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International Journal of Heat and Mass Transfer 50 (2007) 1163–1175

Three-dimensional numerical study of wavy fin-and-tubeheat exchangers and field synergy principle analysis

Y.B. Tao, Y.L. He *, J. Huang, Z.G. Wu, W.Q. Tao

State Key Laboratory of Multiphase Flow in Power Engineering, School of Energy and Power Engineering, Xi’an Jiaotong University,

Xi’an, Shaan xi 710049, China

Received 24 October 2005; received in revised form 17 March 2006Available online 11 December 2006

Abstract

In this paper, 3-D numerical simulations were performed for laminar heat transfer and fluid flow characteristics of wavy fin-and-tubeheat exchanger by body-fitted coordinates system. The effect of four factors were examined: Reynolds number, fin pitch, wavy angle andtube row number. The Reynolds number based on the tube diameter varied from 500 to 5000, the fin pitch from 0.4 to 5.2 mm, the wavyangle from 0� to 50�, and the tube row range from 1 to 4. The numerical results were compared with experiments and good agreementwas obtained. The numerical results show that with the increasing of wavy angles, decreasing of the fin pitch and tube row number, theheat transfer of the finned tube bank are enhanced with some penalty in pressure drop. The effects of the four factors were also analyzedfrom the view point of field synergy principle which says that the reduction of the intersection angle between velocity and fluid temper-ature gradient is the basic mechanism for enhance convective heat transfer. It is found that the effects of the four factors on the heattransfer performance of the wavy fin-and-tube exchangers can be well described by the field synergy principle.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Wavy fin-and-tube heat exchanger; Field synergy principle; Numerical simulation

1. Introduction

Fin-and-tube heat exchangers are employed in a varie-ties of engineering applications, for example, applicationsin such areas like air conditioning units, process gas heaterand coolers, compressor intercoolers, etc. Since majority ofthe thermal resistance of fin-and-tube heat exchangers is onthe air side, improving air side fin configuration, enhancingits heat transfer is the most effective way to improve theperformance of the heat exchangers. One of the conven-tional methods for enhancing air side heat transfer is theadoption of the wavy and slot fins.

There have been a number of numerical studies on theheat transfer and fluid flow characteristics for two-dimen-sional corrugated wavy channel flow. Nishimura et al. [1]

0017-9310/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijheatmasstransfer.2006.03.019

* Corresponding author. Tel.: +86 29 82663851; fax: +86 29 82669106.E-mail address: yalinghe@mail.xjtu.edu.cn (Y.L. He).

used the finite element method to study a two-dimensionalpulsatile flow in a wavy channel with periodically converg-ing–diverging cross-sections. Xin and Tao [2] numericallyanalyzed the laminar fluid flow and heat transfer in two-dimensional wavy channels with uniform cross-sectionalarea. Patel et al. [3,4] numerically investigated the laminarand turbulent boundary layer flows over a wavy wall. Rutl-edge and Sleicher [5] numerically studied the possibility ofimproving heat transfer rates by incorporating small corru-gations into a two-dimensional channel. Comini and Non-ino [6] adopted a simplified two-dimensional approach todeal with convective heat and mass transfer in laminarflows of humid air through wavy finned-tube exchangers.

Experimental studies of heat transfer and pressure dropin two-dimensional corrugated or wavy channels and three-dimensional wavy fin-and-tube heat exchangers have beenperformed by many researchers. Goldstein and Sparrow[7] used the naphthalene sublimation technique to

Nomenclature

A surface or cross-sectional area (m2)Cp specific heat (J/kg K)D tube outer side diameter (m)f friction factorFp fin pitch (m)h heat transfer coefficient (W/m2 K)k thermal conductivity (W/m K)L flow length (m)M module productionN the number of control volume or pointn tube row number~n dimensionless unit normal vectorNu local Nusselt number, (h Æ D)/kp pressure (Pa)Dp pressure drop in flow direction (Pa)Q heat transfer capacity (W)Re Reynolds number based on tube outer side

diameter (UcD/t)s arc length along the arbitrary boundaryS tube row pitch (m)T temperature (K)Tin inlet temperature (K)Tw wall temperature (K)T bulk average temperature (K)�p bulk average pressure (Pa)

u,v,w x,y,z velocity components (m/s)uin frontal velocity (m/s)Uc velocity at the minimum cross-sectional area

(m/s)~U velocity vector (m/s)U,V,W transformed velocity (m/s)x,y,z Cartesian coordinates

