International Journal of Heat and Mass Transfer 122 (2018) 11–21

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Exergy destruction minimization: A principle to convective heat transferenhancement

https://doi.org/10.1016/j.ijheatmasstransfer.2018.01.0480017-9310/� 2018 Elsevier Ltd. All rights reserved.

⇑ Corresponding authors.E-mail addresses: w_liu@hust.edu.cn (W. Liu), zcliu@hust.edu.cn (Z.C. Liu).

W. Liu ⇑, P. Liu, J.B. Wang, N.B. Zheng, Z.C. Liu ⇑School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

a r t i c l e i n f o

Article history:Received 29 September 2017Received in revised form 21 December 2017Accepted 13 January 2018

a b s t r a c t

Enhancing convective heat transfer, by strengthening fluid disturbance, is an effective way of raising theutilization efficiency of an energy system. In this study, in order to introduce a principle for convectiveheat transfer enhancement, a physical quantity called available potential was defined, and an equationof available potential was derived to obtain an expression of local exergy destruction caused by heattransfer and fluid flow. Then, a mathematical model of convective heat transfer optimization was estab-lished by adopting adequate objective function and constraint condition based on exergy destructionminimization, and an optimal flow field of longitudinal whirling flow with multi-vortexes was obtainedby solving the optimization controlling equations. Finally, through numerical simulation and a PIV exper-iment, a heat transfer unit consisting of center-connected awl-shaped slices was studied to realize lon-gitudinal whirling flow with multi-vortexes in a tube, which indicated an excellent effect of heattransfer enhancement.

� 2018 Elsevier Ltd. All rights reserved.

1. Introduction

Heat transfer and flow phenomena are commonly involved inenergy utilization. Enhancing heat transfer for efficient energy uti-lization is an effective way of saving energy. Among three heattransfer modes, i.e., conduction, convection, and radiation, convec-tive heat transfer enhancement has received the most attentionowing to its wide applications [1]. Those techniques that con-tribute positively to the increment in heat transfer coefficient areusually termed as heat transfer enhancement or augmentation,which have been developed to the third generation [2–4]. Gener-ally, convective heat transfer enhancement techniques can bedivided into active and passive ones, according to whether or notextra energy is needed during the process [5]. Compared to theactive enhancement technique, the passive one does not requireexternal energy during implementation, which results in low cost,easy maintenance, and widespread application. The ordinary tech-nique of passive enhancement is to utilize surface or geometricalmodifications to disturb the flow field near the wall. In this cate-gory, the most popular methods include coiled tubes [6–8], fins[9–11], ribs [12–14], and grooves [15–19]. The enhancementmechanism of this method is to strengthen fluid disturbance onthe tube wall by creating a secondary flow or swirling turbulence.Another method is to place various inserts or turbulators in the

tubes, such as coiled wires [20–22], helical screw tapes [23–25],twisted tapes [26–34], and others [35–38]. By applying this tech-nique, the fluid mixing in a tube is significantly strengthened,which leads to the improvement of temperature uniformity insidethe tube, and thereby increasing fluid temperature gradient nearthe tube wall.

There is no doubt that heat transfer performance can beimproved by the enhancement techniques, but the accompanyingpressure drop is also substantially increased. Excessive pumpingpower consumption owing to flow blockage will affect the overallheat transfer performance. Therefore, the heat transfer enhance-ment techniques without excessive pressure drop are of greatinterest. Although significant developments have been achievedin convective heat transfer enhancement techniques during thepast few decades, the correlative theories aimed at revealing thephysical mechanisms behind these enhancement techniques arestill relatively underdeveloped. The existing principles forconvective heat transfer enhancement include entropy generationminimization [39–42], field synergy principle [43–52], entransydissipation extreme principle [53–59], and heat transfer optimiza-tion [60–64]. Until now, however, there still exists the research gapbetween theories and techniques in convective heat transferenhancement, and further theoretical studies ought to be madeto promote technique developments in this area.

