Contents of the presentation

(1)

The porous medium model PORFLO for 3D two-phase flow and its application to BWR fuel bundle simulations

Mikko Ilvonen and Ville Hovi

(2)

VTT TECHNICAL RESEARCH CENTRE OF FINLAND

2

Contents of the presentation

• The team: Thermal-hydraulic calculation methods

• Why porous medium modeling ? A brief overview

• The PORFLO code: history, applications, main principles

• Comparisons of 5-equation and 6-equation approach

• The BFBT benchmark: specification, results, comparison

• The set of closure relations; lift force development

• Near future plans:

• Comparison with CATHARE-3

• Horizontal steam generator, PSBT ?

• TRICOT plan of 2009

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The team

• Nuclear Energy

• Thermal-hydraulic calculation methods

• Persons currently involved in PORFLO work:

• Mikko Ilvonen, team leader

• Ville Hovi

• MSc (Tech) thesis on PORFLO, 2008

• Development of the code

• View into the past:

• Jaakko Miettinen (deceased in August 2008)

• Original author of PORFLO code

• There is more work to be done than working force…

(4)

4

Introduction: Why porous medium ?

• Three-dimensional simulation of two- phase flow in complex geometries is a very challenging task, to which no

completely satisfactory solution has been found.

• The current modelling tools are always a trade-off between CPU-time and

spatial resolution.

• Porosity models attempt to capture the best of both: the speed, and the ability to model complex 3D geometries.

• Objectives of PORFLO: independent code, easier to set up & faster to compute than typical CFD simulation

1D System analysis codes

3D Porosity models

3D CFD CPU-time

Spatial resolution of results

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A brief overview of porosity modeling

• As opposed to common practice in CFD, in which the grid is usually body-fitted, in porosity models the domain can be split regardless of the interfaces between solid and fluid.

• The fraction of the total volume occupied by the two-phase fluid is defined as porosity:

36.4 mm

12.3 mm 16.2 mm

total fluid V

=V

ε

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6

PORFLO: history, applications, current state

• Original idea and coding by Jaakko Miettinen

• The first applications:

• Particle bed dry-out investigations (fuel debris particles)

• Two-phase flow of isolation condenser secondary side

• Current main emphasis:

• BFBT benchmark (BWR Full-size fine-mesh bundle tests)

• Appr. 50 000 lines of f77 & f90; most written by Ville Hovi

• Potential future applications:

• Horizontal steam generator

• PSBT benchmark (PWR fuel bundle)

• PWR open reactor core flow simulations

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PORFLO: 3D porous medium model

• Proper 3D solution of pressure and flow field

• Cartesian grid, staggered arrangement of variables

• Porous medium approach: ε = V

_fluid

/ V

_total

1. Solution based on the five-equation model:

¾

Direct method

Mixture continuity and momentum equations are combined and solved for pressure

¾

Iterative methods:

SIMPLE, SIMPLEC and SIMPLER algorithms 2. Solution based on the six-equation model:

¾

Phase-Coupled SIMPLE algorithm

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8

Conservation equations for the five-equation model

• Mass conservation equations

• steam volume fraction

• vaporization (kg/m³s)

• Mixture momentum equation

• viscous forces

• gravity term

• body forces (friction, …)

• Enthalpy equations

⋅T

∇

( )

[ ] [₍ ₎ ]

( )

^αρ

(

^αρ

)

^γ

γ ρ

ρ α α

+

=

⋅

∇

∂ +

∂

−

=

−

⋅

∇

∂ +

−

∂

g g g

l l l

: vapor

1 1 :

liquid

u

t t

( û )⁺^∇^⋅[( û )^⊗û ]⁼^−∇ ⁺^∇^⋅^T⁺ ^g⁺^F

∂

m m

m m m

m ρ ρ

ρ p

t

( )

[ ] [₍ ₎ ]

( ) (

g g g

)

wg lg

g g

lg wl l

l l l

l

: vapor

1 1 :

liquid

q q t h

h

q q t h

h

+ ′′′

= ′′′

⋅

∇

∂ +

∂

− ′′′

= ′′′

−

⋅

∇

∂ +

−

∂

u

αρ αρ

ρ ρ α

α

mg ρ

⋅T

∇ F α γ

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Conservation equations for the six-equation model

• Mass conservation equations

• Momentum equations liquid:

vapor:

• Enthalpy equations

( )

