ML19320B136

Fiesta:One Dimensional,Two Group Space Time Kinetics Code for Calculating PWR Scram Reactivities

ML19320B136
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Issue date:	11/30/1979
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O CEN - 133 ( B )

FIESTA A ONE DIMENSIONAL, TWO GROUP SPACE-TlME KINETICS CODE FOR CALCULATlNG PWR SCRAM REACTlVITIES NOVEMBER,1979 l

8007000 3 bNS T MS COMBLST;CN ENGNEEctNG NC

11 a

6 LEGAL NOTICE THIS REPORT WAS PREPARED AS AN ACCOUNT OF WORK SPONSORED BY COMBUSTION ENGINEERING, INC. NEITHER COMBUSTION ENQiNEERING NOR ANY PERSON ACTING ON ITS BEHALF:

A.

MAKES ANY WARRANTY OR REPRESENTATION, EXPRESS OR IMPLIED INCLUOING THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE OR MERCHANTASILITY, WITH RESPECT TO THE ACCURACY, COMPLETENESS, OR USEFULNESS OF THE INFORMATION CONTAINED IN THIS REPORT, OR THAT THE USE OF ANY INFORMATION, APPARATUS, METHOD, OR PROCESS DISCLOSED IN THIS REPORT MAY NOT INFRINGE PRIVATELY OWNED RIGHTS;OR B. ASSUMES ANY LIABILITIES WITH RESPECT TO THE USE OF,OR FOR DAMAGES RESULTING FROM THE USE OF, ANY INFORMATION, APPARATUS, METHOD OR PROCESS DISCLOSED IN THIS REPORT.

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'iii ABSTRACT This report describes FIESTA, a one-dimensional two-group space-time

~

kinetics computer code. The intended use of this code is for PWR scram reactivity calculations which account for the axial space-time variations in the neutron flux.

The use of space-time reactivities reduces the margin requirements for transients sensitive to scram reactivity characteristics compared to the use of static reactivities which do not account for delayed neutron effects.

The FIESTA code is used for these calculations instead of the HERMITE code previously approved by NRC mainly for two reasons: (1) FIESTA calculations are more rapid than HERMITE ones, and (2) FIESTA produces kinetics para-meters which are readily used for point kinetics calculations.

The FIESTA code has been s'rified with HERMITE and with comparisons of standard benchmark problems. Although the code has been written to incorporate feedback on fuel temperature and moderator density, such feedbacks are not needed for scram reactivity calculations.

A code input description is included in the Appendix.

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iv TABLE OF CONTENTS Section Title Page 1.0 Introduction 1

2.0 FIESTA model 2

2.1 The Neutronics Model 2

2.1.1 Time Dependent Neutron Diffusion Equation a

2.1.2 Two Group Reactor Kinetics Equations 9

2.2 Cross Section Representation 12 2.3 References for Section 2.0 15 3.0 FIESTA Computer Code 16 3.1 Code Structure 16 3.2 Numerical Solution Strategy 16 3.3 References for Section 3.0 18 4.0 Verification of the FIESTA Code 25 4.1 Comparisons with Benchmark Problems 25 4.1.1 Subcritical transient 25 4.1.2 Supercritical transient 26 4.2 FIESTA - HERMITE Comparison 26 4.3 References for Section 4.0 27 5.0 Space-Time Results 40 5.1 Space-Time and static scram reactivity 40 comparisons 5.2 Benefits of Space-Time Reactivities 41 APPENDIrES Aooendix Title Page A

FIESTA Input Description 58

1 1.0 Introduction It is standard practice in the nuclear industry to calculate the time dependent reactivity insertion by calculating the critical eigenvalue as a function of rod position. This type of calcula-tion assumes that the neutron flux shape during a scram rod insertion is equal to the critical flux shape at each rod position.

Such an assumption is conservative since the critical (or static) flux shape tends to shift away from the rods more than space-time calculation would predict, and hence result in smaller reactivity insertion. The primary reason for the difference in the flux shape is that the delayed neutron precursors are distributed according to the initial neutron flux shape, and as the neutron population decreases the importance of the precursors increases. The neutron precursors provide a source of neutrons which tends to tilt the neutron flux shape toward the rods compared to static methods, and hence leads to greater reactivity insertion at intermediate rod positions.

This effect tends to become large at the end of a reactor cycle when the power shape tends to be axially flat. At certain inter-mediate rod positions, for example, the reactivity change predicted by static methods may be only half or a third of the reactivity pre-dicted by space-time methods. At full rod insertions both methods yield nearly the same reactivity.

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s

2 2.0 FIESTA Model FIESTA is a one-dimensional, 2 group spaca-time kinetics code which allows up to six delayed neutron precursor groups. The model is based on a space-time factorization method which divides the neutron flux into a time-deper. dent amplitude function times a time-dependent shape function. This method is useful since it reduces the number of times that the flux hape must be re-evaluated and permits ready computations of reactivity components.

Section 2.1 describes the neutron kinetics model.

The first part of which is a general derivation of the kinetics equations and a development of the equations which are solved by the space-time factorization method. The second part applies the resulting equations to the two energy group model.

2.1 Time Decendent Neutron Diffusion Ecuation Soace-Time Factorization Metnod The space-time factorization method presented in this section is identical to the improved quasistatic method presented by Ott et al (Ref. 2.1). The term "quasistatic" is avoided in this report since it is sometimes used to refer to the adiabatic method * (Ref.

2.2).

The general time dependent neutron diffusion equation may be written as follows:

hao E,t) = 7 0(r,E,t) 7e(r,E,t)

T(r,E,t)c (r,E,t) +/ dE'I (r,E",t) f(r,E'-E) o(r,E',t)

-I s

/dE'(1 - S ) v (E') If(r,E',t) o(r,E',t) d d

3

+ I f (E) p J

+Iff(E)AC(r,t) j$

i 2.1.1 aC (r,t) d j

/dE'afv(E')I(r,E',t)c(r,E',t) i

= -\\ C$ $ (r,t) +

I f

37 J

• The adiabatic metnod assumes that flux shapes correspond to the static flux shapes of the perturbed reactor state.

The space-time factorization method is an exact method, in that the solution I

approaches the exact solution for sufficiently small time steps since all time derivatives are retained in this formulation.

The adiabatic method does not have this property since the solution for the flux shape neglects all time derivatives.

3 v:here the definition of terms is as follows:

o(r,E,t) -

is the. neutron flux at position r, energy E and time t 0(r,E,t) is the diffusion coefficient I (r,E,t) is the total cross section T

vI (r,E,t) is nu times the fission cross section f

I(r,E,t) - is the scattering cross section s

f(r,E'+E) is the scattering kernel at position r s

df (E) is the prompt neutron spectrum p

denotes the spatial gradient 7

ff(E) is the delayed neutron spectrum of delayed group (i)

SY - is the delayed neutron fraction of I

group (i) resulting from fissionable isotope (j) is the decay constant of delayed group (i)

A j

C(r,t) is the concentration of precursors of g

delayed group (i)

The following definition is applied in the space-time factorization method o(r,E,t) a T(t)

?(r,E,t) 2.1.2 The motivation for this definition is to place most of the time variation of the neutron' flux into the amplitude function T(t).

This definition introduces an additional variable, and therefore an additional equation is required. The following normalization equation is chosen:

IdE /dr Y (r,E) ?(r,E,t) h= constant 2.1.3 O

where ? (r,E) is the initial steady state adjoint flux.

The value of the onstant is determined by the initial flux, since the initial amplitude is assumed to be equal to unity

?(r,E,t)l

$ ("'

)

=

0 The time dependent neutron diffusion equation may be written in the following short notation

4 hh=La+

AC 9j aC

• - A C; + H )

2.1.4 i

9 at

$(r,E,t) where t

=

j C (r,E,t) f (E)

C9=

9 d

L and H are operators _ whose form tray be obtained from 4

Equation 2.1.1 and Equation 2.1.2.

Substituting Equation 2.1.2 into the above equation, multiplying by the adjoint and integrating over space and energy, the following is obtained:

h <Yh h>

= T<?0 '#"

+

• EAi <v+ C;>

h<v

= - <v+CA>

+

<v+ H 7> T C>

0 $

0 4 T(t) is the flux amplitude where T

=

A(r,E,t) is the flux shape function A =

+

+

Y (r,E) is the initial adjoint flux A

0 O

signifies integration over space and energy These equations may be arranged into a form identical to the Point-Kinetics equations.

h =

,~g 3 T+IAKjj aK

• ~AKii+

T at

<7(r,E)ff(E)C(r,t)>

+

0 g

where K (t) =

j

<Y(r,E)hE)?(r,E,t)>

0 l

l l

l

5

<T H7>

[(t) -_

O 4

<9+9, 0v

<9 li #

I

<#*H9>

0 0 9 kCt)

=

+ I

+,

+y

<797>

i=1

<v7>

g Space-time behavior is retained with these equations since the shape function is time varying.

The Point-Kinetics restriction (T(r,E,t) = Y (r,E)) is not applied with this method (i.e., p/A O

and s/Adepend on the flux shape).

These equations are non-linear and cannot be solved directly since the shape function is not known.

To obtain the shape function, Equation 2.1.4 is considered again.

The precursor concentration may be solved directly since it depends only on the flux history prior to the time of interest.

-A t t

-A (t T) j j

C;(r,t)

C;o(r)e

+ fdE f dtH (r,E,t) ?(r,E,1)e

=

j 0

2.1.6 The neutron diffusion equation may be written as follows:

fT 8

LTP + IA C;

+

9

=

j Since the flux shape is a slowly varying function, the following approximation is made f

~I 8_Y f

-9 at I 2

at t=t 2.1.7 This approximation results in the imoroved quasistatic method which has been shown to be accurate compared to exact results by Meneley, et al (Ref. 2.3).

The equation for the shape may then be written at time level (2) f 2

f M (r,E) 7 (r,E) + S (r,E) = 0 2.1.8 O

I where M#=TE {L - II 2 at lt=t f

1. }

v T vat E 2-1 T Y 6g* 1ACij+7 g

1 at

6 A

L is the diffusion operator at the new time level (2) which is Known unless there is feecback.

For feedback an iteration procedure is required.

T ' dT 2

2 are the amplitude and the time derivative of the amplitude at the new time level.

These quantities are unknown, but approximate values may be used which yield good results.

The solution to the shape equation is then reduced to a fixed source problem.

It can be seen that the space-time factorization method is an alternate method of time differencing the kinetics equations.

