Optimization method to determine mass transfer variables in a PWR crud deposition risk assessment tool

Figures & data

Figure 1. Centered parameter studies of the effect of (a) particle deposition, (b) out-of-core particle release, (c) in-core non-boiling particle release, and (d) mass transfer coefficient for escape probability of boiling on the cycle core boron mass for Unit A with prior source terms adjustments.

Figure 2. Effect of the multiplier on the core boron mass for Unit A. A multiplier value 0.79 effectively “tunes” the unit.

Table 1. Nuclear steam supply system (NSSS) characteristics of Unit A and Unit B.

Unit	NSSS	Cycle length (months)	CIPS	Grid type	IFM	Max SR, (kg/hr m^2)	Core boron threshold (kg)
A	WEC	18	Yes	Mixing vane	No	2808	0.15
B	WEC	18	Yes	Mixing vane	Yes	3052	0.15

Note: S_R: steaming rate, a measure of energy transfer in the vaporization of the fluid. WEC: Westinghouse Electric Company.

Table 2. Plant CIPS history for Unit A and Unit B.

Cycle	SG type	Core boron (kg)	CIPS
Unit A
2	Alloy 600	<0.15	No
3	Alloy 600	0.345	Yes
4	Alloy 600	<0.15	No
Unit B
9	Alloy 600	>0.409	Yes
10	Alloy 600	>0.409	Yes
11	Alloy 600	0.345	Yes
12	Alloy 600	<0.15	No

Table 3. Output constraints used for defining convergence criteria.

Cycle	Label	Lower	Upper
Unit A
2	Boron, kg	0	0.136
2	Crud fraction	0	∞*
3	Boron, kg	0.254	0.435
3	Crud fraction	0	1.0
4	Boron, kg	0	0.136
4	Crud fraction	0	1.0
Unit B
9	Boron, kg	0.136	0.726
9	Crud fraction	0	∞*
10	Boron, kg	0	∞*
10	Crud fraction	0	1.0
11	Boron, kg	0.254	0.435
11	Crud fraction	0	1.0
12	Boron, kg	0	0.136
12	Crud fraction	0	1.0

*An upper limit of infinity means that the response is unconstrained. Crud fraction of the first cycle in each sequence is typically very large and also unconstrained.

Table 4. Input constraints used for limiting ranges of possible solution.

Variables	Lower	Initial	Upper
Mass transfer coefficients
PD (m/s)	1.00e-6	1.00e-5*	1.00e-4
CKSG (s^−1)	1.00e-8	2.40e-7*	1.00e-6
CKNB (s^−1)	1.00e-8	1.20e-7*	1.00e-6
CKBL (m/s)	1.00e-4	6.00e-4*	1.00e-2
Source terms adjustment^+
A – CREL/CRELFE/FM Multi.	0.70	1.00	1.30
B – CREL/CRELFE/FM Multi.	0.70	1.00	1.30

* These values are the best estimated values derived by the development team.

⁺ Because this is a concurrent optimization of two plants, a separate source terms multiplier is used for Unit A and Unit B. These multiplier terms scale the nickel and iron release rate together with the crud retention factor.

Figure 3. Implementation of multilevel optimization. Arrows on the left are input while arrows on the right are collected outputs fed back to the optimizer.

Figure 4. Illustration of the setup for improving runtime and parallelization, (a) shows the connection mechanism through PBS and MPI calls and (b) shows the thread splitting.

Table 5. Mass transfer coefficients after DAKOTA optimization runs.

Mass transfer coefficients	Run 1	Run 2
PD (m/s)	1.00e-5	1.58e-5
CKSG (s^−1)	2.40e-7	2.40e-7
CKNB (s^−1)	1.20e-7	2.09e-7
CKBL (m/s)	6.28e-4	3.14e-4

Figure 5. Change in objective function over iteration for (a) Run 1 and (b) Run 2.

Table 6. Percentage of constraints met for Unit A.

Constraint	Run 1 (%)	Run 2 (%)
C2Boron	82.4	76.9
C2CrudFraction	100	100
C3Boron	89.7	84.6
C3CrudFraction	98.7	98.4
C4Boron	98.7	98.1
C4CrudFraction	83.7	84.0

Table 7. Percentage of constraints met for Unit B.

Constraint	Run 1 (%)	Run 2 (%)
C9Boron	87.1	65.1
C9CrudFraction	100	100
C10Boron	98.7	97.1
C10CrudFraction	98.7	98.4
C11Boron	86.7	83.3
C11CrudFraction	98.7	98.4
C12Boron	6.4	13.8
C12CrudFraction	96.1	97.1

Table 8. Combined constraint frequency

Combined total	Run 1 (%)	Run 2 (%)
6	1.3	1.6
8	0	1.0
9	6.4	5.1
10	4.3	6.7
11	8.2	8.0
12	10.3	32.1
13	69.5	45.5
14	0	0

Figure 6. Unit 2, Cycle 12 core boron difference from CIPS threshold of iterations with 13 constraints met for (a) Run 1 and (b) Run 2.

Table 9. Core boron mass comparison.

Cycles	Target (kg)	Best estimate (kg)	Run 1 (kg)	Run 2 (kg)
Unit A
3	0.345	0.345	0.345	0.345
4	<0.150	0.009	0.010	0.021
Unit B
11	0.345	0.310	0.312	0.345
12	<0.150	0.176	0.176	0.163