Showing papers on "Mathematical model published in 1989"

Conservation equations and physical models for hypersonic air flows in thermal and chemical nonequilibrium

Abstract: The conservation equations for simulating hypersonic flows in thermal and chemical nonequilibrium and details of the associated physical models are presented These details include the curve fits used for defining thermodynamic properties of the 11 species air model, curve fits for collision cross sections, expressions for transport properties, the chemical kinetics models, and the vibrational and electronic energy relaxation models The expressions are formulated in the context of either a two or three temperature model Greater emphasis is placed on the two temperature model in which it is assumed that the translational and rotational energy models are in equilibrium at the translational temperature, T, and the vibrational, electronic, and electron translational energy modes are in equilibrium at the vibrational temperature, T sub v The eigenvalues and eigenvectors associated with the Jacobian of the flux vector are also presented in order to accommodate the upwind based numerical solutions of the complete equation set

Internal combustion engine modeling

Abstract: Air standard cycles and equilibrium charts mathematical models of rotary engines mathematical models of diesel engines mathematical models of spark-ignition engines mathematical models of gas exchange processes in two-stroke engines.

Water Distribution System Design Under Uncertainties

Abstract: A chance constrained model is presented for the minimum cost design of water distribution networks. This methodology attempts to account for the uncertainties in required demands, required pressure heads, and pipe roughness coefficients. The optimization problem is formulated as a nonlinear programming model which is solved using a generalized reduced gradient method. Details of the mathematical model formulation are presented along with example applications. Results illustrate that uncertainties in future demands, pressure head requirements, and pipe roughness can have significant effects on the optimal network design and cost.

Modelling liquid-solid phase changes with melt convection

Abstract: SUMMARY Methods are described for modelling of phase change processes using the finite element method to simulate freezing and melting including convection in the melt. Evaluation of several enthalpy/specific heat methods and time marching schemes is also included. Suppression of velocities in the solid region is described, and example problems are given. Comparison is made to simulations performed by other researchers using finite difference methods. Substantially different results were found for one of these problems, and this result is shown to be caused by numerical problems in the earlier work. strong effect on the resulting microstructure. A number of researchers have shown reorientation of columnar grains,' alteration of the size and location of equiaxed zones' and macro~egregation,~. all due to melt convection. Mathematical models have been used in attempts to better understand the processes and thus control them. Although the most convenient mathematical models would use analytical solutions to the coupled heat and momentum transport equations, very few such solutions exist for these problems, and none would extend to the realistic problems where complicated geometries and temperature dependent material properties are included. For this reason, nearly all of the efforts in this area have been numerical. There are different types of numerical methods which are appropriate to phase change problems, depending on the kind of material involved. In pure materials, eutectics or congruent melting phases, the liquid-solid interface is sharp and corresponds to an isotherm. For these kinds of problems it may be appropriate to have part of the mesh coincide with the solidification front at all times, and distort the mesh in both phases as the boundary moves. A number of these front- tracking methods have been de~eloped,~, but none exists for three-dimensional problems. For alloys which freeze over a range of temperatures, front-tracking methods are no longer applicable. Instead, what is normally done is to specify the evolution of latent heat over a freezing range as part of the material properties. The phase boundaries are then recovered from the solution as the isotherms corresponding to the beginning and end of the transformation. The two

Power law and fractional calculus model of viscoelasticity

Abstract: Mathematical relationships between the power law model and the fractional calculus model of viscoelastic behavior are developed. It is demonstrated that the two models have mathematically similar relaxation spectra. This similarity is seen to produce an asymptotic equivalence of the models at long relaxation times and correspondingly low frequencies of motion in the lower transition and rubbery regions. The divergent behavior of the models at higher frequencies in the transition and glassy regions is presented. Porposals to take advantage of the similarities between power law and fractional calculus models are discussed

Comparison of 1-, 2-, and 3-Parameter IRT Models

Abstract: This module discusses the 1-, 2-, and 3-parameter logistic item response theory models. Mathematical formulas are given for each model, and comparisons among the three models are made. Figures are included to illustrate the effects of changing the a, b, or c parameter, and a single data set is used to illustrate the effects of estimating parameter values (as opposed to the true parameter values) and to compare parameter estimates achieved though applying the different models. The estimation procedure itself is discussed briefly. Discussions of model assumptions, such as dimensionality and local independence, can be found in many of the annotated references (e.g., Hambleton, 1988).

