laminar fully developed mixed convection with … · pa tion on fully de vel oped mixed con vec...

LAMINAR FULLY DEVELOPED MIXED CONVECTION

WITH VISCOUS DISSIPATION IN A UNIFORMLY

HEATED VERTICAL DOUBLE-PASSAGE CHANNEL

by

Mostafa M. Salah El-Din

Original scientific paperUDC: 536.25:532.517.2

BIBLID: 0354-9836, 11 (2007), 1, 27-41

Fully de vel oped lam i nar mixed con vec tion has been in ves ti gated nu mer i -cally in a ver ti cal dou ble-pas sage chan nel by tak ing into ac count the ef fectof vis cous dis si pa tion. The chan nel is di vided into two pas sages by means ofa thin, per fectly con duc tive plane baf fle and the walls are uni formly heated.The ef fects of Brink man num ber and the ra tio be tween Grashof andReynolds num bers on the ve loc ity and tem per a ture pro files and the Nusseltnum ber on the hot wall have been an a lyzed. The re sults show that these ef -fects de pend mainly on the baf fle po si tion.

Key words: convection, channel, double-passage, uniform heat flux

Introduction

En hance ment of the heat trans fer in a ver ti cal chan nel is a ma jor aim be cause ofits prac ti cal im por tance in many en gi neer ing sys tems, such as the so lar en ergy col lec tion[1, 2] and the cool ing of elec tronic sys tems [3, 4]. The con vec tive heat trans fer may been hanced in a ver ti cal chan nel by us ing rough sur face, in serts, swirl flow de vice, tur bu -lent pro moter, etc. [5]. Candra et al. [6, 7] in ves ti gated the use of ribbed walls, Han et al.[8] used V-shaped tur bu lence pro mot ers, Lin et al. [9] and Beitelmal et al. [10] dem on -strated the ef fect of jet im pinge ment mech a nism. Re cently, Dutta and Hossain [11] in ves -ti gated the heat trans fer and the fric tional loss in a rect an gu lar chan nel with in clined solid and per fo rated baf fles.

Un for tu nately, most of these meth ods cause a con sid er able drop in the pres sure.Guo et al. [12] sug gested that the con vec tive heat trans fer could be en hanced by us ingspe cial in serts, which can be spe cially de signed to in crease the in cluded an gle be tweenthe ve loc ity vec tor and the tem per a ture gra di ent vec tor rather than to pro mote tur bu lence. So, the heat trans fer is con sid er ably en hanced with as lit tle pres sure drop as pos si ble. Aplane baf fle may be used as an in sert to en hance the rate of heat trans fer in the chan nel. To avoid a con sid er able in crease in the trans verse ther mal re sis tance into the chan nel, a thinand per fectly con duc tive baf fle is used. The ef fect of such baf fle on the heat trans fer in aver ti cal chan nel can be found else where [13, 14].

DOI:10.2298/TSCI0701027S 27

In a pre vi ous work [15], the ef fect of vis cous dis si pa tion on fully de vel oped lam -i nar mixed con vec tion in a ver ti cal dou ble-pas sage chan nel with uni form wall tem per a -tures has been in ves ti gated nu mer i cally. Al though vis cous dis si pa tion is usu ally ne -glected in low-speed and low-vis cos ity flows through con ven tion ally sized chan nels ofshort lengths, it may be come im por tant when the length-to-width ra tio is large [16]. As an il lus tra tion, the work ing di men sions are: length = 1.2 m, width = 0.2 m and the vol umeflow rate = 1×10–5 m3/s [17]. For dou ble–pas sage chan nels, the length-to-width ra tio be -comes larger as the baf fle be comes near the wall. So, vis cous dis si pa tion may be come im -por tant.

