[PPT] - Thermohydraulic of nuclear core reactor: construction, study and PowerPoint Presentation

SLIDE 1

Paris, 9 February 2018

Thermohydraulic of nuclear core reactor: construction, study and dicretisation of models

Asymptotic vs compressible models

LMNC projects (2014-2017)

Team Needs-2018 Team ❶ Gloria Faccanoni, Cédric Galusinski, Mehmet Ersoy, Moustoifa Rafiou Team ❷ Bérénice Grec, Samuel Kokh, Olivier Lafitte, Yohan Penel, (Stéphane Dellacherie) Team ❸ Jean-Marc Hérard, Olivier Hurisse, Lucie Quibel Team ❹ Hélène Mathis, Hala Ghazi, Nicolas Seguin, Benjamin Boutin, Thuong Nguyen, Jonathan Jung Team ❺ Olivier Doche, Pablo Rubiolo

SLIDE 2

1. Context
2. Past
3. 2018. . .

Outline

1

Context

2

Needs 2014-2017

3

Needs 2018. . .

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 2 / 23 :

SLIDE 3

1. Context
2. Past
3. 2018. . .

Section 1 Context

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 3 / 23 :

SLIDE 4

1. Context
2. Past
3. 2018. . .

Context

Modelisation and numerical simulation of heat transfers in a core of a nuclear reactor (or steam generator) where the coolant is water (or sodium or molten salt) with or without phase transition.

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 4 / 23 :

SLIDE 5

1. Context
2. Past
3. 2018. . .

Core of a Pressurized Water Reactor

Control Rods Fuel Elements Φ ≃170 MW Liquid Water (290 ◦C and 5 m · s−1) Mixture liquid-steam (330 ◦C) Water : coolant and moderator p0 ≃ 155 bar

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 5 / 23 :

SLIDE 6

1. Context
2. Past
3. 2018. . .

Core of a Pressurized Water Reactor

Nominal regime Inlet velocity : |u| ≃ 5 m · s−1 Speed of sound at p0 = 155 bar and T = 300 ◦C : c∗

ℓ ≃ 1.0 × 103 m · s−1

Mach number M = |u| c∗

ℓ

≃ 5 × 10−3 ≪ 1 LOFA The Loss of Flow Accident is an accidental scenario induced by a coolant pump trip event with phase change Acoustics negligible (no shock waves) BUT high heat transfers : div u = 0 LOCA This is not the case for very fast depressurizations such as a Loss of Coolant Accident

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 6 / 23 :

SLIDE 7

1. Context
2. Past
3. 2018. . .

Core of a Pressurized Water Reactor

Nominal regime Inlet velocity : |u| ≃ 5 m · s−1 Speed of sound at p0 = 155 bar and T = 300 ◦C : c∗

ℓ ≃ 1.0 × 103 m · s−1

Mach number M = |u| c∗

ℓ

≃ 5 × 10−3 ≪ 1 LOFA The Loss of Flow Accident is an accidental scenario induced by a coolant pump trip event with phase change Acoustics negligible (no shock waves) BUT high heat transfers : div u = 0 LOCA This is not the case for very fast depressurizations such as a Loss of Coolant Accident

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 6 / 23 :

SLIDE 8

1. Context
2. Past
3. 2018. . .

Core of a Pressurized Water Reactor

Nominal regime Inlet velocity : |u| ≃ 5 m · s−1 Speed of sound at p0 = 155 bar and T = 300 ◦C : c∗

ℓ ≃ 1.0 × 103 m · s−1

Mach number M = |u| c∗

ℓ

≃ 5 × 10−3 ≪ 1 LOFA The Loss of Flow Accident is an accidental scenario induced by a coolant pump trip event with phase change Acoustics negligible (no shock waves) BUT high heat transfers : div u = 0 LOCA This is not the case for very fast depressurizations such as a Loss of Coolant Accident

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 6 / 23 :

SLIDE 9

1. Context
2. Past
3. 2018. . .

Core of a Pressurized Water Reactor

Nominal regime Inlet velocity : |u| ≃ 5 m · s−1 Speed of sound at p0 = 155 bar and T = 300 ◦C : c∗

