Conjugate Heat Transfer of a Finned Oval Tube Part b Flow Patterns

8/13/2019 Conjugate Heat Transfer of a Finned Oval Tube Part b Flow Patterns

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CONJUGATE HEAT TRANSFER OF A FINNED OVAL TUBEPART B: HEAT TRANSFER BEHAVIORSY. Chen a , M. Fiebig a & N. K. Mitra aa Institut fr Thermo- und Fluiddynamik, Ruhr-Universitt Bochum , D-44780 Bochum,GermanyPublished online: 28 Mar 2007.

To cite this article: Y. Chen , M. Fiebig & N. K. Mitra (1998) CONJUGATE HEAT TRANSFER OF A FINNED OVAL TUBE PART B:HEAT TRANSFER BEHAVIORS, Numerical Heat Transfer, Part A: Applications: An International Journal of Computation andMethodology, 33:4, 387-401

To link to this article: http://dx.doi.org/10.1080/10407789808913945

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CONJUG TE H E T T R N SF ER OF FINNEDOV L TUBE P RT B HE TTR NSFER EH VIORS

Chen M. Fiebig n N. K. MitraInstitut jUr Thermo- un Fluiddynamik Ruhr-Uniuersitdt BochumD44780 Bochum Germany

Conjugate heat transfer in a high-performance finned oval tube heat exchanger element hasbeen calculated fo r a thermally n hydrodynamicaUy developing three-dimensional lamina rflow. The infloence of Reynolds number in the range 100-500 and of f in parameter Fi theproduct of the dimensionless fin thickness and its thermal conductivity on the heat transferbehavior have been studied. In part I, the computed velocityfield was presented. Here wepresent the heat transfer behavior including heat flux on the tube fi n temperatures finNusse/l numbers and heat flux distributions fm efficiencies Colburn j factor n apparentfriction factor f . For the investigated configuration the ratio of heat transfer on the tubeto that on the fin remains under 10 The fin temperature and fin efficuncy depend weakly i

INTRODUCTION

Analysis of the flow field in part 1 of this article [1]showsthe weak horseshoevortex in front of the tube, the recirculating zone in the form of a hyperbolicparaboloid where the helical vortex is found. The introduction to the heat transferin a finned tube, the mathematical formulation, and the solution method are

presented in part A of this article. In this part B, heat transfer distribution on thefin and on the tube will be presented. Besides the Reynolds number, the parameterFi has been varied. For typical finned tube heat exchangers, Fi generally lies between 10 2 and 10 3 For example, if air [thermal conductivity k = 0.026 W / m K)at 2 C] is used as the fluid and 8 H is assumed to be 0.1, fins of aluminum[thermal conductivity k = 221 W / m K) at 20C]have Fi = 850; fins of steel with0.6 carbon content [thermal conductivity k = 50 W / m K) at 2 C] haveFi = 192; and fins of stainless steel [k 15 W / m K) at 20C] have Fi = 58 [2]In this study, Reynolds numbers of 100, 200, 300, 400, and 500 and fin parametersof 100, 500, 1000, and 00 are used. The flow medium is air (Pr = 0.7). The fluidtemperatures at the entrance and on the tube surface are To = 0 and TT = 1,respectively.

Received 22 May 1997; accepted 7 October 1997.Dr.vlng, Y. Chen thanks DLR-IB for granting a scholarship of the German Federal Ministry of

Education, Science and Technology.Address correspondence to Prof. N. K. Mitra, Institut r Thermo- und Fluiddynamik, Ruhr-Uni

versitiit Bochum, 44780 Bochum, Germany. E-mail: [email protected]

Numerical Heat Transfer, Part A, 33:387-401, 1998Copyright 1998 Taylor & Francis

1040-7782/98 12.00 .00 387


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388 Y. CHEN ET AL.

NOMEN L TURE

a thermal diffusivity, coefficient fl fluidA re frt frontal width ht heat transferD diameter, characteristic length j k indexh convective he t tr nsfer coefficient m me nv lueH channel height nb neighboringI flux p central control volumeJ Jacobian sp spanwise averagedk thermal conductivity T tubeL length w wallp pressure 0 reference v luer r dius r tioI time SuperscriptT temperature v w Cartesian velocity components dimension l volumex y z rtesi n coordin tes Dimensionless Parameters and Variablesa central angle cofactor Bi Biot number h 6 k l6 fin thickness f.pp friction factor I fin efficiency [ l i p / p ~ u ~ 2 / 2 X A t , , / A ~ , ) ]v kinem tic viscosity Fi fin parameter I [ ~ k i / k ~X 6 / H ) ] 1 C generalized coordinates Ft fin parameter 2 [ at / a ~ X l / P e ]p density j Colburn j factor Num/Re Pr 1/ 3

general variable Nu Nusselt number{ h W / k ~ = [tj/2 T, Ta ]}

Subscripts Pe Peeler number Re PrjPr Prandtl numberI = v6laV

B bulk q fin heat fluxcs cross section [


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HE T TRANSFER OF A F IN NED O V L TUBE, PART B

