow" components are identified, several of these methods use the quasi-steady state assumption (QSSA) as the basis for reduction. Together with a... 

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