ABSTRACT

 

In this paper we will discuss the most likely cause of most (but not all) occurrences of flare pulsation.  The periodic surges of flame that occasionally burst forth from elevated flares into the quietude of the nighttime sky are often caused by seal drum sloshing.  We will explain how seal drum sloshing arises, how to tell if seal drum sloshing is the problem and what to do about it if it is. We will show that the diagnosis, cause and elimination of flare pulsation due to seal drum sloshing is most easily understood in terms of The Fluid Mechanics of Beer Steins.  The exact analytical solution will be recounted briefly and several verifying experiments, including one you can do in your coffee cup or in a large mason jar or in a bathtub with a laboratory partner of your choice, will be suggested.  Finally several real world examples that prove the theory will be discussed.

 

INTRODUCTION

 

In petroleum refineries, petrochemical and chemical plants, elevated flares are the ultimate safety device.  They are used to safely dispose of excess process gas that is generated from time to time by minor and occasionally major plant upsets.  A complex engineered flare tip is mounted on a tall stack that is often provided with a water seal at its base.  The water seal’s function is to prevent the entry of air and the development of an explosive mixture in the main plant piping.

 

This paper discusses flare pulsation due to seal drum sloshing.  The problem is one of fluid mechanics but its manifestation is the disturbing light and sound of combustion pulsation in an elevated flare.  Sloshing of the water level in a seal drum can lead to periodic surges in the gas supplied to the combustion tip resulting in a periodic combustion “whump.”  While the sound pulsations easily protrude more than 10 decibels above the steady combustion noise levels, it is the intermittent nature of the disturbance that annoys people.  For typical seal drums, the period of the fundamental sloshing motion is of the order of one second and pulsations noticed by remote observers occur at periods that can be explained precisely by fluid mechanical sloshing within the seal drum.

 

WHAT HAPPENS

 

Before turning to the analytical solution that provides an exact prediction of the liquid sloshing period in a right circular cylindrical domain such as a flare seal drum, we will discuss the types of liquid motion with which we are concerned and illustrate why they produce flare pulsation.

 

Consider the sloshing that so easily arises when you walk down the hall with your coffee cup.  This is “1st mode” sloshing, up on one side of the cup and down on the other; then, a half-period later, down on the formerly up side and up on the other.  This is the mode with which the authors are most familiar and which we have seen the world over.  It is important to understand that this fundamental (longest period, lowest frequency) sloshing motion that so easily arises is not symmetrical.

 

(Visualization from Van Dyke, M., An Album of Fluid Motion, Parabolic Press, Stanford, 1982, Plate 191.)

 

As is so easily proven in your coffee cup, the fundamental sloshing mode is to-and-fro. The streamlines illustrated above are not circularly symmetric.  There is a nodal diameter along which the equilibrium liquid level remains unchanged.  Experimental observations reveal that the 1st mode sloshing nodal diameter precesses, probably in different directions depending upon the hemisphere in which     the observations are carried out, but we are     only familiar with northern hemisphere experiments.  In any event, the 1st mode’s important distinction of antisymmetry leads us to consider two other types of liquid sloshing in a right circular cylindrical domain.

 

The authors are not entirely sure that they have ever actually observed flare pulsation due to sloshing modes other than the fundamental antisymmetric sloshing mode illustrated above.  Nevertheless we believe that there are at least two other types of sloshing motions that must be accounted for in any proposed fix of flare pulsation due fundamental (1st mode) seal drum sloshing.

 

The reason is that, done wrong, the so-called “fix” might get rid of 1^st mode sloshing and its attendant flare pulsation but cause one of two symmetrical modes to arise to produce flare pulsation at somewhat shorter but also predictable periods.  That would be like, as Mark Twain said, “… catching the cure and dying.”

 

Below is a nice visualization of symmetric 2nd mode sloshing in a cylindrical tank.  It can be thought of as the “first harmonic” of the antisymmetric fundamental sloshing mode that arises so easily in your coffee cup.

 

But try to produce this one in your coffee cup!  You will only produce a mess for your trouble which illustrates that, unless the other modes are aided as we will discuss presently, the antisymmetric fundamental sloshing mode is generally the culprit in flare pulsation due to seal drum sloshing.

 

The 2nd mode sloshing visualized in the next illustration is circularly symmetric.  Not to-and-fro like the 1st mode but up in the center, down at the wall; and then a half-period later, down at the center, up at the wall.  There is a node but it is a nodal circle as shown.  If you imagine rotating the visualization around its central axis you will have a good picture of the streamlines and how the liquid would move if only you could get that mode going in your coffee cup.  Good luck and get some paper towels ready! 

 

(Visualization from Van Dyke, M., An Album of Fluid Motion, Parabolic Press, Stanford, 1982, Plate 191.)

