Review of Surge Theory

Before attempting to deal with pressure surges, it would be appropriate to review some basic principles regarding the cause, magnitude and prevention of surge pressures. Pressure Transients or Surge, sometimes called waterhammer when a noise is associated with it, results when any attempt is made to alter the velocity of a column of water. It matters not if the column of water is at rest and is being put into motion, or if the water is in motion and is being brought to a stop or slowed down. The magnitude of the resulting pressure surge is directly proportional to the rate of change of velocity produced on the water column. It is not the total velocity change produced over a period of time which is the prime concern, but the rate of change. This factor will be elaborated on later in this writing. By applying Joukowsky¦#39;s equation for determining the pressure rise in a pipe,

hmax = a dV / g

In English units, “hmax” equals the headrise in feet of water, “dV” being the change in the water velocity in
feet per second, and “g” being gravity or 32.14. The “a” denoting the velocity at which the surge wave travels once created. The surge wave travels at the speed of sound in that particular pipe. The velocity “a” is not the same for every pipe. For example, in a typical small diameter Ductile Iron pipe “a” is nearly four times the speed of sound in air, or approximately 4000 fps. The surge wave velocity is a function of the pipe diameter, the pipe wall thickness, and the modulus of the pipe material. The other factor affecting the velocity of the surge wave is the water bulk modulus, which itself is affected by the water temperature and its pressure. Changing water pressure and temperature, however, have little effect on the surge wave velocity. Dissolved air however, can have a significant effect on the wave speed. It has been written that 1 part of air in 10000 parts of water by volume at standard conditions can reduce the wave speed by about 50 percent. The equation for determining the surge wave velocity is expressed by various equations. One such equation is shown here. (A pipe restraint factor sometimes mentioned being omitted.)

a = 4720 / sq.. rt. (1 + k d / E t)

K represents water bulk modulus which is nearly 300,000 psi. (NOTE: the modulus for steel is nearly 30,000,000 psi which means that water is 100 times more compressible than steel. All liquids have their
own bulk modulus, 230,000 PSI for crude oil, 130,000 psi for gasoline, etc. Illustrating that liquids are very
elastic substances.) “d” being the pipe diameter in inches. “t” is the pipe wall thickness in inches. “E” is
the pipe material modulus. A few examples are given.

(All values are in psi.)

Steel ............................... 30,000,000
Ductile Iron ................... 24,000,000
Cast Iron ........................ 15,000,000
Asbestos Cement ............. 3,400,000
PVC .................................... 400,000
HDPE ................................. 110,000

A graphical chart is included which can solve for surge wave velocities for any liquid or pipe material up to 96 inch dia. (See Figure 1) For example, a 48 inch Ductile Iron pipe with a 0.51 inch wall thickness would have a surge wave velocity of 3228 fps. The equation given earlier for hmax is derived from Newton’s force and impulse equations, or rate of change of momentum. This should not be confused with Newton’s F=MA expression which applies to solid or rigid bodies. Water is elastic, and there is no physical process known where a force can be applied to all particles of an elastic substance simultaneously. The equation assumes the velocity change was produced instantly. It is commonly known that nothing can be done instantly, however, it is not commonly understood that a flow need not be stopped instantly to obtain the same result as an instantaneous flow stoppage. It so happens in hydraulics, that any flow stoppage occurring within one surge period is equivalent to an instantaneous flow stoppage. A surge period must now be defined. To simplify this concept, visualize a long straight pipeline, for example, a 48 inch D.I. pipeline from a reservoir with a valve at its discharge end. Assume also that water is discharging from the pipe and valve at say ten fps neglecting line friction. The potential surge or headrise would be hmax = 3228 x 10 / 32.2 = 1002 feet. You may have noticed the headrise in feet is 100 times the flowing velocity in fps. This will be true for this pipe at any flowing velocity because a constant can be made for 3228 / 32.2 which is nearly 100. At 5 fps, the resulting potential headrise would be 500 feet. For each pipe diameter, wall thickness, and pipe material, a similar constant can be determined.

When the discharge valve flowing at 10 fps is closed suddenly, the flow at the valve inlet is suddenly stopped
with a pressure wave created of 1000 feet which begins its travel up the pipe at the speed of sound or 3228 fps.
The water at the valve’s inlet is of course stopped, but water at the reservoir end continues to enter the pipe because water there feels no effect of the discharge valve being closed, and it will not feel any effect until the surge wave arrives at the reservoir. The time required for the surge wave to arrive at the reservoir depends on the pipeline length. If we assume, for simplicity, the pipe is 32280 feet long, then the surge wave will require 32280 / 3228 or 10 seconds to reach the reservoir.

At this time it should be remembered that the surge resulting from closing the discharge valve is only the increase
in pipeline pressure not the total pressure. The static pipeline head must be added to the surge pressure
to obtain the total pressure. To re-emphasize, surge is independent of the existing pressure. An upsurge is the
increase in pressure, not the total pressure. Assuming the reservoir static head is 100 feet then, the total head in the pipe would be (Capital H) Hmax or 1000 plus 100 equaling 1100 feet. At time equals 10 seconds after valve closure, the surge wave arrives at the reservoir, the pipeline is now pressurized to 1100 feet of head. With a reservoir head of only 100 feet, the pipeline water in a compressed state (which is what pressure is), immediately begins to relieve its excess pressure back into the reservoir creating a reversing pressure wave traveling at back towards the closed valve at the same speed of sound. This returning surge wave will require 10 seconds to reach the closed valve. At that time the head in the pipe for an instant is equal to the static head, or 100 feet. The required time for the round trip of the initial upsurge is therefore, 20 seconds.

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