ℓ ≃ 1.0 × 103 m · s−1

Mach number M = |u| c∗

ℓ

≃ 5 × 10−3 ≪ 1 LOFA The Loss of Flow Accident is an accidental scenario induced by a coolant pump trip event with phase change Acoustics negligible (no shock waves) BUT high heat transfers : div u = 0 LOCA This is not the case for very fast depressurizations such as a Loss of Coolant Accident

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 6 / 23 :

SLIDE 10

1. Context
2. Past
3. 2018. . .

All Mach number flow : Compressible vs incompressible

Large Mach number

Compressible flow : hyperbolic systems of conservation laws Shock discontinuities are generic Conservation schemes necessary in order to guarantee consistency for weak solutions CFL restriction physiological (we are in general interested in acoustic waves)

Small Mach number

Quasi-incompressible flow Often one is not interested in resolving acoustic waves Material shocks do not form from smooth initial data and acoustic shocks have negligible amplitude Classical CFL restriction is pathological : it is due to the stiffness of the problem and should be avoided

Zero Mach number

Incompressible flow No acoustic waves

Compressible and incompressible flows are usually treated by different techniques. Not

bvious how to unify the treatment.

Goal : design numerical method for compressible flows that can handle both the compressible regime and the incompressible limit

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 7 / 23 :

SLIDE 11

1. Context
2. Past
3. 2018. . .

All Mach number flow : Compressible vs incompressible

Large Mach number

Compressible flow : hyperbolic systems of conservation laws Shock discontinuities are generic Conservation schemes necessary in order to guarantee consistency for weak solutions CFL restriction physiological (we are in general interested in acoustic waves)

Small Mach number

Quasi-incompressible flow Often one is not interested in resolving acoustic waves Material shocks do not form from smooth initial data and acoustic shocks have negligible amplitude Classical CFL restriction is pathological : it is due to the stiffness of the problem and should be avoided

Zero Mach number

Incompressible flow No acoustic waves

Compressible and incompressible flows are usually treated by different techniques. Not

bvious how to unify the treatment.

Goal : design numerical method for compressible flows that can handle both the compressible regime and the incompressible limit

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 7 / 23 :

SLIDE 12

1. Context
2. Past
3. 2018. . .

All Mach number flow : Compressible vs incompressible

Large Mach number

Compressible flow : hyperbolic systems of conservation laws Shock discontinuities are generic Conservation schemes necessary in order to guarantee consistency for weak solutions CFL restriction physiological (we are in general interested in acoustic waves)

Small Mach number

Quasi-incompressible flow Often one is not interested in resolving acoustic waves Material shocks do not form from smooth initial data and acoustic shocks have negligible amplitude Classical CFL restriction is pathological : it is due to the stiffness of the problem and should be avoided

Zero Mach number

Incompressible flow No acoustic waves

Compressible and incompressible flows are usually treated by different techniques. Not

bvious how to unify the treatment.

Goal : design numerical method for compressible flows that can handle both the compressible regime and the incompressible limit

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 7 / 23 :

SLIDE 13

1. Context
2. Past
3. 2018. . .

All Mach number flow : Compressible vs incompressible

Large Mach number

Compressible flow : hyperbolic systems of conservation laws Shock discontinuities are generic Conservation schemes necessary in order to guarantee consistency for weak solutions CFL restriction physiological (we are in general interested in acoustic waves)

Small Mach number

Quasi-incompressible flow Often one is not interested in resolving acoustic waves Material shocks do not form from smooth initial data and acoustic shocks have negligible amplitude Classical CFL restriction is pathological : it is due to the stiffness of the problem and should be avoided

Zero Mach number

Incompressible flow No acoustic waves

Compressible and incompressible flows are usually treated by different techniques. Not

bvious how to unify the treatment.

Goal : design numerical method for compressible flows that can handle both the compressible regime and the incompressible limit

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 7 / 23 :

SLIDE 14

1. Context
2. Past
3. 2018. . .

Which model wrt regime ?

1

Mach number M = 0

Compressible Navier-Stokes system: model with acoustics and with heat transfers, if M ≪ 1 standard FV shock-capturing hyperbolic solvers have difficulties.

2

Mach number M ≪ 1 and high heat transfers div u = 0

Asymptotic low Mach model: model without acoustics but with heat transfers

3

Mach number M ≃ 0

Incompressible Navier-Stokes system: model with no acoustics and no heat transfers

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 8 / 23 :

SLIDE 15

1. Context
2. Past
3. 2018. . .

Which model wrt regime ?

