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Investigation on the RPV structural behaviors caused by various cooling water levels under severe accident
Engineering Failure Analysis 79 (2017) 274–284
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Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Investigation on the RPV structural behaviors caused by various cooling water levels under severe accident
MARK
Jianfeng Maoa,b,⁎, Linghui Hua, Shiyi Baoa,b,⁎⁎, Lijia Luoa, Zengliang Gaoa,b a b
Institute of Process Equipment and Control Engineering, Zhejiang University of Technology, Hangzhou, Zhejiang 310032, PR China Engineering Research Center of Process Equipment and Re-manufacturing, Ministry of Education, PR China
AR TI CLE I NF O
AB S T R A CT
Keywords: Structural behavior Severe accident Failure assessment IVR RPV
In the severe accident (SA) event of core meltdown, the ‘In-vessel retention (IVR)’ mitigation has been certified as an effective strategy for accident management in most advanced nuclear power plants. The traditional concept of IVR was based on the idea that the reactor pressure vessel (RPV) was fully submerged into the cooling water. However, the Fukushima accident showed that the cooling water was insufficient due to the malfunction of water supply system, so the RPV structural behavior had not been appropriately assessed. Therefore, the paper tries to address the structure-related issue on determining whether RPV safety can be maintained or not with the effect of various water levels created by the SA condition. In achieving this goal, the structural behaviors were numerically investigated in terms of several field parameters, such as temperature, deformation, stress (strain) and damage. Due to the melting pool on the inside and water cooling on the outside, the high temperature gradient was formed across the wall thickness, so RPV failure was found to be the consequence of creep, plasticity and thermal expansion. According to the requirement on RPV safety during the prescribed time, it must be ensured that the water cooling takes effect in preventing the structural failure under SA condition. Through vigorous investigation, it is found that the RPV safety is secured within the 100 creep h. Furthermore, the structural failure site, time and mode are predicted with consideration of the effect of various water levels. Most importantly, the failure is found to take place at the site aside around the water level.
1. Introduction In the severe accident (SA) of core meltdown, a famous strategy called ‘In-vessel (IVR) mitigation’ is widely used by most advanced nuclear power plant (NPP) to maintain the safety of the reactor pressure vessel (RPV) [1]. In fact, the IVR mitigation has been certified by nuclear regulatory commission (NRC) as a standard measure in USA for SA management since 1996 [2]. On most occasions, the IVR mitigation is to provide long-term water cooling on the outside wall of the RPV, so the decay heat is removed without the need for any active actions and assistance measurement during the SA [3]. Accordingly, it can be known that the external water cooling is the most important characteristic for accident management, as shown in Fig. 1. In traditional concept of IVR, the lower head (LH) of the RPV is assumed to be fully submerged into the water flooding, prior to the arrival of core debris on the inside [4]. Therefore, the melting pool is formed within the RPV with the temperature of approximate 1327 °C, while the temperature on the outer vessel wall is close to the water saturation temperature of around 150 °C [5]. With the basic assumption of critical heat flux
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Correspondence to: J. Mao, Institute of Process Equipment and Control Engineering, Zhejiang University of Technology, Hangzhou, Zhejiang 310032, PR China. Corresponding author. E-mail addresses: [email protected] (J. Mao), [email protected] (S. Bao).
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http://dx.doi.org/10.1016/j.engfailanal.2017.04.029 Received 15 January 2017; Received in revised form 26 March 2017; Accepted 22 April 2017 Available online 24 April 2017 1350-6307/ © 2017 Elsevier Ltd. All rights reserved.
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Fig. 1. The scheme of RPV with various water levels in severe accident and its typical failure segments [6,22].
(CHF) on the RPV during the accident period, it can be learnt from previous studies that the RPV safety can be secured within a prescribed time [6]. The above conventional IVR concept isn't seriously challenged until the Fukushima accident on 2011 [7]. It showed that the malfunction took place in the cooling water supply system, resulting in insufficiency of water supply. Furthermore, the consequence of the disaster indicated that the structural behavior had not been appropriately assessed [8]. Therefore, the related key issue is that whether LH structural integrity can be maintained with various cooling water levels under severe accident. As indicated before, the structural safety is in the top priority for the operation of nuclear power plants, especially for the event of severe accident, so the IVR mitigation should be robust and capable of relieving the accident consequence. In most cases, the RPV integrity should be ensured during the first 72 h, while the operator may have sufficient time to take some emergence measures [9]. However, due to the complexity of the thermal-mechanical loadings, some of the failure mechanisms are not fully understood due to the low possibility of the occurrence [10,11]. In dealing with the issue of structural safety under SA, numerous efforts have been made by researchers and engineers. For the past several years, many geometrical scaled tests had been performed to characterize the failure site, time and mode of the RPV structure under the programs [12–17] of FOREVER, USNRC/SNL, LIVE and CORVIS. Under the loading of severe accident, one of 1:10 scaled test specimens can be seen in Fig. 1, and the LH failed at the site around the high water level. Besides, a number of finite element (FE) computations had been carried out for investigating the structural failure behaviors under the simulated thermal-mechanical loadings [18], like the loadings of critical heat flux (CHF) and low internal pressure. As indicated in previous studies [19,20], both experimental and theoretical results showed that the predictability of the failure time, site and mode varied with the field parameters, such as temperature, stress, strain. Due to core meltdown on the inside and external water cooling, the high temperature gradient is formed across the wall thickness. Consequently, it is found that creep damage accumulates over a wide area on the inside wall, while the local plastic damage is concentrated on the outside wall [21]. With the increase of creep time, both creep and plastic damages are increasing very significantly and interacted with each other at the site of geometric discontinuity [22]. As pointed out in some literatures [23,24], the CHF loading may be considered as a limit thermal boundary on the inside wall before the occurrence of melt-through, the failure process was found to be very complex, including wall bulge, thinning and even necking at the failure site. Moreover, the plastic and viscoplastic straining was the major contributors to the global or local failure. So far, the complete assessment on Fukushima accident had been submitted to the IAEA by Japan government, one of the critical issues was to answer whether the IVR mitigation takes effect or not [25]. Actually, the water cooling was the essential measure for ensuring the RPV integrity, as given in the report of the accident event. However, this kind of research on the effect of various water levels is scarcely found in the previous studies [26], especially for the era before Fukushima accident. The typical example of water levels is depicted in Fig. 1, indicating that the cooling water can be maintained at low, mid and high level respectively. Accordingly, the assumption of neglecting the water level effect on structural failure in traditional IVR concept is seriously challenged nowadays. In dealing with the above issue, the in-depth understanding of the RPV structural behavior is urgently necessary with the consideration of the effect of various water levels. Therefore, the main task of the paper is to numerically investigate the RPV structural behaviors with the effect of various water levels under the severe accident of core meltdown, which is seldom concerned in previous studies. Toward this end, the 2D FE-models of the RPV with various water levels were developed on the platform of ABAQUS. In order to make a comparison, three typical water levels were employed in the analysis, including low, mid and high water levels. Due to the high temperature gradient across the wall thickness, the time- and temperature-dependent material properties were considered in the FEM, which were also highly nonlinear relationships for creep and plasticity. In accounting for the most dangerous situation before the melt-through, the critical heat flux (CHF) was regarded as the limit thermal boundary condition on the FEM. In order to make sure whether the RPV safety was maintained or not, the 100 creep h was taken as a basis for the FE calculation. Through vigorous investigation, it was found that the RPV safety can be secured within the 100 h, and the RPV structural failure took place at the site aside around the water level. Finally, the structural behaviors of the RPV with various water levels were analyzed carefully by means of field parameters, such as temperature, deformation, stress (strain) and damage. Most importantly, the phenomenon of deformation incompatibility and stress 275
