The solitary wave in advanced nuclear energy system
** Corresponding author : Jin Feng Huang, [email protected] The solitary wave in advanced nuclear energy system
Jin Feng Huang * Department of Nuclear science and Engineering, East China University of Technology, Nanchang 330000, China
Abstract
The solitary wave naturally arises in many areas of mathematical physics, including in nonlinear optics, plasma physics, quantum field theory and fluid mechanics. For advanced nuclear energy system, travelling wave reactor or CANDLE reactor, has been proposed for couples of years. However, the exact solution has not been discovered. In this study, according to the perspective of solitary wave, the exact solution of this advanced nuclear energy system is demonstrated through coupling one group neutron diffusion equation with burnup equation. The tanh-function method is applied to solve that nonlinear partial differential equation. The relationship between velocity of solitary wave, wave amplitude or neutron flux and the evolution of nuclide is revealed by analytical method.
Key word : Nonlinear system, Solitary wave, CANDLE burnup Introduction
The solitary wave in non-linear system had been discovered by Russell John Scott in 1834. Solitary waves are particle-like waves that own to a balance between nonlinear and dispersive effects. Solitary wave propagates without any temporal evolution in shape or size when it moves at constant speed and conserves amplitude, shape, and velocity after a collision with another soliton. Solitons naturally arise in many areas of mathematical physics, including in nonlinear optics, plasma physics, quantum field theory and fluid mechanics. The classical example of an equation yielding solitary wave solutions is the Korteweg-de Vries equation, which is model of waves on shallow water surfaces. The solitary wave also can be observed in advanced nuclear energy system, travelling wave reactor or CANDLE (Constant Axial shape of Neutron flux, nuclide number densities and power shape During Life of Energy producing reactor) reactor. A concept of completely automated nuclear reactor for long-term operation was proposed by Teller (Teller et al., 1996). This reactor core which comprises of ignition region and breeding region is quite different from the conventional reactor. The neutrons leaking from ignition region are captured by fertile fuel converted the fissile fuel in breeding region. The breeding region is adopted with thorium material but the fuel burnup can be as high as 50% marvellously. The self-stabilizing criticality waves in such reactor were presented by Hugo Van Dam (H. Dam, 2000). The analytical model with reactivity feedback was illustrated through introducing a parabolic burnup function which is most simple form, and the ignition condition for a criticality wave was provided. Base on the same neutronic model with reactivity feedback, a two-group diffusion model coupled with simplified burn-up equations is investigated (X.-N. Chen et al., 2007) for a one-dimensional burn-up drift wave problem. The feasibility of creating self-organizing breeding/burning waves was analyzed by S. Fomin (S. Fomin et al., 2005). At the starting phase, the requirements for wave initiation and evolution in space and time were discussed through coupling diffusion transient model with burnup equations. Breed-and-burn strategy in a fast reactor with optimized starter fuel was carried out by J. Huang (Jin Feng Huang et al., 2015). The optimization of the starter fuel was performed to reduce the initial positive excess reactivity swing and to flatten the power distribution, and the results show that elaborating the ignition region is effective to reduce fuel enrichment and improve the operation performance during the starting phase. A particular class of travelling wave reactor, called the CANDLE reactor, was illustrated firstly by Sekimoto (Sekimoto et al., 2001). The equilibrium state of the CANDLE burnup was discussed through solving numerically neutronic diffusion equations coupled with burnup equations. In the equilibrium state, distributions of nuclide densities, neutron flux, and power density are remained axially constant shapes and the same constant speed for constant power operation. The equilibrium state of the CANDLE also can be considered as solitary wave formed by neutron flux starting to propagate steadily. For the solitary wave, only nonlinear and dispersive effects retain balance that solitary wave would not disappear. The dispersive effects represent in leakage term in the nonlinear partial differential equation. Consequently, dealing with the nonlinear effects caused by neutron fission and absorption in medium is the key point to produce and propagate the solitary wave. In this study, the nonlinear partial differential equation (PDE) was constructed through coupling neutron diffusion equation with burnup equation. Usually, nonlinear PDE is difficult to attain the analytical solution. However, if the nonlinear term in above PDE was expanded by Taylor’s series, then the tanh-function method could be applied to solve this PDE. The necessary physical condition was analyzed to get the exact solution. The results show that the neutron flux, neutron fluence, fuel burnup and the evolution of nuclide density can be represented as the form of solitary wave. Nonlinear partial differential equation 2.1
Burnup equations Where N , Φ , , , γ denote atomic density of nuclide I, neutron flux, microscopic absorption cross section of nuclide I, microscopic capture cross section of nuclide I, radioactive decay constant of nuclide I, respectively. In an actual situation, for U- Pu conversion chains, the production of
Pu (e.g., by the neutron capture of
U) and decay processes may be considered, but they are omitted here for the sake of simplicity. Therefore, U- Pu conversion chains can be considered as U → Pu → Pu (or fission productions). This simplification would not change the physical inherence and make the issue more easily. For
U, the burnup equation can be expressed as: = − (1) The solution of this differential equation is as followed: = , (2) = (3) Where , denotes the initial atomic density of U. Similarly, for
Pu, burnable poisons (BP) and fission productions (FP) the burnup equation can be expressed as: = − + (4) = − (5) = − + ∑ (6) The solutions of these differential equations are as followed: = , + , − [ − ] (7) = , (8) = (9) Where , denotes the initial atomic density of Pu. The value of , can be equal to zero for CANDLE reactor in breeding region since the production of Pu is only through the neutron capture of
U. One of commons for these solutions should be emphasized that they can be expressed as exponential functions even if take account into
Pu and
Pu in the burnup chain. These exponential functions can be expanded as the Taylor’s series, which provides one way of analytical solving the nonlinear PDE.
