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

Numerical modelling of heat transfer in a channel
with transverse baffles
M.A. Louhibi 1, N. Salhi1, H. Bouali 1 , S. Louhibi 2
1

Laboratoire de Mécanique et Energétique, Faculté des Sciences, BP. 717 Boulevard Mohamed VI - 60000
Oujda, Maroc.
2
Ecole National des sciences Appliquées, Boulevard Mohamed VI - 60000 Oujda, Maroc.

Abstract:- In t hi s wo r k, fo r ced co n ve ct io n h ea t tra ns fer i n s id e a c h a n ne l o f rec ta n g u lar
sec tio n , co n ta i ni n g so me r e cta n g u lar b a ffle p lat es , i s n u mer ica ll y an al yz ed . W e ha v e
d ev elo p ed a n u me r i cal mo d e l b a sed o n a fi n ite - vo l u me me t ho d , a nd we ha v e so l ved t h e
co up li n g p r e s s ur e - v elo c it y b y t he SI MP LE al go rit h m [6 ]. W e s ho w t he effec ts o f vario u s
p ara me ter s o f t he b a f fl e s, s u c h as, b a ffle ’s hei g ht , lo ca tio n a nd n u mb er o n t he i so t h er ms ,
str ea ml i ne s, t e mp er at ur es d i str ib ut io n s a nd lo c al N u ss el t n u mb er v al u es. It i s co nc l ud ed
th at : I ) t he b a f fl es lo c atio n a nd he i g ht h as a m ea ni n g fu l e ffect o n i so t her ms , s tre a ml i ne s
and to ta l he at tr a n s f e r t hr o u g h t he c ha n n el . ( I I ) T he h eat tr a n s fer e n ha n ce s wi t h
in cre as i n g b o t h b a f f le ’ s he i g ht a nd n u mb e r.
K ey w o rd s: - Fo r c ed co n ve ct io n, B a f f le s, Fi n it e vo l u me me t ho d , S i mp l e, Q ui c k, k -

I.

INTRODUCTION

In r ec e nt ye ar s, a lar ge n u mb er o f e xp eri m en ta l a nd n u mer ica l wo rk s wer e
p er fo r med o n t ur b u le nt fo r c ed co n vec ti o n i n hea t e xc ha n g er s wi t h d iffere n t t yp e o f
b affle s [1 -4 ] . T h is i n ter es t is d ue to t h e v ario u s i nd u str ial ap p li cat io n s o f t h i s t yp e o f
co n fi g ura tio n s u c h a s co o li n g o f n u cl ear p o wer p la nt s a nd a ircra ft e n g i ne ... e tc.
S.V P a ta n kar a nd E M S p ar r o w [ 1 ] ha ve ap p li ed a n u mer ica l so l ut io n p r o ced ure i n o rd e r to
trea t t he p r o b le m o f f l uid flo w a nd he at tr a n s fer i n ful l y d e v elo p ed h ea t e x c ha n ger s.
T hes e o ne wa s eq u ip p ed b y i so t her ma l p la te p la ced tra n s ver se l y to t h e d irec tio n o f flo w.
T he y fo u nd t hat t he flo w f i eld i s c har ac ter ized b y s tro n g re cir c ula tio n zo ne s ca u s ed b y
so l id p la te s. T h e y co nc lud ed t ha t t he N u ss el t n u mb e r d ep e nd s stro n g l y o n t h e Re yn o ld s
n u mb e r, a nd it is hi g he r in t he ca se o f ful l y d e v e lo p ed t h e n t h at o f l a mi n ar flo w re g i me.
De mar ti n i et a l [ 2 ] co nd uc ted n u mer ica l a nd e xp eri me n ta l st ud ie s o f t ur b ul e nt flo w
in s id e a r ec ta n g ul ar c h an n el co n ta i ni n g t wo r ecta n g u lar b a ffl es . T he n u me ric al r es u lt s
we re i n go o d a gr ee me n t wi t h t ho se o b t ai ned b y e xp er i me n t. T h e y no te d th at t h e b a ffl es
p la y a n i mp o r ta n t r o le i n t h e d yn a mic e xc h an g er s st ud ied . I nd eed , re gio n s o f hi g h
p res s ure 're cir c ula tio n r eg io n s ' a r e fo r me d nea rl y to c h ica n es .
Rec e nt l y, Na sir ud d i n a nd Sid d iq ui [3 ] st ud ied n u me ri cal l y e ffec t s o f b a ffl es o n fo rc ed
co n v ect io n flo w i n a he at e xc h a n ger. T h e e ffe ct s o f s ize a nd i n cl in at io n a n gl e o f
b affle s wer e d eta il ed . T he y co ns id er ed t hre e d iffere n t arra n g e me n t s o f b a ffl e s. T he y
fo u nd t h at i n cr ea s i n g t he si ze o f t he ve rt ica l b affle s ub sta n ti al l y i mp ro ve s t h e N u s sel t
n u mb e r. Ho we ver , t h e p r es s ur e lo s s is a lso i m p o rta nt . Fo r t he ca se o f i nc li n ed b a ffle s ,
th e y fo u nd t ha t t he N u s sel t n u mb er i s ma x i mu m fo r a n g le s o f i n cli n at io n d ire cted
d o wn s tre a m o f t he b a f f l e, wi t h a mi n i mu m o f p r es s ure lo ss .
Mo re rece n tl y, Sa i m et al [ 4 ] p r ese n ted a n u me ric al st ud y o f t he d y na mi c b e h a vio r o f
tu rb ule n t a ir flo w i n ho r izo nt al c h a n ne l wi t h t r an s ve rs e b a ffle s. T he y a d ap ted n u me ric al
fi ni te vo l u me me t ho d b ased o n t he SIMP LE a l go r it h m a nd c ho se
K- mo d el fo r
trea t me n t o f t ur b u le nc e. Re s ul t s o b ta i ned fo r a ca se o f s uc h t yp e, at lo w R e yn o ld s
n u mb e r, wer e p r es e nte d in te r ms o f velo ci t y and te mp era t ure fi eld s. T he y fo u nd t he
ex i ste n ce o f r el at i ve l y str o n g r e cir c ula tio n zo ne s nea r t he b a ffle s. T he ed d y zo n es a re
resp o n sib le o f lo cal v ar i atio n s i n t h e N u s se lt s n u mb e rs alo n g t he b a ffl e s and wa ll s.
W e k no w t h at t h e p r i ma r y he at e x c ha n ge r go al is to e ffici e nt l y tr a ns fer he at fro m
o ne fl u id to a no t her s e p ar at ed , i n mo st p rac ti c al ca se s, b y so lid wa l l. T o in cre as e he at
tra n s fer, se ver al ap p r o a ch es ha v e b ee n p ro p o s e d . W e ca n cit e t h e sp ec i fic t rea t me n t o f
so l id sep ar at io n s u r fac e ( r o u g h n es s, t ub e wi nd i n g, v ib ra tio n, etc .). T h i s tr a ns fer c a n a lso
b e i mp ro ved b y cr eat i o n o f lo n gi t ud i na l vo r tic es i n t h e c ha n ne l. T hes e ed d ie s are