Instead of time differencing the total flux, the amplitude and the flux shape are time differenced, separately, with different time step sizes.

Since the flux shape is more slowly varying than the amplitude, a larger time step size may be chosen for the former.

In addition, since the flux shape between time levels may be assumed to be constant, the Point-Kinetics approximation is used in this time range.

The space-tima factorization method, therefore, intro-duces an additional variaDie (T) and an additional equation (Point-Kinetics Equation).

An overall saving may be realized, however, since the shape equation, which is the most time consuming calcula-tion, may be evaluated less frequently than with the normal space-time differencing procedures.

Kinetics Parameters and Adioint Weichting The general form of the kinetics parameters is:

p(t) = p g) <1*(r,E) L (r,E,t) 7(r,E,t)>

0 p

Sj(t) = F t) #v (r,E) L (r,E,t) 7(r,E,t)>

0 r,E) v(E) 9(r,E,t)>

2.1.9 A(t) = F t) 0 where F(t) 15. an arbitrary function which is normally taken as the fission source.

F(t)=I<t[(r,E)vIf(r,E,t)9(r,E,t)>

j i

L (r,E,t) is the reactivity operator p

L (r,E,t) is the delayed neutron operator o

7 L (r,E,t) 1(r,E,t) a 7 0(r,E,t) VT(r,E,t)

-I (r,E,t) P(r,E,t) + fdE' I (r,E',t) f (r,E'-E) P(r,E',t)

T s

j

+If(E)fdE'(1-s)vIf(r,E',t)1(r,E',t) d d

p j

+ 2 I ff(E) fdE' sf v1f(r,E',t) V(r,E',t) i J Iffd(E)fdE'sfvIf(r,E',t)T(r,E',t) g$(r,E,t) V(r,E,t)

L s

J 2.1.10 The reactivity operator may be simplified somewhat by making use of the initial steady state condition L

(r,E) 7 (r,E) = 0 2.1.11 p

0 A similar expression may be written for the adjoint flux.

+

L* (r,E) 1 (#'E) = 0 0

PO 2.1.12 where the adjoint ope ator is defined by:

<9*L T>

s <TL 9 >

2.1.13 The reactivity operator and the flux may be expanded.

L (r,E,t) + Lp (r,E) + 6L (r,E,t) p p

P(r,E,t) = 7 (r,E) + 6Y(r,E,t) 0 The reactivity may then be written as the sum of four terms p(t) = 1

+

F(t) < 9 'p0 90, 0

p(t) < 90 'p0 t

6L 70*

+ p(g) < 70 p

6L 69 >

+ p(g) <TO p

e 8

The first term vanishes from the steady state condition Equation 2.1.1.

The second term vanishes since

< 69t* 90>

<9*L 69 > =

= 0 O p0 PO Only two terms remain

~

1 1

+ Ol Y'

g <TO p O T('tJ "O+ 6L 69 >

p(t)

=

p 2.1.15 The first term is the normal Point Kinetics

• Reactivity Comoonent and the second term is the space-time contribution to tne reac-tivity (Reactivity introduced due to flux shape changes).

The latter will be referred to as the Flux-Shace-Reactivity-Comoonent throughout this report.

The above discussion illustrates the reason why the kinetics parameters are usually adjoint weighted.

An arbitrary weighting factor could be chosen without introducing any error.

The choice of the adjoint, however, eliminates first order errors in the calculated reactivity due to errors in the flux shape.

Space-time effects are minimized, or conversely, the time range over which the Point-Kinetics approximation is valid has been extended.

The reactivity components can be sub-divided further into the various contributors.

rod moderator Doppler motion feedback feedback L

+L

+L

+L p

p p

The Point-Kinetics-Reactivity-Comoonent and the Flux-Shace-Reactivity Comoonent can each be subdivided into these three separate reactivity components.

• Normally the Point Kinetics Reactivity is calculated with F(t) = F.

The above definition is retained, however, to 0

preserve the additive property of the reactivity components.

This is a good approximation since only small variations in F(t) are expected.

I

)

9 2.1.2 Two Grouo Reactor Kinetics Ecuations Soace-Time Factorization Method The general kinetics results are reduced to two energy groups in this section.

The diffusion equation is reduced to two energy groups by integrating over a finite energy range.

j 00)(r,t)

• 7'0 (r,t) VQ)(r,t) - {Ia (r,t) + I (r,t c)(r,t)

T at 1 p 1 1 + (1 - s) vIf (r,t) Q)(r,t) + (1 - s) vIf (r,t) c (#'D) 2 + I A C;(r,t) j i 89 (r,t) j 2 0 (#'t) Y'2(r,t) - 2 (r,t) c (#'D) = T at 2 3 2 2 2 + I (r,t) c)(r,t) p 8C;(r,t) = -A C;(r,t) + Sj{vIf (r,t) c)(r,t) + vIf (r,t) c (r,t)} j 2 gg l 2 2.1.16 whe re:. $ (r,t) = f dE $(r,E,t) f dE v(E) o(r,E,t) (r,t) =, ,gy f dE O(r,E,t) 79(r,E,t) D (r,t) = y 7,g) g Z f#'t) * $ (r,t) f dE I,(r,E,t) &(r,E,t) ag 9 9 1 I (r,t) = 0)(r,t) f dE f dE' I (r,E',t) f (r,E'+E) o(r,E',t) p s s 92 91 l

10 vIg (r,t) = g g)IfdEvId(r,E,t)9(r,E,t) g 9 j 9 I dE' sf vIf(r,E',t) f(r,E',t) s9(r,t) = 3 ) I dE' v1 (r,E',t) 9(r,E',t) f J It is assumed that the spectrum within each group is constant, so that these e.ross-sections are not time dependent as a result of spectrum cnanges. (The cross-sections are, however, time depend-ent as a result of feedback.) so has been assumed that all delayed neutrons are identical in spectrum to the prompt neutrons and that all prompt and delayed neutrons are born in the fast energy group. d dEff(E)=f,,dEf(E)=1 f p gj f dEff(E)=f dEf (E) = 0 g g The difference in spectrum between prompt and delayed neutrons is not distinguishable in a two energy group model. (The difference in spectrum between the prompt and the delayed neutrons is often taken into account by calculating a modified delayed neutron fraction (s-effective). Detailed discussion of this calculation may be found elsewhere (Ref. 2.4). Equation 2.1.16 may be written in a short notation 091"'11*"12iAC 1 0 0 I at i 9 1 1 0*2 * '2 2 * "2@l 1 9 T at 2 aC

• ~ACi $ + sj {R)e) + R 0 }

2.1.17 22 at The Space-Time Factorization method is applied $ (r,t) = T(t) ?)(r,t) 3

11 0 (r,t) = T(t) P (r,t) 2.1.18 2 2 and substituted into the above equation. This process yields the amplitude equation (Equation 2.1.5) as shown in the previous section. The kinetics parameters are modified from the previous results to reflect the use of the two group approximation p(t)=h{<t W)> - <T{0 l 0 a) # # 1 10 3 -<P{0 I Y # # ## VI r 1 0 f) 1 +<9{0 0 Y # VI 2'+< 20 2 2 f2 -<T I

1. }

20 a 2 2 Sj(t)=F(t)k0 I< VI Y # **Y VI Y #l 10 1 10 2 J l 2

1. # #Y
2. #}

A(t) = p(t) {< 710 l 20 2 F(t) = I {< 9 VI

1. # **Y VI
2. #l 10 1

10 f 2 j 1 2 2.1.19 It is assumed the delayed neutron fractions may be pracalculated for specific fissioning isotopes in the core and that the delayed neutron fraction-i's time independent. The shape equation may be written in two energy groups .~, .,e,

12 1-1 {- LE+ I l 2} ~M 2 2 a v)T 2

1. at v)at t=t 1

3-1 2}vj** vat -M Y

• 5' ' +

2 2 2 vT 2 v at g z=g g g 2.1.20 At steady state the equation reduces to the following: L) 910

• di 20 = 0 Y

2.1.21 M #10 + '2 #20 = 0 2 or A9=0 0 in matrix form. Similarly, the adjoint equation may be written as follows: A+ t =0 2.1.22 and it can be shown that A' = gT 2.1.23 The adjoint operator is obtained by transposing the matrix operator of the initial flux solution Ref. 2.5. 2.2 Cross Section Reoresentation This section summarizes the cross sectional representation that is incorporated into the numerical reactor model. Functional forms were chosen for each cross section parameter as a function of the local moderator density and fuel temperature. (It is noted that feedback is not necessary for scram reactivity calculations. A description of feedback is included in this section for completeness). The cross section functions were assumed to be separable as follows: I(p,T,B,C) = 1 (p,T) + I (p,B) + I;(p,C) 2.2.1 0 g l I l

13 where I signifies a two group cross section p is the moderator density (lb/ft ) T is the effective fuel temperature ( F) 8 is the baron concentration (ppm) C is the f raction of control rod insertion The uncontrolled (and unborated) reactor cross sections (I ) **"* O least square fit to one of the following functions 0 = a + bp + c 6 + dp d (for I ) 1) 2 p 2) 20 = a + bin (p + p0) (f r 0 ' I f} 2 a' 2 2 = a + b 6 + c2n(p + p0) + d d in(p + p0) 3) Ig (for D), I , vIf) 2.2.2 These functions were chosen due to an obse,r,xpd approximate linear variation of the fast cross sections with lT and due to an approx-imately logarithmic variation with moderator density for most other cross sections. A nectly linear cross section variation of I with moderatur density was observed and a different function wSsthereforechosenforthisparameter. The change in the reactor cross sections due to boron concentration I (p,8) was assumed to depend only on the moderator density and the g b5ron concentration. The assumed function is as follows: I (p,B) = B(aB + b p) 2.2.3 B g where a and b are constants. B 8 This relation assumes a cross section magnitude which is pro-portional to the boron concentration and linear with moderator density. The constants a and b are evaluated at two moderator B g densities and at some reference coron concentration as follows: I

14 B(P ' Oref) P1 I (P), Bref) 92 ~ 2 g a8* (P ~P)B 2 1 ref I (P ' Oref) ~ I (P, Bref) B 2 B 1 B (P ~ P ) 8,f l 7 2 The change in cross section due to control rod insertion is evaluated as follows I (P' )" I c c c0 is the fraction of control rod insertion. where C W - is a normalization constant which adjusts the C rod worth to measured values (Control-Rod-Weighting Factor) Ic() is the reference control rod cross section 0 Ic (P), is soecified as a The reference control rod cross section, function of local moderator density. Thefubctionsemployedare as follows: 1) I +b