For the fully de vel oped lam i nar duct flow, Guo et al. [12] ob served that Nu forthe case of the isoflux ther mal bound ary con di tion is greater than Nu for the case of iso -ther mal bound ary con di tion. This can be ex plained based on the con cept of the in cludedan gle be tween the ve loc ity and tem per a ture gra di ent vec tors. This an gle is larger atisoflux ther mal bound ary con di tion than at iso ther mal bound ary con di tion. There fore,they stated that chang ing the ther mal bound ary con di tion could en hance the con vec tiveheat trans fer. Fully de vel oped flow is closely ap proached in a chan nel whose length islarge com pared with its width. The study of this flow is, there fore, use ful; it is also in -struc tive be cause it yields the lim it ing con di tions for the de vel op ing flow. Aung et al.[18] have found that when L ³ 10–2 (L = Gr l/b, l is the length) the case is fully de vel oped.There fore, the pres ent work is de voted to study, nu mer i cally, the ef fect of vis cous dis si -pa tion on fully de vel oped mixed con vec tion in a ver ti cal dou ble-pas sage chan nel withuni form wall heat flux.

Analysis

The chan nel shown in fig. 1 is di vided into two pas sages by means of a per fectlycon duc tive and thin baf fle. Lam i nar, two-di men sional, in com press ible, steady flow iscon sid ered. The fluid prop er ties are as sumed to be con stant ex cept for the buoy ancy termof the mo men tum equa tion. The chan nel walls have uni form heat fluxes.

For fully de vel oped flow, it is as sumed that the trans verse ve loc ity is zero andthe pres sure de pends only on x. The gov ern ing equa tions, ex press ing the above con di -tions, are:

vu

yT T

p

xi id

dg

d

d

2

2 00

1= - - +b

r( ) (1)

kT

y

u

yc u

T

x

¶

¶

¶

¶

2

2

2

0+æ

èçç

ö

ø÷÷ =m r

d

dp (2)

The sub script “i” de notes stream 1 or stream 2.It should be noted that many au thors [13, 19, 20] used the en ter ing fluid tem per -

a ture as a ref er ence tem per a ture. This al lows com par i son with the nu mer i cal so lu tion of

28

THERMAL SCIENCE: Vol. 11 (2007), No. 1, pp. 27-41

the de vel op ing flow prob lem at great dis -tances from the en try, but is rather bi zarreas far as the fully de vel oped flow is con -cerned. The mo men tum bal ance eq. (1)has been writ ten ac cord ing to theBoussinesq ap prox i ma tion by in vok ing alinearization of the equa tion of state r(T)around the ref er ence tem per a ture T0.Barletta and Zanchini [21] rec om mendedthe choice of the mean fluid tem per a tureas the ref er ence tem per a ture in the fullyde vel oped re gion. This choice en suresthat at each chan nel sec tion, the av er agesquare de vi a tion from the lo cal tem per a -ture is min i mum and, as a con se quence,yields the best con di tions for the va lid ityof the equa tion of state r(T). Re cently,Boulama and Galanis [22] adopted thetem per a ture of the cooler wall as ref er -ence, be cause it leads to sim ple ex pres -sions for the ther mal bound ary con di -tions. In par tic u lar, the value of thedimensionless tem per a ture at this wall equals zero. In the pres ent work, the choice of Barletta and Zanchini is used. So the ref er ence tem per a ture is de fined as:

Tb

T yb

00

1= ò d (3)

The bound ary con di tions are:

y uT

y

q

k

y b u u T T

y b uT

y

= = = -

= = = =

= = =

0 0

0

0

11

1 2 1 2

2

: ,

: ,

: ,

*

¶

¶

¶

¶-

q

k2

In te gra tion of eq. (2) with re spect to y yields:

kT

yk

T

y

u

yy c

T

xu dy

y b y

b

p¶

¶

¶

¶

¶

¶= =

- +æ

èçç

ö

ø÷÷ =ò

0

2

0

00

m rd

dd

b

ò (4)

29

Salah El-Din, M. M.: Laminar Fully Developed Mixed Convection with Viscous...