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concentration was found at the site of water level. 2. Mathematical model description 2.1. Creep and plastic damage approaches Due to the high temperature of the melting pool within the RPV, the inner RPV wall is exposed to the environment of T = 1327 °C (melting point). As for the outer RPV wall, it is submerged into the cooling water. According to the ‘thermal failure criterion’, the critical heat flux (CHF) is formed before the melt-through of the RPV wall, which is also considered as the limit thermal boundary condition. The maximum temperature on the outside is considered as 150 °C (nucleate boiling), so the high temperature gradient is established. As well known, the creep failure mechanism occurs when the temperature exceeds 0.4 time of the melting point. In accounting for the creep effect, the constitutive equation is formulated with a number of coefficients [9,16,22],
⎡ d ⎤ εċ = d1 σ d2 εcd3 exp ⎢ − 4 ⎥ ⎣ T⎦
(1)
where, the d1, d2, d3, d4 are material coefficients in fitting creep constitutive equation. Regarding the RPV material, the creep experiences three stages, and the nonlinearity in the tertiary stage is the severest. In describing the nonlinearity, the d1 > 0 and d3 > 0 must be satisfied. In fact, the material coefficients in Eq. (1) are time- and temperature-dependent. In order to achieve the convergence, a newly-proposed scheme with nonlinear characteristics is employed, which is an approach for weighting coefficients fitting. For plasticity, the Ramberg-Osgood (R-O) equation is used to describe the nonlinear relationship between stress and strain. The RO equation is very useful in describing the strain hardening phenomenon for the RPV material, and the original form of R-O equation can be expressed as follows,
εp =
⎛ σ ⎞n σ + α⎜ ⎟ E ⎝ σy ⎠
(2)
Wherein, σ and εp are the true stress and true strain respectively, E is Young's modulus. Moreover, α and n are the material constants, which can be identified using the data ranging from the yield point to the necking point, n = [ln(eus / 0.002)] / [ln(σu / σy)], herein, eus is uniform strain at maximum load, σu and σy are the ultimate and yield strength respectively. In order to raise the fitting accuracy, the option of multi-linear isotropic hardening is activated in ABAQUS software for the plasticity. In order to give a proper assessment on the RPV structural safety, the so called ‘ductility failure criterion’ is adopted for predicting the damage. In the criterion, both creep and plastic damage are incorporated into the following Eq. (3). Therefore, the total damage increment ΔD is the sum of the plastic damage increment and creep damage increment. The increment ΔD is accumulated at the end of each sub-step [12], and ΔD = 0 means “no damage increment”. The expression of the ΔD is formulated as follows [9,22], pl ⎤ ⎡ Δε cr Δεeqv eqv ⎥ Rv ΔD = ⎢ cr + pl ⎢⎣ εfrac (σ , T) εfrac (T) ⎥⎦
(3)
where, the εfrac and εfrac are creep and plastic fracture strain respectively, and they can be obtained by the tensile test with fixed thermal-mechanical loads. For simplicity, the εfraccr and εfracpl are obtained separately, according to the experimentally found material behavior [9,12,15,21]. Most notably, in order to account for the effect of multiaxial stress state, the triaxiality factor Rv is incorporated in the ‘ductility failure criterion’, cr
Rv =
pl
⎛ σ ⎞2 2 (1 + v ) + 3(1 − 2v ) ⎜ H ⎟ ⎝ σM ⎠ 3
(4)
where, σH is the hydrostatic stress, σM is the von-Mises equivalent stress, v is the Poisson's ratio. With the linear accumulation rule, the final total damage D can be computed by the sum of each damage increment ΔD, nStep
D=
∑
ΔDi
(5)
i =1
Actually, the above damage increment ΔDi is computed by averaging its node equivalent damages for each element. When the D is equal to 1, the element is correspondingly set to be inactive using element death technique, so the death element does no longer contribute to the material stiffness. 2.2. Description of FE modeling In order to evaluate the RPV structural behaviors, the two-dimensional (2D) FE-models are established for the RPV with various water levels, which can be seen in Fig. 2. Three FE-models are considered herein, which are RPVs with low, mid and high water level respectively. Due to the axisymmetric characteristics for geometry and load, the half of the RPV is sufficient for the FE calculation. As for the RPV configuration, the thickness of the LH is 165 mm, and the radius of the LH is 2180 mm. The geometric size is taken from 276
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Fig. 2. The geometric size of RPV and its FE-model with applied boundary conditions.
the real component of AP600-type power plant. Due to the possibility of high water level, the LH is extended by 3578 mm into the cylindrical segment. The cylindrical segment is so high that the effect from other component to the LH can be neglected. As shown in Fig. 2, there exists a transition portion between the LH and cylindrical segment, and the wall thickness is gradually increasing from 165 mm to 180 mm. In order to account for the thermal-mechanical behaviors, the element type of C3D20R is used in the FE-models, which is known as ‘20-node quadratic thermal-coupled brick with the reduced integration scheme’. The selection of this element is due to its suitability of computing creep and plasticity simultaneously. Besides, this type of element provides more flexibility for considering the geometric nonlinearity, e.g. large strain and defection. As shown in Fig. 2, the RPV model is carefully meshed, and the grid density is slightly increased for the transition portion. The grid of LH wall thickness is made of 20 nodes, and the element size is approximate 8 mm. Through repeated validation, the total element number is determined to be 4200. As for constraint boundary conditions, the X-oriented freedom is constrained on the axis, and meanwhile the Y-oriented freedom is fixed on the top surface of the cylindrical segment. According to different cooling water level, the thermal boundary condition is extracted from corresponding experimental test on critical heat flux (CHF). As a matter of fact, the CHF is considered as the most dangerous thermal boundary before the complete meltthrough of the RPV wall. With the above assumed limit boundary, the structural behavior is accordingly investigated for the RPV with various water levels. Within the RPV, the melting temperature (1327 °C) is obtained from the thermal calculation, while the temperature is set to 150 °C for the outside wall below the water level. The temperature on the outside is corresponding to the transition point from nucleate boiling to film boiling. For the RPV wall above the water level, the adiabatic condition is assumed. Moreover, the thermal radiation above the melting pool is transferred into an equivalent temperature on the inside wall. As shown in Fig. 2, the top surface of cylindrical segment is far away from the melting pool, so the temperature on the cylindrical segment is assumed to be linearly decreasing from 1320 °C to 500 °C. After the purely thermal calculation, the structural behavior is simulated with incorporating the previous temperature field. Since the RPV material may experience high temperature gradient and huge deformation, the time- and temperature-dependent material properties [16,27] are considered in the FEM. In order to make an assessment on the structural safety under the SA condition, the thermal-elastic-visco-plasticity calculation is achieved for the RPV with various water levels.
3. Results and discussions 3.1. The temperature and deformation fields During the severe accident of core meltdown, the so called ‘IVR mitigation’ is assumed to be able to arrest the degraded melting core within the RPV, and maintain the RPV integrity within a period of prescribed time, usually 72 h. Consequently, the melting pool is formed in the lower head (LH) of RPV. According to the thermal failure criterion, the basic concept of ‘critical heat flux (CHF)’ is applied on the FE-models of the RPV, and regarded as the most dangerous thermal boundary condition before the complete meltthrough of the RPV wall. Accordingly, the temperature on the inner is approaching the melting point (1327 °C) of the RPV material. As for the outer surface of the RPV, the temperature is various for the three water levels, which is shown in Fig. 3. It is clear to see in Fig. 3 that the water cooling plays an important role in lowering the temperature, and the water level significantly changes the 277
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Fig. 3. Temperature distributions for RPV with various water levels.