• P) c c

c 0 0 vZI) (for aD), AI,, aD ' OIa' 2 2 2 5) c f 0 i 2.2.5 W

15 2.3 References for Section 2.0 2.1 Ott, K.O., Meneley, 0.A., " Accuracy of the Quasistatic Treatment of Spa tial Reactor Kinetics" NSE-36, pp. 402-411 (1969). 2.2 Yasinsky, J.B. " Notes on Nuclear Reactor Kinetics" WAPD-TM-1960 (July 1970). 2.3 Menely, D.A., Ott, K.0., Wiener, E.S., " Influence of the Shape-Function 7 1e Oerivative on Spatial Kinetics Calculations in Fast Reacia s" Trans. Am. Nuc. Soc., Vol. II, No. 1 (June 10-13, 1968). 2.4 Henry, A.. .cs s stion of Parameters Appearing in the Kinetics Equations", -.' t', nc (Decembec 1955). 2.5 Lamarsh, J.R., " Introduction

• .o Nuclear Reactor Theory" Addison-1 Wesley (1966).

t

16 3.0 FIESTA Comouter Code This section describes the space-time code FIESTA. A general discussion of the solution strategy of the Space-Time Factorization method is presented along with the subroutine structure of the FIESTA code. The FIESTA code is written in FORTRAN and is operational on the CDC-7600 computer. The neutronics model is identical to a previously published model (References 3.1, 3.2 and 3.3). 3.1 Code Structure The structure of the FIESTA code is presented in figure 3.1, where the function of each subroutine is briefly described in Table 3.1. Figure 3.1 also shows the subroutines which are used only for feedback calculations. These subroutines are not used in scrar reactivity calculations, but are presented here for completeness. 3.2 Numerical Solution Strategy One of the features of the Space-Time-Factorization method is that the Point-Kinetics approximation is simply a special case of the more general method. The Point-Kinetics approximation results if the neutron flux shape changes during a transient may be neglected. The method therefore has the potential of having a rapid numerical solution during those portions of the transient in which flux shape changes are not important. The space-time factorization method separ stes variables of the neutron diffusion equation according to their characteristic time constants. The flux amplituce is calculated by the Point-Kinetics equations using very small time steps (the FIF.STA code reduces or expands these time steps as needed in order to achieve a converged Point-Kinetics solution). The flux shape is calculated by the shape equation using a time step specified by the user. For the space-time factorization method, a general time differencing numerical scheme can be envisioned wherein three separate types of time steps are used for time-differencing transient equations of the reactor model as shown in figure 3.2.

17 The flux shape is assumed constant during a time interval aT whereas the feedback solution and the Poin- 'inetics solutickHP' are solved with smaller time steps. In FIESTA the time step for the flux shape solution is user specified and is equal to tne time step for tne thermal-hydraulic solution ATKinetics solution is k$omatib$. The time step for the Point = aT 11y specified by the code. The code solution strategy is as follows: 1. It is assumed that 'P 'I is constant over the time steo A O SHP 2. The amplitude equation (Point Kinetics equation) is solved over time step AT using estimated variations of the k'netic parameteh$pover the time step AT iareextrapolatedforwardusingk"quadratictime p ( and pende ce from previously known values at time points 2-1, 2-2 and 2-3). 3. The flux shape equation is solved at time point 2 using new rod position. 4 The kinetics parameters are solved at time point 2 using the new flux shape. 5. A quadratic fit is performed for the kinetics parameters using naw values at time point 2 and previous time points 2-1 and 2-2. 6. The amplitude equation is solved again over time step AT,.3p. 7. The flux shape equation is solved at time point 2 using rod position and cross section changes due to feedback, where feedback is calculated during the time step using the average power P(x) = P (x) + PA~I(x) A 2 A 8. Steps 4 to 7 are repeaced untii the amplitude T is converged.

18 3.3 References for Section 3.0 3.1 Ferguson, K.L., " Development and Examination of Data Handling Schemes for Reactiv'Ly Measurements in the Pressence of Spatial Effects in large f Jclear Power Reactors". Phd Thesis, Carnegie-Mellon Uriversity (1973). 3.2 Eisenhart, L.D., Poncelet, C.G., "Quasistatic Applications to Parabolic Equations with Application to Nuclear Reactor Kinetics", Proc. Int. Symposium on Numerical Solutions of Partial Differential Equations", University of Maryland (1970). 3.3 Decher, U. " Light Water Reactor Accidents: A consistent Analysis of the importance of Space-time Effects" Phd Thesis, Carnegie-Mellon University (1975). l l

19 TABLE 3.1 Subroutine Descriotion 1. FIESTA main controlling program 2. INREAD initializes the problem 3. REDLIST namelist input routine 4. FLXCNT control of steady state flux solution 5. PROP computes the water property table based on the reactor pressure 6. PINDIPP calculates fuel pin geometrical ft.ctors and relative radial power density in the pin 7. SETR00 computes position of the control rod tip 8. CALMAC calculates reactor cross sections based on control rod location, coolant density, fuel temperature and boron concentration 9. R00F0 control rod cross section calculation based an the local water density 10. COEF computes coefficients of til steady state finite differenced flux equitions 11. GUESS provides an initial guess of the fast and thermal flux equations 12. INTEG a generalized spatial integration routine 12. CLXSWP fast and thermal flux calculation 14. POWERI spatial power calculation 15. FEEDBK steady state TWIGL feedback equations FEEDBK2 steady state thermo-hydraulic calculation with detailed fuel pin thermal model and two phase flow analysis 16. DENS water density calculation for subcooled liquid 17. DNB3 W-3 DNBR correlation DNB4 Janssen Levy DNBR correlation 18. FILM coolant film coefficient calculation

20 MBLE 3.1 (Continued) 19. FUNT water temperature calculation from enthalpy 20. PINHEAT calculates the coefficients of the tridiagonal matrix for the steady state and for the transient heat trans-fer within the fuel rod. 21. PROPTH UO and Zirconium heat transfer properties 2 22. GAP computes the clad pellet gap conouctance 23. TRIDA tri-diagonal matrix solution 24. AVTEMPF computes average fuel temperature 25. PICA, POCK fuel temperature retrieval and storage routines 26. SUBVOID coolant void model for two phase flow 27. ADJCNT controls the adjoint flux solution 28. ADJSWP solves for the adjaint flux 29. PARA solves the kinetics parameters p, s, A and performs the initial normalization of the neutron amplitude function 30. DIFFUSE integrates the spatial neutron diffusion operator

31. OUTEDT writes all output results 32.

PTOEDT reduces all vi cross sections by the steady stata f eigenvalue 33. RH0 EDIT detailed reactivity edit 34. DELTK spatial k, edit 35. DRIVER controls the transient solution 36. INDLNT computes the initial precursor concentra :ons 37. PRINTO controls output print 38. INTERPO computes inlet enthalpy, reactor pressure, boron con-centration, and inlet flow based on input tables; also calculates enthalpy mixing in the plenum, and computes the inlet enthalpy for the time-delay flow model

39. MOVR00 computes control rod tip position from control rod velocity equation

n 21 TABLE 3.1 (Continued) 40. PITKIN Point-Kinetics calculation 41. THTR transient TWIGL feedback equations THTR2 transient thermo-hydraulic calculation with detailed fuel pin thermal model and two phase flow 42. DELAY time dependent delayed neutron precursor calculation 43. WEIGHT spatial weighting coefficient calculation 44. TRCOEF computes coefficients of the transient finite differenced flux equations 45. SOLVE computes transients neutron flux shape 4 e S i l

22 FIGURE 3.1 FIESTA Subroutine Structure FIESTA fINREAD I REDLIST ~ FLXCNT ! PROP

• l PINDIPP*

!SETROD l lCALMAC l lRODFD* !COEF l ' UESS l j I, FLXSWP l DENS

• JDNB3, 4 l IPOWERl l

--} FILM HPROPTH* i MFUNT* HGAP* l --- - 4 FEEEK* l MPINHEAT*l HTRIDA* l H PICK,P0CK* -l AVTEMP*! !FEEBK2*l !SUBVOID l lADJCNTl lADSWP l PARA l DIFFUSE l i [INTEG l OUTEDT ' ' PT0EDT l l RHOEDIT l DELTK i DRIVER lINDLNT l l INTERPO l !PRINTO l ! PROP

• l

_A l si i (*These subroutines are needed only for feedback calculations)

23 FIGURE 3.1 (Continued) FIESTA Subroutine Structure p b-FIESTA DRIVER MOVR00 CALMAC H INTEG l PARA l l OIFFUSE l lPITKINl H DENS

• l H DN83,4* l l POWER 1l

-i FILM

• l

-j PROPTH* l -i FUNT* l -] GAP

• l

- - - - -l THTR* l d PINHEAT* l lTRIDA* l H PICK,PCCKj* AVTEMP*l l THTR2*l l SUBVOID* l l OELAY l l WEIGHT l l TRCOEFl l SOLVE l lOUTEDTj lRH0 EDIT l (*These subroutines are needed only for feedback ca'culations)

24 FIGURE 3.2 Gena-al Time Differencing Model t-1 t .y .p neutron flux shace solution at3gp T 'K I 1 7 Point Xinetics Solution --I l'atPK (TH)1'd (TH)1 Thermal Hydraulics Selution 4 at k 39 = Time O e

25 4.0 Verification of the FIESTA Code The FIESTA code has been verified by comparison with several benchmark problems discussed in Reference 4.1, and by comparison with HERMITE (Ref. 4.2). Results of these comparisons are presented in this section. o 4.1 Comparisons with Benchmark Problem This section describes the results of the benchmark proulem comparisons. The benchmark problems described in reference 4.1 consist of an infinite slab reactor model with three regions as shown in figure 4.1. Region i Region 2 Region 3 40 cm --> 0=0 0=0 - +-- 40 cm 160 cm - l l ax = 2 cm Figure 4.1 Infinite Slab Reactor Model Cross sections and delayed neutron parameters are presented in Table 4.1. 4.1.1 Subcritical Transient Fgr the subcritical transient, the thermal absorption cross section, Results I, in region 1 is linearly increased by 3% in one second. afe shown in table 4.2 for various codes. Table 4.2 shows that FIESTA agrees well with other codes, eventhough the mesh structure had to be altered slightly in FIESTA due to a ~ mesh limitation in FIESTA. Table 4.3 shows the kinetics parameters that are calculated by FIESTA for this transient. These kinetics parameters are not usually available in other space-time codes therefore no comparison is possible. The kinetics parameters are shown in equations 2.1.19. i l i I

