1. International Journal of Applied Engineering Research
ISSN 0973-4562 Volume 9, Number 18 (2014) pp. 5125-5139
© Research India Publications
http://www.ripublication.com
Pressure and Heat Transfer over a Series
of In-line Non-Circular Ducts in a
Parallel Plate Channel
S. D. Mhaske #1
, S. P. Sunny #2
, S. L. Borse #3
, Y. B. Parikh#4
#1
Department of Mechanical Engineering, Symbiosis Institute of Technology,
Pune, India.
#2
Department of Mechanical Engineering, Symbiosis Institute of Technology,
Pune, India.
#3
Department of Mechanical Engineering, Rajarshi Shahu College of Engineering,
Pune, India.
#4
Department of Mechanical Engineering, Symbiosis Institute of Technology,
Pune, India.
1
mhaskesiddharth@gmail.com, 2
sobzsunny@gmail.com,
3
sachinlb@yahoo.co.uk, 4
yash.parikh@sitpune.edu.in
Abstract
Flow and heat transfer for two-dimensional laminar flow at low Reynolds number
for five in-line ducts of various cross-sections in a parallel plate channel are
studied in this paper. The governing equations were solved using finite-volume
method. A commercial CFD software, ANSYS Fluent 14.5 was used to solve this
problem. A total of three different non-conventional cross-sections and their
characteristics are compared with that of circular cross-section. Shape-1, Shape-2
and Shape-3 performed better for heat transfer rate than the circular cross-section,
but also offered higher resistance to the flow. Shape-1 offered less resistance to
flow at Re < 200 but post Re = 200 the resistance equalled to that of the Shape-3.
In overall, Shape-2 performed better when the heat transfer and resistance to flow
were considered.
Keyword- CFD, Flow over non-circular ducts, Heat transfer, Pressure drop,
Parallel Plates
I. INTRODUCTION
The quest for heat transfer enhancement in most of the engineering applications is a
never ending process. The need for better heat transfer and low flow resistance has led to
extensive research in the field of heat exchangers. Higher heat transfer and low pumping

2. 5126 S. D. Mhaske et al
power are desirable properties of a heat exchanger. Tube shape and its arrangement highly
influences flow characteristics in a heat exchanger. Flow past cylinders, esp. circular, flat, oval
and diamond arranged in a parallel plate channel were extensively studied by Bahaidarah et al.
[1-3]. They carried domain discretization in body-fitted co-ordinate system while the governing
equations were solved using a finite-volume technique. Chhabra [4] studied bluff bodies of
different shapes like circle, ellipse, square, semi-circle, equilateral triangle and square
submerged in non-Newtonian fluids. Kundu [5] [6] investigated fluid flow and heat transfer
coefficient experimentally over a series of in-line circular cylinders in parallel plates using two
different aspect ratios for intermediate range of Re 220 to 2800. Grannis and Sparrow [7]
obtained numerical solutions for the fluid flow in a heat exchanger consisting of an array of
diamond-shaped pin fins. Implementation of the model was accomplished using the finite
element method. Tanda [8] performed experiments on fluid flow and heat transfer for a
rectangular channel with arrays of diamond shaped elements. Both in-line and staggered fin
arrays were considered in his study. Jeng [9] experimentally investigated pressure drop and
heat transfer of an in-line diamond shaped pin-fin array in a rectangular duct. Ota [10] studied
the Heat transfer characteristics and flow behaviours around an elliptic cylinder at high
Reynolds number. Chen et al. [11] [12] calculated flow and conjugate heat transfer in a high-
performance finned oval tube heat exchanger element have been calculated for a thermally and
hydrodynamically developing three-dimensional laminar flow. Computations have been
performed with a finite volume method based on the SIMPLEC algorithm for pressure
correction. Zdravistch [13] numerically predicted fluid flow and heat transfer around staggered
and in-line tube banks and found out close agreement experimental test cases. Tahseen et al.
[14] conducted numerical study of the two-dimensional forced convection heat transfer across
three in-line flat tubes confined in a parallel plate channel, the flow under incompressible,
steady-state conditions. They solved the system in the body fitted coordinates (BFC) using the
finite volume method (FVM). Gautier and Lamballais [15] proposed a new set of boundary
conditions to improve the representation of the infinite flow domain.
Conventional tube/duct shapes have been investigated for fluid flow and heat transfer
characteristics. In this paper two-dimensional, steady state, laminar flow over non-
conventional non-circular (Shape-1, Shape-2 and Shape-3) in-line ducts confined in a parallel
plate channel is considered. The different geometries investigated are Circular, Shape-1,
Shape-2 and Shape-3. Only representative cases are discussed in this paper.
NOMENCLATURE
A area of heat transfer surfaces in a module (m2
)
Cp specific heat (J/kg.K)
D blockage height of a tube inside the channel (m)
f module friction factor
f0 module friction factor for circular tubes
H height of the channel (m)
L module length (m)
k thermal conductivity (W/m.K)
l length of the tube cross-section along the flow direction (m)
Nu module average Nusselt number (=
𝑚̇ 𝐶 𝑝 𝛥𝑇 𝑏 𝐻
𝑘𝐴(𝑇 𝑤−𝑇 𝑚)
)
Nu0 module average Nusselt number for the reference case of circular tubes
Nu+
heat transfer enhancement ratio (= Nu/Nu0)

