ORNL Seminar19.05.11

 

Reactor Point Kinetics--Then and Now Barry D. Ganapol

     Fellow        Advanced Institute of Studies Unibo

and  DIENCA Visiting Professor

andUniversity of Arizona

UTK

v Errors and missing information in papers

v Unsubstantiated claims of accuracy, simplicity,      usefulness and elegance

v Lack of benchmarks and benchmarking strategyv The “simple” algorithm is missingv  Extreme accuracy has always been achievablev  No ultimate PKE algorithm currently exists

Some thoughts concerning PKE algorithms--

Sketch of Point Kinetics Equations

N(t)

dN(t)

N(t)-

N(t)+ +S(t)

Separability:

Adjoint weighting:m

i = 1,…,m

A Survey of Past Solutions to the PKEs

Nuclear Reactor Kineticsby G.R. Keepin, 1965

1

1

, 1,...,

n

i n i n

n

t m

n i i nit

tt t t ti

i i nt

N t N t dt t N t C t C tl

C t e C t dt N t e i ml

 + Laplace transform and inversion + Requires extensive tables of poles and residuals + RTS code advanced at the time + Too difficult to use routinely

+ Integral form (RTS Code 1960)

A New Solution of the Point Kinetics EquationsJ. A. W. da Nóbrega

NSE 46, 366-375 (1971)

v Consider constant reactivity insertion and     constant source 

d ttdt

yAy S

0

0

0

1

10 ...m

m

NN

N

y

10t e e At AtA I Sy y

1 2

11

2

....

0 0 .... 0 ....

.... 0

0 .... 0

m

mm

t

A

1

...m

N tC tt

C t

y

v Require inverse of A which is argued to be too     computationally expensive (at the time)     - Advocates Padé approximant, e.g., P(2,0)

3 32 / 6f At he At A

12 22 1 / 2 , , ,i if At h h fcn

A A

Note: All eigenvalues are not necessary but            still require extreme eigenvalues

v At best 5x10-5 relative error

Unnecessarily complicated for outcome

Solution of the Reactor Kinetics Equations by Analytical Continuation

John Vigil NSE 29, 292-401 (1967)

Taylor Series

Recurrence

+ Time step control

+ Continuous Analytical Continuation

Time Discretization

+ A method ahead of its time

 On the Numerical Solution of the Point KineticsEquations by Generalized Runge-Kutta Methods

J. SanchezNSE 103, 94-99 (1989)

0, 0y t f y t y y

*1

1

1* *

1

1*

1

, 1,...,

s

n n k kk

k

n k n kj jj

k

n kj jj

y y c f

I hf y f hf y f

hf y f k s

All coefficients are specified

Method Underperforms

t GRK Claimed

Exact Exact

0.003 1 2.20985 2.2098 2.20984045698 10 8.01891 8.0192 8.01919997323 20 2.82948e+1 2.8351e+1 2.82973997810e+1 0.0055 0.1 5.21 5.21 5.21002839001 2 4.3022e+1 4.3025e+1 4.30251435157e+1 10 1.388e+5 1.3886e+5 1.38860225602e+5 0.007 0.01 4.50885 4.5088 4.50885848635 0.5 5.3445e+3 5.3459e+3 5.34588761204e+3 2 2.05697e+11 2.0591e+11 2.05915601782e+11 0.008 0.01 6.20276 6.2029 6.20285357509 0.1 1.4101e+3 1.4104e+3 1.41042180359e+3 1 6.1486e+23 6.1634e+23 6.16333374991e+23

L = 2e-05s

A new integral method for solving the pointreactor neutron kinetics equations

Li, Chen, Luo, Zhu, ANE 36, 427-432 (2009)

v Start from integral equation and assume

21

0

10

, 1,...,i

kjh

n jj

kh

i n i n j ijj

N t a e

C t e C t a G i m

t BBF Claimed Exact Exact =-0.003

0.2 4.809750E-01 4.809743E-01 4.80973210584E-01 0.4 4.652904E-01 4.652903E-01 4.65289326117E-01 0.6 4.519648E-01 4.519650E-01 4.51963975793E-01 0.8 4.402738E-01 4.402732E-01 4.40272277652E-01 1.0 4.297825E-01 4.297830E-01 4.29782046265E-01

