heat flux gradient limits for liquid-protected divertors s. i. abdel-khalik, s. shin, and m. yoda...
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Heat Flux Gradient Limits for Liquid-Protected
Divertors
S. I. Abdel-Khalik, S. Shin, and M. Yoda
ARIES Meeting, San Diego (Nov 4-5, 2004)
G. W. Woodruff School ofMechanical Engineering
Atlanta, GA 30332–0405 USA
2
• Work on Liquid Surface Plasma Facing Components and Plasma Surface Interactions has been performed by the ALPS and APEX Programs
• Operating Temperature Windows have been established for different liquids based on allowable limits for Plasma impurities and Power Cycle efficiency requirements
• This work is aimed at establishing limits for the maximum allowable heat flux gradients (i.e. temperature gradients) to prevent film rupture due to thermocapillary effects
Problem Definition
3
• Maximum Temperature gradients to prevent film rupture were previously established -- Analysis was extended to determine the maximum allowable heat flux gradients (based on comment made by Don Steiner)
• Spatial Variations in the wall and Liquid Surface Temperatures are expected due to variations in the wall loading
• Thermocapillary forces created by such temperature gradients can lead to film rupture and dry spot formation in regions of elevated local temperatures
• Initial Attention focused on Plasma Facing Components protected by a “non-flowing” thin liquid film (e.g. porous wetted wall)
Problem Definition
4
Problem Definitionopen B.C. at top, 0
y
T
y
v
y
u
yl>>hoperiodic B.C. in x direction
)2
-(2
cos2
)(L
xL
TTxT s
ms
Specified Non-uniform Wall Temperature
• Two Dimensional Cartesian (x-y) Model (assume no variations in toroidal direction)
• Two Dimensional Cylindrical (r-z) Model has also been developed (local “hot spot” modeling)
L
Gas
Liquid y Wall o initial film thickness
x
Convective cooling ])([)( TxThxq s
)
2(
2cos1 l
low
xx
xqq
Non-uniform heat flux
L
Gas
Liquid y Wall o initial film thickness
x
Non-uniform wall temperature
Specified Non-uniform Surface Heat flux
Initially, quiescent liquid and gas (u=0, v=0, T=Tm at t=0)
5
Variables definition
• Non-dimensional variables
La o
o
yy
o
ax
L
xx
)/( L
uu
LL
)/( La
vv
LL
)/( 2LLL
tt
oL
LgV
s
m
T
TTT
32
22
FroL
L
o
g
gg
V
ooL
L
o
ogLV
22
We
L
LL
k
cPr
LL
oso T
M
y
Tkq L
oLo
Tkq
~
L
o
k
qT 0~
L
o
L
o
k
h
k
hL
LaNu
)/( Loo kq
TTT
2
12cos1 xQ
Specified Non-uniform Wall TemperatureSpecified Non-uniform
Surface Heat flux
6
• Energy
• Momentum
• Conservation of Mass
Governing Equations
0
y
v
x
u
y
uv
x
uu
t
ua 2
y
vv
x
vu
t
va 4
y
Tk
yx
Tk
x
a
y
Tcv
x
Tcu
t
Tca
Pr
1
Pr
22
jtn ˆ2Fr
224
dss
ay
v
ya
y
u
xa
x
v
xa
y
p
itn ˆ2 22
dssx
v
ya
y
u
yx
u
xa
x
p
T M/Pr)(We/1
7
Asymptotic Solution
• Asymptotic solution used to analyze cases for Lithium, Lithium-lead, Flibe, Tin, and Gallium with different mean temperature and film thickness
• Asymptotic solution produces conservative (i.e. low) temperature gradient limits
• Limits for “High Aspect Ratio” cases analyzed by numerically solving the full set of conservation equations using Level Contour Reconstruction Method
0FrPr)/(M2
