investigation on laminar convection heat transfer in...

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Applied Thermal Engineering 27 (2007) 2609–2617

Investigation on laminar convection heat transfer in fin-and-tubeheat exchanger in aligned arrangement with longitudinal

vortex generator from the viewpoint of field synergy principle

J.M. Wu a,b, W.Q. Tao a,*

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

Xi’an City 710049, Chinab School of Environment and Chemical Engineering, Xi’an University of Engineering Science and Technology, Xi’an City 710048, China

Received 21 October 2005; accepted 17 January 2007Available online 9 February 2007

Abstract

3-D numerical simulation results are presented for laminar flow heat transfer of the fin-and-tube surface with vortex generators. Theeffects of Reynolds number (from 800 to 2000), the attack angle (30� and 45�) of delta winglet vortex generator are examined. The numer-ical results are analyzed from the viewpoint of field synergy principle. It is found that the inherent mechanism of heat transfer enhance-ment by longitudinal vortex can be explained by the field synergy principle, the second flow generated by the vortex generators results inthe reduction of the intersection angle between the velocity and fluid temperature gradient. In addition, the heat transfer enhancement ofdelta winglet with the attack angle of 45� is larger than that of 30�, while the delta winglet with the attack angle of 45� results in anincrease of the pressure drop, however, the delta winglet with the attack angle of 30� results in a slight decrease.� 2007 Published by Elsevier Ltd.

Keywords: Fin-and-tube heat exchanger; Aligned arrangement; Longitudinal vortex; Field synergy principle

1. Introduction

The compact heat exchanger is widely used in manyfields such as automobile, air conditioning, power system,chemical engineering, electronic chip cooling and aero-space, etc. The main subject to design the compact heatexchanger is how to enhance the heat transfer so that itsintegral performance may be improved to meet the demandof high efficiency(energy saving) and low cost with the vol-ume as small as possible and the weight as light as possible.Many studies have been carried out and many methodshave been applied to the heat transfer enhancement inthe compact heat exchanger since 1960s.

As we know, how to reduce the thermal resistance in theairside of the compact heat exchanger is the key for the

1359-4311/$ - see front matter � 2007 Published by Elsevier Ltd.

doi:10.1016/j.applthermaleng.2007.01.025

* Corresponding author. Tel./fax: +86 29 82669106.E-mail address: [email protected] (W.Q. Tao).

heat transfer enhancement. A widely adopted technique isto use finned surface. For the finned surface an interestingmethod is to make some minor changes on the surface sothat the heat transfer coefficient of the gas side may beincreased. If so, under the condition of the same heat trans-fer rate, the fin surface area may be decreased, the volumeand manufacturing costs may be reduced, and the requiredenergy to pump the heat transfer medium can also bedecreased. Based on this thinking, the fin surface may beperiodically changed in the main flow direction such as lou-ver fin and slotted fin to disturb the flow field and then toenhance the heat transfer. This is the main flow manipula-tion method. The vortex generator (VG) is another kind ofsuch effective methods. It can be directly punched out fromthe fin surface to generate the secondary flow deliberatelyand the heat transfer may also be enhanced. In the conven-tional point of view, this method swirls flow and makesflow destabilization. This is the secondary flow manipula-tion method.

mailto:[email protected]

Nomenclature

A total area of heat transfer (m2)Ac minimum flow area (m2)cp specific heat of fluid (J/(kg K))D tube outer diameter (m)Dc fin collar outside diameter (m)Dh hydraulic diameter (m)f Fanning frictional factorFp fin pitch (m)h height of DWLVG (m)H net fin pitch (=Fp � t, m)j Colburn factorl length of DWLVG (m)L length of heat exchanger in the air flow direction

(m)N number of tube rowNu Nusselt numberp pressure (Pa)Pl longitudinal tube pitch (m)Pr Prandtl numberPt transverse tube pitch (m)

Re Reynolds numbert thickness of fin (m)T temperature (K)u, v, w velocity components (m/s)x, y, z Cartesian coordinates (m)

Greek symbols

b angle of attack (�)k thermal conductivity (W/m K)l dynamic viscosity (Pa s)h synergy angle (�)q density (kg/m3)