1

Mach number M = 0

Compressible Navier-Stokes system: model with acoustics and with heat transfers, if M ≪ 1 standard FV shock-capturing hyperbolic solvers have difficulties.

2

Mach number M ≪ 1 and high heat transfers div u = 0

Asymptotic low Mach model: model without acoustics but with heat transfers

3

Mach number M ≃ 0

Incompressible Navier-Stokes system: model with no acoustics and no heat transfers

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 8 / 23 :

SLIDE 16

1. Context
2. Past
3. 2018. . .

Outline of the project

Compressible Navier-Stokes systems Hierarchy of models depending on (dis)equilibriums Closures (interfacial quantities. . .) Numerical schemes Mach regime depended Asymptotic low Mach models Hierarchy of asymptotic models Notion of low Mach number when two velocities Closures Schemes with divergence constrained (≃ “incompressible”) Equations of state Disequilibriums Relaxations phase transitions Verification : analytical formula (NA-SG, VdW, PR. . .) parameters ? range validity ? saturation ? Validation : tabulated values ? IAPWS (XSteam . . .), “Fitting”, FLT, “splines” . . . performances ? Neutronics simplified description

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 9 / 23 :

SLIDE 17

1. Context
2. Past
3. 2018. . .

Outline of the project

Compressible Navier-Stokes systems Hierarchy of models depending on (dis)equilibriums Closures (interfacial quantities. . .) Numerical schemes Mach regime depended Asymptotic low Mach models Hierarchy of asymptotic models Notion of low Mach number when two velocities Closures Schemes with divergence constrained (≃ “incompressible”) Equations of state Disequilibriums Relaxations phase transitions Verification : analytical formula (NA-SG, VdW, PR. . .) parameters ? range validity ? saturation ? Validation : tabulated values ? IAPWS (XSteam . . .), “Fitting”, FLT, “splines” . . . performances ? Neutronics simplified description

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 9 / 23 :

SLIDE 18

1. Context
2. Past
3. 2018. . .

Outline of the project

Compressible Navier-Stokes systems Hierarchy of models depending on (dis)equilibriums Closures (interfacial quantities. . .) Numerical schemes Mach regime depended Asymptotic low Mach models Hierarchy of asymptotic models Notion of low Mach number when two velocities Closures Schemes with divergence constrained (≃ “incompressible”) Equations of state Disequilibriums Relaxations phase transitions Verification : analytical formula (NA-SG, VdW, PR. . .) parameters ? range validity ? saturation ? Validation : tabulated values ? IAPWS (XSteam . . .), “Fitting”, FLT, “splines” . . . performances ? Neutronics simplified description

Coupling / Benchmark

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 9 / 23 :

SLIDE 19

1. Context
2. Past
3. 2018. . .

Outline of the project

Compressible Navier-Stokes systems Hierarchy of models depending on (dis)equilibriums Closures (interfacial quantities. . .) Numerical schemes Mach regime depended Asymptotic low Mach models Hierarchy of asymptotic models Notion of low Mach number when two velocities Closures Schemes with divergence constrained (≃ “incompressible”) Equations of state Disequilibriums Relaxations phase transitions Verification : analytical formula (NA-SG, VdW, PR. . .) parameters ? range validity ? saturation ? Validation : tabulated values ? IAPWS (XSteam . . .), “Fitting”, FLT, “splines” . . . performances ? Neutronics simplified description

M

d

e l c

u

p l i n g M

d

e l c

u

p l i n g Coupling / Benchmark

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 9 / 23 :

SLIDE 20

1. Context
2. Past
3. 2018. . .

Outline of the project

Compressible Navier-Stokes systems Hierarchy of models depending on (dis)equilibriums Closures (interfacial quantities. . .) Numerical schemes Mach regime depended Asymptotic low Mach models Hierarchy of asymptotic models Notion of low Mach number when two velocities Closures Schemes with divergence constrained (≃ “incompressible”) Equations of state Disequilibriums Relaxations phase transitions Verification : analytical formula (NA-SG, VdW, PR. . .) parameters ? range validity ? saturation ? Validation : tabulated values ? IAPWS (XSteam . . .), “Fitting”, FLT, “splines” . . . performances ? Neutronics simplified description

NEEDS 2014-2017

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 9 / 23 :