The local mean fluid tempera ture over the height of the channel is def ined as

j lu x , y, z IT x, y, z dzTm x,y = j lu x ,y,z ldz

389

The spanwise averaged Nusselt number Nus/x and heat flux 4s p x are used tocompare different Re and Fi. They are def ined as

and

1 1 o .


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390 Y. CHEN ET AL.

a

2

3

4

~ ~ 3 :2

o 2 4 6

b

c

6 10 X 12 14

2

- - - ~ - - - - - - - - L . . . - _ - - - - - - - - - - - - - - _ . . J . . J o

o 2 4 6 8 10 12X

14

d

Figure I, Distribution of a the fin temperatures, b the mean fluidtemperatures, c the fluid temperatures adjacent to the fin, d thefluid temperatures on the middle plane of the channel for Re ~ 300,Fi 500.

emperature istributions

Figure 1 compares the fin temperature, the local mean fluid tempera ture

Tm x,.y defined in Eq. 3 , the fluid tempera ture adjacent to the fin o onand the fluid temperature on the midplane of the duct z = 0.5 for Re = 300 andFi 500. Generally, the lowest temperatures of the fin or the fluid are located atthe point x 0, y 0 . The temperature increases with increasing x and yexcept in the tube wake. Comparing Figures 1 and 1b it is evident that thetemperature distribution in the fin is smoother than tha t in the fluid. Near the


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HEAT TRANSFER OF A FINNED OVAL TUBE, PART B 391

tube, the temperature gradient of the fluid is much larger than that of the fin.From Figure 1b the fluid temperature field can be briefly divided into four regionswith different distribution:

the entry flow region from the entrance to a short distance upstream of thetube where the temperature field develops something like in a flat duct;

the area near the tube (the heat source) where there are large tempera turegradients normal to the tube surface; the isotherms run almost parallel tothe x direction, i.e., the temperature gradient in the y direction dominates;

the area away from the tube where tempera ture gradients in the main flowdirect ion dominate, as in a flat duct; and

the tube wake where the fluid temperatures are high because of therecirculation and the temperature gradient in the x direction is very small.

Comparing Figures 1e and 1d we find a clear difference of the temper aturedistribution between the fluid near the fin (z = 0.01) and that in the middle of theduct z 0.5). Th e temperature distribution near the fin resembles the fin temperature (see Figure l.c), while that in the middle of the duct resembles the mean fluidtemperature (see Figure 1 b except in the entrance of the duct in both cases. Also,there are large tempera ture differences between the fluid near the fin and that inthe middle of the duct, except in a small area around the tube. From Figures1b-1d we notice tha t the fluid temperature remains at a low level in a large a reaof the duct. Beginning with x 5 the minimum mean fluid tempera ture in a crosssection with x const turns out to be in y 2-2 .5 , not in y 0, and so does thefluid temperature in the midplane of the duct from x 3. This does not appear forthe fluid temperature adjacent to the fin.

Th e mean fluid temperature at any cross section increases with decreasing Reand increasing Fi. Figure 2 compares the mean fluid temperatures for different Fi

with constant Re (Figures 2a and b and for different Re with constant Fi(Figures 2e and Zd . For a constant Re 300) the temperature gradient in the ydirection for Fi 100 is larger than that for Fi 1000. Th e global difference ofthe mean fluid temperature between Fi = 100 and Fi = 1000 is small (Figures 2aand Zb . This is also true for o ther investigated Re. From Figures 2e and 2d it isev ident tha t for a constant Fi 500), the mean temperature level for Re = 100 ismuch higher than that for Re 500. Th e mean temperature depends strongly onRe and only weakly on Fi. The temperature gradient in the y direction forRe 500 is much larger than that for Re 100, especially in the area near thetube.

Th e fin temperature distribution is much smoother than tha t of the fluidtemperature. Figure 3 compares the fin temperatures for different Fi with constantRe (see Figures 3a and 3b and for different Re with constant Fi (see Figures 3c

and sd . Evidently, the largest temperature difference between different parameters appears in the region near the point x = 0, y = 0), while the smallestdifference exists in the wake region. A change of Re from 100 to 500 with Fi = 500(Figures 3e and 3d influences the fin temperature much more than a change of Fifrom 100 to 1000 for Re 300. With increasing Re, the fin temperature leveldecreases quickly, especially in the area near x = 0, y = 0).


