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Numerical Heat Transfer, Part A, 37:373393, 2000

Copyright Q 2000 Taylor & Francis

10407782 rrrrr00 $12.00 H .00

NUMERICAL SIMULATION OF LAMINARFORC ED CONVECTION IN AN AIR-C OOLEDHORIZONTAL PRINTED CIR CUIT

BOARD ASSEMBLY

C. W. Leung, S. Chen, and T. L. ChanDepartm ent of Mechanical En gineering, The Hong Kong Polytechnic University,

Hung Hom, Kowloon, Hong Kong

A numerical solution of the steady-state forced convection for air flowing through a

( )horizontally oriented simulated printed circuit board PCB assembly under laminar flow

condition has been developed. The considered assembly consists of a channel formed by two

parallel plates. The upper plate is thermally insulated, whereas the bottom plate is attachedwith uniformly spaced identical electrically heated square ribs perpendicular to the mean air

flow. The bottom plate is used to simulate the PCB, and the ribs with heat generation are

used to simulate the electronic components. A second-order upwind scheme is adopted in the

calculation and a very fine mesh density is arranged near the obstacle and the channel

( )surface to achieve higher calculation accuracy. Four Nusselt numbers Nu are of particular

interest in this analysis: local distribution along the ribs surfaces, mean value for

individual surfaces of the rib, overall obstacle mean value, and overall PCB mean value

between the central lines of two obstacles. The effect of the obstacle size and the separation

between two obstacles is discussed systematically.

INTRODUCTION

Heat dissipation capability of an ele ctronic syste m has become one of the

primary limiting factors for circuit miniaturization. Effective cooling of ele ctronic

components is crucial to maintain their normal operation and hence reliability. It is

desirable that the heat generated per unit volume of a device can be dissipated to

the cooling fluid rapidly to avoid its temperature from rising significantly, which

may lead to malfunction and bre akdown of the entire device.Heat transfer analysis in channels formed by a parallel PCB is challenging.

The problem is complicated because of complex geometry of the PCB assembly

and different thermal propertie s of the board mate rials. Both experimental and

w xnumerical methods have been employed exte nsively. Incropera 1 and Peterson

w xand Ortega 2 compiled a comprehe nsive review of the relevant literature .Numerically, a simulation of combined forced and free convection in horizon-

w xtal parallel plates had been studied by Kennedy and Zebib 3, 4 using the vorticityw xstream function formulation. Braaten and Patankar 5 assessed the he at transfe r

enhance ment by free convection for flow in channel fixed uniformly with heatedw xblocks. Kim and Boehm 6 used a numerical method to predict the rate of heat

Received 15 July 1999; accepted 4 November 1999.

The authors wish to thank The Hong Kong Polytechnic University for the support of this

research project.

Address correspondence to Prof. C. W. Leung, Department of Mechanical Engineering, Hong

Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong.

373

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C. W. LEUNG ET AL.374

NOMENCLATURE

A exposed obstacle area of interest T inlet temperature of the fluidi 0( ) ( )A v step function ambient temperature

c* ratio of solid to fluid unit thermal T truncation erroru( ) ( )capacities s rC r rC U, u* dimensional and dimensionless axialP s P f

H, h dimensional and dimensionless ve locity, u* s U rU0height of obstacle, h s H rL V, v* dimensional and dimensionless axial

k* ratio of solid to fluid thermal ve locity, v* s V rU0conductivity s k rk X, x dimensional and dimensionlesss f

k thermal conductivity of the fluid x coordinate, x s X rLfk thermal conductivity of the solid Y, y dimensional and dimensionlesssL , l dimensional and dimensionless plate y coordinate, y s Y rLt t

length, l s L rL a convective heat transfer coefficientt t cn normal coordinate s* ratio of the viscosity of the material inNu local Nusse lt number the domain to the viscosity of the fluidx

Nu mean Nusselt number for each n kinetic viscosityiobstacle surface u* dimensionless temperature

( ) ( )Nu overall obstacle mean Nusselt number s T y T r Q rkim 0 f Nu overall PCB mean Nusselt numbertotalP, p* dimensional and dimensionless

