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Evaluation of the thermal strain of an NPP containment structure during leakage rate tests
Engineering Structures 201 (2019) 109761
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Evaluation of the thermal strain of an NPP containment structure during leakage rate tests
T
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Xuguang Wanga, Xu Huanga, Oh-Sung Kwona, , Evan Bentza, Julia Tchernerb, Mark DeMerchantc, Mark Molandc a
Department of Civil and Mineral Engineering, University of Toronto, Canada Candu Energy, Inc. a member of SNC Lavalin Group, Canada c NB Power, Canada b
A R T I C LE I N FO
A B S T R A C T
Keywords: Thermal strain Containment structure Solar radiation
A strategy for isolating the thermal strain due to solar radiation from the total measured strain during the Integrated Leak Rate Test of a nuclear containment structure is presented in this paper. In the proposed method, a series of numerical models are developed to estimate the temperature change on the prestressed concrete containment structure and to predict the strain due to temperature and applied pressure. The effects of the solar radiation, heat exchange with internal and external air and heat conduction are considered in the proposed method. The developed method is validated by reproducing the total strain profiles measured from the leak rate tests conducted at the Point Lepreau CANDU®1 6 containment structure.
1. Introduction The prestressed concrete containment structures for nuclear power plants are designed to meet the required leak-tightness under operating and test conditions [1]. To ensure the structure meets the leak-tightness requirement over its service life, an Integrated Leak Rate Test (ILRT) is performed periodically subsequent to the first in-service use of the containment system [2]. During the ILRT, the concrete containment structure is pressurized from inside. The containment is expected to deform due to the pressure and develop corresponding strains, which can be used to estimate the uncracked stiffness of the structure and to determine if partial cracking has occurred in parts of the structure. Because the applied pressure is within the linear elastic range of the containment structure, the developed strains are expected to be proportional to the applied pressure. However, non-negligible level of strain fluctuations have been observed in the ILRTs. Based on the daily pattern of strain fluctuation and smaller fluctuation at the north face of the containment structure, the source of the strain fluctuations are expected to be the temperature changes on the exterior surfaces of the containment structure, particularly due to the solar irradiance. Although some of the strain gauges and sensors used for measuring strains during the ILRTs can compensate for the thermal effects, such as
the fiber Bragg grating sensors used by Li et al. [3], only the effects of temperature on the strain gauges are compensated. Consequently, the measured strain is the total strain of concrete which consists of mechanical strain resulting from internal pressure increase and thermal strain resulting from the temperature variation during the tests. It is difficult to directly measure the thermal strain of the containment structure due to the non-uniform distribution of temperature through the structure’s thickness and height. Therefore, heat transfer analysis is required for obtaining an estimate of the temperature history and structural analysis is required for calculating the thermal strains during the ILRT. In the past, heat transfer analysis was used to obtain the thermal properties of the concrete containment structures [4,5] and investigate long-term and short-term behaviors of the structures subjected to thermal loads during the operational period [6–9]. However, few studies have been conducted on the analysis of the thermal effects with the consideration of the realistic thermal boundary conditions during the IRLT. To bridge the gap, a set of numerical models for predicting the thermal effects and isolating the thermal strains from the mechanical strains of the containment structure during the IRLT is developed in this study. The influence of the historical weather conditions and the solar radiation is taken into account in the numerical model. The developed
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Corresponding author. E-mail address: [email protected] (O.-S. Kwon). 1 CANDU (CANada Deuterium Uranium Reactor) is a registered trademark of the Atomic Energy of Canada Limited (AECL), used under exclusive license by Candu Energy Inc. https://doi.org/10.1016/j.engstruct.2019.109761 Received 17 April 2019; Received in revised form 24 August 2019; Accepted 5 October 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.
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the simulation of convection and heat radiation. In addition to convection and heat radiation, solar radiation is another source of temperature change on the exterior surface of the containment structure. Radiation from the Sun influences the surface temperature of the containment structure significantly and causes nonuniformly distributed temperature on the exterior surface of the containment structure due to the temporal changes of the projection angle and the intensity of the radiation. Thus, both the intensity and the relative angles between the Sun and points on the exterior surface of the cylindrical containment structure should be considered when calculating the heat influx from solar radiation. To get the relative angles, the information of the Sun’s path which is described by the altitude, α , and azimuth, θ , at the coordinate of the containment structure needs to be considered. The solar angle information, for example, can be obtained from The United States Naval Observatory (USNO) website. The intensity of heat irradiation from the Sun can be calculated using empirical models. In this approach, the model developed by IES [12] is employed. In the model, the intensity of direct solar irradiation, Id , that reaches the ground after passing through the atmospheres is expressed as:
method is applied to the CANDU 6 reactor system at Point Lepreau and compared to the test results of ILRTs conducted in 2000, 2004, 2012 and 2014. 2. Modeling the containment structure during ILRT To analyze and isolate the thermal effects of the containment structure during an ILRT, a set of three numerical models are required. A heat transfer analysis is performed in one of the numerical models while structural analyses are performed in the other two. The heat transfer model uses the air temperature, weather condition, and solar irradiation during the test period to predict the temperature history of the containment structure. In order to isolate the strain caused by the thermal effects from the total strain, the result of the heat transfer analysis and the pressure are applied to the second and third models, respectively, for structural analysis. With the assumption that the containment structure behaves within the linear elastic range, the analysis results from the second and the third models can be superimposed to get the total stresses and strains over the ILRT. The details of the modeling strategy in each model are presented in the following sections.
Id = I0 exp (−cm)
2.1. Heat transfer analysis
where c is the atmospheric extinction coefficient that represents the sky clearness conditions, m is the optical air mass, and I0 is the extraterrestrial solar illuminance. In the model, the atmospheric extinction coefficient is suggested as 0.21 for clear sky and 0.8 for partly cloudy conditions. The optical air mass can be approximated as:
The heat transfer analysis considers the heat exchange between the containment structure and the internal/external environment through radiation and convection, as well as the heat conduction through the thickness of the containment structure. The overall heat flow rate on the surface of the containment structure is the summation of the heat flow rates from the convection, heat radiation, and solar irradiance effects. The total heat flow rate, qtot , can be expressed as:
qtot = qc + qrad + qsolar
m=
(5)
2π I0 = Isc ⎛1 + 0.034cos ⎛ (n − 2) ⎞ ⎞ ⎝ 365 ⎠⎠ ⎝
(6) 2
where Isc , the solar constant is 1366 W/m , and n is the day of the year, for example, n = 1 for January 1st. From the geometry, the solar radiation normal to a panel can be expressed as:
(2)
In = Idsin(αp−α )sin(θp − θ)
where Tc is the temperature at the surface of the containment, Ta is the temperature of the surrounding air, A is the surface area, qc is the heat transferred per unit time, and h is the convection coefficient depending on the Rayleigh number and Prandtl number of the surrounding air. The values of the Rayleigh number and the Prandtl number are affected by the wind speed and humidity. Thus, the convection coefficient can also be obtained by using the regression model of the experimental work using wind speed and humidity as variables [10]. The other heat exchange mechanism is thermal radiation where the temperature of an object changes due to the transfer of energy via electromagnetic waves. The mathematical expression of the rate of heat flow due to the radiation, qrad , is:
qrad = εσA (Tc 4 − Ta4 )
1 sinα
and I0 is expressed as:
(1)
where qc , qrad , and qsolar are the rate of heat flow caused by convection, heat radiation, and solar irradiance. The rate of heat flow due to heat convectionqc , is mathematically expressed as:
qc = hA (Tc − Ta)
(4)
(7)
where αp is the angle of the panel respect to the ground and θp is the angle between the panel surface and the true north as shown in Fig. 1. Hence, the rate of heat flow caused by the solar radiation, qsolar , is:
qsolar = αab I n A
(8)
where A is the surface area of an element and αab is the solar absorptivity of the concrete containment, which can be taken as 0.65 as recommended by [13]. Although the containment structure is surrounded by the service building, the height of the service building is generally shorter than one quarter of the height of the containment structure and the gauges to measure the circumferential strain are equipped at the mid-height of the containment structure. The shading and albedo effects may influence the strain change at the bottom of the structure, but the influence to the strain measurement at the mid-height is minor. Thus, the shading and albedo effects from surrounding buildings are not considered in the model. The change of the surface temperature, ΔT , can be related using the following equation:
(3)
where ε is the emissivity of the material, qrad is the heat flow rate, A is the surface area, Tc and Ta are the surface temperature of the object that emits heat and the temperature of the surrounding environment respectively, and σ is the Stefan-Boltzmann constant which is W 5.670 × 10−8 2 ° 4 [11]. The total heat flux via radiation is the sum of m C absorption and emission. Since the containment structure is not surrounded by any high temperature objects, the heat influx from absorbing the radiation of the surrounding objects is negligible compared with the heat radiation from the Sun. Therefore, the radiation effect in the model only refers to the heat emission from the containment structure when the surrounding air temperature is lower than its surface temperature. The record of actual air temperature history, wind speed, and relative humidity of the air, if available, should be used for
ΔT =
qtot Δt Vρcp
(9)
where Δt is the time step, V is the volume of an element, ρ and cp are the density and the specific heat of concrete, respectively. The heat transfers through the thickness of the containment wall via heat conduction. For a small element of the containment structure, the general heat equation for heat conduction is 2
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Fig. 1. Illustration of the solar radiation on the containment.
