How is the polytropic exponent n derived from k and η_p?

From the GPSA Section 13 identity n/(n−1) = (k/(k−1))·η_p, solve to get n = (k·η_p) / (k·η_p − k + 1). Example: k = 1.28, η_p = 0.80 → n = 1.024 / 0.744 = 1.376. The polytropic exponent always falls between 1 (isothermal limit) and k (isentropic limit).

Why is polytropic efficiency preferred over isentropic efficiency for compressor selection?

Polytropic efficiency is independent of pressure ratio for a given machine, so it lets you compare vendors and operating points consistently. Isentropic efficiency drops as ratio rises (because the ideal compressed-gas temperature climbs faster than the actual one), so a single η_isen number only describes one operating point. ASME PTC-10 and API 617 both use polytropic basis for performance testing.

How do you convert between isentropic and polytropic efficiency?

η_p = ln(r^((k−1)/k)) / ln[1 + (r^((k−1)/k) − 1) / η_isen]. The reverse: η_isen = (r^((k−1)/k) − 1) / (r^((n−1)/n) − 1) where n is found from the polytropic identity. At r = 2.5, k = 1.28, η_isen = 0.75: η_p = 0.210 / 0.272 ≈ 0.77. The conversion always yields η_p ≥ η_isen for compression.

What temperature limits apply to compressor discharge and what triggers them?

API 618 reciprocating ≈ 350°F (valve seat life and lubricant film breakdown), API 617 centrifugal ≈ 450°F (rotor metallurgy and seal-oil), API 619 screw ≈ 300°F (oil-flooded lubricant viscosity). For discharge above the limit, options are: add intercooling, increase stages, pre-cool suction, or specify high-temperature metallurgy and synthetic lube.

What's the difference between gas horsepower (GHP), brake horsepower (BHP), and driver power?

GHP = ṁ·H_p / (33,000·η_p) is the aerodynamic power transferred to the gas (GPSA Eq. 13-30). BHP = GHP / η_mech includes mechanical losses (bearings, seals, gears, oil pump). Driver HP = BHP × (1 + margin) adds the API minimum 10% safety factor for transients and motor service factor; round up to the next NEMA MG-1 standard size.

Compressor Sizing Fundamentals

1. Compressor Types

Compressors increase gas pressure by imparting mechanical energy. Selection depends on flow rate, pressure ratio, gas properties, and operational requirements.

Positive Displacement

Reciprocating (API 618)

50–10,000 ACFM, ratio 2–4:1/stage. High efficiency, variable flow capability.

Dynamic

Centrifugal (API 617)

2,000–200,000+ ACFM, ratio 1.5–3:1/stage. High reliability, continuous duty.

Positive Displacement

Screw (API 619)

500–12,000 ACFM, ratio 2–6:1. Simple, tolerates dirty gas.

Dynamic

Axial

50,000–1,000,000+ ACFM, ratio 1.1–1.2/stage. Highest efficiency at large scale.

Selection Summary

Type	Flow Range	Max Ratio/Stage	η_p Range	Best Application
Reciprocating	50–10,000 ACFM	4.0:1	82–88%	High ratio, variable flow
Centrifugal	2,000–200,000 ACFM	3.0:1	75–85%	High flow, continuous
Screw	500–12,000 ACFM	6.0:1 (oil-flooded)	70–78%	Moderate flow, simplicity
Axial	>50,000 ACFM	1.2:1	85–90%	Very high flow, LNG

Rule of thumb: Use reciprocating for high pressure ratios and variable loads; centrifugal for large continuous flows; screw for simplicity and dirty gas; axial for very high flows (>50,000 ACFM).

2. Compression Thermodynamics

Gas compression can follow three idealized paths. Real compressors approximate polytropic compression.

Isentropic (Adiabatic Reversible) Compression

No heat transfer, reversible process. Represents ideal dynamic compressor behavior.

Isentropic Relations: PV^k = constant Temperature ratio: T₂/T₁ = (P₂/P₁)^(k-1)/k = r^(k-1)/k Where: k = Cₚ/Cᵥ (specific heat ratio) r = P₂/P₁ (compression ratio) T = absolute temperature (°R) P = absolute pressure (psia) For natural gas: k ≈ 1.26–1.31 (function of composition and T)

Polytropic Compression (Real Process)

Accounts for heat transfer and irreversibilities. The polytropic exponent n lies between 1 (isothermal) and k (isentropic).

Polytropic Relations: PVⁿ = constant Temperature ratio: T₂/T₁ = (P₂/P₁)^(n-1)/n = r^(n-1)/n Polytropic exponent from efficiency: n/(n-1) = (k/(k-1)) × η_p Solving for n: n = (k × η_p) / (k × η_p - k + 1) For η_p = 0.80 and k = 1.28: n = (1.28 × 0.80) / (1.28 × 0.80 - 1.28 + 1) = 1.024/0.744 = 1.376

Discharge Temperature

Actual Discharge Temperature: Method 1 (from isentropic efficiency): T₂ = T₁ + (T₂,isen - T₁) / η_isen Where: T₂,isen = T₁ × r^(k-1)/k Method 2 (from polytropic exponent): T₂ = T₁ × r^(n-1)/n Example: T₁ = 80°F = 539.67°R, r = 3.0, k = 1.28, η_isen = 0.75 T₂,isen = 539.67 × 3.0^0.219 = 539.67 × 1.278 = 690°R T₂ = 539.67 + (690 - 539.67) / 0.75 = 539.67 + 200.4 = 740°R = 280°F

Temperature limits: Reciprocating (API 618): 350°F max (valve life). Centrifugal (API 617): 450°F typical. Exceeding limits requires intercooling.

