COMPARISON OF DIRECT AND QUASI-STATIC METHODS FOR NEUTRON KINETIC CALCULATIONS WITH THE EDF R&D COCAGNE CODE

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1 PHYSOR 2012 Advances in Reactor Physics Linking Research, Industry, and Education, on CD-ROM, American Nuclear Society, LaGrange Park, IL (2012) COMPARISON OF DIRECT AND QUASI-STATIC METHODS FOR NEUTRON KINETIC CALCULATIONS WITH THE EDF R&D COCAGNE CODE E. Girardi and P. Guérin Electricité de France R&D 1 av. du Général de Gaulle, 92141, Clamart (France) enrico.girardi@edf.fr; pierre.guerin@edf.fr S. Dulla, M. Nervo and P. Ravetto Dipartimento di Energetica Politecnico di Torino 24, c.so Duca degli Abruzzi, 10129, Torino (Italy) sandra.dulla@polito.it; marta.nervo@tiscali.it; piero.ravetto@polito.it ABSTRACT Quasi-Static (QS) methods are quite popular in the reactor physics community and they exhibit two main advantages. First, these methods overcome both the limits of the Point Kinetic (PK) approach and the issues of the computational effort related to the direct discretization of the time-dependent neutron transport equation. Second, QS methods can be implemented in such a way that they can be easily coupled to very different external spatial solvers. In this paper, the results of the coupling between the QS methods developed by Politecnico di Torino and the EDF R&D core code COCAGNE are presented. The goal of these activities is to evaluate the performances of QS methods (in term of computational cost and precision) with respect to the direct kinetic solver (e.g. θ scheme) already available in COCAGNE. Additionally, they allow to perform an extensive cross-validation of different kinetic models (QS and direct methods). Key Words: Quasi-static method, neutron kinetic, neutron transport, predictor-corrector. 1. INTRODUCTION The quasi-static (QS) method introduced by A. F. Henry at the end of the 1950 s is a standard tool for the solution of space-time neutron transport problems [1]. It is often used in the algorithmic form developed by Ott and Meneley called Improved Quasi-Static method (IQS) [2]. More recently, a new form of the QS approach, taking advantage from a predictor-corrector algorithm (PCQM), was proposed [3,4]. Quasi-Static methods are quite popular in the reactor physics community and they exhibit two main advantages. First, these methods overcome both the limits of the Point Kinetic (PK) approach and the issues of the computational effort related to the direct discretization of the time-dependent neutron transport equation. Thus they represent a good trade-off between precision and computing time. Second, QS methods can be implemented in such a way that they can be easily coupled to different external spatial solvers. corresponding authors: enrico.girardi@edf.fr, pierre.guerin@edf.fr

2 Enrico Girardi, Pierre Guérin et al In this paper, the results of the R&D activities carried out at EDF, in collaboration with Politecnico di Torino [8], on the coupling between the QS methods and the SP N core code COCAGNE [9] are presented. The goal of these activities is to evaluate the performance of QS methods (in term of computational cost and precision) with respect to the direct kinetic solver (e.g. θ scheme) already available in COCAGNE. Additionally, they allow to perform an extensive cross-validation of different kinetic models (QS and direct methods) Kinetic Equations 2. KINETIC EQUATIONS AND QUASI-STATIC METHODS When dealing with the time-dependent neutron transport equation, one has to modify the stationary Boltzmann equation in order to take into account the contribution of prompt and delayed neutrons to the chain reaction. While the prompt neutrons are emitted almost instantaneously after the fission event, the delayed neutrons are emitted with a certain time delay by some fission products called delayed neutron precursors. Thus, one has to define a set of coupled equations, called kinetic equations, which describes both the evolution of the neutron population and the delayed neutron precursors concentrations: 1 φ v t = L(t)φ(t) + 1 4π χ i (E)λ i C i (r, t) + S(r, E, Ω, t) i χ i (E) C i 4π t = χ i(e) 4π λ ic i (r, t) + P d,i (t)φ(t), i = 1,..., I (1) where φ(t) φ(r, E, Ω, t) is the neutron angular flux, S the external source (if any), I the number of delayed neutrons precursors, P d,i (t) denotes the delayed neutron emission operator and L(t) denotes the time-dependent transport operator including streaming, absorption, scattering and prompt fission. A direct approach for the solution of the time dependent system of equations (1) is already available in the COCAGNE core code. In this work, an alternative way of solving the kinetic equations by using the COCAGNE flux solver in the quasi-static framework is investigated. In the next sections the generic Quasi-Static approach is discussed and the Improved Quasi-Static method (IQS) and Predictor-Corrector Quasi-Static Method (PCQM) are briefly introduced (see Ref. [3,4,7] for a more detailed description) The Quasi-Static Approach The basic idea of the Quasi-Static method is to split the neutronic flux dependence in two parts: an amplitude function depending only on time and a shape function, which includes the dependence on space, energy and angle (and more slowly on time): φ(r, E, Ω, t) = A(t) ψ(r, E, Ω, t) (2) There is a degree of arbitrariness in the choice of A(t) and ψ(r, E, Ω, t), as only the product of the two really counts. In order to make the factorization unique, a normalization constraint is 2/12