temperature distribution. Furthermore, the high temperature zone is decreasing with increasing the water level. Actually, without water cooling, the local temperature field is homogenized as soon as the melting pool is formed on the inside, the temperature of which is maintained at approximate melting point of 1327 °C. On the contrary, the high temperature gradient is formed from 1327 °C on the inside to 150 °C (max. nucleate boiling) on the outside, due to the existence of water cooling. Close observation of Fig. 3 shows that the high temperature gradient still exists at the transition aside around the water levels. From Fig. 3, it can be inferred that the existence of cooling water can prevent the RPV from immediate structural failure. Since the outer surface of RPV is submerged into the water, the high material strength is correspondingly maintained for the outer layer of RPV wall thickness. As a matter of fact, the thermal expansion is temperature-dependent as well as creep deformation, so the various temperature distributions will inevitably result in a phenomenon of deformation incompatibility. In accounting for the effect of various water levels, Fig. 4 presents the corresponding distribution of total displacement for the lower head (LH). General observation of Fig. 4 shows that the LH stretches outward very significantly, and the bulge at the bottom (along Y direction) is much more significant than the one at the transition (along X direction) of spherical segment to cylindrical segment. Taking RPV with low water level for example (see Fig. 4(a)), the bulge along Y direction is 47.3 mm, while the bulge along X direction is only 21.5 mm. Among the three adopted water levels, the low water level results in the most significant deformation for the LH. As displayed in Fig. 4, the max. deformation is as high as 71 mm for the case with low water level. General speaking, the deformation is increasing with decreasing the water level, and the significance of deformation highly depends on the size of high-temperature homogenized zone. Moreover, Fig. 4(a) discloses that the max. deformation is located at the site of water level for the RPV with low water level, the behavior of which is similar to the one in Fig. 4(b). However, the max. deformation for the RPV with high water level is located at the bottom, as indicated in Fig. 4(c). In fact, the stiffness of the RPV with high water level is the biggest among the three cases, due to the best performance of water cooling. It is clear to see in Fig. 4 that for low to mid water level, the most significant deformation definitely occurs at the hightemperature homogenized zone. As indicated in Fig. 4, the outer surface layer is performed in the state of tension for the highlydeformed zone, which is more detrimental to RPV safety than the compression distributed on the inside. More interestingly, Fig. 4 discloses that a significant highly-squeezed zone is formed with the bulge of the LH, and an additional bending moment can also be found at the transition of spherical segment to cylindrical segment. Furthermore, the comparison of Fig. 4 reveals that the RPV with
Fig. 4. Comparison of total displacement distributions for RPV with various water levels.
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Fig. 5. Mises stress distributions for RPV with various water levels at pre-creep stage.
high water level has most significant bending moment. 3.2. The RPV structural behavior at pre-and post-creep stage As well known, the RPV material strength (e.g. yield stress) is decreasing with increasing temperature, so the Mises stress is much lower at high temperature zone than the one at low temperature zone. As shown in Fig. 5, the highest stress is concentrated at the site aside around the water level, the height of which is increasing with the increase of the water level. Fig. 5 shows that the max. Mises stress is as high as 580.3 MPa, which is approaching the yield strength. However, due to the high temperature distribution on the RPV above water level, the stress on the corresponding zone is as low as 4.03 MPa. Clear, the difference of the Mises stresses is huge between the high temperature zone and low temperature zone. At this point, the load-carrying capability of the RPV can be reflected by the water level, and the limit load must be determined by the ‘weakest link’. As discussed before, the deformations aren't compatible at the site aside around the water level, so huge equivalent plastic strain (PEEQ) is found to be accumulated on the wall thickness in the proximity of water level. As indicated in Fig. 6, the PEEQ accumulated on the inside wall is significantly bigger than the one on the outside wall. For the condition of low to mid water level, there exist two highly-concentrated zones of the PEEQ, one is located at the site at proximate water level, and the other is located at the transition of spherical segment to cylindrical segment. The observation of Fig. 6 shows that the inside wall below the water level suffers significant PEEQ, indicating that the water level changes the distribution of the PEEQ. Further observation of Fig. 6 reveals that there exists a small elastic core in proximate middle layer of the wall thickness, especially for the RPV with mid water level. Actually, the existence of elastic core is essential for ensuring the RPV safety under severe accident (SA) condition, since the elastic core has some load-carrying capability in preventing immediate collapse. From Fig. 6, it can be inferred that the structural failure has to initiate at the site of water level, when the thermalmechanical loads are further increasing. As the incompatible deformation occurs, the failure site may suffer ductile tearing, like shear failure mode. As for the RPV with high water level, the elastic core is squeezed into a very small size at the transition site, as shown in Fig. 6. Actually, the local plastic yielding is greatly aggravated with increasing the water level, especially for the inner surface. Overall speaking, it can be concluded that the global failure (collapse) is not a feasible mode for the RPVs with various water levels,
Fig. 6. Equivalent plastic strain distributions for RPV with various water levels at pre-creep stage.
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Fig. 7. Comparison of Mises stress distributions along the LH between pre- and post-creep stage.
due to the concentration phenomenon of field parameters. By looking into the distribution of equivalent stress along the angular position, it can be found in Fig. 7 that the peak value of the stress occurs at the site of respective water level. The comparison shows that the max. stress for the RPV with mid water level is slightly higher than the one for the RPV with other two levels. Fig. 7 also shows that the equivalent stress experiences a significant decrease before the max. stress is reached, indicating that the structural behavior for the RPV wall below the water level is different from the one for the RPV wall above the water level. The observation of Fig. 7 displays that the equivalent stress experiences a sharp decrease immediately after the max. value. Accordingly, it can be inferred in Fig. 7 that most of the loads are supported by the RPV wall below the water level. With the creep time increasing, this phenomenon becomes more significant, since the stress of the RPV wall above the water level is relaxed to be near stress-free. As for the RPV wall below the water level, the stress relaxation is also very significant, and the peak value of the stress is almost reduced by approximate 40% after 100 creep h. The comparison in Fig. 7 indicates that the RPV with mid water level suffers the severest stress relaxation. Furthermore, with the stress redistribution, the stress profile is also changed into a new one along the angular position after 100 creep h. By looking into the distribution of equivalent stress along the wall thickness, it can be found in Fig. 8 that the stress on the outside is significantly higher than the one on the inside. Due to the occurrence of high temperature from the melting pool, the stress on the inside is significantly low, most of which is approaching the material strength. With the effect of stress relaxation, the equivalent stress is reduced down to the near-zero level within the wall thickness range of 0 ≤ t ≤ 105 mm. Moreover, Fig. 8 demonstrates that the creep effect is almost ceased at the site of T = 425 °C. However, due to the stress redistribution, the outer surface layer of 105 mm < t ≤ 165 mm suffers stress reduction, and it is found in Fig. 8 that the max. stress is relaxed by around 44% on the outside. In the middle of the wall thickness, there exists an elastic core that is below the material strength, as shown in Fig. 8. In the elastic core, the equivalent stress is decreasing with the increase of the water level for the condition at both pre- and post-creep stages. Close
Fig. 8. Comparison of Mises stress distributions across the wall thickness between pre- and post-creep stage.
