Numerical Heat Transfer, Part A, 61: 735–753, 2012
Copyright # Taylor & Francis Group, LLC
ISSN: 1040-7782 print=1521-0634 online
DOI: 10.1080/10407782.2012.667649
EVALUATION OF ENHANCED HEAT TRANSFER
WITHIN A FOUR ROW FINNED TUBE ARRAY
OF AN AIR COOLED STEAM CONDENSER
Rupak K. Banerjee1, Madhura Karve1, Jong Ho Ha2,
Dong Hwan lee3, and Young I. Cho4
1
Department of Mechanical Engineering, University of Cincinnati,
Cincinnati OH, Ohio, USA
2
Korea Heat Exchanger Co. Ltd, Gunsan, Jeonbuk, Korea
3
Chonbuk National University, Jeonju, Joenbuk, Korea
4
Department of Mechanical Engineering, Drexel University, Philadelphia,
Pennsylvania, USA
Air cooled steam condensers (ACSC) consist of finned-tube arrays bundled in an A-frame
structure. Inefficient performance under extreme temperature operating conditions is a
common problem in ACSCs. The purpose of this study was to improve the heat transfer
characteristics of an annular finned-tube system for better performance in extreme climatic
conditions. Perforations were created on the surface of the annular fins to increase heat
transfer coefficient (h). Mesh generation and finite volume analyses were performed using
Gambit 2.4.6 and Fluent 6.3 with an RNG k–e turbulent model to calculate pressure drop
(DP), heat flux (q), and heat transfer coefficient (h). Solid (no perforations) finned-tubes
were simulated with free stream velocity ranging between 1 m=s–5 m=s and validated with
the published data. Computations were performed for perforations at 30 interval starting
at 60 , 90 , 120 , 150 , and 180 from the stagnation point. Five cases with single
perforation and three cases with multiple perforations were evaluated for determining the
maximum q and h, as well as minimum DP. For the perforated case (perforations starting
from 60 at interval of 30 ), the fin q and h performance ratios increased by 5.96% and
7.07%, respectively. Consequently, the fin DP performance ratio increased by 11.87%.
Thus, increased q and h is accompanied with a penalty of higher DP. In contrast, a single
perforation location at 120 provided favorable results with a 1.70% and 2.23% increase in
q and h performance ratios, respectively, while there was a relatively smaller increase (only
1.39%) of DP performance ratio. Perforations in the downstream region at 120 , 150 ,
Received 18 February 2011; accepted 3 February 2012.
Rupak K Banerjee and Madhura Karve have contributed equally to this article. Madhura Karve
implemented the mathematical analyses, numerical calculations, and drafted initial versions of the
article under the supervision of Rupak K: Banerjee. Dong Lee, Jong Ha, and Young Cho provided
technical insights into this research during model design and numerical data analysis. Madhura Karve
could not effectively complete the rebuttal and implement the modifications based on the reviewer’s
comments within the stipulated time because of her commitment with her new employer. In light of
this, Rupak K. Banerjee, with the assistance of Marwan Al-Rjoub, Anup Paul and Kranthi Kolli,
completed the rebuttal that was acceptable to the other co-authors.
Address correspondence to Rupak K. Banerjee, Mechanical Engineering Program, School of
Dynamic Systems, 593, Rhodes Hall, ML 0072, University of Cincinnati, Cincinnati, OH 45220, USA.
E-mail: Rupak.Banerjee@uc.edu
735
736
R. K. BANERJEE ET AL.
NOMENCLATURE
A
Ac
Cp
D
E
f
h
j
k
l
n
p
q
s
T
t
v
total surface area, m2
frontal area, m2
specific heat at constant pressure,
J=kg-K
outer tube diameter, m
total internal energy, m2=s2
friction factor
heat transfer coefficient, W=m2-K
thermal conductivity, W=m-K
turbulent kinetic energy, m2=s2
fin length, m
number of tube rows
static pressure of air, Pa
heat flux (q ¼ h DT), W=m2
fin pitch, m
temperature, K
fin thickness, m
velocity of air, m=s
Nu
Pr
Rein
SL
ST
a
e
t
q
m
h
DP
r
Nusselt number ¼ h D=j
Prandtl number ¼ m CP=j
Reynolds number ¼ q V D=m
longitudinal pitch of tubes, m
transverse pitch of tubes, m
thermal diffusivity, m2=s
turbulent energy dissipation, m2=s3
specific volume, m3=kg
density of air, kg=m3
viscosity of air, kg=m-s
angular location, degrees
differential pressure, Pa
d
d
d
i þ dy
j þ dz
k
¼ dx
Subscripts
eff
effective
max
maximum
t
turbulent
and 180 also resulted in a similar favorable outcome. Furthermore, the spacing of the fins
along the arms of an A-frame ACSC was altered to decrease DP across the finned-tube
array. Fin spacing in the A-frame structure with sparsely spaced fins in the center resulted
in a 1.80% reduction in DP. Thus, penalty in DP for a perforated fin can possibly be offset
by changing the fin spacing along the arms of an A-frame structure.
1. INTRODUCTION
Finned-tube heat exchangers are commonly employed as air cooled steam condensers (ACSC) in power plants or as radiators in automobiles. The design of such
an array has been investigated by previous researchers due to the complexities
involved in the cross flow of air over the tube bundles. The efficiency of finned-tubes
depends on geometric parameters such as fin and tube spacing, operating conditions,
and material properties. Efficient fins help to reduce the number of tube rows,
resulting in compactness of design as well as saving energy.