[ ] [₍ ₎ ]

( )

αρ

(

αρ

)

γ

γ ρ

ρ α α

+

=

⋅

∇

∂ +

∂

−

=

−

⋅

∇

∂ +

−

∂

g g g

l l l

: vapor

1 1 :

liquid

u

t t

( )

[ û ]⁺^∇^⋅{[₍ ⁻ ₎ û ]^⊗û } (⁼⁻ ⁻ )^∇ ⁺^∇^⋅^T⁺( ⁻ ) ^g⁺^F ⁺^F ⁺^F

∂

−

∂

IF PC l

l l l l

l 1 1 1

1 α ρ α ρ α α ρ

t p

(

^u

)

⁺^∇^⋅

[ (

^u

)

^⊗^u

]

⁼⁻ ^∇ ⁺^∇^⋅^T⁺ ^g⁻^F ⁻^F ⁺^F

∂

IF PC g

g g g g

g αρ α αρ

αρ p

t

( )

[ ] [₍ ₎ ]

( ) (

g g g

)

wg lg

g g

lg wl l

l l l

l

: vapor

1 1 :

liquid

q q t h

h

q q t h

h

+ ′′′

= ′′′

⋅

∇

∂ +

∂

− ′′′

= ′′′

−

⋅

∇

∂ +

−

∂

u

αρ αρ

ρ ρ α

α

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Case specific input &

initialization

Particle bed BWR bundle

Isolation condenser

Steam generator

3D core

Common input, initialization & restart

Advance time step Calculate interfacial heat transfer

Calculate structure heat transfer

Solve pressure and volumetric flow distributions

Direct method:

- Combined mass & momentum

Iterative methods:

- SIMPLE, SIMPLEC & SIMPLER Phase separation by drift-flux model

Void fraction prediction Integrate liquid & vapour masses

Solve enthalpy equations for liquid & vapour

Calculate mixture densities and void fractions from liquid & vapour masses

Yes START

No STOP

New time step

All the steps of the solution can be performed inside the

iteration loop of the iterative methods

5-equation

model: flow of

computation

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VTT TECHNICAL RESEARCH CENTRE OF FINLANDCase specific input &

initialization

Particle bed BWR bundle

Isolation condenser

Steam generator

3D core

Common input, initialization & restart

Advance time step Calculate interfacial heat transfer

Calculate structure heat transfer START

Solve pressure and volumetric flow distributions using the Phase-Coupled SIMPLE algorithm:

Step 1: Solve the momentum equations for vapor & liquid Step 2: Solve the pressure correction equations

Step 3: Correct pressure and velocity fields

Step 4: Calculate a prediction for the void fraction at the end of the time step (-Solve void fractions implicitly from vapor mass equations)

Step 5: Calculate the mass flow rates explicitly

Converged?

No

Integrate liquid & vapour masses Solve enthalpy equations for liquid & vapour Calculate mixture densities and void fractions

from liquid & vapour masses

6-equation

model: flow of computation

Pressure and

velocity by PC

SIMPLE (Phase-

Coupled SIMPLE)

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New linear iterative solvers and preconditioning methods

• Several Krylov subspace methods have been tried:

• SSOR (Symmetric Successive Over-Relaxation) preconditioned Conjugate Gradient method (SSORPCG) was developed for symmetric systems of equations, such as pressure correction equations.

¾The overall computation time was reduced by 86%

• SSOR preconditioned BiCGSTAB method was developed for asymmetric systems of equations, such as momentum equations.

¾Only a slight reduction was achieved

• Preconditioning based on the Incomplete Cholesky Factorization using various degrees of fill-in was tested with the Conjugate Gradient method.

¾A slight reduction in overall computation time compared to SSORPCG

¾Not as robust as SSORPCG

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Some preliminary comparisons: 5-eq. vs. 6-eq. model

36.4 mm

12.3 mm 16.2 mm

Parameter Value Unit

System pressure

(pressure at outlet) 60 bar

Mass flux at inlet 1500 kg/m²s

Inlet enthalpy 1100 kJ/kg

Inlet density 778,7 kg/m³

Heating power Linearly increased to 220 kW

Time step 0.001 s

Friction factor for

horizontal flow 0.01 -

Friction factor for

vertical flow 0.01 -

• Simulations of a 2x2 test bundle, length 3.6 m (note: no corresponding experiment!)

