PREDICTING UNBALANCED FORCES IN LIQUID LINES CAN AVERT DAMAGE

R. R. Burnett
Flour Daniel Williams Bros.
Tulsa

Predicting the unbalanced forces caused by pressure surges in petroleum liquid pipelines can protect line pipe and equipment. These forces occur when vapor pockets in liquid pipelines collapse.

Presented here is an explanation of these forces and a method for predicting their magnitudes. With this knowledge, designs and operating procedures can be developed to mitigate damage caused by these forces.

PRESSURE-SURGE FRONT

In cross-country liquid pipelines, internal pressures are lower at high points where the pipeline traverses hills or mountain passes. When pressures fluctuate because of operational changes at pump stations, the internal pressures at these high points may fall below the vapor pressure of the liquid, causing liquid to boil and form vapor pockets in the pipeline.

When pipeline pressures are once again raised above vapor pressure, the liquid rushes into the vapor pocket. The pocket collapses, flow stops, and an instantaneous pressure rise occurs.

This starts a pressure-surge front moving along the pipeline with the speed of sound in the liquid, 2,000 mph or more. Because the pressure rise at the wave front is instantaneous, the low pressure before passage of the wave front is followed closely by high pressure trailing the front.

These two different pressures, closely coupled, when situated in a straight pipe segment with changes in direction at each end, cause a large imbalance in the thrust forces acting on the ends of the straight segment. This imbalance tends to move the pipe segment in the direction of the larger force.

Another cause of large, instantaneous unbalanced forces results from a rupture during hydrostatic test. In that situation, the rupture releases the high internal pressure.

Then a rarefaction-wave front travels with the speed of sound in the system, with high pressure preceding and low pressure following the wave front. Here too, unbalanced forces are closely coupled and can displace a straight pipeline segment.

The profile of the traveling surge wave caused by such an instantaneous change in velocity and pressure is fundamentally different from the usual pressure wave caused when a pump is stopped or a control valve position is changed.

In the latter situation, the profile is much longer and consequently the unbalanced force is applied over several miles of the pipeline. As a result, normal anchoring of the pipeline by the surrounding soil can resist any movement of the pipeline.

Commercial software used in the pipeline industry for computing transient pressures fail to take into account the unbalanced forces created by collapse of vapor pockets. And only instantaneous pressure and velocity changes are computed in the more sophisticated software which recognizes the condition of vapor in a pipeline.

This article concludes that the formation of vapor pockets should be avoided insofar as possible; damage caused by the collapse of vapor can be minimized through careful pipeline operating practices; and the potential for damage should be acknowledged in the design of pipeline facilities.

PRESSURE CHANGES

A hypothetical pipeline elevation profile and steady flow hydraulic gradient are shown in Fig. 1 (35703 bytes).

The pressures along the pipeline are proportional to the difference in height of the hydraulic gradient above the elevation profile at each point along the pipeline. The lowest pressures are mainly at the highest elevations where the hydraulic gradient is near the elevation profile.

The hydraulic gradient is sloped downward in the direction of flow, and the flow rate can be increased by an increase in the slope. For this hypothetical pipeline, a pump must be imagined at the left end of the pipeline to raise the pressure to the level shown.

Also, at the right end, where the liquid is delivered, the pressure is being controlled by some control system.

A change in the pressure at either end can change the flow rate. For example, lowering the controlled pressure at the right end would increase the slope and thus increase the flow rate.

In Fig. 2 (38054 bytes), the controlled pressure at the right end has been raised rapidly and a traveling pressure-surge wave is moving from right to left. The pressure-surge wave raises the pressure along the pipeline above the original level of the steady flow gradient.

Each small change in pressure causes a surge wave to start traveling along the pipeline with the speed of sound, 2,000-3,000 mph, or roughly 3,000-4,000 fps in a liquid pipeline.

If the total change of pressure is thought of as a series of small changes taking place in sequence, with the resulting waves all traveling at the same speed, then the accumulated wave front is stretched out over a distance proportional to the time interval over which the original pressure changes took place.

The distance over which the pressure-surge wave extends is called the "pressure-wave front." The faster the change, the shorter the pressure-wave front.

With reference again to Fig. 2 (38054 bytes): as the change at the right end took place over several seconds, the pressure-wave front is extended over several miles.