6

Numerical modelling of heat transfer in a channel with …
p ro d uc ed b y i ntr o d uc i n g o ne o r mo re tra n s ver s e b arr ier s (b a ffle p lat es) i n sid e t h e
ch a n nel . T he fo r ma tio n o f t h es e vo r ti ce s d o wn s t rea m o f b a ffl es c a u se s recir c ula tio n zo n es
cap ab l e o f r ap id a nd e f f ici e nt he at tr a ns fer b et we e n so l id wa ll s a nd fl uid flo w. I t i s t h i s
ap p ro ac h t ha t we wi ll f o llo w i n t h is s t ud y. I nd ee d , we are i nt ere st ed i n t hi s wo r k o n t h e
n u me ric al mo d e li n g o f d yna mi c a nd t her ma l b eh a vio r o f t urb u le n t fo r ced co n vec tio n i n
ho r izo nt al c ha n n el wh e r e t wo wal l s ar e ra i sed to a hi g h te mp era t ure. T hi s c ha n n el ma y
co n ta i n o ne o r s e ver a l r ecta n g u lar b a ffle s.
A sp ec ial i n ter e s t is gi v e n to t h e i n fl ue n ce o f d i ffere n t p ar a me ter s, s uc h a s he i g ht ,
n u mb e r a nd b a f fl e p o sit i o n s o n he at tra n s fer a nd fl uid flo w.

II.

MATHEMATICAL FORMULATION

T he geo met r y o f t h e p r o b le m i s s ho wn sc h e ma ti cal l y i n Fi g u re 1 . It i s a rect a n g u lar
d uc t wi t h i so t h er ma l ho r izo nt al wa l ls , cro ss ed b y a sta tio n ar y t ur b ul e nt flo w. T h e
p h ys i cal p ro p er tie s ar e c o n sid er ed to b e co ns ta n t s.

Figure 1: The studied channel
At e ac h p o i n t o f t h e f lo w t h e ve lo ci t y ha s c o mp o ne n t s ( u, v) i n t h e x a nd y
d ir ec tio n s, a nd t he te m p er at ur e i s d e no ted T . T he t urb u le nc e mo d el i n g i s ha nd led b y t he
cla s sic al mo d e l ( k - ) . k i s t he t ur b u le nt ki n et ic e ner g y a nd  t he v is c o u s d i ss ip a tio n o f
tu rb ule n ce.
T he T r an sp o r t eq ua tio n s ( co n ti n u it y, mo me n t u m, te mp era t ure, t u rb ul en t ki n et ic
en er g y a nd d i ss ip a tio n o f t ur b u le nc e) go v er ni n g t he s ys te m, are wr it te n i n t h e fo llo wi n g
ge n era l fo r m :
      
 
 

 ( u )  ( v )
x 
   y   S

 

x
y
x
y

(1)

W her e ρ i s t he d e n s it y o f t he f l uid p as s i n g t hro u g h t he c h a n nel a nd ɸ, F φ an d S φ ar e gi v e n
b y:
Eq u at io ns





S

Co nt i n ui té
Q ua nt it é d e
mo u ve me nt s elo n x

1
u

0

0

Q ua nt it é d e
mo u ve me nt s elo n y

v

  t

En er gie to tal e

T



En er gie ci n ét iq ue
tu r b ule n te

k

Di s sip a tio n t ur b u le nt e



  t

t
T

 t
k

 t

7

P
x
P

y


0

 .  G
(C1 G 
C 2  )


k

2
2
  u  2
 v 
 v u  


W it h : G   t 2   2   
   µ a nd µ t
  x 
 y 
 x y  



rep re se n t, re sp ec ti v el y, t he

d yna mi c a nd t u r b ule n t v is co s it ie s.

