Heat Transfer Asian Research, 41 (1), 2012
Turbulent Heat Transfer Enhancement in a Triangular Duct
Using Delta-Winglet Vortex Generators
M.A. Althaher,1 A.A. Abdul-Rassol,1 H.E. Ahmed,1 and H.A. Mohammed2
1
Department of Mechanical Engineering, College of Engineering, University of Anbar, Anbar, Iraq
2
Department of Mechanical Engineering, College of Engineering, Universiti Tenaga Nasional,
Selangor, Malaysia
The augmentation of convective heat transfer of a turbulent flow using delta-
winglet vortex generators (VG) in a triangular duct was experimentally investigated.
Two side walls of the heated test section are electrically heated with a constant heat
flux while the lower wall is indirectly heated. Single, double, and triple pairs of VG
are utilized. Each pair of VG was punched on one wall of the test duct. The effects of
the number of VG pairs, the VG angle of attack, the VG location from the leading edge
of the test duct, the VG geometry, and Reynolds number are examined in this paper.
The results indicate that the Nusselt number and friction factor are relatively propor-
tional to the size, number, and the inclination angle of the VG. The Nusselt number
increases and the friction factor decreases as the Reynolds number increases. The
present results were compared with the available literature and they show good
agreement. Correlation equations of Nusselt number and friction factor for turbulent
flow are developed, for the cases studied, as a function of Reynolds number and VG
angle of attack. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res, 41(1): 43 62,
2012; Published online 9 August 2011 in Wiley Online Library (wileyonlineli-
brary.com/journal/htj). DOI 10.1002/htj.20382
Key words: turbulent flow, vortex generator, delta-winglet, triangular duct,
heat transfer enhancement
1. Introduction
The fluid flow and heat transfer characteristics of the non-circular duct have been extensively
studied in the past decades. This is due to its wide applications in compact heat exchangers, solar
collectors, laser curtain seals, and cooling of electrical devices. However, in these applications, gases
were used instead of liquids as the heat transfer medium. Gases inherently have a lower heat transfer
rate than liquids. Therefore, heat transfer enhancement techniques that maximize heat transfer are of
particular interest. Previous investigators have studied the techniques of heat transfer augmentation
theoretically and experimentally. They found that the only effective way to increase the heat transfer
coefficient is to decrease the thermal resistance of the sublayer immediately adjacent to the wall (i.e.,
the viscous effects of the sublayer are dominant).
© 2011 Wiley Periodicals, Inc.
43
This can be achieved by increasing the turbulence in the main stream so that the turbulent
eddies can penetrate deeper into this layer [1, 2]. The thermal resistance can be reduced and the heat
transfer can be augmented by using a VG that can be punched or attached to the internal surface of
the channel wall. The VG generates longitudinal vortices along their leading edge. These vortices
turn the flow field perpendicular to the main flow direction and then enhance the fluid mixing and
penetrate deeply in the corners. In addition, these vortices disrupt the growth of the thermal boundary
layer and serve ultimately to enhance the heat transfer between the fluid and the duct wall.
One relevant application using such flow configurations is the heat transfer between flowing
fluid and plates in the case of plate-fin heat exchangers as shown in Fig. 1. Fins are commonly used
between the plates in such a heat exchanger to increase their compactness and this often results in
plate-fin flow passages of triangular cross-section with a small hydraulic diameter. Vortex generators,
as turbulators, are used to enhance the heat transfer rate of the contact surface between the working
fluid and the conductor channel walls. However, the existence of the VG as an obstacle in the flow
stream promotes the pressure drop penalty at the trailing edge of the VG because of the introduction
of form drag. There are generally four types of vortex generators; rectangular wing, triangular wing,
rectangular winglet pair, and delta winglet pair as shown in Fig. 2.
Many researchers have investigated the pressure drop and heat transfer characteristics in a
duct with a non-circular cross-section. They found that the pressure drop in a fully developed turbulent
flow can be estimated from a turbulent flow correlation of circular duct if the diameter is replaced by
the hydraulic diameter of a particular cross-section. This can also be applied for turbulent flow in
triangular ducts in which only small deviation from circular tube correlations with 21% overestimated
at 4° apex angle and 5% at 38° angle [3]. Chegini and Chaturvedi [3] examined analytically and
experimentally the friction factor for fully developed flow in internally finned triangular ducts without
VG. They have found that the Nusselt number for the isosceles triangular duct was the highest for an
apex angle of 60°. A significant deviation obtained from the circular tube correlation for the sharp
cornered ducts such as isosceles triangular ducts with narrow apex angle and without VG was reported
by Rohsenow et al. [4]. It was found that the deviation in a friction factor correlation decreases as the
apex angle of triangular duct increases.
