forced convection in wavy duct

8/12/2019 Forced Convection in Wavy duct

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9200,7.No,27.Vol,Journal.Tech&.Eng

* Mechanical Engineering Department, University of Technology / Baghdad 1385

Numerical Study of Forced Convection in Wavyand Diverged-Converged Ducts

Dr. Waheed S. Mohamad*,Dr. Sattar J. Habeeb*& Mr. Anmar M. Basheer*

Received on: 28/9/2008Accepted on:26/1/2009

AbstractA three-dimensional study of developing fluid flow and heat transfer through

wavy and diverged-converged ducts were studied numerically for a Prandlt number 0.7and 5.85 and compared with flow through corresponding straight duct. The Navier-Stokes and energy equations are solved by using control finite volume method.Development of the Nusselt number in wavy and diverged-converged ducts are

presented for different flow rates (50

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Eng. & Tech. Journal ,Vol.27, No.7,2009 Numerical Study of Forced Convection in Wavyand Diverged - Converged Ducts

1386

Greek symbols

Density kg/m 3 Dynamic viscosity kg/m.s! Dependent variable

" diffusion coefficient ,, computational coordinates m

Subscriptsmin Minimummax Maximumin Inlet conditionw Condition at the wall

IntroductionAn effective design for a heat

exchanger is the one which maximized theheat transfer while reducing the powerexpended. Increase in surface area, whichis the primary feature of many compactheat exchangers, invariably increases theheat transfer. However, this also increasesthe power expended and cost. There areseveral books and papers published on the

basic design procedures of heatexchangers based on experimental andtheoretical study of the past. However,some challenges of heat exchanger designwill remain for fore-seeable future. This is

due to the fact that the heat exchangerdesigns need to adopt itself to the evergrowing process, power and aerospaceindustries. As these industries grow, more

precise design of heat exchangers andsuitable materials become crucial. Thereduction in space occupied, better

performance and cost effective designwould be future objectives of the heatexchanger industry. Hwang et.al, 2006[1] investigated the flow and heat/masstransfer characteristics of wavy duct for

the primary surface heat exchangerapplication. Local heat/mass transfercoefficients on the corrugated duct sidewalls are measured using a naphthalenesublimation technique. Takachiro andHaruo, 2002[2] applied the spectralelement and finite difference methodsusing numerical studies on the three-dimensional laminar forced convectionheat transfer in a channel with span wise-

periodic fluted parts. Blomerius et.al,2000[3] investigated flow field and heattransfer in sine-wave crossed-corrugatedducts by numerical solution of the Navier-

Stokes and energy equations in the laminarand transitional flow regime betweenRe=170 and 2000. Stone and Vanka,1999[4] presented numerical results on

developing flow and heat transfercharacteristics in a furrowed wavy channelThe flow was assumed to be two-dimensional, and the Navier-Stokesequations governing a time-dependentflow were numerically solved. Mamum etal, 2007[5] investigated a natural-convection boundary layer along a verticalcomplex wavy surface with uniform heatflux. The complex surface study combinestwo sinusoidal functions, a fundamentalwave and its first harmonic . Mohamed et.al, 2006[6 ] determinedlaminar forced convection in the entrancezone in a two dimensional converging-diverging channel with sinusoidal wallcorrugation by using numerical simulation.The wavy surface of the channel lowerwall is maintained to a temperature whichis twice the upper wall temperature.

Numerical solutions are obtained using thecontrol-volume finite-difference method.Manglik et.al, 2005[7] presented detailedunderstanding forced convection in

periodically developed low Reynoldsnumber (10 # Re # 1000) air (Pr = 0.7)flows in three-dimensional wavy-plate-fincores. Constant property computationalsolutions are obtained using finite-volumetechniques for a non-orthogonal non-staggered grid. Hossian and SadrulIslam, 2007[8] investigated fluid flow andheat transfer characteristics in wavychannels by using numerical solution ofthe Navier-Stokes and energy equations.

