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Applied Thermal Engineering 31 (2011) 2981e2991

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Applied Thermal Engineering

journal homepage: www.elsevier .com/locate/apthermeng

Experimental study on convective heat transfer and flow resistance characteristicsof water flow in twisted elliptical tubes

Sheng Yang, Li Zhang*, Hong XuState-Key Laboratory of Chemical Engineering, School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, PR China

a r t i c l e i n f o

Article history:Received 15 January 2011Accepted 21 May 2011Available online 30 May 2011

Keywords:Twisted elliptical tubeHeat transfer enhancementFlow resistanceField synergy principle

* Corresponding author. Tel./fax: þ86 21 64253810E-mail address: lzhang@ecust.edu.cn (L. Zhang).

1359-4311/$ e see front matter � 2011 Elsevier Ltd.doi:10.1016/j.applthermaleng.2011.05.030

a b s t r a c t

Heat transfer and flow resistance characteristics of water flow inside the twisted elliptical tubes (TETs)with different structural parameters were experimentally investigated. Effects of tube structuralparameters (aspect ratio and twist pitch) on the performance of TETs were analyzed and the overallthermal-hydraulic performance of TETs was evaluated. Experimental results showed that the TETs canprovide considerable heat transfer augmentation and also high pressure drop inside tube. Larger tubeaspect ratios and smaller twist pitches resulted in higher heat transfer coefficients and friction factors.The best operating regime for TETs is at lower Reynolds numbers. It was also discovered that theexperimental Nusselt numbers/friction factors can be expressed with one unified equation for entireReynolds number range, which confirms the early flow transition from laminar to turbulent in TETs.Experimental results were compared with some existing correlations, and the causes for the differencesbetween them were analyzed. Heat transfer enhancement mechanism of TETs was discussed from theviewpoint of field synergy. The longitudinal vortex induced by the twisted tube wall improves thesynergy between the velocity vector and temperature gradient, which in turn results in a better heattransfer performance.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Shell and tube heat exchangers are widely used in many engi-neering fields. Heat transfer enhancement techniques inside tubesare very important for improving the efficiency of heat exchangersandmany important progresses have beenmade during the past 30years. In general, heat transfer enhancement techniques can bedivided into two groups, namely active and passive techniques. Theactive technique requires external forces such as an electric field,acoustic or surface vibration, whereas the passive techniquerequires special surface geometries such as a rough surface,extended surface, or fluid additives. Both active and passive tech-niques were used by researchers over a century ago to increase theheat transfer rate in heat exchangers [1e3].

The twisted tube is one of the passive heat transfer enhance-ment tubes. It is formed into an elliptical or oblate cross sectionwith a superimposed twist as illustrated in Fig. 1 and is left round attwo ends for conventional fixing into the tube sheet. The geomet-rical features of a twisted tube include the 360� twist pitch (S),

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major dimension of the cross section (A), and minor dimension ofthe cross section (B). Besides, the severity of tube twisting isdescribed by the dimensionless twist ratio (S/d or S/de), where d isthe diameter of the smooth round tube used to produce the twistedtubes and de is the equivalent hydraulic diameter of the twistedtube. The severity of tube flattening is described by dimensionlessaspect ratio (A/B). Twisted tubes are sometimes chosen in thedesign of industrial shell and tube heat exchangers, which canconsiderably reduce the size of the heat exchangers.

2. Brief review of previous work

Thermal-hydraulic performance of twisted tubes has beenstudied by a few researchers as shown in Table 1. Ievlev et al. [4]presented the experimental results of heat transfer and hydraulicresistance of a heat transfer agent flowing inside the twisted flattubes and inter-tube space. The working fluid used in their studywas air, and Reynolds numbers ranged from 6000 to 100,000 forthe tube side and from 3000 to 60,000 for the inter-tube side. Theeffectiveness of using these tubes in heat exchange equipment wasanalyzed, and the results showed that the replacement of smoothtubes by twisted ones leads to a 20%e40% increase in heat transferwith a 50%e80% increase in hydraulic resistance which allows

Nomenclature

A major axis of the twisted elliptical tube, mB minor axis of the twisted elliptical tube, mc constant in Eq. (6)cp isobaric specific heat, J/(kg K)d diameter of a circular tube, mde equivalent hydraulic diameter, mF effective surface area of tested tube, m2

f friction factor, dimensionless, f ¼ 2d$Dp=ðL$ru2Þh heat transfer coefficient, W/(m2 K)k thermal conductivity, W/(m K)L effective length of the tested tube, m_m mass flow rate, kg/sNu Nusselt number, dimensionless, Nu ¼ hde=kPr Prandtl number, dimensionless, Pr ¼ mcp=kDp tube side pressure drop, PaRe Reynolds number, dimensionless, Re ¼ ud=vS twist pitch of a twisted elliptical tube, mT temperature, KDT temperature difference, KU overall heat transfer coefficient, W/(m2 K)U!

velocity vector, m/s

u velocity, m/s

Greek symbolsm dynamic viscosity, Pa sy kinematic viscosity, m2/sr density, kg m�3

q synergy angle, �

Subscriptsc cold watere enhanced tubeepp equal pumping powerf fluidh hot wateri innerin inletLMTD logarithmic mean temperature differencem mean or average valueo outerout outlets smooth tubew wall

S. Yang et al. / Applied Thermal Engineering 31 (2011) 2981e29912982

a reduction in the heat exchanger volume of up to 30%. Thecorrelations for Nusselt number and friction factor were proposed.