3 Comparison of Direct and Quasi-Static Methods for Neutron Kinetic Calculations imposed on the shape function: ψ 0, 1 v ψ = ψ0(r, E, Ω) 1 v ψ(r, E, Ω, t) dr de dω = γ 0 ψ0, 1 v φ 0 (3) where γ 0 is a constant, determined by the projection of the steady-state flux φ 0 over the solution of the corresponding adjoint transport problem (ψ 0). Introducing (2) and its first derivative with respect to time into the first equation of the system (1), one gets the shape equation: L(t)ψ(r, E, Ω, t)+ 1 4π 1 χ i (E)λ i C i (r, t) A(t) i + S(r, E, Ω, t) A(t) = 1 v ψ t + 1 v ψ(r, E, Ω, t) 1 da A(t) dt (4) Then, the amplitude equations are derived using a projection technique involving the steady-state adjoint flux as a weight function. Finally, after some algebraic manipulations, one gets a system of (I + 1) ordinary differential equations for the amplitude and the neutron precursors concentrations (the appropriate definition of the kinetic parameters can be found in [3]): da(t) dt dc i (t) dt = ρ β Λ A(t) + i = λ i c i (t) + β i Λ A(t) λ i c i (t) + S i = 1,..., I (5) Equations (4) and (5) form a set of non-linear equations, as the solution of the shape equation (4) requires the knowledge of A(t), while the kinetic parameters in the amplitude equation (5) depend on ψ(r, E, Ω, t). As the shape is supposed to change more slowly than the amplitude over time, an efficient way to solve the previous coupled system is to use a two-level nested iterative procedure, dealing with different time-scales. The shape equation is solved over a slow time scale, associated to a large time-step t (macro-time-step), while the amplitude is computed on a fast time scale (micro-time-step) δt. Ideally, the ratio t/δt should be as large as possible Algorithmic structure of IQS The IQS algorithm [2] can be summarized as follows: Step 1. Direct and adjoint fluxes are evaluated in the steady-state condition and the value of the normalization constraint (3) is determined. The PK parameters of the amplitude equation system (5) are evaluated, assuming the shape function has not been modified from its steady-state value ψ 0. Once the PK parameters are known, the amplitude equations (5) may be solved using a stiff ODE system solver with time-step δt, through the time interval t. 3/12

