ML20206Q125

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Rev 0 to Methods for Generation of Core Kinetics Data for RETRAN-02
ML20206Q125
Person / Time
Site: Oyster Creek
Issue date: 02/25/1987
From: Cabrilla D, Fu H, Furia R
GENERAL PUBLIC UTILITIES CORP.
To:
Shared Package
ML20206Q111 List:
References
TR-033, TR-033-R00, TR-33, TR-33-R, NUDOCS 8704210346
Download: ML20206Q125 (70)
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Text

I TR-033 I

i Methods for the Generation of Core Kinetics Data fo: RETRAN-02 TR-033 (Rev. 0)

BA No.: 335430 D. E. Cabrilla I H. Fu R. V. Furia C. B. Mehta Authors Date: February 25, 1987 l Approvals: l I .

l S 2-% -87 Nuclear Analysis & Fuels Director Date GPU Nuclear 100 Interpace Parkway Parsippany, New Jersey 07054 8704210346 870415 PDR ADOCK 05000219 P PDR k-

ABSTRACT I The methods for calculating the point and 10 kinetics input for the Oyster Creek RETRAN-02 model are presented in this report. The point kinetics methods use the adiabatic approximation with the nodal code, NODE-8, for j generating void, Doppler and scram reactivity. EPRI developed methods, using I

the SIMULATE-E and SIMTRAN codes, are used for the 10 input. The methods were verified against plant startup test data, by comparison of 10 and 3D calculations and with sensitivity studies.

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t ACKNOWLEDGEMENTS I Several individuals contributed to the preparation of this report. The

' authors would like to especially thank Prof. A. F. Henry of MIT for his valuable contribution, comments and review of this effort.

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I Table of Contents Section Page I

1.0 INTRODUCTION

9 2.0 P0 INT KINETICS 11 2.1 RETRAN Point Kinetics Model 11 2.2 Delayed Neutron Fraction and Prompt Neutron Lifetime 17 2.3 Reactivity 20 2.3.1 Doppler Reactivity 20 2.3.2 Void Reactivity 21 2.3.3 Scram Reactivity 23 I 3.0 1D KINETICS 24 3.1 Cross Section Collapsing Procedure 26 3.2 Perturbation Procedure 28 3.3 Reflector Cross Section Procedure 29 3.4 Radial Leakage Correction Procedure 31 3.5 Control Rod Model 31 I

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l Table of Contents Page Section i

e 4.0 Verification 38 4.1 Comparison to Measured Data 38 4.2 Comparison of 10 and 3D Calculations 40 4.3 Sensitivity Studies 42 I 5.0 References 69 TOTAL PAGES 70 I

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I LIST OF TABLES Table Title Page I 3-1 Collapsing Equations Used in SIMTRAN 36 I 4-1 SIMULATE /RETRAN Comparison: Static Rod Bank Eigenvalue 46 I 4-2 Void and Doppler Reactivity Coefficient Variation During 47 Core Life I 4-3a Thermal-Hydraulic Perturbation and Resulting Void 48 Coefficient at B0C 4-3b Thermal-Hydraulic Perturbation and Resulting Void 48 Coefficient at E0C i

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I LIST OF FIGURES  ;

I Figure Title Page 4-1 Dome Pressure Perturbation for EPR Setpoint Test (STP-19) 49 4-2 Normalized Power for EPR Setpoint Test (STP-19): Point 50 Kinetics Model 4-3 Normalized Power for EPR Setpoint Test (STP-19): 10 51 I

Kinetics Model 4-4 Normalized Power for Reactor Water Level Setpoint Test (STP-21): 52 I Point Kinetics Model 4-5 Normalized Power for Reactor Water Level Setpoint Test (STP-21): 53 10 Kinetics Model 4-6 Axial Power Comparison Cycle 1 SP309: 0 Ft Rod Bank 54 4-7 Axial Power Comparison Cycle 1 SP309: 2 Ft Rod Bank 55 4-8 Axial Power Comparison Cycle 1 SP309: 4 Ft Rod Bank 56 4-9 Axial Power Comparison Cycle 1 SP309: 6 Ft Rod Bank 57 4-10 Axial Power Comparison Cycle 1 SP309: 8 Ft Rod Bank 58 4-11 Axial Power Comparison Cycle 1 SP309: 10 Ft Rod Bank 59 4-12 Axial Power Comparison Cycle 1 SP309: 12 Ft Rod Bank 60 4-13 Axial Power Comparison E0C 10: 0 Ft Rod Bank 61 4-14 Axial Power Comparison E0C 10: 2 Ft Rod Bank 62 4-15 Axial Power Comparison E0C 10: 4 Ft Rod Bank 63 I

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I I LIST OF FIGURES I Page Figure Title 4-16 Axial Power Comparison E0C 10: 6 Ft Rod Bank 64 4-17 Axial Power Comparison E0C 10: 8 Ft Rod Bank 65 4-18 Axial Power Comparison EOC 10: 10 Ft Rod Bank 66 4-19 Axial Power Comparison E0C 10: 12 Ft Rod Bank 67 4-20 Scram Reactivity Curve 68 I

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1.0 INTRODUCTION

This report describes the methods used by GPUN to calculate the kinetics input to the RETRAN-02" ' computer code. RETRAN is the transient anal-ysis code used by GPUN for its Oyster Creek reload core analysis. It can accept input for either point kinetics or one-dimensional (ID) space-time kinetics. GPUN has methods for providing both types of input.

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.The point kinetics parameters are determined from standard definitions inherent in the point kinetics formalism using one-group nodal equa-tions. The one-group simulator code N00E-8"' is used to calculate void and Doppler reactivity feedback and scram reactivity for input to RETRAN. The delayed neutron fraction and prompt neutron lifetime are calculated using the computer code BELLEROPHON"' which uses power and fuel exposure distributions from N00E-8. This methodology is discussed in Section 2.0, For the 10 kinetics input to RETRAN, GPUN uses an EPRI developed methodology consisting of the computer codes SIMULATE-E"' and SIMTRAN"' to generate the required input. A discussion of the EPRI methods is contained in Section 3.0.

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The verification of the methods presented in this report is the demonstration that the kinetics input to RETRAN are correctly generated.

This is done by showing that the kinetics input, both point and 10, I

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provide the correct response to a core perturbation. The 10 kinetics input generation is further vertfled by comparing the 10 results in RETRAN to the 30 simulator results for different steady-state

, conditions. Finally, sensitivity studies are included to show that the point kinetics parameters are being applied in a conservative manner.

The verification work is presented in Section 4.0.