 t   .C 

k2



T he co n st a nt s u sed i n t h e t ur b u le nc e mo d e l ( k -Ɛ ) Are t ho se ad o p ted b y C hi e n g a nd
La u nd er (1 9 8 0 ) [ 8 ] :

C

C1

C2

T

k



0 ,0 9

1 ,4 4

1 ,9 2

0 ,9

1

1 ,3

B o un da ry co n d it io n s
At t he c h a n ne l i n le t :
u  U in ; v  0 a n d T  Tin
2
k  0,005 U in and   0,1 k 3 / 2

At so l id wal l s :
u  v  0 ; k    0 and T  Tw  Tin
At t h e c ha n n el e x it s t he gr ad ie n t o f a n y q u an ti t y, wi t h r esp ect to th e lo n g it ud i nal
d irec tio n x i s n ul .
O ur go al i s to d ete r mi n e v elo ci t y a nd t e mp er at ure fie ld s, a s we l l a s t h e t urb ul e nce
p ara me ter s. P ar tic u lar a tte n tio n i s gi ve n to t h e q ua n ti fic ati o n p ara me t ers r e fle ct i n g hea t
ex c ha n ge s s uc h a s t h e N u s sel t n u mb er (lo cal a nd av era g e).

III.

NUMERICAL FORMULATION

T he co mp ut er co d e t ha t we ha v e d e velo p ed is b a sed o n t he fi ni te vo l u me
me t ho d . Co mp ut at io nal d o ma i n i s d i v id ed i n to a n u mb er o f st it c he s. T o cho o se t he n u mb er
o f ce ll s us ed i n t hi s s tud y, we p er fo r me d se ver al si mu l a tio n s o n a cha n n el ( wi t h o r
wi t h o ut b a ff le s) . F i na ll y , we h a ve o p ted fo r a 2 1 0 × 9 0 me s he s.
T he me s h d i me n s io ns ar e var iab le s; t he y t i g hte n at t he so l id wa ll s nei g hb o r ho o d s .
Co n seq ue n tl y, t he s ti tc h d e ns it y i s hi g h er ne ar t he h o t wal l s a nd b a ffl es .
W e co ns id er a me s h ha vi n g d i me n s io ns ∆ x a nd ∆ y. I n t he mi d d l e o f e ac h vo l u me
we co n s id er t he p o i n t s P , cal led c e nte rs o f co nt ro l vo l u me s . E, W , N, S are t h e c e nter s o f
th e ad j ace n t co n tr o l vo l u me s. W e al so co n sid er cen ter s, E E; W W , N N, SS. T h e fac es o f
eac h co n tro l vo l u me ar e d e no ted e, w, n, s.

Figure 2: Mesh with P at center
B y i nt e gr at i n g t he tr a n sp o r t eq ua tio n (1 ) o f ɸ o n t he co n tro l vo l u me , we fi nd a
rela tio n to t he x d ir ec tio n:

 u  e   u  w   d 



 d 
    S P
 dx  e  dx  w
8

W her e

S 

 P

is t he so u r ce ter m.

O ne no te F   u , t he co n ve ct io n flo w a nd
wi t h ta k i n g :

D   / x , t he d i ffu sio n co e ffic ie n t. And

Fe   u e ; Fw   u w

De   / x e ; Dw   / xw

 e   E

An d : d

 P

W e fo u nd :

;

d w   P  W

Fe e  Fw w  De (E  P )  Dw (P  W )  S P

T o esti mat e ɸ on t he fa ces " e " a nd " w " we o p t ed fo r t he c la s s ica l q ui c k s c he me
[5 ] wh ic h is a q u ad r at ic fo r ms u si n g t hr ee no d e s . T he c ho ic e o f t he se no d es i s d i ct ated b y
th e d i rec tio n o f flo w o n th e se fa ce s (u  0 or u  0) .
Fi n all y, t h e tr a nsp o r t eq ua tio n ( 1 ) i s d i scre ti zed o n t he me s h wi t h P a t c e nt er a s :

aP P  aW W  aE E  aWW WW  aEE EE  S P

( 2)

W it h :
6
1
3

 aW  D w  8  w Fw  8  e Fe  8 (1   w ) Fw

1
W her e :

aWW    w Fw
8


a E  De  3  e Fe  6 (1   e ) Fe  1 (1   w ) Fw

8
8
8

1
a EE  (1   e ) Fe

8

a P  aW  aWW  a E  a EE  S  P


 e  1 ou 0 , si

Fe  0 ou Fe  0

 w  1 ou 0 , si Fw  0 ou Fw  0
T he sa me t h i n g is d o ne fo r t h e ver ti ca l " y" d ir ect io n, u si n g t h e fa ce s " n" and " s"
and b y i n tro d uc i n g t h e Q uic k d ia gr a m no d e s N, NN, S a nd SS.
i f co n v ect io n f lo w a nd d i f f us io n co e ffici e nt o f t he s id e s " e" , " w" , " s " , " n" are k no wn
An d e sp ec ia ll y t he so ur ce t er m S  , t he so l u tio n o f eq u at io n (2 ) t he n gi v e s u s t h e va l ue s

