NUMERICAL STUDY OF FLUID FLOW AND HEAT TRANSFER …faculty.kfupm.edu.sa/ME/haithamb/NUMERICAL STUDY...

NUMERICAL STUDY OF FLUID FLOW AND HEATTRANSFER OVER A SERIES OF IN-LINE NONCIRCULARTUBES CONFINED IN A PARALLEL-PLATE CHANNEL

Haitham M. S. BahaidarahDepartment of Mechanical Engineering, King Fahd University of Petroleumand Minerals, Dhahran, Saudi Arabia

M. Ijaz and N. K. AnandDepartment of Mechanical Engineering, Texas A&M University,College Station, Texas, USA

Two-dimensional steady developing fluid flow and heat transfer across five in-line tubes con-

fined in a channel were studied numerically for a fluid with a Prandlt number of 0.7. The

tube cross-sectional shapes studied were circular, flat, oval, and diamond. Domain disrecti-

zation was carried out in body-fitted coordinate system. The contravariant components of

velocities were used as the dependent variables. The governing equations were solved using

a finite-volume technique. Grid independence study was carried out by running the

developed code for several different grid sizes for each of the four geometric configurations

and monitoring module average Nusselt number and normalized pressure drop.

The results for flat, oval, and diamond tubes were compared with each other and with those

for circular tubes. Flat and oval tubes offered greater flow resistance and heat transfer rate

when compared to circular tubes for all values of Reynolds number (Re) considered in this

study. Diamond tubes offered less resistance to flow compared to circular tubes for

Ree 250. For all values of Re, diamond tubes exhibited the lowest heat transfer rate. When

both the flow resistance and heat transfer rate were considered, diamond tubes were better

than flat and oval tubes for Re < 40. For Re > 50, flat and oval tubes performed better. Both

geometry and flow field were major factors affecting the heat transfer performance

for Re < 50, whereas geometric shape was found to affect performance for Re > 50 more

significantly.

INTRODUCTION

Some studies have shown that the tube shapes and their arrangement in heatexchangers have positive influence on heat transfer [1–3]. The effect of flow pastbluff bodies, especially cylinders, has been a major attraction for fluid mechanicsinvestigators for a long time. Most of these studies were concerned with the flow overa circular cylinder. Williamson [4] and Zdrakovich [5] wrote comprehensive reviews

Received 2 October 2005; accepted 7 December 2005.

Support for this research by King Fahd University of Petroleum and Minerals is gratefully

appreciated.

Address correspondence to N. K. Anand, Department of Mechanical Engineering, Texas A&M

University, College Station, TX 77843-3123, USA. E-mail: [email protected]

97

Numerical Heat Transfer, Part B, 50: 97–119, 2006

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7790 print=1521-0626 online

DOI: 10.1080/10407790600599041

on this topic. Other researchers considered many different tube shapes, such as rec-tangular, circular, elliptical, diamond, and flat. Kundu et al. [6, 7] and Kundu [8]studied heat transfer and fluid flow numerically and experimentally over a row ofin-line cylinders placed between two parallel plates. Incompressible two-dimensionallaminar flow was considered. In general, the pressure drop and heat transfer werespatially periodic, indicating periodically fully developed characteristics. Numericalsolutions were obtained by Grannis and Sparrow [9] for the fluid flow in a heatexchanger consisting of an array of diamond-shaped pin-fins. The solutions werebased on the periodically fully developed regime. The solution domain was discre-tized by subdividing it into two-dimensional, nine-noded quadrilateral elements toimplement the finite-element method. Chen et al. [10, 11] studied the effect ofReynolds number on flow and conjugate heat transfer in a high-performance finnedoval tube heat exchanger element for a thermally and hydrodynamically developingthree-dimensional laminar flow. Computations were performed with a finite-volumemethod based on the SIMLPEC algorithm for pressure correction. Breuer et al. [12]investigated in detail the confined flow around a cylinder of square cross sectionmounted inside a plane channel by two entirely different numerical techniques, lat-tice-Boltzmann automata and the finite-volume method. Both numerical methodsare second-order-accurate in space and time. Flat tube designs have been recentlyintroduced for use in modern heat exchanger applications such as automotive radia-tors [13]. Availability of heat transfer and pressure drop characteristics of flat tubeheat exchangers in open literature is rather limited. Bahaidarah [14] and Bahaidarahet al. [15] have studied these characteristics numerically for various tube layouts.Recently numerical analysis was conducted to study the laminar two-dimensionalsteady forced convection of air crossflow over square and nonsquare in-line circulartube arrangements [16]. A finite-volume method with a nonorthognal body-fitted

NOMENCLATURE

A area of heat transfer surfaces in a module

cp specific heat

D blockage height of a tube inside the

channel

f module friction factor

f0 module friction factor for circular tubes

H height of the channel

k thermal conductivity

l length of the tube cross section along the

flow direction

L module length

_mm mass flow rate

Nu module average Nusselt number

[¼ _mmcp DTb H=kAðTw � TmÞ]Nu0 module average Nusselt number for the

reference case of circular tubes

Nuþ heat transfer enhancement ratio

(¼Nu=Nu0)

Nu� heat transfer performance ratio

[¼Nuþ=ðf =f0Þ1=3]

p pressure

Re Reynolds number (¼UinH=n)

T temperature

Tm average of module inlet bulk

temperature and module outlet bulk

temperature

Tw temperature of heat transfer surfaces

un, ug contravariant components of velocity

Uin velocity at the channel inlet

V!

velocity field

y vertical distance from the channel

bottom

Dp pressure drop across a module

Dp� normalized pressure drop (¼Dp=qU2in)

DTb change in bulk temperature across a

module

n kinematic viscosity

n, g curvilinear coordinates

q density

r del operator

98 H. M. S. BAHAIDARAH ET AL.

grid and collocated variables was used to solve the Navier-Stokes and energy conser-vation equations for a tube bundle with five longitudinal rows, including inlet andoutlet sections. Comparison of the results with well-established experimental dataand empirical correlations [17, 18] showed good overall agreement.

