Abdessalem Abidi
email: abdessalemabidi9@gmail.com

Samar Tawbi

Christophe Guyeux

Belgacem Bouallègue

Mohsen Machhout

Summary of Topological Study of Chaotic CBC Mode of Operation

In cryptography, block ciphers are the most fundamental elements in many symmetric-key encryption systems. The Cipher Block Chaining, denoted CBC, presents one of the most famous mode of operation that uses a block cipher to provide confidentiality or authenticity. In this research work, we intend to summarize our results that have been detailed in our previous series of articles. The goal of this series has been to obtain a complete topological study of the CBC block cipher mode of operation after proving his chaotic behavior according to the reputed definition of Devaney.

I. INTRODUCTION

Cryptography is the mathematics of secret codes. Block ciphers are the most fundamental elements in many symmetric-key encryption systems, it means that both encryption and decryption utilize the same key. Block cipher algorithm divides the message into separate blocks of fixed size, and encrypts/decrypts each block individually. The encrypted block and the original one have the same size. Block ciphers provide confidentiality that is used in a large variety of applications, such as: protection of the secrecy of login passwords, email messages, video transmissions and many other applications [START_REF] Rezaeipour | The block cipher algorithm-properties, encryption efficiency analysis and security evaluation[END_REF].

At the programming level, it is not sufficient to put anyhow a block cipher algorithm. In fact, this latter can be used in various ways depending on their specific needs. These ways are called the block cipher modes of operation. There are several modes and each one of them possesses own characteristics in addition to its specific security properties. In this paper, the mode on which we will focus is the cipher block chaining one, and we will study it according to chaos.

The chaos theory that we consider in our various research works is the Devaney's topological one [START_REF] Devaney | An Introduction to Chaotic Dynamical Systems[END_REF]. In addition to being recognized as one of the best mathematical definition of chaos, this theory offers a framework with qualitative and quantitative tools to evaluate the notion of unpredictability [START_REF] Jacques M Bahi | Efficient and cryptographically secure generation of chaotic pseudorandom numbers on gpu[END_REF]. As an application of our fundamental results, we are interested in the area of information safety and security, and more precisely we focus on symmetric-key encryption systems. In this research work, which is a summary of our previous papers, we intend to preview the different results, which have been detailed respectively in [START_REF] Abidi | Proving chaotic behavior of cbc mode of operation[END_REF], [START_REF] Abidi | Quantitative Evaluation of Chaotic CBC Mode of Operation[END_REF], and [START_REF] Abidi | The dynamics of the CBC Mode of Operation[END_REF]. These results allowed us to deepen a complete topological study of the CBC mode of operation after proving its chaotic behavior according to Devaney.

The remainder of this research work is organized as follows. In Section II, we will recall some basic definitions concerning chaos and CBC mode of operation. Section III is devoted to sum up the various demonstrations that allowed us to proof the chaotic behaviour of the CBC mode. Moreover, new cases will be studied here. In Section IV, quantitative topological properties for chaotic CBC mode of operation will be summarized, while Section V resumes the main contribution that has been detailed in our last article [START_REF] Abidi | The dynamics of the CBC Mode of Operation[END_REF]. This research work ends by a conclusion section in which our contribution is recalled and some intended future work are proposed.

II. BASIC RECALLS

A. Devaney's Chaotic Dynamical Systems

In the remainder of this paper, m n denotes the n th block message of a sequence S while m j stands for the j -th bit of integer of the block message m ∈ 0, 2 N -1 , expressed in the binary numeral system and x i stands for the i th component of a vector x.

X N is the set of all sequences whose elements belong to X .

f •k = f • ... • f
is for the k th composition of a function f . N is the set of natural (non-negative) numbers, while N * stands for the positive integers 1, 2, 3, . . . Finally, the following notation is used:

1; N = {1, 2, . . . , N }.
Consider a topological space (X , τ ), where τ represents a family of subsets of X , and a continuous function f : X → X on (X , τ ) [START_REF] Abidi | The dynamics of the CBC Mode of Operation[END_REF].

Definition 1 The function f is topologically transitive if, for any pair of nonempty open sets U, V ⊂ X , there exists an integer k > 0 such that f

•k (U) ∩ V = ∅. Definition 2 An element x is a periodic point for f of period n ∈ N, n > 1, if f •n (x) = x and f •k (x) = x, 1 ≤ k ≤ n.
Definition 3 f is regular on (X , τ ) if the set of periodic points for f is dense in X : for any point x in X , any neighborhood of x contains at least one periodic point.

Definition 4

The function f has sensitive dependence on initial conditions on the metric space (X , d) if there exists δ > 0 such that, for any x ∈ X and any neighborhood V of x, there exist y ∈ V and n > 0 such that the distance d between the results of their n th composition, f •n (x) and f •n (y), is greater than δ:

d (f •n (x), f •n (y)) > δ.
δ is called the constant of sensitivity of f . Definition 5 (Devaney's formulation of chaos [START_REF] Devaney | An Introduction to Chaotic Dynamical Systems[END_REF]) The function f is chaotic on a metric space (X , d) if f is regular, topologically transitive, and has sensitive dependence on initial conditions. Banks et al. have proven in [START_REF] Banks | On devaney's definition of chaos[END_REF] that when f is regular and transitive on a metric space (X , d), then f has the property of sensitive dependence on initial conditions. This is why chaos can be formulated too in a topological space (X , τ ): in that situation, chaos is obtained when f is regular and topologically transitive. Note that the transitivity property is often obtained as a consequence of the strong transitivity one, which is defined below. Definition 6 f is strongly transitive on (X , d) if, for all point x, y ∈ X and for all neighborhood V of x, it exists n ∈ N and x ∈ V such that f n (x ) = y.

Finally, a function f has a constant of expansivity equal to ε if an arbitrarily small error on any initial condition is always magnified until ε [START_REF] Guyeux | A topological study of chaotic iterations. application to hash functions[END_REF]. Mathematically speaking,

Definition 7 The function f is said to have the prop- erty of expansivity if ∃ε > 0, ∀x = y, ∃n ∈ N, d(f n (x), f n (y)) ε.
Then, ε is the constant of expansivity of f. We also say that f is ε-expansive.

Definition 8 A discrete dynamical system is said topologically mixing if and only if, for any couple of disjoint open set U, V = ∅, there exists an integer n 0 ∈ N such that, for all n > n 0 , f

•n (U) ∩ V = ∅.

B. CBC mode characteristics

In cryptography, Cipher Block Chaining is a block cipher mode that provides confidentiality but not message integrity. Similar to some other modes, the input to the encryption processes of the CBC mode includes not only a plaintext, but also a data block called the initialization vector, which is denoted IV.

In particular, the CBC mode offers a solution to most of the problems presented by the Electronic Code Book (ECB). Indeed, thanks to this mode, the encryption will depend not only on the plaintext, but also on all preceding blocks. More precisely, each block of plaintext is XORed immediately with the previous cipher text block before being encrypted (i.e., the binary operator XOR is applied between two stated blocks).

For the first block, the initialization vector acts as the previous cipher text block. For decryption, we proceed in the same way, but this time, we start from the encrypted text to obtain the original one using now the decryption algorithm instead of the encryption function, see Figure 1.

The IV needs not to be secret; however, for any particular execution of the encryption process, it must be unpredictable. In the next section, we will summarize the results of our first article [START_REF] Abidi | Proving chaotic behavior of cbc mode of operation[END_REF].

III. PROVING CHAOTIC BEHAVIOR OF CBC MODE

In this section, we prove that CBC mode of operation behaves as Devaney's topological chaos if the iteration function used is the vectorial Boolean negation. This function has been chosen here, but the process remains general and other iterate functions g can be used. The sole condition is to prove that G g satisfies the Devaney's chaos property. To do this, we have began by modeling CBC mode as a dynamical system.

A. Modeling CBC as dynamical system

Our modeling follows a same canvas than what has been done for hash functions [START_REF] Bahi | Hash functions using chaotic iterations[END_REF], [START_REF] Guyeux | A topological study of chaotic iterations. application to hash functions[END_REF] or pseudorandom number generation [START_REF] Bahi | Evaluating quality of chaotic pseudo-random generators. application to information hiding[END_REF]. Let us consider the CBC • S N = 0, 2 N -1 N , the set of infinite sequences of natural integers bounded by 2 N -1, or the set of infinite N-bits block messages, in such a way that X is constituted by couples of internal states of the mode of operation together with sequences of block messages. Let us consider the initial function:

i : S N -→ 0, 2 N -1 (m i ) i∈N -→ m 0
that returns the first block of a (infinite) message, and the shift function:

σ : S N -→ S N (m 0 , m 1 , m 2 , ...) -→ (m 1 , m 2 , m 3 , ...)
which removes the first block of a message. Let m j be the j-th bit of integer, or block message, m ∈ 0, 2 N -1 , expressed in the binary numeral system, and when counting from the left. We define:

F f : B N × 0, 2 N -1 -→ B N (x, m) -→ (x j m j + f (x) j m j ) j=1..N
This function returns the inputted binary vector x, whose m j -th components x mj have been replaced by f (x) mj , for all j = 1..N such that m j 1. In case where f is the vectorial negation, this function will correspond to one XOR between the clair text and the previous encrypted state. So the CBC mode of operation can be rewritten as the following dynamical system:

X 0 = (IV, m) X n+1 = (E k • F f0 (i(X n 1 ), X n 2 ) , σ(X n 1 )) (1)
For any given g : 0, 2 N -1 × B N -→ B N , we denote G g (X) = (g(i(X 1 ), X 2 ); σ(X 1 )) (when g = E k • F f0 , we obtain one cypher block of the CBC, as depicted in Figure 1). So the recurrent relation of Eq.( 1) can be rewritten in a condensed way, as follows.