The front is traveling at the speed of sound in the liquid. For example, if a pump is shut down and its rundown time is 20 sec and if the speed of sound is 3,200 fps, then the wave front generated by the pump shut down is 20 x 3,200 or 64,000 ft in length, more than 12 miles.

Over these long distances, the soil backfill provides adequate longitudinal anchoring for a buried pipeline. If there are aboveground sections, the gradual application of force imbalance between the ends of straight segments is readily restrained by friction on the supports.

SLACK LINE EFFECT

In Fig. 3 (36354 bytes), the controlled pressure at the right end has been lowered so that the pressures at certain high elevation points have dropped below vapor pressure and boiling is occurring. The pipeline is said to be in a "slack" condition with vapor pockets located at these high points.

Fig. 4 (33551 bytes) represents a typical high point with "column separation," a ten-n to describe the condition in which low pressure causes the liquid to boil and create pockets of vapor between columns of liquid in a pipeline.

In Fig. 5 (38478 bytes), the pressure has been raised, and the vapor pocket formed at the downstream high point has been collapsed. The figure shows one example of a wave front after it started traveling along the pipeline.

In this wave front, the pressure rise is instantaneous as a result of the sudden collapse of a vapor pocket.

Liquid column rejoining occurs when a vapor pocket is collapsed as the pressure rises above vapor pressure. The liquid velocity at that time is changed instantaneously.

The pressure rise also occurs instantaneously, therefore, its magnitude being proportional to the change in velocity. The axial length of the pressure-wave front is practically zero as it starts traveling at the speed of sound along the pipeline in both directions.

This means that the distance from the low pressure in front of the traveling wave and the high pressure in the wave's wake is nearly zero and the unbalanced force which accompanies the wave is exerted on very short straight segments of pipe and acts almost like a hammer blow to each pipeline segment as the wave front passes.

As this unbalanced force travels through the pipeline, the hammer-blow effect is applied to any straight segments of the system having bends at each end. The more vulnerable segments are in any locations in which there is flexible aboveground piping. This is discussed further presently.

The unbalanced force is exerted at any change in direction made by the piping. For example, the aboveground flexible piping shown in Fig. 6 (40729 bytes) has an elbow at each end of a straight segment of pipe parallel to the main buried pipeline.

At the instant the wave front enters this straight segment of piping, the lower pressure is exerted at one elbow with a force equal to the pressure multiplied by the pipe cross-sectional area while the other elbow is subjected to the force of the higher pressure.

The difference between these two forces acts along the straight segment tending to move the pipe toward the elbow with the high pressure. The force continues as long as the wave front remains in the straight segment.

The time over which the force acts on the segment equals the segment length divided by the wave travel speed. Therefore, if the length is 100 ft and the speed is 3,200 fps, the duration of the force in the segment is 100/3,200, or X2 of a second (0.0313 sec).

In buried portions of the pipeline, the unbalanced force is exerted along all straight segments between direction changes. The force at these changes tends to move the pipeline along its axis toward the bend or direction change and can cause a small amount of damage to the coating.

This type of damage can accumulate over years of operation and hasten the need for coating repairs and an increase in the cathodic-protection loads.

If there are rocks lying against the pipe, some gouges in the pipe may occur and degrade the strength of the pipe in addition to causing coating damage.

Where underground pipelines pass through congested areas, turning numerous angles along the route, the unbalanced force delivers an impact load every time a direction change is made.

The instantaneous wave-front pressure rise is diminished by the system friction and elasticity as it travels along the line, a phenomenon analyzed elsewhere.,

The difference between the usual pressure wave and the surge caused by the collapse of a vapor pocket must be made clear. When a vapor pocket collapses, there is an almost instantaneous stoppage of the velocity of the liquid column as the vapor condenses into liquid.

This surge differs from the more familiar pressure surges caused by pump station shutdowns or an accidental closure of a main line valve. In either situation, the surge pressure rise takes place over a period of several seconds and the unbalanced force is extended over several miles.

For aboveground piping resting on supports, the effect can be more serious. Transient lateral and longitudinal loads on the supports may easily exceed the bases for design unless the unbalanced forces described here are taken into consideration.

Aboveground cross-country pipelines would be subject to damage with supports being especially vulnerable. These designs are characterized by relatively short segments having direction changes at each end. Certain of these flexible designs are standard in areas such as Alaska, Iran, or other parts of the Middle East and South America. On the other hand, an axially restrained aboveground pipeline, such as the Trans-Arabian Pipeline, would suffer little or no damage in the aboveground portions.