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392 Y. CHEN ET AL.

a

o

o

o

2

2

8

8

b

c

d

8

8

X 2

X 2

14

14

2

3

2

3

3

2

Figure 2. Comparison of the mean fluid temperature for differentRe and Fi. a Re = 300, Fi = 100; b Re 300, Fi = 1000; c Re = 100, Fi = S d Re = 500, Fi = 500.

Figure 4 shows the temperature distributions in eight y-z cross section withx = 1.5, 3.5, 5.6, 7.7., 9.7, 11.8, 13.8, and 15.4, respectively. Near the tube thetemperature gradients perpendicular to the tube dominate, while .in other regions,

the temperature gradients normal to the fin dictate. The lowest temperature of thefluid does not appear in the plane of y = 0 the farthest from the tube, the heatsource , but around 2-2.5 with z = 0.5 see Figures b e and h forinstance . The high tempera ture near the tube leads to the poor heat transfer ofthe fin in that area, as will be discussed below.


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HE T TRANSFER OF A FINNE OV L TUBE PART B

0

14

2

3

4

4

3

2

_ l ~ _ _ __ 0

393

o 2 4 6

b

c

8 10 X12 14

o 2 4 6 8 10 12 14

2

3

4

4

3

2

L-- - - - - . . . . . . . ; l -........

_ L

o 2 4 6

d

8 10 X 12 14

Figure 3. Comparison of the fin temperature for different Re and Fi. 0 Re = 300 Fi = 100; b Re = 300 Fi = 1000; c Re = 100Fi = 500; d Re = 500 Fi = 500.

e at ran sfer on the ube

Heat transfer on the tube is often ignored because it can be very smallcompared to tha t on the fin. Figure 5 compares hea t transfe r on the tube for thr ee

Re with constant Fi Heat transfer near the middle of th e tube z = 0.5 is muchlarger than th at n ear the fin. In the junction of the tube and fin the flow isdominated by the combined boundary layer of fin and tube. Th e velocity andtemperature gradient is much smaller than that near the middle of the tube. Th emaximum heat transfer appear s on the front arc an d decreases sharply with the


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4 3 2 o

Y CH EN ET AL

Figure 4. Tempera ture distribution oneight y z planes with constant x forRe 300, Fi = 500.

beginning of the side arcs. On the front are, he at flux for Re 500 is more thantwice that for Re = 100. Heat flux in the rear part of the tube is very small forall Re.

f the heat fluxes on the tube are averaged over the height, the span-averagedheat flux along the circumference of the tube ih defined in Eq. (6) is obtained; seeFigure 6. Clearly, heat fluxes on the front arc of the tube are much largercompared to those of the rear part, which are negligibly small. Also, the maximumdifference of hea t flux for different Re appears on the front of the tube , while theabsolute difference of heat flux in the rear of th e tube is negligible. Th e averagedmaximum heat fluxes do not occur directly at the f ront stagnat ion point X = 0),but in the neighborhood of that point 0.4). This phenomenon becomes moreevident with increasing Re. Heat flux decreases abruptly after the peak values untilan inflection point is achieved, which shifts to larger X for larger Re. Th einflection point is located at X 1.2 for Re = 100 and at X T 1.8 for Re = 500.After the inflection point, decreases slowly bu t remains low.

Integrating the heat flux on the tube and dividing this by the total amount of

heat transfer on the fin yields the ratio of he at transfer on the tube to tha t on thefin defined in Eq. (7) (see Figure 7). increases with increasing Re anddecreasing Fi. practical application (Fi = 100-1000), the dependence of onFi is small ( < 0.5 ). or the configuration and Reynolds number range used here, remains < 10 . t is possible from the tendency of the curves that exceeds10 at higher Re R e > 500) and smaller Fi (Fi < 100).


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HEAT TRANSFER OF A FINNED OVAL TUBE, PART B 395

upperfin t flux on the lube Re=100 FI=5

Z 0.5 0.5

lowerfi0.0

Iheat flux on the lUbe Re=300 FI=500 I1.0

0.1

Z 0.5

0.0

\heatflux onthe tube Re=5 Fi=5 I

1.0

Z 0.5

0.00 2 4 6 8 10 12

Figure 5. Heat flux on the surface of a half tube for three Reynoldsnumbers. Here xT = 0 corresponds to the front stagnat ion point ,X 13.7 the rear stagnat ion point , XT = 0.9 the junc tion of thefront and the side are, and xT = 12.8 the junct ion of the side andthe re r rc z is the coordin te in the ch nnel height direction the z coordinate is 4X enlarged .

e t Transfer on th e Fin

Heat transfer on the fin is determined by the flow structure in the duct andheat conduction in the fin. Figures 8 and 9 compare heat fluxes and Nusseltnumbers for different Fi with constant Re Figures 8a 8b 9a and 9b and fordifferent Re with constant Fi Figures 8e 8d ge and 9d . Generally, heat transferin the leading edge of the fin is very good, while in the tube wake it is very poor.

