Showing papers on "Mathematical model published in 1996"

Modelling Aspects of Wastewater Treatment Processes

Abstract: Wastewater treatment processes are inherently dynamic because of the large variations in the influent wastewater flow rate, concentration and composition. Moreover, these variations are to a large extent not possible to control. The adaptive behaviour of the involved microorganisms imposes further difficulties in terms of time-varying process parameters. Mathematical models and computer simulations are essential to describe, predict and control the complicated interactions of the processes. The number of reactions and organism species that are involved in the system may be very large. An accurate description of such systems can therefore result in highly complex models, which may not be very useful from a practical, operational point of view. A reduced order dynamic model, describing an activated sludge process performing carbonaceous removal, nitrification and denitrification with reasonable accuracy, is presented. The main objective is to combine knowledge of the process dynamics with mathematical methods for estimation and identification. The identifiability of the model is investigated using both off-line and on-line methods, and its dynamic behaviour is validated by simulations of a recognized model. The information required by the identification algorithms is based on directly measurable real-time data. The simplified model may serve as a tool for predicting the dynamic behaviour of an activated sludge process, since the parameters can be tracked on-line during varying operating conditions. The model is aimed for operation and control purposes as an integral part of a hierarchical control structure. The main objective of the work on settler modelling is to enlighten recent theoretical results. A new one-dimensional settler model is compared to a traditional layer model by means of numerical simulations. Emphasis is put on the numerical solution¹s ability to approximate the analytical solution of the conservation law written as a non-linear partial differential equation. The new settler model is consistent in this respect. Several problems that occur when integrating a model of the biological reactor with a model of the settler are also discussed. In particular, the concentrations of the biological components of the particulate material are of importance for an accurate description of the sludge that is recycled to the biological reactor. Two one-dimensional algorithms have been evaluated. The first algorithm is commonly used and some of its inherent problems are discussed. The second algorithm is a new analytically derived method. Few attempts have been made to take into account the influence of higher order organisms in biofilm systems when developing or applying mathematical models. This work describes a simplified modelling approach to include some possible effects of higher order organisms on nitrification, based on a proposed hypothesis of their oxygen consumption in the biofilm. Three different models are developed and investigated. Model simulations are validated using data from a laboratory experiment using continuous-flow suspended-carrier biofilm reactors, where the predators were selectively inhibited. The proposed models are capable of reproducing several of the observed effects. They are primarily aimed at capturing the steady-state behaviour of the biofilm but may also prove to be a useful basis for describing the dynamics. (Less)

Models for network evolution

Abstract: This paper describes mathematical models for network evolution when ties (edges) are directed and the node set is fixed. Each of these models implies a specific type of departure from the standard null binomial model. We provide statistical tests that, in keeping with these models, are sensitive to particular types of departures from the null. Each model (and associated test) discussed follows directly from one or more socio‐cognitive theories about how individuals alter the colleagues with whom they are likely to interact. The models include triad completion models, degree variance models, polarization and balkanization models, the Holland‐Leinhardt models, metric models, and the constructural model. We find that many of these models, in their basic form, tend asymptotically towards an equilibrium distribution centered at the completely connected network (i.e., all individuals are equally likely to interact with all other individuals); a fact that can inhibit the development of satisfactory tests.

Model Reduction via the Karhunen-Loeve Expansion Part I: An Exposition

Abstract: In formulating mathematical models for dynamical systems, obtaining a high degree of qualitative correctness (i.e. predictive capability) may not be the only objective. The model must be useful for its intended application, and models of reduced complexity are attractive in many cases. In Part I of this paper we provide an exposition of some techniques that are useful in finding models of reduced complexity for dynamical systems involving flows. The material presented here is not new. The techniques we discuss are based on classical theory such as the Karhunen-Loeve expansion and the method of Galerkin, and the more recent concept of “coherent structures”. They have been heavily exploited in a wide range of areas in science and engineering. The attempt here is to present this collection of important methods and ideas together, at a high level of detail, in coherent form, and in the context of model reduction for simulation and control. In this manner we lead in to Part II which illustrates their usefulness in model reduction by applying them to some elementary examples of distributed parameter systems which are related to processes found in semiconductor manufacturing.

