Nuclear Engineering and Design 384 (2021) 111422

Available online 27 September 20210029-5493/© 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Development of a novel two-phase flow solver for nuclear reactor analysis: Validation against sodium boiling experiments

Stefan Radman a,*, Carlo Fiorina a, Andreas Pautz a,b

a Ecole Polytechnique Federale de Lausanne, Laboratory for Reactor Physics and Systems Behaviour, 1015 Lausanne, Switzerland b Paul Scherrer Institut, Nuclear Energy and Safety Department, 5232 Villigen, Switzerland

A R T I C L E I N F O

Keywords: Two-phase Porous-medium Sodium boiling OpenFOAM GeN-Foam

A B S T R A C T

Recent numerical advancements at the Laboratory for Reactor Physics and Systems Behaviour (LRS) have led to the development of a novel two-phase porous medium-based thermal–hydraulic simulation tool, integrated in the broader multi-physics framework of the GeN-Foam code. This simulation tool takes advantage of the modern programming paradigm offered by the finite volume-oriented OpenFOAM numerical library. It enables the 3-D investigation of thermal-hydraulic phenomena in nuclear systems via a dispersed Euler-Euler 6-equation formalism for the representation of two-phase flow. Stemming from previous work related to the presentation and verification of the solution algorithms, this paper is focused on the presentation of the modelling capabilities and their validation against different sets of sodium boiling experiments. These experiments were performed under both quasi-steady-state and transient conditions at former experimental facilities at the Joint Research Centre, Ispra and Kernforschungszentrum Karlsruhe.

1. Introduction

The simulation of two-phase flow dynamics is generally required for the investigation of a variety of reactor concepts in a variety of cir-cumstances. These range from steady-state operation of traditional reactor concepts such as Boiling Water Reactors (BWRs) to transients in Pressurized Water Reactors (PWRs). With regards to fast reactor tech-nologies, two-phase flow modelling capabilities are required for the modelling of accident scenarios in e.g. Sodium Fast Reactors (SFRs) beyond their initiation phase. Of particular relevance is the fact that in these fast systems, the generally positive (at least locally) void reactivity coefficient (Sun et al., 2011; Tommasi et al., 2010) poses a further layer of risk which calls for adequate multi-physics tools for the of modelling transient phenomena in which boiling might arise. While the overall impact of positive void coefficients can be mitigated via particular core and fuel element design choices (Beck et al., 2017; Sciora et al., 2020), the general consensus is that such phenomena should not be excluded from safety analyses of SFRs.

At the Laboratory for Reactor Physics and Systems Behaviour (LRS) at the EPFL, computational activities have been focused in recent years on the development of open-source computer codes for the analysis of several aspects of reactor dynamics. These efforts resulted in the

OFFBEAT code (Scolaro et al., 2020) for 3-D fuel performance analysis and the GeN-Foam code (Fiorina et al., 2015) for internally coupled multi-physics analyses of single-phase thermal–hydraulics, neutronics and thermal-mechanics applicable to a variety of reactor concepts, ranging from SFRs (Fiorina et al., 2016, 2017; Radman et al., 2019; Fiorina et al., 2019; Radman et al., 2018) to Molten Salt Fast Reactors (MSFRs) (Fiorina, 2019). Both codes take advantage of the modern programming paradigm offered by the C++ programming language and the OpenFOAM numerical library (OpenCFD, 2021; Weller et al., 1998). This offers several advantages in terms of code flexibility, ease of maintenance and scalability for High Performance Computing (HPC) applications, thus extending the application range of said codes to computational models consisting of several tens of millions of cells. With particular regards to the GeN-Foam code, its thermal–hydraulics capa-bilities were until recently constrained to single-phase scenarios that, while of significant interest, preclude the analysis of a significant number of systems and scenarios as mentioned earlier. This ultimately motivated the development of a novel OpenFOAM based solver for two- phase flows through porous media at sub-channel level resolution capable of modelling mass-transfer phenomena. The code has recently been integrated in the GeN-Foam multi-physics package.

This work is meant to be a stepping stone of said integration process, and it focuses primarily on the validation of the single-physics

* Corresponding author. E-mail address: stefan.radman@epfl.ch (S. Radman).

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Nuclear Engineering and Design

journal homepage: www.elsevier.com/locate/nucengdes

https://doi.org/10.1016/j.nucengdes.2021.111422 Received 13 December 2020; Received in revised form 21 May 2021; Accepted 12 August 2021

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thermal–hydraulics component against a variety of two-phase sodium boiling scenarios. In particular, the present work stems from previous extensive verification efforts of the developed numerics and algorithms (Radman et al., 2021). The primary reasons for the development of a novel OpenFOAM thermal–hydraulic solver rather than employing existing ones are covered in greater detail in said work, yet are hereby summarized as: 1) more flexible flow regime map representation of transfer models (i.e. drag, heat, mass transfer); 2) tensorial approach to modelling of fluid–structure drag as well as modelling of two-phase flow effects of fluid–structure drag; 3) fluid-coupled models for the treatment of structure heat sources; 4) solution algorithms (with regard to segre-gated pressure–velocity coupling) for the improved stability of certain two-phase flow scenarios, often well represented by sodium boiling. Nonetheless, a verification process deals exclusively with the sanity of the implementation of the mathematical models and algorithms, but it is fundamentally oblivious to how well the code is capable of reproducing experimental data, which is primarily dependent on the choice of spe-cific correlations for specific phenomena. Thus, comparison against experimental data and validation are the primary objectives of this work.

In Section 2 the governing equations as well as the principal modelling capabilities with regards to mass, momentum and heat transfer are presented. With regards to the solution algorithms, these were the topic of a previous verification work via the Method of Man-ufactured Solutions (Radman et al., 2021). In Section 3 a validation of the solver capabilities is performed against quasi-steady-state boiling experiments that were performed at the Joint Research Centre in Ispra, Italy, (Kottowski and Savatteri, 1984) in heated tubular test sections. Because of the quasi-steady-state nature of these experiments, the outcome of calculation results is mostly dependent on just a handful of physical models, namely two-phase pressure drop multipliers, latent heat and saturation temperature models. In Section 4, the results ob-tained by the code are compared against a transient boiling experiment carried out at the KNS experimental facility at the Kernforschungszen-trum Karlsruhe (Bottoni et al., 1990). The comparison allowed to perform a preliminary qualitative sensitivity study of the calculation results against a variety of both geometric and physical modelling choices. Conclusions regarding the results presented in this paper, along side future perspective work directions are presented in Section 5.

2. Code overview

The code relies on a Finite Volume, Euler-Euler, 6-equation model for the description of dispersed two-phase flows in porous media, with the porous medium being represented by a third, stationary phase. For the sake of clarity, by porous medium we mean the mathematical entity that arises subsequently to the application of volume-averaging techniques (Whitaker, 1985) for the modelling of a macroscopically non-porous structure (e.g. a fuel pin bundle). Thus, we are outside the traditional domain of Darcy-Forchheimer flows in microscopically porous media (e. g. sands) (Kozelkov et al., 2018). An overview of the governing equa-tions is provided in sub-section 2.1, while a more detailed presentation concerning the solution algorithms is deferred to previous work (Rad-man et al., 2021). The modelling capabilities of the code, specifically for the treatment of mass, momentum and heat transfer and their depen-dence on a user-selectable flow-regime map are discussed in greater detail in sub-section 2.2.

2.1. Governing equations

2.1.1. Continuity equation The continuity equation for the i-th phase can be written as:

∂∂t(αiρi)+∇⋅(αiρiui) = Γi (2.1)

It should be noted that the phase fraction αi is absolute in the sense that it is not normalized to the total porous structure void fraction. In practice, for a system of two phases labelled 1 and 2, and a porous structure that occupies a volume fraction αs, one has α1 + α2 + αs = 1, and by structure void fraction we mean 1 − αs. From a numerical perspective, due to the potential dependence of the density ρ on the system pressure for the generic case of compressible fluids (such as for the case of a vapour in a boiling two-phase flow), the continuity equa-tion is numerically treated as:

∂∂t

αi +∇⋅(αiui) = Sα,i (2.2)

Sα,i =1ρi

(

Γi − αi∂∂t

ρi − αiui⋅∇ρi

)

(2.3)

Nomenclature

α Volumetric phase fraction (–) Γ Volumetric mass transfer ( kg

m3s)

ε Specific turbulent kinetic energy dissipation rate (m2

s3 )

∊ Relative surface roughness (–) κ Thermal conductivity ( W

m K)

μ Molecular viscosity (Pa s)ν Kinematic diffusivity (m2

s )

ρ Density (kgm3)

τ Structure tortuosity tensor when modelled by porous medium (–)

ϕ2 Two-phase drag multiplier (–) ξ Continuity error ( kg

m3s)

Ω Computational domain a Thermal diffusivity (m2

s )

a Fluid interfacial area density (m2

m3)

as Structure surface area density (m2

m3)

Dh Hydraulic diameter or phase characteristic dimension (m)

Dp Pin diameter (m)

Dw Wire diameter (m)

g Gravitational acceleration (ms2)

h Specific enthalpy ( Jkg)

H Heat transfer coefficient ( Wm2K)

i, j Phase indices, either 1 or 2 k Specific turbulent kinetic energy (m2

s2 )

K, K Dimensional drag factor, scalar for fluid-fluid drag, tensor for fluid-structure drag ( kg

m3s)

Lw Pin-wrapping-wire lead length (m)

Pp Pin pitch (m)

Pt = Dp + 1.0444 Dw Wire-adjusted pin pitch (m)

p Pressure (Pa)q Volumetric heat source, sink or transfer term (W

m3)

T Temperature (K)u Velocity (m

s )

V Volume (m3)

Vm Virtual mass coefficient (–) Wp Wet pin bundle perimeter (m)

Ww Wet wrapper perimeter (m)

x Flow quality (–)

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The term Sα,i represents the phase fraction source term, which can be thought of as the combined source due to compressibility and mass transfer effects and is evaluated explicitly. This explicit evaluation characterizes the form of the continuity Eq. 2.3 as fundamentally non- conservative, and this treatment is employed due to the fact that in a general 2-D or 3-D domain, the implicit discretization of a density gradient would yield vector coefficients in the discretized coefficient matrix (compared to scalar coefficients that arise e.g. in 1-D system codes), which the underlying OpenFOAM framework does not support by default. In particular, the continuity equation is solved via the Multidimensional Universal Limiter with Explicit Solution (MULES) al-gorithm (Weller, 2008), which is a particular instance of a Flux Cor-rected Transport (FCT) technique (Boris and Book, 1997). Further details to the implementation are deferred to (Radman et al., 2021).

2.1.2. Momentum conservation equation The momentum conservation equation for the i-th phase is written

as:

∂∂t(αiρiui)+∇⋅(αiρiui ⊗ ui) = − αi∇p+∇⋅σ +Su,i (2.4)

where the divergence of the deviatoric stress tensor ∇⋅σ is treated via the Stokes hypothesis:

∇⋅σ = ∇⋅(

αiρiνiτi⋅(

∇ui + (∇ui)T−

23(∇⋅ui) I

))

(2.5)

The porous structure tortuosity tensor figures as τ, and, for ease of visualization, one can think of the whole effective kinematic viscosity term being modelled no longer as scalar but as a tensor in the form νiτ. Other momentum source terms have been grouped in Su,i:

these terms consist of, respectively, the fluid–fluid drag, fluid–structure drag, virtual mass force, momentum source and sink due to mass transfer, and a numerical correction term, namely ξiui,to account for possible continuity errors. The continuity error ξi is obtained via an explicit evaluation of the conservative formulation of the continuity Eq. 2.1 and should be physically null at all times. However, partly due to the segregated solution approach, partly due to the non-conservative formulation of the discretized continuity Eq. 2.3, these might not be always null though the iterative solution process. The resulting spurious momentum and enthalpy sources are accounted for via terms in the form ξi•, with • being either specific momentum (i.e. velocity) or specific enthalpy. The fluid–fluid drag factor is scalar by nature while the flu-id–structure drag factor is a tensor that accounts for the potential anisotropy of the structure. This is treated semi-implicitly as:

Kis⋅ui =13

tr(Kis)ui +

(

Kis −13

tr(Kis)I)

⋅ui

in which the first term on the left-hand-side is treated semi-implicitly (as the drag tensor depends on velocity) and the second term (i.e. the off- diagonal components) is treated explicitly. Important implementation and modelling aspects related to the calculation of the fluid–structure drag tensors Kis and how two-phase pressure drop multipliers act on them will be discussed in sub-section 2.2.