Greek symbols

a wavy angle (�)l dynamic viscosity (Pa s)t kinematic viscosity (m2/s)q density (kg/m3)k thermal conductivity (W/m K)h the local intersection angle (�)n,g,f body-fitted coordinates

Subscriptscal calculation resultsexp experiment resultsin inlet parametersm mean or average valueout outlet parametersw at wall conditionsC boundary segment

1164 Y.B. Tao et al. / International Journal of Heat and Mass Transfer 50 (2007) 1163–1175

determine the local and average heat transfer characteris-tics for flow in a two-dimensional corrugated wall channel.The effects of rounding of protruding edges of a two-dimensional corrugated wall duct were investigated bySparrow and Hossfeld [8]. Ali and Ramadhyani [9] experi-mentally studied the convective heat transfer in theentrance region of two dimensional corrugated channels.Snyder et al. [10] investigated forced-convection heat trans-fer rates and pressure drops in the thermally fully devel-oped region of a two-dimensional wavy channel. Yoshii[11] presented dry-surface Nusselt number data for twoeight-row coils with a wavy pattern . Later, Yoshii et al.[12] reported wet and dry surface data for two wavy-finnedcooling coils. Beecher and Fangan [13] experimentallytested 27 fin-and-tube heat exchangers with 21 of them hav-ing wavy-fin geometry. Webb [14] used a multiple regres-sion technique to correlate Beecher and Fangan [13] data.An experimental study was conducted by Mirth and Rama-dhyani [15] to determine the Nusselt number and frictionfactor on the air-side of wavy-finned, chilled-water coolingcoil. Xin et al. [16] experimentally investigated the air-sideheat transfer and pressure drop performances of nine trian-gular wavy fin-and-tube heat exchanger coils. The Nussultnumber and friction factor correlations were provided in awide range of Reynolds number. Wang et al. [17,18] madeextensive experiments on the heat transfer and pressuredrop characteristics of wavy fin and tube heat exchangers.

Somchai and Yutasak [19] experimentally investigated theeffect of fin pitch and number of tube rows on the air sideperformance of herringbone wavy fin and tube heatexchangers.

Jang and Chen [20] numerically studied the heat transferand fluid flow in a three-dimensional wavy fin-and-tubeheat exchanger. The effects of tube row numbers, wavyangles and wavy heights were investigated. Manglik et al.[21] analyzed the effects of fin density on low Reynoldsnumber forced convection in three-dimensional wavy-plate-fin compact channels by numerical simulation.

The foregoing literature review shows that only a littlerelated numerical works on the three-dimensional wavyfin-and-tube heat exchangers have been published in openliteratures, and the main emphasis of these studies wasplaced on the parametric study only. In all the previous stud-ies, these results were just described, or with some explana-tions from convective heat transfer and fluid flowconsiderations, not enough in explaining why things hap-pens the way it happened. In this study, the focus will beon the physics of the reasons for enhancement or deteriora-tion of the convective heat transfer with different parametercombinations. In 1998, Guo and his co-workers [22] pro-posed a novel concept about the enhancement of convectiveheat transfer for parabolic flow and showed that the reduc-tion of the intersection angle between the velocity and tem-perature gradient can effectively enhance the heat transfer.

(a) single-tube row

(b) two-tube rows

Y.B. Tao et al. / International Journal of Heat and Mass Transfer 50 (2007) 1163–1175 1165

Later, Tao et al. [23,24] demonstrated that this idea is alsovalid for elliptic flow if the flow Peclet number is not toosmall. This concept is now called as field synergy principle,since the word ‘‘synergy’’ means cooperative action of twoforces or the like [25]. An extension of the field synergy prin-ciple to more general transport phenomena was made by He[26] and He et al. [27], and a comprehensive review on therecent development of the study on the field synergy princi-ple was conducted by Tao and He [28], He and Tao [29].

This paper examines the effects of four parameters onthe heat transfer and fluid flow of wavy fin-and-tube heatexchangers from the point view of field synergy principle.Numerical simulation for wavy fin-and-tube heat transfersurface is conducted with body-fitted coordinates systemand the volume average intersection angle between thevelocity and temperature gradient within the computa-tional domain is determined. Then compare the trend ofheat transfer and the trend of the average intersection anglewith the same parameter to see if the two trends are consis-tent with the new concept. Numerical results show that thetwo trends are consistent with the new concept very well.

The details of the field synergy principle can be foundfrom the above-referenced papers, and for the simplicityof presentation, it is not re-stated here.

