The Crucial Role of the The Crucial Role of the Lewis No. in Jet IgnitionLewis No. in Jet Ignition

Nika Rezaeyan, Luc BauwensNika Rezaeyan, Luc BauwensUniversity of CalgaryUniversity of Calgary

Matei RadulescuMatei RadulescuUniversity of OttawaUniversity of Ottawa

Fernando Fachini FilhoFernando Fachini FilhoInstituto Nacional de Pesquisas EspaciaisInstituto Nacional de Pesquisas Espaciais

ICHS 2011ICHS 2011 San Francisco CASan Francisco CA

OutlineOutline

MotivationMotivation Jet ignitionJet ignition Physical ModelPhysical Model Magnitude Analysis and PerturbationMagnitude Analysis and Perturbation ResultsResults ConclusionConclusion

MotivationMotivation

Jet ignition: key unresolved issue in hydrogen safety Jet ignition: key unresolved issue in hydrogen safety

May hurt or help?May hurt or help?

Review by Astbury & Hawksworth (2009)Review by Astbury & Hawksworth (2009)

Original study: Wolanski & Wojicki (1973)Original study: Wolanski & Wojicki (1973)

Jet ignitionJet ignitionHydrogen known to ignite in transient jets in leaks from Hydrogen known to ignite in transient jets in leaks from

high pressure (Wolanski and Wojcicki, 1973).high pressure (Wolanski and Wojcicki, 1973).

Formation of high pressure jet, Radulescu & Law Formation of high pressure jet, Radulescu & Law (2007)(2007)

Issues under focusIssues under focus

Interplay between diffusion and chemistry? Interplay between diffusion and chemistry?

Effect of expansion (Radulescu)? Effect of expansion (Radulescu)?

Lewis number: Mass diffusivity vs. heat diffusivity?Lewis number: Mass diffusivity vs. heat diffusivity?

Hydrogen: mass diffusivity > heat –> Low Lewis numberHydrogen: mass diffusivity > heat –> Low Lewis number

Analysis by Liñan & Crespo (1976) and Liñan & Williams Analysis by Liñan & Crespo (1976) and Liñan & Williams (1993) (1993)

Physical ModelPhysical Model One dimensional One dimensional

frame of reference attached to contact surface initially frame of reference attached to contact surface initially separating shock-heated air from cold, expanded hydrogenseparating shock-heated air from cold, expanded hydrogen

In that (nearly inertial) frame, low Mach numberIn that (nearly inertial) frame, low Mach number

Single step Arrhenius chemistrySingle step Arrhenius chemistry

Negligible cross diffusionNegligible cross diffusion

Prescribed expansion rate Prescribed expansion rate

Ideal gas, constant specific heat and Lewis number Ideal gas, constant specific heat and Lewis number

Shock tube problemShock tube problem

Physical ModelPhysical Model Diffusion problem (heat, fuel, oxidant) with sources: Diffusion problem (heat, fuel, oxidant) with sources:

chemistry and expansion chemistry and expansion

Initial conditions: jump at contact surfaceInitial conditions: jump at contact surface Boundary conditions at infinity consistent with jumpBoundary conditions at infinity consistent with jump

Assumptions/magnitudesAssumptions/magnitudes Key physical processes: reaction, diffusion and expansion.Key physical processes: reaction, diffusion and expansion.

Time short compared with chemical timeTime short compared with chemical time

High activation energyHigh activation energy

• Frozen flow regime: chemistry negligible at leading Frozen flow regime: chemistry negligible at leading orderorder

• Ignition as a perturbation of the order of inverse Ignition as a perturbation of the order of inverse activation energy.activation energy.

Frozen FlowFrozen FlowFrozen flow: diffusion and expansion (which causes a Frozen flow: diffusion and expansion (which causes a

temperature drop in time)temperature drop in time) Mass-weighed coordinateMass-weighed coordinate Self-similar solution:Self-similar solution:

Frozen FlowFrozen Flow

Frozen FlowFrozen Flow

Lewis NumberLewis Number Lewis number: ratio between heat and mass diffusionLewis number: ratio between heat and mass diffusion

Lewis NumberLewis Number Chemistry peaks close to maximum temperature Chemistry peaks close to maximum temperature Peak larger for smaller fuel Lewis numberPeak larger for smaller fuel Lewis number

PerturbationPerturbation Chemistry strongest when departure from maximum Chemistry strongest when departure from maximum

temperature is small. So, introduce rescalingtemperature is small. So, introduce rescaling

Asymptotic expansion of order of inverse activation energyAsymptotic expansion of order of inverse activation energy

PerturbationPerturbation

PerturbationPerturbation

Negligible transient and expansion term lead to quasi-Negligible transient and expansion term lead to quasi-steady formulation.steady formulation.

