Modeling of Natural Convection Modeling of Natural Convection in Electronic Enclosuresin Electronic Enclosures

P. Teertstra, M.M. Yovanovich, J.R. P. Teertstra, M.M. Yovanovich, J.R. CulhamCulham

Microelectronics Heat Transfer LaboratoryMicroelectronics Heat Transfer LaboratoryDepartment of Mechanical EngineeringDepartment of Mechanical Engineering

University of WaterlooUniversity of WaterlooWaterloo, Ontario, CanadaWaterloo, Ontario, Canada

June 2, 2004June 2, 2004

11

OutlineOutline•• Introduction and problem descriptionIntroduction and problem description•• Model developmentModel development•• Numerical simulationsNumerical simulations•• ValidationValidation•• SummarySummary

22

IntroductionIntroduction•• Current design practice for sealed electronic enclosuresCurrent design practice for sealed electronic enclosures

Numerical CFD simulationsNumerical CFD simulationsExperimental prototype testingExperimental prototype testingTime consuming, expensiveTime consuming, expensive

•• AnalyticallyAnalytically--based modelingbased modelingQuick, easy to implementQuick, easy to implementIdeal for preliminary design, parametric studiesIdeal for preliminary design, parametric studies

•• Objective: Objective: to develop and validate a natural convection model for to develop and validate a natural convection model for simple, sealed enclosuressimple, sealed enclosures

Vertical rectangular flat plate at center of a Vertical rectangular flat plate at center of a cuboidcuboid shaped enclosureshaped enclosureFull range of Full range of RayleighRayleigh number from laminar natural convection to number from laminar natural convection to conductionconduction

33

Problem DescriptionProblem Description•• Enclosure dimensions Enclosure dimensions

•• Isothermal boundary conditionsIsothermal boundary conditions

•• Total heat transfer rateTotal heat transfer rate

•• RayleighRayleigh number number

oo

o

i

i

i

o

Lb

WL

WL

LL ,,,

( )0for* →=

−= RaS

TTAkQNu

ii Aoii

A

( )( )αν

β3

ioiA

ATTgRa

i

−=

oi TT >iT

44

General Model FormulationGeneral Model Formulation

•• Combination of three asymptotic solutions Combination of three asymptotic solutions (Teertstra, 2003)(Teertstra, 2003)

212211

−

∗

+

+=

bltrAA NuNu

SNuii

==

=∗

bl

tr

A

NuNu

Si

conduction shape factorconduction shape factor

transition flow convectiontransition flow convectionlaminar boundary layer laminar boundary layer flow convectionflow convection

Ra√Ai

Nu √

Ai S*√Ai

Nutr

Nubl

Nutr Nubl

1[( ) + ( ) ]-n -n 1/n

Nutr Nubl

1[( ) + ( ) ]-n -n 1/nS*

√Ai +

55

Conduction Shape FactorConduction Shape Factor•• Composite model Composite model (Churchill and (Churchill and UsagiUsagi, 1972), 1972)

•• one dimensional conduction in gapone dimensional conduction in gap

•• shape factor independent ofshape factor independent of

( ) ( )[ ] 3223*23*0

*∞→→ += LbLbA SSS

i δe →∞( )

bA

ASS

bAS i

iLb

i === ∗→0

= isothermal flat plate in full space region= isothermal flat plate in full space region

= equivalent sphere in full space region= equivalent sphere in full space regionsphere

plate

R

R

0→Lb

∞→Lb Lb

sphereplatei

Lb RRRRAk

S −==∞→1*

( ) 2ooeff WLd +=

66

Laminar Boundary LayerLaminar Boundary Layer•• AssumptionsAssumptions

uniformuniformNonNon--intersecting boundary layersintersecting boundary layers

•• Series combination of resistancesSeries combination of resistances

•• Convection at boundaries modeled using Lee, Yovanovich and Convection at boundaries modeled using Lee, Yovanovich and JafarpurJafarpur (1991)(1991)

Rconv = Ri + Ro

( )( )io

i

iconvibl RR

RAkRAk

Nu+

==1

1111

ii

bii NuAkQ

TTR 11=

−=

oo

obo NuAkQ

TTR 11=

−=

Tb

( ) 41Pr AAA RaGFNu =

77

Laminar Boundary LayerLaminar Boundary Layer

•• PrandtlPrandtl number functionnumber function

•• Body gravity functionsBody gravity functions

( ) 513.0Pr =F

( ) 81812 iiA LWGi=

( )4554107

41

1

Pr

+

=

o

i

ii

A

A

o

i

AAbl

G

G

AA

RaGFNu

for air at STPfor air at STP

( ) ( )( )( )

