Mixture fraction

Mixtures can be identified with the help of computer software that subtracts the spectra of pure compounds from that of the sample. For complex mixtures, fractionation may be needed as part of the analysis. Commercial instmments are available that combine ftir, as a detector, with a separation technique such as gas chromatography (gc), high performance Hquid chromatography (hplc), or supercritical fluid chromatography (96,97). Instmments such as gc/ftir are often termed hyphenated instmments (98). Pyrolyzer (99) and thermogravimetric analysis (tga) instmmentation can also be combined with ftir for monitoring pyrolysis and oxidation processes (100) (see Analytical methods, hyphenated instruments). 

Crotyl bromide [29576-14-5] M 135.0, b 103-105 /740mm, n s 1.4792. Dried with MgS04, CaC03 mixture. Fractionally distd through an all-glass Todd column. [Todd column. A column (which may be a Dufton type, fitted with a Monel metal rod and spiral, or a Hempel type, fitted with glass helices) which is surrounded by an open heating jacket so that the temperature can be adjusted to be close to the distillation temperature (Todd Ind Eng Chem (Anal Ed) 17 175 1945)]. 

Introducing a concept of gradient diffusion for particles and employing a mixture fraction for the non-reacting fluid originating upstream, / = c Vc O) and a probability density function for the statistics of the fluid elements, /(/), equation (2.100) becomes... 

Because p-xylene is the most valuable isomer for producing synthetic fibers, it is usually recovered from the xylene mixture. Fractional crystallization used to be the method for separating the isomers, but the yield was only 60%. Currently, industry uses continuous liquid-phase adsorption separation processes.The overall yield of p-xylene is increased... 

Mixture Fraction, Dissipation, and Finite-Rate Chemistry... 

The state of mixing between the fuel and oxidizer streams in nonpremixed flames is quantified by the mixture fraction, Conceptually, the mixture fraction is the fraction of mass that originates in the fuel stream, with 0.0 corresponding to the oxidizer stream and 1.0 corresponding to the pure fuel stream. The stoichiometric mixture fraction, indicates the condition for which the fuel... 

Scatter plots of temperature atx/d = 15 in turbulent Cl-14/air jet flames with Reynolds numbers of 13,400 (Flame C) and 44,800 (Flame F). The stoichiometric mixture fraction is = 0.351. The line shows the results of a laminar counterflow-flame calculation with a strain parameter of a = 100 s and is included as a visual guide. (From Barlow, R.S. and Frank, J.H., Proc. Combust. Inst, 27,1087,1998. With permission.)... 

One of the most challenging aspects of modeling turbulent combustion is the accurate prediction of finite-rate chemistry effects. In highly turbulent flames, the local transport rates for the removal of combustion radicals and heat may be comparable to or larger than the production rates of radicals and heat from combustion reactions. As a result, the chemistry cannot keep up with the transport and the flame is quenched. To illustrate these finite-rate chemistry effects, we compare temperature measurements in two piloted, partially premixed CH4/air (1/3 by vol.) jet flames with different turbulence levels. Figure 7.2.4 shows scatter plots of temperature as a function of mixture fraction for a fully burning flame (Flame C) and a flame with significant local extinction (Flame F) at a downstream location of xld = 15 [16]. These scatter plots provide a qualitative indication of the probability of local extinction, which is characterized... 

Qualitative comparison of the inclined structure of thin layers of high scalar dissipation in a piloted CH4/air jet flame as revealed by (a) mixture fraction imaging, (b) LES with a steady flamelet library (a and b are adapted from Kempf, A. Flemming, F., and Janicka, ]., Proc. Combust. Inst, 30, 557, 2005. With permission.), and (c) LES with unsteady flamelet modeling. (Adapted from Pitsch, H. and Steiner, H., Proc. Combust. Inst., 28, 41, 2000. With permission.)... 

Figure 7.2.12 shows scatter plots of instantaneous measurements of temperature and CH4 mole fraction obtained at a height of 5 mm and at several radial locations, which are color-coded in the figure. The foremost observable characteristics are that there are no samples richer than 0.2 in the mixture fraction (1.0 being pure fuel) and that many samples remain at room temperature even within the limits of flammability. Many... 

Scatter plots of temperature and CH4 mole fraction versus mixture fraction in a model gas turbine combustor. (From Meier, W., Duan, X.R., and Weigand, R, Combust. Flame, 144, 225, 2006. With permission.)... 