X n+1 = G E k •F f 0 (X n ) . (2)
With such a rewriting, one iterate of the discrete dynamical system above corresponds exactly to one cypher block in the CBC mode of operation. Note that the second component of this system is a subshift of finite type, which is related to the symbolic dynamical systems known for their relation with chaos [START_REF] Lind | An introduction to symbolic dynamics and coding[END_REF]. We now define a distance on X as follows: d((x, m); (x, m)) = d e (x, x) + d m (m, m), where:

         d e (x, x) = N k=1 δ(x k , xk ) d m (m, m) = 9 N ∞ k=1 N i=1 |m i -mi | 10 k .
This distance has been introduced to satisfy the following requirements:

• The integral part between two points X, Y of the phase space X corresponds to the number of binary components that are different between the two internal states X 1 and Y 1 .

• The k-th digit in the decimal part of the distance between X and Y is equal to 0 if and only if the k-th blocks of messages X 2 and Y 2 are equal. This desire is at the origin of the normalization factor 9 N .

B. Proof of chaos

As mentioned in Definition 5, a function f is chaotic on (X , τ ) if f is regular and topologically transitive. We have began in [START_REF] Abidi | Proving chaotic behavior of cbc mode of operation[END_REF] by stating some propositions that are primarily required in order to proof the chaotic behavior of the CBC mode of operation.

Proposition 1 Let g = E κ • F f0
, where E κ is a given keyed block cipher and f 0 : B N -→ B N , (x 1 , ..., x N ) -→ (x 1 , ..., x N ) is the Boolean vectorial negation. We consider the directed graph G g , where:

• vertices are all the N-bit words.

• there is an edge m ∈ 0, 2 N -1 from x to x if and only if g(m, x) = x.

If G g is strongly connected, then G g is strongly transitive.

We have then proven that, Proposition 2 If G g is strongly connected, then G g is regular.

According to Propositions 1 and 2, we can conclude that, depending on g, if the directed graph G g is strongly connected, then the CBC mode of operation is chaotic according to Devaney, as established in our previous research work [START_REF] Abidi | Proving chaotic behavior of cbc mode of operation[END_REF].

In this article and for illustration purpose, we have also given some examples of keyed block ciphers, which can be used by the CBC mode, and which can lead to a chaotic behavior for this mode when they have a strongly connected directed graph. These examples are taken from so-called transposition cipher methods. Among these examples, we considered the Caesar shift one.

C. Caesar shift case

The Caesar shift case is considered as one of the simplest and most widely known substitution cipher. In this method, each symbol in the plaintext is replaced by a symbol some fixed number of positions down the given alphabet. Translated in the N binary digits set of integers, this cypher can be written as follows:

E k (x) : 0, 2 N -1 -→ 0, 2 N -1 x -→ x + k mod 2 N D k (x) : 0, 2 N -1 -→ 0, 2 N -1 x -→ x -k mod 2 N
where k is the shift value acting as secret key. We will now show through examples that the CBC mode of operation embedding the Caesar shift can behave either chaotically or not, depending on k and N. The two following tables I and II contain the g(x, m) values for a shift of 1 and 2 respectively, in Caesar cipher over 3-bit blocks.

IV. QUANTITATIVES MEASURES

In this section, we are interested to recall our previous results related to quantitative measures. They have been detailed in [START_REF] Abidi | Quantitative Evaluation of Chaotic CBC Mode of Operation[END_REF], in which both expansivity and sensibility of symmetric cyphers are regarded, in the case of CBC mode of operation. These quantitative topology metrics, taken from the mathematical theory of chaos, allow to measure in which extent a slight error on the initial condition is magnified during iterations.

In this research work we stated that, in addition to being chaotic as defined in the Devaney's formulation, the CBC mode of operation is indeed largely sensible to initial errors or modifications on either the IV or the message to encrypt. The second important tool that reinforces the chaotic behavior of the CBC mode of operation is the expansivity. This property has been evaluated too, but it is not satisfied, as it has been established thanks to a counter example. For more details and to do this, we have began by announcing and then proving these two following propositions: Proposition 3 The CBC mode of operation is sensible to the initial condition, and its constant of sensibility is larger than the length N of the block size.

Proof: Let X = (x; (m 0 , m 1 , ...)) ∈ X and δ > 0. We are looking for X = (x ; (m 0 , m 1 , ...) and n ∈ N such that d(X, X ) < δ and d(G n g (X), G n g (X )) > N . Let us define k 0 = -log 10 (δ) + 1, in such a way that all X of the form:

(X 1 , (m 0 , m 1 , ..., m k0 , m k0+1 , m k0+2 , ...)) are such that d(X, X ) < δ. In other words, all messages m whose k 0 first blocks are equal to (m 0 , m 1 , ..., m k0 ) are δ-close to X. where D k is the keyed decryption function associated to E k , and z is the negation of z. We thus define X as follow:

•

X 1 = x, • ∀k k 0 , m k = m k , • m k0+1 = m , • ∀k k 0 + 2, m k = m k , so d(G k0+1 g (X), G k0+1 g (X )) = d (G g (y; (m k0+1 , m k0+2 , ...)) , G g (y; (m , m k0+1 , m k0+2 , ...))) = d ((z; (m k0+2 , m k0+3 , ...)) , (E k (y ⊕ m ); (m k0+1 , m k0+2 , ...))) = d ((z; (m k0+2 , m k0+3 , ...)) , (E k (y ⊕ (y ⊕ D k (z))); (m k0+1 , m k0+2 , ...))) = d ((z; (m k0+2 , m k0+3 , ...)) , (E k ((y ⊕ y) ⊕ D k (z)); (m k0+1 , m k0+2 , ...))) = d ((z; (m k0+2 , m k0+3 , ...)) , (E k (0 ⊕ D k (z)); (m k0+1 , m k0+2 , ...))) = d ((z; (m k0+2 , m k0+3 , ...)) , (E k (D k (z)); (m k0+1 , m k0+2 , ...))) = d ((z; (m k0+2 , m k0+3 , ...)) , (z; (m k0+1 , m k0+2 , ...))) = d e (z, z) + d m ((m k0+2 , m k0+3 , ...), (m k0+1 , m k0+2 , ...)) = N + 9 N ∞ k=k0+2 m k -m k 10 k = N + 9 N ∞ k=k0+2 N 10 k = N + 9 ∞ k=k0+2 1 10 k = N + 1 10 k0+1 > N,
which concludes the proof of the sensibility of G g .

Proposition 4

The CBC mode of operation is not expansive.

Proof:

Consider for instance two initial vectors x = (1, 0, . . . , 0) and x = (0, 1, 0, . . . , 0), associated to the messages m = ((0, 1, 0, . . . , 0), (0, . . . , 0), (0, . . . , 0), . . .) and m = ((1, 0, . . . , 0), (0, . . . , 0), (0, . . . , 0), . . .): all blocks of messages are null in both m and m , except the first block. Let X = (x, m) and X = (x , m ).

Obviously, x = x , while x ⊕ m 0 = x ⊕ m 0 . This latter implies that X 0 1 = X 0 1 , and by a recursive process, we can conclude that ∀i ∈ N, X i 1 = X i 1 . So the distance between points X = (x, m) and X = (x , m ) is strictly positive, while for all n > 0, d G n g (X), G n g (X ) = 0, which concludes the proof of the non expansive character of the CBC mode of operation by the mean of the exhibition of a counter example.

To sum up, proving these two propositions claimed previously allowed us to conclude that CBC mode of In conclusion, all of these properties lead to a complete unpredictable behavior for the CBC mode of operation.

VI. CONCLUSION

In this paper, our goal was to summarize our numerous results that have been detailed in our previous series of articles. Hence, the interest of our work is not to provide a collection of secure and complex CBC, but to initiate a complementary approach for studying such modes of operation. Our intention is to show how to model such modes, and that it is possible to study the complexity of their dynamics. Up-to-date block ciphers and modes of operation, together with topological analyses using most recent developments in this field, need to be investigated, while the interest of each topological property of complexity must be related to desired objectives for each mode of operation.

Fig. 1 .

 1 Fig. 1. CBC mode of operation

Fig. 3 .

 3 Fig. 3. k 0 + 1-th iterate of Gg Let y = G k0 g (X) 1 and z = G k0+1 g (X) 1 as defined in Figure 3. We consider the block message m defined by: m = y ⊕ D k (z)

x m F f 0 (x, m) g(m, x) = E k • F f 0 (x, m) 0 (0,0,0) 0 (0,0,0) 0 (0,0,0) 1 0 (0,0,0) 1 (0,0,1) 1 (0,0,1) 2 0 (0,0,0) 2 (0,1,0) 2 (0,1,0) 3 0 (0,0,0) 3 (0,1,1) 3 (0,1,1) 4 0 (0,0,0) 4 (1,0,0) 4 (1,0,0) 5 0 (0,0,0) 5 (1,0,1) 5 (1,0,1) 6 0 (0,0,0) 6 (1,1,0) 6 (1,1,0) 7 0 (0,0,0) 7 (1,1,1) 7 (1,1,1) 0 1 (0,0,1) 0 (0,0,0) 1 (0,0,1) 2 1 (0,0,1) 1 (0,0,1) 0 (0,0,0) 1 1 (0,0,1) 2 (0,1,0) 3 (0,1,1) 4 1 (0,0,1) 3 (0,1,1) 2 (0,1,0) 3 1 (0,0,1)

4 (1,0,0) 5 (1,0,1) 6 1 (0,0,1)

5 (1,0,1) 4 (1,0,0) 5 1 (0,0,1)

6 (1,1,0) 7 (1,1,1) 0 1 (0,0,1)

7 (1,1,1) 6 (1,1,0) 7 2 (0,1,0) 0 (0,0,0) 2 (0,1,0) 3 2 (0,1,0) 1 (0,0,1) 3 (0,1,1) 4 2 (0,1,0) 2 (0,1,0) 0 (0,0,0) 1 2 (0,1,0) 3 (0,1,1) 1 (0,0,1) 2 2 (0,1,0) 4 (1,0,0)

