Validation of a Fluid-Structure Interaction Model against SSEXHY test facility measurments

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TESI

VALIDATION OF A FLUID STRUCTURE

INTERACTION MODEL AGAINST

SSEXHY TEST FACILITY

MEASURMENTS

HALIM Ossama

Università di Pisa

Dipartimento di Ingegneria Civile e Industriale (DICI)

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U

NIVERSITÀ DI

P

ISA

Scuola di Ingegneria

Corso di Laurea Magistrale in INGEGNERIANUCLEARE Dipartimento di INGEGNERIACIVILE EINDUSTRIALE(DICI)

Validation of a Fluid Structure Interaction

model against SSEXHY test facility

measurments

Tesi di

HALIM Ossama

Relatori:

Prof. AMBROSINI Walter

Prof. CARCASSI Marco

Ing. ETIENNE Studer

Ing. KUDRIAKOV Sergey

Sessione di Laurea Magistrale 28 settembre 2020 Anno Accademico 2019/2020

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U

NIVERSITÀ DI

P

ISA

Scuola di Ingegneria

Corso di Laurea Magistrale in INGEGNERIANUCLEARE Dipartimento di INGEGNERIACIVILE EINDUSTRIALE(DICI)

Validation of a Fluid Structure Interaction

Model against SSEXHY test facility

measurments.

Tesi di

HALIM Ossama

Relatori:

Prof. AMBROSINI Walter

...

Prof. CARCASSI Marco

...

Ing. ETIENNE Studer

...

Ing. KUDRIAKOV Sergey

...

Candidato:

HALIM Ossama

...

Sessione di Laurea Magistrale 28 settembre 2020 Anno Accademico 2019/2020

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Acknowledgments

This research work was carried out at the Commissariat à l’Energie Atomique (CEA), Paris Saclay. I would like to thank the Thermo-hydraulics and Fluid Dy-namics Unit (STMF) of the Nuclear Energy Division (DEN) for offering the oppor-tunity to learn and work side by side with pioneer scientists.

I would like to express my gratitude to Studer ETIENNE, my supervisor and men-tor for his support and guidance. I would like also to thank Sergey KUDRIAKOV for our fruitful and constructive discussions and for helping me to integrate into the team in such short time. Special thanks go to Alberto BECCANTINI and Pas-cal GALON for their help and explanations, which helped me to understand the numerical model.

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Abstract

In industry sectors handling hydrogen, explosions are feared because of their harm for on people and property. In the nuclear industry, hydrogen explosions, which are possible during severe accidents, can challenge the containment and potentially release radioactive materials into the environment. The Three Mile Island accident in the United States in 1979 and more recently the Fukushima accident in Japan have highlighted the importance of this phenomenon for a safe operation of nuclear installations and during accident management.

In 2013, the French Research Agency (ANR) launched the MITHYGENE project with the main aim of improving knowledge on hydrogen risk for the benefit of reac-tor safety. In this project, one of the areas of work concerned the effect of hydrogen explosions. In this context, CEA carried out a test program with its SSEXHY fa-cility and CFD computational analysis with the EUROPLEXUS code to build a database on deformations of simple structures following an internal hydrogen ex-plosion. Different regimes of explosion propagation have been studied from deto-nation to slow deflagration. Different targets were tested such as plates of variable thickness and cylinders. Detailed instrumentation was used to obtain data for the validation of coupled CFD models of combustion and structural dynamics.

In this work, a validation of the models addressing Fluid Structure Interaction (FSI) implemented in EUROPLEXUS code was performed. In the validation procedure a step-by-step approach was adopted. Three main steps were involved starting by validating the target boundary conditions with quasi-static tests. Then, in a second step, the dynamics of the structure was studied in a decoupled manner using the verified boundary conditions obtained from the first step; this resulted in obtaining reasonable results compared to the experimental ones. Eventually, the third step

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List of Tables

2.1 Qualitative Differences between Detonation and Deflagration in Gases. 20 3.1 Stainless Steel 304L Mechanical properties . . . 46 5.1 Different FSI algorithms available in EUROPLEXUS [11]. . . 76 6.1 Material properties of the Plate and Gasket . . . 83 6.2 Composition of hydrogen/air mixtures at P0= 1 bar and T0= 293 K. 92

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Chapter 1 Introduction

1.1 Background and motivation

The recent difficulties in relying on fossil fuels for the long-term supply of electric-ity have become apparent. Regarding the fact that reserves of crude oil and natural gas are limited and that the continuing use of fossil fuels is adverse to the envi-ronment, promoting climate changes, seeking for alternatives solutions becomes inevitable. Consequently, the search for a convenient Energy-Mix has emerged, en-couraging the use of all the available energy sources to secure the energy needs. At the same time the nuclear reactors based on the fission reaction represent a well established technology with promises of future developments of advanced, and pro-viding that the under development breeder reactor concepts extending the optimum utilization of the Uranium resources. As a result, the nuclear power reactors are the subject of continuous attention and are major source of clean power.

Nuclear power plants (NPPs) are designed to operate safely. The main goal of safety is to ensure that a NPPs will not contribute significantly to public health risks. This translates into the prevention of the release of radioactive material into the environment from the NPP. Consequently, to keep radioactive material out of the environment, NPPs are constructed with several barriers. The first barriers are the fuel matrix and the sealed cladding which encase the ceramic uranium fuel pellets. While further barriers are the heavy steel reactor vessel and the primary cooling water system piping. The last barrier is finally the reactor building containment, which must be attentively preserved owing to its role of ultimate provision to avoid

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dispersion of radioactivity [37].

In case of a severe accident, the degradation of the reactor core occurs resulting in large quantities of hydrogen to be released into the containment due to zirco-nium–water interaction. Other scenarios may be envisaged, involving the accu-mulation of the molten core in the reactor pressure vessel lower plenum which may eventually lead to the rupture of the vessel and the spilling of the melt in the reactor cavity. A phenomenon known as Molten Corium–Concrete Interaction (MCCI) where a significant part of hydrogen is produced during the early phase of corium–concrete interaction. The main threat caused by the presence of a large amount of hydrogen in a nuclear power plant is its possible combustion [37].

As the Three Miles Island (TMI-2) in 1979 and Fukushima in 2011 accidents re-vealed, hydrogen combustion could cause high pressure spikes (i.e. mechanical loads) and high temperatures (i.e. thermal loads). These explosive loads can dam-age the essential equipment necessary to mitigate the accident or cause immediate failure of the containment, as demonstrated in Fukushima, thus breaking the last safety barrier and allowing the release of fission products that have accumulated in the containment to the environment.

Both these accidents proved the importance of a correct management of the risk of hydrogen explosions for the safety of nuclear reactors. The dynamics of combustion reactions is strongly influenced by boundary conditions: confined geometries tend in fact to promote flame acceleration. Under certain circumstances, the overpressure generated by the lead shock ahead of the flame may cause structural failure.

1.2 Contributions of this work

In this work, we address the modelling of experimental data collected in a middle-scale combustion facility named SSEXHY (Structure Subjected to an EXplosion of HYdrogen) [36]. The facility was used to generate detonation loads causing a dy-namic elastoplastic response of the examined structure (circular plate), with a large deflection in the center which corresponds to the primary shock wave reflection. The data showed that the secondary and tertiary reflections had negligible effects

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1.2. Contributions of this work 17

on plate deflection.

