International Journal of Engineering Research and Development (IJERD)

International Journal of Engineering Research and Development

e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com

Volume 8, Issue 12 (October 2013), PP. 06-18

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Numerical modelling of heat transfer in a channel

with transverse baffles

M.A. Louhibi1, N. Salhi

1, H. Bouali

1, S. Louhibi

2

1Laboratoire de Mécanique et Energétique, Faculté des Sciences, BP. 717 Boulevard Mohamed VI - 60000

Oujda, Maroc. 2Ecole National des sciences Appliquées, Boulevard Mohamed VI - 60000 Oujda, Maroc.

Abstract:- In this work, forced convect ion hea t t ransfer ins ide a channel o f rec tangular

sec tion, conta ining some rectangular baff le plates , i s numer ica lly analyzed. We have

developed a numerical model based on a fini te -vo lume method, and we have so lved the

coupling pressure -veloc ity by the SIMPLE algori thm [6] . We sho w the effec ts o f var ious

parameters o f the baffles, such as, baff le ’s height , loca tion and number on the i sotherms,

streamlines, temperatures distr ibut ions and local Nussel t number values. I t i s concluded

that : I ) the baffles location and he ight has a meaningful e ffect on i sotherms, s treamlines

and to ta l heat t ransfer through the channel . ( I I ) The heat t ransfer enhances wi t h

increas ing both baff le ’s he ight and number.

Keywords:- Forced convect ion, Baff les, Finite volume method, Simple, Quick, k-

I. INTRODUCTION In recent years, a large number o f experimenta l and numer ica l works were

per formed on turbulent forced convecti on in hea t exchangers wi th different type of

baff les [1 -4] . This interes t is due to the var ious industr ial app licat ions of this type of

configura tion such as cooling of nuclear po wer p lants and a ircraf t engine . . . e tc .

S.V Patankar and EM Sparrow [1] have app lied a numer ica l so lut ion procedure in order to

trea t the prob lem of f luid flo w and heat t ransfer in ful ly developed hea t exchangers.

These one was equipped by i so thermal p la te p laced transverse ly to the direc t ion of f low.

They found that the flow f ield i s charac ter ized by s trong recircula t ion zones caused by

sol id p la tes. They concluded tha t the Nussel t number depends strongly on the Reynolds

number, and i t is higher in the case o f ful ly developed then that o f laminar flo w regime.

Demar tini et a l [2] conducted numer ica l and experimenta l stud ies o f turbulent flo w

ins ide a rec tangular channel conta ining two rectangular baffles . The numerical result s

were in good agreement wi th those obtained by exper iment. They no ted that the baffles

play an impor tant ro le in the dynamic exchangers stud ied. Indeed, regions o f high

pressure 'recircula t ion regions ' a re formed near ly to chicanes .

Recent ly, Nasiruddin and Sidd iqui [3] stud ied numerical ly e ffec ts o f baffles on forced

convect ion flo w in a heat exchanger . The e ffects o f s ize and incl inat ion angle of

baff les were deta i led. They considered three different arrangements o f baffles. They

found that increas ing the size o f the ver t ica l baff le substantial ly improves the Nussel t

number . Ho wever , the p ressure loss is a lso important . For the case o f inc lined baff les ,

they found tha t the Nussel t number i s maximum for angles o f inclinat ion directed

downstream of the baff le , wi th a minimum of pressure loss .

More recently, Sa im et al [4] presented a numerical study of the dynamic behavior o f

turbulent a ir flo w in horizontal channel wi th t ransverse baff les. They adapted numerical

f ini te vo lume method based on the SIMPLE a lgor i thm and chose K- model for

trea tment o f turbulence. Resul t s ob ta ined for a case o f such type, at lo w Reynolds

number, were presented in te rms of veloci ty and tempera ture fields. They found the

existence of relat ive ly strong recircula t ion zones near the baff les. The eddy zones a re

responsib le o f local var iat ions in the Nusse lt s numbers along t he baffles and walls .

