56326_2

8/11/2019 56326_2

1/14

Al-KhwarizmiEngineering

JournalAl-Khwarizmi Engineering Journal, Vol. 8, No. 3, PP 81 - 94 (2012)

Natural Convection Heat Transfer in an Inclined Open-EndedSquare Cavity with Partially Active Side Wall

Jasim M. MahdiDepartment of Energy Engineering/ College of Engineering/ University of Baghdad

E-mail:[email protected]

(Received 10 November 2011; accepted 7 August 2012)

Abstract

This paper reports a numerical study of flow behaviors and natural convection heat transfer characteristics in aninclined open-ended square cavity filled with air. The cavity is formed by adiabatic top and bottom walls and partially

heated vertical wall facing the opening. Governing equations in vorticity-stream function form are discretized via finite-difference method and are solved numerically by iterative successive under relaxation (SUR) technique. A computerprogram to solve mathematical modelhas beendeveloped and written as a code for MATLABsoftware. Results in the

form of streamlines, isotherms, and average Nusselt number, are obtained for a wide range of Rayleigh numbers 103-10

6

with Prandtl number 0.71 (air) , inclination angles measured from the horizontal direction 0-60, dimensionless lengthsof the active part 0.4-1 ,and different locations of the thermally active part at the vertical wall. The Results show that

heat transfer rate is high when the length of the active part is increased or the active part is located at middle of verticalwall. Further, the heat transfer rate is poor as inclination angle is increased.

Keywords:Natural convection, square cavity, partially active side wall.

1. Introduction

Natural convection in a cavity with onevertical side open has received increasingattention in recent years due to its importance in a

number of practical applications such as heattransfer and fluid flow in solar thermal receiver

systems [1], fire spread from room [2], and

cooling of electronic equipment [3]. Manytheoretical and experimental studies on naturalconvection in the 2-D open square cavity have

been carried out [4]![10]. For the sake of brevityliterature review will be only restricted to studiesof natural convection in open square cavities withadiabatic walls and isothermal at the wall facingthe aperture. Chan and Tien [4] performed anumerical study of laminar natural convection in atwo-dimensional square open cavity with a heated

vertical wall and two insulated horizontal walls.To overcome the difficulties of unknown

conditions at the opening, they made theircalculations in an extended computational domain

beyond the opening. Results obtained for

Rayleigh numbers ranging 103- 10

6were found to

approach those of natural convection over avertical isothermal flat plate. Later on, the sameauthors [5] investigated open shallow cavity forRayleigh numbers up to 10

6to test the validity of

approximate boundary conditions at the opening

for a computational domain restricted to the cavityinstead of using an extended domain. They

concluded that for a square open cavity having anisothermal vertical side facing the opening and

two adiabatic horizontal sides, satisfactory heattransfer results could be obtained by the restricted

domain. Polat and Bilgen [6] numerically studiedinclined fully open shallow cavities in which theside facing the opening was heated by constantheat flux, and two adjoining walls being adiabatic

over Rayleigh number values ranging 103 - 1010.The opening was in contact with a reservoir at

constant temperature and pressure; thecomputational domain was restricted to the cavity.

They observed that heat transfer approachesasymptotic values at Rayleigh numbersindependent of the aspect ratio and asymptotic
mailto:[email protected]:[email protected]

8/11/2019 56326_2

2/14

Jasim M. Mahdi Al -Khwari zmi Engineer ing Journal, Vol. 8, No. 3, PP 81- 94 (2012)

82

values are close to that for a flat plate withconstant heat flux. Chakroun [7] performed anexperimental investigation to study the effect of

wall conditions as well as its tilt angle on heattransfer for a fully open tilted cavity.

His study contains seven different wall

configurations over an inclination angle measuredfrom the vertical direction range "90 to +90. Itwas concluded that thetilt angle, wall

configuration, and number of hot walls are allfactors that strongly affect the natural convectioninside the fully open cavity. Nateghi and Armfield[8] numerically studied natural convection flow ina two dimensional inclined open cavity for both

transient and steady-state flow over Rayleighnumbers ranging 105 - 1010 with Prandtl number

0.71 and inclination angles 10- 90. Their resultsshowed that the flow is steady at the low Rayleigh

number for all angles and becomes unsteady at thehigh Rayleigh number for all angles and the

critical Rayleigh number decreases as theinclination angle increases. Hinojosa et al. [9]

presented results for natural convection andsurface thermal radiation in a square tilted open

cavity. Effects of Rayleigh number 103 - 107 andinclination angles 0- 180are investigated for afixed Prandtl number (0.7). The results show thatthe convective Nusselt number changessubstantially with the inclination angle of thecavity, while the radiative Nusselt number isinsensitive to the orientation change of the cavity.