Simulation approach to structural optimization

Abstract: A unified approach to various problems of structural optimization is presented. It is based on a combination of mathematical models of different complexity. The models describe the behaviour of a designed structure. From the computational point of view, it is connected with the sequential approximation of design problem constraints and/or an objective function. In each step, a subregion of the initial search region in the space of design variables is chosen. In this subregion, various points (designs) are selected, for which response analyses are carried out using a numerical method (mostly FEM). Using the least-squares method, analytical expressions are formulated, which then replace the initial problem functions. They are used as functions of a particular mathematical programming problem. The size and location of sequential subregions may be changed according to the result of the search. The choice of one particular form of the analytical expressions is described. The application of the approach is shown by means of test examples and comparison with other optimization techniques is presented.

Mathematical modeling of landfill gas extraction

Abstract: A model, incorporating Darcy’s law, is developed for single-species gas flow inside a nonisotropic porous medium, within which gas is continuously generated. A linear parabolic equation is derived to express the internal pressure variations in terms of the permeability, gas production rate, and boundary conditions of a site when a series of extraction wells are present. The model is used to determine the gas fluxes within a site of rectangular cross section, into which an arbitrary number of horizontal extraction pipes have been installed. This solution is used to predict the consequences of cumulative failures within a complex pumping system, and hence estimate the circumstances under which gas is first able to escape from a site. Using the results of the model some brief guidelines are offered for designing such installations. The analytical solution technique is contrasted with purely numerical methods for similar problems, and the speed and accuracy of each are compared. An efficient computer implementation for preliminary design studies is suggested.

Traffic flow theory and chaotic behavior

Abstract: Many commonly occurring natural systems are modeled with mathematical expressions and exhibit a certain stability. The inherent stability of these equations allows them to serve as the basis for engineering predictions. More complex models, such as those for modeling traffic flow, lack stability and thus require considerable care when used as a basis for predictions. In 1960, Gazis, Herman, and Rothery introduced their generalized car-follow (or GHR) equation for modeling traffic flow. Experience has shown that this equation may not be continuous for the entire range of input parameters. The discontinuous behavior and nonlinearity of the equation suggest chaotic solutions for certain ranges of input parameters. Understanding the chaotic tendencies of this equation allows engineers to improve the reliability of models and predictions based on those models. This paper describes chaotic behavior and briefly discusses the methodology of the algorithm used to detect its presence in the GHR equation. It also discusses two systems modeled with the GHR equation and their associated chaotic properties.

Experimental study of instability modes in a three-dimensional boundary layer

Abstract: Travelling waves in an unstable three-dimensional boundary layer are studied experimentally with the use of hot-wire anemometry. For the sake of realistic comparisons with stability theory, the tests were performed on a swept flat plate where infinite swept-wing conditions were approximated by means of contoured end plates. The required pressure gradient was imposed by a displacement body. The Reynolds number for the first appearance of travelling waves is roughly the same as that of stationary vortices. The frequencies of the most amplified waves depend on the Reynolds number. It is shown with the aid of a twin probe that in a more strongly amplified state, the travelling waves propagate in a direction different from that of the mean flow. Further upstream, where stationary waves first become visible in the oil-flow pattern, a uniform direction could not be identified. Under certain conditions, travelling waves of two frequency ranges are amplified that propagate in different directions. The present work is part of the transition experiment started at the DFVLR. It is closely connected to the theoretical work by Dallmann and Bieler.

A singular perturbation approach to modeling and control of manipulators constrained by a stiff environment

Abstract: Mathematical models for robot manipulator dynamics which incorporate the effects of interaction forces between the manipulator end effector and a stiff environment are presented. With appropriate assumptions, the model is transformed into a standard singularly perturbed form expressed in terms of a small parameter which is inversely proportional to the stiffness and the damping of the environment. The model for the slow time scale dynamics, corresponding to the case where the environment is rigid, is shown to be equivalent to the differential-algebraic equations which have been developed previously for mechanical systems with holonomic constraints. the model for the fast time scale dynamics is also derived. A feedback structure based on control of the slow time scale dynamics is proposed and justified. A simple example is presented to illustrate the concepts. >

A critical comparison of turbulence models for homogeneous shear flows in a rotating frame

