A comprehensive flow-induced vibration model to predict crack growth and leakage potential in steam generator tubes

Nuclear Engineering and Design 292 (2015) 17–31

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Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes

A comprehensive flow-induced vibration model to predict crack growth and leakage potential in steam generator tubes Salim El Bouzidi a , Marwan Hassan a,∗ , Jovica Riznic b a b

School of Engineering, University of Guelph, Guelph, Ontario N1G 2W1, Canada Operational Engineering Assessment Division, Canadian Nuclear Safety Commission, Ottawa, Ontario K1P 5S9, Canada

h i g h l i g h t s • • • • •

Comprehensive flow induced vibrations time domain model was developed. Simulations of fluidelastic instability and turbulence were conducted. Nonlinear effect due to the clearances at the supports was studied. Prediction of stresses due to fluid excitation was obtained. Deterministic and stochastic analyses for crack and leakage rate were conducted.

a r t i c l e

i n f o

Article history: Received 28 August 2014 Received in revised form 14 March 2015 Accepted 28 April 2015

a b s t r a c t Flow-induced vibrations (FIVs) are a major threat to the operation of nuclear steam generators. Turbulence and fluidelastic instability are the two main excitation mechanisms leading to tube vibrations. The consequences to the operation of steam generators are premature wear of the tubes, as well as development of cracks that may leak hazardous fluids. This paper investigates the effect of tube support clearance on the integrity of tube bundles within steam generators. Special emphasis will be placed on crack propagation and leakage rates. A crack growth model is used to simulate the growth of surface flaws and through-wall cracks of various initial sizes due to a wide range of support clearances. Leakage rates are predicted using a two-phase flow leakage model. Nonlinear finite element analysis is used to simulate a full U-bend subjected to fluidelastic and turbulence forces. Monte Carlo simulations are then used to conduct a probabilistic assessment of steam generator life due to crack development. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Tube-in-shell steam generators are widely used components in power plants. They provide a closed interface between the heat source and the steam cycle. Their integrity is very important as they provide a barrier between the two fluids. This is especially important in the case of nuclear power plants where the mixing of the two fluids may constitute an environmental hazard. A great deal of research has been conducted over the past half century to investigate and understand the main causes of premature wear and shortened life of steam generators. The main excitation sources in steam generators are turbulence, vortex shedding, and fluidelastic instability. Turbulence is a

∗ Corresponding author. Tel.: +1 519824 4120. E-mail address: [email protected] (M. Hassan). http://dx.doi.org/10.1016/j.nucengdes.2015.04.040 0029-5493/© 2015 Elsevier B.V. All rights reserved.

random excitation mechanism resulting in low amplitude vibrations. On the other hand, vortex shedding is a well characterized periodic excitation whose frequency is linearly related to the flow velocity. In liquid flows, it becomes a serious issue if the periodicity of the flow coincides with the natural frequency of the structure; at such point, there is a lock-in between the fluid and structure resulting in structural resonance. Finally, fluidelastic instability (FEI) is a self excited phenomenon that manifests itself in a large amplitude displacement when a certain critical velocity threshold (Ucr ) is exceeded. The latter has been the subject of an extensive amount of research over the past half century, resulting in various types of models to predict its onset (Weaver, 2008). While the effects of vortex induced vibration (VIV) can be avoided by operating the steam generator away from known resonance peaks, turbulence cannot be suppressed as it is essential to the operation of a steam generator since it enhances heat transfer. Turbulence can hence result in long term wear due to fretting and impact at the tube supports

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Nomenclature A0 A(s, t) a a0 [C] Cc Cri COD dh E {Fimp } {FFEI } {Fturb } h2 K Kth L K [K] Kc KI [M] N P Pi Po Pw rm s U Ur

vm

{w} ˙ {w} ¨ {w} ıe ıp  b m t w

flow channel inlet area flow channel total area surface crack depth through-wall crack initial half length damping matrix contact damping radial clearance crack opening displacement hydraulic diameter Young’s modulus contact force vector fluidelastic force vector turbulence force vector plastic influence function stress intensity factor threshold stress intensity factor surface crack length threshold minimum stress intensity factor for crack growth stiffness matrix contact stiffness crack tip stress intensity factor mass matrix number of sinusoidal stress cycles applied to crack pressure inlet pressure exit pressure crack perimeter mean tube radius curvilinear position along the channel flow velocity reduced flow velocity specific volume position vector velocity vector acceleration vector elastic crack opening displacement component plastic crack opening displacement component crack half angle membrane stress membrane stress normal tensile stress wall shear stress

