Tuan Anh Pham
email: tuan-anh.pham@inria.fr

Nhan Le Thanh
email: nhan.le-thanh@inria.fr

A Ontology-based Approach for Business Process Compliance Checking

Keywords: Ontology Reasoning, Business Process Compliance, Business Rule

The early detection of flaws and errors has become a significant feature of a business process modeling tool. This paper proposes an ontology-based approach for business process compliance checking. The business processes and the business rules are represented in a machine understandable form, a reasoner is used to reason on this knowledge base for detecting the potential semantic error by using a set of predefined rules.

INTRODUCTION

Business Process Management (BPM) has become more and more important with the methods, techniques and software developed to design, control, and analyze operational processes involving humans, organizations, applications, documents and other sources of information [START_REF] Ryan | Business process management (BPM) standards: a survey[END_REF]. Organizations have understood the cost effectiveness of BPM that emerges from flaw detection and automation of business processes [START_REF] Fasbinder | Why model business processes[END_REF] [START_REF] Hommes | The Evaluation of Business process Modeling Techniques[END_REF].

To be able to validate a business process, one of the challenges is to check the compliance of business process with a set of business rules and to create the correspondence between them automatically. The other challenge is that the business process and the business rules may be modified during the runtime because of changing in enterprise's policies. Therefore, the challenges for system designers are to build a flexible intelligent system which accepts and verifies the change on the business process and the business rules automatically. It can also work as a rule-based system which can reason and deduce a new knowledge or new decision based on a set of rules and facts.

In this paper, an ontology-based approach for business process modeling and checking is proposed, it gives full play to the excellence of ontology's semantic representation ability and automation reasoning. The business process model has been designed by using Coloured Petri Net (CPN) [START_REF] Feng | Colored Petri net extended with price information and its applicaion[END_REF] and translated into a set of axioms in Business Process Ontology (BPO). The business rules have been also translated into a set of axioms in Business Rules Ontology (BRO). After finishing the translation, the consistency of the business process model and the business rule has been validated by a reasoner. The system can detect potential flaws automatically either at design time or during runtime. If the user designs or modifies a business process and this action causes a conflict with the business rules, the system will notify automatically.

Our contributions in this paper are:



An ontology-based method for modeling a domain ontology and business rules.



Proposing a method for detecting the potential error of business processes automatically during design time and runtime.

The rest of the paper is organized as follows. Section 2 analyses the differences with previous works. Section 3 presents some basic definition to help the reader to understand this paper. Section 4 presents the methodology for modeling a business process. Section 5, we will introduce about the content of business rule ontology (ontology domain). A methodology of potential semantic error detection in a business process is introduced in section 6. Section 7, we provide an example. We will do some evaluation in section 8. In section 9, some discussions about the addressed topics, some conclusions and future research directions are presented.

RELATED WORKS

Detecting the potential semantic error of business processes is a challenging task: the number and complexity of business rules is increasing and the rules are subject to constant change. This calls for a holistic approach to manage compliance. Becker et al. define business process compliance management (BPCM) as "the steady modeling, refinement, and analysis of business processes regarding the fulfillment of regulatory compliance" [START_REF] Becker | Generalizability and Applicability of Model-Based Business Process Compliance-Checking Approaches -A State-of-the-Art Analysis and Research Roadmap[END_REF]. Ramezani et al. outline the interdependencies between BPM and Compliance Management (CM) and describes CM as a "methodology to elicit, specify and formalize, implement, check and analyze, and optimize compliance requirements in organizations" [START_REF] Ramezani | Separating Compliance Man gement and Business Process Management[END_REF]. Works on Business process compliance have focused on examining whether a given process model is compliant with a certain reference model/pattern. On the technical aspect, the business process pattern initiative has identified various patterns for the specification of control-flow [START_REF] Van Der Aalst | Workflow patterns[END_REF], data-flow [START_REF] Russell | Workflow data patterns: Identification, representation and tool support[END_REF], and resources [START_REF] Russell | Workflow data patterns: Identification, representation and tool support[END_REF] in busi-ness process management systems. The work in [START_REF] Tran | Broadening the use of process patterns for modeling processes[END_REF] deals also with the planning layer by formalizing process patterns using UML concepts. These compliance works have focused on the structural level of process models, while another line of works focuses on the combination of data and structure [START_REF] Awad | Towards Resolving Compliance Violations in Business Process Models[END_REF][START_REF] Weber | Semantic business process validation[END_REF][START_REF] Governatori | Compliance checking between business processes and business contracts[END_REF][START_REF] Thom | Inventing Less, Reusing More, and Adding Intelligence to Business Process Modeling[END_REF][START_REF] Kharbili | Towards a framework for semantic business process compliance management[END_REF]. The frameworks in [START_REF] Arbab | Towards using reo for compliance-aware business process modeling[END_REF][START_REF] Kharbili | Towards a framework for semantic business process compliance management[END_REF][START_REF] Gschwind | Applying patterns during business process modeling[END_REF], for example, provide general compliance criteria for assessing the compliance of processes with semantic constraints. In addition, some compliance works aimed at supporting specific purposes, for example: correcting process models at design time [START_REF] Arbab | Towards using reo for compliance-aware business process modeling[END_REF], verifying changes in existing models [START_REF] Arbab | Towards using reo for compliance-aware business process modeling[END_REF], identifying compliance in the context of process mining [START_REF] Gschwind | Applying patterns during business process modeling[END_REF], and identifying violations of execution order compliance rules [START_REF] Awad | Towards Resolving Compliance Violations in Business Process Models[END_REF].

In our approach, we take advantage of CPN's color set for checking the compliance of processed data in data-flow with the constraints in BRO. A color set can be defined in many types: int, string or object. When firing a transition, the value of each color can be changed, the new value must respect to the constraint in BRO. This work will be explained in more detail in section 3.

On the other hand, for representing the business rules, some works use an ontology approach [START_REF] Reynares | An automatable approach for SBVR to OWL 2 mappings[END_REF] [9] [10] [START_REF] Bernotaityte | Developing SBVR Vocabularies and Business Rules from OWL2 Ontologies[END_REF]. They translate the Semantic of Business Vocabulary and Rule (SVBR) [16] vocabulary to OWL [29] and Semantic Web Rule Language (SWRL) [START_REF] Horrocks | Swrl: A semantic web rule language combining owl and ruleml[END_REF], they provide the mapping or the rule in order to translate each property of SVBR (the definition of OMG) to a set of axioms in an ontology. In our approach, we classify the business rule into five main type of business rules, and we use Attempto Controlled English (ACE) [START_REF] Auer | OntoWiki -A Tool for Social, Semantic Collaboration[END_REF] for defining the business rule, each ACE phrase will be translated into an axiom in BRO. We consider not only the business rule representation aspect but also the compliance of business process with a set of business rules. The advantage of our approach is to ensure that the business process is well-defined at design time and executed correctly during runtime. We must consider this aspect because a business process must always respect the predefined rules in BRO. Therefore, the difference between our work and the previous ones, is that we allow the system to check not only the consistency of business process and business rule but also the consistency of the integration between them.

BACKGROUND 3.1 OWL ontology and Semantic Web Rule Language

OWL

Ontology is about the exact description of things and their relationships. The Web Ontology Language (OWL) [29] is a family of knowledge representation languages for authoring ontologies. It was designed to provide a common way to process the content of web information (instead of displaying it). OWL was also designed to be read by computer applications (instead of humans).

SWRL

he Semantic Web Rule Language (SWRL) [30] is a proposed language for the Semantic Web that can be used to express rules as well as logic, combining OWL DL or OWL Lite with a subset of the Rule Markup Language.

The rule is represented in this form:

antecendent ⇒ consequent
where both antecedent and consequent are conjunctions of atoms written a1 ∧ ... ∧ an. Variables are indicated using the standard convention of prefixing them with a question mark (e.g., ?x). Using this syntax, a rule asserting that the composition of parent and brother properties implies the uncle property would be written:

parent(?x,?y) ∧ brother(?y,?z) ⇒ uncle(?x,?z)

In this syntax, built-in relations that are functional can be written in functional notation, i.e., op:numeric-add(?x,3,?z) can be written instead as ?x =op:numeric-add(3,?z)

Coloured Petri Nets

Business Process Compliance

Business process compliance checking is the set of activities an enterprise does to ensure that its core business does not violate relevant regulations, in the jurisdictions in which the business is situated, governing the (industry) sectors where the enterprise operates. The activities an organisation does to achieve its business objectives can be understood as business processes, and consequently they can be represented by business process models.

On the other hand a normative document (e.g., a code, a bill, an act) can be understood as a set of clauses, and these clauses can be represented in an appropriate formal language. Business process compliance is a relationship between the formal representation of a process model and the formal representation of a relevant regulation.

BUSINESS PROCESS ONTOLOGY

As mentioned above, we use Color Petri Nets for modeling business processes; In this section, we introduce the method of building BPO, this method helps the user to represent a business process by an ontologies. The advantage of this method is to allow the user to check the consistency business process automatically by the reasoning. The TBox of BPO is defined as follow:

BUSINESS RULE ONTOLOGY 5.1 Classification of Business Rule

The different structural categories of business rules are (Wagner 2005:

1. Integrity (or constraints); For example: Each company must have one and only one director.

Business Rules Ontology

In this section, we introduce the method of building a business rule ontology. As mentioned above, there are five type of rules. For each type of rule, we create a set of axioms in the BRO. We also introduce some addition rules for allowing the reasoner to reason on BRO and BPO to detect the potential semantic error automatically.

Integrity Rule

The integrity rule have the same meaning with a constraint in the relational database. In table 1, we define the cardinality rules. It will be translated into a set of cadinality axiom inside BRO.

Derivation Rule

This kind of rule allow the system to deduce a new knowledge. If a set of fact satisfy the derivation rule, the reasoner will deduce a new fact from the existing facts. We use SWRL rule to represent this kind of rule. This is the advantage of ontology, the information is represent in a machine understandable form, so the system can reason on our information to make some proposition to the user.

Reaction Rule

One of the important rule is the reaction rule which allows the user to define the relationship between a set of actions in a specific domain. We propose six kind of relationships: dependency, parallel execution, choice execution, sequential exlusion, parallel exclusion and choice exclusion.

Production Rule and Transformation Rule

This kind of business rules is represented in the form "IF something DO something". For representing this form with OWL language, we use SWRL [30].

Table 4. Production rule and Transformation rule

Rule Example OWL and SWRL

If something do something PaltiumCustomer(w)-> hasDiscount(x,5)

Addition Rule

In order to detect the potential semantic error inside a business process, we add into the BRO a set of addition rules. It defines the rule which allows the reasoner to reason and find the potetial semantic error. It can dedect not only the neighbour step in a business process, but also the step which can find by using a chain reasoning.

