A piping system can respond far differently to a dynamic load than it would to a static load of the same magnitude. Static loads are those which are applied slowly enough that the system has time to react and internally distribute the loads, thus remaining in equilibrium. In equilibrium, all forces and moments are resolved (that is, the sum of the forces and moments are zero) and the pipe does not move.
A dynamic load changes quickly with time. The piping system does not have time to internally distribute the loads. Forces and moments are not always resolved, resulting in unbalanced loads and pipe movement. Because the sum of forces and moments are not in equilibrium, the internally-induced loads can be different—either higher or lower—than the applied loads.
The software provides several methods for analyzing different types of system response under dynamic loads. Each method provides a trade-off of accuracy versus computing requirements. The methods include modal natural frequency calculations, harmonic analysis, response spectrum analysis, and time history analysis.
- Modal natural frequency analysis measures the tendency of a piping system to respond to dynamic loads. The modal natural frequencies of a system typically should not be too close to equipment operating frequencies. Generally, higher natural frequencies usually cause less trouble than low natural frequencies. CAESAR II provides calculation of modal natural frequencies and animated plots of the associated mode shapes.
- Harmonic analysis addresses dynamic loads that are cyclic in nature, such as fluid pulsation in reciprocating pump lines or vibration due to rotating equipment. These loads are modeled as concentrated forces or displacements at one or more points in the system. To provide the proper phase relationship between multiple loads, a phase angle can also be used. Any number of forcing frequencies can be analyzed for equipment start-up and operating modes. Harmonic responses represent the maximum dynamic amplitude the piping system undergoes and have the same form as a static analysis: node deflections and rotations, local forces and moments, restraint loads, and stresses. For example, if the results show an X displacement of 5.8 cm at a node, then the dynamic motion due to the cyclic excitation is from +5.8 cm. to -5.8 cm. at that node. The stresses shown are one half of, or one amplitude of, the full cyclic stress range.
- Response spectrum analysis allows an impulse-type transient event to be characterized by response versus frequency spectra. Each mode of vibration of the piping system is related to one response on the spectrum. These modal responses are summed together to produce the total system response. The stresses for these analyses, summed with the sustained stresses, are compared to the occasional stress allowables defined by the piping code. Spectral analysis can be used in a wide variety of applications. For example, in uniform inertial loading, ground motion associated with a seismic event is supplied as displacement, velocity, or acceleration response spectra. The assumption is that all supports move with the defined ground motion and the piping system “catches up” to the supports. It is this inertial effect which loads the system. The shock spectra, which define the ground motion, can vary between the three global directions and can even change for different groups of supports (such as independent or uniform support motion). Another example is based on single point loading. CAESAR II uses this technique to analyze a wide variety of impulse-type transient loads. Relief valve loads, water hammer loads, slug flow loads, and rapid valve closure type loads all cause single impulse dynamic loads at various points in the piping system. The response to these dynamic forces can be predicted using the force spectrum method.
- Time history analysis is one of the most accurate methods, because it uses numeric integration of the dynamic equation of motion to simulate the system response throughout the load duration. This method can solve any type of dynamic loading, but due to its exact solution, requires more resources (such as computer memory, calculation speed and time) than other methods. Time history analysis is not appropriate when, for example, the spectrum method offers sufficient accuracy.
Force versus time profiles for piping are usually one of three types: Random, Harmonic, or Impulse. Each profile has a preferred solution method. These profiles and the load types identified with them are described below.
- Random
The major types of loads with random time profiles are wind and earthquake.
- Wind
Wind velocity causes forces due to the decrease of wind momentum as the air strikes the pipe creating an equivalent pressure on the pipe. Wind loadings, even though they can have predominant directions and average velocities over a given time, are subject to gusting, such as sudden changes in direction and velocity. As the time period lengthens, the number of wind changes also increases in an unpredictable manner, eventually encompassing nearly all directions and a wide range of velocities.
