Three-Dimensional Pin-by-Pin Transient Analysis for PWR-Core

Abstract

To ensure the safety of PWR-core operation, three-dimensional whole-core transient analysis needs to be carried out for the sake of the pin-power distribution. For this purpose, this paper presents “Bamboo-Transient 2.0”, a three-dimensional pin-by-pin transient analysis program. The program adopts a fully-implicit backward method with finite difference for time variable discretization, a method of exponential function expansion nodal (EFEN) SP₃ for the neutron transport calculation, and a multi-channel model for the thermal feedback calculation. In addition, Picard iteration is used to couple the neutronics with thermal-feedback, which is intended to guarantee the convergence of coupling iteration at each time step. Moreover, the program can perform parallel computing based on Message Passing Interface (MPI) for the whole-core pin-by-pin transient analysis. This developed program has been applied to two commercial PWRs, viz. AP1000 and CNP1000. Numerical results of this application demonstrate that Bamboo-Transient 2.0 can yield much more refined results than the traditional legacy coarse-mesh neutron-diffusion programs based on assembly homogenization. Its pin-wise distributions of state parameters are reliable and thus can satisfactorily meet the requirements and purpose of safety analysis.

1 Introduction

Compared with the steady-state operation process, the reactor core is more dangerous in transient processes. For PWR, only by determining the hot spots and heat pipes at the core can its safe operation be ensured and the ultra-high temperature-caused core melting be prevented. To find the hot spots and heat pipes at the core, the transient analysis is supposed to be accurate in the fuel rod scale and be capable of providing various state parameters. Thus, it is necessary to track and predict the changes in the key parameters of the core to prevent accidents and reduce the harm after accidents if there is any. As the traditional transient analysis method is based on the diffusion calculation of assembly homogenization, only information in the assembly scale can be retained whereas other relevant information in the pin scale is ignored. Therefore, in order to obtain the power distribution of fuel rod scale, it is necessary to carry out the power reconstruction based on a series of approximations and assumptions. Consequently, the calculation accuracy of the traditional method is far from desirable [1].

It is against such a background that fast and accurate transient analysis methods for nuclear reactors are becoming increasingly important and should be designed and developed. In recent years, a pin-by-pin transient analysis method based on pin homogenization has attracted extensive attention. With the pin transport calculation accomplished, this method can directly homogenize the calculation area and retain all kinds of information in the pin scale. As a result, in the core calculation, the core information in the pin scale can be obtained directly without introducing the error caused by the power reconstruction [2]. In addition, the channel of thermal feedback should be accurate in the rod scale so as to match with neutronics. Additionally, the transient process of PWR is a process of coupling neutronics with thermal-hydraulics [3]. This paper presents how the coupling of neutronics with thermal-feedback is calculated, which is to be suitable for the pin-by-pin transient analysis. In addition, due to the significantly increased number of computational meshes, it is urgent to improve the efficiency of whole-core pin-by-pin transient analysis. In this paper, the parallel technology is also discussed.

The rest of the paper is organized as follows. Section 2 introduces each method in detail. Section 3 introduces the numerical verification and analysis. Section 4, the last part of this paper, sums up the study and concludes the paper.

2 Theoretical Models

The fully implicit method and EFEN method are specially employed to solve the neutron kinetics equation, whereas a multi-channel model in the pin scale is employed to treat the heat transfer and flow process of coolant, and a 1D cylindrical heat conduction model is employed to treat the heat conduction process in fuel rods. Picard iteration is utilized at each time step to guarantee the convergence between neutronics and thermal-feedback. Using MPI of distributed memory, the same spatial domain decomposition is performed for both neutronics and thermal-feedback calculation for parallel computing, which can significantly shorten the computing time needed by transient analysis.