Altemani and Sparrow [5] performed an experimental work in an equilateral triangular duct
without VG. They have found that the hydraulic diameter is not sufficient to rationalize the difference
Fig. 1. Cross flow heat exchanger along with triangular inserts [35].
44
Fig. 2. Longitudinal vortex generators types: (a) delta wing, (b) rectangular wing, (c) delta-winglet
pair, and (d) rectangular-winglet pair [28].
between the tube and triangular duct geometries. The delta-winglet pair of VG was preferred for heat
transfer enhancement application in ducts as reported through an experimental work done by Pauley
and Eaton [6] and Tiggelbeck et al. [7]. Tiggelbeck et al. [7] reported that the Stanton number increases
as the angle of attack increases until it approaches 65° for delta-winglet pair type and then the St
number decreases after this value. They also reported that the delta-winglet pair causes a lower
pressure drop and gives higher heat transfer rate than that of the wing and winglet pair.
The main problem in heat transfer enhancement between the flowing fluid and the conductor
surface of the channel wall is the additional pressure drop associated with the increase of the heat
transfer rate. The fully developed axial pressure drop in an equilateral triangular duct without VG was
experimentally studied by Aly et al. [8]. Their results revealed that the deviation ranged from 5% to
6.5% lower than the values predicted by the Blasius correlation for the friction factor in smooth
circular tubes. Wroblewski and Fibeck [9] and Dep et al. [10] studied analytically the laminar and
turbulent boundary layer in a rectangular channel with a single pair of delta-winglet type vortex
generator. They reported that the increase in peak value of the average skin friction coefficient was
33% and 27% for Re number of 5000 and 15,000, respectively.
Biswas and Chattopadhyay [11] and Biswas et al. [12] performed an analytical solution to
study the effect of the built-in delta-wing and winglet-pair type vortex generators in a rectangular
channel. It is shown that the frictional loss for the built-in delta wing in the channel is about 79%
more than that of a plane channel flow, while it was approximately 65% for the delta-winglet pair at
the same location at an angle of attack of 26° when the size of the VG was kept constant.
Other researchers have studied the effect of the existence of a row (array) of winglet-type
vortex generators organized in a way that was periodically mounted in one wall of a rectangular
channel. Zhu et al. [13] studied numerically the flow structure through a rectangular channel using
VG. They inferred that the increase in flow loss is much larger than the enhancement in Nu number
when the VG was used. Ketcioglu et al. [14] arranged the arrays periodically in interrupted divergent
and convergent channels. The results showed that the friction factor was strongly dependent on Re
number. An increase in heat transfer coefficient was accompanied with a large pressure drop when
the inclination angle is increased.
45
Other studies have demonstrated the influence of the longitudinal and transverse vortex
generators on fluid flow structure and heat transfer enhancement in compact heat exchangers [15 18].
A relationship between the intensity of the secondary flow and the strength of convective heat transfer
in a flat tube bank fin heat exchanger with VG was numerically studied by Chang et al. [15]. Joardar
and Jacobi [16] demonstrated that the heat transfer coefficient of full scale wind tunnel testing of a
compact plain-fin and tube heat exchanger increases by 16.5% 44% for single pair VG and 30%
68.8% for three VG array arrangements. Shi et al. [17] showed that (Nu/f) of a three-row flat tube
bank fin depends on the fin spacing and the thickness of fin. They reported that the optimal fin spacing
is about 2 mm in industrial applications for the configuration of tube bank fin studies.
