Heat and Mass Transfer 30 (1995) 159-166 9 Springer-Verlag x995

Mixed convection in vertical channels with a discrete heat source H. Tiirk~lu, N. Yiicel

159

Abstract Two dimensional laminar mixed convection flow in ver-

Nomenclature

tical channels with a discrete heat source was numerically analyzed. An isoflux discrete heating element was located on the left wall, while the isothermal conditions were imposed on the other walt. The governing equations were solved using a finite difference method based on the control volume approach. The mean Nusselt number was calculated and the maximum component temperature was determined. The computations were carried out for different Grashof number, Reynolds number, heater locations and the channel width. It was observed that the location of the heating element does not play a considerable role on the flow. At low Reynolds numbers (Re<5o), the mean Nusselt number and the maximum temperature are mainly controlled by the Grashof number. However, at higher Reynolds numbers, the Reynold number plays an important role on the flow. It was also found that at low Reynolds numbers, cooling is more effective when the channel width is large (W/H> 1). However, at high Reynolds numbers more effective cooling is obtained in narrow channels.

g Gr H k L Nu P Pr Re S T Tc To u U

gravitational acceleration Grashof number (Gr=gflq'H4/v2k) heater hight thermal conductivity of fluid height of the channel Nusselt number pressure Prandtl number Reynolds number (Re= VoH/v) position of heater center temperature cold wall temperature inlet temperature velocity component in x-direction dimensionless velocity component in x-direction

X

horizontal axis dimensionless horizontal axis (x/H) velocity component in y-direction dimensionless velocity component in y-direction

(V=u/Vo)

X Mischkonvektion in vertikalen Kan~len mit einer lokalen W~irmequelle V Zusammenfassung Die zweidimensionale laminare Mischkonvek- V tion in vertikalen Kan~ilen mit einer lokalen W~irmequelle wird (V=v/Vo) numerisch untersucht. Ein Heizelement konstanter W~irmeleiinlet velocity Vo stung befindet sich auf der linken Kanalwand, die rechte hat W width of the channel konstante Temperatur. Die L6sung der Grundgleichung erfolgte vertical axis Y mit Hilfe der auf dem Kontrollvolumenprinzip basierenden Finit- Y dimensionless vertical axis (y/H) differenzenmethode. Die mittlere Nusselt-Zahl sowie die Maximaltemperatur des Heizelementes wurden berechnet, und zwar Greek symbols unter Variation der Grashof-Zahl, der Reynolds-Zahl, der Lage a thermal diffusivity des Heizelements und der Kanalbreite. Letztere hatte nur gerinfl thermal expansion coefficient gen Einflufl auf den Str6mungsverlauf. Bei kleinen Reynoldsp density of fluid Zahlen (Re<5o) werden Nusselt-Zahl und Maximaltemperatur v kinematic viscosity vorrangig durch die Grashof-Zahl bestimmt, w~ihrend bei hohen 0 dimensionless temperature (O=(T-Tc)/[q"H/k]) Reynolds-Zahlen letztere den StrOmungsvorgang beherrscht. Ferner zeigte sich, daft bei niedrigen Reynolds-Zahlen die Kfihlung ffir groge Kanalbreite (W/H> 1) effektiver wird und bei 1 hohen Reynolds-Zahlen die VerMltnisse gerade umgekehrt lieIntroduction gen. In recent years, electronic industry has been developing rapidly. Parallel with this rapid technological development, the circuit inReceived on August 5, 1994 tegration density and hence the rate of the heat dissipated by an electronic component have increased. Cooling of electronic comDr. Hasmet Tiirko~lu ponents has become more important, since proper cooling is esDr. Nuri Yficel sential for the equipment safety and reliable operation. The main Mechanical Engineering Department objective of cooling of electronic components is to retain a relaFaculty of Engineering and Architecture tively constant component temperature below the maximum opGazi University erating temperature specified by the manufacturer. Investigao657o Maltepe, Ankara Turkey tions have demonstrated that a single component operating to ~ beyond this temperature can reduce the reliability of some Correspondence to: H. Tiirko~lu systems by as much as 50 percent [1].

