1

MODELING DT VAPORIZATION AND MELTING IN A DIRECT

DRIVE TARGET

B. R. Christensen, A. R. Raffray, and M. S. Tillack

Mechanical and Aerospace Engineering Department and Center for Energy Research, University of California,

San Diego, La Jolla, CA 92093-0438, christensen@fusion.ucsd.edu

Presented at the 16th ANS TOFE

Madison, WI

September 14-16, 2004

2

Introduction• Initial perturbations in a direct drive target, which are amplified

during implosion, must be eliminated or minimized in order to maximize the implosion efficiency.

• The initial perturbations could result from the thermal loading of the target (thermal expansion and phase change).

• In the past it has been assumed that the D-T temperature must remain below the D-T triple point temperature (19.79 K) to ensure a viable target.

• This study reports the maximum allowable heat flux for several target configurations (where failure is based on the triple point limit), and investigates the potential of exceeding the triple point (allowing phase change).

• A 1-D integrated thermomechanical model was created to compute the coupled thermal (heat conduction, phase change) and mechanical (thermal expansion, deflection) response of a direct drive target.

3

Modeling the Thermal and Mechanical Response of a Direct-Drive Target

Description of the Model• Constant uniform heat flux at outer surface

of the target.

• The finite difference method is used to model 1-D heat conduction.

• Melting of the DT is accounted for by using the apparent cp method (phase change is assumed to take place over a small temperature interval).

• When studying the effect of DT vapor, the vapor is assumed to be a continuous layer between the polymer an DT solid (maintains a 1-D problem). No helium-3 present.

• The polymer shell and DT solid deflect as the DT changes volume due to thermal expansion and phase change (melting and vaporization).

1-10 m Polymer Shell with Au or Pd Reflective Coating

290 m Solid DT/Foam

190 m Solid DT

DT Vapor Core

~ 4 mm

A typical direct drive IFE target considered in this study

(not to scale).

4

Integrated Thermomechanical Model(1)1-D Heat Conduction Equation in Spherical Coordinates• Discretized and solved using the forward time central space (FTCS) finite

difference method. • Temperature-dependent material properties• Apparent Cp model to account for latent heat of fusion (at melting point)

Boundary Conditions• Heat transfer at solid (and liquid) region interfaces:

• Mass transfer at liquid/vapor interface:

€

∂T∂t

=1

ρcp(T )

∂T

∂r

2k

r+∂k

∂r

⎛

⎝ ⎜

⎞

⎠ ⎟+ k∂ 2T

∂r 2

⎡

⎣ ⎢

⎤

⎦ ⎥

€

−ka∂T

∂r= h Ta

n+1 − Tbn+1

( )

€

j =M

2πR

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2psatTs

1/ 2−pvapTvap

1/ 2

⎡

⎣ ⎢

⎤

⎦ ⎥

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Integrated Thermomechanical Model(1)

Outer Polymer Shell Deflection

• Membrane theory for shell of radius rpol and thickness tpol:

Inner Solid DT Deflection• Thick spherical shell with outer and inner radii, ra and rb :

€

δpolymer =prpol

2 (1− υ pol )

2Epol tpol

€

Δra =− praEDT

(1− υDT )(rb3 + 2ra

3 )

2(r 3 − rb3 )

− υ DT ⎡

⎣ ⎢

⎤

⎦ ⎥

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The Integrated Thermomechanical Model Has Been Validated

• An exact solution for a solid sphere with an initial temperature To = Tm (the melting temperature) was derived.

• The solution converged to the exact solution as the mesh size was decreased.

• The melt layer thickness is correctly modeled.

• Some error exists in the temperature profile.

0

2

4

6

8

10

12

0.0E+00 5.0E-04 1.0E-03 1.5E-03Time (s)

Melt Layer (

m)

Exact

dr = 1e-7 m

dr = 5e-7 m

dr = 1e-6 m

19

20

21

22

23

24

25

1.98E-03 1.99E-03 2.00E-03Position (m)

Temperature (K)

Exact

DT = 0.4 K

DT = 0.2 K

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Reducing the Initial Temperature of a Basic Target Increases the Maximum Allowable

Heat Flux• DT triple point

temperature is assumed as limit.

• Take the required “target survival time” to be 16.3 ms.

• Decreasing the initial temperature from 16 K to 14 K does not have as large of an effect as a decrease from 18 K to 16 K.

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0 1 2 3 4 5Heat Flux (W/cm2)

Time to Reach T.P. (s)

Tinit = 18 K

Tinit = 16 K

Tinit = 14 K

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Insulating the Target Could Allow for Very High Heat Fluxes

• Failure is assumed at the DT triple point temperature.

• Take the required “target survival time” to be 16.3 ms.

• Initial target temperature = 16 K.

• A 150 m, 25% dense insulator would increase the allowable heat flux above 12 W/cm2, nearly an order of magnitude increase over the basic target.