Song et al. [18] found that using a finned flat tube bank fin with vortex generators mounted
on both surfaces of the fin can reduce the height of VG and it can give satisfactory heat transfer
enhancement with reduced pressure drop. Recently, Wu and Tao [19, 20] showed that the average
Nusselt number of a rectangular channel under conditions considering the thickness of a pair of
Rectangular Winglet Longitudinal Vortex Generator (RWLVG) is lower than that of the case
neglecting the thickness of LVG. They reported that the VG with an attack angle of 45° always
provides better heat transfer enhancement, followed by attack angle of 60, 30, 90, and 15°, respec-
tively. Very recently, Munish et al. [21] studied the influence of the rectangular winglet on heat
transfer augmentation in a plate-fin heat exchanger with triangular fins. They found, for a fixed area,
that increasing the length and decreasing the height of the VG was better for heat transfer enhance-
ment. As the RWLVG was moved towards the channel inlet the influence range of the LV was larger
in the streamwise direction. The enhancement potential of triangular fins which are used in a plate-fin
heat exchanger having delta winglets mounted on their slant surfaces was numerically examined by
Vasudevan et al. [22]. The attack angles of the VG, different thermal boundary conditions, and the
combination of triangular fins and VG with stamping on the slant surfaces were evaluated. As a result,
it was observed that about a 20 25% enhancement of heat transfer can be achieved at the expense of
a moderate pressure drop. They stated that the enhancement of the heat transfer is dependent also on
the angle of attack of the delta winglet. They concluded that the triangular fins with mounted delta
winglets (secondary fins) have a lot of promise as inserts (protrusions) within the flow passages
created by the two neighboring plates in a plate-fin heat exchanger.
Additional studies on vortex-enhancement heat transfer using different delta-winglet pairs
configuration was done by Rütten and Krenkel [23]. They arranged one pair on the hot wall Common
Flow Up (CFU) of a rectangular channel, and one on the cold opposite channel wall Common Flow
Down (CFD). They reported that this arrangement leads to a transport of fluid from the hot to the cold
wall with the upper vortex pair feeding the lower vortex pair. Several other studies of winglet VGs
on round, oval, and flat tubes were reported by O Brien et al. [24]. Their results revealed that the
addition of the single winglet pair to the oval-tube heat exchange yielded significant heat transfer
enhancement of 38% higher than the oval-tube no-winglet case. Wang et al. [25] reported that the
delta winglet showed more intense vertical motion and flow unsteadiness compared to the annular
winglet VG. They found that the frictional penalty of the proposed VG was about 10 65% higher
than that of the plain fin geometry. The numerical results of Leu et al. [26] showed that using VG to
improve the heat transfer rate over a 3-row plate-fin and tube heat exchanger not only produces
longitudinal vortices, but also aids the fluid into the wake recirculation zone. They also demonstrated
that VG at 45° attack angle provided the best relative heat transfer enhancement with 11 15% increase
of the fanning friction factor and it gave the greatest area reduction ratio up to 25%.
46
Fiebig et al. [27] studied experimentally the heat transfer enhancement of fin-tube configura-
tions with a small delta-winglet pair punched out of the fins. They found that, for a winglet to fin area
ratio of 0.003, the overall heat transfer enhancement on the fin was 20% resulting with a simultaneous
drag reduction of 7%. Chen et al. [28] found that the 30° angle of attack of a winglet VG and an aspect
ratio of the winglet of 2 provided the best ratio of the heat transfer enhancement of a finned oval tube
heat exchanger to flow loss penalty in their investigated configurations. The effects of the pitch of
the in-line delta winglet VG on heat transfer and pressure drop in a finned three-row flat tube bank
using the naphthalene sublimation method were studied by Zhang et al. [29]. Their results showed
that the pitch of the in-line VG has a significant effect on the heat transfer performance. Chomdee
and Kiatsiriroat [30, 31] studied experimentally the heat transfer enhancement by the delta winglet
VG in air cooling of an in-line and a staggered array of rectangular electronic modules. They pointed
out that the VG with a 20° attack angle, could enhance the adiabatic heat transfer coefficients, and
reduce the thermal wake functions and the module temperatures significantly. Their results revealed,
as in Ref. 31, that the VG enhanced the heat transfer coefficient by 10 30% and reduced the thermal
wake function and the module temperature. They reported that the attack angle should not be more
than 10°.
Joardar and Jacobi [32] studied numerically three different winglet configurations in a CFU
arrangement in a seven-row compact fin-and-tube heat exchanger. They found that the winglet
impingement redirected flow on the downstream tube which is an important heat transfer augmenta-
tion mechanism for the CFU arrangement of vortex generators in the inline-tube geometry.