(c) three-tube rows

(d) four-tube rows

Fig. 2. Schematic of computational domains and geometric parameters ofthe fin surfaces studied.

2. Model description

2.1. Physical model

The schematic diagram of a wavy fin-and-tube heatexchanger is shown in Fig. 1(a). Fig. 1(b) gives a top viewof the computation domain of the two-row heat exchanger.Fig. 2 presents the four computation domains of differenttube rows (1–4) finned tube banks studied. The actual com-putation domain was 7.5 times of the original heat transferzone. The domain was extended 1.5 times of the original

(a) Schematic of wavy fin-an

(b) Schematic of compu

Z

Y

Fig. 1. Schematic diagram of a wavy fin-and-tub

heat transfer zone for the entrance section to ensure theinlet uniformity and at the exit of heat transfer regiondomain was extended 5 times of the original heat transferzone in order to make sure that the local one-way method

Fp

d-tube heat exchanger

tational domain X

α

e heat exchanger and computation domain.

Fig. 3. A pictorial view of the computational domain for two-row case forwavy angle = 0�.

1166 Y.B. Tao et al. / International Journal of Heat and Mass Transfer 50 (2007) 1163–1175

can be used for the numerical treatment of the outer flowboundary condition. In Fig. 3 the pictorial view of the com-putational domain is presented, where the upstream anddownstream parts of the computation domain are not pre-sented in scale with the fin length in order to save the space.In these two parts of the computational domain a muchcoarser grid distribution was adopted to save the comput-ing resource.

2.2. Grid generation technique

Because of the irregularity of the computationaldomain, it is difficult to use simple Cartesian coordinatesto generate grid. In this paper, the body-fitted coordinateswas adopted. The basic idea of the body-fitted coordinate isto numerically generate a curvilinear coordinate systemhaving coordinate lines coincident with each boundary ofthe real computational domain, regardless of the shapesof these boundaries. This is implemented by solving ellipticpartial differential equations. Constant values of one of thecurvilinear coordinates are specified as Dirichlet boundaryconditions on each boundary. Values of the other coordi-nates are either specified by a monotonic variation over aboundary as Dirichlet boundary conditions, or determinedby Neumann boundary conditions. In the latter case, thecurvilinear coordinate lines can be made to intersect theboundary according to some specified conditions, such asbeing normal or parallel to some given directions. It is alsopossible to exercise control over the spacing of the curvilin-ear coordinate lines in the field in order to concentrate linesin regions of expected high gradients. In any case, thenumerical generation of the coordinate system is doneautomatically for any shape boundaries, requiring onlythe input of points on the boundary.

In order to obtain a grid in the transformed space, a gridsystem generating method needs to be developed. The sim-plest equation that could be used to generate the grid isLaplace’s equations:

r2ni ¼ 0; i ¼ 1; 2; 3 ð1Þ

The commonly used grid generation techniques arebased on the Poisson equation proposed in Thompsonet al. [30]. The 3-D Poisson equation in the physical spacecan be expressed as:

o2n

o2xþ o2n

o2yþ o2n

o2z¼ P ðn; g; 1Þ

o2g

o2xþ o

2g

o2yþ o

2g

o2z¼ Qðn; g; 1Þ

o21

o2xþ o

21

o2yþ o

21

o2z¼ Rðn; g; 1Þ

ð2Þ

where P,Q,R are functions for controlling the spacing be-tween coordinate lines. The above partial differential equa-tions are subject to a set of Dirichlet boundary conditions,such as

n

g

f

264375 ¼

n1ðx; y; zÞg1

f1ðx; y; zÞ

264

375; ðx; y; zÞ 2 C ð3Þ

where g1 is a specified constant, n1(x,y,z) and f1(x,y,z) arespecified monotonic functions on a boundary segment C.

The above equations are transformed into the computa-tional space where the Cartesian coordinates are the depen-dent variables. Then we have

