Stress analysis of the TMSR graphite component under irradiation conditions

Stress analysis of the TMSR graphite component under irradiation conditions 1 first-author 1 corresp gaoyantao@sinap.ac.cn gaoyantao@sinap.ac.cn gaoyantao@sinap.ac.cn 1 1 1 corresp Corresponding authors: E-mail address: Zengguangli@sinap.ac.cn; Corresponding authors: E-mail address: Zengguangli@sinap.ac.cn; Zengguangli@sinap.ac.cn Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China TMSR uses nuclear graphite as a neutron moderator, a reflector, and the structural material, and utilizes molten salt as a coolant. When running normally, the graphite components are immersed in the molten salt. Thus, the nuclear graphite comes into direct contact with the molten salt, which infiltrate the open pores of the nuclear graphite. This infiltration may influence the stress analysis of the graphite component. In this study, a User MATerial (UMAT) subroutine was used to analyze the stress distribution of the graphite component, both with and without molten salt infiltration. Many influence factors were taken into consideration, such as the dose gradient, the shape of the permeation zone, and the permeation area. The results show that the dose gradient, shape, and area of the permeation zone all significantly influence the stress distribution. Furthermore, the results of the stress analysis indicate that for a regular graphite component with a square cross-section, the peak maximum principal stress value occurs at the center of the cross-section, and the symmetry of the maximum principal stress distributions was modified by quarter circle and half ellipse permeation zones. Nuclear graphite Stress analysis Irradiation Permeation zone 1. Introduction 1 The concept of a Molten Salt Reactor (MSR) was established in the 1950s at Oak Ridge National Laboratory (ORNL, Tennessee, United States), in which the primary nuclear reactor coolant, or even the fuel itself, was a molten salt mixture. In 1954, the Aircraft Reactor Experiment (ARE), which reached a maximum power of <math id="M1"><mrow><mtext> </mtext><mn>2.5</mn><mtext> MWt</mtext></mrow></math> , was designed for operation at a temperature of <math id="M2"><mrow><mn>1500</mn><mo>°</mo><mtext>F </mtext><mrow><mo>(</mo><mrow><mn>1100</mn><mtext>K</mtext></mrow><mo>)</mo></mrow></mrow></math> , using a circulating fluoride-salt fuel in a heterogeneous core [ 1 1 ] . ARE became critical and operated in a stable and self-regulating manner for nine days, without mechanical or chemical problems. The continuation of this work led to the commissioning of the Molten-Salt Reactor Experiment (MSRE). The MSRE [ 2 2 ] , an <math id="M3"><mrow><mtext> </mtext><mn>8</mn><mtext> MWt</mtext></mrow></math> graphite moderated reactor at ORNL, reached its criticality in 1965, was powered by molten fluoride salt at <math id="M4"><mrow><mn>1200</mn><mo>°</mo><mtext>F</mtext></mrow></math> , which circulated through the core of graphite bars. Though the MSRE program ended in 1969, its result demonstrated the practicality and performance of molten-salt power reactors. The success of this reactor and the expertise gained during this experiment led, in the 1970s, to the initiation of a new power reactor project, the Molten Salt Breeder Reactor (MSBR) [ 3 3 ] . Nowadays, there is a renewed interest in MSRs, which are believed to be a promising Generation IV reactor type [ 4 4 ] . There are several MSR development programs worldwide. For example, Terrestrial Energy Inc. (TEI), a Canadian based company, is developing a small modular reactor called the Integral Molten Salt Reactor (IMSR) [ 5 5 ] , while Seaborg Technologies, a Danish company, is developing a core that utilizes a single high temperature salt, and named it the Molten Salt Waste-Burner (MSW) [ 6 6 ] . In 2011, a preliminary reactor system concept, for a Small modular Advanced High Temperature Reactor (SmAHTR) was described by researchers at ORNL, along with an integrated high-temperature thermal energy storage, or "salt vault" system [ 7 7 ] . The SmAHTR is a 125 MWt, liquid salt-cooled, coated particle-graphite fueled, integral primary low-pressure system, operating at <math id="M5"><mrow><mtext> </mtext><mn>700</mn><mo>°</mo><mtext>C</mtext></mrow></math> . In the same year, China began the Thorium Molten Salt Reactor program (TMSR) [ 8 8 - 9 9 8-9 ] , which was based on the MSRE. Originally, a <math id="M6"><mrow><mn>2</mn><mtext> MWt</mtext></mrow></math> demonstration plant was built, which could then be scaled to commercial size. Within the next year, Japan, in order to prevent nuclear proliferation and terrorism and to minimize nuclear waste, presented a new nuclear program