A typical ACSC has four row finned-tube arrays stacked in an A-frame structure. High pressure and high temperature steam is let into these finned-tubes, which
are cooled by a large fan at the base of the A-frame structure. It is a relatively inexpensive alternative to the water cooled condensers. ACSCs have gained more importance recently as a water conservation measure in regions with limited water
resources. However, under extreme climatic conditions ACSCs are faced with problems such as inefficient performance in high temperature regions, such as the desert,
and freezing and choking of finned-tubes in low temperature regions. Enhancing the
heat transfer coefficient (h) of the fins can help in minimizing the problems associated
with extreme climatic conditions. Researchers have explored several approaches of
reducing air side resistance in order to increase heat transfer from finned-tube
surfaces.
ENHANCED HEAT TRANSFER WITHIN A CONDENSER
737
Heat transfer enhancement techniques, such as surface vibration and magnetic
field, are classified as active techniques based on the use of external power [5]. In contrast, passive techniques included: treated or rough surfaces, and an extended surface
with alterations such as wavy fins, slotted fins, and perforated fins. Webb [1] surveyed
developments in plate-fin and circular-fin heat exchangers and suggested the use of
slotted and punched fins for increasing the heat transfer. Webb [1] recommended
the use of the correlation for heat transfer presented by Briggs and Young [2], and
correlation for pressure drop presented by Robinson and Briggs [3]. Žukauskas [4]
introduced correlations for Nusselt number (Nu) and pressure drop (DP) for an inline
as well as a staggered arrangement of finned-tube bundles. Flow through the tube
bundles was reported to have three circumferential regions: laminar, turbulent, and
separated flows. Žukauskas [4] also mentioned the need to artificially disrupt the
boundary layer to attain better performance of the heat exchanger.
Wang et al. [6] studied wavy fins and presented an empirical correlation for
herringbone plate-fin-tube heat exchangers. Tao et al. [7] studied plate fins with a
slotted X arrangement of strips commonly used in air conditioning. This arrangement resulted in a 97% increase in h and a 63% increase in DP. Cheng et al. [8] studied three different configurations of X protruding strips, positioned along the flow
direction according to the rule of ‘‘front coarse and rear dense.’’ Recent studies of
pin fins by Sahin et al. [9] and of rectangular fins by Shaeri et al. [10] increased
the surface contact area of fluids by perforations. This assisted in improving turbulence and mixing, and in reducing the friction factor.
Jang et al. [11] was the first to study annular finned-tubes numerically. Simulations of a staggered arrangement of annular finned-tubes under dry and wet operating conditions were conducted and validated experimentally. They showed that
isothermal fin approximation overestimates the heat transfer coefficient by 5–35%.
Mon et al. [12] performed numerical calculations of annular finned-tubes to investigate the effect of variation of fin spacing on the rate of heat transfer. Mon et al. [12]
accounted for heat conduction and convection through fin thickness and applied
constant temperature to tube walls. They reported that the boundary layer development on fin and tube surfaces mainly depends on the ratio of fin spacing to fin height.
Abundant literature regarding annular fins has been published, but the idea of
perforated annular fins has not been studied in detail. Perforations disrupt the
boundary layer and enhance mixing of the flow. This motivated the authors to evaluate performance of the annular finned-tubes with perforations. The main objective of
the present study was to assess the increase in heat transfer and pressure drop (DP)
due to the circular perforations. Annular finned-tubes, which typically carry steam
inside, were cooled with forced air flow. This research employed three-dimensional
numerical calculations of four row finned-tubes. Comparison of both the heat transfer rates and changes in DP was performed for solid (no perforations) and perforated
(perforations at 60 to 180 , with 30 interval) fins. Variations in DP were
evaluated for various combinations of perforations.
1.1. Losses in ACSC
Different types of losses such as pressure drop (primary loss) across the tube
bundle and secondary losses such as inlet loss, fan loss, and jetting loss were reported
738
R. K. BANERJEE ET AL.
by the researchers in the A-frame structures. Kroger [13] studied various secondary
losses in A-frame structures and reported that secondary losses were of the same
order as primary pressure losses. Since perforations in a fin increase pressure drop
[7, 12], it is important to assess if such losses could be offset by reducing secondary
losses, such as by geometric modifications of the fin spacing within the fin-tube bundles. Meyer et al. [14] investigated plenum losses and found that heat exchanger inlet
geometry has an influence on plenum losses. Further, Meyer et al. [15] conducted
experiments with four types of eight blade axial flow fans. They used both elliptical
and rectangular cross-sections of tubes and fins to study inlet losses. The
cross-sectional profile of finned-tubes was found to be one of the important factors
influencing inlet pressure drop. Also, losses were observed to increase with a decrease
in angle of incidence of inlet air. However, to the authors’ knowledge, heat exchanger loss due to the changed spacing of fins within the finned-tube bundle has not been
reported in the existing literature. This study altered the spacing of the fins along the
arms of an A-frame structure to assess reduction in pressure drop, which, in turn,
may off-set the increased pressure losses due to perforations.
2. Methodology
2.1. Finned-Tube Analysis
Numerical calculations were conducted by simulating three-dimensional air
flow and heat transfer over solid and perforated finned-tube configurations.
2.1.1. Computational domain. The domain with four finned-tubes in staggered arrangement and symmetric boundaries was developed, as shown in Figure 1a.
