Effect of Discrete Heat Sources on Natural Conv in a Square Cavity By_jaikrishna

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Proceedings of the 37th

National & 4th

International Conference on Fluid Mechanics and Fluid Power

FMFP2010

December 16-18, 2010, IIT Madras, Chennai, India

1

FMFP2010________

EFFECT OF DISCRETE HEAT SOURCES ON NATURAL

CONVECTION IN A SQUARE CAVITY

Jaikrishna. C. R

Student, Department of Mechanical EngineeringPES Institute of Technology,

Bangalore-560 085, Karnataka, INDIA

Rathan Ram. B

Student, Department of Mechanical EngineeringPES Institute of Technology,

Bangalore-560 085, Karnataka, INDIA

[email protected] [email protected]

Aswatha

Department of Mechanical EngineeringBangalore Institute of Technology,

Bangalore-560 004, Karnataka, [email protected]

K. N. Seetharamu

Department of Mechanical Engineering,PES Institute of Technology,

Bangalore - 560 085, Karnataka, [email protected]

ABSTRACTIn the present study a finite volume computational procedure is used to investigate

natural convection in a square cavity. The enclosure used for flow and heat transfer

analysis has been bounded by adiabatic top and bottom walls, constant temperatureright cold wall and discretely heated left wall. Also the computations are carried out

for with 25 % opening of cold wall at the top. The Rayleigh Number (Ra) varying

from 103-107 and Pr = 0.7. When the Rayleigh number is increased, rate of heat

transfer also increases and the maximum temperature at the heater surface decreases.

The effect of convection is more dominant with partial opening. Best heat transfer

was obtained when the heater element is placed at the center with minimum heating

length and opening one.

Key words: Natural convection; square cavity; discrete heat source; electronic cooling

components

1. INTRODUCTION

Natural convection in cavities has gained importance in many electronic applications.

Natural convection cooling is desirable because it doesn’t require energy source for

cooling and hence more reliable. Air is taken as the cooling medium for cooling

electronic components due to its simplicity and low cost. Microprocessors are treated

as heat sources on flat surfaces.

There are numerous studies in the literature regarding natural convection heat transfer

in cavities. Aydin and Yang (2000) numerically investigated natural convection in

enclosures with localized heating from below and symmetrically cooled from side

walls. Nguyen and Prudhomme (2001) have studied convection flows in a rectangular

cavity subjected to uniform heat flux. Salat et al. (2004) studied experimentally andnumerically the turbulent natural convection in a large air-filled cavity.

446



2

NOMENCLATURE

g acceleration due to gravity (m/s2)

k thermal conductivity (W m-1 K -1)

L side length of the square cavity (m)

Nu local Nusselt number p dimensional pressure (Pa)

Pr prandtl number

q″ heat flux (W m-2)

Ra Rayleigh number

T temperature (K)

T c temperature of vertical wall (K)

u x- component of velocity

v y- component of velocity

Greek symbols

α thermal conductivity (m2

s-1

) β volume expansion coefficient (K -1)

θ Non - dimensional temperature

γ kinematic viscosity (m2s-1)

ρ density (kg m-3)

ψ stream function

Subscripts

Bilgen and Oztop (2005) investigated numerically natural convection heat transfer in

partially open inclined square cavities. They made an effort to study the steady state

heat transfer by laminar natural convection in a two dimensional partially open

cavities. Natural convection in cavities with constant flux heating at the bottom wall

and isothermal cooling side walls have been studied by Sharif et al. (2005). The effect

of aspect ratio, inclination angles and heat source length on the convection and heat

transfer process in the cavities are analyzed. Kasayapanand (2007) studied the effect

of electric field on natural convection in the partially opened square cavities by finite

volume technique. Nithyadevi et al. (2007) studied the effect of aspect ratio on thenatural convection of a fluid contained in a rectangular cavity with partially active

Fig.1 .Schematic d iagram of the physical system

B

C

D

A



3

side walls. The active part of the left side wall was selected at a higher temperature

than that of the right side wall. The top and bottom of the cavity and inactive part of

the side walls were thermally insulated. Yasin Varol et al. (2007) have made efforts to

understand the variable protruding heater length on Natural convection in triangular

enclosures. Recently, Ayla DOGAN et al. (2009) studied numerically the heat transfer

by natural convection from partially open cavities with one wall heated. In this study,the steady state investigation deals with natural convection heat transfer inside the

cavity under uniform heat fluxes, different opening ratios, tilt angles and cavity aspect

ratios for top and center opening positions. They found that the average heat transfer

coefficient increases and the average wall temperature decreases, with the increase in

opening ratio and decrease in the tilt angle. Best heat transfer was obtained with the

aspect ratio 1, for opening ratio of 0.75 and tilt angle of 10° in the clockwise

direction.

