Research Article Numerical Investigation of Non-Newtonian Flow...

Research ArticleNumerical Investigation of Non-Newtonian Flow and HeatTransfer Characteristics in Rectangular Tubes with Protrusions

Yonghui Xie Zheyuan Zhang Zhongyang Shen and Di Zhang

School of Energy and Power Engineering Xirsquoan Jiaotong University Xirsquoan Shaanxi 710049 China

Correspondence should be addressed to Yonghui Xie yhxiemailxjtueducn

Received 30 August 2014 Accepted 23 October 2014

Academic Editor Haochun Zhang

Copyright copy 2015 Yonghui Xie et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Flow characteristics and heat transfer performances in rectangular tubes with protrusions are numerically investigated in this paperThe thermal heat transfer enhancement of composite structures and flow resistance reduction of non-Newtonian fluid are takenadvantage of to obtain a better thermal performance Protrusion channels coupled with different CMC concentration solutions arestudied and the results are compared with that of smooth channels with water flow The comprehensive influence of turbulenceeffects structural effects and secondary flow effects on the CMCrsquos flow in protrusion tubes is extensively investigated The resultsindicate that the variation of flow resistance parameters of shear-thinning power-law fluid often shows a nonmonotonic trendwhich is different from that of water It can be concluded that protrusion structure can effectively enhance the heat transfer of CMCsolution with low pressure penalty in specific cases Moreover for a specific protrusion structure and a fixed flow velocity thereexists an optimal solution concentration showing the best thermal performance

1 Introduction

The application of protrusion has been attracting the atten-tion of many researchers due to its advantages for frictionfactors and enhancement for heat transfer Mahmood et al[1] studied the flow characteristics in rectangle tube withprotrusions structures They found that protrusions canstrengthen the vortex secondary flow and unsteadiness ofthe fluid Then Moon et al [2] investigated the phenomenonof flow and heat exchange in narrow channel arrayed withdimples and protrusions It was reported that the region ofenhancing heat transfer was focused on the windward sideand with the increased Reynolds number coefficients of bothfriction and heat transfer are decreased Borisov et al [3]studied the characteristics of flow and heat transfer in similarchannels as Moonrsquos and the average heat transfer coefficientand friction coefficient were obtained They compared theflow in tubes arrayed with dimples tubes arrayed with bothdimples and protrusions and traditional sine wave surfacechannels finding that protrusions can enhance heat transferbut it also results in extra pressure loss which is due to

the destruction of vortex structure produced by dimples Inaddition heat transfer coefficients of different channels allreach the maximum near the transition Reynolds

Simultaneously several researchers have investigatednon-Newtonian flow characteristics and its heat exchangephenomenon Arhuoma et al [4] studied the viscosity ofwater-in-oil emulsion in porousmedia finding that when theconcentration changed from 678 to 3348 the solutionshowed shear-thinning features and belonged to a typicalpower-law fluid They concluded that the degree of dropletsize distribution had little influence on emulsion viscosityand they reported the prediction correlations of the emulsionviscosity Srisamran and Devahastin [5] investigated the flowand mixing behavior of steady laminar confined impingingstreams of shear-thinning fluids which was compared withthat of Newtonian impinging streams Eesa and Barigou [6]using a validated CFDmodel showed that a transverse vibra-tionmotion on a steady laminar flow could generate sufficientchaotic fluid motion which leads to intense radial mixingAs a result the heat transfer is strengthened and a near-uniform radial temperature field with a substantial heating of

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 121048 11 pageshttpdxdoiorg1011552015121048

2 Mathematical Problems in Engineering

Adiabatic

Adiabatic

Heat flux

Flow direction

Entrance section

Exit section

Research domain

x

y

z

Figure 1 Schematic diagram of boundary conditions

Figure 2 Protrusion arrangement in the research domain

the inner flow is achieved Pimenta and Campos [7] carriedout an experimental work to obtain for Newtonian andnon-Newtonian fluids heat transfer coefficients at constantwall temperature in fully developed laminar flow inside ahelical coil Hojjat et al [8] experimentally studied the forcedconvective heat transfer of three kinds of nanofluids flowingin a uniformly heated circular tube under turbulent flowconditions

Although much research has been devoted to protru-sion structure and non-Newtonian fluid respectively littleresearch has been done on the combination of these twoparts In this study a numerical investigation is done tocharacterize the flow and heat transfer of CMC solution inrectangle tubes arrayedwith protrusion structures in order tomake use of both advantages of composite structure and non-Newtonian fluidThepurpose of the present paper is to obtaina new cooling technique for industrial equipment which caneffectively enhance heat transfer with little pressure penalty inorder to improve the efficiency and stability of the equipmentoperation

2 Numerical Method

21 Computational Geometry and Boundary Conditions Theschematic diagram of the protrusion roughened rectangularchannel used in this study is shown in Figure 1 The coordi-nates 119909 119910 and 119911 represent the streamwise normal and span-wise direction respectively The channel is roughened withprotrusions on one wall of the research section in an inlinearrangement The protrusion arrangement in the researchsection is shown inFigure 2Fully developed periodic velocity

R725

8

20

20

2

12060110

Figure 3 Geometry size of the protrusion unit

and temperature can be obtained after certain streamwiserows of protrusions Under this condition 7 rows of protru-sion structures are applied in the computational domain

CMC solutions with three kinds of concentrations(100 ppm500 ppm2000 ppm) and water are employed forcomparison in the present study The baseline is the case ofwater flowing in smooth rectangular channel The followingboundary conditions are specified (a) A uniform heat fluxof 119902 = 5 times 10

5Wsdotmminus2 and no-slip boundary condition areapplied at the four external surfaces of the research section(b) The relative pressure is 1 times 10

5 Pa turbulence intensityis 5 the static temperature at inlet is 300K and the staticpressure at outlet is 0 Pa As shown in Figure 1 the researchsection is placed between entrance section and exit sectionof which the length is more than 30 times the height in orderto ensure that the solutions enter the research section witha fully developed velocity and exit the section without theinfluence of non-Newtonian extrusion swelling effect

