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A Quadrature-Based CFD Model for Single-Phase Turbulent Reacting Flows

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Computational fluid dynamics (CFD) modeling of turbulent reacting flows has a variety of applications, including minimizing the amount of pollutants from internal combustion engines and maximizing the selectivity of a desired product in a chemical reactor. One goal of modeling these flows is to better understand how fluid mixing affects selectivity. A transport equation for the joint composition probability density function (PDF) can be used to model for this purpose (Fox, 2003). Here, the interaction-by-exchange-with-the-mean (IEM) model is used to close the micromixing term in the joint composition PDF transport equation. Both the Conditional Quadrature Method of Moments (CQMOM) (Yuan and Fox, 2011) and the Direct Quadrature Method of Moments (DQMOM) (Fox, 2003) are used in this study to solve the joint PDF transport equation. Compared to direct numerical simulations (DNS), these two methods reduce the computational cost significantly and are applicable to simulate large-scale systems by forcing lower-order moments of a presumed PDF to be exactly preserved (Fox, 2003). Despite reducing the computational cost, statistical methods like these introduce errors that deviate from exact solutions obtained from DNS. In the past, DQMOM has been studied and found comparable results to previous methods (Zucca et al., 2007), but DQMOM introduces correction terms to the transport equations. CQMOM does not introduce these correction terms, and the optimal moments (Fox, 2008) are transported directly rather than as weights and weighted abscissas. Here, a competitive-consecutive reaction system is modeled by the PDF transport equation and solved using DQMOM-IEM and CQMOM-IEM to compare the statistical errors and computational costs of each method. The CQMOM formulation results in a robust solution algorithm for the PDF transport equation with a similar computational cost by avoiding the potentially singular correction terms arising in DQMOM-IEM (Akroyd et al., 2010). References Akroyd, J., Smith, A.J., McGlashan, L.R., Kraft, M., 2010. Numerical investigation of DQMOM-IEM as a turbulent reaction closure. Chemical Engineering Science 65, 1915–1924. Fox, R.O., 2003. Computational Models for Turbulent Reacting Flows. Cambridge University Press. Fox, R.O., 2008. Optimal moment sets for multivariate direct quadrature method of moments. Industrial & Engineering Chemistry Research 48, 9686–9696. Yuan, C., Fox, R.O., 2011. Conditional quadrature method of moments for kinetic equations. Journal of Computational Physics 230, 8216–8246. Zucca, A., Marchisio, D.L., Vanni, M., Barresi, A.A., 2007. Validation of bivariate DQMOM for nanoparticle processes simulation. AIChE Journal 53, 918–931.
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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
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 
  
  
 
       
 
       
 
  
 
   
 
   
 

