In turbomachinery design, from centrifugal pumps and axial compressors to expander turbines, engineers have always looked for a safe shortcut to predict machine performance away from the original test point. For decades, that shortcut has been the affinity laws: simple, elegant, and easy to memorise.
Yet behind their simplicity lies an assumption that is often forgotten: fluid density is treated as constant. Once that assumption breaks down, especially in systems handling real gases whose density changes strongly with pressure and temperature, engineers must move from affinity-based scaling to far more demanding real-gas solvers.
This briefing explains why that transition is necessary, at what point the affinity laws start to fail, and how modern real-gas solvers handle the complexity of variable fluid density. For the simpler case of fans and blowers, see our guides to fan affinity laws and the density correction affinity laws.
Table of Contents
Understanding the Affinity Laws and Their Limits
The affinity laws are scaling relations that connect the flow rate (Q), head (H), and power (P) of a fluid machine to changes in rotational speed (N) or impeller diameter (D). In their most common form:
| Quantity | Relationship |
|---|---|
| Flow rate | Q₂/Q₁ = N₂/N₁ |
| Head | H₂/H₁ = (N₂/N₁)² |
| Shaft power | P₂/P₁ = (N₂/N₁)³ |
These relations are derived from dimensional analysis and dynamic similarity for machines operating in the incompressible regime. They assume that the working fluid has constant density along the flow path, that the head, flow, and power coefficients do not change with Reynolds or Mach number, and that efficiency stays the same across the compared operating range.
For centrifugal pumps handling water or light hydrocarbon liquids across a moderate pressure range, these assumptions are reasonably valid. Liquid density does change with temperature, but the change is small and often ignored in early design. This is where the affinity laws are genuinely useful: fast, requiring no CFD, and accurate enough for scaling within a reasonable operating range.
Problems arise when the same approach is applied to a compressible fluid. For compressors and blowers, density can no longer be treated as fixed. Gas density is a strong function of pressure and temperature, and at high pressure or low temperature, real gas behaviour deviates significantly from the ideal-gas model. The further the system operates from the design reference condition, the larger the error produced by standard affinity scaling.
Why Variable Density Breaks the Underlying Assumptions
When density changes along the flow path inside an impeller or blade row, several physical effects shift at once. The affinity laws were never designed to capture that interaction.
1. Mach number becomes a design parameter
In compressors with high pressure ratios, the flow velocity relative to the blades can approach or even exceed the local speed of sound. As a general rule of thumb, once the relevant Mach number (such as the blade-tip or machine Mach number) rises above roughly 0.3, compressibility effects start to influence the pressure distribution on blade surfaces. Performance coefficients that the affinity laws assume to be constant begin shifting with rotational speed.
2. Specific volume changes along the machine
In a multistage centrifugal compressor, the gas entering the first stage has a much larger specific volume than the gas leaving the last stage, because it has already been compressed several times. The blade geometry that is ideal for the first stage is therefore not ideal for later stages, something a single set of affinity coefficients cannot represent.
3. The compressibility factor deviates from 1
The compressibility factor Z departs from the ideal value (Z = 1), especially near the fluid’s critical point or at high pressure. For natural gas, supercritical CO₂, refrigerants, and light hydrocarbon mixtures common in oil and gas and petrochemical plants, the deviation can range from a few percent at moderate pressures to very large values near the critical point. Because density is linked to Z through the equation of state, any error in estimating Z propagates directly into predicted head, power, and efficiency.
4. Real-gas properties are not constant
Specific heats (Cp, Cv) and the isentropic exponent (k = Cp/Cv) vary with pressure and temperature. Affinity-based compressor methods often rely on a constant polytropic head coefficient, but when k changes significantly along the compression process, that assumption no longer holds with enough accuracy for high-precision design.
The Tipping Point: When to Switch to Real-Gas Solvers
A practical question for design teams is when to abandon the simple approach in favour of real-gas solvers. Four indicators are commonly used in industry:
- Pressure ratio per stage. Above roughly 1.5 to 2.0, compressibility effects are usually too large for simple Mach or Reynolds corrections.
- Deviation of Z. Once Z departs by more than about 5 to 10% from the design reference condition, head and power predictions from standard scaling start producing errors that are unacceptable for contractual guarantees.
- Proximity to the critical point. Near the critical or retrograde region, density can change drastically with small changes in pressure or temperature, and fluid behaviour becomes highly non-linear.
