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A Quadrature-Based CFD Model for

Single-Phase Turbulent Reacting Flows

D. W I LL IA M S 1, E . M AD A D I - K A N DJ A NI 2, A . PAS S A L A C Q UA 1 , 2 , R . O . F OX 1

1Department of Chemical and Biological Engineering, Iowa State University

2Department of Mechanical Engineering, Iowa State University

2015 AIChE Annual Meeting

Turbulent reacting flows

Turbulence enhances mixing

◦Velocity fluctuations

Simulations aid scale-up

◦Identify dead-zones

◦Minimize undesired byproducts

Example applications

◦Reduce air pollutants from

combustion

◦Reduce byproducts in flash

nanoprecipitation used to produce

pharmaceuticals

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

Characteristic scales and mixing

Integral scale characterizes the

largest eddies in the flow.

Kolmogorov scale characterizes

the smallest eddies in a flow,

where the inertial and viscous

forces have the same effect (i.e. Re

= 1).

Batchelor scale shows where

chemical gradients are dissipated

via viscous-diffusion mechanism.

Macromixing brings particles to

close proximity through advection

(e.g. mean velocity)

Mesomixing occurs through

random fluctuations from mean

velocity.

Micromixing describes the

transport at small scales (e.g.

molecular diffusion) and drives

system to homogeneity at

molecular level.

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

Transported PDF method

We consider the evolution equation for the joint composition PDF

:

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

Interaction by Exchange

with the Mean mixing

model (Fox, 2003) Source term due to

reaction

Principles of quadrature-based moment methods

Apply moment definition

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

Finite set of moment transport equations

Closure problem

Direct quadrature method of moments - DQMOM

The PDF is presumed to have the form:

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

where and are quadrature weights and abscissae obtained from

the transported moments by means of a moment inversion algorithm

Moment transport

equations

Transport

equations for

quadrature weights

and abscissae

Presumed PDF

Chemical kinetics

We consider a simple case of two competitive consecutive reactions

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

where

◦R is the desired product

◦S is the byproduct

◦

Fast reaction

Slower reaction

Rewrite PDF in terms of

◦Mixture fraction

◦Reaction progress variables

DQMOM transport equations

Considering two quadrature nodes, we have

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

where:

◦is the mixture fraction in the environment corresponding to weight .

◦and are the reaction progress variables for reaction 1 and 2.

DQMOM Source Terms

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

Challenges with DQMOM

Singularity in the correction terms

◦Numerical instability if untreated

◦Need of smoothing functions (Akroyd

et al., Chem. Eng. Sci, 2010)

◦Removal of correction terms (but it

leads to incorrect predictions of the

variance of the mixture fraction!)

Transports non-conservative

quantities (weights and abscissae)

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

CQMOM:

◦Avoids singularities

◦Transports conserved quantities

(moments) rather than non-

conservative quantities

Conditional quadrature method of moments

We rewrite the joint composition PDF in terms of conditional PDFs:

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

we then consider the conditional moments

and we represent the PDF as

Moment transport equations in CQMOM

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

No singularities

Transported moment set in CQMOM

We consider

◦Two quadrature nodes for the

direction

◦One quadrature node for the

direction

◦One quadrature node for the

direction

The resulting set of moments that

conserved by the CQMOM

procedure is:

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

Test case

One-dimensional reactive mixing problem reproducing Akroyd et al. (Chem.

Eng. Sci, 2010) case:

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

AB

w1= 0

w2= 1

<ξ>(1) = 0

<ξ>(2) = 1

<Y1>(n) = 0

<Y2>(n) = 0

w1= 1

w2= 0

<ξ>(1) = 0

<ξ>(2) = 0

<Y1>(n) = 0

<Y2>(n) = 0

Fluxes are imposed to be zero at walls (Neumann conditions)

Integration performed with MATLAB PDE solver

RK2 integrator

Adaptive time-stepping

Predicted mixture fraction

The mixture fraction for both methods shows the two environments mixing

together, reaching a steady state at mixture fraction = 0.5.

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

DQMOM CQMOM

Predicted variance of the mixture fraction

The mixture fraction variance results show distributions of chemical compositions,

thus showing micromixing in the reactor.

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

DQMOM CQMOM

Concentration of species A

Both DQMOM and CQMOM give similar shapes for each variable tracked.

The side containing initially A has less reaction occurring.

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

DQMOM CQMOM

Concentration of species B

DQMOM CQMOM

Being the limiting reactant, species B depletes and halts the reaction system,

leaving the system as an inert mixing system.

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

DQMOM CQMOM

Concentration of species R

DQMOM CQMOM

The bulk of the first reaction is apparent near the middle of the box, with the crest

in the production of species R.

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

DQMOM CQMOM

Concentration of species S

DQMOM CQMOM

The bulk of the byproduct reaction occurs near the wall of the side initially

containing B.

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

DQMOM CQMOM

DQMOM and CQMOM comparison

CQMOM conserves a larger number of moments than DQMOM, by definition.

Mean Value DQMOM CQMOM

<ξ> 0.5000 0.5000

<CA > (mol/L) 0.5347 0.4883

<CB> (mol/L) 0.0000 0.0000

<CR> (mol/L) 0.4894 0.5820

<CS> (mol/L) 0.4704 0.4240

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

Summary

A new conditional quadrature method of moments was developed for

turbulent reactive mixing problems.

The method avoids problems with singularity in the DQMOM

formulation and preserves a larger number of moments than DQMOM.

The two methods were compared considering a competitive

consecutive reaction in a one-dimensional mixing problems obtaining

satisfactory results.

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

Acknowledgements

Support from the National Science Foundation of the United States,

under the SI2 –SSE award NSF –ACI 1440443 is gratefully

acknowledged.

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

Questions?

WILLIAMS ET AL. - IOWA STATE UNIVERSITY

References

J. Akroyd, A. J. Smith, L. R. McGlashan, and M. Kraft, "Numerical investigation of DQMoM-IEM as a turbulent

reaction closure," Chemical Engineering Science, vol. 65, pp. 1915-1924, Mar 15 2010.

J. Baldyga and J.R. Bourne, Turbulent Mixing and Chemical Reactions. Chichester, England: John Wiley & Sons,

1999, pp. 52, 201.

R.B. Bird, W.E. Stewart, and E.N. Lightfoot, Transport Phenomena. New York: John Wiley & Sons, second ed., 2002.

J. R. Bourne, F. Kozicki, and P. Rys, “Mixing and fast chemical reaction. 1. Test reactions to determine segregation.”

Chemical Engineering Science, vol. 36, pp. 1643-1648, Oct 1981.

J. C. Cheng and R. O. Fox, "Kinetic Modeling of Nanoprecipitation using CFD Coupled with a Population Balance,"

Industrial & Engineering Chemistry Research, vol. 49, pp. 10651-10662, Nov 3 2010.

R.O. Fox, Computational Models for Turbulent Reacting Flows. Cambridge, United Kingdom: Cambridge University

Press, 2003.

D.L. Marchisio, and R. O. Fox, “Reacting flows and the interaction between turbulence and chemistry,” Reference

Module in Chemistry, Molecular Sciences and Chemical Engineering, Ed. J. Reedijk, Elsevier ISBN: 978-0-12-

409547-2 (2015).

C. Yuan and R.O. Fox, “Conditional quadrature method of moments for kinetic equations,” Journal of

Computational Physics, vol. 230, no. 22, pp. 8216-8246, 2011.

WILLIAMS ET AL. - IOWA STATE UNIVERSITY