Non‐premixed simple reaction in homogeneous turbulence

Abstract

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

Modeling of isotropic reacting turbulence by a hybrid mapping‐EDQNM closure

“…Note that the asymptotic value of this normalized variable approaches the constant value of 2/7r. This value is due to an asymptotic Gaussian distribution for the PDF of the random variable 3 and is consistent with that obtained in previous DNS results of Givi and McMurtry (1988) and discussed by Kosaly (1987). This limiting value is valid only if the reactants are completely segregated at the inlet of the reactor.…”

Study of mixing and chemical reaction in RIM

“…From CFD simulation considerable insight into the reactor hydrodynamics can be gained, since all the hydrodynamic variables are known at every instant and every point of the physical domain (Kosály, 1987). Albeit the longstanding importance of RIM process economics', and the actual feasibility for CFD simulations, only very few research works on the subject can be found in the literature.…”

Mixing and Chemical Reactions

“…Ever since Toor and co-workers (Vassilatos and Toor, 1965;Toor, 1969;Mao and Toor, 1971;Li and Toor, 1986) defined methods for relating reaction conversion for non-premixed reactants to their degree of mixing, workers in the field of mixed chemical reactors have attempted to build upon and refine their analysis, which was based on the use of an assumed probability density function (PDF) for reactant concentration (see Donaldson 1975;Brodkey and Lewalle, 1985;Kosaly, 1987;Baldyga, 1994;Baldyga and Henczka, 1995). Pope (1985) presented an extensive review of concentration PDF closure methods for mixed chemical reactions, but only the spiked and β-PDFs are presented here.…”