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Residual lifetime assessment of thermal power plant superheater header
Accepted Manuscript Residual lifetime assessment of thermal power plant superheater header
O. Yasniy, Yu. Pyndus, V. Iasnii, Y. Lapusta PII: DOI: Reference:
S1350-6307(17)30281-9 doi: 10.1016/j.engfailanal.2017.07.028 EFA 3246
To appear in:
Engineering Failure Analysis
Received date: Revised date: Accepted date:
30 March 2017 5 July 2017 26 July 2017
Please cite this article as: O. Yasniy, Yu. Pyndus, V. Iasnii, Y. Lapusta , Residual lifetime assessment of thermal power plant superheater header, Engineering Failure Analysis (2016), doi: 10.1016/j.engfailanal.2017.07.028
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ACCEPTED MANUSCRIPT Residual lifetime assessment of thermal power plant superheater header O. Yasniya,*, Yu. Pyndusa, V. Iasniia, Y. Lapustab
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a) Ternopil Ivan Pul'uj National Technical University, Ruska 56, Ternopil, 46001, Ukraine b) Institut Pascal, UMR 6602 / UBP / CNRS / IFMA, Clermont Université, BP 265, 63175 Aubière Cedex, France
Abstract
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The safe operation of thermal power plants (TPPs) is largely determined by the reliability of main components of steam systems of power units, to which belong the headers of boiler superheaters. The aim of this study is to estimate the residual lifetime of superheater header starting from the initial defect size and up to the maximum allowable one. The above-mentioned lifetime is affected by the fluctuations of temperature under steady mode of operation of TPP superheater header. Superheaters headers operate in a steamy environment under the pressure of 15.5 MPa at temperature of 545 ºС. The header is a thick-walled cylinder made of 12Cr1MoV steel with a length of 2314 mm, outer diameter of 325 mm and thickness of 50 mm. To estimate the residual lifetime of the header, the steam temperature in header was measured and digitized. The temperature range of the header under quasi-stable operational mode was divided into three classes: (1) t < 10 °C; (2) 10 °C < t < 30 °C; (3) t > 30 °C. The local minima and maxima were determined based on the obtained steam temperature history. The residual durability was evaluated taking into account the effect of thermo-mechanical stresses and also the stresses caused by internal steam pressure. The stress intensity factors (SIFs) at the crack tip in the ligament between the holes of superheater collector were estimated by FEM. The SIF of mode I (KI) were determined along the fronts of modelled semi-elliptical cracks. The SIF correction function at the midpoint of the semi-elliptical crack front in the ligament between holes of the superheater header versus the crack depth and the temperature difference between the external and internal surfaces under a constant internal pressure of steam was obtained. Based on the analysis of header defects, the proposed front shape was taken in the form of the semi-ellipse. The crack growth at 500 ° C was modelled by Paris and NASGRO equations. With the increase of temperature difference between the external and internal header surfaces from 10 °С to 50 °С, the number of cycles, that is necessary for the crack to reach 35 mm in depth, decreases approximately in 25 times. It was calculated, that the average value of temperature fluctuations is 15°С for the Class 1, and is 46.2 °С for the Class 2. The lifetime of header can be extended due to the decrease of fluctuations of temperature range and their frequency.
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Keywords: Stress intensity factors; finite element analysis; fatigue crack growth; life assessment; fatigue.
* Corresponding author. Tel.: +38-097-9-55-55-23; fax: +38-0352-254-983.. E-mail address: [email protected] (O. Yasniy)
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minimum stress
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min max yy
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crack depth parameter of Paris equation parameter of NASGRO equation stress intensity factor, mode I minimum stress intensity factor, mode I maximum stress intensity factor, mode I thickness of collector wall number of loading cycles parameter of Paris equation parameter of NASGRO equation parameter of NASGRO equation parameter of NASGRO equation temperature time stress intensity factor range temperature range stress range normal stress range
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a C C1 KI K min K max w N n m1 p q t T K t yy
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Nomenclature
maximum stress
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normal stress
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1. Introduction
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The safe operation of thermal power plants (TPPs) is largely determined by the reliability of main elements of steam power system [1] to which belong the boilers superheaters headers. The problem of ensuring the reliability of TPP equipment is being aggravated with the prolonged operation and with detected damage. The main factors that lead to the emergence of cracks is the inhomogeneous distribution of temperature field along the wall thickness and the high internal steam pressure. The durability of superheaters headers depends on the stress level, that arise in the most heavily loaded areas. Usually the cracks appear on the inner surface of the ligaments between nozzles holes [2,3]. The proper operation of power stations considerably depends on the reliability of the basic elements of steam power systems [4–6] such as superheaters headers in boilers [7–9]. The problems with safety of thermal power plant equipment are aggravated by long-term operation of such equipment, which in many cases exceeds the designed service life, as well as by observed significant damage and multiple cracks in superheaters headers [10]. The main factors which cause the cracking and failure of superheaters include a non-uniform distribution of temperature through the wall thickness and high internal pressure of steam [2,11]. The life of superheaters headers also depends on stresses that occur in the most loaded regions. Typically, multiple defects appear on the internal surface of the ligament between header holes [2,10,12,13]. Taking into account the existing experience with respect to operation modes of TPPs superheater headers [13– 15], the regions which are the most prone to operational damage and defects are the ligaments between the holes of nozzles supplying superheated steam. The stress state in superheater headers was studied, for example, in the papers [2,16].
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2. Temperature fluctuations
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The deformation of the TPP superheater header during operation was modelled by means of FEM [16] in elastic formulation. It was found that due to internal pressure, the equivalent stress intensity in the steady-state conditions of the header is the highest in the vicinity of the holes and reaches only 0.43 of the yield stress y = 290 MPa while the axial stress normal to the plane of possible crack initiation yy is 0.15y. However, the thermal stresses caused by rapid cooling of the hot header due to water penetration with a condensate or significantly colder steam into some holes reached value of yield strength of 12Cr1MoV in the ligament between holes at the temperature difference equal to 186 °C during the stoppage of equipment [16]. The aforementioned and other studies investigate the stress state only under certain operational parameters (internal pressure, temperature difference between the internal and external surfaces of the header, etc.); however, they do not examine the effect of changing these parameters on the stress state. The latter is very important to substantiate the restrictions concerning the operation modes of power plant equipment with operational damage and to investigate the influence of temperature difference on the inner and outer surfaces of header on its stress state, in particular, taking into account the damage in a form of semi-elliptical cracks between the holes, that is observed in practice. The aim of the study is to estimate the residual lifetime of superheater header from the current size defect and up to the maximum allowable one. The above-mentioned lifetime is caused by the fluctuations of temperature under quasistatic operational regime of TPP superheater header.
The information about header, from which the specimens were cut and its modes of operation are given in Table 1.
Value 2 1976 hot 325 12Cr1MoV 1257 14 MPa 545 °С – internal 8. Operational temperature (external and internal walls) 565 °С – external 9. Place of template cut ( distance from the end cap of header) 3.45 m 10. Period of operation, hours 178000 11. Thickness, mm 50
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Parameters Block number Year of operation start Header type (cold, hot) External header diameter, mm Steel grade Number of starts during operation Internal pressure during operation
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# 1. 2. 3. 4. 5. 6. 7.
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Table 1. The information about the header
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The characteristics of mechanical properties of steel in the initial state: yield strength 0.2 = 320 MPa, ultimate tensile strength U = 480 MPa [17]. The chemical composition of 12Cr1MoV steel is 0.12 % С; 0.54 % Mn; 0.26 % Si; <0.015 % P; <0.019 % S; <1.0 % Cr; <1.0 % Ni. The operational data of registered temperature steam, which were provided by TPP, were used for the evaluation of residual durability. The steam temperature during header operation was measured with chromel-alumel thermocouple and was registered during 54.5 hours on analog potentiometer. Fig. 1 shows the example of digitized temperature dependence of steam during operational mode, which is interpreted as typical for superheater header at constant loading without cold, semi-cold and hot starts of block.
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t, ° C
T, s
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3. FEM modeling of the superheater header
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Fig. 1. The digitized temperature dependence of steam in superheater header under stable mode of operation.
3.1. Modeling of a TPP superheater header with cracks by FEM
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A FE model of the TP-100 boiler superheater header in the form of a thick-walled cylinder with holes was prepared in ANSYS Workbench. The modelling involved solving an elastic three-dimensional problem. Taking into account the symmetry conditions [2,16] was analyzed a fragment (quarter) of the superheater header (Fig. 2). The model is built in a rectangular Cartesian system of coordinates. The orientation of the Cartesian coordinate system is shown in Fig. 2. The inside and outside diameters of the collector are 225 mm and 325 mm, respectively. The holes are 22 mm in diameter, the angle between the axes of the holes in the XOY plane is 20 °C. The spacing between the rows of holes along the OY axis is 80 mm. The 10-node 3D finite element SOLID87 was used for numerical analysis. This finite element has one degree of freedom (temperature) in each node and it is suitable for 3D modelling of complex curved shape by static and transient thermal analysis. The model meshing was performed using mapped face meshing (Fig. 1). The finite element mesh was refined between the holes. The total number of finite elements in the model was 40432. The finite element SOLID87 was replaced by an equivalent 10-node structural element, SOLID187, to perform further calculations in the Static Structural module. SOLID187 is a 3D element, which consists of 10 nodes, each node having three degrees of freedom: translation in the directions OX, OY and OZ. This element also has the properties of elasticity, plasticity, hyper-elasticity, creep, stiffness, large deflections and strains. In addition, the element can be loaded with forces, stresses, displacements and temperature, either separately or in combination [18].
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Fig. 3. The distribution of temperature field of header at internal surface temperature of 500°С, and external of 560°С.
Fig. 2. The full-scale FE model of superheater header.
central
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Fig. 4. The distribution of normal stresses yy in the superheater header under internal pressure of 14 MPa and temperature of internal surface of 500 °С, and external of 560 °С.
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The finite element mesh was refined in the vicinity of the crack front. The minimal spacing between the finite element nodes was 0.05 mm. The internal surfaces of the headers cylinder, holes and crack surfaces were loaded with pressure and temperature. The thermo-physical and mechanical properties of 12Cr1MoV steel are given in Table 2. For a more reliable representation of the material behavior under the combined thermo-mechanical effect, the FEM model of the superheater header [14] takes into account only these parameters which vary with the change of temperature. Table 2: Temperature properties of material
Temperature,
Young's modulus,
t, ° C 20 100 200 300 400
E∙105, MPa 1.98 1.93 1.88 1.83 1.75
Poisson's Coefficient of Thermal ratio thermal expansion, conductivity, 0.27 0.28 0.29 0.30 0.32
1.24 1.30 1.36 1.40 1.44
46 45 44 42 40
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500 600
1.67 1.57
0.33 0.34
1.47 1.49
37 35
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The transient thermal analysis results were transferred to the Static Structural module. The stress state of the superheater header was calculated taking into account the non-uniform thermal expansion of the material through the thickness of header wall and the non-linearity of physical and mechanical properties of the material (Table 1). In addition to the temperature effect, the internal surfaces of the cylinder and holes were loaded with a pressure of 14 MPa. Taking into account area of the end cap of the cylinder, the rear end surface of the cylinder was loaded with a tensile pressure of 12.87 MPa. Fig. 3 shows the stress field normal to the XZ plane under an internal pressure of 14 MPa when the internal surface temperature tint is 500 °C and the external surface temperature text is 560 °C. Obviously, the maximum stresses (yy = 230 MPa) are observed in the XZ plane of the ligaments between the holes of the superheater header, which can lead to cracks initiation in real structural elements under operating conditions. 4. Results and discussion
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The 3D model (Fig. 4) shows the "surface" and "central" lines along which the curves of stress distribution were built. The "surface" and "central" lines were determined in the places of maximum stress concentration where fatigue cracks are the most likely to occur. The influence of the internal and external surfaces temperature difference t = |tint − text| on the stress state of the superheater header was studied. Fig. 5 shows the equivalent stress intensity curves int and the normal stresses yy distribution obtained in numerical modelling. These curves are built along the "central" line at tint > text (Fig. 5a) and at tint < text (Fig. 5b). To determine the impact of the temperature on the stress state, the calculations were performed at various ranges of tint and text. The pressure on the internal surfaces of the header and its holes was set to 14 MPa. If tint > text (Fig. 5a), the stresses yy on the internal surface of the header ligament reach 50 MPa and they are approximately three times lower than at tint < text. The results (Fig. 5b) indicate that the highest equivalent stress intensity int and the normal stresses yy equal 150 MPa are observed on the internal surface of the ligament between the holes along the "central" line when the internal surface temperature is lower than the external one, that is tint < text. The normal stresses yy have a significant impact on the equivalent stress intensity int (Fig. 5a,b).
Figure 5: Distribution of stresses int and yy along the line "central" at temperature difference 50 C : a) tint > text; b) tint < text
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along the "central" line at tint < text and different t
Figure 7: Distribution of the normal stress yy
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Figure 6: Distribution of the normal stress yy
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The coincidence of the curves (Fig. 5a,b) indicates that the stresses in the superheater header depend only on the temperature difference t between the internal and external surfaces, and does not depend on their absolute values tint and text. Further examination of the effect of the temperature difference t on the stress state was performed for the condition tint < text when t gradually changes. The FEM results indicate that the stresses increase in the ligament between the holes when the internal surface temperature is lower than the temperature of the external surface (Fig. 6, Fig. 7).
along the "surface" line at tint < text and different
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If the temperature difference reaches t > 50, the maximum value of the normal stress yy along the "surface" line (Fig. 7) begins to shift from the edges of the holes towards the central region between the holes.
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3.2. Modelling of a TPP superheater header with cracks by FEM
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A system of localized semi-elliptical cracks (Fig. 8) was created in the XZ coordinate plane in the ligament between the holes. The cracks were modelled with respect to their geometry and location in real superheater headers [10]. The center of ellipse, which defines the cracks fronts (# 1 - # 8, Fig. 8), is located in the middle of the ligament between the holes on the external surface of the superheater header. The axis Oa of the ellipse was maintained constant, while the size of the axis Ob describing the depth of the crack front varied. The distance from the center of the ligament between the holes on the internal surface of the cylinder to the center of the semi-elliptical crack front was defined as a depth of crack l. This technique allows to model the whole crack front by changing only the coordinate of its middle point (crack depth).
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Figure 8: Geometry of real and modelled crack fronts in the ligament between the superheater header holes
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The surfaces created between the cracks fronts in the XZ plane can simulate solid material or cracks surface. The conditions of XZ plane symmetry were applied to represent the solid material on the planes below the contour of the front of a corresponding crack. The crack surface was described by the planes above the contour of the crack front which were not restricted in displacement. The corresponding element of operated header with a crack was superimposed with the modelled cracked ligament (Fig. 8). Their comparison shows that the modelled crack front is very close to the real geometry of crack front of perforated area.
Figure 9: Refined finite element mesh near the crack front #2
The finite element mesh was refined in the vicinity of the crack front. The minimal spacing between the finite element nodes was 0.05 mm (Fig. 9). The internal surfaces of the header's cylinder, holes and crack surfaces were loaded with pressure and temperature. The stress state was modelled under a constant internal pressure of 14 MPa and different temperatures tint < text of the internal and external surfaces of the superheater header 0 C ≤ t ≤ 60 C with a step of 15 °C. The mode I SIF KI along the front of semi-elliptical cracks was calculated for 11 equidistant points of the front
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for each of 8 semi-elliptical cracks. The direct force method for SIF calculation was used [19]. The results of KI distribution along the fronts of the modelled semi-elliptical cracks # 2 - 5 in the ligament
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between the holes in the TPP superheater header are given in Fig. 10.
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c d Figure 10: SIF distribution at various t along the crack front s: a) # 2, l = 0.0034 m; b) # 3, l = 0.00706 m; c) # 4, l = 0.01244 m; d) # 5, l = 0.01855 m
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The main factor which contributes to the augmentation of KI (Fig.10) is the increase in temperature differences t on the internal and external surfaces when tint ≤ text. 3.3. Determination of the SIF correction function
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It can be assumed that crack front is formed as an ellipse with a constant axis Oa and a variable axis Ob, which determines its depth l and symmetry of KI distribution about the midpoint of the crack front (Fig. 10). As a result, it is sufficient to determine KI and the corresponding crack extension l at the midpoint of the crack front using the fatigue crack growth diagram of material to predict crack propagation along its front. Here, the difficulty with determining KI can be due to the fact that the stress state at the crack tip is defined by a combined effect of internal pressure and thermo-mechanical stresses. Hence, the SIF correction function at constant internal pressure depends on the temperature difference t between the internal and external cylinder surfaces and the crack depth l. A series of numerical simulations was performed to determine the impact of t on the distribution of normal stresses yy(t) along the "surface" line (Fig. 4) of the superheater header model without cracks. The FEM results (Fig. 11) indicate that the stresses on the internal surface of the ligament between the holes in the plane XZ increase with a gradual decrease in temperature on the internal surface of the cylinder compared to its external surface. When t = 0 the stresses in the ligament are only caused by an internal pressure of 14 MPa.
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Figure 12: Dimensionless normal stresses in the midpoint of "surface" line versus the temperature difference t
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Figure 11: Distribution of normal stresses along the "surface" line taking into account the temperature difference t between the external and internal walls of the superheater header
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yy(t)/yy(t = 0) = 1 + 0.048t
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Fig. 12 shows the effect of t on the dimensionless normal stresses yy(t)/ yy(t = 0) in the centre of the ligament ("surface" line in the centre) under an internal steam pressure of 14 MPa. The obtained data (Fig. 12) were approximated by a linear function: (1)
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where yy(t = 0) is the normal stress in the center of the "surface" line at (t = 0). Using formula (1), we determined the relationship between the normal stress and temperature differences t in the centre of the ligament between the holes on the internal surface:
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where
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Fig. 13 shows the dependences of KI on temperature difference t and crack depth l at the midpoint of the crack front in the ligament between holes of the superheater header. When t increases from 0 to 60 °C, KI increases by almost 4 times for the same cracks depth, hence the curves move up. At the given temperature difference t the stresses yy(t) can be calculated using the relation in (Eq. (2)), as well as the SIF at the midpoint of the crack front by the formula:
is the correction function depending on the ratio
(3)
of crack depth to the wall thickness w of the
superheater header and the temperature difference t. Taking into account Equation (3), the following equation was obtained:
(4)
The values of the correction function (Eq. (4)) given in Fig. 14 were set using a power law for different values of t.
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(5)
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where A and m are the parameters to be estimated.
Figure 13: KI versus temperature difference t and crack depth l
Figure 14: Correction function versus crack depth l for different t values
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The parameter m varied insignificantly with changing t (the maximum difference is 16%). Therefore, its value was averaged and taken as a constant m = - 0.295. The values of the parameter A (Fig. 15) for each curve were corrected by the approximation of data using the function from (Eq. (5)) with constant m. The relationship between A and t (Fig. 15) was fitted with the exponential function
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is the A value at t = 60 C; A1 = 0.32 and p = 15.62 are parameters.
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where
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Figure 15: Parameter A from Equation (6) versus temperature difference t
Considering Eq. (5, 6) we can write
(7)
The relation given above is a SIF correction function for Eq. (3). The curves given in Fig. 14 illustrate the correction function (Eq. (7)) depending on the crack depth and temperature difference between the external and internal surfaces of the superheater header. Application of equations (3) and (7) allows determining the SIF in the midpoint of semi-elliptical crack for further crack growth prediction of entire semi-elliptical crack front.
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4. Determination of residual durability of superheater header. To simplify the analysis while assessing the durability, the temperature fluctuation range under quasi-stable mode of header operation was classified similarly to [2]. Table 3 shows the steam temperature range and stresses for each class, respectively. Since the stress range yy, caused by temperature fluctuations of Class 0 is insignificant, it was not taken into account in the numerical analysis.
Temperature range, t,о С
0 1 2
t < 10 t < 30 t > 30
Number of temperature fluctuations, per day 151 87 1
Maximum stress yy, MPa
Stress range yy, MPa
up to 58.4 up to 96.3 up to 127
19.0 56.9 87.5
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Table 3. The ranges of steam temperature fluctuations.
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The local maxima and minima on the digitized steam temperature history in superheater header were found, and afterwards the number of cycles of a certain class was determined. The number of temperature fluctuations was calculated for the period of 50.25 hours, and after that, it was recalculated for the day of operation for each class, presented in Table 2. Table 4 presents the input data for the calculation of residual durability of superheater header with a defect of 25 mm in depth along the central hole. It was assumed, that the crack grows by two laws. In the first case, Paris law was employed [20]: da n C K , dN
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where C , n are the experimentally determined parameters; N is the number of loading cycles; a is the crack depth; K Kmax Kmin is the stress intensity factor range (SIF), Kmax , Kmin are the maximum and minimum SIF of loading cycle, respectively. In the second case, NASGRO equation was used, which is written in the form of equation [21]: da
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Kth 1 Keff
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K max 1 K fc
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where C1 , m1 , p , q are experimentally determined material parameters; Kth is the threshold SIF range; Keff Kmax Kop is the effective SIF range; Kop is the crack opening SIF. The С and n parameters of Paris law for 12Cr1MoV steel at 500° С were taken from [22], according to which С = 1.96·10-10 m/cycle MPa
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, n = 2. Also, the parameters C1, m1, p and q of NASGRO equation were taken
from the same paper [22], as well as Kth and Kfc. Therefore, Kth = 3.98 MPa m , Kfc = 37.6 MPa m , C1 = 2.0·10-9 m/cycle MPa
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, m1 1 , p = 1.95, q = 1.15.
In paper [22], in order to describe the experimental data of FCG rate in a 12CrMoV steel the effective SIF range Keff in the equation (9) was replaced by K . That is why, the same approach was used in the present paper. In the modeling, SIF was calculated by the formula (7).
(9)
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The main parameters of operational loading of superheater header under quasi-stable mode of operation are given in Table 4.
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Table 4. The main parameters of operational loading of superheater header under quasi-stable mode of operation. Operational parameters Quasi-stable temperature fluctuations Loading waveform Saw-tooth Stress range elastic Frequency 5∙10-4 to 5∙10-3 Stress ratio, R 0.3 – 0.6 Number of occurrences per year Class 1: 31755 Class 2: 365
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The number of temperature fluctuations is equal to 87 times a day for Class 1 and 1 time per day for the Class 2 and they occur each 0.28 and 24 hours, accordingly. The number of cycles for a crack to grow from the initial crack depth of 25 mm to a critical defect size of 35 mm by employing Paris law is shown in Fig. 16. With the increase of temperature difference between the external and internal surfaces of header from 10 ° C to 50 ° C, the number of loading cycles required for a crack to grow to a depth of 35 mm is reduced in 24.72 times from 2.2∙10 6 to 8.9∙104, or from 251.17 thousand hours to 10.16 thousand hours. It was calculated, that the average magnitude of temperature fluctuations is 15 °C for Class 1 and is 46.2 °C for Class 2. For a crack to grow from the initial depth of 25 mm to a critical defect size of 35 mm under temperature fluctuations of Class 1 (Class 2) there must be 112 thousand hours and 10.2 thousand hours of operation. When performing the modeling by NASGRO equation, the number of cycles to reach the given before crack depth is in general smaller as compared with Paris equation. This is due to the fact that NASGRO equation takes into account the entire FCG diagram and, therefore, allows to increase the accuracy by providing better model. In this case, for a crack to grow from the initial depth of 25 mm to a critical defect size of 35 mm under temperature fluctuations of Class 2 there must be 2400 thousand hours of operation, that is 4.25 less in compare with the assessment by means of Paris law. It was assumed that the mechanism of crack growth is determined only by fatigue, which can be divided into Class 1 and Class 2 temperature fluctuations.
a Fig. 16. The dependency of crack depth along the hole of pipe on the number of loading cycles: t = 10 ° C – (1); 15 ° C – (2); 20 ° C – (3); 30 ° C – (4); 40 ° C – (5); 46.2 ° C – (6); 50 ° C – (7), modeling by Paris law
Fig. 17. The dependency of crack depth along the hole of pipe on the number of loading cycles: t = 15 ° C – (1); 20 ° C – (2); 30 ° C – (3); 40 ° C – (4); 46.2 ° C – (5); 50 ° C – (6), modeling by NASGRO equation
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Thus, the extension of the header lifetime can be achieved by reducing the temperature fluctuations range and their frequency. 5. Conclusions
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The residual durability superheater header was investigated by analyzing the stresses calculated by the finite element method to determine the maximum allowable size of the defect and the time of its growth up to a critical depth along the center hole. The stresses in the superheater header depend only on the temperature difference between the internal and external surfaces in the studied temperature range (20 - 600 C), and not on their values. The most dangerous
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operation modes of the superheater header occur when the temperature on its internal surface is lower than that on theexternal one. When this temperature difference increases from 0 to 100 C, the normal stress on the internal surface of the ligament between the superheater holes increases by 5 times to 230 MPa, which exceeds the yield stress of the material at a temperature of 500 C. The impact of steam pressure on the stresses in the superheater header is not significant at t = 0 C; it generates stresses int < 75 MPa in the ligament between the holes. The stress state of the superheater header with cracks was calculated considering the operation loading modes: steam pressure and temperature difference t = |text - tint| between the external and internal surfaces of the header when tint text. The SIF distributions of mode I (KI) were determined along the front of semi-elliptical crack by the direct method. When t increases from 0 to 60 C, the KI values increase by almost 4 times at the same depth of cracks and constant internal steam pressure. The SIF correction function at the midpoint of the semi-elliptical crack front in the ligament between holes of the superheater header was obtained. The function depends on the crack depth and the temperature difference between the external and internal surfaces of the superheater header under a constant internal pressure of steam. The dependences of the residual life of superheater heater on temperature fluctuations under the quasi-stable mode of operation were obtained. The loading cycles due to thermal fluctuations were grouped into two classes. It was revealed that the steam temperature fluctuations under quasi-stable mode of operation greatly contribute to the growth of cracks. These loading cycles were identified and determined to be the most influential factor in the growth of cracks. The modeling was performed by Paris law and NASGRO equation. It was shown, the latter one gives more precise results. 6. Acknowledgements
References
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This study is part of the research project no. 0115U002447 financed by the Ministry of Education and Science of Ukraine. The authors kindly acknowledge the support in accomplishment of the research program.
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I.M. Dmytrakh, A.B. Vainman, M.H. Stashchuk, L. Toth, Reliability and durability of structural elements for heat-and-power engineering equipment, Academperiodyka (in Ukrainian), Kyiv, 2005.
[2]
O. Kwon, M. Myers, A.D. Karstensen, D. Knowles, The effect of the steam temperature fluctuations during steady state operation on the remnant life of the superheater header, Int. J. Press. Vessel. Pip. 83 (2006) 349–358. doi:10.1016/j.ijpvp.2006.02.012.
[3]
O.P. Yasniy, Y.I. Pyndus, V.P. Iasnii, Evaluation of stress intensity factors in semi-elliptical cracks of thermal power plant superheater collector, Sci. Notes. 47 (2014) 211–220.
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A.K. Ray, Y.N. Tiwari, P.K. Roy, S. Chaudhuri, S.C. Bose, R.N. Ghosh, J.D. Whittenberger, Creep rupture analysis and remaining life assessment of 2.25Cr–1Mo steel tubes from a thermal power plant, Mater. Sci. Eng. A. 454–455 (2007) 679–684. doi:10.1016/j.msea.2006.11.019.
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Highlights
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frequency.
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There was estimated the residual lifetime of superheater header starting from the initial defect size and up to the maximum allowable one. The registered operational data of steam temperature in header were used. The stress intensity factors at the crack tip in the ligament between the holes of superheater header were estimated by FEM. The lifetime of header can be extended due to the decrease of fluctuations of temperature range and their
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O. Yasniy, Yu. Pyndus, V. Iasnii, Y. Lapusta PII: DOI: Reference:
S1350-6307(17)30281-9 doi: 10.1016/j.engfailanal.2017.07.028 EFA 3246
To appear in:
Engineering Failure Analysis
Received date: Revised date: Accepted date:
30 March 2017 5 July 2017 26 July 2017
Please cite this article as: O. Yasniy, Yu. Pyndus, V. Iasnii, Y. Lapusta , Residual lifetime assessment of thermal power plant superheater header, Engineering Failure Analysis (2016), doi: 10.1016/j.engfailanal.2017.07.028
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ACCEPTED MANUSCRIPT Residual lifetime assessment of thermal power plant superheater header O. Yasniya,*, Yu. Pyndusa, V. Iasniia, Y. Lapustab
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a) Ternopil Ivan Pul'uj National Technical University, Ruska 56, Ternopil, 46001, Ukraine b) Institut Pascal, UMR 6602 / UBP / CNRS / IFMA, Clermont Université, BP 265, 63175 Aubière Cedex, France
Abstract
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The safe operation of thermal power plants (TPPs) is largely determined by the reliability of main components of steam systems of power units, to which belong the headers of boiler superheaters. The aim of this study is to estimate the residual lifetime of superheater header starting from the initial defect size and up to the maximum allowable one. The above-mentioned lifetime is affected by the fluctuations of temperature under steady mode of operation of TPP superheater header. Superheaters headers operate in a steamy environment under the pressure of 15.5 MPa at temperature of 545 ºС. The header is a thick-walled cylinder made of 12Cr1MoV steel with a length of 2314 mm, outer diameter of 325 mm and thickness of 50 mm. To estimate the residual lifetime of the header, the steam temperature in header was measured and digitized. The temperature range of the header under quasi-stable operational mode was divided into three classes: (1) t < 10 °C; (2) 10 °C < t < 30 °C; (3) t > 30 °C. The local minima and maxima were determined based on the obtained steam temperature history. The residual durability was evaluated taking into account the effect of thermo-mechanical stresses and also the stresses caused by internal steam pressure. The stress intensity factors (SIFs) at the crack tip in the ligament between the holes of superheater collector were estimated by FEM. The SIF of mode I (KI) were determined along the fronts of modelled semi-elliptical cracks. The SIF correction function at the midpoint of the semi-elliptical crack front in the ligament between holes of the superheater header versus the crack depth and the temperature difference between the external and internal surfaces under a constant internal pressure of steam was obtained. Based on the analysis of header defects, the proposed front shape was taken in the form of the semi-ellipse. The crack growth at 500 ° C was modelled by Paris and NASGRO equations. With the increase of temperature difference between the external and internal header surfaces from 10 °С to 50 °С, the number of cycles, that is necessary for the crack to reach 35 mm in depth, decreases approximately in 25 times. It was calculated, that the average value of temperature fluctuations is 15°С for the Class 1, and is 46.2 °С for the Class 2. The lifetime of header can be extended due to the decrease of fluctuations of temperature range and their frequency.
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Keywords: Stress intensity factors; finite element analysis; fatigue crack growth; life assessment; fatigue.
* Corresponding author. Tel.: +38-097-9-55-55-23; fax: +38-0352-254-983.. E-mail address: [email protected] (O. Yasniy)
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minimum stress
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min max yy
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crack depth parameter of Paris equation parameter of NASGRO equation stress intensity factor, mode I minimum stress intensity factor, mode I maximum stress intensity factor, mode I thickness of collector wall number of loading cycles parameter of Paris equation parameter of NASGRO equation parameter of NASGRO equation parameter of NASGRO equation temperature time stress intensity factor range temperature range stress range normal stress range
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a C C1 KI K min K max w N n m1 p q t T K t yy
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Nomenclature
maximum stress
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normal stress
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1. Introduction
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The safe operation of thermal power plants (TPPs) is largely determined by the reliability of main elements of steam power system [1] to which belong the boilers superheaters headers. The problem of ensuring the reliability of TPP equipment is being aggravated with the prolonged operation and with detected damage. The main factors that lead to the emergence of cracks is the inhomogeneous distribution of temperature field along the wall thickness and the high internal steam pressure. The durability of superheaters headers depends on the stress level, that arise in the most heavily loaded areas. Usually the cracks appear on the inner surface of the ligaments between nozzles holes [2,3]. The proper operation of power stations considerably depends on the reliability of the basic elements of steam power systems [4–6] such as superheaters headers in boilers [7–9]. The problems with safety of thermal power plant equipment are aggravated by long-term operation of such equipment, which in many cases exceeds the designed service life, as well as by observed significant damage and multiple cracks in superheaters headers [10]. The main factors which cause the cracking and failure of superheaters include a non-uniform distribution of temperature through the wall thickness and high internal pressure of steam [2,11]. The life of superheaters headers also depends on stresses that occur in the most loaded regions. Typically, multiple defects appear on the internal surface of the ligament between header holes [2,10,12,13]. Taking into account the existing experience with respect to operation modes of TPPs superheater headers [13– 15], the regions which are the most prone to operational damage and defects are the ligaments between the holes of nozzles supplying superheated steam. The stress state in superheater headers was studied, for example, in the papers [2,16].
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2. Temperature fluctuations
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The deformation of the TPP superheater header during operation was modelled by means of FEM [16] in elastic formulation. It was found that due to internal pressure, the equivalent stress intensity in the steady-state conditions of the header is the highest in the vicinity of the holes and reaches only 0.43 of the yield stress y = 290 MPa while the axial stress normal to the plane of possible crack initiation yy is 0.15y. However, the thermal stresses caused by rapid cooling of the hot header due to water penetration with a condensate or significantly colder steam into some holes reached value of yield strength of 12Cr1MoV in the ligament between holes at the temperature difference equal to 186 °C during the stoppage of equipment [16]. The aforementioned and other studies investigate the stress state only under certain operational parameters (internal pressure, temperature difference between the internal and external surfaces of the header, etc.); however, they do not examine the effect of changing these parameters on the stress state. The latter is very important to substantiate the restrictions concerning the operation modes of power plant equipment with operational damage and to investigate the influence of temperature difference on the inner and outer surfaces of header on its stress state, in particular, taking into account the damage in a form of semi-elliptical cracks between the holes, that is observed in practice. The aim of the study is to estimate the residual lifetime of superheater header from the current size defect and up to the maximum allowable one. The above-mentioned lifetime is caused by the fluctuations of temperature under quasistatic operational regime of TPP superheater header.
The information about header, from which the specimens were cut and its modes of operation are given in Table 1.
Value 2 1976 hot 325 12Cr1MoV 1257 14 MPa 545 °С – internal 8. Operational temperature (external and internal walls) 565 °С – external 9. Place of template cut ( distance from the end cap of header) 3.45 m 10. Period of operation, hours 178000 11. Thickness, mm 50
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Parameters Block number Year of operation start Header type (cold, hot) External header diameter, mm Steel grade Number of starts during operation Internal pressure during operation
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# 1. 2. 3. 4. 5. 6. 7.
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Table 1. The information about the header
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The characteristics of mechanical properties of steel in the initial state: yield strength 0.2 = 320 MPa, ultimate tensile strength U = 480 MPa [17]. The chemical composition of 12Cr1MoV steel is 0.12 % С; 0.54 % Mn; 0.26 % Si; <0.015 % P; <0.019 % S; <1.0 % Cr; <1.0 % Ni. The operational data of registered temperature steam, which were provided by TPP, were used for the evaluation of residual durability. The steam temperature during header operation was measured with chromel-alumel thermocouple and was registered during 54.5 hours on analog potentiometer. Fig. 1 shows the example of digitized temperature dependence of steam during operational mode, which is interpreted as typical for superheater header at constant loading without cold, semi-cold and hot starts of block.
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t, ° C
T, s
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3. FEM modeling of the superheater header
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Fig. 1. The digitized temperature dependence of steam in superheater header under stable mode of operation.
3.1. Modeling of a TPP superheater header with cracks by FEM
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A FE model of the TP-100 boiler superheater header in the form of a thick-walled cylinder with holes was prepared in ANSYS Workbench. The modelling involved solving an elastic three-dimensional problem. Taking into account the symmetry conditions [2,16] was analyzed a fragment (quarter) of the superheater header (Fig. 2). The model is built in a rectangular Cartesian system of coordinates. The orientation of the Cartesian coordinate system is shown in Fig. 2. The inside and outside diameters of the collector are 225 mm and 325 mm, respectively. The holes are 22 mm in diameter, the angle between the axes of the holes in the XOY plane is 20 °C. The spacing between the rows of holes along the OY axis is 80 mm. The 10-node 3D finite element SOLID87 was used for numerical analysis. This finite element has one degree of freedom (temperature) in each node and it is suitable for 3D modelling of complex curved shape by static and transient thermal analysis. The model meshing was performed using mapped face meshing (Fig. 1). The finite element mesh was refined between the holes. The total number of finite elements in the model was 40432. The finite element SOLID87 was replaced by an equivalent 10-node structural element, SOLID187, to perform further calculations in the Static Structural module. SOLID187 is a 3D element, which consists of 10 nodes, each node having three degrees of freedom: translation in the directions OX, OY and OZ. This element also has the properties of elasticity, plasticity, hyper-elasticity, creep, stiffness, large deflections and strains. In addition, the element can be loaded with forces, stresses, displacements and temperature, either separately or in combination [18].
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Fig. 3. The distribution of temperature field of header at internal surface temperature of 500°С, and external of 560°С.
Fig. 2. The full-scale FE model of superheater header.
central
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Fig. 4. The distribution of normal stresses yy in the superheater header under internal pressure of 14 MPa and temperature of internal surface of 500 °С, and external of 560 °С.
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The finite element mesh was refined in the vicinity of the crack front. The minimal spacing between the finite element nodes was 0.05 mm. The internal surfaces of the headers cylinder, holes and crack surfaces were loaded with pressure and temperature. The thermo-physical and mechanical properties of 12Cr1MoV steel are given in Table 2. For a more reliable representation of the material behavior under the combined thermo-mechanical effect, the FEM model of the superheater header [14] takes into account only these parameters which vary with the change of temperature. Table 2: Temperature properties of material
Temperature,
Young's modulus,
t, ° C 20 100 200 300 400
E∙105, MPa 1.98 1.93 1.88 1.83 1.75
Poisson's Coefficient of Thermal ratio thermal expansion, conductivity, 0.27 0.28 0.29 0.30 0.32
1.24 1.30 1.36 1.40 1.44
46 45 44 42 40
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500 600
1.67 1.57
0.33 0.34
1.47 1.49
37 35
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The transient thermal analysis results were transferred to the Static Structural module. The stress state of the superheater header was calculated taking into account the non-uniform thermal expansion of the material through the thickness of header wall and the non-linearity of physical and mechanical properties of the material (Table 1). In addition to the temperature effect, the internal surfaces of the cylinder and holes were loaded with a pressure of 14 MPa. Taking into account area of the end cap of the cylinder, the rear end surface of the cylinder was loaded with a tensile pressure of 12.87 MPa. Fig. 3 shows the stress field normal to the XZ plane under an internal pressure of 14 MPa when the internal surface temperature tint is 500 °C and the external surface temperature text is 560 °C. Obviously, the maximum stresses (yy = 230 MPa) are observed in the XZ plane of the ligaments between the holes of the superheater header, which can lead to cracks initiation in real structural elements under operating conditions. 4. Results and discussion
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The 3D model (Fig. 4) shows the "surface" and "central" lines along which the curves of stress distribution were built. The "surface" and "central" lines were determined in the places of maximum stress concentration where fatigue cracks are the most likely to occur. The influence of the internal and external surfaces temperature difference t = |tint − text| on the stress state of the superheater header was studied. Fig. 5 shows the equivalent stress intensity curves int and the normal stresses yy distribution obtained in numerical modelling. These curves are built along the "central" line at tint > text (Fig. 5a) and at tint < text (Fig. 5b). To determine the impact of the temperature on the stress state, the calculations were performed at various ranges of tint and text. The pressure on the internal surfaces of the header and its holes was set to 14 MPa. If tint > text (Fig. 5a), the stresses yy on the internal surface of the header ligament reach 50 MPa and they are approximately three times lower than at tint < text. The results (Fig. 5b) indicate that the highest equivalent stress intensity int and the normal stresses yy equal 150 MPa are observed on the internal surface of the ligament between the holes along the "central" line when the internal surface temperature is lower than the external one, that is tint < text. The normal stresses yy have a significant impact on the equivalent stress intensity int (Fig. 5a,b).
Figure 5: Distribution of stresses int and yy along the line "central" at temperature difference 50 C : a) tint > text; b) tint < text
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along the "central" line at tint < text and different t
Figure 7: Distribution of the normal stress yy
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Figure 6: Distribution of the normal stress yy
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The coincidence of the curves (Fig. 5a,b) indicates that the stresses in the superheater header depend only on the temperature difference t between the internal and external surfaces, and does not depend on their absolute values tint and text. Further examination of the effect of the temperature difference t on the stress state was performed for the condition tint < text when t gradually changes. The FEM results indicate that the stresses increase in the ligament between the holes when the internal surface temperature is lower than the temperature of the external surface (Fig. 6, Fig. 7).
along the "surface" line at tint < text and different
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If the temperature difference reaches t > 50, the maximum value of the normal stress yy along the "surface" line (Fig. 7) begins to shift from the edges of the holes towards the central region between the holes.
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3.2. Modelling of a TPP superheater header with cracks by FEM
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A system of localized semi-elliptical cracks (Fig. 8) was created in the XZ coordinate plane in the ligament between the holes. The cracks were modelled with respect to their geometry and location in real superheater headers [10]. The center of ellipse, which defines the cracks fronts (# 1 - # 8, Fig. 8), is located in the middle of the ligament between the holes on the external surface of the superheater header. The axis Oa of the ellipse was maintained constant, while the size of the axis Ob describing the depth of the crack front varied. The distance from the center of the ligament between the holes on the internal surface of the cylinder to the center of the semi-elliptical crack front was defined as a depth of crack l. This technique allows to model the whole crack front by changing only the coordinate of its middle point (crack depth).
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Figure 8: Geometry of real and modelled crack fronts in the ligament between the superheater header holes
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The surfaces created between the cracks fronts in the XZ plane can simulate solid material or cracks surface. The conditions of XZ plane symmetry were applied to represent the solid material on the planes below the contour of the front of a corresponding crack. The crack surface was described by the planes above the contour of the crack front which were not restricted in displacement. The corresponding element of operated header with a crack was superimposed with the modelled cracked ligament (Fig. 8). Their comparison shows that the modelled crack front is very close to the real geometry of crack front of perforated area.
Figure 9: Refined finite element mesh near the crack front #2
The finite element mesh was refined in the vicinity of the crack front. The minimal spacing between the finite element nodes was 0.05 mm (Fig. 9). The internal surfaces of the header's cylinder, holes and crack surfaces were loaded with pressure and temperature. The stress state was modelled under a constant internal pressure of 14 MPa and different temperatures tint < text of the internal and external surfaces of the superheater header 0 C ≤ t ≤ 60 C with a step of 15 °C. The mode I SIF KI along the front of semi-elliptical cracks was calculated for 11 equidistant points of the front
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for each of 8 semi-elliptical cracks. The direct force method for SIF calculation was used [19]. The results of KI distribution along the fronts of the modelled semi-elliptical cracks # 2 - 5 in the ligament
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between the holes in the TPP superheater header are given in Fig. 10.
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c d Figure 10: SIF distribution at various t along the crack front s: a) # 2, l = 0.0034 m; b) # 3, l = 0.00706 m; c) # 4, l = 0.01244 m; d) # 5, l = 0.01855 m
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The main factor which contributes to the augmentation of KI (Fig.10) is the increase in temperature differences t on the internal and external surfaces when tint ≤ text. 3.3. Determination of the SIF correction function
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It can be assumed that crack front is formed as an ellipse with a constant axis Oa and a variable axis Ob, which determines its depth l and symmetry of KI distribution about the midpoint of the crack front (Fig. 10). As a result, it is sufficient to determine KI and the corresponding crack extension l at the midpoint of the crack front using the fatigue crack growth diagram of material to predict crack propagation along its front. Here, the difficulty with determining KI can be due to the fact that the stress state at the crack tip is defined by a combined effect of internal pressure and thermo-mechanical stresses. Hence, the SIF correction function at constant internal pressure depends on the temperature difference t between the internal and external cylinder surfaces and the crack depth l. A series of numerical simulations was performed to determine the impact of t on the distribution of normal stresses yy(t) along the "surface" line (Fig. 4) of the superheater header model without cracks. The FEM results (Fig. 11) indicate that the stresses on the internal surface of the ligament between the holes in the plane XZ increase with a gradual decrease in temperature on the internal surface of the cylinder compared to its external surface. When t = 0 the stresses in the ligament are only caused by an internal pressure of 14 MPa.
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Figure 12: Dimensionless normal stresses in the midpoint of "surface" line versus the temperature difference t
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Figure 11: Distribution of normal stresses along the "surface" line taking into account the temperature difference t between the external and internal walls of the superheater header
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Fig. 12 shows the effect of t on the dimensionless normal stresses yy(t)/ yy(t = 0) in the centre of the ligament ("surface" line in the centre) under an internal steam pressure of 14 MPa. The obtained data (Fig. 12) were approximated by a linear function: (1)
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where yy(t = 0) is the normal stress in the center of the "surface" line at (t = 0). Using formula (1), we determined the relationship between the normal stress and temperature differences t in the centre of the ligament between the holes on the internal surface:
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where
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Fig. 13 shows the dependences of KI on temperature difference t and crack depth l at the midpoint of the crack front in the ligament between holes of the superheater header. When t increases from 0 to 60 °C, KI increases by almost 4 times for the same cracks depth, hence the curves move up. At the given temperature difference t the stresses yy(t) can be calculated using the relation in (Eq. (2)), as well as the SIF at the midpoint of the crack front by the formula:
is the correction function depending on the ratio
(3)
of crack depth to the wall thickness w of the
superheater header and the temperature difference t. Taking into account Equation (3), the following equation was obtained:
(4)
The values of the correction function (Eq. (4)) given in Fig. 14 were set using a power law for different values of t.
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(5)
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where A and m are the parameters to be estimated.
Figure 13: KI versus temperature difference t and crack depth l
Figure 14: Correction function versus crack depth l for different t values
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The parameter m varied insignificantly with changing t (the maximum difference is 16%). Therefore, its value was averaged and taken as a constant m = - 0.295. The values of the parameter A (Fig. 15) for each curve were corrected by the approximation of data using the function from (Eq. (5)) with constant m. The relationship between A and t (Fig. 15) was fitted with the exponential function
(6)
is the A value at t = 60 C; A1 = 0.32 and p = 15.62 are parameters.
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where
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Figure 15: Parameter A from Equation (6) versus temperature difference t
Considering Eq. (5, 6) we can write
(7)
The relation given above is a SIF correction function for Eq. (3). The curves given in Fig. 14 illustrate the correction function (Eq. (7)) depending on the crack depth and temperature difference between the external and internal surfaces of the superheater header. Application of equations (3) and (7) allows determining the SIF in the midpoint of semi-elliptical crack for further crack growth prediction of entire semi-elliptical crack front.
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4. Determination of residual durability of superheater header. To simplify the analysis while assessing the durability, the temperature fluctuation range under quasi-stable mode of header operation was classified similarly to [2]. Table 3 shows the steam temperature range and stresses for each class, respectively. Since the stress range yy, caused by temperature fluctuations of Class 0 is insignificant, it was not taken into account in the numerical analysis.
Temperature range, t,о С
0 1 2
t < 10 t < 30 t > 30
Number of temperature fluctuations, per day 151 87 1
Maximum stress yy, MPa
Stress range yy, MPa
up to 58.4 up to 96.3 up to 127
19.0 56.9 87.5
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Table 3. The ranges of steam temperature fluctuations.
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The local maxima and minima on the digitized steam temperature history in superheater header were found, and afterwards the number of cycles of a certain class was determined. The number of temperature fluctuations was calculated for the period of 50.25 hours, and after that, it was recalculated for the day of operation for each class, presented in Table 2. Table 4 presents the input data for the calculation of residual durability of superheater header with a defect of 25 mm in depth along the central hole. It was assumed, that the crack grows by two laws. In the first case, Paris law was employed [20]: da n C K , dN
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where C , n are the experimentally determined parameters; N is the number of loading cycles; a is the crack depth; K Kmax Kmin is the stress intensity factor range (SIF), Kmax , Kmin are the maximum and minimum SIF of loading cycle, respectively. In the second case, NASGRO equation was used, which is written in the form of equation [21]: da
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Kth 1 Keff
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where C1 , m1 , p , q are experimentally determined material parameters; Kth is the threshold SIF range; Keff Kmax Kop is the effective SIF range; Kop is the crack opening SIF. The С and n parameters of Paris law for 12Cr1MoV steel at 500° С were taken from [22], according to which С = 1.96·10-10 m/cycle MPa
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from the same paper [22], as well as Kth and Kfc. Therefore, Kth = 3.98 MPa m , Kfc = 37.6 MPa m , C1 = 2.0·10-9 m/cycle MPa
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In paper [22], in order to describe the experimental data of FCG rate in a 12CrMoV steel the effective SIF range Keff in the equation (9) was replaced by K . That is why, the same approach was used in the present paper. In the modeling, SIF was calculated by the formula (7).
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The main parameters of operational loading of superheater header under quasi-stable mode of operation are given in Table 4.
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Table 4. The main parameters of operational loading of superheater header under quasi-stable mode of operation. Operational parameters Quasi-stable temperature fluctuations Loading waveform Saw-tooth Stress range elastic Frequency 5∙10-4 to 5∙10-3 Stress ratio, R 0.3 – 0.6 Number of occurrences per year Class 1: 31755 Class 2: 365
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The number of temperature fluctuations is equal to 87 times a day for Class 1 and 1 time per day for the Class 2 and they occur each 0.28 and 24 hours, accordingly. The number of cycles for a crack to grow from the initial crack depth of 25 mm to a critical defect size of 35 mm by employing Paris law is shown in Fig. 16. With the increase of temperature difference between the external and internal surfaces of header from 10 ° C to 50 ° C, the number of loading cycles required for a crack to grow to a depth of 35 mm is reduced in 24.72 times from 2.2∙10 6 to 8.9∙104, or from 251.17 thousand hours to 10.16 thousand hours. It was calculated, that the average magnitude of temperature fluctuations is 15 °C for Class 1 and is 46.2 °C for Class 2. For a crack to grow from the initial depth of 25 mm to a critical defect size of 35 mm under temperature fluctuations of Class 1 (Class 2) there must be 112 thousand hours and 10.2 thousand hours of operation. When performing the modeling by NASGRO equation, the number of cycles to reach the given before crack depth is in general smaller as compared with Paris equation. This is due to the fact that NASGRO equation takes into account the entire FCG diagram and, therefore, allows to increase the accuracy by providing better model. In this case, for a crack to grow from the initial depth of 25 mm to a critical defect size of 35 mm under temperature fluctuations of Class 2 there must be 2400 thousand hours of operation, that is 4.25 less in compare with the assessment by means of Paris law. It was assumed that the mechanism of crack growth is determined only by fatigue, which can be divided into Class 1 and Class 2 temperature fluctuations.
a Fig. 16. The dependency of crack depth along the hole of pipe on the number of loading cycles: t = 10 ° C – (1); 15 ° C – (2); 20 ° C – (3); 30 ° C – (4); 40 ° C – (5); 46.2 ° C – (6); 50 ° C – (7), modeling by Paris law
Fig. 17. The dependency of crack depth along the hole of pipe on the number of loading cycles: t = 15 ° C – (1); 20 ° C – (2); 30 ° C – (3); 40 ° C – (4); 46.2 ° C – (5); 50 ° C – (6), modeling by NASGRO equation
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Thus, the extension of the header lifetime can be achieved by reducing the temperature fluctuations range and their frequency. 5. Conclusions
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The residual durability superheater header was investigated by analyzing the stresses calculated by the finite element method to determine the maximum allowable size of the defect and the time of its growth up to a critical depth along the center hole. The stresses in the superheater header depend only on the temperature difference between the internal and external surfaces in the studied temperature range (20 - 600 C), and not on their values. The most dangerous
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operation modes of the superheater header occur when the temperature on its internal surface is lower than that on theexternal one. When this temperature difference increases from 0 to 100 C, the normal stress on the internal surface of the ligament between the superheater holes increases by 5 times to 230 MPa, which exceeds the yield stress of the material at a temperature of 500 C. The impact of steam pressure on the stresses in the superheater header is not significant at t = 0 C; it generates stresses int < 75 MPa in the ligament between the holes. The stress state of the superheater header with cracks was calculated considering the operation loading modes: steam pressure and temperature difference t = |text - tint| between the external and internal surfaces of the header when tint text. The SIF distributions of mode I (KI) were determined along the front of semi-elliptical crack by the direct method. When t increases from 0 to 60 C, the KI values increase by almost 4 times at the same depth of cracks and constant internal steam pressure. The SIF correction function at the midpoint of the semi-elliptical crack front in the ligament between holes of the superheater header was obtained. The function depends on the crack depth and the temperature difference between the external and internal surfaces of the superheater header under a constant internal pressure of steam. The dependences of the residual life of superheater heater on temperature fluctuations under the quasi-stable mode of operation were obtained. The loading cycles due to thermal fluctuations were grouped into two classes. It was revealed that the steam temperature fluctuations under quasi-stable mode of operation greatly contribute to the growth of cracks. These loading cycles were identified and determined to be the most influential factor in the growth of cracks. The modeling was performed by Paris law and NASGRO equation. It was shown, the latter one gives more precise results. 6. Acknowledgements
References
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This study is part of the research project no. 0115U002447 financed by the Ministry of Education and Science of Ukraine. The authors kindly acknowledge the support in accomplishment of the research program.
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Highlights
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frequency.
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There was estimated the residual lifetime of superheater header starting from the initial defect size and up to the maximum allowable one. The registered operational data of steam temperature in header were used. The stress intensity factors at the crack tip in the ligament between the holes of superheater header were estimated by FEM. The lifetime of header can be extended due to the decrease of fluctuations of temperature range and their
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