ML19327C156

Effective Diffusion Theory Cross Sections for Univ of Virginia Control Rods.

ML19327C156
Person / Time
Site:	University of Virginia
Issue date:	01/31/1990
From:	Wasserman S VIRGINIA, UNIV. OF, CHARLOTTESVILLE, VA
To:
Shared Package
ML19327C150	List: ML19327C149 ML19327C152 ML19327C153 ML19327C154 ML19327C155 ML19327C156
References
NUDOCS 8911210029
Download: ML19327C156 (114)

Text

p . ._... .

iI o Wn.

N I Effective. Diffusion Theory Cross Sections for UVAR Control-Rods LI. M- -

. ACK!10ULEDGEICllT

i This vorh una perforned with nupport fron Departocat of Energy Granto

1. Di1FG0500ER75300 jl #DEFG0506ER75273 A Thesis Presented to the Faculty of the School of Engineering and Applied Science University of Virginia In Partial Fulfillment of the Requirements for the Degree -

Master of Science (Engineering Physics) l l

l i  !

i by  !

Stuart Wasserman i n January 1990 LI

( '

) P

i , s r

i "

, .. -s,  ;

Effective Diffusion Theory Cross Sections for UVAR Control-Rods --

y _ , _ _ _ ,,  ;

3

\

}q ACK110ULEDGE!!311T E Thic vork una perfotned uith nupport frou r

, [h'" Departocat of Energy Grants lI n(

] I and

1. DEFG0500En75300 t

I. fr _ ___ #DEFG0586ER75273 _ _ _ _

I' A Thesis Presented to the Faculty of the School of Engineering and Applied Science '

, , l' University of Virginia ,

i, -

i 2n Partial Fulfillment l

of the Requirements for the Degree i Master of Science (Engineering Physics) 4 by Stuart wesserman g ,

HI , _ ,1,,,

L g-gl -

g2as5%

e 1 j

fi l, l 1

\\

\

\

1 l , l

.. o 1 TABLE OF CONTENTS j

l l- 1 l

1. Introduction . . . . . . . .. . . . . . . ,. . . . . 1 1.1 Background . . . . . . . .. . . . . .'. . . 1 I 1.2- The Problem . . . . . .-. . . .. . . . . . .

2 1.3 ~ The Solution

l. 1.4 Outline 3

5

2. Diffusion Theory Versus Transport Theory . . . . . 7 i

, . 2.1 The Transport Equation . . . . . . . . . . .- 7  ;

2.2- The Diffusion Equation . . . . . . . . . . . 8 2.3 Validity of the Diffusion Approximation . . . '10 I- 3. Description of the Computer Codes . . . . . . . . . 15 3.1 GAMTEC . . . . . . . . . . . . . . . . . . . 15. i 3.1A GAM . . . . . . . . . . . . . . . . . 16

,. 3.1B TEMPEST . . .' . . . . . . . . . . . . 21 3.2 BRT . . . . . . . . . . . . . . . . . . . . 25 5 3.3 Comparison of GAMTEC and 2DB-UM . . . . . . 26 3.4 EXTERMINATOR and 2DB-UM . . . . . . . . . . 26

4. The Transport Computer Models . . . . . . . . . . 29 4.1 The Problem . . . . . . . . . . . . . . . . 29 4.2 The Solution . . . . . . . . . . . . . . . . 30 r 4.3 The Actual Control-Rod Cell . . . . . . . . 32 4.4 Preliminary Calculations . . . . . . . . . . 32 I .

~I I .

v __ :

t l

C '!

s I 4.5 The Models 4.5.1 Modelling Limitations of the codes . .

35 35 4.5.2 Trial case constants. 41 '

4.5.3 Trial Case A - Slab Model . . . . . . 44 I

l 4.5.4 Trial Case B - Slab Model . . . . . . 47 '

M 4.5.5 Trial Case C ~ Slab Model . . . . - . . 50

+g .

4.5.6 Trial Case D - Slab Model . . . . . . 52 4.5.7 Trial case E - Slab Model . . . . . . 54 4.5.8 Trial Case F - Slab Model . . . . . . 56- -

4.5.9 Trial Case G - Slab Model . . . . . . 58- ]

Jg 4.5.10 The cylindrical Models . . . . . . . 59 g 4.6 The Output Parameters . . . . . . . . . . . 61

'4.6.1 The Thermal Group . . . . . . . . . . 61 f

4.6.2 The Fast Group . . . . . . . . . . . 62

. '4. 7 Transport Code Output . . . . . . .. . . . 63 4.7.1 Slab vs. Cylindrical Models 63 I

i 4.7.2 The Composition of the Rod Region . . 70

4.7.3 Modelling the Grooves . . . . . . . . 73 4.7.4 The Fast Group . . . . . . . . . . . 76 c 4.7.5 The Final Transport Representation . . 77

5. The Diffusion and Effective Diffusion Models . . . 79 5.1 The 2DB-UM Model . . . . . . . . . . . . . . 79 5.2 The EXTERMINATOR Cell Model . .. . . . . . 83 5.3 Effective Cross Sections . d. . . . . 85

- 5.4 Effects of the Diffusion Approximation 88 l

6. The Regulating-Rod . . . . . . . . . . r. . . . . 95

~'

I sg -

u i

y i s

_I.

E ,

b

7. Results . . . . . . . . . . . . . . . . . . . . . 97 ,

The Experimental Data 7.1 . . . . . . . . . . . 97 '

~

7.2 Calculated Control-Rod Worths . . . . . . . 98 7.3- Conclusions . . . . . . . . .. . . . . . . 100 )

i E 7.4 Recommendations. . . . . . . . . .. . . . . 100 Appendix 103

8. . . . . . . . . . . . . . . . . . . . . .

E l

LI:

L IE -

I 1I L

'g.

g

I .

Lg 3

1I i

1 l

1' l

..,_.___ _ , _ , , _ . . , , . _ . . _ _ _ . . - _ - - - . . _ . - . . ~ . . . - . ...

, . .~ _. - - . . _ _ .

I

1 I

LIST OF FIGURES I Figure 2-1 Modelling Approximations 9 I

Figure 4-1 The Fuel Elements . . . . . . . .-. 33 I

Figure'4-2 The Control and Regulating Rods . . 34 Figure 4-3 Computer Code Limitations . . . . . 38 Figure 4-4 Slab Model A vs. the Actual cell . 39 Figure 4-5 Conversion of a Slab Model into a cylindrical Model 40 t

Figure 4-6 Slab Model A vs. Slab Model B . . . 49

.g-Figure 4-7 Slab Model B vs. Slab Model C . . . 51 Figure 4-8 Slab Model B vs. Slab Model D . . . 53 ,

1 Figure 4-9 Slab Models E and G . . . . . . . . 57 Figure 4-10 Comparison of the Fluxes Generated by Cylindrical Model D . . . . . 68 l

('I Figure 5-1 Complete TWODB-UM Model . . . . . . 81 l

Figure 5-2 TWODB-UM Control-Rod Cell Model . . 82 Figure 5-3 BRT vs. EXTERMINATOR . . . . . . . 91 Figure 5-4 BRT Predictions vs.

L. TWODB-UM Predictions . . . . . . . 93 '

1 I 1

$ [

l

.. . . _ . . _ _ _ _ _ _ . _ _ _ _ _ . . _ _ _ . ~ . . . _ . _ . . _ _ ..

o 1

I LIST OF TABLES Table I A Hypothetical Boundary . . . . . . . 13

,- l 4

l Table II Summary of Initial Calculations . . 35 Table III Transport Output Parameters . . . . 65

's Table IV Model B vs. Model D . . . . . . . . 72 l (g

i Table V Model D vs. Model G . . . . . . . . 75

)

l Table VI Comparison of the Cylindrical Models 78 Table VII Effective Cross Sections for l Different Transport Models . . . . . 89  ;

! Table VIII Experimental Rod Worths . . . . . . . 97  ;

i I

Table IX Trial Set Results vs. Experimental Data 100 Table X Final Recommended Cross Sections for l TWODB-UM . . . . . . . . . . . . . . 101 *

,I i l l I

~

I .

I I ,

I

- , . ~ . _ . . . _ .

]

T i I LIST OF SYMBOLS l

l

, r - spatial coordinate -

l

! E - energy

{

b - solid angle (direction in which neutron travels)

( = neutron flux Ee

- macroscopic total cross section E, -

macroscopic scattering cross section x(E) - fission spectrum v -

average number of neutron births per fission Eg -

macroscopic fission cross section J -

neutron current I

, D -

diffusion coefficient f -

reaction rate i

Ea -

macroscopic absorption cross rection l h

Efine - fine group macroscopic cross sections l 141 number of neutrons in group i

, a - microscopic cross section s -

number of neutrons entering a group due to a source un - lethargy of nth group RI -

resonance integral Pf (u) - legendre polynomials t 1

I U(E) - Maxwellian distribution Ee -

thermal cutoff energy 1

T(x,x',E). scattering kernel I  !

I l 1

I I  !

I I l

ACKNOWLEDGEMENTS l

The author would like to thank Dr. Roger Rydin for his i

guidance.. Whether the discussion took place in his office j I or on phone calls from Charlottesville to Europe, or from i

New York to Charlottesville, ha'was always generous with his time. Thanks are also due to Mary Fehr, and Dave Freeman for all their comments and suggestions. A special thank you goes to Lisa and Kathy Wasserman. To Lisa, for running all over New York City to help with the final copy at a time when she had plenty of her own work to worry about, and to ,

Kathy for putting up with the always-growing stack of paper ,

which has occupied the corner of her living room for the l I

past year.

I  :

I i P

I I -

I I

l

. I I DEDICATION j i

To my grandfather, Ralph Roimisher. This paper is as much a product of his hard work as it is mine.  ;

I  :

E i i l 5  :

e l l l

1 l'

I e

5 I _

l l- E /

1 .

1 i

I i i

ABSTRACT In order to predict control margins for the proposed I University of Virginia Reactor (UVAR) Low Enrichment Uranium i

(LEU) core, effective control-rod cross sections need to be  !

developed and verified against experimental data. Although diffusion theory is sufficient for most of the core analysis, the strongly absorbing control-rods must be treated using transport theory.

Effective two group absorption cross sections were developed based on parameters taken from two transport theory computer codes, THERMOS and GANTEC. These cross sections were adjusted in the diffusion code, EXTERMINATOR, until cell model transpert parameters were matched.

When the effective cross sections were subsequently input into a full core model in the diffusion code TWODB-UM the calculated control-rod worths were predicted within 15%

of experimental values for each of the three control-rods.

I I

I I

l l l l

I i I 1. INTRODUCTION

1.1 Background

A number of the nuclear reactors in the United States are used exclusively for research purposes. Presently, most l l

of these reactors are powered by high enriched uranium (HEU) fuel.'HEU is considered to be weapons-grade material because j 235 it contains at least 90% 0. As a response to the possible threat of terrorism the government recently ordered all research reactors to convert to fuel powered by low enriched 235 uranium (LEU) which is enriched with no more than 20% U

.l [1]. Because each reactor has different physical

characteristics, each one requires its own redesign study, j l The study being performed on the University of Virginia l l Reactor (UVAR) uses the 2DB-UM (2) computer code to predict neutron multiplication factors during the burnup of both  !

I existing and proposed new cores. 2DB-UM is based upon i

t

. neutron diffusion theory as opposed to the more exact transport theory.

The current UVAR standard fuel elements consist of eighteen HEU fuel plates, while the control-rod elements contain nine HEU fuel plates positioned around a control-

, rod. The new LEU fuel will be enriched with approximately 235 235 20% U compared to 93% 0 in the HEU. If no other changes were made, the LEU's multiplication factor would be less 235 than the HEU's because of the decrease in U as well as the l1 c .

i 1

.I l

)

I increase in absorptions due to the extra rN . Therefore, in i

order to consider the LEU as a direct replacement it is necessary to increase the quantity of uranium in each fuel f I element. The fuel conversion study explored options such as i

~

increasing the mass of uranium per plate, increasing the  !

I number of plates per fuel element, or some combination of j

the two.

1.2 The Problem l l Before any change in a nuclear reactor core may be

implemented, it is first necessary to have the ability to I predict the corresponding effects on the neutron flux dis- f tribution throughout the core. Although obtaining solutions i

to the multi-group diffusion equation is adequate for most of a core analysis, the transport equation must be j considered in the vicinity of a strong absorber. Therefore, )

if a core study is to be done utilizing a diffusion theory computer code, it is necessary to develop a relatively i

simple and inexpensive method of incorporating a transport I

theory study of the control-rod region.

The main tool used in the UVAR conversion study was the l two dimensional diffusion theory computer code 2DB-UM. Most l

of the necessary cross sections were generated and input to l 2DB-UM by the LEOPARD and LINX computer codes, respectively.

LEOPARD was not used to compute cross sections for the control-rod regions. This did not present an inconvenience I-

'f i I ,

I l I because the diffusion theory code could not be directly i

l applied to the rod region, anyway. Therefore, the problem boiled down to choosing " effective" cross sections for the  !

rod regions in such a way that they would cause 2DB-Vit to j yield accurate results in the rod regions.

1.3 The Solution i

The first step of the effective cross section development was to obtain an accurate picture of neutron f behavior in a control-rod fuel element. Neutron transport I theory codes were used for this purpose, because transport theory's validity is not compromised in the vicinity of a f

strong absorber. Two transport codes, Battelle-Revised- l l I THERMOS (BRT) (3) and GAMTEC (4) were both applied to the l control-rod cell. The physical medelling of the control-rod's shape for input to the transport codes proved to be a  !

very significant part of the study. BRT is a one dimensional }

l L code which solves the integral transport equation in the l thermal energy range. GAMTEC solves the neutron balance ,

, equation over the energy range 0-10 Mev by breaking the l

l problem into a thermal and a fast calculation. The thermal  ;

calculation utilizes the transport equation while a  :

spatially flat flux approximation is invoked during the fast calculation. BRT's method of solution is considered to be more accurate, because it does not assume that the space and '

energy dependence of the flux are separable. Therefore the I

i accurate representation of the control-rod combines thermal -;

parameters chosen trom BRT's output with fast parameters  ;

I taken from GANTEc's description of the fast neutron  ;

behavior.  !