The rectangular channel is 8mm times 60mm in cross sec-tion The length of the research section is 140mm Thedetailed parameters of the protrusion unit are shown inFigure 3

The protrusion is a sector of a sphere with radius 119877 =

725mmThe depth and diameter of the protrusion unit arerespectively 2mm and 10mmThe streamwise and spanwisepitch of protrusions is 20mm

22 Solution Method The assumption that the flow is steadyincompressible and turbulent flow Constant fluent proper-ties is adopted for modeling the flow characteristics andheat transfer in the protrusion channel in this paper Theincompressible steady Navier-Stokes equation is solved usingthe finite-volume method based solver CFX and the SSTturbulentmodel coupled with gamma-theta transitionmodelwhich is adopted in the investigation The computations areconsidered to be converged when the residues for continuityenergy velocities and the area-averaged wall temperature areless than 1 times 10minus6 1 times 10minus7 1 times 10minus3 and 1 times 10minus6 respectivelyThe flow velocity in the present paper ranges from 1289msdotsminus1

to 3221msdotsminus1 and the corresponding Newtonian Reynoldsnumber ranges from 20000 to 50000 which is different withthat of non-Newtonian fluids

Mathematical Problems in Engineering 3

Table 1 Grid independency study

Case Total nodes Nu Difference 119891 Difference Mesh 1 1599456 136609 331 0012503 045Mesh 2 2432201 139166 150 0012527 025Mesh 3 3817146 140310 069 0012548 009Mesh 4 5579691 141284 Reference 0012559 Reference

23 Data Reduction The Reynolds number of non-Newto-nian fluids is described by [9]

Re1015840 =119863119899V2119899minus11205888119899minus1

119870

(

4119899

3119899 + 1

)

119899

(1)

where V is the inlet velocity 120588 is the density 119870 is theconsistency coefficient and 119899 is the flow behavior index Thehydraulic diameter119863 is given by

119863 =

4119860

119875

(2)

The Reynolds number of Newtonian fluids is defined by

Re =

120588V119863120583

(3)

where 120583 is the viscosity coefficientThe local Nusselt number in the present study is defined

by

Nu119909=

ℎ119909119863

120582

(4)

where 120582 is the thermal conductivity of solutions The heattransfer coefficient ℎ

119909is defined as

ℎ119909=

119902

Δ119879119909

(5)

where 119902 represents the heat flux and Δ119879119909is the mean

temperature difference between the wall and the fluids whichcan be obtained as follows

Δ119879119909= 119879119908119909

minus 119879119909 (6)

where119879119908119909

and119879119909are the local temperature of wall and fluids

respectivelyThe average Nusselt number for the channel can be cal-

culated by

Nu =

(intNu119909119889119878)

119878

(7)

The Fanning friction factor 119891 is defined as

119891 = minus

(Δ119875119871)119863

2120588V2 (8)

where Δ119875 is the pressure drop and 119871 is the streamwise lengthof the research section

Figure 4 Schematic diagram of mesh on protrusion surface

The friction loss 119862119891 is given by

119862119891=

119883wall(12) 120588V2

(9)

where119883wall is the wall shear stressThe form loss can be obtained as follows

119862119901= 119891 minus 119862

119891 (10)

The thermal performance is

TP = (

NuNu0

) sdot (

119891

1198910

)

minus13

(11)

The baseline Fanning friction factor 1198910 and Nusselt

number Nu0 for the case of water flowing in the smooth

rectangular channel are used to normalize the Fanningfriction factor and Nusselt number of CMC solutions flowingin the protrusion channel

The baseline Fanning friction factor 1198910and Nusselt

number Nu0are given as

1198910=

03164

Re025

Nu0= 0023Re08119875119903119899

(12)

where 119875119903 is the Planck number

24 Grid Independence Study In this paper O-grids are usedfor the protrusion region as shown in Figure 4 and an H-type grid is generated for the flow core The mesh around theprotrusion region is refined Before the numerical simulationthe grid independency study is carried out and four differentgrid systems are investigated Water is used as flow mediumand the inlet velocity is set as 1289msdotsminus1

Table 1 summarizes the comparisons The Nu and 119891

change by 069 and 009 from mesh 3 to mesh 4 respec-tively Hence mesh 3 is used


Table 2 Flow parameters of aqueous CMC solutions [10]

Concentrationppm 119899 119870kgsdots119899minus2sdotmminus1

100 09512 000383500 08229 0008492000 07051 002792

Table 3 Physical properties of CMC solutions used in the simula-tion [11]

Property ValueDensity (120588) 1000 kgsdotmminus3

Specific heat (119862119875) 4100 Jsdotkgminus1sdotKminus1

Thermal conductivity (120582) 07Wsdotmminus1sdotKminus1

25 Physical Properties In this study water and CMC (car-boxyl methyl cellulose) solutions are used as typical Newto-nian and shear-thinning fluids respectively The CMC solu-tions exhibit pseudoplastic (shear-thinning) feature Table 2gives the values of the flow behavior index and the con-sistency coefficient of CMC solutions at different concen-trations and the physical properties of the CMC solutionsare listed in Table 3 The temperature difference of the inletand outlet is lower than 5∘C and the potential effects ofthe temperature-dependent thermophysical properties canbe decoupled Hence the properties of fluids are assumed tobe constant in this study

3 Results and Discussion

31 Flow Characteristics of the Protrusions In the presentpaper the center plane in the flow direction (Figure 5) ischosen for the investigation In order to study the flow ofdifferent CMC solutions the distribution of streamline andturbulence intermittency 120574 for the protrusion regions at V =

1289msdotsminus1 is shown in Figure 6Through an examination of the case at V = 1289msdotsminus1

the streamline distributions of different CMC concentrationsolutions are nearly the same On the protrusion side separa-tion is identified downstream of the protrusion At the sametime a smaller separation zone can be observed as the flowslows down before it impinges on the front of the protrusion

