STUDIL3 o f t h e r e a c t i v i t y TEMPERATURE COEFFICIENT IN LIGHT WATER REACTORS

MALTE EDENIUS

DEPARTMENT OF REACTOR PHYSICS

Studies of the reactivity temperature coefficient in

light water reactors

Malte Edenius

AKADEMISK AVHANDLING

som framlagges till offentlig granskning

vid sektionen for teknisk fysik for av-

laggande av teknisk doktorsexamen i reak-

torfysik den 4 maj 1976, kl 10.00, semi-

narierummet, institutionen for reaktor-

fysik, Gibraltargatan 3.

Goteborg

Mars 1976

AE—RF—76—3160

STUDIES OF THE REACTIVITY TEMPERATURE COEFFICIENT IN

LIGHT WATER REACTORS

Malte Edenius

AB Atomenergi, Studsvik

March 1976

LIST OF CONTENTS

1. Introduction 1

2. Influence of temperature on the neutron

transport 4

2.1 Thermalization 4

2.1.1 Neutron scattering laws 5

2.1.2 Scattering models for water 7

2.2 The Doppler effect 9

2.3 Density effects 14

3. The cell code AE-BUXY 17

3.1 Nuclear data library 17

3.2 Resonance treatment 19

3.3 Micro group calculation 21

3.4 Macro group calculation 23

3.5 Fundamental mode calculation 24

4. Comparison between theoretical results

and experimental information 28

4.1 Description of the measurements 29

4.2 Description of the calcula-

tional methods 30

4.3 The reactivity worth of spacers 31

4.4 Results of comparison between

theory and experiment 33

4.4.1 Lattices with 1.35 % enriched

U 0 2 33

4.4.2 Lattices with 1.9 % enriched

U O 2 rods and PUO 2 rods 35

4.4.3 Comments to the results 39

5. The components of the temperature co­

efficient 44

5.1 Partial temperature, coeffi­cients 46

5.2 Contributions to the tempera­

ture coefficient calculatedby COEFF 52

5.2.1 Description of COEFF 52

5.2.2 Results from calculations with

COEFF 55

6. The influence of approximations in the

theoretical treatment on calculated tempe­rature coefficients

6.1 The crystalline binding in

6.2 The cylindricalization of pincells

6.3 Comparison between pin cell

calculations using isotropic and anisotropic scattering

6.4 The calculation of leakage

6.5 The number of energy groupsand space points

6.5.1 Macro groups and Gauss points in AE-BUXY

6.5.2 Energy groups and mesh points in the diffusion theory calcu­

lation

6.6 The influence of thermal ex­pansion on reactivity

6.7 The plutonium fission neutron spectrum

7. The influence of nuclear data on the

calculated temperature coefficient

7.1 Comparison of temperature co­

efficients calculated by use of ENDF/B and UKNDL data

7.2 The effective resonance inte­gral of U-238

7.3 Thermal scattering data

7-3.1 Scattering in water

7.3.2 Scattering in UC^

7.4 Thermal absorption and fission

cross sections

7.4.1 Group cross sections for a

1/v-absorber in Maxwellian spectra of various temperatures

7.4.2 Thermal data for U-235 and U-238

8. Summary

9. Acknowledgement

10. References

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1. INTRODUCTION

One important quantity in determining the operating charac­

teristics and safety of nuclear reactors is the temperature

coefficient of reactivity. The isothermal temperature coef­

ficient in Light Water Reactors (LWR) varies considerably

with the design, the moderator temperature and the boron con­

centration in the moderator etc. Typical values for a fresh

Boiling Water Reactor (BWR) core is -5 pcm/°C (1 pcm = 10

at 20°C and -25 pcm/°C at 280°C. Typical values for a Pres­

surized Water Reactor (PWR) are in the range -5 to -30 pcm/°C.

Much effort has been devoted to the development of proper

methods for the calculation of temperature coefficients. De­

spite this, the methods in current use to calculate the tem­

perature dependence of reactivity in light water reactors are

not altogether successful [l - 4]. As an example measured and

calculated values of dk ,-,/dT from the Swedish Oskarshamn-Ierr

BWR reactor are shown in Fig 1.1.

The calculated temperature coefficient is usually 2 - 5

pcm/°C too negative compared to the measured value. Going

from room temperature to operating temperature this means an

error in the predicted reactivity of about 1 %. Thus an in­

centive exists for further studies in this field of reactor

physics. The literature lacks information about clean

(simple, well-defined) and precise high temperature experi­

ments. With the extensive series of experiments which have

been performed in the high temperature facility KRITZ at

Studsvik sample data to compare with calculated ones have

been produced.