In this perspective, the main scope of the present work is to set up a validation methodology for the Fluid-Structure Interaction models in EUROPLEXUS code [1] against the experimental data, in which fluid-structure interaction effects were investigated by comparing pressure records from the experiments with rigid mas-sive walls with those ones with deformable plates.

In Chapter 2, an overview of the characteristics of deflagration and detonation phenomena and the balance equations for combustion with a particular focus on the Hugoniot equation and the Chapman-Jouguet theory are provided. Then, the mechanical response of clamped plates are summarized. A brief description of the SSEXHY test facility and the description of the instrumentation used as well as the characteristics of the deformed structures are given in Chapter 3. In Chapter 4 an overview of the Finite Element Method as well as the description of different meshes used in the EUROPLEXUS code [1] are introduced. Chapter 5 gives a description of the numerical model implemented in EUROPLEXUS code, in which the balance and conservation laws of the structure and fluid subdomains are briefly discussed; in addition, the material model, the constitutive equation, and the FSI coupling condition are described. Eventually, the numerical results and their dis-cussion are presented in Chapter 6.

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Addressed Phenomenology

The integrity of the containment during severe accident scenarios must be ensured in safety analyses, as it represents the last barrier against releasing radioactive sub-stance to the environment. Combustion of the hydrogen produced during damage to the core and interaction between the core, overpressure and the concrete are all phenomena that could compromise the tightness of this third barrier. These effects can be quantified, first by ascertaining the relative pressure loads for the different combustion modes and then studying the mechanical response of the structure.

In this chapter, the physics of combustion is discussed, addressing the two main categories of combustion process, i.e. Detonation/Deflagration, as well as high-lighting an approximate closed form solution for detonation thermodynamic quan-tities. Also, a summary on the structure mechanical response including the static and dynamic behavior is given.

2.1 Physics of Combustion

2.1.1 Introduction

Considering a premixed hydrogen/air mixture, different combustion modes can be observed depending upon the initial and boundary conditions such as pressure, tem-perature, gas composition, degree of homogeneity, geometry, and ignition charac-teristics [27]. The zone where very rapid chemical reaction occurs is often called the flame front (also known as reaction wave). Two main classes of the

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combus-2.1. Physics of Combustion 19

tion wave can be observed, either Detonation or Deflagration, based on the speed of wave propagation through a reacting mixture.

These two combustion modes have different properties, which can be easily dis-tinguished. For the sake of simplicity, a one-dimensional planar combustion wave is shown in the following Fig. 2.1. The wave is moving to the left at a constant velocity u1; the unburnt gases ahead of the wave can be thus considered to move

at velocity u1 toward the wave front. The subscript 1 indicates conditions of the

unburnt gases ahead of the wave, and subscript 2 indicates conditions of the burnt gases behind the wave.

Figure 2.1: Schematic of a stationary one-dimensional combustion wave, Adapted from [27].

Deflagration is a combustion wave propagating at a subsonic speed, typically at ve-locities of the order of some meters per second. It is a mechanism which depends on the external conditions as the initial temperature and ambient pressure of the mixture [26]. Deflagration creates an expansion wave, in which both pressure and density decrease across the wave. Moreover, the unburnt gases are heated by heat transfer from the hot burnt gases.

On the other hand, detonation is a combustion wave propagating at a supersonic speed, typically at velocities of the order of few kilometers per second [27]. Det-onation combustion mode creates a compression wave in which both pressure and density increase across the wave. Therefore it is characterized by the reaction front in which the combustion is occurring and the strong shock wave ahead of it. This shock front heats the unburnt gases ahead of the combustion wave to the burning temperature.

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Typical data on the upstream and downstream conditions provide the physical char-acteristics associated with a deftagration and detonation wave. Qualitative differ-ences between deflagration and detonation in premixed gases can be summarized as in Table 2.1 [27].

Table 2.1: Qualitative Differences between Detonation and Deflagration in Gases.

Ratio Deflagration Detonation

u_l/c1 0.0001 − 0.03 5 − 10

u₂/c1 4 − 6 (acceleration) 0.4 − 0.7 (deceleration)

P₂/P1 ≈ 0.98 (slight expansion) 13 − 55 (compression)

T2/T1 4 − 16 (heat addition) 8 − 21 (heat addition)

ρ2/ρ1 0.06 − 0.25 1.7 − 2.6

c₁is the acoustic velocity in the unburnt gases. ul/c1is the Mach number of the wave.

Although the temperature ratio T2/T1 indicates approximately the same amount of

heat addition generation for both detonation and deflagration, other thermodynamic states can be used to distinguish the differences between these two combustion waves. The main difference is the rise in density and pressure across the flame front for a typical detonation wave, which can be illustrated in terms of the internal struc-ture of a deflagration and a detonation wave in Fig. 2.2. The rise in pressure results in compression for detonation while the loss in pressure results in a slight expan-sion for deflagration. Although a detonation wave is composed of three-dimenexpan-sional structures, a simplified one-dimensional analysis can still provide useful insight into a detonation wave, it is going to be summarized in the following sections, and al-lows for qualitative differences between deflagration and detonation to be observed.

2.1.2 Conservation Equations for Reactive Systems

Combustion processes are subject to fluid dynamics combined with chemical reac-tions involving species conversion and release of thermal energy. In principle, they can be described by a set of equations which differ from the usual Navier Stokes equations for non reacting species. A complete discussion and derivation of the con-servation equations for a reactive system based on continuum hydrodynamic can be

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2.1. Physics of Combustion 21

Figure 2.2: Internal structure of a deflagration wave (top) and a detonation wave (bottom), Adapted from [30].

found in [27, 35].

Considering a reactive multi-component gas mixture, composed of K different species, this system can be described by a set of equations, consisting of three conserva-tion equaconserva-tions, mass, momentum, energy, plus continuity equaconserva-tions for the single species, and of an equation of state (EOS). The unknown variables of the system are the thermodynamic parameters; temperature T , pressure p and density ρ, the three components of the velocity field U , and the K − 1 species mass fractions Yk. As a

result, the total number of the variables (and equations) is therefore equal to K + 5.

The mixture’s mass continuity equation for a reactive mixture is given by, ∂ ρ

∂ t + ∇ · (ρU) = 0 (2.1)

where, ρ denotes the mixture density, U denotes the velocity field. The left-hand side indicates the change of mass for a given control volume by either the change of density with time or by the mass flow rate balances.

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While, the mass continuity for a species of a reactive mixture is given by, ∂

∂ t(ρYk) + ∇ · (ρUYk) = −∇ · jk+ ωk (2.2) where, Y_k denotes the species mass fraction and can be computed knowing the molecular weight W_k and the mole fraction X_kfrom the following relation:

Y_k=Wk W Xk (2.3) W = K

∑

k=1 X_kW_k (2.4)

and j_k and ωk denote the diffusion flux of species k and its production rate

respec-tively. According to the Fick’s law, j_kcan be calculated as follows:

j_k= −ρDk∇Yk; Dk: is the mass diffusivity of species k. (2.5)

The total rate of change of linear momentum is given by, ∂

∂ t(ρU) + ∇ · (ρUU) = −∇p + ∇ · τ + Fb (2.6) where, τ denotes the viscous stress tensor, which can be defined for a Newtonian fluid as: τi j = µ ∂ ui ∂ xj +∂ uj ∂ xi −2 3µ ∂ u_k ∂ xk δi j (2.7)

In the above formula, µ denotes the dynamic viscosity, where as the Kronecker delta function, δi j, is defined in such way that

δi j =

(

1, i = j

0, i 6= j (2.8)

and Fb is the volumetric force acting on all the species and can be calculated as

follows: F_b= ρ K

∑

k=1 Y_kf_k (2.9)

where, fkdenotes the volume force acting on species k.