Subscripts

0 inlet parameteri, k indexm mean values solid parameterw wallx local

2610 J.M. Wu, W.Q. Tao / Applied Thermal Engineering 27 (2007) 2609–2617

The vortex may be divided into transverse vortex (TV)and longitudinal vortex (LV) according to the vortex rotat-ing direction. The TV’s rotating direction is normal to themain flow(streamwise) direction. The flow with TV may bea pure two-dimensional flow. Von Karman vortex streetbehind a long cylinder is a typical example of TV. TheLVs rotating direction is the same with the main flow direc-tion. Thus, it is also called the stream-wise vortex. LV isalways three-dimensional due to that the flow spiralsaround the main flow direction, and the flow structure isvery complex.

The first literature reporting the longitudinal vortex inboundary layer control was presented by Schubauuer andSpangenberg [1]. Johnson and Joubert [2] firstly reportedthe research on heat transfer related to VG. After theirwork, the study on the heat transfer enhancement in com-pact heat exchanger by vortex generators has receivedenormous attentions. Fiebig [3] presented a review paperand concluded that longitudinal vortices are more effectivethan transverse vortices in the heat transfer enhancement.With the rapid development of the numerical simulationtechnique, the investigation about it no longer onlydepends on experiment, but also is conducted by numericalsimulation rapidly and economically. The references ofnumerical simulation on the heat transfer enhancement oflongitudinal vortex generators (LVGs) in compact heatexchangers include Fiebig and Sanchez [4], Biswas et al.[5], Jahromi et al. [6], Chen et al. [7,8], Fiebig et al. [9]and Leu et al. [10], etc. However, simplifications to thephysical problem have somewhat been made in the existentreferences, for examples, (1) no reference considers the

effects of thickness of the LVG and the punched holesunder the LVGs at the same time; (2) the fin temperatureis often assumed to be constant and fin efficiency is oftenassumed to be 1, the tube and LVGs are treated as theobstacles in the fin channel [5,6]. Even though it wasdeclared that the fin conduction was considered in the ref-erences [7,8], the actual implementation is that the tubewall and fin (including the winglet) were assigned with dif-ferent temperature; the thickness of fin and winglet wereoften not considered too; (3) only one tube (the first rowor the mid row in the flow direction) or two tubes (the firsttwo rows in the flow direction), not all the rows of tubes(i.e. three tubes)is included in the computational elementin the existing literature except Leu et al. [10]; in addition,the inlet and outlet boundary conditions of the computa-tional element are not the same with each other, and hencethere is no common base to compare the results of differentauthors. Due to the above points, some improvementsshould be made in the numerical study on the fin-tube heatexchanger with LVGs.

The present paper reports the numerical investigationson the gas side heat transfer enhancement of the compactheat exchanger in aligned arrangement with LVGs underthe conditions of taking into account of all physical factors.These include: the heat conduction in the fin and LVG, thethickness of the LVG and the punched hole, and all rows oftubes being in the computational element. To employ theuniform inlet boundary condition and fully developed out-let boundary condition reasonably, the computationaldomain is extended in upstream and downstream direc-tions, respectively. The flow and heat transfer of the channel

J.M. Wu, W.Q. Tao / Applied Thermal Engineering 27 (2007) 2609–2617 2611

with different LVG in different conditions are numericallystudied in [11]. It concludes that the delta winglet vortexgenerator is more effective for heat transfer enhancementthan rectangular winglet vortex generator. Thus, the deltawinglet vortex generator is selected as the longitudinal vor-tex generator in the present paper, and is called hereafterdelta winglet longitudinal vortex generator (DWLVG).

Little research has been made about the inherent mech-anism of heat transfer enhancement by LV in the open lit-eratures. Fiebig [3] has summarized the mechanism of heattransfer enhancement by LV as follows: (1) developingboundary layers on the vortex generator surface; (2) swirl;and (3) flow destabilization. These are the traditionalpoints about it. In the present paper, the numerical resultswill be analyzed from the viewpoint of field synergy princi-ple [15–18] to find out the inherent mechanism of heattransfer enhancement by LV.