The objective of the present paper is to investigate the effect of discrete heat sources

on natural convection in a square cavity with and without opening on the cold wall.

2. MATHEMATICAL FORMULATIONA square cavity illustrated in Fig.1 is chosen for simulating natural convective

flow and heat transfer characteristics. The square cavity of length (L) has left wall

with discrete heat sources of varying lengths and positions. The gravitational force is

acting downwards. A buoyant flow develops because of thermally induced density

gradient.

The governing equations for natural convection flow are conservation of mass,

momentum and energy equations, written as:

Continuity: 0=¶

¶+

¶

¶

y

v

x

u (1)

X-momentum: ÷÷ ø ö

ççè æ

¶¶+

¶¶+

¶¶-=

¶¶+

¶¶

2

2

2

21

y

u

x

uv x

p

y

uv x

uu r

(2)

Y-momentum: ( )cT T g y

v

x

vv

y

p

y

vv

x

vu -+÷

÷

ø

ö

çç

è

æ

¶

¶+

¶

¶+

¶

¶-=

¶

¶+

¶

¶b

r 2

2

2

21 (3)

Energy: =¶

¶+

¶

¶

y

T v

x

T u ÷

÷

ø

ö

çç

è

æ

¶

¶+

¶

¶2

2

2

2

y

T

x

T a (4)

No-slip boundary conditions are specified at all walls.

Left side wall: case 1, 0= x ,4

L y = to

4

3 L =

¶

¶-

x

T k q ″

case 2, 0= x , 0= y to4

L and

4

3 L y = to L =

¶

¶-

x

T k q ″ (5)

case 3, 0= x , 0= y to4

L and

2

L y = to

4

3 L =

¶

¶-

x

T k q ″

case 4, 0= x ,8

3 L y = to

8

5 L =

¶

¶-

x

T k q ″

Remaining lengths of the left side wall are adiabatic

Top and bottom wall: ( ) 0, =¶

¶ L x

y

T and ( ) 00, =

¶

¶ x

y

T

Right side wall: ( ) ( ) cT y LT yT == ,,0

For opening at the right wall: L x = ,

43 L y = to L , 0=P and 0=

¶¶=

¶¶

x

v

x

u



4

Where x and y are the dimensional co-ordinates along horizontal and vertical

directions respectively; u and v are dimensional velocity components in x and y -

directions respectively; T is the temperature; p is the dimensional pressure; Here,

the fluid is assumed to be Newtonian and the properties are constant. Only the

Boussinesq approximation is invoked for the buoyancy term.

The changes of variables are as follows:

,k

LqT

"

=D ,a

n =Pr

and

2

3 Pr

n

b TLg Ra

D= (6)

In the present investigation, the selected geometry has been modeled. The modeled

geometry is discretized using Gambit 2.4. The meshed model is saved as data file and

mesh file separately. The saved Gambit files are read in ANSYS FLUENT 6.3 to give

specified boundary conditions, selection of fluid and fluid properties. The several

cases mentioned earlier are solved for Ra ranging from 103 to 107.

3. Stream function and Nusselt number

3.1 Stream function

The motion of buoyant driven fluid inside the cavity is represented by using the

stream function Ψ obtained from velocity components u and v . The relationship

between stream function, Ψ and velocity components for two dimensional flows are

given by Batchelor (1993):

yu

¶

¶= y

and

xv

¶

¶-= y

(7)

which yields to a single equation:

x

v

y

u

y x ¶

¶-

¶

¶=

¶

¶+

¶

¶

2

2

2

2 y y

(8)

The local heat transfer coefficient is defined as h y = q" / [Ts-Tc] at a given point on the

heat source surface where Ts is the local temperature on the surface. Accordingly the

local Nusselt number is obtained as Nu = (h y W)/k. The trapezoidal rule is used for

numerical integration to obtain the average Nusselt number.

4. Numerical procedure

The set of governing equations are integrated over the control volumes, which

produces a set of algebraic equations. The PISO algorithm developed by Issa (1985) is

used to solve the coupled system of governing equations. The set of algebraic

equations are solved sequentially by ADI method. A second-order upwind

differencing scheme is used for the formulation of the convection contribution to the

coefficients in the finite-volume equations. Central differencing is used to discretize

the diffusion terms. The computation is terminated when the entire residuals one

below 10−5.