4 Enrico Girardi, Pierre Guérin et al Step 2. Once the amplitude and its time-derivative at t = t are known, the shape equation (4) can be integrated over t, approximating the shape time-derivative by an implicit Euler algorithm : ψ/ t [ψ(r, E, Ω, t) ψ(r, E, Ω, 0)]/ t. Step 3. The error on the shape normalization condition (see Eq. 3) is evaluated. The new shape is then normalized to γ 0 and used for a new estimation of the point kinetic parameters. Step 4. The introduction of these kinetic parameters into the amplitude equations (5) allows to update the value of the amplitude A and its time-derivative at t = t which are introduced in the next shape equation resolution (4). Step 5. If the convergence criterion is fulfilled, the algorithm continues on the next macro time-step, else steps 2-4 are repeated Algorithmic structure of PCQM An alternative algorithm, dealing with the couple amplitude angular flux (instead of the amplitude shape couple) may be considered. Such approach, named Predictor-Corrector Quasi-static Method (PCQM) [3], yields a linear integration scheme. The amplitude function and the point kinetic parameters are solved on the fast time-scale, using equations (5). The angular flux is kept as the basic unknown and evaluated at the end of each macro-time-step t, while the shape function is used as an auxiliary tool to fulfill the normalization condition (3) required by the splitting equation (2). The algorithm runs as follows: Step 1. Direct and adjoint fluxes φ 0 and ψ 0 are evaluated in the steady-state condition and the value of the normalization constraint is computed. Step 2. Equations (1) are solved, using an implicit Euler scheme as an approximation of the flux time-derivative. The angular flux and delayed precursors concentrations are evaluated at t = t and denoted, respectively, φ(r, E, Ω, t) and C i (r, t). These are the predicted values with (presumably) some error on the amplitude because of the (large) time-step t. This error, however, will be corrected at the next step. Now the quantity z = 1 ψ γ 0, φ is evaluated. It can be verified that: 0 v ψ(r, E, Ω, t) := 1 z φ(r, E, Ω, t) (6) satisfies the constraint (3) and hence represents an approximation of the shape function evaluated at t = t. Step 3. The shape function (6) is used to compute the PK parameters. Then equations (5) are solved to determine the evolution of the amplitude up to t = t, using a standard method for stiff ODEs and the micro-time-step δt. Step 4. The corrected values of the angular flux and the precursor concentrations at t = t are respectively retrieved through φ(r, E, Ω, t) := ψ(r, E, Ω, t)a( t) and by the algebraic inversion of the implicit Euler approximation applied to the precursor concentration of Eq. (1). 4/12

5 Comparison of Direct and Quasi-Static Methods for Neutron Kinetic Calculations Step 4 completes the determination of an approximate solution of the angular flux and the delayed neutron precursors concentrations at time t. Starting from Step 2 the same procedure must be applied successively to all subsequent macro-time-steps, up to the end of the transient. 3. COUPLING OF THE QUASI-STATIC METHODS TO COCAGNE The quasi-static methods require a shape (IQS) or a flux (PCQM) solver, coupled to a PK module which evaluates the PK parameters and performs the amplitude evolution. In a previous work the coupling between the quasi-static methods and the DRAGON code [10] was carried out [5,7]. Starting from the existing DYNODRAGO code [6], this work was devoted to create a new integrated code performing the coupling between the quasi-static module and the COCAGNE flux solver General Structure of the Integrated Code The integrated code was designed to perform dynamics calculations in agreement with the IQS (see Sec ) and the PCQM (see Sec ) methods. Figure 1 shows the structure of the two integrated quasi-static codes coupled to COCAGNE. Following this structure the main steps are: The steady-state calculation is performed with the shape/flux solver COCAGNE, providing direct and adjoint fluxes. The value of γ 0 is determined. The geometrical configuration and the material description are written in a format understandable by the PK module. The procedure described in Sec (IQS) and Sec (PCQM) is applied. In the IQS method, the shape allows to compute the kinetic parameters and evaluate the amplitude profile. Then, a new input is created for COCAGNE in order to solve the shape equation and evaluate the updated shape. The previous procedure is iterated until the normalization condition is fulfilled or the maximum number of iterations is reached. In the PCQM method, the COCAGNE flux solver is called to determine the predicted value of the flux. The latter allows to determine the PK parameters and the amplitude evolution. The knowledge of the amplitude allows to correct the previous estimation of the flux and of the precursors concentrations. No convergence iteration is required. The previous set of instructions is applied successively to the next macro-time-steps, up to the end of the transient. Finally the full transient power profile is reconstructed Coupling Between IQS Module and COCAGNE The coupling between the IQS module and COCAGNE is achieved by producing the input files for the next COCAGNE calculation. By reordering the terms of the equation (4) in a suitable way, one may obtain the so called pseudo steady-state equation with an appropriate external source: [ L(t) 1 v t 1 va(t) da(t) dt ] ψ(r, E, Ω, t) + Q(r, E, Ω, t) = 0 (7) 5/12

