Components have long promised to neatly encapsulate data and programs into a box that operates predictably without requiring that users know the specifics of how it does so. Many advocates have predicted that components will bring about the bright future of large-scale software reuse, spawning a market for components usable with such mainstream software buses as the Common Object Request Broker Architecture (CORBA) and Distributed Component Object Model (DCOM). In the Windows world, at least, this prediction is becoming a reality.

Yet recent reports [START_REF] Jézéquel | Design by Contract: The Lessons of Ariane[END_REF][START_REF] Weyuker | Testing Component-Based Software: A Cautionary Tale[END_REF] indicate mixed results when using and reusing components in mission-critical settings. Such results raise disturbing questions. How can you trust a component? What if the component behaves unexpectedly, either because it is faulty or simply because you misused it?

Mission-critical systems cannot be rebooted as easily as a desktop computer. We thus need a way of determining beforehand whether we can use a given component within a certain context. Ideally, this information would take the form of a specification that tells us what the component does without entering into the details of how. Further, the specification should provide parameters against which the component can be verified and validated, thus providing a kind of contract between the component and its users.

In real life, contracts exist at different levels, from Jean-Jacques Rousseau's social contract to negotiable contracts, from various forms of license agreements to cash-and-carry. Likewise, we have identified four classes of contracts in the software component world: basic or syntactic, behavioral, synchronization, and quantitative.

Responsibility for contract management must be shared by both client and service provider. In those cases where the contracts are self-explanatory, assigning responsibility is easy. But some contracts rely on external constraints that too often remain implicit. In particular, quantitative contracts such as quality of ser-vice refer to features "outside the box," which reside in the system layer on which client and provider run.

In this article we define a general model of software contracts and show how existing mechanisms could be used to turn traditional components into contractaware ones.

FOUR LEVELS

When we apply contracts to components, we find that such contracts can be divided into four levels of increasingly negotiable properties, as Figure 1 shows. The first level, basic, or syntactic, contracts, is required simply to make the system work. The second level, behavioral contracts, improves the level of confidence in a sequential context. The third level, synchronization contracts, improves confidence in distributed or concurrency contexts. The fourth level, quality-of-service contracts, quantifies quality of service and is usually negotiable.

Basic contracts

Interface definition languages (IDLs), as well as typed object-based or object-oriented languages, let the component designer specify

• the operations a component can perform,

• the input and output parameters each component requires, and • the possible exceptions that might be raised during operation.

Static type checking verifies at compile time that all clients use the component interface properly, whereas dynamic type checking delays this verification until runtime.

Typically, a first-level contract would be implemented in an object bus such as CORBA. The IDL specification of the objects' interface ensures that client and server can communicate. Here, for example, is a possible interface for a component implementing a simple bank account management system:

Antoine Beugnard ENST Bretagne

Jean-Marc Jézéquel Irisa/CNRS Noël Plouzeau Irisa/University of Rennes

Damien Watkins Monash University

Before we can trust a component in mission-critical applications, we must be able to determine, reliably and in advance, how it will behave. This definition states that you can make several operations on a bank account: deposit, withdraw, and set overdraft limit. Further, you can get specific information from the account, including its balance and overdraft limit. Both the Component Object Model (COM) and JavaBeans let you write similar descriptions, albeit with slightly different syntaxes.

Making components contract aware

Behavioral contracts

Unfortunately, because the BankAccount specification does not define precisely the effect of operation executions, the user can only guess at their outcomes. A deposit would probably increase the balance, but what if the deposit amount is negative? What would happen to a withdrawal operation that exceeds the overdraft limit?

Drawing from abstract-data-type theory, we can specify an operation's behavior by using Boolean assertions, called pre-and postconditions, for each service offered, as well as for class invariants. First implemented under the name design by contract in the Eiffel language, 3 this approach has spread to several other programming languages (the Java extension iContract, [START_REF] Kramer | iContract-The Java Design by Contract Tool[END_REF] for instance) and can be found in the Unified Modeling Language as the Object Constraint Language (OCL). [START_REF] Warmer | The Object Constraint Language[END_REF] The example shown in Figure 2 more precisely defines our BankAccount interface's behavior. The assertions' syntax is based loosely on OCL; the operator @pre refers to the value its argument had when entering the method.

The design-by-contract 6 approach prompts developers to specify precisely every consistency condition that could go wrong, and to assign explicitly the responsibility of its enforcement to either the routine caller (the client) or the routine implementation (the contractor). A contract carries mutual obligations and benefits: The client should only call a contractor routine in a state where the class invariant and the precondition of the routine are respected. In return, the contractor promises that when the routine returns, the work specified in the postcondition will be done, and the class invariant will be respected. Depending on the strength of the postcondition, we can imagine negotiable contracts at this level. A client may, for example, require a prohibitively high-level guarantee that requires an expensive server-side check, but other clients-that may only need weak and cheap postconditions-could be implemented at a weaker level 2.

When using a third-party component, contracts offer a specification against which you can validate that component. Still, when working from the speci- fication alone you must bear in mind that you can never be sure the component will perform correctly, only that it will perform as specified.

Synchronization contracts

The level 2 behavioral contract pretends that services are atomic or executed as transactions, which is not always practical or true. The level 3 synchronization contract specifies the global behavior of objects in terms of synchronizations between method calls. The aim of such a contract is to describe the dependencies between services provided by a component, such as sequence, parallelism, or shuffle. This kind of contract can be observed in entity structures or more formal applications. [START_REF] Mchale | Synchronization in Concurrent, Object-Oriented Languages: Expressive Power, Genericity and Inheritance[END_REF] The contract reifies-makes concrete and specificthe ways in which the component serves its clients. A contract is useful to the functioning of a single client, but is significantly more important in an environment of one server and many clients. In such cases, the contract guarantees to each client that, whatever the other clients request, the requested service will be executed correctly. For example, we must know how our BankAccount would behave when a client requests a withdraw() while the BankAccount serves a set-OverdraftLimit().

To express behavior in concurrent contexts, Ciaran McHale 7 proposed attaching to components special elements called synchronization policies. A subset of these synchronization policies can be described using the well-known path-expression formalism. [START_REF] Campbell | The Specification of Process Synchronizations by Path Expressions[END_REF] Many strategies can be defined to manage intracomponent concurrency. For example, a Mutex strategy could force all services to be mutually exclusive, ensuring atomicity. The synchronization contract is reified so that its binding to an object can differ from time to time, and so that various objects can share the same synchronization policy.

Synchronization

Java provides a stripped-down version of synchronization through the keyword "synchronized," which specifies that a given block or method should be run in mutual exclusion with other operations on the same object. While this method lacks the expressive power and versatility of McHale's synchronization policies, it is superior to implementing locks explicitly.

Quality-of-service contracts

Now that we've expressed, qualified, and contractually defined all behavioral properties, we can quantify the expected behavior or offer the means to negotiate these values. You can specify the quality-ofservice contract statically by enumerating the features the server objects will respect. Alternatively, you can implement a more dynamic solution that conducts negotiations between the object client and its server.

Common examples of quality-of-service parameters that may be exposed by a server are • maximum response delay; • average response;

• quality of the result, expressed as precision; and • result throughput for multiobject answers such as data streams.

The main difficulty with satisfying such contracts is that they rely on third parties, which do not offer performance guarantees. Thus, these quantified contracts can be used only if their use is generalized. Several works deal with adding quality-of-service control to component-based software and propose extensions to CORBA with quality-of-service specifications. 9

CONTRACT MANAGEMENT

To a great extent, contract management issues are independent of the contract's level. We focus on the interactions that arise between a component and the client code that binds to the component's interfaces and requests services through them. Contract management deals with contract definition, subscription, application (including handling contract violations), termination, and deletion. Because contract and interface issues closely interlink, many aspects of contract management link to aspects of component management.

Several moments serve as milestones in a component's life: build time, load time, service exposition time, service request time, and unload time. Using the following implementation examples for each contract level-

• a plain CORBA IDL at level 1,

• Eiffel mechanisms for design by contracts at level

2,

• the Service Object Synchronization (SOS) mechanism at level 3, and You can never be sure that a thirdparty component will perform correctly, only that it will perform as specified.

• TAO (the Adaptive Communication Environment object request broker) [START_REF] Schmidt | The Design of the TAO Real-Time Object Request Broker[END_REF][START_REF] Watkins | Using Interface Definition Languages to Support Path Expressions and Programming by Contract[END_REF] and the Rusken computer-aided cooperative work environment [START_REF] Lorcy | Reifying Quality of Service Contracts for Distributed Software[END_REF] at level 4 -we sketch relationships between these particular moments using existing contract-implementation schemes.

Contract definition and subscription

Although typical level 1 and 2 implementations bind these aspects so closely that they become indistinguishable, contract subscription and contract definition differ significantly at levels 3 and 4. At higher levels, defined contracts can be selected and tailored through subscription to accommodate dynamic operating conditions or client needs.

Definition. Contract definition is the feature set that fully describes the service offered by a component; definition time is the time a component makes the feature set available to customers as potential client code.

In level 1 components, such as exposing standard CORBA IDL definitions, contract definition time is the same as component-build time. The same relationship applies to level 2 contracts, which use Eiffellike contracts: Preconditions, postconditions, and invariants are bound syntactically and semantically to method signatures.

Level 3 contracts, such as those provided by the SOS proposal, separate data properties (preconditions, postconditions, invariants) from control properties (invocation paths, concurrency). To retain as much flexibility as possible, this separation must be preserved in implementations.

Data and control constraints must, however, be merged before requests can be issued. This merge could take place at service exposition time, when a component exposes its interfaces. Choosing which control specification should be used with which data service is left to the configuration of the considered component.

Level 4 contracts also separate data constraints from control constraints. Stephane Lorcy and colleagues, [START_REF] Lorcy | Reifying Quality of Service Contracts for Distributed Software[END_REF] for example, reify contracts as instances from subclasses of the Contract class. This process completely separates the definition of contracts from that of the components that provide contract-aware services. The component thus controls contract definition time during the component's execution.

Level 4 contract reification allows dynamic adaptation of the contract to environmental changes, while level 3 contract reification enables a static choice under the service provider's responsibility. This static choice can be updated from time to time. Thus, a shared resource such as a printer can change its priority management policy from "first arrived, first served" to "largest job first." Such updating differs from level 4 contract negotiation in that it involves all clients simultaneously, while level 4 involves only a single client.

Subscription. We define contract subscription to mean the following:

• selection of a contract among a set of possible contracts,

• selection of parameter values when contracts have parameters, and • agreement of both client and component parts regarding use of a contract in later service request execution.