3. Pressure and Heat Transfer over a Series of In-line Non-Circular Ducts 5127
Nu*
heat transfer performance ratio
ṁ mass flow rate (kg/s)
p pressure (Pa)
Re Reynolds number (= ρVH/µ)
T temperature (K)
Tm average of module inlet bulk temperature and module outlet bulk temperature (K)
Tw temperature of heat transfer surfaces (K)
Uin velocity at the channel inlet (m/s)
𝑉⃗ velocity field
y vertical distance from the channel bottom (m)
Δp pressure drop across a module (Pa)
Δp*
normalized pressure drop (=Δp /ρ 𝑈ⅈ𝑛
2
)
ΔTb change in bulk temperature across a module (K)
µ dynamic viscosity (N.s/m2
)
ρ density
𝛻 del operator
II. MATHEMATICAL FORMULATION
The governing mass, momentum, and energy conservation equations [1] for steady
incompressible flow of a Newtonian fluid are expressed as:
Mass conservation:
𝛻 ⋅ 𝑉⃗ = 0 (1)
Momentum conservation:
𝑉⃗ ⋅ (𝛻𝑉⃗ ) = −𝛻𝑝 + 𝑣𝛻2
𝑉⃗ (2)
Energy Conservation:
𝜌𝐶 𝑝 𝑉⃗ ⋅ (𝛻𝑇) = 𝑘𝛻2
𝑇 (3)
A commercial CFD software, ANSYS Fluent 14.5 was used to solve the governing equations.
The pressure-velocity coupling is done by Semi-Implicit Method for Pressure Linked
Equations (SIMPLE) scheme.
III. GEOMETRIC CONFIGURATION
A total of four different duct cross-sections namely Circular, Shape-1, Shape-2andShape-3
were analysed. The five in-line ducts were confined in a parallel-plate channel. The distance
between two consecutive ducts was kept constant. H/D, L/D and l /D ratio (shown in Fig. 1)
kept constant for all cross-sections of the ducts. The unobstructed length of the pipe before the
first and the last module kept as one module length and three module lengths respectively so
that the flow is fully developed. It is also ensured that the flow is fully developed when it enters
the channel. The ducts and the parallel plate channel walls were assumed to be of infinite extent
in the z-direction i. e. perpendicular to the paper. Hence, the flow could be considered as two-
dimensional. Since, the geometry is symmetric along the x-axis, only the lower half portion is
considered for the computation purpose.

4. 5128 S. D. Mhaske et al
(a)
(b)
(c)
(d)
Fig. 1. Various duct cross-sections studied in this work: (a) Circular cross-section ducts,
(b) Shape-1 cross-section ducts, (c) Shape-2 cross-section ducts, (d) Shape-3 cross-section
ducts
IV. GRID GENERATION
The grid for all the cases is generated using the Automatic Method with all Quadrilateral
elements in the Mesh Component System of the ANSYS Workbench 14.5 software. In the
numerical simulation, the grid of 11,000 nodes is found to be fair enough for grid independent
solution. Refining the grid proved to be unnecessary and resulting in utilizing more computer
resources. Fig. 2 shows the grid generated for all the configurations.
(a)
(b)