We kno w that the pr imary heat exchanger goal is to e ff icient ly transfer heat from

one fluid to ano ther separated, in most prac tical cases, by so lid wal l . To increase heat

t ransfer , several approaches have been proposed. We can ci te the spec i fic t rea tment of

sol id separat ion sur face (roughness, tube winding, vibra t ion, etc . ) . This transfer can a lso

be improved by creat ion of longi tudina l vor t ices in the channel . These edd ies are

Numerical modell ing of heat transfer in a channel with…

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produced by introducing one or more transverse barr ier s (b aff le plates) inside the

channel . The formation of these vor t ices downst ream of baffles causes recircula t ion zones

capable o f rapid and eff icient heat t ransfer between sol id walls and fluid f low. I t i s th is

approach tha t we wi l l fol lo w in this s tudy. Indee d, we are interested in this work on the

numerical modeling of dynamic and thermal behavior o f turbulent forced convection in

hor izontal channel where two wal ls are ra ised to a high tempera ture. This channel may

conta in one or severa l rectangular baff les.

A spec ial in teres t is given to the influence of di fferent parameters, such as he ight ,

number and baffle posit ions on heat t ransfer and f luid f lo w.

II. MATHEMATICAL FORMULATION The geo metry of the problem is sho wn schematical ly in Figure 1 . I t i s a rectangular

duc t wi th i sothermal hor izontal wal ls , crossed by a sta t ionary turbulent f low. The

physical proper t ies are considered to be constants.

Figure 1: The studied channel

At each point o f the f low the ve loci ty has components (u, v) in the x and y

direc t ions, and the temperature i s denoted T . The turbulence model ing i s handled by the

classical model (k - ) . k i s the turbulent kinet ic energy and the viscous d iss ipa tion of

turbulence.

The Transpor t equations (continuity, momentum, tempera ture, tu rbulent kinet ic

energy and diss ipa tion of turbulence) governing the sys tem, are wr it ten in the fo l lo wing

genera l form :

)1( )()(

S

y

y

x

x

y

v

x

u

Where ρ is the density o f the f luid pass ing through the channel and ɸ, F φ and S φ are given

by:

Equat ions S

Cont inui té 1 0 0

Quant i té de

mouvement selon x

u t

x

P

Quant i té de

mouvement selon y

v t

y

P

Energie to tale T

T

t

0

Energie cinét ique

turbulente

k

k

t

G .

Dissipa tion turbulente

t

kC

GC

)

(

2

1


8

With :

222

22y

u

x

v

y

v

x

uG t µ and µ t represent , respec tively, the

dynamic and turbulent viscosit ies.

2

.k

Ct

The constants used in the turbulence model (k -Ɛ) Are those adopted by Chieng and

Launder (1980) [8] :

C 1C 2C T k

0 ,09 1,44 1,92 0,9 1 1 ,3

Boundary condit ions

At the channel inle t :

inUu ; 0v and inTT

2/32 1,0005,0 kandUk in

At sol id wal ls :

0 vu ; 0k inw TTTand

At the channel exit s t he gradient o f any quanti ty, wi th respect to the longitud inal

direc t ion x i s nul .

Our goal i s to determine veloci ty and temperature f ie lds, as wel l as the turbulence

parameters. Par t icular a t tention i s given to the quanti f icat ion parameters re flect ing hea t

exchanges such as the Nussel t number ( local and average) .

III. NUMERICAL FORMULATION The computer code tha t we have developed is based on the f ini te volume

method. Computat ional domain i s divided into a number o f st i tches. To choose the number

of ce l l s used in this s tud y, we per formed several simula tions on a channel (wi th or

wi thout baff les) . F ina lly , we have op ted fo r a 210 × 90 meshes.

The mesh dimensions are var iab les; they t ighten at the sol id walls neighborhoods .

Consequently, the s t i tch density i s higher near the hot wal ls and baffles .

We consider a mesh having d imensions ∆x and ∆y. In the middle o f each volume

we consider the points P , cal led centers o f cont rol volumes. E, W, N, S are the centers o f

the adjacent control volumes. We also consider centers, EE; WW, NN, SS. The faces o f

each contro l volume are denoted e, w, n, s .

Figure 2: Mesh with P at center

By integrat ing the transport equation (1) o f ɸ on the control volume , we find a

rela t ion to the x d irec t ion:

P

we

weS

dx

d

dx

duu


9

Where P

S is the source term.

