Keywords

1 Introduction

The main pipelines in power plants play a critical role in transporting coolant from boilers to power generation equipment [1]. These pipelines, which consist of multiple layers, are responsible for simultaneously conveying both cold and hot working fluids. Due to temperature differences between the inner and outer layers, the main pipelines experience significant temperature gradients and high thermal stress. Figure 1 illustrates a schematic diagram of the main pipeline's structure and its operational concept. Operating in high-temperature and high-pressure environments, the main pipelines undergo creep and increasing the risk of leaks or ruptures. Therefore, it is crucial to consider the influence of creep on the main pipelines. Similarly, during the operation of main pipelines, they are subjected to cyclic loading, such as pressure fluctuations caused by temperature variations or changes in operating modes, which result in fatigue failure. Additionally, main pipelines typically are often very long and contain complex local structures which significantly increase the computational requirements for their analyses. Consequently, the assessment of main pipelines becomes challenging and lengthens the development cycle.

Fig. 1
A schematic has a cylindrical structure with the following components. Hot gas duct, reactor pressure vessel, cone-shaped cylindrical pipe, corrugated pipe, hot gas duct casing, thermal insulation layer, and a high temperature fluid is flowing to the right.

Schematic diagram of main pipeline

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There has been a great deal of research on the mechanisms of creep-fatigue. In 1992, Kitamura et al. [2] proposed a simplified stochastic model to predict the distribution of crack initiation and early-stage propagation under creep and creep-fatigue loading conditions. This model was constructed based on the concept of critical damage necessary for damage accumulation and intergranular failure. Wei et al. [3] developed a creep-fatigue damage accumulation model for creep-brittle materials to predict the propagation of creep and fatigue cracks under periodic and continuous creep and fatigue loading. Farrahi et al. [4] performed deterministic calculations on pressurized pipes with external longitudinal semi-elliptical surface cracks to determine the propagation of creep-fatigue cracks. However, these studies have focused primarily on microscale phenomena, limiting their practical guidance for creep-fatigue assessment in engineering applications. In engineering practice, creep-fatigue assessment is typically conducted according to the corresponding formulas specified in the ASME Boiler and Pressure Vessel Code [5], which is a set of standards established by the American Society of Mechanical Engineers (ASME).

Although, many studies have been conducted, however, currently, there lack automated creep-fatigue assessment capabilities for structures with discontinuous pipes in commercial software [6] for engineers. This paper proposes an efficient strategy for evaluating creep-fatigue in main pipelines with complex local structures and large computational requirements. To simplify the assessment process, non-critical irregular sections in the pipelines are approximated using regular pipelines with equivalent stiffness. This simplification improves both the computational accuracy and efficiency of the evaluation, while also identifying critical locations prone to creep-fatigue. Additionally, a post-processing software based on ASME standards has been developed, utilizing existing commercial finite element software to automate the assessment of main pipelines. This approach significantly enhances computational efficiency compared to alternative strategies.

2 Method

To achieve automated and efficient assessment, it is crucial to consider reducing the overall computational requirements while ensuring accuracy from the modelling stage. During the geometric modelling phase, it is essential to identify the components of irregular structures and determine whether they are critical components of the main pipeline. If these irregular structures are non-critical components, an approach for finding equivalent replacements should be employed to simplify the assessment process. This not only reduces the computational requirements for assessing the main pipeline but also eliminates interference caused by irregular components, thus improving computational efficiency.

In the assessment phase following the computational stage, for continuous straight pipelines, evaluation can be limited to key locations such as structural discontinuities or material transitions. This approach eliminates the need to assess all cross-sections exhaustively, greatly reducing the computational workload. Taking the main pipeline with the structure shown in Fig. 1 as an example, the assessment steps and strategies will be introduced.

2.1 Simplified Components of the Main Pipeline

Taking the example of the corrugated tube as an irregular structure within the main pipeline, the corrugated tube is considered a non-critical component. The structure of the corrugated tube is irregular. Existing commercial finite element software does not provide sufficiently fine meshing for this region [7], resulting in reduced computational accuracy and even loss of physical significance in the results. Therefore, it is necessary to simplify the bellows tube by considering it as an equivalent cylindrical pipe. As shown in Fig. 2, Fig. 2a represents the original three-dimensional view of the bellows tube, while Fig. 2b shows the completed equivalent three-dimensional view. Figure 2c and d illustrate the meshing of the insulation layer of the main pipeline before and after simplifying the bellows tube, revealing a noticeable improvement in mesh quality.