In the present paper a survey of the temperature effects in

a nuclear reactor is first given. Then follows a description

of the analysis of the KRITZ experiments and a comparison

between theoretical results and experimental information.

The theoretical methods used in the analysis are described.

Different components of the temperature coefficient are

studied in chapter 5. Uncertainties in the employed theore­

tical methods are discussed in chapter 6 and the influence

of nuclear data on the temperature coefficient is investiga­

ted in chapter 7. A summary is given in chapter 8.

Tem

p co

eff

( pcm

/ °C

)

Fig. 1.1 Oskarshamn I. The isothermal temperature coefficient

versus temperature [1].

2. INFLUENCE OF TEMPERATURE ON THE NEUTRON TRANSPORT

The influence of temperature on the neutron transport is

caused by the thermal movement of nuclei influencing the

scattering of thermal neutrons and the Doppler broadening of

resonances and by the thermal expansion of different mate­

rials.

2.1 Thermalization

In many materials which are present in a nuclear reactor the

atoms may be considered to be free. The energy distribution

of the atoms is then the Maxwell distribution and it is poss­

ible to derive an accurate expression for the scattering of

neutrons. This model can be used e.g. for scattering against

heavy nuclei which does not affect the neutron spectrum very

much.

For light nuclei, however, the treatment of scattering re­

quires a consideration of the chemical binding and for a sa­

tisfactory treatment of the thermalization in water it is

necessary to use scattering cross sections computed according

to a relevant model for the scattering process. A number of

approximations are involved in these scattering models. The

accuracy of the model is of great importance when predicting

the temperature dependence of the neutron spectrum and hence

for predicting the temperature effects in thermal reactors.

It may be noted that an accurate treatment of the thermali­

zation is especially important in systems with only partial

thermalization, i.e. when the neutrons do not obtain a Max­

wellian velocity distribution before they are absorbed. This

is the case in light water reactors.

The influence of binding on absorption is negligible and

absorption cross sections can be taken to be the same as for

free nuclei.

2 . 1.1 Neutron scattering laws

The scattering in a monoatomlc gas is treated in most books

on reactor physics and will not be discussed here. In

applications all scattering except that in the main con­

stituent of the moderator is usually considered to obey

the free atom scattering law.

For neutrons undergoing scattering in a medium containing

bound atoms it has been shown [5] that the scattering function

can be written as the sum of differential coherent and in­

coherent cross sections

E and ft are the energy and the unit vector in the direction

of motion before the collision. E* and ft* are the corre-

are the macroscopic bound coherent and incoherent cross

sections, e = E-E* is the energy change of the neutron and

•tfic = m ( v - v T) is the neutron momentum change vector.

The functions S( k ,c ) and S. (k ,e ) are defined by— inc — J

Z (E-*E 1 1) * Z L---- coh

(E->E *, 1) +1. (E-*E f 1)— m e ------ (2 .1)

(2 .2 )

(2.3)

spending quantities after the collision. Z , and E. r n coh im e

(2.4)

eiUr-xt/fi)

G (r,t)drdt s

(2.5)

The pair distribution functions Gs (£> t) and G^Cr,t) de­

defined as the probability that a nucleus which is at origin

at time t = 0 will be present within a unit volume at posi­

tion _r at time t. G^(r,t) is the probability that a nu­

cleus other than that at origin at time zero will be at po­

sition r at time t.

The interference effects of the scattering are contained in

G^(_r, t). These interference effects are important for elastic

scattering in many materials, but for inelastic scattering

in most liquids and polycrystalline materials they can be neg­

lected, i.e. G^(r,t) can be put equal to zero. Then

which is known as the incoherent approximation. For light water

the incoherent approximation is especially good because

neutron scattering by hydrogen is almost completely inco­

herent (a = 1.8 b and o. = 80 b ) . coh m e

One further approximation is employed in many scattering

models, viz. the Gaussian approximation. A function inter­

mediate between G(^r,t) and S(k ,g ) is defined by

The Gaussian approximation of the intermediate scattering

function is given by

termine the dynamics of the scattering systems. Gg (r^t) is

(2 .6)

(2.7)

(2 .8)

with

f (w)e~fiu/2kT

2ui s i n h ( W 2 k T )-not ,

e -1

A is the mass of the scattering atoms, k the Boltzmann

constant, T the temperature, w is the angular oscillation

frequency and f(w) the frequency spectrum normalized so

that

ff(w)du) = 1

For simple scattering models assumes the Gaussian

form and this form is also applied in many sophisticated

models for scattering in water. In order to determine

>'(t)-y(0) the frequency spectrum, f(ui), is estimated from

physical considerations.

