Partial equilibrium approximation

There are some common characteristics for gas-phase reaction systems that form the basis for understanding and describing the chemical behavior. In this section we will discuss some basic definitions and terms that are useful in kinetics, such as reaction order, molec-ularity, chain carriers, rate-limiting steps, steady-state and partial equilibrium approximations, and coupled/competitive reactions. 

Although numerical integrations of equation (14) usually can be performed accurately now, some systems remain that are too complex or too stiff for the computational methods to succeed. To analyze such systems, as well as to understand better the behaviors of various systems amenable to accurate numerical integrations, rational approximations producing simplifications are wanted. Two important approximations of this type are the steady-state and partial-equilibrium approximations. 

The partial-equilibrium approximation differs from the steady-state approximation in that it refers to a particular reaction instead of to a particular species. The mechanism must include the forward and backward steps of any reaction that maintains partial equilibrium, and the approximation for a reaction k is then expressed by setting = 0 in equation (11). It is not always proper to conclude from this that when equations (6), (10), and (11) are employed in equation (14), the terms may be set equal to zero for each k that maintains partial equilibrium partial equilibria occur when the forward and backward rates are both large, and a small fractional difference of these two large quantities may contribute significantly to dcjdt. The criterion for validity of the approximation is that be small compared with the forward or backward rate. 

The same type of difficulty that is resolved by use of equation (35) for the partial-equilibrium approximation may also arise in connection with the steady-state approximation. For example, part of the sum of terms that contribute to the production rate of a primary species, to which the steady-state approximation is not applied, may be a constant multiple of cz . for an intermediary that is subject to the steady-state approximation, and the remaining terms in the production rate may be smaller than (U- even though (u. is small compared with. Under this condition, inaccurate results for the concentration history of the primary species will be obtained by use of the steady-state approximation for the intermediary unless a substitution... 

The Lindemann mechanism for unimolecular reactions, discussed in Section B.2.2, provides a convenient vehicle for illustrating partial-equilibrium approximations and for comparing them with steady-state approximations, even though this mechanism is not a chain reaction. To use the partial-equilibrium approximation for the two-body production of SRJ, select for example, as the species whose concentration is to be determined by partial equilibrium and use... 

To analyze the rate terms on the right-hand side of equation (2.6), the use of the partial equilibrium approximation is extended to permit evaluation of other unmeasured species concentrations which may enter these terms. This use of the measured main reaction progress to evaluate the concentration of a kinetically significant intermediate is closely analogous to the conventional quasi-steady state approximation in kinetics, but free of the usual restriction on its accuracy or utility that the concentration so evaluated be stoichiometrically minor. 

Kinetic interpretation of the recombination portion of the [OH] profile proceeds by means of the partial equilibrium approximation, by which the course of change of the entire system composition with recombination is computed, using the measured shock wave speed and initial gas composition and known thermochemical properties of the expected species. To place the progress of recombination in perspective, the change in N from its original value of unity to the value at full equilibrium, A, is reckoned in terms of the normalized progress variable defined as... 

Experimental limitations restricted the quantitative analysis of HgO infrared emission profiles to hydrogen-lean (jj = l OandO 5)H2-Oa-Ar mixtures. Under these conditions an empirical calibration, combined with the partial equilibrium approximation and equation (2.18), made possible an evaluation of k/[M]. Correction for the small contribution of Mj = HjO using the ratio kf jkf = 25 from Table 2.2 yielded a series of values of kf that average 3-0 x 10 cm mole sec. As in the OH absorption experiments, kf is essentially independent of v, indicating that reaction (/) is the sole termolecular reaction of significance throughout the recombination regime. 

Apply the partial equilibrium approximation to the mechanism (37)-(39) assuming that the three reactions are at quasi-equilibrium. 

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. 

The pre-equilibrium approximation (PEA also called the partial equilibrium approximation or fast-equilibrium approximation) is applicable when the species participating in a pair of fast-equilibrium reactions are consumed by slow reactions. After the onset of an equilibrium, the rates of the, forward and backward reactions become equal to each other, and therefore the ratios of the concentrations of the participating species can be calculated from the stoichiometry of the reaction steps and the equilibrium constant. According to the pre-equilibrium approximation, if the rates of the equilibrium reactions are much higher than the rates of the other reactions consuming the species participating in the equilibrium reactions, then the concentrations of these species are determined, with good approximation, by the equilibrium reactions only. 

Gou, X., Chen, Z., Sun, W., Ju, Y. A dynamic adaptive chemistry scheme with error control for combustion modeling with a large detailed mechanism. Combust. Flame 160, 225-231 (2013) Goussis, D.A. Quasi steady state and partial equilibrium approximations their relation and their validity. Combust. TheOTy Model. 16, 869-926 (2012)... 

---