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

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Heat Transfer Engineering, 31(5):335–343, 2010
Copyright C Taylor 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 explanation 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 Mechanical and Manufacturing Engineering, Trinity College Dublin, Parsons Building,
Dublin 2, Dublin, Ireland. E-mail:

to quantify the impact on heat transfer enhancement. Starting
with the experimental results of McGrew et al. [5] and followed
by the early numerical work of Larkin [6], thermal Marangoni
convection and its contribution to heat transfer during boiling
became the subject of some debate in the open literature [7]. Recently, it has been established that for certain conditions, such
as microgravity boiling, the thermocapillary induced ﬂow has
associated 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 literature
there is still insufﬁcient information available with regard to
Marangoni ﬂow contribution on the heat transfer in miniature
conﬁned channels [11, 12], especially when it takes place in the
presence of a cross ﬂow under microgravity conditions. To the
best of our knowledge, this conﬁguration has only been investigated by Bhunia and Kamotani [13]. Their numerical study is
focused on the ﬂuid motion due to thermocapillary ﬂow around
a bubble situated on the heated wall of a channel. The ﬂuid
mechanics aspects of the problem were characterized based on

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C. RADULESCU AND A. J. ROBINSON

Reynolds number (Rσ ) and velocity (Vσ ) due to surface tension 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 increasingly 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 inﬂuence 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 consideration 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.

heat transfer engineering

MATHEMATICAL FORMULATION
Figure 1 shows the simpliﬁed schematic of the channel
through which water at a mean inlet temperature Tm ﬂows with
the average velocity Vavg . The ﬂow is assumed to be hydrodynamically fully developed at the inlet with a parabolic velocity

proﬁle. The heated bottom wall is maintained at a constant temperature (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 ﬂow
directed perpendicular to the bubble axis. This is located near
the entrance of the channel, since this is the expected region of
the nucleate boiling ﬂow regime [17], with stratiﬁed or slug ﬂow
regimes being more likely downstream. It was concluded that
the bubble nucleus grows slowly to visible size in the laminar
inlet ﬂow [17]. As a result, the ﬂow and heat transfer problem
has been simpliﬁed by considering steady-state conditions for
three different dimensions of the bubble, starting with the incipient 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 /σT is much less than unity
[13] and also taking into consideration the imposed low bulk
liquid inlet velocities.
Lastly, it is well known that the ﬂow around the bubble within
a small channel is inherently a three-dimensional problem. However, at the mid-plane of the spherical bubble the ﬂow is approximately two-dimensional (2D). In this respect the problem can
be treated qualitatively as a two-dimensional phenomenon consistent with the previous work of Bhunia and Kamotani [13].
Governing Equations
The governing continuity, momentum, and energy equations
for steady ﬂow are as follows [18–21]:
The continuity equation is solved in the following form:
∇ u=0

(1)

Conservation of momentum is described by Eq. (2):
1
u ∇ u = − ∇p + ν∇ 2 u
ρ

(2)

The energy equation without internal heat generation is reduced to
u ∇T = α ∇ 2 T

(3)

The governing equations are solved subject to the following
assumptions and boundary conditions: (i) The bubble is represented as hemispherical, (ii) the heat ﬂux is zero at the bubble
interface, (iii) the top wall is insulated, (iv) the inlet ﬂow is hydrodynamically fully developed, (v) the bottom wall is heated
at a constant temperature, (vi) gravitational effects are negligible, (vii) apart from surface tension, temperature variations in
physical properties are not considered, (viii) the liquid is incompressible, and (ix) the no-slip condition is applied to all surfaces
vol. 31 no. 5 2010

C. RADULESCU AND A. J. ROBINSON

337

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):
∂σ
∂u
=
∇S T
(4)
∂n
∂T

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 simulations 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.
u · n = 0µ

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 × 10−3 N/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:
Ma =

(∂σ/∂T )(Twall − Tm ) Rb2
= Rσ P r
µα
H

(5)

To approximate different stages of bubble growth, i.e., nucleation to bubble sliding [11], the bubble size relative to the channel 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
heat transfer engineering

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

near the top of the bubble and lowest near the heated wall. This
surface tension variation generates thermocapillary ﬂow along
the bubble surface, away from the hot wall toward the bulk
liquid. Figure 2 illustrates clearly that increasing the driving
potential for thermocapillary ﬂow, which in this situation is
T = Twall – Tm , the inﬂuence of the surface tension driven
ﬂow becomes stronger, which is apparent from the increased

deformation of the streamlines as compared with the baseline
Ma = 0 case.
Upstream (front) side of the bubble. In this region the thermocapillary action accelerates the liquid ﬂow along the bubble
surface. The shear driven ﬂow at interface has the effect of drawing the relatively colder bulk liquid downward toward the front
corner of the bubble, as apparent from the deformation of the
near-wall streamlines toward the hot front corner of the bubble
for the Ma = 50 and 100 cases.
Downstream side of the bubble. In this region a sizable vortex is formed even at low Marangoni numbers (Ma = 50) when
the recirculation cell is strong enough to cross over the line of
symmetry of the bubble toward the front region. For the highest
Marangoni number obtained for Rb /H = 0.1 (i.e., Ma = 300)
the high shear rate at the front bubble interface interacts with the
strong 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 penetrate
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C. RADULESCU AND A. J. ROBINSON

Figure 3 Magnitude of x-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 the x 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 detail, 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 separation point position (deﬁned by the angle β) for the same Re =
100 and temperature difference T = 10◦ C. It is noticed that
this point is situated closer to the front region of the bubble
(at higher separation angle β) 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.

heat transfer engineering

Figure 4 Streamlines of steady ﬂow for Rb /H = 0.1 (B1) at Re = 300.