Asmantas et al. [5] experimentally studied the heat transfer andflow resistance characteristics of air flow inside twisted flat tubesover a range of S/do from 6.2 to 12.2, Reynolds number from 7000 to200,000. The heat transfer coefficient for a twisted flat tube is largerthan that for a smooth tube over the entire Reynolds number range.The heat transfer is augmented by asmuch as 40% and the hydraulicresistance also increases by a factor of 1.7 on an average at S/do ¼ 62. The correlations for the Nusselt numbers and frictionfactors were also presented.

Dzyubenko and Yakimenko [6] also presented the formulas forcalculating the Nusselt numbers and friction factors for flow insidetwisted oval tubes and inter-tube space respectively. It wasdiscovered that the heat transfer is enhanced obviously when theReynolds number is smaller than 7000. By means of a method ofeffective parameters, it was discovered that the thermal-hydraulicefficiency of twisted tubes is higher than that of smooth roundtubes and smooth tubes with annular knurling.

Fig. 1. Twisted tube section.

Si et al. [7] experimentally studied the heat transfer and pres-sure drop characteristics of diesel oil flowing in twisted flat tubes.Three kinds of twisted flat tubes with different structural param-eters were tested. It was found that the tube side heat transfercoefficient and friction factors for these tubes were higher thanthose for the smooth tube. The twisted tubes have better heattransfer performance at smaller Reynolds number from around2000 to 7000. And the smaller the twist pitch, the better the heattransfer performance. The flow resistance for the twisted tube hasa greater increase at smaller Reynolds number from 2000 to 7000compared with that of the smooth tube, and with the decrease ofthe twist pitch, the flow resistance increases. The author analyzedthat the swirling flow in the twisted tube increase the fluidturbulence, which enhances the heat transfer while increasing theflow resistance.

Zhang et al. [8] conducted an experimental investigation on theheat transfer and flow resistance of water flow inside differenttwisted tubes. The tests were conducted at Reynolds numbersranging between 7900 and 26,500. Both heat transfer coefficientsand friction factors increasewith the decrease of twist pitch or withthe increase of aspect ratio at the same Reynolds number. Thethermal-hydraulic performance of the twisted tube is better whenthe Reynolds number is smaller than 10,000. Based on the exper-imental results, correlations for the Nusselt number and frictionfactor were also obtained.

Gao et al. [9] experimentally investigated the heat transfer andflow resistance characteristics of water flow inside twisted tubeswith large twist ratio in transition and turbulent flow regime.Nusselt numbers and friction factors for twisted tubes are 1.3e2.5times and 1.2e1.5 times of those for the smooth tube, respectively.It was also discovered that the Nusselt numbers and friction factorsincrease with the decrease of twist pitch or with the increase ofaspect ratio, which is in agreement with the findings in Ref. [8].

Yang et al. [10] studied the heat transfer and flow resistance offluid flowing inside a twisted elliptical tube in laminar flow regimeusing the CFDmethod. The major axis, minor axis and twist pitch ofthe tube are 23 mm, 14 mm and 320 mm, respectively. The Rey-nolds number ranges from 100 to 500, and the Prandtl number isless than 200. Results showed that the twisted elliptical tube

Table 1Investigations on the twisted tubes with various working fluid and tube configurations.

Sources Tube parameters Workingfluid

Tube side correlations

do (mm) S (mm) Ai (mm) Bi (mm) S/Ao

Ievlev [4] e 12.1 e 6.2e16.7 air Nu ¼ 0:019Re0:8h1þ 0:547=ðs=AÞ0:83

if ¼ 0:316

h1þ 3:27ðs=AÞ�0:87

iRe�0:25

6 � 103 < Re < 105

Asmantas [5] e 12.1 e 6.2e12.2 airNu ¼ 0:021Re0:8Pr0:4

h1þ 2:1ðs=AÞ�0:91

i TwTf

!n

n ¼ �0:17� 0:27� 10�5ðx=deÞ1:37hðs=deÞ2:1 � 109:6

if ¼ 0:82ðs=AÞ�0:63Re�0:18

7 � 103 < Re < 2 � 105

Si [7] 19 144, 192, 250 21 e 6.86, 9.14,11.9

dieselNu ¼ 0:396Re0:544

sde

!0:161 sA

!�0:519

Pr0:33

lgf ¼ a1 þ a2lgReþ a3ðlgReÞ21000 < Re < 17,000

Zhang [8] 25 160, 200, 250 24, 26 14, 9 e waterNu ¼ 1:50618Re0:51825Pr�1:2446

AiBi

!1:12252 sde

!�0:32367

f ¼ 0:71497Re0:07777Pr�1:03974

AiBi

!�0:76212 sde

!�0:33393

7900 < Re < 26,500

Gao [9] 19 200, 300, 400 19.8, 21.2,21.8

9.4, 7.0,5.8

e waterNu ¼ 0:034Re0:784Pr0:333

BiAi

!�0:590 sde

!�0:165

f ¼ 4:572Re�0:521

BiAi

!�0:334 sde

!�0:082

5000 < Re < 20,000

Yang [10] 19 e e e 10e17 e Nu ¼ 3:66þ 0:512Re0:477Pr0:975 1� Bi

Ai

!1:532 sB

!�0:609

100 < Re < 500

S. Yang et al. / Applied Thermal Engineering 31 (2011) 2981e2991 2983

provide a remarkable heat transfer enhancement performance, andthe greater the Prandtl number, the better the heat transferperformance. The findings about effects of tube structural param-eters (S and A/B) on the heat transfer and flow resistance charac-teristics of the tube were the same with those in Refs. [8,9]. Reasonfor heat transfer enhancement was concluded that the secondaryflow induced by the fluid swirling directly flushes some part of theinner tube wall, temperature gradient near the tube is higher, thedistribution of fluid velocity and temperature is more uniform,which enhances the heat transfer effectively.