The vertftcation work does not address the adequacy of the method, point or 10, for use in a particular transtant. The adequacy is determined, in part, by the limitations that come from the approximations made in the development of the methods. These 11mitations are accounted for in the application of a particular method to a transient. Beyond these l

limitations, the adequacy of a method, in the context of a reload analysis, is the abtitty to provide a conservative response (for minimum critical power ratto, peak pressure, etc.) in the transient under consideration. The adequacy of the kinetics method is then linked to the performance of the overall RETRAN model. Therefore, the determination of the adequacy of the point or 10 method for a transtant must be addressed with the RETRAN model, which is beyond the scope of this report. The application of the point or 10 model to different transients will be discussed in the GPUN Topical Report on RETRAN.

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I i 2.0 POINT KINETICS 2.1 RETRAN Point Kinetics Model The point kinetics equations as given in the RETRAN manual are:

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$= 0~0+

n+ AM6 (2-1) dt ,

A Ak, a i.i I

and (2-2) 3, d

dt

  1. 3_g A I where the coefficients or " inner products" were defined as:

5 I g = 1 <Y*,

-M + 1(F" + 1,xfFf) Y>

6 -

F k, I

, i.' -

(2-3a) p, i h <Y'. xfFfW>,# = #,

'- 4-1 (2-3b)

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( - V D, 7 + I., + Is 2) 0 '.

M= 3d

.-I t 2 -Y'0 27 +1 2-I I l 1

E

I A = 1 G*,1 > (2-3d)

F v I

F = G *,(F' + d (2-3e) n Ff)W>

I i.: ,

I G*, n dC,> .

(2-3f) n, =

I The conventional point kinetics equations are well known and are presented in Reference 6, and some addlltonal limitations thit apply to the expression for reactivity given in 2-3a.are addressed in Reference 7.

I In these equations, the amplitude function was derived from the source-free, continuous energy diffusion by separating the neutron flux.

I (2-4) f(r,E,t) = n(t)P(r,E,t)

The point kinetics equations make the assumption that since most

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perturbations disturb the shape of the flux in only a small way  !

l I (particularly for small, tightly coupled cores), the time-dependent shape function can be replaced by a time-independent shape func, tion. In other words:

f(r,E,t) = n(t)Y o(r,E) (2-5)

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The time-independent shape function is usually taken to be the unper-turbed flux shape.

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The form of the amplitude function equation is not altered by Eq. 2-5.

The inner products definitions of the relevant kinetics quantitles, however, are. Since no method is available to determine the time-dependent flux, Y , Eq. 2-3a must be modified. Most developments result in the usual perturbation theory reactivity definition. The reactivity definition in this form is I a 1<Y',(-dM + 1 F) d Yo>

e F ko (2-6)

I approximated as where 6 Indicates the difference between the "just critical" reactor parameters and the perturbed parameters. Analogous expressions for the remaining inner product terms can also be developed.

However, the expressions for #. A, and F lead to practically constant quantitles.

I In order to arrive at the RETRAN point kinetics form, some preliminary definitions are required.

let:

c=^

i f

I (2-7a)

I and I R(t) = 0- = reactivityin dollars.

(2-7b)

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Substituting these definitions into Eq. 3-1 and 3-2 yields I  ;

=

- n(t) + EAC,(t) (2-8)

I dC,(t) , 0,n(t) - C,(t). (2-9) dt #1' Power generation from fission sources may be spatially distributed within the point kinetics approximation. The distributions of the power and also of any reactivity feedback effects are fixed during initialization, with each region following the same time behavior. The power level is controlled by the reactivity term, R(t), in Eq. 2-8 and scaled by the initial power level.

B Contributions to the system reactivity include explicit' functions of time which simulate control mechanisms, and feetback reactivity effects from fuel and moderator changes. Spatially dependent reactivity coefficients may be specified for feedback effects. At any time the system reactivity is given by I R(t) - R, + (R(t)- R(0)],,, + KR,(t)-ER,(0) (2-10)

I where I R = initial reactivity (must be zero for steady state)

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I R., = explicit reactivity function I R, = feedback reactivity from i-th spatial node. l 8 Both the explicit reactivity and feedback terms are normalized such that their initial values are zero. Feedback effects considered in RETRAN include moderator density, fuel temperature, and water temperature. The form of the equation for the 1-th region is 0 i i R,(t) = W'; R, , j + W"nRg Tp(t) l + a'p7Tp{t) + a'wrT'w(t) g ,jj)

I where W; = weighting factor for moderator density reactivity R, = water density reactivity function e' = water density in I-th region WFr = weighting factor for fuel temperature reactivity Rp7(T'p) = fuel temperature reactivity function T*p = average fuel temperature in I-th region I a'pr = fuel temperature coefficient of reactivity in I-th region a'wr = moderator temperature coefficient of reactivity in I-th region T', = average water temperature in I-th region I

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I The reactivity functions and temperatures coefficients in Eq. 2-11 are alternate methods of describing the feedback phenomena. GPUN uses the reactivity functions in its methods.

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2.2 Delayed Neutron Fraction and Prompt Neutron Lifetime I The delayed neutron fraction and prompt neutron lifetime are calcu-lated using the computer code BELLEROPHON using data generated by N00E-B. As shown in Reference 3 an expression for #.n , that is consistent with equation 2.3b and employs a one-energy group treat-ment can be reduced to l

I J fdVW(r)IpyM(r)t(r) g g, - i-'

fdVW(r)Idd(r)t(r) (2-12) 1-1 l

I where j is the number of isotopes included.  ;

I In N00E-8, the calculated parameters correspond to values averaged over the entire node. Hence, I N -J -

Iv wn Entn n Ipy En. (2-13) g , ,n.iN .i.i

-J -

X V nWnEntn Ed g n.i .i. i En.

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I where N is the number of nodes, Vn is the volume of node n, and l

~

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an - IIln-i g

Note that (I{n/En) is just the fraction of fissions in node n due to isotope j.

Consistent with this formulation and using the adjoint flux for the weighting factor N(r), BELLEROPHON calculates delayed fractions as I

N J 2

E 11ddF8VP n n g , n Nit-iJ 2

8 11dNV n.ij.i nnP (2-14)

I where F!, is the fission fraction of isotope j in node n. P.

is the relative power produced in node n, and N is the number of nodes. The square of the power results from the one group treatment" ? . The volume V of the node may be handled very easily since NODE-B uses the same volume for all nodes.

An importance factor is applied to the #i's E , pi = (Importance Factor) x pi I

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I and the core average effective delayed neutron fraction is 0*" " , ' (2-15) i BELLEROPHON calculates A in the following fashion. In a one-energy-group treatment, the mean distance traveled by a neutron from its point of birth to its point of absorption is (1/5.),

where f is the average absorption cross section for the reactor as a whole. Its average speed is the one-energy-group neutron speed

v. Since prompt neutrons are emitted instantaneously with fission, the average lifetime of a prompt neutron is 8

(1/2,) 1 (2-16)

I v V I. j I This estimate, of course, is correct only for an infinite medium.

