ACERC
Abstracts - 1994 |
ACERC
Abstracts - 1994 |
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Thrust Area 5: Comprehensive Model Development |
Sikorski, K.; Kowalski,
M. and Stenger, F.
Oxford University Press, Oxford, England, 1995. Funded partially by IBM and
ACERC.
This text covers classical basic results of approximation theory. It also obtains new developments in the theory of moments and Sinc approximation, as well as n-widths, s-numbers, and the relationship of these concepts to computational complexity. In addition, the text also contains several computational algorithms.
Chapter 1 covers basic concepts of classical approximation.
In Section 1.1, the classical and basic concepts of approximation theory are couched in the language of functional analysis. It is thus convenient to cover the concepts of existence and uniqueness of best approximation in a normed space setting.
A very important normed space from the point of both theory and application, in which to study best approximation is that of an inner product space. In this space it is convenient to study approximation in various polynomial settings, such as, via classical orthogonal polynomials, and via Cardinal, or sinc approximation. These concepts are covered in Section 1.2 of the text.
Concepts of approximation in the uniform norm are covered in Section 1.3. These concepts are also conveniently presented in a Banach space setting. Important examples from a classical standpoint include polynomial approximation with respect to the uniform norm.
Chapter 2 deals with spline methods of approximation.
Polynomials. Are historically the most popular tools of approximation, since they are easy to compute. However, a polynomial of high degree does not do a good job of approximating arbitrary given data, since it is then nearly always the case that the polynomial also has large over and undershoots between data points. On the other hand, splines, which are piecewise polynomials discussed in Chapter 2, are very convenient for approximating data, particularly data that is contaminated with noise. This is especially true of the practically important B-spines, which have variation-diminshing properties when used to approximate data, i.e., the total variation of the spline approximant is not more than that of the data. Thus splines provide particularly useful methods of approximation in the important areas of computer aided geometric design and for representing computer graphics displays.
Sinc methods discussed in Chapter 3 are ideal for approximating functions that may have singularities at end-points of an interval. Sline methods are ideal for approximation of data, and polynomial methods are ideal for approximating analytic functions that have no singularities on the interval of approximation. For example, if a function is analytic in a region containing an interval I, then we can achieve an O(exp{-b n}) error in a degree n polynomial approximation of the function in I, where b is some positive constant. This rapid exponential rate of decrease of the error reduces to a drastically slow O(n-b') rate in the case when the function has a singularity on I. Also, it turns out that Sinc methods provide simple to use accurate approximation tools for every operation of calculus, including the approximation of Hilbert transform, the approximation of derivatives, the approximation of definite and indefinite integration, the approximation and inversion of Laplace transforms, and the approximation of indefinite convolution integrals. While an O(exp{-b' n1/2}) error is also possible via spline approximation, by suitable choice of both the mesh and the degree of the spline on each subinterval, the constant b' in this exponential rate of convergence is usually not as large as the corresponding constant b for sinc approximation.
In Chapter 4 we present a family of simple rational functions, which make possible the explicit and arbitrarily accurate rational approximation of the filter, the step, and impulse functions.
Moment problems are conveniently discussed in the setting of approximation theory in Chapter 5. Included among the well known moment problems are the discrete and continuous moment problems named after Hausdorff, Stieltjes, and Hamburger, as well as the discrete and continuous trigonometric moment problems. We also include the Sinc moment problem, whose solution in the appropriate Sinc space is relatively easy, and devoid of the difficulties that one encounters using the usual monomial bases.
Chapter 6 deals with some rather "deep" concepts of approximation, namely, of n-widths and s-numbers. A historically first and practically important problem of approximation theory is to achieve a practically accurate approximation to a function f by e.g., a polynomial of a certain degree. Next, one might want to know the error of best approximation of the function f by polynomials of degree n. Following this, we could identify a class of functions F to which the function f to be approximated belongs, and we could then determine the maximum error of best approximation, as f is varied throughout F. Finally, we could vary out tools of approximation, i.e., the classes of n-dimensional subspaces, (e.g., not just polynomials) and so deduce the best method of approximation of functions in F. We are thus able to explore limits of approximability, a knowledge of which is both practically and theoretically worth while.
Chapter 7 discusses optimal methods of approximation, and optimal algorithms for general, nonlinear approximation problems. It relates approximability with the amount of work required to achieve a certain accuracy. These concepts are discussed in the general setting of normed spaces, and later, they are connected with splines, and also with the concepts of n-widths and s-numbers discussed in the previous chapter.
Chapter 8 illustrates applications of the approximation theory of the previous chapters. We discuss here the solution of Burgers' equation, the approximation of band-limited signals, and a nonlinear zero finding problem.
Each section of the text ends with a set of exercises. Each chapter closes with annotations including historical remarks that indicate the source of the material. The references follow afterwards.
Exercises are numbered from 1 to 180 globally throughout the text. Theorems, lemmas, corollaries, examples, and figures are numbered consecutively within each page. Therefore the notation Theorem 265.1 and Example 265.1 means first theorem and first example on page 265. There are no references to the literature inside of the text. All references are discussed in annotations to each chapter. Numbered formulas are almost eliminated to provide more structured text. In a few cases left they are numbered again according to the page system described above, i.e., formula 15.1 means first formula on page 15. We believe that the special format chosen will best serve the reader by providing more structured and self contained text.