Mohamad et al. [10] simulated the naturalconvection in an open cavity using Lattice

Boltzmann Method. The paper demonstrated thatopen boundary conditions used at the opening ofthe cavity are reliable, where the predicted resultsare similar to that of conventional CFD method.

Prandtl number is fixed to 0.71 while Rayleighnumber and aspect ratio of the cavity are changed

in the range of 104 to 106 and of 0.5 to 10,respectively. The results show that the rate of heat

transfer deceases asymptotically as the aspectratio increases and may reach conduction limit for

large aspect ratio.Most of the above-cited works deal with the

active vertical wall of the cavity at isothermal orisoflux condition. In real life such as in fields likesolar energy collection and cooling of electroniccomponents, the active wall may be subject toabrupt temperature nonuniformities due to

shading or other natural effects[11], therefore onlya part of the wall is either in isothermal or isofluxcondition. The present study deals with the natural

convection in an open square cavity filled with airas working fluid with partially thermally activevertical wall, for different lengths of thermally

active part. The active part is located at the top,middle and bottom to locate the position wherethe heat transfer rate is maximum and minimum.

2. Mathematical Formulation

The geometry, the coordinate system, and theboundary conditions for the problem under

consideration are shown in Fig. (1). A portion ofthe right side wall of the cavity is kept at a

temperature Th and the remaining parts areinsulated. Under the assumptions of a two-

dimensional viscous incompressible fluid withsteady-state conditions, the equations governing

the motion and the temperature distribution insidethe cavity may be written as: [12]Continuity equation

u# Y = 0 (1)Conservation of momentum in x-direction

% !u u# +uY" =#

p

& $%u & +'u () %gsin' (2)

Conservation of momentum in y-direction

% *u v# + vY+ =#p +& ,-v .

/ 01 # %gcos' (3)

Energy Equation

uT#

TY = ( 2

3T 4

5T 67 (4)

As in many investigations of naturalconvection, all fluid properties, except density in

buoyancy term(%g), are assumed constants and theBoussinesq approximation is used to incorporate

the temperature dependence of density in themomentum equation. In this approximation the

density in buoyancy term is assumed to be a linear

function of temperature, % = %)81# *(T#T))9[12].

The use of vorticity!stream functionformulation can eliminate pressure gradient termsfrom momentum equations and simplify thesolution procedure. With the stream function, the

velocity components u and v and vorticity can beexpressed as:

8/11/2019 56326_2

3/14


83

u =#+y,v =+xand , =

uy#

vx

Furthermore by introducing the following no

dimensional variables: (X,Y) = (x,y) L ,: (U,V) = (u,v) (( L: ): - = + (: , . = , (( L;: ),: / = (T#T)) (T< #T)): withT= > T),the governing equations in terms of vorticity-stream function form become:Stream function

>?- =#. (5)Vorticity

>@. = 1Pr

AU.# +V.YB

#RaC/

#

cos' # /

Y

sin'D (6)Energy

>E/ = U /# +V/Y (7)

where

U =#-Y V =-

(8)

The dimensionless parameters appearing inequations (5) through (8) are the Prandtl number

Pr =0 (: and the Rayleigh numberRa=g*(T

F #T

))L

G (0

:.

Other parameters which enter into this study arethe dimensionless length of the active part

H =HI (0.4J HJ 1) and the dimensionless

position of the active part center S=KL (H 2: J SJ 1#H 2: ).The appropriate boundary conditions in

dimensionless form can be formulated as:

at X = 0 , SJYJ S+MN,- = 0 . =#

O-XP / = 1

at X = 0, 0JY < Q # RS , S+TU < V J 1,- = 0 . =#W-XX

/X= 0

at 0 < X< 1 ,Y = 0,1 ,and

- = 0 . =#Y-YZ /Y = 0

at X = 1 ,0< Y < 1.