One no te uF , the convect ion flo w and xD / , the d i ffusion coeff ic ient . And

wi th taking :

wwee uFuF ;

wwee xDxD /;/

And : WPwPEe dd ;

We found :

PWPwPEewwee SDDFF )()(

To est imate ɸ on the faces " e " and " w " we opted for the c lass ica l quick scheme

[5] which is a quadrat ic forms using three nodes . The choice o f these nodes i s dictated by

the di rec t ion of flo w on these faces )00( uoru .

Finally, the transport equation (1) i s d iscre t ized on the mesh wi th P a t center as :

)2(PEEEEWWWWEEWWPP Saaaaa

With :

PEEEWWWP

eeEE

wweeeeeE

wwWW

wweewwwW

Saaaaa

Fa

FFFDa

Fa

FFFDa

)1(8

1

)1(8

1)1(

8

6

8

38

1

)1(8

3

8

1

8

6

Where : 00,01 eee FouFsiou

00,01 www FouFsiou

The same thing is done for the ver t ica l "y" d irect ion, using the faces "n" and "s"

and by introducing the Quick diagram nodes N, NN, S and SS.

i f convect ion f low and di ffus ion coeff icient o f the s ides "e" , "w" , "s" , "n" are kno wn

And espec ia l ly the source term P

S , the solution of equat ion (2) then gives us the va lues

of ɸ in di fferent P nodes .

One no tes tha t to accelerate the convergence of equat ion (2) we have introduced a

relaxat ion fac tor :

)3(1 0

PPPEEEEWWWWEEWWPP aSaaaa

a

Where 0

P is the value of P in the previous s tep.

The source te rm appears especial ly in the conservat ion of momentum equations in

the form of a pressure gradient which i s in pr incip le unknown . To ge t around this

coupling we have chosen to use in our code the "Simple" algor i thm developed by

Pantankar [7] . The basic idea of this algori thm i s to assume a fie ld o f ini t ia l pressure and

inject i t into the equat ions o f conservat ion of momentum. Then we solve the sys tem to find

a f ield o f intermediate speed (which i s not fa ir because the pressure i sn ’ t . ) The continuity

equat ion i s t ransformed into a pressure correct ion equat ion. This las t i s determined to f ind

a pressure correct ion tha t wi l l injec t a new pressure in the equations o f mot ion. The

cycle i s repeated as many t imes as necessary unt i l a pressure correction equal "zero"

corresponding to the algor i thm convergence. In the end we so lve the t ranspor t equations

of T , k and Ɛ.

In this approach, a problem is encountered. I t is known as the checkerboard

problem.

The r i sk i s tha t a pressure f ie ld can be highly d is turbed by the sensed formul a tio n

which comprises per forming a l inear interpo lat ion for es t imate the pressure value on the

face ts o f the contro l vo lume. To circumvent this problem we use the so -cal led staggered

gr ids proposed by Har low and Welch [6] . In this technique, a f ir st gr id p ressure (and other


10

sca lar quant i t ies T , k and Ɛ ) is placed in the center o f the control volume. Whi le o ther

staggered gr ids are adopted for the veloc ity components u and v (see Figure 3) .

F igure 3 : Shi f ted mesh

The sca lar var iab les, including pressure, are s to red at the nodes ( I , J ) . Each node

(I , J ) is surrounded by nodes E, W, S and N. The hor izontal veloc ity component u i s stored

on faces "e" and "w", whi le the ver t ica l component v i s s tored on the faces denoted "n"

and "s" .

So the control vo lume for pressur e and o ther scalar quant i t ies T , k and Ɛ is

),();1,();1,1();,1( jijijiji

For component u cont rol vo lume is

),1();1,1();1,();,( jIjIjIjI .

Whi le for the v component we use : )1,1();,();,1();1,1( JiJiJiJi .

By integrat ing the conservat ion momentum equation in the hor izonta l direct ion on the

volume ),1();1,1();1,();,( jIjIjIjI , we found :

)4( )(),(),1( 1,, jjnbnbJiJi yyJIPJIPuaua

Where :NNNNNNSSSSSSEEEEEEWWWWWWnbnb uauauauauauauauaua

The coeff ic ients anb are determined by the quick scheme ment ioned above.