Next, thermal and fact cross sections generated by BRT l and GAMTEc, respectively, were used as input for EXTERMINATOR (5). EXTERMINATOR is a two-dimensional few-group diffusion theory code. When the actual cross sections i were input to an EXTERMINATOR control-rod element cell model, some expected discrepancies were found between the output of the diffusion and transport models. The I discrepancies arose because the diffusion approximation was not valid in the vicinity of the control-rod.

The basic plan of attack w&s to iteratively change the fast and thermal absorption cross sections in this region  !

I until the discrepancies became negligible. These new cross  ;

j sections were referred to as the effective diffusion theory l cross sections. It is important to note that the rod region  :

had to be treated as a black box. In other words, we could l r not exactly mimic the neutron flux throughout the region, l

but we hoped to create a situation in which most of the more important parameters seen by the rest of the cell were l fairly accurately predicted.

Finally, it was possible to use the effective cross l sections as input data for 2DB-UM. While the EXTERMINATOR l

- , . . , - ~ . . _ _ - -

f I

model was based strictly on one control-rod fuel element, the 2DB-UM model consisted of the full core; sixteen fuel or rod elements plus graphite and water surroundings. 2DB-UM was used to calculate rod worths for each of the four rods.

These rod worths were the primary parameters that were checked against experimental data in the verification of the

, effective cross sections.

1.4 Outline Following the introduction, Chapter 2 gives a short review of the differences between diffusion and transport theory. This section was included to be sure that the reader fully appreciates why the control-rods warrant special treatment in a diffusion theory study, and consequently, why effective cross sections are needed.

The computer codes are introduced in Chapter 3.

Special attention is given to the approximations that each code uses before solving the transport equation. This approach was taken because the computer codes were viewed i only as tools that could predict neutron behavior. The 1

approximations that were required dictated when and how each

" tool" should be applied.

Chapter 4 consists of the transport theory study of the control-rod element. Different physical models were compared, and the best model was selected. All subsequent transport theory data was taken from this model.

- T I -

L i

!I  !

I I chapter 5 consists of a description of the diffusion

• I theory cell model an$ the method used to generate effective j cross sections for this model.

I cuapter 6 describes how and why the less highly l

absorbing regulating rod was treated differently than the l control-rods. j Finally, the effective cross sections are verified against experimental data in Chapter 7.  ;

I I

I i

I  :

I .

I g e I - - - _ _ _ _ _ _ _ _ _ _ _ _ _ _

[: ,

l l

I 2. Diffusion Theory Versus Transport Theory l

i 2.1 The Transport Equation l

The stecdy state neutron transport equation is derived I by performing a balance over all possible neutron production l and destruction reactions in an arbitrary volume.

Productions, at a specific energy, include fission and '

scattering reactions into this energy and spatial region. j 1

l Destructions are caused by absorptions, and scattering out l l i of this energy or spatial region. When the cross sections l 3

may be considered constant throughout the volume of interest, the result of this balance is l i

A A A O.Vd(r, E,0)+I t (r.E)d(r, E,0) - [4, dfydE'I (E'-+E.

s O)d(r,E',O') ,

+ [4, d h (E')E f(E')d(r E',8') (2-1)

The only assumptions made so far are that all fission i

neutrons are emitted simultaneously and isotropically. In I every other respect this is an exact equation for a single '!

energy (energy =E) in a spatially homogenous region. A system of equations, comprised of the transport equation for ,

as many discrete energy groups and spatial regions as desirable, must be solved over the full core to completely i

determine the steady state neutron behavior, j In order to obtain these solutions in a heterogenous cell some approximations are necessary. First, the cell is t l

I - . - - .. . ._

p. , . . . . . - . _ . . . _ . ~ .

I LI I spatially sectioned.

All the nuclides within each section are homogenized, thereby yielding constant cross sections for each region. Consequently, neutron balances need be written only for a relatively small number of homogenous mixtures. The system of equations may be further simplified -

by assuming that the system is infinite in one, two or three  !

directions (Figure 2-1). The flux is then assumed constant  ;

in these directions thereby decreasing the dimensionality of l I the problem. Usually, because of the effort necessary to  !

solve the transport equation, the flux is assumed constant -

in two dimensions when it is solved.

2.2 The Diffusion Equation Another common approximation is the diffusion >

I approximation, which essentially states that neutrons will

travel from regions of high neutron density to regions of lower density. The diffusion approximation is actually the s result of the P3 approximation which states that the  ;

directional dependence of the angular flux ($(r,0,E)) is only linearly anisotropic. When the higher orders of angular dependence are ignored, and Equation (2-1) is l integrated over angle, the identities  !

I fg6(r.E,b)db - d(r,E)

I I

I-

I 5

'I l I ,

g WGMRhwHMMWffilllW l I uma num man hamn v

= X I f# # #### # 0#I#### M B !!

g

{

( % % ( E \\\\\\\\\\\ E b a \ M \\\\\\\\\\\ g y g  :

weerer a use(a e Emu a 0msamua 0h 8

' t g e n e erster< tr e stre t e rrett!

I a) I) 0)

J O J

\MtR mMM \M\\nt LEE \\\\

g e-is 2-1e i fuel fuel fuel fuel fuel l rod rod rod rod rod I  : fuel fuel ,

fuel

l. .. ... .. . . . . ,

fuel fuel a-1C Figure 2-1: Modelling Approximations I. Figure 2-1A shows one possibility for an actual core configuration. Figure 2-1B illustrates how one cell might be modelled in one dimension. Note that the model may have inhomogeneities in only one direction. Figure 2-1C portrays I a boundary implication of assuming the system is infinite parallel to the x-axis. In effect the calculation is performed on the element within the dotted box as if the I element is bordered by identical elements on either side.

Compare this to actual bordering materials shown in 2-1A.

I

J L

e 10 and f,,b'V4(rE,8)dd-V*J(r,E) f may be used to write V*J(r,E)+E(r,E)d(rE)-(dE'I,(E'*E)d(r,E')+x(E)(v(E')E(E')d(r,E')

g g

\

If D(r,E) is taken to be the constant of proportionality, the approximation is written as Fick's law, J(r,E) - D(r)V4(r,E) which leads to the diffusion equation

• VeD(r,E)V4(r,E)+I(r,E)d(r,E)-(dE'E,(E'*E)d(r,E')

g

+x(E) f* v(E')EF(E')d(r.E')dE' 2.3 Validity of the Diffusion Approximation l As mentioned earlier the diffusion approximation is not valid in the vicinity of a strong absorber. To understand this phenomenon, it is informative to know how reactor diffusion theory differs from the diffusion theory of gases.

Gaseous diffusion theory is based on the assumption of many scattering collisions between the gas particles. When a gas molecule travels into a highly populated region it has a greater chance of striking another gas particle, and consequently a greater chance of backscattering. The l

l 1

s-

)

L 11 spatial distribution of particles is flattened,because

(

particles tend to travel to less populated regions.

j Neutron diffusion theory predicts results which are similar to the gas theory results even though the logic is different. The probability of a neutron-neutron interaction is so small relative to that of a neutron-nucleus collision that the neutron-neutron collision can be neglected in nuclear reactor theory, thus making the equations linear.

The reaction rate for a neutron-nucleus reaction is given by f=T4 (2-2)

In a homogenous medium the scattering cross section is constant, so Equation (2-2) shows that the scattering frequency is only dependent upon the flux. Therefore, in a region of higher neutron density the scattering frequency P (as well as the frequency of any other neutron-nucleus reaction) is greater. As a result neutrons, like gas molecules, tend to travel to regions of lower concentration.

Consequently, the neutron's directional dependence is primarily a function of the gradient of the spatial flux.

In other words, the angular flux may be assumed to be linearly anisotropic and the diffusion approximation is valid.  %

The previous paragraph implies that the diffusion approximation should not be valid at the border between any T

/

l 5  !

l I

12 two materials with different cross sections. If this were true the approximation would not be much help at all, j Fortunately, the physical properties of most reactor  !

i I materials are close enough alike, in reference to the l

diffusion approximation, that the behavior at the border is dominated by a boundary condition which only demands that l the bordering fluxes and currents match at the boundary. l When one of the border materials is a control-rod the  !

behavior is no longer dominated by these boundary conditions. To illustrate this point consider the border

[ between a control-rod and a material which has a scattering  ;

cross section exactly equal to that of the control-rod.

l Even if the flux were spatially flat, the scattering probability would be dependent upon position thereby invalidating the diffusion approximation. Because of the I rod's extraordinarily large absorption cross section, the  !

same number of neutrons travelling into the rod create far j fewer scattering collisions (due to the increased

. probability of absorption) than if they were travelling away from the rod.

Table I illustrates this situation. The table shows ,

that even if the scattering cross section is constant, the

• large deviation in absorption cross section at a control-rod

< border invalidates any thought of a constant scattering probability. The scattering probability is no. longer solely

'l.

lE 1

1 13 L Table I A Hypothetical Boundary Neutrons Neutrons Neutrons E, E, entering absorbed scattered the in the in region region the region control-

, rod 10b 190b 500 475 25 Border region 10b 10b 500 250 250 l

a function of the spatial flux, it is also dependent on the medium's composition. The matching boundary conditions i cannot be relied on to compensate for the problem because of the large variations in the spatial flux distribution expected at this point. The angular flux cannot be considered to be only linearly anisotropic, because neutron transport in this vicinity is affected by changes in properties of the medium as well as neutron flux fluctuations.

A related problem occurs even when the diffusion approximation is applied within the control-rod region, away from the boundaries. In the rod, the probability of an

, absorption reaction dominates over the probability of any other reactions. Therefore, it cannot be assumed that a I

neutron will undergo the large number of scattering l

> l l

)

L 14 reactions required in the diffusion theory assumptions. As a

(

result the angular flux cannot be assumed to be a simple function of the spatial gradient. These problems with the diffusion theorf make it necessary to consider the transport equation when treating the control-rod.

f I

I I

I I

I I

I I

'4 15

3. Description of the computer codes Several FORTRAN computer codes are available to solve the transport equation in one dimension. Each code employs a different method of solution and therefore has it's own advantages and disadvantages. A basic understanding of these solution methods will help to maximize the benefits to be derived from the codes. The two transport computer codes chosen for this project were BRT and GAMTEC.

3.1 GAMTEC GAMTEC solves the multi-group Pg equations with sixty eight fast energy groups, as well as a continuous thermal-energy treatment. GAMTEC's principal advantage over BRT is that it allows the user to collapse these groups into one thermal and a few fast groups, and treat each group separately. It is beneficial to treat the groups differently because different reactions should be emphasized in each group. In the fast groups upscattering may be ignored while the fission spectrum, downscattering and resonance absorption determine the neutron behavior. In comparison, the thermal motion of the nuclei and chemical binding play significant roles as thermal neutrons approach thermal equilibrium with their surroundings. GAMTEC possesses the capability to treat fast and thermal groups differently because it is actually a combination of two I

1.

l codes, GAM and TEMPEST. GAM performs the fast calculations while TEMPEST does the thermal calculations.

I 3.1.A GAM The most significant point to consider regarding the fast calculation is that it is done in zero dimensions. This means that all the materials in the cell must be combined to form one homogenous mixture. An important implication of this "zero dimensionalization" is that the fast neutron flux is assumed to be spatially flat. Mathematically this is written as I $(r,E)=C4(E) (3-1) where C is a constant. In preparing the fast-group libraries for GAM, a rough estimate of the flux is made by assuming that the energy dependence is of the form 1/E so that I $(r,E)= C/E (3-2)

Now, weighted fine-group constants are calculated from expressions such as E

Nfn+1ai(E)d(E)dE i

fine (3-3)

E Jn+1(E)dE n

4 where o, are tabulated cross sections taken from a more detailed library. In some cases, which will be discussed

17 shortly, Equation 3-3 is modified to correct for obvious errors caused by the zero-dimensionalization.

These cross sections allow GAM to obtain a more accurate energy-dependent flux by numerically solving the Pg slowing down equations. The slowing-down equation for a discrete energy has the general form E,(E)p(E)+E,(E)d(E)-S(E)+[E,(E'*E)d(E')dE' (3-4)

After solving this equation, the code has access to the fluxes and cross sections for 68 energy groups. This data is used to calculate all the output parameters which are desired by the user.

Since the output parameters obtained from the solution to Equation 3-4 were vital to the remainder of the project it was necessary to verify that the assumptions involved in the equation's solution were applicable to the problem of interest. The homogenization of the cell, which was critical to obtaining the solution to the slowing-down equations,' implied a flat spatial flux. Obviously, if a physical system proved to have large spatial fluctuations, then GAM's approach could not be applied.

The flat flux approximation was supported by four facts. First, the most obvious cause of larcfe, fluctuations was easily eliminated, because there are no strong absorbers of fast neutrons in a thermal reactor. A secEnd

- T

1 1

18 justification may be obtained by noting that the primary source of fast neutrons is fission reactions. If fission events create fast neutrons which travel large distances quickly and isotropically, relative to the cell size, then a spatially flat flux is a fair assumption. Also, in a thermal reactor only a small percentage of the high energy fuel absorptions lead to fission reactions. In other words, a fast group absorption in the fuel is likely to have the same result as any other absorption. Therefore, spreading the fuel evenly throughout the cell should not be detrimental to the calculations. Finally, GAM's primary concern is the slowing down of fast neutrons to thermal energies. The main forces in this process are resonance absorption, and scattering reactions. The fraction of neutrons which slow down is determined by the ratio of absorption to scattering, and the energy loss per collision.

These properties are constant for each particular nuclide regardless of the surrounding materials. The variable is the quantity of flux exposed to each nuclide. Given that we are starting with a fairly flat spatial flux, each nuclide sees an equal number of neutrons and it follows that the absorption-to-scattering ratio may be considered not to be a function of position. Therefore, while tha homogenization of the cell eased the computation it did a minimum of harm to it's validity. _

l

ue 19 Even though there are theoretical grounds for homogenizing the cell, GAM still needs to make corrections for obvious errors due to the neglect of spatial effects.

For example, the fact that the fuel is actually lumped together leads to a higher fast fission factor than that which would be computed for a homogenized cell. In the 1

homogenized model, neutrons are exposed to the moderator immediately after their births. In reality, neutrons born a few path lengths within a fuel lump may undergo a number of collisions in the fuel before they ever reach the moderator.

Because these neutrons have not been slowed down, the probability that their energies are above the fast fission threshold is greater than the probability that would be calculated for a homogenized model.