Depending on the contract level and the typical implementations, contract subscription times differ broadly.

• Level 1 contracts are subscribed when the client code is bound to a component's interface definition.

• Level 2 Eiffel-like contracts are also bound at interface definition time.

• Level 3 SOS-like contracts may be bound at component load time or later, at interface definition time, depending on the component's behavior.

• Level 4 TAO-and Rusken-like contracts can be subscribed at any time prior to a service invocation, and can be changed or negotiated when the need arises.

Acquiring the ability to alter contracts in response to changing needs provided a strong reason for defining contracts as objects. Thus defined, contracts can be tailored at application design time by subclassing or aggregation: Several specialized subclasses of the Contract class provide a starting point for producing specific contracts for specific needs. Contracts can also be tailored at runtime during a negotiation phase.

Before a customer sends a request to a provider component, a contract object must be established between them. The customer selects a contract from the set exported by the customer, sets up parameters such as delays or other properties specific to the contract subclass chosen by the customer, then submits the contract for approval to the service provider. This provider then examines the desired terms of the contract and tunes them to make the request feasible. The contract is then returned to the customer for approval.

Should the customer reject the reconfigured contract, other contracts from other classes, such as those with weaker quality-of-service features, may be tried. The negotiation phase plays an important role in the Rusken scheme because quality of service involves both a customer (which has needs) and a provider (which has means): A scheme in which a customer Acquiring the ability to alter contracts in response to changing needs provided a strong reason for defining contracts as objects.

requires quality-of-service features blindly is not realistic in a world of components loaded with heterogeneity and instability. Further, a provider component also relies on other components as subcontractors and must negotiate subcontracts. Finally, components often include dynamic interface presentation and, thus, quality of service must also obey this property while taking advantage of it.

Contract application

This aspect of contract management involves checking and handling violations.

Checking contracts. Provided there is enough theorem-proving machinery, some level 1 and 2 contracts could be checked statically, but most require some kind of runtime monitoring. This requirement forces us to decide if the check takes place on the client or server side. If dealing with static, third-party components, it is probably more efficient to monitor the contract on the client side. In a concurrent system, however, the component's state could change between the moment we do the check and the moment we call one of its methods. This suggests establishing a reified contract monitor somewhere in the component middleware. The monitor would then cooperate with the request broker, and should be sufficiently intelligent to optimize out some of the message passing when only one client is connected to a given component. [START_REF] Watkins | Using Interface Definition Languages to Support Path Expressions and Programming by Contract[END_REF] Handling violations. When the system detects that a contract has been broken-say, for example, a client tries to withdraw too much money from a BankAccount file-an action can be undertaken. We have identified four possible actions:

1. Ignore. Proceed with the operation, ignoring any adverse effect. This is the effect of disabling the assertion checking in Eiffel, C++, and others. 2. Reject. In modern programming languages like Java, Eiffel, and C++, the reject action involves raising an exception and propagating it to the client. Older environments such as Unix C libraries returned a completion status to the client as the result of the "function call." 3. Wait. This action blocks the client call until the contract becomes valid through, for example, a wait for a deposit() or setOverdraftLimit() operation that could change the truth value of the withdraw() precondition. Clearly this option has no application to sequential programs, but it is the default behavior for so-called separate objects in Eiffel's concurrency extension proposal. 6 4. Negotiate. Failure of the contract may be solved by retrying (in the case of a network failure) or renegotiating its terms.

Contract termination and deletion

These activities share a highly symmetric relationship to contract definition and subscription. Level 1 and 2 contracts are terminated on service interface termination, which in turn occurs upon component unloading. Level 3 and 4 contracts, such as SOS and Rusken schemes, allow unilateral contract termination by a contract partner. Such terminations often occur to cope with changes in the environment, such as quality-of-service variations exported by the component's providers. Level 3 or level 4 contracts can be deleted by removing their definition from the component's interface.

TOWARD A UNIFIED FRAMEWORK

We must make components contract aware if we are serious about developing mission-critical applications based on third-party components. Making contracts explicit improves confidence. The easiest way to achieve this goal is to design a new, component-oriented language that features in its semantics the various concepts we've outlined.

If we want the idea of contractible components to succeed, however, we must strive to integrate contract awareness concepts into popular component frameworks. Thus we propose building on existing technologies such as COM, CORBA, and JavaBeans by making contractible components available on demand for these platforms. This option means you pay for performance overhead only if you use the component.

Having described the many possible times at which contract definition, subscription, and application occur, we now elaborate on our BankAccount example to estimate how several of these techniques could be unified, and give some preliminary form for a contract-aware component technology.

For brevity's sake, we exclude schemes that bind contract subscription to service requests. We focus rather on binding service request times with contract subscription ones. Further, we base our examples on dynamic-interface-binding programming styles. aware interfaces should derive from CtServiceInterface. This interface also defines contract management operations and divides them into the categories previously described.

Discovering contract-aware interfaces

Contract definition. Once a client has a reference to an interface object with a CtServiceInterface, it can get the interface to export these contracts by using its QueryContract() method.

Contract objects contain all the information for level 2, 3, and 4 contracts managed by the service interface; level 1 contract information is available from the interface description itself. BehavioralContract and SynchronizationContract interfaces provide all level 2 and 3 contract information by any suitable means for example, an XMLformatted description of the contracts expressed using UML and OCL formalisms. Although both types of contracts are usually nonnegotiable, the contract interface can be extended to handle negotiable qualityof-service issues, and level 4 contracts are often negotiable.

Contract subscription. Client code can fill in contract parameters provided by QueryContract(), then call Propose() to have the configured contract accepted by the component. If the contract configuration has been updated by the component to propose reasonable values, the method changed() returns a true result. For example, a component may adjust a service delay to some realistic value.

When the client code agrees to the last contract configuration, it can accept it by calling Accept() from the CtServiceInterface. The result value is the contract a customer must use in future request submissions. We supplied this provision because the provider may change the contract internally upon acceptance-to add, for example, implementation details used in future request invocation. Moreover, we need a rejection primitive to cope with the frequent case where a component uses subcontractors. In such a case, client code aborts the negotiation phase by calling Reject().

Thus, the CtServiceInterface takes the following form: Contract application. Once a contract has been subscribed, client code can issue contract-aware service requests. The component maintains a contract context together with the interface it provided as the return value of the QueryInterface() call. Hence no ancillary Contract parameter is necessary to request a service. Indeed, adding contract capabilities to some existing component does not imply alterations of service interface methods. However, we must provide some means to monitor the application of contracts. The predicate Failed() serves this purpose by returning a true result if and only if the last use of the subscribed contract failed. This style of explicit synchronous check can be enhanced to deal with implicit checks by, for example, using exceptions as provided by the environment.

// Returns

Using the BankAccount example

We now sketch out a possible contract definition for the BankAccount example, shown in Figure 3. Next, in Figure 4 we sketch some simple, Java-style code for getting a component and contract definition, subscribing to the contract, applying it by requesting a deposit, and finally terminating the contract. This example shows that contracts offer a powerful tool for enhancing the versatility and functionality of components.

Reusing software components in mission-critical applications cannot succeed if the components do not export clearly stated service guarantees. Indeed, a would-be component user cannot trust a component without strong statements from it. Several aspects of contractible-components technology have already been implemented in various projects. [START_REF] Choukair | Real-time Object-Oriented Distributed Processing with COREMO[END_REF][START_REF] Schmidt | The Design of the TAO Real-Time Object Request Broker[END_REF][START_REF] Watkins | Using Interface Definition Languages to Support Path Expressions and Programming by Contract[END_REF][START_REF] Lorcy | Reifying Quality of Service Contracts for Distributed Software[END_REF] We must now unify these implementations and generalize them to all standard component platforms so that mission-critical application designers can gain the full benefits of contractible components. This gain, however, will be effective only if developers consider their work as one link in a customer/provider chain. Links are hooked together at various levels with contracts. You build trust in the chain with contract-checking mechanisms. Although some techniques for checking contracts already exist, much research is still needed to deal with the palette of contracts we've outlined in this article. A key point would be the definition of a standard contract description language or notation mapped to the already existing contract techniques, as for the level 1 contract with IDL.

Classical component-based software such as network computer systems and interactive, distributed applications that run on the Internet are natural candidates for contract-based components. Any distributed software with soft real-time constraints, such as telecom software, would benefit from this approach. y

Figure 1 .Figure 2 .

 12 Figure 1. The four contract levels, which start with basic syntactic nonnegotiable features and progress to include highly dynamic, negotiable, quantitative properties.

  Then our BankAccount component could use this Mutex policy between the methods withdraw() and setOverdraftLimit(), as follows:Interface BankAccount use Mutex(withdraw, setOverdraftLimit) { // As before. }

  a contract for a given // service interface Contract QueryContract(); void Propose(inout Contract c); void Accept(inout Contract c); void Reject(in Contract c);

Figure 3 .

 3 Figure 3. One possible contract definition for a simple bank account management system.

Nonnegotiable Dynamically negotiable Level 3: Synchronization

	Level 4: Quality-of-service level
	level
	service object synchronization, path expression,
	synchronization counters
	Level 2: Behavioral level
	pre-and postcondition,
	Eiffel, Blue, Sather,
	UML/OCL, iContract for Java
	Level 1: Syntactic level
	interface definition language,
	usual programming languages

  The CtServiceInterface defines the interface base for contract-aware service interfaces. Hence all contract-

	We must make
	components contract
	aware if we are
	serious about
	developing mission-
	critical applications
	based on third-party
	components.
	Contractible components should export the basic
	interface-discovery interface ContractAwareComponent.
	This interface includes a QueryInterface() method for
	browsing contractible service interfaces.
	interface ContractAwareComponent {
	// Basic interface query system,
	// with provision for contracts
	CtServiceInterface QueryInterface(
	String interfaceUniversalName);
	}

true iff last service // execution // did not comply with // contract terms. boolean Failed();

	Contract termination. Client code may terminate a
	contract subscription by calling the Terminate
	method from CtServiceInterface.
	void Terminate(Contract c);
	}

Contact the authors at antoine.beugnard@enst-bretagne.fr, {jezequel, plouzeau} @irisa.fr, damien.watkins@csse.monash.edu.au.


Arnaud Rougirel
email: rougirel@math.univ-poitiers.fr

TIME FRACTIONAL SCHRÖDINGER EQUATION

We propose a time fractional extension of the Schrödinger equation that keeps the main mechanical and quantum properties of the classical Schrödinger equation. This extension is shown to be equivalent to another well identified time first order PDE with fractional hamiltonian.

