Deflagration-to-detonation transition

Deflagration-to-detonation transition (DDT) is the process by which a flame that starts as a subsonic deflagration accelerates until it becomes a detonation, a supersonic combustion wave led by a shock. In a deflagration the pressure rises almost uniformly throughout an enclosure, whereas a detonation produces a sudden and highly non-uniform pressure rise as the shock passes. The transition matters because detonation pressures are roughly twice those of a closed-vessel deflagration, and because explosion vents and suppression systems cannot cope with them.

Accidental ignition sources are far too weak to start a detonation directly, so in practice an accidental detonation develops from a deflagration. The run-up distance is shorter for more reactive mixtures, rough or obstructed pipes, larger diameters, higher initial pressure and temperature, and stronger initial turbulence. Whether a mixture can detonate at all in a given geometry is judged with the detonation cell width, which ranges from about 1 cm for acetylene to about 28 cm for methane. NUREG-1805 notes that the possibility of DDT depends on both the mixture and its environment, and that detonations are very rare. No exact criteria exist that predict a transition, so assessments combine test data with engineering judgement.

Deflagration, detonation and quasi-detonation

A deflagration is flame propagation at subsonic speed away from the ignition site. NUREG-1805 gives the practical boundary: a deflagration travels below the speed of sound in air, about 330 m/s at 20 °C, while a detonation travels at speeds of the order of 2,000 m/s. In a detonation the shock and the reaction zone move together, faster than sound in the unburned gas. Simmons gives a range of 1,500 to 3,000 m/s for detonations in ducts. Between the two lies the quasi-detonation, in which the flame moves slightly slower than sound, so that the pressure rise is non-uniform but no shock waves form.

The difference in speed changes the pressure field. Because a deflagration flame is slow compared with sound, the pressure rises almost uniformly throughout the enclosure while the flame spreads. Calculated worst-case closed-vessel deflagration pressures for methane, propane and hydrogen in air lie between 8 and 9.6 atm absolute. In a detonation the pressure jumps almost instantly when the shock arrives. Its peak follows from the Chapman–Jouguet theory, which combines chemical equilibrium with the conservation laws of gas dynamics across the front. For near-stoichiometric gas–air mixtures at atmospheric pressure, this Chapman–Jouguet pressure lies between 16 and 20 atm, about twice the closed-vessel deflagration pressure.

This difference decides the protection strategy. Explosion vents open after an explosion has started and limit the rise of pressure, and suppression systems must act while the pressure is still low. The sudden, spatially non-uniform pressure rise of a detonation or quasi-detonation therefore rules out both methods. The high peak and short duration of the load also call for special care in the assessment of structural resistance. After an event, fragmented debris from brittle failure points to a detonation, whereas deflagration pressures beyond the yield point tend to make steel structures bulge in ductile failure.

Mechanism of the transition

Simmons describes how a flame travelling along a duct can turn into a detonation. At first the flame moves at a fairly uniform speed, set by the burning velocity of the mixture and by the area of the flame front. The linear flame speed V equals the burning velocity S_u multiplied by the ratio of the flame front area to the cross-sectional area of the duct:

$$V=\frac{S_{\mathrm{u}}{A}_{\mathrm{f}}}{A_{\mathrm{d}}}$$

Because the flame front is larger than the duct cross-section, typically by a factor of two or three, the flame moves correspondingly faster than the burning velocity. If the hot combustion products cannot vent, they push the flame and the unburned gas ahead of it forward with increasing velocity. This flow makes the gas ahead of the flame more turbulent, which in turn increases the burning velocity.

Turbulence wrinkles the flame front and so increases its effective area. The state of a flow is described by the Reynolds number, which is below about 2300 for laminar flow and above about 3200 for flow that is usually turbulent. For propane, ethene and acetylene flames, the burning velocity roughly doubles as the Reynolds number rises to 40,000. A theoretical treatment suggests an increase by a factor of five when the turbulence is very high. In a long enough duct with rapid enough acceleration, the flame front finally acts as an accelerating piston, a shock wave forms ahead of it and the flame becomes a detonation.

Flame acceleration in a duct closed at the ignition end: the expanding products push the unburned gas ahead, turbulence speeds up the flame, a shock forms and the flame becomes a detonation. Based on Simmons (2002), pp. 1-146 to 1-147.

A detonation can also start directly, without a preceding deflagration, but only with a very strong ignition source. The minimum energy for such direct initiation ranges from about 5 kJ for acetylene and hydrogen in air to an estimated 93,000 kJ for methane. These energies are many orders of magnitude larger than those of accidental ignition sources, so direct initiation can be ruled out in almost all accident scenarios. An accidental detonation therefore needs a flame that accelerates, which requires a highly elongated geometry or an exceptionally high level of turbulence.

