Piston driven converging shock waves in a stiffened gas

A one-dimensional performance analysis code has been modified to account for real gas effects in the combustion zone of the thermally choked ram accelerator. Previous computational studies utilized the ideal gas equation of state and underpredicted experimentally measured thrust values as a function of Mach number. The post-combustion pressure in the ram accelerator is on the order of a few ten MPa and in this regime it is no longer appropriate to ignore intermolecular interactions. Including a compressibility factor to account for these real gas effects provides a much better match with experimental data. The current research code incorporates the following equations of state: ideal gas, Boltzmann, Percus-Yevick, and a direct solution to the virial expansion in which the virial coefficients are computed using the Lennard-Jones and Stockmayer potentials. The ideal gas equation of state performs well for initial fill pressures up to 1.0 MPa, while the Boltzmann and direct virial model perform adequately for fill pressures up to 5.0 MPa. The Percus-Yevick model also performs well for ram accelerator fill pressures up to 5.0 MPa and is expected to accurately model fill pressures up to a few ten MPa.

A modified expression of the electron entropy production in a plasma is deduced by means of the Kelly equations of state instead of the ideal gas equations of state. From the Debye–Hückel model which considers the interaction between the charges, such equations of state are derived for a plasma and the entropy is deduced. The technique to obtain the modified entropy production is based on usual developments but including the modified equations of state giving the regular result plus some extra terms. We derive an expression of the modified entropy production in terms of the tensorial Hermitian moments by means of the irreducible tensorial Hermite polynomials.

Operation of the thermally choked propulsive mode provides a high level of acceleration at velocities below the Chapman-Jouguet detonation speed. A one-dimensional modeling of this process has been elaborated which takes into account the real gas effects and yields results that are in good agreement with experimental data at this velocity and acceleration. It also provides an estimate for the pressure level at the thermal choking point that is assumed to occur in the region where the pressure rapidly drops and the oscillatory character of the signal abruptly smoothes out. A fairly good agreement between experimental data and those derived from the one-dimensional modeling based on the ideal gas equation of state was observed. The effects of increasing the initial pressure, however, suggest that a real gas equation of state should be used. The present paper is aimed at providing experimental data on the thermal choking pressure. These values are compared to calculations based on real gas equations of state, i.e. Boltzmann and Percus-Yevick. The thermal choke point pressure in many high-pressure experiments is much lower than the calculated values which brings into question the validity of the criteria for identifying the choke point and/or the accuracy of the pressure measurements under these conditions.

The pulse-tube refrigerator (PTR) is a rather new device for cooling down to extremely low temperatures, i.e. below 4 K. The PTR works by the cyclic compression and expansion of helium that flows through a regenerator made of porous material, a cold heat exchanger, a tube, a hot heat exchanger and an orifice, in series. In a Stirling-type PTR compression and expansion are generated by a piston. The compression increases the temperature of the helium in the tube and makes it flow towards the orifice; the expansion decreases the temperature and makes the helium flow backwards to the regenerator. The net effect of warmer helium flowing in one direction and colder helium in the opposite direction is that of cooling power at the cold heat exchanger. Three PTRs are inter-connected aiming to obtain the desired 4 K lowest temperature. The conservation laws of mass, momentum and energy, and an equation of state, are simplified using asymptotic analysis based on low Mach-numbers. The regenerator is modelled one-dimensionally with Darcy’s law for flow resistance. The tube is modelled either one-dimensionally without resistance or two-dimensionally with axisymmetric laminar viscous flow. The heat transfer in the porous medium of the regenerator and in the solid tube wall is taken into account. The gas can be either ideal or real. All the material properties, including viscosity and conductivity, are taken temperature and pressure dependent. Three single-stage PTRs are connected with the regenerators in series and the tubes in parallel and six flow possibilities at the junctions are considered. Three by-passes (double-inlets) are used to enhance and tune the performance. The governing equations are numerically solved with a finite-difference method of nominally second-order accuracy in space and time. Pressure correction, flux limiter, 1D-2D connections and domain decomposition are the keywords here. Special attention is paid to suitable initial conditions, high resolution in the boundary layers and to the correct calculation of the three-way junctions in multi-stage PTRs. The model describes the fluid dynamics and thermodynamics of the pulse tubes and regenerators. The heat exchangers are assumed to be ideal in the whole analysis. The equation of state for real gas and other real properties of gas and regenerator material, which are temperature and/or pressure dependent, are applied in the three-stage PTR which works with extreme low temperatures, where the ideal gas law does not hold. The numerical methods require special attention. Typically for flow problems we deal with various length scales. Straight forward discretisation will result in unnecessary fine grids and therefore unacceptable computational time. We developed robust and efficient algorithms to deal with boundary layer problems. The employed domain decomposition technique allows for using coarse grids in areas where the solution does not change significantly. It also decouples a larger system into smaller ones, leading to smaller complexities. The objective of computing accurately and efficiently the steady oscillatory flow and heat transfer in a PTR has been achieved. In particular the tiny viscous and thermal boundary layers of a PTR operating at high frequency (20 Hz) could be resolved. The simulated three-stage PTR was able to cool down close to 5 K. The developed software is intended for use in design and optimisation of multi-stage PTRs.