2pressure, p s Pr rU Subscripts0( )Pe Pe cle t number s U rL ra s Re ? Pr0 f

Q he at input pe r unit le ngth in e ach s solid

obstacle f fluid

Re R eynolds nu mber s U L rn L left surface of the obstacle0U average velocity R right surface of the obstacle0T temperature T top surface of the obstacle

transfer from a series of heate d blocks, which were mounte d on the surface of a

ve rtical channel. The mixed conve ction flow around thre e obstacles, in both

w xhorizontal and ve rtical channels, was nume rically studie d by Kim et al. 7 . The

w xwork of Huang and Vafai 8 was of particular relevance to the multiple obstacle

configurations in which the enhance ment of forced conve ction using variousarrange ments of multiple porous obstacles in a channe l were de monstrate d.

w xDavalath and Bayazitoglu 9 simulated the convection heat transfe r nume ricallyw xfrom thre e heated rectangular blocks. Patankar and Schmidt 10 performed

extensive numerical analyses of the heat transfer in the fully developed region of a

duct containing heated, uniformly spaced blocks unde r laminar flow conditions.

w xKim and Anand 11 reporte d a numerical he at transfer study for the fullydeveloped region of a series of parallel plates with surface-mounted discrete heat

w xsources. Ghaddar et al. 12, 13 investigate d incompressible mode rate Reynolds

( )numbe r Re flow in periodically grooved channe ls using the spectral elementmethod of direct nume rical simulation. A detailed inve stigation on the force d

conve ction cooling of a heate d obstacle mounted upon a channe l wall with finite

w xelement formulation had bee n presented by Young and V aifai 14, 15 .Experimental studies on heat transfer in single channels with surface mounted

w x w xblocks were carrie d out 16] 19 . Lehmann and Wirtz 20, 21 visualized the flows inw xa channel containing uniformly spaced rectangular ribs. Tam et al. 22 carried out

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FORCED CONVECTION IN PRINTED CIRCUIT BOARD 375

an investigation with a horizontally orientate d simulated PCB assembly in turbu-

w xlent flow. Leung e t al. 23 studied the ste ady-state convection heat transfe r andpressure drop characteristics for l aminar air flows over a horizontally orientated

simulated PCB assembly experimentally and numerically.

The purpose of the present investigation is to obtain systematic computa-

tional results for forced convective cooling of the horizontal PCB assembly. Toenhance the calculation accuracy, a se cond-order upwind scheme is adopte d and a

sufficient mesh density is arranged near the wall region. Special emphasis is given

to detail the local Nu distributions, mean Nu for the obstacle surfaces, overall

obstacle mean Nu, and overall PCB mean Nu. The effect of the obstacles size and

the separation between two obstacle s are also discusse d in detail.

MATHEMATICAL MODEL

The horizontal PCB assembly is simulated by two horizontal rectangularplate s. These two plate s are assumed to be thermally insulated, whereas the lower

plate is attache d uniformly with he ate d square obstacle s. The flow betwee n two

parallel plates is represente d by a steady, incompressible, Ne wtonian fluid, as

shown in Figure 1. Thermal physical properties of the fluid and solid are taken as

constant. Buoyancy effects are assume d negligible . V iscous heat dissipation in the

fluid is assumed to be negligible when compared with conduction and conve ction.

The governing equations are continuity, momentum, and the energy equations;

their dimensionless forms can be written as

u* v*( )q s 0 1

x* y*

u* u* s* 2 u* 2 u* p*( )u* q v* s q y 2

2 2( ) x* y* Re x* x* y* v* v* s* 2 v* 2 v* p*

( )u* q v* s q y 32 2

( ) x* y* Re y* x* y*

u * u * k* 2u* 2u* 1 1( ) ( )u* q n* s q q ? ? A v 42 2 2( ) x* y* Pe ? c* Pe ? c* x* y* h

Figure 1. The geome try of a paralle l plate channel with two heated obstacle s.

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The ratio s* rRe for the block can be set to an infinitely large value such as 10 30

( )to simulate a solid. In the energy equation, A v is a step function that is set tozero e verywhere except the block.