SLIDE 21

1. Context
2. Past
3. 2018. . .

Outline of the project

Compressible Navier-Stokes systems Hierarchy of models depending on (dis)equilibriums Closures (interfacial quantities. . .) Numerical schemes Mach regime depended Asymptotic low Mach models Hierarchy of asymptotic models Notion of low Mach number when two velocities Closures Schemes with divergence constrained (≃ “incompressible”) Equations of state Disequilibriums Relaxations phase transitions Verification : analytical formula (NA-SG, VdW, PR. . .) parameters ? range validity ? saturation ? Validation : tabulated values ? IAPWS (XSteam . . .), “Fitting”, FLT, “splines” . . . performances ? Neutronics simplified description

M

d

e l c

u

p l i n g

NEEDS 2018

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 9 / 23 :

SLIDE 22

1. Context
2. Past
3. 2018. . .

Outline of the project

Compressible Navier-Stokes systems Hierarchy of models depending on (dis)equilibriums Closures (interfacial quantities. . .) Numerical schemes Mach regime depended Asymptotic low Mach models Hierarchy of asymptotic models Notion of low Mach number when two velocities Closures Schemes with divergence constrained (≃ “incompressible”) Equations of state Disequilibriums Relaxations phase transitions Verification : analytical formula (NA-SG, VdW, PR. . .) parameters ? range validity ? saturation ? Validation : tabulated values ? IAPWS (XSteam . . .), “Fitting”, FLT, “splines” . . . performances ? Neutronics simplified description

M

d

e l c

u

p l i n g M

d

e l c

u

p l i n g Coupling / Benchmark

NEEDS 2018. . .

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 9 / 23 :

SLIDE 23

1. Context
2. Past
3. 2018. . .

Outline of the project

Compressible Navier-Stokes systems Hierarchy of models depending on (dis)equilibriums Closures (interfacial quantities. . .) Numerical schemes Mach regime depended Asymptotic low Mach models Hierarchy of asymptotic models Notion of low Mach number when two velocities Closures Schemes with divergence constrained (≃ “incompressible”) Equations of state Disequilibriums Relaxations phase transitions Verification : analytical formula (NA-SG, VdW, PR. . .) parameters ? range validity ? saturation ? Validation : tabulated values ? IAPWS (XSteam . . .), “Fitting”, FLT, “splines” . . . performances ? Neutronics simplified description

M

d

e l c

u

p l i n g M

d

e l c

u

p l i n g Coupling / Benchmark

NEEDS 2018. . .

Multi-physics Multi-phasic Multi-regime

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 9 / 23 :

SLIDE 24

1. Context
2. Past
3. 2018. . .
1. PDE
2. EoS

Section 2 Needs 2014-2017

Governing equations Equation(s) of State

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 10 / 23 :

SLIDE 25

1. Context
2. Past
3. 2018. . .
1. PDE
2. EoS

Needs 2014-2017

2011 1D monophasic SG Exact steady state solution (any EoS) 2012 1D biphasic SG with phase transition Exact transient solution (SG EoS) 2013 2D (FreeFem++) Coupling with a compressible model 2014 3D (FreeFem++) Tabulated EoS 2015 Thermodynamic pressure variable 2016 Thermal conductivity Asymptotic coupling with a simplified neutronic model 2017 3D (NS with variable density and divergence constrained) Tabulated EoS with exact saturation

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 11 / 23 :

SLIDE 26

1. Context
2. Past
3. 2018. . .
1. PDE
2. EoS

Needs 2014-2017

2011 1D monophasic SG Exact steady state solution (any EoS) 2012 1D biphasic SG with phase transition Exact transient solution (SG EoS) 2013 2D (FreeFem++) Coupling with a compressible model 2014 3D (FreeFem++) Tabulated EoS 2015 Thermodynamic pressure variable 2016 Thermal conductivity Asymptotic coupling with a simplified neutronic model 2017 3D (NS with variable density and divergence constrained) Tabulated EoS with exact saturation

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 11 / 23 :

SLIDE 27

1. Context
2. Past
3. 2018. . .
1. PDE
2. EoS

Needs 2014-2017

2011 1D monophasic SG Exact steady state solution (any EoS) 2012 1D biphasic SG with phase transition Exact transient solution (SG EoS) 2013 2D (FreeFem++) Coupling with a compressible model 2014 3D (FreeFem++) Tabulated EoS 2015 Thermodynamic pressure variable 2016 Thermal conductivity Asymptotic coupling with a simplified neutronic model 2017 3D (NS with variable density and divergence constrained) Tabulated EoS with exact saturation

