Temperature Gradient Limits for Liquid-Protected Divertors S. I. Abdel-Khalik, S. Shin, and M. Yoda...
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Transcript of Temperature Gradient Limits for Liquid-Protected Divertors S. I. Abdel-Khalik, S. Shin, and M. Yoda...
Temperature Gradient Limits for Liquid-Protected
Divertors
S. I. Abdel-Khalik, S. Shin, and M. Yoda
ARIES Meeting (June 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 temperature gradients (i.e. heat flux gradients) to prevent film rupture due to thermocapillary effects
Problem Definition
3
• 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
)2
-(2
cos2
)(L
xL
TTxT s
ms
Problem Definition
Non-uniform wall temperature
Wall
• 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)
periodic B.C. in x direction
open B.C. at top,L
Gas
ho initial film thickness
yl>>ho
Liquid
x
y
0
y
T
y
v
y
u
Initially, quiescent liquid and gas (u=0, v=0, T=Tm at t=0)
5
Variables definition
L
ha o
oh
yy
oh
ax
L
xx
)/( L
uu
LL
)/( La
vv
LL
)/( 2LLL
tt
oL
Lg h
V
32
22
FroL
L
o
g
hggh
V
ooL
L
o
ogL
h
hV
22
We
L
LL
k
cPr
LL
oso hT
Ms
m
T
TTT
•Non-dimensional variables
6
• Energy
• Momentum
• Conservation of Mass
Governing Equations0
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
0FrPr)/(M2
3
We
Fr3
32
x
Th
x
hh
x
ha s
Asymptotic Solution
• Long wave theory with surface tension effect (a<<1)
Governing Equations reduce to: [Bankoff, et al. Phys. Fluids (1990)]
• Generalized Charts have been generated for the Maximum non-dimensional temperature gradient (aM/Pr) as a function of the Weber and Froude numbers
8
Asymptotic Solution
• Similar Plots have been obtained for other aspect ratios
• In the limit of zero aspect ratio
1/Fr
(T
s/L)/
(L2/
Loh
o2)
Max
imum
Non
-di
men
sion
al T
empe
ratu
re
Gra
dien
t 1/We=107
1/We=106
1/We=105
1/We=104
1/We=103
1/We=102
Fr
1
12Pr/M
2crit
a=0.02
9
Results
• 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
10
Property Ranges (ho=1mm)*
2
2
ooL
L
h
Parameter
LithiumLithium-
LeadFlibe 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 ho, 1/Fr ho3
11
Results -- Asymptotic Solution (ho=1mm)
CoolantMean
Temperature [K]
Max. Temp. Gradient(Ts/L)max [K/cm]
Lithium 573 13
Lithium-Lead 673 173
Flibe 673 38
Tin 1273 80
Ga 1073 211
12
• Evolution of the free surface is modeled using the Level Contour Reconstruction Method
• Two Grid Structures Volume - entire computational domain
(both phases) discretized by a standard, uniform, stationary, finite difference grid.
Phase Interface - discretized by Lagrangian points or elements whose motions are explicitly tracked.
Phase 2
Phase 1
F
Numerical Method
13
Force on a line element
NUMERICAL METHOD
)( moo TT
Numerical Method
Variable surface tension :
A single field formulationConstant but unequal material properties Surface tension included as local delta function sources
n : unit vector in normal direction
t : unit vector in tangential direction
: curvature
AtA-BtB
nBtB
e
-CtC
A
B
s
ssss
)( t
tt
tn
)(d)(
d BBAA
A
Bs
e ss
ss
F ttt
tn
thermocapillary force
normal surface tension force
14
•Material property field
Variables definition
L
G
kk
k
L
G
L
G
cc
c
L
G
),()1(1 tI x
),()1(1 tI x
),()1(1 tIcc x
),()1(1 tIkk x
15
• Two-dimensional simulation with 1[cm]0.1[cm] box size, 25050 resolution, and ho=0.2 mm
Dimensional Simulation (Lithium)
x [m]
y [m
]
time [sec]
y [m
]
maximum y location
minimum y location
black :
blue :
red :
]K/cm[20max
L
T
dx
dT s
]K/cm[40max
L
T
dx
dT s
]K/cm[60max
L
T
dx
dT s
a=0.02
16
• Two-dimensional simulation with 1.0[cm]0.1[cm] box size and 25050 resolution
• ho=0.2 mm, Ts=20 K
velocity vector plot
temperature plot
Dimensional Simulation (Lithium)
17
Results (1/Fr=1)
1/We
(T
s/L)/
(L2/
Loh
o2)
Max
imum
Non
-di
men
sion
al T
empe
ratu
re
Gra
dien
t
Pr=40Pr=0.4
Pr=0.004Asymptotic
Solution
a=0.02
18
Results (1/Fr=103)
1/We
(T
s/L)/
(L2/
Loh
o2)
Max
imum
Non
-di
men
sion
al T
empe
ratu
re
Gra
dien
t
Pr=40Pr=0.4
Pr=0.004Asymptotic
Solution
a=0.02
19
Results (1/Fr=105)
1/We
(T
s/L)/
(L2/
Loh
o2)
Max
imum
Non
-di
men
sion
al T
empe
ratu
re
Gra
dien
t
Pr=40Pr=0.4
Pr=0.004Asymptotic
Solution a=0.02
20
Results -- Numerical Solution (ho=1mm)
CoolantMean
Temperature [K]
Max. Temp. Gradient(Ts/L)max [K/cm]
Num. Sol. Asymp. Sol.
Lithium 573 30 13
Lithium-Lead 673 570 173
Flibe 673 76 38
Tin 1273 113 80
Ga 1073 600 211
21
Experimental Validation
22
• Limiting values for the temperature gradients (i.e. heat flux gradients) to prevent film rupture can be determined
• Generalized charts have been developed to determine the temperature 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
• Preliminary results for Axisymmetric geometry (hot spot model) produce more restrictive limits for temperature gradients
Conclusions