1. Power Formulas
Compressor power can be calculated using theoretical thermodynamic formulas or empirical quick-estimation methods. Both have their place in engineering practice.
Adiabatic (Isentropic) Power
Adiabatic Horsepower:
HP = (k/(k-1)) × (P₁ × Q₁ / 229) × [(P₂/P₁)^((k-1)/k) - 1]
Where:
k = Specific heat ratio (Cp/Cv)
P₁ = Suction pressure (psia)
P₂ = Discharge pressure (psia)
Q₁ = Suction volume flow (acfm)
229 = Conversion constant
Polytropic Power
Polytropic Horsepower:
HP = (n/(n-1)) × (P₁ × Q₁ / 229) × [(P₂/P₁)^((n-1)/n) - 1]
Where:
n = Polytropic exponent (typically 1.0 < n < k)
Other variables as above
Note: n accounts for heat transfer during compression
Brake Horsepower
Brake HP:
BHP = HP_theoretical / (η_volumetric × η_adiabatic × η_mechanical)
Typical efficiencies:
η_volumetric = 0.70 - 0.95 (depends on clearance, ratio)
η_adiabatic = 0.80 - 0.90 (compressor process)
η_mechanical = 0.90 - 0.95 (bearings, seals)
Quick Estimation (HP per MMscfd)
For natural gas compression, a quick estimate can be made using empirical factors:
HP per MMscfd Quick Estimate:
HP/MMscfd ≈ 21 × r^(1/n) × n × 1.154
Where:
r = Compression ratio
n = Number of stages
21 = Empirical constant for natural gas
1.154 = Real gas correction
| Compression Ratio | Stages | HP/MMscfd |
|---|---|---|
| 2:1 | 1 | 25-35 |
| 3:1 | 1 | 40-55 |
| 4:1 | 1-2 | 55-75 |
| 6:1 | 2 | 75-100 |
| 10:1 | 2-3 | 100-150 |
2. Rod Load Calculations
Rod load is the critical mechanical limit for reciprocating compressors. Both gas load and inertia load must be calculated and combined to verify frame limits.
Gas Load - Double Acting Cylinder
Gas Load Formulas:
Head End Area: A_HE = π × D² / 4
Crank End Area: A_CE = A_HE - A_rod = π × (D² - d²) / 4
Compression Load = P_d × A_HE - P_s × A_CE
Tension Load = P_d × A_CE - P_s × A_HE
Where:
D = Cylinder bore diameter (in)
d = Piston rod diameter (in)
P_d = Discharge pressure (psia)
P_s = Suction pressure (psia)
Inertia Load
Inertia Load (approximate):
F_inertia = W × R × (RPM/1000)² × 28.4
Where:
W = Reciprocating weight (lb)
R = Crank radius = Stroke/2 (in)
RPM = Rotational speed
28.4 = Constant for these units
Peak inertia occurs at TDC and BDC
Combined Rod Load
Combined Load:
At TDC (crank end): Load = Gas_compression + Inertia
At BDC (head end): Load = Gas_tension - Inertia
Maximum Compression = Max(Gas_compression + Inertia, Gas_tension - Inertia)
Maximum Tension = Max(Gas_tension + Inertia, Gas_compression - Inertia)
Rod Reversal Check
For proper crosshead bearing lubrication, the rod must experience both tension and compression during each revolution:
Reversal Requirement: Minimum reversal is typically 3-5% of the larger load. If reversal is marginal or absent, operating conditions must be adjusted or the cylinder deactivated.
3. Temperature Calculations
Discharge temperature rises with compression ratio. Limiting this temperature is critical for valve life, lubricant performance, and material integrity.
Adiabatic Temperature Rise
Discharge Temperature (Adiabatic):
T₂ = T₁ × (P₂/P₁)^((k-1)/k)
Where:
T₁ = Suction temperature (°R = °F + 459.67)
T₂ = Discharge temperature (°R)
P₂/P₁ = Compression ratio
k = Specific heat ratio
Convert result: °F = °R - 459.67
Polytropic Temperature Rise
Discharge Temperature (Polytropic):
T₂ = T₁ × (P₂/P₁)^((n-1)/n)
Where n = polytropic exponent
For efficiency correction:
T₂_actual = T₁ + (T₂_isentropic - T₁) / η_adiabatic
Temperature Limits
| Limit | Temperature | Concern |
|---|---|---|
| Normal operation | <275°F | Standard materials OK |
| Caution zone | 275-300°F | Reduced valve life |
| High temperature | 300-350°F | Special lubricants needed |
| Maximum limit | >350°F | Consider staging/cooling |
4. Volumetric Efficiency
Volumetric efficiency determines actual capacity relative to cylinder displacement. It's affected by clearance, compression ratio, and gas properties.