The turbulence intermittency 120574 can be used as the criteriaof flow state at fixed point at 120574 = 0 the flow remains laminarand at 120574 ge 1 the flow is fully developed turbulent In therange of 0 lt 120574 lt 1 the flow is in transition condition Asshown in Figure 6 for a specific kind of solution the biggestlaminar area appears in the first protrusion region anddecreases along the flowdirectionMeanwhile for the specificposition of protrusion the laminar area increases as thesolution concentration increases In fact at V = 1289msdotsminus1the flow of water in the protrusion channel has reachedfully developed condition and remains invariable as thevelocity increases However there is also a certain amountof laminar area existing in the flow of CMC solutions It canbe concluded that it is harder for the CMC solution flow

Protrusion 1

Protrusion 2

Protrusion 3

Flow direction

Center plane

Figure 5 Schematic diagram of the center plane in the flow direc-tion

with higher concentration to reach fully developed turbulentcondition at the same velocity due to its higher viscosity

It is worth pointing out that the turbulence intermittencyinstead of Reynolds number is used to estimate flow statebecause it can illustrate the transient process more clearlyand the transition Reynolds numbers of CMC solutions areunknown

In order to understand the influence of different flowconditions two typical protrusion regions are chosen tostudy which are shown by the blocks in Figure 7

Figure 8 shows the distribution of streamline and temper-ature in two typical protrusion regions at V = 2255msdotsminus1 andV = 3221msdotsminus1 As observed the fluids imping on the frontedge of protrusions result in small recirculation zones Thenthe fluids detour around the side edge of protrusions leadingto strong perturbation and high flow velocity Besides theflow separation taking place in the trailing edges producesa large separation zone For the protrusions in symmetricallocation it is worth pointing out that a dominant featureof streamline distribution is that there are two focusessymmetrical about the center plane of themThe two focusesare considered to be the feet of the horseshoe vortex structureIn the region near the flow impingement the temperatureis relatively lower than that in other regions and in theseparation zone the temperature is relatively higher

Through an examination of the flow structures of all thecases the differences in concentration solution and velocitylead to the variation in size and location of recirculation zonesand separation zones Taking the case of V = 2255msdotsminus1as an example as concentration increases the recirculationzone at the front edges of upstream protrusions increaseswhile the separation zone at the trailing edges of midstreamprotrusions decreases as presented in Figure 8

32 Characteristics of Secondary Flow In order to studythe secondary flow in the tube and the interaction betweenprotrusion perturbation and non-Newtonian secondary floweffect two cross sections perpendicular to the flow directionare chosen as shown in Figure 9

For cross section 1 the flow structures in different pro-trusion regions are similar and the streamline distribution issymmetrical as shown inFigure 10As a result the flow struc-ture in the mid-protrusion region is chosen for the followingstudy The streamline distribution of different solutions in


00 01 02 03 04 05 06 07 08 09 1000 01 02 03 04 05 06 07 08 09 10 00 01 02 03 04 05 06 07 08 09 10

CMC2000 CMC2000CMC2000

Water

CMC100 CMC100 CMC100

CMC500CMC500CMC500

Protrusion 1Water

Protrusion 2Water

Protrusion 3

Figure 6 Distribution of streamline and turbulence intermittency

Flow directionFlow direction

Figure 7 Schematic diagram of the protrusion unit

mid-protrusion region at V = 2255msdotsminus1 is shown inFigure 11 As observed the streamline distribution in pro-trusion region shows a dual symmetrical vortex structureThe first one is under protrusions and the second one whichshows ellipse shape is at the protrusion edge side The twopairs of vortices rotate in the opposite directionThe influenceof concentration is mainly reflected by the size and the shapeof the second vortex structures At V = 2255msdotsminus1 thelocation of the second vortex of higher concentration solutionis more closed to the upper surface and the shape of the

second vortex is more oblate This phenomenon is due tothe combined effect of non-Newtonian secondary flow andperturbation from protrusions

Then the present section chose cross section 2 to illustratethe single effect of secondary flow The streamline distri-butions of different solution at V = 1289msdotsminus1 and V =

3221msdotsminus1 are shown in Figure 12 As observed in thesmooth rectangular channel the flow can be divided intotwo parts One is the flow in boundary region which isrelated to turbulent effect and shown in blocks The otheris the secondary flow region which is related to viscosity ofnon-Newtonian fluid The boundary region area increaseswith the increasing inlet velocity and the secondary flowregion area increases with the increasing concentration ofsolutions More specifically in the case of V = 3221msdotsminus1the boundary region ofwater andCMC100 is obviously biggerthan that at V = 1289msdotsminus1 and a boundary region appearsin the flow of CMC500 and CMC2000 which does notexist at V = 1289msdotsminus1 The difference in the streamlinedistribution between the two cross sections indicates that inthe present situation the effect of protrusion dominates theflow characteristics and the variation in solution viscosity just


CMC500

CMC2000

310 332 354 376 398 421 443 465 487 509 531 553 575 597 619 642 664 686 708 730 300 309 319 328 338 347 357 366 376 385 395 404 414 423 433 442 452 461 471 480

CMC500

CMC2000

CMC100

Recirculation zone at thefront edge

Separation zone at the

trailing edge

= 3221mmiddotsminus1 = 2255mmiddotsminus1CMC100

Figure 8 Distribution of streamline and temperature (K) of the protrusion regions

Flow direction

Cross section 1

Cross section 2

Figure 9 Schematic diagram of the research cross sections

changes the flow distribution in a small part of the researchregions

According to the results above the flow in rectangulartubes is influenced by the combined effect of turbulenceeffect secondary flow effect and the perturbation of pro-trusions which results in the variation in dimensional flowcharacteristics for different cases

33 Heat Transfer Characteristics The distribution of localNusselt number of different CMC concentration solutions atV = 2255msdotsminus1 on protrusion surface is shown in Figure 13

As observed the flows of CMC solution in tubes arrayedwith protrusion structures all show high Nusselt number atthe front edge of protrusions and low Nusselt number atthe trailing edge of protrusions which is because the flowimpingement at the front edge strengthens the perturbationand the flow separation at the trailing edge weakens theexchange of mass and energy This is in accordance with theanalysis of flow characteristics above

