Real Gas Effects
Real Gas Effects: Theoretical Foundations
Overview
Professor, "real gas effects" essentially mean cases where the ideal gas assumption doesn't hold, right? In what situations does this become a problem?
Broadly speaking, there are two main situations. One is at high temperatures where gas molecules undergo vibrational excitation, dissociation, or ionization (e.g., hypersonic re-entry, plasma, etc.). The other is at high pressure and low temperature where intermolecular forces and molecular volume can no longer be ignored (e.g., supercritical CO₂ cycles, LNG processes, etc.). In both cases, the deviation from the ideal gas equation of state $pv = RT$ is fundamental.
We measure the deviation using the compressibility factor $Z$, right?
Exactly. The compressibility factor is defined as
For an ideal gas, $Z = 1$. For high-pressure natural gas, $Z \approx 0.8$, and for supercritical CO₂, $Z$ can drop as low as 0.2-0.5.
Equation of State (EOS)
What kind of equations of state are used instead of the ideal gas law?
Let's look at some representative EOS (Equation of State).
van der Waals equation:
Here, $a$ represents intermolecular attraction, and $b$ represents the excluded volume. This is historically important but has limited accuracy. In practice, Peng-Robinson (PR) or Soave-Redlich-Kwong (SRK) are widely used.
Peng-Robinson EOS:
Here, $\kappa = 0.37464 + 1.54226\omega - 0.26992\omega^2$, and $\omega$ is the acentric factor.
So if you have the critical temperature $T_c$, critical pressure $p_c$, and acentric factor $\omega$, you can use it for any substance, right?
Yes. For CO₂, $T_c = 304.1$ K, $p_c = 7.38$ MPa, $\omega = 0.225$. However, the PR-EOS tends to underestimate liquid density by 5-15%. Sometimes the Peneloux volume correction term is added to compensate for this.
High-Temperature Air Thermochemical Model
How are real gas effects on the high-temperature side modeled?
For air, the following phenomena occur sequentially depending on temperature.
| Temperature Range | Phenomenon | Effect |
|---|---|---|
| < 800 K | Ideal gas-like | $\gamma \approx 1.4$ |
| 800-2500 K | Vibrational excitation of O₂ | $\gamma$ decreases |
| 2500-4000 K | Dissociation of O₂ | Generation of O atoms |
| 4000-9000 K | Dissociation of N₂ | Generation of N atoms |
| > 9000 K | Ionization | Generation of e⁻, N⁺, O⁺ |
Assuming chemical equilibrium, the composition of each chemical species is determined by minimizing the Gibbs free energy. For finite-rate reaction models, Arrhenius parameters for each reaction are specified.
The specific heat ratio $\gamma$ changing with temperature seems like it would have a big impact.
Exactly. When the post-shock temperature exceeds 3000 K, $\gamma$ drops from 1.4 to 1.1-1.2. This causes the shock wave angle and density ratio to differ significantly from ideal gas predictions.
What van der Waals realized in 1873—molecules have "size"
The theoretical foundation for real gases was laid by the Dutch physicist Johannes van der Waals. In his 1873 doctoral thesis, he proposed the revolutionary idea for its time that "gas molecules have a finite size and there are attractive forces between them." The van der Waals equation of state derived from this was the first practical model capable of describing gas behavior in high-pressure, low-temperature regions. Modern engineering uses Peng-Robinson and Redlich-Kwong equations, which are refinements of van der Waals' concept. So, a doctoral thesis from 150 years ago directly influences the choice of equations of state in today's compressible CFD.
Computational Methods for Real Gas Effects
Numerical Implementation of EOS
When incorporating a real gas EOS into CFD, are there any computational issues?
There are several important points. For ideal gases, conversion from conservative variables ($\rho, \rho\mathbf{u}, \rho E$) to temperature or pressure can be done analytically, but for PR-EOS and others, iterative calculations become necessary.
Specifically, when internal energy $e$ and density $\rho$ are given, temperature $T$ is found from
using Newton's method. Here, $e_{departure}$ is the departure function derived from the EOS; for PR-EOS it is
This iterative calculation is required for each cell × time step, so computational cost increases.
How much slower does the calculation become compared to ideal gas?
Generally about 2-5 times slower. Since EOS evaluation counts dominate, using a look-up table method for speedup is common in practice. A 2D table of temperature and pressure is pre-computed, and at runtime, only interpolation is performed.
Numerical Difficulties with Supercritical Fluids
What's numerically difficult about the supercritical state?
Near the critical point, thermodynamic properties change rapidly. Constant-pressure specific heat $c_p$ peaks near the pseudo-critical temperature, and density also changes abruptly. These steep changes cause numerical oscillations or divergence.
For example, under supercritical CO₂ conditions (p = 8 MPa, T ≈ 305 K), $c_p$ can jump to more than 10 times its normal value. Density can change drastically from 700 kg/m³ to 200 kg/m³ with just a few Kelvin change.
The changes are that drastic? Special techniques in the numerical scheme are needed then.
Exactly. The recommended approaches are as follows.
- Table method: Generate property tables from the NIST REFPROP database and access them via bilinear interpolation
- Implicit method: Due to large density changes, implicit pressure-density coupling is essential for stability
- Mesh resolution: Refine mesh in regions where pseudo-critical transition occurs
- Time step: Use adaptive time stepping to handle rapid density changes
Extension of Riemann Solvers
Can Roe scheme or HLLC be used for real gases as well?
They can be used, but modifications are needed. The ideal gas assumption must be removed when calculating the Roe average state. In the real gas Roe scheme, the average speed of sound is calculated using a generalized $\Gamma = v/(c_p)(\partial p / \partial T)_v$ as in
(e.g., Vinokur's formulation). HLLC has less dependence on the EOS and is easier to implement, so it is often preferred for real gases.
Considering implementation effort, HLLC is more practical, right?
Yes. OpenFOAM's rhoCentralFoam is based on the KNP scheme and is independent of the EOS, so it has the advantage that real gas calculations can be done simply by swapping the EOS.
The "Not Enough Grid Points" Problem in Real Gas Look-up Tables
Calculating real gas equations of state (van der Waals, Peng-Robinson, etc.) in real-time slows down CFD by tens of times. Therefore, in practice, look-up tables (LUT) are used where thermodynamic quantities are pre-computed on temperature-pressure grid points. The problem is the trade-off: "Too few table grid points lead to large interpolation errors, too many increase memory and loading time." An "adaptive table" that concentrates dense grids in high-temperature, high-pressure regions where shock waves pass and uses coarse grids in calm regions is efficient, but designing it itself becomes a piece of technical know-how.
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