Lemouedda et al. [33] studied the heat transfer performance of plate-fin-and-tube banks consisting
of three circular tube rows with a pair of delta-winglets VG. They examined the angle of attack of the
VG varied from  90° to +90°. Their results showed that the common-flow-down configuration with
the inline tube arrangement was found to be more suitable than the common-flow-up configuration.
It was also revealed that the latter showed better performances for the staggered arrangement.
Yong-Gang et al. [34] reported numerically that the delta-winglet VG with an attack angle of 20° and
a heat exchanger aspect ratio of 2 provides the best integrated performance. The Colburn j-factor of
the optimal configuration is shown to increase by 35.1 45.2% with a corresponding increase of
19.3 34.5% in the friction factor. Sachdeva et al. [35] found that by the use of triangular secondary
inserts in plane rectangular channels, heat transfer can be enhanced at the cost of a greater pumping
power requirement and the bulk temperature increases by 35.46% at Reynolds number of 100.
It is obvious from the literature review that the case of convective heat transfer enhancement
of turbulent flow using the delta-winglet VG in an unsymmetrically triangular duct has received little
attention in the past and this has motivated the present study. Thus, the current study deals experi-
mentally with heat transfer and pressure drop measurements in a triangular duct for a Reynolds number
range from 24,200 to 57,100. Single, double, and triple pairs of VG are utilized and each pair of VG
was punched on one wall of the test duct. The effects of VG pair s number, the VG angle of attack,
the VG location from the leading edge of the test duct, the VG geometry, and Reynolds number are
examined in this paper. Results of Nusslet number and friction factor for turbulent flow are reported
and interpreted along with the developed correlations, for the cases studied, as a function of Reynolds
number and VG angle of attack.
47
Nomenclature
A: cross-sectional area of the duct, m2
a: constant, Eq. (15)
b: constant, Eq. (15)
c: constant, Eq. (10)
d: constant, Eq. (10)
Dh: hydraulic diameter, = 4A/p, m
f: friction factor
fo: friction factor for base case
h: convective heat transfer coefficient, W/m2Å"K
h: height of the vortex generator, m
"H: pressure difference at the orifice plate, m
º: thermal conductivity, W/mÅ"K
l: length of the vortex generator, m
L: length of the duct, m
.
m: mass flow rate, kg/s
Nu: Nusselt number
Nuo: Nusselt number for base case
p: periphery of the duct, m
"P: pressure drop, Pa
q: convective heat transfer rate, W/m2
Re: Reynolds number
S: distance between tips of winglet pair, m
St: Stanton number
T: temperature, K
_
u: mean velocity, m/s
X: distance, m
W: transverse distance, m
Greek Symbols
²: angle of attack of VG, degree
µ: dynamic viscosity, paÅ"s
n: kinematic viscosity, m/s2
Á: density, kg/m3
Subscript
b: bulk
f: film
in: inlet
out: outlet
w: water
x: axial direction
48
Fig. 3. Schematic diagram of the geometry of the duct with VG.
2. Experimental Apparatus
2.1 Facility
The experimental apparatus works in an open-loop air flow circuit which takes air from the
laboratory environment and discharges it to the atmosphere. It consists essentially of a blower, circular
tube, control valve, contraction section, development duct, test duct, thermal mixing chamber, and
heating element as shown schematically in Fig. 4. The blower forces the air through a control valve
which was used to adjust the flow rate. The air then flows through a circular tube provided with an
orifice plate flowmeter to measure the mass flow rate. The upstream end of the circular tube is coupled
with the blower using a rubber tube to minimize the vibration that might occur when the blower is
operated. Then air passes through the hydrodynamic development duct that has an equilateral
triangular cross-section. The circular cross-section of the tube is converted to a triangular cross-section
by using a convergent contraction part made of an aluminum sheet. After that, the air flows through
the test section and the thermal mixing chamber and then it is ultimately exhausted to the atmosphere.
Fig. 4. Schematic diagram of the experimental apparatus.