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Fig. 9. Equivalent creep strain distributions for RPV with various water levels at post-creep stage.
observation of Fig. 8 discloses that the equivalent stress at both inner and outer surface layers is beyond the material strength, so the plastic yielding is correspondingly inevitable. As mentioned before, the immediate RPV failure is not a feasible mode before the complete melt-through of the RPV wall, since the creep mechanism is a time-dependent process. As well known, the creep strain as well as plastic strain is irreversible, so there must be some creep strain accumulation after 100 h. In describing it, Fig. 9 plots the creep strain contours for the RPV with various water levels. Overall observation of Fig. 9 shows that there exists a highly-concentrated region of equivalent creep strain (CEEQ) at the site of around the water level. Clearly, the CEEQ on the inside is significantly higher than that on the outside, due to the high temperature gradient from melting pool to water cooling. The comparison of Fig. 9 reveals the RPV with mid water level suffers slightest CEEQ on the inside wall in the proximity of the water level. As for the transition portion of spherical segment to cylindrical segment, there exists a most significant accumulation of CEEQ for the RPV with high water level. For the low to mid water level, the CEEQ distributed on the segment without water cooling is found to be remarkably lower than the one with water cooling, the reason for that is the corresponding lower stress for the segment without water cooling. From Fig. 9, it can be seen that there exists a discontinuity of high CEEQ at the region below the water level, implying that the incompatible deformation takes place at the corresponding region. From the above analysis, it can be learnt that the RPV failure possibly initiates at the inner surface layer of the transition portion, due to its significant accumulation of both plastic and creep strain. Still, it can be known that the RPV failure can be determined locally at the site in proximity of the water level. 3.3. The multiaxial damage analysis on crust effects As indicated before, due to the various water levels, the field parameters, such as temperature, stress and strain, tend to perform in a multiaxial manner. In describing it, the triaxiality factor Rv defined in the above Eq. (4) is adopted in the analysis, which is plotted
Fig. 10. The evolution of triaxiality factor Rv at two typical nodes during the creep time.
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Fig. 11. Distributions of plastic and creep damage across wall thickness for the RPV with various water levels.
in Fig. 10. Actually, the RPV wall at round the water level may suffers various states of stress, such as tension, shear. The thermal stress at the site above the water level is also different from the one below the water level. In order to reflect the multiaxial state of stress, the Rv is incorporated in the so called ‘ductility failure criterion’. In traditional concept of RPV integrity assessment, the Rv is usually reduced to be equal 1, due to its simplicity of using uniaxial test data. However, this simplification may result in huge conservatism. In reducing it, the Rv should be carefully considered in the temperature- and time-dependent FE calculation. As indicated in Fig. 10, the Rv for the large deformation node has been increased with the creep time, the max. value of which is 3 at 100 h. The Rv for the small deformation node is almost kept at constant level during the creep hours, except for the one at initial stage. The comparison in Fig. 10 reveals that the RPV failure may be accelerated by the multiaxial state of stress, since the Rv is larger than 1, and continuously increasing with the creep time. From this perspective, the assumption of the uniaxiality may be improper when assessing the RPV structural failure with the effect of various water levels, because the distribution of Rv among the lower head (LH) is changing with the creep time, especially for the site around the water level. In order to leave enough time for the operator to take some emergency measures when the severe accident (SA) occurs, the RPV integrity has to be secured within the least 72 h, which has been recognized and certified by NRC [6,8]. Accordingly, the most attractive point is to assess the failure time, site and mode for the RPV with various water levels under SA condition. In disclosing it, Fig. 11 plots the distributions of the plastic and creep damage across the wall thickness in the proximity of the water level. Since creep damage is also a time-dependent parameter, the 100 h is taken as a basis for comparison in Fig. 11. In fact, the creep and plastic damages occur simultaneously during the process, and they must be interacted with each other. Overall observation of Fig. 11 shows that both plastic and creep damages are distributed as a ‘spoon’, the bottom of the damage profile is always located at the region of elastic core. As indicated in Fig. 11, the lowest damage is distributed at approximate midpoint of the wall thickness for the condition of low to mid water level, while the lowest damage is located at the site close to the outside for the RPV with high water level. It can be seen in Fig. 11 that the damage on the inside is larger than the one on the outside at pre- and post-creep stages. Notably, it can be seen in Fig. 11 that the creep damage is maintained at a high level throughout the wall thickness, comparing with the plastic damage distribution. The comparison in Fig. 11 demonstrates that the RPV with high water level suffers the largest damage at the transition portion of spherical segment to cylindrical segment. In order to make an assessment on the RPV structural failure behavior, the final contours of total damage after 100 h are given in Fig. 12 for the RPVs with low, mid and high water levels respectively. After the experimental test, the creep rupture strain of εfraccr = 6% at 100 h is considered as one basis for creep damage calculation, meanwhile the plastic rupture strain of εfracpl = 5% is obtained as the other basis for plastic damage evaluation. Through FEM, the quantitative prediction on the failure site, time and mode is achieved. As shown in Fig. 12, there exists a huge damage concentration on the inside of the RPV wall aside around the water level. It is clear to see in Fig. 12 that the total damage distributed on the inside is much larger than the one distributed on the outside, the reason for that is the great contribution by creep damage. Due to the water cooling on the outside, the total damage is dominated by the plastic damage at the site in the proximity of water level. Still, it can be found in Fig. 12 that the damage on the midpoint of the wall thickness is significantly lower than the one on each side of the RPV wall, which is owing to the existence of the elastic core on the middle layer. As for the effect of water level, it can be observed in Fig. 12 that the high water level induces the severest total damage for the RPV, and the structural failure must initiate at the transition portion of spherical segment to cylindrical segment. After 100 h, almost two thirds of the wall thickness lost its load-carrying capability for the RPV with high water level. The comparison in Fig. 12 shows that the mid water level may be the best option for maintaining the safety among the three adopted water levels, due to its least damage accumulation on the corresponding site. More notably, Fig. 12 reveals that the damage on the segment without water cooling is increasing with the water level, and it isn't as high as the one on the site around the water level. Actually, the damage distribution is determined by the temperature and stress, so the low stress level may be the reason for the low damage on the segment 282
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Fig. 12. Total damage distributions for RPV with various water levels after 100 h.
without water cooling. In concluding this paragraph, the emphasis should be given to the effect of the water level. The RPV structural failure is contributed by the plastic and creep damage, and takes place locally at the site around the water level. 4. Conclusions In the current paper, the RPV with various water levels is chosen for study, due to the possibility of water cooling system malfunction under severe accident of core meltdown. Detailed numerical investigation on the structural behaviors of the RPV has been illustrated. In achieving it, the thermal-elastic-visco-plastic FE-models were developed with time-and temperature-dependent characteristics. Because of the presence of the melting pool (~ 1327 °C) on the inside and cooling water (~ 150 °C) on the outside, the high temperature gradient is formed, and considered in the FE-models. Consequently, the RPV failure is found to be affected by some major factors, such as creep damage, plastic damage and thermal expansion. Due to the existence of various water levels, the temperature fields are quite different from each other, resulting in significant variation of structural failure site, time and mode for the RPVs. Through vigorous investigation, some important concluding remarks are summarized as follows: (1) The various water levels induce deformation incompatibility between the RPV material below and above the water level under severe accident condition. Accordingly, there exists huge damage concentration at the site around the water level. However, the RPV safety can be secured within the period of 100 h for all the conditions of three water levels. (2) The equivalent stress is concentrated at the site around the water level for the RPVs. The plastic strain on each side of wall thickness is much higher than the one on the midpoint, due to the existence of elastic core. Furthermore, the creep strain is mainly distributed on the inside wall, because of the presence of high temperature. (3) Through comparison, it is found that the total damage for the RPV with high water level is the severest among the three adopted cases. The total damage on the inside is much larger than the one on the outside for the RPV wall, since both creep and plastic damage make a great contribution to the total damage. (4) The stress on the high temperature region is relaxed to a near-zero level. Moreover, the maximum stress is also reduced very significantly. After a period of creep time, the stress field is redistributed into a new one. Besides, due to the incompatible deformation, the RPV wall at around the water level suffers severe stress triaxiality, which may significantly accelerate the damage evolution during severe accident. Nomenclature d1–d4 D , ΔD Rv T v ε̇ εfraccr εfracpl α,n θ σH
material parameters in creep equation the damage, damage increment the triaxiality factor temperature Poisson ratio creep strain rate creep fracture strain plastic fracture strain material constants for R-O equation polar coordinate angle along lower header hydrostatic stress
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σM σus σy
von-Mises stress ultimate strength yield strength
Acknowledgements This work supported by National Natural Science Foundation of China (Grant Nos. 51505425; 51575489), Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ15E050007, Public Welfare Technology Key Projects of Zhejiang Province (2014C23001). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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Investigation on the RPV structural behaviors caused by various cooling water levels under severe accident
MARK
Jianfeng Maoa,b,⁎, Linghui Hua, Shiyi Baoa,b,⁎⁎, Lijia Luoa, Zengliang Gaoa,b a b
Institute of Process Equipment and Control Engineering, Zhejiang University of Technology, Hangzhou, Zhejiang 310032, PR China Engineering Research Center of Process Equipment and Re-manufacturing, Ministry of Education, PR China
AR TI CLE I NF O
AB S T R A CT
Keywords: Structural behavior Severe accident Failure assessment IVR RPV
In the severe accident (SA) event of core meltdown, the ‘In-vessel retention (IVR)’ mitigation has been certified as an effective strategy for accident management in most advanced nuclear power plants. The traditional concept of IVR was based on the idea that the reactor pressure vessel (RPV) was fully submerged into the cooling water. However, the Fukushima accident showed that the cooling water was insufficient due to the malfunction of water supply system, so the RPV structural behavior had not been appropriately assessed. Therefore, the paper tries to address the structure-related issue on determining whether RPV safety can be maintained or not with the effect of various water levels created by the SA condition. In achieving this goal, the structural behaviors were numerically investigated in terms of several field parameters, such as temperature, deformation, stress (strain) and damage. Due to the melting pool on the inside and water cooling on the outside, the high temperature gradient was formed across the wall thickness, so RPV failure was found to be the consequence of creep, plasticity and thermal expansion. According to the requirement on RPV safety during the prescribed time, it must be ensured that the water cooling takes effect in preventing the structural failure under SA condition. Through vigorous investigation, it is found that the RPV safety is secured within the 100 creep h. Furthermore, the structural failure site, time and mode are predicted with consideration of the effect of various water levels. Most importantly, the failure is found to take place at the site aside around the water level.
