Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 90 (2014) 497 – 503
10th International Conference on Mechanical Engineering, ICME 2013
Effects of temperature dependent thermal conductivity on natural convection flow along a vertical wavy cone with heat flux Sharaban Thohuraa, Azad Rahmanb, Md. Mamun Mollac, M. M. A. Sarkerd,* a Department of Mathematics, Jagannath University Dhaka-1100, Bangladesh School of Engineering & Technology, Central Queensland University, Australia c Department of Electrical Engineering & Computer Science, North South University, Dhaka, Bangladesh d Department of Mathematics, Bangladesh University of Engineering & Technology, Dhaka, Bangladesh b
Abstract A steady natural convection along a vertical cone with uniform surface heat flux for temperature dependent thermal conductivity k (T) has been investigated numerically. Using the appropriate variables the basic equations are transformed to non-dimensional boundary layer equations and then solved employing marching order implicit finite difference method. In this paper, attention is mainly focused to the evolution of the surface shear stress in terms of local skin-friction and the rate of heat transfer in terms of local Nusselt number, velocity and temperature profiles, velocity vector field, streamlines and isotherms for a wide range of values of thermal conductivity variation parameter. The present numerical results are compared with the available published results which show a good agreement indeed. © 2014 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2014The TheAuthors. Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of the Department of Mechanical Engineering, Bangladesh University of Selection and peer-review under responsibility Engineering and Technology (BUET). of the Department of Mechanical Engineering, Bangladesh University of Engineering and Technology (BUET)
Keywords: Natural convection; vertical wavy cone; temperature dependent thermal conductivity; heat flux; finite difference method.
1. Introduction The problem of laminar natural convection boundary layer flow and heat transfer over a vertical cone obtains considerable attention in various branches of science and engineering. If the surface is wavy, the flow is disturbed
*Corresponding author. Tel.: 880-2-9665650/Ext-7948 (Off), 7114 (Res) E-mail address:
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1877-7058 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of the Department of Mechanical Engineering, Bangladesh University of Engineering and Technology (BUET) doi:10.1016/j.proeng.2014.11.763
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by the surface and this alters the rate of heat transfer. Among the few papers to date which study the effects of such non-uniformities on the vertical convective boundary layer flow of a Newtonian fluid we mention those of Yao [1], Moulic and Yao [2, 3]. Hossain et al. [4, 5, and 6] have studied the problem of natural convection flow along a vertical wavy cone and wavy surface with uniform surface temperature in presence of temperature dependent viscosity and thermal conductivity. Yao [7] has studied natural convection along a vertical complex wavy surface. Molla et al. [8] have studied natural convection flow along a vertical complex wavy surface with uniform heat flux where thermal conductivity of the fluid is constant. Nomenclature a Gr
Amplitude wavelength ratio Grashof number Unit vector normal to the surface Temperature in the boundary layer Temperature of the ambient fluid Temperature at the surface
nˆ
T Tf Tw
E γ
U φ
\
Greek Symbols Volumetric coefficient of thermal expansion thermal conductivity variation parameter Density of the fluid The half angle of the cone Stream function
Assuming the viscosity and thermal conductivity of the fluid to be proportional to linear function of temperature, two semi-empirical formulae were proposed by Charraudeau [9]. Following him, Hossain et al. [10, 11] investigated the mixed convection along a vertical flat plate, the natural convection past a truncated cone and for a wedge flow for the fluid having temperature dependent viscosity and thermal conductivity. Very few of the aforementioned authors have studied natural convection flow for a surface which exhibits the uniform surface heat flux. In the present study, the natural convection boundary layer flow along a vertical wavy cone with uniform heat flux and temperature dependent thermal conductivity has been considered. Here the thermal conductivity of the fluid is taken to be a linear function of the temperature which is appropriate for the small Prandtl number or gaseous fluid. The current problem is solved numerically by using marching order implicit finite difference method. Solutions are obtained for the fluid having Prandtl number Pr = 0.7 (air) with the different values of thermal conductivity variation parameter. 2. Formulation of the problem The boundary layer analysis outlined below allows the shape of the wavy surface Vˆ xˆ to be arbitrary, but our detailed numerical work will assume that the surface exhibits sinusoidal deformations. Thus the wavy surface of the cone is described by V ( xˆ )
yˆ w
aˆ sin S xˆ L
(1)
The physical model of the problem and the two-dimensional coordinate system are shown in Figure 1,
where rˆ xˆ is the local radius of the flat surface of the cone which is defined by xˆ sin M
rˆ
(2)
Under the Boussinesq approximation, we consider the flow to be governed by the following equations: w ( rˆ uˆ ) wxˆ
w ( rˆ vˆ ) wyˆ
0
(3)
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uˆ
wuˆ wxˆ
wuˆ
vˆ
1 wpˆ
wyˆ
U wxˆ
P
U
2 uˆ gE T Tf cos M
(4)
aˆ
qw
g
rˆ( xˆ )
yˆ L
xˆ
M
Tf
Vˆ ( xˆ )
ˆw y 0
Fig. 1: Physical model and the coordinate system
uˆ
uˆ
wvˆ wxˆ
vˆ
wT
vˆ
wxˆ
wvˆ
wyˆ
1 wpˆ
U wyˆ
wT
1
wyˆ
U Cp
P U
2 vˆ gE T Tf sin M
(5) (6)
.( k T)
where k is the thermal conductivity of the fluid which is defined as a linear function of the temperature.
