Non-ideal Equation Of State Research Articles

Modeling of the quenching of blast products from energetic materials by expansion into vacuum

Condensed phase energetic materials include propellants and explosives. Their detonation or burning products generate dense, high pressure states that are often adjacent to regions that are at vacuum or near-vacuum conditions. An important chemical diagnostic experiment is the time of flight mass spectroscopy experiment that initiates an energetic material sample via an impact from a flyer plate, whose products expand into a vacuum. The rapid expansion quenches the reaction in the products so that the products can be differentiated by molecular weight detection as they stream past a detector. Analysis of this experiment requires a gas dynamic simulation of the products of a reacting multi-component gas that flows into a vacuum region. Extreme computational difficulties can arise if flow near the vacuum interface is not carefully and accurately computed. We modify an algorithm proposed by Munz [1], that computed the fluxes appropriate to a gas–vacuum interface for an inert ideal gas, and extend it to a multi-component mixture of reacting chemical components reactions with general, non-ideal equations of state. We illustrate how to incorporate that extension in the context of a complete set of algorithms for a general, cell-based flow solver. A key step is to use the local exact solution for an isentropic expansion fan, for the mixture that connects the computed flow states to the vacuum. Regularity conditions (i.e. the Liu–Smoller conditions) are necessary conditions that must be imposed on the equation of state of the multicomponent fluid in the limit of a vacuum state. We show that the Jones, Wilkins, Lee (JWL) equation of state meets these requirements.

Multirelaxation-time lattice Boltzmann model for droplet heating and evaporation under forced convection.

We investigate the evaporation of a droplet surrounded by superheated vapor with relative motion between phases. The evaporating droplet is a challenging process, as one must take into account the transport of mass, momentum, and heat. Here a lattice Boltzmann method is employed where phase change is controlled by a nonideal equation of state. First, numerical simulations are compared to the D(2) law for a vaporizing static droplet and good agreement is observed. Results are then presented for a droplet in a Lagrangian frame under a superheated vapor flow. Evaporation is described in terms of the temperature difference between liquid-vapor and the inertial forces. The internal liquid circulation driven by surface-shear stresses due to convection enhances the evaporation rate. Numerical simulations demonstrate that for higher Reynolds numbers, the dynamics of vaporization flux can be significantly affected, which may cause an oscillatory behavior on the droplet evaporation. The droplet-wake interaction and local mass flux are discussed in detail.

MINIMUM CORE MASSES FOR GIANT PLANET FORMATION WITH REALISTIC EQUATIONS OF STATE AND OPACITIES

Giant planet formation by core accretion requires a core that is sufficiently massive to trigger runaway gas accretion in less that the typical lifetime of protoplanetary disks. We explore how the minimum required core mass, M_crit, depends on a non-ideal equation of state and on opacity changes due to grain growth, across a range of stellocentric distances from 5-100 AU. This minimum M_crit applies when planetesimal accretion does not substantially heat the atmosphere. Compared to an ideal gas polytrope, the inclusion of molecular hydrogen (H_2) dissociation and variable occupation of H_2 rotational states increases M_crit. Specifically, M_crit increases by a factor of ~2 if the H_2 spin isomers, ortho- and parahydrogen, are in thermal equilibrium, and by a factor of ~2-4 if the ortho-to-para ratio is fixed at 3:1. Lower opacities due to grain growth reduce M_crit. For a standard disk model around a Solar mass star, we calculate M_crit ~ 8 M_Earth at 5 AU, decreasing to ~5 M_Earth at 100 AU, for a realistic EOS with an equilibrium ortho-to-para ratio and for grain growth to cm-sizes. If grain coagulation is taken into account, M_crit may further reduce by up to one order of magnitude. These results for the minimum critical core mass are useful for the interpretation of surveys that find exoplanets at a range of orbital distances.

Study of Kinetic Equation for Non-Ideal Gases Using Lattice Boltzmann Method: Interparticle Forces from BBGKY Hierarchy

The first equation of the BBGKY hierarchy is closed with a truncation scheme which conserves mass, momentum and total energy. Resulting kinetic equation is solved numerically using the lattice Boltzmann method over an isothermal \(D2Q9\) lattice. The inter particle forces are derived using a density gradient approximation to the collision term of the BBGKY hierarchy and the resulting non-ideal equation of state is calculated. In this paper, we show analytically that the time rate of change of Boltzmann’s H-function i.e.,\(\frac{dH \left( t \right) }{dt}\) is zero for this truncation scheme, implying reversibility in time. Therefore, a BGK term is added to the collision integral to describe the macroscopic irreversibility. Both the local and global \(H\)-functions are calculated numerically for significant variations in the density. We also have simulated a switching experiment over this system and obtained the non-equilibrium work distributions.

