RELAP5 Water Hammer Benchmarking via a Theta-Implicit Finite - - PowerPoint PPT Presentation

RELAP5 Water Hammer Benchmarking via a Theta-Implicit Finite Element Algorithm

Stuart Walker UT CFD Lab Colloquium May 18, 2010

NE 697: Analysis of RELAP5

RELAP5

• Analyze slow thermal hydraulic transients in 1D 2-Phase

systems, i.e. Nuclear Power plant

• Area Averaged 6-Equation Model
• Best-estimate code used for NRC licensing of new power plant

design and current power up-rates

• NOT validated for many accident based transients

Course Topics

• RELAP5 heat transfer in a single HFIR channel with HEU and

LEU power profiles using a dense nodalization

• Water hammer benchmarking via a FE algorithm

{ }

T g f g f

h h v v P q , , , , , α = 

S x q B t q A    = ∂ ∂ ⋅ + ∂ ∂ ⋅ ~ ~

RELAP5 Semi-Implicit Numerics

• Pressure and velocities are calculated implicitly
• Energy fluxes are evaluated explicitly from donor-cell

enthalpies

• Interfacial processes are evaluated implicitly
• Advection Discretization Example:

1st Order Accurate

• Transient time-scales >> Acoustic time-scales

1 1 n n n n i i i i

T T T T T T v v t x t x ∆ ∆

+ −

− − ∂ ∂ + = → + = ∂ ∂

Water Hammer: A benchmarking

• pportunity

RELAP5: Built to analyze transients on time-scales characteristic

• f the convective velocity
• Water hammer phenomena act on time-scales shorter than that

associated with the propagation of acoustic energy Benchmarking literature compares 2nd Order Accurate Upwind schemes with OS (WAHA) to RELAP5 1st Order Accurate with OS FE: GWS+θTS allows a full suite of 2nd Order accurate schemes to be examined

Water Hammer Introduction

• Water hammer phenomena characterize a rapid

transfer from kinetic to potential energy in a closed fluid system

• Pressure surges in piping systems can lead to

component failure and liquid flashing

• Abrupt Valve Closure: 1D Liquid Solid System

Water Hammer Theory

• 1D Adiabatic Liquid-Solid Mass and Momentum

2

= ∂ ∂ ⋅ + ∂ ∂ x v c t P ρ

(Re) 1 = ⋅ ⋅ + ∂ ∂ + ∂ ∂ v v f x P t v ρ

{ }

T

v P q , = 

S x q B t q A    = ∂ ∂ ⋅ + ∂ ∂ ⋅ ~ ~

( )

      = v v f S Re 

        ⋅ = 1 ~

2

ρ ρ c B

I A ~ ~ =

⇔

Weak Formulation with Newton Time-Stepping

• 1D Adiabatic Inviscid Water-Solid Pipe
• MatLab implementation via FEMLIB toolbox

S x q B t q A    = ∂ ∂ ⋅ + ∂ ∂ ⋅ ~ ~

1

=         ⋅ ⋅ ∂ Ψ ∂ ⋅ Ψ ⋅ + ∂ ∂ ⋅ ⋅ Ψ ⋅ Ψ =∑ ∫

∫

= Ω Ω

e e e

N e T e e e e T e e N

Q d x B t Q d GWS

µ

τ τ

{ }

T

v P q , = 

        ⋅ = 1 ~

2

ρ ρ c B

I A ~ ~ =

  = S

⇔

( ) ( )

( )

( )

θ

θ θ

f n n n n

t O Q Q t Q Q ∆ + ′ ⋅ − + ′ ⋅ ⋅ ∆ + =

+ +

1

1 1

⇔

s m c 1500 ≈

3

1000 m kg ≈ ρ

[ ] [ ]

201 ~ 200 = ⋅ ⋅ + ∂ ∂ ⋅ =

A A A A N A

Q L A B t Q L A GWS

RELAP5 and FE Geometry

Transient Results: Open Valve

• RELAP5 solution exhibits

2Δx oscillations

• GWS+θTS produces

monotone solution for smooth ICs

,

= ∂ ∂

= L x

x P

,

= ∂ ∂

= L x

x P

⇔

( ) t

s m v t L P t v ⋅ = + ⋅ ⋅ ∆ =

2

05 . ) ( ρ

Pa P

5

10 1 . × = ∆

= ∂ ∂

=VALVE x

x v

( )