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 11 / 23 :

SLIDE 28

1. Context
2. Past
3. 2018. . .
1. PDE
2. EoS

Needs 2014-2017

2011 1D monophasic SG Exact steady state solution (any EoS) 2012 1D biphasic SG with phase transition Exact transient solution (SG EoS) 2013 2D (FreeFem++) Coupling with a compressible model 2014 3D (FreeFem++) Tabulated EoS 2015 Thermodynamic pressure variable 2016 Thermal conductivity Asymptotic coupling with a simplified neutronic model 2017 3D (NS with variable density and divergence constrained) Tabulated EoS with exact saturation

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 11 / 23 :

SLIDE 29

1. Context
2. Past
3. 2018. . .
1. PDE
2. EoS

Needs 2014-2017

2011 1D monophasic SG Exact steady state solution (any EoS) 2012 1D biphasic SG with phase transition Exact transient solution (SG EoS) 2013 2D (FreeFem++) Coupling with a compressible model 2014 3D (FreeFem++) Tabulated EoS 2015 Thermodynamic pressure variable 2016 Thermal conductivity Asymptotic coupling with a simplified neutronic model 2017 3D (NS with variable density and divergence constrained) Tabulated EoS with exact saturation

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 11 / 23 :

SLIDE 30

1. Context
2. Past
3. 2018. . .
1. PDE
2. EoS

Needs 2014-2017

2011 1D monophasic SG Exact steady state solution (any EoS) 2012 1D biphasic SG with phase transition Exact transient solution (SG EoS) 2013 2D (FreeFem++) Coupling with a compressible model 2014 3D (FreeFem++) Tabulated EoS 2015 Thermodynamic pressure variable 2016 Thermal conductivity Asymptotic coupling with a simplified neutronic model 2017 3D (NS with variable density and divergence constrained) Tabulated EoS with exact saturation

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 11 / 23 :

SLIDE 31

1. Context
2. Past
3. 2018. . .
1. PDE
2. EoS

Needs 2014-2017

2011 1D monophasic SG Exact steady state solution (any EoS) 2012 1D biphasic SG with phase transition Exact transient solution (SG EoS) 2013 2D (FreeFem++) Coupling with a compressible model 2014 3D (FreeFem++) Tabulated EoS 2015 Thermodynamic pressure variable 2016 Thermal conductivity Asymptotic coupling with a simplified neutronic model 2017 3D (NS with variable density and divergence constrained) Tabulated EoS with exact saturation

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 11 / 23 :

SLIDE 32

1. Context
2. Past
3. 2018. . .

2.1. PDE

2. EoS

An Asymptotic Low Mach Model

p(t, x) = p0(t) + ¯ p(t, x) with ¯ p(t, x) p(t, x) = O(M2)            div u = − p′

0(t)

̺(h, p0)(c∗(h, p0))2 + β(h, p0) p0(t) [Φ + div(λ(h, p0)∇T(h, p0))] ̺(h, p0)

∂th + u · ∇h
= Φ + p′

0(t) + div(λ(h, p0)∇T(h, p0))

̺(h, p0)

∂tu + (u · ∇)u
+ ∇¯

p = div(σ(u)) + ̺(h, p0)g Unknowns Given quantities Equation Of State Boundary conditions

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 12 / 23 :

SLIDE 33

1. Context
2. Past
3. 2018. . .

2.1. PDE

2. EoS

An Asymptotic Low Mach Model

p(t, x) = p0(t) + ¯ p(t, x) with ¯ p(t, x) p(t, x) = O(M2)            div u = − p′

0(t)

̺(h, p0)(c∗(h, p0))2 + β(h, p0) p0(t) [Φ + div(λ(h, p0)∇T(h, p0))] ̺(h, p0)

∂th + u · ∇h
= Φ + p′

0(t) + div(λ(h, p0)∇T(h, p0))

̺(h, p0)

∂tu + (u · ∇)u
+ ∇¯

p = div(σ(u)) + ̺(h, p0)g Unknowns

(t, x) → u velocity (t, x) → h enthalpy (t, x) → ¯ p dynamic pressure

Given quantities Equation Of State Boundary conditions

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 12 / 23 :