160

The cooling systems of electronic equipment may be based on free and forced gaseous and liquid convection, as well as radiation and conduction. As surveyed by Peterson and Ortega [1] several strategies have been developed for controlling and removing the heat generated in electronic components. These include natural and forced air cooling and natural and forced liquid cooling. Natural convection air cooling is good for the cooling of low heat flux electronic packages. It is inexpensive, reliable and maintenance free. Consumer electronics, electronic test equipment and low end computer packages are often cooled by natural convection in air. However, the electronic equipment with denser circuits require more effective and reliable cooling systems. As a consequence, cooling system of many electronic devices is based on both natural and forced convection. The electronic components are often mounted on a circuit board. The circuit boards are placed parallel to each other in vertical, inclined or horizontal orientations. Hence, the boards form a series of channels with discrete heat sources on the walls. The objective of analysis of such problems has been the prediction of the maximum component temperature and the heat removal rate (Nusselt number) from the component, under different operating and geometrical conditions. Investigation of the effect of channel inclination angle, Grashof and forced flow Reynolds numbers have received attention in the literature. The problem of a discrete heat source located on the upper or lower wall of a horizontal channel was studied by Kennedy and Zebib [2, 3], Incropera et al. [4] and Hasnaoui et al. [5]. It was observed that placement of the heat source, on the upper wall will realize the temperatures almost as high as twice those that the heat source being located on the lower wall. Mixed convection flow and heat transfer were studied in inclined channels with localized heat sources by Yficel et al. [6], Choi and Ortega [7], and Tomimura and Fujii [8]. In these studies, an isothermal or isoflux heat sources located on one wall and the opposite wall was assumed to be at a constant lower temperature. The effect of inclination angle on the flow and temperature fields were investigated for different Grashof and Reynolds numbers. They observed that the Nusselt number increases with increasing Reynolds and Grashof numbers. The influence of the channel angle disappears gradually as the Reynolds number increases. A reverse flow was observed at low Reynolds and high Grashof numbers. The best heat transfer performance was obtained in the vertical channel case. Free and forced convection in a vertical channel with a discrete heat source located on one vertical wall was investigated by Ravine and Richard [9] and Elpidorou et al. [lO]. In their numerical study, the channel width was taken to be equal to the height of heating element, and the location of the heating element was held the same. They found that the heat transfer rate is strongly function of Grashof and Reynolds numbers. The Nusselt number increases with the increasing Grashof number at mixed convection regime. However, Nusselt number is independent from the Grashof number at forced convection regime. In the present study, a vertical channel with a flush mounted isoflux heat source located on the left vertical wall was considered. Rest of the left wall was assumed to be insulated. The opposite vertical wall was kept at a constant temperature. A numerical model based on the control volume approach was used to predict flow and temperature fields. Effects of rate of heat flux (Grashof number), the location of heat source, channel width and the forced convection Reynolds number on the flow and temperature

fields were investigated. Using the temperature fields, the mean Nusselt number was calculated. The variation of the mean Nusselt number and the maximum temperature with the location of heater, Grashof number, Reynolds number and the channel gap was investigated.

2 Mathematical formulation The geometry considered and the coordinate system are shown in Fig. 1. A constant flux discrete heat source is placed on one side wall while the opposite wall is isothermal. Total height of the channel is fifteen times that of the heater (L=15 H). It is assumed that the inlet and the cold wall temperatures are the same, i.e. To= Tc. Using the Boussinesq approximation, the steady state dimensionless governing equations for the laminar two dimensional flow are written as au + av = o

3X

(1)

3Y

3u u3U +v 3X OY

_

c-)P 4-1l ~ 3X

U 3V + v - V _ OX 3Y

Re LOG2

+ a2U]-]

(z)

aN 2 I

3P q llO2V .{_ 02V + Gr 0 Re LOX2 3y z Re 2

(3)

3Y

UOO+VO0_ 1 I ~ 2 0 + 020 1 OX OY RePr L OX 2 OY2 _l

(4)

where X and Y are the dimensionless distance in the x andy directions defined as x/H andy/H, respectively. U and V are dimensionless velocity components in x andy directions, respectively. The velocities are non dimensionalized as U= u/Vo and V= v/Vo.

