--`,,```,,,,````-`-`,,`,,`,`,,`---3.1.3 Effects of PulsationPulsation, which is produced by a moving element such as the piston of a positive displacement pump or compressor, propagates
Overview of Pulsation Concepts
This section focuses on pressure variations caused by the oscillatory flow of positive displacement machinery, known as pulsation Pulsation affects systems that handle both gases and liquids, leading to issues such as high vibration, support degradation, and fatigue failures due to dynamic forces To mitigate the adverse effects of pulsation and vibration during the design phase, it is essential to grasp several key technical concepts.
Section 3.1.1 discusses excitation mechanisms, followed by an explanation of acoustic response in section 3.1.2 In section 3.1.3, common results of excessive pulsation are reviewed, along with an explanation of acoustic-mechanical coupling Finally, section 3.1.4 concludes the discussion by reviewing techniques for pulsation control.
In systems with positive displacement machinery, the flow of gas or liquid is characterized by a series of dynamic flow pulses rather than a steady stream, resulting in a time-varying movement through the piping that overlays the average flow.
The flow pulses in a reciprocating compressor cylinder are influenced by various physical, geometrical, and mechanical characteristics, including rotational speed, bore, stroke, loading, and compression ratio These flow pulses generate excitations that lead to pressure and flow modulations, or acoustic waves, which propagate through the process fluid in the piping system Typically, the primary pressure and flow modulations produced by a reciprocating compressor can be modeled as one-dimensional waves at specific frequencies.
A key aspect of acoustic analysis is creating a compressor model that effectively forecasts the dynamic flow excitation (flow over time) produced by the compressor Simplified examples illustrating this concept can be found in Figures 1, 2, 3, and 4.
The flow pulses caused by the reciprocating action of the compressor or pump create pressure pulses or waves that move through the piping system as shown in Figure 5
The flow pulse frequencies produced by the compressor depend on its mechanical properties, while the acoustical response in the piping is influenced by the compressor's mechanical characteristics, the thermophysical properties of the gas, and the acoustical network established by the connected piping.
When a specific harmonic of running speed aligns with an acoustical natural frequency, the dynamic pressure amplitude experiences amplification This phenomenon, known as resonance, occurs when the excitation frequency matches a natural frequency, resulting in either simple organ-pipe type resonances or more complex modes that involve the entire piping system.
In simple constant diameter pipes with open or closed boundary conditions, specific lengths can trigger acoustical natural frequencies When the pipe length aligns with integer multiples of half or quarter wavelengths, acoustical resonance occurs based on the end conditions For half wave resonances, both ends must share the same condition, either open-open or closed-closed, while quarter wave resonances require opposite conditions, such as one open end and one closed end These configurations are illustrated in Figures 6 and 7 and are governed by Equations (1) and (2).
Figure 1—Piston Motion and Velocity for a Slider Crank Mechanism
Figure 2—Single Acting Compressor Cylinder with Rod Length/Stroke = ∞ and No Valve Losses
Figure 3—Symmetrical, Double Acting Compressor Cylinder with Rod Length/Stroke = ∞ and No Valve
Figure 4—Unsymmetrical, Double Acting Compressor Cylinder with Rod Length/Stroke = 5 and No Valve
Figure 5—Traveling Wave in Infinite Length Pipe
Figure 6—Mode Shapes of Half Wave Responses where
O is the wavelength = a/f; a is the acoustic velocity; f is the frequency = 1/t;
T is the time period of one cycle; t is the time. t = 1/2( O/a) t = 0 t = ( O /a) t
Pipe closed at both ends Pipe open at both ends
Formula for half wave (closed-closed and open-open acoustic response frequency):
(1) where f is the acoustical natural frequency (subscript indicates order); n is 1, 2, 3,…; a is the speed of sound;
L is the length of pipe.
Formula for quarter wave (open-closed acoustic response frequency):
(2) where f is the acoustical natural frequency (subscript indicates order); n is 1, 3, 5,…(odd integers); a is the speed of sound;
L is the length of pipe.
Figure 7—Mode Shapes of Quarter Wave Responses
Pipe open at one end and closed at the other end f na
Pulsation generated by moving components like pistons in positive displacement pumps or compressors travels through various system elements, including valves, gas passages, and piping The issues arising from pulsation are influenced by its magnitude and frequency, leading to problems such as vibration, fatigue failures, performance degradation, driver overload, and inaccuracies in flow metering Each of these concerns will be explored in greater detail in the subsequent sections.