6 Enrico Girardi, Pierre Guérin et al Steady-state block Steady-state block Input steady COCAGNE Steady state computation Output steady Input steady COCAGNE Steady state computation Output steady Input dynamics Input dynamics Quasi - Static block Quasi - Static block δt PK parameters calculation Amplitude calculation COCAGNE Flux prediction Shape generation Power profile Shape recomputation Δt δt PK parameters calculation Amplitude calculation Δt COCAGNE Flux correction Normalization condition New flux Updating A and da/dt test Power profile Next macrotime-step Next macrotime-step End of the transient End of the transient (a) IQS Integrated Code Block Diagram (b) PCQM Integrated Code Block Diagram Figure 1. IQS and PCQM Integrated Code Block Diagrams The operator in brackets is defined as generalized total operator, while Q is the generalized source including the contributions of the external source, the shape at the previous macro-time-step and the precursor evolution. Q(r, E, Ω, t) = + I i=1 S(r, E, Ω, t) A(t) + ψ(r, E, Ω, t t) v t 1 [ t χi ] λ i A(t) 4π C i(r, t t)e λi t + ψ(r, E, Ω, t t) P d,i (τ)a(τ)e λi(t τ) dτ t t + (8) To solve the equation (7), the generalized source and the virtual cross-section are provided to the shape solver COCAGNE. The latter is defined as: Σ t (r, E, t) = Σ t (r, E, t) 1 v t 1 da va(t) dt (9) where Σ t is the virtual total cross-section, Σ t is the total cross-section. 6/12

7 Comparison of Direct and Quasi-Static Methods for Neutron Kinetic Calculations 3.3. Coupling Between PCQM Module and COCAGNE Equation (1) can be recast in a pseudo steady-state equation form: [ ] L(t) 1 v t + t I λ i P d,i (t) φ(r, E, Ω, t) + 2 Q(r, E, Ω, t) = 0 (10) i=1 where the operator in brackets is the generalized total operator and Q is the generalized source including all contributions due to the external source, the shape at the previous macro-time-step and the precursor evolution: Q(r, E, Ω, t) = + S(r, E, Ω, t) + I i=1 ψ(r, E, Ω, t t) v t [ λ i e λ χi i t 4π C i(r, 0) + t 2 P d,i(t) φ(r, E, Ω, t t) + ] (11) From equation (10), the virtual total cross-sections to be given to the flux solver COCAGNE are: Σ t (r, E, t) = Σ t (r, E, t) 1 v t + t 2 I λ i P d,i (t) (12) i=1 4. NUMERICAL RESULTS This section is devoted to verify the integrated code and to evaluate the performances of the QS methods (in term of computational cost and precision) with respect to the full-kinetic solver available in COCAGNE A Simple Dynamic Reactor Model In order to verify the code, the simple two-dimensional reactor model described in Ref. [6] is used. It simulates a very simple source-driven system: a square fuel region surrounded by a reflector. One-group sources and cross-sections are given in the Table I. It is important to remark that the values adopted were chosen in a suitable way to reproduce a subcritical system. Thanks to the symmetry of the problem, only a quarter of the system is simulated in the code (see Figure 2). The computational grid adopted in the system is purely regular, with the grid lines equally spaced by 2.5 cm, both for x and y axis. The dark gray region represents the area where the perturbation is inserted at t = 0 +, which is used to initiate a transient. Symmetrical boundary conditions are imposed on all sides of the domain. Input data used in the simulation are: T trans = 5 ms, is the total duration of the transient; δt = 10 3 ms, is the micro-time-step used in the amplitude equation; β = 0 is the delayed neutron fraction, that means there are no delayed neutrons. 7/12

8 Enrico Girardi, Pierre Guérin et al Region Parameter Value Σ t 0.20 REFLECTOR Σ s 0.19 S 1.0 Σ t 0.40 Σ s 0.20 FUEL νσ f 0.20 χ 1.0 S 1.0 Σ t 0.10 SOURCE Σ s 0.09 S e 1.0 S 1.0 Table I. Source and Cross-Section Figure 2. Computational Model and Perturbation Localization (Dark Grey Area) To simulate a transient, a fission cross-section perturbation is introduced. At the beginning of the transient (t = 0 + ), the fission cross section is increase by a factor 1 + δ f in the dark gray area of the geometry described in the Figure 2. As a consequence, the total and production cross-section become, respectively: Σ t = Σ t + 1 ν δ fνσ f ν Σ f = (1 + δ f ) νσ f (13) where ν is the average number of neutrons per fission. We now fix the value δ f = 0.1. We define the reactivity injection as the difference between the reactivity obtained solving the critical problem with the perturbed cross-sections and the reactivity of the initial steady-state configuration. The increase of the fission cross-section induces an increase of reactivity equal to 609 pcm Comparison Between COCAGNE, IQS and PCQM methods In this section, the results of the fission perturbation benchmark are presented, in terms of accuracy and efficiency. All the results are obtained with the SP 1 approximation, which is equivalent to a diffusion calculation with D = 1/(3Σ t ). The kinetic module available in COCAGNE uses a θ-scheme. We used θ = 1, which is equivalent to the well-known Implicit Euler scheme. For the IQS method (see section 2.2.1), 3 iterations are used in the shape normalization loop. Indeed, a larger number of iterations does not increase significantly the precision. We use a very strict fission source criterion (10 10 ), in order to be sure that the spatial flux is well converged. 8/12

