International Journal of Heat and Mass Transfer 70 (2014) 734–744

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International Journal of Heat and Mass Transfer

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Heat transfer and pressure performance of a plain fin with radiantlyarranged winglets around each tube in fin-and-tube heat transfer surface

0017-9310/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.11.024

⇑ Corresponding author. Tel.: +86 29 82669106.E-mail address: wqtao@mail.xjtu.edu.cn (W.Q. Tao).

M.J. Li, W.J. Zhou, J.F. Zhang, J.F. Fan, Y.L. He, W.Q. Tao ⇑Key Laboratory of Thermo-Fluid Science and Engineering, Ministry of Education, School of Energy & Power Engineering, Xi’an Jiaotong University, 28 Xianning West Road,Xi’an 710049, China

a r t i c l e i n f o

Article history:Received 11 July 2013Received in revised form 8 November 2013Accepted 10 November 2013Available online 12 December 2013

Keywords:Longitudinal vortex generatorFin-and-tube heat exchangerWavy finField synergy principlePerformance evaluation plotWaste heat recoveryNumerical study

a b s t r a c t

A radiantly arranged LVGs enhanced fin-and-tube structure is numerically investigated in this paper toenhance air side heat transfer. The arrangement of LVGs is totally different from existing publications.In the proposed structure there exist 12 winglets around each tube. The attack angles are 50, 50, 50,50, 70 and 110�, respectively. The height ends of the winglets are further away from the tube, whilethe closed point ends of winglets are close to the tube wall. Heat transfer and pressure drop performanceis compared with other three structures: an arc-shaped wavy fin-and-tube surface, a common-flow-down LVGs enhanced fin-and-tube surface and a plain plate fin-and-tube surface. The simulation resultsshow that the 12 winglets form five local passages which can guide the moving fluid from the main flowto the tube wall, leading to some impinging effect or reducing the wake region behind the tube. The per-formance evaluation of the four structures is conducted by using the newly proposed ln (Nue/Nuo) vs. ln(fe/fo) plot based on energy saving. It is found that the proposed radiantly arranged LVGs enhanced fin-and-tube surface is the best. The field synergy principle is adopted to analyze the four structures andit is found that the domain averaged synergy angle of the proposed radiantly arranged LVGs enhancedstructure is significantly less than that of other three cases. Finally characteristics of the proposed fin-and-tube surface with five tubes are investigated at five fin pitches and compared with the wavy struc-ture of six tubes at the same other conditions. It is found that the proposed structure of five tubes canreplace the wavy structure of six tubes.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Fin-and-tube heat exchangers are widely employed in indus-tries such as heating, ventilation, air-conditioning and refrigeration(HVAC&R) systems. Their efficiency directly determines the energyconsumption of heat exchangers for both manufacturing and oper-ating processes. Air is a common used working fluid in fin-and-tube heat exchangers due to its cleanness and low cost, but theheat transfer capability of air is quite low, which leads to high ther-mal resistance on the air side of fin-and-tube heat exchanger. Intypical applications, thermal resistance of the air side takes up over90% of all [1], so the main approach for improvement of such heatexchangers is to enhance the air side heat transfer. Researchershave developed a lot of types of fin surfaces to enhance air-sideheat transfer without introducing tremendous penalty of pressuredrop and material consumption. Wavy fin and longitudinal vortexgenerator (LVG) are two usually employed enhancement tech-niques. The wavy structure periodically interrupts the growth of

the thermal boundary layers on the heat transfer surfaces and in-duces transverse vortex in wavy trough thus increasing fluid mix-ing and vortices. For the LVG enhanced fin surfaces, when fluidflows over LVG set at an appropriate attack angle, longitudinal vor-tices are generated [2] leading to the enhancement of heat transfer.In this paper a new-type arrangement of vortex generator will beproposed to enhance heat transfer of plain-fin surface and its per-formance is compared with an existing wavy fin surface. Hence, inthe following previous studies on wavy and longitudinal vortexgenerator (LVG) enhanced fin surfaces will be briefly reviewed.

In general, wavy or corrugated fins have different specific forms,including herringbone wavy fin, sinusoid or co-sinusoid wavy fin,v-shaped or triangular wavy fin, cambered corrugated fin, curvedwavy fin and so on. Heat transfer and fluid flow characteristics ofthis fin configuration were reported in detail in literatures [3–5].Jang and Chen [3] numerically investigated the effects of geomet-rical parameters especially the wavy angle on the triangular wavyfin performance, and concluded that the average Nusselt numberNu and friction factor f increase with the increase of wavy anglefor equal wavy height. Savino et al. [4] identified that f alwaysincreases with the Reynolds number Re, while Nu increases signif-icantly with Re only above a critical value of Re for herringbone

Nomenclature

Latin SymbolsA air side heat transfer areaAc cross-section area at the inletb1,b2,b3 lnCU, P, lnCU, 4p and lnCU, u

cp specific heatL effective length of finP pump powerPr Pr numberqm mass flow rateT temperatureu,v,w fluid velocity in x, y, and z direction

Greek Symbolsam modulus average intersection anglek thermal conductivity of fluidq density of fluidg dynamic viscosity

Subscriptse enhanced fino reference finw wall

M.J. Li et al. / International Journal of Heat and Mass Transfer 70 (2014) 734–744 735

wavy fin. Furthermore, according to the study of [5], there exists anoptimum fin pitch at which Nu is the maximum, and the increaseof Re leads to the increase of Nu and the decrease of f. The abovestudies show that wavy fin will possess a satisfactory performanceonly at some certain circumstances like high inlet velocity and spe-cific geometrical parameters.

Longitudinal vortex generator is the fourth generation of en-hanced measures [6]. In earlier studies, Jacobi and Shah [7] andFiebig [8] gave thorough reviews of the development of LVG.Many kinds of generators in different shapes have been devel-oped during the past two decays, such as wedge type, ploughtype, rectangular wing, delta winglet and so on. Among thesetypes of LVGs triangular wing, rectangular wing, delta wingletand rectangular winglet are the most widely used ones [7]. Mostof the typically used LVGs are summarized in [9]. Fiebig and hiscoworkers [10–12] made a great contribution to the developmentand application of LVG. They found that delta winglet has the bestperformance. So in this paper delta winglet generators areadopted, but the orientation and arrangement are different fromall previous studies.