This type of arrangement is typical for ACSCs. Geometrical dimensions were similar
to those by Jang et al. [11], as shown in Table 1. Thickness of the tubes was considered
with the inner diameter of the tube as the boundary of the domain. Circular perforations, 4 mm in diameter, were located at a pitch interval of 30 around the annular fin,
starting at 60 (Figure 1b), to 60 in the clockwise direction. Since the domain
showed geometrical symmetry, only the top part from 0 to 180 of the finned-tube
was developed. Thus, a perforated finned-tube with a symmetric surface had four perforations at 60 , 90 , 120 , and 150 and a half perforation at 180 . Domain boundaries were extended to minimize the effect of the uniform flow boundary condition
at the inlet on the flow velocity near the finned-tube walls. The inlet plane of the flow
domain was located 1.5 fin diameters upstream from the first finned-tube, while the
outlet plane was 5 fin diameters downstream from the last finned-tube.
2.1.2. Governing equations. Conjugate fluid flow and heat transfer calculations were carried out to solve conduction in the tube and fin walls and forced
convection over the finned-tubes.
Continuity and momentum equations for the air side flow were solved as
follows.
r ðq vÞ ¼ 0
ð1Þ
r ðq v vÞ ¼ rp þ r mðrvÞ
ð2Þ
ENHANCED HEAT TRANSFER WITHIN A CONDENSER
739
Figure 1. Geometry of finned-tube array. (a) Staggered arrangement of fins labeled fin-1–fin-4 from left to
right and with a perforated finned surface; (b) angular locations of perforations on any fin surface; and (c)
3-D view of finned-tube array.
Table 1. Dimensions of finned-tube array for computational and analytical calculations
Geometrical parameter
Inside tube diameter (Di)
Outside tube diameter (D)
Fin outer diameter (Df)
Fin thickness (t)
Fin spacing (s)
Transverse tube pitch (ST)
Longitudinal tube pitch
Tube material
Fin material
Dimensions in mm
23.2
27
41
0.5
3.5
21.5
73
Steel
Aluminum
740
R. K. BANERJEE ET AL.
Energy equation [16] for the convection over the finned-tubes was solved as follows.
r ½vðqE þ pÞ ¼ r ðjeff rTÞ
ð3Þ
In the above equation the effective conductivity is as follows.
jeff ¼ ða cP Þ=meff
ð4Þ
Here, the effective viscosity is as follows.
meff ¼ m þ mt
ð5Þ
The Reynolds number (Re) was calculated using the maximum velocity in the
domain and the outer diameter of the tube as the length parameter
(Re ¼ q D vmax =m). Re was found to be in the lower turbulent scale within the
range of 4,000 to 24,000. Steady state numerical calculations were carried out using
the RNG k–e turbulent model to account for small scale vortices and the effects of
swirl on turbulence. Standard equations were solved for the RNG k–e model [16]:
r ðq k vÞ ¼ rðak meff r kÞ qe þ Gk
r ðq e vÞ ¼ r ðae meff r eÞ þ C1e
2
e
Gk C2e q
Re
k
k
hei
ð6Þ
ð7Þ
where ak ¼ ae ¼ 1.393 are the inverse effective Prandtl numbers for k and e, respectively. The model constants C1e (¼ 1.42) and C2e(¼ 1.68) are derived analytically by
the RNG theory [16]. The term Gk represents the generation of turbulence kinetic
energy due to the mean velocity gradients and is calculated as follows.
Gk ¼ qvi vj
qvj
qxi
ð8Þ
mt is turbulent viscosity and is defined based on the turbulence model being
used. For the RNG k–e turbulence model [16],
mt ¼ q cm k2 =e; where cm ¼ 0:0845
Re ¼
cm qg3 1 gg e2 0
1 þ b g3
k
ð9Þ
ð10Þ
where b ¼ 0.012, g ¼ Sek, and g0 ¼ 4.38; here, S is strain rate magnitude.
Energy equation [16] for the convection over the finned-tubes was solved using
the following equation.
ENHANCED HEAT TRANSFER WITHIN A CONDENSER
741
where E is defined as
p v2
E¼Hþ þ
q 2
ð11Þ
and H ¼ specific enthalpy, defined as
H¼
Z
cp qT
ð12Þ
Tref
In solid, the energy equation that was solved has the following form [20].
r ðvqhÞ ¼ r ðjrTÞ
ð13Þ
2.1.3. Boundary conditions. Solid (no perforations) and perforated
(perforations at 60 to 180 with 30 interval) cases were simulated for inlet free
stream air velocities of 1, 3, and 5 m=s at 300 K for validation of Nusselt number
(Nu) and friction factor (f). Other combinations of perforated cases were simulated
for free stream velocity of 3 m=s. Density of the inlet air is calculated during
numerical calculations based on incompressible ideal gas law [16], where density is
dependent only on the operating pressure and local temperature. No slip condition
was applied to the outer tube and the fin walls. The inner tube wall was assumed to
have a constant temperature of the condensing steam (350 K). This model considered
conduction within the fin and the tube thickness. Symmetric planes, as shown in
Figure 1a, were assumed to have zero diffusive flux. All the normal gradients of flow
variables were set to zero at the plane of symmetry. Free stream air properties were
specified as viscosity (m) ¼ 1.798 105 kg=m-s, specific heat (Cp) ¼ 1006.43 J=kg-K,
and thermal conductivity (j) ¼ 0.0242 W=m-K.
