Accuracy and efficient solution of helical coiled once-through steam generator model using JFNK method
Introduction
Pebble-bed High-Temperature Gas-cold Reactor (HTGR) is one of the 4th generation nuclear power plants, featured with a high temperature at the reactor outlet and high thermal efficiency. As a key equipment in HTGR system, the Helical coiled tube Once-Through Steam Generator(H-OTSG) couples the primary circuit and the secondary circuit, transferring the fission energy from the hot helium at the first side to the water at the second side. Compared with the traditional U-tube natural circulation steam generator, the water in the H-OTSG is not only heated reaching the water-steam two-phase flow but also further reaching the superheated steam to pursue higher thermal efficiency. The centrifugal forces acting on the fluid inside the helical ducts tend to laminarize the flow, enhance the heat transfer coefficient (Xia et al., 2017). The accurate simulation of the H-OTSG behavior, especially the complicated the two-phase flow phenomenon, is an important step for a successful HTGR design, as well as the safety analysis. A continuous effort has been carried out to achieve the accurate simulation of the steam generators in the HTGR plant.
The fully three-dimensional CFD-based simulation is a useful numerical tool to provide detailed information in the steam generator (Ferng and Chang, 2008), but still suffering from the too-exhausted computational cost. The one-dimensional model is the most widely used numerical tool for steam generator simulation in the engineering application due to its practicality and efficacy. Numerous studies have been carried out on inveatigating single-phase and two-phase pressure drop and heat transfer characteristics of H-OTSG. BLAST is a specially designed one-dimensional computational code for the H-OTSG in HTGR plant (Hedrick and Cleveland, 1976), which has been successfully coupled with the reactor core simulation codes, such as THERMIX (Zheng et al., 2012) and TINTE (Gerwin et al., 2009), and has been widely used in HTGR engineering design and safety analysis. In the BLAST code, the homogeneous flow model is used to describe the two-phase flow behavior in the waterside, which assumes that the steam and water share with the same velocity and the velocity difference between the two phases could not be distinguished. Moreover, the conservation equations of the helium side, tube component and waterside are sequentially solved by the traditional semi-implicit coupling method in the BLAST code. Therefore, the coupling information from the waterside is lagged result in inconsistent convergence in each time step, which leading to the material Courant time step limit. To assess the design decisions and investigate the transient behavior, a lot of work on H-OTSG has been conducted. Ahn et al. (2017) validated the H-OTSG model using the system transient analysis program TASS/SMR developed by Lee et al. (2009). Xiaowei Li developed a lumped-parameter H-OTSG model based on a movable boundary method to perform the steady-state calculation and the simulation results agree well with design data (Li et al., 2011). Zhe Dong proposed the simplified one-dimensional H-OTSG models as a component in the whole HTGR system for designing the automatic control system (Dong et al., 2020). Reactor safety analysis codes based on the two-phase flow model may also have been capable to simulate the behaviors of H-OTSG after moderate modification, such as RELAP5 (U.S. NRC, 1995a), TRAC (U.S. NRC, 2010), RETRAN (Moore et al., 1977) and TRACE (U.S. NRC, 2010). However, these safety analysis codes use the traditional semi-implicit coupling method and the time step is also limited by the material Courant number.
To the authors’ best knowledge, most existing work focus on the detailed analysis of the two-phase flow behavior within H-OTSG. However, only a few attempts are made from the viewpoint of the numerical solution method, as well as their efficiency and accuracy. The fully implicit, super-linear convergent Jacobian-Free Newton Krylov (JFNK) method is used in this work to pursue higher computational performance. JFNK method is an advanced fully implicit coupling method and has been successfully applied in the multiphase flow (Zou et al., 2016) and multi-physics coupling issues (Zhang et al., 2018). Different from the traditional semi-implicit method, the highly nonlinear conservation equations of the helium side, tube component and waterside are solved in a tightly nonlinear form as a whole system in JFNK method. All the physical quantities are updated simultaneously and consistent convergence in time. In this work, the two-phase drift-flux model is used to consider the velocity difference between the two phases in the waterside. The second-order accuracy temporal and spatial numerical scheme is utilized to achieve a higher numerical accuracy, compared with the first-order accuracy scheme in most existing codes. Furthermore, please note that, JFNK method could provide a powerful mathematical freamework which has the potential to solve the whole HTGR coupled system within one uniform algorithm (Zhang et al., 2018). Several efforts have been made in our team to solve the neutron kinetic and thermal hydraulic coupled issue by JFNK method (Zhang et al., 2018, Zhang et al., 2019, Wu et al., 2020, Lu et al., 2018), and we plan to extend the coupling system from the reactor core to the whole primary circuit, and further to the whole HTGR nuclear power plant. In this work, we focus on the prediction of H-OTSG behavior using JFNK, which is an important and necessary part for the whole HTGR coupled system simulation under the uniform JFNK framework. The main contributions of this work are:
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An efficient physical-based preconditioner is proposed for the H-OTSG model, which is derived from the original semi-implicit based method. The preconditioner plays an important role in the JFNK method to ensure high computational efficiency.
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The numerical techniques of the phase generation/disappearance and the global convergence algorithms of JFNK method are implemented and discussed for the H-OTSG model.
Section snippets
Descriptions and assumptions in the steam generator
The H-OTSG is a helical coil heat exchanger, where the hot helium flows on the heat transfer tube surface and the secondary side feed water flows inside the helically coiled tubes. After a countercurrent heat transfer, the single-phase water is heated to superheated steam by the hot helium. Moreover, the helium in the primary side, the metal helical coil tube, and the secondary side feed water are coupled together due to the conjugate heat transfer process.
Several widely used assumptions are
Fully implicit discretization of governing equations
In most existing thermal–hydraulic codes, the finite volume method(FVM) with a staggered grid is commonly applied for the spatial discretization. The scalar variables such as pressure, void fraction and internal energy are obtained at the mesh cell center, while the vector variables velocities are obtained at the mesh cell faces. The staggered grid mesh is selected because it is flexible to handle the two-phase equations in primitive forms and it is compatible with most existing system analysis
Result
A thermal–hydraulic code for once-through steam generator has been developed using the fully implicit JFNK method. The steady-state and transient behaviors for the H-OTSG in HTR-10 reactor have been simulated to analyze the performance of the code. The key parameters and initial/boundary conditions for this steam generator are summarized in reference Ju et al. (2001). An efficient numeric library-Portable, Extensible Toolkit for Scientic Computation (PETSc) is implemented in this work (Balay et
Conclusion
The H-OTSG is the largest heat exchanger in the HTGR system, together with the complicated conjugate phase change heat transfer phenomenon. In this work, the high-order time and spatial integration scheme with a staggered grid are used in the discretization of HTR-10 H-OTSG. The JFNK method is applied to solve fully implicit nonlinear equations. Deriving from the semi-implicit discretized governing equations, a conceptually simple and computational efficient preconditioner is constructed. The
Data availability
The data used to support the findings of this study are available from the corresponding author upon request.
CRediT authorship contribution statement
Yingjie Wu: Conceptualization, Methodology, Software, Writing - original draft. Baokun Liu: Methodology, Software. Han Zhang: Conceptualization, Methodology, Writing - review & editing. Kaijie Zhu: Software. Boran Kong: Software. Jiong Guo: Writing - review & editing. Fu Li: Conceptualization, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work is supported by Beijing Natural Science Foundation 1212012, Chinese National Natural Science Foundation Project 11505102 and 11375099, Chinese National ST Major Project 2018ZX06 902013, and IAEA CRP I31020.
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