The non-leakage probab111ty Pu for a finite medium is given simply by I

I l Put = k/k.

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I and therefore E ,

1 (2-18)

^ " {kg I

2.3 Reactivity 2.3.1 Doppler Reactivity I

' The Doppler reactivity is calculated using N00E-8. A refer-ence case is established usually at full power and full flow with the full power control rod pattern corresponding to the

, time in core life. A second case is then run with the fuel temperature at each node perturbed by fixed ratto (T,/T o). The core average Doppler reactivity coefficient is calculated by the following I

I ,, . x,< - w e>,<x,x (T,-To) e > m ,,

I where k - Core k-effective 1 - Co,e a,e, age fee, tem,e, ate,e ,

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I I The coefficient is then used to calculate the RETRAN fuel temperature versus reactivity table.

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'N00E-8 normally uses a constant value for the Doppler coef-ficient for each fuel lattice. In order to account for ex-posure effects on the Doppler coefficient, an exposure dependent Doppler coefficient is used for this calculation.

2.3.2 Vold Reactivity The void reactivity is calculated in a similar manner as the Doppler reactivity. However, the second case is run with a change in either feedwater temperature, system pressure or recirculation flow holding all other inputs constant to ef-fact a change in volds. The core average vold reactivity coefficient is calculated by E

, , (k,- ko)/(ki ko) , g (V, - Vo) AV

. I where k - Core -effective v - Core average fuel volds l

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I Since the void reactivity is strongly influenced by the power distribution, the void reactivity coefficient is not directly used to calculate the density versus reactivity input table into RETRAN. The axial variation in a v is calculated using the perturbation theory expression for reactivity (Eq. 2-6) in a form consistent with one group nodal code solution k}

given by the following equation I ANg 1S ufijg ij I ' "gv " 61 ES ufijg ij kn ii Aq,

" gy, (2-21)

I Where I Siin - Node-B Source at node ijk kijn - k-infinity at node ijk tijg - one group flux at node ijk 8 The axial coefficient is multiplied by a constant, N,, such that Eq. 2-21 is normalized to the adiabatic result.

N, " avg = '-

AVu k 3ko (2-22)

I E The normalized al's, are then used to generate the density versus reactivity table in RETRAN.

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2.3.3 Scram Reactivity The scram reactivity is also calculated by N00E-8 using a i

number of steady state calculations to simulate the control rod bank movement. The reference case is set up in the same j manner as the void and Doppler calculations followed by a l series of dependent cases with the control bank inserting in I two foot intervals. If any control rods are partially inserted, they will move in at the same rate as rods which are fully withdrawn. The void distributton calculated in the reference case is held constant for subsequent bank insertion cases. The reactivity due to scram is calculated at each bank position by I

4= (2-23)

E The scram bank position is correlated to time by assuming a scram rate that bounds the slowest insertion rate allowed by Oyster Creek technical specifications. A 3.3 ft/sec rate is used for the first two seconds followed by a rate of 1.67 ft/sec for the remainder of the scram. The scram table for RETRAN is input as reactivity vs time.

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I 3.0 10 KINETICS I The point kinetics model discussed in Section 3 is usually adequate for transients with small reactivity perturbation. For some transients, the change of shape function is so significant that the point kinetics assumption of time-independent shape function may not be adequate. For these transients, it is more desirable to have both time and space dependency in the shape function. The 10 kinetics option available in RETRAN-02 provides a more accurate shape representation for most transients. The limitation of the 10 approach is the inadequacy of modeling transients which require multi-dimensional representations such as rod drop, rod ejection, or asymmetrical core inlet water temperature.

I The RETRAN-02 kinetics solution is based on space-time factorization method. It treats the reactor core as an axial slab and solves the ID neutron diffusion equation using quasi-static approximation. The effects of thermal feedbacks and control rod motions are introduced through the change of cross sections. The initial cross sections are specified and the feedback is determined by any deviation from the initial thermal hy-draulic state. Each feedback parameter is represented by a polynomial expression of moderator density, fuel temperature, and moderator tempera-I ture. A control function is defined from the SIMULATE-E control fraction variable. As the rod moves, the cross sections will be calculated by interpolating between those obtained from the 30 simulator cases using control function as the independent variable.

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I The cross section data is supplied to RETRAN as an external set. At GPU Nuclear, this is done by using EPRI developed codes; CPM"),

NORGE"', SIMULATE-E, and SIMTRAN-E. CPM is the lattice physics code and NORGE fits the fuel lattice cross sections into the SIMULATE format.

SIMULATE is a 3D nodal code for reactor core simulation. SIMTRAN is a linkage code between SIMULATE and RETRAN-02. It translates the 3D cross The section information in SIMULATE format into the 10 RETRAN-02 format.

processing sequence of generating RETRAN-0210 kinetics input using SIMULATE and SIMTRAN is summarized as follows:

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1. Submit a SIMULATE run for the initial core configuration, followed by dependent cases corresponding to rod insertion and rod withdrawn cases. The dependent cases will have all the independent variables held fixed so that they reflect the effects of control rods. Each I SIMULATE case generates a restart file.

E 2. Read the SIMULATE restart file for the initial configuration.

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3. Collapse the SIMULATE 3D cross sections into 10.
4. Perturb the SIMULATE 30 cross sections by varying the moderator den-sity and fuel temperature and collapse the perturbed 30 cross sec-tions to 10.
5. Fit the 10 cross sections to the RETRAN-02 format.

I 6. Collapse the 30 SIMULATE control fraction radially ~and normalize it I to obtain the control function for ID kinetics.

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7. Repeat steps 2-6 for all SIMULATE dependent cases These SIMTRAN procedures are described in the sections below.

3.1 Cross Section Collapsing Procedure I The two group 3D cross sections from SIMULATE need to be collapsed to 10 for RETRAN-02 usage. In SIMTRAN, the collapsing method preserves reactivity using importance weighting. The ID analysis, using the collapsed set of 10 axial cross section and related parameters, would produce agreement with the corresponding 3D results in an average sense. Based on this constraint, Ref. 5 derives that I J f dxdyY+(z)$(z)Y g 4z) = f f dxdy(g*(x,y,z)I(x,y,z) to.(x,y,z) g (3-1) axay Axav i to (x,y,z): group g' 3D neutron flux; i tg*(x,y,z) : group g 3D adjoint neutron flux; Yg4z): group g' 1D neutron flux; Y+(z):

g group g 1D adjoint flux; I(x,y,z) : 3D cross section parameter;

$(z) : collapsed 1D cross section parameter.