- Earthquake
Seismic (earthquake) loadings are caused by the introduction of random ground motion, such as accelerations, velocities, and displacements and corresponding inertia loads (the mass of the system times the acceleration) into a structure through the structure-to-ground anchorage. Random ground motion is the sum of an infinite number of individual harmonic (cyclic) ground motions. Two earthquakes can be similar in terms of predominant direction (for example, along a fault), predominant harmonic frequencies (if some underlying cyclic motions tend to dominate), and maximum ground motion, but their exact behavior at any given time can be quite different and unpredictable.
2. Harmonic
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.
3. Impulse
- Relief Valve
When system pressure reaches a dangerous level, relief valves are set to open in order to vent fluid and reduce the internal pressure. Venting through the valve causes a jet force to act on the piping system. This force ramps up from zero to its full value over the opening time of the valve. The relief valve remains open (and the jet force remains relatively constant) until enough fluid is vented to relieve the over-pressure condition. The valve then closes, ramping down the jet force over the closing time of the valve.
- Fluid Hammer
When the flow of fluid through a system is suddenly halted through valve closure or a pump trip, the fluid in the remainder of the system cannot be stopped instantaneously. As fluid continues to flow into the area of stoppage (upstream of the valve or pump), the fluid compresses causing a high-pressure situation. On the other side of the restriction, the fluid moves away from the stoppage point, creating a low pressure (vacuum) situation. Fluid at the next elbow or closure along the pipeline is still at the original operating pressure, resulting in an unbalanced pressure force acting on the valve seat or the elbow.
The fluid continues to flow, compressing (or decompressing) fluid further away from the point of flow stoppage, causing the leading edge of the pressure pulse to move through the line. As the pulse moves past the first elbow, the pressure is now equalized at each end of the pipe run, leading to a balanced (that is, zero) pressure load on the first pipe leg. The unbalanced pressure, by passing the elbow, has now shifted to the second leg. The unbalanced pressure load continues to rise and fall in sequential legs as the pressure pulse travels back to the source, or forward to the sink.
The ramp up time of the profile roughly coincides with the elapsed time from full flow to low flow, such as the closing time of the valve or trip time of the pump. Because the leading edge of the pressure pulse is not expected to change as the pulse travels through the system, the ramp-down time is the same. The duration of the load from initiation through the beginning of the down ramp is equal to the time required for the pressure pulse to travel the length of the pipe leg.
- Slug Flow
Most piping systems are designed to handle single-phase fluids (that is, fluids that are uniformly liquid or gas). Under certain circumstances, a fluid may have multiple phases and is susceptible to slug flow. For example, liquid slugs may be entrained in a wet steam line. These slugs of liquid create an out-of-balance load when the slugs change direction in bends or tees.
In general, fluid changes direction in a piping system through the application of forces at elbows. This force is equal to the change in momentum with respect to time, or
Fr = dp / dt = Dr v2 A [2(1 - cos q)]1/2
Where:
dp = change in momentum
dt = change in time
Dr = liquid density - vapor density
v = fluid velocity
A = internal area of pipe
q = inclusion angle at elbow
With constant fluid density, this force is normally constant and is small enough that it can be easily absorbed through tension in the pipe wall. The force is then passed on to adjacent elbows with equal and opposite loads, zeroing the net load on the system. Therefore, these types of momentum loads are usually ignored in analysis. If the fluid velocity or density changes with time, this momentum load will also change with time, leading to a dynamic load which may not be canceled by the load at other elbows.
For example, consider a slug of liquid in a gas system. The steady state momentum load is insignificant because the fluid density of a gas is effectively zero. The liquid suddenly slug hits the elbow, increasing the momentum load by orders of magnitude. This load lasts only as long as it takes for the slug to traverse the elbow, and then suddenly drops to near zero again with the exact profile of the slug load depending upon the shape of the slug. The time duration of the load depends upon the length of the slug divided by the velocity of the fluid.
Where:
F1 = rv2 A(1 - cos q)
Fr = rv2 A [2(1 - cos q)]½
F2 = rv2 A sin q
Dynamic analysis is crucial to ensure the piping system can withstand these forces without failure. It involves studying the system’s response over time, considering factors like acceleration, frequency, and unbalanced forces.
