1. Overview
Pulsation analysis predicts pressure oscillations generated by reciprocating compressors and their interaction with piping system acoustics. When a compressor excitation frequency coincides with a piping acoustic natural frequency, resonance occurs, causing dangerously amplified pressure pulsations that lead to piping fatigue failure, valve damage, and equipment malfunction.
Excitation
Compressor Harmonics
Multiples of running speed create pulsation
Acoustic Response
Piping Natural Freq
Standing waves in pipe segments
Resonance
Amplification
10-50x pressure magnification possible
Mitigation
Dampeners & Orifices
Bottles, choke tubes, acoustic filters
Consequences of Inadequate Pulsation Control
| Failure Type | Mechanism | Typical Time to Failure | Repair Cost |
|---|---|---|---|
| Small-bore connection fatigue | High-cycle vibration at branch | Weeks to months | $10K-50K |
| Main piping fatigue | Acoustic resonance stress | Months to years | $50K-500K |
| Valve plate failure | Flutter from reflected waves | Hours to weeks | $5K-50K per event |
| Orifice meter error | Square-root error amplification | Continuous | Revenue loss |
| Foundation damage | Transmitted pulsation forces | Years | $100K-1M |
Industry experience: Over 80% of reciprocating compressor piping failures are caused by pulsation-induced vibration. Proper analysis before construction is far less expensive than retrofit solutions.
2. Acoustic Theory
Sound propagation in piping follows the one-dimensional wave equation. The speed of sound, pipe geometry, and boundary conditions determine the acoustic natural frequencies.
Speed of Sound in Gas
Speed of sound:
c = sqrt(k * Z * R_u * T / MW) (ft/s)
Where:
k = Cp/Cv (specific heat ratio)
Z = Gas compressibility factor
R_u = 1545.35 ft-lbf/(lbmol-R)
T = Absolute temperature (R = F + 459.67)
MW = Molecular weight (lb/lbmol)
Practical form:
c = 68.1 * sqrt(k * Z * T / (MW * gamma_g))
Where gamma_g = specific gravity (air = 1.0)
Pipe wall compliance correction:
c_pipe = c / sqrt(1 + (D * c^2 * rho) / (E * t))
Where:
D = Pipe internal diameter
E = Pipe elastic modulus
t = Pipe wall thickness
rho = Gas density
Typical reduction: 1-5% for standard steel pipe
Acoustic Natural Frequencies
Boundary conditions determine mode shapes:
Open-Open (pipe between two large volumes):
f_n = n * c / (2 * L) (n = 1, 2, 3, ...)
Pressure nodes at both ends, antinode at center
Open-Closed (pipe with one closed end):
f_n = (2n - 1) * c / (4 * L) (n = 1, 2, 3, ...)
Only odd harmonics; pressure antinode at closed end
Closed-Closed (pipe with both ends closed):
f_n = n * c / (2 * L) (n = 1, 2, 3, ...)
Pressure antinodes at both ends
Where:
L = Effective pipe length (ft)
c = Speed of sound (ft/s)
n = Mode number (integer)
Effective length corrections:
End correction for flanged opening: +0.425 * D
End correction for unflanged opening: +0.3 * D
Tee branch: model as open end with correction
Acoustic Impedance
Characteristic acoustic impedance of a pipe:
Z_0 = rho * c / A
Where:
rho = Gas density (slug/ft^3)
c = Speed of sound (ft/s)
A = Pipe cross-sectional area (ft^2)
Impedance change at area discontinuity:
Reflection coefficient: R = (Z_2 - Z_1) / (Z_2 + Z_1)
Transmission coefficient: T = 2*Z_2 / (Z_2 + Z_1)
At pipe expansion (A_2 > A_1): Partial reflection
At pipe contraction (A_2 < A_1): Partial reflection
At closed end (Z_2 = infinity): Full reflection, R = 1
At open end (Z_2 = 0): Full reflection, R = -1
Pulsation bottle effectiveness:
Area ratio A_bottle/A_pipe > 4:1 for significant attenuation
Ideal: > 9:1 for 90%+ reflection of incoming wave
3. Excitation Spectrum
The compressor generates pulsation at discrete frequencies (harmonics of running speed). The amplitude and frequency content depend on the number of cylinders, crank arrangement, and whether single or double-acting.
Harmonic Content by Configuration
| Configuration | Dominant Orders | 1x Amplitude | 2x Amplitude | 3x Amplitude |
|---|---|---|---|---|
| 1-cyl, single-acting | 1, 2, 3, 4, ... | 100% | 50% | 33% |
| 1-cyl, double-acting | 1, 2, 3, 4, ... | 15%* | 100% | 15% |
| 2-cyl, 180 deg, DA | 2, 4, 6, ... | 0% | 100% | 0% |
| 3-cyl, 120 deg, DA | 3, 6, 9, ... | 0% | 0% | 100% |
| 4-cyl, 90 deg, DA | 4, 8, 12, ... | 0% | 0% | 0% |
*1x content in DA cylinders due to rod area difference between HE and CE.