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t Using the QX1 model"" and the assumption in RETRAN-02 that the

' diagonal matrix of inverse neutron speed is time independent, Ref. 5 showed that the 10 collapsed neutron flux and adjoint flux can be expressed, respectively, as Y(z)=

g ((dxdy fg(x,y,z)

Ax ay and f f dxdy f *(x,y,z) 9 fg(x,y,z) (3-3)

Y+(z) g

= (Ax)(Ay)"##

f f dxdy fg(x,y,z)

I AxAy where ax and ay are the planar dimensions. Substituting Eq.

(3-2) and Eq. (3-3) into Eq. (3 1), one have g { ^f f dxdy 4g*(x,y,z)I(x,y,z)9 4 **Y

,(x,y,z)9 } { *f f dxdy 4 (x,y,z) }

5 E(z)= (3-4) f f dxdy9 4 +(x,y,z) 4g(x,y,z)} { Axayf f dxdy fg(x,y,z)}

{Axar For the parameters which do not appear in the diffusion equations, they are collapsed to conserve physically measurable properties such as power, temperature, or enthalpy. Table 3-1 lists the collapsing equations used.

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Importance weighting requires the use of adjoint fluxes as weight factors. In SIMTRAN, the adjoint fluxes are approximated from the two group fluxes. Following the method developed in Ref. 13, it is shown in Ref. 5, that 4f(x,y,z) = fi(x,y,z). _

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  • *Y'*

42 (x,y,z) = $3(x,y,z) -

I The adjoint fluxes are computed from the base cross sections for each rod configurations and used for all perturbed cross section set for that configuration.

3.2 Perturbation Procedure The cross sections used in SIMULATE and SIMTRAN may be functions of many independent variables such as exposure, void history, moderator density, etc. However, the RETRAN-02 thermal feedback calculation depends only on moderator density, fuel temperature, and moderator temperature. To prepare 10 cross sections for RETRAN-02, SIMTRAN neglects the moderator temperature dependence. It calculates the dependence of each cross section on moderator density and fuel I

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temperature by varying these two independent variables about their initial values while holding all other variables constant at their respective values from the SIMULATE base case calculation. A 3D set of cross sections is obtained for each perturbation of the indepen-dent variables. Each of these sets is then radially collapsed using adjoint weighting to obtain a 10 cross section set. The collapsing procedure is described in Sec. 3.1. The averaged cross section set is associated with the averaged values of the perturbed independent variables.

Average values of the moderator density and fuel temperature for the base cases are obtained by volume averaging the 30 distributions

, available from the SIMULATE restart file. For perturbed cases, average planar delta for the variables are defined by input data and the corresponding 30 distributions constituted by incrementing the base distribution with weighted values of the average deltas. The weight function used at GPU Nuclear is the one that applies a constant delta to all nodes in a plane.

I 3.3 Reflector Cross Section Procedure The treatment of boundary conditions differs in the SIMULATE and RETRAN-02 core models. The SIMULATE core model uses albedo boundary conditions at the top and bottom reflector interfaces while RETRAN-02 core model permits explicit cross sections for the re-lt I E

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flector region. The reflector cross sections are input separately to SIMTRAN rather than from the SIMULATE restart file. Therefore, the reflector cross sections may not be totally consistent with the SIMULATE albedoes. To remedy the situation, SIMTRAN provides a method of adjusting the fast group diffusion coefficient in each reflector to agree with the respective SIMULATE albedo values. A second parameter is also adjusted iteratively until the normalized power in the surface node matches the averaged 10 value from the SIMULATE surface plan. The adjustable parameter may be specified as either the fast group radial leakage from the surface node or the thermal group absorption cross sections in the reflector region.

The static, one-dimensional, two-group diffusion equation is solved to obtain the group fluxes. Details of this method is described in Ref. 5.

The regional power is determined from the converged fluxes and nor-malized in thc same manner as the ID axial power distribution from SIMULATE. If the calculated power in the core region adjacent to each reflector does not match the corresponding SIMULATE value, the specified cross section parameter is adj'usted and the iteration re-peated. The iteration process terminates normally when the conver-gence criteria is met. It terminates abnormally after maximum num-ber of iterations have been taken without convergence. Correction factors calculated for the initial reflector cross sections are ap-plied to all subsequent values for the all rods withdrawn and all

,eds inse,ted cases.

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I 3.4 Radial Leakage Correction Procedure The spatial collapsing procedure, described in Sec. 3.1 does not contain terms representing radial neutron leakage losses. Neglect-ing of such losses in the ID diffusion equations results in an over-prediction of the eigenvalue and distortion of the flux shape. A separate radial leakage calculation is performed in SIMTRAN to ac-count for these losses.

I SIMTRAN assumes that the radial leakage loss of neutrons are from fast group flux only. The leakage correction terms are calculated on a planar basis from the discrete form of the ID neutron balance equations. The resulting radial buckling values are assumed to de-pend only on the controlled state and not on the thermal-hydraulic condition. Therefore, the radial buckling values were evaluated only from the base case cross section set and applied to the per-I turbed cases.

l E 3.5 Control Rod Model I

The RETRAN-02 M04BSPL"*' control rod model is used. EPRI developed this model after it learned that for some transient analyses the control rod model of RETRAN-02 M003' causes positive reactivity insertion at the start of a scram"'). The model l t

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l i i restricts the description of control rod positions in a one dimensional representation of the reactor core. Indt.vidual rod l motions cannot be followed, but each set of averaged group parameters for the kinetics equations includes the effect of any control rod present in the core. The rod position information in one dimension is limited to an axial profile of the number of control rods present in the core. All of this information is defined as a neutronic state and consists of the complete set of group parameters necessary to solve the time-dependent diffusion equations together with control rod fraction profiles used to generate the group parameters.

The current model is based on the control rod model of the SIMULATE-E code, The presence or absence of a control rod in a node is denoted by a variable with values ranging between -1 and 1 Indicating complete to no control for the node. For the 10 kinetics model, the 3D simulator control variable is averaged over the radial node resulting in 10 control variable, CT (z), where n is the neutronics state index. The control variable is renormalized so that it has values ranges from 0 to 1 corresponding to control rods fully withdrawn and fully inserted cases. This defines the control function for the one dimensional kinetics model, CF (z), as follows:

CF,,(z) = 10 - Wz) 1.0 - CTu(z) ' (3-7)

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I where CFn(z) = 1-D Kinetics model control function, I

CTn(z) = simulator 1-D control fraction, and n = neutronic state sequencer number (n=1, N).

l The model restricts all control rods, even if initially inserted to different depths, tp move together during the transient. The model simulates the motion of a control rode bank by translating the base case configuration, CFo(z),

through the core with no change in shape. The dynamic control function is then defined as CF(z,t) = CF (z-d(t)) ,

i (3-8)

I where CF(z,t) = dynamic control function, CF (z) = base case control function, 3

d(t) = distance (ft.) that rods have moved in time t (sec) after trip, The boundary condition for CF is CF(z,t) = 0.0 for z > height of core CF(z,t) = 1 for z < 0 I The reactor core is modelled by a finite number of neutronic states. These l

neutronic states have embedded within them the axial profile of control I

5 l

g_ -

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fractions by region. Cross sections from the SIMULATE-E are evaluated to provide the information for the corresponding neutronic state. A minimum of two simulator neutronic states are necessary to utilize the rod model where the first state is the base case. Additional neutronic states may contain more or less control rods than the base case representing the rod movement.