Anomalous wave structure in magnetized materials described by non-convex equations of state

We analyze the anomalous wave structure appearing in flow dynamics under the influence of magnetic field in materials described by non-ideal equations of state. We consider the system of magnetohydrodynamics equations closed by a general equation of state (EOS) and propose a complete spectral decomposition of the fluxes that allows us to derive an expression of the nonlinearity factor as the mathematical tool to determine the nature of the wave phenomena. We prove that the possible formation of non-classical wave structure is determined by both the thermodynamic properties of the material and the magnetic field as well as its possible rotation. We demonstrate that phase transitions induced by material properties do not necessarily imply the loss of genuine nonlinearity of the wavefields as is the case in classical hydrodynamics. The analytical expression of the nonlinearity factor allows us to determine the specific amount of magnetic field necessary to prevent formation of complex structure induced by phase transition in the material. We illustrate our analytical approach by considering two non-convex EOS that exhibit phase transitions and anomalous behavior in the evolution. We present numerical experiments validating the analysis performed through a set of one-dimensional Riemann problems. In the examples we show how to determine the appropriate amount of magnetic field in the initial conditions of the Riemann problem to transform a thermodynamic composite wave into a simple nonlinear wave.

Positivity-preserving Lagrangian scheme for multi-material compressible flow

Robustness of numerical methods has attracted an increasing interest in the community of computational fluid dynamics. One mathematical aspect of robustness for numerical methods is the positivity-preserving property. At high Mach numbers or for flows near vacuum, solving the conservative Euler equations may generate negative density or internal energy numerically, which may lead to nonlinear instability and crash of the code. This difficulty is particularly profound for high order methods, for multi-material flows and for problems with moving meshes, such as the Lagrangian methods. In this paper, we construct both first order and uniformly high order accurate conservative Lagrangian schemes which preserve positivity of physically positive variables such as density and internal energy in the simulation of compressible multi-material flows with general equations of state (EOS). We first develop a positivity-preserving approximate Riemann solver for the Lagrangian scheme solving compressible Euler equations with both ideal and non-ideal EOS. Then we design a class of high order positivity-preserving and conservative Lagrangian schemes by using the essentially non-oscillatory (ENO) reconstruction, the strong stability preserving (SSP) high order time discretizations and the positivity-preserving scaling limiter which can be proven to maintain conservation and uniformly high order accuracy and is easy to implement. One-dimensional and two-dimensional numerical tests for the positivity-preserving Lagrangian schemes are provided to demonstrate the effectiveness of these methods.

ALFVÉN WAVES IN SIMULATIONS OF SOLAR PHOTOSPHERIC VORTICES

Using advanced numerical magneto-hydrodynamic simulations of the magnetised solar photosphere, including non-grey radiative transport and a non-ideal equation of state, we analyse plasma motions in photospheric magnetic vortices. We demonstrate that apparent vortex-like motions in photospheric magnetic field concentrations do not exhibit "tornado"-like behaviour or a "bath-tub" effect. While at each time instance the velocity field lines in the upper layers of the solar photosphere show swirls, the test particles moving with the time-dependent velocity field do not demonstrate such structures. Instead, they move in a wave-like fashion with rapidly changing and oscillating velocity field, determined mainly by magnetic tension in the magnetised intergranular downflows. Using time-distance diagrams, we identify horizontal motions in the magnetic flux tubes as torsional Alfv\'en perturbations propagating along the nearly vertical magnetic field lines with local Alfv\'en speed.