Fuel concentration contains two terms:Fuel concentration contains two terms:1. Mass diffusion1. Mass diffusion2. Fuel consumption due to chemistry2. Fuel consumption due to chemistry

Then expansion only appears in the Arrhenius termThen expansion only appears in the Arrhenius term

LeLe close to unity close to unity Perturbation problem reduced to ODE:Perturbation problem reduced to ODE:

Fuel mass diffusion of same order as fuel consumption Fuel mass diffusion of same order as fuel consumption

Max value of the perturbation function of ratio initial Max value of the perturbation function of ratio initial temperatures difference/ adiabatic flame temperature, temperatures difference/ adiabatic flame temperature, times O(1) factor depending upon small difference times O(1) factor depending upon small difference Le - 1Le - 1..

LeLe close to 1, close to 1, < 1 < 1

LeLe close to 1, close to 1, < 1 < 1

Ignition happens at turning Ignition happens at turning point.point.

LeLe close to 1, close to 1, < 1 < 1

for uniform pressure (for uniform pressure (pp00'=0'=0) ignition always occurs ) ignition always occurs (Liñan)(Liñan)

If turning point at If turning point at * < * < maxmax, ignition occurs. For , ignition occurs. For stronger expansion, no ignitionstronger expansion, no ignition

LeLe close to 1, close to 1, > 1 > 1

Solution Solution (1)(1)(() increases monotonically with) increases monotonically with so no so no turning point: so no thermal explosionturning point: so no thermal explosion

Front from warm side toward cold sideFront from warm side toward cold side

Unconditionally quenched by expansionUnconditionally quenched by expansion

LeLe – 1 negative and of O(1) – 1 negative and of O(1)

Fuel supplied by mass diffusion > fuel consumptionFuel supplied by mass diffusion > fuel consumption Ignition at turning point.Ignition at turning point. Ignition time shorter for smaller Lewis number.Ignition time shorter for smaller Lewis number. Similar to Similar to LeLe of O(1), of O(1), < 1. < 1.

LeLe > 1 by O(1) > 1 by O(1) Mass difussion no longer supplies fuel concentration at Mass difussion no longer supplies fuel concentration at

order order . So, chemistry now limited by fuel. Need to rescale: . So, chemistry now limited by fuel. Need to rescale:

Then, problem becomes:Then, problem becomes:

Temperature increase due to chemistry now negligible.Temperature increase due to chemistry now negligible. Equilibrium region propagating towards fuel rich regionEquilibrium region propagating towards fuel rich region Eventually expansion quenches ignitionEventually expansion quenches ignition Similar to Similar to LeLe of O(1), of O(1), > 1 > 1

LeLe > 1 by O(1) > 1 by O(1)

Conclusions from AnalysisConclusions from Analysis

Reaction rate peaks close to hot air side.Reaction rate peaks close to hot air side.

For Lewis numbers greater than threshold close to unity, no For Lewis numbers greater than threshold close to unity, no ignition (jet ignition only observed for hydrogen)ignition (jet ignition only observed for hydrogen)

For Lewis numbers below that threshold, ignition occurs at For Lewis numbers below that threshold, ignition occurs at finite time as long as expansion rate < a critical ratefinite time as long as expansion rate < a critical rate

No ignition for expansion rates faster than the critical rateNo ignition for expansion rates faster than the critical rate

ConclusionsConclusions

““Ignition source” in jet ignition: likely interplay between Ignition source” in jet ignition: likely interplay between diffusion and reactiondiffusion and reaction

Occurs with hydrogen because hydrogen diffuses easily Occurs with hydrogen because hydrogen diffuses easily

Ignition may get killed by expansionIgnition may get killed by expansion

Since there is a clear relationship between leak size and Since there is a clear relationship between leak size and expansion rate, current results consistent with experimentsexpansion rate, current results consistent with experiments