Except the shape discussed above, the arrangement of LVGs isalso an important factor influencing the comprehensive character-istics of enhanced surfaces. Chen et al. [13,14] employed punchedwinglet longitudinal vortex generators in staggered and inlinearrangements to enhance heat transfer of oval tube heat exchan-ger, and found that winglets in staggered arrangement can bringlarger heat transfer enhancement than in in-line arrangement. Tor-ii et al. [15] referred to a pair of delta winglets as common-flow-up(in this winglets orientation flows between two adjacent wingletsaccelerate) and found it effective in reducing form drag andenhancing heat transfer of the wake region. Allison and Dally[16] also investigated common-flow-up winglets and found thatthe heat transfer of the winglet surface is 87% of a standard louverfin surface while the pressure drop is only 53%. Kwak et al. [17] andBiswas et al. [18] studied the common-flow-down configuration(in this winglets orientation flows between two adjacent wingletsdecelerate) and found it more effective for higher Re than for lowerRe. He et al. [19] proposed a vortex generator array of ‘‘V’’ configu-ration inspired by the locomotion formation of bird and fish. Thearray is composed of two delta-winglet pairs placed at the attackangle of 10 degree or 30 degree. It is found that VG array with30� is more efficient than two conventional single-pair designs atlow Re representative of many HVAC&R applications. Fan et al.[20] combined LVGs with slotted protruding parallel strips andtried different variations of arrangement, finally substituted twotube-rows of the combined structure for an air-side wavy surfacewith three tube-rows.

Researchers also conducted lots of studies on the effects of geo-metric parameters of LVGs. Wu and Tao [21] investigated geomet-ric shape, size and the location of LVGs in a channel, and found thatthe overall Nu of the channel is higher with larger space betweenthe LVG pair and larger area of LVG, and decreases with the LVG’slocation away from the inlet of the channel. Lemouedda et al. [22]utilized a CFD analysis, response surface methodology and geneticalgorithms to investigate the optimal attack angle of delta-wingletLVGs. They concluded that common-flow-up configuration showsbetter performance for the staggered arrangement, while com-mon-flow-down is better for inline arrangement. Zeng et al. [23]studied and optimized the parameters of vortex-generator by theTaguchi method. They revealed that fin pitch has the greatest effecton the comprehensive performance, and then does the transversetube pitch, attack angle, length of vortex generator, longitudinaltube pitch, and height of vortex generator. The friction factor andNusselt number of heat transfer surface are almost independentof fin thickness and fin material.

As far as the basic mechanism of enhancing heat transfer byLVG is concerned, it is often attributed to the disturbing of thethermal boundary layer, swirling and flow destabilization causedby LVG [2,7,8]. Wu and Tao [24,25] made comprehensive studieson this issue and their results definitely show that the fundamentalreason that LVG can enhance heat transfer is the improvement ofsynergy between velocity and fluid temperature gradient.

For more information of the recent developments and applica-tions of LVG the review paper of [26] is recommended.

The winglets adopted in the enhanced surfaces in open litera-tures are mostly placed in the line along the main flow direction,for either staggered or parallel arrangement. Considering that theflow passing a tube bank periodically changes its local direction:sometime towards the tube wall sometime leaving the tube wall,if we arrange individual winglet orientation to accommodate thelocal fluid flow direction, we may get some profits in enhancingheat transfer. The major goals of this paper is to find a more efficientstructure to replace a wavy fin-and-tube surface already used in air-coolers widely adopted in air-conditioning equipment for clean airspace in China. The present authors propose a new type of wingletorientation arrangement with 12 winglets being radiantly arrangedaround each tube. Considering that ‘common-flow-down’ structureof longitudinal vortex generator is one of the earliest and most rep-resentative LVG arrangements in literatures and the plain fin oncewas undoubtedly the most widely used fin in heat exchanger forits simple structure, we choose these four kinds of structure tomake a comparison study in this paper. Numerical simulation isconducted and the results are compared with 3 referenced configu-rations (a corrugated fin in use, a conventional single-pair LVGs

(a) Radiantly arranged LVGs enhanced fin

(b) Wavy fin

(c) Common-flow-down LVGs enhanced fin

(d) Plain plate fin

Fig. 2. Physical model of the four simulated structures.

Table 1Geometric dimensions of computational zone.

Items Dimensions (mm)

Fin thickness df 0.115Tube diameter Dc 13Transverse tube pitch Pt 27.5Longitudinal tube pitch Pl 15.875

736 M.J. Li et al. / International Journal of Heat and Mass Transfer 70 (2014) 734–744

placed at 45� (common-flow-down) enhanced fin and a plain platefin with the same geometrical parameters) to illustrate that theproposed one possesses a satisfactory comprehensive performance.The influence of the winglets size is also investigated. The resultsare shown in an evaluation plot newly proposed by Fan et al. [27].

2. Physical and numerical models for simulated fin-and-tubeheat transfer surface

2.1. Introduction to enhanced structures studied

The schematic view of the proposed enhanced fin surface is par-tially shown in Fig. 1. As shown there we define x, y the streamwiseand spanwise coordinates, respectively, and z the fin pitch direc-tion. The entire heat transfer surface has five heat transfer unitswith each unit being composed of one tube and 12 surroundingwinglets (see Fig. 2a). The delta winglets are punched out fromthe plain fin around each tube. The chord of the winglets l is5 mm and will be optimized in Section 5.1. The number of attackangles, defined as the angle between the chord and the tube bankcenterline, is 50, 50, 50, 50, 70 and 110�, respectively. These attackangles are obtained through some preliminary simulation for abetter performance. The height of winglets hw is 2.425 mm. Whenpunched out the delta winglets are just stuck perpendicular to thebase surface and act as an obstacle to the coming flow, hence, canincrease the disturbance of air flow. Since the delta winglet ispunched from the plain fin, the thickness of the delta winglet isthe same as the fin. The space between two neighboring wingletsis not less than 1 mm for mechanical strength of the fin. The finpitch Fp varies in different cases. Other geometric parameters arelisted in Table 1.

