Content uploaded by Alberto Passalacqua
Author content
All content in this area was uploaded by Alberto Passalacqua on Nov 14, 2015
Content may be subject to copyright.
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