Assessment of CFD hydrogen deflagration models for application in light water cooled nuclear reactor containments

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UNIVERSITÀ DI PISA

SCUOLA DI INGEGNERIA

CORSO DI LAUREA MAGISTRALE IN INGEGNERIA ENERGETICA

Assessment of CFD hydrogen deflagration models for application in light water cooled nuclear reactor containments

Master Thesis

Supervisors Candidate

Prof. Walter Ambrosini Andrei Cutrono Rakhimov Prof. Leonardo Tognotti

Mr. Ed Komen

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ABSTRACT

Large quantities of hydrogen can be generated during a severe accident in a water-cooled nuclear reactor due to degradation of the core. Hydrogen is a flammable gas that can form a combustible mixture with air. When released in the air-filled containment of a pressurized water reactor, the hydrogen can thus create a potential deflagration risk. The dynamic pressure loads resulting from hydrogen combustion can be detrimental to the structural integrity of the reactor safety systems and the reactor containment, hence pose a great threat to the environment. Therefore, accurate prediction of these pressure loads is an important safety issue. During a severe accident, cooling water from the primary circuit will enter the containment as steam. Therefore, it is important to capture the effect of steam on hydrogen deflagration in a Computational Fluid Dynamics (CFD) model.

This thesis work follows from a larger framework research concerning the hydrogen deflagration issue developed at the Nuclear Research and Consultancy Group (NRG) in The Netherlands. In past NRG assignments, a CFD-based method has been validated to determine the pressure loads from a fast deflagration for uniform hydrogen-air mixtures, hydrogen-air mixtures with diluents and non-uniform hydrogen-air mixtures. The combustion model applied in the CFD method is based on the Turbulent Flame Speed Closure (TFC) of Zimont and is implemented in the CFD software ANSYS Fluent using user defined functions. In a more recent step, the extension of the above combustion model, Extended Turbulent Flame Speed Closure (ETFC) of Lipatnikov, which includes the laminar source term, has been studied, and its validation against hydrogen deflagration experiments in the slow deflagration regime for uniform hydrogen-air mixtures has been evaluated.

The primary objectives of the present work are to further validate the TFC and ETFC combustion models in the slow deflagration regime for uniform hydrogen-air-steam mixtures, and investigate their capability to predict the effect of steam on hydrogen deflagration. Experiments conducted in a medium-scale Thermal-Hydraulics Hydrogen Aerosols and Iodine (THAI) test facility are used for validation purpose. In particular, three THAI hydrogen deflagration (HD) experiments with vertical flame propagation and increasing concentration of steam are considered, namely, THAI HD-15, HD-22 and HD-24.

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This thesis work describes the HD experiments from the medium scale THAI facility, the applied combustion models, and the most relevant results of the code validation.

The peak pressures, the trends of the flame velocity, and the pressure rise with an increase in the initial steam dilution are captured reasonably well by both combustion models. In addition, the ETFC model appeared to be more robust to mesh resolution changes. The mean pressure rise is evaluated with 18% under-prediction and the peak pressure is evaluated with 5% inaccuracy, when steam is involved.

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List of Figures

2.1 (a) Premixed and (b) non-premixed flame configurations (Poinsot and Veynante,

2001) 15

2.2 Laminar flame structure (Ciccarelli and Dorofeev, 2008) 17

2.3 Laminar flame speed versus equivalence ratio at 100kPa and 298K. The symbols represent experimental data while solid lines represent simulations. The legend, reported in Sathiah et al., 2012b, is summarized as follows: 1) Aung et al. (1997); 2) Dowdy et al. (1993); 3) Wu and Law (1984); 4) Takahashi et al. (1983); 5) Vagelopoulos et al. (1994); 6) Conaire et al. (2004); 7) Taylor (1991) 8) Kwon and Faeth (2001); 9) Iijima and Takeno (1986); 10) Law (1993); 11) Tse et al. (2000); 12) Liu and MacFarlane (1983); 13) Lamoureux et al. (2003); 14) Egolfopoulos and Law

(1991); and 15) Malet(2005) and Bleyer et al. (2012) 18

2.4 The variation of laminar flame speed with diluent concentrations measured by Bentaib and Chaumeix (2012) and Liu and MacFarlane (1983) for a hydrogen

concentration of 13% 20

2.5 Evolution of PDT instability. Gradual growth and branching of large cracks on the flame surface indicates growth of PDT instability (Fushui, 2012) 24

2.6 The Borghi Diagram (Lipatnikov and Chomiak, 2002) 25

3.1 a) THAI test vessel (inner cylinder and condensate trays removed for HD-tests. b) Schematic representation of the THAI facility (Kanzleiter and Langer, 2010) 30 3.2 Short term instrumentation used to record the deflagration phenomena (Kanzleiter

and Langer, 2010) 33

3.3 Measured pressure evolution during THAI experiments HD-15, HD-22 and HD-24 (Kanzleiter and Langer, 2010). The symbols represent the raw experimental data and

the lines the corresponding moving averages 35

3.4 Experimentally measured axial flame front positions during THAI experiments HD-15, HD-22 and HD-24. The symbols represent the first flame front arrival at elevation and

the lines the corresponding moving averages 35

3.5 Flame front propagation as timelines, test HD-15, HD-22 and HD-24 (Kanzleiter and

Langer, 2010) 36

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4.2 Progress variable contour and mesh refinement for THAI HD-22 test at t = 2.22s using

ETFC combustion model and 1 AMR 52

5.1 a) Borghi diagram with the domain of possible regimes in case of a nuclear power plant accident (Goulier et al., 2016). b) Borghi diagram for the simulated THAI

experiments using ETFC model 57

5.2 Comparison of the measured (symbols) and predicted (lines) pressure evolution for

THAI HD-15 59

5.3 Comparison of the measured (symbols) and predicted (lines) flame front position for

THAI HD-15 59

THAI HD-22 60

THAI HD-22 61

THAI HD-24 62

THAI HD-24 63

THAI HD-15, HD-22 and HD-24. Predictions by the TFC model 64

THAI HD-15, HD-22 and HD-24. Predictions by the ETFC model 64

5.10 Progress variable contours for THAI HD-15 experiment using ETFC combustion model

and 1 AMR 66

5.11 Progress variable contours for THAI HD-22 experiment using TFC combustion model

and 1 AMR 66

and 1 AMR 67

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List of Tables

2.1 Values of Liu and MacFarlane of constants 19

3.1 THAI HD experiments considered for validation (Kanzleiter and Langer, 2010) 31

4.1 General features of the CFD analyses 55

5.1 Peak pressure 68

5.2 Maximum local pressure rise 69

5.3 Mean pressure rise 69

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Nomenclature

Acronyms

AMR Adaptive Mesh Refinement

AICC Adiabatic Isochoric Complete Combustion AS Asymptotically Steady limit

BWR Boiling Water Reactor

CNRS Centre National de la Recherché Scientifique

CFD Computational Fluid Dynamics

CFL Courant-Friedrichs-Lewy condition DDT Deflagration to Detonation Transition

DO Discrete Ordinates

ETFC Extended Turbulent Flame Speed Closure FDS Flux Difference Splitting

HD Hydrogen Deflagration

ISP Intermediate Steady Propagation LWR Light Water Reactor

LP Lumped Parameter

NEA Nuclear Energy Agency

NRG Nuclear Research and Consultancy Group

OECD Organization for Economic Co-operation and Development PAR Passive Autocatalytic Recombiner