SLIDE 34

1. Context
2. Past
3. 2018. . .

2.1. PDE

2. EoS

An Asymptotic Low Mach Model

p(t, x) = p0(t) + ¯ p(t, x) with ¯ p(t, x) p(t, x) = O(M2)            div u = − p′

0(t)

̺(h, p0)(c∗(h, p0))2 + β(h, p0) p0(t) [Φ + div(λ(h, p0)∇T(h, p0))] ̺(h, p0)

∂th + u · ∇h
= Φ + p′

0(t) + div(λ(h, p0)∇T(h, p0))

̺(h, p0)

∂tu + (u · ∇)u
+ ∇¯

p = div(σ(u)) + ̺(h, p0)g Unknowns Given quantities

(t, x) → Φ ≥ 0 power density g gravity t → p0 thermodynamic pressure (constant) λ thermal conductivity

Equation Of State Boundary conditions

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 12 / 23 :

SLIDE 35

1. Context
2. Past
3. 2018. . .

2.1. PDE

2. EoS

An Asymptotic Low Mach Model

p(t, x) = p0(t) + ¯ p(t, x) with ¯ p(t, x) p(t, x) = O(M2)            div u = − p′

0(t)

̺(h, p0)(c∗(h, p0))2 + β(h, p0) p0(t) [Φ + div(λ(h, p0)∇T(h, p0))] ̺(h, p0)

∂th + u · ∇h
= Φ + p′

0(t) + div(λ(h, p0)∇T(h, p0))

̺(h, p0)

∂tu + (u · ∇)u
+ ∇¯

p = div(σ(u)) + ̺(h, p0)g Unknowns Given quantities Equation Of State : (h, p0) → ̺ density = ⇒        (h, p0) → β

def

= −

p0 ̺2(h,p0) ∂̺ ∂h

p0

compressibility coefficient (h, p0) → T temperature (h, p0) → c∗ speed of sound Boundary conditions

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 12 / 23 :

SLIDE 36

1. Context
2. Past
3. 2018. . .

2.1. PDE

2. EoS

An Asymptotic Low Mach Model

p(t, x) = p0(t) + ¯ p(t, x) with ¯ p(t, x) p(t, x) = O(M2)            div u = − p′

0(t)

̺(h, p0)(c∗(h, p0))2 + β(h, p0) p0(t) [Φ + div(λ(h, p0)∇T(h, p0))] ̺(h, p0)

∂th + u · ∇h
= Φ + p′

0(t) + div(λ(h, p0)∇T(h, p0))

̺(h, p0)

∂tu + (u · ∇)u
+ ∇¯

p = div(σ(u)) + ̺(h, p0)g Unknowns Given quantities Equation Of State Boundary conditions

Free outflow + Adiabatic

σ(u)n − ¯

pn = 0 λ∇T · n = 0 Free-slip + Adiabatic      u · n = 0 (σ(u)n) · τ = 0 λ∇T · n = 0 Inflow

h = he(t, x),

̺u = (0, De(t, x))

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 12 / 23 :

SLIDE 37

1. Context
2. Past
3. 2018. . .
1. PDE

2.2. EoS

Diphasic EOS

Liquid κ = ℓ and vapour κ = g are characterized by their thermodynamic properties : (h, p0) → ̺κ In the mixture, full equilibrium between liquid and vapour phases : T = T s(p0) and we define values at saturation : hs

κ(p0)

def

= hκ

p0, T s(p0)
,

̺s

κ(p0)

def

= ̺κ

p0, T s(p0)
= ̺κ
hs

κ, p0

.

̺(h, p0) =      ̺ℓ(h, p0), if h ≤ hs

ℓ(p0),

̺m(h, p0) if hs

ℓ(p0) < h < hs g(p0),

̺g(h, p0), if h ≥ hs

g(p0),

h ̺ ̺ℓ(h, p0) ̺g(h, p0) hs

ℓ(p0)

̺s

ℓ(p0)

hs

g(p0)

̺s

g(p0)

̺m(h, p0) LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 13 / 23 :

SLIDE 38

1. Context
2. Past
3. 2018. . .
1. PDE

2.2. EoS

Mixture EoS

̺ = α̺s

g(p0) + (1 − α)̺s ℓ(p0)

