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

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

Overview

● Motivation● Requirements for simulation of high speed flow

(review)● Chemical reaction● Hydrogen: chain-branching● Two examples

– Hydrolyser– Tunnel

Motivation

● Hydrogen: detonates over wide range of concentrations

● Detonations: quite violent and destructive● Ignition: still unpredictable● Enclosed environment may be problematic

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

Conservation● Conservation laws

– Example: 1-D mass: for arbitrary xl and xr,

xl xr

y

x

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.

Burger's equation

● Similar (simpler) problem:

● Solved on characteristics: on we have

since

Burger's Equationu

0

1● Consider IC at t = 0:● Solution: 1 x

y

x

u = 1u = 0

u = ?

0 1

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

Burger's Equation

● But note that Burger's equation is consistent not only with the conservation above, but more generally with:

● So,

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

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

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

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

Entropy conditionu

● Burger's eqn, with IC:

● Solution:

1

1 x0

y

u = 0u = 1

u = ?

x0 1

Entropy conditionu

● What if IC ( small):● Shock: “invent information”● Solution: fan

1

x0

y

u = 0u = 1

u = ?

x0 1

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...

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

Chemical Kinetics (example)

– (Miller et al. 1982. Cited in Powers & Paolucci200 )

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)

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...)

Chain-Branching

● Typical of hydrogen-oxygen

(From Dainton, 1965)

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”

Simple Chain-Branching

Four step

explosion

no explosion

1st limit

2nd lim

it

3 rd limit

Dainton 1965, H2+O2

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]

Simulation vs. Experiments

Simulations

[R]

1 2flow

12

H2+O2+85% Ar

Austin 2003

[OH]

Four steps vs. single step

Single stepFour step Single step Four step

Mass

Pressure

Temperature

Chain-branching coefficient a

explosion

no explosion

explosion

a

a > 1

explosion

no explosion

Reaction length

explosion

no explosion

explosion

10-4

10-2

100

102

104

L

100

101

102

103

104

p

1

2Smoke foils

1

2

10-4

10-2

100

102

104

L

100

101

102

103

104

p

32

Smoke foils

2

3

10-4

10-2

100

102

104

L

100

101

102

103

104

p

56Smoke foils

5

6

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

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,...?

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

Issues (continued)

– Length: typical 1st order termination: infinite reaction length

– Even “free propagating:” subsonic at exit

– CJ plane?

Hydrogen Safety applications (quite preliminary)

– Electrolyzer (with Andrei Tchouvelev)(Dispersion data from Andrei)

0.02 s

Electrolyzer

0.06 s

Electrolyzer

0.1 s

Electrolyzer

0.2 s

Electrolyzer

0.5 s

Electrolyzer

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

Electrolyzer

Electrolyzer

Electrolyzer

Electrolyzer

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

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.

Tunnel

Tunnel

Pressure gradient Temperature gradient

Tunnel

Pressure gradient Temperature gradient

Tunnel

● Does not look too good

● Pressure on vehicles will be high

● Would be worth looking at longer section

● Resolution may be insufficient

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

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.

References

● Del Alamo, G. and Williams, F. A., “Thermal-Runaway Approximation for Ignition Times of Branched-Chain Explosions,”AIAA J. Vol. 43, No. 12, pp. 2599-2605, 2005.

● Gamezo, V.N., Ogawa, T. & Oran, E.S., “Numerical Simulations of flame Propagation and DDT in Obstructed Channels Filled with Hydrogen-Air mixtures,” Proc. Combust. Inst., Vol. 31, 2006.

● Godunov, S. K., "A Difference Scheme for Numerical Computation of discontinuous Solutions of the Hydrodynamic Equations,” Mat. Sb., Vol. 47, pp. 271-306, 1959.

● Hwang, P., Fedkiw, R., Merriman, B., Karagozian, A. R., and Osher, S. J., “Numerical resolution of pulsating detonation waves”, Combustion Theory and Modeling, Vol. 4, No. 3, pp. 217-240, 2000.

● Kapila, A.K., “Homogeneous Branch-Chain Reactions: Initiation to completion,” J. Eng. Math. Vol. 12, pp. 221-235, 1978.