L r~

iVo

, To

Fig. t. Geometry and coordinate system of the problem

Where Vo is the velocity at the channel inlet. Reynolds number is defined on the basis of the heater height H and the inlet velocity

3 Numerical solution procedure

v0,

A finite-difference method (SIMPLE) based on the control volume approach was employed [11] to solve the governing equations. The upwind scheme was used to approximate convection and diffusion terms. After preliminary computations, a uniform mesh of (30 x 15o) (x, y) was chosen. The computations were terminated when the continuity for each control volume was satisfied within the magnitude of error less than lO-3. The code was validated by comparing the simulations with the results of Choi and Ortega [7] and Elpidorou et al. [lO]. The comparison showed that the mean Nusselt numbers and the dimensionless maximum temperatures agreed well. Also, comparison was made for the square cavities. The mean Nusselt number is in agreement within less than 2 percent of those of Wilkes and Churchill [12] and Ahmed and Yovanovich [13].

R e - HVo v 0

represents the dimensions temperature which is defined as

o= T - T c q"H/k where T~is the cold wall temperature, q" is source heat flux (W/m 2) and k is the thermal conductivity of the fluid in the channel. Grashof number is defined as

Gr = gfiq"H4 kv 2 where fi is the thermal expansion coefficient of the fluid. Equations (1), (2), (3) and (4) are subject to the following boundary conditions. At the left wall (X=0) U=0 for O<_Y<_L/H V=0 for O<_Y<_L/H

S_l
O~0=-i 0X

at

3~0=0 3X

at 0_
H

2

H

2 s +!
At the right wall (X= W/H) U=0,

V=0,

0=0

At the inlet (Y=0)

V=I,

U=0,

0=0

At the outlet (Y=L/H) OU =0, OX

3V =0 3Y

O0 --=0

if

V_>0

0=0

if

V<0

0y

Equations (1), (2), (3) and (4) were solved together with the relevant boundary conditions to determine the velocity and temperature distributions in the channel. Using the velocity field, values of the streamlines over the flow domain were calculated. Overall Nusselt number was calculated from the temperature distribution as S I ---t--

Nu = f

1

S

1

H

2

dY X=0

4 Results and discussion Numerical results were obtained for Pr= 0.707 and Reynolds numbers of 1, lO, 5o, lOO and 5oo. At each Reynolds number the computations were performed for Grashof numbers of lo 3, lO4, 5 • lO4 and lO5. Five different heater locations were considered, i.e. S/H=o.5, 4, 7.5, 11 and 14.5. In order to investigate the effect of the channel width on the heat transfer, the computations were carried out for different channel widths (W/H=o.5, 1.o, 1.5, 2.o and 2.5). In Fig. 2, streamlines and isotherms are shown for Re=lo and Gr=lo 4 for all the heater locations considered. It is seen that a cell near the opposite wall and slightly above the heater forms. Formation of the cell near the cold wall indicates that the fluid heated by the heater accelerates near the heater and the fluid near the cold walt tends to flow toward the heated surface. This shows that natural convection is effective on the flow field. Similar patterns are seen in the flow fields for all heater locations considered, except the case with the heater positioned near the channel outlet. Since, the streamline patterns are similar, the isotherm contours also show similar behavior. A steep temperature gradient near the heater exists. However, below and far above the heater the temperature distributions are uniform. In Fig. 3, streamlines and isotherms are plotted for Gr=lo 4 and S/H=7.5 to investigate the effects of Reynolds number on the flow and heat transfer. It is seen that at low Reynolds numbers (Re=l and lO) a circulating cell forms in the channel. However, at high Reynolds numbers, the fluid flows along the straight streamlines. This indicates that as the Reynolds number increases, the effect of the natural convection decreases. When the Reynolds number is equal or greater than 5o, the natural convection has no significant effects on the flow. Hence it can be said that at low Reynolds numbers natural convection regime, and at increased Reynolds numbers mixed and forced convection regimes exist in the channel. Below the heater, the temperature is unchanged. With the increasing Reynolds number, the temperature of the fluid near the cold wall also remains unchanged. At high Reynolds numbers, the temperature gradient is seen only near the heater and above it. The effects of the Grashof number on the flow and temperature fields are illustrated in Fig. 4. The flow fields in this figure were obtained by keeping the heater at the same position at S/H=4, Re=l and W/H=I. It is seen that a circulating zone forms