PDT Preferential Diffusion Thermal instability PWR Pressurized Water Reactor

RMS Root Mean Square

THAI Thermal-Hydraulics Hydrogen Aerosols and Iodine

TFC Turbulent Flame Speed Closure

URANS Unsteady Reynolds Averaged Navier Stokes equations

UDF User Defined Function

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9 Overbars, superscripts and subscripts

̅̅̅ Reynolds averaged mean component

̃ Favre-averaged mean component

fluctuating component

adiabatic

burned state of the mixture

deficient reactant excess reactant fuel laminar oxidizer steam turbulent

turbulent, time dependent turbulent, fully developed

unburned state of the mixture

Greek letters

Overall thermokinetic index

Flame brush thickness [ ]

Kronecker symbol

Heat of combustion [ ]

Turbulent dissipation rate [ ]

Length of smallest eddies [ ]

Activation temperature [ ] Thermal diffusivity [ ] Thermal conductivity [ ] Dynamic viscosity [ ] Kinematic viscosity [ ] Density [ ]

Chemical time scale [ ]

Critical chemical time scale [ ]

Viscous force tensor [

Lagrangian time scale [ ] Turbulent time scale [ ] Kolmogorov time scale [ ]

Equivalence ratio

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10 Roman letters

Time averaged mean flame surface area [ ] Instantaneous flame surface area [ ] Combustion progress variable

Specific heat capacity at constant pressure Specific heat capacity at constant volume Molecular mass diffusivity [ ]

Damkohler number Total energy

Body force tensor Enthalpy ⃗⃗ Diffusion flux

Turbulent kinetic energy [ ] Karlovitz number

Integral length scale Laminar source term Lewis number

̇ Mass flow rate [ ] Molecular weight Pressure [ ]

Adiabatic Isochoric Complete Combustion (AICC) pressure [ Mean maximum pressure [ ]

Reference pressure [ ] Prandtl number

̇ Energy production rate Flame radius [ ] Ignition radius [ ] Gas constant Reynolds number Radiative source [ ] Schmidt number Time scale [ ] Temperature [ ]

Flame development time ] Reaction time scale [ ] Inner layer temperature [ ] Reference temperature [ ]

Turbulent intensity, RMS of the turbulent velocity Flame speed [ ]

Flow velocity component [ Flow velocity component ] Velocity of the smallest eddies ] Mole fraction

Coordinate in -direction [ ] Coordinate in -direction [ ] Mass fraction

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Chapter 1 Introduction

1.1 Motivation

The risks of hydrogen release and combustion during a severe accident in a Light Water Reactor (LWR) have received considerable attention after the accidents at Three Mile Island, USA in 1979 (Alvares et al., 1982) and at Fukushima, Japan in 2011 (ANS Committee, 2012). When mixed with air in the containment, the hydrogen can form a flammable or even explosive gas mixture. After ignition, a combustion process is initiated resulting in a propagation of a turbulent flame and correlated build-up of pressure waves that may damage relevant safety systems and may even compromise the integrity of the containment walls. This is critically dangerous because nuclear containment is the last barrier between radioactive chemicals and the environment. The recent accident in Fukushima confirmed the destructive power of a hydrogen deflagration and therewith the importance of hydrogen control, which is a key safety issue for nuclear power plants.

“Small” leaks can potentially be controlled through ventilation. This is the general safety concept employed to prevent the build-up of an explosive atmosphere inside an environment. Hydrogen mitigation systems, such as Passive Autocatalytic Recombiners (PARs) can be designed and installed to reduce the risk of hydrogen combustion. Despite the installation of PARs, it has been generally recognized that the temporary existence of flammable gas clouds cannot be fully excluded during certain postulated accident scenarios (Bentaib et al., 2010). Therefore, reliable numerical modelling is needed to assess the associated residual risk of possible hydrogen deflagrations. In addition, numerical models can be employed to optimize the design of the hydrogen mitigation systems in order to reduce this residual risk as far as possible. For this purpose, CFD codes offer a detailed representation of the combustion phenomena involved although they are relatively more CPU demanding compared to the containment system codes, which belong to the lumped parameter (LP) codes.

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12 1.2 Role of CFD modelling

In the nuclear research community, there is currently an ongoing debate about whether CFD is really necessary to predict hydrogen deflagration phenomena occurring in real-scale containments. This issue is often raised, since Lumped Parameter (LP) codes can reasonably predict the mean maximum pressure or the Adiabatic Isochoric Complete Combustion (AICC) pressure in case of a hydrogen explosion. However, during fast deflagrations, intermediate peak pressures occur, which can be substantially higher than . Estimation of these maximum pressures are required in order to assess possible structural damage due to a hydrogen explosion. Furthermore, the evaluation of possible structural damage of the containment does not involve solely the maximum pressure, but also the dynamic loads, that is the rate of pressure change versus time, , must be considered as well. Furthermore, the containment can undergo structural damage, if the residual pressure that remains after complete burn oscillates with eigen-frequencies corresponding to those of the containment structure or the safety systems inside the containment. Therefore, both the frequency and the amplitude of the pressure oscillations may also be important. This essentially means that four parameters are important for the evaluation of the possible structural damage:

 Maximum pressure;

 Rate of pressure increase ;

 Eigen-frequencies;

 Amplitudes corresponding to the residual pressure waves.

The latter three parameters cannot be obtained using an LP code, since they depend on the flame acceleration. This flame acceleration in turn depends on the generation of turbulence in the flame front and the interaction of this flame with obstacles. Overall, flame acceleration due to turbulence generation by local obstacles cannot be computed using LP codes. Furthermore, the propagation of the pressure wave phenomena induced by the flame acceleration also cannot be computed by LP codes. This essentially means that CFD codes should be used in order to assess the consequences of a possible hydrogen deflagration.

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13 1.3 Objectives

This thesis work raises from a larger framework research concerning hydrogen deflagration developed at the Nuclear Research and Consultancy Group (NRG) in The Netherlands. The NRG in-house CFD-based model for hydrogen deflagration analysis (Sathiah et al., 2012a) will be the main instrument of this work and it is based on:

 Density-based coupled solver (ANSYS Fluent, 2008) for accurate tracking of the induced pressure wave phenomena;

 Application of an advanced Turbulent Flame Speed Closure (TFC) combustion model based on the Zimont model (Zimont, 1979) via User Defined Functions (UDF);

 Application of Adaptive Mesh Refinement (AMR) for accurate and efficient tracking of the turbulent flame propagation (Sathiah et al., 2012b).