A single geometric tube shape has been studied for fluid flow and heat transferby many researchers in the past. In this article we study heat transfer over four dif-ferent geometries for the purpose of comparison. Consideration is given to 2-Dsteady laminar incompressible flow over noncircular (flat, oval, and diamond) andcircular in-line tubes confined in a channel (Figure 1). The four different geometriesstudied are circular, flat, oval, and diamond, as shown in Figure 2.

MATHEMATICAL FORMULATION

The governing mass, momentum, and energy conservation equations for steadyincompressible flow of a Newtonian fluid with constant thermophysical properties,no heat generation, and negligible viscous dissipation are

Mass conservation : r � V!¼ 0 ð1Þ

Figure 1. Flow domains studied in this work, showing five in-line tubes in a channel: (a) circular cross-

section tubes; (b) flat-cross-section tubes; (c) oval-cross-section tubes; (d) diamond-cross-section tubes.

FLOW OVER IN-LINE NONCIRCULAR TUBES 99

Momentum conservation : V!� ðrV

!Þ ¼ �rpþ nr2 V! ð2Þ

Energy conservation : qcp V!� ðrTÞ ¼ kr2T ð3Þ

A FORTRAN code was developed to solve these equations numerically. Forcomplex geometries, as mentioned by Napolitano and Orlandi [19], the simplicityand efficiency of finite-difference techniques can be exploited by coordinate trans-formation. However, because of developments in grid generation techniques,finite-volume methods are increasingly being used for computing incompressibleflow in arbitrary geometries. In the current work, a finite-volume techniqueproposed by Karki [20] is used to discretize the governing conservation equationsfor body-fitted coordinates, n and g. The flux balances are represented in terms ofprimary and secondary fluxes. The secondary flux is solely diffusive in nature andcomes into the picture because of nonorthogonality. For orthogonal grids the

Figure 2. Geometric modules for various tube cross sections (shaded area in each module is the calculation

domain): (a) circular; (b) flat; (c) oval; (d) diamond.


secondary flux is zero. The secondary flux is treated explicitly to avoid nine-pointformulation. The power law is used to represent the solution as a one-dimensionalconvection-diffusion equation. The contravariant components of velocity, un andug, are used as the dependent variables. In this formulation, the discretized momen-tum equations are expressed in terms of parallel velocities instead of actual velocities.The velocity and pressure fields are linked by the Semi-Implicit Method for PressureLinked Equations (SIMPLE) algorithm. The resulting set of discretization equationsis solved iteratively using a line-by-line procedure which is a combination of thetri-diagonal matrix algorithm (TDMA) and the Gauss-Seidel scheme.

Mathematical details of the finite-volume technique used in this study can befound in Bahaidarah [14], Bahaidarah et al. [15], and Karki [20], and are notrepeated here.

GEOMETRIC CONFIGURATIONS

Four different geometric shapes of tube cross section were investigated—circular, flat, oval, and diamond (Figure 1). Calculations were made for five in-linetubes in a channel. The longitudinal distance between two consecutive tubes waskept constant. Tube modules were defined as shown in Figure 1. A tube module con-tained one tube and was bounded by two fictitious surfaces located midway betweencenters of the two consecutive tubes. The ratio of area of cross section of the channelto the blockage area of the tube (H=D) was kept constant for all geometric shapes oftube cross section considered in this study. The case of circular-cross-section tubeswas taken as a reference. Geometric similarity was ensured for the other three casesby taking constant values for the ratio of module length to tube height normal to theflow, the ratio of channel height to tube height, and the ratio of tube width in theflow direction to the tube height. Unobstructed length of the channel upstream ofthe first module was equal to one module length. To ensure a fully developed flowat the channel exit, an unobstructed channel length equal to three modules lengthwas included downstream of the last module. The tubes and the channel walls wereassumed to be of infinite extent in the direction perpendicular to the paper, so thatthe flow could be considered as two-dimensional. Owing to the symmetric behaviorof the laminar flow in the Reynolds number range investigated, only the lower halfof the channel was used as the calculation domain, as shown in Figure 2.

GRID GENERATION

In each configuration, the domain was discretized into a structured grid. Thecomputational domain was divided into three individual regions. These regions werethe entry region, the tube modules, and the exit region (Figure 1). A uniform orthog-onal grid was used for both entry and exit regions. However, to handle the arbitrarilyshaped domain of the tube modules, the Geometry and Mesh Building IntelligentToolkit (GAMBIT) was used. The irregular physical domain was discretized intonumerous quadrilateral grid elements and a body-fitted coordinate system was gen-erated. Typical single-module grids for all four geometries considered in this studyare illustrated in Figure 3. The grid distribution for a single tube module can be


repeated successively to generate the domain of any number of tube modules. In thisstudy, five consecutive tubes were included in the computational domain.

