Concentration dependence of the rate of diffusion-controlled reactions

Felderhof, B. U.; Deutch, J. M.

doi:10.1063/1.432087

Cited by 207 publications

(80 citation statements)

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“…In a macroscopic System, the perturbation of the concentration caused by particle j is screened by the presence of the other particles, 9 so that c(r) -c 0 goes to zero exponentially far from particle j. As a consequence, the integral of Vc over the large surface S" in Eq.…”

Section: General Properties Of the Transport Matrixmentioning

confidence: 99%

“…9 Consider to this end the hypothetical Situation that c eq (a, } = c 0 for all i^j, so that ö, = Z,j [c eq K) -c o] · If S", i s a I ar 8 e surface which completely encloses the System, then we obtain…”

Section: General Properties Of the Transport Matrixmentioning

confidence: 99%

“…This remarkable irrelevance of the external concentration has its physical origin in the phenomenon ofscreening 9 (see See. III); the bulk of the system is, in effect, screened from the solute concentration field in the reservoir by the diffusive interactions between the particles.…”

Section: Equivalence Of Open and Closed Systemsmentioning

confidence: 99%

“…These properties of the transport matrix, proven in See. III, reflect the tendency toward minimalization of the surface to volume ratio of the precipitate, äs well äs a drastic decrease of the effective ränge of diffusive couplings (screening 9 }. In See.…”

Section: Introductionmentioning

confidence: 99%

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Theory of Ostwald ripening for open systems

Beenakker

Ross

1985

The Journal of Chemical Physics

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Diffusion-controlled coarsening (Ostwald ripening) of a precipitate is analyzed for the case of an open System, immersed in a reservoir of constant solute concentration. Equivalence of the evolution of such open Systems and closed Systems studied previously is established in the limit of infinite Systems. The cause of this equivalence is screening of the bulk of the System from the reservoir by diffusive interactions between the precipitated particles. The applicability to large, but finite Systems is discussed. l. INTRODUCTIONOstwald ripening, or coarsening, of a precipitate is the final stage of phase Separation in a solution, during which the larger particles of the precipitate grow at the expense of the smaller particles, which disappear. As recognized by Ostwald,' the driving force for the process which now bears his name is the increased solubility of the smaller particles due to surface tension between the precipitate and the solute. If we assume that the solute is in local equilibrium with the precipitate, then this solubility difference induces a solute concentration gradient and leads to a diffusive flux from the smaller to the larger particles. One speaks of diffusion-controlled growth (äs opposed to growth controlled by slow deposition of solute atoms at the particle surfaces).The theory of Ostwald ripening began with the works of Lifshitz and Slyosov 2 and of Wagner. 3 Since then a detailed theoretical understanding of this phenomenon has been obtained (see the recent review article by Voorhees 4 ). The present paper deals with one aspect of the problem which, however, has received only little attention. This is the issue of the process occurring in closed vs open Systems.The assumption of a closed System plays an essential role in the Lifshitz-Slyosov-Wagner theory 2 · 3 (and subsequent theories, see Ref. 4), since it allows a determination of the (average) solute concentration from properties of the precipitate, by invoking the law of conservation of mass in the closed system. Obviously, such a procedure is not possible in the case of an open System in contact with a reservoir, since the possibility of transport of solute between the System and the reservoir has to be accounted for (cf. In See. II we formulate the equations which determine the coarsening of the precipitate on a microscopic level of description. 4 · 6 " 8 The Solutions of these equations are determined by a "transport matrix," which is Symmetrie and positive definite and which in the limit of an infinite System satisfies a certain sum rule. These properties of the transport matrix, proven in See. III, reflect the tendency toward minimalization of the surface to volume ratio of the precipitate, äs well äs a drastic decrease of the effective ränge of diffusive couplings (screening 9 }. In See. IV the equivalence of Ostwald ripening in open and closed infinite Systems is established, on the basis of the results of the previous section. In large, but finite Systems this equivalence is estimated to hold for periods much longer than the ...

