Rodney O. Fox

On the relationship between Lagrangian micromixing models and computational fluid dynamics

The relationship between Lagrangian micromixing models, which are widely employed in chemical reaction engineering, and Eulerian computational fluid dynamic (CFD) models based on the Reynolds-averaged species conservation equation is explored. A general modeling methodology which combines the strengths of both approaches is developed in the form of a multi-environment CFD micromixing model. The formulation is shown to be equivalent to a presumed multi-scalar probability density function (PDF) approach. The four-environment generalized mixing model (GMM) model originally proposed by Villermaux and Falk (Villermaux and Falk, Chem. Eng. Sci. 49 (5127) (1994)) is used to illustrate the methodology by applying it to model a series-parallel reaction in a tubular reactor.

On velocity‐conditioned scalar mixing in homogeneous turbulence

Scalar mixing models are required to model turbulent molecular mixing in full probability density function (pdf) simulations of turbulent reacting flows. Despite the existence of direct numerical simulation (DNS) data suggesting the contrary, most scalar mixing models assume that molecular mixing is independent of the instantaneous velocity, i.e., 〈D∇2φ|V,ψ〉=〈D∇2φ|ψ〉. Since in a joint velocity, composition pdf calculation the velocity is known, this assumption is unnecessary and leads to a lack of local isotropy in the scalar field. Moreover, since velocity conditioning offers a numerically tractable approach for including the effects of local anisotropy and mean velocity gradients on scalar mixing, it should be of considerable interest for the numerical simulation of scalar mixing in inhomogeneous turbulent flows. An efficient numerical implementation of velocity‐conditioned scalar mixing for full pdf simulations is proposed and verified against DNS data for homogeneous turbulence (isotropic and shear fl...

Improved Fokker–Planck model for the joint scalar, scalar gradient PDF

The joint scalar, scalar gradient probability density function (PDF) of an inert nonpremixed scalar diffusing in a one‐dimensional system of random‐sized lamellas is investigated by numerical simulation. The form of the scalar PDF, at a given RMS value, is nearly identical to that predicted by direct numerical simulation (DNS) of scalar mixing in isotropic turbulence and the mapping closure, and the moments of both the scalar and the scalar gradient suggest that their limiting marginal PDF are Gaussian. The joint scalar, scalar gradient PDF is found to be restricted to a bounded region in the scalar–scalar gradient plane, whose form is independent of the initial mixing ratio. These results are incorporated into the Fokker–Planck (FP) model for the joint scalar, scalar gradient PDF, and the improved model shows good agreement with numerical simulation data. An extension of the FP model that includes random stretching of the scalar gradient in isotropic turbulence is formulated.

The Fokker–Planck closure for turbulent molecular mixing: Passive scalars

A turbulent‐molecular‐mixing closure for passive scalar mixing is derived based on the theory of diffusion in layerlike lamellar structures. The closure is formulated in terms of the Fokker–Planck (FP) equation (or an equivalent stochastic differential equation), and is to be employed in conjunction with the probability density function (pdf) balance equation appearing in the pdf methods for modeling turbulent reactive flows. Like the mapping closure, the FP closure predicts a limiting Gaussian pdf for the passive scalar concentration in isotropic turbulence. In addition, the FP closure models the joint pdf of the scalar concentration and the scalar gradient and thus predicts the scalar dissipation rate. The closure predictions for the scalar rms concentration, the marginal pdf, and the joint pdf as well as other relevant statistics have been studied by Monte Carlo simulation. The shape and temporal evolution of the marginal pdf for the scalar concentration compare favorably with published results found f...

The spectral relaxation model of the scalar dissipation rate in homogeneous turbulence

A model for the effect of scalar spectral relaxation on the scalar dissipation rate of an inert, passive scalar (Sc≥1) in fully developed homogeneous turbulence is presented. In the model, wave‐number space is divided into a finite number [the total number depending on the turbulence Reynolds number Reλ and the Schmidt number (Sc)] of intermediate stages whose time constants are determined from the velocity spectrum. The model accounts for the evolution of the scalar spectrum from an arbitrary initial shape to its fully developed form and its effect on the scalar dissipation rate for finite Reλ and Sc≥1. Corrsin’s result [AIChE J. 10, 870 (1964)] for the scalar mixing time is attained for large Reλ in the presence of a constant mean scalar gradient and a stationary, isotropic turbulence field. Comparisons with DNS results for stationary, isotropic turbulence and experimental data for decaying, homogeneous grid turbulence demonstrate the satisfactory performance of the model.

