1. Efficiency Definitions
Compressor efficiency quantifies how much of the shaft power input is converted into useful compression work on the gas. The remainder becomes heat due to friction, turbulence, incidence losses, and disk windage. Two definitions are used in the industry, each suited to different purposes.
Isentropic Efficiency
η_isen
Ideal work / Actual work; varies with pressure ratio
Polytropic Efficiency
η_p
Small-step efficiency; constant for same aerodynamics
Mechanical Efficiency
η_mech
Bearing, seal, and windage losses; typically 97–99%
Overall Efficiency
η_overall
η_thermo × η_mech; total shaft-to-gas
Isentropic Efficiency:
η_isen = (Isentropic Work) / (Actual Shaft Work)
η_isen = H_isen / H_actual
η_isen = (T₂s - T₁) / (T₂ - T₁)
Where:
T₂s = Isentropic discharge temperature
T₂ = Actual discharge temperature
T₁ = Suction temperature (all in °R)
Polytropic Efficiency:
η_p = (Polytropic Work) / (Actual Work)
η_p = [(k-1)/k] / [(n-1)/n]
Or equivalently:
η_p = ln(P₂/P₁) × [(k-1)/k] / ln(T₂/T₁)
Mechanical Efficiency:
η_mech = (Gas Power) / (Shaft Power)
Losses: bearings (0.5–1.5%), seals (0.2–0.5%), windage (0.1–0.5%)
Overall Efficiency:
η_overall = η_thermo × η_mech
For centrifugal: typically 0.73–0.83
Important distinction: Isentropic efficiency is a function of pressure ratio for the same machine. Polytropic efficiency is approximately constant regardless of pressure ratio, making it the preferred metric for centrifugal compressor specification per API 617.
2. Polytropic vs Isentropic
The choice between polytropic and isentropic efficiency is not merely academic; it fundamentally affects how compressors are compared, specified, and tested. Using the wrong definition leads to incorrect performance expectations and driver sizing errors.
Mathematical Relationship
Converting Between Efficiencies:
Given η_p and pressure ratio r = P₂/P₁:
η_isen = [r^((k-1)/k) - 1] / [r^((k-1)/(k×η_p)) - 1]
Given η_isen and r:
η_p = ln(r^((k-1)/k)) / ln(1 + [r^((k-1)/k) - 1]/η_isen)
Key Properties:
• For the same machine: η_p > η_isen (always)
• Difference increases with pressure ratio
• At r = 1.0: η_p = η_isen (converge)
• Typical difference: 2–5 percentage points
Why Polytropic is Preferred for Centrifugal
| Criterion | Isentropic | Polytropic |
|---|---|---|
| Constant with r? | No — varies with r | Yes — constant for same aero |
| Stage comparison | Unfair (high r stages look worse) | Fair (same aero = same η_p) |
| Multi-stage analysis | Complex (stages interact) | Simple (stages additive) |
| Vendor comparison | Misleading at different r | Apples-to-apples |
| API 617 | Acceptable for recip | Required for centrifugal |
| ASME PTC-10 | Type 1 test basis | Type 2 test basis |
Numerical Comparison at Different Pressure Ratios
| Pressure Ratio (r) | η_p = 0.80 | Equivalent η_isen | Difference |
|---|---|---|---|
| 1.5 | 80.0% | 79.1% | 0.9% |
| 2.0 | 80.0% | 78.3% | 1.7% |
| 2.5 | 80.0% | 77.6% | 2.4% |
| 3.0 | 80.0% | 77.0% | 3.0% |
| 4.0 | 80.0% | 75.9% | 4.1% |
| 6.0 | 80.0% | 74.3% | 5.7% |
| 10.0 | 80.0% | 72.0% | 8.0% |
Specification trap: A vendor quoting η_p = 80% at r = 3.0 is performing identically to a vendor quoting η_isen = 77%. If one vendor quotes polytropic and another quotes isentropic, you must convert to the same basis before comparison.
3. Schultz Method
The Schultz method corrects polytropic efficiency calculations for real-gas behavior. For ideal gases (Z = 1.0 everywhere), the standard polytropic formulas are exact. For real gases near their critical point, the Schultz correction can change calculated efficiency by 2–5 percentage points.
When to Apply Schultz Correction
| Condition | Schultz Needed? | Typical Cases |
|---|---|---|
| T_r > 2.0 and P_r < 0.3 | No — gas behaves ideally | Lean gas at moderate pressure |
| 1.5 < T_r < 2.0, P_r < 0.5 | Recommended | Pipeline gas at high pressure |
| T_r < 1.5 or P_r > 0.5 | Required | Rich gas, CO₂, refrigerants |
| Z varies > 5% across path | Required | High-ratio compression |
Schultz Correction Factors X and Y:
X = (T/v) × (∂v/∂T)_P - 1
Y = -(P/v) × (∂v/∂P)_T
For an ideal gas: X = 0, Y = 1
For real gases: X ≠ 0, Y ≠ 1
Modified Polytropic Exponent:
n_s/(n_s - 1) = [k/(k-1)] × η_p × [1/(1+X)] × Y
Where n_s is the Schultz-corrected polytropic exponent.
