1. Overview
Polytropic head is the fundamental performance parameter for centrifugal compressors. Unlike isentropic head (used for reciprocating compressors), polytropic head provides a more consistent basis for comparing centrifugal compressor performance because polytropic efficiency remains nearly constant with changing compression ratio.
Polytropic Head (Hp)
Energy Input
Work per unit mass; independent of gas at constant Z, T
Polytropic Efficiency
Stage Performance
Remains constant with ratio; unlike isentropic
Impeller Tip Speed
U = pi D N / 720
Head ~ U^2; sets mechanical limits
Head Coefficient
mu = Hp / U^2
Dimensionless; typically 0.45-0.55
Why Polytropic for Centrifugal?
| Property | Isentropic | Polytropic |
|---|---|---|
| Efficiency constancy | Varies with compression ratio | Nearly constant with ratio |
| Multi-stage analysis | Cannot simply add stage heads | Stages can be analyzed independently |
| Performance comparison | Misleading at different ratios | Valid comparison across conditions |
| API 617 standard | Not the preferred basis | Required basis for centrifugal |
| Typical values | 2-5% lower than polytropic | 75-85% for centrifugal |
Key distinction: Polytropic efficiency represents the efficiency of an infinitesimally small compression step. Because centrifugal compressors achieve compression through many small increments across the impeller, polytropic analysis matches their physics more accurately than isentropic analysis.
2. Polytropic Process
A polytropic process follows the path PV^n = constant, where n is the polytropic exponent. For real compression, n is always greater than k (the isentropic exponent) due to irreversibilities.
Polytropic Exponent (n):
n / (n-1) = [k / (k-1)] * eta_p
Solving for n:
n = 1 / [1 - (k-1) / (k * eta_p)]
Where:
n = Polytropic exponent (always > k for compression)
k = Isentropic exponent (Cp/Cv)
eta_p = Polytropic efficiency
Relationship between n and k:
Ideal (eta = 1.0): n = k
Typical (eta = 0.80): n = 1.375 for k = 1.27
Poor (eta = 0.70): n = 1.455 for k = 1.27
Polytropic Head Equation
Ideal Gas Polytropic Head (GPSA Eq. 13-18):
Hp = (Z_avg * R * T1 / MW) * [n/(n-1)] * [(P2/P1)^((n-1)/n) - 1]
Where:
Hp = Polytropic head (ft-lbf/lb)
Z_avg = Average compressibility (Z1 + Z2) / 2
R = 1545.35 ft-lbf/(lbmol-degR)
T1 = Suction temperature (degR)
MW = Molecular weight (lb/lbmol)
P2/P1 = Compression ratio
Gas Horsepower:
GHP = (mass_flow * Hp) / 33,000
Brake Horsepower:
BHP = GHP / eta_p + mechanical losses
Mechanical losses typically 1-3% of GHP.
Discharge Temperature
Polytropic Discharge Temperature:
T2 = T1 * (P2/P1)^((n-1)/n)
Key difference from isentropic method:
The polytropic discharge temperature is the ACTUAL discharge temperature
(not an ideal value requiring efficiency correction).
Temperature limits:
Normal operation: T2 < 300 deg F
High-performance seals: T2 < 350 deg F
Special materials: T2 < 400 deg F (requires API 617 review)
Practical note: The polytropic exponent n can also be determined from test data using: n = ln(P2/P1) / ln(v1/v2), where v is specific volume. This is useful for back-calculating efficiency from field measurements.
3. Schultz Method
The Schultz method corrects the ideal gas polytropic head equation for real gas behavior. This is essential for gases near their critical point or at high pressures where Z varies significantly through compression.
Schultz Correction Factor
Schultz Corrected Polytropic Head:
Hp_real = f * Hp_ideal
Where f is the Schultz correction factor.
Schultz Factor (f):
f = (h2s - h1) / [(n/(n-1)) * (P2*v2 - P1*v1)]
Simplified form using X and Y factors:
X = (T/v) * (dv/dT)_P (isothermal volume expansivity)
Y = -(P/v) * (dv/dP)_T (isothermal compressibility)
f = (Y - 1) / (X - 1) approximately
When f matters:
f = 1.00: Ideal gas (Z constant through compression)
f = 0.95-1.00: Light hydrocarbons at moderate pressure
f = 0.85-0.95: Heavy gases or near critical point
f < 0.85: Very high pressure or near-critical; use enthalpy method
X and Y Factor Calculation
From Equation of State:
X = 1 + (T1/Z1) * (dZ/dT)_P
Y = 1 - (P1/Z1) * (dZ/dP)_T
Practical evaluation using finite differences:
X = 1 + T1 * [Z(T1+dT, P1) - Z(T1-dT, P1)] / [Z1 * 2 * dT]
Y = 1 - P1 * [Z(T1, P1+dP) - Z(T1, P1-dP)] / [Z1 * 2 * dP]
Use dT = 5 degR, dP = 5 psi for numerical stability.
