Transcript ARIES: Fusion Power Core and Power Cycle Engineering

1. Liquid Wall Ablation under IFE Photon Energy
Deposition
2. Flibe Properties from EoS
A. René Raffray and Mofreh Zaghloul
University of California, San Diego
ARIES Town Meeting on Liquid Wall Chamber Dynamics
Hilton Garden Inn, Livermore, CA
May 5-6, 2003
May 5-6, 2003/ARR
1
Physical Processes in X-Ray Ablation
Surface
Vaporization
Energy
Deposition &
Transient Heat
Transport
Liquid
Film
X-Rays
Impulse
y
Induced
Thermal- Spikes
Mechanical
Response
Fast Ions
Phase
Transitions
•Stresses and Strains
and Hydrodynamic
Motion
•Fractures and Spall
x
Slow Ions
• Surface Vaporization
•Heterogeneous Nucleation
•Homogeneous Nucleation
(Phase Explosion)
Spall Fractures
Impulse
Ablation Processes
Expansion,
Cooling and
Condensation
May 5-6, 2003/ARR
z
Phase Explosion
Liquid/Vapor
Mixture
2
High Photon Heating Rate Could Lead to Explosive Boiling
• Effect of free surface vaporization is
reduced for very high for heating rate
(photon-like)
• Vaporization into heterogeneous nuclei
is also very low for high heating rate
Ion-like
heating
rate
Photon-like
heating rate
• Rapid boiling involving homogeneous
nucleation leads to superheating to a
metastable liquid state
• The metastable liquid has an excess free
energy, so it decomposes explosively
into liquid and vapor phases.
- As T/Ttc increases past 0.9, BeckerDöhring theory of nucleation indicate an
avalanche-like and explosive growth of
nucleation rate (by 20-30 orders of
magnitude)
May 5-6, 2003/ARR
From K.
Song and X.
Xu, Applied
Surface
Science 127129 (1998)
111-116
3
Analysis Based on Photon Spectra for 458 MJ Indirect
Drive Target
(25%)
(1%)
• High photon energy for indirect drive target case (25% compared to 6% ion energy))
• More details on target spectra available on ARIES Web site:
http://aries.ucsd.edu/ARIES/
May 5-6, 2003/ARR
4
Photon Energy Deposition Density Profile in Flibe Film
and Explosive Boiling Region (for Rchamber=6.5m)
Energy deposition (J/m 3)
1x10 12
Sensible
energy based
on uniform
vapor
pressure
following
photon
passage in
chamber and
including
evaporated
Flibe from
film
1x10 11
Cohesion energy (total evaporation energy)
1x10 10
0.9 Tcritical
Sensible energy (energy to reach saturation)
1x10 9
1x10 8
1x10 7
Evap.
region
0
Explo.
boil.
region
2-phase region
2.5
4.1
5
10 10.4
15
Penetration depth (micron)
Bounding estimates of thermally-induced aerosol source term from photon energy deposition:
(1)Upper bound: the whole 2-phase region; (2)Lower bound: explosive boiling region
May 5-6, 2003/ARR
5
Photon Energy Deposition Density Profile in Pb
Film and Explosive Boiling Region (for Rchamber=6.5m)
1x10 13
Energy deposition (J/m 3)
1x10 12
1x10 11
Cohesion energy (total evaporation energy)
1x10 10
0.9 Tcritical
1x10 9
Sensible energy (energy to reach saturation)
1x10 8
Evaporated region
1x10 7
1x10 6
0
1
Explo.
boil.
region
2-phase region
1.92
2.5
3
3.94
5
Penetration depth (micron)
May 5-6, 2003/ARR
6
Explosive Boiling Results for Rchamber=3.5 m
Explosive Boiling Ablated Thickness for:
Rchamber= 6.5 m
Rchamber= 3.5 m
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Flibe
Pb
4.1 mm
10.9 mm
2.5 mm
3.83 mm
7
Mechanical Response to Induced Shock
•
•
Rapid increase in internal energy due to x-ray energy deposition and
ablation impulse creates high pressure within the material
-
The induced shock wave propagates to the wall, gets reflected back to the free
surface where a rarefaction wave is produced (creating tensile stresses) and
propagates back through the material. The process is repeated.
-
If the magnitude of the rarefaction wave is greater than the tensile strength of
the material, fracture or spall will occur establishing a new surface.
Evolution of spall in a body subject to transient stresses is complex
-
Material dependent: brittle, ductile or liquid
-
Small perturbations can lead to opening of voids and initiation of spall process
-
A reasonable prediction of the dynamic spall strength, time to failure, and
some measure of the nominal fragment size created in the spall event are
needed to characterize the spall process
-
An upper bound theoretical spall strength can be derived from intermolecular
potential
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8
Theoretical (Maximum) Spall Strength Provides an Upper
Bound Estimate in the Absence of Appropriate Spall Data
• Based on intermolecular potential reflecting dependence on
cohesive energy and bulk modulus with inherent energy balance
- Using a three-parameter potential
such as the Morse potential

