ARIES: Fusion Power Core and Power Cycle Engineering

1
Target Survival Effort

• Presented by A.R. Raffray
• Other contributors B. Christensen, Z.
  Dragojlovic, J. Pulsifer,
• M. S. Tillack, X. Wang
• University of California, San Diego
• Target Mini Workshop
• Los Alamos National Laboratory
• July 8, 2003

2
Outline

• Target survival requirements
• Means of enhancing target thermal robustness
• The need for a numerical model
• Results from ANSYS and beyond ANSYS
• Description and validation of the target model
  (1-D vs 2-D)
• Initial results
• Future work
• RD needs

3
Review of Target Survival Requirements

• What is target survival?
• The target must remain intact during injection
• The continuity of the target when fired upon by
  the driver must meet the physics requirements for
  optimum compression and burn (maximum gain)
• The target survival requirements can be met if DT
  temperature doesnt reach the triple point, 19.79
  K ( i.e., no phase change)
• Using this criterion the target could only be
  subjected to 0.6 W/cm2 if injected into a 6 m
  radius chamber with an initial temperature of 18
  K
• To provide a reasonable design window for gas
  protection and power core performance
• - Gas pressure up to 50 mtorr at 1000-4000 K
  (qcond 4 -10 W/cm2 for Xe)
• - Chamber wall temperature 1000-1500 K
  (qrad 0.2 -1.2 W/cm2)
• - Total q to be accommodated by target 5 -11
  W/cm2
• Need means to increase thermal robustness of
  target

4
Three Options for Increasing the Thermal
Robustness of the Target Have Been Proposed
Foam Insulator

• Design modifications
• Foam insulator
• Shell thickness
• Allow phase change to occur
• This solution must accommodate target physics
  requirements
• Inject the target at a lower base temperature

5
ANSYS Model Shows that Decreasing the Temperature
at which the Target is Injected Delays Phase
Change
Flight time for 400 m/s target in 6 m chamber
0.015 s
q Constant
DT ice initially at uniform temperature Tinit
2 mm radius DT sphere
Decreasing target temperature helps but is not
sufficient
6
Example DT Interface Temperature History for
Different Thicknesses(mm)of 25 Dense Outer Foam
Region

• Transient analyses performed using ANSYS
• - q 2.2 W/cm2 for example case
• (10 mTorr/4000 K Xe)
• - Outer foam region density 25
  (Consistent with J. Sethians guideline)

• 130 mm (32 mm of equivalent solid
  polystyrene) would be sufficient to
  prevent DT from reaching the triple point
  after 0.015 s (corresponding to flight time of
  400 m/s target in 6 m radius chamber)
• As comparison, DT would reach the triple
  point after 0.002 s in the absence of the
  outer foam layer

7
Summary of Thermal Analysis Results on
Effectiveness of Insulating Outer Foam Layer
To increase target thermal
robustness - maximize both thickness and
porosity of outer foam layer
while - accommodating target physics and
structural integrity requirements.
8
Additional Measure to Enhance Target Thermal
Robustness Needed to Relieve Challenge of
Fabricating Relatively Thick, High Porosity Foam
Layer on Target
Allow Phase Change to Occur
Bond quality between DT/foam and plastic seal
coat is key factor affecting vapor formation in
DT targets

• For high quality bond, evaporation would only
  occur through nucleation
• Homogeneous nucleation very low under
  typical conditions (0 for Tlt26 K and takes off
  at 34 K)
• Melting only
• If localized micro-defects are present,
  heterogeneous nucleation is possible (gt 1 mm)
• If micro-gap present, surface evaporation will
  occur (worst case scenario considered here)

9
Mechanisms Affecting Vapor Growth in DT Targets
Include
Thermal interaction between DT/foam and plastic
seal coat - Evaporation/condensation - Thermal
resistance of vapor gap Mechanical interaction
of DT/foam and plastic seal coat - This
interaction could prevent significant vapor
production - Need to determine the details of
the foam-plastic seal coat bond, foam-DT bond,
DT-plastic seal coat bond and foam
properties - Will the DT flow through the
foam? Localized phase change
effect - Buckling of DT ice, or bulging of
shell? Transient response of the target - How
much vapor will exist after 15 ms? - Will the DT
ice have time to flow or buckle?
10
An Integrated Thermo-Mechanical Model is Being
Developed To Explore the Effects of Foam
Insulation and Phase Change

