Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
362 value of the forced convection 4.364. This increase is
all the more important when the Grashof number is raised, Nguyen and Galanis [3], Boufendi and Afrid [4].
When the wall is of finite thickness, thermal conductivity is not remainder without influence. Its importance in the
azimuth and axial directions justified by important variations in gradients of the temperature was well
highlighted in work of Newell and Bergles [5], Baughn [6], Ouzzane and Galanis [7],[8]. The influence of the
variation of the physical properties with the temperature was introduced into the numerical work of Shome and
Jensen [9] and Shome [10]. Recently, Boufendi and Afrid [11] take account of the whole of these effects, the
thermal dependent properties and the conduction in the solid, and solve numerically the problem in 3D, for the
case of a uniform volumetric heating within the solid thickness, a incompressible laminar fluid flow with a
parabolic profile and a
constant temperature at the entry of the pipe. Their results, which are in perfect agreement
with those of an experimental study of Abid and al. [12], show clearly the non-uniformity of the heat flux at the
interface fluid-wall, the reduction of about 70 of the viscosity of the fluid at fluid at. the exit and a Nusselt
number who reaches at the exit of the pipe a value from 8 to 10 times higher than the asymptotic forced
convection value, 4.364.
In accordance with these developments and in particular with those used the method of the volumetric
heating, this investigation examines the influence of the parietal conduction on the dynamic and thermal fields of
the system. Four materials of different thermal conductivities were selected: Inconel, tantalum, steel and
aluminum. In this model the variation of the conductivity of the solid involves a change of the characteristic
difference temperature which has a direct influence on the number of Grashof based on this difference. This last
will thus have a value appropriate to each studied case while the other parameters of control, geometrical and
dynamic, as well as the generation of volumetric heat, are maintained fixed.
II. The Problem Formulation
Fig. 1 illustrates the geometry of the problem. size. It is a horizontal duct of length L=1m, inner and outer
diameters, respectively D
i
=0.96cm and D
o
=1cm of which the thickness is the seat of an internal generation of
uniform heat produced by Joule effect, with a rate of 4 10
7
Wm
3
. At the entry of the pipe, arise a distilled water laminar flow of parabolic profile with a mean velocity
equal to1.710
-2