̺h = α̺s

g(p0)hs g(p0) + (1 − α)̺s ℓ(p0)hs ℓ(p0)

for h ∈ [hs

ℓ(p0); hs g(p0)]

⇓ ̺m(h, p0) = p0/βm(p0) h − qm(p0)

h ̺ hs

ℓ(p0)

̺s

ℓ(p0)

hs

g(p0)

̺s

g(p0)

̺m(h, p0)

where βm(p0)

def

= p0

1 ̺s

g − 1

̺s

ℓ

hs

g − hs ℓ

= − p0 ̺m(h, p0) ∂̺m ∂h

p0

qm(p0)

def

= ̺s

ghs g − ̺s ℓhs ℓ

̺s

g − ̺s ℓ

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 14 / 23 :

SLIDE 39

1. Context
2. Past
3. 2018. . .
1. PDE

2.2. EoS

Pure phase EoS : Noble Abel Stiffened Gas law

1 ̺κ (h, p0) = γκ − 1 γκ h − qκ p0 + πκ + bκ γκ > 1 adiabatic coefficient πκ reference pressure qκ binding energy bκ covolume ⇓ βκ(p0) = − p0 ̺2

κ(h, p0)

∂̺ ∂h

p0

= γκ − 1 γκ p0 p0 + πκ independent of h ⇓ ̺κ(h, p0) = p0/βκ(p0) h − ˆ qκ(p0), ˆ qκ(p0)

def

= qκ − p0 βκ(p0)bκ

h ̺ ̺ℓ(h, p0) ̺g(h, p0) hs

ℓ(p0)

̺s

ℓ(p0)

hs

g(p0)

̺s

g(p0)

̺m(h, p0) LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 15 / 23 :

SLIDE 40

1. Context
2. Past
3. 2018. . .
1. PDE

2.2. EoS

Pure phase EoS : Tabulated laws at p = p0

κ h [kJ/kg] ̺κ [kg/m3] Tκ [K] c∗

κ [m · s−1]

βκ ℓ 15.608 1007.5 273.16 1427.4 ✗ ℓ 30.678 1007.5 276.79 1445.0 ✗ . . . . . . . . . . . . . . . . . . ℓ 1602.8 609.10 614.77 659.56 ✗ ℓ hs

ℓ

594.38 T s 621.43 ✗ g hs

g

101.93 T s 433.40 ✗ g 2602.6 101.06 618.41 435.61 ✗ . . . . . . . . . . . . . . . . . . g 2.5299 35.139 996.37 747.83 ✗ g 2.5290 34.985 1000.0 749.37 ✗ Source : http://webbook.nist.gov/chemistry/fluid/

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 16 / 23 :

SLIDE 41

1. Context
2. Past
3. 2018. . .
1. PDE

2.2. EoS

Pure phase EoS : Tabulated laws at p = p0

First approach Approximation of h → ̺κ(h) by least squares on {hi, ̺i}i∈Iκ Deduction of of h → βκ(h) = −

p0 ̺2

κ(h)̺′

κ(h)

Second approach Approximation of h → ̺κ(h) by least squares on {hi, ̺i}i∈Iκ Construction of the tabulated values βi by finite differences Approximation of h → βκ(h) by least squares on {hi, βi}i∈Iκ Third approach Construction of the tabulated values βi using relation β =

p c∗√ T

1

cv − 1 cp

Approximation of h → βκ(h) by least squares on {hi, βi}i∈Iκ Deduction of

1 ̺κ (h, p) = 1 ̺s

κ(p) +

h

hs

κ(p)

βκ(h) p

dh h → ̺κ(h) is continuous, positive, strictly decreasing. h → βκ(h) = −

p0 ̺2

κ(h)̺′

κ(h) and saturation exactly holds.

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 16 / 23 :

SLIDE 42

1. Context
2. Past
3. 2018. . .
1. PDE

2.2. EoS

Pure phase EoS : Tabulated laws at p = p0

First approach Approximation of h → ̺κ(h) by least squares on {hi, ̺i}i∈Iκ Deduction of of h → βκ(h) = −

p0 ̺2

κ(h)̺′

κ(h)

Second approach Approximation of h → ̺κ(h) by least squares on {hi, ̺i}i∈Iκ Construction of the tabulated values βi by finite differences Approximation of h → βκ(h) by least squares on {hi, βi}i∈Iκ Third approach Construction of the tabulated values βi using relation β =

p c∗√ T

1

cv − 1 cp

Approximation of h → βκ(h) by least squares on {hi, βi}i∈Iκ Deduction of

1 ̺κ (h, p) = 1 ̺s

κ(p) +

h

hs

κ(p)

βκ(h) p

dh h → ̺κ(h) is continuous, positive, strictly decreasing. h → βκ(h) = −

p0 ̺2

κ(h)̺′

κ(h) and saturation exactly holds.