∂T k 2 = ∇T ∂t ρcp
3. Solar thermal-stress analysis for nuclear power plants (10) 3.1. Background and the previous ILRTs
where ∂T is the rate of change of temperature with respect to time at of ∂t the point of interest, k , ρ and cp are the thermal conductivity, density, and specific heat of the material, and ∇2 T is the second derivative of the temperature in all directions in the 3-D space. As shown in the equation, the rate of temperature change depends on the properties of the material. The containment structure is composed of different materials including concrete, steel reinforcement, and tendons. Therefore, an accurate thermal conductivity of the composite material should be obtained through experiments.
The Point Lepreau CANDU® 6 reactor system has been in service for more than 30 years. The modelling approach described previously is used for replicating the strain measurement from the ILRTs conducted in the years 2000, 2004, 2012 and 2014. To measure the strain changes during the tests, four 27.5 m long-gauge fiber optic sensors (LGFOS) are installed in the circumferential direction, and eleven (eight primary and three secondary) clusters of strain gauges were installed before the year 2000 [14]. The LGFOS readings are based on the changes of the refractive index of the embedded Fibber Bragg Grating (FBG). The measurements during the ILRTs are recorded in mm and converted to microstrain as follows:
2.2. Structural analysis The results from the heat transfer analysis are temperature histories at various points of the containment structure, and through the thickness of the containment wall. The temperature histories of the containment structure over the testing period are applied to the structural analysis model as thermal boundary conditions. For the purpose of isolating the thermal strain from the total strain, other loads, such as gravity load and the internal pressure, are applied in a separate structural analysis model without considering the thermal effects. When no cracks are observed during the test, the total strain, εtot , can be expressed as:
εtot = εth + εp
εm =
R1 − R2 × 106 27.5
(12)
where εm is the measured strain, R1 is the initial reading at the beginning of the ILRT, and R2 is the incremental reading during the test. The measured results showed that the strains during each test period significantly fluctuated over the course of the test. It was concluded that the observed fluctuations were due to the strain component resulting from the thermal effects [14]. In a previous numerical analysis of the containment structure subjected to the ILRT in 2000, the timedependent strain component resulting from temperature changes were not considered [15]. Thus, the numerically predicted strain history did not show any fluctuation over the testing period. If the thermal strain component of the observed total strain can be accurately predicted, the strain due to the pressure increase can be isolated, which is expected to lead to a better understanding of the structural performance of the containment structure. In the following sections, the strain history of the CANDU® 6 containment structure during the ILRTs, including both thermal and mechanical strain components, are evaluated based on the analysis method discussed in Section 2. The leak-rate test in 2000 is studied as a benchmark to verify the modeling technique for thermal and mechanical analyses. Numerical models for the tests in other years are developed by changing pressure and external weather histories corresponding to the testing conditions of each ILRT. The year 2000 ILRT started on August 23, 2000, around 7:00 p.m. and lasted for about 80 h. The weather of the first two days was cloudy and wet, so the solar radiation did not significantly affect the measuring results; however, over the last two days, the influence of the heat radiation from the Sun
(11)
where εth is the thermal strain due to temperature variation during the test and εp is the strain due to all other loads. The εp of the containment structure can be calculated from the pressure model as described in Section 3.4. The thermal strain in Eq (11) is calculated by imposing the temperature history from the heat transfer analysis as thermal boundary conditions on a structural analysis model as described in Section 3.3. The thermal expansion of the material is specified in the model for computing thermal strains. Ideally, the thermal expansion coefficient of the composite material should be obtained from material tests, but the values are expected to be close to the thermal expansion coefficient of the concrete due to the significantly larger volume of concrete in comparison to the volume of reinforcement bars or tendons. The ILRT typically takes less than a week, so the strains due to longterm parameters, such as shrinkage, creep and relaxation, are not considered in the model. The main input parameters and analysis framework for predicting the total strain are summarized in Fig. 2. 3
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Fig. 2. Variables for the Proposed Analysis Strategy.
directions of the containment structure are expected to be small due to the low thermal conductivity of concrete while the temperature gradient through the wall thickness is more pronounced due to the temperature difference on the exterior and interior surfaces. Thus, the structure is coarsely meshed in the vertical and circumferential directions and meshed much more finely through the wall thickness to evaluate heat transfer through the thickness of the structure and to determine the corresponding strain profile. At each location on the perimeter wall and the dome, there are 20 layers through the thickness to capture the temperature gradient. The numerical model simulates the heat transfer effects from two days (48 h) before the test starting time to ensure the temperature of the structure is in a steady state condition. It is assumed that, from the time the test is started, the internal air temperature decreases linearly over the test period from 40 °C to 20 °C due to the cooling of the reactor building. Historical air temperature is used for simulating the convection and radiation effects on the exterior surface. The convection coefficient, h , in Eq. (2) is calculated using an empirical regression
was significant. 3.2. Heat transfer model and assumptions The heat transfer analysis was conducted using ABAQUS [16]. The geometry and the global coordinate system of the model are shown in Fig. 3(a). The containment structure has a cylindrical shape. On the top of the perimeter wall, there is a ring beam which supports a dome. On the perimeter wall, there are four buttresses as shown in Fig. 3(b). For simplification, the buttresses are not included in the numerical model as they are far away from the location of the strain measurements during the ILRTs. The orientation of the model is defined for calculating the relative angle when simulating the solar radiation. The base slab of the containment structure lays in the x-z plane. The center of the top of the base slab is defined as the origin with the coordinates of (0, 0, 0) and the positive z-direction is defined as due south. The heat transfer analysis was performed with 8-node linear heat transfer brick elements. The temperature gradients along the height and circumferential
Fig. 3. Geometry and coordinate system of the numerical model. (a) Elevation view of the containment structure, (b) Plan view of the containment structure. 4
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model by Guo et al. [10] as follows:
h=
0.9337v 2
− 1.15v + 7.365Rh + 3.937
Table 1 Thermal properties of the heat transfer model.
(13)
where v is the wind speed in m/s, and Rh is the relative humidity of air in percent. The air temperature, wind speed and the relative humidity of air are adopted from the hourly historical data collected by Point Lepreau CS weather station (Climate ID: 8104201) [17]. The wind speed is assumed to be uniform along the height of the containment structure and the direction of the wind is not taken into account. The calculated convection coefficient, h , during the test varies in a wide range. During windy days, the convection coefficient can be much higher than the average value during the test. For example, during the test in 2000, the peak value of convection coefficient reaches 7833 J/ min/m2, but the average value is 1057 J/min/m2. An ABAQUS user subroutine was developed to simulate the direction of solar radiation using Eqs. (4)–(8). In the subroutine, the solar radiation is defined as heat influx at each node on the exterior surface in the model. The αp for all nodes on the perimeter wall is set to 90° while the αp for the dome is calculated using the following equation:
αp = tan−1
(14)
where x , y, z are the coordinates of a node on the dome and x 0,y0,z 0 are the coordinates of the point on the center line of the containment structure at some elevation from the base slab. The θp of each node is calculated as:
θp =
π x + tan−1 2 −z
Value
Unit
Conductivity, k Density, ρ Specific heat, cp Emissivity, ε
120 2400 800
J/min/m/°C kg/m3 J/kg/°C
0.95
–
stress and strain do not influence the results from heat transfer analysis. Therefore, the strain on the containment structure due to the thermal effects can be obtained by imposing the results from the heat transfer model mentioned in Section 3.2. From the results of the heat transfer analysis, the temperature history at each node on the containment structure has been determined. The mesh for the structural analysis model is not necessarily the same as the heat transfer model. When the mesh of structural analysis model and heat transfer model are different, the nodal temperature on the structural analysis model can be gained by linear interpolation. In this practice, the same model geometry and mesh from the heat transfer model was used for the thermal stress analysis. An 8-node linear brick element was selected for the analysis. The temperature history of each node from the heat transfer analysis is applied as the thermal boundary condition. A fixed boundary condition is assigned at the bottom of the containment wall. With such a boundary condition, the predicted stress and strain near the bottom will be higher than the partially fixed boundary condition, but the impact on the global behavior is minor. Because the objective of the study is to predict global fluctuation of thermal strain at an elevation far from the support, the above assumption on the support deemed appropriate. At each time step, the structure analysis model calculates the thermal strain based on the thermal expansion caused by nodal temperature and mechanical properties of the containment structure. The deformations caused by the thermal effects are assumed to be within the linear elastic range. Due to the large volumetric ratio of concrete to steel, the material properties of concrete are used in the model. Thus, the elastic modulus and the thermal expansion coefficient are 34.5 GPa and 10−5/ °C respectively.