,0,0) 5 4 (1,0,0) 0 (0,0,0) 4 (1,0,0) 5 4 (1,0,0) 1 (0,0,

(1,0,0) 3 (0,1,1) 7 (1,0,0) 0 4 (1,0,0) 4 (1,0,0) 0 (0,0,0) 1 4 (1,0,0) 5 (1,0,1) 1 (0,0,1) 2 4 (1,0,0) 6 (1,1,0) 2 (0,1,0) 3 4 (1,0,0) 7 (1,1,1) 3 (0,1,1) 4 5 (1,0,1) 0 (0,0,0) 5 (1,1,0) 6 5 (1,0,1) 1 (0,0,1) 4 (1,0,0) 5 5 (1,0,1) 2 (0,1,0) 7 (1,0,0) 0 5 (1,0,1) 3 (0,1,1) 6 (1,1,0) 7 5 (1,0,1) 4 (1,0,0) 1 (0,0,1) 2 5 (1,0,1) 5 (1,0,1) 0 (0,0,0) 1 5 (1,0,1) 6 (1,1,0) 3 (0,1,1) 4 5 (1,0,1)

0 (0,0,0) 0 (0,0,0) 2 0 (0,0,0) 1 (0,0,1) 1 (0,0,1) 3 0 (0,0,0) 2 (0,1,0) 2 (0,1,0) 4 0 (0,0,0) 3 (0,1,1) 3 (0,1,1) 5 0 (0,0,0) 4 (1,0,0) 4 (1,0,0) 6 0 (0,0,0) 5 (1,0,1) 5 (1,0,1) 7 0 (0,0,0) 6 (1,1,0) 6 (1,1,0) 0 0 (0,0,0) 7 (1,1,1) 7 (1,1,1) 1 1 (0,0,1) 0 (0,0,0) 1 (0,0,1) 3 1 (0,0,1) 1 (0,0,1) 0 (0,0,0) 2 1 (0,0,1)

4 (1,0,0) 0 (0,0,0) 2 4 (1,0,0) 5 (1,0,1) 1 (0,0,1) 3 4 (1,0,0) 6 (1,1,0) 2 (0,1,0) 4 4 (1,0,0) 7 (1,1,1) 3 (0,1,1) 5 5 (1,0,1) 0 (0,0,0) 5 (1,1,0) 7 5 (1,0,1) 1 (0,0,1) 4 (1,0,0) 6 5 (1,0,1) 2 (0,1,0) 7 (1,0,0) 1 5 (1,0,1) 3 (0,1,1) 6 (1,1,0) 0 5 (1,0,1)

4 (1,0,0) 1 (0,0,1) 3 5 (1,0,1) 5 (1,0,1) 0 (0,0,0) 2 5 (1,0,1) 6 (1,1,0) 3 (0,1,1) 5 5 (1,0,1)

, for its part, presents the graph of iteration of the Caesar based CBC mode of operation, with the same kind of shifts, operating on blocks of size 3. We can verify that, at each time, the cipher block chaining behaves chaotically. In that situation, we can guarantee that any error on the IV (starting state) or on the message to encrypt (edges to browse) may potentially lead to a completely different list of visited states, that is, of a completely different ciphertext.

In this section, it has been proven that some well chosen block ciphers can lead to a chaotical behavior for the CBC mode of operation. In the following section, we will recall some quantitative measures of chaos that have already been proven in one of our previous research work. operation is sensible to the initial conditions. But, on the other side, it is not expansive.

Let us now investigate new original aspects of chaos of the CBC mode of operation.

V. TOPOLOGICAL MIXING AND TOPOLOGICAL ENTROPY

A. Topological mixing

As mentioned in Definition 8, a discrete dynamical system is said topologically mixing if and only if, for any couple of disjoint open set U, V = ∅, there exists an integer n 0 ∈ N such that, for all n > n 0 , f •n (U) ∩ V = ∅. In [START_REF] Abidi | The dynamics of the CBC Mode of Operation[END_REF], we have deepened the topological study for the CBC mode of operation. Indeed, we have regarded if this tool possesses the property of topological mixing. In order to proof this property, we have began by stating the following proposition:

which is an immediate consequence of the lemma below.

Lemma 1 For any open ball

Proving this lemma led us to conclude the proposition claimed previously. Hence, this dynamical system owns the property of topological mixing.

In addition to this property, other quantitative evaluations have been performed, and the level of topological entropy has been evaluated too.

B. Topological entropy

Another important tool to measure the chaotic behavior of a dynamical system is the topological entropy, which is defined only for compact topological spaces. Therefore, before studying the entropy of CBC mode of operation, we must then check that (X , d) is compact.

1) Compactness study: In this section, we will prove that (X , d) is a compact topological space, in order to study its topological entropy later. Firstly, as (X , d) is a metric space, it is separated. It is however possible to give a direct proof of this result: Proposition 6 (X , d) is a separated space.

Proof: Let (x, w) = (x, ŵ) two points of X . 1) If x = x, then the intersection between the two balls B (x, w), 1 2 and B (x, ŵ), 1 2 is empty. 2) Else, it exists k ∈ N such that w k = ŵk , then the balls B (x, w), 10 -(k+1) and B (x, ŵ), 10 -(k+1) can be chosen.

Let us now prove the compactness of the metric space (X , d) by using the sequential characterization of compactness.

Proposition 7 (X , d) is a compact space.

2) Topological entropy: Let (X, d) be a compact metric space and f : X → X be a continuous map. For each natural number n, a new metric d n is defined on X by

Given any ε > 0 and n 1, two points of X are ε-close with respect to this new metric if their first n iterates are ε-close (according to d).

This metric allows one to distinguish in a neighborhood of an orbit the points that move away from each other during the iteration from the points that travel together. A subset E of X is said to be (n, ε)-separated if each pair of distinct points of E is at least ε apart in the metric d n .

Definition 9 Let H(n, ε) be the maximum cardinality of a (n, ε)-separated set, the topological entropy of the map f is defined by (see e.g., [START_REF] Adler | Topological entropy[END_REF] or [START_REF] Bowen | Entropy for group endomorphisms and homogeneous spaces[END_REF]) So for all n ∈ N, the maximal number H(n, 1) of (n, 1)-separated points is greater than or equal to e n 2 , and then h top (G g , 1) = lim 1 n log (H(n, 1)) > lim 1 n log e n 2 = lim (n) = +∞.

But h top (G g , ε) is an increasing function when ε is decreasing, then h top (G g ) = lim ε→0 h top (G g , ε)

> h top (G g , 1) = +∞, which concludes the evaluation of the topological entropy of G g .


M Bertrand

P Pochon

N Cedat

O Baudouin

R Sardeing

Modelling of



  Currently noble gases separated by cryogenic distillation  expensive process with safety constraints due to the cryogenic temperatures used  Adsorptive separation (temperature/pressure swing adsorption)  an energy, safety and cost effective alternative method  Different selective materials used from inorganic adsorbents based on physical adsorption to new metal-organic frameworks (MOFs) based on size and chemistry Experimental set-up: fixed-bed adsorption column OBJECTIVES  Development of a Kr/Xe separation process by selective adsorption on a chabazite zeolite in a fixed bed column  Experimental study of the Kr and Xe adsorption dynamics at different temperatures and under different operating conditions  Modelling of the Kr/Xe adsorption separation from a gas mixture Zeolite AW500 suitable for Xe/Kr separation process from a gas mixture flow as their breakthrough times are significantly different Development of an adsorption modelling using software ProSim DAC The model developed can be applied to design PSA/TSA cycles to separate Kr and Xe The authors greatly acknowledge orano for sponsoring this study Modelling using ProSim Murielle Bertrand 1 , Patrick Pochon 1 , Nadège Cédat 1 , Olivier Baudouin 2 , Rodolphe Sardeing 2 1-CEA, Nuclear Energy Division, Research Department on Mining and Fuel Recycling Processes BP 17171 F-30207 Bagnols-sur-Cèze, France 2-ProSim SA Immeuble Stratège A, 51, rue Ampère F-31670 Labège, France Choice of the adsorbent  A dozen of zeolites experimentally tested   selection of a synthetic chabazite AW500 of general formulae M x [(AlO 2 ) x (SiO 2 ) y ],zH 2 Adsorbent assumed to be thermodynamically inert  Radial concentration gradients negligible  Gas mixtures supposed to behave as an ideal gas Adsorbent characteristics  no impact on the adsorption and separation performances  Process parameters: temperature, feed flow or initial concentration  high impacts Example of breakthrough curves of Kr and Xe in N


Bastien Faure
email: bastien.faure@cea.fr

Pascal Archier

Laurent Buiron

A 2D/1D METHOD FOR CONSISTENT BURNUP PARAMETRIZATION OF CROSS SECTIONS IN SFR FUEL DEPLETION CALCULATIONS

Keywords: Fuel depletion, Neutron Transport, 2D/1D Method

The accuracy of neutronic calculations in reactor physics is determined by the quality of the averaged cross sections used to solve the Boltzmann transport equation. As the reactor burns its fuel, a change occurs in the neutronic properties of the media so the averaged cross sections become timedependent. Several transport calculations are therefore required at the cross section generation stage to perform a burnup (or fluence) parametrization of the data. This paper proposes to use the now well known 2D/1D method to perform this parametrization. The idea of such a strategy is to avoid computationally expensive 3D simulations while overcoming the drawbacks of standard 2D models. The algorithm is applied to produce time-dependent effective cross sections for a SFR fuel assembly with CFV design. The fuel depletion analysis is then conducted in a "core environment" and results are compared to independent Monte Carlo simulations. Good performances are found both for the evolution of the spatial distribution of isotopic concentrations and the assembly reactivity loss.

INTRODUCTION

The neutronic calculation of a nuclear reactor core consists in solving a Boltzmann transport equation whose solution is the neutron flux. Its knowledge gives access to several quantities of interest such as the power map, the multiplication factor or the evolution of the fuel inventory.