[-\] =

^._` = 0,a

/ = 0 if UJ 0 (inblow )c/de = 0 if U > 0(outflow)

Local Nusselt number on thermally active part isdefined by:

Nu =#hgLk

=L

Th #T) iTxjklm

Where the local heat transfer coefficient hy is

defined by equating heat transfer by convection tothat by conduction at the active part:

h(Tn #T)) =#k oTxpqrsby introducing the dimensionless variablesdefined above, local Nusselt number will be:

Nu = tu/vwxyz{The average Nusselt number is obtained byintegrating local Nusselt number over the active

part:

Nu| = Nu.dY!"(# $: )%&(' (: )

Fig. 1. Schematic of the Square Cavity, the

Coordinate System and Boundary Conditions.

3. Numerical Technique

A finite difference method (FDM) with centraldifference scheme is used to discretize the systemof the partial differential equations (5 - 8). The

new algebraic equations system will be solvedusing iterative under relaxation method (URM), to

give approximate values of the dependentvariables at a number of discrete points called(grid points or nodes) in the computational

domain. These nodes are formed by subdividingthe computational domain in the X and Ydirections with vertical and horizontal uniformly

T

L

g

)*

-./

01234567

8/11/2019 56326_2

4/14


84

spaced grid lines. Based on a grid independencestudy a grid of 61 161 nodes is adopted typicallyin this study. However, a careful check for the

grid independence of the numerical solution hasbeen made to ensure the accuracy and validity of

the numerical schemes. For this purpose, four grid

systems, 45145, 61161, 81181, and 1011101 aretested. Results show the average Nusselt numberprediction for 61161 grid points nearlycorrespond to that obtained from other grid points,the maximum relative error (% 2) obtainedthrough the five grid systems is no more than 1%as shown in Table (1). The accuracy of thenumerical code was assessed by applying it to a

case studied by Mohamad et al. [10], whoconsidered a cavity configuration similar to that

being considered in this paper with a fully activehot vertical wall, and good agreement was

observed, as shown in Table (2). The maximumdifference obtained for the average Nusselt

number at the hot vertical wall was 1.7 %.Furthermore, in order to lend more confidence in

the present numerical results, the results in formof isothermals contours were compared with those

of Nateghi and Armfield [8] and Mohamad et al.[10]; and an excellent agreement was achieved,

Fig.(2).The convergence criteria employed toterminate the computations and reach the solutionwere preassigned as (-3 - -3-1)/-3 4 10-6,(/3 - /3-1)//3410-6, and (.3 - .3-1)/ .3410-6, theindices 3 and 3 "1 represent the current andprevious iteration, respectively.

Table 1,

Grid Independence Study Results for Average

Nusselt Number.

% 101!10181!8161!6145!45Size

1.01.1891.1881.1801.173Ra = 103

0.53.3183.3273.3193.308Ra = 104

0.37.2987.3017.3017.284Ra = 105

0.714.2714.2314.2214.32Ra = 10

Table 2,

Comparison of the Obtained Results for Average

Nusselt Number with Literature.

Difference (%)Nu [this study]Nu [10]Ra

1.73.3193.377104

0.27.3017.323105

1.114.2214.3810

4. Results and Discussion

In this section, the numerical results in form ofthe streamlines, isotherms, and average Nusselt

numbers for various values of the parametersgoverning the heat transfer and fluid flow will be

presented and discussed. The parameters arelocation of thermally active part, thedimensionless length of thermally active part, H =

0.4 - 1.0, Rayleigh number, Ra = 103

- 106, and

inclination angle, '=0- 60. Prandtl number, Pr= 0.71 for air was kept constant. In Figures (3-6),the darkened area of side wall indicates theposition of thermally active part while the symbol( ) represents the center of the circulation which

defined as the point of the extremum of the streamfunction -max.

The scenario of development naturalconvective currents inside open cavity beginswhen the air in contact with the active part of side

wall heated by conduction, becomes lighter, andascends in adjoin to the side wall to face the

adiabatic upper wall and then through which itmoves horizontally to leave the cavity from theupper part of the aperture. In order to retrievefluid left outside the cavity, fresh air entrained

from the lower part of the aperture toward heatedportion of side wall to be heated and resulting inclockwise recirculation inside cavity, as shown in

left sides of Figs. (3-6).