The energy balance equation: ∂

∂ t(ρht) + ∇ · (ρUht) = −∇ · jq+ ∇ · (τ U) + UFb+ ∂ p

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where, the total specific enthalpy, ht, is the sum of the specific enthalpy and the

kinetic energy, defined in such way

h_t= h +U 2 2 (2.11) h= K

∑

k=1 Y_kh_k (2.12) h_k= h0_k+ Z T T0 C_p,k(T ) dT (2.13)

where hkis the enthalpy of species k and it is the summation of the heat of formation,

h0_k, and the sensible heat. Moreover, jqdenotes the heat flux and it includes a heat

diffusion term expressed by Fourier’s Law and a second term associated with the diffusion of species with different enthalpies which is specific of multi-species gas.

j_q= −λ∂ T ∂ x + ρ K

∑

k=1 h_kD_k∇Yk (2.14)

where λ is the thermal conductivity.

Eventually, the equation of state for an ideal gas is expressed as follows: p= ρRT K

∑

k=1 Y_k W_k (2.15)

where R is the specific gas constant.

2.1.3 1-D Flame Propagation

Consider a premixed combustible mixture in a long tube, see Fig. 2.1, that under-goes a complete burning. A combustion wave will travel down the tube starting from the ignition point. It is possible to relate the thermodynamic properties up-stream and downup-stream of the flame, simplifying the burning front to a moving dis-continuity, neglecting all the physical and transport processes and reaction kinetics inside the flame structure. Such goal can be accomplished assuming:

• One-dimensional steady flow. • Constant tube area.

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• Inviscid flow.

• Adiabatic conditions and irreversible process. • Body forces are negligible.

The system of conservation Eqs. (2.1), (2.6) and (2.10) now reduces to: d dx(ρU) = 0 (2.16) d dx p+ ρU 2_{= 0} _(2.17) d dx(ρUht) = 0 (2.18)

The energy Eq. (2.18) can be reduced using the product rule, and yields to d (ρU ht) dx = dht dx(ρU) + ht d(ρU ) dx = 0 (2.19)

According to Eq. (2.16), the energy Eq. (2.19) yields to ht= constant and follows

Eq. (2.11). Then, integrating the above reduced equations over certain volume con-taining the flame front, yields to

ρ1u1= ρ2u2 (2.20) p1+ ρ1u21= p2+ ρ2u22 (2.21) h₁+u 2 1 2 = h2+ u2₂ 2 (2.22)

The known variables for such a system are the thermodynamic conditions of the unburnt gases; p1, T1 and ρ1 can be calculated from EOS, Eq. (2.15). While the

unknowns are the burnt gas state variables; p2, T2, ρ2and the burnt and unburnt gas

velocities in relation to the combustion wave; u1 and u2. Consequently, there are 5

unknown variables in 4 equations and the system cannot be solved without giving the value of one of the unknowns.

The continuity, Eq. (2.20) and momentum, Eq. (2.21), can be combined and the re-sulting equation is known as the Rayleigh line, which is a straight line in a (p, 1/ρ) diagram always with a negative slope. Thus from the inlet conditions at point A, one can have the outlet conditions located along the straight line as shown in Fig. 2.3. p₂− p1= − (ρ1u1)2 1 ρ2 − 1 ρ1 (2.23)

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The energy, Eq. (2.22), and Rayleigh, Eq. (2.23), can be combined resulting in the Rankine - Hugoniot relation:

h₂− h1= 1 2(p2− p1) 1 ρ2 + 1 ρ1 (2.24) where the enthalpy jump, h2− h1, can be expressed in terms of Eq. (2.13) to yield:

h0₂(T0) − h01(T0) + Z T₂ T0 C_p2dT− Z T₁ T0 C_p1dT = 1 2 1 ρ1 + 1 ρ2 (p2− p1) (2.25)

According to the assumption of constant and equal specific heats, and making use of the following relations;

h◦₂(T₀) − h◦₁(T₀) = q (2.26) where q denotes the heat released due to formation, knowing the relationship be-tween the specific heat at constant pressure and the specific heat at constant volume:

C_p−Cv= R Cp/Cv= γ → Cp=

Rγ

γ − 1 (2.27)

From the EOS, P = ρRT , CpT can be expressed as follows;

C_pT = γ

γ − 1p/ρ (2.28)

Substituting Eq. (2.28) and Eq. (2.26) into Eq. (2.25), the Rankine-Hugoniot rela-tion becomes: γ γ − 1 p₂ ρ2 −p1 ρ1 −1 2 1 ρ1 + 1 ρ2 (p2− p1) = q (2.29)

The Rankine-Hugoniot curve, as shown in Fig. 2.3, presents the possible values of the thermodynamic variables of the combustion products (1

ρ2, p2) for given thermo-dynamic properties of the reactants (_ρ1

1, p1) and the heat release q. The point (1

ρ1, p1) is the characteristic of the unburnt gases and usually called the origin of Rankine-Hugoniont curve, as indicated in Fig. 2.3 with the symbol A. Through this point, all the Rayleigh lines pass, which represents a condition that must be fulfilled by the burnt gases state variables. The burnt gases state will con-sequently be represented by the intersection of one Hugoniot curve, Eq. (2.24) with one Rayleigh line, Eq. (2.23).

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Figure 2.3: Rankine-Hugoniot curve. Adapted from [31].

Starting from point A, and drawing a vertical line, AC line, which represents the Isochoric state, and a horizontal line, AD line, which represents the Isobaric state and in addition, drawing two tangent lines that touch the Rankine-Hugoniont curve in two points B and E, we divide the Rankine-Hugoniont curve into five regions.

Although these five regions represent all the possible solutions of the Hugoniot re-lation, not all of these solutions are physically valid. As the C-D region requires that p2> p1 and _ρ1₂ > _ρ1₁, which violates the constraint imposed by the Rayleigh

line following Eq. (2.23), (i.e. any line connects point A with the region C-D will result in a positive slope, "this region is impossible").

The line AD represents the Adiabatic IsoBaric complete Combustion (AIBC) pro-cess which takes place at zero slope (i.e. the velocity would approach zero), follow-ing Eq. (2.23). Such transformation is characterized by total consumption of the reactants for which the enthalpy is conserved. The final temperature, TAIBC, can be

defined following the first law of thermodynamics for constant pressure process and holding the equilibrium conditions, that yields to:

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∆H = ∆H0_f +

Z T_AIBC

T1

∑

_j

njCp, j(T ) dT = 0 (2.30)

where ∆H0_f denotes the difference between the enthalpy of formation of the prod-ucts and the reactants at the reference temperature. And nj is the number of moles

of the chemical species j in the products.