Unfortunately, time dependent three-dimensional fine calculations are still out of range of even the most powerful super-computers. When the reactor is in normal operating conditions though, the change in the flux shape is slow compared to the evolution of the fuel composition. Is is then licit to use an adiabatic approximation which enables transforming the time dependent Boltzmann transport equation into a succession of steady state problems [START_REF] Hébert | Applied Reactor Physics[END_REF]. Between two time steps, isotopic compositions are updated by solving a set of Bateman equations [START_REF] Bateman | Solution of a System of Differential Equations Occurring in the Theory of Radioactive Transformations[END_REF].

Even so, solutions of steady state problems often require long hours of calculation over large-scale parallel computers. For practical applications, full core calculations are therefore performed over both spatial and energy coarse meshes. Thus, they rely on homogenized multigroup effective cross sections which are flux-weighted average of the input fine group nuclear data.

As the reactor operates, the fuel depletion induces a change in the neutronic properties of the media and consequently in the flux spectrum and spatial distribution. As a result, effective microscopic cross sections become time-dependent. The burnup parametrization of cross sections provides an efficient way to account for that change. It also allows to manage the core reloading strategy (i.e. the fact that each fuel assembly has his own local burning rate).

Traditionally, the burnup parametrization of cross sections relies on two-dimensional assembly calculations for which the flux level is normalized to the averaged reactor power. This strategy however fails when the flux presents sharp axial gradients. It is the case in modern Sodium Fast Reactor (SFR) designs that tend to present axial heterogeneities such as fertile blankets and even sodium plenums. This paper proposes to take into account the three-dimensional geometry of a fuel assembly to perform the burnup parametrization of cross sections. In order to avoid expensive 3D flux calculations, a 2D/1D approximation -inspired of Cho's fusion method [START_REF] Cho | A Fusion Technique of 2D/1D Methods for Three-Dimensional Whole-Core Transport Calculations[END_REF] -is carried out to solve the neutron transport equation. The 1D solution is used both to supply axial leakage for 2D calculations and to normalize the flux to the true assembly power. Time-dependent consistent effective cross sections are therefore produced for the entire fuel assembly including fertile blankets, structures and shields.

The organization of this paper is as follows. Section 2 presents how the 2D/1D method can be used to produce time-dependent effective cross sections for SFR applications while Section 3 is devoted to numerical results obtained with the APOLLO3 R code [START_REF] Schneider | APOLLO3: CEA/DEN Deterministic Multi-Purpose Code for Reactor Physics Analysis[END_REF]. The benchmark chosen for application is a reflected 3D fuel assembly representative of the CFV (coeur a faible vidange or low sodium void effect core) which is under study at CEA in France as part of the ASTRID project [START_REF] Beck | Conceptual Design of ASTRID Fuel Sub-assemblies[END_REF].

BURNUP PARAMETRIZATION OF CROSS SECTIONS WITH THE 2D/1D METHOD

The Transport Equation

In the scope of fuel depletion calculations, the time-dependent Boltzmann neutron transport equation is often replaced by a system involving Bateman equations and the steady state form of the transport equation. While the former rules the evolution of materials under irradiation, the latter handles the level and spatial distribution of the nuclear reaction rates. This system reads:

   (Ω • ∇ + Σ) ψ = q (1a) dN dt = A(λ, τ )N (1b)
where Eq. (1a) is the steady state Boltzmann transport equation and Eq. (1b) is the matrix form of the Bateman equations. The macroscopic total cross section Σ = k N k (t)σ is a sum over all the isotopes of their concentration N k times their microscopic total cross section σ. q is the source term that includes contributions from fission and scattering and ψ is the neutron flux. N is the vector of isotopic concentrations and A the evolution matrix that depends on decay constants λ and reaction rates τ .

The solution of the above system is obtained by successively solving Eq. (1a) to get fluxes and reaction rates and Eq. (1b) to update the isotopic concentrations.

The 2D/1D Algorithm for Cross Section Generation

This section focuses on the resolution of Eq. (1a). It presents how the 2D/1D method can be used to avoid full 3D calculations when producing effective cross sections. The ideas of section 2.2.1 have already been exposed in our previous work [START_REF] Faure | A 2D/1D Algorithm for Effective Cross-Section Generation in Fast Reactor Neutronic Transport Calculations[END_REF] but they are recalled here for the sake of clarity.

General Equations

In the scope of cross section generation, the geometrical domain over which Eqs. (1a) and (1b) need to be solved is typically a fuel assembly with reflective radial boundary conditions. If axial heterogeneities are present, the entire (3D) geometry should be considered.

But if cross sections are axially invariant in a domain [z i -; z i +] and if a flat shape is assumed for axial leakage, which is not so far from reality in SFR, it is shown in reference [START_REF] Faure | A 2D/1D Algorithm for Effective Cross-Section Generation in Fast Reactor Neutronic Transport Calculations[END_REF] that the successive integration of Eqs. (1a) in the axial and radial directions yields the following set of 2D/1D equations:

   (Ω • ∇ xy + Σ) ψ i = q i -L i (2a) µ ∂ ∂z + Σ r ψ r = q r (2b)
where ψ i , q i (resp. ψ r , q r ) are the axially (resp. radially) integrated flux and source and Σ r is the radially homogenized total cross section. L i is the axial leakage:

L i = µ A xy ψ r | z i + z i - (3)
with µ the polar cosine and A xy = Dr dxdy the radial area of the pattern. The fact that radial leakage do not appear in Eq. ( 2b) is a direct consequence of the reflective radial boundary condition.

The solution of the above system is obtained by iteratively solving Eq. (2a) in each 2D layer of the fuel assembly (with index i) using axial leakage given by Eq. ( 3) and retrieved from Eq. (2b). Iterations are required because Σ r is homogenized with fluxes ψ i :

Σ r,i = < Σψ i > Dr < ψ i > Dr . (4)

Transverse Leakage Splitting

In the iterative resolution of the 2D/1D equations, instabilities may arise when the right hand side of Eq. (2a) is negative. In our previous work, the problem was avoided by neglecting those sources that have a negative isotropic component i.e. when 0 < L i,00 = 4π d 2 Ω L i .

In this paper however, another solution inspired by the work of the MPACT development team [START_REF] Collins | Stability and Accuracy of 3D Neutron Transport Simulations using the 2D/1D Method in MPACT[END_REF] is proposed: the "'transverse leakage splitting technique" (TLS). It consists in replacing negative leakage by an additional contribution in the macroscopic total cross section. In that case, whenever the isotropic component of the axial leakage is negative, Eq. ( 2a) is replaced by:

               Ω • ∇ xy + Σ ψ i = q i -L i (5a) Σ = Σ + L i,00 ψ i,00 (5b)
L i = L i -ψ i L i,00 ψ i,00 (5c)
where ψ i,00 = 4π d 2 Ω ψ i . It can be seen that the system of Eqs. (5a), (5b) and (5c) is fully consistent with Eq. (2b) and that it still defines a transport equation over an isotropic medium.

With those definitions, the modified leakage source L i has null isotropic component so most instabilities are avoided. The high angular orders of L i can still pose some problems but our experience showed that the method is stable in nearly every case. The only exception we found so far is the case of "over-critical" planes (i.e. planes for which the number of neutrons produced by fission exceeds the number of neutrons being absorbed). The way the issue is dealt with is explained in reference [START_REF] Faure | A 2D/1D Algorithm for Effective Cross-Section Generation in Fast Reactor Neutronic Transport Calculations[END_REF].

For practical application of the TLS in the APOLLO3 R code, it is worth mentioning that flux ψ i is replaced by its mean value in the expressions of the modified cross section Eq. (5b) and source Eq. (5c) i.e. ψ i ≈ 1 Axy Dr dxdy ψ i = 1 Axy z i + z i -dz ψ r . This allows to compute the correction term directly from the solution of Eq. (2b).

Cross Section Collapsing

When the system is converged, cross sections are homogenized in each 2D layer (index i). A summation is also performed over the energy groups in order to reduce their number to a few tens maximum (33 in this work). The averaged cross sections are stored in a library for future core calculations.

In this paper, the (angular) "flux moments" collapsing technique proposed in reference [START_REF] Vidal | APOLLO3 Homogenization Techniques for Transport Core Calculations -Application to the ASTRID CFV Core[END_REF] is used. It is preferred to the standard flux-volume weighting formulas because these do not preserve the angular information (flux anisotropy in the calculation domain). On the other hand, the flux moments collapsing technique uses the angular moments of the flux to weight cross sections in a consistent way. This theory is relevant in the scope of the 2D/1D equations since angle dependent leakage are used in Eq. (2a).

Burnup Parametrization

The 2D/1D framework provides a natural way to perform the coupling of the Boltzmann and Bateman equations.

Starting from a fresh fuel assembly, Eqs. (2a) and (2b) are first solved (with TLS eventually). The flux is then normalized to the assembly power and normalized reaction rates are supplied to Eq. (1b). Once the Bateman equations are solved, the vector of isotopic concentrations N is updated and the process is repeated. At each step, effective cross sections are stored in a parametric library referenced by the local burnup (or fluence for structures and shields). Time-dependent fully consistent cross sections are thus produced for the entire fuel assembly. This iterative procedure is summarized in Figure 1.

The advantage of this strategy is that it uses the local assembly power, which is usually a realistic (and even measurable) parameter of the core, whereas standard 2D calculations use an average value of the power to normalize the flux. In addition, it consistently takes into account any change in the axial flux shape which can be significant in heterogeneous core designs.

Flux calculation

N n → ψ n

Power normalization

ψ n = arg [P (ψ n ) = P assembly ] Depletion calculation (ψ n , N n ) → N n+1 n → n + 1

Local burnup XS generation

Figure 1. Fuel depletion calculations

Finally, it should be said that solving Eqs. (2a) and (2b) at each time step might be time consuming especially for those planes that do not contain depleting materials (structures, shields...). To avoid doing so, our implementation of the algorithm allows to selectively choose planes for which Eq. (2a) is to be solved at each time. This will be discussed in section 3.