49
The test section and development duct are horizontal, collinear, and shared a common internal
hydraulic diameter of 35.7 mm. The development duct is long enough to reduce and disperse the
turbulence of airflow because the blower and the orifice plate cause a high turbulence in the flow
stream. The lengths of the development and test duct are 33.6 Dh and 42 Dh, respectively. The heated
walls of the test duct are made of aluminum to obtain a uniform wall temperature distribution at the
circumference of any cross-section of the heated wall. The unheated wall (lower wall) is made of
Perspex. The inner surface of the heated walls was polished by abrasive paper to a high degree of
smoothness. The test duct was heavily insulated by glass wool with thickness of 25 mm to minimize
the heat losses.
2.2 Vortex geometry
In this work, a delta winglet pair of VG was used to examine its effect on heat transfer and
pressure drop in a triangular duct. The geometrical configuration of the test duct and VG with its
location from the leading edge is shown in Fig. 3. One side of each winglet-pair was fixed on the
bottom wall while the trailing edge of each winglet was left free. The angle of attack of VG was varied
in the range of 6, 18, 24, 30, 40, and 50° to study its effect on the Nu number and friction factor values.
Two different geometries of VG were used having a 10 mm height and 40 mm length referenced as
(I, II, and III) for a single, double, and triple pair of VG. The second geometry has a 5.0 mm height
and 27.5-mm length referenced as (IV, V, and VI) for a single, double, and triple pair of VG. The
space between the pair of VG is kept at 35 mm. Both large and small sizes of VG were examined with
angles of attack of 6, 18, and 24° to investigate the influence of VG on the heat transfer and pressure
drop, while the small size VG was only examined with the last three angles of attack of 30, 40, and
50° to study its effects on the friction factor.
2.3 Instrumentation
The airflow rate was measured by using an orifice plate flowmeter which is placed in the
midsection of the circular tube (at a sufficient length to ensure that the fluctuation of the manometer
reading of the pressure difference between the two sides of the orifice plate does not exceed 0.4 mm
for the highest flow rates). The inner diameter ratio of the orifice plate to the tube diameter (Do/Dt)
is 0.7. The orifice plate flowmeter is designed according to ISO 5167 orifice plate specifications [36]
with an accuracy of Ä…0.2 mm H2O for the highest flow rate. The static pressure drop through the test
duct was measured for each wall by drilling one hole at the upstream and downstream ends of the test
duct and then fixed with an inclined manometer using PVC tubes with an accuracy of Ä…0.2 mm H2O.
The test duct was provided with 28 K-type thermocouples to measure the heated wall surface
temperature and another two thermocouples located at the upstream and downstream end of the test
duct to measure the inlet and outlet air bulk temperatures. The local bulk air temperature was
calculated by fitting straight line interpolation between the measured inlet and outlet bulk air
temperatures. In addition, 24 thermocouples were distributed at the circumference of the heated walls
uniformly at four stations to monitor the distribution of the circumferential temperatures. A digital
data logger (type CHINO with digital recorder 030) was used to record the temperature with accuracy
Ä…0.1 °C and it was connected to a computer. The thermocouple was calibrated when the junction of
the thermocouple was coupled with the bulb of a glass mercury thermometer with an accuracy of Ä…1
°C.
50
3. Uncertainty Analysis
The accuracy of the experimental results depends upon the accuracy of the individual
measuring instruments and the manufacturing accuracy of the test duct. The accuracy of any
instrument is limited by its minimum division. The calculation of the error, precision, and the general
validity of the experimental measurements has been calculated. The probable errors in the experimen-
tal results are those values that have some uncertainty. This uncertainty varies a great deal depending
upon the circumstances of an experiment. In fact, the magnitude of the experimental error is always
uncertain. In the present work, the uncertainties in heat transfer coefficient (Nusselt number), and
Reynolds number were estimated following the differential approximation method as reported by
Holman [38]. In fact, the measurements should be combined to calculate particular results, which are
desired. Therefore, it should be known the uncertainty in the final result is due to the uncertainties in
the measurements. The differential approximation method was considered to evaluate the uncertainty
in a result  Rs that is a function of the independent parameters: X1, X2, X3, . . . , Xn,
(1)
Rs = Rs (X1, X2, X3, . . . , Xn)
At the same time it may perturb the variables by "X1, "X2, "X3, . . . , "Xn and then
Rs(X1 + "X1) = Rs(X1 + "X1, X2, X3, . . . , Xn)
(2)
Rs(Xn + "Xn) = Rs(X1, X2, X3, . . . , Xn + "Xn)
Therefore, for small enough values of the quantities "X1, "X2, "X3, . . . , "Xn, the partial
derivatives can be well approximated by:
(3)
where i = 1, 2, 3, . . . , n.
