Bird R.B., Stewart W.E., Lightfoot E.N. Transport Phenomena
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d2vz d2v, d2vZ
[V2v],
=
[V Vv],
=
-
+-
ax2
+
3
az2
dvx dvx dvx
[v
Vv],
=
vx
-
+
v
-
+
v,
--
dx
Y
dy dz
dvz dv, dvz
[v
'
Vv],
=
vx
-
+
v
-
+
v,
-
dx
Y
dy dz
~(v,v,)
a(vyvx) d(v,vX)
[V
vv],
=
-
+
------
+
-
dx dy dz
a(vXvy) a(vYvy)
~(v,v,)
[V
.
vv],
=
-
+-+-
dx dy dz
a(vXvz) d(vyvz)
~(v,v,)
[V
vv],
=
----
+-+-
dx
dy
dz
dvx dux dux
(T
:
Vv)
=
rxx
-
+
r
-
+
rxz
-
dx dy dz
Note:
the differential operations may
not
be simply generalized to curvilinear coordi-
nates; see Tables
A.7-2
and
A.7-3.
l
*MOLECULAR FLUX EXPRESSIONS (SEE APPENDIX B.l, B.2, AND B.3)
Momentum (p
=
constant, Newtonian fluid):
m
=
p6
-
p(Vv
+
(Vv)+)
or
T-
11
=
pa-
11
-
p
Heat (pure fluid only):
q
=
-kVT
or
q.
=
-kg
dxi
Mass (for a binary mixture
of
A
and
B):
d@A
j~
=
-P~AB~@A
Or
A
=
P~AB
dx,
l l
.CONVECTED FLUX EXPRESSIONS (SEE
§§I
.7,9.7,17.7)
Momentum:
PVV
or pvpj
Energy:
+
$v2)v or
+
$vZ)vi
Mass:
P@A~
or PUAVi
l
l
COMBINED FLUX EXPRESSIONS
Momentum:
Energy:
e
=
p(U
+
fv2)v
+
q
+
[n0v]
=
p(ii
+
f~)v
+
+
[vv]
Mass:
nA
=
pwAv
+
jA
Note:
The quantity
[.rr
v]
is the molecular work flux (see
g9.81,
and
n
=
pa
+
T
(see
Table 1.2-21).
All
fluxes obey the same sign convention: they are positive when the
entity being transported is moving from the negative side of a surface to the positive
side.
***EQUATIONS OF CHANGE IN TERMS OF THE COMBINED FLUXES
These equations are valid only for systems in which gravity is the only external
force. More information may
be
found in
519.2.
Momentum:
apv= -[V-+I +pg
dt
(Eq.
3.2-8)
Energy:
dA
-p(~+
fv2)
=
-(V.e)
+
p(v*g)
dt
(Eq. 11.1-6)
Mass:
d
-P@A
=
-(V
nA)
+
rA
dt
(Eq. 19.1-6)
l
l
*EQUATIONS OF CHANGE (SPECIAL FORMS)
-
-
Momentum (for Newtonian fluids with constant
p
and
p):
(53.6)
Energy (for Newtonian fluids with constant p and k):
(SB.9)
Mass (for binary mixtures of
A
and
B
with constant pgAB):
(SB.11)
l
l
DIMENSIONLESS GROUPS
(1,
and
vo
are
a
characteristic length and
a
characteristic velocity, respectively)
Re
=
&v,p/p
Pr
=
CPp/k
SC
=
p/p%B
Ra
=
GrPr
Gr
=
g@AT/v2 Gr,
=
gll:Aw,/
v2
Nu
=
hlo/k
P6
=
RePr
PkAB
=
ReSc
Sh
=
kcIo/9AB
j,
=
NU/R~P~'/~
j,
=
s~/R~sc'/~