It is worth presenting at this point the mentioned framework from where this thesis work starts from, in order to understand its objectives. The TFC combustion model has been validated at NRG against fast deflagration experiments from the ENACCEF experimental facility operated by CNRS (Centre National de la Recherché Scientifique, France) (Bleyer et al., 2012; Chaumeix and Bentaib, 2010, 2011; ISP-49, 2011), for the following mixtures:

 Homogeneous hydrogen-air mixtures (Sathiah et al., 2012b);

 Homogeneous hydrogen-air-steam mixtures (Sathiah et al., 2015a);

 Non-homogeneous hydrogen-air mixtures (Sathiah et al., 2015b).

The flame velocity reached the maximum value of 150 m/s in ENACCEF facility, that is fast deflagration. The general conclusion was that the applied CFD model predicted all the considered ENACCEF experiments reasonably well. That means, for example, the maximum pressure was predicted within 12-15% inaccuracy for the considered tests.

Recently, Sathiah et al., (2016) presented an extension of the TFC combustion model that was validated for slow deflagrations in homogeneous hydrogen-air experiments conducted in a medium-scale Thermal-Hydraulics Hydrogen Aerosols and Iodine (THAI) test facility (Kanzleiter and Langer, 2010; ISP-49, 2011). The flame velocity reached the maximum value of 10 m/s in THAI facility, that is slow deflagration. This model, referred to as the Extended TFC (ETFC), is based on the improvements made by Lipatnikov and Chomiak (2002) to the

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Zimont model. It was concluded that the ETFC model shows improvement in the initial laminar-like regime of slow deflagration compared to the TFC model.

In order to continue the development and validation study for both the TFC model and the ETFC model within the slow deflagration regime, this thesis work studies the steam dilution effect for slow deflagrations in homogeneous mixtures. For that purpose, the experimental data from the THAI facility are used again. Three THAI hydrogen deflagration (HD) experiments with different initial steam concentrations are considered, namely THAI HD-15 with no steam, THAI HD-22 with 25 vol.% steam, and THAI HD-24 with 48 vol.% steam.

The objectives of the current work are defined as follows:

 To further validate the TFC model from Zimont (1979) and the ETFC model from Lipatnikov and Chomiak (2002) against three slow deflagration experiments performed in the THAI facility, with different initial steam amount;

 To investigate if the combustion models are able to predict the effect of steam on hydrogen deflagration;

 To perform grid sensitivity of the aforementioned combustion models.

1.4 Structure of the thesis

The thesis work is structured as follows:

 First, the combustion and deflagration phenomena are presented (Chapter 2);

 The experimental facility and the experiments used for the validation are described in Chapter 3;

 The applied CFD models, with all the issues concerning the hydrogen-air-steam uniform mixture are described in Chapter 4;

 Next, the validation results are presented and discussed in Chapter 5;

 The summary and conclusions are then presented in Chapter 6, including the discussion of possible future steps for further validation of the presented CFD-based combustion modelling approach.

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Chapter 2 Combustion and deflagration processes

2.1 Introduction

The term combustion refers to a very fast chemical reaction between the fuel and the comburent, which leads to heat release and light emission. The fuel can be any substance able to release energy when oxidized, while the combustive agent is a substance which contains oxygen. The reaction takes often place in a very restricted zone called the flame front. Combustion alone, without turbulence, is an intrinsically complex process involving a large range of chemical time scales. Turbulent combustion involves a large number of physical and chemical time scales, therefore it is a result from two way interaction of chemistry and turbulence. Turbulent combustion is a much more complicated phenomenon which is challenging the science community for the last few decades.

Fig. 2.1: (a) Premixed and (b) non-premixed flame configurations (Poinsot and Veynante, 2001).

Depending on how the fuel and the combustive agent interact, it is possible to divide the combustion into three sub-categories, premixed, non-premixed and partially premixed combustion. In premixed combustion, the fuel and oxidizer are mixed before they enter the combustion chamber (Fig. 2.1a). In non-premixed combustion, also known as diffusion flames, the fuel and oxidizer are introduced separately into the combustion chamber through two (or more) inlets where flow rates and mass fractions are controlled separately (Fig. 2.1b). Partially premixed combustion is defined as a combustion process in which species are not perfectly mixed before combustion takes place but are better mixed than in

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a pure diffusion flame. Partially-premixed combustion can thus be interpreted as a combination of non-premixed and premixed combustion. This thesis work is focused on premixed combustion of flames, and from here on we refer to premixed flames simply as flames, until and unless mentioned otherwise.

During premixed flame propagation, the speed at which the flame front travels relative to the unburned mixture is known as the flame speed. This speed may be affected by many parameters such as the concentration of the fuel in the mixture, the geometry of obstacles along path of the flame, dimensions of the enclosures, etc. Depending on flame speed, premixed flame can propagate in three modes:

 Deflagration: It is a process in which the flame front travels at subsonic speed relative to the unburned gas;

 Deflagration to Detonation transition (DDT): During a deflagration wave propagation, a flame can accelerate under turbulent conditions and transform into a detonation wave. Turbulence enhances combustion to a point where a shock wave is formed just ahead of the flame front;

 Detonation: It is a process in which the flame front travels at a velocity greater than the speed of sound. The unburned mixture is heated by a shock wave behind which the mixture reaches a high temperature.

2.2 Laminar flames

The one-dimensional laminar flame propagation within a premixed gas is the basic of combustion configurations, both for theory and for numerical techniques The structure of a laminar premixed flame consists of four distinct regions as shown in Fig. 2.2:

 Unburned region;

 Preheat region;

 Reaction zone;

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17 Fig. 2.2: Laminar flame structure (Ciccarelli and Dorofeev, 2008).

Initially, fresh unburned mixture at the ambient conditions is transported to the flame zone. As the mixture approaches the flame front, it is heated by conduction and radiation from the flame zone upstream. Chemical reaction and heat release are negligible at this stage. Once temperatures are high enough to sustain combustion, chemical reaction takes place in the reaction zone. The gases emerging from this zone enter the burned gas zone where concentration and temperature are once again constant.

2.3 Laminar flame speed

The laminar flame speed of a fuel-air mixture is an important physical-chemical property, which depends on the pressure, temperature, equivalence ratio, and type of the mixture. A wide spread in the values of laminar flame speeds of hydrogen-air mixtures are observed in the combustion literature. Sathiah et al., (2012b) compared many correlations available in the combustion literature based on experimental results and numerical simulations. Fig. 2.3 displays the spread of these results as a function of the equivalence ratio.

