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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
3Compact Analytical Models For Effective Thermal Conductivity of Rough Spheroids
Packed Beds, IMECE 2004, Nov. 13-19, 2004, Anaheim, California, USA.
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
4Compact Analytical Models For Effective Thermal Conductivity of Rough Spheroids
Packed Beds, IMECE 2004, Nov. 13-19, 2004, Anaheim, California, USA.
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
5Compact Analytical Models For Effective Thermal Conductivity of Rough Spheroids
Packed Beds, IMECE 2004, Nov. 13-19, 2004, Anaheim, California, USA.
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
6Compact Analytical Models For Effective Thermal Conductivity of Rough Spheroids
Packed Beds, IMECE 2004, Nov. 13-19, 2004, Anaheim, California, USA.
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
7Compact Analytical Models For Effective Thermal Conductivity of Rough Spheroids
Packed Beds, IMECE 2004, Nov. 13-19, 2004, Anaheim, California, USA.
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
8Compact Analytical Models For Effective Thermal Conductivity of Rough Spheroids
Packed Beds, IMECE 2004, Nov. 13-19, 2004, Anaheim, California, USA.
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
=
9Compact Analytical Models For Effective Thermal Conductivity of Rough Spheroids
Packed Beds, IMECE 2004, Nov. 13-19, 2004, Anaheim, California, USA.
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]
10Compact Analytical Models For Effective Thermal Conductivity of Rough Spheroids
Packed Beds, IMECE 2004, Nov. 13-19, 2004, Anaheim, California, USA.
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
11Compact Analytical Models For Effective Thermal Conductivity of Rough Spheroids
Packed Beds, IMECE 2004, Nov. 13-19, 2004, Anaheim, California, USA.
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
12Compact Analytical Models For Effective Thermal Conductivity of Rough Spheroids
Packed Beds, IMECE 2004, Nov. 13-19, 2004, Anaheim, California, USA.
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
13Compact Analytical Models For Effective Thermal Conductivity of Rough Spheroids
Packed Beds, IMECE 2004, Nov. 13-19, 2004, Anaheim, California, USA.
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
14Compact Analytical Models For Effective Thermal Conductivity of Rough Spheroids
Packed Beds, IMECE 2004, Nov. 13-19, 2004, Anaheim, California, USA.
ACKNOWLEDGMENTS
• Natural Sciences and Engineering Research Council of Canada (NSERC)
• The Center for Microelectronics Assembly and Packaging (CMAP)
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