2. Computational models and numerical methods

2.1. Computational domain

The schematic diagram of fin-and-tube heat exchangerin aligned arrangement with DWLVGs is shown inFig. 1. A pair of delta winglets is punched out from thefin symmetrically behind every round tube to delay the sep-aration of the boundary layer from the tube surface, andthe wake region behind the tube may be reduced, even beeliminated by the LVs [3]. The height of delta winglet isequal to the net pitch of fins and the delta winglets serveas the fin pitch holders at the same time. Due to the sym-metric arrangement, the region enclosed by the dashed linesin Fig. 1a is selected as the computational element, and the

Fig. 1. The scheme of the fin-and-tube heat exchanger in alignedarrangement with DWLVGs. (a) Diagram of fin-and-tube heat exchangerin aligned arrangement, (b) side view of a DWLVG, (c) top view of a pairof DWLVGs.

neighboring two fins’ centric surfaces are selected as theupper and lower boundary of the computational element.Because of the fin thickness, the air velocity profile is notuniform at the entrance of the channel formed by the fins’centric surfaces. The computational domain is extendedupstream 10 times of the net fin pitch so that a uniformvelocity distribution can be assigned at the domain inlet.The computational domain is also extended downstream30 times of the net fin pitch so that fully developed bound-ary condition can be used at the domain outlet. The com-putational domain is shown schematically in Fig. 2, inwhich x, y, z are streamwise, normal, and spanwise coordi-nates, respectively.

2.2. Governing equations and boundary conditions

Due to the low air velocity and the small fin pitch, theflow in the fin channel of compact heat exchanger is lami-nar. In our computation, the Reynolds number based onthe inlet average velocity and 2 times of net fin pitch is below2000. The flow is taken as steady, incompressible one. Thegoverning equations include continuity, momentum andenergy equations for fluid region, conduction equation forsolid region. The equations can be expressed as follows.

Continuity equation:

o

oxiðquiÞ ¼ 0 ð1Þ

Momentum equation:

o

oxiðquiukÞ ¼

o

oxil

ouk

oxi

� �� op

oxkð2Þ

Energy equation:

o

oxiðquiT Þ ¼

o

oxi

kcp

oToxi

� �ð3Þ

The conductive equation for solid region:

o

oxiks

oToxi

� �¼ 0 ð4Þ

where the doubled index means summation.The required boundary conditions are described as fol-

lows: Uniform velocity and temperature boundary condi-tions are used at the inlet of the computational domain.Fully developed boundary condition is employed at theoutlet of the computational domain. Symmetry conditionis used to the side boundaries of extended domain.

For the fin-and-tube zone without LVG, no-slip andadiabatic boundaries are assigned on the upper and lowerfin surfaces, symmetry condition is employed at the fluidregion of the front/back sides, no-slip and fixed tempera-ture boundary conditions are used at the tube wall regionsof the front/back sides, no-slip and adiabatic boundariesare assigned at the fin regions of the front/back sides.

For the fin-and-tube zone with LVG, no-slip and peri-odic conditions are imposed on the upper and lower fin

Fig. 2. Three-dimensional schematic diagram of the computational domain.

Fig. 3. Grid system around the tube and delta winglet.

Table 1Geometric sizes and computational conditions for the heat exchanger instaggered arrangement without DWLVG used to verify the models andnumerical methods

Parameter Symbol/unit Size or value

Outside diameter of tube D/mm 10.2Longitudinal tube pitch Pl/mm 22.0Transverse tube pitch Pt/mm 25.4Number of tube row N 3Fin pitch FP/mm 315 per meterFin thickness t/mm 0.33Flow passage hydraulic diameter Dh/mm 3.63Wall temperature Tw/K 300Inlet temperature of air To/K 340Frontal velocity u0/m/s 1.3–5.5


surfaces, the boundary conditions at the front/back sidesare same as the ones in the case without LVG. In orderto take account of the effect of the punched holes underthe winglet pair, periodic conditions are assigned.

2.3. Numerical methods

The above set of equations with the prescribed bound-ary conditions is solved by a computational fluid dynamicscode (FLUENT). An unstructured quadrilateral mesh isgenerated. To improve the accuracy of the numericalresults, the grid around the DWLVGs and tubes is refined.The computational domain contains about four millionscells. The grid around the DWLVG and tube is shown inFig. 3.