GAM accounts for these additional fast fissions by modifying the atom densities of U 2" seen by high energy neutrons, and consequently changing the effective absorption and fission cross sections. To obtain the appropriate correction GAM relates the fuel and cell radii to calculate the probability of a neutron's first and successive collisions occurring in the fuel. This method's major restriction is that the calculation of n-flight collision probabilities is only applied to concentric geometries.

Another modification is necessary to account for the increase in the resonance escape probability due to fuel l

20.

lumping. Because the absorption crosis sections are so large for the resonance energies most neutrons in these energy ranges are absorbed near the surface of the fuel. This leads to a scarcity of neutrons in the resonance ranges in the center portions of the fuel'. Therefore, resonance absorbers located away from the fuel surface absorb fewer neutrons than they would in the homogenized model.

GAM treats resonance absorption and the spatial self-shielding effect by modifying the absorption cross section for any energy group containing a resonance peak (6). The first step in this process is to split the absorption cross section into resonant and non-resonant components. The smoothly varying non-resonant cross section is determined by Equation (3-3) as described earlier. The resonant component requires a different approach, because the resonance invalidates the assumption that the flux is of the form 1/E by significantly perturbing the flux. In order to treat the perturbation, spatial self shielding must be considered.

Therefore 'I/" is determined by E[" = N au E (RI)g ,

n i~" .

where (RI), is the resonance integral of the 1" group and un is the lethargy width of the group. The resonance integral is a measure of absorptions within a given

7 9 . ._ .

t- 's

(

21 resonance peak based upon the fuel'.s average chord length and Doppler broadening. GAM uses tables developed by Adler and Nordheim to obtain resonance integrals based upon these parameters.

GAM substitutes the modified cross sections into ths

.

fine-group slowing down equations and solves them using the method which was previously outlined. In addition to determining broad-group cross sections, the solution yields information which will be valuable to TEMPEST during the thermal calculation. The most important bit of information obtained from GAM will be the fraction of fast neutrons which survive to reach the thermal group.

3.1.B TEMPEST There are significant differences between the methods applied by GAM and TEMPEST to determine fluxes and effective cross sections. The foremost difference being that while GAM solves the energy-dependent Pg slowing down equations, TEMPEST must first solve the more complicated one-speed P 3 equations. The P3 equations are basically a set of one-dimensional transport equations which incorporate the ussumption that the angular flux may be written as the first four terms of 9(r.p,E) j[0 f f (r,E)Pj(p) l

r s

I where p (g) are the Imgendre polynomials.

g The presence of strong thermal absorbers combined with the vast differences f

between thermal neutron behavior in the fuel and moderator prevent TEMPEST from disregarding the spatial dependence of the flux. Therefore, TEMPEST must seek a solution which is dependent on both energy and position.

To simplify the problem TEMPEST assumes that the spatial and energy dependencies of the solution are separable. Mathematically,

$(r.E) = f(r)g(E) (3-5) which leads to the creation of two separate differential equations. Solving the equation in this manner implies that the spatial solution shape will'have absolutely no effect on the energy-dependent solution. This is a questionable statement with regard to the thermal group. Consider the spatial self-shielding effect discussed earlier which would lead to a dip in the flux of any highly absorbed energy in the center of that absorber. In order to account for self-shielding it is necessary to consider spatial and energy effects simultaneously. In situations where self-shielding is not a major concern the space-energy separability assumption may be applied with greater confidence.

TEMPEST's next simplification is to treat the problem in only one spatial dimension. As GAM's zero-

I ~

23 o dimensionalization implied a homogenous flux throughout the core, TEMPEST's one-dimensionalization implies a. flat fivx in two dimensions. In order to be consistent with GAM, TEMPEST must solve the transport equation in cylindrical geometry, cenerally, the core composition changes most dramatically in the radial direction so TEMPEST chooses to perform a radial calculation while assuming flat fluxes in the e and z planes. By treating the problem in only one dimension TEMPEST distinguishes between the fuel and moderator regions while still keeping the difficulty of the problem reasonable.

To begin, TEMPEST assumes that all thermal neutrons are in thermal equilibrium with their surroundings. Each concentric region is assigned a temperature by the user, and the initial flux is a Maxwellian distribution, W(E), based upon these temperatures. Thermal cross sections for each spatial region are determined by combining the flux and material composition using the expression

• E dE I" 6 "(E)V(E)

I.[Co E

f," dE W E)

A one speed P3 approximation is then used to solve the space-dependent portion of Equation 3-5 analytically for f(r). The number of separate regions, and mesh points per region at which results are to be printed are, selected by

- 1 1

\

-- - -

I' j

.t t

the user. The important results from this portion of tha calculation are the relative fluxes of each region. The re.tios of the average fluxes of each region are known as the 5 disadvantage' factors.

The disadvantage factors enable data obtained from the ,

space-dependent P3 solution to be incorporated into the f energy-dependent equation. The disadvantage factors are used to weigh the spatial atom densities according to the spatial flux before the volumetric homogenization. The I modified atom densities are used as input to the energy- l dependent equation.

The method used to solve the energy-dependent portion I of the equation is similar to the method used by GAM, but 1 the equation to be solved is the Wigner-Wilkins equation.

The Wigner-Wilkins equation is more complicated than the slowing-down equation because it must account for things like the motion of the nucleus, and upscattering.

Macroscopic crose sections, based on the P3 spatial homogenization, are substituted into the Wigner-Wilkins equation. An analytical solution yields the energy-dependent flux g(E). The final flux, from which all output data is based, is obtained by combining g(E) with f(r) as in t Equation 3-5.

LB 7

5 8

I l I .

3.2 BRT ,

An alternate, and more accurate treatment of the )

thermal group is employed by the BRT code. BRT's method allows Equation 2-1 to be solved after making only two notable approximations. By con'sidering the problem to be symmetric about one dimension, and assuming isotropic scattering, Equation 2-1 may be simplified to E

b iM + E (x,E)p(x,E,0) - 1_ f,cdE'fdO'E,(x , E'*E)u(x ,E,b) + S /x . E'(3-6) c 5

dx 4, 4, where E, is the cutoff of the thermal energy group. Because all fission births occur in the fast group, the source term for any thermal group is S(x,E) - dE'Is (r E'*E)d(r,E') (3-7)

Equation.3-7 is substituted into Equation 3-6 and the resulting equation is integrated over space and angle to

yield E

((x,E) .- fdx T(x,x' ,E)[f, dET W 'MN' s (3-8) where T(x,x',E) is a scattering kernel. The scattering kernel represents the uncollided flux at point x of energy E

~

due to a source of the same energy at x'. BRT solves Equation 3-8 numerically using up to 30 space and 30 energy  ;

points while employing either slab or cylindrical geometry.

l 3

.5

1 l

26 L 3.3 Comparison of GAMTEC and BRT Theoretically, more confidence is placed in BRT relative to TEMPEST because it performs it's calculation without the questionable assumption of space-energy l

~ separability. Computationally,~BRT utilizes 30 fine energy groups withf.n the range of 0-0.683 niection volts while the same range is covered analytically in GAMTEC. BRT's method of solution also has greater flexibility which allows the problem to be solved in different geometries.

BRT's primary disadvantage is one of cost. The iterative multigroup calculation consumes larger amounts of computer time and money. Another drawback is that BRT does not contain any type of internal calculation for the fast group.

3.4 EXTERMINATOR and 2DB-UM The major assumption invoked by both EXTERMINATOR and 2DB-UM is the diffusion approximation. This is such a great simplification that it makes it feasible to perform the calculation in two dimensions. This allows the diffusion code computer models to have much greater spatial detail than the transport models. Figure 2-1 illustrates the advantages of using a code which has the capability to treat a two-dimensional model. A one-dimensional model would not even be able to differentiate between a control-rod element and a fuel element, because this would require inhomoge-I

. . . . . . . . , . . . .. , m,,

I w ]

I neities in two directions. Therefore, whenever the applicability between a diffusion and a transport treatment is similar, the diffusion treatment is usually used.

EXTERMINATOR solves the staady-state multi-group diffusion equation. A big difference between EXTERMINATOR and the transport codes is that EXTERMINATOR demands that all cross sections be input manually. EXTERMINATOR does not treat the thermal and fast groups differently, as GAMTEC does, because it expects that the differences are accounted for in the selection of cross sections. This approach gives I the user the flexibility to choose the number and range of l the energy groups as he likes. Once the size and space points per region are chosen, the code may substitute the cross sections into the diffusion equation and solve for the flux, multiplication factor, etc. numerically. 5 I 2DB-UM has an added advantage in that it solves the l

time-dependent multi-group diffusion equation. This gives 2DB-UM the ability to calculate the burnup of the fuel, and subsequently, fluxes and multiplication factors at different times throughout the core's life. Cross sections are l calculated by the LEOPARD computer code, and input to 2DB-UM by a code called LINX. LEOPARD could not be used to

~

calculate control-rod cross sections, because it demanded that the fuel be in the center of any model.

I g

1

I LI EXTERMINATOR was used as a link between the transport codes and 2DB-UM for a few reasons. One reason was that at the time it seemed to be an easier code to use than 2DB-UM. I The second, and the real deciding factor, was that EXTERMI-NATOR was up and running on our' computer system while kinks in 2DB-UM were still being worked out. Finally, EXTERMINATOR has special numer1 cal features that allow it to run cell problems with symmetry conditions on all sides, whereas some codes do not converge well under these conditions.

LI-I I-

I I

_ Y Lg.

I I

I I

B

I 29

4. The Transport Computer Models 4.1 The Problem' i The transport codes were used to obtain an accurate description of neutron behavior in the control-rod fuel

~ element. The key to achieving this goal was an appropriate conversion of.the actual three-dimensional cell into a one- +

dimensional model to which the transport codes could be applied. An accurate conversion was essential, because the i

E model was actually the problem statement. Even though the

, transport codes were known to be applicable to'the rod i

regions, their outputs were worthless unless they were asked to solve the proper problem.

The conversion'of the cell into a one-dimensional model presented a problem, because it required that some cell characteristics be input incorrectly. For example, it was 1

impossible to simultaneously use the rod's actual length, density, and exposed surface area. Consequently, a choice had to be made between which one (or two) of these l parameters would have the least effect on the calculation if

' it were to be " fudged". The parameter which proved to have the greatect effect on the calculation, and therefore would l

'. be input accurately, will be referred to asthe " primary parameter". N The second nalf of the problem was to form a cell representation based on the output of the trafisport codes.

-i B /

/

30 Two parameters-had to be chosen from the codes' output which

[

told as much information about the cell's behavior as was possible (these will be referred to as the " output parameters"). only two parameters could be chosen because 1

effective cross sections were to be developed for two groups. Each cross section was to be developed by forcing the diffusion model to match one of the chosen " output parameters" from the transport model. Therefore, the use of these cross sections insured that, at the very least, the control-rod region of the 2DB-UM model would behave as the transport codes predicted with respect to these two parameters.

4.2 The Solution The plan of attack for obtaining an HEU control-rod cell model was to construct a few different models and choose the best one. Simple logic suggested why one model might highlight the actual cell's more important characteristics better than the other models. A comparison

.of the codes' treatment of different models actually confirmed this logic. of courre, the final check involved a comparison to experimental data. The effective cross sections for each case were used to calculate control-rod worths which were compared to measured rod worths. Since the final cell representation combined output fron BRT and GAMTEC, each trial case actually consisted of a, group of

I I

I

.three models. For each case, a slab BRT model was

. constructed first. Next, cylindrical BRT and GAMTEC models were constructed consistent with the logic used in developing the BRT slab model. For example, if the mole nur.bers in-each region had been preserved throughout the I. slab model construction, then they would also be preserved throughout the construction of the remaining models for that

~

trial case.

Next, a few values from the transport code outputs were ,

chosen as reasonable possibilities for being the " output I. parameters", the parameters which contained the most information about the cell's behavior. Two sets of effective cross sections were generated for each trial case by forcing the EXTERMINATOR model to match the output parameters. Each set of effective cross sections was based on a different set

,I of possible " output parameters". Knowledge of each case's effective cross sections made it possible for 2DB-UM to calculate control-rod worths for each trial set of models.

l The control-rod worths were compared to experimental data to help choose the best representation of the cell. The logic that proved to lead to the best representation of the HEU l

l cell could then be confidently applied in the development the LEU cell models.

I LI l

L I-

4.3'The Actual Control-Rod Cell An actual HEU control-rod fuel element is shown in >

Figure 4-1B. It consists of nine curved HEU fuel plates,

'I two aluminum guide plates, two outer aluminum side plates, a ,

grooved control-rod, and the water between all the components. A single fuel plate is shown in Figure 4-lD.

An alloy of U-Al comprises the fuel. The uranium is 93.7%

235 23a U and 6.3% 0 by mole number. The alloy is enclosed in an aluminum clad. The aluminum clad on the two outer plates is slightly thicker than on the others.  :

p I The control-rod's site and shape can be seen clearly in Figure 4-2. The rod's composition is a mixture of 1.5% boron  ;

and 98.5% stainless steel (ss304) by weight. It is important to make a note of the rod's shape. This shape, which significantly increases the rod's surface area, would prove lI 1

to be the major stumbling point in modeling the rod. The entire rod was enclosed in a thin layer (0.0'54 e cm) of A1.

4.4 Preliminary Calculations The first step in developing the computer models of the cell was to calculate the volumes, and mole numbers of each 1I component of the actual cell which were to be included in the model. Most of the initial data was found in the UVAR Safety Analysis Report [7] while the remainder of the data was taken from the UVAR blueprints (8]. A summary of the results is shown in Table II. Note that aLthough the outer

-3;

,-  ;

'I 33 p ....... >  : ......_  :

, I.

^ , , n-c ,

' **=

r r l r .-

n r .

ri

, r o..sv.. i I.. r 2 1

.... . 4

+

r i o **7==

i ,

~6 r r

]

r .

,I- .

%/

2 _

r v

. T -. c sseine.re sw. a s .=ns control noe Pues us m.ns 4-1A 4-1B

( v... . )

r- a. 39e= 7- ^

I U i

'  : i c.~~~.~.

i o .... '- ,

i

....- E

• • ***** i gg f

.....- q  ;

1 I

.1 J L v

4

.. ..s:-. ... w., ... ,s -  ;

I U

i

~

4-1C ww.i e i . s.

i' 1

4-1D

.R Figure 4-1: The Fuel Elements  ;

l W-3' Figures 4-1A, 4-1B, and 4-1C are overhead views of standard, control-rod, and re9ulatins fual elements, reSPectively.