Bishara et al. [11] also used the CFD method to obtain thetemperature and velocity distribution of fluid in the cross section ofa twisted elliptical tube with a twist ratio of 6.0 and an aspect ratioof 1.43. According to the temperature and velocity plots, the paperindicates that the secondary flows promote mixing in the planenormal to the bulk flow direction. This mixing works to maintaina high temperature gradient close to tube wall and then increasethe Nusselt number.

In the above literatures, heat transfer and flow resistance char-acteristics of fluid flow inside the twisted tubes and the inter-tubespace have been studied, and wherein tube side is the researchemphasis. Experimental and CFD methods were employed, and theworking fluid covered air, diesel oil and water. Effects of tubestructural parameters on heat transfer and flow resistance perfor-mance were analyzed, and correlations to predict Nusselt numberand friction factor were presented. However, little experimentalresearchwas associatedwith the thermal-hydraulic performance oftwisted tubes in laminar flow regime. Equations presented forcalculating Nusselt numbers and friction factors had various formsand often there were significant differences among the predictedvalues for the same twisted tube under the same operational

condition. Mechanism of heat transfer enhancement is exploredfrom the traditional point of view, which can be summarized byincreasing flow turbulence or decreasing thickness of the devel-oping thermal boundary layer. In view of this, the present workconducts an experimental research on the thermal-hydraulicperformance of five twisted elliptical tubes with different struc-tural parameters. The working fluid is water, and the Reynoldsnumber ranges from around 600 to 55,000, covering the laminar,transition and turbulent flow regime. Correlations for heat transferand flow resistance are obtained and compared with those existingones. In addition, temperature field and velocity field of water inTETs are numerically investigated; and the heat transfer enhance-ment mechanism of TETs is further explored from the viewpoint offield synergy principle.

3. Experimental apparatus and data reduction

3.1. Experimental apparatus

A schematic diagram of the experimental apparatus is shown inFig. 2. The experimental apparatus mainly consisted of a coldmedium loop, a hot medium loop, a test section and a measure-ment system. The test section was a double tube exchanger. Eachtested tube had an effective length of 1.75 m. The working fluid onboth tube side and shell side was deionized water. The hot watertank had a maximum heating power of 20 kW and a temperaturecontrol system was equipped for it to adjust the heating power tokeep a constant water temperature at a given flow rate, and the hotwater temperature can range from 30 to 95 �C. Hot water wasconveyed to the test section by a centrifugal pump and cooled in thetested tube via counter flow heat exchange with the cold water

Table 2Specifications of the tested TETs.

Tube No. Twist pitch, S (mm) Aspect ratio, A/B

1 104 1.602 152 1.903 192 2.154 192 1.765 192 1.49

Cold water tank Hot water tank

Test section Tested tube

cooling waterhot watertemperature measuring point

Flow transducer Flow transducer

Plate heat exchangerData acquisition

systemDifferential pressure

transducer

Back to hot water tank

Back to cold water tank

measurementpressure measuring point

Fig. 2. Schematic diagram of the experimental apparatus.

S. Yang et al. / Applied Thermal Engineering 31 (2011) 2981e29912984

flowing on shell side. The upstream tube section of the tested tubewas made long and straight in order to ensure a fully developedflow in tube. An inverted U-bend was mounted on the downstreamside to ensure that the tested tube can always be flooded with fluidat any given flow rate. Cold water was pumped through shell sideand a plate heat exchanger in a closed loop, heated outside thetested tube and then cooled in the plate heat exchanger by meansof water from an independent external cooling tower. A thermo-static control valve adjusts the flow rate of the cooling water tocontrol the inlet temperature of cold water to be constant. The coldwater temperature ranged from 30 �C to 60 �C. Finally, to reduce theheat loss to the ambient, the whole apparatus was well insulated.

The experimental apparatus was equipped with a data acquisi-tion system (DAS) based on an Agilent Data Acquisition Unit34970A and a PC. Parameters required to be measured includesflow rates of hot/cold water, bulk temperatures of hot/cold water atthe inlet/outlet of the tested tube/shell, and pressure drop acrossthe tested tube. Two sheathed T-type thermocouples were utilizedto measure the hot/cold water bulk temperatures. All thermocou-ples were calibrated in the same constant temperature bathwith anaccuracy of �0.1 �C within the tested range. Two turbine flowme-ters were employed to measure the flow rates of hot/cold water,and the uncertainty was estimated as �0.2%. Pressure drop acrossthe tested tube is measured using the differential pressure trans-ducer GE Druck PMP 4170 with an accuracy of �0.04%.

For each test run, the whole systemwas first stabilized for about30 min by continuously running the working fluid. Steady stateconditions were assured by checking both temperature anddifferential pressure readings. The flow rates of hot water and coldwater were set to certain values during each test run. The hot watertank was thermostatic with adjustable heating power, thus the hotwater could be maintained at a desired temperature at the inlet ofthe tested tube. Cold water flowing on shell side was heated andthen conveyed back to the cold water tank passing through a plateheat exchanger. By adjusting the control valve of the plate heatexchanger, constant temperature of water that flowed back to thecold water tank could be obtained, as well as a constant inlettemperature of the cold water. Besides, the differential pressureacross the tested tube was monitored. If the fluctuation of water

temperature at the inlet/outlet of tube/shell side did not exceed�0.1 �C within 2 min, and the fluctuation of the differential pres-sure was less than �5%, a steady state for heat transfer and fluidflow was achieved, and then the data collection began. Experi-mental data were acquired by the DAS every 3 s, and the averagevalues over 2 min were used for calculating both friction factor andheat transfer coefficient.