The differences of q x, y and Nutx, y between Fi = 100 and Fi = 1000 forRe = 300 are much smaller than those between Re = 100 and Re = 500 withFi = 500. Comparing the vector plots Figure 7 in part 1 [1] , the mean fluidtemperature Tm x y Figure 2 , and the q x, y and Nut r y distribution, we seethat the area of maximum velocity Figure 7 in part 1 or minimum Tm x yFigure 2 in a cross section x = const in the range of the tube corresponds to the


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Y. CHEN ET AL.

span ver ged heatfluxon lIle tIlba

420

lrearjunetlJra of lila arcs

8

-- Re=500- Re=400- Ra=300 Re=200 Re=100

396

20

16

12

8

4

00

Figure 6. Span-averaged heat flux along the circumference of a half tube a ll with Fi = 500). Here XT = 0 corresponds to the front stagnation point,x = 13.7to the rear stagnation point, XT = 0.9to the junct ion of the frontand the side are, and = 12.8 to the junct ion of the side and the rear arc.

maximum of tj x, y and Nuf,r, y) , In any cross section downstream of 4,minimum heat transfer on the fin occurs in the vicinity of the tube. Two factorslead to this occurrence: 1) the combined boundary layers of the fin and tube resultin a weak transport ability of the fluid, and 2) the fluid temperature in that area ishigh due to heat transfer on the outer surface of the tube. The span-averaged heatfluxes q and Nusselt numbers Nu,p defined in Eqs. 4) and 5) for different Reare shown in Figures 10 and 11. There are very large gradients of heat flux and Nuin the leading edge of the fin. The disturbance of the tube with the formation ofhorseshoe vortices slows down this tendency. Small local peaks of q and Nubetween = 1 and = 2 result from the horseshoe vortices. However, the amountand area influenced by these vortices are very small in comparison with those of acircular tube fin [3, 4]. Heat flux decreases slowlyin the range of the tube. The Nufor Re = 100 is almost constant in the range of the tube F igure 11). A furtherdecrease of the heat transfer occurs in the wake of the tube, where fluid recirculates, and thus heat transfer is very weak. Because of the very small dead waterzone, decrease in the span-averaged value is not evident.

Figure 7. Dependence of ratio ofthe total heat t ransfer on the tube to thaton the fin, on the Reynolds numbe r Reand the fin parameter Fi.


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HEAT TRANSFER OF A FINNED OVAL TUBE, PART B

a

397

U L . . _ L . . . . . . 0o 4 6

b

c

10

6 10X

12

12

14

14

3

4

4

3

3

4

4

3

LU..--- --- ------ ----- ------ oo 2 4 6

d

8 10X

12 14

Figure 8. Comparison of heat flux distribution on the fin showingthat the heat flux has a, b a weak dependence on Fi for aco nstan t Re of 300 and c, d a stronger dependence on Re for aconstant Fi of 500. a Re = 300, Fi 100; b Re = 300,Fi = 1000; c) Re = 100, Fi = 500; d Re = 500, Fi = 500.

The fin efficiency l f defined in Eq. 8) becomes unity for Fi --> 00 , i.e., infinitefin thermal conductivity or an infinitely thick fin. Both cases correspond to an

isothermal fin having the same temperature as the tube. Figure 12 shows that l fdecreases with decreasing Fi and increasing Re. For the investigated configurationand parameters, l f remains high. For Fi = 1000, l f remains almost > 90 . Thedifference of l f between Fi = 100 and Fi = 1000 is surprisingly small < 4 ). Twofactors lead to this result: (1 ) the use of an oval tube, which ensures a relativelyhomogeneous temperature distribution of the fin, and 2) the use of a moderate


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398 Y. CHEN ET AL.

a

12 14

4

3

u......J..L.....l..- -- - o

o 2 4 6 b

c

6 10 X 12 14

o 4 6 6 10 12 14

L l . . . . . L . . L ~ 0

o 4 6

d

8X

10 12 14

Figure 9. Comparison of Nussel t number distr ibut ion on the fmshowing that the Nussel t number has a, b a weak dependenceon the Fi for a constant Re of 300 and (c , d a s trongerdependence on Re for a constant Fi of 500. a Re 300,Fi - 100; b Re - 300, Fi 1000; c Re 100, Fi 500; d Re 500, Fi 500.