2.1 Calculation of Neutron Dynamics

In the transient process of PWR, considering the influence of delayed neutrons and adopting multi-group approximation, the neutron flux distribution at the core meets the spatiotemporal neutron transport equations, which is shown in Eq. (1). Where, ν_g is the neutron velocity of group g/cm·s⁻¹, r the spatial location, Φ_g the neutron angular flux of group g/(cm3·s)⁻¹, Ω the neutron motion direction, t the time/s, Σ_{t, g}, Σ_{f, g} the total, fission cross sections of group g/cm⁻¹, Σ_{s, g’→ g} the scattering cross section from group g’ to group g/cm⁻¹, χ_{p, g} and χ_{d, g, i} the prompt neutron fission spectrum of group g and the delayed neutron fission spectrum of group g, delayed group i, v the number of neutrons per fission, C_i the precursor concentration of delayed group i, λ_i the decay constant of precursor delayed group i/s⁻¹, β_i the delayed neutron fractions of group i/pcm, g = 1, 2, …, G; i = 1, 2, …, Nd the neutron energy group index and the delayed neutron precursor group index.

$$\left\{ {\begin{array}{*{20}c} \begin{gathered} \frac{1}{{\upsilon_{{\text{g}}} ({\mathbf{r}})}}\frac{{\partial {{\varvec{\Phi}}}_{g} \left( {{\mathbf{r}},{{\varvec{\Omega}}},t} \right)}}{\partial t} = - {{\varvec{\Omega}}} \cdot \nabla {{\varvec{\Phi}}}_{g} \left( {{\mathbf{r}},{{\varvec{\Omega}}},t} \right) - \Sigma_{{{\text{t}},g}} \left( {{\mathbf{r}},t} \right){{\varvec{\Phi}}}_{g} \left( {{\mathbf{r}},{{\varvec{\Omega}}},t} \right) \\ + \int {\Sigma_{{{\text{s}},g^{^{\prime}} \to g}} \left( {{\mathbf{r}},{{\varvec{\Omega}}}^{\prime} \to {{\varvec{\Omega}}},t} \right){{\varvec{\Phi}}}_{g} \left( {{\mathbf{r}},{{\varvec{\Omega}}}^{\prime},t} \right)d{{\varvec{\Omega}}}^{^{\prime}} } \\ + \left( {1 - \beta ({\mathbf{r}})} \right)\frac{{\chi_{{{\text{p}},g}} \left( {\mathbf{r}} \right)}}{4\pi }\sum\limits_{{g^{^{\prime}} = 1}}^{G} {\int {\nu ({\mathbf{r}})\Sigma_{{{\text{f}},g^{^{\prime}} }} \left( {{\mathbf{r}},t} \right){{\varvec{\Phi}}}_{{g^{^{\prime}} }} \left( {{\mathbf{r}},{{\varvec{\Omega}}}^{\prime},t} \right)d{{\varvec{\Omega}}}^{\prime}} } \\ + \frac{1}{4\pi }\sum\limits_{i = 1}^{Nd} {\chi_{{{\text{d}},g,i}} \left( {\mathbf{r}} \right)\lambda_{i} C_{i} \left( {{\mathbf{r}},t} \right)} \\ \end{gathered} \\ {\frac{{\partial C_{i} \left( {{\mathbf{r}},t} \right)}}{\partial t} = \beta_{i} ({\mathbf{r}})\sum\limits_{{g^{^{\prime}} = 1}}^{G} {\int {\nu ({\mathbf{r}})\Sigma_{{{\text{f}},g^{^{\prime}} }} \left( {{\mathbf{r}},t} \right){{\varvec{\Phi}}}_{{g^{^{\prime}} }} \left( {{\mathbf{r}},{{\varvec{\Omega}}}^{\prime},t} \right)d{{\varvec{\Omega}}}^{\prime}} } - \lambda_{i} C_{i} \left( {{\mathbf{r}},t} \right)} \\ \end{array} } \right.$$

(1)

As Eq. (1) cannot be solved straightforwardly, it has to be discretized. First, the fully implicit backward difference method is used for the time term [4] to obtain the equation of the angular flux at t_n+1, which is shown in Eq. (2):

