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 493
τ
φφ
=−μ
2
1
r sin θ
∂u
φ
∂φ
+
u
r
r
+
u
θ
r
cot θ
−
2
3
1
r
2
∂
∂r
(r
2
u
r
) +
1
r sin θ
∂
∂θ
(u
θ
sin θ) +
1
r sin θ
∂u
φ
∂φ
(D.3-6)
τ
rθ
= τ
θr
=−μ
r
∂
∂r
u
θ
r
+
1
r
∂u
r
∂θ
(D.3-7)
τ
θφ
= τ
φθ
=−μ
sin θ
r
∂
∂θ
u
φ
sin θ
+
1
r sin θ
∂u
θ
∂φ
(D.3-8)
τ
φr
= τ
rφ
=−μ
1
r sin θ
∂u
r
∂φ
+ r
∂
∂r
u
φ
r
(D.3-9)
For the special case of an incompressible Newtonian fluid with constant viscos-
ity, equations (D.3-1) through (D.3-3) combined with equations (D.3-4) through
(D.3-9) simplify to
ρ
∂u
r
∂t
+ ρu
r
∂u
r
∂r
+ ρ
u
θ
r
∂u
r
∂θ
+ ρ
u
φ
r sin θ
∂u
r
∂φ
− ρ
u
2
θ
+ u
2
φ
r
=−
∂P
∂r
+ μ
1
r
2
∂
2
∂r
2
(r
2
u
r
) + μ
1
r
2
sin θ
∂
∂θ
sin θ
∂u
r
∂θ
+ μ
1
r
2
sin
2
θ
∂
2
u
r
∂φ
2
+ ρg
r
(D.3-10)
ρ
∂u
θ
∂t
+ ρu
r
∂u
θ
∂r
+ ρ
u
θ
r
∂u
θ
∂θ
+ ρ
u
φ
r sin θ
∂u
θ
∂φ
+ ρ
u
r
u
θ
r
− ρ
u
2
φ
cot θ
r
=−
1
r
∂P
∂θ
+ μ
1
r
2
∂
∂r
r
2
∂u
θ
∂r
+ μ
1
r
2
∂
∂θ
1
sin θ
∂
∂θ
(u
θ
sin θ)
+ μ
1
r
2
sin
2
θ
∂
2
u
θ
∂φ
2
+ μ
2
r
2
∂u
r
∂θ
− μ
2cosθ
r
2
sin
2
θ
∂u
φ
∂φ
+ ρg
θ
(D.3-11)
ρ
∂u
φ
∂t
+ ρu
r
∂u
φ
∂r
+ ρ
u
θ
r
∂u
φ
∂θ
+ ρ
u
φ
r sin θ
∂u
φ
∂φ
+ ρ
u
φ
u
r
r
+ ρ
u
θ
u
φ
r
cot θ
=−
1
r sin θ
∂P
∂φ
+ μ
1
r
2
∂
∂r
r
2
∂u
φ
∂r
+ μ
1
r
2
∂
∂θ
1
sin θ
∂
∂θ
(u
φ
sin θ)
+ μ
1
r
2
sin
2
θ
∂
2
u
φ
∂φ
2
+ μ
2
r
2
sin θ
∂u
r
∂φ
+ μ
2cosθ
r
2
sin
2
θ
∂u
θ
∂φ
+ ρg
φ
(D.3-12)
APPENDIX E
Equations of Motion for Porous Media
E.1 RECTANGULAR COORDINATES
The following forms of the x-, y-, and z-components of the equations of motion
in rectangular coordinates for flow through porous media are based on Brinkman’s
empirical modification of Darcy’s law and assume a body force due to a grav-
itational field, and an incompressible fluid having constant viscosity μ and per-
meability k
p
;theu
i
denote the components of the superficial velocity based on
considering a porous medium to be homogeneous
1
:
0 =−
∂P
∂x
−
μ
k
p
u
x
+ μ
∂
2
u
x
∂x
2
+ μ
∂
2
u
x
∂y
2
+ μ
∂
2
u
x
∂z
2
+ ρg
x
(E.1-1)
0 =−
∂P
∂y
−
μ
k
p
u
y
+ μ
∂
2
u
y
∂x
2
+ μ
∂
2
u
y
∂y
2
+ μ
∂
2
u
y
∂z
2
+ ρg
y
(E.1-2)
0 =−
∂P
∂z
−
μ
k
p
u
z