>
kf 1 H T Tf
k
@
(7)
where k∞ is the thermal conductivity of ambient fluid outside the boundary layer and ε is a constant. The boundary condition for the present problem is uˆ
0, vˆ
kf ( nˆ . T )
0, q w
at yˆ
yˆ w
V ( xˆ ) and uˆ
0, T
Tf
as yˆ o f
(8)
To convert the problem in a non-dimensional system we have used the following non-dimensional variable: xˆ
x
Gr1/ 5 , r
,y
yˆ V xˆ
rˆ
L
L
L
u
UL P
Gr
2 / 5
UL
uˆ , v
P
Gr
1/ 5
,a
aˆ
, V ( x)
L
vˆ V xuˆ , T
V ( xˆ ) L
,V x
T Tf
qw L
kf
dVˆ
dV
dxˆ
dx
15 Gr , Gr
,p
2 L
UQf
2 Gr
gE q w cos M 4 2 L k fQ f
4 5
pˆ ,
(9a) (9b)
On introducing the above dimensionless dependent and independent variables into the equations (3)-(6) the following dimensionless form of the governing equations are obtained at leading order in the Grashof number, Gr: w(r u ) wx
w rv wy
0
(10)
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Sharaban Thohura et al. / Procedia Engineering 90 (2014) 497 – 503
wu
u
wx
§ ©
v
V x¨u wT
u
wx
wu wy
wu
v
wx
v
wu ·
¸ V xxu wy ¹
wT
1
wy
Pr
PC p
where Pr
2 2 w u 1 / 5 wp V x Gr 1 Vx 2 T wx wy wy
wp
kf
1 Vx
2 2 w u 1 5 wp Gr Vx 1 Vx 2 T tan M wy wy
2
2
(11)
2
1 JT
and J
H
w T2 wy
qw L
Gr
1 Pr
J 1 Vx
2
§¨
· ¸ w y © ¹ wT
(12)
2
(13)
1 5
(14)
kf
Here J is a parameter which controls the values of H and hence the temperature dependent thermal conductivity k as it is defined by equations (7) and (14).Equation (11) represents that the pressure gradient along the x direction is 1 5).
In the present problem this pressure gradient is zero because, no externally induced free stream exists. The elimination of wp/wy from equations (11) and (12) leads to O (Gr
u
wu wx
v
wu
1 Vx
wy
2 1 V x tan M 2 w u V xV xx 2 T 2 u 2 2 wy 1 Vx 1 Vx
(15)
The corresponding boundary conditions for the present problem then turn into u
0, v
wT wy
0,
2 1Vx
1
at y
0
and u
T
0,
0
as y o f
(16)
3. Numerical Methods Investigating the present problem we have employed the marching order implicit finite difference method. Firstly we introduce the following transformations to reduce the governing equation to a convenient form:
1 5 `, R r , U X , Y u ^ 5 x 3 5 `, 15 15 V X , Y 5 x v, 4 X , Y T ^ 5 x ` X
x, Y
^
y
(17a)
5x
(17b)
Introducing the transformations given in equation (17) into the equations (10), (15) and (13) we have,
wU Y wU wV
8U 5 X
wX
wY
V xV xx
½
¯
1 Vx
¿
5 X U wU V YU wU ®3
5 X U
wX
wY
w4
w4
wX
V YU
wY
(18)
0
wY
U4
1 Pr
2 2 5 X ¾U
1 Vx
2
^
1 Vx
1 J 5X
15
2 1 V x tan M 2 w U 4 2 2 wY 1 Vx
`
2 1 w 4 15 2 J 5X 1 Vx 4 2 Pr wY
(19)
§ w4 · ¨ ¸ © wY ¹
2
(20)
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Sharaban Thohura et al. / Procedia Engineering 90 (2014) 497 – 503
The boundary conditions (16) now take the following form: U
0,
0,
V
w4 wY
1
2 1 Vx
at Y
0 and U
4
0,
0
as Y o f
(21)
Solutions of the system of partial differential equations given by (18)-(20) and subject to the boundary conditions (21) are obtained by using the marching order implicit finite difference method developed by L. S. Yao [1, 2, 12]. It is important to calculate the values of the average Nusselt number, Num from the following relation which is obtained by using the set of transformations according to Yao [2]:
Nu m 5 Gr
15
15 X 2 ³0 1 V x dX 15 2 15 X 4 .4 X , 0 dX 1 J 5X ³0 1 V x X X
(22)
Also the skin friction coefficient is defined as
Cf
x
Gr 1 5 ^2 5 X 2 5 `
2 wU º 1Vx » wY ¼ Y 0
(23)
In order to validate the present numerical results, the skin friction coefficient and the surface temperature have been compared with those of Lin [13] and Pullepu et.al.[14]. The comparative results were summarized on Table 1 which shows that the present results have excellent agreement when the effect of thermal conductivity variation parameter was passed over. Table 1: Comparison of Surface temperature and Surface shear stress Equivalent terms of Surface temperature Pr
Lin [13] results
209 1 5T 0
Pullepu et. al.