34 Flow Resistance Characteristics Figure 14 describes thevariation of flow resistance parameters at different CMCconcentrations as the inlet velocity increases from 1289msdotsminus1to 3221msdotsminus1 Four flow resistance parameters of Fanningfriction coefficient normalized friction coefficient frictionloss and form loss are selected As observed the flowresistance parameters of CMC solution in protrusion tubesoften show nonmonotonous variation tendency as velocityincreases The variations of flow resistance parameters ofdifferent CMC concentrations solution are also different inthe range of present study To be specific the flow resistanceparameters of CMC100 and CMC500 increase at first andthen decrease while the tendency for CMC2000 decreasesat first and then increases The protrusion structures destroythe development of the boundary layer thus producing theform loss due to the adverse pressure gradient inside theseparation zone Although the shear stress in the boundary


Figure 10 Schematic diagram of the protrusion region in the cross section 1

Water CMC100

CMC500 CMC2000

The second vortex

structure

Figure 11 Streamline distribution of the protrusion region in the cross section 1

layer increases the thickness of the boundary layer decreasesat the same time which leads to the increase of friction lossTo sum up the protrusion structure produces an extra pres-sure penalty However the pressure penalty can be reduced bythe effect of non-Newtonian secondary flow For CMC2000when the velocity is lower than approximately 25msdotsminus1 theeffect of the decrease in pressure penalty caused by non-Newtonian secondary flow dominates variation of the totalfriction At V gt 25msdotsminus1 the effect of increasing the pressurepenalty by protrusion structures dominates the variation oftotal friction coefficientHowever forCMC100 andCMC500the domination order is opposite and the turning point is atabout V = 20msdotsminus1 Also the 119891119891

0values are always larger

than 1 which indicates that the effect of protrusion is strongerthan that of the secondary flow As a result the pressurepenalty of CMC solution flowing in protrusion channels ishigher than that of water flowing in smooth channels

35 Heat Transfer Performance Being different from theparameters of flow characteristics the average Nusselt num-ber and normalized Nusselt number vary monotonouslyamong the range of the inlet velocity studied Compared withthe flow in smooth channels the protrusion structures caneffectively strengthen the exchange of mass and energy in the

boundary region resulting in the heat transfer enhancementin channels At the same time the non-Newtonian secondaryflow in the center regions of channels has a negative effecton the heat transfer performance However as shown inFigure 15 the NuNu

0values are always larger than 1 which

illustrates that the effect of protrusion dominates the heattransfer characteristics and leads to the heat transfer enhance-ment

36 Overall Thermal Performance Figure 16 shows the over-all thermal performance of different concentration solutionsflowing in protrusion rectangular tubes and smooth rectan-gular tubes In Figure 16 ldquoprrdquo means the protrusion tubes andldquosmrdquo means smooth tubes As observed the overall thermalperformance parameter of each combination increases withinlet velocity ranging from 1289msdotsminus1 to 3221msdotsminus1 At thesame inlet velocity the CMC500 solution in protrusion tubesshows the best thermal performance while CMC100 solutionin smooth tubes shows the worst thermal performanceHowever at a low inlet velocity (V lt 18msdotsminus1) the values ofldquoCMC500 + prrdquo are lower than 1 in the present study whichmeans that the combination of protrusion structures andshear-thinning fluids improves the thermal performance onlyin the specific situations In other words the specific channelstructure combined with corresponding concentration of


Water Water

CMC100 CMC100

CMC500 CMC500

CMC2000 CMC2000

= 1289mmiddotsminus1 = 3221mmiddotsminus1

Figure 12 Streamline distribution of the research cross section 2

CMC20001860167614921308112494075657238820420

CMC100 CMC5001860167614921308112494075657238820420

1860167614921308112494075657238820420

Figure 13 Distribution of Nusselt number on the protrusion surface

solution and flow velocity is expected to possess an optimalthermal performance

4 Conclusion

Based on the analytical and numerical investigations pre-sented in this paper the following conclusions can beobtained

(1) The turbulence intermittency is adopted to estimateflow state of different CMC solutions It can beconcluded that it is harder for the CMC solution flowwith higher concentration to reach fully developedturbulent condition at the same velocity due to itshigher viscosity At the same time as observed on theprotrusion side separation is identified downstreamof the protrusion a smaller separation zone can be

observed as the flow slows down before it impingeson the windward side of the protrusion

(2) Through the observation of the flow structures on theprotrusion surface the differences in concentrationsolution and velocity lead to the variation in size andlocation of recirculation zones and separation zoneswhich can be attributed to the combined effect ofsolution viscosity and perturbation from protrusions

(3) The flow patterns of the cross section in the protru-sion channel are compared with those in the smoothchannel In the protrusion rectangular channel thestreamline distribution shows a dual symmetricalvortex structure and the influence of concentrate ismainly reflected by the size and the shape of thesecond vortex structures and the effect of protrusiondominates the flow characteristics and the variation


(ms)10 15 20 25 30 35

f

0054

0051

0048

0045

0042

0039

CMC100CMC500CMC2000

(a) Fanning friction coefficient

CMC100CMC500CMC2000

(ms)10 15 20 25 30 35

ff

0

24

23

22

21

20

19

18

17

(b) Normalized friction coefficient

(ms)10 15 20 25 30 35

0009

0008

0007

0006

0005

Cf

CMC100CMC500CMC2000

(c) Friction loss

(ms)10 15 20 25 30 35

Cp

0045

0042

0039

0036

0033

CMC100CMC500CMC2000

(d) Form loss

Figure 14 Variation of flow resistance parameters of CMC solution (a) Fanning friction coefficient (b) normalized friction coefficient (c)friction loss and (d) form loss

in solution viscosity just changes the flow distributionin a small part of the research regions In the smoothrectangular channel the flow region can be dividedinto the boundary region and the secondary flowregion The difference in the streamline distributionbetween the two cross sections indicates that in thepresent situation the effect of protrusion dominatesthe flow characteristics

(4) The characteristics of CMC solution flow in pro-trusion tubes often show nonmonotonous variationtendency as velocity increases The flow resistanceparameters of CMC100 and CMC500 increase at firstand then decrease while tendency for CMC2000decreases at first and then increases It is mainly

due to the combined effect of turbulence effectsstructural effects and secondary flow effects The1198911198910values are always larger than 1 which illustrates

that the pressure penalty of CMC solution flowingin protrusion channels is higher than that of waterflowing in smooth channels