BOUNDARY CONDITIONS

In Figure 3, the surface BC is the bottom wall of the tube and the surface EFrepresents the lower wall of the channel. A no-slip boundary condition was assignedfor these surfaces, where both velocity components were set to zero at that boundary(u ¼ v ¼ 0). The tubes and the channel walls were considered to be at the sameconstant temperature (T ¼ Tw, Dirichlet condition). AB and CD are the lines ofsymmetry where no flow crosses these boundaries, and the normal component ofvelocities and the normal gradient of the parallel component of velocity are set tozero. Finally, the lines AF and DE are recognized as the module inlet and moduleoutlet, respectively. A uniform inlet velocity profile (u ¼ Uin) and a constant inlettemperature (T ¼ Tin), different from the walls temperature, were assigned at thechannel inlet. At the exit, fully developed flow boundary condition was applied.The streamwise gradients of all variables were set to zero at the outlet boundary.

CONVERGENCE CRITERIA

In the momentum and energy equations, the sum of relative residuals for all thenodes was monitored for the variables un, ug, and T. For the pressure equation, it isappropriate to check for mass imbalance in the continuity equation. Convergencewas declared when the normalized residuals of mass, momentum, and energy becameless than 10�6 for all cases considered in this study.

Figure 3. Grids for the geometries studied: (a) circular-cross-section tube; (b) flat-cross-section tube;

(c) oval-cross-section tube; (d) diamond-cross-section tube.


VALIDATION

The developed code was validated by reproducing solutions for some of thebenchmark problems.

The fluid flow and heat transfer in a parallel-plate channel subjected to con-stant wall temperature was predicted. The Nusselt number for the fully developedregion between two parallel plates subjected to constant wall temperature usingthe developed code is 7.56, which agrees favorably with the Nusselt number valueof 7.54 reported by many authors such as Incropera and DeWitt [21].

Normalized pressure drop (Dp�) was computed for the third module of aparallel-plate channel with circular tubes using the developed code and comparedwith the numerical predictions of Kundu et al. [6] and the experimental data ofKundu [8]. It is evident from Figure 4 that there is a very good agreement betweenthe results obtained by the developed code and results in the literature [6, 8]. Theaverage heat transfer coefficient and the corresponding Nusselt number across themodules were also computed for the geometric parameter values L=D ¼ 3 andH=D ¼ 2. Table 1 shows the average Nusselt number (Nu) at two different Reynoldsnumbers. As shown in Table 1, the numerical predictions of heat transfer using the

Figure 4. Validation: normalized pressure drop across module 3.


developed code agree very well with the numerical predictions of Kundu et al. [6],thus further validating our code.

GRID INDEPENDENCE

The grid independence test was performed for each configuration for severalgrid sizes at the highest Reynolds number considered in this study (i.e., Re ¼ 350).

Table 1. Validation: module average Nusselt number for L=D ¼ 3 and H=D ¼ 2

Second module Third module Fourth module

Re ¼ 50

Kundu et al. [6] 9.4 9.4 9.8

Present work 9.23 9.23 9.23

Re ¼ 200

Kundu et al. [6] 12.5 12.6 12.8

Present work 12.44 12.43 12.42

Table 2. Grid independence study for module 3 for Re ¼ 350, Pr ¼ 0.7

Grid

Number

of

nodes

Normalized

pressure

drop (Dp�)Relative %

difference

Module average

Nusselt

number (Nu)

Relative %

difference

Circular tubes

35� 57 1,995 1.244 14.621

40� 125 5,000 1.163 6.451 14.169 3.092

50� 160 8,000 1.143 1.755 13.999 1.198

59� 188 11,092 1.132 0.938 13.915 0.602

70� 200 14,000 1.126 0.584 13.853 0.446

80� 212 16,960 1.120 0.488 13.803 0.356

Flat tubes

35� 57 1,995 1.455 16.713

40� 125 5,000 1.358 6.630 16.128 3.497

50� 160 8,000 1.335 1.761 15.932 1.214

59� 188 11,092 1.324 0.809 15.837 0.596

70� 200 14,000 1.315 0.630 15.769 0.434

80� 212 16,960 1.310 0.437 15.713 0.354

Oval tubes

35� 57 1,995 1.258 15.621

40� 125 5,000 1.184 5.887 15.153 2.997

50� 160 8,000 1.166 1.541 14.993 1.059

59� 188 11,092 1.157 0.732 14.910 0.550

70� 200 14,000 1.151 0.495 14.855 0.373

80� 212 16,960 1.157 0.497 14.910 0.375

Diamond tubes

35� 58 1,995 1.266 13.821

40� 126 5,000 1.190 6.052 13.476 2.499

50� 160 8,000 1.169 1.759 13.333 1.061

59� 188 11,092 1.160 0.765 13.265 0.510

70� 200 14,000 The solution did not converge.

80� 212 16,960


The grids used for grid independence study were 40� 125, 50� 160, 59� 188,70� 200, and 80� 212 with 5,000, 8,000, 11,092, 14,000, and 16,960 nodes, respect-ively. The grid was refined successively by adding about 3,000 nodes at each step.Relative percentage changes in normalized pressure drop and module averageNusselt number for module 3 were monitored as the grid was refined successively.The results are presented in Table 2 and plotted in Figure 5. It is evident fromTable 2 and Figure 5 that the relative percentage change in normalized pressure dropand module average Nusselt number between two successive grid sizes is insignificantas the grid is refined beyond 59� 188 (bolded in Table 2). Therefore, to optimizeCPU resources with an acceptable level of accuracy, all parametric runs were madewith the 59� 188 grid (11,092 nodes).