5. Pressure and Heat Transfer over a Series of In-line Non-Circular Ducts 5129
(c)
(d)
Fig. 2. Grid for geometries considered: (a) Circular cross-section ducts, (b) Shape-1
cross-section ducts, (c) Shape-2 cross-section ducts, (d) Shape-3 cross-section ducts
V. BOUNDARY CONDITIONS
Fig. 3 represents the schematic of the boundary conditions for all the cross-sections used. A
fully developed flow with velocity profile (u = Uin) and with temperature (T = Tin) was
assigned for the flow at inlet. The temperature was kept different than that of the plate and duct
walls. A no slip (u = v = 0) boundary condition was given to the plate and duct surface. The
surface temperature of the plates and the ducts were taken to be constant (T = Tw). At the
symmetry normal component of velocity and normal gradient of other velocity components
were taken as zero. At the outflow boundary there is no change in velocity across boundary.
Fig. 3. Flow domains studied in this study with the boundary condition
VI. VALIDATION
The parallel-plate channel problem corresponds to the rectangular duct with infinite aspect
ratio. The calculated value of Nusselt number is 7.541. Comparing the calculated Nusselt
number with that of the theoretical Nusselt number value of 7.54 [16], shows that our computed
values are in good agreement with that of the theoretical values.
Using the geometric parameters L/D = 3 and H/D = 2 for circular cross-section, the
normalized pressure drop (Δp*
) and Nusselt number for third module of the parallel plate
channel, when compared with the values of Kundu et al. [5] shows good agreement between
both the values.

6. 5130 S. D. Mhaske et al
Fig. 4. Validation with previous work Kundu et al. [5]: normalized pressure drop
across third module
TABLE I: Validation with previous work of Kundu et al. [6]: module average Nusselt
number for L/D= 3, H/D= 2
Re Second Module Third Module Fourth Module
Kundu et al. [6]
50
9.4 9.4 9.8
Present Work 9.51 9.48 9.48
Kundu et al. [6]
200
12.5 12.6 12.8
Present Work 12.68 12.63 12.64
VII. RESULTS AND DISCUSSIONS
The numerical simulation was carried out for Reynolds number ranging from 25-350 and
Prandtl number considered as 0.74.
A. Onset of Recirculation:
For any duct shape no separation of flow was observed below Re = 30. Circular duct displayed
the signs of recirculation at lowest Re value (Re = 32) [1]. The Shape-1 and Shape-2 ducts
displayed the onset of recirculation at Re value 45 while the Shape-3 showed the signs of
recirculation just under Re = 35, almost similar to the value of Circular duct.
As the values of Reynolds number is increased the recirculation region starts increasing.
At Re = 150 the length of the recirculation region is slightly smaller than the distance between
the two consecutive ducts in the flow direction while at Re = 350 the length of the recirculation
region is greater than the distance between the two consecutive ducts. Since there is no
obstruction to the flow downstream of the last duct the elliptic behaviour of the flow is observed
as shown in Fig. 5-8.

7. Pressure and Heat Transfer over a Series of In-line Non-Circular Ducts 5131
(a)
(b)
(c)
Fig. 5: Effect of Reynolds number on the streamline over Circular cross-section ducts:
(a) Re = 25, (b) Re = 150 and (c) Re = 350
(a)
(b)
(c)
Fig. 6: Effect of Reynolds number on the streamline over Shape-1 cross-section ducts:
(a) Re = 25, (b) Re = 150 and (c) Re = 350
(a)