Fig. 2
A 4-part illustration. A depicts a tightly coiled copper wire with a shiny surface. B depicts a hollow cylinder. C depicts a cross-section of a horizontal cylindrical structure with four vertical lines on it. D depicts a horizontal cylindrical structure.

Meshing diagrams of insulation layer before and after simplification

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In order to maintain similar properties to the original bellows expansion joint, an equivalent method is required to determine the elastic modulus E of the simplified bellows expansion joint. The equivalence is achieved by ensuring that the simplified cylindrical pipe and the original bellows expansion joint exhibit the same macroscopic displacement under identical loads, thereby achieving stiffness equivalence. Based on the axial stiffness formula [8], the following equation can be derived.

The axial force of the corrugated tube before simplification is represented by Eq. (1).

$$F_{P} = K_{Z} \times \Delta L$$
(1)

The equivalent axial force of the simplified tube is represented by Eq. (2).

$$F_{J} = \sigma A = E\varepsilon A$$
(2)

The axial force before and after simplification is represented by Eq. (3).

$$F_{P} = F_{J}$$
(3)

Substituting Eqs. (1) and (2) into Eq. (3), we obtain:

$$E = \frac{{K_{Z} l_{0} }}{A}$$
(4)

\(K_{Z}\) is axial stiffness of the bellows expansion joint. \(F_{P}\) is axial force applied to the bellows expansion joint. \(F_{J}\) is axial force of the simplified bellows expansion joint. \(\Delta L\) is axial elongation of the expansion joint. E is elastic modulus of the simplified expansion joint. \(l_{0}\) is initial length of the expansion joint. \(A\) is area of the simplified expansion joint.

In the calculation of the temperature field distribution of the Main Pipeline, the actual operating conditions of the conduit are taken into account. The inner wall of the hot air duct carries high-temperature fluid, while the outer wall of the hot air duct and the inner surface of the duct shell carry low-temperature fluid. Therefore, the boundary conditions for the temperature field of the Main Pipeline are set as convective heat transfer with high-temperature fluid on the inner surface of the inner wall of the hot air duct, and convective heat transfer with low-temperature fluid on the inner surface of the outer wall of the hot air duct and the inner surface of the duct shell. The boundary conditions for the temperature field analysis are illustrated in Fig. 3.

Fig. 3
A schematic with labeled components, some of which are as follows. A hot gas duct, reactor pressure vessel, cone-shaped cylindrical pipe, corrugated pipe, hot gas duct casing, thermal insulation layer, convective heat transfer with high-temperature fluid, and a high-temperature fluid is flowing to the right.

Boundary condition settings for the temperature field and structural field

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For the structural analysis of the Main Pipeline, it is necessary to apply the results obtained from the temperature field analysis of the pipeline to the nodes of the structural field. In the analysis of thermal stress, in order to avoid insufficient degrees of freedom constraints leading to rigid body displacement in the finite element method, the three directions of freedom at both ends of the Main Pipeline are restricted. In the analysis considering the structural field under self-weight, a standard gravity acceleration along the negative Y-axis direction is applied, while also restricting the three directions of freedom at both ends of the Main Pipeline. The boundaries for the structural field analysis are illustrated in Fig. 3.

After the simplification steps described above, the temperature distribution of the main pipeline is obtained, as shown in Fig. 4. It can be observed that the inner layers of the pipeline have higher temperatures, while the outer layers have lower temperatures. The conical barrel in the middle experiences significant temperature gradients. The thermal stress distribution of the main pipeline is illustrated in Fig. 5. Due to the slip joint structure, the stress in the inner layers is released. In order to obtain the resultant moment \(M_{A}\) caused by self-weight, it is necessary to calculate the stress field distribution of the main conduit under self-weight.

Fig. 4
A temperature contour plot has a cylindrical structure with a scale on the left that ranges from 514.67 to 743.47. 743.47 has the maximum coverage on the contour plot.

Temperature contour plot of the main pipeline

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Fig. 5
A temperature contour plot has a cylindrical structure with a scale on the left that ranges from 0.010009 to 2684.7. 0.010009 has the maximum coverage on the contour plot.