26 It is noted that $(t) = $ a constant, since the function F(t) has been chosen to be an integral over the fission cross section, so that it is identical to the nunierator of the integral defining S. The flux amplitude is also shown in Table 4.3. This flux amplitude is the solution to the Point Kinetics equations. It is nearly equal to the power (Table 4.2). The reactor power may be written as follows:

1. I (r,E,t) 9(r,E,t)>

f T(t) 4.1.1 P(t) _ Po - <Ifg(r,E) T,(r,E)> Since the neutron flux shape is normalized such that <Y[(r,E)h(E)9(r,E,t)>= constant a good agreement between power, and flux agglitude is expected whenever the space-energy distribution of - closely approximates the space-energy distribution of 1 or wheh the flux shape change 7 is small. 4.1.2 Supercritical Transient The second benchmark problem consists of a supercritical transient using the same initial problem goemetry. A supercritical transient is induced by linearly decreasing the regional thermal absorption cross section by 1% in one second. Results of the code comparison are shown in Table 4.4. The kinetics parameters computed by FIESTA are shown in Table 4.5. 4.2 FIESTA-HERMITE comparisons. In order to verify the FIESTA code for the specific use of cal-culating scram reactivities an additional comparison was performed between FIESTA and HERMITE. The problem analyzed consisted of an axial reactor scram, in which control rods are inserted linearly into the core in 3.0 seconds. Results are shown in Figure 4.2 for the power comparisons. The power shape comparison during the transient are shown in figures ~ 4.3 through figures 4.8. The excellent agreement between FIESTA and HERMITE verifies FIESTA for calculating scram reactivities.

44 +& Ar ?>g, v*Q.'%f'e;f i.. l*9 pfp t'? g)gs /q,, <g gt"(g / fte \\ IMAGE EVALUATION N TEST TARGET (MT-3) 1.0 Ea m k ll;4 llg2 u L"WLE I.8 1.25 I.4 r I.6 I-6" MICROCOPY RESOLUTION TEST CH ART ey$Ps%)pP 44+m/s

• r m

y $ y>d q y #syy:4 <> a / w n - o

d> vc,4 *&v'>o, 4', ,. ~ ?lg

sf~ f

, i' f//// ffQ IMAGE EVALUATION TEST TARGET (MT-3) 1.0

W S M y $] Ha i.i

[": lHe jj l.8 1.25 1.4 1.6 MICROCOPY RESOLUTION TEST CH ART sa %y/7/,Al /// /$ <> > #b ,s # w v 4% y o

27 4.3 References For Section 4.0 4.1 "Argonne Code Center: Numerical Determination of the Space-Time, angle or energy distribution of particles in an assembly" ANL-7416 supplement 1, December 1972. 4.2 CENPD-188, "HERMITE-A 3-0 Space-Time Neutronics Code with Thermal Feedback". P l l r I i 1 l i 4 4 r c---- --v-- w s ---w w --

26 TABLE 4.1 Cross Section and Delayed Neutron Parameters for the Benchmark Problems legion Region 1 and 3 2 1 and 3 2 DI (cm) 1.50 1.00 I1+2 (cm-I) .015 .010 ~ 02 (cm) .50 .50 X 1.00 1.00 I I (cm-I)a .026 .02 X 0 0 2 I (cm-I)a .18 .08 v I (cm/sec) 1.0 x 10 1.0 x 10 7 7 vIl (cm-I) .010 .005 v2 (cm/sec) 3.0 x 10 3.0 x 10 5 f 2'(cm'I) v1 .200 .099 2 " Total removal cross section, including I ' I, and I c f Additional Data Delayed Neutron Parameters Effective Decay Effective Decay Tyoe Delay Fraction Constant (sec )) Tyoe Delay Fraction Constant (sec )) 1 .00025 .0124 4 .00296 .3010 2 .00164 .0305 5 .00086 1.1400 3 .00147 .1110 6 .00032 3.0100 ~1

29 TABLE 4.2 Succritical Benchmark Problem Results C4*I) QX1(4*I) HERMITE( * ) FIESTA Code RAUMZEIT (4 l) WIGLE k,ff .9015507 .9015507 .9015507 .9015162 .9015473 Power / Initial Power tggconds 1.0 1.0 1.0 1.0 1.0 .1 .9299 .9298 .9298 .9296 .9295 .2 .8733 .8732 .8733 .8732 .8730 .5 .7597 .7596 .7597 .7598 .7595 1.0 .6588 .6588 .5588 .6590 .6587 1.5 .6432 .6432 .6433 .6434 .6431 2.0 .6307 .6306 .6307 .6309 .6306 number of mesh intervals 120 120 120 120 96* aX 2 cm 2 cm 2 cm 2 cm 36 @ 2 cm 24 @ 4 cm 36 @ 2 cm

• FIESTA is currently limited to 100 mesh intervals.

6 9 5

30 TABLE 4.3 Kinetics r?rameters for the Subcritical Benchmark Problem (FIESTA Results) Delayed Neutron Prompt Neutron Flux Time Reactivity Fraction Lifetime Amplitude (seconds) p s T _4 -8* -2 0 .6538 x 10 .75 x 10 .21299 1.0 -2 -2 .1 .0580 x 10 .75 x 10 .21324 .92928 -2 -2 .2 .10788 x 10 .75 x 10 .21345 .87265 ~2 -2 .5 .22009 x 10 .75 x 10 .21394 .75885 -2 -2 1.0 32797 x 10 .75 x 10 .21441 .65784 ~2 -2 1.5 .31476 x 10 .75 x 10 .21436 .64235 -2 -2 2.0 .30414 x 10 .75 x 10 .21432 .62983 o is a measure of the convergence to a critical solution. g l i -n.

37 TABLE 4.4 Supercritical Bencnmark Problem Results (Ref. 4.1) (Ref. 4.1) (Ref. 4.1) (Ref. 4.2) Code RAUMZEIT WIGLE QX1 HERMITE FIESTA k l eff .9015507 .9015507 .9015507 .9015162 9015473 1 Power / Initial Power t=0 seconds 1.0 1.0 1.0 1.0 1.0 .1 1.028 1.028 1.028 1.029 1.029 .2 1.063 1.062 1.063 1.063 1.063 .5 1.205 1.205 1.205 1.204 1.206 1.0 1.740 1.740 1.740 1.738 1.741 1.5 1.959 1.959 1.9f9 1.955 1.959 2.0 2.166 2.165 2.125 2.159 2.165 3.0 2.606 2.605 2.606 NA 2.604 4.0 3.108 3.107 3.108 NA 3.106 4 1 4 a wr w ~ w n er

32 TABLE 4.5 Kinetics Darameters for the i Supercritical Benchmark Prnblem (FIESTA Results) Delayed Neutron Prompt Neutron Flux Time Reactivity Fraction Lifetime Amplitude (seconds) p s T a _.4 (10 sec) ~ -8 * -2 0 .6538 x l0 .75 x 10 .21299 1.0 -2 -2 .1 .02141 y. 10 .75 x 10 .21289 1.0280 -2 -2 2 .04403 x 10 .75 x 10 .21280 1.0631 -2 -2 .5 .12084 x 10 .75 x 10 .21247 1.2058 -2 -2 i 1.0 .29050 x 10 .75 x 10 .21173 1.7426 -2 -2 1.5 .30054 x 10 .75 x 10 .21168 1.9608 -2 -2 2.0 .30801 x 10 .75 x 10 .21164 2.1674 -2 -2 3.0 .3198 x 10 .75 x 10 .21158 2.6073 -2 -2 4.0 .32870 x 10 .75 x 10 .21153 3.1097 l 1 I

• c is a measure of the convergence to a critical solution.

o l e 1 i +-r y .r-

33 Figure 4.2 COMPARISON OF FIESTA AND HERMITE POWER DURING A REACTOR SCRAM (6%.'.o TOTAL ROD WORTH) j 1.0 ; 3 g g g 0.9 0.8 0.7 0.6 0.5 0.4 0.3 ~

0.2 w3o e-2 5

t E2 w $0.1 e 0.09 0.08 0.07 0.06 n 0.05 JL LEGEND xX 0.04 X x = FIESTA x4 XX XX = HERMITE 0.03 RODS FULLY INSERTED 1 I I I 0.02 0 1 2 3 4 5 TIME, SECONDS 1

3a

== l l l l t vs 5 em 2 C I l-g O 1r m d = C \\ g g w\\ ~$ w 3 C<3 = x G l 2 w C< z_ ~ w

J M,

= ::. cZ 4 m- "C m -W d w m C. 2 ) gm- < w = a xd H M C m cL C / N: X w _a 2 NC M, 3<C 2 u Cpw c-w c. m w M 3, m Z = r - [o eo 22 w -u H e w 5 -c H 2 C$ -m r-m c C" w 3 za u = 3$ < H 'I 8 c <y w j z C e-w X w C o mC W X -2 y u. y J x w = C C X o r a

u. -

) C = g m_ -c { X = x 2 C u n X I I I I I M c-m N o es c h e t.1 H J J J c d d d c c -~ 3dVHS 83 mod G3Zl"lVWh!ON

l l

1 g

0 P ( 4O ( 1 I 13 [ 6 M 3 I [ ARC 4 I S [ 3 R O 2 l I T 3 C N [ AOI E T 0 I R R 3 [ AES ) G N 8 S 2 E I N I l l l ) l l I C l U O s N 6 l Df 1 I l 2 E % 7 P 0 6 2 A 4 3 1 1 T 2 l l S A S i D T R E N 2l E P 1 4 WlA O 2i I G C I 4 O E E iS l' S 0I a ) i I 2 I 6 E r E l ug T E 0 R W" 8O I iF MO 1 C I RP E I ( M E ) R N l I T 6E l I I D 1 / B I N lG M A3 U 1 4 N A I I 1 T = E E T S T A D I E t T M 2O I i F G S I I 1 R N E l E E E I I l l F I 0 i l T D = = 1 I l E N F R A E X OO y G NC E X I 11 O( L X S I R X 6 I A P x M O X 1 4 C 2 1 X M 0O I 6 t. 4 3 2 1 0 9 8 7 0 4 3 2 3 T i i 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 g OB p<b ru"gno$ $g O ts

