background

Combustion is an unsteady process that characteristically produces a broadband haystack-shaped sound spectrum that is often referred to as “combustion roar.” On purely theoretical grounds, S. L. Bragg deduced that something of the order of one-millionth of the chemical energy in the fuel is converted to sound.  Perhaps that doesn’t sound like much but the heat release of a typical 10-million Btu/hr burner is equivalent to nearly 3-million watts.  If something like one-millionth is converted, that gives around 3-watts of sound or in terms of sound power about 125 decibels.  A typical industrial furnace might contain up to 40 such burners yielding a prodigious 140 decibels of sound power.

As eddy-like globs of fuel-air burn, the combustion products expand locally and then collapse.[1]  We can roughly estimate the period required to consume a typical glob as its diameter (say about 10^-2 ft) divided by the combustion wave speed (say 1-5 ft/sec) or 2-10 milliseconds.  Inverting this we get 100-500 Hz, so it comes as no surprise that those are the frequencies of ordinary combustion roar.

The natural result of combustion is a low pitched roar easily exceeding the OSHA 90 dBA 8-hour exposure limit.  Today we handle this moderately intense sound source with absorptive burner plenums.  But this paper discusses a rare but much more intense and dangerous source of noise and vibration, combustion-driven oscillation. End Note 1

The Singing Flame

The general features of “vibrations maintained by heat” were understood by the mid-1800s.  Rayleigh recognized the crucial phase relationship between the communication of heat and the vibration in the resonator.  The classical demonstration consisted of a hydrogen flame burning inside an open tube. Pressure variations in the tube cause the flow of gas, and therefore the heat release, volume expansion, and backpressure on the nozzle to vary during the vibration.

If the product of the fluctuating parts of the heat release and the backpressure, integrated over a cycle of the vibration, is positive, in the absence of damping the vibration will be maintained.  In other words, if the fluctuating heat release is more in phase than out of phase with the vibration in the resonator, conditions are right for feeding energy into the vibration.

That is often called the “Rayleigh Criterion” and what Rayleigh observed is often referred to as the “singing flame.”

Singing flames are analogous to other, more familiar, self-excited vibrations in which the vibrating system extracts its sustaining energy from another system, usually a flowing stream.  Singing telephone cables are a familiar example.  The Tacoma Narrows bridge failure is an oft-told classic.   In a singing flame, combustion provides the energy to sustain gas vibrations inside the combustion chamber.  Thus, the gas inside the combustion chamber plays the role of the telephone cable, and the heat energy released in the combustion process plays the role of the wind.

Burning Delay Time

To understand better how combustion-driven oscillation arises, we need the concept of a “burning delay time.”  This is the period of time that elapses between the release of an eddy-like glob of reactant from an orifice until that glob actually reaches the flame front and burns.

Think of a premixed conical flame like that of a Bunsen burner.  It is conical because the jet velocity exceeds the flame speed.  If you increase the flame speed, or decrease the jet velocity, the flame gets shorter, which means the burning delay time gets shorter.  Eventually, when the flame speed is the same as the jet velocity, the flame retreats all the way back to the nozzle.  There is no delay time at all.  The moment a glob of reactant emerges, it burns.

Now suppose that, for some reason, the nozzle sees a sinusoidally varying backpressure.  What happens? Naturally, the reactant emerges at the same sinusoidally varying rate.  Think of it as a train of progressively larger-than-normal reactant globs followed by progressively smaller-than-normal globs and let’s follow the fate of one of those larger-than-normal globs of reactant.

Depending on the delay time, it takes a while before that larger-than-normal glob burns to release its larger-than-normal amount of heat, which produces a larger-than-normal volume expansion, which produces a pressure pulsation.  In free space this doesn’t amount to much.  But if it occurs in the limited volume of a combustion chamber, and during a positive half-cycle of the sinusoidally varying backpressure in the combustion chamber, this extra heat release augments the backpressure variation.  If not, it suppresses it.  By varying the flame speed and jet velocity, the flame can be “tuned” as it were to build up the sinusoidally varying backpressure.

That’s about all there is to it.  All you have to do is stick the nozzle into the chamber, tune up the flame, and presto!  Combustion-driven oscillation.  Naturally it isn’t quite that simple but you get the idea.

You can try for yourself how this works by sticking a common home-plumbing propane torch into a piece of pipe.  The pipe or rather the air inside the pipe, as we all know, has “organ pipe” resonance frequencies.  To get the vibration going you will have to diddle with the fuel valve to tune up the flame.  But once you get it going, the sound, the tone of which can be calculated exactly based on the length of the pipe, is prodigious indeed.

Ground Flare

Is this merely a cute demonstration or does this happen in real life?  On rare occasions it certainly does.  Monster organ pipes exist in some refinery and petrochemical complexes.  Called “ground flares,” many are about 20 ft in diameter and about 100 ft tall.  Many years ago one my first encounters with combustion-driven oscillation occurred in just such a unit.

An organ pipe of this length has a very low frequency and of this size can generate a lot of sound.  If you consider that the thermal energy (Btu/hr) being dissipated in such a unit is the power equivalent of about 100-million watts, then if even a small fraction is converted to vibrational energy, shaking down the Walls of Jericho or the refractory bricks inside the steel shell is not out of the question at all.  For this one the fix was a minor nozzle modification that changed the burning delay time.