 

P

o f ɸ i n d i ffer e nt P no d e s .
O ne no te s t ha t to a cc eler ate t h e co n v er ge nc e o f eq uat io n (2 ) we ha v e i ntro d u ced a
rela x at io n fac to r

aP



W her e

:

P  aW W  aE E  aWW WW  aEE EE  S P 

1 



0
aPP

(3)

0
 P is t he v al u e o f  P in t he p re v io us s tep .

T he so ur ce te r m ap p ear s e sp e ci al l y i n t h e co n s erv at io n o f mo me n t u m e q ua tio n s i n
th e fo r m o f a p r e s s ur e gr ad ie n t wh i c h i s i n p rin cip le u n k no wn . T o ge t aro u nd t h is
co up li n g we ha v e c ho se n to u se i n o ur co d e t he " Si mp le" al go r it h m d e v elo p ed b y
P an ta n kar [ 7 ] . T he b a si c id e a o f t hi s al go rit h m i s to a s s u me a fie ld o f i n iti al p re s s ure a nd
inj e ct it i nto t he eq u at io n s o f co n ser v at io n o f m o me n t u m. Then we so l ve th e s ys te m to fi nd
a field o f i n ter med i at e s p eed ( wh i c h i s no t fa ir b eca u se t h e p re ss ur e i s n ’ t.) T he co n ti n u it y
eq u at io n i s tr a n s fo r me d in to a p r e s s ure co rre ct io n eq uat io n. T h is la s t i s d eter mi n ed to fi nd
a
p re s s ur e co r r e ct io n t ha t wi ll i nj ec t a ne w p res s ure i n t he eq ua tio n s o f mo t io n. T he
c ycle i s rep e ated as m an y ti me s a s n ece s sar y u nt il a p re s s ure co rr e ctio n eq ual " zero "
co rre sp o nd i n g to t h e al go r it h m co n v er ge nc e. I n the e nd we so l ve t h e t ran sp o r t eq ua tio n s
o f T , k a nd Ɛ.
In t h is ap p r o ac h, a p r o b le m i s e nco u nt ered . It is k no wn a s t he c h ec kerb o ard
p ro b le m.
T he r i s k i s t ha t a p r e ss u r e f ie ld ca n b e hi g h l y d is t urb ed b y t he se n sed fo r mu l a tio n
wh ic h co mp r i se s p er fo r mi n g a li n ear i n terp o lat io n fo r e s ti ma te t he p re s s ure val u e o n t he
fa ce ts o f t he co ntr o l vo lu me. T o cir c u mv e nt t h i s p ro b l e m we u se t h e s o -c al led st a g gered
gr id s p ro p o sed b y Har lo w a nd W el c h [6 ]. I n t h i s tec h n iq ue, a fir st gr id p res s ure ( a nd o t he r

9

sca lar q u a nt it ie s T , k a nd Ɛ ) is p la ced i n t h e ce nte r o f t h e co n tro l v o lu me. W hi le o t her
st a g gered g r id s ar e ad o p ted fo r t he v elo c it y co m p o ne n t s u a nd v (s ee Fi g ure 3 ).

F i g ure 3 : S h i ft ed me s h
T he s ca lar var iab le s, i n cl ud i n g p r e ss ur e, ar e s t o red at t he no d es ( I, J ). Eac h no d e
(I, J ) is s urr o u nd ed b y n o d es E, W , S a nd N. The ho r izo nt al v elo c it y co m p o ne n t u i s sto r ed
o n fac e s " e" and " w" , wh i le t he ve r t ica l co mp o ne n t v i s s to red o n t he fac e s d e no ted " n"
and " s" .
So t h e co n tr o l vo l u me f o r p r es s ur e a nd o t her sc alar q u a nt it ie s T , k a nd Ɛ is

(i  1, j ); (i  1, j  1); (i, j  1); (i, j )

Fo r co mp o n e nt u co nt r o l vo l u me i s

( I , j ); ( I , j  1); ( I  1, j  1); ( I  1, j ) .
W hi le f o r t h e v co mp o n en t we u se : (i  1, J  1); (i  1, J ); (i, J ); (i  1, J  1) .

B y i n te gr ati n g t he co n s er v at io n mo me n t u m eq ua tio n i n t he ho r izo n ta l d ire ct io n o n t he
vo l u me ( I , j ); ( I , j  1); ( I  1, j  1); ( I  1, j ) , we fo u nd :



W her e :



ai , J ui , J  anbunb  P(I  1, J )  P(I , J )( y j 1  y j )

a

(4)

u  aWWuWW  aW uW  aEEuEE  aEuE  aSS uSS  aS uS  aNN uNN  aN uN

nb nb

T he co e ffic ie n ts anb ar e d eter mi n ed b y t he q uic k s ch e me me n t io ned ab o ve.
Si mi l ar l y, t he i n te gr a ti o n o f co n ser va tio n mo me n t u m eq u at io n i n t h e ver ti ca l
d irec tio n o n t he vo l u me
(i  1, J  1); (i  1, J ); (i, J ); (i  1, J  1) ,



gi v e s u s :

a I , jvI , j   a nb vnb  P(I,J 1)  P(I,J)(xi 1  xi )



(5)