3
We
Fr3
32
x
T
xx
a s 0)(
1
)/(
)/(
2
3
L3
3
2L
x
Q
aNux
Q
gka
LQ
xxgL o
o
Specified Non-uniform Wall Temperature * Specified Non-uniform Surface Heat flux
* [Bankoff, et al. Phys. Fluids (1990)]
• Long wave theory with surface tension effect (a<<1). Governing Equations reduce to :
8
• Similar Plots have been obtained for other aspect ratios
• In the limit of zero aspect ratio
Asymptotic Solution
1/Fr
(T
s/L)/
(L2/
Lo
o2)
Max
imum
Non
-dim
ensi
onal
T
empe
ratu
re G
radi
ent
1/We=107
1/We=106
1/We=105
1/We=104
1/We=103
1/We=102
a=0.02aNu=100
aNu=10-1
aNu=10-2
aNu=10-3
aNu=10-4
(Q/L
)/(a
Lgk/
o)
o/LgL2
Max
imum
Non
-dim
ensi
onal
H
eat F
lux
Gra
dien
t
Specified Non-uniform Wall Temperature Specified Non-uniform Surface Heat flux
Fr
1
12Pr/M
2crit
9
2
2
ooL
L
ParameterLithium Lithium-Lead Flibe Tin Gallium
573K 773K 573K 773K 573K 773K 1073K 1473K 873K 1273K
Pr 0.042 0.026 0.031 0.013 14 2.4 0.0047 0.0035 0.0058 0.0029
1/Fr 1.2104 2.2105 1.9105 6.3105 1.2103 3.8104 5.3105 6.7105 5.7105 8.2105
1/We 7.8105 1.3106 9.4105 3.0106 1.3104 4.0105 4.2106 4.8106 6.9106 1.0107
[K/m] 2.8 1.5 4.4 1.3 140 4.3 0.72 0.55 1.6 1.1
* 1/We o, 1/Fr o3
Results -- Asymptotic Solution– Specified Non-uniform Wall Temp (o=1mm)
• Property ranges
CoolantMean Temperature
[K]
Max. Temp. Gradient : (Ts/L)max [K/cm]
Asymptotic Solution Numerical Solution
Lithium 573 13 30
Lithium-Lead 673 173 570
Flibe 673 38 76
Tin 1273 80 113
Ga 1073 211 600
10
CoolantMean Temperature
[K]o/LgL2
Max. Heat Flux. Gradient : (Q/L)max [(MW/m2)/cm]
aNu=1.0 aNu=0.1 aNu=0.01
Lithium 573 2.5410-4 0.61 3.110-2 3.010-3
Lithium-Lead 673 2.0410-5 4.9 0.24 2.410-2
Flibe 673 4.3510-5 6.410-2 3.210-3 3.210-4
Tin 1273 3.0410-5 6.8 0.34 3.310-2
Ga 1073 4.8110-5 19 0.97 9.510-2
o=10 mm, a=0.02
CoolantMean
Temperature [K]
Max. Heat Flux. Gradient : (Q/L)max [(MW/m2)/cm]
a=0.05 a=0.02 a=0.01 a=0.005 a=0.002
Lithium 573 1.9100 6.310-1 3.110-1 1.510-1 6.110-2
Lithium-Lead 673 1.2101 4.9100 2.4100 1.2100 4.910-1
Flibe 673 1.710-1 6.510-2 3.210-2 1.610-2 6.410-3
Tin 1273 1.7101 6.8100 3.4100 1.7100 6.810-1
Ga 1073 5.0101 1.9101 9.7100 4.8100 1.9100
o=1 mm, aNu=1.0
Results -- Asymptotic Solution– Specified Non-uniform surface heat flux
11
• Limiting values for the heat flux gradients (i.e. temperature gradients) to prevent film rupture can be determined
• Generalized charts have been developed to determine the heat flux gradient limits for different fluids, operating temperatures (i.e. properties), and film thickness values
• For thin liquid films, limits may be more restrictive than surface temperature limits based on Plasma impurities limit
• Experimental Validation of Theoretical Model has been initiated
Conclusions
13
Results -- Asymptotic Solution (o=10 mm, a=0.02)
CoolantMean
Temperature [K]
o/LgL2
Max. Heat Flux. Gradient(Q/L)max [(MW/m2)/cm]
aNu=1.0 aNu=0.1 aNu=0.1
Lithium 573 2.5410-4 0.61 3.110-2 3.010-3
Lithium-Lead
673 2.0410-5 4.9 0.24 2.410-2
Flibe 673 4.3510-5 6.410-2 3.210-3 3.210-4
Tin 1273 3.0410-5 6.8 0.34 3.310-2
Ga 1073 4.8110-5 19 0.97 9.510-2
14
Results -- Asymptotic Solution (o=1 mm, a=0.02)
CoolantMean