Apart from the proposed enhanced structure, we also take threereferenced structure for comparison. Fig. 2b–d give the physicalmodel of the three referenced structures studied in this paper.Fig. 2b is a collared wavy fin with 5 tubes in staggered arrange-ment, which is actually used in air-coolers. Further enhancementof heat transfer of the air-cooler is demanded in order to meetsome practical requirements. The wavelength of the fin is10.334 mm, the amplitude of the fin is 2 mm, and the diameterof the step is 18.0 mm. Fin pitch, fin thickness, tube diameter,transverse tube pitch and longitudinal tube pitch are identical withthe proposed fin structure shown in Fig. 2a and Table 1; Fig. 2cshows a model with a common-flow-down arranged delta-wingletpair placed at the wake region of the tube. This structure is themost original arrangement of LVGs thus taken for comparison.The attack angle is 45� recommended by Fiebig and Chen [28];Fig. 2d presents a plain fin-tube structure, which is the earliestfin-and-tube structure and still widely used now in industry.

It’s worth pointing out further that the winglets configurationemployed in this paper are different from that in open literaturesin two ways. One difference indicated above is about the globalarrangement of winglets: in conventional arrangement they arealong the line of major flow direction, while in our arrangement

Fig. 1. Schematic view of radiantly arranged winglets enhanced fin.

they are grouped and winglets in each group are around a tube;the other is related to the orientation of the two ends (heightend and its counterpart point end) of each winglet. Taking a pairof winglet arranged in common-flow-up pattern as an example,in the conventional arrangement the height end of each wingletis close to tube wall while its counterpart away from the tube wall(Fig. 3a); on the contrast, in the present arrangement the heightend of the winglet is away from the tube wall while the pointend is close the tube wall (Fig. 3b).

2.2. Computational domain

Because of the obstruction of fin, the air velocity profile at theentrance of the channel is not uniform. So the computational do-main has to be extended upstream by 2� the streamwise fin lengthto ensure a uniform velocity distribution at the domain inlet. Withregard to the outlet, it is extended 10� the fin region to avoid recir-culation and finally outflow boundary condition can be employed.The neighboring two fins’ center planes are chosen as the upperand lower boundaries of the computational domain due to theperiodical repetition of the air channels in z direction in the wholefin-and-tube heat exchanger [20]. In the spanwise direction, thecomputational region is bounded by two adjacent center lines oftubes, also because of the periodicity in y-direction.

2.3. Physical assumptions and governing equations

The inlet Reynolds number based on the outside tube diameterranges from 1335 to 3116, and according to literatures [21,27,28]the flow is considered laminar and incompressible. The fluid flow

(a) Conventional common-flow-up

(b) Present common-flow-up

Fig. 3. Two kinds of common-flow-up arranged LVGs.

M.J. Li et al. / International Journal of Heat and Mass Transfer 70 (2014) 734–744 737

and heat transfer are in steady state. The fluid thermophysical prop-erties are constant. The heat transfer coefficient on the inner wall ofthe tube and the thermal conductivity of the tube wall are high en-ough, so the tube is assumed to be at constant temperature.

Based on the foregoing assumptions, the governing equations ofcontinuity, momentum and energy for three dimensional, constantmaterial properties, laminar and steady forced convection flow canbe described as follows:

Continuity equation,

@

@xiðquiÞ ¼ 0 ð1Þ

Momentum equation,

@

@xiðquiukÞ ¼

@

@xig@uk

@xi

� �� @p@xk

; k ¼ 1;2;3 ð2Þ

Energy equation,

@

@xiðquiTÞ ¼

@

@xiC@T@xi

� �ð3Þ

where C ¼ k=cp:

Boundary conditions for the former equations are listedbelow:

Boundary conditions in the x coordinate direction are:

Domain inlet:

u ¼ constantð1:5� 3:5m=sÞ; m ¼ w ¼ 0; Tin ¼ 305K ð4aÞ

Domain outlet:

@u@x¼ @m@x¼ @w@x¼ @T@x¼ 0 ð4bÞ

Boundary conditions in the y coordinate direction are:Tube region:

Tw ¼ 280K; u ¼ m ¼ w ¼ 0; ð4cÞFin surface region:

u ¼ m ¼ w ¼ 0;@T@y¼ 0 ð4dÞ

Fluid region:

@m@y¼ @w@y¼ 0; v ¼ 0;

@T@y¼ 0 ð4eÞ

All the top and bottom boundaries (z direction): periodicboundary conditions.

2.4. Numerical methods

In the present study, the computational domain is discretizedby non-uniform grids with software GAMBIT. The governing equa-tions are discretized by the finite-volume method [29,30] and aresolved using commercial code FLUENT (version 12). The tempera-tures in the solid fin surface and in the fluid are determined simul-taneously, i.e., solved in the conjugated way. The convection termsfor momentum and energy equations are discretized by QUICKscheme. The coupling between pressure and velocity is executedby SIMPLEC algorithm [30]. Iteration convergence is consideredto be achieved, if following conditions are all satisfied:

The residual of the continuity is less than 10�4.The residual of velocity component is less than 10�6.The residual of the energy is less than 10�8.

