Heat transfer engineering an international journal, tập 31, số 5, 2010

As a result, this article presents a two-dimensional 2D numerical study that investigates the inﬂuence of steady thermal Marangoni convection on the ﬂuid dynamics and heat transfer aroun

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CopyrightTaylor and Francis Group, LLC

ISSN: 0145-7632 print / 1521-0537 online

DOI: 10.1080/01457630903359784

Mixed Convective Heat Transfer Due

to Forced and Thermocapillary Flow Around Bubbles in a Miniature

Channel: A 2D Numerical Study

CRISTINA RADULESCU and ANTHONY J ROBINSON

Department of Mechanical and Manufacturing Engineering, Trinity College Dublin, Ireland

Marangoni thermocapillary convection and its contribution to heat transfer during boiling has been the subject of some

debate in the literature Currently, for certain conditions, such as microgravity boiling, it has been shown that Marangoni

thermocapillary convection has a signiﬁcant contribution to heat transfer Typically, this phenomenon is investigated for the

idealized case of an isolated and stationary bubble resting on a heated surface, which is immersed in a semi-inﬁnite quiescent

ﬂuid or within a two-dimensional cavity However, little information is available with regard to Marangoni heat transfer in

miniature conﬁned channels in the presence of a cross ﬂow As a result, this article presents a two-dimensional (2D) numerical

study that investigates the inﬂuence of steady thermal Marangoni convection on the ﬂuid dynamics and heat transfer around

a bubble during laminar ﬂow of water in a miniature channel with the view of developing a reﬁned understanding of boiling

heat transfer for such a conﬁguration This mixed convection problem is investigated under microgravity conditions for

channel Reynolds numbers in the range of 0 to 500 at liquid inlet velocities between 0.01 m/s and 0.0 5m/s and Marangoni

numbers in the range of 0 to 17,114 It is concluded that thermocapillary ﬂow may have a signiﬁcant impact on heat transfer

enhancement The simulations predict an average increase of 35% in heat ﬂux at the downstream region of the bubble, while

an average 60% increase is obtained at the front region of the bubble where mixed convective heat transfer takes place due

to forced and thermocapillary ﬂow.

INTRODUCTION

Based on the experimental results published in 1855 by

Thomson [1], Marangoni [2] later offered a viable

explana-tion of the effect of surface tension on drops of one liquid

spreading upon another Subsequent to this several numerical

and experimental studies [3, 4] established that thermocapillary–

Marangoni convection is a physical phenomenon that takes place

at gas–liquid and liquid–liquid interfaces The thermocapillary

ﬂow that forms is the result of surface tension gradients, which

can be brought about by variations in the liquid concentration

or temperature Once the existence of this phenomenon was

conﬁrmed, the main focus of the scientiﬁc research work was

We gratefully acknowledge the support from the Science Foundation Ireland

that sponsored this research.

Address correspondence to Dr Anthony Robinson, Department of

Mechan-ical and Manufacturing Engineering, Trinity College Dublin, Parsons Building,

Dublin 2, Dublin, Ireland E-mail: arobins@tcd.ie

to quantify the impact on heat transfer enhancement Startingwith the experimental results of McGrew et al [5] and followed

by the early numerical work of Larkin [6], thermal Marangoniconvection and its contribution to heat transfer during boilingbecame the subject of some debate in the open literature [7] Re-cently, it has been established that for certain conditions, such

as microgravity boiling, the thermocapillary induced ﬂow hasassociated with it a signiﬁcant enhancement of heat transfer due

to the liquid flow in the vicinity of the bubble interface [8–10] Despite the research conclusions presented in the literaturethere is still insufficient information available with regard toMarangoni flow contribution on the heat transfer in miniatureconfined channels [11, 12], especially when it takes place in thepresence of a cross flow under microgravity conditions To thebest of our knowledge, this configuration has only been inves-tigated by Bhunia and Kamotani [13] Their numerical study isfocused on the fluid motion due to thermocapillary flow around

a bubble situated on the heated wall of a channel The ﬂuidmechanics aspects of the problem were characterized based on

335

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336 C RADULESCU AND A J ROBINSON

Reynolds number (Rσ) and velocity (Vσ) due to surface

ten-sion effect As a result, with this work as a reference, our study

aims to present supplementary information regarding the heat

transfer enhancement based on quantifying heat ﬂux distribution

around the bubble The work is focused on the mixed convective

heat transfer due to forced and thermocapillary ﬂow It is aimed

to offer a baseline case for comparison and together with the

generalized mathematical formulation presented by Bhunia and

Kamotani [13] to provide information related to the ﬂuid ﬂow

and to quantify the impact on heat transfer enhancement due to

the thermocapillary effect The particular conﬁguration selected

for our investigation is the laminar ﬂow of water in a miniature

channel at low liquid inlet velocities 0.01 m/s≤ Vavg ≤ 0.05

m/s for 100≤ Re ≤ 500 as deﬁned in the literature by Shah et

al [14] A single isolated bubble is located near the entrance

and the heat transfer enhancement due to thermocapillary ﬂow

is quantiﬁed taking into account the conﬁnement effects The

cases under study have a signiﬁcant importance for a wide range

of applications where boiling heat transfer needs to be clearly

understood

Boiling heat transfer in minichannels has become an

increas-ingly important topic due to its application in the compact heat

exchanger design such as those required for electronics thermal

management or for miniature power generators, to give just two

examples Typically, nucleate boiling is the preferred regime

of operation for such applications because the small increase

in wall superheat is accompanied by a disproportionately large

increase in the wall heat ﬂux [15, 16] Apart from the high

rates of heat transfer at relatively low volumetric ﬂow rates, the

isothermal nature of two-phase convective boiling makes this a

very attractive technology in contrast with single-phase channel

cooling

The objective of this article is to provide qualitative and

quantitative information regarding the ﬂuid motion and the

in-ﬂuence on heat transfer enhancement due to thermocapillary

ﬂow around the centerline of a bubble placed on the bottom

heated wall of a rectangular-section minichannel (1× 20 mm)

in a cross-ﬂow conﬁguration as illustrated in Figure 1 This

mixed convection problem is investigated for laminar ﬂow of

water for increasing the inlet mass ﬂow rate For a ﬁxed inlet

temperature the Marangoni number (Ma) is varied in the range of

0≤ Ma ≤ 17,114 by increasing the temperature of the channel

bottom heated wall Furthermore, the inﬂuence of the bubble

dimension on the ﬂow pattern and heat transfer is taken into

con-sideration for the following geometrical characteristics: Rb/H=

0.1 (B1), Rb/H= 0.5 (B2), and Rb/H= 0.75 (B3)

Figure 1 Physical domain showing a hemispherical bubble near the entrance

of the miniature channel.

MATHEMATICAL FORMULATION

Figure 1 shows the simplified schematic of the channelthrough which water at a mean inlet temperature Tmflows withthe average velocity Vavg The flow is assumed to be hydrody-namically fully developed at the inlet with a parabolic velocityprofile The heated bottom wall is maintained at a constant tem-perature (Twall), while the top wall is considered to be insulated.The hemispherical gas bubble is situated on the heated wall,creating a cross-flow configuration due to the bulk liquid flowdirected perpendicular to the bubble axis This is located nearthe entrance of the channel, since this is the expected region ofthe nucleate boiling flow regime [17], with stratified or slug flowregimes being more likely downstream It was concluded thatthe bubble nucleus grows slowly to visible size in the laminarinlet flow [17] As a result, the flow and heat transfer problemhas been simplified by considering steady-state conditions forthree different dimensions of the bubble, starting with the incip-ient stage of growth and at the final stage before bubble sliding

is anticipated [11] The bubble shape deformation is neglected

as the capillary number Ca= µVavg /σ T is much less than unity[13] and also taking into consideration the imposed low bulkliquid inlet velocities

Lastly, it is well known that the ﬂow around the bubble within

a small channel is inherently a three-dimensional problem ever, at the mid-plane of the spherical bubble the ﬂow is approx-imately two-dimensional (2D) In this respect the problem can

How-be treated qualitatively as a two-dimensional phenomenon sistent with the previous work of Bhunia and Kamotani [13]

at a constant temperature, (vi) gravitational effects are ble, (vii) apart from surface tension, temperature variations inphysical properties are not considered, (viii) the liquid is incom-pressible, and (ix) the no-slip condition is applied to all surfacesheat transfer engineering vol 31 no 5 2010

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negligi-except the Marangoni stress boundary condition for the bubble

[6, 22, 23]; the shear stress applied at its interface is assumed to

be balanced by viscous effect as quantiﬁed in Eq (4):

u · n = 0µ ∂u ∂n =∂T∂σ∇S T (4)Here,n is the unit vector normal to the bubble interface and

∇S T is the temperature gradient.

The governing equations were solved numerically with the

computational ﬂuid dynamics (CFD) package Fluent Version

6.3.26 [18, 24], and the physical domain and grid were created

in Gambit Version 2.2.30 [19] Cartesian coordinates were used

with a nonuniform grid of 14,400 cells In order to resolve the

ﬂow and temperature ﬁelds accurately, grid clustering near the

bubble was implemented The accuracy of the resulting

simula-tions has been conﬁrmed by reproducing the numerical results

of Bhunia and Kamotani [13] and Radulescu and Robinson [25],

as well as assuring that the solutions were grid independent

RESULTS AND DISCUSSION

The results presented in this study have been carried out

for water, which has a surface tension gradient of dσ/dT = –

0.1477× 10−3N/mK The problem is investigated for channel

Reynolds numbers in the range of 0≤ Re ≤ 500 by increasing

the inlet average velocity for 0.01 m/s ≤ Vavg ≤ 0.05 m/s

The Marangoni number is in the range of 50≤ Ma ≤ 17,114

obtained due to variations of the difference between the liquid

inlet temperature and the wall temperature of the channel (T =

Twall– Tm) for 1◦C≤ T ≤ 30◦C Consistent with [9], [13],

and [22] the Marangoni number has been deﬁned as:

To approximate different stages of bubble growth, i.e.,

nucle-ation to bubble sliding [11], the bubble size relative to the

chan-nel height has been investigated for Rb/H= 0.1 (B1), Rb/H=

0.5 (B2), and Rb/H= 0.75 (B3)

The primary objective of this study is to provide a qualitative

description of the effect of thermocapillary convection on the

ﬂow ﬁeld and to quantify the heat transfer enhancement during

bubble growth in a miniature channel

The Effect of Marangoni Convection on the Flow Field

Figure 2 presents the streamlines for steady ﬂow around the

bubble for the case Rb/H= 0.1, Re = 100, and Ma = 0, 50,

100, and 300 To provide a baseline case for comparison, steady

ﬂow around a bubble with no thermocapillary effect has been

simulated for this test case and each test case to follow [26] This

is equivalent to imposing a constant surface tension (∂σ/∂T =

0) such that Ma= 0 even though there are temperature gradients

along the interface Due to these, the surface tension is highest

Figure 2 Streamlines of steady ﬂow for R b /H = 0.1 (B1) at Re = 100.

near the top of the bubble and lowest near the heated wall Thissurface tension variation generates thermocapillary ﬂow alongthe bubble surface, away from the hot wall toward the bulkliquid Figure 2 illustrates clearly that increasing the drivingpotential for thermocapillary ﬂow, which in this situation is

T = Twall – Tm, the inﬂuence of the surface tension drivenﬂow becomes stronger, which is apparent from the increaseddeformation of the streamlines as compared with the baseline

Ma= 0 case

Upstream (front) side of the bubble In this region the

ther-mocapillary action accelerates the liquid ﬂow along the bubblesurface The shear driven ﬂow at interface has the effect of draw-ing the relatively colder bulk liquid downward toward the frontcorner of the bubble, as apparent from the deformation of thenear-wall streamlines toward the hot front corner of the bubblefor the Ma= 50 and 100 cases

Downstream side of the bubble In this region a sizable

vor-tex is formed even at low Marangoni numbers (Ma= 50) whenthe recirculation cell is strong enough to cross over the line ofsymmetry of the bubble toward the front region For the highestMarangoni number obtained for Rb/H= 0.1 (i.e., Ma = 300)the high shear rate at the front bubble interface interacts with thestrong rear vortex to form a recirculation cell near front region

of the bubble With regard to the rear vortex itself, increasing

Ma has the effect of increasing the strength of the vortex, as

is evident from the higher concentration of the streamlines,

as well as increasing the vortex size, as it is seen to penetrateheat transfer engineering vol 31 no 5 2010

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Figure 3 Magnitude ofx-velocity and separation point position for B1, B3

at Re= 100 and T = 10◦C.

deeper into the bulk liquid and along the heated wall, as well as

moving forward along the bubble surface In order to visualize

the velocity magnitude on thex coordinate, Figure 3 shows the

case of Ma= 100 together with information for the geometrical

dimensions of the recirculation thermocapillary cell

Near the downstream end the surface velocity due to the bulk

liquid ﬂow is in an opposite direction, with the recirculation cell

created by the thermocapillary ﬂow The point on the bubble

surface where the forward and reverse ﬂow meet is known in

the literature as the separation point [13]

As is evident in Figure 2, and discussed later in more

de-tail, this separation point appears to move closer to the front

region of the bubble with increasing Ma Figure 3 presents the

inﬂuence of the bubble dimension (B1 and B3) on the

separa-tion point posisepara-tion (deﬁned by the angleβ) for the same Re =

100 and temperature differenceT = 10◦C It is noticed that

this point is situated closer to the front region of the bubble

(at higher separation angleβ) for the smaller bubble dimension

(B1)

Figure 4 illustrates the inﬂuence of inertial effects of the

channel ﬂow by considering the identical conﬁguration as in

Figure 2 but for the higher Reynolds number case of Re= 300

As one would expect, the increase in the cross-ﬂow velocity has

an important impact on the ﬂow structure by suppressing the

inﬂuence of the Marangoni ﬂow This is clear considering that

for Re= 300 it takes a Marangoni number of Ma = 300 to

roughly reproduce the ﬂow structure that a Marangoni number

of Ma= 100 was able to produce for Re = 100

With the view of future development of a dynamic bubble

growth model, it seemed instructive to investigate the inﬂuence

of the bubble size on the ﬂow and heat transfer within the channel

for this idealized case of steady two-dimensional ﬂow Figures

5 and 6 show the simulated ﬂow patterns around bubbles with

aspect ratios Rb/H = 0.5 and 0.75, respectively, for a ﬁxed

Reynolds number of Re= 100

Increasing the relative size of the bubble from Rb/H= 0.1

to Rb/H= 0.5 has a notable influence on the flow pattern, asevident from Figure 5 It must first be noted that the Ma numberincreases disproportionately compared with the increase inT

and Rb/H, since the length scale has been chosen as R2b/H for thisstudy to incorporate the influence of confinement on the flowand heat transfer Compared with the relatively unconfined casedepicted in Figure 2 for Rb/H= 0.1, the flow structure within theliquid for the more confined case of Rb/H= 0.5 indicates thatthe thermocapillary induced convection has a more profoundinfluence on flow for a like driving temperature differential and

a significantly higher Ma The presence of the confining upperwall tends to form elongated yet more concentrated recirculationzones at the downstream end of the bubble, compared with themore unconfined case at identicalT For T = 30◦C, Figure

5 shows that conﬁnement effects result in the formation of threerecirculation zones—in particular, an elongated vortex spanning

a considerable portion of the top region of the bubble interface.Figure 6 illustrates the extreme situation of Rb/H = 0.75where the bubble is at its maximum growth dimension beforesliding in the minichannel due to the inertial forces of the bulkliquid flow [11] It is noticed that the flow pattern is alterednotably compared with the previous two cases In this case theconfining and adiabatic top wall tends to restrict the rear vor-tex from encroaching on the frontal region with increasingT.