Figure 4.5 COMPARISON OF Tile FIESTA AND llEllMITE POWER SilAPE DUlllNG A REACTOlt SCllAM (COllE IIEIGitT = 136.7 IN) POWER SilAPE AT 40% HOD INSEllTION TIME - 1.2 SECONDS 2.8 i i i i i i i i i i i l g i g i 2.6 s 2.4 2.2 2.0 LEGEND ~ r 'w <1 a FIESTA 5 ,g un 1.6 XXXX " llEllMITE cc Eo g 1.4 til 31.2 4scc 1.0 o z 0.11 0.6 0.4 l NOD

---

N 0.2 m*

I I i i i i l I I I i l I I I i I i i g,o O 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 3 11 40 m llOTTOM NODE NUMllEll (COllE llElGitT " 136.1 INCllES) IOP

u ~ 0 P 4 O g T 1 I 3 gI i t M 's g g-ARC m_ _ 4 I S 3 g g R O 2 y-T I 3 g C N AO g E I g 0 R T I 3 g l l AE m_ S ) G 8S N E M_ I N 2E g T I A l I I l R D T M C s U 0 S H I 6N 1 D1 E E 2 I g I l E

• F 7

I P 0 6 =- = 3 A 6 D I -21 g l T N X i S s S A E D G X t R E N 2 I l E 2i E P O L X G g ~ 6 WA C l I 4 O E X E I P S S 0I I o 2I y r E R 1 t 1 D E a iu ~l E U-H 1 iW M = 8O F O I 1 C R P E ( E MI ) l l N T 6E l l [ I I D 1 l i N 7 M i A U (3 4N I A 1 1 T = E S D l EI l 2O i I 1 F G N l E E l I i 0 I I T g 1 E F l Of O 8 N C I O( } S l 6 l I l g A P M O 4 I g C 2 I g M O O T f 8 6 4 2 0 8 6 4 7, 0 f. 6 4 2 0 i 2 2 2 2 2 1 1 1 j 1 0 0 0 0 0 0 1( 42tOz c N yo j gOoOlN l a t [ t

mm P O \\_ __ 0 4l 8 N l i 3 M 6 A l g 3 I x_ I CS x 4 l 3 R x_ O T x_ 2 C N l g 3 AO x_ I E T l 0 R x I l g 3 AE S x G N ) S N x i 8 I 2 i E l D E l l T U O A I 4 C l l D R T M 6N M S l I g 2I l E E P 0 E 7 l I 8 A D 6 3 I 43 T N u = l I S i 21 iS A D E a E N G X l l T E e E P O L X 2t A I g 7 Wl C 2i G E X 4 O iS S I P E e X E R 4 g I 0l 4 r 2l u E 2 g T E iW = i l F M O E MI N l E 8Ol P I l 1 C ) ( l N T R l I DN I GE 7 i 1 B G A3 M A U 1 T =- 4N g I 1 S T D_ E E t D O_ 2O I i F G i I R_ I 1 y l N E E l T I 0 I E F l 1 l O O NC O( g I 8 Sl tf 6 A i I P M O x g 4 C I y 2 i M O 0 I 8 6 4 2 0 1 6 4, 2 0 8 6 4.2 o. T 1 0 0 0 0 o O 2 2 2 2 2 1 1 j 1 i l i E$en cmEon. n]4sg2 c

i 1 U 0 P ( 4 O T 1 1 1 l 3 1 G M 3 l A I IC x l 4 l S 3 t N l O l 2 T N M 1 3 C O AI V E T 0 l M 3 R l 1 f 4 E A S G N ) 1 8 S 4 I l N l 2 E 1 l D t i 0 C l U 1 6 N i 1 l D 2 I E 0 7 P 0 6 A I 4 3 1 I 2 1 l T S iS AD = E N 2 T l E P O l I I t 2 i 8 WA C G 4 O E l l i E 0I P S S u I I E R 0 2 I r E uiuT E 3 I l F M W = l l o I I i OE C l l l P M E ( l ) I f N T i l l l D I E I I I N7 M A 6 E 4 U 3 T l A1 A I 1 N I T = T M E S S T D l E l E t E 2 O I I l i I F G F I N 1 I E D - = I E l N i l 0 T E X i I i 1 E F G : Ol E. X l O X I 8 NC l I O( X D-S l ) O-l 6 I I A l I P M O 4 l I C 2 I I M O 0 -l I 2 1 0 0 1 7 6 6 4 3 1 o. O 1 1 1 1 0 0 0 0 0 0 0 0 0 o lI Lj4IMrubon.Oy $rO2 i it r

40 4 5.0 Soace-Time Results This section presents typical results for a space-time scram reactivity analysis. The first section (5.1) presents scram reactitivities for a typical cycle, using space-time f lition of reactivity, and compares them to the static reactivity ition. The second section (5.2) discusses the impact of using t , pace-time definition of scram reactivities on the results of analyses of design basis transients. 5.1 Scace-Time and Static Scram Reactivity Comparisons Space-time calculations more accurately predict the axial flux shape during a rod insertion than dc critical static calculations. The reason for this difference is that space-time calculations account for the presence of delayed neutron precursors, which are neglected in static calculations. The delayed neutron precursors reduce the axial tilt away from the rods, compared to static calculations, and hence increase the negative reactivity which is inserted during a scram. This effect is largest during the early portion of the scram and is nearly nonexistent at full rod insertion when both definitions of reactivity agree. Figures 5.1 and 5.2 show the results for two different initial power shapes. These figures show that at intermediate rod insertions space-time negati/e reactivity insertions due to rods can be much la ger than the predictad static reactivities. The reactivities used in the safety analysis are based on a large number of calculations such as those shuwn in Figures 5.3 through 5.6. These figures show the scram reactivities as a function of the initial power shape (ASI). Typical results are shown for both static and space-time definitions of reactivity. The lower bounds of these calculations which envelorze the scram reactivities as a function of initial ASI are used in safety analyses. These lower bound reactivities are shown in Table 5.1 for a case with 6% ao total rod worth. In order to calculate the scram reactivity for a total rod worth other than 6% ac, correction factors are developed. Scram reactivities are calculated at off nominal reactivities (such as 2, 4', 8 and 10%ao) to calculate reactivity ratios relative to the base case (base 6%). These reactivity ratios are shown in Figures 5.7 through 5.10 for the example given. Lower bound values of these calculated ratios are used to calculate scram reactivity ratios shown in Table 5.2 The values presented in Tables 5.1 and 5.2 define the negative space-time reactivity insertion as a function of initial ASI, rod insertion and total rod worth. e -n

41 5.2 Benefits of Space-Time Reactivities The benefit of using Space-Time varsus quasi-static calculated scram reactivities is the insertien of a larger amount of negative reactivity at a given CEA position. The added reactivity will slow down tha core power rise and result in a lower peak power. Also, the additional scram reactivity will cause both the core power and the core average heat flux to decrease at a faster rate following the initiation of the scram. The net result is to lower peak powers and heat fluxes and hence reduce the margin requirements (i.e., a gain in operating flexibility). This is true for all Design Bases Events (DBE's) where a trip is initiated. As an example of the benefits of using Space-Time reactivities versus the quasi-static scram reactivities, one can consider the Loss of Flow transient. For the 4-pump Loss of Flow DBE, the reduction in the margin requirement is typically 1 to 3% when calculated using the standard CESEC-TORC methodology.

42 TABLE 5.1 Reactivity insertion (%2p) during a scram from all rods out, for a 6%ap rod worth for various axial shape indices (ASI) (reactivities computed by space time methods) core height = 136.7 inches PCT ASI INSERTION .6 .4 .2 0.0 +.2 +4 +6 5 .19 .14 .10 .07 .03 .01 .002 10 .47 .35 .25 .14 .07 .02 .0045 15 .79 .62 .4 .21 .10 .04 .0075 20 1.15 .80 .5 .27 .13 .05 .012 25 1.60 1.07 .65 .33 .16 .06 .02 30 2.14 1.37 .80 .41 .19 .08 .03 35 2.70 1.72 1.0 .5 .23 .10 .043 40 3.50 2.14 1.17 .59 .27 .13 .07 45 4.25 2.50 1.37 .7 .33 .18 .10 50 4.82 2.95 1.62 .81 4 .24 .15 55 5.25 3.35 1.95 1.0 .52 .34 .24 60 5.42 3.90 2.33 1.22 .68 48 .37 65 5.50 4.35 2.75 1.51 .85 .66 .57 70 5.68 4.77 3.27 1.91 1.15 .93 .85 75 5.75 5.15 3.85 2.50 1.75 1.38 1.25 80 5.80 5.47 4.47 3.24 2.55 2.07 1.95 85 5.89 5.73 5.13 4.10 3.55 3.09 2.95 90 5.89 5.89 5.57 5.07 4.70 4.47 4.25 95 5.94 5.94 5.87 5.77 5.61 5.55 5.3 100 6.0 6.0 6.0 6.0 6.0 6.0 6.0 e

43 TABLE 5.2 Ratio of the reactivity inserted during a scram at various rod positions relative to the reactivity inserted for a 6%ap total rod worth scram fro *m ARO core height 136.7 inches space time method 5% rod insertion Total rod worth ASI %ao .6 .4 .2 0 .2 .4 .6 10 1.40 1.38 1.36 1.34 1.32 1.33 1.42 8 1.20 1.20 1.19 1.17 1.16 1.20 1.40 4 .69 .70 . 71. .72 .73 .72 .71 2 .38 .39 41 .42 .41 .40 .38 10% rod insertion ASI .6 .4 .2 0 .2 .4 .6 10 1.30 1.26 1.23 1.20 1.18 1.19 1.30 8 1.15 1.14 1.12 1.10 1.09 1.12 1.19 4 .75 .78 .80 .81 .80 .80 .78 2 .45 .47 .49 .52 .54 .52 .48 15% rod insertion ASI .6 .4 .2 0 .2 .4 .6 10 1.25 1.22 1.17 1.14 1.12 1.14 1.25 8 1.14 1.12 1.11 1.06 1.05 1.07 1.17 4 .76 .81 .83 .85 .86 .85 .79 2 .46 .51 .55 .59 .61 .59 .52 20% rod insertion ASI .6 .4 .2 0 .2 .4 .6 10 1.25 1.20 1.15 1.11 1.10 1.12 1.22 0 8 1.13 1.12 1.08 1.06 1.05 1.07 1.15 4 .76 .81 .85 .86 .86 .85- .81 2 .47 .53 .57 .61 .61 .61 .55