Figure 1. Physical configuration of thedouble-passage channel

The first two terms of LHS of eq. (4) rep re sent the heat fluxes on the walls andthe in te gra tion in RHS rep re sents the mass flow rate within the chan nel, which is con stant at any cross-sec tion. The ref er ence ve loc ity u0 is de fined as:

ub

udyb

00

1= ò (5)

Thus, eq. (4) can be re writ ten as:

¶

¶

T

x c buq q

u

yy

p

b

= + +æ

èçç

ö

ø÷÷

é

ë

êê

ù

û

úúò

1

0 01 2

2

0r

md

dd (6)

Now, the en ergy equa tion can be re writ ten as:

kT

y

u

y

u

u bq q

u

yy

¶

¶

2

2

2

01 2

2

0

+æ

èçç

ö

ø÷÷ = + +

æ

èçç

ö

ø÷÷m m

d

d

d

dd

b

òé

ë

êê

ù

û

úú

(7)

To make the gov ern ing equa tions dimensionless, the fol low ing dimensionlessquan ti ties are in tro duced:

Xx

bY

y

bU

u

u

T T

q b

k

q b

kR

q

q

Pp

= = =

=-

= =

=

Re, ,

, ,

0

0

1

14

2

2

1

qb

nGr

g

r n

m

0 02

0 02

1u

u b u

q b, Re ,= =Br

Hence, the dimensionless gov ern ing equa tions are:

d

d

Gr d

di

ii

2

2

U

Y

P

X= - +

Req (8)

¶

¶

2

2

2 2q

Y

U

Y

U

YY+

æ

èç

ö

ø÷ =

æ

èç

ö

ø÷

é

ë

êê

ù

û

úòBrd

d1+ + Br

d

dd

0

1

Rú

=U Uh (9)

where

h= + +æ

èç

ö

ø÷ò1

2

RU

YYBr

d

dd

0

1

(10)

30


The pres sure gra di ent in eq. (8) is as sumed to be con stant, i. e.:

d

di

iP

X=g (11)

In tro duc tion of the dimensionless pa ram e ters into eq. (3) gives:

qdY =ò 00

1

(12)

The dimensionless bound ary con di tions are:

Y UY

Y Y U U

Y UY

R

= = = -

= = = =

= = =

0 0 1

0

1 0

1

1 2 1 2

2

: ,

: ,

: ,

*

¶

¶

¶

¶

q

q q

q

whereY

b

b

**

=

Con ser va tion of mass con sid ered at any sec tion of the chan nel pas sages gives:

U Y YY

10

d =ò **

(13)and

U Y YY

2

1

1d = -ò *

*(14)

The dimensionless bulk tem per a ture in pas sage 1 is:

q

q q

b

d

d

d

1

10

10

10= =

ò

ò

òU Y

U Y

U Y

Y

Y

Y

Y*

*

*

*(15)

The Nusselt num ber on the hot wall is:

Nub

10 1

1=

-=q qY

(16)

Numerical methodology

In ves ti ga tion of the gov ern ing equa tions for mu lated shows that the en ergy equa -tion and the mo men tum equa tion are cou pled and non-lin ear. Fur ther more, for uni formwall heat fluxes the wall’s tem per a tures are not spec i fied. There fore, the so lu tions can be

31


ob tained by use of an it er a tive scheme with suc ces sive ap prox i ma tions. The equa tionsare first ap prox i mated by fi nite dif fer ence equa tions. Then, the fol low ing so lu tion pro ce -dure is used:(1) guess the walls’ temperatures,(2) guess a temperature field at the interior points inside the whole channel,(3) solve the momentum equation, using eqs. (13) and (14), to obtain the velocity field

and the pressure gradient for the two streams separately,(4) solve the energy equation, using eq. (12) and the values of the velocity of the

previous step, to obtain new values of the temperature at the interior points inside thewhole channel and the value of h,

(5) by means of a three-point derivative formula, using the temperature values computed at step 4, compute new values for the wall’s temperatures, and

(6) return to step 3 and repeat until convergence.A uni form grid in the Y-di rec tion is used with equal num bers of grid points for

each pas sage (41 points). Thus, a smaller grid size in the nar rower pas sage will be ob -tained. The so lu tion pro ce dure and grid sizes were val i dated by com par i son with the casewhen the ef fect of vis cous dis si pa tion is ne glected. For this case eq. (10) gives:

h = +1 R (17)

The nu mer i cal re sults of h against the val ues from eq. (17) are shown in tab. 1for dif fer ent val ues of R. It is found that the de vi a tions in h are al ways less than 1.4%.