• A 1-D numerical model being developed first,
  coupling heat transfer, phase change, and solid
  mechanics
• Includes key mechanisms
• Latent heat of evaporation/condensation
• Pressure increase, and hence the deflection of
  the target, due to sublimation and/or evaporation
• Increased thermal resistance of vapor-filled gap
• Why 1-D?
• Complexity of model
• Can simulate 2-D case with large enough gap
• Use as test bed to fine tune model
• Possibility of upgrading to 2-D later if required

11
Effect of 2-D Heat Flux Distribution

• 1-D ANSYS results very close to 2-D ANSYS results
  for typical cases considered
• - Reasonably conservative to use 1-D analysis

12
Example 2-D ANSYS Analysis Results for
10mtorr/4000K Xe case (q2.2 W/cm2)
13
2-D ANSYS Model Used to Study the Effects of
Local Vapor Gap Formation on Heat Transfer

• 3 mm thick vapor gap
• Gap arc length varied and results compared to 1-D
  case

Local Vapor Gap
3 mm
DT Vapor Core
Plastic Shell
Rigid DT
14
Comparison of the Results For a Small Gap to
those Obtained for a Continuous Vapor Gap Show
that a 2-D Heat Transfer Model may be Important
for Small Gaps
15 mm arc length
 
Entire arc length
Time (s)
Time (s)
Outer Surface of the Plastic Shell Vapor-Plastic
Interface Vapor-DT Interface
15
Comparison of the Results for a Finite Gap to
those Obtained for a Continuous Vapor Gap Show
that a 1-D Heat Transfer Model is Sufficient for
Large Gaps (15-50 mm arc length for 3 mm gap)
50 mm arc length
 
Entire arc length
Time (s)
Time (s)
Outer Surface of the Plastic Shell Vapor-Plastic
Interface Vapor-DT Interface
16
The Heat Conduction Equation can be Used to Model
Solid-to-Liquid Phase Change

• 1-D heat conduction equation in spherical
  coordinates with no heat generation and variable
  properties
• Variable properties are needed to account for the
  DT solid to liquid phase change
• DT thermal conductivity experiences a jump during
  phase change
• Solid to liquid phase change can be modeled by
  defining an apparent Cp

17
In Order for the Apparent Cp to be Defined
Everywhere, DT is Assumed to Undergo Solid to
Liquid Phase Change over a Small Temperature
Range
for TltTmelt and Tgt TmeltDTmelt
for DTmelt region
DTmelt
Actual Path
Path Used for Calculating Cpap
T.P.
18
Numerically Modeling Heat Transfer and Phase
Change

• Finite Difference Scheme
• Thermal conductivity at the n1 time step is
  extrapolated using the equation

19
The Accuracy of the Finite Difference Model,
Using the Apparent Cp Approach to Model Phase
Change, is Shown by Comparison to the Exact
Solution

• An exact solution exists for the melting of a
  solid slab, where initially the solid is at the
  melting temperature and the boundary is raised to
  some temperature To gt Tm at time t0 (N. Ozisik
  1993).

Where l is given by the transcendental equation
The exact solution for a spherical geometry can
be obtained by transforming the governing
equations and boundary conditions according to
the equation (N. Ozisik 1993)
Liquid
Solid
To
Tl(x,t)
Ts Tm
Tm
Interface
A set of equations similar to those above are
obtained using the transformation
x
0
S(t)
20
The Finite Difference Model and The Exact
Solution Produce Similar Results for a Uniform
Sphere DT
Temperature Profile after 0.015 s
Note Only a solid sphere of DT is considered
here the plastic shell and foam are not included
21
A Closer Look at the Portion of the Target that
Experienced Phase Change Shows Very Good
Agreement Between the Numerical and Exact
Solutions
22
The Heat Flux from an IFE Chamber to the Target
is Modeled as a Uniform Heat Flux

• Constant uniform heat flux applied for outer
  boundary condition
• Adiabatic solid-vapor interface applied for inner
  boundary condition

q Constant
Foam
Plastic Shell
q 0
DT Vapor Core
DT Ice
DT/Foam
Target thermal boundary conditions
23
The Jump in Thermal Conductivity at the
Foam-Plastic Interface Requires Modification