2.1.3. Energy conservation equation The energy conservation equation for the i-th phase is solved in terms

of phase enthalpy hi, while the phase specific kinetic energy eK,i =12ui⋅ui

is treated explicitly. The energy conservation equation is formulated as:

∂∂t(αiρi(hi + eK,i))+∇⋅(αiρiui(hi + eK,i)) − ∇⋅(αiaiτ⋅∇hi)

= − αi∂∂t

p+ αiρiui⋅g+ qiδ + qis + qi,Γ + ξi(hi + eK,i) (2.7)

The source terms on the left-hand side consist of the pressure-work term, gravitational work, interfacial heat transfer, source due to struc-ture heat sources, mass transfer and a numerical correction term to ac-count for numerical deviations from mass conservation. Some clarification on the heat transfer source terms are introduced here, which will be later expanded in Section 2.2. The interfacial heat transfer between the two fluids is modelled via a two-resistance approach, meaning that qiδ represents the volumetric heat source due to heat transfer between the bulk of the i-th fluid and the interface. If the interface is at a temperature Tδ, then:

qiδ = aHiδ(Tδ − Ti) (2.8)

which, numerically, can be treated semi-implicitly by linearizing the temperature with respect to enthalpy by considering that, by definition of enthalpy ∂hi

∂T |p = cp,i. For the case of two-phase flows in which both phases belong to the same chemical species (e.g. flows where mass transfer between the phases is possible via phase change), the interfacial temperature is set to the saturation temperature according to specie- specific models, as it will be discussed in sub-section 2.2. Possible im-balances in the heat flux on each side of the interface (which cannot store energy) are what drives phase change and are at the basis of how the mass transfer term Γ is calculated, as it will be discussed later. Otherwise, if the two phases belong to chemically different species and

no thermally-driven phase change is to be modelled, the interfacial temperature is calculated so that the heat flux on both sides of the interface is conserved. With regards to the fluid–structure heat transfer term qis, this is treated as:

qis = fiasHis(Ts − Ti) (2.9)

in which the additional term 0⩽fi⩽1 term quantifies the contact fraction between the i-th fluid and the structure surface. This heat transfer contribution is treated semi-implicitly as the previous one. The structure surface temperature field Ts is updated according to the selected struc-ture thermal models, discussed later, and is updated after solving for the fluid temperature equations. The mass-transfer-related heat transfer term qi,Γ will be discussed in Section 2.2.

2.2. Modelling capabilities

The modelling framework of the code is presented hierarchically in Fig. 2.1. The most relevant aspects will be hereby discussed.

2.2.1. Structure modelling The porous structure modelling consists in a hydraulic aspect and a

thermal aspect. On the hydraulic side, relevant quantities consist of the structure volume fraction, hydraulic diameter, surface area density and tortuosity (Ghanbarian et al., 2013). By tortuosity we mean the tensor τ characterising the inverse of the average lengthening of the diffusion lines through a porous medium. For the sake of clarity and as an example, let us consider a pin bundle in a certain 3-D reference frame

Su,i = Kij(uj − ui) − Kis⋅ui +Vm ρc

(∂∂t

ui + ui⋅∇ui −∂∂t

uj

)

+ Γj→iuj − Γi→jui + αiρig+ ξiui (2.6)

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composed of some axes labelled 0, 1 and 2. If the bundle axis were to be aligned along any of the axes, the tortuosity tensor describing the anisotropy of the volume-averaged porous structure representing the bundle would be a diagonal tensor. In particular, if the bundle axis were to be aligned along e.g. axis 2, then the three main components of the tortuosity tensor would be τ0 < 1, τ1 < 1, τ2 = 1. The first two compo-nents correspond to the directions perpendicular to the bundle axis, and these are < 1 as the fluid distance that separates any two points lying on this perpendicular plane is increased due to the presence of the pin compared to a clear-fluid scenario (i.e. if the pin bundle was not there). Conversely, as any two fluid points along axis 2 can still be reached via a straight path (as the bundle axis is aligned along axis 2), said tortuosity component is unity. If the bundle were not to be aligned along any of the reference frame axes, the strategy in deriving a tortuosity tensor would be to define it locally in a reference frame that is aligned with the bundle, so that the resulting tortuosity tensor is diagonal. The tensor in the global reference frame can then by obtained via tensor base-change operations. The code currently supports the option to define local

reference frame axes in different regions of the computational domain along the three main diagonal tortuosity components, with the mentioned base-change operations used to construct the actual tortu-osity tensor in the global reference frame being performed automatically at run-time. This discussion leads to the question of how to compute the tortuosity components for a given structure geometry, yet said discus-sion is deferred to (Ghanbarian et al., 2013) for brevity. What is important to recall is that the tortuosity tensor is used exclusively for the calculation of the tortuosity-adjusted momentum τνi and thermal τai diffusivity tensors, employed in Eqs. 2.5 and 2.7 respectively. For strongly advection dominated-problems, the effect of diffusion and thus tortuosity are limited.

By thermal structure modelling we mean the representation of some possibly power-producing underlying structures with specific geome-tries (e.g. nuclear fuel pins), whose internal temperature profiles can be of interest to resolve. Some of these models thus rely on a secondary sub- mesh, which exists within each global mesh cell, to resolve these pro-files. For this reason, the energy conservation equations associated with these structures are generally solved on a global mesh cell-by-cell basis. From the perspective of thermally coupling the structures to the fluid, regardless of the specifics of the structure thermal model, the heat flux Sfs represents the heat exchange between the fluid mixture and the structure. In particular:

Sfs = asHfs(Tf − Ts) (2.10)

where Ts is a field representing structure surface temperature. For a two phase-scenario, the quantities Hfs and Tf represent the average fluid-–structure heat transfer coefficient and the average fluid mixture tem-perature. For clarity, let us label the two fluids as 1 and 2. A meaningful choice for Hfs is to assume that the two fluids both transfer heat with the structure in parallel, so that:

Hfs = f1H1s + f2H2s (2.11)

Energy conservation (i.e. Sfs + S1s + S2s = 0) then provides a constraint on how to compute the average mixture temperature Tf :

Tf =f1H1,sT1 + f2H2,sT2

f1H1,s + f2H2,s(2.12)

After having discussed the form of the coupling heat flux, let us discuss the differences in structure thermal treatment between the currently available models.

0-D structure Within each mesh cell where it exists, this model rep-resents a structure that can be characterized uniquely by an average temperature value, density ρs and heat capacity Cp,s, and an optional power density source term qs. Thus, this model does not rely on a sub- mesh to update the structure temperature. In particular, this model as-sumes that the surface temperature is representative of the average structure temperature Ts, so that energy conservation is expressed as:

ρsCp,s∂Ts

∂t= qs + Sfs (2.13)

Heat conduction between adjacent cells in the global mesh is neglected. The term Sfs is treated as discussed earlier.

1-D uniform pin structure Within each mesh cell where it exists, this model represents a cylindrical pin of radius R, density ρs and heat ca-pacity Cp,s, and an optional power density source term qs. This model relies on a sub-mesh within each global mesh cell, with the sub-mesh representing the radial domain of the pin, and the fuel pin tempera-ture profile T = T(r) being defined on such sub-mesh. An energy equa-tion is solved in the following form:

ρsCp,s∂Tr

∂t−

ks

r∂∂r

(

r∂T∂r

)

= qs (2.14)

A convective (i.e. Robin) boundary condition on the outer pin sur-face (i.e. at r = R) ensures thermal coupling with the fluid:

Fig. 2.1. Logical representation of the modelling components of the code and their hierarchical dependency. Black lables represent the main logical sub-divisions of the modelling: 1) structure-related models; 2) fluid-related models; 3) Phase interaction-related models. The main variables that are handled by each model are reported to the right of each entry. Numbers in brackets mean the number of models of that kind required to describe the system.

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(ks

r∂∂r

(

r∂T∂r

))

|R = Sfs (2.15)

with the fluid–structure heat flux being defined as in Eq. 2.10, with Ts =

T(R). Conversely, a zero-gradient boundary conditions is applied at r =

0. The energy Eq. 2.14 is solved via a finite-volume discretization within each global mesh cell. As for the 0-D model discussed earlier, heat conduction between different sub-meshes in adjacent global mesh cells is neglected.

1-D nuclear fuel pin Within each mesh cell where it exists, this model represents a nuclear fuel pin consisting of two layers: fuel and cladding. Each layer is characterised by an inner and outer radius, density and heat capacity, and an optional power density source term in the fuel. A gap layer separates the fuel from the cladding as is characterised uniquely by a certain gap conductance value. In spite of these additional modelling layers, the modelling via a radial sub-mesh and the coupling to the fluid is the same as discussed for the 1-D uniform pin. The specific implementation of the fuel pin model (which is mathematically equiv-alent to the uniform heated pin model discussed earlier) can be found in (Fiorina et al., 2015).

2.2.2. Interaction models By interaction we mean any process that results in an exchange of

momentum or energy between the fluids and the structure, or of mo-mentum, energy or mass between the two fluids. Physically-speaking, all these processes take place at the boundaries layers at the interfaces between the phases, (i.e. fluid–fluid or fluid–structure), and their eval-uation by a computer code is possible based on first-principles as long as the computational mesh is resolved enough at said interfaces. However, one of the main goals of a porous medium-based coarse-mesh approach is to avoid the large computational burdens that would be associated with such a geometrically detailed system description. In particular, it can be shown that the application of volume averaging techniques (Whitaker, 1985) to the microscopically governing equations yields new volumetric source terms that represent the effect of the aforementioned interactions. These new terms representative of momentum, heat and mass transfer effects require closure, which is generally achieved via the use of available experimental correlations. Some the most relevant de-tails concerning the role and implementation of said models will be discussed with reference to Fig. 2.1. With regards to the specific corre-lations used within the scope of the present work, these will be reported in the validation-related Sections 3 and 4 alongside the geometric modelling choices.

Flow regime map modelling. In principle, the correlations can include some type of flow-regime dependence themselves. For the sake of flex-ibility, however, the code supports the definition of flow regime maps, currently parametrizable with respect to a single code variable of choice (e.g. the volume fraction of one of the two phases). Within the code, flow regimes act as containers for other models, and store the infor-mation relative to the mesh cells in which said models should be applied. Interpolation regimes are created automatically if the parameter bounds provided for each regime mismatch. For clarity, an example is made. Let us consider a flow regime map parametrized with respect to a generic code variable x, and let us consider two regimes, labelled 1 and 2, with bounds x1, lo, x1, hi and x2, lo, x2, hi respectively. For a given variable of interest y that is handled by the interaction models (e.g. a heat transfer coefficient Hij), so that flow regime 1 would evaluate y as y1 and flow regime 2 as y2, y will be set on a cell-by-cell basis as:

y =

⎧⎨

⎩

y1 x1, lo⩽x < x1, hiyi x1, hi⩽x < x2, loy2 x2, lo⩽x < x2, hi

with yi being either a linearly or quadratically interpolated value be-tween the two values predicted by the models prescribed for each of the

two flow regimes.