Introduction

We would like to address the issue of what are suitable time fractional extensions of the Schrödinger equation. Our approach consists in, first, selecting some (important) properties of the Schrödinger equation; secondly, finding a time fractional equation which conserves these properties.

We have selected two properties: (i) the conservation of the L 2 -norm of wave functions. That point is central w.r.t. the probabilistic character of quantum mechanics objects. (ii) The dynamics of the Schrödinger equation, i.e. the time reversibility of solutions which is related to the fact that the related solution operator is a group. Indeed, in any reasonable mechanical theory, trajectories of autonomous systems must be described by equations generating a group.

It is clear that the use of time fractional operators as D α 0,t , do not allow to match the above requirements. We refer to Remark 4.1 below for details, and to [START_REF] Diethelm | The analysis of fractional differential equations[END_REF] or [START_REF] Samko | Fractional integrals and derivatives[END_REF] for more information on fractional derivatives. Regarding the dynamics of time fractional equations, we refer to [START_REF] Emamirad | Solution operators of three time variables for fractional linear problems[END_REF].

In this paper, the real number α is always supposed to range between 0 and 1.

In order to recover the above properties, we consider time fractional operators with lower bound -∞, i.e. D α -∞,t . More precisely, if, for simplicity, we restrict our attention to the Hamiltonian Ĥ := -∆, then we will show that the problem

D α -∞,t u = -i α ∆u, u(0) = v, (1.1)
admits a unique solution u in some suitable function space. Theorem 4.1 states that the solution to (1.1) is given by u(t) = e it(-∆) 1/α v, hence the L 2 -norm of u is conserved and the solution operator of (1.1) is a group. Then, according to Corollary 4.3, (1.1) is equivalent to

d dt u = i(-∆) 1/α u, u(0) = v.
(1.2)

The coefficient i α in (1.1) is not surprising since we know from the work of Naber, in [START_REF] Naber | Time fractional Schrödinger equation[END_REF], that the operator D 0,t + i∆ has a parabolic behavior (see Remark 4.1).

In this paper, we consider more generally abstract time fractional Schrödinger equations of the form

d dt u = i α Au, (1.3)
where A is a positive self-adjoint operator. The outline is as follows. The forcoming section is dedicated to preliminaries regarding fractional derivatives, in particular time fractional weak derivatives. In section 3, we solve (1.3) in the case where the underlying Hilbert space in R. That turns out to be the corner stone of our work since, in section 4, we solve (1.3) by diagonalisation using spectral theory.

Fractional derivatives with lower bound -∞

We start with the convolution of functions defined on R, with the fractional kernel, whose definition is as follow.

Definition 2.1. For β ∈ (0, ∞), let us denote by g β the function of L 1 loc ([0, ∞)) defined for a.e. t > 0 by

g β (t) = 1 Γ(β) t β-1 .
Let X be a complex Banach space with norm • .

Proposition 2.1. Let α ∈ (0, 1) and u ∈ L 1 (R; X) ∩ L ∞ (R; X). Then, for all t ∈ R, the function (-∞, t) → X, y → g α (t -y)u(y) is integrable on (-∞, t) and sup t∈R t -∞ g α (t -y) u(y) dy < ∞.
Proof. For all t ∈ R, we have

t -∞ g α (t -y) u(y) dy ≤ t-1 -∞ g α (t -y) u(y) dy + t t-1 g α (t -y) u(y) dy ≤ g α (1) u L 1 (R;X) + g α+1 (1) u L ∞ (R;X) .
Under the assumptions and notation of Proposition 2.1, we put, for all t ∈ R,

g α * u(t) := t -∞ g α (t -y)u(y)dy = ∞ 0 g α (y)u(t -y)dy.
Also, we define

g α * u(t) := ∞ t g α (y -t)u(y)dy = ∞ 0 g α (y)u(t + y)dy.
In some sense, these convolutions are adjoints. More precisely, we have the following result, whose easy proof is left to the reader.

Proposition 2.2. Let α ∈ (0, 1), u ∈ L 1 (R; X) ∩ L ∞ (R; X) and ψ ∈ L 1 (R) ∩ L ∞ (R). Then R g α * u(t) ψ(t) dt = R u(t) g α * ψ(t) dt.
(2.1)

Then we may give the following definition of fractional derivatives.

Definition 2.2. Let α ∈ (0, 1) and u ∈ L 1 (R; X) ∩ L ∞ (R; X). We say that u admits a (forward) derivative of order α in L ∞ (R; X) if

g 1-α * u ∈ W 1,∞ (R; X).
In this case, its (forward) derivative of order α is the function of L ∞ (R; X) defined by

D α -∞,t u := d dt g 1-α * u .
Definition 2.3. Let α ∈ (0, 1) and u ∈ W 1,1 (R; X) ∩ W 1,∞ (R; X). Then we say that u admits a backward derivative of order α in L ∞ (R; X) if

g 1-α * d dt u ∈ L ∞ (R; X).
In this case, its backward derivative of order α is the function of L ∞ (R; X) defined by

D α t,∞ u := g 1-α * d dt u.
Let S(R) denote the Schwartz space of rapidly decreasing complex functions defined on R.

Proposition 2.3. Let α ∈ (0, 1), u ∈ L 1 (R; X) ∩ L ∞ (R; X) and ϕ ∈ S(R). Assume that u admits a fractional derivative of order α in L ∞ (R; X). Then R D α -∞,t u(t)ϕ(t) dt = - R u(t)D α t,∞ ϕ(t) dt. (2.2)
Proof. Integrate by parts and use Proposition 2.2.

We will now introduce fractional derivatives in the sense of distributions. The following result makes possible such a definition. Proposition 2.4. Let α ∈ (0, 1) and ϕ ∈ S(R).

Then D α t,∞ ϕ ∈ L 1 (R) and D α t,∞ ϕ L 1 (R) ≤ C ϕ L 1 (R) + ϕ L 1 (R) , (2.3)
where ϕ := d dt ϕ and the constant C depends only on α. Proof. For any fixed time t in R, let v(y) := ϕ(y + t) -ϕ(t). Integrating by part, we get

α ∞ 0 y -α-1 v(y)dy = ∞ 0 y -α ϕ (y + t)dy. Hence D α t,∞ ϕ(t) = 1 |Γ(-α)| ∞ 0 y -α-1 ϕ(y + t) -ϕ(t) dy, and
|Γ(-α)D α t,∞ ϕ(t)| ≤ 1 0 y -α ϕ(y + t) -ϕ(t) y dy + ∞ 1 y -α-1 |ϕ(y + t) -ϕ(t)|dy =: I 1 (t) + I 2 (t),
with obvious notation for I 1 (t) and I 2 (t).

Let us show that I 1 is integrable on R. For, since

ϕ(y + t) -ϕ(t) y = 1 0 ϕ (ys + t) ds, ∀y ∈ (0, 1], we derive R |I 1 (t)| dt ≤ 1 0 y -α dy 1 0 ds R |ϕ (ys + t)| dt = ϕ L 1 (R) 1 -α .
Regarding I 2 (t), we have

R |I 2 (t)| dt ≤ ∞ 1 y -α-1 R |ϕ(y + t)| + |ϕ(t)| dt = 2 α ϕ L 1 (R) .
That estimate allows us to define fractional derivatives in the sense of distributions. Indeed, (2.3) shows that, for each u ∈ L ∞ (R; X), the linear map

S(R) → X, ϕ → - R u(t)D α t,∞ ϕ(t) dt
is a tempered distribution. The set of tempered distributions with values in X is denoted by S (R; X). That allows us to set the following definition.

Definition 2.4. Let α ∈ (0, 1), X be a complex Banach space and u ∈ L ∞ (R; X).

Then the weak derivative of u is the X-valued tempered distribution, denoted by D α -∞,t u, and defined, for all ϕ ∈ S(R), by

D α -∞,t u, ϕ = - R u(t)D α t,∞ ϕ(t) dt.
If we want to highlight the duality taking place in the above bracket, we will write

D α -∞,t u, ϕ S (R;X),S(R) or D α -∞,t u, ϕ S (R;X) instead of D α
-∞,t u, ϕ . In this paper, we will need to compute D α -∞,t u for bounded functions u. For such functions, the integral t -∞ g 1-α (t -y)u(y)dy is, in general, not absolutely convergent. That point turns out to be a major drawback; and we will explain the reason in the sequel.

Works on fractional calculus with lower terminal -∞ and not absolutely convergent integrals, go back, at least to 1938 with the smart paper [START_REF] Love | Fractional Integration, and Almost Periodic Functions[END_REF]. The difficulties when dealing with non absolutely convergent integral are illustrated by Theorem 5 in [START_REF] Love | Fractional Integration, and Almost Periodic Functions[END_REF], which states, roughly speaking that each u ∈ L ∞ (R) such that

sup t∈R t 0 u(y)dy < ∞, (2.4)
satisfies, for all t ∈ R, the following (hard to prove) identity

g 1-α * (g α * u)(t) = g 1-α * (g α * u)(0) + t 0 u(y)dy.
That result is quiet surprising since, extending g α by 0 outside of (0, ∞), we have

g 1-α * g α (t) = 1 if t > 0 0 if t < 0 ;
so that the convolution

(g 1-α * g α ) * u(t) = t -∞ u(y)dy
is not defined in general.

In this paper, we deal with bounded functions satisfying typically (2.4). However, Love's approach (see also [START_REF] Bernardis | Maximum principles, extension problem and inversion for nonlocal one-sided equations[END_REF]) allowed us to obtain only partial results. That is, we must assume that α = 1/n for some positive integer n.

Time fractional equations in R

We will use the framework of Section 2 with X = C. For a ∈ R and α ∈ (0, 1), we consider this equation:

Find u ∈ L ∞ (R) such that D α -∞,t u = (ia) α u, in S (R). (3.1)
Of course, in (3.1), D α -∞,t u is understood in the sense of Definition 2.4. For ψ ∈ L 2 (R d ), we denote by Fψ or ψ its Fourier transform. More precisely, for ϕ ∈ L 1 (R), we have

φ(σ) = c F R e -iσt ϕ(t) dt, F -1 ϕ(t) = c F -1 R e itσ ϕ(σ) dσ, (3.2)
where c F , c F -1 denote reader's favorite constants of normalization, whose product equals 1 2π . Remark 3.1. Solving the fractional equation

D α -∞,t u = (ia) α u (3.3) is a touchy business. First, it is known that the function u 1 : t → exp(iat) satisfies ∞ 0 g 1-α (y)u 1 (t -y)dy = (ia) α-1 u 1 (t).
Thus, u 1 solves (3.3) in some "classical sense". However, the above integral is not absolutely convergent. Thus it is not clear how to get uniqueness for (3.3) in a space containing u 1 (typically C b (R)). This is the reason why we use time fractional derivatives in the sense of distributions to solve (3.3). But then, existence for (3.1) is tricky since, we have not been able to show directly that u 1 solves (3.1). Indeed, Fubini's Theorem being not applicable, the path from

R e iat D α t,∞ ϕ(t) dt = R e iat ∞ t g 1-α (y -t)ϕ (y)dy to R ϕ (y)dy y -∞ e iat g 1-α (y -t) dt
is not clear. We overcome this difficulty by using Fourier transforms.