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, as
evident from Figure 5. It must ﬁrst be noted that the Ma number
increases disproportionately compared with the increase in T
and Rb /H, since the length scale has been chosen as R2b /H for this
study to incorporate the inﬂuence of conﬁnement on the ﬂow
and heat transfer. Compared with the relatively unconﬁned case
depicted in Figure 2 for Rb /H = 0.1, the ﬂow structure within the
liquid for the more conﬁned case of Rb /H = 0.5 indicates that
the thermocapillary induced convection has a more profound
inﬂuence on ﬂow for a like driving temperature differential and
a signiﬁcantly higher Ma. The presence of the conﬁning upper
wall tends to form elongated yet more concentrated recirculation
zones at the downstream end of the bubble, compared with the
more unconﬁned case at identical T. For T = 30◦ C, Figure
5 shows that conﬁnement effects result in the formation of three
recirculation 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 = 0.75
where the bubble is at its maximum growth dimension before
sliding in the minichannel due to the inertial forces of the bulk
liquid ﬂow [11]. It is noticed that the ﬂow pattern is altered
notably compared with the previous two cases. In this case the
conﬁning and adiabatic top wall tends to restrict the rear vortex from encroaching on the frontal region with increasing T.
This pinching of the rear vortex is strong enough that for the
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339

Figure 6 Streamlines for steady ﬂow for Rb /H = 0.75 (B3) at Re = 100.
Figure 5 Streamlines of steady ﬂow for Rb /H = 0.5 (B2) at Re = 100.

highest driving potential of T = 30◦ C no secondary recirculation zones develop since the separation 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 nonlinear 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
heat transfer engineering

presence of the bubble has a small inﬂuence on the thermal ﬁeld,
acting as a simple obstruction to the ﬂow [26]. However, for
the same wall temperature but with thermocapillary Marangoni
convection, a nonsymmetric jet of warm ﬂuid is forced into the
bulk of the ﬂow. The size of the warm jet is consistent with the
size of the vortices observed in Figure 2, and the penetration
depth of the warm jet increases with Ma in the same way as the
size of the recirculation regions is increased in Figure 2. Near the
front edge of the bubble it is clearly evident that the deformation
of the streamlines observed in Figure 2 has associated with it the
drawing in of the cooler bulk liquid, which will have important
implications with regard to the wall heat transfer.
Figure 8a and b presents the wall heat ﬂux distribution for
Rb /H = 0.1, Re = 100 and 300, with Ma = 30, 50, 100, and 200.
The corresponding heat ﬂux distribution for the situation of no
Marangoni ﬂow is also plotted for each temperature differential.
It is clear that the thermal and ﬂow ﬁelds resulting from thermocapillary convection have a direct impact on the heat transfer
in the vicinity of the bubble, as is evident from the peaks in the
wall heat ﬂux that appear around it.
Upstream (front) side of the bubble. In this region the increase in the heat ﬂux is stronger due to the combined effect of
forced and thermocapillary convection with the thermocapillary
component drawing the cooler bulk liquid toward the heated
wall and thinning the thermal boundary layer as depicted in
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C. RADULESCU AND A. J. ROBINSON

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

Figure 7
and 300.

Temperature proﬁle for Rb /H = 0.1 at Re = 100 and Ma = 0, 50,

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 substantial.
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 with T = 5◦ C, it is clear
that Marangoni convection tends to diminish the thermal gradients 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 convecheat transfer engineering

tion as cold bulk ﬂuid is drawn toward the surface and accelerated along it. The surface temperature decreases to a minimum at the separation point. Behind the separation point the
temperature gradients along the bubble surface are less steep
as the ﬂow transitions from a combined convection region to a
dominantly thermocapillary-driven recirculation region where

the average liquid temperature is generally much higher than
the bulk liquid. Increasing the Marangoni number has two notable 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.

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C. RADULESCU AND A. J. ROBINSON

341

stagnation point moves closer to the bubble front region with the
increase 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 ﬂatten
the temperature proﬁle along the bubble interface, with large
gradients isolated to the front and rear of the bubble.
Figure 10 shows the effect of the top conﬁning wall on the
thermal ﬁeld around the bubble for which Rb /H = 0.5 for the
same T values in Figure 7.
Compared with the small bubble, the conﬁnement effects
tend to keep the warm recirculation zone near the rear of the
bubble with strong effects on the front region, where the shear
ﬂow causes entrainment of the cold bulk ﬂuid toward the wall,
which thins the thermal boundary layer in this region more so
than for the Rb /H = 0.1 case.
Figure 11 illustrates this point more clearly where the heat
ﬂux proﬁles for Rb /H = 0.1, 0.5, and 0.75 are plotted for Re =

100 and T = 10◦ C. It is clear that the peak heat ﬂux at the
front of the bubble increases notably with increasing Rb /H. The
peak local heat transfer enhancement at the rear side also tends
to increase at higher Rb /H but to a much lesser extent.

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

Figure 11 Heat ﬂux distribution for Rb /H = 0.1, 0.5, and 0.75 at Re = 100
and T = 10◦ C.

heat transfer engineering

This article presents a two-dimensional numerical model 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 rectangular minichannel. At
the downstream side of the bubble a sizable vortex is formed
even at low Marangoni numbers ( T = 5◦ C, i.e., Ma = 50
for B1). The recirculation cell is strong enough to cross over
the line of symmetry of the bubble toward the front region. For
the high Marangoni numbers obtained at the maximum T =
30◦ C under consideration (i.e., Ma = 300 for B1) the shear
rate at the front bubble interface interacts with the strong rear
vortex to form a recirculation cell near the front region of the
bubble. With regard to the rear vortex itself, increasing Ma
has the effect of increasing the strength as well as the vortex
size, which penetrates deeper into the bulk liquid and along
the heated wall as well as moving forward along the bubble
surface.
It is concluded that thermocapillary ﬂow has a signiﬁcant impact on heat transfer enhancement for this conﬁguration, with

an average increase of 35% in the heat ﬂux ﬁgures at the downstream of the bubble, while the mixed convective heat transfer
due to forced and thermocapillary ﬂow results in an average of
60% increase at the front side of the bubble. As presented, the 2D
numerical approach employed in this work provides information
related with the ﬂow pattern, thermal ﬁeld, and heat ﬂux at the
mid-plane of spherical bubbles of several dimensions. Future
work will involve simulations which include three-dimensional
effects as well as unsteady effects due to bubble growth.
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C. RADULESCU AND A. J. ROBINSON

