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Differential equations, Partial -- Numerical solutions. Numerieke wiskunde.

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Matematica da computacao. Differential equations Numerical methods Summary This volume is designed as an introduction to the concepts of modern numerical analysis as they apply to partial differential equations. The book contains many practical problems and their solutions, but at the same time, strives to expose the pitfalls--such as overstability, consistency requirements, and the danger of extrapolation to nonlinear problems methods used on linear problems.

Numerical Methods for Partial Differential Equations, Third Edition reflects the great accomplishments that have taken place in scientific computation in the fifteen years since the Second Edition was published. This new edition is a drastic revision of the previous one, with new material on boundary elements, spectral methods, the methods of lines, and invariant methods. At the same time, the new edition retains the self-contained nature of the older version, and shares the clarity of its exposition and the integrity of its presentation.

Elliptic Equations; Introduction. Chapter 4. Bibliographic record and links to related information available from the Library of Congress catalog Information from electronic data provided by the publisher. May be incomplete or contain other coding. Parabolic Equations. Elliptic Equations.

Hyperbolic Equations. Special Topics.

Author Index. Subject Index. Library of Congress subject headings for this publication: Differential equations, Partial Numerical solutions. Notes Also available in print.

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Book; Illustrated English Monash University. Monash University Library. Open to the public ; Mos Murdoch University Library. Gardens Point Campus Library. Open to the public ; University of Queensland Library. Open to the public ; QA Barr Smith Library. Dixson Library. These are of proprietary design, typically consisting of a perforated ring at the circumference of the flare tip. The gas flow is divided by the ring into small streams thereby increasing air-gas mixing in a portion of the gas stream Ref.

Large pilot flames can also be used to stabilize the flame. Small amounts of gas having a relatively high burning rate, such as hydrogen, may be added to the flared stream in order to widen the stability limits Ref.

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  • The instability at the lower velocity limit can be avoided by the use of a purge gas which may be either a flammable or inert gas. The low flow instability is not a problem when vapor purging is employed, for safety reasons, to prevent the formation of flammable mixtures in the flare stack at low or no flow. Vapor purging is discussed further in Section 4.

    There is evidence that flame stability can be maintained at Mach numbers up to 0. It is doubtful whether a model exists for turbulent flames which is satisfactory for estimation of the burning velocity. It has been determined Ref. It is probable that mixing controls the burning velocity in flare systems. Recent flare tip designs for smokeless burning have included tangential discharge of either the flare stream or steam to stabilize the flame at high discharge velocities, but such developments appear to be based on empirical observation rather than analysis.

    It is the purpose of this section to discuss the probable causes of emissions, the state of the art in quantifying and con- trolling these emissions, and the extent to which flare design has been affected. As in the case of thermal radiation, it is probable that most of the visible radiation is the result of radiation from hot carbon particles.

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    The distribution of radiation fre- quencies from hot carbon particles is predicted from Planck's radiation law and requires a knowledge of the flame temperature. For practical use, a close approximation is given by Wien's law Ref. Since the intensity at this wavelength is directly proportional to area, it follows that control of the emission of visible light is closely related to the con- centration and surface area, of participates and the flame temperture.

    For hotter flames, the radiation is shifted toward the visible portion of the spectrum. In flaring practice therefore, injection of steam to reduce carbon formation decreases both the flame temperature and the area for emissions and therefore the emission of visible light. Increasing the steam beyond the amount needed to prevent soot formation causes a further reduction in luminosity Ref.

    Smokeless flaring achieved by pre-mix burning or multijet burning should result in a higher flame temperature and a higher luminosity than would be observed during steam injection. No design modification has been developed which will completely eliminate luminosity, and in practice the tendency in populated areas has been to enclose the flame at ground level.

    This requires a special type of ground flare and has several disadvantages and limitations. Such flares are essentially ground level distributed burners to reduce flame height en- closed within a refractory shield to reduce thermal and light emissions. Air is supplied by a natural draft, therefore turndown is limited and an initial time lag between initial fuel firing and air supply is inevitable Ref. Capital costs for these units are higher than those for con- ventional flares of the same capacity by about a factor of 10, and main- tenance costs are also higher.

    Because of the relatively low discharge height, such flares are not suitable for flaring toxic or hazardous gases. Because of the limited turndown and inability to respond to sudden flow changes, low-level flares are more suitable for flaring when normal flows are continuous. Elevated flares are recommended for use in addition to the ground flare whenever protection against sudden upsets is required. Jet noise is caused by a fluid passing through a constriction and is directly proportional to the pressure drop Ref. The intensity of jet noise is also a function of the fluid properties.

    Combustion noise is a function of flame turbulence and is directly proportional to the amount of air mixed with the flare gas Ref.