The Science of Defoaming

555

Frequently Used Symbols

and Abbreviations

SYMBOLS

B, B

e

: Unspecied and equilibrium bridging coefcients

Ca: Capillary number



c

i

: Concentration of antifoam i



ct

i

()

: Time-dependent concentration of antifoam

D: Diffusion coefcient

F: Ratio of volume of air in foam in presence of antifoam/volume of air in foam in

absence of antifoam.

E, E

i

, E

si

, E

e

: Unspecied, initial, semi-initial, and equilibrium classical entry

coefcients

E

g

: Generalized entry coefcient

g: Gravitational constant

h: Thickness of liquid lm

h

i

: Thickness of lm i

h

crit

: Critical rupture thickness of liquid lm

H

o

: Initial foam height without antifoam

H

o

e

: Equilibrium foam height after foam drainage has ceased

H

t

: Foam height at time t

H

E

: Henry’s law constant

k

: Boltzmann constant

N

i

(r

b

): Number of antifoam entities in lm of a bubble of radius r

b

at the top of a

foam

P

atm

: Atmospheric pressure

P(r

b

,t): Bubble size distribution as a function of r

b

and t

p

c

i

: Capillary pressure at i (with respect to some reference pressure)

p

c

crit

: Critical applied capillary pressure for rupture of pseudoemulsion lm

p

c

PB

: Plateau border capillary pressure

∆p

c

ij

: Capillary pressure jump across surface ij where i and j are different uids

r

AF

: Radius of antifoam entity (oil drop or oil/particle mixture)

r

f

: Radius of cylindrical foam lm

r

b

: Radius of spherical bubble or radius of equivalent spherical bubble

S, S

i

, S

si

, S

e

: Unspecied, initial, semi-initial, and equilibrium classical spreading

coefcients

SSS S

gg

i

g

si

g

e

,, ,

: Unspecied, initial, semi-initial, and equilibrium generalized spread-

ing coefcients

T: Temperature

556 Frequently Used Symbols and Abbreviations

t: Time

u

y

: Fluid velocity in the y direction

u

AF

: Mean velocity of antifoam entity in draining lm

u

AF

: Velocity of antifoam in gravity eld

V

b

: Bubble volume

V

G

: Volumetric gas ow rate

V(t): Volume of foam generated after time t

V

T

: Total volume of air in a foam

GREEK SYMBOLS

Γ

k

ij

: Surface excess of species k at the surface ij where i and j are different uids



γ

i

: Shear rate in uid i

ε: Surface dilatational modulus (of elasticity)

ε

G

: Gibbs elasticity of foam lm

η

i

: Viscosity of uid i

θθθθ

ij ki

**

,

: Angles determining shape of lens of liquid i at the surface of uids j and k by

the surfaces ij and ki according to Neumann’s triangle

θ

*

: Angle formed at a lens of liquid i at the surfaces of uids j and k by the interfaces

ij and jk

θ

AW

: Contact angle of a solid against the air–water surface measured through the

aqueous phase

θ

OW

: Contact angle of a solid against the oil–water surface measured through the

aqueous phase

κ

t

: Curvature of the plateau borders at the top of a draining foam

μ

i

: Chemical potential of species i



νν

: Frequency of sound

Π

ALA

(h): Disjoining pressure in an air–liquid–air foam lm of thickness h

Π

AWA

(h): Disjoining pressure in an air–water–air foam lm of thickness h

Π

ALO

(h): Disjoining pressure in an air–liquid–oil pseudoemulsion lm of thickness h

Π

AWO

(h): Disjoining pressure in an air–water–oil pseudoemulsion lm of thickness h

ρ

i

: Density of uid i (where i = W = water)

σσσσσσ

ij ij

i

ij

e

,,

: Unspecied, initial, and equilibrium surface tensions between uids i

and j

σ

AWO

(h): Air–water oil pseudoemulsion lm tension

σ

AOW

(h): Air–oil–water spread lm tension

∆σσ

ij

sp

: Classical spreading pressure of liquid at ij surface

Σ

BIK

: Foaminess as dened by Bikerman

∆

σσ

ij

sp e

()hh

=

: Generalized spreading pressure at ij surface to form an equilibrium

lm of thickness h

e

τ: Line tension

ω: Angular rotational velocity (radians unit time

–1

)

ΦΦ

G

foam

: Gas volume fraction in foam

ΦΦ

L

f

oam

: Liquid volume fraction in foam

Φ

lp

: Volume fraction of gas in liquid phase below foam in bubbling method of foam

generation (the gas hold up)

557Frequently Used Symbols and Abbreviations

ABBREVIATIONS

CAC: Critical aggregation concentration

CMC: Critical micelle concentration

FTT: Film trapping technique

CHEMICALS

AOT: Sodium bis-dioctyl sulfosuccinate

C

10

TAB: Decyl tetra-ammonium bromide

C

12

TAB: Dodecyl tetra-ammonium bromide

C

14

TAB: Tetradecyl tetra-ammonium bromide

C

16

TAB: Hexadecyl tetra-ammonium bromide

C

n

EO

m

: C

n

H

2n+1

(CH

2

CH

2

O)

m

OH

C

12

6ΦSO

3

Na: Sodium 6-phenyl dodecyl benzene sulfonate

C

12

ΦSO

3

Na: Commercial sodium dodecyl benzene sulfonate (branched chain

blend)

C

12

APB: Commercial dodecyl aminopropyl betaine

C

12

EO

2.5

SO

4

Na: Commercial sodium dodecyl ether sulfate

PDMS: Polydimethylsiloxane

SAS: Commercial secondary alkyl sulfate

SDS: Sodium dodecyl sulfate

Triton X-100: Octylphenyl (CH

2

CH

2

O)

10

OH