(y − y0 ) (x − x 0)2 + (z − z 0 )2
Thermal Properties
(15)
According to the test report and the historical temperature record, the first two days of the test (August 23 and August 24, 2000) were cloudy and wet, and the sky was clear for the following two days. Therefore, the solar intensity of the following two days is expected to be higher than the solar intensity of the previous days. The atmosphere extinction coefficient in Eq. (4) is used to adjust the intensity. Instead of setting the extinction coefficient to the suggested values by IES, it is calibrated to produce a final result that best matches the measurement for the year 2000 IRLT. The main reason for this calibration is due to the lack of knowledge of actual cloud conditions during the ILRT. Heat conduction through the thickness of the structure is calculated by solving Eq. (10) numerically. Compared to the solar irradiance and the heat exchange with the surrounding air, the heat exchange between the bottom of the containment structure and soil has a small influence on the temperature history over the test period especially at the locations where the sensors are installed. Therefore, by assuming no heat transfer between the containment structure and the soil, an insulated thermal boundary condition is assigned at the bottom of the containment structure. The thermal properties from material tests are not available. The typical concrete conductivity and specific heat, as suggested in [18], are 84 J/min/m/oC and 900 J/kg/oC, while typical values for steels are 2700 J/min/m/oC and 600 J/kg/oC respectively. Considering the high percentage of the embedded steel, the specific heat and the conductivity values used in the numerical models are calibrated with trial-and-error approach. Both values are between the suggested values of concrete and steel and able to produce reasonable results. The emissivity of the concrete surface is based on the value suggested in [19] and calibrated to simulate more active thermal transfer effects. The general thermal properties for concrete for the analysis are summarized in Table 1.
3.4. Analysis of pressure dependent strain component The strain changes due to temperature-independent parameters, such as gravity loads, prestressing forces, and the internal pressure increase were predicted separately from a detailed finite element model of the containment structure using the VecTor4 program. The program is based on the Modified Compression Field Theory (MCFT) [20] and is specialized for nonlinear analysis of reinforced concrete shells [21,22]. In the numerical model, the containment was modelled with heterosis shell elements, each of which had 9 nodes and 42 degrees of freedom. The in-plane transverse and longitudinal reinforcement and tendons were defined as smeared layers which were superposed onto the concrete layers, while the out-of-plane reinforcement was defined as a property of the concrete layers. Fig. 4 shows example elements at the dome and the perimeter wall where the layers of the reinforcement and tendons are shaded. More details about the program can be found in [21]. As proposed by the authors [23], the aging effects due to time-dependent parameters, such as shrinkage and creep of concrete, and relaxation of prestressing force were taken into account in the VecTor4 model. Specifically, strain changes due to concrete creep and shrinkage were firstly predicted from the CEB/FIP 90 model [24] and then were applied to the model as equivalent temperature loads. The relaxation of prestressing steel was modelled by applying an equivalent pre-strain for each tendon layer. These time-dependent parameters, however, were not expected to influence the mechanical strain during the ILRTs. These parameters were modelled for long-term performance prediction of an
3.3. Analysis of thermal strain component Although it is possible to run the fully coupled thermomechanical analysis which can determine the temperature history and strain history in the same model, the computational cost is much higher than the sequentially coupled analysis. For most civil structures, the changes in 5
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Nodal coordinate system
z
y
x
Vertical tendon Transverse bars are smeared in concrete Nodal coordinate system
Radial bars
y
x
Circumferential tendon
Circumferential bars 3 tendon layers
Vertical and horizontal bars Vertical and horizontal bars
z Circumferential bars Radial bars
(a) A shell element for upper dome
(b) A shell element for the perimeter wall
Fig. 4. Containment elements in VecTor4.
inputs for the heat transfer analysis. The weather history from Climate Canada during the test days are presented in Fig. 5. The temperature history of the containment through the thickness during the test days was calculated by the heat transfer model. The temperature history of north, south, east, and west at the mid-height of the perimeter wall are presented in Fig. 6. It can be observed that the internal surface temperature gradually decreases from 40 °C to 20 °C while the external surface temperature fluctuates over time due to the irradiation from the Sun. The fluctuating temperature of the external surface propagates less than halfway through the thickness of the containment wall. The fluctuation is lowest on the north side wall where the structure remains in the shade. The predicted temperature on the exterior surface fluctuates during the test days for all four locations. However, due to the azimuth of the Sun, the range of the fluctuations on each side is different. The surface temperature in all directions shows an overall increasing trend in the last two days and the fluctuations are more pronounced in the last two days due to the weather conditions. In the first two days, the exterior surface of the containment structure lost more heat due to the lower air temperature and higher wind speed as shown in Fig. 5, and gained less solar energy due to the cloudy sky condition. As a result, the temperature on the exterior surface kept increasing in the last two days and reached peaks at noon time. The thermal conductivity of concrete is relatively low, so neither the temperature fluctuation on the external surface nor the assumed decrease of internal surface temperature can penetrate all the way through the thickness of the cylindrical wall. At the point halfway through the wall thickness, the influence of the temperature fluctuation
aged containment structure. In addition to these time-dependent parameters, the containment was subjected to a load combination of gravity loads, pressure from the dousing water, prestressing forces and the internal pressure. In the numerical model, the compressive strength of concrete was assumed to be 34.5 MPa and the Poisson’s ratio was taken as 0.15. The yield strength for the rebars and the tendons were 414 MPa and 1400 MPa, respectively. More details about the VecTor4 model of the containment structure can be found in [23]. Due to the prestressing forces, the containment structure before and after considering the time-dependent parameters was still mainly in compression. It was found that tensile stresses were only predicted at the exterior surface of the dome elements close to the ring beam and the interior surface of the wall elements close to the top and bottom of the wall after the time parameters were included in the model [25]. In addition, only minor cracks were predicted at the bottom of the wall mainly due to the fixed base assumption in the VecTor4 model. These cracks had little impact on the global behavior of the containment structure. 4. Result and discussion 4.1. Heat transfer analysis results The numerical analysis results for the ILRT conducted in the year 2000 is discussed in detail in this section as a benchmark study. The weather history for calculating the heat exchange between the containment structure and the surrounding air is required as one of the
Fig. 5. Weather history during the test days of the year 2000. 6
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a) East Side
b) West Side
c) South Side
d) North Side
Fig. 6. The temperature profiles through wall thickness at mid-height for ILRT in 2000.