44 TABLE 5.2 (Continued) 25% rod insertion Total rod worth ASI %ap .6 .4 .2 0 .2 .4 .6 4 10 1.27 1.20 1.13 1.11 1.09 1.11 1.19 8 1.13 1.12 1.06 1.05 1.04 1.05 1.13 4 .74 .80 .83 .87 .87 .86 .79 2 .45 .51 .58 .63 .67 .62 .52 30% rod insertion ASI .6 .4 .2 0 .2 .4 .6 10 1.28 1.22 1.14 1.10 1.08 1.11 1.19 8 1.16 1.12 1.09 1.05 1.03 1.05 1.15 4 .75 .80 .83 .87 .87 .86 .75 2 .44 .51 .56 .64 .65 .63 .54 35% rod insertion ASI i .6 .4 .2 0 .2 .4 .6 10 1.36 1.24 1.15 1.11 1.11 1.12 1.19 8 1.19 1.12 1.07 1.04 1.04 1.05 1.1 4 .69 .80 .85 .88 .89 .85 .80 2 .39 .49 .56 .61 .67 .62 .50 40% rod insertion ASI .6 .4 .2 0 .2 .4 .6 10 1.41 1.27 1.17 1.11 1.10 1.12 1.19 8 1.20 1.13 1.07 1.05 1.05 1.06 1.12 4 .69 .76 .80 .85 .87 .84 .78 2 .38 .46 .54 .60 .65 .61 .51

45 TABLE 5.2 (Continued' 45% rod insertion Total rod worth ASI %ao .6 .4 .2 0 .2 .4 .6 10 1.50 1.30 1.21 1.12 1.12 1.13 1.23 8 1.24 1.15 1.09 1.04 1.04 1.05 1.12 4 .67 .72 .78 .84 .85 .84 .78 2 .35 .43 .51 .61 .63 .60 .48 50% cod insertion ASI .6 .4 .2 0 .2 .4 .6 10 1.52 1.34 1.23 1.15 1.15 1.15 1.21 8 1.25 1.15 1.10 1.07 1.06 1.07 1.10 4 .66 .72 .78 .83 .84 .83 .80 2 .33 .43 .50 .57 .62 .53 .50 55% rod insertion ASI .6 .4 .2 0 .2 .4 .6 10 1.55 1.37 1.26 1.17 1.15 1.14 1.17 8 1.27 1.22 1.13 1.08 1.07 1.07 1.10 4 .65 .69 .75 .81 .80 .82 .77 2 .30 .38 .48 .54 .57 .62 .57 60% rod insertion ASI .6 .4 .2 0 .2 .4 .6 10 1.60 1.45 1.30 1.18 1.14 1.17 1.20 8 1.28 1.25 1.15 1.08 1.07 1.09 1.12 4 .65 .68 .73 .80 .82 .81 .78 2 .32 .34 .39 .52 .58 .55 .50 e

46 TABLE 5.2 (Continued) 65% rod insertion (space-time method) Total rod worth ASI %So .6 .4 .2 0 .2 .4 .6 10 1.62 1.51 1.35 1.21 1.18 1.17 1.23 8 1.30 1.26 1.17 1.11 1.09 1.09 1.12 4 .65 .67 .71 .77 .75 .75 .77 2 .32 .33 .40 .5 .52 .51 47 70% rod insertion ASI .6 .4 .2 0 .2 .4 .6 10 1.66 1.56 1.42 1.27 1.21 1.2 1.25 8 1.30 1.28 1.21 1.10 1.08 1.11 1.14 4 .63 .65 .68 .72 .77 .76 .72 2 .33 .33 .37 .44 .48 .48 .47 75% rod insertion ASI . 6, .4 .2 0 .2 .4 .6 10 1.64 1.60 1.48 1.33 1.26 1.23 1.26 i 8 1.30 1.28 1.23 1.15 1.12 1.12 1.15 4 .64 .65 .67 .72 .73 .73 .72 2 .3 .32 .34 42 .46 47 46 80% rod irsertion ASI .6 .4 .2 0 .2 .4 .6 10 1.65 1.61 1.54 1.43 1.38 1.38 1.38 8 1.3 1.31 1.27 1.21 1.20 1.20 1.20 i 4 .64 .62 .64 .66 .68 .70 .72 2 .30 .315 .33 .36 .40 .43 .40

47 TABLE 5.2 (Continued) 85% rod insertion (space-time method) Total rod worth ASI %ap .6 4 .2 0 .2 .4 .6 10 1.65 1.64 1.60

1. 5T.

1.46 1.43 1.43 8 1.30 1.31 1.29 1.25 1.22 1.21 1.4 4 .63 .65 .65 .68 .68 .7 .7 2 .31 .32 .32 .34 .36 .36 .35 90% rod insertion ASI .6 -.4 .2 0 .2 .4 .6 ,10 1.65 1.65 1.64 1.61 1.59 1.56 1.55 8 1.3 1.3 1.3 1.3 1.28 1.27 1.26 4 .65 .66 .66 .67 .67 .67 .68 2 .30 .32 .32 .33 .30 .33 .35 95% rod insertion ASI .6 .4 .2 0 .2 4 .6 10 1.65 1.65 1.65 1.65 1.64 1.64 1.64 8 1.33 1.31 1.30 1.30 1.30 1.30 1.30 4 .65 .65 .65 .65 .65 .65 .65 2 .32 .32 .32 .32 .32 .32 .32 100% rod insertion 3 ASI .6 -.4 .2 0 .2 .4 .6 10 1.66 1.66 1.66 1.66 1.66 1.66 1.66 8 1.33 1.33 1.33 1.33 1.33 1.33 1.33 4 .66 .66 .66 .66 .66 .66 .66 2 .33 .33 .33 .33 .33 .33 .33 J 4

00 48 77 % o e w w m n eo I I I I I m C =< A -DC m CC Si A a CT CW C 5C <O C a cC g 6 C

• • =

< = a w U E Z W < W.- C "a tri C T d m W 6 w p m a e c = o- . _m C_.4 a C wy W C e c, = ~ N _a C Cp C c w< a W o g 2 2 0 - I" -W

- =

=w fA Z_ H C Cli w l I I I I = N. C. c. o n. m. c. e o e = c 77 %

N 00 1 )C I TA ) O T l S l ( A 0 ( l T f U A O S00 1 1 L_ Y l A G MO OL R O F D ) O E Ml M Ai I T T l E N l C M E O S C I E A T A M P l l 2 R S E I 5 O T ( O S F E O SN I a C l I N l r A A D ug O P O i I F T S l l l D lE N S A N C I I D T OA HT s S v G N N OI S I T U l l ESN I YT IV I T 2 C 0 A E I I IS A ~ 0 2 1 0 6 s 4

Fiuino S.3 SPACE-TIME AND S FATIC REACTIVITY COMPAlllSONS FOR A SCRAM FilOM Allo (20% flOD INSERTION) 1.200 x LEGEND DATA POINT T HIPLETS flEACTOR MODEL OulX FIESTA FIESTA 1.000 (CYCLE 2) (STA l'lC) (STATIC) (SPACE-TIME) - FIESTA 8000 MWD /T" O O o ,] _ SPACE-TIME SPACE-TIME 4000 MWD /T + A t y. REACTIVITY z DATA POINTS 8000 MWD /T 0 + x ENVELOPE x f 500 MWD /T O x z I 0.800 f l + ((i ~

• " DEPLETED WITil ALL HODS m

OUT. ALL OTilEll CASES WEllE Z DEPLETED WITil LEAD 43 IIANK 25% INSERTED INCHEASE IN REACTIVITY O \\ y' INSERTION DUE TO "8 0 SPACE-TIME EFFECTS ] 6 O 0.400 y e ~ STATIC 'I. I OINTS 0.200 S 0.000 1 i i 0.600 -0.400 -0.200 0.000 0.200 0.400 0.600 INITIAL AXIAL SilAPE INDEX y

SI - c j E d ( 1 w I <2 t 4 HH t $U Q&KN i m A H G l w w< l J = l m 1- ~ i c I e T C O Z d= ~ V o = H w 0 3 l ' '8 C M h.- <U 2 H-2 mH c44x Cm f C O. W< CWmc y o -w + - 14 < c 3 < wm sgww x" W ~ JH tm g a 23: 5 ^ w C S

• H C

4 X 8 5E + +co 5$5$ Og C J Cg .1 8 .s** N w a w e C-H<wHN y. 6 w W l x = H =J 2 .,J m a = HwHw m N WUU< O 'C 'C 'C 'C v* 29 C 2 G.Yyya 33E3 E C C C c3 N r - M2

- c.,

<c o 2222 = a H.- m2> CCCC w O CCCC + g = ?. ecem m 2.- 2@d awm g f g,,,, 2, 6 .h > C 4 $x m u-o 5c 5 $k r H = +* 94 b w C g H-N g x - 1 3 P N-m i C N I 4 c + eu a s< x w d 2 i 1 P + G yo 2 C N y, g 9 w X e l c g e 1 x+ s +* i l N c e I e '1+ l i N i i o n c 5 5 BC 5 e N K O. P ' O. m N c = l d7% 'NOl183SNI AllAllOV3B lV101

'f 0 ~. 0 vP ) .h 6 0 ) ah E M AI l TT S-r '3 E O+xZ S EI C TE FA g"4 L P P S 0 E 0 O l ( l a R i l A'S' T E 0 l ) S A T AC D W M N TI OS S T oAI x O I O EA HESD R D P T I F S LAAD F N A LCEE ( M E T Al LT A G A l R ) l l E iEi R D C E L Tl T z C XI S i I I T T UA s+0O W WN 0 l S O I 0 A QT gD L D % t S E L E 5 r~ 2 R O T A T2 0 ( E E F L .L K X S TTTT PTPN E N R ) 2 //// EUEA x 4 b-D O) OL DDDD DODB x l N SN TEE WWWW O CDL I l l I AO C MMMM 4 E l Y 0000 P AT EM e-A 5 P R R C 0000 5 ME 0005 ( l 0i O S 848 S 0 aC N 3