- Mach number. High blade-tip or machine Mach numbers mean the performance coefficients can no longer be treated as constant.
These thresholds are industry rules of thumb, not universal limits, so confirm them against the manufacturer’s data and the applicable test code. The most challenging case occurs near the critical region, as in supercritical CO₂ cycles, mixed-refrigerant refrigeration, or high-pressure natural gas processing. There, the ideal-gas equation of state, or even simple corrections to it, cannot capture the physics.
As a summary, the table below shows which approach fits which condition.
| Condition | Suitable approach |
|---|---|
| Very low pressure ratio (fans and blowers, a few percent), Z close to 1 | Affinity laws with density correction |
| Moderate pressure ratio, Z close to 1 | Affinity laws with empirical Mach and Reynolds corrections, or manufacturer performance maps |
| High pressure ratio per stage, large Z deviation, or near the critical point | Real-gas solvers with an equation of state (1D mean-line first, then 3D CFD) |
Anatomy of Real-Gas Solvers
Real-gas solvers replace the ideal-gas assumption with an equation of state (EOS) that represents real molecular behaviour far more accurately. Common choices in industry include Peng-Robinson, Soave-Redlich-Kwong (SRK), and Benedict-Webb-Rubin (BWR). For specific fluids, reference equations of state are used instead, such as Span-Wagner for CO₂ and IAPWS-95 (or the faster industrial formulation IAPWS-IF97) for water and steam. Open-source tools such as CoolProp implement many of these models and are widely used for property calculations.
The EOS is then integrated into a Computational Fluid Dynamics (CFD) solver as the fluid property model, replacing the default ideal-gas model. Every mesh cell inside an impeller passage or blade row has its density, viscosity, specific heat, and local speed of sound recalculated from local pressure and temperature, rather than assumed uniform.
This carries real computational consequences. The solver must resolve the Navier-Stokes equations together with the energy equation and a turbulence model, where fluid properties are no longer constants but depend on the solution itself. The extra non-linearity makes convergence harder and often calls for more conservative under-relaxation or more sophisticated coupled solvers.
To balance accuracy and speed, many real-gas solvers use property lookup tables. Instead of evaluating the EOS at every iteration, the solver builds tables of density, enthalpy, entropy, and speed of sound as functions of pressure and temperature at the start, then interpolates during the solution. This speeds up computation without sacrificing much accuracy, provided the table resolution is fine enough around critical operating regions.
Impact on Head, Power, and Efficiency Predictions
One of the most important consequences of real-gas solvers is a shift in the very definition of head. For incompressible fluids, head is energy per unit weight of fluid, expressed as a column height. For compressible gases, head splits into polytropic head and isentropic head, both of which depend on the actual thermodynamic path of compression, not just the start and end points.
When density varies significantly along that path, polytropic efficiency can no longer be assumed constant across the operating range. Shaft power calculations usually require numerical integration along the actual compression path using real-gas property data, rather than a simple closed-form formula.
This means two compressors with identical geometry and speed, operating on different gas compositions (for example, natural gas with different CO₂ content), can have significantly different performance curves. The standard affinity laws cannot predict that.
A frequent industrial case is acid gas injection or CO₂ compression for carbon capture and storage (CCS). The CO₂ is often compressed to near or beyond supercritical conditions, where density changes drastically within a narrow pressure range. Designing for such duty with the affinity laws alone is not realistic. Real-gas solvers with an accurate EOS are necessary from the conceptual design stage.
Implications for the Design Process and Performance Guarantees
Moving to real-gas solvers is not only a technical decision. It also affects the whole design process and the commercial side of a turbomachinery project.
- Aerodynamic design. Blade and impeller geometry must be re-optimised for density variation along the flow path, not at a single average point. This often drives fully three-dimensional blade design, with inlet and outlet angles tailored for each stage.
- Validation and testing. Factory acceptance tests for real-gas compressors often use a surrogate gas, because the actual process gas may be unavailable or unsafe in a test facility. Engineers use EOS-based similarity methods to translate the test results into predicted performance with the actual gas, in line with test codes such as ASME PTC 10. This needs specialised thermodynamic software and is far more complex than applying the affinity laws.
