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CAESAR II Users Guide

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CAESAR II Version
13

With this type of profile, the load changes direction and/or magnitude following a harmonic profile, ranging from its minimum to its maximum over a fixed time period. For example, the load can be described by a function of the form:

F(t) = A + B cos(w t + f)

Where:

F(t) = force magnitude as a function of time

A = mean force

B = variation of maximum and minimum force from mean

w = angular frequency (radian/sec)

f = phase angle (radians)

t = time (sec)

Loads with harmonic force/time profiles are best solved using a harmonic method. The major types of loads with harmonic time profiles are equipment vibration, acoustic vibration, and pulsation.

Equipment Vibration

If rotating equipment attached to a pipe is slightly out-of-tolerance (for example, when a drive shaft is out-of-round), it can impose a small cyclic displacement onto the pipe at the point of attachment. This is the location where the displacement cycle most likely corresponds to the operating cycle of the equipment. The displacement at the pipe connection can be imperceptibly small but could cause significant dynamic-loading problems. Loading versus time is easily predicted after the operating cycle and variation from tolerance is known.

Acoustic Vibration

If fluid flow characteristics are changed within a pipe (for example, when flow conditions change from laminar to turbulent as the fluid passes through an orifice), slight lateral vibrations may be set up within the pipe. These vibrations often fit harmonic patterns, with predominant frequencies somewhat predictable based upon the flow conditions. For example, Strouhal’s equation predicts that the developed frequency (Hz) of vibration caused by flow through an orifice will be somewhere between 0.2 V/D and 0.3 V/D, where V is the fluid velocity (ft./sec) and D is the diameter of the orifice (ft). Wind flow around a pipe sets up lateral displacements as well (a phenomenon known as vortex shedding), with an exciting frequency of approximately 0.18 V/D, where V is the wind velocity and D is the outer diameter of the pipe.

Pulsation

During the operation of a reciprocating pump or a compressor, the fluid is compressed by pistons driven by a rotating shaft. This causes a cyclic change over time in the fluid pressure at any specified location in the system. Unequal fluid pressures at opposing elbow pairs or closures create an unbalanced pressure load in the system. Because the pressure balance changes with the cycle of the compressor, the unbalanced force also changes. The frequency of the force cycle is likely to be some multiple of that of the equipment operating cycle, because multiple pistons cause a corresponding number of force variations during each shaft rotation. The pressure variations continue to move along through the fluid. In a steady state flow condition, unbalanced forces may be present simultaneously at any number of elbow pairs in the system. Load magnitudes can vary. Load cycles may or may not be in phase with each other, depending upon the pulse velocity, the distance of each elbow pair from the compressor, and the length of the piping legs between the elbow pairs.

For example, if the pressure at elbow a is Pa(t) and the pressure at elbow b is Pb(t), then the unbalanced force acting along the pipe between the two elbows is:

F(t) = (Pa(t) - Pb(t)) A

Where:

A = internal area of the pipe

Assuming the pressure peak hits the elbow "a" at time t = 0, Pa(t) is:

Pa(t) = Pavg + 0.5 (dP) cos w t

Where:

Pavg = average pressure in the line

dP = alternating component of the pressure

w = driving angular frequency of pulse

If the length of the pipe between the elbows is L, then the pressure pulse reaches elbow bts after it has passed elbow a:

ts = L / c

Where:

c = speed of sound in the fluid

Therefore, the expression for the pressure at elbow b is:

Pb(t) = Pavg + 0.5(dP) cos (w t - Q)

Where:

Q

= phase shift between the pressure peaks at a and b

= w ts

Combining these equations, the unbalanced pressure force acting on an elbow pair is:

F(t) = 0.5(dP)A * [ cos w t - cos w (t - L/c) ]

Under steady-state conditions, a similar situation exists at all elbow pairs throughout the piping system.