From the classical kinetic theory of gases we know the equation of state of the ideal gas, βp = n (see Chap. 1). For real gases, the interaction forces between the molecules lead to corrections to the ideal gas equation of state. We mention the classical theory by van der Waals and the systematic expansions with respect to density, called virial expansions.

view Abstract Citations (12) References (18) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Vortex funnels in accretion flows Scott, H. A. ; Lovelace, R. V. E. Abstract A study is carried out of the axisymmetric gravitational accretion of a perfect fluid with angular momentum. Flows are calculated with the aid of two coordinate expansions of the poloidal velocity potential. By way of example, the generalization of spherically symmetric inflow is presented in detail. The flows possess empty vortex cores that become cylindrical far from the accreting object with a width proportional to the fluid's angular momentum. In the study, the flow is assumed to be nonrelativistic, inviscid, isentropic, and axially symmetric. An ideal gas equation of state is adopted, and the self-gravity of the gas is neglected. The poloidal flow is assumed to be irrotational and the velocity of the fluid is assumed to approach zero at large distances from the accreting object. Publication: The Astrophysical Journal Pub Date: January 1982 DOI: 10.1086/159599 Bibcode: 1982ApJ...252..765S Keywords: Angular Momentum; Axisymmetric Flow; Gravitational Effects; Ideal Gas; Stellar Mass Accretion; Vortices; Axial Flow; Equations Of State; Flow Equations; Flow Velocity; Inviscid Flow; Isentropic Processes; Potential Flow; Astrophysics full text sources ADS |

The semiempirical method was suggested for construction of a simple equation of state for liquid and gas. With this purpose, some new conditions and restrictions, which should be satisfied by the “right” equation of state, were formulated. In particular, the preferable structure of the equation of state was determined. The equations for description of thermodynamic properties of real gases and liquid phase were derived with involvement of additional restrictions. The second virial coefficient calculated on the basis of the modified step potential of intermolecular interactions was used as the temperature functions in the derived equations, and the density functions were determined from analysis of experimental data. A new equation of state for liquid and gas with six individual adjustable parameters was suggested. This new equation describes the thermodynamic properties of substances much better than the known Benedict−Webb−Rubin equation of state. Some caloric properties of substances were calculated.

Summary Some preliminary experiments are first described relating to the determination of the variable specific heat function for highly superheated steam. It is emphasized that the neglect of any explicit incorporation of such a function in the thermodynamic equations for vapours obscures the reasons for many otherwise apparently anomalous results. The experimental data so far obtained are few, owing to difficulties encountered in the experimental method, but they indicate the probable existence of an inflexion point on the constant-pressure specific heat-temperature curve at the critical temperature. An analytical investigation into the problem of incorporating the variable specific heat function into the equations of state for gases and vapours occupies the into the equations of state for gases and vapours occupies the latter portion of the paper. By a combination of fundamental thermodynamic relations with the most probable adiabetic relation for gases and vapours, as revealed by experiment, an equati...

To describe the thermodynamic properties of xenon, a new fundamental low-parametric equation of state (in the form of reduced Helmholtz energy) is obtained with the help of the methods and approaches developed by the authors. It allows us to describe the thermal properties of gas, liquid, and fluid with a sufficiently high accuracy close to the accuracy of experiment in a range from the density in the ideal gas state to the density at the triple point, excluding the critical region. The caloric properties and speed of xenon sound are calculated without involving any caloric data, with the exception of ideal gas enthalpy. The values of isobaric heat capacity, sound speed, and other thermodynamic properties obtained by calculations are in good agreement with the experimental data.

