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
The finite volume method conduct according to the power law numerical scheme, well described by Patankar
[15], is used to discretize the system of non-linear and strongly coupled partial derivative equations, Eqs. 1 -
6 with the viscosity and the conductivity correctly estimated at the volumes interfaces The sequential
solution of the systems of the linearized algebric equations is obtained with the SIMPLER algorithm [15]
and the iterative solution is achieved by the line by line sweeping using the Thomas algorithm in the radial and
axial directions and the tri diagonal cyclic algorithm in the angular direction. With the step time of 10
-3
, the time marching is continued until the reached convergence
confirmed by the satisfaction of the global mass and energy balances and the temporal invariance of the local
and average dependent variables. In order to ensure itself of the reliability of the developed code and the precision
of the results which it provides, this last was lengthily tested for various grids and perfectly validated by
comparison with numerical and experimental works published in the specialized bibliography. Moreover
fuller details on these tests and these validations are deferred in the references [4], [11]. This reveals that the
grid adopted in this study is 26x44x162 nodes according
to each direction respectively
r , , z
θ
∗ ∗
including 5 nodes in the radial direction representatives the small solid part.
Among the validations which were made, this, for example, in Fig. 2 show a good agreement between ours
results and the numerical results of Ouzzane and Galanis [7].
IV. Results and Discussion
The results were produced for four different materials whose parameters are as follows Table I.
Fig. 2. Comparison between our Nusselt number results and those obtained by Ouzzane and Galanis [7]
TABLE I
V
ALUES
O
F
D
IMENSIONLESS
C
ONDUCTIVITIES
O
F
U
SED
M
ATERIALS
A
ND
A
SSOCIATES
G
RASHOF
N
UMBERS
K =0.5893
Materials Dimensionless
conductivity Grashof number
Inconel 25.45 2.57
10
5
Stainless Steel 82.30
9.37 10
3
Tantalum 97.57 8.97
10
3
Aluminum 402.17 4.63
10
2
Their exploitation will relate to the influence of the thermal conductivity of the solid on the dynamic and
thermal fields and the effect on the evolution of the Nusselt number. Also the effect on the variation of
viscosity and conductivity of the fluid will be highlighted.
IV.1. The Secondary Flow
In the Figs. 3.1-3.3 we represent the evolutions of the secondary flow in the transverse plan in two judiciously
selected axial stations for the three materials: inconel, tantalum and steel. The selected axial stations chosen
represent respectively where the secondary flow is most intense like that at the exit of the pipe. In all cases the
symmetry compared to the vertical diameter is respected showing a secondary flow represented by two counter
rotating cells. Because of this symmetry the stream lines are traced only on one half of the cross-section
considered of the duct. Compared to tantalum and steel, these figures show that the secondary flow intensifies
more quickly for inconel. The fluid traverses a distance two to three times larger in the case of tantalum and steel
to reach a maximum intensification. In addition, in these stations the maxima reached are different for each case
with nets deviations of centers of vortices. For inconel the maximum angular component velocity is
0 12712 V
.
θ
∗
=
at
0 4375 r
.
∗
=
and
1 428 .
θ
=
whereas for tantalum it is worth
0 06884 V
.
θ
∗
=
at
0 4375 r
.
∗
=
and
1 428 .
θ
=
, and for steel
0 05789 V
.
θ
∗
=
at
0 4375 r
.
∗
=
and
1 57 .
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
365
a z=5.53025
b z=104.17 Figs. 3.1. a-b: Streamlines and isotacs: inconel case
a z=13.99
b z=104.17 Figs. 3.2. Streamlines and isotacs: tantalum case
At the exit
104 17 z
.
∗
=
these maximum of velocity are equal to:
0 09034 V
.
θ
∗
=
at
0 4625 r
.
∗
=
and
1 991 .
θ
=
for inconel, and
0 06344 V
.
θ
∗
=
at
0 4375 r
.
∗
=
and
4 569 .
θ
=
for tantalum while
for steel
0 04987 V
.
θ
∗
=
at
0 4375 r
.
∗
=
and
1 856 .