during the night of second day (between 24 h and 30 h after the test started) overestimate the decreasing of thermal strain. This error may have resulted from the assumption on the internal air temperature. During the leak rate test, the pressure was applied by blowing external air into the containment structure. The air from outside lowered the internal air temperature. During the time the pressure was held at a constant level the internal temperature should have a slower cooling rate than that during the period when the internal pressure was increasing. Therefore, the thermal strain is not expected to decrease when the pressure was held constant, or when the pressure was decreased. The thermal strain on the north side shows an overall decreasing trend. The north side is not directly exposed to the Sun, so it is more strongly influenced by the decrease of the internal temperature. There is a discrepancy between the numerical result and the measured result after the morning of the second day (13 h after the test started). In the numerical model low solar radiation and high convection coefficient is used due to the rainy and windy weather condition. As a consequence, the thermal strain on that day shows no peak. However, due to the wind direction, the convection coefficient on the north side may not be as high as other sides. With more appropriate coefficients and internal air temperature history, the discrepancy is expected to be diminished. On the south side, the numerical result does not show the peak on the second test day (20 h after the test started) due to the same reason as the north side. The measured strain shows a very sharp peak in the afternoons of the last two test days (around 45 h and 70 h after the test started). The numerical model captures the similar behavior but the peak values are lower. Material thermal properties and internal air temperature changing rate may be the reasons for the difference. On all three sides, there is a small discrepancy between the numerical result and the measured result for the points of thermal strain start to increase and decrease. This might be due to the shading effect of the surrounding structures blocking the solar beam and creating albedo effects. Also, due to the internal stress of the structure, inaccurate temperature history on one side can influence the result on the other
on the exterior surface is very minimum and the temperature at the point is almost a constant. Moreover, the temperature on the interior surface is not influenced by the temperature on the exterior surface but only depends on the interior air temperature. 4.2. Structural analysis results Although the temperature change on the exterior surface can not influence the temperature on the interior surface due to the low heat conductivity of concrete, the thermal effect on both surfaces interferes with the thermal strain of each other due to the internal stress in the concrete. The strain on the exterior surface was measured with LGFOS at the 2/3 height of the perimeter wall between the buttresses. The circumferential strains of elements at the corresponding locations from the numerical models are averaged and compared to the measurements. The predicted strain from the pressure model, the total strain (the sum of the strain from the pressure model and the train due to the thermal effects) from the numerical models and the measured strain from each LGFOS are presented in Fig. 7(a)–(c). The LGFOS located between the buttress 3 and 4 was not functional during the test, so there is no available measurement at this location. The thermal strain from the numerical analysis of each side is presented in Fig. 7(d). In the plots, the elongation in circumferential direction is noted as positive while shortening is noted as negative. Although the material properties and some thermal boundary conditions used in the numerical models are assumed, the numerical results show a very similar pattern to the measured results. Different locations along the circumference of the structure show distinctively different behavior due to the hourly change of the heat-influx from the Sun. Between buttress 1 and 2 (approximately facing east), the thermal strain started increasing around 8:00 am and started decreasing around 1:00 pm on each testing day. This pattern is clearly shown on the last two testing days (40 h after the test started) when the wind speed is relatively low and the sky is clear. However, the numerical results 7
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100
80
80 Micro Strain
Micro Strain
X. Wang, et al.
60 40 20 0 -20 0
60 40 20 0 -20 0
10 20 30 40 50 60 70 80 Elapsed Time (h)
10 20 30 40 50 60 70 80 Elapsed Time (h)
a) The strain between buttress 1 and 2
b) The strain between buttress 2 and 3
(Approximately facing east)
(Approximately facing south)
60 Micor Striain
Micro Strain
80
40 20 0 -20
0
10 20 30 40 50 60 70 80
30
East Side
20
South Side North Side
10 0 -10
-40
c) The strain between buttress 4 and 1
40
60
80
Elapsed Time (h)
d) Thermal Strain from Numerical Model for
(Approximately facing north) Numerical Result
20
-20 -30
Elapsed Time (h)
0
Each Side Measurement
Strain from Pressure Model
Fig. 7. Strain histories during the ILRT in the year 2000.
predict the strain on the containment structure due to both mechanical and thermal effects. It is also expected that the model can be used to predict the thermal effects during normal operation of the nuclear station. To further improve the accuracy of the temperature-induced strain using the procedure proposed in the paper, the following tasks are suggested:
three sides. From the above results, it can be concluded that the modeling strategy is capable of simulating the strain history, but the following assumptions can cause variances in results:
• The material properties of the containment structure. • Influence from the surrounding obstacles. • Weather condition, such as wind speed, humidity, and sky clearness. • The air temperature inside the containment during the ILRT.
– If further refinement of input data is required, conduct material tests to determine the thermal properties of the containment structure such as thermal expansion coefficient, thermal conductivity, specific heat, and solar absorptivity. – Monitor environmental data parameters during ILRTs to reduce the assumption on the environmental conditions.
The other three tests conducted in the years of 2004, 2012 and 2014 are also simulated numerically, but without calibrating the solar intensity, internal air temperature history or the material properties. The averaged circumferential strain from the numerical models and the site measurements are compared and presented in Fig. 8. Although some discrepancies can be observed between the numerical analysis result and the measured data, the patterns are similar. The matching in the pattern of the strain history indicates that the modelling strategy is correct, but the numerical predictions have to be further improved by eliminating the assumptions on the material properties and weather conditions in the numerical models.
It is expected that the pressure-induced strains, which is obtained by eliminating the temperature-induced strains from total measured strains during ILRTs, can be used to evaluate response of the containment structure to the applied pressure thereby confirming integrity of a containment structure. Declaration of Competing Interest
5. Conclusion and future works
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
During an ILRT, the strain gauges measured the total strain caused by the internal pressure and the thermal effects on the containment structure. The developed modeling strategy confirms that the fluctuation in the measurement is caused by thermal effect. Due to a lack of information, the numerical results and the measured data show some discrepancies, but the overall trend of the fluctuation is replicated. It is proven that the developed modeling technique has the potential to
Acknowledgment This study is sponsored by CANDU Owners Group (COG), Inc and the Collaborative Research and Development Grant of the Natural 8
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Sciences and Engineering Research Council of Canada. The involvement of COG Concrete Working Group is acknowledged.
[12] DiLaura DL, Burkett RJ, Clark F, Fink WL, Gillette G, Goodbar I, et al. Recommended practice for the calculation of daylight availability. J Illum Eng Soc 1984;13:381–92. https://doi.org/10.1080/00994480.1984.10748791. [13] Steven Winter Associates I. The passive solar design and construction handbook. John Wiley & Sons; 1998. [14] Gray M, Khan AA, Shenton B, Elgohary M. Measuring deformations in a candu 6 concrete containment structure. SMiRT 16, Washington DC, USA. 2001. [15] Huang X, Kwon O, Bentz E, Tcherner J. Effects of time-dependent parameters on the performance of a concrete containment structure. SMiRT 23, Manchester, UK. 2015. [16] Simulia DS. Abaqus 6.12 documentation. Provid Rhode Island, US; 2012, p. 6. https://doi.org/10.1097/TP.0b013e31822ca79b. [17] Government of Canada Environment and Natural Resources. Hourly Data Report for June 17, 2012; 2018. [18] Long T, Therese P, John L, Morgan J. Best practice guidelines for structural fire resistance design of concrete and steel buildings; 2010. [19] Bentz DP. A computer model to predict the surface temperature and time of wetness of concrete pavements and bridge decks. United States Dep Commer Technol Adm Natl Inst Stand Technol; 2000. [20] Vecchio FJ, Collins MP. The modified compression-field theory for reinforced concrete elements subjected to shear. ACI J 1986;83:219–31. [21] Hrynyk TD. Behavior and modelling of reinforced concrete slabs and shells under static and dynamic loads PhD. Thesis Department of Civil Engineering, University of Toronto; 2013. [22] Caputo RA. Non-Linear finite element analysis for an offshore oil reserve tank. University of Toronto; 2011. [23] Huang X, Kwon O-S, Bentz E, Tcherner J. Evaluation of CANDU NPP containment structure subjected to aging and internal pressure increase. Nucl Eng Des 2017;314:82–92. [24] Comité euro-international du béton. CEB-FIP model code 1990: design code. London: Thomas Telford; 1993. [25] Huang X, Kwon OS, Bentz E, Tcherner J. Method for evaluation of concrete containment structure subjected to earthquake excitation and internal pressure increase. Earthq Eng Struct Dyn 2018;47:1544–65. https://doi.org/10.1002/eqe. 3029.
References [1] CSA. CSA N287.3-14: Design requirements for concrete containment structures for nuclear power plants; 2014, p. 40. [2] CSA. CSA N287.7-17: In-service examination and testing requirements for concrete containment structures for nuclear power plants; 2017. [3] Li J, Liao K, Kong X, Li S, Zhang X, Zhao X, et al. Nuclear power plant prestressed concrete containment vessel structure monitoring during integrated leakage rate testing using fiber Bragg grating sensors. Appl Sci 2017;7. https://doi.org/10.3390/ app7040419. [4] Noh HG, Lee JH, Kang HC, Park HS. Effective thermal conductivity and diffusivity of containment wall for nuclear power plant OPR1000. Nucl Eng Technol 2017;49:459–65. https://doi.org/10.1016/j.net.2016.10.010. [5] Shin K-Y, Kim S-B, Kim J-H, Chung M, Jung P-S. Thermo-physical properties and transient heat transfer of concrete at elevated temperatures. Nucl Eng Des 2002;212:233–41. https://doi.org/10.1016/S0029-5493(01)00487-3. [6] Bhat PD, Vecchio F. Design of reinforced concrete containment structures for thermal gradients effects. SMiRT 7 Conf., Chicago, USA. 1983. [7] England GL. Creep and temperature effects in concrete structures: reality and prediction. Appl Math Model 1980;4:261–7. https://doi.org/10.1016/0307-904X(80) 90193-6. [8] Gurfinkel G. Thermal effects in walls of nuclear containments-elastic and inelastic behavior. SMiRT 1, Berlin, Germany. 1971. [9] Könönen M. Temperature induced stresses in a reactor containment building - a case study of Forsmark F1. Stockholm, Sweden: Royal Institute of Technology (KTH); 2012. [10] Guo L, Guo L, Zhong L, Zhu Y. Thermal conductivity and heat transfer coefficient of concrete. J Wuhan Univ Technol Sci Ed 2011;26:791–6. https://doi.org/10.1007/ s11595-011-0312-3. [11] Krane KS. Modern physics. 3rd ed. John Wiley & Sons; 1996.
