In the world of turbomachinery design — from centrifugal pumps and axial compressors to expander turbines — engineers have always looked for a safe shortcut to predict machine performance under operating conditions that differ 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 significantly with pressure and temperature — engineers must abandon the affinity-based approach and move to far more computationally and conceptually demanding real-gas solvers. This article examines why that transition is necessary, at what point the Affinity Laws start to fail, and how modern real-gas solvers handle the complexity introduced by variable fluid density in turbomachinery systems.

Understanding the Affinity Laws and Their Limits

The Affinity Laws are a set of scaling relations that connect 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:

 Q₂/Q₁ = N₂/N₁
 H₂/H₁ = (N₂/N₁)²
 P₂/P₁ = (N₂/N₁)³

These relations are derived from dimensional analysis and dynamic similarity for fluid machines operating in the incompressible regime. The underlying assumptions are that the working fluid has constant density along the flow path, that performance coefficients (head coefficient, flow coefficient, power coefficient) remain unchanged with Reynolds number 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, this assumption is reasonably valid. Liquid density does change with temperature, but the change is small and often ignored in early-stage design. This is where the Affinity Laws are genuinely useful: fast, requiring no CFD simulation, and accurate enough for scaling purposes within a reasonable operating range.

Problems arise when the same machine type is applied to a compressible fluid — gas. For compressors and blowers, fluid 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 the standard affinity approach.

 Why Variable Density Breaks the Underlying Assumptions

When density changes along the flow path inside an impeller or compressor blade row, several physical phenomena shift simultaneously — an interaction the Affinity Laws were never designed to capture.

First, local Mach number becomes a parameter that cannot be ignored. In gas compressors with high pressure ratios, the flow velocity relative to the blades can approach or even exceed the local speed of sound. Once the Mach number exceeds roughly 0.3, compressibility effects begin to significantly affect the pressure distribution on blade surfaces, and performance coefficients once assumed constant under the Affinity Laws start shifting with rotational speed.

Second, changing density causes the gas’s specific volume to vary along the flow path in multistage machines. In a multistage centrifugal compressor, the gas entering the first stage has a much larger specific volume than the gas leaving the final stage, since the gas has already been compressed multiple times. This means the optimal blade geometry for the first stage is no longer optimal for later stages — something a single set of affinity coefficients simply cannot represent.

Third, the compressibility factor Z deviates from the ideal value (Z = 1), especially near the fluid’s critical point or at high pressure. For natural gas, supercritical CO₂, refrigerants, or light hydrocarbon mixtures commonly found in oil and gas and petrochemical industries, the deviation in Z can reach 20–40% from the ideal value. Because density is directly linked to Z through the equation of state, any error in estimating Z is directly propagated into errors in predicted head, power, and efficiency.

Fourth, real-gas thermodynamic properties such as specific heats (Cp, Cv) and the isentropic exponent (k = Cp/Cv) are not constant; both vary with pressure and temperature. Compressor-oriented Affinity Law approaches often rely on a “constant polytropic head coefficient” assumption, but when k varies significantly along the compression process, this assumption no longer holds with sufficient accuracy for high-precision design.

The Tipping Point: When Engineers Must Switch Approaches

A practical question for design teams is: at what point should the simple affinity approach be abandoned in favour of a real-gas solver? Several practical indicators are commonly used in industry, including pressure ratio per stage, deviation of the compressibility factor, and proximity to the fluid’s thermodynamic critical point.

For compressors with a per-stage pressure ratio above roughly 1.5–2.0, compressibility effects are usually already large enough that simple corrections (such as Mach or Reynolds number correction factors) are no longer adequate, making real-gas equation-of-state-based modelling a necessity. Similarly, once the compressibility factor Z deviates by more than about 5–10% from the design reference condition, head and power predictions based on standard affinity scaling begin to produce errors that are unacceptable for contractual performance guarantees.

The most challenging case occurs when the operating point approaches the critical or retrograde region on the fluid’s phase diagram — a situation commonly encountered in supercritical CO₂ cycle applications, refrigeration with mixed refrigerants, or high-pressure natural gas processing. In this region, density can change drastically with relatively small changes in pressure or temperature, and fluid behavior becomes highly non-linear. The ideal-gas equation of state, or even simple corrections to it, is completely unable to capture this phenomenon.

