Turbulent motion, mixing, and kinetics in a chemical reactor configuration

In turbulent flow the equations governing the development of conversion vs. distance are not closed because of the contribution of the concentration fluctuations to the average reaction rate. Toor's hypothesis permits closure from data obtained for mixing without reaction. This paper presents a critical review of the derivation of the hypothesis and discusses its validity. IntroductionThe study of chemical reaction in spatially homogeneous turbulent flow is of practical importance for plug flow type reactors. If variations in the cross-stream direction are negligible in the reactor, time averages taken at a given axial position z Toor (1969, 1975) characterized turbulent mixing by the ratio of unmixedness, defined as and proved that +bit) and d ( t ) are equal in the slow chemistry limit. In Eq. 2 subscript m refers to pure mixing; ( a ; ) , is the mean square fluctuation of species A in the absence of chemical reaction.In The indexfindicates infinitely fast reaction. Since in this limit Toor concludes that (4)that is, $ ( t ) and d ( t ) are equal in the fast chemistry limit as well.It was hypothesized therefore that is valid for arbitrary reaction rates; this is Toor's hypothesis. (Toor, 1969(Toor, , 1975):If Eq. 3 is valid, then Eq. 6 can be written equivalently as (7)Toor's hypothesis found general acceptance in the chemical engineering literature (Hill, 1976;Brodkey, 1981). It is presently under investigation whether Toor's hypothesis can be generalized for more complicated cases than second-order reactions (Brodkey and Lewalle, 1985).If Eqs. 6 and 7 are valid, either of them can be applied to close the equations satisfied by the average concentrations. The present paper investigates the validity of estimating $*(t) via Eq. 6 or Eq. 7.Our rapid reaction results differ from Eqs. 3 and 5. The results show that Eq. 6 becomes valid in the limit when the reaction rate is much smaller than the rate of mixing. It is concluded that Eqs. 6 and 7 are not equivalent but represent different estimates of $ ( t ) . While we refer to Eq. 6 as Toor's hypothesis, we AIChE Journal emphasize the difference between Eqs. 6 and 7 by referring to Eq. 7 as the fast conversion estimate of + ( t ) . This estimate becomes valid in the limit when the reaction rate is much larger than the rate of mixing.The derivations refer to stoichiometric conditions. It is suggested that further progress requires more detailed plug flow measurements and direct numerical simulation work in homogeneous turbulence. BackgroundWe consider a second-order, irreversible isothermal reaction

“…It is concluded that the measurements of McKelvey et al (1975) support the application of Eq. 7 for closure.…”

Section: Discussionmentioning

confidence: 74%

“…7 is valid for fast chemistry and deteriorates for slow chemistry, its application for closure involves an inherently favorable tendepcy. McKelvey et al (1975) measure conversion vs. distance in…”

Section: =064mentioning

confidence: 99%

Non‐premixed simple reaction in homogeneous turbulence

Kosály

1987

“…The multiinjection head with 97 parallel stainless-steel tubes in a square array was expected to yield a uniform flow field with a relatively flat velocity profile, although McKelvey et al (1975) showed that this was not the case near the inlet, due to back flow near the wall. Mao (1 969) studied a multijet head similar to that of Vassilatos, but used 188 tubes in a close-packed arrangement.…”

Section: Results and Discussion: One-dimensional Reactormentioning

confidence: 98%

Turbulent micromixing parameters from reactive mixing measurements

Shenoy

Toor

1989

Previously published conversion data for turbulent diffusion-controlled acid-base reactions in two different multijet reactors are used to extract the mixing parameters, tracer probability density function, and variance. The PDF is obtained with this instantaneous reaction method by differentiating twice the reactive mixing plot of mean reactant concentrations. The limiting behavior for the cases of complete segregation and complete mixing is derived and turns out to be piecewise linear on the reactive mixing plot. The variance is shown to be twice the area between the actual reactive mixing curve and the limiting curve for complete mixing; it can be computed by numerical integration without any foreknowledge of the PDF shape. The method, in principle, allows detection of Dirac delta functions (which represent unmixed virgin fluid) and indeed shows significant amounts of virgin fluid close to the inlet of one of the reactors.

“…Mao and Toor (1971) successfully tested the hypothesis for near stoichiometric liquid-phase reactions in a 2014 December 1989 multijet tubular reactor for rapid reactions and further found that the dependence of G on feed stoichiometry can often be neglected; similar conclusions for a gas-phase reaction were obtained by Ajmera (1970). McKelvey et al (1975) reported successful prediction of stoichiometric and nonstoichiometric intermediate and slow rate chemistry data using stoichiometric fast reaction data for closure. Recent direct numerical simulation results (Leonard and Hill, 1987) are alsoconsistent with the hypothesis.…”

Section: R + B -Smentioning

confidence: 99%

Closure models for turbulent reacting flows

Dutta

Tarbell

1989

In this paper, a simple procedure based on fast and slow reaction asymptotics has been employed to derive first-order closure models for the nonlinear reaction terms in turbulent mass balances from mechanistic models of turbulent mixing and reaction. The coalescence-redispersion (CRD) model, the interaction by exchange with the mean (IEM) model, the three-environment (3E) model, and the four-environment (4E) model have been used to develop closure equations. The closure models have been tested extensively against experimental data for both single and multiple reactions. The closures based on slow asymptotics for the CRD, 3E and 4E models provide very good predictions of all of the experimental data, while other models available either in the literature or derived here are not adequate. The simple new closure equations developed in this paper may be useful in modeling systems involving turbulent mixing and complex chemical reactions. showed that all of these models possess an analogy to theories of isotropic turbulent micromixing as expressed by AlChE Journal -_ cm'/s rnol/m' k , = 1.3 x 10' m'/mol/s; k2 = 2.3 m'/mol/s; 7" = 2.0 s; U = 1.6 cm/s AlChE Journal