NOMENCLATURE

REFERENCES

B1
B2
B3
Ca
Dh
hMa
H
lM
L
Ma
n

p
Pr
Rb

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[13] Bhunia, A., and Kamotani, Y., Flow Around a Bubble on a Heated
Wall in a Cross-Flowing Liquid Under Microgravity Condition,
International Journal of Heat and Mass Transfer, vol. 44, no. 20,
pp. 3895–3905, 2001.
[14] Shah, R. K., Kraus, A. D., and Metzger, D., Compact Heat Exchangers, Hemisphere Publishing Corporation, Taylor & Francis
Group, New York, pp. 123–149, 1990.
[15] Bintoro, J. S, Akbarzadeh, A., and Mochizuki, M., A Closed-Loop
Electronics Cooling by Implementing Single Phase Impinging Jet
and Mini Channels Heat Exchanger, Applied Thermal Engineering, vol. 25, no. 17–18, p. 2740, 2005.
[16] Narumanchi, S., Troshko, A., Bharathan, D., and Hassani, V.,
Numerical Simulations of Nucleate Boiling in Impinging Jets:
Applications in Power Electronics Cooling, International Journal
of Heat and Mass Transfer, vol. 51, no. 1–2, pp. 1–12, 2008.

Re

ﬁrst bubble under consideration
second bubble under consideration
third bubble under consideration
capillary number
hydraulic diameter [14]
height of Ma recirculation cell [m]
height of the channel [m]

length of Ma recirculation cell [m]
length of the minichannel [m]
Marangoni number (thermocapillary)
unit normal vector (height of the boundary)
static pressure [N/m2 ]
Prandtl number
bubble radius (RB1 , RB2 , RB3 radius of B1, B2, B3, respectively) [m]
ρV Dh
Reynolds number avg
[14]
µ

Rσ
T
Tm
Twall
T∗
Vavg
Vσ
u
x, y
X*
T

surface tension Re number l µref b [13]
l
temperature [◦ C]
avg bulk liquid temperature at inlet [◦ C]
heated wall temperature [◦ C]
dimensionless temperature (T − Tm )/ T

liquid inlet average velocity [m/s]
σ
T
thermocapillary velocity Tµ| [13] [m/s]
l
velocity vector
Cartesian coordinates
dimensionless x coordinate by Rb (x/Rb )
temperature difference Twall − Tm [◦ C]

ρV

R

Greek Symbols
α
β
µ
σ
σT
ρ
ν
∇
∇S T

thermal diffusivity [m2 /s]
stagnation point angle
dynamic viscosity [Ns/m2 ]
liquid surface tension [N/m]
surface tension gradient [N/mK]

density [kg/m3 ]
kinematic viscosity [m2 /s]
Laplace divergence operator
temperature gradient at the bubble interface

Subscripts
avg
b
m
Ma
ref

average
bubble
bulk liquid
Marangoni
reference
heat transfer engineering

vol. 31 no. 5 2010

C. RADULESCU AND A. J. ROBINSON
[17] Wen, D. S., Youyou, Y., and Kenning, D. B. R., Saturated Flow
Boiling of Water in a Narrow Channel: Time-Averaged Heat
Transfer Coefﬁcients and Correlations, Applied Thermal Engineering, vol. 24, no. 8–9, pp. 1207–1233, 2004.
[18] Fluent software, http://www.ﬂuent.com
[19] Gambit, http://www.ﬂuent.com/software/gambit/index.htm
[20] Tritton, D. J., Physical Fluid Dynamics, Oxford Science Publication, Clarendon Press, Oxford, pp. 162–172, 1998.
[21] Panton, R. L., Incompressible Flow, John Wiley & Sons, New

York, pp. 102–106, 1993.
[22] Petrovic, S., Robinson, A. J., and Judd, L. R., Marangoni Heat
Transfer in Subcooled Nucleate Pool Boiling, International Journal of Heat and Mass Transfer, vol. 47, no. 23, pp. 5115–5128,
2004.
[23] O’Shaughnessy, S., and Robinson, A. J., Numerical Investigation
of Marangoni Convection Caused by the Presence of a Bubble
on a Uniformly Heated Surface, Proceedings of ITP2007 Interdisciplinary Transport Phenomena V: Fluid, Thermal, Biological,
Materials and Space Sciences, Bansko, Bulgaria, paper ITP-0744, 2007.
[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-

heat transfer engineering

343

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 mechanics and heat transfer at Trinity College Dublin, Ireland. He received his Ph.D. at McMaster University,
Canada, in 2002. His research interests are in the ﬁeld
of two-phase ﬂow and heat transfer.

vol. 31 no. 5 2010

Heat Transfer Engineering, 31(5):344–349, 2010
Copyright C Taylor and Francis Group, LLC
ISSN: 0145-7632 print / 1521-0537 online
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 typically 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 developments in the electronic industry, microfabrication technologies, 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 behavior and macroscale Stokes flow theory. Sobhan and Garimella
[3] and Obot [4] reviewed the experimental results in the exThe 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 Mechanical Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey.
E-mail:

isting literature for the convective heat transfer in microchannels. Rostami et al. [5, 6] presented reviews for flow and heat
transfer of liquids and gases in microchannels. Gad-el-Hak [7]
broadly surveyed available methodologies to model and compute transport phenomena within microdevices. Guo and Li
[8, 9] reviewed and discussed the size effects on microscale
single-phase fluid flow and heat transfer. Morini [10] presents
an excellent review of the experimental data for the convective heat transfer in microchannels in the existing literature.
He critically analyzed and compared the results in terms of the
friction factor, laminar-to-turbulent transition, and the Nusselt
number.
There is a scarcity of experimental data and theoretical analysis available in the existing literature, many of which are contradictory and conflicting, yielding different correlations with
opposite characteristics. Therefore, the mechanisms of flow and
heat transfer in microchannels are still not understood clearly.
Recently, Aydin and Avci [11–14] conducted a series of theoretical studies on various microgeometries. Experimental studies
on the microscale geometries have mostly used liquids as working fluid, while studies investigating gas flows are comparatively

scarce. Table 1 summarizes studies in the open literature, which
includes the geometry, the hydraulic diameter, and the working
fluid used for each study. It also includes the observation of each
study for the friction coefficient.
To the authors’ best knowledge, this is the first study in
the existing literature investigating air flow in a microchannel
between two parallel plates. The aim of the study is focused on

344

M. AVC˙I AND O. AYD˙IN

345

Table 1 Summary of friction results of gas flow through microchannels
Authors
Wu and Little [15]
Pfahler et al. [16]
Choi et al. [17]
Arkilic et al. [18]
Pong et al. [19]
Liu et al. [20]
Yu et al. [21]
Harley et al. [22]
Shih et al. [23]
Li et al. [24]
Lalonde et al. [25]
Turner et al. [26]
Araki et al. [27]

Yang et al. [28]
Hsieh et al. [29]
Kohl et al. [30]

Cross-section

Dh [µm]

Trapezoidal
Rectangular
Circular
Rectangular
Rectangular
Rectangular
Circular
Rectangular Trapezoidal
Rectangular
Circular
Circular
Rectangular
Trapezoidal
Circular
Rectangular
Rectangular

130–300
1.6–65
3–81
2.6
1.94–2.33

2.33
19–102
1.01–35.91
2.33
128.8–179.8
52.8
4–100
3–10
173–4010
80
25–100

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 regulator, 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 laminar flow element (LFE) technology having ultrafast response
time (20 ms).

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

heat transfer engineering

Test fluid

f < ftheory

N2
He, N2
N2
He
He, N2
He, N2
N2
He, N2 , Ar
He, N2
N2
Air
Air, He, N2
He, N2
Air
N2
Air

√
√
√
√

√
√
√
√
√
√
√

f ∼
= ftheory

f > ftheory
√

√
√
√
√

For small values of Reynolds number, the volumetric flow
rates 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 the
top 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 the
microchannel were measured by differential capacitance cell
manometers and K-type thermocouples installed at the inlet and
outlet plenums, respectively (see Figure 2b). Three different

ranges of transducers were used to meet the pressure drop requirements in the experiment (WZ-68035-00, EW-10400-02,
Comark C9505/IS). The characteristics and uncertainties of the
measurement instruments are given in Table 2.

Test Section
Figure 2a and b, shows an exploded and a sectioned view of
the test section, respectively. All parts of the test section were
fabricated 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.

vol. 31 no. 5 2010

M. AVC˙I AND O. AYD˙IN

346

Table 2 Characteristics and uncertainties of the measurement instruments
Instrument

Range

Volumetric flow controller (CZ-32907-25)
0–100 [L/min]
Digital manometers
WZ-68035-00
0–5 [inches H2 O]
EW-10400-02 R

0–15 [inches H2 O]
Comark C9505/IS
0–830 [inches H2 O]
Thermocouple (K type)
–250 to 1372 [◦ C]
Marsurf Profilometer M2
0–150 [µm]

Uncertainty
±0.8%∗
±0.05%
±0.1%∗
±0.2%∗
±0.1%∗
±0.012µm

*Full scale.

Plexiglas plate with a thickness of 5 mm. The test section consists of three regions, which are the inlet plenum, microchannel,
and outlet plenum.
Basically, the microchannel is composed of two parallel Plexiglas 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).

The hydrodynamic development length of a smooth parallel
plate duct for a fully developed laminar flow is given by [32]
0.315
l
+ 0.011Re
=
Dh
0.0175 Re + 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, Dh , is equal to 2h. Considering this fact,
the length of the channels tested in the study is ensured to be
higher 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] and
Celata et al. [34]. According to this method, the fully developed
pressure drop is given by

The laminar-to-turbulent flow transition in microchannels is
an important topic that was analyzed by a number of investigators. 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 influenced by the channel surface roughness [31].

In this study, the mean surface roughness of the each microchannel 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.

Pl −

P =

(2)

Ps

for a certain length, l (= ll − ls ), where ll and ls are the
lengths of the long and the short channel, respectively. Note that
these two lengths are higher than the hydrodynamic development length calculated based on Eq. (1). Pl and Ps are the
total pressure drops of the long and the short channel, including
the same inlet, outlet, and developing section pressure losses
between the inlet and out plenums.
The Poiseuille number, P o(= f Re), is given by
P o(= f Re) =

Data Reduction

(1)

P Dh2
=
l 2µUm

P 2h3 w

l µQ

(3)

where f is the Fanning friction factor, Q is the volumetric flow
rate, Um is the mean velocity, w is the width of channel, h is the
height of channel, µ is the dynamic viscosity of working fluid,
and Re is the Reynolds number, given by
Re =

Um Dh
2Q
=
υ
wυ

(4)

During the experiments, volumetric flow rate, pressure, and temperature were measured for the steady-state case.