RESULTS AND DISCUSSION

The code was run for different values of Re in the range 25–350. The value ofPrandtl number in this study was fixed at 0.7. Only representative cases are chosenfor discussion in this article.

Figure 5. Grid independence study—variation of relative percent difference in normalized pressure drop

and module average Nusselt number with grid density for module 3: (a) circular tubes; (b) flat tubes;

(c) oval tubes; (d) diamond tubes.


Onset of Recirculation

For any given tube cross section, no flow separation was observed below a cer-tain value of Re. As Re increases, the flow separation occurs downstream of thetubes and a recirculation region develops behind every tube. Circular tubes displayedrecirculation at the lowest Re value (Re ¼ 32). Flat-cross-section tubes showed noseparation up to Re values just under 35. Flow across the oval tubes started separat-ing downstream of the tubes at Re ¼ 45. Diamond tubes were the most gentle to theflow and showed no separation up to Re values just under 55. As shown by thestream-function contour plots in Figures 6–9, for all tube cross sections the size ofrecirculation region increases along the flow direction with increase in Re.

Elliptic Behavior of the Flow

For higher values of Re (Re ¼ 350) the recirculation region is larger than thedistance between consecutive tubes in the flow direction. For modules 1–4 therecirculation region interferes with the downstream tube. However, there is nointerference from other objects downstream of the last tube. Hence, the shape of

Figure 6. Streamlines over circular-cross-section tubes: (a) Re ¼ 25; (b) Re ¼ 150; (c) Re ¼ 350.

Figure 7. Streamlines over flat-cross-section tubes: (a) Re ¼ 25; (b) Re ¼ 150; (c) Re ¼ 350.


recirculation region downstream of the last tube is considerably different from thatof the recirculation region downstream of the other tubes (Figures 6–9). These fig-ures confirm the presence of strong elliptic effects.

Periodically Fully Developed Flow

It is evident from Figures 6–9 that the shapes of the streamlines behind tubes1–4 are similar. This shows that the flow is hydrodynamically periodically developedfor the inner modules (modules 2–4).

Periodic development of the flow is also confirmed from u-velocity profiles atthe inlet of modules 2–5. Figures 10–12 show normalized u-velocity profiles at theinlet of modules 2–5 for Reynolds numbers 50, 150, and 350 for the lower half ofthe channel. All four velocity profiles at the inlets of modules 2–5 coincide with eachother for all Reynolds number values considered in this study.

Table 3 shows normalized pressure drop (Dp�) and module average Nusseltnumber (Nu) for all modules. Dp� and Nu values shown in Table 3 remain almostthe same for modules 2–4 for all Re values considered in this study. Thus streamwisevelocity distributions at the module inlets (Figures 10–12), and normalized pressure

Figure 8. Streamlines over oval-cross-section tubes: (a) Re ¼ 25; (b) Re ¼ 150; (c) Re ¼ 350.

Figure 9. Streamlines over diamond-cross-section tubes: (a) Re ¼ 25; (b) Re ¼ 150; (c) Re ¼ 350.


drop and module average Nusselt number for each module (Table 3) confirm theexistence of periodically fully developed flow downstream of the second module.

Flow Resistance Characteristics

Since the flow is periodically developed for the inner modules, study of oneinner module is sufficient to capture essential flow physics. Nondimensionalizedpressure drop (Dp�) as a function of Re for module 3 is shown in Figure 13 andTable 4. As Reynolds number increases, module Dp� decreases for all tube cross sec-tions. Decrease in Dp� is very sharp at low Reynolds numbers, up to Re ¼ 100. Overthe entire Reynolds number range studied, flat tubes exhibited the highest Dp� for allRe values. Dp� for oval tubes was lower than that for flat tubes but higher than thatfor circular tubes. Diamond tubes offered the least resistance to flow, as their Dp�

was even less than that for circular tubes. However, Dp� for diamond tubes decreasesmore gradually than for oval and circular tubes. At high Re values (Re > 250), cir-cular, oval, and diamond tubes exhibit almost the same Dp�. The ratio of normalizedpressure drop for noncircular tube module (Dp�) to that for the circular tube module(Dp�0) is presented in Figure 14 and Table 4 as a function of Re for module 3 for flat,oval, and diamond tubes. For flat and oval tubes the ratio decreases with Re at simi-lar rates. The slope of these curves is very small for Re > 200. The ratio, however, is

Figure 10. Normalized u-velocity profiles at the inlet of modules at Re ¼ 50.


greater than 1 for both the flat and oval tubes over the entire Reynolds number rangestudied. Flat and oval tubes, therefore, offer more flow resistance when compared tocircular tubes. A heat exchanger with flat or oval tubes will cause more pressuredrop, hence require more pumping power, than one with circular tubes. If the useof flat or oval tubes is dictated by some other considerations, then the heat exchan-ger should be designed for high fluid velocities, as the ratio Dp�=Dp�0 is low at highReynolds numbers. However, going beyond Re ¼ 200 does not help substantially, asthe Dp�=Dp�0 curves become nearly horizontal. For diamond tubes, the ratio Dp�=Dp�0increases with Re, as opposed to flat and oval tubes. This result can be explainedwith the help of Figure 13, which shows that Dp� for diamond tube module decreaseswith Re at a lower rate compared to that for circular tubes. For Re < 250 the ratioDp�=Dp�0 for diamond tubes is less than unity (Figure 14). Hence, if pumping power isa concern, diamond tubes should be preferred over circular tubes only in the lowReynolds number range. In the Re > 250 range, circular tubes are better than dia-mond tubes from the viewpoint of pressure drop.