0. L n

I 0A Fiu YD Z i I T O X IVR A IT% L C 0 Z X AI I A (6 T E I o. N R I C 7 <4 I e, 0 I 0 -A x T g 2 S 0 z D + .w NA r O E z i M r I 4 T e + E C a A 0 P 2 x e 0 S 4 \\ x O 0 + C + g = t ee m*- 0 1. 0 0 0 0 0 0 0 6 0 0 0 0 0 00 3 3 3 3 3 3 - 1' 0 5 4 3 2 o4# dol $s$u3 >b> o4y Np

i $3 l _ g i i Ti = l ! c i I i -m <E E S wH i $N Q&xN 5 R g M s e -O t w w< l J L I 3 m I i 52 e e 4 5 g@a l a x A c H <a = m 5 o <! + x @d mC y I c A -H w m J<<C cl w Z .,a g m m ~ H as 2 W U -Gpwc W p -m c C C N R X-a 3k 4+CO m o3E l e CH _CaCg e' c a N = w a w n' l m R o a o H4 Hm m W w M JAHJLZ x M _Hwww y m 2 n wDw< C CC CJ CCCC CCCc g* =s' gg$ mama E =- <o u 2222 m $g b hhhh +/ o El / 8m m u- ,a o' S .9 > oC s v w H n X p8 a $ O j I N X V c g 3 + + 0 N 3

== c A 1 8 x l v C N S Z + S x .o m N f 2 1 Tw + a O< 1 N! g

• /

c + c e 1 a 4,x s d + I in i i E e e e e c = c' 8 e e o e c-w o = = m w d 4 6 d n W% 'NOliB3SNI AllA110V38 Tf101

$4 e Cc. e + e o \\ Q u m =s4 5g 3m c' o e + Co 9=- 1e <o e +3 e o N ro e o + 6 mO = x o< w C

,8 e

o + <w 2

e n

>w w m cQ- =- 5 owz 'E2 CG ( e' n z w r e = c l C. 5 5 m= c E$>zw I - Ej X w UN i>Oo 5 8 + + >H ro Z > w< C: i e aw e e g +4 o N. <= l c wr = -o n Cic w r z: $o = c e

• w 5 $888 kp wgo nwxc

~ -10 o a< = pw a

• +cC

<$I v. e el e g o m o +, +\\ l I i l i I i I 5 e O C C C C C C C4 m a = = c x = xw o e = n c e e d 4 o o a c' o o NOll'd3SNI AllAl10V38 3H130 Ol1VM e.

sc ~x l/ 3 V 6 a = I 8, l e

== !- 1 = 2< c e a. o o m Cc c-

p J

<c -a e N c-i a = + e ey m- { x =- m e< E=g = = + c <cw e > v. m a

==c-cWz e u. - e l = 3 e= z5p a2 cm= 9J.< 5 -- / e-gzw-m x = .z m > e- =- a< E>c c o + >. - = = u >w p= ae e ea i N wH =i =c e- -a e o = + ~> u. 8 c,<- cJ p p W.! Cic:: 2:o = "'E 8888 8 blc v. Nece +e e o yo = a .+aa He e a + e I l-I I e _ 8 8 8 8 8s 8 ) v. N. c. =. e- = c 5 NOl183SNI AllAllOV3M 7V1013H1:!O Ol1V8

5b 8 e. O o e + g ,= 7 = _-=4 5 at E* c e ? C O C_ e-ac 5o 8 o5 ~. Hc e c + c wa

== 5 o< e - e. e o + <5o ez > u. o m w =c- = o

m. u =: z I

$. m = = a ,zg-a =c .B r === es =

u. e >- m x

z w se c a e

• no e

o + s > t- = t-CE z 2m -c e a e + c < w- ~ 5P ~C er gm o e +

u. ?

g c< = c a pw Cl : e <= z:e s'E g ae g g 8 c e a w o mwac +\\ w C a o

• +oa

/ <HC + e o 8 1 t i i i i i 8 e 8 8 8 8 8 8 8# m. m. e n o n e o c = c NOl183SNI AllAllCV38 SH140 Ol1VB

57 g-1 .e e o t i 1 -x 4 u 0 S; <s 5# 3* c o + + Oo o_ c-W an$6 1 g l o3 t i n I +1 6 Ho e e + mo 3: X o< w n at e o e<e o e > o "_ u. w i a 5wz i 5

• u. H Q wg I

8m o 2e-o< e ca Cw-I, p z w g-m X -'w>2 es@ a 25 c e >P= [ t-su -e E w< se e a c e o_ <a n I d WP ~o w. 3w e, e

u. 2 or o<

a w pw cle <e 2:o $l5

1. sse l

= nwmo wlo e a + +1 a0 c

• +ca a<

Ho e o H + + I l l I I I i i 8 8 8 8 8 8 8 o n = n m. e,. J d = o c n NOllW3SNI AllAl10V38 3H130 Ol1V8 i i I i

l 58 APPENDIX A FIESTA Code - Inout Descriotion Namelist if usao for input of all data except the title. The title may consist of only one card, and for steady state continuation cases it must immediately. follow the namelist arrays of previous steady state cases with a punch in column 1. For all cards except the title card the first column of every card must be blank. 1) Title 80 characters (The first character must non-blank for continuation cases, a "+" indicates new input is added to reference case input, otherwise all input must be respecified for new reference case) 2) $ FIESTA 3) Array name KNTL (1) - number of meshblocks - N0 PTS (1100) (2) - left boundary condition - IBCL 0 = Zero Flux 1 = Zero Current (3) - right boundary condition - IBCR (4) geometry of neutron flux solution - IGEOM 0 - slab 1 - cylinder 2 - sphere (5) - type of flow - ITYPE O - no flow I parallel to neutron solution 2 perpendicular to neutron solution (6) - number of delayed neutron precursors (up to six are allowed) - IDELAY (7) - number of perturbable rods (120) - NR00s 3 (8) - numoer of time steps (no limit) - NTS [ (9) - number of changes in the time step length - NPT (120) I

59 (10) - 0,1 for short or long edit. - NEDIT (0 is recommended with detailed print specified by MPRINT) (11) - cross section feedback functions - NFEEDY 0 - linear cross section variation with feedback variables 1 - special, functional relation between cross sections and feedback variables (12) - type of control rods - NRTYPE O - top entry rods 1 - bottom entry rods (applies only when flow is parallel to neutron solution) (13) - mesh point at which cross sections are calculated for input checking purposes - NFMAC (for NFMAC = 0, no cross section check edits) (14) - type of DN8 calculation - ;10NB (ITH = 1) 0 - none (used for this application) 1 - Janssen Levy 2 - TONG (W-3) (15) adjoint calculated for every steady-state case 1 - adjoint calculated only for the first steady-state case transient is allcwed from only the first steady state case if KNTL(15)=1) (Default value = 0) (19) - time dependent reactivity table - NRHO O - no 1 yes (17) - number of mesh intervals in the fuel pellet for thermal calculation - NFUEL (Ignore unless ITH = 1) (18) - nureber of mesh intervals in the clad - NCLAD (NFUEL+NCLAD < 18) (if either of the above are less than one, the code switches to a homogeneous thermal pin model) (Ignore unless ITH = 1) (19) normal feedback (default value) 1 - Skip Feedback calculation for transient solution or calculate Feedback for only the first steady state calculation (20) no burnout (heat transfer coefficient for film boiling is not used) 1 - with burnout (IDN8) (Ignore if NDNB = 0) i

60 (21) - feedback option - ITH 0 - simplified feedback model based on enthalpy and homogeneous fuel pin model with no two phase flow calcelation (TWIGL Podel) 1 - feedback model with detail fuel pin model and 2 phase flow. (22) - number of input values for time dependent inlet enthalpy table - ITHIN 1100 (if 0 temperature (*F) instead of enthalpy is input) (23) - number of input values for time dependent inlet flow table - ITFLW 120 (24) - number of input values for time dependent baron table - ITBOR 120 (25) - number of input values for time dependent pressure table ITPRS (26) - sub-cooled void model - IVOID (ITH = 1) 0 - fog flow model 1 - Tong model with subcooled boiling (27) - film coefficient correlation for boiling - IBdIL (ITH = 1) 0 - Jens Lottes 1 - Thom (28) - flux guess - IGUESS 0 - none 1 - use input value or (for continuation cases) previously computed value (29) punch option - NPUNCH 0 - no 1 yes (30) point kinetics option - NPTKIN 0 - no 1 yes (bypasses flux shape calculation during tra,'sient) (31) - number of load demand values for time dependent tables - IDMD 120 (32) - recirculation option (ICIRC) 0 - no 1 yes (inlet enthalpy is calculated based on time delayed load and outlet enthalpy) (33) - axial power shape for void model (for ITYPE = 2) 1 - uniform 2 - cosine i ._.m,

61 (34) - axial weighting of void (for ITYPE = 2) 1 - uniform 2 - cosine (35) - for transverse flow (ITYPE = 2), the mesh interval at which axial thermo-hydraulic edits are desired - ITR (36) - for transverse flow (ITYPE = 2) the number of axial nodes for integration of the void correlation - IAXIAL (default value = 40) 4) Array name PR (1) - inverse fast velocity (sec/cm) - VI (2) - inverse thermal velocity (sec/cm) - V2 (3) - integral convergence criterion - EPSE (10E-6 default value) (4) - inlet system temperature - TIN * (for inlet enthalpy use minus sign ( F or BTV/lb) (5) - system pressure (psia)* - PRES (6) - thermo-hydraulic fuel rod - DE equivalent diameter (ft.) (7) - fuel density in either lb/ft or fract. of theoretical density of 683.4 lb/ft - F0 ENS (8) - initial core flow rate" - WZERO (lb/hr) (9) power dip in the center of the pin; power is distributed according to a parabolic fit - PDIP (ITH = 1) (10) - initial boron concentration * (PPM) - 80R0 (11) - fraction of the pin power which is generated directly in the clad - PCLA0 (ITH = 1) (power generated in clad = PCLAD* (1-FRAC (I))* P(I) -g (12) - convergence criteria for the thermo-hydraulic equations - EPSSO (ITH = 1) (13) - reactor steady state power (MWT) - PZERO "For time depencent variations of these parameters the above values are the initial values which are automatically placed in the tables as the time-zero entry.