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)
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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.
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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.
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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.
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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.
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DQMOM CQMOM
Concentration of species S
DQMOM CQMOM
The bulk of the byproduct reaction occurs near the wall of the side initially
containing B.
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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
Future work
Implement the CQMOM method into OpenQBMM
(www.openqbmm.org), based on OpenFOAM.
WILLIAMS ET AL. - IOWA STATE UNIVERSITY
Apply CQMOM to simulating flash nanoprecipitation in a multi-inlet vortex
reactor for nanoparticle coating.
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
... The Method of Moments has been successfully applied to various applications, including nano-particles and aerosols [35], sprays [36] and combustion-related problems, such as soot formation [37][38][39]. For the closure of the joint PDF, similarly to Monte-Carlo transported PDF methods, the Method of Moments was primarily applied to turbulent non-premixed flames [40][41][42][43][44][45]. In [46], the Quadrature Method of Moments (QMOM) [35] was applied to premixed turbulent flames. ...
... In a coupled simulation, the algorithm presented in [45,46,60] can be used to solve the moment transport equations. ...
... • CQMOM: In the a priori CQMOM closure, only part of the algorithm described in [45,46,60] needs to be used. In particular, instead of solving transport equations for the moments, they are directly calculated from the quasi-DNS samples (step i. below) and used as input parameters for the CQMOM algorithm. ...
Preprint
Full-text available
Presumed probability density function (PDF) and transported PDF methods are commonly applied to model the turbulence chemistry interaction in turbulent reacting flows. However, little focus has been given to the turbulence chemistry interaction PDF closure for flame-wall interaction. In this study, a quasi-DNS of a turbulent premixed, stoichiometric methane-air flame ignited in a fully developed turbulent channel flow undergoing side-wall quenching is investigated. The objective of this study is twofold. First, the joint PDF of the progress variable and enthalpy that needs to be accounted for in turbulence chemistry interaction closure models is analyzed in the quasi-DNS configuration, both in the core flow and the near-wall region. Secondly, a transported PDF closure model, based on a Conditional Quadrature Method of Moments approach, and a presumed PDF approach are examined in an a priori analysis using the quasi-DNS as a reference both in the context of Reynolds-Averaged Navier Stokes (RANS) and Large-Eddy Simulations (LESs). The analysis of the joint PDF demonstrates the high complexity of the reactive scalar distribution in the near-wall region. Here, a high correlation between the progress variable and enthalpy is found, where the flame propagation and quenching are present simultaneously. The transported PDF approach presented in this work, based on the Conditional Quadrature Method of Moments, accounts for the moments of the joint PDF of progress variable and enthalpy coupled to a Quenching-Flamelet Generated Manifold. In the a priori analysis both turbulence chemistry interaction PDF closure models show a high accuracy in the core flow. In the near-wall region, however, only the Conditional Quadrature Method of Moments approach is suitable to predict the flame structures.
... The Method of Moments has been successfully applied to various applications, including nano-particles and aerosols (McGraw, 1997), sprays (Pollack et al., 2016) and combustion-related problems, such as soot formation (Salenbauch et al., 2019;Wick et al., 2017;Ferraro et al., 2021). For the closure of the joint PDF, similarly to Monte-Carlo transported PDF methods, the Method of Moments was primarily applied to turbulent non-premixed flames (Raman et al., 2006;Tang et al., 2007;Koo et al., 2011;Jaishree and Haworth, 2012;Donde et al., 2012;Madadi Kandjani, 2017). In (Pollack et al., 2021), the Quadrature Method of Moments (QMOM) (McGraw, 1997) was applied to premixed turbulent flames. ...
... In a coupled simulation, the algorithm presented in (Madadi Kandjani, 2017;Fox, 2018;Pollack et al., 2021) can be used to solve the moment transport equations. ...
... where the β-PDF P (C) is defined by the first and second moment (C , C'' 2 ) , and P(H) ≈ δ(H −H) is a δ-PDF centered at H . • CQMOM: In the a priori CQMOM closure, only part of the algorithm described in (Madadi Kandjani, 2017;Fox, 2018;Pollack et al., 2021) needs to be used. In particular, instead of solving transport equations for the moments, they are directly calculated from the quasi-DNS samples (step i. below) and used as input parameters for the CQMOM algorithm. ...
Article
Presumed probability density function (PDF) and transported PDF methods are commonly applied to model the turbulence chemistry interaction in turbulent reacting flows. However, little focus has been given to the turbulence chemistry interaction PDF closure for flame-wall interaction. In this study, a quasi-DNS of a turbulent premixed, stoichiometric methane-air flame ignited in a fully developed turbulent channel flow undergoing side-wall quenching is investigated. The objective of this study is twofold. First, the joint PDF of the progress variable and enthalpy that needs to be accounted for in turbulence chemistry interaction closure models is analyzed in the quasi-DNS configuration, both in the core flow and the near-wall region. Secondly, a transported PDF closure model, based on a Conditional Quadrature Method of Moments approach, and a presumed PDF approach are examined in an a priori analysis using the quasi-DNS as a reference both in the context of Reynolds-Averaged Navier Stokes (RANS) and Large-Eddy Simulations (LESs). The analysis of the joint PDF demonstrates the high complexity of the reactive scalar distribution in the near-wall region. Here, a high correlation between the progress variable and enthalpy is found, where the flame propagation and quenching are present simultaneously. The transported PDF approach presented in this work, based on the Conditional Quadrature Method of Moments, accounts for the moments of the joint PDF of progress variable and enthalpy coupled to a Quenching-Flamelet Generated Manifold. In the a priori analysis both turbulence chemistry interaction PDF closure models show a high accuracy in the core flow. In the near-wall region, however, only the Conditional Quadrature Method of Moments approach is suitable to predict the flame structures.