Fluid–fluid drag modelling. These models are responsible for calculating the fluid–fluid drag term Kij, which is scalar in nature (as the interfacial drag force will always be either parallel or anti-parallel to the slip ve-locity depending on the phase under consideration). The generic form of Kij is:

Kij = (1 − αs)ρd

2Dh, c|ui − uj|fd, ij (2.16)

with the c, d subscripts denoting either continuous or dispersed phase properties. The information of which phase is continuous and which one is dispersed is specified within each flow regime. The term fd, ij is the Darcy friction factor. Generally, but not necessarily, this factor is a function of the interfacial Reynolds number Reij, which is calculated as:

Reij =Dh, d |ui − uj|

νc(2.17)

Fluid–structure drag modelling and two-phase multipliers. The overall frictional pressure gradient experienced by the two-phase mixture can be expressed in terms of a superficial frictional pressure gradient expe-rienced by one phase, adjusted by a two-phase multiplier ϕ2

i , so that:

∇p|totfric = ϕ2

i ∇p|ifric, sup (2.18)

The superficial frictional pressure gradient ∇p|ifric, sup can be defined in two different ways. With reference to a 1-D pipe flow for simplicity, it is the frictional pressure gradient that the i-th phase would experience if it occupied the entirety of the available pipe cross-sectional area either with: 1) a mass flux equal to its real mass flux in the two-phase scenario; or 2) a mass flux equal to the total mixture mass flux. These two different definitions of the superficial frictional pressure gradient lead to different definitions of the two-phase multiplier. Within the scope of this com-puter code, the first definition is used, as it is the one used by many established two-phase pressure drop multiplier correlations (No and Kazimi, 1989) (e.g. Lockhart-Martinelli). The superficial frictional pressure drop is then calculated as:

∇p|ifric, sup = Kis, sup⋅(α*i ui) (2.19)

with α*i ui being the superficial velocity, Kis,sup being a drag factor that we

will refer to as superficial drag factor, which can be defined as a tensor with its k-th component computed as:

Kis, sup, k = (1 − αs)ρi

2Dhα*

i ui, kfd, is, k(Reis, sup) (2.20)

with:

α*i =

αi

1 − αs(2.21)

in which α*i ui, k is the k-th component fo the superficial phase velocity

and fd, is, k the fluid–structure drag coefficient for the component under consideration, which is computed starting from the superficial Reynolds Reis, sup defined as:

Reis, sup =Dh α*

i |ui|

νi(2.22)

The fluid–structure drag is treated in a tensorial way to account for potential structure anisotropy, so that different drag coefficients fd, is, k

can be defined for each main tensor component k. Only three of these components need to be specified alongside a local reference frame in which the drag tensor is diagonal, analogously to what was discussed for the tortuosity tensor. Nonetheless, the option to specify a single drag coefficient for all components and to treat drag isotropically still exists.

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While the theoretical framework of tho-phase multipliers was developed for 1-D flow analyses, it is applied to all drag tensor components for lack of a more meaningful alternative, as highlighted by other computer codes with a similar domain of applicability (e.g. SABENA (Ninokata and Aki Okano, 1990)). Returning to the point at issue, we recall that the actual frictional pressure gradient experienced by the i-th phase is:

∇p|ifric = Kis⋅ui (2.23)

The frictional pressure gradient in the form of Kis⋅ui should also be related to the total mixture pressure gradient. Assuming that the total pressure gradient is divided between the two phases according to the fluid–structure contact fraction fi, the following holds:

Kis⋅ui = fi ∇p|totfric (2.24)

If we label with M the fluid that was used for the calculation of the total pressure drop and by combining Eq. 2.24 with 2.18, 2.19, Kis can finally be calculated as:

Kis = fi α*M ϕ2

M KMs, sup|uM |

|ui|(2.25)

It is important to state that in the derivation of 2.25, it was assumed that ui and uM are parallel, in line with the 1-D nature of the underlying two-phase multiplier approach. For the sake of clarity, for a liquid-–vapour mixture with phases labelled l and v, if the total pressure drop is calculated via a liquid-based two-phase pressure drop multiplier, one would have: ⎧⎪⎨

⎪⎩

Kls = fl α*l ϕ2

l Kls, sup

Kvs = fv α*l ϕ2

l Kls, sup|ul|

|uv|

(2.26)

which is how the drag factors are calculated. Note that since the total pressure drop was calculated on the basis of the liquid phase superficial frictional gradient (i.e. M = l), the assumption that ul is parallel to uM is tautologically true. However, this is not necessarily true for ug. In short, the distribution of the total drag between the liquid and gas phases as-sumes that both the liquid-structure and gas-structure frictional pressure gradients are parallel. This limitation is however deemed acceptable as, as highlighted earlier, this is a 3-D adaptation of an otherwise 1-D model. Please note that in a purely single-phase scenario were only phase i is present (yet still potentially in the presence of a porous me-dium), one has α*

i = 1, and if models for fi,ϕ2i are also set to yield 1 in the

absence of a secondary phase, the entire framework simplifies to Kis =

Kis, sup, in which Kis, sup as defined in Eq. 2.20 leads to the conventional way calculating frictional pressure gradients for single-phase flows, albeit in tensorial form.

Heat transfer modelling. Heat transfer models deal with the imple-mentation of the fluid-interface Hiδ and the fluid structure His heat transfer coefficients that figure in Eqs. 2.8, 2.9. Currently, the only model that can be prescribed for both Hiδ and His consists of a Nusselt- based model that computes the heat transfer coefficients as:

Hi • =Nui • ki

Dh, •(2.27)

in which • represents either the fluid–fluid interface δ or the fluid-–structure interface s. In particular, Dh, • stands either for the structure hydraulic diameter or the dispersed phase characteristic dimension, depending on •. In turn, the Nusselt can be specified via a correlation in the form:

Nui • = A+B ReCi • PrD

i (2.28)

with A,B,C,D being user-selectable coefficients, as this correlation covers the vast majority of heat transfer coefficient models for flow

through tubes and fuel pin bundles (Mikityuk, 2009). The Reynolds number Rei • is computed in different ways depending on the pair under consideration. For the fluid–structure heat transfer coefficient, the same definition is used as the one provided in Eq. 2.22, while for the fluid- interface coefficient the one provided in Eq. 2.17 is used. A further fluid–structure specific heat transfer model exists for the treatment of sub-nucleate boiling. In general, sub-nucleate boiling is modelled as a superposition of purely-convective and purely pool-boiling related heat transfer mechanisms. While there are a number of approachs to model it (TRACE Theory Manual, 2012), the form currently adopted in the code is a linear superposition model (Chen, 1966) in the form:

His = Fi His, C + Si His, PB (2.29)

in which His, C and His, PB the purely convective and the purely pool- boiling-related heat transfer coefficients, while Fi and Si are a flow factor and a suppression factor respectively. Physically, the flow factor represents the enhancement of the convective component due to the added near-wall turbulence caused by departing bubbles. Conversely, the suppression factor exists to blend pool boiling between the nucleate boiling regime to the transition boiling regime, in which the pool boiling contribution decreases significantly. A convective heat transfer coeffi-cient in the form of Eq. 2.27 can be prescribed while separate models for the flow factor, suppression factor and pool boiling heat transfer co-efficients can be provided.

Mass transfer modelling and mass transfer-related enthalpy transfer. Mass transfer is thermally-driven and is calculated based on the heat con-duction limited model, which is commonly employed in sub-channel and other system codes (e.g. TRACE (TRACE Theory Manual, 2012)) in which inter-phase heat transfer phenomena are modelled via a two- resistance approach. In particular, a heat-conduction limited approach ensures that energy is conserved at the inter-phase interface (i.e. the interface cannot store any energy), so that any imbalance between the heat fluxes at each side of the interface is absorbed in the volumetric mass transfer rate Γ. For a system of two phases labelled i and j:

Γi→j =qiδ + qjδ

QL,ij(2.30)

with the latent heat QL,ij being calculated according to a user-provided latent heat model which generally depends on the saturation tempera-ture and system pressure. It should be noted that one should obtain Γi→j = − Γj→i by swapping indices. However, on the right-hand-side, the nominator is insensitive to index swapping, so that, from an imple-mentation perspective, we implemented QL,ij = − QL,ji. For ease of dis-cussion, the rest of the paragraph deals in terms a liquid phase l and a vapour phase v with i = l,j = v. Some important aspects of how the mass transfer related energy contributions are added to the enthalpy equa-tions need to be discussed. These contributions are the latent heat contribution, associated with the mass transfer process itself, and the intrinsic contribution, associated with the disappearance/appearance of a mass (per unit volume and time) of fluid i Γi→j. With regards to the former, it is easy to see that as the mass transfer is computed based on an interfacial enthalpy balance, the heat transferred purely due to phase change will consist of the interfacial heat fluxes qiδ for each phase i. Let us make an explicit example in terms of a liquid and a vapour phase. If the vapour is at saturation and the liquid is super-heated, then qlδ > 0 and Γl→v =

qlδQL, lv

. The heat contribution that figures on the right-hand- side of the liquid enthalpy equation is − qlδ, which by definition of how the mass transfer term is computed, results in qlδ = Γl→vQL, lv. Physically, this corresponds to the heat transferred associated with Γl→v. Conversely, as the solver operates in terms of sensible enthalpy (rather than absolute), the heat contribution on the vapour side is qvδ = 0 as there is (physically) no change in vapour temperature during boiling. The same logic applies in the opposite scenario with the condensation of

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sub-cooled vapour in a liquid at saturation, with Γv→l = − Γl→v = −qvδ

QL, lv.

Thus, the interfacial heat terms that figure in the enthalpy equations already constitute the latent heat contribution of the phase change process. On a different note is the intrinsic enthalpy change contribu-tion. Let us focus on a fluid mass dm. The enthalpy contribution asso-ciated with said mass is dm hl.Thus, when such a mass of liquid is being removed e.g. due to phase change, in addition to the latent heat QL lv, one needs to account for this intrinsic contribution hl. In particular, for the case of boiling, it is easy to see that − Γl→vhl represents such contribution per unit time and volume. During condensation however, the contribution is − Γl→vhl, sat > 0 (recall that during condensation Γl→v < 0). Physically, both these contributions should be evaluated at saturation enthalpy, as it is at saturation that the fluid is disappearing or reappearing. However, numerically, a super-heat of the liquid phase is required for boiling to take place, meaning that one always has hl >

hl, sat at boiling. If one where to remove fluid from a cell at an enthalpy lower than the saturation one, theremaining fluid will inevitably remain at a higher specific enthalpy and thus, at a higher temperature (as it is also explained in the TRACE system code guide (TRACE Theory Manual, 2012), pp. 12 − 13). Thus, to prevent thermal run-aways, the intrinsic enthalpy change contribution is evaluated at the actual liquid enthalpy hl rather than at the saturation one. This also applies to the vapour phase during sub-cooled vapour condensation scenarios. With respect to the enthalpy Eq. 2.7 defined for the generic phase i, one thus has:

qi, Γ =

{− Γi→j hi Γi→j⩾0− Γi→j hi, sat Γi→j < 0 (2.31)

In order to be consistent with this approach of managing the intrinsic enthalpy change contribution, the latent heat should be reduced by hl − hl,sat during boiling and by hv, sat − hv during condensation. By reconnecting us to the generic case with phases i and j, one has:

QL, ij =

{QL0, ij − |hi − hi, sat| Γi→j⩾0QL0, ij − |hj − hj, sat| Γi→j < 0 (2.32)

with QL0, ij being the latent heat computed by the selected latent heat model. This is the same approach that is adopted in the TRACE system code (TRACE Theory Manual, 2012).

3. Steady state validation

The present section discusses the validation efforts performed against quasi-steady-state boiling scenarios. In particular, the experi-ments under consideration are those performed at the Joint Research Centre in Ispra, Italy, in the 1980s, concerning sodium boiling (Kot-towski and Savatteri, 1984). These were carried in the wider then- existing European framework of SFR development (a notable example being the German SNR-300 SFR), yet also to further knowledge con-cerning a fluid, Sodium, whose use in thermo-vector applications was enjoying increasing popularity outside of nuclear-related applications.

3.1. Experiments under consideration

The experimental setup at Ispra consisted of a closed loop with interchangeable heated test sections so that boiling in different flow geometries could be investigated. The overall description of the test loop can be found in (Kottowski and Savatteri, 1984). A number of different test sections were experimented with, ranging from tubular and annular ones, to hexagonal wire-wrapped pin bundles. With regards to the set of experiments that were used for this initial validation stage, these consist in quasi-steady-state boiling in the tubular test-sections. These were performed for a number of test section input power levels, and were used to assess the dependence on the total two-phase pressure drop experi-enced over the test section with respect to the inlet test section sodium velocity. For a decreasing sodium velocity past a certain threshold, boiling onsets and the overall pressure drop will increased, which is the

phenomenon that two-phase multipliers are meant to model. These tests are referred to as quasi-steady-state due to the fact that while boiling is an inherently transient phenomenon, the experimental conditions were such to allow time-averaged quantities, such as pressure and velocity, to reach a steady-state. These experiments represents a great opportunity for validation purposes as most two-phase multiplier models depend exclusively on the liquid and fluid vapour flow qualities, which, physi-cally speaking, depend exclusively on the thermo-physical properties of the fluid, on the liquid inlet properties, and on the power.