This pinching of the rear vortex is strong enough that for theheat transfer engineering vol 31 no 5 2010

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highest driving potential ofT = 30◦C no secondary

recircula-tion zones develop since the separarecircula-tion point is far enough back

on the bubble interface that it does not signiﬁcantly obstruct the

frontal shear ﬂow This is also illustrated in Figure 3, where it

is shown that the separation point on the bubble surface, where

the forward and reverse ﬂows meet, has the tendency to occur at

smaller separation angles,β, for the larger bubbles at identical

T and Re.

The Effect of Marangoni Convection on the Thermal Field

and Wall Heat Transfer

The ﬂow and thermal ﬁelds are directly coupled via the

non-linear convection term in Eq (3) and more indirectly through

the Marangoni stress boundary condition given in Eq (4) As

a result, the interaction of the ﬂow and thermal ﬁelds must be

understood in relation to one another in order to elucidate the

effect on less global parameters such as the stagnation angle,

bubble surface temperature distribution and resulting wall heat

transfer in the vicinity of the bubble

Figure 7 illustrates the thermal proﬁle for the Rb/H= 0.1,

Re= 100, and Ma = 0, 50, and 300 case For Ma = 0 the

Figure 6 Streamlines for steady ﬂow for R b /H = 0.75 (B3) at Re = 100.

presence of the bubble has a small influence on the thermal field,acting as a simple obstruction to the flow [26] However, forthe same wall temperature but with thermocapillary Marangoniconvection, a nonsymmetric jet of warm fluid is forced into thebulk of the flow The size of the warm jet is consistent with thesize of the vortices observed in Figure 2, and the penetrationdepth of the warm jet increases with Ma in the same way as thesize of the recirculation regions is increased in Figure 2 Near thefront edge of the bubble it is clearly evident that the deformation

of the streamlines observed in Figure 2 has associated with it thedrawing in of the cooler bulk liquid, which will have importantimplications with regard to the wall heat transfer

Figure 8a and b presents the wall heat flux distribution forRb/H= 0.1, Re = 100 and 300, with Ma = 30, 50, 100, and 200.The corresponding heat flux distribution for the situation of noMarangoni flow is also plotted for each temperature differential

It is clear that the thermal and ﬂow ﬁelds resulting from mocapillary convection have a direct impact on the heat transfer

ther-in the victher-inity of the bubble, as is evident from the peaks ther-in thewall heat ﬂux that appear around it

Upstream (front) side of the bubble In this region the

in-crease in the heat ﬂux is stronger due to the combined effect offorced and thermocapillary convection with the thermocapillarycomponent drawing the cooler bulk liquid toward the heatedwall and thinning the thermal boundary layer as depicted inheat transfer engineering vol 31 no 5 2010

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Figure 7 Temperature proﬁle for R b /H = 0.1 at Re = 100 and Ma = 0, 50,

and 300.

Figures 2, 4, and 7 For Re= 100 a two- to threefold increase

in the local heat ﬂux is evident As expected, the enhancement

decreases somewhat with increasing Re; however, it is still

sub-stantial

Downstream of the bubble The increase in the wall heat

transfer is only evident for the higher Ma number because it is

primarily the warm liquid that was extracted from the frontal

region that is being recirculated in this area Here the local heat

ﬂux increases by a factor of nearly 1.2 for Re= 100 and Ma =

200 and tends to improve with increasing Re, with a 1.4 times

improvement for Re= 300 and Ma = 200

The bubble dimensionless interface temperature distributions

for Rb/H= 0.1, Re = 100, and Ma = 0, 30, 50, 100, 200, and

300 are plotted in Figure 9 Considering the Ma= 0 and Ma =

50 cases, which both correspond withT = 5◦C, it is clear

that Marangoni convection tends to diminish the thermal

gra-dients over the majority of the bubble surface For Ma > 0

the general shape of the temperature proﬁles are similar In the

front region of the bubble the gradients are steep due to the

combined inﬂuences of the forced and thermocapillary

convec-Figure 8 Wall heat ﬂux distribution for B1 at Re = 100 and 300.

tion as cold bulk ﬂuid is drawn toward the surface and erated along it The surface temperature decreases to a mini-mum at the separation point Behind the separation point thetemperature gradients along the bubble surface are less steep

accel-as the ﬂow transitions from a combined convection region to adominantly thermocapillary-driven recirculation region wherethe average liquid temperature is generally much higher thanthe bulk liquid Increasing the Marangoni number has two no-table effects on the bubble surface temperature proﬁle First, the

Figure 9 Bubble B1 interface temperature for Re = 100 and Ma = 0, 30, 50,

100, 200, and 300.

heat transfer engineering vol 31 no 5 2010

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Figure 10 Temperature distribution for B2 at Ma = 0, 1260, and 7600.

Figure 11 Heat ﬂux distribution for R b /H = 0.1, 0.5, and 0.75 at Re = 100

andT = 10◦C.

stagnation point moves closer to the bubble front region with theincrease in Ma numbers However, further variations at a higher

Ma beyond Ma = 200 have a minimal effect on the position

of the separation point Second, increasing Ma tends to ﬂattenthe temperature proﬁle along the bubble interface, with largegradients isolated to the front and rear of the bubble

Figure 10 shows the effect of the top conﬁning wall on thethermal ﬁeld around the bubble for which Rb/H= 0.5 for thesameT values in Figure 7.

Compared with the small bubble, the confinement effectstend to keep the warm recirculation zone near the rear of thebubble with strong effects on the front region, where the shearflow causes entrainment of the cold bulk fluid toward the wall,which thins the thermal boundary layer in this region more sothan for the Rb/H= 0.1 case

Figure 11 illustrates this point more clearly where the heatﬂux proﬁles for Rb/H= 0.1, 0.5, and 0.75 are plotted for Re =

100 andT = 10◦C It is clear that the peak heat ﬂux at thefront of the bubble increases notably with increasing Rb/H Thepeak local heat transfer enhancement at the rear side also tends

to increase at higher Rb/H but to a much lesser extent

CONCLUSIONS

This article presents a two-dimensional numerical model thatinvestigates the influence of steady thermal Marangoni convec-tion on the fluid dynamics and heat transfer around a bubbleduring laminar flow of water in a rectangular minichannel Atthe downstream side of the bubble a sizable vortex is formedeven at low Marangoni numbers (T = 5◦C, i.e., Ma= 50for B1) The recirculation cell is strong enough to cross overthe line of symmetry of the bubble toward the front region Forthe high Marangoni numbers obtained at the maximumT =

30◦C under consideration (i.e., Ma = 300 for B1) the shearrate at the front bubble interface interacts with the strong rearvortex to form a recirculation cell near the front region of thebubble With regard to the rear vortex itself, increasing Mahas the effect of increasing the strength as well as the vortexsize, which penetrates deeper into the bulk liquid and alongthe heated wall as well as moving forward along the bubblesurface

It is concluded that thermocapillary flow has a significant pact on heat transfer enhancement for this configuration, with

im-an average increase of 35% in the heat flux figures at the stream of the bubble, while the mixed convective heat transferdue to forced and thermocapillary flow results in an average of60% increase at the front side of the bubble As presented, the 2Dnumerical approach employed in this work provides informationrelated with the flow pattern, thermal field, and heat flux at themid-plane of spherical bubbles of several dimensions Futurework will involve simulations which include three-dimensionaleffects as well as unsteady effects due to bubble growth.heat transfer engineering vol 31 no 5 2010

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down-342 C RADULESCU AND A J ROBINSON

NOMENCLATURE

B1 ﬁrst bubble under consideration

B2 second bubble under consideration

B3 third bubble under consideration

Ca capillary number

D h hydraulic diameter [14]

h Ma height of Ma recirculation cell [m]

H height of the channel [m]

l M length of Ma recirculation cell [m]

L length of the minichannel [m]

Ma Marangoni number (thermocapillary)

n unit normal vector (height of the boundary)

p static pressure [N/m2]

Pr Prandtl number

R b bubble radius (RB1, RB2, RB3radius of B1, B2, B3,

re-spectively) [m]

Re Reynolds number ρV avgµD h [14]

Rσ surface tension Re number ρl Vµref R b

l [13]

T temperature [◦C]

T m avg bulk liquid temperature at inlet [◦C]

T wall heated wall temperature [◦C]

σ liquid surface tension [N/m]

σT surface tension gradient [N/mK]

ρ density [kg/m3]

ν kinematic viscosity [m2/s]

∇ Laplace divergence operator

∇S T temperature gradient at the bubble interface

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[24] FLUENT User Guide, User Service Center, http://www.ﬂuent.

com

[25] Radulescu, C., and Robinson, A J., The Inﬂuence of Gravity

and Conﬁnement on Marangoni Flow and Heat Transfer Around

a Bubble in a Cavity: A Numerical Study, Microgravity

Sci-ence and Technology Journal, vol 20, no 3–4, pp 253–259,

2008

[26] Straub, J., The Role of Surface Tension for Two-Phase Heat and

Mass Transfer in the Absence of Gravity, Experimental Thermal

and Fluid Science, vol 9, no 3, pp 253–273, 1994.

Cristina Radulescu is a postdoctoral researcher at

Trinity College Dublin She received her Ph.D in

2005 from Galway-Mayo Institute of Technology, Ireland Her main research work at present is related to the mechanism of heat transfer during nucleate pool boiling and the inﬂuence of thermocapillary convection on heat transfer.

Anthony Robinson is a lecturer in ﬂuid

mechan-ics and heat transfer at Trinity College Dublin, land He received his Ph.D at McMaster University, Canada, in 2002 His research interests are in the ﬁeld

Ire-of two-phase ﬂow and heat transfer.

Trang 11

CopyrightC Taylor and Francis Group, LLC

DOI: 10.1080/01457630903359800

Frictional Pressure Drop for Gas

Flows in a Microchannel Between

Two Parallel Plates

METE AVC˙I and ORHAN AYD˙IN

Department of Mechanical Engineering, Karadeniz Technical University, Trabzon, Turkey

In this study, air flow through a microchannel between two parallel plates of height ranging from 100 to 710 µm was

investigated experimentally Each channel was made of Plexiglas and had a large cross-sectional aspect ratio to supply the

microplaneduct geometry The flow rate and pressure drop across the microchannel were measured at steady state to obtain

the friction factor The Reynolds number ranged from 30 to 2300 The experimental friction factor values were found in good

agreement with an existing analytical solution for an incompressible, fully developed, laminar flow under no-slip boundary

conditions.

INTRODUCTION

Microelectromechanical systems (MEMS) have gained a

great deal of interest in recent years Such small devices

typi-cally have characteristic size ranging from 1 mm to 1 µm, and

may include sensors, actuators, motors, pumps, turbines, gears,

ducts, and valves

The interest in the area of microchannel flow and heat transfer

has increased substantially during the last decade due to

devel-opments in the electronic industry, microfabrication

technolo-gies, biomedical engineering, etc In general, there also seems

to be shift in the focus of published articles, from descriptions

of the manufacturing technology to discussions of the physical

mechanisms of flow and heat transfer [1]

Readers are referred to see the following excellent reviews

related to transport phenomena in microchannels Ho and Tai [2]

summarized discrepancies between microchannel flow

behav-ior and macroscale Stokes flow theory Sobhan and Garimella

[3] and Obot [4] reviewed the experimental results in the

ex-The authors greatly acknowledge the financial support of this work by the

Scientific and Technological Research Council of Turkey (TUBITAK) under

grant 104M436 The second author of this article is also indebted to the Turkish

Academy of Sciences (TUBA) for the financial support provided under the

Programme to Reward Success Young Scientists (TUBA-GEBIT).

Address correspondence to Professor Orhan Aydın, Department of

Me-chanical Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey.