- Performance guarantees. Clients and vendors should agree from the outset on the method and EOS software used to verify performance. Different equations of state (for example, Peng-Robinson versus BWR) produce slightly different density and Z-factor predictions, which in turn change the guaranteed head and power figures.
Toward a Hybrid Approach: When Affinity Still Matters
Even though real-gas solvers become a necessity under extreme operating conditions, the affinity laws are not obsolete. They remain a valid and efficient tool for quick estimates in early feasibility studies, for scaling between units of similar geometry over a narrow operating range, and for fluids with stable density, such as water, light oil, and low-pressure air.
An increasingly common approach is a hybrid one. Use the affinity laws with empirical corrections (Mach number, Reynolds number, or average compressibility factor) for rapid estimates at the conceptual stage, then validate and refine the design using real-gas solvers during detailed engineering. This balances the speed of early iteration with the accuracy required at the final stage, without running expensive CFD for every concept. For the fan-and-blower end of the scale, you can try our affinity law calculator, and for background on fans, read the significance of fans in turbomachinery.
Conclusion
The affinity laws remain one of the most elegant tools in the turbomachinery engineer’s toolbox: simple, fast, and accurate enough for many incompressible applications. But once a system handles gas whose density varies significantly with pressure and temperature, particularly at high pressure ratios or near the critical point, the assumptions behind the affinity laws collapse one by one.
Real-gas solvers, using equations of state that represent real molecular behaviour, are the answer. They bring far greater computational and conceptual complexity, but they are the only way to predict and guarantee the performance of natural gas compressors, supercritical CO₂ systems, and high-pressure refrigeration with the accuracy modern industry demands.
For design teams, the key is not to choose one approach exclusively. It is to understand precisely when the validity limits of the affinity laws are exceeded, and when the time and computing resources for real-gas solvers are truly warranted.
Frequently Asked Questions
What is the difference between the affinity laws and real-gas solvers?
The affinity laws are simple scaling relations that assume constant fluid density, making them suitable for incompressible fluids or gases over a narrow pressure range. Real-gas solvers calculate density, enthalpy, and other thermodynamic properties locally at every point in the flow domain using an equation of state, so they can capture large density variation caused by changes in pressure and temperature.
Can the affinity laws still be used for gas compressors?
Only in limited cases. For very low pressure ratios with Z close to 1, as in fans and blowers, the affinity laws with density correction work well. For moderate pressure ratios, add empirical Mach and Reynolds corrections or use manufacturer performance maps. Beyond roughly a 1.5 to 2.0 pressure ratio per stage, or with a large Z deviation, use real-gas solvers. The table in this article summarises the three cases.
How do I know whether a gas behaves ideally or as a real gas?
The most practical indicator is the compressibility factor Z. If Z stays close to 1 across the whole operating range of pressure and temperature, the gas can be treated as ideal. If Z deviates significantly from 1, especially at high pressure or near the critical temperature, a real-gas approach is required.
Which equations of state are most commonly used in turbomachinery?
Peng-Robinson and Soave-Redlich-Kwong are the most popular for hydrocarbons and natural gas because they balance accuracy and simplicity. For specific fluids such as CO₂ or water and steam, reference equations such as Span-Wagner, IAPWS-95, or the industrial IAPWS-IF97 are used for their higher accuracy across the full phase range.
Do real-gas solvers always require full CFD?
No. At the conceptual stage, a one-dimensional mean-line solver using EOS-based property tables is often enough. Full 3D CFD with a real-gas model is typically reserved for detailed engineering or final validation.
Why do factory acceptance tests often use a surrogate gas?
The actual process gas, such as high-pressure natural gas or acid gas mixtures, is often unavailable, unsafe, or impractical to use in a test facility. A surrogate gas (for example, a nitrogen-CO₂ mixture) is used instead, and the results are translated into predicted performance with the actual gas using EOS-based similarity methods, in line with test codes such as ASME PTC 10.
What is the biggest risk of ignoring real-gas effects?
Predicted head, power, and efficiency can deviate from actual operation. This can lead to a machine that misses its contractual performance targets, a drive motor of the wrong size, or even surge and stall risk caused by an inaccurately predicted performance curve.
Can a hybrid approach fully replace real-gas solvers?
Not entirely. A hybrid approach speeds up iteration at the conceptual stage, but for extreme operating conditions, high pressure ratios, near-critical operation, or contractual performance guarantees, final validation with real-gas solvers is still required.