Neutron diffusion theory
The one-group neutron diffusion model plays an important role in reactor theory even through it is sufficiently simple. This simple diffusion model coupled with burnup equation also sufficiently realistic to reveal the producing and propagating the solitary wave. The one-group neutron diffusion (Duderstadt and Hamilton, 1976) can be expressed here, D + ( − )Φ = (10) Where Φ is neutron flux, D denotes the neutron diffusion coefficient, k ∞ the infinite medium multiplication factor, the macroscopic absorption cross section, average neutron number per fission, macroscopic fission cross section, neutron speed. Furthermore, = ∑ , = ∑ , D = denotes different nuclide number i. could be substituted the solution of burnup equations, such as, , , etc. All the expressions of contacting with the factor multiplied Φ are nonlinear terms in PDE. Only these nonlinear effects cancelling out dispersive effects, the solitary wave can propagate over large distances without dissipation. The ( − ) was equal to: ( − ) = + + − ( + + + + ) (11) All the ( i =238, 239, 240, FP, BP) are the function of exponential form and can be expanded as Taylor’s series. The Pu fissions caused by fast neutrons are dominant compared with
U fissions and
Pu fissions in fast neutron spectrum. Therefore,
U fissions and
Pu fissions are neglected. All the terms in function (12) taken account into for calculation is fine and can improve the accuracy only need to repeat expansion behavior, but such behaviors increase the complication of solving nonlinear PDE. For the sake of simplicity, only the , , are remained to reveal the rule of solitary wave clearly. Consequently, the nonlinear PDE can be rewritten: D ∂ Φ ∂ + ( 9 − 8 )Φ = 1 Φ (12) ( ) ≡ ( − ) = , − − − − (13) Analytical solution and discussion 3.1 Analytical solution
In this section, the processes of pursuing analytical solution of nonlinear PDE are illustrated in detail. Usually, the nonlinear PDE is difficult to have analytical solution. It is discovered that this type PDE would have analytical solution by using the tanh-function method (
Huibin Lan and Kelin Wang, 1989 ) if the nonlinear terms are expanded by Taylor’s series.
D ∂ Φ ∂ + ( )Φ = 1 Φ (14) Step 1.
Consider following the travelling wave transformation,
Φ( ,t) = Φ( − ut) = Φ(η) (15)
Where u is phase speed of wave, and η means coordinate. ( , ) = − ( ) , = ( ) (16) Therefore, Eq.(15) can be transformed into the ordinary differential equation (ODE): Φ(η) η + u Φ(η) η + ( )Φ(η) = 0 (17) Step 2 . Assume that the solution of Eq.(18) has the form,
Φ = ∑ , ≡ ℎ( ) (18) Where is constant coefficient, denotes hyperbolic tangent function. The integer j is depended on the balance between the highest order derivatives and the nonlinear terms. Step 3 . and can be expanded by second order Taylor’s series and here omit the higher order terms. = 1 + (− )+ 12 (− ) = 1 + (− )+ 12 (− ) The highest order derivatives terms is + 2 , and the highest order nonlinear terms ( )Φ is . The proof would be represented in Appendix. Consequently, we obtain = 2 and then Φ = + + . Now the boundary conditions ( Φ(± ∞ ) = 0 ) are taken account into this expression. For a solitary wave, the function value must be close to a constant number if coordinate is away enough. In the CANDLE reactor, solitary wave which is limited in narrow region other than distributes entire region, the constant number should be zero since the neutrons have the finite mean free path in medium. Therefore, according to Fig. 1, the coefficient should be zero, not only this, but also should be equal to − then the boundary conditions could be yielded. Consequently, Φ = − + (19) The neutron flunce can be obtained by the integral of Φ , = ∫ (η) = − (20) Where is an arbitrary constant, for the simplification, could be set up to zero since the neutron flux needs to be normalized due to different power level. Substitute Eq.(20) into Eq.(14), the expression can be written, ( ) = , ( − )+ − Tu ( − )+ 12 Tu (( ) − ( ) ) (21) = − − = − = + Fig. 1.