62 2 (14) - cross sectional area of the reactor (cm ) - SCAREA (axial l problems) (15) - height of the reactor - HIGH (radial proolems) (cm) a (16) - depth of the reactor - WIDE (slab problems) (cm) (17) - reference water density - REFOR (this value is only used to calculate cross gections of control rods for input checking purposes) (lb/ft ) (18) - fuel pellet radius (cm) - RPELLET (ITH = 1) (19) - radial gap thickness (cm) - TGAP (ITH = 1) (20) - clad thickness (cm) - TCLA (ITH = 1) (21) pointwise convergence criterion - EPip (defaults to 100 X EPSE) (22) - water density for cross section calculation for input checking 3 purposes - (ib/ft ) - RH0W1 (23) - same as above - RHOW2 (24) - same as above - RHOW3 (25) - same as above - RHOW4 (26) - reactor scram power (MWT) - PSCRAM (rod scram is initiated where the power exceeds this value) (27) - fuel temperature for cross section calculation for input checking purposes (*F) - TEMP 1 (28) - same as above - TEMP 2 (29) - same as above - TEMP 3 (30) - same as above - TEMP 4 (31) - velocity coefficient of the scram rods ASCRAM (cm/sec) (32) - velocity coefficient of the scran rods - BSCRAM (cm/sec ) 2 rod position = (ASCRAM)

• t + (BSCRAM) *t (33) - maximum change in the amplitude function (default = 10) between flux shape recalculations - RMAX (34) - minimum time step (default.001 sec) - this value is only used if a very small time step is calculated when PR(33) is near unity. Otherwise it is not used.

63 (35) - velocity coefficient of the scram rods in a second time interval - ASCRAM2 (36) - same as above - BSCRAM2 (37) - reference boron concentration - REFBOR o (38) - reference water density for baron input cross sections - DENS 1 4 (39) - reference water density for boron input cross sections - DENS 2 2 (40) gap conductance limit - GPLIM (BTU /ft F hr) (41)*- water volume between outlet of the reactor and the load (FT3) - VOL1 (ICIRC = 1) (42)*- water volume between outlet of the reactor and the inlet of the reactor (FT3) - VOL2 (ICIRC = 1) (43) - fraction of non-moderator volume which is fuel - FUELRF (default = 1.0) (44) - volume of the reactor inlet plenum (FT3) - VOLPLN ideal mixing is assumed to occur with the water entering the plenum; this model is used to calculate the inlet enthalpy only (for ICIRC = 1)

• These volumes are used to calculate the flow time delay by the following formula:

I = 3600*V* /Wc The density is computed from the inlet enthalpy; the volume therefore need not be the actual physical volume (as in steam lines for example) as long as T is computed correctly. The input volume should be as follows: 1W I V=V 9-E 1 p 9 I I I where V p ai.. W are the actual volume, density and flow in the pipe p is the density at core inlet conditions W is the core flow rate e

64 5) Array name BETA (I - i, IDELAY) Delayed neutron ft.ction for each precursor group. 6) Array name LAMn (I = 1, IDELAY) Oecay constants of the delayed precursors (sec'I). 7) MESH I = 1, NOPTS special' mesh intervals (cm) 8) BUKF I = 1, NOPTS pointwise transverse fast buckling (cm~2) 9) BUKS I = 1, NOPTS Pointwise transverse thermal buckling (cm-2) The followina are all oointwise arrays I = 1, N0 PTS

10) RD1 reference group one diffusion constant

11) RAl reference group one absorption cross section 1

12) RR reference removal cross section

13) RNFl reference nu sigma fission in group one

14) RFl reference sigma fission in group one

15) R02 reference therma' diffusion constant

16) RA2 reference thermal absorption cross section

17) RNF2 reference thermal nu sigma fission

18) RF2 reference thermal sigma fission The following are the feedback definitions if NFEEDY = 0 (if all the following are zero no feedback is performed) 4 I = 1, NOPTS

19) 001 pointwise change in 0), per change in moderator density (cm/g/cm#)

20) DAl same for Ial

65

21) DR same for Ir

22) DNFl same for vI )

f

23) 002 same for D 2

24) DA2 same for Ia2

25) DNF2 same for vI f2

26) DDIDF pointwise change in D per change in the square root of the j

absolute fuel temperature (cm/ R)

27) TAl same for Ial

28) TR same for Ir

29) TNFl same for vI )

f The following are the feedback definitions if NFEEDY = 1 I = 1, N0 PTS the following functions are used FUNCTION 1) I = a + bp + c U + d p i T ' 2) I = a + b in (p + po) 3) I = a + b -i T'+ c in (p + po) + d i T'In (p + po) 3 where p = water density in g/cm for all functions the value of a, b, c and d is computed from the reference cross sections

30) DDIDF pointwise value of constant "b" in equ. 3 for D j

31) 001 "c" equ. 3 for 0)

32) DDIDQ "d" equ. 3 for D)

33) DDIDR (g/cc) "po" equ. 3 for D)

34) TAl "b" equ. 3 for Ial

35) DAl "c" equ. 3 for Ial

36) DSIDQ "d" equ.-3 for Ial

37) DS1DR (g/cc) "po" equ. 3 for I al t

i

66

38) OR "b" equ.1 for I r

39) TR "c" equ. I for 2 r

40) OROQ "d" equ. 1 for I p

41) ONF1 "c" equ. 3 for vI )

f

42) TNF1 "b" equ. 3 for vI )

f

43) ONT10Q "d" equ. 3 for vI )

f

44) DNF10R "po" (g/cc) equ. for vi )

f

45) 002 "b" equ. 2 for 02

46) 0020R "po" equ. 2 for 02

47) DA2 "b" equ. 2 for Ia2

48) 0520R "po" equ. 2 for Ia2

49) DNF2 "b" equ. 2 for vIf2

50) ONF20R "po" equ. 2 for vI f2 Control Rod Cross Sections The following functions are used to evaluate control rod cross sections 1) a2 = a + b in (p + po) 2 2) a2=a+p+cp

51) R0001 rodwise value of "a" in equ. I for a0) I = 1,NrcOS

52) OlFR00 "b" equ. 1 for a0)

53) 02FR00 "po" equ. 1 for a0)

54) R0051 "a" equ. I for AIal

55) 'AlFR00 "b" equ. 1 for al )

g

56) A2FR00 "po" equ. 1 for aI,)

57) RODR "A" in equ. 2 for aI 1

r

58) RlFR00 "b" in equ. 2 for AIr

59) R2FR00 "c" in equ. 2 for aIp

67

60) R00NF1 "a" in equ. 2 for avI )

f

61) FIFR00 "b" in equ. 2 for avI )

f

62) F2FR00 "c" in equ. 2 for av2 fj

63) R0002 "a" in equ. 1 for a0 2

64) OlSR00 "b" equ. 1 for 102

~ 02SR00 "po" equ. 1 for a0 SS) 2

66) R00S2 "a" equ. 1 for aIa2

67) A15R00 "b" equ. 1 for AIa2

68) A2SR00 "po" equ. 1 for ala2

69) R00NF2 "a" equ. 1 for avIf2

70) FIFR00 "b" equ.1 for avIf2

71) F2FR00 "po" equ. 1 for avIf2

72) R0WGT rodwise weighting factor; all macroscopic control rod cross sections are multiplied by this value; this allows easy calibration of each rod to the measured value

73) POSINT rodwise initial position of the tip of the rod from the bottom of the core (cm) for ITYPE = 1 and NRTYPE = 0 the rod cross sections are overlayed between POSINT and the top of the Core for ITYPE = 1 and NRTYPE = 1 the rod cross sections are overlayed between POSINT and the bottom of the core

74) NESHLT -

left and right intervals MESHRT - boundary for rod overlay when ITYPE = 2 the fraction of rod (k) which is inserted is equal to *POSINT (K) - HIGH)/HIGH

75) TSTART - time (sec) at which the rod (k) starts moving; negative value indicates time delay for scram rods a

76) R00 RAT (1) linear velocity coefficient of rod 1 (2) quadratic velocity coefficient of rod (3) linear velocity coefficient of rod 2 (4) quadratic velocity coefficient of rod 2 etc. (2 x NR005 values 140) l

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77) TSTART2 - time (sec) at which the rod (k) starts moving according to the following coefficients, for scram rods this is a time delay

78) RdORAT2 - velocity coefficients in the second time interval K - 1, 2.....

2 x NR005 140

79) DIST -

the maximum distance (cm) that each rod is allowed to move

80) TMSTEP - time step in seconds (NPT values 120)

81) NOSTEP - time limit (sec) to which TMSTEP applies (NPT values 120)

82) CVOL -

pointwise coolant volume fraction 0<CV0L<1.0

83) AREA -

poig}wiseheattransferareadividedbycoolantvolume (ft ) (for detailed fuel pin thermal model this quantity is computed) AREA (I) = 60.96 (1 - CVOL (I))/(CVOL (I) *Rclad) R = clad outside radius (cm) clad

84) COND -

point wise gap conductance BTU /hr ft2 op if a homogeneous fuel pin model is used, this value is the overall conductance if a detailed fuel pin model is used, this value need not be specified; however, if it is specified, it overrides the cal-culated value of the gap conductance

85) FRAC -

pointwise fraction of the power which is generated directly in the coolant

86) RTF -

pointwise reference fuel temperature ( F) 3

87) RDEN -

pointwise reference coolant density (1b/ft )

88) HINTA3 - inlet enthalpy (BTU /lb), input table (or teraperature ( F) for KNTL(22) < 0)

TMHIN - time (sec)

89) FLWTAB -

inlet flow (lb/hr) table time (sec) TMFLW

90) 80RTAB - Boron (PPM) table time (sec)

TMB0R i l

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91) PRSTAB - pressure (psia) table time (sec)

TMPRS

92) OMOTAB - load demand table (fraction of initial) time (sec) e THOMO

93) 0180R1 AlBORI RBOR1 pointwire change in macroscopic NFlBORl parametars for 0280R1 refereace boron concentration A280R1 at water density (DENS 1)

NF280R1

94) 01802 AlBOR2 RBOR2 pointwise change in macroscopic NF180R2 parameters for reference boron D280R2 concentration at water A280R2 density (DENS 2)

NF2BOR2

95) NPRINT -

for NEDIT = 0 this array specifies the time points at which a detailed print is desired If MPRINT(1) = -KK signifies a detailed print every KKth time step

96) FLXS - thermal and fast flux guess FLXF - for IGUESS = 1 97) last card of namelist has "S" anywhere except in column 1 A blank card must follow the input if a transient calculation is desired.

If several transient calculations are to be performed, each case must be followed by a blank card. If only steady state cases are desired, no blank card should appear between cases. For continuation cases, if a "+" appears in column one of the title card, only the changes need to be specified. If a "+" does not appear in column one, the entire input is initialized to zero and all the input values must be re-specified.

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