Real-gas Equations-of-State for the GASFLOW CFD code
Highlights
► We have implemented Leachman's NIST hydrogen EOS. ► A modified van der Waals EOS and a modified Nobel-Abel EOS were also implemented. ► A numerically exact benchmark problem was used to test and verify the real-gas EOS implementation.
Introduction
In a future hydrogen economy, hydrogen will be a secondary energy carrier which requires storage and transportation. In order to have sufficient energy, hydrogen should be stored in high pressure reservoirs; in a hydrogen vehicle, for instance, high-pressure tanks are required to provide a reasonable fuel autonomy. Many safety issues need to be addressed for such high pressure reservoirs such as the consequences of unintended leak scenarios and accidents during refueling operations. Hydrogen discharges from high pressure reservoirs were object of previous analyses [1], [2]. Hereafter, we focus on hydrogen storage for vehicular applications, and in particular on the simulation of cryogenic hydrogen tanks. At ambient temperatures, hydrogen is commonly stored as a high-pressure gas and the ideal gas relationship is not suitable to model such cases. For this reason, the real gas Equation of State was implemented in the GASFLOW [3] simulation code for modeling cryogenic hydrogen tanks.
An ideal gas obeys the general (or ideal) gas equation PV = nRT, where P is the pressure of the gas, V is the volume of the gas, n is the amount of substance of gas (i.e. number of moles), R is the ideal gas constant and T is the temperature of the gas. A real gas does not obey the ideal gas equation and other gas laws at all conditions of temperature and pressure. If we introduce the compressibility factor Z, we can state that a real gas obeys the following generalized law PV = Z·nRT, with Z > 0. The coefficient Z is derived from experimental data and depends on pressure, temperature and gas type. In the case of ideal gases, Z is equal to 1.
Alternatively, the van der Waals Equation of State (by Johannes D. van der Waals, 1873) can be utilized to describe a real gas. This reads [P + a(n/V)2]·(V − nb) = nRT, where P, V, n, R and T are defined as for the ideal gas equation and the introduced coefficients a and b are positive for real gases and approach zero when the ideal gas conditions are met. For real hydrogen gas we have [4]:
The constant a provides a correction for the intermolecular forces (which are neglected in the case of real gas), while constant b represents the correction for finite molecular size (which is assumed negligible in the ideal gas case).
The standard for real-gas modeling is provided by the National Institute for Standards and Technology (NIST) in Boulder, Colorado. In the GASFLOW simulation code, we have implemented Leachman's NIST hydrogen EoS as well as two simpler models for single-phase, one-component flows: (1) a modified van der Waals EOS and (2) a modified Nobel-Abel EoS. The purpose of this paper is to formulate the numerical procedure to implement generalized real-gas equations of state for the GASFLOW [3] semi-implicit-ALE CFD Code.
Section snippets
Simplified GASFLOW equation set
For this discussion, a simplified GASFLOW equation set is employed with the following assumptions: 1. Single fluid specie, 2. No phase change, 3. No heat and/or mass transfer, and 4. No chemical reactions. These assumptions led to the following set of equations:
Volume Equationwhere u is the fluid velocity vector and V is the discretized fluid volume.
Mixture Mass Equationwhere ρ is the fluid density.
Mixture Momentum Equationswhere p is the
The ICEd-ALE numerical methodology
The ICEd-ALE Numerical Methodology is extensively presented in Ref. [5]. This particular technique, developed for the solution of the Navier–Stokes equations, is both Lagrangian and Eulerian: the method uses a finite difference mesh with vertices that may move with the fluid (Lagrangian), be held fixed (Eulerian), or be moved in other arbitrary specified ways. Due to this flexibility, the method is referred to as an Arbitrary-Lagrangian–Eulerian (ALE) technique. This technique is applicable to
Some example equations of state for the above analysis
The National Institute for Standards and Technology (NIST) [7] is considered the real-gas reference basis in this paper. Where approximate models to the NIST real-gas are referenced in this section, notably the modified Nobel-Abel and modified van der Waals EoS models, some coefficients are described and can be found by fitting relevant data points from NIST [7].
A numerically exact benchmark problem for model verification
The implementation of real gas equations of state in CFD codes is a difficult task. For this reason, testing and verifying such implementation is important to provide the code developers and users confidence that the required development and coding has been correctly accomplished. This section provides a numerically exact benchmark problem to help developers and users gain that confidence.
One can describe a time-dependent, one-dimensional, filling problem. A schematic drawing of the problem is
Example of an ideal (no structural heat transfer) cryo-compressed tank filling simulation
A cryo-compressed hydrogen tank, shown schematically in Fig. 6, can be filled with 35 K hydrogen from initial to final conditions, see Table 1: Stage 0. Decompressing the tank from the initial conditions to 2.5 MPa, Stage 1. Cooling the tank by filling 35 K hydrogen with constant pressure of 2.5 MPa to an average tank temperature equaling 44 K, and Stage 2. Charging the tank with 35 K hydrogen to 25 MPa. The thermodynamic paths using the NIST para-hydrogen EOS are presented in Fig. 7. Table 1
Conclusions
The generalized NIST real-gas EoS and related real-gas models for single-phase, one-component flow have been implemented into the GASFLOW CFD code. A numerically exact benchmark problem, a one-dimensional tube filling with hydrogen, has been used to test and verify the NIST real-gas EoS implementation. A comparison of temperature and density profiles found from the exact numerical solution of the benchmark and from the GASFLOW simulations shows good agreement and demonstrates that the real-gas
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