The geometry of the test section is illustrated in Fig. 3.1. In partic-ular, it consisted of a tube of 6 mm in inner diameter and 29 mm in outer diameter. Different input power levels resulting in (inner) wall heat flux densities ranging from 106 W/cm2 to 250 W/cm2 were investigated. For each power level, different inlet mass flows were prescribed by adjusting the inlet pressure, and for each velocity, a the overall pressure drop over the test assembly was measured. Data points span both single-phase and two-phase flow regimes.

3.2. Modelling of the experiments

The system was represented as a 1-D model spanning the entire ge-ometry as reported in Fig. 3.1. With regards to the condenser, it was modelled as well via a tapered pyramid of progressively increasing volumes, yet its meshing is still 1-D in the axial direction. Boundary conditions for each case consisted of fixed outlet pressure, fixed liquid inlet temperature and velocity, and inlet pressure gradient prescribed to be consistent with the inlet velocity (i.e. fixedFluxPressure). All other quantities were prescribed to have a null gradient at system boundaries. The modelling of the condenser is necessary as, if the model stopped short at the tube outlet (where two-phase flow exists), it would be inaccurate to prescribe a fixed pressure at such location. With regards to tallied results, the overall pressure drop (exclusive of the hydrostatic component) between points P11 and P14 as reported in Fig. 3.1 was monitored.

As mentioned, the advantage of a quasi-steady-state scenario is that the final results are expected to be independent of a number of param-eters, as the final vapour quality depends exclusively on an enthalpy balance. As a result, heat transfer coefficients and inter-phase drag co-efficients play little role in shaping the final results. The parameters that matter the most are the liquid-structure drag coefficient fD, ls, two-phase drag multiplier for the liquid phase ϕ2

l , as well as the thermo-physical properties of the fluid, inclusive of its pressure–temperature saturation curve and the latent heat of evaporation L.

With regards to the liquid-structure drag coefficient fD, ls, due to the tubular geometry of the system, the Blasius correlation was used:

fd, ls =0.316Re− 25

ls

For the liquid two-phase pressure drop multiplier ϕ2l , the following

five models were investigated:

• Lockhart-Martinelli (Lockhart and Martinelli, 1949), a well-known correlations which relates the two-phase multiplier of the i-th phase ϕ2

i to a dimensionless parameters X2i , called the Lockhart-

Martinelli parameter, defined as:

X2i =

(μi

μj

)0.2(1 − xj

xj

)1.8ρj

ρi(3.1)

and:

ϕ2i = 1+

CXi

+1

X2i

with C being a parameter that ranges between 5 and 20 depending on

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the postulated flow regime of each phase (whether turbulent or laminar, allowing for 4 possible combinations). Generally C = 20 is used for most applications.

• Lottes-Flinn, which predicts:

ϕ2i =

1(1 − α*

j )2

This was proposed for small values of the gas phase flow quality. With specific regards to sodium boiling however, Nguyen proposed the following relationship between α*

j and the Lockhart-Martinelli parameter of the i-th phase to extend it to high-phase fraction boiling flows (Nguyen, 1985):

α*j = (1 + X0.8

i )− 0.378

• Kaiser (Kaiser et al., 1988), a correlation developed for sodium- boiling in pin bundle geometries that relies on the Lockhart- Martinelli parameter. This was developed for specific application to the liquid phase, so the l and g subscripts will be used to denote the liquid and gas (i.e. vapour) phase:

ln ϕl = 1.48 − 1.05 lnXl

√+ 0.09

(ln

Xl

√ )2

• Kottowski-Savatteri (Kottowski and Savatteri, 1984), a correlation developed for Sodium boiling in pin bundle geometries that relies on the Lockhart-Martinelli parameter. Similarly to the work by Kaiser et al., this was developed for specific application to the liquid phase l:

log10 ϕl = 0.6252 − 0.5098 log10 Xl + 0.1046 (log10 Xl)2

• Chenk-Kalish (Chen and Kalish, 1988), a correlation developed for Potassium boiling that relies on the Lockhart-Martinelli parameter, developed for the liquid phase l:

ln ϕl = 1.59 − 0.518 ln Xl + 0.0867 (ln Xl)2

With regards to the thermo-physical properties of the liquid and vapour sodium, the proposed data compiled by Fink and Leibowitz (Fink and Leibowitz, 1996) was used for density, heat capacity, molecular viscosity, latent heat and the pressure–temperature saturation curve. In particular, the latent heat of evaporation is evaluated as:

QL0, lv = 393370(

1 −Tsat

Tc

)

+ 4398600(

1 −Tsat

Tc

)0.29302

with QL0, ij in J/kg, temperatures in K,Tc = 2503.7K the critical tem-perature of sodium and Tsat its saturation temperature at the current system pressure. In particular, the following correlation was used to describe the pressure–temperature saturation curve:

ln p = 11.9463 −12633.7

T− 0.4672 ln T

with Tsat in K, the pressure p in MPa. A summary of the employed models that were maintained constant

throughtout the different investigated quasi-steady-state cases is pre-sented in Table 3.1. No particular regime maps were used.

3.3. Results and discussion

The simulations involved the reproduction of the experiments per-formed at four different input power levels, resulting in tube wall heat

Fig. 3.1. Overview of the test section of the ISPRA sodium boiling loop..

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fluxes of 138 W/cm2, 159 W/cm2, 175 W/cm2, 250W/cm2, each with varying levels of inlet liquid flow velocities. Each set of simulations performed at a different power level will be referred to as case. The results are summarized in Fig. 3.2. For all cases, the flow at the highest simulated inlet velocity results in a single-phase flow, and thus the overall pressure drop decreased as the velocity is reduced. However, below a certain threshold that depends on the input power level, the liquid velocity will be insufficient to carry away enough heat without preventing the liquid to exceed its saturation temperature, after which boiling starts, and the pressure drop increases due to the two-phase flow

circumstances. In particular, the lower the velocity, the higher the heat that is carried away as sensible heat, the larger the heat that results in a phase change, and thus, the higher the vapour quality. Physically though, the pressure drop is due to reach a maximum as the pressur-e–velocity curve approaches that of an almost pure single-phase vapour flow. This can be clearly observed in the cases with wall heat fluxes of 158 W/cm2 and 175 W/cm2, as they where run for inlet velocity profiles resulting in power-to-flow rations even higher than those investigated in the other two cases. With regards to the actual comparison between the experiments and the calculation results, a good agreement can be observed, with the two-phase pressure drop most accurately reproduc-ing the results being that by Kottowski and Savatteri, which, unsur-prisingly, was developed specifically starting from the data of these experiments (Kottowski and Savatteri, 1984). The second best per-forming correlation is the one developed by Kaiser, also based on sodium boiling experiments in tubular and pin bundle geometries. The corre-lations by Lockhart and Martinelli, Lottes and Flinn also prove to yields very comparable results. With regards to the latter however, the Nguyen correlation for the vapour phase fraction was used instead of the vapour fraction itself when calculating the multiplier, as described in the pre-vious section. It was in fact observed that using the original correlation by Lottes and Flinn tends to overestimate the two-phase pressure drop at high vapour volume fractions, while yielding results in line with the other correlations at lower phase fractions. The Nguyen correlation was instead devised to adjust the Lottes-Flinn model specifically for sodium boiling scenarios. Similar finding were observed by Ninokata with the SABENA code (Ninokata and Aki Okano, 1990). On the other hand of the spectrum, the correlation by Chen and Kalish was developed for

Table 3.1 Summary of the employed correlations and/or values for a variety of different quantities in the steady-state simulations. The only element of variance consists of the two-phase drag multiplier models, which are discussed outside the scope of this Table. The liquid, vapour and wall names are shortened as L, V, W.

Quantity Correlation

L-V friction coefficient Autruffe (Autruffe et al., 1979) L-W friction coefficient Blasius Virtual mass coefficient Constant (0) L-V heat transfer coeff. (both sides) Constant Nusselt (10) L-W heat transfer coefficient Mikityuk (Mikityuk, 2009) L-V interfacial area density Assumes annular topology L-W contact area fraction Constant (1.0) Dispersed phase Constant (V) Dispersed phase diameter Constant (6 mm) Mass transfer model Heat conduction limited Latent heat model Fink-Leibowitz (Fink and Leibowitz, 1996) Saturation T-p relation Browning-Potter (Fink and Leibowitz, 1996)

Fig. 3.2. Comparison between the calculation results obtained by the various pressure drop multiplier correlations against experimental results for each of the four different wall heat fluxes. For each wall heat flux, calculations were performed by varying the inlet velocity in steps of 0.25 m/s.

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experimental data regarding potassium alloys, and proves to over-estimate the overall pressure drop in all circumstances.

The latent heat and saturation pressure–temperature relations were found to be important quantities in shaping the calculated curves. In particular, the higher the pressure, the higher the corresponding satu-ration temperature, which thus reduces the overall vapour production, its mass flow rate, and ultimately the overall pressure drop than what would be expected at lower inlet velocities if the saturation temperature were to be set as constant. Conversely, the evaporation latent heat model predicts smaller latent heats for higher saturation temperatures (consistently with the fact that, at the critical point at higher tempera-tures, the latent heat is null by definition), and thus, for higher system pressures. A smaller latent heat will results in a larger vapour produc-tion, yet the latent heat effects on the overall resulting pressure drop were found to be significantly smaller than those associated with the saturation pressure–temperature dependence.

4. Comparison against transient

In order to assess the transient performance of the code, the focus is now shifted on the modelling of a sodium boiling experiment that was performed at the Kompakter Natriumsiede Kreislauf (KNS) experimental facility at the Kernforschungszentrum Karlsruhe (now the Karlsruhe Institute of Technology) in the 1980s. The experimental setup and de-tails pertaining the test itself is presented in sub-section 4.1. The computational modelling choices in terms of model geometry and physics model choices are presented in sub-section 4.2. The computa-tional results will be presented alongside the experimental ones in sub- section 4.3.

4.1. Description of the experimental setup and test

The KNS was originally set up within the broader framework of fast reactor development, specifically for the German SNR-300 Fast Breeder Reactor development program. The KNS was a sodium loop that allowed

for the investigation of boiling in a variety of assembly geometries under Uncontrolled Loss Of Flow (ULOF) conditions. A schematic of the facility in its KNS-37 configuration is presented in Fig. 4.1. This configuration was dedicated to the investigation of boiling phenomena in a 37-pin electrically-heated mock-up bundle.

Among the tests that were performed on the KNS facility in this configuration, the one chosen for this analysis consists of test L22. This test was about the investigation of sodium boiling subsequent to a fast pump trip transient representative of possible ULOF conditions. Quali-tatively, the test proceeded as follows. For a certain axial input power profile with no radial tilt and inlet conditions, a steady state single-phase flow was reached. The pump was then tripped resulting in a rapid loss of flow, eventually inducing the onset of boiling. Due to the appreciable difference in the power-to-flow ratio between the assembly bulk and its periphery, the void region is initially limited to the bulk of the assembly and only minor inlet mass flow oscillations are observed. As the boiling progresses and the void regions extends towards the assembly periphery, the increase in pressure drop results in a flow excursion. Among different quantities that were monitored at some locations throughout the test, pin surface cladding temperatures proved of great importance. In particular, a sharp rise in said temperatures was taken as indicative of

Fig. 4.1. Schematic of the KNS test loop facility (right) and assembly details (left).

Table 4.1 Pin bundle specifications.

Assembly data

Number of pins 37 Pin diameter (mm) 6 Pin pitch (mm) 7.6

Flow area (cm2) 11.7

Heated pin length (mm) 900 Unheated pin length (mm) - top 450 - bottom 200

Maximum linear power (W/cm) 320

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dryout conditions, which was used as a criterion to power off the as-sembly. After power off, the transient proceeded until flow restoration and eventual end of boiling. The quantitative technical and transient details are presented in Tables 4.1 and 4.2.

4.2. Computational model

As mentioned, the evolution of the transient is initially shaped by the radial vapour expansion. Because of this, and as it has already been highlighted in other works (Chenu et al., 2009; Radman et al., 2019), a 1-D analysis is insufficient as flow excursion would be significantly anticipated. This is due to the fact that in a 1-D model, right after boiling onset, the entirety of the available flow cross section is already occupied by some fraction of vapour. A 2-D axial-symmetric wedge model has then been adopted, as it allows to refine the radial domain with a much smaller mesh when compared to a full 3-D model.

