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Applied Thermal Engineering 28 (2008) 1502–1513

Entropy production due to free convection in partially heatedisosceles triangular enclosures

Yasin Varol a,*, Hakan F. Oztop b, Ahmet Koca a

a Department of Mechanical Education, Firat University, 23119 Elazig, Turkeyb Department of Mechanical Engineering, Firat University, 23119 Elazig, Turkey

Received 24 April 2007; accepted 31 August 2007Available online 8 September 2007

Abstract

A numerical analysis of the entropy production has been performed due to natural convection heat transfer and fluid flow in isoscelestriangular enclosures with partially heated from below and symmetrically cooled from sloping walls. Governing equations are solved byfinite difference method. Governing parameters on flow and temperature fields are Rayleigh number (103

6 Ra 6 8.8 · 105), dimension-less length of heater (0.25 6 (‘ 0 = ‘/L) 6 1.0), dimensionless location of heater (0.25 6 (c 0 = c/L) 6 0.75) and inclination angle of slop-ping walls (30� 6 b 6 60�). Heat transfer results are presented in terms of local and mean Nusselt numbers (Nu) while entropy productionresults are shown with entropy production number (Ns) and Bejan number (Be). Isotherms, streamlines, contours of entropy productiondue to heat transfer and fluid friction irreversibility are plotted. It is observed that entropy production number increases but Bejan num-ber decreases with increasing of Rayleigh number. However, both entropy production due to heat transfer and fluid friction irreversibil-ity are affected by higher inclination angle of triangle and length of heater.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Natural convection; Entropy; Isosceles; Triangular cavities

1. Introduction

Natural convection heat transfer in triangular enclosurecan be applied for roof/attics of buildings or some prob-lems of electronical cooling. Right-angle triangular enclo-sure or isosceles enclosures are used in these kinds ofproblems. Solution of natural convection equations inright-angle triangular enclosures is widely investigated inearlier studies [1–6]. However, little work has been reportedon natural convection in isosceles triangular enclosures.Holtzman et al. [7] investigated the natural convection inisosceles triangular enclosures heated from below and sym-metrically cooled from above. They studied the problemboth numerically and experimentally to compare the flowvisualization with each other. They reported that a pitch-

1359-4311/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.applthermaleng.2007.08.013

* Corresponding author. Tel.: +90 424 237 0000x4219; fax: +90 424 2367064.

E-mail address: ysnvarol@gmail.com (Y. Varol).

fork bifurcation is observed in which two steady asymmet-ric mirror image solutions can be obtained. Ridouane andCampo [8] compared the isosceles, right-angled trianglesand rectangular enclosure to maximize the convection heattransport in cavities and minimize their sizes. Theyobserved that the airflow remains symmetric for the isosce-les triangular cavity even at high Grashof numbers. Karya-kin et al. [9] considered two cases to analyse the naturalconvection in isosceles triangular enclosures. In first case,the horizontal base is adiabatic, while the inclined wallsare isothermal (cold and hot); in the second, surfaces areisothermal (hot inclined surfaces and cold bottom). Theyobserved that, at high Grashof number, gradient regionsof the type of dynamic and thermal boundary layers areformed on all the solid surfaces and the temperature distri-bution in the central part of the enclosure approaches theconditions of fluid stratification. Recently, Ridouaneet al. [10] compared the solution of two different boundaryconditions for isosceles triangular enclosures. In first case,

Nomenclature

Be Bejan numberc 0 dimensionless location of heaterg gravitational acceleration, m/s2

Gr Grashof numberH maximum height of triangle, mL length of bottom wall, m‘ 0 dimensionless length of heaterNu Nusselt numberNs dimensionless entropy production numberPr Prandtl numberRa Rayleigh numberT temperature, Ku, v velocities, m/s

X, Y non-dimensional coordinates

Greek letters

t kinematic viscosity, m2/sh non-dimensional temperatureb sloping wall inclination angle of cavityX non-dimensional vorticitya thermal diffusivity, m2/sW non-dimensional streamfunction

Subscript

c coldh hot

Y. Varol et al. / Applied Thermal Engineering 28 (2008) 1502–1513 1503

the base is cooled and the two inclined walls are symmetri-cally heated and in the second case, the base is heated andthe two top inclined walls are cooled. They observed that inthe first case flow field is symmetric but in the second case itis non-symmetric.