Factors that shorten the run-up distance

The distance that a flame travels before it becomes a detonation is called the DDT distance or transition length. For a weak, typically accidental, ignition source in a pipe or another elongated enclosure, Zalosh lists five factors that govern it. A more reactive mixture accelerates faster to detonation. A rougher pipe wall or more obstructions shorten the transition length, and so does a larger pipe or enclosure diameter. Higher initial pressure and temperature also shorten it, as does a higher level of initial turbulence or gas velocity in the enclosure. Obstacles matter because the interaction between a flame and an obstacle generates turbulence.

Experiments in a smooth pipe of 5 cm diameter show the effect of reactivity. The tests used methane, ethane and propane in mixtures of oxygen and nitrogen, at fuel-to-oxygen ratios close to the worst case. The ratio of transition length to pipe diameter rose steeply as the nitrogen-to-oxygen ratio increased, with propane giving the shortest transition length and methane the longest. A transition was only observed at nitrogen-to-oxygen ratios below 3.76, the ratio in air. For these alkanes in air, the transition length in such a pipe therefore exceeds 120 diameters, the largest ratio the experiments could measure.

Wall roughness can change the outcome completely. Flessner and Bjorklund saw no transition with gasoline–air mixtures in a smooth pipe of 15 cm diameter and 16 m length. When they inserted an expanded metal liner at the pipe entrance to simulate a rough section, the flame turned into a detonation after 74 pipe diameters. The smooth pipe was more than 100 diameters long, so the rough section made the difference between no detonation and a detonation within the length of the pipe.

Detonability and cell size

Not every mixture can sustain a detonation in every geometry. Zalosh uses the detonation cell width as the length scale for detonability. The front of a detonation is a complex network of curved shock segments that move sideways across the direction of travel. These transverse waves leave a diamond-shaped or fish-scale pattern on a smoke foil inserted on the inner wall of a detonation tube, and the cell width is measured from that pattern. Cell widths can also be related to the length of the reaction zone in a one-dimensional model of the detonation, the ZND model. That model needs chemical kinetics data and assumptions about heat loss and drag at the tube wall.

Structure of a detonation front: transverse shock waves meet at triple points, whose paths trace a fish-scale pattern of cells on a smoke foil; the cell width is the detonability length scale. Based on Zalosh (2002), p. 3-412.

Chapman–Jouguet Mach numbers and cell widths of stoichiometric gas–air mixtures

Gas	Chapman–Jouguet Mach number	Cell width (cm)
Acetylene	5.46	0.98
Hydrogen	4.89	1.5
Ethylene	not given	2.8
n-Butane	not given	5.0 to 6.2
Propylene	not given	5.4
Ethane	not given	5.4 to 6.2
Propane	5.38	6.9
Hydrogen sulphide	not given	10
Methane	5.17	28

The cell widths in the table apply at atmospheric pressure and room temperature. Higher initial pressures and temperatures give smaller cells, and some fuel-rich mixtures have cells up to 40 % smaller than the stoichiometric values. Smaller cells mean a more sensitive mixture.

The cell width sets two minimum sizes. A detonation will not propagate along a pipe whose diameter is less than about one third of the cell width. It will not pass from the open end of a pipe into the surrounding mixture if the pipe diameter is less than 13 cell widths. A pipe of 1 cm diameter can therefore carry a hydrogen or acetylene detonation, but it is far too narrow for methane, which would need a pipe of more than 9 cm. NUREG-1805 makes the same point in general terms: detonable limits depend on the environment as well as on the mixture.

Critical pipe diameters derived from the detonation cell width: below one third of the cell width a detonation cannot propagate along a pipe, and below 13 cell widths it cannot pass from a pipe into a larger volume. The line marks a 1 cm fuel line. Based on Zalosh (2002), p. 3-412.

Pressure and impulse of a detonation

Behind the Chapman–Jouguet front the pressure falls to a plateau of about 0.35 times the peak. The front moves at the Chapman–Jouguet Mach number times the speed of sound in the unburned mixture. For several stoichiometric mixtures this Mach number lies between 4.9 and 5.5, and the speed of sound is 330 to 350 m/s in hydrocarbon–air mixtures and 403 m/s in hydrogen–air. A propane–air detonation, with a Mach number of 5.38, therefore travels at about 1,780 to 1,880 m/s. This agrees with the order of 2,000 m/s given in NUREG-1805.