THE LAGRANGIAN SPECTRAL RELAXATION MODEL OF THE SCALAR DISSIPATION IN HOMOGENEOUS TURBULENCE

Lagrangian pdf methods are employed to extend the spectral relaxation (SR) model of the scalar dissipation of an inert, passive scalar (1⩽Sc) in homogeneous turbulence. The Lagrangian spectral relaxation (LSR) model divides wavenumber space into a finite number (the total number depending on the Taylor-scale Reynolds number Rλ and the Schmidt number Sc) of wavenumber bands whose time constants are determined from the mean turbulent kinetic energy and instantaneous turbulent energy dissipation rate. The LSR model accounts for the evolution of the scalar spectrum (viz., pdf) from an arbitrary initial shape to its fully developed form. The effect of turbulent-frequency fluctuations on the instantaneous scalar dissipation rate following a Kolmogorov-scale fluid particle is incorporated into the LSR model through a Lagrangian pdf model for the turbulent frequency. Model results are compared with DNS data for passive scalar mixing in stationary, isotropic turbulence. Two distinct causes of non-Gaussian scalar s...

CFD analysis of micromixing effects on polymerization in tubular low-density polyethylene reactors

A novel multi-environment CFD micromixing model is used to describe the small-scale mixing of chemical species inside a tubular low-density polyethylene (LDPE) reactor under different operating conditions. The model is coupled with a comprehensive kinetic scheme describing ethylene polymerization that includes kinetic mechanisms describing polymer properties and ethylene decomposition. The simulation results show that imperfect mixing between initiator and monomer reduces monomer conversion and increases the polydispersity index. Insufficient micromixing also causes local hot spots, which may lead the reactor to thermal runaway. Thus, the small-scale mixing has a significant impact on the reactor stability. The study not only illustrates the importance of mixing effects on LDPE polymerization but also provides important insights into the physical phenomena occurring inside the reactor, which are extremely helpful in evolving a criteria for stable operation of the reactor while controlling the product quality in the plant-scale tubular LDPE reactor. Compared to full probability density function (PDF) methods used in the literature for similar studies, the multi-environment CFD micromixing model offers a computationally highly efficient description of the turbulent reacting flow inside the LDPE reactor.

PDF simulation of a turbulent series—parallel reaction in an axisymmetric reactor

Abstract Because both the turbulent mixing and chemical reaction terms are modeled in the traditional moment approach, it is very difficult to isolate the effect of turbulent mixing on chemical reactions. In contrast, these two terms appear in closed form in the joint velocity—composition probability density function (PDF) formulation. A joint velocity—composition PDF code was to study the effect of the scalar mixing rate on a series—parallel reaction (A + B → R, R + B → S) in a single-jet tubular reactor and the results are compared with experimental data. The turbulent flow field was obtained from thek-ɛ model and the chemical reaction terms were efficiently dealt with using a look-up table. Using PDF simulations, the effects of different scalar mixing rate models on the chemical reaction were investigated under different flow conditions. Since the first reaction step is highly sensitive to the scalar mixing rate, by comparing the yield of the desired product (R) from simulations with the experimental data, the best scalar mixing rate model is determined.

Modeling multiple reactive scalar mixing with the generalized IEM model

An outstanding feature of the amplitude mapping closure is its ability to relax an arbitrary initial probability density function (PDF) to a Gaussian PDF asymptotically. Due to the difficulties in computing either the analytical or numerical solution, the mapping closure has never been applied to multiple scalars with finite reaction rates. In this work, the generalized IEM (GIEM) model is combined with the mapping closure to model the molecular mixing terms in the PDF balance equation. The GIEM model assumes a linear relationship between the rates of change of the reactive scalars and an inert scalar (shadow scalar) during the mixing step. By applying the mapping closure for binary mixing to the shadow scalar, the GIEM model yields excellent agreement for both one‐ and two‐step reactions with DNS data, the conditional moment closure (CMC) and reaction–diffusion in a random lamellar system for a wide range of initial volume ratios and reaction rates.

Reactive mixing in a tubular jet reactor: a comparison of PDF simulations with experimental data

Abstract The simulation of turbulent reactive flows in chemical reactors can be extremely useful for practical applications. In particular, both industrial and academic chemical reaction engineers are keenly interested in the exact description of the composition field in a reactor volume. Results from two different probability density function (PDF) descriptions of a laboratory tubular jet reactor are compared with experimental data for acid—base neutralization in turbulent liquid media. The first description is based on the Lagrangian joint PDF of velocity and composition, while the second is based on the Eulerian composition PDF. In both descriptions chemical reactions are treated exactly, but molecular mixing must be modeled. A computational fluid dynamics code (FLUENT) provides the mean velocity field and turbulence quantities. Results are found to be sensitive both to turbulent diffusivity and to the model for molecular mixing. A new model for the effect of scalar integral-length scale relaxation on the molecular mixing rate is proposed. By comparison with experimental data, the superior performance of a new model over existing molecular mixing models is demonstrated.