Partially heated or cooled enclosures are simulated forsquare or rectangular enclosures in earlier studies by Aydinand Yang [11], Yucel and Turkoglu [12,13]. The effects ofpartial heating are also analysed by right-angle triangularenclosure by Koca et al. [3]. They investigated positionand length of heaters and coolers on flow and temperaturedistributions.

Entropy generation minimization is a method for mod-eling and optimizing of energy systems. It is result of sec-ond law of thermodynamics analysis. In earlier studiesrelated to the natural convection problem in isosceles trian-gular enclosures, only First-law of thermodynamics wasused. However, the method of entropy generation com-bines from the start the most important parameters of ther-modynamics, heat transfer, and fluid mechanics.Fundamentals of entropy generation are presented byBejan [14], San et al. [15], Rosen [16] and Narusawa [17].Mourad et al. [18] investigated the entropy productiondue to heat transfer, mass transfer and fluid friction insteady state for laminar double diffusive convection in aninclined square enclosure. Their results show that the totalentropy production increases with the thermal Grashofnumber and the buoyancy ratio for moderate Lewis num-bers. Erbay et al. [19] numerically analysed the entropyproduction during transient laminar natural convection ina square enclosure which is heated either completely orpartially from the left side wall and cooled from the oppo-site wall. They found that the active sites in the completelyheated case are at the left bottom corner of the heated walland the right top corner of the cooled wall at the samemagnitudes. In the case of partial heating, the active siteis observed at the top corner of the heated section. Yilbaset al. [20] analysed the entropy production due to natural

convection in an enclosure heated from bottom and cooledfrom top wall with insulated vertical walls. Baytas [21]made a study on the entropy production in an inclinedenclosure. He showed that entropy production stronglydepends on the inclination angle of the enclosure. Magher-bi et al. [22] analysed the time dependent entropyproduction in a differentially heated enclosure. Magnetohy-drodynamic (MHD) free convection and entropy produc-tion were investigated by Mahmud and Fraser [23]. Theyshowed that both Rayleigh number and Hartmann numberare strongly effective parameters on flow, heat transfer andentropy production. Recently, Dagtekin et al. [24] showedthe active and passive sites in a C-shaped enclosure usingentropy production method. They observed that geometri-cal parameters are strongly affects the fluid friction andheat transfer, namely entropy production.

The main objective of this study is to analyse the numer-ical results for entropy production in partially heated isos-celes triangular enclosures due to natural convection. Theabove literature survey clearly shows that entropy produc-tion method applied for natural convection in rectangular/square enclosure and isosceles triangular enclosures has notyet been encountered. Thus, the study is the first attemptand it will help to designer to prevent energy losses for athermal system in isosceles triangular enclosures.

2. Theoretical model

The considered schematical configuration is sketchedwith coordinates and boundary conditions in Fig. 1a. How-ever, Fig. 1b shows the grid arrangement for isosceles trian-gular enclosure and numerical solution. The height oftriangle is shown by H and length of bottom wall is depictedby L. The length of heater is given by ‘ and dimensionlesslength of heater is shown by ‘ 0 = (‘/L). The center oflocation of heater is c 0 = (c/L). Tilt angle of the inclinationof sloping walls is given by b (apex angle). The gravita-tional acceleration acts perpendicular to the bottom wall.

H

βu=0, v=0, Th

c

L

u=0, v=0 , 0y/T =∂∂Insulated wall

g

u=0, v=0, Tc

x

y Δx

Δy

Fig. 1. (a) Physical model of isosceles triangular enclosure and (b) grid arrangement.