The last difficulty is technical and concerns the use of Fourier transforms. More precisely, starting from (3.3), we obtain formally by Fourier transform, (iσ) α û = (ia) α û.

However, if u = u 1 then (iσ) α û = 1 c F -1 (iσ) α δ a , which has no meaning since the function σ → (iσ) α , being not smooth at σ = 0, cannot multiply δ a , the Dirac mass at the point a.

In order to overcome this difficulty, we first determine the support of û, which turns out to be discrete, as expected; and then compute u by inverse Fourier transform.

Theorem 3.1. Let a ∈ R, α ∈ (0, 1) and u ∈ L ∞ (R). Then u is a solution of (3.1) iff there exists k ∈ C such that u(t) = k exp(iat),
a.e. t ∈ R.

The following lemma will be usefull in the proof of Theorem 3.1.

Lemma 3.2. Let ϕ ∈ S(R). Then, for all σ ∈ R,

F D α -∞,t ϕ (σ) = (iσ) α φ(σ) (3.4) D α σ,∞ φ(σ) = -F (it) α ϕ (σ). (3.5)
Proof. For the proof of (3.4), we refer to [SKM93, Section 7.1]. We establish (3.5) by a duality argument. Indeed, for all ϕ and ψ in S(R), there holds, by Proposition 2.3,

R D α σ,∞ φ(σ)ψ(σ) dσ = - R φ(σ)D α -∞,σ ψ(σ) dσ = - R ϕ(σ)F D α -∞,t ψ (σ) dσ,
by Plancherel's formula. Thus with (3.4) and Plancherel's formula once again, we get

R D α σ,∞ φ(σ)ψ(σ) dσ = - R ϕ(σ)(iσ) α ψ(σ) dσ = - R F (it) α ϕ (σ)ψ(σ) dσ.
That proves (3.5).

Proof of Theorem 3.1. (i) Let us show that the function u : t → e iat is solution to (3.1). For, we compute

F D α -∞,t u , ϕ S (R),S(R) = D α -∞,t u, φ = - R u(σ)D α σ,∞ φ(σ) dσ = R e iaσ F (it) α ϕ (σ) dσ,
by (3.5). Besides, the Fourier transform of t → (it) α ϕ(t) belongs to L 1 (R) by (3.5) and Proposition 2.4. Then the latter integral is equal, by inverse Fourier transform (see (3.2)), to

1 c F -1 (ia) α ϕ(a) = (ia) α δ a c F -1 , ϕ = (ia) α F e iat , ϕ = (ia) α Fu, ϕ .
Thus by invertibility of the Fourier transform in S (R), we deduce that t → e iat is solution to (3.1).

(ii) Conversely, let us show that each solution u to (3.1) is proportional to some exponential function. Computing as above, we get, for each test-function ϕ in S(R),

F D α -∞,t u , ϕ S (R),S(R) = R u(σ)F (it) α ϕ (σ) dσ.
Thus, since u solves (3.1),

0 = R u(σ)F {(it) α -(ia) α }ϕ (σ) dσ.

Now, set

I a := {0} if a = 0 {0, a} if a = 0 .
Then, for each ψ ∈ S(R) with support, denoted by suppψ, in R \ I a , we may find a function ϕ ∈ S(R) so that

ψ(t) = (it) α -(ia) α ϕ(t), ∀t ∈ suppψ.
There results that û, ψ S (R),S(R) = 0.

Thus the support of the distribution û lies in I a . By standard results in the theory of distributions, we infer that there exist some integer n ≥ 0 and c k ,

d k ∈ C (k = 0, . . . , n) such that û = n k=0 c k δ (k) 0 + d k δ (k) a ,
where δ a denotes the Dirac mass at a. Hence, for other constants still labeled c k and d k ,

u(t) = n k=0 c k t k + d k t k e iat , ∀t ∈ R.
Since u is bounded, u(t) = c 0 + d 0 e iat . Finally, according to the existence part (i) of this proof, we have

D α -∞,t (c 0 + d 0 e iat ) = d 0 (ia) α e iat in S (R).
On the other hand, since u solves (3.1), there holds

D α -∞,t (c 0 + d 0 e iat ) = (ia) α (c 0 + d 0 e iat ).
Thus, for all t ∈ R, we have u(t) = d 0 exp(iat).

Time fractional Schrödinger equation

Let α ∈ (0, 1), d ≥ 1 be an integer and A : D(A) ⊆ L 2 (R d ) → L 2 (R d ) be a unbounded operator on L 2 (R d ) with domain D(A). For each v ∈ D(A), we consider the following Time Fractional Schrödinger Equation Remark 4.1. There are many ways to extend free Schrödinger equation, i.e.

     Find u ∈ C b R, D(A) such that D α -∞,t u = i α Au, in S (R, L 2 (R d )) u(0) = v.
d dt u = -i∆u, (4.2)
into a time fractional equation (see for instance [START_REF] Naber | Time fractional Schrödinger equation[END_REF], [START_REF] Laskin | Fractional Schrödinger equation[END_REF], [START_REF] Luchko | Fractional Schrödinger equation for a particle moving in a potential well[END_REF]). First, we may consider D α 0,t u = -i∆u. However, as pointed out in [START_REF] Emamirad | Time fractional linear problems on L 2 (R d )[END_REF], that equation has regularizing effect and dissipative properties, in a great contrast with (4.2). In order to recover a hyperbolic behavior, it is better to extend (4.2) by

D α 0,t u = -i α ∆u. (4.3)
Indeed, according to [ER18, Theorem 4.4 & Example 4.6], this equation has no regularizing effect and possesses an asymptotic conservation law. The drawback of (4.3) regarding the dynamics, is that it does not generate a semi-group. However, (4.1) does, as we will show in the sequel.

Remark 4.2. The equation of (4.1) holds equivalently in C b R, L 2 (R d ) . In any case, D α -∞,t u is understood in the sense of Definition 2.4. Theorem 4.1. Let α ∈ (0, 1), A : D(A) ⊆ L 2 (R d ) → L 2 (R d ) be a positive self adjoint operator on L 2 (R d ) and v ∈ D(A). Then (4.1) has a unique solution u. Moreover, for all t ∈ R,

u(t) = e itA 1/α v, in D(A). (4.4)
Before to prove this theorem, let us precise the meaning of (4.4). According to the Spectral Theorem in multiplication form (see [START_REF] Reed | Methods of modern mathematical physics[END_REF]Theorem VIII.4] or [START_REF] Hall | Quantum theory for mathematicians[END_REF]Theorem 10.10]), there exist a measure space (X, µ) with finite measure µ, a unitary map U : L 2 (R d ) → L 2 (X, µ) and a measurable real-valued function h on X which is finite µ-a.e., such that

U D(A) = ψ ∈ L 2 (X, µ) | hψ ∈ L 2 (X, µ) (4.5) and UAU -1 (ψ) = hψ, ∀ψ ∈ U D(A) . (4.6)
Under the assumption of Theorem 4.1, we claim that h ≥ 0 µ-a.e. on X.

(4.7)

Indeed, let us denote by (•, •) L 2 (X,µ) the standard inner product of L 2 (X, µ). We put, for each positive integer n,

X n := ξ ∈ X | -n < h(ξ) < 0 ,
and denote by 1I Xn the indicator function of X n (i.e. 1I Xn = 1 on X n and 0 elsewhere). Then h1I Xn lies in L 2 (X, µ) since µ is finite. Hence

h1I Xn , 1I Xn L 2 (X,µ) = UAU -1 (1I Xn ), 1I Xn L 2 (X,µ) (by (4.6)) = AU -1 (1I Xn ), U -1 (1I Xn ) L 2 (R d ) (since U is unitary) ≥ 0 (since A is positive).
Thus, µ(X n ) = 0; and since h is finite µ-a.e., we obtain (4.7).

We are now in position to define the operator e itA 1/α . For each t ∈ R, denote by f t , the function

f t : [0, ∞) → C, x → e itx 1/α .
Then e itA 1/α is defined through bounded functional calculus for self adjoint operators (see [RS80, Theorem VIII.5] or [Kow09, Corollary 4.43]), by

e itA 1/α := f t (A) := U -1 M ft•h U, (4.8)
where

M ft•h : L 2 (X, µ) → L 2 (X, µ)
is the multiplication operator defined by

M ft•h ψ = f t (h)ψ, ∀ψ ∈ L 2 (X, µ).
Proof of Theorem 4.1. (i) Uniqueness of the solution u. By linearity, it is enough to show that v = 0 in (4.1) implies u = 0. For each ϕ ∈ S(R), one has

U D α -∞,t u, ϕ S (R;L 2 (R d )) = -U R u(t)D α t,∞ ϕ(t) dt (by Definition 2.4) = - R U u(t) D α t,∞ ϕ(t) dt (by [ABHN11, Prop. 1.1.6]) = D α -∞,t û, ϕ S (R;L 2 (X,µ)) in L 2 (X, µ),
according to Definition 2.4 and with the notation û(t) := U(u(t)).

On the other hand,

U i α Au, ϕ S (R;L 2 (R d )) = U R i α Au(t)ϕ(t) dt = R i α hû(t)ϕ(t) dt (since UA(u(t)) = hû(t) by (4.6)) = i α hû, ϕ S (R;L 2 (X,µ)) .
Since u solves (4.1), we get

D α -∞,t û = i α hû, in S R; L 2 (X, µ) .
In particular, for µ-a.e. ξ ∈ X, there holds h(ξ) ≥ 0 and

D α -∞,t û(•, ξ) = i α h(ξ)û(•, ξ), in S (R). (4.9)
Besides, û(0, ξ) = U(u(0))(ξ) = 0 for µ-a.e. ξ in X. Thus Theorem 3.1 leads to û(•, ξ) = 0; so that u = 0. That completes the uniqueness part of the proof.

