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 { } ∂ ∂ ~ q q ~ = α ⋅ + ⋅ = T q , P , v , v , h , h A B S f g f g ∂ ∂ t x • 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
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: + ∂ ∂ − − n 1 n n n T T T T T T + = → + = − i i i i 1 v 0 v 0 ∂ ∂ ∆ ∆ t x t x 1 st Order Accurate • Transient time-scales >> Acoustic time-scales
Water Hammer: A benchmarking opportunity RELAP5: Built to analyze transients on time-scales characteristic of the convective velocity • Water hammer phenomena act on time-scales shorter than that associated with the propagation of acoustic energy Benchmarking literature compares 2 nd Order Accurate Upwind schemes with OS (WAHA) to RELAP5 1 st Order Accurate with OS FE: GWS+ θ TS allows a full suite of 2 nd 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 ∂ ∂ P v + ρ ⋅ = 2 c 0 ⇔ ∂ ∂ ∂ ∂ ~ q q ~ t x ⋅ + ⋅ = A B S ∂ ∂ ∂ ∂ v 1 P t x + + ⋅ ⋅ = f (Re) v v 0 ∂ ρ ∂ t x ⋅ ρ 2 0 0 c ~ = ~ = { } ~ = = S ( ) T B q P , v A I 1 0 f Re v v ρ
Weak Formulation with Newton Time-Stepping • 1D Adiabatic Inviscid Water-Solid Pipe { } = = T q P , v S 0 ∂ ∂ ρ ≈ 3 ~ q ~ q 1000 kg m ⋅ + ⋅ = A B S ⋅ ρ ~ = 2 ≈ ∂ ∂ 0 c ~ c 1500 m s ~ t x = A I B 1 0 ρ ⇔ ∂ ∂ Ψ N T Q = ∑ ∫ e ∫ Ψ ⋅ Ψ ⋅ τ ⋅ + ⋅ Ψ ⋅ ⋅ τ ⋅ = N T e e GWS d B d Q 0 e ∂ ∂ e e e e t x µ = Ω Ω 1 e e ∂ [ ] [ ] Q ~ = ⋅ + ⋅ ⋅ = N GWS A 200 L B A 201 L Q 0 ∂ A A A A A t ⇔ ( ) ( ( ) ) ( ) ′ ′ θ = + ∆ ⋅ θ ⋅ + − θ ⋅ + ∆ f 1 Q Q t Q Q O t + + n 1 n n 1 n • MatLab implementation via FEMLIB toolbox
RELAP5 and FE Geometry
∂ ∂ P v = = 0 0 ∂ ∂ x x = = VALVE x 0 , L x Transient Results: Open Valve ∂ P ∆ ( ) t = P 0 = ⋅ + = ⋅ 2 ∂ ⇔ v ( t ) t v 0 . 05 m s x ρ ⋅ 0 = L x 0 , L ∆ ( ) t = ⋅ = P x , 0 P 0 x ( ) ∆ = × 5 c 0 . 375 P 0 . 1 10 Pa ∆ ( ) x = = , 0 ( ) 0 v x v x 0 RELAP vs. FE: Open Pipe -3 x 10 5 • RELAP5 solution exhibits Velocity (m/s) 4 2 Δ x oscillations 3 GWS+ θ TS produces • 2 monotone solution for t=0s t=0.05s (RELAP5) 1 smooth ICs t=0.10s (RELAP) t=0.05s (FE) 0 t=0.10s (FE) 0 50 100 150 200 Axial Distance (m)
Transient Results: Abrupt Valve Closure RELAP vs FE: Closed Pipe, Velocity Profile 0.1 ∂ v = 0 t=0s ∂ Velocity (m/s) x ⇔ t=0.03s (RELAP5) = VALVE x t=0.06s (RELAP) 0.05 ∆ = ρ ⋅ ⋅ t=0.03s (FE) P v c = v ( t ) 0 t=0.06s (FE) max = valve x ( ) − = 0 open valve IC q x , t = t 2 s ∆ t ⋅ = 0 . 375 c -0.05 ∆ 0 50 100 150 200 x Axial Distance (m) RELAP vs. FE: Closed Pipe, Pressure Profile • Both algorithms fail to 5 5x 10 generate monotone solutions t=0s t=0.03s (RELAP5) for step IC Pressure (Pa) t=0.06s (RELAP) 4 t=0.03s (FE) • RELAP5 solution is some t=0.06s (FE) 3 what diffusive 2 1 0 50 100 150 200 Axial Distance (m)
Full GWS+θTS Implementation of a 6-Equation Model { } ∂ ∂ ( ) ( ) ( ) 1 ~ q ~ q ~ = α ⋅ + ⋅ = ⋅ T q , P , v , v , h , h A q B q S q ∂ ∂ f g f g t x ∂ ∂ Ψ N T ( ) ( ) ( ) ~ Q ~ ~ = ∑ ∫ e ∫ ∫ Ψ ⋅ ⋅ Ψ τ ⋅ + Ψ ⋅ ⋅ τ ⋅ − Ψ ⋅ τ ⋅ = N T e e GWS A q d B q d Q S q d 1 0 ∂ ∂ e e e e e t x µ = Ω Ω Ω 1 e e e ~ ( ) ~ ~ = + ⋅ + ⋅ T T A q A A q A q q 1 2 3 ~ ( ) ~ ~ = + ⋅ + ⋅ T T B q B B q B q q 1 2 3 ~ ( ) ~ ~ = + ⋅ + ⋅ T T S q S S q S q q 1 2 3 • 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
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 opportunity currently being examined using 2 nd Order schemes with OS (WAHA) • The GWS+ θ TS 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+ θ TS 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.
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