Heat Transfer Characteristics

Variation of module average Nusselt number with the Reynolds number formodule 3 is shown in Figure 15 and Table 4 for all the four tube cross sections.



Module average Nusselt number curves tend to merge at very low Reynoldsnumbers. Below Re ¼ 50, module average Nusselt number increases rapidly withRe, with almost a linear trend. The curves are again almost linear for Re > 150,but the slope is much less than that for Re < 50. Therefore, if heat transfer rate isa major design consideration, fluid velocity should be maximized in the low-Rerange (i.e., Re < 50), because heat transfer rate is very sensitive to the fluid velocityin this range regardless of the tube shape. However, in the high-Re range (i.e.,Re > 150), a better tube cross section should be chosen instead of increasing the fluidvelocity, in order to maximize heat transfer rate. The rapid increase in module aver-age Nusselt number for Re < 50 can be explained as follows. At low Re values, thereis no recirculation. Advection effects are significant, and heat transfer is very sensi-tive to the fluid velocity. Therefore, a small increase in Re value causes a largeincrease in module average Nusselt number. When recirculation starts, the fluid inthe recirculation region keeps swirling in the form of vortices. Heat transfer occursfrom the tube surface adjacent to the recirculation region to the swirling fluid. How-ever, these recirculation pockets do not transfer heat energy effectively to the mainstream of fluid. Thus they have little contribution to advective heat transfer. Flattubes have highest module average Nusselt number for all values of Re. Oval tubesare less effective from a heat transfer point of view than flat tubes, but are better in



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813

.64

81

.30

71

2.9

71

1.1

92

1.0

90

3.0

86

1.1

06

12

.31

11

.13

11

.09

42

.57

00

.92

11

0.5

480

.96

90

.99

6

52

.71

01

0.8

593

.54

51

.30

81

2.9

00

1.1

88

1.0

86

2.9

97

1.1

06

12

.25

81

.12

91

.09

22

.49

40

.92

01

0.5

260

.96

90

.99

7

Re¼

15

0

12

.96

51

2.0

183

.60

61

.21

61

4.8

03

1.2

32

1.1

54

3.0

72

1.0

36

13

.85

71

.15

31

.13

92

.95

20

.99

61

2.0

211

.00

01

.00

2

22

.08

61

1.7

322

.61

71

.25

51

3.9

03

1.1

85

1.0

99

2.2

39

1.0

73

13

.11

31

.11

81

.09

21

.98

80

.95

31

1.2

720

.96

10

.97

6

32

.08

41

1.7

292

.60

91

.25

21

3.8

85

1.1

84

1.0

98

2.2

28

1.0

69

13

.09

91

.11

71

.09

22

.00

30

.96

11

1.2

900

.96

30

.97

5

42

.08

31

1.7

282

.60

81

.25

21

3.8

84

1.1

84

1.0

98

2.2

27

1.0

69

13

.09

81

.11

71

.09

22

.00

30

.96

11

1.2

890

.96

30

.97

5

51

.99

41

1.6

862

.52

31

.26

51

3.5

76

1.1

62

1.0

74

2.1

39

1.0

73

12

.91

81

.10

51

.08

01

.92

50

.96

61

1.1

220

.95

20

.96

3

(Co

nti

nu

ed)

111

Ta

ble

3.