In addition, the effects of heat transfer enhancement need to beevaluated in terms of energy conservation. As we know, thermalenergy conversion and transfer are two aspects of energy

12 W. Liu et al. / International Journal of Heat and Mass Transfer 122 (2018) 11–21

utilization. Although the efficiency is a general criterion for ther-mal energy conversion [65–68], is there an efficiency for thermalenergy transfer? The answer is no if we follow the first law of ther-modynamics. However, based on the second law of thermodynam-ics, there should be an efficiency that reflects the effectiveness ofthermal energy transfer in an irreversible process. Then, how canwe mathematically define and derive the efficiency expression toevaluate the utilization rate of thermal energy in the irreversibletransfer process? This is also what we aim to find in this study.

2. Effectiveness of thermal energy transfer

2.1. Available potential of working fluid

Not only the irreversibility of a transfer process characterizedby entropy, exergy, and other thermodynamic quantities, but alsothe unavailability of thermal energy in a transfer process relatedto ambient temperature should be considered to describe the effec-tive transfer of thermal energy.

In a thermodynamics or fluid dynamics system, the availablepotential of a working fluid can be defined as

e ¼ h� T0s; ð1Þwhere h is enthalpy, s is entropy, T0 is ambient temperature, and e isa thermodynamic function called available potential. In Eq. (1), theunavailable thermal potential T0s is deducted from the total poten-tial h, that is to say, only thermal energy above ambient tempera-ture could be utilized in a thermal energy conversion or transferprocess. Obviously, the available potential is different from thermo-dynamic exergy, which is defined as

ex ¼ ðh� T0sÞ � ðh0 � T0s0Þ ¼ e� e0; ð2Þwhere h0 and s0 are enthalpy and entropy, respectively, at environ-mental temperature T0 and pressure p0; ex represents maximumwork when the working fluid reaches the atmospheric environmentduring a reversible process; and e0 represents a reference potentialat the environmental state, which is called thermodynamic deadstate.

According to the definition of enthalpy, Eq. (1) can be written as

e ¼ g þ ðT � T0Þs; ð3Þwhere g is Gibbs free enthalpy. In Eq. (3), the available potential of aworking fluid includes two parts: the work potential g, representingthe ability of the working fluid to do work, and the available ther-mal potential (T � T0)s, in which only a part is capable of being uti-lized in energy conversion. Thus, the thermodynamics function ecan represent the energy potential of a working fluid that is avail-able to do work.

2.2. Available potential equation

Now, let us focus on the thermal energy transport in an uncom-pressible fluid. By substituting the differential expression ofenthalpy into Eq. (1), we can have a differential expression of avail-able potential,

de ¼ dh� T0ds ¼ Tdsþ vdp� T0ds ¼ ðT � T0Þdsþ 1qdp: ð4Þ

Then, the total derivative of available potential can be corre-lated to those of entropy and pressure,

qDeDt

¼ qðT � T0ÞDsDt þDpDt

: ð5Þ

In Eq. (5), the total derivative of entropy can be written as

qDsDt

¼ �r � qT

� �þ kðrTÞ2

T2 þUT; ð6Þ

where q is the heat flux vector, and entropy generation caused bythe heat transfer and fluid friction is

Sgen ¼ kðrTÞ2T2 þU

T; ð7Þ

and the viscous dissipation function in the Cartesian system can bewritten as

U ¼ l 2@u@x

� �2

þ @v@y

� �2

þ @w@z

� �2" #

þ @u@y

þ @v@x

� �2(

þ @u@z

þ @w@x

� �2

þ @v@z

þ @w@y

� �2): ð8Þ

Substituting Eq. (6) into Eq. (5), we can have

qDeDt

¼ �r � 1� T0

T

� �q

� �þ 1� T0

T

� �Uþ T0rT

T2 � qþ DpDt

; ð9Þ

where the total derivative of pressure can be written as

DpDt

¼ @p@t

þ U � rp: ð10Þ

For an uncompressible fluid in a fluid dynamics system, Eq. (10)becomes

DpDt

¼ �U � ðqU � rU � lr2UÞ: ð11Þ

Substituting Eq. (11) into Eq. (9), we can obtain the followingavailable potential equation,

qDeDt

¼ �r � 1� T0

T

� �q

� �þ 1� T0

T

� �U� T0

kðrTÞ2T2 � U

� ðqU � rU � lr2UÞ: ð12ÞEq. (12) shows that the change of available potential is equal to

the exergy flux plus local exergy destruction, that is

exd ¼ T0kðrTÞ2

T2 þ U � qU � rU � lr2U� �

: ð13Þ

Then, the total exergy destruction of the fluid caused by heattransfer and fluid flow can be integrated in the flow region X,