Table 5. Reaction rule

Rule Example OWL and SWRL

Task A is depended on task B

DETECTING THE POTENTIAL SEMANTIC ERROR

In Fig 1, we introduce the sketch of our solution. Business processes (CPN graph) are represented by an individual of the correspondence concept in BPO. Business rules are created and modified by an editor. Each rule is represented by a set of axioms and SWRL rule inside BRO. In order to check the compliance of business process with business rules, we merge BRO and BPO into one ontology; two concepts Transition in BPO and Task in BRO are defined as two equivalence concepts. The business term individuals can be used as a color and a token in CPN graph (business process). During the execution of business process, the value of individual can be changed but it must respect the constraints inside BRO (TBox and Properties).

At design time, when a user defines a business process, the business term will be used to name an item. Each transition individual in BPO is equivalent to an action individual in BRO. Depending on the user's given order, BPO editor will generate a set axioms inside BPO.

For example: there are two tasks inside BRO, which is defined that b depends on a as follow:

ObjectPropertyAssertion(:hasDependencyTask ∶b∶a)
It means that a must be executed before b, but at design time a user define that a executes after b, and the rule is generated as follow:

ObjectPropertyAssertion(:hasDependencyTask ∶a∶b) Two rules above are opposite, so the merged ontology of BRO and BPO will be inconsistent. It can be checked by a reasoner (Pellet, Helmit). Because the property hasDependencyTask is defined as a TransitiveObjecProperty in the reaction rule 1 in table 3.

At runtime, we use the same approach to verify the consistency of merged ontology. If a user modifies a business process, the modification will be generated and insert into BPO; for each modification, the reasoner will check the consistency of merged ontology and notify the result to the user automatically.

In table 6, the list of potential semantic error which can be detected by our approach is presented. It allows the designer to avoid the semantic error at the design-time. It help to reduce the time of building a system and to ensure the correctness of this system in the early step of development.

EVALUATION

We implemented this approach and did some unit test on our business process repository. In table 7, we provide three main test which were done by a computer Core I7, RAM 16 GB.

CONCLUSION

In this paper, an ontology-based approach for detecting the potential semantic error of business process and business rules is proposed. It takes important features of the ontology which are the reasoning capabilities, the possibility to express complex actions, and its declarative semantics to validate not only the consistency of business rules and business process but also the compliance of business process with a set of business rules. The advantage of this approach is to allow the system to detect the flaws of business process automatically at design time and run time by using the ontology's reasoning capabilities. Nevertheless, by using this approach, if BRO has many concepts and properties, the reasoning may take long time for checking the consistency of BPO and BRO ontology. According to that, future theoretic works involve three main issues. The first one is to focus on the distributed reasoning.

The second one will be achieved by selecting the related rule of an action for the validation. And the last goal is to consider the business process execution and work with a data source.

  Task A exclude task B in sequential hasDependencyTask(A,B) Λ hasExSequentialTask(B,A) depended on task B Task A must be executed in choice with task B hasDependencyTask(A,B) Λ hasChoiceTask(B,A) ->Error(A) Task A must be executed in parallel with task B Task A is depended on task B hasDependencyTask(A,B) Λ hasParallelTask(B,A) ->Error(A) Task A must be executed in choice with task B Task A must be executed in parallel with task B hasChoiceTask(A,B) Λ hasParallelTask(B,A) ->Error(A) Define class Tast is a disjoint class of the class Error DisjointClasses(ekiip:Action ekiip:Error)

Figure 1 .

 1 Figure 1. Checking the compliance of business process with a set of business rules.

  is an arc expression function. It maps each arc a∈A into the expression e. The input and output types of the arc expressions must correspond to type of nodes the arc connected to.

	Use of node function and arc expression function allows multiple
	arcs connect the same pair of nodes with different arc expressions.
	G is a guard function. It maps each transition t∈T into guard
	expression g. The output of the guard expression should evaluate to
	Boolean value true or false.
	I is an initialization function. It maps each place p into an
	initialization expression i. The initialization expression must
	evaluate to multiset of tokens with a color corresponding to the
	color of the place C(p).

Definition 1. A net is a tuple N = (P, T, A, Σ, C, N, E, G, I ) where:  P is a set of places.  T is a set of transitions.  A is a set of arcs In CPNs sets of places, transitions and arcs are pairwise disjoint P ∩ T=P ∩ A=T ∩ A=∅ Σ is a set of color sets defined within CPN model. This set contains all possible colors, operations and functions used within CPN. C is a color function. It maps places in P into colors in Σ.

N is a node function. It maps A into (P × T)∪(T × P).

E

  CPN is the concept for representing all CPN graphs. A CPN graph is well defined if and only if it has at least one place, one transition, one input arc and output arc. We define in BPO following classes: CPN, Place, Transition, OutputArc, InputArc and some properties which define the relations between them, hasPlace, hasTransition, hasInputArc, hasOutputArc. Place represents the properties of place, we define a concept Place. A place may have a token or not, it has also at least one InputArc or one OutputArc. The concept Transition is defined for all transitions. A transition must have at least one InputArc and one OutputArc. It's one of the minimum conditions for having a well-defined CPN graph. A transition may have only one guard function or not. The concept InputArc defined for all input arcs. An input arc has only one source place and one target transition. It may be marked by only one expression or not. An OutputArc has only one source transition and only one target place. It also may have only one expression or not.

	1. CPN ⊑
	≧1hasPlace.Place⊓≧1hasTransition.Transition⊓≧1
	hasInputArc.InputArc⊓≧1hasOutputArc.OutputArc
	2. Place⊑≧0hasToken.Token⊓≧1hasArc.(InputArc⊓OutputAr
	c)
	3. Transition ⊑
	≧1hasInputArc.InputArc⊓≧1hasOutptArc.OutputArc⊓≦1h
	asGuardFunction.Expression
	4. InputArc ⊑
	1hasSourcePlace.Place⊓=1hasTargetTransition.Transition
	⊓≦1hasExpression.Expression
	5. OutputArc ⊑
	=1hasTargetPlace.Place⊓=1hasSourceTransition.Transitio
	n⊓≦1hasExpression.Expression

Table 1 . Integrity rules Rule Example OWL and SWRL

 1

	something that owns at least	ObjectMinCardinality(n R C)
	2 cars
	something that owns at most	ObjectMaxCardinality(n R C)
	2 cars
	something that owns exactly	ObjectExactCardinality(n R C)
	2 cars

Table 2 . Derivation rule

 2

	Rule Example	OWL and SWRL
	platinum customers receive a	PaltiumCustomer(w)->
	5% discount. John Doe is a	hasDiscount(x,5)
	platinum customer. As a
	conclusion, John Doe receives
	a 5% discount

Table 3 . Reaction rule

 3

	Rule Example	OWL and SWRL
	Task A is depended on task B hasDependencyTask(B,A)
		TransitiveObjectProperty
	Task A exclude task B in	hasExSequentialTask(B,A)
	sequential	SymmetricObjectProperty
	Task A exclude task B in	hasExParallelTask(B,A)
	parallel	SymmetricObjectProperty
	Task A exclude task B in	hasExChoiceTask(B,A)
	choice	SymmetricObjectProperty
	Task A must be executed in	hasParallelTask(B,A)
	parallel with task B	SymmetricObjectProperty
	Task A must be executed in	hasChoiceTask(B,A)
	choice with task b	SymmetricObjectProperty

Table 6 . Type of detected errors

 6

	Test case	BPO	BRO
	1	T0 depend Ti with 1 ≤ i ≤ n	T0 seqExc T1 and Ti depends T1 with 2 ≤ i ≤ n
	2	T0 depend Ti with 1 ≤ i ≤ n	T0 seqExc T1 and Ti parallel T1 with 2 ≤ i ≤ n
	3	T0 parallel Ti with 1 ≤ i ≤ n	T0 paraExc T1 and Ti parallel T1 with 2 ≤ i ≤ n
	4	T0 parallel Ti with 1 ≤ i ≤ n	T0 choiceExc T1 and Ti parallel T1 with 2 ≤ i ≤ n
	5	T1 paralExc Ti with 1 ≤ i ≤ n	T1 parallel Ti with 2 ≤ i ≤ n
	6	T1 sequenExc Ti with 1 ≤ i ≤ n	T1 depends Ti with 2 ≤ i ≤ n

Table 7 . Evaluation BPO + BRO Ontology Inconsistency Consistency

 7

	20 concepts	1s	2s
	15 properties
	30 reaction rules (7
	SWRL)
	30 tasks
	20 concepts	1s	1,2 minute
	15 properties
	80 reaction rules (7
	SWRL)
	80 tasks
	20 concepts	2s	48 minutes
	15 properties
	4OO reaction rules (7
	SWRL)
	400 tasks
	In

table 7 ,

 7 in order to verify the consistency of business process model and there are no potential error, it mays take long times and depend on the size of BPO and BRO.


Thi-Hoa-Hue Nguyen

Nhan Le-Thanh
email: nhan.le-thanh@unice.fr

Ensuring the Correctness of Business Workflows at the Syntactic Level: An Ontological Approach

Keywords: Business Workflow, Correctness, OWL DL, SPARQL, Verification

High quality business workflow definitions play an important role in the organization. An incorrectly defined workflow may lead to unexpected results. Therefore, each business workflow definition should be carefully analyzed before it is put into use. In this paper, we introduce an ontological approach which is suitable for ensuring the syntactic correctness of business workflows. In details, we first introduce the CPN Ontology that is developed for representing CPNs with OWL DL. Then, we define axioms, which are added to the CPN Ontology to provide automated support for establishing the correctness of business workflows. Finally, by relying on the CORESE semantic engine, SPARQL queries are implemented to detect shortcomings in concrete workflows. To the best of our knowledge, this is a novel approach for representing and verifying business workflows based on ontologies.

Introduction

The current tendency in e-business has resulted in more complex business processes. However, specifying a real-world business process is generally manual and is thus prone to human error. An incorrectly designed workflow may lead to failed workflow processes, execution errors or not meet the requirements of customers, etc. In fact, existing techniques applied to check the correctness of workflows are particularly used in commercial business workflow systems. Most of them assume that a workflow is correct if it complies with the constraints on data and control flow during execution [START_REF] Lu | Automatic workflow verification and generation[END_REF]. Whether the workflow is in conformity with the design requirements is neither specified nor proved. There is thus a great need for developing a thorough and rigorous method that automatically supports workflow designers to ensure workflows being well-formed.