1504 Y. Varol et al. / Applied Thermal Engineering 28 (2008) 1502–1513

Temperature of inclined walls is colder than heater whileremaining walls of the bottom wall are adiabatic. This situ-ation is associated with the flush mounted electronicalequipment.

2.1. Governing equations

The system was considered to be laminar, incompress-ible and steady-state. The dimension perpendicular to theplane of the diagram is assumed to be long enough, so thatthe airflows may be conceived with 2D motion. Dimension-less governing equations in streamline-vorticity form canbe obtained via introducing dimensionless variable asfollows:

X ¼ xL; Y ¼ y

L; W ¼ wPr

t; X ¼ xðLÞ2Pr

t; h ¼ T � T c

T h � T c

;

u ¼ owoy; v ¼ � ow

ox; x ¼ ov

ox� ou

oy

� �;

Ra ¼ bgðT h � T cÞL3Prt2

; Pr ¼ ta: ð1Þ

The simultaneous effects of the density, dynamic viscos-ity, specific heat capacity, and thermal conductivity of theair changing with temperature are assumed to be negligible.Thus, the standard Boussinesq approximation adopted andall fluid properties are assumed to be constant. Based onthe dimensionless variables above governing equationsusing vorticity–streamfunction formulation together withenergy equation, the Navier–Stokes equations for a Newto-nian fluid can be written as

� X ¼ o2W

oX 2þ o2W

oY 2ð2Þ

o2X

oX 2þ o

2X

oY 2¼ 1

ProWoY

oXoX� oW

oXoXoY

� �� Ra

ohoX

� �ð3Þ

o2h

oX 2þ o

2h

oY 2¼ oW

oYohoX� oW

oXohoY

ð4Þ

2.2. Equations for entropy production

Local entropy production is given according to localthermodynamic equilibrium with linear theory by [15–18]

S000gen ¼k

T 02

oTox

� �2

þ oToy

� �2" #

þ lT 0

2ouox

� �2

þ 2ovoy

� �2

þ ouoyþ ov

ox

� �2" #

ð5Þ

By using he same dimensionless parameters given in Eq.(1), dimensionless entropy generation is given as

N s ¼ohoX

� �2

þ ohoY

� �2" #

þ / 4o

2woXoY

� �2

þ o2w

oY 2� o

2w

oX 2

� �2" #

ð6Þ

In this equation, / is defined as irreversibility

/ ¼ lT 0

ka

LðDT Þ

� �2

ð7Þ

Y. Varol et al. / Applied Thermal Engineering 28 (2008) 1502–1513 1505

where / value is taken as 10�2 for all calculations. Finally,Bejan number (Be), which is the ratio of heat transfer irre-versibility to the total entropy production (Ns), can bemathematically expressed as

Be ¼ohoX

� �2 þ ohoY

� �2h i

N s

ð8Þ

This dimensionless number ranges from 0 to 1. Accord-ingly, Be = 1 is the limit at which the heat transfer irrevers-ibility dominates, Be = 0 is the opposite limit at which theirreversibility is dominated by fluid friction effects, and

0

500

1000

1500

2000

2500

0 2 4

R

Dim

ensi

onle

ss T

otal

Ent

ropy

Gen

erat

ion

Present StudyMagherbi et al. (200

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.E+03

Be

Present StudyMagherbi et al. (20

=1.E-2

0

10

20

30

40

50

60

0 0.2 0.4

Nux

Present study

Tzeng et al. (2005)

Asan and Namli (2000)

Akinsete and Coleman (1

Fig. 2. (a) Comparison of results with literature for dimensionless entropy geneand (c) comparison of results of local Nusselt numbers for triangular enclosur

Be = 0.5 is the case in which the heat transfer and fluid fric-tion entropy production rates are equal [21].