Exd ¼ZZZ

XexddV

¼ZZZ

XT0

kðrTÞ2T2 þ U � ðqU � rU � lr2UÞ

" #dV ð14Þ

2.3. Efficiency of thermal energy transfer

According to Eq. (13), the total exergy destruction of the fluidconsists of two parts: thermal dissipation and power consumption,

Exd ¼ Exd;DT þ Exd;Dp; ð15Þwhere thermal dissipation owing to temperature difference is

Exd;DT ¼ZZZ

XT0

kðrTÞ2T2 dV ð16Þ

and power consumption owing to pressure drop is

Exd;Dp ¼ZZZ

XU � ðqU � rU � lr2UÞdV ð17Þ

Fig. 1. Schematic diagram of a two-region flow model in a circular tube.

W. Liu et al. / International Journal of Heat and Mass Transfer 122 (2018) 11–21 13

In addition, according to Gauss’ divergence theorem, the totalexergy flux entering the system from the boundary C can beexpressed as

Ex ¼ZZZ

X�r � 1� T0

T

� �q

� �dV ¼ tC � qe � dS; ð18Þ

where qe is exergy flux vector, and S is integral surface vector.Consequently, we can define the efficiency of thermal energy

transfer based on the second law of thermodynamics,

ge ¼ 1� Exd

Ex¼ 1� Exd;DT þ Exd;Dp

Ex: ð19Þ

Substituting Eqs. (16)–(18) into Eq. (19), we can finally obtainan efficiency expression of thermal energy transfer process,

ge ¼ 1�RRR

XT0kðrTÞ2

T2dV þ RRR

XU � ðqU � rU � lr2UÞdVtC � qe � dS

; ð20Þ

Besides, to evaluate the comprehensive performance of a heattransfer unit or device, we have defined a calculation formulacalled efficiency evaluation criterion (EEC for short),

EEC ¼ Q=Q0

Pw=Pw0; ð21Þ

where Q and Q0 represent heat fluxes, and Pw and Pw0 representpower consumptions for the heat transfer enhanced tube and plaintube, respectively. If the value of Reynolds number is the same inboth tubes, we then have

EEC ¼ Q=Q0

DP=DP0; ð22Þ

where DP and DP0 represent pressure drops in the heat transferenhanced tube and plain tube, respectively. For a simplified calcula-tion, Eq. (22) becomes

EEC ¼ Nu=Nu0

f=f 0; ð23Þ

where Nu and Nu0 represent the Nusselt numbers, and f and f 0 rep-resent resistance coefficients for the heat transfer enhanced tubeand plain tube, respectively. Actually, either Eq. (22) or Eq. (23)can be used to evaluate heat transfer and flow characteristics in aheat transfer enhanced tube, but Eq. (22) can be applied to bothtube and shell sides in a shell-and-tube heat exchanger.

3. Convective heat transfer optimization

3.1. Physical model

In order to reveal the physical mechanism for reducing the irre-versibility of heat transfer and fluid flow, convective heat transferwas optimized by establishing and solving a mathematical modelbased on exergy destruction minimization to enhance convectiveheat transfer in a tube.

We take a circular tube as a physical model. As shown in Fig. 1,the flow area in a tube cross section can be divided into tworegions: core flow and boundary flow. In these regions, as the irre-versible losses of heat transfer and fluid flow are different, we needto choose a suitable optimization objective so as to reduce theexergy destruction of the fluid as much as possible.

3.2. Mathematical equations

To obtain the optimal flow field, we constructed the Lagrangefunctional by selecting appropriate objective function and con-straint condition in the core and boundary flow regions of the tube,respectively,

Ji ¼Z Z Z

XpiþC0;i/iþAir�UþBiðkr2T�qcpU �rTÞh i

dV ; i¼1;2;