Basic VE Formula
Volumetric Efficiency:
VE = 1 - C × [(P₂/P₁)^(1/k) - 1]
Where:
C = Clearance as fraction of swept volume
P₂/P₁ = Compression ratio
k = Specific heat ratio
Example: C = 0.15, ratio = 3:1, k = 1.27
VE = 1 - 0.15 × [3^(1/1.27) - 1]
VE = 1 - 0.15 × 1.27 = 0.81 = 81%
Corrected VE
VE with Corrections:
VE = α - Slip - C × [(Z₁/Z₂) × r^(1/k) - 1]
Where:
α = Valve loss factor (typically 0.96-0.98)
Slip = Ring/packing leakage (0.02-0.08)
Z₁/Z₂ = Compressibility ratio (suction/discharge)
Capacity Calculation
Actual Capacity:
Q_actual = PD × VE × N × (1 or 2)
Where:
PD = Piston displacement (ft³/stroke) = π × D² × L / (4 × 1728)
VE = Volumetric efficiency
N = Speed (RPM)
1 or 2 = Single or double acting
For MMscfd: Q_MMscfd = Q_acfm × (P_s/P_std) × (T_std/T_s) × 1440 / 1,000,000
5. Pulsation Basics
Reciprocating compressors create pulsating flow that can cause vibration, damage, and measurement errors. Understanding pulsation is essential for proper system design.
Pulsation Frequency
Fundamental Frequency:
f = (RPM × N_throws) / 60 Hz
For a single-acting, single-throw: f = RPM/60
For double-acting: dominant frequency = 2 × RPM/60
Higher harmonics at 2f, 3f, 4f... are also present
Acoustic Resonance
Piping can resonate when pulsation frequency matches pipe acoustic frequency:
Pipe Acoustic Frequency:
f_pipe = c / (2L) for closed-open pipe
f_pipe = c / L for open-open pipe
Where:
c = Speed of sound in gas (ft/s) ≈ 68.1 × √(k × T / MW)
L = Pipe length (ft)
Pulsation Control
- Dampeners: Volume bottles attenuate pulsations
- Orifices: Create acoustic resistance
- Piping design: Avoid resonant lengths
- Support: Proper pipe support prevents vibration damage
API 618: New compressor installations typically require a pulsation and mechanical analysis per API 618 to ensure safe operation and prevent piping failures.
6. Rules of Thumb
These practical guidelines provide quick checks and estimates for common situations. They should be verified with detailed calculations for final design.
Sizing Guidelines
| Parameter | Rule of Thumb |
|---|---|
| Max ratio per stage | 3.5:1 to 4:1 (temperature limited) |
| Number of stages | n = log(r_overall) / log(3.5) |
| HP per MMscfd | 80-120 HP/MMscfd typical |
| Clearance range | 10-25% normal, 40-100% pipeline |
| Adiabatic efficiency | 80-90% (higher at low ratio) |
Operating Limits
| Parameter | Typical Limit |
|---|---|
| Discharge temperature | 275-300°F standard, 350°F max |
| Valve temperature rise | <20°F above discharge temp |
| Rod load | Per frame rating (never exceed) |
| Rod reversal | Minimum 3-5% of larger load |
| Piston speed | 700-1200 ft/min typical |
Quick Conversions
| From | To | Factor |
|---|---|---|
| psig | psia | Add 14.7 |
| °F | °R | Add 459.67 |
| HP | kW | × 0.746 |
| MMscfd | scfm | × 694.4 |
| acfm | scfm | × P/14.7 × 520/T |
Verification: Rules of thumb are for preliminary estimates and quick checks. Always verify critical parameters with detailed calculations or manufacturer data before finalizing designs.
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