9 Comparison of Direct and Quasi-Static Methods for Neutron Kinetic Calculations Analysis of the IQS and PCQM power profiles Figure 3 presents the power evolution for IQS and PCQM methods, in function of the number of macro-time steps used. The Point Kinetic (PK) approximation is also reported in the figure, as a term of comparison. It can be easily observed that the power profile calculated with the PCQM is less affected by the flux update than the IQS method. Moreover, the equilibrium power is closer to the asymptotic value. The lack of the shape update is compensated by the predictor calculation, which gives a better estimation of the kinetic parameters and consequently reflects a more accurate evaluation of the power profile. This is very coherent with the conclusions found in the literature [5]. (a) PK and IQS methods (b) PCQM method Figure 3. Normalized Power Evolution with Respect to Different Macro-Time-Steps N step (the Blue Dashed Line Represent the Equilibrium Power Value) Time convergence of IQS, PCQM and COCAGNE This section is devoted to the analysis of the time convergence of IQS, PCQM algorithms and the kinetic module COCAGNE. The results are compared to a reference calculation. This reference is obtained by a COCAGNE simulation characterized by an accurate convergence constraint (fission source criterion=10 10 ) and a very fine time step (δt = s). The parameters involved in our analysis are: the relative error on the power evolution profile, err(t) = P ref(t) P (t) ; P ref (t) the relative mean squared error on the power, ε 2 = P ref(t) P (t) 2 P ref 2 ; the relative maximum error on the power, ε = P ref(t) P (t) P ref ; 9/12

10 Enrico Girardi, Pierre Guérin et al the number of flux iterations N φ, which takes into account all the flux iterations involved in the computational procedure and which is a good indicator of the computational cost. Figure 4a shows that a good convergence is already reached with 500 time steps. Thus, with δt = ms, the solution is fully converged. Observing Figure 4b, it is possible to compare the evolution of the power calculated by IQS and PCQM algorithms. It can be observed that the PCQM algorithm is closer to the reference value than the IQS. The first part of the transient is the part more affected by the introduction of the fission cross-section perturbation. The weak point of the IQS algorithm is the accuracy of the power evolution in the evaluation of the first macro-time-step. Indeed the PK parameters are evaluated with the initial steady-state flux and the spatial effect of the perturbation is not taken into account. This is clearly shown when observing the percentage error in Fig. 4b. In fact, the first part of the transient is the one affected by the biggest relative percentage error. In the PCQM algorithm the issue of the first evaluation of PK parameters is overcome thanks to the predictor technique and the results show that the method presents a better estimation of the power profile, in particular in the first part of the transient. For example, in the case of two macro-time-steps, it is clear that the PCQM algorithm gives better results than the IQS one. In order to have comparable performances the IQS method needs to involve sixteen or twenty macro-time-steps. (a) COCAGNE (b) PCQM and IQS Figure 4. The Relative Percentage Error on the Power Evolution Profile Performance comparison In Fig. 5 the performance of PCQM, IQS and COCAGNE algorithms are compared: the errors with respect to the number of flux iterations are presented. The aim is to investigate which simulation method offers the smaller computational effort with approximately the same precision. We now use a fission source criterion of 10 4, which is more compatible with an industrial application. Also, the number of iteration for the shape normalization in the IQS is set to 1. One can observe that IQS is the algorithm which presents the highest values of errors. The kinetic 10/12