2.3. Boundary conditions

It is assumed that walls of enclosures are solid andimpermeable. Thus, all velocities and streamfunction arezero on the walls. The trapped air does not slip at the cavitywalls. The temperature boundary conditions are estab-lished with a prescribed hot temperature Th at the hot por-tion of the bottom wall and a prescribed cold temperature

6 8 10 12

a (1.E-4)

3) =1.E-2

=3.E-3

=1.E-3=1.E-4

1.E+04 1.E+05

Ra

03)

=1.E-3

=1.E-4

=1.E-5

0.6 0.8 1X

982)

ration number, (b) comparison of results with literature for Bejan numberes.

1506 Y. Varol et al. / Applied Thermal Engineering 28 (2008) 1502–1513

Tc at the cold sloping walls. On the other hand, the rest ofthe bottom wall is adiabatic. Thus, the physical boundaryconditions are illustrated in the physical model (Fig. 1a)and they can be defined as follows:

On the bottom wall; on the heater; U ¼ V ¼ 0; h ¼ 1;

on adiabatic wall; oh=oY ¼ 0 ð9ÞOn inclined walls; U ¼ V ¼ 0; h ¼ 0 ð10Þ

2.4. Heat transfer relations

The average heat transfer coefficient h, for natural con-vection can be represented in dimensionless form by theproper Nusselt number. Calculation of the local Nusseltnumber was performed over partially heated active wall by

max= 0.18 (x=0.2min=-1.01 (x=0.5

max= 0.71 (x=0.min=-8.53 (x=0.4

max= 6.07 (x=0.4min=-21.1 (x=0.5

Fig. 3. Streamlines (on the left) and isotherms (on the right) at b = 60�,

Nux ¼ �ohoY

����Y¼0

ð11Þ

and mean Nusselt number is given by

Nu ¼Z ‘

0

NuxdX ð12Þ

3. Numerical model

3.1. Procedure for numerical solution

The numerical method used in the present study is basedon finite difference method to discretize the governingequations (Eqs. (2)–(4)) and the set of algebraic equationsare solved using Successive Under Relaxation (SUR) tech-

0.1

0.3

0.5

0.70.9

0.1

0.3

0.5

0.7

0.9

0.1

0.3

0.7

0.9

0.5

20, y=0.155)60, y=0.236)

141, y=0.098)91, y=0.252)

93, y=0.525)14, y=0.221)

c 0=0.25, ‘ 0 = 0.50, (a) Ra = 104, (b) Ra = 105 and (c) Ra = 8.8 · 105.

Y. Varol et al. / Applied Thermal Engineering 28 (2008) 1502–1513 1507

nique. Thus, the velocity and temperature fields areobtained in the cavity. Central difference method is appliedfor discretization of equations. The convergence criterion ischosen as 10�4 for all depended variables and 0.1 is takenfor under-relaxation parameter. In the study, a regular gridis used and grid dimension was taken as 41 · 81. Rectangu-lar mesh size of Dx by Dy is presented in Fig. 1b. Theuppermost grid-point on each vertical grid line coincidedwith the top wall of the triangular enclosure. The inclinedwall was approximated with staircase-like zigzag lines. Thismethod was applied in earlier studies by Asan and Namli[1] and Varol et al. [5]. Based on earlier studies of Riduoaneet al. [10], Briggs and Jones [25], we limited our numericalcalculations of value of Rayleigh number 6 106.

Fig. 4. Entropy generation due to heat transfer (on the left) and friction (on(c) Ra = 8.8 · 105.