A smooth tube and five twisted tubes were tested, and all tubeswere made from commercially pure copper. The mean outsidediameter, wall thickness and length of the smooth tube are19.02 mm, 1.01 mm and 1880 mm, respectively. The twisted tubeswere manufactured from smooth tubes and have an elliptical crosssection. Geometrical specifications and profile of the twistedelliptical tubes are shown in Table 2 and Fig. 3.

3.2. Data reduction

Convective heat transfer coefficients of tube side are obtainedby the method suggested by J.A. Meng [12]. First, the exponentialcorrelation of Nusselt numbers, Nu, for the outer surface of thetested tube is obtained by varying the flow rate of cold wateroutside tube and keeping the hot water flow rate inside the tubeat its maximum value. Then set the cold water flow rate outsidetube at its maximum value and measure the overall thermalresistance. The heat transfer coefficients for hot water inside tubecan be determined by subtracting the convection resistanceoutside the tube and the conduction resistance through tube wallfrom the overall thermal resistance. Following are the detailedprocedure.

Fig. 3. Photograph of tested TETs.

S. Yang et al. / Applied Thermal Engineering 31 (2011) 2981e2991 2985

The first step is to obtain the correlation for calculation of theheat transfer coefficients outside the tube. The logarithmic meantemperature difference, DTLMTD, between the hot and cold fluid canbe derived from Eq. (1)

DTLMTD ¼�Th;in � Tc;out

�� �Th;out � Tc;in�

lnTh;in � Tc;outTh;out � Tc;in

(1)

Where Th,in and Th,out are the temperatures of hot water at the inletand outlet of the tested tube, respectively. Tc,in and Tc,out are thetemperatures of cold water at the inlet and outlet of the shell side,respectively. This temperature difference is kept 10e25 �C in theexperiments. Heat transfer rates of tube side and shell side, Qh andQc are calculated by Eqs. (2) and (3), and their arithmetic meanvalue is considered as the heat transfer rate of the experiment.

Qh ¼ _mhcp;h�Th;in � Th;out

�(2)

Qc ¼ _mccp;c�Tc;out � Tc;in

�(3)

Q ¼ Qh þ Qc

2(4)

Where _mh and _mc are the mass flow rates of hot and cold water,respectively. The overall heat transfer coefficient is determined byEq. (5)

U ¼ QFoDTLMTD

(5)

The hotwater flow rate inside tube is fixed at its maximumvaluewhile the cold water flow rate outside tube is regulated. Fit the heattransfer correlation for the shell side with an exponential relationas the following form of Eq. (6):

Nuo ¼ cRe0:8Pr0:4 (6)

In the present work, the fouling resistance is neglected since allthe tubes are cleaned before each experiment. Then, the overallthermal resistance can be expressed as the sum of the partialthermal resistances corresponding to the inner convection, the heatconduction through tube wall and the outer convection, accordingto Eq. (7)

1U

¼ 1ho

þ do2kw

lndodi

þ dohidi

(7)

Submitting Eq. (6) into Eq. (7), the overall thermal resistance canbe written as:

1U

¼ 1c

�dokcRe�0:8Pr�0:4

�þ�

do2kw

lndodi

þ dodihi

�(8)

Where hi and ho are the heat transfer coefficient for the tube sideand shell side, respectively. di and do are the inner diameter andouter diameter of the tube, respectively. kc is the thermal conduc-tivity of the cold water, and kw is the thermal conductivity of tubematerial. do

2kwlndo

diis the conductive thermal resistance across the

tube wall. In the experiments, the flow rate of hot water inside thetested tube is fixed at its maximumvalue and the bulk temperatureof the hot water is basically kept constant, therefore the heattransfer coefficient for the tube side is almost constant. It could beseen from Eq. (8) that 1/U only varies with Re�0.8Pr�0.4 linearly.Take X ¼ do=kcRe�0:8Pr�0:4 as abscissa and Y¼ 1/U as ordinate, onecan obtain a linear relation between Y and X by changing the flowrate outside the tube. The constant c can be determined by a curvefitting method. Once c is obtained, the heat transfer coefficientoutside the tube can be calculated by Eq. (9)

ho ¼ ckcdo

Re0:8Pr0:4 (9)

The internal heat transfer performance can be determinedexperimentally after the thermal resistance outside tube is known.In such experiments, the flow rate of the cold water outside tube isfixed at its maximum to reduce the external convection resistancewhile the flow rate of the hot water inside tube is regulated. Theoverall heat transfer coefficient U can be decided by Eqs. (1)e(5).Then the heat transfer coefficient for the tube side, hi, can beobtained by Eq. (10)

1hi

¼ dido

�1U� 1ho

� do2kw

lndodi

�(10)

The Nusselt number inside the tube is

Nu ¼ hidikh

(11)

The friction factor for the tube side is easy to be calculatedaccording to Eq. (12):

f ¼ diL$2Dpru2

(12)

Where Dp is the pressure drop of hot water across the tested tube;L is the tube length; u is the mean flow rate of water inside tube.