Schultz Polytropic Head:
H_p = Z_avg × R × T₁ × [n_s/(n_s-1)] × [(P₂/P₁)^((n_s-1)/n_s) - 1] / MW
Schultz Polytropic Efficiency:
η_p_Schultz = H_p_ideal / H_actual
Where H_p_ideal uses Schultz-corrected exponents.
Practical Procedure:
1. Calculate X, Y at average suction/discharge conditions
2. Determine Z_avg = (Z₁ + Z₂)/2
3. Compute Schultz-corrected n_s
4. Calculate head and efficiency using n_s
Impact of Schultz Correction
| Gas / Condition | η_p (uncorrected) | η_p (Schultz) | Difference |
|---|---|---|---|
| Lean NG, 500 psia | 78.0% | 78.2% | +0.2% |
| Rich NG (C₃+ = 8%), 800 psia | 78.0% | 79.5% | +1.5% |
| CO₂ at 1200 psia | 78.0% | 81.2% | +3.2% |
| Propane refrigerant | 78.0% | 82.0% | +4.0% |
ASME PTC-10 requirement: Performance tests must use the Schultz correction when the gas departs significantly from ideal behavior. Without correction, the calculated efficiency understates actual performance, potentially leading to unjust rejection of conforming machines.
4. Efficiency Curve Shape
The efficiency-vs-flow curve for a centrifugal compressor has a characteristic bell shape. Understanding this shape is essential for selecting operating points, sizing drivers, and setting control strategies.
Key Points on the Efficiency Curve
| Operating Region | % Design Flow | Relative η | Characteristics |
|---|---|---|---|
| Surge region | < 60–70% | Unstable | Flow reversal, mechanical damage, must avoid |
| Left of BEP | 70–95% | 90–98% of peak | Positive incidence, increased recirculation |
| BEP (design) | 95–105% | 100% (peak) | Minimum losses, zero incidence angle |
| Right of BEP | 105–115% | 90–98% of peak | Negative incidence, increasing Mach |
| Stonewall / Choke | > 115–125% | Drops rapidly | Sonic velocity, head collapses |
Factors Affecting Curve Shape
Impeller Design Parameters:
Specific Speed (Ns):
Ns = N × Q^0.5 / H_ad^0.75
Where:
N = Rotational speed (RPM)
Q = Inlet volume flow (ft³/min)
H_ad = Adiabatic head (ft)
Low Ns (< 500): Narrow curve, steep head rise
Higher peak η, but less range
Medium Ns (500–1000): Moderate curve width
Good compromise of η and range
High Ns (> 1000): Wide curve, flat head rise
Lower peak η, but more range
Flow Coefficient (φ):
φ = V_meridional / U_tip
Head Coefficient (ψ):
ψ = H / U_tip²
Efficiency Relationship:
η = f(flow coefficient, head coefficient, Re, Mach)
Peak efficiency occurs at the design φ/ψ ratio.
Factors That Reduce Efficiency
| Factor | Efficiency Impact | Mitigation |
|---|---|---|
| High Mach number | -1 to -5 points | Reduce tip speed, add stages |
| Low Reynolds number | -1 to -3 points | Surface finish, larger machine |
| High MW gas | -1 to -4 points | Lower tip speed, more stages |
| Surface roughness | -1 to -3 points | Polish impeller and diffuser |
| Clearance gaps | -0.5 to -2 points | Tight clearances, labyrinth seals |
| Fouling / deposits | -2 to -8 points | Online washing, filtration |
| Wet gas / liquids | -3 to -10 points | Scrubbers, knockout drums |
Operating strategy: Design the system to operate within 90–110% of the BEP flow. Operating consistently below 70% or above 115% of design flow wastes energy and accelerates mechanical wear.
5. Typical Efficiency Ranges
Efficiency varies significantly with machine size, number of stages, gas properties, and manufacturer design philosophy. The following tables provide realistic ranges for engineering estimates.