5 Fuel plates are represented by the shaded regions. A grooved boron-stainless steel rod is located between two Al guide ,

plates in the control-rod element. Note the different shape I- of the regulating rod which is composed of 100% stainless-steel. A. front view of one fuel plate is shown in Figure 4-

~

1D. The shaded portion represents the U-Al alloy which is enclosed in a coat of A1.

I

I u i

i 34

/ N N 5.6S3cm f 1 Hg C ,

Wi B

/N l I' R Vl Vl V V -

A O 2.21Som

.I

( o n n n I \

< I

\/

l R=1.342cm Areo of one groove - O.511 cm 2 4 - 2f:1 I

h I

6 .

4-28 il Figure 4-2: The Control and Regulating Rods Figure 4-2A shows the control-rod before the outer layer of aluminum was applied. 11ote that the grooves result in a I decrease in volume and an increase in surface area. Although the regulating-rod (Figure 4-2B) has a greater volume, it has less surface area. The regulating-rod is also a weaker absorber because it does not contain any boron, as the control-rod does.

'I I

n . . . .. .

1

l, I .- )

-fuel elements contain more Al clad than the inner plates, the meat ~ portions of the different plates are identical. ,

t i

5 Table II: Summary of Initial Calculations 3

Component Volume (cm ) Composition 1.630 moles A One fuel slab 16.916 0.0463 moles (meat) 0.0031 moles. .I 5.227 moles S

One inner fuel plate 52.865 0.0463 moles U I 0.0031 moles 6.815molesAg5 One outer fuel plate 68.727 0.0463 moles 2

0.0031 moles One guide plate 140.060 14.015 moles A1 4.584 moles B The control-rod 460.102 3.836 moles Al

.I- 58.432 moles steel The eight grooves 224.500 I- The regulating rod 662.892 96.467 moles steel ,

I q 4.5 The Models 4.5.1 Modelling Limitations of the Codes -

Both BRT and GAMTEC had similar limitations with regard to the models that they would accept. The only notable difference was that BRT would accept slab or cylindrical I' /

,- -- - - _  ;

I 36 mo dels while GAMTEC would accept only cylindrical model's.

~

This fact combined with the rectangular shape of the actual  ;

cell added support to the theory that the BRT output i provided.a more accurate description than GAMTEC's.

As mentioned earlier, BRT and GAMTEC's major i

I' restriction was that they would only accept inhomogeneities

in one dimension. Therefore the first step in building the  ;

models was to decide how to divide the cell into regions which would be considered to be, both, homogenous and infinite in two of the three spatial dimensions. Obviously, l.I '

L the objective was to split the cell in such way that the code would best be able to predict the fluctuations in the #

neutron flux.

Because the codes were only capable of handling inhomogeneities in one dimension, the volumes above and below the fuel alloy were excluded (see Figure 4-1D) . This choice allowed the codes to focus on predicting the neutron population in the fuel without diluting the region with the Al and H2 O volumes above and below the fuel (relative to the ,

z-axis). The basis for this decision was that in order to predict changes in the reactor behavior, knowledge of the neutron flux in the fuel portion of the core was deemed more important than knowledge of the flux in the clad or the water above and below the fuel. Therefore, it was better to I r I- l

?I LI-I 37 ignore the extra region altogether rather than dilute the ,

information which was considered vital.

Another restriction of the transport codes was that they both placed an upper-limit on the number of spatial >

,. homogenous regions that they would accept. BRT could handle eight different regions while GAMTEC had the capability to

, accept up to ten regions. Because there are nine fuel plates as well as the control-rod in one cell, BRT clearly did not have the capability to calculate every dip and rise l in the flux. Therefore, the cell had to be divided with the  ;

1 L

objective of predicting the large fluctuations while l-

'~

ignoring the small perturbations in the spatial flux as

• illustrated in Figure 4-3. This premise led to BRT models in which all the fuel plates on one side of the rod were homogenized into one fuel region. This fuel region's -

homogenous mixture also included water, and side plate l' materials. The remainder of the BRT models consisted of two guide-plate regions, a control-rod region, and another fuel region as tchown in Figure 4-4.

i Before GAMTEC could be applied, it was necessary to i ., convert the slab model into an equivalent cylindrical model.

1 As Figure 4-5 shows, the conversion was achieved by combining the slab model's two fuel, and two guide-plate regions into one perimeter fuel region, and a c.oncentric guide-plate region, respectively. Therefore, the cylindrical i I

1 f .

I LI  !

38 l

g- 70  ;

I 63 a o3),,giy, 00tu01

/

for the colculction

,,90 I 50 -

40 . .

/<oI -

)

1 l-3 39 .

p

/

I -

rod ..A

/ fueI 20

'g. a/

10

/

u-a/ ,

0 5 10 15 20 position ,

Figure 4-3: Computer Code Limitations

(

Theoretically, it is known that the actual thermal flux will

. fluctuate in the fuel region corresponding to the alternating regions of fuel and water (Figure 4-1). It was impossible to simulate each peak because-BRT can only handle l

up to eight regions. A workable objective _was to try to I simulate an average flux through this region.

I:

LI 1

1l -

lI c

1' u _

s F. ,

s.g

( .

39 s -

~_ a.nese. F I NA 3

o. eve.. M

,*,,,,, c

, R ,a R,G,S a a.tTeam c...e.- s i --

o I . . ~ .

T -

5 .F e.cose.

( 7.4eeem l 7,,3go, The Actuot Control Model A Rod F:ve i E1ement 8 Figure 4-4: Slab Model A vs. the Actual Cell Figure 4-4: In this and the figures which follow the following names will be used to refer to the given sets of materials in I the actual control-rod element in order to simplify the descriptions of the different models.

control rod- The Boron-ss-Al mixture which makes up the rod.

I Shown as component R in the figure of the actual cell.

groove water- The volume of water which fills the rod I grooves. One of the eight grooves is shown as component G in the figure of the actual cell.

side water- The remaining water and Al which occupy the same length as the rod. This. region is marked I S in the figure, and it's boundaries are the two dotted lines above and below the S.

guide mixture- A mixture composed of the water above and below the rod, the guide plates, as well I as the Al side plates which border the region. One of the two . guide mixture

, . regions is marked M, and again is outlined by a dotted line below and the top of the I guido plate above.

fuel mixture- A mixture made up of all the materfals above and below the top and bottom guide plates, respectively. This mixture includes the fuel, I

water, and side plates.

There are two impot-ant points to note regarding model A.

First, the length of the rod region is equal to the length of the actual rod before calorization. The second point is that the side and groove water are included in the rod region.

These facts are consequences of choosing the length as the

" primary parameter".

1 l

I 1 40

'l I 1 I- -

I ,

/ / -

.I j .

I -

I I / .

l // -

. Figure 4-5: Conversion of a Slab Model into a Cylindrical Model The slab models were converted to cylindrical models as I illustrated above. The materials in each slab region were inserted into the corresponding cylindrical region which can be identified by the shading.

LI

'I.

i d

i I

P' . ij ~ . -......m_ , _ . ..

-

/3 l

u .-

l model consisted of three regions instead of the slab model's 1 five. The control-rod region was considered to be the center of the cell. The rod region was surrounded by the guide-plate region, and then the fuel region.

L 4.5.2 Trial case Constants Before discussing how the trial cases differ from each other, it is important to mention some of the parameters

i. which were held-constant for the different cases. The most significant input data which fell into this category were the boundary conditions. Reflecting boundary conditions'were i used in all trial cases. Because BRT and GAMTEC are one-dimensional codes they automatically assumed that the cell was bordered by identical control-rod cells in their two homogenous directions (Figure 2-1). Of course, this is I

irrelevant in the cylindrical model's theta direction.

Reflecting boundary conditions implied that the model cell was also bordered by identical cells in the direction in which inhomogeneities were allowed.

Reflecting boundary conditions were chosen because in most cases the actual cell was bordered by fuel cells in the inhomogeneous direction (Figure 5-1). Even though one would expect the average fuel-cell neutron flux to be greater than that of a control-rod element, reflecting boundary conditions were valid. They were valid because the control elements border region contained at least four fuel plates.

~ '

I_

I -

.+

42 Therefore, reflecting boundary conditions correctly implied that the cell was bordered by fuel plates. ~

Another decision which was based on the presence of

. bordering fuel plates pertained to the composition of the fuel regions. The actual element contained one region with four fael plates, and a second region with five plates. The

,; second region was. larger so that there was an equal volume of water between every pair of plates. Instead of creating unsymmetric models, all the models were constructed with the

.\

contents of four and one-half fuel plates in each of two f

.I equally sized regions.

The basis for constructing the models in this manner can be seen from Figure 2-1. The figure shows that in any ,

i configuration, each rodded element will be borderod by at i'

least one fuel element along it's y-axis. It is well known I

1 I that each control-rod element is influenced by the materials which border it. In a case where there are fuel elements ll l

both above and below the control-rod element, the control-

, rod would see an equal number of fuel plates per unit length in either direction. Therefore, the placement of four and

'I one-half plates on either side of the rod should lead to more realistic predictions.

GAMTEC required four additional input parameters for which the same values were used in each trial case. A geometric buckling was required to allow GAMTEC to estimate

x W F

e L

43 the flux shape in the z direction. The buckling was calculated by B'= (w/H) 2=0. 002 8 3 The larger distances travelled by fast, in comparison to thermal, neutrons suggests that the geometric buckling is more important to the fast calculation. Without a correction for buckling the flux on the upper edge of the cell is assumed to be equal to the flux in the center. This was acceptable for the thermal calculation because most of the collisions were expected to occur in the center of the cell (with respect to the z-axis). The high collision rate was due to the combination of the high flux in the x-y plane of the fuel, which was located in the center, with the average path length of I

A =l/E =0.0026cm.

g These facts support the assumption that thermal neutrons located in the fuel had a low probability of reaching the edge of the cell before undergoing a collision.

The effectiveness of the moderator was included by a logarithmic energy-decrement factor. This factor gave the code information pertaining to the fraction of energy lost per collision with a moderator molecule. The energy-decrement factor for water is 0.92.

ema i i n __

~

~

k p

44 The effect of neutrons originating in other fuel, cells that were scattered into the cell of interest were taken into account by using the Dancoff shielding factor. The value was taken from a tabulated chart which related the radius of the cell, the distance between neighboring cells,

- and the moderator scattering cross section (9). The shielding factor was found to be 0.0001. GAMTEC also asks for the circumference of the outermost region of the cell so that it may calculate the fast fission effect. The circumference was 27.85 cm.

4.5.3 Trial case A - Slab Model The slab model was constructed first because it most resembled the actual cell. Therefore, it required fewer approximations, and consequently, there were fewer chances to create errors. The five slab regions were constructed slightly differently for each trial case. The slab model used in trial A was the result of a first impulse to make the appearance of the model resemble the actual cell as closely as possible. The relative length, and molecular composition of each region were chosen as the input parameters which would have the greatest effect on the cell behavior. Therefore, these two parameters were taken exactly from the actual cell while other parameters, such as surface area of components, were given values which were l

l 45

[' dictated by the conservation of region length and composition.

( While the length of the cell, and of each region, were included in the one-dimensional slab model, the cell height and width were indirectly implied. Both GANTEC and THERMOS require the molecular composition to be input in units of

t. atoms / barn cm, a three-dimensional quantity. Therefore, before computing the molecular composition it was necessary to define the other two spatial dimensions of the actual j cell which were to be included in the one-dimensional model.

The actual cell height was taken to be 59.055cm while the width was taken as 7.468cm. The height corresponded to the s

height of the fuel portion of the fuel plate (Figure 4-lD) while the width corresponded to the width of the actual cell (Figure 4-1B).

1 Figure 4-4 shows slab model A and the actual cell side j by side. The actual cell's components were grouped according to the dotted lines in the figure of the actual  ;

cell. Next, the sizes of the corresponding model regions were chosen based on the length of the regions shown in the I

actual cell. The model description was completed by i

homogenizing each group of components into the appropriate i volume. 1 The length of the model's two equivalent fuel regione (regions I, and V) was arrived at by averaging the actual i

46 cell's unequal fuel regions. The rod region's (region III) length was taken to be equivalent to the length of the boron-ss mixture in the actual cell. The' remaining length of the actual cell was then split between the two equivalent guide-plate regions (regions II'and IV) which separated the rod from the fuel regions. Note that this implies that although the rod's outside layer of Al was homogenized into the red region, the rod region's volume was chosen as if this layer was not part of the rod. Instead this volume was incorporated into the more weakly absorbing guide-plate region.-

It was now possible to calculate mole number densities for every nuclide (the chemical elements will be referred to as nuclides to avoid confusion with the fuel elements) present in each model region. After mole numbers were l

obtained for each group of components the number density was found by number density = molecules of nuclide in region / region volume As mentioned earlier the two actual fuel regions were combined for this calculation and the resulting mole number density was used for both model regions. Therefore, each model fuel region contained a homogenous mixture composed of 4\ fuel plates, 2 aluminum side plates each 2.235cm in length, and water. Each guide-plate region contained the i

molecular content of one guide-plate, two Al side plates each 0.678cm long, and water. The molecular content of the Al layer which enclosed the rod was included with the boron-ss mixture, two Al side plates each 2.22cm long, and water in the rod region. Each region's water volume was determined by computing the total volume of all components, and then taking the remaining volume as water.

4.5.4 Trial Case B- Slab Model The construction of model B was based on two premises.

The first premise was that the volume of the control-rod relative to rest of the cell was the " primary parameter".

The second assumption was actually a modification to the first. It stated that the eight grooves (see Figure 4-2) actually increased the rod's absorbing strength.

The presence of the grooves changed two characteristics of the rod. They significantly increased the rod's surface area, and simultaneously decreased the rod's total volume.

