PROBLEM:
The cooling water supply line shown as follows suffers a pressure surge when the turbine driven pump drops offline due to a bearing temperature problem. The elbow at node 45 is observed to “jump” 6 to 8 in. in the “X” direction when the turbine trip occurs.
Design an alternative support scheme to eliminate the large field displacements associated with the turbine trip.
Fluid Properties: 250 psi @ 140°F Flow Velocity: 6 fps
Water Bulk Modulus: 313000 psi
SOLUTION:
The magnitude of the pump supply side pressure wave, which emanates from the pump discharge at node 5, can be estimated from
dp = c dv Where:
- the fluid density
c - the speed of sound in the fluid dv - the change in velocity of the fluid The speed of sound in the fluid can be estimated from
c = [Ef/ ( + (Ef/ E) (d/t) )] 0.5
Where:
Ef - is the bulk modulus of the fluid (313000 psi)
E - is the modulus of elasticity of the pipe (30E6 psi) d - is the pipe mean diameter
t - is the pipe wall thickness - is the fluid density (62.4 lbm/ft3)
+ (Ef/ E)(d/t) = 62.4 lbm/ft3 [1 + (313000/30E6) (8.625 -0.322)/0.322 ] = 79.1875 lbm/ft3
c = (313000 lbf/ in2) (ft3/79.1875 lbm) (32.2 lbm ft/lbf sec2) (144in2/ft2)1/2 = 4281 ft/sec
Note See the PIPING HANDBOOK, Crocker & King, Fifth Edition, McGraw-Hill pages 3-189 through 3-191 for a more detailed discussion and evaluation of the speed of sound.
Apply the equation above for the magnitude of the water hammer pressure wave. dp = c dv = (62.4 lbm/ft3) (4281 ft/sec) (6.0 ft/sec)
= (62.4 lbm/ ft3) (4281 ft/sec) (6.0 ft/sec) (lbf sec2/32.2 lbm ft) ( ft2/144 in2)
= 345.6 psi
There are two distinct pressure pulses generated when a flowing fluid is brought to a stop. One pulse originates at the supply side of the pump, and the other pulse originates at the discharge side of the pump. This example only deals with the supply side water hammer effect, but the magnitude and impact of the discharge side water hammer load should likewise be investigated when in a design mode.
Ps - Source (tank or static) pressure
Pos - Suction pressure (while running)
dp - Pressure fluctuation due to the instantaneous stoppage of flow through the pump Pv - Liquid vapor pressure at flow temperature
There will be an unbalanced load on the piping system due to the time it takes the pressure wave to pass successive elbow- elbow pairs. The magnitude of this unbalanced load can be computed from:
F unbalanced = dp * Area
The duration of the load is found from t = L/c, where L is the length of pipe between adjacent elbow-elbow pairs. For this problem the elbow-elbow pairs most likely to cause the large deflections at node 45 are 45-75 and 90-110.
The rise time for the unbalanced dynamic loading should be obtained from the pump manufacturer or from testing and can be determined from graphs such as those shown above. For this problem a rise time of 5 milliseconds is assumed.
CALCULATIONS: L 45-75 = 7 + 4(20) + 4 = 90 ft. L 90-110 = 3(20) + 15 = 75 ft. Area = /4 di2; di = 8.625 - (2) (0.322) = 7.981 in. Area = /4 (7.981)2= 50.0 in2 F unbalanced = dp * Area = (345.6) (50.0) = 17289 lbf t duration = L/c
= (90) / (4281) = 21 milliseconds, on leg from 45 to 75 = (75) / (4281) = 17.5 milliseconds, on leg from 90 to 110
t rise = 5.0 milliseconds
Because the piping in this example is ductile low carbon steel, the major design variable will be the large displacement; i.e. the problem will be assumed to be solved when the restraint system is redesigned to limit the large displacements due to water hammer without causing any subsequent thermal problem due to over-restraint.
Next we define the Spectrum:
Then we define the force sets as follows:
The sustained static load case is now combined with each dynamic load case for code stress checks. Note that for operating restraint loads the static operating case would be combined with each dynamic load case as well. That is left for the user to investigate.
On the pump or valve supply side the magnitude of the pressure wave is calculated as shown in this example using: dp = c dv.
On the pump or valve discharge side the maximum magnitude of the pressure wave is the difference between the fluid vapor pressure and the line pressure.
On the supply side a positive pressure wave moves away from the pump at the speed of sound in the fluid. The magnitude of the pressure wave is equal to the sum of the suction side pressure and dp.
On the discharge side a negative pressure wave moves away from the pump at the speed of sound in the fluid. The maximum magnitude of this negative pressure wave is the difference between the pump discharge pressure and the fluid vapor pressure. Once the pump shuts down, the pressure at the discharge begins to drop. The momentum of the fluid in the downstream piping draws the discharge pressure down. If the fluid reaches its vapor pressure the fluid adjacent to the pump flashes. As the negative pressure wave moves away from the pump these vapor bubbles collapse instantly. This local vapor implosion can cause extremely high pressure pulses. In addition, there may be a fluid backflow created due to the rapid drop in pressure. In this case the backflow slap at the idle pump can be accentuated by the collapse of created vapor bubbles, resulting in an extremely large downstream water hammer loading.
Water hammer loadings will cycle to some extent. The pressure wave passes through the system once at full strength. Reflections of the wave may then cause secondary pressure transients. Without a transient fluid simulation or field data the usual procedure is to assume one or two significant passes of the pressure wave.
Where critical piping is concerned or where the maximum loads on snubbers and restraints is to be computed, the independent effect of a single pass of the pressure wave should be analyzed for each elbow-elbow pair in the model. A separate force spectrum load set is defined for the elbow with the highest pressure as the wave passes between the elbow- elbow pair. The direction of the applied force is away from the elbow-elbow pair. An individual dynamic load case is run for each separate force set, combinations of different force sets are usually not run. This approach has proved
satisfactory when applied to large, hot steam piping systems that have very few fixed restraints, and a high number of low modes of vibration. Extrapolation to other types of piping systems should be made at the designer's discretion.
CAESAR II does not check the integrity of the piping system due to the local increase in hoop stress that occurs as the fluid