named FUJI [ 10 10 ] . The primary advantage of MSRs is the continuous adjustment of the fuel-salt composition, which ensures its operation over time, with no reactivity reserve. Moreover, it is possible to reach very high temperatures and power densities without large internal pressures, and therefore, the structural elements are unconstrained. However, in an MSR, the molten salt comes into direct contact with the graphite moderator and reflector, which may lead to changes to the graphite properties, such as the coefficient of thermal expansion and irradiation resistance, and may damage the graphite structure in particular circumstances [ 11 11 ] . The material test by ORNL [ 12 12 ] showed that salt permeation into the MSRE-specified graphite type was confined mostly to surface pits or to penetration along cracks, usually microscopic, that intersected the surfaces. Cracks and open pores intersect the surface of the graphite component increasing the total contact area between the molten salt and the graphite, which could cause the irradiation dose to increase inside the graphite. The material test also showed that almost any distribution of salt within the graphite will cause the local temperature to rise, even in regions of maximum heat flux. In this study, the cracks in the permeation zone were assumed to be longitudinal and propagating the full length of the graphite bar. In addition, the graphite material properties are changed by rapid neutron irradiation as a function of the irradiation temperature and dose in the reactor. The fast neutron dose and the irradiation temperature decrease with distance from the center of the core to the boundary of the vessel; the graphite components are, therefore, subject to significant irradiation and temperature gradients. The irradiation dose and temperature variations across the large cross-sections of graphite components, in normal operation, generate internal stresses that could ultimately lead to component failure. In order to analyze the internal stress of the TMSR graphite component during irradiation and infiltration of molten salt, a User MATerial (UMAT) subroutine, containing irradiation-dependent constitutive relationships for graphite dimensional changes and the material properties of graphite, for example, the elastic modulus, the coefficient of thermal expansion, and irradiation creep, was developed by Tsang based on previous work [ 13 13 - 15 15 13-15 ] . This subroutine can be used together with the ABAQUS finite element program to preform time-integrated non-linear irradiated graphite stress analyses, and the UMAT can be used for high temperatures up to <math id="M7"><mrow><mtext> </mtext><mo>~</mo><mn>1000</mn><mo>°</mo><mtext>C</mtext></mrow></math> . The effects of molten salt on stress distribution were analyzed under different conditions in this study. 2. The UMAT constitutive equations 1 In this section, the irradiated graphite constitutive equations are briefly presented. The new TMSR UMAT is designed for anisotropic materials. The total strain within irradiated graphiteε total is the sum of five different strain components: <math display="block" id="M8"><mrow><msub><mtext>ε</mtext><mrow><mtext>total</mtext></mrow></msub><msub><mrow><mtext>=ε</mtext></mrow><mtext>e</mtext></msub><msub><mrow><mtext>+ε</mtext></mrow><mrow><mtext>pc</mtext></mrow></msub><msub><mrow><mtext>+ε</mtext></mrow><mrow><mtext>sc</mtext></mrow></msub><msub><mrow><mtext>+ε</mtext></mrow><mrow><mtext>dc</mtext></mrow></msub><msub><mrow><mtext>+ε</mtext></mrow><mrow><mtext>th</mtext></mrow></msub></mrow></math> where ε e is the elastic strain, ε pc is the primary creep strain, ε sc is the secondary creep strain, ε dc is the dimensional change strain, and ε th is the thermal strain, with or without stress. 2.1 Elastic strain 2 The elastic strain, ε e , is related to the stress, σ, by Hooke’s law of linear elasticity, i.e. <math display="block" id="M9"><mrow><msub><mrow><mtext>σ=Dε</mtext></mrow><mtext>e</mtext></msub></mrow></math> where D is the material matrix. The material matrix is assumed to be a function of E e , the Young’s modulus, and v e , Poisson ratio. 