Modified polytropic exponent:
n_s / (n_s - 1) = [k/(k-1)] * eta_p * (1/f)
Use n_s in place of n in the head equation for Schultz-corrected results.
| Gas / Condition | Typical f | Notes |
|---|---|---|
| Methane at P < 500 psia | 0.99-1.00 | Nearly ideal |
| Natural gas (SG=0.65), 800 psia | 0.96-0.99 | Moderate correction |
| Natural gas (SG=0.65), 1200 psia | 0.92-0.96 | Significant correction |
| CO2 at 1000 psia | 0.85-0.92 | Near critical; large correction |
| Propane at 200 psia | 0.90-0.95 | Heavier hydrocarbon |
| Hydrogen at any pressure | 0.99-1.00 | Behaves nearly ideal |
When to use Schultz: Always apply the Schultz correction when reduced pressure (Pr = P/Pc) exceeds 0.5 or reduced temperature (Tr = T/Tc) is below 2.0. For light gases (H2, He) at moderate pressures, the correction is negligible.
4. Gas Property Effects
Gas properties directly influence polytropic head and the resulting compressor selection. Understanding these effects is critical for multi-service or variable-composition applications.
Molecular Weight Effects
Head is inversely proportional to MW:
Hp ~ (Z * R * T1) / MW
For same compression ratio and conditions:
Light gas (MW = 16): High head per stage, many stages needed
Heavy gas (MW = 30): Low head per stage, fewer stages needed
Head per impeller (typical):
Backward-leaning: 8,000 - 12,000 ft-lbf/lb
Radial: 10,000 - 15,000 ft-lbf/lb
Number of impellers:
N_impellers = Hp_total / Hp_per_impeller
Example: 60,000 ft-lbf/lb total head
Backward-leaning: 60,000 / 10,000 = 6 impellers
Radial: 60,000 / 12,500 = 5 impellers
Property Sensitivity
| Property | Effect on Head | Effect on Power | Effect on T2 |
|---|---|---|---|
| MW increase | Decreases (1/MW) | Increases (mass flow up) | Slight decrease |
| k increase | Increases (more work) | Increases | Increases |
| Z decrease | Decreases | Approximately constant | Slight decrease |
| T1 increase | Increases (proportional) | Approximately constant | Increases (proportional) |
| P1 increase | No change (ratio based) | Increases (density up) | No change |
Compressibility Factor (Z)
| Gas Condition | Typical Z Range | Z Calculation Method |
|---|---|---|
| Light gas, low P (< 300 psia) | 0.95 - 1.00 | Ideal gas (Z = 1.0) acceptable |
| Natural gas, moderate P (500-1000) | 0.85 - 0.95 | Standing-Katz, Peng-Robinson |
| Natural gas, high P (> 1000) | 0.70 - 0.90 | Equation of state required |
| CO2 near critical | 0.20 - 0.80 | Span-Wagner EOS recommended |
| H2 at any condition | 1.00 - 1.05 | Z slightly above 1.0 |
Average Z: Use Z_avg = (Z1 + Z2) / 2 for the head equation. For high compression ratios (r > 3), this average can introduce errors exceeding 2-3%. In such cases, evaluate Z at multiple intermediate pressures or use the Schultz method.
5. Multi-Stage Analysis
Multi-stage centrifugal compressors contain multiple impellers on a single shaft within one or more casings. Polytropic analysis allows each stage to be analyzed independently and the results summed.
Stage-by-Stage Calculation
Total Polytropic Head:
Hp_total = Sum of Hp for each stage
For N stages with equal work distribution:
Hp_per_stage = Hp_total / N
Stage Pressure Ratios:
r_total = Product of r_i for each stage
For equal ratios: r_per_stage = r_total^(1/N)
Intercooling Benefit:
Without intercooling: T2_final = T1 * r_total^((n-1)/n)
With intercooling to T1 between stages:
T2_final = T1 * r_per_stage^((n-1)/n)
Power savings from intercooling: 8-20% depending on overall ratio.
Multi-Stage Configuration
| Configuration | Stages | Head Range | Application |
|---|---|---|---|
| Single casing, inline | 1-8 impellers | Up to 100,000 ft-lbf/lb | Pipeline, refrigeration |
| Single casing, back-to-back | 4-10 impellers | Up to 120,000 ft-lbf/lb | Balanced thrust loads |
| Two casings, series | Up to 16 impellers | Up to 200,000 ft-lbf/lb | High ratio with intercooling |
| Integrally geared | 2-8 stages | Up to 300,000 ft-lbf/lb | Air, process gas, high ratio |
Impeller Selection Factors
| Impeller Type | Head/Stage | Flow Coefficient | Efficiency | Stability |
|---|---|---|---|---|
| 2D backward-lean | 8,000-10,000 | 0.01-0.08 | 78-83% | Wide range |
| 3D backward-lean | 9,000-12,000 | 0.02-0.10 | 82-87% | Wide range |
| Radial (straight vane) | 10,000-15,000 | 0.01-0.06 | 75-80% | Narrow range |
| Mixed flow | 5,000-8,000 | 0.08-0.20 | 80-85% | Wide range |
Back-to-back configuration: Arranging impellers in opposing directions on the same shaft significantly reduces net axial thrust. This allows smaller thrust bearings and higher operating pressures. The tradeoff is more complex internal flow paths and slightly lower efficiency.