E-mail: oaydin@ktu.edu.tr

isting literature for the convective heat transfer in nels Rostami et al [5, 6] presented reviews for flow and heattransfer of liquids and gases in microchannels Gad-el-Hak [7]broadly surveyed available methodologies to model and com-pute transport phenomena within microdevices Guo and Li[8, 9] reviewed and discussed the size effects on microscalesingle-phase fluid flow and heat transfer Morini [10] presents

microchan-an excellent review of the experimental data for the tive heat transfer in microchannels in the existing literature

convec-He critically analyzed and compared the results in terms of thefriction factor, laminar-to-turbulent transition, and the Nusseltnumber

There is a scarcity of experimental data and theoretical ysis available in the existing literature, many of which are con-tradictory and conflicting, yielding different correlations withopposite characteristics Therefore, the mechanisms of flow andheat transfer in microchannels are still not understood clearly.Recently, Aydin and Avci [11–14] conducted a series of theoret-ical studies on various microgeometries Experimental studies

anal-on the microscale geometries have mostly used liquids as ing fluid, while studies investigating gas flows are comparativelyscarce Table 1 summarizes studies in the open literature, whichincludes the geometry, the hydraulic diameter, and the workingfluid used for each study It also includes the observation of eachstudy for the friction coefficient

work-To the authors’ best knowledge, this is the first study inthe existing literature investigating air flow in a microchannelbetween two parallel plates The aim of the study is focused on

344

Trang 12

Table 1 Summary of friction results of gas flow through microchannels

Authors Cross-section Dh [µm] Test fluid f < ftheory f ∼ = ftheory f > ftheory

Wu and Little [15] Trapezoidal 130–300 N 2 √ Pfahler et al [16] Rectangular 1.6–65 He, N 2 √

Choi et al [17] Circular 3–81 N 2 √

Arkilic et al [18] Rectangular 2.6 He √

Pong et al [19] Rectangular 1.94–2.33 He, N 2 √

Liu et al [20] Rectangular 2.33 He, N 2 √

Yu et al [21] Circular 19–102 N 2 √

Harley et al [22] Rectangular Trapezoidal 1.01–35.91 He, N 2 , Ar √

Shih et al [23] Rectangular 2.33 He, N 2 √

Li et al [24] Circular 128.8–179.8 N 2 √

Lalonde et al [25] Circular 52.8 Air √

Turner et al [26] Rectangular 4–100 Air, He, N 2 √

Araki et al [27] Trapezoidal 3–10 He, N 2 √

Yang et al [28] Circular 173–4010 Air √

Hsieh et al [29] Rectangular 80 N 2 √

Kohl et al [30] Rectangular 25–100 Air √

determining friction factors for various Reynolds numbers in

the laminar regime

EXPERIMENTAL APPARATUS AND PROCEDURE

A photograph of the experimental apparatus used in flow

resistance measurement of the parallel-plate microchannel is

shown in Figure 1 The experimental apparatus consists of a

compressor, an air tank, a filter, an air dryer, a pressure

regu-lator, a microfilter, a microvalve, a volumetric flow controller

with setpoint module, and a test section The working gas is

supplied from a compressor and it flows through an air dryer

to condition the relative humidity of the air at a proper level A

pressure regulator is connected to the air dryer, cutting off the

flow of the gas at a certain pressure To prevent the microchannel

from possible clogging by foreign materials in the gas a 0.1 µm

microfilter is installed between the microvalve and volumetric

flow controller The flow rate is measured by a high-precision

volumetric flow controller (CZ-32907-25), which uses the

lam-inar flow element (LFE) technology having ultrafast response

time (20 ms)

Figure 1 Photograph of (a) experimental apparatus and (b) test section.

For small values of Reynolds number, the volumetric flowrates were also controlled by versatile flow meters having ranges

of 0–1 LPM (FF-32460-42) and 0–10 LPM (EW-32460-46)

attached at the outlet plenum

The gas enters the test section from two holes drilled at thetop piece of the inlet plenum, passes through the parallel passage(microchannel), and exits from the outlet plenum, respectively.The pressure drop and the temperature of the gas through themicrochannel were measured by differential capacitance cellmanometers and K-type thermocouples installed at the inlet andoutlet plenums, respectively (see Figure 2b) Three differentranges of transducers were used to meet the pressure drop re-quirements in the experiment (WZ-68035-00, EW-10400-02,Comark C9505/IS) The characteristics and uncertainties of themeasurement instruments are given in Table 2

Test Section

Figure 2a and b, shows an exploded and a sectioned view ofthe test section, respectively All parts of the test section werefabricated by traditional mechanical machining technique from

Figure 2 Schematic diagram of the test section: (a) an exploded view, (b) a sectioned view, and (c) cross-section view of the microchannel.

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346 M AVC˙I AND O AYD˙IN

Table 2 Characteristics and uncertainties of the measurement instruments

Instrument Range Uncertainty

Volumetric flow controller (CZ-32907-25) 0–100 [L/min] ±0.8%∗

Plexiglas plate with a thickness of 5 mm The test section

con-sists of three regions, which are the inlet plenum, microchannel,

and outlet plenum

Basically, the microchannel is composed of two parallel

Plex-iglas plates divided by a pair of thickness gaugea at different

heights ranging from 100 to 710 µm Each channel has a large

cross-sectional aspect ratio, w/ h, to supply the microplaneduct

geometry The geometric parameters of the microchannel are

measured by a digital compass with a resolution of 1 µm and

are given in Table 3

A 1 mm thick gasket is placed on the Plexiglas covers to seal

the gaps between the microchannel and the other components of

the test section To measure the pressure drop and temperature,

two taps are installed at the top and the bottom surfaces of

the each plenum The surface roughness of the microchannel is

measured by a perthometer (Marsurf Perthometer M2)

Data Reduction

The laminar-to-turbulent flow transition in microchannels is

an important topic that was analyzed by a number of

investiga-tors Some of the initial studies indicated an early transition to

turbulent flow in microchannels However, a number of recent

studies showed that the laminar-to-turbulent transition in smooth

microchannels is not influenced by the channel dimensions and

remains unchanged (2000 < Re < 2300) The transition is

in-fluenced by the channel surface roughness [31]

In this study, the mean surface roughness of the each

mi-crochannel is about 0.09 µm Considering the relative roughness

is about 0.09 µm, the effect of the roughness on the transition

Reynolds number has been neglected

Table 3 Geometric parameters of microchannels

Length (mm), l± 0.05 Height Width (mm),

Note Average roughness of the microchannel surface (µm)= ±0.09.

The hydrodynamic development length of a smooth parallelplate duct for a fully developed laminar flow is given by [32]

l

D h = 0.315

0.0175 Re+ 1 + 0.011Re (1)

where l is the hydrodynamic development length and Re is the

Reynolds number based on the mean velocity, and hydraulic

diameter of the duct, D h , is equal to 2h Considering this fact,

the length of the channels tested in the study is ensured to behigher than the hydrodynamic development length (for instance,

for h = 710 µm and Re = 2300 the hydrodynamic development

length is nearly 36 mm)

The fully developed pressure drop through the microchannel

is calculated by the method described by Mala and Li [33] andCelata et al [34] According to this method, the fully developedpressure drop is given by

for a certain length, l (= l l − l s ), where l l and l s are thelengths of the long and the short channel, respectively Note thatthese two lengths are higher than the hydrodynamic develop-

ment length calculated based on Eq (1) P l and P s are thetotal pressure drops of the long and the short channel, includingthe same inlet, outlet, and developing section pressure lossesbetween the inlet and out plenums

The Poiseuille number, P o(= f Re), is given by

P o(= f Re) = P

l

D2

h 2µ U m

uncertain-by following the method described uncertain-by Holman and Gajda [35]

If the result R is a given function of the independent variables

x1, x2, x3, , xnas

R = R(x1 , x2, x3, , x n) (5)heat transfer engineering vol 31 no 5 2010

Trang 14

Table 4 Uncertainties of relevant parameters

Note The repeatability test deviation,±4%.

and u1 , u2, u3, , u nare the uncertainties in these independent

variables, the uncertainty in the result, u R, can be evaluated by

h

2+u w

w

2+u l

l

2+

In the same way, the uncertainty in pressure drop, P , and

Reynolds number, Re, can be expressed as, respectively,

w

2+u υ

υ

21/2

(9)

Using Eqs (7–9), the calculated uncertainties in Po, P,

and Re are given in Table 4 As can be noticed from Eq (7),

the experimental uncertainty in Po is dominated by the error in

the measurement of channel height h, since the uncertainty in

height is multiplied by a factor of 9 Therefore, the height of the

microchannel should be measured very precisely

Experiments were conducted for seven different heights of

the microplaneduct: 100 µm, 150 µm, 200 µm, 300 µm, 400 µm,

500 µm, and 710 µm The variations of the Poiseuille number,

Po, with the Reynolds number, Re, is determined for each of the

heights tested and is presented in Figure 3, a and b As seen, the

friction factors obtained agree well with the usual continuum

flow value defined for the macroscale case, which is Po= 24,

since data obtained are well distributed around this value This

observation confirms the applicability of the classical continuum

theory for the problem studied under the range covered The

magnitude of the compressible effects on the friction factor is

also tested for the smallest height (100 µm) microduct For Re=

30–1024, the Mach number (Ma) is changed in the range of

Figure 3 Influence of Reynolds number and channel height on Poiseuille number.

0.06–0.21 that the flow is assumed to be incompressible (Ma <

CONCLUSION

An experimental study on the fully developed flow resistance

in a microchannel between two parallel plates has been done.heat transfer engineering vol 31 no 5 2010

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348 M AVC˙I AND O AYD˙IN

The results show that the friction factors obtained agree well

with the usual continuum flow value defined for the macroscale

case, Po= 24, within ±10% for uncertainty

Po Poiseuille number (=f Re)

Q volumetric flow rate, m3/s

R function of independent variables

Re Reynolds number (=U m D h /ν)

u uncertainty in measured variable

l difference in channel length (=l l − l s), m

λ molecular mean free path, m

[1] Palm, B., Heat Transfer in Microchannels, Microscale

Thermo-physics, Eng., vol 5, pp 155–175, 2001.

[2] Ho, C M., and Tai, C Y., Micro-Electro-Mechanical Systems

(MEMS) and Fluid Flows, Annual Review of Fluid Mechanics,

vol 30, pp 579–612, 1998

[3] Sobhan, C B., and Garimella, S V., A Comparative Analysis

of Studies on Heat Transfer and Fluid Flow in Microchannels,

Microscale Thermophysical Engineering, vol 5, pp 293–311,

2001

[4] Obot, N T., Toward a Better Understanding of Friction and

Heat/Mass Transfer in Microchannels—A Literature Review,

Mi-croscale Thermophysical Engineering, vol 6, pp 155–173, 2002.

[5] Rostami, A A., Mujumdar, A S., and Saniei, N., Flow and Heat

Transfer for Gas Flowing in Microchannels: A Review, Heat and

Mass Transfer, vol 38, pp 359–367, 2002.

[6] Rostami, A A., Saniei, N., and Mujumdar, A S., Liquid Flow

and Heat Transfer in Microchannels: A Review, Heat Technology,

vol 18, pp 59–68, 2000

[7] Gad-el-Hak, M., Flow Physics in MEMS, M´ecanique &

indus-tries, vol 2, pp 313–341, 2001.

[8] Guo, Z Y., and Li, Z X., Size Effect on Single-Phase Channel

Flow and Heat Transfer at Microscale, International Journal of

Heat and Fluid Flow, vol 24, no 3, pp 284–298, 2003.

[9] Guo, Z Y., and Li, Z X., Size Effect on Microscale Single-Phase

Flow and Heat Transfer, International Journal of Heat and Mass

Transfer, vol 46, pp 59–149, 2003.

[10] Morini, G L., Single-Phase Convective Heat Transfer in

Mi-crochannels: A Review of Experimental Results, International

Journal of Thermal Science, vol 43, pp 631–651, 2004.

[11] Aydın, O., and Avcı, M., Analysis of Micro-Graetz Problem in

a Microtube, Nanoscale and Microscale Thermophysical

[16] Pfalher, J., Harley, J., Bau, H H., and Zemel, J N., Liquid and Gas

Transport in Small Channels, ASME DSC, vol 19, pp 149–157,

1990

[17] Choi, S B., Barron, R F., and Warrington, R O., Fluid Flow and

Heat Transfer in Microtubes, Micromechanical Sensor Actuator,

vol 32, pp 123–134, 1991

[18] Arkilic, E B., Breuer, K S., and Schmidt, M A., Gaseous Flow in

Microchannels, Applied Microfabrication and Fluid Mechanics,

[22] Harley, J., Huang, Y., Bau, H H., and Zemel, J N., Gas Flow in

Microchannels, Journal of Fluid Mechanics, vol 284, pp 257–

274, 1995

[23] Shih, J C., Ho, C M., Liu, J., and Tai, Y C., Monatomic and

Polyatomic Gas Flow through Uniform Microchannels, ASME

DSC, vol 59, pp 197–203, 1996.

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[24] Li, Z X., Du, D X., and Guo, Z Y., Characteristics of

Fric-tional Resistance for Gas Flow in Microtubes, Proceedings of

Symposium on Energy Engineering in the 21st Century, vol 2,

pp 658–664, 2000

[25] Lalonde, P., Colin, S., and Caen, R., Mesure de Debit de Gaz dans

les Microsystemes, M´ecanique & industries, vol 2, pp 355–362,

2001

[26] Turner, S E., Sun, H., Faghri, M., and Gregory, O J.,

Com-pressible Gas Flow through Smooth and Rough

Microchan-nels, Proceedings of IMECE New York, USA, HTD-24145,

2001

[27] Araki, T., Kim, M S., Iwai, H., and Suzuki, K., An Experimental

Investigation of Gaseous Flow Characteristics in Microchannels,

2002

[28] Yang, C Y., Wu, J C., Chien, H T., and Lu, S R.,

Fric-tion Characteristics of Water, R-134a and Air in Small Tubes,

2003

[29] Hsieh, S S., Tsai, H H., Lin, C Y., Huang, C F., and Chien,

C M., Gas Flow in a Long Microchannel, International Journal

of Heat Mass Transfer, vol 47, pp 3877–3887, 2004.