43

67

343481

222625.02

++++

=oooo

oooA WLbWL

WbLWbGo

(Lee et al., 1991)(Lee et al., 1991)

((JafarpurJafarpur and Yovanovich, 1993)and Yovanovich, 1993)

88

Transition FlowTransition Flow•• Boundary layers merge at low Boundary layers merge at low

RayleighRayleigh numbersnumbers•• Linear temperature distribution in core Linear temperature distribution in core •• Convective heat transfer in top and Convective heat transfer in top and

bottom recirculation regionsbottom recirculation regions•• Enthalpy balance in end regionsEnthalpy balance in end regions

( ) iAiio

iitr Ra

ALLLW

Nu3

eff

13602

+=

δ

io LL ,effδ = gap spacing of equivalent spherical cavity= gap spacing of equivalent spherical cavity

= effective flow length on outer, inner wall= effective flow length on outer, inner wall

99

Numerical SimulationsNumerical Simulations

•• Conduction shape factorConduction shape factorFlothermFlotherm simulations of 56 simulations of 56 enclosure geometriesenclosure geometries •• Natural convectionNatural convection

IcepakIcepak simulations of 42 simulations of 42 enclosure geometriesenclosure geometries

05.012,1,5.0

2,6.1,2.1,05.12040

→===

===

o

ooii

io

oi

LbWLWL

LLCTCT oo

05.012,1

2,6.1,2.1,05.1

→===

=

o

ooii

io

LbWLWL

LL

1010

Conduction Model ValidationConduction Model Validation

b / Lo

S* √Ai

10-1 100 101100

101

102Lo / Li1.051.21.62.01.051.21.62.0

model

data

Li / Wi =

Lo / Wo =

11

1111

Convection Model ValidationConvection Model Validation

Ra√Ai

Nu √

Ai

102 103 104 105 106 107

25

75

125

175 10.50.20.10.05

10.50.20.10.05

model

data

b / LoLo / Li =

Lo / Wo =

Li / Wi =

1.05

1

1

1212

Convection Model ValidationConvection Model Validation

Ra√Ai

Nu √

Ai

102 103 104 105 106 107

25

75

125

175 10.50.20.10.05

10.50.20.10.05

model

data

b / LoLo / Li =

Lo / Wo =

Li / Wi =

1.2

0.5

0.5

1313

Convection Model ValidationConvection Model Validation

Ra√Ai

Nu √

Ai

102 103 104 105 106 107

25

75

125

175 10.50.20.10.05

10.50.20.10.05

model

data

b / Lo

Lo / Li =

Lo / Wo =

Li / Wi =

1.6

2

2

1414

Convection Model ValidationConvection Model Validation

Ra√Ai

Nu √

Ai

102 103 104 105 106 107

25

75

125

175 10.50.20.1

0.051

0.50.20.1

0.05

model

data

b / Lo

Lo / Li =

Lo / Wo =

Li / Wi =

2

1

1

1515

SummarySummary•• Analytical model developed for natural convection for a Analytical model developed for natural convection for a

vertical plate in a sealed, vertical plate in a sealed, cuboidcuboid shaped enclosureshaped enclosure

•• Validated with data from CFD simulationsValidated with data from CFD simulations10 % average difference with numerical data10 % average difference with numerical data

•• Demonstrates trends in data as function of geometry Demonstrates trends in data as function of geometry and and RayleighRayleigh numbernumber

•• Future workFuture workIsofluxIsoflux inner boundary conditioninner boundary condition

Array of vertical platesArray of vertical plates

Experimental validation of analytical modelsExperimental validation of analytical models

1616

AcknowledgementsAcknowledgements

•• NSERC NSERC –– Natural Sciences and Engineering Research Natural Sciences and Engineering Research Council of CanadaCouncil of Canada

•• MMO MMO –– Materials and Manufacturing OntarioMaterials and Manufacturing Ontario

•• CMAP CMAP –– Centre for Microelectronics Assembly Centre for Microelectronics Assembly and Packagingand Packaging