Krantz W.B. Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach to Model Building and the Art of Approximation
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SPHERICAL COORDINATES 503
∂ρ
A
∂t
+ u
r
∂ρ
A
∂r
+ u
θ
1
r
∂ρ
A
∂θ
+ u
φ
1
r sin θ
∂ρ
A
∂φ
= D
AB
1
r
2
∂
∂r
r
2
∂ρ
A
∂r
+ D
AB
1
r
2
sin θ
∂
∂θ
sin θ
∂ρ
A
∂θ
+ D
AB
1
r
2
sin
2
θ
∂
2
ρ
A
∂φ
2
+ G
A
(G.3-5)
The following form of the equation of continuity or species balance in spherical
coordinates for component A in a binary system allows for nonconstant physical
properties and is expressed in terms of the molar concentration c
A
and the molar
flux components N
Ai
:
∂c
A
∂t
+
1
r
2
∂
∂r
(r
2
N
Ar
) +
1
r sin θ
∂
∂θ
(N
Aθ
sin θ) +
1
r sin θ
∂N
Aφ
∂φ
=
ˆ
G
A
(G.3-6)
where
ˆ
G
A
is the molar generation rate of component A per unit volume. The
components of the molar flux are given for a binary system by Fick’s law of
diffusion in the form
N
Ar
= x
A
(N
Ar
+ N
Br
) − cD
AB
∂x
A
∂r
= c
A
ˆu
r
− cD
AB
∂x
A
∂r
(G.3-7)
N
Aθ
= x
A
(N
Aθ
+ N
Bθ
) − cD
AB
1
r
∂x
A
∂θ
= c
A
ˆu
θ
− cD
AB
1
r
∂x
A
∂θ
(G.3-8)
N
Aφ
= x
A
(N
Aφ
+ N
Bφ
) − cD
AB
1
r sin θ
∂x
A
∂φ
= c
A
ˆu
φ
− cD
AB
1
r sin θ
∂x
A
∂φ
(G.3-9)
in which x
A
is the mole fraction of component A and ˆu
i
is the molar-average
velocity component in the i-direction. The form of the equation of continuity for
a binary mixture or species balance given by equation (G.3-6) is particularly use-
ful for describing mass transfer in gas systems for which the molar density c
is constant at a fixed temperature and pressure. This equation is also used to
describe reacting systems for which the generation rate of species is dictated
by the reaction stoichiometry in terms of molar concentrations. For the special
case of a fluid having a constant molar density and a constant binary diffu-
sion coefficient, equation (G.3-6) combined with equations (G.3-7) through (G.3-9)
simplifies to
∂c
A
∂t
+ˆu
r
∂c
A
∂r
+ˆu
θ
1
r
∂c
A
∂θ
+ˆu
φ
1
r sin θ
∂c
A
∂φ
= D
AB
1
r
2
∂
∂r
r
2
∂c
A
∂r
+ D
AB
1
r
2
sin θ
∂
∂θ
sin θ
∂c
A
∂θ
+ D
AB
1
r
2
sin
2
θ
∂
2
c
A
∂φ
2
+
ˆ
G
A
(G.3-10)
APPENDIX H
Integral Relationships
H.1 LEIBNITZ FORMULA FOR DIFFERENTIATING AN INTEGRAL
Moving boundary problems usually involve determining the time derivative of an
integral of a scalar quantity ϕ over a system whose volume V(t) can be changing
in time. The scalar quantity ϕ can be a function of time as well as the spatial
coordinates. Leibnitz’s formula permits recasting this volume integral into a more
convenient form given by
d
dt
V
ϕdV =
V
∂ϕ
∂t
dV +
S
ϕ(u
B
·n) dS (H.1-1)
where S is the time-dependent area enclosing the volume, u
B
the local velocity
of the bounding surface enclosing the volume, and n a unit normal vector to
this surface. For a system having whose surfaces move in only one direction,
equation (H.1-1) simplifies to
d
dt
L
2
L
1
ϕds =
L
2
L
1
∂ϕ
∂t
ds + ϕ(L
2
,t)
dL
2
dt
− ϕ(L
1
,t)
dL
1
dt
(H.1-2)
where s denotes an unspecified spatial coordinate for which L
1
and L
2
are lower
and upper limits that can be time-dependent.