The line AC represents the Adiabatic IsoChoric complete Combustion (AICC) pro-cess which takes place at an infinite slope (i.e. the velocity would approach an infinite value). Such transformation is characterized by total consumption of the re-actants for which the internal energy is conserved. The final temperature, TAICC, and

pressure, pAICC, are completely defined by thermodynamics considerations [18]:

∆E = ∆H0_f− ∆nRT1 + Z T_AICC T1

∑

_j n_jCv, j(T ) dT = 0 (2.31) p_AICC= n2RTAICC V₁ (2.32)

The two tangent points, B and E are considered the upper C-J point and lower C-J point, respectively for the combustion waves. The upper C-J point, B, characterizes conditions for the wave speed of a stable detonation, while the lower C-J point, E, characterizes conditions for the wave speed of a stable deflagration.

The area above point B represents the strong detonation region. This region is higher on the pressure axis than the upper C-J point. Solutions for this region re-quire experimental setup for generating over driven shock waves. [27].

Region B-C represents the weak detonation region. These pressure values are less than that of a C-J detonation. These weak detonations are rarely seen and typically stabilize to a stable deflagration in a short period of time [39].

Region D-E represents weak deflagrations, which are often observed. In passing through a weak deflagration, the gas velocity relative to the wave front is acceler-ated from a subsonic velocity to a greater subsonic velocity. In most experimental conditions, the pressure in the burnt-gas zone is less than the pressure of the un-burnt gases. Thus, solutions in this region are the most commonly observed form

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of deflagrations [27].

The region below point E represents strong deflagrations. In passing through a strong deflagration, the gas velocity must be accelerated significantly from sub-sonic to supersub-sonic. Since this situation cannot occur in a duct of constant area, this solution is nearly impossible to observe due to the wave structures prohibiting a change from subsonic to supersonic speed [27].

The thermodynamic characteristics for the burnt gases at C-J points can be defined taking into account that the Rayleigh line is tangent to the Rankine-Hugoniot curve at these points. Assuming q to be constant and differentiating the Hugoniot relation with respect to 1/ρ2yields;

d p₂ d(1/ρ2) = (p2− p1) − 2γ γ −1 p2 2γ γ −1 1 ρ2 − 1 ρ1+ 1 ρ2 (2.33)

While the slopes at the tangent points; upper C-J and lower C-J point following Rayleigh relation Eq. (2.23) can also be denoted as

d p₂ d(1/ρ2) C−J = p₁2− p1 ρ2− 1 ρ1 = − (ρ1u1)2 (2.34)

Imposing the equality for Eq. (2.33) and Eq. (2.34) would yield to γ p2ρ2= − p₂− p₁ 1 ρ2− 1 ρ1 (2.35)

Comparing Eq. (2.35) with the Rayleigh line expression Eq. (2.23), the outlet ve-locity of the burned gases yields to

u2₂= γp2 ρ2

(2.36) Knowing that the acoustic velocity in gas can be computed as follows,

c_s≡pγ RT = s γ p ρ (2.37) By comparing the outlet velocity of the burnt gas given by Eq. (2.36) to the acoustic velocity given by Eq. (2.37), we note that the burnt gas velocity, at both Chapman-Jouget points, equals to the acoustic velocity in the burnt gas.

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2.1.4 Chapman-Jouget detonation velocity approximation

In the detonation branch of the Rankine-Hugoniot curve (regions BC and above B), 1/ρ2< 1/ρ1; therefore from Eqs. (2.22) and (2.23) the velocity difference can be

expressed as follows u₂− u1= ˙m 1 ρ2 − 1 ρ1 < 0 i.e. u₂< u1

where ˙mis the mass flux of the unburnt gases. The C-J detonation is very special as the outlet gas velocity is sonic and the burnt gas expansion waves will not catch up with the combustion wave and weaken its strength. Experimentally, it is found that the observed detonations are often C-J detonations and it is therefore of interest to calculate their characteristics such as the detonation velocity and the change in pressure, temperature and density over the wave. Although, an analytical solution is not possible, with some approximations an estimate of the outlet conditions can be, nevertheless, obtained [27].

Let us assume constant heat capacities of the reactants and products. The conserva-tion Eqs. (2.20) to (2.22) can be written in terms of u2₂= cs2= γ2R2T2as follows;

ρ1u1= ρ2u2= ρ2 p γ2R2T2 (2.38) p1+ ρ1u21= p2+ γ2p2 (2.39) 1 2u 2 1+ h01(T0) + (T1− T0)Cp1 = 1 2u 2 2+ h02(T0) + (T2− T0)Cp2 (2.40) Assuming Cp1T0= Cp2T0and expressing the heat of formation in the form of Eq. (2.26), the energy Eq. (2.40) can be written as follows,

1 2u 2 1+ q +Cp1T1= 1 2u 2 2+Cp2T2 (2.41)

In order to obtain an approximate value for the Chapman-Jouget detonation velocity, u₁, it is necessary to find the density ratio ρ2/ρ1as well as T2. Assuming p2>> p1,

the term p1 in the momentum Eq. (2.39) can be neglected and it can be expressed

as follows

ρ1u21= p2+ γ2p2 (2.42)

imposing the equality condition in the continuity Eq. (2.38), the term ρ1u1in Eq. (2.42)

can be eliminated and the density ratio ρ2/ρ1is obtained as follows

ρ2/ρ1=

γ2+ 1

γ2

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Using Eqs. (2.37) and (2.43) and knowing that the specific gas constant of the burnt gas is obtained by follows

R2=

γ2− 1

γ2

Cp2 (2.44)

T₂can be determined from the energy Eq. (2.41) as follows T2= 2γ₂2 γ2+ 1 q C_p₂ + C_p₁ C_p₂T1 (2.45) Substituting Eqs. (2.43) and (2.45) into Eq. (2.38), an approximate value of u1=

u_CJ can be obatained

u_CJ = q

2 γ₂2− 1 (q +Cp1T1) (2.46)

2.2 Structure Mechanical Response

The load generated on structures due to the combustion process might vary from quasi-static to a dynamic one, depending on the flame propagation mechanisms (i.e. Deflagration or Detonation) and structure characteristics.

On one hand, Deflagrations cause a slow pressurization of the combustion cham-ber, as shown in Fig. 2.2. As a result, the pressure load is maintained for such a sufficiently long period that it can be considered as quasi-static [42]. On the other hand, Detonations are instead characterized by supersonic waves and they result in a nonuniform dynamic structural loading. For steady-state detonation, the local mechanical response of the structure can be determined by considering the dynamic load resulting from a mono-dimensional Chapman-Jouguet detonation. Consider-ing a long combustion tube with closed ends, the maximum pressure is found to be at the tube end where the detonation wave is reflected. In this case, thermodynamic equilibrium codes (such as CHEMKIN [24]) can be used to calculate CJ state char-acterization.

When a detonation strikes a surface normally, a shock wave is transmitted into the structure and another is reflected back into the gas, which complicates the me-chanical response of the structure. Tieszen [42] studied the effect of initial ther-modynamic conditions on the loads generated by the combustion of homogeneous hydrogen-air-steam mixtures. It was found that the reflected pressure is a function

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2.2. Structure Mechanical Response 31

of two material properties; being the ratio of specific heats, γ, and the molecular weight of the material, W , as well as, the initial temperature, T1, and the particle

velocity, U2. The peak reflected shock pressure from a detonation wave striking a

wall normal to its surface is given by:

p₂ p₁ = 2γ γ + 1   U₂ C₁ γ + 1 4 + U2 C₁ γ + 1 4 2 + 1 !1/2  2 − γ − 1 γ + 1 (2.47) C₁= γ RT1 W 1/2 (2.48)

where the subscripts 1 and 2 here refer to incident and reflected shocks respectively. γ , W , and T1, are evaluated at the CJ condition. Since the reflected shock

deceler-ates the incident particle velocity to zero, the reflected shock particle velocity, U2,

is equal to the CJ particle velocity, UCJ.