• When the thermal conductivity jumps at the
  foam-plastic interface, erroneous results are
  obtained (the model does not conserve energy)
• A linear change in properties over a small region
  corrects this problem

kp
q Constant
kf
Thermal Conductivity, k
q h(Tinf-Tp)
Position
L
Plastic to foam transition
Where L is the distance over which the thermal
conductivity changes
24
Conservation of Energy is Satisfied if the
Thermal Conductivity is Allowed to Change over a
Sufficient Number of Nodes
2 Node Transition
Property change over 10 nodes
0.5 mm Node Spacing
Error
Error
0.5 mm Node Spacing
0.1 mm Node Spacing
10 Node Transition
Time
Time
The relative change of k from one node to the
next is more important than the node spacing
The model conserves energy when k is changed over
a sufficient number of nodes
25
Using the Appropriate Mesh Configuration For the
Finite Difference Equations is Essential for
Obtaining Useful Results

• Since the finite difference equations are
  accurate to O(d2) it is desirable to use a fine
  mesh
• Applying a fine mesh over the entire target
  becomes computationally expensive
• Through experience it was discovered that the
  fine mesh is essential in the region where phase
  change occurs
• A coarse mesh is used in the foam and plastic
  shell portions of the target
• A fine mesh transitioning to a coarse mesh in
  the DT

Representative fine mesh points in phase change
section
Fine mesh depth
tf
Representative coarse mesh points in solid DT
section

DT Vapor Core
DT Ice
26
By Decreasing Mesh Spacing the Solution Converges
Effect of Mesh Size on the Numerical Model
40
mesh 5e-6 m
mesh 1e-6 m
mesh 0.5e-6 m

35
mesh 0.1e-6 m
Mesh Spacing
DT Surface Temperature K)
30
25
20
15
0
0.005
0.01
0.015
Time (s)
27
Mesh Spacing Appears to Directly Influence the
Step Behavior of the Surface Temperature
Effect of Mesh Size on the Numerical Model
30
mesh 5e-6 m
mesh 1e-6 m
mesh .5e-6 m
mesh 0.1e-6 m
28

26
Mesh Spacing
24
DT Surface Temperature K)
22
20
18
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
time (s)
-3
x 10
28
If a Sufficient Number Fine Mesh Points are not
used the Solution Appears to Saturate at the
Melting Temperature
Fine Mesh Depth
Fine Mesh Depth, tf 40 microns
tf
20
19.8
19.6
19.4

19.2
Apparent Saturation
DT Surface Temperature (K)
19
18.8
18.6
18.4
18.2
18
0
0.005
0.01
0.015
Time (s)
29
By Increasing the Number of Fine Mesh Points the
Saturation Behavior is Eliminated
Fine Mesh Depth
tf

30
Sublimation/Evaporation at the DT Outer Layer
can be Investigated

• Assumptions
• A small gap ( 0.1mm) is initially present around
  entire target
• Gap is filled with DT vapor at the saturation
  pressure corresponding to the vapor temperature
• The volume of the gap is approximately constant
  over a time step (0.1ms)
• DT vapor can be modeled as an ideal gas

Vapor Gap
Plastic Shell
DT Vapor Core
DT Solid/Liquid
Simplified Target Cross Section
31
The Mass Flux of Vapor and hence the Total Vapor
Mass in the Gap can be Calculated

• The mass flux from a surface is given by

(kg/m2-s)
Evaporation
Condensation
Assuming a constant volume over a time step the
mass flux equation can be integrated, giving
(kg)
Mass if Vapor is Saturated
Mass if Vapor is Saturated
Time Constant, t (s-1)
Mass at the End of the Last Dt
Time Step (s)
For A/V, T, and Dt applicable to this problem
For the time steps used in the numerical model
the vapor is saturated by the end of the time step
32
Sublimation/Evaporation Creates an Apparent Heat
Flux at the Surface of The Sublimation/Evaporatio
n Interface

• How will the latent heat associated with
  sublimation/evaporation effect the heat transfer
  into the DT solid/liquid?
• Using the mass flux equation and using the
  assumption that the vapor is saturated at each
  time step an average mass flux per time step can
  be found