Simi lar ly, the integra t ion of conserva tion momentum equat ion in the ver t ica l

direc t ion on the volume

)1,1();,();,1();1,1( JiJiJiJi ,

gives us : (5) )x(xJ)P(I,1)JP(I,va va i1inbnbj,Ij,I

One considers pr imari ly an ini t ial pressure f ie ld P*. The provisional so lut ion of the

equat ions (4) and (5) wi l l be denoted u* and v

*. We note tha t u

* and v

* doesn’t checks the

continuity equat ion, and:

)6()(),(),1( 1

***,

*

, ayyJIPJIPuaua jjnbnbJiJi

)6()(),()1,( 1

***,

*

, bxxJIPJIPvava iinbnbjIjI

At this stage any one of the three var iables i s correct . They require correction :

)7(

'

'

'

*

*

*

vvv

uuu

PPp

Injec t ing (7) in equations (4) , (5) an d (6) we find :

)8()(),('),1(''' 1,, ayyJIPJIPuaua jjnbnbJiJi

)8()(),(')1,(''' 1,, bxxJIPJIPvava iinbnbjIjI


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At this leve l , an approximation is in troduced. In order to l inear ize the

equat ions (8) , the terms nbnbua ' and nbnbva ' are s imply neglected .

Normally these terms must cance l a t the procedure convergence. That i s to say that

th is omission does not affec t the final result . However , the convergence rate i s changed

by this s impl i fica t ion. I t turns out tha t the cor re ct ion P ' i s overest imated by the S imple

algori thm and the calculat ion tends to diverge . The remedy to s tab il ize the calcula t ions i s

to use a relaxat ion factor .

We Note tha t a fur ther treatment o f these te rms i s proposed in the so -cal led

algori thms "SIMPLER" and "SIMPLEC" [7] .

The equations (8) become:

)9(),('),1('' , aJIPJIPdu JiJi

)9(),(')1,('' , bJIPJIPdv jIjI

With

Ji

jj

Jia

yyd

1 and

Ij

ii

jIa

xxd

1

Equat ions (9) give the correc tions to app ly on veloc it ies through the formulas (7) . We

have therefore :

)10(),('),1(',*

, aJIPJIPduu JiJiJi

)10(),(')1,(',*

, bJIPJIPdvv jIjIjI

Now the d iscre t ized cont inui ty equat ion on the contro l vo lume of scalar quant i t ies i s

wr i t ten as fol lows :

0)()()()( ,1,,,1 jIjIJiJi vAvAuAuA

Where an are the s ize o f the corresponding faces .

The introduction of the veloci ty correct ion equat ions (10) in the continuity

equat ion gives a final equation a l lo wing us to determine the scope of pressure correct ions

P:

)11()1,('')1,(''),1(''),1(''),('' 1111 JIPaJIPaJIPaJIPaJIPa JIJIJIJIJI

The a lgori thm can be summar ized as fol lo ws : one s tar t s f rom an ini t ial

f ield**** ,, andvuP , wi th represents the sca lar quant i t ies T, k and Ɛ. Then the sys tem (6)

is so lved to have new values o f ** vandu .

Then the sys tem (11) i s solved for the correc tions f ield pressure P '.

Thereaf ter , pressure and ve loc ity are cor rec ted by equat ions (7) and (10) to have P , u and

v.

We so lve the fo l lo wing transport equation of scalar quant i t ies (2) , ɸ = T, K and Ɛ .

Finally, we consider :

**** ;;; vvuuPP .

And the cyc le i s repeated unti l convergence.

IV. RESULTS AND DISCUSSIONS The dimensions o f the channel presented in this work a re based on exper imenta l

data publ ished by Demartini et a l [2] . The air flo w is carr ied out under the fol lowing

condit ions.

Channel length: 𝐿 = 0 .554 m ;

Channel diameter D = 0.146 𝑚 ;

baffle he ight : 0 <ℎ <0 1. 𝑚 ;

baff le thickness: δ = 0 .01 𝑚 ;

Reynolds number : 𝑅𝑒 = 8 .73 104

;

The hydrodynamic and thermal boundary condit ions are given by

At the channel inle t

inuu =7,8m/s;

0v ; inTT =300 K


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On the channel walls

0 vu and KTT w 373

At the channel exit the sys tem is assumed to be ful ly developed, ie

0

x

Tx

vx

u

Fir s t , we compared the structure o f s treamlines and i sotherms in a channel wi thout

baff le wi th tho se ob tained when we introduce baff le having a he ight h = 0 .05

m (Case 2 -1) at the absc ise 𝐿1 = 0 .218 𝑚 . Indeed , in figures 4 and 5 we present s treamlines

and i sotherms for two configurat ions.