In the computation, the fin and DWLVG regions aredefined as solid in which the velocity is 0 and heat conduc-tion equation is solved only to consider the fin efficiency.The convection terms in Eqs. (2) and (3) are discretizedby the second upwind scheme. The coupling between pres-sure and velocity is implemented by SIMPLEC algorithm.The convergence criterion for the velocity is that the maxi-mum mass residual of the cells divided by the maximumresidual of the first 5 iterations is less than 10�7. Ournumerical practice found that once the above referencedcondition was satisfied the residuals of momentum andenergy equations were all less than 10�8.

3. Verifications of models and numerical method

Incropera and Dewitt [12] presented benchmark solu-tions of Nusselt numbers and friction factors for fullydeveloped laminar flow in the rectangular duct with differ-ent radio of width to height. For example, for a duct with a

ratio of width to height of 4.0, and with constant wall tem-perature, the Nusselt number based on hydraulic diameteris 4.44 and the product of fRe = 73. The numerical solu-tions for such a duct by using the above method are as fol-lows: Nusselt number is 4.476 and fRe = 71.6. Thenumerical results have relative errors of 0.8% and1.9%compared with the benchmark solutions.

Kays and London [13] presented the benchmark data inform of figure for Colburn factor j = Nu/Re Pr1/3 and fric-tion factor f for the plain plate fin-and-tube heat exchangerin staggered arrangement with surface 8.0-3/8T. Based onthe geometric sizes of Kays and London’s fin-and-tube heatexchanger, the air flow and heat transfer characteristics arenumerically simulated using the present method. Its geo-metric sizes and the computational conditions are tabu-lated in Table 1. The hydraulic diameter of the finchannel is defined as

Dh ¼4AcL

Að5Þ

where Ac is the minimum heat exchanger flow area in unitof m2, A is the total heat transfer area in unit of m2, and L

is the length of heat exchanger in the air flow direction, thatis, the length of fin, in unit of m.

Some empirical correlations of j and f factors for stag-gered fin-and-tube heat exchanger are also available. Toour knowledge, Wang et al.’s [14] correlations for Colburnand friction factors are most accurate and reliable withwide applicable ranges. Their correlations are developedbased on 74 samples of plain-plate fins and can describe88.6% of the database for heat transfer and 85.1% of thedatabase for pressure drop, both within ±15%. The charac-

Solid lines: Num; Dash lines: Pressure drop

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

0 500 1000 1500 2000 2500

Re

Nu m

0.0

20.0

40.0

60.0

80.0

100.0

120.0

140.0

Pre

ssur

e dr

op/P

a

Wang,2000

Kays,1998

Present

Fig. 4. Comparisons of Nusselt number and pressure drop between thecomputed results and references’ predictions.


teristic dimension of Wang et al.’s correlations is the fincollar outside diameter, that is, Dc = D + 2t.

The present numerical computational results are com-pared with the results from Kays and London’s figureand Wang et al.’s correlations to validate the computa-tional models and numerical method. The Nusselt numberand pressure drop are compared in Fig. 4. Fig. 4 shows thatthe present Nusselt numbers are between the predictedresults of Wang et al. and Kays and London in our compu-tational Reynolds number (based on the hydraulic diame-ter) range. The relative error of the present Nusseltnumber to the Wang et al.’s result is less than 8%. The pres-ent pressure drops are very close to the results from Wanget al.’s correlation, the relative error is less than 6%. Thegood agreement between the computed and the testedresults shows the reliability of the present models andnumerical methods.

4. Numerical computation of the fin-and-tube heat exchanger

with three rows of tubes in aligned arrangement and with

DWLVG

The geometric sizes of the fin-and-tube heat exchangerand computational conditions are listed in Table 2. Thelongitudinal tube pitch is larger than the transverse tube

Table 2Geometric sizes and computational conditions for the computed heatexchanger with DWLVG in aligned arrangement