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Fig. 2.3: Laminar flame speed versus equivalence ratio at 100kPa and 298K. The symbols represent experimental data while solid lines represent simulations. The legend, reported in Sathiah et al., 2012b, is summarized as follows: 1) Aung et al. (1997); 2) Dowdy et al. (1993); 3) Wu and Law (1984); 4) Takahashi et al. (1983); 5) Vagelopoulos et al. (1994); 6) Conaire et al. (2004); 7) Taylor (1991) 8) Kwon and Faeth (2001); 9) Iijima and Takeno (1986); 10) Law (1993); 11) Tse et al. (2000); 12) Liu and MacFarlane (1983); 13) Lamoureux et al. (2003); 14) Egolfopoulos and Law (1991); and 15) Malet(2005) and Bleyer et al. (2012).

The equivalence ratio of a given mixture is defined as the ratio of the fuel mass fraction to the oxidizer mass fraction in a mixture to that required for ideal stoichiometric conditions. That is, it is given by:

( ) ( )

⁄

Where is the fuel mass fraction, and is oxidizer mass fraction. For rich mixtures, the fuel is in excess, and . For lean mixtures, the oxidizer is in excess, and .

In nuclear applications we generally deal with low equivalence ratio mixtures with to . An enlarged zoom out of this area is also shown in Fig. 2.3. It can be observed that the spread in values of laminar flame speed is much less in this region. Therefore, we have a conservative estimate of laminar flame speed for the work performed in this thesis. However, for high equivalence ratios, i.e., rich fuel mixtures, the laminar flame speed values are widely spread. Therefore, for applications that need laminar flame speed in this range, more experiments are required for its accurate estimation.

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Liu and MacFarlane (1983) measured the laminar flame speed of hydrogen-air-steam mixtures in a Bunsen burner. The experiment covers hydrogen concentrations in the range of 18-65 vol.%, steam concentrations in the range of 0-15 vol.%, a temperature range of 296-523 K, and pressure of 1 atm. They fitted their experiments by an empirical formula as follows:

where, , and are constants given by:

( ) ( )

( ) and

where, is the mole fraction of steam, is the mole fraction of hydrogen, (K) is the temperature of the unburned gas, and , , , , and are constants, summarized in the following table.

Table 2.1: Values Liu and MacFarlane of constants

4.644e-04 -2.119e-03 2.344e-03 1.571 3.839e-01 -2.21

Bentaib and Chaumeix (2012) measured the laminar flame speed for hydrogen-air-CO2-He mixtures for different diluent mole fraction. They claimed that a mixture of 60% CO2 and 40% He can mimic H2O as diluent, since both of these diluents have similar physical properties. The measured results are fitted into the following correlation for the laminar flame speed:

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where is the equivalence ratio and is the mole fraction of the diluent such as steam. The measurements were performed at a reference pressure of 100 kPa and reference temperature of 298 K. The experiment covers a temperature range of 296-353 K and a pressure range of 100-500 kPa.

From the following Fig. 2.4, we can see the variation of the laminar flame speed with diluent concentration as calculated using the two correlations given in Eqs. (2.2-2.5). It can be observed that an increase in the diluent mole fraction decreases the laminar flame speed. The laminar flame speed values obtained by Liu and Macfarlane (1983) are higher because their measurements were not corrected for stretch effects.

Fig. 2.4: The variation of laminar flame speed with diluent concentrations measured by Bentaib and Chaumeix (2012) and Liu and MacFarlane (1983) for a hydrogen concentration of 13%.

2.4 Turbulent flame speed

When flows entering a flame are turbulent, the laminar flame mode is replaced by a regime where turbulence and combustion interact. Unlike a laminar flame, where the flame speed depends on the thermal and chemical properties of the unburned mixture, the turbulent flame speed also depends on the behavior of the flow, as well as on the mixture properties. The effect of turbulence is to wrinkle and distort a laminar flame front. This flame front locally propagates with laminar flame speed consistent with a plane laminar flame. For an observer traveling with the flame, a turbulent flame speed can be defined as the velocity at

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which reactants enter the flame zone in a direction normal to the flame front. Since the direct measurement of unburned gas velocity at a point near a turbulent flame is exceedingly difficult, flame velocities usually are at best determined from measurements of reactant mass flow rates. Thus, the turbulent flame speed is expressed as:

̇

where, ̇ is the mass flow rate of unburned gas, is the unburned gas density, is the time averaged mean flame surface area, is the turbulent flame speed, is the instantaneous flame surface area and is the laminar flame speed. Therefore, the ratio of the turbulent flame speed to the laminar flame speed is given by:

Before further discussing the turbulent flame speed, it is useful to define certain terms and non-dimensional numbers relevant for turbulent combustion.

Turbulent time scale: The integral length scale is a physical quantity describing the size of the large energy containing eddies in a turbulent flow. The turbulent time scale is the time scale associated with these large eddies and is given by:

where ( )

where, is RMS of the turbulent velocity, is the turbulent length scale constant, and is the turbulent dissipation rate.

Kolmogorov time scale: The time scale of the smallest eddies is associated with the Kolmogorov time scale which is defined as:

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where is the kinematic viscosity.

Chemical time scale: The chemical time scale, , is defined as,

⁄ , where

in which is the thermal diffusivity, is laminar flame speed, is the thermal conductivity, is the density, and is the specific heat at constant pressure.

Flamelet: In premixed combustion flows, chemical activity often occurs in thin (in comparison with the integral length scale) and strongly wrinkled sheet like fronts. These fronts are known as the flamelets.

Flame brush thickness: The flame brush thickness is a characteristic measure of the transition zone between the unburned and burned states of a premixed flame.

Damkohler number: The Damkohler number is defined as the ratio of the integral time scale to the chemical time scale, that is,

Small values of correspond to slow reactions and large values of corresponds to fast reactions.

Karlovitz number: It is defined as the ratio of the chemical time scale to the Kolmogorov time scale and is given by:

where, is the flame brush thickness, is the length of smallest eddies in turbulence cascade, is the velocity of the smallest eddies. If , the chemical reactions occur much faster than all turbulent time scales. Turbulence does not alter the flame structure. As is increased the flame is more prone to alterations by turbulent fluctuations.

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23 Turbulent Reynolds number: The Reynolds number represents the ratio of the inertia forces to the viscous forces. It characterizes the extent of the turbulent energy cascade. The turbulent Reynolds number is based on the integral length scale of turbulence and the turbulence intensity and is given by:

The Damkohler number, the Karlovitz number and the Turbulent Reynolds number can be combined to show that:

Prandtl Number: The Prandtl number is defined as the ratio of momentum diffusion and thermal diffusion and is given by:

Schmidt Number: The Schmidt number is defined as the ratio of momentum diffusion and mass diffusion and is given by:

where, is the mass diffusivity.

Lewis Number: The Lewis number is the ratio of thermal diffusion to mass diffusion given by:

2.5 Preferential Diffusion Thermal instability

The Preferential Diffusion Thermal (PDT) instability is a combination of thermal and diffusive instability. During flame propagation both mass and thermal diffusion exist across the flame surface (Fushui, 2012).

