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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