• Grid: (18 x 18 x 36) = 11664 cells

• Boundary conditions:

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Maximum void fraction as a function of time

• Note: Location of the maximum shown here is changing dynamically

¾Local variations are probably smaller

Maximum Void Fraction (5-eq)

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

0 1 2 3 4 5

Time [s]

Void Fraction

0 50 100 150 200 250

Power [kW]

Maximum Void Fraction (6-eq)

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

0 1 2 3 4 5 6

Time [s]

Void Fraction

0 50 100 150 200 250

Power [kW]

---Power ---Void fraction

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Maximum velocities as a function of time

---Power ---Velocity

Maximum Mixture Velocity (5-eq)

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5

Time [s]

Velocity [m/s]

0 50 100 150 200 250

Power [kW]

Maximum Vapor Velocity (6-eq)

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6

Time [s]

Velocity [m/s]

0 50 100 150 200 250

Power [kW]

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Void fraction at bundle outlet height

• First steps were taken with the 6-eq. model; no specific transport of void fraction normal to the main flow direction:

¾Steam remains near the heated surface, where it was produced.

• No turbulence model in either case!

• Note that in the lateral direction, the drift flux model effectively evens the void fraction distribution like a diffusion process.

6-equation model 5-eq. model + drift flux

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Void fraction at bundle outlet height: comparison in 2D

• The bundle is 3.5 cm x 3.5 cm in cross-section

• Note: The cells inside the ’big blue squares’ are fuel-rod-only cells

• In the 6-eq. case, a column of water has remained in the middle subchannel

• Even in the 5-eq. case, most steam volume is seen right between the rods

• The distributions are not completely symmetrical, contrary to expectations

• 7 days of CPU time (note: before major development of solver) 6-equation model 5-eq. model + drift flux

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BFBT benchmark: equipment & simulation parameters

Parameter Value Unit

System pressure

(pressure at outlet) 60 bar

Mass flux at inlet 1500 kg/m²s Mass flow rate at inlet 14.2 kg/s

Inlet enthalpy 1100 kJ/kg

Inlet density 778.7 kg/m³

Heating power 3.52 MW

Time step 0.003 s

Friction factor for

horizontal flow 0.01 -

Friction factor for

vertical flow 0.01 -

Simulation parameters (note: not exactly as experiment in the case shown here):

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BFBT benchmark: power distribution and the simulation

Radial power distribution (axial: cosine):

• NUPEC (Nuclear Power Engineering

Corporation) BFBT (BWR Full-size Fine-mesh Bundle Tests) benchmark steady-state exercises were performed on a BWR (8 x 8) fuel bundle using the five-equation SIMPLE algorithm of PORFLO, as a transient simulation.

• Nodalization: (83 x 83 x 36) => 250 000 nodes

• The steady-state results were obtained at the end of a 9.5 s transient.

• The computation time spent on a single core of a 2.992 GHz quad-core processor was 25 days.

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BFBT results: temporal trends (1)

Maximum Mixture Velocity

0 2 4 6 8 10 12

0 2 4 6 8

Time [s]

Velocity [m/s]

0 500 1000 1500 2000 2500 3000 3500 4000

Power [kW]

Maximum Void Fraction

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

0 1 2 3 4 5 6 7 8 9

Time [s]

Void Fraction

0 500 1000 1500 2000 2500 3000 3500 4000

Power [kW]

Total mass contents

15 17 19 21 23 25 27 29

0 1 2 3 4 5 6 7 8 9

Time [s]

Liquid Mass [kg]

-0,05 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5

Vapour Mass [kg]

Maximum void fraction [-]

Total void fraction [-]

Total mass [kg]

Maximum mixture velocity [m/s]

Total vapor fraction

0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45

0 1 2 3 4 5 6 7 8 9

Time [s]

Vapor Fraction [-]

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BFBT results: temporal trends (2)

Maximum Pressure

0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45

0 2 4 6 8

Time [s]

Pressure [bar]

0 500 1000 1500 2000 2500 3000 3500 4000

Power [kW]

Minimum Pressure

0 0,001 0,002 0,003 0,004 0,005 0,006

0 2 4 6 8

Time [s]

Pressure [bar]

0 500 1000 1500 2000 2500 3000 3500 4000

Power [kW]

Wall Heat Flux

0 500 1000 1500 2000 2500 3000 3500 4000

0 1 2 3 4 5 6 7 8 9

Heat Flux

0 500 1000 1500 2000 2500 3000 3500 4000

Power [kW]