ð24Þwhere C0,i is the constant Lagrange multiplier, and Ai and Bi are vari-able Lagrange multipliers. The objective function pi and constraintcondition /i can be expressed as,

pi ¼ ½p1;p2�; /i ¼ ½/1;/2�; ð25a;bÞwhere

p1 ¼ /2 ¼ T0kðrTÞ2

T; p2 ¼ /1

¼ U � ðqU � rU � lr2UÞ: ð26a;bÞIn Eqs. (24)–(26), subscripts 1 and 2 represent the core and

boundary flow regions, respectively. In the core flow region, weneed to minimize the exergy destruction caused by heat transfer,for maintaining a uniform fluid temperature, and we choose p1

and /1 as the optimization objective and constraint condition,respectively. However, in the boundary flow region, we aim atminimizing the exergy destruction caused by fluid flow, for reduc-ing fluid momentum and friction losses, and we choose p2 and /2

as the optimization objective and constraint condition, respec-tively. In this way, we hope to obtain an optimal flow field struc-ture for convective heat transfer in the tube.

Then, a functional variation with respect to velocity and tem-perature was conducted, and the following equation set wasobtained,

r � U ¼ 0; ð27Þ

qcpU � rT ¼ kr2T; ð28Þ

qU � rU ¼ �rpþ lr2U þ F i; i ¼ 1;2; ð29Þwhere the source term F i represents an additional thermal effectexerted on the fluid to form an optimal flow field, which can bewritten as,

F i ¼ F1; F2½ �; ð30Þwhere

F1 ¼ qcpB1rTC0;1

; F2 ¼ qcpB2rT; ð31a;bÞ

where scalars B1 and B2 are variable Lagrange multipliers corre-sponding to the core and boundary flow regions, respectively,which can be solved by the following partial differential equationsthat had been derived as,

qcpU � rB1 ¼ �kr2B1 þ 2T0kr2T

T2 � 2T0kðrTÞ2T3 ; ð32aÞ

14 W. Liu et al. / International Journal of Heat and Mass Transfer 122 (2018) 11–21

qcpU � rB2 ¼ �kr2B2 þ 2C0;2T0kr2T

T2 � 2C2T0kðrTÞ2T3 ; ð32bÞ

where C0,1 and C0,2 are constant Lagrange multipliers correspondingto the core and boundary flow regions, respectively.

3.3. Optimized flow field

A circular tube with 20 mm in diameter and 1700 mm in length,which is widely used in the heat exchangers, is adopted as the cal-culation model. Water is chosen as the working fluid and the tem-perature of the inlet fluid is set as 300 K, while the tube wall ismaintained at a constant temperature of 310 K. The Reynolds num-bers for the inlet are 400, 800 and 1200. For the boundary condi-tion of constant wall temperature, the value of B1 and B2 is 0 atthe tube wall, which is deduced in Ref. [62]. An outflow conditionwas used at the outlet.

In order to obtain the optimized results, the controlling equa-tion set, namely Eqs. (27), (28), (29) and (32), was solved numeri-cally in a circular tube. The computational fluid dynamics (CFD)software FLUENT 15.0 is adopted in this calculation. The SIMPLEalgorithm is used for pressure-velocity coupling, and the second-order upwind scheme is adopted to discretize the convection anddiffusion terms. The user defined function (UDF) is utilized toadd the source term into the momentum equation. The userdefined scalar (UDS) is utilized to solve the constraint equations(32a, b) for variable Lagrange multipliers, namely B1 and B2. Theconvergent solutions are obtained when the values of residualsare less than 10�4.

A grid independent testing has been performed for this physicalmodel. The hexahedral grid is adopted in the calculation, and thegrid in the boundary region is refined. Three mesh systems with1,780,818, 3,059,392, and 5,918,747 grids are adopted to calculatea baseline case. By the calculation comparison among the threemesh systems, the 3,059,392-grid system with 1.4% deviation ofthe Nusselt number and 1.0% deviation of the friction factor isfound to be dense enough to result in the grid independentsolutions.

As the flow region in the tube cross section is divided into twoparts, i.e., core and boundary flows, we should define the propor-tion of the core and boundary flow regions. In our previous work[62], the proportion was described as a dimensionless thickness

Table 1Performance parameters of optimized field at different dimensionless thicknesses(C0,1 = �1 � 107; C0,2 = 3 � 10�7; Re = 400).

e Nu/Nu0 f/f0 EEC

0.10 3.61 1.07 3.380.15 2.94 1.07 2.740.20 2.54 1.05 2.41

Fig. 2. Schematic diagram of fields (Re = 400, C0,1 = �1 � 108, C

e, which was defined by the ratio of boundary flow thickness andcircular tube radius. In order to decide the proper dimensionlessthickness, we calculated the cases with different values of e, whenC0,1 is �1 � 107, C0,2 is 3 � 10�7, and Re number is 400. The resultsare shown in Table 1, in which we can found that when the edecreases, the EEC increases. However, if the e is further decreasedmuch lower than 0.1, the calculation convergence will be influ-enced. So the e is set at 0.1.

