Fast chemistry

In any circumstances, it can be expected that and (5x are algebraic functions of turbulence length scale and kinetic energy, as well as chemical and molecular quantities of the mixture. Of course, it is expedient to determine these in terms of relevant dimensionless quantities. The simplest possible formula, in the case of very fast chemistry, i.e., large Damkohler number Da = (Sl li)/ SiU ) and large Reynolds Re = ( Ij)/ (<5l Sl) and Peclet numbers, i.e., small Karlovitz number Ka = sjRej/Da will be Sj/Sl =f(u / Sl), but other ratios are also quite likely to play a role in the general case. 

Let us recapitulate. We have achieved a solution to boundary-layer-like burning of a steady liquid-like fuel. A thin flame or fast chemistry relative to the mixing of fuel and oxygen is assumed. All effects of radiation have been ignored - a serious omission for flames of any considerable thickness. This radiation issue cannot easily be resolved exactly, but we will return to a way to include its effects approximately. 

The first factor occurs even in homogeneous flows with two inert scalars, and is discussed in Section 3.4. The second factor is present in nearly all turbulent reacting flows with moderately fast chemistry. As discussed in Chapter 4, modeling the joint scalar dissipation rate is challenging due to the need to include all important physical processes. One starting point is its transport equation, which we derive below. 

In general, liquid-phase reactions (Sc > 1) and fast chemistry are beyond the range of DNS. The treatment of inhomogeneous flows (e.g., a chemical reactor) adds further restrictions. Thus, although DNS is a valuable tool for studying fundamentals,4 it is not a useful tool for chemical-reactor modeling. Nonetheless, much can be learned about scalar transport in turbulent flows from DNS. For example, valuable information about the effect of molecular diffusion on the joint scalar PDF can be easily extracted from a DNS simulation and used to validate the micromixing closures needed in other scalar transport models. 

Figure 5.1. Closures for the chemical source term can be understood in terms of their relationship to the joint composition PDF. The simplest methods attempt to represent the joint PDF by its (lower-order) moments. At the next level, the joint PDF is expressed in terms of the product of the conditional joint PDF and the mixture-fraction PDF. The conditional joint PDF can then be approximated by invoking the fast-chemistry or flamelet limits, by modeling the conditional means of the compositions, or by assuming a functional form for the PDF. Similarly, it is also possible to assume a functional form for the joint composition PDF. The best method to employ depends strongly on the functional form of the chemical source term and its characteristic time scales.

For a one-step reaction, most commercial CFD codes provide a simple fix for the fast-chemistry limit. This (first-order moment) method consists of simply slowing down the reaction rate whenever it is faster than the micromixing rate ... 

The determination of a mixture-fraction basis is a necessary but not a sufficient condition for using the mixture-fraction PDF method to treat a turbulent reacting flow in the fast-chemistry limit. In order to understand why this is so, note that the mixture-fraction basis is defined in terms of the conserved-variable scalars pcv without regard to the reacting scalars pT. Thus, it is possible that a mixture-fraction basis can be found for the conserved-variable scalars that does not apply to the At reacting scalars. In order to ensure that this is not the case, the linear transformation Mr defined by (5.30) on p. 149 must be applied to the (K x VIM ) matrix... 

The irradiation of water is immediately followed by a period of fast chemistry, whose short-time kinetics reflects the competition between the relaxation of the nonhomogeneous spatial distributions of the radiation-induced reactants and their reactions. A variety of gamma and energetic electron experiments are available in the literature. Stochastic simulation methods have been used to model the observed short-time radiation chemical kinetics of water and the radiation chemistry of aqueous solutions of scavengers for the hydrated electron and the hydroxyl radical to provide fundamental information for use in the elucidation of more complex, complicated chemical, and biological systems found in real-world scenarios. 

The velocity of the front increases linearly with the initial reactant concentration and with the square root of the rate constant and the diffusion coefficient. Fast chemistry and high diffusion both increase the speed of the wave. 

These approaches can be divided into two groups. In the first group, fast chemistry (approaches 1 and 2), it is assumed that the rate of chemical conversion is not kinetically controlled. The second group,finite rate chemistry (approaches 3-5), allows for kinetically controlled processes, in that restrictions are put on the chemical reaction rate. Below we discuss these different approaches in more detail. 

The fast chemistry approximation implies that chemical reaction is instantaneous. Consequently the conversion from reactants to products is limited only by mass transport. This way of describing the chemistry is useful in many applications, but obviously it must be used with caution. 

