Free convection from a cone with a closed end

LEITERS IN HEAT A N D M A S S TRANSFER 0094-4548/82/060455-08503.00/0 Vol. 9, pp. 455-462, 1982 ©Pergamon Press Inc. Printed in the United States

FREE CONVECTION FROM A CONE WITH A CLOSED END

Akira Nakayama Department of Mechanical Engineering Shizuoka University, Hamamatsu, 432 Japan Sei-ichi Ohsawa Tokyo Sanyo Electric Corporation Oh-lzumi, Gunma, Japan Hitoshi Koyama Department of Mechanical Engineering Shizuoka University, Ham-matsu, 432 Japan

(CoL,t~nicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT Full Navier-Stokes equations have been solved numerically to investigate the laminar free convective heat transfer from a cone with a closed end. It is shown that the elliptic nature of the flow contributes significantly toward the augmentation of the local heat transfer rate.

Introduction A number of theoretical studies on laminar free convection have been reported for the two-dimenslonal and axisymmetric situations. Merk and Prins [ 1 ] developed the general relations for similar solutions in the case of axisymmetric flow. Hering and Grosh [ 2 ] showed that the similar solutions to the boundary layer equations for a cone exist when the wall temperature distribution is in a form of a power function of distance along a cone ray. Later, it was pointed out that the radial curvature ( which has been neglected to obtain similar solutions ) can have rather strong effects on the processes of the free convective heat transfer from vertical cones. The effects were first investigated by Kuiken [ 3 ] employing a perturbation technique to solve full boundary layer equations for axisymmetric flow. It was found that the heat transfer rate from a cone is augmented significantly for slender cones compared to the case when no account of the radial curvature is taken. All these studies have been carried out within the scope of the boundary layer 455

456

A. Nakayama, S. Ohsawa and H. Koyama

Vol. 9, No. 6

approximations. In the present study, a vertical cone with a closed end is treated to simulate the actual free convection flow induced in a sealed laboratory. Unlike the free convection studies reported to date, the present study directly deals with the full Navier-Stokes equations by means of the finite difference calculation procedure. problem,

Thus, the phenomenon is considered to be an elliptic

in which the recirculation region may exist behind the end surface

of the cone. It is shown that the elliptic nature of the flow may significantly influence the processes of the local heat transfer from the cone with a closed end. Analysis The flow configuration in the present study is indicated in the figure i with the cylindrical coordinates

( x, r ) in which the positive x is aligned

in the direction opposing to the gravitational force ( In the actual situations, the cone may have a sting, the present configuration should represent a cone with a very small sting ). Upon employing the Boussinesq approximation for the buoyancy force, the full governing equations in the cylindrical coordinates may be expressed in a general form as

~xr

( u~ -

)

+

r

( v~

-

)

=

so

( 1 )

Each governing equation is then given by setting ~, F and so as follows: For the continuity equation, = i,

F = O,

so = 0

( 2a )

For the u momentum equation, = u,

F = ~,

so =

r ~

~x ( p + pgx ) + rgS( T - T

r O

~ ( P + 0gx ) - F -~-v ~r r

)

( 2b )

For the v momentum equation, = v,

r = ~,

so = _

( 2c )

For the energy equation, = T,

r = v/Pr,

so = 0

( 2d )

In the above equations, u and v are the velocity components in the x and r directions while T and p denote the temperature and the pressure. The density, kinematic viscosity and the Prandtl number are indicated by 0, ~ and Pr respectively. The buoyancy term contains the thermal expansion coefficient and the acceleration due to gravity g.

Vol. 9, NO. 6

I

/

FREE--ON

/,.

/

/

,,

/

/

/

/

/

I

FI~MACONE

,,

/

:

/

/

l

I

457

I

I,,/1

t'

i

/ / /

-:

g

/ /

~0 I r ~

End5uc fr1¢e

/ / /

/ .