4.2.1. Geometrical modelling and volume averaging The model consists of a 2-D axial symmetric wedge of 2 degrees of

aperture. The pin bundle was treated as a porous structure, whose hy-draulic properties are modelled via three parameters, namely, for each mesh cell: the structure volume fraction; the hydraulic diameter; and the tortuosity. Because of differences between the bulk of the pin bundle and its edge, different properties were obtained for these two regions. With reference to Fig. 4.2, the inner bundle volume fraction and hydraulic diameter were estimated on the basis of the innermost triangular sub-channel. Conversely, properties for the bundle edges were estimated on the basis of the highlighted trapezoidal region. With regard to tortuosity, this was estimated as the ratio between the straight line distance be-tween two points in subchannels opposite to a given pin and the actual in-fluid distance between said points, along a circumference, thus resulting in a tortuosity of 2/π. While in principle the bundle pitch- diameter ratio affects tortuosity (e.g. for an infinite pitch-diameter ratio the tortuosity should reduce to clear-fluid tortuosity, i.e. unity), the current choice was deemed satisfactory for the purpose of this work. It should be also mentioned that certain phenomena in SFRs (e.g. azimuthal heat conduction through fuel pins (Ro and Todreas, 1988)) can be captured via adequate modifications to the value of the radial tortuosity component, which will be the subject of future studies for the

authors. Axially, the model extends from the bottom of the lower unheated

pin region to 50 cm above the top of the upper unheated pin section. The condenser was not modelled as, experimentally, the vapour volume remained confined within the bundle, condensing while traversing the cooler upper unheated pin bundle region. The 2-D axial-symmetric wedge model is thus composed of seven regions: 1) two regions (inner and outer, i.e. edge, bundle) for the lower unheated pins; 2) two regions (inner and outer bundle) for the heated pins); 3) two regions (inner and outer bundle) for the upper unheated pin; 4) the region in between the outlet and the bundle top, which we will refer to as pipe region. The volume-averaged porous properties as well as the thermal properties of these regions are summarized in Table 4.3.

With regard to the thermal modelling, the pins were treated via a 1-D cylindrical model and coupled to the fluid as discussed in Section 2.2.1, with a power density source prescribed according to the experimental axial power distribution in both the inner and the outer heated bundle regions. The interfacial area density as of the pin cladding was derived from the volume averaging process to be related to the pin volume fraction and pin radius ro as:

as =2αs

ro

4.2.2. Constitutive relations and closure In place of the i and j subscripts to represent the two fluids, the

subscripts l and v will be here used to denote the liquid and vapour phase respectively.

Fluid–fluid drag With respect to Eq. 2.16, the following drag coef-ficient fD, lv models were investigated:

• Autruffe et al. (Autruffe et al., 1979), a correlation for two-phase sodium flows:

fd, lv = 4.31 ((1 − α*g)(1 + 75 (1 − α*

g)))0.95

Table 4.2 Test L22 data. The variables z and t represent the axial distance from the bottom of the heated pin section in mm and the elapsed time since pump trip in s respectively.

Test L22 data

Steady state conditions Average linear power (W/cm) 215.4 Normalized axial power profile

0.38 + 0.62 sin(

π z + 25900

)

Bundle pressure (bar) - inlet 2.241 - outlet 1.045

Bulk coolant temperatures (K) - inlet 653 - outlet 812

Inlet mass flow (kg/s) 3.41

Boiling dynamics Normalized inlet mass flow (1 + 0.3003 t)− 1.297

ramp down until boiling Time elapsed until (s) - onset of boiling 6.11 - onset of flow excursion 8.30 - onset of dryout 9.25 - power off 9.45 - end of boiling 12.31

Fig. 4.2. Cross section of the KNS-37 assembly (credits to (Bottoni et al., 1990)), not to scale. Inner bundle porous properties were calculated based on averaging on the highlighted triangular subchannel. Outer bundle porous properties were calculated based on the trapezoidal region instead. Tortuosity was evaluated as the ratio between the shortest straight-line path between the highlighted points to the in-fluid path between said points, along .a circumference.

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• Wallis, a correlation for two-phase sodium flows used in the SABENA code (Ninokata and Aki Okano, 1990):

fd, lv = 0.02α*

g

√(1+ 150 (1 −

α*

g

√))

Fluid–structure drag As it was discussed in Section 2.2.2, the fluid-–structure drag depends on the fluid–structure superficial drag co-efficients, the fluid–structure contact fractions and the two phase pressure drop multiplier. For all flow regimes, the liquid was used as the basis fluid for the calculation of the drag coefficients via the two phase multiplier method (i.e. M = l with reference to Eq. 2.25, which results in Eq. 2.26). With respect to Eq. 2.25, the following drag coefficient fD, is

models were investigated:

• Blasius correlation in the form:

fd, ls =0.316Re0.25

ls

which was applied to both (i.e. axial and radial) flow directions in the pipe region of the model.

• Rehme (Rehme, 1973) correlation for the friction factor for flows in wire-wrapped pin bundles with an hexagonal lattice:

fd, ls =

(64

RelsF0.5 +

0.0816Re0.133

lsF0.9335

)Wp

Wp + Ww

F =

Pt

Dp

√

+

(

7.6Dp + Dw

Lw

(Pt

Lw

)2)2.16

which was applied to the axial flow direction in all the pin bundle regions. It should be noted that the KNS-37 pins are spaced by a spacer grid rather than a wrapping wire. However, within a coarse- mesh finite-volume context, we tried to capture the volume- averaged pressure loss effect of said grids by adjusting the Rehme model wire lead length (Lw, i.e. the axial pitch of the wounding) in order to obtain the measured steady-state pressure drop.

• Gunter-Shaw correlation in the form:

fd, ls =0.96Rels

which was applied to the transverse flow direction in all the pin bundle regions.

Recent developments in the field of friction factors for both wire- wrapped and bare hexagonal fuel assemblies, such as the further expansion of the original work of Cheng and Todreas by Chen et al. (Chen et al., 2018), which offer ways to model differences in friction factors between bulk and edge sub-channels, have not been presently considered. This is due to the fact that the pressure losses come primarily from spacer grids, as previously mentioned, and lacking dedicated models for the KNS spacer grids, the choice of such advanced correla-tions would still require adjustments to some quantities, such as the edge hydraulic diameter, in order to reproduce the experimental data. The tuning of pressure losses for the modelling of the KNS transients is not novel, as testified e.g. by the ”artificial friction” model adopted in pre-vious investigations with the TRACE system code (Chenu et al., 2009). With regard to the two-phase pressure drop multiplier models that were presented in the ISPRA experimental analysis section were investigated as well.

Fluid–fluid interface heat transfer coefficients The heat transfer co-efficients between fluid bulk and fluid–fluid interface htci, δ that govern boiling-related energy and mass transfer as per Eqs. 2.8, 2.30 were calculated from a prescribed, constant Nusselt, thus using a correlation in the form of Eq. 2.27. The Nusselt numbers used both for the liquid and vapour side of the heat transfer coefficients were set to 10, as per rec-ommendations by Schor et al. (Schor et al., 1984).

Fluid–structure heat transfer coefficients The heat transfer coeffi-cient between liquid and structure was computed via the superposition model as reported in Eq. 2.29, thus relying on four sub-models for the convective heat transfer component, pool boiling heat transfer compo-nent, a flow enhancement factor and a flow suppression factor. In particular:

• for the convective component, a correlation in the form of Eq. 2.27 was used with the Nusselt computed as:

Nu = 7.485+ 0.030 Re0.77ls Pr0.77

l

which was proposed by Mikityuk after a review of pin bundle to liquid sodium heat transfer coefficient data (Mikityuk, 2009);

• for the pool boiling component, the correlation by Shah (Mohammed Shah, 1992) for liquid metals was used:

Hls, PB = C q0.7ls pm

r

C =

{ 13.7 pr < 0.0016.8 pr⩾0.001

m =

{ 0.22 pr < 0.0010.12 pr⩾0.001

with qls being the fluid–structure heat flux and pr the reduced pres-

Table 4.3 2-D axial-symmetric wedge model data with regions listed in order, inwards from outwards, bottom to top. With regard to the porous properties columns, αs refers to the structure volume fraction, Dh to the hydraulic diameter, τr to the tortuosity component in the radial direction. The tortuosity is a 2-D tensor in this scenario, with the axial component being unity in all regions.

Region Dimensions Porous properties

Physical (mm) Mesh ( − ) αs ( − ) Dh (mm) τr ( − )

Number Name Axial Radial Axial Radial

1 Inner lower bundle 200 21.553 10 5 0.523 5.469 2π

2 Outer lower bundle 200 5.007 10 2 0.374 12.305 2π

3 Inner heated bundle 900 21.553 23 5 0.523 5.469 2π

4 Outer heated bundle 900 5.007 23 2 0.374 12.305 2π

5 Inner upper bundle 450 21.553 12 5 0.523 5.469 2π

6 Outer upper bundle 450 5.007 12 2 0.374 12.305 2π

7 Pipe 500 26.560 13 7 0 53.120 0

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sure (i.e. the pressure divided by the critical pressure of the liquid in exam. Among other metals, this correlations was verified for Sodium in a wide range of pressures. The numerical inconvenience of pool boiling correlations is that they generally related the heat transfer coefficient to the heat flux, which is what the heat transfer co-efficients is needed for in the first place. Within the scope of this code, this dependence is treated explicitly;

• for the flow and suppression factor models, the well known corre-lations by Chen (Chen, 1966) were employed.

With regard to the structure-vapour heat transfer coefficient, this was deemed of secondary importance due to the fact that vapour and structure are expected to come into contact only for a very short period of time around dryout. The heat transfer was calculated assuming a constant Nusselt of unity. As for the fluid–structure drag coefficients, the final heat transfer effect depends on the fluid–structure contact fraction fis between liquid and structure, vapour and structure.

Fluid geometry models By fluid geometry models we mean any correlations that set the liquid–vapour interfacial area density a, dispersed phase characteristic dimension Dh, d and the structure contact fractions fls, fvs that quantify what proportion of the structure (in this case, the pins) is wetted by the liquid and vapour respectively. With regard to the fluid–fluid interfacial area, both the correlations by Schor (Schor et al., 1984) and by No and Kazimi (No and Kazimi, 1987) were investigated. For the remaining quantities, these were set via a regime map with two flow regimes, which we will refer to as annular flow regime and mist flow regime. No bubbly flow regime was modelled as said regime is virtually non-existent in sodium due to the near- atmospheric operational pressure of SFRs and the subsequently very high vapour-liquid density ratio (Kottowski and Savatteri, 1984). This regime map was parametrized with respect to vapour volume fraction (normalized with respect to the structure volume fraction) α*

v . For each regime, these quantities were treated as constants. Each of these regimes exists for certain values of the vapour volume fraction. In this case, the regime map can be entirely described by two parameters, α*

v,an and α*v,ms

with α*v,an < α*

v,ms, namely the vapour volume fraction below which the annular regime is assumed to exist and the vapour volume fraction above which the mist regime is assumed to exist. For vapour volume fractions in between the two, all the parameters and the other models discussed so far are quadratically interpolated. With regard to the con-tact fractions, these were set to fl = 1.0 and fv = 0.0 in the annular flow regime, assuming that the liquid is at all times in contact with the structure. Conversely, for the mist flow regime we assumed fl = 0.0 and fv = 1.0. Furthermore, the vapour was treated as the dispersed phase in the annular regime, and the liquid was treated as the dispersed phase in the mist regime. Practically, this affects all quantities that rely on tracking which phase is dispersed and which is continuous in any given cells. Chief among these are continuous phase density and disperse phase volume fraction, which, among others, were found to be most relevant in shaping virtual mass effects.

A summary of the core set of correlations that were used as basis for the different models presented in Section 4.3 is reported in Table 4.4. With regard to the choice of the liquid–vapour friction coefficients as well as two-phase friction multipliers, these are not reported in said table as they vary between the different models presented in the next section. With particular regard to the liquid–vapour friction coefficients, each model that employs a certain correlation for said quantity, employs it both in the annular and the mist flow regime. While this choice is debatable as the liquid–vapour friction coefficient generally increases in the mist regime, the final effect on the overall flow was expected to be marginal due to the narrow region in both space and time in which a mist regime was observed to exist.