1. Introduction In the severe accident (SA) of core meltdown, a famous strategy called ‘In-vessel (IVR) mitigation’ is widely used by most advanced nuclear power plant (NPP) to maintain the safety of the reactor pressure vessel (RPV) [1]. In fact, the IVR mitigation has been certified by nuclear regulatory commission (NRC) as a standard measure in USA for SA management since 1996 [2]. On most occasions, the IVR mitigation is to provide long-term water cooling on the outside wall of the RPV, so the decay heat is removed without the need for any active actions and assistance measurement during the SA [3]. Accordingly, it can be known that the external water cooling is the most important characteristic for accident management, as shown in Fig. 1. In traditional concept of IVR, the lower head (LH) of the RPV is assumed to be fully submerged into the water flooding, prior to the arrival of core debris on the inside [4]. Therefore, the melting pool is formed within the RPV with the temperature of approximate 1327 °C, while the temperature on the outer vessel wall is close to the water saturation temperature of around 150 °C [5]. With the basic assumption of critical heat flux
⁎
Correspondence to: J. Mao, Institute of Process Equipment and Control Engineering, Zhejiang University of Technology, Hangzhou, Zhejiang 310032, PR China. Corresponding author. E-mail addresses: [email protected] (J. Mao), [email protected] (S. Bao).
⁎⁎
http://dx.doi.org/10.1016/j.engfailanal.2017.04.029 Received 15 January 2017; Received in revised form 26 March 2017; Accepted 22 April 2017 Available online 24 April 2017 1350-6307/ © 2017 Elsevier Ltd. All rights reserved.
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Fig. 1. The scheme of RPV with various water levels in severe accident and its typical failure segments [6,22].
(CHF) on the RPV during the accident period, it can be learnt from previous studies that the RPV safety can be secured within a prescribed time [6]. The above conventional IVR concept isn't seriously challenged until the Fukushima accident on 2011 [7]. It showed that the malfunction took place in the cooling water supply system, resulting in insufficiency of water supply. Furthermore, the consequence of the disaster indicated that the structural behavior had not been appropriately assessed [8]. Therefore, the related key issue is that whether LH structural integrity can be maintained with various cooling water levels under severe accident. As indicated before, the structural safety is in the top priority for the operation of nuclear power plants, especially for the event of severe accident, so the IVR mitigation should be robust and capable of relieving the accident consequence. In most cases, the RPV integrity should be ensured during the first 72 h, while the operator may have sufficient time to take some emergence measures [9]. However, due to the complexity of the thermal-mechanical loadings, some of the failure mechanisms are not fully understood due to the low possibility of the occurrence [10,11]. In dealing with the issue of structural safety under SA, numerous efforts have been made by researchers and engineers. For the past several years, many geometrical scaled tests had been performed to characterize the failure site, time and mode of the RPV structure under the programs [12–17] of FOREVER, USNRC/SNL, LIVE and CORVIS. Under the loading of severe accident, one of 1:10 scaled test specimens can be seen in Fig. 1, and the LH failed at the site around the high water level. Besides, a number of finite element (FE) computations had been carried out for investigating the structural failure behaviors under the simulated thermal-mechanical loadings [18], like the loadings of critical heat flux (CHF) and low internal pressure. As indicated in previous studies [19,20], both experimental and theoretical results showed that the predictability of the failure time, site and mode varied with the field parameters, such as temperature, stress, strain. Due to core meltdown on the inside and external water cooling, the high temperature gradient is formed across the wall thickness. Consequently, it is found that creep damage accumulates over a wide area on the inside wall, while the local plastic damage is concentrated on the outside wall [21]. With the increase of creep time, both creep and plastic damages are increasing very significantly and interacted with each other at the site of geometric discontinuity [22]. As pointed out in some literatures [23,24], the CHF loading may be considered as a limit thermal boundary on the inside wall before the occurrence of melt-through, the failure process was found to be very complex, including wall bulge, thinning and even necking at the failure site. Moreover, the plastic and viscoplastic straining was the major contributors to the global or local failure. So far, the complete assessment on Fukushima accident had been submitted to the IAEA by Japan government, one of the critical issues was to answer whether the IVR mitigation takes effect or not [25]. Actually, the water cooling was the essential measure for ensuring the RPV integrity, as given in the report of the accident event. However, this kind of research on the effect of various water levels is scarcely found in the previous studies [26], especially for the era before Fukushima accident. The typical example of water levels is depicted in Fig. 1, indicating that the cooling water can be maintained at low, mid and high level respectively. Accordingly, the assumption of neglecting the water level effect on structural failure in traditional IVR concept is seriously challenged nowadays. In dealing with the above issue, the in-depth understanding of the RPV structural behavior is urgently necessary with the consideration of the effect of various water levels. Therefore, the main task of the paper is to numerically investigate the RPV structural behaviors with the effect of various water levels under the severe accident of core meltdown, which is seldom concerned in previous studies. Toward this end, the 2D FE-models of the RPV with various water levels were developed on the platform of ABAQUS. In order to make a comparison, three typical water levels were employed in the analysis, including low, mid and high water levels. Due to the high temperature gradient across the wall thickness, the time- and temperature-dependent material properties were considered in the FEM, which were also highly nonlinear relationships for creep and plasticity. In accounting for the most dangerous situation before the melt-through, the critical heat flux (CHF) was regarded as the limit thermal boundary condition on the FEM. In order to make sure whether the RPV safety was maintained or not, the 100 creep h was taken as a basis for the FE calculation. Through vigorous investigation, it was found that the RPV safety can be secured within the 100 h, and the RPV structural failure took place at the site aside around the water level. Finally, the structural behaviors of the RPV with various water levels were analyzed carefully by means of field parameters, such as temperature, deformation, stress (strain) and damage. Most importantly, the phenomenon of deformation incompatibility and stress 275
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concentration was found at the site of water level. 2. Mathematical model description 2.1. Creep and plastic damage approaches Due to the high temperature of the melting pool within the RPV, the inner RPV wall is exposed to the environment of T = 1327 °C (melting point). As for the outer RPV wall, it is submerged into the cooling water. According to the ‘thermal failure criterion’, the critical heat flux (CHF) is formed before the melt-through of the RPV wall, which is also considered as the limit thermal boundary condition. The maximum temperature on the outside is considered as 150 °C (nucleate boiling), so the high temperature gradient is established. As well known, the creep failure mechanism occurs when the temperature exceeds 0.4 time of the melting point. In accounting for the creep effect, the constitutive equation is formulated with a number of coefficients [9,16,22],
