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Predicting the preferential sites to liquid droplet erosion of the bellows assemblies by CFD
Nuclear Engineering and Design 241 (2011) 2295–2306
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Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Predicting the preferential sites to liquid droplet erosion of the bellows assemblies by CFD H.X. Hu, Y.G. Zheng ∗ , C.B. Liu State Key Laboratory for Corrosion and Protection, Institute of Metal Research, Chinese Academy of Science, 62 Wencui Road, Shenyang 110016, PR China
a r t i c l e
i n f o
Article history: Received 13 August 2010 Received in revised form 16 March 2011 Accepted 21 March 2011
a b s t r a c t Bellows assemblies are placed within extraction steam lines to absorb the differential displacement (driven primarily by thermal growth) between the turbine lower casing and the condenser shell in nuclear power plants. The thin wall thickness and service environmental conditions of the bellows predetermine their vulnerable endurance to the liquid droplet erosion (LDE). A three-dimensional computational fluid dynamics (CFD) bellows model proposed in this paper consists of a hydraulic model and a LDE model aiming to investigate the LDE on the external surface of the bellows assemblies. The result shows that the most prone site to LDE locates at the upwind side of the first bellows convolution (USFBC). The droplet diameter has significant effect on the LDE extent, rather than the LDE distribution. Moreover, the addition of the weld seam before the bellows can mitigate the LDE of the bellows downstream. In addition, the protecting cover (PC) effectively prevents the bellows from the LDE, but causes the droplet accumulation on the convolution, which possible leads to corrosive damage. The PC structure was designed optimization according to the requirements of the displacement compensation, the manufacture cost and the LDE resistance. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The bellows expansion joints provide a flexible pressure retaining connection designed to absorb motion in a system caused by thermal expansion and low levels of vibration. They are widely applied in many fields such as chemical plants, nuclear and fossil power systems, and heat exchange systems. By necessity, the bellows is one of the thinnest and most fragile pressure-carrying components in the piping system. Any behavior such as improper handling, installation and other careless practices possibly causes bellows failure. Bellows are also subject to deterioration from corrosion, erosion or a variety of other service related problems. A major bellows failure consists of one or more extraction joints failing in an explosive out-rush of expansion joint material (shrapnel) and extraction steam. This explosive failure would likely destroy adjacent expansion joints, compounding the effect with the possibility of damaging the internal condenser structure and other process lines. Worse yet, in extraction steam line systems installed in a free-hung configuration (no engineered hangers for the extraction lines), failure of a bellows and the assemblies’ support tie rods would lead to a guillotine break in the extraction line. The unsupported line would fall, with the likelihood of destroying addi-
∗ Corresponding author. Tel.: +86 24 23928381; fax: +86 24 23894149. E-mail addresses: [email protected], [email protected] (Y.G. Zheng). 0029-5493/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2011.03.045
tional extraction lines, damaging internal condenser components and/or rupturing the condenser tube sheets. This catastrophic failure mechanism, while not very likely, would lead to a lengthy forced outage. Therefore, it is essential to investigate the reasons causing bellows assemblies failure and take corresponding actions. Many researches focused on the bellows failure mechanism mostly resulting from mechanical force (Garion and Skoczen, 2008; Li and Sheng, 1990; Qian and Chen, 1994). Some studies also aimed at the corrosive behavior of the bellows materials mainly in the medium containing chloride ions (Shaikh et al., 2001; Zhu et al., 2006), apart from the foundation of Osborne et al. (2002) obtained in hydrofluoric acid (HF). However, few attentions are paid on the bellows failure caused by liquid droplet erosion (LDE) except our recent study (Hu et al., 2010). LDE is a serious phenomenon of materials damage in various important technological fields such as large steam turbines, high-speed fixed-wing aircraft and helicopter rotors. The damage produced by one or more of these loading conditions on a material surface exposed to single or multiple water drop impact, which is responsible for initiating damage and subsequent material removal. Fig. 1 obtained from a nuclear power plant shows the obvious eroded scar on the bellows convolution as labeled. Three common countermeasures for LDE can be taken at present, i.e. substituting for the bellows periodically or using more LDE resistant materials (Hu et al., 2010), bellows protection by LDE resistant coatings, and optimization of bellows structure by
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dimensional flow characteristics and localized wear sites. Ferng et al. (Ferng, 2008; Ferng et al., 2000, 2008) proposed a multidimensional CFD model to predict the localized wear sites inside pipes of the nuclear power plant, which consisted of a threedimensional two-phase hydrodynamic model and a erosion model. Bozzini et al. (2003) studied the erosion–corrosion inside the elbow in four-phase flow via CFD and experimental analysis. Most of these work focused on the prediction of internal wall of different geometrical pipes (T-shaped pipes and elbows), but little work referred to the LDE on the external surface of the complicated bellows with many convolutions. Furthermore, few studies concern on the effect of the droplet size on the distribution of the LDE. The purpose of this paper is to predict the preferential wear sites due to LDE on the external surface of the bellows assemblies by CFD, analyze the effect of the droplet size on the LDE, and optimize the structure of the protecting cover (PC). The hydrodynamic model is used to capture the characteristics of the two-phase flow outside the bellows assemblies. The appropriate erosion model coupled with hydrodynamic model is utilized to predict the localized wear sites. Based on the wear sites predicted, the model proposed in this paper can significantly reduce the detecting labor work and subsequently save unnecessary cost. Moreover, the influence of the droplet diameter on the LDE and the optimization of the PC can guide the operators and designers in the direction of reducing LDE and designing PC to some extent.
Nomenclature ˛
v u ε f () f HV N Re Cd F S D m n
volume fraction density (kg/m3 ) velocity (m/s) velocity along the wall (m/s) turbulent viscosity (kg/m s) turbulent energy dissipation turbulent Prandtl numbers characteristic function friction factor impact angle wall hardness impacting frequency Reynolds number drag coefficient interface drag force (N) momentum source terms droplet diameter (m) erosion rate (kg/s) distance normal to the wall (m)
Subscript g vapor phase l liquid phase
2. Model descriptions computational fluid dynamics (CFD). The bellows replacement is time-consuming and cost-consuming especially for the nuclear power plants where the outage time is very short from the viewpoint of economy. While, bellows repair by spraying LDE resistant coatings needs less time, but it is very hard to do that with so thin and complex-shape bellows. Moreover, once the coating is damaged, the pure water in power plant may be contaminated and results in new problems. Therefore, predicting the most prone sites to LDE and taking countermeasures, i.e. optimization of bellows structure by CFD, become a relative feasible way, which can not only save cost, but also avoid the occurrence of terrible accidents. Previous CFD model aimed to predict the wear rate of pipelines with empirical or semi-empirical correlations (Blatt et al., 1989; Nesic and Postlethwaite, 1991a,b; Remy and Bouchacourt, 1992; Zeisel and Durst, 1990), which failed to capture the multi-
2.1. Hydrodynamic models Hydrodynamic models consist of a continuity equation, a momentum equation, a k–ε two-equation turbulence model (Launder and Spalding, 1974), constitutive models for inter-phase exchange phenomena, and appropriate numerical scheme and boundary conditions. The assumptions are as follows: Assumptions: to derive the governing equations, constitutive equations, and appropriate boundary conditions the assumptions are made as follows: 1. No heat and mass transfer is considered in this model. 2. Momentum and continuity equations are solved for each phase. 3. Pressure is the same for both phases.
Fig. 1. Pictures of LDE of the bellows obtained at the scene.
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4. Droplets in the two-phase flow are considered as ideal spheres with a constant diameter of 1.0, 0.1 and 0.01 mm, respectively. 5. The standard k–ε turbulent model for single-phase flow is modified to adapt to the two-phase flow. 6. The steady fully developed flow condition is achieved through adding further calculation domain on both the inlet and outlet of the bellows. 7. The flow is assumed to be viscid, uncompressible flow. 8. The fully developed flow is set at the outlet and no special boundary conditions are needed. As for the selection of the droplet diameter in hydraulic models, most previous investigations (Ferng, 2008, 2009; Ferng and Hung, 2008; Ferng et al., 2000, 2008) set it as 1 mm artificially. Vapor in the steam condensed primary droplets with typical sizes of 10−5 to 10−3 mm in the last stages of steam turbines when the steam expanded below the saturation line (Ni et al., 2006). A fraction of these primary droplets deposits on the guide vanes where it may eventually form rivulets or water films. These structures grow in size, become unstable due to aerodynamic forces and eventually convert to a spray of coarse secondary droplets of up to tens or hundreds microns (Ahmad et al., 2009; Crane, 2004). It is reported that there is minor effect on the LDE when the diameter of the droplet is greater than 1 mm and this erosion was reduced as the decreasing of the droplet diameter (Springer, 1976). However, it is unclear that how the droplet size affects the distribution of the LDE sites. Therefore, a rank of different droplets in size (D = 1, 0.1, 0.01 mm) is selected to investigate the LDE of the bellows assemblies. 2.1.1. Two-phase models 2.1.1.1. Continuity equation. ∂ (˛i i ) + ∇ · (˛i i vi ) = 0 ∂t
(1)
where ˛i is the volume fraction for i phase, i is the density of i phase, vi is the velocity of i phase. Subscript i = g for the continuous phase of vapor, i = l for the dispersed phase of liquid droplet.
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Table 1 Empirical constants for turbulent models. C
Cε1
Cε2
k
ε
0.09
1.44
1.92
1.0
1.3
The turbulence kinetic energy, k, and its dissipation rate, ε, can be obtained from the following transport equations: ∂(˛i i ki ) + ∇ (˛i i ki ui ) = ∇ (˛i k,i ∇ ki ) + Sk,i ∂t
(5)
∂(i εi ) + ∇ (i εi ui ) = ∇ (ε,i ∇ εi ) + Sε,i ∂t
(6)
where k,i and ε,i represent the effective diffusivity of k and ε, respectively, Sk,i and Sε,i represent the source terms of k and ε, respectively, k,i , ε,i , Sk,i and Sε,i can be described as follows: k,i = l,i +
t,i i
(7)
ε,i = l,i +
t,i ε
(8)
Sε,i = ˛i
εi (Cε1 Gk,i − Cε2 i εi ) ki
(9)
where k and ε represent the turbulent Prandtl numbers for k and ε, respectively. Gk,i is the generation of turbulence kinetic energy due to mean velocity gradients, which is described as follows: Gk = −i u i uj ∇ uj
(10)
Cε1 , Cε2 , k and ε are empirical constants for turbulent models and are illustrated in Table 1 (Launder and Spalding, 1974). Continuative equations: The constitutive equations that account for interactions between the two phases include: Void fraction: ˛g + ˛l = 1
(11)
Inter-phase drag between the two phases: →−− → Fi = f (− u ul ) g
2.1.1.2. Momentum equation. − →U
∇ · (˛i i v2i ) = −˛i ∇ P + ∇ · [˛i (l,i + t,i )∇ vi + S i ]
(2)
where l,i is the molecular viscosity for i phase, t,i is the turbulent viscosity for i phase, SiU = −i ˛i g + Fi is the momentum source terms for i phase, g is the gravitational acceleration vector, and Fi is the interphase drag force for i phase.
where f is the friction factor between the vapor and liquid phases. The total drag force per unit volume was evaluated as the sum of the drag forces on each individual spherical droplet contained in that volume. Then, the inter-phase friction factor can be expressed as follows: f =
2.1.2. Turbulent model In the current model, the well-known standard k–ε twoequation model of single phase (Launder and Spalding, 1972) was employed for the two-phase droplet flow. The turbulence-induced shear stress for both phases expressed by Boussinesq concept is as follows: −u v = t ·
∂u ∂n
(3)
where u and v are the velocity fluctuations, n is the distance normal to the wall, t is the turbulent viscosity, which can be evaluated by the traditional k–ε model. t = c
k2 ε
(4)
where c is the turbulent model constant, which can be seen in Table 1.
(12)
3 g ˛g Cd − → →−− ul u g 4 Dl
(13)
where Dl is the droplet diameter and Cd is the drag coefficient.
⎧ ⎪ ⎨ 24/Re
Cd
⎪ ⎩
0.2 < Re ≤ 1000
0.04
1000 < Re
Re =
Re ≤ 0.2
24(1 + 0.15Re0.687 )/Re
D d U d l l,d
(14)
(15)
2.2. Liquid droplet erosion models The two-phase system simulated in the present study is a high quality (90.7%) steam system, in which the size of the liquid droplet is small enough to be carried by the high-velocity steam. The high energy produced by continuous impact of the droplet is sufficient to cause erosion to the target material (Bowden and Field, 1964;
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Heymann, 1969; Lesser, 1981; Lesser and Field, 1983). This kind of damage on the metal surface is named as LDE, which is assumed in current model to dominate the mechanical erosion phenomenon in the two-phase system. Many parameters affect the complicated LDE process, such as the impingement angle, the hardness of the pipe material and the frequency of impingement. A model simply describes this phenomenon as one that the wall surface material is removed by the action of numerous individual impacts of liquid droplets (Remy and Bouchacourt, 1992), which can be expressed as m = Cs Nf ()
l Ul2 HV
(16)
where m is the wear rate, Cs is the system constant dominated by fabrication and installation of piping, N is the impacting frequency, f is a characteristic function of the impact angle, is the impingement angle, l is the liquid density, Ul is the velocity normal to the pipe wall, and HV is the material hardness. From the equations, it can be known that the material loss caused by impingement erosion is proportional to the quantity of the liquid droplet, the droplet density, and the square of its normal impingement velocity, and inversely proportional to the hardness of target materials. The similar relationship has been confirmed in previous experimental and analytical works as for the erosion wear is proportional to Uln , where the exponent n ranges from 2 to 3 for steels (Finnie, 1972; Finnie and Mcfadden, 1978; Sargent et al., 1981; Sheldon and Kanhere, 1972; Sundararajan and Shewmon, 1983; Tilly and Sage, 1970). The concentration of liquid droplet, the characteristic function, and the hardness of target pipe material given are in terms of the practical model proposed in this study. As a result, a simplified parameters, ˛ll Ul Ul , is adopted to be an appropriate indicator to predict the distribution of severe erosion sites induced by droplet impingement. In the form, ˛l is the volume fraction of the liquid phase.
3. Calculative conditions Fig. 2 shows the geometrical model of the naked bellows assemblies and the dimensions of the bellows with orthogonal protecting cover (OPC). The connecting pipes upstream and downstream of the bellows are used to construct the calculation domain. The one upstream aims to make fully developed flow conditions before the bellows and the downstream is to avoid the possible recirculation paths leading to numerical converged errors or unphysical results at the outlet of the domain. The support ring as the connecting component is employed to weld the bellows to the connecting pipe because it is easy to burn through the thin bellows convolutions.
The present study is performed by adopting a three-dimensional mesh for the calculative domain. A body-fitted coordinate method is adopted to deal with this multidimensional geometry (Spalding, 1988) since the geometry of the bellows assemblies is not simple rectangular of cylindrical configuration. The governing equations describing the two-phase flow are solved by a finite differencing approach. In addition, the wall function method is adopted to avoid the need of fine grids near the wall (Sha and Launder, 1979). Three kinds of the calculative domain of bellows assemblies simulated in present study are naked bellows, the bellows with weld seam and the bellows with PC, respectively. The system in the calculative domain is high quality steam where the pressure is 7500 Pa, the enthalpy is 2350.7 kJ/kg, and the steam quality is 90.7%. 4. Results and discussion 4.1. Grid independency study A sensitivity study of different mesh numbers was performed for each bellows assembly model. A composite mesh system of structured and unstructured mesh was employed to generate the meshes in the complicated bellows system. In order to reduce the deviation of velocity distribution around the wall of the bellows assemblies, all meshes in models generated were along the normal direction of the component walls. Smaller meshes were adopted to capture the local flow characteristics around the convolutions of the bellows assemblies. The straight lines used to monitor the fluid parameters were in Z–Y plane. The relative coordinate in the following grid sensitivity study and the data comparison was based on the reference frame as shown in Fig. 2c. A sensitivity study of different mesh numbers of the naked bellows without weld seam was shown in Figs. 3 and 4. The velocity distribution of the three mesh systems on the straight line of z = 0.304 in y direction was almost the same except for a slight deviation of the active mesh * 267% (Fig. 3). However, the velocity of the active mesh * 267% greatly deviated from that of the smaller mesh pattern (Fig. 4). It indicated that the active mesh system was fine enough to capture the flow characteristics. The grid independency of the bellows with weld seam was also examined and shown in Fig. 5. When the mesh size increased to 1.2 times of the active mesh, the velocity distribution in z direction is obviously different from that of the active mesh. when the mesh size reduced to 40% of the active mesh, the velocity distribution showed minor difference indicating a grid independency. As a result, the active mesh system was sufficient to catch the flow properties around the bellows assemblies. Figs. 6 and 7 display the comparison of the distribution of the velocity and the liquid fraction, respectively. The velocity distribution of different mesh sizes in the straight line of z = 0.4 in y direction
Fig. 2. Schematic of the bellows assemblies and the main dimensions.
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Fig. 3. Mesh sensitivity validation of the naked bellows: velocity distribution in y direction. Fig. 6. Mesh sensitivity validation of the bellows with OPC: velocity distribution in y direction.
4.2. Naked bellows
Fig. 4. Mesh sensitivity validation of the naked bellows: velocity distribution in z direction.
are almost the same as shown in Fig. 6. The results in liquid fraction distribution were almost the same, although the mesh size greatly varied from half to 125% of the active mesh, which proved that the active mesh system used in this paper was available. Similar results were also obtained in grid sensitivity study of the bellows with fillet protecting cover (FPC) and not presented here again.
Fig. 5. Mesh sensitivity validation of the bellows with weld seam: velocity distribution in y direction.
Fig. 8 shows the mesh model of the calculative domain and the partial magnification. There should be a weld seam upstream generated in welding the bellows to the connecting pipe in practice. The effect of the weld seam on the LDE of the bellows will be considered individually in the next section. This section focuses on the characteristics of the LDE under different droplet diameter conditions. Only one-fourth of the computational domain was modeled by considering its geometrical symmetry. The gravitational orientation is consistent with the flow direction, i.e., the negative direction of y-axis. Fig. 9a, b and c represents the distribution of the liquid fraction on the surface of naked bellows under different conditions of droplet diameters of 1, 0.1 and 0.01 mm, respectively. The color index shows a range of the liquid fraction. Red indicates the maximum value of the liquid phase fraction and blue is the minimum. The high-quality stream flows over the naked bellows and impacts on the upwind of the bellows. As shown in Fig. 9, red on the upwind side of the first bellows convolution (USFBC) indicates more frequency of the droplets impingement there, where the site is the most prone to LDE and subsequently leakage. Meanwhile, the retard effect of the first wave changes the local flow field and reduces the liquid fraction on the rest convolutions. Moreover, the liquid fraction on USFBC is decreasing as the reducing of the diame-
Fig. 7. Mesh sensitivity validation of the bellows with OPC: liquid fraction distribution in y direction.
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Fig. 8. Mesh model of the calculative domain of the naked bellows assemblies. The right one is the magnification of the bellows.
ter of droplets and no significant erosion occurs on the other three convolutions. It indicates that the LDE on the USFBC is alleviated under the condition of the relative small droplets. The comparison of liquid velocity in y direction of the naked bellows between different droplet diameters is shown in Fig. 10. The coordinate of the straight line monitored in Y–Z plane is as labeled. The velocity distribution of the liquid droplet with diameter of 1 and 0.1 mm are almost the same. It was significantly different from that with diameter of 0.01 mm especially in the stage from −0.625 to −0.75, where is the position of the bellows convolutions. One can deduce that smaller liquid phase is disturbed more apparently by the turbulence. It is because those small droplets are easier to be entrained by the steam with high velocity under the pressure difference. Bigger droplets with higher gravity are difficult to be carried by the steam well, thus big droplets impact more on the bellows. The velocity before the convolutions (before −0.25) reduced for the same reason. As a result, the LDE on the convolution was mitigated as the decrease of the impingement velocity (Fig. 10) and liquid phase (Fig. 9), which is accordance with previous study (Springer, 1976). It is crucial to find the distribution of wear sites from these local flow parameters after obtaining the three-dimensional two-phase characteristics outside the naked bellows. Based on the results illustrated in Figs. 9 and 10, one can deduced that the most prone region to the LDE locates at the USFBC. The predicted wear sites well agree with the practical eroded points obtained at scene (Fig. 1). Therefore, the methodology proposed in this study can help construct the monitoring project of wall thickness measurement for bellows assemblies of nuclear power plants, especially in scheduling the measuring ranges on the convolutions of the bellows. Furthermore, based on the effect of the liquid droplet size on the LDE observed above, any conditions causing droplets nucleation and growth should be avoided to the greatest extent. Fig. 9 and Fig. 1 show a reasonable correspondence between the predicted results and the realistic image, which reveals that the methodology proposed in this study can capture the characteristics of the severe LDE sites of the bellows. Therefore, the methodology was further used to simulate the flow field around the more complicated bellows assemblies below.
Fig. 9. Distribution of liquid fraction on external surface of the bellows under different droplet diameters of (a) D = 1 mm, (b) D = 0.1 mm and (c) D = 0.01 mm.
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Fig. 10. Velocity distribution in y direction of the naked bellows under different droplet diameter conditions of 1, 0.1 and 0.01 mm.
4.3. Naked bellows with weld seam Fig. 12. Distribution of liquid fraction on the wall of the bellows with weld seam.
Fig. 11 presents the mesh system of the bellows with weld seam. The height of the weld seam is 5 mm, the width is 10 mm and the distance from the support ring is 94 mm. The total mesh numbers is 2586344. The LDE location, rate and extent of thinning or material loss strongly depend on the nature of the flow regime and fluid-structure interactions. The introduction of the weld seam may disturb the distribution of the flow field and then affect the LDE on the bellows downstream. Whether the affection alleviates or aggravates the LDE of the bellows is not clear. Whether the erosion sites change after introducing the weld seam is unknown, either. Therefore, the flow field around the bellows with weld seam was simulated, and the effect of the weld seam on the LDE to the bellows was investigated with questions aforementioned. The distribution of the liquid fraction on the external surface of the bellows with weld seam is revealed in Fig. 12. The droplet diameter in the present simulation is 1 mm. The color index indicates the magnitude of the droplet fraction. It can be found that the high liquid fraction appears on the USFBC, similar to the phenomenon of the bellows without weld seam (Fig. 9a). The calculated
Fig. 11. Mesh model of the bellows with weld seam. The right one is its partial magnification.
result demonstrates that the severe LDE site does not change yet, although the protruding weld seam changed the flow field around the bellows downstream. However, the amount of the liquid fraction on the USFBC is lower than that of the naked bellows (Fig. 9a), which indicates that the introducing of the weld seam alleviates the LDE of the bellows. Fig. 13 shows the comparison of the liquid phase velocity in y direction between the bellows with and without weld seam. It can be observed that the effect of the weld seam on the liquid velocity in y direction is minor. In the zone readers noticed as labeled (USFBC) by a rectangular, the velocity raised slightly resulting from the disturbing of the weld seam to the flow there. Although the velocity increased after the bellows convolutions, it had tiny impact on the LDE of the bellows. Moreover, the weld seam played a limited role in the liquid velocity in y direction because of its minor dimension. However, it did reduce the liquid fraction (Fig. 12) and the impingement velocity (Fig. 13) on the USFBC. Therefore, it is reasonable to deduce that the weld seam with some height and width is beneficial to alleviate the LDE of the bellows downstream.
Fig. 13. Liquid velocity distribution in y direction of the bellows assemblies with and without weld seam.
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Fig. 14. Dimension of the OPC and mesh model of the calculative domain of the bellows with the OPC.
Fig. 15. Distribution of the liquid fraction on the external surface of bellows with OPC under different droplet diameters of (a) D = 1 mm, (b) D = 0.1 mm and (c) D = 0.01 mm.
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Now it is clear that the protrusion of the weld seam assists in decreasing the impact frequency of the droplet, but it does not change the most prone site to LDE. In addition, it is unnecessary to grind the weld to be flat with the pipe. Meanwhile, the results also give some useful information to welders in the welding processes. 4.4. Bellows with protecting cover Fig. 14 shows the configuration and mesh model of the calculative domain for the geometric model of the bellows assemblies including an orthogonal protecting cover (OPC). The mesh numbers of the mesh model is 1561317. The structure of the OPC consists of a ring baffle and a protecting sleeve with the height of 73 mm. It is 16 mm from the top of the ring baffle to the opposite internal surface of the protecting sleeve. The thickness of both the protecting sleeve and the ring baffle is 10 mm as well as the main pipe. The PC in this style is a primary one, which has not been widely applied in practice yet. It is known that the PC is neither too high nor too low. Too high, the weight of the PC will threaten the security of the pipe systems. Too low, it will collide with the interior bellows convolutions when compensating the pipe displacement. What is the best dimension of the PC? How is the protection efficiency of the PC? The prediction of the preferential LDE sites to the bellows assemblies with PC was conducted under different droplet diameter conditions, and the geometry optimization of the PC was also performed with these questions. 4.4.1. Naked bellows with orthogonal protecting cover The effect of the droplet size on the LDE was analyzed in previous section. The results show that the smaller droplets cause lighter LDE and the droplet size has no effect on the distribution of the LDE sites. However, whether it is appropriate to the bellows assemblies with PC is still unclear, especially for the complicated semi-closed zone between the bellows wall and the PC. With this question, the hydraulic model and the LDE model aforementioned were utilized to reproduce the flow field around the bellows assemblies including PC under different droplet diameter conditions. Through the droplet impact indicator, the most prone site to LDE was predicted. Fig. 15 presents the distribution of liquid phase fraction on the external surface of the bellows with OPC under different droplet diameter conditions. The color index represents the fraction magnitude. Red indicates the maximum droplet fraction and blue is the minimum. Compared to Fig. 9, the amount of liquid phase on the bellows convolutions inside the PC was much less, which indicated a good shelter of the PC to the interior bellows. The comparison of the liquid velocity in y direction of the bellows with OPC is shown in Fig. 16. The velocity distribution of the diameter of 1 and 0.1 mm are almost the same, which significantly differ from that of the diameter of 0.01 mm. The rules are also applied to the liquid fraction distribution on the wall (Fig. 15). Moreover, the velocity of the liquid with the diameter of 0.01 mm is the highest in the stage approximately from −0.45 to −0.7 where the OPC locates. It can be explained that the lighter liquid droplets are easier to be disturbed by the flow changes. The preferential erosion sites located in the position where the velocity sharply change, like the position of y = −0.4 on y-axis. After it flowed through the OPC in the position, the velocity sharply increased as the pressure reduced (Fig. 16). In addition, the velocity distribution did not vary with the change of droplet diameter, which is in agreement with that in the case of naked bellows. The characteristic of the wet steam flow around the bellows with OPC is captured by the three-dimensional two-phase hydraulic model. Coupled with the droplet impingement indicator, the vulnerable site to LDE locates at the upwind side of the PC. The first convolution of the bellows is prone to the corrosive damage because of the accumulation of the droplet. The LDE extent on PC
Fig. 16. Velocity distribution of the liquid phase in y direction of the bellows with OPC under different droplet diameter conditions of 1, 0.1 and 0.01 mm.
varied little as the reducing of the droplet diameter. These results can guide the wall thickness measurement of PC and the corrosive damage detection of the bellows convolutions, which is also useful for the PC designers to optimize the construction. 4.4.2. Optimization of the protecting cover Although the PC played a role in protecting the thin bellows convolutions inside, it was still difficult to avoid the LDE on PC wall because its shape did not match the flow pattern. Moreover, the droplet accumulation on the bellows convolutions is much higher than that on the surface of PC (Fig. 15), which is easy to lead corrosion there in high-pressure environment. Therefore, it is essential to optimize the construction of the PC to reduce the damage to the least. Fig. 17 presents a set of schemes of the optimized PC through changing the size of the turning angle and the distance between the protecting sleeve and the ring baffle, which are named as FPC-1,
Fig. 17. Schematic of the bellows with (a) FPC-1, (b) FPC-2, (c) FPC-3 and (d) FPC-4.
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FPC-2, FPC-3 and FPC-4, respectively. The height of FPC-1 and FPC-2 is as high as that of the OPC (Fig. 15). The thickness of the protecting sleeve and ring baffle are both 10 mm, which is the same as that of the main pipe. The mesh numbers of the bellows with FPC-3 model is 1154505 and shown in Fig. 18. The mesh schematic of the other three structures is the same as that of the FPC-3, which were not present here. Fig. 19 shows the contours of the liquid fraction on the external surface of the bellows with different FPC. The color index presents the magnitude of the liquid fraction. Red indicates the highest droplet fraction and blue is the lowest. In comparison of Fig. 19a with b, it can be seen that the narrow gap between the protecting sleeve and the ring baffle causes more droplets accumulation on the ring baffle, which maybe unfavorable to heat transfer. If there is corrosive media in the steam, it is prone to accumulate there and causes corrosive damage when it goes up to some concentration. Moreover, the weld is vulnerable site to corrosion because it right locates within the accumulation zone of the liquid phase. Therefore, it is crucial to avoid the storage of the liquid phase inside the PC. Neither taller nor lower PC do not played positive role in draining the droplet out of the FPC with the same gap conditions (Fig. 19a, c, d). In contrast, FPC-3 eliminates the droplet storing on the inside of the ring baffle and the potential corrosion damage
Fig. 18. Mesh model of the bellows with FPC-3.
Fig. 19. Distribution of the liquid phase fraction on the external surface of the bellows with (a) FPC-1, (b) FPC-2, (c) FPC-3 and (d) FPC-4.
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1. The site on the naked bellows, which is the most prone to encounter the LDE, locates at the USFBC. The predicted LDE sites for the naked bellows correspond well with the practical damage, which indicates the feasibility of using the CFD model in further simulation. 2. The weld seam assists in alleviating the LDE of the bellows downstream, which indicates that it is unnecessary to grind the weld seam to be flat with the pipe. 3. The smaller the droplet diameter is, the lighter damage the LDE causes, but the location of the LDE site does not depend on the droplet size. 4. Taking into account of the requirements of displacement compensation, manufacture cost and resistance to LDE, the elementary optimized dimension of the PC is 70 mm in inner chamfer diameter and 16 mm in distance from the top of the ring baffle to the protecting sleeve. Fig. 20. Liquid phase velocity distribution in y direction of the bellows with OPC and FPC-3.
there. In addition, the adoption of the fillet angle of the protecting sleeve (Fig. 19) greatly decreases the impact frequency on the upwind of the protecting sleeve in contrast with that of the OPC (Fig. 15). Fig. 20 displays the comparison of the liquid phase velocity in y direction between the bellows with OPC and FPC-3. The position of the PC was also labeled in the picture. The flow characteristics around the FPC-3 are different from that around the OPC. The velocity of the liquid droplet there is lower than that before the OPC because of the streamline outline of the FPC-3. It reduced the impingement erosion to the upwind side of the PC. As a result, coupled with Fig. 19 and Fig. 20, it is reasonable to draw the conclusion that FPC-3 has high efficiency in protecting the bellows convolutions inside and be more resistant to the LDE. It can be deduced that the FPC-3 with the inner fillet diameter 70 mm and gap 16 mm is the optimized dimensions for the PC design through the reasonable comparison above. It not only satisfied the requirement of the displacement compensation, but also took into account of the load of the weight of the PC to the bellows. The most important was that the FPC-3 greatly prevents the bellows and the PC from the severe LDE. Therefore, the present methodology combining the hydraulic CFD model and the droplet impingent indicator can help the design and the wall thickness measurement of the PC. Many nuclear power plants begin to adopt the CHECWORKS code to perform their pipe-wall monitoring program (EPRI, 1993). However, CHECWORKS is mainly based on the mechanism of FAC (Flow-Accelerated Corrosion) of carbon steels or low alloy steels rather than LDE and it cannot be used to predict the LDE of the bellows assemblies. The model proposed in this paper can provide appropriate indicators for the local distribution of severe LDE, and then greatly help the monitoring of the pipe wall program, the design of the PC, and the set of the operation conditions.
5. Conclusions The local distribution of LDE sites for the bellows assemblies of the nuclear power plant was predicted by the present CFD model, which included the three-dimensional two-phase hydrodynamic model and the mechanical erosion indicator. The optimization design of the PC was carried out in view of mitigating the LDE through the model. The effect of the droplet diameter on the LDE of the bellows assemblies was analyzed according to the calculated near-wall distribution of liquid fraction and velocity. The main conclusions are listed below based on the results:
Acknowledgement The authors acknowledge the Financial Support of the Special Funds for the Major State Basic Research Projects (2011CB610504). References Ahmad, M., Casey, M., Sürken, N., 2009. Experimental assessment of droplet impact erosion resistance of steam turbine blade materials. Wear 267, 1605–1618. Blatt, W., Kohley, T., Lotz, U., Heitz, E., 1989. The influence of hydrodynamics on erosion–corrosion in two-phase liquid/particel flow. Corrosion 45, 793–804. Bowden, F.P., Field, J.E., 1964. The brittle fracture of solids by liquid impact, solid impact and by shock. Proc. R. Soc. Lond. A 282, 331–352. Bozzini, B., Ricotti, M.E., Boniardi, M., Mele, C., 2003. Evaluation of erosion–corrosion in multiphase flow via CFD and experimental analysis. Wear 255, 237–245. Crane, R., 2004. Droplet deposition in steam turbines. Proc. IME. C. J. Mech. Eng. Sci., 859–870. EPRI, 1993. CHECWORKSTM Computer Program User Guide, ed. Ferng, Y.M., 2008. Predicting local distributions of erosion–corrosion wear sites for the piping in the nuclear power plant using CFD models. Ann. Nucl. Energy 35, 304–313. Ferng, Y.M., 2009. CFD investigating the influence of power uprate on impingement wear sites for the feedwater heater in the nuclear power plant. Nucl. Eng. Des. 239, 231–238. Ferng, Y.M., Hung, C.T., 2008. Predicting distributions of wear sites on shell wall of feedwater heater in nuclear power plant using two phase models. Energy Convers. Manage. 49, 1182–1192. Ferng, Y.M., Ma, Y.P., Chung, N.M., 2000. Application of local flow models in predicting distributions of erosion–corrosion locations. Corrosion 56, 21. Ferng, Y.M., Tseng, Y.S., Pei, B.S., Wang, S.L., 2008. A two-phase methodology to predict FAC wear sites in the piping system of a BWR. Nucl. Eng. Des. 238, 2189–2196. Finnie, I., 1972. Some observations on erosion of ductile metals. Wear 19, 81–90. Finnie, I., Mcfadden, D.H., 1978. Velocity dependence of erosion of ductile metals by solid particles at low angles of incidence. Wear 48, 181–190. Garion, C., Skoczen, B., 2008. Influence of micro-damage on reliability of cryogenic bellows in the LHC interconnections. In: Adv. Cryog. Eng. Mater. Trans. Inter. Cryog. Mater. Conf. , pp. 100–107. Heymann, F.J., 1969. High-speed impact between a liguid drop and a solid surface. J. Appl. Phys. 40, 5113–5122. Hu, H.X., Zheng, Y.G., Qin, C.P., 2010. Comparison of Inconel 625 and Inconel 600 in resistance to cavitation erosion and jet impingement erosion. Nucl. Eng. Des. 240, 2721–2730. Launder, B.E., Spalding, D.B., 1972. Lectures in Mathematical Models of Turbulence, third ed. Academic Press, London. Launder, B.E., Spalding, D.B., 1974. The numerical computation of turbulent flows. Comput. Methods Appl. Mech. 3, 269–289. Lesser, M.B., 1981. Analytic solutions of liquid-drop impact problems. Proc. R. Soc. Lond. A 337, 289–308. Lesser, M.B., Field, J.E., 1983. The impact of compressible liquids. Annu. Rev. Fluid Mech. 15, 97–122. Li, Y., Sheng, S., 1990. Strength analysis and structural optimization of U-shaped bellows. Int. J. Pres. Ves. Pip. 42, 33–46. Nesic, S., Postlethwaite, J., 1991a. Hydrodynamics of disturbed flow and erosion–corrosion. Part I. Single-phase flow study. Can. J. Chem. Eng. 69, 698–703. Nesic, S., Postlethwaite, J., 1991b. Hydrodynamics of disturbed flow and erosion corrosion. Part II. A two-phase flow study. Can. J. Chem. Eng. 69, 704–710. Ni, Y.-j., Hua-hun, Q., Guo-rong, Q., Jing-bo, L., et al., 2006. Erosion developing precess of the steam turbing blade. Turb. Technol. 48, 460–461 (in Chinese). Osborne, P.E., Icenhour, A.S., Cul, G.D.D., 2002. Corrosion Test Results for Inconel 600 vs Inconel–stainless UG Bellows. Nuclear Science and Technology Division.
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Qian, Y., Chen, L., 1994. Failure analysis of bellows under external pressure. Int. J. Pres. Ves. Pip. 57, 321–325. Remy, F.N., Bouchacourt, M., 1992. Flow-assisted corrosion: a method to avoid damage. Nucl. Eng. Des. 133, 23–30. Sargent, G.A., Keshavan, M.K., Mehrotra, P.K., Conrad, H., 1981. Proceedings of International Conference on Wear of Materials ,. ASME, New York, NY, p. 613. Sha, W.T., Launder, B.E., 1979. A General Model for Turbulent Momentum and Heat Transfer in Liquid Metal. ANL-77-87. Argonne National Laboratory. Shaikh, H., George, G., Singh Khatak, H., 2001. Failure analysis of an AM 350 steel bellows. Eng. Fail. Anal. 8, 571–576. Sheldon, G.L., Kanhere, A., 1972. An investigation of impingement erosion using single particles. Wear 21, 195–209.
Spalding, D.B., 1988. Modelling a Body-Fitted-Coordinate Grid During a PHOENICS Computation. ChHAM/PRE/88/1. Springer, G.S., 1976. Erosion by Liquid Impact. Scripta Publishing Company, Washington, DC. Sundararajan, G., Shewmon, P.G., 1983. A new model for the erosion of metals at normal incidence. Wear 84, 237–258. Tilly, G.P., Sage, W., 1970. The interaction of particle and material behaviour in erosion processes. Wear 16, 447–465. Zeisel, H., Durst, F., 1990. Computations of eosion-corrosion processes in separated two-phase flows. In: Proceedings of the Conference on Corrosion , Houston, Texas, p. 29. Zhu, Y.Z., Wang, H.F., Sang, Z.F., 2006. The effect of environmental medium on fatigue life for u-shaped bellows expansion joints. Int. J. Fatigue 28, 28–32.
Contents lists available at ScienceDirect
Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Predicting the preferential sites to liquid droplet erosion of the bellows assemblies by CFD H.X. Hu, Y.G. Zheng ∗ , C.B. Liu State Key Laboratory for Corrosion and Protection, Institute of Metal Research, Chinese Academy of Science, 62 Wencui Road, Shenyang 110016, PR China
a r t i c l e
i n f o
Article history: Received 13 August 2010 Received in revised form 16 March 2011 Accepted 21 March 2011
a b s t r a c t Bellows assemblies are placed within extraction steam lines to absorb the differential displacement (driven primarily by thermal growth) between the turbine lower casing and the condenser shell in nuclear power plants. The thin wall thickness and service environmental conditions of the bellows predetermine their vulnerable endurance to the liquid droplet erosion (LDE). A three-dimensional computational fluid dynamics (CFD) bellows model proposed in this paper consists of a hydraulic model and a LDE model aiming to investigate the LDE on the external surface of the bellows assemblies. The result shows that the most prone site to LDE locates at the upwind side of the first bellows convolution (USFBC). The droplet diameter has significant effect on the LDE extent, rather than the LDE distribution. Moreover, the addition of the weld seam before the bellows can mitigate the LDE of the bellows downstream. In addition, the protecting cover (PC) effectively prevents the bellows from the LDE, but causes the droplet accumulation on the convolution, which possible leads to corrosive damage. The PC structure was designed optimization according to the requirements of the displacement compensation, the manufacture cost and the LDE resistance. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The bellows expansion joints provide a flexible pressure retaining connection designed to absorb motion in a system caused by thermal expansion and low levels of vibration. They are widely applied in many fields such as chemical plants, nuclear and fossil power systems, and heat exchange systems. By necessity, the bellows is one of the thinnest and most fragile pressure-carrying components in the piping system. Any behavior such as improper handling, installation and other careless practices possibly causes bellows failure. Bellows are also subject to deterioration from corrosion, erosion or a variety of other service related problems. A major bellows failure consists of one or more extraction joints failing in an explosive out-rush of expansion joint material (shrapnel) and extraction steam. This explosive failure would likely destroy adjacent expansion joints, compounding the effect with the possibility of damaging the internal condenser structure and other process lines. Worse yet, in extraction steam line systems installed in a free-hung configuration (no engineered hangers for the extraction lines), failure of a bellows and the assemblies’ support tie rods would lead to a guillotine break in the extraction line. The unsupported line would fall, with the likelihood of destroying addi-
∗ Corresponding author. Tel.: +86 24 23928381; fax: +86 24 23894149. E-mail addresses: [email protected], [email protected] (Y.G. Zheng). 0029-5493/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2011.03.045
tional extraction lines, damaging internal condenser components and/or rupturing the condenser tube sheets. This catastrophic failure mechanism, while not very likely, would lead to a lengthy forced outage. Therefore, it is essential to investigate the reasons causing bellows assemblies failure and take corresponding actions. Many researches focused on the bellows failure mechanism mostly resulting from mechanical force (Garion and Skoczen, 2008; Li and Sheng, 1990; Qian and Chen, 1994). Some studies also aimed at the corrosive behavior of the bellows materials mainly in the medium containing chloride ions (Shaikh et al., 2001; Zhu et al., 2006), apart from the foundation of Osborne et al. (2002) obtained in hydrofluoric acid (HF). However, few attentions are paid on the bellows failure caused by liquid droplet erosion (LDE) except our recent study (Hu et al., 2010). LDE is a serious phenomenon of materials damage in various important technological fields such as large steam turbines, high-speed fixed-wing aircraft and helicopter rotors. The damage produced by one or more of these loading conditions on a material surface exposed to single or multiple water drop impact, which is responsible for initiating damage and subsequent material removal. Fig. 1 obtained from a nuclear power plant shows the obvious eroded scar on the bellows convolution as labeled. Three common countermeasures for LDE can be taken at present, i.e. substituting for the bellows periodically or using more LDE resistant materials (Hu et al., 2010), bellows protection by LDE resistant coatings, and optimization of bellows structure by
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dimensional flow characteristics and localized wear sites. Ferng et al. (Ferng, 2008; Ferng et al., 2000, 2008) proposed a multidimensional CFD model to predict the localized wear sites inside pipes of the nuclear power plant, which consisted of a threedimensional two-phase hydrodynamic model and a erosion model. Bozzini et al. (2003) studied the erosion–corrosion inside the elbow in four-phase flow via CFD and experimental analysis. Most of these work focused on the prediction of internal wall of different geometrical pipes (T-shaped pipes and elbows), but little work referred to the LDE on the external surface of the complicated bellows with many convolutions. Furthermore, few studies concern on the effect of the droplet size on the distribution of the LDE. The purpose of this paper is to predict the preferential wear sites due to LDE on the external surface of the bellows assemblies by CFD, analyze the effect of the droplet size on the LDE, and optimize the structure of the protecting cover (PC). The hydrodynamic model is used to capture the characteristics of the two-phase flow outside the bellows assemblies. The appropriate erosion model coupled with hydrodynamic model is utilized to predict the localized wear sites. Based on the wear sites predicted, the model proposed in this paper can significantly reduce the detecting labor work and subsequently save unnecessary cost. Moreover, the influence of the droplet diameter on the LDE and the optimization of the PC can guide the operators and designers in the direction of reducing LDE and designing PC to some extent.
Nomenclature ˛
v u ε f () f HV N Re Cd F S D m n
volume fraction density (kg/m3 ) velocity (m/s) velocity along the wall (m/s) turbulent viscosity (kg/m s) turbulent energy dissipation turbulent Prandtl numbers characteristic function friction factor impact angle wall hardness impacting frequency Reynolds number drag coefficient interface drag force (N) momentum source terms droplet diameter (m) erosion rate (kg/s) distance normal to the wall (m)
Subscript g vapor phase l liquid phase
2. Model descriptions computational fluid dynamics (CFD). The bellows replacement is time-consuming and cost-consuming especially for the nuclear power plants where the outage time is very short from the viewpoint of economy. While, bellows repair by spraying LDE resistant coatings needs less time, but it is very hard to do that with so thin and complex-shape bellows. Moreover, once the coating is damaged, the pure water in power plant may be contaminated and results in new problems. Therefore, predicting the most prone sites to LDE and taking countermeasures, i.e. optimization of bellows structure by CFD, become a relative feasible way, which can not only save cost, but also avoid the occurrence of terrible accidents. Previous CFD model aimed to predict the wear rate of pipelines with empirical or semi-empirical correlations (Blatt et al., 1989; Nesic and Postlethwaite, 1991a,b; Remy and Bouchacourt, 1992; Zeisel and Durst, 1990), which failed to capture the multi-
2.1. Hydrodynamic models Hydrodynamic models consist of a continuity equation, a momentum equation, a k–ε two-equation turbulence model (Launder and Spalding, 1974), constitutive models for inter-phase exchange phenomena, and appropriate numerical scheme and boundary conditions. The assumptions are as follows: Assumptions: to derive the governing equations, constitutive equations, and appropriate boundary conditions the assumptions are made as follows: 1. No heat and mass transfer is considered in this model. 2. Momentum and continuity equations are solved for each phase. 3. Pressure is the same for both phases.
Fig. 1. Pictures of LDE of the bellows obtained at the scene.
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4. Droplets in the two-phase flow are considered as ideal spheres with a constant diameter of 1.0, 0.1 and 0.01 mm, respectively. 5. The standard k–ε turbulent model for single-phase flow is modified to adapt to the two-phase flow. 6. The steady fully developed flow condition is achieved through adding further calculation domain on both the inlet and outlet of the bellows. 7. The flow is assumed to be viscid, uncompressible flow. 8. The fully developed flow is set at the outlet and no special boundary conditions are needed. As for the selection of the droplet diameter in hydraulic models, most previous investigations (Ferng, 2008, 2009; Ferng and Hung, 2008; Ferng et al., 2000, 2008) set it as 1 mm artificially. Vapor in the steam condensed primary droplets with typical sizes of 10−5 to 10−3 mm in the last stages of steam turbines when the steam expanded below the saturation line (Ni et al., 2006). A fraction of these primary droplets deposits on the guide vanes where it may eventually form rivulets or water films. These structures grow in size, become unstable due to aerodynamic forces and eventually convert to a spray of coarse secondary droplets of up to tens or hundreds microns (Ahmad et al., 2009; Crane, 2004). It is reported that there is minor effect on the LDE when the diameter of the droplet is greater than 1 mm and this erosion was reduced as the decreasing of the droplet diameter (Springer, 1976). However, it is unclear that how the droplet size affects the distribution of the LDE sites. Therefore, a rank of different droplets in size (D = 1, 0.1, 0.01 mm) is selected to investigate the LDE of the bellows assemblies. 2.1.1. Two-phase models 2.1.1.1. Continuity equation. ∂ (˛i i ) + ∇ · (˛i i vi ) = 0 ∂t
(1)
where ˛i is the volume fraction for i phase, i is the density of i phase, vi is the velocity of i phase. Subscript i = g for the continuous phase of vapor, i = l for the dispersed phase of liquid droplet.
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Table 1 Empirical constants for turbulent models. C
Cε1
Cε2
k
ε
0.09
1.44
1.92
1.0
1.3
The turbulence kinetic energy, k, and its dissipation rate, ε, can be obtained from the following transport equations: ∂(˛i i ki ) + ∇ (˛i i ki ui ) = ∇ (˛i k,i ∇ ki ) + Sk,i ∂t
(5)
∂(i εi ) + ∇ (i εi ui ) = ∇ (ε,i ∇ εi ) + Sε,i ∂t
(6)
where k,i and ε,i represent the effective diffusivity of k and ε, respectively, Sk,i and Sε,i represent the source terms of k and ε, respectively, k,i , ε,i , Sk,i and Sε,i can be described as follows: k,i = l,i +
t,i i
(7)
ε,i = l,i +
t,i ε
(8)
Sε,i = ˛i
εi (Cε1 Gk,i − Cε2 i εi ) ki
(9)
where k and ε represent the turbulent Prandtl numbers for k and ε, respectively. Gk,i is the generation of turbulence kinetic energy due to mean velocity gradients, which is described as follows: Gk = −i u i uj ∇ uj
(10)
Cε1 , Cε2 , k and ε are empirical constants for turbulent models and are illustrated in Table 1 (Launder and Spalding, 1974). Continuative equations: The constitutive equations that account for interactions between the two phases include: Void fraction: ˛g + ˛l = 1
(11)
Inter-phase drag between the two phases: →−− → Fi = f (− u ul ) g
2.1.1.2. Momentum equation. − →U
∇ · (˛i i v2i ) = −˛i ∇ P + ∇ · [˛i (l,i + t,i )∇ vi + S i ]
(2)
where l,i is the molecular viscosity for i phase, t,i is the turbulent viscosity for i phase, SiU = −i ˛i g + Fi is the momentum source terms for i phase, g is the gravitational acceleration vector, and Fi is the interphase drag force for i phase.
where f is the friction factor between the vapor and liquid phases. The total drag force per unit volume was evaluated as the sum of the drag forces on each individual spherical droplet contained in that volume. Then, the inter-phase friction factor can be expressed as follows: f =
2.1.2. Turbulent model In the current model, the well-known standard k–ε twoequation model of single phase (Launder and Spalding, 1972) was employed for the two-phase droplet flow. The turbulence-induced shear stress for both phases expressed by Boussinesq concept is as follows: −u v = t ·
∂u ∂n
(3)
where u and v are the velocity fluctuations, n is the distance normal to the wall, t is the turbulent viscosity, which can be evaluated by the traditional k–ε model. t = c
k2 ε
(4)
where c is the turbulent model constant, which can be seen in Table 1.
(12)
3 g ˛g Cd − → →−− ul u g 4 Dl
(13)
where Dl is the droplet diameter and Cd is the drag coefficient.
⎧ ⎪ ⎨ 24/Re
Cd
⎪ ⎩
0.2 < Re ≤ 1000
0.04
1000 < Re
Re =
Re ≤ 0.2
24(1 + 0.15Re0.687 )/Re
D d U d l l,d
(14)
(15)
2.2. Liquid droplet erosion models The two-phase system simulated in the present study is a high quality (90.7%) steam system, in which the size of the liquid droplet is small enough to be carried by the high-velocity steam. The high energy produced by continuous impact of the droplet is sufficient to cause erosion to the target material (Bowden and Field, 1964;
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Heymann, 1969; Lesser, 1981; Lesser and Field, 1983). This kind of damage on the metal surface is named as LDE, which is assumed in current model to dominate the mechanical erosion phenomenon in the two-phase system. Many parameters affect the complicated LDE process, such as the impingement angle, the hardness of the pipe material and the frequency of impingement. A model simply describes this phenomenon as one that the wall surface material is removed by the action of numerous individual impacts of liquid droplets (Remy and Bouchacourt, 1992), which can be expressed as m = Cs Nf ()
l Ul2 HV
(16)
where m is the wear rate, Cs is the system constant dominated by fabrication and installation of piping, N is the impacting frequency, f is a characteristic function of the impact angle, is the impingement angle, l is the liquid density, Ul is the velocity normal to the pipe wall, and HV is the material hardness. From the equations, it can be known that the material loss caused by impingement erosion is proportional to the quantity of the liquid droplet, the droplet density, and the square of its normal impingement velocity, and inversely proportional to the hardness of target materials. The similar relationship has been confirmed in previous experimental and analytical works as for the erosion wear is proportional to Uln , where the exponent n ranges from 2 to 3 for steels (Finnie, 1972; Finnie and Mcfadden, 1978; Sargent et al., 1981; Sheldon and Kanhere, 1972; Sundararajan and Shewmon, 1983; Tilly and Sage, 1970). The concentration of liquid droplet, the characteristic function, and the hardness of target pipe material given are in terms of the practical model proposed in this study. As a result, a simplified parameters, ˛ll Ul Ul , is adopted to be an appropriate indicator to predict the distribution of severe erosion sites induced by droplet impingement. In the form, ˛l is the volume fraction of the liquid phase.
3. Calculative conditions Fig. 2 shows the geometrical model of the naked bellows assemblies and the dimensions of the bellows with orthogonal protecting cover (OPC). The connecting pipes upstream and downstream of the bellows are used to construct the calculation domain. The one upstream aims to make fully developed flow conditions before the bellows and the downstream is to avoid the possible recirculation paths leading to numerical converged errors or unphysical results at the outlet of the domain. The support ring as the connecting component is employed to weld the bellows to the connecting pipe because it is easy to burn through the thin bellows convolutions.
The present study is performed by adopting a three-dimensional mesh for the calculative domain. A body-fitted coordinate method is adopted to deal with this multidimensional geometry (Spalding, 1988) since the geometry of the bellows assemblies is not simple rectangular of cylindrical configuration. The governing equations describing the two-phase flow are solved by a finite differencing approach. In addition, the wall function method is adopted to avoid the need of fine grids near the wall (Sha and Launder, 1979). Three kinds of the calculative domain of bellows assemblies simulated in present study are naked bellows, the bellows with weld seam and the bellows with PC, respectively. The system in the calculative domain is high quality steam where the pressure is 7500 Pa, the enthalpy is 2350.7 kJ/kg, and the steam quality is 90.7%. 4. Results and discussion 4.1. Grid independency study A sensitivity study of different mesh numbers was performed for each bellows assembly model. A composite mesh system of structured and unstructured mesh was employed to generate the meshes in the complicated bellows system. In order to reduce the deviation of velocity distribution around the wall of the bellows assemblies, all meshes in models generated were along the normal direction of the component walls. Smaller meshes were adopted to capture the local flow characteristics around the convolutions of the bellows assemblies. The straight lines used to monitor the fluid parameters were in Z–Y plane. The relative coordinate in the following grid sensitivity study and the data comparison was based on the reference frame as shown in Fig. 2c. A sensitivity study of different mesh numbers of the naked bellows without weld seam was shown in Figs. 3 and 4. The velocity distribution of the three mesh systems on the straight line of z = 0.304 in y direction was almost the same except for a slight deviation of the active mesh * 267% (Fig. 3). However, the velocity of the active mesh * 267% greatly deviated from that of the smaller mesh pattern (Fig. 4). It indicated that the active mesh system was fine enough to capture the flow characteristics. The grid independency of the bellows with weld seam was also examined and shown in Fig. 5. When the mesh size increased to 1.2 times of the active mesh, the velocity distribution in z direction is obviously different from that of the active mesh. when the mesh size reduced to 40% of the active mesh, the velocity distribution showed minor difference indicating a grid independency. As a result, the active mesh system was sufficient to catch the flow properties around the bellows assemblies. Figs. 6 and 7 display the comparison of the distribution of the velocity and the liquid fraction, respectively. The velocity distribution of different mesh sizes in the straight line of z = 0.4 in y direction
Fig. 2. Schematic of the bellows assemblies and the main dimensions.
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Fig. 3. Mesh sensitivity validation of the naked bellows: velocity distribution in y direction. Fig. 6. Mesh sensitivity validation of the bellows with OPC: velocity distribution in y direction.
4.2. Naked bellows
Fig. 4. Mesh sensitivity validation of the naked bellows: velocity distribution in z direction.
are almost the same as shown in Fig. 6. The results in liquid fraction distribution were almost the same, although the mesh size greatly varied from half to 125% of the active mesh, which proved that the active mesh system used in this paper was available. Similar results were also obtained in grid sensitivity study of the bellows with fillet protecting cover (FPC) and not presented here again.
Fig. 5. Mesh sensitivity validation of the bellows with weld seam: velocity distribution in y direction.
Fig. 8 shows the mesh model of the calculative domain and the partial magnification. There should be a weld seam upstream generated in welding the bellows to the connecting pipe in practice. The effect of the weld seam on the LDE of the bellows will be considered individually in the next section. This section focuses on the characteristics of the LDE under different droplet diameter conditions. Only one-fourth of the computational domain was modeled by considering its geometrical symmetry. The gravitational orientation is consistent with the flow direction, i.e., the negative direction of y-axis. Fig. 9a, b and c represents the distribution of the liquid fraction on the surface of naked bellows under different conditions of droplet diameters of 1, 0.1 and 0.01 mm, respectively. The color index shows a range of the liquid fraction. Red indicates the maximum value of the liquid phase fraction and blue is the minimum. The high-quality stream flows over the naked bellows and impacts on the upwind of the bellows. As shown in Fig. 9, red on the upwind side of the first bellows convolution (USFBC) indicates more frequency of the droplets impingement there, where the site is the most prone to LDE and subsequently leakage. Meanwhile, the retard effect of the first wave changes the local flow field and reduces the liquid fraction on the rest convolutions. Moreover, the liquid fraction on USFBC is decreasing as the reducing of the diame-
Fig. 7. Mesh sensitivity validation of the bellows with OPC: liquid fraction distribution in y direction.
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Fig. 8. Mesh model of the calculative domain of the naked bellows assemblies. The right one is the magnification of the bellows.
ter of droplets and no significant erosion occurs on the other three convolutions. It indicates that the LDE on the USFBC is alleviated under the condition of the relative small droplets. The comparison of liquid velocity in y direction of the naked bellows between different droplet diameters is shown in Fig. 10. The coordinate of the straight line monitored in Y–Z plane is as labeled. The velocity distribution of the liquid droplet with diameter of 1 and 0.1 mm are almost the same. It was significantly different from that with diameter of 0.01 mm especially in the stage from −0.625 to −0.75, where is the position of the bellows convolutions. One can deduce that smaller liquid phase is disturbed more apparently by the turbulence. It is because those small droplets are easier to be entrained by the steam with high velocity under the pressure difference. Bigger droplets with higher gravity are difficult to be carried by the steam well, thus big droplets impact more on the bellows. The velocity before the convolutions (before −0.25) reduced for the same reason. As a result, the LDE on the convolution was mitigated as the decrease of the impingement velocity (Fig. 10) and liquid phase (Fig. 9), which is accordance with previous study (Springer, 1976). It is crucial to find the distribution of wear sites from these local flow parameters after obtaining the three-dimensional two-phase characteristics outside the naked bellows. Based on the results illustrated in Figs. 9 and 10, one can deduced that the most prone region to the LDE locates at the USFBC. The predicted wear sites well agree with the practical eroded points obtained at scene (Fig. 1). Therefore, the methodology proposed in this study can help construct the monitoring project of wall thickness measurement for bellows assemblies of nuclear power plants, especially in scheduling the measuring ranges on the convolutions of the bellows. Furthermore, based on the effect of the liquid droplet size on the LDE observed above, any conditions causing droplets nucleation and growth should be avoided to the greatest extent. Fig. 9 and Fig. 1 show a reasonable correspondence between the predicted results and the realistic image, which reveals that the methodology proposed in this study can capture the characteristics of the severe LDE sites of the bellows. Therefore, the methodology was further used to simulate the flow field around the more complicated bellows assemblies below.
Fig. 9. Distribution of liquid fraction on external surface of the bellows under different droplet diameters of (a) D = 1 mm, (b) D = 0.1 mm and (c) D = 0.01 mm.
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Fig. 10. Velocity distribution in y direction of the naked bellows under different droplet diameter conditions of 1, 0.1 and 0.01 mm.
4.3. Naked bellows with weld seam Fig. 12. Distribution of liquid fraction on the wall of the bellows with weld seam.
Fig. 11 presents the mesh system of the bellows with weld seam. The height of the weld seam is 5 mm, the width is 10 mm and the distance from the support ring is 94 mm. The total mesh numbers is 2586344. The LDE location, rate and extent of thinning or material loss strongly depend on the nature of the flow regime and fluid-structure interactions. The introduction of the weld seam may disturb the distribution of the flow field and then affect the LDE on the bellows downstream. Whether the affection alleviates or aggravates the LDE of the bellows is not clear. Whether the erosion sites change after introducing the weld seam is unknown, either. Therefore, the flow field around the bellows with weld seam was simulated, and the effect of the weld seam on the LDE to the bellows was investigated with questions aforementioned. The distribution of the liquid fraction on the external surface of the bellows with weld seam is revealed in Fig. 12. The droplet diameter in the present simulation is 1 mm. The color index indicates the magnitude of the droplet fraction. It can be found that the high liquid fraction appears on the USFBC, similar to the phenomenon of the bellows without weld seam (Fig. 9a). The calculated
Fig. 11. Mesh model of the bellows with weld seam. The right one is its partial magnification.
result demonstrates that the severe LDE site does not change yet, although the protruding weld seam changed the flow field around the bellows downstream. However, the amount of the liquid fraction on the USFBC is lower than that of the naked bellows (Fig. 9a), which indicates that the introducing of the weld seam alleviates the LDE of the bellows. Fig. 13 shows the comparison of the liquid phase velocity in y direction between the bellows with and without weld seam. It can be observed that the effect of the weld seam on the liquid velocity in y direction is minor. In the zone readers noticed as labeled (USFBC) by a rectangular, the velocity raised slightly resulting from the disturbing of the weld seam to the flow there. Although the velocity increased after the bellows convolutions, it had tiny impact on the LDE of the bellows. Moreover, the weld seam played a limited role in the liquid velocity in y direction because of its minor dimension. However, it did reduce the liquid fraction (Fig. 12) and the impingement velocity (Fig. 13) on the USFBC. Therefore, it is reasonable to deduce that the weld seam with some height and width is beneficial to alleviate the LDE of the bellows downstream.
Fig. 13. Liquid velocity distribution in y direction of the bellows assemblies with and without weld seam.
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Fig. 14. Dimension of the OPC and mesh model of the calculative domain of the bellows with the OPC.
Fig. 15. Distribution of the liquid fraction on the external surface of bellows with OPC under different droplet diameters of (a) D = 1 mm, (b) D = 0.1 mm and (c) D = 0.01 mm.
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Now it is clear that the protrusion of the weld seam assists in decreasing the impact frequency of the droplet, but it does not change the most prone site to LDE. In addition, it is unnecessary to grind the weld to be flat with the pipe. Meanwhile, the results also give some useful information to welders in the welding processes. 4.4. Bellows with protecting cover Fig. 14 shows the configuration and mesh model of the calculative domain for the geometric model of the bellows assemblies including an orthogonal protecting cover (OPC). The mesh numbers of the mesh model is 1561317. The structure of the OPC consists of a ring baffle and a protecting sleeve with the height of 73 mm. It is 16 mm from the top of the ring baffle to the opposite internal surface of the protecting sleeve. The thickness of both the protecting sleeve and the ring baffle is 10 mm as well as the main pipe. The PC in this style is a primary one, which has not been widely applied in practice yet. It is known that the PC is neither too high nor too low. Too high, the weight of the PC will threaten the security of the pipe systems. Too low, it will collide with the interior bellows convolutions when compensating the pipe displacement. What is the best dimension of the PC? How is the protection efficiency of the PC? The prediction of the preferential LDE sites to the bellows assemblies with PC was conducted under different droplet diameter conditions, and the geometry optimization of the PC was also performed with these questions. 4.4.1. Naked bellows with orthogonal protecting cover The effect of the droplet size on the LDE was analyzed in previous section. The results show that the smaller droplets cause lighter LDE and the droplet size has no effect on the distribution of the LDE sites. However, whether it is appropriate to the bellows assemblies with PC is still unclear, especially for the complicated semi-closed zone between the bellows wall and the PC. With this question, the hydraulic model and the LDE model aforementioned were utilized to reproduce the flow field around the bellows assemblies including PC under different droplet diameter conditions. Through the droplet impact indicator, the most prone site to LDE was predicted. Fig. 15 presents the distribution of liquid phase fraction on the external surface of the bellows with OPC under different droplet diameter conditions. The color index represents the fraction magnitude. Red indicates the maximum droplet fraction and blue is the minimum. Compared to Fig. 9, the amount of liquid phase on the bellows convolutions inside the PC was much less, which indicated a good shelter of the PC to the interior bellows. The comparison of the liquid velocity in y direction of the bellows with OPC is shown in Fig. 16. The velocity distribution of the diameter of 1 and 0.1 mm are almost the same, which significantly differ from that of the diameter of 0.01 mm. The rules are also applied to the liquid fraction distribution on the wall (Fig. 15). Moreover, the velocity of the liquid with the diameter of 0.01 mm is the highest in the stage approximately from −0.45 to −0.7 where the OPC locates. It can be explained that the lighter liquid droplets are easier to be disturbed by the flow changes. The preferential erosion sites located in the position where the velocity sharply change, like the position of y = −0.4 on y-axis. After it flowed through the OPC in the position, the velocity sharply increased as the pressure reduced (Fig. 16). In addition, the velocity distribution did not vary with the change of droplet diameter, which is in agreement with that in the case of naked bellows. The characteristic of the wet steam flow around the bellows with OPC is captured by the three-dimensional two-phase hydraulic model. Coupled with the droplet impingement indicator, the vulnerable site to LDE locates at the upwind side of the PC. The first convolution of the bellows is prone to the corrosive damage because of the accumulation of the droplet. The LDE extent on PC
Fig. 16. Velocity distribution of the liquid phase in y direction of the bellows with OPC under different droplet diameter conditions of 1, 0.1 and 0.01 mm.
varied little as the reducing of the droplet diameter. These results can guide the wall thickness measurement of PC and the corrosive damage detection of the bellows convolutions, which is also useful for the PC designers to optimize the construction. 4.4.2. Optimization of the protecting cover Although the PC played a role in protecting the thin bellows convolutions inside, it was still difficult to avoid the LDE on PC wall because its shape did not match the flow pattern. Moreover, the droplet accumulation on the bellows convolutions is much higher than that on the surface of PC (Fig. 15), which is easy to lead corrosion there in high-pressure environment. Therefore, it is essential to optimize the construction of the PC to reduce the damage to the least. Fig. 17 presents a set of schemes of the optimized PC through changing the size of the turning angle and the distance between the protecting sleeve and the ring baffle, which are named as FPC-1,
Fig. 17. Schematic of the bellows with (a) FPC-1, (b) FPC-2, (c) FPC-3 and (d) FPC-4.
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FPC-2, FPC-3 and FPC-4, respectively. The height of FPC-1 and FPC-2 is as high as that of the OPC (Fig. 15). The thickness of the protecting sleeve and ring baffle are both 10 mm, which is the same as that of the main pipe. The mesh numbers of the bellows with FPC-3 model is 1154505 and shown in Fig. 18. The mesh schematic of the other three structures is the same as that of the FPC-3, which were not present here. Fig. 19 shows the contours of the liquid fraction on the external surface of the bellows with different FPC. The color index presents the magnitude of the liquid fraction. Red indicates the highest droplet fraction and blue is the lowest. In comparison of Fig. 19a with b, it can be seen that the narrow gap between the protecting sleeve and the ring baffle causes more droplets accumulation on the ring baffle, which maybe unfavorable to heat transfer. If there is corrosive media in the steam, it is prone to accumulate there and causes corrosive damage when it goes up to some concentration. Moreover, the weld is vulnerable site to corrosion because it right locates within the accumulation zone of the liquid phase. Therefore, it is crucial to avoid the storage of the liquid phase inside the PC. Neither taller nor lower PC do not played positive role in draining the droplet out of the FPC with the same gap conditions (Fig. 19a, c, d). In contrast, FPC-3 eliminates the droplet storing on the inside of the ring baffle and the potential corrosion damage
Fig. 18. Mesh model of the bellows with FPC-3.
Fig. 19. Distribution of the liquid phase fraction on the external surface of the bellows with (a) FPC-1, (b) FPC-2, (c) FPC-3 and (d) FPC-4.
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1. The site on the naked bellows, which is the most prone to encounter the LDE, locates at the USFBC. The predicted LDE sites for the naked bellows correspond well with the practical damage, which indicates the feasibility of using the CFD model in further simulation. 2. The weld seam assists in alleviating the LDE of the bellows downstream, which indicates that it is unnecessary to grind the weld seam to be flat with the pipe. 3. The smaller the droplet diameter is, the lighter damage the LDE causes, but the location of the LDE site does not depend on the droplet size. 4. Taking into account of the requirements of displacement compensation, manufacture cost and resistance to LDE, the elementary optimized dimension of the PC is 70 mm in inner chamfer diameter and 16 mm in distance from the top of the ring baffle to the protecting sleeve. Fig. 20. Liquid phase velocity distribution in y direction of the bellows with OPC and FPC-3.
there. In addition, the adoption of the fillet angle of the protecting sleeve (Fig. 19) greatly decreases the impact frequency on the upwind of the protecting sleeve in contrast with that of the OPC (Fig. 15). Fig. 20 displays the comparison of the liquid phase velocity in y direction between the bellows with OPC and FPC-3. The position of the PC was also labeled in the picture. The flow characteristics around the FPC-3 are different from that around the OPC. The velocity of the liquid droplet there is lower than that before the OPC because of the streamline outline of the FPC-3. It reduced the impingement erosion to the upwind side of the PC. As a result, coupled with Fig. 19 and Fig. 20, it is reasonable to draw the conclusion that FPC-3 has high efficiency in protecting the bellows convolutions inside and be more resistant to the LDE. It can be deduced that the FPC-3 with the inner fillet diameter 70 mm and gap 16 mm is the optimized dimensions for the PC design through the reasonable comparison above. It not only satisfied the requirement of the displacement compensation, but also took into account of the load of the weight of the PC to the bellows. The most important was that the FPC-3 greatly prevents the bellows and the PC from the severe LDE. Therefore, the present methodology combining the hydraulic CFD model and the droplet impingent indicator can help the design and the wall thickness measurement of the PC. Many nuclear power plants begin to adopt the CHECWORKS code to perform their pipe-wall monitoring program (EPRI, 1993). However, CHECWORKS is mainly based on the mechanism of FAC (Flow-Accelerated Corrosion) of carbon steels or low alloy steels rather than LDE and it cannot be used to predict the LDE of the bellows assemblies. The model proposed in this paper can provide appropriate indicators for the local distribution of severe LDE, and then greatly help the monitoring of the pipe wall program, the design of the PC, and the set of the operation conditions.
5. Conclusions The local distribution of LDE sites for the bellows assemblies of the nuclear power plant was predicted by the present CFD model, which included the three-dimensional two-phase hydrodynamic model and the mechanical erosion indicator. The optimization design of the PC was carried out in view of mitigating the LDE through the model. The effect of the droplet diameter on the LDE of the bellows assemblies was analyzed according to the calculated near-wall distribution of liquid fraction and velocity. The main conclusions are listed below based on the results:
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