The axial force anchors which prevent expansion due to temperature rise also prevent movement due to the unbalanced forces described here.

EXAMPLES

The following are actual situations in which the collapse of vapor pockets or instantaneous pressure changes have occurred with startling results.

Pump starting, stopping. On a large diameter pipeline (200 miles long), an intermediate pump station was being added and the new pumps were being tested for correct rotation.
The elevation profile was generally flat except for a rise about 40 miles downstream from the new station and 10 miles from the next existing pump station.
First one new pump was started, then stopped after rotation was verified. Immediately afterward, the next pump was started. Within approximately 1 min, the downstream pump station started shaking violently.
Although pumps at the existing station did not damage their foundations, their alignments were disturbed, requiring realignment of each PUMP.
Damage to the piping was confined to several instrument lines connected to the main piping. Except for the pumps, all piping and valve supports were free to slide on concrete foundations.
Analysis led to the conclusion that stoppage of the first new pump had lowered the pipeline pressure to vapor pressure at the rise in elevation, 10 miles from the next pump station, and created a large vapor pocket in that region.
Starting the second pump had then raised the pressure and caused the vapor pocket to collapse and start a pressure surge and unbalanced force to travel to the next pump station.
"Slack Line." A large diameter pipeline traversed a pass through a low mountain range, through a pressure reduction station, then to a storage terminal which it entered through a final pressure-reduction station.
Distance from the last PUMP station to the summit was 55 miles, then 10 miles farther to the first pressure-reduction station, then another 60 miles to the terminal.
Piping at the first pressure- reduction station was mounted on steel frame supports bolted to concrete foundations. The supports were designed essentially to support the weight of the piping.
Before flow in the pipeline could be shut down, the incoming pressures at both pressure-reduction stations had to be raised to prevent pressures at the intermediate high point from dropping to vapor pressure and forming vapor pockets.
Any unscheduled stoppage of a pump at the last pump station would cause vapor pockets to form at the summit unless the incoming pressures at the pressure-reduction stations were first raised. These precautions may not have been taken in every case of a shutdown. One or more of these undesirable events appeared to occur from time to time.
In any case, the structural supports at the first pressure-reduction station suffered severe damage several times. The logical conclusion was that the collapse of vapor pockets in combination with light-weight inflexible supports resulted in failures.
Valve closure. A valve manifold is located on a hill with storage tanks from which marine tankers are loaded by gravity through 2 miles of 30-in. pipeline.

Preliminary to the loading of a ship, a valve at a control station at the seawater's edge was opened, allowing liquid to drain from the valve manifold into the tanker and filling the 30-in. pipe with vapor for about 1,000 lineal ft starting from the uphill manifold. Then a manifold valve was opened to start loading.

The liquid from a storage tank rushed into the vapor space and collapsed the pocket. This caused the manifold containing fifty-seven 24-in. gate valves to move violently back and forth across its concrete slab support.

The control station down at the edge of the seawater also jumped suddenly. Pressure gauges with 500 psig ranges had their pointers bent, indicating an unknown excess pressure.

Fortunately, both the manifold and the control station were flexible enough to allow for the violent movement. Otherwise, supports would have been severely damaged. There was no damage to the main piping.

Hydrotest rupture. A flexible aboveground cross-country liquid pipeline was provided with expansion loops at intervals of approximately 2,000 ft.

The pipe and expansion loops were 24-in. OD, and the entire configuration was provided with saddles resting on sliding supports. Several other similar pipelines were mounted on the same supports.

Being in a cold climate, all of the pipelines were insulated then covered with an outer metal sheathing.

The pipeline was being hydraulically tested with water, and pressure had reached approximately 1,000 psig when a rupture occurred in an expansion loop.

The result was considerable damage to supports over a distance of approximately 2 miles. The main carrier pipe was not seriously damaged but jumped off its supports in several places. The insulation and sheathing required extensive repairs due to general damage.

In this case, the rupture caused a nearly instantaneous drop in the internal pressure so that a rarefaction wave started traveling along the pipeline. The pipeline configuration consisted mainly of straight segments with 90 elbows at each end.

The expansion loops also were made up of three straight segments, each about 40-50 ft long and with 90 elbows.

