Reference problem ===================== Geometry --------- The geometry studied is that of the core zone of a generic 1300 MWe level tank. Defect considered ---------------- Within the framework of the simplified methodology for fault toxicity analysis, the method for calculating the elastic stress intensity factor via the influence coefficient method intrinsically assumes the absence of effective modeling of the defect in the mesh. In fact, the mesh makes it possible to first calculate the stresses at the nodes and post-processing is then applied to calculate the intensity factor of the elastic stresses by the method of influence coefficients based precisely on these constraints at the nodes (the method is detailed in [:ref:`R7.02.10 `]). For this test, the sub-coating defect in question is semi-elliptical and has a longitudinal orientation at a first stage and a circumferential orientation at a second time. Its tip on the coating side rests rigorously on the interface between the coating and the base metal. Its dimensions are as follows (see figure below): * depth: :math:`{\mathit{prof}}_{\text{def}}\mathrm{=}\mathrm{5mm}` * width: :math:`\mathrm{2b}\mathrm{=}\mathrm{25mm}` .. image:: images/Shape1.gif .. _RefSchema_Shape1.gif: General description of thermo-mechanical chaining ------------------------------------------------- The study consists of thermo-mechanical chaining. The thermal calculation as well as the mechanical calculation are carried out on the same mesh. The mesh used for these two calculations consists only of elements of the 2nd order (so-called "quadratic" mesh). Taking into account the invariances of the problem, the model chosen is axisymmetric and only one mesh is used in the direction of the height of the tank. Material properties ----------------------- **For thermal calculation:** Two properties are specified, they are: * LAMBDA: isotropic thermal conductivity as a function of temperature, expressed in :math:`{\mathit{W.m}}^{\mathrm{-}1}\mathrm{.}{K}^{\mathrm{-}1}`, * BETA: volume enthalpy as a function of temperature, expressed in :math:`{\mathit{J.m}}^{\mathrm{-}3}`. For the coating: .. csv-table:: "**Temperature (** :math:`°C` **)**", "**LAMBDA**" "0", "14.7" "20", "14.7" "50", "15.2" "100", "15.8" "150", "16.7" "200", "17.2" "250", "18" "300", "18.6" "350", "19.3" .. csv-table:: "**Temperature (** :math:`°C` **)**", "**BETA**" "0", "0.000000.E+00" "50", "1.102100.E+08" "100", "3.013300.E+08" "150", "5.014300.E+08" "200", "7.081300.E+08" "250", "9.188800.E+08" "300", "1.132910.E+09" "350", "1.348980.E+09" For the base metal: .. csv-table:: "**Temperature (** :math:`°C` **)**", "**LAMBDA**" "20", "37.7" "50", "38.6" "100", "39.9" "150", "40.5" "200", "40.5" "250", "40.2" "300", "39.5" "350", "38.7" .. csv-table:: "**Temperature (** :math:`°C` **)**", "**BETA**" "0", "0.000000.E+00" "50", "1.061900.E+08" "100", "2.903300.E+08" "150", "4.829100.E+08" "200", "6.832800.E+08" "250", "8.921600.E+08" "300", "1.109440.E+09" "350", "1.335060.E+09" **For mechanical calculation:** Five parameters are filled in, they are: .. csv-table:: "* :math:`E`:", "Young's modulus, expressed in :math:`\mathit{Pa}`," "* :math:`\mathit{nu}` ", "Poisson's ratio (n=0.3)," "* ALPHA:", "isotropic thermal expansion coefficient, expressed in :math:`°C`," "* TEMP_DEF_ALPHA = 20:", "value of the temperature at which the values of the thermal expansion coefficient ALPHAont were determined, expressed in :math:`°C`." "* VALE_REF = 280", "Reference temperature :math:`{T}_{\mathit{Réf}}` for which there is no thermal deformation, expressed in :math:`°C`." For the coating: .. csv-table:: "**Temperature (** :math:`°C` **)**", ":math:`E`" "0", "1.985E+11" "20", "1.97E+11" "50", "1.95E+11" "100", "1.915E+11" "150", "1.875E+11" "200", "1.84E+11" "250", "1.8E+11" "300", "1.765E+11" "350", "1.72E+11" .. csv-table:: "**Temperature (** :math:`°C` **)**", "**ALPHA**" "0", "1.756E-05" "20", "1.764E-05" "50", "1.7787E-05" "100", "1.8019E-05" "150", "1.8225E-05" "200", "1.8575E-05" "250", "1.8568E-05" "300", "1.8768E-05" For