Partial equilibrium assumption

As will be discussed in the following chapter, most combustion systems entail oxidation mechanisms with numerous individual reaction steps. Under certain circumstances a group of reactions will proceed rapidly and reach a quasi-equilibrium state. Concurrently, one or more reactions may proceed slowly. If the rate or rate constant of this slow reaction is to be determined and if the reaction contains a species difficult to measure, it is possible through a partial equilibrium assumption to express the unknown concentrations in terms of other measurable quantities. Thus, the partial equilibrium assumption is very much like the steady-state approximation discussed earlier. The difference is that in the steady-state approximation one is concerned with a particular species and in the partial equilibrium assumption one is concerned with particular reactions. Essentially then, partial equilibrium comes about when forward and backward rates are very large and the contribution that a particular species makes to a given slow reaction of concern can be compensated for by very small differences in the forward and backward rates of those reactions in partial equilibrium. 

A specific example can illustrate the use of the partial equilibrium assumption. Consider, for instance, a complex reacting hydrocarbon in an oxidizing medium. By the measurement of the CO and C02 concentrations, one wants to obtain an estimate of the rate constant of the reaction... 

Thus, one observes that the rate expression can be written in terms of readily measurable stable species. One must, however, exercise care in applying this assumption. Equilibria do not always exist among the II2 02 reactions in a hydrocarbon combustion system—indeed, there is a question if equilibrium exists during CO oxidation in a hydrocarbon system. Nevertheless, it is interesting to note the availability of experimental evidence that shows the rate of formation of C02 to be (l/4)-order with respect to 02, (l/2)-order with respect to water, and first-order with respect to CO [17,18], The partial equilibrium assumption is more appropriately applied to NO formation in flames, as will be discussed in Chapter 8. 

If one invokes the steady-state approximation described in Chapter 2 for the N atom concentration and makes the partial equilibrium assumption also described in Chapter 2 for the reaction system... 

Once the skeletal mechanism is established, a reduced mechanism is developed by applying steady-state and partial-equilibrium assumptions. The criteria for assuming steady-state or partial-equilibrium are discussed in Section 13.2.5. The concentration of species, typically radicals, that can be assumed in steady state can be estimated based on concentrations of other species and rate constants for relevant reactions. Thereby the steady-state species can be eliminated from the reaction mechanism. After elimination of steady-state species, the required number of multi-step reactions is determined. The reaction rate for these multi-step reactions can be calculated from the reaction rates of the original mechanism. The multi-step reaction rates depend on the concentration of the eliminated steady-state species. Partial equilibrium assumptions are often applied to the fastest elementary reactions to simplify the estimation of the steady-state concentrations. 

Below are some examples of chain-propagating and chain-branching systems. These examples are used to illustrate the different stages of a gas-phase reaction and to introduce the steady-state and partial equilibrium assumptions as tools for analysis. 

To eliminate the second unknown variable, the oxygen atom concentration, we introduce another simplifying assumption, the partial-equilibrium assumption. Reactions that are fast in both the forward and reverse direction may be assumed in partial equilibrium. Usually this assumption can only be applied to reactions that involve radicals both as reactants and as products, such as the reaction N + OH NO + H (R11). However, here we will assume that the oxygen atom is in partial equilibrium with molecular oxygen,... 

Thermal dissociation of 02 has a high activation energy and is usually quite slow. However, at the high temperatures in combustion systems where thermal NO is important, both O and 02 are involved in a number of reactions that are fast compared to the thermal NO formation. Due to the fast exchange between O and 02 in these reactions, the partial equilibrium assumption for (R12) is a reasonable approximation. Assuming partial equilibrium, we can relate the oxygen atom concentration to the 02 concentration and the equilibrium constant for the reaction,... 

They found the heat release rate to be proportional to the product [H] [O21 [H2O], and the dependence of H on pressure and mass flow to be also consistent with the removal of H by reaction (iv). Similar conclusions about the recombination were reached by Getzinger and Schott [181] from shock tube experiments, in which OH concentrations were measured and used to calculate total radical concentrations by means of the partial equilibrium assumption. 

Comparison of the computed profiles with experiment may in principle be used to establish values of some of the unknown rate coefficients. The radical pool in this computation includes molecular oxygen as a bi-radical. The validity of the partial equilibrium assumptions will be discussed in Sect. 5.4.4. 

The range of validity of the partial equilibrium assumptions in specific flames may now be examined by comparison of the H, OH, O and Oj profiles computed on this assumption with those computed by means of the q.s.s. condition. The p.e. assumption gives profiles which continue to rise indefinitely on integration backwards from the hot boundary of the flame. It can also be shown that the q.s.s. overall radical profile, represented by (ATh + 2A"o + Xq h ) approaches the similar p.e. profile (i.e. 