Co

nti

nu

ed

Mo

du

le

Cir

cula

rF

lat

Ov

al

Dia

mo

nd

Dp� 0

Nu

0D

p�

Dp�

Dp� 0

Nu

Nuþ

Nu�

Dp�

Dp�

Dp� 0

Nu

Nuþ

Nu�

Dp�

Dp�

Dp� 0

Nu

Nuþ

Nu�

Re¼

20

0

12

.80

81

3.4

593

.26

21

.16

21

6.2

92

1.2

10

1.1

51

2.8

14

1.0

02

15

.189

1.1

28

1.1

28

2.9

20

1.0

40

13

.752

1.0

22

1.0

08

21

.70

91

2.4

562

.08

61

.22

11

4.5

70

1.1

70

1.0

94

1.8

05

1.0

56

13

.730

1.1

02

1.0

82

1.6

56

0.9

69

11

.837

0.9

50

0.9

60

31

.69

01

2.4

362

.06

11

.21

91

4.5

18

1.1

67

1.0

93

1.7

73

1.0

49

13

.675

1.1

00

1.0

82

1.6

70

0.9

88

11

.902

0.9

57

0.9

61

41

.68

91

2.4

312

.06

01

.21

91

4.5

14

1.1

68

1.0

93

1.7

71

1.0

49

13

.670

1.1

00

1.0

82

1.6

70

0.9

88

11

.901

0.9

57

0.9

61

51

.62

51

1.8

212

.00

21

.23

21

3.7

18

1.1

61

1.0

83

1.7

06

1.0

50

13

.099

1.1

08

1.0

90

1.6

20

0.9

97

10

.962

0.9

27

0.9

28

Re¼

25

0

12

.70

31

4.9

603

.04

71

.12

71

7.7

12

1.1

84

1.1

38

2.6

54

0.9

82

16

.560

1.1

07

1.1

14

2.8

99

1.0

73

15

.497

1.0

36

1.0

12

21

.47

11

3.0

251

.76

71

.20

11

5.0

91

1.1

59

1.0

90

1.5

43

1.0

49

14

.227

1.0

92

1.0

75

1.4

29

0.9

72

12

.286

0.9

43

0.9

52

31

.43

71

3.0

121

.72

11

.19

71

5.0

19

1.1

54

1.0

87

1.4

89

1.0

36

14

.145

1.0

87

1.0

74

1.4

44

1.0

05

12

.415

0.9

54

0.9

53

41

.43

41

3.0

051

.71

81

.19

81

5.0

10

1.1

54

1.0

87

1.4

86

1.0

36

14

.136

1.0

87

1.0

74

1.4

43

1.0

06

12

.415

0.9

55

0.9

53

51

.39

31

1.7

201

.67

91

.20

51

3.6

92

1.1

68

1.0

98

1.4

41

1.0

34

13

.064

1.1

15

1.1

02

1.4

16

1.0

16

10

.633

0.9

07

0.9

02

Re¼

30

0

12

.62

61

6.3

612

.89

71

.10

31

9.0

70

1.1

66

1.1

28

2.5

42

0.9

68

17

.891

1.0

94

1.1

05

2.8

84

1.0

98

17

.157

1.0

49

1.0

16

21

.30

61

3.4

811

.55

51

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11

5.5

35

1.1

52

1.0

87

1.3

68

1.0

48

14

.638

1.0

86

1.0

69

1.2

61

0.9

66

12

.646

0.9

38

0.9

49

31

.26

11

3.4

921

.49

01

.18

11

5.4

48

1.1

45

1.0

83

1.2

96

1.0

28

14

.547

1.0

78

1.0

68

1.2

81

1.0

16

12

.862

0.9

53

0.9

48

41

.25

51

3.4

841

.48

51

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31

5.4

35

1.1

45

1.0

82

1.2

90

1.0

28

14

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78

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68

1.2

78

1.0

18

12

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0.9

54

0.9

48

51

.22

81

1.6

001

.45

51

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41

3.5

89

1.1

71

1.1

07

1.2

57

1.0

24

12

.961

1.1

17

1.1

09

1.2

63

1.0

28

10

.310

0.8

89

0.8

81

Re¼

35

0

12

.56

81

7.6

892

.78

61

.08

52

0.4

02

1.1

53

1.1

23

2.4

60

0.9

58

19

.182

1.0

84

1.1

00

2.8

75

1.1

20

18

.753

1.0

60

1.0

21

21

.18

41

3.8

851

.40

41

.18

61

5.9

40

1.1

48

1.0

85

1.2

43

1.0

50

15

.019

1.0

82

1.0

64

1.1

33

0.9

56

12

.950

0.9

33

0.9

47

31

.13

21

3.9

151

.32

41

.16

91

5.8

37

1.1

38

1.0

80

1.1

57

1.0

22

14

.910

1.0

72

1.0

64

1.1

60

1.0

24

13

.265

0.9

53

0.9

46

41

.12

31

3.9

071

.31

61

.17

21

5.8

22

1.1

38

1.0

79

1.1

47

1.0

21

14

.896

1.0

71

1.0

64

1.1

53

1.0

24

13

.265

0.9

53

0.9

45

51

.10

21

1.4

651

.28

81

.16

91

3.4

34

1.1

72

1.1

12

1.1

20

1.0

17

12

.806

1.1

17

1.1

11

1.1

48

1.0

42

10

.022

0.8

74

0.8

62

112

the same respect than circular tubes. Diamond tubes showed the lowest moduleaverage Nusselt number over the entire Reynolds number range studied.

Heat Transfer Enhancement Ratio

Figure 16 and Table 4 present variation of heat transfer enhancement ratio(Nuþ ) with Re for module 3 for flat, oval, and diamond tubes. Nuþ values for allthese cases seem to converge to a value of unity at very low Reynolds numbers. Thismeans that all four tube cross sections should exhibit the same heat transfer charac-teristics at very low fluid velocities. For flat and oval tubes, Nuþ is greater thanunity for all values of Re, implying that flat and oval tubes are better than circulartubes from a heat transfer point of view. However, diamond tubes have Nuþ lessthan unity over the whole range of Re studied. This shows that diamond tubes donot perform better than circular tubes from a heat transfer point of view. Flat andoval tubes exhibit a sharp increase in Nuþ up to Re ¼ 50. This shows that the heattransfer rate for flat and oval tubes is more sensitive to change in the fluid velocity inthe low-Re range (i.e., Re < 50) than that for circular tubes. For diamond tubes,Nuþ decreases gradually with Re. Therefore, sensitivity of heat transfer rate tochange in fluid velocity is comparable for diamond and circular tubes. ForRe > 100, Nuþ starts decreasing for flat tubes. Similarly, for oval tubes, Nuþ startsdecreasing for Re > 70. At high Reynolds numbers (Re > 350), Nuþ plots for alltubes level out, maintaining almost constant difference from each other. Therefore,

Figure 13. Variation of normalized pressure drop (Dp�) with Re for module 3.


Ta

ble

4.