Hexahedral elements, as shown in Figure 2, were used to mesh the flow
domain. Total element count in the domain was 426,500 and the maximum skewness
was 0.76. Mesh with finer element size was used near the fins and outer tube walls to
Figure 2. Hexagonal mesh. (a) Mesh across the domain with fine mesh elements near the finned-tube walls
and relatively coarser mesh elements away from the walls; and (b) mesh over the fin and tube surfaces.
742
R. K. BANERJEE ET AL.
Table 2. Heat transfer coefficient and Nusselt number results for grid independence study
Cases
Fine case
Present case
Coarse case
Heat transfer coefficient
Nusselt number
72.5
72.6
67.3
2,994.9
3,001.8
2,782.4
capture the sharp velocity gradient accurately. The size of mesh elements increased
and became coarser away from the walls. Two additional cases of solid fins, coarser
mesh with 15% less mesh count, and finer mesh with 15% more mesh count were
assessed. Nu and h values of all three cases (fine, coarse, and present) were compared. Nu and h values (Table 2) for the present solid fin case were within 0.2%
of the finer mesh case.
2.2. A-Frame Structure Analysis
A-frame structure of the ACSC was simulated to reduce the overall pressure
drop in the structure by altering the fin spacing across the tube.
2.2.1. Computational domain. A two-dimensional geometry of the right
symmetric half side of the A-frame was developed, as shown in Figure 3a. Fins representing four row finned-tube arrays along the arms of the A-frame were created. Perforations on the finned-tubes were not considered in this representative fin array.
Figure 3. Uneven distribution of fins along the arms of the A-frame. (a) Computational domain and
boundary conditions, and (b) uneven fin distribution cases.
ENHANCED HEAT TRANSFER WITHIN A CONDENSER
743
Outlet boundary of the domain was located at a distance of 25 times fin length. Four
cases, with change in spacing of the fins over the entire arms of the A-frame, were
simulated to assess the reduction in DP, as described in Figure 3b. Total number
of fins in each of the cases was the same.
Continuity and momentum equations were solved, as described in Eqs. (1) and
(2), respectively, to calculate the pressure drop (DP) across the fins. The DP in the
system was calculated at three locations, marked as 1, 2 and 3 in Figure 3a. Steady
state lower turbulent scale (Re ranges from 4,000 to 24,000) simulations were performed using k–x shear stress transport model (SST) equations, as shown below.
r ðq k vÞ ¼ r ðCk r kÞ þ Gk Yk
r ðq x vÞ ¼ r ðCx r xÞ þ Gx Yx þ 2ð1 F1 Þqrx;2
ð14Þ
1 qk qx
x qxj qxj
ð15Þ
where rk and rw are the turbulent Prandtl numbers of k and x, respectively
[16]. The Yk and Yw represent the effective dissipation of k and x, respectively
[16]. Similarly, Gk and Gx are generation of k and x, respectively [16]. Turbulent viscosity required to calculate, Ck (effective diffusivity of k) and Cw (effective diffusivity
of x) is defined as follows.
mt ¼
qk
1
x max 1 ; SF2
a
ð16Þ
a1 x
where a is the turbulent viscosity damping factor for low-Reynolds-number
correction, S is the strain rate magnitude, and F1, F2 are the blending function for
modeling near-wall and far-field zones in the k–x SST turbulence model. Other constants that are used in the calculation for the turbulence model [16] are ak;1 ¼
1:176; ak;2 ¼ 1:0; rx;1 ¼ 2:0; rx;2 ¼ 1:168; a1 ¼ 0:38; bi;1 ¼ 0:075; and bi;2 ¼ 0:0828.
2.2.2. Boundary conditions. An inlet boundary represented forced air velocity of 3 m=s from the fan. A pressure outlet boundary condition was used at the outlet boundary of the domain. A no slip wall boundary condition was applied on the
fin array. A wall boundary condition was also specified at boundaries, representing
the steam duct and other walls of the A-frame. A symmetry condition was imposed
on the left side boundary by specifying all normal gradients as zero. Geometry was
meshed with finer quadrilateral elements between the fins and with relatively coarser
triangular elements in the remaining domain. The interface type of boundary was
used to account for the transition from quadrilateral to triangular mesh elements.
Total element count was about 100,000 and had maximum skewness of 0.75.
2.3. Validation with Analytical Results
Computational results of solid fins were validated for the Nusselt number (Nu)
with Briggs and Young correlation [2], and for friction factor (f) with Robinson and
Briggs correlation [3], as mentioned in Eqs. (17) and (18), respectively.
744
R. K. BANERJEE ET AL.
Nu ¼ 0:134 Re0:681 Prð1=3Þ f ¼ 18:93 Re0:361 ST
D
hsi0:2 hsi0:1134
l
t
0:927 0:515
ST
SL
ð17Þ
ð18Þ
The Reynolds number (Re) in Eqs. (17) and (18) were calculated at the minimum cross section area, perpendicular to the flow. According to the continuity equation, the area of minimum cross section has maximum flow velocity. For finned-tube
configurations, the area of minimum cross section is the thin space between the two
fins. Hence, Eq. (19), reported by Zukauskas [4], was used to calculate the maximum
velocity.
vmax ¼
ST v
ST D
ð19Þ
Friction factor (f) for the numerical simulations was calculated as
f ¼
DP
nqv2
ð20Þ
Figure 4. Comparison of computational and analytical results for the solid case. (a) Comparison of Nusselt number with Briggs and Young correlation, and (b) comparison of friction factor with Robinson and
Briggs correlation.