 Anatomy of a Real-Gas Solver

Real-gas solvers used in modern turbomachinery design replace the ideal-gas assumption with an equation of state (EOS) that far more accurately represents real molecular gas behavior. Commonly used EOS models in industry include Peng-Robinson, Redlich-Kwong-Soave, Benedict-Webb-Rubin (BWR), and, for specific fluids such as CO₂ or water, high-accuracy reference equations of state such as Span-Wagner developed specifically for accuracy across the entire phase range.

This equation of state is then integrated into a Computational Fluid Dynamics (CFD) solver as the fluid property model, replacing the default ideal-gas model. Every point in the computational domain — every mesh cell inside an impeller passage or blade row — has its density, viscosity, specific heat, and local speed of sound recalculated based on local pressure and temperature, rather than assumed uniform.

This approach carries significant computational consequences. The solver must resolve a coupled system of Navier-Stokes equations together with the energy equation and a turbulence model, where fluid properties are no longer constants but variables dependent on the solution itself. This introduces additional non-linearity that makes numerical convergence harder to achieve, and often requires more conservative under-relaxation schemes or more sophisticated coupled solvers compared to conventional segregated solvers.

In addition, fluid property lookup tables are often used as a compromise between accuracy and computational speed. Instead of evaluating the EOS directly at every iteration (which is computationally expensive), the solver builds tables of density, enthalpy, entropy, and speed of sound as functions of pressure and temperature at the start of the simulation, then interpolates during the iterative solution process. This table-based approach can significantly speed up computation without sacrificing too much accuracy, provided the table resolution is sufficiently fine around critical operating regions.

 Impact on Head, Power, and Efficiency Predictions

One of the most important consequences of using a real-gas solver is a shift in the very definition of “head.” For incompressible fluids, head is simply defined as energy per unit weight of fluid (expressed as a column height of fluid). For compressible gases, however, the concept of head must be split into polytropic head and isentropic head, both of which depend on the actual thermodynamic path of the compression process — 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, and shaft power calculations require numerical integration along the actual compression path based on real-gas property data, rather than a simple closed-form formula. This means two compressors with identical geometry operating on two different gas compositions (for example, natural gas with different CO₂ content) can have significantly different performance curves even with identical speed and geometry — something the standard Affinity Laws cannot predict.

A case frequently encountered in industry is acid gas injection or CO₂ compression for carbon capture and storage (CCS) applications. In these applications, CO₂ is often compressed to near or beyond supercritical conditions, where density changes drastically within a narrow pressure range. Designing a compressor for such an application is practically impossible using the Affinity Laws alone; a real-gas solver with an accurate EOS becomes an absolute necessity from the conceptual design stage onward.

Implications for the Design Process and Performance Guarantees

Shifting to a real-gas solver is not merely a technical decision — it also carries implications for the entire design process and the commercial aspects of a turbomachinery project.

From an aerodynamic design perspective, blade and impeller geometry needs to be re-optimised considering density variation along the flow path, rather than at a single average operating point. This often drives the use of fully three-dimensional (3D) blade design, with inlet and outlet angles tailored per stage, rather than a two-dimensional blade design scaled uniformly.

From a validation and testing perspective, factory acceptance testing for real-gas compressors often requires a surrogate gas, since the actual process gas is not always available or safe to use in test facilities. Engineers must use EOS-based similarity methods to translate test results obtained with the surrogate gas into predicted performance with the actual process gas — a process far more complex than simply applying the Affinity Laws, and one that requires specialised thermodynamic simulation software validated against standards such as ASME PTC-10.

From a contractual performance guarantee perspective, clients and vendors need to agree from the outset on the method and EOS software used to validate performance, because choosing a different EOS (for example, Peng-Robinson versus BWR) can produce slightly different density and Z-factor predictions, which in turn affect the calculated head and power figures being guaranteed.

 Toward a Hybrid Approach: When Affinity Still Matters

Even though real-gas solvers become a necessity under extreme operating conditions, it’s important to note that the Affinity Laws are not entirely obsolete. For quick estimation during early feasibility studies, for scaling between units of similar geometry over a narrow operating range, or for fluids with relatively stable density (water, light oil, low-pressure air), the Affinity Laws remain a valid and efficient tool.