3. Data reduction, grid independence examination and modelvalidation

Parameters used to evaluate and compare the performance ofheat transfer surfaces are defined as follows:

Total heat transfer rate:

U ¼ qmcpðTout � T inÞ ð5Þ

The logarithmic mean temperature difference:

DT lg ¼DTmax � DTmin

lnðDTmax=DTminÞð6Þ

DTmax ¼maxðTw � Tin; Tw � ToutÞ ð7Þ

The heat transfer coefficient :

h ¼ UADT lg

ð8Þ

Nusselt number:

Nu ¼ hDc

kð9Þ

Reynolds number:

Re ¼ quinDc

gð10Þ

Pressure drop:

Dp ¼ pin � pout ð11Þ

The friction factor:

f ¼ Dp12 qu2

in

� Dc

Lð12Þ

The Colburn j factor:

j ¼ Nu=ðRe � Pr1=3Þ ð13Þ

Table 2Simulation results of different grid numbers.

Grid number 1.23 million, 1.97 million 2.65 million

Nu 45.663 47.453 47.548f 0.00360 0.00373 0.00382

738 M.J. Li et al. / International Journal of Heat and Mass Transfer 70 (2014) 734–744

In the above equations Tin and Tout are the values of mean airinlet and outlet cross-section of the fin zone, Tw is the tube walltemperature, Dc is the tube outside diameter, uin is the mean veloc-ity at the inlet cross-section. The heat transfer coefficient h is de-fined in terms of the heat transfer rate U and the log-meantemperature difference, the heat transfer rate U is determined bythe aid of FLUENT.

Grid independence assessment is conducted with radiantly ar-ranged LVGs enhanced fin at channel height of 10 fins per inch at2.5 m/s. Three grid systems of 1.23 million, 1.97 million and2.65 million are adopted. The predicted averaged Nusselt numbersfor the three grid systems are shown in Table 2. The average Nus-selt number difference between 1.97 million and 2.65 millionspaced meshes is only 0.2%, thus the result of grid system 1.97 mil-lion meshes is regarded grid independent. Finally 1.97 million isused for radiantly arranged LVGs enhanced fin. The grid systemsaround each tube of the four types of fin-and-tube surfaces areshown in Fig. 4

In order to validate the computational model, numerical simu-lation is performed at the same geometric sizes and operatingconditions as the test data provided in [31,32], which is a rectangu-lar channel with a delta winglet pair of vortex generators in com-mon-flow-down arrangement. The attack angle is 30�. Thecomparison of numerical and experimental results is provided inFig. 5. The mean deviation is 6.5% and the maximum deviation is13.4%. Here the mean deviation is defined as the average differencebetween the experiment result and the numerical value of thiswork. Such agreement between numerical and experimental re-sults show the reliability of the model and method used in thepresent study.

(a) Radiantly arranged LVGs enhanced fin

(c) Common-flow-down LVGs enhanced fin

Fig. 4. Grid

4. Simulation results and discussion

4.1. Temperature distribution

Fig. 6 shows temperature contour distribution in the middleplane of channel height direction for the four simulated cases at2.5 m/s. The inlet air temperature is 305 K, and the tube tempera-ture 280 K. The fin pitch is 10 fins per inch. It is well-known thatthe heat transfer in wake region is always weak owing to vortexand circulation. Hence, the wake region should be reduced as muchas possible. From the figure the wake region is very obvious aftereach tube for the plain fin in Fig. 6a, while for the common-flow-down LVG enhanced fin and the wavy fin shown in Fig. 6b and cits size is reduced, and for the proposed structure in Fig. 6d it al-most disappears. The local flow channel (passage) formed betweenthe fifth winglet and sixth winglet for each tube in case D is bene-ficial to the reduction of wake region. In addition the local flowpassages formed by the first to fourth pair of winglets can guidethe moving fluid from the main flow towards the tube wall, leadingto some impinging effect on the tube wall .The above two functionsof the proposed arrangement of winglets make the fluid tempera-ture distribution being much more uniform at each cross section incase D than other three cases, which also contributes to heat trans-fer enhancement [33]. It is also noticeable that the outlet temper-ature of case D is the lowest, which means the total heat transferrate of the proposed fin is the highest.

4.2. Flow field

Stagnation or wake region is always unfavorable in heat ex-changer because in these zones fluid velocity is low, and foul iseasy to deposit, leading to weak heat transfer. On the contrary, highvelocity in zones with high temperature gradient is expectedaccording to field synergy principle [31–34]. Fig. 7 presents thevelocity contour distribution of the 4 types of structures at2.5 m/s. In Fig. 7a there are large scope of wake region and highestvelocity lies in the main flow; in Fig. 7b although velocity near thetube wall is improved, wake region is still obvious after the tubes;

(b) Wavy fin

(d) Plain plate fin

system.

Fig. 5. Comparison of heat transfer coefficient between the computed results andexperimental data of common-flow-down LVG pair in rectangular channel.

M.J. Li et al. / International Journal of Heat and Mass Transfer 70 (2014) 734–744 739

and in Fig. 7c wake region is reduced and velocity is also increased,but fluid gathers at the trough of the wavy fin and velocity is lowconsequently. In Fig. 7d, stagnant zones with low velocity andweak heat transfer are significantly reduced; vortices are gener-ated by the winglets and the fluid field is much more disorderedcompared with other 3 cases. The velocity of fluid near the tubein Fig. 7d is also much higher than that in main flow. This is be-cause the winglets construct convergent channels between eachwinglet and the tube wall so that air flowing through the channelis accelerated and directed rightly towards the tube wall. On theother hand, the 2 latter winglets enhance heat transfer in the wakeregion where velocity is low, and the 4 former ones guide themainstream flow with high velocity toward the tube wall wherethe temperature gradient is great and largely broaden the area ofimproved heat transfer zones. All the above points contribute tointensify heat transfer in the channel with radiantly arranged LVGs.

Fig. 8 shows the velocity vector on four cross-sections (thespecific positions are marked in Fig. 1) perpendicular to the flowdirection. It can be seen in the figure that winglets arranged as sug-gested in this paper can generate longitudinal vortices as well asthe conventional ones. On the cross-section A there is no obviousvortex after winglet A; as the vortex develops the leading edge vor-tex arises due to separation along the leading edge on cross-section

Unit: K

Fig. 6. Temperature distr

B; the corner vortex in the stagnation region of the winglet can befound on cross-section C and the partial enlarged view is shown inFig. 8B; both of the two vortices disappear on cross-section D. Thesame development of vortices occurs for winglet B on cross-sectionC and D. These vortices lead to destabilization of the streamwisevelocity and initiates higher disturbance [8]. This is also responsi-ble for the pressure penalty.