At any instant, it is likely that molecular diffusion of the deficient reactant, , is larger than thermal diffusion, i.e. . In that case, chemical energy supplied to the positively curved

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parts of the flame surface (upstream-pointing bulges) exceeds the heat losses due to molecular conductivity. Also, if the molecular diffusion of the deficient reactant is greater than the molecular diffusion of the excess reactant, i.e. , the mixture component in the bulges tends to stoichiometric composition. Both these processes increase the flame speed locally in upstream pointing bulges. The opposite phenomenon occurs in downstream pointing bulges, that is the flame speed is decreased. This results in growing amplitude of flame front perturbations (bulges), (Lipatnikov, 2005).

Fig. 2.5: Evolution of PDT instability. Gradual growth and branching of large cracks on the flame surface indicates growth of PDT instability (Fushui, 2012).

In Fig. 2.5, the evolution of unequal diffusion instability is shown. We can observe gradual growth and branching of large cracks in PDT instability. Very lean flames have a slow burning speed. Such slow burning speed gives cracks, which are formed due to initial spark defects and local heterogeneity or unequal diffusion effects, plenty of time to grow and branch. Thus, very lean flames are often born with large cracks due to the PDT instability.

Lean hydrogen-air mixtures have low Lewis numbers, i.e. , and therefore, high thermal instability. Thus, it is important to take into account the PDT instability effects when considering lean hydrogen-air mixtures, or in general mixtures with low Lewis numbers.

2.6 Turbulent premixed flames

Due to the presence of initial turbulence levels and combustion instabilities (discussed in previous section), the laminar premixed flame can transform into a turbulent premixed

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flame. The governing physical mechanism of premixed turbulent flame propagation are commonly described using the Borghi diagram (see Fig. 2.6).

In the Borghi diagram, the lines , , and represent the boundaries between the different regimes of premixed turbulent combustion. Another boundary of interest is (for our nomenclature, flame velocity is ) which separates the wrinkled flamelets from the corrugated flamelets. These regimes are explained below (Peters, 1988).

Fig. 2.6: The Borghi Diagram (Lipatnikov and Chomiak, 2002).

Laminar flames: The line separates the laminar flames from the turbulent flames .

Wrinkled flamelets: In this regime , i.e., even the large eddies cannot convolute the flame front. Therefore, the flame displacement by is faster than the displacement by turbulence with in this regime. That is, flame propagation is dominating over the turbulence. The flamelet structure remains close to that of laminar flames.

Corrugated flamelets: In this regime we have and . Therefore, using Eq. (2.12):

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The velocity of larger eddies is larger than the burning velocity, thus, these eddies will push the flame front around causing convolution of the flame surface. The smallest eddies have a velocity less than the burning velocity, and will not wrinkle the flame front. Therefore, the larger eddies interact with the flame and unburned pockets are formed that penetrate inside the flame brush and are eventually consumed by the flame advancement. At sufficiently low turbulence levels, the mean thickness of a turbulent flame should be increased by this mechanism.

Distributed combustion: In this regime, , i.e. (using Eq. (2.12)). Therefore, even the smallest eddies can enter into the flamelet structure. Thus, the flame is thickened by turbulent eddies. This results in intensified heat and mass transfer across the flame. This regime is also known as thickened flame regime or distributed reaction zone.

Well-stirred combustion: In this regime , that is the chemistry is slow. Turbulence homogenizes the flow field by rapid mixing, leaving the slow chemistry to be the phenomenon determining the rate of the process. No specific interaction between turbulence and combustion can occur in this regime.

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Chapter 3 Hydrogen deflagration experiments

Since severe accidents became a more and more part of licensing of nuclear reactors, a variety of reactor containments has been or will be equipped with systems to mitigate the threat of uncontrolled hydrogen combustion. The state of the art of such mitigation systems is the containment inertisation for Boiling Water Reactors (BWRs) and passive autocatalytic recombiners (PARs) for Pressurized Water Reactors (PWRs), as stated in Bentaib, 2010. Passive autocatalytic recombiners can cope with a wide range of hydrogen release scenarios, but for extreme accident scenarios hydrogen combustion at then relatively low concentrations is possible.

Experimental data on hydrogen combustion exists to a large extent, but there is a lack of data for combustion in vertically oriented, sufficiently large test facilities and under accident and containment typical conditions. More specifically, conditions to be examined are moderate hydrogen concentration, concentration gradients, elevated initial temperature, elevated initial pressure, and relatively high steam concentration.

In such a context, the OECD-THAI project was started in January 2007 under the auspices of the OECD-NEA to fill knowledge gaps and to deliver suitable data for evaluation and simulation. The Nuclear Energy Agency (NEA) is a specialized agency within the Organization for Economic Co-operation and Development (OECD), having as mission to assist its member countries in maintaining and further developing, through international co-operation, the scientific, technological and legal bases required for a safe, environmentally sound and economical use of nuclear energy for peaceful purposes.

The Thermal-Hydraulics Hydrogen Aerosols and Iodine (THAI) test facility (Kanzleiter and Langer, 2010) is particularly appropriate for this purpose, since it is of relatively large scale, designed for combustion pressures and temperatures, and has installed an advanced and sophisticated instrumentation and data acquisition system. This facility is located in Eschborn, Germany and the measurements for a large number of hydrogen deflagration

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(HD) experiments were performed by Becker Technologies GmbH under the sponsorship of the German Federal Ministry of Economics and Technology.

3.2 THAI-HD tests objectives and justification

The general objective of the hydrogen deflagration experiments is to provide additional data for an improved understanding of hydrogen combustion phenomena, and for the further development and validation of containment system codes and Computational Fluid Dynamics (CFD) codes. This requires experiments under conditions typical for severe accidents in a relatively large test facility which allows:

 Combustion in upward and downward direction;

 Variation of hydrogen and steam concentration and initial pressure;

 High initial temperatures up to 140 °C;

 Combustion of mixtures without and with steam and hydrogen concentration gradients, that is the coupling of hydrogen distribution and combustion;

 Detailed recording of flame propagation, pressure transients, temperature transients and combustion completeness.

Particular objectives of the experiments are achieved by a systematic variation of the test parameters, i.e. mainly the initial conditions; hence, the influence of these parameters on combustion behavior is to be determined. The following parameters were varied:

 Initial pressure;

 Initial temperature;

 Vertical temperature gradient;

 Hydrogen concentration (homogeneous and stratified);

 Steam concentration (homogeneous and stratified);

 Burn direction (upwards and downwards).

By comparing the results of the tests, the influence of these parameters on pressure build-up, temperature increase, flame front propagation (velocity and shape), and on combustion completeness is investigated.

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The justification of the tests can be outlined as follows (Kanzleiter and Langer, 2010):

 Hydrogen deflagrations cannot be ruled out completely even by the use of Passive autocatalytic recombiners (PARs) and therefore have to be the subject of further investigations;

 Experimental data mainly exist for horizontal flame propagation and/or small geometries;

 Deflagrations in upward direction and in large facilities are faster (stronger) than in horizontal geometries;

 Current modeling of low-concentration hydrogen deflagrations is still inadequate and needs to be improved in order to predict the effects reliably. Experimental data are needed for further model development.