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 16 / 23 :

SLIDE 43

1. Context
2. Past
3. 2018. . .
1. PDE

2.2. EoS

Pure phase EoS : Tabulated laws at p = p0

First approach Approximation of h → ̺κ(h) by least squares on {hi, ̺i}i∈Iκ Deduction of of h → βκ(h) = −

p0 ̺2

κ(h)̺′

κ(h)

Second approach Approximation of h → ̺κ(h) by least squares on {hi, ̺i}i∈Iκ Construction of the tabulated values βi by finite differences Approximation of h → βκ(h) by least squares on {hi, βi}i∈Iκ Third approach Construction of the tabulated values βi using relation β =

p c∗√ T

1

cv − 1 cp

Approximation of h → βκ(h) by least squares on {hi, βi}i∈Iκ Deduction of

1 ̺κ (h, p) = 1 ̺s

κ(p) +

h

hs

κ(p)

βκ(h) p

dh h → ̺κ(h) is continuous, positive, strictly decreasing. h → βκ(h) = −

p0 ̺2

κ(h)̺′

κ(h) and saturation exactly holds.

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 16 / 23 :

SLIDE 44

1. Context
2. Past
3. 2018. . .
1. PDE

2.2. EoS

Influence of EOS : asymptotic 1d solution

Schematic comparison of the phases repartition in the core for different EoS (exact steady state solution) L

NIST-p NIST-0 SG NASG Liquid Mixture Vapour Legend

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 17 / 23 :

SLIDE 45

1. Context
2. Past
3. 2018. . .
1. 2D
2. 3D
3. Project

Section 3 Needs 2018. . .

2D Numerical tests with FE+convect scheme 3D Numerical tests with pressure correction scheme Outline of the project

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 18 / 23 :

SLIDE 46

1. Context
2. Past
3. 2018. . .

3.1. 2D

2. 3D
3. Project

2D case with phase transition

2D test case with phase transition (appearance of mixture) Power density compactly supported in a disc in the lower part of the domain EoS : NIST-p FreeFem++ with convect With thermal diffusion Enthalpy Mach number

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 19 / 23 :

SLIDE 47

1. Context
2. Past
3. 2018. . .

3.1. 2D

2. 3D
3. Project

(Non) influence of thermal diffusion

Same test case Relative difference between the enthalpy with λ = 0 and λ = 0 at time t = 0.55 s

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 20 / 23 :

SLIDE 48

1. Context
2. Past
3. 2018. . .

3.1. 2D

2. 3D
3. Project

Influence of gravity

Same test case Variation of the orientation of the gravity Rayleigh-Taylor instability / Gravity in competition with convection

(a) g = (0, −g) (b) g = (0, 0) (c) g = (0, g) (d) g = (g, 0) Figure – Enthalpy at time 0.55 s for several orientations of the gravity field

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 21 / 23 :

SLIDE 49

1. Context
2. Past
3. 2018. . .
1. 2D

3.2. 3D

3. Project

3D case with phase transition

3D test case with phase transition (appearance of mixture and pure vapour) Power density compactly supported in a sphere EoS : phase-wise SG Pressure-correction scheme (originally developed for incompressible NS) with divergence coupled to the enthalpy, MAC grid, OpenMP, big ratio of density Enthalpy Volume fraction

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 22 / 23 :

SLIDE 50

1. Context
2. Past
3. 2018. . .
1. 2D
2. 3D

3.3. Project

Outline of the project

Derivation of a new hierarchy of models (compressible and asymptotic) Enrichment of the modelling by taking into account neutronics Development of numerical methods (compressible, all Mach number, asymptotic) Accurate equations of state (closure for compressible and asymptotic models) Coupling of codes Extension to other reactors

LMNC projects (2014-2017) Thermohydraulic: LOFA, LOCA, vapor explosion 23 / 23 :