161

f

Illlllll

Jt

II111111 IIIIII]1 !!!!!!!! Illflll~

162

II~llll

IIIII

liHmJl

Iflrlfllt

.,,,H.,

!!!!~.~,\\

IIt\l\t/ IIIIIII

Illtl~ll

IIIIIII

a

9 Fig.

b

2a-e.

c

/1/11/t

Streamline and isotherm contours (Re=]o and

d ]~Jll. r Gr=lo4).

a

e

S/H=o.5; b S/H= 4.o; c S/H=76; d S/H= ll; e S/H=14. 5

ir illll~

I/I/

IIIIII1~

lillllll

I[111111 I~H.ll

II H m L

b I///tl/1

Fig. 3 a - e .

Streamline and i s o t h e r m contours

c

d

e

(S/H=7.5 and G r = l o 4). a Re=l; b Re=~o; c Re=5o; d R e = z o o ; e R e = 5 o o

163

a

b

e

d

Fig. 4a-d. Streamline and isotherm contours (Re=l and S/H=4). a Gr=lo3; b Gr=lo4; C Gr=5xl04; d Gr=lo 5

for all the Grashof numbers considered due to low Reynolds number (Re = 1). With the increasing Grashof number, the circulating zone elongates, and finally the edge of it reaches the channel exit and hence the fluid starts to enter into the channel through the outlet near the cold wall This indicates that at low Reynolds numbers the flow is mainly dependent on the Grashof number. The effect of the heater location on the mean Nusselt number and the maximum temperature are illustrated in Figs. 5 a and 5 b, respectively, for Re=lo and G r = l o 3, lO 4, 5• 4 and lo 5. As seen in these figures, except the case that the heater is located adjacent to the channel inlet, the heater location has negligible effect on the Nusselt number and the maximum component temperature. It is observed that when the heater is located near the entrance of the channel, the Nusselt number is higher than the Nusselt number of the other cases. This can be related to the channel entrance effect. Since the thermal and hydraulic boundary layer are yet to develop at the entrance section (on the heater) of the channel, more heat can be imparted from the heater by the fluid flowing in the channel. Variation of the mean Nusselt number and the maximum dimensionless temperature with the heater location are shown in Fig. 6 a and 6 b, respectively, for Gr= lO4 and Re= 1, lO, 50, lOO and 500. As seen in these figures, increase in the Reynolds number results in an increase in the Nuseslt number, and hence a decrease in the maximum component temperature. Again neither

the mean Nusselt number nor the maximum temperature are dependent on heater location, at different Reynolds numbers. However, when the heater is located adjacent to the inlet, higher Nusselt numbers and lower component temperatures are seen at all Reynolds numbers, as a consequence of the entrance effect. The variations of the mean Nusselt number with the Grashof number are drawn in Fig. 7 for different Reynolds numbers (Re=l, lO, 50, lOO and 500). It is seen that as the Reynolds number increases, the mean Nusselt number increases as well. Dependence of the Nusselt number on the Grashof number decreases as Reynolds number increases. At high Reynolds numbers (Re >_5o), Nusselt number is almost independent from Grashof number. At low Reynolds numbers (Re < 5o), Nusselt number increases with increasing Grashof number. This variation of the Nusselt number indicates that at low Grashof numbers, heat is transferred mainly by conduction, at high Grashof numbers the heat transfer is dominated by natural convection. At high Reynolds numbers (Re >__50) heat transfer is mainly controlled by the forced convection. The variation of the Nusselt number with the channel width at different Reynolds numbers (Re=l, lO, 50 and loo) are shown in Fig. 8 for Gr= 5 • lO4 and the heater lo cation of S/H= 4. When this figure is analyzed, it is seen that at the Reynolds numbers considered, the Nusselt number does not change with the channel gap to a considerable extend. However, at low Reynolds numbers, the