(ii) Existence. Let us show that the function u : t → e itA 1/α v solves (4.1). For, let t ∈ R. With the notation ŵ := U(w) for w ∈ L 2 (R d ), we get in view of (4.8)

û(t) = f t (h)v. (4.10)
Since f t is bounded µ-a.e. on X, we deduce that û belongs to L ∞ (R; L 2 (X, µ)). Then, thanks to Definition 2.4, we infer

D α -∞,t û, ϕ S (R;L 2 (X,µ)) = - R û(t)D α t,∞ ϕ(t) dt, in L 2 (X, µ).
Thus for µ-a.e. ξ in X,

D α -∞,t û, ϕ S (R;L 2 (X,µ)) (ξ) = - R f t h(ξ) v(ξ)D α t,∞ ϕ(t) dt (by (4.10)) = D α -∞,t f t h(ξ) v(ξ) , ϕ S (R) = i α h(ξ)û(•, ξ), ϕ S (R)
(by Th. 3.1).

Hence we have proved that, for µ-a.e. ξ in X,

D α -∞,t û(•, ξ) = i α h(ξ)û(•, ξ), in S (R). (4.11)
Moreover, by (4.10),

hû(t, •) 2 L 2 (X,µ) = X h 2 |v| 2 dµ < ∞,
since v lies in U(D(A))-see (4.5). Hence t → hû(t) belongs to L ∞ (R, L 2 (X, µ)); so that, with (4.11), we derive that D α -∞,t û = i α hû, in S (R, L 2 (X, µ)).

(4.12)

Let us go back in the direct space L 2 (R d ) and write down the expected equation for u. In view of (4.6), we have

hû(t) = UAu(t), in L 2 (X, µ).
Also (see above, the uniqueness part of the proof)

D α -∞,t û = U D α -∞,t u , in S (R, L 2 (X, µ)).
Thus, with (4.12)

D α -∞,t u = i α Au, in S (R; L 2 (R d )).
Finally, û(0) = v, thus u(0) = v. Hence u is solution to (4.1), which completes the proof of the theorem. Theorem 4.1 allows us to define the solution operator associated to (4.1). Indeed, under the assumptions of that theorem, for each t in R, we put S α (t) := e itA 1/α . (4.13)

Then, for each v in D(A), t → S α (t)v is the solution to (4.1), according to Theorem 4.1. Thus, by definition, S α is the solution operator associated to (4.1).

With these notation, we may give some properties of S α .

Theorem 4.2. Let α ∈ (0, 1) and A be a positive self adjoint operator on L 2 (R d ).

Then S α defined by (4.13), is a strongly continuous unitary group on L 2 (R d ). Moreover, its infinitesimal generator is iA 1/α .

Above, A 1/α is defined through unbounded functional calculus. More precisely, denoting by L(L 2 (R d )) the space of linear and continuous maps from L 2 (R d ) into itself, Theorem 10.4 in [START_REF] Hall | Quantum theory for mathematicians[END_REF] states that there exists a unique projection-valued measure µ

A on [0, ∞) with values in L(L 2 (R d )) such that ∞ 0 λ dµ A = A. Then A 1/α := ∞ 0 λ 1/α dµ A . The domain of A 1/α is D(A 1/α ) := v ∈ L 2 (R d ) | ∞ 0 λ 2/α dµ A v (λ) < ∞ , (4.14)
where

µ A v is the Borel measure on [0, ∞) defined for each Borel set E ⊆ [0, ∞) by µ A v (E) = v, µ A (E)v) L 2 (R d ) . (4.15) We recall that µ A (E) is an orthogonal projection of L 2 (R d ). Proof of Theorem 4.2. Recalling that f t (x) := e itx 1/α , it is clear from (4.8) that S α (t) is bounded on L 2 (R d ). Also, S α (0) is equal to Id L 2 (R d ) , the identity operator of L 2 (R d ).
Moreover, for each real numbers t, s, the properties of the functional calculus featured in [RS80, Theorem VIII.5] lead to

S α (t)S α (s) = f t (A)f s (A) = f t+s (A) = S α (t + s).
Let us show that S α (t) is unitary. For, again by functional calculus, we have

S α (t)S α (t) * = f t (A)f t (A) = |f t | 2 (A) = Id L 2 (R d ) ,
since |f t (x)| = 1 for x ≥ 0. In the same way, there holds S α (t) * S α (t) = Id L 2 (R d ) , so that S α (t) is unitary. Furthermore, by (4.6) and (4.8), for each v in L 2 (R d ),

S α (t)v -v 2 L 2 (R d ) = M ft•h v -v 2 L 2 (X,µ) = X e ith(ξ) 1/α -1 2 |v(ξ)| 2 dµ(ξ) --→ t→0 0,
by the Lebesgue dominated convergence theorem. Hence, S α is strongly continuous. Finally, let us compute its infinitesimal generator. According to [RS80, Theorem VIII.6], there holds

S α (t) = ∞ 0 f t (λ) dµ A .
Hence, for each v in D(A 1/α ), Theorem 5.9 in [Sch12, Chap 5] leads to

S α (t)v -v t -iA 1/α v 2 L 2 (R d ) = ∞ 0 e itλ 1/α -1 t -iλ 1/α 2 dµ A v (λ).
Using the fact that λ → λ 2/α lies in L 1 (0, ∞; µ A v ) (by (4.14)), we obtain with the dominated convergence theorem, that the latter integral goes to zero as t → 0. That is to say iA 1/α is the infinitesimal generator of S α .

The next result states that the time fractional problem (4.1) may be reduced to a time first order PDE. Notice that the latter PDE is obtained formally by raising the differential operators involved in (4.1) to the power 1/α. (4.17)

The requirements given in the introduction are satisfied, namely (i) the L 2 -norm of u(t) is conserved, i.e.

u(t) L 2 (R d ) = v L 2 (R d ) , ∀t ∈ R,
and (ii) S α is a group, i.e. S α (t)S α (s) = S α (t + s), ∀t, s ∈ R.

Also, if v belongs to the domain of (-∆ + V ) 1/α then Corollary 4.3 yields that u given by (4.17) is the unique solution to

d dt u = i(-∆ + V ) 1/α u, u(0) = v.

  1), C b R, D(A) is the space of bounded and continuous functions defined on R with values in D(A).

Corollary 4. 3 .

 3 Let α ∈ (0, 1), A be a positive self adjoint operator on L 2 (R d ) and v ∈ D(A 1/α ). Then (4.1) is equivalent to the following problem Find u ∈ C 1 R, L 2 (R d ) ∩ C R, D(A 1/α ) such that d dt u = iA 1/α u, in C R, L 2 (R d ) u(0) = v.(4.16)Proof. If u solves (4.1) then u(t) = exp(itA 1/α )vby Theorem 4.1. Then u is solution to (4.16) according to Theorem 4.2. Moreover, functional calculus tells us that A 1/α is self adjoint on L 2 (R d ). Thus (see for instance [Sch12, Proposition 6.5]), (4.16) has a unique solution. Consequently, (4.1) and (4.16) are equivalent. Finally, as an example, let us consider the caseA := -∆ + V, where V ∈ L ∞ (R d ) is non negative a.e. on R d , and D(A) = H 2 (R d ).Then, for α ∈ (0, 1) and v in H 2 (R d ), the time fractional Schrödinger problemD α -∞,t u = i α (-∆ + V )u, u(0) = v,hasa unique solution u, according to Theorem 4.1. Moreover, u(t) = S α (t)v = e it(-∆+V ) 1/α v, ∀t ∈ R.

This research was in part supported by a grant from IPM # 91470221.


Elias Daouk
email: elias.daouk@mines-albi.fr

Rababe Sani

Doan Pham

Ange Nzihou

Thermo-conversion of Solid Recovered Fuels under inert and oxidative atmospheres: Gas composition and chlorine distribution

Keywords:

The interest in thermo-conversion of Solid Recovered Fuels (SRFs) processes for syngas production has increased in the last decades. However, the monitoring and control of trace contaminants in SRFs such as chlorine in syngas still meet difficulties. The aim of this study is to determine the gas composition and to track the chlorine content in the products from the pyrolysis and oxidation of SRFs. Two types of SRFs from two different industrial sites in France were chosen for this study. The first one, SRF1, was chosen for its low chlorine content (0.3 wt% of Cl) comparing with the second one, SRF2 (1.1 wt% of Cl). SRF1 with low chlorine content showed that the most of Cl was transferred into the gas phase during thermo-conversion process. SRF2 with high chlorine content showed more NaCl in the residual solids after the reactions than its initial inorganic chlorine content.

Introduction

In 2012, the EU-28 generated 2 414.4 million tonnes of non-hazardous waste and 100.7 million tonnes of hazardous waste. Germany, France and the United Kingdom made up a 39% share of total EU-28 waste generation. More than 45% of the waste treated was subject to recovery, other than energy recovery, and almost half, more than 48%, was subject to disposal operations other than incineration, mostly landfilling [START_REF]Energy, transport and environment indicators -2015 edition -ESS News display page[END_REF].

The EU aims to minimize disposal methods that do not contribute to a valorisation of MSW such as landfilling. Thus, a waste recovery option that offers business opportunities for the waste management companies is to produce the so-called Solid Recovered Fuels (SRF). Referring to the standard EN 15357 [START_REF]Solid recovered fuels : terminology, definitions and descriptions: CEN/TS 15357 Technical Specification : English Version[END_REF], Solid Recovered Fuels or SRFs are "solid fuels prepared from non-hazardous waste to be utilised for energy recovery in incineration or co-incineration plants and meeting the classification and specification requirements laid down in EN 15359" [START_REF]Solid recovered fuels : specification and classes: CEN/TS 15359 Technical Specification : English Version[END_REF]. SRFs offer the opportunity to produce energy from processed fractions of MSW that cannot be recycled, which is an acceptable waste recovery option. In 2012, the remaining waste, that was not recovered nor landfilled, was incinerated with energy recovery (4% of waste) and without energy recovery (2% of waste) [START_REF]Energy, transport and environment indicators -2015 edition -ESS News display page[END_REF].

