Design Constraints for Liquid-Protected

Divertors

S. Shin, S. I. Abdel-Khalik, M. Yoda and ARIES Team

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

• 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

3

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 ho initial film thickness

x

Convective cooling )( TThq sgc

)

2(

2cos1 l

low

xx

xqq

Non-uniform heat

flux

L

Gas

Liquid y Wall ho 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)

4

• Energy

• Momentum

• Conservation of Mass

Governing Equations

0

y

v

x

u

itn ˆ2 222

dssx

v

ya

y

u

yx

u

xa

x

p

y

uv

x

uu

t

ua

jtn ˆ2Fr

2244

dss

ay

v

ya

y

u

xa

x

v

xa

y

p

y

vv

x

vu

t

va

y

Tk

yx

Tk

x

a

y

Tcv

x

Tcu

t

Tca

Pr

1

Pr

22

T M/Pr)(We/1

L

ha o

oh

yy

oh

ax

L

xx

)/( L

uu

LL

)/( La

vv

LL

)/( 2LLL

tt

oL

Lg h

V

s

m

T

TTT

32

22

FroL

L

o

g

hggh

V

ooL

L

o

ogL

h

hV

22

We

L

LL

k

cPr

LL

oso hT

M

• Non-dimensional variables

y

Tkq L

oLo h

Tkq

~

L

o

k

qhT 0~

L

go

L

go

k

hh

k

Lh

L

haNu

)/( Loo kqh

TTT

2

12cos1 xQ

Specified Non-uniform Wall Temperature Specified Non-uniform Surface Heat flux

5

Asymptotic Solution (Low Aspect Ratio Cases)

• 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

• Long wave theory with surface tension effect (a<<1). Govering Equations reduce to :

0FrPr)/(M2

3

We

Fr3

32

x

Th

x

hh

x

ha s 01

)/(

)/(

2

3

L3

3

2L

x

Q

Bx

hQ

gka

LQh

x

hh

x

h

gL o

o

Specified Non-uniform Wall Temperature * Specified Non-uniform Surface Heat flux

* [Bankoff, et al. Phys. Fluids (1990)]

6

• Similar Plots have been obtained for other aspect ratios

• In the limit of zero aspect ratio

Asymptotic Solution (Low Aspect Ratio Cases)

1/Fr

(T

s/L)/

(L2/

Loh

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

tSpecified Non-uniform Wall Temperature Specified Non-uniform Surface Heat flux

Fr

1

12Pr/M

2crit

7

2

2

ooL

L

h

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 1.210

3

3.8104 5.3105 6.710

5

5.7105 8.2105

1/We 7.8105 1.3106

9.4105 3.0106 1.310

4

4.0105 4.2106 4.810

6

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

Asymptotic Solution (Low Aspect Ratio Cases)– Specified Non-uniform Wall Temp (ho=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

• ho=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

• ho=1 mm, aNu=1.0

Asymptotic Solution (Low Aspect Ratio Cases)– Specified Non-uniform Surface Heat Flux

9

• 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 Solution (High Aspect Ratio Cases)

A single field formulation Constant but unequal material properties Surface tension included as local delta function

sources

Force on a line element

)( moo TT Variable surface tension :

)(d)(

d BBAA

A

Bs

e ss

ss

F ttt

tn

ssss

)( t

tt

tn

thermocapillary

force

normal surface

tension force

AtA-BtB

nBtB

e

-CtC

A

B

s

n : unit vector in normal direction

t : unit vector in tangential direction

: curvature

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• Two-dimensional simulation with 1[cm]0.1[cm] box size, 25050 resolution, and ho=0.2 mm

Numerical Solution (High Aspect Ratio Cases)- Specified Non-uniform Wall Temperature case using 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

velocity vector plot

temperature plot

11

1/We

(T

s/L)/

(L2/

Loh

o2)

Pr=40 Pr=0.4 Pr=0.004 Asymptotic Solution

Max

imum

Non

-dim

ensi

onal

T

empe

ratu

re G

radi

ent

1/We 1/We

1/Fr=103 1/Fr=1051/Fr=101

Numerical Solution (High Aspect Ratio Cases)- Specified Non-uniform Wall Temperature case

a=0.02

12

• Limiting values for the temperature gradients (or heat flux gradients) to prevent film rupture can be determined

• Generalized charts have been developed to determine the temperature (or 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 (evaporation) constraint

• Experimental Validation of Theoretical Model has been initiated

• Preliminary results for Axisymmetric geometry (hot spot model) produce more restrictive limits for temperature gradients

Conclusions