H.2 GAUSS–OSTROGRADSKII DIVERGENCE THEOREM
For a closed volume V surrounded by a surface S, the Gauss–Ostrogradskii or
divergence theorem permits converting a volume integral into a surface integral.
This theorem can be applied to scalars as follows:
V
∇ϕdV =
S
nϕ dS (H.2-1)
Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach
to Model Building and the Art of Approximation, By William B. Krantz
Copyright © 2007 John Wiley & Sons, Inc.
504
GAUSS–OSTROGRADSKII DIVERGENCE THEOREM 505
where ϕ denotes an unspecified scalar quantity and n is a unit normal vector to the
surface S. The divergence theorem can be applied to vectors as follows:
V
(∇·u) dV =
S
(n ·u) dS (H.2-2)
where u denotes an unspecified vector quantity. The divergence theorem can be
applied to tensors as follows:
V
(∇·σ )dV =
S
(n · σ )dS (H.2-3)
where σ
denotes an unspecified second-order tensor quantity.
NOTATION
Dimensions are given in terms of mass (M), moles (mol), length (L), time (t),
force (F ), and temperature (T ).
A unspecified constant, variable dimensions
A amplitude of axial oscillation, L
a integer or fractional quantity, dimensionless
a surface area per unit volume of contacting device, L
−1
a
C
surface area per unit volume of carbon particles, L
−1
B unspecified constant, variable dimensions
b integer or fractional quantity, dimensionless
C
n
coefficients defined by equations (4P.12-2), (4.P.13-2), and (4.P.14-2)
C
p
heat capacity at constant pressure, L
2
/t
2
· T
C
pi
heat capacity at constant pressure of region, medium, or component i, L
2
/t
2
· T
C
v
heat capacity at constant volume, L
2
/t
2
·T
c integer or fractional quantity, dimensionless
c molar density or molar concentration, mol/L
3
c speed of sound, L/t
c
Ai
molar concentration of component A at a point or plane defined by i,mol/L
3
c
i
molar concentration of component i,mol/L
3
ˆc
i
molar concentration of component i in the bulk fluid on the macroscale, mol/L
3
c
E
i
equilibrium or maximum molar concentration of component i,mol/L
3
c
◦
i
molar concentration of component i at the interface, mol/L
3
c
◦
ir
reaction equilibrium molar concentration of component i,mol/L
3
c
is
scale factor for the derivative of the concentration with respect to the i-coordinate,
mol/L
4
c
lm
log-mean concentration, mol/L
3
c
w
molar concentration at the wall, mol/L
3
c
0
initial or upstream molar concentration, mol/L
3
D
i
binary diffusion coefficient in medium i, L
2
/t
D
ij
binary diffusion coefficient for components i and j, L
2
/t
D
L
axial dispersion coefficient, L
2
/t
D
0
binary diffusion coefficient at infinite dilution, L
2
/t
F
f
force vector exerted by fluid particle on surrounding fluid, M · L/t
2
F
p
force vector exerted by surrounding fluid on a fluid particle, M · L/t
2
f frequency in cycles per unit time, t
−1
Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach
to Model Building and the Art of Approximation, By William B. Krantz
Copyright © 2007 John Wiley & Sons, Inc.
