Scaling solutions of Smoluchowski's coagulation equation

Dongen, P. G. J. van; Ernst, M. H.

doi:10.1007/bf01022996

Cited by 145 publications

(203 citation statements)

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“…In both cases the expected behaviour is of self-similar form (except for some particular kernels with homogeneity 1) but the time and mass scales are only well identified for non-gelling kernels and for the multiplicative kernel K 2 (x, y) = xy, see the survey articles [18,51] and the references therein. Existence of mass-conserving self-similar solutions for a large class of nongelling kernels have been constructed recently [29,31] and their properties studied in [11,26,32,60,65].…”

Section: Introductionmentioning

confidence: 99%

Weak Compactness Techniques and Coagulation Equations

Laurençot

2014

Lecture Notes in Mathematics

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Abstract. Smoluchowski's coagulation equation is a mean-field model describing the growth of clusters by successive mergers. Since its derivation in 1916 it has been studied by several authors, using deterministic and stochastic approaches, with a blossoming of results in the last twenty years. In particular, the use of weak L 1 -compactness techniques led to a mature theory of weak solutions and the purpose of these notes is to describe the results obtained so far in that direction, as well as the mathematical tools used.

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Section: Introductionmentioning

confidence: 99%

Weak Compactness Techniques and Coagulation Equations

Laurençot

2014

Lecture Notes in Mathematics

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“…However, as τ c > 2, ψ τ has infinite mass for any τ ∈ [τ c , 3). A similar property was already observed for the Smoluchowski coagulation equation (1) [2,3,12] and is related to the fact that the self-similar behaviour (2) is rather expected to be relevant for large values of y, see [10] for a more detailed discussion. In that connection, let us point out that, while the second moment of f τ blows up at time T , it follows from (12) that f τ (t, y) has a finite limit γ 0 y −τ as t → T for each y ∈ (0, ∞).…”

Section: Introductionmentioning

confidence: 56%

“…For that particular case, the homogeneity degree of a is λ = 1 and the value 5/2 = (λ + 3)/2 of τ suggested in [2] indeed lies within the range of values of τ for which a self-similar solution does exist. It is nevertheless a peculiar value since it is the only value of τ for which ϕ τ has a finite third moment (and actually decays exponentially fast as y → ∞).…”

Section: Introductionmentioning

confidence: 99%

“…For homogeneous coagulation kernels with homogeneity degree λ ∈ (1, 2] (i.e. a(ξy, ξy * ) = ξ λ a(y, y * ) for (ξ, y, y * ) ∈ (0, ∞) 3 ), the value τ = (λ + 3)/2 has been proposed in [2] but numerical simulations performed in [9,11] seem to indicate a different value of τ . We refer to the review article [10] for a thorough discussion of this issue.…”

Section: Introductionmentioning

confidence: 99%

“…A first task is then to look for self-similar solutions f S to (1) as described in (2). This problem seems however to be of considerable difficulty and is only completely solved for the multiplicative coagulation kernel a(y, y * ) = y y * [3,12].…”

Section: Introductionmentioning

confidence: 99%

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Self-similar solutions to a coagulation equation with multiplicative kernel

Laurençot

2006

Physica D: Nonlinear Phenomena

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Existence of self-similar solutions to the Oort-Hulst-Safronov coagulation equation with multiplicative coagulation kernel is established. These solutions are given by s(t) −τ ψ τ (y/s(t)) for (t, y) ∈ (0, T ) × (0, ∞), where T is some arbitrary positive real number, s(t) = ((3 − τ )(T − t)) −1/(3−τ ) and the parameter τ ranges in a given interval [τ c , 3). In addition, the second moment of these self-similar solutions blows up at time T . As for the profile ψ τ , it belongs to L 1 (0, ∞; y 2 dy) for each τ ∈ [τ c , 3) but its behaviour for small and large y varies with the parameter τ . MSC 2000: 45J05, 34C11

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Monte Carlo Simulation of Coalescence Processes in Oil Sands Slurries

Friesen

Dąbroś

2004

Can J Chem Eng

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A n essential step in the extraction of bitumen from oil sands is the mixing of ore, water, and air prior to bitumen recovery in the p r i m a ry separation vessel (PSV). The purpose of this conditioning step is first to liberate bitumen from the oil sands matrix and then to allow coalescence of bitumen drops and air bubbles so that large, aerated drops can be recovered easily (Shaw et al., 1996). Process variables determining the efficiency of conditioning include ore properties (grade and fine solids content), temperature, mixing time, shear rate, and water chemistry.The conventional method of preparing an oil sands slurry has been to mix oil sands with hot water at pH ≈ 8.5 in a rotating vessel (tumbler). With the recent implementation of hydrotransport technology, in which an oil sands slurry is pipelined directly from the mine site to the PSV, it is expected that the conditioning occurring in the pipeline is sufficient to ensure good bitumen recovery in the PSV. For either method, a knowledge of how the slurry evolves in time for given operating conditions would be invaluable in predicting the physical state of the bitumen and solid phases at the end of the process.Bitumen recovery and froth quality in the PSV depend in large part on the size and density of the bitumen drops. During conditioning, the size distribution of drops at any time is a function of their initial sizes and their subsequent growth due to coalescence with other drops and air bubbles entrained in the slurry. Drop density increases if solid particles are present in or attached to the bitumen and decrease when air bubbles become attached or engulfed. If the kinetics of the particulate phases in the slurry are understood, the process can be controlled to produce a slurry that is conditioned for optimized separation in the PSV.The objective of modelling the kinetics of aggregation and fragmentation processes in a flowing oil sands slurry is to describe the distribution of bitumen in the drops and bitumen-sand aggregates in the system. This information can then serve as input to a model for predicting bitumen recovery and product quality in the PSV.One common method of modelling a particulate system is to solve population balance equations (PBE). In this approach, the motions of individual particles or clusters are ignored; instead, their average behaviour is described by rate equations for mass transfer between classes of aggregates, where each class is defined by a unique composition and structure. The result is a system of coupled integro-differential equations that are analytically intractable except for a small number of simple expressions for the rate constants (kernels) describing particle aggregation and fragmentation. * Author to whom correspondence may be addressed. E-mail address: friesen @nrcan.gc.ca.Conditioning of an oil sand slurry is a critical step in the extraction of bitumen from oil sand ore. To model the conditioning process, a constant-number Monte Carlo algorithm is used to simulate the mean-field kinetics of coalescing ...

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Scaling solutions of Smoluchowski's coagulation equation

Cited by 145 publications

References 16 publications

Weak Compactness Techniques and Coagulation Equations

Weak Compactness Techniques and Coagulation Equations

Self-similar solutions to a coagulation equation with multiplicative kernel

Monte Carlo Simulation of Coalescence Processes in Oil Sands Slurries

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