3.3 THAI-HD tests configuration

The THAI test facility is a cylindrical stainless steel vessel, completely enveloped by a 120-mm rockwool thermal insulation. The vessel is 9.2 m high and has an internal diameter of 3.2 m with a total volume of 60 m3 (Kanzleiter and Langer, 2010). The maximum admissible overpressure is 14 bar at 180 °C, or 16 bar at 20 °C wall temperature. The setup of the THAI facility during the deflagration experiments is shown in Fig. 3.1b.

During the HD experiments there are no internal structures present in the THAI vessel that can generate extra turbulence: the inner cylinder and the condensate trays are removed (Fig. 1a). Access into the vessel for installation of components and instrumentation is provided by the top flange (1540 mm wide) and by a lower and an upper man hole. Supporting structure for instrumentation can be assumed not to have any significant effect on turbulence generation and thus on the deflagration phenomena and flame propagation.

In total, 29 hydrogen-air and hydrogen-air-steam flame propagation experiments have been performed in the THAI facility within the OECD-THAI project (2007-2009), reported in ISP-49, 2011:

 HD-1R through HD-14 (15 experiments) correspond to deflagrations in homogeneous hydrogen-air mixtures at ambient temperature;

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30  HD-15 through HD-21 (6 experiments) correspond to deflagrations in homogeneous

hydrogen-air mixtures at elevated temperature;

 HD-22 through HD-24 (3 experiments) correspond to deflagrations in homogeneous hydrogen-air-steam mixtures at superheated and saturated conditions;

 HD-25 through HD-29 (5 experiments) correspond to deflagrations in non-uniform hydrogen-air-steam mixtures.

All test types have been performed with either upward or downward burn direction, i.e. initiated by either vessel bottom ignition or vessel top ignition.

a) b)

Fig. 3.1: a) THAI test vessel (inner cylinder and condensate trays removed for HD-tests. b) Schematic representation of the THAI facility (Kanzleiter and Langer, 2010).

As presented in the state of art review in section 1.3, CFD model validation in the slow hydrogen deflagration regime has been started by Sathiah and Holler (Sathiah et al., 2016) on the basis of the THAI-HD experiments. Sathiah and Holler have used the THAI tests HD-12, HD-7 and HD-3 in order to validate the CFD model in the slow hydrogen deflagration regime

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for different uniform hydrogen-air mixtures. Future modelling steps have been indicated and therefore this current thesis work aims to continue the research. In particular, a steam effect on the hydrogen deflagration phenomena is to be investigated, hence uniform hydrogen-air-steam mixtures in the slow deflagration regime are considered here. In order to investigate the steam effect, three experiments with upward propagation and similar initial conditions but different steam concentrations (at saturated state) were selected, namely: HD-15 with no steam, HD-22 with 25 vol.% steam, and HD-24 with 48 vol.% steam. In these experiments, the initial temperatures, the initial pressures, and the initial hydrogen concentrations are very similar. The main characteristics of experiments HD-15, HD-22 and HD-24 are summarized in Table 3.1. It can be noted that the main varied parameter is the steam concentration.

Table 3.2: THAI HD experiments considered for validation (Kanzleiter and Langer, 2010) RUN P0 [bar] T0 [K] H2 [vol.%] Steam [vol.%] Burn direction Mixture

HD-15 1.504 366.0 9.9 0 Upward Uniform

For the HD tests, the test facility has been equipped with the following test specific installations and components:

 Hydrogen feed system;

 Steam feed system;

 Recirculation fan;

 Igniters.

A hydrogen distributor located in the sump was used to inject the hydrogen in the cylindrical vessel before each experiment of our interest started. The steam is at 8 bar and 170 °C at steam generator exit. An axial fan located near the hydrogen distributor was used to create a homogeneous mixture before it was ignited. Remote controlled spark igniters, one near the vessel bottom and one near the vessel top are applied to initiate either an upward or downward directed deflagration.The ignition of our interest is located 0.5 m from the bottom of the facility and the mixture was ignited, about 10 to 15 minutes after the end of fan operation.

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32 3.4 Instrumentation

Long term instrumentation is mainly used for preparing the conditions in the test vessel prior to ignition, whereas the short term instrumentation, shown in Fig. 3.2, is used to record the deflagration phenomena.

Hydrogen concentration measuring system

Fifteen continuously operating gas sampling lines take gas samples from the vessel atmosphere at different locations prior and after hydrogen combustion. The sampled gas is fed to a 15-channel gas concentration analyzer which determines the hydrogen concentration in dry air by heat conductivity measurement. In experiments with steam, the steam is removed by coolers and condensate traps upstream of the analyzers.

Temperature measurement

Temperature of the vessel gas atmosphere prior to ignition is measured by five (tests HD-1 through HD-21) and by 13 (tests HD-22 through HD-29) calibrated thermocouples distributed along the vessel height. In addition, a large number of thermocouples is installed to measure the wall temperature during the experiment.

Flame front detection

A grid of 43 fast sheathed thermocouples (outer diameter 0.25 mm), is installed at different elevations and radial positions in the vessel to monitor flame propagation (“flame front arrival”) and flame temperature during hydrogen combustion.

The flame position was determined by the arrival of the flame at the locations of the fast thermocouples by measuring a steep temporal increase of the temperature. Only measurements of “first arrival” were used in order to exclude the influence of hot burnt gas.

Pressure measurement

The pressure signal was measured using four fast pressure transducers of the strain gauge type which are mounted at the inner wall of the vessel.

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33 Fig. 3.2: Short term instrumentation used to record the deflagration phenomena (Kanzleiter and Langer, 2010).

Further details about instrumentation accuracy, sensitivity, time resolution and the repeatability of the experiments are available in the technical report (Kanzleiter and Langer, 2010).

3.5 Test results

The measured pressure evolution and the axial flame front positions during experiments HD-15, HD-22 and HD-24 are shown in Fig. 3.3 and Fig. 3.4. Scattered clouds with all experimental points are represented in Fig. 3.3 and the corresponding moving averaged pressure profiles. The window for the moving average is 0.1 s and it is applied for the

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quantitative analysis as well. In all the following figures in this paper, a reduced number of experimental points will be used in order to have better overall visibility. Clearly, the peak pressure is the highest for test HD-15, without steam, and decreases with increasing steam content. In test HD-15, the peak pressure is also reached faster. The effect of increasing steam content is also evident in the plot of flame front position. A slower flame propagation is observed at higher steam concentration.