Va

ria

tio

no

fh

eat

tra

nsf

era

nd

pre

ssu

red

rop

wit

hR

efo

rm

od

ule

3

Re

Cir

cula

rF

lat

Ov

al

Dia

mo

nd

Dp� 0

Nu

0D

p�

Dp�

Dp� 0

Nu

Nuþ

Nu�

Dp�

Dp�

Dp� 0

Nu

Nuþ

Nu�

Dp�

Dp�

Dp� 0

Nu

Nuþ

Nu�

25

8.6

57

6.5

79

12

.603

1.4

56

7.0

93

1.0

78

0.9

51

10

.55

81

.22

06

.98

41

.06

20

.99

47

.24

40

.83

76

.54

70

.99

51

.05

6

30

7.3

29

7.3

61

10

.597

1.4

46

8.1

17

1.1

03

0.9

75

8.8

75

1.2

11

7.9

47

1.0

80

1.0

13

6.1

72

0.8

42

7.3

12

0.9

93

1.0

52

35

6.3

90

7.9

84

9.1

70

1.4

35

8.9

70

1.1

23

0.9

96

7.6

80

1.2

02

8.7

37

1.0

94

1.0

29

5.4

18

0.8

48

7.9

14

0.9

91

1.0

47

40

5.6

92

8.4

83

8.1

03

1.4

24

9.6

73

1.1

40

1.0

14

6.7

88

1.1

93

9.3

81

1.1

06

1.0

43

4.8

60

0.8

54

8.3

91

0.9

89

1.0

43

45

5.1

54

8.8

87

7.2

77

1.4

12

10

.252

1.1

54

1.0

28

6.0

98

1.1

83

9.9

07

1.1

15

1.0

54

4.4

31

0.8

60

8.7

72

0.9

87

1.0

38

50

4.7

26

9.2

20

6.6

18

1.4

00

10

.731

1.1

64

1.0

40

5.5

49

1.1

74

10

.337

1.1

21

1.0

63

4.0

91

0.8

66

9.0

80

0.9

85

1.0

33

60

4.0

87

9.7

33

5.6

32

1.3

78

11

.465

1.1

78

1.0

59

4.7

29

1.1

57

10

.991

1.1

29

1.0

76

3.5

86

0.8

78

9.5

47

0.9

81

1.0

25

70

3.6

31

10

.115

4.9

28

1.3

57

11

.994

1.1

86

1.0

71

4.1

45

1.1

42

11

.459

1.1

33

1.0

84

3.2

28

0.8

89

9.8

86

0.9

77

1.0

16

10

02

.79

21

0.8

813

.64

81

.30

71

2.9

711

.19

21

.09

03

.08

61

.10

61

2.3

111

.13

11

.09

42

.57

00

.92

11

0.5

480

.96

90

.99

6

15

02

.08

41

1.7

292

.60

91

.25

21

3.8

851

.18

41

.09

82

.22

81

.06

91

3.0

991

.11

71

.09

22

.00

30

.96

11

1.2

900

.96

30

.97

5

20

01

.69

01

2.4

362

.06

11

.21

91

4.5

181

.16

71

.09

31

.77

31

.04

91

3.6

751

.10

01

.08

21

.67

00

.98

81

1.9

020

.95

70

.96

1

25

01

.43

71

3.0

121

.72

11

.19

71

5.0

191

.15

41

.08

71

.48

91

.03

61

4.1

451

.08

71

.07

41

.44

41

.00

51

2.4

150

.95

40

.95

3

30

01

.26

11

3.4

921

.49

01

.18

11

5.4

481

.14

51

.08

31

.29

61

.02

81

4.5

471

.07

81

.06

81

.28

11

.01

61

2.8

620

.95

30

.94

8

35

01

.13

21

3.9

151

.32

41

.16

91

5.8

371

.13

81

.08

01

.15

71

.02

21

4.9

101

.07

21

.06

41

.16

01

.02

41

3.2

650

.95

30

.94

6

114

Figure 14. Variation of ratio Dp�=Dp�0 with Re for module 3.

Figure 15. Variation of module average Nusselt number with Re for module 3.


Figure 16. Variation of heat transfer enhancement ratio (Nuþ ) with Re for module 3.

Figure 17. Variation of heat transfer performance ratio (Nu�) with Re for module 3.


in the low-Re range (i.e., Re < 50), advantage in heat transfer over circular tubes canbe maximized by increasing the fluid velocity for any of the flat and oval tubes. How-ever, diamond tubes are not a good replacement for circular tubes from the view-point of heat transfer, especially at high fluid velocities. At high Reynoldsnumbers, change in the geometric shape of the tube cross section rather than increasein fluid velocity should be considered in equipment design for better heat transfer rate.

Heat Transfer Performance Ratio

Variation of heat transfer performance ratio (Nu�) with Re for module 3 is pre-sented in Figure 17 and Table 4. Nu� signifies heat transfer enhancement per unitincrease in pumping power. This parameter is useful to compare various tube crosssections if both the heat transfer rate and pumping power are of major concern in thedesign of equipment. For Re < 40, diamond tubes are better than both the flat andoval tubes. However, for Re > 50, flat and oval tubes have substantial advantageover diamond tubes. Again, below Re ¼ 50 (low-Re range), increase in fluid velocityhelps substantially. However, at high Re values, geometric shape is more importantthan fluid velocity.

CONCLUSIONS

Flat and oval tubes offer more flow resistance compared to circular tubes. Aheat exchanger with flat or oval tubes will cause more pressure drop, and hencerequire more pumping power, than one with circular tubes. Oval tubes, however,offer less flow resistance when compared to flat tubes. Diamond tubes cause lesspressure drop than circular tubes for Re < 250. Hence, if pumping power is a con-cern, diamond tubes should be preferred over circular tubes only in the low Reynoldsnumber range (Re < 250). For Re > 250, circular tubes are better than diamondtubes from the viewpoint of pressure drop.