If there are uncertainties W1, W2, W3, . . . , Wn in the independent variables and WR is the
uncertainty in the result on the same odds, then the uncertainty in the result can be given as
(4)
Since the values of the partial derivative and the errors in the measuring parameters may be
positive or negative, then the absolute values are considered to obtain the maximum absolute
uncertainty in the result WR. For a typical experiment, the total uncertainty in measuring the heater
input power, temperature difference (Tw - Tb), the heat transfer rate, the test duct surface area, and
the air flow rate were Ä…0.25%, Ä…0.23%, Ä…1.4%, Ä…1.2%, and Ä…0.53%, respectively. These were
combined to give a maximum error of 1.63% in Nusselt number and maximum error of 1.31% in
Reynolds number.
4. Data Reduction Method
The measured axial pressure gradient yielded a straight line on a pressure versus axial
coordinate (X) where it was computed from the following equations:
(5)
51
The thermophysical properties of the air were estimated at the film temperature using the
following equation:
(6)
The Re number is defined as
(7)
The friction factor is computed from the Fanning friction factor equation [40]:
(8)
The Darcy friction factor equals four times the Fanning friction factor. The following equation
was used to calculate the friction factor as a function of Re number:
(9)
The local heat transfer coefficient of the heated wall was computed as follows:
(10)
The local Nu number of the heated wall was calculated as follows:
(11)
The average Nu number was computed using Simpson s rule [37]:
(12)
_
_
where R represents the average Nu number along the test duct.
5. Results and Discussion
A high pressure drop occurs in the immediate neighborhood behind the winglet-body junction
which eventually causes an increase in the friction factor downstream of the location. The functionality
of the developed experimental apparatus was checked by performing pre-tests without using the VG
(base case), and the results of friction factor obtained from these tests were compared with the
experimental data of the previous work as shown in Fig. 4(a). It should be noted that the friction factor
correlations of Blasius [38], Petkhov and Popov [39], and Prandtl [40] are of a circular tube, whereas
the correlations of Chegini et al. [3] and Altemini and Sparrow [5] are of equilateral triangular ducts.
The results of Refs. 3, 38, 39, and 40 overestimated the present data by approximately 6.5, 9.7, 8.8,
and 7.6%, respectively. However, the present data are in good agreement with the results of Altemini
and Sparrow [5] with a deviation of 1.7%. Furthermore, another comparison is done between the
present Nu number for the base case and the results of the previous work of Refs. 5, 39, and 41 as
52
Fig. 5. Comparison of the present experimental results with the previous work: (a) friction factor,
(b) Nu number.
shown in Fig. 5(b). The deviation does not exceed 7.5% between the present data and the results of
Altemani and Sparrow [5].
The temperatures were measured at four axial locations along the test duct which are
approximately symmetric as shown in Fig. 6. The temperature at the lower position of the heated
wall-side (at the corner between heated and unheated wall) is slightly less than that at the upper
position of the heated wall-side (the corner at an apex angle). This is because the temperature
difference between the heated and unheated wall is large due to the conduction heat transfer, whereas
the heat balance occurs at the corner of an apex angle of the duct. This means that there is a slight
conduction heat transfer. This exhibits good agreement with the experimental results of Hassan et al.
[42].
Fig. 6. Transverse wall temperature distribution at different axial locations at Re = 29,950.
53
Fig. 7. Axial distribution of Nu number for base case.
Figure 7 shows the local Nu number against the dimensionless axial distance (X/Dh) for the
base case (X = 0 represents the beginning point from the leading edge toward the other end of the test
duct). It is observed that the Nu number is relatively proportional to Re number and the thermal
development is more rapid at a lower Re number and slower at a higher Re number. The Nu number
at Re = 24,258 shows a shorter entrance length (X/Dh = 9.8) while it becomes greater as the Re number
increases until it becomes constant after a value of Re = 37,727. This is because the increase in the
flow rate causes an increase in the flow turbulence and this would destroy and penetrate deeper into
the boundary layer especially at the corners. It is also observed that the Nu number increases at the
downstream end of the test duct. This may be attributed to the heat losses within the last portion of
the test duct. For the base case, the values of Nu number of Refs. 5 and 39 and Dittus Boelter [41]
overestimated the present data by about 7.8, 17.6, and 28.1%, respectively. The deviation of 7.8% is
expected because the cross-section of the present test duct is similar to that used by Altemini and
Sparrow [5]. However, the other two deviations are high because the difference in the cross-section
of the duct used by Refs. 39 and 41. This comparison adds another confirmation that the results of
Ref. 5 have a superiority of comparison and it demonstrates that the hydraulic diameter does not
provide an adequate rationalization of the noncircular geometry. This fact is also strongly confirmed
by Chegini and Chaturvedi [3].