DEPLETION OF A CFV FUEL ASSEMBLY

The CFV is a core concept that presents strong axial heterogeneities. In this section, the algorithm described in Sec. 2 is applied to produce burnup parametrized 33 group homogenized cross sections for a fuel assembly representative of a CFV core. Such cross sections are then used in a fuel depletion calculation. The APOLLO3 R code which is being developed at CEA in France is used. Results are compared to reference Monte Carlo depletion calculations. Some comparisons with standard 2D calculations, which constitute the previous methodology used at CEA to get a fuel assembly cross sections, are also provided.

Description of the Benchmark

The benchmark, whose geometry is depicted in Fig. 2, has already been introduced in reference [START_REF] Faure | A 2D/1D Algorithm for Effective Cross-Section Generation in Fast Reactor Neutronic Transport Calculations[END_REF]. We recall that C1 stands for fissile (U,Pu)O 2 material, FCA for fertile depleted UO 2 media and PLN, PNS, SVE are French acronyms for sodium plenum, axial B 4 C neutronic protection and expansion tank (container for gaseous fission products) respectively. Some other internal structures do not appear in the figure to avoid burdening it.

In this study, the depletion duration is set to 1080 days and the assembly power P = 5.4 MW is kept constant during this time.

The axial partition of the domain chosen for the 2D/1D algorithm follows the axial layout of the assembly resulting in thirteen 2D layers. In particular, this division leads to five distinct depleting material (two C1 and three FCA, one of which do not appear in Fig. 2 because it is an isolated UO 2 pellet located at the top of the upper C1 zone). Even though this partition is probably inadequate to accurately model the axial concentration gradients, this might not be so troublesome as long as the 2D/1D algorithm is only used to produce burnup parametrized cross sections. A finer axial mesh will be used in the core calculations. This partition being defined, the 2D/1D depleting equations are solved using a 1968 group cross section library processed from the JEFF-3.1.1 evaluation. The decay chain includes 18 heavy nuclides and 126 fission products [START_REF] Archier | Validation of a Multi-Purpose Depletion Chain for Burnup Calculations through TRIPOLI-4 Calculations and IFP Perturbation Method[END_REF]. Eqs. (2a) and (2b) are solved respectively with TDT and IDT characteristics solvers of APOLLO3 R . Eq. ( 1b) is solved with the MENDEL [START_REF] Lahaye | First Verification and Validation Steps of MENDEL Release 1.0 Cycle code system[END_REF] depletion solver. To ensure a converged solution, a relatively high number of time steps is chosen leading to 17 points of tabulation for the homogenized cross sections library. At the end of the 2D/1D depletion calculations, we obtain an effective cross section library parametrized with local burnups (or fluence). Using this library, the depletion calculation is repeated in a "core environment" featuring a simplified version of the assembly in which radial heterogeneities are homogenized. The number of energy groups is also reduced from 1968 to 33. Taking advantage of the simplicity of the model, a fine axial mesh (5 cm) is used to distinguish depleting materials. At this stage, the MINARET S N solver of APOLLO3 R ("core solver") is used together with MENDEL.

Results are compared to Monte Carlo calculations performed with a validated version of the TRI-POLI4 R code coupled to the MENDEL depletion solver [START_REF] Lee | SFR Whole Core Burnup Calculations with TRIPOLI-4 Monte Carlo Code[END_REF]. They feature the heterogeneous 3D geometry of the fuel assembly and a continuous energy description. A fine time mesh (10 time steps), a second order time scheme and the simulation of 24.10 6 particles per time step ensure an acceptable convergence of the results. In the following, they will be trusted as reference values.

Numerical Results

The numerical results presented in this section include evolution of the assembly reactivity and isotopic concentrations. Reactivity differences are computed as ∆ρ = 10 5 (1/k T4 -1/k AP3 ) where k is the multiplication factor handled by the code. We also insist on the fact that APOLLO3 R results always stand for results of the "core environment" featuring 33 group homogenized cross sections (i.e. the 2D/1D algorithm is only used to produce cross sections even though it also handles good results). Fig. 3 presents results for the reactivity loss which is evaluated to -3180 pcm (i.e. ≈ 3 pcm/day). The reactivity difference between APOLLO3 R and the reference results clearly shows that 2D/1D cross sections improve the calculation accuracy compared to the standard 2D methodology previously used at CEA. More precisely, the latter presents a 300 pcm difference at t = 0 while all 2D/1D results are within a 50 pcm range from reference results. In addition, the evolution of the reactivity difference follows a monotonous pattern with 2D/1D cross sections meanwhile for 2D results the behavior is more irregular.

Benefits of the burnup parametrization of cross sections are less impressive but they show that a 50 pcm extra accuracy can be achieved when the impact of the flux dependency on time is taken into account to produce cross sections. If the transverse leakage splitting correction (TLS) of section 2.3 is applied, all comparisons between APOLLO3 R and TRIPOLI4 R agree within the statistical 2σ error bar. Fig. 4 now shows how some calculation time can be saved on the resolution of the 2D/1D equations. The impact of the number of tabulated points (i.e. time steps) is evaluated showing that the difference between a cross section library that contains seventeen tabulated burnup points and another one that only contains two such points is less than 20 pcm. In other words, no intermediate calculations are in fact required but most of the effect can be catch with only two transport calculations performed at the initial and final stages of the depletion. Furthermore, Fig 4 also shows that the re-evaluation of non depleting planes (i.e. structures such as the sodium plenum or the neutronic protections) is not necessary. In fact, we observe no difference with the case in which Eq. ( 2a) is solved for all planes including those ones.

All in one, those results mean that the generation of the cross section library only requires one full 2D/1D transport calculation at t = 0, a single isotopic depletion calculation over the full time range and a second "partial" 2D/1D calculation in which Eq. (2a) is not solved for structure planes. Tab. I shows that the final concentrations of actinides (Pu239 and Np237) and fission products (Cs133) are well predicted when the 2D/1D method is used to produce the effective cross sections. Most discrepancies are below 1% except in the upper fertile zone (FCA) for which differences can reach a few % (with both 2D and 2D/1D methods). Further investigation is required to understand why such values are found in this zone. On the other hand, we observe that the standard 2D method systematically underestimates the concentration values in fertile layers and especially at the bottom of the assembly. In particular, a 9% discrepancy is found for the concentrations of Cs133 and Np237 in the lower fertile blanket.

CONCLUSIONS

This paper presents an algorithm based upon a 2D/1D method to consistently produce timedependent effective cross sections for SFR depletion calculations. The algorithm is applied to a radially reflected 3D CFV fuel assembly.

The method couples 2D and 1D flux solvers to solve the 3D neutron transport equation and a depletion solver to solve the Bateman equations. Few groups homogenized cross sections are produced at distinct burnup steps and used in a core calculation featuring a simple model of the assembly.

Good performances are found against reference Monte Carlo simulations both for the evolution of the reactivity and isotopic compositions. On the contrary, it is shown than cross sections produced with standard 2D models present a bias on the spatial distribution of isotopic concentrations and larger discrepancies on reactivity. Finally, the number of points of tabulation needed for the construction of the cross sections library can be optimized to decrease the overall calculation time.

The method has recently been used in a realistic full core calculation in which radial heterogeneities are correctly taken into account (the CFV core presents two enrichment zones for fuel assemblies and a reflector). Encouraging results have been found and shall be presented in a near future.

Figure 2 .

 2 Figure 2. ASTRID internal fuel assembly: (a) axial (b) fuel pins mesh (1/12 th ), (c) sodium plenum mesh, (d) axial protection mesh

Figure 3 .

 3 Figure 3. Reactivity loss for fuel assembly depletion (left) and reactivity discrepancy with 2D or 2D/1D cross sections (right). Note: T 0 stands for time-independent cross sections and T for burnup parametrized

Figure 4 .

 4 Figure 4. Reactivity discrepancy with 2D/1D cross sections: influence of the number of tabulated burnup points in the cross section library

Proceedings of the PHYTRA4 Conference, Marrakech, Morocco, September 17-19, 2018

ACKNOWLEDGEMENTS

The authors acknowledge the CEA of Cadarache for funding this work, FRAMATOME and EDF for their long term support and the APOLLO3 R development team for the provided support.


M Allinei

Maulin

Accumulation of nuclear material in nuclear facilities an iterative approach in order to develop measuring stations L. Loubet, P-G.

Accumulation of nuclear material in nuclear facilities:

an iterative approach in order to develop measuring stations Laurent LOUBET 1 , Pierre Guy ALLINEI 1 , Maeva MAULIN 2

1 CEA, DEN Cadarache F-13108 Saint-Paul-Lez-Durance, France 2 Université Clermont Auvergne, Département de Physique F-63171 Aubière, France

Abstract

Measuring the deposits of nuclear material accumulated during processes in nuclear facilities is a major challenge in terms of safety and criticality. The characterization of the nuclear material, "holdup", has to be taken into account since the design of new facilities, but also during their operation, and finally for the dismantling of historic equipment and facilities. Considering the diversity of encountered configurations, the holdup measurement is specific to each case. In this context, the Nuclear Measurement Laboratory of CEA Cadarache is specialized in developing and implementing gamma and neutron measuring stations, based on preliminary design and performance assessment by numerical simulation, then on iterative calculations taking into account the feedback of field measurements. In this paper, we illustrate this approach on different case studies, such as glove boxes in hot labs, covering the design, exploitation and dismantling phases of nuclear equipment and facilities.

Context and need

French and international safety authorities require that facilities using sensitive nuclear material (U, Pu, Th) guard against the risks of loss, theft and diversion of these nuclear materials [START_REF]Passive Nondestructive Assay of Nuclear Materials" -United States Nuclear Regulatory Commission 2[END_REF]. Many solutions such as weighing, measurement, physical monitoring can meet this absolute need in operation. However, over time, the deposits of low amounts of nuclear materials that accumulate during processes in nuclear facilities, hot labs, hot cells, and glove boxes can lead to the retention of significant quantities of nuclear materials. This "holdup" has to be taken into account both in old installations, in order to dismantle them properly, in installations currently in operation, and preventively in installations under construction. Therefore, measuring low amount of deposits of nuclear material is a major challenge. Today, the possibilities offered by non-destructive nuclear measurement provide solutions for many installations [2].

