Applied Scientific Research 46: 335-345, 1989. © 1989 Kluwer Academic Publishers. Printed in the Netherlands.
335
N a t u r a l convection flow over a vertical frustum o f a cone for c o n s t a n t wall heat flux
P. S I N G H , 1 V. R A D H A K R I S H N A N 1 & K.A. N A R A Y A N 2'* tDepartment of Mathematics; 2Department of Chemical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India (*author for correspondence) Received 2 February 1989; accepted 1 April 1989 Abstract. Laminar natural convection flow and heat transfer over a vertical frustum of a cone has been studied. The governing boundary layer equations are solved using local non-similarity method for constant wall heat flux. The local similarity and the local non-similarity two and three-equation models are constructed and the resulting equations are solved numerically. Results obtained from two and three-equation models are in good agreement. The numerical values of the flow and temperature functions required to calculate the surface skin friction and heat transfer rate are reported for various values of Prandtl numbers.
1. Introduction
The problem o f laminar natural convection flow and heat transfer over a cone is o f p a r a m o u n t importance in various branches o f engineering. While the natural convection flow over a full cone has been studied extensively [1-4], flow and heat transfer over a vertical frustum o f a cone has not received m u c h attention. Free convection over a vertical cone can be dealt with similarity methods whereas that over a frustum o f a cone c a n n o t be solved by usual similarity methods and hence some alternate m e t h o d has to be used. The local similarity m e t h o d is one o f the available technique for treating non-similar problems. In this method, the terms containing the stream-wise directional derivatives are discarded. The transformed and simplified b o u n d a r y layer equations then resemble with those o f similarity equations. F o r prescribed values o f the stream-wise coordinate, these equations can be treated as ordinary differential equations and can be solved by well k n o w n techniques. However, the local similarity m e t h o d provides numerical results that are o f uncertain accuracy because there is not positive way to justify the omission o f certain terms in the m o m e n t u m and energy equations. T o correct the drawbacks o f the local similarity m e t h o d Sparrow, Quack and Boerner [5] developed a local non-similarity method. This m e t h o d has been applied successfully to m a n y fluid flow problems [6-8].
336
P. Singh et al.
In this article the natural convection flow and heat transfer over a frustum of a cone has been studied by using the local non-similarity method. Since the flow and heat transfer fields are coupled, the conservation equations were solved simultaneously. For the case of constant wall heat flux the local similarity and local non-similarity two and three equation models have been constructed and the resulting equations are solved by Runge-Kutta-Gill method.
2. Governing equations The physical model and the coordinate system are shown in Fig. 1. The boundary layer is assumed to develop at the leading edge of the frustum 2 = 20. Due to the difference in the temperature between the surface and the surrounding fluid, an upward flow is created as a result of buoyancy force. The flow is assumed to be steady and laminar and the fluid is
Fig. 1. Physical model and co-ordinate system.
Natural convection f l o w
337
incompressible. Except density variation in b o d y force term the fluid is assumed to have constant physical properties. The b o u n d a r y layer equations for the natural convection over the frustum of a cone are: c~ (~?) + oV
(~3~) =
~/~
(~/~
0,
(2. la)
(~2~
ft Ux + ~ ~--fi =
fi-~-o.~ + ~3 @
v ~fiy2 + gfi cos 7(T -- iP~),
-
~ @~,
(2.1b)
(2. l c)
where v, ~, g, fl are respectively the kinematic viscosity, thermal diffusivity, acceleration due to gravity and the coefficient of volume expansion, iPoo denotes the temperature of the fluid at infinity. In the above equations the b o u n d a r y layer has been assumed to be very thin relative to the local radius of the frustum and hence the local radius to a point in the b o u n d a r y layer has been replaced by the radius of the frustum ~. The pressure gradient across the boundary layer has been taken as negligible and hence the analysis is ~¢alid only when ? is small. The boundary conditions of the problem are
35
=
O:fi
=
0, z3
=
0,-k
~f
-
qo, (2.1d)
? -~ ~ : ~
--* O , f - - ~
3. Mathematical
f~.
analysis
Introducing the non-dimensional variables:
x -
L
'
Y =
L
'
fi U
m
V
--
Uc
r-z
Uc
'
r=
] P - OriP°~ '
(3.1)
P. Singh et al.
338 where
ucL ReL
=
--
,
V
= [gflcosT(~--~-)(vL)l/2]2/5
blc
and ~/oL Or
--
If ~0 is taken as the characteristic length L, then
[
gfl cos 7
=
ReL
2
= Gr2_/5 - - x0 "
V2
The governing equation and b o u n d a r y conditions reduce to
~---~(ur) + ~0 (vr) = O, ~U
~U
(3.2a)
~2U
u ~x ÷ v ~y
Oy2 + T,
OT OT U ~ x + v Oy
o- Oy2 ,
1 02
y =
O:u
=
O,v
y ~
oo:u ~
(3.2b)
T
=
(32c)
~T O,c~y
1, (3.2d)
0, T ~
0.