The optimized fields are obtained by solving Eqs. (27), (28),(29), and (32), when Re number is 400, C0,1 is �1 � 108, and C0,2is 3 � 10�7. As shown in Fig. 2(b), the optimized velocity field isa longitudinal whirling flow with six vortexes. The pairs of vor-texes are in the core flow region near the boundary flow. The vor-ticity field in the z-direction is shown in Fig. 2(c), in which we canfind that the vorticity of optimized flow field is relatively weak atRe number 400. Owing to this longitudinal whirling flow withmulti-vortexes, the temperature field is different from the tubeflow without longitudinal whirling flow, and the temperature con-tour varies from the concentric circles to a ‘‘three-petal” shape, asshown in Fig. 2(a). Thus, the fluid temperature in the core flowregion becomes more uniform compared with the tube flow with-out multi-vortexes, and the exergy destruction caused by thermaldissipation is reduced. As the results, the fluid temperature gradi-ent in the boundary flow is increased, and the convective heattransfer is enhanced.

In the cases at different Re numbers, similar longitudinal whir-ling flows with multi-vortexes are found by solving the optimiza-tion equation set. Fig. 3 shows the flow fields with different Renumbers. Different combinations of multipliers C0,1 and C0,2 implydifferent intensities of the extra thermal effect exerted on the fluid,which form different optimal flow fields. A pattern of longitudinalwhirling flow with eight vortexes is observed when the extra ther-mal effect becomes stronger, as showed in Fig. 3. Once the multi-pliers C0,1 and C0,2 are set properly, similar flows with eightvortexes are obtained in these cases when the Re numbers are400, 800 and 1200, as shown in Fig. 3(a)–(c). Compared with theflow field in Fig. 2(b), the flow fields in Fig. 3 have more vertexes,and the intensity of these vortexes is stronger. Besides, the locationof the vortexes is near to the boundary flow.

The performance parameters of optimized fields is shown inTable 2. In this table, Ur,max is the maximum radial velocity, Um isthe mean velocity in the cross section, and their ratio can be usedto express the relative intensity of vortexes. The performanceparameters of Case 1 are illustrated in Fig. 2, while that of Cases2–4 in Fig. 3. Owing to the difference in the Lagrange multiplier,i.e., the intensity of the extra thermal effect exerted on the fluid,the flow fields are different, as mentioned above. Besides, the per-formances of optimized fields in Cases 1 and 2 are different. For thelongitudinal whirling flow with six vortexes in Case 1, its relativeintensity is 0.94 and heat transfer is enhanced with a little increaseof resistance coefficient, compared with the tube without longitu-

0,2 = 3 � 10�7): (a) Temperature; (b) velocity; (c) vorticity.

Fig. 3. Schematic diagram of flow fields with different Re numbers: (a) 400; (b) 800; (c) 1200.

Table 2Performance parameters of optimized field for different cases.

Case C0,1 C0,2 Re Ur,max/Um Nu/Nu0 f/f0 EEC

1 �1 � 108 3 � 10�7 400 0.94 1.79 1.01 1.782 �1 � 107 3 � 10�7 400 2.08 3.61 1.07 3.383 �4 � 107 3 � 10�7 800 2.12 3.74 1.10 3.424 �4 � 107 1 � 10�8 1200 2.06 3.63 1.12 3.24

W. Liu et al. / International Journal of Heat and Mass Transfer 122 (2018) 11–21 15

dinal whirling flow. The efficiency evaluation criterion (EEC) is1.78. From the Cases 2–4, it can be found that longitudinal whirlingflows with eight vortexes are formed in the optimized fields whenthe Re numbers are 400, 800, and 1200, shown in Fig. 3. The rela-tive intensity of vortexes is above 2 for all Re numbers, and theincrease amplitude of Nusselt number is much bigger than thatof resistance coefficient, compared with the tube without longitu-dinal whirling flow. As the results, The EECs of these heat transferoptimized tubes are all greater than 3, which shows a great com-prehensive performance.