Three different types of two dimensionalwavy geometries (e. g. sine-shaped,triangular, and arc-shaped) are considered Bahaidarah et.al, 2005[9] presented detailed numerical steady two dimensionfluid flow and heat transfer through a

periodic wavy passage with a Prandltnumber 0.7 and compared to flow througha corresponding straight (parallel-plate)channel. Sinusoidal and arc-shapedconfigurations were studied for a range ofgeometric parameters. At low Reynolds

number, the two geometric configurationsshowed little or no heat transfer

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consecutive field values of variables is lessthan or equal to 10 -6. For furtherstabilization of numerical algorithm, underrelaxation factor with range from 0.4 to

0.7 for u, v, w, T, and P are used. Thecalculations used non-orthogonalcurvilinear body fitted grid containingthree dimensions 31*60*31 internal cells.The procedure used the power low schemeand simple algorithm method for pressureand velocity correction developed byPatankar [12].

The numerical solutions usedelliptic grid generation technique (Poison'sequation type) so as to transform geometryfrom curvilinear grid in physical domainin term (x, y, z) to rectangular grid incomputational domain in term ),,( followed the work of Thompson,.1999et.al [13]. Computations were performedfor wavy duct with Reynolds number (100,200 and 300), length ratio (L/2a=10, 6.67,5) and aspect ratio (L/H = 4) and for thediverged-converged duct used Reynoldsnumber (50, 100, 300) with height ratio(Hmin/Hmax = 0.46, 0.39, 0.33) and aspectratio (L/H min = 4).

The local Nusselt number on thewavy walls of the duct is defined by: :

)(

.

inw

h

loc T T

DT

Nu

=

........ (4)

2/122

+

=

z

T

x

T T

.... (5)

The average Nusselt number is defined bythe following relation:

=

==

1

2

1 Nj j

jlocavg Nu N

Nu

.. (6)

While the Reynolds number is defined as:

avg h v D=Re (7)

where D h is the hydraulic diameter.

While the shear stress is defined as:

p

pw x

U = . (8)

where222 W V U U p ++=

andx p is the normal distance between the

near-wall grid mode (p) and the wall.

Model Validation

The model validation is anessential part of a numerical investigation.Validation of the present study has been

performed for the laminar flows in adiverged-converged duct and comparedwith pervious work for Bahaidarah et.al[9]. The comparison was conducted for(Re = 25 and 100) and Pr = 0.7. Goodagreement was achieved as illustrated infig (2) and fig (3) that show the results ofthis work together with the numericalresults of Bahaidarah et.al [9] in

streamlines and temperature contoursrespectively. Table (1) shows thecomparison of average Nusselt number onthe wavy wall with the numerical resultsof Bahaidarah et.al [9]. Result and DiscussionFlow field and temperature

Streamlines and velocity vectorfor flows with Re=100, 200,300 andlength ratio (L/2a=10, 6.67 and 5) areshown in figures (4, 5, 6) respectively forwavy duct. Lateral vortex is trigged at theRe=300 and (L/2a=6.67). With increasingflow rate and length ratio led to increase inthe lateral swirl. At a very low Reynoldsnumber, however, viscous forces dominateto produce undisturbed streamline flowsand vortex is not developed, irrespectiveof the length ratio. The temperaturecontour distribution in water (Pr=5.85)flows shown in figures (4, 5, 6). Theincreasing special coverage of the troughvortices with Re and length ratio resultsmore uniform core region temperature



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1389

distributions and the overall convectionheat transfer increases.

Streamlines and velocity vectorfor flows with Re=100,200,300 and height

ratio (H min/Hmax=0.46, 0.39, 0.33) areshown in figures (7, 8, 9) for diverged-converged duct. Lateral swirl is trigged atthe Re=100 and (H min/Hmax=0.46). Withincreasing Reynolds number anddecreasing height ratio led to increaselateral vortex. The temperature couturedistribution at Pr = 0.7 shown in figures(7, 8, 9), the increase of flow rate anddecrease of height ratio results more coreregion temperature distribution andincreasing of overall heat transfer.