Temperature [K]
o/LgL2
Max. Heat Flux. Gradient(Q/L)max [(MW/m2)/cm]
aNu=1.0 aNu=0.1 aNu=0.1
Lithium 573 2.5410-2 0.63 3.210-2 3.110-3
Lithium-Lead
673 2.0410-3 4.9 0.24 2.410-2
Flibe 673 4.3510-3 6.510-2 3.310-3 3.210-4
Tin 1273 3.0410-3 6.8 0.34 3.310-2
Ga 1073 4.8110-3 19 0.98 9.610-2
15
Results -- Asymptotic Solution (o=0.1 mm, a=0.02)
CoolantMean
Temperature [K]
o/LgL2
Max. Heat Flux. Gradient(Q/L)max [(MW/m2)/cm]
aNu=1.0 aNu=0.1 aNu=0.1
Lithium 573 2.54100 2.6 0.11 1.110-2
Lithium-Lead
673 2.0410-1 6.2 0.3 2.910-2
Flibe 673 4.3510-1 0.1 4.810-3 4.710-4
Tin 1273 3.0410-1 9.5 0.46 4.410-2
Ga 1073 4.8110-1 31 1.5 0.14
16
Results -- Asymptotic Solution (o=1 mm, aNu=1.0)
CoolantMean
Temperature [K]
Max. Heat Flux. Gradient(Q/L)max [(MW/m2)/cm]
a=0.05 a=0.02 a=0.01 a=0.005 a=0.002
Lithium 573 1.9100 6.310-1 3.110-1 1.510-1 6.110-2
Lithium-Lead
673 1.2101 4.9100 2.4100 1.2100 4.910-1
Flibe 673 1.710-1 6.510-2 3.210-2 1.610-2 6.410-3
Tin 1273 1.7101 6.8100 3.4100 1.7100 6.810-1
Ga 1073 5.0101 1.9101 9.7100 4.8100 1.9100
17
Results -- Asymptotic Solution (o=1 mm, aNu=0.1)
CoolantMean
Temperature [K]
Max. Heat Flux. Gradient(Q/L)max [(MW/m2)/cm]
a=0.05 a=0.02 a=0.01 a=0.005 a=0.002
Lithium 573 9.210-2 3.210-2 1.510-2 7.710-3 3.010-3
Lithium-Lead
673 6.210-1 2.410-1 1.210-1 6.110-2 2.410-2
Flibe 673 8.410-3 3.310-3 1.610-3 8.110-4 3.210-4
Tin 1273 8.710-1 3.410-1 1.710-1 8.510-2 3.410-2
Ga 1073 2.5100 9.810-1 4.910-1 2.410-1 9.710-2
18
Results -- Asymptotic Solution (o=1 mm, aNu=0.01)
CoolantMean
Temperature [K]
Max. Heat Flux. Gradient(Q/L)max [(MW/m2)/cm]
a=0.05 a=0.02 a=0.01 a=0.005 a=0.002
Lithium 573 8.910-3 3.110-3 1.510-3 7.510-4 3.010-4
Lithium-Lead
673 6.110-2 2.410-2 1.210-2 6.010-3 2.410-3
Flibe 673 8.210-4 3.210-4 1.610-4 8.010-5 3.210-5
Tin 1273 8.510-2 3.310-2 1.710-2 8.310-3 3.310-3
Ga 1073 2.510-1 9.610-2 4.810-2 2.410-2 9.510-3
19
Results -- Asymptotic Solution (L=5 cm, aNu=1.0)
CoolantMean
Temperature [K]
Max. Heat Flux. Gradient(Q/L)max [(MW/m2)/cm]
a=0.05 a=0.02 a=0.01 a=0.005 a=0.002
Lithium 573 1.6100 6.310-1 3.210-1 1.610-1 6.410-2
Lithium-Lead
673 1.2101 4.9100 2.5100 1.3100 5.210-1
Flibe 673 1.610-1 6.510-2 3.310-2 1.710-2 8.510-3
Tin 1273 1.4101 6.8100 3.4100 1.7100 6.810-1
Ga 1073 4.8101 1.9101 9.5100 4.8100 1.9100
20
Results -- Asymptotic Solution (L=5 cm, aNu=0.1)
CoolantMean
Temperature [K]
Max. Heat Flux. Gradient(Q/L)max [(MW/m2)/cm]
a=0.05 a=0.02 a=0.01 a=0.005 a=0.002
Lithium 573 8.010-2 3.210-2 1.610-2 8.010-3 3.210-3
Lithium-Lead
673 6.010-1 2.410-1 1.210-1 6.010-2 2.410-2
Flibe 673 8.310-3 3.310-3 1.710-3 8.510-4 3.410-4
Tin 1273 8.510-1 3.410-1 1.710-1 8.510-2 3.410-2
Ga 1073 2.5100 9.810-1 4.910-1 2.510-1 1.010-1
21
Results -- Asymptotic Solution (L=5 cm, aNu=0.01)
CoolantMean
Temperature [K]
Max. Heat Flux. Gradient(Q/L)max [(MW/m2)/cm]
a=0.05 a=0.02 a=0.01 a=0.005 a=0.002
Lithium 573 7.810-3 3.110-3 1.610-3 8.010-4 3.210-4
Lithium-Lead
673 6.010-2 2.410-2 1.210-2 6.010-3 2.410-3
Flibe 673 8.010-4 3.210-4 1.610-4 8.010-5 3.210-5
Tin 1273 8.310-2 3.310-2 1.710-2 8.510-3 3.410-3
Ga 1073 2.410-1 9.610-2 4.810-2 2.410-2 9.610-3