Dynamic systems, variational inequalities and control theoretic models for predicting time-varying urban network flows

Abstract: In this paper, we set forth certain axioms for a positive theory of dynamic urban network flows. We then show that mathematical models which fulfill these axioms may be created by adapting and extending certain fundamental notions from microeconomics and nonlinear systems theory. We further show that models created in this fashion, using concepts of fast and slow dynamic processes, may be manipulated into a variety of mathematical forms, thereby providing a synthesis of dynamic systems, variational inequality and control theoretic perspectives for predicting dynamic urban network flows. We close with a discussion of the implications of this synthesis for route guidance and intelligent vehicle infrastructure. Throughout, our presentation is at a conceptual level; the mathematical arguments are purposely not rigorous to embrace the widest possible readership.

Application of black-box models to HVAC systems for fault detection

Abstract: This paper describes the application of black-box models for fault detection and diagnosis (FDD) in heating, ventilat-ing, and air-conditioning (HVAC) systems. In this study, mul-tiple-input/single-output (MISO) ARX models and artificial neural network (ANN) models are used. The ARX models are examined for different processes and subprocesses and compared with each other. Two types of models are established--system models and component mod-els. In the case of system models, the HVAC system as a whole is regarded as a black box insteati of as a collection of compo-nent models. With the component model type, the components of the HVAC system are regarded as separate black boxes

Optimal Campaign Planning/Scheduling of Multipurpose Batch/Semicontinuous Plants. 2. A Mathematical Decomposition Approach

Abstract: In the companion paper, a single-level mathematical formulation has been presented describing the multiple campaign planning/scheduling problem in considerable generality. However, the resulting mixed integer linear programming model is too large to be computationally tractable for many cases of practical interest. In this paper, we present a rigorous decomposition approach to the solution of this problem and demonstrate its effectiveness by applying it to a number of illustrative examples. In addition, we consider ways in which the structure of the constituent mathematical models of the decomposition scheme can be exploited to reduce their sizes and the associated integrality gaps. Examples illustrating the applicability of the overall approach are also presented.

A simple and accurate added mass model for hydrodynamic fluid—Structure interaction analysis

Abstract: A theoretical model of an added mass representation for a flexible cylinder vibrating in a fluid medium is presented. To accomplish this, the fluid-structure interaction problem under the influence of harmonic ground and inertia dominated hydrodynamic loading, is first studied by solving the coupled differential equations exactly. Explicit expressions for computing the hydrodynamic interaction pressure and eigenquantities like natural frequencies and mode shapes are given here. However, this analytical model, as in many other mathematical models, suffers from a severe handicap; its expressions are too complicated and require the use of a computer program to generate the results. One solution which is of particular interest, is the computation of natural frequencies. Using the added mass representation, a simple formula for evaluating the natural frequency is proposed. The formula is very simple to use, requiring only a minimal computational effort on a standard calculator. Comparison with the analytical solutions shows that the formula is extremely accurate, with errors under 0.5% or less, in nearly all the cases tested. Also, more importantly, this accuracy does not appear to deteriorate in the computation of higher natural frequencies, and thus should be very useful for designers working in the dynamics of submerged structures, taking into account their hydrodynamic interactions.

Mathematical models of hysteresis

Abstract: The concept of hysteresis operator is outlined, and some simple models are illustrated. Some differential equations with hysteresis are also briefly discussed.

A queueing theory approach to speed-flow-density relationships

Abstract: A model for speed-flow-density relationships of traffic flow is developed that is completely based on queueing theory. Thus the queueing theory is shown to provide a unified frame both for roads (uninterrupted flow) and intersections (interrupted flow). The parameters of the model are the jam density, the mean desired speed of the drivers and the coefficient of variation of the time for passing a given piece of roadway at the desired speeds. The model is discussed and its properties are investigated. Numerical results obtained from the model are compared with empirical data. (A) For the covering abstract see IRRD 886400.