The profile of Tanh function
Step 4 . Substitute Eq.(22) along Eq.(20) and Eq.(21) into Eq.(18) and collecting all the terms of the same power i , i = 0,1,2,3,4. (1 − )[− 2 ( + 2 )+ 2 (1 − )]+ (1 − )( + 2 )+ ( ) = 0 (22) :(2 − 2 + , + , − , − , ) :(6 + , − , ) Equating each coefficient of this polynomial to zero yields a set of algebraic equations for a n (n = 0, 1, 2). Due to a determined by Tanh( ) boundary condition, and coefficient of this polynomials was selected. Solving the equation system, we can construct a variety of exact solutions for Eq. (23), = √ , (23) = 3 , ( − ) − + (24) = ( ) (25) Consequently, the neutron flux Φ has the analytical solution, Φ(η) = 6 ( − )− + 1 − ℎ ( ) (26) It should be mentioned that a critical reactor can operate at any flux level, hence, the magnitude of flux is depended on the power level of the core. There is still a coefficient B should be multiplied Eq. (27) to normalize flux. For the neutron flunce, if the wave travels toward positive direction, such boundary condition ( (− ∞ ) = 0 ) should be yielded. Therefore, the neutron flunce can be represented: = (1 + ℎ( )) = 2√3 , − + (1 + ℎ( )) (27) Eq. (25) reveals that the phase speed of solitary wave formed by neutron flux is proportional to neutron velocity and square root of diffusion coefficient and initial
U density. Eq. (27) implies that the amplitude of neutron flux is linearly proportional to the diffusion coefficient and neutron velocity, but is inversely proportional to initial
U density since diffusion coefficient D is inversely to initial U density and other materials. This illustrates that reducing the fuel density increases the amplitude of neutron flux, but the total reaction rate should keep the same if the power is constant, therefore, the changing makes the solitary wave lanky. The larger amplitude of solitary wave is not desirable since it increases the power peaking factor notably. Oppositely, the lower fuel density would broaden the distribution of power. The one of concerns for CANDLE reactor is that the very high power peaking factor takes great challenge for reactor safety. This study provides insight into solving this issue. Fig. 2. The neutron flux profile of bell-shaped solitary wave drifting
Eq. (28) indicates that the neutron fluence is inversely proportional to initial
U density. It denotes that the lower fuel density would have higher neutron fluence. Φ,× 6 ( − )− + ,× 2√3 , − + Fig. 3. The neutron fluence profile of Anti-Kink solitary wave drifting
One of the notable merits for CANDLE reactor is that the fuel burnup can be as high as 400 Gwd/t only loading with depleted uranium or natural uranium in breeding region. The fuel burnup is linearly proportional to neutron fluence. Therefore, the fuel burnup should have same profile with Fig. 3 but with different coefficient. The trend of burnup solitary wave profile coincides well with the previous study (Jin Feng Huang et al., 2015) which is performed by Monte-Carlo method coupled ORIGEN burnup code. On the other hand, the very high fuel burnup also takes great challenge due to the fuel and cladding radiation damages for the nuclear engineering. This analytical solution of solitary wave point out that the fuel burnup can be reduced through increasing the initial fuel density. The evolutions of nuclides play a vital role in maintaining solitary wave. The solutions of burnup equations reveal such evolutions. Applying the Taylor’s series, the nuclide
U and
Pu can be written, = , (1 − + 12 ) (28) Substitute Eq.(29) into Eq.(28), = , = , 1 + − ℎ( ) + 12 ℎ( ) (29) Similarly, the evolution of
Pu can be expressed as, = , + − ℎ( ) ( , − , )+ 12 − ℎ( ) (− , − , + , ) The evolutions of
U and
Pu profiles are displayed as Fig. 4 and Fig. 5. The trends of evolutions of
Pu are also matched well with the previous studies (Jin Feng Huang et al., 2015; Xue-Nong Chen et al., 2012). The
U, or burnable poison density is depended on microscopic absorption cross section. The evolution of
Pu, breeding from
U, shows that there is a peak value toward the propagating direction. Fig. 4 The Kink solitary wave profile of evolution of U A r b it r a r y U n it Fig. 5 The profile of evolutions of Pu Conclusions
The analytical solutions of advanced nuclear energy system, travelling wave reactor or CANDLE reactor, is represent through coupling neutron diffusion equation with burnup equation, and the tanh-function method is employed to solve this nonlinear partial differential equation. The analytical solutions provide important insights into the physical phenomena of advanced nuclear energy system clearly. The neutron flux, neutron fluence, fuel burnup and the evolution of nuclide density can be represented as the different solitary waves.
Acknowledgements A r b it r a r y U n it This work was supported by the Ignition Research Funds for East China University of technology.
The authors are also grateful to the China Scholarship Council (CSC No. 201706310058) for their support, education department project in Jiangxi province of China (No. GJJ180402) and Special high-level talent research start-up (No. DHBK2017132 and No.110-1410000994).