Local Nusselt NumberFor wavy duct, Fig (10) shows

the variation of Nusselt number with axialdirection for Pr = 0.7, while fig (11) showsthe same relation but for Pr = 5.85. It isobvious from fig (10) that the maximum

Nusselt number takes place at the inlet forlength ratio equal 10. A decrease in thelength ratio causes a decrease in Nusseltnumber at the inlet. The maximum valueof the Nusselt number decreases withdecreasing the length ratio. It is clear fromfig (10) that the location of the minimumlocal Nusselt number remains almost atthe same place. It is clear from fig (11),that the maximum local Nusselt numbertakes place at about 25% from the inletlocation for all length ratios. A decrease inthe length ratio causes an increase in themaximum Nusselt number. Regarding theminimum local Nusselt number for Pr =5.85, fig (11) shows a slight decrease with

decreasing length ratio.For diverged-converged duct, Fig(12) shows the variation of Nusseltnumber with axial direction for Pr = 0.7,while Fig (13) shows the same relation butfor Pr = 5.85. It is obvious from figure(12) that the local Nusselt numberdecreases rapidly with axial direction and

becomes minimum approximately at thecenter of the duct. Thereafter, it increasesslightly with axial direction for any heightratio. A decrease in the height ratio causes

a slight shift of the curves downward.Figure (13) shows that the lowest Nusselt

number takes place when the Reynoldsnumber approaches 50. An increase inReynolds number to 300 leads that toincrease in the value of the minimum local

Nusselt number. This phenomenon occursfor all height ratios. However, a decreasein the height ratio causes the curves to becloser to each other especially for heightratio equal 0.33. Also for a given Reynoldsnumber, the minimum value of the local

Nusselt number increases with decreasingof the height ratio.

Shear StressFor wavy duct, figure (14) and

(15) show the variation of the shear stresswith axial direction for a Prandtl numberof 0.7 and 5.85 respectively. It is clearfrom these figures, that the effect ofdecrease in length ratio is quite

pronounced at the location of swirl nearthe axial direction at 70% from the inletlocation. At this location, the shear stressincreases with increasing Reynoldsnumber for any length ratio. However, thedecrease in length ratio causes asignificant increase in shear stress. It isworthwhile mentioning that the shearstress peak in fig (15) for Pr=5.85 in abouttwice that for Pr=0.7 shown in fig (14).

Figures (16) and (17) show thevariation of shear stress with axialdirection for a Prandtl number of 0.7 and5.85 respectively for diverged-convergedduct. It is obvious from fig (16) that the

behavior of shear stress versus axialdirection for Prandtl number of 0.7 is quitesimilar to that of fig (17) for Pr=5.85. It isobvious from fig (16) that for a given

height ratio, an increase in Reynoldsnumber causes the curves to be shiftedupward. It is interesting to note that for aReynolds number of 300, a decrease inheight ratio causes the curve for shearstress to have a small peak at the axialdirection at end of the swirl. However, themagnitude of shear stress for Pr=0 .7 isabout half that for Pr=5.85.

Average Nusselt NumberFigure (18) for wavy duct with

Pr=0.7 shows that the average Nusseltnumber increases with Reynolds number.



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1390

However, when the ratio (L/2a) decreasesfrom infinity to (10) causes the curve to beshifted upward resulting into an increasein the average Nusselt number. A further

decrease in (L/2a) ratio from 6.67 to 5causes a decrease in the average Nusseltnumber. Figure (19) for wavy duct withPr=5.85 is similar to figure (18) but therate of increase of the average Nusseltnumber with Reynolds number is muchlower than that for figure (18). However,the average Nusselt number in fig (19) forthe same (L/2a) ratio is much higher thanthat of figure (18).

Figure (20) for diverged-converged duct with Pr=0.7 and heightratio (H min/Hmax) equal 0.46, the average

Nusselt number increases with increasingReynolds number. A decrease of(Hmin/Hmax) ratio leads to a decrease inaverage Nusselt number. The average

Nusselt number at (H min/Hmax=1) becomesmaximum. Figure (21) for diverged-converged duct with Pr=5.85 and heightratio (H min/Hmax) equal 0.46, the average

Nusselt number increases with increasingReynolds number. A decrease in(Hmin/Hmax) ratio leads to a decrease in theaverage Nusselt number. The average

Nusselt number becomes minimum at(Hmin/Hmax=1).

Conclusion1. For air heat exchanger it can be

concluded that the wavy duct perform better than the diverged-convergedduct in term of average Nusseltnumber.

2. The water heat exchanger performance

was found to be approximately similarfor wavy and diverged-convergedducts.