[30] Kohl, M J., Abdel-Khalik, S I., Jeter, S M., and Sadowski,

D L., An Experimental Investigation of Microchannel Flow with

Internal Pressure Measurements, International Journal of Heat

Mass Transfer, vol 48, pp 1518–1533, 2005.

[31] Kandlikar, S G, Garimella, S., Li, D., Colin, S., and King, M R.,

Heat Transfer and Fluid Flow in Minichannels and

Microchan-nels, Elsevier, Oxford, UK, 2006.

[32] Chen, R Y., Flow in the Entrance Region at Low Reynolds

Numbers, Journal of Fluids Engineering, vol 95, pp 153–158,

1973

[33] Mala, G M., and Li, D., Flow Characteristics of Water in

Mi-crotubes, International Journal of Heat Fluid Flow, vol 20,

[35] Holman, J P., and Gajda, W J., Experimental Methods for

Engi-neers, McGraw-Hill, New York, 1989.

Mete Avcı is an assistant professor of mechanical

engineering at Karadeniz Technical University, bzon, Turkey He received his Ph.D in mechanical engineering in 2008 from Karadeniz Technical Uni- versity His research interests include heat and mass transfer in microchannels, non-Newtonian fluid dynamics, and transport in porous media.

Tra-Orhan Aydın is a professor of mechanical

engi-neering at Karadeniz Technical University, Trabzon, Turkey His research interests cover microfluidics, electronics cooling, pulsating biological flows, heat and mass transfer, micropolar fluids, thermal energy storage, transport phenomena in porous media, non- Newtonian fluid dynamics, natural and mixed convection in enclosures, gas radiation, and computational fluid dynamics (CFD) He has co-authored more than

100 refereed journal and conference publications He

is the recipient of the Successful Young Scientist Reward from the Turkish Academy of Sciences (TUBA) and the Junior Science Award from the Scien- tific and Technological Research Council of Turkey (T¨uB˙ITAK).

Trang 17

DOI: 10.1080/01457630903373132

Experimental Investigations on

Natural Convection Heat Transfer

Around Horizontal Triangular Ducts

MOHAMED E ALI and HANY AL-ANSARY

Mechanical Engineering Department, King Saud University, Riyadh, Saudi Arabia

Experimental investigations have been reported on steady-state natural convection from the outer surfaces of horizontal

ducts with triangular cross sections in air Two different horizontal positions are considered; in the first position, the

vertex of the triangle faces up, while in the other position, the vertex faces down Five equilateral triangular cross-section

ducts have been used with cross-section side length of 0.044, 0.06, 0.08, 0.10, and 0.13 m The ducts are heated using

internal constant-heat-flux heating elements The temperatures along the surface and peripheral directions of the duct

wall are measured Longitudinal (perimeter-averaged) heat transfer coefficients along the side of each duct are obtained for

natural convection heat transfer Total overall averaged heat transfer coefficients are also obtained Longitudinal

(perimeter-averaged) Nusselt numbers and the modified Rayleigh numbers are evaluated and correlated using different characteristic

lengths Furthermore, total overall averaged Nusselt numbers are correlated with the modified Rayleigh numbers Moreover,

a dimensionless temperature group was developed and correlated with the modified Rayleigh number For the upward-facing

case, laminar and transition regimes are obtained and characterized However, for the downward-facing vertex case, only

the transition regime is observed The local (perimeter-averaged) or the overall total Nusselt numbers increase as the

modified Rayleigh numbers increase in the transition regime However, Nusselt numbers decrease as the modified Rayleigh

numbers increase in the laminar regime.

INTRODUCTION

Steady-state natural convection from triangular ducts has

many engineering applications, e.g., cooling of electronic

com-ponents, design of solar collectors, and heat exchangers Survey

of the literature shows that correlations for natural convection

from a vertical plate (McAdams [1] and Churchill and Chu [2]),

a horizontal surface (Goldstein et al [3] and Lloyd and Moran

[4]), a long horizontal cylinder (Morgan [5] and Churchill and

Chu [6]), and spheres (Churchill [7]) have been reported for

different thermal boundary conditions Recently, fluid flow and

heat transfer from an infinite circular cylinder has been reported

for both isothermal and isoflux boundary conditions in

Newto-nian and power-law fluids by Khan et al [8] and [9], respectively

This experimental investigation is supported by Saudi Arabian Basic

Indus-trial Company (SABIC) and the Research Center, College of Engineering at

King Saud University under project 22/427 This support is highly appreciated

and acknowledged.

Address correspondence to Professor Mohamed E Ali, Mechanical

Engi-neering Department, King Saud University, PO Box 800, Riyadh 11421, Saudi

Arabia E-mail: mali@ksu.edu.sa

Free convection simulation from an elliptic cylinder was ied by Badr and Shamsher [10] and by Mahfouz and Kocabiyik[11], and correlations for natural convection from helical coilswere reported by Ali [12–15] for different Prandtl numbers

stud-On the other hand, there are limited correlations available

in the literature for natural convection from the outer surface

of triangular ducts, which motivates the current investigation.The approximation method suggested by Raithby and Hollands[16] to predict heat transfer from cylinders of various crosssections and for wide ranges of Prandtl and Rayleigh numberswas simplified by Hassani [17] for two-dimensional cylinders ofarbitrary cross section The laminar free convection from a hor-izontal cylinder with cross section of arbitrary shape had beentheoretically analyzed for uniform surface temperature and uni-form surface heat flux by Nakamura and Asako [18] Nakamuraand Asako [18] also checked their analytical results by experi-ments for a short modified triangular prism in water However,their results showed that the experimental mean heat transfercoefficient is about 10 to 30% higher than their analytical value.Recently, Zeitoun and Ali [19] have reported numerical sim-ulations of natural convection heat transfer from isothermal

350

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horizontal rectangular cross-section ducts in air Their results

showed that as the aspect ratio increases, separation and

circu-lation occur on the top surface of the duct at fixed Rayleigh

num-ber, and the corresponding behavior has been observed through

the isotherms Zeitoun and Ali have also obtained a general

cor-relation using the aspect ratio as a parameter Most recently, Ali

[20] has reported an experimental study for natural convection

heat transfer from rectangular and square ducts in air His study

has shown that there are two distinct regimes of heat transfer:

laminar and transition These modes are fully reported and

cor-related using the modified Rayleigh numbers and the overall

correlation is also obtained

This article presents the results of an experimental

investiga-tion of natural convecinvestiga-tion heat transfer from the outer surface

of triangular ducts with their axis oriented horizontally The

study focuses on the determination of local axial

(perimeter-averaged) and overall averaged heat transfer coefficient in

di-mensionless form of Nusselt numbers Furthermore, general

correlations using Nusselt numbers as function of the modified

Rayleigh numbers are obtained A new dimensionless surface

temperature group was also obtained and correlated with the

modified Rayleigh number

EXPERIMENT SETUP AND PROCEDURE

Figure 1 shows a schematic cross-section view of the duct

(D) and the thermocouple locations in the longitudinal (axial)

direction (TCW) on three sides of the duct The ducts are

po-sitioned such that the vertex of the triangle faces down in one

position and faces up in the second position Five ducts are used

with equilateral cross-section side length of 0.044, 0.06, 0.08,

0.10, and 0.13 m, with the duct length being 1 m The ducts (D)

were made from steel (polished mild steel) An electrical

heat-ing element (H) (0.0066 m outer diameter) was inserted into the

center of the duct Bakelite end plates (Bk, thermal conductivity

= 0.15 W/mK [21]) that are 0.0206 m thick were attached at

both ends of each test duct (D) to reduce the rate of heat loss

from the duct ends

The surface temperature was measured at 11 points in the

longitudinal direction of each duct at the three equilateral

sur-faces as seen in Figure 1a Thirty-five calibrated chromel–alumel

(type K) self-adhesive thermocouples (0.3 s time response with

flattened bead) were affixed to the duct surfaces 0.1 m apart and

two of them were affixed to the outer surface of the Bakelite

end plates; one for each plate Two thermocouples (0.01 inch

or 0.25 mm diameter, one at each plate) were inserted through

the Bakelite end plates in the axial direction and leveled with its

inside surface as seen in Figure 1a The ambient air temperature

was measured by one more thermocouples mounted in the room

The duct was oriented horizontally using two vertical stands in

a room away from air conditioning and ventilation openings to

minimize any possible forced convection Those thermocouples

were connected to a 40 channel data acquisition system (DA),

DA TCW

Ac

VR W

(a)

D H Bk Power line

Figure 1 (a) Schematic of the experimental system showing the ple locations in the longitudinal (TCW) direction (see text for details) (b) Two randomly selected temperature time dependence signals at various distances on the duct surface showing steady-state condition.

thermocou-which in turn was connected to a computer where the measuredtemperatures were stored for further analysis

The input electrical power (Ac) to the heating element (H)

is controlled by a voltage regulator (VR) The power consumed

by the duct is measured by a wattmeter (W) and assumed to beuniformly distributed along the duct length The heat flux perunit surface area of the duct is calculated by dividing the con-sumed power (after deducting the heat loss by axial conductionthrough the Bakelite end plates) over the duct outer surface area.The input power to the duct is increased for each duct fromabout 2 to 680 W in two stages up to the duct dimensions Inthe first stage, the power is increased by increasing the voltage

in 4-V steps up to 26-V for possible laminar natural convectionregime However, in the second stage, which represents the tran-sition regime, the voltage is increased in 10-V steps These stepsare stopped once the surface temperature reaches 160◦C, whichrepresents the thermocouples’ limit As a result, the experiment

is repeated about 17 times for each duct to account for the ious input power levels Temperature measurements are takenheat transfer engineering vol 31 no 5 2010

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var-352 M E ALI AND H AL-ANSARY

after 2 h of setting a new input power to ensure that steady-state

conditions have been reached as shown in Figure 1b The

pro-cedure just outlined is used to generate natural convection heat

transfer data in air (Prandtl number≈0.69)

ANALYSES OF THE EXPERIMENT

The heat generated inside the duct wall dissipates from the

duct surface by convection and radiation in addition to the heat

lost by axial conduction through the Bakelite end plates:

EIP= Electrical input power = As(qc+ qr)+ ABkqBk (1)

where Asis the duct total surface area, ABkis the Bakelite

sur-face area normal to the direction of heat transfer by conduction

through the end plates, and qcand qrare the fraction of the heat

flux dissipated from the duct surface by convection and

radi-ation, respectively The heat flux lost by radiation (qr) and by

axial conduction through the Bakelite end plates (qBk) can be

calculated respectively by:

It should be noted that qr is estimated using the total overall

averaged surface temperature ¯T at each experimental run on a

given duct and ε is the surface emissivity of the duct, which

was taken to be 0.27 for polished mild steel [22] Measurements

show that the fraction of radiated heat transfer is about 16 to

20% of the total input power, while the axial conduction heat

lost through the Bakelite end plates is 0.5 to 1.1% In the second

term of Eq (2), TiBand ToBare the measured inside and outside

surface temperatures of the Bakelite end plates, respectively,

and kBk and δ represent the Bakelite thermal conductivity and

thickness, respectively The heat transferred by convection is

assumed to dissipate uniformly from the outer surface of the

duct It is also assumed that the duct’s surface behaves as a

gray surface such that the first part of Eq (2) can be used for

estimating the radiative heat lost

Axial (Perimeter-Averaged) Heat Transfer Coefficient

In this case the perimeter-averaged surface temperature at

any station x in the longitudinal direction for each constant heat

flux (run) is determined by:

Tx=3

j=1

where j represents the thermocouples in the perimeter direction

at any station x along the surface of the duct The arithmetic

mean surface temperature is calculated along the axial direction

for each run by:

θx = 0.5 (Tx + T∞), x = 1, 2, , 11 (4)

Therefore for each heat flux (run) there are 11 Tx longitudinaltemperature measurements Consequently, once the electricalinput power to the duct is measured, qr and qBk can be calcu-lated from Eq (2) and the corresponding value of qccan then

be calculated from Eq (1) Using this information, the axial(perimeter averaged) heat transfer coefficient hx can be calcu-lated from:

Tx− T∞, x= 1, 2, 3, , 11 (5)Hence, the nondimensional axial (perimeter averaged) Nusseltnumber and the modified Rayleigh number are obtained from:

tem-χ =(T x −T ∞ ) k

q c L c in a correlated form of Rayleigh number, as shown

in the Results section

Total Overall Averaged Heat Transfer Coefficient

In this case the perimeter-averaged heat transfer coefficient

hxis first evaluated at each station x as in Eq (5) and then theoverall longitudinal average ¯h1is obtained as:

¯h1=9

to be used in the equation

Therefore, each heat flux qcis represented by only one all averaged heat transfer coefficient, in contrast to the axialcase where qcis presented by 11 hxterms along the longitudi-nal direction given by Eq (5) All perimeter-averaged physicalproperties are first obtained at θx; then the overall averagedproperties are obtained the same way following Eq (7) Thenondimensional overall averaged Nusselt number and the mod-ified Rayleigh number are defined using either the duct length

over-as a characteristic length (L = 1 m) or one side of the lateral cross-section triangle Lc as a characteristic length asfollows:

Trang 20

In order to compare the present results with similar previously

published results, another way of averaged results using the

overall averaged temperature is also used In this way the

tem-perature is first perimeter-averaged following Eq (3), and then

the overall average temperature is obtained as:

¯

T=9

x=3

All physical properties are obtained in this case at the arithmetic

mean surface temperature,

Tmean= 0.5 ( ¯T + T∞) (10)The overall averaged heat transfer coefficient for constant heat

flux is determined from:

¯h2 = qc

and Nusselt and Rayleigh numbers using one side length of the

equilateral cross-section triangle Lc as a characteristic length

are defined in this case as:

NuLc = ¯h2Lc

k , RaLc=g β ( ¯T− T∞) L3

Experimental Uncertainty

In this section, the experimental uncertainty is to be estimated

for the calculated results on the basis of the uncertainties in

the primary measurements It should be mentioned that some

of the experiments are repeated more than twice to check the

calculated results and the general trends of the data, especially

in the laminar range of the experiment The error in measuring

the temperature, estimating the emissivity, and calculating the

surface area is±0.2◦C,±0.02, and ±0.003 m2, respectively

The accuracy in measuring the voltage is taken from the manual

of the wattmeter as 0.5% of reading±2 counts with a resolution

of 0.1 V, and the corresponding one for the current is 0.7% of

reading±5 counts +1 mA with a resolution of 1 mA

For each run, 40 scans of the temperature measurement are

made by the data acquisition system for each channel and the

mathematical average of these scans is obtained Furthermore,

since the input power, as mentioned earlier, has two stages, one

for laminar and the other for transition, then using the already

mentioned errors produces the maximum itemized uncertainties

of the calculated results shown in Table 1 for each range using

the method recommended by Moffat [23] Table 1 shows, in

general, that the uncertainty of the quantities in the laminar

regime is higher than that in the transition regime, which is

expected since both the input power and the temperature range

are very small

Table 1 Maximum percentage uncertainties of various quantities in the laminar and transition regimes

Transition range (%) Laminar range (%),

Quantity duct facing up Duct facing up Duct facing down EIP 3.49 2.73 2.74

Experimental data are obtained for triangular ducts orientedhorizontally in air Figure 2a and b shows the axial perimeter av-eraged surface temperature normalized by the ambient temper-ature t∞versus the dimensionless axial (longitudinal) distancealong the duct for some selected values of qcusing duct number

5 (Lc = 0.13 m) when the vertex of the triangle faces up Asseen in Figure 2b, the temperature distribution at low heat flux isleast affected by the end effects where the heat lost through theBakelite end plates is minimal Therefore, the thermal boundarylayer thickness is expected to be limited and the convection ve-locity to be small As the heat flux inside the duct increases, thesurface temperature increases and the end effects become morenoticeable In this case, both the thermal layer thickness andthe natural convection velocity increase, leading to an intensiveconvection plume, which in turn enhances the heat transfer co-efficient as seen in Figure 3a The distance between the dashedlines in Figure 2a and b shows that the temperature is almostuniform and minimally affected by the end effects Therefore,

in order to avoid the end effects, the test section of the duct ischosen to be between these two dashed lines where the study

is focused Figure 2a shows the axial temperature distributions

as in Figure 2b but for higher values of heat flux where the endeffects are noticeable

The heat transfer coefficients corresponding to the ature profiles given by Figure 2a are presented in Figure 3a.This figure shows a comparison between the axial perimeter-averaged heat transfer coefficient and the overall averaged heattransfer coefficient Figure 3a shows the transition regime whereheat transfer coefficient increases as the heat flux increases It isworth mentioning that the data in this figure are for duct number

temper-5 (Lc= 0.13 m) and correspond to the temperature distributionsgiven in Figure 2a It should be noted that other ducts give sim-ilar effects On the other hand, Figure 3b shows that the heattransfer coefficient corresponding to the temperature profiles inFigure 2b decreases as the heat flux increases, confirming thatthe laminar regime is achieved

The axial perimeter averaged Nusselt numbers versusthe modified Rayleigh numbers are shown in Figure 4,heat transfer engineering vol 31 no 5 2010

Trang 21

354 M E ALI AND H AL-ANSARY

1021.31 835.97

Figure 2 Perimeter-averaged dimensionless axial temperature distributions

along the duct surface for selected heat fluxes for duct number 5 (L c = 0.13 m):

(a) transition regime and (b) laminar regime.

corresponding to the test section defined by the dashed lines

in Figure 2 using all ducts for all heat fluxes Since the

modi-fied Rayleigh number is a function of qcand x4, the following

observations can be drawn from this figure:

(i) At any fixed station x along the duct length, as the heat

flux increases the Nusselt number decreases down to a

minimum critical value, then increases as qcincreases

x/L

9 10 11 12

(ii) The decrease in Nux at fixed x and at different heat fluxcorresponds to an increase in Ra∗xas indicated by the down-ward inclined arrow

(iii) At fixed heat flux, as x increases along the duct surface,Nux increases This corresponds to an increase in Ra∗x asindicated by the upward inclined arrow

(iv) Below the solid line, all the data are less sensitive to be tinguished either for qcor x and are collapsed on each otherwith general trends of increasing Nux as Ra∗x increases.heat transfer engineering vol 31 no 5 2010

Trang 22

dis-1E+7 1E+8 1E+9 1E+10 1E+11 1E+12

x

Figure 4 Local perimeter averaged Nusselt numbers versus the modified

Rayleigh numbers; solid line separates the laminar data (above the line) and the

transition data (below the line) The inclined upward arrow shows the transition

direction, while the downward arrow presents the laminar direction up to the

solid line.

Therefore, this region can be characterized as a transition

region, as seen in Figure 5

(v) The data above the solid line, as mentioned earlier, can be

identified either by qcor x and cannot be collapsed on each

other Hence, the general trend is Nux decreases as Ra∗x

increases at fixed location on the duct surface for different

heat flux Therefore, this region is defined as a laminar

x

18 %

Figure 5 Local perimeter-averaged Nusselt numbers versus the modified

Rayleigh numbers for all ducts in the transition regime; solid line presents

the correlation obtained by Eq (14).

Ra Lc

2

4 6 8

0.01

0.10

χ

Duct # 1 (4.4 cm) Duct # 2 (6 cm) Duct # 3 (8 cm) Duct # 4 (10 cm) Duct # 5 (13 cm)

18%

Figure 6 Local perimeter-averaged dimensionless surface temperature tributions for up-facing triangular ducts; solid line presents the fitting through the data given by Eq (15).

dis-region, and it should be noted that this region has a highexperimental uncertainty as seen in Table 1

The correlation of the solid line in this figure segregating thelaminar and transition data is

Nux= 0.325(Ra∗

x)0.261 , 1.0× 107≤ Ra∗

x≤ 1.0 × 1011 (13)with a correlation coefficient of R= 98.2% This line representsthe best curve fit through all the critical points of all the usedducts

Figure 5 is constructed to obtain a more general correlation

in the transition region for the data below the solid line in Figure

4 A least-squares power-law fit through the data set yields thefollowing correlation:

Nux= 0.429(Ra∗

x)0.241 , 2.0× 108≤ Ra∗

x ≤ 1.0 × 1012 (14)with a correlation coefficient of R= 97.5%, and an error band

of±18% where 92.4% of the data fall within this band and theerror limits of the exponent inside this band are±0.008 Sincethe current problem involves a uniform surface heat flux, it isimportant to investigate the surface temperature distribution.Consequently, Figure 6 was developed using the dimensionless

surface temperature χ vs the modified Rayleigh numbers using

one side of the equilateral cross-section triangle side length

Lc as a characteristic length The axial perimeter-averaged χ

plotted in Figure 6 for all up-facing triangular ducts shows adecreasing trend as the modified Rayleigh numbers increase

A least-squares power-law fit through the data set yields thefollowing correlation:

Trang 23

Figure 7 The overall averaged Nusselt numbers for the transition regime: (a)

using L = 1 m as a characteristic length; solid line presents the fitting through

the data given by Eq (16); (b) using L c as a characteristic length; solid line

presents the fitting through the data given by Eq (17).

with a correlation coefficient of R= 95.2% and an error band

of±18% where 87.8% of the data points fall within this band

and the error limits of the exponent inside this band are±0.009

The overall averaged results using the definitions of NuL and

Ra∗L given by Eq (8) are shown in Figure 7a for the transition

region The fitting curve through these data is obtained by the

following correlation, which is presented by the solid line in

Figure 7a with a correlation coefficient of R= 84.8%

a characteristic length on the behavior of the overall averagedNuLcand Ra∗Lc The fitting curve through the data points in thiscase is obtained as

NuLc= 0.789Ra∗Lc0.203

, 3.0× 105≤ Ra∗

Lc≤ 5.5 × 108 (17)with a correlation coefficient of R= 96.4% and an error bandwidth of±18% where 84.8% of the points fall within the bandand the error limits of the exponent inside this band are±0.009.Now we turn to the case where the vertex angle of the trianglefaces down, i.e., the duct is turned by 180 degrees Figure 8ashows the axial perimeter-averaged surface temperature normal-ized by the ambient temperature t∞versus the dimensionlessaxial (longitudinal) distance along the duct for some selectedvalues of qcusing duct number 5 (Lc = 0.13 m) As seen inFigure 8a, the temperature distribution is almost uniform at themiddle (test section) between the dashed lines, but the tempera-ture decreases significantly near the ends of the ducts from eitherside due to heat loss through the Bakelite end plates It can also

be seen that this heat loss increases as the heat flux increases.Figure 8b shows the heat transfer coefficient corresponding tothe temperature profiles given in Figure 8a In this figure, as theheat flux increases the heat transfer coefficient increases for allvalues of the heat flux, in contrast to the case where the vertexangle is facing up as discussed in Figure 3b Therefore, in thisposition of the duct the mode of heat transfer is characterized astransition to turbulent for the studied range of heat flux It should

be mentioned that lower values of heat flux have been tried andgave no indication of the presence of a laminar range Since theexperimental uncertainty is very high in the case of low heatflux due to the low duct surface temperature, results cannot betrusted and therefore are not shown here Comparison betweenthe two duct orientations indicates that the laminar regime isobserved only when the vertex angle faces up This observationcould be attributed to the fact that the convection plume is go-ing up in a smooth way with no eddies or vortices (Figure 9a).However, the orientation where the vertex angle faces down andthe flat surface of the triangle faces up is expected to give rise

to eddies and vortices, which make the convection plume in thetransition regime In this case, the laminar regime could not beobserved as shown in the flow pattern suggested by Figure 9b.Figure 10 shows all the local axial perimeter-averaged datafor this orientation of the duct to obtain a correlation in the transi-tion region between Nusselt number and the modified Rayleighnumber as follows:

Nux= 0.688Ra∗x0.222

, 9.0× 107≤ Ra∗

x≤ 1.0 × 1012 (18)where the solid line represents this correlation with a correlationcoefficient of R= 97.3%, with the dashed lines showing an errorband of±18% where 91.2% of the data fall within this bandand the error limits of the exponent inside this band are±0.008.heat transfer engineering vol 31 no 5 2010

Trang 24

Figure 8 Perimeter-averaged parameters along the down-pointing duct

sur-face for selected heat fluxes for duct number 5 (L c = 0.13 m) (transition

regime): (a) dimensionless axial temperature distributions and (b) heat transfer

coefficients.

As seen in this figure, most of the data, especially at high heat

fluxes, collapse on each other Figure 11 was developed in a

manner similar to Figure 6 to show the dimensionless surface

temperature χ vs the modified Rayleigh numbers for the

down-facing triangular ducts A least-squares power-law fit through

the data set yields the following correlation:

χ = 1.27Ra∗Lc−0.204

, 3.0× 105 ≤ Ra∗

Lc≤ 6.0 × 108 (19)

Figure 9 Possible changes in flow pattern with the position of the duct vertex.

with a correlation coefficient of R= 96.1% and an error band

of±18% where 90% of the data points fall within this band andwith the same error limits of exponent as Eq (15)

The overall averaged results using the definitions of NuLand Ra∗Lgiven by Eq (8) are shown in Figure 12a for the transi-tion region The fitting curve through these data is obtained bythe following correlation, which is represented by the solid linewith a correlation coefficient of R= 93.1%:

NuL= 4.672Ra∗L0.156

, 5.5× 1010 ≤ Ra∗

L≤ 2.5 × 1012 (20)Dashed lines in Figure 12a represent an error band of ±15%where all the data fall within this band with error limits of

Ra*

10 100

1000

Nux

x

Duct # 1 (4.4 cm)Duct # 2 (6 cm)Duct # 3 (8 cm)Duct # 4 (10 cm)Duct # 5 (13 cm)

18 %

Figure 10 Local perimeter-averaged Nusselt numbers versus the modified Rayleigh numbers for all ducts in the transition regime for the triangle facing down; solid line presents the correlation obtained by Eq (18).

Trang 25

Figure 11 Local perimeter-averaged dimensionless surface temperature

dis-tributions for down-facing triangular ducts; solid line presents the fitting through

the data given by Eq (19).

exponent similar to those in Eq (16) Figure 12b is constructed

in a manner similar to Figure 7b to see the effect of using Lcas

a characteristic length on the behavior of the overall averaged

NuLc and Ra∗Lc when the ducts face down The fitting curve

through the data points in this case is obtained as:

NuLc= 0.803Ra∗Lc0.203

, 3.0× 105≤ Ra∗

Lc≤ 5.5 × 108

(21)with a correlation coefficient of R= 97% and an error band

width of±18% where 84.8% of the points fall within the band

and with error limits of exponent similar to those in Eq (17)

It should be noted that by inspecting Figures 6 and 11, which

show the dimensionless surface temperature for the two duct

orientations as well as the corresponding Eqs (15) and (19),

one could notice that those experimental data are less sensitive

to the duct orientation Therefore, those data are gathered in

one curve as shown in Figure 13 with the following best fitting

correlation, which covers both orientations:

χ = 1.21Ra∗Lc−0.201

, 3.0× 105≤ Ra∗

Lc≤ 6.0 × 108 (22)with a correlation coefficient of R= 95.6% with the same error

band width of±18% and with the same error limits of exponent

as Eq (15)

The same remarks could apply to Figures 7a and 12a for

the overall average parameter with the following correlation

covering both orientations:

Duct # 1 (4.4 cm)Duct # 2 (6 cm)Duct # 3 (8 cm)Duct # 4 (10 cm)Duct # 5 (13 cm)L

15 %

(a)

Duct # 1 (4.4 cm)Duct # 2 (6 cm)Duct # 3 (8 cm)Duct # 4 (10 cm)Duct # 5 (13 cm)

Trang 26

1E+5 1E+6 1E+7 1E+8 1E+9

Figure 13 Local perimeter-averaged dimensionless surface temperature

dis-tributions for both orientations of the ducts; solid line represents the fitting

through the data given by Eq (22).