(5) The average Nusselt number and normalized Nusseltnumber vary monotonously among the range ofthe inlet velocity studied The heat transfer can bestrengthened by the protrusion structures and weak-ened by the effect of non-Newtonian secondary flowThe NuNu


illustrates that the effect of protrusion dominates theheat transfer characteristics


(ms)10 15 20 25 30 35

(ms)10 15 20 25 30 35

Nu

Nu

Nu 0

16

15

14

13

12

11

10

550

500

450

400

350

300

250

200

150

CMC100CMC500CMC2000

CMC100CMC500CMC2000

Figure 15 Variation of Nu and NuNu0at different CMC concentrations

TP

CMC100 + smCMC500 + smCMC2000 + sm

CMC100 + prCMC500 + prCMC2000 + pr

12

11

10

09

08

07

(ms)10 15 20 25 30 35

Figure 16 Overall thermal performance of CMC solutions

(6) In this paper at the same inlet velocity the CMC500solution in protrusion tubes shows the best thermalperformance while CMC100 solution in smoothtubes shows the worst thermal performance It canbe concluded that the specific channel structure com-bined with corresponding concentration of solutionand flow velocity will possess optimal thermal perfor-mance

Nomenclature

120591 Shear stressN120574 Shear ratesminus1119881 Velocity vectormsdotsminus1

119901 PressurePa119891V Body forceN120583 Viscosity coefficientNsdotssdotmminus2

119870 Consistency coefficientkgsdots119899minus2 sdotmminus1119899 Flow behavior index119863 Hydraulic diameterm119860 Inlet cross section aream2

119878 Target surface aream2119875 Wet perimeterm120588 Densitykgsdotmminus3

V Inlet velocitymsdotsminus1

119902 Heat fluxWsdotmminus2

ℎ119909 Heat transfer coefficientWsdotmminus2 sdot Kminus1

119879119908119909

Local wall temperatureK119879119909 Local fluid temperatureK

Δ119875 Pressure dropPaTP Overall thermal performanceNu Average Nusselt numberNu0 Reference Nusselt number

NuNu0 Normalized Nusselt number

119891 Fanning friction coefficient1198910 Reference Nusselt number

1198911198910 Normalized friction coefficient

119862119891 Friction loss

119862119901 Form loss

Re Newtonian Reynolds numberRe1015840 Non-Newtonian Reynolds number119875119903 Planck number119871 Streamwise lengthm119883wall Wall shear stressN120582 Thermal conductivity coefficientWsdotmminus1 sdot

Kminus1

120582 Time constant


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G IMahmoodM Z Sabbagh andPM Ligrani ldquoHeat transferin a channel with dimples and protrusions on opposite wallsrdquoJournal of Thermophysics and Heat Transfer vol 15 no 3 pp275ndash283 2001

[2] H K Moon OT Connell and R Sharma ldquoHeat transferenhancement using a convex-patterned surfacerdquo in Proceedingsof the ASME Turbo Expo 2002 Power for Land Sea and Air pp887ndash895 American Society of Mechanical Engineers 2002

[3] I Borisov A Khalatov S Kobzar and B Glezer ldquoHeat transferand pressure losses in a narrow dimpled channel structuredwith spherical protrusionsrdquo in Proceedings of the ASME TurboExpo 2006 Power for Land Sea and Air pp 191ndash197 AmericanSociety of Mechanical Engineers 2006

[4] M Arhuoma M Dong D Yang and R Idem ldquoDeterminationof water-in-oil emulsion viscosity in porous mediardquo Industrialamp Engineering Chemistry Research vol 48 no 15 pp 7092ndash7102 2009

[5] C Srisamran and S Devahastin ldquoNumerical simulation of flowand mixing behavior of impinging streams of shear-thinningfluidsrdquo Chemical Engineering Science vol 61 no 15 pp 4884ndash4892 2006

[6] M Eesa and M Barigou ldquoEnhancing radial temperature uni-formity and boundary layer development in viscous Newtonianand non-Newtonian flow by transverse oscillations a CFDstudyrdquo Chemical Engineering Science vol 65 no 6 pp 2199ndash2212 2010

[7] T A Pimenta and J B LM Campos ldquoHeat transfer coefficientsfrom Newtonian and non-Newtonian fluids flowing in laminarregime in a helical coilrdquo International Journal of Heat and MassTransfer vol 58 no 1-2 pp 676ndash690 2013

[8] MHojjat S G Etemad R Bagheri and JThibault ldquoConvectiveheat transfer of non-Newtonian nanofluids through a uniformlyheated circular tuberdquo International Journal of Thermal Sciencesvol 50 no 4 pp 525ndash531 2011

[9] A B Metzner and J C Reed ldquoFlow of non-newtonian fluidsmdashcorrelation of the laminar transition and turbulent-flowregionsrdquo AIChE Journal vol 1 no 4 pp 434ndash440 1955

[10] H J Poh K Kumar H S Chiang and A S Mujumdar ldquoHeattransfer from a laminar impinging jet of a power law fluidrdquoInternational Communications in Heat and Mass Transfer vol31 no 2 pp 241ndash249 2004

[11] A Kumar and M Bhattacharya ldquoTransient temperature andvelocity profiles in a canned non-Newtonian liquid food duringsterilization in a still-cook retortrdquo International Journal of Heatand Mass Transfer vol 34 no 4-5 pp 1083ndash1096 1991

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Stochastic AnalysisInternational Journal of


Adiabatic

Adiabatic

Heat flux

Flow direction

Entrance section

Exit section

Research domain

x

y

z

Figure 1 Schematic diagram of boundary conditions

Figure 2 Protrusion arrangement in the research domain

the inner flow is achieved Pimenta and Campos [7] carriedout an experimental work to obtain for Newtonian andnon-Newtonian fluids heat transfer coefficients at constantwall temperature in fully developed laminar flow inside ahelical coil Hojjat et al [8] experimentally studied the forcedconvective heat transfer of three kinds of nanofluids flowingin a uniformly heated circular tube under turbulent flowconditions

