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Thermodynamic and Diffusion Model Estimates on Metamorphic Temperatures and Timescales for Basaltic Eucrite GRA 98098

Introduction: HED meteorites are thought to rep-resent igneous rocks from Vesta’s basaltic crust and preserve evidence of early crustal metamorphism. Determining the temperatures and timescales of thermal metamorphism is important for reconstructing crustal evolution in the early solar system. Here, we study basaltic eucrite Graves Nunataks (GRA) 98098, which has been identified as a highly metamorphosed eucrite [1]. We present new estimates on metamorphic temperatures determined via thermodynamic modeling as well as the initial results from diffusion models constraining timescales of thermal metamorphism. Sample Description: GRA 98098 is an unbrecciated eucrite with a granoblastic plagioclase and pyroxene mineralogy. Millimeter to cm-long lathes of tridymite cross-cut and poikilitically enclose plagioclase and pyroxene [1,this work]. Pyroxene grains have exsolved into Ca-rich (~Wo38En29Fs33) and Ca-poor (~Wo4.5En36Fs59.5) lamellae. Both unzoned and zoned plagioclase grains are observed. Unzoned plagioclase grains are found solely with tridymite laths. These grains have ~An92 compositions. The cores of the zoned plagioclase grains have the same composition and thin, relatively sodic rims (~An67), (Fig. 1). The bulk sample is unusually enriched in highly in-compatible elements and has one of the most fractionated REE patterns reported [1]. Maximum metamorphic temperatures of 985±78°C have been estimated using two-pyroxene thermometry [2]. Methods: Thermodynamic modeling. Thermodynamic models were constructed using the software Perple_X, which employs a Gibbs free energy minimization in order to determine the most stable phase assemblage for a given bulk rock composition [3]. Bulk composition was calculated using mineral com-positions acquired via EMPA (this study) and the observed abundancies present in the thin section. Two bulk compositions were estimated; 1) includes all phases present in the thin section, (assumes that all phases are present during metamorphism), 2) excludes tridymite from the bulk calculation (assumes that tridymite was not present during metamorphism). In order to determine whether metamorphic equilibria was achieved and estimate temperatures of metamorphism, we compared measured pyroxene compositions with thermodynamically predicted compositions [4, Fig. 2]. Diffusion Modeling. Several time-temperature de-pendent diffusion profiles were calculated in order to determine the best match for XAn chemical profiles observed at the edges of the zoned plagioclase (Fig. 3). We assumed that the start condition was a stepwise gradient at the plagioclase/pyroxene interface. We also assumed an average diffusion coefficient (D) and a constant temperature using the equation in [5]. D was determined for two temperatures (T = 1060ºC; near eucrite solidus [6] and T = 985ºC; metamorphism reported in [4]) and then XAn was calculated as a function of distance from plagioclase core to rim using an error function solution to Fick’s second law. Results: Thermodynamic model results are summarized in Fig. 2. For a bulk composition that includes all phases in the thin section, pyroxene endmember compositions plot in the following temperature ranges: Fs ~660-860ºC, En~1000ºC & 1150ºC, and Wo~760-900ºC (Fig. 2a). For a bulk composition that excludes tridymite from the peak metamorphic assemblage (i.e., the bulk composition minus the contribution from tridymite), a temperature range could not be determined for the Fs component of pyroxene. For Wo, T~760-900ºC and En, T~1000ºC & 1150ºC (Fig. 2b). Fig. 3 summarizes the diffusion model results. For T = 1060°C & 985°C, the most appropriate time interval was estimated based on which diffusion curve most matched (solid lines, Fig. 3) the EMPA data. For T = 1060°C, the best looking match was t = 500 ka. For T = 985°C, the best match was t = 7 Ma. Discussion and future work: Temperature estimates from thermodynamic models are not conclusive because the temperature ranges determined for pyroxene endmember stability do not overlap (colored fields in Fig. 2), thus implying that there is disequilibrium between pyroxene crystals and the bulk composition considered [4]. Thus, additional exploration is needed to define a metamorphically equilibrated do-main that accurately records peak temperature. The utility of defining metamorphically equilibrated do-mains to improve the accuracy and level of detail elucidated regarding the petrogenetic history of metamorphose samples has been demonstrated previously [4,7]. We suggest that in the case of Fig. 2a, the thin section composition is not representative of the length scales over which metamorphic equilibrium was achieve and in the case of Fig. 2b, the assumption that tridymite was not present during metamorphism was incorrect. However, results from thermodynamic models can provide insight into the relative timing of mineral and compositional textures. For example from texture alone, it is unclear whether tridymite was igneous in origin and represents the last bits of melt in a crystallizing magma chamber, or if it formed during (and possibly initiated) open system thermal metamorphism. The latter could be consistent with a partial melt hypothesis [8,9] while the former implies that simple fractional crystallization can yield the textures present in GRA 98098. The lack of coincidence be-tween pyroxene endmember compositions in Fig. 2b suggest that the bulk composition minus tridymite was not the assemblage in equilibrium with the pyroxene, suggesting that tridymite was present during metamorphism and formed during igneous crystallization. We conclude that the development of the Na-rich plagioclase rims likely occurred during or immediately after peak thermal metamorphism, because eucrites of similar metamorphic grade and texture have unzoned plagioclase (~An92) [2,4,8], and Na zoning is only observed in the plagioclase not included in the tridymite. This suggests that the zoning formed after tridymite formation, and therefore after igneous crystallization. Thus, the timescales calculated via diffusion modeling possibly represent the time interval over which thermal metamorphism occurred. Cooling rates approximated for the Vestan crust predict that the crust cooled below 300°C around 35-40 Ma after formation[10]. This is consistent with our modeling results that predict formation of the Na rich plagioclase rims occurring at higher temperatures over a period of 0.5 to 7 Ma years. Future work. Additional thermodynamic modeling work will focus on selecting an equilibrated bulk rock domain in which to elucidate metamorphic conditions. Diffusion models currently provide a minimum time-scale, since diffusion slows down as the system cools. Future work with will focus on integrating cooling into the diffusion models and constraining the depth at which thermal metamorphism occurs because it could be used to determine whether the range of time-scales calculated for thermal metamorphism are consistent with the geologic environment.