Figure 14 is constructed to compare our results of isoflux

surface with the analytical isothermal surface temperature

ex-pression developed by Hassani [17] and with the empirically

obtained correlation for short triangular prism in water by

Naka-mura and Asako [18] The experimental data points of overall

averaged Nusselt numbers versus Rayleigh numbers, defined

by Eq (12), are shown in Figure 14 for all up-facing triangular

ducts used in this study Although the comparison is not one

Nakamura and Asako [18]

Hassani [17]

Figure 14 Comparison of the overall averaged Nusselt numbers for the

isoflux surface ducts with the analytical expression of Hassani [17] and the

empirical correlation of Nakamura and Asako [18] of a triangular prism.

Ra*

10 100

to one, there is an agreement with the well-known fact that anisoflux surface’s Nusselt numbers are always higher than those

of an isothermal surface (Kays and Crawford [24] and Ali [20]).Figure 15 shows another comparison between Eq (14), whichrepresents the triangular ducts facing up, and Eq (16), whichrepresents the ducts facing down, using the local perimeter-averaged Nusselt numbers and the modified Rayleigh numberswith the empirical correlation obtained by Ali [20] for horizon-tal rectangular ducts using the same technique It can be seenthat using triangular ducts enhances heat transfer by about 30 to35% when the ducts are facing up and by about 42 to 29% whenthe ducts are facing down These percentages are estimated at

Ra∗x = 3 × 108and 7× 1011, respectively

CONCLUSIONS

An experimental study has been made on natural tion heat transfer from horizontal equilateral triangular ducts inair Nusselt numbers are observed to increase with the modi-fied Rayleigh numbers as the axial distance x increases alongthe duct’s surface for any value of heat flux Laminar regimesare obtained only at very low heat flux when the vertex faces

convec-up and this is characterized by a decrease in Nusselt number

as the modified Rayleigh number increases However, in thecase where the vertex is facing downward, only the transitionregime is observed and it is observed that Nusselt number in-creases as the modified Rayleigh number increases Generalcorrelations are obtained for the two duct orientations usingthe local perimeter averaged heat transfer data given by Eqs.(14) and (18) Furthermore, dimensionless perimeter-averagedheat transfer engineering vol 31 no 5 2010

Trang 27

axial surface temperatures are defined and correlated with the

modified Rayleigh number for each orientation and for both of

them [Eqs (15), (19), and (22)] Moreover, more correlations

are obtained using the overall average data when L or Lcis used

as a characteristic length for each duct orientation and for both

of them It was found that the dimensionless surface

tempera-ture χ and the overall average parameters are less sensitive to

the duct’s orientation Finally, a critical correlation is obtained

to segregate the laminar and transition regimes when the duct

vertex faces up [Eq (13)]

NOMENCLATURE

As duct total surface area, 3LcL, m2

ABk end plate cross-section area, m2

EIP electrical input power, W

qc convection heat flux, W/m2

qr radiation heat flux, W/m2

RaLc Rayleigh number, gβ ( ¯T− T∞)L3cν−1α−1

Ra∗ Modified Rayleigh number, gβ qcx4ν−1α−1k−1 or

Tmean arithmetic mean temperature, K

x axial or longitudinal distance, m

Greek Symbols

α thermal diffusivity, m2s−1

β coefficient for thermal expansion, K−1

χ dimensionless surface temperature,(Tx −T ∞ ) k

1 used for overall longitudinal-averaged heat transfer

co-efficient defined by Eq (7)

2 used for overall averaged heat transfer coefficient

de-fined by Eq (11)

j indices in the perimeter direction ranging from 1 to 3

L characteristic length, msur surroundings

x indices in the axial direction ranging from 1 to 11

[2] Churchill, S W., and Chu, H H., Correlating Equations for

Lami-nar and Turbulent Free Convection from a Vertical Plate,

Interna-tional Journal of Heat and Mass Transfer, vol 18, pp 1323–1329,

1975

[3] Goldstein, R., Sparrow, E M., and Jones, D C., Natural

Convec-tion Mass Transfer Adjacent to Horizontal Plates, InternaConvec-tional

Journal of Heat Mass Transfer, vol 16, pp 1025–1035, 1973.

[4] Lloyd, J R., and Moran, W R., Natural Convection Adjacent

to Horizontal Surfaces of Various Planforms, ASME Paper

74-WA/HT-66, 1974

[5] Morgan, V T., The overall convective heat transfer from smooth

circular cylinders in: Advances in Heat Transfer, vol 11, eds T F.

Irvine and J P Hartnett, Academic Press, New York, pp 199–264,1975

[6] Churchill, S W., and Chu, H H., Correlating Equations for inar and Turbulent Free Convection From a Horizontal Cylinder,

Lam-International Journal of Heat and Mass Transfer, vol 18, pp.

1049–1053, 1975

[7] Churchill, S W., Free convection around immersed bodies in:

Heat Exchanger Design Handbook, ed E U Schl¨under,

Hemi-sphere, New York, section 2.5.7, 1983

[8] Khan, W A., Culham, J R., and Yovanovich, M M., Fluid FlowAround and Heat Transfer From an Infinite Circular Cylinder,

ASME Journal of Heat Transfer, vol 127, no 7, pp 785–790,

[10] Badr, H M., and Shamsher, K., Free Convection From an Elliptic

Cylinder with Major Axis Vertical, International Journal of Heat

and Mass Transfer, vol 36, no 14, pp 3593–3602, 1993.

[11] Mahfouz, F M., and Kocabiyik, S., Transient Numerical lation of Buoyancy Driven Flow Adjacent to an Elliptic Tube,

Simu-heat transfer engineering vol 31 no 5 2010

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International Journal of Heat and Fluid Flow, vol 24, pp 864–

873, 2003

[12] Ali, M E., Experimental Investigation of Natural Convection from

Vertical Helical Coiled Tubes, International Journal of Heat and

Mass Transfer, vol 37, no 4, pp 665–671, 1994.

[13] Ali, M E., Laminar Natural Convection from Constant Heat Flux

Helical Coiled Tubes, International Journal of Heat and Mass

Transfer, vol 41, no 14, pp 2175–2182, 1998.

[14] Ali, M E., Free Convection Heat Transfer from the Outer Surface

of Vertically Oriented Helical Coils in Glycerol Water Solution,

Heat and Mass Transfer, vol 40, no 8, pp 615–620, 2004.

[15] Ali M E., Natural Convection Heat Transfer from Vertical Helical

Coils in Oil, Heat Transfer Engineering, vol 27, no 3, pp 79–85,

2006

[16] Raithby, G D., and Hollands, K G T., A general method of

obtaining approximate solutions to laminar and turbulent free

convection problems in: Advances in Heat Transfer, vol 11, eds.

T F Irvine and J P Hartnett, Academic Press, New York, pp

265–315, 1975

[17] Hassani V., Natural Convection Heat Transfer from Cylinders of

Arbitrary Cross Section, ASME J Heat Transfer, vol 114, pp.

768–773, 1992

[18] Nakamura, H., and Asako, Y., Laminar Free Convection From

a Horizontal Cylinder with Uniform Cross Section of Arbitrary

Shape, Bulletin of the JSME, vol 21, no 153, pp 471–478, 1978.

[19] Zeitoun, O., and Ali, M., Numerical Investigation of Natural

Con-vection Around Isothermal Horizontal Rectangular Ducts,

Numer-ical Heat Transfer Part A, vol 50, pp189–204, 2006.

[20] Ali, M, Natural convection heat transfer from horizontal

rectan-gular ducts, ASME Journal of Heat Transfer, vol 129, no 9, pp.

1195–1202, 2007

[21] William, D., and Callister, Jr., Materials Science and Engineering,

An Introduction, 6th ed, John Wiley & Sons, New York, chap 19.

p 660, 2003

[22] Siegel, R., and Howell, J R., Thermal Radiation Heat Transfer,

3rd ed., McGraw-Hill, New York, 1992

[23] Moffat, R J., Describing Uncertainties in Experimental Results,

Experimental Thermal And Fluid Science, vol 1, no 1, pp 3–7,

1988

[24] Kays, W M., and Crawford, M E., Convective Heat and Mass

Transfer, 3rd ed., McGraw-Hill, Singapore, chap 17, p 403, 1993.

Mohamed E Ali is a professor of heat transfer in the

Mechanical Engineering Department at King Saud University, Riyadh, Saudi Arabia He received his Ph.D in 1988 from University of Colorado, Boul- der His main research interests are stability of fluids with heat transfer, numerical and semi-analytical heat transfer of stretched plate, experimental and numerical natural convection heat transfer from different objects (coils, rectangles, squares, and triangular ducts), and nano-fluid heat transfer He has published more than 50 articles in well-recognized journals and proceedings He is a referee for most international journals in his field He has collaborated in research with professors at University of Colorado at Boulder, Northwestern University at Evanston, IL, and Swiss Federal Institute of Technology, Z¨urich, Switzerland.

Hany Al-Ansary is an assistant professor in the

Me-chanical Engineering Department at King Saud versity, Riyadh, Saudi Arabia He received his Ph.D from the Georgia Institute of Technology, Atlanta, in

Uni-2004 His research interests are in the areas of lar thermal energy utilization, heating/ventilation/air conditioning (HVAC), energy conservation, and heat- operated refrigeration systems He is currently involved in a number of research projects funded by King Saud University, King Abdulaziz City for Sci- ence and Technology, and the Saudi Telecom Company His research work has

so-so far resulted in two patent applications During his course of study at Georgia Tech, he received more than two years of cooperative training at GE Power Systems in Atlanta, GA, that involved analysis of the performance of heavy- duty gas turbines and selection of appropriate conversion and modification.

He is currently the director of the Intellectual Property Program at King Saud University.

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DOI: 10.1080/01457630903373165

Numerical Study of Laminar Natural Convection in a Side-Heated

Trapezoidal Cavity at Various

Inclined Heated Sidewalls

KHALIL LASFER, MOUNIR BOUZAIANE, and TAIEB LILI

D´epartement de Physique, Facult´e des Sciences de Tunis El-Manar, Campus Universitaire, El-Manar II, Tunisie

Steady natural convection of air flow in a two-dimensional side-heated trapezoidal room was investigated numerically using

a non-orthogonal, collocated finite-volume grid system The considered geometry has an inclined left heated sidewall, a

vertical right cooled sidewall, and two insulated horizontal upper and lower walls Computations are performed for seven

values of the heated sloping wall angle, three different values of aspect ratio, and five Rayleigh number values Results

are displayed in terms of streamlines, isotherms, and both local and average Nusselt number values The principal result

of this work is the great dependence of the flow fields and the heat transfer on the inclination angle, the aspect ratio, and

the Rayleigh number A correlation between the average Nusselt number, Rayleigh number, heated sloping wall angle, and

aspect ratio is proposed.

INTRODUCTION

Natural convection in enclosures is of great importance in

many engineering applications, such as energy transfer in rooms

and buildings, nuclear reactor cooling, solar collectors, and

elec-tronic equipment cooling Numerous experimental and

numeri-cal studies of natural convection in a cavity have been conducted

by a great number of researchers In most cases regular-shaped

enclosures have been considered However, a moderate

num-ber of studies have been interested in enclosures of irregular

geometry, such as trapezoidal ones, although they are widely

used in several practical applications The natural convection

flow problems within trapezoidal enclosures may be divided

into two main classes: enclosures heated from below, as in the

classical Rayleigh–B´enard problem, and those heated from the

side, as in the current investigation However enclosures heated

from the side have received less attention than those heated from

below

The authors thank the referees and the editor (Dr Afshin J Ghajar), who

contributed to the improvement of this work Also, they are grateful to

Profes-sor Mosbah Amlouk (Faculty of Science of Bizerte) for reading the article in

manuscript and introducing some corrections in English style.

Address correspondence to Mr Khalil Lasfer, Laboratoire de M´ecanique

des fluides, D´epartement de Physique, Facult´e des Sciences de Tunis El-Manar,

Campus Universitaire, 2092 El-Manar II, Tunisie E-mail: lasferkhalil@yahoo.fr

Among the earliest and interesting representative studies intrapezoidal enclosures were those presented by Iyican et al [1],who investigated natural convection motion and heat transferwithin an inclined trapezoidal enclosure with parallel cylindri-cal top and bottom walls The study was conducted for differ-ent temperatures and plane adiabatic sidewalls for a range ofRayleigh numbers, up to 2 7× 106, and enclosure tilt anglesvarying from 0 to 180 degrees measured from the vertical Also,Iyican et al [2] reported experimental data for natural convec-tion of air in an inclined trapezoidal enclosure The small side

of the trapezoid was electrically heated, while the opposinglarge side was cooled to a uniform temperature Their resultswere reported for a Rayleigh number ranging from∼2 × 103

to∼5 × 107 The effect of tilt angle from 0 to 90 degrees (fromhorizontal) was investigated Data were also obtained for 180degrees (hot surface facing down) Their experimental data are

correlated by an equation of the form Nu = C · Ranover a widerange of Rayleigh numbers Computations in the same geome-try using the finite-element method (FEM) and finite-differencemethod (FDM) were carried out by Baliga et al [3] and Karkiand Patankar [4]

By 1989, Karyakin [5] reported two-dimensional laminarnatural convection in enclosures of arbitrary cross section Thestudy reported on transient natural convection in an isosceles

trapezoidal cavity inclined at angle from the vertical where a

362

Trang 30

single circulation region is found to increase with the increase

in angle.