4.2.3. Boundary conditions and simulation remarks This transient necessitates the prescription of a pressure field at the

inlet and outlet in order to resolve the pressure-driven flow oscillations and excursion. At the outlet, a pressure equal to the experimentally measure pressure of 1bar was imposed throughout the simulation. At the inlet, a time-dependent pressure profile is imposed in order to reproduce the experimentally reported pump mass flow rate ramp, as reported in Table 4.2. All other fields have trivial boundary conditions: fixed values for the liquid temperature at the inlet and null gradient boundary con-ditions for all other quantities at both the inlet and outlet. On the wedge faces, a null gradient is imposed for all quantities in the direction normal to the faces. The power is turned off at the same time as the one reported experimentally.

4.3. Results

4.3.1. Single-phase results The calculated and experimental results concerning the axial and

radial temperature distribution at steady state and at boiling onset are reported in Fig. 4.3. The overall agreement between the calculated and experimental values is good, yet the calculated radial temperature profile is somewhat under-estimated at the bundle-edges at steady-state. The main issue lies in the fact that, at this stage of modelling, we relied on the same drag coefficient correlation, the one by Rehme, both for the inner and outer pin bundles. However, the outer bundle hydraulic diameter had to be adjusted to obtain the right flow redistribution be-tween inner and outer bundle, which is paramount for the prediction of

Table 4.4 Summary of the core set of correlations used for the transient analysis. The quantities that are not reported (e.g. liquid–vapour friction coefficients and liquid-wall two-phase drag multipliers) were parametrically investigated, as discussed in Section 4.3. The liquid, vapour and wall names are shortened as L, V, W. If no notes are made regarding a quantity, the corresponding correlation was applied to all flow regimes and all domain regions.

Quantity Correlation Notes

L-W friction coefficient Blasius Region 7, annular regime, all flow directions

Rehme Regions 1–6, annular regime, axial flow

Gunter-Shaw Regions 1–6, annular regime, radial flow

Constant (0) All regions, mist regime, all flow directions

L-W contact fraction Constant (1.0) Annular regime Constant (0.0) Mist regime

V-W friction coefficient Constant (0) All regions, annular regime, all flow directions

Blasius All regions, mist regime, all flow directions

Virtual mass coefficient Constant (0.1) L-V heat transfer coeff.

(both sides) Constant Nusselt (10)

L-W heat transfer coefficient

Linear superposition

Annular regime, sub- correlations are Mikityuk (convective), Shah (pool boiling), Chen (flow enhancement and suppression factors)

Constant (0) Mist regime V-W heat transfer

coefficient Dittus-Boelter Mist regime Constant (0) Annular regime

L-V interfacial area density

Schor Correlation spans both flow regimes

Dispersed phase Constant (V) Annular regime Constant (L) Mist regime

Dispersed phase diameter

Constant Equal to structure hydraulic diameter (depends on model region)

Mass transfer Heat conduction limited

Latent heat Fink-Leibowitz Saturation T-p

relationship Browning-Potter

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the radial temperature distribution. Ideally, one would require a different model for the drag coefficient in the outer bundle regions. While a number of models where experimented with, with the geometrically calculated outer bundle hydraulic diameter, these efforts were met with little success, so that it was preferred to use a single drag coefficient model and vary the outer hydraulic diameter until the radial temperature distribution (prioritizing agreement with the boiling- inception one over the steady state one) could be reproduced. This is of fundamental importance for the transient evolution prediction, as the radial temperature gradient governs the radial temperature vapour expansion rate, and thus the flow excursion dynamics.

4.3.2. Two-phase results A number of different correlations for different quantities and

parameter combinations were investigated. The results obtained by the application of four different sets of correlations and relevant parameters are presented and compared against the experimental ones, while the effects of changing other parameters will be discussed qualitatively. However, it should be noted that the overall outcome of modifying any given model or model parameters is itself dependent on the choice for correlations governing other quantities, yet a thorough sensitivity and co-variance analysis of all the models was out of scope for the present work. For clarity, the four presented sets of results will be referred to as models I through IV. All of the presented results relied on the Schor interfacial area correlation, a virtual mass coefficient of 0.1, an upper

Fig. 4.3. Experimental measurements and calculated results for the axial (left) and radial (right) temperature profiles at steady state (smaller values) and at the onset of boiling (larger values). The axial temperatures are evaluated along the bundle centerline, wile the radial temperature profiles are evaluated at an axial elevation of z = 779 mm from the bottom of the heated section of the pin bundle. A continuous line was used to report the calculated values for the sake of clarity.

Fig. 4.5. Model I, calculated results in time for: normalized inlet mass flow (top left); total vapour volume (top right); absolute pressure change at the axial location z = 779 mm (bottom left); maximum and minimum axial vapour extent (defined as α*

v⩾0.01, bottom right). The time is measured with respect to the pump trip time at t = 0 s.

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annular regime normalized vapour fraction threshold α*v,an = 0.94 and a

lower mist regime normalized vapour fraction threshold α*v,ms = 0.99.

As it will be later discussed in further detail, the choice of a virtual mass coefficient of 0.1 was based mostly out of numerical convenience to reduce calculation times, as the results were found to be largely insen-sitive to it.

Fig. 4.5 presents the results obtained with what we will refer to as model I. In particular, the Autruffe correlation for the liquid–vapour interfacial friction and the Kottowski-Savatteri two-phase multiplier were used. Figs. 4.6 and 4.7 present the results obtained by model II and III respectively, which used the same correlations as model I, except model II relied on the Lockhart-Martinelli two-phase pressure drop multiplier and model III on the Chen-Kalish one. Fig. 4.5 finally presents the results obtained by model IV, which used the Wallis liquid–vapour interfacial friction correlation alongside the Kottowski-Savatteri two- phase multiplier model. All models predicts the same time of the onset of boiling, which is a consequence of the inlet mass flow ramp down and pin power which were the same in all cases. In particular, the boiling onset was predicted at t = 6.14 s after pump trip, comparable to the experimental t = 6.11 s. In the time window between boiling onset and the start of flow excursion, all models yield equivalent results in terms of total vapour volume evolution (top right plot of each figure), and axial vapour expansion (bottom right plot of each figure), and they compare well against the experimental data.Fig. 4.8.

With regard to the inlet mass flow rate (top left plot of each figure) and pressure (always evaluated at z = 779 mm, bottom left plot of each figure), the average trend of all models except for model III are com-parable between each other and comparable to experimental results. The main difference results in the magnitude of the high-frequency (in the 3 − 6 Hz range) oscillations, which was found to be higher for model II, that employs the Wallis correlation. This is primarily due to the fact that the Wallis model predicts a lower value of the drag coefficient across all vapour void fraction ranges, so that the decreased momentum coupling between the fluid decreases oscillation dampening. Another important aspect that was later on found to reduce the inlet mass flow oscillations, regardless of the models, was to extend the overall simu-lation domain, e.g. lengthening the assembly. This effectively allows to model the inertia of the rest of liquid sodium in the loop, which is otherwise neglected if limiting the analysis to the assembly domain and

imposing von Neumann boundary conditions for the velocity at the inlet. Nonetheless, an extended investigation is beyond the scope of the pre-sent work. The reason why model III predicts an overall faster inlet mass flow reduction is due to the fact that, as it was observed for the steady- state ISPRA experiments, it predicts a larger two phase drag multiplier factor.

With regard to the vapour axial and radial expansion in this initial transient phase (bottom right plot of each figure), all models agree well both among each other and with experimental results, especially With regard to the downward vapour expansion. In particular, boiling starts in the innermost cells at the axial location between z = 860 mm and z =

900 s, that is, at the top of the heated bundle. The large axial temper-ature gradient in the upper unheated bundle region combined with the much smaller axial temperature gradient within the top of the heated bundle itself, below the unheated part, results in an overall downward vapour axial expansion in the initial phase of the transient. As the upper bundle is heated by the condensing vapour, the upper front starts rising as well. Radially, while the results are not shown, the vapour reached the outer bundle in the 8.25 − 8.35 s range for all models except model III, which anticipates it at 7.30 s. The start of the flow excursion follows shortly after for all models and can be recognized clearly from the sharp increase in pressure and decrease in inlet mass flow. All models except for model III predicted a somewhat delayed start of the flow excursion at 8.90 s, compared to the experimentally measured 8.30 s. As an example, the radial vapour distribution predicted by model I at three different instances in time is reported in Fig. 4.4. From the perspective of inlet mass flow evolution however, the marginally faster rate at which the flow excursion takes place makes it so that the overall inlet flow profile is well resolved by all models except model III. As it was highlighted earlier, the larger two phase drag multiplier predicted by the Chen- Kalish model is higher than that predicted by other models, so that the flow excursion is much more severe, thus resulting in a larger vapour volume production and no flow recovery within the time bounds of the transient. For this reason, model III will not be referred any further in the discussion.

Concerning the pressure profiles, while all of the three models under exam under-predict the pressure increase, model I predicts the smallest out of all of them. When comparing models I and II, this result can be apparently unexpected since, as it was observed in the ISPRA

Fig. 4.6. Model II, calculated results in time for: normalized inlet mass flow (top left); total vapour volume (top right); absolute pressure change at the axial location z = 779 mm (bottom left); maximum and minimum axial vapour extent (defined as α*

v⩾0.01, bottom right). The time is measured with respect to the pump trip time at t = 0 s.

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experiments re-analysis, the Kottowski-Savatteri and the Lockhart- Martinelli correlations for the two phase drag multiplier yielded com-parable results in all investigated scenarios. The main difference lies in the fact that during the KNS transient, much larger vapour flow qualities were attained in some mesh cells, resulting is much smaller values of the liquid Lockhart-Martinelli parameter Xl, on which both the Kottowski- Savatteri and the Lockhart-Martinelli multipliers depend. In particular, at very low values of Xl, the Lockhart-Martinelli two-phase multiplier is larger than the Kottowski-Savatteri one, which explains the more pro-nounced pressure increase observed in model II compared to model I. Model III also uses the Kottowski-Savatteri correlation, yet the use of the Wallis interfacial drag model results in a higher pressure increased

compared to model I itself. The reason for this not fully clear, yet we assume that this is due to the increase in the vapour flow quality due to the lower drag, on which the two-phase drag models depend. Shortly after flow excursion, at 9.45 s, the power is turned off in all models. While experimentally this was performed in response to a dryout detection at 9.25 s following a rapid rise in cladding temperatures (of 90 K/s as measured at z = 870 mm), no such reliable dryout criterion could be identified by the code. In fact, while all models predicted a surface pin temperature rise between the onset of boiling and the onset of flow excursion, they predicted a relatively slow rise of pin surface temperature of 50 K/s. For this reason, we preferred to impose a power off time to be compatible with the experimental ones.

Fig. 4.7. Model III, calculated results in time for: normalized inlet mass flow (top left); total vapour volume (top right); absolute pressure change at the axial location z = 779 mm (bottom left); maximum and minimum axial vapour extent (defined as α*

v⩾0.01, bottom right). The time is measured with respect to the pump trip time at t = 0 s.

Fig. 4.8. Model IV, calculated results in time for: normalized inlet mass flow (top left); total vapour volume (top right); absolute pressure change at the axial location z = 779 mm (bottom left); maximum and minimum axial vapour extent (defined as α*

v⩾0.01, bottom right). The time is measured with respect to the pump trip time at t = 0 s.

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With regard to the total vapour volume evolution (top right plot of each figure), regardless of the models, the same total vapour production is observed to the point where vapour reaches the outer bundle region. After that, behaviour varies significantly between the models, with model IV predicting a vapour volume evolution that is the closest to the experimentally measured one (which was measured indirectly via a time-integrated difference between inlet and outlet liquid mass flows). In this sense, model I and II predict that the upper axial vapour front reaches the outlet of the bundle, while model IV predicts an upper axial vapour front evolution that resembles experimental data points more closely.

With regard to the inlet mass flow re-establishment, all the three models agree well with experimental results. Concerning the end of boiling, it varied between 12.0 s and 13.0 s for all models, compared to the experimentally determined 12.30 s. The overall rates at which the vapour condenses compare well between all models and the experiment, in spite of starting from different maxima, which explains the different end-of-boiling times.