Now that we have a physical feel for it, let’s see what a little mathematics tells us about this problem.  Since the fluctuating component of heat release provides the sustaining energy when a singing flame occurs, we have to consider how waves form in the fuel supply line.  By inserting sinusoidal representations of the pressure and velocity waves into the Rayleigh Criterion, one obtains the following necessary condition for maintenance of combustion-driven oscillation:

2(1+β)sin(2ωS/C)sin(ωτ) – β(β+2)cos(ωτ) > 0

The damping is represented by β, ω is the circular frequency, S is the length of the fuel supply line, C is the speed of sound in the supplied gas and τ is the burning delay time.

In practical situations, generally the damping (β) is very small and can be neglected.  Furthermore, the burning delay time (τ) often turns out to be short enough that sin(ωτ) is always positive.  Under these conditions the Rayleigh Criterion becomes even simpler:

sin(2ωS/C) > 0 or sin(4πS/λ) > 0

The wavelength of the standing wave in the supply pipe (λ) is determined by dividing the speed of sound in the supplied gas by the frequency of pulsation in the combustion chamber.

The mathematics suggest that for the common case of a supply line open at the supply end as, for example, in a large receiver or knockout drum, small damping and small burning delay time, vibrations may be sustained whenever the length of the supply pipe is equal to or less than one-quarter of the wavelength in the supplied gas of the vibration in the combustion chamber.  On the other hand, if the supply pipe is greater than one-quarter wavelength, up to one-half wavelength, conditions are unfavorable and the vibration evidently should not be maintained.  In the next quarter wavelength (i.e., beyond a half wavelength), conditions are again favorable, and so on.

Let us now apply these ideas to the ground flare pictured above.  If we express the wavelength in the supply pipe (λ) as the speed of sound in the gas supply (C_S) divided by the frequency of vibration (ƒ) in the combustion chamber, and remember that the fundamental natural frequency is C_L/2L (assuming the combustion chamber is acoustically open at both ends), where C_L is the speed of sound in the combustion gases, the Rayleigh Criterion can be related to the physical parameters of this example:

                           n < (C_L/C_S)(S/L) < (2n+1)/2; n = 0,1,2, …

Going one step further, the sonic velocities can be related to specific heat ratios (α) and absolute temperature (T), and the Rayleigh Criterion becomes even simpler.

In the actual case of the ground flare pictured above, the flared gas was a propane/butane mixture (α_S ≈ 1.1 and MW_S ≈ 55) supplied at 100°F.  The combustion gases can be assumed to be mostly carbon dioxide and nitrogen (α_L ≈ 1.3 and MW_L ≈ 30) at a temperature that depends upon the flare load.  The length of the supply pipe was approximately 80 ft and the combustion chamber was 100 ft tall.  Substituting these values, the Rayleigh Criterion expressed in terms of the combustion chamber temperature becomes quite simple:

n² < T_L/404 < (2n+1)²/4; n = 0,1,2, …

Finally then, the mathematics suggest to us that as the system comes into resonance, the absolute temperature of the combustion gases should just be crossing the value 404n².  In the combustion range of interest, this gives about 1150°F as the temperature at which it should begin to be possible to sustain combustion-driven oscillation.

The ground flare pictured above was 100 ft tall and 18 ft in diameter.  It was open at the top, and at the bottom there were five vertical slots (about 2 ft wide by 10 ft high) into which protruded the gas supply pipes, and through which passed the combustion air. In operation, this system occasionally exhibited strong vibrations at a frequency of about 7 Hz.  The foregoing application of the Rayleigh Criterion predicted that combustion-driven oscillation should not be sustained until the flue gas temperature reached approximately 1150°F, corresponding to a hydrocarbon flow rate about 50% of the maximum design load. 

Naturally those were exactly the conditions that marked the onset of combustion-driven oscillation or I wouldn’t be sharing this example with you.  Ain’t science wunaful?

Hydrogen Reformer

Some years ago another instance of combustion-driven oscillation arose in a hydrogen manufacturing plant.  The reforming furnace cavity is shaped like that in an electric toaster and has end-to-end standing wave natural frequencies just like the ground flare does. The steam methane reformer furnace’s performance also fit neatly into the Rayleigh Criterion picture painted above.

Remember that the burner flames can be tuned up by, for example, varying the flame speed and the jet velocity.  So how do you do that in a process furnace that operates at constant heat release? 

Well, the flame speed increases as you approach stoichiometric combustion; i.e., as you reduce the excess air.  And, at constant heat release, the jet velocity decreases as the specific gravity (heating value) increases.  Thus, reduced excess air and increased specific gravity lead to shorter burning delay times.  So there you have two purely operational means of tuning up the burner flames.

And tune them we did!  The frequency and intensity details of the combustion-driven oscillation are not particularly important for the purposes of this discussion.  Suffice it to say that at excess oxygen below about 2%, or at specific gravity above about 0.8, the reformer sounded like an idling diesel truck, a monstrous idling diesel truck, and exactly at the frequency of a standing half-wave in the long dimension of the furnace cavity.

Outside this oxygen-gravity envelope the reformer operated normally.  The operational solution cost the shareholders nothing.  The fix was to red-line the controllable operational parameters so as to avoid the sensitive regime.  A more “clever” (i.e., expensive) fix would have been needed if it were impossible to operate within a safe excess oxygen – specific gravity envelope.

With the foregoing as background we are now finally ready to turn to the discussion promised in the abstract.  Perhaps it can now mercifully be truncated, too!