Similar results for a trapezoidal cavity composed by two

vertical adiabatic sidewalls, a horizontal hot bottom wall, and

an inclined cold top wall were reported by Lam et al [6]

Re-sults were presented for a cavity with aspect ratio equal to 4,

Rayleigh numbers varying in the range 103–107, and top surface

inclination sweeping from 0◦to 25◦ The obtained results have

illustrated that the onset of natural convection in a trapezoidal

cavity occurs progressively at lower Ra as the angle of

inclina-tion was increased from 0◦ These experimental heat transfer

re-sults were in good agreement with those computed numerically

and showed that the Nusselt number decreases with increasing

of the inclination angle

Computational study of convection in an inclined trapezoidal

enclosure of moderate aspect ratio with thermally insulated

bases and isothermal walls inclined at an angle of 45◦has been

investigated by Lee [7] The solution of the problem is obtained

using the stream function vorticity variables in 102 ≤ Ra ≤

105range at different enclosure orientations with respect to the

gravity vector The reported results indicate that the heat transfer

and fluid motion within the enclosure are very sensitive to both

the Rayleigh number and the cavity orientation angle

Also Lee [8] carried out both experimental and numerical

investigations on an incompressible fluid contained in a tilted

nonrectangular enclosure His results are reported for a Rayleigh

number ranging from 102to 105, and Prandtl numbers sweeping

from 0.001 to 100 The wall angles are 22.5◦, 45◦, and 77.5◦with

aspect ratios of 3 and 6, respectively Such results point out that

the heat transfer and fluid motion within the enclosure are strong

functions of Rayleigh number, Prandtl number, and orientation

angle of the enclosure It is shown that the transition in the

mode of circulation occurred at the angles corresponding to the

minimum or maximum rate of heat transfer Peric [9] conducted

a similar study to that found by Lee [8] and concluded then that

Lee’s results were not accurate

Using the vorticity-stream function formulation and the

con-trol volume-based finite-element method, Sadat and Salagnac

[10] examined laminar natural convection in the same geometry

for Rayleigh numbers ranging from 103to 2× 105

Kuyper and Hoogendoorn [11] have presented a

numeri-cal study of laminar natural convection flow of air in

differen-tially heated trapezoidal enclosures, for inclination angles of the

isothermal walls varying from−45◦(trapezoidal enclosure) to

0◦ (square enclosure) and for a Rayleigh number varying

be-tween 104 and 108 The effect of the inclination angle on the

flow and Nusselt number and the dependence of the averaged

Nusselt number on the Rayleigh number have been discussed

by them

Moukalled and Darwish [12, 13] have studied numerically

natural convection heat transfer in a partially divided trapezoidal

cavity with partial divider attached to the lower horizontal base

[12] or upper inclined surface [13] of the cavity Two thermal

boundary conditions have been considered In the first, the left,

short vertical wall is heated while the right, long vertical wall

is cooled (buoyancy assisting mode along the upper inclinedsurface of the cavity) In the second, the right long vertical wall

is heated while the left short vertical wall is cooled (buoyancyopposing mode along the upper inclined surface of the cavity).For all considered cases, the presence of the baffles resulted in

a relative decrease of heat transfer across the cavity up to 70%.Recently, Natarajan et al [14] carried out a study of naturalconvection flow in a trapezoidal cavity with uniformly heatedbottom wall and linearly heated and/or cooled vertical wall(s)

in the presence of an insulated top wall For linearly heatedside walls, symmetry in flow patterns is observed, whereas asecondary circulation is then noted for the linearly heated leftwall and cooled right wall The local Nusselt number indicatesreversal of heat flow at the side walls or the left wall Theaverage Nusselt number versus Rayleigh number illustrates thatthe overall heat transfer rate at the bottom wall is larger for boththe linearly heated left wall and cooled right wall

The majority of studies cited earlier are limited to eithertrapezoidal cavities heated from below with inclined cold walls

or those heated from symmetric inclined sidewalls In some plications, it is suitable to keep one of the sidewalls inclined,whereas the other remains in the vertical position However, toour knowledge, the case of natural convection in trapezoidalcavities with inclined heated sidewall has not extensively taken

ap-up for investigation previously Hence, the main purpose of thisarticle is to deepen the understanding of the flow and heat trans-fer mechanisms in a room presenting a large inclined surface

at summer day’s boundary conditions In the present work, thefinite-volume method (FVM) based on a non-orthogonal gridwith a collocated arrangement of variables is used Here, specialattention is accorded to the effects of the inclination angles ofthe heated sidewall Furthermore, effects of the thermal strengthand the aspect ratio of the trapezoidal shape on the flow structure

as well as the temperature field and heat transfer are studied nally, a correlation for the average Nusselt number as a function

Fi-of the Rayleigh number, the aspect ratio and the sloping heatedsidewall is provided to suit for the best configuration of heattransfer modes

MATHEMATICAL FORMULATION

A schematic representation of the studied configurations

is depicted in Figure 1 and refers to a steady-state, dimensional incompressible natural convection flow of air

two-within trapezoidal enclosures of height H and of lower zontal base length W The sidewalls are isothermal, while the

hori-horizontal walls are insulated The left inclined sidewall is set

at a higher temperature T h, while the right vertical sidewall is

set at a lower temperature T c ϕ is the angle between the heated

wall of length L and the horizontal Hence, the case ϕ= 90◦corresponds to the situation of rectangular cavity Inclination

angle ϕ, aspect ratio H /W , and Rayleigh number are modified

in order to analyze their effects on the resulting flow structureand heat performance of the trapezoidal room

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364 K LASFER ET AL.

Figure 1 Model of the cavity geometry.

The Boussinesq approximation [15],

ρ = ρc (1 − β (T − T c)) (1)was considered while solving the momentum equation

The natural convection is described by the equations of

con-servation of mass, momentum, and energy After applying the

Boussinesq approximation and neglecting viscous dissipation

and radiative exchange terms, the governing equations are

In the energy equation, the viscous dissipation terms are

ne-glected and the reference temperature in the buoyancy term was

taken equal to T c

The boundary conditions for the governing equations are the

non-slip, impermeable surfaces at all sides, with the horizontal

ones being adiabatic, whereas the left and right surfaces are

isothermal The boundary conditions can be therefore written as

follows:

On all solid walls, u = v = 0

On adiabatic walls, ∂T ∂y = 0

On left inclined side wall, T = T h

On right side wall, T = T c

Initial fluid temperature inside the cavity is taken as:

T =T c +T h

2 .

A dimensionless form of the governing equations can

be obtained via introducing the following dimensionlessvariables:

U = 0, V = 0, θ = 1 on the left inclined wall, which is denoted

Trang 32

where n denotes the direction normal to the inclined wall of the

enclosure, and is defined as:

n = −x sin(ϕ) + y cos(ϕ) (15)The gradient term ∂T ∂n is the temperature gradient normal to

the hot inclined wall

Local Nusselt numbers based on the height of the enclosure

are integrated to calculate the average value of Nusselt number

The domain of interest is subdivided into a finite number of

control volumes, or cells The finite-volume method based on

non-orthogonal grid with a collocated arrangement of variables

is used to discretize the governing equations [16] The

non-staggered grid arrangement is applied and the Rhie and Chow

interpolation [17] is employed to obtain a suitable coupling

between pressure and velocity A second-order central

differ-encing scheme is used to discretize the diffusion terms, whereas

a blending of upwind and central differencing is used for the

convection terms The convective fluxes at the control volume

faces are calculated as F = F L + γ(F H – F L)old where F L

stands for an interpolation of a lower order scheme (e.g.,

“first-order upwind differencing scheme”) and F H a higher order

ap-proximation (e.g., “second-order central differencing scheme”)

The term in parentheses with the superscript “old” indicates

the value at the previous iteration level, which is calculated

explicitly and added to the source terms The contribution of

the central differencing scheme is controlled by the weighting

factor γ that ranges between 0 and 1, where the two limiting

values correspond to the pure upwind and pure central

differenc-ing schemes, respectively The use of this “deferred correction”

scheme damps oscillations and improves the diagonal

domi-nance of the matrix coefficients In this way, the stability of

the algorithm can be enhanced with maintaining sufficient

ac-curacy The source terms in the governing transport equations

are not functions of the respective transported variables and are

calculated explicitly In the collocated variable arrangement,

all the dependent variables are calculated at the center of each

control volume The pressure–velocity coupling is handled by

means of the well-known SIMPLE algorithm as introduced by

Patankar [18] However, the SIMPLE algorithm needs to be

modified when the grid is non-orthogonal The systems of

alge-braic equations are solved using the strongly implicit procedure

of Stone [19] The convergence of the sequential iterative

solu-tion is achieved when the sum of the absolute differences of the

solution variables between two successive iterations falls below

a prefixed small number, which is chosen as 10−6in this study

In order to reduce eventual errors due to numerical diffusion, a

Table 1 Grid independence study for Ra= 10 5 and ϕ = 120 ◦

(Ar) Grid size N u Error % max Error % 0.5 40 × 20 3.7007 2.65 15.8566 1.30

The grid independence study was carried out at Ra= 105and

ϕ = 120◦ The grid size corresponding to a grid-independentsolution was selected such that the change in either one of themaximum values of absolute values of the stream function, or theaverage Nusselt number was less than 0.04% with considered

finest grid For the case where the aspect ratio Ar = H/W =

1, grid convergence was studied using six different grid sizes

of 40× 40, 60 × 60, 80 × 80, 100 × 100, 120 × 120, and

140 × 140 The test results of averaged Nusselt numbers andthe maximum of absolute values of the stream function are listed

in Table 1 It is observed that grid independence is achieved with

a 100× 100 grid Thus, the two-dimensional meshes were found

to be sufficiently accurate Similar grid dependency studies are

conducted for Ar = 0.5 and Ar = 1.5 and an optimum grid size

is obtained for each aspect ratio The results are summarized inTable 1 For numerical solutions, all the iterations are made on

a PC with Intel Pentium processor of 3.2 GHz clock speed and

512 MB of RAM The maximum computer time consumed inthe solution of one case is 15 min and 33 s The minimum time

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366 K LASFER ET AL.

Table 3 Comparison of results for an air filled trapezoidal cavity with

isothermal vertical walls

Moukalled and Darwish [13] Present study

The numerical accuracy of the present computer code was

validated by comparing results from the present study with those

from the correlation proposed by Iyican et al [2] and with those

obtained by Baliga et al [3] and Karki and Patankar [4] for

natu-ral convection in a trapezoidal cavity; see Table 2 Furthermore,

we have compared our numerical code results with those found

by Moukalled and Darwish [13] for the case where the

trape-zoidal enclosure is baffle-free; see Table 3 We note that good

agreement with the preceding numerical and experimental

re-sults was obtained for both stream function|ψmax| and averaged

Nusselt number N u.

A numerical study is made to simulate natural convection

flow and thermal fields inside a trapezoidal enclosure for a

sum-mer day’s boundary conditions Air is used as the working fluid,

where the Prandtl number, Pr, is fixed at 0.71 The angle of the

sloping wall, ϕ, was varied through 60◦, 70◦, 80◦, 90◦, 100◦,

110◦, and 120◦ For each value of ϕ, the Rayleigh number, Ra,

was varied as 103, 104, 105, 106, and 107, whereas the aspect

ratio, Ar, was varied as 0.5, 1.0, and 1.5 In order to reveal the

ef-fects of these parameters on the flow and heat transfer

character-istics, results are presented in terms of streamline and isotherm

plots and both local and average Nusselt number values

Flow and Temperature Fields

Samples of streamlines and isotherms are shown in Figures

2 and 3 for different Rayleigh numbers Through these figures,

the aspect ratio Ar is kept constant and equal to 1.0 while two

representative sloping wall angles of 60◦ and 120◦ were taken

into account In each case the flow rises along the heating side as

a result of buoyancy forces, and gets blocked at the top adiabatic

wall, which turns the flow horizontally toward the isothermal

cold wall The flow then falls down along the vertical cold wall

and turns back horizontally to the heated wall after striking the

bottom wall Therefore, the flow exhibits a main clockwise

rotat-ing cell, and its strength increases when Ra increases This can

be seen from absolute values of the minimum stream function

Such results present some common features as for the

differen-tially heated rectangular enclosure Similar behavior of the flow

Figure 2 Streamlines and isotherms plots for Ar = 1.0 and ϕ = 60 ◦ at

different values of Rayleigh number.

and thermal fields is observed at other angles of the sloping walland aspect ratios, for which the plots are not shown here forbrevity

Figure 2 shows the streamlines and isotherms for the case ofinclination angle 60◦ At lowest Rayleigh numbers, Ra= 103and

Ra= 104, the flow pattern is observed to be one cell with lowermagnitudes and the isotherms are uniformly distributed overthe domain implying conduction as the important heat trans-

fer mode For Ra > 104, the buoyant convection flow in theenclosure distorts the isotherms field This distortion increases

with enhanced buoyancy as Ra increases When Ra increases to

106 convective roll becomes flatter For Ra= 107, the tion continues to strengthen with the creation of two secondaryheat transfer engineering vol 31 no 5 2010

Trang 34

convec-Figure 3 Streamlines and isotherms plots for Ar= 1.0 and φ = 120 ◦ at

different values of Rayleigh number.

vortices close to the heated and cooled walls, respectively Also,

a strong recirculating cell is observed at the lower right corner

The temperature gradient near the side walls is further increased

indicating that the convection is the dominating heat transfer

mechanism in the enclosure

Figure 3 shows streamlines and isotherms contour for Ar=

1.0 and ϕ= 120◦ The flow patterns are almost same as for the

case of inclination angle of 60◦, with the main flows marked

as one vortex but with slightly thicker boundary layers The

isotherms are slightly less compressed compared to the case for

inclination angle of 60◦ Conduction heat transfer is dominant

for the Rayleigh numbers Ra= 103 At Ra= 104the core of the

streamlines begins to elongate diagonally with a clear distortion

of isotherms Two circulating cells are observed at Ra= 105,

with one circulating cell near the upper left and bottom right

corners inside the main circulating cell We also note that in

comparison with the previous inclination angle of 60◦ where

the flow is retarded in lower left corner; in this case, the flow

is retarded in the upper left corner, presumably due to difficulty

turning the sharper corner

Moreover, the effect of the aspect ratio on the temperature and

flow fields is shown in Figures 4 and 5 for Ra= 105and for two

representative cases of the sloping wall angle, ϕ= 60◦and 120◦.