⎡ d ⎤ εċ = d1 σ d2 εcd3 exp ⎢ − 4 ⎥ ⎣ T⎦
(1)
where, the d1, d2, d3, d4 are material coefficients in fitting creep constitutive equation. Regarding the RPV material, the creep experiences three stages, and the nonlinearity in the tertiary stage is the severest. In describing the nonlinearity, the d1 > 0 and d3 > 0 must be satisfied. In fact, the material coefficients in Eq. (1) are time- and temperature-dependent. In order to achieve the convergence, a newly-proposed scheme with nonlinear characteristics is employed, which is an approach for weighting coefficients fitting. For plasticity, the Ramberg-Osgood (R-O) equation is used to describe the nonlinear relationship between stress and strain. The RO equation is very useful in describing the strain hardening phenomenon for the RPV material, and the original form of R-O equation can be expressed as follows,
εp =
⎛ σ ⎞n σ + α⎜ ⎟ E ⎝ σy ⎠
(2)
Wherein, σ and εp are the true stress and true strain respectively, E is Young's modulus. Moreover, α and n are the material constants, which can be identified using the data ranging from the yield point to the necking point, n = [ln(eus / 0.002)] / [ln(σu / σy)], herein, eus is uniform strain at maximum load, σu and σy are the ultimate and yield strength respectively. In order to raise the fitting accuracy, the option of multi-linear isotropic hardening is activated in ABAQUS software for the plasticity. In order to give a proper assessment on the RPV structural safety, the so called ‘ductility failure criterion’ is adopted for predicting the damage. In the criterion, both creep and plastic damage are incorporated into the following Eq. (3). Therefore, the total damage increment ΔD is the sum of the plastic damage increment and creep damage increment. The increment ΔD is accumulated at the end of each sub-step [12], and ΔD = 0 means “no damage increment”. The expression of the ΔD is formulated as follows [9,22], pl ⎤ ⎡ Δε cr Δεeqv eqv ⎥ Rv ΔD = ⎢ cr + pl ⎢⎣ εfrac (σ , T) εfrac (T) ⎥⎦
(3)
where, the εfrac and εfrac are creep and plastic fracture strain respectively, and they can be obtained by the tensile test with fixed thermal-mechanical loads. For simplicity, the εfraccr and εfracpl are obtained separately, according to the experimentally found material behavior [9,12,15,21]. Most notably, in order to account for the effect of multiaxial stress state, the triaxiality factor Rv is incorporated in the ‘ductility failure criterion’, cr
Rv =
pl
⎛ σ ⎞2 2 (1 + v ) + 3(1 − 2v ) ⎜ H ⎟ ⎝ σM ⎠ 3
(4)
where, σH is the hydrostatic stress, σM is the von-Mises equivalent stress, v is the Poisson's ratio. With the linear accumulation rule, the final total damage D can be computed by the sum of each damage increment ΔD, nStep
D=
∑
ΔDi
(5)
i =1
Actually, the above damage increment ΔDi is computed by averaging its node equivalent damages for each element. When the D is equal to 1, the element is correspondingly set to be inactive using element death technique, so the death element does no longer contribute to the material stiffness. 2.2. Description of FE modeling In order to evaluate the RPV structural behaviors, the two-dimensional (2D) FE-models are established for the RPV with various water levels, which can be seen in Fig. 2. Three FE-models are considered herein, which are RPVs with low, mid and high water level respectively. Due to the axisymmetric characteristics for geometry and load, the half of the RPV is sufficient for the FE calculation. As for the RPV configuration, the thickness of the LH is 165 mm, and the radius of the LH is 2180 mm. The geometric size is taken from 276
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Fig. 2. The geometric size of RPV and its FE-model with applied boundary conditions.
the real component of AP600-type power plant. Due to the possibility of high water level, the LH is extended by 3578 mm into the cylindrical segment. The cylindrical segment is so high that the effect from other component to the LH can be neglected. As shown in Fig. 2, there exists a transition portion between the LH and cylindrical segment, and the wall thickness is gradually increasing from 165 mm to 180 mm. In order to account for the thermal-mechanical behaviors, the element type of C3D20R is used in the FE-models, which is known as ‘20-node quadratic thermal-coupled brick with the reduced integration scheme’. The selection of this element is due to its suitability of computing creep and plasticity simultaneously. Besides, this type of element provides more flexibility for considering the geometric nonlinearity, e.g. large strain and defection. As shown in Fig. 2, the RPV model is carefully meshed, and the grid density is slightly increased for the transition portion. The grid of LH wall thickness is made of 20 nodes, and the element size is approximate 8 mm. Through repeated validation, the total element number is determined to be 4200. As for constraint boundary conditions, the X-oriented freedom is constrained on the axis, and meanwhile the Y-oriented freedom is fixed on the top surface of the cylindrical segment. According to different cooling water level, the thermal boundary condition is extracted from corresponding experimental test on critical heat flux (CHF). As a matter of fact, the CHF is considered as the most dangerous thermal boundary before the complete meltthrough of the RPV wall. With the above assumed limit boundary, the structural behavior is accordingly investigated for the RPV with various water levels. Within the RPV, the melting temperature (1327 °C) is obtained from the thermal calculation, while the temperature is set to 150 °C for the outside wall below the water level. The temperature on the outside is corresponding to the transition point from nucleate boiling to film boiling. For the RPV wall above the water level, the adiabatic condition is assumed. Moreover, the thermal radiation above the melting pool is transferred into an equivalent temperature on the inside wall. As shown in Fig. 2, the top surface of cylindrical segment is far away from the melting pool, so the temperature on the cylindrical segment is assumed to be linearly decreasing from 1320 °C to 500 °C. After the purely thermal calculation, the structural behavior is simulated with incorporating the previous temperature field. Since the RPV material may experience high temperature gradient and huge deformation, the time- and temperature-dependent material properties [16,27] are considered in the FEM. In order to make an assessment on the structural safety under the SA condition, the thermal-elastic-visco-plasticity calculation is achieved for the RPV with various water levels.
3. Results and discussions 3.1. The temperature and deformation fields During the severe accident of core meltdown, the so called ‘IVR mitigation’ is assumed to be able to arrest the degraded melting core within the RPV, and maintain the RPV integrity within a period of prescribed time, usually 72 h. Consequently, the melting pool is formed in the lower head (LH) of RPV. According to the thermal failure criterion, the basic concept of ‘critical heat flux (CHF)’ is applied on the FE-models of the RPV, and regarded as the most dangerous thermal boundary condition before the complete meltthrough of the RPV wall. Accordingly, the temperature on the inner is approaching the melting point (1327 °C) of the RPV material. As for the outer surface of the RPV, the temperature is various for the three water levels, which is shown in Fig. 3. It is clear to see in Fig. 3 that the water cooling plays an important role in lowering the temperature, and the water level significantly changes the 277
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Fig. 3. Temperature distributions for RPV with various water levels.