The gas equation of state (EOS) serves as a critical tool for analyzing the thermal effects within the hydrogen storage tank during refueling processes. It quantifies the dynamic relationships among pressure, temperature and volume, playing a vital role in numerical simulations of hydrogen refueling, the development of refueling protocols, and ensuring refueling safety. This study first establishes a lumped-parameter thermodynamic model for the hydrogen refueling process, which combines a zero-dimensional gas model with a one-dimensional tank wall model (0D1D). The model’s accuracy was validated against experimental data and will be used in combination with different EOSs to simulate hydrogen temperature and pressure. Subsequently, parameter values are derived for the van der Waals EOS and its modified forms—Redlich–Kwong, Soave, and Peng–Robinson. The accuracy of the modified forms is evaluated using the Joule–Thomson inversion curve. A polynomial EOS is formulated, and its parameters are numerically determined. Finally, the hydrogen temperatures and pressures calculated using the van der Waals EOS, Redlich–Kwong EOS, polynomial EOS, and the National Institute of Standards and Technology (NIST) database are compared. Within the initial and boundary conditions set in this study, the results indicate that among the modified forms for van der Waals EOS, the Redlich–Kwong EOS exhibits higher accuracy than the Soave and Peng–Robinson EOSs. Using the NIST-calculated hydrogen pressure as a benchmark, the relative error is 0.30% for the polynomial EOS, 1.83% for the Redlich–Kwong EOS, and 17.90% for the van der Waals EOS. Thus, the polynomial EOS exhibits higher accuracy, followed by the Redlich–Kwong EOS, while the van der Waals EOS demonstrates lower accuracy. This research provides a theoretical basis for selecting an appropriate EOS in numerical simulations of hydrogen refueling processes.

An efficient procedure for evaluating the thermodynamic and transport properties of real fluids was presented. Block-structured adaptive mesh refinement was used to construct the equations of state look-up table, which provides uniform accuracy over a selected thermodynamic plane with fewer grids. An overlapping uniform index domain was developed to provide fast search capability. Properties can be evaluated by bilinear, bicubic, and biquintic interpolation to achieve different levels of function continuity and thermodynamic consistency. Reconstruction discontinuity may appear across the interface of unequally sized blocks, but the level of discontinuity decreases when reconstruction is based on structured grids on each block. The effectiveness and efficiency of the method was demonstrated by comparing the storage, data retrieval, and computational fluid dynamics samples. The storage requirement decreases by 20 to 80% compared to the table constructed with uniform mesh. The overlapping uniform index domain is 14 to 19% faster than the tree data-structure in computational fluid dynamics calculations with the table of 0.1% accuracy. When reconstructed property is within 0.1% of the exact value, the degree of discontinuity is small enough to obtain a convergent solution in computational fluid dynamics cases. An equation of state table look-up method is as efficient as ideal gas equations of state and two orders of magnitude faster than the complex thermodynamic database.

There is currently a great amount of interest in the outcomes and astrophysical implications of mergers of double degenerate binaries. In a commonly adopted approximation, the components of such binaries are represented by polytropes with an index of n = 3/2. We present detailed comparisons of stellar mass-transfer and merger simulations of polytropic binaries that have been carried out using two very different numerical algorithms—a finite-volume “grid” code and a smoothed-particle hydrodynamics (SPH) code. We find that there is agreement in both the ultimate outcomes of the evolutions and the intermediate stages if the initial conditions for each code are chosen to match as closely as possible. We find that even with closely matching initial setups, the time it takes to reach a concordant evolution differs between the two codes because the initial depth of contact cannot be matched exactly. There is a general tendency for SPH to yield higher mass transfer rates and faster evolution to the final outcome. We also present comparisons of simulations calculated from two different energy equations: in one series, we assume a polytropic equation of state and in the other series an ideal gas equation of state. In the latter series of simulations, an atmosphere forms around the accretor, which can exchange angular momentum and cause a more rapid loss of orbital angular momentum. In the simulations presented here, the effect of the ideal equation of state is to de-stabilize the binary in both SPH and grid simulations, but the effect is more pronounced in the grid code.