2.2 Creep strains 2 The primary creep strain, ε pc , is defined as <math display="block" id="M10"><mrow><mfrac><mrow><msub><mrow><mtext>dε</mtext></mrow><mrow><mtext>pc</mtext></mrow></msub></mrow><mrow><mtext>d</mtext><mi>γ</mi></mrow></mfrac><mo>=</mo><msub><mi>k</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>α</mi><msub><mtext>D</mtext><mrow><mtext>pc</mtext></mrow></msub><msub><mrow><mtext>σ-ε</mtext></mrow><mrow><mtext>pc</mtext></mrow></msub><mo stretchy="false">)</mo></mrow></math> where γ is the radiation fluence, k 1 is the primary creep dose constant, α is the magnitude of asymptotic primary creep strain in elastic strain units, and D pc is the primary creep material matrix, which is a function of the primary creep modulus, E pc , and Poisson’s ratio for primary creep, v pc . The secondary creep strain, ε sc , is defined as <math display="block" id="M11"><mrow><mfrac><mrow><msub><mrow><mtext>dε</mtext></mrow><mrow><mtext>sc</mtext></mrow></msub></mrow><mrow><mtext>d</mtext><mi>γ</mi></mrow></mfrac><mo>=</mo><mi>β</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>k</mi><mn>2</mn><mi>γ</mi></mrow></msup><mo stretchy="false">)</mo><msub><mtext>D</mtext><mrow><mtext>sc</mtext></mrow></msub><mtext>σ,</mtext></mrow></math> where k 2 is the secondary creep rate incubation dose constant, β is the saturated secondary creep rate parameter, and D sc is the secondary creep material matrix, which is a function of the secondary creep modulus, E sc , and Poisson’s ratio for secondary creep, v sc . In the new TMSR UMAT, the material properties E e , E pc , E sc , v e , v pc , and v sc are all independent variables, that may have values different from their corresponding elastic properties. 2.3 Dimensional change strain 2 For a given irradiation fluence and temperature, the unrestrained dimensional change rate, subject to irradiation and weight loss, is given by <math display="block" id="M12"><mrow><mfrac><mrow><mtext>d</mtext><msub><mi>ε</mi><mrow><mtext>dc</mtext></mrow></msub></mrow><mrow><mtext>d</mtext><mi>γ</mi></mrow></mfrac><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>ϑ</mi><mo>,</mo><mi>T</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></math> Where <math id="M13"><mi>ϑ</mi></math> is the dose ratio, T is the temperature and ω is weight loss. 2.4 Thermal strain 2 The thermal strain is defined as <math display="block" id="M14"><mrow><msub><mtext>ε</mtext><mrow><mtext>th</mtext></mrow></msub><mo>=</mo><msub><mtext>α</mtext><mrow><mtext>(</mtext><msub><mi>T</mi><mrow><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub><mo>-</mo><mi>T</mi><mtext>)</mtext></mrow></msub><mo stretchy="false">(</mo><mi>T</mi><mo>−</mo><msub><mi>T</mi><mrow><mtext>ref</mtext></mrow></msub><mo stretchy="false">)</mo><mo>,</mo></mrow></math> where <math id="M15"><mrow><msub><mtext>α</mtext><mrow><mtext>(</mtext><msub><mi>T</mi><mrow><mtext>ref</mtext></mrow></msub><mtext>-</mtext><mi>T</mi><mtext>)</mtext></mrow></msub></mrow></math> is the mean coefficient of thermal expansion between temperatures T ref °C and T °C. A comprehensive irradiated material property database is required for any graphite component stress analysis. The data required includes irradiated elastic material properties, irradiation creep properties, irradiation induced dimensional changes, and changes to the coefficient of thermal expansion. These properties are required as a function of fast neutron fluence, irradiation temperature, oxidation, and molten salt infiltration. Empirical irradiated material property data trends were derived from the analysis of data obtained by irradiating small samples in a Material Test Reactor (MTR). The units for the irradiation fluence and irradiation temperature are usually expressed in terms of equivalent DIDO nickel dose × 10 20 n/cm 2 (EDND) [ 16 16 ] and temperature (°C), respectively. As the TMSR project is in progress, no irradiation material properties for TMSR graphite could be obtained. For this study, the irradiated material property database was obtained from a gas-cooled reactor, PBMR. As the irradiation properties are commercially confidential, detailed information cannot be provided in this paper. 3. Finite element models of graphite component 1 The graphite components used in the MSRE measured 2 inches square by about 6 feet long, were oriented vertically and rested on a grid of lattice support bars [ 11 11 ] . According to this, a primary graphite component was designed for the TMSR, with a cross-section of <math id="M16"><mrow><mn>50</mn><mtext> mm </mtext><mo>×</mo><mtext> </mtext><mn>50</mn><mtext> mm</mtext></mrow></math> and a length of <math id="M17"><mrow><mn>1800</mn><mtext> mm</mtext></mrow></math> . Two assumptions were made to simplify the analyses of the nuclear graphite in the TMSR. First, supposing the graphite bars are completely immersed into the molten salt, and the fast neutron dose is a constant value within the surface which directly contacts the molten salt, the neutron dose varies with distance from the center to the boundaries of the graphite bar, but every cross-section along its length has the same dose distribution. Second, a molten salt permeation zone, which includes numerous cracks or small open