Semi wavelet-based improved quasi static method for the analysis of PHWR transients
Introduction
Safety analysis tools demand accurate prediction of reactor core behaviour under normal operating conditions as well as during accidental situations like ejection of control rods from the reactor core. It is important from safety point of view that careful consideration is given to develop methods for the analysis of reactor core behaviour with sufficient accuracy. The proposed methods and the solution should reflect the true behaviour of reactor core under normal operating as well as postulated accidental scenarios. The success of any model, predicting the core behaviour, lies in solving the space and time dependent neutron diffusion equation with thermal hydraulics feedback. The development of various methods for solving the space–time neutron diffusion equation remains a challenging job. One gross simplification of the space–time neutron diffusion equation is the point reactor kinetics. The point reactor kinetics is applicable for predicting the transient in small reactors. Rapid transients of reactor power in large reactors, caused by sudden reactivity insertion, have strong space-dependent effect associated with them and in such cases the point reactor model alone cannot predict the true behaviour of reactor transients. The inadequacy of the point-reactor model for the analysis of large thermal reactor transients was numerically demonstrated by Yasinsky and Henry (1965).
Many numerical methods for the solution of time dependent neutron diffusion equations have been of interest for decades (Clark and Hansen, 1964, Ash, 1979, Henry, 1969 Henry, 1972). The improved quasi static (IQS) method (Ott and Meneley, 1969) is one of the widely used methods to solve the time dependent neutron diffusion equation and it is an efficient method in estimating the reactor transients (Jain and Gupta, 1986, Dahmani et al., 2001; Razak and Devan, 2017). The IQS method can be used to estimate the transients in high temperature gas-cooled reactor (HTR) (Chunlin et al., 2014). In the IQS method, the flux is factored into two functions, i.e. amplitude function and shape function. The amplitude function forms a stiff differential equation and it describes the fast time evolution of neutron flux in the reactor. It requires a very small time step (micro time step) to numerically solve the equation. The shape function weakly depends on time and it can be solved using large time step (macro time step). In the IQS method, two different time scales are adopted, i.e. micro time step and macro time step to solve the neutron diffusion equation in space and time.
In this proposed computational method, wavelets are incorporated in the IQS scheme and the space–time neutron diffusion equation is solved using only one time step, i.e. single time step and the concept of micro time step and macro time step is avoided. Nowadays wavelets find their applications in the numerical solution of partial differential equations (Lepik, 2005, Lepik, 2007, Sokhal and Verma, 2021, Saha Ray and Behera, 2020) and they are very useful in solving stiff differential equations (Lepik, 2009). The wavelet method is extremely useful in solving fractional order delay differential equations and integro-differential equation (Amin et al., 2021; Saemi et al., 2021, Aruldoss et al., 2021). Wavelet method can be used to solve the neutron transport equation (Zheng et al., 2010) as well as diffusion equation (Cho and Park, 1996). When wavelets are used for solving neutron transport or diffusion equation, it is hard to satisfy the interface and boundary conditions (Nasif et al., 1999; Nasif et al., 2001). In this proposed method, wavelets are used to solve the amplitude function and the shape function is solved using implicit scheme. Here the transient estimation methodology is different from the conventional IQS method.
This computational method is applied to estimate the transients in CANDU 3D-PHWR benchmark (Judd and Rouben, 1981), CANDU-2D (Gupta, 1980, Fernando, 2003) and TWIGL-2D reactors (Aboanber and Nahla, 2007, Nahla et al., 2012). The transient in CANDU 3D-PHWR, estimated by this method, is compared with CERKIN code (Judd and Rouben, 1981). The transients in CANDU-2D and TWIGL-2D reactors are estimated by this method and they are compared with the reference. This method is capable of estimating the transient with temperature feedback. To prove this a Doppler temperature feedback is introduced in the absorption cross section in TWIGL reactor and the transient is estimated with feedback and compared with the reference. The results are found to agree to a good accuracy. From the comparison of results, it is established that this new computational method is capable of estimating the transients in PHWR and other reactors to a good accuracy by adopting only one time step, i.e. single time step. Here the accuracy of the transient estimation depends on the order (level) of resolution of wavelet being used. It is also shown that as the order of the wavelet is increased, the accuracy in the estimation of reactor transient increases.
Section snippets
Improved quasi static (IQS) method
The time dependent neutron diffusion equation (in energy group ‘g’) with delayed neutron precursor groups is written as (Stacey, 1969)
In the above equations, is the time dependent neutron flux in group , is the diffusion coefficient, is the removal cross section, denotes the prompt fission spectrum, is the fission
Solution of amplitude function using Haar wavelets
In the recent years wavelet approach has become more popular in the field of numerical solution of ordinary and partial differential equations. Different types of wavelets and approximating functions have been used in the numerical solution of initial and boundary value problems (Lepik, 2005, Lepik, 2007). Wavelets are useful in solving stiff differential equations (Lepik, 2009, Patra and Saha Ray, 2014) as well as time evolution of partial differential equations (Lepik, 2007). Wavelets are
Transient estimation methodology using Haar wavelet
First the order of resolution of wavelet is fixed and the transient duration is divided into collocation points in time as , , ,… etc. The state of the reactor, i.e. control device position, cross section etc. at collocation points is obtained and a prior estimate of the amplitude and shape functions at these collocation points is made. Using these a priori estimates, the kinetic parameters , , are calculated at collocation points (step 1). The point kinetics
CANDU 3D-PHWR transient estimation
CANDU 3D-PHWR is a realistic space–time kinetics benchmark problem (Judd and Rouben, 1981). It contains inner core and outer core, surrounded by heavy water reflector. The core is shown in Fig. 2. The length of the core in Z direction is 600 cm. The height and width of the core are 780 cm. The vertical cross section () of the core is shown in Fig. 3. The horizontal cross section of the core at is shown in Fig. 4. The two group cross section data for the core are given in Table 1
CANDU 2D PHWR transient estimation and discussion
This benchmark problem (Gupta, 1980, Fernando, 2003) is a realistic representation of CANDU PHWR in two dimensions. The reactor geometry is shown in Fig. 10. The reactor contains two cores, inner core and outer core, surrounded by heavy water reflector. The cross section data for various regions in the core are given in Table 3. The delayed neutron precursor data are given in Table 4. Here the transient is generated by coolant voiding in one half of the core. The coolant voiding is generated by
Conclusions
A new computational method is developed by incorporating the Haar wavelet in the improved quasi static (IQS) scheme to estimate the transients in CANDU 3D PHWR benchmark, CANDU-2D and TWIGL-2D reactors. Here the transient estimation methodology is different from the IQS scheme. In this computational method, the concept of micro time step and macro time step is avoided and the transient is estimated by adopting only one time step, i.e. single time step. In this method, the time dependent neutron
CRediT authorship contribution statement
M. Mohideen Abdul Razak: Conceptualization, Methodology, Formal analysis, Writing - original draft. K. Obaidurrahman: Resources, 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.
Acknowledgements
The authors greatly acknowledge the help rendered by Mr. S. Venkatesan, BARC-F, Kalpakkam.
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