*

O ne co ns id er s p r i ma r i l y a n i ni ti al p re s s ure fie ld P . T he p ro v i sio na l so l ut io n o f t he
eq u at io ns (4 ) a nd ( 5 ) wi ll b e d e no ted u * a nd v * . W e no te t ha t u * a nd v * d o es n ’t c h ec k s t h e
co n ti n u it y eq ua t io n, and:



 P (I , J 1)  P (I , J )( x

ai ,J u*i ,J   anbu*nb  P* (I  1, J )  P* (I , J ) ( y j1  y j ) (6  a)
*

aI , j v

I,j

  anbv

*

nb

*

*

i1

 xi ) (6  b)

At t hi s st a ge a n y o ne o f th e t h r ee v ar i ab le s i s co rrect. T he y req u ire co rre ctio n :

 p'  P  P *

*
 u'  u  u
 v'  v  v *


(7 )

Inj ec ti n g (7 ) i n eq ua tio n s ( 4 ) , ( 5 ) a n d (6 ) we fi n d :

ai ,J u'i ,J   anbu'nb P' ( I  1, J )  P' (I , J )( y j1  y j ) (8  a)

aI , j v'I , j   anbv'nb P' (I , J 1)  P' (I , J )( xi1  xi ) (8  b)
10

At

t hi s

l e ve l,

eq u at io ns (8 ) , t he ter ms

an

a

a p p r o x i ma tio n
nb

is

i n tro d uc ed . In o rd er

u' nb and  a nb v' nb are s i mp l y n e gl ect e d .

to

li ne ari ze

the

No r ma ll y t he se t er ms m u st ca n ce l a t t he p ro ced ure co n v er ge nc e. T hat i s to s a y t hat
th i s o mi s sio n d o e s no t af f ec t t he fi n al re s u lt. Ho we v er, t he co n v er ge n ce rat e i s c ha n ged
b y t hi s s i mp l i fi ca tio n. I t t ur n s o ut t ha t t he co r re ct io n P ' i s o ver e st i ma t ed b y t he S i mp le
al go ri t h m a nd t he c alc u l atio n t e nd s to d i ver ge . T he re me d y to s tab ili ze t he c alc u la tio n s i s
to us e a rel a xat io n f acto r .
W e No te t ha t a f ur t he r tr e at me n t o f t he se te r ms i s p ro p o sed i n t h e so - cal led
al go ri t h ms " SI M P LE R" and " SI MP LE C" [7 ].
T he eq ua tio n s ( 8 ) b eco m e:

u 'i , J  d i J P ' ( I  1, J )  P ' ( I , J ) (9  a )
v' I , j  d I j P' ( I , J  1)  P' ( I , J ) (9  b)

W it h

di J 

y j 1  y j
ai J

and

dI j 

xi 1  xi
a Ij

Eq u at io ns ( 9 ) g i ve t he co r r ec tio n s to ap p l y o n v elo c it ie s t hro u g h t he fo r mu la s (7 ). W e
ha v e t h ere fo r e :

ui ,J  u*i ,J  di J P' ( I  1, J )  P' ( I , J )

(10  a)

vI , j  v

(10  b)

*

 d I j P' ( I , J  1)  P' ( I , J )

I,j

No w t he d is cr e tiz ed co nt i n ui t y eq u at io n o n t h e co ntro l vo l u me o f sc alar q u a nt it ie s i s
wr i tte n a s fo l lo ws :

( uA)

i 1, J





 ( uA)i , J   ( vA) I , j 1  ( vA) I , j  0

W her e a n ar e t he s ize o f t he co r r e sp o nd i n g face s .
T he i n tr o d uc tio n o f t h e v elo ci t y co rre ct io n eq u at io ns (1 0 ) i n t he co n ti n u it y
eq u at io n g i ve s a fi n al e q ua tio n a llo wi n g u s to d eter mi n e t he sco p e o f p r es s ure co rr ect io n s
P:

a 'I J P' ( I , J )  a 'I 1 J P' ( I  1, J )  a 'I 1J P' ( I  1, J )  a 'I J 1 P' ( I , J  1)  a ' I J 1 P' ( I , J  1) (11)
T he a l go ri t h m ca n b e s u m m ar ized a s fo llo ws : o ne s tar t s fro m a n i n it ial
field P* , u* , v* and  * , wi t h  r ep r e se n t s t he s ca lar q u a nt it ie s T, k and Ɛ. T he n t he s ys t e m (6 )
is so l ved to ha v e ne w va lu e s o f u *and v* .
T he n t h e s ys t e m ( 1 1 ) i s so l v ed fo r t he co rrec tio n s field p re ss u re P '.
T herea fter, p r e ss u r e a nd ve lo c it y ar e co r rec ted b y eq u at io ns (7 ) a nd (1 0 ) to ha ve P , u a nd
v.
W e so l ve t he fo llo wi n g tr a n sp o rt eq ua tio n o f s c alar q u a nt it ie s (2 ), ɸ = T, K and Ɛ .
Fi n all y, we co n sid er :

P*  P ; u *  u ; v*  v ;  *   .
An d t h e c yc le i s r ep e at e d u n ti l co n ve r ge nc e.

IV.