The theoretical implication of the second premise was that the consequences of the additional surface area outweighed those of the volume loss. Due to the rod's high absorption cross section most neutrons were absorbed in the boundary layer of the rod. Therefore, a change in surface area would be expected to have a much greater effect thaq a change in the interior volume. In regard to the model, this implies

/

i. ' l 5  ;

L L

L 48

I that it would have been a mistake to shrink the rod volume in response to the presence of the grooves.

l This line of reasoning led to the model portrayed in Figure.4-6. The control-rod region (slab region III) volume t-was takeln to.be equal to the rod volume prior to the grooves p being cut into it (see Figures 4-2 and 4-6) . Because of the desire to keep the model width identical to the width of the ,

L actual cell, the length of the region was adjusted to 1.551cm to obtain the desired volume. The material composition included the boron-ss mixture, the enclosing Al L5 l

layer, and water.-The quantity of water was chosen to exactly fill the difference in the volumes of the region and the rod (including the Al layer). From here on this volume l will be referred to as the groove vo:.ume.

The water and side plates which had been included in slab model A's region III, but had been excluded in model L

ll3 - B's, were divided equally between model B's regions II and l-5- IV. Thus, model B's guide-plate regions contained the same materials as model A's guide-plate regions, as well as the additional water and Al. The length of these regions were adjusted to 0.990cm to account for the additional volume.

Slab model B's fuel regions were identical to those of model A.

A comparison of the first two models shows that the primary differences were the size of the rod region, and the i

I

Y  %

[-

49 i Ei

] W ^

-F 2.t36mm  % g,g3,,,

2

) I T -

1 I U e y M

e I

0.95asm l I O."66asen WI .

= =i* -

f44p , ... ..e g g 1 -.m-- a..mii m g si, a j n .. - ._ . . . .

. i ...

c.esdem , , o,. , ,

.s i i ,--- , - ,

El 7 =i- r=

2.spoem  ? 3.nenem 3 -5r

? -r 3 2I -

-L '  :

".- s E,,,. ,," 7 s ,,,.,

f.

The shaded portions of the two figures shown above represent the volumes included in the rod regions of models A an'd B, respectively.

R.Deecm 2.DeSem F O.c? Gem M o,..o., M.S R,G,S ,,,,,em R.G

,,,,,,, l

• • ** M.S C.e7som M F 2.300cm F a.neecm l 7.4S8cm 7.4SBem Mode 1 R Model B  !

Figure 4-6: Slab Model A vs. Slab Model B  !

.This figure illustrates the differences between models A and B. The length of model B's rod region has been shortened so i that it's volume is equal to that of the actual rod plus the groove water (the shaded portion in the figure above model i B). The rod region's homogenous mixture is composed of the rod and the groove water. The side water has been included in the mixture of regions II and IV, which have had their lengths increased to preserve the distance between the guide plates.

l l

50 density of the rod materials. While model A spread the rod composition of 6.730 moles of boron and 91.707 moles of SS304 over a volume of 957.718cm 3 , model B spread the same 3

number of moles over a volume of 684.152cm. A neutron travelling through the models lengthwise would encounter 2.172cm of control-rod material in model A, and 1.551cm in model B.

t

<4.5.5 Trial Case C- Slab Model In contrast to model B, the construction of model C (Figure 4-7) stuck strictly to the premise that the volume of the rod region relative to that of the entire cell was the " primary parameter" in the model construction. .

According to model C's premise, both models.A and B incorrectly increased the rod's' absorbing power by spreading the boron-ss mixture over larger volumes than the actual rod I had. Although this statement sounds reasonable by itself, recall from section 4.5.4 that it implies that the rod's increased surface area was neglected in model C.

The implied volume of the rod region was made to equal the actual rod volume by adjusting the region's length to i 1.043cm. The implied width and height were held at 7.468cm and 59.055cm, respectively. The composition of this region consisted solely of the aluminum, horon and ss which comprised the control-rod.

l L.

s i 3 51 g

-?  ;

, . . . ~

,  ; _ _,$-

{ == g

• O .Ne~m" I hA '

i.

' " S S

~ ~"-'

)

T o.... , , ..

,- - i s = _

p ......  ;; m =

q l - - - _ _ _ [

r_ 7. - 2 s

Shaded portion of the figure shown above represents volume in'cluded in rod region of model C i

h'r -.