Dynamics of impact oscillator with dry friction

Abstract: Our investigations of the dynamics of a simple mechanical system show that (i) the structure of the bifurcation diagram seems to be independent of the friction models, and (ii) small changes of experimentally estimated parameters like friction and restitution coefficients (within the error of estimation) can lead to significant qualitative changes in the system's dynamics. The last observation shows that for such systems it is very difficult (if not impossible) to build mathematical models which can qualitatively describe experimental results for all possible values of system parameters.

Analytical visco-elastic relaxation models

Abstract: In the past, relaxation processes employing PREM or 1066B-stratified earth models have been solved numerically, either directly in the time domain (initial value models) or in the Laplace-transformed domain (normal mode models). Solving N-layer stratified models analytically has only been performed for small values of N. We present a brief outline of the analytical method of general radially stratified N-layer, self-gravitating, Maxwell rheological models. The analytical models allow for the relaxation times of the myriad of relaxation modes to be determined with very high accuracies, as the secular determinant from which they derive is an analytical function. We show by explicitly comparing results on a 5 and 30-layer incompressible Maxwell model with a convex mantle viscosity profile, that the differences in the results between the two models are small when the 5-layer model has the volume-averaged structural and rheological properties of the 30-layer model. The fact that in both models the fluid values of the Love numbers reach the same expected limits leaves no room for hypothetical so-called non-modal contributions. The application to compressible linear rheologies of this generalized analytical normal mode method is more involved, but straightforward.

Chance-Constrained Optimal Monitoring Network Design for Pollutants in Ground Water

Abstract: A mathematical model for designing a ground-water-quality monitoring network is developed that links a ground-water pollution-transport simulation model and an optimization model. Tritium is considered as the (radioactive) pollutant. The model is formulated using chance constraints and solved by using a mixedinteger programming algorithm. It incorporates uncertainties in the prediction of pollutant movement in the saturated zone. Nonlinearities due to the inclusion of cumulative distribution functions (CDFs) of actual spatial concentrations are accommodated in the optimization model through a piecewise linearization scheme. The design of the optimal monitoring network is based on the solution of two mathematical models: a simulation model for the prediction of radioactive pollutant transport in the saturated zone, and an optimization model. Constraints of the optimization model are formulated by incorporating results from the prediction-simulation model. The simulation model provides information about pollution transport with respect to time and space. The chance-constrained optimization model solution specifies the optimal location of the monitoring wells subject to the maximum limit on the number of such wells. Performance evaluation of the developed model demonstrates potential applicability of this model for designing ground-water-quality monitoring networks.

Large Strain Finite-Element Analysis of Snow

Abstract: The modified Cam-clay model and the classical Drucker-Prager plasticity model are used for the numerical representation of snow. Both models are recast within an algorithmic framework for multiplicative finite deformation plasticity. A fully implicit return project algorithm for the Cam-clay model in conjunction with a modification of the elastic stress-strain law and the computation of the exact algorithmic tangent moduli provide an efficient implementation of this model. Both material models are calibrated according to the experimental data obtained from hydrostatic and shear-box tests of snow and verified by reanalyses of these tests. Numerical results obtained from finite-element analyses of the mechanical interaction between a single tread rubber block from an automobile tire moving over a snow-covered surface are presented.