11 Comparison of Direct and Quasi-Static Methods for Neutron Kinetic Calculations COCAGNE solver allows to have a more accurate evaluation of the power with less iterations compared to the IQS method. The PCQM simulations present the most advantageous performances when looking at the relative mean squared error. It is also in general the best choice from the maximum error criterion, even if in some cases the kinetic COCAGNE solver represents the more accurate solution with the less number of flux iterations. These results suggest the idea of the introduction of an adapted time grid, which allows to improve the accuracy in the evaluation of the power in the part of the transient more affected by the perturbation, and also to avoid unnecessary shape calculations when the spatial form of the neutron flux has already reached the equilibrium. Mean Squared Error Maximum Error Figure 5. Errors with Respect to the Number of Flux Iterations 5. CONCLUSIONS The analysis of IQS and PCQM algorithms on a particular benchmark shows that quasi-static methods coupled with the COCAGNE core solver give quite accurate results in the estimation of the power evolution profile. As expected, the accuracy in the estimation of the equilibrium power increases when the number of macro-time-steps is bigger, meaning that the shape is updated more frequently. The IQS code has to involve more shape recomputations than PCQM for the same accuracy. The main computational effort of the IQS code is related to the normalization condition constraint, which is not need by PCQM algorithm thanks to the introduction of a predictor-corrector technique. Compared with the kinetic module of the COCAGNE solver which solves the full transport problem at each time step, the IQS performances are not very satisfactory because the shape updates are not efficient. For the transient considered in this paper (step perturbation introduced at t = 0), the main difficulty is a good evaluation of the power in the first macro-time-step. Even if the size of the macro-time-step is small, the first shape calculation in the IQS method does not take into account the spatial features of the perturbation phenomenon. On the other hand, the PCQM method uses a predictor-corrector technique that allows to obtain a better ratio accuracy/number of iterations than the COCAGNE kinetic module. 11/12

12 Enrico Girardi, Pierre Guérin et al The quasi-static approach was tested with a simple two dimensional reactor model with one energy group. The encouraging results of the benchmarks provide several perspectives. It will be interesting to perform further benchmark experiments using more complex models in multigroup energy environment. To improve the computational performances the introduction of an adaptive time grid could be very appropriate. The adaptive technique would increase the accuracy of the evaluation of the power where the transient is more influenced by the spatial and spectral distortions of the flux. Therefore it would avoid shape calculations when the effect of the transient is lessened. REFERENCES [1] A.F. Henry, The application of reactor kinetics to the analysis of experiments, Nuclear Science and Engineering, 3, pp (1958). [2] K.O. Ott, D.A. Meneley, Accuracy of the quasi-static treatment of spatial reactor kinetics, Nuclear Science and Engineering, 36, pp (1969). [3] S. Dulla, E.H. Mund, P. Ravetto, Accuracy of the predictor-corrector quasi-static method for space-time reactor dynamics, Proceeding of Advances in Nuclear Analysis and Simulation, PHYSOR 06, Vancouver, Canada, (2006). [4] S. Dulla, E.H. Mund, P. Ravetto, The Quasi-Static Method Revisited, Progress in Nuclear Energy, 50, pp (2008). [5] P. Picca, S. Dulla, E.H. Mund, P. Ravetto, G. Marleau, A quasi-static transport module using the DRAGON code, Proceeding of PHYSOR 08, Interlaken, Switzerland, (2008). [6] F. Alcaro, Development of approximate dynamic models for neutron transport kinetic calculations, Master thesis, Politecnico di Torino, Italy, (2008). [7] F. Alcaro, S. Dulla, G. Marleau, E.H. Mund, P. Ravetto, Development of dynamic models for neutron transport calculations, Il Nuovo Cimento C, 33, pp (2010). [8] S. Dulla, E. Girardi, P. Guerin, M. Nervo, P. Ravetto, Neutron Kinetic Calculations using a Quasi-Static Method with the COCAGNE Code, Transactions of the American Nuclear Society, 105, pp (2011). [9] L. Plagne, A. Ponçot, Generic Programming for Deterministic Neutron Transport Codes, Proceeding of Mathematics and Computation, Supercomputing, Reactor Physics and Nuclear and Biological Applications, M&C 2005, Avignon, France, September [10] G. Marleau, A. Hébert, R. Roy, A user s guide for DRAGON 3.05, Report IGE-174 Rev. 6, Institut de Génie Nucléaire, Ecole Polytechnique de Montréal (2006). 12/12