As stated in the technical report on the tests (Kanzleiter and Langer, 2010), an increased combustion time means enhanced heat transfer to the vessel walls, resulting in lower peak temperatures and peak pressures for the experiments with steam. Apart from this, replacing air by steam means an increased heat capacity of the gas ( kJ/kgK; kJ/kgK) which will also result in a lower peak temperature and peak pressure for the same amount of energy released as has been the case for the three selected experiments. The mass difference of steam and air compensates this effect only partially. In Fig. 3.3, scattered clouds of experimental data are the consequence of acoustic effects (high frequency pressure waves). This effect can be observed in tests HD-15 and HD-22, but not in test HD-24 with high steam content. The reason for this is not known from the technical report, either it is due to the longer combustion time, or the steam changes the acoustic properties, e. g. the damping behaviour and/or resonance frequency of the system. The relative low steam content in the test HD-22 reduces flame front velocity, but not as significantly as for the test HD-24. Combustion is erratic and slow in test HD-24 with the flame front bypassing the vessel centreline, leaving there portions of unburned gas, which are ignited later during the deflagration. This high steam content brings the mixture relatively close to the flammability limit, and this apparently results in an unsteady combustion.

As shown in Fig. 3.5, the flame propagates rather steady in tests HD-15 and HD-22, but in an extremely chaotic way in test HD-24 where it burns in a kind of “hoses”, producing separated fire balls. However, at the end, combustion was also complete in this test (as in tests HD-15 and HD-22), indicating that the mixture was ignitable throughout the vessel.

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35 Fig. 3.3: Measured pressure evolution during THAI experiments HD-15, HD-22 and HD-24 (Kanzleiter and Langer, 2010). The symbols represent the raw experimental data and the lines the corresponding moving averages.

Fig. 3.4: Experimentally measured axial flame front positions during THAI experiments 15, HD-22 and HD-24 (Kanzleiter and Langer, 2010). The symbols represent the first flame front arrival at elevation and the lines the corresponding moving averages.

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36 Fig. 3.5: Flame front propagation as timelines, test HD-15, HD-22 and HD-24 (Kanzleiter and Langer, 2010).

It is worth stressing here that initial turbulence levels were not measured in these experiments, heat transfer to the walls and heat losses from the domain boundary are also not specified. However, from the gradual decrease in the pressure after complete burning in the experimental results, it is evident that some heat loss occurred. In the Technical Report Kanzleiter and Langer (2010) it is only stated that the initial turbulence level in the gas at the time of ignition, caused by the prior operation of the mixing fan, appears to be very low. Ignition of the mixture by a spark igniter was done 10 to 15 minutes after the end of fan operation.

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Chapter 4 Numerical methodology

The presented hydrogen deflagration models in this chapter are implemented in the commercial CFD code ANSYS Fluent, 2008. The unsteady Favre-averaged (density-weighted) Navier-Stokes equations are solved with a 2-D axisymmetric approach, using the standard turbulence model. Within the CFD code, the equations for the conservation of mass, momentum, energy and a combustion progress variable are solved.

The combustion model is here discussed. First, the differences between the Turbulent Flame Speed Closure (TFC) combustion model based on Zimont (1979), and the Extended Turbulent Flame Speed Closure (ETFC) combustion model based on Lipatnikov and Chomiak (2002) will be explained. Next, combustion sub-models for preferential diffusion and compression effects on the laminar flame speed are described. Finally, the laminar flame speed, additional models, the mesh, boundary conditions, and the numerical schemes applied in the CFD model are described.

4.2 Conservation equations

The equations for the conservation of mass, momentum and energy are given as follows (Poinsot and Veynante, 2001).

Mass:

̅ ( ̅ ̃)

Momentum:

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38 Energy: ( ̅ ̃) ( ̅ ̃ ̃) ( ̃ ̅) . ̅ ̅ ̃ ∑ ⃗⃗ ̅̅̅̅̅̅̅̅̅̅̅̅ / ( ̅̅̅̅̅̅) ( ̅ ̃) ̅̅̅̅̅ ̅ ̇̃ Reynolds averages denoted by overbars ̅̅̅, as well as the Favre averages ̃, are used. Here, is time, and are the flow velocity components, and are the coordinate components. The gas density is , is the pressure, and are the body force and viscous force tensor respectively, is the total energy, is the thermal conductivity, is the enthalpy of the th species, ⃗⃗ is the diffusion flux, is the radiative source term. Finally, ̇̃ term in Eq. (4.3) represents the energy production rate, which is closed as follows:

̇̃ | ̃|

where, is the unburned gas density, is the turbulent flame speed, ̃ is the progress variable and is the heat of combustion of the mixture in J/kg.

For multi-dimensional computations of perfectly premixed, adiabatic, turbulent combustion under slight pressure variations, it is a common practice to characterize the combustion process by a combustion progress variable ( ̃ in the unburned gas and ̃ in the burned products) following the well-known Bray-Moss method (Bray KNC, 1977). The balance equation for the Favre-averaged progress variable is as follows:

̅ ̃ ( ̅ ̃ ̃) ( ̅ ̃ ) ̅ ̇̃

Here, ̅ ̇̃ represents the reaction rate.

Under the assumption of a perfect gas, Lewis number , and a single step irreversible and adiabatic chemical reaction, the progress variable ̃ is defined as follows:

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39 ̃ ̃ ̃ ̅

Here, , is the gas temperature and represents the fuel mass fraction. Indexes and stand for unburned and burned state of the mixture, respectively. For non-unity Lewis number, as for hydrogen-air mixtures, the progress variable is defined as follows:

̃ ̃

The density in Eq. (4.1), (4.2), (4.3) and (4.5) is obtained using the ideal gas law as follows:

̅ ̅

( ⁄ ) ̃

where is the gas constant, is the molecular weight of the mixture, ̃ is the Favre-averaged temperature obtained by solving Eq. (4.3) and using the caloric equation of state relating temperature and composition to enthalpy.

4.3 Turbulence model

The standard model is used as the turbulence mode. The transport equations for turbulent kinetic energy and turbulent dissipation rate follows from Launder and Spalding, 1972. ̅ ̅ ̃ [( ) ] ̅ ̅ ̅ ̃ [( ) ] ̅ where , , and are model constants and the production term is given by:

̅ ̃ ̃

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The Reynolds stress term ( ̅ ̃) in Eq. (4.2) is closed using the isotropic eddy viscosity formulation as follows: ̅ ̃ ( ̃ ̃ ̃ ) ̅

while the energy flux in Eq. (4.3) and the progress variable flux in Eq. (4.5) are closed using the classical simple gradient assumption as follows:

̅ ̃ ̃ ̅ ̃ ̃

Here, , are the turbulent Prandtl number and Schmidt number and is the turbulent viscosity and is given as follows:

̅

where, is a constant.

4.4 TFC combustion model

The commercial CFD code ANSYS Fluent, 2008, has the Zimont model implemented only in its pressure based solver. In order to use the density based solver, the model needs to be implemented using a user-defined function (UDF). Differences among the solvers is described in section 4.11.