UNCERTAINTY ANALYSIS
Table 3 Geometric parameters of microchannels
Height
(µm), h ± 3
100
200
300
400
500
710

Length (mm), l± 0.05

w ± 0.05

Width (mm),
w/ h

Shorter, ls

Longer, ll

37
37
37
37
37
47

370.0
185.0
123.3
92.5
74.0
66.2

30
30
30
30

30
40

50
50
50
50
50
60

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

heat transfer engineering

Experimental studies are not free of errors and uncertainties originating from measuring instruments, environment, observer, friction, and oscillations during the running of the system. Therefore, in order to indicate the quality of the measurements carried out, an uncertainty analysis has been performed
by following the method described by Holman and Gajda [35].
If the result R is a given function of the independent variables
x1 , x2 , x3 ,. . . , xn as
R = R(x1 , x2 , x3 , ........, xn )
vol. 31 no. 5 2010

(5)

M. AVC˙I AND O. AYD˙IN

347

Table 4 Uncertainties of relevant parameters
±2.9%

±1.8%
±8.6%

P
Re
Po
Note. The repeatability test deviation, ±4%.

and u1 , u2 , u3 ,. . . , un are the uncertainties in these independent
variables, the uncertainty in the result, uR , can be evaluated by
uR =

∂R
u1
∂x1

2

2

∂R
u2
∂x2

+

∂R
un
∂xn

+ .......... +

2 1/2

(6)
Using the Poiseuille number relation of Eq. (3), the uncertainty
in Po can be expressed as
u

uP o
=
Po

P

2

P
2

uQ
+
Q

+ 3

uh
h

2

uw
w

+

2

+

u

l

2

l

+

uµ
µ

2

1/2

(7)

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

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

P

P

=

uRe
=
Re

u

Pl

2

P
uQ
Q

2

+

u

uw

+
w

Ps

2 1/2

(8)

P
2

uυ
+
υ

2

1/2

(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.

RESULTS AND DISCUSSION

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
heat transfer engineering

Figure 3
number.

Influence of Reynolds number and channel height on Poiseuille

0.06–0.21 that the flow is assumed to be incompressible (Ma <
0.3).
The use of geometric parameters given in Table 3 and measured inlet and outlet pressures ensure that the highest Knudsen
number (Kn) in the test section is lower than 0.002, which closely
corresponds to the upper limit of continuum flow (0.001 < Kn).
The temperature in the inlet and exit of the test section is
also measured. It is shown that the difference between the inlet and outlet temperatures is below 1%. Therefore, it is disclosed that the viscous dissipation effect can be considered to be
negligible.

CONCLUSION

An experimental study on the fully developed flow resistance
in a microchannel between two parallel plates has been done.
vol. 31 no. 5 2010

M. AVC˙I AND O. AYD˙IN

348

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.

NOMENCLATURE
c
Dh
f
h
Kn
Ma
l
Po
Q
R
Re
u
U
w
x

speed of sound, m/s
hydraulic diameter (= 2h), m
Fanning friction factor
channel height, m
Knudsen number (= λ/Dh )
Mach number (= U/c)
channel length, m
Poiseuille number (=f Re)
volumetric flow rate, m3 /s
function of independent variables
Reynolds number (=Um Dh /ν)
uncertainty in measured variable
velocity, m/s
channel width, m
independent variable

Greek Symbols
P
l
λ
µ
υ

pressure drop (= Pl – Ps ), Pa
difference in channel length (=ll − ls ), m
molecular mean free path, m
dynamic viscosity, Pa
kinematic viscosity, m2 /s

Subscripts

l
m
s

long
mean
short

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heat transfer engineering

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[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 Microchannels, 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.

heat transfer engineering

349

[33] Mala, G. M., and Li, D., Flow Characteristics of Water in Microtubes, International Journal of Heat Fluid Flow, vol. 20,
pp. 142–148, 1999.
[34] Celata, G. P., Cumo, M., McPhail, S., and Zummo, G., Characterization of Fluid Dynamic Behavior and Channel Wall Effects
in Microtube, International Journal of Heat and Fluid Flow, vol.
27, pp. 135–143, 2006.
[35] Holman, J. P., and Gajda, W. J., Experimental Methods for Engineers, McGraw-Hill, New York, 1989.
Mete Avcı is an assistant professor of mechanical
engineering at Karadeniz Technical University, Trabzon, Turkey. He received his Ph.D. in mechanical
engineering in 2008 from Karadeniz Technical University. His research interests include heat and mass
transfer in microchannels, non-Newtonian fluid dynamics, and transport in porous media.

Orhan Aydın is a professor of mechanical engineering 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, nonNewtonian 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 Scientific and Technological Research Council of Turkey (T¨uB˙ITAK).

vol. 31 no. 5 2010

Heat Transfer Engineering, 31(5):350–361, 2010
Copyright C Taylor and Francis Group, LLC
ISSN: 0145-7632 print / 1521-0537 online
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 (perimeteraveraged) 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 components, 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 Newtonian and power-law fluids by Khan et al. [8] and [9], respectively.
This experimental investigation is supported by Saudi Arabian Basic Industrial 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 Engineering Department, King Saud University, PO Box 800, Riyadh 11421, Saudi
Arabia. E-mail:

Free convection simulation from an elliptic cylinder was studied by Badr and Shamsher [10] and by Mahfouz and Kocabiyik
[11], and correlations for natural convection from helical coils
were reported by Ali [12–15] for different Prandtl numbers.
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 cross
sections and for wide ranges of Prandtl and Rayleigh numbers
was simplified by Hassani [17] for two-dimensional cylinders of
arbitrary cross section. The laminar free convection from a horizontal cylinder with cross section of arbitrary shape had been
theoretically analyzed for uniform surface temperature and uniform surface heat flux by Nakamura and Asako [18]. Nakamura
and Asako [18] also checked their analytical results by experiments for a short modified triangular prism in water. However,
their results showed that the experimental mean heat transfer
coefficient is about 10 to 30% higher than their analytical value.
Recently, Zeitoun and Ali [19] have reported numerical simulations of natural convection heat transfer from isothermal

350

M. E. ALI AND H. AL-ANSARY

horizontal rectangular cross-section ducts in air. Their results
showed that as the aspect ratio increases, separation and circulation occur on the top surface of the duct at fixed Rayleigh number, and the corresponding behavior has been observed through
the isotherms. Zeitoun and Ali have also obtained a general correlation 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 correlated using the modified Rayleigh numbers and the overall
correlation is also obtained.
This article presents the results of an experimental investigation of natural convection heat transfer from the outer surface
of triangular ducts with their axis oriented horizontally. The
study focuses on the determination of local axial (perimeteraveraged) and overall averaged heat transfer coefficient in dimensionless 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.