Here o-(= v/a) is the Prandtl number. N o w we define a stream function ~ such that
rH
~---
ay'
rv
=
c~x
(3.3a)
N a t u r a l convection f l o w
339
and introduce the transformations:
=
x,
q
-
Y xl/5
'
(3.3b) f ( ~ , q) =
T
0(~, q) -
rx4/5 ,
Xl/5 •
The governing equations (3.2) reduce to the form f"
+ ( R + ~ ) f f " -- -3# ( f , )2 + 0
1
-0"
+ ( R + 4)fO' -- } f ' O
=
=
~ ( f ' g' - f " g),
~ ( f ' cp -- O"g),
(3.4a)
(3.4b)
G
with the b o u n d a r y conditions f(~,0)
= f'(~,0
=
0,
=
-1,
(3.4c) f ' ( ¢ , oo) =
0(~, oo) =
0,
where R = ~/(~ + 1), g = Of/O~, ~o = ~0/0~ and prime (') indicates differentiation with respect to q(~/c?t/). At ~ = 0, equations (3.4) become f,,, + ~ f f , , _ ~ ( f , ) 2 + 0 = 1
-0" ff
=
+ ~ f O ' -- ½ f ' O
0,
(3.5a)
0.
(3.5b)
With the b o u n d a r y conditions: f(0)
= f'(0)
=
0,
0'(0)
=
-
1,
(3.5c) f'(oo)
=
g(oo) =
0.
340
P. Singh et al.
As ~ --* o% R ~ 1 another similarity solution exists and the equations (3.4a) and (3.4b) reduce to f,,, + ~ f f , , _ } ( f , ) 2 + 0 = 1 - O " + 9sfO" - ~ f ' O
=
0,
(3.6a)
0,
(3.6b)
along with the b o u n d a r y conditions f(0) f'(oo)
= f'(0)
=
0,
=
=
0.
g(oo)
0'(0)
=
-1, (3.6c)
The equations (3.6) are the similarity equations for the flow over a full cone. The solution of equations (3.4) are expected to change gradually from the solution of equations (3.5) to that of (3.6) as ~ increases from zero to infinity. Local similarity model To derive the governing equations for the local similarity model, we neglect terms containing the ~ derivatives in equations (3.4a) and (3.4b) i.e. the terms g, g', q~. Thus we get f"
+ (R + ~ ) f f " -- 3s(f')2 + 0 =
1 -0" ly
+ (R + 4) fO' - ~ f ' O
f(~,0)
= f'(~,0)
f ' ( ~ , oo) =
=
0,
0(4, oo) =
0.
=
0,
(3.7a)
0,
0'(~,0)
(3.7b)
=
-1, (3.7c)
The reduction of equations (3.4) to (3.7) is clearly justified for ~ values that are close to zero. On the otherhand, when ~ is not small, local similarity is based on the assumption that the derivatives with respect to ~ are very small. In solving the system of equations (3.7) ~ may be regarded as a parameter and hence they can be treated as ordinary differential equations and solved by well-established techniques. Moreover, for a given value of 4, the solution obtained will be independent of the solutions for any other value of 4.
Natural convection flow
341
Thus, by assigning successively different values to 4, the ~ dependence o f f and 0 can be determined. Local non-similarity models
To derive the governing equations for the local non-similarity models, we differentiate equations (3.4) with respect to ~ and obtain the first set of subsidiary equations and the corresponding boundary conditions. These are
g" + (R + ~)(g~f 4 ,, + + f,,g
_
1 + (1 + 4) 2i f , ,
gf , , )
¢(g,2 _ gg,,) + O =
~(f'h"
_ ½1f,g,
(3.8a)
-f"h),
1 1 -~r O" + (R + 4)(0' f + O'g) + (1 4- ~)~fO' -- ~ f ' o l g, O + gO' -- ~.(g'o -- g o ' )
g(~, O) = g'(~, O) =
=
O'(~, O) =
~(f'~b -
O'h),
(3.8b)
0,
(3.8c) g'(¢, oo) =
O(~, oo) =
O,
where
h -
8g 8~ -
82f 842 and
0
-
80 8~ -
820 8~2
To get the equations for the two equation model, we retain all the terms in equations (3.4a) and (3.4b). In the subsidiary equations (3.8a) and (3.8b) the terms ~h, ~h' and ~0 are neglected. Thus the governing equations for the two-equation model are: (i) equations (3.4a) and (3.4b); (ii) equations (3.8a) and (3.8b) with terms on their respective right hand sides deleted; (iii) boundary conditions (3.4c) and additional boundary conditions (3.8c). Since the set of equations derived above for the four functions f, g, 0 and O are coupled, these equations must be solved simulataneously. If ~ is regarded as a prescribable parameter, these equations may be treated as a system of ordinary differential equations.