3. The local Nusselt number, average Nusselt number and shear stressincrease with increasing of Reynoldsnumber.

4. the maximum average Nusselt numberfor diverged-converged ducts took

place at a height ratio (H min/Hmax=1) atPr=0.7, while the maximum took placeat height ratio (H min/Hmax=0.46) for

Pr=5.85.

5. For wavy duct at Pr=0.7 and Pr=5.85the maximum average Nusselt numbertakes place at length ratio (L/2a=10).

6. For diverged-converged and wavy

duct at Pr=0.7 and 5.85, the shearstress increases with increasing heightand length ratio.

7. It is clear from study that the transfer by diffusion are more significant inwater (Pr=5.85) than in air (Pr=0.7).

8. There is good agreement between theresults of this work with thecorresponding numerical results

presented by Bahaidarah [9]

Reference[1] S.D.Hwang, I.H.Jang and H.H.cho,

"Experimantal study on flowand local heat/mass transfercharacteristics inside corrugatedduct" , International Journal of heatand fluid flow 27(2006)21-32.

[2] T.Adachi and H.Uehara, " pessuredrop and heat transfer inspainwise periodic fluted channels :,

Numercal heat transfer, part A, 43;47-64,2002

[3] H.Blomerius and N.K.Mitra,"Numerical investigation ofconvective Heat transfer andpressure drop in wavy duct" ,

Numerical Heat transfer, Part aA,37;37-54, 2000

[4] K.stone and S.P.Vanka, "Numericalstudy of Developing Flow and heattransfer in a wavy passage", Journalof fluid Engineering, December 1999,Vol.121.

[5] M.Molla, A.Hossain and L.yao,

"Natural-convection flow along avertical complex wavy surface withuniform heat flux" , Journal of heattransfer, October 2007, Vol.129.

[6] N.Mouhamad, B.Khedidja andZ.Belkacem, "Numerical Analysis offlow and heat transfer in theentrance region of a wavy channelwith thermal Asymmetric walls" ,International Journal of appliedEngineering, research IssNo973-4562Volume 1 Number 2(2006) pp.273-

293.



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[7] Raj M.Manglik, Jiehai Zhang, ArunMuleg , " Low Reynolds numberforced convection in three-dimensional wavy-plate-fin compact

channels fin density effects", International Journal of heat transfer48(2005) 1439-1449.

[8] M.Z.Hossain and A.K.M.Sadrul Islam,"Numeral investigation of fluid flowand heat transfer characteristics insine, triangular, and Arc-shapedchannels", Thermal science: Vol,11(2007), No.1, pp.17-26.

[9] Haitham M.S.Bahaidarah, N.K.Anand,and H.C.Chen, "Numerical study ofheat and momentum transfer inchannels with wavy walls" ,

Numerical Heat Transfer, part A,47:417-439,2005.

[10] M.Gradeck, B.Hoaread andM.Lebouche "Local analysis of heattransfer inside corrugated channel" ,International Journal of heat andmass transfer 48(2005)1909-1915.

[11] Versteeg H.K., Masasekera W.,"An introduction to computationalfluid mechanics" , Longman GroupLtd, 1995.

[12] Patankar S.V., "Numerical Heattransfer and fluid Flow" ,Hemisphere Puplishing Corporation,1980.

[13] Joe F. Thompson, bharat K. soniand Nigel P weatherill, " Gridgeneration", New York, Washington,London, 1999.

Table (1) shows the comparison ofaverage Nusselt number on the wavy wall

with the numerical results of Bahaidarahet.al [9] at H min/Hmax=0.3 and L/2a=4.

Reference Re=25 Re=100 Re=400Bahaidarah 8.7739 9.1764 11.5474Presentwork

7.72 8.21 9.98

a

b

Figure (1) Geometric configurationsof a: wavy duct, b: diverged-converged duct



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0 . 2 0

0 . 3

0

0 . 3

6

0 . 0 2

0 . 1

1

0 . 4 2

0 . 0

7

0 . 2

0

0 . 4 9

0 . 1

1 0 .