J S Gorce

Thermodynamic models in cosmochemical systems.

Generalized computer methods are developed for inferring details of the formation of cosmochemical systems. Compositions of ideal gas mixtures existing in equilibrium with multicomponent solid and liquid phases are calculated. A comparison of computed results with experimental data is made for the ternary system MgO-FeO-SiO2. While the ideal-solution approximation is shown to be inaccurate in dealing with the silicate melts, the stable phases and compositions can be accurately calculated in a system where there are only solids and gas. A model system containing the elements H, O, Si, Mg, S, C, Cl, and F is investigated over a range of compositions involving the gas and ten solid phases, to show the power of the technique in dealing with complex gas-solid equilibria. Systems close to cosmic composition are next considered, both with and without iron.

Griffiths, P. R.

Non-LTE line-blanketed model atmospheres of hot stars. 1: Hybrid complete linearization/accelerated lambda iteration method

A new munerical method for computing non-Local Thermodynamic Equilibrium (non-LTE) model stellar atmospheres is presented. The method, called the hybird complete linearization/accelerated lambda iretation (CL/ALI) method, combines advantages of both its constituents. Its rate of convergence is virtually as high as for the standard CL method, while the computer time per iteration is almost as low as for the standard ALI method. The method is formulated as the standard complete lineariation, the only difference being that the radiation intensity at selected frequency points is not explicity linearized; instead, it is treated by means of the ALI approach. The scheme offers a wide spectrum of options, ranging from the full CL to the full ALI method. We deonstrate that the method works optimally if the majority of frequency points are treated in the ALI mode, while the radiation intensity at a few (typically two to 30) frequency points is explicity linearized. We show how this method can be applied to calculate metal line-blanketed non-LTE model atmospheres, by using the idea of 'superlevels' and 'superlines' introduced originally by Anderson (1989). We calculate several illustrative models taking into accont several tens of thosands of lines of Fe III to Fe IV and show that the hybrid CL/ALI method provides a robust method for calculating non-LTE line-blanketed model atmospheres for a wide range of stellar parameters. The results for individual stellar types will be presented in subsequent papers in this series.