ENHANCED HEAT TRANSFER WITHIN A CONDENSER
745
Table 3. Fin performance ratios and angular location (theta in degrees) of perforations along the fin surface for various perforation cases
Cases
Solid
Perforated
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8
Perforation
location
Heat flux,
q (Q=m2)
Fin q
performance
ratio (%) (A)
Pressure
drop,
DP (Pa)
Fin DP
performance
ratio (%) (B)
Relative
q-DP factor
(A=B)
—
60 to 180
60
90
120
150
180
120 , 150
120 , 180
120 , 150 , 180
3,136.11
3,323.05
3,167.73
3,172.90
3,189.43
3,189.06
3,170.78
3,229.02
3,223.49
3,261.08
—
5.96
1.01
1.17
1.70
1.69
1.11
2.96
2.79
3.99
115.4
129.1
120.9
122
117
117.4
116.9
119.8
119.1
119.8
—
11.87
4.77
5.72
1.39
1.73
1.30
3.81
3.21
3.81
—
0.50
0.21
0.21
1.23
0.97
0.85
0.78
0.87
1.05
Indicates favorable fin performance values.
Values of DP in Eq. (20) and Nu values are obtained from a post processing
step in Fluent. A reference temperature of 300 K was used for calculating Nu.
Comparison of analytical and computational results of Nu and f is shown in
Figure 4. Computational results for Nu number were within 15.7% of the analytical
correlation (Figure 4a). From Figure 4b, it is observed that velocity changes do not
alter the general decreasing trend of the friction factor, and the difference between
analytical and numerical results remained nearly the same for all velocity values.
A 43.7% difference for a velocity of 5 m=s was observed between numerical and analytical results. Though the results for f are not close to the analytical results, they are
consistent with the data reported by Mon et al. [12]. They showed that the numerical
results for Nu and f were within 25% and 40%, respectively, with those obtained
by analytical results. Other perforation combinations, as shown in Table 3, were also
simulated in order to evaluate the variation of DP with the location of perforation.
3. RESULTS
3.1. Solid and Perforated Finned-Tubes
Numerical calculations were conducted for the solid fins (no perforations) to
assess the variation of velocity and temperature distributions across the domain.
The velocity and temperature distribution on the surface of solid fins were then
compared with those of perforated fins (perforations at 60 to 180 with 30
interval). Other cases with single, double, and triple perforations, as described in
Table 3, were simulated to assess the effect of perforation location on velocity and
temperature profiles. Percentage reductions in perforated fin surface area for heat
transfer calculation for various perforation combinations are shown in Table 4.
The case of solid fins was used as the baseline case to compare subsequent cases
involving various combinations of perforations.
3.1.1. Comparison of solid and perforated cases. Velocity vectors and
temperature of the solid finned-tubes for the free stream velocity of 3 m=s are shown
746
R. K. BANERJEE ET AL.
Table 4. Fin h performance ratios for various perforation cases
Cases
Solid
Perforated
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8
Perforation
location
Reduction in
area of fins (%)
Heat transfer
coefficient,
h (W=m2-K)
—
60 to 180
60
90
120
150
180
120 , 150
120 , 180
120 , 150 , 180
—
10.79
2.40
2.40
2.40
2.40
1.20
4.80
3.60
5.99
76.4
81.8
77.3
77.4
78.1
78
77.5
78.9
79.1
80.1
Fin heat transfer
performance
ratio (%)
—
7.07
1.18
1.31
2.23
2.09
1.44
3.27
3.53
4.84
Indicates favorable fin performance values.
in Figure 5. It was observed that free stream air flow approaching the finned-tube
array encountered stagnation at the angular location (h) of 0 at the first row
(Figure 1b). At this point, flow velocity reduced to zero and total pressure, being
inversely proportional to velocity, was the maximum in the domain. Pressure started
decreasing as the flow advanced, creating favorable flow conditions [dp=dx < 0] in the
upstream region of all fins. Maximum velocity was observed at about h ¼ 90 . Thereafter, velocity started decreasing and pressure started increasing, creating a positive
pressure gradient [dp=dx > 0]. This resulted in flow separation from the fin wall, creating wake areas. Formation of a recirculation zone was observed in the downstream
regions of the first, second, and third finned-tube rows followed by a large wake area
at the downstream region of the fourth finned-tube row. Fluid particles in this region
lost energy in overcoming the shear forces in the boundary layer. The remaining
energy was insufficient to resist increasing pressure forces, resulting in a reversal
of fluid particles. Because of the slower air flow, high temperatures persisted in wake
areas. Thus, the upstream half of the region experienced higher heat removal than
the downstream region.
Subsequent to the solid fins, perforations were introduced in the downstream
region with an attempt to improve the flow characteristics and increase the heat
transfer efficiency. Four and one-half perforations were created on a symmetric
model of the fin surface (equivalent to nine perforations on a complete fin surface),
as discussed earlier (Figure 1b). Solid and perforated fins were compared at a free
stream velocity of 3 m=s, using velocity vectors (Figure 5a for solid fin and
Figure 5b for perforated fins) and temperature contours. Temperature contours of
solid fins (Figure 5c) and perforated fins (Figure 5d) showed a reduction in size of
the high temperature zones in the downstream region of the tubes. The average temperature of the fins and tube walls in the perforated case was 4 K less than the solid
case. In general, a significant reduction in temperature was observed in the downstream region of the fourth row of finned-tubes. Comparison of velocity contours
showed that perforations altered the velocity gradients in the domain. Recirculation
zones in the downstream region of the first and subsequent finned-tubes were disrupted by the perforations. As a result, these recirculation zones shrank in size.
