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
1
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.
2
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 mm Polymer
Shell with Au or Pd
Reflective Coating
290 mm Solid
DT/Foam
190 mm
Solid DT
DT Vapor
Core
~ 4 mm
A typical direct drive IFE target
considered in this study (not to
scale).
3
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)
T
1 T 2k k   2T 

    k 2 
t cp (T ) r  r r  r 
Boundary Conditions
• Heat transfer at solid (and liquid) region interfaces:
T
ka
 hTan1  Tbn1
r

• Mass transfer at liquid/vapor interface:

 M 1/ 2 psat pvap 
j  
  1/ 2  1/ 2 
2R  Ts
Tvap 
4
Integrated Thermomechanical Model(1)
Outer Polymer Shell Deflection
• Membrane theory for shell of radius rpol and thickness tpol:
2
prpol
(1 pol )
 polymer 
2Epol tpol
Inner Solid DT Deflection
• Thick spherical shell with outer and inner radii, ra and rb :


 pra (1 DT )(rb3  2ra3 )
ra 
 DT 

3
3
EDT 
2(r  rb )

5
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.
25
• The melt layer thickness is
correctly modeled.
• Some error exists in the
temperature profile.
Temperature (K)
24
Exact
DT = 0.4 K
23
DT = 0.2 K
22
21
20
19
1.98E-03
1.99E-03
Position (m)
2.00E-03
6
Reducing the Initial Temperature of a Basic
Target Increases the Maximum Allowable
Heat Flux
0.02
Time to Reach T.P. (s)
• 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.
0.018
0.016
Tinit = 18 K
Tinit = 16 K
Tinit = 14 K
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
0
1
2
3
2
Heat Flux (W/cm )
4
5
7
Insulating the Target Could Allow for
Very High Heat Fluxes
High-Z coat
0.03
Insulating foam
0.289 mm
0.19 mm
DT + foam
DT solid
0.025
Dense plastic
overcoats
(not to scale)
0.02
Time (s)
x
100 microns, 10%
150 microns, 10%
100 microns, 25%
150 microns, 25 %
No Insulator
0.015
0.01
DT gas
0.005
1.5 mm
0
0
5
10
2
15
Heat Flux (W/cm )
•
•
•
•
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 mm, 25% dense insulator would increase the
allowable heat flux above 12 W/cm2, nearly an order of
magnitude increase over the basic target.
8
Allowing Phase Change: Melting Only
• Possible failure criteria:
Time to 0.8Tc
0.03
Survival Time (s)
– 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).
0.035
Time to Tc
Time to Polymer
Ultimate Stress
0.025
0.02
0.015
0.01
0.005
Tinit = 16 K
0
4
5
6
7
8
Heat Flux (W/cm2)
9
10
• 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.
9
Calculating the Superheat (with Melting Only) Indicates
the Potential for Nucleating and Growing Vapor Bubbles
liq
sat
10
9
Maximum Super Heat (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
  T to
 Toccur (at
nucleation sites), the liquid
must be superheated by 2-3
K. Where the superheat is
defined as:   Tliq  Tsat
8
7
6
Tinit = 14 K
5.5 W/cm2
5
4
3
2.5 W/cm2
2
1
0
1.0 W/cm2
-1
-2
0.0E+00
4.0E-03
8.0E-03
1.2E-02
1.6E-02
Time (s)
• 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).
10
Allowing Phase Change: A Uniform Vapor Layer
• Possible failure criteria:
• For targets with initial
temperatures of 14 K, 16 K,
and 18 K, The polymer
ultimate strength was
reached before the DT
ultimate strength.
Tinit = 14 K
Time to Polymer
Ultimate Strength (s)
– Ultimate strength of the
DT solid or polymer shell
is exceeded.
– Vapor layer thickness
exceeds a critical value
(unknown).
0.03
0.025
Tinit = 16 K
Tinit = 18 K
0.02
0.015
0.01
0.005
0
1
2
3
4
5
6
Heat Flux (W/cm2)
• 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.
11
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.
Tinit = 18 K
Tinit = 14 K
12
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.
13