351
Ac
Power line

W

VR
Bk
D
H
TCW
DA

(a)

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 positioned 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 heating 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 surfaces 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),
heat transfer engineering

Figure 1 (a) Schematic of the experimental system showing the thermocouple 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.

which in turn was connected to a computer where the measured
temperatures 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 be
uniformly distributed along the duct length. The heat flux per
unit surface area of the duct is calculated by dividing the consumed power (after deducting the heat loss by axial conduction
through the Bakelite end plates) over the duct outer surface area.
The input power to the duct is increased for each duct from
about 2 to 680 W in two stages up to the duct dimensions. In

the first stage, the power is increased by increasing the voltage
in 4-V steps up to 26-V for possible laminar natural convection
regime. However, in the second stage, which represents the transition regime, the voltage is increased in 10-V steps. These steps
are stopped once the surface temperature reaches 160◦ C, which
represents the thermocouples’ limit. As a result, the experiment
is repeated about 17 times for each duct to account for the various input power levels. Temperature measurements are taken
vol. 31 no. 5 2010

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 procedure 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:

Therefore for each heat flux (run) there are 11 Tx longitudinal
temperature measurements. Consequently, once the electrical
input power to the duct is measured, qr and qBk can be calculated from Eq. (2) and the corresponding value of qc can then
be calculated from Eq. (1). Using this information, the axial
(perimeter averaged) heat transfer coefficient hx can be calculated from:
qc
, x = 1, 2, 3, . . . , 11
(5)

hx =
Tx − T∞
Hence, the nondimensional axial (perimeter averaged) Nusselt
number and the modified Rayleigh number are obtained from:

EIP = Electrical input power = As (qc + qr ) + ABk qBk (1)
where As is the duct total surface area, ABk is the Bakelite surface area normal to the direction of heat transfer by conduction
through the end plates, and qc and qr are the fraction of the heat
flux dissipated from the duct surface by convection and radiation, respectively. The heat flux lost by radiation (qr ) and by
axial conduction through the Bakelite end plates (qBk ) can be
calculated respectively by:
qr = εσ T¯ 4 −, T4sur ,

qBk = kBk

(TiB − ToB )
δ

(2)

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), TiB and ToB are 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:
3

Tx =

Txj /3,

(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)

heat transfer engineering

Nux =

hx x
,
k

Ra∗x =

g β qc x4
νkα

(6)

All physical properties are evaluated at the axial perimeter averaged mean temperature θx for each qc .
Since the duct surface is uniformly heated, the surface temperature distribution becomes another important parameter. A
suitable way to account for the axial surface temperature distribution is by using a dimensionless surface temperature group
∞) k
in a correlated form of Rayleigh number, as shown
χ = (Txq−T
c Lc
in the Results section.

Total Overall Averaged Heat Transfer Coefficient
In this case the perimeter-averaged heat transfer coefficient
hx is first evaluated at each station x as in Eq. (5) and then the
overall longitudinal average h¯ 1 is obtained as:
9

h¯ 1 =

hx /7

(7)

x=3

It should be mentioned that to obtain the overall average heat
transfer coefficient by Eq. (7), two stations from each end are
excluded to eliminate end effects, allowing only seven stations
to be used in the equation.
Therefore, each heat flux qc is represented by only one overall averaged heat transfer coefficient, in contrast to the axial
case where qc is presented by 11 hx terms along the longitudinal direction given by Eq. (5). All perimeter-averaged physical
properties are first obtained at θx ; then the overall averaged
properties are obtained the same way following Eq. (7). The
nondimensional overall averaged Nusselt number and the modified Rayleigh number are defined using either the duct length
as a characteristic length (L = 1 m) or one side of the equilateral cross-section triangle Lc as a characteristic length as
follows:
NuL =

h¯ 1 L
,
k

Ra∗Lc =

g β qc L4c
νkα

vol. 31 no. 5 2010

Ra∗L =

h¯ 1 Lc
g β qc L4
or NuLc =
,
νkα
k
(8)

M. E. ALI AND H. AL-ANSARY

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 temperature is first perimeter-averaged following Eq. (3), and then
the overall average temperature is obtained as:
9

T¯ =

Tx /7

(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:
h¯ 2 =

qc
(T¯ − T∞ )

(11)

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 =

h¯ 2 Lc
,
k

RaLc =

g β (T¯ − T∞ ) L3c
να

(12)

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.
heat transfer engineering

353

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

Quantity
EIP
qBk
qr
qc
h
Nux
Ra∗x
NuL
Ra∗L

Transition range (%)

Laminar range (%),
duct facing up

Duct facing up

Duct facing down

3.49
17.93
17.87
6.70
14.32
14.33
6.87
16.20
6.74

2.73
12.09
10.76
4.49
7.56
7.57
4.73
8.7
4.52

2.74
12.85
11.95
4.67
7.69
7.71
4.91
8.69
4.71

RESULTS AND DISCUSSION
Experimental data are obtained for triangular ducts oriented
horizontally in air. Figure 2a and b shows the axial perimeter averaged surface temperature normalized by the ambient temperature t∞ versus the dimensionless axial (longitudinal) distance
along the duct for some selected values of qc using duct number
5 (Lc = 0.13 m) when the vertex of the triangle faces up. As
seen in Figure 2b, the temperature distribution at low heat flux is
least affected by the end effects where the heat lost through the
Bakelite end plates is minimal. Therefore, the thermal boundary
layer thickness is expected to be limited and the convection velocity to be small. As the heat flux inside the duct increases, the
surface temperature increases and the end effects become more

