Lesson 4: Introduction

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Lesson 4 Introduction

• Buckling equivalence method
• Surface density method
• Density analog method
• Validation
• Calculation of subcritical limits

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Hand calculation methods

• Buckling shape conversion
• Surface density method
• Analog density method
• Solid angle method (not studied)
• Usefulness
• Analyst Starting point for your model
• Analyst/Reviewer Approximate check of results

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Buckling equivalence

• From one-speed diffusion theory, we have

which (ultimately) gives us
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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 d2 cm (bare) or d5 cm (H2O reflected) if
  better information not available

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Buckling equivalence (3)

• Example
• 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)

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Surface density method

• Basic idea
• What is the minimum spacing of a 2D array of
  unstacked units for criticality?
• Express as an areal density
• An infinite array of basic units with spacing d
  can be limited by a compressed, water reflected
  arrangement, corrected for unit reactivity
• s, s0 Areal densities (g/cm2)
• f fraction critical (MUST be lt 0.73)

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Surface density method (2)

• Equation
• Conservatism applied to optimum SLAB density
  (found using Fig. 7.2)
• Conservatism applied to most reactive single unit
  (with worst-case shape and reflection)

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Surface density method (3)

• Procedure
• Given a unit fissile mass and H/U ratio, find the
  unreflected spherical critical mass from Fig. 7-1
  (25 mm curve).
• Using this value, find f(unit mass)/(uscm)
• If fgt0.73, you cannot use the method.
• For H/U ratio, use Fig. 7-2 to get BOTH the
  critical thickness (300 mm curve) AND the
  concentration. The product of these is s0.
• Use the formula to get your limiting s value (or
  other values from it).
• Apply intelligently (floor or one of the walls).

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Surface density method (4)
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Surface density method (5)
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Density analog method

• Basic idea
• 3D version of previous
• Basic idea What is the minimum spacing of a 3D
  array of units for criticality?

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Density analog method

• Basic idea
• 3D version of previous Finite cubic array of N
  units limited by a volumetrically smeared sphere
  of same material
• Uses the same parameters as SD methodVolume of
  unit (V) and mass of unit (m)
• Compared to SD method
• Arrays can be 4x taller (but finite in number)
• Vertical spacing must be maintained (e.g.,
  cabinet)

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Validation

• Testing and sharpening calculational tools
• Evaluation of method and cross section data
• Assure yourself that methodsdata can calculate
  the relevant experiments
• Find the bias
• Find the trends
• Establish range of applicability

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Validation Definition

• Indication of reliability of calculation vs.
  reality
• How far do we trust the calculational method and
  under what conditions?
• vs. Verification Indication of reliability that
  the mathematical operations are being performed
  as expected
• Use sample problems from vendor

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Validation Def. (contd)

• Applies to code AND data
• WRONG We validated KENO-Va.
• RIGHT We validated KENO-Va using the standard
  27 group library.
• Bottom line Conditions predicted to be
  subcritical would be expected to be subcritical.
• NOT vice versa.
• Not accuracy, but reliability (conservatism) in a
  particular direction.

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Other terms of interest

• Upper subcritical limit (k-safe) highest
  value of calculated k-eff 2s that is considered
  subcritical in evaluation.
• U.S.L. 1 bias - bias uncertainty - MSM
• Bias systematic disagreement between calculated
  and experimental data.
• Single sided lower tolerance limit
• Single sided lower tolerance band
• Bias is calculated as the average calculated
  k-effective - 1

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Other terms (contd)

• Bias uncertainty Measure of precision of
  calculation and accuracy of experiment
• Uncertainty in experiment
• Lack of precision in calculational method
• (Extension to conditions out range of
  experiments)
• Calculated as
• 1. Some multiple (usually 3)of the standard
  deviation of the population of calculated
  k-effectives (i.e., consider the k-effs as a
  series of estimates and calculate the SD NOT the
  SD of the calculations) (Statistical approach)
• 2. A value that is below the lowest calculated
  k-effective for any benchmark (Non-statitical
  approach)
• NOTE Both the bias and bias uncertainty can vary
  with a parameter (usually enrichment)

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AOA

• Area of applicability (AOA) Range covered by
  experiments in variable space
• Traditional procedure
• 1. Decide on the important nuclides in the
  problem based on
• Fissile material
• Enrichment ( range)
• Absorbers
• Moderation (HX range)
• Moderating material
• Reflection material
• 2. Find benchmark problems that contain these
  nuclidespreferably several of each to BOUND the
  concentration that the application has
• 3. Pay particular attention to spectrum based on
• H/U ratio (or its equivalent for moderation and
  fissile material in the problem
• EALFenergy of the average lethargy of fission
• 4. Take a USL penalty (added to the MSM) if
  outside AOA

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Other terms (contd)

• Minimum Subcritical Margin (MSM) Extra safety
  margin applied to U.S.L.
• Does a thorough validation exist?
• Have independent experimenters been used?
• How do conditions compare to AOA?
• Is the system/process simple or complex?
• Does fissile material maintain its shape?
• Are the physics/chemistry well understood?
• How sensitive is reactivity to changes?
• Usually some minimum (e.g., 3)

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Sample problem

• Using an MSM of 3 and a 3s bias uncertainty,
  determine an USL for the following calculated
  k-effectives
• 1.01 0.98 0.995 1.005 1.000
• Average 0.998 (sum of values/5)
• Bias -0.002 1-average (or 0 if
  positive)
• Variance 0.000133
• Average of error-squaredN/(N-1)
• Sigma 0.0115 Square root of variance
• Resulting USL 0.933 1 bias 3s MSM

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Picking variables for regression
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Example Linear regression