Thus, by solving the optimization equation set, we find that thelongitudinal whirling flow with multi-vortexes is an optimizedflow field structure in the tube. The comprehensive performance,i.e., efficiency evaluation criterion (EEC), is highly satisfactory,and convective heat transfer is enhanced with a cost of smallincrease of resistance coefficient in the tube.

Fig. 4. Schematic diagram of a vortex generator with center-connected awl-shapedslices: (1) geometry model; (2) PIV testing unit; (3) simulation unit; (4) tube fittedwith inserts.

4. Convective heat transfer enhancement

4.1. Method and principle

The goal of enhancing convective heat transfer is to improve theperformance of heat exchange device with low thermal dissipationand low power consumption by means of reducing the irreversibil-ity of heat transfer and fluid flow. The methods for enhancing con-vective heat transfer can be divided into two categories: the wall-based and fluid-based heat transfer enhancements. The former hasbeen widely used to develop heat transfer techniques from the firstto the third generations. All extended bodies on the tube walls,such as fins and surface vortex generators, belong to the former.For a different approach, the latter can be technically regarded asthat the units of disturbing the fluid are totally immersed in thefluid, and there is no temperature difference of convective heattransfer between the disturbing units and the fluid in a tube. Allinserts inside the tubes, such as twisted tapes and metal meshes,belong to the latter.

To further develop the heat transfer techniques, exergydestruction minimization can be used as a principle for convectiveheat transfer enhancement. It is necessary to minimize exergydestruction caused by thermal dissipation and power consumptionin the fluid to achieve a better comprehensive performance of heat

transfer unit. In fact, the method of fluid-based heat transferenhancement is more beneficial to exergy destruction minimiza-tion, which is testified by the flow field structure predicted by con-vective heat transfer optimization.

4.2. Technical realization

To verify the optimal flow field and the effect of this pattern oflongitudinal whirling flow, a type of vortex generator with center-connected awl-shaped slices has been designed to generate a lon-gitudinal whirling flow with multi-vortexes in a tube, as shown inFig. 4. The tilt angle of the awl-shaped slice is 30�, the diameters of

Fig. 5. Physical mesh model applied in the calculation.

Fig. 6. Nusselt number and friction factor calculated by different grid numbers.

Fig. 7. Flow structures in the tube with inserts: (a) 3-D streamlines; (b) tangential velocity vectors in the cross section.

Fig. 8. Comparisons of exergy destructions caused by heat transfer and fluid flowbetween the enhanced tube and plain tube.

16 W. Liu et al. / International Journal of Heat and Mass Transfer 122 (2018) 11–21

W. Liu et al. / International Journal of Heat and Mass Transfer 122 (2018) 11–21 17

the big end (D1) and small end (d) of the awl-shaped slices are 18mm and 6 mm, respectively, and the central angle (a) is 40�. Thediameter of the center-connected rod is 2 mm, and the pitch (p)is 40 mm. The diameter (D) and the length (L) of the circular tubeare 20 mm and 500 mm, respectively.

Fig. 9. Variations of Q/Q0 ratio, DP/DP0 ratio, and EEC with Re number.

Fig. 10. Comparisons of the synergy angle b and h between the enhanced tube andplain tube.

Fig. 11. Comparisons of the efficiency between the enhanced tube and plain tube.

The flow status in the tube was assumed as steady laminar flow,and water was selected as the working fluid. On the tube wall, aconstant temperature condition (Tw = 333 K) was specified. Inorder to eliminate the inlet effect, the fully developed profiles ofvelocity and temperature were imposed at the inlet. An outflowcondition was used at the outlet, and a no slip condition wasapplied on the tube wall and the surfaces of awl-shaped slices.All the governing equations were solved by using the CFD softwareFLUENT 15.0, which is based on the finite volume method. The

Fig. 12. Schematic diagram of stereoscopic-PIV system: 1 computer, 2 PIV supplyunit, 3 laser, 4 lenses, 5 CCD cameras, 6 water prisms, 7 light sheet, 8 test section, 9inserts, 10 control valves, 11 water pump, 12 water tank, 13 electromagneticflowmeter, 14 upstream tube.

Table 3Overview of the relevant experimental parameters.