In earlier work [START_REF] Nguyen | An ontology-enabled approach for modelling business processes[END_REF][3], we concentrated on machine-readable knowledge bases that are designed to take advantage of automated deductive reasoning. We proposed an ontological approach to represent Control flow-based Business Workflow Templates (CBWTs) in a knowledge base, which is designed to share and reuse among process-implementing software components. In this study, we extend our previous work in designing well-formed CPNs-based business workflow templates and checking their correctness. The approach is based on Knowledge Engineering, Coloured Petri Nets (CPNs) and Semantic Web technologies which provide semantically rich business process definitions and automated support for CBWTs verification. Our contributions are:

-Presenting a classification of syntactic constraints in modelling business processes and creating their related axioms using Description Logic (DL) in order to support workflow designers; -Showing the SPARQL [4] query language is able to verify workflow templates.

The rest of this paper is structured as follows: In Section 2, we give a short introduction to our CPN Ontology used to represent Coloured Petri Nets (CPNs) with OWL DL. We then present syntactic constraints and create their related axioms added to the CPN Ontology to support designers in establishing wellformed workflow templates in Section 3. In Section 4, we introduce the SPARQL query language used to ensure the correctness of CBWTs at the syntactic level. Related work is given in Section 5. Finally, Section 6 concludes the paper with an outlook on the future research.

Modelling Business Processes with Coloured Petri

Nets -The CPN Ontology

Coloured Petri Nets (CPNs) [START_REF] Kristensen | The practitioner's guide to coloured petri nets[END_REF] have been developed into a full-fledged language for the design, specification, simulation, validation and implementation of large-scale software systems. CPN is a well-proven language which is suitable for modelling workflows or work processes [START_REF] Jørgensen | From task descriptions via colored petri nets towards an implementation of a new electronic patient record workflow system[END_REF]. Therefore, CPNs are chosen as the workflow language in our work to transform a business process into a control flow-based business workflow template. Although CPNs have been widely studied and successfully applied in modelling workflows and workflow systems, the lack of semantic representation of CPN components can make business processes modelled with CPNs (i.e., business workflows) difficult to interoperate, share and reuse. Besides, an ontology with its components, which provides machine-readable definitions of concepts, can play a pivotal role in representing semantically rich workflow definitions. Once workflow definitions are stored as semantically enriched workflow templates, IT experts can easily develop their appropriate software systems from the workflow templates.

In this section, we introduce the CPN Ontology defined for business processes modelled with CPNs, which is first proposed in [START_REF] Nguyen | An ontology-enabled approach for modelling business processes[END_REF]. The main purpose is to facilitate business process models to be easily shared and reused.

Our CPN ontology is developed to represent Coloured Petri Nets with OWL Description Logic (OWL DL). Each element of CPNs is translated concisely into a corresponding OWL concept. Figure 1 depicts the core concepts of our CPN ontology. The ontology is described based on DL syntax and the axioms supported by OWL. The CPN Ontology comprises the concepts: CPNOnt defined for all possible CPN-based process models; Place defined for all places; Transition defined for all transitions; InputArc defined for all directed arcs from places to transitions; OutputArc defined for all directed arcs from transitions to places; Token defined for all tokens inside places (We consider the case of one place containing no more than one token at one time); GuardFunction defined for all transition expressions; CtrlNode defined for occurrence condition in control nodes; ActNode defined for occurrence activity in activity nodes, Delete and Insert defined for all expressions in input arcs and output arcs, respectively; Attribute defined for all attributes of individuals); Value defined for all subsets of I 1 ×I 2 ×. . .×I n where I i is a set of individuals.

Properties between the concepts in the CPN Ontology are also indicated in Figure 1. For example, a class Transition has two properties connectsPlace and hasGuardFunction. Consequently, the concept Transition can be glossed as 'The class Transition is defined as the intersection of: (i) any class having at least one property connectsPlace whose value is equal to the class Place and; (ii) any class having one property hasGuardFunction whose value is restricted to the class GuardFunction'.

Taxonomy of Constraints in Modelling Business Processes

To provide automated support for workflow designers in establishing the correctness of ontology-based workflow representations, in this Section we introduce a set of syntactic constraints. The constraints are categorized into two groups. Axioms related to the constraints are also defined using a DL as SHOIN (D) to complete the CPN Ontology.

As mentioned earlier, we aim at representing the correct CBWTs in a knowledge base. Therefore, at first, we define the soundness property that is used as the criterion to check the correctness of workflow processes at the syntactic level.

Definition 1 (Sound). A CPN-based process model

P M is sound iff: (i) P M is connected. (ii) P M is well-formed. (iii)

Syntactic Constraints Related to Uses of Control Nodes

A poorly designed workflow due to improper uses of control nodes can result in deadlock, infinite cycle or missing synchronization. However, these errors can be detected when designing a workflow template and therefore, we can get rid of them. To do that, we next introduce Constraint 12 and the symptoms related to deadlock, infinite cycle or missing synchronization. Constraint 12. There is no deadlock, no infinite cycle and no missing synchronization.

-Deadlock: A deadlock is a situation in which a process instance falls into a stalemate such that no more activity can be enabled to execute. Figure 2 shows three simple deadlock simulations. We next create the axioms related to the control nodes (one of two types of transitions), including And -split, And -join, Xor -split and Xor -join, used to detect deadlock, infinite cycle or missing synchronization.

-And-split is connected to at least two output places. Every output place contains one token. We create the axiom corresponding to And-split as follows: AndSplit T ransition connectsP lace.hasM arking.T oken connectsT rans -.hasM arking.T oken hasGuardF unction.hasControl. CtrlN ode = 1 connectsT rans -.P lace ≥ 2 connectsP lace.P lace -And-join: There are at least two input places connected to And-join. In order to activate And-join, every input place has to contain one token. We create the axiom corresponding to And-join as follows: AndJoin T ransition connectsP lace.hasM arking.P lace connectsT rans -.hasM arking.T oken hasGuardF unction.hasControl. CtrlN ode ≥ 2connectsT rans -.P lace = 1 connectsP lace.P lace -Xor-split is connected to at least two output places. Unlike And-split, at any time, one and only one output place of Xor-join can contain a token. We create the axiom corresponding to Xor-split as follows: XorSplit T ransition ¬AndSplit hasGuardF unction.hasControl. CtrlN ode = 1 connectsT rans -.P lace ≥ 2 connectsP lace.P lace connectsT rans -.hasM arking.T oken -Xor-join: There are at least two input places connected to Xor-join. Unlike

And-join, Xor-join is activated if one and only one input place contains a token. We create the axiom corresponding to Xor-join as follows: XorJoin T ransition ¬AndJoin connectsP lace.hasM arking.T oken ≥ 2 connectsT rans -.P lace. hasGuardF unction.hasControl.CtrlN ode = 1 connectsP lace.P lace

A Wrong Workflow Example

An example of a wrongly designed business process modelled with CPNs is illustrated in Figure 4. The air ticket agent first requires a customer to provide some information related to the flights that he or she wants to book, including name(s), depart, destination, date and class. It then looks for the requested ticket(s) on its partner websites. For simplicity, we assume that two websites are utilized. The obtained results, which may consist of no results, some results or time out, are then evaluated in order to make a decision.

As shown in Figure 4, the example model contains syntactic errors. There are three end points, i.e., T ime out, End 2 and End 3. Besides, the combination of a Xor-split (the transition t2 -P repare to look f or a f light) and an Andjoin (the transition t5 -Collect results) causes a deadlock. Assuming that the place Request verif ied contains a token that makes the transition t2 to be enabled. If the transition Xor-split t2 fires, it consumes the token from its input place Request verif ied and then produces one token for only one of its output places. Consequently, either t3 or t4 may be activated. Since only one of the two transitions t3 and t3 can fire, not all input places of the transition And-join t5 can get its token. As a result, a deadlock occurs because the transition t6 will never be enabled to fire.

We have introduced the CPN ontology represented in OWL DL and axioms which are defined to support designers in verifying CPNs-based process models. It is necessary to note that, to develop or modify CBWTs (i.e., CPN models), manipulation operations [START_REF] Nguyen | An ontology-enabled approach for modelling business processes[END_REF], such as inserting new elements, deleting existing elements, etc., on business process models are required. Therefore, at design time, workflow templates stored in RDF format need to be verified before they are put into use. In the next Section, we present the SPARQL query language used to detect shortcomings in workflow templates represented in RDF syntax. We rely on the CORESE [START_REF] Corby | Corese/kgram[END_REF] semantic search engine for answering SPAQRL queries asked against an RDF knowledge base. The SPAQRL query language is chosen in our work because: (i) It is an RDF query language; (ii) It is a W3C Recommendation and is widely accepted in the Semantic Web and also AT community; (iii) Its syntax is quite simple which allows for a query to include triple patterns, conjunctions, disjunctions and optional patterns; and(iv) It can be used with any modelling language.

In order to verify a workflow template, SPARQL verification queries are created based on the syntactic constraints. Two query forms are used in our work, including ASK and SELECT. According to [4], SELECT query is used to extract values, which are all, or a subset of the variables bound in a query pattern match, from a SPARQL endpoint. The variables that contain the return values are listed after a SELECT keyword. In the WHERE clause, one or more graph patterns can be specified to describe the desired result. ASK query is used to return a boolean indicating whether a query pattern matches or not.

The following query3 , for example, is used to check whether there exist errors related to improper uses of control nodes or not. This query is used to detect if there are any deadlocks caused by the combination of pairs of control nodes, Xor-split and And-join.

SELECT distinct ?xorsplit ?andjoin WHERE { ?xorsplit rdf:type h:Xor-split ?andjoin rdf:type h:And-join ?t1 h:hasGuardFunction/h:hasActivity _:b1 ?t2 h:hasGuardFunction/h:hasActivity _:b2 ?xorsplit h:connectsPlace/h:connectsTrans ?t1 ?xorsplit h:connectsPlace/h:connectsTrans ?t2 ?t1 h:connectsPlace/h:connectsTrans ?andjoin ?t2 h:connectsPlace/h:connectsTrans ?andjoin FILTER(?t1!=?t2) } As a result of the execution of each SPARQL query created based on the syntactic constraints, we obtain an XML file which results in nodes consisting of required information (e.g., the name) and causes shortcomings. For example, Figure 5 shows the result of the execution of the above query applied to check whether the workflow, depicted in Figure 4, contains deadlocks or not. The sample query does not only demonstrate that the SPARQL query language is able to check the syntactic correctness of workflow processes, but also the usage of terminological background knowledge provided by the CPN Ontology, such as Xor-split and hasGuardFunction.