Contribution of the Nuclear Measurement Laboratory of CEA Cadarache on hold-up measurements

Analysis context and methodology

The Nuclear Measurements Laboratory (LMN) of CEA Cadarache is specialized in developing and implementing gamma and neutron measuring stations, based on preliminary design and performance assessment by numerical simulation, then on iterative calculations taking into account the feedback of field measurements.

In our methodology, the first step in designing such measuring stations, in a glove box or in a hot cell for example, is to establish the list of radiations emitted by the studied nuclear material in order to select the more suitable nuclear measurement. This makes then possible to choose the sensor and the detection chain whose performances are the most adapted for a given configuration. For this, the laboratory relies on its long experience in nuclear gamma and neutron measurements, and photon imaging as well.

Finally, numerical modelling of the selected detector in its operating environment is carried out in order to determine, after several iterations, the optimal measurement configuration (type, size and position of the detector) and to define the sensitivity of the measurement (to different parameters such as the type, mass and distribution of nuclear materials).

Recent examples of measuring stations design by the LMN

Our lab is solicited for different studies ranging from the simplest case of new installations, where the design of the measurement system is open, to the most difficult case of historic installations, where it is necessary to adapt to existing implementation constraints, and to more or less known history leading in some instance to poor knowledge on nuclear materials.

The two examples below illustrate both cases. The first situation is concerning a hot cell under construction where nuclear material will be reconditioned. The nuclear material is well known, the geometry of the equipment is still modifiable. The choice and the position of the detector can be optimized by successive numerical modelling comforted by experimental campaigns. The second case concerns an old equipment used to characterize nuclear materials. This equipment had to be moved in a safe way. In this case, numerical simulation has been used in order to select the best measurement configurations. Then experimental results have been interpreted using numerical simulation to obtain quantitative results on the nuclear material hold-up. The methodology used by our lab for those cases can easily been applied to other situations.

Figure 1 .

 1 Figure 1. Numerical modelling of a holdup measuring station for a reconditioning glove box.

Figure 2 .

 2 Figure 2. Measurements and quantitative results for holdup in an equipment in view of its transport.


Widloecher

E Studer

S Abe

D Abdo

Ph Debesse

J.-L Widloecher
email: jean-luc.widloecher@cea.fr

Erosion of a stratified layer by a diffuse buoyant jet: controlled experimental parametric variations

Keywords: Containment, Atmosphere mixing, Hydrogen distribution

Erosion of a stratified layer by a diffuse buoyant jet controlled experimental parametric variations


B Valentin

M Lainet

Michel Pelletier

Nathalie Chauvin

A Gallais-During
email: annelise.gallais-during@cea.fr

Pre

Pre-design of an AmBB pin loaded with 10% of americium

Keywords:

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. Introduction

High activity wastes are currently vitrified and planned to be stored in deep geological repositories. In order to reduce the radiotoxic inventory of vitrified wastes and the footprint of deep storage [START_REF] Hoorelbeke | Impact of minor actinide transmutation options on geological disposal. The French case[END_REF], research concerning solutions that could separate the most radiotoxic and long-lived elements from spent fuel and transmute them into non-radioactive or short-lived ones in nuclear reactors is being carried out on an international level. Transmutation being only reasonably applicable for Minor Actinides (MAs), (chiefly americium, neptunium, and curium) and the best transmutation performance being obtained in fast neutron reactors, MA incorporation into the fuel has become a prerequisite for Generation IV reactors to bring benefits in the disposal requirements by reducing the MA content in the high activity wastes [START_REF] Pillon | Actinide-bearing fuels and transmutation targets[END_REF]- [START_REF] Delage | Outcomes on oxide fuels developments for Minor Actinides recycling[END_REF]. Based on historical experience and knowledge, oxide fuels have emerged in Europe as the shorter term solution to meet the Generation IV assigned performances and reliability goals. Two main MA-recycling options have been under consideration:

 the homogeneous recycling mode, or Minor Actinide Driver Fuel (MADF) concept, where MAs are diluted in (U,Pu)O2 standard driver fuel at a low enough content (<3%) to limit the MA impact on the performance of the fuel and on the core safety as well as on the fuel cycle facilities, as far as possible;  the heterogeneous recycling mode on UO2 fuel located in radial core blankets, or Minor Actinide Bearing Blanket (MABB) concept, where MAs are concentrated in UO2 based fuels at a content of ~10% into the radial breeder blankets of Sodium-cooled Fast Reactors (SFRs) core.

In this concept, leakage neutrons allow MA transmutation. Due to the lower flux in these radial locations, heterogeneous recycling is inherently less efficient than homogeneous recycling in the internal core. However, its main advantage is a low impact on operating parameters and core safety [START_REF] Buiron | SFR Minor actinides transmutation in SFR depleted uranium radial blanket, neutron and thermal-hydraulic evaluation[END_REF]. In addition, the use of the UO2 matrix as a support for MAs should ease developments, as UO2 behaviour under irradiation as well as UO2 reprocessing are well known, even if MA addition during fuel manufacturing will have to be optimized.

At the beginning of investigations, experimental data on MABB fuels were scarce, with the unique irradiation SUPERFACT, performed in the PHENIX reactor in the 80's [START_REF] Prunier | Some specific aspects of homogeneous Am and Np based fuel transmutation through the outcomes of the SUPERFACT experiment in PHENIX[END_REF], [START_REF] Walker | Transmutation of neptunium and americium in a fast neutron flux:EPMA resultsand KORIGEN predictions for the SUPERFACT fuels[END_REF]. In this experiment, the irradiation of U0.6Am0.2Np0.201.926 pellets irradiated at rather high linear heat rate led to a complete release of helium during irradiation and a highly porous fuel microstructure, consistent with high temperature operating conditions (calculated to be between 1500°C to 1900°C, not fully representative of MABB operating conditions). In addition, the occurrence of a weak Fuel-Cladding Mechanical Interaction (FCMI) and the absence of a central hole were noticed.

Complementarily, a comprehensive R&D program of MABB fuel qualification started in 2008, including, as a first stage, two separate-effect irradiation tests: MARIOS [START_REF]The results of the irradiation experiment MARIOS on americium transmutation[END_REF], irradiated within FAIRFUELS [START_REF] Hania | FAIRFUELS : FAbrication, Irradiation and Reprocessing of FUELS and targets for transmutation[END_REF] and examined within PELGRIMM [START_REF] Delage | Investigation of pelletized and sphere-packed oxide fuels for minor actinide transmutation in sodium fast reactors, within the FP-7 European project PELGRIMM[END_REF] FP7 European projects, and DIAMINO [START_REF] Bejaoui | Status of the DIAMINO experiment irradiated in the OSIRIS reactor[END_REF], implemented within the French national nuclear program. They were completed by MARINE [START_REF]The MARINE experiment: Irradiation of sphere-pac fuel and pellets of UO2 for americium breading blanket concept[END_REF], the first semi-integral irradiation of AmBB fuels, carried out within PELGRIMM FP7 European project.

In parallel of these experimental tests, in 2008, a first thermo-mechanical and thermal-hydraulic pre-design of a MABB pin loaded with 20% of MA on a UO2 matrix was performed [START_REF] Valentin | Heterogeneous minor actinide transmutation on a UO2 blanket and on (U,Pu)O2 fuel in a Sodium Fast Reactor -Preliminary design of pin & sub-assembly[END_REF]. From a thermomechanical point of view, the main results of this first study were:

 a high amount of produced and released helium leading to high pressurization in the pin: the criterion on the cladding primary membrane stress was dimensioning,  to respect this criterion, and decrease the pin pressure, the plena heights had to be significantly increased.

Consequently, the resulting volume fraction of MABB fuel in the core (as a percentage of fuel in the neutronic mesh, considering a 100% density fuel) was calculated to be between 28% to 38% (instead of 41% in the neutronics scenario studies), so that the effective performances of the system had to be re-assessed by neutronics studies.

Finally, along with other transmutation studies (neutronics, fuel…), the analysis of this first predesign study [START_REF] Valentin | Heterogeneous minor actinide transmutation on a UO2 blanket and on (U,Pu)O2 fuel in a Sodium Fast Reactor -Preliminary design of pin & sub-assembly[END_REF], led to consider the following new scenario:

 only americium transmutation to be considered as a first stage (AmBB fuel instead of MABB fuel),  reduction of the Am content down to 10% of heavy atoms (initially 20%),  reduction of the irradiation duration down to 1940 EPFD (initially 4100 EPFD),  no 180-degree rotation of the MABB sub-assembly at mid-irradiation.

The objective of this paper is to update the first MABB pin pre-design study with these new considerations. The first part presents the pin modelling implemented in the fuel performance code GERMINAL V2.2 that has been adapted to the MABB specificities. The second part comments the calculation assumptions and the third part provides and discusses the results.

II. Modelling and simulation

GERMINAL V2.2 [START_REF] Lainet | Current status and progression of GERMINAL fuel performance code for SFR oxide fuel pins[END_REF], part of the PLEIADES platform [START_REF] Marelle | Validation of PLEIADES/ALCYONE 2.0 fuel performance code[END_REF], is CEA's fuel performance code dedicated to the simulation of the irradiation behaviour of SFR (U,Pu)O2 mixed oxide fuel pins. The simulation of transmutation fuels in heterogeneous mode ((U,Am)O2 fuels), with a significant content of americium, requires adaptations of the models implemented in GERMINAL. The combined effect of high helium production, coming from alpha decays of MA isotopes formed from americium, with low temperatures, due to the use of a depleted UO2 matrix (no or few fissile isotopes), can potentially lead to high gaseous swelling related to high retention of helium and fission gases at low temperatures. The specific model and laws lastly implemented in GERMINAL and their verification are presented in the first paragraph of this section. The second paragraph describes the simulation parameters constituting the reference case for the pre-design study of the AmBB pin.

