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This article aims to simulate the sudden and fast pressure drop of VVER-1000 reactor core coolant, regarding acoustic phenomenon. It is used to acquire a more accurate method in order to simulate the various accidents of reactor core. Neutronic equations should be solved concurrently by means of DRAGON 4 and DONJON 4 coupling codes. The results of the developed package are compared with WIMS/CITATION and final safety analysis report of Bushehr VVER-1000 reactor (FSAR). Afterwards, time dependent thermal-hydraulic equations are answered by employing Single Heated Channel by Sectionalized Compressible Fluid method. Then, the obtained results were validated by the same transient simulation in a pressurized water reactor core. Then, thermal-hydraulic and neutronic modules are coupled concurrently by use of producing group constants regarding the thermal feedback effect. Results were compared to the mentioned transient simulation in RELAP5 computer code, which show that mass flux drop is sensed at the end of channel in several milliseconds which causes heat flux drop too. The thermal feedback resulted in production of some perturbations in the changes of these parameters. The achieved results for this very fast pressure drop represent accurate calculations of thermoneutronic parameters fast changes.

One of the most important aspects in design of different safety systems with sufficient preparation is simulation and analysis of transient states in reactor core. For these kinds of analysis basic equations in neutronic and thermal-hydraulic modules have to be coupled. Coupling of neutronic and thermal-hydraulic modules is different from each other, considering numerical solution methods and time and body meshing size. Thus, written codes for different transient states are mostly used for study of fuel and coolant temperature changes, power peak level, coolant pressure, stability time, and other parameters. Computer coding submits models with different degree of accuracy and validity. Most codes are not able to analyze coupled conditions of very fast transient (FT) states in very short time. This deficiency is associated with neutronic and thermal-hydraulic calculations or both. Therefore designing a code which is capable of analyzing FT conditions is highly needed.

One of the efficient ways in analyzing FT is using waveform method. Chan (1991) studied asymptotic waveform evaluation in analysis of time-dependent calculations [

Various kinds of thermal-hydraulic and neutronic calculating models are put to use in transient calculating code development studies. Reducing costs and runtime and achieving required accuracy are three main purposes of them. Leung et al. (1981) studied acoustic impact techniques for increasing the accuracy of FT states modeling in CODA code [

In order to decrease complicated solving parameters in using compressible fluid method, geometry of every fuel assembly could be turned into a single heated channel (SHC). Four different methods are used to solve SHC transient equations. They are channel integral (CI), single mass velocity (SV), momentum integral (MI), and sectionalized compressible fluid (SC). SC model considers both sound impact and thermal expansion, while the other three models only consider thermal expansion [

The SHC compressible fluid method is one of the ways that make it possible to use wave propagation method and acoustic phenomenon. Shapiro (1953) considered the derivation and properties of the dynamics and thermodynamics of compressible fluid flow which is along with acoustic theory [

The mechanism of compressible fluid method was published by Bar-Meir in 2007 [

Numerical considerations (i.e., the stability and/or accuracy) of the difference solution require that the time step of integration be less than the time interval for sonic wave propagation across the spatial grid points. Compared with the transport velocity, the fluid sonic velocity is large. It causes limitation of the time step in most numerical schemes to very small values. Acoustic phenomenon is used in accommodation of very short time and very small body meshing. This accommodation is determined by Courant’s criterion [

FT pressure drop is a type of loss of coolant accident (LOCA). That is well known as the double ended guillotine break. When double ended guillotine break occurs (main coolant pipeline cold leg breaks at the reactor inlet), suddenly reactor coolant pressure decreases with leak coolant flow rate of 45000 Kg/s [

The coolant fast pressure drop accident can lead the fluid to become two-phased and thermoneutronic parameters to change. Gonzalez-Santalo and Lahey Jr. (1972) investigated this matter by study of pressure drop transient in two-phase condition [

Calculation program of VVER-1000 reactor core FT pressure drop by means of SC method and acoustic phenomenon was developed in this investigation. In order to use the mentioned method every one of the 28 fuel assemblies should be considered as one SHC. Fuel assembly conversion into SHC and meshing method toward

