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

Can a Direct-Drive Target Survive Injection into an IFE Chamber?

Japan-US Workshop on IFE Target Fabrication, Injection and TrackingOsaka, Japan

18-19 October 2004

Mechanical and Aerospace Engineering Department

and the

Center for Energy Research

Degradation of targets in the chamber must not exceed requirements for successful

implosion

• Spherical symmetry

• Surface smoothness

• Density uniformity

• TDT (<19.79 K?)

• Better definition is needed

Physics

requirements:

IFE Chamber (R~6 m)

Protective gas (Xe, He) at ~4000 K heating up target

Chamber wall ~ 1000–1500 K, causing q’’rad on target

Target Injection (~400 m/s)

Target Implosion Point

We have characterized target heat loads and the resulting thermomechanical behavior in

order to help define the operating parameter windows

• Energy transfer from impinging atoms of background gas Enthalpy transfer (including condensation) or convective loading

Recombination of ions (much uncertainty remains regarding plasma conditions during injection)

• Radiation from chamber wall Dependent on reflectivity of target surface and wall temperature

Estimated as 0.2 – 1.2 W/cm2 for = 0.96 and Twall = 1000 – 1500 K

Heat loads:

1. Convective loading using DSMC

2. Integrated thermomechanical model developed at UCSD, including phase change behavior of DT

Analyses performed:

1. Computation of energy transfer from background gas using DS2V

The DSMC method has been used to study targets in a low density (3x1019 – 3x1021 m–3) protective gas where Kn is moderately high

(0.4–40)

Assumptions

• Axially symmetric flow

• Stationary target

• Xe stream

velocity = 400 m/s

T = 4000 K

density=3.22x1021 m-3

• Target surface fixed at T = 18 K

• Sticking coefficient 0<

• Accommodation coefficient 0<

• No target rotation

Temperature field around a direct drive target

Flow

= 0= 0

If the stream density is high, the number flux at the target surface

increases when the sticking coefficient () decreases

1.E+22

1.E+23

1.E+24

1.E+25

0.0 1.0 2.0 3.0Angle from Trailing Edge (rad)

Number Flux (atoms/m

2s)

T = 4000 K, sigma = 0T = 1300 K, sigma = 0T = 4000 K, sigma = 1T = 1300 K, sigma = 1Kinetic Theory, T = 4000 KKinetic Theory, T = 1300 K

n = 3.22x1021 m-3

• Instead of screening incoming particles, stagnated particles add to the net particle flux

• Kinetic theory and DS2V show good agreement

decreasing

Conversely, if the stream density is high, the heat flux at the target surface

decreases when the sticking coefficient decreases

1.E+03

1.E+04

1.E+05

1.E+06

0.E+00 5.E-01 1.E+00 2.E+00 2.E+00 3.E+00 3.E+00Position on Surface (m)

Heat Flux (W/m

2)

T = 4000 K,sigma = 1T = 4000 K,sigma = 0T = 1300 K,sigma = 1T = 1300 K,sigma = 0

The strong dependence of heat flux on position suggests that the time-averaged peak heat flux could be reduced significantly by rotating the target.

n = 3.22x1021 m-3 • The heat flux decreases

when = 0 due to the shielding influence of low temperature reflected particles interacting with the incoming stream

• For the low density cases there is less interaction between reflected and incoming particles.

decreasing

The sticking coefficient () and accommodation coefficient () both have a large impact on the maximum heat flux at

the leading edge

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Normalized Heat Flux

α = 0

α = 0.5

α = 1

Region of no screening

Parameters: – 400 m/s injection into– Xe @ 3.22x1021 m-3 – 4000 K – max. heat flux = 27

W/cm2 (with = 1 and =1)

Experimental determination of the sticking coefficient and accommodation coefficient is needed

2. Integrated thermomechanical modeling of targets during injection

• 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

• The maximum allowable heat flux was analyzed for several target configurations where failure is based on the triple point limit

• The potential of exceeding the triple point (allowing phase change) was explored

• In the following, we discuss: a description of the model validation of the model the effect of initial target temperature the effect of thermal insulation the effect of injection velocity the effect of allowing a melt layer to form the effect of allowing a vapor layer to form

Background

The 1D transient energy equation is solved in spherical coordinates

• Discretized and solved using 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)

Interface Boundary Condition

€

∂T

∂t=

1

ρc p (T)

∂T

∂r

2k

r+

∂k

∂r

⎛

⎝ ⎜

⎞

⎠ ⎟+ k

∂ 2T

∂r2

⎡

⎣ ⎢

⎤

⎦ ⎥

€

j =M

2πR

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2psat

Ts1/ 2

−pvap

Tvap1/ 2

⎡

⎣ ⎢

⎤

⎦ ⎥

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 =− pra

EDT

(1− υ DT )(rb3 + 2ra

3 )

2(r 3 − rb3 )

− υ DT

⎡

⎣ ⎢

⎤

⎦ ⎥

Deflection of polymer shell and DT nonlinearly affects the pressure and vapor

layer thickness

The model was validated using an exact solution for a solid sphere

• Initial temperature T=Tm (the melting point)

• Surface suddenly raised to Ts=25 K at t=0

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

• The melt layer thickness is correctly modeled.

• Slight 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

Ts

19.825t=0

• 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

Reducing the initial temperature of a basic target increases the maximum allowable heat

flux

An insulating foam on the target could allow very high heat fluxes

• Failure is assumed at the DT triple point temperature

• Required “target survival time” assumed = 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

For a basic target, using the TP limit, there is an optimum injection velocity

when = 1

0.0E+00

5.0E+19

1.0E+20

1.5E+20

2.0E+20

2.5E+20

3.0E+20

3.5E+20

100 150 200 250 300 350 400

Injection Velocity (m/s)

Maximum Density (m

-3) Tinit = 18 K

Tinit = 16 K

Tinit = 14 K

0.0E+00

5.0E+19

1.0E+20

1.5E+20

2.0E+20

2.5E+20

100 200 300 400 500 600Injection Velocity (m/s)

Maximum Density (m

-3) Tinit = 18 K

Tinit = 16 K

Tinit = 14 K

= 1 = 0

• DS2V is used to predict heat flux, and the integrated thermomechanical model is used to predict the response

• This optimum occurs due to a competition between increasing heat flux vs. lower thermal penetration

For an insulated target, higher injection velocity significantly

increases the maximum allowable gas density

100 mm, 10% dense insulator, (sticking coefficient) = 1

0.00E+00

5.00E+20

1.00E+21

1.50E+21

2.00E+21

2.50E+21

100 200 300 400 500 600Injection Velocity (m/s)

Maximum Density (m

-3)

Tinit = 18 KTinit = 16 KTinit = 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

criterion

• 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

• 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)

Tinit = 16 K

However, the amount of superheat (with melting only) indicates a potential for nucleating & growing

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).

-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

If a vapor layer is 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 criterion

• 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)

For some initial temperatures and heat fluxes, the vapor layer closes, suggesting that bubbles

can be minimized or eliminated in some circumstances

• 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 occurs 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).

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

Future model development activities are guided by the desire to plan and analyze

experiments

• The coupled thermal and mechanical response of a direct drive target has helped us understand the behavior of the target and limiting factors on target survival

• However, the simple 1D vapor model does not account for real-world heterogeneities

• Future numerical model improvements will include a prediction of the nucleation and growth (homogeneous or heterogeneous from He3) of individual vapor bubbles in the DT liquid

• We are evaluating the feasibility of a 2D model of the energy equation