Table 1. Calculations of h without viscous dissipation (Gr/Re = 100, Br = 0, Y* = 0.5)

R 1 0.5 0

h (eq. 17) 2 1.5 1

h (numerical) 1.97217 1.496515 0.9977519

Fur ther more, the cal cu la tions were car ried out for the flow with out buoy ancyforces. For this case the an a lyt i cal so lu tion of the mo men tum equa tion gives:

g g12

2212 12 1= - = - -/ / ( )* *U Uand (18)

The nu mer i cal cal cu la tions give very close val ues to those cal cu lated us ing eq.(18). For ex am ple, at Y* = 0.5 eq. (18) gives g1 = g2 = –48 while the nu mer i cal cal cu la -tions give g1 = g2 = –47.99993. This con firms the ac cu racy and con ver gence of the nu -mer i cal cal cu la tions.

Results and discussions

The gov ern ing equa tions for mu lated with the given bound ary con di tions show thatthe nu mer i cal cal cu la tions de pend on the fol low ing pa ram e ters: R, Br, Y*, and Gr/Re.

32


When the pa ram e ter Gr/Re ex ceeds athresh old value, flow re ver sal may oc cur caus ing downflow along the cooler wall. Since the equa tions are par a bolic in their na ture, un sta ble so lu tions are ex pectedwhen the flow re ver sal oc curs. Ex am i -na tion of the flow re ver sal phe nom e nonis be yond the scope of this study. There -fore, the so lu tion pro ce dure was im me -di ately stopped when a neg a tive ve loc ity has been at tained. For more dis cus sionsof the flow re ver sal phe nom e non seeAung and Worku [19, 23].

In the fol low ing sec tions the ef fectof Br and Gr/Re on the Nu on the hotwall will be pre sented at dif fer ent po si -tions of the baf fle with sym met ric heat -ing (R = 1), asym met ric heat ing (R ==j0.5), and one wall is adi a batic (R = 0).The cho sen val ues for Br and Gr/Re arecom monly found in the ap pli ca tion ofcool ing of elec tronic sys tems. First, theve loc ity and tem per a ture pro files will be pre sented be cause of their use ful ness inclar i fi ca tion of the heat trans fer mech a nism in the chan nel.

The ef fect of Br on the ve loc ity pro files for sym met ric heat ing is shown in fig. 2 atdif fer ent po si tions of the baf fle. The fig ure re veals that the ef fect of Br ap pears clearly inthe wider pas sage, while in the nar rower pas sage its ef fect is not no tice able. When thebaf fle is in the mid dle of the chan nel very slight ef fects of vis cous dis si pa tion can be no -ticed in both pas sages. The max i mum ve loc ity point moves to wards the baf fle as Br in -creases.

The en ergy equa tion shows that the tem per a ture pro file de pends on the ve loc itypro file. Since the ve loc ity pro files de pend on the baf fle po si tion, it is ex pected that thetem per a ture pro files de pend on the baf fle po si tion too. Figure 3 pres ents the ef fect of Bron the tem per a ture pro files for the case of sym met ric heat ing at dif fer ent po si tions of thebaf fle. The fig ure re veals that an in crease in Br in creases the tem per a ture sig nif i cantly inthe nar rower pas sage. This is due to the in creas ing in vis cous heat ing since thelength-to-width ra tio is large in the nar rower pas sage. This in creas ing in tem per a ture inthe nar rower pas sage is ac com pa nied with a sig nif i cant de crease in tem per a ture next tothe wall of the wider pas sage. This can be ex plained with the aid of eq. (12) which im plies that the in te gra tion of the tem per a ture through the whole chan nel is zero.