Fuel assembly conversion into SHC and meshing method toward

A computational program for simulation of VVER-1000 reactor core FT state of sudden pressure drop has been developed in this study. It includes the two thermal-hydraulic and neutronic basic models. Group constants of the fuel assemblies and reflectors are produced by DRAGON 4 code [

In this study, the integral transport equation is solved by the SYBILT module, self-shielding calculations are performed by the SHI module by means of generalized Stamm’ler method, and the CPO module is utilized for production of equivalent fuel assembly parameters in consistent format that can be used in forgoing calculation. After that, time-dependent multigroup neutron diffusion equations are solved by DONJON 4 models [

In thermal-hydraulic model, conservation of mass and momentum and energy dependent equations are solved by applying considered transients and use of channel axial division in MATLAB software. The channel is regarded as a typical coolant subchannel inside an assembly that only receives coolant through its bottom inlet. The fuel and clad heat transport equations are solved separately from coolant equations. One-dimensional transient transport equations of the coolant with radial heat input from the clad surfaces are resolved. The flow area is assumed axially uniform but pressure loss due to local area changes is still regarded. Any axial position of flow area could be considered as control area in order to derive radially averaged coolant flow equations. Mentioned equation is envisioned for SHC state and SC method of every 28 fuel assemblies (1/6 core). From the four SHC different methods only SC model considers both sound impact and thermal expansion, while the other three models considering thermal expansion [

Several approximations can be used to decouple the momentum and energy equations to facilitate solution of this transient. Additionally, the numerical solution of the discussed transient problem in this study is particularly simplified following the work of Meyer [

In SC model, numerical solution approach involves a set of difference equations representing the differential transport equations, arranged to consider

Equations (

Sound velocity is different in various spaces; also the fluid is running in a specified direction with a specified speed. Fluid velocity is accordingly dependent on its other parameters. Regime pressure drop transient numerical considerations (i.e., the stability and/or accuracy) of the difference solution need that the time step of integration be less than the time interval for sonic wave propagation across the spatial grid points, that is, (

Different codes have developed various methods for predicting theses states. CATHARE code [

All these codes, in which Courant criterion is not used, consider different ways in order to decrease the time of calculations but still they do not have the accuracy of the codes using Courant criterion, in time of critical flow conditions and very short times. Therefore Courant criterion was taken into account in the simulating codes of this study. Time mesh amount is obtained about 10

Very FT coolant pressure drop in VVER-1000 reactor core is explained in

It is highly mentioned that axial fluid motion is considered steady in SC model which eliminates cross flow effect [

Coupling algorithm of the two thermal-hydraulic and neutronic modules is as follows. Primarily every fuel assembly is modeled by DRAGON 4 cell calculating code according to Table

VVER-1000 geometry and operating conditions [

Operating condition | Value |
---|---|

Core height in the working state (cm) | 355 |

Rod diameter (mm) | 9.1 |

Pitch (mm) | 12.75 |

Fuel pellet material | UO_{2} |

Number of fuel rods in the fuel assembly | 311 |

Average linear heat rate (W/cm) | 166.7 |

Coolant flow rate (m^{3}/h) |
84800 |

Inlet pressure (MPa) | 16 |

Outlet pressure (MPa) | 15.70 |

Temperature at the inlet (°C) | 291 |

In order to evaluate the authenticity of thermal-hydraulic module, the mentioned transient is modeled in PWR core fuel assembly (conditions are noted in Table

PWR geometry and operating conditions [

Operating condition | Value |
---|---|

Channel length (m) | 3.66 |

Rod diameter (mm) | 9.70 |

Pitch (mm) | 12.80 |

Flow area for rod (mm^{2}) |
90.00 |

Equivalent diameter (mm) | 12.00 |

Linear heat (kW/m) | 17.50 |

Mass flux (kg/m^{2}s) |
4125 |

Inlet pressure (MPa) | 15.50 |

Outlet pressure (MPa) | 15.42 |

Inlet enthalpy (kJ/kg) | 1337.2 |

Neutronic and thermal-hydraulic coupling calculation algorithm.