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Evaluation of the thermal strain of an NPP containment structure during leakage rate tests
T
⁎
Xuguang Wanga, Xu Huanga, Oh-Sung Kwona, , Evan Bentza, Julia Tchernerb, Mark DeMerchantc, Mark Molandc a
Department of Civil and Mineral Engineering, University of Toronto, Canada Candu Energy, Inc. a member of SNC Lavalin Group, Canada c NB Power, Canada b
A R T I C LE I N FO
A B S T R A C T
Keywords: Thermal strain Containment structure Solar radiation
A strategy for isolating the thermal strain due to solar radiation from the total measured strain during the Integrated Leak Rate Test of a nuclear containment structure is presented in this paper. In the proposed method, a series of numerical models are developed to estimate the temperature change on the prestressed concrete containment structure and to predict the strain due to temperature and applied pressure. The effects of the solar radiation, heat exchange with internal and external air and heat conduction are considered in the proposed method. The developed method is validated by reproducing the total strain profiles measured from the leak rate tests conducted at the Point Lepreau CANDU®1 6 containment structure.
1. Introduction The prestressed concrete containment structures for nuclear power plants are designed to meet the required leak-tightness under operating and test conditions [1]. To ensure the structure meets the leak-tightness requirement over its service life, an Integrated Leak Rate Test (ILRT) is performed periodically subsequent to the first in-service use of the containment system [2]. During the ILRT, the concrete containment structure is pressurized from inside. The containment is expected to deform due to the pressure and develop corresponding strains, which can be used to estimate the uncracked stiffness of the structure and to determine if partial cracking has occurred in parts of the structure. Because the applied pressure is within the linear elastic range of the containment structure, the developed strains are expected to be proportional to the applied pressure. However, non-negligible level of strain fluctuations have been observed in the ILRTs. Based on the daily pattern of strain fluctuation and smaller fluctuation at the north face of the containment structure, the source of the strain fluctuations are expected to be the temperature changes on the exterior surfaces of the containment structure, particularly due to the solar irradiance. Although some of the strain gauges and sensors used for measuring strains during the ILRTs can compensate for the thermal effects, such as
the fiber Bragg grating sensors used by Li et al. [3], only the effects of temperature on the strain gauges are compensated. Consequently, the measured strain is the total strain of concrete which consists of mechanical strain resulting from internal pressure increase and thermal strain resulting from the temperature variation during the tests. It is difficult to directly measure the thermal strain of the containment structure due to the non-uniform distribution of temperature through the structure’s thickness and height. Therefore, heat transfer analysis is required for obtaining an estimate of the temperature history and structural analysis is required for calculating the thermal strains during the ILRT. In the past, heat transfer analysis was used to obtain the thermal properties of the concrete containment structures [4,5] and investigate long-term and short-term behaviors of the structures subjected to thermal loads during the operational period [6–9]. However, few studies have been conducted on the analysis of the thermal effects with the consideration of the realistic thermal boundary conditions during the IRLT. To bridge the gap, a set of numerical models for predicting the thermal effects and isolating the thermal strains from the mechanical strains of the containment structure during the IRLT is developed in this study. The influence of the historical weather conditions and the solar radiation is taken into account in the numerical model. The developed
⁎
Corresponding author. E-mail address: [email protected] (O.-S. Kwon). 1 CANDU (CANada Deuterium Uranium Reactor) is a registered trademark of the Atomic Energy of Canada Limited (AECL), used under exclusive license by Candu Energy Inc. https://doi.org/10.1016/j.engstruct.2019.109761 Received 17 April 2019; Received in revised form 24 August 2019; Accepted 5 October 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.
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the simulation of convection and heat radiation. In addition to convection and heat radiation, solar radiation is another source of temperature change on the exterior surface of the containment structure. Radiation from the Sun influences the surface temperature of the containment structure significantly and causes nonuniformly distributed temperature on the exterior surface of the containment structure due to the temporal changes of the projection angle and the intensity of the radiation. Thus, both the intensity and the relative angles between the Sun and points on the exterior surface of the cylindrical containment structure should be considered when calculating the heat influx from solar radiation. To get the relative angles, the information of the Sun’s path which is described by the altitude, α , and azimuth, θ , at the coordinate of the containment structure needs to be considered. The solar angle information, for example, can be obtained from The United States Naval Observatory (USNO) website. The intensity of heat irradiation from the Sun can be calculated using empirical models. In this approach, the model developed by IES [12] is employed. In the model, the intensity of direct solar irradiation, Id , that reaches the ground after passing through the atmospheres is expressed as:
method is applied to the CANDU 6 reactor system at Point Lepreau and compared to the test results of ILRTs conducted in 2000, 2004, 2012 and 2014. 2. Modeling the containment structure during ILRT To analyze and isolate the thermal effects of the containment structure during an ILRT, a set of three numerical models are required. A heat transfer analysis is performed in one of the numerical models while structural analyses are performed in the other two. The heat transfer model uses the air temperature, weather condition, and solar irradiation during the test period to predict the temperature history of the containment structure. In order to isolate the strain caused by the thermal effects from the total strain, the result of the heat transfer analysis and the pressure are applied to the second and third models, respectively, for structural analysis. With the assumption that the containment structure behaves within the linear elastic range, the analysis results from the second and the third models can be superimposed to get the total stresses and strains over the ILRT. The details of the modeling strategy in each model are presented in the following sections.
Id = I0 exp (−cm)
2.1. Heat transfer analysis
where c is the atmospheric extinction coefficient that represents the sky clearness conditions, m is the optical air mass, and I0 is the extraterrestrial solar illuminance. In the model, the atmospheric extinction coefficient is suggested as 0.21 for clear sky and 0.8 for partly cloudy conditions. The optical air mass can be approximated as:
The heat transfer analysis considers the heat exchange between the containment structure and the internal/external environment through radiation and convection, as well as the heat conduction through the thickness of the containment structure. The overall heat flow rate on the surface of the containment structure is the summation of the heat flow rates from the convection, heat radiation, and solar irradiance effects. The total heat flow rate, qtot , can be expressed as:
qtot = qc + qrad + qsolar
m=
(5)
2π I0 = Isc ⎛1 + 0.034cos ⎛ (n − 2) ⎞ ⎞ ⎝ 365 ⎠⎠ ⎝
(6) 2
where Isc , the solar constant is 1366 W/m , and n is the day of the year, for example, n = 1 for January 1st. From the geometry, the solar radiation normal to a panel can be expressed as:
(2)
In = Idsin(αp−α )sin(θp − θ)
where Tc is the temperature at the surface of the containment, Ta is the temperature of the surrounding air, A is the surface area, qc is the heat transferred per unit time, and h is the convection coefficient depending on the Rayleigh number and Prandtl number of the surrounding air. The values of the Rayleigh number and the Prandtl number are affected by the wind speed and humidity. Thus, the convection coefficient can also be obtained by using the regression model of the experimental work using wind speed and humidity as variables [10]. The other heat exchange mechanism is thermal radiation where the temperature of an object changes due to the transfer of energy via electromagnetic waves. The mathematical expression of the rate of heat flow due to the radiation, qrad , is:
qrad = εσA (Tc 4 − Ta4 )
1 sinα
and I0 is expressed as:
(1)
where qc , qrad , and qsolar are the rate of heat flow caused by convection, heat radiation, and solar irradiance. The rate of heat flow due to heat convectionqc , is mathematically expressed as:
qc = hA (Tc − Ta)
(4)
(7)
where αp is the angle of the panel respect to the ground and θp is the angle between the panel surface and the true north as shown in Fig. 1. Hence, the rate of heat flow caused by the solar radiation, qsolar , is:
qsolar = αab I n A
(8)
where A is the surface area of an element and αab is the solar absorptivity of the concrete containment, which can be taken as 0.65 as recommended by [13]. Although the containment structure is surrounded by the service building, the height of the service building is generally shorter than one quarter of the height of the containment structure and the gauges to measure the circumferential strain are equipped at the mid-height of the containment structure. The shading and albedo effects may influence the strain change at the bottom of the structure, but the influence to the strain measurement at the mid-height is minor. Thus, the shading and albedo effects from surrounding buildings are not considered in the model. The change of the surface temperature, ΔT , can be related using the following equation:
(3)
where ε is the emissivity of the material, qrad is the heat flow rate, A is the surface area, Tc and Ta are the surface temperature of the object that emits heat and the temperature of the surrounding environment respectively, and σ is the Stefan-Boltzmann constant which is W 5.670 × 10−8 2 ° 4 [11]. The total heat flux via radiation is the sum of m C absorption and emission. Since the containment structure is not surrounded by any high temperature objects, the heat influx from absorbing the radiation of the surrounding objects is negligible compared with the heat radiation from the Sun. Therefore, the radiation effect in the model only refers to the heat emission from the containment structure when the surrounding air temperature is lower than its surface temperature. The record of actual air temperature history, wind speed, and relative humidity of the air, if available, should be used for
ΔT =
qtot Δt Vρcp
(9)
where Δt is the time step, V is the volume of an element, ρ and cp are the density and the specific heat of concrete, respectively. The heat transfers through the thickness of the containment wall via heat conduction. For a small element of the containment structure, the general heat equation for heat conduction is 2
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Fig. 1. Illustration of the solar radiation on the containment.