In deed the in crease in Br in creases the tem per a ture and hence the ve loc ity in creasesdue to the in crease in ther mal buoy ancy force, but at the same time the in crease in Br in -creases the pres sure gra di ent, which de creases the ve loc ity. For nar rower pas sage, in

33


Figure 2. Effect of Br on velocity profiles forsymmetric heating(a) Br = 0, (b) Br = 0.05, (c) Br = 0.1(Gr/Re = 100)

which the length-to-width ra tio is large, thein crease in the pres sure gra di ent is highenough to re sist the in crease in ther malbuoy ancy force. So, Br has in sig nif i cant ef -fect on the ve loc ity pro file. For the widerpas sage, the high in crease in the tem per a ture near the baf fle makes the ther mal buoy ancyforce the dom i nant force. Hence, the ve loc -ity in creases with Br near the baf fle. Con se -quently, the ve loc ity de creases with Br nearthe wall, since the mass flow is con stant atany cross-sec tion of the chan nel pas sages.

The ef fect of Br on the ve loc ity pro -files for R = 0.5 and 0 are sim i lar to that forthe case of sym met ric heat ing ex cept that aslight de crease in ve loc ity is no ticed near the cool wall when the heat flux ra tio be comessmaller. This is due to the de crease in thebuoy ancy forces there. The ef fects of Br onthe tem per a ture pro files at dif fer ent po si -tions of the baf fle for the cases of asym met -ric heat ing and one wall is adi a batic areshown in figs. 4 and 5, re spec tively. For Br =0, the tem per a ture on the hot wall is greaterthan that on the cool wall. For Br > 0, the ef -

fect of vis cous dis si pa tion on the tem per a ture pro files is sim i lar to that for the case ofsym met ric heat ing.

Ta ble 2 and fig. 6 pres ent the vari a tions of Nu1 with Br at dif fer ent po si tions ofthe baf fle for sym met ric heat ing. The ta ble shows that for Y* = 0.5, Nu1 is a de creas ingfunc tion of Br. This can be ex plained with the aid of fig. 3 which shows that for Y* = 0.5,q|Y=0 is an in creas ing func tion of Br. So, the de nom i na tor of eq. (16) in creases and as acon se quence, Nu1 de creases. Ta ble 2 shows also that the rate of de creas ing in Nu1, due tothe in creas ing in Br, in creases as the baf fle is nearer to wall 1. As men tioned be fore, thisis be cause of the in creas ing of the ef fect of vis cous dis si pa tion as the length-to-width ra -tio be comes large. It may be noted that at a given value of Br, Nu1 in creases as the baf fleis nearer to wall 1. This is due to the in crease in the in cluded an gle be tween the ve loc ityvec tor and the tem per a ture gra di ent vec tor.

When the baf fle is nearer to wall 2, Nu1 in creases up to a very high value as Br in -creases. With more in creas ing in Br, Nu1 changes its sign. Fig ure 3 shows that thedimensionless tem per a ture on the wall from which the baf fle is far is a de creas ing func -tion of Br. So, the dif fer ence be tween the wall tem per a ture and the bulk tem per a ture inthe de nom i na tor of eq. (16) be comes very small and as a con se quence, the value of Nu1

be comes very high. More in creas ing in Br makes the de nom i na tor of eq. (16) neg a tiveand as a con se quence, Nu1 changes its sign.

34


jFigure 3. Effect of Br on velocity profilesjjjfor symmetric heating (a) Br = 0, (b) Br = 0.05, (c) Br = 0.1 (Gr/Re = jjj=j100)

Table 2. Variations of Nu1 with Br at defferent positions of the bafflefor symmetric heting (R = 1, Gr/Re = 100)

Br ×102 Y* = 0.2

Nu1

Y* = 0.5 Y* = 0.8

0 11.51 5.63 4.57

1 9.58 5.24 5.18

2 8.18 4.95 6.12

3 7.29 4.67 7.49

4 6.7 4.43 11.1

5 6.26 4.2 19.11

6 5.8 4 78.6

7 5.35 3.8 –35.6

8 4.98 3.62 –14.7

9 4.65 3.46 –8.75

10 4.37 3.29 –6.11

35


Figure 4. Effect of Br on temperatureprofiles for asymmetric heating(a) Br = 0, (b) Br = 0.05, (c) Br = 0.1(Gr/Re = 100)

jjjjFigure 5. Effect of Br on temperaturejjjjprofiles with an adiabatic walljjjj(a) Br = 0, (b) Br = 0.05, (c) Br = 0.1 (Gr/Re = 100)