16 Design of spatial experiments: Model fitting and prediction

Abstract: Publisher Summary This chapter focuses on the design of spatial experiments. Gribik et al. were the first to use the optimal experimental design methods for environmental monitoring. They analyzed the problem of allocating measuring resources to aid in accurately estimating ground level pollution concentrations throughout a region X . The regression model was the linearized version of the diffusion model for four pollution sources and unknown background source. Because the diffusion model used in the study was a large scale model, measurements separated by distances smaller than a threshold value distance appeared to be correlated in the corresponding parameter estimation problem. In most spatial experiments, after the sites are selected measurements are usually taken on some regular schedule, for instance, several times a day, or they are continuously recorded. Generally, the response function may depend upon time. Random errors can be correlated both in time and space. When time is included explicitly in a model, then the concept of sensor allocation can be extended and “mobile” sensors may be introduced in the model.

Application of first-order and Monte Carlo analysis in watershed water quality models

Abstract: To achieve effective environmental control, it is important to develop methodologies for dealing with uncertainties in model simulation of pollution behaviour and effects. Several procedures have been proposed to quantify uncertainties in modelling studies. This paper utilizes the two methods that are widely applied, i.e. functional analysis and Monte Carlo Simulation. The first-order part of the functional analysis method provides a measure of uncertainties in dependent variables in terms of uncertainties in independent variables. The procedure is based on first-order terms in the Taylor series expansion of the dependent variable about its mean value with respect to one or more independent variables. The major assumption in this procedure is that all independent and dependent variables are the second moment variables (SMV), which means that the behaviour of any SMV is completely described by its mean and standard deviation. The mathematical simplicity of the procedure allows application by simple input-output models. Consequently, it has been applied to many environmental simulators, e.g. hydrological models, stream water quality models, lake water quality models and ground water pollution models. The Monte Carlo Simulation (MCS) method uses a large number of repeated trials or simulations with the values for stochastic inputs or uncertain variables selected at random from their assumed parent probability distributions to establish an expected range of model uncertainty.

A numerical method for power plant simulations

Abstract: This paper describes a highly flexible computerized method of calculating operating data in a power cycle. The computerized method presented here permits the study of steam, gas and combined plants. Its flexibility is not restricted by any defined cycle scheme. A power plant consists of simple elements (turbine, compressor, combustor chamber, pump, etc.). Each power plant component is represented by its typical equations relating to fundamental mechanical and thermodynamic laws, so a power plant system is represented by algebraic equations, which are the typical equations of components, continuity equations, and data concerning plant conditions. This equation system is not linear, but can be reduced to a linear equation system with variable coefficients. The solution is simultaneous for each component and it is determined by an iterative process. An example of a simple gas turbine cycle demonstrates the applied technique. This paper also presents the user interface based on MS-Windows. The input data, the results, and any characteristic parameters of a complex cycle scheme are also shown.

Stability of Gyroelastic Beams

Abstract: An idealized mathematical model of a linear elastic Bernoulli-Euler beam, in which each element of the beam has an infinitesimal quantity of stored angular momentum, is presented. This continuous distribution of angular momentum is termed gyricity. The governing equations of motion are derived when the system is subject to conservative external loads. It is shown that these systems can display both static instabilities (divergence) and dynamic instabilities (flutter), that the structure of the stability regions depends on the distribution of stiffness, and that gyric stabilization is sometimes possible.

Mathematical Modeling of Gravity Drainage After Gas Injection into Fractured Reservoirs

Abstract: This paper addresses mathematical modeling of free-fall gravity drainage which is believed to occur in naturally fractured reservoirs after depletion of oil in the fractures or gas injection into the fractured system. Comparison of wetting phase recoveries calculated using existing mathematical models with experimental data indicates the inaccuracy of these models. The causes of error are identified to be the unrealistic assumptions made in formulation of the models. Based on Darcy's law and film flow theory, we have developed a new mathematical model to describe the free-fall gravity drainage process. A simple non-linear governing equation for phase demarcator in dimensionless form was formulated and solved numerically as a function of dimensionless time. Based on the dimensionless demarcator, fluid recovery during free-fall gravity drainage is calculated. Comparisons of wetting phase recoveries given by the new model with 20 sets of experimental data obtained under thermodynamic equilibrium conditions for a variety of fluids and cores show much better accuracy of the model over the existing models. Using the dimensionless time, t D = k e Δρgt/μL, fluid recovery obtained from laboratory studies can be scaled to field applications for estimation of projected oil recoveries in oil fields. We have also applied the new model to simulation of free-fall gravity drainage under non-equilibrium conditions where molecular diffusion between phases is considered. Experimental oil recovery data obtained from CO 2 injection into a Berea core and a reservoir sandstone core, which were saturated with separator oil, have been matched by the model using empirical correlations for fluid properties. The objective of this paper is to provide reservoir engineers with a useful tool for estimating the oil recovery from fractured reservoirs after gas injection.