As the control fraction in a particular region changes due to rod motion, cross sections are evaluated by linear interpolation between the neutronic I states that have control fractions bounding the dynamic control fraction for the region, i.e.

L(z ,t) = g,(t)L,g.3(z) + (1.0-g3(t))h,g(z) ,

3 3 3 (3-10) 1 I Where I

((z ,t) j = cross-section set for region z; at time t, L,g.3(z) = cross-section set for control state k+1 and region z,j L,g(z)3 = cross-section set for control state k and region z 3, and g,(t) = interpolating variable between control states k and k+1 for region zj t

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I The interpolating variable g3(t) is calculated from the equation 3 l CF(z,, t)- CFg(z,)

. g,(t) - )

CFg .3(z)-CFx(z)j 3

I Where CF(z3 ,t) = control fraction for region z; at time t, CFg(z) 3 = control fraction from neutronic state k for region zj , and CFu.i(z)3 = control fraction from neutronic state k+1 for region z3 1

I The MOD 4 control model has been shown to better represent the core reactivity change due to rod motion, especially for the case with high control rod density at the beginning of the transient.

I E

I  !

l l

8 I I

\

TABLE 3-1 EQUATIONS TO COLLAPSE CROSS SECTION PARAMETERS I

d (z) = f f D fi dxdy I

i i

fftifidxdy d(2)=

2 ft2 2fdxdy fft2 t2dxdy I.

8'+'dxdy E,3(z)=II+

fffft,dxdy

, ffffI tid dxdy4 I

ri fftifixdy

  • I 42 dxdy Ea2(z) = fft2
  • a2 fft2 tadxdy

,ff42* Ir34 dxdyfft2 dxdy I 1 fft2 *+2dxdYfftidxdy II*I"I"+'d*dY I' 'In(z) fftifixdy d

~I,2(z) SSNI"*** d**Y SSN'd*dY l ffti fidxdyfft2

+

xdy d

I 'In(z)

II*k+'d*dY fftsdxdy

  • "+*d*dy

~I,2(z) =

I fft2dxdy I f tid xdy 9'(z)= fff *[V r'tidxdy i

fft2 42dxdY 1 0*(z)- fft2[V T'+2 dxdy 2

(k E - as -

I

I i 1 m e 3.i (CONTINUED)

I T, = u+u.s,+,exey g

I l (T, + T 2)

I*'"

fftivInfi xdy d + II+ivE2+2dxdy I E(z) = ff P dxdy/A h 1,(z) = ff+,dxdy/A l T2(z) = If+2dxdy/A c o(z) = ff U odxdy/A g

u (n = u r - dviA E = Averaged power 0, = Averaged reference density g

h = Averaged reference fueltemperature A = ffdxdy.

I I

8 32

I 4.0 VERIFICATION E The verification of the point and 10 kinetics methods is done through the comparison of the transient model results to measured data. The ability of the method to duplicate core behavior following a core perturbation demonstrates that the methods are reasonable and correctly applied. This is especially important in the case of the ID kinetics where the transfer of data from SIMULATE to RETRAN involves the collapsing, fitting and interpolating procedures for the cross sections.

In addition to the comparison with measured data, the 10 procedures are further tested by comparing the 3D and ID results for eigenvalue and power shape. This is done at different rod bank positions to demonstrate correct control rod worth calculations. The agreement of the 10 and.3D solutions measures the adequacy of the methods used.

I Finally, results of sensitivity studies are presented to show that the application of the point kinetics methods is done in a conservative manner. This involves both the way the kinetics parameters are calculated and the time in core life that they are applied. This is to provide the most limiting transient response for the methods presented.

4.1 Comparison to Measured Data For Oyster Creek the measured data available is from the startup test data ("'. The startup tests used in the verification are tests which consisted of setpoint changes to the pressure regulator I

I I

(STP 19) and reactor water level setpoint (STP 21). Both of these transients provide a mild power change. These were selected to evaluate the kinetics methods since the core response is primarily dependent upon the reactor kinetics rather than system modeling.

These tests were performed at 1600 MW, full flow and at operating pressure.

In the first test, the pressure regulator was decreased by a 10 psi step change and then raised by 10 pst. The subsequent pressure perturbation (Figure 4-1) resulted in a power decrease and return to its initial value followed by a power increase and return to initial power. Both the point model (Figure 4-2) and ID model (Figure 4-3) closely tracked the power changes.

In the second test, the level setpoint was decreased by 10 inches.

The decreased setpoint causes the feedwater flow to initially drop, thus reducing core subcooling. The power drops and then returns to its initial level. The comparison of the power level response for the point and 10 models are shown in Figures 4-4 and 4-5 respectively.

E The results show that both models give the correct response to a l core perturbation. Since these were mild transients and the power distributton did not change significantly, the test is consistent with the assumptions used in the point kinetics model and I l I 1 I i 4

l

I y

demonstrates the adequacy of the model in this context. For the ID model the correctness of the collapsing and fitting procedures is shown.

l 4.2 Comparison of 10 to 3D Calculations I As a further test of the collapsing and fitting procedures and to show the adequacy of the control rod worth calculation, a number of cases were run to compare the eigenvalue and axial power shape between the 30 and 10 solutions. The cases presented are for the Cycle 1 data used in the startup test analysis, which had an initial control rod density of 20.35%, and an EOC all rods out case which is typical of the conditions used in a safety analysis.

I Comparisons were made between SIMULATE and RETRAN at different control rod bank positions. SIMULATE runs were made at the initial control rod configuration and with rods banked in at two foot intervals ending with all rods inserted. Two SIMULATE runs were used to provide the 10 kinetics input to RETRAN, the initial rod I configuration and the all rods inserted case. The interpolating procedures described in Section 3.5 were used to calculate the intermediate rod bank cross sections.