Pressure along a tube during a detonation of stoichiometric propane-air, using the approximations in the text: peak about twice the closed-vessel deflagration pressure, a plateau of 0.35 times the peak, and 2.76 times the peak after reflection from a rigid end. Based on Zalosh (2002), pp. 3-407 and 3-411.

The Chapman–Jouguet pressure can be calculated with chemical equilibrium computer codes such as STANJAN and CET89. Burgess and co-workers proposed a much simpler estimate. It takes twice the pressure of a closed-vessel deflagration of the same mixture:

$$P_{\mathrm{C}\mathrm{J}}\approx 2P_{\mathrm{m}}$$

For propane, with a maximum deflagration pressure of 7.9 bar g, or about 8.9 bar absolute, this gives roughly 18 bar absolute, inside the range of 16 to 20 atm quoted above. Reflection makes the load worse. If a detonation reaches a rigid closed end, the reflected shock for many gas–air mixtures is 2.76 times the Chapman–Jouguet pressure, while an open end reflects the wave as a rarefaction. For a mixture initially at atmospheric pressure, Simmons gives up to 20 bar directly behind the front of a detonation. Where the detonation reflects from the end of a duct, the pressure can reach 100 bar. NUREG-1805 estimates detonation pressure rises at two to four times those of a deflagration.

Peak pressure alone does not describe the threat to a structure, because a detonation load is very short. The structural load depends on the impulse per unit area, the integral over time of the overpressure at a point. The detonation front arrives at a point x at the time x divided by the speed of the front. Neglecting reflected waves, Sichel showed that the impulse at a distance x from the point of initiation is approximately:

$$I_{\mathrm{D}}=\frac{0.35{\rho }_1{a}_1{M}_{\mathrm{D}}x}{{\gamma }_2+1}$$

Symbol	Description	Unit
$V$	linear speed of a flame along a duct	m/s
$S_{\mathrm{u}}$	burning velocity of the unburned mixture	m/s
$A_{\mathrm{f}}$	area of the flame front	m²
$A_{\mathrm{d}}$	cross-sectional area of the duct	m²
$P_{\mathrm{C}\mathrm{J}}$	Chapman–Jouguet detonation pressure	bar or atm (absolute)
$P_{\mathrm{m}}$	pressure at the end of a closed-vessel deflagration	bar or atm (absolute)
$I_{\mathrm{D}}$	impulse per unit area of wall	Pa·s
${\rho }_1$	density of the unburned mixture	kg/m³
$a_1$	speed of sound in the unburned mixture	m/s
$M_{\mathrm{D}}$	detonation Mach number	none
$x$	distance from the point of initiation	m
${\gamma }_2$	ratio of specific heats of the burned gas, often 1.2	none

The impulse grows in proportion to the distance from the point of initiation. Burgess and co-workers observed that pipeline detonations often cause periodic ruptures along a pipeline. Each break relieves the pressure, and the impulse must build up again over a further length of pipe before the next failure.

Worked example: a burner fuel line

Zalosh analyses a large process oven heated by a burner, whose fuel line is 1 cm in diameter and 5 m long and contains a stoichiometric fuel–air mixture. The question is whether a delayed ignition, after the oven has been filled with a fuel–air mixture by mistake, could lead to a detonation in the line or in the oven. For propane the cell width is 6.9 cm, so a detonation needs a pipe of at least 6.9/3 = 2.3 cm. The 1 cm line is too narrow, and a propane–air detonation cannot be sustained in it.

With hydrogen the cell width is 1.5 cm and the minimum diameter only 0.50 cm, so a detonation in the line becomes possible. The length of 500 pipe diameters makes this more likely. To pass into the oven, however, the detonation would need a pipe wider than 13 cell widths, about 20 cm. The oven itself should therefore not experience a detonation, and explosion venting could still protect it.

If a hydrogen detonation does run along the line, Sichel's equation estimates the load on the pipe wall. The stoichiometric hydrogen–air mixture contains 29.5 % hydrogen by volume, which gives a molecular weight of 20.9 kg/kmol and a density of 0.87 kg/m³. With a speed of sound of 400 m/s, a Mach number of 4.89, a pipe length of 5 m and γ₂ = 1.2, the impulse at the end of the pipe is:

$$I_{\mathrm{D}}=\frac{0.35\times 0.87\times 400\times 4.89\times 5}{1.2+1}\approx 1360\mathrm{P}\mathrm{a}\mathrm{\cdot }\mathrm{s}$$

The value is rounded up because it is a load; the handbook gives 1354 Pa·s. The detonation needs 5/(4.89 × 400) = 2.56 ms to run the length of the pipe, so the average pressure on the wall is about 530 kPa. This is roughly one third of the Chapman–Jouguet pressure, and a structural engineer can use these values to judge whether the line would survive.