Although much research has been devoted to protru-sion structure and non-Newtonian fluid respectively littleresearch has been done on the combination of these twoparts In this study a numerical investigation is done tocharacterize the flow and heat transfer of CMC solution inrectangle tubes arrayedwith protrusion structures in order tomake use of both advantages of composite structure and non-Newtonian fluidThepurpose of the present paper is to obtaina new cooling technique for industrial equipment which caneffectively enhance heat transfer with little pressure penalty inorder to improve the efficiency and stability of the equipmentoperation

2 Numerical Method

21 Computational Geometry and Boundary Conditions Theschematic diagram of the protrusion roughened rectangularchannel used in this study is shown in Figure 1 The coordi-nates 119909 119910 and 119911 represent the streamwise normal and span-wise direction respectively The channel is roughened withprotrusions on one wall of the research section in an inlinearrangement The protrusion arrangement in the researchsection is shown inFigure 2Fully developed periodic velocity

R725

8

20

20

2

12060110

Figure 3 Geometry size of the protrusion unit

and temperature can be obtained after certain streamwiserows of protrusions Under this condition 7 rows of protru-sion structures are applied in the computational domain

CMC solutions with three kinds of concentrations(100 ppm500 ppm2000 ppm) and water are employed forcomparison in the present study The baseline is the case ofwater flowing in smooth rectangular channel The followingboundary conditions are specified (a) A uniform heat fluxof 119902 = 5 times 10

5Wsdotmminus2 and no-slip boundary condition areapplied at the four external surfaces of the research section(b) The relative pressure is 1 times 10

5 Pa turbulence intensityis 5 the static temperature at inlet is 300K and the staticpressure at outlet is 0 Pa As shown in Figure 1 the researchsection is placed between entrance section and exit sectionof which the length is more than 30 times the height in orderto ensure that the solutions enter the research section witha fully developed velocity and exit the section without theinfluence of non-Newtonian extrusion swelling effect

The rectangular channel is 8mm times 60mm in cross sec-tion The length of the research section is 140mm Thedetailed parameters of the protrusion unit are shown inFigure 3

The protrusion is a sector of a sphere with radius 119877 =

725mmThe depth and diameter of the protrusion unit arerespectively 2mm and 10mmThe streamwise and spanwisepitch of protrusions is 20mm

22 Solution Method The assumption that the flow is steadyincompressible and turbulent flow Constant fluent proper-ties is adopted for modeling the flow characteristics andheat transfer in the protrusion channel in this paper Theincompressible steady Navier-Stokes equation is solved usingthe finite-volume method based solver CFX and the SSTturbulentmodel coupled with gamma-theta transitionmodelwhich is adopted in the investigation The computations areconsidered to be converged when the residues for continuityenergy velocities and the area-averaged wall temperature areless than 1 times 10minus6 1 times 10minus7 1 times 10minus3 and 1 times 10minus6 respectivelyThe flow velocity in the present paper ranges from 1289msdotsminus1

to 3221msdotsminus1 and the corresponding Newtonian Reynoldsnumber ranges from 20000 to 50000 which is different withthat of non-Newtonian fluids





Re1015840 =119863119899V2119899minus11205888119899minus1

119870

(

4119899

3119899 + 1

)

119899

(1)


119863 =

4119860

119875

(2)


Re =

120588V119863120583

(3)


by

Nu119909=

ℎ119909119863

120582

(4)


119909is defined as

ℎ119909=

119902

Δ119879119909

(5)



Δ119879119909= 119879119908119909

minus 119879119909 (6)

where119879119908119909



culated by

Nu =

(intNu119909119889119878)

119878

(7)


119891 = minus

(Δ119875119871)119863

2120588V2 (8)




119862119891=

119883wall(12) 120588V2

(9)


119862119901= 119891 minus 119862

119891 (10)


TP = (

NuNu0

) sdot (

119891

1198910

)

minus13

(11)






1198910=

03164

Re025

Nu0= 0023Re08119875119903119899

(12)








100 09512 000383500 08229 0008492000 07051 002792











Protrusion 1

Protrusion 2

Protrusion 3

Flow direction

Center plane










00 01 02 03 04 05 06 07 08 09 1000 01 02 03 04 05 06 07 08 09 10 00 01 02 03 04 05 06 07 08 09 10


Water


CMC500CMC500CMC500

Protrusion 1Water

Protrusion 2Water

Protrusion 3









CMC500

CMC2000

310 332 354 376 398 421 443 465 487 509 531 553 575 597 619 642 664 686 708 730 300 309 319 328 338 347 357 366 376 385 395 404 414 423 433 442 452 461 471 480

CMC500

CMC2000

CMC100



trailing edge



Flow direction

Cross section 1

Cross section 2









Water CMC100

CMC500 CMC2000

The second vortex

structure











Water Water

CMC100 CMC100

CMC500 CMC500

CMC2000 CMC2000



CMC20001860167614921308112494075657238820420

CMC100 CMC5001860167614921308112494075657238820420

1860167614921308112494075657238820420



4 Conclusion







(ms)10 15 20 25 30 35

f

0054

0051

0048

0045

0042

0039

CMC100CMC500CMC2000


CMC100CMC500CMC2000

(ms)10 15 20 25 30 35

ff

0

24

23

22

21

20

19

18

17


(ms)10 15 20 25 30 35

0009

0008

0007

0006

0005

Cf

CMC100CMC500CMC2000

(c) Friction loss

(ms)10 15 20 25 30 35

Cp

0045

0042

0039

0036

0033

CMC100CMC500CMC2000

(d) Form loss










(ms)10 15 20 25 30 35

(ms)10 15 20 25 30 35

Nu

Nu

Nu 0

16

15

14

13

12

11

10

550

500

450

400

350

300

250

200

150

CMC100CMC500CMC2000

CMC100CMC500CMC2000


TP



12

11

10

09

08

07

(ms)10 15 20 25 30 35



Nomenclature








119879119908119909







119862119901 Form loss


Kminus1





References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Re1015840 =119863119899V2119899minus11205888119899minus1

119870

(

4119899

3119899 + 1

)