ms and a constant temperature of 15°C. The external surface of the tube, of emissivity
9 .
= ε
, is subjected to convective and radiative losses with the
ambient air. The viscosity and the thermal conductivity of the fluid are functions of the temperature while the
thermal depending of the density is expressed according to the Boussinesq approximation.
Fig. 1. Geometry and dimensions: 1
i
D
∗
= , 1 04
o
D .
∗
= and
104 17 L
.
∗
=
We are thus in the presence of a conjugate heat transfer problem modeled by the dimensionless
conservation equations and the initial and boundary conditions:
At
t
∗
=
:
r z
V V
V T
θ
∗ ∗
∗ ∗
= =
= =
1 At
t
∗
: Mass conservation equation:
1 1
z r
V V
r V r
r r
r r
θ
∗ ∗
∗ ∗ ∗
∗ ∗
∗ ∗
∂ ∂
∂ +
+ =
∂ ∂
∂
2 Radial momentum conservation equation:
2 2
1 1
1 1
1
r r
r r
z r
rr r
rz
V r V V
V V r
r r
r V
Gr P
V V cos T
z r
r Re
r r
r r
Re r
z
θ θ
θ θθ
θ θ
τ τ
θ τ
τ
∗ ∗ ∗ ∗
∗ ∗ ∗
∗ ∗
∗ ∗
∗ ∗
∗ ∗ ∗
∗ ∗
∗ ∗ ∗
∗ ∗
∗ ∗
∗ ∗
∗ ∗
∂ ∂
∂ +
+ +
∂ ∂
∂ ∂
∂ +
− = −
+ +
∂ ∂
∂ ∂
⎡ ⎤
+ +
⎢ ⎥
∂ ∂
⎢ ⎥
+ ⎢
⎥ ∂
− +
⎢ ⎥
∂ ⎣
⎦
3
Angular momentum conservation equation:
2 2
2
1 1
1 1
1 1
r r
z r
z
V r V V
V V t
r r
r V V
Gr P
V V sin
T z
r r
Re r
r r
r Re
z
θ θ
θ θ θ
θ θ
θθ θ
θ θ
θ τ
τ θ
τ
∗ ∗ ∗ ∗
∗ ∗ ∗
∗ ∗
∗ ∗ ∗
∗ ∗
∗ ∗ ∗
∗ ∗
∗ ∗ ∗
∗ ∗
∗ ∗
∗ ∗
∂ ∂
∂ +
+ +
∂ ∂
∂ ∂
∂ +
+ = −
− +
∂ ∂
∂ ∂
⎡ ⎤
+ +
⎢ ⎥
∂ ∂
⎢ ⎥
+ ∂
⎢ ⎥
+ ⎢
⎥ ∂
⎣ ⎦
4
∗ o
D
∗
r
∗
z
∗ i
D
∗
L
θ
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
363 Axial momentum conservation equation:
1 1
1 1
1
z r
z z
z z
rz z
zz
V r V V
V V r
r r
r P
V V z
z r
Re r
r r
z
θ
θ
θ
τ τ
τ θ
∗ ∗ ∗ ∗
∗ ∗ ∗
∗ ∗
∗ ∗
∗ ∗ ∗
∗ ∗ ∗
∗ ∗
∗ ∗
∗ ∗
∂ ∂
∂ +
+ +
∂ ∂
∂ ∂
∂ +
= − +
∂ ∂
∂ ∂
∂ ⎡
⎤ +
+ +
⎢ ⎥
∂ ∂
∂ ⎣
⎦ 5
Energy conservation equation:
1 1
1 1
1
r z
r z
T r V T
r r
r V T
V T r
z r q
r r
G Re Pr
q q
r z
θ
θ
θ
θ
∗ ∗ ∗ ∗
∗ ∗
∗ ∗ ∗
∗ ∗ ∗
∗ ∗ ∗
∗ ∗
∗ ∗
∗ ∗
∗
∂ ∂
+ +
∂ ∂
∂ ∂
+ +
= ∂
∂ ∂
⎡ ⎤
+ ⎢
⎥ ∂
⎢ ⎥
= −
∂ ∂
⎢ ⎥
+ +
⎢ ⎥
∂ ∂
⎣ ⎦
6
The viscous stress components are:
2
r rr
V r
τ µ
∗ ∗
∗ ∗
∂ =
∂
, 1
2
r
V V
r r
θ θθ
τ µ
θ
∗ ∗
∗ ∗
∗ ∗
⎡ ⎤
⎛ ⎞
∂ =
+ ⎢
⎥ ⎜
⎟ ⎜
⎟ ∂
⎢ ⎥
⎝ ⎠
⎣ ⎦
2
z zz
V z
τ µ
∗ ∗
∗ ∗
∂ =
∂ ,
z r
zr rz
V V
r z
τ τ
µ
∗ ∗
∗ ∗
∗ ∗
∗
⎡ ⎤
∂ ∂
= =
+ ⎢
⎥ ∂
∂ ⎢
⎥ ⎣
⎦ 1
z z
z
V V
z r
θ θ
θ
τ τ
µ θ
∗ ∗
∗ ∗
∗ ∗
∗
⎡ ⎤
∂ ∂
= =
+ ⎢
⎥ ∂
∂ ⎢
⎥ ⎣
⎦
7
1
r r
r
V V
r r
r r
θ θ
θ
τ τ
µ θ
∗ ∗
∗ ∗
∗ ∗
∗ ∗
∗
⎡ ⎤
⎛ ⎞
∂ ∂
= =
+ ⎢
⎥ ⎜
⎟ ⎜
⎟ ∂
∂ ⎢
⎥ ⎝
⎠ ⎣
⎦
and the heat fluxes are:
r
T q
K r
∗ ∗
∗ ∗
∂ = −
∂
,
K T
q r
θ
θ
∗ ∗
∗ ∗
∂ = −
∂
,
z
T q
K z
∗ ∗
∗ ∗
∂ = −
∂
8 The boundary conditions are:
- At the inlet of the duct
z
∗
=
: In the fluid domain:
r
V V
T
θ
∗ ∗
∗
= =
=
,
2
2 1 4
z
V r
∗ ∗
= −
9 In the solid domain:
r z
V V
V T
θ
∗ ∗
∗ ∗
= =
= =
10 -
At the outlet of the duct
104 17 z
.