In the following sections, a brief review is given on structure response to static and transient pressure loads. In this thesis, the focus was given to the mechani-cal response of circular plates. Considering an isotropic, homogeneous and elastic material, internal stresses and strain are related via Hooke’s law up to the yielding condition. The yielding condition is defined as the value of the stress that corre-sponds to ε = 0.2%. Once the stress exceeds the value of σYS, yielding stress, the

strain is no longer proportional to the stress applied and the strain-rate models need to be considered in the plastic region to correlate the internal stress to the strain.

In this work, all circular plates are made of stainless-steels, which tends to absorb energy while deforming. Following such process, elevated levels of strain may be reached before rupture occurs. Due to the complexity of the problem, finite element methods (FEM) are usually addressed to model plastic response and structure fail-ure. Strain hardening, material in-homogeneity and pre-existing flaws can then be included in model to predict structure deformation in the plastic regime.

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2.2.1 Clamped Plates

In literature, many authors studied the problem of clamped circular plates. Timo-shenko [43], for example, studied the behavior of circular plates and provided an analytical solution based on the elasticity theory assuming that the material is per-fectly homogeneous and its behavior obeys Hooke’s law [43]. The linear bending theories are based on the assumption that the deflections are small compared to the plate thickness; however,in practical cases such as structures subjected to detona-tion load, the deflecdetona-tions may be of the same order of magnitude as the thickness. Thus, nonlinear analysis is inevitably more appropriate to model actual physical phenomena [44, 40].

2.2.1.1 Linear Elastic Theory of Plates

Kirchhoff theory of thin plates can be applied to those solids with two dimensions (i.e. width, length) much grater than the third one (i.e. thickness). Considering the Cartesian coordinate system, letting the z coordinate to identify the thickness, then the deformation of the plate is defined by the deflection in the xz and yz planes. Since the thickness is much smaller than the other dimensions, only the elastic surface (i.e. the mid-surface, z = 0 ) is taken into account for the balance equations. Moreover, only loads perpendicular to the surface and bending ones are considered [43, 8]. The model is based on the following hypotheses:

1. plate material is homogeneous and plate thickness, h, is constant and it does not change during deformation;

2. plate material obeys Hooke’s law for plane stress;

3. when unloaded, the plane z = 0 is not deformed; εx= εy= 0;

4. the displacement, w, in the z direction is small compared to the thickness of the plate.

For circular plates subjected to an axisymmetric load, the stress tensor in polar coordinates is denoted by:

σ =     σr 0 τrz 0 σθ 0 τrz 0 0     (2.49)

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where τrθ = τθ z= τθ r= τzθ = 0 is imposed to keep the symmetry of the problem.

While the σz can be neglected since it is equal to zero at the inner plate surface

while it is equal to the applied load at the outer plate surface, whose value is much lower than the stress components. The deformation of the plate, see Fig. 2.4, is then described by the strain vector (εr, εθ), whose components are defined as:

εr= z ρr = −zd 2_w dr2 (2.50) εθ = z ρθ = −z r dw dr (2.51)

Figure 2.4: Circular plate strain due to an axisymmetric load.

where w denotes the displacement in the z direction. since σz = 0, the bending

moments per unit of length [N · m/m] can be related to the surface curvatures κrand

κ_θ as in case of pure bending: mr= D (κr+ νκθ) = D 1 ρr + ν 1 ρθ = −D d 2_w dr2 + ν r dw dr (2.52) m_θ = D (κθ+ νκr) = D 1 ρ_θ + ν 1 ρr = −D 1 r dw dr + ν d2w dr2 (2.53)

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where we have made the hypotheses that r ' ρ_θϕ while dr = ρrdϕ. Here the

bend-ing angle is defined as ϕ = −(dw/dr). Eqs. (2.52) and (2.53) represent plates constitutive equations. Bending stiffness, D, is defined as follows:

D = Eh

3

12 (1 − ν2₎ (2.54)

Consider a slice of a circular plate, as shown in Fig. 2.5, subjected to a uniform load p. The rotating equilibrium in the rz plane results in:

Figure 2.5: Equilibrium of a circular plate volume element subjected to a uniform load p. Adapted from [8].

mr+ ∂ mr ∂ r dr (r + dr)dθ − mrrdθ + trrdθ dr − 2mθdr θ 2 = 0 (2.55) While the equilibrium in the z-direction gives:

2πrtr= πr2p (2.56)

Substituting Eq. (2.56) into Eq. (2.55) we get: ∂ (mrr)

∂ r − mθ = − r2p

2 (2.57)

Taking into account the constitutive relations in Eqs. (2.52) and (2.53), we can rewrite Eq. (2.57) to be:

d dr 1 r d(rϕ) dr = −pr 2D (2.58)

Under the hypothesis that the plate is clamped at the edges, the boundary conditions for that case are:

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r= 0 ⇒ ϕ = 0 r= R ⇒ ϕ = 0 r= R ⇒ w = 0

(2.59)

The solution for Eq. (2.58) under an axisymmetric load can be obtained:

w(r) = p 64D R

2_{− r}22

(2.60) Hence the maximum displacement in z-direction, at the central point (r = 0), is

w(r = 0) = w0=

pR4

64D (2.61)

Once the displacement is calculated, Eq. (2.60), the moment distribution along the radius can be computed from Eqs. (2.52) and (2.53) substituting the first and the second derivatives of the displacement along the radius. The solutions derived so far can be applied with sufficient accuracy if the deflections of plate from its initial plane are small in comparison with the thickness of the plate [43]. Way [44] has shown that for clamped circular plates the above described theory is applicable for maximum deflections less then 0.4 of the thickness, where in this case the error is less than 10%.

However, for structures subjected to detonation loads, large deflections are more likely to occur and non-linear effects must be taken into account. Way [44] derived a solution for large deflections 0.5 < w_h < 1 taking into account a power series decomposition for the lateral displacement w. A membrane solution for a uniform thickness circular isotropic elastic plate clamped at its boundary and subjected to uniform lateral loading was first studied by Hencky in 1915 [21]. The solution is obtained by imposing a no-slip condition at the edge (i.e. zero radial displacement; u(R) = 0 and u(0) = 0 ). The membrane deflection at the center is given by the following formula: w(0) h = 3 4 1 − ν2 pR4 Eh4 !1₃ (2.62) The non-linear corrections to the elastic theory presented above were studied by Li [29]. A strain model was added to the middle plane in order to take into account

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the stretching effect of the neutral plane when its central deflection is larger then the plate thickness: w(r) = j0 1 − r R 22 (2.63) j₀= w0     1 1 + 0.4442 (1−ν2₎ w2₀ h2     (2.64)

In summary, the following Fig. 2.6 presents a comparison of the non-dimensional maximum deflection for a circular plate with clamped edge obtained with the linear theory Eq. (2.61), the power series solution (Way’s solution), the membrane theory Eq. (2.62) and the strain model Eq. (2.63) for ν = 0.3.