The unbalanced force traveling along with the rarefaction wave front possessed the same capability for damage as from the pressure wave due to the collapse of a vapor pocket.

CALCULATIONS

A severe condition can be created if the liquid is allowed to drain downhill from the summit for a short distance during a flow stoppage. The vapor begins at the summit and goes as far as the pipeline is drained.

When the flow is restarted from the last pump station, the velocity of the liquid running downhill can easily exceed the velocity of the pumping rate. The downhill velocity accelerates to a maximum or "terminal" value which depends on the angle of the downward slope.

The downhill velocity can be calculated with reasonable accuracy using Newton's law that force equals mass multiplied by acceleration: F = ma.

In this case, the force of gravity on the slope equals the weight multiplied by the sine of the slope angle. This downslope force is resisted by liquid friction in the pipeline which is simplified to a constant multiplied by the square of velocity (Equation 1 (33183 bytes) in the accompanying equations box).

The pressure drop (P) may be found with the form of the Darcy flow formula found in Equation 2 (33183 bytes).

The frictional resistance acts on the slug with a force of the magnitude shown in Equation 3a (33183 bytes).

Equation 1 (33183 bytes), therefore, contains the slug weight (W in each term; Equation 3b (33183 bytes)). This simplifies to Equation 3c (33183 bytes).

This equation is integrated to obtain the solution for the velocity vs. time, giving Equation 4 (33183 bytes).

The maximum or "terminal" velocity (Vm) is approached asymptotically as the hyperbolic tangent approaches 1.0; therefore, velocity approaches the value yielded by Equation 5 (33183 bytes).

Let "M" be the coefficient of time (t; Equation 4 (33183 bytes)) in the argument of the hyperbolic tangent (Equation 4a (33183 bytes)).

Equation 4 (33183 bytes) is integrated to obtain the solution for distance vs. time, the result being Equation 6 (33183 bytes). The only information required for the solution is the pipe ID, the fluid friction factor, and the slope.

As an example, assume a 24-in. pipeline with 23.25-in. ID and a fluid friction factor of f = 0.012 and consider three slopes: 5, 10, and 20.

With Equations 4a (33183 bytes)and 6 (33183 bytes) used to obtain velocities and distances vs. time, Fig. 7 (122601 bytes) shows the gain in velocity of a slug along with its distance from the starting point for 5, 10, and 20 downhill slopes, respectively.

For example, if the slope is 5 downhill, Fig. 7a (122601 bytes) shows that in 30 sec, the velocity reaches nearly its maximum value of 30 fps while traveling a distance of 345 ft.

The time and travel distance for the slug velocities to reach 99% of their maximum values are summarized Table 1 (10017 bytes). This illustrates in general that the slug-traveling velocities are high, and that they increase quickly within short distances.

The velocity is stopped when the liquid columns rejoin, and the energy in the moving stream, when it collides with the stationary liquid column, is converted to wave-front pressure by the relation, from Equation 2 (33183 bytes), shown in Equation 7 (33183 bytes). And the initial wave-front force equals the pressure multiplied by the cross-sectional area.

Continuing with the example with a cross-sectional area of 424.6 sq in. and under the assumption that the speed of sound is 3,200 fps and the density is 53 lb/cu ft, the initial wave-front pressures from Equation 7 (33183 bytes) and unbalanced forces corresponding to the terminal velocities are shown in Table 2 (12072 bytes).

With these large forces, support designs which are rigid, allowing for no movement, can fail. On the other hand, conventional station designs, which allow for movement restrained by the main piping and the frictional resistance of sliding supports, have generally proven to be adequate to avoid damage.

The presence of equipment such as pumps, tanks, or control systems which may have to resist these large forces should be considered in designing the piping and anchoring systems.

WHAT'S BEEN LEARNED

1. Complete avoidance of the release of vapor or slack line operation is desirable to minimize potential damage even though numerous pipelines operate in a slack condition at least part of the time.

2. Since the existence of vapor in a pipeline may be unavoidable, however, the defense against severe jolts due to column rejoining is to change pressures slowly. Prudent pipeline operators have learned that slow changes in pressure and flow give best results over the long term.

3. Finally, designs should provide for the worst case of unbalanced forces based on terminal velocities.

Allowing for some flexibility in aboveground piping supports to handle certain axial movements has been generally successful in avoiding damage.