506
NOTATION 507
f
i
(···) denotes a function of the variables or parameters in parentheses, variable
dimensions
G
e
energy generation rate per unit volume, M/L · t
2
G
i
mass generation rate of component i per unit volume, M/L
3
·t
G
0
amplitude of periodic energy generation rate per unit volume, M/L · t
2
˜
G heating rate, M ·L
2
/t
3
ˆ
G total molar generation rate per unit volume, mol/L
3
· t
ˆ
G
i
molar generation rate of component i per unit volume by homogeneous chemical
reaction, mol/L
3
·t
ˆ
G
m
i
maximum molar generation rate of component i by homogeneous chemical
reaction, mol/L
3
·t
g gravitational acceleration, L/t
2
g
c
dimensional constant in Newton’s law of motion, M · L/t
2
· F
g
i
component of the gravitational acceleration in the i-direction, L/t
2
g
i
(···) denotes a function of the arguments in parentheses, variable dimensions
H thickness, depth, or location of fixed boundary, L
H Henry’s law constant, T
2
/L
2
H
i
enthalpy per mole of component i, M · L
2
/t
2
· mol
H
i
thickness of layer i, L
h heat-transfer coefficient, M/t
3
· T
h
P
heat-transfer coefficient between particles and gas phase, M/t
3
· T
h
W
heat-transfer coefficient between wall and gas phase, M/t
3
· T
h
1
,h
2
thickness of layers 1 and 2, L
K integration rate constant, variable dimensions
K crystallization rate constant, t
−n
K
i
distribution coefficient of component i between two phases, variable units
k thermal conductivity, M · L/t
3
· T
k an integer, dimensionless
k
G
thermal conductivity of gas phase, M · L/t
3
· T
k
•
G
gas-phase mass-transfer coefficient based on the molar flux relative to the molar
average velocity, L/t
k
•
G0
gas-phase mass-transfer coefficient in the absence of chemical reaction based on
the molar flux relative to the molar average velocity, L/t
k
•
i
mass-transfer coefficient of region, medium, or component i, L/t
k
L
liquid-phase mass-transfer coefficient based on the molar flux relative to a
stationary observer, L/t
k
•
L
liquid-phase mass-transfer coefficient based on the molar flux relative to the
molar average velocity, L/t
k
•
L0
liquid-phase mass-transfer coefficient in the absence of chemical reaction based
on the molar flux relative to the molar average velocity, L/t
k
m
membrane permeability, L
2
· t/M
k
n
nth-order homogeneous reaction-rate parameter, mol
(1−n)
· L
3(n−1)
/t
ˆ
k
n
nth-order heterogeneous reaction-rate parameter, mol
(1−n)
·L
(3n−2)
/t
k
p
permeability coefficient in Darcy’s law for flow through porous media, L
2
L length in a particular coordinate direction, L
L location of fixed or moving boundary, L
L
d
final depth corresponding to draining time t
d
,L
L
e
entrance length required for flow to become fully developed, L
L
i
initial width of pulse of injected particles in field-flow fractionation, L
508 NOTATION
L
m
maximum thaw depth, L
L
s
unspecified length scale, L
L
0
initial depth corresponding to t = 0,L
˙
L time rate of change of the thickness L, L/t
l
i
equilibrium adsorption distribution coefficient for component i, L · t
2
/M
M mass, M
M
i
molecular weight of component i, M/mol
m an integer, dimensionless
m
ad
total mass of adsorbent, M
m
O
2
total mass of oxygen produced per day in the pulsed PSA process, M
N
Ai
molar flux of component A in the i-direction or at surface i,mol/L
2
·t
N
i
molar flux of component i,mol/L
2
· t
N
iw
molar flux of component i at the wall, mol/L
2
· t
N
iw
time-averaged molar flux of component i at the wall, mol/L
2
· t
n an integer, reaction order, or Avrami exponent, dimensionless
n
Ai
mass flux of component A in the i-direction, M/L
2
· t
n
G
mass flux in the gas phase, M/L
2
· t
n
i
mass flux of component i, M/L
2
·t
n unit normal vector, dimensionless
P pressure, M/L · t
2
˙
P time rate of change of the pressure, M/L · t
3
P
d
dynamic contribution to the total pressure, M/L · t
2
P
h
hydrostatic contribution to the total pressure, M/L · t
2
p an integer, dimensionless
p(···) denotes a function describing the pressure, M/L · t
2
p
◦
i
vapor pressure of component i, M/L · t
2
p
i
partial pressure of component i, M/L · t
2
Q volumetric flow rate, L
3
/t
Q
L
volumetric flow rate of liquid, L
3
/t
q heat-transfer flux, M/t
3
q
i
heat-transfer flux at surface i, M/t
3
q
i
moles of component i per unit volume of adsorbent, mol/L
3
q
e
i
moles of component i per unit volume of adsorbent at equilibrium, mol/L
3
q
∞
i
value of q
e
i