∂T k 2 = ∇T ∂t ρcp
3. Solar thermal-stress analysis for nuclear power plants (10) 3.1. Background and the previous ILRTs
where ∂T is the rate of change of temperature with respect to time at of ∂t the point of interest, k , ρ and cp are the thermal conductivity, density, and specific heat of the material, and ∇2 T is the second derivative of the temperature in all directions in the 3-D space. As shown in the equation, the rate of temperature change depends on the properties of the material. The containment structure is composed of different materials including concrete, steel reinforcement, and tendons. Therefore, an accurate thermal conductivity of the composite material should be obtained through experiments.
The Point Lepreau CANDU® 6 reactor system has been in service for more than 30 years. The modelling approach described previously is used for replicating the strain measurement from the ILRTs conducted in the years 2000, 2004, 2012 and 2014. To measure the strain changes during the tests, four 27.5 m long-gauge fiber optic sensors (LGFOS) are installed in the circumferential direction, and eleven (eight primary and three secondary) clusters of strain gauges were installed before the year 2000 [14]. The LGFOS readings are based on the changes of the refractive index of the embedded Fibber Bragg Grating (FBG). The measurements during the ILRTs are recorded in mm and converted to microstrain as follows:
2.2. Structural analysis The results from the heat transfer analysis are temperature histories at various points of the containment structure, and through the thickness of the containment wall. The temperature histories of the containment structure over the testing period are applied to the structural analysis model as thermal boundary conditions. For the purpose of isolating the thermal strain from the total strain, other loads, such as gravity load and the internal pressure, are applied in a separate structural analysis model without considering the thermal effects. When no cracks are observed during the test, the total strain, εtot , can be expressed as:
εtot = εth + εp
εm =
R1 − R2 × 106 27.5
(12)
where εm is the measured strain, R1 is the initial reading at the beginning of the ILRT, and R2 is the incremental reading during the test. The measured results showed that the strains during each test period significantly fluctuated over the course of the test. It was concluded that the observed fluctuations were due to the strain component resulting from the thermal effects [14]. In a previous numerical analysis of the containment structure subjected to the ILRT in 2000, the timedependent strain component resulting from temperature changes were not considered [15]. Thus, the numerically predicted strain history did not show any fluctuation over the testing period. If the thermal strain component of the observed total strain can be accurately predicted, the strain due to the pressure increase can be isolated, which is expected to lead to a better understanding of the structural performance of the containment structure. In the following sections, the strain history of the CANDU® 6 containment structure during the ILRTs, including both thermal and mechanical strain components, are evaluated based on the analysis method discussed in Section 2. The leak-rate test in 2000 is studied as a benchmark to verify the modeling technique for thermal and mechanical analyses. Numerical models for the tests in other years are developed by changing pressure and external weather histories corresponding to the testing conditions of each ILRT. The year 2000 ILRT started on August 23, 2000, around 7:00 p.m. and lasted for about 80 h. The weather of the first two days was cloudy and wet, so the solar radiation did not significantly affect the measuring results; however, over the last two days, the influence of the heat radiation from the Sun
(11)
where εth is the thermal strain due to temperature variation during the test and εp is the strain due to all other loads. The εp of the containment structure can be calculated from the pressure model as described in Section 3.4. The thermal strain in Eq (11) is calculated by imposing the temperature history from the heat transfer analysis as thermal boundary conditions on a structural analysis model as described in Section 3.3. The thermal expansion of the material is specified in the model for computing thermal strains. Ideally, the thermal expansion coefficient of the composite material should be obtained from material tests, but the values are expected to be close to the thermal expansion coefficient of the concrete due to the significantly larger volume of concrete in comparison to the volume of reinforcement bars or tendons. The ILRT typically takes less than a week, so the strains due to longterm parameters, such as shrinkage, creep and relaxation, are not considered in the model. The main input parameters and analysis framework for predicting the total strain are summarized in Fig. 2. 3
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Fig. 2. Variables for the Proposed Analysis Strategy.
directions of the containment structure are expected to be small due to the low thermal conductivity of concrete while the temperature gradient through the wall thickness is more pronounced due to the temperature difference on the exterior and interior surfaces. Thus, the structure is coarsely meshed in the vertical and circumferential directions and meshed much more finely through the wall thickness to evaluate heat transfer through the thickness of the structure and to determine the corresponding strain profile. At each location on the perimeter wall and the dome, there are 20 layers through the thickness to capture the temperature gradient. The numerical model simulates the heat transfer effects from two days (48 h) before the test starting time to ensure the temperature of the structure is in a steady state condition. It is assumed that, from the time the test is started, the internal air temperature decreases linearly over the test period from 40 °C to 20 °C due to the cooling of the reactor building. Historical air temperature is used for simulating the convection and radiation effects on the exterior surface. The convection coefficient, h , in Eq. (2) is calculated using an empirical regression
was significant. 3.2. Heat transfer model and assumptions The heat transfer analysis was conducted using ABAQUS [16]. The geometry and the global coordinate system of the model are shown in Fig. 3(a). The containment structure has a cylindrical shape. On the top of the perimeter wall, there is a ring beam which supports a dome. On the perimeter wall, there are four buttresses as shown in Fig. 3(b). For simplification, the buttresses are not included in the numerical model as they are far away from the location of the strain measurements during the ILRTs. The orientation of the model is defined for calculating the relative angle when simulating the solar radiation. The base slab of the containment structure lays in the x-z plane. The center of the top of the base slab is defined as the origin with the coordinates of (0, 0, 0) and the positive z-direction is defined as due south. The heat transfer analysis was performed with 8-node linear heat transfer brick elements. The temperature gradients along the height and circumferential
Fig. 3. Geometry and coordinate system of the numerical model. (a) Elevation view of the containment structure, (b) Plan view of the containment structure. 4
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model by Guo et al. [10] as follows:
h=
0.9337v 2
− 1.15v + 7.365Rh + 3.937
Table 1 Thermal properties of the heat transfer model.
(13)
where v is the wind speed in m/s, and Rh is the relative humidity of air in percent. The air temperature, wind speed and the relative humidity of air are adopted from the hourly historical data collected by Point Lepreau CS weather station (Climate ID: 8104201) [17]. The wind speed is assumed to be uniform along the height of the containment structure and the direction of the wind is not taken into account. The calculated convection coefficient, h , during the test varies in a wide range. During windy days, the convection coefficient can be much higher than the average value during the test. For example, during the test in 2000, the peak value of convection coefficient reaches 7833 J/ min/m2, but the average value is 1057 J/min/m2. An ABAQUS user subroutine was developed to simulate the direction of solar radiation using Eqs. (4)–(8). In the subroutine, the solar radiation is defined as heat influx at each node on the exterior surface in the model. The αp for all nodes on the perimeter wall is set to 90° while the αp for the dome is calculated using the following equation:
αp = tan−1
(14)
where x , y, z are the coordinates of a node on the dome and x 0,y0,z 0 are the coordinates of the point on the center line of the containment structure at some elevation from the base slab. The θp of each node is calculated as:
θp =
π x + tan−1 2 −z
Value
Unit
Conductivity, k Density, ρ Specific heat, cp Emissivity, ε
120 2400 800
J/min/m/°C kg/m3 J/kg/°C
0.95
–
stress and strain do not influence the results from heat transfer analysis. Therefore, the strain on the containment structure due to the thermal effects can be obtained by imposing the results from the heat transfer model mentioned in Section 3.2. From the results of the heat transfer analysis, the temperature history at each node on the containment structure has been determined. The mesh for the structural analysis model is not necessarily the same as the heat transfer model. When the mesh of structural analysis model and heat transfer model are different, the nodal temperature on the structural analysis model can be gained by linear interpolation. In this practice, the same model geometry and mesh from the heat transfer model was used for the thermal stress analysis. An 8-node linear brick element was selected for the analysis. The temperature history of each node from the heat transfer analysis is applied as the thermal boundary condition. A fixed boundary condition is assigned at the bottom of the containment wall. With such a boundary condition, the predicted stress and strain near the bottom will be higher than the partially fixed boundary condition, but the impact on the global behavior is minor. Because the objective of the study is to predict global fluctuation of thermal strain at an elevation far from the support, the above assumption on the support deemed appropriate. At each time step, the structure analysis model calculates the thermal strain based on the thermal expansion caused by nodal temperature and mechanical properties of the containment structure. The deformations caused by the thermal effects are assumed to be within the linear elastic range. Due to the large volumetric ratio of concrete to steel, the material properties of concrete are used in the model. Thus, the elastic modulus and the thermal expansion coefficient are 34.5 GPa and 10−5/ °C respectively.
