numerical simulation of hydrogen detonations and ...€¦ · requirements for simulation of high...
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Numerical Simulation of Hydrogen Detonations and Application in EnclosedEnvironments
Luc BauwensLuc BauwensUniversity of Calgary, Mechanical & Mfg EngineeringUniversity of Calgary, Mechanical & Mfg EngineeringFirst European Summer School on Hydrogen SafetyFirst European Summer School on Hydrogen SafetyBelfast, August 2006Belfast, August 2006
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Thanks
● To Vladimir Molkov for the invitation
● To Elaine Oran for her considerable help (movies, pictures, papers, advice)
● To Koichi Hayashi & N. Tsuboi for their pictures
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Overview
● Motivation● Requirements for simulation of high speed flow
(review)● Chemical reaction● Hydrogen: chain-branching● Two examples
– Hydrolyser– Tunnel
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Motivation
● Hydrogen: detonates over wide range of concentrations
● Detonations: quite violent and destructive● Ignition: still unpredictable● Enclosed environment may be problematic
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Gas dynamics
● Perhaps better mathematical background than for low speed flow?
● Looks easy: just unroll time-derivatives?
● Real issue: dealing with shocks and expansion waves – Conservation– Flux limiting– Entropy condition
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Conservation● Conservation laws
– Example: 1-D mass: for arbitrary xl and xr,
xl xr
y
x
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Conservation paradox
● Expanding etc., recover usual continuity ● Remember, for inviscid non-conducting
conservation (mass, momentum, energy) -> isentropic
● However, from the Rankine-Hugoniot equations (continuity of fluxes across shock), a shock is NOT isentropic.
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Burger's equation
● Similar (simpler) problem:
● Solved on characteristics: on we have
since
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Burger's Equationu
0
1● Consider IC at t = 0:● Solution: 1 x
y
x
u = 1u = 0
u = ?
0 1
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Burger's Equation
● What happened? Multiple solutions?● What if the equation actually were the
conservation law:
● Then we can construct a shock solution: x <xs: u = ul but x > xs, u = ur
● Replacing, we find xs=(ul + ur)/2=1/2
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Burger's Equation
● But note that Burger's equation is consistent not only with the conservation above, but more generally with:
● So,
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Burger's Eqn: summarizing
● Differential equation could have come from infinitely many conservation laws
● Shock speed depends upon the specific conservation law
● So shock speed not determined by the differential equation
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Conservative schemes
● Exact discrete conservation (on the grid)● Arbitrary schemes: error depends upon mesh
size both– Directly– Through the jump in the function in
neighboring grid points● If conservative: only the former. So, error
becomes small with mesh even if jump
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Flux Limiting
● Issue is oscillations close to shocks● Avoiding oscillations: monotone schemes● Godunov theorem: linear conservative
monotone schemes at most first order accurate● First order: quite poor, not good enough
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Flux Limiting
● Oscillations arise because fluxes oscillate from one time step to next
● Crucial idea: limit fluxes where they oscillate– Near shocks, accuracy deteriorates to 1st order– Elsewhere, higher order
● Proposed independently by Jay Boris and Bram van Leer in the early seventies
● Nonlinear schemes
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Entropy conditionu
● Burger's eqn, with IC:
● Solution:
1
1 x0
y
u = 0u = 1
u = ?
x0 1
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Entropy conditionu
● What if IC ( small):● Shock: “invent information”● Solution: fan
1
x0
y
u = 0u = 1
u = ?
x0 1
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Gas Dynamics-Summary
● Pretty good
● But near shocks: first order at best
● Handles shocks and expansions transparently
● Explicit: CFL (Courant-Friedrichs-Lewy) condition for stability
● However even implicit: near-CFL for accuracy
● Various schemes: TVD, Godunov, ENO, FCT...
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Chemical Kinetics
● Two key issues:– How good is the model?– Stiffness
● How good?– Hydrogen air is the simplest and best known– However, typically not calibrated against
detonations– Uncertain at high pressures
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Chemical Kinetics (example)
– (Miller et al. 1982. Cited in Powers & Paolucci200 )
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Kinetic schemes
● Various levels of simplification– “Full” kinetics (some more “ful” than others)– Various levels of reduced schemes– Single (or two, or three) steps
● Even for oxygen-air (simplest)– Different schemes -> different results– None really matches cell size– (Typically calibrated for flames)
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Stiffness
● Kinetics: very slow and very fast rates– (many orders of magnitude)
● “Regular” integration– Either unreasonably (impossibly) small step– Or step too big for fast rates -> instability
● “Stiff solvers” avoid instability
● No miracle: fast steps are NOT resolved
● (= inadvertent scheme reduction...)