= 0.007 0.2 1.597867E+02 1.597257E+02 1.59725769863E+02 0.4 1.665854E+03 1.667386E+03 1.66728769255E+03 0.6 1.709542E+04 1.713190E+04 1.71319342103E+04 0.8 1.752925E+05 1.758905E+05 1.75890975931E+05 1.0 1.797286E+06 1.805726E+06 1.80573163423E+06

L = 2e-05sMethod performs poorly

Aboanber Methods:PWS: an efficient code system for solving

space-independent nuclear reactor dynamicsA.E. Aboanber*, Y.M. Hamada

Annals of Nuclear Energy 29 (2002) 2159–2172

2.000000000E+00 1.338200050E+00 4.000000000E+00 2.228441897E+00 6.000000000E+00 5.582052449E+00 8.000000000E+00 4.278629573E+01 9.000000000E+00 4.875200217E+02 1.000000000E+01 4.511636239E+05 1.100000000E+01 1.792213607E+16

Ramp

Claimed “exact” solution is not so

Solution of the point kinetics equations in the presence

of Newtonian temperature feedback by Pad´eapproximations via the analytical inversion method

A E Aboanber and A A NahlaJ. Phys. A: Math. Gen. 35 (2002) 9609–9627

Same method as da Nóbrega and Sanchez

2.000000000E+00 1.338200050E+00 4.000000000E+00 2.228441897E+00 6.000000000E+00 5.582052449E+00 8.000000000E+00 4.278629573E+01 9.000000000E+00 4.875200217E+02 1.000000000E+01 4.511636239E+05 1.100000000E+01 1.792213607E+16

Inaccurate at large time

A Resolution of the Stiffness Problem ofReactor KineticsY. Chao, A. Attard 

NSE 90, 40-46 (1985)

0 0( ) , ( ) ( )

t t

dt w t dt u t

i iN t e C t f t e

Define:

Choose u and w to confine most variation to N

t SCM Claimed

Exact Exact

0.003 1 2.2254 2.2098 2.20984045698 10 8.0324 8.0192 8.01919997323 20 2.8351e+1 2.8297e+1 2.82973997810e+1 0.0055 0.1 5.2057 5.2100 5.21002839001 2 4.3024e+1 4.3025e+1 4.30251435157e+1 10 1.3875e+5 1.3886e+5 1.38860225602e+5 0.007 0.01 4.5001 4.5088 4.50885848635 0.5 5.3530e+3 5.3459e+3 5.34588761204e+3 2 2.0627e+11 2.0591e+11 2.05915601782e+11 0.008 0.01 6.2046 6.2029 6.20285357509 0.1 1.4089e+3 1.4104e+3 1.41042180359e+3 1 6.1574e+23 6.1634e+23 6.16333374991e+23

 Confusing mathematical physics modeling 

An analytical solution of the point kinetics equations with time-variable reactivity

by the decomposition methodClaudio Z. Petersen , Sandra Dulla, Marco T.M.B. 

Vilhena, Piero RavettoProgress in Nuclear Energy xxx (2011)

Adomian Decomposition Method

 Inappropriate claims of accuracy, utility and simplicity 

t 0.003 0.007 0.008

1.0000E-05 1.02940792213E+00 1.07000000355E+00 1.08040134076E+00 1.0000E-03 1.73656448024E+00 8.00355370443E+00 1.47531198687E+01 1.0000E-01 1.80476704702E+00 1.50688886725E+04 2.89485876592E+44 1.0000E+00 2.21520297121E+00 2.85050383437E+25 3.25148973415+436 1.0000E+01 8.05231886873E+00 1.66961653500+238 1.32953281684+872* *t = 2 (QP)

L = 10-6s

A simple reliable, robust algorithm to solve the PKEs is lacking.

+ Must think numerically and use   - new computational architectures   - robust numerical methods   - experimental numerical methods

+ Must abandon the outmoded ideas of time   step control and the “minimum time step   competition”.

….hence….

A Survey of New Solutions to the PKEs                            + GPCA                            + TS

1

,

, 1,...,

m

l ll

l ll l

dN t t NN t C t

dt

dC tN t C t l m

dt

m GPCA G(?)Piecewise Constant Approximation      to the Solution of Point Kinetics Eqns (PKEs)

Recall:

,d t

t N t tdt y

A y

1 2

11

2

, ....

0 0, .... 0 ....

