The Crucial Role of the Lewis No. in
Jet Ignition
Nika Rezaeyan, Luc Bauwens
University of Calgary
Matei Radulescu
University of Ottawa
Fernando Fachini Filho
Instituto Nacional de Pesquisas Espaciais
ICHS 2011
San Francisco CA
Outline
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Motivation
Jet ignition
Physical Model
Magnitude Analysis and Perturbation
Results
Conclusion
Motivation
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Jet ignition: key unresolved issue in hydrogen safety
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May hurt or help?
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Review by Astbury & Hawksworth (2009)
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Original study: Wolanski & Wojicki (1973)
Jet ignition
Hydrogen known to ignite in transient jets in leaks from high pressure
(Wolanski and Wojcicki, 1973).
Formation of high pressure jet, Radulescu & Law (2007)
Issues under focus
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Interplay between diffusion and chemistry?
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Effect of expansion (Radulescu)?
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Lewis number: Mass diffusivity vs. heat diffusivity?
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Hydrogen: mass diffusivity > heat –> Low Lewis number
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Analysis by Liñan & Crespo (1976) and Liñan & Williams (1993)
Physical Model
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One dimensional
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frame of reference attached to contact surface initially separating shockheated air from cold, expanded hydrogen
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In that (nearly inertial) frame, low Mach number
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Single step Arrhenius chemistry
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Negligible cross diffusion
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Prescribed expansion rate
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Ideal gas, constant specific heat and Lewis number
Shock tube problem
Physical Model
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Diffusion problem (heat, fuel, oxidant) with sources: chemistry and expansion
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Initial conditions: jump at contact surface
Boundary conditions at infinity consistent with jump
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Assumptions/magnitudes
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Key physical processes: reaction, diffusion and expansion.
Time short compared with chemical time
High activation energy
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Frozen flow regime: chemistry negligible at leading order
Ignition as a perturbation of the order of inverse activation energy.
Frozen Flow
Frozen flow: diffusion and expansion (which causes a temperature drop in
time)
 Mass-weighed coordinate
 Self-similar solution:
Frozen Flow
Frozen Flow
Lewis Number
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Lewis number: ratio between heat and mass diffusion
Lewis Number
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Chemistry peaks close to maximum temperature
Peak larger for smaller fuel Lewis number
Perturbation
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Chemistry strongest when departure from maximum temperature is small.
So, introduce rescaling
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Asymptotic expansion of order of inverse activation energy
Perturbation
Perturbation
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Negligible transient and expansion term lead to quasi-steady formulation.
Fuel concentration contains two terms:
1. Mass diffusion
2. Fuel consumption due to chemistry
Then expansion only appears in the Arrhenius term
Le close to unity
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Perturbation problem reduced to ODE:
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Fuel mass diffusion of same order as fuel consumption
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Max value of the perturbation function of ratio initial temperatures
difference/ adiabatic flame temperature, times O(1) factor depending upon
small difference Le - 1.
Le close to 1,  < 1
Le close to 1,  < 1
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Ignition happens at turning point.
Le close to 1,  < 1
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for uniform pressure (p0'=0) ignition always occurs (Liñan)
If turning point at * < max, ignition occurs. For stronger expansion, no
ignition
Le close to 1,  > 1
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Solution (1)() increases monotonically with  so no turning point: so no
thermal explosion
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Front from warm side toward cold side
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Unconditionally quenched by expansion
Le – 1 negative and of O(1)
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Fuel supplied by mass diffusion > fuel consumption
Ignition at turning point.
Ignition time shorter for smaller Lewis number.
Similar to Le of O(1),  < 1.
Le > 1 by O(1)
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Mass difussion no longer supplies fuel concentration at order e. So, chemistry
now limited by fuel. Need to rescale:
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Then, problem becomes:
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Temperature increase due to chemistry now negligible.
Equilibrium region propagating towards fuel rich region
Eventually expansion quenches ignition
Similar to Le of O(1),  > 1
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Le > 1 by O(1)
Conclusions from Analysis
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Reaction rate peaks close to hot air side.
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For Lewis numbers greater than threshold close to unity, no ignition (jet ignition
only observed for hydrogen)
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For Lewis numbers below that threshold, ignition occurs at finite time as long as
expansion rate < a critical rate
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No ignition for expansion rates faster than the critical rate
Conclusions
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“Ignition source” in jet ignition: likely interplay between diffusion and reaction
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Occurs with hydrogen because hydrogen diffuses easily
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Ignition may get killed by expansion
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Since there is a clear relationship between leak size and expansion rate, current
results consistent with experiments