 

Before discussing the one other sloshing motion, the “liquid pendulum” mode, which we believe has to be accounted for in fixing flare pulsation due to seal drum sloshing, given the fluid mechanics background thus far presented perhaps we should explain how the flare pulsations arise.

 

Consider the flare seal drum sketched below.  It has an imaginary handle to remind us that what we are really discussing is The Fluid Mechanics of Beer Steins.

Suppose that the plant pressure has depressed the water level in the dip leg just down to the bottom of the dip leg.  The next increment of plant pressure would produce a gas release.  Suppose further that a sudden process burp is sufficient to start first mode sloshing as shown.

 

That is not much of a supposition.  It has been demonstrated in ¼-scale plastic models that this sloshing motion quite naturally arises even in a steady release of gas or, in the case of the plastic models, air.  Virtually inevitable just like in your coffee cup, it is like a self-starting “Look Ma’ … no moving parts!” washing machine.

 

With the liquid level depressed inside the dip leg just to the bottom of the dip leg, lowering the hydrostatic head on one side of the dip leg produces a gas surge to the flare tip as shown above.  A half-period later another surge is produced on the opposite side of the dip leg.  Thus, for the fundamental 1st mode antisymmetric sloshing motion, the flare pulsation period is half that of the sloshing period; i.e., there are two flare surges during each complete sloshing cycle.  The result is shown below.

In the case of 2nd mode symmetrical sloshing, should it arise, the depression of the liquid level outside the dip leg would be uniform all around the dip leg.  Thus, in the case of 2nd mode sloshing, should it arise, the flare pulsation and the sloshing period would be the same.

Finally, not quite as an afterthought, we need to introduce one further sloshing motion.  The reason is that, while as far as the authors are aware they have never observed it, we believe that this mode could arise if aided, for example, by an improper antislosh baffle design that suppresses 1st mode antisymmetric sloshing but perhaps favors or even guides symmetric sloshing.  That mode is the “liquid pendulum” motion illustrated above. We do not have a visualization of this mode.

 

EXACT ANALYTICAL SOLUTION

 

Assuming the sloshing motion is inviscid, incompressible and irrotational, the continuity equation must be satisfied throughout the liquid contained within any right circular cylindrical domain such as the one pictured below.  Expressed in terms of the velocity potential Φ, the continuity equation is the Laplace equation

Ñ²Φ = 0.

In polar cylindrical coordinates, the natural coordinates for a right circular cylinder, the Laplace equation has the form

Φ_rr + (1/r)Φ_r + (1/r²)Φ_θθ + Φ_zz = 0.

Partial differentiation of the velocity potential in any coordinate direction gives (minus) the velocity in that direction.  That is the definition of the velocity potential; thus,

u = - Φ_r , v = - (1/r)Φ_θ , w = - Φ_z.

Imposing the boundary condition of no flow through the walls,

u = - Φ_r = 0 at r = a

w = - Φ_z = 0 at z = - h

it remains to impose a boundary condition on the free surface.

 

Assuming the motion of the liquid is irrotational, the Bernoulli equation applies throughout the liquid and particularly on the free surface; thus,

p/ρ + v²/2 + gs = Φ_t

where p is the pressure in the liquid at the surface and v is the velocity at the surface.    Interestingly perhaps, this is the only point at which dynamics enters the solution.

 

In a linearized analysis, we assume that the liquid surface remains flat and the motion is small; then p is constant and v² can be neglected.  Differentiating the linearized free surface condition partially with respect to time and introducing the kinematic condition that particles in the free surface are constrained to move in the free surface,

s_t = w = - Φz ,

we arrive at the free surface boundary condition

Φ_tt = gs_t = - gΦ_z at z = 0 .

We assume that the time and space parts of the solution can be represented as a product

Φ(r,θ,z,t) = Ψ(r,θ,z)sinωt.

Substituting, we arrive at the formulation of the eigenvalue problem for small sloshing motions in a right circular cylindrical domain such as that of a beer stein.

Ψ_rr + (1/r)Ψ_r + (1/r²)Ψ_θθ + Ψ_zz = 0

Ψ_r = 0 at r = a; Ψ_z = 0 at z = - h;

ω²Ψ = gΨ_z at z = 0.

Only for certain values of ω² (“eigenvalues”) does this problem have any solution at all.  Those ω are the circular frequencies (2π¦) of the natural sloshing modes (“eigenmodes”) of the liquid surface.