By Compressor Type
| Compressor Type | η_p Range | η_isen Range | Best Achievable |
|---|---|---|---|
| Single-stage overhung | 72–78% | 70–76% | 80% |
| Multi-stage beam (2–4 stages) | 75–82% | 72–79% | 84% |
| Multi-stage beam (5–8 stages) | 77–85% | 73–80% | 87% |
| Integrally geared | 80–86% | 77–83% | 88% |
| Axial | 85–90% | 82–88% | 92% |
| Pipeline booster | 82–87% | 79–84% | 89% |
By Application
| Application | Typical η_p | Gas | Notes |
|---|---|---|---|
| Gas pipeline | 82–87% | Lean NG | Large flow, optimized design |
| Gas gathering | 72–78% | Rich NG | Variable conditions, smaller machines |
| Gas processing | 75–82% | Various | Mid-size, moderate ratios |
| Refrigeration | 78–84% | Propane/MR | Constant conditions, optimized |
| CO₂ injection | 72–78% | CO₂ | High MW, real-gas effects |
| Hydrogen recycle | 68–75% | H₂-rich | Very low MW, high head |
| FCC wet gas | 75–80% | Mixed | Fouling, variable composition |
By Inlet Volume Flow (Machine Size Effect)
Size-Efficiency Correlation:
Larger machines achieve higher efficiency because:
• Lower relative clearance losses
• Higher Reynolds number (→ lower friction)
• Better surface finish relative to passage size
• More sophisticated aerodynamic design justified
Approximate η_p by inlet flow (ACFM):
< 2,000 ACFM: 70–75% (small, overhung)
2,000–10,000: 75–80% (medium beam)
10,000–50,000: 78–84% (large beam)
50,000–200,000: 82–87% (pipeline class)
> 200,000 ACFM: 85–90% (axial / large pipeline)
Reynolds Number Correction (ASME PTC-10):
η_corrected = 1 - (1 - η_test) × (Re_test / Re_field)^0.1
This correction is small (< 1 point) for most applications
but significant for high-pressure, dense gas services.
Guarantee values: Manufacturers typically guarantee polytropic efficiency within ±2 percentage points of the predicted value. API 617 requires the guarantee point to be within the surge-to-stonewall operating range.
6. Worked Examples
Example 1: Calculating Polytropic Efficiency from Test Data
Given (field test data):
Gas: Natural gas, k = 1.27, MW = 18.9
P₁ = 400 psia, T₁ = 85°F (544.67°R)
P₂ = 1000 psia, T₂ = 280°F (739.67°R)
Step 1: Pressure Ratio
r = 1000/400 = 2.5
Step 2: Temperature Ratio
T₂/T₁ = 739.67/544.67 = 1.3580
Step 3: Polytropic Efficiency
η_p = ln(r) × [(k-1)/k] / ln(T₂/T₁)
η_p = ln(2.5) × (0.27/1.27) / ln(1.3580)
η_p = 0.9163 × 0.2126 / 0.3063
η_p = 0.1948 / 0.3063
η_p = 0.636 (63.6%)
Step 4: Evaluate
η_p = 63.6% is below normal range (75–85%).
This indicates compressor degradation:
• Fouled impellers or diffusers
• Increased seal clearances
• Damaged labyrinth seals
Recommend maintenance inspection.
Example 2: Polytropic to Isentropic Conversion
Given:
η_p = 0.80, k = 1.27, r = 3.0
Step 1: Isentropic Temperature Ratio
(k-1)/k = 0.27/1.27 = 0.2126
r^((k-1)/k) = 3.0^0.2126 = 1.2615
Step 2: Polytropic Temperature Ratio
(n-1)/n = (k-1)/(k × η_p) = 0.27/(1.27 × 0.80) = 0.2657
r^((n-1)/n) = 3.0^0.2657 = 1.3280
Step 3: Isentropic Efficiency
η_isen = [r^((k-1)/k) - 1] / [r^((k-1)/(k×η_p)) - 1]
η_isen = (1.2615 - 1) / (1.3280 - 1)
η_isen = 0.2615 / 0.3280
η_isen = 0.7973 (79.7%)
Verification: η_p - η_isen = 0.3% = 0.3 points
At r = 3.0 with k = 1.27, the difference is small
because k is relatively low.
Example 3: Efficiency Impact on Power
Given:
50 MMSCFD natural gas, r = 3.0, k = 1.27
T₁ = 90°F, Z = 0.95, MW = 18.9
Case A: η_p = 0.82 (new, clean machine)
(n-1)/n = 0.27/(1.27 × 0.82) = 0.2593
H_p = (0.95 × 1545.35 × 549.67/18.9) × [1/(1-0.2593)] × [3.0^0.2593 - 1]
H_p = 42,705 × 1.3501 × 0.3195 = 18,415 ft·lbf/lb
BHP_A = (ṁ × 18,415) / 33,000 = 965 HP
Case B: η_p = 0.72 (degraded machine)
(n-1)/n = 0.27/(1.27 × 0.72) = 0.2952
H_p = 42,705 × 1/(1-0.2952) × [3.0^0.2952 - 1]
H_p = 42,705 × 1.4189 × 0.3726 = 22,566 ft·lbf/lb
BHP_B = (ṁ × 22,566) / 33,000 = 1,183 HP
Power Penalty:
ΔHP = 1,183 - 965 = 218 HP (+22.6%)
At $0.05/kWh, annual cost = 218 × 0.746 × 8,000 × $0.05
= $65,100/year in excess energy cost.
Economic impact: A 10-point drop in polytropic efficiency (82% to 72%) increases power consumption by 20–25% and can cost $50,000–$150,000/year per machine in excess energy. Regular monitoring of efficiency trends justifies maintenance investment.
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