Choice Of Equation Of State Research Articles

Simulation of shockwave propagation characteristics in nuclear reactor cavity during external steam explosion using unified SPH

Steam explosions in nuclear reactors pose significant risks to reactor safety and containment integrity during severe accidents. This study addresses the challenges of accurately simulating shockwave propagation and structural impact in such events by establishing a unified Smoothed Particle Hydrodynamics (SPH) framework. The proposed SPH model was optimized using GPU parallelization and validated against experimental results from shock tube, underwater explosion and high-velocity impact tests, demonstrating its efficacy in capturing shockwave-structure interactions. The model was applied to simulate steam explosions in the APR1400 reactor cavity under various conditions, examining factors such as the equation of state (EOS) for water, structural materials and explosive forces. The study further found that the choice of EOS and structural material greatly influenced the peak pressure and the extent of damage, with the Mie-Grüneisen EOS producing higher peak pressures and reinforced concrete showing higher damage resistance. These findings suggest that the unified SPH approach provides a comprehensive and robust framework for assessing steam explosion risks, offering crucial insights for optimizing reactor design and safety measures against such events.

Effect of Equation of State and Cutoff Density in Smoothed Particle Hydrodynamics Simulations of the Moon-forming Giant Impact

The amount of vapor in the impact-generated protolunar disk carries implications for the dynamics, devolatilization, and moderately volatile element isotope fractionation during lunar formation. The equation of state (EoS) used in simulations of the giant impact is required to calculate the vapor mass fraction (VMF) of the modeled protolunar disk. Recently, a new version of M-ANEOS (Stewart M-ANEOS) was released with an improved treatment of heat capacity and expanded experimental Hugoniot. Here, we compare this new M-ANEOS version with a previous version (N-SPH M-ANEOS) and assess the resulting differences in smoothed particle hydrodynamics (SPH) simulations. We find that Stewart M-ANEOS results in cooler disks with smaller values of VMF and in differences in disk mass that are dependent on the initial impact angle. We also assess the implications of the minimum “cutoff” density (ρ c ), similar to a maximum smoothing length, that is set as a fast-computing alternative to an iteratively calculated smoothing length. We find that the low particle resolution of the disk typically results in >40% of disk particles falling to ρ c , influencing the dynamical evolution and VMF of the disk. Our results show that the choice of EoS, ρ c , and particle resolution can cause the VMF and disk mass to vary by tens of percent. Moreover, small values of ρ c produce disks that are prone to numerical instability and artificial shocks. We recommend that future giant impact SPH studies review smoothing methods and ensure the thermodynamic stability of the disk over simulated time.

The EOS/resolution conspiracy: convergence in proto-planetary collision simulations

ABSTRACT We investigate how the choice of equation of state (EOS) and resolution conspire to affect the outcomes of giant impact (GI) simulations. We focus on the simple case of equal-mass collisions of two Earth-like 0.5-M⊕ proto-planets showing that the choice of EOS has a profound impact on the outcome of such collisions as well as on the numerical convergence with resolution. In simulations where the Tillotson EOS is used, impacts generate an excess amount of vapour due to the lack of a thermodynamically consistent treatment of phase transitions and mixtures. In oblique collisions this enhances the artificial angular momentum (AM) transport from the planet to the circum-planetary disc reducing the planet’s rotation period over time. Even at a resolution of 1.3 × 106 particles, the result is not converged. In head-on collisions, the lack of a proper treatment of the solid/liquid-vapour phase transition allows the bound material to expand to very low densities, which, in turn, results in very slow numerical convergence of the critical specific impact energy for catastrophic disruption $Q_{\rm {RD}}^{*}$ with increasing resolution as reported in prior work. The simulations where ANalytic Equation Of State (ANEOS) is used for oblique impacts are already converged at a modest resolution of 105 particles, while head-on collisions converge when they evidence the post-shock formation of a dense iron-rich ring, which promotes gravitational re-accumulation of material. Once sufficient resolution is reached to resolve the liquid-vapour phase transition of iron in the ANEOS case, and this ring is resolved, the value of $Q_{\rm {RD}}^{*}$ has then converged.