a11xnn þ a22xgg þ a33xff þ 2a12xng þ 2a13xnf

þ 2a23xgf þ J 2ðPxn þ Qxg þ RxfÞ ¼ 0

a11ynn þ a22ygg þ a33yff þ 2a12yng þ 2a13ynf

þ 2a23ygf þ J 2ðPyn þ Qyg þ RyfÞ ¼ 0

a11znn þ a22zgg þ a33zff þ 2a12zng þ 2a13znf

þ 2a23zgf þ J 2ðPzn þ Qzg þ RzfÞ ¼ 0

ð4Þ

where ajk ¼P3

m¼1bmkbmk, and bmk is the cofactor of the(m,k) element in the matrix M

M ¼xn xg xf

yn yg yf

zn zg zf

264

375 and J ¼ det jM j ð5Þ

Thus,

a11 ¼ b211 þ b2

21 þ b231

a22 ¼ b212 þ b2

22 þ b232

a13 ¼ b213 þ b2

23 þ b233

a12 ¼ b11b12 þ b21b22 þ b31b32

a13 ¼ b11b13 þ b21b23 þ b31b33

a23 ¼ b12b13 þ b22b23 þ b32b33

ð6Þ

The transformed boundary conditions are

x

y

z

264375 ¼

f1ðn; g1; fÞf2ðn; g1; fÞf3ðn; g1; fÞ

264

375; ðn; g1; fÞ 2 C� ð7Þ

where f1(n,g1,f), f2(n,g1,f) and f3 (n,g1,f) are determinedby the known shape of the boundary segment C and thespecified distribution of n thereon. C* is the boundary seg-ment of C in the transformed space.

Y.B. Tao et al. / International Journal of Heat and Mass Transfer 50 (2007) 1163–1175 1167

In order to improve the quality of the generated grid sys-tem, especially for the parts of wave crest and wave trough,in this paper the final grid systems were generated by theblock structured method with body-fitted coordinates[31]. The pictures of the generated grid systems for the finpart will be presented in Section 3.

Now attention is turned to the governing equations andthe boundary conditions for the physical problem studied.

2.3. Governing equations

The governing equations in Cartesian coordinates are

Continuity equation :

o

oxiðquiÞ ¼ 0 ð8Þ

Momentum equation :

o

oxiðquiukÞ ¼

o

oxil

ouk

oxi

� �� op

oxkð9Þ

Energy equation :

o

oxiðquiT Þ ¼

o

oxi

kCp

oToxi

� �ð10Þ

The governing equations in the computational space are

o

onðqUÞ þ o

ogðqV Þ þ o

o1ðqW Þ ¼ 0 ð11Þ

o

onðqUUÞ þ o

ogðqV UÞ þ o

o1ðqW UÞ

¼ o

onaJ

CU oUon

� �þ o

ogbJ

CU oUog

� �þ o

o1cJ

CU oUo1

� �þ JS

ð12Þ

where U,V,W are velocity component in transformedspace.

U ¼ a1uþ a2vþ a3w; V ¼ b1uþ b2vþ b3w;

W ¼ c1uþ c2vþ c3w;

J ¼ xnygzf þ xgyfzn þ xfynzg � xfygzn � xgynzf � xnyfzg ð13Þ

2.4. Boundary conditions

The fluid is assumed to be incompressible with constantproperty and the flow is laminar and in steady state. Inorder to simplify the generation of grid and the calculation,further simplification is made for the fin. The fin surface isconsidered to be a constant temperature surface. Thisimplies that the fin efficiency is assumed to be equal to 1.This simplification can also be understood as the assump-tion of isothermal wall boundary condition.

Inlet boundary condition: u = uin,v = w = 0,T = Tin.Outlet boundary condition: local one-way method [31].Front and back boundary condition:on the tube surface, u = v = w = 0, T = Tw;in the rest part,

v ¼ 0;ouoy¼ 0;

owoy¼ 0;

oToy¼ 0:

Top and bottom boundary condition:on fin surface, u = v = w = 0, T = Tw;in extended surface

w ¼ 0;ouoz¼ 0;

ovoz¼ 0;

oToz¼ 0:

3. Numerical methods and parameter definitions

The governing equations are discretized by using thecontrol volume method [31]. The SIMPLE algorithm isused to ensure the coupling between velocity and pressure.

The definitions of Re, average Nu number and frictionfactor are as follows.

Re ¼ U cD=t; Nu ¼ hD=k; f ¼ DpD=½ð1=2ÞqU 2cL� ð14Þ

where Uc, t, and k are the air velocity in the minimum flowcross-section of the tube row, kinetic viscosity and thermalconductivity, D is the outside diameter of the tube, Dp isthe pressure drop along the air flow direction, and L isthe fin length along the air flow direction.