(y − y0 ) (x − x 0)2 + (z − z 0 )2
Thermal Properties
(15)
According to the test report and the historical temperature record, the first two days of the test (August 23 and August 24, 2000) were cloudy and wet, and the sky was clear for the following two days. Therefore, the solar intensity of the following two days is expected to be higher than the solar intensity of the previous days. The atmosphere extinction coefficient in Eq. (4) is used to adjust the intensity. Instead of setting the extinction coefficient to the suggested values by IES, it is calibrated to produce a final result that best matches the measurement for the year 2000 IRLT. The main reason for this calibration is due to the lack of knowledge of actual cloud conditions during the ILRT. Heat conduction through the thickness of the structure is calculated by solving Eq. (10) numerically. Compared to the solar irradiance and the heat exchange with the surrounding air, the heat exchange between the bottom of the containment structure and soil has a small influence on the temperature history over the test period especially at the locations where the sensors are installed. Therefore, by assuming no heat transfer between the containment structure and the soil, an insulated thermal boundary condition is assigned at the bottom of the containment structure. The thermal properties from material tests are not available. The typical concrete conductivity and specific heat, as suggested in [18], are 84 J/min/m/oC and 900 J/kg/oC, while typical values for steels are 2700 J/min/m/oC and 600 J/kg/oC respectively. Considering the high percentage of the embedded steel, the specific heat and the conductivity values used in the numerical models are calibrated with trial-and-error approach. Both values are between the suggested values of concrete and steel and able to produce reasonable results. The emissivity of the concrete surface is based on the value suggested in [19] and calibrated to simulate more active thermal transfer effects. The general thermal properties for concrete for the analysis are summarized in Table 1.
3.4. Analysis of pressure dependent strain component The strain changes due to temperature-independent parameters, such as gravity loads, prestressing forces, and the internal pressure increase were predicted separately from a detailed finite element model of the containment structure using the VecTor4 program. The program is based on the Modified Compression Field Theory (MCFT) [20] and is specialized for nonlinear analysis of reinforced concrete shells [21,22]. In the numerical model, the containment was modelled with heterosis shell elements, each of which had 9 nodes and 42 degrees of freedom. The in-plane transverse and longitudinal reinforcement and tendons were defined as smeared layers which were superposed onto the concrete layers, while the out-of-plane reinforcement was defined as a property of the concrete layers. Fig. 4 shows example elements at the dome and the perimeter wall where the layers of the reinforcement and tendons are shaded. More details about the program can be found in [21]. As proposed by the authors [23], the aging effects due to time-dependent parameters, such as shrinkage and creep of concrete, and relaxation of prestressing force were taken into account in the VecTor4 model. Specifically, strain changes due to concrete creep and shrinkage were firstly predicted from the CEB/FIP 90 model [24] and then were applied to the model as equivalent temperature loads. The relaxation of prestressing steel was modelled by applying an equivalent pre-strain for each tendon layer. These time-dependent parameters, however, were not expected to influence the mechanical strain during the ILRTs. These parameters were modelled for long-term performance prediction of an
3.3. Analysis of thermal strain component Although it is possible to run the fully coupled thermomechanical analysis which can determine the temperature history and strain history in the same model, the computational cost is much higher than the sequentially coupled analysis. For most civil structures, the changes in 5
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Nodal coordinate system
z
y
x
Vertical tendon Transverse bars are smeared in concrete Nodal coordinate system
Radial bars
y
x
Circumferential tendon
Circumferential bars 3 tendon layers
Vertical and horizontal bars Vertical and horizontal bars
z Circumferential bars Radial bars
(a) A shell element for upper dome
(b) A shell element for the perimeter wall
Fig. 4. Containment elements in VecTor4.
inputs for the heat transfer analysis. The weather history from Climate Canada during the test days are presented in Fig. 5. The temperature history of the containment through the thickness during the test days was calculated by the heat transfer model. The temperature history of north, south, east, and west at the mid-height of the perimeter wall are presented in Fig. 6. It can be observed that the internal surface temperature gradually decreases from 40 °C to 20 °C while the external surface temperature fluctuates over time due to the irradiation from the Sun. The fluctuating temperature of the external surface propagates less than halfway through the thickness of the containment wall. The fluctuation is lowest on the north side wall where the structure remains in the shade. The predicted temperature on the exterior surface fluctuates during the test days for all four locations. However, due to the azimuth of the Sun, the range of the fluctuations on each side is different. The surface temperature in all directions shows an overall increasing trend in the last two days and the fluctuations are more pronounced in the last two days due to the weather conditions. In the first two days, the exterior surface of the containment structure lost more heat due to the lower air temperature and higher wind speed as shown in Fig. 5, and gained less solar energy due to the cloudy sky condition. As a result, the temperature on the exterior surface kept increasing in the last two days and reached peaks at noon time. The thermal conductivity of concrete is relatively low, so neither the temperature fluctuation on the external surface nor the assumed decrease of internal surface temperature can penetrate all the way through the thickness of the cylindrical wall. At the point halfway through the wall thickness, the influence of the temperature fluctuation
aged containment structure. In addition to these time-dependent parameters, the containment was subjected to a load combination of gravity loads, pressure from the dousing water, prestressing forces and the internal pressure. In the numerical model, the compressive strength of concrete was assumed to be 34.5 MPa and the Poisson’s ratio was taken as 0.15. The yield strength for the rebars and the tendons were 414 MPa and 1400 MPa, respectively. More details about the VecTor4 model of the containment structure can be found in [23]. Due to the prestressing forces, the containment structure before and after considering the time-dependent parameters was still mainly in compression. It was found that tensile stresses were only predicted at the exterior surface of the dome elements close to the ring beam and the interior surface of the wall elements close to the top and bottom of the wall after the time parameters were included in the model [25]. In addition, only minor cracks were predicted at the bottom of the wall mainly due to the fixed base assumption in the VecTor4 model. These cracks had little impact on the global behavior of the containment structure. 4. Result and discussion 4.1. Heat transfer analysis results The numerical analysis results for the ILRT conducted in the year 2000 is discussed in detail in this section as a benchmark study. The weather history for calculating the heat exchange between the containment structure and the surrounding air is required as one of the
Fig. 5. Weather history during the test days of the year 2000. 6
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a) East Side
b) West Side
c) South Side
d) North Side
Fig. 6. The temperature profiles through wall thickness at mid-height for ILRT in 2000.