The vari a tions of Nu1 with Br are pre sented in tab. 3 and fig. 7 for asym met richeat ing (R = 0.5) at dif fer ent po si tions of the baf fle. From the ta ble it is ev i dent that Nu1

var ies with Br in a man ner sim i lar to that for the case of sym met ric heat ing ex cept that the change in sign of Nu1 at Y* = 0.8 oc curs at higher value of Br than that for sym met richeat ing. As men tioned be fore, this is due to that the heat flux on the hot wall is greaterthan that on the cool wall. With an adi a batic wall (R = 0) more vis cous heat ing is re quiredto change the sign of Nu1 at Y* = 0.8 as shown in tab. 4 and fig. 8. To show the ef fect ofthe baf fle on heat trans fer in the chan nel, it is use ful to know the value of Nu1 for a sin -gle-pas sage chan nel. From a pre vi ous work [24] Nu1 » 1 for Br = 0.

Table 3. Variations of Nu1 with Br at different positions of the bafflefor asymmetric heating (R = 0.5, Gr/Re = 100)

Br ×102 Y* = 0.2Nu1

Y* = 0.5 Y* = 0.8

0 11.36 5.16 3.97

1 9.64 4.87 4.36

2 8.58 4.6 4.96

3 7.79 4.36 5.9

4 6.83 4.14 7.25

5 6.37 3.93 10.82

6 5.29 3.74 19.1

7 4.77 3.56 93.23

8 4.72 3.4 –31.53

9 4.69 3.25 –13.33

10 4.4 3.1 –7.97

36


,Figure 6. Variations of Nu1 with Br at,different positions of the baffle for,symmetric heating (R = 1, Gr/Re = 100)

Figure 7. Variations of Nu1 with Br atdifferent positions of the baffle forasymmetric heating (R = 0.5, Gr/Re = 100)

Table 4. Variations of Nu1 with Br at different positions of the baffle with an adiabatic wall (R = 0, Gr/Re = 100)

Br ×102 Y* = 0.2Nu1

Y* = 0.5 Y* = 0.8

0 11.75 4.83 3.54

1 10.21 4.55 3.79

2 8.87 4.32 4.17

3 7.86 4.09 4.76

4 7.05 3.9 5.76

5 6.41 3.72 7.47

6 5.88 3.55 10.62

7 5.12 3.4 19.28

8 4.57 3.25 126.36

9 4.17 3.1 –28.01

10 4.04 2.96 –11.74

Fig ures 9 and 10 il lus trate the ef fectof Gr/Re on the ve loc ity and tem per a -ture pro files, re spec tively. Buoy ancyforces have in sig nif i cant ef fect on theve loc ity pro file in the nar rower pas sage, but for the wider pas sage sig nif i cant in -crease in the ve loc ity near the coolerwall (the baf fle in this case) is ob served. Gr/Re has less ef fect on the tem per a turepro file as shown in fig. 10 since the en -ergy equa tion does not in clude this pa -ram e ter.

Ta ble 5 pres ents the vari a tions ofNu1 with Gr/Re at dif fer ent po si tions ofthe baf fle for sym met ric heat ing. The ta -ble shows that when the baf fle is in themid dle of the chan nel, Nu1 is a de creas ing func tion of Gr/Re. Oth er wise; Nu1 is an in -creas ing func tion of Gr/Re. Fig ure 10 shows that the tem per a ture on wall 1 in creaseswith Gr/Re when the baf fle is in the mid dle of the chan nel and de creases at the otherpo si tions. So, Nu1 changes its trend with Gr/Re as ex plained be fore. The ef fect ofGr/Re on Nu1 for R = 0.5 and 0 is sim i lar to its ef fect for R = 1.