As a result of time period selection limitations (short time periods), which leads to increase in the number of nodes, the rise in parameter numbers of defined study case and complicated equations or scales should not be considered. Therefore program solving time is lowered.

Relative radial power distribution in 1/6 of VVER-1000 reactor core is shown in Table

Relative power radial distribution in 1/6 of VVER-1000 reactor core.

Fuel assembly number | Power peaking factor (FSAR) [ |
Power peaking factor |
Power peaking factor |
---|---|---|---|

1 | 1.01 | 1.02 | 0.99 |

2 | 0.80 | 0.79 | 0.78 |

3 | 1.02 | 1.01 | 1.00 |

4 | 0.75 | 0.75 | 0.72 |

5 | 0.86 | 0.85 | 0.83 |

6 | 0.92 | 0.89 | 0.90 |

7 | 1.16 | 1.17 | 1.18 |

8 | 0.92 | 0.93 | 0.90 |

9 | 0.78 | 0.78 | 0.76 |

10 | 0.87 | 0.88 | 0.84 |

11 | 0.86 | 0.85 | 0.84 |

12 | 1.24 | 1.22 | 1.24 |

13 | 0.96 | 0.97 | 0.98 |

14 | 0.78 | 0.77 | 0.76 |

15 | 1.03 | 1.04 | 1.02 |

16 | 0.89 | 0.88 | 0.87 |

17 | 1.11 | 1.10 | 1.11 |

18 | 1.18 | 1.17 | 1.24 |

19 | 0.87 | 0.86 | 0.84 |

20 | 0.89 | 0.87 | 0.87 |

21 | 1.29 | 1.27 | 1.31 |

22 | 1.24 | 1.25 | 1.31 |

23 | 0.86 | 0.85 | 0.84 |

24 | 1.11 | 1.09 | 1.11 |

25 | 1.24 | 1.23 | 1.31 |

26 | 1.24 | 1.22 | 1.24 |

27 | 1.18 | 1.17 | 1.24 |

28 | 0.96 | 0.96 | 0.98 |

The maximum coupling calculation errors of DRAGON 4/DONJON 4 is 2.8% while it is 5.64% in WIMS/CITATION [

Relative axial power distribution in 1/6 of VVER-1000 reactor core.

After the defined model is validated, thermal-hydraulic module preparation is started based on SHC by SC method. In order to evaluate the authenticity of newly made module, coolant fast pressure drop accident in PWR core fuel assembly is simulated by SC method (PWR core features are noted in Table

Short-term response of the PWR inlet pressure transient using the SC model.

For

The written program accuracy in neutronic states is compared to Bushehr VVER-1000 reactor FSAR data [

Changing diagrams of thermal-hydraulic parameters by using sound effect in pressure propagation, for very FT coolant pressure drop in VVER-1000 reactor HFA, are being studied. Pressure drop causes the mass flux reduction; therefore temperature increases and power drops in every node. The transfer of each node’s thermal data to DRAGON 4 module leads to the decrease of cross section moderation. Mentioned data and axial power profile are used as DONJON 4 module entrance. Finally lowered power distribution and pressure drop transient are applied as thermal-hydraulic module and thermal distribution of every one of 360 nodes is obtained again. The process continues until the convergence criteria of each node thermal difference are available. Considering the central pressure of each node and reactor core pressure changes, which is 0.3 MPa, alterations of pressure per time for the first node is shown in Figure

Alterations of pressure per time for the first node.

Figure ^{2}s, occurs up to the beginning of fuel assembly. However, these changes are not sensed by the upper nodes. After 1 ms, mass flux drop arrives at the middle of fuel assembly, which increases from 0.35 Kg/m^{2}s to 12 Kg/m^{2}s for the first node and decreases to 25 Kg/m^{2}s after 2 ms. The channel transient time of sonic wave is about 4 ms. The rapid decrease of the mass flux has not yet reached the end of the channel in this period of time; that means the upstream region is not yet affected by the pressure wave.

Mass flux changes during fuel assembly in different three times.