The uncertainty is estimated with the method suggested byMoffat [13] and Kline [14]. As mentioned above, the measurementerror of temperature is 0.1 �C, the turbine flow meter has a preci-sion of 0.2%, and the measurement accuracy of the differentialpressure meter is 0.04%. In addition, the measurement uncer-tainties of tube diameter and tube length are about 0.6% and 0.08%,respectively. The relative error of each physical parameter (e.g.,density, specific heat of fluid) is about 0.5%. According to theuncertainty propagation equation, the maximum relative uncer-tainties for the experimental Nusselt number and friction factor are11.4% and 8.2%, respectively.

500 1000 1500 2000 25000

5

Nu/

[Pr1/

3 (µf/µ

w)1/

3 ]

Re

Sieder-Tate Equation Experimental

0 10000 20000 30000 40000 50000 600000

10000

20000

30000

40000

50000

60000Gnielinski EquationExperimental

Nu/

F(f

,Pr)

Re

0.10

a

b

c

S. Yang et al. / Applied Thermal Engineering 31 (2011) 2981e29912986

4. Experimental results

4.1. Verification of smooth tube

In order to validate the reliability of the experimental system,heat transfer and flow resistance characteristics of water flowinginside a smooth tube were first investigated. The Nusselt numbersfor laminar, transition and turbulent flow in a smooth tube obtainedfrom the present experiments were compared with predictions ofthe SiedereTate correlation [15,16], Eq. (13), and Gnielinski corre-lation [11,17], Eq. (14), respectively. The experimental friction factorswere compared with the Darcy friction factors (f ¼ 64=Re) andpredictions of the Petukhov correlation [18], Eq. (15).

Nu ¼ 1:86Re1=3Pr1=3f

�diL

�1=3�mfmw

�0:14(13)

Nu ¼ ðf =8Þ$ðRe� 1000Þ$Prf1þ 12:7

ffiffiffiffiffiffiffiffif =8

p$�Pr

2=3

f � 1� 1þ

�diL

�2=3!�

PrfPrw

�0:11

ð14Þ

f ¼ ð1:82lgRe� 1:64Þ�2 (15)

Fig. 4 shows the comparison of experimental Nusselt numbersand friction factors with those from correlations of Eqs. (13)e(15).The experimental Nusselt numbers for laminar flow agree with theSiedereTate correlation with a deviation between �6.5% and 11%,and the average deviation is 1.6%. The experimental Nusseltnumbers for transition and turbulent flowagreewith the Gnielinskicorrelation with a deviation between �6.1% and 10.6%, and theaverage deviation is �2.6%. For the friction factors, the deviation isbetween �9.3% and 3.5%, and the average deviation is 1.8%. Theresults prove the reliability of the experimental system.

0 10000 20000 30000 40000 50000 600000.00

0.02

0.04

0.06

0.08

Re

f

Experimental

f=(1.82lgRe-1.64)-2

f=64/Re

Fig. 4. Experimental results of heat transfer and flow resistance for smooth tube: (a),(b) Nusselt number; (c) friction factor. (a) Comparison of Nusselt numbers in laminarflow regime (b) Comparison of Nusselt numbers in transition and turbulent flowregime (c) Comparison of friction factors.

4.2. Heat transfer performance of the TETs

The heat transfer performance of TETs is presented in the formof Nusselt number. Variation of Nusselt numbers with Reynoldsnumbers for the tested TETs and smooth tube are given in Fig. 5.Reynolds numbers are from around 600 to 55,000, which coversthe three flow patterns of laminar flow, transition flow andturbulent flow. Referring to Fig. 5, one can observe that the Nusseltnumbers for all the tested tubes increase as the rise of Reynoldsnumbers. The increase in Nusselt number indicates an enhance-ment in heat transfer coefficient due to the increase of convection.All the tested TETs have a better heat transfer performance than thesmooth tube. TET No. 1 has the best heat transfer performanceamongst the five TETs and followed successively by TET No. 2, TETNo. 3, TET No. 4 and TET No. 5. It should be noted that the Nusseltnumber for TET No. 5 is only slightly higher than that for thesmooth tube especially when Reynolds numbers exceed around10,000. Comparing the structural parameters of TET No. 3, TET No. 4and TET No. 5 listed in Table 1, one can discover that these threetubes have same twist pitch of 192 mm but different aspect ratios.The Nusselt numbers increase with the rise of tube aspect ratios.TET No. 3 has the largest aspect ratio of 2.15 among these threetubes and has the highest Nusselt number, or best heat transferperformance. This finding about effect of tube aspect ratio on tubeheat transfer performance is in agreement with the reports inliteratures [8,9]. Comparing with TET No. 3, both TET No. 1 and TETNo. 2 have smaller tube aspect ratio but better heat transferperformance, which is due to the smaller twist pitch of TET No. 1and TET No. 2. The conventional view believes that the swirlingflow, that is the secondary flow induced by the twisted tube wall,

intensifies the turbulence and promotes the radial mixing of thebulk flow. The mixing works to maintain a high temperaturegradient close to the tube wall and in turn enhance the heattransfer. Both a larger tube aspect ratio and a smaller twist pitch canlead to a more intensive turbulence and effective mixing to obtaina better heat transfer performance. From the experimental results,one can also discover that the effect of the twist pitch on the heattransfer performance of a TET is more notable than that of the tubeaspect ratio.

0 10000 20000 30000 40000 50000 600000.00

0.05

0.10

0.15

0.20

0.25

0.30

Re

f

Smooth tubeTube No.1Tube No.2Tube No.3Tube No.4Tube No.5

Fig. 7. Variation of friction factor with Reynolds number for the TETs.

0 10000 20000 30000 40000 50000 60000

0

40

80

120

160

200Smooth tube

TET No.1TET No.2TET No.3TET No.4TET No.5

Re

Nu/

[Pr1/

3 (µf/µ

w)0.

