Review MAGICMERV Buckling equivalence method Areal density concept Surface density method Review MAGICMERV Which ones can stand alone M A G I C M E R V 2 Hand calculation methods Buckling shape conversion ID: 213506
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Slide1
Lesson 3: Hand calculations
Visiting lecturer: Chris Haught
Review: Lesson 2 slides
Review:
MAGICMERV
Buckling equivalence method
Homework assignedSlide2
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SCALE 6.x or SCALE 6.x-EXE
Slide3
Hand calculation methods
Buckling shape conversion
Surface density method (not studied
)
Analog density
method (not studied)
Solid angle method (not studied)
Usefulness:
Analyst: Starting point for your model
Analyst/Reviewer: Approximate check of resultsSlide4
Buckling equivalence
From one-speed diffusion theory, we have:
which (ultimately) gives us:Slide5
K-effective
Re-arranging and integrating over all volume:
So:
(Roughly) For a given material, the integrals will all increase and decrease proportionally.
The bigger
B
2
is, the LOWER k-effective is
(Roughly) For a given material, two unit shapes with the same
B2 will have the same k-effectiveSlide6
Buckling
equivalence (2)Slide7
Buckling equivalence (2)
For different geometric arrangements, the buckling reduces to Table 8-I in text:
Sphere of radius r
Cylinder (r,h)
Cuboid (a,b,c)
Use
d
=2 cm (bare) or
d
=5 cm (H2O reflected) if better information not availableSlide8
Buckling equivalence
(3)
Sample problem:
You have determined a volume limit of 10 liters for a cylinder with a H/D ratio of 2.5.
What would be the volume of a sphere with the same k-effective?
Answer: 5.2 litersSlide9
Homework
Homework 3-1
You fill a shoebox (15 cm x 20 cm x 30 cm) with a fissile solution and find that it is exactly critical. What would be the approximate radius of a critical sphere of the same material? (Ans. = 11.645 cm)Slide10
Homework
Homework
3-2
It is common practice to assume that a cylinder with a height/diameter ratio of 1.00 has the highest reactivity (lowest buckling). Use your calculus to show that based on the
formula (ignoring the extrapolation distance),
the actual value for the most reactive H/D (for fixed volume) is 0.924.
(You may use a spreadsheet to show that this is optimum.) Slide11
Think like a neutron
What separates good NCS engineers from great NCS engineers is to
see
a situation
Understand the process being evaluated [multi-disciplinary]
Understand the risks by understanding how neutrons behave [
Feynmann
K-25]
This gives you credibility because you can explain why different rules are in place without having to look them up [crit vs fire as you get close]
NOT having to say: “Wait, let me calculate that” [8.26 hands-on course]Slide12
K-effective: Take 1
Generational
What is a generation?
Most unsophisticated
Surprisingly, this is the one that KENO uses!Slide13
K-effective: Take 2
Four factor formula (yeah, I know it has five!):
Fermi’s invention: One of the most amazing analysis tools ever
He knew how to put together experiments to determine each of them
When the multiplied to give >1, he knew it was physically possible!Slide14
K-effective: Take 3
Balance:
Neutron PRODUCTION from fission exactly balances neutron LOSS from absorption and leakage
Most useful for us because it ties to the
MAGICMERV
parametersSlide15
K-effective: Take 4
Eigenvalue
The eigenvalue
l
is k-effective
Its represents how much
n
would have to decrease to make the equation balanceSlide16
Criticality: Neutron balance (2)
Our focus is a little different from reactor physics because we are much more influenced by LEAKAGE
In this regard, we are much closer to Fermi, et al., because of the UNIQUENESS of our situations and our strong dependence on SIZE and SHAPE of the system being consideredSlide17
U-235 SphereSlide18Slide19Slide20Slide21Slide22
Parametric overview: MAGICMERVSlide23
MAGICMERV
Simple checklist of conditions that MIGHT result in an increase in k-eff.
Mass
Absorber loss
Geometry
Interaction
Concentration
Moderation
Enrichment
ReflectionVolume23Slide24
Parameter #1: Mass
Mass: Mass of fissile material in unit
More is worse -- higher k-eff (usually).
Possible maximization problem. (Example?)
Should allow for measurement uncertainties (e.g., add 10% for assay accuracy)
Parametric studies?