+ μ
∂
2
u
z
∂x
2
+ μ
∂
2
u
z
∂y
2
+ μ
∂
2
u
z
∂z
2
+ ρg
z
(E.1-3)
E.2 CYLINDRICAL COORDINATES
The following forms of the r-, θ-, and z-components of the equations of motion
in cylindrical coordinates for flow through porous media are based on Brinkman’s
empirical modification of Darcy’s law and assume a body force due to a grav-
itational field, and an incompressible fluid having constant viscosity μ and per-
meability k
p
;theu
i
denote the components of the superficial velocity based on
considering a porous medium to be homogeneous
2
:
0 =−
∂P
∂r
−
μ
k
p
u
r
+ μ
∂
∂r
1
r
∂
∂r
(ru
r
)
+ μ
1
r
2
∂
2
u
r
∂θ
2
− μ
2
r
2
∂u
θ
∂θ
+ μ
∂
2
u
r
∂z
2
+ ρg
r
(E.2-1)
1
H. C. Brinkman, Appl.Sci.Res., A1, 27–34, 81–86 (1947).
2
Ibid.
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.
494
SPHERICAL COORDINATES 495
0 =−
1
r
∂P
∂θ
−
μ
k
p
u
θ
+ μ
∂
∂r
1
r
∂
∂r
(ru
θ
)
+ μ
1
r
2
∂
2
u
θ
∂θ
2
+ μ
2
r
2
∂u
r
∂θ
+ μ
∂
2
u
θ
∂z
2
+ ρg
θ
(E.2-2)
0 =−
∂P
∂z
−
μ
k
p
u
z
+ μ
1
r
∂
∂r
r
∂u
z
∂r
+ μ
1
r
2
∂
2
u
z
∂θ
2
+ μ
∂
2
u
z
∂z
2
+ ρg
z
(E.2-3)
E.3 SPHERICAL COORDINATES
The following forms of the r-, θ-, and φ-components of the equations of motion
in spherical coordinates for flow through porous media are based on Brinkman’s
empirical modification of Darcy’s law and assume a body force due to a grav-
itational field, and an incompressible fluid having constant viscosity μ and per-
meability k
p
;theu
i
denote the components of the superficial velocity based on
considering a porous medium to be homogeneous
3
:
0 =−
∂P
∂r
−
μ
k
p
u
r
+ μ
1
r
2
∂
2
∂r
2
(r
2
u
r
) + μ
1
r
2
sin θ
∂
∂θ
sin θ
∂u
r
∂θ
+ μ
1
r
2
sin
2
θ
∂
2
u
r
∂φ
2
+ ρg
r
(E.3-1)
0 =−
1
r
∂P
∂θ
−
μ
k
p
u
θ
+ μ
1
r
2
∂
∂r
r
2
∂u
θ
∂r
+ μ
1
r
2
∂
∂θ
1
sin θ
∂
∂θ
(u
θ
sin θ)
+ μ
1
r
2
sin
2
θ
∂
2
u
θ
∂φ
2
+ μ
2
r
2
∂u
r
∂θ
− μ
2cosθ
r
2
sin
2
θ
∂u
φ
∂φ
+ ρg
θ
(E.3-2)
0 =−
1
r sin θ
∂P
∂φ
−
μ
k
p
u
φ
+ μ
1
r
2
∂
∂r
r
2
∂u
φ
∂r
+ μ
1
r
2
∂
∂θ
1
sin θ
∂
∂θ
(u
φ
sin θ)
+ μ
1
r
2
sin
2
θ
∂
2
u
φ
∂φ
2
+ μ
2
r
2
sin θ
∂u
r
∂φ
+ μ
2cosθ
r
2
sin
2
θ
∂u
θ
∂φ
+ ρg
φ
(E.3-3)
3
Ibid.
