Jason E. Huffer, M.S. Washington Safety Management Solutions (WSMS) Jason.Huffer@wsms
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Transcript of Jason E. Huffer, M.S. Washington Safety Management Solutions (WSMS) Jason.Huffer@wsms
Jason E. Huffer, M.S.Washington Safety Management Solutions (WSMS)[email protected]
Jason E. Huffer, M.S.Washington Safety Management Solutions (WSMS)[email protected]
Optimization of Water-to-Fuel Ratios (W/F) in Clad Cylinder Arrays
2
Simplified Derivation: N=3, t=0Simplified Derivation: N=3, t=0
From geometry:
Solve for h:
Cell Area:
Fuel Area:
22216 PhR +=
( )2216 PRh −=
( )⎥⎦⎤
⎢⎣⎡ −+= 22162 PRRPAreaCell
23 RAreaFuel π=
3
Water-to-Fuel RatioWater-to-Fuel Ratio
Water-to-Fuel Ratio:
( )2
222
3
3162/
R
RPRRPFW
π
π−⎥⎦⎤
⎢⎣⎡ −+
=
With algebra manipulate into a function of P/R:
( )( ) ( )( ) 132163
1/ 2
2−+⎟
⎠⎞⎜
⎝⎛ −= R
PR
PR
PFW ππ
This function can be graphed over the range of 2< P/R < 4:
4
Where’s the Peak? Estimated at 0.5.Where’s the Peak? Estimated at 0.5.
Water to Fuel Ratio Vs. Pitch to Radius Ratio
0.0
0.1
0.2
0.3
0.4
0.5
0.6
2.0 2.5 3.0 3.5 4.0
Pitch to Radius Ratio (P/R)
Water to Fuel Ratio
(W/F)
5
•Derivative from W/F Equation:
Setting the above derivative to zero, multiply by the radical, and simplify:
⎟⎠⎞⎜
⎝⎛ −+⎟
⎠⎞⎜
⎝⎛−= 2
2
2
21622160
RP
RP
Derive the Mathematical SolutionDerive the Mathematical Solution
( ) ( )( ) RPRR
P
R
PR
Pd
FWd
πππ 3
2
1633
16/222
2
2
22
+−
−−
=
6
Isolating the radical to one side, squaring both sides and simplifying yields:
48150 2
2
4
4+⎟⎠⎞⎜
⎝⎛−=
RP
RP
Solving with the quadratic equation:
2
33152
2 ±=
RP
Tests reveal that a Pitch-to-Radius Ratio of 3.2205 results in the maximum possible water to fuel ratio of 0.4941, justifying the bounding assumption of P/R=0.5.
Now Do N= 4, 5, 6, 7, 8, 9, 10 ... N and add a clad thickness (t):
Quadratic AttackQuadratic Attack
7
General SolutionsGeneral Solutions
Handley/Huffer Pitch Optimization Equation:
( )( ) ( )
91681
2
1
28/ 2
2222 +−
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−+
−−=+ NN
NNtRPopt
• W/F Ratio
( ) ( )2
414 ⎟⎠⎞⎜
⎝⎛
+−= tRPhwith • Note: This h is not in the Paper
( ) ( )( )
( )( )
2
2112/
RtRhtR
PN
NtR
PNFW +
⎥⎦⎤
⎢⎣⎡ −⎟
⎠⎞⎜
⎝⎛
+⎟⎠⎞⎜
⎝⎛ −+⎟
⎠⎞⎜
⎝⎛
+= ππ
8
One complaint about the above approach was that it is just looking at the “EDGE EFFECT”.
Yes, and now we can Quantify the Edge Effect!
Demonstration of Edge Effect
0.200
0.300
0.400
0.500
0.600
0.700
0 5 10 15 20 25
Stack Height (N)
Maximum W/F Ratio for
Stack
Infinite Stack = 0.273
Maximum Water-to-Fuel Ratio
9
LEU ExampleLEU Example
10
Varying the pitch for the above configuration of pellets with LEU (5.0 wt.% U-235) yields:Varying the pitch for the above configuration of pellets with LEU (5.0 wt.% U-235) yields:
0.10
0.30
0.50
0.70
1.5 2 2.5 3 3.5
Pitch/Radius Ratio
keff+2*sigma (also W/F Ratio) W/F Ratio
keff+2*sigma for LEU
Max. W/F=0.393
OPT. P/R=3.049
11
SS Fuel Bin Example: 23 LEU Rods HighSS Fuel Bin Example: 23 LEU Rods High
12
Varying the pitch for the above configuration of rods with LEU (5.0 wt.% U-235) yieldsVarying the pitch for the above configuration of rods with LEU (5.0 wt.% U-235) yields
Notice that the effect is damped with the addition of cladding.
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
1.5 2 2.5 3 3.5
Pitch/(Radius+ th.) Ratio
keff + 2
sigma (and W/F Ratio)
W/F Ratio
keff+2*sigma for LEU
Opt. P/(R+t)=2.873
Max. W/F = 0.1231
13
Alternate FuelsAlternate Fuels
Weapons Grade (WG-MOX) Plutonium (35 wt.%) and Natural Uranium (65 wt.%) Reactor Grade (RG-MOX) Plutonium (35 wt.%) and LEU Uranium w/3 wt.% U-235 (65 wt.%)
Modeling the Molybdenum Boat Stacking Incident:
14
The Pu Fuel still responds to a varying W/F Ratio.The Pu Fuel still responds to a varying W/F Ratio.
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.5 2 2.5 3 3.5Pitch/Radius Ratio
keff + 2*sigma (and W/F Ratio)
Opt. P/R
W/F Ratio
LEU
RG-MOX
WG-MOX
keff+2*sigma for:
15
ConclusionConclusion
Applies to any stacked cylinder array.
Accident scenarios where spilled cylinders could pass through optimum pitch.
The analysis will have to optimize the pitch to determine maximum array reactivity.
16
Stephanie Corinne Huffer Memorial Scholarship Endowment Fund
Stephanie Corinne Huffer Memorial Scholarship Endowment Fund
North Carolina State University-Animal Sciences
No Alcohol or Illegal Drug use
Involvement in abstinence programs on campus
Tax deductible donations Yearly “Meet and Greet” for
Donors and Recipients http://www.cals.ncsu.edu/giv
enow Keyword: Huffer
http://www.cals.ncsu.edu/advancement/Stephanie%20Huffer.htm