∗
=
: Fluid domain:
r z
V V
V T
K z
z z
z z
θ
∗ ∗
∗ ∗
∗ ∗
∗ ∗
∗ ∗
⎛ ⎞
∂ ∂
∂ ∂
∂ =
= =
= ⎜
⎟ ⎜
⎟ ∂
∂ ∂
∂ ∂
⎝ ⎠
11 Solid domain:
r z
T V
V V
K z
z
θ
∗ ∗
∗ ∗
∗ ∗
∗
⎛ ⎞
∂ ∂
= =
= =
⎜ ⎟
⎜ ⎟
∂ ∂
⎝ ⎠
12 -
At the duct axis, the dynamical condition is applied. -
At the duct outer wall
0 5208 r
.
∗
=
:
r z
r c
i
V V
V h
h D
T K
T K
r
θ
∗ ∗
∗ ∗
∗ ∗
∗
⎧ = =
= ⎪
+ ⎨
∂ −
= ⎪
∂ ⎩
13
with
2 2
r
h T
T T
T
ε σ
∞ ∞
= +
+
and
c
h
for air is locally approximated from the correlation of Churchill
and Chu [13] valid for all Pr and Ra numbers in the range
6 9
10 10
Ra
−
≤ ≤
:
[ ]
2 8 29
1 6 9 16
0 6 0 387
1 0 559
c i
air air
air
h D K .
. Ra
. Pr
= ⎡
⎤ ⎛
⎞ =
+ +
⎜ ⎟
⎢ ⎥
⎝ ⎠
⎣ ⎦
with
3 o
o air
air
Ra g
T R , , z T
D β
θ ν α
∞
⎡ ⎤
= −
⎣ ⎦
. The air thermophysical properties are evaluated at the
local film temperature: 2
film o
T T R , , z
T θ
∞
⎡ ⎤
= +
⎣ ⎦
. In this study the aspect ratio LD
i
is equal to 104.17, and the Re, Pr and Gr
numbers are calculated with the physical properties of the water evaluated at the entrance
reference temperature
288 T
K =
: 143 2836
Re .
= ,
8 082 Pr
. =
. The functions
T µ
∗ ∗
and K
T ,
∗ ∗
clarified in [11], are obtained by fittings of the tabulated values cited in
[14] and the solid dimensionless viscosity
s
µ
∗
is infinite, valued to
30
10 . The local Nusselt number at the interface
solid-fluid is:
0 5
0 5
i r
. b
h , z D
Nu , z
K K
T r
T . ,
, z T
z
θ θ
θ
∗
∗ ∗
∗ ∗
= ∗
∗ ∗
∗
= =
⎡ ⎤
∂ ∂
⎢ ⎥
= ⎢ ⎥
− ⎢
⎥ ⎣
⎦
14
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
364 where the dimensionless bulk fluid temperature is:
1 2 2 0 0
1 2 2 0 0
b
V r , , z
T r , , z
r dr d T
z V r , , z
r dr d
π π
θ θ
θ θ
θ
∗ ∗
∗ ∗
∗ ∗
∗ ∗ ∗
∗ ∗
∗ ∗ ∗
=
∫ ∫ ∫ ∫
15
The local axial mean peripheral Nu number is:
2
1 2
Nu z Nu
, z d
π
θ θ
π
∗ ∗
=
∫
16 and the average Nu number for the whole interface is:
104 17
1 104 17
.
Nu Nu z
dz .
∗ ∗
=
∫
17
III. The Numerical Resolution