Upstream pipe Diameter 20 mmLength 2.5 mMaterial Steel

Test pipe Diameter 20 mmMaterial GlassWall thickness 2 mm

Center-connectedawl-shaped slices

Length 500 mmTilt angle h 30�Central angle a 45�Pitch p 40 mmDiameter of the big end (D1) 18 mmDiameter of the big end (d) 6 mm

Water pump Type Centrifugal pumpFlowmeter Type Electromagnetic

flowmeterFlow Fluid Water

Re 900Seeding Type Hollow glass sphere

Specific weight 1 g/cm3

Diameter 12 lmConcentration 5 g/m3

Light sheet Laser type Nd:YAGMaximum energy 200 mJ/pulseWave length 532 nmPulse duration 6 nsThickness �1 mm

Camera Type CCDResolution 2048 � 2048 pxDiscretization 12 bitRepetition rate 7.5 HzLens focal length 60 mm

Imaging f-number 11Viewing angles ±45�Viewing area 25 � 25 mmExposure time-delay 1.0 ms

PIV analysis Reconstruction method Three-dimensionalcalibration

Interrogation area (IA) 32 � 32 pxOverlap IA 50%

18 W. Liu et al. / International Journal of Heat and Mass Transfer 122 (2018) 11–21

SIMPLE algorithm was used to obtain a solution for the couplingbetween pressure and velocity.

A physical mesh model, shown in Fig. 5, has refined grids in theboundary flow region and hexahedron grids in the core flowregion. The grids in the regions near the insert surfaces are alsorefined. Three mesh systems with different grid numbers(2,634,127, 6,325,318, and 10,087,596) were applied for the gridindependence testing. The results indicate that a mesh systemwith6,325,318 grids is sufficiently dense for the calculation with thedeviations of 1.9% and 1.0% for Nusselt number and friction factor,respectively, as shown in Fig. 6.

Fig. 13. Schematic diagram of flow fields in a circular tube tested by a PIV experiment atdirection; (c) velocities in the z-direction.

As shown in Fig. 7, we can observe three pairs of vortexes in thecross section of the tube at different Reynolds numbers, which aredisturbed by the vortex generator with center-connected awl-shaped slices, and the vortex strength increases with the Reynoldsnumber. It can be found that a piece of awl-shaped slice yields apair of vortexes in the view of the cross section of the tube. Thisplays a role of forming optimal flow field, which is numerically pre-dicted by solving the minimization controlling equations (27), (28),(29) and (32).

As shown in Fig. 8, the simulation results indicated that,compared with the plain tube, exergy destruction caused by heat

Re = 900: (a) tangential velocity vectors in the cross-section; (b) vorticities in the z-

W. Liu et al. / International Journal of Heat and Mass Transfer 122 (2018) 11–21 19

transfer was reduced in the enhanced tube with vortex generators,but exergy destruction caused by fluid flow increased at the sametime. The main reason of strengthening convective heat transfer byapplying this vortex generator is duo to the strengthening of fluiddisturbance in the tube. So the fluid temperature becomes uniformin the core flow of the tube, which is in favor of reducing thermalresistance between the fluid and the tube wall, and results in a rel-atively larger temperature gradient in the boundary flow of thetube. At the same time, however, fluid power consumption is notlargely increased owing to this pattern of longitudinal whirlingflow.

As shown in Fig. 9, thanks to the effect of heat transfer enhance-ment in the enhanced tube, the heat fluxes are increased to 3.28–8.74 times those of the plain tube, while the pressure drops areonly increased to 2.87–5.94 times those of the plain tube. Thus,the values of EEC reach to 1.14–1.47, which indicates the increaseamplitude of heat transfer quantity is larger than that of fluidpower consumption.

4.3. Evaluation for convective heat transfer

The performances of heat transfer and flow in the heat transferenhanced tube can be evaluated by the following synergy anglesbetween the velocity vector and temperature or pressure gradientof the fluid,

b ¼ arccosU � rTjUjjrTj ; h ¼ arccos

U � rpjUjjrpj : ð33a;bÞ

From Eq. (33), it can be seen that synergy relations betweenheat transfer and fluid flow are determined by three vectors, i.e.,velocity, temperature gradient, and pressure gradient, whichreflect the mutually affected mechanism between heat transferand fluid flow. The smaller the synergy angle b, the less the thermaldissipation in a heat transfer process, and the smaller the synergyangle h, the less the power consumption in a flow process. There-fore, the synergy angles can be used to evaluate the performanceof heat transfer enhanced tube.