1 v s block( ) ( )A v s 5

w 0 v s fluidh2 stands for the volume of the heat source in the source term of the energy

equation. In the fluid domain, k* and c* are both set e qual to one. In the solid

region, as the solid ve locity is identically zero the effect of c* can be eliminated.

w xTherefore, c* is set equal to one for both regions 9 .To assess the effects of the change s in governing parame te rs on the obstacle

heat transfer, the local Nu is evaluated as

a H y 1 uU

c f( )Nu s s 6x U

k u nf w all

The mean values of the Nu for the three exposed surfaces, AB, BC, and CD( ) ( )Nu , Nu , Nu , and the ove rall obstacle mean Nu Nu , are calculate d by usingL T R m

H Nu dxA xi ( )Nu s 7iA i

p Nu Ai s L, T , R i i( )Nu s 8m

A q A q AL T R

where A is an individual expose d surface area of the obstacle. The overalliobstacle mean value is thus an area weighted average of the exposed surface mean

values. In the same way, the ove rall PCB Nu, Nu , between the central lines oftotaltwo obstacles can be calculated.

Boundary Conditions

The fluid enters the channe l with a parabolic velocity profile from one end

and leaves at the other end of the plate s carrying the heat dissipated by the

obstacles. At the outlet the streamwise gradie nts of the velocity compone nts are

assumed to be zero. It is assumed that the flow is nearly fully developed at the exit

plane. It is ensured that the computational outflow boundary conditions have no

effect upon the physical domain solution by choosing an exte nde d computational

domain. At the entrance, the fluid is assumed to be at the ambient temperature. Inthe same way, the temperature gradient in the axial direction is set to zero at the

outlet. The choice of an exte nded computational domain ensures also that the

thermal boundary condition at the exit has no significant effe ct upon the solution

near the region of interest. The uppe r and lower plate s are thermally insulate d.

This condition will yield the maximum expe cted temperature in both the solid and

the fluid regions. To handle the abrupt change d in thermophysical properties

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( )within the calculation domain solid and fluid , the harmonic mean is used to

evaluate the properties at different interfaces. No-slip boundary conditions are

applied at the two paralle l plate s. The boundary conditions are expressed as

follows:

( )1. Along the upper plate 0 - x - l , y s 1 :t

( )u s 0 v s 0 u r y s 0 9

( )2. Along the lower plate 0 - x - l , y s 0 :t

( )u s 0 v s 0 u r y s 0 10

( )3. At the entrance x s 0, 0 - y - 1 :

( ) ( )u s 6 y 1 y y v s 0 us 0 11

( )4. At the outlet x s l , 0 - y - 1 :t

( ) u r x s 0 u r x s 0 v s 0 12

NUMERICAL METHOD

w xIn this study, the SIMPLE algorithm 24 is adopted as the computationalalgorithm. A control volume method was utilized to solve the conservation equa-tions using a pressure -velocity formulation. A stagge red grid was conside red such

that the ve locity compone nts are located at the control volume faces, whereas

pressure and te mpe rature are locate d at the cente rs of control volume s to avoid

the velocity-pressure decoupling. The sudden change in the diffusion coefficients( )viscosity or the rmal conductivity was handled by use of the harmonic mean to

ensure conservation and uniqueness of mass and fluxes at each control volume face

w x25 . The algebraic equations were solve d using a line-by-line technique, combining

the tridiagonal matrix algorithm and the Gauss ] Seidel method. Because of thenonline arities in the momentum equations, the ve locity compone nts were under-

relaxed.

Calculation Scheme

For two-dimensional problems, the discrete equation is shown as below:

( )a f s a f q a f q a f q a f q b 13P P E E W W N N S S

where a is the coefficient of the variable at the grid point under consideration andpthe as on the right side of the equation are the coefficients of the four immediate

neighbors. The term b is refe rred to as the source term and contains the explicit

source terms.