(kg/m2-s)
qConstant
The apparent heat flux due to sublimation/evaporat
ion is then given by
Foam
qevap
(W/m2)
Vapor
Where Ls/e (J/kg) is the latent heat of
sublimation or evaporation of the DT evaluated at
the surface temperature.
DT Solid/Liquid
A portion of the target
33
The Mechanical Response of the Target to
Evaporation/Sublimation

• Assumptions
• Each of the assumptions used for determining the
  thermal response (small gap, gap constant width
  over Dt, etc.)
• DT solid is rigid
• The shell is the only structure resisting
  deflection (i.e., no foam-DT-shell interaction)

Vapor Gap
Plastic Shell
DT Vapor Core
Rigid DT Solid
Simplified Target Cross Section
34
If a Vapor Gap is Present Around the Entire
Target, Membrane Theory can be Used to Model
Deflection

• Deflection of the plastic shell due to vapor
  pressure is modeled using membrane theory (valid
  for r/t gt 10) for a sphere with a uniform
  internal pressure
• Deflection due to thermal expansion
• Total deflection

Uniform Internal Pressure
t
r
35
Local Vapor Gap Growth Could Affect the Heat
Transfer and the Deflection

• Effect on heat transfer was discussed earlier
• Methods of calculating the deflection of the
  shell
• Membrane theory
• Bending theory with simple boundary conditions
• 2-D finite element modeling
• The first two methods will be examined here, the
  third is a likely addition to the future model

Local Vapor Gap
Plastic Shell
DT Vapor Core
Rigid DT Solid
Simplified Target Cross Section
36
Membrane Theory is Applicable to Uniformly Loaded
Spherical Sections When the Bending Resistance is
Low

• Assumptions for the use of membrane theory
• The ratio r/t gt 10
• The edges of the spherical section only resist
  loading tangentially.
• Small deflection relative to the radius of
  curvature
• Using these assumptions the deflection of the
  plastic shell under local bubble growth, with
  pressure P, is identical to the deflection of a
  full spherical shell with a constant internal
  pressure P

Tangential b.c.
Tangential b.c.
Uniform Internal Pressure, P
Portion of the target under loading by a local
bubble
37
Bending Theory with Clamped Boundary Conditions
Shows the Possible Effect of Foam-Shell
Interaction

• Assumptions for the use of bending theory
• Uniform local pressure P
• The edges of the spherical section are clamped.
• Small deflection relative to the radius of
  curvature
• Equations of deflection

Uniform Internal Pressure, P
Spherical Section
f
a
Membrane Theory
38
A Comparison of ModelsIf fixed Conditions Exist,
there is no Deflection for Small Gap Sizes
Spherical Section
f
a
39
The Coupling of Deflection and Heat Transfer

• As the vapor pressure increases due to
  evaporation/sublimation the thermal resistance of
  the vapor region increases
• Since the deflection is small for each time step
  the thermal resistance is based on the size of
  the vapor gap at the beginning of the time step.
  This eliminates the need for iteration.

Vapor Gap
Plastic Shell
DT Vapor Core
Rigid DT Solid
Simplified Target Cross Section
40
The Target must be Separated for Numerical
Modeling Purposes

• The heat transfer model is non-linear due to the
  presence of the evaporation/sublimation heat
  flux
• Even if the evaporation/sublimation heat flux
  can be neglected, the drastic differences in the
  properties of the vapor and other materials,
  require the problem to be separated into three
  linear problems
• This problem requires iteration to ensure the
  consistency of qo, qI, To, and TI

qConstant
qConstant
Foam
Foam
To
qo
qevap
TI
Vapor
qI qo- qevap
DT Solid/Liquid
DT Solid/Liquid
41
The Model is in its Final Development Stages

• Initial runs indicate that the model correctly
  predicts the expected behavior for cases with a
  fixed vapor gap thickness
• However, the model needs to be fine-tuned for
  cases when the shell is allowed to deflect to
  make sure that energy is conserved
• Difficulty is linked with the sensitivity of the
  evaporation/condensation heat and mass fluxes to
  small changes in temperature

42
Initial Results from the Model Exhibit the
Expected Behavior as the Outer Foam Insulating
Layer is Increased ( for a Constant 0.1 mm Vapor
Gap)
The vapor-plastic interface temperature history
decreases with increasing foam thickness
The vaporization interface temperature history
decreases with increasing foam thickness
43
The Pressure in the Gap is Decreased
Significantly by Utilizing Insulating Foam
(Constant 0.1 mm Vapor Gap)
44
Future Modeling Effort and Issues to be Addressed