0 7

0 . 5 7

0 .8 0

0 . 6

6

0 . 4 2

0 . 0

2

0 . 1

1

0 . 6

6

0 . 1 1

0.20

0 . 1

1 0

. 0 7

0 . 2 0

0 . 3 0

0 . 5

7

0 . 6

1

0 . 3

0

0 . 3 6

0 . 4

2

0 . 4 9

0 . 7

0

Re = 25Re = 25

Re =100Re =100

Figure (2) Comparison of the streamlines between the present work and that ofBahaidarah et.al [9] using Re = 25 and Re = 100, H in /H ax=0.3 and L/2a=4

Present results Bahaidarah et.al 1

Present results Bahaidarah et.al 1

Re =25

Re =100

Re =25

Re =100

Figure (3) Comparison of the temperature contours between the present workand that of Bahaidarah et.al [9] using Re = 25 and Re = 100, H min /H max =0.3 and



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Figure (4) Velocity vector, stream line and contour for Re 100, 200 and 300 with aPr=5.85, (L/2a=10), (L/H=4) and (W/H=2) at (Z=0.5W).

Re=100

Re=200

Re=300

Re=100

Re=200

Re=300

4 0

3 3

2 7

3 8

3 8

3 3

3 3

Re=100

4 0

3 8

3 3

2 7

3 8

3 3

2 7

3 3

Re=200

4 0

3 8

3 3

27

3 8

3 3

2 7

Re=300

Relative (Cm/Magnitude) = 35

X

Y

Z

X

Y

Z

X

Y

Z



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Figure (5) Velocity vector, stream line and contour for Re 100, 200 and 300 with aPr=5.85, (L/2a=6.67), (L/H=4) and (W/H=2) at (Z=0.5W).

Re=100

Re=200

Re=300

Re=100

Re=200

Re=300

3 8

4 0

3 8

3 5

2 7

3 5

2 7

3 5

Re=100

2 7

3 8

3 8

3 5 2 7

4 0

3 8

3 5

2 7

Re=200

3 5

3 8

2 7

3 5

3 8

2 7

4 0

3 5

Re=300


X

Y

Z

X

Y

Z

X

Y

Z



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Figure (6) Velocity vector, stream line and contour for Re 100, 200 and 300 with aPr=5.85, (L/2a=5), (L/H=4) and (W/H=2) at (Z=0.5W).

Re=100

Re=200

Re=300

Re=100

Re=200

Re=300

3 5

3 8

3 8

2 7

4 0

3 5

2 7

3 8

Re=100

2 7 3 8

3 8

35

2 7

3 8

3 5

2 7

4 0

Re=200

3 5

3 5

2 7

3 8

3 5

3 2

2 7

40

Re=300


X

Y

Z

X

Y

Z

X

Y

Z



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Figure (7) Velocity vector, stream line and contour for Re 100, 200 and 300 with aPr=0.7, (Hmin/Hmax=0.46), (L/Hmin=4) and (W/H=2) at (Z=0.5W).

Re = 50

Re = 100

Re = 300

Re =50

Re = 100

Re =300

2 8

2 4

2 4

3 2

2 8

3 2

3 8

Re = 50

2 8

2 4

2 8

3 2

3 8

Re = 100

2 8

2 4

28

3 2

3 8

2 4

3 2

3 8

Re = 300

Relative (Cm/Magnitude) = 3.5

X

Y

Z

X

Y

Z

X

Y

Z



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Re = 50

Re = 100

Re = 300

Re =50

Re =100

Re =300

3 8

3 2

2 8

2 4

3 2

2 8

2 4

Re = 50

3 2

2 8

2 4

3 2

2 8

2 4

3 8

3 8

Re = 100

2 4

3 2

3 8

3 2

2

8 3 8

2 4

2 8

Re = 300


X

Y

Z

X

Y

Z

X

Y

Z



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Re = 50

Re =100

Re =300

Re=100

Re=50

Re =300

3 2

3 2

2 4

2 8

2 4

3 8

2 8

Re = 50

3 2

3 2

2 4

2 8

2 4

28 3 8

Re = 100

3

2 3 2 2 4

2 82 8

2 4

3 2

3 8

Re = 300


X

Y

Z

X

Y

Z

X

Y

Z



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0.01 0.03 0.00 0.02 0.04

Axial Direc t ion (m)

