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Compact Analytical Models For Effective Thermal Conductivity of Rough Spheroids Packed Beds, IMECE 2004, Nov. 13-19, 2004, Anaheim, California, USA.

COMPACT ANALYTICAL MODELS FOR EFFECTIVE THERMAL CONDUCTIVITY OF

ROUGH SPHEROID PACKED BEDS

Majid BahramiM. M. Yovanovich

J. R. Culham

Microelectronics Heat Transfer LaboratoryDepartment of Mechanical Engineering

University of WaterlooOntario, Canada

2Compact Analytical Models For Effective Thermal Conductivity of Rough Spheroids

Packed Beds, IMECE 2004, Nov. 13-19, 2004, Anaheim, California, USA.

OVERVIEW

• Introduction

• Motivations and Objectives

• Conduction Through Contact Spots

• Conduction Through Interstitial Gas

• Present Model

• Comparison with Experimental Data

• Conclusions



INTRODUCTION

• high ratio of solid surface area to volume.

• packed beds applications: – Catalytic reactors, heat recovery systems, heat exchangers, heat

storage systems, and insulators

• regular packing: Simple Cubic (SC), Body Center Close (BCC), andFace Center Close (FCC)

S2 DD

FCCD

SC



MOTIVATIONS AND OBJECTIVES

• existing models can be categorized into: – numerical (FEM) models:

• Buonanno et al. : time consuming, B.C. must be fed into the code for thermal contact resistance

– analytical models:• Slavin et al. : a point contact between spheres assumed• Ogniewicz & Yovanovich and Turyk & Yovanovich: limited to

smooth spheres

• develop compact models for determining effective thermal conductivity that account for:– roughness– gas rarefaction effect– contact load– gas temperature and pressure



REGULAR PACKED BED ARRANGMENTS

• solid fraction is defined

VVs /=ε

D

D

2 3 D / 3

2S3 D / 3 S2 D

Simple Packing (SP) Body Center Close (BCC) Face Center Close (FCC)e= 0.524 e= 0.680 e= 0.740

S2 D

0.740FCC0.680BCC0.524SC

Solid fractionPacking



HEAT TRANSFER MECHANISMS IN PACKED BEDS

two main paths for transferring thermal energy in packed beds are:

• conduction through microcontacts• heat transfer through interstitial gas

Q

Q

F

F

bL

microgaps and microcontactsheat flow lines

Q Qgs

k

1

gk r

2

QGLa

macrogap heat flow lines

GQ

gkgas ρ

gk

s

T

T

ks

isothermal plane

isothermal plane

1

2aL



THERMAL RESISTANCE NETWORK

g

i,1

i,2

s jRG

1R

2R

1T

2T

RL,1

L,2

T1 1T

T2 2T

R

R

RRGR

T

T <<

<<

GR

RG

thermal joint resistance network components:

• macro-constriction, RL

• micro-constriction, Rs

• microgap resistance, Rg

• macrogap, RG

( )1

11

/1/11

−

−⎥⎥⎦

⎤

⎢⎢⎣

⎡+

++=

GLgsj RRRR

R



CONDUCTION THROUGH CONTACT SPOTS

• macro-constriction resistance, R, Bahrami et al. [17]

T

Z∆T

T

T

temperature profile

microcontacts constriction resistance

body 1

body 2

resistance

Q

Q

F

F

aL

Qi

Q = ΣQi

1

2

( ) ( )

LsL

H

L

HH

H

akR

PPPP

aa

aa

PPP

PP

21

147.051.251.347.001.0/605.1

/and/

37.111

1

'0

'0

'0

'0

'

2

075.0,0

0'0

20

=

⎪⎩

⎪⎨⎧

≤≤−≤≤=

==

+==

−=

−

ρτσρα

τα

ξξ γ

• micro-constriction resistance, RsBahrami et al. [14]

( )Fk

mHRs

s/565.0 * σ

=



CONDUCTION THROUGH INTERSTITIAL GAS

dr

r

aL

F

P(r)ρ

b

Q

L

dqG

Y(r) σ, m,

F

E'isothermal plane Ti,1

T1

T2

heat flow lines

H'

⎟⎟⎠

⎞⎜⎜⎝

⎛

++

=

σπ

σ

2/1ln

2

1

22

2

Maaak

aR

Lg

g

101-

201-

1 '03.0erfcand

'2erfc a

HPa

HPa −⎟

⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛=

conduction regimes in a gas layer between two parallel plates:• continuum• temperature-jump or slip• transition• free-molecular