Related Work

Today, software systems that automate business processes have become more and more advanced. Various researchers have paid attention to the problem of ensuring the correctness of process models. They mainly focused on checking the compliance of models concerning aspects of the syntax and formal semantics. To process modelling, there exist some formal criteria, such as "soundness", "completeness", " well-structureness". These criteria are used to examine anomalies, e.g., deadlock, livelock, missing synchronization and dangling references. There are some methods have been proposed to verify workflow models, such as Petri Nets-based [START_REF] Van Der Aalst | Verification of workflow nets[END_REF], [START_REF] Van Der Aalst | The application of petri nets to workflow management[END_REF], [START_REF] Verbeek | Diagnosing workflow processes using woflan[END_REF], logic-based [START_REF] Bi | Applying propositional logic to workflow verification[END_REF], [START_REF] Wainer | Logic representation of processes in work activity coordination[END_REF], graph reduction-based [START_REF] Sadiq | Analyzing process models using graph reduction techniques[END_REF] methods. However, most of them check the conformance of a workflow process based on the principle that if the constraints on data and control flow are met during execution, the workflow is correct.

We know that the ontology-based approach for modelling business process is not a new idea. There are some works made efforts to build business workflow ontologies, such as [START_REF] Gasevic | Interoperable petri net models via ontology[END_REF], [START_REF] Hepp | An ontology framework for semantic business process management[END_REF], [START_REF] Koschmider | Ontology based business process description[END_REF], [START_REF] Salimifard | Petri net-based modelling of workflow systems: An overview[END_REF], [START_REF] Sebastian | Customizable workflow support for collaborative ontology development[END_REF] to support (semi-)automatic system collaboration, provide machine-readable definitions of concepts and interpretable format. However, the issue that relates to a classification of constraints and workflow verification at the syntactic level is not mentioned.

By extending the state-of-the-art, we use the Web Ontology Language to define the CPN Ontology for representing CPNs-based process models. Our ontological approach enables the formulation of constraints added to the CPN Ontology to ensure the soundness of workflow patterns. The constraints are then applied to concrete CBWPs using an RDF engine in order to automatically verify workflow processes at the syntactic level. To the best of our knowledge, this is a novel approach for representing business process definitions along with ensuring the syntactic correctness of workflow processes based upon ontologies.

Conclusion

In this paper, we present an ontological approach to support designers in verifying workflow processes. We first introduce the CPN Ontology, a representation of Coloured Petri Nets and OWL DL, which is defined to take advantage of powerful reasoning systems. Then, we describe two groups of syntactic constraints that ensure the soundness of well-formed business workflow templates. We concentrate on defining axioms corresponding to the constraints and introduce some axioms involving the use of control nodes.

To check the soundness of concrete CBWTs represented in RDF syntax, we specify the syntactic errors and errors related to improper uses of control nodes as SPARQL queries. By relying on the CORESE semantic engine, we show that the SPARQL query language is usable to workflow verification.

We know that verifying workflow templates at build-time is not enough to ensure a workflow template can be executed correctly. The ability to check the correctness of workflow execution is also needed. In future work, we plan to develop a run-time environment for validating CBWTs.

Fig. 1 :

 1 Fig. 1: CPN ontology expressed in a description logic

Fig. 2 :

 2 Fig. 2: Deadlock simulations

Fig. 3 :

 3 Fig. 3: Infinite cycle simulation

Fig. 4 :

 4 Fig. 4: A wrongly designed workflow model for the airline booking process

  For every state M j reachable from state Start M 0 , there also exists another firing sequence starting from state M j to state End M e . (iv) State End M e is the only state which is reachable from state Start M 0 with one token in place e. (v) There is no deadlock, no infinite cycle and no missing synchronization in P M . Constraint 2. There is one and only one start point in a process model. We create the axiom corresponding to Constraint 2 as follows: CP N Ont ¬(= 1 hasP lace.(connectsT rans.hasGuardF unction.hasActivity. There is one and only one end point in a process model. We create the axiom corresponding to Constraint 3 as follows: CP N Ont ¬(= 1 hasP lace.(connectsP lace -.hasGuardF unction.hasActivity. ActN ode ¬(∃ connectsT rans.hasT rans -.CP N Ont))) ⊥ Constraint 4. A place has no more than one leaving arc. If a place is connected to a transition, there exists only one directed arc from the place to the transition. Constraint 5. A place has no more than one entering arc. If a transition is connected to a place, there exists only one directed arc from the transition to the place. We create the axioms corresponding to Constraint 5 as follows: P lace ¬(≤ 1 connectsP lace -.(= 1hasT rans -.OutputArc)) ⊥ Constraint 6. There are no pairs of activity nodes connected via a place. We create the axiom corresponding to Constraint 6 as follows: P lace ∃connectsT rans.hasGuardF unction.hasActivity.ActN ode ∃connectsP lace -.hasGuardF unction.hasActivity.ActN ode ⊥ Constraint 7. There are no pairs of control nodes connected via a place. We create the axiom corresponding to Constraint 7 as follows: P lace ∃connectsT rans.hasGuardF unction.hasControl.CtrlN ode ∃connectsP lace -.hasGuardF unction.hasControl.CtrlN ode ⊥ -Constraints related to transitions. Constraint 8. A transition is on the path from the start point to the end point of a process model. -If a transition has no input place, it will never be enabled. -If a transition has no output place, it will not lead to the end. Consequently, each transition in a workflow must have at least one entering arc and at least one leaving arc. We create the axiom corresponding to Constraint 8 as follows: T ransition ≥ 1 connectsP lace.P lace ≥ 1 connectsT rans -.P lace Constraint 9. An activity node has only one entering arc and one leaving arc. We create the axiom corresponding to the Constraint 9 as follows: hasGuardF unction.hasActivity.ActN ode = 1 connectsP lace.P lace = 1 connectsT rans -.P lace Constraint 10. A control node does not have both multi-leaving arcs and multi-entering arcs. We create the axiom corresponding to the Constraint 10 as follows: ≥ 2 connectsP lace.P lace ≥ 2 connectsT rans -.P lace hasGuardF unction.hasControl.CtrlN ode ⊥ -Constraints related to directed arcs. Constraint 11. Directed arcs connect places to transitions or vice versa. We create the axioms corresponding to the Constraint 11 as follows: hasP lace

	ActN oce ¬(∃ connectsP lace We create the axiom corresponding to Constraint 4 as follows:
	P lace	¬(≤ 1 hasP lace -.InputArc) ⊥

3.1 Syntactic Constraints related to the Definition of Process Model

-Constraints related to places. Constraint 1. For every place p ∈ P , p connects and/or is connected with transitions via arcs.

We create the axiom corresponding to Constraint 1 as follows: hasP lace -.CP N Ont ¬(∃connectsT rans.hasT rans -.CP N Ont ∃connectsP lace -.hasT rans -.CP N Ont) ⊥ -.hasT rans -.CP N Ont))) ⊥ Constraint 3. -.InputArc ≡ connectsT rans.hasT rans -.CP N Ont hasT rans -.OutputArc ≡ connectsP lace.hasP lace -.CP N Ont

The prefix is assumed as: P REF IXh :< http : //www.semanticweb.org/CP N W F # >


domain: math nlin

text:
H Lochon

F Daude

P Galon

J.-M Hérard

Computation of fast depressurization of water using a two-fluid model: revisiting Bilicki modelling of mass transfer

This paper is devoted to the computation of the fast depressurization of water using a twofluid model. Such application, which is extensively studied in the nuclear field, involves many interactions between two phenomena, the mass transfer and the propagation of pressure waves. A simple but physically-based modelling of the mass transfer for the depressurization of water is proposed, which relies on the work of Bilicki & Kestin [4] in the homogeneous frame. Four different experiments have been chosen to assess the proposed model. Three of them study the depressurisation of hot water in a pressurized pipe. The comparison between converged numerical results and the experimental data shows a good agreement and demonstrates the ability of the twofluid-model to capture the proper mass transfer for a wide range of thermodynamical conditions. The last test-case is the HDR experiment which considers the depressurization of a full-scale vessel under the hypothesis of a Loss Of Coolant Accident. The results of an ALE computation show the ability of the proposed model to retrieve experimental data in both structure and fluid.

Introduction

Compressible two-phase flows are involved in many industrial applications such as naval engineering, petroleum industry, nuclear power generation, automative or aerospace technologies. In the nuclear field, two-phase flows are mainly steam-water flows in Pressurized Water Reactors (PWR). Since the 1960s, many two-phase flow models have been proposed for nuclear thermal hydraulics. Most of the modelling efforts have been made to predict the Loss-Of-Coolant-Accident (LOCA) which is one of the basic failure mode of PWR. In such transient steam-water flows, two main physical phenomena are involved and highly interact with each other, the propagation of pressure waves and the mass transfer. The interactions with the structure surrounding the fluid should also be taken into account when the amplitudes of pressure waves are strong.

Two different approaches are usely distinguished to model two-phase flows: the homogeneous approach and the two-fluid approach. Models based on the first approach consider the mixture of the two phases as one fluid assuming different equilibria between both phases. In the second approach, no phasic equilibrium is assumed and two different fluids are considered. As a consequence, many transfers between the two phases, including the mass transfer, require proper modelling. Ransom and Hicks [START_REF] Ransom | Hyperbolic two-pressure models for two-phase flow[END_REF] proposed one of the first models based on this second approach. Since then, the two-fluid approach has gained interest and many models can be found in the literature, see [START_REF] Baer | A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials[END_REF][START_REF] Kapila | Two-phase modeling of DDT: Structure of the velocity-relaxation zone[END_REF][START_REF] Glimm | Two-phase modelling of a fluid mixing layer[END_REF][START_REF] Saurel | A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows[END_REF][START_REF] Gavrilyuk | Mathematical and Numerical Modeling of Two-Phase Compressible Flows with Micro-Inertia[END_REF][START_REF] Gallouët | Numerical Modeling Of Two-Phase Flows Using The Two-Fluid Two-Pressure Approach[END_REF] among others.

