Geoscience Reference
In-Depth Information
The temperature profile in fully developed laminar pipe flow depends on the
temperature boundary conditions. We'll consider the analytically simple case where
the fluid and wall temperatures vary linearly with
x
but their difference, and the
wall heat flux, are independent of
x
. Its temperature profile is
(Problem 1.2)
r
2
2
R
2
1
4
R
2
,
R
2
u
ave
α
r
2
∂T
∂x
T(
0
,x)
−
T(r,x)
=−
−
(1.11)
with
α
k/(ρc
p
)
the thermal diffusivity of the fluid. The relation between the wall
heat flux and
∂T/∂x
is
(Problem 1.2)
=
Du
ave
ρc
p
4
∂T
∂x
,
H
wall
=−
(1.12)
so the temperature profile
(1.11)
can be rewritten as
r
2
2
R
2
1
4
R
2
.
r
2
H
wall
D
k
T(
0
,x)
−
T(r,x)
=
−
(1.13)
The wall heat flux made dimensionless with pipe diameter
D
, fluid thermal
conductivity
k
, and a temperature difference
T
is called a Nusselt number
Nu
:
H
wall
D
kT
Nu
=
.
(1.14)
T
is defined through the wall temperature
T
w
(x)
and the “bulk fluid temperature”
at that position,
T
b
(x)
:
0
u(r) T (r, x)
2
πr dr
πR
2
u
ave
H
wall
D
k(T
b
−
T
b
(x)
=
, Nu
=
T
w
)
.
(1.15)
Turns
(
2006
) shows that in the laminar case in this problem
Nu
4
.
4.
In the turbulent case the heat flux at a point on the wall, like the stress there,
fluctuates chaotically in time. The turbulent mixing makes the mean temperature
gradient relatively small over most of the cross section; it is large only near the wall,
as for velocity
(Figure 1.4)
.
The mean wall heat flux, the product of the fluid thermal
conductivity
k
and the mean temperature gradient at the wall, is much larger than
in the laminar case.
From
Eq. (1.14)
in this problem we can write the ratio of wall heat fluxes as, for
given values of
D, k,
and
T
,
H
wall
Nu
Nu(
laminar flow
)
=
Nu
4
.
4
.
H
wall
(
laminar flow
)
=
(1.16)