Parameter Symbol/unit Size or value

Outside diameter of tube D/mm 10.2Longitudinal tube pitch Pl/mm 28.0Transverse tube pitch Pt/mm 25.4Number of tube row N 3Net fin pitch H/mm 3.0Fin thickness t/mm 0.3Wall temperature Tw/K 300Inlet temperature of air To/K 340Frontal velocity u0/m/s 2.0–6.0Length of DWLVG l/mm 6.0Height of DWLVG h/mm 3.0Angle of attack b/degr 30� and 45�

pitch so that the effect of LVs may spread in the far down-stream region. The locations of the DWLVGs have beenshown in Fig. 1c. The Reynolds number ranges from 800to 2000 in the computation. The cases with attack anglesof 30� and 45� are calculated, respectively. The same heattransfer exchanger without DWLVG is also computed tocompare the effects of DWLVG on the performances offlow and heat transfer. The air-side heat transfer areas ofthe heat exchangers with DWLVGs are the same with thoseof plain-plate fin-and-tube heat exchangers because theDWLVGs are directly punched from the fins. To be conve-nient for representation, the plain-plate fin-and-tube heatexchanger without DWLVG will be denoted by the phrase‘‘case 1’’, and those with DWLVGs will be denoted by thephrases ‘‘case 2’’ and ‘‘case 3’’ for different attack angles of30� and 45�, respectively.

4.1. Effect of DWLVG on the heat transfer

Compared to the heat exchanger in staggered arrange-ment, the tubes of the heat exchanger in aligned arrange-ment offer fewer disturbances to the fluid flow; thus, thepressure drop is also lower relatively. However, a widerwake region behind the tube in aligned arrangementoccurs. The velocity between the two neighboring rows oftubes is very low, where the stagnation region may occurif the longitudinal tube pitch is too small. With the deltawinglets behind the aligned tubes, strong LVs may be gen-erated, and the heat transfer behind the tubes may beimproved greatly. Fig. 5 shows the velocity profiles for case1 on the middle plane of the computational channel, whichis parallel to the x–z-coordinate plane, under the conditionof Re = 2000. Fig. 6 shows the velocity profiles for case 3on the same plane and under the same Reynolds number.Comparing Fig. 5 with Fig. 6, we can know that the bound-ary separation occurs later for case 3 than for case 1, andthe wake regions with low velocity of case 3 are smallerthan those of case 1. Strong longitudinal vortices are gen-erated in the zones between the two neighboring rows oftubes. LVs result in the increase of the disturbance and

Fig. 5. Velocity profiles in the middle plane of the computational domainin y direction for case 1 (Re = 2000).

Fig. 6. Velocity profile in the middle plane of the computational domainin y direction for case 3 (Re = 2000).

Fig. 8. Temperature profile in the middle plane of the computationaldomain in y direction for case 1.

Fig. 9. Temperature profile in the middle plane of the computationaldomain in y direction for case 3.


mixing effects of the downstream air, especially, the airbetween the tubes. The heat transfer between the air andthe trailing edge of tubes and fins is enhanced. Fig. 7 showsthe zoomed longitudinal vortex generated by the deltawinglet located on the left-behind side of the first row oftube in the streamwise direction. Due to the alignedarrangement, the LVs behind every row of tube are verysimilar with the same rotating direction, but the rotatingintensity increase slightly from the first to the third row.Numerical results show that the maximum secondary flowvelocity is larger than 2 times of the average inlet velocity.Figs. 8 and 9 show the temperature profiles in the sameplanes as those of Figs. 5 and 6, respectively. ComparingFig. 8 with Fig. 9, we can find that for case 1 the air tem-perature is very high and close to the inlet temperature inthe whole zones except in the wake regions behind thetubes, however, due to the generation of LVs, the air tem-perature behind the DWLVGs for case 3 is obviously lowerthan that in the same position of case 1, and the air temper-ature decreases rapidly in the main flow direction. The tem-perature difference between the inlet and outlet of the heatexchanger increases and the total heat transfer rate alsoincreases. The numerical results also show that the average

Fig. 7. Zoomed longitudinal vortices in the cross-sections after the firstrow tube.

temperature of fin surface for case 3 increases 2–4 K thanthat for case 1 in the same Reynolds number. Thus, theaverage temperature difference between the wall(includingfin and tube) and air decreases. Therefore, the heat transfercoefficient of the air-side for the heat exchanger withDWLVGs is increased because of the increase of total heattransfer rate and the decrease of the average temperaturedifference between the wall and air. The airside averageNusslet numbers for cases 1, 2 and 3 are shown inFig. 10. It can be seen from Fig. 10 that the DWLVGsdo enhance the heat transfer. In the Reynolds numberrange of the present study, the average Nusselt numberincreases 16–20% for case 2, and 20–25% for case 3 com-pared with case 1. It is concluded that the heat transferenhancement caused by DWLVG is efficient for the heatexchanger in aligned arrangement. The case 3 is better thancase 2 from the point of heat transfer enhancement.