The present work is dedicated to the modelling of the fast depressurization of water following the two-fluid approach. Thus, the governing equations of the model are obtained using a classical statistical averaging of the local conservation laws [START_REF] Ishii | Thermo-fluid dynamic theory of two-phase flows[END_REF][START_REF] Drew | Theory of Multicomponent Fluids[END_REF]. The corresponding model is hyperbolic but cannot be written in a fully conservative form. It also requires closure laws for interfacial quantities involved in the non-conservative terms and for the source terms. Based on the comparison of different two-fluid models on steam-water transients proposed in [START_REF] Lochon | Comparison of two-fluid models on steam-water transients[END_REF], the closure laws of the Baer-Nunziato model [START_REF] Baer | A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials[END_REF] have been chosen for the interfacial pressure and velocity. As in previous work [START_REF] Hérard | A fractional step method to compute a class of compressible gasliquid flows[END_REF][START_REF] Crouzet | Validation of a two-fluid model on unsteady liquid-vapor water flows[END_REF], the modelling of the source terms is based on relaxation phenomena to comply with the entropy inequality. Thus, several relaxation time scales are involved in the transfers between phases and require proper modelling. Evaluations of the pressure and velocity relaxation time scales can be found in [START_REF] Lochon | Comparison of two-fluid models on steam-water transients[END_REF] whereas the chemical potential relaxation, which corresponds to the mass transfer, requires further work. In order to propose simple but physically-based evaluation of this time scale, we focus on the work of Bilicki and co-authors [START_REF] Bilicki | Physical aspects of the relaxation model in two-phase flow[END_REF][START_REF] Bilicki | Evaluation of the relaxation time of heat and mass exchange in the liquid-vapour bubble flow[END_REF][START_REF] Downar-Zapolski | The non-equilibrium relaxation model for one-dimensional flashing liquid flow[END_REF] on the Homogeneous Relaxation Model. Based on the same approach, a simple evaluation of the temperature relaxation time scale is also proposed. As in [START_REF] Crouzet | Validation of a two-fluid model on unsteady liquid-vapor water flows[END_REF], particular attention is paid to the validation of the proposed model. Four different experiments studying the fast depresussurisazion of water have been chosen to that end.

Hence the paper is organized as follows. We first present the governing equations of the two-fluid model with the complete sets of closure laws including the non-conservative terms and the source terms. The main mathematical properties of the model are briefly recalled. Then, the evaluation of the four relaxation time scales involved in the source terms are discussed. The modelling of the mass transfer in the current model is compared, under some hypothesis, to the one proposed by Bilicki et al. [START_REF] Bilicki | Physical aspects of the relaxation model in two-phase flow[END_REF] in the Homogeneous Relaxation Model. Thus, an evaluation of the chemical potential relaxation time scale is obtained. The second section of the paper is dedicated to the validation of the proposed model with the Bilicki-like mass transfer. The numerical methods used for the computation of the two-fluid model are briefly presented with associated references. Four experiments have been chosen to assess the proposed model. Three of them study the fast depressurization of hot water from a pressurized pipe: the Canon, Super-Canon and Edwards pipe experiments. 1D converged numerical results of the two-fluid model with and without the Bilicki-like mass transfer model are compared to the experimental data. The comparison shows that the proposed Bilicki-like mass transfer significantly improves the results of the two-fluid model for a wide range of thermodynamical states. The last test-case is the HDR experiment which studies the depressurization of a full-scale vessel under the hypothesis of a LOCA. Both pressures in the fluid and diplacements of internal structures are measured in the vessel during the experiment allowing to focus on the mechanical consequences of a LOCA. A 3D Arbitrary Lagrangian-Eulerian computation taking into account the Fluid-Structure Interactions is carried out and numerical results show a fair agreement with experimental measurements in both structure and fluid.

The two-fluid model

The derivation of two-fluid models is based on classical statistical averaging of local conservation laws [START_REF] Ishii | Thermo-fluid dynamic theory of two-phase flows[END_REF][START_REF] Drew | Theory of Multicomponent Fluids[END_REF]. Relevant closure laws are required in the construction and the entropy inequality is a major tool to derive them [START_REF] Gavrilyuk | Mathematical and Numerical Modeling of Two-Phase Compressible Flows with Micro-Inertia[END_REF][START_REF] Gallouët | Numerical Modeling Of Two-Phase Flows Using The Two-Fluid Two-Pressure Approach[END_REF][START_REF] Jin | Entropy of averaging for compressible two-pressure two-phase flow models[END_REF][START_REF] Müller | Closure conditions for non-equilibrium multi-component models[END_REF].

Governing equations

The two-fluid model considered here that does not assume any equilibrium between the two phases. Throughout the paper, the subscript k refers to the phase (k = v for the vapor and k = l for the liquid). α k ∈ ]0, 1[ is the statistical fraction of phase k and complies with the following relation:

α v + α l = 1
We denote ρ k the mean density of phase k, u k its mean velocity, p k its mean pressure and e k its mean specific total energy complying with:

e k = ε k + 1 2 u 2 k ( 1
)
ε k is the mean specific internal energy of phase k which is linked to the mean density and pressure of phase k by an Equation Of State (EOS):

ε k = ε k (ρ k , p k ) (2)
Based on the EOS, the celerity c k of acoustic waves as well as the specific entropy s k of phase k can be defined. They comply with the following relations:

ρ k c 2 k = (∂ p k ε k ) -1 p k ρ k -ρ k (∂ ρ k ε k ) (3) c 2 k (∂ p k s k ) + (∂ ρ k s k ) = 0 (4)
Thus, the governing equations of the two-fluid model read:

       ∂ t (α k ) + u I ∂ x (α k ) = S 1,k ∂ t (α k ρ k ) + ∂ x (α k ρ k u k ) = S 2,k ∂ t (α k ρ k u k ) + ∂ x α k ρ k u 2 k + α k p k -p I ∂ x (α k ) = S 3,k ∂ t (α k ρ k e k ) + ∂ x (α k ρ k e k u k + α k p k u k ) -p I u I ∂ x (α k ) = S 4,k (5)
Many closure laws for the interfacial velocity and pressure (u I , p I ) have been proposed in the literature [START_REF] Baer | A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials[END_REF][START_REF] Coquel | Closure laws for a two-fluid two-pressure model[END_REF][START_REF] Glimm | Renormalization group solution of two-phase flow equations for Rayleigh-Taylor mixing[END_REF][START_REF] Saurel | A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows[END_REF] and some of them have been compared for steam-water transients in [START_REF] Lochon | Comparison of two-fluid models on steam-water transients[END_REF]. In the sequel, we consider the closure laws proposed by Baer & Nunziato [START_REF] Baer | A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials[END_REF], given by the following:

(u I , p I ) = (u l , p v ) (6)
The closure laws for the source terms S j,k (j = 1, 4) on the right-hand side are the following:

S 1,k = Φ k S 2,k = Γ k S 3,k = D k + UΓ k S 4,k = Q k + UD k + HΓ k -p I Φ k (7)
where U = 1 2 (u v + u l ) and H = 1 2 (u v u l ). The different contributions are written as relaxation phenomena:

Φ k = 1 τ p p ref α k α j (p k -p j ) Γ k = 1 τ µ µ ref m k m j m k + m j (µ j -µ k ) D k = 1 τ u m k m j m k + m j (u j -u k ) Q k = 1 τ T m k C V k m j C V j m k C V k + m j C V j (T j -T k ) , j = k (8)
where

C V k is the heat capacity at constant volume of phase k, m k = α k ρ k and µ k = g k T k
. T k is the temperature of phase k and g k its specific Gibbs enthalpy:

1 T k = (∂ p k ε k ) -1 (∂ p k s k ) (9)
g k = ε k + p k ρ k -T k s k (10)
Each relaxation phenomenum involves a relaxation time scale τ ϕ which drives the quantity ϕ (ϕ = p, µ, u, T ) towards the phasic equilibrium. Finally p ref and µ ref are respectively a reference pressure and a reference potential.

Main properties of the model

We may now recall the main properties of the two-fluid model ( 5) with the Baer-Nunziato closure law [START_REF] Coquel | Closure laws for a two-fluid two-pressure model[END_REF]. The first property focuses on the entropy of the global model whereas the other properties deal with the convective part of the model which was first studied by Embid & Baer [START_REF] Embid | Mathematical analysis of a two-phase continuum mixture theory[END_REF].

• Entropy inequality Smooth solutions of system (5) comply with an entropy inequality:

∂ t k m k s k + ∂ x k m k s k u k = (µ l -µ v ) Γ v + 1 T l (p v -p l ) Φ v + 1 2T v + 1 2T l (u l -u v ) D v + 1 T v - 1 T l Q v ≥ 0 (11)
• Hyperbolicity and structure of waves The convective part of system (5):

       ∂ t (α k ) + u I ∂ x (α k ) = 0 ∂ t (α k ρ k ) + ∂ x (α k ρ k u k ) = 0 ∂ t (α k ρ k u k ) + ∂ x α k ρ k u 2 k + α k p k -p I ∂ x (α k ) = 0 ∂ t (α k ρ k e k ) + ∂ x (α k ρ k e k u k + α k p k u k ) -p I u I ∂ x (α k ) = 0 (12)
is hyperbolic. It admits six real eigenvalues:

λ 1 = u v -c v , λ 2 = u v , λ 3 = u v + c v , λ 4 = u l -c l , λ 5,6 = u l , λ 7 = u l + c l ( 13
)
and associated righteigenvectors span the whole space R 7 if |u l -u v | = c v . Fields associated with eigenvalues λ 2,5,6 are linearly degenerate (LD) whereas fields associated with eigenvalues λ 1,3,4,7 are genuinely non linear (GNL).

• Jump conditions Unique jump conditions hold within each isolated field. Moreover, classical single phase jump relations hold in the GNL fields:

       [α k ] = 0 -σ [ρ k ] + [ρ k u k ] = 0 -σ [ρ k u k ] + ρ k u 2 k + p k = 0 -σ [ρ k e k ] + [ρ k e k u k + p k u k ] = 0 (14)
where [ϕ] = ϕ R -ϕ L is the jump between the Left and Right states on each side of a shock wave traveling at speed σ.

The three properties are crucial from both theoretical and numerical points of view. We underline the fact that the 5,6-field is LD ; thus the non-conservative products of the model are properly defined and not active in GNL fields. More details on those properties could be found in [START_REF] Coquel | Closure laws for a two-fluid two-pressure model[END_REF][START_REF] Gallouët | Numerical Modeling Of Two-Phase Flows Using The Two-Fluid Two-Pressure Approach[END_REF][START_REF] Guillemaud | Modélisation et simulation numérique des écoulements diphasiques par une approche bifluide à deux pressions[END_REF].

Evaluation of relaxation time scales

This section is dedicated to the evaluation of the four relaxation time scales τ ϕ (ϕ = p, µ, u, T ) in the case of the fast depressurization of water. Such an evaluation is difficult to provide and most available work in the literature deals with instantaneous relaxation [START_REF] Métayer | Dynamic relaxation processes in compressible multiphase flows. Application to evaporation phenomena[END_REF][START_REF] Pelanti | A mixture-energy-consistent six-equation two-phase numerical model for fluids with interfaces, cavitation and evaporation waves[END_REF][START_REF] Daude | Numerical experiments using a HLLC-type scheme with ALE formulation for compressible two-phase flows five-equation models with phase transition[END_REF]. Since the underlying physical phenomena are far from being instantaneous in many applications, we only consider non-instantaneous relaxations in the present work. For pressure and velocity relaxations, we consider the time scales used in [START_REF] Lochon | Comparison of two-fluid models on steam-water transients[END_REF] for steam-water transients. The chemical potential relaxation corresponds to the mass transfer Γ k and a new evaluation of its time scale is proposed based on the work of Bilicki et al. [START_REF] Bilicki | Physical aspects of the relaxation model in two-phase flow[END_REF][START_REF] Downar-Zapolski | The non-equilibrium relaxation model for one-dimensional flashing liquid flow[END_REF]. Finally, we propose a simple evaluation of the temperature relaxation time scale.