In the case of energy recovery, SRFs can thus contribute to the preservation of natural resources as they can substitute fossils fuels. Another major benefit of SRFs is their biogenic contents of the initial waste stream (wood, paper, cardboard…), a carbon dioxide (CO 2 ) neutral. This biogenic fraction varies with regions and economics and it can be estimated by the determination of biomass content using the standard EN 15440 [START_REF]Solid recovered fuels[END_REF]. Many studies [START_REF] Séverin | The biogenic content of process streams from mechanical-biological treatment plants producing solid recovered fuel. Do the manual sorting and selective dissolution determination methods correlate?[END_REF][START_REF] Wagland | Comparison of coal/solid recovered fuel (SRF) with coal/refuse derived fuel (RDF) in a fluidised bed reactor[END_REF][START_REF] Garg | An integrated appraisal of energy recovery options in the United Kingdom using solid recovered fuel derived from municipal solid waste[END_REF][START_REF] Agraniotis | Numerical investigation of Solid Recovered Fuels' co-firing with brown coal in large scale boilers -Evaluation of different co-combustion modes[END_REF] showed that SRFs derived from MSW have a high biogenic fraction (50-70%).

For thermochemical conversion processes (incineration, co-combustion, pyrolysis and gasification), there is a big debate about the best technology dealing with biomass and waste. To answer this question, Life Cycle Assessment (LCA) has been used extensively these last decade to study the environmental loads of thermal Waste-to-Energy (WtE) technologies [START_REF] Astrup | Life cycle assessment of thermal Wasteto-Energy technologies: Review and recommendations[END_REF]. LCA makes it possible to compare the different technologies, to choose the most adapted thermo-conversion process based on energy, environment and economy criteria. Arena et al. [START_REF] Arena | A life cycle assessment of environmental performances of two combustion-and gasification-based waste-to-energy technologies[END_REF] found that combustion and gasification WtE technologies had sustainable environmental performances, but the combustion unit was better for most of the selected impact categories. In contrary, Consonni and Vigano [START_REF] Consonni | Waste gasification vs. conventional Waste-To-Energy: A comparative evaluation of two commercial technologies[END_REF] showed that waste gasification has lower generation of some pollutants comparing to conventional WtE. The energy performances of combustion and gasification were very similar.

The gasification of different wastes, including SRFs, has been studied in detail by other authors [START_REF] Recari | Gasification of two solid recovered fuels (SRFs) in a lab-scale fluidized bed reactor: Influence of experimental conditions on process performance and release of HCl, H2S, HCN and NH3[END_REF][START_REF] Berrueco | Experimental investigation of solid recovered fuel (SRF) gasification: effect of temperature and equivalence ratio on process performance and release of minor contaminants[END_REF][START_REF] Arena | Gasification of a solid recovered fuel in a pilot scale fluidized bed reactor[END_REF][START_REF] Dunnu | The solid recovered fuel Stabilat®: Characteristics and fluidised bed gasification tests[END_REF]. Some of these studies have also investigated the influence of process conditions on syngas quality [START_REF] Berrueco | Experimental investigation of solid recovered fuel (SRF) gasification: effect of temperature and equivalence ratio on process performance and release of minor contaminants[END_REF][START_REF] Björkman | Release of chlorine from biomass at pyrolysis and gasification conditions[END_REF][START_REF] Pinto | Gasification improvement of a poor quality solid recovered fuel (SRF). Effect of using natural minerals and biomass wastes blends[END_REF]. By far, the most case-studies assessed combustion and gasification, while relatively few studies focused on waste pyrolysis. From these few studies, pyrolysis characteristic of SRFs were studied at laboratory scale using a thermogravimetric analyser [START_REF] Montané | Volatilization characteristics of solid recovered fuels (SRFs)[END_REF][START_REF] Grammelis | Pyrolysis kinetics and combustion characteristics of waste recovered fuels[END_REF] or using a fixed bed reactor [START_REF] Phan | Characterisation of slow pyrolysis products from segregated wastes for energy production[END_REF]. In the both cases, tests were performed in nonisothermal conditions: the SRFs samples were heated from room temperature to a fixed high temperature at a constant heating rate (between 1 and 20 °C/min). Those researchers have focused on the behaviour of SRFs samples from a pure thermogravimetric point of view which involves weight loss as a function of temperature of the sample. It is observed that the volatiles from plastics generally start to evolve at a higher temperature range (300-500 °C) as compared to those from biomass or coals (200-400 °C) [START_REF] Ahmed | Characteristics of syngas from co-gasification of polyethylene and woodchips[END_REF]. Considering the product yields, the results showed that both light and heavy liquid fractions increased with the presence of plastic material in the waste [START_REF] Sharypov | Co-pyrolysis of wood biomass and synthetic polymer mixtures. Part I: influence of experimental conditions on the evolution of solids, liquids and gases[END_REF]. They supposed that the olefinic products from plastics thermal conversion react with some products from the biomass depolymerisation to result in the formation of light liquids.

Nevertheless, due to their nature or preparation method, SRFs can contain high levels of chlorine, sulphur, and heavy metals. For thermal processes, they cause fouling, corrosion of pipes and down-stream units, besides poisoning of catalysts [START_REF] Recari | Gasification of two solid recovered fuels (SRFs) in a lab-scale fluidized bed reactor: Influence of experimental conditions on process performance and release of HCl, H2S, HCN and NH3[END_REF]. Referring to EN 15508 [START_REF] Agraniotis | Numerical investigation of Solid Recovered Fuels' co-firing with brown coal in large scale boilers -Evaluation of different co-combustion modes[END_REF], the technological criterion for SRFs is chlorine content. The chlorine nature in SRF is dual, organic from chlorinated polymers (e.g., PVC) and inorganic, for instance salts (NaCl and/or KCl) from food waste [START_REF] Wagland | Comparison of coal/solid recovered fuel (SRF) with coal/refuse derived fuel (RDF) in a fluidised bed reactor[END_REF][START_REF] Garg | An integrated appraisal of energy recovery options in the United Kingdom using solid recovered fuel derived from municipal solid waste[END_REF][START_REF] Ma | Chlorine characterization and thermal behavior in MSW and RDF[END_REF]. In general, the Cl content should be below 0.5 wt% or 1.0 wt% depending on the application.

Worldwide, in term of polymer consumption, polyvinyl chloride (PVC) is ranked second behind polyethylene [START_REF] Scheirs | Polymer recycling: science, technology and applications[END_REF]. Thus, PVC is omnipresent in SRFs composition and they will affect the thermo-conversion and gas composition. Around a half of the chlorine in municipal solid waste (MSW) comes from PVC, which means a half of the HCl in the combustion gases from MSW incinerators [START_REF] Ma | Study of the low temperature pyrolysis of PVC[END_REF]. In addition to the formation of HCl, other forms of chlorine such as chlorides of Na, K, Zn, Pb, Sn, Sb, Fe, and other elements in the gas phase can cause high temperature corrosion of incinerators and reactors. To minimize the costly chlorine-associated problems, it is important to study the chlorine concentration in waste, chlorine species and thermal behaviour of chlorine [START_REF] Ma | Chlorine characterization and thermal behavior in MSW and RDF[END_REF].

In this study, we will focus on the fate of chlorine during SRFs pyrolysis and oxidation in an isothermal furnace. The operational conditions of the isothermal furnace, used in this study, imitate the industrial scale operation conditions. The main objective of this study is to determine gas composition and to track the chlorine content during pyrolysis and oxidation for both SRFs.

Materials and methods

Samples preparation and characterization

Two types of SRFs from two industrial producers in France were chosen for this study. During the preparation of the first one (SRF1), an optical detector was used in order to minimize its chlorine content. The second one (SRF2) was prepared without optical detector, and contains obviously more chlorine than that in the first SRF.

Before the characterization and thermo-conversion tests, the two SRFs samples were dried and ground under 1 mm using a rotary cutting mill equipped with a 1 mm sieve (Fig. 1).

The proximate analyses of the two SRFs were carried out according to the EN 15402 [26] for volatile matter and EN 15403 [START_REF]Solid recovered fuels[END_REF] for ash content. The fixed carbon was deduced by difference. The elemental composition of the samples was carried out according to:

• EN 15407 [28] for the determination of carbon (C), hydrogen (H) and nitrogen (N) content, with a CHN analyser Flash 2000.

• EN 15408 [29] for Sulphur (S) and chlorine (Cl) content. This method consists of an oxygen combustion procedure of the sample in a bomb. Sulphur and chlorine compounds are converted respectively to chloride and sulphate which are absorbed and dissolved in an absorption solution (KOH 0.2 ml/l solution). After each sample's combustion in the calorimetric bomb IKA C500, the bomb was washed out to recover the solution, which were subsequently analysed by ionic chromatography to determine S and Cl contents.

• EN 15410 [30] for the determination of the content of major and trace elements. The method consists of using an inductively coupled plasma-optical emission spectrometer (ICP-OES). Before analysis, the SRFs samples were digested at 220 °C with an adapted acid mixture using a special closed reactor. The dissolution conditions were optimized to dissolve completely these SRFs samples. The most adapted acid mixture was: sulfuric (3 ml), nitric (3 ml), hydrofluoric (1 ml) and hydrogen peroxide (3 ml), with a SRF sample of about 200 mg.

The calorific value was measured following the standard EN 15400 [START_REF]Solid recovered fuels[END_REF], using the calorimetric bomb IKA C500.

Table 1 depicts the results for the characterization of the two SRFs samples. The data of the proximate analysis, C, H, N, O, S, Cl and the LHV are the average of at least triplicate analyses, and the data for the other major and trace elements are the average of at least 5 analyses. The proximate analysis shows that the volatile matters content of the SRF2 is slightly higher than that of SRF1. The ash content was high for both SRFs and reached more than 17%. Finally, the fixed carbon of these two SRF samples accounts for 9.4 and 7.5 wt%, respectively and is in accordance with the literature data [START_REF] Grammelis | Pyrolysis kinetics and combustion characteristics of waste recovered fuels[END_REF]. The very low content of the fixed carbon measured in this work is commonly found for this kind of SRFs, which mainly are composed of plastics, refused paper and biomass. According to elemental analysis, there was no notable difference between the two SRFs except for chlorine content and some other elements (i.e. C, S, Ca, Al, and Zn). The chlorine content of these SRFs (0.30 ± 0.20 for SRF1 and 1.09 ± 0.15 for SRF2) confirmed the effect of PVC elimination step during their preparation. Knowing that plastic materials present the highest elemental carbon content (63-91 wt%) among the SRFs samples [START_REF] Grammelis | Pyrolysis kinetics and combustion characteristics of waste recovered fuels[END_REF], this could explain the highest C content observed for SRF2 (49.9 wt%) in comparison with SRF1 (46.2 wt%). The higher plastics content in SRF2 comparing to SRF1 can explain the slightly higher amount of H observed in SRF2.

mm

Chlorine content in SRFs 2.2.1. Total chlorine

One of the parameters to assess the thermo-conversion performance is the tracking of the evolution of chlorine from the SRF to the products (solid residue and gas). This paragraph is dedicated to detail the method adapted and used in this study to measure the chlorine content in SRF and solid residue (char and ash) after thermo-conversion process.