8.0

25.0 Or = r u * m r m Or ..... Cr 6 6 6 ~ 0 Or

1 0~ 10 ~ ~;~04

ooooo

0 ~6.0 Z

o3

uooooRe ~ R e ~_ 2 0 . 0 o9 _o

---e

•

Z

..~

1.0 10.0

50.0

0~r

108.0

Re =

15.0

03 o3

5.0

Z

Z

10.0

g

o~4.0

164

= =

A4&4~ R~ =

5.0

5.0

I], 2,0

&

0 0.4

0,5

o

0J L 0.4 c3_

/ r t

~-0,3

--

o

--~

&

E

~0.~
g

f

E

,o_ 0 , 2

0

03

cmO. k5

0

0

E k5 E ~0,1

0- - ~ - - - - - 4

s

oouooRe m ~ A ~ Re ~ Re 0 0 0 0 0 Re ~ Re

_E

,~ 0.1

auuumGr = ru~RfU~Gr ~ a Or o 0 6 6 0 Gr

x

10 ~ 10~ 4 5x~O 10 ~

0 0

b

4,0

8.0

12.0

0

16,0

Heoter Locotion ( S / H )

Fig. 5a, b. Variation of the mean Nusselt number and the maximum temperature with the heater location for Re= ~o at different Grashof numbers

Nusselt number curves reach an asymptotic value by an increase with increasing channel width, but at higher Reynolds numbers (Re>_ 50) it reaches the asymptotic value by a decreasing behavior with the increasing channel width. The effects of the channel gap on the Nusselt number at low Reynolds number can be explained by analyzing the stream line contours given in Fig. 9 for Re= lo. As seen in this figure, at all the channel gaps, a circulating cell forms in the channel. However, the structure of the cells are different at small and large channel gaps. As seen in Fig. 9 a, when the channel is narrow, the cell is totally confined. With the increasing channel width, the cell occupies the upper part of the channel, and causes the fluid to entrain into the channel through the outlet. Hence, bulk of the fluid circulating in the cell increases. As seen in these figures, due to the cells, the effective flow cross-sectional area at the heater level decreases. The decrease in the effective flow cross-sectional area is more when the channel width is large since there is fluid entrainment into the channel through the outlet. This decrease in the flow cross-sectional area shows itself on the flow as a jet effect which is stronger at wider channel cases. This results in a more effective cooling of the heater surface and hence a higher Nusselt number. The explanation of the behavior of the Nusselt number variation with the channel gap also lies in the flow and temperature field structures. As pointed out earlier (see Fig. 3), at higher Rey-

i~l,llllllll~l,l~llll,lb,JJ~lflll,l~l]l 4.0 8.0

= = = = =

1.0 10.0 50.0 100.0 500.0

12.0

16,0

Heeter Loection (S/H)

b

Fig. 6a, b. Variation of the mean Nusselt number and the maximum temperature with the heater location for Gr= 104 at different Reynolds numbers

14.o 12.o

~ z

lO.O

~ 8.0 z oc ~.0 03 z 4.o 2.6 2,0

4,0

6.0

8,0

Grashof Number

Fig. 7.

10.0

12,0

(xlO4)

Variation of the mean Nusselt number with Grashof number for

S/H=7.5 at different Reynolds numbers

nolds numbers (Re > 5o), the fluid particles flow along straight streamlines, no circulating cells form. Also, a temperature gradient is seen only near the wall on which the heater is located. The velocity gradient near the wall, when the channel is narrow, is higher than that when the channel is wide. Hence, when the

10.0

L

8.0

E Z

• 6,0

/

o3 Z

4,0

o3

"5

..... a a ~ ..... ~**~

2.0

0

llllIJLll,lb,'hl*lll

[3,5

Re Re Re Re

H IllLJllllllll~lll

1,0

Width

1,5

of

Channel

= = = =

1.0 10,0 50.0 100.0

J'll~'JIlllbl~llLI

2.0

2.5

3.0

(W/H)

Fig. 8. Variation of the mean Nusselt n u m b e r with the channel width for Gr= 5 x zo4 at different Reynolds numbers

channel is narrow, more fluid flows near the wall where there is a temperature gradient, and hence the heater is cooled more effectively. This results in a decrease in the Nusselt number with increasing channel width at high Reynolds numbers (Re >_5o), as seen in Fig. 8.