.... 0

0 .... 0

m

mm

R t N t

t N t

A

,,

t N tR t N t

v First consider constant reactivity insertion      (without source)

d ttdt

yAy

,0t e Aty y 1

1

1

0 ...m

m

y

Diagonalize  A : A = UWU-1

1 0Det

I A+ Eigenvalues from

+ Eigenvectors form U from

0 k UI A 1k k

+ U-1 = VT from transpose0 T T

k VI A

+ W = diag{k; k=1,…,G}T VUWA

Attributable to da Nobrega, Sanchez, Allen, Aboanber

1

1

2,..., 1/ , 1,..., 11

; , 1,..., 1

llk k

k l

lk

l mk m

l k m

u

U u

2

1

/11

ml l

k kl k l

1

1

22

1

1/ 1

1

; , 1,..., 1

k llk

k k l

ml l

k kl k l

lk l k m

v

vV

In-hour Equation:

0Tt e VWtUy y

+ Exact solution for step insertion

HQR Algorithm for k

- Algorithm Summary

Explicit eigenvector representation

Note: All done through linear algebra

1

11

1 , i

i

t

i i ii i t

dt t t t tt t

11

0j ji i

iT T

i i i i j jj

e e

v vW tW tu uy y y

v Now consider prescribed reactivity insertion 

- Note: must introduce a time step and  solve for eigenvalues for each time interval

Efficient numerical solution of the point kineticsequations in nuclear reactor dynamics

M. Kinard and E. AllenANE 31 1039-1051 (2004)

An Implicit Method for Solving the Lumped Parameter Reactor-Kinetics Equations by

Repeated ExtrapolationM. Izumi and T NodaNSE 41 299-303 (1970)

Combined R-K with repeated Richardsonextrapolation to improve the FD/RK scheme

Can  this concept be generalized ?

Another article method ahead of its time

v Convergence acceleration    + True solution based on the limit

1

0lim 0j

iT

j jj

hhi

t e

vWuy y

+ Form a sequence of solutions

1

; , , 0,1,....2

0j ni

T ii n j j n n

j

h tt h h ne

vWuy y

;limi i nnt t h

y y

+ Apply a convergence accelerator to    to find a new sequence    such that 

0;

lim i n i

i i nn

t tt t h

y yy y

n ity

n ity is found from the asymptotic behavior  (in n) of original sequence

Some accelerators are: Romberg, Aitkin                                           Wynn-epsilon (W-e),                                           Euler transformation 

+GPCA Ganapolized Piecewise Constant Reactivity Approximation

tj-1 tj

Sequentially halve interval Build a sequence of solutions over all gridsAccelerate convergence via Romberg or W-eBegin each interval with converged IC

 - Goal is extreme accuracy 

lr (n = 2lr)

2 4 6 8 10 12 14 16

Rel

ativ

e Er

ror

1e-13

1e-12

1e-11

1e-10

1e-9

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

1e+0

1e+1

No AccelerationRombergW-e

Ramp 0.1/sFor all edits

Approach to Prompt Jump

Time (s)0 2x100 4x100 6x100 8x100 10x100 12x100

Neu

tron

Den

sity

0

2

4

6

8

10

12

14

 = 10-7,10-8,….,10-19

Approach to Prompt Jump

Time (s)-24-23-22-21-20-19-18-17-16-15-14-13-12-11-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3

Neu

tron

Den

sity

0

2

4

6

8

10

12

14

 =10-19

 = 10-7

$0.50 Step Insertion in a fast reactor

2.0000E+00 1.3382000E+00 4.0000E+00 2.2284419E+00 6.0000E+00 5.5820524E+00 8.0000E+00 4.2786295E+01

Claudio Z. Petersen , Sandra Dulla, Marco T.M.B. Vilhena, Piero Ravetto

Progress in Nuclear Energy xxx (2011)

GPCA

Ramp

1.0000E+00 1.1239405E+00 2.0000E+00 1.1688896E+00 3.0000E+00 1.0744847E+00 4.0000E+00 9.5382929E-01 5.0000E+00 9.0735349E-01 1.0000E+01 9.8468032E-01

0.00073sint t 19652011

Here is robust for ramp considered earlier 2.0000E+00 1.338200050E+00 4.0000E+00 2.228441897E+00 6.0000E+00 5.582052449E+00 8.0000E+00 4.278629573E+01 9.0000E+00 4.875200217E+02 1.0000E+01 4.511636239E+05 1.1000E+01 1.792213607E+16

