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