Lattice-Boltzmann-based two-phase thermal model for simulating phase change

A lattice Boltzmann (LB) method is presented for solving the energy conservation equation in two phases when the phase change effects are included in the model. This approach employs multiple distribution functions, one for a pseudotemperature scalar variable and the rest for the various species. A nonideal equation of state (EOS) is introduced by using a pseudopotential LB model. The evolution equation for the pseudotemperature variable is constructed in such a manner that in the continuum limit one recovers the well known macroscopic energy conservation equation for the mixtures. Heats of reaction, the enthalpy change associated with the phase change, and the diffusive transport of enthalpy are all taken into account; but the dependence of enthalpy on pressure, which is usually a small effect in most nonisothermal flows encountered in chemical reaction systems, is ignored. The energy equation is coupled to the LB equations for species transport and pseudopotential interaction forces through the EOS by using the filtered local pseudotemperature field. The proposed scheme is validated against simple test problems for which analytical solutions can readily be obtained.

Thermodynamic analysis of an integrated gasification solid oxide fuel cell plant combined with an organic Rankine cycle

A 100 kWe hybrid plant consisting of gasification system, solid oxide fuel cells and organic Rankine cycle is presented. The nominal power is selected based on cultivation area requirement. For the considered output a land of around 0.5 km2 needs to be utilized. Woodchips are introduced into a fixed bed gasification plant to produce syngas which fuels the combined solid oxide fuel cells – organic Rankine cycle system to produce electricity. More than a hundred fluids are considered as possible alternative for the organic cycle using non-ideal equations of state (or state-of-the-art equations of state). A genetic algorithm is employed to select the optimal working fluid and the maximum pressure for the bottoming cycle. Thermodynamic and physical properties, environmental impacts and hazard specifications are also considered in the screening process. The results suggest that efficiencies in the region of 54–56% can be achieved. The highest thermal efficiency (56.4%) is achieved with propylcyclohexane at 15.9 bar. A comparison with the available and future technologies for biomass to electricity conversion is carried out. It is shown that the proposed system presents twice the thermal efficiency achieved by simple and double stage organic Rankine cycle plants and around the same efficiency of a combined gasification, solid oxide fuel cells and micro gas turbine plant.

A physical model for predicting the sound speed and attenuation coefficient in Titan's atmosphere based on Cassini-Huygens data

NASA and ESA are discussing plans for a collaborative mission to use montgolfieres to gather long-duration data in Titan’s atmosphere. Acoustic sensors can listen for thunder, bolide explosions, wind noise, cryo-volcanoes, and many other phenomena. This emphasizes the need for accurate acoustic predictions for Titan. In 2005, during the descent of the Huygens probe on Titan, an active ultrasonic sensor measured the speed of sound over the last 12 km. Using the ambient pressure, density, temperature, and methane concentration measured by Huygens as inputs, as well as temperature- and pressure-dependent transport parameters extracted from NIST, a theoretical model has been developed to predict the sound speed and attenuation coefficient in Titan's atmosphere. Based upon non-ideal equations of state, the sound speed predictions agree quite well with Huygens measurements in the lower troposphere. The effect of measured zonal winds on tropospheric propagation is presented via ray-tracing, showing quiet zone predictions. The model can be extended to the upper atmospheric layers (since ambient data are available); nevertheless care must be taken to account for altitude dependent processes such as winds, clouds, aerosols, chemistry, gravity waves, etc. in order to increase the accuracy.

Magneto-convection

Convection is the transport of energy by bulk mass motions. Magnetic fields alter convection via the Lorentz force, while convection moves the fields via the curl(v×B) term in the induction equation. Recent ground-based and satellite telescopes have increased our knowledge of the solar magnetic fields on a wide range of spatial and temporal scales. Magneto-convection modelling has also greatly improved recently as computers become more powerful. Three-dimensional simulations with radiative transfer and non-ideal equations of state are being performed. Flux emergence from the convection zone through the visible surface (and into the chromosphere and corona) has been modelled. Local, convectively driven dynamo action has been studied. The alteration in the appearance of granules and the formation of pores and sunspots has been investigated. Magneto-convection calculations have improved our ability to interpret solar observations, especially the inversion of Stokes spectra to obtain the magnetic field and the use of helioseismology to determine the subsurface structure of the Sun.

Relativistic (lattice) Boltzmann equation with nonideal equation of state

The relativistic Boltzmann equation for a single particle species generally implies a fixed, unchangeable equation of state that corresponds to that of an ideal gas. Real-world systems typically have more complicated equations of state which cannot be described by the Boltzmann equation. The present work derives a 'Boltzmann-like' equation that gives rise to a conserved energy-momentum tensor with an arbitrary (but thermodynamically consistent) equation of state. Using this, a Lattice Boltzmann scheme for diagonal metric tensors and arbitrary equations of state is constructed. The scheme is verified for QCD in the Milne metric by comparing to viscous fluid dynamics.