37


Figure 8. Variations of Nu1 with Br atdifferent positions of the baffle with anadiabatic wall (R = 0.5, Gr/Re = 100)

Table 5. Variations of Nu1 with Gr/Re at different positions of the baffle for symmetric heating (Br = 0.05)

Gr/Re Y* = 0.2Nu1

Y* = 0.5 Y* = 0.8

0 5.93 4.34 14.83

50 6.2 4.28 16.85

100 6.26 4.2 19.11

150 6.32 4.12 22.4

Conclusions

The ef fect of vis cous dis si pa tion on fully de vel oped lam i nar mixed con vec tion in aver ti cal dou ble-pas sage chan nel has been stud ied nu mer i cally. Uni form wall heat fluxeswith sym met ric (R = 1), asym met ric (R = 0.5), and one wall is adi a batic (R = 0) have beencon sid ered. The ve loc ity and tem per a ture pro files have been pre sented and the Nusseltnum ber on the hot wall has been eval u ated at dif fer ent po si tions of the baf fle. It can becon cluded that:

38


Figure 9. Effect of Gr/Re on velocityprofiles for symmetric heating(a) Gr/Re = 0,(b) Gr/Re = 100, (c) Gr/Re =j200(Br = 0.05)

Figure 10. Effect of Gr/Re on temperatureprofiles for symmetric heating(a) Gr/Re = 0, (b) Gr/Re = 100, (c) Gr/Re = 200 (Br = 0.05)

– Br and Gr/Re have significant effects on the velocity profiles in the wider passage for all cases. In the narrower passage, these effects are insignificant,

– an increase in Brinkman number leads to a significant increase in the temperature inthe narrower passage for all cases. This is accompanied with a significant decrease in the temperature in the wider passage. When the baffle is in the middle of the channelthe effect of Brinkman number on the temperature in the whole channel decreasesclearly,

– Gr/Re has less effect on the temperature profiles than Br,– for Y* £ 0.5, the Nusselt number on the hot wall is a decreasing function of Brinkman

number. When the baffle is far from the hot wall Nu1 becomes an increasing functionof Br. At a certain value of Brinkman number Nu1 changes its sign. For symmetricheating this change in sign of Nu1 occurs at less value of Br than for other cases, and

– when the baffle is in the middle of the channel, Nu1 is a decreasing function of Gr/Re. Otherwise; Nu1 is an increasing function of Gr/Re.

Nomenclature

b – channel width, [m]b* – width of passage 1, [m]Br – Brinkman number (= mu q b0

21/ ) , [–]

cp – specific heat at constant pressure, [Jkg–1K–1]g – gravitational acceleration, [ms–2]Gr – Grashof number (= g b nqb k4 2/ ), [–]h – coefficient of heat transfer, [Wm–2K–1]k – thermal conductivity, [Wm–1K–1]n – number of grid points, [–]Nu – Nusselt number (= hb/k), [–]P – dimensionless pressure (= p u/ ), [–]r0 0

2

p – pressure, [Pa]q – heat flux, [Wm–2]R – heat flux ratio (= q2/q1), [–]Re – Reynolds number (= u0b/n), [–]T – temperature, [K]u – axial velocity, [ms–1]x – axial coordinate, [m]y – transverse coordinate, [m]Y* – (= b*/b), [–]

Greek symbols

b – volumetric coefficient of thermal expansion, [K–1]g – pressure gradient (= dP/dX), [–]q – dimensionless temperature [= (T – T0)/(q1b/k)], [–]m – dynamic viscosity, [Pa×s]n – kinematic viscosity, [m2s–1]r – density, [kgm–3]

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Subscripts

b – bulk0 – reference1 – value in stream 12 – value in stream 2

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Author's address:

M. M. Salah El-DinP. O. Box 2574, Al-Sarai 21411Alexandria, Egypt

E-mail: msalaheldin [email protected]

Paper submitted: June 13, 2006Paper revised: September 9, 2006Paper accepted: September 15, 2006

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