Jason E. Huffer, M.S.Washington Safety Management Solutions (WSMS)Jason.Huffer@wsms.com

Jason E. Huffer, M.S.Washington Safety Management Solutions (WSMS)Jason.Huffer@wsms.com

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

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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:

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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.

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Stephanie Corinne Huffer Memorial Scholarship Endowment Fund

Stephanie Corinne Huffer Memorial Scholarship Endowment Fund

North Carolina State University-Animal Sciences

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Involvement in abstinence programs on campus

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