$$\begin{gathered} {{\varvec{\Omega}}} \cdot \nabla {{\varvec{\Phi}}}_{g} \left( {{\mathbf{r}},{{\varvec{\Omega}}},t_{n + 1} } \right) + \overline{\Sigma }_{{{\text{t}},g}} \left( {{\mathbf{r}},t_{n + 1} } \right){{\varvec{\Phi}}}_{g} \left( {{\mathbf{r}},{{\varvec{\Omega}}},t_{n + 1} } \right) = \hfill \\ \sum\limits_{{g^{^{\prime}} = 1}}^{N} {\int {\Sigma_{{{\text{s}},g^{^{\prime}} \to g}} \left( {{\mathbf{r}},{{\varvec{\Omega}}}^{\prime} \to {{\varvec{\Omega}}},t_{n + 1} } \right){{\varvec{\Phi}}}_{g} \left( {{\mathbf{r}},{{\varvec{\Omega}}}^{\prime},t_{n + 1} } \right)d{{\varvec{\Omega}}}^{^{\prime}} } } \hfill \\ + \frac{1}{4\pi }\overline{\chi }_{g} \left( {\mathbf{r}} \right)\sum\limits_{{g^{^{\prime}} = 1}}^{N} {\int {\nu ({\mathbf{r}})\Sigma_{{{\text{f}},g^{^{\prime}} }} \left( {{\mathbf{r}},t_{n + 1} } \right){{\varvec{\Phi}}}_{g^{\prime}} \left( {{\mathbf{r}},{{\varvec{\Omega}}}^{\prime},t_{n + 1} } \right)d{{\varvec{\Omega}}}^{\prime}} } + Q_{g} \left( {{\mathbf{r}},{{\varvec{\Omega}}}} \right) \hfill \\ \end{gathered}$$

(2)

where,

$$\begin{gathered} \overline{\Sigma }_{{{\text{t}},g}} \left( {{\mathbf{r}},t_{n + 1} } \right) = \Sigma_{t,g} \left( {{\mathbf{r}},t_{n + 1} } \right) + \frac{1}{{\upsilon_{g} ({\mathbf{r}})\Delta t}} \hfill \\ \overline{\chi }_{g} \left( {\mathbf{r}} \right) = \left( {1 - \beta ({\mathbf{r}})} \right)\chi_{p,g} \left( {\mathbf{r}} \right) + \sum\limits_{i = 1}^{Nd} {\chi_{d,g,i} \left( {\mathbf{r}} \right)\beta_{i} ({\mathbf{r}})} - \sum\limits_{i = 1}^{Nd} {\chi_{d,g,i} \left( {\mathbf{r}} \right)\frac{{\beta_{i} ({\mathbf{r}})}}{{1 + \lambda_{i} \Delta t}}} \hfill \\ Q_{g} \left( {{\mathbf{r}},{{\varvec{\Omega}}}} \right) = \frac{1}{4\pi }\sum\limits_{i = 1}^{Nd} {\chi_{d,g,i} \left( {\mathbf{r}} \right)\lambda_{i} \frac{{C_{i} \left( {{\mathbf{r}},t_{n} } \right)}}{{1 + \lambda_{i} \Delta t}}} + \frac{1}{{\upsilon_{g} ({\mathbf{r}})}}\frac{{{{\varvec{\Phi}}}_{g} \left( {{\mathbf{r}},{{\varvec{\Omega}}},t_{n} } \right)}}{\Delta t} \hfill \\ \end{gathered}$$

Next, the P₁ approximation is used for the angle to obtain the diffusion fixed source equation as shown in Eq. (3):

$$- D\nabla^{2} \psi_{g}^{0} \left( x \right) + \overline{\Sigma }_{{{\text{t}},g}} \psi_{g}^{0} \left( x \right) = S_{g}$$

(3)

The SP₃ approximation is used for the angle to obtain the SP₃ fixed source equations as shown in Eq. (4):

$$\left\{ {\begin{array}{*{20}c} { - D\nabla^{2} [\psi_{g}^{0} \left( x \right){ + }2\psi_{g}^{2} \left( x \right)] + \overline{\Sigma }_{r,g} [\psi_{g}^{0} \left( x \right){ + }2\psi_{g}^{2} \left( x \right)] = S_{g} + 2\overline{\Sigma }_{r,g} \psi_{g}^{2} \left( x \right)} \\ { - \frac{27}{{35}}D\nabla^{2} \psi_{g}^{2} \left( x \right) + \overline{\Sigma }_{t,g} \psi_{g}^{2} \left( x \right) = S_{2} + \frac{2}{5}(\overline{\Sigma }_{r,g} \psi_{g}^{0} \left( x \right) - S_{g} )} \\ \end{array} } \right.$$

(4)

The derivation process is described in detail in Reference [4]. Equation (3) and (4) can be solved by using EFEN method, whose process is described in detail in Reference [5]. By solving the diffusion or SP₃ fixed source equation at each time step, the neutron dynamics is calculated.