) ( ,

0 x

P x P = ( )

) ( , = = x v x v

50 100 150 200 1 2 3 4 5 x 10

• 3

Axial Distance (m) Velocity (m/s)

RELAP vs. FE: Open Pipe

t=0s t=0.05s (RELAP5) t=0.10s (RELAP) t=0.05s (FE) t=0.10s (FE)

375 . = ∆ ∆ ⋅ x t c

Transient Results: Abrupt Valve Closure

• Both algorithms fail to

generate monotone solutions for step IC

• RELAP5 solution is some

what diffusive

= ∂ ∂

=VALVE x

x v

⇔

) ( =

=valve x

t v

( )

valve

• pen

s t

t x q IC

− =

=

2

,

c v P ⋅ ⋅ = ∆ ρ

max

50 100 150 200

• 0.05

0.05 0.1

Axial Distance (m) Velocity (m/s) RELAP vs FE: Closed Pipe, Velocity Profile

t=0s t=0.03s (RELAP5) t=0.06s (RELAP) t=0.03s (FE) t=0.06s (FE)

50 100 150 200 1 2 3 4 5x 10

5

Axial Distance (m) Pressure (Pa) RELAP vs. FE: Closed Pipe, Pressure Profile

t=0s t=0.03s (RELAP5) t=0.06s (RELAP) t=0.03s (FE) t=0.06s (FE)

375 . = ∆ ∆ ⋅ x t c

Full GWS+θTS Implementation of a 6-Equation Model

• Quadratic form generates hyper-matrix structure

i.e. [MASS] <=> [A500000]

• Time-stepping scheme remains unchanged
• Semi-implicit time-stepping using OS method remains an option

{ }

T g f g f

h h v v P q , , , , , α = 

( ) ( ) ( ) 1

~ ~ ~       ⋅ = ∂ ∂ ⋅ + ∂ ∂ ⋅ q S x q q B t q q A

( ) ( ) ( )

1 ~ ~ ~

1

=         ⋅ ⋅ Ψ − ⋅ ∂ Ψ ∂ ⋅ ⋅ Ψ + ∂ ∂ ⋅ Ψ ⋅ ⋅ Ψ =∑ ∫

∫ ∫

= Ω Ω Ω

e e e e

N e e T e e e T e e N

d q S Q d x q B t Q d q A GWS

µ

τ τ τ           

( )

T T

q q A q A A q A      ⋅ + ⋅ + =

3 2 1

~ ~ ~

( )

T T

q q B q B B q B      ⋅ + ⋅ + =

3 2 1

~ ~ ~

( )

T T

q q S q S S q S      ⋅ + ⋅ + =

3 2 1

~ ~ ~

Conclusion

• Inter-phase exchange source terms lead to

characteristic time scales which raise questions concerning the efficacy of legacy best-estimate codes like RELAP5

• Water hammer phenomena remains a benchmarking
• pportunity currently being examined using 2nd

Order schemes with OS (WAHA)

• The GWS+θTS solution process is exhibited with

attention to the 6-Equation Model

• A 1D adiabatic inviscid water solid benchmark is

presented with closed form solutions

Future Work

• Implementation of a 6-Equation model via a

GWS+θTS would provide a numeric environment to study the relaxation source terms associated with accident based transients

• Shift from RELAP5

1. Active acoustic void measurements using a cross-correlation technique (ANS-2010) 2. Mechanistic modeling of swirling jet micro-bubblers (FEDSM- ICNMM2010-30534) 3. Validation of commercial and open-source CFD turbulence modeling using time-resolved 3D PET scans of a scale 4-rod fuel bundle

References

• Tiselj, I., Horvat, A. Accuracy of the Operator Splitting

Technique for Two-Phase Flow with Stiff Source Terms. Proceedings of ASME FEDSM. 2002.

• Shieh, A., Ransom, V., Krishnamurthy, R. RELAP5/MOD3 Code

Manual Vol. 3: Validation of Numerical Techniques in RELAP5/MOD3. Information Systems Laboratories. 1994.

• Moody, F. Introduction to Unsteady Thermo-fluid Mechanics.

John Wiley & Sons. 1990.

• Baker, A.J. The Computational Engineering Sciences. J-

Computek Press, Loudon, TN. 2006. ISBN 0-9790459-0-8.