8. 5132 S. D. Mhaske et al
(b)
(c)
Fig. 7: Effect of Reynolds number on the streamline over Shape-2 cross-section ducts:
(a) Re = 25, (b) Re = 150 and (c) Re = 350
(a)
(b)
(c)
Fig. 8: Effect of Reynolds number on the streamline over Shape-3 cross-section ducts:
(a) Re = 25, (b) Re = 150 and (c) Re = 350
B. Fluid Flow Characteristics:
From the Fig. 5-8 and the data from Table II we can observe that the streamlines past the ducts
1-4 are similar in nature. Also the Fig. 9 shows the normalized u-velocity profiles at module
inlet for modules 2,3,4 and 5 for the lower portion of the channel for various cases at Re = 150.
We can observe that the normalized u-velocity profiles at module inlet for modules 2, 3, 4 and
5 matches well with each other at all the Reynolds number. Hence, we can say that the flow is
periodically developed for inner modules. As the flow is periodically developed, the pressure
drop and Nusselt number values for inner modules are same we can say that the study of any
module from 2-5 is sufficient. In present study we have selected module 3.
Fig. 10 shows the normalized pressure drop (Δp*
) as a function of Reynolds number. We
can observe that, as the value of Re increases there is a steep descend in normalized pressure
drop for all the ducts presented in this study. All the ducts show higher normalized pressure
drop than the Circular ducts. The Shape-3 duct shows highest normalized pressure drop. There
is geometric similarity between Shape-2 and Shape-3. Hence there is no significant normalized
pressure drop difference between Shape-2 and Shape-3 ducts until Re = 50, but as the Re value
increases normalized pressure drop difference between Shape-2 and Shape-3 ducts starts
increasing. Recirculation of flow plays a vital role in this behaviour of normalized pressure
drop.

9. Pressure and Heat Transfer over a Series of In-line Non-Circular Ducts 5133
The ratio of normalized pressure drop for noncircular duct module (Δp*
) to that for the
circular duct module (Δp0
*
) is shown in Fig. 11. The Shape-1 ducts presents lower pressure
drop ratio than its counterparts for Re < 200. On the other hand the pressure drop ratio for
Shape-2 ducts dips considerably. The pressure drop ratio shown in Fig. 11 of Shape-1, Shape-
2 and Shape-3 ducts is higher than 1 for all the values of Reynolds number which means that
they offer higher flow resistance than Circular ducts. This will lead to higher pumping power
for heat exchanger with any of the above duct cross-section.
(a) (b)
(c) (d)
Fig. 9. Normalized u-velocity profiles at inlet of modules at Re=150
Fig. 10. Variation of normalized pressure drop (Δp*) with Re for third module

10. 5134 S. D. Mhaske et al
Fig. 11.Variation of Δp*/Δp0
* with Re for third module
C. Heat Transfer Characteristics:
Fig. 12 presents the module average Nusselt number as a function of Re for module 3 for Shape-
1, Shape-2, Shape-3 and Circular ducts. The Nusselt number for Re < 50 increases rapidly as
the Re increases. The duct shape has a very little effect on heat transfer rate at Re < 50. Post
Re = 50 the Nusselt number varies exponentially with the cross-section of the duct. Shape-1,
Shape-2 and Shape-3 ducts showed higher heat transfer rate than the circular ducts. At higher
values of Re the recirculation of the flow takes place. This recirculation vortices do not transfer
heat energy to the main stream of flow. Hence, their contribution to heat transfer is very less.
Fig. 13 presents the heat transfer enhancement ratio (Nu+
) as a function of Re for module
3 for Shape-1, Shape-2 and Shape-3 ducts. Shape-1, Shape-2 and Shape-3 ducts performed
better than the Circular ducts. Shape-2 ducts showed highest heat transfer enhancement ratio
(Nu+
) post Re = 50.
Fig. 14 presents the heat transfer performance ratio (Nu*
) as a function of Re for module
3 for Shape-1, Shape-2 and Shape-3 ducts. Nu*
represents heat transfer enhancement per unit
increase in pumping power. At Re < 40 Shape-1 ducts performed better than its counterparts,
but post Re = 40 Shape-2 performs better.
Fig. 12. Variation of Nu with Re for third module

11. Pressure and Heat Transfer over a Series of In-line Non-Circular Ducts 5135
Fig. 13. Variation of Nu+ with Re for third module
Fig. 14. Variation of Nu* with Re for third module