Solid lines: Num; Dash lines: Pressure drop

8.0

12.0

16.0

20.0

24.0

28.0

32.0

36.0

40.0

600 1000 1400 1800 2200Re

Nu m

40.0

80.0

120.0

160.0

200.0

240.0

Pre

ssur

e dr

op/P

a

Without LVG

=30°=45°

ββ

Fig. 10. Effect of DWLVGs on Nusselt number and pressure drop (3 tuberows).

Solid lines:Nu m /Nu0 dash lines: Pressure drop

2.0

6.0

10.0

14.0

18.0

600 1000 1400 1800 2200

Re

Nu m

20.0

60.0

100.0

140.0

180.0

Pre

ssur

e dr

op/P

a

Plain fin

Fin with LVG

Fig. 11. Effect of DWLVGs on Nusselt number and pressure drop (2 tuberows).


4.2. Effect of DWLVG on pressure drop

As we know, the airside friction characteristics of thefin-and-tube heat exchanger in aligned arrangement willbe influenced by the DWLVGs too. It is interesting thatwe find the pressure drop from inlet to outlet of the heatexchanger decreases about 8–10% for case 2 compared withcase 1 in the studied range of Reynolds number. However,the pressure drop for case 3 increases about 10–12% com-pared with case 1 in the same Reynolds number range. Thepressure drops for the three cases are shown in Fig. 10 too.The reasons leading in the results of Fig. 10 may beexplained as follows. For case 1, the air flow resistancefrom the inlet to outlet of the heat exchanger comes fromthe local resistance of the tubes and the friction resistanceof the fin surfaces. And the local resistance of the tubesplays dominant part of the total pressure drop. When theDWLVGs are installed, the airflow resistance from the inletto outlet of the heat exchanger not only comes from theabove two aspects, but also comes from the local resistanceof the DWLVGs. On the one hand, DWLVG brings aboutadditional form drag; on the other hand, the DWLVGdelays the separation of boundary layer from the circulartube and decrease the wake region behind the tube so thatthe form drag from the tubes decreases. Therefore, whetherthe total pressure drop will increase or decrease for case 2and case 3 is determined by the combined effects of thesetwo aspects. The DWLVG with attack angle of 45�offersrather larger form drag than that with attack angle of30� [11]. Therefore, the DWLVG with attack angle of 30�results in the decrease of the total pressure drop andDWLVG with attack angle of 45� results in the increaseof the total pressure. From this point, case 2 is more attrac-tive than case 3,especially in higher Reynolds number.

86.0

86.4

86.8

87.2

87.6

88.0

88.4

600 1000 1400 1800 2200Re

Syne

rgy

angl

e/ D

eg.

Without LVG=30º=45º

ββ

Fig. 12. Effect of DWLVGs on synergy angle (3 tube rows).

5. Effect of DWLVG on the fin-and-tube heat exchanger

with two rows of tubes in aligned arrangement

Up to now, the air heat exchanger(cooler or heater) withtwo rows of tubes is widely used in the fields of air condi-tioning and refrigeration to decrease fan power, and fur-ther to decrease the noise. As described above, theDWLVG with attack angle of 30� may cause the decreaseof pressure drop. Thus, the case of fin-and-tube heatexchanger with two rows of tubes in aligned arrangementis computed in this section to check its advantage.

The geometric sizes and computational conditions arethe same as those listed in Table 2 except the number oftube row is 2 rather than 3. The case of heat exchangerwithout DWLVG is also computed to compare with eachother. The average Nusselt number and pressure drop areshown in Fig. 11 for these two cases. It is found that theNusselt number in the air-side increases and pressure dropdecreases after the DWLVG is installed. In the presentrange of Reynolds number, the Nusselt number increasesabout 13–18% and pressure drop decreases about 9–11%.