Thermal photons in QGP and non-ideal effects

We investigate the thermal photon production-rates using one dimensional boost-invariant second order relativistic hydrodynamics to find proper time evolution of the energy density and the temperature. The effect of bulk-viscosity and non-ideal equation of state are taken into account in a manner consistent with recent lattice QCD estimates. It is shown that the non-ideal gas equation of state i.e ε − 3 P ≠ 0 behaviour of the expanding plasma, which is important near the phase-transition point, can significantly slow down the hydrodynamic expansion and thereby increase the photon production-rates. Inclusion of the bulk viscosity may also have similar effect on the hydrodynamic evolution. However the effect of bulk viscosity is shown to be significantly lower than the non-ideal gas equation of state. We also analyze the interesting phenomenon of bulk viscosity induced cavitation making the hydrodynamical description invalid. It is shown that ignoring the cavitation phenomenon can lead to erroneous estimation of the photon flux.

SIMULATION OF FLUX EMERGENCE FROM THE CONVECTION ZONE TO THE CORONA

Here, we present numerical simulations of magnetic flux buoyantly rising from a granular convection zone into the low corona. We study the complex interaction of the magnetic field with the turbulent plasma. The model includes the radiative loss terms, non-ideal equations of state, and empirical corona heating. We find that the convection plays a crucial role in shaping the morphology and evolution of the emerging structure. The emergence of magnetic fields can disrupt the convection pattern as the field strength increases, and form an ephemeral region-like structure, while weak magnetic flux emerges and quickly becomes concentrated in the intergranular lanes, i.e. downflow regions. As the flux rises, a coherent shear pattern in the low corona is observed in the simulation. In the photosphere, both magnetic shearing and velocity shearing occur at a very sharp polarity inversion line (PIL). In a case of U-loop magnetic field structure, the field above the surface is highly sheared while below it is relaxed.

Laboratory astrophysics and non-ideal equations of state: the next challenges for astrophysical MHD simulations

Laboratory astrophysics holds great promise not only as a highly effective validation tool for astrophysical magneto-hydrodynamics (MHD) codes but it also presents a unique challenge for these codes. The high-density plasmas found in these experiments are not well modeled by the ideal equations of state (EOS) found in most astrophysical simulation codes. To solve this problem, we replaced the ideal EOS scheme in an existing MHD code, AstroBEAR, with a non-ideal EOS method and validated our implementation with van der Waals shock tube tests. The improved code is also able to model flows that contain more than one material, as required in laboratory experiments. Simulations of jet experiments performed at the OMEGA Laser reproduce the morphology of the jet much better than when the code used a single material and an ideal EOS.

Lattice Boltzmann method with self-consistent thermo-hydrodynamic equilibria

Lattice kinetic equations incorporating the effects of external/internal force fields via a shift of the local fields in the local equilibria are placed within the framework of continuum kinetic theory. The mathematical treatment reveals that in order to be consistent with the correct thermo-hydrodynamical description, temperature must also be shifted, besides momentum. New perspectives for the formulation of thermo-hydrodynamic lattice kinetic models of non-ideal fluids are then envisaged. It is also shown that on the lattice, the definition of the macroscopic temperature requires the inclusion of new terms directly related to discrete effects. The theoretical treatment is tested against a controlled case with a non-ideal equation of state.

A lattice Boltzmann framework to simulate boiling water reactor core hydrodynamics

This paper presents a consistent LBM formulation for the simulation of a two-phase water–steam system. Results of initial model validation in a range of thermodynamic conditions typical for Boiling Water Reactors (BWRs) are also shown. The interface between the two coexisting phases is captured from the dynamics of the model itself, i.e., no interface tracking is needed. The model is based on the Peng–Robinson (P–R) non-ideal equation of state and can quantitatively approximate the phase-coexistence curve for water at different temperatures ranging from 125 to 325 ∘ C. Consequently, coexisting phases with large density ratios (up to ∼1000) may be simulated. Two-phase models in the 200–300 ∘ C temperature range are of significant importance to nuclear engineers since most BWRs operate under similar thermodynamic conditions. Simulation of bubbles and droplets in a gravity-free environment of the corresponding coexisting phase until steady state is reached satisfies Laplace law at different temperatures and thus, yield the surface tension of the fluid. Comparing the LBM surface tension thus calculated using the LBM to the corresponding experimental values for water, the LBM lattice unit (lu) can be scaled to the physical units. Using this approach, spatial scaling of the LBM emerges from the model itself and is not imposed externally.