With regard to other models and parameters, a qualitative note of interest is made regarding virtual mass effects. As expected, the virtual mass coefficient was found to significantly reduce maximum liquid and vapour flow velocities in the boiling region, without significantly affecting overall results (as most parameters of relevance, e.g. the vapour flow quality, are defined in relative and not absolute terms). However, the reduction in flow velocities had the overall effect of reducing inlet mass flow oscillations and, most importantly, calculation times. In particular, simulations performed without virtual mass effects were found to take up to 4 times as long as calculations with a virtual mass coefficient as low as 0.1. While this is also due to the aforemen-tioned fluid velocity reduction, better convergence of the numerical solution of the pressure–velocity system was also observed. These findings appear to be in line with extensive numerical analyses

performed by Lahey on virtual mass effects in accelerating two-phase flows (Lahey et al., 1980).

4.3.3. Qualitative mesh convergence assessment One of the advantages of a porous medium-based formulation of

thermal–hydraulics is that it generally offers greater flexibility in the assessment of mesh convergence of the obtained results when compared to sub-channel codes, in spite of both approaches having comparable intended spatial resolutions. While mesh-convergence analyses are not generally undertook in the domain of system-level or sub-channel-level simulations, the investigation of complex phenomena such flow boiling at said resolutions might nonetheless benefit from such analyses, which are otherwise standard-practice in the domain of fine-mesh Computa-tional Fluid Dynamics (CFD) computer codes.

While very rigorous practices exist in order to assess the degree of mesh convergence of the results, such as those devised by Roache (Roache, 1994) and employed by us in previous single-phase in-vestigations (Radman et al., 2019), the present investigation is better served by a qualitative mesh convergence assessment due to its transient and highly parameter-sensitive nature. More formal mesh-convergence efforts where applied during the verification process of the developed two-phase thermal–hydraulics solver via the Method of Manufactured Solutions in previous investigations (Radman et al., 2021).

The considered mesh refinement consisted of radial mesh refine-ment. We recall that 1-D models are fundamentally incapable of capturing the excursion timing. This is due to the fact that 1-D models would predict an expanding vapour region occupying the entirety of the available cross-sectional flow area from the onset of boiling, thus immediately resulting in some form of flow excursion. We also recall that all of the results presented in the previous section were obtained with a mesh of seven radial cells, namely five for the assembly bulk and two for the assembly edge. The core motivation was that a resolution of approximately two cells per sub-channel was deemed sufficient as it is comparable to typical radial resolutions of sub-channel codes. This qualitative mesh convergence assessment then focused on comparing the results previously obtained to the results obtained with two other radial mesh choices for a constant axial mesh size. The first choice consists of a radial mesh of one cell per sub-channel, namely three cells for the assembly bulk and one for the assembly edge, which will be referred to as ”coarse” mesh; the second choice consists of a radial mesh of three cells per sub-channel, namely nine cells for the assembly bulk and three for the assembly edge, which will be referred to as ”fine” mesh.

Representative results for this qualitative mesh convergence assess-ment are reported in Figs. 4.9 and 4.10 for the reference-to-coarse and reference-to-fine mesh comparisons involving results obtained with the previously discussed Model I. These have been split in separate figures for clarity.

With regard to the coarse mesh results as presented in Fig. 4.9, a number of observations can be made. In the first place, regarding the timing, the onset of boiling is delayed from 9.1 s to 9.3 s, the onset of flow excursion is anticipated by ≃ 1 s and the overall boiling duration is reduced by 0.7 s. Furthermore, the duration of the excursion is signifi-cantly extended as it can be observed from the inlet mass flow rate profile. This behaviour is nonetheless consistent with the observation that with a single cell in the radial direction (i.e. a 1-D mesh), the excursion is significantly anticipated (and prolonged) for the reasons discussed earlier. In spite of this, the predicted total vapour volume evolution in time and the axial vapour extent are in reasonable agree-ment with the reference model. In particular, the total vapour volume represents the arguably more meaningful parameter to be compared in the present qualitative mesh convergence assessment due to it being the most time-integral of all the reported variables.

With regard to the fine mesh results as presented in Fig. 4.10, both the onset and the end of boiling occur at the same times as in the reference case. The average behaviour of the inlet mass flow rate, namely the time of the onset of the excursion, excursion duration and

Fig. 4.4. Distribution of the normalized vapour phase fraction α*v at three

different moments of the transient, when (from left to right): at t = 8.30s, the vapour reaches outer bundle region; at t = 8.90s, the flow excursion starts; at t = 9.45, the power is turned off. Please recall that the mesh domain consists of a 2-D axial-symmetric wedge, with the wedge axis being the leftmost edge of the domain in each of the three images. The heated bundle section upper and lower boundaries are highlighted with red lines. The mesh and domain are axially scaled by a factor 0.075 for compactness.

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magnitude are comparable and are overall much closer to the reference case when compared to the coarse case results, although the intensity of the short-time-scale oscillations appears to be larger in the fine case. The evolution of the total vapour volume in the fine case is also significantly closer to the reference case, when compared to the coarse case. How-ever, differences arise in the vapour condensation phase, as the lower vapour front appears to condense at a slower pace than the upper vapour front. However, the absolute amount of vapour in the system when differences in the vapour axial extent start to grow significant is small compared to the maximum vapour volumes attained during boiling, so secondary importance is given to this.

Overall, this qualitative mesh convergence assessment shows that results obtained in section 4.3.2 can be considered as a good

compromise between accuracy and computational footprint. In partic-ular, the total vapour volume evolution, which can be argued to provide the most meaningful metric for this assessment, does not change for finer meshes.

4.3.4. Comparison of two-phase results against TRACE system code results For the sake of completeness, the results obtained via the code, which

will be referred to simply as GeN-Foam due to the inclusion of the developed thermal–hydraulic solver in it, have been compared against results for KNS-37 Test L22 available in the literature. In particular, these are compared against the results obtained via a similar 2-D wedge domain with a modified version of the TRACE system code, modified so to enable sodium boiling simulations (Chenu et al., 2009).

Fig. 4.9. Comparison between the results obtained by Model I on the reference and the coarse meshes. Results consist of: normalized inlet mass flow (top left); total vapour volume (top right); absolute pressure change at the axial location z = 779 mm (bottom left); maximum and minimum axial vapour extent (defined as α*

v⩾0.01, bottom right). The time is measured with respect to the pump trip time at t = 0 s.

Fig. 4.10. Comparison between the results obtained by Model I on the reference and the fine meshes. Results consist of: normalized inlet mass flow (top left); total vapour volume (top right); absolute pressure change at the axial location z = 779 mm (bottom left); maximum and minimum axial vapour extent (defined as α*

v⩾0.01, bottom right). The time is measured with respect to the pump trip time at t = 0 s.

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While a significant portion of the work presented by Chenu et al. in (Chenu et al., 2009) deals with 1-D models, the ultimate results show that such models fail in predicting the onset of the flow excursion with accuracy due to the reasons previously discussed. For this reason, the results obtained via our 2-D wedge domain are compared against the 2-D TRACE results, which were unfortunately explored less extensively by Chenu. When it comes to the selection of models. With regard to the extent of the TRACE model domain, only the radial meshing choice, namely 4 radial cells, is presented, while no information on the axial extent nor the axial meshing of the TRACE model are provided. With regard to the choice of correlations and flow regime map modelling, a complete picture as the one provided in Table 4.4 is similarly not pro-vided, with the only information presented consisting of the choice for the liquid-wall and liquid–vapour drag coefficients, fluid-vapour inter-facial area density, fluid-vapour and vapour-fluid interfacial heat transfer coefficients. In particular, the results of the TRACE code pre-sented in Fig. 4.11 were obtained via the utilization of the Churchill and Wallis correlations for the liquid-wall and liquid–vapour drag co-efficients respectively, the interfacial area density predicted via the Schor model, and the liquid–vapour heat transfer coefficient treated via the No-Kazimi correlation (No and Kazimi, 1983) on the liquid side and with a constant, large value of 1⋅108 W/m2K on the vapour side.

With regard to the choice of correlations, these are reasonably similar to the correlations used in model IV with our code as previously discussed, with the largest differences lying in the liquid-wall drag co-efficient and the liquid–vapour heat transfer coefficients. It is further assumed that no particular regime map was employed for the modelling of the liquid-wall and vapour-wall contact fractions, as any mention in that regard is absent from their work (Chenu et al., 2009).

With regard to the differences in the liquid-wall drag coefficients, the Churchill correlation adopted by the TRACE model was used in conjunction with an ”artificial friction model”, which the authors use as an additional variable to adjust the vapour axial expansion in time, and is justified on the basis of the need to better model the hydraulic resis-tance of the overall experimental loop. This approach is consistent with our choice of treating the hydraulic diameter for the drag coefficient calculations as an unknown parameter. However, the overall wall drag modelling choices, i.e. Churchill with artificial friction in TRACE against to Rehme with adjusted hydraulic diameters in GeN-Foam, make it

difficult to make a detailed comparison. With regard to the differences in the liquid–vapour heat transfer

coefficients, as both the TRACE and the GeN-Foam models rely on a heat conduction limited heat-exchange model, it can be argued that as long as the liquid–vapour heat transfer coefficient on the liquid side is sufficient high to avoid non-physical fluid super-heating during the two-phase regime (yet after the onset of boiling, which is owed a separate discus-sion (Kottowski and Savatteri, 1984)), then the actual choice of the correlations is not particularly impactful. It is for this precise reason that the TRACE model employs a very high liquid–vapour heat transfer co-efficient on the vapour side, to avoid vapour super-heating.

With regard to the comparison presented in Fig. 4.11, it can be observed that the average inlet mass flow rate predicted via the TRACE model does not appear to reach an excursion of the same magnitude as that predicted by GeN-Foam, which is in greater agreement with the experimental data. Nonetheless, the results predicted by GeN-Foam are otherwise appreciably more oscillatory in nature, and this could be due either to differences in the wall-drag heat transfer coefficient modelling as discussed early and/or geometric modelling choices altogether (e.g. size of the overall domain). We assume that the TRACE model consisted of a closed loop of 1-D components representative of the overall facility, with the assembly modelled in 2-D. The modelling of the additional fluid inertia in the system might help significantly reduce oscillations in the observed mass flow rate (given pressure oscillations of otherwise similar magnitude between the two models).

5. Conclusions

Within the framework of computational activities at the LRS, EPFL, a novel solver for nuclear reactor thermal–hydraulic analysis was recently developed based on the OpenFOAM numerical library. The solver itself relies on a porous medium representation and a dispersed Euler-Euler framework for the treatment of the two-phase flow. It is based on a six-equation model to aim at the highest degree of accuracy in terms of conservation of mass, momentum and energy for each phase. A variety of correlations is available to treat the exchange of momentum and energy between the phases themselves and each phase and the structure, as well as for the thermally-driven mass transfer between the phases. While most of the implemented correlations pertain to liquid and vapour

Fig. 4.11. Comparison between the results obtained by Model IV via our code (GeN-Foam) against results obtained by a similar model with the TRACE system code in 2-D. Compared quantities consist of: normalized inlet mass flow (top); absolute pressure change at the axial location z = 779 mm (bottom left). The time is measured with respect to the pump trip time at t = 0 s.

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sodium, the framework was developed to avoid hard-coding of the available models, which in turn enables a large degree of flexibility when implement new correlations.

This paper focused on a preliminary validation effort against steady- state and transient sodium boiling experiments. The steady-state ex-periments, performed at JRC, ISPRA, consisted in sodium boiling in heated tube geometries at constant inlet flow velocities and inlet power levels until a quasi-steady state is reached. This proved an excellent ground for the validation of some of the models of code, as quasi-steady state scenarios are insensitive to a variety of other models that are otherwise relevant in transient scenarios. The results of this validation phase were satisfactory. Test L22 at the KNS-37 facility at Kernfor-schungszentrum Karlsruhe was then modelled and reanalyzed to pre-liminary validate the transient capabilities of the solver. Different correlations for a variety of quantities where investigated, some of which were proven to yield satisfactory results when compared to the experimental ones.

The solver was integrated within the broader GeN-Foam multi- physics platform to enable coupled analysis of more complex geometries and transients. Future work will focus on the analysis of further boiling experiments and tests with emphasis on other working fluids, as well as on the application of the solver for multi-physics simulations.