ENHANCED HEAT TRANSFER WITHIN A CONDENSER
747
Figure 5. Comparison of solid and perforated fins for free stream velocity of 3 m=s. (a) Velocity vectors for
the solid fin case; (b) velocity vectors for the perforated fin case; (c) temperature contours for solid fins; and
(d) temperature contours for perforated fins (color figure available online).
Perforations in this region also increased the mixing of air in the downstream areas.
An overall enhancement in the values of heat flux (q) and heat transfer coefficient (h)
were observed for the perforated fins as compared to the solid fins.
A comparison of the Nusselt number (Nu) number along the fin surface for h
ranging from 0 to 180 is plotted in Figure 6 for solid and perforated fins. As shown
in Figure 6a, the first row of finned-tubes was directly exposed to free stream air flow
and was generally unaffected by perforations. However, the value of Nu number for
the perforated finned-tubes was observed to increase gradually as the flow advanced
from fin-1 to fin-4. As discussed earlier, the flow over the solid fin was detached from
the wall at about 90 , resulting in a recirculation zone in the downstream region. Nu
number at h ¼ 90 (near the point of detachment) was observed to have almost the
same value for perforated and solid fins (Figures 6a and 6b). As angular locations
from 120 to 180 were in the recirculation areas, perforations at these locations
improved mixing due to enhanced turbulence. Thus, Nu numbers were observed to
increase for every perforated finned-tube for h values starting from 90 to 180 .
The flow in the upstream region of the inner finned-tubes (rows 2 and 3) was influenced by the increased mixing and turbulence of air in the upstream region. Hence,
748
R. K. BANERJEE ET AL.
Figure 6. Plot of Nusselt number versus h (theta in degrees) along the fin surface shows enhanced heat
transfer. (a) Fin 1, first finned-tube facing the free stream air flow at 3 m=s; (b) fin-2; (c) fin-3; and (d)
fin-4, the last finned-tube in the direction of free stream.
a considerable rise in Nu was observed at the upstream locations for the third
(Figure 6c) and fourth (Figure 6d) finned-tubes. It was evident that downstream
finned-tubes benefited from the perforations of upstream fins. The increase in the area
averaged Nusselt number (Nu) for the third and the fourth perforated finned-tube
was 6.7% and 9.5%, respectively. The percentage increase of Nu number for perforated fins was defined with respect to the solid fins as [(Nuperforated Nusolid) 100=Nusolid]. An increase in Nu for the entire array of finned-tubes was observed to
be 7.07%.
Žukauskas [4] reported that hydrodynamic forces arising due to the turbulent
fluctuations of the flow and pressure was one of the causes of flow-induced vibrations.
Thus, limiting the pressure drop (DP) becomes one of the primary challenges in the
design of finned-tubes. Perforations, which increased h and the Nu number, also
increased the pressure drop by about 11.87% [(DPperforated DPsolid) 100=DPsolid].
Thus, for better optimization perforations should be placed only at the locations
which result in a minimum increase in DP and the maximum Nu (or h).
3.1.2. Performance ratios. Symmetric fin combinations were assessed to
find the location of perforation that result in higher h and q, but lower the increase
in DP, as shown in Table 3. Four combinations of single perforation at 60 (case 1),
90 (case 2), 120 (case 3), and 150 (case 4) were evaluated. Due to the geometric
symmetry, half perforation was created at h ¼ 180 (case 5). Based on the results
of the best single perforation location, additional cases with multiple perforations,
ENHANCED HEAT TRANSFER WITHIN A CONDENSER
749
120 –150 (case 6), 120 –180 (case 7) and 120 –150 –180 (case 8), were assessed to
obtain a combination that has a better balance between enhanced q and minimal
increase in DP. Three fin performance ratios were introduced to calculate a percentage increase in DP, q, and h values for perforated fin cases in relation to the solid fin.
Fin DP performance ratio ¼
DPperfo DPsolid
100
DPsolid
ð21aÞ
Fin q performance ratio ¼
qperfo qsolid
100
qsolid
ð21bÞ
Fin h performance ratio ¼
hperfo hsolid
100
hsolid
ð21cÞ
Fin q and DP performance ratios for single and half perforation cases in the
semi-circular fin domain with reference to the solid case are shown in Figure 7. It
was observed that case 3 (h ¼ 120 ) resulted in the least increase in DP and maximum
increase in q values. The results for case 4 (h ¼ 150 ) and 5 (h ¼ 180 ) showed a somewhat lower increase in q when compared to the increase in DP. However, cases 1
(h ¼ 60 ) and 2 (h ¼ 90 ) showed a reverse trend with a maximum increase in DP
and the least increase in q values. Thus, perforation at h ¼ 60 and 90 were not beneficial and were of little practical use.
3.1.3. Relative q-DP factor. The efficiency of perforations was determined to
find the optimum case having the least increase in DP with the maximum increase in
q. Calculations, as shown in the last column of Table 3, were conducted to find the
relative q-DP factor, a dimensionless quantity defined as the ratio between fin q performance ratio and fin DP performance ratio.