4.1. Effect of Active Part Length

The effects of active part length H on flow and

temperature fields are shown in figure (3) forRayleigh number, Ra = 10

5, inclination angle, '

=05 and position of active part is at middle. Thevalues of maximum stream functions -max showthat higher values are obtained with increasing theactive part length from H=0.4 to H=1. In fact,when the active part is wider major portion of the

cavity is occupied by circulating cells (seestreamlines in Fig. (3)) and flow strengthincreases making the cavity more significantlyaffected by the buoyancy-driven flow. The

corresponding isotherms indicate that as Hincreases the isotherms are denser near the active

part that means thermal boundary layer becomesthinner and temperature gradients becomes

steeper due to strong convection. The amount ofheat transferred from the active part across the

cavity obviously increases as the length H of thethermally active part increasesdue to higher heattransfer surface. Correspondingly, average Nusseltnumber which would describe the thermal

8/11/2019 56326_2

5/14


85

behavior of the cavity should increase withincreasing H as shown in Fig. (7).

4.2. Effect of Active Part Location

Figure (4) shows the effect of active part

position on the streamlines and isothermspatternsat Ra = 10

5, '=05, and dimensionless length of

active part, H=0.4. In fact, due to impingement of

circulated fluid to the middle of the vertical walland impingement of hot fluid to the top adiabatic

wall, fluid at right top and bottom corner becomesmotionless and also, the half bottom of the cavitybecomes cooler than the upper one. Thus strengthof circulation becomes larger, as clearly denoted

by the higher value of -max, and major portion ofthe cavity is occupied by circulating cells as theactive part occupies a position lower in the cavity(see streamlines in Fig. (4)). In contrast, when the

active part is located at the bottom or the top ofvertical wall the rising fluid by buoyancy force

cannot wash the entire surface of the active partand thermal boundary layer around the active partgets thicker (see isotherms in Fig. (4)), thus theamount of heat exchanged is smaller than thatexchanged when the active part is located at

middle of vertical wall. Subsequently, the averageNusselt number which describes the overall heat

transfer process through the cavity is higher when

the active part located at middle as shown in Fig.(8).

4.3. Effect of Inclination

The effect of inclination angle ' in the rangeof 0 to 606which covering practical cases in solarcollectors [6] is examined for Ra=10

5and H=0.6.

Fig. (5) shows the effect of inclination angle on

streamlines and isotherms patterns. Streamlinesshow that as the inclination is increased flow of

the hot fluid from the aperture is choked and a

smaller part of the upper region of the cavity isoccupied by discharged fluid. As a result, thecenter of the circulation is impelled upward

indicating significant increase in fluid velocities.The isotherms show a thicker thermal boundarylayer is formed on the active part of side wall, andalso a thinner hot intrusion on the upper insulatedwall formed as inclination angle increases.Increasing the inclination angle in the range 04' 4 60 leads to a decrease in the averageNusselt number for the same dimensionlesslength of active part as shown in Fig. (9). This

because as ' increases, the thermal boundarylayer on active part get thicker and fluid thermal

stratification in the upper region of the cavity getpoorer (see isotherms in Fig. (5)), since flow ofthe hot fluid from the aperture is choked as the

inclination is increased and that leads to slowerreplacement of the hot air by fresh air and that

makes convection less rigorous.

4.4. Effect of Rayleigh Number

Figure (6) shows the effect of increasingRayleigh number in the range (103-106) on

streamlines and isotherms patterns at H=0.8 and '=05. Usually, the Rayleigh number, is defined by(Ra = g*L37T/0(), and in case of usingBoussinesq approximation with air as a working

fluid, the 0, ( and g will be constants. Thus,only *and 7T will be the factors that govern

the change in value of Rayleigh number. In caseof Ra =10

3 , the circulation inside the cavity is so

weak that the viscous forces are dominant over thebuoyancy force and that leads to rather weakconvection, also the effect of the open side does

not seem deep in this case. Consequently, thecorresponding isotherms exhibit rather slight

difference as compared to those of pure heatconduction which are parallel to the surface of thethermally active part. On the other hand, whenRayleigh number increases, the buoyancy forceaccelerates the circulation of fluid flow and the

effect of the open side penetrates further into theflow. As a result, the corresponding isotherms are

greatly deformed and crowded near the active partleaving the cavity core empty. Therefore, asRayleigh number increases, the flow becomesfully convective dominated, the fresh fluid is

entrained right to the left vertical wall wherehigh temperature gradients are created, and thedischarging fluid from the upper part of the cavityoccupies smaller section of the opening. It is

interesting to note in case of Ra = 106, the

formation of vortex in the upper portion of the

cavity due to the high heat transfer rate, also theeffect of the open side is quite significant in thiscase.