  2 ( v  v0 ) 
  ( v  v0 ) 
U ( v )  U coh  exp 
  2 exp 

a
a





- The cold pressure is given by:
P( v )  
dU 2 U coh

dv
a
   2 ( v  v0 ) 
  ( v  v0 ) 
exp

exp





a
a



 
Ucoh= Specific cohesive energy
v = 1/ = Specific volume
v0 = Specific volume at zero pressure
a=(2v0 Ucoh / B0 )1/2
B0 = Bulk modulus
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Theoretical spall strength, Pth,
given by minimum of P(v):
Pth 
U coh B0
8 v0
9
Spall Strength is Highly Temperature Dependent
• Using the Soft Sphere EOS to Estimate Temperature-Dependent Spall Strength
1/ 3
3




  

1

  ( n  4 ) Q  n / 9 
   m 

U ( v , T )  U coh  N K B T   C n  n / 3 
2
K
T
6
K
T
K
T

 B 
 B 
 B 
1/ 3
N KB T 
1
  1
 
 
n/ 3 
n/ 9 
m 
 
  m  

P ( v ,T ) 
n ( n  4 ) Q  
 1  n C n  
V
3
K
T
18
K
T
K
T

 B 
 B 
 B 
n, m, Q, , and  are adjustable parameters
to satisfy the available experimental data
N: Number of molecules,
V: Specific volume,
= N 3 / (21/2) V,
 : the sphere diameter,
Cn: FCC Madelung constant.
• Theoretical spall strength is
then calculated from:
d P ( v ,T )
 0.
dv
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10
Temperature-Dependent Spall Strengths of Example Materials
May 5-6, 2003/ARR
T (K)
Pb (GPa)
Li (GPa)
Flibe (GPa)
750
-2.0014
-1.4401
-2.4914
1450
-1.4098
-0.8950
-1.4212
2250
-0.8981
-0.4267
-0.6848
2999
-0.5221
-0.1010
-0.2814
3749
-0.2235
Gas
-0.0657
11
Parameters of Different IFE Reactor Design Studies &
Pressure Pulse Profile
Parameters
Prometheus-L
Osirus
Hiball
Liquid/Structure
Pb/SiC
Flibe/C
Pb17Li/SiC
Pb/SiC
Flibe/SS
X-ray yield (MJ)
31
129*
95
115
115
Closest distance from target
(m)
5
3.5
5
6.5
3.5
6.5
3.5
Vaporized mass (kg/m2)
0.0125
0.0278
0.0234
0.026
0.04
0.008
0.022
Reactive impulse** (Pa s)
22
59.0
60
15.2
23.31
17.3
45.77
* X-ray and debris
** Ablated material velocity ~ sonic velocity ~ 586/2094 m/s for Pb/Flibe at Tcrit (~5100/4500 K)
•
The pressure wave is steady (no
change in shape)
•
Parameters of different IFE
reactor design studies and the
present study are comparable
•
Osirus profile scaled according to
the relative impulses and used for
the present analysis.
May 5-6, 2003/ARR
Present Study
12
Illustration of Spalling Following a Pressure Wave