2.Deeom

~~

. o.cocom M,S i . =, e2 MSG

~

R,G o'3om R i... ..

~~~""~'

'**"a* M,S,G

.**L2*

M'S -

i F ,, .. .... p  !

m.nesom ,

i 7.4SBem 7.4Seem i

Model B Model C  !

l i

Figure 4-7: Slab Hodel B vs. Slab Model C The composition and volume of model C's rod region reflects only the boron-ss. The side and groove water lave been incorporated into regions II, and IV.

. l

i I

i i  !

52 I i

As a result of decreasing region III's volume from the l

F,nvious models, the volumes of regions II and IV were l increased. The length of regions II and IV was increased to  ;

3 1.244cm which implied a combined volume of 548.632cm. This volume corresponded exactly to the volume of the actual cell [

I

! between the two guide-plates' outer edges after subtracting j out the control-rod's 'rofume. It in11 owed that then molecular camposition for these regions u s a mixture of all i the materia 3a betwet.n cuter edut;s of the guide-plates excluding the control-rod.  !

W 4.5.6 Trial' Case D- Slab Nodal f Slab model D (Figure 4 8) was based on the same two j premises as model B with respect to the actual cell. It l differed from model D in that it proposed to portray these i premises differently in the computer model. Like snodel B,  ;

model D assumed that in the actual cell the rod's absorbing j l

power varied directly with the rod's volume, and was }

increased by the presence of grooves. Whereas model B depicted the situation by increasing the rod region's volume from the actual cell by including the groove water in the rod region, model D replaced the groove water with an equivalent volume of control-rod material. The result was a control-rod region composed solely'of the boron-ss mixture. [

Model D helped to answer the question of whether the control-rod could be homogenized with water without

!a . - -- .. - - . - _ - -- .

l s

r 53

e. . F ..... 6

.;

-- ' . o... .

MS o. no.. M.s i.e ic. R,G i .ni . D + R C G *> l o . ..o- M.S u

.... . __ti,. S _

F F .

e . o*** e. ..

7.4SBcm 7.4SSom }

I .

Mocei e Meemi o i

I Figure 4-8 Slab Model B vs. Slab Model D Models B and D are structurally equivalent. The difference between the models is that model D attempts to account for the grooves by replacing the groove water with an equivalent volume of. boron-ss.

l.

9 3

54 influencing it's properties. The logic which led to model D said that since the rod was a solid block of boron-ss, it should not be portrayed as a mixture of boron-ss and water.

The fuel regions were not changed even though this line of thought soumed to demand it. Recull frem Figurt- 4-1 that I the actual fuel regions contain thin fuel plates separated by water. Therefore, homogentJing the fuel and water wac deemed allowable while thc; wixing of the rod and we.ter was not.

Regions I, II, IV, and V of model D were identical to those of model B. The spatial dimensions of model D's rod region were also identical to those of model B. The only difference between the two models occurred in the molecular I make up of the rod region. Whereas region III of model B f

contained 4.584 moles of boron, 58.432 moles of ss and 6.200

! moles of water, the same region of model D contained 7.024 moles of boron and 89.472 moles of ss. Therefore, a rod region was created with the volume, and molecular  !

l composition that the rod had before having any grooves cut into it, i 4.5.7 Trial Case E- Slab Model

slab model D attempted to account for the additional j

,

absorbing power caused by the grooves by filling in the grooves with boron-ss. Effectively, this resulted in the

rod being portrayed as if the grooves had never been cut.

lI .

8

~

55 Therefore, model D did not satisfactorily account for the

'. grooves. At best, it prevented the absorbing power from being decrear,ed because of the grooves. In order to 3

itscrease the absorbing power the three remaining models used the surface areas of the rod c.s their " prime.rf pacaneter",

i k Up to this point the rod surf aco area had not been tried as the " primary parar.eter" because it required that the length and vidth of the trancport model differ from that i

S. w of the 2CB-UM model. At *irst the thought of altering the model sito seemed to contradict the purpose of the project.

This viewpoint changed when it was realized that the

transport model could be altered without changing the diffusion model. The purpose of the transport codes was to obtain an accurate picture of the cell's behavior. The only values from the " transport portion' which were important to the remainder of the project were the two " output parameters" chosen to represent the cell. Therefore, the size of the transport model was irrelevant. Any sized model was acceptable as long as it led to the most accurata

=

prediction of the " output parameters".

At this point, slab regions I, II, IV and V of the first four models were considered to be satisfactory, but there was still a desire to improve region III.

Consequently, the objective of slab model E was to create a rod region with the perimeter of the actual rod without 1

N - - - -

changing the fuel, nor the guide-plate regions from models r

A-D. To accomplish this objective the implied model width had to be held at 7.468cm while the perimeter of the rod region was adjusted to 25.554cm (the perimeter of the actual rod excluding the aluminum coating). The only way to obtain a kodal with these charactaristics was to incrosse the length of the rod region to 5.309cm (see rigure 4~9). The resciting rod region had a " fudged" volume of 2340.55cm 3 which was filled with 2240.55cu3 of the boron-ss mixture.

4.5.8 Hodel F- Slab Model In slab model F, the restriction which demanded that regions I, II, IV and V be preserved was abandoned. Model F explored the suggestion that more could be gained by

- improving region III than would'ba lost by slight changes in the other regions. This proposition was actually a direct consequence of choosing the rod surface area as the most important parameter in the model.

Although model E did preserve the surface area of the rod, a quick glance at Figure 4-9 shows how model E missed the point. The surface area was chosen as a significant characteristic because a greater surface area allowed more water to be in contact with the rod, and consequently the rod became a more effective absorber. The diagram shows that model increased region III's perimeter by stretching it's 1

57 I -

3 .-

..... 1 = ===:p

f  ; __ --

2.Deecm p

.. .., t u i e .3 s.sem M.S.G s n s I

r o[I 6 1 L_

[,n-,-----..-m,.--_m hAWIW":T2.%M-e n -

e. coco.

3 W= tM -

qc.

!< v....c. jl b

The ActucI ControI i.e..

M,g,g lp Rcc F ve 1 5Iement a.ceec. F i 7 . ** S S e m

-e , e

i.esee , F o.aovem M.S.G o.eveem R o.co7cm M.S,G t.cosem ,

I '

O -

ti.4eeem ModeI G Figure 4-9: Slab Models E and G Models E and G both attempt to simulate the control-rod's grooves by creating rod regions which preserNe the actual rod's perimeter. Note that model G is more effective, because it actually increases the area in contact with regions II and IV. _

58 length. It did not increase the area of region III which was in contact with the other regions.

In contrast to model E, when model F increased region III's perimeter the area in contact witn regions II and IV uns increased. This capabillty was a cor.sequance of allowing model F to vary it's width from the other models.

The model width was chosen to coincide with the " effective" width of the rod. The " effective" rod width can best be i describod by looking at Figure 4-2. In this case the effective width was taken as the distance trave?. led if the rod boundary was traced from point B to C. This distance was 9.370cm. Next, the region length was chosen to be 0.8315cm which caused the area of the rod region equal to that of the actual boron-ss mixture. Because the cell width had been changed, the lengths of regions I, II, IV and V were also changed so that they would cover the same arec that they had in model C. Therefore, the number densities for the cell were identical to those of model C.

4.5.9 Model G -Slab Model The logic behind model G (Figure 4-9) was identical to the logic behind model F. The only difference was in the selection of the " effective width". Figure 4-1 shows that instead of tracing the distance from B to C, model G used the distance from A to D. Again, the region length was chosen so that the region area would match the actual area.

i

1 59 Therefore, the surface area of region III which was exposed to regions II and IV exactly matched the exposed surface area of the actual rod. In this case the " effective" width was 11.488ca, and the region length was 0.678cm. Following the same logic as was used in model F the length of the fuel and guide-plate regions were 1.538cm and 0.907en, respectively.

4.5.10 The cylindrical Models Ir. order to apply GANTEC, the slab models were converted into equivalent cylindrical models. As Figure 4-5 shows, the cylindrical models consisted of three regions instead of five, as in the the slab models. The control-rod region was considered to be the center of the cell. The rod region wss surrounded by a guide-plate region, and then a fuel region. In each case, the molecular contents of slab regions II and IV were combined to form the guide-plate region while slab regions I and V comprised the fuel region.

The size of a region in a cylindrical model was dictated by the choice of radius. In models A-D the region radii were chosen so that the area of each region would equal the implied area of the corresponding region in the slab model. Therefore, the region radii were dictated by wrI ,-rr g=

I (7. 4 68cm)

• L

i 60 where r, and r, were the inner and outer radii, respectively, of the region, 7.468cm was the implied width l of the slab model, and L was the combined length of the slab regions that were being converted. The resulting regions had radii of 1.9205ca, 2.897cm, and'4.432cm for cylindrica3 regions I, II, and III respectively.

Only one cylindrical model was developed which attempted to conserve surface area. The primary reason for developing only one model was that the cylindrical model was being developed in order to perform calculations for the fast group. Since the rod's absorbing power was much weaker with respect to the fast group, the reasoning which suggested that the grooves would cause many more thermal absorpt. ions (see section 4.5.4A) did not apply to the fast group.

A secondary reason for constructing only one model was practicality. The redius of the center region was chosen so that the rod region had the desired perimeter. This fact meant that the region volume was automatically chosen. It could not be adjusted as it had been in the slab model by varying the region length. This was a problem because the preservation of the actual surface area led to rod regions with volumes three to four times the actual volume. At the same time, the large radius of the center region forced the fuel region to be very thin relative to the rod region.

I l

s 61

/ The one model which was developed most resembled slab model F. The rod region perimeter was taken to be twice the effective length which was used in slab model F. The resulting radius and volume were 2.983cm and 1650.388cm 3, respectively. The actual rod mole numbers were spread over this volume. The radii of the guide-plate and fuel regions were selected to be 3.849cm and 5.105cm so that their volumes would be the same as in model C. Therefore the number densities from cylindrical model c could bu used here.

The significant point here was that in each case, a cylindrical cell was constructed in which the " primary parameter" was preserved from the corresponding slab model.

In models A-D, each cylindrical region had an implied volume, and contained a molecular content exactly the same as it's corresponding slab region. Cylindrical model F corresponded to slab models E-G because it preserved rod surface area.

4.6 The output Parameters 4.6.1 The Thermal Group After applying BRT and GAMTEC to each model, the next problem was to choose the two values from the codes' output which would give the most information about the cell. One value was taken from BRT to describe the thermal group l

i 62 behavior while the other parameter was chosen from GANTEC to describe the fast group behavior.

Early in the project, the fraction of thermal absorptions in the fuel region relative to the entire cell was chosen as the thermal parameter. Throughout the remainder of the project no other parameter was suggested which could challenge t'he absorption fraction in the fuel as the one thermal pare.neter which told the most about cell behavior, Very simply, the most important information pertaining to a thermal neutron was knowing whether or not it caused a fission reaction. Combined with knowledge of the cross sections in the fuel, the fraction of absorptions in the fuel told what fraction of neutrons would cause fissions.

An argument could have been made for attempting to match the shape of the thermal flux instead of the fuel absorption fraction. Although the flux shape was considered, it was never used. The main reason that the flux shape was of interest was to gain insight to the fraction of neutrons being absorbed in the diflerent regions. Since the codes gave direct access to these fractions, knowledge of the flux was of secondary importance.

I 4.6.2 The Fast Group At first, the cell multiplication factor was taken from GAMTEC to be the fast group " output parameter". The l

63 multiplication factor was chosen to take advantage of the fact that GANTEC could perform a calculatio'n which included both groups. What was overlooked was that in the process used to develop effective cross sections this choice actually nullified the BRT calculation. This was definitely not desired because the BRT calculation was considered the l more accurate of the two.

l The second suggestion for a fast " output parameter" was the fraction of fast neutrons which survived to reach the thermal group without being absorbed. Like thermal

'- neutrons, the most vital knowledge regarding a fast neutron is whether or not it causes a fission in the future. By combining knowledge of the fraction of fast neutrons which reach the thermal group witu the fraction of thermal neutrons which cause a fission, an adequate picture of cell behavior would be obtained. The advantage of this parameter over the multiplication factor was that it had no effect on l

the more accurate thermal calculations.

4.7 Transport Code Output 4.7.1 Slab vs. Cylindrical Models '

Section 4.5 described the different approaches taken to model a control-rod element in one dimension'As some of the 1

i 64

) explanations of 4.5 hinted, the later models were inspired by drawing conclusions based on the output of the earlier models. In some cases, comparisons between the output of different trial cases yielded definite answers as to whethor or not the selected model should have certain characteristics. Section 4.7 will concentrate on how these comparisons led to the creation of model G, and why that model was expected to yield the best representation of the control-rod element.

One question which had to be answered before proceeding was how well did the data generated by the BRT slab, BRT cylindrical, and GAMTEC cylindrical models agree. Earlier discussions (sections 3.3 and 4.5.3) explained why it was assumed that the BRT slab model would lead to the best results with respect to the thermal group. Recall that because of this assumption, the objective in developing a cylindrical model was to make it behave like the equivalent slab modele Therefore, the cylindrical models were judged by comparing their output to that of the BRT slab models.

All the relevant information has been compiled in Tuble III. The fraction of all thermal neutrons absorbed in the fuel, (vE,/E,) ,,g g (all cross sections referred to in this chart are thermal cross sections), and I,g ,gA were chosen as three parameters which summarized the predictions of cell

~N

65

> behavior made by each model. Two comparisons were made from the chart.

Table III: Transport Output Tarameters I'raction absorbed in Eg,g I the fuel region vE,/E, BRT BRT CAM BRT BRT CAM BRT BRT CAM Slab Cy1 cyl Slab Cy1 Cy1 Slab Cy1 Cy1 Model A 0.421 0.391 0.380 0.697 0.648 0.624 17.46 17.79 31.51 Model B 0.455 0.455 0.537 0.755 0.754 0.723 17.72 18.06 31.43 Model C 0.502 0.546 0.580 0.833 0.906 0.962 19.11 19.90 31.38 Model D 0.492 0.487 0.533 0.815 0.807 0.881 27.27 28.36 48.05

)

I The first comparisen looked at the results of performing the same calculation on two different models. The application of two different codes to the same model was the source of the second comparison.

Applying BRT to both the slab and cylindrical models allowed a judgement to be made on whether equivalent models had been constructed in the two different geometries. Table III shows that in nach trial case, every value calculated from the BRT cylindrical output was within 10% of the same value generated by the slab model. That is to say when the same calculation was conducted on two different mod 21s the i

66 results were similar. This fact verified that the method used to transform the slab model into a cylindrical model did indeed create an " equivalent" model.

The next step was to apply both BRT and GANTEC to the same cylindrical model. The table shows that in every case the application of BRT to the cylindrical model came significantly closer to reproducing the BRT slab output than when GAMTEC was applied to the same model. This fact con-firmed our earlier suspicion that the differences in the.

solution methods of the codes could not be ignored. If the differences had been negligible then BRT would not have been necessary, because it had already been decided that the cylindrical model was " equivalent" to the slab model.

It is interesting to note that when the rod element was treated in it's entirety, GAMTEC's results were all in the same ballpark as BRT's, but when the individual regions were examined the differences became quite large. The table showe.

that for the two parameters which pertain to the entire cell, (that is the fraction or neutrons absorbed in the fuel and VEp/E,), the GAMTEC values verr always within 18% of the BRT slab values. In contrast, GAMTEC values for E,gh ,g, which only pertained to the rod region, differed from the BRT values by as much as 76%.

Some insight into how BRT's and GAMTEC's region values could differ by so much while the cell values came fairly 4

I 67 close can be found in Figure 4-10. This figure shows the different flux shapes predicted by BRT and GANTEC when they were applied to cylindrical model D. The figure shows that while GAMTEC predicted a lower thermal flux in the rod tagion, it predicted n higher flux in the fuel region.

Therefore, parameters which pertained to indivi'Jual regions reflected these differences in flux, and parameters which pertain to the whole cell reflect the fact that these

~

differences declined when the flux was averaged over the entire cell.

The differences in region fluxes can easily be explained by recalling the primary difference between BRT's and GAMTEC's solution methods. Chapter three explained that the major difference with regard to the thermal group was that TEMPEST failed to separate the spatial and energy dependences of the flux. This fact led to a neglect of self-shielding effects, thereby making both the rod and fuel regions into stronger absorbers than they actually were. In the rod region this is portrayed by a lower flux while in the fuel region the additional absorptions lead to more fission reactions and therefore a higher flux. To reiterate, this illustrates that GAMTEC's assumption of space-energy separability does lead to unnecessary errors, and supports the decision to select the thermal " output parameter" from the BRT output.

L

-<I o - ,n h US 'v. O . 68

c. ya

\

o - v l

f $ g%.il g,g Y h ~

4 oi

• a I .

< o o 5 g2&v m*e 2 8 h b g ,l t : 5 's b

l b b S e J y.j ?dl l El n

3 k b -

4 l k 'oN 9  ? a: 26 Qk O O

.?

c o g e s '= H o

- \ e % 20 L$ h

'k - 'os

• 5 'E $b g I e

< O .c 8 1. E .g .s ~

a -

, s#na a-2 N n s l < .

o' E o o

N.,' '<..\o, O OU g $

v g$ o

, n' ... .,<o d' N e . . . .o. .

pu3eg 3sgy .!

. %'%.. \<,

hb