H----

FIG. i Coordinates and Flow Configuration

A half the meridian plane above the axis of symmetry is taken for the calculation domain. The no-slip conditions are applied on the solid walls indicated by hatching in the figure i. The wall temperatures of the solid walls including the end surface of the cone are set to a constant temperature T

while that of the lateral surface of the cone is set to a higher temperature

Tw ( The problem, of course, dose not have to be restricted to the isothermal wall ). Usual symmetry conditions are imposed along the axis of symmetry. Before the discretization, the equations are normalized by the cone slant height L, the temperature difference ( T

- T

) and the reference velocity

W

Ure f = [ gSL( Tw - T

) ]1/2. Obviously, it is sufficient to give Pr, (UrefL/~)

and the apex half angle ~ in order to specify the problem ( provided the room walls are located far enough from the cone ). The boundary layer coordinates ( Xb' Yb ) are also indicated in the figure. The local Grashof number based on xb

is then given by Grx = GrL(Xb/L)3 where Gr L E (UrefL/9)2cos The left hand side of the general conservation equation ( I ) has been

discretized following the procedure employed in the innovative work on the three-dimensional parabolic flow by Patankar and Spalding [ 4 ]. Thus, the general finite difference equation has been obtained once for all dependent variables. The continuity equation, however, has been replaced by the pressure correction equation which is obtained by substituting an abbreviated momentum balance relationship into the continuity equation. The computer code actually used in the present study is a modified version ( without coordinate transformations ) of the versatile two-dimensional code employed in the calculations

458

A. Nakayamm, S. Ohsawa and H. Koyama

of the fully developed The calculations system

turbulent flows in non-circular

Vol. 9, No. 6

ducts

[ 5

].

have been performed on the air, Pr = 0.7 with the grid

(40 x 40) which has

finer meshes near the cone. The complexity asso-

ciated with the boundary conditions along the cone ray has been avoided by maintaining

the spacing of nodes in the two directions

in the ratio of i : tan

For the matter of the influence of the room walls on the solutions, ence gained in the preliminary calculation domain becomes

calculations has indicated

~,-,=terial

the experi-

that the size of the

to both the velocity and temperature

fields around the cone as the domain as large as -i ~ x/L ~ 8 and 0 ~ r/L i0 sin ~ is employed.

The requirement

for convergence has been taken as

satisfied when the maximum change in each variable during an iteration becomes less than a prescribed value,

10 -6 . Results

The velocity vector plot for GrL = 104 and ~ = 45 ° is shown in the figure 2 where the recirculation

region may b ~ observed clearly behind

the

end surface of the cone. The reclrculatlon bubble is seen to entrain the fluid turning around the trailing edge

( the intersection between the lateral surface

and the end surface ). As a consequence, surface accelerates

the flow passing over the cone lateral

further to the trailing ~dge, and the boundary

becomes thinner toward the edge due to the continuity principle.

0.2 uref

.. -. ~ . -. 2 2 C . - - ~ J i l . / ; / f / / ~ /

.2 2 Z ~ / / : . I / / / , /

~Z__-~/i.~ff/,/ ~.~..,.~JJl[/// ,/

,\-

i

\

Y.~-..';/-<'//,/ ~

\ \

',

"~///,./

\ \

\ \ \ \ \ \

-!V FIG. 2 Velocity Vector Plot

layer even

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FREE CONVEL~ION F R 3 M A OONE

459

In order to confirm this trend, the velocity vector distribution has been re-processed in terms of the boundary layer coordinates ( Xb, Yb ) through the interpolative procedure. The figure 3 shows the profiles of the xb direction velocity component u b at Xb/L = 0.51, 0.74 and 0.96. The point of the maximum velocity first moves away from the wall as in the usual boundary layer development ( see those at Xb/L = 0.51 and 0.74 ), but then, comes back near to the wall as the flow approaches toward the trailing edge ( Xb/L = i ) as a result of the strong acceleration ( see those at Xb/L = 0.74 and 0.96 ). This phenomenon also reflects on the temperature field which is of the primary interest in the present study. T h e

temperature gradient initially

decreases as the thermal boundary layer diffuses in the normal direction Yb" It, however, becomes steep again toward the trailing edge as observed in the figure 4.