For the base case, the following formula is used to determine the Nu number as a function of
Re number [5]:
(13)
The above equation can be simplified as follows:
(14)
where c and d are constants.
The fully developed region is based on 5% approach of the heat transfer coefficient to its fully
developed values. Thus, the following correlation was developed from the present data:
(15)
54
Fig. 8. Thermal entrance length versus Re number for the base case.
The entrance length may be defined as the axial location at which the heat transfer coefficient
approaches to within 5% of its fully developed values [38]. The value of Nu number changes slightly
at Re = 50,000. This means that the entrance length is nearly constant at this value of Re number as
shown in Fig. 8 which represented the dimensionless axial distance (X/Dh) versus Re number.
5.1 Effect of Reynolds number
The effect of Reynolds number on the friction factor is presented in Fig. 9 at ² = 18° for the
case of single, double, and triple pair (small size) which referred to (IV), (V), (VI). At a high Re
number a higher penetration of the air in the hydraulic boundary layer would occur (i.e., less thickness
of the boundary layer). The highest pressure drop occurred in case VI because of a small area of the
cross-section of the flow channel at the trailing edge of the VG. It clearly seems that the friction factor
is inversely proportional to Re number.
Fig. 9. Effect of Re number on the friction factor at ² = 18°.
55
The friction factor used by the previous investigators was calculated by [41]:
(16)
For the base case, Eq. (12) can be rewritten as follows:
(17)
The friction factor correlations of Blasius [38] for a smooth tube and Chegini and Chaturvedi
[3] and Nikuradse et al. [43] for the equilateral triangular duct are written as follows, respectively:
(18)
(19)
(20)
The deviation between the present data and data of Eqs. (18) to (20) is about 9.96, 6.61, and
9.32%, respectively.
The Nu number is significantly influenced by Re number. As the Re number increases, the
Nu number increases as shown in Fig. 7.
5.2 Effect of vortex generator number
The effect of the VG number on the dimensionless friction factor (f/fo) is shown in Fig. 10 for
different angles of attack of 30, 40, and 50°, respectively. The friction factor ratio for the cases (IV),
(V), and (VI) increases by about 27.31, 55.83, and 64.35% compared to the base case (without VG)
at ² = 40° and Re = 53,000. The results show a significant difference between the cases (IV) and (V)
especially at high Re number values. There is a slight change in pressure drop penalty between cases
(V) and (VI).
Generally, the existence of the VG would induce the creation of the streamwise longitudinal
vortices behind it. The spiral flow of these vortices serves to entrain the fluid from their outside into
their core. These vortices disrupt the growth of the thermal boundary layer and lead to enhance the
heat transfer between the fluid and the heated walls of the duct [3]. The increase of the number of VG
leads to increase the turbulence in fluid flow through the channel. Thus, many VG are used to examine
their influence on increasing the turbulators area on the Nusselt number. The average fully developed
Nu number of the three cases (IV, V, VI) versus Re number is depicted in Fig. 11 and compared with
the base case at ² = 18°. For example, the Nu number increases with the airflow rate and the number
of VG when the angle of attack is kept constant.
5.3 Effect of angle of attack
The effect of the VG angle of attack is one of the major parameters that affect the friction
factor. As the VG angle of attack increases the static pressure drop increases and thereby the friction
factor increases. The ratio of the (f/fo) for the cases (IV), (V), and (VI) to that of the base case is
presented in Fig. 12 for different angles of attack ranging from 6° to 50°. For small values of attack
angle, the (f/fo) ratio increases slightly but at higher values of attack angle (more than 18°) the (f/fo)
56
Fig. 10. The effect of VG number on the friction factor ratio for various angles of attack:
(a) ² = 30°, (b) ² = 40°, (c) ² = 50°.