14]

Fig. 5. Variation of Nusselt number with Reynolds number for various tubes.

S. Yang et al. / Applied Thermal Engineering 31 (2011) 2981e2991 2987

The effectiveness of heat transfer enhancement for the TETsrelative to the smooth tube was compared in Fig. 6. The effective-ness is indexed by a ratio of the Nusselt number for the enhancedtube to that for the smooth tube, in terms ofNue=Nus

. As shown in

Fig. 6, the Nusselt number ratios for all the tested TETs tend toincreasewhen Reynolds number increase from around 600 to 2300,then to decrease with the rise of Reynolds number from around2300 to 10,000, and finally shows a slightly decrease for higherReynolds numbers. This indicates that the heat transfer enhance-ment of the TETs is more significant at lower fluid velocities cor-responding to the laminar and transition flow regime. The Nusseltnumber ratios for TET No. 1 that has the best heat transfer perfor-mance are around 3.9e4.8 for Reynolds numbers from 600 to 2300and 2.8e1.4 for Reynolds number higher than 2300. The corre-sponding Nusselt number ratios are 3.6e4.6, 2.5e1.3 for TET No. 2,3.4e4.3, 2.2e1.2 for TET No. 3, 2.8e3.5, 2.0e1.1 for TET No. 4 and2.5e3.0, 1.5e1.0 for TET No. 5.

For the TETs, it is found that the experimental results for theNusselt number can be correlated with an unified equation, whichis a function of the Reynolds number (Re), Prandtl number (Pr),twist ratio (S/de) and aspect ratio(Ai/Bi), for Reynolds numbers from600 to 55,000 as shown in Eq. (16):

Nu ¼ 0:3496Re0:615Pr1=3�AiBi

�0:490� Sde

��0:394

(16)

1000 10000

1

2

3

4

5 TET No.1

TET No.2

TET No.3

TET No.4

TET No.5

Re

Nu e/N

u s

Fig. 6. Variation of Nusselt number ratio (Nue/Nus) with Reynolds number for the TETs.

4.3. Hydraulic performance of the TETs

The hydraulic performance of the TETs is presented in the formof friction factor. Variation of the friction factor with Reynoldsnumbers for the TETs and the smooth tube is presented in Fig. 7. Inthe figure, it is apparent that the friction factors for all the TETs aregreater than those for the smooth tube. Friction factors are higher atlower Reynolds number and tend to decrease with the rise ofReynolds numbers from around 600 to 15,000, then show a slightlydecrease for higher Reynolds numbers. The difference in frictionfactors for TET No. 1 to TET No. 4 are trivial, therefore a partialenlarged drawing was employed to show the details of difference.From the partial enlarged drawing, one can clearly discover that theTET No. 3 has the greatest friction factors and followed successivelyby the TET No. 2, TET No. 4, TET No. 1 and TET No. 5, which is just inaccordance with the order of the tube aspect ratios from large tosmall. This demonstrates that the TET with a larger tube aspectratio has a larger friction factor, which is in agreement with thefinding in previous literatures, Refs. [8,9]. The increase of flowresistance in TET can be attributed to the swirling flow caused bythe twist channel and the blocking effect of the special tube wall.Just as mentioned above, when fluid flows in the TET with largertube aspect ratio, a more intensive turbulence is induced toincrease the flow resistance. In theory, a smaller twist pitch canimprove the blocking effect to cause a greater friction factor.However, the effect of twist pitch on the friction factor for the TETsis not obvious in the present experiment. The effect of tube aspect

1000 10000 100000

1

2

3

4

5

Re

Tube No.1Tube No.2Tube No.3Tube No.4Tube No.5

f e/f s

Fig. 8. Variation of friction factor ratio (fe=fs) with Reynolds number for the TETs.

0.0 0.1 0.2 0.30.0

0.1

0.2

0.3

-60%

-20%

-10%

+10%Eq.(6)Zhang et al.[8]Gao et al.[9]

Experimental f

Pred

icte

df

Fig. 10. Comparison of experimental flow resistance data for TET No. 3 with existingcorrelations.

S. Yang et al. / Applied Thermal Engineering 31 (2011) 2981e29912988

ratio on the flow resistance is more notable than that of tube twistpitch.

Fig. 8 presents the variation of friction factor ratio fe=fs, withReynolds numbers. The friction factor ratio first increaseswith Reynolds numbers from around 600 to 2300, then decreaseswith rise of Reynolds number from around 2300 to 8000, andfinally shows a slightly decrease, which is similar to the variationtrend of the Nusselt number ratio. The improved turbulenceprovided by the swirling flow increase the flow resistance at lowReynolds number more remarkably, however, after a fully devel-oped turbulent flow occurring at higher Reynolds numbers, theeffect of secondary flowweakens relatively. Friction factor ratios forTET No.3 are around 2.1e4.6 for Reynolds numbers from 600 to2300 and 3.5e2.4 for Reynolds numbers higher than 2300. Thecorresponding friction factor ratios are 1.9e4.4, 3.0e2.3 for TET No.2, 1.75e3.75, 2.9e2.2 for TET No. 4, 1.5e3.4, 2.5e2.0 for TET No. 1and 1.0e2.25, 1.75e1.3 for TET No. 5.