GPCA Correct to 9-places in comparisonwith FD (Ganapol/NSE Letter)

v Test by manufactured solution

1 0

1 ii

tmt tt

i ii

dN tt

N t dt

e dt e N tN t

+ Solve for reactivity

+ Specify N(t)+ Input             imply    t N t

1 1 tN t f e

1

1 11

i i

i

t

t tm

i tti

i

t eN t

e f e

fN t e e

Assume exponential to power level

Implies

Manufactured Solution f = 2

X Data

0 1 2 3 4 5 6

Den

sity

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Time(s)0 1 2 3 4 5 6

Rea

ctiv

ity

0.000

0.001

0.002

0.003

0.004

0.005

Time(s)0 1 2 3 4 5 6

Num

ber o

f Cor

rect

Dig

its

8

9

10

11

12

13

14

15

Error Measure forall 3 cases

 = 10-10

m Taylor series (TS) solution to PKEs (J. Vigil/1967)       + GPCA        - requires discretization solution and In-hour       - not analytical       - iteration to include non-linear reactivity

    + TS solution most natural solution and        gives an analytical solution

,

, , 10

,

k jk

i i k j jk

k j

N t nc t c t t

t

+ Form TS in interval [tj-1,tj]

+ Naturally generate the following recurrence:

1, , , ,1

, 1, , , ,

11

1

m

k j k j i i k ji

ii k j k j i i k j

k n c

k c n c

, 0, ,

1

, ,0

k j j k j

k

k l j l jl

n

n

+ Numerical implementation   - Must proceed with caution        TS slowly converging and therefore        sensitive to round off (from “swell”)   - Use Continuous Analytical Continuation (CAC)

, 1

0

0, 1

k

k j jk

j j

N t n t t

n N t

Choose interval [tj-1,tj] to limit number of terms in TS to KAccelerate convergence of partial sums via W-eat original and added time edits if necessary

tj-1 tj

- Form sequence for N(tj) by refining grids

then accelerate grid endpoint with W-e

Called “effort based time step control”

1.0000E+00 1.130832298566E+00 2.0000E+00 1.338200050049E+00 3.0000E+00 1.668749984373E+00 4.0000E+00 2.228441896810E+00 5.0000E+00 3.277251197891E+00 6.0000E+00 5.582052448674E+00 7.0000E+00 1.216798285040E+01 8.0000E+00 4.278629573112E+01 9.0000E+00 4.875200217231E+02 1.0000E+01 4.511636239090E+05 1.1000E+01 1.792213607343E+16

1.130832298566E+00 1.338200050049E+00 1.668749984373E+00 2.228441896810E+00 3.277251197891E+00 5.582052448673E+00 1.216798285040E+01 4.278629573111E+01 4.875200217231E+02 4.511636239090E+05 1.792213607342E+16

GPCA TS

Comparison for 0.1$/s Ramp

1

0

1 1

0, 1

1, 1

, 1,

, 2

j

t

t

j jt

j j

j j

k j k j

t at B dt N t

t t a t t B dt N t

t

a BN t

Bn k

+ Demonstration with feedback    - Adiabatic temperature feedback (Keepin)

Adiabatic with Doppler

Time(s)0 1 2 3 4 5 6

Neu

tron

Den

sity

10-1

100

101

102

103

104

105

106

107

108

109

1010

1011

1012

a = 0.003, 0.005,      0.01, 0.05,0.1

0 1 2 3 4 5 610-1

100

101

102

103

104

105

106

107

108

109

1010

1011

0 1 2 3 4 5 6

Der

ivat

ive

-1e+11

-8e+10

-6e+10

-4e+10

-2e+10

0

2e+10

4e+10

6e+10

8e+10

1e+11

1

, m

l ll

dN t t NN t C t

dt

Determination of time a first peak

 2.91058215.114160E+09TS

Most accurate calculation to date

My Message:+ Two highly accurate solutions to PKE presented   

+ All based on previous work

+ Achievable in 1970

+ Methods far outperform any existing algorithm   for standard problems of prescribed and    non-linear reactivities

+ Simple and elegant and sets the standard

+ Nearly completes the numerical solution to PKEs