Hobbs, M.L.; Radulovic,
P.T. and Smoot, L.D.
Prog. Energy Combustion Science, 19:505-586, 1993. Funded by US Department
of Energy/Morgantown Energy Technology Center through Advanced Fuel Research
Co. and ACERC.
Fixed-bed processes are commercially used for the combustion and conversion of coal for generation of power or production of gaseous or liquid products. Coal particle sizes in fixed-bed processes are typically in the mm to cm diameter range, being much larger than in most other coal processes. This review provides a broad treatment of the technology and the science related to fixed-bed systems. Commercialized and developmental fixed-bed combustion and gasification processes are explored, including countercurrent, concurrent, and crosscurrent configurations. Ongoing demonstrations in the U.S. Clean Coal Technology program are included. Physical and chemical rate processes occurring in fixed-bed combustion are summarized, with emphasis on coal devolatilization and char oxidation. Mechanisms, rate data and models of these steps are considered with emphasis on large particles. Heat and mass transfer processes, solid flows, bed voidage, tar production and gas phase reactions were also considered. Modeling of fixed-bed processes is also reviewed. Features and assumptions of a large number of one- and two-dimensional fixed-bed combustion and gasification models are summarized while the details of a recent model from this laboratory are presented and compared with data. Research needs are also discussed.
Radulovic, P.T.; Ghani,
M.U. and Smoot, L.D.
Fuel, 1994 (in press). Funded by US Department of Energy/Morgantown Energy
Technology Center and ACERC.
An improved one-dimensional model for countercurrent oxidation and gasification of coal in fixed beds has been developed. The model incorporates an advanced devolatilization submodel that can predict the evolution rates and the yields of individual gas species and tar. A split, back-and-forth, shooting methods is implemented to exactly satisfy the boundary conditions for both the feed coal and the feed gas streams. An option to switch between equilibrium and non-equilibrium gas phase composition has been added. The model predictions are compared with the experimental data for two coals; a Jetson bituminous coal and a Rosebud subbituminous coal. An illustrative simulation for an atmospheric, air-blown, dry ash, Wellman-Galusha gasifier, fired with the Jetson bituminous coal, is presented. Areas that need additional improvements are identified.
Ueng, S.-K. and Sikorski,
K.
SIAM Proceedings: Parallel Computation for Scientific Computing, 1994
(in press). Funded by ACERC.
In this paper, parallel volume-rendering algorithms for 3D finite element analysis (FEA) data are presented. These algorithms are designed for shared memory machines and distributed memory machines. Data space division and image space division are performed. The whole volume-rendering task is decomposed into sub-tasks called working units. Working units are dynamically assigned to processors. Data exchange and image composition are not necessary. Three kinds of images are produced: semi-transparent volume cloud, iso-surface and cross section images.
Hill, S.C. and Cannon, J.N.
Proceedings of the Joint AFRC/JFRC Pacific Rim International Conference on
Environmental Control of Combustion Processes, Maui, HI, October 1994. Funded
by New York State Electrical & Gas Corp. and Empire State Electrical Energy
Research Corp.
A comprehensive combustion code, PCGC-3, is used to simulate the flow, combustion, and NO pollutant formation processes in an 85 MWe coal-fired utility boiler. The code is used to predict NO emissions from the boiler under various operating conditions. The conditions tested in this study are: over-fire air, % excess air, and burner tilt. Code predictions are compared with effluent NO measurements made in this boiler. These comparisons show good agreement between model predictions for some observed trends, and demonstrate that the computer code is a useful tool that can provide insights into boiler operation. Comparisons that do not show the correct trend suggest that a finer grid resolution is required to correctly predict some trends.
Solomon, P.R.; Serio, M.A.;
Hamblen, D.G.; Smoot, L.D.; Brewster, B.S. and Radulovic, P.T.
Proceedings of the Coal-Fired Power Systems 94 - Advances in IGCC and PFBC
Review Meeting, Morgantown, West Virginia, June 1994. Funded by US Department
of Energy/Morgantown Energy Technology Center.
The main objective of this program is to understand the chemical and physical mechanisms in coal conversion processes and incorporate this technology for the purposes of development, evaluation in advanced coal conversion devices. To accomplish this objective, this program will: 1) provide critical data on the physical and chemical processes in fossil fuel gasifiers and combustors; 2) further develop a set of comprehensive codes; and 3) apply these codes to model various types of combustors and gasifiers (fixed-bed transport reactor, and fluidized-bed for coal and gas turbines for natural gas).
To expand the utilization of coal, it is necessary to reduce the technical and economic risks inherent in operating a feedstock which is highly variable and which sometimes exhibits unexpected and unwanted behavior. Reducing the risks can be achieved by establishing the technology to predict a coal's behavior in a process. This program is creating this predictive capability by merging technology developed at Advanced Fuel Research, Inc. (AFR) in predicting coal devolatilization behavior with technology developed at Brigham Young University (BYU) in comprehensive computer codes for modeling of entrained-bed and fixed-bed reactors and technology developed at the U.S. DOE-METC in comprehensive computer codes for fluidized-bed reactors. These advanced technologies will be further developed to provide: 1) a fixed-bed model capable of predicting combustion and gasification of large coal particles, 2) a transport reactor model, 3) a model for lean premixed combustion of natural gas, and 4) an improved fluidized-bed code with an advanced coal devolatilization chemistry submodel.
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