4.3. Friction and heat transfer characteristics comparison

Fig. 9a presents Nu number of the four configurations versus Renumber and Fig. 9b provides friction factor f versus Re number ofthe four heat exchange surfaces. The Reynolds number ranges fromabout 1335 to 3116, corresponding to the inlet velocity from 1.5 m/s to 3.5 m/s, which is the usual velocity range in HVAC & R systems.

As can be expected, vortex generators not only enhance heattransfer but also cause additional pressure loss. In the figure, radi-antly arranged LVGs enhanced fin has the highest Nu number com-pared with other three cases, but also the highest friction factor f.For example, at inlet velocity of 2.5 m/s, Nu of radiantly arrangedLVGs enhanced fin is 16.7, 38.5 and 100.1% higher than wavy fin,common-flow-down LVGs enhanced fin and the plain plate fin,respectively, while the friction factor f is 19.4, 86.5 and 144.0%higher than the other three cases

4.4. Performance evaluation

It is essential to determine whether the enhanced technique iseconomical and advisable in practical applications, with both heattransfer and pressure drop being increased compared with a refer-enced structure. Aiming at energy saving, Fan et al. [27] proposed aunified, easy and clear log–log plot for the performance compari-son of enhanced techniques with a referenced structure underthe most widely adopted three constraints: identical pumpingpower, identical pressure drop and identical mass flow rate of fluid.

The derivation procedure is quite lengthy and can be found indetail in [27]. For the simplicity of presentation, it will not be re-peated here. Only the major results and the understanding of theplot are described below for reader’s convenience.

Based on some assumptions widely adopted in engineeringpractice, the ratio of enhanced heat transfer rate over that of thereferenced structure, Qe/Qo, can be expressed as follows:

(a) Case A

(b) Case B

(c) Case C

(d) Case D

ibution of four cases.

Unit: m/s

(a) Case A

(b) Case B

(c) Case C

(d) Case D

Fig. 7. Velocity distribution of four cases.

Unit: m/s Cross-sectionD

Cross-sectionC

Cross-sectionB

Cross-sectionA

(a) Longitudinal vortex on four cross-sections along x direction (See Fig. 1 for the positions of cross sections A,B,C and D)

(b) Partial enlarged view of vortices in cross-section (C)

Fig. 8. Longitudinal vortices generated by winglets at different cross-sections along x direction.

740 M.J. Li et al. / International Journal of Heat and Mass Transfer 70 (2014) 734–744

CQ ;i ¼Q e

Q o¼ Nue

Nuo

� �Re

fe

fo

� �ki

Reði ¼ P; Dp; VÞ

,ð14Þ

where for the identical pumping power (i = P), identical pressuredrop (i = Dp) and identical flow rate (i = V) the exponent ki are:

kP ¼m2

3þm1; kDp ¼

m2

2þm1; kV ¼ 1:0 ð15Þ

where m1 and m2 are exponents in correlations of f � Rem1 andNu � Rem2 for the referenced structure respectively. Taking the log-arithm of (14) and setting ln(Nue/Nuo)Re, ln(fe/fo)Re as the ordinateand abscissa, respectively, we get Eq. (16), which represents astraight line in such a coordinate system:

lnNue

Nu0

� �Re¼ bi þ ki ln

fe

f0

� �Re

ð16Þ

bi is the intercept and ki is the slope of straight line in log–log coor-dinate system. The constant bi takes the values of ln CQ ;P , ln CQ ;Dp,and ln CQ,V for the three constraints, respectively.

The two coordinates, ln(Nue/Nuo) and ln(fe/fo) divide a plan intofour quadrants: in the first quadrant both Nue/Nuo and fe/fo are lar-ger than 1, and it is the most frequently encountered situation; inthe second quadrant, Nue/Nuo is larger than 1 while fe/fo less than 1.This is the most perfect situation, but no enhanced techniquesdeveloped so far can meet such condition; in the third quadrantboth Nue/Nuo and fe/fo are less than 1, which sometimes can be

Fig. 9. Nu and f of the four structures at different Re numbers.

Fig. 10. Performance evaluation plot of three structures with wavy fin serving asbaseline (Tin = 305 K, uin = 1.5,2,2.5,3,3.5 m/s, Tw = 280 K).

M.J. Li et al. / International Journal of Heat and Mass Transfer 70 (2014) 734–744 741

encountered in engineering; finally in the fourth quadrant, Nue/Nuo

is less than 1 but fe/fo larger than 1. Enhanced techniques with suchperformance will never be adopted in practice.

The first quadrant is divided into four regions by the three con-straints: the first region, Region 1, is bounded by the abscissa andthe constraint of identical pumping power whose straight lineslope kP ¼ m2

3þm1. Any combination of Nue/Nuo and fe/fo within this re-

gion can enhance heat transfer but cannot save energy; the secondregion is bounded by the two constraints, identical pumping powerand identical pressure drop whose slope kDp ¼ m2

2þm1. Techniques

whose values of Nue/Nuo and fe/fo are in this region can save energyunder the identical pumping power constraint; the third region isbounded by the diagonal where Nue/Nuo = fe/fo and the constraintof identical pressure drop. In this region heat transfer is enhancedat the constraint of identical pressure drop; finally, in the fourth re-gion, any combination of Nue/Nuo and fe/fo implies that the heattransfer enhanced ratio is larger than the ratio of friction factorat the same flow rate. Fig. 10 shows such a plot. The three straightlines corresponding to the three constraints are baselines. Theapplication of this plot for evaluating performance of enhancedtechniques is very clear. Each combination of fe/fo and correspond-ing Nue/Nuo can be represented by a point (hereafter, workingpoint) in the figure. The third quadrant is also divided into 4 re-gions and the space below the baseline of a certain constraint

means the working points lie in area within which the heat trans-fer has deteriorated from view point of this constraint.