In chemically reacting flow systems, the overall reaction rate may be limited by the mixing rate of the reactants or by the rate of the chemical reaction upon mixing. If mixing is slow compared to chemical reaction, the system is diffusion or mixing controlled, while fast mixing and slow reaction results in a kinetically controlled system (Fig. 13.1). The assumption of fast chemistry is valid if the system is mixing controlled. 

One way to assess the validity of the assumption of fast chemistry is to estimate the Damkohler number (see also Section 6.8.1.1). This number is an important dimensionless parameter that quantifies whether a process is kinetically or diffusion controlled. The... 

The chemical equilibrium assumption often results in modeling predictions similar to those obtained assuming infinitely fast reaction, at least for overall aspects of practical systems such as combustion. However, the increased computational complexity of the chemical equilibrium approach is often justified, since the restrictions that the equilibrium constraint places on the reaction system are accounted for. The fractional conversion of reactants to products at chemical equilibrium typically depends strongly on temperature. For an exothermic reaction system, complete conversion to products is favored thermodynamically at low temperatures, while at high temperatures the equilibrium may shift toward reactants. The restrictions that equilibrium place on the reaction system are obviously not accounted for by the fast chemistry approximation. 

The values predicted by the equilibrium calculations can be compared with exhaust concentrations observed in practical combustion systems. The major species (i.e., CO2, H2O, and O2) are well predicted by thermal equilibrium. In most of the temperature range covered in the figure, the fuel is fully oxidized. The fast chemistry assumption would also be sufficient to predict the exhaust concentrations of these species. The problem arises if the chemical equilibrium assumption is also used to estimate the concentration level of minor species, such as carbon monoxide, nitrogen oxides, and sulfur oxides. 

The assumption of fast chemistry is useful in systems with diffusion limitation. However, if the chemistry is kinetically limited or if the chemistry involves competing product channels, only finite rate chemistry provides a good representation. Finite rate chemistry can be represented on different levels of complexity, ranging from a single global reaction to a detailed reaction mechanism involving perhaps hundreds of species. These approaches are described in more detail in the following sections. 

Kaiser, N.F.K., Bremberg, U., Larhed, M., Moberg, C. and Hallberg, A., Fast, convenient, and efficient molybdenum-catalysed asymmetric allylic alkylation under noninert conditions an example of microwave-promoted fast chemistry Angew. Chem., Int. Ed. Engl., 2000,39, 3596-3598. 

These are the quantities to which we are giving our attention. Vibrational Raman scattering is being used for the temperature and density data, and, when taken simultaneously with velocity data from coupled LV instrumentation (.8), provides also the fluctuation mass flux through use of fast chemistry assumptions and the ideal gas law for atmospheric pressure flames. 

Conceptually, it is the atomic number and the electronic configuration of an element that define its position in the Periodic Table. Since they cannot be measured for the very heavy elements, information on its chemical behavior is often used to place an element in a chemical group. Unfortunately, with increasing nuclear charge the cross sections and the production rates drop so rapidly that such chemical information can be accessed only for elements with a half-life of the order of at least few seconds and longer. In this case, some fast chemistry techniques are used. They are based on the principle of chromatographic separations either in the gas phase exploiting the differences in volatility of heavy element compounds, or in the aqueous... 

The combination of microwave heating and fluorous chemistry is of considerable interest, as this approach combines fast chemistry and easy separation. 

For initially nonpremixed reactants, two limiting cases may be visualized, namely, the limit in which the chemistry is rapid compared with the fluid mechanics and the limit in which it is slow. In the slow-chemistry limit, extensive turbulent mixing may occur prior to chemical reaction, and situations approaching those in well-stirred reactors (see Section 4.1) may develop. There are particular slow-chemistry problems for which the previously identified moment methods and age methods are well suited. These methods are not appropriate for fast-chemistry problems. The primary combustion reactions in ordinary turbulent diffusion flames encountered in the laboratory and in industry appear to lie closer to the fast-chemistry limit. Methods for analyzing turbulent diffusion flames with fast chemistry have been developed recently [15], [20], [27]. These methods, which involve approximations of probability-density functions using moments, will be discussed in this section. 

Turbulence models are generally limited to fully developed high-Reynolds number flows. Gas-phase flows are normally characterized by 5c 1, while for liquid phase flows, 5c > 1. The value of this Damkohler number indicates the relative rates of the mixing and chemical reaction rate time scales. Reactive flows might thus be divided into three categories Slow chemistry (Da/ -C 1), fast chemistry (Da/ 1), and finite rate chemistry (Da/ 1). 

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