Models and methods summary for the FEHMN application

Abstract: This models and methods summary provides a detailed description of the mathematical models and numerical methods employed by the finite element heat and mass tranfer code (FEHMN) application.

Design of spatial experiments: Model fitting and prediction

Abstract: The main objective of the paper is to describe and develop model oriented methods and algorithms for the design of spatial experiments. Unlike many other publications in this area, the approach proposed here is essentially based on the ideas of convex design theory.

A model for closing the inviscid form of the average passage equation system

Abstract: A mathematical model is proposed for closing or mathematically completing the system of equations which describes the time average flow field through the blade passages of multistage turbomachinery. These equations referred to as the average passage equation system govern a conceptual model which has proven useful in turbomachinery aerodynamic design and analysis. The closure model is developed so as to insure a consistency between these equations and the axisymmetric through flow equations. The closure model was incorporated into a computer code for use in simulating the flow field about a high speed counter rotating propeller and a high speed fan stage. Results from these simulations are presented.

Comparison and Analysis of Diffusion Models

Abstract: A real diffusion process consists of a huge amount of interrelated variables This complexity can be modelled by diffusion models building a simplified mathematical representation of the main features of the process as a time series of indicators describing the phenomenon in interest Mathematical models are mostly used for technological forecasting purposes; forecasting is based on the best fit of the empirical data to the model formula and trend extrapolation outside the empirical period The fit gives numeric values to the parameters of the model Some of the models include also explanative factors — values of parameters describe behavioral properties of the process The paper concentrates on mathematical diffusion models The variety of models derived from the literature is introduced and analyzed As an application of mathemat ical approach to the diffusion the diffusion of mobile phones is discussed The focus of this paper is to point out problems in mathematical analysis and to discuss about alternative approaches

Macroscopic balance model for wave rotors

Abstract: A mathematical model for multi-port wave rotors is described. The wave processes that effect energy exchange within the rotor passage are modeled using one-dimensional gas dynamics. Macroscopic mass and energy balances relate volume-averaged thermodynamic properties in the rotor passage control volume to the mass, momentum, and energy fluxes at the ports. Loss models account for entropy production in boundary layers and in separating flows caused by blade-blockage, incidence, and gradual opening and closing of rotor passages. The mathematical model provides a basis for predicting design-point wave rotor performance, port timing, and machine size. Model predictions are evaluated through comparisons with CFD calculations and three-port wave rotor experimental data. A four-port wave rotor design example is provided to demonstrate model applicability. The modeling approach is amenable to wave rotor optimization studies and rapid assessment of the trade-offs associated with integrating wave rotors into gas turbine engine systems.

Thermal-Hydraulic Performance Prediction in Fluid Power Systems

Abstract: This paper presents a modelling approach to the study of thermal-hydraulic performance in fluid power systems. A set of lumped parameter mathematical models are developed which are based on conservation of mass and energy for the system. The theoretical basis and modelling strategy are discussed for an open circuit containing a hydraulic pump, loading valve, heat exchanger and reservoir. Simulation results are presented which show a comparison of model/rig performance, and the agreement obtained demonstrates the validity of the modelling approach. It is shown that the thermal response is dominated by the reservoir heat capacity and that close correspondence between the model and rig is only achievable with accurate hydraulic performance models.