The comparison of the eigenvalues is shown in Table 4-1. The I comparison of the power shapes are shown in Figures 4-6 through 4-19. The comparisons show good agreement between the ID and 3D I

I .

v . , -

I  ;

calculations for both eigenvalue and power shape, with better agreement for the all rods out case. The average difference between the 1D and 3D eigenvalue is -0.00266 1 0.00222 for Cycle 1

~

and -0.00128 2 0.00192 for EOC 10. The power shape comparisons are very good at peak nodes. The maximum peak node error from all cases is 2.8%. But as the rod bank inserts, the average error and standard deviation increase. This is especially apparent in the Cycle 1 case. This is due to the strong shift in power to the top of the core. The normalized power is quite low (less than 0.05) in the bottom of the core and this results in large percentage differences in the power. These errors are not significant due to the low power. The good agreement in the higher power regions is evident from the figures.

I The good agreement between SIMULATE and RETRAN in eigenvalue and power shape demonstrates the adequacy of the 10 kinetics input generation. It also demonstrates that the control rod interpolating procedure provides proper control rod worth during a scram. However in the case where all rods are out and the control rod bank has all rods at the same location, the procedure is more accurate.

I l I

1 l

. i

l l

4.3 Sensitivity Studies Sensitivity studies were performed to determine the most limiting combination of the point kinetics parameters. The most limiting combination is that which leads to conservative results in transient analyses. The studies looked at different time in core life and in the case of void reactivity, the thermal-hydraulic perturbation.

I The most limiting transients for OC, in terms of critical power ratio, are the power increase transients. For conservatism this would mean the smallest negative Doppler coefficient and the .

largest negative vold coefficient. For the scram curve, the conservative characteristics are the same for power increase or decrease transients. This would be a scram shape function that is low during the initial phase (about 2 seconds) of the control rod bank movement relative to the total reactivity insertion.

I The Doppler coefficient becomes less negative with core burnup.

This can be seen in Table 4-2. The EOC condition would be the most conservative. The table also shows the change in vold coefficient i with core burnup. In this particular case the peak void l coefficient occurs at a cycle energy of 4.0 GH0/MT. Since the void coefficient is dependent upon many factors, including power shape ano control rod density, the time in core burnup that peak void l

reactivity occurs can change from cycle to cycle.

I

r ,

l l I i The reactivity change due to a scram depends primarily on the power and vold distributton. The worth of a control rod is greater in regions of high power or high void fraction. The initial position l of the control rods is important since control rods that are partially inserted reach the higher power and void' regions sooner than in the case when all rods are withdrawn. This makes the end of cycle condition the most limiting for a BWR, when typically all rods are out. This can be seen in Figure 4-20. The scram worth is significantly lower at EOC than at BOC which has a control rod density of 18.6% in this case. Figure 4-20 also shows that the EOC scram curve generated using a Haling power distribution is more conservative than the one using an EOC power distribution based on l

depletion steps using a projected control rod pattern. In this case the depletion steps resulted in an axial power distribution l

l less bottom peaked than the Haling power distributton.

The neutron lifetime does not vary significantly for the different reactor conditions addressed and was not considered in determining the most limiting time in core life. On the other hand, the total delayed neutron fraction does vary with core burnup. The core average Beta value decreases as the PU-239 builds up. A smaller delayed neutron fraction results in the more rapid power increase in response to a given reactivity increase.

I

Hith the exception of the void reactivity, the EOC conditions result in the most conservative response to a power increase transient. Since the scram reactivity curve has a predominant l effect on the course of the transient, the EOC core conditions are used to analyze transients which rely on the scram to terminate the transient. For transients which do not involve a scram, the time  ;

in core life that is most limiting must be determined.

The Haling power distribution is used to generate EOC core -

conditions, since it provides a conservative scram curve. It is also the target power shape for operation. Core burnup is monitored to ensure the Haling power distribution is not exceeded such that the core is burned with a power shape that results in a less conservative analysis.

I Although the 10 kinetics does not generate reactivity coefficierits and a scram reactivity curve, the EOC conditions generated by a Haling power distribution will also be limiting for ID kinetics for .

l the same reasons. The same EOC conditions for point and ID l I kinetics models will be used in the safety analysis.

The method by which the voids are perturbed effects the void coefficient. A different void coefficient is calculated if the perturbation is due to a change in index subcooling, core flow or reactor pressure. The void reactivity coefficient was calculated

, g using each method at BOC and EOC conditions. The results are 3

I

shown in tables 4-3a and 4-3b. As can be seen the change in core flow yields the largest negative void coefficient at both BOC and EOC. The core flow perturbation will be used to determine the void reactivity in a reload core safety analyses for Oyster Creek, since it is conservative for the most limiting transients.

I I  ;

i I

I 1 I

I I

I -

1 l

TABLE 4-1 I SIMULATE /RETRAN COMPARISON:

STATIC R00 BANK EIGENVALUE I CYCLE 1 SP 309 EOC 10 ROD BANK IN RETRAN SIMULATE RETRAN SIMULATE K-EFF K-EFF K-EFF I FT K-EFF 0 .98333 .98330 .99880 .99892 2 .95504 .95708 .99590 .99598 I 4 .92449 .92866 .99310 .99339 6 .89089 .89592 .98729 .98792  ;

I 8 .84714 .85267 .97179 .97334 10 .83066 .83160 .90861 .91410 g 12 .830ee .83iei .78ei, .7ee9e i

I I - 4e -

I -

i l i

i SE TABLE 4-2 ,

1 l 1 I VOID AND DOPPLER REACTIVITY COEFFICIENT VARIATION 1

DURING CORE LIFE Cycle Exposure Void Coefficient Doppler Coefficient GWD ap/7. Vx10* __

ap/*F x 10*

0.0 -10.47 -0.2046 1.0 -10.91 -0.2043 2.0 -11.25 -0.1981 3.0 -11.32 -0.1835 4.0 -11.79 -0.1956 5.0 -10.89 -0.1803 6.0 -10.35 -0.173 I

I  ;

i I

. I I

I I

TABLE 4-3a I THERMAL-HYDRAULIC PERTURBATION AND RESULTING V0ID COEFFICIENT AT BOC Thermal-Hydraulic Vold Coefficient Percent Perturbations ao/% Vx10*_ Difference Core flow increase (3%) -

-10.18 -

Core pressure increase (10 psi) -9.97 2.08 FW Temp. decrease (20*) -8.70 14.58 TABLE 4-3b I THERMAL-HYDRAULIC PERTURBATION AND RESULTING

' VOID COEFFICIENT AT EOC Thermal-Hydraulic Vold Coefficient Percent Perturbations ap/% Vx10* Difference I

Core flow increase (3%) -8.0; -

Core pressure increase (10pst) -7.04 12.73 FH Temp. decrease (20*) -6.41 20.52 lI I

I . - . -

I I

Figure 4-1 I. DOME PRESSURE PERTURBATION FOR EPR SETPOINT STARTUP TEST (STP-19)

I 20

- Measured I ==.... Calculated I 2 m 10 -

E I E R

e 2

I

  • j a.