As mentioned previously, the method used for chlorine determination is based on the standard EN 15408 [START_REF]EN 15408:2011 -Solid recovered fuels[END_REF]. This method consists of an oxidation of the sample in a calorimetric bomb followed by the analysis of the resulting solution in an ionic chromatography. In order to check the reliability and to validate our method of determination of chlorine content in the SRFs, a control test was carried out. For this purpose, a model sample consisting of a mixture of wood with a well-known content of polyvinyl chloride (PVC) was used. The average recovery for chlorine from the calorimetric bomb to the solution for ionic chromatography analysis was around 94%. This value is in the range recommended by the standard EN 15408 [START_REF]EN 15408:2011 -Solid recovered fuels[END_REF], which allows validating the analysis method.

Organic and inorganic chlorine

The total chlorine content in the SRFs can be separated into two main categories:

• Organic chlorine from plastics, mainly PVC; • Inorganic chlorine from salts, mainly kitchen and garden wastes.

Organic and inorganic chlorine have a distinctly different thermal behaviour. It is important to quantify the proportion of each chlorine type in SRFs in order to understand and to track their evolutions during thermo-conversion tests. For this purpose, an elution test was used which provides the possibility to distinguish organic and inorganic chlorine [START_REF] Ma | Chlorine characterization and thermal behavior in MSW and RDF[END_REF]. Fig. 2 shows the steps of the elution procedure used in this study to determine the inorganic and organic chlorine contents in SRFs. This method is based on the fact that, for SRFs, water-soluble chlorine is referred to as inorganic chlorine and non-water-soluble chlorine, which remains in the solute after elution, is referred to as organic chlorine.

Fig. 3 shows the results of chlorine content of both SRFs studied in this study (SRF1 and SRF2), and the distribution of organic and inorganic chlorine for each SRF from the elution method. The amount of inorganic chlorine was similar for both SRFs (about 0.12 wt%). On the other hand, the organic chlorine fraction was much higher for SRF2 (0.98 wt%) than for SRF1 (0.19 wt%). In fact, for the SRF1, PVC, which was the main organic chlorine source, was mostly eliminated during its preparation by an optical detector as mentioned above.

Experimental device and procedure

Fig. 4 shows the experimental setup used to study SRFs pyrolysis and oxidation under isothermal conditions. It is mainly composed of a reactor tube (20 mm internal diameter) made by inert ceramic and heated by an electrical furnace, and a gas trap system.

Thermo-conversion experiments were performed under both inert atmosphere of nitrogen, and oxidant atmosphere of air. The experiments were carried out with a fixed inlet gas flow rate of 200 ml/min, under isothermal conditions. After heating the reactor to the desired temperature (800 °C), the inert crucible with the SRF sample (700-800 mg) was rapidly introduced in the reactor, for less than 15 s to start the thermos-conversion process. The temperature evolution around the sample was measured, and we observed that the heating Fig. 2. Elution procedure for the determination of organic and inorganic chlorine contents. At the outlet of the reactor, the produced gas passes through a series of condensers containing isopropyl alcohol in order to remove particulates and condense water and tars. After, the permanent gas is recovered along the test, during 20 min, in a sampling bag for subsequent analyses. Each test was repeated at least threefold.

A micro-GC (MyGC Agilent) was used to quantify the major components in the permanent gas (H 2 , O 2 , N 2 , CO, CO 2 , CH 4 , C 2 and C 3 ) recovered in the sampling-bag. Nitrogen, which is an inert gas, was used as internal standard to calculate the total mass of permanent gas (m PG ), from the sample, at reactor output according to Eq. ( 1):

= - - m m Y Y Y (1
)

PG N i N o N o O o , , , , 2 2 2 2 (1)
where m N i are respectively the nitrogen and oxygen mass fractions in the gas recovered in the bag (calculated using the micro-GC results). The mass of condensates (tars, particulates and water) was deduced taking into account the mass of permanent gases and the mass of the solid fraction that remained in the reactor. In the case of pyrolysis, the sum of the products is equal to the SRF sample mass. In the case of oxidation, a part of the oxygen from the introduced air reacts with the SRF sample. Thus, the total mass of all the products is calculated from the mass of SRF sample and the oxygen consumed during the reaction.

The residual solid product after test (char after pyrolysis test or ash after oxidation test) was collected and analysed to measure its chlorine content. For chars, the method based on the standard EN 15408 [START_REF]EN 15408:2011 -Solid recovered fuels[END_REF] was directly used. Because ashes are of an incombustible nature, a mixture of wood and ash was prepared. Then, the standard EN 15408 [START_REF]EN 15408:2011 -Solid recovered fuels[END_REF] was applied for this mixture to measure its chlorine content. The crystalline structure of Cl containing compounds in the produced chars or ashes was also analysed by X-ray Diffraction (XRD, PanAnalytical X'PertPhilips). XRD patterns were analysed by using software X4Pert HighScore based on ICDD database.

Thermodynamic equilibrium model

In order to develop an idea on the distribution and forms of chlorine in both gaseous and solid phases under the thermos-conversion conditions used, thermodynamic equilibrium calculations were carried out using FactSage.

FactSage is a thermodynamic program developed by Thermfact in Canada and GTT-Technologies in Germany, and it is one of the largest fully integrated database computing systems in chemical thermodynamics. Equilibrium calculations use the Gibbs energy minimization method at fixed temperature and pressure. In our case, the pressure was fixed at 1 atmosphere and the temperature was varied from 500 to 800 °C.

Results and discussion

SRFs pyrolysis

Product yields and permanent gas composition

Fig. 5 shows the permanent gas, char and condensable yields obtained from pyrolysis experiments of both SRFs at 800 °C. For each experiment, the product yields were calculated as the percentage of mass of compound per mass of dried SRF sample. The results show the similar char yield (22-23 wt%) for both SRFs. The permanent gas yield was slightly higher for SRF2 (51 wt%) compared to SRF1 (45 wt%), and the condensable yield was slightly lower for SRF2 (27 wt%) compared to SRF1 (32 wt%). These differences presumably derived from the feedstock composition and/or can be related to some tar cracking toward permanent gas products for SRF2.

The permanent gas compositions for both SRFs pyrolysis are presented in Fig. 6. In general, despite the feedstocks differences, the permanent gas composition varies slightly. The major gases were always CO (30-33%vol.) followed by H 2 (16-18%vol.), CO 2 (12-19% vol.), CH 4 (13-16%vol.) and C 2 H 4 (11-18%vol.). Some traces of light hydrocarbons (C 3 , C 2 H 6 and C 2 H 2 ) were also detected. The volume ratio of H 2 /CO was close to 0.54 for both SRFs. Considering the volume ratio of CO/CO 2 , we observed that it was higher for SRF2 (2.5) than for SRF1 (1.74). This trend can be related to the more intense cracking of the hydrocarbons present in the gas during SRF2 pyrolysis as mentioned above [START_REF] Boroson | Product yields and kinetics from the vapor phase cracking of wood pyrolysis tars[END_REF][START_REF] Morf | Mechanisms and kinetics of homogeneous secondary reactions of tar from continuous pyrolysis of wood chips[END_REF][START_REF] Rath | Cracking reactions of tar from pyrolysis of spruce wood[END_REF].

The monomer forming the polyethylene plastic PE, ethylene (C 2 H 4 ) is an indicator of PE thermal degradation [START_REF] Ahmed | Characteristics of syngas from co-gasification of polyethylene and woodchips[END_REF]. This can explain the relatively high percentage of ethylene in the permanent gas produced by both SRFs pyrolysis. Fig. 6 shows that the ethylene is higher for SRF2 compared to SRF1. This result cannot be related directly and only to the amount of PE in the feedstock. In fact, Ahmed et al. [START_REF] Ahmed | Characteristics of syngas from co-gasification of polyethylene and woodchips[END_REF] showed that blends of PE with another matter, as woodchips (WC), always yielded higher values of hydrocarbons and ethylene compared to PE alone. This indicates the synergic interaction effect between PE and WC, or other components, during the thermal degradation. The presence of acetylene (C 2 H 2 ) in the gas for SRF2, but not for SRF1, can be attributed to the thermal decomposition of PVC as shown in Eq. ( 2). PVC was much more present in SRF2 than in SRF1.

→ + CH CHCl nHCl nC H ( ) n 2 2 2
(2)

3.1.2. Chlorine distribution Fig. 7 shows the chlorine distribution between the gas phase (volatile chlorine) and the solid residual phase (non-volatile chlorine) after the pyrolysis test at 800 °C for both SRFs. We observe that the distinction between inorganic and organic chlorine (Fig. 3) does not correlate with the differentiation between volatile and non-volatile chlorine after pyrolysis (Fig. 7). For SRF1, the majority of the chlorine, around 0.22 wt%, was transferred into the gas phase, and only 0.08 wt % remained in the solid phase. For SRF2, the majority of chlorine was transferred to the gas phase (0.69 wt%), but in this case an important part of the chlorine, nearly 0.41 wt%, remained in the char. For SRF2, the amount of non-volatile chlorine is much higher than inorganic chlorine initially present in the sample (0.12 wt%).

In order to understand and to track the chlorine distribution between the gas and solid phases, the elemental composition of the two SRFs studied (Table 1) was inputted in the thermodynamic equilibrium calculations using FactSage software. The results of the thermodynamic calculations are shown in Fig. The result showed that the chlorine content of the fuels has a significant influence on the production of gaseous HCl, KCl, NaCl, Na 2 Cl 2 and K 2 Cl 2 , and of the solid phases of NaCl and KCl. This result is in a good agreement with the literature [START_REF] Dionel | Effects of feed composition on boiler corrosion in waste-to-energy plants, 12th North[END_REF]. In the case of SRF1, below 700 °C, chlorine exists totally in the solid phase as NaCl and KCl. Above 700 °C the solid phase starts to decrease sharply, leading in high NaCl, KCl, Na 2 Cl 2 and K 2 Cl 2 in the gas phase. The thermodynamic calculation shows that at 800 °C all the chlorine is transferred to the gaseous phase.