12. 5136 S. D. Mhaske et alTABLEII:ComparisonofnormalizedpressuredropandmoduleaverageNusseltnumberforallthegeometrictubeshapes
Shape-3
Nu*
0.976
0.977
0.977
0.976
0.980
1.019
1.020
1.019
1.020
1.021
1.043
1.031
1.033
1.032
1.039
1.062
1.029
1.032
1.031
1.034
Nu+
1.047
1.047
1.047
1.047
1.050
1.087
1.085
1.084
1.085
1.084
1.100
1.092
1.089
1.090
1.089
1.111
1.090
1.084
1.089
1.076
Nu
7.207
7.216
7.205
7.207
7.252
10.313
10.330
10.281
10.283
10.311
12.473
12.254
12.128
12.128
12.166
13.979
13.150
12.984
13.000
12.900
Δp*/Δp0*
1.234
1.232
1.232
1.233
1.229
1.216
1.205
1.203
1.204
1.196
1.173
1.186
1.173
1.179
1.151
1.143
1.188
1.162
1.179
1.128
Δp*
9.896
9.997
9.992
9.989
9.992
5.519
5.432
5.428
5.419
5.378
3.761
3.191
3.200
3.180
3.064
3.282
2.400
2.406
2.391
2.264
Shape-2
Nu*
0.976
0.976
0.977
0.974
0.983
1.020
1.017
1.019
1.017
1.018
1.047
1.029
1.037
1.031
1.038
1.064
1.030
1.041
1.036
1.037
Nu+
1.048
1.046
1.046
1.044
1.053
1.089
1.082
1.083
1.082
1.082
1.100
1.090
1.090
1.091
1.087
1.106
1.095
1.087
1.095
1.078
Nu
7.211
7.210
7.202
7.188
7.276
10.325
10.298
10.267
10.259
10.285
12.474
12.232
12.133
12.131
12.140
13.916
13.215
13.017
13.072
12.918
Δp*/Δp0*
1.238
1.231
1.230
1.232
1.231
1.215
1.205
1.201
1.206
1.199
1.158
1.189
1.160
1.182
1.147
1.122
1.200
1.138
1.181
1.121
Δp*
9.927
9.992
9.978
9.984
10.006
5.513
5.430
5.419
5.426
5.389
3.715
3.198
3.165
3.188
3.053
3.222
2.423
2.358
2.395
2.251
Shape-1
Nu*
0.996
0.996
0.996
0.995
0.977
1.016
1.013
1.014
1.012
1.015
1.026
1.010
1.012
1.004
1.024
1.034
1.007
1.007
1.001
1.023
Nu+
1.026
1.025
1.025
1.025
1.005
1.045
1.042
1.042
1.043
1.042
1.052
1.044
1.044
1.047
1.042
1.061
1.048
1.045
1.056
1.031
Nu
7.059
7.063
7.057
7.059
6.945
9.915
9.912
9.882
9.883
9.908
11.934
11.716
11.622
11.648
11.638
13.360
12.647
12.506
12.609
12.361
Δp*/Δp0*
1.093
1.091
1.090
1.093
1.090
1.088
1.088
1.086
1.094
1.083
1.080
1.104
1.097
1.135
1.054
1.082
1.128
1.114
1.177
1.024
Δp*
8.766
8.854
8.846
8.858
8.858
4.939
4.902
4.901
4.924
4.870
3.463
2.970
2.994
3.061
2.806
3.106
2.279
2.309
2.387
2.057
Circular
Nuo
6.882
6.890
6.883
6.885
6.908
9.486
9.517
9.482
9.479
9.509
11.340
11.224
11.134
11.123
11.172
12.588
12.070
11.973
11.937
11.988
Δp0*
8.020
8.118
8.113
8.104
8.128
4.538
4.507
4.510
4.500
4.496
3.208
2.690
2.729
2.697
2.662
2.871
2.020
2.071
2.028
2.008
Module
Re=25
1
2
3
4
5
Re=50
1
2
3
4
5
Re=100
1
2
3
4
5
Re=150
1
2
3
4
5