θ
=
. It should be specified that in the case of aluminum there is almost no secondary flow
from the entry to the exit. The initial Poiseuille profile is maintained to the exit without being disturbed by any
gradient of the velocity in the azimuth direction. The stream lines are circular with a circular with a maximum
velocity at the axis center of the duct.
a z=15.2996
b z=104.17 Figs. 3.3. Streamlines and isotacs: steel case
IV.2. The Temperature Field
To the same selected axial stations for the representation of the secondary flow, one reports in the
Figs. 4.1-4.3 the distributions of the temperatures in these stations. In all cases these figures show important
angular and axial variations in the temperature of the fluid and solid system through each section as well as a
monotonous increase from a station to another. The unstable stratification generated in the vicinity of the
wall and in bottom of the duct upwards involves a circulation of the heated fluid of bottom. In the top of the
duct it is rather the reverse which occurs thus creating a stable stratification of the temperatures. The maximum of
the temperatures will be always at the top of the duct while the minimum is thorough towards the lower part of
the pipe. For the case of tantalum the minimum
temperature will move from
0 2375 r
.
∗
=
to
0 3375 r
.
∗
=
and that for steel it will move from
0 2375 r
.
∗
=
to
0 3125 r
.
∗
=
while for inconel it will move from
0 2625 r
.
∗
=
to
0 3625 r
.
∗
=
.
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
366
.0 8
3 0.0083
0.0 16
6 .0
2 4
9 .0
2 4
9 .0
33 2
.0 3
3 2
0. 04
14 0.04
14 0.0497
0.0497 0.0
58 0.0580
0. 06
63 0.0
746 0.0828
a z=5.53025
.1 5
9 7
0.1714 0.18
31 .1
8 3
1 0.
19 48
0.2065 0.2183
0.230 0.241
7 0.2417
0.2534 0.2651
b z=104.17 Figs. 4.1. Isotherms field: Inconel case
.0 3
8 3
.0 7
4 5
0.1106 .1
1 6
.1 4
6 8
0.1468
0.1829 0.
21 91
0.2191 0.2553
0.25 53
0.2914 0.2
914 0.3276
0.3 63
7
a z=13.99
0.6 078
0.6516
0.695 5
.7 3
9 3
0.7393 .7
8 3
1 .7
8 3
1 0.8269
0. 82
69 0.87
07 .9
1 4
5 0.9
58 4
1.0022
b z=104.17 Figs. 4.2. Isotherms field: Ta ntalum case
0.0 34
1 .0
6 5
4 0.0967
.0 9
6 7
0. 12
79 0.1279
.1 5
9 2
.1 5
9 2
0.1905 0.1905
0.2 21
7
0.2 21
7 .2
5 3
0.28 43
0.3155
a z=15.2996
0.4721 0.5070
0.5418
0. 54
18 0.5767
0.5 76
7 0.6115
0.6464
.6 8
1 2
0.6812 0.
71 61
0. 75
09 0.7858
b z=104.17 Figs. 4.3. Isotherms field: Stainless steel case
As for the secondary flow, the case relating to aluminum was not represented: it was found that the
mixed convection is not creates within the fluid and that the distribution of the temperatures in any cross section
between the entry and the exit is radial from the surface of the pipe hottest towards the axis of the pipe
coldest. The contours lines are circular. It is thus a forced convection. In our interpretation one represents in
the Fig. 5 the axial evolutions of the temperatures at the top
θ
=
and the bottom θ π
=
of the duct for the studied materials. The monotonous increases in the
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
367 temperatures beyond some
z
∗
are consistent with the constant heating over the entire length of the tube.
The effect of the material conductivity is clearly distinguished since the levels of the temperatures reached
vary from a material to another as well as the gradient of the temperatures between the top and the bottom of the
duct for each studied case.
IV.3. The Nusselt Number
The convective heat transfer between the interface fluid-solid is quantified by the local axial and
circumferentially averaged Nusselt number, Nu, Eq. 17. This axial variation of the Nu for the studied
materials is represented on Fig. 6.
Fig. 5. Axial profiles of the temperatures at the top θ=0 and the
bottom θ=π of the outer wall duct
Fig. 6. Evolution and comparaison of the local axial and peripherally averaged Nusselt number for the studied materials
We have also added more in this figure the pure
forced convection case
Gr
∗
=
for inconel. Globally,
the qualitative profile is similar for the three materials.
On a short axial distance near the entrance of the tube, the Nu presents a rapid reduction to reach a minimum
and then, beyond this, the Nu starts an important increase up to the exit. In this zone the behavior of three materials
is different. For Inconel the Nu exhibit an monotonous increase up to value 39.69 whereas for steel and tantalum
the Nu one rather presents an asymptotic behavior to the exit of the duct whose values are respectively 12.07 and
11.12. While for aluminum the mean Nu is very weak by presenting a brutal reduction in a short zone very close to
the entry and an asymptotic behavior on almost the whole of the duct. Also, the Fig. 6 show that the Nusselt
numbers are very high comparatively to one obtained by forced convection.
V. Conclusion