Equivalent terms of Surface shear stress Present results
Lin [13] results
[14] results T(0)
5 1 5 4(0)
209 2 5 f cc 0
Pullepu et. al. [14] results
WX
Present results
5 2 5 >wwUY @Y
0.72
1.7864
1.7796
1.78649
1.22396
1.2154
1.22391
1
1.6327
1.6263
1.63277
1.07966
1.0721
1.07962
2
1.3633
1.3578
1.36336
0.82926
0.8235
0.82921
100
0.5675
0.5604
0.56755
0.18403
0.1813
0.18397
0
4. Results and Discussion The numerical results are presented for the different values of thermal conductivity variation parameter J for a suitable fluid having Prandtl number Pr = 0.7 (air). To examine the effect of J we have considered that a = 0.3 and φ = 30o remain constant. Attention is focused on the thermal conductivity variation parameter J and its effect on the average Nusselt number Num(5/Gr)1/5, skin friction Cfx as well as velocity and temperature distribution.
Sharaban Thohura et al. / Procedia Engineering 90 (2014) 497 – 503
a
0.30
J J J J J
0.25 0.20
1
b
J J J J J
0 -1 -2
2.5
1.5
-3
0.10
0.5
-5
0.00 0
1
2
3
4
5
-6 0
6
y
1.0
-4
0.05
J J J J J
2.0
v
u
0.15
c
Temperature
502
1
2
3
4
y
5
0.0 0
6
1
2
3
y
4
5
Fig. 2: (a) Tangential velocity distribution and (b) Normal velocity distribution (c) Temperature distribution at X =1.0 for Pr =0.7
Figure 2 (a), (b) and (c) represent the non-dimensional tangential and normal velocity distribution and temperature distribution for different values of J respectively at a fixed point X 1.0 . As thermal conductivity is considered linearly proportional to the temperature, it affects the surface temperature (Fig. 3(a)) to increase significantly with increasing value of J . Fig. 3(b) and 3(c) illustrate the effect of J on the surface shear stress in terms of skin friction coefficient and on the average rate of heat transfer in terms of average Nusselt number respectively. Figures 4(a)-4(c) show the isotherm for a wavy cone, while the values of J are taken as 0.0, 0.5 and 1.0 respectively. The figures indicate that the increasing the value of J affect isotherm and leads to thicker thermal boundary layer
3.5 3.0
b
Skin friction coefficient
Surface Temperature
a
3.5
2.5
J J J J J
4.0
J J J J J
2
Average rate of heat transfer
4.5
3.0
c
2.5
1.5
2.5 2.0
1.0 0
2
4
x
6
8
10
4
x
2.0 1.5
1
1.0
0.5
1.5
J J J J J
0.5
0 0
2
4
6
x
8
0.0 0
10
2
6
8
10
Fig. 3: (a) Surface temperature distribution (b) skin friction coefficient (c) Average rate of heat transfer for Pr =0.7
5. Conclusion From the present investigation the following conclusions may be drawn. x The skin friction increases within the computational domain for increasing value of thermal conductivity variation parameter J . x The average rate of heat transfer increases significantly when the values of J increases x Tangential velocity increases considerably with the increasing value of thermal conductivity variation parameter. x As the thermal conductivity variation parameter increases, the rate of fluid flow and temperature distribution increase significantly in the boundary layer. x One important finding is that, the effect of J increases the temperature of the fluid causing the thermal boundary layer thickness to increase. 40
b
35 30
20
J
25
Y
20
15 10 5 0 0
c
35
40 35 30
J
25
Y
25
40
30
Y
a
15
10
10 5
5 2
4
X
6
8
10
0 0
J
20
15
2
4
X
6
8
10
00
2
4
X
6
8
10
Sharaban Thohura et al. / Procedia Engineering 90 (2014) 497 – 503 Fig. 4: Isotherm for different values of
J
for a wavy cone and Pr =0.7
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