... EQMOM was shown to provide an accurate PDF reconstruction. A first proof of concept that a reacting system could be closed using CQMOM was provided in [22], while the underlying general idea for the closure of different systems was additionally discussed in [23]. Similarly to MC transported PDF methods, which have been primarily applied to turbulent nonpremixed reacting flows, most QbMM studies have investigated non-premixed flames [14,15,[17][18][19]22]. ...
... A first proof of concept that a reacting system could be closed using CQMOM was provided in [22], while the underlying general idea for the closure of different systems was additionally discussed in [23]. Similarly to MC transported PDF methods, which have been primarily applied to turbulent nonpremixed reacting flows, most QbMM studies have investigated non-premixed flames [14,15,[17][18][19]22]. Following these recent developments, this work aims to evaluate the QbMM methods in turbulent premixed flames using a tabulated chemistry approach. ...
... If EQMOM uses a single Beta-KDF, the closure is the same as the presumed Beta-PDF approach [23]. The algorithm used here coupled with the CFD solver is [22,23]: ...
Preprint
Full-text available
Transported probability density function (PDF) methods are widely used to model turbulent flames characterized by strong turbulence-chemistry interactions. Numerical methods directly resolving the PDF are commonly used, such as the Lagrangian particle or the stochastic fields (SF) approach. However, especially for premixed combustion configurations, characterized by high reaction rates and thin reaction zones, a fine PDF resolution is required, both in physical and in composition space, leading to high numerical costs. An alternative approach to solve a PDF is the method of moments, which has shown to be numerically efficient in a wide range of applications. In this work, two Quadrature-based Moment closures are evaluated in the context of turbulent premixed combustion. The Quadrature-based Moment Methods (QMOM) and the recently developed Extended QMOM (EQMOM) are used in combination with a tabulated chemistry approach to approximate the composition PDF. Both closures are first applied to an established benchmark case for PDF methods, a plug-flow reactor with imperfect mixing, and compared to reference results obtained from Lagrangian particle and SF approaches. Second, a set of turbulent premixed methane-air flames are simulated, varying the Karlovitz number and the turbulent length scale. The turbulent flame speeds obtained are compared with SF reference solutions. Further, spatial resolution requirements for simulating these premixed flames using QMOM are investigated and compared with the requirements of SF. The results demonstrate that both QMOM and EQMOM approaches are well suited to reproduce the turbulent flame properties. Additionally, it is shown that moment methods require lower spatial resolution compared to SF method.
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A model study has been conducted for Flash Nanoprecipitation (FNP)—a novel approach to produce functional nanoparticles. A population balance equation with the FNP kinetics has been integrated into a computational fluid dynamics (CFD) simulation of a custom-designed microscale multi-inlet vortex reactor (MIVR) to yield conditions comparable to the real experimental settings. In coping with the complicated aggregation model in the CFD code, a new numerical approach, the conditional quadrature method of moments (CQMOM), has been proposed, which is capable of solving the multivariate system efficiently and accurately. It is shown that the FNP process is highly influenced by mixing effects in the microreactor, and thus coupling CFD with the kinetics model is essential in obtaining valid comparisons with experiments.
Article
Kinetic equations arise in a wide variety of physical systems and efficient numerical methods are needed for their solution. Moment methods are an important class of approximate models derived from kinetic equations, but require closure to truncate the moment set. In quadrature-based moment methods (QBMM), closure is achieved by inverting a finite set of moments to reconstruct a point distribution from which all unclosed moments (e.g. spatial fluxes) can be related to the finite moment set. In this work, a novel moment-inversion algorithm, based on 1-D adaptive quadrature of conditional velocity moments, is introduced and shown to always yield realizable distribution functions (i.e. non-negative quadrature weights). This conditional quadrature method of moments (CQMOM) can be used to compute exact N-point quadratures for multi-valued solutions (also known as the multi-variate truncated moment problem), and provides optimal approximations of continuous distributions. In order to control numerical errors arising in volume averaging and spatial transport, an adaptive 1-D quadrature algorithm is formulated for use with CQMOM. The use of adaptive CQMOM in the context of QBMM for the solution of kinetic equations is illustrated by applying it to problems involving particle trajectory crossing (i.e. collision-less systems), elastic and inelastic particle–particle collisions, and external forces (i.e. fluid drag).
Book
This book presents the current state of the art in computational models for turbulent reacting flows, and analyzes carefully the strengths and weaknesses of the various techniques described. The focus is on formulation of practical models as opposed to numerical issues arising from their solution. A theoretical framework based on the one-point, one-time joint probability density function (PDF) is developed. It is shown that all commonly employed models for turbulent reacting flows can be formulated in terms of the joint PDF of the chemical species and enthalpy. Models based on direct closures for the chemical source term as well as transported PDF methods are covered in detail. An introduction to the theory of turbulent and turbulent scalar transport is provided for completeness. The book is aimed at chemical, mechanical, and aerospace engineers in academia and industry, as well as developers of computational fluid dynamics codes for reacting flows.