Figure 4 Streamlines and isotherms plots for Ra= 10 5 and ϕ = 60 ◦ at

different values of aspect ratio.

At ϕ= 60◦, it is observed (Figure 5) that at Ar= 0.5, the flowpatterns exhibit two vortices communicating through a very thinoverall rotating eddy, indicating stronger convection; however,

at Ar= 1.0 and 1.5 there is only one circulating cell In addition,the strength of the flow is decreased with increasing aspect ratio,

as seen by the magnitude of the stream function This is due tothe presence of a small domain and short distance of hot and coldboundaries In this case, isotherms are almost parallel to the sidewalls, especially at the upper part of the enclosure Besides, thestrength of the flow decreases with increasing the aspect ratio.Further, with the increase of the sloping wall angle to 120◦

Figure 5 Streamlines and isotherms plots for Ra= 10 5 and ϕ = 120 ◦at

different values of aspect ratio.

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368 K LASFER ET AL.

Figure 6 Streamlines and isotherms plots for Ar = 1 and Ra = 105 at different

values of inclination angle.

as illustrated in Figure 5, one circulating cell is observed at

Ar= 0.5 The eye of this recirculating vortex is close to the

heated sloping wall of the enclosure, where the largest velocities

are located Two circulating cells are observed at Ar= 1.0 and

1.5, which are shifted near the upper left corner and near the right

lower one On the contrary, when ϕ= 60◦, the flow strength is

not strongly affected by varying the aspect ratio

In the same way, the effects of the side wall inclination angle

ϕ on the temperature and flow fields are presented in Figure

6 for Ar = 1.0, and Ra = 105 It is seen that the parameter

ϕdoes not have any influence on the flow strength or rotating

direction of the cells However, as can be seen from this figure,

the structure of the flow is a strong function of the sloping wall

angle ϕ The temperature distribution near the boundary is also

affected by changing the sloping wall angle ϕ The contours are

cumulated near the wall for lower values of ϕ When the sloping

wall angle becomes smaller, the flow at the lower left corner of

the enclosure becomes stagnant due to the sharp inclination,

while at higher sloping wall angle the stagnant flow is detected

at the upper left corner At these zones, conduction dominates

Table 4 Correlation coefficients for each aspect ratio Variable Ar= 0.5 Ar= 1.0 Ar= 1.5

Regression Variable Results

j −9.087523 E-06 −1.190409 E-04 −3.604831 E-02

Therefore, heat is transferred mainly by conduction in thesezones

increasing function of Ra and Ar At low Rayleigh numbers (Ra

≤ 104), especially for lower sloping wall angle (ϕ= 60◦and

80◦) and higher aspect ratio (Ar= 1.5), the influence of Rayleighnumber is not significant, due to the quasi-conductive regime.The influence becomes progressively stronger as the Rayleighnumber increases further than 104due to the convection mode

of heat transfer Hence, the slopes of the curves increase asthe aspect ratio decreases, implying a considerable aspect ratioeffect However, the increase of the aspect ratio on heat transferbecomes insignificant when ϕ increases, due to the increase ofdistance between the hot and cold walls Also, it is noticeable

that the separation between the Nu curves corresponding to each aspect ratios reduces as Ra expands from 103 to a high 107.Values at different aspect ratio are closer to each other with theincrease of Rayleigh number, as indicated in the figure Thus,

Table 5 Comparison of our correlation results with those from the published correlations [20, 21]

Ra

Present results

Markatos and Pericleous [20]

Henkes and Hoogendooren [21]

Trang 36

Figure 7 Averaged Nusselt number N u versus Rayleigh number Ra: (a) ϕ= 60 ◦, (b) ϕ= 80 ◦, (c) ϕ= 100 ◦, (d) ϕ= 120 ◦.

the aspect ratio becomes not effective on heat transfer when

Rayleigh number increases

To show how the inclination angle of the heated wall affects

the heat transfer rate through the cavity, the averaged Nusselt

number at the heated wall is discussed Variations of the

aver-aged Nusselt number versus the sloping wall angle at different

Rayleigh numbers, Ra, and for three values of the aspect ratios,

Ar, are presented in Figure 8 At lower Rayleigh number values

of 103 and 104, this figure shows a quasi-independence of the

averaged Nusselt number on the sloping wall angle at the aspect

ratios of Ar = 0.5 and Ar = 1.0 due to the conductive regime

and the long distance between the isothermal walls However,

for higher aspect ratio of 1.5 (Figure 8), average Nusselt number

marked a significant enhancement at the lower inclination angles

due to the strong decrease of distance between the isothermal

walls

As the Rayleigh number increases, the convective effect is

enhanced and the profiles of the averaged Nusselt number

ver-sus inclination become quas-symmetrical with respect to ϕ=

90◦, indicating domination of heat transfer by convection due to

the buoyancy effect The buoyancy component along the heated

wall is equivalent to “g sin(ϕ)”; therefore, the buoyancy force

along an inclined wall is different from that along a vertical wallwith a sinus of the inclined angle Consequently, the averagedNusselt number decreases with increasing or decreasing incli-nation angle from 90◦ Thus, closing the inclination angle to 90◦contributes to strengthening the convection At an angle of 90◦,the force of gravity is parallel to the isothermal hot wall Thus,the body force exerted on the fluid is greater and the flow veloc-ity increases, yielding a stronger convection flow However, thevalue of the average Nusselt number at each sharper inclinationangle was slightly higher compared to the value obtained atits symmetrical inclination angle This observed dissymmetry

is due to the additional conduction effect at lower inclinationangles induced by the decrease of distance between the hot andcold walls As can be seen in Figure 8, the quasi-conductiveregime is validated up to 106for the highest aspect ratio Ar=1.5 and up to the inclination angle of ϕ = 90◦ due to thesmall distance between hot and cold walls Thus, in thesecases conduction heat transfer is stronger than convection heattransfer However, it was observed in all considered cases thatthe conductive effect is not effective on heat transfer for higherheat transfer engineering vol 31 no 5 2010

Trang 37

isother-The computed data for N u distributions are correlated by

using a nonlinear multiple regression analysis The correlationfit takes the following form:

N u = a + b × Ln(Ra) + c × ϕ + d × Ln(Ra)2+ e × ϕ2

+f × Ln(Ra) × ϕ + g × Ln(Ra)3+ h × ϕ3

+ i × Ln(Ra) × ϕ2+ j × Ln(Ra)2× ϕ (17)

The correlation constants a − j are listed in Table 4 The

range of validity of Eq (17) is obtained for 103 ≤ Ra

≤ 107 and 60◦ ≤ ϕ ≤ 120◦ In order to enhance the curacy of the correlation, in total 105 points were consid-ered The maximum error between the numerical and predicted

ac-N u is less than 13% However the averaged error is equal to2.6%

The results of the present correlation could not be comparedagainst other ones due to the lack of published correlation datafor the specific configurations investigated here However, thepresent correlation was tested and verified via comparing theresults in the limiting case where ϕ = 90◦ and Ar= 1 withthose obtained from published correlations data [20, 21] The

comparison was made at four values of Ra equal to 103, 104,

105, and 106 As illustrated in Table 5, an excellent agreementwas found

The variation of local Nusselt number along the heated wallfor different aspect ratios and sloping wall angles is presented in

Figure 9 for Ra= 103(on the left) and Ra= 107(on the right),corresponding to the lowest and highest Rayleigh number in

this study, respectively At the lower Rayleigh number Ra =

103, the distance between hot and cold walls plays a significant

Trang 38

Figure 9 Variation of local Nusselt number along the heated wall for different aspect ratios and inclination angles: (a) ϕ = 60 ◦, (b) ϕ= 80 ◦, (c) ϕ= 100 ◦,

(d) ϕ = 120 ◦.

role in the transfer of heat by conduction In this situation, due

to the low Rayleigh number, conduction heat transfer is greater

than convection When ϕ= 60◦, for each aspect ratio, the figure

shows that the local Nusselt number increases slightly in a linear

way up to the end of the heated wall, and then it increases

dramatically at that point The maximum local Nusselt number

value is decreased with decreasing aspect ratio Also, it can

be seen that the sloping wall angle affects the location of the

maximum value of the local Nusselt number For example, when

ϕ= 120◦, the distance between hot and cold walls increases

with the height of the enclosure; thus, the local Nusselt number

presents a slight decrease at the starting point of the heated wall,

and then it marks an almost quasi-constant value due to the long

distance between the hot and cold walls Consequently, the local

Nusselt number Nu reaches the highest value for ϕ= 60◦ and

Ar = 1.5 Essentially, the peculiar behavior of Nu must then be

attributed to increased conductive heat transfer due to the small

separation between the hot and cold walls and not to the increase

in vortex strength

At a Rayleigh number of Ra = 107, as indicated earlier,

natural convection becomes dominant to conduction due to the

higher Rayleigh number Thus, higher local Nusselt numbervalues are obtained, as expected A peak of the local Nusseltnumber is observed near the leading edge of the heated wall forthe cases ϕ= 60◦and ϕ= 80◦ These behaviors are attributed

to the trapped warm fluid on the lower part of the hot wall,inhibiting the penetration of the cold fluid However, for ϕ=

60◦, the flow tends to be more trapped in the lower left cornerthan that for ϕ= 80◦, due to difficulty in turning the sharpercorner Thus, the peak for φ= 60◦ is more pronounced thanthat for ϕ= 80◦ Furthermore, for ϕ= 60◦and at the end part

of the heated wall, the local Nusselt number exhibits a slightenhancement This is explained as a result of the transition ofthe moving fluid from a hot region to a relatively colder fluid

at the shorter adiabatic wall For the inclination angle of ϕ=

120◦, the lower left corner has a favorable effect, allowing sometangentiality to the flow moving along the inclined heated wallwith no trapped fluid, resulting in an almost monotonic decrease

of the local Nusselt number along the heated wall without ence of peaks However, in contrast to the lower Rayleigh num-

pres-ber case, the local Nusselt numpres-ber Nu is the highest for ϕ=

120◦and Ar= 1.5

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372 K LASFER ET AL.

Figure 9 (Continued)

CONCLUSIONS

We could highlight the behavior of the air in laminar mode at

the interior of a differentially heated trapezoidal enclosure with

an inclined heated sidewall The obtained results presented here

have shown that the flow and the heat transfer depend strongly

on the sloping wall angle, aspect ratio, and thermal strength The

heat transfer mode is conduction at small Rayleigh numbers,

es-pecially at both low inclination angles and high aspect ratios

Thus, the Nusselt number becomes constant for the small values

of Rayleigh number and high values of aspect ratio The Nusselt

number increases with the increasing Rayleigh number

How-ever, it decreases with increasing or decreasing the inclination

angle of the heated wall from ϕ= 90◦due to the buoyancy effect.

Moreover, the aspect ratio effect is more effective at both lower

inclination angles and lower Rayleigh numbers Therefore,

in-clination angle and aspect ratio can be considered as control

parameters for heat transfer In the same way, we noticed that in

spite of the increase in the heat-transferring surface (left inclined

wall), generated by the variation of the inclination from ϕ=

90◦, the transfer of heat decreases, except for lower inclination

and higher aspect ratio, where the conductive regime dominates

L length of the heated wall, m

n direction normal to the inclined wall, m

Nu local Nusselt number

N u average Nusselt number

u,v velocity components along (x, y) axes, m/s

U, V dimensionless velocity components

W base length, m

x,y Cartesian space coordinate, m

X, Y dimensionless coordinateheat transfer engineering vol 31 no 5 2010

Trang 40

Greek Symbols

α thermal diffusivity of fluid, m2/s

β coefficient of thermal expansion, 1/K

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Khalil Lasfer is a Ph.D student in energetic

engi-neering in the fluid mechanics research laboratory in the Physics Department of Tunis University He received his master’s degree from Faculty of Science

of Tunis, Tunisia, in 2003 and his engineering gree from engineering school of Monastir, Tunisia, in

de-2001 His research interest is in computational fluid dynamics in trapezoidal geometries.

Mounir Bouzaiane is a professor of fluid mechanics

at the Faculty of Science of Bizerte, Tunisia He tained his thesis of doctorate in 1998 from Faculty of Science of Tunis, Tunisia His main research interests are natural convection and second-order modeling of turbulent flow He has published more than 30 articles

ob-in journals and conference proceedob-ings.

Taieb Lili is a professor of physics and director of

fluid mechanics laboratory at Faculty of Science of Tunis, Tunisia His main research interests are simulation and modeling of variable density turbulent flows (binary mixture flows, compressible flows) and environment flows.

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