119899

(1)


119863 =

4119860

119875

(2)


Re =

120588V119863120583

(3)


by

Nu119909=

ℎ119909119863

120582

(4)


119909is defined as

ℎ119909=

119902

Δ119879119909

(5)



Δ119879119909= 119879119908119909

minus 119879119909 (6)

where119879119908119909



culated by

Nu =

(intNu119909119889119878)

119878

(7)


119891 = minus

(Δ119875119871)119863

2120588V2 (8)




119862119891=

119883wall(12) 120588V2

(9)


119862119901= 119891 minus 119862

119891 (10)


TP = (

NuNu0

) sdot (

119891

1198910

)

minus13

(11)






1198910=

03164

Re025

Nu0= 0023Re08119875119903119899

(12)








100 09512 000383500 08229 0008492000 07051 002792











Protrusion 1

Protrusion 2

Protrusion 3

Flow direction

Center plane










00 01 02 03 04 05 06 07 08 09 1000 01 02 03 04 05 06 07 08 09 10 00 01 02 03 04 05 06 07 08 09 10


Water


CMC500CMC500CMC500

Protrusion 1Water

Protrusion 2Water

Protrusion 3









CMC500

CMC2000

310 332 354 376 398 421 443 465 487 509 531 553 575 597 619 642 664 686 708 730 300 309 319 328 338 347 357 366 376 385 395 404 414 423 433 442 452 461 471 480

CMC500

CMC2000

CMC100



trailing edge



Flow direction

Cross section 1

Cross section 2









Water CMC100

CMC500 CMC2000

The second vortex

structure











Water Water

CMC100 CMC100

CMC500 CMC500

CMC2000 CMC2000



CMC20001860167614921308112494075657238820420

CMC100 CMC5001860167614921308112494075657238820420

1860167614921308112494075657238820420



4 Conclusion







(ms)10 15 20 25 30 35

f

0054

0051

0048

0045

0042

0039

CMC100CMC500CMC2000


CMC100CMC500CMC2000

(ms)10 15 20 25 30 35

ff

0

24

23

22

21

20

19

18

17


(ms)10 15 20 25 30 35

0009

0008

0007

0006

0005

Cf

CMC100CMC500CMC2000

(c) Friction loss

(ms)10 15 20 25 30 35

Cp

0045

0042

0039

0036

0033

CMC100CMC500CMC2000

(d) Form loss










(ms)10 15 20 25 30 35

(ms)10 15 20 25 30 35

Nu

Nu

Nu 0

16

15

14

13

12

11

10

550

500

450

400

350

300

250

200

150

CMC100CMC500CMC2000

CMC100CMC500CMC2000


TP



12

11

10

09

08

07

(ms)10 15 20 25 30 35



Nomenclature








119879119908119909







119862119901 Form loss


Kminus1





References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












100 09512 000383500 08229 0008492000 07051 002792











Protrusion 1

Protrusion 2

Protrusion 3

Flow direction

Center plane










00 01 02 03 04 05 06 07 08 09 1000 01 02 03 04 05 06 07 08 09 10 00 01 02 03 04 05 06 07 08 09 10


Water


CMC500CMC500CMC500

Protrusion 1Water

Protrusion 2Water

Protrusion 3









CMC500

CMC2000

310 332 354 376 398 421 443 465 487 509 531 553 575 597 619 642 664 686 708 730 300 309 319 328 338 347 357 366 376 385 395 404 414 423 433 442 452 461 471 480

CMC500

CMC2000

CMC100



trailing edge



Flow direction

Cross section 1

Cross section 2









Water CMC100

CMC500 CMC2000

The second vortex

structure











Water Water

CMC100 CMC100

CMC500 CMC500

CMC2000 CMC2000



CMC20001860167614921308112494075657238820420

CMC100 CMC5001860167614921308112494075657238820420

1860167614921308112494075657238820420



4 Conclusion







(ms)10 15 20 25 30 35

f

0054

0051

0048

0045

0042

0039

CMC100CMC500CMC2000


CMC100CMC500CMC2000

(ms)10 15 20 25 30 35

ff

0

24

23

22

21

20

19

18

17


(ms)10 15 20 25 30 35

0009

0008

0007

0006

0005

Cf

CMC100CMC500CMC2000

(c) Friction loss

(ms)10 15 20 25 30 35

Cp

0045

0042

0039

0036

0033

CMC100CMC500CMC2000

(d) Form loss










(ms)10 15 20 25 30 35

(ms)10 15 20 25 30 35

Nu

Nu

Nu 0

16

15

14

13

12

11

10

550

500

450

400

350

300

250

200

150

CMC100CMC500CMC2000

CMC100CMC500CMC2000


TP



12

11

10

09

08

07

(ms)10 15 20 25 30 35



Nomenclature








119879119908119909







119862119901 Form loss


Kminus1





References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00 01 02 03 04 05 06 07 08 09 1000 01 02 03 04 05 06 07 08 09 10 00 01 02 03 04 05 06 07 08 09 10


Water


CMC500CMC500CMC500

Protrusion 1Water

Protrusion 2Water

Protrusion 3









CMC500

CMC2000

310 332 354 376 398 421 443 465 487 509 531 553 575 597 619 642 664 686 708 730 300 309 319 328 338 347 357 366 376 385 395 404 414 423 433 442 452 461 471 480

CMC500

CMC2000

CMC100



trailing edge



Flow direction

Cross section 1

Cross section 2









Water CMC100

CMC500 CMC2000

The second vortex

structure











Water Water

CMC100 CMC100

CMC500 CMC500

CMC2000 CMC2000



CMC20001860167614921308112494075657238820420

CMC100 CMC5001860167614921308112494075657238820420

1860167614921308112494075657238820420



4 Conclusion







(ms)10 15 20 25 30 35

f

0054

0051

0048

0045

0042

0039

CMC100CMC500CMC2000


CMC100CMC500CMC2000

(ms)10 15 20 25 30 35

ff

0

24

23

22

21

20

19

18

17


(ms)10 15 20 25 30 35

0009

0008

0007

0006

0005

Cf

CMC100CMC500CMC2000

(c) Friction loss

(ms)10 15 20 25 30 35

Cp

0045

0042

0039

0036

0033

CMC100CMC500CMC2000

(d) Form loss










(ms)10 15 20 25 30 35

(ms)10 15 20 25 30 35

Nu

Nu

Nu 0

16

15

14

13

12

11

10

550

500

450

400

350

300

250

200

150

CMC100CMC500CMC2000

CMC100CMC500CMC2000


TP



12

11

10

09

08

07

(ms)10 15 20 25 30 35



Nomenclature








119879119908119909







119862119901 Form loss


Kminus1





References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










CMC500

CMC2000

310 332 354 376 398 421 443 465 487 509 531 553 575 597 619 642 664 686 708 730 300 309 319 328 338 347 357 366 376 385 395 404 414 423 433 442 452 461 471 480