RESULTS AND DISCUSSIONS

T he d i me n s io n s o f t h e ch a n nel p re se n ted i n t h is wo r k a r e b a sed o n e xp er i me n ta l
d ata p ub l is h ed b y De m ar ti n i et a l [ 2 ]. T he air flo w i s carri ed o u t u n d er t he fo ll o wi n g
co nd it io n s.
 C ha n n el le n g t h: 𝐿 = 0 .5 5 4 m;
 C h a n ne l d i a me ter D = 0 .1 4 6 𝑚 ;
 b a ffl e he i g ht : 0 < ℎ < 0 1 . 𝑚 ;
 b affle t h ic k ne s s: δ = 0 .0 1 𝑚 ;
 Re yn o ld s n u mb er : 𝑅 𝑒 = 8 .7 3 1 0 4 ;
 T he h yd r o d yn a mi c a nd t her mal b o u nd ar y co nd it i o n s ar e gi v e n b y
At t he c h a n ne l i n le t
u  uin =7 ,8 m/ s;

v  0 ; T  Tin =3 0 0 K
11

O n t h e c ha n n el wa ll s
u  v  0 an d T  Tw  373 K
At t he c h a n ne l e x it t he s ys t e m i s as s u me d to b e ful l y d e v elo p ed , i e

u

x

 v

x

 T

x



0

Fir s t, we co mp ar ed t he str u ct ur e o f s trea ml i ne s and i so t her ms i n a c h a n ne l wi t ho u t
b affle wi t h t ho se o b tai n ed wh e n we i n tro d uce b affle ha v i n g a he i g ht
h = 0 .0 5
m ( Ca se 2 -1 ) at t h e ab s c is e 𝐿 1 = 0 .2 1 8 𝑚 . Ind e ed , in fi g ur es 4 a nd 5 we p res e nt s trea ml i ne s
and i so t h er ms fo r t wo c o n f i g ur at io n s.
T hes e r e s u lt s cle ar l y s ho w t h e i mp o rt a nce o f p re se nc e o f b a ffl e (a cti n g a s a
co o li n g fi n) . I nd e ed t he b af f le i ncr e as e s h ea t tr an s fer b et wee n t he wal l an d t he fl u id . This
in cre as e i s ca u sed b y r e cir c u la tio n zo ne s d o wn s trea m o f t h e b a ffle.

(a)

(b)
Fi g ur e 4 : s tr ea ml i ne s: ( a ) ca se 1 ; (b ) ca s e 2 -1

(a)

(b)
Fig u re 5 : I so t her ms : (a) ca se 1 ; (b ) ca se 2 -1
W e t he n s t ud i ed t he in f l ue n ce o f b a ffle he ig h t o n flo w s tr uc t ure and he at
tra n s fer. W e c ho s e t h e h ei g ht s ( h = 0 .0 5 m; c as es 2 -1 ) h = 0 , 0 7 3 𝑚 ( Ca se 2 -2 ) a nd
h
= 0 , 1 𝑚 (ca s e 2 -3 ) .
T he fi g ure s 6 a nd 7 p r e s en t s tr e a ml i ne s a nd i so t her ms fo r a c ha n ne l co n t ai ni n g b a ffle wi t h
d i ffere n t he i g ht s .

(a)

12


(b)
Fi g ur e 6 : str ea ml i n es : (a ) ca se 2 -2 ; (b ) ca se 2 -3

(a)

(b )
Fi g ur e 7 : I so t her ms : (a) cas e 2 -2 ; (b ) c as e 2 -3
It i s c le ar t h at t he i so t her ms are mo re co nd e n sed n ear b a ffle , a s a nd wh e n t he
he i g ht h i n cr ea se s. T h i s i nd i cat es a n i ncr ea se o f h eat tr a ns fer ne ar b affle. I nd eed , t he
in cre as e i n h e xp a nd e d ex c ha n ge are a b et we en fl uid a nd wa l l s. I n ad d it io n, wh e n h
in cre as es , t he r ec ir c ul at io n zo n e b e co me s i ncr e as i n gl y i mp o r ta n t ( se e Fi g ur es 4 (b ) a nd
fi g ure s 6 (a) a nd ( b ) ) . T hi s ca u s es a n acc ele ra tio n o f flo w, wh i c h i mp ro ve s hea t t ra n s fer
wi t h i n t he c h a n ne l.
W e p r e se nt fi g ur es 8 a n d 9 i n o rd er t o id e n ti fy th e i n fl ue n ce o f h o n ve lo ci ti es a nd
te mp er at ur e p r o f il e s al o n g y at x = 0 .4 5 fr o m c ha n n el i n le t. Fo ur val u e s o f h are
co n s id ered .

Fi g ur e 8 : Ho r izo nt al v el o cit y P ro fi le s alo n g y at x = 0 .4 5 « i n fl ue n ce o f h »

13


Fi g ur e 9 : Te mp er at ur e p r o file s alo n g y at x = 0 .4 5 m « i n fl ue nc e o f h »
T hes e r e s ul ts a llo w u s to co n cl ud e t h at t he i ncr ea se o f b a ffle he i g ht ha s t wo
co n trad ic to r y e f fe ct s. I t is tr ue t hat t her e i s s ub sta n ti al i ncre as e i n he at e x c ha n ge
(ap p eara n ce o f r ec ir c u l atio n zo ne s mo s t co m m o n), b u t t here i s a l so a lo s s o f p re s s ure
(flo w b lo c k a ge) .
In t h e p ur p o s e o f mea s ur i n g t he i n fl ue n ce o f 'h ' o n lo ca l h ea t tra n s fe r, we ha ve
p res e nted i n fi g ur e 1 0 , t he lo cal N u s se lt n u mb er alo n g t h e c h a n nel fo r t he fo ur ca se s o f h
co n s id ered . I t s ho ws t ha t up s tr ea m o f t he b a ffle (x <0 .2 m) c ur v es are co n fu s ed , wh i le,
j us t a fter b a f f le lo c atio n, e f fe ct o f b a ffle hei g ht o n N us s el t n u mb er b eco me i ncr ea si n g l y
i mp o rt a nt a wa y f r o m t he b a f fle s.