3.2. Validation of numerical code

For the case of differentially heated square enclosure,the numerical code was validated against the results ofthe benchmark results of Magherbi et al. [22], in terms ofdimensionless total entropy production and Bejan num-bers. Good agreement was obtained in the range of uand Rayleigh number as can be seen Fig. 2a and b. Also,the same code was tested against those obtained by Tzenget al. [26] by comparing the local Nusselt number forRa = 2772 with right-angle triangular enclosure. It is seenin this comparison that the agreement between theobtained results and literature was excellent as seen inFig. 2c.

the right) at b = 60�, c 0 = 0.25, ‘ 0 = 0.50, (a) Ra = 104, (b) Ra = 105 and

1508 Y. Varol et al. / Applied Thermal Engineering 28 (2008) 1502–1513

4. Results and discussion

A numerical analysis has been performed to show heattransfer, fluid flow and entropy production results due tonatural convection in partially heated isosceles triangularenclosures. The influencing parameters are Rayleigh num-ber (103

6 Ra 6 8.8 · 105), length of heater (0.25 6(‘ 0 = ‘/L) 6 1.0), location of heater (0.25 6 (c 0 = c/L) 6 0.75) and inclination angle of triangle (30� 6b 6 60�). The thermophysical properties of the air(Pr = 0.71) were evaluated at the reference temperature.Results obtained from the present code and literature wascompared in Fig. 2a–c to validate the code. Thus, resultswere tested with the results of benchmark study as differen-tially heated square cavity. As can be shown from the fig-ures that dimensionless entropy production number,Bejan number and heat transfer show good agreement withliterature. Maximum value of Rayleigh number wasrestricted to prevent three-dimensional and unsteady flowas suggested in the literature [10,25].

Flow and temperature fields are presented in terms ofstreamlines and isotherms, respectively. Fig. 3 presents

max= 16.4 (x=0min=-13.8 (x=0

max= 11.0 (x=0min=-23.2 (x=0

max= 8.83 (x=0min=-24.4 (x=0

max= 9.10 (x=0min=-24.5 (x=0

Fig. 5. Streamlines (on the left) and isotherms (on the right) at b = 45�, c 0 = 0.5

the streamlines (on the left) and isotherms (on theright) for different Rayleigh numbers and parameters ofb = 60�, c 0 = 0.25, and ‘ 0 = 0.50. At lower Rayleigh num-bers, double cells were formed in different rotation direc-tions and isotherms show semi-circle distribution due toconduction dominant heat transfer regime. For bothstreamlines and isotherms, 15 equally spaced contours havebeen used to define the corresponding structure. Withincreasing Rayleigh number, the intensity of the recircula-tion inside the enclosure also increases. This can be seenfrom the values of streamfunctions which is written top-right corner of the streamline figures. And, the isothermsare deformed with increasing temperature gradient andheat transfer due to convection mode begins to play a sig-nificant role with increasing the flow velocity. Multiple cellcenters and plumelike temperature distribution wereformed due to strong convection at the highest Rayleighnumber as Ra = 105 (Fig. 3b) and Ra = 8.8 · 105

(Fig. 3c). This results in a significant increment in the meanconvective heat transfer coefficient. Fig. 4 shows the effectsof Rayleigh number on entropy production due to heattransfer (on the left) and fluid flow irreversibility (FFI)

0.10.30.

5

0.70.9

0.10.3

0.5

0.7

0.9

0.10.30.5

0.7

0.9

0.10.3

0.5

0.7

0.9

.391, y=0.194) .642, y=0.180)

.286, y=0.130) .550, y=0.199)

.241, y=0.109) .535, y=0.191)

.241, y=0.111) .535, y=0.197)

0, Ra = 8.8 · 105, (a) ‘ 0 = 0.25, (b) ‘ 0 = 0.50, (c) ‘ 0 = 0.75 and (d) ‘ 0 = 1.0.

Y. Varol et al. / Applied Thermal Engineering 28 (2008) 1502–1513 1509

(on the right). Entropy production due to FFI spreads allover the cavity but entropy production due to heat transferis active only near the heater. Entropy generates at a highermagnitude near the cavity walls where a large temperaturegradient occurs. It may be observed from entropy produc-tion contours that as the wall temperature increases,entropy production increases due to an enhancement inthe circulation and temperature gradient depends on con-centration of temperature gradient. Entropy productiondue to viscous effect increases with the buoyancy ratio.Since all of the walls are no-slip boundaries, the contoursof the entropy production due to fluid friction irreversibil-ity are observed at the center of sloping and bottom walls.

