Turbulent reactive mixing

Traditionally reactor engineers analyze the mixing of fluids in terms of the degree of segregation (Levenspiel, 1972), a measure of mixing on the molecular scale. 

FIGURE 5.1 Macrofluid and microfluid Idealizations (from Levenspiel, 1972). 

In order to gauge the relative importance and possible interaction between turbulence and chemical reactions, it is necessary to evaluate the various processes involved in reactive mixing. When a fluid element of different component (tracer) is added to the turbulent flow field, molecular mixing (and reaction, if possible) proceeds through several steps/mechanisms, some of which are listed below  

Step 1 convection by mean velocity Step 2 turbulent dispersion by large eddies Step 3 reduction of segregation length scale Step 4 laminar stretching of small eddies Step 5 molecular diffusion and chemical reaction 

FIGURE 5.2 Energy and concentration spectra for isotropic turbulence (from Bakker, 1996). 

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Van Vliet, E., Turbulent reactive mixing in process equipment , Ph.D. Thesis, Delft University of Technology, Delft, Netherlands (2003). 

Li, K. T and Toor, H. L., Turbulent reactive mixing with a series-parallel reaction Effect of mixing on yield. AIChE. J. 32,1312 (1986). 

Li, K.T. and Toor, H.L. (1986), Turbulent reactive mixing with series parallel reaction, effect of mixing on yield,A/C/i 7.,32, 1312. 

Bretherick, L., 1990, Handbook of Reactive Chemical Hazards , 4th ed., Butterworths, London. Brodkey, R.S., 1975, Turbulence in Mixing Operations , Academic Press, New York. 

Amerja, P. V., M. Singh, and H. L. Toor (1976). Reactive mixing in turbulent gases. Chemical... 

Subramaniam, S. and S. B. Pope (1998). A mixing model for turbulent reactive flows based on Euclidean minimum spanning trees. Combustion and Flame 115,487-514. 

Comparison of mixing model performance for nonpremixed turbulent reactive flow. Combustion and Flame 117, 732-754. 

Rhodes, P. R. 1975. A probability distribution function for turbulent flows. In Turbulent mixing in nonreactive and reactive mixing. Ed. S. N. B. Murthy. New York, NY Plenum Press. 235-41. 

The Eulerian (bottom-up) approach is to start with the convective-diffusion equation and through Reynolds averaging, obtain time-smoothed transport equations that describe micromixing effectively. Several schemes have been proposed to close the two terms in the time-smoothed equations, namely, scalar turbulent flux in reactive mixing, and the mean reaction rate (Bourne and Toor, 1977 Brodkey and Lewalle, 1985 Dutta and Tarbell, 1989 Fox, 1992 Li and Toor, 1986). However, numerical solution of the three-dimensional transport equations for reacting flows using CFD codes are prohibitive in terms of the numerical effort required, especially for the case of multiple reactions with... 

The probability density function, written as pif), describes the fraction of time that the fluctuating variable/ takes on a value between/ and/ + A/. The concept is illustrated in Fig. 5.7. The fluctuating values off are shown on the right side while p(f) is shown on the left side. The shape of the PDF depends on the nature of the turbulent fluctuations of/. Several different mathematical functions have been proposed to express the PDF. In presumed PDF methods, these different mathematical functions, such as clipped normal distribution, spiked distribution, double delta function and beta distribution, are assumed to represent the fluctuations in reactive mixing. The latter two are among the more popular distributions and are shown in Fig. 5.8. The double delta function is most readily computed, while the beta function is considered to be a better representation of experimentally observed PDF. The shape of these functions depends solely on the mean mixture fraction and its variance. The beta function is given as... 

The most difficult term to close in (7.73) is the reaction rate terms denoted by Sc uj). To simulate turbulent reactive flows accurate modeling of this term is very important. For slow reactions (i.e., Dai 1) the turbulent mixing is completed before the reaction can take place thus an adequate closure is available ... 

There is also axial dispersion. Again this is not mixing but a mechanism that introduces a residence time distribution. If a pulse is added to a turbulent pipeline, it will gradually lengthen with time. The best discussion of this is in Levenspiel s book (1967), where the work of Levenspiel and Bischoff is discussed. Mixing time, an important concept for reactive mixing, is given by... 

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