6. Mechanism of the heat transfer augmentation with LVG

As mentioned in the introduction, the traditional view-point of the reasons of the heat transfer augmentation withLVG is attributed to that the generated longitudinal vorti-ces disturb, swirl and mix the fluid flow, break the bound-ary layer developing and thin it. As a result, the secondaryflow generated by LVs change the flow and temperaturefield, that is to say, the inherent flow and heat transferstructure is changed. Thus the inherent mechanism of theheat transfer enhancement may be explained by the newtheory of field synergy.

The field synergy principle for the boundary-layer flowwas firstly proposed by Guo and co-workers [15]. Its mainidea is that reducing the intersection angle between thevelocity and the temperature gradient is the basic mecha-nism for enhancing convective heat transfer. This ideawas extended from parabolic flow to elliptic flow in [16],and numerical verifications were provided in [17] showingthat the existing convective heat transfer enhancementmechanisms could be unified under the field synergy princi-ple. A comprehensive review of the recent studies of thefield synergy principle is provided in [18]. According to[17], the domain-average intersection angles (called synergyangle) between velocity and temperature gradient for case


1–3 in different Reynolds number are calculated, and thecomparisons between them are presented in Fig. 12. Itcan be seen that the synergy angle for case 1 increasesslightly with the increase of Reynolds, while the synergyangles for case 2 and case 3 decrease slightly with theincrease of Reynolds. These mean that the synergy betweenvelocity and temperature gradient becomes worse withincreasing Reynolds number for plain fin case, butimproves for the case with LVG. Comparing Figs. 12and 10, we can see that case 3 always has the smallest syn-ergy angle and the largest Nusselt number in different Rey-nolds number, while case 1 always has the largest synergyangle and the smallest Nusselt number. The values for case2 are in between. Thus we can conclude that the function ofDWLVG on the fin surface is to improve the synergybetween the velocity and temperature gradient. This isthe most fundamental reason why the longitudinal vortexgenerator can enhance the heat transfer.

7. Conclusions

The present paper reports the numerical investigationson the gas side heat transfer enhancement of a compactheat exchanger in the aligned arrangement with DWLVGsunder the conditions of taking into account of all physicalfactors such as the heat conduction in the fin andDWLVG, the thickness of the DWLVG and the punchedhole, including all rows of tubes in the computational ele-ment. To employ the uniform inlet boundary conditionand fully developed outlet boundary conditions, the com-putational domain is extended in upstream and down-stream directions, respectively.

The remarkable conclusions go to the follows:

(1) The computational models and numerical methodsare validated by comparing the numerical results withexperimental data or correlation in existing refer-ences. The good agreement between the computa-tional and the tested results shows the reliability ofthe present models and numerical methods.

(2) Longitudinal vortices with strong secondary flowvelocity generated by delta winglet pairs punchedout from the fins can enhance the air side heat trans-fer of the fin-and-tube heat exchanger greatly com-pared with the plain plate fin-and tube heatexchanger, and the average Nusselt number isincreased 16–20% by DWLVG with attack angle of30� and is increased 20–25% by DWLVG with attackangle of 45�.

(3) It is interesting that the delta winglet pairs withattack angle of 45� result in the increase of the totalpressure drop, however the delta winglet pairs withattack angle of 30� bring about the decrease of thetotal pressure drop. The pressure drop for the formeris about 10–12% higher than that for the plain-platefin-and-tube heat exchanger, and the pressure dropfor the later is about 8–10% lower than that for the

plain-plate fin-and-tube heat exchanger. From thispoint, the delta winglet pairs with attack angle of30� are more attractive than delta winglet pairs withattack angle of 45�, especially in higher Reynoldsnumber.

(4) For the aligned heat exchanger with two rows oftube, the delta winglet pairs with attack angle of30� may increase the heat transfer about 13–18%and decrease the pressure drop about 9–11%.

(5) An analysis for the three cases is carried out with fieldsynergy principle. The numerical results of the pres-ent study show that reducing the synergy anglealways brings about the heat transfer enhancementby DWLVG, demonstrating that the field synergyprinciple reveals the inherent mechanism of heattransfer enhancement with LVG.

Acknowledgements

This work was supported by the National NaturalScience Foundation of China (50476046) and the Funda-mental Key Project of R&D of China (2007BC206902).

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