In this study the wavy fin shown in Fig. 2b is taken as the refer-ence. Actually the plain fin can serves as the reference in [27], butas indicated above in this paper we want to find out a more effi-cient structure to replace the wavy fin. By taking the wavy fin asreference in the log–log comparison plot [27] not only the advan-tage and disadvantage of the proposed structure can be clearlyfound, but also other two fins can be obviously presented. The per-formance comparisons of the three structures over wavy fin areshown in Fig. 10 by the points of square, circle and triangle, respec-tively. In the figure, each structure has five working points corre-sponding to inlet velocity of 1.5 m/s, 2 m/s, 2.5 m/s, 3 m/s, and3.5 m/s from left to right, respectively. The five working points ofthe radiantly arranged LVGs enhanced fin lie almost on the ‘fixedflow’ line. This means the proposed fin has better performancethan the wavy fin under ‘fixed pump power’ and ‘fixed pressuredrop’ constraints, and nearly better under ‘fixed flow’ condition.That is under the same flow rate the enhanced ratio of heat transferis nearly the same as the increased ratio of friction factor, a casewhich very hardly occurs for gas heat transfer enhancement. Forexample, for the off-set fin the heat transfer enhancement ratioover the friction increase ratio is only 0.8 [36]. For the plain finand the common flow down LVGs, their working points are inthe 3rd quadrant, which implies that compared with the wavyfin, the heat transfer is deteriorated and the friction factor is de-creased, too. As shown in the figure for the ‘‘common-flow-down’’LVGs the ratio of Nue/Nuo is much larger than friction factor ratiofe/fo, while for the plain plate these two ratios become closer. Thisindicates that the ‘common-flow-down’ fin has better performancethan the plain plate fin. Thus it can be concluded that for the fourstructures compared the proposed radiantly arranged LVGs en-hanced fin is the best, the wavy-fin is next, and then comes thecommon-flow-down LVGs enhanced fin, and the plain fin is theworst.

4.5. Field synergy principle analysis

Field synergy principle (FSP) has been proved quite useful to ex-plain the mechanism responsible for the heat transfer enhance-ment of many kinds of heat transfer surfaces in recent years. Itstates that reducing the intersection angle between velocity andtemperature gradient can effectively enhance convective heattransfer. This concept was first put forward by Guo et al. in [34],later further enhanced in [35,37,38]. According to FSP, the localintersection angle between velocity and temperature gradient re-flects local heat transfer intensity: the smaller the angle the better

Fig. 11. Average intersection angles of four structures.

Fig. 12. Heat transfer rate and pressure drop of radiantly arranged LVGs enhancedfin with different winglet lengths.

Fig. 13. Heat transfer rate and pressure drop of radiant LVGs enhanced fin withdifferent fin pitches.

742 M.J. Li et al. / International Journal of Heat and Mass Transfer 70 (2014) 734–744

the local heat transfer. For the entire heat transfer region the mod-ulus average intersection angle, which is the most suitable deter-mination method for the averaged synergy angle according to themeaning of synergy re-stated in [35], can be used to indicate theglobal synergy between velocity and temperature gradient, whichreads [39]:

am ¼X j u!ji � jgradtji � dViP

j u!ji � jgradtji � dVi

� ai ð17Þ

In this study the modulus average intersection angles are calcu-lated by a self-developed UDF program with FLUENT. The numeri-cal results of am vs. Re number is presented in Fig. 11. It can be seenfrom the picture that the average intersection angle of radiantly ar-ranged LVGs enhanced fin is appreciably lower than that of theother three structures and ranks the smallest of all. The synergydegrees of wavy fin, common-flow-down LVG and plain plate finthen follow up in order. This tendency is just the opposite to thatof Nu number.

5. Effect of winglet size and fin pitch on proposed structure

Among many geometric parameters of fin-and-tube surfacesthe fin pitch often is the most sensitive parameter to the perfor-mance [5]. For the proposed LVGs enhanced surface the attack an-gles of the winglets have been fixed, the height of winglet isaffected by fin pitch; hence, the only parameter of LVG whichcan be easily changed is its length. In the following the effect ofwinglet size and fin pitch on proposed structure are presented.

5.1. Effect of winglet length

The simulation results of the influence of the length of wingletare displayed in Fig. 12. The height of the LVGs is 2.425 mm(10 fins per inch); the studied chord length is 4 mm, 5 mm and6 mm, respectively. Numerical simulation indicates that pressure

M.J. Li et al. / International Journal of Heat and Mass Transfer 70 (2014) 734–744 743

drop increases with the length of winglets, while heat transfer ratereaches its maximum value at a length of 5 mm. At 2.5 m/s heattransfer rate of the fin with 5 mm LVGs is 4.6% and 1.5% higherthan 4 mm and 6 mm, respectively; while pressure drop is 16.3%and �8.35% (negative means less) higher than 4 mm and 6 mm.Thereason can be described as below: the additional pressure drop ofthe winglet is mainly caused by its form drag, and the longer thewinglet, the larger its form drag. With respect to heat transfer, asindicated above, winglets in the fin surface form a kind of channelwhich can direct fluid with higher velocity from the main flow to-wards the tube wall. Fluid near the centerline of the channel isthen mixing with fluid near the wall where the temperature differ-ence is higher so that the heat transfer is much more intensified.Numerical results show that such channel function is most effec-tive when the winglet chord equals to 5 mm.