APPENDIX F
Thermal Energy Equation
F.1 RECTANGULAR COORDINATES
The following form of the thermal energy equation in rectangular coordinates allows
for nonconstant physical properties, energy generation, and conversion of mechan-
ical to internal energy through viscous dissipation, which is expressed in terms of
unspecified viscous stress–tensor components τ
ij
:
ρC
v
∂T
∂t
+ ρC
v
u
x
∂T
∂x
+ ρC
v
u
y
∂T
∂y
+ ρC
v
u
z
∂T
∂z
=
∂
∂x
k
∂T
∂x
+
∂
∂y
k
∂T
∂y
+
∂
∂z
k
∂T
∂z
− T
∂P
∂T
ρ
∂u
x
∂x
+
∂u
y
∂y
+
∂u
z
∂z
− τ
xx
∂u
x
∂x
− τ
yy
∂u
y
∂y
− τ
zz
∂u
z
∂z
− τ
xy
∂u
x
∂y
+
∂u
y
∂x
− τ
xz
∂u
x
∂z
+
∂u
z
∂x
− τ
yz
∂u
y
∂z
+
∂u
z
∂y
+ G
e
(F.1-1)
where C
v
is the heat capacity at constant volume, k the thermal conductivity, and G
e
the energy generation rate per unit volume. Equation (F.1-1) can be applied to non-
Newtonian fluids if the appropriate constitutive equation relating the viscous stress to
the rate of strain is known. For the special case of an incompressible Newtonian fluid
with constant thermal conductivity for which the components of the viscous stress
tensor are given by equations (D.1-4) through (D.1-9), equation (F.1-1) simplifies to
ρC
p
∂T
∂t
+ ρC
p
u
x
∂T
∂x
+ ρC
p
u
y
∂T
∂y
+ ρC
p
u
z
∂T
∂z
= k
∂
2
T
∂x
2
+ k
∂
2
T
∂y
2
+ k
∂
2
T
∂z
2
+ 2μ
∂u
x
∂x
2
+ 2μ
∂u
y
∂y
2
+ 2μ
∂u
z
∂z
2
+ μ
∂u
x
∂y
+
∂u
y
∂x
2
+ μ
∂u
x
∂z
+
∂u
z
∂x
2
+ μ
∂u
y
∂z
+
∂u
z
∂y
2
+ G
e
(F.1-2)
where C
p
is the heat capacity at constant pressure.
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.
496
SPHERICAL COORDINATES 497
F.2 CYLINDRICAL COORDINATES
The following form of the thermal energy equation in cylindrical coordinates allows
for nonconstant physical properties, energy generation, and conversion of mechan-
ical to internal energy using viscous dissipation, which is expressed in terms of
unspecified viscous stress–tensor components τ
ij
:
ρC
v
∂T
∂t
+ ρC
v
u
r
∂T
∂r
+ ρC
v
u
θ
r
∂T
∂θ
+ ρC
v
u
z
∂T
∂z
=
1
r
∂
∂r
rk
∂T
∂r
+
1
r
2
∂
∂θ
k
∂T
∂θ
+
∂
∂z
k
∂T
∂z
− T
∂P
∂T
ρ
1
r
∂
∂r
(ru
r
) +
1
r
∂u
θ
∂θ
+
∂u
z
∂z
− τ
rr
∂u
r
∂r
− τ
θθ
1
r
∂u
θ
∂θ
+ u
r
− τ
zz
∂u
z
∂z
− τ
rθ
r
∂
∂r
u
θ
r
+
1
r
∂u
r
∂θ
− τ
rz
∂u
z
∂r
+
∂u
r
∂z
− τ
θz
1
r
∂u
z
∂θ
+
∂u
θ
∂z
+ G
e
(F.2-1)
where C
v
is the heat capacity at constant volume, k the thermal conductivity, and G
e
the energy generation rate per unit volume. Equation (F.2-1) can be applied to non-
Newtonian fluids if the appropriate constitutive equation relating the viscous stress
to the rate of strain is known. For the special case of an incompressible Newtonian
fluid with constant thermal conductivity for which the components of the vis-
cous stress tensor are given by equations (D.2-4) through (D.2-9), equation (F.2.1)
simplifies to
ρC
p
∂T
∂t
+ ρC
p
u
r
∂T
∂r
+ ρC
p
u
θ
r
∂T
∂θ
+ ρC
p
u
z
∂T
∂z
=
k
r
∂
∂r
r
∂T
∂r
+
k
r
2
∂
2
T
∂θ
2
+ k
∂
2
T
∂z
2
+ 2μ
∂u
r
∂r
2
+
2μ
r
2
∂u
θ
∂θ
+ u
r
2
+ 2μ
∂u
z
∂z
2
(F.2-2)
+ μ
∂u
θ
∂z
+
1
r
∂u
z
∂θ
2
+ μ
∂u
z
∂r
+
∂u
r
∂z
2
+ μ
1
r
∂u
r
∂θ
+ r
∂
∂r
u
θ
r
2
+ G
e
where C
p
is the heat capacity at constant pressure.