Fig. 11. Effect of Re number on Nu number for different VG cases at ² = 18°.
57
Fig. 12. Effect of VG angle of attack on the friction factor ratio at Re = 54,000.
ratio increases significantly because the area of the VG across the flow direction increases. In other
words, the increase in the area of the VG across the flow direction serves as an obstacle in the flow
direction. This leads to increasing the pressure drop between the leading edge and the trailing edge
of the VG. Increasing the angle of attack affects the vortex circulation and a higher value of friction
factor is obtained. Of course, this is associated with a high rate of heat transfer. The friction factor
correlation for the cases IV, V, and VI as affected by the angle of attack can be written as
(21)
The values of the variables a and b in Eq. (21) are given in Table 1.
Fig. 13. Effect of angle of attack of VG on the Nu number at Re = 54,000: (a) large size, (b) small
size.
58
Table 1. Values of Variables a and b in Eq. (21)
Figure 13 displays the effect of the attack angle of the VG on the Nu number ratio for the small
size (IV, V, VI) and large size (I, II, III) at Re = 54,000. Cases (III) and (VI) have good enhancement
along the flow direction compared to other geometries at all ranges of Re number. Figure 13(a) shows
that the Nusselt number increases by about 17.8% and 24% compared to the base case (without VG)
when the angles of attack varied from 6° to 24°.
5.4 Effect of the vortex generator size
The height of the VG should be of the same order as the turbulent boundary layer thickness
to disrupt the growth of the boundary layer and to generate a perturbed flow behind the VG. Therefore,
two sizes were chosen during the experiments to investigate the VG size. The constant parameter (c)
of the Nu number correlation in Eq. (14) of the cases (IV) and (I) is presented in Fig. 14 where the
parameter (d) is considered constant. This figure shows that there is insufficient tip length of VG to
generate a turbulence flow which can penetrate deeply in the regions that have a thick boundary layer
as in the corners.
Fig. 14. Parameter of Nusselt number correlation versus Re number.
59
5.5 Effect of the vortex generator period
The period of the VG plays an important role in heat transfer augmentation. Near the leading
edge of the test duct, the thermal boundary layer is thin and it cannot be significantly perturbed by
the VG. Therefore, a 33.6 Dh of the fully developed duct is used to allow the thermal boundary layer
to grow to the center line of the duct before it reaches the leading edge of the heated duct. In other
words, the hydraulic and thermal boundary layers are fully developed at the entrance of the test duct.
For all cases studied, the strength of the longitudinal vortices is reduced to a great extent although a
spiral flow still exists after a certain length behind the VG due to the recirculation of the flow to repeat
the mixing of the cooler stream of the core with the hot fluid from the wall. Therefore, another array
of the VG is almost needed after a distance of 25 Dh from the leading edge of the test duct.
6. Conclusions
In this study, the effect of delta-winglet pairs of a vortex generator on the heat transfer and
fluid flow characteristics in an unsymmetrically heated triangular duct are experimentally studied.
The effects of Reynolds number, VG number, VG angle of attack, the VG geometry, and the VG
location were examined. The following conclusions can be drawn from this study:
" When Re number increases, the Nu number increases and the friction factor decreases.
" The Nu number increases when the number of VG increases. The pressure drop explicitly
increases at the trailing edge of VG.
" The Nu number and friction factor are strongly affected by the VG angle of attack. The
maximum heat transfer augmentation obtained is 28.5% for case III at ² = 24° and Re = 57,700.
The friction factor slightly increases when the angle of attack varied from 6° to 18°, whereas it
apparently increases at ² = 18°.
" The results exhibit that the large geometry of the VG has a significant effect on the Nu number
rather than the small VG geometry. The friction factor is slightly increased for both geometries.
" The turbulent flow obviously vanished after 25 Dh from the leading edge of the test duct. Thus,
another array of VG is almost needed to spiral the fluid flow to mix the cooler fluid at the core
with the hot fluid at the wall.
Acknowledgments
This work was performed at the Mechanical Engineering Department at the University of
Anbar. The authors would highly like to express their gratitude to this Department for providing the
required equipment and facilities. Dr. Raad Al-Nuaimi s contribution in constructing the experimental
rig is gratefully acknowledged.
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