0 I .5 2

3 n

I "

I 10 -

I I I I I  !

-20 0 20 40 60 80 100 120 140 I Elapsed Time (sec)

I I 4e -

I

1 l

l lI I Figure 4-2 POWER CHANGE FOR EPR SETPOINT TEST (STP-19): POINT KINETICS MODEL  :

I 20

- Measured

...... Calculated I.

5 -

E.

l o . \.

m  ;

c \

2  ! i..,

O I u 3

0 6.

I "tf .

e ,

G- , .

I

\

  • g .. -

'I I -20 0 20 40 60 80 100 120 140 Elapsed Time (sec)

I g . ee .

I .

I -

1 I '

g Figure 4-3

! POWER CHANGE FOR EPR SETPOINT TEST (STP-19) 1D KINETICS MODEL i

)

I 10 Measured

...... Calculated I e cm 5 - A C .!

a

, . I, . o  ;

g u . .

I I I I 2 e

g .+'. .  !

! i s **

g 0 . ,.,. , . . . . . . . * " ~ ' ' . . . . . . . .

g ,f........,,.............................,;

I .,

t I

5 -

I i l

l l 1 l I I I 10 J 0 20 40 60 80 100 120 140 Elapsed Time (sec) l l

I

I I

I Figure 4-4 I POWER CHANGE FOR LEVEL SETPOINT TEST (STP-21)

POINT KINETICS MODEL I

I 20

. - Measured

>=***** Calculated I 10 -

I &

I i e --- -

I g 0 :

o e

I # 3

/ p .*......- ,

l

/

'/

10 -

k/#

I l

I 20 0 20 I I 40 I

60 I

80 I

100 l

120 140 Elapsed Time (sec)

I I .-. --

I I

Figure 4-5 I POWER CHANGE FOR LEVEL SETPOINT TEST (STP-21) 1D KINETICS MODEL 2a

, l

...... Calculated I 10 -

I a o .

I 2

. / ... ..

1

\ .

1 !

I -10 -

1/

tl

't I

, , , , i g .,o 0

i 20 40 so 80 100 120 140 Elapsed Time (sec)

I I

t AX A Chl R COVJARSC\

1 CYC =  :. S 330E  : O -

CD BAN <

RETRAN  : SIMULATE NORMALIZED POWER 2.0 4

1.5 - PEAK NODE ERROR. .2%

E i.0 -

NERAGE EFROFt0.1%

STANDARD DEMATION 2.0%

, f 0.0 i 2 3 4 5 6 7 8 9 10 11 12 AXIAL NODE l

GPUN 1986 FIGURE 4-6 4

l AX A_. JCW R DDV3 ARSON

=  :. S3309 : 2 DD BAN <

CYC -

i i =

RETRAN SIMULATE NORMALIZED POWER 2.0 PEAK NODE EFIRORI.7%

15 -

/

1.0 ./

/ A/ERAGE EFEOFt0.3%

STANDED DEVIATIOtt7.2% \

0.5 -

0.0 i 2 3 4 5 6 7 8 9 10 11 12 A X I A l_. M FIGURE 4-7 m 1988

1 AX A 3CW-R DCVJARSEN

! CYC_E  :. S330E  : I -

~OJ 3AN<

4 4

=

RETRAN  : SIMULATE NORMALIZED POWER 2.5 PEAK NODE ERROR -2.3%

2.0 -

4 i

l,,

c. 1.5 -

f i

i l

i M N E m 1.6 %

1.0 -

STANDARD DEVIATION 4.7%

f 0.5 -

0.0  !

6 7 8 9 10 11 12 i 2 3 4 5 i

" l

""" AXIAL NODE

' FIGURE 4-8 seuN 1986 i

l i

AX A 30^l R COVJAR SDN CY = .' S330E  : 5- 0] BAN <

l

=

RETRAN l SIMULATE i

NORMALIZED POWER 3.0 PEAK NCOE ERROR:-2.1%

2.5 -

i

, 2.0 -

i.5 -

NERAGE ERROR 4.0%

1.0 -

o.5 -

o.o

. M , , , , , , ,

1 2 3 4 5 6 7 8 9 10 11 12

! AXIAL NODE

, FIGURE 4-9 L

GPUN 1986

.l

! AX'A 3CW R CCv3ARSCN CYC =  :. S330E  : 8 -

O] BAN <

1

=

RETRAN  : SIMULATE NORMALIZED POWER

! 4 1

' PEAK NODE ERROR-1.3%

1 i

j 3 -

/

i ,

/

2 - /

A/ERN3E ERROR 14.8%

i -

STANDARD DEVIATICM17.8%

i 1 -: , , , ,

o 8 9 12 1 2 3 4 5 6 7 10 11

  • "* AXIAL NME FIGURE 4-10 GPUN 1986

i I

l

! AX A_ 3CW R CCV3AR SON CY _E  : S3309:  :.0 -

1)) 3AN<

=

FtiHAN  : SIMULATE

, NORMALIZED POWER -

A PEAK NOOE ERROR-1.3%

3 -

g -

A/ERAGE ERROR-6.6%

i -

STANDARD DEVIATICM14.5%

0- 7 8 9 12 1 2 3 4 5 6 10 11

  • "" AXIAL NOJE i FIGURE 4-11 GPUN 1986

AX A_ 3CW R DEV3AR SCN CYC_E :. S 330E :  :.2 -

0] BAN <

=

FeiHAN SIMULATE NORMALIZED POWER 4

i PEAK NOOE ERROR-1.3%

3 -

t b

i 2 -

NERAGE ERROR-0.7%

1 STANDAR3 DEVIATICM4.3%

1 1 1 1 I 0 I I I I i

o 6 7 8 9 10 11 12 i

i 2 3 4 5

""" AX1AL NODE FIGURE 4-12 i GPUN 1986 I _ _ _ _ _ _ _ _ _ _

4

! AX A DEW R CEV JAR SEN DJ l _CC :.0 : 0 -I BAN <

=

RETRAN  : SIMULATE NCH4ALIZED POWER 1.6 4

g,4 - PEAK NODE ERROR 2.8%

ii, 1.2 -

0 1.0 -

0.8 -

0.6 -

A/ERAGE ERROR-0.6%

STANDAFU DEVIATICMS.0%

0.4 j

0.2 -

0.0 234 5678 9 10 1112131415161718192021222324 1

AXIAL NODE l

FIGURE 4-13 i

GPUN 1986 1

.l i .