Minimum pressure [bar]

Maximum cladding temperature [C]

Wall heat flux [kW]

Maximum pressure [bar]

Maximum Cladding Temperature

240 260 280 300 320 340 360 380 400

0 1 2 3 4 5 6 7 8 9

Temperature [C]

0 500 1000 1500 2000 2500 3000 3500 4000

Power [kW]

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BFBT results: variables at bundle outlet (1)

Vertical slip velocity [m/s]

Vapor

velocity [m/s]

Liquid velocity [m/s]

Mixture velocity [m/s]

(23)

BFBT results: variables at bundle outlet (2)

Temperature [C]

[kg/m³s]

Evaporation rate

[kg/m³s]

Void fraction [-]

fg wl/h q ′′′

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State of closure law development in PORFLO

• Momentum equation

• Wall friction

• Pressure loss (when necessary)

• Interphase friction

• Lift force

• Virtual mass force

• Drift-flux model (with 5-eq. model)

• Turbulence model

• Heat transfer & phase change

• Wall heat transfer / distribution on phases, heat-up / boiling

• Interphase heat transfer

• Conduction (in rods)

• Radiation

• Phase distribution / interface information

• Flow regime

• Bubble variables (nucleation density, detachment diameter, …)

• Droplet variables (entrainment, diameter, deposition, …)

• CHF correlations

• Red = not done yet

• Others: mostly very simple & preliminary

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Lift force

• 1 = liquid

• 2 = vapor

• v₁₂ = velocity difference between liquid and vapor, v₁ – v₂

• C_L = Lift coefficient

( ) ( )

( )

⎪⎭

⎪⎬

⎥ ⎫

⎦

⎢ ⎤

⎣

⎡ ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛

∂

− ∂

∂

− ∂

⎟⎠

⎜ ⎞

⎝

⎛

∂

− ∂

∂ + ∂

⎥⎦

⎢ ⎤

⎣

⎡ ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

∂

− ∂

∂

− ∂

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

∂

− ∂

∂ + ∂

⎪⎩

⎪⎨

⎧ ⎥

⎦

⎢ ⎤

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝

⎛

∂

−∂

∂

− ∂

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

∂

− ∂

∂

= ∂

⇔

×

∇

×

=

⇔

×

∇

×

−

=

z k v y

v w x

w z

u u

y j u x u v

z v y

w w

x i w z

w u y

u x v v

C C C

1 1

12 1

1 12

1 1

12 1

1 12

1 1

12 1

1 12 1

L lift,2

1 12

1 L lift,2

1 2

1 1 L lift,2

α ρ

α ρ α ρ

F

v v

F

v v

v F

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26

Lift force experimentation (1): no lift force

Void fraction [-]

Liquid

velocity [m/s]

Vapor velocity [m/s]

(27)

Lift force experimentation (2): with lift force

Void fraction [-]

Liquid

velocity [m/s]

Vapor velocity [m/s]

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PORFLO development plans in TRICOT in 2009

• The source terms (forces) of the momentum equations of the 6-equation model developed in 2008 will be extended & tailored for the simulation of various

applications with their proper physics.

• The remarkable solver development made in 2008 will be continued to improve probability and speed of convergence even further.

• Coarse-level (domain splitting) and fine level parallelization of the code will be studied by using MPI library on the new Linux cluster.

• A post-processing tool tailored for PORFLO will be developed in order to facilitate visualization and thus speed up the development of the code, particularly application specific models.

• A basic turbulence model (k-ε) will be added in the code.

• With VTT funding only: The secondary side of the steam generator of a VVER type NPP will be simulated using PORFLO

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Planned code comparisons

• Objective: Facilitate PORFLO development by comparing BFBT simulations by PORFLO & CATHARE-3

• CATHARE-3 has many features that PORFLO does not yet have

• It may be possible to learn from C-3 which models are appropriate and should be developed further in PORFLO

• EU project ’NURISP’: BFBT simulations by 3D, 3 field module of CATHARE-3

• CATHARE-3 is developed by CEA Grenoble (France)

• Cartesian staggered grid, porosity, droplet field

• Predict void fraction distribution & CHF

• A mechanistic approach to CHF by C-3: continuous water phase on rod surfaces disappears, wall surface temperatures will be elevated

• Additionally, a new PWR oriented benchmark exercise is starting: PSBT

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