Abstract: A variety of turbulence models, including five second-order closure models and four two equation models, are tested for the problem of homogeneous turbulent shear flow in a rotating frame. The model predictions for the time evolution of the turbulent kinetic energy and dissipation rate, as well as those for the equilibrium states, are compared with the results of physical and numerical experiments. Most of the two-equation models predict the same results for all rotation rates (omega/S) in which there is an exponential time growth of the turbulent kinetic energy and dissipation rate. The second-order closures are qualitatively superior since, consistent with physical and numerical experiments, they only predict this type of unstable flow for intermediate rotation rates in the range -0.1 less than or equal to omega/S less than or equal to 1.6. For rotation rates outside this range, there is an exchange of stabilities with a solution whose kinetic energy and dissipation rate decay with time. Although the second-order closures are superior to the two-equation models, there are still problems with the quantitative accuracy of their predictions.

A mathematical model for pressure evaluation in an infinite-conductivity horizontal well

Abstract: A mathematical model was developed to study the transient pressure behavior in a well with an infinite-conductivity horizontal drainhole in an infinite slab reservoir. The physical model includes a fluid of small and constant compressibility flowing through an infinitely large anisotropic reservoir with upper and lower impermeable boundaries. The analytical solution is obtained by applying the concepts of instantaneous sources and Green's functions. The authors discuss the uniform-flux model, a special case of the infinite-conductivity model, and present a simple way to use it to calculate the pressure at the wellbore face. They suggest that the pressure for the infinite-conductivity case can be evaluated with the uniform flux model at a fixed point along the wellbore with an error of less than 1% for combinations of the various parameters that may be encountered in real situations. They show to what extent the accuracy of the model will be affected by neglecting gravitational effects.

A computer model of the secondary arc in single phase operation of transmission lines

Abstract: A computer model of the secondary arc is described which has been developed for use in system simulation studies. The computer model is based on the mathematical model described by A.T. Johns et al., which is largely empirical and was obtained by fitting mathematical equations to experimental data. The computer model utilizes the Electromagnetic Transients Program (EMTP). Block diagrams of the computer model as well as a listing are provided. Simulation results utilizing the computer model compare favorably with results of a previous simulation and field test results. >

Nonequilibrium lattice models: Series analysis of steady states

Abstract: A perturbation theory for steady states of interacting particle systems is developed and applied to several lattice models with nonequilibrium critical points near an absorbing state. The expansion is expressed directly in terms of the kinetic parameter (creation rate), rather than in powers of the interaction. An algorithm for generating series expansions for local properties is described. Order parameter series (16 terms) and precise estimates of critical properties are presented for the one-dimensional contact process and several related models.

Optical waveguide preform fabrication: Silica formation and growth in a high‐temperature aerosol reactor

Abstract: Silica particle formation and growth in a high‐temperature aerosol reactor has been studied. A novel sampling probe has been used to obtain the evolution of the silica particle size distribution in the reactor. A simple model approximating the particle size distribution by a unimodal lognormal function was used to describe the aerosol behavior in the reactor. Good agreement of the integral properties of the aerosol size distribution was obtained between theory and experiment. Reasonable agreement was also obtained with the prediction of more sophisticated models for aerosol dynamics.

Dynamic centrifuge model tests on clay embankments

Abstract: Centrifuge modelling is a valuable means for obtaining data to study the response of geotech-nical structures to infrequent or extreme events such as earthquakes. The results of a study of five clay embankments subjected to 23 simulated earthquakes on the Cambridge geotechnical centrifuge are presented and analysed. The influence of earthquake intensity on dynamic magnification, the existence of a yield acceleration, and a delayed failure were observed in this test series. Deformation patterns observed by X-ray radiography are presented. The stress history and soil shear strength data for the models are presented to enable other researchers to test their ability to predict the observed embankment response. The mechanisms of deformation and failure of the embankments are observed and discussed. Well-documented data of this type are not available from natural earthquakes, since large earthquakes are infrequent and unrepeatable, and it is difficult to predict where they will strike. Un modele a la centrifuge...