• Correct the problems that occur when the plastic
  shell is allowed to deflect
• Compare the results of the new model and the
  former simple model
• Incorporate a 2-D model of the deflection of the
  shell due to localized vapor bubble growth
• Effect of foam and plastic shell thickness
• How will the solid DT and the foam interact ?
• Investigate possible modes of heat transfer
  through vapor gap
• Continuum conduction
• Molecular conduction
• Continuum convection
• Investigate the response of the DT solid when a
  load is created by increasing vapor pressure
• Check to ensure that the model is conserving
  energy
• 1-D or 2-D thermo-mechanical model ?

45
RD Requirements (experimental other)

• DT/foam and plastic outer coating bond quality
  characterization
• Thermal/mechanical behavior at DT/outer coat
  interface
• DT mechanical properties
• Plastic (solid and foam) thermo-mechanical
  properties at cryogenic temperatures
• Foam insulation fabrication (graded porosity)
• Target physics guidance on acceptable vapor
  region and foam region thicknesses (including
  porosity grading)
• Propose to help with modeling experimental
  results to support pre and post experimental
  analyses and to help better understand target
  thermo-mechanical behavior

46
Reminder of Action Items from Last Target
Survival Workshop (NRL, Dec. 2002)

• 1) NRL to evaluate the insulating foam target for
  stability (both uniformly dense and graded) (A.
  Schmitt, D. Colombant, S. Obenschain)
• 2) Schafer to look up the data on a "graded
  density" foams and see if this could be feasible
  (D. Schroen)
• 3) NRL to confirm that a uniform DT vapor region
  thickness below the outer seal (of about 3
  microns) is acceptable and, in the actual case of
  non-uniform heating to provide guidance on how
  much variation is acceptable between the
  thickness of the vapor regions on opposite ends
  of the target (i.e. corresponding to the highest
  and lowest heat fluxes) (A. Schmitt, D.
  Colombant, S. Obenschain)
• 4) GA/UCSD to evaluate how much temperature
  drop there is to keep the insulated target cold
  (with beta decay heat) and determine how
  beneficial this temperature drop is with respect
  to survival estimates (R. Raffray, R. Petzoldt)
• 5) GA/UCSD to evaluate the effect of asymmetric
  heating in particular on local phase change
  behavior. A new multi-dimensional model being
  developed for the thermo-mechanical behavior of
  the target will help better understand this (R.
  Raffray, R. Petzoldt)
• 6) GA/UCSD to evaluate whether the insulated
  target with an outer seal that is permeable could
  actually be filled and "dryed" of DT in the outer
  foam (R. Petzoldt, R. Raffray)
• While not specifically discussed at the
  workshop, two additional action items came out of
  subsequent discussions and are listed below
• 7) Measure the compressive strength of DT/foam at
  relevant temperatures (J. Hoffer).
• 8) Investigate possibility of layering at lower
  temperature (18, 17, 16 K) to provide a means of
  accommodating higher heat fluxes during
  injection. (J. Hoffer). The effect of the
  correspondingly lower gas pressure on the target
  physics should be assessed (NRL).

47
Follow-Up on GA/UCSD Action Items from Last
Target Survival Workshop

• GA/UCSD to evaluate how much temperature drop
  there is to keep the insulated target cold (with
  beta decay heat) and determine how beneficial
  this temperature drop is with respect to survival
  estimates (R. Raffray, R. Petzoldt)
• - q from beta decay small 5.42 x 104 W/m3
• - q through insulating layer 20 W/m2
• - DT through 72 mm 25 dense insulating layer
  0.07 K
• - DT though DT ice even lower 0.02 K
• 5) GA/UCSD to evaluate the effect of asymmetric
  heating in particular on local phase change
  behavior. A new multi-dimensional model being
  developed for the thermo-mechanical behavior of
  the target will help better understand this (R.
  Raffray, R. Petzoldt)
• 6) GA/UCSD to evaluate whether the insulated
  target with an outer seal that is permeable could
  actually be filled and "dryed" of DT in the outer
  foam (R. Petzoldt, R. Raffray)