Following the steps for formulation of a CFD model introduced earlier, we begin by determining the set of state variables needed to describe the flow. Because the density is constant and we are only interested in the mixing properties of the flow, we can replace the chemical species and temperature by a single inert scalar field (x, t), known as the mixture fraction (Fox, 2003). If we take = 0 everywhere in the reactor at time t — 0 and set / = 1 in the first inlet stream, then the value of (x, t) tells us what fraction of the fluid located at point x at time t originated at the first inlet stream. If we denote the inlet volumetric flow rates by qi and q2, respectively, for the two inlets, at steady state the volume-average mixture fraction in the reactor will be... 

Thus, the reactor will be perfectly mixed if and only if = at every spatial location in the reactor. As noted earlier, unless we conduct a DNS, we will not compute the instantaneous mixture fraction in the CFD simulation. Instead, if we use a RANS model, we will compute the ensemble- or Reynolds-average mixture fraction, denoted by ( ). Thus, the first state variable needed to describe macromixing in this system is ( ). If the system is perfectly macromixed, ( ) = < at every point in the reactor. The second state variable will be used to describe the degree of local micromixing, and is the mixture-fraction variance (maximum value of the variance at any point in the reactor is ( )(1 — ( )), and varies from zero in the feed streams to a maximum of 1/4 when ( ) = 1/2. 

Note that when solving the CFD transport equations, the mean velocity and turbulence state variables can be found independently from the mixture-fraction state variables. Likewise, when validating the CFD model predictions, the velocity and turbulence predictions can be measured in separate experiments (e.g., using particle-image velocimetry [PIV]) from the scalar field (e.g., using planar laser-induced fluorescence [PLIF]). 

The basis idea behind multi-environment models is that the mixture fraction at any location in the reactor can be approximated by a distribution function in the form of a sum of delta functions as follows ... 

In other words, if we know pn and at every point in the reactor, then we can compute up to 2N— 1 independent mixture-fraction moments. 

In theory, this model can be used to fix up to three moments of the mixture fraction (e.g., (c), ( 2), and (c3)). In practice, we want to choose the CFD transport equations such that the moments computed from Eqs. (34) and (35) are exactly the same as those found by solving Eqs. (28) and (29). An elegant mathematical procedure for forcing the moments to agree is the direct quadrature method of moments (DQMOM), and is described in detail in Fox (2003). For the two-environment model, the transport equations are... 

As discussed earlier, acid-base reactions are always nonpremixed. For example, a semi-batch reactor could initially be filled with base at concentration B0 and acid is added with concentration A0. Likewise, a continuous reactor could be run with two feed streams one for acid and one for base. For both of these examples, the degree of mixing between the acid stream and the rest of the reactor contents can be quantified by introducing the mixture fraction , which obeys... 

The modeling approaches discussed below for a single mixture fraction component can thus be extended to to treat more complex flow configurations (Fox, 2003). 

The mixture fraction as defined above describes turbulent mixing in the reactor and does not depend on the chemistry. However, by comparing Eqs. (45) and (46), we can note that they have exactly the same form. Thus, for the acid-base reaction, the mixture fraction is related to rA—B by... 

Thus, the CFD simulation need to only treat the turbulent mixing problem for the mixture fraction. Once (or its statistics) are known, the acid and base concentrations can be found from Eqs. (49) and (50), respectively. 

The acid-base reaction is a simple example of using the mixture fraction to express the reactant concentrations in the limit where the chemistry is much faster than the mixing time scales. This idea can be easily generalized to the case of multiple fast reactions, which is known as the equilibrium-chemistry limit. If we denote the vector of reactant concentrations by and assume that it obeys a transport equation of the form... 

In fact, we are only interested in the value of for t — oo subject to initial conditions that depend on the mixture fraction as follows ... 

In a turbulent flow, the local value (i.e., at a point in space) of the mixture fraction will behave as a random variable. If we denote the probability density function (PDF) of by f - Q where 0 < ( < 1, the integer moments of the mixture fraction can be found by integration ... 

For example, (A) and (B) can be computed using Eqs. (49) and (50), respectively. Note that instead of Eq. (55), we could use the simpler expression for given by Eq. (33), which avoids the need for numerical quadrature. In both cases, the mean and variance of the mixture fraction are identical (and thus both models account for finite-rate mixing effects.) In practical applications, the differences in the predicted values of () can often be small (Wang and Fox, 2004). 

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