Figure 2.6: Comparison of the non-dimensional maximum deflection for a circular plate with clamped edge. Adapted from [36].

2.2.1.2 Plates subjected to transient loads

In case of detonation and many practical cases, in which plate structures are sub-jected to transient loads and large amplitude motions occur, if the magnitude of the

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displacement is of the same order as the thickness of the plate, then for the math-ematical description of motions the classical linear plate theory is inadequate. The use of non-linear plate theory is thus inevitable which takes into account the inter-action between the bending and stretching of the mid-plane.

Considering the case of circular plates and assuming that loads generated due to detonation are axisymmetric dynamic loads, the governing equations reduce to two coupled non-linear partial differential equations.

D∇4w+γh∂ 2_w ∂ t2 +γhkv ∂ w ∂ t − Eh 1 − ν2 " ∂2w ∂ r2 + 1 r ∂ w ∂ r ∂ u ∂ r+ 1 2 ∂ w ∂ r 2 + νu r ! +∂ w ∂ r ∂2u ∂ r2+ ν ∂ u r∂ r − ν r2u+ ∂ w ∂ r ∂2w ∂ r2 = p(r,t) (2.65) ∂2u ∂ r2+ 1 r ∂ u ∂ r− u r2+ ∂ w ∂ r ∂2w ∂ r2 + (1 − ν) 2r ∂ w ∂ r 2 = 0 (2.66)

where D is the flexural rigidity, p(r,t) is the applied load, E is the plate modulus of elasticity, t is time, h is the plate thickness, u is the radial displacement, kv is the

viscous damping constant, w is the vertical deflection, γ is the density of the plate material, ν is the poisson ratio for the plate material. These equations, Eqs. (2.65) and (2.66), because of their complexity are mostly solved by using approximate methods of various types such as finite difference schemes [23, 2].

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SSEXHY Facility

SSEXHY facility [36] is a modular structure composed of multiple sections that can be easily assembled and dismantled to serve different purposes, installed at Commissariat à l’énergie atomique, CEA-Saclay. It is composed of two parts: a combustion chamber to characterize the flame propagation mechanisms and a mod-ule to examine the dynamic response of structures to combustion generated loads. A complete description of the facility and the description of the instrumentation used as well as the characteristics of the deformed structures are given in the following sections.

3.1 Combustion chamber

The combustion chamber is a modular stainless-steel obstructed duct, composed of multiple sections, aimed to examine the acceleration mechanisms of premixed hy-drogen/air flames. The duct consists of four modular sections connected by flanges. Each section is 1310 mm long with an internal diameter of 120 mm. The design pressure of the tube is 100 bar. In order to investigate the targeted phenomena; det-onation/deflagration, three sections are assembled to form the combustion chamber while performing the experiment. Two blind flanges are used to seal the combustion tube at its extremities. A schematic of the experimental setup is shown in Fig. 3.1.

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3.1. Combustion chamber 39

Figure 3.1: Picture of the combustion chamber. Adapted from [36].

3.1.1 Gas injection and venting system

Gas injection lines are connected to one end-flange via an isolation valve. At the center of this flange, a threaded hole houses an automotive spark plug. Mixture ignition is provided by the electrical discharge between the two electrodes of the plug. The energy released by an ordinary spark plug is around 25 mJ [45]. While the minimum ignition energy (MIE) for a stoichiometric hydrogen/air dry mixture is 0.019 mJ [28].

On the other extremity, venting lines ensure burnt gas evacuation from the combus-tion tube at the end of the experiment. A primary vacuum pump is also connected at both sides of the tube. A sketch of the combustion tube is shown in Fig. 3.2. A set of three static pressure sensors with different operating ranges, used to control gas injection procedure; a vacuum gauge (P1) in the range 0-133 mbar, a pressure transmitter (P2) in the range 0-1000 mbar, and a third sensor (P3) covering the range 0-5 bar. Further details on preparing the desired mixture pressure and filling process are discussed in [36].

3.1.2 Sampling system and recirculation loop

Once hydrogen and air are injected inside the tube, a non-homogeneous mixture is formed. Homogenization is then achieved using a gas recirculation pump. Two sampling volumes are installed to the recirculation loop connections to the tube wall, see Fig. 3.2, in order to verify the flammable mixture concentration. The mixture is forced to recirculate via an external loop for about 30 minutes before

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Figure 3.2: Injection and venting system of the combustion tube and external recir-culation loop. Adapted from [36].

reaching the homogeneous condition. This period of time was calibrated from pre-tests using helium instead of hydrogen and measuring the local concentration with thermal conductivity gauges, as described in [36].

3.1.3 Tube internals: obstacles array

An array of equally spaced annular obstacles are placed inside the tube with the aim of promoting turbulence at the wall. Two obstacle blockage ratios (i.e. the ra-tio between the area obstructed by the obstacle and the tube cross secra-tion area) are available ranging from 0.3 to 0.6.

The structure supporting the obstacles is formed by three thin threaded rods. The stainless steel annular obstacles ( 5 mm thick) have three holes, 120◦spaced from each other, for rods insertion. Obstacles are 120 mm spaced from each other, as shown in Fig. 3.3. According to [17], this corresponds to the optimal configuration for annular obstacles in terms of potential for flame acceleration.

In order to increase the accuracy of the optical measurements using the PMT, ob-stacles surface, as well as tube inner wall, was thermally treated with black oxides to prevent light reflection.

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3.1. Combustion chamber 41

Figure 3.3: Schematic of the combustion duct. PMT, photomultiplier tube; PP, pressure sensor; CC, shock sensor. Adapted from [36].

3.1.4 Gas-chromatographic analysis

At the end of the homogenization process, the flammable mixture is sampled and analyzed via gas chromatography. The analysis is performed in a Agilent 490 Micro GC System equipped with a thermal conductivity detector (TCD). Argon is used as carrier gas. This allows detecting helium, oxygen, nitrogen, hydrogen and methane in a single GC module.

To handle the gas samples, two sampling volumes are attached to the recirculation loop. Each volume features a stainless steel cylinder with a leak tight piston ac-tivated by a lead screw (see Fig. 3.4). The cylinders are also equipped with an isolation valve in order to connect and disconnect them from the recirculation loop (as shown in Fig. 3.2). The moving piston and the pressure gauge installed on the gas chromatograph inlet (as shown in Fig. 3.4) allow controlling the GC inlet pres-sure. Gas analysis is then performed at constant pressure ensuring a linear response of the instrument.

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Figure 3.4: Schematic of the gas chromotograph. Adapted from [36].

3.2 Diagnostics Instrumentation

Along the duct wall, several ports are available for the diagnostics. In addition, there are threaded central holes of the two end-flanges, which can be used to house special bolts equipped with pressure transducer and the spark plug. A cross sectional view of the tube (A - A section) in Fig. 3.3 shows the standard position of the sensors:

• horizontally, at ϑ = 0 , the photomultiplier tubes (PMT) for flame time-of-arrival measurements;

• on the vertical position, at ϑ = π

2 , the dynamic pressure sensors (PP);

• facing the PMT, at ϑ = π, the piezoelectric shock sensors (CC).