CMC500

CMC2000

CMC100



trailing edge



Flow direction

Cross section 1

Cross section 2









Water CMC100

CMC500 CMC2000

The second vortex

structure











Water Water

CMC100 CMC100

CMC500 CMC500

CMC2000 CMC2000



CMC20001860167614921308112494075657238820420

CMC100 CMC5001860167614921308112494075657238820420

1860167614921308112494075657238820420



4 Conclusion







(ms)10 15 20 25 30 35

f

0054

0051

0048

0045

0042

0039

CMC100CMC500CMC2000


CMC100CMC500CMC2000

(ms)10 15 20 25 30 35

ff

0

24

23

22

21

20

19

18

17


(ms)10 15 20 25 30 35

0009

0008

0007

0006

0005

Cf

CMC100CMC500CMC2000

(c) Friction loss

(ms)10 15 20 25 30 35

Cp

0045

0042

0039

0036

0033

CMC100CMC500CMC2000

(d) Form loss










(ms)10 15 20 25 30 35

(ms)10 15 20 25 30 35

Nu

Nu

Nu 0

16

15

14

13

12

11

10

550

500

450

400

350

300

250

200

150

CMC100CMC500CMC2000

CMC100CMC500CMC2000


TP



12

11

10

09

08

07

(ms)10 15 20 25 30 35



Nomenclature








119879119908119909







119862119901 Form loss


Kminus1





References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Water CMC100

CMC500 CMC2000

The second vortex

structure











Water Water

CMC100 CMC100

CMC500 CMC500

CMC2000 CMC2000



CMC20001860167614921308112494075657238820420

CMC100 CMC5001860167614921308112494075657238820420

1860167614921308112494075657238820420



4 Conclusion







(ms)10 15 20 25 30 35

f

0054

0051

0048

0045

0042

0039

CMC100CMC500CMC2000


CMC100CMC500CMC2000

(ms)10 15 20 25 30 35

ff

0

24

23

22

21

20

19

18

17


(ms)10 15 20 25 30 35

0009

0008

0007

0006

0005

Cf

CMC100CMC500CMC2000

(c) Friction loss

(ms)10 15 20 25 30 35

Cp

0045

0042

0039

0036

0033

CMC100CMC500CMC2000

(d) Form loss










(ms)10 15 20 25 30 35

(ms)10 15 20 25 30 35

Nu

Nu

Nu 0

16

15

14

13

12

11

10

550

500

450

400

350

300

250

200

150

CMC100CMC500CMC2000

CMC100CMC500CMC2000


TP



12

11

10

09

08

07

(ms)10 15 20 25 30 35



Nomenclature








119879119908119909







119862119901 Form loss


Kminus1





References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Water Water

CMC100 CMC100

CMC500 CMC500

CMC2000 CMC2000



CMC20001860167614921308112494075657238820420

CMC100 CMC5001860167614921308112494075657238820420

1860167614921308112494075657238820420



4 Conclusion







(ms)10 15 20 25 30 35

f

0054

0051

0048

0045

0042

0039

CMC100CMC500CMC2000


CMC100CMC500CMC2000

(ms)10 15 20 25 30 35

ff

0

24

23

22

21

20

19

18

17


(ms)10 15 20 25 30 35

0009

0008

0007

0006

0005

Cf

CMC100CMC500CMC2000

(c) Friction loss

(ms)10 15 20 25 30 35

Cp

0045

0042

0039

0036

0033

CMC100CMC500CMC2000

(d) Form loss










(ms)10 15 20 25 30 35

(ms)10 15 20 25 30 35

Nu

Nu

Nu 0

16

15

14

13

12

11

10

550

500

450

400

350

300

250

200

150

CMC100CMC500CMC2000

CMC100CMC500CMC2000


TP



12

11

10

09

08

07

(ms)10 15 20 25 30 35



Nomenclature








119879119908119909







119862119901 Form loss


Kminus1





References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(ms)10 15 20 25 30 35

f

0054

0051

0048

0045

0042

0039

CMC100CMC500CMC2000


CMC100CMC500CMC2000

(ms)10 15 20 25 30 35

ff

0

24

23

22

21

20

19

18

17


(ms)10 15 20 25 30 35

0009

0008

0007

0006

0005

Cf

CMC100CMC500CMC2000

(c) Friction loss

(ms)10 15 20 25 30 35

Cp

0045

0042

0039

0036

0033

CMC100CMC500CMC2000

(d) Form loss










(ms)10 15 20 25 30 35

(ms)10 15 20 25 30 35

Nu

Nu

Nu 0

16

15

14

13

12

11

10

550

500

450

400

350

300

250

200

150

CMC100CMC500CMC2000

CMC100CMC500CMC2000


TP



12

11

10

09

08

07

(ms)10 15 20 25 30 35



Nomenclature








119879119908119909







119862119901 Form loss


Kminus1





References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(ms)10 15 20 25 30 35

(ms)10 15 20 25 30 35

Nu

Nu

Nu 0

16

15

14

13

12

11

10

550

500

450

400

350

300

250

200

150

CMC100CMC500CMC2000

CMC100CMC500CMC2000


TP



12

11

10

09

08

07

(ms)10 15 20 25 30 35



Nomenclature








119879119908119909







119862119901 Form loss


Kminus1





References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Numerical Investigation of Non-Newtonian Flow...

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