ratio of fin to tube area, which is common in practice. Areas with local fin efficiencies larger than unity, which is called heat transfer excess, are observed [ ]

Heat transfer and flow characteristics of a compact hea t exchanger aretypically presented with two correlations: the Colburn j factor defined in Eq. 9)and the friction factor f defined in Eq. 10) [6] Figure 13 shows that j and fdecrease with increasing Re. Their gradient in the range Re 100-300 is largerthan that in the range Re = 300-500. The ratio of j is 0.27 for Re = 100 and0.20 for Re 500.


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HEAT TRANSFER OF A FINNED OVAL TUBE, PART B

T lbSpanaveraged heat flUl

399

20

Re=500 A Re=300 A Re=100 Fi

o 2 8 8 10x

12 ,.Figure 10. Spanwise averaged heat flux on the fin for three

different Reynolds numbers.

Span.veraged Nusselt mmber

20

Nu ,

Re=500Re=300Re=100 Fi

,

o 2 8 8

x

12

Figure n. Span averaged Nusselt number on the fin for threedifferent Reynolds numbers.

100 . .

fI=o

flo5OO R 1005 It )

90

100 200

Re

500Figure 12. Dependence of the fin effi-ciency on the Reynolds number and thefin parameter Fi,


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400 Y. CHEN ET AL

0.20 . . . . ........-T - . - - . . . - - . . . - - . . - - . . . - - . . . - - - -

18

0.12

0.08

~ F i = 5 ~

0.04 0.00 . .. .. .- . .. .. . - - - - - - - - - - - - - - - - - - - - - -

100 200

Re300 400 5

Figure 13. Dependence of apparent friction factor f pp and Col-

burn j factor of a finned oval tube element on Reynolds number.

Heat transfer reversal as found in a finned circular tube [ was not found inthe FOT.

ON LUDING REM RKS

Flow and conjugate heat t ransfer in an FO T element have been investigatedby three-dimensional simulation with finite-volume method in curvilinear coordinates. Air with Pr = 0.7 is used as a working medium. Parameter variat ion of Reand Fi has been performed. Thermal analysis reveals tha t the fin t empera tu resremain high in the investigated range of parameters, and so do the fin efficiencies,which de pend weakly on Fi. This means tha t material can be saved by makingthinner fins without a significant decrease in fin efficiency.

Heat t ransfer on the outer surface of the tube increases with decreasing Fiand increasing Re, Fo r the investigated configuration and Re range, the rat io ofheat transfer on the tube to tha t on the fin R Q remains under 10 . Judging fromthe tendency of the curves, it is possible that R Q may exceed 10 , at higher Reand smaller Fi.

With an oval tube, heat transfer on a large area of the fin is primarilydetermined by the fin boundary layers. is therefore necessary to improve theheat transfer in that area, which is in the scope of a no the r article, where a wingtype vortex generator is used as a tool for heat t ransfer enhancement .

REFEREN ES

1. Y. Chen, M. Fiebig and N. K Mitra, Conjugate Heat Transfer of a Finned Oval Tube,Part 1: Flow Patterns, Numer. Heat Transfer Part this issue.

VDI Wiirmeatlas 7. Auflage, VDI Verlag Dusseldorf, 1994.


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HEAT TRANSFER OF A FINNED OVAL TUBE PART B 401

3. M. Fiebig A. Grosse Gorgemann Y. Chen and N. Mitra Conjugate e t Transfer ofa Finned Tube Part A: H ea t Tran sfer Behavior and Occurrence of Heat TransferReversal Numer Heat Transfer PartA vol. 28 pp. 133 146 1995.

4. M. Fiebig Y. Chen A. Grosse Gorgemann and N. Mitra Conjugate Heat Transfer ofa Finned Tube Part B: Heat Transfer Augmentation and Avoidance of e t TransferReversal by LVG Numer Heat Transfer Part vol. 28 pp. 147 155 1995.

5. Y. Chen M. Fiebig and N. Mitra Numerical Investigation of the Fin Efficiencies in aFinned Oval Tube in Proc 15th lMACS World ongr ss on Scientific ComputationModelling and Applied Mathematics Berlin Germany vol. 5 pp. 761 766 August 24 291997.

6. W. M. Kays and A. L. London Compact Heat Exchangers 3rd ed. McGraw Hili NewYork 1984.

Conjugate Heat Transfer of a Finned Oval Tube Part b Flow Patterns

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