Hubeny, I.

Simple estimate of critical volume

Method for estimating critical molar volume of materials is faster and simpler than previous procedures. Formula sums no more than 18 different contributions from components of chemical structure of material, and is as accurate (within 3 percent) as older more complicated models. Method should expedite many thermodynamic design calculations.

Fedors, R. F.

Modeling Thermodynamics Of Charring Of A Polymer

Improved method of predicting and of nondestructively analyzing pyrolytic effects in, and thermomechanical properties of, polymers combines theories and techniques from several engineering and scientific disciplines. Evolved in effort to understand charring and erosion of carbon phenolic ablative material in nozzle of rocket engine. Also applicable to other polymers and particularly to nondestructive engineering analysis of specimens heated, charred, or burned.

Funaro, Gregory V.

Steady and transient performance calculation method for prediction, analysis, and identification

The detailed design and development of turbofans Involves the prediction and identification, by means of test analysis, of the performance of the engine and its components. The thermodynamic simulation and analysis codes integrate existing knowledge and Interpretations of the detailed operating procedure of the components of the engine being developed. The relevance of the predicted performance depends on the quality of the representation of the various physical phenomena affecting the characteristics of the components and, consequently, on the incorporation of experimental correlation in the modelling. In this context, the representation of compressor and turbine characteristics is particularly important. Firstly, we will analyze the ability of corrected parameters to represent MACH similitude at the component inlet under various conditions. The various measurements achieved on the engine during development are used to enhance the modelling. The methods of identifying the thermodynamic calculation code with the various measurements, considered here with their uncertainties, are then presented and described. The analysis of the tests performed on the powerplant, designed to Identify the real characteristics of the engine components, can be undertaken, considering one or more engine test points and incorporating knowledge acquired through experimentation or on the component test bench. There are many possible fields of use for these identification methods ranging from the rematching of components to the optimization of the control system.

Jean Pierre Duponchel

Evaluation of the Munich Method for modeling rocket engine performance

A new procedure, dubbed the Munich Method, has been proposed recently for the modeling of rocket engine performance. The author of the Munich Method claims it to be an extension and improvement of the thermodynamic procedures used to model rocket engines in the NASA-Lewis chemical equilibrium program. An examination of the Munich Method shows that it contains several flaws. If these defects are corrected then the Munich Method will produce results identical to those generated by the NASA-Lewis Code.

Zeleznik, Frank J.

An internal variable assisted formulation of moisture sorption models for polymers and composites

The purpose of the paper is to demonstrate that constitutive equations for moisture sorption and swelling can be obtained with an internal variable method which is rooted in the theory of irreversible thermodynamics. Realism is injected into the model through the concept of free volume. The power of the method is illustrated by the solution of the formidable nonlinear boundary value problem for diffusion with swelling-stress dependent diffusivity.

Hiel, C. C.

