On the pump or valve supply side, the magnitude of the pressure wave is calculated as shown in this example using the following formula:
dp = ρ 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. After 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 can 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 cycle to some extent. The pressure wave passes through the system once at full strength. Reflections of the wave can 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 is 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 discretion of the piping designer.
CAESAR II does not check the integrity of the piping system due to the local increase in hoop stress that occurs as the fluid pressure wave passes each pipe cross-section. Slowing the mechanism that tends to reduce the flowrate can reduce the magnitude of the water hammer loads. In the case of valve closing, it means slowly closing the valve. In the case of a pump going off line, it means slowly removing power from the pump. Slowly in each of these instances can be estimated from:
T = 2L/c
T = Time of one wave cycle sec.
L = Characteristic length of piping system. This is usually the length between the pump or valve and the source or sink.
c = Speed of sound in the fluid.
If the pump or valve stops in a time shorter than T, then the water hammer should be analyzed as shown in this example for instantaneous closure. Calculations for this problem are given below.
Of primary interest is the largest time segment that must be used to close a valve or bring a pump flowrate to a halt such that water hammer type pressure pulses are not generated. Calculations using the lengths of several reflecting systems are made to determine the variation of the computed Ts. The longest time is for the wave to leave the supply side at node 5 and move to the tank connection at node 125. This represents a total L of about 270-feet.
T = (2) (270) ft./(4281)ft/sec = 126 milliseconds
The length through which the wave passes that causes the most trouble is the length between nodes 45 and 75:
T = (2) (90)/(4281) = 42 milliseconds
If the pump or valve can slow down in greater than 126 milliseconds, the tendency for water hammer in the piping system is usually abated. If the pump or valve can slow down in greater than 42 milliseconds then the tendency for water hammer in the 45-75 length is abated.
Water hammer excitation initially produces axial acoustic waves in the steel pipe wall that can induce locally very high, very short duration forces and stresses. These short duration loads are usually not a design problem in ductile steel piping systems. Where crack propagation in welds and material due to water hammer loads is a concern, use the following rules:
A very high number of natural frequencies must usually be included in the analysis. Cutoff frequencies of 300 Hz are not unusual. These are the axial natural modes of the pipe between the excited elbow-elbow pairs. Higher modes must be computed until the inclusion of extra modes does not produce an appreciable change in the force/stress response. The maximum frequency cutoff can be estimated using
E = Pipe material modulus of elasticity
p = Pipe material density
L = Length of a single pipe element in the primary run that is to have accurate stresses computed due to the passing of the water hammer originated acoustic stress wave.
Calculation of the maximum cutoff frequency for the 45-75 elbow-elbow pair for the 20-foot pipe lengths is given as follows:
fcutoff = SQRT (E/ρ)/L
= SQRT ((30E6)(32.2)(12)/(0.283))/20
= (202388 in./sec) / (20 ft. 12 in/ft)
= (843.3 rad./sec) / (2 p rad./cycles)
= 134.2 Hz
Alternatively, including the Missing Mass Correction approximates the contribution from the omitted modes.
The length of any element in the primary axial runs should not be greater than about ct/4, where c equals the speed of sound in the pipe and t equals the duration of the water hammer load. Calculation of the greatest element length for the 45-75 elbow-elbow pair is given as follows:
Lmax = ct/4
= (4281) ft/sec (0.021) sec/(4)
= 22.5 ft
To get an accurate estimate of the stresses due to the passing of the stress wave in the pipe, individual element lengths should be smaller than about 20 feet. Shorter duration loads require shorter elements to monitor the passing of the stress wave.
The inclusion of the response due to the higher modes does not affect the displacement results (only the force and stress results). Displacement results, such as the 6- to 8-inches in this example, can usually be computed accurately after the inclusion of the low frequency modes with participation factors greater than about 0.01.