Same as Nusselt numbers, the friction factors for the TETs alsocan be expressed with one unified equation, a function of theReynolds number (Re), twist ratio (S/de) and aspect ratio (Ai/Bi), asshown in Eq. (17):

f ¼ 1:529Re�0:350�AiBi

�1:686� Sde

��0:366

(17)

The critical Reynolds number for laminar flow transition toturbulent flow inside a circular smooth tube is 2300. However, itmust be smaller than 2300 for the TETs due to the improvedturbulence caused by the swirling flow. Usually the correlations forthe calculation of heat transfer coefficients and friction factors forlaminar, transition and turbulent flow are different. The fact thatthe experimental heat transfer and flow resistance data for the TETscan be correlatedwith one unified expression respectively confirmsthe early transition from laminar to turbulent flow.

4.4. Comparison of experimental data with existing correlations

Figs. 9 and 10 compare the experimental heat transfer and flowresistance data for TET No. 3 with the predictions by the fittedempirical equations, Eqs. (5) and 6, and the existing correlationsalso for convective heat transfer of water inside tube proposed byZhang et al. [8] and Gao et al. [9], respectively. Results in Figs. 9 and10 show that the fitted empirical equations can well predict theexperimental heat transfer and flow resistance datawithmaximum

0 50 100 150 200 250 3000

50

100

150

200

250

300

-60%

-20%

Pred

icte

dN

u

Experimental Nu

Eq.(5)Zhang et al.[8]Gao et al.[9]

+20%

Fig. 9. Comparison of experimental heat transfer data for TET No. 3 with existingcorrelations.

deviations of �20% and �10% respectively. The correlationsproposed by Zhang et al. [8] predict the present experimental heattransfer and flow resistance data with maximum deviations ofaround �60% and �20% respectively. The correlations proposed byGao et al. [9] predict the experimental heat transfer data reasonablywell with a maximum deviation of �5%, however, underestimatethe friction factors with a maximum deviation of around 60%. Twomain reasons are contributing to considerable deviations betweenthe experimental results and the predictions by the existingcorrelations. First, there are differences in the concrete structuralparameters of tested TETs. The TETs tested by Gao have larger twistpitches, and TETs tested by Zhang have larger smooth tube diam-eters, both which can lead to smaller Nusselt numbers and frictionfactors. Second, the tube wall temperature measuring methods aredifferent. Zhang and Gao used a direct measurement method,whereas a indirect measurement method was adopted in thepresent study.

4.5. Overall performance of the TETs

The overall performance of the TETs is evaluated using the heattransfer performance evaluation criteria factor (performancefactor) h, defined as the ratio of the tube side Nusselt number, Nue,for the TET to that for the smooth tube, Nus, for equal pumpingpower, as shown in Eq. (18):

1000 10000

1

2

3

4

TET No.1

TET No.2

TET No.3

TET No.4

TET No.5

Re

Fig. 11. Variation of performance factor with Reynolds number for the TETs.

S. Yang et al. / Applied Thermal Engineering 31 (2011) 2981e2991 2989

h ¼ Nue

Nus

epp

¼ Nue=Nusðfe=fsÞ1=3

(18)

Fig. 11 shows the variation of the performance factor h, withReynolds numbers for all the TETs. It can be seen that the perfor-mance factors are greater than 1.0 over the entire Reynolds numberrange for TET No. 1, TET No. 2 and TET No. 3 whereas lower than 1.0for TET No. 4 and TET No. 5 when Reynolds numbers are higherthan around 10,000. For all the TETs, when Reynolds numbers arehigher than 2300, the performance factors tend to decrease withthe increase in the Reynolds number. This means that theiradvantages associated with energy saving degrade as the Reynoldsnumber is raised. In some extreme cases, the overall performancesof TET No. 4 and 5 are even inferior to that of a smooth tube at highReynolds numbers. The performance factors are higher at lowerReynolds numbers, which also indicates that the role of the twistedchannel in increasing the turbulence intensity is more significant atlower velocities and the best operating regime for the TETs is atlower Reynolds numbers. TET No. 1 that presented the greatestperformance factor is regarded to have the best overall heattransfer enhancement performance.

Fig. 12. Isotherms and streamlines in the cross section of the TET No. 3 at Re ¼ 1000,5000 and 20,000.

0 10000 20000 30000 40000 5000089.0

89.2

89.4

89.6

89.8

m

Re

Smooth tube TET No.1 TET No.2 TET No.3 TET No.4 TET No.5

θ

Fig. 13. The synergy angle versus Reynolds number for the smooth tube and TET No. 3.

5. Analysis and discussions

Although the heat transfer enhancement has been studied formore than half a century, the basic mechanism of enhancingconvective heat transferwasnot clearly revealedup to the endof lastcentury [19]. Recently, Guo and co-workers [20,21] proposed anovelconceptof enhancing convectiveheat transfer for theparabolicflow:the convection term can be expressed as the dot product of velocityvector and temperature gradient, U

!$gradT ¼ jU!jjgradT jcos q, and

the convective heat transfer rate is proportional to the integration ofthedotproductover the thermalboundary layer. Theobjectiveof theproposed approach was to improve the uniformity of the velocityand temperature profiles and to reduce the intersection angle q, alsoknown as the synergy angle, between the velocity vector, U

!, and the

temperature gradient, gradT . This concept was extended to ellipticflow by both theoretical analysis and numerical computations in2002 [22,23]. After that, this principle was further confirmed bymany experimental and numerical studies [24e33] and now istermed field synergy principle [34].