3.2.1 Photomultiplier tubes

In a mixture of hydrogen and oxygen, it is presumed feasible that the presence of a free radical in the form of an OH radical or H atom should result in the following reaction cycle [19]:

H + O2→ O + OH

O + H2→ H + OH

The emission band of OH (306 nm < λOH < 310 nm) presents a strong head at

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3.3. Module for the dynamic response 43

Along the combustion tube wall, 15 photomultiplier tubes (PMT) can be installed in order to measure the axial velocity of the flame front propagating along the tube. These detectors collect the UV light emitted by OH radicals located at the reaction front within a very narrow solid angle. As the flame tip passes through the PMT solid angle, a negative voltage is recorded. This signal allows us to extrapolate the time-of-arrival of the flame tip and monitor the flame propagation along the tube axis. Once observing the time of arrival of two consecutive PMTs, the flame front speed can be calculated knowing the distance between the PMTs. For further details see [36].

3.2.2 Pressure sensors

Three types of Kistler-piezo electric sensors were tested (0 - 250 bar pressure range): 601A, 6001 and 7001. Types 601A and 6001 feature a high natural frequency (FN = 150 kHz); the 6001 present a better resistance to high temperatures up to 350◦C. The 7001 has a lower natural frequency (FN = 70 kHz), but at the same time its sensitivity is higher (the diameter of the sensitive area of the 7001 sensor is 9.5 mm, while the diameter of 601A and 6001 types is 5.55 mm).

3.2.3 Shock sensors

Chimimetal piezo-electic sensors allow the detection of a shock wave in the un-burnt gas. Thanks to their fast response (3 ns pulse rise time and 2.5 MHz natural frequency), they are usually employed for time-of-flight measurements of shock waves. Moreover, they can be used to extrapolate the velocity of the pressure wave ahead of the flame in a more accurate way because of their small sensitive area (2 mm diameter).

3.3 Module for the dynamic response

To investigate the effect of combustion loads on structures, the combustion tube de-scribed above is coupled with a supplementary module, the FSI module, designed for this purpose. The connection between the combustion tube and the FSI module occurs at the end-flange of the tube. Here, the end sealing flange is replaced by an intermediate flange that ensures the coupling. The intermediate flange also serves

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as support for the testing specimens. Moreover, it hosts the connections for nitro-gen injection and venting lines (to inert FSI module atmosphere). It has an outer diameter of 895 mm and an internal diameter of 120 mm, that allows the perfect continuity with the combustion tube internal wall. A schematic view of the cou-pling between the combustion tube at the FSI module is shown in Fig. 3.5.

Figure 3.5: Schematic view of the experimental device for the study of structure response to combustion originated loads: (1) combustion tube; (2) intermediate flange; (3) safety dome; (4) frame for tube anchoring; (5) movable frame for safety dome handling. Adapted from [36].

The FSI module is composed by the following interconnected parts: 1. Intermediate flange;

2. Adapter or support for the specimen to test; 3. Stainless steel specimen;

4. Counter-flange for specimen flxation; 5. Safety dome.

The FSI module is enclosed by a safety dome. It is a stainless steel pressure vessel with a nominal diameter of 700 mm and a total length of 1800 mm, designed for a nominal pressure of 8 bar. The dome is designed to serve multiple purposes:

1. to contain possible missiles that can be generated and ejected following spec-imens failure and fragmentation;

2. to provide an inert atmosphere surrounding the specimens (the dome is filled with nitrogen);

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3.4. Description of the specimens 45

3. to avoid pressure differences across specimen thickness during venting and filling procedures;

4. to provide optical accesses on the specimens (in order to use optical diagnos-tics to characterize specimens deformation in time);

5. to provide electrical ports for the different diagnostics connections.

To avoid any direct impact between specimen fragments and the dome, the internal wall of the dome is shielded by a deformable sheet metal. The optical access to the specimen is provided by three glass windows:

• two lateral rectangular windows with a visualization area of 400 x 80 mm

• one circular window on the convex end of the dome, facing the dome axis, with Φ = 80 mm diameter.

The two rectangular windows are located 275 mm from the intermediate flange and 90◦spaced to each other. They provide the direct optical access to the testing zone of the cylindrical specimen.

A pressure transducer is installed on the dome to record the static pressure during filling and venting procedure. It allows to manage the pressure differences between the combustion tube and the dome during these phases in order to avoid excessive pressure loads on the specimens (avoiding plastic deformations of the specimen due to the establishment of a ∆p across the specimen thickness).

3.4 Description of the specimens

SSEXHY facility designed to house different test specimens; (1) small circular plates; (2) large circular plates; (3) cylinders. In this work, the main focus will be on the small circular plates and further details on the other specimens can be found in [36].

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3.4.1 Specimen material

Specimens are made of 304L austenitic stainless steel, which is one of the most commonly used steel in the nuclear industry. Mechanical properties of the SS 304L are listed in Table 3.1.

Table 3.1: Stainless Steel 304L Mechanical properties

E[GPa] 200 ν 0.3 ρkg/m3 7930 σ0.2%[MPa] 180 σm[MPa] 500 c_sl[m/s] 5790

3.4.2 Specimen geometry

Specimen characteristic dimensions are as follows: the thickness h = 0.5 mm, the external diameter Φe = 175 mm and the deformable area has diameter Φi= 120

mm. Schematic of the circuital plate specimen is shown in Fig. 3.6. Once installed in the facility, the circular plate is clamped at the edge. The clamped area, Ac,

ex-tends from the external diameter Φeup to the the diameter Φi, Ac= π Φ2e− Φ2i /4.

Therefore, the plate area that is subjected to pressure loads is Ap= πΦ2i/4.

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3.4. Description of the specimens 47

3.4.3 Specimen fixation

In order to fix the specimen to the combustion tube, at one side the adapter is fixed to the intermediate flange, while at the other side the end-flange has a groove to accommodate the specimen. Specimen edges are then clamped between the adapter end-flange and a counter-flange. The sealing is ensured by two flat gaskets at each side that extent over the whole clamped area Ac.

3.4.4 Small plate adapter

Small circular plates can be installed directly between the intermediate flange and the counter-flange. The counter-flange is a ring with an internal diameter equal to Φi= 120 mm.

It is possible to extend the length of the combustion tube inside the safety dome of the FSI module by using a cylindrical adapter, as shown in Fig. 3.7. This adapter features a thick stainless steel flanged tube 500 mm long with an inner diameter Φi

= 120 mm. As previously mentioned, the circular plate is installed at the end-flange. On the adapter wall four ports are available for instrumentation diagnostics: two are located at 167 mm from the first flange and two at 333 mm. The two ports at the same location along the tube axis are 90◦spaced to each other.

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Finite Element Formulations

In this chapter a brief description of the Finite Element Method and the main steps involving the finite element analysis is given. Three well known mesh descriptions are presented here; the Lagrangian, Eulerian and Arbitrary Lagrangian-Eulerian. The key factors and advantages of each method as well as the mapping between the three descriptions are summarized below.

4.1 Finite Element Method

The Finite Element Method (FEM), is a numerical technique for solving a system of differential/integral equations. Basically, the method consists of dividing a domain into a number of smaller elements and applying the numerical modeling on each one of them. Knowing the characteristics and behavior of the individual elements, the behavior of the whole domain can be obtained [4]. The Finite Element Analysis (FEA) is achieved using the aid of computer to solve sets of partial differential equations (PDEs) and it can be employed in a wide variety of application areas [20]. FEA in engineering problems consists of the following three main steps:

1. Describing the domain and choosing the mathematical model; 2. Discretising and solving the governing equation;

3. Interpretation of the results.

The FEM is employed to solve complex mathematical models, but it is important to realize that the finite element solution can never give more information than that

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4.1. Finite Element Method 49

contained in the mathematical model. Thus, the user has to pay great attention to that.