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Section: General Properties Of the Transport Matrixmentioning

confidence: 99%

Section: General Properties Of the Transport Matrixmentioning

confidence: 99%

Section: Equivalence Of Open and Closed Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Theory of Ostwald ripening for open systems

Beenakker

Ross

1985

The Journal of Chemical Physics

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“…Therefore, it is assumed that these inclusions are statistically distributed particles interacting through the inorganic vapor field within the char. Because of the low volume fraction of inclusions in coal (~0.01), a "mean field approximation" [Felderhof and Deutch, 1976] has been employed to evaluate the profile for the 'macroscopic' mole fraction, X M , of inorganic vapor in the porous char that results from the generation of ash vapor from a group of sources. If ρ I is the number density of inclusions in a char particle, then the vapor mole fraction X M , with respect to the char's radial coordinate, is determined by Equation ( Thus the vaporization rate of an inclusion is, due to interaction among a large number of inclusions, dependent on its position in the char .…”

Section: Rementioning

confidence: 99%

A Virtual Engineering Framework for Simulating Advanced Power System

Bockelie¹,

Swensen²,

Denison³

et al. 2008

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In this report is described the work effort performed to provide NETL with VE-Suite based Virtual Engineering software and enhanced equipment models to support NETL's Advanced Process Engineering Co-simulation (APECS) framework for advanced power generation systems. Enhancements to the software framework facilitated an important link between APECS and the virtual engineering capabilities provided by VE-Suite (e.g., equipment and process visualization, information assimilation). Model enhancements focused on improving predictions for the performance of entrained flow coal gasifiers and important auxiliary equipment (e.g., Air Separation Units) used in coal gasification systems. In addition, a Reduced Order Model generation tool and software to provide a coupling between APECS/AspenPlus and the GE GateCycle simulation system were developed. CAPE-Open model interfaces were employed where needed. The improved simulation capability is demonstrated on selected test problems.As part of the project an Advisory Panel was formed to provide guidance on the issues on which to focus the work effort. The Advisory Panel included experts from industry and academics in gasification, CO2 capture issues, process simulation and representatives from technology developers and the electric utility industry. To optimize the benefit to NETL, REI coordinated its efforts with NETL and NETL funded projects at Iowa State University, Carnegie Mellon University and ANSYS/Fluent, Inc.The improved simulation capabilities incorporated into APECS will enable researchers and engineers to better understand the interactions of different equipment components, identify weaknesses and processes needing improvement and thereby allow more efficient, less expensive plants to be developed and brought on-line faster and in a more cost-effective manner. These enhancements to APECS represent an important step toward having a fully integrated environment for performing plant simulation and engineering. Furthermore, with little effort the modeling capabilities described in this report can be extended to support other DOE programs, such as ultra super critical boiler development, oxy-combustion boiler development or modifications to existing plants to include CO2 capture and sequestration.iii Table of Executive SummaryThe US DOE National Energy Technology Laboratory (NETL) is facilitating the development of future zero-emission, high-efficiency energy plants which could include co-production, from fossil fuels and opportunity fuels, a broad mix of power, heat, transportation fuels, and chemical products, using a wide range of plant sizes and configurations. Many of the proposed plant concepts are based on coal -an abundant, secure, stably priced fuel -and will use Integrated Coal Gasification Combined Cycle (IGCC) technology. Some plants, such as FutureGen, will include carbon capture. Computer simulation will play an important role in developing and deploying these advanced power generation systems. To reduce the time, cost and technical risk of evalua...

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Kinetic Theory of Chemical Reactions in Liquids

Kapral

1981

Advances in Chemical Physics

117

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Concentration dependence of the rate of diffusion-controlled reactions

Cited by 207 publications

References 16 publications

Theory of Ostwald ripening for open systems

Theory of Ostwald ripening for open systems

A Virtual Engineering Framework for Simulating Advanced Power System

Kinetic Theory of Chemical Reactions in Liquids

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