2.2 Calculation of Thermal Feedback

The transient analysis of PWR entails that the coupling of neutron dynamics with transient thermal feedback is calculated. The solution to the neutron dynamics equation requires the cross-sections of all materials in the core. To obtain these cross-sections, the state parameters at the core are needed. Thus, thermal feedback calculation should provide the distributions of these state parameters. Calculation of the thermal feedback of the reactor core consists of two parts, viz. Fluid calculation of the coolant and heat conduction calculation of the fuel rod, in which the fluid calculation is 1D calculation in the axial direction of the flow area and the heat conduction calculation is 1D calculation in the radial direction of the cylindrical rod.

Fluid Model

Transient mass and energy conservation equations of the coolant are shown in Eq. (5) and (6). Where, ρ is the density of coolant/kg·cm⁻³, h the enthalpy of coolant/J, u the velocity of coolant/cm·s⁻¹, A_c the circulation area/ cm⁻², q_c the heat release in fuel and q_w the fuel surface heat flux.

$$\frac{{\partial \rho \left( {z,t} \right)}}{\partial t} + \frac{{\partial \left( {\rho \left( {z,t} \right)u\left( {z,t} \right)} \right)}}{\partial z} = 0$$

(5)

$$\frac{{\partial \left( {\rho \left( {z,t} \right)h\left( {z,t} \right)} \right)}}{\partial t} + \frac{{\partial \left( {\rho \left( {z,t} \right)u\left( {z,t} \right)h\left( {z,t} \right)} \right)}}{\partial z} = P_{h} \frac{{q_{{\text{w}}} \left( {z,t} \right)}}{{A_{c} }} + q_{{\text{c}}} \left( {z,t} \right)$$

(6)

The two equations are discretized by θ difference in time to obtain the equation of the nodal enthalpy rise. Given the parameters of the coolant at the channel inlet, the heat flux can be obtained via the heat conduction calculation. Hence, the temperature of the coolant can be solved in the axial direction from the inlet to the outlet.

Heat Conduction Model

1D transient heat conduction model of cylinder is shown in Eq. (7). Where, c_p is the specific heat capacity at constant pressure/(J/kg·K), k the thermal conductivity/(W/m·K), and T the temperature/K.

$$\rho \left( {r,t} \right)c_{{\text{p}}} \left( {r,t} \right)\frac{{\partial T\left( {r,t} \right)}}{\partial t} = \frac{1}{r}\frac{\partial }{\partial r}\left( {k\left( {r,t} \right) \cdot r \cdot \frac{{\partial T\left( {r,t} \right)}}{\partial r}} \right) + q\left( {r,t} \right)$$

(7)

In the radial direction, the fuel pellet is divided into n meshes, the gas gap divided into 1 mesh, and the clad divided into 3 meshes. Equation (7) is discretized by finite difference in space and θ difference in time. The radial temperature distribution of the fuel is obtained by Gauss-Seidel iterative calculation.

2.3 Coupling of Neutronics with Thermal-Hydraulics

As the mesh size of the whole-core pin-by-pin neutron dynamics calculation can be either that of a fuel rod, or a control rod or a water tunnel, the corresponding thermal feedback adopts multi-channels in pin scale division to simulate the coolant flow between the rods. The channel is the coolant area between fuel rods and guide tubes. The channels along the axis can be divided into several meshes, which can exchange mass and energy with each other, ignoring the exchange of momentums and the exchange between the radial channels. In addition, different physical fields adopt various meshing methods while the neutron and thermal fields uses rod-centered meshing methods and the flow field uses channel-centered meshing method. The various methods are shown in Fig. 1.

Fig. 1.