The Turbulent Flame Speed Closure (TFC) combustion model based on Zimont model (Zimont, 1979) solves the following progress variable equation:

̅ ̃ ( ̅ ̃ ̃) ( ̅ ̃

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Here and are the thermal diffusivity of the unburned mixture and the turbulent diffusivity respectively, is the turbulent flame speed, and is the unburned gas density and ̃ is the Favre-averaged progress variable, given by Eq. (4.7). This is true for flame propagation in closed vessels as indicated by Cant and Bray (1989).

The turbulent diffusivity is evaluated as:

Here, is the turbulent kinematic viscosity and the turbulent Schmidt number. The turbulent kinematic viscosity is calculated using the standard turbulence model presented in section 4.3.

Although widely validated and used, the Zimont model has limitations discussed and addressed by Lipatnikov and Chomiak (2002). These relate to the fact that turbulent combustion can appear in several regimes and it is difficult to cover all of them efficiently in one model. It is worth defining at this point the flame brush thickness that will be used from now on as a characteristic measure of the transition zone between the unburned and burned states of a premixed flame. More specifically, limitations of the Zimont model are:

 First, the TFC model simplifies the flame development process to the growth of a mean flame brush thickness, , fully determined by the turbulent diffusion law.This is a reasonable zero-level approximation, employing the turbulent diffusivity , which offers the opportunity to account for the effective turbulent diffusivity increase with flame development time;

 Second, the TFC model assumes a steady turbulent burning velocity in a flame of a growing thickness. Such a flame is called in literature as Intermediate Steady Propagation (ISP) flame. Again, this is a reasonable zero-level approximation where the burning velocity tends to approach an Asymptotically Steady (AS) limit faster than the mean flame brush thickness , because the turbulent flame speed is mainly controlled by small-scale eddies. Moreover, large-scale eddies, which increase , also increase instantaneous flame surface area and therewith the burning velocity. Thus, a first-order approximation for must depend, although weakly, on time when the thickness grows and has to account for a weak growth of when approaching the AS regime ( turbulent time scale);

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42  Third, at small and moderate ⁄ , effects of the laminar flame speed and mixture properties on the development of the mean flame brush thickness are possible, mostly for expanding flames;

 Fourth, in case of weak turbulence, when , the model predicts zero flame speed rather than .

The ETFC model addresses the aforementioned limitations.

4.5 ETFC combustion model

To overcome the Zimont model limitations, Lipatnikov and Chomiak have applied the following modifications. First, the introduction of an additional laminar-like source term ( ) and second, the inclusion of time-dependent expressions for the turbulent diffusivity ( ) and the turbulent flame speed ( ). The Extended Turbulent Flame Speed Closure (ETFC) combustion model based on Lipatnikov and Chomiak (2002), solves the following progress variable equation:

̅ ̃ ( ̅ ̃ ̃) ( ̅( ) ̃

) | ̃|

where the additional laminar-like source term on the right-hand-side is given by:

̅ ̃ ( ⁄ )

(

̃)

The activation temperature is , and is the reaction time scale. As shown by Lipatnikov and Chomiak (2002), this formulation gives the correct burning velocity in the limit of weak turbulence, namely , rather than as for TFC model. We can expect this model to be more suitable for simulating slow deflagration, especially in the initial phase following ignition. However, the model requires the evaluation of the reaction time scale , which is not straightforward. Therefore, we rely on an alternative formulation, from the quoted authors, for the laminar-like source term. The alternative progress variable equation has a slightly different laminar source term, namely:

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43

( ) ̃ ̃

This formulation is computationally somewhat less demanding and does not require evaluation of the reaction time scale. It is worth mentioning that the turbulent flame speed does not exactly converge to the laminar flame speed in the limit of weak turbulence using the proposed alternative equation.

The first model modification discussed, namely the laminar-like source term insertion, offers the opportunity to obtain the steady mean flame brush thickness in the fully developed case instead of a growing as for the TFC model. It includes neither new constants nor unknown parameters. When , the laminar-like source term dominates and controls the mean burning rate and thereby directly the turbulent flame speed . When is markedly higher than , the laminar-like source term is strongly reduced and its effects on the burning rate are negligible. Moreover, the flame speed computation dependency on flame thickness by numerical errors should be reduced to a minimum thanks to an adaptive mesh refinement applied in the flame front region.

The time-dependent turbulent diffusivity in Eq. (4.18), (4.19), (4.20) and the time-dependent turbulent burning velocity in Eq. (4.18) follow from Lipatnikov and Chomiak (2002): [ ( )] , [ ( )

where represents the flame development time, which is simply equal to the time counted from spark ignition for expanding flames. For typical stationary flames stabilized in parabolic flows, should be replaced by ∫ where is the mean gas flow velocity and is the axial distance from the flame holder. Thus, for these flows the development of the mean flame brush thickness with time is equivalent to the increase in with distance from the flame holder.

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where is the turbulent intensity, according to Taylor’s theory of turbulent diffusion (Brodkey, 1967) and the fully developed turbulent diffusivity, given by:

̃

Here, is a turbulence model constant, ̃ and ̃ are the Favre-averaged turbulent kinetic energy and turbulent dissipation rate, respectively. This definition is analogous to ⁄ previously shown for the TFC model. The same analogy is valid for from ETFC and from TFC. The closure for this turbulent flame speed will be explained in the following section.

The second model modification discussed, namely the time-dependent turbulent diffusivity the time-dependent turbulent burning velocity introduction offers the opportunity to simulate the transition from a laminar to a turbulent flame kernel, as well as the turbulent flame development. The modification includes neither new constants nor unknown parameters. The time-dependence of turbulent diffusivity accounts for the fact that, initially, only small fast scales can participate in the dispersion of an admixture cloud, whereas larger, slow scales are effective with a delay characterized by their time scales. Then, the effective turbulent diffusivity increases with time.

From a physical perspective, the growth of and is associated with the concept that the larger eddies wrinkle the instantaneous flame surface as the turbulent flame brush grows. It is worth emphasizing that Eq. (4.22) is consistent with other parts (Eq. (4.21) and (4.25)) of the whole model. Finally, thanks to the aforementioned extensions that include the laminar-like source term and address the time-dependence of the turbulent flame speed and turbulent diffusivity, we can expect this ETFC model to be more suitable for simulating slow deflagrations, especially in the initial phase following ignition.

It is worth reminding that the aim of this thesis work is to investigate the steam effect prediction capabilities of the codes as well. Indeed, the ETFC model includes differentiation of the laminar flame speed due to steam dilution, according to the different experiments from Table 3.1. Hence, this model is expected to be more capable for predicting the steam

Assessment of CFD hydrogen deflagration models for application in light water cooled nuclear reactor containments

UNIVERSITÀ DI PISA

ABSTRACT

Contents

List of Figures

List of Tables

Nomenclature

Chapter 1

Introduction

Chapter 2

Combustion and deflagration processes

Chapter 3

Hydrogen deflagration experiments

Chapter 4

Numerical methodology