as well as accelerated crack growth. On the other hand, fluidelastic instability can significantly shorten a steam generator’s life. Hence, when assessing a steam generator’s operational life due to flow induced vibration, both turbulence and fluidelastic instability need to be considered. An understanding of the sources of excitation is necessary at the design stage of modern steam generators, in order to minimize maintenance and shutdown costs over the life of the device. This would also help extend the operational life of the steam generator. Additionally, periodic inspection of the steam generators using non-destructive techniques may reveal the existence of cracks. Therefore, it is also important to be able to predict crack propagation and leakage scenarios that may occur under normal operating conditions. During the operational life of steam generators, cracking and leakage are frequent events that will typically require costly shutdowns in order to plug the tubes (Païdoussis, 2006). An extensive study by Green and Hetsroni (1995) regarding pressurized water

reactor (PWR) steam generators suggests that the leading cause of tube plugging is stress corrosion cracking (52% of plugging instances) followed by fretting wear (19.6% of plugging instances). More recently, Gorman et al. (2009) reported that typical wear and failure modes in nuclear steam generators were pitting at tube support plates, fretting wear at tube support plates, and stress corrosion cracking of the tubes. Fretting wear results from the repetitive impact and sliding of the tubes against their supports. The impact occurs because of the clearances that are allowed between the tubes and their supports for manufacturing and assembly reasons, and to allow for material expansion due to thermal cycling. The excitation level, support clearance and friction, and the type and location of the supports affect the resulting fretting wear. With the proper design of the supports and their clearances, fretting wear can be controlled. The local stress concentrations at the microscopic level due to imperfections in manufacturing provide a starting point for surface flaws. These may progress under cyclic loading, mainly due to the dynamic loads applied by turbulence and fluidelastic instability. There are two types of cracks that are encountered in steam generators. Surface cracks are superficial and semi-elliptical in cross-section with a length 2L and a crack depth a. The quantity 2L/a is referred to as the crack aspect ratio. Through-wall-cracks are characterized by their length 2a. These cracks may either be along (axial cracks) or transverse to the tube (circumferential cracks). This study focuses on circumferential cracks, as the stresses that would allow axial cracks to progress were negligible. While a previous study by Hassan and Riznic (2014) focused on quantifying wear due to flow induced vibration (FIV) in a typical CANDU1 steam generator, the current paper provides a comprehensive model that can be used to assess the likelihood of crack propagation and leakage in nuclear steam generators. The model includes a full U-bend with loose supports subjected to typical flow profiles. A time domain model for fluidelastic instability and turbulence was used to represent the fluid excitation. A tube–support contact model was used to predict the interaction between the tubes and the loose supports. The time varying stresses around the U-bend are then used to predict surface flaws and through-wall-crack propagation as well as leakage rates. The first section presents a parametric study to understand what effect varying clearances has on surface and through wall crack propagation and leakage rates. In the second, a Monte Carlo study is presented in order to account for the uncertainty in the support clearances and tube offsets while providing steam generator life predictions. This model and investigation procedure can be used to assess the crack and leakage life of any nuclear steam generator given its geometry and flow conditions.

2. Modelling A finite element (FE) model was developed using the in-house finite element code INDAP (Hassan and Dokainish, 2011). These simulations provide the prediction of tube response and the stress level. This requires the knowledge of fluid excitation, nonlinear effects due to the support gaps, and the stress calculations. The crack growth and leakage calculations were conducted using an in-house crack modelling code PreTAP.

1

CANDU is a trade-mark of Atomic Energy of Canada Limited.

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1 Paris−Tada Model Zahoor E.P. Model Zahoor E. Model Experimental Data Points

0.9

Crack Opening Displacement [m]

0.8

0.7

0.6

0.5 0.4

0.3

0.2

0.1

0 40

60

80

100

120

140

160

180

200

220

Bending Stress [MPa] Fig. 3. Crack opening displacement vs. bending stress prediction comparison.

Fig. 1. The Lever and Weaver model as applied to a flow cell of the U-bend.

in Hassan et al. (2003). Briefly, the dynamics of the structure are expressed as: ¨ + [C]{w} ˙ + [K]{w} = {Fimp (w, w, ˙ t)} + {FFEI (w, t)} [M]{w}

2.1. U-bend structural modelling The current study focused on nonlinear configurations of the steam generator, as is the case with loose supports. Modal analysis results for the linear configurations investigated can be found in Hassan and Riznic (2014). The structural analysis is based on the finite element framework and has been developed in detail

˙ t)} + {Fturb (w, w,

(1)

where [M] is the mass matrix, [C] is the damping matrix, and ˙ and {w} ¨ are the displacement, [K] is the stiffness matrix. {w}, {w}, velocity, and acceleration vectors, respectively. {FFEI } is the vector containing the fluidelastic force at each node at time t. {Fturb } is the

Fig. 2. The flat tube support interaction model used for supports of complex geometries.

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Fig. 4. Finite element representation of a full U-bend.

turbulence excitation vector. The tube support impact and friction ˙ t)}. forces are represented by the force vector {Fimp (w, w, 2.2. Fluidelastic force model The fluidelastic force vector {FFEI } is computed at each time step at each node using the flow cell model. The development of this model was described in detail by Hassan and Hayder (2008). However, for the sake of completeness a brief description of the model will be presented here. Based on the Lever and Weaver (1986) postulation, the flow around the tube bundle organizes itself in well behaved channels. The flow inside these channels can be assumed to be one-dimensional, inviscid, and unsteady. In addition, it is sufficient to represent the system with a single flexible tube at the centre of the flow cell. This is illustrated for a flow cell in Fig. 1, whereby the area A(s, t), velocity U(s, t), and pressure P(s, t) along the channel as a function of time can be expressed as: ¯ A(s, t) = A(s) + (−1)i+1 a(s, t) ¯ + (−1)i+1 u(s, t) U(s, t) = U(s) ¯ P(s, t) = P(s) + (−1)i+1 p(s, t)

(2)

¯ ¯ ¯ where A(s), U(s), and P(s), are respectively the steady-state channel area, velocity, and pressure, a(s, t), u(s, t), and p(s, t) are respectively the area, velocity, and pressure perturbation components, and i is the channel index. The area and velocity perturbations’ relationship can be expressed through the unsteady continuity equation in an integral from the inlet s = − s0 to a curvilinear position s as:



s

−s0



∂A (s,t) ds + A(s, t)U(s, t) (−s ,t) = 0 0 ∂t

(3)

Further simplifications involve stating that the area, velocity, and pressure at the inlet are constant. Hence, it is possible to solve for the velocity perturbation resulting in: u(s, t) =

−1 A0 + a(s, t)





s

U0 a(s, t) + −s0

∂A ds ∂t



(4)

Obtaining the velocity perturbation u(s, t) is thus possible so long as the area perturbation is known, which is related to the tube displacement through: a(s, t) = f (s)w(t − )

(5)

where f(s) is the area perturbation decay function along the channel, for which several forms have been postulated, including Lever and

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Fig. 5. RMS node displacements.

Fig. 6. RMS node bending stresses.

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Fig. 7. Surface flaw growth for Case 1.

Weaver (1982), Lever and Weaver (1986), and Yetisir and Weaver (1993). Later, it was formally investigated by El Bouzidi et al. (2014) for a normal triangular configuration using CFD. w(t − ) is the tube displacement shifted by the time lag . The original formulation of the Lever and Weaver (1982) model stated that the time lag took a form  = s/U0 for which suggested values of  were given for different array geometries. Later other forms of  were postulated (Lever and Weaver, 1986; Yetisir and Weaver, 1993). A recent investigation by El Bouzidi et al. (2014) revealed that the time lag followed trends similar to the ones assumed by Lever and Weaver (1986) for reduced flow velocities Ur > 10, but that the trends were more complex for low reduced flow velocities (Ur ≤ 10). Once the velocity perturbation is resolved, the pressure can be calculated using the unsteady Bernoulli equation. An integration around the tube then provides a means of computing the net force on the centre tube Fnet (t). 2.3. Turbulence excitation modelling Turbulence within a tube bundle is a random distributed excitation force that can be modelled based on a turbulence bounding spectra. The bounding spectra by Oengören and Ziada (2000) will be utilized in this current work. These power spectral density (PSD) curves can be transformed into a time-domain force signal using an inverse Fast Fourier Transform (FFT) algorithm. The desired correlation length can be maintained by adjusting the number of elements that have the same random force set. In the current work the

turbulent forces are assumed to be correlated along each span of the tube, which is considered to be a conservative assumption. 2.4. Loose support contact model The tube–support contact model adopted here follows the approach proposed by Hassan et al. (2002). This technique allows the modelling of any tube–support geometry by a set of massless bars with a radial clearance, contact stiffness Kc , and contact damping Cc arranged around the tube as shown in Fig. 2. For the ith flat bar, the radial clearance Cri , as well as the normal and tangential unit vectors eˆ ti and eˆ ni need to be defined. Impact occurs when the  ni is greater than the tube displacement normal to the support w radial clearance, resulting in an overlap ınj given by: ınj = wni − Cri

(6)

This overlap allows the computation of the contact force Fci through: Fci = −(Kci ıni + 1.5˛ sign(ı˙ ni )|Kci ıni |) · eˆ ni

(7)

Arbitrary support types can hence be simulated by a number of flat bars arranged around the tube and used to compute net contact force at a given location: Fc,net =

N  i=1

Fci

(8)

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Fig. 8. Surface flaw growth for Case 2.

2.5. Crack modelling The crack model used throughout this study is based on work conducted by Vincent et al. (2009) and follows the approach taken by Sauvé et al. (1999). Estimating the crack growth requires computing the crack tip stress intensity factor, using: √ KI = a(Fm m + Fb b ) (9) where  m and  b are the tube membrane and bending stresses, while Fm and Fb are geometry factors computed through empirical curve fitting of experimental data presented by Sauvé et al. (1999) for surface flaws and through-wall-cracks with certain ranges of applicability. Fatigue tests performed by Sauvé et al. (1999) for Inconel 600 and Incoloy 800 have resulted in empirical relationships for I600/I800 crack growth based on the best data fit given by: da 2.39 2 (K − Kth ) = √ dN E2 1 − R

(10)

where N is the number of cycles, a the crack growth, R the loading ratio, and Kth the threshold minimum stress intensity factor for crack growth. There are multiple models to predict crack opening displacement (COD), such as those developed by Zahoor (1989) and Paris and Tada (1983). These models were utilized in this work and verified against experimental data as shown in Fig. 3. It is seen

that for small bending stresses (≤100 MPa RMS) all three models (Zahoor elastic, Zahoor elasto-plastic, and Paris and Tada) agree very well with the experimental data available. For larger stresses, Zahoor’s elastic model and the Paris and Tada model diverge from the experimental data, whereas Zahoor’s elastic–plastic model’s validity extends to 160 MPa RMS. Beyond 160 MPa, all models significantly underestimate the COD. Although most stresses encountered throughout the current investigation are fairly small (below 10 MPa RMS), Zahoor’s elastoplastic model was selected as it is accurate over a wide range of bending stresses. The elasto-plastic model decomposes the total COD into an elastic component ıe and a plastic component ıp . The elastic and plastic terms are given by: ıe =

4rm  (˛t t + ˛b b ) E



ıp = ˛rm 0 rm h2

 rm , n,  t

(11)



M M0

nrm (12)

where E is Young’s modulus,  is the crack half angle, and rm is the mean tube radius. ˛b and ˛t are geometry factors computed based on empirical equations provided by Zahoor (1989).  b and  t are the nominal bending and tensile stresses, respectively. ˛rm and nrm are constants from the Ramberg–Osgood relation (Ramberg and Osgood, 1943). M0 is a reference moment determined empirically and is a function of the crack half angle . 0 is the ratio of the

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Fig. 9. Through-wall-crack growth for Case 1.

0.2% offset yield strength and Young’s modulus. h2 is a tabulated function called the plastic influence function and some of its data is provided by Zahoor (1989). The crack growth model is intended for periodic loadings with a single frequency and amplitude. To deal with variable amplitude loadings as is usually the case when dealing with turbulence and nonlinear supports, the rainflow counting technique is used to obtain an equivalent closed loop hysteresis for the stress–strain curve, and an equivalent number of cycles. There are various ready algorithms in the literature, such as the one provided in great detail by Downing and Socie (1982).

dU U dvm −U dA − = vm dz A dz dz

(14)

where U is the flow velocity, vm is the specific volume, and A is the cross-sectional area. The momentum equation yields: U dvm −Pw dP w + = vm dz A dz

(15)

where Pw is the crack perimeter, and w is the wall shear stress. This model has been validated extensively against experimental data (Vincent et al., 2009).

2.6. Leakage modelling To model the leakage rate, several considerations were made. Initially, the primary fluid inside the tube is assumed to be single phase. It then expands while flowing through the crack (Matsumoto et al., 1991; Vincent et al., 2009). Briefly, the pressure drop through the crack can be expressed as follows: Pi − Po = Pi + Po + Pf

Parameters for this model are listed in Table 1. To model multiphase flow through the crack, equilibrium expansion is assumed. The continuity equation yields:

(13)

where Pi is the inlet pressure, Po is the exit pressure, and the P terms are losses due to the entrance, exit, and friction. The inlet and exit loss terms depend on their respective loss coefficients. The friction loss (Pf ) is a function of the local hydraulic diameter dh , flow density, flow velocity, and friction coefficient.

3. U-bend simulation model In order to illustrate the use of this modelling technique, various case studies were conducted and provided as a sample to the type of analysis that is useful in a case of a typical steam generator under typical flow conditions. Fig. 4 shows the finite element model of the U-bend tube utilized in this simulation. The model represents one of the CANDU steam generator designs which includes 7 cold-leg supports (broached holes) and 6 hot-leg supports (broached holes). The U-bend is supported by 3 pairs of scallop-bar supports (S1–S3) and two pairs of flat-bar supports (F1 and F2). In addition, a flat-bar support pair

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Fig. 10. Through-wall-crack growth for Case 2.

(CM) is at the top of the hot leg. The flat-bars represent a retrofit in order to stabilize the system after the potential loss of the top support in the hot leg due to corrosion. The node numbers are assigned such that node 1 is at the bottom of the hot leg and node 100 is at the bottom of the cold leg. The U-bend begins at node 29, with an apex at node 50, and ends at node 70. The current analysis is readily applicable to any combination of tube–support types. However, for the purpose of illustrating this approach, the aforementioned support type combination has been chosen. This configuration is a challenging one since it contains almost all of the different types of supports. The flow velocity distribution is also shown in Fig. 4. It is shown that the highest flow velocities are concentrated around the U-bend region, as well as the lowest section of the hot leg.

Damping is an important parameter that affects the nonlinear computation. It manifests itself in several forms such as structural damping, fluid viscous damping, impact and friction damping at the supports. A value of 0.5% was selected for structural damping, which is a conservative value. The impact and friction damping is not imposed on the simulation. However, it comes as a result of the numerical impact and friction computations. Similarly, the added fluid damping is accounted for in the fluidelastic instability modelling. The choice of a time step is an important factor that has to meet three requirements. Such requirements are the time integration stability, the nonlinear impact computation, and the fluidelastic time history resolution. Newmark’s time integration method was used which is unconditionally stable for the set of coefficients

Table 1 Leakage model parameter values. Quantity

Description

Value

Validity range

Ki Ko

Entrance loss coefficient Exit loss coefficient

0.35 0.65

– –

C1

Correlation constant

2.0 3.39

C2 rc Pi Po Ti To

Correlation constant

1.74

Surface roughness Inlet pressure Exit pressure Inlet temperature Exit temperature

0.86 8.84 ␮m 9.31 MPa 4.43 MPa 303 ◦ C 243 ◦ C

dh 2rc dh 2rc dh 2rc dh 2rc

– – – – –

> 100 < 100 > 100 < 100

26

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Fig. 11. Leakage rate predictions for Case 1.

chosen. For reasonable convergence rates of the impact iteration process it was found that the time step should be less than 1/10th of the period of the highest participating mode. Fluidelastic computations require that at least 75 previous response history points be saved. Therefore, the impact convergence is the critical factor that controls the choice of time step. Using the above criteria, a time step of 10−5 s was selected. 4. Simulation results 4.1. Deterministic simulations It may be useful to assess the sensitivity of the crack growth to the support clearance conditions around the U-bend. For this purpose an extensive set of deterministic simulation results is provided that present a thorough parametric study. Three support variation cases were studied. In these cases the same clearances profile along the hot leg was maintained. In this profile the clearance at the bottom support and top supports was set to 0.25, and 4.15 mm, respectively. The intermediate support clearances were set to follow this profile. This was done in order to simulate an accidental support loss at the top hot leg support due to corrosion. As the effects of corrosion are less severe along the cold leg, clearances were linearly increased from 0.25 mm at the bottom support to 0.55 mm at the top support. Three cases were considered to assess the effect of support clearance on crack growth. In Case 1 the hot-side scallop bar support clearance was varied from 0.1 mm to 1.5 mm while keeping the remaining support clearances at a base clearance Crb . After each set, the base clearance was increased by

0.1 mm from a lower bound clearance of 0.1 mm to an upper bound of 1.5 mm. Cases 2 and 3 consist of permutations of Case 1. In Case 2 the top scallop bar clearances were varied while in Case 3 the coldside scallop bar clearances were taken as the variable. For all three configurations, 5 crack locations were assumed around the U-bend, where location 1 was set at the bottom of the U-bend on the hot leg side, and location 5 was placed similarly on the cold leg side. Location 3 was placed at the apex of the U-bend. Locations 2 and 4 were placed at midpoint between locations 1–3 and 3–5, respectively. Based on stress histories, surface and through-wall cracks of various properties were assumed for each location, and the crack behaviour was investigated. For surface flaws, the crack half angle  was set to 30◦ at all locations, and the crack ratio a/tw was varied between 0.45 and 0.75 with 0.05 increments. For through-wall cracks at all locations, the initial half crack length was varied from 1.5 mm to 3.5 mm with half millimetre increments. For the leakage study, the predictions were made based on the through-wall crack analysis using parameters from Table 1. The deterministic study resulted in 675 nonlinear FE simulations and 8100 crack and leakage analysis cases. While simulations are conducted at 0.1 mm increments over a clearance range from 0.1 mm to 1.5 mm, results only show samples at specific clearances (0.2 mm, 0.5 mm, 1.0 mm, and 1.3 mm) for the sake of clarity. 4.1.1. Tube displacement and RMS stress From the simulation results it was observed that Cases 1 and 2 behave rather similarly, even though the symmetry is not complete due to wider clearances on the hot leg side. As a result, displacements and stresses are only shown for Cases 1 and 2. Fig. 5 shows

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Fig. 12. Leakage rate predictions for Case 2.

4.1.2. Crack propagation and leakage From the numerous simulations performed, it was found that the crack located at the apex (location 3) is the critical one. Figs. 7 and 8 show surface flaw growth predictions for Cases 1 and 2 with an initial crack ratio of 45%. Each subfigure is for a specific base clearance (0.2 mm, 0.5 mm, 1.0 mm, and 1.5 mm) and the variable clearance is increased from 0.1 mm to 1.5 mm. It is seen that for base clearances less than or equal to 0.5 mm, surface cracks may propagate but their depth will not reach 80% of the tube wall within 12 years. For the 1.0 mm and 1.5 mm base clearance cases, the crack depth may become unstable. Figs. 9 and 10 show results for through-wall cracks. These cracks appear to be more unstable

and may reach large sizes at clearances as little as 0.2 mm for Case 2. Figs. 11 and 12 show leakage rate predictions for Cases 1 and 2. As expected, the leakage rate behaviour is consistent with throughwall crack growth. Consequently Case 2 remains the critical case. Additionally, leakage rates remain stable below a threshold value of approximately 0.02 kg/s, after which they tend to grow out of

Final crack growth. 1000 cases. SCC, Location 3 0.8

0.7

0.6

Final Crack Ratio a/t

the nodal displacements for the latter configurations while Fig. 6 shows the stresses. The effect of base and variable clearance is also included. It is seen that while the displacements and stresses at the bottom of the hot leg (nodes 1–15) are not very sensitive to the clearance variations, the top of the hot leg (nodes 15–20) is greatly affected. It is also worth noting that while the largest displacements are concentrated in the hot leg region, the highest stresses can be found in the U-bend region. Note that while what is shown here is the RMS stress at each node, for crack growth predictions a full stress time history is required. For Case 1 (Fig. 6a and b) neither the hot leg support (S1) clearance nor the base clearance have a significant effect on the predicted stresses. For Case 2 (Fig. 6c and d), the bending stresses are greatly influenced by variations in the clearance at the scallop-bar support at the apex (S2).

0.5

0.4

0.3

0.2

0.1

0 0.45

0.5

0.55

0.6

0.65

0.7

0.75

Initial Crack Ratio a/t Fig. 13. Final crack ratio vs. initial crack ratio for surface circumferential cracks.

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Fig. 14. Through-wall-crack final length vs. initial length.

the bounds of the leakage model. In base clearance cases exceeding 1.0 mm, leakage life can be as small as a week. 4.2. Probabilistic simulations In order to make useful predictions regarding steam generator life based on realistic steam generator clearance conditions, which are most often known as statistical distributions due to manufacturing tolerances rather than deterministic values, a probabilistic assessment (Monte Carlo simulations) needs to be conducted. In this case study the support clearances were randomly generated according to preset clearance distributions. The clearance distributions were determined by setting the mean clearance to observed support clearance values, such as the measured enlarged clearances due to corrosion on the hot and cold legs. The standard deviation was set to manufacturing tolerances. The parameters used to generate the data set are summarized in Table 2. The crack analysis was also randomized, as the initial crack length is typically unknown. There was little indication in the

literature regarding the distribution of crack occurrences or the likelihood of certain crack geometries. The cracks were hence uniformly generated over the range of validity of the crack models, as indicated in Table 3. To accomplish this study, 2000 finite simulation cases were considered for the nonlinear structural analysis, while 20,000 crack cases were investigated. Fig. 13 shows 1000 cases for surface crack simulations for nominal steam generator clearance distributions at the critical crack location (U-bend apex). It is seen that all the simulation cases remain at their initial crack depth, and are unaffected by the stresses applied. On the other hand, it is shown in Fig. 14 that certain observed through-wall crack cases may grow significantly if their initial half crack length is above 2.5 mm for locations 1–5, and as little as 1.5 mm for location 3. The leakage rate predictions corresponding to the cases shown in Fig. 14 are presented in Fig. 15. It is seen that although throughwall cracks with an initial half length of above 3 mm may not be of concern at location 1 they may still result in significant leakage. Location 3 remains the critical location and a non-negligible fraction of the cases reached significant leakage levels over the entire range studied (1.5–3.5 mm).

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Fig. 15. Probabilistic final leakage rate vs. through-wall-crack initial half length.

While Fig. 14 provides a direct assessment of crack growth instances due to randomized initial conditions, it is difficult to obtain an overall sense of the likelihood of unstable crack growth from such figures. Consequently, probabilistic life predictions are made for cracks that exceed 3, 5, 9, and 12 year thresholds, and are

shown in Fig. 16. It is seen that for location 1, the probability of tube life exceeding 12 years is well above 80% even for initial half crack lengths of 3.5 mm. This is in contrast to location 3 which shows that initial half crack lengths of 1.5 mm have a 90% likelihood of exceeding 3 years, 80% likelihood of exceeding 5 years, 70% likelihood of

Table 2 Support clearance distribution. Support

Distribution

(mm)

 (mm)

Hot leg support Cold leg support Flat bars Scallop bars

Gaussian Gaussian Gaussian Gaussian

0.25–4.15 0.25–0.55 0.21 0.3

0.1 0.1 0.02 0.1

Table 3 Crack parameter distribution. Variable

Distribution



Range

CSF, a0 /tw CSF, 2L/a TWC, a0

Uniform Uniform Uniform

0.60 7.5 2.5 mm

0.45–0.75 3–12 1.5–3.5 mm

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Fig. 16. Probabilistic through-wall-crack life prediction.

exceeding 8 years, and 60% likelihood of exceeding 12 years. On the other hand, initial cracks of 3.5 mm only have a 5% chance of exceeding 12 years and a 20% chance of exceeding 3 years. 5. Conclusion This paper presented a comprehensive flow induced vibration model to simulate tube arrays. The model includes the turbulence excitation which is modelled as random forces that are correlated along each tube span. The fluid force due to the fluidelastic effect was taken into account through the use of the time domain flow cell model. The developed model was utilized to conduct an in-depth investigation of U-tube steam generator crack life and leakage potential. The predicted stress-time history due to the flow-induced vibrations was instrumental in predicting the crack growth in both surface and through-wall cracks. In addition, the leakage rate was predicted for the case of through-wall cracks. The crack growth and leakage rate models were validated against the experimental data. A thorough sensitivity analysis consisting of 675 nonlinear FE simulations and 8100 crack and leakage analysis cases provided an understanding of the effect of clearance on crack growth in CANDU steam generators that were subjected to a catastrophic support loss. It was determined that clearances below 0.3 mm are crucial to restricting crack growth rates to acceptable levels. It was also seen that leakage rates tend to grow very rapidly once they reach mass flow rates of 0.02 kg/s in all observed instances.

In order to assess the crack growth and life of actual CANDU steam generators, a Monte Carlo study was conducted. Randomized input clearance distributions and initial crack conditions based on manufacturing tolerances and crack model ranges were used to obtain realistic life predictions. Based on typical clearances for this specific U-bend geometry, surface flaws are less likely to grow in any significant manner due to the flow-induced vibrations effects alone. On the other hand, through-wall cracks exhibit a significant probability of being unstable in a short period of time as a result of the stress fluctuation due to flow induced vibrations. Acknowledgement The authors thankfully acknowledge the support of the Canadian Nuclear Safety Commission (CNSC). References Downing, S.D., Socie, D., 1982. Simple rainflow counting algorithms. Int. J. Fatigue 4 (1), 31–40. El Bouzidi, S., Hassan, M., Fernandes, L.L., Mohany, A., 2014. Numerical characterization of the area perturbation and timelag for a vibrating tube subjected to cross-flow. In: Oshkai, P. (Ed.), Proceedings of the 8th International Symposium on Fluid-Structure Interaction, Flow-Sound Interactions and Flow-Induced Vibration and Noise, ASME Pressure Vessel and Piping Division. Anaheim, CA. Gorman, J., Moroney, V., White, G., 2009. Alloy 800 steam generator tube performance. In: Proceedings of the 6th CNS International Steam Generator Conference, vol. 11, November 8, Toronto, Canada. Green, S.J., Hetsroni, G., 1995. PWR steam generators. Int. J. Multiph. Flow 21, 1–97.

S. El Bouzidi et al. / Nuclear Engineering and Design 292 (2015) 17–31 Hassan, M., Dokainish, M., 2011. Incremental Nonlinear Dynamic Analysis Program: Flow Induced Vibration Version. University of Guelph, Guelph, ON, Canada. Hassan, M., Hayder, M., 2008. Modelling of fluidelastic vibrations of heat exchanger tubes with loose supports. Nucl. Eng. Des. 238 (10), 2507–2520, ISSN: 00295493. Hassan, M., Riznic, J., 2014. Evaluation of the integrity of steam generator tubes subjected to flow induced vibrations. J. Press. Vessel Technol. 136, http://dx.doi. org/10.1115/1.4026982, 11 pp. Hassan, M., Weaver, D., Dokainish, M., 2002. A simulation of the turbulence response of heat exchanger tubes in lattice-bar supports. J. Fluids Struct. 16 (8), 1145–1176. Hassan, M., Weaver, D., Dokainish, M., 2003. The effects of support geometry on the turbulence response of loosely supported heat exchanger tubes. J. Fluids Struct. 18 (5), 529–554. Lever, J.H., Weaver, D.S., 1982. A theoretical model for the fluidelastic instability in heat exchanger tube bundles. In: Chen, S.S., Païdoussis, M.P., Au-Yang, M.K. (Eds.), Flow-Induced Vibration of Circular Cylindrical Structures. , pp. 87–108. Lever, J., Weaver, D., 1986. On the stability of heat exchanger tube bundles. Part I: Modified theoretical model. J. Sound Vib. 107 (3), 375–392. Matsumoto, K., Nakamura, S., Gotoh, N., Narabayashi, T., Miyano, H., Furukawa, S., Tanaka, Y., Horimizu, Y., 1991. Study on crack opening area and coolant leak rates on pipe cracks. Int. J. Press. Vessels Pip. 46 (1), 35–50.

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