noticeable. In this case, both the thermal layer thickness and
the natural convection velocity increase, leading to an intensive
convection plume, which in turn enhances the heat transfer coefficient as seen in Figure 3a. The distance between the dashed
lines in Figure 2a and b shows that the temperature is almost
uniform and minimally affected by the end effects. Therefore,
in order to avoid the end effects, the test section of the duct is
chosen 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 end
effects are noticeable.
The heat transfer coefficients corresponding to the temperature profiles given by Figure 2a are presented in Figure 3a.
This figure shows a comparison between the axial perimeteraveraged heat transfer coefficient and the overall averaged heat
transfer coefficient. Figure 3a shows the transition regime where
heat transfer coefficient increases as the heat flux increases. It is
worth mentioning that the data in this figure are for duct number
5 (Lc = 0.13 m) and correspond to the temperature distributions
given in Figure 2a. It should be noted that other ducts give similar effects. On the other hand, Figure 3b shows that the heat
transfer coefficient corresponding to the temperature profiles in
Figure 2b decreases as the heat flux increases, confirming that
the laminar regime is achieved.
The axial perimeter averaged Nusselt numbers versus
the modified Rayleigh numbers are shown in Figure 4,
vol. 31 no. 5 2010

354

M. E. ALI AND H. AL-ANSARY
12

7.0

qc= 1396.96 W/m2

qc= 1396.96 W/m2

1210.42

6.0

hx(W/m2K)

t/t

11

1021.31
5.0

835.97

1210.42
1021.31
10

835.97

4.0

9

3.0
0.0

0.2

0.2

0.4

0.6

0.8

0.4

0.6

0.8

x/L

1.0

x/L

(a)

(a)

7.2

qc= 22.86 W/m2

6.8

1.20

hx(W/m2K)

t/t

1.15

14.42
1.10

qc= 7.47 W/m2

6.4

14.42

6.0

22.86

5.6

7.47

1.05

5.2
0.2

1.00
0.0

0.4

0.6

0.8

x/L
0.2

0.4

0.6

0.8

1.0

x/L
(b)
Figure 2 Perimeter-averaged dimensionless axial temperature distributions
along the duct surface for selected heat fluxes for duct number 5 (Lc = 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 modified Rayleigh number is a function of qc and 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 qc increases.
heat transfer engineering

(b)
Figure 3 Axial perimeter averaged heat transfer coefficient (solid lines) along
the duct surface for selected heat fluxes for duct number 5 (Lc = 0.13 m). Dashed
lines present the overall averaged heat transfer coefficient given by Eq. (7): (a)
transition regime and (b) laminar regime.

(ii) The decrease in Nux at fixed x and at different heat flux
corresponds to an increase in Ra∗x as indicated by the downward inclined arrow.
(iii) At fixed heat flux, as x increases along the duct surface,
Nux increases. This corresponds to an increase in Ra∗x as
indicated by the upward inclined arrow.
(iv) Below the solid line, all the data are less sensitive to be distinguished either for qc or x and are collapsed on each other
with general trends of increasing Nux as Ra∗x increases.
vol. 31 no. 5 2010

M. E. ALI AND H. AL-ANSARY

355

1000

18%

0.10
8
6

Nux

χ

100

Duct # 1 (4.4 cm)
Duct # 2 (6 cm)
Duct # 3 (8 cm)
Duct # 4 (10 cm)
Duct # 5 (13 cm)
10
1E+7

1E+8

1E+9

1E+10

1E+11

1E+12

Rax*
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 qc or 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
1000

18 %

4

2

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

0.01

1E+5

1E+6

1E+7

1E+8

1E+9

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

region, and it should be noted that this region has a high
experimental uncertainty as seen in Table 1.
The correlation of the solid line in this figure segregating the
laminar 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 represents
the best curve fit through all the critical points of all the used
ducts.
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 the
following correlation:
Nux = 0.429(Ra∗x )0.241 , 2.0 × 108 ≤ Ra∗x ≤ 1.0 × 1012 (14)

Nux

100

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

1E+9

1E+10

1E+11

1E+12

Rax*
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).

heat transfer engineering

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 the
error limits of the exponent inside this band are ±0.008. Since
the current problem involves a uniform surface heat flux, it is
important 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 a
decreasing trend as the modified Rayleigh numbers increase.
A least-squares power-law fit through the data set yields the
following correlation:
∗
χ = 1.15 RaLc

−0.198

vol. 31 no. 5 2010

,

∗
3.0 × 105 ≤ RaLc
≤ 6.0 × 108 (15)

356

M. E. ALI AND H. AL-ANSARY
1000

15 %

Nu

L

Dashed lines in Figure 7a represent an error band of ±15%
where all the data fall within this band and the error limits of
the exponent inside this band are ±0.006 where the average
Nusselt numbers increase as the modified Rayleigh numbers
increase. Figure 7b is constructed to see the effect of using Lc as
a characteristic length on the behavior of the overall averaged
NuLc and Ra∗Lc . The fitting curve through the data points in this
case is obtained as
NuLc = 0.789 Ra∗Lc

100
1E+10

1E+11

RaL*

1E+12

1E+13

(a)
100
80
60

Duct # 1 (4.4 cm)
Duct # 2 (6 cm)

Duct # 3 (8 cm)
Duct # 4 (10 cm)
Duct # 5 (13 cm)

40

NuLc
20

10

18%
1E+005

1E+006

1E+007

1E+008

1E+009

RaLc
(b)
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 Lc 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%.
NuL = 5.472 Ra∗L