Modelling magnetohydrodynamic equilibrium in magnetars with applications to continuous gravitational wave production

ABSTRACT Possessing the strongest magnetic fields in our Universe, magnetars mark an extremum of physical phenomena. The strength of their magnetic fields is sufficient to deform the shape of the stellar body, and when the rotational and magnetic axes are not aligned, these deformations lead to the production of gravitational waves (GWs) via a time-varying quadrupole moment. Such gravitational radiation differs from signals presently detectable by the Laser Interferometer Gravitational-Wave Observatory. These signals are continuous rather than the momentary ‘chirp’ waveforms produced by binary systems during the phases of inspiral, merger, and ringdown. Here, we construct a computational model for magnetar stellar structure with strong internal magnetic fields. We implement an n = 1 polytropic equation of state (EOS) and adopt a mixed poloidal and toroidal magnetic field model constrained by the choice of EOS. We utilize fiducial values for magnetar magnetic field strength and various stellar physical attributes. Via computational simulation, we measure the deformation of magnetar stellar structure to determine upper bounds on the strength of continuous GWs formed as a result of these deformations inducing non-axisymmetric rotation. We compute predictions of upper limit GW strain values for sources in the McGill Magnetar Catalog, an index of all detected magnetars.

Piston driven converging shock waves in a stiffened gas

The problem of a one-dimensional (1D) cylindrically or spherically symmetric shock wave converging into an inviscid, ideal gas was first investigated by Guderley[Starke kugelige und zylinrische verdichtungsstosse in der nahe des kugelmitterpunktes bzw. Der zylinderachse,” Luftfahrtforschung 19, 302 (1942)]. In the time since, many authors have discussed the practical notion of how Guderley-like flows might be generated. One candidate is a constant velocity, converging “cylindrical or spherical piston,” giving rise to a converging shock wave in the spirit of its classical, planar counterpart. A limitation of pre-existing analyses along these lines is the restriction to flows in materials described by an ideal gas equation of state (EOS) constitutive law. This choice is of course necessary for the direct comparison with the classical Guderley solution, which also features an ideal gas EOS. However, the ideal gas EOS is limited in its utility in describing a wide variety of physical phenomena and, in particular, the shock compression of solid materials. This work is thus intended to provide an extension of previous work to a nonideal EOS. The stiff gas EOS is chosen as a logical starting point due to not only its close resemblance to the ideal gas law but also its relevance to the shock compression of various liquid and solid materials. Using this choice of EOS, the solution of a 1D planar piston problem is constructed and subsequently used as the lowest order term in a quasi-self-similar series expansion intended to capture both curvilinear and nonideal EOS effects. The solution associated with this procedure provides correction terms to the 1D planar solution so that the expected accelerating shock trajectory and nontrivially varying state variable profiles can be obtained. This solution is further examined in the limit as the converging shock wave approaches the 1D curvilinear origin. Given the stiff gas EOS is not otherwise expected to admit a Guderley-like solution when coupled to the inviscid Euler equations, this work thus provides the semianalytical limiting behavior of a flow that cannot be otherwise captured using self-similar analysis.

The influence of CO2 mixture composition and equations of state on simulations of transient pipeline decompression

In a CO2 transport pipeline, decompression can occur either due to planned maintenance or accidental rupture. We investigate the impact of modelling simplifications and assumptions regarding the purity of CO2 on the decompression of an industrial scale transport pipeline. Using industrially relevant compositions of impurities we both calculate simplified (isentropic) decompression curves as well as perform full-scale fluid simulations using models for heat transfer, friction and choked flow. Herein, we compare the use of the Peng–Robinson cubic equation of state (EOS) with the use of the highly accurate EOS-CG and GERG-2008 EOS.We find that the saturation pressure can change significantly when certain impurities are present, even when in small quantities. A simplifying assumption that CO2 is pure can therefore lead to significant underestimation of the fluid pressure during the decompression, with consequences for the prediction of running ductile fractures. The choice of EOS was found to mainly affect the velocities of the initial decompression wave, with the long-time evolution of the pipeline decompression remaining relatively unchanged. The temperature minimum did neither depend significantly on the choice of EOS nor the presence of impurities.

The Equation of State and the Temperature, Pressure, and Shear Dependence of Viscosity for a Highly Viscous Reference Liquid, Dipentaerythritol Hexaisononanoate

Measurements are reported for dipentaerythritol hexaisononanoate (DiPEiC9) of pressure–volume–temperature (pVT) response to pressures to 400 MPa and temperatures to 100 °C, and of viscosity at pressures to 700 MPa and temperatures to 90 °C and shear stress to 18 MPa. These data complement the low-shear viscosities published by Harris to pressures to 200 MPa and the compressions by Fandiño et al. to 70 MPa. The improved Yasutomi correlation reproduces all viscosity measurements with accuracy better than the Doolittle free volume and the Bair and Casalini thermodynamic scaling models which require an equation of state (EoS). The interaction parameter for thermodynamic scaling, γ = 3.6, is less than that reported by Harris (γ = 4.2) and the difference is primarily in the choice of EoS. The shear stress at the Newtonian limit, about 6 MPa, is exceptionally large given the high molecular weight of DiPEiC9. The large Newtonian limit is also seen in the oscillatory shear response.