342
P. Singh et al.
Next, the governing equations for the three-equation model are obtained as follows. First equations (3.8a) and (3.8b) are differentiated with respect to ~ and terms involving ~(Oh/~), ~(Oh'/O~) and ~(0~/0~) are then deleted from the resulting equations. However, the governing equations (3.4) and the subsidiary equations (3.8) are left intact. Thus the resulting equations for the three-equation model are: (i) equations (3.4); (ii) equations (3.8); (iii) the following truncated equations: 2 h" + (R + 4 ) ( h ' f + hf" + 2gg") + (1 + 4) 2 ( g ' f + g f ' )
2 (1 + 3) 3f f " _ 1~ (g,Z + f ' h ' ) + 2(gg" + f " h ) - ~(3g'h" - gh" - 2g"h) + ~
(3.9a)
= O,
1 1 -G ~" + (R + 4)(O,f + 2q)'g + O'h) + (1 + 4)2 (~of + O'g)
2 ~)3fO" -- J~ ffJf' -- &~ g' q) Jr 2(q~'g + hO') (1 + h'O - ~(2g'O - gO' + h'cp' - 2h~0') =
0,
(3.9b)
with boundary conditions 0)
=
0)
=
oo)
=
0)
=
0, (3.9c)
=
0.
In the above model, the equations (3.4), (3.8) and (3.9) are again coupled and they are to be solved for six unknown functions f, g, h, 0, q~ and 0. If is prescribed these equations can be treated as a system of coupled partial differential equations. The local non-similarity methods described above preserve the two most attractive features of the local similarity method. Firstly the system of equations can be treated as ordinary differential equations and the solution for any streamwise location can be obtained independently. Secondly in the
Natural convection flow
343
local non-similarity method, all nonsimilar terms in conservation equations are retained and the terms are deleted from the subsidiary equations only. This is an obvious improvement over the local similarity method where certain terms are deleted from conservation equation itself.
4. Numerical results and discussion
It is worthwhile to mention that although the solutions of the flow and thermal fields provide information about the functions f, g, 0 and q~ for the two-equation model and the functions f, g, h, 0, q~ and ~ for the three-equation model, it is only f and 0 and their r/ derivatives that are physically relevant. For example, the dimensionless velocity along the x direction is given by u = ~1/2f,(~, q) and the temperature distribution by 0(~, q). Since the heat flux at the surface is specified, the quantities of main physical interest are the surface temperature and the skin-friction variations with respect to ~. The wall shear stress is given by
-cw =
#. ~=0
and from this the dimensionless skin-friction can be obtained as
sx.
-
1 Q%u ,
=
2
G rx. -1/4 -fH(:: , . , 0),
where cos
-
:T
)x . 3
V2
is the Grashof number based on x* and x* = 2 - 20. The governing equations for the local similarity model and local nonsimilarity two and three-equation models are solved numerically using the fourth order Runge-Kutta-Gill method. In the numerical computations the effects of the step size Atl and tloo on the final results are examined in detail. The values off'(~, 0) and 0(4, 0) for a = 10.0 along with the finite difference solution [9] are shown in Table 1. As seen from Table 1, the results obtained from the three-equation model are in good agreement with those given in Ref. [9]. The conjecture that the three-equation model should provide results that are more accurate than the two-equation models was also verified from
344
P.
Singh et al.