Pressure relaxation

Based on [START_REF] Gavrilyuk | The structure of pressure relaxation terms : one-velocity case[END_REF], the pressure relaxation time scale is evaluated using the Rayleigh-Plesset equation, which considers the evolution of a bubble in an infinite medium. It provides the following evaluation:

τ p p ref =                4 3 η l if α v < α min 4 3 η v if α v > α max α max -α v α max -α min 4 3 η l + α v -α min α max -α min 4 3 η v otherwise ( 15
)
where η k is the dynamic viscosity of phase k.

Velocity relaxation

The behavior of a bubble in an infinite medium is also considered for the velocity relaxation. Thus the evaluation of the corresponding time scale is based on the drag equation. More precisely, we consider the following model proposed in the NEPTUNE-CFD code [START_REF] Lavieville | NEPTUNE CFD V1.0 -Theory Manual[END_REF] for liquid-gas separated flows:

D k = α k α j F D (α v ) (u j -u k ) , j = k ( 16
)
where the function F D of α v is defined as follows:

F D (α v ) =                F D,v (1 -α v ) if α v < α min F D,l (α v ) if α v > α max α max -α v α max -α min F D,v (1 -α min ) + α v -α min α max -α min F D,l (α max ) otherwise
The function F D is equal to the function F D,p (α q ) in the case of isolated spherical inclusions of phase p in the continuous phase q. The following classic correlations are used [START_REF]RELAP5-3D Code Manual Volume IV: Models and Correlations[END_REF][START_REF] Lavieville | NEPTUNE CFD V1.0 -Theory Manual[END_REF]:

F D,p (α q ) = 3 4 ρ q α q C D,p d p |u p -u q | , q = p with: C D,p = 24 Re p 1 + 0.15 (Re p ) 0.687 and Re p = ρ q d p η q |u p -u q |
where d p is a characteristic diameter of the inclusion. It is defined using the critical Weber number W e = 10 (see [START_REF]RELAP5-3D Code Manual Volume IV: Models and Correlations[END_REF]) and the surface tension σ of the liquid phase:

W e = ρ q d p σ |u p -u q |
Thus, the evaluation of the velocity relaxation time scale is obtained using ( 8) and ( 16):

τ u = ρ v ρ v m v + m l 1 F D (α v ) (17)

Chemical potential relaxation

The chemical potential relaxation corresponds to the mass transfer Γ k . Therefore we focus on the modelling of the mass transfer used by Bilicki et al. [START_REF] Bilicki | Physical aspects of the relaxation model in two-phase flow[END_REF][START_REF] Downar-Zapolski | The non-equilibrium relaxation model for one-dimensional flashing liquid flow[END_REF] for the depressurization of water in order to propose an evaluation of the chemical potential relaxation time scale. The Homogeneous Relaxation Model (HRM) of Bilicki & Kestin [START_REF] Bilicki | Physical aspects of the relaxation model in two-phase flow[END_REF] is detailed in 4. Downar-Zapolski et al. [START_REF] Downar-Zapolski | The non-equilibrium relaxation model for one-dimensional flashing liquid flow[END_REF] proposed a correlation for the relaxation time scale involved in this model based on the results of the Moby Dick experiments [START_REF] Reocreux | Contribution à l'étude des débits critiques en écoulement diphasique eau-vapeur[END_REF]. Although this correlation has been determined in the stationary case, it has been used in the transient case with both homogeneous [START_REF] Faucher | Computation of flashing flows in variable cross-section ducts[END_REF] and two-fluid [START_REF] Gale | Two-fluid model of the WAHA code for simulations of water hammer transients[END_REF] models.

In order to propose an evaluation of the time scale τ µ based on the model proposed by Bilicki & Kestin [START_REF] Bilicki | Physical aspects of the relaxation model in two-phase flow[END_REF] in the homogeneous frame, we assume that that both phases are in an homogeneous state (∂ x (ϕ) = 0, ∀ϕ) at rest (u k = 0) with the following thermodynamical equilibrium: p v = p l and T v = T l . Based on those hypothesis, the governing equations of the two-fluid model ( 5) become:

       ∂ t (α k ) = 0 ∂ t (α k ρ k ) = Γ k ∂ t (α k ρ k u k ) = 0 ∂ t (α k ρ k e k ) = 0 (18)
We can recast this system in the following way:

       ∂ t (α k ) = 0 ∂ t (m k ) = Γ k ∂ t (m k u k ) = 0 ∂ t (m k ε k ) = 0 (19)
Thus, the thermodynamical variable µ k , whose desequilibrium drives the mass transfer Γ k , can be written as a function of the partial mass m k only. Indeed:

µ k = µ k (ρ k , ε k ) = µ k m k α 0 k , m 0 k ε 0 k m k
where we use the notation ϕ 0 = ϕ(t = 0). To provide a deeper analysis of the mass transfer, we assume that both phases are governed by the Stiffened Gas EOS:

             ε k = ((γ k -1)ρ k ) -1 (p k + γ k p ∞ k ) + q k T k = (C V k (γ k -1)ρ k ) -1 (p k + p ∞ k ) s k = C V k ln (C V k (γ k -1) ρ k ) -γ k (p k + p ∞ k ) + q k ( 20
)
We also assume that q k = 0. Based on this particular choice of the EOS, the function µ k (m k ) can be expressed in the following way:

µ k (m k ) = C P k + C P k ln (m k ) -S 0 k ( 21
)
where we use the following constants :

C P k = γ k C V k and S 0 k = C V k ln p 0 k + p ∞ k α 0 k γ k (C V k (γ k -1)) γ k + q k
Let us now focus on the difference µ j -µ k . An important remark is that those quantities are equal at the thermodynamical and chemical equilibrium since both temperatures and chemical potentials would be equal: µ j = µ k . Therefore, this difference could be written in the following way:

µ j -µ k = µ j -µ j -µ k -µ k = µ j (m j ) -µ j (m j ) -µ k (m k ) -µ k (m k ) = C P j ln m j m j -C P k ln m k m k
Finally, a last hypothesis is being made, we assume that the two-phase flow is close to the thermodynamical and chemical equilibrium:

m k ≈ m k . Noticing that m k + m j = m k + m j , we get: µ j -µ k = C P j ln m j m j -C P k ln m k m k ≈ C P j m j -m j m j -C P k m k -m k m k ≈ - C P j m k + C P k m j m j m k m k -m k
Thus, we obtain the following approximation of the mass transfer, defined in equation ( 8):

Γ k ≈ - 1 τ µ µ ref C P j m k + C P k m j m k + m j m k -m k , j = k (22)
If we compare it to the closure law proposed by Bilicki and Kestin [START_REF] Bilicki | Physical aspects of the relaxation model in two-phase flow[END_REF] and given in equation ( 32), we have:

τ µ µ ref ≈ C P j m k + C P k m j m k + m j θ (23)
Therefore, we propose the following evaluation of the time scale τ µ and the reference potential µ ref :

τ µ = θ and µ ref =                C P v if α v < α min C P l if α v > α max α max -α v α max -α min C P v + α v -α min α max -α min C P l otherwise ( 24
)
where θ is defined as follows (see [START_REF] Downar-Zapolski | The non-equilibrium relaxation model for one-dimensional flashing liquid flow[END_REF] and the recent study in [START_REF] Lochon | Modélisation et simulation d'écoulements transitoires eau-vapeur en approche bifluide[END_REF]):

θ = 6.51 × 10 -7 α -0.257 v p sat -p p sat -2.24 (25)
using the mean pressure p = α v p v + α l p l . The use of this time scale will be referred to as Bilicki-like mass transfer in the sequel.

Temperature relaxation

In order to propose a simple evaluation of the temperature relaxation time scale, we assume that this phenomenum is mainly driven by the vaporisation. This hypothesis is relevant for steam-water transients such as the depressurization of water since the heat diffusion can be neglected in those fast applications. Thus, we have the following evaluation of the time scale:

τ T = τ µ ( 26
)
3 Numerical results

In order to assess the proposed modelling of the mass transfer, four different experiments have been selected. Three of them (the Canon experiment, the Super-Canon experiment [START_REF] Riegel | Contribution à l'étude de la décompression d'une capacité en régime diphasique[END_REF] and the Edwards pipe experiment [START_REF] Edwards | Studies of phenomena connected with the depressurization of water reactors[END_REF]) study the rapid depressurization of hot liquid water from an horizontal pipe with similar characteristics (a length of about 4 m and a constant inner diameter of about 10 cm). The pipe has a closed end at one side and a membrane on the other side. At time zero, the membrane is broken and a rarefaction wave starts propagating along the pipe. At the final time, water is fully turned into vapor therefore these are excellent test-cases to study the mass transfer. Those experiments are classical test cases for nuclear safety codes in order to study the Loss-Of-Coolant-Accident (LOCA).

The main difference between the three experiments lies in the initial conditions, summarized in table 1. The last experiment, which has been carried out by the HDR (Heißdampfreaktor) Safety Program in Germany [START_REF] Wolf | Experimental results of coupled fluid-structure interactions during blowdown of the HDR-vessel and comparisons with pre-and post-test predictions[END_REF], studies the depressurization of a full-scale reactor vessel under the hypothesis of a LOCA. Many measurements of pressure and displacement in the vessel are provided so that mechanical consequences of the depressurization can be studied unlike the three other test-cases. A fractionnal step method is used for the computation of the two-fluid model with source terms (system (5)). A key property of this method, proposed in [START_REF] Hérard | A fractional step method to compute a class of compressible gasliquid flows[END_REF] without mass transfer, is to be consistent with the entropy inequality of the model (equation ( 11)). It was then extended to the case with mass transfer in [START_REF] Crouzet | Validation of a two-fluid model on unsteady liquid-vapor water flows[END_REF][START_REF] Lochon | Comparison of two-fluid models on steam-water transients[END_REF]. We also use the first-order HLLC-type convective solver recently proposed in [START_REF] Lochon | HLLC-type riemann solver with approximated two-phase contact for the computation of the baer-nunziato two-fluid model[END_REF], with a Courant number C CF L = max(λ k ) ∆t ∆x = 0.5 . The Equation Of State used in the computations is the Stiffened Gas EOS:

             ε k = ((γ k -1)ρ k ) -1 (p k + γ k p ∞ k ) + q k T k = (C V k (γ k -1)ρ k ) -1 (p k + p ∞ k ) s k = C V k ln (C V k (γ k -1) ρ k ) -γ k (p k + p ∞ k ) + q k (27)
Thus 5 thermodynamical constants are required for each phase:

γ k > 1, C V k > 0, p ∞ k > 0,
q k and q k . We also need (p k + p ∞ k ) > 0 in order to guarantee the positivity of some thermodynamical quantities. Finally, we use the estimations of the relaxation time scales τ ϕ (ϕ = p, µ, u, T ) provided in the previous section (equations ( 15)-( 17)-( 24)-( 26)) with α min = 0.2 and α max = 0.8. All the following computations have been performed with the fast transient dynamics software Europlexus [START_REF]Commissariat à l'énergie atomique et aux énergies alternatives (CEA)[END_REF].