A physically-based Mie–Grüneisen equation of state to determine hot spot temperature distributions

A physically-based form of the Mie–Grüneisen equation of state (EOS) is derived for calculating 1d planar shock temperatures, as well as hot spot temperature distributions from heterogeneous impact simulations. This form utilises a multi-term Einstein oscillator model for specific heat, and is completely algebraic in terms of temperature, volume, an integrating factor, and the cold curve energy. Moreover, any empirical relation for the reference pressure and energy may be substituted into the equations via the use of a generalised reference function. The complete EOS is then applied to calculations of the Hugoniot temperature and simulation of hydrodynamic pore collapse using data for the secondary explosive, hexanitrostilbene (HNS). From these results, it is shown that the choice of EOS is even more significant for determining hot spot temperature distributions than planar shock states. The complete EOS is also compared to an alternative derivation assuming that specific heat is a function of temperature alone, i.e. cv(T). Temperature discrepancies on the order of 100–600 K were observed corresponding to the shock pressures required to initiate HNS (near 10 GPa). Overall, the results of this work will improve confidence in temperature predictions. By adopting this EOS, future work may be able to assign physical meaning to other thermally sensitive constitutive model parameters necessary to predict the shock initiation and detonation of heterogeneous explosives.

Modelling Dam Break Evolution over a Wet Bed with Smoothed Particle Hydrodynamics: A Parameter Study

When investigating water flow in spillways and energy dissipation, it is important to know the behavior of the free surfaces. To capture the real dynamic behavior of the free surfaces is therefore crucial when performing simulations. Today, there is a lack in the possibility to model such phenomenon with traditional methods. Hence, this work focuses on a parameter study for one alternative simulation tool available, namely the meshfree, Lagrangian particle method Smoothed Particle Hydrodynamics (SPH). The parameter study includes the choice of equation-of-state (EOS), the artificial viscosity constants, using a dynamic versus a static smoothing length, SPH particle spatial resolution and the finite element method (FEM) mesh scaling of the boundaries. The two dimensional SPHERIC Benchmark test case of dam break evolution over a wet bed was used for comparison and validation. The numerical results generally showed a tendency of the wave front to be ahead of the experimental results, i.e. to have a greater wave front velocity. The choice of EOS, FEM mesh scaling as well as using a dynamic or a static smoothing length showed little or no significant effect on the outcome, though the SPH particle resolution and the choice of artificial viscosity constants had a major impact. A high particle resolution increased the number of flow features resolved for both choices of artificial viscosity constants, but at the expense of increasing the mean error. Furthermore, setting the artificial viscosity constants equal to unity for the coarser cases resulted in a highly viscous and unphysical solution, and thus the relation between the artificial viscosity constants and the particle resolution and its impact on the behavior of the fluid needed to be further investigated.

What does a measurement of mass and/or radius of a neutron star constrain: Equation of state or gravity?

Neutron stars (NSs) are thought to be excellent laboratories for determining the equation of state (EoS) of cold dense matter. Their strong gravity suggests that they can also be used to constrain gravity models. The mass and radius (M-R) of a NS both depend on the choice of EoS and gravity, meaning that NSs cannot be simultaneously good laboratories for both of these questions. A measurement of M-R would constrain the less well known physics input. The assumption that M-R measurements can be used to constrain EoS-presumes general relativity (GR) is the ultimate model of gravity in the classical regime. We calculate the radial profile of compactness and curvature (square root of the full contraction of the Weyl tensor) within a NS and determine the domain not probed by the Solar System tests of GR. We find that, except for a tiny sphere of radius less than a millimeter at the center, the curvature is several orders of magnitude above the values present in Solar System tests. The compactness is beyond the solar surface value for r>10 m, and increases by 5 orders of magnitude towards the surface. With the density being only an order of magnitude higher than that probed by nuclear scattering experiments, our results suggest that the employment of GR as the theory of gravity describing the hydrostatic equilibrium of NSs is a rather remarkable extrapolation from the regime of tested validity, as opposed to that of EoS models. Our larger ignorance of gravity within NSs suggests that a measurement of M-R constrains gravity rather than EoS, and given that EoS has yet to be determined by nucleon scattering experiments, M-R measurements cannot tightly constrain the gravity models either. Near the surface the curvature and compactness attain their largest values, while EoS in this region is fairly well known. This renders the crust as the best site to look for deviations from GR.