5. Verification of the present methodology

The verification is made with reference to the results of Sharif and Taquiur (2005).

The cavities used to study are bounded by uniform temperature vertical side walls and

adiabatic top wall. The bottom wall is subjected to a uniform heat flux spread over

from 20 % to 80 % of the length from centre and the remaining length is considered

adiabatic. The Grashof number (Gr) is varied from 10

3

to 10

6

. In order to obtain gridindependent solution, a grid refinement study is performed for a square cavity

T

T T c

D

-=q



5

(AR = 1) with heating length = 0.6. Fig. 2(a) shows the convergence of the average

Nu at the heated surface with grid refinement for Gr = 105 of Sharif and Taquiur

(2005). Different grid sizes of 31 x 31, 41 x 41, 51 x 51 and 61 x 61 with uniform

mesh as well as biasing have been studied. The grid 41 x 41 biasing ratio (BR) of 2

(The ratio of maximum cell to the minimum cell is 2, thus making cells finer near the

wall) gave results identical to that of 61 x 61 uniform mesh. In view of this, 41 x 41

grid with biasing ratio 2 is used in all further computations. Fig. 2(b) shows variation

of the Average Nusselt number with Sharif and Taquiur (2005). The percentage of

error was within 2.4 %. This is found to be a good agreement with Sharif and Taquiur

(2005).

6. RESULTS AND DISCUSSION

6.1 Cavity without openingThe flow and heat transfer characteristics in a square cavity have been studied for

four different cases as illustrated in Fig. 1 without opening. Computations are carried

out for Rayleigh number ranging from 103 to 107. The results are presented in the

form of stream lines, isotherms, local Nu and average Nu for all four different cases.

Fig. 3 Contour plots for case 1 without opening of cold wall with Ra = 105.

Fig.2 Conve rgence o f ave rage Nusse l t number w i th (a ) Gr id re f inemen t and

(b ) Sharif and Taquiur (2005).



6

Fig. 4 Contour plots for case 2 without opening of cold wall with Ra =105.

Fig.5 Contour plots for case 3 without opening of cold wall with Ra = 105.

Fig. 6 Contour plots for case 4 without opening of cold wall with Ra = 105.



7

Figs. 3 to 6 shows the streamlines and isotherms plots for all four different cases. For

case 3 yields higher values of magnitudes of the stream functions as compared to first

two cases and last one. The cells extend towards the bottom right corner of the cavity

for all cases. From the temperature profiles it is observed that the contours are spread

out the entire cavity except case 4 (25% of heating length). The temperature contours

are concentrated at heating length as expected for all the cases. For case 4, the majortemperature contours are settled at the top half of the cav ity, except θ = 0.2 and 0.1.

Figs. 4 & 5 show the plots for the discrete heat sources (Case 2 &3). It is observed

that, the temperature profiles are closer to reach open at the top right portion

indicating higher heat transfer at that location

The variation of local Nusselt number, along the heater element for four different

cases and Rayleigh number 103, 105 & 107 is shown in Fig. 7. The heat transfer rate is

higher at the center of the cavity and reduces towards the heater surface. The local

Nusselt numbers are lower for split heat sources compared to continuous heating.

Case 1 and case 4 yields the similar Nu because of heaters at the centre. In discrete

heating (case 2 and 3) Heater A is giving the maximum Nu as compared to the otherheaters.

Fig.7 Variation of local Nusselt number along the heating length



8

Fig. 12 and Table 1 combinedly show

the average Nu for all the cases. It is

observed that the average Nu increase

monotonically with increase of Ra. The

average Nu is maximum for heater (C)

Ra up to 105

, but it is more for Ra = 106

and 107 for heater (B).

Table.1 Average Nusselt Number V/s. Rayleigh Number

6. 2 Cavity with openingIn order to increase the heat transfer rate, the 25 % of the length of the cold wall

is opened at the top. Figs. 9-12 Show the streamlines and temperature plots for the

same.