Partial equilibrium assumptions. At present, the only systematic method for the truncation of QSSA expressions is the use of partial equilibrium assumptions which may be applied in the case of fast reversible reactions. If a steady-state species has many reactions, but two of these are fast and reversible with rates much higher than the other reactions involved, e.g.. 

These partial equilibrium assumptions are equivalent to truncating the QSSA expression after the largest term. The study of relative reaction rates is, therefore, important in order to find such fast reversible reactions. 

The rates of the last two reactions were obtained by partial equilibrium assumptions, while the rates of the first two reactions were tuned to reproduce both the measured flame speed and the observed CO to H2... 

Sarofim and Pohl (16) used this same technique and found fair agreement with their data on premixed, atmospheric pressure flat flames. Iverach et al, 17) used a similar partial equilibrium assumption to correlate their data on hydrocarbon flames and found good agreement under fuel-lean (excess air) conditions. Poor agreement was observed under fuel-rich conditions unreasonably large radical concentrations were required to make the Zeldovich mechanism account for the measured NO. Iverach, therefore, suggested that reactions such as those proposed by Fenimore may be important under fuel-rich conditions. 

Contrast the applicability of equilibrium and kinetic models to some environmental problems. Explain local and partial equilibrium assumptions. 

A point that often receives insufficient appreciation is that a system in the laboratory often behaves differently in sometimes critical ways from what is nominally the same system in the field. If one is studying rocks that are the products of natural hydrothermal alteration, the partial equilibrium assumption is more likely to be valid than it is in a laboratory hydrothermal apparatus in which one attempts to recreate such alteration. Such differences are commonly manifested in the appearance of different mineral assemblages, though changes in fluid chemistry may be very similar. An example is hydrothermal reaction of seawater and basalt, a process which occurs naturally at midocean ridges (sec refs, i, M, and many sources cited therein). The naturally altered basalts become rich in chlorite or chlorite plus epidote. In experimental systems, smectite clays appear instead. Time appears to be the limiting factor. 

The use of partial equilibrium assumption is justified when the coefficients of the reaction rates of the forward and backward reactions are much larger than aU the other coefficients of the reaction mechanism [14]. 

Consider the mechanism of hydrogen presented in Table 2.4 [15], A numerical or experimental analysis shows that for high temperatures, T > 1800K and p = 1 bar, the reaction rate coefficients of forward and backward reactions are so high that the partial equilibrium assumption can be considered for the reactions 1, 2, and 3. In this case, each reaction is in equilibrium and, therefore, the coefficients of the reaction rates of the forward and backward reactions are equal, that is,... 

The other extreme is to evaluate the remaining terms at time n + 1)((50 the fully implicit or backward differencing approach. It leads to a set of algebraic equations from which the dependent variables at time (n -h 1)( 0 can be calculated. This approach is unconditionally stable (Richtmyer and Morton, 1967), and is the approach used here. We may of course also use other schemes in which intermediate weights are given to the forward and backward differences. These partially implicit schemes lead to improved accuracy. However, if attempts are made to use them on systems of stiff equations, the latter must be treated by asymptotic techniques. In chemical situations such techniques are equivalent to the use of the chemical quasi-steady-state or partial equilibrium assumptions at long times. They will be considered again in Section 9. 

Chemical quasi-steady-state and partial equilibrium assumptions in reactive flow modeling... 

The partial equilibrium assumptions by themselves in conjunction with Eqs. (9.1), (9.2), and (9.3), and a reaction mechanism as outlined above, do not permit construction of a complete model from which an eigenvalue burning velocity and full profiles may be computed ab initio. On the other hand the assumptions are extremely useful when dealing with H-N-C-O ffame systems, since their application to reactions (i), (ii), (iii), and (xviii) above allows us to calculate many of the species profiles, and particularly the free radical profiles, on close approach to full equilibrium. The time-dependent computation does not economically do this directly. The computations require an input mass flux or burning velocity which must be either a measured or a separately calculated value. For composite flux calculation purposes the overall radical pool is chosen so as to represent a total flux of free electron spins, that is, spins belonging to H, O, OH, and O2 (Dixon-Lewis et al, 1975). 

Flame profiles based on partial equilibrium assumptions are comparatively easy to compute, with a single integration through the zone being sufficient. The initial values of the dependent variables must represent a partial equilibrium condition which is a perturbation of full equilibrium (Dixon-Lewis and Greenberg, 1975). Such computations were used by Dixon-Lewis (1979) in connection with the analysis of measured radical concentration... 

Figure 13. Computed temperature profile and mole fraction profiles of stable species and hydrogen atoms in 60% (24.1% hydrogen + 75.9% carbon monoxide) + 40% air flame at atmospheric pressure and = 298 K (from Cherian et al., 1981b). Solid lines time-dependent flame calculation broken lines calculation with partial equilibrium assumptions on reactions (i), (ii), (iii), and (xxi). (By courtesy of The Royal Society.)...

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