Fig. 5 shows the effects of length of heaters on flow andtemperature fields for b = 45�, c 0 = 0.50 and Ra =8.8 · 105. This illustration is very similar to study of Aydinand Yang [11] in the case of square enclosure. Mushroomshaped temperature distribution is formed and it showssymmetrical behavior in the case of the smallest heaterlength. As heater length increases, the right cell becomesbigger and it rotates in clockwise direction. Plumelike dis-

Fig. 6. Entropy generation due to heat transfer (on the left) and friction ((c) ‘ 0 = 0.75 and (d) ‘ 0 = 1.0.

tribution of temperature moves toward to the left slopingwall. As can be seen form the figure that absolute valuesof maximum streamfunction decreases with increasing ofheater length. On the contrary, values of minimum stream-function are increased. For the case of isosceles triangularenclosure with fully heated from the bottom wall, a hugemain cell and a small cell were formed. This result alsoshows us a comparison with benchmark literature withHoltzman et al. [7] and Ridouane et al. [10]. Fig. 6 showsentropy production contours for different heat length asconsidered in Fig. 5. Thus, entropy production due to heattransfer (on the left) and fluid friction irreversibility (on theright) are presented in the figure. Both flow strength andheat transfer increase with increasing of heater length. Thiscorresponds also to the fact that when the length of heaterincreases, the convective flow is enhanced and the resultingheat transfer increases generating an increase of entropyproduction.

Effects of angle of sloping walls (b, apex angle) on flowand temperature distribution are presented in Fig. 7 forRa = 105, c 0 = 0.75 and ‘ 0 = 0.25. Double circulation cells

on the right) at b = 45�, c 0 = 0.5, Ra = 106, (a) ‘ 0 = 0.25, (b) ‘ 0 = 0.50,

Fig. 7. Streamlines (on the left) and isotherms (on the right) at Ra = 105, c 0 = 0.75, ‘ 0 = 0.25, (a) b = 30�, (b) b = 45� and (c) b = 60�.

Fig. 8. Generation due to heat transfer (on the left) and friction (on the right) at Ra = 105, c 0 = 0.75, ‘ 0 = 0.25, (a) b = 30�, (b) b = 45� and (c) b = 60�.

1510 Y. Varol et al. / Applied Thermal Engineering 28 (2008) 1502–1513

Y. Varol et al. / Applied Thermal Engineering 28 (2008) 1502–1513 1511

were formed for the cases of b = 30� and 45�. Main cellrotates in counterclockwise and it becomes bigger as valueof b increases. For the smallest value of b, conductionmode of heat transfer is effective due to short distancebetween hot and cold walls. However, both multiple cellswere formed and plumelike temperature distribution begins

010002000300040005000600070008000

1.0E+04 1.0E+05 1.0E+06

Ra

Ns

′ =0.25, c'=0.50 ′ =0.50, c'=0.50 ′ =0.75, c'=0.50 ′ =1.00, c'=0.50

0

0.2

0.4

0.6

0.8

1

1.0E+04 1.0E+05 1.0E+06

Ra

Be

′ =0.25, c'=0.50′ =0.50, c'=0.50′ =0.75, c'=0.50′ =1.00, c'=0.50

Fig. 9. (a) Variation of dimensionless entropy production (Ns) with Rayleigh nuof heater, (b) Variation of Bejan Number (Be) with Rayleigh Number at b =