Pulsation alone does not cause vibration in piping systems; instead, dynamic forces arise from points of acoustical-mechanical coupling Common force-coupling points in compressor piping systems include geometric discontinuities like elbows, reducers, tees, and capped ends.
Elbows in piping systems create differential areas due to variations in outside and inside radii, leading to dynamic forces during pulsation To minimize these forces, it is advisable to reduce the number of elbows in the system Additionally, placing dynamic restraints, such as clamps, near elbows is beneficial, as this is where dynamic forces are most likely to occur.
Force coupling points arise in reducers, tees, and other piping areas with unbalanced forces, with capped ends being a clear example This situation is common at the ends of vessels, such as compressor pulsation suppression devices The forces exerted on the closed ends of these volume bottles can lead to significant vibrations in the bottles themselves.
Equipment, vessels, and pipe runs are generally regarded as rigid along their axis, leading to an effective shaking force that is the total of pulsation multiplied by the unbalanced area at each geometric discontinuity It is essential to take into account the pulsation magnitude, phase angle, and force direction at each unbalanced area, as demonstrated in Figures 8 through 14, which illustrate examples of dynamic shaking forces at points of acoustic-mechanical coupling.
P dyn is the dynamic pressure;
F Result is the resultant shaking force; φ is the elbow angle
D 1 is internal diameter of larger pipe;
D 2 is internal diameter of smaller pipe
Figure 8—Elbow with Dynamic Forces
Figure 9—Reducer with Dynamic Forces
P dyn is the dynamic pressure;
D is the internal diameter of pipe
F A is the shaking force in direction A;
F B is the shaking force in direction B;
P A is the dynamic pressure at elbow A;
P B is the dynamic pressure at elbow B;
F Total is the resultant shaking force; id is the internal diameter of the projected area.
Figure 10—Tee with Dynamic Forces
F max is the maximum shaking force;
P dyn is the dynamic pressure;
A head is the projected internal area of head;
A baffle is the projected area of baffle with choke tube;
Figure 11—Elbow with Dynamic Forces
Figure 12—Pulsation Suppression Device with Dynamic Forces
A choke is the area of choke tube bore;
P 1 is the dynamic pressure at first head;
P 2 is the dynamic pressure at first head-side of baffle;
P 3 is the dynamic pressure at second head-side of baffle;
P 4 is the dynamic pressure at second head
P 1 is the dynamic pressure at pipe tee;
P 2 is the dynamic pressure at pipe reducer;
P 3 is the dynamic pressure at elbow;
A tee is the area of pipe tee bore;
A reducer is the differential area of pipe reducer;
A ell is the bore area at elbow
Figure 13—Shaking Force for Sample Pulsation Damper
High shaking forces from pulsation can lead to significant vibration in piping systems However, even low dynamic forces can cause excessive vibration if the excitation frequency is near or matches the mechanical natural frequency In such instances, vibration can be amplified, often by a factor of 5.
The amplitude of a system at resonance is constrained by its damping characteristics This article will further explore the concepts of resonance and the amplification of vibrations that occur at this condition in section 3.2.
3.1.3.3 Fatigue Failure, Piping, and Support Degradation
Fatigue failures of main piping small bore attachments and piping support degradation are common problems associated with vibration caused by pulsation induced forces.
To discuss compressor performance and overload, it is helpful to understand how required compression power is determined.
Overview of Mechanical Concepts
The mechanical characteristics of a compressor or pump system, along with its associated piping system, encompass the natural frequency and forced vibration response In accordance with Design Approach 3 of API 618, a mechanical analysis is conducted to assess these characteristics, ensuring that vibration and stress levels throughout the system remain within acceptable limits.
Understanding the evaluation of beam characteristics necessitates a grasp of fundamental beam theory and vibration concepts, such as resonance, amplification factor, separation margin, and stress Additionally, it is crucial to recognize the differences between ideal theoretical conditions and the non-ideal actual conditions typically encountered in compressor, pump, and piping systems.
The vibratory behavior of piping spans is governed by the differential equations of motion for uniform beams, along with specific boundary conditions that facilitate the calculation of their natural frequencies Section 3.2.7.2.1 elaborates on the application of fundamental beam theory in the design of piping systems.
From simple beam theory, the basic relation can be summarized as:
(21) where f is the natural frequency (Hz); λ is the frequency factor;
L is the length of span (m);
E is the pipe material modulus of elasticity (N/m 2 ); w is the pipe mass per unit length (kg/m);
I is the moment of inertia (m 4 ).