5 Conclusions In this study, the flow and temperature fields in an open ended vertical channel with a discrete heat source located on the left wall are numerically simulated. At the inlet, the fluid enters the channel with a uniform velocity. The effects of the heater location, Grashof and Reynolds numbers and the channel gap on the flow and temperature distributions were investigated. From the computed temperature fields, the mean Nusselt numbers were calculated and the maximum temperatures were found. The results are of the interest from the practical point of view, i.e. electronic cooling. At low Reynolds numbers (Re<5o), a circulating cell forms in the channel near the cold wall, slightly above the heater. As Reynolds number increases, the circulating cell diminishes and the fluid particles flow along straight streamlines. This implies that at low Reynolds numbers, the natural convection is the dominant flow mode, but at the higher Reynolds numbers (Re->5o), flow is dominated by the forced convection. It was observed that at all the Grashof and Reynolds numbers considered, the effect of the heater location on the Nusselt number and the maximum temperature is negligible. However, when the heater is located near the channel entrance, as a consequence of the entrance effect, the Nusselt number is higher than those that were obtained at other heater locations. At low Reynolds numbers, the Nusselt number increases with the increasing

\ Fig. 9 a - e . Streamline contours (Re=in, S/H=4 and Gr=5• a W/H=o.5; b W/H=I.o; c W/H=Ls; a

b

c

d

e

d W/H=2.o; e W/H=2.5

165

Grashof numbers. On the other hand, at high Reynolds numbers (Re>-5o), Grashof number does not play any considerable role on the Nussett number and the maximum component temperature. The channel width does not affect the Nusselt number and the maximum component temperature to a considerable extent. At low Reynolds numbers, the Nusselt number reaches an asymptotic value by an increase with the increasing Reynolds number. However, at high Reynolds numbers, the Nussdt number decreases with the increasing Reynolds number.

166

References 1. Peterson, G. P.; Ortega, A.: Thermal Control of Electronic Equipment and Devices. Advances in Heat Transfer 20 (199o) I81-314 2. Kennedy, K. L; Zebib, E.: Combined Forced and Free Convection between Parallel Plates. Grigul et al. (eds.) in 7th International Heat Transfer Conference 3 (1982) 447-451 3. Kennedy K. l.; Zebib, E.: Combined Free and Forced Convection between Parallel Plates. Int. J. Heat and Mass Transfer 26 (1983) 471-474 4. Incropera, E P.; Kerby, J. S.; Moffat, D. F.; Ramadhyani, S.: Convective Heat Transfer from Discrete Heat Sources in a Rectangular Channel. Int. J. Heat and Mass Transfer 29 (1986) lO51-1o58 5. Hasnaoui, M.; Bilgen, E.; Vasseur, P.; Robillard, L.: Mixed Convective Heat Transfer in a Horizontal Channel Heated Periodically from Below. Numer. Heat Transfer, Part A, 20 (1991) 297-315

6. Yiicel, C.; Hasnaoui, M.; Robillard, L.; Bilgen, E.: Mixed Convection Heat Transfer in Open Ended Inclined Channels with Discrete Isothermal Heating. Numer. Heat Transfer, Part A, 24 (1993) lO9-126 7. Choi, C. Y.; Ortega, A.: Mixed Convection in an Inclined Channel with a Discrete Heat Source. Int. J. Heat and Mass Transfer 36 (1993) 3119-3134 8. Tomimura, T.; Fujii, M.: Laminar Mixed Convection Heat Transfer between Parallel Plates with Localized Heat Sources. W. Aung (ed.), in Proc. Int. Syrup. on Cooling TechnoIogy for Electronic Equipment, Honolulu (1988) 233-247 9. Ravine, T. L.; Richards, D. E.: Natural Convection Heat Transfer from a Discrete Thermal Source on a Channel Wall. J. Heat Transfer 11o (1988) lOO4-1oo7 lo. Elpidorou, D.; Prasad, V.; Modi, V.: Convection in Vertical Channel with a Finite Wall Heat Source. Int. J. Heat and Mass Transfer 34 (1991) 573-578 11. Patankar, S. V.: Numerical Heat Transfer and Fluid Flow. Hemisphere Pub. Corp., New York 198o lZ. Wilkes, J. O.; Churchill, S. W.: The Finite-Difference Computation of Natural Convection in a Rectangular Enclosure. A.I.Ch.E. Journal 12 (1966) 161-166 13. Ahmed, G. R.; Yovanovich, M. M.: Numerical Study of Natural Convection from Discrete Heat Sources in a Vertical Square Enclosure. ]. Thermophysics 6 (199z) 121-127