5

15

25

0

10

20

30

L o c a

l N

u s s e

l t n u m

b e r

L/2a=10

Re=100

Re=200

Re=300

0.01 0.03

0.00 0.02 0.04 Axia l Di rec t i on (m)

10

30

0

20

L o c a

l N

u s s e

l t n

u m

b e r

L/2a=6.67

Re=100

Re=200

Re=300

0.01 0.03 0.00 0.02 0.04

Axial Direc t ion (m)

10.00

30.00

0.00

20.00

L o c a

l N

u s s e

l t n

u m

b e r

L/2a=5

Re=1 0 0

Re=2 0 0

Re=3 0 0

0.01 0.03 0.00 0.02 0 .04

Axial Direction (m)

30.00

50.00

20.00

40.00

60.00

L o c a

l N

u s s e

l t n u

m b

e r

L/2a=10

Re=100

Re=200

Re=300

0.01 0.03

0.00 0.02 0.04Axial Direction (m)

20.00

60.00

0.00

40.00

80.00

L o c a

l N

u s s e

l t n

u m

b e r

L/2a=6.67

Re=100

Re=200

Re=300

0.01 0.030.00 0.02 0.04

Axial Direction (m)

20.00

60.00

0.00

40.00

80.00

L o c a

l N

u s s e

l t n

u m

b e r

L/2 a=5

R e=1 0 0

R e=2 0 0

R e=3 0 0

Figure (11) Local Nusseltnumber distribution in wavyduct at Pr = 5.85 and length

ratio equal 10, 6.67 and 5.

Figure (10) Local Nusselt numberdistribution in wavy duct at

Pr = 0.7 and length ratio equal10, 6.67 and 5.



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0.01 0.03 0.00 0.02 0.04

Axial Direction (m)

5.00

15.00

0.00

10.00

20.00

L o c a

l N

u s s e

l t n

u m

b e r

H m i n / H m a x = 0 . 3 9

Re=50

Re=100

Re=300

0.01 0.03 0 .00 0.02 0.04

Axial Direction (m)

5.00

15.00

0.00

10.00

20.00

L o c a

l N

u s s e

l t n

u m

b e r

Hmin/Hmax=0.33

Re=50

Re=100

Re=300

0.01 0.03 0.00 0.02 0.04

Axial Direction (m)

20.00

60.00

0.00

40.00

80.00

L o c a

l N

u s s e

l t n

u m

b e r

Hm in/Hmax =0.33

Re = 5 0

Re = 100

Re = 300

0.01 0.03 0.00 0.02 0.04

Axial Direction (m)

20.00

60.00

0.00

40.00

80.00

L o c a

l N

u s s e

l t n

u m

b e r

Hmin/Hmax=0.46

Re=50

Re=100

Re=300

0.01 0.03 0.00 0.02 0.04

Axial Direction (m)

5.00

15.00

0.00

10.00

20.00

L o c a

l N

u s s e

l t N

u m

b e r Hmin /Hmax=0 .46

Re=50

Re=100

Re=300

0.01 0.03 0.00 0.02 0.04

Axial Direction (m)

20.00

60.00

0.00

40.00

80.00

L o c a

l N

u s s e

l t n

u m

b e r

Hmin/Hmax=0.39

Re=50

Re=100

Re=300

Figure (12) Local Nusselt numberdistribution in diverged-

converged duct at Pr = 0.7 andheight ratio equal 0.46, 0.39 and

0.33.

Figure 13: Local Nusseltnumber distribution in

diverged-converged duct at Pr =5.85 and height ratio equal 0.46,

0.39 and 0.33.



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0.01 0.03 0.00 0.02 0.04

Axia l d i rec t i on (m)

0.005

0.015

0.000

0.010

0.020

s h

e a r s

t r e s s

( N / m 2 )

L/2 a = 1 0

Re = 1 0 0

Re = 2 0 0

Re = 3 0 0

0.01 0.03 0.00 0.02 0.04 Axia l d i rec t i on (m)

0.005

0.015

0.000

0.010

0.020

s h

e a r s

t r e s s

( N / m 2 )

L/2 a = 6 .6 7

Re = 1 0 0

Re = 2 0 0

Re = 3 0 0

0.01 0.03 0.00 0.02 0.04


0.005

0.015

0.000

0.010

0.020

s h e

a r s

t r e s s

( N / m 2 )