The mean temperature and pressure of a cross-sectionare defined as:

T ¼R R

A uT dAR RA u dA

; �p ¼R R

A p dAR RA dA

ð15Þ

The total heat transfer and pressure drop and log-meantemperature difference are expressed as:

Q ¼ _mCpðT in � T outÞ; DP ¼ �pin � �pout;

DT ¼ ðT w � T inÞ � ðT w � T outÞln½ðT w � T inÞ=ðT w � T outÞ�

: ð16Þ

The heat transfer coefficient is defined as: h = Q/(ADT).For the presentation of numerical results in terms of

the field synergy principle, the following parameter isintroduced:

M ¼Xj~U jjgrad T j=N ð17Þ

where N is the number of the control volume covering thefin region. Obviously, where the intersection angle betweenvelocity and temperature gradient becomes zero, the pro-duction of velocity vector and temperature gradient,j~U jjgrad T j, is the largest. For simplicity it will be calledmodule production.

The local intersection angle is determined by the follow-ing equation:

h ¼ cos�1u oT

ox þ v oToy þ w oT

oz

j~U jjgrad T jð18Þ

And from the local intersection angle, the average intersec-tion angle of the computation domain of the fin area can beobtained by using numerical integration,

0 20000 40000 60000 80000 100000

19

20

21

22

23

24

25

142× 32× 20

142× 22× 10

142× 12× 10

Nu

Grids number

78× 12× 10

Fig. 4. Variation of the predicted Nusselt number with grid numbersystems.

(a) Top view of singl

(b) Top view of two

(c) Top view of three

(d) Top view of four

(e) Front view of two

Fig. 5. Grid systems generated

1168 Y.B. Tao et al. / International Journal of Heat and Mass Transfer 50 (2007) 1163–1175

hm ¼P

hi;j;kdvi;j;kPdvi;j;k

ð19Þ

where dvi,j,k is the volume element of the control volume(i, j,k).

The grid independence test has been made. Taking thetwo-row case as an example. In order to validate the solu-tion independency of the grid number, four different gridsystems are investigated. They are 78 · 12 · 10, 142 · 12 ·10, 142 · 22 · 10, 142 · 32 · 20. The predicted averagedNusselt numbers for the four grid system are shown inFig. 4. From the figure it can be seen that the solution ofthe grid system of 142 · 22 · 10 can be regarded as grid-independent. Similar examinations were also conductedfor the other three cases. The final grid numbers adoptedare 102(x) · 22(y) · 10(z), 142 · 22 · 10, 182 · 22 · 10,192 · 22 · 10, respectively for the four cases shownin Fig. 2. The grid systems generated by body-fitted

e-tube row

-tube row

-tube row

-tube row

-tube row

by body-fitted coordinates.

0 1000 2000 3000 4000 500070.0

70.5

71.0

71.5

72.0

72.5

73.0

73.5

/ Deg

.

Re

(a) Variation of intersection angle

10

θ

Y.B. Tao et al. / International Journal of Heat and Mass Transfer 50 (2007) 1163–1175 1169

coordinates are illustrated in Fig. 5. Only the fin region ispresented in the figure for simplicity.

4. Simulation results and discussions

The effect of the Reynolds number, the fin pitch, thewavy angle, and the tube row number on heat transferand fluid flow were analyzed by the self-developed code.The major results are presented in the following section.

4.1. Code validation and Re number effect

In order to validate the reliability of the numerical sim-ulation procedure and the self-developed code, numericalsimulation was carried out at the same operating condi-tions and fin geometrical configurations as presented in[16]. The computation is conducted for the two-row casewith the tube outside diameter of 10.55 mm, fin pitch2.096 mm, wavy angle at 17.44�. The Re number ranges

0 1000 2000 3000 4000 500010

15

20

25

30

35

40

45

50

55

Nu

Re

Experimental Numerical

Experimental Numerical

(a) Effect of Re on Nu

0 1000 2000 3000 4000 50000.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

f

Re

(b) Effect of Re on f

Fig. 6. Effects of Re on Nu and f.

0 1000 2000 3000 4000 5000

0

2

4

6

8

M /

Ks

-1

Re

(b) Variation of module production

Fig. 7. Variations of intersection angle and module production with Re.

from 500 to 5000, the corresponding frontal air velocityranges from 0.464 to 4.64 m/s. The results of the simulationand the comparisons between the simulation and experi-ment are shown in Fig. 6(a) and (b), where the experimen-tal correlations of the Nu and f are adopted from [16]:

As can be seen from the figures, the agreements are gen-erally good, with the mean deviation in Nu and f being8.0% and 9.7%, respectively.