II.1. Specific models and laws

II.1.1 Gaseous swelling

The main evolution implemented in GERMINAL V2.2 concerns a gaseous swelling model correlated to the high helium and fission gas retentions. This model aims to provide an "envelope" estimate of the gaseous swelling, which maximizes the pellet-cladding mechanical interaction resulting from the closure of the gap caused by the swelling of the pellet. This conservative approach is well adapted for a pre-design type study.

The gaseous swelling of the (U,Am)O2 fuel is taken into account by the mechanical model as a stress-free strain, in addition to that induced by the solid swelling. The new gaseous swelling model implemented in the AmBB calculation scheme is largely inspired by the one describing the swelling of (U,Pu)O2 mixed oxide fuels in transient regime [START_REF] Roche | Modelling of the thermomechanical and physical processes in FR fuel pins using the GERMINAL code[END_REF]. The main interest of this model is that it correlates the local maximum gaseous swelling to local retention of fission gases.

For standard (U,Pu)O2 fuel pins and their high operating conditions, the assumption commonly adopted is that the produced helium is completely released. This assumption is well founded and verified by post-irradiation examinations for fissile pins, operating at high temperatures, but for AmBB pins operating at lower temperatures, it is no longer appropriate and it is necessary to estimate helium retention. The solution to assess the contribution of helium to gaseous swelling is to apply the fission gas (xenon, krypton) retention rate to the amount of produced helium. Thus, the helium contributing to gaseous swelling is that produced in the pellet regions where the fission gases are retained. This assumption is conservative, thus consistent with a pre-design approach, because it leads to maximize the amount of helium contributing to gaseous swelling, since the fission gas retention rate maximizes that of helium.

In addition, for a pin pre-design study, a total helium release is still considered to determine changes in gas composition and pressure of the plenum. This assumption may seem inconsistent with the previous one adopted for helium retention evaluation, but the reason is that the combination of these two choices leads to maximize both the gaseous swelling induced by helium retention, and potentially the resulting fuel-cladding mechanical interaction, and also the plenum gas pressure evolution. In other words, the principle consists in maximizing the mechanical loadings on cladding, in an "envelope" pre-design approach.

II.1.2. Conductivity law

A specific thermal conductivity law was developed and implemented in GERMINAL for (U,Am)O2 fuels. It consists of a law determined for a (U,Am)O2-x mixed oxide sub-stoichiometric fuel sample (CP-ESFR4 sample [START_REF] Jp Ottaviani | CP-ESFR : properties measurement of uranium americium mixed oxide fuels[END_REF]) with the following composition: U/M=85%, Am/M=15%, 1.93<2-x<1.95.

The variation of the conductivity of this fuel sample as a function of temperature is described by the following correlation: 0(T)=1/(3.369 .10 -1 +1.966 .10 -4 xT) This correlation is valid (i.e. deduced from thermal diffusivity experimental measurements) in the temperature range 800 K ≤ T ≤ 1600 K. However, as Figure 1 shows, given the slow evolution of the values, extrapolation of the correlation outside the validity range is reasonably acceptable between [700K-2000K], temperature range of interest for the present study.

II.1.3. Fuel restructuring and actinides redistribution

Fuel restructuring models (migration of porosities and formation of the central hole) and redistribution of actinides, used for standard (U,Pu)O2 fuels, are disabled in the AmBB calculation scheme by appropriate parameters assignments. Actually, these models (determining oxygen potential, chemical potential of oxides in vapour phase, partial pressures of these oxides and associated fluxes in the pellet) are adapted to (U,Pu)O2. As the different oxide species and thermodynamic balances are materialspecific, these models are therefore not applicable to (U,Am)O2 fuel, given the differences in material composition, the high amount of minor actinides and for which other species and balances should be considered.

In addition, AmBB fuels contain throughout their stay in the core a low proportion of fissile isotopes, thus the operating thermal regime will be significantly lower than that of a fissile element. Therefore, it is reasonable to consider that the operating regime of AmBB fuels is significantly lower than the thermal activation thresholds of the transport/diffusion mechanisms leading to the migration of porosities and species. In other words, these mechanisms of fuel restructuring and actinides redistribution will not activate in AmBB fuels.

II.1.4. Verification

The verification of the AmBB calculation scheme is based on the simulation of the irradiation in PHENIX of SUPERFACT pin N°6 [START_REF] Prunier | Some specific aspects of homogeneous Am and Np based fuel transmutation through the outcomes of the SUPERFACT experiment in PHENIX[END_REF], [START_REF] Walker | Transmutation of neptunium and americium in a fast neutron flux:EPMA resultsand KORIGEN predictions for the SUPERFACT fuels[END_REF]. This pin is constituted of (U,Am,Np)O2-x fuel with (Am/M) = 19%, (Np/M) = 21% et 2-x = 1.926. The objective is to verify the predictions of the new gaseous swelling model. Figure 2 shows the calculated radial profile of gaseous swelling in the pellet at the maximum flux plane at end of irradiation (EOI), and the corresponding macrograph of the pellet. The prediction for the maximum gaseous swelling is very close to 15%, considered as "envelope" value in order to increase the pellet-cladding mechanical interaction resulting from the closure of the gap caused by the swelling of the pellet. It occurs at a pellet radius of 1.4 mm (~ 0.5 r/r0). This is consistent with the fact that there is an intermediate area of the pellet with an optimum of gas retention and temperature leading to maximum swelling: the swelling decreases  in the centre of the pellet, due to gas release which is predominant in high temperature regions,  and at the periphery of the pellet, due to a lower volume fraction of precipitated gases at lower temperature.

The predictions of the gaseous swelling model are therefore physically consistent, and its amplitude in accordance with the targeted envelope value.

II.2. Simulation parameters of the AmBB pin reference case

The scenario, neutronics characteristics, sub-assembly / pin geometry and materials detailed in the following paragraphs constitute the reference case for the AmBB pin pre-design study.

II.2.1. Scenario and neutronics characteristics

Americium transmutation is supposed to take place in a 3600 MWth system core. A schematic diagram of a typical ASTRID-type core considered for this study is represented on Figure 3. This study concerns the hottest AmBB sub-assembly : red circle on Figure 3. The fuel neutronics characteristics are:

 americium in a uranium oxide matrix : (U,Am)O2-x,  fuel fraction (U,Am)O2-x in the sub-assembly = 41.6%,  chemical composition: Am/(U + Am) = 10%,  americium isotopic composition: 80% of 241 Am and 20% of 243 Am,  cross dose, linear power evaluated by neutronic calculations and provided on input to GERMINAL V2.2

Inner core

Outer core

AmBB

Neutron shielding Reflector

II.2.1. Sub-assembly / pin geometry and materials

The geometrical data of the sub-assembly and pin considered for the study are summarized in Table 1. The cladding is in austenitic steel 15-15 Titanium grade D4, precursor of AIM1 [START_REF] Courcelle | Evolution under Irradiation of Optimized Austenitic Steel For Gen-IV Reactors. Impact on Fuel Cladding Properties and Performances[END_REF].

The (U,Am)O2-x fuel has the following characteristics:

 density of the pellet compared to its theoretical density: 88%,  porosity: 12%, essentially open porosity,  O/M ratio: 1.995.

III. Results of pre-design of an Am-BB pin

The evolution of parameters with irradiation, fuel and cladding behavior are discussed in the first paragraph of this section, for the AmBB pin reference case (without uncertainties). Then, the predesign criteria and corresponding calculated pre-design results are detailed. Finally, in the last paragraph of this section, the uncertainties on the data related to the geometry and the behaviour of the pellet and cladding are taken into account and the results are presented.

III.1. Evolution of parameters with irradiation, fuel and cladding behavior

III.1.1. Linear power, dose and burn-up

The linear power, burn-up and dose evolution with irradiation are represented on Figure 4. The maximum Linear Heat Rate (LHR) at the Maximum Flux Plane (MFP) is 70.2 W/cm at the End Of Irradiation (EOI). This value is low compared to that of a standard fuel (> 400 W/cm). The consequence of this low value is that the fuel remains "cold", which limits both the internal pressure in the fuel pin and the release of helium and gaseous fission products. This could lead to an increased swelling of the pellet due to this high gas retention (see section III.1.3).

At the end of irradiation (1940 EFPD), the maximum dose is 63.1 dpa and the averaged burn-up over the height of the fissile zone is only 1.52 at% (14,6 GWj/tHM). This low value, related to the assumptions on irradiation duration and Am content, will lead to a low solid swelling of the fuel. Its averaged value is estimated to be about 1%, but is quite dependent on temperature. The maximum burn-up at MFP at EOI is 2.4 at%, corresponding to 45 µm increase of pellet diameter.

III.1.2. Helium and fission gas production and release, internal pressure

Figure 5 shows helium and Fission Gases (FG) Xe and Kr production and release, as well as the internal pressure of the pin as a function of irradiation time. The volume of gas formed at EOI (at hot state) is 194.5 cm 3 for FG (green curve) and 598 cm 3 for helium (blue curve). As specified above, as the fuel remains cold, the FG release is negligible (red curve), and the consequence is an almost total gaseous retention with a potential gaseous swelling of the pellets.

The maximum value of the internal pressure in the fuel pin is 27.4 bar (orange curve) with the assumption of total helium release. With this conservative assumption, this value is compatible with quite high margins with regards to the RAMSES thermo-mechanical criteria (see III.2.2).

III.1.3. Strain and swelling of the fuel and cladding, pellet-cladding gap

Figure 6 shows the main mechanisms contributing to the fuel total deformation (purple curve): gaseous swelling (blue curve) and inelastic deformation (green curve), as a function of the pin height.

The closure of the pellet-to-cladding gap (orange curve) can be seen between 622 mm and 958 mm (0.52 and 0.8 h/h0), which is consistent with Figure 7 where the fuel surface temperature matches the cladding internal surface temperature (see III.2.1). In all the following graphs representing the evolution of the parameters as a function of the pin axial height, 0 mm represents the bottom and 1200 mm the top of the pin.