at infinite partial pressure of component i,mol/L
3
q heat flux averaged over the surface, M/t
3
˙q
i
rate of molar adsorption of component i per unit volume, mol/L
3
·t
˙q
e
i
equilibrium-controlled rate of molar adsorption of component i per unit volume,
mol/L
3
· t
R radius, L
R gas constant, M · L
2
/t
2
· T · mol
R regression coefficient, dimensionless
R
i
rate of homogeneous reaction of component i per unit volume, mol/L
3
·t
R
P
particle or pore radius, L
R
1
,R
2
radii at surfaces defined by 1 and 2, L
r radial coordinate, L
S surface area, L
2
S
c
cross-sectional area, L
2
S
t
energy generation rate per unit volume, M/L · t
2
˙
S rate of entropy generation per unit volume, M/L · t
2
·T
NOTATION 509
T temperature, T
T
c
crystallization temperature, T
T
f
freezing or front temperature, T
T
G
temperature of gas phase, T
T
i
temperature evaluated at surface i, T
T
is
scale factor for derivative of temperature in the i-direction,T/L
T
◦
reference temperature, T
T average temperature, T
T
m
maximum value of the spatially averaged temperature, T
˙
T time rate of change of temperature, T/t
t time, t
t unit tangential vector, dimensionless
t
a
time at which rate of phase separation is a maximum, t
t
ad
characteristic adsorption time, t
t
c
contact or cooking time, t
t
d
characteristic axial dispersion, draining, or depressurization time, t
t
g
time for growth of dispersed phase, t
t
H
time required for the temperature to penetrate thickness H,t
t
o
observation time, t
t
p
pressurization time, t
t
sp
time scale corresponding to the periodic motion, t
t
st
time scale corresponding to the observation time, t
t
sv
time scale corresponding to the viscous penetration, t
U superficial velocity in porous media, L/t
U
G
linear velocity of gas, L/t
U
m
maximum velocity, L/t
U
t
terminal velocity, L/t
U
w
axial velocity at the wall, L/t
U
0
specified constant velocity, L/t
U average velocity, L/t
U
w
average velocity in the wall region, L/t
u mass-average velocity vector, L/t
ˆu molar-average velocity vector, L/t
u
B
local mass-average velocity of a surface enclosing a volume, L/t
u
i
mass-average velocity component in the i-direction, L/t
ˆu
i
molar-average velocity component in the i-direction, L/t
1
u
i
mass-average velocity component in the i-direction in a porous media, L/t
V volume, L
3
V
−H
,V
+H
velocity at which fluid is injected and withdrawn at the surfaces at ±H,L/t
V
s
scale for velocity of a moving boundary, L/t
V
T
total volume, L
3
V
u
volume of upper chamber, L
3
V
0
specified blowing or suction velocity, L/t
W width, L
W
G
molar flow rate of gas, mol/t
ˆ
W
H
molar injection rate of hydrogen per unit length, mol/t · L
W
i
molar flow rate of component i,mol/t
W
0
i
molar flow rate of component i in the feed stream, mol/t
˜
W
m
mass injection or removal rate per unit area, M/t · L
2
510 NOTATION
w gap width, L
X
c
fractional dissociation or conversion, dimensionless
x spatial coordinate, L
x
i
mole fraction of component i, dimensionless
y spatial coordinate, L
y
BL
product purity measured in terms of the mole fraction of component B,
dimensionless
y
i
gas-phase mole fraction of component i, dimensionless
˜y translated spatial coordinate, L
y
∗
m
dimensionless y-coordinate in the equations of motion
y
∗
t
dimensionless y-coordinate in the energy equation
z spatial coordinate, L
GREEK SYMBOLS
α thermal diffusivity, L
2
/t
α undetermined constant, variable units
α arbitrary quantity considered in dimensional analysis, variable units
α
i
thermal diffusivity of medium i, L
2
/t
β arbitrary quantity considered in dimensional analysis, variable units
β unspecified time constant, t
−1
β constant coefficient for temperature dependence of vapor pressure, M/L · t
2
·T
β constant coefficient for temperature dependence of thermal conductivity,
M · L/t
2
· T
2
β
s
scale factor for the spatial derivative of the velocity, L/t
β
s
coefficient of solutal volume expansion, L
3
/mol
β
t
coefficient of thermal volume expansion, T
−1
γ ratio of heat capacity at constant pressure to that at constant volume, dimensionless
γ arbitrary quantity considered in dimensional analysis, variable units
γ
t
second-order coefficient of thermal volume expansion, T
−2
c
A
c
A0
−c
A∞
, concentration difference between control surfaces defined by 0 and ∞,
mol/L
3
c
lm
log-mean concentration driving force, mol/L
3
E activation energy for reaction, M · L
2
/t
2
· mol
H
f
latent heat of fusion per unit mass, L