Modeling deflagration-to-detonation transition in granular explosive pentaerythritol tetranitrate

Based on an approach suggested by Stewart et al. [Phys. Fluids 6, 2515 (1994)] we develop a model to simulate deflagration-to-detonation transition (DDT) in pentaerythritol tetranitrate (PETN) powders. The model uses a continuum mechanics formulation of conservation laws for a mixture of solid reactants and gas products, written in terms of mixture quantities plus two independent variables used to account for exothermic conversion of solid reactants into gas products, and compaction associated with pore collapse and grain rearrangement. We propose a simple empirical dependence of the reaction rate on the initial bed compaction that allows us to calibrate the model for a wide range of initial conditions. For the solid reactants we use a wide-ranging equation of state (EOS) developed by Davis and co-workers in a series of papers [Proceedings of the Tenth International Symposium on Detonation, 1993, pp. 369–376; Explosive Effects and Applications (Springer, New York, 1998), Chap. 1, Combust. Flame 120, 399 (2000); Proceedings of the 12th International Symposium Detonation, San Diego, CA, 2002, pp. 624–631; . ONR 333-05-2; Proceedings of the Eighth Detonation Symposium, 1985, pp. 785–795; Proceedings of the 11th International Symposium on Detonation, 1998, pp. 303–308]. The EOS for powder uses the P-α model of Herrmann [J. Appl. Phys. 40, 2490 (1969)] and Carrol and Holt [J. Appl. Phys. 43, 759 (1972)]. To close the system, we suggest phenomenological closure relations, consistent with the limit of a compressible inert material and of a solid fully reactive material, such that the EOS can be posed only in terms of mixture quantities and the reaction and compaction variables. We demonstrate the model’s ability to capture DDT in PETN powders by matching transients typically observed in experiments through simulation. We show that for flows calculated using nonideal EOSs and complex reaction kinetics such as those formulated in our model, it is possible to define a separatrix, i.e., the C+ characteristic that separates the C+ characteristics that evolve into the detonation front from those that evolve away from it. We comment on the effects that the variability in the grain microstructure in PETN explosive powder beds can have on the overall mechanics of DDT and discuss possible ways to model this.

Transport coefficients of multi-particle collision algorithms with velocity-dependent collisionrules

Detailed calculations of the transport coefficients of a recently introduced particle-basedmodel for fluid dynamics with a non-ideal equation of state are presented. Excluded volumeinteractions are modeled by means of biased stochastic multi-particle collisions whichdepend on the local velocities and densities. Momentum and energy are exactly conservedlocally. A general scheme to derive transport coefficients for such biased, velocity-dependentcollision rules is developed. Analytic expressions for the self-diffusion coefficientand the shear viscosity are obtained, and very good agreement is found withnumerical results at small and large mean free paths. The viscosity turns out to beproportional to the square root of temperature, as in a real gas. In addition, thetheoretical framework is applied to a two-component version of the model, andexpressions for the viscosity and the difference in diffusion of the two speciesare given.

High-order numerical simulation and modelling of the interaction of energetic and inert materials

We present an integrated algorithm on a Eulerian grid, for multimaterial simulations of energetic and inert materials modelled by non-ideal equations of state. We employ high-resolution shock capturing numerical algorithms for each material inside its domain and use an overlap domain method across the interface, maintained by a recently developed, hybrid, level-set algorithm. For applications to condensed explosives we implement a non-ideal, wide-ranging equation of state and reaction rate law. For inert materials, like plastic, metal, water, etc., we implement a (linear in the pressure) Mie–Grüneisen, (U p −U s ), equation of state. We present a series of verifications of the integrated multimaterial code and show validations against experiment. We show examples of simulations of various experiments associated with real or planned experiments, some of which contain energetic materials (specifically the condensed explosives PBX-9502 and PBX-9501).