temperature distribution. Furthermore, the high temperature zone is decreasing with increasing the water level. Actually, without water cooling, the local temperature field is homogenized as soon as the melting pool is formed on the inside, the temperature of which is maintained at approximate melting point of 1327 °C. On the contrary, the high temperature gradient is formed from 1327 °C on the inside to 150 °C (max. nucleate boiling) on the outside, due to the existence of water cooling. Close observation of Fig. 3 shows that the high temperature gradient still exists at the transition aside around the water levels. From Fig. 3, it can be inferred that the existence of cooling water can prevent the RPV from immediate structural failure. Since the outer surface of RPV is submerged into the water, the high material strength is correspondingly maintained for the outer layer of RPV wall thickness. As a matter of fact, the thermal expansion is temperature-dependent as well as creep deformation, so the various temperature distributions will inevitably result in a phenomenon of deformation incompatibility. In accounting for the effect of various water levels, Fig. 4 presents the corresponding distribution of total displacement for the lower head (LH). General observation of Fig. 4 shows that the LH stretches outward very significantly, and the bulge at the bottom (along Y direction) is much more significant than the one at the transition (along X direction) of spherical segment to cylindrical segment. Taking RPV with low water level for example (see Fig. 4(a)), the bulge along Y direction is 47.3 mm, while the bulge along X direction is only 21.5 mm. Among the three adopted water levels, the low water level results in the most significant deformation for the LH. As displayed in Fig. 4, the max. deformation is as high as 71 mm for the case with low water level. General speaking, the deformation is increasing with decreasing the water level, and the significance of deformation highly depends on the size of high-temperature homogenized zone. Moreover, Fig. 4(a) discloses that the max. deformation is located at the site of water level for the RPV with low water level, the behavior of which is similar to the one in Fig. 4(b). However, the max. deformation for the RPV with high water level is located at the bottom, as indicated in Fig. 4(c). In fact, the stiffness of the RPV with high water level is the biggest among the three cases, due to the best performance of water cooling. It is clear to see in Fig. 4 that for low to mid water level, the most significant deformation definitely occurs at the hightemperature homogenized zone. As indicated in Fig. 4, the outer surface layer is performed in the state of tension for the highlydeformed zone, which is more detrimental to RPV safety than the compression distributed on the inside. More interestingly, Fig. 4 discloses that a significant highly-squeezed zone is formed with the bulge of the LH, and an additional bending moment can also be found at the transition of spherical segment to cylindrical segment. Furthermore, the comparison of Fig. 4 reveals that the RPV with
Fig. 4. Comparison of total displacement distributions for RPV with various water levels.
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Fig. 5. Mises stress distributions for RPV with various water levels at pre-creep stage.
high water level has most significant bending moment. 3.2. The RPV structural behavior at pre-and post-creep stage As well known, the RPV material strength (e.g. yield stress) is decreasing with increasing temperature, so the Mises stress is much lower at high temperature zone than the one at low temperature zone. As shown in Fig. 5, the highest stress is concentrated at the site aside around the water level, the height of which is increasing with the increase of the water level. Fig. 5 shows that the max. Mises stress is as high as 580.3 MPa, which is approaching the yield strength. However, due to the high temperature distribution on the RPV above water level, the stress on the corresponding zone is as low as 4.03 MPa. Clear, the difference of the Mises stresses is huge between the high temperature zone and low temperature zone. At this point, the load-carrying capability of the RPV can be reflected by the water level, and the limit load must be determined by the ‘weakest link’. As discussed before, the deformations aren't compatible at the site aside around the water level, so huge equivalent plastic strain (PEEQ) is found to be accumulated on the wall thickness in the proximity of water level. As indicated in Fig. 6, the PEEQ accumulated on the inside wall is significantly bigger than the one on the outside wall. For the condition of low to mid water level, there exist two highly-concentrated zones of the PEEQ, one is located at the site at proximate water level, and the other is located at the transition of spherical segment to cylindrical segment. The observation of Fig. 6 shows that the inside wall below the water level suffers significant PEEQ, indicating that the water level changes the distribution of the PEEQ. Further observation of Fig. 6 reveals that there exists a small elastic core in proximate middle layer of the wall thickness, especially for the RPV with mid water level. Actually, the existence of elastic core is essential for ensuring the RPV safety under severe accident (SA) condition, since the elastic core has some load-carrying capability in preventing immediate collapse. From Fig. 6, it can be inferred that the structural failure has to initiate at the site of water level, when the thermalmechanical loads are further increasing. As the incompatible deformation occurs, the failure site may suffer ductile tearing, like shear failure mode. As for the RPV with high water level, the elastic core is squeezed into a very small size at the transition site, as shown in Fig. 6. Actually, the local plastic yielding is greatly aggravated with increasing the water level, especially for the inner surface. Overall speaking, it can be concluded that the global failure (collapse) is not a feasible mode for the RPVs with various water levels,
Fig. 6. Equivalent plastic strain distributions for RPV with various water levels at pre-creep stage.
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Fig. 7. Comparison of Mises stress distributions along the LH between pre- and post-creep stage.
due to the concentration phenomenon of field parameters. By looking into the distribution of equivalent stress along the angular position, it can be found in Fig. 7 that the peak value of the stress occurs at the site of respective water level. The comparison shows that the max. stress for the RPV with mid water level is slightly higher than the one for the RPV with other two levels. Fig. 7 also shows that the equivalent stress experiences a significant decrease before the max. stress is reached, indicating that the structural behavior for the RPV wall below the water level is different from the one for the RPV wall above the water level. The observation of Fig. 7 displays that the equivalent stress experiences a sharp decrease immediately after the max. value. Accordingly, it can be inferred in Fig. 7 that most of the loads are supported by the RPV wall below the water level. With the creep time increasing, this phenomenon becomes more significant, since the stress of the RPV wall above the water level is relaxed to be near stress-free. As for the RPV wall below the water level, the stress relaxation is also very significant, and the peak value of the stress is almost reduced by approximate 40% after 100 creep h. The comparison in Fig. 7 indicates that the RPV with mid water level suffers the severest stress relaxation. Furthermore, with the stress redistribution, the stress profile is also changed into a new one along the angular position after 100 creep h. By looking into the distribution of equivalent stress along the wall thickness, it can be found in Fig. 8 that the stress on the outside is significantly higher than the one on the inside. Due to the occurrence of high temperature from the melting pool, the stress on the inside is significantly low, most of which is approaching the material strength. With the effect of stress relaxation, the equivalent stress is reduced down to the near-zero level within the wall thickness range of 0 ≤ t ≤ 105 mm. Moreover, Fig. 8 demonstrates that the creep effect is almost ceased at the site of T = 425 °C. However, due to the stress redistribution, the outer surface layer of 105 mm < t ≤ 165 mm suffers stress reduction, and it is found in Fig. 8 that the max. stress is relaxed by around 44% on the outside. In the middle of the wall thickness, there exists an elastic core that is below the material strength, as shown in Fig. 8. In the elastic core, the equivalent stress is decreasing with the increase of the water level for the condition at both pre- and post-creep stages. Close
Fig. 8. Comparison of Mises stress distributions across the wall thickness between pre- and post-creep stage.
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Fig. 9. Equivalent creep strain distributions for RPV with various water levels at post-creep stage.
observation of Fig. 8 discloses that the equivalent stress at both inner and outer surface layers is beyond the material strength, so the plastic yielding is correspondingly inevitable. As mentioned before, the immediate RPV failure is not a feasible mode before the complete melt-through of the RPV wall, since the creep mechanism is a time-dependent process. As well known, the creep strain as well as plastic strain is irreversible, so there must be some creep strain accumulation after 100 h. In describing it, Fig. 9 plots the creep strain contours for the RPV with various water levels. Overall observation of Fig. 9 shows that there exists a highly-concentrated region of equivalent creep strain (CEEQ) at the site of around the water level. Clearly, the CEEQ on the inside is significantly higher than that on the outside, due to the high temperature gradient from melting pool to water cooling. The comparison of Fig. 9 reveals the RPV with mid water level suffers slightest CEEQ on the inside wall in the proximity of the water level. As for the transition portion of spherical segment to cylindrical segment, there exists a most significant accumulation of CEEQ for the RPV with high water level. For the low to mid water level, the CEEQ distributed on the segment without water cooling is found to be remarkably lower than the one with water cooling, the reason for that is the corresponding lower stress for the segment without water cooling. From Fig. 9, it can be seen that there exists a discontinuity of high CEEQ at the region below the water level, implying that the incompatible deformation takes place at the corresponding region. From the above analysis, it can be learnt that the RPV failure possibly initiates at the inner surface layer of the transition portion, due to its significant accumulation of both plastic and creep strain. Still, it can be known that the RPV failure can be determined locally at the site in proximity of the water level. 3.3. The multiaxial damage analysis on crust effects As indicated before, due to the various water levels, the field parameters, such as temperature, stress and strain, tend to perform in a multiaxial manner. In describing it, the triaxiality factor Rv defined in the above Eq. (4) is adopted in the analysis, which is plotted
Fig. 10. The evolution of triaxiality factor Rv at two typical nodes during the creep time.