Canon experiment

The Canon experiment was carried out by Riegel [START_REF] Riegel | Contribution à l'étude de la décompression d'une capacité en régime diphasique[END_REF] and studied the fast depressurization of hot liquid water (220 • C and 32 bar). Figure 1 presents the experimental set-up, consisting in a 4.389 m long horizontal pipe, with an inner diameter of 102.3 mm. Pressure is measured at different locations along the pipe (from P1 to P5). The void fraction is also measured at location Pt (between P2 and P3 on figure 1) using a neutron scattering method. The parameters of the Stiffened Gas EOS for both phases are chosen to recover the steam-water phase diagram as in [START_REF] Daude | Numerical experiments using a HLLC-type scheme with ALE formulation for compressible two-phase flows five-equation models with phase transition[END_REF] and recalled in table 2. The computational domain is a 1D grid with a closed end at one side, and a tank, where atmospheric conditions are applied, on the other side. The initial conditions for the pipe and the tank are provided in table 3. Before comparing the experimental data and the numerical results obtained with the Bilicki-like mass transfer, we study the influence of the mesh refinements on the numerical results. Four different 1D grids with 10 3 to 10 4 cells are used in this study. The history of the void fraction at Pt obtained with those different grids is provided in figure 2. The gap between the results obtained with the coarsest grid and the results obtained with the finest one appears to be small. Refining the mesh only slightly delays the vaporisation process. We notice that grid independance is obtained from 5000 cells therefore the results obtained on this grid will be considered as converged results. The comparison between the converged numerical results and the experimental data on the void fraction at Pt is shown on figure 3. We also add a reference numerical result without the Bilicki-like mass transfer obtained on the same grid with constant time scales, τ µ = 5 × 10 -5 s and τ T = 10 -7 s, as in a previous work [START_REF] Lochon | Comparison of two-fluid models on steam-water transients[END_REF]. An overall good agreement between the numerical and experimental results can be observed, with a complete vaporisation of the water. Although this vaporisation process happens earlier for both computations, the use of the Bilicki-like mass transfer delays it, which significantly improves the numerical results when compared with the experimental ones. Figure 4 provides the comparison between the measured pressure at P1 and the mean pressure p = α v p v + α l p l obtained with the different computations. The experimental results show a sudden drop from the initial pressure 32 bar to 21 bar. Then the pressure remains constant at a saturation plateau until 400 ms and finally slowly decreases towards the atmospheric pressure. The reference computation that uses constant time scales shows a plateau at 23 bar, the exact saturation pressure of the initial temperature, due to the fact that time scales remain very small. When compared with the experiment, this plateau appears to be slightly too high and too short. The results obtained with the Bilicki-like mass transfer show an important pressure undershoot before a pressure plateau at 20 bar. Unlike the reference computation, this plateau is slightly below the experimental one. We also notice that it is longer thanks to the proposed mass transfer, although the vaporisation process is predicted earlier in both computations. Apart from the undershoot before the plateau, the overall behavior of the pressure given by the two-fluid model with the Bilicki-like mass transfer is really similar to the measured pressure. We underline the fact that it delays the vaporisation and no longer imposes the saturation pressure when compared to the reference computation. Thus the Bilicki-like mass transfer appears to bring a significant improvement on the modelling of the vaporisation process with the two-fluid model ; nonetheless it needs to be assessed on other experimental cases.

p ∞ k q k C V k q k (Pa) (J.kg -1 ) (J.kg -1 .K -1 ) (J.kg -1 .K -1 )

Super-Canon experiment

The Super-Canon experiment was also carried out by Riegel [START_REF] Riegel | Contribution à l'étude de la décompression d'une capacité en régime diphasique[END_REF] with the same set-up as the Canon experiment (figure 1). The main difference between the two experiments lies in the initial conditions of the liquid water in the pipe which are the standard conditions of the primary loop in Pressurized Water Reactors (PWR): 300 • C and 150 bar. Based on the initial temperature, the EOS parameters, gathered in table 4, are determined as before to retrieve the steam-water diagram [START_REF] Daude | Numerical experiments using a HLLC-type scheme with ALE formulation for compressible two-phase flows five-equation models with phase transition[END_REF]. Table 5 provides the initial conditions used in the computations. As in the previous case, a convergence study is carried out and suggests that numerical results obtained on the 5000-cell grid can be considered as converged results. Thus, we compare those numerical results obtained with the Bilicki-like mass transfer to the experimental results and to the results of a reference computation that uses constant time scales (τ µ = 5 × 10 -5 s and τ T = 10 -7 s). Figure 5 shows the comparison between those three results on the history of the void fraction at location Pt. If we first focus on the experimental data, we notice that the vaporisation process is similar to the one observed in the Canon experiment, despite the fact that it appears to be faster. We also observe that the measurements are more scattered in this case, particularly before 100 ms. Later on, it is easier to compare the experimental data with the numerical results. As it occurs in the Canon experiment, both computations show an earlier vaporisation but Bilicki-like mass transfer significantly reduces the gap between the computed results and the experimental ones. Then we compare the measured pressure at location P1 to the mean pressure p = α v p v + α l p l from the computations (figure 6). The overall behavior of the pressure in the Super-Canon experiment is the same as in the Canon experiment: a sudden drop of pressure from the initial pressure to a plateau and a final slow decrease towards the external pressure. It is important to notice that the pressure level of the plateau, i.e. 60 bar, is very different from the saturation pressure of the initial temperature 86 bar unlike in the Canon experiment. Thus, the metastable states of water must be taken into account in the modelling process in order to retrieve the experimental data. When comparing the results of the reference computation to the experimental ones, we observe that the plateau of the computation is higher, around 80 bar, and really shorter. It shows that the use of small constant time scales is not relevant to capture metastable states since the pressure plateau is too close to the saturation pressure of the initial temperature. On the contrary, the computation with the Bilicki-like mass transfer provides a plateau with the same pressure level as in the experiment. The main differences between those two results are the important numerical pressure undershoot before the plateau and a shorter plateau obtained by the computation. Apart from those differences, the overall behavior of the pressure history obtained with the Bilicki-like mass transfer is really close to the experimental history. Thus it confirms the modelling choice of the mass transfer, especially regarding metastable states of water.

p ∞ k q k C V k q k (Pa) (J.kg -1 ) (J.kg -1 .K -1 ) (J.kg -1 .K -1 )

Edwards pipe experiment

We study a third experiment of rapid depressurization of water carried out by Edwards and O'Brien [START_REF] Edwards | Studies of phenomena connected with the depressurization of water reactors[END_REF]. The horizontal pipe used in the experiment is 4.097 m long and its inner diamater is 7.315 cm. One end of the pipe is closed and the other end has a membrane which is broken at the initial time. Unlike the previous experiments, the cross section of the break is 12.5% smaller than the cross section of the pipe. Different sensors are located around the pipe as shown in figure 7. Pressure is measured at both ends of the pipe (GS1 and GS7) and temperature is measured at GS5. A X-ray densitometer is also located at GS5 to measure the void fraction. 6. The initial conditions for the computations are provided in table 7. For simplicity, we consider a full break and use 1D grids. As previously, different mesh refinements are used showing that the results on the 5000-cell grid can be considered as mesh independent. Figure 8 provides the comparison of the experimental and numerical results on the void fraction. We first notice that only a few void fraction measurements are available in this experiment unlike the previous ones. Thus we only compare the overall behavior of the void fraction and we draw the same conclusion as for the Super-Canon experiment. Indeed, we observe that both numerical results have the same behavior, although the Bilicki-like mass transfer delays the vaporisation process which drives the results really close to the experimental data. At the same location, the temperature is also measured in the experiment so we can also compare it to the numerical results. Figure 9 provides the comparison between those measurements and the mean temperature T = mvC V v Tv+m l C V l T l mvC V v +m l C V l , corresponding to the total internal energy, obtained in the computations. We can see that all curves first show a plateau between 500 K and the initial temperature 515 K. This plateau appears to be really short for the reference computation and longer when Bilicki-like mass transfer is used, though not as long as the experimental one. Then we observe a rapid decrease for both computations and the temperature finally remains steady around 380 K whereas a slower regular decrease is observed on the experimental results until the final time. This faster decrease of the temperature on the computational results is linked to a faster vaporisation. Moreover, the final constant temperature observed on the numerical results is due to the fact that no heat diffusion is taken into account in the numerical model. Finally, we compare the experimental pressure history at GS7 with the numerical mean pressure p = α v p v + α l p l (figure 10). This comparison appears to be really similar as the comparison of the pressure history on Super-Canon experiment. On the one hand, the reference computation with constant time scales provides a short pressure plateau at 34 bar, really close to the saturation pressure 34.5 bar at the initial temperature. On the other hand, the experimental results show a longer plateau at 27 bar and the use of Bilicki-like mass transfer leads to a pressure plateau at 28 bar. As observed on the void fraction history, it also delays the vaporisation and we obtain a longer pressure plateau, which is closer to the experimental one. Thus the proposed modelling of the mass transfer provides a significant improvement of the numerical results obtained with the Baer-Nunziato model when compared with the experimental results. Some differences are still observed, particularly a slightly earlier vaporisation and a pressure undershoot in the first milliseconds of the computation.

p ∞ k q k C V k q k (Pa) (J.kg -1 ) (J.kg -1 .K -1 ) (J.kg -1 .K -1 )