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Chain-Branching
● Typical of hydrogen-oxygen
(From Dainton, 1965)
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Simple Chain-Branching
● Four-step model (three-step: Short 1997)
Termination
Initiation
Branching
– Competition for specie X– Production/consumption a = (rI+rB)/(rW+rG)– A > 1: “explosion”
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Simple Chain-Branching
Four step
explosion
no explosion
1st limit
2nd lim
it
3 rd limit
Dainton 1965, H2+O2
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Four Step: ZND Profiles
Mass Fraction
P, T
0 1 2 3 4
0.8
1.0
1.2
1.4
p/ps
T/Ts
Inductionzone
Chain explosion zone
Recombination zone
Shock
0 1 2 3 40.0
0.5
1.0
[X]
[R]
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Simulation vs. Experiments
Simulations
[R]
1 2flow
12
H2+O2+85% Ar
Austin 2003
[OH]
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Four steps vs. single step
Single stepFour step Single step Four step
Mass
Pressure
Temperature
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Chain-branching coefficient a
explosion
no explosion
explosion
a
a > 1
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explosion
no explosion
Reaction length
explosion
no explosion
explosion
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10-4
10-2
100
102
104
L
100
101
102
103
104
p
1
2Smoke foils
1
2
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10-4
10-2
100
102
104
L
100
101
102
103
104
p
32
Smoke foils
2
3
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10-4
10-2
100
102
104
L
100
101
102
103
104
p
56Smoke foils
5
6
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Kinetics: Summary
– A hierarchy of models
● Single step
● A few steps
● Reduced models (10-30 steps)
● Full kinetics
– Arguably there is use for all. None is perfect
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Simulation of detonations: Issues
– Obviously kinetics and stiffness
– Size of the domain (width, length)
– Resolution especially in hot spots (Quirk)
– Grid refinement vs. parallel optimization
– Downstream BC● Reflective, but how? Length?● “overdrive” but what? P, u, T,...?
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Issues (continued)
– Convergence of the inviscid model?
● Kelvin-Helmholtz: convergence requires viscous model on the scale of the thickness of the splitter plate
● Here: thickness of the triple point! (Mean free path, non-equilibrium?)
● Gamezo and Oran do that.
● If reaction fronts Rayleigh-Taylor unstable (or RM), same issue
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Issues (continued)
– Length: typical 1st order termination: infinite reaction length
– Even “free propagating:” subsonic at exit
– CJ plane?
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Hydrogen Safety applications (quite preliminary)
– Electrolyzer (with Andrei Tchouvelev)(Dispersion data from Andrei)
0.02 s
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Electrolyzer
0.06 s
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Electrolyzer
0.1 s
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Electrolyzer
0.2 s
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Electrolyzer
0.5 s
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Electrolyzer
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Electrolyzer
Model: single step (Gamezo et al. 2006)
(Reaction zone not resolved anyway – Heat release is what matters)
BCs: solid walls. Ignore equipment inside
Show movies next
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Electrolyzer
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Electrolyzer
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Electrolyzer
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Electrolyzer
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Electrolyzer
Initially the detonation propagates
When mixture becomes lean it fails
(Reaction front decouples from shock)
Shock weakens as it moves toward walls
Reflections: unphysical
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Tunnel
Assumed 8 kg of hydrogen spilled
Cloud: half-parabolic, stoichio on top, zero hydrogen along bottom parabola (see picture)
Section: 50 m2, height 5.1 m, bottom 1.5 m is vehicles so domain is 3.6 m.
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Tunnel
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Tunnel
Pressure gradient Temperature gradient
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Tunnel
Pressure gradient Temperature gradient
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Tunnel
● Does not look too good
● Pressure on vehicles will be high
● Would be worth looking at longer section
● Resolution may be insufficient
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Summary
● Simulation of detonations
– Not perfect– But not too bad either
● Situation is better than for example– turbulent combustion – hydrogen dispersion
● Much closer to actual physics
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References
● Austin J.M. and Shepherd, J.E., “Detonations in Hydrocarbon Fuel Blends,” Combust. Flame, Vol. 132, pp. 73-90, 2003.
● Birkan, M and Kassoy, D.R., “The Unified Theory for Chain-Branching Thermal Explosions with Dissociation-Recombination and Confinement Effects,” Combust. Sci. Technology, Vol. 44, Nos. 5-6, pp. 223-256, 1986.
● Boris J.P. and Book, D.L., “Flux-Corrected-Transport, I. SHASTA, A Fluid Transport Algorithm That Works,” J. Comp. Phys., Vol. 11, pp.38-69, 1973.
● Browne, S., Liang, Z., & Shepherd, J.E., “Detonation Front Structure and the competition for radicals,” Proc. Combust. Inst., Vol. 31, 2006.
● Courant, R, Friedrichs, K.O., and Lewy, H., “Uber die partiellen Differenzgleichungen der mathermatischen Physik,” Mathematischen Annalen, Vol. 100, pp. 32-74, 1920.
● Dainton, F.S., “Chain Reactions, An Introduction,” London, Methuen● & Co. Ltd, p.123, 1965.
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