III.2. Design criteria

III.2.1. Thermal design criteria and temperature of the fuel and cladding

In this study, the thermal criterion for the fuel pellet consists in the verification that the margin to melting is higher than 300 °C; this margin is usually considered for (U,Pu)O2 fuel [START_REF] Guerin | Fuel Performance of Fast Spectrum Oxide Fuel[END_REF]. Another thermal criterion is the Nominal Cladding Temperature (NCT), which is the maximal temperature of the hottest cladding of the sub-assembly. It is imposed to be NCT = 620°C to avoid cladding thermal-activated creep and to preserve margin to melting in fuel. The respect of this criterion is guaranteed by the sub-assembly cooling (see III.2.1).

We can see on Figure 7 that at EOI, the fuel centre temperature does not exceed 825 °C. The margin to melting is considerable (~ 2000 °C). Another consequence of this low thermal regime is that no central hole could form in the pellet. On either side of the MFP, the fuel surface temperature matches the cladding internal surface temperature, showing a closure of the pellet-to-cladding gap, probably leading to fuel cladding mechanical interaction (see also Figure 6).

III.2.2. Thermo-mechanical criteria (RAMSES criteria) and thermo-mechanical behaviour of the cladding

Three criteria define the margins to failure of the cladding. They are the so-called RAMSES criteria.

 R1 criterion relates to the primary membrane stress (Pm); it has to be higher than 1 at any time during irradiation in order to validate the design of the cladding (for this criterion):

with Rm: tensile strength, Rp0.2 : yield stress for 0.2% plastic deformation and Pm : general primary membrane stress  R2 criterion shows, in addition to the primary membrane stress, the general flexion stress, |Pl+Pb|; it has to be higher than one at any time during irradiation in order to avoid reaching the limits of the cladding material:

with Pl+Pb: general primary bending stress and K = 1.5 for the cladding in nominal operating conditions  R3 criterion compares the total local Von Mises stress () with the local material properties (0.8Rp0.2); it has to be higher than one to avoid reaching the limits of the cladding material:

 An additional criterion for creep usage fraction, U, has been defined; it has to be less than 1 at any time during irradiation in order to avoid reaching the creep limits of the cladding material.

Figure 8 shows the evolution of R1, R2 and R3 criteria along the pin, for the studied AmBB pin reference case. The three criteria are minimum at EOI, at the top of the pin: R1min = 3.7, R2min = 5 and R3min = 8.3. The three RAMSES criteria are widely respected. In addition, the U criterion for creep usage fraction is U ~ 7.3 10 -9 << 1. The criterion is respected and the margin is huge.

III.2.3. Thermal-hydraulic pre-design criteria and results

The AmBB sub-assembly must be coolable. Since the nominal cladding temperature is 620°C in the GERMINALV2.2 calculations, the thermal-hydraulics calculations first determine the necessary flow rate to guarantee this temperature when the highest power of the sub-assembly is reached. Once the flow rate known, the pressure loss in the pin bundle of the AmBB sub-assembly is deduced. The calculated values are then compared with the flow rate and pressure loss of a typical ASTRID bundle in order to check that:

 the cooling flow rate is compatible with the flow rate range of SFR sub-assemblies, among which ASTRID ones (~4 kg/s to ~40 kg/s),  the bundle pressure loss is acceptable and compatible with the loss of total core load (i.e. lower and limited to a maximum between ~1.5 and ~1.8 bar, bundles with lower pressure may require the use of a diaphragm to reach these values).

For these thermal-hydraulic calculations, the evaluation of cooling flow rate and pressure loss of the bundle is performed with the assumption of independent sub-channels. This approach is conservative since it maximizes the flow rate and pressure loss. The sub-assembly is supposed to be adiabatic.

The following results are obtained:

 the dimensioning time is EOI,  the required flow rate to cool the bundle is ~ 7 kg/s,  the pressure loss related to the bundle is ~ 0.013 bar,  at the beginning of the cycle the average bundle temperature is ~ 74 °C, and the NCT is 534 °C,  at the end of the cycle the average bundle temperature is ~ 123 °C, and the NCT is 620°C.

In conclusion, the required flow rate to cool the bundle is consistent with an ASTRID sub-assembly, but the bundle pressure loss is much lower than that of the ASTRID core. Such a sub-assembly will require the use of a diaphragm to achieve a higher pressure loss.

III.3. Influence of uncertainties

Uncertainties are not taken into account in the results of the previous section (reference case). Uncertainties to be considered should be high due to high uncertainties on:

 gap size (linked to gaseous swelling),  gaseous release,  linear heat rate (due to neutron flux gradient),  fuel composition and thermal conductivity.

The influence of some of them are taken into account in the present section. The objective of this study is not to assess the influence of all uncertainties since they are not yet all known at this stage. The conservatisms taken into account in this study are the following:

 minimum cladding thickness = 0.46 mm (nominal value used for fuel sub-assemblies = 0.5 mm),  diminution of the expansion volume height (EVH) in the ratio as follows: x0.99 for the lower EVH and x0.96 for the upper EVH,  complete release of helium and fission gases,  maximization of internal and external corrosion speed (maximum standard laws in GERMINAL V2),  maximum cladding temperature = 630 °C.

The results show that these different conservatisms don't modify the conclusions obtained for the reference case. The thermal and thermo-mechanical (RAMSES criteria) pre-design margins are still high, as shown in Table 2, which summarises the results obtained for the reference case and for the case with uncertainties.

IV. Conclusion

This paper presents the results of a pre-design study of an AmBB pin performed with the GERMINAL V2.2 fuel performance code of the PLEIADES simulation platform.

A calculation scheme dedicated to the AmBB pin was developed and implemented with the following specificities:

 gaseous swelling model correlated to fission gas retention and helium production,  internal pressure calculation assuming total helium release,  thermal conductivity law specific to (U,Am)O2 fuel,  no activation of the standard (U,Pu)O2 fuel restructuring (migration of porosities and formation of the central hole) and redistribution of the actinides.

The data used in this study are the following:

 geometry of the fuel sub-assembly of ASTRID-type outer core,  10% americium loaded in a UO2-x matrix,  irradiation time: 1940 EFPD in five operating cycles of 388 EFPD,  no 180 degree rotation of the sub-assembly at mid-life.

The results of this study show that all the pre-design criteria are respected and the thermal and thermo-mechanical margins are high, even when taking into account the uncertainties. A first thermal-hydraulic pre-design of the reference case, performed with the assumption of independent sub-channels (envelope approach) shows that the required flow rate and pressure loss remain compatible with that of SFR, including ASTRID. These conclusions open the path, from the AmBB pin behaviour point of view, to improvement possibilities. The AmBB pin pre-design shows high margins, suggesting that increasing both the Americium content and the irradiation time could be possible (without considering the constraints related to the fuel cycle). Considering the fuel thermal margins and the cladding thermomechanical margins (low internal pressure), a promising improvement could be to increase the pin diameter. The consequence would be an increase of the fuel proportion in the sub-assembly, leading to an increase of the linear power at the end of irradiation. Thus, the fuel temperature would be higher, which would enhance gas release and reduce swelling and risk of pellet-cladding mechanical interaction. Another potential advantage would be a lower number of pins to fabricate, but with a different geometry than standard driver fuel pin. In parallel, R&D efforts on fuel fabrication have been pursued to increase open, interconnected fuel porosity in order to favour gas release.

Figure 1 :

 1 Figure 1: Thermal conductivity of a (U,Am)O2 sample as a function of temperature

Figure 2 :

 2 Figure 2: Gaseous swelling of SUPERFACT pin N°6: a. Linear Heat Rate, fuel temperature and pellet-to-cladding gap as a function of time; b. Fuel temperature and gaseous swelling at EOI, as a function of pellet radius

Figure 3 :

 3 Figure 3: Position of the dimensioning sub-assembly in the core

Figure 4 :

 4 Figure 4: Linear power, dose and burn-up as a function of irradiation time

Figure 5 :

 5 Figure 5: Helium and fission gas production and release, internal pressure

Figure 6 .

 6 Strain and swelling of the fuel and cladding, pellet-cladding gap

Figure 7 .

 7 Temperature of the fuel and cladding

Figure 8 .

 8 Figure 8. RAMSES criteria along the pin at EOI

Table 1 : Geometrical data of the AmBB pin and sub-assembly

 1

	Data	Value	Data	Value
	Pellet diameter	8.45 .10 -3 m	Internal width of hexagonal wrapper tube	0.2083 m
	Initial pellet-cladding gap	125 µm	External width of hexagonal wrapper tube	0.1993 m
	Cladding internal diameter	8.7 .10 -3 m	Inter-assembly gap	4.5 .10 -3 m
	Cladding external diameter	9.7 .10 -3 m	Neutronic mesh	2.25 .10 -3 m
	Spacer wire diameter	1.0 .10 -3 m	Number of rows	8
	Spacer wire step	0.225 m	Inferior fertile zone height	0.3 m
	Pin mesh	13.18 .10 -3 m	Na section of the bundle	1.82 .10 -2 m²
	Inferior plenum height	0.830 m	Wet perimeter of the bundle	7.99 m
	Superior plenum height	0.071 m	Hydraulic diameter of the bundle	9.1 .10 -3 m

Table 2 : Comparison between results obtained for the reference case and for the case with uncertainties Data Reference case Case with uncertainties

 2

	Initial pellet diameter	8.45 .10 -3 m	8.45 .10 -3 m
	He volume produced	598 cm 3	598 cm 3
	FG volume produced	194 cm 3	194 cm 3
	Pellet-cladding gap at EOI	Gap closure	No gap closure
	Pressure	27.4 bars	36.3 bars
	R1 criterion	3.7	3.3
	R2 criterion	5.0	4.5
	R3 criterion	8.3	7.5
	Creep usage rate	7.3 .10 -9	2.1 .10 -8