Mass flux changes in 5 ms indicate the wave reflux from end of the channel. From now on, for a few milliseconds, the reversible wave causes more mass flux drop in the end of channel than the beginning of it. Mass flux changes in 5 and 9 ms, which are calculated by RELAP5 [

The observed difference between DRAGON 4 and DONJON 4/SHC diagram (in 5 ms) and RELAP5 diagram (in 5 ms) is because of the difference between neutronic and thermal-hydraulic modules accuracy. Also the effects of thermal feedback cause perturbation in mass flux alterations, in DRAGON 4 and DONJON 4/SHC code. RELAP5 [

Driving force drop caused by inlet pressure drop that reduces the channel average mass flux.

The fluid thermal expansion due to heating which causes a small slope change; that is,

The fluids being considered uncompressible (i.e.,

Weak neutronic model and not using sufficient time meshes sizes for the mentioned transient.

The obtained results from mass flux changes simulation in primary, middle, and ending points of relative length of fuel assembly, in 9 ms, in DRAGON 4 and DONJON 4/SHC and RELAP5 code are shown in Figure ^{2}s). Mass flux in central point of fuel assembly starts dropping after whereabouts 1.2 ms. However severe drop for ending points occurs after 4 ms. The profiles at ^{2}s. The net mass flux is the superposition of the forward wave and the reflected wave.

Mass flux changes during fuel assembly in different three nodes.

RELAP5 results show mass flux profile changes, with steady slope, in primary, middle, and ending points and does not present adequate description of the passing pressure wave.

HFA heat flux changing per time for beginning, middle, and ending nodes of fuel assembly, in “0–3 ms” period of time, is shown in Figure ^{2} in that amount of time. However, mass flux of fuel assembly middle node drops whereabouts 30 KW/m^{2} in 1.7 ms. It also occurs after 2.3 ms for ending node. Temperature changes per time for the first node are shown in Figure

HFA heat flux changing per time for beginning, middle, and ending nodes of fuel assembly.

Temperature changes per time for the first node.

Simulation of VVER-1000 reactor core FT pressure drop, by using acoustic phenomenon, was studied in this article. Coupling of the two DRAGON 4 and DONJON 4 neutronic modules, along with thermal-hydraulic module by means of SC method, was utilized in this simulation. It is the first time that neutronic and thermal-hydraulic modules are externally coupled, assuming that the coolant fluid is compressible, in order to obtain the best simulation for this transient.

In this transient, at the channel inlet, the mass flux begins to decrease because of the pressure reduction; it was observed in calculations that a perturbation was produced and propagates into the channel as time elapses. After pressure waves arrive at the end of the channel, reflected pressure waves affect the mass flux profile for short time. The incoming rarefaction wave will be reflected as a compression wave, due to the assumption of constant outlet pressure, at the exit boundary. Therefore a wave travels in the opposite direction with the same amplitude but opposite sign. The reverse wave causes more mass flux drop in upper nodes and as time passes, it decreases the difference between upper nodes mass flux drop and the ones in beginning of the channel. Thus it is concluded that, in such transients, mass flux drop rate does not follow stable conditions.

This transient (double ended guillotine break) is very fast, and therefore DRAGON 4 and DONJON 4 codes are used in neutronic module (comparing to WIMS/CITATION) considering their high speed and accuracy in analyzing the problem. Also SC method is utilized in thermal-hydraulic module for solutions of conservation equations by forward implicit method and Courant’s criterion in a SHC. Thus the very short time step (

The obtained results from DRAGON and DONJON/SHC for the times that a great volume of coolant is lost in a short time, assuming coolant fluid is compressible, are highly acceptable and they show its ability to effectively and accurately calculate the fast transient. On account of that, the results are not desirable if the sonic effect in coolant is not regarded.

Fast transient

Finite element method

Pressurized water reactor

Channel integral

Single mass velocity

Momentum integral

Sectionalized, compressible fluid

Nuclear power plant

Final safety analysis report

Loss of coolant accident

Single heated channel

Hot fuel assembly

Millisecond

Microsecond

International Atomic Energy Agency.

All authors have no conflicts of interest to declare.

This study was performed as part of long-term research into nuclear safety supported by the Nuclear Engineering Department of Science and Research Branch, Islamic Azad University. The valuable researches into various accidents of VVER-1000 reactor led us into performing this study.

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