Figure 7. Plot of fin q- and h- performance ratios for single perforation configurations (cases 1–5).
750
R. K. BANERJEE ET AL.
fin q performance ratio
fin Dp performance ratio
ð½qperfo qsolid =qsolid Þ
¼
ð½DPperfo DPsolid =DPsolid Þ
Relative q DP factor ¼
ð22Þ
A higher value of relative q-DP factor implies a combination of maximum
increase in q values in conjunction with a minimum increase in DP values. Thus, a
relative q-DP factor value greater than unity indicates a better performing fin configuration.
Amongst the single perforated cases, perforation at h ¼ 120 (case 3) provided
the best result with a 1.70%increase in fin q performance ratio, while an increase in
fin DP performance ratio was only 1.39%. Thus, the relative q-DP factor of 1.23 for
case 3 was better as compared to cases 4 (0.97) and 5 (0.85). Relative q-DP factor was
even lower (0.21) for the perforations at h ¼ 60 (case 1) and h ¼ 90 (case 2). These
combinations resulted in a significant increase in DP causing a lower relative q-DP
factor. It was evident that perforations placed in the downstream half portion
(between h ¼ 90 to 180 ) were more advantageous than in the upstream half portion
(between h ¼ 0 to 90 ). Hence, for better relative q-DP factor, it was recommended
to have perforations only in the downstream region beyond the point of detachment
(h ¼ 90 ). As discussed earlier, the perforated case (perforations at 60 to 180
with 30 interval) had high pressure drop and, hence, the relative q-DP factor was
less than 1. Results of fin h performance ratio, tabulated in Table 4, also confirm that
perforations in the downstream region result in better heat transfer.
An increase in all three fin performance ratios (q, h, and DP) and relative q-DP
factor for multiple perforation cases are also shown in Tables 3 and 4. Case 8 showed
a maximum increase in q (3.99%) performance ratio coupled with a relatively lower
increase in DP (3.81%) performance ratio among all the combinations. Since the relative q-DP factor was greater than 1 for case 8 (1.05), it was considered more favorable than the other multiple perforation cases. Cases 6 (relative q-DP factor ¼ 0.78)
and 7 (relative q-DP factor ¼ 0.87) can be incorporated if higher heat transfer rates
are desired, while an increase in pressure drop is allowable for the structure. However, it must be noted that from the perspective of relative q-DP factor, case 3 (perforation at h ¼ 120 ; relative q-DP factor ¼ 1.23) is better than case 8 (perforations at
h ¼ 120 , 150 , and 180 ; relative q-DP factor ¼ 1.05).
3.2. Fin Spacing in an A-Frame Structure
Four cases of fin spacing were configured, as shown in Figure 3b. Analysis was
initiated with case A, the base line case, with evenly spaced fins at a pitch of 3.5 mm.
Velocity vectors and pressure contours were plotted to study the flow field in the
domain. Pressure drop was measured across all three locations and tabulated in
Table 5. It was observed that velocity at the upstream side of the fins increased from
the inlet end towards the top end. Cases B and C were configured such that fins were
clustered at the inlet end and the fin spacing was increased towards the top end.
However, DP across the fins for case B increased by 2.1% (location 2). Case C
showed a favorable change with a decrease in DP by 1% (location 2). Case C showed
ENHANCED HEAT TRANSFER WITHIN A CONDENSER
751
Table 5. Pressure drop (DP) characteristics for combinations of fin spacing
Case
Location 1
DP (Pa)
2.85
2.91
2.82
2.80
A
B
C
D
% change in DP
–
2.11
1.16
1.75
Location 2
DP (Pa)
2.87
2.93
2.84
2.82
% change in DP
–
2.1
1.03
1.86
Location 3
DP (Pa)
1.19
1.26
1.18
1.18
% change in DP
–
5.88
0.88
1.28
indicates favorable change in DP.
that more air flow occurred through the central region of the fin array. Based on this
observation, case D with clustered fins at both inlet and top ends and sparsely spaced
fins at the center was developed.
A comparison of velocity vectors between cases A and D is shown in Figure 8a.
The sparsely placed fins at the center facilitated greater air flow with reduced air
resistance. Pressure magnitude (Figure 8b) was observed to be reduced on the
upstream side of the fin array. Thus, DP across the fins was decreased by 1.9%
Figure 8. Comparison of velocity vectors for cases A and D. (a) Velocity vectors and (b) contours of pressure (color figure available online).
752
R. K. BANERJEE ET AL.
(location 2). This additional reduction in DP, provided by the use of unequal fin spacing along the arms of the A-frame, can facilitate the use of multiple perforations for
yielding higher heat transfer.
4. DISCUSSION
The h and DP are important parameters in the design of an annular finned-tube
array. Heat transfer enhancement with perforations on the annular fin surface was
studied numerically using various combinations of perforations. These arrays, commonly used in air cooled steam condensers (ACSC), are known for high pressure
losses. The fin spacing along the arms of the A-frame was changed to reduce the
DP across the fin bundle. Fin spacing is one of the important geometric factors influencing fluid flow and heat transfer. Unequal spacing reduced the pressure loss in the
A-frame structure, but its effects on heat transfer in a perforated annular finned-tube
array are unknown. Perforated finned-tube array with unequal fin spacing needs to
be studied together to assess the change in heat transfer and DP characteristics.