3. Power Calculations

Compressor power is calculated from head (energy per unit mass) and mass flow rate. GPSA Section 13 provides standard methods.

Head Calculation

Polytropic Head (GPSA Eq. 13-4): H_p = (Z_avg × R × T₁ / MW) × (n/(n-1)) × [r^(n-1)/n - 1] Where: H_p = polytropic head (ft-lbf/lbm) Z_avg = average compressibility factor R = 1545.35 ft-lbf/(lbmol·°R) T₁ = suction temperature (°R) MW = molecular weight (lb/lbmol) n = polytropic exponent r = compression ratio Isentropic Head: H_isen = (Z_avg × R × T₁ / MW) × (k/(k-1)) × [r^(k-1)/k - 1]

Gas Horsepower

Gas Horsepower (GPSA Eq. 13-30): GHP = (ṁ × H_p) / (33,000 × η_p) Where: GHP = gas horsepower (HP) — actual aerodynamic shaft power ṁ = mass flow rate (lb/min) H_p = polytropic head (ft-lbf/lbm) η_p = polytropic efficiency 33,000 = ft-lbf/min per HP The η_p divisor converts the "ideal" polytropic work (per Schultz / GPSA Eq. 13-22) into the actual shaft work the gas absorbs. Omitting it under-predicts BHP by 15-25%. Mass flow from standard flow: ṁ = Q_std × (P_std × MW) / (R × T_std) Where Q_std in SCF/min at 14.696 psia, 60°F

Brake Horsepower

Shaft Power: BHP = GHP / η_mech Driver Power (with API 10% margin): Driver HP = BHP × 1.10 Typical mechanical efficiencies: • Centrifugal: 0.96–0.99 • Reciprocating: 0.90–0.95 • Screw: 0.92–0.96

GPSA Simplified Power Equation

GPSA Direct Power Calculation (k-form approximation): HP = (Q × Z × T₁ × k × [r^(k-1)/k - 1]) / (C × (k-1) × η_p) Where: Q = flow rate (MMSCFD at 14.696 psia, 60°F) T₁ = suction temperature (°R) Z = compressibility factor at suction C = 11.667 (constant for units shown — derived from P_std·144 / (R · ṁ-conversion · 33,000)) η_p = polytropic efficiency Rule of thumb (natural gas centrifugal, η_p ≈ 0.80): HP/MMSCFD ≈ 50 (r=2) · 75 (r=3) · 95 (r=4) · 110 (r=5) These are single-stage values at T₁ ≈ 80°F. For multi-stage with intercooling, use ratio-per-stage in the lookup.

Example Calculation

Given: Q = 50 MMSCFD natural gas (SG = 0.65, MW = 18.8) P₁ = 400 psig = 414.7 psia P₂ = 900 psig = 914.7 psia T₁ = 90°F = 549.67°R k = 1.28, Z = 0.88, η_p = 0.80, η_mech = 0.97 Step 1: Compression ratio r = 914.7 / 414.7 = 2.206 Step 2: Polytropic exponent n = (1.28 × 0.80) / (1.28 × 0.80 - 1.28 + 1) = 1.024 / 0.744 = 1.376 Step 3: Polytropic head H_p = (0.88 × 1545 × 549.67 / 18.8) × (1.376/0.376) × [2.206^0.273 - 1] H_p = 39,800 × 3.66 × [1.244 - 1] = 39,800 × 3.66 × 0.244 H_p = 35,500 ft-lbf/lbm Step 4: Mass flow SCFM = 50 × 10⁶ / 1440 = 34,722 SCF/min ṁ = SCFM × P_std × 144 × MW / (R × T_std) ṁ = 34,722 × 14.696 × 144 × 18.8 / (1545.35 × 519.67) = 1,720 lb/min Step 5: Gas horsepower (per GPSA Eq. 13-30) GHP = (1,720 × 35,500) / (33,000 × 0.80) = 2,313 HP Step 6: Brake and driver power BHP = 2,313 / 0.97 = 2,384 HP Driver = 2,384 × 1.10 = 2,623 HP → Select 3,000 HP motor (next NEMA MG-1 size)

4. Efficiency Definitions & Conversions

Polytropic efficiency is preferred for compressor selection as it remains constant regardless of pressure ratio. Isentropic efficiency varies with ratio.

Polytropic Efficiency

Definition: η_p = Ideal work (differential) / Actual work (differential) From measured data: η_p = [(k-1)/k] × ln(r) / ln(T₂/T₁) Typical values: • Large centrifugal (>10,000 HP): 78–85% • Small centrifugal (<3,000 HP): 72–78% • Reciprocating: 82–88% • Screw (oil-flooded): 70–78% • Axial: 85–90% Polytropic efficiency is independent of pressure ratio— same value applies whether ratio is 1.5 or 3.0.