Frequency factors are obtained from theoretical end conditions for straight spans or through finite element calculations for spans with bends The use of this expression for idealized pipe spans and bends is detailed in section 3.2.7.2.1.
Resonance occurs when the frequency of a periodic forcing function aligns with a system's natural frequency, leading to amplified vibration amplitudes In a resonant state, forced vibrations from mechanisms like pulsation-induced shaking are significantly enhanced compared to static deflection, influenced by the system's damping level Key indicators of resonance include a marked increase in vibration amplitude and a phase angle shift between the exciting force and the system's vibrational response.
To prevent vibration issues in compressor, pump, and piping systems, it is crucial to avoid resonance These systems possess multiple mechanical natural frequencies and typically exhibit low damping Effective design strategies include minimizing the magnitude of harmonic forcing functions and modifying the piping support system or layout to alter the mechanical natural frequencies of the piping.
As previously explained, resonance is typically identified by a substantial vibration amplitude increase and a shift in phase angle The amplification factor indicates how substantial that vibration amplitude increase will be.
Analyzing single-degree-of-freedom response curves is essential for understanding resonance effects This involves examining a basic spring-mass-damper system to calculate the amplification factor based on the frequency ratio.
The amplification factor (AF) quantifies the relationship between static deflection and dynamic deflection, indicating how much the static deflection must be multiplied to obtain the vibration amplitude For a single-degree-of-freedom system, the equation for calculating the amplification factor is provided below.
The dynamic deflection (\$x\$) represents the vibration, while the static deflection (\$x_o\$) is calculated as \$x_o = \frac{F}{k}\$, where \$F\$ is the applied force and \$k\$ is the effective stiffness The ratio of the forcing frequency to the natural frequency is denoted as \$\frac{\omega}{\omega_n}\$, with \$\omega_n\$ representing the natural frequency Additionally, \$m\$ signifies the mass, and \$\zeta\$ indicates the damping ratio, which is the percentage of critical damping.
At resonance, where the frequency ratio equals one, the amplification factor is determined by the damping ratio, expressed as \$\frac{1}{2\zeta}\$ For instance, a system with 1% critical damping achieves an amplification factor of 50, while a system with 2% critical damping has an amplification factor of 25, and one with 5% critical damping results in an amplification factor of 10 Typically, piping and compressor systems exhibit damping ratios ranging from 1% to 5%, leading to common amplification factors between 10 and 50 A plot illustrating the relationship between amplification factor and frequency ratio for damping ratios from 2.5% to 50% is presented in Figure 30.
The amplification factor starts at 1 and rises at a rate influenced by the damping ratio as it nears resonance At the point of resonance, the amplification factor reaches a value of \$\frac{1}{2\zeta}\$.
As frequency surpasses resonance, the amplification factor drops below 1 and approaches zero as frequency increases, a phenomenon known as isolation However, the application of isolation in piping and compressors, which are multi-degree of freedom systems, is inherently limited Further discussion on multi-degree of freedom systems will follow.
Significant amplification of vibrations occurs when the forcing frequency is within 10% above or below the mechanical natural frequency Figures 30 and 31 illustrate that the effective amplification factor at these frequencies is 5:1 for critical damping ratios below 5% Consequently, the excitation frequency does not have to precisely match the mechanical natural frequency to cause substantial increases in vibration levels.
Separation margin is defined as margin between the harmonic forcing function and the natural frequency of a compressor, pump or piping system element.
Figure 31 illustrates that the amplification factor at resonance is influenced by damping, whereas the amplification factor for responses that are 10% or more away from resonance is solely determined by the distance from resonance To eliminate damping as a source of uncertainty, a minimum separation margin of 10% is necessary.
Shifting from resonance to 10% off resonance can decrease vibration by a factor of five to ten, depending on the level of damping Further, moving from 10% to 20% off resonance results in an additional reduction of vibration by a factor of two However, the transition from 20% to 30% off resonance only leads to a modest reduction of 1.5 times This indicates that the most significant decrease in vibration occurs near resonance, with the majority of the reduction happening within the first 10% shift away from it.
Field experience generally shows that a 10 % shift of the natural frequency away from resonance results in acceptable vibration when the primary cause of the high vibration is resonance.
Considering the above theoretical and practical observations, a 10 % actual separation margin is recommended Further, allowing for 10 % uncertainty in natural frequency predictions, a 20 % design separation margin is recommended.