CRediT authorship contribution statement

Stefan Radman: Conceptualization, Methodology, Software, Vali-dation, Formal analysis, Investigation, Data curation, Writing - original draft, Writing - review & editing, Visualization. Carlo Fiorina: Conceptualization, Methodology, Resources, Data curation, Writing - review & editing, Supervision, Project administration, Funding acqui-sition. Andreas Pautz: Resources, Writing - review & editing, Supervi-sion, Project administration, Funding acquisition.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by a grant from the Swiss National Supercomputing Centre (CSCS) under project ID sm34.

References

Sun, K., Krepel, J., Mikityuk, K., Pelloni, S., Chawla, R., 2011. Void reactivity decomposition for the sodium-cooled fast reactor in equilibrium fuel cycle. Ann. Nucl. Energy 38 (7), 1645–1657. https://doi.org/10.1016/j.anucene.2011.02.018.

United States Nuclear Regulatory Commission, Trace theory manual, url: https://www. nrc.gov/docs/ML1200/ML120060218.pdf, 2012.

Whitaker, S., 1985. A simple geometrical derivation of the spatial averaging theorem. Chemical Engineering Education 19.

Tommasi, J., Archier, P., Ruggieri, J., 2010. Validation of the sodium void reactivity effect prediction using JEFF-3.1 nuclear data, Annals of Nuclear Energy 37 (11) (2010) 1534–1553. doi:10.1016/j.anucene.2010.06.010.

Beck, T., Blanc, V., Escleine, J.-M., Haubensack, D., Pelletier, M., Phelip, M., Perrin, B., Venard, C., 2017. Conceptual design of astrid fuel sub-assemblies. Nucl. Eng. Des. 315, 51–60. https://doi.org/10.1016/j.nucengdes.2017.02.027 url:http://www. sciencedirect.com/science/article/pii/S0029549317300857.

Sciora, P., Buiron, L., Varaine, F., 2020. The low void worth core design (’cfv’) based on an axially heterogeneous geometry. Nucl. Eng. Des. 366, 110763 https://doi.org/ 10.1016/j.nucengdes.2020.110763 url:http://www.sciencedirect.com/science/ article/pii/S0029549320302570.

Scolaro, A., Clifford, I., Fiorina, C., Pautz, A., 2020. The offbeat multi-dimensional fuel behavior solver. Nucl. Eng. Des. 358, 110416 https://doi.org/10.1016/j. nucengdes.2019.110416 url:http://www.sciencedirect.com/science/article/pii/ S0029549319304479.

Fiorina, C., Clifford, I., Aufiero, M., Mikityuk, K., 2015. GeN-Foam: a novel OpenFOAM based multi-physics solver for 2D/3D transient analysis of nuclear reactors. Nucl. Eng. Des. 294, 24–37. https://doi.org/10.1016/j.nucengdes.2015.05.035.

Fiorina, C., Kerkar, N., Mikityuk, K., Rubiolo, P., Pautz, A., 2016. Development and verification of the neutron diffusion solver for the GeN-Foam multi-physics platform. Ann. Nucl. Energy 96, 212–222. https://doi.org/10.1016/j.anucene.2016.05.023.

Fiorina, C., Hursin, M., Pautz, A., 2017. Extension of the GeN-Foam neutronic solver to SP3 analysis and application to the CROCUS experimental reactor. Ann. Nucl. Energy 101, 419–428. https://doi.org/10.1016/j.anucene.2016.11.042.

Fiorina, C., Radman, S., Koc, M.-Z., Pautz, A., 2019. Detailed modelling of the expansion reactivity feedback in fast reactors using openfoam, in: M&C Conference 2019, American Nuclear Society.

S. Radman, C. Fiorina, K. Mikityuk, A. Pautz, A simplified numerical benchmark for pool- type sodium fast reactors, in: 26th International Conference on Nuclear Engineering, American Society of Mechanical Engineers, 2018, pp. V004T15A018–V004T15A018. doi:10.1115/ICONE26-82260.

OpenCFD, Ltd. Radman, S., Fiorina, C., Mikityuk, K., Pautz, A., 2019. A coarse-mesh methodology for

modelling of single-phase thermal-hydraulics of esfr innovative assembly design. Nucl. Eng. Des. 355, 110291 https://doi.org/10.1016/j.nucengdes.2019.110291 url: http://www.sciencedirect.com/science/article/pii/S0029549319303085.

Fink, J., Leibowitz, L., 1996. A consistent assessment of the thermophysical properties of sodium. High Temp. Mater. Sci. 35 (1).

Fiorina, C., 2019. Impact of the volume heat source on the rans-based cfd analysis of molten salt reactors. Ann. Nucl. Energy 134, 376–382. https://doi.org/10.1016/j. anucene.2019.06.024 url:http://www.sciencedirect.com/science/article/pii/ S0306454919303421.

Weller, H.G., Tabor, G., Jasak, H., Fureby, C., 1998. A tensorial approach to computational continuum mechanics using object-oriented techniques, Computers in Physics 12 (6) (1998) 620–631. arXiv:https://aip.scitation.org/doi/pdf/10.1063/ 1.168744, doi:10.1063/1.168744. url:https://aip.scitation.org/doi/abs/10.1063/ 1.168744.

Radman, S., Fiorina, C., Pautz, A., 2021. Development of a novel two-phase flow solver for nuclear reactor analysis: algorithms, verification and implementation in openfoam. Nucl. Eng. Des. 379, 111178 https://doi.org/10.1016/j. nucengdes.2021.111178 url:https://www.sciencedirect.com/science/article/pii/ S0029549321001308.

Kottowski, H., Savatteri, C., 1984. Fundamentals of liquid metal boiling thermohydraulics. Nucl. Eng. Des. 82 (2), 281–304. https://doi.org/10.1016/0029- 5493(84)90216-4.

Bottoni, M., Dorr, B., Homann, C., Huber, F., Mattes, K., Peppler, F.W., Struwe, D., 1990. Experimental and numerical investigations of sodium boiling experiments in pin bundle geometry, Nuclear Technology 89 (1) (1990) 56–82. arXiv:https://doi.org/ 10.13182/NT90-A34359, doi:10.13182/NT90-A34359. url:https://doi.org/ 10.13182/NT90-A34359.

Kozelkov, A., Lashkin, S., Efremov, V., Volkov, K., Tsibereva, Y., Tarasova, N., 2018. An implicit algorithm of solving Navier-Stokes equations to simulate flows in anisotropic porous media. Computers Fluids 160, 164–174. https://doi.org/ 10.1016/j.compfluid.2017.10.029. url:http://www.sciencedirect.com/science/ article/pii/S0045793017303948.

Weller, H.G., 2008. A new approach to vof-based interface capturing methods for incompressible and compressible flow, Tech. Rep. Report, OpenCFD Ltd., London (2008).

Boris, J.P., Book, D.L., 1997. Flux-corrected transport. J. Comput. Phys. 135 (2), 172–186. https://doi.org/10.1006/jcph.1997.5700 url:http://www.sciencedirect. com/science/article/pii/S0021999197957004.

Ghanbarian, B., Hunt, A.G., Ewing, R.P., Sahimi, M., 2013. Tortuosity in porous media: A critical review, Soil Science Soc. Am. J. 77 (5) (2013) 1461–1477. arXiv:https:// acsess.onlinelibrary.wiley.com/doi/pdf/10.2136/sssaj2012.0435, doi:10.2136/ sssaj2012.0435. url:https://acsess.onlinelibrary.wiley.com/doi/abs/10.2136/ sssaj2012.0435.

No, H., Kazimi, M., 1989. Nuclear Systems I: Thermal Hydraulic Fundamentals, 1st Edition. CRC Press.

Ninokata, H., Aki Okano, T., 1990. Sabena: Subassembly boiling evolution numerical analysis, Nuclear Engineering and Design 120 (2) (1990) 349 – 367. doi:10.1016/ 0029-5493(90)90386-C. url:http://www.sciencedirect.com/science/article/pii/ 002954939090386C.

Mikityuk, K., 2009. Heat transfer to liquid metal: Review of data and correlations for tube bundles. Nucl. Eng. Des. 239 (4), 680–687. https://doi.org/10.1016/j. nucengdes.2008.12.014.

Chen, J.C., 1966. Correlation for boiling heat transfer to saturated fluids in convective flow, Industrial & Engineering Chemistry Process Design and Development 5 (3) (1966) 322–329. arXiv:https://doi.org/10.1021/i260019a023, doi:10.1021/ i260019a023. url:https://doi.org/10.1021/i260019a023.

Lockhart, R.W., Martinelli, R.C., 1949. Proposed correlation of data for isothermal two- phase two-component flow in pipes. Chem. Eng. Prog. 45 (1), 39–48.

Nguyen, H., 1985. A finite element method for one-dimensional sodium boiling simulation, in: Proceedings of the 3rd Int. Topical Meeting on React Thermal- hydraulics, p. 17.

Kaiser, A., Huber, F., Bottoni, M., Dorr, B., 1988. Contribution to sodium boiling heat transfer, pressure drop, and void distribution in multi-pin geometry. In: Proceedings of the 13th Liquid Metal Boiling Working Group (LMBWG), pp. 71–98.

Chen, J.C., Kalish, S., 1988. An experimental investigation of two-phase pressure drop for potassium with and without net vaporization, in: Proceedings of the 4th Int. Heat Transfer Conference, Vol. 6.

Autruffe, M., Wilson, G., Stewart, B., Kazimi, M., 1979. A proposed momentum exchange coefficient for two-phase modeling of sodium boiling. In: Proceedings of the International Meeting on Fast Reactor Safety Technology, pp. 2515–2521.

S. Radman et al.

Nuclear Engineering and Design 384 (2021) 111422

21

Radman, S., Fiorina, C., Pautz, A., 2019. Preliminary development of a coarse-mesh sodium boiling model for openfoam-based multi-physics solvers. In: 18th International International Topical Meeting on Nuclear Reactor Thermal Hydraulics. American Nuclear Society.

Chenu, A., Mikityuk, K., Chawla, R., 2009. TRACE simulation of sodium boiling in pin bundle experiments under loss-of-flow conditions. Nucl. Eng. Des. 239 (11), 2417–2429. https://doi.org/10.1016/j.nucengdes.2009.07.015.

Ro, T.S., Todreas, N.E., 1988. Energy transfer mechanisms in lmr rod bundles under mixed convection conditions. Nucl. Eng. Des. 108 (3), 343–357. https://doi.org/ 10.1016/0029-5493(88)90223-3 url:https://www.sciencedirect.com/science/ article/pii/0029549388902233.

Rehme, K., 1973. Pressure drop correlations for fuel element spacers. Nucl. Technol. 17, 12–23.

Chen, S., Chen, Y., Todreas, N., 2018. The upgraded cheng and todreas correlation for pressure drop in hexagonal wire-wrapped rod bundles. Nucl. Eng. Des. 335, 356–373. https://doi.org/10.1016/j.nucengdes.2018.05.010 url:https://www. sciencedirect.com/science/article/pii/S0029549318301389.

Schor, A., Kazimi, M., Todreas, N., 1984. Advances in two-phase flow modeling for lmfbr applications. Nucl. Eng. Des. 82 (2), 127–155. https://doi.org/10.1016/0029-5493

(84)90208-5 url:http://www.sciencedirect.com/science/article/pii/ 0029549384902085.

Mohammed Shah, M., 1992. A survey of experimental heat transfer data for nucleate pool boiling of liquid metals and a new correlation, International Journal of Heat and Fluid Flow 13 (4) (1992) 370–379. doi:10.1016/0142-727X(92)90007-V. url: http://www.sciencedirect.com/science/article/pii/0142727X9290007V.

No, H., Kazimi, M., 1987. Investigation of the physical foundations of two-fluid representation of sodium boiling in the liquid-metal fast breeder reactor. Nuclear Sci. Eng. 97 (4), 327–343. https://doi.org/10.13182/NSE87-A23516 cited By 11.

Lahey, R., Cheng, L., Drew, D., Flaherty, J., 1980. The effect of virtual mass on the numerical stability of accelerating two-phase flows. Int. J. Multiph. Flow 6 (4), 281–294. https://doi.org/10.1016/0301-9322(80)90021-X url:http://www. sciencedirect.com/science/article/pii/030193228090021X.

Roache, P.J., 1994. A method for uniform reporting of grid refinement studies. J. Fluids Eng. 116, 405–413. https://doi.org/10.1115/1.2910291.

No, H., Kazimi, M., 1983. An investigation of the physical and numerical foundations of two-fluid representation of sodium boiling with application to lmfbr experiments.

S. Radman et al.