F.3 SPHERICAL COORDINATES
The following form of the thermal energy equation in spherical coordinates allows
for nonconstant physical properties, energy generation, and conversion of mechan-
ical to internal energy through viscous dissipation, which is expressed in terms of
unspecified viscous stress–tensor components τ
ij
:
498 APPENDIX F
ρC
v
∂T
∂t
+ ρC
v
u
r
∂T
∂r
+ ρC
v
u
θ
r
∂T
∂θ
+ ρC
v
u
φ
r sin θ
∂T
∂φ
=
1
r
2
∂
∂r
r
2
k
∂T
∂r
+
1
r
2
sin θ
∂
∂θ
k sin θ
∂T
∂θ
+
1
r
2
sin
2
θ
∂
∂φ
k
∂T
∂φ
− T
∂P
∂T
ρ
1
r
2
∂
∂r
(r
2
u
r
) +
1
r sin θ
∂
∂θ
(u
θ
sin θ) +
1
r sin θ
∂u
φ
∂φ
− τ
rr
∂u
r
∂r
− τ
θθ
1
r
∂u
θ
∂θ
+
u
r
r
− τ
φφ
1
r sin θ
∂u
φ
∂φ
+
u
r
r
+
u
θ
cot θ
r
− τ
rθ
∂u
θ
∂r
+
1
r
∂u
r
∂θ
−
u
θ
r
− τ
rφ
∂u
φ
∂r
+
1
r sin θ
∂u
r
∂φ
−
u
φ
r
− τ
θφ
1
r
∂u
φ
∂θ
+
1
r sin θ
∂u
θ
∂φ
−
cot θ
r
u
φ
+ G
e
(F.3-1)
where C
v
is the heat capacity at constant volume, k the thermal conductivity, and G
e
the energy generation rate per unit volume. Equation (F.3-1) can be applied to non-
Newtonian fluids if the appropriate constitutive equation relating the viscous stress
to the rate of strain is known. For the special case of an incompressible Newtonian
fluid with constant thermal conductivity for which the components of the viscous
stress tensor are given by equations (D.3-4) through (D.3-9), equation (F.3-1) sim-
plifies to
ρC
p
∂T
∂t
+ ρC
p
u
r
∂T
∂r
+ ρC
p
u
θ
r
∂T
∂θ
+ ρC
p
u
φ
r sin θ
∂T
∂φ
=
k
r
2
∂
∂r
r
2
∂T
∂r
+
k
r
2
sin θ
∂
∂θ
sin θ
∂T
∂θ
+
k
r
2
sin
2
θ
∂
2
T
∂φ
2
+ 2μ
∂u
r
∂r
2
+ 2μ
1
r
∂u
θ
∂θ
+
u
r
r
2
+ 2μ
1
r sin θ
∂u
φ
∂φ
+
u
r
r
+
u
θ
cot θ
r
2
+ μ
r
∂
∂r
u
θ
r
+
1
r
∂u
r
∂θ
2
+ μ
1
r sin θ
∂u
r
∂φ
+ r
∂
∂r
u
φ
r
2
+ μ
sin θ
r
∂
∂θ
u
φ
sin θ
+
1
r sin θ
∂u
θ
∂φ
2
+ G
e
(F.3-2)
where C
p
is the heat capacity at constant pressure.