In terms of a straightforward finite difference approximation, the finite

differe nce approximation of the conve ction term for the second-orde r upwind

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scheme can be written as the following:

For u ) 0:

( ) uf 1( ) ( )s 3u f y 4 u f q u f q T 14 aP P W W WW WW u x 2 dP

For u - 0:

( ) uf 1( ) ( )s y u f q 4 u f y 3u f q T 14 bEE E E E E P P u x 2 dP

This scheme use s a linear extrapolation of the two upwind neighbors to

determine f. It is derived exactly by integrating the fluxes over the control volume

and is, hence, completely conse rvative and consiste nt with the control volumeformulation. The discretized equations for second-orde r upwind scheme are given

w xby the following 26 :

w <

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( ) ( )Figure 2. a Schematic of the control volume. b The mesh distribution of the typical( )computational domain h s 0.25 .

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RESULTS AND DISC USSION

An inlet length of L s 3 was confirmed to ade quate ly isolate the obstacle1region from any entrance effects, and an outlet length of L s 10 was found to2ensure that the large recirculation zone downstre am of the obstacle reattached

well ahe ad of the outlet and that the fluid exited the computational domain in a

parabolic, fully developed profile.

In the cooling of PCB asse mblies, a very important range of Re lies under

w x500; another refere nce 28 may be nee ded. Figure 3 shows a typical single obstaclecase with Re s 500, h s 0.25, and k rk s 1000 to illustrate the effects of thes fsingle obstacle on the flow and temperature. The basic characteristics of the

streamlines and isotherms at Re s 500 are shown in Figures 3 a and 3b. It is worthnoting that Figure 3c shows there is a clockwise vortex near the front of the lower

left corner. Perhaps because of the use of coarse meshes, many researchers except

( ) ( ) ( )Figure 3. Flow around the single obstacle in the channel. a Streamlines, b isotherms, c upstream( )recirculation zone, d downstream recirculation zone.

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w x14 had not paid attention to the vorte x that developed in an upstre am recircula-tion zone. A clockwise recirculation is also deve loped downstre am from the

obstacle. In the same way as the very fine meshes are arranged near the obstacle

and the channel surfaces in the present work, Figure 3d shows a ve ry we ak

anticlockwise vorte x that is de veloped in the back of the lower right corner.

Figure 4 shows the effects of Re on the flow. As Re increases, the length andstrength of the downstre am recirculation zone incre ase. A comparison of the

recirculation zone and reattachment length behind the obstacle with those pre-

w xdicted by Young and V afai 14 has be en made and is within ; 2% .Figure 5 shows the variation of upstream recirculation zone with Re and

obstacle sizes. As Re incre ases, the strength and the size of upstre am recirculation

zone also increase. When Re incre ases from 200 to 500, the vortexs height

increases from 0.4 h to 0.5h, and its width increases from 0.5h to 1.0 h for h s 0.25.This situation also occurs when h s 0.5. It can be concluded that for the upstream

recirculation zone , its height changes from 0.4h to 0.5h, whereas its width variesfrom 0.5h to 1.0h under different Re. The magnitudes of the ve locities in this

region are two- to three-fold less than those in the main flow region. The

computational results of the upstre am recirculation zone are consiste nt with those

w xpresented in 14 .

Figure 4. The variation of streamlines around a single obstacle with Re for h s 0.25.( ) ( ) ( )a Re s 200, b Re s 350, c Re s 500.

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Figure 5. The variation of upstream recirculation zone with different Re and obstacle sizes.( ) ( ) ( ) ( )a Re s 200, h s 0.25; b Re s 350, h s 0.25; c Re s 500, h s 0.25; d Re s 200, h s 0.5;( ) ( )e Re s 350, h s 0.5; f Re s 500, h s 0.5.

The variation of the vortex in back of the lower right corner with Re is as

shown in Figure 6; the presence of the very weak anticlockwise vorte x behind the

w xobstacle at the lower right corne r has bee n seen in recent literature 29 . Themagnitudes of the velocities within this zone are much lower than those within

the other two vortice s. On the other hand, the area of the vortex is almost

unchanged with the Re. When h s 0.25, the height and width of the vorte x arearound 0.05h. When h s 0.5, its height and width are around 0.1h. As the velocitywithin this region is ve ry small, it could be predicte d that the heat dissipation in

this zone is extremely slow. For the sake of brevity, it is not presented here.

Figure 6. The variation of the vorte x in back of the lower right corner with different( ) ( ) ( )Re . a Re s 200, b Re s 350, c Re s 500.