pores, allows molten salt to enter the graphite bar. As previously discussed, the permeation zone has a higher temperature than the other zone and the fast neutron dose is as high as the area in immediate contact with the molten salt. 3.1 2D finite element models 2 A series of two-dimensional (2D) models were built based on the assumptions previously discussed. These 2D models were used to analyze the influence of the permeation zones and the irradiation dose gradient on the stress distribution of the graphite component. Three types of permeation zones were analyzed: a quarter circle in one corner, a half ellipse at the center of one edge, and a frame surrounding the boundaries. These are shown in Fig. 1 Fig. 1 . The size of the permeation zone was equated to the permeation area ratio (PAR) which was defined as the permeation zone area divided by the total model area. In these models, each type of permeation zone has four PAR values: 0.005, 0.02, 0.045 and 0.08. Moreover, the irradiation dose gradient was defined as the dose difference divided by the dose value at the boundaries of the model. The dose difference is the value at the boundaries minus the dose value at the center of the model; the boundary dose for all the models was the same. Each permeation area ratio model was assigned four irradiation dose gradients (10%, 15%, 20% and 25%) to build different dose gradient models. As previously discussed, the dose value of the permeation zone is equal to the value at the model boundaries, which come in direct contact with the molten salt. The irradiation temperature in the rest zone was set to <math id="M18"><mrow><mn>650</mn><mo>°</mo><mtext>C</mtext></mrow></math> , which is similar to the working temperature in the TMSR, while the temperature in the permeation zone was set at <math id="M19"><mrow><mn>750</mn><mo>°</mo><mtext>C</mtext></mrow></math> due to molten salt retention in the permeated areas. The three models with PAR values of 0.005 are shown in Fig. 2 Fig. 2 . The finite elements used in this simulation were the 8 nodes Generalization Plan Strain Element (CPEG8), which is typically used to model a section of a long structure that is free to expand axially, or is subjected to axial loading. The total numbers of elements for the three models were 2280 (quarter circle at one corner), 2704 (half ellipse at the center of one edge) and 2524 (frame surrounding the boundaries). The total numbers of nodes for the three models were 7037, 7775, and 8321, respectively. A group of models were built without any permeation area present in the graphite component. The neutron dose was assumed as space-dependent. Fig. 3 Fig. 3 shows the dose distributions of different dose gradient models at the end of ten full power years. The dose gradients for the models (a) through (d) are 10%, 15%, 20%, 25%, respectively. The tensile strength of graphite is much less than its compressive strength. According to the theory of maximum stress, the main cause of rupture in graphite should be the maximum tensile stress. In this study, the maximum tensile stress was equated to the maximum principal stress and used to analyze the internal stress of the graphite component. The maximum principal stress results were normalized by the peak maximum principal stress value from all the models analyzed in this paper. The stress results are shown in Fig. 4 Fig. 4 , with the same dose gradients for (a) through (d) as in Fig. 3 Fig. 3 . It is clear that the maximum principal stress of the graphite component increases as the dose gradient increases. The peak maximum principal stress value exists at the center of the graphite component, meaning damage is initiated from this point when the stress reaches the rupture limit. The stress distributions within the graphite component were found to be symmetrical and the stress decreases from their central point to their boundaries. The outer surface was compressive. Taking the permeation zones into consideration, a series of models with different permeation areas were built to explore the influence of various shapes and areas on the stress of the nuclear graphite component. Figs. 5 Figs. 5 , 6 6 , and 7 7 display the maximum principal stress distribution of the graphite component with a quarter circle, half ellipse and a frame permeation zone, respectively. The permeation area ratios for these models are 0.045, and the dose gradients are as in Fig. 3 Fig. 3 . Some details agreed well with the results from the permeation zone-free models: for any given permeation area, the maximum principal stress increases as the dose gradient increases, and the peak value of the maximum principal stress always occurs at the central point