APPENDIX G
Equation of Continuity
for a Binary Mixture
G.1 RECTANGULAR COORDINATES
The following form of the equation of continuity or species balance in rectangular
coordinates for component A in a binary system allows for nonconstant physical
properties and is expressed in terms of the mass concentration ρ
A
and the mass
flux components n
Ai
:
∂ρ
A
∂t
+
∂n
Ax
∂x
+
∂n
Ay
∂y
+
∂n
Az
∂z
= G
A
(G.1-1)
where G
A
is the mass generation rate of component A per unit volume. The com-
ponents of the mass flux are given for a binary system by Fick’s law of diffusion
in the form
n
Ax
= ω
A
(n
Ax
+ n
Bx
) − ρD
AB
∂ω
A
∂x
= ρ
A
u
x
− ρD
AB
∂ω
A
∂x
(G.1-2)
n
Ay
= ω
A
(n
Ay
+ n
By
) − ρD
AB
∂ω
A
∂y
= ρ
A
u
y
− ρD
AB
∂ω
A
∂y
(G.1-3)
n
Az
= ω
A
(n
Az
+ n
Bz
) − ρD
AB
∂ω
A
∂z
= ρ
A
u
z
− ρD
AB
∂ω
A
∂z
(G.1-4)
in which ω
A
is the mass fraction of component A, D
AB
the binary diffusion coef-
ficient, and u
i
the mass-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.1-1) is particularly useful for describing mass transfer in incompress-
ible liquid and solid systems for which the mass density ρ is constant. For the
special case of an incompressible fluid or fluid having a constant mass density
and a constant binary diffusion coefficient, equation (G.1-1) when combined with
equations (G.1-2) through (G.1-4) simplifies to
∂ρ
A
∂t
+ u
x
∂ρ
A
∂x
+ u
y
∂ρ
A
∂y
+ u
z
∂ρ
A
∂z
= D
AB
∂
2
ρ
A
∂x
2
+ D
AB
∂
2
ρ
A
∂y
2
+ D
AB
∂
2
ρ
A
∂z
2
+ G
A
(G.1-5)
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.
499
500 APPENDIX G
The following form of the equation of continuity or species balance in rect-
angular 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
+
∂N
Ax
∂x
+
∂N
Ay
∂y
+
∂N
Az
∂z
=
ˆ
G
A
(G.1-6)
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
Ax
= x
A
(N
Ax
+ N
Bx
) − cD
AB
∂x
A
∂x
= c
A
ˆu
x
− cD
AB
∂x
A
∂x
(G.1-7)
N
Ay
= x
A
(N
Ay
+ N
By
) − cD
AB
∂x
A
∂y
= c
A
ˆu
y
− cD
AB
∂x
A
∂y
(G.1-8)
N
Az
= x
A
(N
Az
+ N
Bz
) − cD
AB
∂x
A
∂z
= c
A
ˆu − cD
AB
∂x
A
∂z
(G.1-9)
in which x
A
is the mole fraction of component A. The form of the equation of
continuity for a binary mixture or species balance given by equation (G.1-6) is
particularly useful 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 diffusion coefficient,
equation (G.1-6) combined with equations (G.1-7) through (G.1-9) simplifies to
∂c
A
∂t
+ˆu
x
∂c
A
∂x
+ˆu
y
∂c
A
∂y
+ˆu
z
∂c
A
∂z
= D
AB
∂
2
c
A
∂x
2
+ D
AB
∂
2
c
A
∂y
2
+ D
AB
∂
2
c
A
∂z
2
+
ˆ
G
A
(G.1-10)
G.2 CYLINDRICAL COORDINATES
The following form of the equation of continuity or species balance in cylindrical
coordinates for component A in a binary system allows for nonconstant physical
properties and is expressed in terms of the mass concentration ρ
A
and the mass
flux components n
Ai
:
∂ρ
A
∂t
+
1
r
∂
∂r
(rn
Ar
) +
1
r
∂n
Aθ
∂θ
+
∂n
Az
∂z
= G
A
(G.2-1)
where G
A
is the mass generation rate of component A per unit volume. The com-
ponents of the mass flux are given for a binary system by Fick’s law of diffusion
in the form
CYLINDRICAL COORDINATES 501
n
Ar
= ω
A
(n
Ar
+ n
Br
) − ρD
AB
∂ω
A
∂r
= ρ
A
u
r
− ρD
AB
∂ω
A
∂r
(G.2-2)
n
Aθ
= ω
A
(n
Aθ
+ n
Bθ
) − ρD
AB
1
r
∂ω
A
∂θ
= ρ
A
u
θ
− ρD
AB
1
r
∂ω
A
∂θ
(G.2-3)
n
Az
= ω
A
(n
Az
+ n
Bz
) − ρD
AB
∂x
A
∂z
= ρ
A
u
z
− ρD
AB
∂x
A
∂z
(G.2-4)
in which ω
A
is the mass fraction of component A, D
AB
the binary diffusion
coefficient, and u
i
the mass-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.2-1) is particularly useful for describing mass transfer in incom-
pressible liquid and solid systems for which the mass density ρ is constant. For
the special case of an incompressible fluid or fluid having a constant mass den-
sity and a constant binary diffusion coefficient, equation (G.2-1) combined with
equations (G.2-3) through (G.2-4) simplifies to
∂ρ
A
∂t
+ u
r
∂ρ
A
∂r
+ u
θ
1
r
∂ρ
A
∂θ
+ u
z
∂ρ
A
∂z
= D
B
1
r
∂
∂r
r
∂ρ
A
∂r
+ D
B
1
r
2
∂
2
ρ
∂θ
2
+ D
AB
∂
2
ρ
A
∂z
2
+ G
A
(G.2-5)
The following form of the equation of continuity or species balance in cylindrical
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
∂
∂r
(rN
Ar
) +
1
r
∂N
Aθ
∂θ
+
∂N
Az
∂z
=
ˆ
G
A
(G.2-6)
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.2-7)
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.2-8)
N
Az
= x
A
(N
Az
+ N
Bz
) − cD
AB
∂x
A
∂z
= c
A
ˆu
z
− cD
AB
∂x
A
∂z
(G.2-9)
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.2-6) is particularly use-
ful for describing mass transfer in gas systems for which the molar density c
502 APPENDIX G
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.2-6) combined with equations (G.2-7) through (G.2-9)
simplifies to
∂c
A
∂t
+ˆu
r
∂c
A
∂r
+ˆu
θ
1
r
∂c
A
∂θ
+ˆu
z
∂c
A
∂z
= D
AB
1
r
∂
∂r
r
∂c
A
∂r
+ D
AB
1
r
2
∂
2
c
A
∂θ
2
+ D
AB
∂
2
c
A
∂z
2
+
ˆ
G
A
(G.2-10)
G.3 SPHERICAL COORDINATES
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 mass concentration ρ
A
and the mass
flux components n
Ai
:
∂ρ
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
= G
A
(G.3-1)
where G
A
is the mass generation rate of component A per unit volume. The com-
ponents of the mass flux are given for a binary system by Fick’s law of diffusion
in the form
n
Ar
= ω
A
(n
Ar
+ n
Br
) − ρD
AB
∂ω
A
∂r
= ρ
A
u
r
− ρD
AB
∂ω
A
∂r
(G.3-2)
n
Aθ
= ω
A
(n
Aθ
+ n
Bθ
) − ρD
AB
1
r
∂ω
A
∂θ
= ρ
A
u
θ
− ρD
AB
1
r
∂ω
A
∂θ
(G.3-3)
n
Aφ
= ω
A
(n
Aφ
+ n
Bφ
) − ρD
AB
1
r sin θ
∂ω
A
∂φ
= ρ
A
u
φ
− ρD
AB
1
r sin θ
∂ω
A
∂φ
(G.3-4)
in which ω
A
is the mass fraction of component A, D
AB
the binary diffusion
coefficient, and u
i
the mass-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.3-1) is particularly useful for describing mass transfer in incom-
pressible liquid and solid systems for which the mass density ρ is constant. For
the special case of an incompressible fluid or fluid having a constant mass den-
sity and a constant binary diffusion coefficient, equation (G.3-1) combined with
equations (G.3-2) through (G.3-4) simplifies to