13. Pressure and Heat Transfer over a Series of In-line Non-Circular Ducts 5137TABLEII:Continued
Shape-3
Nu*
1.076
1.026
1.025
1.033
1.015
1.079
1.020
1.016
1.031
1.007
1.081
1.010
1.007
1.025
1.004
1.082
1.000
0.998
1.020
1.001
Nu+
1.122
1.086
1.077
1.092
1.052
1.124
1.078
1.067
1.091
1.043
1.127
1.065
1.059
1.087
1.041
1.131
1.052
1.051
1.082
1.040
Nu
15.769
13.776
13.604
13.706
13.087
17.657
14.254
14.063
14.323
13.062
19.558
14.606
14.432
14.868
12.943
21.418
14.883
14.722
15.352
12.790
Δp*/Δp0*
1.133
1.187
1.159
1.183
1.115
1.131
1.180
1.160
1.188
1.111
1.134
1.172
1.163
1.192
1.113
1.139
1.163
1.168
1.196
1.121
Δp*
3.039
1.978
1.979
1.969
1.860
2.886
1.712
1.710
1.704
1.615
2.779
1.526
1.525
1.521
1.450
2.700
1.388
1.391
1.387
1.329
Shape-2
Nu*
1.074
1.035
1.036
1.043
1.024
1.081
1.032
1.029
1.046
1.016
1.088
1.026
1.023
1.043
1.019
1.095
1.025
1.015
1.040
1.015
Nu+
1.112
1.105
1.079
1.104
1.061
1.118
1.106
1.071
1.109
1.056
1.126
1.103
1.063
1.107
1.065
1.136
1.106
1.055
1.107
1.067
Nu
15.622
14.011
13.625
13.858
13.199
17.560
14.629
14.108
14.550
13.230
19.539
15.136
14.490
15.147
13.243
21.515
15.648
14.774
15.699
13.120
Δp*/Δp0*
1.108
1.215
1.129
1.185
1.115
1.106
1.230
1.125
1.191
1.124
1.109
1.244
1.123
1.198
1.142
1.114
1.256
1.121
1.205
1.162
Δp*
2.975
2.026
1.928
1.973
1.860
2.821
1.784
1.659
1.710
1.634
2.716
1.620
1.472
1.529
1.486
2.642
1.499
1.334
1.398
1.378
Shape-1
Nu*
1.046
1.007
1.000
1.003
1.007
1.057
1.007
0.990
0.992
1.003
1.068
1.006
0.982
0.976
1.008
1.079
1.004
0.974
0.960
1.012
Nu+
1.078
1.054
1.042
1.065
1.011
1.097
1.057
1.036
1.057
1.009
1.115
1.057
1.029
1.042
1.019
1.135
1.056
1.024
1.025
1.028
Nu
15.154
13.361
13.162
13.369
12.575
17.222
13.982
13.643
13.873
12.643
19.350
14.499
14.027
14.257
12.676
21.500
14.939
14.339
14.543
12.648
Δp*/Δp0*
1.096
1.146
1.130
1.200
1.014
1.116
1.156
1.144
1.211
1.019
1.139
1.161
1.154
1.216
1.033
1.163
1.162
1.161
1.219
1.050
Δp*
2.942
1.910
1.930
1.997
1.691
2.848
1.677
1.685
1.738
1.482
2.791
1.512
1.512
1.553
1.345
2.756
1.387
1.382
1.414
1.245
Circular
Nu0
14.053
12.681
12.630
12.548
12.435
15.706
13.225
13.175
13.125
12.525
17.352
13.717
13.625
13.681
12.435
18.945
14.149
14.009
14.185
12.299
Δp0
*
2.684
1.667
1.707
1.664
1.668
2.551
1.450
1.474
1.435
1.454
2.450
1.302
1.311
1.277
1.302
2.370
1.194
1.190
1.160
1.186
Module
Re=200
1
2
3
4
5
Re=250
1
2
3
4
5
Re=300
1
2
3
4
5
Re=350
1
2
3
4
5

14. 5138 S. D. Mhaske et al
VIII. CONCLUSION
In an environment wherein the heat transfer rate is critical, any of the above mentioned non-
conventional cross-section of the ducts (Shape-1, Shape-2 and Shape-3) will perform better
than that of the circular cross-section ducts. The duct shape has a very little influence on the
heat transfer rate at low Reynolds number. For optimal design of a heat exchanger for heat
transfer enhancement per unit increase in pumping power, Reynolds number and cross-section
of the duct must be carefully chosen. In overall, Shape-2 performed better when both, the heat
transfer rate and resistance to flow were considered.
ACKNOWLEDGEMENT
The authors would like to present their sincere gratitude towards the Faculty of Mechanical
Engineering in Symbiosis Institute of Technology, Pune and Rajarshi Shahu College of
Engineering, Pune.
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