 

The solution for the spatial part of the velocity potential Ψ, arrived at by the method of separation of variables, is

Ψ = J_m(αr)cos(mθ)cosh[α(z+h)]

where J_m is the Bessel function of the first kind and order m.  The α are determined by

J_m′(αa) = 0

which comes from the no flow condition at     r = a.  The lowest zero corresponding to the lowest natural frequency occurs for m = 1 at   αa = 1.8412.  The sloshing frequency is obtained by substituting the solution for the spatial part of the velocity potential, Ψ, into the free surface boundary condition

ω² = g(Ψ_z/Ψ)|_{z = 0} = gαtanh(αh)

= g(1.8412/a)tanh(1.8412h/a).

 

Finally, we can express the exact solution for the 1st mode sloshing period as

                                                             T = 2π/ω =

MODIFICATION FOR ANNULAR TANKS

 

The presence of the central dip leg in a flare seal drum allows the liquid surface to be discontinuous at the center.  This in turn permits Bessel functions of the second kind to enter the solution of the eigenvalue problem for the sloshing period.  As a result, the diameter ratio now enters the formula for the sloshing period.

In this case the no flow boundary condition at the wall produces

J′_m(β)N′_m(βD_i/D_o) - J′_m(βD_i/D_o)N′_m(β) = 0

in which J′  and N′ are the first derivatives of the Bessel functions of the first order and first and second kind, respectively; D_i is the inner diameter of the tank and D_o its outer diameter.

 

The modified formula together with a chart for the now diameter-ratio dependent values of β is shown below.  We see that when the inner diameter (D_i ) approaches zero, the solution for the 1st mode sloshing period approaches that for a simple cylindrical tank (β = 1.8412), just as it should.

For more on this the reader should consult Horace Lamb.  He is dead but he wrote a nice book.*  In the interim hydrodynamics has not changed much as far as the authors are aware.

 

SIMPLE EXPERIMENTAL PROOFS

 

Inserting into the formula above typical values for a common coffee cup, D_o = 2¾" and h = 2¼", we get T = 0.290 seconds, or about 17 sloshes in 5 seconds.  Pass the coffee.  Nudge your cup and count.  You will get about 17 sloshes in 5 seconds.  Ain’t science wunaful?  For a little bigger experiment, something your kid might want to do for a science fair, try a large mason jar.

Alternatively, if you are of an analytical bent, carry out the foregoing analysis in rectangular coordinates.  This is the “Professor’s Trick,” of course.  Solve the easy one and leave the hard one for the students.  The solution in a rectangular domain is not more complex but it is more complicated.

 

If you just want the answer, the formula for the period of the fundamental sloshing wave in a rectangular domain such as, for example, a bath tub, is exactly the same as for coffee cups, beer steins and flare seal drums except that 1.8412 is replaced by π/2 and D_o is the length of the tub.  You might like to check that out with a laboratory partner of your choice.

 

Ever see a Bessel function?  Solutions for waves in rectangular enclosures like bath tubs come out in terms of sines and cosines.  In a coffee cup, beer stein, mason jar or flare seal drum, waves come out in terms of Bessel functions of which there are two kinds, as well.  Appropriately if not interestingly, Bessel functions are also called “Cylinder” functions and that is why.

 

Bessel functions of the first kind are the only ones that are continuous at the origin, so they are the only ones that work in a beer stein.  Unless, of course, you drink your beer with a straw.  In that case the wave form could be discontinuous, Bessel functions of the second kind would be admissible and that realization is what brought us to the foregoing modification of the theory for annular tanks like a flare seal drum and dip leg.

 

The first two Bessel functions of the first kind look like this:

If the shape of the first part of J₁ looks familiar, it should; multiplied by cosθ, it is the shape of the sloshing wave in a beer stein.  If you want to see a J₀ , watch as you set your coffee cup down sharply.  Its rings will appear in all their glory, gleaming in the reflected light on the surface!

 

REAL WORLD PROOFS

 

In a 1972 episode illustrated below, flame pulsations were timed and counted while the flare was operating.  The seal drum was 8-ft in diameter with a central dip leg of 3-ft diameter and a water depth of 5-ft.  From the formula given above, the 1st mode sloshing period would be expected to be about 1.91 seconds resulting in a calculated flare pulsation frequency of 1.05 per second.  The observed pulsation frequency was 1.1 per second.

In the more recent episode in 2000 illustrated above, the calculated pulsation frequency of 0.78 per second was again bang-on with the observed 0.75-0.80 per second flare pulsation.  

 

SUPPRESSION

 

Now let us think about how we might prevent or interfere with these perfectly natural sloshing motions.  Below we have superimposed on the sloshing visualizations an indication of the equilibrium water lever, a dip leg and a perforated cylindrical antislosh baffle.

 

Perforated baffles have long been used to suppress sloshing in aircraft wing tanks and spacecraft propellant tanks.  Should the sloshing motion arise, the perforated baffles cause the sloshing wave to spend its energy in viscous dissipation in consequence of which the sloshing motion simply does not arise.