l

( AX~A 3CW R DCV JAR SON ECC :.0 : 2 -

DD BAN <

=

FeiHAN  : SIMULATE NOFNALIZED PCLER l 2.0 i PEAK NOOE EFHORO.02%

^

I l l 1.5 -

s i.0 -

NERAGE ERROR-1.4%

0.5 -

i g,o i 234 56 78 9 10 1112131415161718192021222324 AXIAL NODE GPUN 1986 FIGURE 4-14 i

l AX A 3Chl-R DDVJARSDN

ECC :.0 : z -

OD BA\<

FeiHAN SIMULATE

=

1 NOFNALIZED PCH R l E5 i

PEAK NODE EFHORO.2%

2.0

- i, j

$ 1.5 -

1.0 -

NEFUGE EFHOR-3.4%

STAMMFO DEVIATION 5.9%

i 0.5 -

g,g , ,

i234 5 6 7 8 9 10 1112131415161718192021222324 "

" AXIAL NME l FIGUFE 4-15

GPUN 1986

E...

i

! AX'A_. 30W R DOVJAR SDN 3

ECC :.0 : 5 -

O] 3A\<

i

=

FETRAN  : SIMULATE NORMALIZED POWER i 3.0 PEAK NODE ERRORO.9%

2.5 -

4 i

EO -

2 1.5 -

i

-6.3%

1.0 -

STAND 40 DEVIATION 7.5%

I O.5 -

a g,o 3 i234 5 6 7 8 9 10 1112131415161718192021222324 A X I A l._ m GPUN 1986 FIGURE 4-16 i

i 1

E-- --. . --- --

l AX A_. RChl R DEVJARSON l ECC :.0 : 8 - l 4]D BAN <

=

FETRAN l SIMULATE

! NOFNALIZED PCWER 4

PEAK NODE ERRORO.6%

! 3 -

. 8 i 2 -

A/ERAGE ERROR-8.9%

l

! STAN[WO DEVIATICM7.9%

! 1 i

0  ?

I 1 234 567 8 9 10 1112131415161718192021222324 "

" AXIAL NOOE

' FIGURE 4-17 GPUN 1986 I

i

i i

l l AX'A 3CW-R DEV3ARSDN i ECC :.0 :  :.0 -

OD BAN <

1

=

FETRAN  : SIMULATE l NCFNAL!7Fn M

. 7

' PEAK NOOE ERRORO.5%

6 -

5 -

l'

. 4 -

\

+

i 3 -

NERAGE EFROR-5.6%

2 - STANDEU DEVIATICM4.4%

]

1 1

l o: : : : : : : : : : : : :

i23 4 56 7 8 9 10 1112131415161718192021222324 I AXIAL NODE GPUN 1986 FIGURE 4-18 I

i AX A_ 3"W R ~"ov-)AR.So\

l ~

TJD BA\<

5

ECC  :.0  :  :.2 -

l FETRAN  : SIMLLATE ,

i i NCFNALIZED M S

i, i

4 _ PEAK NODE EFNOR-0.5%

l .

S 3 -

2 -

A/ERAGE ERROR-1.5% '

srANDED DEVIATIOtt2.3%

I

! 1-i

,,,,,,,,4:  : 3 3 3 3 3 3 g , , ,

1 2 3 4 5 6 7 8 9101112131415161718192021222324 "

" AXIAL NOOE-

! FIGUFE 4-19

GPUN 1986

"'M EE ME 6 m Emi e meer amis e auW m ur en as as - m I

4 i

I Sv" V ^

Mu v

.V . vu vV

=

BOC  ! EOC

  • EOC-HALING i

SCRAM FEACTIVITY. DOI I ARS l @ .

50 -

i

./

40 -

EOC BER=o.oo5Mp-f -

1 0 F l

7 30 -

soc sEn-o.oosos -

I 20 - 4 I

I 10 -

... g.,,. ['

m

- **f,,,g...::11F I I I f Gs, i 2 3 4 5 6 O

TIME SEOJNOS GPUN 1986 FIGUFE 4-20 l

l l

I

5.0 REFERENCES

E 1. J. H. McFadden et al., "RETRAN A Program for Transient Thermal-Hydraulic Analysis of Complex Fluid Flow Systems," EPRI NP-1850-CCM, Rev. 2, Electric Power Research Institute (1984).

I

2. R. V. Furia, " Methods for the Analysis of Bolling Hater Reactors Steady State Physics", GPUN TR 021, January 31, 1986 i 3. K. M. Smolinske, "Bellerophon A Reactory Physics Code for Calculat-Ing Core Averaged Delayed Neutron Fractions, Delayed Group Con-stants, and Reactivities as Functions of Nodal Fission Fractions, NAI-77-28, May 6, 1977.

- 4. D. M. Ver Planck et al, " SIMULATE-E: A Nodal Core Analysis Program for Light Hater Reactors," EPRI NP-2792-CCM, Electric Power Research Institute (1983).

I G. C. Gose et al., "SIMTRAN - A SIMULATE-to-RETRAN Datalink," EPRI 5.

I NP-2506-LO, Electric Power Research Institute (1982).  :

8 6. A. Henry, Nuclear Reactor Analysis, MIT Press 1976. l

7. D. E. Cabrilla, "On the Formulation of the Kinetics Parameters and the Methods Adopted at GPUN for their Calculation", GPUN TOR-730, September 10, 1986.

i I

1 l

I

8. H. Fu, R. V. Furia, " Methods for the Analysis of Bolling Hater Reactors Lattice Physics", GPUN TR 020, July 25, 1985.
9. "NORGE Code Description," EPRI NP-2019-CCH, Prepared by Science Ap-p11 cations, Inc., August 1981.
10. D. A. Meneley, K. Ott, and E. S. Heiner, " Space-Time Kinetics, The QX1 Code," ANL-7310, Argonne National Laboratory (1967).

I 11. D. C. Wade, "An Approximation to the Adjoint Constructed from the Flux." ANS Transactions, Vol . II, No.1,1968.

12. "RETRAN-02 HD4BSPL Manual," Prepared by Energy Inc., Draft, March 1986.
13. J. A. Naser, C. Lin, R. C. Gose, J. A. McClure and Y. Matsui,

" Proceedings of 3rd international RETRAN Conference," EPRI l NP3803-Sr, P.9-1, Feb. 1985.

14. Oyster Creek Startup Test Results, May 1, 1970 (NEDE-13109).

E I

i i

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h e

i f

k. .a . cw ~ _ _ - - - - . .

-. g . ,

Nf 9:

m ,. ..

'. ' r.; . 'c:. 's.

,-l. . s,',, . :

y ., ', . #-

p; ;

[. <

k's ..i.'.'.

-x . .

.f E:;:?l

v. j e 4

.?.. Efl

$l?l0W

?

N Nuclear OtMRAL MJELC UTElits

---..--i.--- . --

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