Table 1. Values o f f " ( ¢ , 0) and 0(~, 0) for a = 10.0
f"(~, 0)
0* 0.1 0.25 0.75 1.75 3.75 7.75 15.75 21.75 63.75 127.75 oot
0(4, 0)
Local similarity model
Twoequation model
Threeequation model
Local similarity model
Twoequation model
Threeequation
Ref. [9]
0.5832 0.5648 0.5452 0.5108 0.4852 0.4690 0.4597 0.4547 0.4521 0.4508 0.4501 0.4494
0.5832 0.5739 0.5613 0.5293 0.5002 0.4789 0.4660 0.4576 0.4544 0.4517 0.4503 0.4494
0.5832 0.5752 0.5639 0.5341 0.5051 0.4809 0.4677 0.4592 0.4553 0.4521 0.4503 0.4494
0.0590 1.0425 1.0246 0.9922 0.9674 0.9512 0.9418 0.9367 0.9341 0.9327 0.9321 0.9314
0.0590 1.0476 1.0292 1.0091 0.9797 0.9597 0.9441 0.9378 0.9339 0.9322 0.9319 0.9314
0.0590 1.0501 1.0329 1.0121 0.9820 0.9621 0.9459 0.9387 0.9337 0.9320 0.9316 0.9314
1.0589 1.0507 1.0363 1.0142 0.9855 0.9642 0.9473 0.9393 0.9333 0.9317 0.9295 0.9336
* Solution of equations (3.5). t Solution of equations (3.6).
comparison of the numerical values between Of/O~ and OglO~, ctf'13~ and ~g'/O~ and ~30/0~ and 0~o/ct~. The magnitudes of the quantities ~g/O~, Og'/~ and ~ o / ~ are respectively much smaller than those of Of/O~, ~f'/?~ and ~30/~ and this clearly shows that the three-equation model solutions are more accurate than the two-equation model. Further, it was noticed that the values of Of/O~,~0/~ etc., decrease with ~ and hence the local non-similarity models can be expected to yield more accurate results for larger values of 4. As there is no significant difference between the two-equation and threeequation model solutions, we have obtained the results only from the two-equation model for o- = 0.72 and 0.01, which are shown in Table 2. It is observed that the solution of equations (3.4) starts from those of (3.5) and approaches to the similarity solution of (3.6) and the presence of the nonsimilar terms in equations (3.4) is felt upto 2 -- 50 2o approximately.
Conclusions
In this analysis a well-known approximation method of Sparrow et al. [5] is used to solve the boundary layer equations for a non-similar flow along the frustum of a cone. It is found that the local non-similar results by two and three equation models are close to each other and compare well with the known finite difference results. It shows that the approximation does not necessarily become better if more equations are introduced. However, it is
Natural convectionflow
345
Table 2. Values o f f " ( ~ , 0) and 0(~, 0) obtained from the two-equation model for a = 0.72 and 0.01. ~ = 0.72
0* 0.1 0.25 0.75 1.75 3.75 7.75 15.75 31.75 63.75 127.75 oet
a = 0.01
f " ( ~ , 0)
0(~, 0)
f " ( ~ , 0)
0(~, 0)
1.55002 1.52731 1.49678 1.41969 1.34434 1.29197 1.25906 1.24169 1.23250 1.22735 1.22553 1.22395
2.05209 2.03107 2.00381 1.93986 1.88000 1.83823 1.81357 1.80011 t.79285 1.78938 1.78754 1.78647
6.70007 6.67886 6.63401 6.47796 6.28498 6.12506 6.02102 5.96199 5.93004 5.91366 5.90510 5.89733
8.69804 8.60295 8.47388 8.17903 7.91299 7.72910 7.62022 7.56111 7.53018 7.51435 7.50633 7.49865
* Solution of equations (3.5). * Solution of equations (3.6).
interesting to see that if wall heat flux is not directly introduced at the leading point of the cone (x = O) but at non-zero distance x0, a deviation o f the cone similarity solution is found close to x0 where it is replaced by the similarity solution for a vertical plate with constant wall heat flux. It is worth to note that a simple approximation method gives rather accurate results.
Acknowledgement The authors are grateful to the referees for their valuable comments and useful suggestions.
References I. 2. 3. 4. 5. 6. 7. 8. 9.
Hering, R.G. and Grosh, R.J.: Int. J. Heat Mass Transfer 5 (1962) 1059. Hering, R.G.: Int. J. Heat Mass Transfer 8 (1965) 1333. Roy, S.: J. Heat Transfer, Trans. A S M E 96 (1974) 115. Kuiken, H.K.: Int. J. Heat Mass Transfer 11 (1968) 1141. Sparrow, E.M., Quack, H. and Boerner, C. J.: A I A A 8 (1970) 1936. Sparrow, E.M. and Yu, H.S.: J. Heat Transfer, Trans. A S M E 93 (1971) 328. Minkowycz, W.J. and Sparrow, E.M.: J. Heat Transfer, Trans. A S M E 96 (1974) 178. Chen, T.S. and Mucoglu, A.: J. Heat Transfer, Trans. A S M E 97 (1975) 198. Na, T.Y. and Chiou, J. P.: Appl. Sci. Res. 35 (1979) 409.