8/11/2019 56326_2

6/14


86

(a) Pr = 0.71 Ra = 105 "= 0# H=1

(b) Pr = 0.71 Ra = 105 "= 10# H=1

Fig. 2. Comparison of Isotherms with those of (a) Mohamad et al.[10] and (b)Nateghi and Armfield [8].

8/11/2019 56326_2

7/14


87

Streamlines Isotherms

a)

H = 0.4

b) H = 0.6

c) H = 0.8

d)

H = 1Fig. 3. Streamlines and Isothermal Contours for "=0#, location of Active Part at Middle and Ra =105, withDifferent Lengths of Thermally Active Part.

1.6

1.6

1.61.6

3.2

3.2

3.2

3.2

4.8

4.8

4.8

4.8

6.4

6.4

6.4

6.4

8

8

8

9.6

9.6

9.6

11.2

11.2

11.2

12.8

12.8

14.4

0.1

0.1 0.1

0.

2

0.2 0.2

0.3

0.3 0.3

0.

4

0.4

0.4

0.

5

0.5

0.

6

0.6

0.

7

0.

8

0.

9

1.7

1.7

1.7

1.7

3.4

3.4

3.4

3.4

5.1

5.1

5.1

5.1

6.8

6.8

6.8

6.8

8.5

8.5

8.5

10.2

10.2

10.2

11.9

11.9

11.9

13.6

13.6

15.3

0.

1

0.10.1

0.

2

0.2 0.2

0.

3

0.3 0.3

0.

4

0.4 0.4

0.

5

0.5

0.5

0.

6

0.6

0.

7

0.7

0.

8

0.

9

1.8

1.8

1.8

1.8

3.6

3.6

3.6

3.6

5.4

5.4

5.4

5.4

7.2

7.2

7.2

7.2

9

9

9

9

10.8

10.8

10.8

12.6

12.6

12.6

14.4

14.4

16.2

0.10.1

0.

1

0.2

0.2

0.2

0.

3

0.30.3

0.

4

0.40.4

0.

5

0.5 0.5

0.

6

0.6

0.6

0.

7

0.7

0.

8

0.8

0.

9

0.

9

1.8

1.8

1.8

1.8

3.6

3.6

3.6

3.6

5.4

5.4

5.4

5.4

7.2

7.2

7.2

7.2

9

9

9

9

10.8

10.8

10.8

12.6

12.6

12.6

14.4

14

.4 14.4

16.2

0.10.1

0.1

0.2

0.2

0.

2

0.3

0.3

0.

3

0.40.4

0.

4

0.50.5

0.

5

0.5

0.

6

0.6

0.6

0.

7

0.7

0.

8

0.8

0.

9

0.9

$max= 15.4

$max= 16.5

$max= 17.5

$max= 18.0

8/11/2019 56326_2

8/14


88


a)

Bottom location

b) Middle location

c) Top location

Fig. 4. Streamlines and Isothermal Contours for "=0#, H=0.4 and Ra =105with Different Locations of ThermallyActive Part.

1.1

1.1

1.11 .1

2.2

2.2

2.22 .2

3.3

3.3

3.33.3

4.4

4. 4 4. 4

5.5

5.5

5.5

6.6

6.6

7.7

7.7

8.8

9.90.1 0 .1

0.2

0.2

0.3

0.3

0 .3

0.4

0.4

0.4

0.5

0.5

0.6

0.6

0.70 .8

0.9

1.6

1.6

1.61.6

3.2

3.2

3.2

3.2

4.8

4.8

4.8

4.8

6.4

6.4

6.4

6.4

8

8

8

9.6

9.6

9.6

11.2

11.211.2

12.8

12.8

14.4

0.

1

0. 1 0.1

0.

2

0.2 0.2

0.

3

0.3 0.3

0.

4

0.4

0.4

0.

5

0.5

0.

6

0.6

0.

7

0.

8

0.