In order to further understand the mechanism of heat transferenhancement of the TETs in terms of the field synergy principle, theflow field and temperature distribution in the TETs are numericallyanalyzed. The commercial CFD software FLUENT is used for thenumerical simulations. In the simulation, the fluid flow and heattransfer are considered as fully developed, and the tube walltemperature is constant. Fig. 12 demonstrates the typical flow andtemperature fields in the cross section of TET No. 3, whereTw ¼ 320 K, Tin ¼ 300 K, Re¼ 1000/5000/20,000. It can be seen thatlongitudinal vortex is induced in the tube cross section, and thevortex results in a fluid flow toward the tube wall surface. Thetemperature field in the cross section has a remarkable changeunder the influence of the vortex. The isotherms of a smoothcircular tube and straight elliptical tube are nearly circles andellipses, respectively [35]. However, those of the TET are distortednear the two ends of the long-axis, and with the increase of Rey-nolds number such distortion become more significant. Thetemperature gradient normal to the tube wall is increased, and thedot product of the velocity vector and the temperature gradient willbe increased. In addition, the combination of distorted isothermsand longitudinal vertex leads to some regions in the cross sectionwhere the isotherms are almost perpendicular to the local velocity.This implies that the local temperature gradient is almost parallel

to the local velocity, which just is an expected situation to achievethe heat transfer enhancement from the point viewof field synergy.

For simplicity of comparison, the most commonly used domainaverage synergy angle between the velocity vector and the

S. Yang et al. / Applied Thermal Engineering 31 (2011) 2981e29912990

temperature gradient in the whole flow field, qm, is adopted toshow the synergy level. The synergy angles for the five TETs anda smooth tube are presented in Fig.13. It is revealed that the smoothtube has the largest average synergy angle among all tubes, whilethe synergy angle of TET No. 1 is the smallest. By comparing thetrend of Nusselt number in Fig. 5 with the trend of the domainaverage synergy angle in Fig. 13, it is seen that the order of theNusselt number corresponds to that of the synergy angle. Thesmaller the synergy angle is, the larger the Nusselt number is. Thisis completely consistent with the field synergy principle. In otherwords, when the longitudinal vortex induced by the twisted tubewall changes the velocity and temperature distributions, thesynergy angle between velocity vector and temperature gradient isreduced, hence the heat transfer enhancement. A larger tube aspectratio and a smaller twist pitch will result in a higher synergy leveland thereby lead to the improvement of heat transfer, which canwell explain the effects of tube structural parameters on the heattransfer performance of the TETs. In addition, from Fig. 13 one alsocan see that the difference between the synergy angle of the TETsand the smooth tube is larger at a low Reynolds number than that ata large Reynolds numbers. This explains why the heat transferenhancement of TETs is more significant at a lower fluid velocity. Inconclusion, the special twisted tube wall yields a better synergybetween velocity and temperature gradient, and the TETs can,therefore, enhance heat transfer.

6. Conclusions

In the present work, an experimental study has been carried outto investigate the heat transfer and flow resistance characteristicsof water inside twisted elliptical tubes (TETs) with different struc-tural parameters. The unified correlations to predict the heattransfer coefficient and friction factor are proposed for Reynoldsnumber from around 600 to 55,000, and compared with existingcorrelations. Effects of tube structural parameters on the heattransfer and flow resistance performance of TETs is analyzed andthe heat transfer enhancement mechanism is discussed based onthe field synergy principle. Major findings are summarized asfollows:

(1) The TETs can enhance the heat transfer, especially at lowerReynolds numbers. The Nusselt number ratio Nue/Nus, firstincreases and then decreases with the rise of Reynolds number.The TET No. 1 with the smallest twist pitch has the best heattransfer performance. Its Nusselt number ratios is 3.9e4.8 atReynolds numbers ranging from 600 to 2300, and is 2.8e1.4 atReynolds numbers higher than 2300. Both a larger tube aspectratio and a smaller twist pitch can lead to a better heat transferperformance, and the effect of twist pitch on heat transferperformance is more notable than that of tube aspect ratio.

(2) The friction factor for the TETs is higher than that of the smoothtube. The variation trend of the friction factor ratio, fe=fs, issimilar to that of the Nusselt number ratio. TET No. 3 with thelargest aspect ratio has the largest friction factor ratio of around2.1e4.6 at Reynolds numbers from 600 to 2300, and 3.5e2.4 atReynolds numbers higher than 2300. Both a larger tube aspectratio and a smaller twist pitch can lead to a higher flow resis-tance in the TET, and the effect of tube aspect ratio on the flowresistance is more notable than that of tube twist pitch.

(3) The overall thermal-hydraulic performance of TETs is evaluatedusing the standard performance factor for equal pumpingpower. The overall performances of TETs are not so much goodat high Reynolds numbers as at low Reynolds numbers. Eventhe performance factors for TET No. 4 and TET No. 5 are lower

than 1.0 at high Reynolds numbers. Thereby, the best operatingregime for TETs is at lower Reynolds numbers.

(4) Both Nusselt numbers and friction factors for the TETs can beexpressed with unified equations containing various processparameters (Re, Pr, S/de and Ai/Bi) for the entire Reynoldsnumber range, which confirms the early flow transition fromlaminar to turbulent in TETs. The experimental results agreewell with the fitted correlations and have been compared withpredictions of some existing correlations, and the causes for thedifference between the experimental results and the predic-tions have been discussed.

(5) A good understanding of the heat transfer enhancement of theTETs can be derived from the field synergy principle.The longitudinal vortex inside the TET induced by the twistedtubewall improves the synergy between the velocity vector andtemperature gradient, resulting in the better heat transferperformance. More importantly, a smaller average synergyangle between the velocity vector and the temperature gradientproduces a higher Nusselt number.

Acknowledgement

Supports from the Shanghai Leading Academic DisciplineProject of China (B503) are greatly appreciated.

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