MSbBaA

ABASBSS

Rk

LL

Gg

+−=−=−=

−+⎟⎠⎞

⎜⎝⎛

−−

=

02222

ln

12

ωρρρ

π

microgap resistance, Rg, Bahrami et al. [26]

macrogap resistance, RG, Bahrami et al. [25]



CONDUCTION IN BASIC CELLS

steps to determine the bed conductivity:

• calculate the relation between the apparent load and contact load

• break up the unit cell into contact regions

• calculate the thermal joint resistance of a contact region

• determine the effective conductivity

cc

cc AR

Lk =

Rc

RBR

RBR

top copper plate

bottom copper plate

Fb

bF

D

L

r

basiccell

Q

Buonanno et al. [3, 5] experimental apparatus100Cr6 s. steel spheresLength of the bed = 15 cmD = 19.05 mmspheres H = 8.32 GPaspheres E = 200 GPaspheres k = 60 W/mK

air at 20 C, 1 atm.air k = 0.027 W/mK

copper k = 398 W/mKcopper E = 117 GPa

SC packing is shown.

bed



COMPARISON WITH EXPERIMENTAL DATA

♥ ♥

♥ ♥ ♥ ♥ ♥ ♥

Pg mm Hg

Rj

K/W

10-5 10-4 10-3 10-2 10-1 100 101 102 103

20

40

60

80

100F = 16 N

F = 22.5 N

F = 55.6 N

F = 195 N

F = 267 N

F = 467 N

Kitscha and Yovanovich [23] datacarbon steel sphere of radius 12.7 mmflat steel 1020, σ = 0.13 µm, bL = 12.7 mmair, Pr = 0.70, γg = 1.4, αT = 0.87, Λ0 = 64 nmkg (W / m K) = 0.0021+ 8 x10−5 T (K)

present modelSC unit cell

Pg mmHgR

jK

/W10-2 10-1 100 101 102 103

20

40

60

80

100

F= 16 N

F= 196 N

Kitscha and Yovanovich [23] datacarbon steel sphereof radius 12.7 mmflat steel 1020, σ = 0.13 µm, bL = 12.7 mmargon, Pr = 0.67, γg = 1.67, αT = 0.90, Λ0 = 66.6 nmkg (W/ mK) = 0.0159 + 4 x10−6 T (K)

present modelSC unit cell

F = 56 N

F= 467 N

Kitscha and Yovanovich (1974) SC basic cell data

air argon



COMPARISON WITH EXPERIMENTAL DATA

σ µm

k eW

/mK

10-3 10-2 10-1 100 1010.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Buonanno et al. [3] data, SC packing100Cr6 stainless steel spheres of radius 19.05 mmks = 60 W / mKEs = 200 GPa, νs = 0.3Hmic = 8.32 GPaair at 1 atm pressure, T = 20 °Ckg = 0.027 W / mK, Pr = 0.7, γg = 1.4αT ≈ 0.78Fc = 0.983 N

F N

k eW

/mK

10-1 100 1010

0.1

0.2

0.3

0.4

0.5

0.6

Buonanno et al. [5] data, SC packing100Cr6 stainless steel spheres of radius 19.05 mmks = 60 W / mKEs = 200 GPa, νs = 0.3Hmic = 8.32 GPaair at 1 atm pressure, T= 20 °Ckg = 0.027 W / mK, Pr = 0.7, γg = 1.4Fc = 0.983 N

σ = 0.03 µm

σ = 1.7 µm

SC packed bed, Buonanno et al. (2003) data

atmospheric air



SUMMARY AND CONCLUSIONS

• compact models are proposed for determining effective thermal conductivity in regularly packed beds, SC and FCC arrangements

• present model accounts for thermophysical properties of spheres and gas, load, roughness, gas temperature and pressure, and gas rarefaction effects

• the present model is compared against experimental data, both SC and FCC, over a variety of packed bed conditions and good agreement is observed



ACKNOWLEDGMENTS

• Natural Sciences and Engineering Research Council of Canada (NSERC)

• The Center for Microelectronics Assembly and Packaging (CMAP)

thermal conductivity of packed beds - university of · pdf file3 compact analytical models for...

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