These result s clear ly sho w the importance of presence of baffle (act ing as a

cooling f in) . Indeed the baff le increases hea t t ransfer between the wal l an d the fluid . This

increase i s caused by recircula t ion zones downs tream of the baff le .

(a)

(b)

Figure 4: s t reamlines: (a ) case 1 ; (b) case 2 -1

(a)

(b)

Figure 5: Iso therms : (a) case 1 ; (b) case 2 -1

We then s tudied the inf luence of baff le he igh t on flo w s truc ture and heat

t ransfer . We chose the heights (h = 0.05 m; cases 2 -1) h = 0 , 073 𝑚 (Case 2 -2) and h

= 0 , 1 𝑚 (case 2 -3) .

The figures 6 and 7 present s treamlines and i so therms for a channel containing baffle wi th

di fferent he ights .

(a)


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(b)

Figure 6: streamlines : (a ) case 2 -2 ; (b) case 2 -3

(a)

(b)

Figure 7: Iso therms: (a) case 2 -2; (b) case 2 -3

I t i s c lear that the i sotherms are more condensed near baff le , as and when the

he ight h increases. This indicates an increase of heat t ransfer near baff le . Indeed, the

increase in h expanded exchange area between f luid and wal ls . In addit ion, when h

increases , the rec irculat ion zone becomes increas ingly impor tant ( see Figures 4 (b) and

f igures 6 (a) and (b)) . This causes an accelera t ion of flo w, which improves hea t t ransfer

wi thin the channel .

We present figures 8 and 9 in order t o identi fy the inf luence of h on ve loci t ies and

temperature prof i les along y at x = 0.45 from channel inle t . Four values of h are

considered.

Figure 8: Hor izontal velocity Profi les along y at x = 0.45 « inf luence of h »


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Figure 9: Temperature p rof i le s along y at x = 0.45 m « inf luence of h »

These resul ts a l lo w us to conclude that the increase o f baff le he ight has two

contrad ic tory effects. I t is t rue that there i s substantial increase in heat exchange

(appearance of rec irculat ion zones most common), but there i s a lso a loss o f pressure

( f low b lockage) .

In the purpose of measur ing the influence of 'h ' on loca l hea t t ransfer , we have

presented in figure 10, the local Nusse lt number along the channel for the four cases o f h

considered. I t sho ws tha t ups tream of the baff le (x <0.2 m) curves are confused , whi le ,

jus t a f ter baff le location, e ffect o f baff le height on Nussel t number become increasingly

important away from the baffles.

Figure 10 : Local number Nu along the channel « inf luence of h »

We then s tudied the inf luence of baff les number and posi t ion 'F igure 11 and 12 '.

(a)

(b)


15

(c)

Figure 11 : S treamlines : (a) cas 3 -1; (b) cas 3 -2; (c) cas 3 -3

We consider two baff les of same he ight h = 0.08 m. The f ir st baff le i s placed a t the

dis tance 𝐿 1 = 0 .15 𝑚 , whi le the second i s posi t ioned at the distance d 1 = 0 .05 m from the

f irs t (case 3 -1 ) , d 1 = 0 .10 m (case 3 -2) and d 1 = 0 .15 m (case 3 -3) . We no te the

existence of two recircula t ion zones downst ream of the fir st baff le . Also the f ir st

recircula t ion zone 'defined by baffles ' becomes increas ingly important , which contr ibute

to an increase in heat t ransfer in this area, as sho wn in Figure 12. Indeed, when the

baff les are c lose to each other 'd 1 smal l ' , the fluid is blocked in the chimney de l imi ted by

x = L1 and x = L1 + d 1 . This reduces the flo w speed a t that loca tion and thus there wi l l be

a decrease in the heat t ransfer in this area. Or when d 1 increases, the f luid has sufficient

space to move rapid ly, hence heat t ransfer increases in this zone 'see f igure15 ' .

