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Hydrodynamic Aspects of Flocculation

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The Scientific Basis of Flocculation

Part of the book series: NATO Advanced Study Institutes Series ((NSSE,volume 27))

Abstract

In the classical analyses of both Brownian (perikinetic) and velocity gradient (orthokinetic) flocculation by Smoluchowski1, particle encounters in sufficiently dilute systems are treated as binary collisions between rigid spheres. In these analyses it is assumed that the relative motions between particle pairs can be described by superposition of the isolated particle motions, each particle behaving as though the others were not present. With this assumption the only permitted interactions are those of the external force fields resulting from combined attraction and repulsion. According to the far-reaching DLVO theory, the field forces are a consequence of London-van der Waals attraction and electrical double layer repulsion2.

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Abbreviations

a:

particle radius

m

A:

Hamaker constant

J

A(s), B(s), C(s):

hydrodynamic functions

dimensionless

B(v, v):

breakup function

m3 s-1

C:

floc breakup coefficient

CA :

attraction group

dimensionless

CR :

repulsion group

dimensionless

ds :

maximum stable diameter

m

D:

absolute isolated particle diffusivity

m2s-1

Di :

absolute diffusivity of particle i

m2s-1

D12 :

relative diffusivity between unequal particles

m2s-1

Dr :

relative diffusivity between equal particles

m2s-1

D:

relative diffusivity tensor

m2s-1

Fe :

external field force

N

f(p):

electromagnetic retardation function

dimensionless

G:

velocity gradient

s-1

h:

gap between particles

m

I:

rate of encounters with reference particle

s-1

k:

Bolzmann’s constant

J °K-1

K:

coefficient in Eq (33)

m3

KB :

floc breakup rate coefficient

ℓ:

exponent in Eq (31)

m:

exponent in Eq (32)

ni :

number concentration of particle i

m-3

n(v):

number concentration distribution

m-6

N:

total number concentration

m-3

N12 :

binary encounter frequency per unit volume

s-1 m-3

P:

parameter in Eq (14)

P:

Pressure

N m-2

r:

radial separation

m

R11 :

velocity correlation function

dimensionless

s:

dimensionless radial separation

dimensionless

T:

temperature

°K

→ ur :

relative velocity vector

ms-1

ui :

Cartesian velocity component

m s-1

u′i :

turbulent velocity fluctuation

m s-1

ui :

mean velocity

m s-1

v:

floc volume

m3

V:

interaction potential energy

J

xi :

Cartesian coordinate

m

αo :

orthokinetic aggregation efficiency

dimensionless

αp :

perikinetic aggregation efficiency

dimensionless

α f p :

Fuchs’ perikinetic aggregation efficiency

dimensionless

β(v, v):

binary aggregation coefficient

m3 s-1

ε:

energy dissipation rate per unit mass

J s -1 kg-1

εf :

fluid dielectric coefficient

NV-2

η:

turbulence length microscale

m

κ:

reciprocal Debye length

m-1

λ:

wavelength for dispersion force

dimensionless

μ:

absolute viscosity

kg m-1 s-1

ν:

kinematic viscosity

m2 s-1

ρ:

fluid density

kg m-3

ι:

Debye length group

dimensionless

φ:

total suspended volume concentration

dimensionless

ψ0 :

potential at onset of diffuse double layer

V

ω:

rotor angular velocity

s-1

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Authors and Affiliations

  1. Departments of Civil and Chemical Engineering, University of Delaware, Newark, Delaware, USA

    Lloyd A. Spielman

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  1. Lloyd A. Spielman
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Editor information

Editors and Affiliations

  1. Department of Civil and Municipal Engineering, University College London, UK

    Kenneth J. Ives (Professor of civil engineering) (Professor of civil engineering)

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© 1978 Sijthoff & Noordhoff International Publishers B.V.

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Spielman, L.A. (1978). Hydrodynamic Aspects of Flocculation. In: Ives, K.J. (eds) The Scientific Basis of Flocculation. NATO Advanced Study Institutes Series, vol 27. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-9938-1_4

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  • DOI: https://doi.org/10.1007/978-94-009-9938-1_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-009-9940-4

  • Online ISBN: 978-94-009-9938-1

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