A substitute-kernel approximation for radiative transfer in a nongrey gas near equilibrium, with application to radiative acoustics

Gilles, Scott E.; Cogley, Allen C.; Vincenti, Walter G.

doi:10.1016/0017-9310(69)90140-9

Cited by 29 publications

(5 citation statements)

References 4 publications

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“…The method used in the present study to solve Equations (6) and (7) is believed to be different from any described in the literature, and has application in other problems also, so it will be described in some detail here. Basically, the problem is to find F'(0) and G'(0) such that upon integrating Equations (6) and (7), and by using…”

Section: A N a L Y S I Smentioning

confidence: 94%

Convective diffusion in rotating disk systems with an imperfect semipermeable interface

Zeh

Gill

1968

AIChE Journal

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Solutions to the momentum and diffusion equations are obtained for rotating disk systems with an imperfect semipermeable interface, with direct application made to the reverse OSmosis or hyperfiltmtion process of salt water purification. The equations are solved exactly, and o new tcchnique for solving the momentum equations is described. An approximate solution to the diffusion equation is olso obtoined which is also applicable to the energy equation, and is shown to be accurate for Prandtl and Schmidt numbers 1 1, for a wide range of interfacial mass transfer, for 011 wedge-type flows US well as the rotating disk system.One objective of this paper is to discuss the process of reverse osmosis, or hyperfiltration, in a rotating disk system; both exact and very accurate approximate solutions will be presented. The approximate solution developed is similar to well known solutions applicable for high Schmidt and/or Prandtl number systems. The solution developed herein, however, is applicable for lower Schmidt and Prandtl numbers (-1) and for a wider range of system geometries, including interfacial mass transfer, as will be demonstrated by application to flat plate, rotating disk and plane stagnation flow systems.A great deal of time, effort, and money is currently being expended to find means of insuring an adequate supply of pure water for future consumption. Among the potentially important processes for saline water purification, and especially appealing because of its simplicity, i s that of reverse osmosis or hyperfiltration. This process consists essentially of separating pure water from a saline solution by a membrane which is permeable to water but not salt, and by maintaining the pressure in &he saline solution sufficiently greater than that in the pure water so as to overcome the osmotic pressure, and to cause water to flow through the membrane from. the saline side to the pure water side. A major problem in this process is that the rejected salt accumulates near the membrane surface, thereby increasing the osmotic pressure and decreasing the driving force for pure water production. Analytical studies have been published which predict the polarization and water production to be expected in a constant pressure cell (11), in turbulent flow in round tube systems ( 1 3 ) and between concentric rotating cylinders (14), in laminar flow in parallel plate systems (2, 4, 5, 13), in tubes and annuli (6, 13) and in plane stagnation flow (18).A significant advantage of the last system mentioned above is that the defining equations may be expressed in terms of a single independent variable, which greatly simplifies the analysis. Further, incomplete salt rejection may be easily and exactly treated since the salt concentration at the membrane surface is a constant, which is not true of the other systems mentioned above. However, in real systems, such as a cylinder in cross flow, the area described by the stagnation flow equations may not be large. Another system having the same advantages is that of a rotating disk in an...

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Section: A N a L Y S I Smentioning

confidence: 94%

Convective diffusion in rotating disk systems with an imperfect semipermeable interface

Zeh

Gill

1968

AIChE Journal

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“…(3) qR --Eb, 7" -+ oo (4) 3r (For the introduction of Ku and 7? into a formulation for arbitrary optical thickness, see Traugott [18,19], Cogley et al, [20], Gilles et al [21], Arpaci and G6ziim [22], and Arpaci and Baya-zito~lu [23] ).…”

Section: T3rmentioning

confidence: 99%

Radiation-affected laminar flame propagation

Arpacı

Tabaczynski

1982

Combustion and Flame

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Increased laminar flame thickness and flame speed under the influence of radiation is shown in terms of an original heat transfer number H = ~rP Total radiation 1 + 3r2/(1 -to) Conduction where ~ = (Kp/rR ) '~ is the weighted nongreyness, K p and r R are the Planck mean and the Rosseland mean of the absorption coefficient, r = rM6 K is optical thickness, r M = (rprR)'/~ the mean absorption, 8 K the conduction flame thickness, P = 4o TM3/(~5 K) the Planck number, T M the adiabatic flame temperature, ~ the thermal conductivity, and w the albedo, the ratio of scattering to extinction.

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“…The governing equation for acoustic propagation in a non-grey radiating and absorbing gas has been derived by Gilles, Cogley & Vincenti (1969). As usual, radiative scattering, pressure, and energy density are taken to be negligible.…”

Section: Basis For Application To Radiative Acousticsmentioning

confidence: 99%

“…The four boundary conditions needed to form the general well-posed problem for the exact equation (7) can be written as follows (cf. Gilles et al 1969…”

Section: A C Cogley and W G Vincentimentioning

confidence: 99%

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Application to radiative acoustics of Whitham's method for the analysis of non-equilibrium wave phenomena

Cogley

Vincenti

1969

J. Fluid Mech.

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An approximate method due originally to Whitham is applied to the study of acoustic waves propagating in a non-grey radiating and absorbing gas, assumed in local molecular equilibrium. The method, which has general applicability in the study of non-equilibrium wave phenomena, replaces the exact governing equation by a set of lower-order equations that can be solved analytically in many cases. The use of the method is demonstrated by reconsidering the onedimensional problems of (i) harmonic waves driven by a harmonic variation in either position or temperature of a planar wall and (ii) the discrete wave produced by the impulsive motion of a constant-temperature wall. The method greatly simplifies the mathematics for these problems, and comparaison of the results with those of earlier investigators shows the approximate method to be accurate. Moreover, the method allows us to obtain a more systematic and complete analytical solution of the second problem than has been obtained by more conventional methods.

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A substitute-kernel approximation for radiative transfer in a nongrey gas near equilibrium, with application to radiative acoustics

Cited by 29 publications

References 4 publications

Convective diffusion in rotating disk systems with an imperfect semipermeable interface

Convective diffusion in rotating disk systems with an imperfect semipermeable interface

Radiation-affected laminar flame propagation

Application to radiative acoustics of Whitham's method for the analysis of non-equilibrium wave phenomena

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