HDR experiment

The last test-case considered here is the HDR (Heißdampfreaktor) experiment [START_REF] Wolf | Experimental results of coupled fluid-structure interactions during blowdown of the HDR-vessel and comparisons with pre-and post-test predictions[END_REF] which studies the LOCA on a full-scale reactor vessel unlike the previous test-cases. The experimental setup is presented in figure 11. It consists of a rigid pressure vessel with a single discharge nozzle and a simplified representation of the internals, including a flexible core barrel firmly clamped at its upper flange and a massive ring attached to the core barrel bottom to simulate the core mass. Different tests have been carried out on this vessel and we select the test V32 since it involves the most severe depressurization. In this case, the initial pressure in the vessel is 110 bar and the initial temperature varies from 221.6 • C in the lower plenum to 308 • C in the upper one. A membrane at the end of the nozzle is broken at the initial time, generating a rarefaction wave propagating along the pipe toward the vessel. Pressures in the vessel and core barrel displacements are measured at different locations. Thus the mechanical consequences of the depressurization can be studied in order to assess the two-fluid model. 8. The structures are considered as linear elastic materials whose characteristics are detailed in table 9. Finally, table 10 provides the initial conditions for the computation. The temperature gradient in the vessel is neglected and the initial conditions of the nozzle (110 bar and 240 • C) are applied in the whole vessel. As in the previous test-cases, atmospheric conditions are used in the tank representing the exterior medium. Since the physical time of the experiment is very short (80 ms), mass transfer is only considered in the nozzle therefore only pressure and velocity relaxations are applied in the vessel.

p ∞ k q k C V k q k (Pa)
(J.kg -1 ) (J.kg -1 .K -1 ) (J.kg -1 .K Figure 13 provides the comparison between the experimental measurements and the numerical mean pressure histories p = α v p v + α l p l at different locations in the vessel. For all the selected sensors, the results issuing from the experiment show a decrease of the pressure, from 110 bar to about 90 bar. The pressure decrease is really smooth for sensors far from the nozzle (see BP8302 on the bottom right). For sensors closer to the nozzle, a greater pressure decrease is observed between 10 and 20 ms, then pressure remains constant around 100 bar, or even increases (see BP9136 on the top right), and finally decreases again after 55 ms. If we focus on the numerical results, similar trends are observed, however pressure decreases towards a lower value, slightly under 80 bar. Moreover, for sensors close to the nozzle, we notice that the mean pressure remains constant between 20 and 55 ms around 96 bar ; no increase of pressure is observed. When comparing experimental and numerical results, the main difference lies in the fact that pressure is underestimated in the numerical simulations due to a faster decrease. It results in a relative error of 10% at the end of the computation. This difference may be due to the fact that an instantaneous full break is considered in the numerical simulation resulting in a faster discharge of the vessel in the computation.

The histories of both pressure difference and displacement are provided in figure 14. A similar behavior of the pressure differences can be observed on the different experimental histories. Indeed, it first decreases to reach a minimum value at 25 ms due to the fact that the rarefaction wave is propagating in the downcomer before reaching the internal part of the core barrel. Then, pressure differences increase towards a maximum at 55 ms and finally decrease to reach a slightly negative value at the final time. The numerical results show a very good agreement with those experimental results on pressure difference histories. Similar maximum and minimum values can be observed and occur around the same time. This agreement shows that the behavior of the pressure wave is accurately captured in the computation, despite the underestimation of the absolute pressure mentionned earlier. Unlike pressure measurements, the behavior of the radial displacement of the core barrel really differs from one sensor to another. Due to the pressure difference between the internal part of the core barrel, we first observe that the radial displacement increases to reach a maximum value, then it decreases before increasing again. The value of the maximum, as well as the corresponding time, differs a lot depending on the location of the sensor. When compared with the experimental measurements, the numerical results show a rather good agreement. Some differences can be observed on the second half of the computation for KS1023 (at the top) and displacement is underestimated for KS1030 (in the middle) despite a similar behavior. On the whole, a good agreement is observed between numerical and experimental results during the first half of the computation for both pressure difference and displacement histories. More discrepancies appear during the second half and they may be linked to numerical diffusion since first order schemes are used in the computation. Thus, this last test-case shows that the proposed two-fluid model is also relevant for steam-water applications with Fluid-Structure Interactions and can be used on industrial-sized applications.

Conclusion

A two-fluid model has been used to compute fast depressurization of water. Based on the work of Baer & Nunziato [START_REF] Baer | A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials[END_REF], this two-fluid model does not assume any equilibrium between the phases. Thus the exchanges between phases must be taken into account and they are modelled as relaxation phenomena. Evaluations of the corresponding relaxation time scales must be provided. For pressure and velocity, previous work has been used [START_REF] Lochon | Comparison of two-fluid models on steam-water transients[END_REF], nevertheless the chemical potential relaxation, which corresponds to the mass transfer, requires particular attention. In order to model it in a simple but physicallyrelevant way, the work of Bilicki et al. [START_REF] Bilicki | Physical aspects of the relaxation model in two-phase flow[END_REF][START_REF] Downar-Zapolski | The non-equilibrium relaxation model for one-dimensional flashing liquid flow[END_REF], based on the Homogeneous Relaxation Model, has been revisited. Thus, a new modelling of the mass transfer in the two-fluid frame is proposed. Four different experiments have been selected in order to assess the corresponding model. Three of them, the Canon, Super-Canon and Edwards pipe experiments, study the depressurization of a hot liquid water in a pipe. The comparisons between grid-converged computational results and experimental data show that the proposed Bilicki-like mass transfer significantly improves the results of the twofluid model, particularly regarding pressure measurements. Moreover, it appears to be relevant for a wide range of thermodynamical states, including metastable states of water. The last test-case is the HDR experiment where pressures and displacements are measured in a full-scale vessel under the hypothesis of a Loss Of Coolant Accident. The results of an ALE computation show the ability of the proposed model to retrieve experimental data in both the structure and the fluid on an industrial-sized transient application.

The assessment of the proposed modelling of the mass transfer should be extended to other steamwater transients. Although the use of the Stiffened Gas Equation Of State appears to be quite relevant in the present work, the proposed modelling should also be tested with more realistic Equations Of State. To that end, the schemes dedicated to the treatment of the source terms should be extended and the convective solver proposed in previous work [START_REF] Lochon | HLLC-type riemann solver with approximated two-phase contact for the computation of the baer-nunziato two-fluid model[END_REF] could be used. Finally, extension of the current numerical methods to higher order should be considered in order to lower the computational cost of the two-fluid model on industrial applications.

The Homogeneous Relaxation Model

The Homogeneous Relaxation Model (HRM) proposed by Bilicki and Kestin [START_REF] Bilicki | Physical aspects of the relaxation model in two-phase flow[END_REF] consists in the usual three conservation laws for the two-phase mixture and the mass conservation law for one of the phases. Thus ρ denotes the mean density of the two-phase mixture, u its mean velocity, p its mean pressure and e its mean specific total energy. We also denote α k the mean volume fraction of phase k and ρ k its mean density, complying with:

   α v + α l = 1 m v + m l = ρ m k = α k ρ k (28)
The specific total energy of the mixture is given by the following relation:

e = ε + 1 2 u 2 (29)
where an Equation Of State for the two-phase mixture is required:

ε = ε (α k , ρ, p) (30)
Thanks to the different notations, the governing equations of the HRM model are the following:

       ∂ t (α k ρ k ) + ∂ x (α k ρ k u) = Γ HRM k ∂ t (ρ) + ∂ x (ρu) = 0 ∂ t (ρu) + ∂ x ρu 2 + p = 0 ∂ t (ρe) + ∂ x (ρeu + pu) = 0 (31)
Bilicki and Kestin [START_REF] Bilicki | Physical aspects of the relaxation model in two-phase flow[END_REF] propose a simple form of closure law for the mass transfer:

Γ HRM k = - m k -m k θ ( 32
)
where the notation ϕ stands for the quantity ϕ at the thermodynamical and chemical equilibrium. This simple relaxation equation gives its name to the model and requires a relaxation time scale θ.

Figure 1 :

 1 Figure 1: Schematic of Canon test facility

Figure 2 :

 2 Figure 2: Influence of the grid refinement on the history of the void fraction at Pt in Canon experiment

Figure 3 :

 3 Figure 3: Void fraction vs time at Pt in Canon experiment: comparison between the experimental data, converged numerical results obtained with and without Bilicki-like mass transfer

Figure 4 :

 4 Figure 4: Pressure vs time at P1 in Canon experiment: comparison between the experimental data, converged numerical results obtained with and without Bilicki-like mass transfer

  model without Bilicki-like mass transfer Two-fluid model with Bilicki-like mass transfer

Figure 5 :

 5 Figure 5: Void fraction vs time at Pt in Super-Canon experiment: comparison between the experimental data, converged numerical results obtained with and without Bilicki-like mass transfer

Figure 6 :

 6 Figure 6: Pressure vs time at P1 in Super-Canon experiment: comparison between the experimental data, converged numerical results obtained with and without Bilicki-like mass transfer

Figure 7 :

 7 Figure 7: Schematic of the Edwards pipe

  model without Bilicki-like mass transfer Two-fluid model with Bilicki-like mass transfer

Figure 8 :

 8 Figure 8: Void fraction vs time at GS5 in Edwards pipe experiment: comparison between the experimental data, converged numerical results obtained with and without Bilicki-like mass transfer

Figure 9 :

 9 Figure 9: Temperature vs time at GS5 in Edwards pipe experiment: comparison between the experimental data, converged numerical results obtained with and without Bilicki-like mass transfer

Figure 10 :

 10 Figure 10: Pressure vs time at GS7 in Edwards pipe experiment: comparison between the experimental data, converged numerical results obtained with and without Bilicki-like mass transfer

Figure 11 :

 11 Figure 11: Schematic of the HDR pressure vessel

Figure 12 :

 12 Figure 12: 3D mesh for the HDR experiment (Fluid mesh on the left and structure mesh on the right)

Figure 14 :

 14 Figure 14: Pressure difference (on the left) and displacement (on the right) histories at different locations in the vessel

Table 1 :

 1 Initial conditions of liquid water in the different experiments

	Experiment	Pressure Temperature (Bar) ( • C)
	Canon	32	220
	Super-Canon	150	300
	Edwards pipe	70	242

Table 2 :

 2 EOS parameters for Canon experiment γ k

Table 4 :

 4 EOS parameters for Super-Canon experiment γ k

Table 6 :

 6 EOS parameters for Edwards pipe experiment γ k

Table 7 :

 7 Initial conditions for Edwards pipe experiment

	vapor 1.34	0.00	2032350.00	1162.00	2351.11
	liquid 1.66 769317123.86 -1359570.00	2807.61	11671.61
			α k	ρ k	p k	u k
				(kg.m -3 ) (bar) (m.s -1 )
	Pipe	vapor liquid 1 -10 -3 10 -3	35.03 809.40	70 70	0.00 0.00
	Tank	vapor 1 -10 -3 liquid 10 -3	0.50 802.20	1 1	. .

Table 8 :

 8 EOS parameters for HDR experiment γ k

Acknowledgements The first author received a financial support through the EDF-CIFRE contract 0459-2013. Computational facilities were provided by EDF and CEA. Numerical simulations have been performed with the Europlexus software.