On averaging approximations of a polydisperse fuel spray in the autoignition problem

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On averaging approximations of apoly-disperse
fuel spray in the auto-ignition problem
V. Bykov
Institut für Technische Thermodynamik Universität
Karlsruhe (TH), Germany
Collaboration BGU V. Goldshtein, I.
Goldfarb Technion J.B. Greenberg
Queens Hotel, Brighton, UK, July 16-18, 2007
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Overview

• Motivation Modelling of poly-disperse spray
  combustion
• Parcel approach, a simplified mathematical model
  and standard analysis
• Averaging and approximation of the system
  dynamics
• Modification, analysis and comparisons
• Conclusions

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Cell and parcel approach

• Lagrange approach is combined with cell method to
  model and simulate the spray combustion phenomena
• Parcels are used to replace the
• actual size distribution of the spray
• radiuses by its discrete approximation

(Example Ignition of a Cluster of Heptane
Droplets by Oliver Desjardins, http//www.stanford
.edu/group/pitsch/Research/Multiphase.htm)
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Simplified model

• Energy
• Mass
• Concentration

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Non-dimensional form

• Energy
• Mass
• Concentration

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Reduction

• Mass conservation equations can be integrated
• Energy integral
• Reduced model

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Preliminary analysis

• Slow curve on the dimensionless maximal
  radius/temperature plane is
• Critical regimes and classification
• explosion limit of Semenovs type!
• Explosive regime
• Delayed regime

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Time histories

• System profiles of the delayed
• regime

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Phase plane

• Typical behavior of delayed regime in projection
  to the phase plane of dimensionless temperature
  and maximal radius

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Approximate mono-disperse model

• Approximate mono-disperse system
• Overall volume
• Equilibrium value

Idea Replace the poly-disperse model by a
mono-disperse one in such a way that it
reproduces same dynamical scenarios for
considered parametric range!
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Comparative analysis

• Quantitative similarity of critical regimes

Critical parameters can be applied for
construction of a qualitative test of an
approximation!
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Standard definitions

• Standard parametrical averaging
• In dimensionless form it reads

This modification guaranties the same equilibrium
point of both poly-disperse and its approximate
mono-disperse models!
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Standard definitions

• 1 - full poly-disperse system, 2 approximating
  mono-disperse (standard SMD) system

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Modified averaging

• Dynamically consistent definition

(Auto-ignition of a polydisperse fuel spray, 31st
Combustion Symposium, Bykov et. al.)
This modification guaranties the same equilibrium
point of both poly-disperse and its approximate
mono-disperse models!
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Modified definitions

• 1 - full poly-disperse system, 2 approximating
  mono-disperse (conserved SMD) system D32

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Further comparisons and thoughts

• Digits correspond to the averages (conserved) 1
  D31, 2 D32,
• 3 D21, 4 - D20

Depending on the regime different parameters can
be used in order to insure the best approximation!
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Conclusions
A poly-disperse spray model in a combustible gas
medium has been considered and compared with the
dynamics of an equivalent mono-disperse spray
based on different approximations of an average
droplet diameter The Sauter mean diameter
(SMD) has been taken and analyzed for this
purpose Preliminary computed results suggest
that the use of the usual SMD-based mono-disperse
spray leads to quite a significant over-estimate
of the ignition time! An alternative modified
definition of the SMD is suggested. It reduces
significantly the discrepancy between the
ignition time for the poly-disperse spray and
that of the equivalent mono-disperse spray.
Further studies Parametric analysis More
realistic models Extension to models with
transport Algorithms and implementation schemes
for DNS