Oval tubes are less effective from a heat transfer point of view compared to flattubes, but are better in the same respect than circular tubes. Circular tubes are inturn better than diamond tubes with respect to heat transfer rate. Rate of heat trans-fer is very sensitive to fluid velocity in the low-Re range (i.e., Re < 50). Therefore, ifheat transfer rate is the major design consideration, fluid velocity should be maxi-mized in this Re range. However, in the high-Re range (i.e., Re > 150), fluid velocityhas less effect on heat transfer rate than the tube cross-section geometry does.A better tube cross section should be chosen instead of increasing the fluid velocity,in order to maximize heat transfer rate in the high Reynolds number range (i.e.,Re > 150).

At very low fluid velocity, the geometric shape of the tube cross section doesnot affect the heat transfer rate significantly.

Diamond tubes are better than both flat and oval tubes for Re < 40 from theviewpoint of heat transfer enhancement per unit increase in pumping power (Nu�).However, for Re > 50, flat and oval tubes perform better than diamond tubes. ForRe < 50, increase in fluid velocity causes Nu� to increase steeply for oval andflat tubes, and to decrease rapidly for diamond tubes. Oval tubes remain superiorto flat tubes for Re� 100. For Re > 150, flat tubes are better than oval tubes,


whereas oval tubes are superior to diamond tubes from the viewpoint of heat trans-fer performance ratio. If the design of a system is to be optimized for heat transferenhancement per unit increase in pumping power, both geometry and fluid velocitymust be carefully chosen for Re < 150. However, for Re > 150, fluid velocity is notas important as the geometric shape of the tubes. For Re > 350, Nu� is far moresensitive to geometric shape than fluid velocity.

REFERENCES

1. T. Ota, H. Nishiyama, and Y. Taoka, Heat Transfer and Flow around an Elliptic Cylin-der, Int. J. Heat Mass Transfer, vol. 27, no. 10, pp. 1771–1779, 1984.

2. T. Ota, H. Nishiyama, J. Kominami, and K. Sato, Heat Transfer from Two Elliptic Cylin-ders in Tandem Arrangement, J. Heat Transfer, vol. 108, pp. 525–531, 1986.

3. T. Wung, J. Niethammer, and C. Chen, Measurements of Heat-Mass Transfer and Press-ure Drop for Some Non-standard Arrays of Tubes in Cross Flow, in C. Tien, V. Carey,and J. Ferrell (eds.), Heat Transfer, Proc. Eighth Int. Heat Transfer Conf., pp. 1041–1046,Hemisphere, San Francisco, 1986.

4. C. H. K. Williamson, Vortex Dynamics in the Cylinder Wake, Annu. Rev. Fluid Mech,vol. 28, pp. 477–539, 1996.

5. M. M. Zdrakovich, Flow Around Circular Cylinders, 1: Fundamentals, Oxford UniversityPress, New York, 1997.

6. D. Kundu, A. Haji-Sheikh, and D. Y. S. Lou, Heat Transfer Predictions in Cross Flowover Cylinders between Two Parallel Plates, Numer. Heat Transfer A, vol. 19, pp. 361–377, 1991.

7. D. Kundu, A. Haji-Sheikh, and D. Y. S. Lou, Pressure and Heat Transfer in Cross Flowover Cylinders between Two Parallel Plates, Numer. Heat Transfer A, vol. 19, pp. 345–360, 1991.

8. D. Kundu, Computational and Experimental Studies of Flow Field and Heat Transferfrom a Row of In-line Cylinders Centered between Two Parallel Plates, Ph.D. thesis,University of Texas at Arlington, Arlington, TX, 1989.

9. V. B. Grannis and E. M. Sparrow, Numerical Simulation of Fluid Flow through an Arrayof Diamond-Shaped Pin Fins, Numer. Heat Transfer A, vol. 19, pp. 381–403, 1991.

10. Y. Chen, M. Fiebig, and N. K. Mitra, Conjugate Heat Transfer of a Finned Oval TubePart A: Flow Patterns, Numer. Heat Transfer A, vol. 33, pp. 371–385, 1998.

11. Y. Chen, M. Fiebig, and N. K. Mitra, Conjugate Heat Transfer of a Finned Oval TubePart B: Heat Transfer Behaviors, Numer. Heat Transfer A, vol. 33, pp. 387–401, 1998.

12. M. Breuer, J. Bernsdorf, T. Zeiser, and F. Durst, Accurate Computations of the LaminarFlow past a Square Cylinder based on Two Different Methods: Lattice-Boltzmann andFinite-Volume, Int. J. Heat Fluid Flow, vol. 21, pp. 186–196, 2000.

13. R. L. Webb, Principles of Enhanced Heat Transfer, Wiley, New York, 1993.14. H. M. Bahaidarah, A Numerical Study of Heat and Momentum Transfer over a Bank of

Flat Tubes, Ph.D. thesis, Texas A&M University, College Station, TX, 2004.15. H. M. Bahaidarah, N. K. Anand, and H. C. Chen, Numerical Study of Fluid Flow

and Heat Transfer over a Bank of Flat Tubes, Numer. Heat Transfer A, vol. 48,pp. 359–385, 2005.

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