The selection of an appropriate mesh description, i.e. whether a Lagrangian, Eu-lerian or Arbitrary Lagrangian-EuEu-lerian (ALE), is a key factor for applications en-countering large deformation and failure analysis.

The conservation equations make up the basis for a group of fundamental equa-tions of continuum mechanics. These conservation equaequa-tions are often expressed by PDEs, which are referred to as the strong form. When applying the conserva-tion laws on to a domain of the body, an integral relaconserva-tion appears, and this results in the so-called weak form. The goal in FEA is to describe a physical system in which these fundamental equations are satisfied. For a thermomechanical system, the following four conservation laws apply [46, 7]:

1. Conservation of mass;

2. Conservation of linear momentum; 3. Conservation of angular momentum; 4. Conservation of energy.

4.1.1 Description of Motion

Let us consider the motion of a continuum body within which a material particle moves from its original position P at time t = 0 to its current position p at time t, as shown in Fig. 4.1, The region of three-dimensional (3D) Euclidean space occupied by the body at time t = 0 is called the initial or undeformed configuration (Ω0) ,

while the region of 3D Euclidean space occupied by the body at time t is called the current or deformed configuration (Ω). To measure the motion of the body, one particular configuration should be selected as the reference configuration to which the motion of the body will be referred. The motion of the body can be described by the following:

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Figure 4.1: Schematic of the initial and current configurations.

where x is the position of the material point, X, at time t. The function φ(X,t) is often referred to as a mapping function, since it maps the reference configuration onto the current configuration at time t. Eq. (4.1) gives the spatial position of the body as a function of the time, t. The displacement of a material point is defined as the difference between the current and the original position. Thus, it can be expressed as [7]:

u(X,t) = φ(X,t) − X (4.2)

The velocity of a material point is defined as the rate of change of the belonging position vector. Thus, it is the time derivative with X held constant. The velocity can be given as:

v(X,t) = ∂ φ(X , t)

∂ t =

∂ u(X , t)

∂ t ≡ ˙u (4.3)

Acceleration is defined as the rate of change of the velocity of a material point, and is thus defined by:

a(X,t) = ∂ v(X , t)

∂ t =

∂2u(X,t)

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4.2. Lagrangian Finite Elements 51

Another important variable in the process of calculating deformations and strains in nonlinear continuum mechanics, is the deformation gradient, F . Mathematically, it is on the form of a Jacobian matrix, which is a matrix of all first order derivatives of a vector valued function. Thus, the deformation gradient can be seen as the Jacobian matrix of the motion φ(X,t).

F = ∂ φ ∂ X ≡

∂ x

∂ X (4.5)

Another relation related to the deformation gradient comes from its determinant, J= det(F ). By finding the determinant of F , a relation between integrals in the current and the reference configurations can be established:

Z Ω f(x,t)dΩ = Z Ω0 f(φ(X,t),t)JdΩ0 (4.6)

4.2 Lagrangian Finite Elements

For solid mechanics modeling, Lagrangian meshes are widely used. Their attrac-tiveness stems from the ease with which they handle complicated boundaries and their ability to follow material points, in fact, nodes and elements move according to the material, as illustrated in Fig. 4.2. Since there is no advection between the grid and material, no advection term appears in the governing equations, which signifi-cantly simplifies the solution process. In addition, boundaries and interfaces remain coincident with element edges, thus history-dependent materials can be treated ac-curately [7, 46].

Figure 4.2: Schematic of the Lagrangian mesh. Adapted from [46].

The limits of a Lagrangian formulation lie in the element’s capabilities of dealing with large distortions. Severe element distortion results in significant errors in nu-merical solution and even leads to a negative element volume or area, which would

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cause abnormal termination of the computation. Numerically stable solution can be achieved with an explicit time integration scheme, in which the time step must be smaller than a critical time step which is controlled by the minimum characteristic length of all elements in the grid. Decreased characteristic element length, may lead to smaller time steps and accordingly much more time to complete the calcula-tions. Another way could be introduced to complete a Lagrangian computation for an extreme loading case, a distorted grid must be remeshed and its result must be interpolated to the remeshed grid [7, 46].

In literature three approaches of Lagrangian finite elements are commonly used; Updated Lagrangian formulation, Total Lagrangian formulation and Corotational formulation. In the updated Lagrangian formulation, the derivatives are taken with respect to the spatial (Eulerian) coordinates, x. As a result of this, integrals are taken over the deformed (current) configuration, in contrast to the total Lagrangian formulation, where the derivatives are taken with respect to the undeformed (initial) configuration [7].

4.2.1 Governing Equations for Total Lagrangian Formulation

The balance equations for the total Lagrangian formulation are presented below. The mass conservation can be written as follows:

ρ (X , t)J(X , t) = ρ0(X) (4.7)

Conservation of Linear momentum:

ρ0

∂ v(X , t)

∂ t = ∇0· P + ρ0b (4.8)

Conservation of Angular momentum:

F · P = PT · FT (4.9) Conservation of Energy: ρ0w˙int= ρ ∂ wint(X,t) ∂ t = ˙F T _{: P − ∇} 0· ˜q + ρ0s (4.10)

where ρ0 and ρ denote original and current density. wint denotes internal energy

Validation of a Fluid-Structure Interaction Model against SSEXHY test facility measurments

VALIDATION OF A FLUID STRUCTURE

INTERACTION MODEL AGAINST

SSEXHY TEST FACILITY

MEASURMENTS

HALIM Ossama

Università di Pisa

Dipartimento di Ingegneria Civile e Industriale (DICI)

U

NIVERSITÀ DI

P

ISA

Scuola di Ingegneria

Validation of a Fluid Structure Interaction

model against SSEXHY test facility

measurments

Tesi di

HALIM Ossama

Relatori:

Prof. AMBROSINI Walter

Prof. CARCASSI Marco

Ing. ETIENNE Studer

Ing. KUDRIAKOV Sergey

U

NIVERSITÀ DI

P

ISA

Scuola di Ingegneria

Validation of a Fluid Structure Interaction

Model against SSEXHY test facility

measurments.

Tesi di

HALIM Ossama

Relatori:

Prof. AMBROSINI Walter

Prof. CARCASSI Marco

Ing. ETIENNE Studer

Ing. KUDRIAKOV Sergey

Candidato:

HALIM Ossama

Acknowledgments

Abstract

Contents

List of Tables

Chapter 1

Introduction

1.1

Background and motivation

1.2

Contributions of this work

Addressed Phenomenology

2.1

Physics of Combustion

2.1.1

Introduction

2.1.2

Conservation Equations for Reactive Systems

∑

∑

∑

∑

∑

2.1.3

1-D Flame Propagation

∑

∑

2.1.4

Chapman-Jouget detonation velocity approximation

2.2

Structure Mechanical Response

2.2.1

Clamped Plates

SSEXHY Facility

3.1

Combustion chamber

3.1.1

Gas injection and venting system

3.1.2

Sampling system and recirculation loop

3.1.3