Linear and non-linear appraisal using extremal models of bounded variation

Abstract: SUMMARY Model features may be appraised by computing upper and lower bounds for the average value of the model over a specified region. The bounds are computed by constructing extremal models which maximize and minimize this average. In order to compute the most meaningful bounds, it is important that the allowed models are geophysically realistic. In this paper, the appraisal analysis of Oldenburg (1983) is extended to incorporate a bound on the total variation of the extremal models. Restricting the variation discriminates against highly oscillatory models and, as a consequence, the difference between upper and lower bounds is often considerably reduced. The original presentation of the funnel function bound curves is extended to include the variation of the model as another dimension. The interpreter may make use of any knowledge or insight regarding the variation of the model to generate realistic extremal models and meaningful bounds. The appraisal analysis is extended to non-linear problems by altering the usual linearized equations so that a global norm of the model can be used in the objective function. The method is general, but is applied here specifically to compute bounds for localized conductivity averages of the Earth by inverting magnetotelluric measurements. The variation bound may be formulated in terms of conductivity or log conductivity. The appraisal is illustrated using synthetic data and field measurements from southeastern British Columbia, Canada. Bounding the total variation may be viewed as constraining the flatness of the model. This suggests a new method of calculating (piecewise-constant) l1 flattest models by minimizing the norm of the total variation. Unlike l2 flattest models which vary in a smooth, continuous manner, the l1 minimum-variation model is a least-structure model that resembles a layered earth with structural variations occurring at distinct depths.

The mathematical model of induction heating of ferromagnetic pipes

Abstract: A mathematical model of coupled heat transfer and electromagnetic phenomena in induction heaters of ferromagnetic, steel pipes is described. The model takes into account the nonlinearity of all coefficients, the characteristics of the supply source, and the thermal influence of the lining. A method of partial decoupling of fields is proposed. Experimental verification of the computed results is also presented. >

The Stochastic Flamelet Model of turbulent premixed combustion

Abstract: A new stochastic model is presented and used to calculate the properties of turbulent premixed flames in the flame-sheet regime. The flame sheet is represented statistically by infinitestimal flamelets, each characterized by its position, its unit normal vector, and its (in finitesimal) area. The evolution of the position and normal are completely determined by the fluid velocity and its spatial derivatives following the flamelet, which are modeled by stochastic processes. The flamelet area changes by stretching caused by velocity gradients, by the propagation of cusps, and because of curvature. An additional model is developed to account for the latter two mechanisms. The stochastic Flamelet Model is used in conjunction with the joint pdf approach to make calculations of non-stationary, statistically plane turbulent premixed flames. These calculations demonstrate the practicality of the method and illustrate its attributes Because it contains a natural and comprehensive statistical description of the flame sheet, the model allows the essential physical processes to be incorporated in a straigtforward manner.

Power plant models for operator training simulators

Abstract: Models for power plants have been developed over several decades for use in long term dynamic simulation of large power systems. These models are derived from, but are significantly simpler than, models used for studying power plants. This is because in system studies, as opposed to power plant studies, only the trajectories of the output variables of many plants are of interest as opposed to the many variables internal to the plant. The first use of these models was in the simulation and study of automatic generation control. This was done by interconnecting the power plant models through a uniform frequency dynamic model of the power system. Later, the addition of the network model and models of other system components provided the capability to simulate all aspects of power system long term dynamics. The power system simulation required in an operator training simulator is very similar to that needed for long term dynamic studies with the added requirement that it must be run in real time. These available power plant models have been used in operator training simulators and the suitability for such application of these models has also been shown. The fast solution requirement of the operator training simulator, however, has often led to the use of simple low order models whose accuracy, though suspect, has never been reported. In this paper, an attempt is made to provide models of varying complexity.

Testing linear models of sea-floor topography

Abstract: Stochastic models can generate profiles that resemble topography by taking uncorrelated, zero-average noise as input, introducing some correlation in the time series of noise, and integrating the resulting correlated noise. The output profile will depict a nonstationary, randomly rough surface. Two models have been chosen for comparison: a fractal model, in which the noise is correlated even at large distances, and an autoregressive model of order 1, in which the correlation of the noise decays rapidly. Both models have as an end-member a random walk, which is the integration of uncorrelated noise. The models have been fitted to profiles of submarine topography, and the sample autocorrelation, power spectrum and variogram have been compared to the theoretical predictions. The results suggest that a linear system approach is a viable method to model and classify sea-floor topography. The comparison does not show substantial disagreement of the data with either the autoregressive or the fractal model, although a fractal model seems to give a better fit. However, the amplitudes predicted by a nonstationary fractal model for long wavelengths (of the order of 1000 km) are unreasonably large. When viewed through a large window, ocean floor topography is likely to have an expected value determined by isostasy, and to be stationary. Nonstationary models are best applied to wavelengths of the order of 100 km or less.