Macroscopic and bulk-controlled elastic modes in an interaction of interstitial alcali metal cations within a face-centered cubic crystalline fullerine

The calculations in a majority of previous works for the fulleride (AqC-60) crystals were performed within the framework of the rigid-lattice model, neglecting the distortion relaxation of the host fullerene (C-60) crystal caused by the interstitial alkali-metal (A) cations. However, an each cation is a source of a static distortion field, and the resulting field is a superposition of such fields generated by all cations. This is a reason why the host-crystal distortions depend on the A-cations configurations, i.e. on a type of a spatial bulk distribution of interstitial cations. The given paper seeks to find a functional relation between the amplitudes of the doping-induced structure-distortion waves and of static concentration ones. A semiphenomenological model is constructed here within the scope of statistical-thermodynamic treatment and using the lattice-statistics simulation method(*). In this model the effects due to the presence of q solute A cations over available interstices (per unit cell) on the static inherent reorientation and/or displacements of the solvent molecules from the 'average-lattice' sites' as well as on the lattice parameter a of a elastically-anysotropic 'cubic' C-60 crystal are taken into account.

Tatarenko, Valentine A.

A flowfield coupled excitation and radiation model for nonequilibrium reacting flows

A second-order method is developed to correct a radiative transfer analysis for possible local thermodynamic nonequilibrium (LTNE) effects. It uses a two-species excitation model for nitrogen with chemical raction rates obtained from the Kunc and Soon (1989) atomic transition method. A vibrational energy equation is added to calculate a third temperature, which describes the average vibrational energy state of all the diatomic species. A new diffusional model is developed to improve the calculation of the diffusional fluxes of mass and energy, and second-order radiative correction factors are developed for a two-step excitation model for atomic nitrogen. Radiation reaching the wall with the second-order LTNE model shows a greater IR-line contribution and less UV-line center absorption than the first-order one.

Gally, Thomas A.

Solution influence on biomolecular equilibria - Nucleic acid base associations

Various attempts to construct an understanding of the influence of solution environment on biomolecular equilibria at the molecular level using computer simulation are discussed. First, the application of the formal statistical thermodynamic program for investigating biomolecular equilibria in solution is presented, addressing modeling and conceptual simplications such as perturbative methods, long-range interaction approximations, surface thermodynamics, and hydration shell. Then, Monte Carlo calculations on the associations of nucleic acid bases in both polar and nonpolar solvents such as water and carbon tetrachloride are carried out. The solvent contribution to the enthalpy of base association is positive (destabilizing) in both polar and nonpolar solvents while negative enthalpies for stacked complexes are obtained only when the solute-solute in vacuo energy is added to the total energy. The release upon association of solvent molecules from the first hydration layer around a solute to the bulk is accompanied by an increase in solute-solvent energy and decrease in solvent-solvent energy. The techniques presented are expectd to displace less molecular and more heuristic modeling of biomolecular equilibria in solution.

Pohorille, A.

The structure and evolution of Jupiter - The fluid contraction stage

The complete evolution of a contracting star of Jovian mass consisting of a convective adiabatic homogeneous fluid is determined using stellar structure methods, improved model atmosphere calculations, and substantially improved thermodynamic properties for hydrogen and hydrogen-helium fluids. The model atmospheres are calculated in the form of time-averaged vertical temperature structures, including all relevant sources of opacity and a solar energy deposition component, and the thermodynamic properties are modified to obtain better agreement with Monte Carlo results for metallic fluids. The resultant gravitationally contracting evolutionary models are found to have two phases: an early stellar phase similar to a typical low-mass pre-main-sequence body and a later phase constituting an approach to a degenerate-dwarf cooling curve. The first phase is shown to have high luminosities and internal temperatures, while the second gives excellent agreement with the observed radius and luminosity of Jupiter. Analysis indicates that the equation of state and superadiabaticity have the strongest influence on evolution over planetary time scales.

Graboske, H. C., Jr.

Non equilibrium flow in an arc-jet

A method for simulating the flow of a nitrogen plasma in thermodynamic nonequilibrium is described. The method relies on the modeling of the relevant physical and chemical phenomena and on a fully coupled implicit finite volume upwind numerical approximation of the governing equations. The results agree with experimental data.

Rostand, P.