the base metal: .. csv-table:: "**Temperature (** :math:`°C` **)**", ":math:`E`" "0", "2.05E+11" "20", "2.04E+11" "50", "2.03E+11" "100", "2E+11" "150", "1.97E+11" "200", "1.93E+11" "250", "1.89E+11" "300", "1.85E+11" "350", "1.8E+11" .. csv-table:: "**Temperature (** :math:`°C` **)**", "**ALPHA**" "0", "1.122E-05" "20", "1.122E-05" "50", "1.145E-05" "100", "1.179E-05" "150", "1.247-05" "200", "1.278E-05" "250", "1.308E-05" "300", "1.34E-05" Boundary conditions and loads ------------------------------------- Step 1: non-linear thermal calculation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The limit conditions applied to the thermal calculation are summarized in the Figure below and are broken down as follows: * temperature imposed on the inner wall, * thermal insulation on the external wall. .. image:: images/100000000000020A000000EBB34A80969D7BA2A2.png :width: 6.1319in :height: 2.7602in .. _RefImage_100000000000020A000000EBB34A80969D7BA2A2.png: An analytical thermal transient :math:`T(t,z,\theta )`, with an initial temperature of 280° C., is imposed on the inner wall of the tank. This transient describes the temporal evolution of the temperature :math:`{T}_{i}` on the inner wall of the tank as a function of the azimuthal position :math:`\theta` and the longitudinal position :math:`z`, thus offering a three-dimensional analytical transient quite close to a real transient. The analytic expression for the transient is provided in Equation {1}. :math:`T(t,z,\theta )\mathrm{=}{T}_{\mathit{is}}+{T}_{1}\mathrm{\times }{e}^{\frac{\mathrm{-}t}{\mathrm{[}(\mathrm{\sum }_{i\mathrm{=}0}^{6}{a}_{k}{\theta }^{k})({t}_{\mathit{rg}}(1\mathrm{-}\frac{z}{{H}_{\mathit{cuve}}})+{t}_{\mathit{rgcuve}}\mathrm{\times }\frac{Z}{{H}_{\mathit{cuve}}})\mathrm{]}}}+{T}_{2}\mathrm{\times }\mathrm{sin}({f}_{\mathrm{2nd}}\mathrm{\times }t){e}^{\frac{\mathrm{-}t}{{t}_{\mathit{r2nd}}}}` (1) with the following notation: * :math:`{T}_{\mathit{is}}`: safety injection temperature * :math:`{T}_{1}`: temperature of the amplitude of the decrease between the initial temperature and the final temperature (safety injection temperature) * :math:`{T}_{2}`: temperature of the secondary oscillations around the decay * :math:`{T}_{\mathit{rg}}` overall response time * :math:`{T}_{\mathit{rgcuve}}` overall response time from the bottom of the tank * :math:`{t}_{\mathit{r2nd}}` secondary oscillations response time * :math:`{f}_{\mathrm{2nd}}` frequency of secondary oscillations * :math:`{H}_{\mathit{cuve}}` tank height .. csv-table:: ":math:`{T}_{\mathit{is}}(°C)` "," :math:`{T}_{1}(°C)` "," :math:`{T}_{\mathit{rg}}(S)` "," :math:`{T}_{\mathit{rgcuve}}(S)` "," "," :math:`{f}_{\mathrm{2nd}}(\mathit{Hz})` "," :math:`{T}_{2}(°C)` "," :math:`{t}_{\mathit{r2nd}}(S)` "," :math:`{H}_{\mathit{cuve}}`" "10", "270", "200", "700", "700", "0.05", "20", "1000", "5000" .. csv-table:: ":math:`{a}_{0}` "," :math:`{a}_{1}` "," :math:`{a}_{2}` "," :math:`{a}_{3}` "," "," :math:`{a}_{4}` "," :math:`{a}_{5}` "," :math:`{a}_{6}`" "2.01373", "-1.45143E-2", "1.321E-2", "1.321E-3", "-8.07773E-5", "1.60275E-6", "-1.26618E-8", "3.51716E-11" In the present test case, we are interested in the position of azimuth :math:`\theta \mathrm{=}0` and longitude :math:`z\mathrm{=}0`. Step 2: mechanical calculation in linear elasticity ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Given the number of important points constituting the pressure transient, this one is not presented. The boundary conditions of the mechanical problem are summarized in the figure below and are broken down as follows: * fluid pressure on the inner wall, * symmetry along the :math:`\mathit{Oz}` axis on the lower segment, * shape effect as well as uniform movement along the :math:`\mathit{Oz}` axis on the upper segment. .. image:: images/1000020000000453000002C676CAC5DF5E1A0639.png :width: 6.889in :height: 4.4161in .. _refImage_1000020000000000453000002c676 CAC5DF5E1A0639 .png: