Nomenclature
Roman letters
a
Amplitude of wave-like disturbance
A
Cross-sectional area or cloud radius
A
Attenuation
b
Power law index
Ba
Bagnold number,
ρ
S
D
2
̇
γ/μ
L
c
Concentration
c
Speed of sound
c
κ
Phase velocity for wavenumber
κ
c
p
Specific heat at constant pressure
c
s
Specific heat of solid or liquid
c
v
Specific heat at constant volume
C
Compliance
C
Damping coefficient
C
D
Drag coefficient
C
ij
Drag and lift coefficient matrix
C
L
Lift coefficient
C
p
Coefficient of pressure
C
pmin
Minimum coefficient of pressure
d
Diameter
d
j
Jet diameter
d
o
Hopper opening diameter
D
Particle, droplet or bubble diameter
D
Mass diffusivity
D
m
Volume (or mass) mean diameter
D
s
Sauter mean diameter
11
D
(
T
)
Determinant of the transfer matrix [
T
]
D
Thermal diffusivity
e
Specific internal energy
E
Rate of exchange of energy per unit volume
f
Frequency in
Hz
f
Friction factor
f
L
,f
V
Liquid and vapor thermodynamic quantities
F
i
Force vector
Fr
Froude number
F
Interactive force per unit volume
g
Acceleration due to gravity
g
L
,g
V
Liquid and vapor thermodynamic quantities
G
Ni
Mass flux of component
N
in direction
i
G
N
Mass flux of component
N
h
Specific enthalpy
h
Height
H
Height
H
Total head,
p
T
/ρg
He
Henry’s law constant
Hm
Haberman-Morton number, normally
gμ
4
/ρS
3
i, j, k, m, n
Indices
i
Square root of
−
1
I
Acoustic impulse
I
Rate of transfer of mass per unit volume
j
i
Total volumetric flux in direction
i
j
Ni
Volumetric flux of component
N
in direction
i
j
N
Volumetric flux of component
N
k
Polytropic constant
k
Thermal conductivity
k
Boltzmann’s constant
k
L
,k
V
Liquid and vapor quantities
K
Constant
K
∗
Cavitation compliance
Kc
Keulegan-Carpenter number
K
ij
Added mass coefficient matrix
K
n
,K
s
Elastic spring constants in normal and tangential directions
Kn
Knudsen number,
λ/
2
R
K
Frictional constants
Typical dimension
12
t
Turbulent length scale
L
Inertance
L
Latent heat of vaporization
m
Mass
̇
m
Mass flow rate
m
G
Mass of gas in bubble
m
p
Mass of particle
M
Mach number
M
∗
Mass flow gain factor
M
ij
Added mass matrix
M
Molecular weight
Ma
Martinelli parameter
n
Number of particles per unit volume
̇
n
Number of events per unit time
n
i
Unit vector in the
i
direction
N
(
R
)
,N
(
D
)
,N
(
v
) Particle size distribution functions
N
∗
Number of sites per unit area
Nu
Nusselt number
p
Pressure
p
T
Total pressure
p
a
Radiated acoustic pressure
p
G
Partial pressure of gas
p
s
Sound pressure level
P
Perimeter
Pe
Peclet number, usually
WR/α
C
Pr
Prandtl number,
ρνc
p
/k
q
General variable
q
i
Heat flux vector
Q
General variable
Q
Rate of heat transfer or release per unit mass
Q
Rate of heat addition per unit length of pipe
r, r
i
Radial coordinate and position vector
r
d
Impeller discharge radius
R
Bubble, particle or droplet radius
R
∗
k
Resistance of component,
k
R
B
Equivalent volumetric radius, (3
τ/
4
π
)
1
3
R
e
Equilibrium radius
Re
Reynolds number, usually 2
WR/ν
C
R
Gas constant
13
s
Coordinate measured along a streamline or pipe centerline
s
Laplace transform variable
s
Specific entropy
S
Surface tension
S
D
Surface of the disperse phase
St
Stokes number
Str
Strouhal number
t
Time
t
c
Binary collision time
t
u
Relaxation time for particle velocity
t
T
Relaxation time for particle temperature
T
Temp erature
T
Granular temperature
T
ij
Transfer matrix
u
i
Velocity vector
u
Ni
Velocity of component
N
in direction
i
u
r
,u
θ
Velocity components in polar coordinates
u
s
Shock velocity
u
∗
Friction velocity
U, U
i
Fluid velocity and velocity vector in absence of particle
U
∞
Velocity of upstream uniform flow
v
Volume of particle, droplet or bubble
V, V
i
Absolute velocity and velocity vector of particle
V
Volume
V
Control volume
̇
V
Volume flow rate
w
Dimensionless relative velocity,
W/W
∞
W, W
i
Relative velocity of particle and relative velocity vector
W
∞
Terminal velocity of particle
W
p
Typical phase separation velocity
W
t
Typical phase mixing velocity
We
Weber number, 2
ρW
2
R/S
W
Rate of work done per unit mass
x, y, z
Cartesian coordinates
x
i
Position vector
x
Mass fraction
X
Mass quality
z
Coordinate measured vertically upward
14
Greek letters
α
Volume fraction
β
Volume quality
γ
Ratio of specific heats of gas
̇
γ
Shear rate
Γ
Rate of dissipation of energy per unit volume
δ
Boundary layer thickness
δ
d
Damping coefficient
δm
Fractional mass
δ
T
Thermal boundary layer thickness
δ
2
Momentum thickness of the boundary layer
δ
ij
Kronecker delta:
δ
ij
=1for
i
=
j
;
δ
ij
=0for
i
=
j
Fractional volume
Coefficient of restitution
Rate of dissipation of energy per unit mass
ζ
Attenuation or amplification rate
η
Bubble population per unit liquid volume
θ
Angular coordinate or direction of velocity vector
θ
Reduced frequency
θ
w
Hopper opening half-angle
κ
Wavenumber
κ
Bulk modulus of compressibility
κ
L
,κ
G
Shape constants
λ
Wavelength
λ
Mean free path
λ
Kolmogorov length scale
Λ
Integral length scale of the turbulence
μ
Dynamic viscosity
μ
∗
Coulombfrictioncoefficient
ν
Kinematic viscosity
ν
Mass-based stoichiometric coefficient
ξ
Particle loading
ρ
Density
σ
Cavitation number
σ
i
Inception cavitation number
σ
ij
Stress tensor
σ
D
ij
Deviatoric stress tensor
Σ(
T
)
Thermodynamic parameter
15
τ
Kolmogorov time scale
τ
i
Interfacial shear stress
τ
n
Normal stress
τ
s
Shear stress
τ
w
Wall shear stress
ψ
Stokes stream function
ψ
Head coefficient, ∆
p
T
/ρ
Ω
2
r
2
d
φ
Velocity potential
φ
Internal friction angle
φ
Flow coefficient,
j/
Ω
r
d
φ
2
L
,φ
2
G
,φ
2
L
0
Martinelli pressure gradient ratios
φ
Fractional perturbation in bubble radius
ω
Radian frequency
ω
a
Acoustic mode frequency
ω
i
Instability frequency
ω
n
Natural frequency
ω
m
Cloud natural frequencies
ω
m
Manometer frequency
ω
p
Peak frequency
Ω
Rotating frequency (radians/sec)
Subscripts
On any variable,
Q
:
Q
o
Initial value, upstream value or reservoir value
Q
1
,Q
2
,Q
3
Components of
Q
in three Cartesian directions
Q
1
,Q
2
Values upstream and downstream of a component or flow structure
Q
∞
Value far from the particle or bubble
Q
∗
Throat values
Q
A
Pertaining to a general phase or component,
A
Q
b
Pertaining to the bulk
Q
B
Pertaining to a general phase or component,
B
Q
B
Value in the bubble
Q
C
Pertaining to the continuous phase or component,
C
Q
c
Critical values and values at the critical point
Q
D
Pertaining to the disperse phase or component,
D
16
Q
e
Equilibrium value or value on the saturated liquid/vapor line
Q
e
Effective value or exit value
Q
G
Pertaining to the gas phase or component
Q
i
Components of vector
Q
Q
ij
Components of tensor
Q
Q
L
Pertaining to the liquid phase or component
Q
m
Maximum value of
Q
Q
N
Pertaining to a general phase or component,
N
Q
O
Pertaining to the oxidant
Q
r
Component in the
r
direction
Q
s
A surface, system or shock value
Q
S
Pertaining to the solid particles
Q
V
Pertaining to the vapor phase or component
Q
w
Value at the wall
Q
θ
Component in the
θ
direction
Superscripts and other qualifiers
On any variable,
Q
:
Q
,Q
,Q
∗
Used to differentiate quantities similar to
Q
̄
Q
Mean value of
Q
or complex conjugate of
Q
`
Q
Small perturbation in
Q
̃
Q
Complex amplitude of oscillating
Q
̇
Q
Time derivative of
Q
̈
Q
Second time derivative of
Q
ˆ
Q
(
s
)
Laplace transform of
Q
(
t
)
̆
Q
Coordinate with origin at image point
δQ
Small change in
Q
Re
{
Q
}
Real part of
Q
Im
{
Q
}
Imaginary part of
Q
17
NOTES
Notation
The reader is referred to section 1.1.3 for a more complete description of
the multiphase flow notation employed in this book. Note also that a few
symbols that are only used locally in the text have been omitted from the
above lists.
Units
In most of this book, the emphasis is placed on the nondimensional pa-
rameters that govern the phenomenon being discussed. However, there are
also circumstances in which we shall utilize dimensional thermodynamic and
transport properties. In such cases the International System of Units will be
employed using the basic units of mass (
kg
), length (
m
), time (
s
), and ab-
solute temperature (
K
).
18