Range reference atmosphere models

A description is given of the methods used to establish the statistical parameters and models for wind and various thermodynamic quantities at an altitude of 0-70 km for nine geographical locations. It is noted that wind is modeled as a vector quantity using the bivariate normal probability function. With the five parameters of the bivariate normal distribution, the distribution for wind speed is derived as a generalized Rayleigh distribution. In addition, the frequency of wind direction is derived, and the conditional distribution of wind speed given the wind direction is derived. It is pointed out that these and other wind models are consistent with the rigorous mathematical properties of the bivariate normal probability theory. The thermodynamic quantities are consistent with the hydrostatic equation and the equation of state for the mean values. With these methods, many statistical relationships can be derived.

Smith, O. E.

Heat transfer in rocket engine combustion chambers and nozzles

The complexities of liquid rocket engine heat transfer which involve the injector faceplate and regeneratively and film cooled walls are being investigated by computational analysis. A conjugate heat transfer analysis will be used to describe localized heating phenomena associated with particular injector configurations and coolant channels and film coolant dumps. These components are being analyzed, and the analysis verified with appropriate test data. Finally, the component analysis will be synthesized into an overall flowfield/heat transfer model. The FDNS code is being used to make the component analyses. Particular attention is being given to the representation of the thermodynamic properties of the fluid streams and to the method of combining the detailed models to represent overall heating. Unit flow models of specific coaxial injector elements have been developed and will be described. Film cooling simulations of film coolant flows typical of the subscale Space Transportation Main Engine (STME) being experimentally studied by Pratt and Whitney have been made, and these results will be presented. Other film coolant experiments have also been simulated to verify the CFD heat transfer model being developed. The status of the study and its relevance as a new design tool are covered. Information is given in viewgraph form.

Anderson, P. G.

Heat transfer in rocket engine combustion chambers and nozzles

Complexities of liquid rocket engine heat transfer which involve the injector faceplate and regeneratively and film cooled walls are being investigated by computational analysis. A conjugate heat transfer analysis will be used to describe localized heating phenomena associated with particular injector configurations and coolant channels and film coolant dumps. These components are being analyzed, and the analyses verified with appropriate test data. Finally, the component analyses will be synthesized into an overall flowfield/heat transfer model. The FDNS code is being used to make the component analyses. Particular attention is being given to the representation of the thermodynamic properties of the fluid streams and to the method of combining the detailed models to represent overall heating. Unit flow models of specific coaxial injector elements have been developed and will be described. Since test data from the NLS development program are not available, new validation heat transfer data have been sought. Suitable data were obtained from a Rocketdyne test program on a model hydrocarbon/oxygen engine. Simulations of these test data will be presented. Recent interest in the hybrid motor have established the need for analyses of ablating solid fuels in the combustion chamber. Analysis of a simplified hybrid motor will also be presented.

Anderson, P. G.

A Solution Methodology and Computer Program to Efficiently Model Thermodynamic and Transport Coefficients of Mixtures

A solution methodology has been developed to efficiently model multi-specie, chemically frozen, thermally perfect gas mixtures. The method relies on the ability to generate a single (composite) set of thermodynamic and transport coefficients prior to beginning a CFD solution. While not fundamentally a new concept, many applied CFD users are not aware of this capability nor have a mechanism to easily and confidently generate new coefficients. A database of individual specie property coefficients has been created for 48 species. The seven coefficient form of the thermodynamic functions is currently used rather then the ten coefficient form due to the similarity of the calculated properties, low temperature behavior and reduced CPU requirements. Sutherland laminar viscosity and thermal conductivity coefficients were computed in a consistent manner from available reference curves. A computer program has been written to provide CFD users with a convenient method to generate composite specie coefficients for any mixture. Mach 7 forebody/inlet calculations demonstrated nearly equivalent results and significant CPU time savings compared to a multi-specie solution approach. Results from high-speed combustor analysis also illustrate the ability to model inert test gas contaminants without additional computational expense.

Ferlemann, Paul G.