0

4000

8000

12000

16000

20000

1.0E+04 1.0E+05 1.0E+06

Ra

Ns

β=30º

β=45º

β=60º

0.0

0.2

0.4

0.6

0.8

1.0

1.0E+04 1.0E+06

Ra

Be

β=30º

β=45º

β=60º

1.0E+05

Fig. 10. (a) Variation of dimensionless entropy production (Ns) with Rayleighright) for different sloping wall inclination angle, (b) Variation of Bejan Num‘ 0 = 1.0, c 0 = 0.50 (on the right) for different sloping wall inclination angle.

at b = 60�. Almost, left half of the enclosure becomes colddue to higher distance to the heater. A small cell sits to thetop of the enclosure and it rotates in clockwise directions.The coordinates of the main cell becomes constant atx-direction and it goes to the top of the corner with increas-ing of sloping wall inclination angle. Flow strength also

0

6000

12000

18000

24000

30000

1.0E+04 1.0E+06

Ns

Ra

′ =0.25, c'=0.50 ′ =0.50, c'=0.50 ′ =0.75, c'=0.50 ′ =1.00, c'=0.50

0

0.2

0.4

0.6

0.8

1

1.0E+04 1.0E+05 1.0E+06

Ra

Be

′ =0.25, c'=0.50 ′ =0.50, c'=0.50 ′ =0.75, c'=0.50 ′ =1.00, c'=0.50

1.0E+05

mber at b = 30� (on the left) and b = 60� (on the right) for different length30� (on the left) and b = 60� (on the right) for different length of heater.

0

6000

12000

18000

24000

30000

1.0E+04 1.0E+05 1.0E+06

Ra

Ns

β=30º

β=45º

β=60º

0.0

0.2

0.4

0.6

0.8

1.0.

1.0E+04 1.0E+05 1.0E+06

Ra

Be

β=30º

β=45º

β=60º

number at ‘ 0 = 0.50, c 0 = 0.50 (on the left) and ‘ 0 = 1.0, c 0 = 0.50 (on theber (Be) with Rayleigh number at ‘ 0 = 0.50, c 0 = 0.50 (on the left) and

0

4

8

12

16

20

1.0E+04 1.0E+05 1.0E+06

Ra

Nu

′ =0.25, c'=0.50

′ =0.50, c'=0.50

′ =0.75, c'=0.50

′ =1.00, c'=0.50

0

4

8

12

16

20

1.0E+04 1.0E+05 1.0E+06

Ra

Nu

′=0.25, c'=0.50

′=0.75, c'=0.50

′ =0.50, c'=0.50

′=1.00, c'=0.50

0

4

8

12

16

20

1.0E+04 1.0E+05 1.0E+06

Ra

Nu

β=30º

β=45º

β=60º

Fig. 11. Variation of mean Nusselt numbers with Rayleigh number (a)effects of length of heater at b = 30�, (b) effects of length of heater atb = 60� and (c) effects of inclination angle at ‘ 0 = 1.0, c 0 = 0.50.

1512 Y. Varol et al. / Applied Thermal Engineering 28 (2008) 1502–1513

increases with increasing of b. In the similar manner,entropy production contours are shown in Fig. 8. Figureshows that inclination angle of sloping wall are more effec-tive on entropy production due to fluid friction irreversibil-ity than heat transfer. As inclination angle increases activesides locate near the middle of the walls. However, entropygenerates at the edges of the heater. Fig. 9 shows the vari-ation of entropy production number (Fig. 9a) and Bejannumber (Fig. 9b) with Rayleigh number for different incli-nation angle of sloping wall as b = 30�(on the left column)and b = 60� (on the right column). As can be seen from thefigures that dimensionless entropy production values arealmost zero due to domination of conduction mode of heattransfer. However, length of heater is more effective onhigher values of inclination angle of sloping wall (b) dueto domination of effects of convection. Thus, higher dimen-sionless entropy production was formed with increasing offluid velocity. In both cases, values of dimensionlessentropy production increase exponentially with increasingof Rayleigh number. As we indicated above that the Bejannumber is a measure of the relative magnitude of the heattransfer and fluid friction irreversibility as described in Eq.(8). Bejan number decreases with increasing of Rayleighnumber, rapidly. This is caused by the onset of convectionheat transfer as indicated by Magherbi et al. [22]. The dom-ination of one of the two different effects as thermal or vis-cous on total entropy production is highly depends on theRayleigh number and the irreversibility distribution ratio.For a given value of irreversibility distribution ratio, theBejan number decreases with increasing Rayleigh numbertherefore viscous effects irreversibility becomes significantand begins to dominate heat transfer irreversibility. Also,length of heater, which is directly related with heat transfersurface, is more significant on variation of Bejan number.As seen in Fig. 9b (left figure) that Be = 1 for Ra = 104,‘ 0 = 1.0 and b = 30�is the limit at which the heat transferirreversibility dominates. For higher length of heater, itbecomes smaller due to increasing of fluid frictionirreversibility.