Fig. 7(a) shows the variation of the average intersectionangle between velocity and temperature gradient andFig. 7(b) provides the relation of M � Re. It can be seenfrom Fig. 7(a) that the average intersection angle increaseswith increasing Re number, which implies the deteriorationof the synergy between velocity and temperature gradient,but Fig. 7(b) indicates that the value of M is almost pro-portional to the Re number. Thus, it becomes clear thatthe decrease of the increasing trend of average Nu withRe with the increasing of Re is resulted from the increasein the intersection angle between the velocity and tempera-ture gradient which means that with the increase in

(a) isothermal

(b) streamline

Fig. 8. Distributions of isothermal, streamline for Re = 1000 anda = 17.44�.

0 1 2 3 4 5 615

20

25

30

35

40

45

Nu

Fin pitch /mm(a) Variation of average Nu with fin pitch

0 1 2 3 4 5 6

0

2

4

6

8

10

12

14

16

f

Fin pitch /mm(b) Variation of average f with fin pitch

Fig. 9. Variations of average Nu and f with fin pitch.

0 1 2 3 4 5 6

50

55

60

65

70

75

80

An

gle

Th

eta

/Deg

.

Fin pitch /mm

Fig. 10. Variations of average intersection angle with fin pitch.

1170 Y.B. Tao et al. / International Journal of Heat and Mass Transfer 50 (2007) 1163–1175

Reynolds number the synergy between velocity and tem-perature gradient becomes worse and worse.

Fig. 8 presents the local distributions of the temperature,and streamline for the middle cross-section in the x–y planebetween the upper and lower sides of the fin for Re = 1000and the wavy angle a = 17.44�. It can be seen from the fig-ure that in the inlet part of the fin surface the isotherms aremore or less normal to the local streamlines, indicating avery good synergy between the velocity and temperaturefield. In the rest part of the fin surface, the synergy in somepart of the region becomes worse, characterized by thealmost parallel distributions of the streamlines and the iso-therms, and in the backward part of the two tubes, flowseparation occurs and vortex is formed. Thus, from thesynergy point of view the local heat transfer intensity basi-cally decreases streamwisely. Such variation pattern isqualitatively the same for the different cases studied in thispaper.

4.2. Fin pitch effect

For examining the effect of fin pitch, the two-tube-rowconfiguration is adopted. The fin pitch varies from 0.4 to5.2 mm with Re = 1000, wavy angle at 17.44� and the otherparameters kept the same.

Fig. 9(a) shows the relation of the Nu number and thefin pitch. It can be seen that the Nu number reaches themaximum at the fin pitch of 0.6 mm. Departure away fromthis fin pitch will leads to the decrease of the average Nunumber. When the fin pitch is larger than 2.8 mm, the finpitch has little effect on the average Nu number. The effectof fin pitch on the friction factor is presented in Fig. 9(b),which indicates that with the increase of the fin pitch, f willalways decrease, and when fin pitch is larger than 2.8 mm,the fin pitch has little effect on f either.

The field synergy presentations are provided inFig. 10. It can be concluded from the figure that the

average intersection angle almost equal to the same valuewhen the fin pitch is less than 0.6 mm. The increase of fin

38

Y.B. Tao et al. / International Journal of Heat and Mass Transfer 50 (2007) 1163–1175 1171

pitch from this value leads to the increase in the angle,which is very sensitive to the fin pitch close to this value,and once the fin pitch is greater than 1.5 mm the depen-dence becomes weaker with increasing fin pitch. Such vari-ation pattern of hm � Fp is basically consistent with thevariation of Nu � Fp shown in Fig. 9(a). It is noticed thatthe variation of Nu with the fin pitch shown in Fig. 9(a)seems a little too abrupt. The computations were repeatedfor several times and the same results were obtained. Onepossible reason is the isothermal assumption of the fin sur-face. In practical situation the fin surface temperature distri-bution also depends on the flow velocity. The variation ofthe fin pitch affects the flow field and hence the fin surfacetemperature distribution, which in turn affects the fluid tem-perature distribution. Such mutual influences may lead to amore mild variation of the heat transfer coefficient andintersection angle vs. fin pitch. Efforts are now being paidto solve the temperature fields in both fluid and fin sheetin a conjugated way in our group and the results will bereported elsewhere.

The distribution of the temperature and the streamlineat the middle cross-section for fin pitch equals 1.0 mm

(a) isothermal for fin pitch=1.0mm

(b) streamline for fin pitch=1.0mm

(c) isothermal for fin pitch=5.2mm

(d) streamline for fin pitch=5.2mm

Fig. 11. Distributions of isothermal, streamline for fin pitch = 1.0mm and5.2 mm.

and 5.2 mm are provided in Fig. 11. From the figure itcan be clearly observed that the velocity–temperature syn-ergy for Fp = 5.2 mm is worse than that of 1.0 mm. In addi-tion, for the case of Fp = 5.2 mm, two vortices occur at thebackward of the two tubes, which further deteriorate theheat transfer.

