Study of the Soot PSDF: Numerical and Conceptual

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Study of the Soot PSDFNumerical and Conceptual

• Computational Modelling Group
• Department of Chemical Engineering
• University of Cambridge

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FLAME Parameters for Definition

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Why study Premix flames ?

• Facilitates detailed experimental measurements of
  temperature and species.
• Effectively one dimensional and can be made
  steady
• Used to characterize fuel-oxidizers combinations
• Convenient to study chemical kinetics in a
  combustion environment

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Why study SOOT distribution ?
Soot is a carbon-based material comprising mainly
of Polyaromatic hydrocarbons (PAHs) and
resulting from incomplete combustion

• Medical Importance Recent studies show the
  cause of lung cancer as small soot particles
• Automobile Industry Reduction of soot
  emissions
• Carbon Black Industry Soot production /
  particle properties
• Environmental Issue Pollution caused by
  combustion in industry and private sector

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SOOT MODEL

• Following events are possible for soot formation
• Inception of two Pyrenes.
• Coagulation of two soot molecules
• Condensation of Pyrene on soot molecule.
• Oxidation by O2
• Oxidation by OH
• Addition by C2H2 by HACA mech.

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Numerical Approachesto solve SOOT distribution

• Naïve Deterministic
• To solve population balance equation for each
  size class
• Run the code on the fastest super computer and
  wait for your 15th generation to give you the
  result.
• Smart Method of Moments
• The set of huge equations are reduced to five to
  ten moment equations and is solved as a set of
  ODEs.
• Computational advantage
• Loss of accuracy in distribution
• Closure problem with Differential equations
• Also Smart Stochastic Approach

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STOCHASTIC ALGORITHM
Generate the initial particle system
Wait an exponentially distributed time step t
Choose one of the possible events based on
their rates
Particle inception
Surface growth
Coagulation
Condensation
O2 Oxidation
OH Oxidation
Choose particle(s) from the particle system
Update the particle system
Next time step
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Advantages of Stochastic Algorithm

• Efficiency Efficient with respect to Discrete
  sectional and Galerkin Methods.
• Convergence The algorithm converges to solution
  of PBE.
• Resolution It resolves for the entire particle
  size distribution.
• Model It can be applied to model higher
  dimensional PSDFs.
• Oxidation Oxidation of smallest particle can be
  traced and no closure problem.
• Adaptability It is adaptable to any changes in
  the reaction and kinetic soot mechanism.
• Assumption There is no assumption of presumed
  distribution as in case of MoM and other methods.

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THE MATHEMATIC BITS !
Smoluchowski Master equation

• Properties of Majorant kernel
• Simple
• Always greater than the kernel
• Probability of Acceptance

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Regime and Kernels
Coagulation and Majorant kernel
Knudsen
Regime
10 lt Kn
Free molecular
1 lt Kn lt 10
Transition
0.1lt Kn lt 1
Slip flow
Kn lt 0.1
Continuum
? Mean free path of particles d diameter of
soot particle d1 diameter of a carbon molecule
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Low Pressure Flame
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Comparison with MoM
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Experimental validation and Error
Error in first five Moments M0 25 M1 5 M2
13, M3 20 M4 25
Computational Time P4 1.2 GHz 1 min
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PSDF as function of residence time
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Atmospheric Pressure flame
Case Study To understand the Bi-modal
behaviour Study will involve Experimental
observations Form of the Coagulation
Kernel Numerical Experiments by switching off
various events
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Experimental Observations
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Coagulation Kernel
Observation Probability of Coagulation of small
with larger particles is high as compared to
high-high or low-low combinations Hypothesis The
concentration of the small soot particles will be
very low
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PSDF as a function of residence time
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Experiment 1 Switching off Nucleation
Observation Uni-modal distribution
Absence of small soot particles Conclusion The
first peak is due to the Nucleation
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Experiment 2 Switching off Coagulation
Observation Peak distribution
Absence of large soot particles Conclusion The
second peak is due to the Coagulation
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High Pressure flame (Low Sooting)
Direct Simulation Algorithm Weighted
Algorithm _at_ Operator Splitting Simulations run on
P4, 1.2 GHz
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High Pressure Flame (High Sooting)
Direct Simulation Algorithm Age
Algorithm Simulations run on P4, 1.2 GHz
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High Pressure Flame (High Sooting)
Direct Simulation Algorithm Weight
Algorithm Simulations run on P4, 1.2 GHz
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Achievements of the StudyPresent Conclusive
Status

• PSDF for all the pressure regimes
• Significant decrease in computational time to
  match MoM without compromising on Accuracy

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Acknowledgements

• Jasdeep Singh (2nd Year Ph.D Student)
• Michael Balthasar (Post Doc)
• Markus Kraft (Supervisor)
• Wolfgang Wagner
• (Collaborating scientist at WIAS Berlin)
• Grant Body
• Churchill College, University of Cambridge

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Variation in Stochastic Algorithms

• Weighted Algorithm
• Each event is weighted based on the ratio of
  rates of events to each other.
• Probability distribution should not be disturbed
  significantly
• Operator Splitting
• Surface events are performed on the whole
  particle ensemble deterministically after a
  non-surface event takes place.
• Age Algorithm
• Age of each particle is stored together with its
  size
• On selection of particles from non-surface
  events, surface events are performed
  deterministically

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Structure of PREMIX code
GAS PHASE CHEMISTRY
PREMIX THE ODE SOLVER
PREMIX INPUT
SPECIES AND TEMPERATURE PROFILES
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PREMIX Code

• Boundary value problem solving for species and
  temperature profile in a single space
  co-ordinate.
• Mass Conservation, Species transport equation and
  Energy balance are used.
• Vanishing gradient at the hot end and explicit
  information of feed and temperature form the
  boundary conditions.

• Robust numerical techniques e.g. Damped Newtons
  method and Backward Euler method are used to
  solve the ODEs
• Extensive Chemical reaction library (CHEMKIN) and
  transport properties (TRANFIT) are used in
  conjunction to the ODE solver.