Case 2 Case 3

Heater - (A) Heater - (B) Heater - (C) Heater - (D)Ra

Closed Open%

increaseClosed Open

%

increaseClosed Open

%

increaseClosed Open

%

increase

103 1.63 1.65 1.21 1.80 1.85 2.70 2.00 2.02 0.99 1.87 1.89 1.06

104 2.16 2.25 4.00 2.81 2.90 3.10 2.83 2.86 1.05 2.84 2.90 2.07

105 3.44 3.69 6.78 4.77 5.09 6.29 4.63 4.79 3.34 4.79 4.99 4.01

106 5.87 6.82 13.93 8.54 9.13 6.46 7.88 8.85 10.96 8.53 9.05 5.75

10

7

10.18 14.06 27.60 15.36 17.07 10.02 13.22 16.92 21.87 15.27 17.03 10.33

Fig.9 Contour plots for case 1 with 25% of opening at top cold wall with Ra = 105.

Fig. 8 Average Nusselt number Vs

Rayleigh Number for cases 1 and 4.



9

Fig. 10 Contour plots for case 2 with 25% of opening at top cold wall with Ra = 105.





10

It can be clearly seen from the plots drawn the streamlines and the temperature plots flow

out of the cavity at the opening. Fig. 8 and Table 1 combinedly show the variation of

average Nusselt number along the heated surface with Rayleigh Number for constant

heating. The opening at the right cold wall has noticeable effect only after Ra > 5×10 4.

For lower Rayleigh numbers the effect of opening is negligible. For higher Rayleigh

number (107) the opening has a significant effect that can be visualized by the increase in

the local Nusselt number. Table.1. shows the Average Nusselt number for discrete heat

sources (Case 2 & 4) with Rayleigh number. The heater (B) and (D) in case 2 and case 3

respectively have the same values due to its position. Heater (C) has a higher heat

transfer rate than (A). This is because the heat transfer rate is higher at the center and

gradually reduces towards the top wall.

7. Conclusions

The flow and heat transfer characteristics for discrete heating configurations

illustrated in the Fig.1 have been investigated. The following conclusions are drawn from

the present study.

(a) The conduction is dominated, for Ra ≤ 104 for all the cases of with and without

opening in the cold right vertical wall.(b) For higher Rayleigh numbers where convection is dominated the opening has a

significant effect on the average Nusselt number.

(c) With the four different boundary conditions for the vertical wall the maximum heat

transfer occurs for the case of heat source concentrated at the centre.

8. REFERENCES

Aydin, O., Yang, J., 2000. Natural convection in enclosures with localized heating

from below and symmetrical cooling from sides, Int. J. Numer. Methods Heat Fluid

Flow, 10, 518-529.

Ayla DOGAN., BAYSAL, S., Senol BASKAYA., 2009. Numerical analysis of

natural convection heat transfer from partially open cavities heated at one wall, J. of

Thermal Science and Technology 29, 1, 79-90.

Batchelor, G.K., 1993. An introduction to fluid dynamics, Cambridge University

Press, Cambridge, UK. Bilgen, E., Oztop, H., 2005. Natural convection heat transfer in partially open

inclined square cavities, Int. J. of Heat and Mass Transfer, 48, 1470-1479.

Issa, R.I., 1985. Solution of the implicitly discretised fluid flow equations by operator-

splitting, J. Comput. Phys. 62, 40-65.

Kasayapanand, N., 2007. Numerical modeling of natural convection in partially open

cavities under electric field, Int. Comm. Heat Mass Transfer 34, 630-643.

Muhammad, A.R. Sharif., Taquiur Rahman Mohammad., 2005. Natural convection in

cavities with constant flux heating at the bottom wall and isothermal cooling from thesidewalls, Int. J. of Thermal Sciences, 44, 865–878.

Nguyen, T.H., Prudhomme, M., 2001. Bifurcation of convection flows in a

rectangular cavity subjected to uniform heat fluxes, Int. Comm. Heat Mass Transfer, 28,

23-30.

Nithyadevi, N., Kandaswamy, P., Lee J., 2007. Natural convection in a rectangular

cavity with partially active side walls, Int. J. Heat Mass Transfer 50, 4688-4697.

Salat J., Xin S., Joubert P., Sergent A., Penot F., Le Quere P., 2004. Experimental and

numerical investigation of turbulent natural convection in a large air-filled cavity, Int. J.

Heat Fluid Flow 25, 824-832.

Yasin Varol., Hakan, Oztop, F., Tuncay Yilmaz., 2007. Natural convection in

triangular enclosures with protruding isothermal heater, Int. J. of Heat and MassTransfer, 50, 2451-2462.

Documents

Effect of Discrete Heat Sources on Natural Conv in a Square Cavity By_jaikrishna