Propagation in the Flibe Layer on a Perfectly Stiff Wall
  2000 kg/m3,
1. For a perfectly stiff wall,
the pressure wave is
reflected from the wall and
returns to the free surface
as a pressure pulse
Cs  3300 m/s,
Tin = 885.7 K,
2. Pfree-surface= Pchamber and the
pressure pulse arriving at
the free boundary must be
reflected back as a tensile
wave
Pth = -1.887 GPa
3. If the net tensile stress >
the spall strength of the
material, rupture occurs
establishing a new
surface
Spalled Thickness  2.1 µm & Spall Time (t3 – t1)  16.9 ns
Spall time from the beginning of the pressure pulse = 2 L/Cs+ (t3-t1) 200 ns for a 0.3 mm flibe layer
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13
Summary for the Case of Perfectly Stiff Wall
Flibe
Pb
Explosive boiling
thickness
(µm)
R=3.5 m
10.9
3.8
R=6.5 m
4.1
2.5
Spallation thickness
(µm)
R=3.5 m
4.1
1.8
R=6.5 m
2.1
1.1
R=3.5 m
15.0
5.6
R=6.5 m
6.2
3.6
Total fragmented
thickness
(µm)
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14
BUT No Wall is Perfectly Stiff !
A pressure pulse with initial pressure P0 will be reflected at
the mismatched interface with a new pressure, P where:
P   P0 
(Z wall  Z film )
(Z wall  Z film )
P0
With the acoustic impedance Z =  Cs
E.g. For Pb/SiC and 3.5 m Cavity Radius  = -0.541
• Pressure wave is reflected as a reduced tensile wave.
 • The max. pres., Pmax, of the reflected wave = -6.67 GPa
• Spall tensile strength, Pth, = -1.875 GPa
•
Pmax>Pth  Spall occurs (at 0.94 µm from the wall)
May 5-6, 2003/ARR
Material
Z (Mkg/m2 s)
Flibe
6.84
Pb
19.31
SiC
5.75
SS
46.7
C
6.09
In principle, combination of
liquid and structural material
can be chosen to eliminate or
minimize spalling
15
Spallation Analysis for Different Proposed
Liquid/Wall Combinations
Flibe/SS
Flibe/C
Pb/SiC
R = 3.5
m
0.745
(reflected as a
reduced pressure
pulse )
-0.058
(reflected as a
reduced tensile
pulse)
-0.541
(reflected as a
reduced tensile
pulse)
R = 6.5
m
0.745
(reflected as a
reduced pressure
pulse)
-0.058
(reflected as a
reduced tensile
pulse)
-0.541
(reflected as a
reduced tensile
pulse)
R = 3.5
m
18.2 > |Pth|
1.4 < |Pth|
6.67 > |Pth|
R = 6.5
m
6.8 > |Pth|
0.53< |Pth|
4.35 > |Pth|
R = 3.5
m
3.585
0.0
L-0.96
R = 6.5
m
1.784
0.0
L-0.24