9

1.8

1

.8

1.81.8

3.6

3.6

3.6

3.6

5.4

5.4

5.4

5.4

7.2

7.2

7.2

7.2

9

9

9

9

10.8

10.8

10 .8

12.6

12.6

12.6

14.4

14.4

14 .4

16.2

16. 2

0.1

0.1

0

.1

0.2

0.2

0.

2

0.

3

0.3

0.3

0

.4

0.4

0.

5

0.

6

0.

7

0 .

8

0 .

9

$max= 17.8

$max= 15.4

$max= 10.5

8/11/2019 56326_2

9/14

8/11/2019 56326_2

10/14


90

0.16

0.16

0.16

0.16

0.32

0.32

0.32

0.32

0.48

0.48

0.48

0.64

0.64

0.64

0.8

0.8

0.8

0.96

0.96

0.96

1.12

1.12

1.12

1.28

1.28

1.44

1.44

1.6

0.

1

0.2

0.2

0.3

0.

3

0.4

0.4

0.5

0.5

0.6

0.6

0.7

0.7

0.

8

0.

8

0.

9

0.9


a) Ra = 103

b) Ra = 104

c)

Ra = 105

d) Ra = 106

Fig. 6. Streamlines and Isotherms for "=0#, Location of Active Part at Middle and H =0.8, with Different Valuesof Rayleigh Number.

0.7

0.7

0.7 0.7

1.4

1.4

1.4 1.4

2.1

2.1

2.1 2.1

2.8

2.8

2.8

3.5

3.5

3.5

4.2

4.2

4.2

4.9

4.9

4.9

5.6

5.

6

5.6

6.3

6.3

0.1

0.1

0.2

0.2

0.2

0.3

0.3

0.

3

0.4

0.4

0.

4

0.5

0.5

0.

5

0.

6

0.6

0.6

0.

7

0.7

0.

8

0.8

0 .

9

0.9

1.8

1.8

1.8

1.8

3.6

3.6

3.6

3.6

5.4

5.4

5.4

5.4

7.2

7.2

7.2

7.2

9

9

9

9

10.8

10.8

10.8

12.6

12.6

12.6

14.4

14.4

16.2

0.10.1

0.

1

0.2

0.2

0.2

0.

3

0.30.3

0.

4

0.40.4

0.

5

0.5 0.5

0.

6

0.6

0.6

0.

7

0.7

0 .

8

0.8

0.

9

0.

9

3.6

3.

6

3.6

3.6

7.2

7.2

7.2

7.2

10.8

10.8

10.8

10.8

14.4

14.4

14.4

14.4

18

18

18

18

21.6

21.6

21.6

21.6

25.2

25.2

25.2

28.8

28.8

32.4

0.

1

0.1 0.1

0.

2

0.2 0.2

0.

3

0.

3

0.3 0.3

0.

4

0.

4 0.40.4

0.

5

0.5

0.5

0.5

0.

6

0

.6

0.6

0 .

7

0 .7

0.

8

0.

8

.

0.9

$max= 35.2

$max= 17.5

$max= 6.9

$max= 1.7

8/11/2019 56326_2

11/14


91

Fig. 7. Average Nusselt Number as a Function of Rayleigh Number for Different Dimensionless Lengths of

Active Part at "=0#and Location of Active Part at Middle.

Fig. 8. Average Nusselt Number as a Function of Rayleigh Number for Different Locations of Thermally Active

Part at "=0#and H=0.4.

Fig. 9. Average Nusselt Number as a Function of Rayleigh Number for Different Inclination Angles at H=0.4

and Location of Active Part at Middle.

103

104

105

106

0

2

4

6

8

10

12

14

16

18

20

Ra

Nu

H = 0.4

H = 0.6

H = 0.8

H = 1.0

103

104

105

106

0

1

2

3

4

5

6

7

8

9

10

Ra

Nu

Bottom position

Middle position

Top position

103

104

105

106

0

2

4

6

8

10

12

Ra

Nu

= 0

= 30

= 45

= 60

8/11/2019 56326_2

12/14


92

5. Conclusions

A steady, two-dimensional natural convectionheat transfer in a square open cavity with partially

active side wall has been studied numerically fordifferent angles of inclination, different Rayleigh

numbers and different lengths and locations ofthermally active part. The finite-differencemethod was employed for the solution of the

present problem. Graphical results in form ofstreamlines and isotherms for various parameters

governing heat transfer and fluid flow werepresented and discussed. The results of the presentstudy lead to the following conclusions:

Location and length of the active part, Rayleighnumber and inclination angle are important

parameters on flow and temperature fields and

heat transfer in square cavity.Heat transfer increases with the increasing ofactive part length or Rayleigh number. Values ofstream function (flow strength) also increase withthe increasing of Rayleigh number or active part

length.

