In this chapter we derive the general equations and discuss the solution methodology for simulation of zero-dimensional systems that are not well mixed. These "partially stirred" systems include both open (with flow) and closed systems and the equations are solved in transient form, using time-integration methods. The following Ansys Chemkin reactor models are addressed:

Many practical applications deviate significantly from an ideally mixed situation, including gas turbines and internal combustion engines. When the turbulent mixing rate is not fast compared to chemical kinetics, the degree of mixing can have a profound impact on the reactor characteristics. The PaSR model allows us to relax the perfectly stirred reactor (PSR) assumption of fast turbulent mixing. Since the most salient feature of a PaSR is the unmixed nature of the reactive fluids at the molecular level, the modeling focuses on the influence of an unmixed state on the reactor properties. The mean thermo-chemical properties inside a PaSR are assumed to be spatially homogeneous, but imperfectly mixed at the molecular level. That is, the reactive fluids are not completely diffused into each other at the molecular level but their mean values are uniform throughout the reactor by turbulent stirring.

The mixing process in the PaSR is characterized by the mixing frequency, which is often modeled by the reciprocal of the turbulence time scale. Because fluid dynamics inside the PaSR are not resolved, the mixing frequency will be prescribed as an input parameter. Therefore, in addition to the mean reactor residence time, the mixing time is another fluid mechanical time scale that controls the properties of the PaSR.

The composition and temperature in the PaSR are described by a probability density function (PDF). This composition PDF is a subset of the joint velocity-composition PDF, because the flow field in the PaSR is assumed to be spatially homogeneous. Velocity fluctuations are also ignored; that is, the PDF is over scalars only, but is not a delta-function in scalar space because reactants, intermediates, and products are not mixed at the molecular level.

The PaSR is related to and bounded by other models commonly used in combustion. When the mixing time scale approaches zero, the mixing process becomes fast enough that the properties inside the PaSR are homogeneously mixed at the molecular level. In this limit, the PaSR becomes a PSR, for which the joint scalar PDF degenerates to a delta function in the composition space and the mean residence time is the sole controlling time scale. In the other extreme limit (large mixing time), there is no mixing among the pockets of gas in the PaSR; consequently, the PaSR consists of segregated reactive mixtures. The average statistics are the sum of the properties of the reactive mixture pockets weighted by the PDF of their ages inside the PaSR. In this case the PaSR acts like a plug-flow reactor, for limit of no mixing and relatively large mean residence times (closed system).

The PaSR may be used as a stand-alone model for studying turbulent combustion or other reactor systems where mass transport may be a rate-limiting factor. Or a PaSR can be used to simulate the sub-grid turbulent mixing and chemical reactions in a computational fluid dynamic (CFD) cell.

One of the important concerns of turbulent reactive flows, especially turbulent flames, is the coupling between chemical reactions and turbulence. The interaction between chemical reactions and fluid dynamics is best described by the Damköhler number, which is defined as the ratio of characteristic flow time and the characteristic chemical reaction time. Due to the large spectrum of chemical times in a multicomponent chemical system, the Damköhler number can also span a large spectrum. As the Damköhler number corresponding to a specific chemical reaction approaches infinity, the reaction responds much faster to the flow so that it approaches equilibrium conditions. On the other hand, if the Damköhler number of a reaction is small, the reaction is considered frozen. Only when the Damköhler number is of the order of unity, are the interactions between the reaction and the fluid dynamics strong. In this case, the reaction becomes one of the "controlling" steps of the process. Depending on the flow time, the same chemical process can be controlled by different sets of reactions.

A PaSR addresses the interaction between chemical reactions and turbulence. [78], [79] The basic assumptions for the PaSR are similar to other zero-dimensional models. The major difference between a PSR and a PaSR lies in the treatment of the molecular mixing inside the reactor. In a PSR, the contents of the reactor are well mixed, by assuming very diffuse conditions, high-intensity turbulent stirring action, or some other active "stirring" mechanism. The only influence of fluid dynamics in a PSR is introduced by the reactor residence time . Unlike the PSR, a PaSR allows fluid dynamics to control the extent of the molecular mixing and consequently the chemical reactions, by means of an additional parameter: the scalar mixing time, . The turbulent mixing time scale is often considered to be proportional to the turbulent eddy turnover time as

(10–1)

where is usually treated as a constant but its value varies for different flow configurations. The ratio of turbulent kinetic energy to its dissipation rate, , represents the time scale of the energy-containing eddies, which characterize the turbulent mixing action.