As shown in Fig. 10, it can be observed that the synergy angle bis reduced while the synergy angle h is increased in the enhancedtube, which implies lower thermal dissipation and relativelyhigher power consumption in the enhanced tube, compared tothe plain tube. This is consistent with the analysis of exergydestructions represented by temperature difference and pressuredrop in Fig. 8.

As shown in Fig. 11, a comparison of the efficiency of thermalenergy transfer between the enhanced tube and plain tube is made.It is clear that the efficiency of thermal energy transfer in theenhanced tube is much higher than that in the plain tube, owingto the reduction in exergy destruction. This is a perfect case, whichindicates that the comprehensive performance of heat transferenhanced tube can be improved by using the method of exergydestruction minimization.

4.4. PIV testing verification

To confirm the longitudinal whirling flow with multi-vortexesgenerated by the insert of center-connected awl-shaped slices,we carried out a measurement of the flow field by usingstereoscopic-PIV (Particle Image Velocimetry). Fig. 12 shows aschematic diagram of the stereoscopic-PIV system. The blue1 linerepresents the water cycle and the red line represents the controlcircuit. In the test section, two cameras take a 45� shooting angle

1 For interpretation of color in Fig. 12, the reader is referred to the web version ofthis article.

from both sides, and two water prisms are applied to minimize opti-cal deformation of the images. In the upstream of the test section, atube of 2500 mm in length is used for allowing the flow to be fullydeveloped. A centrifugal pump is used to drive the fluid, and an elec-tromagnetic flowmeter is applied to measure the flow rate. The PIVmeasurement was carried out at a cross section, 10 mm downstreamthe inserts at Re = 900. The relevant experimental parameters arelisted in Table 3. Before the measurement, we have carried out thecalibration by using self-calibration in commercial PIV-software(Lavision, Davis 8.3.0) to ensure the accuracy of the PIV measure-ment. More details about the PIV experimental method are describedin our previous work [69].

Three pairs of vortexes were observed by the PIV measurement,and the distributions of the multi-vortex structure in the experi-mental and numerical results are basically the same, as shown inFig. 13(a). Furthermore, to quantitatively compare the vortexstructures in the experimental and numerical results at the samecross section, the distributions of vorticity in the z-direction aredisplayed in Fig. 13(b). It can be observed that the vorticities ofnumerical calculation agree well with that of experimental datain terms of distribution values. Fig. 13(c) shows the comparisonof the z-direction velocity component between the PIV experimen-tal and numerical results. The results indicated that the z-directionvelocity distributions of the experimental and numerical resultsare almost the same. Therefore, in a circular tube, the flow fieldobtained by the PIV experiment is quite consistent with that calcu-lated by the mathematical model. In order to determine the accu-racy of the experiment, the correlation statistics methods [70]were used for PIV uncertainty quantification. The average relativeerrors of fluid velocity components in the three directions arewithin 10% in this experiment.

In general, the overall heat transfer performance of a heat trans-fer enhanced tube could be improved by reducing exergy destruc-tions caused by heat transfer and fluid flow, which is significant toexplore various heat transfer techniques by accepting the methodbased on exergy destruction minimization.

5. Conclusion

In this study, an efficiency expression for the thermal energytransfer process was derived to verify the principle of convectiveheat transfer enhancement based on exergy destruction minimiza-tion. A tube insert of center-connected awl-shaped slices wasinvestigated through numerical simulation and tested by a PIVexperiment to achieve an optimal field of longitudinal whirlingflow with multi-vortexes, which was predicted by convective heattransfer optimization. The study proves that the most effectivemethod for heat transfer enhancement is properly disturbing thefluid inside a tube, rather than only strengthening the fluid distur-bance on the tube wall. By doing so, the exergy destruction causedby heat transfer and fluid flow can be minimized, which may pro-vide a guide to develop new heat transfer techniques.

Conflict of interest

The authors do not have any possible conflicts of interest.

Acknowledgement

This work was supported by the National Natural Science Foun-dation of China (Grant No. 51736004).

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