5.2. Effect of fin pitch

Numerical study for the effect of fin pitch is conducted for theproposed structure of 5 tubes with five different fin numbers perinch: 12, 11, 10, 9 and 8, corresponding to fin pitch of 2.117 mm,2.309 mm, 2.54 mm, 2.822 mm, and 3.175 mm, respectively. Theheight of the LVGs is selected according to the smallest channelheight and takes the value of 2.002 mm. The results are comparedwith a wavy fin of six tubes at the same conditions to demonstratewhether the 6-tube wavy fin in use can be replaced by 5-tube pro-posed fin. The inlet velocity is 2.5 m/s. Fig. 13a and b present heattransfer rate and pressure drop of the 2 configurations. In Fig. 13athe heat transfer rate of radiantly arranged LVGs enhanced fin is abit less than the wavy fin at larger fin pitch, while at smaller finpitch heat transfer of the former gets equal or even a bit larger.The reason for such variation trend may be attributed to the rela-tive effect of the height of the vortex generator. With the fixedheight (2.002 mm) at larger fin pitches the height of the LVGs isa bit too small with respect to the channel, and can’t bring enoughdisturbances to the air flow; with the decreasing fin pitch the effectof disturbance gradually increases and finally leads to a higher heattransfer rate than that of the wavy fin. Thereupon there’s an inter-section on the plot. In Fig. 13b pressure drop of radiantly arrangedLVGs enhanced fin of five tubes is lower than that of wavy fin of sixtubes at all fin pitches investigated. Therefore it can be concludedthat based on the performance comparison shown in Fig. 10 andthe comparison shown in Fig. 13 the radiantly arranged LVGs en-hanced fin has a better comprehensive performance and the 5-tubewith radiantly arranged LVGs enhanced fin can be used to replacethe 6-tube wavy fin, reducing 1/6 of the use of material as well asthe volume of the heat exchanger.

6. Conclusion

A new arrangement of LVGs on the plain fin-and-tube surface isproposed and its heat transfer and fluid flow characteristics aresimulated. Performance evaluation and comparison with otherthree structures are conducted. The effects of winglet length andfin pitch on the proposed structure are investigated. Conclusionsare summarized as follows:

(1) Radiantly arranged winglets on fin surface constitute severalconverging passages which can guide the moving fluid fromthe main flow towards the tube wall, leading to someimpinging effect on the tube wall and reducing the wakeregion behind the tube. Both are beneficial to heat transferenhancement.

(2) Winglet with its height end in the upstream direction can gen-erate longitudinal vortex as the conventional arrangement.

(3) Radiantly arranged winglets enhanced fin possesses the bestcomprehensive performance compared with other threestructures studied in this paper according to the newly pro-posed performance evaluation plot; the average intersectionangle is also the minimum of all, which reveals the essencewhy the proposed fin has the best performance.

(4) The pressure drop of radiantly arranged LVGs enhanced finincreases with the length of LVG chord, while heat transferrate reaches the maximum value at 5 mm, thus the proposedfin structure has an optimum performance at the chordlength of 5 mm.

(5) The total heat transfer rate of the proposed fin with fivetubes are almost the same as that of way fin with six tubes,while the pressure drop of the proposed structure is lowerthan the wavy structure. So the proposed structure with 5-tube can replace the wavy structure with 6-tube.

Acknowledgements

This study was supported by the National Basic Key ResearchProgram of China (973 Program) and the key project of the Na-tional Natural Science Foundation of China (51136004).

References

[1] C.C. Wang, J. Lo, Y.T. Lin, C.S. Wei, Flow visualization of annular and deltawinglet vortex generators in fin-and-tube heat exchanger application, Int. J.Heat Mass Transfer 45 (18) (2002) 3803–3815.

[2] M. Fiebig, Embedded vortices in internal flow: heat transfer and pressure lossenhancement, Int. J. Heat Fluid Flow 16 (5) (1995) 376–388.

[3] J.Y. Jang, L.K. Chen, Numerical analysis of heat transfer and fluid flow in athree-dimensional wavy-fin and tube heat exchanger, Int. J. Heat MassTransfer 40 (16) (1997) 3981–3990.

[4] S. Savino, G. Comini, C. Nonino, Three-dimensional analysis of convection intwo-dimensional wavy channels, Numer. Heat Transfer Part A 46 (9) (2004)869–890.

[5] Y.L. He, Y.B. Tao, J. Huang, Z.G. Wu, W.Q. Tao, Three-dimensional numericalstudy of wavy fin-and-tube heat exchangers and field synergy principleanalysis, Int. J. Heat Mass Transfer 50 (5–6) (2007) 1163–1175.

[6] A.E. Bergles, ExHFT for fourth generation heat transfer technology, Exp. Therm.Fluid Sci. 26 (2–4) (2002) 335–344.

[7] A.M. Jacobi, R.K. Shah, Heat transfer surface enhancement through the use oflongitudinal vortices: a review of recent progress, Exp. Therm. Fluid Sci. 11 (3)(1995) 295–309.

[8] M. Fiebig, Vortices, generators and heat transfer, Chem. Eng. Res. Des. 76 (2)(1998) 108–123.

[9] C.C. Wang, J. Lo, Y.T. Lin, M.S. Liu, Flow visualization of wave-type vortexgenerators having inline fin-tube arrangement, Int. J. Heat Mass Transfer 45 (9)(2002) 1933–1944.

[10] M. Fiebig, P. Kallweit, N. Mitra, S. Tiggelbeck, Heat transfer enhancement anddrag by longitudinal vortex generators in channel flow, Exp. Therm. Fluid Sci. 4(1) (1991) 103–114.

[11] M. Fiebig, A. Valencia, N.K. Mitra, Wing-type vortex generators for fin-and-tube heat exchangers, Exp. Therm. Fluid Sci. 7 (4) (1993) 287–295.

[12] S. Tiggelbeck, N.K. Mitra, M. Fiebig, Comparison of wing-type vortexgenerators for heat transfer enhancement in channel flows, ASME J. HeatTransfer 116 (4) (1994) 880–885.

[13] Y. Chen, M. Fiebig, N.K. Mitra, Heat transfer enhancement of a finned oval tubewith punched longitudinal vortex generators in-line, Int. J. Heat Mass Transfer41 (24) (1998) 4151–4166.

[14] Y. Chen, M. Fiebig, N.K. Mitra, Heat transfer enhancement of finned oval tubeswith staggered punched longitudinal vortex generators, Int. J. Heat MassTransfer 43 (3) (2000) 417–435.