Fi g ur e 1 0 : Lo c al n u mb er N u alo n g t he c ha n ne l « i n fl u e nce o f h »
W e t he n s t ud i ed t h e i n f lu e nce o f b a ffle s n u mb e r a nd p o si tio n 'F i g ure 1 1 a nd 1 2 '.

(a)

(b )

14


(c)
Fi g ur e 1 1 : S tr e a ml i ne s : (a) c a s 3 -1 ; (b ) ca s 3 - 2 ; (c) ca s 3 -3
W e co n s id er t wo b a f f le s o f s a me he i g ht h = 0 .0 8 m. T he fir st b a ffle i s p l aced a t t he
d is ta nc e 𝐿 1 = 0 .1 5 𝑚 , whi le th e seco nd i s p o si ti o ned at t he d i st a nce d 1 = 0 .0 5 m fro m t h e
firs t (ca se 3 -1 ) , d 1 = 0 .1 0 m ( c as e 3 -2 ) a nd d 1 = 0 .1 5 m (c as e 3 -3 ). We no te t he
ex i ste n ce o f t wo r e cir cu la tio n zo ne s d o wn s t rea m o f t he fir st b a ff le. Al so t h e fir st
r ecir c ula tio n zo ne 'd e fi ned b y b a f fle s ' b eco me s in crea s i n gl y i mp o rt a nt, wh ic h co ntr ib ute
to an i n cre as e i n h eat tr a n s fer i n t h i s area, as sho wn i n F i g ure 1 2 . I nd e ed , wh e n t h e
b affle s are c lo se to ea c h o the r 'd 1 s mal l ', t he fl u id is b lo c ked i n t h e c hi mn e y d e l i mi ted b y
x = L 1 a nd x = L 1 + d 1 . T hi s r ed uc es t he flo w sp eed a t t h at lo ca tio n a nd th u s t her e wi l l b e
a d ecre as e i n t he he at tr an s f er i n t h i s are a. Or wh e n d 1 i n crea s es, t he flu id ha s s u ffi ci e nt
sp ac e to mo ve r ap id l y, h en ce he at tr a n s fer i n crea se s i n t h is zo ne 's ee fi g u re1 5 '.

(a)

(b )

(c)
Fi g ur e 1 2 : I so t her ms ( a ) : « c a s 3 -1 » ; (b ) : « c as 3 -2 » ; ( c) « ca s 3 -3 »
In t h e fo llo wi n g f i g ur e s 1 3 and 1 4 , we co mp are th e p ro fil e o f ho r izo n t al velo ci t y
and te mp era t ur e d i str ib ut io n. C alc u la tio n i s d o ne o n t h e y- a x is a t x = 0 .4 5 m fro m t he
ch a n nel i nle t fo r t hr ee d i f fer e n t b a f fl e s sp a ci n g. W e fi nd t h at mo re sp aci n g i s i mp o r ta n t
mo r e hea t e x c ha n ge i s i mp o r ta n t i n area s li mit e d b y t wo b a ffle s.

15


Fi g ur e 1 3 :P r o f il es o f t h e h o r izo n ta l ve lo c it y a t x = 0 .4 5 m « I n fl u e nce o f t he sp a ci n g
b et we e n t wo b a ffl es »

Fi g ur e 1 4 : P r o f ile s o f t he to t al te mp era t ure at x = 0 .4 5 m « I n fl ue n ce o f t he sp a ci n g
b et we e n t wo b a ffl es »
Fi g ur e 1 5 p r e se nt t h e lo cal N u s sel t n u mb er a lo n g t he c ha n n el fo r t h ree co n s id ered
sp ac i n g d 1 = 0 .0 5 m, 0 .1 m a nd 0 .1 5 m .

Fi g ur e 1 5 : Lo ca l N u s se l t n u mb er alo n g t he c ha n ne l « I n fl u e nce o f b a ffle s n u mb e r a nd
p o si tio n »

16

We d ist i n g ue t wo ar ea s :
T he f ir st d e fi n ed b y 0 < x <0 .3 m wh i l e t he s eco n d b y x> 0 .3 m. Fo r t he fi rst re g io n,
an i ncr ea se i n n u mb er o f b a f fl es i mp ro v e s t h e h eat tr a ns fer; wh er ea s th e re v er se is tr u e
fo r t he se co nd zo n e.
W e c ho o se s to s ho w t h e i n f l ue nc e o f b a ffl es n u mb e r o n he at tra n s fe r alo n g t he
ch a n nel fo r t hr ee ca se s: C ha n ne l wi t ho u t b a ffl e s, c ha n n el co n ta i ni n g a si n g le b a ffl e a nd
ch a n nel co nt ai n i n g t wo b af f le s 'F i g ure 1 6 '.
It s ho u ld b e no ted t h at ge n er a ll y, h eat tra n s fer is p ro p o r tio n al to b a ffl es n u mb er.
Ind e ed , t he N us s el t n u m b er c har ac teri zi n g h ea t t ran s fer wi t h i n t he c ha n n el i ncre as es wi t h
in cre as i n g b a f f le s n u mb er .