6. ASME PTC-10 Testing
ASME PTC-10 (Performance Test Code for Compressors and Exhausters) defines the procedures for acceptance testing of centrifugal compressors. It provides methods for comparing tested performance to guaranteed performance using polytropic analysis.
Test Types
| Test Type | Gas Used | Facility | Accuracy |
|---|---|---|---|
| Type 1 | Specified (field) gas | Shop or field | Highest; direct comparison |
| Type 2 | Substitute gas (N2, CO2, R134a) | OEM test stand | Good; requires similarity |
PTC-10 Similarity Requirements (Type 2)
Equivalence Conditions:
Machine Mach number: Mm_test / Mm_spec = 0.8 to 1.2
Machine Reynolds: Re_test / Re_spec > 0.1 (with correction)
Volume ratio: (v1/v2)_test / (v1/v2)_spec = 0.95 to 1.05
Specific heat ratio: k_test / k_spec = 0.95 to 1.05
Flow coefficient: phi_test / phi_spec = 0.96 to 1.04
Machine Mach Number:
Mm = U_tip / a_inlet
Where:
U_tip = Impeller tip speed (ft/s)
a_inlet = Speed of sound at inlet = sqrt(k * Z * R * T1 * 32.174 / MW)
Reynolds Number Correction:
eta_corrected = 1 - (1 - eta_test) * (Re_spec / Re_test)^0.1
This corrects for the difference in boundary layer losses between
test gas and specified gas conditions.
Performance Guarantees
| Parameter | Typical Guarantee | PTC-10 Tolerance |
|---|---|---|
| Polytropic head | Within -2% to +5% of rated | +/- 2% measurement uncertainty |
| Polytropic efficiency | Within -2 points of rated | +/- 1.5 points uncertainty |
| Power | Within +4% of rated | +/- 2% measurement uncertainty |
| Surge point | 10% margin below design flow | Demonstrated by test |
| Discharge temperature | Within +5 deg F of predicted | +/- 1 deg F measurement |
Test gas selection: For Type 2 tests, the substitute gas must achieve similar Mach number and volume ratio. Common choices are nitrogen (for light gases), CO2 (for heavier gases), or R134a (for very heavy gases or high Mach numbers).
7. Worked Examples
Example 1: Basic Polytropic Head
Given:
Natural gas: MW = 18.5, k = 1.28, Z_avg = 0.92
P1 = 400 psia, P2 = 1,000 psia, T1 = 90 deg F (549.67 degR)
Polytropic efficiency eta_p = 0.78
Step 1: Compression ratio
r = 1,000 / 400 = 2.50
Step 2: Polytropic exponent
n = 1 / [1 - (1.28-1) / (1.28 * 0.78)]
n = 1 / [1 - 0.28 / 0.9984]
n = 1 / [1 - 0.2804]
n = 1 / 0.7196 = 1.3896
Step 3: Polytropic head
(n-1)/n = 0.3896 / 1.3896 = 0.2804
Hp = (0.92 * 1545.35 * 549.67 / 18.5) * [1.3896/0.3896] * [2.50^0.2804 - 1]
Hp = 42,241 * 3.567 * 0.2899
Hp = 43,696 ft-lbf/lb
Step 4: Discharge temperature
T2 = 549.67 * 2.50^0.2804
T2 = 549.67 * 1.2899 = 709.0 degR = 249 deg F (OK, < 300 deg F)
Example 2: Schultz Correction
Given (same as Example 1, plus):
Z1 = 0.935, Z2 = 0.905
Schultz factor f = 0.965 (from EOS evaluation)
Step 1: Corrected head
Hp_real = f * Hp_ideal = 0.965 * 43,696 = 42,167 ft-lbf/lb
Step 2: Difference
Error without Schultz correction = (43,696 - 42,167) / 42,167 = 3.6%
This 3.6% error would result in:
Over-predicting head (fewer impellers selected - risk of not meeting duty)
Under-predicting power (driver may be undersized)
Conclusion: Schultz correction is significant for this case
(Pr = 400/670 = 0.60, Tr = 550/370 = 1.49).
Example 3: Multi-Stage Head Distribution
Given:
Total polytropic head required: 85,000 ft-lbf/lb
Impeller type: 3D backward-leaning (10,000-12,000 ft-lbf/lb per impeller)
Use average of 11,000 ft-lbf/lb per impeller
Step 1: Number of impellers
N = 85,000 / 11,000 = 7.7 -> round up to 8 impellers
Step 2: Actual head per impeller
Hp_per_impeller = 85,000 / 8 = 10,625 ft-lbf/lb (within range)
Step 3: Configuration selection
8 impellers -> single casing, back-to-back (4 + 4) arrangement
with intercooler between sections if T2_section1 > 250 deg F
Step 4: Verify tip speed
Hp = mu * U^2 / gc
U = sqrt(Hp * gc / mu) = sqrt(10,625 * 32.174 / 0.50)
U = 827 ft/s (acceptable; below 1,000 ft/s material limit for steel)
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