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w xFigure 7. Comparison of local Nu with Young and V afai 14 and

w xanalytical solution 30 , h s 0.25, Re s 500, and k rk s 1000.s f

w xFigure 7 shows the comparison of local Nu with those presented in 14 . The

w xNu on the upstre am and downstre am facing surfaces in 9 show relative ly smalland nearly constant values when compared with those on the upper surface. In the

present work, Nu distributions on the left and right surfaces are found to havexrelatively greater values near the upper corners, which agrees well with the findings

w x w xin 14 and the analytical solution in 30 . The peak value along ABCD occurs at theupper left corne r of the obstacle, which is the leading edge of hydrodynamic

boundary layer and the thermal boundary layer. The thermal boundary layer at this

point is the thinne st, and there fore, the local Nu is the highest. Around the upper

right corne r of the obstacle , the local Nu has a slight increase. This effect is

because of the relatively lower surface temperature at this corner, and downstream

the fluid is not furthe r heate d by the wall. This phenomenon has bee n reported

w xalso in 14 .Figure 8 shows the variation of local Nu with Re. The local Nu increases withincreased Re. As Re incre ases, the magnitude of the temperature gradie nt in-

creases near the upstream upper corner. The more rapid fluid movement reduces

the thickne ss of the the rmal boundary layer; the magnitude of the temperature

gradie nt the refore is incre ased. Based on this reason, the local Nu near the

upstre am upper corne r has a highe r value . Along the right surface Nu incre ases

slightly with increased Re.

Figure 9 shows variation of the surface mean Nu and the overall obstacle

mean Nu for different obstacle sizes with Re. All of the Nu increases as Reincreases. The ave rage Nu on the left surface is found to be comparable with that

of the top surface , whereas that on the right surface is lower by almost 70% .

Figure 10 shows the effect of different obstacle sizes on surface mean Nu and

overall obstacle mean Nu at Re s 300. The greatest effect occurs on the topsurface mean Nu; howeve r, the magnitude of Nu and Nu decreases slightly withL Rthe incre asing obstacle sizes. With increase in the width of the top surface from

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Figure 8. Variation of local Nu with different Re for h s 0.25.

h s 0.25 to h s 0.5, the magnitude of Nu will decrease by nearly 30% bec ause aTlarge r portion of the surface is remote from the leading edge and therefore has a

lower local Nu. The Nu for h s 0.5 is almost 18% lower than that for h s 0.25,mwhereby Nu is 7% lower. The values of Nu remain almost unchange d. This isL Rbecause the flow directly behind the obstacle is dominant in which the heat

transfer mechanism is the recirculation; therefore the immediate downstream

region is only affe cted slightly by changes in the obstacle size or flow rate. For all

the expose d obstacle surfaces, an increase in height increases the internal thermal

resistance, which leads to a decrease in heat transfer rate. This produces a similar

trend for Nu : At the higher Re, the larger obstacle size has a decrease in Nu ofm mabout 20% .

Figure 9. Variation of the surface mean Nu and the overall obstacle

mean Nu with the Re for different obstacle sizes, h.

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Figure 10. V ariation of the surface mean Nu and the overall

obstacle mean Nu with the obstacle size h when Re s 300.

To inve stigate the effect of separation be tween two obstacles, the flow and

heat transfer is also calculate d by considering two obstacle s in the channe l. Fig-

ure 11 shows the stre amlines with the two-obstacle arrangeme nt with h s 0.25 forRe s 200, 300, and 500, respectively. The distance between central lines is fixed at

( )s s 2.5. At a low Re Re s 200 , the stre amlines be twee n the two obstacle s fall

( ) ( )Figure 11. The streamlines around two heate d obstacles with h s 0.25. a Re s 200, b Re s 350,( )c Re s 500.

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toward the lower surface of the channel, and the recirculation is not touching the

right surface of the second obstacle . So near the right surface of the second

obstacle, the streamlines are rathe r rapid. As Re increases, the recirculation

touches the right surface of the second obstacle. The stre amlines in the main flow

zone are in a direction almost parallel to the plate, and the recirculation almost

coincides with the top of the obstacles. The strength of the vortex increases and thecenter of the recirculation core moves to the right as Re increases.