Hydrogen and vapour clouds

Within hydrogen safety, several properties set hydrogen apart from the hydrocarbon gases in the table. Its cell width of 1.5 cm is about a fifth of that of propane. Its direct initiation energy of about 5 kJ lies at the low end of the range quoted in the handbook, together with acetylene. Its maximum laminar burning velocity in air is 320 cm/s, against 41 cm/s for propane, and its KG value in a closed vessel is 550 bar·m/s, against 100 bar·m/s for propane. NUREG-1805 gives a flammable range in air of 4 to 75 %, but notes that overpressures are not observed below about 8 %, because a flame cannot spread downwards below that concentration.

These properties explain why hydrogen appears in several of the cases above, such as the fuel line of the worked example. In nuclear power plants, hydrogen can collect in battery rooms, because overcharged vented batteries release hydrogen and oxygen in stoichiometric proportions. Sherman and Berman applied their method for categorising detonation probability to a hypothesised hydrogen detonation in a nuclear plant containment building.

Outside equipment, the corresponding hazard is the vapour cloud explosion. Such explosions occur when a large cloud of flammable gas or vapour is ignited in a highly obstructed area, such as a chemical plant with closely spaced process equipment and storage vessels. The strength of the blast wave depends on the flame speed that develops and on the degree of confinement in and around the cloud. NUREG-1805 notes that an unconfined vapour cloud explosion can produce pressures of the same order as an explosion inside a building, and that a detonation gives even higher pressures, although detonations are very rare. Current methods either estimate the flame speed empirically or calculate the flame acceleration with computational fluid dynamics.

Well-documented detonation accidents are rare but instructive, since most accidental explosions are deflagrations. Jacobs and co-workers described a destructive detonation in a refinery in which a mixture of 3 % naphtha, 19 % oxygen and 78 % inert gas entered several large process vessels. The mixture was at 105 psig, about 724 kPa above atmospheric pressure, and the vessels were connected by more than 1000 ft, about 300 m, of piping. Pressures of 3000 to 4000 psi, roughly 21 to 28 MPa, developed at some locations.

Assessment and limitations

There are no exact criteria that predict whether a deflagration will undergo transition, only guidance. The cell width shows whether a detonation can be sustained in a given geometry, and the run-up data show how far a flame may travel before it detonates. Both rest on experiments in tubes of a few sizes, mostly at atmospheric pressure and room temperature. Sherman and Berman therefore developed a semi-quantitative method that classifies mixture reactivity by cell size and geometry by size, confinement, obstructions and venting. Their method extrapolates the experimental data subjectively to the case being assessed.

Standard fire models cannot simulate the transition. The Fire Dynamics Simulator assumes low-speed flow, with Mach numbers below about 0.3, which rules out its use for explosions, choked flow at nozzles and detonations. For gas explosions, specialised computational fluid dynamics models exist, with their own descriptions of turbulence generation and turbulent combustion. When the handbook was written they were too complex and computationally demanding for routine vent design, but they were used for special cases such as offshore modules crowded with equipment.

The criteria in this article also show where the defence lies. A pipe narrower than one third of the cell width cannot carry a detonation, and a pipe narrower than 13 cell widths cannot pass one into a connected volume. Preventing the formation of a flammable mixture, for example by ventilation or inerting, removes the hazard of both deflagration and detonation. In nuclear power plant battery rooms, for example, the ventilation system typically limits hydrogen to less than 2 % of the room volume.

See also

• Explosion
• Explosion venting
• TNT equivalence
• Inerting
• Deflagration index

References

• Iqbal, N., & Salley, M. H. (2004). Fire Dynamics Tools (FDTs): Quantitative fire hazard analysis methods for the U.S. Nuclear Regulatory Commission Fire Protection Inspection Program (NUREG-1805). U.S. Nuclear Regulatory Commission, Office of Nuclear Reactor Regulation.
• McGrattan, K. (2007). Verification and validation of selected fire models for nuclear power plant applications: Vol. 7. Fire Dynamics Simulator (NUREG-1824; EPRI 1011999). U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research; Electric Power Research Institute.
• Simmons, R. F. (2002). Premixed burning. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 1-144–1-154). National Fire Protection Association; Society of Fire Protection Engineers.
• Zalosh, R. (2002). Explosion protection. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 3-402–3-421). National Fire Protection Association; Society of Fire Protection Engineers.