Higher heat transfer rate was obtained when theactive part is at middle location in vertical wall

compared to top and bottom locations.

Increase of inclination angle increases flowstrength and decreases average Nusselt number.

Nomenclature

L length of the square cavityg gravitational accelerationhH

length of the active partdimensionless length of the active part

sS

position of the active part centerdimensionless position of the active part

hy local heat transfer coefficient

Nu local Nusselt number

Nu average Nusselt numberp pressureT dimensional temperaturePr Prandtl number (0/()Ra Rayleigh number (g*L7T/0()x,y

X,Y

dimensional Cartesian coordinates

dimensionless coordinatesu

Uv

dimensional velocity in the x-direction

dimensionless velocity in the X-directiondimensional velocity in the y-direction

V dimensionless velocity in the Y-direction

Greek symbols

( thermal diffusivity*&

thermal expansion coefficient

dynamics viscosity0

,

kinematic viscosity

dimensional vorticity. dimensionless vorticity+ dimensional stream-function- dimensionless stream-function/ dimensionless temperature% density7 laplacian in Cartesian coordinates

Subscripts

) ambient

in inflowout outflow

Superscripts

3 current iteration number3-1 previous iteration number

6. References

[1] A. M. Clausing, Convective losses from

cavity solar receivers: comparisons betweenanalytical prediction and experimentalresults, J. Solar energy 105 (1983) 29-33.

[2] Abdulkarim H. Abib, Yogesh Jaluria,penetrative convection in a partially openenclosure, HTD 198 ASME (1992) 74!83.

[3] S. S. Cha and K. J. Choi. An interferometricinvestigation of open cavity natural

convection heat transfer. Exp. J. Heat transfer2 (1989) 27!40.

[4] Y.L. Chan, C.L. Tien, A numerical study oftwo-dimensional natural convection in square

open cavities, Numer. Heat Transfer 8 (1985)65!80.

[5] Y.L. Chan, C.L. Tien, A numerical study oftwo-dimensional laminar natural convection

in shallow open cavities, Int. J. Heat MassTransfer 28 (1985) 603!612.

[6] O. Polat, E. Bilgen, Laminar naturalconvection in inclined open shallow cavities,Int. J. Thermal Sciences 41 (2002) 360!368.

[7] Chakroun, W., Effect of Boundary WallConditions on Heat Transfer for Fully

Opened Tilted Cavity, ASME J. HeatTransfer, 126 (2004) 915!923.

8/11/2019 56326_2

13/14


93

[8] M. Nateghi, S. W. Armfield, Naturalconvection flow of air in an inclined opencavity, ANZIAM Journal 45 (2004) 870!890.

[9] J.F. Hinojosa, R.E. Cabanillas, G. Alvarez,C.E. Estrada, Nusselt number for the natural

convection and surface thermal radiation in a

square tilted open cavity , Int.Communications in Heat and Mass Transfer32 (2005) 1184!1192.

[10] A.A. Mohamad a, M. El-Ganaoui b, R.Bennacer c, Lattice Boltzmann simulationof natural convection in an open endedcavity, Int. J. Thermal Sciences 48 (2009)1870!1875.

[11] N. Nithyadevi, P. Kandaswamy, J. Lee,Natural convection in a rectangular cavitywith partially active side walls, Int. J. Heat

Mass Transfer 50 (2007) 4688!4697.[12] P.J. Roach, Computational Fluid Dynamics,

Hermosa, Albuquerque, New Mexico,

1985.

8/11/2019 56326_2

14/14

"!$&'( )*,!**-/.012&*4,.06!7809;.01-/!8+*=.0>31"?@>94 - 81)2012(

94

%&'()*+,.0135675798:+7;=>:'&?08@AC&>D9*>E'7GQLJ>RT&5 !"#%&'(*,-/!/'(*0%&134/)&(6789":[email protected] 8:;=?@ABC;DF?H@:

:UVWK)

IJ@ V;b`iXbjBXl?@nb pqDrbAsrJ C tqsu n:_XMvbV

56326_2

Documents