A	area, ${\mathrm{m}}^2$
$a_i$	interfacial area concentration (IAC), 1/m
C	coefficient
$C_D$	drag coefficient
$C{D}_{c1}^{*}{}^2$	inter-group transfer coefficient
Ca	Capillary number
${\mathrm{c}}_{\mathrm{bv}}$	${\mathrm{f}}_{\mathrm{bv}}^{2/3}+{\left(1-{\mathrm{f}}_{\mathrm{bv}}\right)}^{2/3}-1$
${\mathrm{c}}_1$-${\mathrm{c}}_{23}$	adjustable parameters
D	bubble size, m
${\mathrm{D}}_{\mathrm{c}}$	critical diameter for transition between bubble groups, m
${\mathrm{D}}_{\mathrm{cr},\mathrm{ph}}$	bubble critical size due to phase change, m
${\mathrm{D}}_{\mathrm{H}}$	hydraulic diameter of the flow channel, m
${\mathrm{D}}_{\min ,\mathrm{e}}$	an arbitrarily defined minimum eddy size, m
${\mathrm{D}}_{\mathrm{sc}}$	surface equivalent diameter, m
${\mathrm{D}}_{\mathrm{sm}}$	Sauter mean diameter, m
${\mathrm{D}}_1$	diameter of bubble 1, m
${\mathrm{D}}_2$	diameter of bubble 2, m
d	parent bubble diameter, m
${\mathrm{d}}_{\mathrm{j}}$	daughter bubble diameter, m
${\mathrm{d}}_{\mathrm{e}}$	eddy size, m
E	kinetic energy, kg ${\mathrm{m}}^2{\mathrm{s}}^2$
F(α)	turbulence damping factor due to the presence of the gas phase
f	frequency, 1/s
${\mathrm{f}}_{\mathrm{bv}}$	volume fraction, ${\mathrm{V}}_{\mathrm{i}}/{\mathrm{V}}_{\mathrm{j}}$
G	gap in a rectangular/annular channel, m
g	gravitational acceleration, m/${\mathrm{s}}^2$
I	turbulent intensity
j	superficial velocity, m/s
K	constant
k	the ratio of the average turbulent eddy size to the bubble Sauter mean diameter
$〈{\mathrm{l}}_{\mathrm{w}}〉$	average distance between bubbles, m
N	average number of eddies arrive at the surface of a drop in unit time
n	number density of bubbles or eddies, 1/${\mathrm{m}}^3$
R	source/sink rates per unit mixture volume, 1/${\mathrm{m}}^{3}s$
${\mathrm{R}}_t$	radius of the tube, m
r	bubble radius, m
S	source/sink rates per unit mixture volume, 1/${\mathrm{m}}^{6}s$
t	time, s
${\mathrm{u}}_{\mathrm{rel}}$	relative velocity between bubbles, m/s
${\mathrm{u}}_{\mathrm{rw}2}$	wake velocity of Group II bubbles, m/s
${\mathrm{u}}_{\mathrm{w}12}$	wake velocity of Group I bubbles, m/s
V	volume, ${\mathrm{m}}^3$
v	velocity, m/s
${\mathrm{V}}_1^{\mathrm{BOX}}$	volume influenced by a wake, ${\mathrm{m}}^3$
${\mathrm{v}}_i$	interfacial velocity, m/s
${\mathrm{v}}_{pm}$	the average local particle velocity, m/s
We Weber number
Greek symbols
$\mathsf{\alpha },{ \mathsf{\alpha }}_{\max }$	volume fraction, packing limit
$\mathsf{\Lambda }$	factor considering the effect of the size ratio between bubbles and eddies
$\dot{\mathsf{\gamma }}$	shear rate, 1/s
$\mathsf{{\rm Y}}$	factor considering the effect of the bubble that reduces the free space for bubble movement and causes an increase in the collision frequency.
$\epsilon$	energy dissipation rate, ${\mathrm{m}}^2$/${\mathrm{s}}^3$
η	efficiency
$\zeta$	rate of volume source per unit mixture volume, 1/s
$\mathsf{\Gamma }$	mass generation rate, kg/${\mathrm{m}}^3$s
$\mathsf{\Phi }$	IAC source/sink rate, 1/ms
${\mathsf{\mu }}_{\mathrm{f}}$	liquid viscosity, Pa·s
ν	kinetic viscosity, ${\mathrm{m}}^2$/${\mathrm{s}}^2$
ρ	density, kg/${\mathrm{m}}^3$
Π	factor considering the ratio of the mean bubble distance to the turbulent path length
$\sigma$	surface tension, N/m
${\mathsf{\tau }}_{\mathrm{s}}$	surface tension forces, N
${\mathsf{\tau }}_{\mathrm{v}}$	viscous stress, N
$\mathsf{\psi }$	factor depending on bubble shape
$\mathsf{\chi }$	inter-group transfer coefficient
Subscripts
1	Group I
2	Group II
b	breakup
c	coalescence
coll	collision
cr	critical value
D	drag force
d	dispersed
e	eddy
f	liquid phase
g	gas phase
i	i-th component
j	j-th component
kin	kinetic
m	mixture value
max	maximum
max2	maximum in Group II
min	minimum
nuc	nucleation
ph	phase change
RC	random collision mechanism
SO	shearing-off mechanism
TI	turbulent impact mechanism
WE	wake entrainment mechanism
SI	surface instability mechanism
Superscripts
* non-dimensional value
(1)	interactions within Group I bubbles
(11, 2)	coalescence of a Group I bubble with another Group I bubble to generate a Group II bubble
(12, 2)	coalescence of a Group I bubble with a Group II bubble to generate a new Group II bubble
(2)	interactions within Group II bubbles
(2, 1)	Group I bubbles generated from breakup of a Group II bubble
(2, 2)	Group II bubbles generated from breakup of a Group II bubble
(2, 11)	breakup of a Group II bubble to generate two Group I bubbles
(2, 12)	breakup of a Group II bubble to generate a (or multiple) Group I bubble(s) and a Group II bubble