Assessment study about the risk of Wilks’ formula for uncertainty quantification of design extension condition scenarios in prototype Gen-IV sodium fast reactor

Figures & data

Figure 1. Schematic of PGSFR.

Table 1. Phenomena identification and ranking table (PIRT)

			Phenomena importance
System	Subsystem/component	Phenomena	UTOP	ULOF	Knowledge level
Reactor core	Core	Fuel rod heat transfer	H	H	M
		Rate of reactivity insertion	H	N/A	H
		Coolant density effect	M	M	H
		Radial core expansion	H	H	M
	Fuel assembly including CRDL	Axial expansion of fuel and cladding	H	H	M
		Control rod drive line expansion	H	H	H
		Doppler reactivity feedback	M	M	H
		Inter assembly heat transfer	M	H	M
	Fuel rod	Core pressure drop	L	H	H
		Fission gas generation	L	L	M
		Fuel-clad eutectic formation	L	L	M
Reactor vessel	PHTS pump	Pump coastdown	N/A	H	M
		Pump heat generation	L	L	H
	PHTS	Natural circulation (one-dimensional global flow)	N/A	H	M
		Primary system pressure drop	L	H	H
		Thermal stratification	L	L	M
		Reactor vessel heat loss	L	M	L
		Internal heat structure heat transfer	L	H	H
		Three-dimensional flow in reactor vessel	L	M	M
System	Subsystem/component	Phenomena	UTOP	ULOF	Knowledge level
IHX	IHX tube-side (secondary-side)	Tube-side pressure drop	L	L	H
		Tube-side heat transfer	H	H	H
	IHX shell-side (primary-side)	Shell-side pressure drop	L	M	H
		Shell-side heat transfer	H	H	H
IHTS	Expansion tank	Sodium volume expansion	N/A	N/A	H
	IHTS pipe	Pressure drop	L	L	H
	IHTS pump	Electro Magneti (EM) pump characteristic curve	N/A	N/A	M
SGS	SG tube-side	SG tube heat transfer	H	H	L
	SG shell-side	SG shell heat transfer	H	H	H
DHRS	ADHRS	Blower characteristic curve	L	L	H
		EM pump characteristic curve	L	L	M
		FHX air tube heat transfer	L	L	M
		FHX air shell heat transfer	L	L	H
		DHX Na–Na tube heat transfer	L	L	H
		DHX Na–Na shell heat transfer	L	L	H
	PDHRS	AHX air tube heat transfer	L	L	H
		AHX air shell tube heat transfer	L	L	M
		DHX Na-Na tube heat transfer	L	L	H
		DHX Na-Na shell heat transfer	L	L	H
		Natural circulation	L	L	M

FHX : Forced-draft sodium-to-air heat exchanger, DHX : Decay heat exchanger, AHX : Natural-draft sodium-to-air heat exchanger

Table 2. Model identification and ranking table (MIRT) of ULOF accident

				Distribution	Uncertainty band	Sr for coolant
System	Phenomena		Related model	function	[2σ for normal distribution]	temperature (min/max)
Reactor core	Fuel rod heat transfer	M1	Fuel conductivity	Normal	±0.58 W/m K	−0.00434/−0.00303
		M2	Convection (modified-Schad)	Normal	±20%	−0.00087
	Coolant density effect	M3	Sodium density reactivity	Normal	±32.6%	−0.01516/−0.01463
	Core radial expansion	M4	GP strain coefficient	Uniform	±10%	0.0
		M5	ACLP strain coefficient	Uniform	±10%	−0.0685/−0.06503
		M6	Reactivity coefficient	Normal	±30.6%	−0.07424/−0.06092
	Axial expansion of fuel and cladding	M7	Fuel density reactivity	Uniform	±10%	0.0
		M8	Cladding strain coefficient	Uniform	±10%	−0.00347
		M9	Reactivity coefficient	Normal	±30.6%	−0.01615/−0.01530
	Control rod drive line expansion	M10	CRDL expansion reactivity coefficient	Uniform	±10%	−0.01734
		M11	RV expansion reactivity coefficient	Uniform	±10%	0.0
		M12	Control and shutdown rod worth	Normal	±19.8%	−0.01752
	Doppler reactivity feedback	M13	Doppler reactivity	Normal	±30%	−0.02052/−0.01936
	Inter assembly heat transfer	M14	HT-9 conduction	Uniform	±10%	−0.00694/−0.00607
	Core pressure drop	M15	Friction model (simplified Cheng and Todreas)	Normal	±30%	0.009827/0.007515
PHTS	Pump	M16	Coastdown curve	Uniform	±10%	−0.0685/−0.07197
		M17	Pump heat generation	normal	±10%	0.000867
	Internal structure heat transfer	M18	Heat capacity	Uniform	±10%	0.0
		M19	Convection (Aoki)	Normal	±20%	−0.00347/−0.00303
IHTS	Tube-side heat transfer	M20	Convection (Aoki)	Normal	±20%	−0.00737/−0.00434
	Shell-side pressure drop	M21	Wall roughness	Uniform	10^−5–2.0 × 10^−4	0.0
		M22	Spacer grid form loss	Uniform	0.5–1.5	0.0
	Shell-side heat transfer	M23	Convection (Graber–Reiger)	Normal	±12.2%	−0.00924/−0.00640
SGS	SG tube heat transfer	M24	Convection (steam)	Normal	±20%	−0.02124/−0.01474
	SG shell heat transfer	M25	Convection (Graber–Reiger)	Normal	±12.2%	−0.00498/−0.00426