References

● Kapila, A.K., “Homogeneous Branch-Chain Reactions: Initiation to completion,” J. Eng. Math. Vol. 12, pp. 221-235, 1978.

● Lax, P.D., “Weak Solutions of Nonlinear Hyperbolic conservation Laws and their Numerical Computation,” Comm. Pure Appl. Math., Vol. 7, pp. 159-193, 1954.

● Lax. P.D., “Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia, 1973.

● Liang, Z. and Bauwens, L., “Detonation Structure under Chain Branching Kinetics,” Shock Waves, Vol. 15, pp. 247-257, 2006.

● Liang, Z. and Bauwens, L., “Cell Structure and Stability of Detonations with a Pressure Dependent Chain-Branching Reaction Rate Model,” Combust. Theory Modelling, Vol. 9, pp. 93-112, 2005.

● Liang, Z and Bauwens, L., “Detonation Structure with Pressure Dependent Chain-Branching Kinetics,” Proc. Combust. Inst., Vol. 30, pp. 1879-1887, 2005.

References

● Miller, J. A., Mitchell, R. E., Smooke, M. D., and Kee, R. J., “Toward a Comprehensive Chemical Kinetic Mechanism for the Oxidation of Acetylene: Comparison of Model Predictions with Results from Flame and Shock Tube Experiments,” Proc. Combustion Inst., Vol. 19, pp. 181–196, 1982.

● Oran, E. S, Boris, J. P., Flanigan. M., Burks, T., and Picone, M., “Numerical Simulations of Detonations in Hydrogen-Air and Methane-Air Mixtures”, Proc. Combust. Inst., Vol. 18, pp. 1641-1649, 1981.

● Oran, E., Young, T. and Boris, J., “Application of Time-Dependent Numerical Methods to the Description of Reactive Shock Waves,”Proc. Combust. Inst. Vol. 17, 1978, 43-54.

● Powers, J.M. and Paolucci, S., “Accurate Spatial Resolution Estimates for Reactive Supersonic Flow with Detailed Chemistry,”AIAA J., Vol. 43, pp. 1088-1099, 2005.

References

● Quirk, J. J., “Godunov-Type Schemes Applied to Detonation Flows,” NASA ICASE Report No. 93-15, Hampton, VA, April 1993.

● Quirk, J.J., “A Contribution to the Great Riemann Solver Debate,” I. J. Num. Meth. Fluids, Vol. 18, pp. 555-574, 1994.

● Short, M. and Sharpe, G.J., “Pulsating instability of detonations with a two-step chain-branching reaction model: theory and numerics,” Combust. Theory Modelling, Vol. 7, pp. 401–416, 2003.

● Taki, S. and Fujiwara, T., “Numerical Analysis of Two Dimensional Nonsteady Detonations,” AIAA J., Vol. 16, 1978, pp. 73-77.

● Tsuboi, N., Katoh, S., and Hayashi, A. K., “Three-Dimensional Numerical Simulation for Hydrogen/Air Detonation: Rectangular and Diagonal Structures,” Proc. Combust. Inst. Vol. 29, pp. 2783-2788, 2002.

● van Leer, B., “Towards the Ultimate Conservative Difference Scheme II: Monotonicity and Conservation Combined in a Second Order Scheme,” J. Comp. Phys, Vol. 14, pp. 361-370, 1974.

References

● Williams, D. N., Bauwens, L., and Oran, E. S., “A Numerical Study of the Mechanisms of Self-Reignition in Low-Overdrive Detonations,”Shock Waves, Vol. 6, pp. 93-110, 1996.

● Williams, D. N., Bauwens, L., and Oran, E. S., “Detailed Structure and Propagation of Three-Dimensional Detonations,” Proc. Combust. Inst. Vol. 26, 2991-2998, 1996.

● Xu, S.J., Aslam, T. and Stewart, D.S., “High Resolution Numerical Simulation of Ideal and Non-ideal Compressible Reacting Flows with Embedded Internal Boundaries,” Comput. Theory Modelling, Vol. 1, pp. 113-142, 1997.