(a)

(b)

(c)

Figure 12 : Iso therms (a) : « cas 3 -1 » ; (b) : « cas 3 -2 » ; (c) « cas 3 -3 »

In the fo l lo wing f igures 13 and 14, we compare the prof i le o f hor izontal veloci ty

and tempera ture d istr ibut ion. Calcula t ion i s done on t he y-axis a t x = 0 .45 m from the

channel inle t for three di fferent baffles spacing. We find that more spacing i s impor tant

more hea t exchange i s impor tant in areas l imited by two baff les.


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Figure 13 :Prof i les o f the horizonta l ve loc ity a t x = 0.45 m « Influence of the spacing

between two baffles »

Figure 14: Prof i les o f the total tempera ture at x = 0.45 m « Inf luence of the spacing

between two baffles »

Figure 15 present the local Nussel t number a long the channel fo r three considered

spac ing d 1 = 0 .05m, 0 .1m and 0.15m.

Figure 15 :Local Nusse l t number along the channel « Influence of baff les number and

posi t ion »


17

We dist ingue two areas :

The f ir st defined by 0 <x <0.3m whi le the second by x> 0.3m. For the f i rst region,

an increase in number of baffles improves the heat t ransfer ; whereas the reverse is t rue

for the second zone.

We chooses to show the inf luence of baffles number on heat t ransfer along the

channel for three cases: Channel wi thout baffles, channel conta ining a single baff l e and

channel containing two baff les 'F igure 16 ' .

I t should be no ted that genera l ly, heat t ransfer is propor t ional to baff les number.

Indeed, the Nussel t number charac ter izing hea t t ransfer wi thin the channel increases wi th

increas ing baff les number.

Figure 16: Local Nusse lt number a long the channel

V. CONCLUSION The thermal behavior of a s tat ionary turbulent forced convect ion flow wi thin a

baff led channel was ana lyzed. The resul t s sho w the ab il i ty o f our code to predic t dynamic

and thermal f ie lds in var ious geometr ic si tua tions. We s tudied mainly the inf luence of

baff les height and spacing on heat t ransfer and f luid f low. One can conclude tha t:

1 - Increase in the baff le he ight improves heat t ransfer between channel wal ls and

f luid pass ing through i t ;

2 - Heat t ransfer becomes increasingly impor tant wi th adding baffles .

3 - The spac ing d 1 be tween baffles has di fferent effects on local heat t ransfer . Any

t ime d 1 has not a lo t o f inf luence on the overal l heat t ransfer in the channel .

In perspec tive, we in tend deepen and c lar i fy our resul ts . Indeed, we wi l l adap t our code to

others geometr ic cases (non -rectangular baffles or baff les inc lined) . F inally, we wi l l a l so

try to re fine more the turbulence model .

REFERENCES [1] . S.V.PATANKAR, E.M.SPARROW, « Fully deve loped flo w and heat t ransfer in

ducts having s tream wise -per iod ic var iat ions o f cross -sect ional a rea », Journa l o f

Heat Transfer , Vol. 99 , p (180 -6) , 1977.

[2] . L.C.DEMARTNI, H.A.VIELMO and S.V.MOLLER, « Numer ica l and exper imenta l

analys is o f the turbulent f low through a channel wi th baff le plates », J . o f the

Braz. Soc. o f Mech. Sc i . & Eng. , Vol. XXVI, No. 2 , p (153 -159) , 2004 .

[3] . R.SAIM, S.ABBOUDI, B.BENYOUCEF, A.AZZI, « Simula tion numérique de la

convect ion forcée turbulente dans les échangeurs de chaleur à fa is ceau et ca landre

munis des chicanes transversa les », Algérian journa l o f appl ied f luid mechanics /

vol 2 / 2007 (ISSN 1718 – 5130) .


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[4] . M. H. NASIRUDDIN, K. SEDDIQUI « Heat transfer t augmentat ion in a heat

exchanger tube us ing a baff le», Interna tional journa l o f Heat and Fluid Flow, 28

(2007) , 318 -328.

[5] . H.K. Vers teeg ; W. Malalasekera « An int roduction to computat ional fluid

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