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Fig. 11. Distributions of plastic and creep damage across wall thickness for the RPV with various water levels.
in Fig. 10. Actually, the RPV wall at round the water level may suffers various states of stress, such as tension, shear. The thermal stress at the site above the water level is also different from the one below the water level. In order to reflect the multiaxial state of stress, the Rv is incorporated in the so called ‘ductility failure criterion’. In traditional concept of RPV integrity assessment, the Rv is usually reduced to be equal 1, due to its simplicity of using uniaxial test data. However, this simplification may result in huge conservatism. In reducing it, the Rv should be carefully considered in the temperature- and time-dependent FE calculation. As indicated in Fig. 10, the Rv for the large deformation node has been increased with the creep time, the max. value of which is 3 at 100 h. The Rv for the small deformation node is almost kept at constant level during the creep hours, except for the one at initial stage. The comparison in Fig. 10 reveals that the RPV failure may be accelerated by the multiaxial state of stress, since the Rv is larger than 1, and continuously increasing with the creep time. From this perspective, the assumption of the uniaxiality may be improper when assessing the RPV structural failure with the effect of various water levels, because the distribution of Rv among the lower head (LH) is changing with the creep time, especially for the site around the water level. In order to leave enough time for the operator to take some emergency measures when the severe accident (SA) occurs, the RPV integrity has to be secured within the least 72 h, which has been recognized and certified by NRC [6,8]. Accordingly, the most attractive point is to assess the failure time, site and mode for the RPV with various water levels under SA condition. In disclosing it, Fig. 11 plots the distributions of the plastic and creep damage across the wall thickness in the proximity of the water level. Since creep damage is also a time-dependent parameter, the 100 h is taken as a basis for comparison in Fig. 11. In fact, the creep and plastic damages occur simultaneously during the process, and they must be interacted with each other. Overall observation of Fig. 11 shows that both plastic and creep damages are distributed as a ‘spoon’, the bottom of the damage profile is always located at the region of elastic core. As indicated in Fig. 11, the lowest damage is distributed at approximate midpoint of the wall thickness for the condition of low to mid water level, while the lowest damage is located at the site close to the outside for the RPV with high water level. It can be seen in Fig. 11 that the damage on the inside is larger than the one on the outside at pre- and post-creep stages. Notably, it can be seen in Fig. 11 that the creep damage is maintained at a high level throughout the wall thickness, comparing with the plastic damage distribution. The comparison in Fig. 11 demonstrates that the RPV with high water level suffers the largest damage at the transition portion of spherical segment to cylindrical segment. In order to make an assessment on the RPV structural failure behavior, the final contours of total damage after 100 h are given in Fig. 12 for the RPVs with low, mid and high water levels respectively. After the experimental test, the creep rupture strain of εfraccr = 6% at 100 h is considered as one basis for creep damage calculation, meanwhile the plastic rupture strain of εfracpl = 5% is obtained as the other basis for plastic damage evaluation. Through FEM, the quantitative prediction on the failure site, time and mode is achieved. As shown in Fig. 12, there exists a huge damage concentration on the inside of the RPV wall aside around the water level. It is clear to see in Fig. 12 that the total damage distributed on the inside is much larger than the one distributed on the outside, the reason for that is the great contribution by creep damage. Due to the water cooling on the outside, the total damage is dominated by the plastic damage at the site in the proximity of water level. Still, it can be found in Fig. 12 that the damage on the midpoint of the wall thickness is significantly lower than the one on each side of the RPV wall, which is owing to the existence of the elastic core on the middle layer. As for the effect of water level, it can be observed in Fig. 12 that the high water level induces the severest total damage for the RPV, and the structural failure must initiate at the transition portion of spherical segment to cylindrical segment. After 100 h, almost two thirds of the wall thickness lost its load-carrying capability for the RPV with high water level. The comparison in Fig. 12 shows that the mid water level may be the best option for maintaining the safety among the three adopted water levels, due to its least damage accumulation on the corresponding site. More notably, Fig. 12 reveals that the damage on the segment without water cooling is increasing with the water level, and it isn't as high as the one on the site around the water level. Actually, the damage distribution is determined by the temperature and stress, so the low stress level may be the reason for the low damage on the segment 282
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Fig. 12. Total damage distributions for RPV with various water levels after 100 h.
without water cooling. In concluding this paragraph, the emphasis should be given to the effect of the water level. The RPV structural failure is contributed by the plastic and creep damage, and takes place locally at the site around the water level. 4. Conclusions In the current paper, the RPV with various water levels is chosen for study, due to the possibility of water cooling system malfunction under severe accident of core meltdown. Detailed numerical investigation on the structural behaviors of the RPV has been illustrated. In achieving it, the thermal-elastic-visco-plastic FE-models were developed with time-and temperature-dependent characteristics. Because of the presence of the melting pool (~ 1327 °C) on the inside and cooling water (~ 150 °C) on the outside, the high temperature gradient is formed, and considered in the FE-models. Consequently, the RPV failure is found to be affected by some major factors, such as creep damage, plastic damage and thermal expansion. Due to the existence of various water levels, the temperature fields are quite different from each other, resulting in significant variation of structural failure site, time and mode for the RPVs. Through vigorous investigation, some important concluding remarks are summarized as follows: (1) The various water levels induce deformation incompatibility between the RPV material below and above the water level under severe accident condition. Accordingly, there exists huge damage concentration at the site around the water level. However, the RPV safety can be secured within the period of 100 h for all the conditions of three water levels. (2) The equivalent stress is concentrated at the site around the water level for the RPVs. The plastic strain on each side of wall thickness is much higher than the one on the midpoint, due to the existence of elastic core. Furthermore, the creep strain is mainly distributed on the inside wall, because of the presence of high temperature. (3) Through comparison, it is found that the total damage for the RPV with high water level is the severest among the three adopted cases. The total damage on the inside is much larger than the one on the outside for the RPV wall, since both creep and plastic damage make a great contribution to the total damage. (4) The stress on the high temperature region is relaxed to a near-zero level. Moreover, the maximum stress is also reduced very significantly. After a period of creep time, the stress field is redistributed into a new one. Besides, due to the incompatible deformation, the RPV wall at around the water level suffers severe stress triaxiality, which may significantly accelerate the damage evolution during severe accident. Nomenclature d1–d4 D , ΔD Rv T v ε̇ εfraccr εfracpl α,n θ σH
material parameters in creep equation the damage, damage increment the triaxiality factor temperature Poisson ratio creep strain rate creep fracture strain plastic fracture strain material constants for R-O equation polar coordinate angle along lower header hydrostatic stress
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σM σus σy
von-Mises stress ultimate strength yield strength
Acknowledgements This work supported by National Natural Science Foundation of China (Grant Nos. 51505425; 51575489), Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ15E050007, Public Welfare Technology Key Projects of Zhejiang Province (2014C23001). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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