The thermodynamics of ideal gases in the presence of quasistatic electromagnetic fields is considered. It is shown that thermodynamic properties of continua, in the presence of these fields, can be characterized by two conjugate sets of variables consisting of three extensive field-independent, and three intensive field-dependent variables. These sets are entropy, volume, and mass, i.e., {S,V,N}, and field-dependent temperature, pressure, and chemical potential, i.e., {T^,P^,\ensuremath{\zeta}}, respectively. The second set has the same thermodynamic role as {T,P,\ensuremath{\zeta}} that prevails in the absence of fields. In this context, T^, P^, and \ensuremath{\zeta}^ must be uniform at equilibrium and, consequently, T, P, and \ensuremath{\zeta} can have discontinuous jumps across interfaces that separate materials of different electromagnetic properties. Ideal gases that follow the Langevin equation are affected by a fixed B field, so that T^ increases, whereas both P^ and \ensuremath{\zeta}^ decrease as compared to their values in the absence of the field. The equation of state of ideal gases in fields has been formulated in terms of T^ and P^. Using this equation, it is shown that the change in the pressure P, which is induced by the field, is positive at fixed B, whereas it is negative, but smaller, if H is fixed. Magnetic susceptibilities are defined at either fixed density, or at fixed pressure as two distinct and different thermodynamic variables. The susceptibility at fixed density follows the Curie-Wiess law. In contrast, the one defined at fixed pressure, being inversely proportional to the temperature squared, follows a different law. The fundamental equation of ideal gases in the presence of magnetic fields is derived. The field-dependent energy \^U is shown to be a function of S, V, and N and of the field-dependent entropy \ifmmode \hat{S}\else \^{S}\fi{}, and vice versa. Both \^U and \ifmmode \hat{S}\else \^{S}\fi{} are functions of S, V, and N and the magnetic induction B. Field-dependent specific heats of ideal gases, and relations between them, have been formulated under different constraints. At fixed B, the field-induced increase in the specific heat is proportional to the ratio of magnetic to thermal energy squared. Ideal gases that are contained in discrete systems, under the action of magnetic fields, follow an equation of state that can be different than the one which is applicable for cases involving continua. This equation of state is not unique in the sense that it consists of field-dependent variables which are functions of the geometry of the discrete system, and their forms vary according to the constraints imposed on the discrete system and its surroundings. In the presence of fixed intensity magnetic fields, mixtures of permeable ideal gases are shown to satisfy the Gibbs theorem, and, at fixed B, their entropy of mixing is larger than the value prevailing when the field is absent. Finally, the formulas and results obtained for ideal gases in magnetic fields can be applied to ideal gases in electric fields, through appropriate replacement of magnetic variables by their electric counterparts. \textcopyright{} 1996 The American Physical Society.

A spherical or cylindrical cavity containing quiescent gas begins to contract at high constant radial speed, driving an axisymmetric shock wave inward to collapse at the centre. We analyse the flow field by expanding the solution in powers of time, and calculate 40 terms by delegating the arithmetic to a computer. Analysis of the series for the radius of the shock wave confirms Guderley's local self-similar solution for the focusing, including recent refined values for his similarity exponent, and yields higher terms in his local expansion. In the range of adiabatic exponent where the Guderley solution has been shown not to be unique we find, in accord with a conjecture of Gel'fand, that the smallest admissible similarity exponent is realized.

The current attempt is aimed to honor the first centennial of Johannes Diderik van der Waals (VDW) awarding Nobel Prize in Physics. The VDW theory of ordinary fluids is reviewed in the first part of the paper, where special effort is devoted to the equation of state and the law of corresponding states. In addition, a few mathematical features involving properties of cubic equations are discussed, for appreciating the intrinsic beauty of the VDW theory. A theory of astrophysical fluids is shortly reviewed in the second part of the paper, grounding on the tensor virial theorem for two-component systems, and an equation of state is formulated with a convenient choice of reduced variables. Additional effort is devoted to particular choices of density profiles, namely a simple guidance case and two cases of astrophysical interest. The related macroisothermal curves are found to be qualitatively similar to VDW isothermal curves below the critical threshold and, for sufficiently steep density profiles, a critical macroisothermal curve exists, with a single horizontal inflexion point. Under the working hypothesis of a phase transition (assumed to be gas-stars) for astrophysical fluids, similar to the vapour-liquid phase transition in ordinary fluids, the location of gas clouds, stellar systems, galaxies, cluster of galaxies, on the plane scanned by reduced variables, is tentatively assigned. A brief discussion shows how van der Waals' two great discoveries, namely a gas equation of state where tidal interactions between molecules are taken into account, and the law of corresponding states, related to microcosmos, find a counterpart with regard to macrocosmos. In conclusion, after more than a century since the awarding of the Nobel Prize in Physics, van der Waals' ideas are still valid and helpful to day for a full understanding of the universe.