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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
6
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…
7
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
Numerical modell ing of heat transfer in a channel with…
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
Numerical modell ing of heat transfer in a channel with…
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
Numerical modell ing of heat transfer in a channel with…
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
Numerical modell ing of heat transfer in a channel with…
11
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
Numerical modell ing of heat transfer in a channel with…
12
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)
Numerical modell ing of heat transfer in a channel with…
13
(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 »
Numerical modell ing of heat transfer in a channel with…
14
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)
Numerical modell ing of heat transfer in a channel with…
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.
Numerical modell ing of heat transfer in a channel with…
16
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 »
Numerical modell ing of heat transfer in a channel with…
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
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