1. Introduction to Compressor Thermodynamics
Understanding thermodynamics is essential for accurate centrifugal compressor design and performance evaluation. Unlike ideal gas behavior, real gases exhibit deviations that significantly impact head calculations, power requirements, and discharge temperature predictions.
The fundamental principle of compression involves adding energy to gas molecules, increasing their pressure and temperature. The relationship between pressure, temperature, and volume during this process depends on the thermodynamic path taken.
Why Thermodynamics Matters
Incorrect thermodynamic assumptions can lead to undersized compressors, inadequate cooling systems, and inefficient operations. For natural gas at typical pipeline conditions, real gas effects can cause 10-20% deviation from ideal gas calculations.
2. Thermodynamic Processes
Isentropic (Adiabatic Reversible) Process
An isentropic process is an idealized compression where no heat is transferred (adiabatic) and the process is reversible (no friction losses). This represents the theoretical minimum work required for compression.
Isentropic Relationship
T₂/T₁ = (P₂/P₁)^((k-1)/k)
Where k = Cp/Cv (ratio of specific heats)
For an ideal gas undergoing isentropic compression:
- Entropy remains constant (ΔS = 0)
- Temperature rises predictably with pressure
- PVᵏ = constant along the compression path
Polytropic Process
A polytropic process better represents actual compression in centrifugal compressors. It accounts for heat transfer and irreversibilities through the polytropic exponent (n), which differs from the isentropic exponent (k).
Polytropic Relationship
T₂/T₁ = (P₂/P₁)^((n-1)/n)
Where n = polytropic exponent
The relationship between polytropic and isentropic exponents:
n = k × ηₚ / (1 - k(1 - ηₚ))
ηₚ = polytropic efficiency
Isothermal Process
An isothermal process maintains constant temperature throughout compression. While theoretically requiring the least work, it's impractical for actual compressors due to the need for continuous heat removal.
3. Enthalpy and Entropy
Enthalpy (H)
Enthalpy represents the total heat content of a gas, including internal energy and flow work. In compressor calculations, the change in enthalpy equals the work input:
First Law for Open Systems
W = ṁ × (h₂ - h₁)
For adiabatic flow with no change in kinetic or potential energy
Entropy (S)
Entropy is a measure of molecular disorder and energy quality. During compression:
- Isentropic: No entropy change (ideal)
- Actual: Entropy increases due to friction, turbulence, and irreversibilities
The T-S Diagram
Temperature-entropy diagrams visualize compression paths. The vertical distance between the actual and isentropic paths represents the inefficiency of the process.
4. Real Gas Behavior
Compressibility Factor (Z)
Real gases deviate from ideal gas behavior, particularly at high pressures and low temperatures. The compressibility factor Z accounts for these deviations:
Real Gas Equation of State
PV = ZnRT
Z = 1 for ideal gas; Z < 1 means gas is more compressible than ideal
Reduced Properties
Z-factor depends on reduced pressure and temperature:
- Reduced Pressure: Pᵣ = P / Pᶜ (actual / critical)
- Reduced Temperature: Tᵣ = T / Tᶜ (actual / critical)
The Standing-Katz correlation and equations of state (SRK, Peng-Robinson) calculate Z from these reduced properties.
Impact on Compressor Calculations
Real gas effects influence:
- Head calculation: Use average Z (Z₁ + Z₂)/2
- Density: ρ = PM/(ZRT)
- Power: Higher than ideal gas predictions
- Temperature rise: Different from isentropic prediction
5. Head and Work
Polytropic Head
Polytropic head represents work per unit mass added to the gas. It's the primary design parameter for centrifugal compressors:
GPSA Polytropic Head Equation
Hₚ = (Zₐᵥg × R × T₁ / MW) × (n/(n-1)) × [(P₂/P₁)^((n-1)/n) - 1]
Zₐᵥg = average compressibility
R = 1545.35 ft-lbf/(lbmol-°R)
MW = molecular weight (lb/lbmol)
Isentropic Head
Isentropic head uses k instead of n:
Hₛ = (Zₐᵥg × R × T₁ / MW) × (k/(k-1)) × [(P₂/P₁)^((k-1)/k) - 1]
Relationship Between Heads
Hₚ = Hₛ × (ηₛ / ηₚ)
ηₛ = isentropic efficiency, ηₚ = polytropic efficiency
6. Efficiency Concepts
Polytropic Efficiency (ηₚ)
Polytropic efficiency represents the ratio of reversible work to actual work for a small (infinitesimal) compression step. It remains constant regardless of pressure ratio, making it ideal for comparing compressors:
ηₚ = [(n-1)/n] / [(k-1)/k]
Typical polytropic efficiencies:
- Modern centrifugal: 75-85%
- High-efficiency designs: 80-88%
- Older or off-design: 65-75%
Isentropic Efficiency (ηₛ)
Isentropic efficiency compares actual work to ideal isentropic work for the entire compression process:
ηₛ = (h₂ₛ - h₁) / (h₂ - h₁)
h₂ₛ = enthalpy at discharge if process were isentropic
Key Difference
For the same compressor, isentropic efficiency decreases with pressure ratio while polytropic efficiency remains constant. This makes polytropic efficiency the preferred metric for centrifugal compressor specification.
Conversion Between Efficiencies
ηₛ = [(P₂/P₁)^((k-1)/k) - 1] / [(P₂/P₁)^((k-1)/(k×ηₚ)) - 1]
7. Discharge Temperature
Predicting discharge temperature is critical for equipment design, material selection, and downstream process requirements:
Polytropic Discharge Temperature
T₂ = T₁ × (P₂/P₁)^((k-1)/(k×ηₚ))
Temperature Limitations
- Seals: Typically limit to 300-350°F
- Bearings: Oil temperature limits apply
- Casing: Material-dependent limits
- Process: May require intercooling above certain temperatures
High Temperature Considerations
Discharge temperatures above 350°F often require special materials, enhanced cooling, or splitting compression into multiple stages with intercooling.
8. Equations of State
Modern compressor analysis uses equations of state (EOS) to model real gas behavior more accurately than the simple Z-factor approach:
Soave-Redlich-Kwong (SRK)
Widely used for hydrocarbon systems:
P = RT/(V-b) - a(T)/(V(V+b))
Peng-Robinson (PR)
Better for liquid density predictions:
P = RT/(V-b) - a(T)/(V²+2bV-b²)
When to Use EOS vs. Correlations
- Simple correlations: Quick estimates, single-component gases
- SRK/PR: Multi-component mixtures, high accuracy needs
- Specialized EOS: Acid gas, CO₂, near-critical conditions
References
- GPSA, Section 13 - Compressors and Expanders
- API 617 - Axial and Centrifugal Compressors and Expander-Compressors
- Campbell, J.M. - Gas Conditioning and Processing, Volume 2
- Bloch, H.P. - A Practical Guide to Compressor Technology