[15] K. Torii, K. Kwak, K. Nishino, Heat transfer enhancement accompanyingpressure-loss reduction with winglet-type vortex generators for fin-tube heatexchangers, Int. J. Heat Mass Transfer 45 (18) (2002) 3795–3801.

[16] C. Allison, B. Dally, Effect of a delta-winglet vortex pair on the performance of atube-fin heat exchanger, Int. J. Heat Mass Transfer 50 (25–26) (2007) 5065–5072.

[17] K. Kwak, K. Torii, K. Nishino, Simultaneous heat transfer enhancement andpressure loss reduction for finned-tube bundles with the first or twotransverse rows of built-in winglets, Exp. Therm. Fluid Sci. 29 (5) (2005)625–632.

[18] G. Biswas, N.K. Mitra, M. Fiebig, Heat transfer enhancement in fin-tube heat-exchangers by winglet type vortex generators, Int. J. Heat Mass Transfer 37 (2)(1994) 283–291.

744 M.J. Li et al. / International Journal of Heat and Mass Transfer 70 (2014) 734–744

[19] J. He, L. Liu, A.M. Jacobi, Air-side heat-transfer enhancement by a new winglet-type vortex generator array in a plain-fin round-tube heat exchanger, ASME J.Heat Transfer 132 (7) (2010) 071801-1–071801-9.

[20] J.F. Fan, Y.L. He, W.Q. Tao, Application of combined enhanced techniques fordesign of highly efficient air heat transfer surface, Heat Transfer Eng. 33 (1)(2011) 52–62.

[21] J.M. Wu, W.Q. Tao, Numerical study on laminar convection heat transfer in achannel with longitudinal vortex generator. Part B: parametric study of majorinfluence factors, Int. J. Heat Mass Transfer 51 (13–14) (2008) 3683–3692.

[22] A. Lemouedda, M. Breuer, E. Franz, T. Botsch, A. Delgado, Optimization of theangle of attack of delta-winglet vortex generators in a plate-fin-and-tube heatexchanger, Int. J. Heat Mass Transfer 53 (23–24) (2010) 5386–5399.

[23] M. Zeng, L.H. Tang, M. Lin, Q.W. Wang, Optimization of heat exchangers withvortex-generator fin by Taguchi method, Appl. Therm. Eng. 30 (13) (2010)1775–1783.

[24] J.M. Wu, W.Q. Tao, Numerical study on laminar convection heat transfer in arectangular channel with longitudinal vortex generator. Part A: verification offield synergy principle, Int. J. Heat Mass Transfer 51 (5–6) (2008) 1179–1191.

[25] J.M. Wu, W.Q. Tao, Investigation of laminar convection heat transfer in fin-and-tube heat exchanger in aligned arrangement with longitudinal vortexgenerator from the viewpoint of field synergy principle, Appl. Therm. Eng. 27(14–15) (2007) 2609–2617.

[26] Y.L. He, Y.W. Zhang, Advances and outlooks of heat transfer enhancement bylongitudinal vortex generators, Advances in Heat Transfer, vol. 44, Elsevier,Inc., New York, 2012, pp. 119–185.

[27] J.F. Fan, W.K. Ding, J.F. Zhang, Y.L. He, W.Q. Tao, A performance evaluation plotof enhanced heat transfer techniques oriented for energy-saving, Int. J. HeatMass Transfer 52 (1–2) (2009) 33–44.

[28] M. Fiebig, Y. Chen, A. Grossegorgemann, N.K. Mitra, Conjugate heat transfer ofa finned tube. Part B: heat transfer augmentation and avoidance of heat

transfer reversal by longitudinal vortex generators, Numer. Heat Transfer PartA Appl. 28 (2) (1995) 147–155.

[29] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, NewYork, 1980.

[30] W.Q. Tao, Numerical Heat Transfer, second ed., Xi’an Jiaotong University Press,Xi’an, 2001, pp. 220–221.

[31] J.M. Wu, Research on the enhancement of natural and forced convective heattransfer by longitudinal fins and by longitudinal vortex generator (Ph.D.thesis), Xi’an Jiaotong University, Xi’an, 2006.

[32] J.M. Wu, W.Q. Tao, Effect of longitudinal vortex generator on heat transfer inrectangular channels, Appl. Therm. Eng. 37 (5) (2012) 67–72.

[33] Z.Y. Guo, S.Q. Zhou, Z.X. Li, L.G. Chen, Theoretical analysis and experimentalconfirmation of the uniformity principle of temperature difference field in heatexchanger, Int. J. Heat Mass Transfer 45 (10) (2002) 2119–2127.

[34] Z.Y. Guo, D.Y. Li, B.X. Wang, A novel concept for convective heat transferenhancement, Int. J. Heat Mass Transfer 41 (2) (1998) 2221–2225.

[35] Z.Y. Guo, W.Q. Tao, R.K. Shah, The field synergy (coordination) principle and itsapplications in enhancing single phase convective heat transfer, Int. J. HeatMass Transfer 48 (9) (2005) 1797–1807.

[36] R.K. Shah, D.P. Seculik, Fundamentals of Heat Exchanger Design, John Wiley &Sons Inc., New York, 2003, p. 696.

[37] W.Q. Tao, Z.Y. Guo, B.X. Wang, Field synergy principle for enhancingconvective heat transfer-its extension and numerical verifications, Int. J.Heat Mass Transfer 45 (18) (2002) 3849–3856.

[38] W.Q. Tao, Y.L. He, Q.W. Wang, Z.G. Qu, F.Q. Song, A unified analysis onenhancing single phase convective heat transfer with field synergy principle,Int. J. Heat Mass Transfer 45 (24) (2002) 4871–4879.

[39] J.J. Zhou, Research on the mechanisms of heat transfer enhancement andsaving energy of compact heat exchanger surface and its optimization (Ph.D.thesis), Xi’an Jiaotong University, Xi’an, 2005.