during the night of second day (between 24 h and 30 h after the test started) overestimate the decreasing of thermal strain. This error may have resulted from the assumption on the internal air temperature. During the leak rate test, the pressure was applied by blowing external air into the containment structure. The air from outside lowered the internal air temperature. During the time the pressure was held at a constant level the internal temperature should have a slower cooling rate than that during the period when the internal pressure was increasing. Therefore, the thermal strain is not expected to decrease when the pressure was held constant, or when the pressure was decreased. The thermal strain on the north side shows an overall decreasing trend. The north side is not directly exposed to the Sun, so it is more strongly influenced by the decrease of the internal temperature. There is a discrepancy between the numerical result and the measured result after the morning of the second day (13 h after the test started). In the numerical model low solar radiation and high convection coefficient is used due to the rainy and windy weather condition. As a consequence, the thermal strain on that day shows no peak. However, due to the wind direction, the convection coefficient on the north side may not be as high as other sides. With more appropriate coefficients and internal air temperature history, the discrepancy is expected to be diminished. On the south side, the numerical result does not show the peak on the second test day (20 h after the test started) due to the same reason as the north side. The measured strain shows a very sharp peak in the afternoons of the last two test days (around 45 h and 70 h after the test started). The numerical model captures the similar behavior but the peak values are lower. Material thermal properties and internal air temperature changing rate may be the reasons for the difference. On all three sides, there is a small discrepancy between the numerical result and the measured result for the points of thermal strain start to increase and decrease. This might be due to the shading effect of the surrounding structures blocking the solar beam and creating albedo effects. Also, due to the internal stress of the structure, inaccurate temperature history on one side can influence the result on the other
on the exterior surface is very minimum and the temperature at the point is almost a constant. Moreover, the temperature on the interior surface is not influenced by the temperature on the exterior surface but only depends on the interior air temperature. 4.2. Structural analysis results Although the temperature change on the exterior surface can not influence the temperature on the interior surface due to the low heat conductivity of concrete, the thermal effect on both surfaces interferes with the thermal strain of each other due to the internal stress in the concrete. The strain on the exterior surface was measured with LGFOS at the 2/3 height of the perimeter wall between the buttresses. The circumferential strains of elements at the corresponding locations from the numerical models are averaged and compared to the measurements. The predicted strain from the pressure model, the total strain (the sum of the strain from the pressure model and the train due to the thermal effects) from the numerical models and the measured strain from each LGFOS are presented in Fig. 7(a)–(c). The LGFOS located between the buttress 3 and 4 was not functional during the test, so there is no available measurement at this location. The thermal strain from the numerical analysis of each side is presented in Fig. 7(d). In the plots, the elongation in circumferential direction is noted as positive while shortening is noted as negative. Although the material properties and some thermal boundary conditions used in the numerical models are assumed, the numerical results show a very similar pattern to the measured results. Different locations along the circumference of the structure show distinctively different behavior due to the hourly change of the heat-influx from the Sun. Between buttress 1 and 2 (approximately facing east), the thermal strain started increasing around 8:00 am and started decreasing around 1:00 pm on each testing day. This pattern is clearly shown on the last two testing days (40 h after the test started) when the wind speed is relatively low and the sky is clear. However, the numerical results 7
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predict the strain on the containment structure due to both mechanical and thermal effects. It is also expected that the model can be used to predict the thermal effects during normal operation of the nuclear station. To further improve the accuracy of the temperature-induced strain using the procedure proposed in the paper, the following tasks are suggested:
three sides. From the above results, it can be concluded that the modeling strategy is capable of simulating the strain history, but the following assumptions can cause variances in results:
• The material properties of the containment structure. • Influence from the surrounding obstacles. • Weather condition, such as wind speed, humidity, and sky clearness. • The air temperature inside the containment during the ILRT.
– If further refinement of input data is required, conduct material tests to determine the thermal properties of the containment structure such as thermal expansion coefficient, thermal conductivity, specific heat, and solar absorptivity. – Monitor environmental data parameters during ILRTs to reduce the assumption on the environmental conditions.
The other three tests conducted in the years of 2004, 2012 and 2014 are also simulated numerically, but without calibrating the solar intensity, internal air temperature history or the material properties. The averaged circumferential strain from the numerical models and the site measurements are compared and presented in Fig. 8. Although some discrepancies can be observed between the numerical analysis result and the measured data, the patterns are similar. The matching in the pattern of the strain history indicates that the modelling strategy is correct, but the numerical predictions have to be further improved by eliminating the assumptions on the material properties and weather conditions in the numerical models.
It is expected that the pressure-induced strains, which is obtained by eliminating the temperature-induced strains from total measured strains during ILRTs, can be used to evaluate response of the containment structure to the applied pressure thereby confirming integrity of a containment structure. Declaration of Competing Interest
5. Conclusion and future works
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
During an ILRT, the strain gauges measured the total strain caused by the internal pressure and the thermal effects on the containment structure. The developed modeling strategy confirms that the fluctuation in the measurement is caused by thermal effect. Due to a lack of information, the numerical results and the measured data show some discrepancies, but the overall trend of the fluctuation is replicated. It is proven that the developed modeling technique has the potential to
Acknowledgment This study is sponsored by CANDU Owners Group (COG), Inc and the Collaborative Research and Development Grant of the Natural 8
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Sciences and Engineering Research Council of Canada. The involvement of COG Concrete Working Group is acknowledged.
[12] DiLaura DL, Burkett RJ, Clark F, Fink WL, Gillette G, Goodbar I, et al. Recommended practice for the calculation of daylight availability. J Illum Eng Soc 1984;13:381–92. https://doi.org/10.1080/00994480.1984.10748791. [13] Steven Winter Associates I. The passive solar design and construction handbook. John Wiley & Sons; 1998. [14] Gray M, Khan AA, Shenton B, Elgohary M. Measuring deformations in a candu 6 concrete containment structure. SMiRT 16, Washington DC, USA. 2001. [15] Huang X, Kwon O, Bentz E, Tcherner J. Effects of time-dependent parameters on the performance of a concrete containment structure. SMiRT 23, Manchester, UK. 2015. [16] Simulia DS. Abaqus 6.12 documentation. Provid Rhode Island, US; 2012, p. 6. https://doi.org/10.1097/TP.0b013e31822ca79b. [17] Government of Canada Environment and Natural Resources. Hourly Data Report for June 17, 2012; 2018. [18] Long T, Therese P, John L, Morgan J. Best practice guidelines for structural fire resistance design of concrete and steel buildings; 2010. [19] Bentz DP. A computer model to predict the surface temperature and time of wetness of concrete pavements and bridge decks. United States Dep Commer Technol Adm Natl Inst Stand Technol; 2000. [20] Vecchio FJ, Collins MP. The modified compression-field theory for reinforced concrete elements subjected to shear. ACI J 1986;83:219–31. [21] Hrynyk TD. Behavior and modelling of reinforced concrete slabs and shells under static and dynamic loads PhD. Thesis Department of Civil Engineering, University of Toronto; 2013. [22] Caputo RA. Non-Linear finite element analysis for an offshore oil reserve tank. University of Toronto; 2011. [23] Huang X, Kwon O-S, Bentz E, Tcherner J. Evaluation of CANDU NPP containment structure subjected to aging and internal pressure increase. Nucl Eng Des 2017;314:82–92. [24] Comité euro-international du béton. CEB-FIP model code 1990: design code. London: Thomas Telford; 1993. [25] Huang X, Kwon OS, Bentz E, Tcherner J. Method for evaluation of concrete containment structure subjected to earthquake excitation and internal pressure increase. Earthq Eng Struct Dyn 2018;47:1544–65. https://doi.org/10.1002/eqe. 3029.
References [1] CSA. CSA N287.3-14: Design requirements for concrete containment structures for nuclear power plants; 2014, p. 40. [2] CSA. CSA N287.7-17: In-service examination and testing requirements for concrete containment structures for nuclear power plants; 2017. [3] Li J, Liao K, Kong X, Li S, Zhang X, Zhao X, et al. Nuclear power plant prestressed concrete containment vessel structure monitoring during integrated leakage rate testing using fiber Bragg grating sensors. Appl Sci 2017;7. https://doi.org/10.3390/ app7040419. [4] Noh HG, Lee JH, Kang HC, Park HS. Effective thermal conductivity and diffusivity of containment wall for nuclear power plant OPR1000. Nucl Eng Technol 2017;49:459–65. https://doi.org/10.1016/j.net.2016.10.010. [5] Shin K-Y, Kim S-B, Kim J-H, Chung M, Jung P-S. Thermo-physical properties and transient heat transfer of concrete at elevated temperatures. Nucl Eng Des 2002;212:233–41. https://doi.org/10.1016/S0029-5493(01)00487-3. [6] Bhat PD, Vecchio F. Design of reinforced concrete containment structures for thermal gradients effects. SMiRT 7 Conf., Chicago, USA. 1983. [7] England GL. Creep and temperature effects in concrete structures: reality and prediction. Appl Math Model 1980;4:261–7. https://doi.org/10.1016/0307-904X(80) 90193-6. [8] Gurfinkel G. Thermal effects in walls of nuclear containments-elastic and inelastic behavior. SMiRT 1, Berlin, Germany. 1971. [9] Könönen M. Temperature induced stresses in a reactor containment building - a case study of Forsmark F1. Stockholm, Sweden: Royal Institute of Technology (KTH); 2012. [10] Guo L, Guo L, Zhong L, Zhu Y. Thermal conductivity and heat transfer coefficient of concrete. J Wuhan Univ Technol Sci Ed 2011;26:791–6. https://doi.org/10.1007/ s11595-011-0312-3. [11] Krane KS. Modern physics. 3rd ed. John Wiley & Sons; 1996.
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