Elliptical shaped finned-tubes, such as studies by Rocha et al. [17], will provide
more aerodynamic properties than the circular fins. Perforated finned-tube array
with an elliptical cross section can be studied in the future to evaluate h and DP characteristics. Heat transfer and pressure drop in a three-dimensional A-frame structure
can be evaluated using the optimum number of perforations on annular fins and fin
spacing along the arms of the A-frame.
5. CONCLUSION
An annular finned-tube array with perforations was studied numerically for
assessing the enhancement of heat transfer while minimizing an increase in pressure
drop across the domain. To calculate the efficiency of the location of a perforation, a
relative q-DP factor was defined as the ratio of increase in fin q performance ratio
and increase in fin DP performance ratio. A perforation in the wake area (120 ,
150 , or 180 ) beyond the point of detachment was found to be more efficient, particularly for perforation at h ¼ 120 . Multiple perforations at h ¼ 120 , 150 , and 180
also showed a favorable relative q-DP factor. The number of perforations was limited by the allowable DP in the system.
Unequal spacing along the A-frame structure of ACSCs could decrease the
pressure drop across the finned-tube bundle. Clustered spacing at the inlet and top
ends and large spacing at the center assisted in reducing the upstream pressure of
the air and DP in the system. Thus, penalty in DP for a perforated fin can possibly
be offset by changing the fin spacing along the arms of the A-frame structure.
REFERENCES
1. R. L. Webb, Air Side Heat Transfer in Finned Tube Heat Exchangers, Heat Transfer
Eng., vol. 1, pp. 33–89, 1980.
2. D. E. Briggs and E. H. Young, Convection Heat Transfer and Pressure Drop of Air Flowing across Triangular Pitch Banks of Finned Tubes, Chem. Eng. Prog. Symp. Ser., vol. 59,
pp. 1–10, 1963.
ENHANCED HEAT TRANSFER WITHIN A CONDENSER
753
3. K. K. Robinson and D. E. Briggs, Pressure Drop of Air Flowing across Triangular Pitch
Banks of Finned Tubes, Chem. Eng. Prog. Symp. Ser., vol. 62, pp. 177–184, 1966.
4. A. Zhukauskas, High-Performance Single-Phase Heat Exchangers, chaps. 14–15, pp. 291–
346, Hemisphere Publishing, New York, 1989.
5. A. E. Bergles, R. L. Webb, and G. H. Junkan, Energy Conservation via Heat Transfer
Enhancement, Midwest Energy Conf., vol. 4, pp. 193–200, 1979.
6. C. C. Wang and K. Y. Chi, Heat Transfer, and Friction Characteristics of Plain Fin-andTube Heat Exchangers, Part I: New Experimental Data, Int. J. Heat and Mass Transfer,
vol. 43, pp. 2681–2691, 2000.
7. W. Q. Tao, Z. G. Qu, and Y. L. He, Experimental and 3-D Numerical Study of Air Side
Heat Transfer and Pressure Drop of Slotted Fin Surface, Int. Conf. on Enhanced, Compact
and Ultra-Compact Heat Exchangers: Sci. Eng. and Tech., Whistler, British Columbia,
Canada, paper 15, 2005.
8. Y. P. Cheng, Z. G. Qu, and W. Q. Tao, Y. L. He, Numerical Design of Efficient Slotted
Fin Surface based on the Field Synergy Principle, Numer. Heat Transfer A, vol. 45, pp.
517–538, 2004.
9. B. Sahin and A. Demir, Thermal Performance Analysis and Optimum Design Parameters
of Heat Exchanger Having Perforated Pin Fins, Energy Conversion and Management, vol.
49, pp. 1684–1695, 2008.
10. M. R. Shaeri and M. Yaghoubi, Numerical Analysis of Turbulent Convection Heat
Transfer from an Array of Perforated Fins, Int. J. of Heat and Fluid Flow, vol. 30,
pp. 218–228, 2009.
11. J. Y. Jang, J. T. Lai, and L. C. Liu, The Thermal-Hydraulic Characteristics of Staggered
Circular Finned-Tube Heat Exchangers under Dry and Dehumidifying Conditions, Int.
J. of Heat and Mass Transfer, vol. 41, pp. 3321–3337, 1998.
12. M. S. Mon, and U. Gross, Numerical Study of Fin-Spacing Effects in Annular-Finned
Tube Heat Exchangers, Int. J. of Heat and Mass Transfer, vol. 47, pp. 1953–1964, 2004 .
13. D. G. Kroger, Fan Performance in Air-Cooled Steam Condensers, Heat Recovery Systems
and CHP, vol. 14, pp. 391–399, 1994.
14. C. J. Meyer and D. G. Kroeger, Plenum Chamber Flow Losses in Forced Draught
Air-Cooled Heat Exchangers, Appl. Therm. Eng., vol. 18, pp. 875–893, 1998.
15. C. J. Meyer and D. G. Kroeger, Air Cooled Heat Exchanger Inlet Flow Losses, Appl.
Therm. Eng., vol. 21, pp. 771–786, 2000.
16. FLUENT Incorporated, FLUENT 6.3 User’s Guide, Fluent Inc., Lebanon, NH, USA,
1998.
17. L. A. O. Rocha, F. E. M. Saboya, and J. V. C. Vargas, A Comparative Study of Elliptical
and Circular Sections in One- and Two-Row Tubes and Plate Fin Heat Exchangers, Int.
J. Heat and Fluid Flow, vol. 18, pp. 247–252, 1997.