Various meshing methods of physics fields

As the mapping relationships across neutron field, thermal field and flow field are complicated, five types of conversion relationships are considered for different physical fields as follows, which is shown in Fig. 2 as follows:

1) conversion from the nodal power of neutron field to the channel power of flow field;2) conversion from the coolant temperature of flow field to the cladding surface temperature of thermal fields;3) conversion from the nodal power of neutron field to the mesh power of thermal field;4) conversion from the coolant temperature of flow field to the nodal average coolant temperature of the neutron field;5) conversion from the fuel temperature of thermal field to the nodal effective fuel temperature of neutron field.

Fig. 2.

Conversion of various physical fields

Traditional transient analysis calculation programs mostly use explicit coupling method. This method can solve the neutronics equation and thermal feedback calculation separately, but cannot perform the iteration between them. Thus, the convergence of the coupling cannot be guaranteed. To ensure the convergence, a shorter time step is needed for this method. However, due to lack of iteration, the cost of calculation at each time step is also low. However, implicit coupling, or rather, Picard iteration, solves the neutronics and thermal-feedback separately. In essence, Picard iteration is intended to solve the various physical fields in different ways through operator splitting, and then iterates between the two physical fields to converge the coupled parameters. However, it takes a longer time step and more convergence than the explicit coupling. In view of this, the current study employs Picard iteration to calculate the coupling of neutronics with thermal-feedback to accurately calculate the coupling. The neutron dynamics and thermal feedback are calculated at each time step. Only when the neutronics and thermal-hydraulics coupling iteration are converged can the next time step be calculated. The iterative process is shown in Fig. 3.

Fig. 3.

Neutronics and thermal-hydraulics coupling iteration.

2.4 Parallel Calculation

The parallel efficiency is mainly affected by the following factors. 1) Communication overhead. As different threads need to communicate, it takes longer time. Thus, the longer communication lasts, the lower the parallel efficiency. 2) Parallel computing may degrade the iterative format, and as a result, increase the amount of computation. 3) Multithreading may cause worthless waiting to the processes caused by load imbalance. 4) Redundant computation can be caused by parallel algorithm per se.

Given the factors above, the parallelization of whole-core pin-by-pin transient calculation consists of the following aspects:

• 1) Broadcast of input parameters. To prevent multiple CPUs from using the same input channel at the same time, a designated CPU reads the input file and then uses the broadcast function of sending multiple data of the same type in batch at one time to send the input information to all CPUs.

• 2) Domain decomposition. The whole area on average is divided according to the number of CPUs and the scale of calculation problems to keep the load balance across different CPUs. Each CPU stores only the area information responsible for calculation.

• 3) Node sweep and communication. To improve the parallel efficiency, the Red-Black Gauss-Seidel node sweep method [6], which is suitable for parallel computing, is selected to avoid the degradation in iterative format caused by parallel computing. In addition, the fractional neutron currents only need to communicate once after the red and black node sweep to reduce the communication overhead.

• 4) Post-processing of calculation results. To prevent multiple CPUs from using the same output channel at the same time, the calculation results of all CPUs are exported uniformly by the designated CPU.

2.5 Interim Summary

Based on the theoretical models mentioned above, a pin-by-pin program called “Bamboo-Transient 2.0” is developed for the 3D whole-core transient analysis. This program can be automatically coupled with both the lattice-calculation program “Bamboo-Lattice 2.0” and the 3D whole-core pin-by-pin steady-state calculation program “Bamboo-Core 2.0”. This strongly suggests that Bamboo-Transient 2.0 can improve the function of the software package called “NECP-Bamboo 2.0” and enables the software package to perform Pin-by-pin transient analysis for PWR-core. The calculation flow chart of Bamboo-Transient 2.0 is shown in Fig. 4. The multi-physical coupling iteration is shown in detail in Fig. 3.

Fig. 4.

The calculation flow chart of Bamboo-Transient 2.0.

3 Numerical Verification and Analysis

To verify the accuracy and the analytical ability of Bamboo-Transient 2.0, the program is applied to the transient analysis of two commercial PWRs. This section will show the numerical verification and analysis of CNP1000 and AP1000 reactor core by using Bamboo-Transient 2.0.