In the case of SRF2, below 700 °C, about a half of the chlorine is transferred into the gaseous phase as HCl and other half remains in the solid phase mainly as NaCl and a small part as KCl. Above 700 °C, solid chlorine compounds start to volatilize sharply leading to higher ratio of vaporised chlorides including NaCl, Na 2 Cl 2 and KCl. HCl remains constant in the range of temperatures between 500 and 800 °C for SRF2. So, according to the thermodynamic calculation, chlorine in the initial SRFs can be released to the gas phase under the form of HCl and inorganic salts of Na, K and Ca which are the main inorganic elements of both SRFs.

In order to confirm the results of the thermodynamic calculation, Fig. 9 reports the XRD patterns for both chars produced after pyrolysis tests of SRFs. The presence of the crystalline structure of NaCl in both pyrolysis chars was confirmed, as predicted by the thermodynamic calculation above. The absence of other crystalline phases of Cl containing compounds such as KCl could be due to their low content below the detection limit of XRD, which are in line with the results in Fig. 8 by the thermodynamic calculation. Other crystalline phases were also observed. SiO 2 was commonly found in both chars. On the other hand, Ca(OH) 2 and Ca 3 Al 2 O 6 were found only with the char from SRF1, while CaTiO 3 was only found with the char from SRF2. These results could be explained by the elemental analysis with high contents of Ca and Al in SRF1, and high content of Ti in SRF2 (Table 1).

From these results, it is highly recommended to eliminate chlorine, in particular organic chlorine (PVC), during SRF preparation before thermo-conversion processes. It appears to be simpler and cheaper to manage PVC eliminated from SRF preparation than to treat volatile chlorine compounds in the gas phase.

3.2. SRFs oxidation 3.2.1. Product yields and permanent gas composition Fig. 10 shows the permanent gas, char and condensable yields obtained from oxidation experiments for both SRFs at 800 °C. As a result of using the mass of dry SRF sample basis, the sum of permanent gas, condensable and ash yields was higher than 100%. This is explained by the reaction of O 2 with SRFs components. The mass ratio of O 2 /SRF for consumed oxygen during the reaction is defined according to Eq. (3).

= - O SRF m m m O O o SRF 2 , i 2, 2 (3)
where m O i 2, is the mass of introduced oxygen during the 20 min of recovering gas in the sampling-bag, m O o 2, is the mass of oxygen in recovered in the sapling-bag. The same mass ratio O 2 /SRF of 0.19 was obtained for both SRFs. The results showed the similar product yields for both SRFs during oxidation. Exactly the same condensable yield of 9 wt% was obtained. The ash yield was higher for SRF1 (21 wt%) compared to SRF2 (15 wt%), leading to higher permanent gas yield for SRF2 (95 wt%) compared to SRF1 (89 wt%).

The permanent gas compositions for both SRFs oxidation are presented in Fig. 11. Under oxidation atmosphere, the permanent gas compositions for both SRFs were nearly similar. CO 2 represents the majority of the permanent gas (45-50%vol. of permanent gas) followed by CO (18-22%vol.). Despite the oxidation atmosphere, high concentration of CO was observed which could be explained by the short residence time, less than 10 s, of the gas phase inside the reactor. Other permanents gases including C 2 H 4 (8-13%vol.), CH 4 (8-12%vol.) and H 2 (5-7%vol.) were also observed even under oxidation conditions.

Chlorine distribution

Fig. 12 shows the chlorine content in the gas phase and the ash after oxidation of the SRFs samples. In the two cases, most of chlorine was transferred into the gas phase as HCl and vaporised metal chlorides. For SRF1, less than 0.06 wt%, remained in the solid phase. This was smaller than the inorganic chlorine content which was originally present in the SRF1 (0.12 wt%). For SRF2, the measured non-volatile chlorine was of 0.17 wt%, which was higher than the inorganic chlorine content originally present in the SRF2 (0.12 wt%). So, the initial organic chlorine in SRF2 was partially remained in the solid fraction.

Thermodynamic calculations under oxidative atmosphere were also performed and the results obtained were similar compared to those under inert atmosphere (Fig. 8). At 800 °C, the calculated yield of chlorine in the solid phase for SRF2 (0.3 wt%) is higher than the experimental measured yield presented in Fig. 12 (0.18 wt%). These differences may be due to the real temperatures during oxidation that can reach values higher than the furnace temperatures, because of the local exothermic reactions. The higher is the temperature, the higher is the chlorine in the gas phase [START_REF] Wang | HCl formation from RDF pyrolysis and combustion in a spouting-moving bed reactor[END_REF]. Fig. 13 reports the XRD patterns for both ashes recovered after oxidation tests of SRFs. NaCl was identified as the main chlorinated crystalline compound for both ashes. For the ash of SRF1, a very high peak for CaO was observed. This could be explained by the high Ca content in SRF1 (Table 1) under CaCO 3 form which decomposed to form CaO at 800 °C.

Conclusion

Thermo-conversion of two different SRFs was performed under inert and oxidative atmosphere in an isothermal furnace at 800 °C. The SRFs were mostly different from their chlorine contents (0.3 wt% for SRF1 and 1.1% for SRF2). The composition of permanent gas was determined and the distribution of chlorine in the products was established.

Under inert atmosphere of nitrogen, the product yields were nearly similar for both SRFs: about 22-23 wt% of char, 27-32 wt% of condensable and 45-51 wt% of permanent gases. The major produced gas were CO (30-33%vol.) followed by H 2 (16-18%vol.), CO 2 (12-19% vol.), CH 4 (13-16%vol.) and C 2 H 4 (11-18%vol.). Some traces of 2 and C 3 were also detected.

Under oxidative atmosphere of the air, both SRFs reacted with the same amount of oxygen during the reaction: the mass ratio of O 2 /SRF was around 0.19. The same condensable yield of 9 wt% was obtained for both SRFs. The ash yield was higher for SRF1 (21 wt%) compared to SRF2 wt%), leading to a higher permanent gas yield for SRF2 (95 wt %) compared to SRF1 (89 wt%). CO 2 represented the majority of the permanent gas (45-50%vol.) followed by CO (18-22%vol.), C 2 H 4 (8-13%vol.), CH 4 (8-12%vol.) and H 2 (5-7%vol.).

The chlorine initially present in the SRFs was mostly transferred into the gas phase, under inert and oxidative atmospheres, as HCl and vaporised metal chlorides (NaCl, Na 2 Cl 2 , KCl and K 2 Cl 2 ). Thermodynamic calculations and XRD analysis showed that the crystalline structure of chlorine stacked in the residual solid after reaction was mainly in form of NaCl. In the case of SRF2, with high chlorine content, the amount of non-volatile chlorine was higher than inorganic chlorine initially present in the sample. This indicates that organic chlorine (PVC) partially contributed to the chlorine content of the solid fraction during the thermo-conversion of SRF2. From the results obtained, it is recommended to eliminate chlorine, in particular organic chlorine (PVC), before thermo-conversion processes.

Acknowledgement

The authors thank the technicians at the RAPSODEE centre UMR CNRS 5302 for their technical help. We also thank ADEME for financial

Fig. 1 .

 1 Fig. 1. Illustration of SRF sample for laboratory analysis and experiments.

Fig. 3 .

 3 Fig.3. Chlorine content and its distribution between organic and inorganic chlorine for both studied SRFs.

, 2

 2 is the mass of introduced nitrogen during the 20 min of recovering gas in the sampling-bag. Y N o , 2 and Y O o , 2

Fig. 4 .Fig. 5 .

 45 Fig. 4. Diagram of experimental device. (1) Furnace; (2) reactor; (3) isopropyl alcohol impengers; (4) gas sampling bag; (5) three-way valve; (6) crucible with SRF sample.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Permanent gas composition of SRFs pyrolysis.

Fig. 8 . 9 .Fig. 10 .

 8910 Fig. 8. Percentage of HCl, NaCl, KCl, Na 2 Cl 2 and K 2 Cl 2 based on dry SRF in both gaseous and solid phases calculated with FactSage.

Fig. 11 .Fig. 12 .Fig. 13 .

 111213 Fig. 11. Permanent gas composition of SRFs oxidation.

Table 1

 1 Proximate and ultimate analyses of of SRF1 and SRF2.

	Parameters		SRF 1	SRF 2
	Moisture content (wt%, w.b.)		7.3 ± 0.2	8.3 ± 0.2
	Proximate analysis (wt%, d.b.)	Volatiles	73.3 ± 0.1	74.8 ± 0.5
		Fixed	9.4 ± 0.3	7.5 ± 0.7
		carbon *
		Ash	17.3 ± 0.3	17.7 ± 0.2
	Elemental composition (wt%,	C	46.20 ± 0.82	49.90 ± 0.67
	d.b.)	H	6.20 ± 0.42	6.63 ± 0.35
		N	0.60 ± 0.04	0.66 ± 0.01
		S	0.10 ± 0.02	0.51 ± 0.10
		Cl	0.30 ± 0.20	1.09 ± 0.15
		Al	0.99 ± 0.51	0.18 ± 0.05
		Ca	3.35 ± 0.59	1.14 ± 0.37
		Fe	0.24 ± 0.06	0.18 ± 0.01
		K	0.14 ± 0.06	0.12 ± 0.03
		Mg	0.22 ± 0.10	0.21 ± 0.01
		Na	0.46 ± 0.01	0.47 ± 0.01
		P	0.15 ± 0.02	0.09 ± 0.04
		Si	1.00 ± 0.03	1.19 ± 0.11
		Sn	0.08 ± 0.02	0.10 ± 0.01
		Ti	0.14 ± 0.01	0.24 ± 0.06
		Zn	0.03 ± 0.01	0.23 ± 0.07
	Trace elements (mg/kg, d.b.)	As	39.0 ± 23.3	26.1 ± 14.0
		Cd	13.6 ± 10.6	5.6 ± 1.4
		Co	9.2 ± 6.4	10.2 ± 8.0
		Cr	77.0 ± 5.8	126.4 ± 19.8
		Cu	45.0 ± 10.8	610.2 ± 105.6
		Mn	78.4 ± 5.4	97.5 ± 48.8
		Ni	46.7 ± 30.4	49 ± 11.6
		Pb	27.7 ± 30.9	1.4 ± 1.2
		Sb	45.7 ± 7.1	28.8 ± 4.8
		Tl	1.0 ± 8.7	1.5 ± 0.5
		V	28.9 ± 3.5	89.1 ± 12.6
		Hg	0.10 ± 0.01	0.74 ± 0.03
	LHV (MJ/kg, d.b.)		23.2 ± 0.2	22.6 ± 0.3

* By difference; w.b.: wet basis, d.b.: dry basis.