Effects of all sloping wall angle studied in this work ondimensionless entropy production is given at ‘ 0 = 0.5 (onthe left) and ‘ 0 = 1.0 (on the right) for different Rayleighnumber in Fig. 10. As illustrated in the figure, higherdimensionless entropy production was formed for highersloping wall inclination angle due to increasing of flowstrength as shown in Fig. 10a. On the contrary, Bejan num-ber decreases with decreasing of b in both value of lengthof heater (Fig. 10b). Thus, the sloping wall angle can beused as a parameter to control heat transfer, energy savingand entropy generation. Energy saving becomes moreimportant in low entropy generation cases.

Finally, effects of length of heater on heat transfer at dif-ferent Rayleigh number and inclination angle of slopingwall are presented in Fig. 11 via Nusselt number which iscalculated using Eq. (12). As expected, Nusselt numberincreases with increasing of Rayleigh number with increas-ing of convection dominated regime. As can be seen from

the figure, heat transfer mode is conduction at small Ray-leigh number especially at low sloping wall inclinationangle. It also increases with increasing of heat transfer area,namely heater length as shown in Fig. 11a and b. Heattransfer decreases with increasing of sloping wall inclina-tion angle due to increasing of distance of hot and coldwalls. As shown in Fig. 11c, heat transfer decreases withincreasing of inclination angle of sloping wall. As observedagain from this figure, conduction regime is dominant forlow inclination angles. Thus, the inclination angle can bea control parameter for heat transfer for fixed heater lengthand Rayleigh number.

5. Conclusions

Using finite difference method, the entropy productiondue to natural convection and flow in an partially heatedisosceles triangular enclosure was studied for differenceparameters of inclination angle of triangle, length of heaterand Rayleigh number. Some important conclusions fromstudied results can be listed as follows:

Y. Varol et al. / Applied Thermal Engineering 28 (2008) 1502–1513 1513

(1) Dimensionless entropy production number increaseswith increasing of Rayleigh number. It is observedthat dimensionless entropy production numberincreases with increasing of sloping wall inclinationangle of the triangular enclosure. The sloping wallangle can be used as a parameter to control heattransfer, energy saving and entropy generation.Energy saving becomes more important in lowentropy generation cases.

(2) An increasing in Rayleigh number decreases theBejan number. It also decreases with decreasing ofheater length and increasing of sloping wall inclina-tion angle.

(3) Nusselt number increases with increasing of Rayleighnumber and length of heater. For the smallest valueof sloping wall inclination angle, Nusselt numberbecomes almost constant with Rayleigh number.But it is increased with increasing of Rayleigh num-ber for higher inclination angle.

(4) Double and multiple circulation cells were formed atlower and higher Rayleigh number, respectively.

(5) Entropy production contours showed that active citesof the enclosure is isothermal walls. They cumulatedmiddle of the sloping walls in any case.

A future work on this problem can be numerical analy-ses of unsteady turbulent flow. An optimization study canbe performed by using more governing parameters. Both ofthem are out of the scope of this work.

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