Figure 12 shows the nondimendisional velocity distributions at the cente r

region between two obstacles and at the center of the second obstacle. It is noted

that the velocity distribution is affected more at lower Re by the existence of the

cavity between two obstacles. This causes the decrease in the maximum velocity in

the main flow to satisfy the mass continuity. The velocity distribution over the

( )Figure 12. Nondimensional velocity distribution a at the center( )between two cavities, b at the center of the second obstacle.

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obstacle is shown to be independe nt of the Re. A similar result was observed

w xin 11 .Figure 13 shows the te mperature distribution with the two-obstacles arrange -

ment for h s 0.25 at different Re. The isotherms around the two obstacle s havenoticeable difference. Notice that the value of k rk is relatively higher; these twos f

obstacles are almost isothermal. The te mperature distribution betwee n the obsta-cles is similar to temperature distribution in the flow ove r a cavity.

Figure 14 shows the local Nu for the first and the second obstacle s at

differe nt Re. The first obstacle has a much l arger Nu value along its left surfacexthan the second obstacle because of the impact of the core flow as it is redirected

into the bypass region.

Figure 15 shows the variations of the surface mean Nu, overall obstacle me an

Nu and overall PCB mean Nu with the Re as the central distance betwee n two

obstacles fixed at 2.5. For the first obstacle, the Nu and Nu are comparable . AtT L

higher Re, Nu is large r than Nu ; this is also bec ause the core flow is redirectedL Tinto the bypass region. Nu is about 30% of Nu and Nu and is nearly equal to aR T Lconstant. For the second obstacle , the Nu is distorte d since the enlarging of theLvorte x with increased Re. The total Nu increases slightly with the incre ased

obstacle size. Howeve r, as the heat transfer areas incre ase with increased obstacle

size, it could be concluded that the effect of enhanced heat transfer is because of

the increased obstacle size.

Figure 16 shows the Nu at the different central distance between twototalobstacles. The stre amline s between two obstacle s fall toward the lower surface of

the channe l at a low Re. As Re incre ase s, the streamlines in the main flow zone

are in the direction almost parallel to the plate, and the recirculation can fill fully

Figure 13. The temperature distribution with two obstacles for h s 0.25,( ) ( ) ( )a Re s 200, b Re s 350, c Re s 500.

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Figure 14. The local Nu for the first and second he ated obstacle.( ) ( )a first obstacle, h s 0.25, b second obstacle, h s 0.25.

the cavity between two obstacles. At this time, the Nu nearly rises up to atotalconstant. From Figure 15, it is noted the Nu can achieve a large r value astotals s 2.5 compared with s s 1.25 or 3.75. When s is changed from 1.25 to 2.5, thewider spacing allows the core flow to further mix with the fluid in the cavities. This

increases the transfer of thermal energy out from the cavities and into the core

flow, reducing the transport toward the upstre am obstacle s. As the he at transfe r

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Figure 15. The variation of the face mean Nu, overall obstacle mean

Nu, and total mean Nu with different Re as the central distance( )between two obstacles fixed at s s 2.5. a First obstacle, h s 0.25;

( ) ( )b first obstacle, h s 0.375; c first obstacle, h s 0.5.

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( ) ( ) ( )Figure 15. Continued d second obstacle, h s 0.25; e second( )obstacle, h s 0.375; f second obstacle, h s 0.5.

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Figure 16. The total mean Nu Nu at different central distancestotalbetween two obstacles, h s 0.25.

along the bottom of the plate is very weak, when s is further increased from 2.5 to

3.75, the Nu decreases.total

CONCLUSIONS

The present study is a simulation of the horizontal PCB assembly, with the

obstacles simulating the electronic components. A second-orde r upwind scheme

was adopted and a fine mesh density was distribute d ne ar the obstacle and the

channel surfaces to find three vortices within the calculation domain. A compari-

w xson with those schemes in 14 and an analytical solution shows it can provide areasonable estimate for the Nu on the top surface of the obstacle . The depende nce

of flow and temperature fields on Re, obstacles sizes, and the separation between

two obstacles is documented and the results show that these parame ters have a

significant effect on the flow and temperature fields.

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