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

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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

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• 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

12

Extra Slides

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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

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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

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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

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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

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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

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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

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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