Isentropic Efficiency

Definition: η_isen = Isentropic work / Actual work From temperatures: η_isen = (T₂,isen - T₁) / (T₂,actual - T₁) From heads: η_isen = H_isen / H_actual Isentropic efficiency DECREASES as pressure ratio increases for the same machine. Not suitable for comparing compressors at different operating points.

Efficiency Conversion

Isentropic to Polytropic: η_p = ln(r^(k-1)/k) / ln[1 + (r^(k-1)/k - 1) / η_isen] Polytropic to Isentropic: η_isen = [r^(k-1)/k - 1] / [r^(n-1)/n - 1] Where n is found from: n/(n-1) = (k/(k-1)) × η_p Example: Given η_isen = 0.75, r = 2.5, k = 1.28 r^0.219 = 1.234 η_p = ln(1.234) / ln[1 + (0.234/0.75)] η_p = 0.210 / ln(1.312) = 0.210 / 0.272 = 0.77 (77%)

Always use polytropic efficiency when comparing compressors or evaluating multi-stage machines. Isentropic efficiency is acceptable only for quick single-stage estimates at known operating point.

5. Compressor Selection Criteria

API Standards Summary

Standard	Type	Key Requirements
API 617	Centrifugal	Rotor dynamics analysis, 8-10% surge margin, mechanical running test, dry gas seals
API 618	Reciprocating	Pulsation analysis, rod load limits, 3-year valve life, CSPF limits
API 619	Screw	Rotor clearances, bearing life, oil separation (if flooded)

When to Choose Each Type

Select Centrifugal When:

Flow > 2,000 ACFM (optimal > 5,000 ACFM)
Continuous, steady operation
Moderate pressure ratio (1.5–3.0 per stage)
High reliability required (>98% availability)
Minimal maintenance capacity

Select Reciprocating When:

Flow < 5,000 ACFM
High pressure ratio needed (> 3:1 per stage)
Variable flow requirements
High discharge pressure (> 1,500 psig)
Best efficiency critical

Select Screw When:

Flow 500–10,000 ACFM
Dirty or wet gas (oil-flooded)
Simple operation, minimal training
Lower capital cost priority
Short delivery time needed

Staging Guidelines

Optimal Staging (Equal Work): For overall ratio r_total with N stages: r_stage = r_total^1/N Stage count estimate: N = ln(r_total) / ln(r_max,stage) Example: r_total = 10:1, centrifugal (r_max = 2.5:1) N = ln(10) / ln(2.5) = 2.30 / 0.92 = 2.5 → Use 3 stages r_stage = 10^1/3 = 2.15:1 per stage Intercooling: Between stages, cool discharge back to near suction T. Power savings: 10–20% vs. no intercooling.

6. Practical Design Considerations

Surge Protection (Centrifugal)

Surge occurs when flow drops below the minimum stable point, causing flow reversal, vibration, and potential damage.

Surge Prevention: Operate at: Q_operating ≥ Q_surge × 1.10 (10% margin) Anti-surge control: 1. Monitor flow, head, and speed 2. Calculate surge parameter: σ = Q / (N × √H) 3. If σ approaches surge limit → open recycle valve 4. Recycle from discharge to suction Turndown methods: • Variable speed: Best efficiency, 60–100% flow • Inlet guide vanes (IGV): 70–100% flow • Recycle: Minimum flow protection (inefficient)

Capacity Control (Reciprocating)

Clearance pockets: Add volume to reduce volumetric efficiency → 40–100% capacity
Valve unloaders: Hold suction valve open → 25%, 50%, 75%, 100% steps
Speed control: VFD or variable speed driver → 10–100% continuous

Volumetric Efficiency (Reciprocating)

Volumetric Efficiency: η_v = 1 - C × [r^1/k - 1] - L Where: C = clearance ratio (typically 0.05–0.15) r = compression ratio k = specific heat ratio L = losses (typically 0.03–0.05) Higher ratio → lower volumetric efficiency. Limits practical single-stage ratio to ~4:1.

Actual Volumetric Flow

ACFM at Suction Conditions: ACFM = SCFM × (P_std/P₁) × (T₁/T_std) × Z₁ Where: SCFM = standard flow (14.696 psia, 60°F) P₁ = suction pressure (psia) T₁ = suction temperature (°R) Z₁ = compressibility at suction Note: Higher Z → MORE actual volume (real gas less dense). Always use actual volume for compressor sizing!

Driver Selection

Driver	Efficiency	Best Application
Electric motor	94–97%	Plant with power, constant speed
Gas turbine	28–38%	Remote pipeline stations, variable speed
Gas engine	35–42%	Gathering, <3,000 HP
Steam turbine	30–40%	Refinery with steam system

Design checklist: (1) Size driver for 110% BHP, (2) Verify discharge temp < material limits, (3) Install suction scrubber, (4) Provide anti-surge or capacity control, (5) Specify pulsation dampeners for reciprocating.