Pulsation Amplitude Estimation
Approximate cylinder-flange pulsation (undampened):
P_puls / P_avg = (V_swept / V_total) * F_config
Where:
P_puls = Peak-to-peak pulsation pressure
P_avg = Average line pressure
V_swept = Cylinder swept volume
V_total = Total connected volume (cylinder + clearance + bottle)
F_config = Configuration factor
F_config values:
1-cyl SA: 0.65
1-cyl DA: 0.35 (reduced by rod area partial cancellation)
2-cyl 180 DA: 0.25
3-cyl 120 DA: 0.15
Pulsation-induced shaking force:
F_shaking = P_puls * A_pipe
Where A_pipe = pipe cross-sectional area
These forces excite piping mechanical natural frequencies
and must be evaluated against allowable stress.
4. API 618 Approach Levels
API 618 defines three levels of pulsation and vibration analysis with increasing rigor. The selected approach depends on compressor criticality, size, and complexity.
Approach 1: Pulsation Suppression Devices
Scope: Design pulsation bottles using analog methods
or simplified acoustic guidelines.
Requirements:
- Volume-ratio sizing for suction and discharge bottles
- Choke tube sizing for acoustic filtering
- Nozzle location to avoid acoustic resonance
- Basic acoustic length checks on critical piping
- No digital simulation required
Deliverables:
- Bottle dimensions (diameter, length, internals)
- Choke tube specifications
- Nozzle orientations
- Orifice plate requirements (if any)
Applicability:
- Small compressors (< 500 HP)
- Simple piping configurations
- Non-critical service with spare unit
Approach 2: Digital Acoustic Simulation
Scope: Approach 1 plus 1-D acoustic simulation of
the entire piping system.
Requirements:
- Time-domain or frequency-domain acoustic model
- Model includes: cylinders, bottles, all piping, branches,
valves, orifices, vessels, and boundary conditions
- Predict pressure pulsation at all points in the system
- Evaluate against API 618 allowable pulsation levels
- Iterate bottle/orifice design until criteria are met
Additional deliverables:
- Predicted pulsation spectrum at key locations
- Acoustic natural frequency map
- Resonance identification and mitigation report
- Orifice plate sizing and pressure drop summary
Allowable pulsation (API 618):
At cylinder flange: 7% P-P of line pressure
At line connection to piping: 2% P-P of line pressure
Approach 3: Mechanical Response Analysis
Scope: Approach 2 plus forced mechanical response of
the piping system.
Additional requirements:
- FEA model of piping system (mechanical natural frequencies)
- Shaking forces from acoustic analysis applied to FEA model
- Predict vibration displacement, velocity, and stress at all points
- Evaluate against allowable stress (ASME B31.3/B31.8 fatigue)
- Evaluate small-bore connections per EI Guidelines
Additional deliverables:
- Piping mechanical natural frequency list
- Forced response displacement and stress plots
- Small-bore connection assessment
- Support/clamp recommendations
- Spring hanger and snubber specifications
Allowable vibration velocity (typical):
Main piping: 0.5-1.0 in/s peak
Small-bore (< 2"): 0.25-0.5 in/s peak
Near resonance: derated by amplification factor
Approach Selection Guide
| Factor | Approach 1 | Approach 2 | Approach 3 |
|---|---|---|---|
| Power | < 500 HP | 500-5,000 HP | > 5,000 HP |
| Service | Non-critical, spared | Process, pipeline | Critical, unspared |
| Speed | < 600 RPM | 600-1,200 RPM | > 1,200 or variable |
| Piping complexity | Simple (short runs) | Moderate | Complex (long runs, headers) |
| Study cost | $5K-15K | $15K-50K | $50K-150K |
| Study duration | 1-2 weeks | 3-6 weeks | 6-12 weeks |
5. Resonance Avoidance
The primary goal of pulsation analysis is ensuring no acoustic natural frequency of any pipe segment coincides with any compressor excitation frequency. Several strategies exist for avoiding resonance.
Frequency Separation Criteria
Required separation between excitation and acoustic natural:
|f_excitation - f_acoustic| / f_acoustic > 0.20 (20%)
For variable-speed compressors:
Check all harmonics (1x-10x) across the entire speed range.
Construct a Campbell diagram (interference diagram):
- Y-axis: Frequency (Hz)
- X-axis: Compressor speed (RPM)
- Plot lines for each harmonic order (1x, 2x, 3x, ...)
