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Nuclear Reaction Pressure Vessel Safety Assessment

Name Affiliation Phone Number Email Address
Matt Butchers matt.butchers@ktn-uk.org
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This use case is taken from Ref [1] The nuclear reactor pressure vessel (NRPV)

  • A key component
  • Height: 13 m, Internal diameter: 4 m, thickness: 0,2 m, weight: 270 t
  • Contains the fuel bars
  • Where the thermal exchange between fuel
  • bars and primary fluid takes place
  • It is the second “safety barrier”
  • It cannot be replaced !
  • Nuclear Unit Lifetime < Vessel Lifetime
  • Extremely harsh operating conditions
  • Pressure: 155 bar
  • Temperature: 300 deg C
  • Irradiation effects: the steel of the vessel becomes progressibly brittle, increasing the failure risk dueing an accidental situation
  The problem formulation is typical in most nuclear safety problems:
  • Given some hard (and indeed very rare) accidental conditions, what is the “failure probability” of the component?
  • It is the case of “structural reliability analysis” (SRA)
  • The physical phenomenon is described by a computer code
Safety Assessment Example [Muoz-Zuniga et al., 2009] Step A   Accidental conditions scenario: cooling water (about 20 °C) is injected in the vessel, to prevent over-warming
  • Thermal cold choc ->  Risk of fast fracture around a manufacturing flaw
  Thermo-mechanical fast fracture model:
  • thermo-hydraulic representation of the accidental event (cooling water injection, primary fluid temperature, pressure, heat transfer coefficient)
  • thermo-mechanical model of the vessel cladding thickness, incorporating the vessel material properties depending on the temperature t
  • a fracture mechanics model around a manufacturing flaw
  • Stress Intensity \(K_{CP}(t)\) in the most stressed point
  • Steel toughness, \(K_{IC}(t)\) in the most stressed point
Goal: Evaluate the probability that for at least one t, the function \(G = K_{IC} - K_{CP}\) is negative screen-shot-2016-12-13-at-12-19-52  

2.1 Process Inputs

A huge number of physical variables. In this example, three are considered as random. Penalised values are given to the remaining variables

  1. Toughness low limit, playing in the steel toughness law $$K_{IC}$$ . Normal dispersion around a reference value $K_{IC}^{RCC}$
  2. Dimension of the flaw h
  3. Distance between the flaw and the interface steel-clad d
  Table 1. Uncertain parameters.   Table 1. Uncertain parameters.
Variable Distribution Distribution Parameters Comments
\(u_{K_{I}_{C}_1\) Normal \(K^{RCC-M}_{I_C}\) and variation coefficient : \(c_K_I_C = 15 %\) Support truncated at [\(-4\sigma ; +4\sigma\)]
h Weibull Scale parameter \(\alpha\) = 3.09 mm and shape \(\Beta\) = 1.08 mm Distribution estimated by fitting exercise over inspection data
d Uniform [0.1; 100] (mm) The flaw is supposed to be in the inner half-thickness
  screen-shot-2016-12-13-at-12-20-39 A more complex example with 7 randomised inputs is given in [Munoz-Zuniga et al, 2010]  

2.2 Propagation

A numerical challenge

  • High CPU time consuming model
  • Standard Monte Carlo Methods are inappropriate to give an accurate estimate of \(P_f\)
  • An innovative Monte Carlo sampling strategy has been developed: ''ADS-2'' (Advanced Directional Stratification)
A numerical challenge:
  • Standard transformation
  • Directional sampling
  • Adaptive strategy to sample more ''useful'' direction
Table 2: Example of results: NB Pf is here conditional to the occurrence of very rare accidental conditions
Method p n \(f_1(n)\) \{\hat{P}_f (X 10^{-6})\) 95 % CI Length (X 10\(^{-6}\)) Nb. of Calls to \(G_{Min}\)
1 DS 3 50 / 4.3 13 208
2 2-ADS 3 50 10 6.3 7.0 820
3 DS 3 200 / 3.7 16 822
4 2-ADS 3 200 40 8.0 5.0 3290
5 DS 3 1000 / 7.66 7.0 4028

2.3 Interpretation and Communication of Results