[h 0 t

1, $.$. d 4

J

@g e n.

4o a

II H 1 Y J

Z O<

CD o V -

o <

v

, , ug 8 8 @ 8 R S 8 86 d d d 6 xnjj e^o/xnjj I

a isi i

69 The fact that BRT's thermal calculation was more accurate than GAMTEC's was also the reason that GAMTEC's au-ltiplication factor was rejected as a potential fast group

" output parameter". The multiplication factor was originally chosen to taAo advantage r.,f the' fact that GAMTEC treated both the fast and thermal groups. A little scre thought shows why this fact was actually the major flaw in using the multiplication factor.

In order to calculate a cell multiplication factor, GAMTEC needed to make use of parameters calculated by both TEMPEST and GAM. Therefore, any attempt to force the diffu-sion model to match GAMTEC's multiplication factor not only implied an acceptance of the TEMPEST calculation, but could also force the diffusion model's thermal output to move away from the more accurate BRT thermal " output parameter".

This problem was easily solved by choosing a fast-group parameter which was calculated by GAM and would only affect the diffusion model's fast group. GAMTEC calculated a reso-pence escape probability which was defined as the frSction of fast neutrons born which survived to reach the thermal group. The use of this value as the fast group " output parameter" took advantage of GAMTEC's fast-group calculation while neglecting it's thermal calculation, for which a better code was available.

6 i

\

?

U 70 g

4.7.2 The Composition o' the Rod Region one signitaa?nt question regarding the model configu-ration was whether or not water should be included in the rod region. An examination of the output parameters from models 5 and D provided the answer. Recall that these two acdels were identical except that model D replaced the groove water in the rod region with an equivalent volume of boron-FR. Therefore, any differences betwetn the models' output paranctors could be attributed to the inclusion of water in the rod region.

An obvious expectation was that the excess of absorbing material would cause model D's rod region to absorb a higher 1.Totion of the total neutrons than the same region in model B. Table IV shows that the codes predicted a result which was just the opposite of that expectation. The table is a summary of data generated by BRT and GANTEC which was con-sidered relevant to choosing a proper rod-region compool-tion. The first two lines consist of output from BRT's thermal calculation. They show that BRT predicted that model B's mixture of rod and water would be a stronger thermal ab-sorber than model D's solid rod even though the solid rod contained more boron in an equal volume. In contrast, the Table shows that GAMTEC predicted that model B's rod would allow a greater number of fast neutron to slow down to the thermal energy range before being absorbed. Therefore, the

71 inclusion of water in the rod region seemed to multiply the thermal absorbing power of the rod, while having little or no effect on the fast grcup calcul a _sn.

one hypothesis, which explains why the addition of water would increase the rod's thermal absorbing power more J than the addition of boron, was formed. The hypothesis was that the primary ettact et adding water to the rod region was that probability of scattering increand. The increase in the number of scatteririg reactions cauJed two things to happen. Each scatte9:ing reaction slowed the neutron down to energ j es where boron's absorption cross section was higher, and each reaction allowed a neutron to penetrate further into the rod where the absorption probability was higher due to the lower flux. Remember that because BRT did not assume space-energy separability it accounted for the fact that not many neutrons survived to reach the center of the rod.

This hypothesis was supported by examining some of the macroscopic cross sections for the rod region calculated by BRT. Table IV shows that even though BRT predicted that model B's rod region would be a stronger absorber than model D's, it also calculated a smaller absorption cross section.

If the addition of water to the rod region was not increas-ing the absorption cross section, the only way it could increase the absorbing power was to make more neutrons available to be absorbed. A look at the scattering cross um m

(

72 sections shows that the addition of water led to an increase by a factor of about 1.5. Although these facts do not prove the hypothesis, thhy do make a vary good case to support it.

Table IV: Model B vs. Model D slab slab Cylindrical cylindrical Model B Model D Model B Model D BRT toernal fuel absorptior.

I fraction 0.4552 0.4915 0.4547 0.4867 BRT thermal vr,/E, 0.7549 0.8151 0.7540 0.8071 BRT thermal E, ,g 1.53 2.35 1.56 2.45 BRT E,,g 1.31 0.889 1.32 0.889 i GANTEC thermal absorption fraction ~~~- ----

0.7232 0.8812 GAMTEC thermal VE,/I, ---- ----

0.5369 0.5326 GANTEC resonance escape probability ---- ----

0.7464 0.6448 GANTEC mult.

i factor ..... .---

0.5838 0.6104

l I .

1 i

i 73

,I The reason that the rod's absorbing power with respect l to the fast group is not increased by the presence of water can be found by returning to the methods employed by each ,

code to solve the transport equation. The fast group f behavior is easily explained by recalling that GAM solves f the sciation in zero-dimensions. Therefore, the additional water was not addee ta the red region but it was added to  !

the homogenized cell. In thiu can,s the rod has already been l homogeniz6d with water, so adding no';e we,ter to the cell will only have a sr.all etfact on the red. On the other hand, l the additica o. more boron has e. granter effect because it lg is exposed to all the neutrons throughout the cell.

i lg I

This discussion was relevant because it allows an

{

intelligent decision to be made about the rod region's composition. Since the actual rod was indeed a solid mass  !

with no water mixed into it, it was clear that the actual rod did not experience any additional absorbing strength due [

to the effects described. Therefore, it was clear that the '

final model should not have any water homogenized into the  ;

rod region.

4.7.3 Modelling the Grooves Some of the problems related to modelling the grooves were mentioned in section 4.5. It was a fact that the grooves increased the rod's absorbing power by increasing it's surface area. Basically, the problem was to create a

'I l

lE

l I  !

I i I logical algorithm for increasing the abarbing power of the 74 1

computer model's rod region by the same amount as the grooves increased the actual rod's absorbing power. An  ;

easier solution might have been to just keep adding boron to the region until the computer output matched the expe- ,

B. risental results. The problem with this solution was that it did not create any guidelines which could be followed when developing the LEU computer nodels.

As discussed earlier the first attemF.t at a rolution, model D, was disqualified on theoretical reasons. The basis.  !

of.model D was to increase the absorbing power by filling in .

the groove volume with the boron-se mixture. The result was j that the rod was portrayed as it was prior to having the ,

grooves cut into it. Now, if it'was known that the grooves were inserted in order to increase the rod's absorption str-ength, then this portraya2. would necessarily lead to a rod ,

region with less absorbing power than the actual rod's.

Therefore, a good chock on any solution would be to make -

sure that it's rod region absorbed a higher fraction of neutrons than model D's region did.

! A better line of reasoning noted that the grooves in-creased absorbing power by increasing surface area. The-refore, if any logical system was to be developed, it should be based on matching the surface area. Two approaches were taken. In model E the length of the rod region was changed g

75 L

while in model G the width of the region was changed in

order to create a region with the actual surface ~ area. As mentioned earlier, a consequence of changing the region's width was that the width of all the other regions had to be changed accordingly.

BRT predicted that the rod regions of both models E and G would absorb a higher fraction of the thermal neutrons than the sam 9 region in model D. Sinca both nodels fulfilled this requirement, it could not be used as the sole criterion in deciding which nodel was a better representation. Table V shows that the approach taken in model G was much more ef-factive at increasing the rod's absorbing power, but this did not necessarily prove th4t it was a more accurate

! picture of the actual cell.

Table V: Model D vs. Model G Slab model D Slab model E Slab model G Fraction of thermal neutrons absorbed in the rod region 0.5085 0.5196 0.6180 Volume fraction of the cell in the j rod region 0.4273 0.6223 0.4270 thermal absorption fraction 1.190 0.8349 N 1.447 rod volume fraction i

---

i

l i

I

'I  !

76  !

I A review of sections 4.5.8 and 4.5.9 shows that the I main assumption made while developing model G claimed that model D did not really conserve surface area. If this assumption was substantiated by the output data, then the choice of a better representation would be clear. Recall l that the phenomena which the models were attempting to  !

simulate was that the rod was made into a stronger absorber g by cutting grooves into it. Therefore, when the final [

model's rod region was compared to that of model D, it {

should reflect both an increase in the absorption fraction, I and a decreaae in volume. j The tat).o of the fraction of all thennel absorptions which occurred in the rod te, the fraction of the cell's volume made up by the rod region was calculated in Table V.

I As the table shows, BRT predicted that only model G would l produce a relative increase in absorptions per unit area i from model D. This fact proved that although model E did j I. increase the rod's absorption fraction, this increase was not caused by an increase in surface area. on the other hand, model G seemed to fulfill all of the requirements [

expected of a model which was simulating an increase in rod ,

surface area.

I 4.7.4 The Fast Group Knowledge of GAM's solution method made the fast group analysis quite simple. Almost all of the points considered I

g

l L

77 in the thermal analysis were ignored because GAM performed it's N eulation in zero-dimensions. For example, the question of cor. trol-rod region composition or surface area worre irrelevant in a zero-dimension calculation. The only factors which could be controlled were the density and composition of the homogenous mixture.

cylindrical models A, B, and C were all equivalent with r2spect to the fast group. The came mixture was obtained whether medel A, B, or c was homogenized. There was no+ much of a caso to be made against this mixture because it was the mixture n ich would have resulted had the actual cell been ameared into a hnuogenous col)..

Model C was used because che occput suggcated that the thermal calculation did have an irpact on thu fast calculation. Recall that model C was the only model of the three which fulfilled the requirement that no water be included in the rod region. Table VI illustrates the extent of this impact.

4.7.5 The Final Transport Representation Sections 4.7.1-4.7.4 provided the reasoning which led to the " final transport representation". Basically, this final representation consisted of the two parameters which were expected to predict the cell's behavior most completely and accurately. According to the last few sections, these two parameters were the fraction of all thermal absorptions

i I

78 I

which took place in the rod region predicted by model G, and

p. model c's prediction of the fraction of fast neutrons which  ;

reach the thermal group. The final representation was that l 0.382 of all thermal neutrons were absorbed in the fuel, and ,

that 0.765 of all fast neutrons' survived to reach the thermal group.

The diffusion model was also forced to match a few .

other combinations of output parameters, because this final ,

representation was only an educated guess at how the cell behaved. Therefore, the results obtnined from the other [

combitaticns could also be compared to the experimental data If thv logic uuec to form the final representation was good, then the controbrod worths calculated for the diffusion model based on it should cotne closer to the experimental data than the control-rod worths based on the other diffusion :nodels.

3 Table VI: Comparison of the cylindrical Models Cylindrical cylindrical cylindrical model A model B model C i I Resonance escape probability 0.746 0.746 0.765 Multiplication factor 0.510 0.504 0.777 thermal absorption fraction in the 0.380 0.537 0.580 fuel

l I

I 79

5. The Diffusion and Effective Diffusion Models 5.1 The 2DB-UM Model j i

Whereas the transport codes were used to accurately l predict the actual cell's behavior, the diffusion codes were  !

I used to predict how different control-rod element models I would be treated during the core analysis. The core analysis I applied the time-dependent diffusion code 2DB-UM to f calculate nultiplication factors and fuel depletion. Our l objective was to develop effective. absorption cross sections which would force 2DB-bM to treat the contro1~ rod cell as l t:ransport th Jory predicteG it should. Thio uns accotrplf shed bV using the steady-state diffusion code EXTLRMINATOR to i simuh te 2DD-UM's treatment ')* a control-rod eh aent. Tha rod's absorption cross sections were varica in the EXTERMINATOR model until EXTERMINATOR produced the output parameters which had been predicted by the transport codes.

These cross sections then became part'of the 2DB-UM model with the hope that they would cause 2DB-UM to yield proper results.

If this reasoning was to hold up, it was absolutely

. imperative that the 2DB-UM and EXTERMINATOR control-rod models be identical. Not only did the models need to have the same size regions, but the regions needed to have mesh spacings which were as near alike as was possible. After all, the algorithm was based on the assumption that both I

l 1,

80 t

3; j {~ EXTERMINATOR and 2DB-UM applied the same theory to the same model, and therefore should produce identical results.

EXTERMINATOR was used, because it was very easy to calculate the output parameters from it's output.

The 2DB-UM model (Figure 5-1) was developed by Mary

- Fahr (10) as part of her core analysis. The control-rod element is portrayed in Figure 5-2 as nine regions arranged in a 3 X 3 matrix. A fuel region, rod, and then another fuel j region made up the middle column. On either side of these j regions were placed thinner regions consisting of an aluminum and water mixture. The region sizes and mesh-spaco configuration are shown in the figure. The fuel elements in the 2DB-UM model were set up in an identical manner except that the rod region was replaced with a third fuel region.

1 The compositions of 2DB-UM's regions were described by their macroscopic cross sections, in contrast to the transport codes where the region composition was described by their nuclide densities. 2DB-UM obtained the cross sections from the LEOPARD and LINX codes. Nuclide densities were input to the LEOPARD-LINX combination which calculated

_- cross sections for each region and fed them to 2DB-UM. Since

a. LEOPARD and LINX demanded that fuel be located in the center of their model, they could not be applied to the control-rod

} element. This fact created the opportunity to manually input

" effective cross sections" in this region.

g

+

n2 81 6 s >D66 DD6s' DDDD<

r -

>O s OOO<> 90<>Os O 000

>O<

OC

>Ot

<>C E >O:

- : OC c: $ ><><

>O

~

2 c

__OC OC $ ><><

>O  : -- 3 '

Q l Oc: $

><-- s s, Oc:-------- >Ot

>O  :

OC I

e l -

GRAPHITE $ WATER 1

Figure 5-1 (10): Complete TWODB-UM Model l The TWODB-UM core model includes the graphite and water g which are located around the core to act as a reflector.

The core is comprised of 16 elements in a 4 x 4 array. The core.is surrounded on all sides by the equivalent of three Ia cells of reflector material. The location of the three control-rods, and the regulating-rod can alsb be seen in the figure.

l 1,

~

i  !

Il .

1 5

I 82

<'

. D_ 2+ -

i  ;

J.... . . .. ...,

1 -

I }

~  !

v' s/ v

{~ v' -

,A rg A A I. ._ L_ .

>,. , ,i. ....

I

'i 7- _- _ ,

i= '

W p

. ic  !

r - _ , - ,,

l-

.I. Y 'R ,

I I

(I 1.

1.- Figure 5-2: TwoDB-t:.M control-Rod Cell Model l

The dotted lines show how the control-rod cell was split into nine homogenous regions in the TWoDB-UM model. Our objective I

j was to create effective cros.s. sections for the center region.

I I

al cg

.I L I.

L i

83 5.2 The EXTERMINATOR Cell Model l

The purpose of using EXTERMINATOR was to simulate the l 2DB-UM calculation on the rodded element. This implied that the ideal EXTERMINATOR model would have been identical to )

the 2DB-UM model. Therefore, the size, as well as the region I and mesh configurations of the EXTERMINATOR model were identical to those used in the 2DB-UM model's control-rod element representation.

The only differences between the two models were their macroccopic cross sections. The LINX-LEOPARD ccmbination I calculated cross sections for all 2DB-UM's regions except the rod region. Although we planned to " fudge" the absorption cross section for this region, we still needed a method to calculate the transport and scattering cross sections. Therefore, the EXTERMINATOR cross sections could not all be taken from the 2DB-UM model, as all the other model characteristics had been.

EXTERMINATOR gave the user the option of either inputting macroscopic cross sections, or inputting the combination of nuclide densities and microscopic cross sections. EXTERMINATOR's initial run utilized the second option to generate the needed macroscopic cross sections. In this case, EXTERMINATOR was applied to a model in which the structure and nuclide densities were identical to those of the control-rod element in the 2DB-UM model. The fast group

l L

I 84

!!' . I microscopic cross sections were taken from the output of the GAMTEC model C run while thermal microscopic cross sections were taken from THERMOS' slab model G output. To summarize, the purpose of EXTERMINATOR's first run was to obtain macroscopic cross sections generated by a diffusion theory I code which was applied to a model strongly resembling the '

rodded element in the 2DB-UM model. Therefore, these cross i sections would be used in future runs to approximate the cross sections that the LINX-LEOPARD combination would have generated had it been applied to the rod region.

The next, as well as all of the remaining, EXTERMINATOR runs used the macroscopic cross section input option. The purpose of the second run was to obtain an idea of how 2DB-I UM would have treated the rodded element had the diffusion theory cross sections been used without any modifications.

Therefore, the macroscopic cross sections generated in the first run were used as input to each of the nine regions. As expected, the output of the first two runs were identical.

Employing the macroscopic cross section input option was advantageous because it al.' ved us to study the effects of varying one macroscopic cross section. Once we found a value which caused the EXTERMINATOR model to behave as the transport representation predicted, it was ery simple to insert this value for the macroscopic cross section in the 2DB-UM model. In contrast, no simple mechanism existed for I

i

._ - - ._. . -- - ~_. - - - -- -

3 ~ )

LI )

l 85 )

1 changing th'a nuclide density or microscopic cross sections )

in the rod region of the 2DB-UM model.

5.3 The Effective Cross Sections As mentioned in the introduction, the crux of our problem was that the output of these preliminary EXTERMINA-TOR runs differed from the output produced by the transport i codes. For our purposes, the most important difference was L that when the output parameters were calculated based on the EXTERMINATOR data, they were not' equal to those chosen to

[ constitute the transport representation. EXTERMINATOR I predicted that 0.355 of all thermal absorptions occurred in I fuel regions and that 0.725 of all fast neutrons survived to reach thermal energy levels while the transport representation predicted 0.382 and 0.765, respectively.