0~ O5

0.5

°I 0k

o.4 ~L

~L

0,3

0~

0.2

0.2

0.1

0.1

0.1

0.2 0.3 '-%/Ure~

0.4

0.5

FIG. 3 Development of Streamwise Velocity

0

i

i

I

I

0.2

0A

O6

0~

~"hl

tO

(T-T=~/(Tw-T=~ FIG. 4 Development of Temperature

The effects of this elliptic behavior on the local heat transfer is indicated in the figure 5, where Nux/Grx I/4 is plotted against the streamwise coordinate Xb/L. The local Nusselt number is defined by Nux E h xb/k where h and k are the local heat transfer coefficient and the thermal conductivity. In the figure, the value 0.451 from the similar solution obtained by Hering and Grosh [ 2 ] is also indicated along with the curve based on Kuiken's perturbation solution ( up to the third order term ) to the boundary layer equations [ 3 ], which accounts for the radial curvature effects. As observed in the figure, the radial curvature effects on the local heat transfer rate are, in fact,

460

A. Nakayama, S. O h s a w a a n d H .

Koy~'na

Vol. 9, No. 6

significant near to the apex of the cone. It is also interesting to note that Kuiken's perturbation solution is in good accord with the present finite difference calculation result despite that the perturbation solution is intended to be valid only for the region away from the apex where the perturbation parameter becomes sufficiently small. It is also clearly seen that the boundary layer approximations as employed in Kuiken's approach, become strictly invalid beyond Xb/L = 0.7 in this flow case.

1.0 0.9 0.8

]

0.7

~

~

Kuiken (31

0.6

0.~

T 0

I 0.2

I 0.4

xb/L

I 0.6

I 0.8

I 1.0

FIG. 5 Elliptic Effects on Heat Transfer Rate

The calculations have been also performed for ~ = 4.45 ° at GrL = 3.4 x 106 and 3.6 x 107 . In this case of small apex angle, the recirculation bubble has not been observed within the resolution of the present grid system ( 40 x 40 ), yet, the elliptic nature of the flow again has been confirmed upstream near the trailing edge, resulting a higher local heat transfer rate. The calculation results of these three different flow cases are plotted together in the figure 6 in terms of Nux/Grx I/4 against Kuiken's perturbation parameter defined by E 2/Grxl/4tan a. The tail-up behavior observed in the figure, is obviously associated with the elliptic nature of the flow ( note the singularity at Xb/L = i, the temperature double-valued along the trai±ing edge ). It is again confirmed that Kuiken's perturbation solution, Nux/Grx I/4 = 0.451 + 0.2066 + 0.0028~ 2, happens to be still a good approximation even for ~ > i in which the

Vol.

9, No. 6

FREE CONVECTION FROM A CONE

461

series expansions would diverge.

1.2

~GrL=10~.(X=45 ° /GrL= 3.6x 107,(X=4.45~

i

O.8 N

GrL;3.4 xl0 8, IX=4.4~

,~1/4

ux/~rx

0.4

I

I

I

I

0.5

1.0

1.5

2.0

Radial Curvature

FIG. 6 Effects on Heat transfer Rate

Concludin$ Remarks As pointed out in this study,

the elliptic nature of the flow over a cone

with a closed end, has rather significant convective heat transfer. tion flows,

effects on the processes of the free

One should be aware that, in the actual free convec-

the radial curvature effects may not be negligible near to the apex

of the cone, while, near to the closed end surface, the flow may come to contribute

the elliptic behavior of

toward the augmentation of the local heat

transfer rate. Acknowledgement The authors would like to express for a number of suggestions

their sincere thanks to Mr. E. Makita

given during the course of this study.

462

A. Nakayama, S. Ohsawa and H. Koyama

Vol. 9, No. 6

Nomenclature g

acceleration due to gravity

GrL, Grx

Grashof number

h

local heat transfer coefficient

k

thermal conductivity

L

cone slant height

Nux

local Nusselt number

P

pressure

Pr

Prandtl number

T

temperature

u~ v

velocity components in cylindrical coordinates

%

velocity component in xb direction

Uref x, r

reference velocity

Xb' Yb

boundary layer coordinates

cylindrical coordinates

apex half angle

B

coefficient of thermal expansion Kuiken's perturbation parameter kinematic viscosity

P

density References

i.

H. J. Merk and J. A. Prins, Applied Scientific Research, Sec. A ~, ii, 195 and 207 ( 1953 ).

2.

R. G. Herlng and R. J. Grosh, Int. J. Heat Mass Transfer ~, 1059 ( 1962

3.

H. K. Kulken, Int. J. Heat Mass Transfer i_~i, 1141 ( 1968 ).

4.

S. V. Patankar and D. B. Spalding, Int. J. Heat Mass Transfer 15, 1787 ( 1972 ).

5.

A. Nakayama, Ph.D. Thesis, University of Illinois at Urbana-Champaign ( 1981 ).