4.3. Wavy angle effect

The effects of the wavy angle on the average Nu numberand f factor are shown in Fig. 12(a) and (b). The wavyangle varies from 0� to 50� with Re = 1000, fin pitch2.0 mm and the other parameters remain the same.

Fig. 12(a) shows the relation of the Nu number and thewavy angle. It can be seen that the Nu number increaseswith the increase of wavy angle, and the trend becomesstronger at larger wavy angles. Fig. 12(b) shows the rela-tion of f � a which has the same variation trend as Nu � a.

18

20

22

24

26

28

30

32

34

36N

u

Wavy Angle /Deg.

(a) Effect of wavy angle on Nu

0 10 20 30 40 50

Wavy Angle /Deg.

0 10 20 30 40 50

1

2

3

4

5

6

7

f

(b) Effect of wavy angle on f

Fig. 12. Effects of wavy angle on Nu and f.

0 10 20 30 40 50

50

55

60

65

70

75

80

85

Ang

le T

heta

/Deg

.

Wavy Angle /Deg.

(a) Variation of average intersection angle

0 10 20 30 40 500.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

M /K

s-1

Wavy Angle /Deg.

(b) Variation of module production

Fig. 13. Variations of average intersection angle and module productionwith wavy angle.

(a) isothermal for wavy angle=0º

(b) streamline for wavy angle=0º

(c) isothermal for wavy angle=30º

(d) streamline for wavy angle=30º

Fig. 14. Distributions of isothermal, streamline for wavy angle = 0� and30�.

1172 Y.B. Tao et al. / International Journal of Heat and Mass Transfer 50 (2007) 1163–1175

Fig. 13(a) presents the variation of the average intersec-tion angle between velocity and temperature gradientand Fig. 13(b) provides the relation of M � Re. FromFig. 13(a) it can be seen that the average intersection angledecreases with increasing wavy angles, which implies thebetter synergy between velocity and temperature gradient,and Fig. 13(b) indicates that the value of M decreases withthe increasing of the wavy angle too. And from further com-parison between the effects of intersection angle decreaseand the module production decrease, it can be found thatthe positive effect resulted from the intersection angledecrease is much larger than the negative effect of the mod-ule production decrease. Thus, it becomes clear that theincrease of average Nu with the increase of a is resulted fromthe improvement of synergy between velocity and tempera-ture gradient.

The distribution of the temperature and the streamlineat the middle cross-section for wavy angle equals 0� and

30� are provided in Fig. 14. It can be observed from the fig-ure that at a = 30� the synergy becomes better and the twovortices occurring at a = 0� disappear, leading to a betterheat transfer intensity.

4.4. Tube row number effect

For examining the effect of tube row number on the heattransfer and flow characteristics of the wavy fin-and-tubeheat exchanger, the Re number was chosen as 1000, finpitch of 2.0 mm and wavy angle at 17.44�.

Fig. 15(a) shows the variation of the average Nu numberwith the tube row number n. The Nu number decreases withthe increase of tube row number. The difference of Nu num-ber of two and three rows is very distinct, but the differencebetween three and four rows is very subtle. These are fullyconsistent with available test results [16]. The effect of tuberow number on f is shown in Fig. 15(b). According to thefigure, the f decreases with the increase of tube row numbern. The difference of f factor of two and three rows is verysubtle, but the difference between one and two rows is verydistinct.

1.0 1.5 2.0 2.5 3.0 3.5 4.0

21.2

21.4

21.6

21.8

22.0

22.2

22.4

22.6

22.8N

u

(a) Variation of Nu ~ n

1.0 1.5 2.0 2.5 3.0 3.5 4.01.20

1.22

1.24

1.26

1.28

1.30

1.32

1.34

1.36

1.38

1.40

f

Tube Row

Tube Row

(b) Variation of f ~ n

Fig. 15. Variations of Nu and f with tube row number.

1 2 3 4

70.4

70.6

70.8

71.0

71.2

71.4

71.6

71.8

72.0

An

gle

Th

eta

/Deg

.