An increasingly common approach in industry is a hybrid one: using the Affinity Laws with empirical correction factors (such as Mach number, Reynolds number, or average compressibility factor corrections) for rapid estimation at the conceptual stage, then validating and refining the design using a full real-gas CFD-based solver during detailed engineering. This layered approach balances the speed of design iteration in the early stages with the high accuracy required at the final stage, without needing to run expensive CFD simulations for every conceptual iteration.

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 fluid applications. But once a system shifts to gas with density that varies significantly with pressure and temperature — particularly at high pressure ratios, extreme operating pressures, or conditions near the thermodynamic critical point — the fundamental assumptions behind the Affinity Laws begin to collapse one by one.

Real-gas solvers, using equations of state that accurately represent real molecular gas behaviour, are the answer to this limitation. Although they bring far greater computational and conceptual complexity, this approach is the only way to ensure that turbomachinery designs — whether natural gas compressors, supercritical CO₂ systems, or high-pressure refrigeration — can be predicted and guaranteed with the accuracy demanded by modern industry.

For design teams, the key to success is not choosing one approach exclusively, but understanding precisely when the validity limits of the Affinity Laws are exceeded, and when the investment of time and computational resources in a real-gas solver is truly warranted.

 FAQ

1. What is the fundamental difference between the Affinity Laws and a real-gas solver?
The Affinity Laws are simple scaling relations that assume constant fluid density, making them suitable only for incompressible fluids or gases within a narrow pressure range. A real-gas solver calculates density, enthalpy, and other thermodynamic properties locally at every point in the flow domain using an equation of state (EOS), allowing it to capture significant density variation caused by changes in pressure and temperature.

2. Can the Affinity Laws still be used for gas compressors?
Yes, as long as the pressure ratio per stage is low (typically below 1.5) and the deviation of the compressibility factor Z from the reference condition is small (below about 5%). Beyond that range, head and power predictions based on standard affinity scaling start to produce significant errors.

3. How do you know whether a gas behaves “ideally” or as a “real gas”?
The most practical indicator is the value of the compressibility factor Z. If Z stays close to 1 across the entire operating range of pressure and temperature, the gas can be approximated as ideal. If Z deviates significantly from 1 — especially at high pressure or temperatures near the critical point — a real-gas approach is required.

4. Which equation of state (EOS) is most commonly used in the turbomachinery industry?
Peng-Robinson and Redlich-Kwong-Soave are the most popular for hydrocarbons and natural gas because they balance accuracy with computational simplicity. For specific fluids such as CO₂ or water/steam, reference equations of state such as Span-Wagner or IAPWS-IF97 are used because of their much higher accuracy across the full phase range.

5. Does a real-gas solver always require a full CFD simulation?
Not always. For quick estimates, a one-dimensional (1D mean-line) solver using EOS-based property tables is often sufficient at the conceptual design stage. Full three-dimensional CFD simulation with a real-gas model is typically only needed during detailed engineering or final design validation.

6. Why do factory acceptance tests for real-gas compressors often use a surrogate gas?
Because the actual process gas (such as high-pressure natural gas or acid gas mixtures) is often unavailable, unsafe, or impractical to use in a factory test facility. Instead, a surrogate gas (for example, a nitrogen-CO₂ mixture) is used, and the test results are then translated into the actual gas’s predicted performance using EOS-based similarity methods, in accordance with standards such as ASME PTC-10.

7. What is the most significant risk of ignoring real-gas effects in design?
The main risk is that predicted head, power, and efficiency deviate from actual operating conditions, which can result in the machine failing to meet contractual performance targets, incorrectly sized drive motors, or even surge/stall risk in compressors due to an inaccurately predicted performance curve.

8. Can a hybrid approach (affinity plus empirical correction) fully replace a real-gas solver?
Not entirely. A hybrid approach is useful for speeding up iteration at the conceptual stage, but for extreme operating conditions (high pressure ratios, near-critical conditions, or contractual performance guarantees), final validation with a full real-gas solver is still required to ensure adequate accuracy.