DT gas

1.5 mm

DT solid0.19 mm

DT + foam

x

Dense plastic overcoats (not to scale)

0.289 mm

Insulating foam

High-Z coat

0

0.005

0.01

0.015

0.02

0.025

0.03

0 5 10 15Heat Flux (W/cm2)

Time (s)

100 microns, 10%150 microns, 10%100 microns, 25%150 microns, 25 %No Insulator

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Allowing Phase Change: Melting Only• Possible failure criteria:

– Homogeneous nucleation of vapor bubbles in the DT liquid (0.8Tc).

– Ultimate strength of the DT solid or polymer shell is exceeded.

– Melt layer thickness exceeds a critical value (unknown).

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

4 5 6 7 8 9 10Heat Flux (W/cm2)

Survival Time (s)

Time to 0.8Tc

Time to Tc

Time to PolymerUltimate Stress

Tinit = 16 K

• For targets with initial temperatures of 14 K, 16 K, and 18 K, 0.8Tc was reached before the ultimate strength of the polymer was exceeded.

• The maximum allowable heat fluxes were found to be (@ 16.3 ms):– 5.0 W/cm2 (Tinit = 18 K).– 5.5 W/cm2 (Tinit = 16 K).– 5.7 W/cm2 (Tinit = 14 K).

• If only melting occurs, the allowable heat flux is increased by ~ 3-8 times over the cases where the DT triple point temperature is used as the failure criteria.

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Calculating the Superheat (with Melting Only) Indicates the Potential for Nucleating and Growing Vapor Bubbles

• For a basic target with initial temperatures of 16 K, the super heat is > 2-3 K for input heat fluxes > 2.5 W/cm2.

• For a initial temperature of 14 K, the superheat is negative when the heat flux is 1.0 W/cm2 (see figure to the right).

liqsatTTφ=−

-2-1

012

345

678

910

0.0E+00 4.0E-03 8.0E-03 1.2E-02 1.6E-02Time (s)

Maximum Super Heat (K)

5.5 W/cm2

2.5 W/cm2

1.0 W/cm2

Tinit = 14 K

• Due to the presence of dissolved He-3 gas, and small surface defects (nucleation sites), vapor formation is expected to occur before 0.8Tc.

• For bubble nucleation and growth to occur (at nucleation sites), the liquid must be superheated by 2-3 K. Where the superheat is defined as:

€

φ =Tliq − Tsat

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Allowing Phase Change: A Uniform Vapor Layer• Possible failure criteria:

– Ultimate strength of the DT solid or polymer shell is exceeded.

– Vapor layer thickness exceeds a critical value (unknown).

• For targets with initial temperatures of 14 K, 16 K, and 18 K, The polymer ultimate strength was reached before the DT ultimate strength.

0

0.005

0.01

0.015

0.02

0.025

0.03

1 2 3 4 5 6

Heat Flux (W/cm2)

Time to Polymer Ultimate Strength (s)

Tinit = 14 K

Tinit = 16 K

Tinit = 18 K

• The maximum allowable heat fluxes were found to be (@ 16.3 ms):– 2.2 W/cm2 (Tinit = 18 K).– 2.5 W/cm2 (Tinit = 16 K).

– 2.9 W/cm2 (Tinit = 14 K). • If a vapor layer was present, the allowable heat flux is increased by ~ 1.5-3

times over the cases where the DT triple point temperature is used as the failure criteria.

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The Vapor Layer Thickness as a Function of Time is Dependent on the Initial Temperature and Heat Flux

• For a target with an initial temperature of 18 K the vapor layer thickness increases rapidly for heat fluxes > 2.5 W/cm2.

• For a target with an initial temperature of 14 K the vapor layer thickness goes to zero when the heat flux <= 1.0 W/cm2.

• This vapor layer closure occur because the DT expands (due to thermal expansion and melting) faster that the polymer shell expands (due to thermal expansion and the vapor pressure load).

• The vapor layer closure suggests that bubbles could be eliminated or minimized in certain situations. 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 0.004 0.008 0.012 0.016Time (s)

Vapor Layer Thickness (

m)

q'' = 4.0 W/cm2

q'' = 2.5 W/cm2

q'' = 1.0 W/cm2

0

5

10

15

20

25

0 0.004 0.008 0.012 0.016Time (s)

Vapor Layer Thickness (

m) q'' = 5.5 W/cm2

q'' = 2.5 W/cm2

q'' = 1.0 W/cm2

Tinit = 18 K

Tinit = 14 K

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Summary

• The coupled thermal and mechanical response of a direct drive target can be approximately modeled using the 1-D finite difference model created at UCSD.

• Insulating a target could increase the maximum allowable heat flux by nearly an order of magnitude over the basic target design.

• Allowing phase change could increase the maximum allowable heat flux by 2 to 8 times over the basic target design where the DT triple point temperature is used as the failure criteria.

• The results from the 1-D model created at UCSD suggest that vapor bubble growth or formation might be prevented or minimized under certain circumstances.

• A future numerical model is proposed that will predict the growth of individual vapor bubbles in the DT liquid.