of the graphite component. However, the maximum principal stress was no longer symmetrical in the quarter circle and half ellipse permeation zone models. Moreover, the larger the permeation area was, the greater the damage was. While the stress distributions of the frame permeation zone models are similar to the preliminary permeation-free models, this could be because the frame permeation area is uniformly distributed around the graphite component. The peak values of the maximum principal stress for all permeation models are shown in Fig. 8 Fig. 8 . It can be seen that for any given dose gradient, the peak value of the maximum principal stress increases as the permeation area increases. For the cases estimated here, half ellipse permeation zones influence the stress distribution the most, and the influence from the frame permeation zones was the least significant. This could be because the half ellipse and quarter circle zones allow for the most intensive salt permeation. 3.2 3D finite element models 2 Four three-dimensional (3D) finite element models were built to verify the stress results from the 2D finite element models. These models’ cross-sections measured <math id="M20"><mrow><mn>50</mn><mtext> </mtext><mstyle mathvariant="bold" mathsize="normal"><mi>m</mi><mi>m</mi></mstyle><mtext> </mtext><mo>×</mo><mtext> </mtext><mn>50</mn><mtext> </mtext><mstyle mathvariant="bold" mathsize="normal"><mi>m</mi><mi>m</mi></mstyle></mrow></math> and their length were <math id="M21"><mrow><mn>1800</mn><mtext> </mtext><mstyle mathvariant="bold" mathsize="normal"><mi>m</mi><mi>m</mi></mstyle></mrow></math> . All the permeation area ratios of these models were 0.045, and their dose gradients were 25%. These models include the three previously discussed permeation zones. The graphite brick is a deformable body and so C3D20 elements were used to model its behavior. Rigid body motion was performed from the center of the body. The upper and lower surfaces were modeled as free boundaries. Fig. 9 Fig. 9 shows the stress results from these permeation models. The peak value of the maximum principal stress still occurs at the center of the graphite component in each permeation model. These different permeation zones produced different maximum principal stress modification amounts and shapes, affecting the symmetry to different degrees. The modification from the quarter circle was the largest and that from the half ellipse was found to be the second most influential, while the stress distribution from the frame permeation zone is similar to the models which were free of permeation zones. 4. Conclusion 1 In this paper, a User MATerial (UMAT) subroutine, containing irradiation-dependent constitutive relationships for graphite dimensional changes and the material properties of graphite, was utilized to perform initial stress analysis of a graphite component. The irradiated material property database was obtained from a gas-cooled reactor, due to the lack of an irradiated properties database for the TMSR-specific graphite. The stress analysis shows that for a regular graphite component with a square cross section, the peak maximum principal stress value occurs at the center of the cross section, which suggests that the damage will begin from this point if the stress reaches the rupture limit. Furthermore, the dose gradient, the shape, and the area of the permeation zone all significantly influence the stress distribution. First, for each permeation zone modeled, the larger the dose gradient was, the larger the maximum stress was. Second, for each dose gradient of any shape permeation zone, the greater the permeation area was, the higher the peak maximum principal stress value was at the center of the graphite component. Lastly, the symmetry of the maximum principal stress distributions were significantly modified by the quarter circle and the half ellipse permeation zones. For the same permeation zone shape, the larger the permeation area was, the more significant the damage was. 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Dimensional change, irradiation creep and thermal/mechanical property changes in nuclear graphite .  Int. Mater. Rev ., 2016 , 61 ( 3 ): 155 - 182 . doi: 10.1080/09506608.2015.1136460 http://doi.org/10.1080/09506608.2015.1136460 Dimensional change, irradiation creep and thermal/mechanical property changes in nuclear graphite 10.1007/s41365-018-0516-8 2018-01-05 2018-06-08 2018-06-16 2018-11-15 2018-12-01 2021-12-20 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2018 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-018-0516-8 http://dx.doi.org/10.1007/s41365-018-0516-8 . 2018 Nuclear Science and Techniques 12 29