Stress contour plot of the main pipeline

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According to Eq. (4), the equivalent elastic modulus E of the structurally irregular corrugated tube (i.e., expansion joint) after simplification can be determined, ensuring consistency between the results before and after simplification. By employing the approach of utilizing non-critical and irregular components in the pipeline, the quality of the mesh is enhanced, and there is a significant reduction in computational requirements, resulting in a 50% reduction in computation time. Therefore, it is evident that the adoption of the aforementioned strategy of equivalent simplification is highly necessary.

2.2 Creep-Fatigue Assessment of the Main Pipeline

In the assessment of creep-fatigue for the main pipeline, creep refers to the deformation phenomenon that occurs in a material under prolonged high-temperature conditions. When the main pipeline operates at high temperatures, the material may experience creep, resulting in changes in the shape and dimensions of the pipeline. This deformation can lead to stress concentration and material fatigue, increasing the risk of leakage or rupture in the pipeline. Therefore, creep behavior must be considered in the design and material selection of the main pipeline to ensure structural stability throughout its operational lifespan [9].

Fatigue refers to the progressive damage and failure that occurs in a material under repeated loading or stress cycles. In nuclear power plants, the main pipeline is often subjected to cyclic loading, such as pressure fluctuations caused by temperature changes or variations in operating modes. These cyclic loads can cause stress concentration and the formation of cracks in the main pipeline. Over time, these cracks may propagate and lead to pipeline failure. Therefore, fatigue analysis and assessment are essential for the design and operation of the main pipeline to ensure reliability and safety under repeated loading [10].

After the aforementioned simplification steps, the structure becomes more regular, and the mesh quality significantly improves. To quantify the assessment of creep-fatigue on the safety of the main pipeline, the following sections describe the ASME standard provisions used in the mechanical analysis and assessment of the main pipeline.

In the assessment of thermal expansion stress considering creep-fatigue, Formula (5) from ASME Specification HCB-3634(b) is required. This formula is used for the design of nuclear secondary pipelines. The formula is presented as follows [5].

$$\begin{aligned} {\text{S}}_{{{\text{TE}}}} =&\frac{{{\text{PD}}_{{0}} }}{{{\text{4t}}_{{\text{n}}} }}{ + 0}{\text{.75i}}\left( {\frac{{{\text{M}}_{{\text{A}}} }}{{\text{Z}}}} \right){\text{ + i}}\left( {\frac{{{\text{M}}_{{\text{C}}} }}{{\text{Z}}}} \right) \hfill \\& \le \,\,\left[ {{\text{lesser}}\,\,{\text{of}}\left( {{\text{S}}_{{\text{A}}} {\text{ + S}}_{{\text{h}}} } \right)\,\,{\text{or}}\,\,{0}{\text{.75S}}_{{{\text{yc}}}} { + 0}{\text{.25S}}_{{\text{h}}} } \right] \hfill \\ \end{aligned}$$
(5)

In Eq. (5), \(M_{A}\) represents the resultant moment of the pipeline cross-section caused by self-weight. \(M_{C}\) is the resultant moment of the pipeline caused by thermal stress.

It can be observed that the aforementioned equation requires the \(M_{C}\) and \(M_{A}\) data on the cross-sections of the pipeline. However, employing an exhaustive method to obtain the resultant moment data for each cross-section would undoubtedly lead to excessive computational requirements and increased computation time. Therefore, this paper proposes an assessment strategy that utilizes the binary search method to identify critical locations (such as structural discontinuities or material transitions) for evaluation. Compared to the original assessment method, this approach significantly improves computational efficiency. This is of great significance for pipeline evaluations in the engineering development process, as it shortens the development cycle.

3 Conclusion

This paper proposes an efficient strategy for creep-fatigue assessment of pipelines with complex local structures and high computational requirements. The strategy involves simplifying non-critical irregular sections of the pipeline by replacing them with equivalent regular sections in terms of stiffness. This approach enhances both the computational accuracy and efficiency. Additionally, a post-processing software based on the ASME standards has been developed using existing commercial finite element software. This software automatically assesses the creep-fatigue conditions at critical locations, enabling the automation of the main pipeline assessment tasks. Compared to other computational strategies, this approach significantly improves computational efficiency. In summary, the developed program in this study is of great significance for pipeline development assessments in engineering projects, as it shortens the pipeline development cycle.