L/2 a = 5

Re = 1 0 0

Re = 2 0 0

Re = 3 0 0

0.01 0.03 0.00 0.02 0.04

Axi al di rec t ion (m)

0.02

0.06

0.00

0.04

s h

e a r s

t r e s s

( N / m 2 )

L/2 a = 1 0

Re=100

Re=200

Re=300

0.01 0.03 0.00 0.02 0.04 Ax ial di rec t ion (m)

0.02

0.06

0.00

0.04

s h

e a r s

t r e s s

( N / m 2 )

L/2 a = 6 .6 7

Re = 1 0 0

Re = 2 0 0

Re = 3 0 0

0.01 0.03 0.00 0.02 0.04

Ax ial di rec t ion (m)

0.02

0.06

0.00

0.04

s h e

a r s

t r e s s

( N / m 2 )

L/2 a = 5

Re=100

Re=200

Re=300

Figure (14) Shear stress versusaxial direction for wavy duct at

Pr=0.7

Figure (15) Shear stress versusaxial direction for wavy duct at

Pr=5.85



18/19


0.01 0.03 0.00 0.02 0.04


0.002

0.006

0.000

0.004

s h

e a r s

t r e s s ( N

/ m 2 )

H m in /H m a x = 0 .4 6

Re = 5 0

Re = 1 0 0

Re = 3 0 0

0.01 0.03 0.00 0.02 0.04


0.002

0.006

0.000

0.004

s h

e a r s

t r e s s

( N / m 2 )

Hmin/Hmax=0.39

Re=50

Re=100

Re=300

0.01 0.03 0.00 0.02 0.04


0.002

0.006

0.000

0.004

s h

e a r s

t r e s s

( N / m 2 )

H m in /H m a x = 0 .3 3

Re = 5 0

Re=100

Re=300

0.01 0.03 0.00 0.02 0.04


0.005

0.015

0.000

0.010

s h

e a r s

t r e s s ( N

/ m 2 )

H m in /H m a x = 0 .4 6

Re = 5 0

Re = 1 0 0

Re = 3 0 0

0.01 0.03 0.00 0.02 0.04


0.005

0.015

0.000

0.010

s h

e a r s

t r e s s

( N / m 2 )

H m in /H m a x = 0 .3 3

Re = 5 0

Re = 1 0 0

Re = 3 0 0

0.01 0.03 0.00 0.02 0.04


0.005

0.015

0.000

0.010

s h

e a r s

t r e s s

( N / m 2 )

H m in /H m a x = 0 .3 9

Re = 5 0

Re=100

Re=300

Figure (17) Shear stress versusaxial direction for diverged-converged duct at Pr=5.85

Figure (16) Shear stress versusaxial direction for diverged-

converged duct at Pr=0.7



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100 300 200

Reynolds number

3

8

0

5

10

a v e r a g e

N u s s e

l t n

u m

b e r

Hmin/Hmax=1

Hmin/Hmax=0.46

Hmin/Hmax=0.39

Hmin/Hmax=0.33

100 300 200

Reynolds number

30

50

20

40

60

a v e r a g e

N u s s e

l t n

u m

b e r

Hmin /Hmax=1

Hmin/Hmax=0.46

Hmin/Hmax=0.39

Hmin/Hmax=0.33

Figure 20: Distributions of theaverage Nusselt number for

diverged-converged duct at Pr =

Figure 21: Distributions of theaverage Nusselt number for

diverged-converged duct at Pr =5.85

Figure (18) Distributions of theaverage Nusselt number forwav duct at Pr = 0.7

Figure (19) Distributions of theaverage Nusselt number forwav duct at Pr = 5.85

150 250 100 200 300 Reynolds number

6

10

4

8

12

a v e r a g e

N u s s e

l t n

u m

b e r

L/2a=infinity

L/2a=10

L/2a=6.67

L/2a=5

150 250 100 200 300

Reynolds number

30

50

20

40

60

a v e r a g e

N u s s e

l t n

u m

b e r

L /2a=in fin ity

L /2a=10

L /2a=6.67

L /2a=5

forced convection in wavy duct

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