Fig u re 1 6 : Lo c al N u ss e lt n u mb er a lo n g t h e c h a n ne l

V.

CONCLUSION

T he t her mal b e ha vio r o f a s tat io nar y t urb ul e nt fo rc ed co n ve ct io n flo w wi t hi n a
b affled c ha n ne l wa s a na l yzed . T h e r e s ul t s s ho w th e ab il it y o f o ur co d e t o p red ic t d yn a mi c
and t h er ma l f ie ld s i n v ar io u s g eo me tric si t ua ti o n s. W e s t ud i ed mai n l y the i n fl ue n ce o f
b affle s h ei g h t a nd sp a ci n g o n h eat tr a ns fer a nd flu id flo w. O ne ca n co nc l ud e t ha t:
1 - I ncr ea se i n t he b a f f l e he i g ht i mp ro ve s he at t ran s fer b et wee n c ha n ne l wal ls a nd
fl uid p as s i n g t hr o u g h i t;
2 - He at tr a n s fer b eco m es i ncre a si n g l y i mp o r ta n t wi t h ad d i n g b a ffl es .
3 - T he sp ac i n g d 1 b e t wee n b a ffl es ha s d i ffere n t effect s o n lo c al he at tr an s fer. An y
ti me d 1 ha s no t a lo t o f i n f l ue nc e o n t h e o ver al l he at tra n s fer i n t he c ha n ne l.
In p er sp ec ti v e, we i n te n d d eep e n a nd c lar i fy o ur re s ul ts . I nd e ed , we wi l l ad ap t o ur co d e to
o th er s g eo me tr ic c a se s ( no n -r ect a n g ula r b a ffle s o r b affle s i nc li n ed ). F i n all y, we wi l l al so
tr y to r e fi n e mo r e t he t u r b u le nc e mo d e l .

REFERENCES
[1 ].

[2 ].

[3 ].

S.V .P AT AN K AR, E. M. SP AR ROW , « F u ll y d e ve lo p ed flo w a nd h eat tran s fer i n
d uc t s ha v i n g s tr ea m wi s e -p er io d ic vari at io n s o f cro ss - s ect io na l a rea », J o ur na l o f
Hea t T r an s f er , Vo l. 9 9 , p ( 1 8 0 -6 ) , 1 9 7 7 .
L. C .DE M ART NI , H. A.V I E LM O a nd S .V. MO LL ER, « N u mer ica l a nd e xp er i me n ta l
an al ys i s o f t he t ur b ul e nt f lo w t h ro u g h a c ha n ne l wi t h b a ffle p l ate s », J . o f t he
B raz. So c. o f Me c h. Sc i. & E n g., Vo l. X XVI, No . 2 , p (1 5 3 -1 5 9 ), 2 0 0 4 .
R. S AI M, S. AB B O UDI , B .B ENY OU CE F, A. AZ ZI, « Si mu la tio n n u mé riq ue d e la
co n v ect io n fo r c ée t ur b ul en te d a n s le s éc h a n ge ur s d e c h ale u r à fa i s cea u et ca la nd r e
mu n i s d e s c h ica n e s tr a n s ver sa le s », Al géri a n j o ur na l o f ap p l ied fl uid me c ha n ic s /
vo l 2 / 2 0 0 7 ( I SS N 1 7 1 8 – 5 1 3 0 ).

17

[4 ].

[5 ].
[6 ].

[7 ].
[8 ].

M. H. N AS I R U DDI N, K. SE DDI Q UI « Hea t tra ns fert a u g me nt at io n in a he at
ex c ha n ger t ub e u s i n g a b af f le », I n ter na tio n al j o ur na l o f Hea t a nd Fl u id Flo w, 2 8
(2 0 0 7 ) , 3 1 8 -3 2 8 .
H.K. Ve r s tee g ; W . Mal ala se k era « An i nt ro d uc tio n to co mp ut at io n al fl u id
d yna mi c s » I SB N 0 .5 8 2 - 2 1 8 8 4 -5 .
F. H ar lo w; J . W el s h ; « Sta g g ered gr id », N u me rica l ca lc ul at io n o f ti me d ep end e nt
vi s co us i nco mp r e s sib le f lo w wi t h fre e s ur face. P h ys ic s o f fl u id s, vo l. 8 ; p p 2 1 8 2 2189; 1965.
P ata n kar , S. V. ( 1 9 8 0 ) , «N u mer ica l H ea t T ran s fe r a nd F l uid F lo w»,
C hi e n g C. C. a nd La u n d er B .E. «O n t he ca lc ul at io n o f t urb u le n t h e at tr a nsp o r t
d o wn s tr e a m fr o m a n ab r up t p ip e e xp a n sio n », N u me ric al Hea t T ran s fer, vo l . 3 , p p .
1 8 9 -2 0 7 .

18

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