2
/t
2
H
v
latent heat of vaporization per unit mass, L
2
/t
2
P P
2
− P
1
, pressure difference between control surfaces defined by 1 and 2, M/L · t
2
ρ
0
AB
ρ
0
A
− ρ
0
B
, difference between pure component mass densities, M/L
3
δ region of influence or boundary-layer thickness, L
δ arbitrary quantity considered in dimensional analysis, variable units
δ
identity tensor, dimensionless
δ
c
characteristic film thickness, L
δ
i
unit vector in the i-direction, dimensionless
δ
m
momentum boundary-layer thickness, L
δ
p
thickness of region of influence in porous media, L
δ
r
thickness of reaction boundary layer, L
δ
s
solutal boundary-layer thickness, L
NOTATION 511
δ
t
thermal boundary-layer thickness, L
ε fractional void space or porosity, dimensionless
ε small distance, L
ε radiative emissivity of surface, dimensionless
ς dimensional constant, M/L
2
η local film thickness, L
θ angular coordinate, radians
θ angle of inclination or angle between horizontal and unit tangential vector,
dimensionless
κ bulk viscosity, M/L · t
κ ratio of the mass fluxes for binary diffusion, dimensionless
κ stoichiometric coefficient, dimensionless
λ
n
roots of equation (4.P.12-3), (4.P.13-3), or (4.P.14-3), dimensionless
μ shear viscosity, M/L ·t
ν kinematic viscosity, L
2
/t
ξ a dimensionless constant having a value of either 0 or 1, depending on the local
concentration relative to that at reaction equilibrium
i
ith dimensionless group whose definition depends on the particular problem
ρ mass density, M/L
3
ρ average mass density, M/L
3
˜ρ mass density of the gas phase, M/L
3
ρ
i
mass density of region, medium, or component i, M/L
3
ρ
0
i
mass density of pure component i, M/L
3
ρ
p
mass density of pure polymer, M/L
3
ρ
w
mass density of liquid water, M/L
3
σ Stefan–Boltzmann constant, M/t
3
· T
4
σ total stress tensor, M/L · t
2
σ
AB
collision diameter for molecules A and B,L
σ
ij
ij -component of the total stress tensor, M/L · t
2
τ viscous stress tensor, M/L · t
2
τ
ij
ij -component of the viscous stress tensor, M/L · t
2
φ angular coordinate, radians
φ unspecified constant, dimensionless
φ volume of liquid per unit interfacial area, L
ϕ unspecified scalar having variable units
ϕ dimensionless derivative of the concentration
χ ratio of the total absorption rate relative to its maximum value for purely physical
absorption, dimensionless
r
reaction-rate parameter, dimensionless
ψ liquid volume fraction in a multiphase system, dimensionless
˙
ψ time rate of change of the liquid volume fraction, t
−1
ψ
bf
bed-size factor equal to the ratio of the total mass of adsorbent to the total mass of
oxygen produced per day, dimensionless
ψ
r
product recovery equal to the ratio of the volumetric flow of the product to that of
the feed, dimensionless
collision integral, dimensionless
ω angular frequency or velocity, radians/t
ω
i
mass fraction of component i, dimensionless
˜ω
i
mass fraction of component i in the gas phase, dimensionless
512 NOTATION
SUPERSCRIPTS
d depressurization or diluent
E equilibrium value
p pressurization
0 initial value
◦
saturation or reference value
− average value
ˆ variable evaluated in the bulk fluid when using microscale-macroscale modeling
variable evaluated in a porous medium
∗ dimensionless variable
† transpose of a second-order tensor
· time derivative
˜ dummy integration variable or a translated spatial coordinate
→ vector quantity
SUBSCRIPTS
A component A
atm atmospheric conditions
B component B
b evaluated at the boundary
C component C or carbon
c variable in the species-balance equation
D property of turkey dressing or the diluent phase
d dynamic pressure contribution
f frozen region or feed concentration
G property of the gas phase
h hydrostatic pressure contribution
I iron or iron phase
L evaluated at L, downsteam, or in the liquid
l evaluated in the lumen of a hollow fiber membrane
m maximum value, denotes a variable or parameter associated with the fluid or a with
a membrane
o observation time
P property of the polymer or particulate phase
R rust or rust phase
r reference factor or component in the r-direction
s scale factor or denotes a solutal quantity
T property of turkey
t variable for heat transfer or terminal velocity
t time derivative
u unfrozen region
W water or water phase
w evaluated at the wall or in the wall region
x component in the x-direction
y component in the y-direction
z component in the z-direction