Support for real-fluid sCO₂ thermophysical properties in heat-transfer simulations

[ztdepztdep]

I am planning to use FLEXI to simulate heat transfer and fluid flow of supercritical carbon dioxide (sCO₂), particularly near the pseudo-critical temperature.

In this region, the thermophysical properties of CO₂ vary very strongly and nonlinearly with temperature and pressure. For example, density, specific heat capacity, thermal conductivity, viscosity, compressibility, and speed of sound may change rapidly over a narrow temperature interval. Therefore, a constant-property model or a calorically perfect-gas model would not be appropriate.

My initial target case is single-phase sCO₂ flow at approximately 8 MPa, withthe temperature range crossing the pseudo-critical region.

The required properties include:

• density, ρ
• specific internal energy, e
• specific enthalpy, h
• specific heat capacities, cp and cv
• speed of sound, a
• dynamic viscosity, μ
• thermal conductivity, k

I am considering two possible approaches:

1. Implementing a real-fluid thermodynamic model, such as a Helmholtz-energy
   equation of state, Span–Wagner EOS, Peng–Robinson EOS, or an interface to
   an external property library such as CoolProp.

2. Generating a thermophysical-property table in advance and obtaining the
   required properties during the simulation through interpolation.

I would appreciate guidance on the following questions.

1. Existing real-fluid support

Does FLEXI currently provide any built-in support for:

• general real-gas equations of state;
• tabulated thermodynamic properties;
• user-defined thermodynamic models;
• temperature- and pressure-dependent transport properties;
• external property libraries such as CoolProp or REFPROP?

2. Thermodynamic consistency

If a tabulated approach is used, is it sufficient to interpolate each propertyindependently, or must all properties be derived from a common thermodynamicpotential or equation of state?

For example, independently interpolating ρ, h, cp, and the speed of soundmay violate thermodynamic identities and could produce an inconsistent pressureJacobian or nonphysical acoustic speeds.

Does FLEXI require thermodynamic derivatives such as:

• (∂p/∂ρ)e;
• (∂p/∂e)ρ;
• (∂h/∂T)p;
• isothermal compressibility;- thermal expansion coefficient?
  If so, where are these derivatives used in the flux evaluation, state recovery,time-step calculation, or implicit Jacobian?

3. Interpolation near the pseudo-critical regionNear the pseudo-critical temperature, some properties, especially cp, have

very sharp peaks.

Would FLEXI allow the use of interpolation methods such as:

• bilinear or bicubic interpolation;
• monotone cubic interpolation;
• PCHIP;
• spline interpolation;
• locally refined or nonuniform property tables?

Are there any monotonicity, differentiability, or smoothness requirements for
the property model?

4. Software extension point

If this functionality is not currently available, which module or interface
should be extended to implement a new sCO₂ material model?

A minimal example showing how FLEXI evaluates:

• pressure and temperature from conservative variables;
• thermodynamic derivatives;
• viscosity and thermal conductivity;

would be very helpful.

My goal is to develop a thermodynamically consistent and numerically robustsCO₂ property model rather than simply prescribing several independentlyinterpolated material-property curves.

Thank you for any guidance.

[flexi-framework]

Thanks for reaching out and your detailed request.

In fact, you are addressing a challenging topic that is not accounted for by the public version of FLEXI, which relies on the ideal gas EOS with constant material properties (except for Sutherland's law for the viscosity).

However, earlier work of our group considers real gas EOS for transcritical flows by using a tabulation approach (doi.org/10.1007/s00193-019-00896-1), which was also implemented efficiently for HPC applications by means of MPI3 shared memory trees (doi.org/10.1007/978-3-030-66792-4_25). This tabulation approach employs h-refinement to accomodate the mentioned regions of steep gradients and cut-cells to represent saturation lines. The implementation included, among others, the stiffened gas EOS for simple liquids and the Peng-Robinson EOS for real-gas behavior, but also featured an interface to the CoolProp library.

While the tabulation approach is efficient, its accuracy can depend on the specific test case, particularly for binary mixtures, where slight interpolation errors can accumulate over long simulation times. In these cases, evaluating the EOS directly per data point using a suitable initial guess yields higher precision with only a small computational overhead. So, we recommend your first strategy in the end.
If you prefer using a tabulation approach, we have found that tabulating the temperature as a function of density and specific internal energy, and then evaluating all other state variables and transport properties from density and this tabulated temperature, provides the most robust results. We also suggest this strategy for the iterative procedure, i.e., solve for the temperature and store it for subsequent EOS calls.

The mentioned thermodynamic derivatives are only required by the block-Jacobi preconditioner of implicit time discretizations, as it relies on the (analytical) partial derivative of the conservatives with respect to the primitives, for example. The Newton method itself is implemented matrix-free by approximating the Jacobian of the spatial operator via finite differences. Details can be found in the dissertation by Serena Vangelatos (dx.doi.org/10.18419/opus-10767).