Table 3. Model identification and ranking table (MIRT) of UTOP accident

				Distribution	Uncertainty band	Sr for coolant
System	Phenomena		Related model	function	[2σ for normal distribution]	temperature (min/max)
Reactor core	Fuel rod heat transfer	M1	Fuel conductivity	Normal	±0.58 W/m K	0.005509/0.003725
		M2	Convection (modified-Schad)	Normal	±20%	0.001889/−0.01734
	Rate of reactivity insertion	M3	Reactivity insertion	Normal	±19.8%	0.126501/0.129893
	Coolant density effect	M4	Sodium density reactivity	Normal	±32.6%	−0.00892/−0.0085
	Core radial expansion	M5	GP strain coefficient	Uniform	±10%	0.0
		M6	ACLP strain coefficient	Uniform	±10%	−0.0341/−0.03221
		M7	Reactivity coefficient	Normal	±30.6%	−0.03662/−0.03038
	Axial expansion of fuel and cladding	M8	Fuel density reactivity	Uniform	±10%	0.0
		M9	Cladding strain coefficient	Uniform	±10%	−0.00273/−0.00283
		M10	Reactivity coefficient	Normal	±30.6%	−0.01584/−0.01447
	Control rod drive line expansion	M11	CRDL expansion reactivity coefficient	Uniform	±10%	−0.02172/−0.02130
		M12	RV expansion reactivity coefficient	Uniform	±10%	0.0
		M13	Control and shutdown rod worth	Normal	±19.8%	−0.02194/−0.02104
	Doppler reactivity feedback	M14	Doppler reactivity	Normal	±30%	−0.01917/−0.01745
	Inter assembly heat transfer	M15	HT-9 conduction	Uniform	±10%	−0.0086/−0.00808
	Core pressure drop	M16	Friction model	Normal	±30%	0.020147/0.008395
	Pump	M17	Pump heat generation	normal	±10%	0.001154/0.001364
	Internal structure heat transfer	M18	Heat capacity	Uniform	±10%	0.0
		M19	Convection (Aoki)	Normal	±20%	−0.00782/−0.00624
IHTS	Tube-side heat transfer	M20	Convection (Aoki)	Normal	±20%	−0.00881/−0.00546
	Shell-side pressure drop	M21	Wall roughness	Uniform	10^−5–2.0 × 10^−4	0.0
		M22	Spacer grid form loss	Uniform	0.5–1.5	0.0
	Shell-side heat transfer	M23	Convection (Graber–Reiger)	Normal	±12.2%	−0.01084/−0.00774
SGS	SG tube heat transfer	M24	Convection (steam)	Normal	±20%	−0.02513/−0.01716
	SG shell heat transfer	M25	Convection (Graber–Reiger)	Normal	±12.2%	−0.00636/−0.00482

Figure 2. Comparison of FOM distribution between the nominal case and uncertainty quantification methods in ULOF accident.

Figure 3. Trends of 95% upper limit of FOM during Monte Carlo histories in (a) ULOF and (b) UTOP accident.

Figure 4. Comparison of 95% upper FOM with value of third-order Wilks’ formula in ULOF accident.

Figure 5. Trends of 95% upper limit of FOM during Monte Carlo histories in UTOP accident.

Table 4. Comparison of maximum temperature between Monte Carlo and Wilks’ formula of ULOF and UTOP accident

			All parameters in MIRT
	Nominal		ULOF			UTOP
FOM	ULOF	UTOP	Monte Carlo		Wilks	Monte Carlo	Wilks
Coolant temperature (K)	1153.2	953.0	Max.	1179.6	1187.6	977.9	986.7
			Min.	1130.8	1126.9	931.1	926.4
Fuel temperature (K)	1172.5	1050.5	Max.	1199.4	1208.4	1078.8	1089.8
			Min.	1149.8	1145.4	1019.4	1013.7
Cladding temperature (K)	1155.8	960.8	Max.	1182.3	1190.2	986.7	995.6
			Min.	1133.2	1129.4	938.2	932.5

Table 5. Number of Wilks’ formula cases with under-predicted 95th percentile FOM of ULOF and UTOP accident

	The order of	1000 trials
FOM	the formula	ULOF	UTOP
Coolant temperature (K)	First	45	47
	Second	42	35
	Third	42	55
Fuel temperature (K)	First	46	42
	Second	42	42
	Third	46	55
Cladding temperature (K)	First	44	48
	Second	43	42
	Third	49	61

Figure 6. Comparison of 95% upper FOM with value of third-order Wilks’ formula in UTOP accident.