(a) Variation of ~ n

1 2 3 40.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

M /K

s-1

Tube Row

Tube Row

(b). Variation of M ~ n

θ

Fig. 16. Variations of average intersection angle and module productionwith tube row number.

Y.B. Tao et al. / International Journal of Heat and Mass Transfer 50 (2007) 1163–1175 1173

The variation of intersection angle with tube row num-ber is provided in Fig. 16(a). The angle increase with theincreasing tube row number, and after n P 3, the effectof n on h is small. The product of modules of velocityand temperature gradient is shown in Fig. 16(b). Withthe tube row number increasing, M decreases and thedecrease trend slows down gradually. These explains whythe average Nu number decrease with the tube row numberincrease and why the decrease tendency slows down aftern P 3. The above findings show that the smaller the tuberow number the better the field synergy (for tube rown > 1), and the better the heat transfer performance. Soin practical applications, the tube row number less than 3should be recommended.

Attention is now turned to the velocity distribution inthe corrugated channel with tube inserted. The velocity dis-tributions in x–z plane at three y-positions are selected toshow the major characters of the three-dimensional flow.In Fig. 17 such distributions are provided for the two-row tube case at Re = 1000 and a = 17.44�. In all the

figures presented the dimensions are scale. Fig. 17(a)–(c)presented the velocity in x–z plane at front, middle andback section. In order to improve the resolution, inFig. 17(d)–(g) the velocity distributions in the front andmiddle section are separated into two parts. From these fig-ures, following features may be noted. First, in the back-side of the two tubes vortices are formed, characterizedby the small values of velocities in that region. Second, inthe middle section where there is not any inserted tube,the flow field is similar to that in a corrugated channel,characterized by the continuing change of the flow direc-tion and part of the corrugated surface is impinged bythe oncoming flow. Third, by carefully inspecting the flowfields in the middle section, it can be found that for the fourcorrugations, in the first part of the first corrugation, sec-ond part of the second corrugation and second part ofthe third corrugation the main stream is basically parallelto the channel, while in the rest of the corrugated channel,more or less impingement occurs either at the top wall or atthe bottom wall. When impingement occurs at one wall the

(a) Velocity vector on frontal section

(b) Velocity vector on middle section

(c) Velocity vector on back section

(d) Velocity vector on frontal section -1

(e) Velocity vector on frontal section -2

(f) Velocity vector on middle section-1

(g) Velocity vector on middle section-2

Fig. 17. Velocity distributions at different x–z sections for two-row case.

1174 Y.B. Tao et al. / International Journal of Heat and Mass Transfer 50 (2007) 1163–1175

fluid leave the opposite wall. For the case studied, nostrong recirculation is observed.

5. Conclusions

In this paper, 3-D numerical simulations are conductedto explore the effect of the Reynolds number, fin pitch,wavy angle and the tube row number on the heat transferand fluid flow characteristics of wavy fin-and-tube heattransfer surfaces. The numerical results are analyzed fromthe point view of the field synergy principle. The numericalresults obtained in this study show that the heat transferperformance of the wavy fin-and-tube heat exchanger canwell be described by the field synergy principle. The follow-ing conclusions can be made.

(1) The increase of Re number leads to the increase of theNu number and the decrease of the f. The enhance-ment of heat transfer is due to the increase of themodule product of velocity and temperature gradient.However, the synergy between the velocity and thetemperature gradient becomes worse and worse withthe increase of the Reynolds number, leading to a lessincreasing tendency of Nu with Re.

(2) There exists an optimum fin pitch at which the Nu

number is the maximum, but f always decreases withthe increase of fin pitch. The average intersectionangle shows an asymptotic tendency with the increaseof fin pitch, leading to the insensitivity of the Nu

number with the increase of fin pitch beyond a certainvalue of the fin pitch.

Y.B. Tao et al. / International Journal of Heat and Mass Transfer 50 (2007) 1163–1175 1175

(3) Both the Nu number and f increase with the increaseof wavy angle, and the increasing trend of Nu numberand f increases with the wavy angle increasing. Theincrease of average Nu is resulted from the decreasein the intersection angle between the velocity andtemperature gradient.

(4) The average Nu number and f factor decreases withthe increase of tube row number. The less the tuberow number, the better the field synergy. For practi-cal applications, a tube row number less than 3 isrecommended from the heat transfer performanceby the field synergy principle.

Acknowledgements

The present work is supported by the National ScienceFund for Distinguished Young Scholars from the NationalNatural Science Foundation of China (No. 50425620), andthe Key grant Project of Chinese Ministry of Education(No. 306014).

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