- Plot horizontal lines for each acoustic natural frequency
- Intersection points indicate potential resonance
Separation criteria:
No intersection within the normal operating speed range
If intersection unavoidable, verify amplitude is acceptable
Mitigation Strategies
| Strategy | Mechanism | Effect on Frequency | Cost |
|---|---|---|---|
| Pulsation bottle (volume) | Acoustic impedance mismatch | Changes boundary condition | Moderate |
| Choke tube | Acoustic resistance/filtering | Creates low-pass filter | Low |
| Orifice plate | Acoustic resistance | Damps resonance amplitude | Very low |
| Pipe length change | Shifts acoustic natural frequency | f = c/(2L); change L to avoid | Variable |
| Accumulator (side branch) | Helmholtz resonator | Targeted frequency absorption | Moderate |
| Speed avoidance | Operational restriction | Avoid specific RPM ranges | Zero (but limits flexibility) |
Small-Bore Connection Protection
Small-bore connections (< 2" NPS) are highly vulnerable to
fatigue failure from pulsation-induced vibration.
Energy Institute (EI) Guidelines assessment:
Likelihood of Failure (LOF) scoring based on:
- Connection type (thermowell, gauge, vent, drain)
- Main pipe diameter and NPS
- Proximity to pulsation source
- Bracing/support adequacy
Mitigation for high-LOF connections:
1. Brace to main pipe (welded gusset)
2. Use heavy-wall fittings (Sch 160 or XXH)
3. Add bracing clamps within 2 diameters of branch
4. Eliminate unnecessary connections
5. Replace threaded with socket-welded connections
Field verification: After commissioning, measure actual pulsation levels at key locations with dynamic pressure transducers and compare to predicted values. If measured pulsation exceeds predictions by more than 25%, investigate root cause and consider retrofit solutions.
6. Worked Examples
Example 1: Acoustic Natural Frequency Check
Given:
Compressor: 2-cylinder, double-acting, 720 RPM
Suction pipe: 8" NPS, 50 ft between bottle and header
Gas: Natural gas, c = 1,350 ft/s at suction conditions
Step 1: Excitation frequencies
Fundamental: f_1 = 2 * 2 * 720/60 = 48 Hz (4 events/rev)
Harmonics: 96, 144, 192, 240, 288, 336, 384, 432, 480 Hz
Step 2: Pipe acoustic natural frequencies (open-open)
f_n = n * 1,350 / (2 * 50)
f_1 = 13.5 Hz, f_2 = 27.0, f_3 = 40.5, f_4 = 54.0,
f_5 = 67.5, f_6 = 81.0, f_7 = 94.5, f_8 = 108.0, ...
Step 3: Check for coincidence
Excitation 48 Hz vs acoustic 54 Hz: separation = 6/54 = 11% -- TOO CLOSE
Excitation 96 Hz vs acoustic 94.5 Hz: separation = 1.5/94.5 = 1.6% -- RESONANCE
Step 4: Mitigation
Option A: Change pipe length to 45 ft
f_n = n * 1,350 / 90 = 15.0, 30.0, 45.0, 60.0, 75.0, 90.0, 105.0
48 Hz vs 45 Hz: sep = 3/45 = 6.7% -- still marginal
96 Hz vs 90 Hz: sep = 6/90 = 6.7% -- still marginal
Option B: Change pipe length to 40 ft
f_n = 16.9, 33.8, 50.6, 67.5, 84.4, 101.3
48 Hz vs 50.6: sep = 2.6/50.6 = 5.1% -- marginal
96 Hz vs 84.4: sep = 11.6/84.4 = 13.8% -- borderline
96 Hz vs 101.3: sep = 5.3/101.3 = 5.2% -- marginal
Option C: Add orifice plate at 25 ft point to damp
7th acoustic mode and add choke tube in bottle.
This is why digital simulation (Approach 2) is needed.
Example 2: Campbell Diagram for Variable Speed
Given:
Variable-speed compressor: 600-1,000 RPM operating range
1-cylinder, double-acting
Discharge piping acoustic naturals: 35, 70, 105, 140 Hz
Excitation lines (harmonic order vs speed):
1x: f = RPM/60 = 10.0 to 16.7 Hz
2x: f = 2*RPM/60 = 20.0 to 33.3 Hz
3x: f = 30.0 to 50.0 Hz
4x: f = 40.0 to 66.7 Hz
5x: f = 50.0 to 83.3 Hz
6x: f = 60.0 to 100.0 Hz
Intersections (resonance crossings):
2x = 35 Hz at RPM = 35*60/2 = 1,050 -- outside range, OK
3x = 35 Hz at RPM = 35*60/3 = 700 -- IN RANGE, check amplitude
4x = 35 Hz at RPM = 35*60/4 = 525 -- outside range
4x = 70 Hz at RPM = 70*60/4 = 1,050 -- outside, barely
5x = 35 Hz at RPM = 35*60/5 = 420 -- outside
5x = 70 Hz at RPM = 70*60/5 = 840 -- IN RANGE
6x = 35 Hz at RPM = 350 -- outside
6x = 70 Hz at RPM = 700 -- IN RANGE
6x = 105 Hz at RPM = 1,050 -- outside
Critical speeds: 700, 840 RPM require amplitude verification.
If amplitudes are acceptable, no restriction needed.
If not, either restrict speed range or add damping elements.
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