Magnitude of
the maximum
reflected
pressure
Pmax(GPa)
Spallation
thickness
(µm)
May 5-6, 2003/ARR
16
Summary of Study of Wall Ablation Mechanisms as
Aerosol Source Term
• Integrated effect of liquid wall thermal and mechanical responses
to X-ray energy deposition to provide bounding estimates of
ablation as source term for aerosol analysis
-
First principle consideration
-
Ablation depths of liquid wall from explosive boiling and spalling
-
Spall strength of materials as compared to anticipated IFE shocks
-
Acoustic impedances of structural material and liquid can be chosen to
avoid or minimize spalling
• Some key remaining questions
-
Form (vapor conditions, size and distribution of liquid droplets) of the
removed material?
-
What happens to the spalled material (not enough energy to push it
out…where does it go?)
-
Spall time scale
May 5-6, 2003/ARR
17
Flibe Properties from EoS
M. Zaghloul
University of California, San Diego
ARIES Town Meeting on Liquid Wall Chamber Dynamics
Hilton Garden Inn, Livermore, CA
May 5-6, 2003
May 5-6, 2003/ARR
18
Flibe Thermodynamic Properties
 The soft-sphere equation of state for liquid flibe
takes into account the dependence on  and T(*).
Liquid
+
Vapor
Temperature
Tcrit ~ 4500 K
Vapor
chemical
decomposition
of molecules
 Vapor pressure correlations from ORNL & Olander
et al., Fusion Technology,41, 141 (2002). (Revisited)
 Chemical equilibrium codes that use data from
JANAF tables are used (e.g. the computer code
STANJAN, Stanford Univ.)
T ~ 1 eV
 Fitted equations of state for flibe gas(**).
 Inaccuracies include
Plasma
multiple
ionization
- Use of Post-Jensen ionization data calculated from the
Corona equilibrium (valid only for extremely low
densities and very high temperatures-optically thin
plasma-and show no dependence on density).
- Post-Jensen data give no information on the individual
populations needed for the internal energy computation
and so further approximations have to be made.
Improved Modeling and
Computations in the HighTemperature Region are Needed
- No Coulomb corrections.
- No excitation energies included in the computations of
internal energies.
(*) Xiang M. Chen , Virgil E. Schrock, and Per F. Peterson “ The soft-sphere equation of state for liquid Flibe,” Fusion Tech. 21, 1525 (1992).
(**) May
Xiang
Chen, Virgil E. Schrock, and Per F. Peterson, “Fitted Equation of State for Flibe Gas,” Fusion Technology 26, 912 (1994).
5-6,M.
2003/ARR
19
Flibe Vapor Pressure and Stability of LiBeF3(*)
(*) Mofreh Zaghloul, D. K. Sze, A. R. Raffray, “Thermo-physical Properties and Equilibrium Vapor
Composition of Lithium-Fluoride Beryllium-Fluoride (LiF/BeF2) Molten Salt ,” 15th Topical Meeting on the
Technology of Fusion Energy, TOFE, Washington, DC, Nov. 2002.
•
Based on previous experimental results by Buchler and Stauffer(***), Olander
et al(**) assumed a LiBeF3 contribution of ~10% to flibe vapor, and estimated
the LiBeF3 vapor pressure based on a negative energy of formation (G0 < 0)
LiF ( g )  BeF 2 ( g )  LiBeF 3 ( g )
• However, from Knack et al. and using NIST-JANAF tables, G0 for LiBeF3 > 0
for T > 555 K (Meta-stable)
• Accordingly, Olander et al. overestimated the vapor pressure of LiBeF3 by
~ 30 orders of magnitude
•
However the effect on their estimation of the overall vapor pressure of flibe is
much smaller since LiBeF3 contribution was assumed to be ~10%
(**) Olander et al., Fusion Technology,41, 141 (2002).
(***) O. KNACKE, O. KUBASCHEWSKI and K. HESSELMANN (Eds.), Thermochemical Properties of Inorganic
Substances I, second edition, Springer-Verlag, Berlin, Germany (1991).
May 5-6, 2003/ARR
20
Proposed Correlation for Flibe Vapor Pressure
Over a Wide Range of Temperature
One can combine the values
computed for flibe vapor
pressure using thermochemical data from NISTJANAF tables in the low
temperature region to
ORNL measurements
(adequately accurate for T
>1000 C) and use a best fit
to obtain a correlation valid
over a wide range of temp.
log10 PTorr  9.424  11026.208 / T ( K )
May 5-6, 2003/ARR
21
Ionization Equilibrium and Thermodynamic Functions(*)
(*)Mofreh
R. Zaghloul, “A consistent model for the equilibrium thermodynamic functions of partially
ionized Flibe plasma with Coulomb Corrections,” Physics of Plasmas, 10, No. 2 (2003) 527-538.
Assumptions
I- Fully-Dissociated Flibe-Gas (T> 1 eV) (Mixture of Monatomic Gases);
3
II- Local Thermodynamic Equilibrium LTE
(Validity Criterion by Fujimoto(**))
nc ( cm )  1.5  10
18
Z
2
nuc