24Slide25
Figure 7: Effects of Mass on a Fission Chain Reaction Slide26
Parameter #2: Loss of absorbers
Loss of absorbers: Losing materials specifically depended on for crit. control
More (loss) is worse
Not usually a problem because not usually used
We specifically avoid this situation by removing all absorbers we can identify (e.g., can walls, boron in glass)
BE CAREFUL: Fruitful area for contention
Parametric studies?
26Slide27
Parameter #3: Geometry
Geometric shape of fissile material
Worst single unit shape is a sphere: Lowest leakage
Worst single unit cylindrical H/D ratio ~ 1.00
0.94 in a buckling homework problem
Do not depend on either of these in situations with multiple units
Parametric studies?
27Slide28
Figure 9: Typical ContainersSlide29
Figure 10: Favorable vs. Unfavorable GeometrySlide30
Parameter #4: Interaction
Interaction: Presence of other fissile materials
More is usually worse. (Counterexample?)
Typical LATTICE study:
Number
Arrangement
Stacking
Other processes (e.g., material movement) in same room
Hold-up
Parametric studies?30Slide31
Figure 11: Neutron InteractionSlide32
Figure 12: Example of Physical Controls on InteractionSlide33
Parameters #5: Concentration
Concentration
Solution concentration
Considered in addition to mass, volume, moderation because of CONTROL possibilities
No new physics here
33Slide34
Parameter #6: Moderation
Moderation: Non-fissile material that is intermingled with fissile material
Slows down the neutrons [elastic on board]
Affects absorption (up) and leakage (down)
More is usually worse.
Simultaneously a reflector
Usual cases:
Other material in vicinity of unit (structure,
equip’t
)Water from sprinklersOperator body partsParametric studies?34Slide35
Moderation metric: H/U
The common measure of moderation is the ratio of the number of hydrogen atoms to uranium atoms, the H/U ratio.
Often, when other fissile elements (like Pu) are possibly present, this will be called the H/X ratio
But in SCALE, we have to specify VOLUME fractions of water and uranium
How do we get from one to the other?
Answer: A little algebra
35Slide36
Moderation metric: H/U (2)
Here is how it goes: If water is the ONLY source of H and uranium is the ONLY source of U, we must have:
for some constant k
How do we find k?
Simple: Run a simple problem with the volume fractions equal (each 0.5) and see what the H/U fraction is (SCALE reports it as H/92235)
1.36765
36Slide37
Moderation metric: H/U (3)
Now it is just algebra:
By the way---On a test, I will NOT believe that you can solve for in your head, so don’t just memorize the formula!
37Slide38
Figure 14: Energy Losses in
Neutron Collisions Slide39
U-235 Cross sectionsSlide40
100% enriched, H/U=0Slide41
U-235 Cross sectionsSlide42
100% enriched, H/U=1Slide43
U-235 Cross sectionsSlide44
100% enriched, H/U=0Slide45
U-235 Cross sectionsSlide46
100% enriched, H/U=0Slide47
U-235 Cross sectionsSlide48
100% enriched, H/U=0Slide49
U-235 Cross sectionsSlide50
100% enriched, H/U=0Slide51
Critical mass curveSlide52
Parameter #7: Enrichment
Enrichment: % fissile in matrix
U-235, Pu-239, U-233 (?)
Higher is worse. (Counterexamples?)
Source of problem in Tokai-mura accident
Parametric studies?
52Slide53
Parameter #8: Reflection
Reflection: Non-fissile material surrounding the fissile unit
Effect of interest: Bouncing neutrons back
More is worse. (Counterexamples?)
Usual cases:
People: 100% water without gap
Floors
Walls: Assume in corner
Worse than water: Poly, concrete, Be, steel, lead
Do not underestimate nonhydrogenous reflect’nParametric studies?
53Slide54
Figure 15: Nuclear ReflectionSlide55
Parameter #9: Volume
Volume: Size of container holding fissile material
Usually of concern for:
Spacing of arrays (Less is worse.)
Flooding situations. (More is worse.)
Very sensitive to fissile mass
Parametric studies?
55Slide56
Mental exercise
Which ones can provide COMPLETE protection alone?
F, A, or L?
M
A
G
I
C
M
ERV56