3.1 CNP1000

The rated power and the rated operating pressure of CNP1000 are 2895MWt and 15.5 MPa, respectively. There are 157 boxes of fuel assemblies at the reactor core and 57 control rod assemblies in the first cycle. The reactor core is divided into 9 groups according to the needs of the problem. The control rods are 362.49 cm in length, which is divided into 225 steps. The grouping of the control rods is shown in Fig. 5. The program calculates the rod ejection in the following case: the initial power level of the core is 1%. The control rods of the ninth group are fully inserted while the rest are all lifted. The control rods of the ninth group are all ejected in 0–0.1 s. It takes 0.4 s to accomplish the transient process, and the time step is divided into 0.001 s.

Fig. 5.

The grouping of the control rods in CNP1000

The normalized power is shown in Fig. 6. As the control rods are gradually ejected from the core, the normalized power of the core increases rapidly. When all the control rods are ejected from the core, the core enters a stable state. The 3D distribution of power, the effective temperature of fuel and the temperature of coolant are shown in Fig. 7. At the beginning, due to the insertion of control rods, the power is distributed precipitously, and the distribution is high outside but low inside the core. With the control rods ejected, the power distribution is flattened. While the temperature distributions show the same regularity, the effective temperature of the fuel and the temperature of the coolant are more uniform than at the beginning in the radial direction after the rod ejection.

Fig. 6.

Normalized power of CNP1000

Fig. 7.

Three-dimensional distribution at different time

3.2 AP1000

AP1000 is the third-generation advanced passive PWR designed by Westinghouse. Compared with the traditional PWR nuclear reactor, AP1000 adopts a passive safety system, which further simplifies the structure of the power station and improves the safety of the reactor. There are 69 groups of control rods in AP1000, including 53 groups of black rods and 16 groups of gray rods. Some of the grey rods and black rods are employed to form the MSHIM system of the core, which controls the reactivity and power distribution of the core with the boron containing coolant, while the remaining black rods are employed to form the shutdown rod group [7]. The pattern of the control rods is shown in Fig. 8.

Fig. 8.

The pattern of the control rods of AP1000

AP1000 has designed a rapid power reduction system (RPR system). To analyze the system, this study calculates the simulation of rod drop working condition of M1 and S2 rod groups under full power. M1 and S2 rod groups simultaneously fall into the core within 0.1 s while the program simulates the transient process within 0.2 s. The normalized power of the core is shown in Fig. 9. It can be seen that the core power decreased rapidly due to the insertion of control rods into the core. At 0.1 s, the normalized power is reduced to 40% of that at the initial time. Then, the core power increases slowly as the fuel temperature decreases. The 3D distribution at the beginning and at the end is shown in Fig. 10. The figure shows that the power distribution is distorted by the insertion of the control rod and that the power around the control rod is significantly decreased. Besides, the effective fuel temperature also shows this regularity. At the same time, the axial distribution of power and the fuel temperature become increasingly uneven while the temperature of the coolant changes little and decreases slightly.

Fig. 9.

Normalized power of AP1000

Fig. 10.

Three-dimensional distribution at different time

4 Conclusion

In order to perform the 3D whole-core pin-by-pin transient analysis, the theoretical model of the transient analysis is established in this study as follows:

• 1) The pin-by-pin neutron dynamics is calculated by using the fully implicit method and the exponential function expansion nodal method;

• 2) The model of 3D whole-core pin-by-pin transient thermal feedback is established by using multi-channel in the pin scale to simulate the heat transfer and flow process of coolant, and by using a 1D cylindrical heat conduction model to simulate the heat conduction process in fuel rods;

• 3) In each time step, Picard iteration is used to perform the iterative calculation of the coupling of neutronics with thermal-feedback;

• 4) Based on MPI, the parallel calculation of the 3D whole-core pin-by-pin transient analysis is accomplished, which significantly shortens the calculation time.

To sum up, in this study, the program system that couples the pin-by-pin transient neutronics with thermal-feedback is applied to the PWR-core transient analysis of the second-generation nuclear power CNP1000 and the third-generation nuclear power AP1000. Both the condition for rapid rod drop power reduction and that for rod ejection accident are analyzed. The rod power distribution and fuel temperature distribution are more refined than the traditional diffusion program of assembly homogenization. Bamboo-transient 2.0 makes full use of the ability of pin-by-pin transient analysis and calculation, which is more accurate than the traditional program. The working conditions of the reactor core under transient conditions are simulated and speculated finely, which provides a guarantee for the safe operation of PWR.