 Te

a

 ,
7
Z nuc
10 6 
a  0.55  ( 0.49 / Z nuc )1.5
III- No Nuclear Reactions (Constant Number of Heavy Particles ) nH= Const.;
3
IV- Electro-neutrality.
Z max, j
 i 
j 1
i 1
3
i,j
  Z e , j  Z av ,
j  Li , Be , F
j 1
V- Coulomb Corrections
Continuum Lowering(***)
I  Ir, j   Ir, j
eff
r, j
(**) T. Fujimoto et al, Phys. Rev. A 42, 6588 (1990).
Terminated Partition Function
Ur, j 
Er , j , n  I reff, j
g
n 1
r, j,n
 Er , j , n 

exp  
 KB T 
(***)
W. Ebeling et al, Theory of Bound States and Ionization Equilibrium in Plasmas and Solids (Akademic-Verlag, 1976)
May 5-6, 2003/ARR
22
Thermodynamic Relations
1- Pressure
F 

P   
 nH K B T (1  Z av )  PCoul

V

T , { N e , N k }
e
2- Internal energy
3- Contribution of
excitation energies of
ion i of element j to
internal energy
4- Enthalpy
5- Adiabatic exponent
1
1
1
 F
E  (F TS)  T2
 


  T  T V , { N e , N k }
1


 3
 n H ( 1  Z av ) K B ( T  T0 )  nH
 2

W j ,i  K B T 2

 g
j ,i ,
1

 j ,i , exp   j ,i , K B T 
( I
z 1
z


 I z )  W j ,i   ECoul 



 j ,i ,  I j ,i  I j ,i
 g
j ,i ,
exp   j ,i , K B T 
(  eP)
  C P / Cv ,
1/ 2
6- Sound Speed
i

ln U j ,i
T
 j ,i ,  I j ,i  I j ,i
he 

 i 
j1 
i 1

J Z max, j
P
vs  
 


S
 P 
 
 
 
T 
 
1/ 2
The most rigorous expressions for thermodynamic functions have been used
May 5-6, 2003/ARR
23
Results of Detailed Composition & Average Ionization State
• As the temperature increases, the molar
fractions of neutral species (Li, Be, and F)
decrease monotonically as a result of the
progressive ionization.
• With further increase of the temperature,
higher ionized ionic species appear on the
expense of lower-fold ionized species.
May 5-6, 2003/ARR
• The lower the density the lower
recombination rate, and the higher <Z>.
• The lower the density the more prominent is
the step-like structure of the <Z>.
• At very high T, the value of <Z> approaches
its expected value of a fully-dissociated,
fully-ionized flibe-plasma , <Z> = 6.57.
24
Results for Flibe Pressure & Internal Energy
•
•
Substantial deviation from the ideal-gas linear behavior.
Ionization and Coulomb interactions among charged particles are responsible
for the deviation in the pressure curves, while for internal energy another
contribution comes from excitation.
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25
Results for Flibe Specific Heat, Adiabatic Exponent & Sound Speed
• Non-linearity characteristics of high
temperature plasma are most prominent.
• At very high T, Cp increases to its expected
value of a fully-dissociated, fully-ionized flibeplasma (Cp = 1.1138104 J/kg K) and so for the
other properties.
May 5-6, 2003/ARR
26
Results
Validity of LTE Assumption & Effect of Coulomb Corrections
• <Z> is the most affected parameter at
• LTE assumption is applicable for
densities  0.001 kg/m3 for the
temperature range shown.
May 5-6, 2003/ARR
low temperatures.
• Percentage relative differences of
other parameters at the same density
are bounded within  15 %.
27
Comparison with Chen’s et al Results (*).
(*) Xiang M. Chen, Virgil E. Schrock, and Per F. Peterson, Fusion Technology 26, 912 (1994).
• Differences of up to ~ 65%
May 5-6, 2003/ARR
• Differences of up to ~ 150%
28
Optical Characteristics of Flibe Multi-Frequency Absorption
Coefficient
In the evaluation of the multi-frequency absorption coefficient, 
• Molecular absorption is neglected (the plasma temperature is considered to be
sufficiently high > 1 eV that the plasma is fully dissociated);
• Photon energies are low enough (< 1MeV) that high-energy absorption processes such
as pair production and nuclear photoabsorption can be ignored;
• Only electronic transitions in the field of the ions are considered (bound-bound, boundfree, and free-free) as these constitute the major absorption processes for most
conditions of interest.
May 5-6, 2003/ARR
29