The algorithm's next step was to determine a set of fast and thermal cross sections for the rod region which would cause the EXTERMINATOR code to yield output parameters equivalent to those in the transport representation. The EXTERMINATOR code was run repeatedly, with slight adjustments made to the model before each run. The fast and thermal macroscopic cross sections were varied while all the other model characteristics were held constant with respect to the preliminary EXTERMINATOR model. The absorption cross sections were s,o much larger than any of the other rod cross ,

sections that it was cafe to assume that they were the two

.I

LI parameters which dictated neutron behavior in the rod region. Consequently, the variation of absorption cross sections seemed to provide the largest range and greatest flexibilit?/ with regard to possible output parameters than variationt- in any other set of input parameters.  ;

I The iterative process was ve'y simple. The first step was to compare the transport representation's fast group output parameters with those of tle preliminary EXTERMINATOR model. This comparison allowed an educated guess to be made at the fast group absorption cross section value which would lI l'- cause EXTERMINATOR to generate a fast group output parameter equivalent to that of the transport representation. The original cross section was then replaced with the new estinate in the EXTERMINATOR input deck and the code was applied again. This process was repeated until an acceptable cross section was found.

Ne'ct, the same procedure which had been applied to the fast group was applied to the thermal group. At this point, we had to return to the fast group to make sure that the change in the thermal cross section had not had a large effect on the fast group. This step was vital because the effective cross sections were to be input to 2DB-UM as a set, and therefore we had to be sure that the set performed the desired function. If the thermal group changes did have i a significant effect on the fast output parameter, both I

I

I  :

3 cross sections were adjusted again using the same process.

The only difference from the first adjustment was that the second one began with the cross sections obtained in first <

adjustment. This led to much smaller adjustments to the cross sections than in the previous iteration, and consequently the changes in the thermal cross section had ,

~

leus of an effect on the fast group. The process was repeated until the changes in the thermal cross section led

~

to inconsequential changes in the fast group output parameter. The resulting cross sections were the effective

l. cross sections.

l Chapter Four contained the explanations for assuming ,.

l.I that the fraction of neutrons which reached the thermal .

. group was the best choice to be the fast group output parameter, and that slab model G was the best one-dimensional model. Since these assumptions certainly had not l been proven beyond a shadow of a doubt, the iterative l

l process was applied to a few different sets of models and l

output parameters. If the effective cross sections generated from the combination of slab model G and the fraction of fast neutrons reaching the thermal group had the best agreement with the experimental data then this would be another bit of evidence for assuming that this combination should be used in the LEU modeling.

O 3

)

88 i

Details of the development of effective cross sections i

for diffarent sets of models and output parameters are given in Appendix A. A summary of the results is shown in Table VII.

l 5.4 Effects of the Diffusion Approximation Although it was not critical to obtaining the proper l

" effective" cross sections, it was interesting to look at the direct effects of improperly applying the diffusion approximation to a strong absorbing region. Recall that the preliminary EXTERMINATOR run predicted that a smaller fraction of all thermal absorptions took place in the fuel region than the transport representation had predicted. This data seemed to imply that diffusion theory over-estimated the absorber's power. Although this statement may be true, at this point it would have been incorrect to attribute the over-estimation solely to the diffusion approximation.

Instead, it had to be taken into account that the two theories had been applied to significantly different models.

For example, the BRT model had thirty mesh points in the x-direction while the EXTERMINATOR model had only nine mesh points in the same direction.

The simple solution to this problem was to apply both theories to the same medel. Since it was impossible to apply BRT to a two-dimensional model, EXTERMINATOR was applied to the one-dimensional slab BRT model G. The comparison of the

89 Table VII: Effective Cross Sections for Different Transport Models Cross Thersal Output Fast Output Rod Ze2 Rod E,3 Section Set Parameter Parameter Fuel absorption I

l Set I fraction Resonance escape prob.

BRT slab model A CAMTEC model A 1.33 x 10'1 1.77 x 10 2 Set II Fuel absorption Multiplication B fraction factor BRT slab model A CAMTEC model A 1.34 x 10'1 2.64 x 10 2 L Set III Fuel absorption Resonance fraction escape prob.

BRT slab model B CAMTEC model B 1.11 x 10'1 1.78 x 10 2 Set IV Fuel absorption Multiplication fraction factor BRT slab model B CAMTEC model B 1.11 x 10'1 2.09 x 10 2 Set V Fuel absorption Resonance fraction escape prob.

j. BRT slab model C CAMTEC model C 8.90 x 10 2 1.58 x 10 2 Set VI Fuel absorption Multiplication I- fraction factor BRT slab model C CAMTEC model C 8.88 x 10 2 5.84 x 10'3 Set VII Fuel absorption Resonance I- fraction escape prob.

2.64 x 10 2 BRT slab model C CAMTEC model D 1.01 x 101 Set VIII Fuel absorption Multiplication fraction factor BRT slab model C CAMTEC model D 1.01 x 101 2.03 x 10 2 Set IX Fuel absorption Resonance fraction escape prob.

BRT slab model D CAMTEC model C 1.71 x 10'1 1.78 x 10 2 I

3 5

8

t 90 thermal flux shapes predicted by the two theories, displayed in Figure 5-3, shows that the two codes predicted similar shapes. BRT did predict that 0.382 of all thermal absorptions took place in the fuel compared to EXTERMINATOR's prediction of 0.355. Basically, there were no vast differences between the two codes' output with respect to this model.

Two possible conclusions can be reached based on this information. One possibility is that invoking the diffusion approximation really does not have a negative effect. The second possibility is that the fine mesh spacing combined with the rod region's shortened width help to. nullify the negative effects of the approximation. My first reaction was that some insight into these two possibilities could gained by comparing the small differences in the flux shapes to the expectations of the effects of the approximation. The only problem with this approach was that just about any flux shape could be explained by emphasizing different effects of the diffusion approximation.

The only notable differences were that the transport code predicted a slightly higher relative flux in the guide plate region while the diffusion code predicted a slightly higher flux in the fuel region. In order to decide if these differences were due to the approximation, we should review the expected errors which would be caused by incorrectly

f ,

91

)! .

I '

r l \.;

h H

. 8 -

[ "

1 I '.. ,

85 ,

n, l

. 8 4

I .

m

..... g l . ....... ....

ge.O I "am 8.

g

~

l

o

. O l

• l l

lB i l

8 E

F.

E 8 R S R 86 d 6 6 d XnlJ 8AD/Xnl}

Figure 5-3: BRT vs. EXTERMINATOR This graph illustrates the thermal flux shapes predicted by BRT and EXTERMINATOR when they are both applied to the same model. The one-dimensional model used had thirty mesh spaces.

92 invoking it. Recall that the major problem with applying the approximation to the rod region was that the approxination assumed that the probability of a scattering reaction occurring was fairly constant. In fact, the scattering probability severely dropped at'the border of the rod region. Therefore, we expected that the diffusion code would over-estimate the number of neutrons leaving the rod region, on the other hand, the diffusion approximation assumed that the number of neutrons travelling in any direction was directly related to the flux gradient. Therefore, in a region with a very large gradient we would c):pect that the diffusion code would over-estimate the number of neutrons travelling towards the region of lower neutron population.

At the rod border these two effects work in opposite directions. For different models, or mesh-space configurations one effect may outweigh the other. Therefore, any judgement on the effects of the approximation in regard to these models would require more effort than it's worth to this project.

In either case, the necessity of developing effective cross sections can be seen in Figure 5-4 where the flux shape predicted by the application of EXTERMINATOR to the 2DB-UM model has been included in the graph. Recall that this was the data which would have been included in the core analysis in the absence of the effective cross sections.

I

) O 93

)  % 0 1 '% D D 1 D. N,

) 's s'9 s i- g

@y D

g,N O,

N D

.O s

- ao s,

O N

'b 's U

g 'N ~

, .8 O Q Em 9

\

~ %.a' s

,' cl 8

i. o o g,;

9 I O

O g) h,' U 8.

' / *...al l

C @

,/..D j -

,0 je i P /

? /

O /

/,

/O Y..'P 8

a e U

. /,

c' 8 R S 8 8 R S 8 8 a  :  :  : 6 5 5 d XnlJ SAD /Xnl1 Figure 5-4: BRT Predictions vs. TWODB-UM Predictions This graph illustrates the different flux shapes predicted by BRT and TWODB-tim. Recall that the BRT output has been accepted as the most accurate predictions available. The TWODB-UM calculation is simulated by applying EXTERMINATOR to the TWODB-UM model. This calculation yields the data which would have been used in the core study if no corrections were made to the control-rod region. <

t 9

o

s.

]

94 This graph also clearly shows that there was no hope of simulating the flux shape of the transport code because of a shortage of mesh points available in the 2DB-UM model. This fact was the reason that the flux shapes were totally ignored while the effective cross sections were used to forced the 2DB-UM code to match the output parameters of the transport representation.

I I

I I

I I

I B

B

?

95

6. The Regulating-Rod The last order of business before control-rod worths were calculated was to develop effective cross sections for the regulating-rod. Following the same procedure which had been applied to the control-rods, the LEOPARD and LINX codes were not used to calculate macroscopic cross sections for

'the regulating-rods. Therefore, effective cross sections were necessary so that they could be manually input to the 2DB-UM code.

Two characteristics differentiated the regulating-rod from the control-rods. The principal difference was that the regulating-rod was composed of one hundred per cent stainless steel. A quick glance at the control-rod's microscopic cross sections shows that it was the presence of boron which made the rod such a strong absorber. The second

l difference was that the regulating-rod did not have any grooves cut into it (Figure 4-2). The combination of decreased surface area and lack of boron severely reduced the regulating-rod's absorbing power relative to that of a control-rod. Therefore, the logic used to develop the transport models of the control-rods was not applicable to the regulating-rod.

Actually, the development of a regulating-rod model was much simpler than the development of the control-rod model.

Recall that the control-rod's high absorption cross section

l.'

i I  :

ll led us to believe that most neutron absorptions would take

~

place within a small area along the rod's border. Therefore it was imperative to construct a model which accounted for the grooves by conserving surface area. In the absence of the grooves and'high absorption cross section it was assumed that the rod's volume and nuclide densities became the factors which dictated the cell behavior. Therefore, the nodel of the regulating-rod fuel cell conserved these parameters.

After the model was constructed, the procedure was exactly analogous to that outlined for the control-rod. The

macroscopic absorption cross sections were varied in the EXTERMINATOR model until the output parameters matched those predicted by the transport models. The set of cross sections which forced the diffusion code to match the transport codes' cutput was deemed to be the effective cross sections.

I I

I

I~

I I

I

u =. . :a.: ,

I ~

l8

97 >

l

7. Results

.7.1 The Experimental Data The only remaining task was to check how well the effective cross sections had served their purpose. This task included the comparison of the 2DB-UM output with the experimental data, and the verification of the chosen transport representation as the most accurate.

Experimentally measured control-rod worths were

~

available for the HEU core. These worths were taken from Farrar, - J . P. (1975) and are shown in Table VIII. The W experimental error was estimated to be at least ten percent.

, It should be noted that these worths were determined from rod calibrations. In other words, the worth of one rod was measured while the others were partially inserted into the I

core. The presence of three other rods was expected to slightly affect the rod worth being measured.

1 Table VIII: Experimental Rod Worths Rod # Measured Worth j 1 $4.75 2 $5.00 3 $3.06 Regulating $0.57 All in $13.38 I

I  ;

~

98 Table VIII shows that each of the control-rods had different worths even though they were physically identical.

This fact makes it very clear that the rod worths were very dependent upon the rod's position. That is to say, the higher fluxes and importances in the center of the core led

--! to higher worths for the rods placed in the center as 1

opposed to along the core border.

This is an important point because the flux was included in the effective cross section development.

Therefore, the differences in the fluxes would result in

__ slightly different actual weighted cross sections for each

rod. We cannot hope to duplicate these differences because r the effective cross sections were developed for each rod as if each rod element was surrounded by other control-rod elements (recall the reflecting boundary conditions). An argument was made that the neighboring control-rod element would do an acceptable job of simulating other fuel cells.

/ Therefore, we might expect the simulated rod worths of rods 4 1 and 2 to come a little closer to the experimental worths than the regulating-rod or rod 3 which both are located on the core border.

7.2 Calculated Control-Rod Worths In Chapter Five, nine different sets of possible effective cross sections were developed. Each set was based on a different combination of transport model and output

y ,

i

'I' .

99  ;

I parameters. A theoretical argument was made that set IX would lead to the most accurate control-rod worths. The best

~

way to check the validity of that argument was to insert each set of cross sections into the 2DB-UM model, and to see I which set yielded control-rod worths which agreed with the experimental data. The results of the nine 2DB-UM runs are s

given-in Table IX, where the calculated worths were convertert to $ using g,=0.0074.

Sets II, VII, IX all produce calculated rod worths within the ten per cent estimated error of the measured worths. The major difference between the sets is that set IX predicts a higher thermal and lower fast cross section

. than the others. The flaws in the development of sets II l

(' and VII have already been discussed in detail. Set II was

~

based on model A which incorrectly included water in the rod region, and set VII was based on model D which simulated the rod without grooves. Therefore, we have to assume that the l ,

accuracy of these sets' predictions is due more to luck than a carefully thought out procedure. It follows that we cannot hope to achieve the same accuracy when the same procedure is applied to the HEU core.

i The method which led to set IX was chosen as the method L

! to be applied to the LEU core, because the HEU rod worths were all within 10% of the measured values and there were not any obvious flaws in the method's logic. -Therefore,

- 7

.I .

J I

I )

100 ,

Table IX: Trial Set Results vs. Experimental Data )

I' Effective cross section calculated Measured Per cent set # Rod f Worth Worth Error I 1 $4.74 $4.75 0.3 ,

2 $4.93 $5.00 1.4 ';

3 $2.82 $3.06 7.8 II 1 $5.38 $4.75 13.3 ,

2 $5.53 $5.00 10.6 I 3 $3.15 S3.06' 2.9 III 1 $4.51 $4.75 5.1 2 $4.64 $5.00 7.2 '

3 $2.65 $3.06 13.4 I IV 2

3 1 $4.72

$4.86

$2.78

$4.75

$5.00

$3.06 0.6 2.8 9.2 V 1 $4.00 $4.75 15.8

! 2 $4.11 $5.00 17.8 F l- 3 $2.36 $3.06 29.7 VI 1 $3.23 $4.75 32 2 $3.32 $5.00 34 '

3 $1.90 $3.06 38 VII 1 $4.75 $4.75 0.0 ,

i 2 $4.89 $5.00 2.2 3 $2.76 $3.06 9.8 VIII 1 $4.32 $4.75 9.1

'l 2 $4.44 $5.00 11.2

,W 3 $2.52 $3.06 17.6 IX 1 $5.14 $4.75 8.2 I' 2 3

$5.29

$3.15

$5.00

$3.06 5.8 2.9 Reg Rod $0.57 $0.75 31.0 I

I t

I

F l

.g l 101 slab model G and cylindrical model C will be the models used to perform all calculations needed for the LEU core study.

The final recommended set of 2-group constants for-the control-rod. regions are given in Table X.

5 Table X: Final Recommended Cross Sections for 2DB-UM l

.- Group I, E tr E snn E s12 Control-Thermal 1.71 E-1 2.06 E-0 1.56 E-0 ------- -

Control-Fast 1.78 E-2 2.70 E-1 5.73 E-1 2.29 E-2 Regulating-Thermal 3.68 E-2 1.24 E-0 1.37 E-0 -------

Regulating-

l. Fast 2.24 E-3 2.81 E-1 5.46 E-1 1.60 E-2 7.3 Conclusions The most important statement that can be at this point 1.

is that the method worked. The effective cross sections make I it possible to apply 2DB-UM to the entire core. The same ,

. method can now be applied to an LEU control-rod element. The cross sections generated will make it possible to apply 2DB-UM to different cores (by changing fuel plates per element, number of elements, etc.). 2DB-UM's predictions will be used to choose the most practical core.

An important point to keep in mind is that The EXTERMINATOR and 2DB-UM control-rod models should be I

I

s l s

102 I identical. Very simply, in order for the effective cross sections to produce the proper results, they must be used in the manner that they were developed.

'7.4 Recommendations .1 The effective cross section development could be checked further by applying it to different stage in the depletion of both LEU and HEU cores. It might also be checked against other methods, such as Argonne Nati.onal l

Laboratory's Monte Carlo Method. It would also be ,

interacting to attempt to apply the algorithm to other reactors. The only drawback to this suggestion is that it

would require that the other reactor have control-rods shaped similarly to the UVAR's rods.

I -

I

I <

I

. I-B Lg lI

xg 1-REFERENCES

,a (1) " Limitations on the Use of Highly Enriched Uranium in Domestic Hon-power Reactors," 1Q._pFR 50.64, Jan. 1 1987 5 Ed- -

(2) "2DB-UM (APOLLO VERSION)," (input manual for 2DB-UM, Version #10, July 86) , Sept. 1986. )'

[3] Dannett, C.L., Purcell, W.L.,

"BRT-7: Bataile-Revised-

I THERMOS," BNWL-1434, Batelle Memorial Institute, Pacific Northwest Laboratories, June 1970.

[4] Carter, L.L, Richey, C.R., Hughey, C.E.,

"GAMTEC II: A

'I Code for Generating Consistent Multigroup constants Utilized in D!.ffusion and Transport Theory Calculations,3 BNHL-35, Pacific Northwest Laboratory, March.1965.

[5] Fowler, T.B., Tobias, M.L., Vondy, D.R.,

I " EXTERMINATOR 2: A FORTRAN IV Code for Solving Multigroup Neutron Diffusion Equations In Two

-Dimensions," ORNL-4078, Oak Ridge National Laboratory, j

j 1961.

.I [6] Dresner, L.

i

.i

" Resonance Absorotion in Nuclear Reactors, Pergamon Press, 1960. i' I "

(7) ' University Of Virginia Reactor Safety Analysis Report,"

(8) " Fuel Element - Control Rod," Blueprint # NE-34, l Nuclear Science Center; A&M College of Texas, Department of Nuclear Engineering, Sept. 6, 1960.

[9] Rydin, R., " Nuclear Reactor Theory and Deslan,"

University Publications, 1977.

[10] Fehr, M., " Design , Optimization of a Low Enrichment University of Virginia Nuclear Reactor," University of Virginia, Jan. 1989.

I I $

't