0.150

, 7.0 × 1010 ≤ Ra∗L ≤ 2.5 × 1012 (16)
heat transfer engineering

0.203

, 3.0 × 105 ≤ Ra∗Lc ≤ 5.5 × 108 (17)

with a correlation coefficient of R = 96.4% and an error band
width of ±18% where 84.8% of the points fall within the band
and the error limits of the exponent inside this band are ±0.009.
Now we turn to the case where the vertex angle of the triangle
faces down, i.e., the duct is turned by 180 degrees. Figure 8a
shows the axial perimeter-averaged surface temperature normalized by the ambient temperature t∞ versus the dimensionless
axial (longitudinal) distance along the duct for some selected
values of qc using duct number 5 (Lc = 0.13 m). As seen in
Figure 8a, the temperature distribution is almost uniform at the
middle (test section) between the dashed lines, but the temperature decreases significantly near the ends of the ducts from either
side 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 to
the temperature profiles given in Figure 8a. In this figure, as the
heat flux increases the heat transfer coefficient increases for all
values of the heat flux, in contrast to the case where the vertex
angle is facing up as discussed in Figure 3b. Therefore, in this
position of the duct the mode of heat transfer is characterized as
transition to turbulent for the studied range of heat flux. It should
be mentioned that lower values of heat flux have been tried and
gave no indication of the presence of a laminar range. Since the
experimental uncertainty is very high in the case of low heat
flux due to the low duct surface temperature, results cannot be
trusted and therefore are not shown here. Comparison between
the two duct orientations indicates that the laminar regime is
observed only when the vertex angle faces up. This observation
could be attributed to the fact that the convection plume is going up in a smooth way with no eddies or vortices (Figure 9a).
However, the orientation where the vertex angle faces down and
the flat surface of the triangle faces up is expected to give rise
to eddies and vortices, which make the convection plume in the
transition regime. In this case, the laminar regime could not be
observed as shown in the flow pattern suggested by Figure 9b.
Figure 10 shows all the local axial perimeter-averaged data
for this orientation of the duct to obtain a correlation in the transition region between Nusselt number and the modified Rayleigh
number as follows:
Nux = 0.688 Ra∗x

0.222

, 9.0 × 107 ≤ Ra∗x ≤ 1.0 × 1012 (18)

where the solid line represents this correlation with a correlation

coefficient of R = 97.3%, with the dashed lines showing an error
band of ±18% where 91.2% of the data fall within this band
and the error limits of the exponent inside this band are ±0.008.
vol. 31 no. 5 2010

M. E. ALI AND H. AL-ANSARY

357

7.0

qc= 1402.74 W/m2
6.5

1169.93

tx / t

6.0

991.53

5.5

5.0

819.25
4.5

4.0

(a)
Figure 9

3.5

0.0

0.2

0.4

0.6

0.8

1.0

x/L

with a correlation coefficient of R = 96.1% and an error band
of ±18% where 90% of the data points fall within this band and
with the same error limits of exponent as Eq. (15).
The overall averaged results using the definitions of NuL
and Ra∗L given by Eq. (8) are shown in Figure 12a for the transition region. The fitting curve through these data is obtained by
the following correlation, which is represented by the solid line
with a correlation coefficient of R = 93.1%:

(a)

12.0

qc= 1402.74 W/m2

hx(W/m2K)

11.5

(b)

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

NuL = 4.672 Ra∗L

11.0

1169.93

0.156

, 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

10.5

991.53
1000

18 %

10.0

819.25

9.5
0.2

0.4

0.6

0.8

x/L

Nux

(b)
Figure 8 Perimeter-averaged parameters along the down-pointing duct surface for selected heat fluxes for duct number 5 (Lc = 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 downfacing triangular ducts. A least-squares power-law fit through
the data set yields the following correlation:
χ = 1.27 Ra∗Lc

−0.204

, 3.0 × 105 ≤ Ra∗Lc ≤ 6.0 × 108 (19)
heat transfer engineering

100

Duct # 1 (4.4 cm)
Duct # 2 (6 cm)
Duct # 3 (8 cm)
Duct # 4 (10 cm)
Duct # 5 (13 cm)
10
1E+7

1E+8

1E+9

1E+10

*
Rax

1E+11

1E+12

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).

vol. 31 no. 5 2010

358

M. E. ALI AND H. AL-ANSARY
1000

15 %

18%

0.10
8
6

Nu

χ

L

4

2

Duct # 1 (4.4 cm)

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

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

0.01
1E+5

1E+6

1E+7

100
1E+10

1E+8

1E+9

Ra
Lc

,

1E+12

1E+13

100

exponent similar to those in Eq. (16). Figure 12b is constructed
in a manner similar to Figure 7b to see the effect of using Lc as
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:
0.203

RaL*
(a)

Figure 11 Local perimeter-averaged dimensionless surface temperature distributions for down-facing triangular ducts; solid line presents the fitting through
the data given by Eq. (19).

NuLc = 0.803 Ra∗Lc

1E+11

3.0 × 105 ≤ Ra∗Lc ≤ 5.5 × 108

80
60

Duct # 1 (4.4 cm)
Duct # 2 (6 cm)

Duct # 3 (8 cm)
Duct # 4 (10 cm)
Duct # 5 (13 cm)

40

NuLc
20

(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 = 1.21 Ra∗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:
NuL = 5.014 Ra∗L

0.153

,

5.5 × 1010 ≤ Ra∗L ≤ 2.5 × 1012
(23)
heat transfer engineering

10

18%
1E+005

1E+006

1E+007

1E+008

1E+009

RaLc
(b)
Figure 12 The overall averaged Nusselt numbers for the transition regime

for down-facing ducts: (a) using L = 1 m as a characteristic length; solid
line presents the fitting through the data given by Eq. (20); (b) using Lc as a
characteristic length; solid line presents the fitting through the data given by
Eq. (21).

for the same error band width of ±15 % and for R = 89.4 %.
Furthermore, Figures 7b and 12b reveal the same pattern when
Lc is used as a characteristic length in the overall averaged
parameter with the following correlation:
NuLc = 0.797 Ra∗Lc

0.203

,

3.0 × 105 ≤ Ra∗Lc ≤ 5.5 × 108
(24)

with a correlation coefficient of R = 96.7% with the same error
band width of ±18%.
vol. 31 no. 5 2010

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

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