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 2011San Francisco CA

Outline Motivation Motivation Jet ignition Jet ignition Physical Model Physical Model Magnitude Analysis and Perturbation Magnitude Analysis and Perturbation Results Results Conclusion Conclusion

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

Issues 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 number Hydrogen: mass diffusivity > heat –> Low Lewis number Analysis by Liñan & Crespo (1976) and Liñan & Williams (1993) Analysis by Liñan & Crespo (1976) and Liñan & Williams (1993)

Physical Model One dimensional One dimensional frame of reference attached to contact surface initially separating shock- heated air from cold, expanded hydrogen frame of reference attached to contact surface initially separating shock- heated air from cold, expanded hydrogen In that (nearly inertial) frame, low Mach number In that (nearly inertial) frame, low Mach number Single step Arrhenius chemistry Single step Arrhenius chemistry Negligible cross diffusion Negligible 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 problem

Physical Model Diffusion problem (heat, fuel, oxidant) with sources: chemistry and expansion Diffusion problem (heat, fuel, oxidant) with sources: chemistry and expansion Initial conditions: jump at contact surface Initial conditions: jump at contact surface Boundary conditions at infinity consistent with jump Boundary conditions at infinity consistent with jump

Assumptions/magnitudes Key physical processes: reaction, diffusion and expansion. Key physical processes: reaction, diffusion and expansion. Time short compared with chemical time Time short compared with chemical time High activation energy High activation energy Frozen flow regime: chemistry negligible at leading order Frozen flow regime: chemistry negligible at leading order Ignition as a perturbation of the order of inverse activation energy. 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 Mass-weighed coordinate Self-similar solution: Self-similar solution:

Frozen Flow

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

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

Perturbation Chemistry strongest when departure from maximum temperature is small. So, introduce rescaling Chemistry strongest when departure from maximum temperature is small. So, introduce rescaling Asymptotic expansion of order of inverse activation energy Asymptotic expansion of order of inverse activation energy

Perturbation

Perturbation Negligible transient and expansion term lead to quasi-steady formulation. Negligible transient and expansion term lead to quasi-steady formulation. Fuel concentration contains two terms: Fuel concentration contains two terms: 1. Mass diffusion 2. Fuel consumption due to chemistry Then expansion only appears in the Arrhenius term Then expansion only appears in the Arrhenius term

Le 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 temperatures difference/ adiabatic flame temperature, times O(1) factor depending upon small difference Le - 1. 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

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

Le close to 1,  < 1 for uniform pressure (p 0 '=0) ignition always occurs (Liñan) for uniform pressure (p 0 '=0) ignition always occurs (Liñan) If turning point at  * <  max, ignition occurs. For stronger expansion, no ignition If turning point at  * <  max, ignition occurs. For stronger expansion, no ignition

Le close to 1,  > 1 Solution  (1) (  ) increases monotonically with  so no turning point: so no thermal explosion Solution  (1) (  ) increases monotonically with  so no turning point: so no thermal explosion Front from warm side toward cold side Front from warm side toward cold side Unconditionally quenched by expansion Unconditionally quenched by expansion

Le – 1 negative and of O(1) Fuel supplied by mass diffusion > fuel consumption Fuel 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 Le of O(1),  < 1. Similar to Le of O(1),  < 1.

Le > 1 by O(1) Mass difussion no longer supplies fuel concentration at order . So, chemistry now limited by fuel. Need to rescale: Mass difussion no longer supplies fuel concentration at order . 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 region Equilibrium region propagating towards fuel rich region Eventually expansion quenches ignition Eventually expansion quenches ignition Similar to Le of O(1),  > 1 Similar to Le of O(1),  > 1

Le > 1 by O(1)

Conclusions 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 ignition (jet ignition only observed for hydrogen) For Lewis numbers greater than threshold close to unity, no ignition (jet ignition only observed for hydrogen) For Lewis numbers below that threshold, ignition occurs at finite time as long as expansion rate < a critical rate For Lewis numbers below that threshold, ignition occurs at finite time as long as expansion rate < a critical rate No ignition for expansion rates faster than the critical rate No ignition for expansion rates faster than the critical rate

Conclusions “Ignition source” in jet ignition: likely interplay between diffusion and reaction “Ignition source” in jet ignition: likely interplay between diffusion and reaction Occurs with hydrogen because hydrogen diffuses easily Occurs with hydrogen because hydrogen diffuses easily Ignition may get killed by expansion Ignition may get killed by expansion Since there is a clear relationship between leak size and expansion rate, current results consistent with experiments Since there is a clear relationship between leak size and expansion rate, current results consistent with experiments