2^1.2 _____§cattering_models_for_water

One of the early models is the Nelkin model [6]. In this

model the hindered rotational motion is approximated by a

torsional oscillation. The incoherent approximation is used

for the scattering by hydrogen and the Gaussian approxima­

tion is employed with a spectrum representing a set of

discrete oscillator frequencies. The spectrum is written

I 1f(u>) = I -±- 6(u>-w.) (2.10)i=l i L

A 1 =18 ' = 0

2.32 •Ftu>2 = 0.06 eV

A 3 =5.84

fiw3= 0.205 eV

> Ji­ll 2.92 ■fiu>.

4= 0.481 eV

The first term in the summation represents the translational

motion of the free gas molecules and the second term the

hindered rotation which is assumed to be a torsional oscilla­

tion. The remaining two terms represent vibrational modes.

It is possible in the framework of the Gaussian approximation

to reconstruct the complete S(a,|3) function from p(6) .

Egelstaff and Schofield [7] have proposed a scattering model,

which is known as the effective width model, with

where K^(x) is the Bessel function of the second kind.

The effective width, q, is a measure of the width of p(3).

The model degenerates to that of a free gas as q-K) . The

values of q to be used are determined by correlation to

integral experimental data.

Aj, is an effective mass of the i:th quantum state.

Defining

S(a,3) H kTe6/2 S(k ,e ) (2 .11)

with

2(2 .12)

6-G

(2.13)kT

the spectral density function can be introduced as

p (8) s 26 sinh(8/2) Him S(a,B)] rv-vn a Ja-’-O

(2.14)

(2.15)

The Haywood model [8] is based on experimental values of

p(B) for water.

The three scattering models for water mentioned above are

the ones most frequently used in reactor physics calcula­

tions. They are also the models which have been used in the

present work.

The complete scattering cross section for the water moleeule

is obtained by the addition of oxygen as a free gas.

2 . 2 The Doppler effect

The resonance cross section is for nuclei at rest given by

the Breit-Wigner formula

E x ( E ) = N o o rx /ITE 2 2

4 (E-E ) +r o

/IT 2 rrI (E) = No V - J ---- — - ■ - -S.

E 4 (E-E ) +r2 L1

F_ 4(E-Eo ) R

+ r *+ No

(2.16)

(2.17)

The index x denotes absorption or fission. N is the atom

number density. T , T and T are, respectively, the x n

width for reaction x , the width for neutron emission and

the total width of the resonance. E is the resonanceo

energy. K is the reduced de Broglie wave length of the

neutron and R is the nuclear radius, o is the potential

scattering cross section and a denotes the peak value ofo

the total resonance cross section which is given by

? T'n(EJ. n o°o = *o 8 r—

(2.18)

The statistical factor g is expressed by

2J+1

g ~ 2(21+1)

where I is the spin of the target nucleus and J the

spin of the compound state.

When the nuclei are in thermal motion, the resonances are

broadened as a result of the Doppler effect. The Doppler

broadened cross section can be written [9 ]

S(£,e) is the scattering law function defined in Eq. (2.4)

and £ is the momentum of the neutron. In most applications

it is assumed that the absorbing nuclei have a Maxwellian

velocity distribution at the temperature of the medium. At

temperatures above the Debye temperature this approximation

is a good one, because the velocity distribution of the

nuclei is then insensitive to the chemical binding. The

Debye temperature is about 620 K [10] for UO^ and 200 K

for metallic uranium. The effects of crystalline binding on

the resonance absorption will be discussed in § 6.1.

Assuming that the nuclear velocities have a Maxwellian

distribution one obtains [11]

where the Doppler functions t|>(£,Y) and x(?>Y) are

defined by

(2 .20)

(2 .21)

Is (E) = N o q - p *(C,Y)+Noo | x(?,Y)+Nap (2 . 22)

g

2/tT

exp

/iT ■

- \ g(X-Y)2]

i+x2dX (2.24)

and

A = / H

2 1 2 X e — ( E ~ E ) : E = — m v where v is the

T r o r 2 r rrelative neutron-nucleus speed.

The Doppler width, A, represents the effect of temperature

on the shape of the resonance and is a measure of the width

of the Doppler broadened resonance. A is proportional to

the square root of the temperature, T, and the energy, E.

At low temperature A is small so that C is large and

the integral (2.23) becomes

*(C,Y) » — K (2-25)1+Y

Inserting (2.25) into (2.21) one obtains the Breit-Wigner

formula for the unbroadened resonance. At the other extreme,

i.e. at very high temperature, t, is small and

U.(?,Y) « ^ x. exp(- \ C2Y2) (2.26)

so that near the resonance peak

. . X / O / nSc(E) = Nao — V E ~ T exp

In this Gaussian expression A determines the width.

The shape of a resonance is changed markedly by the Doppler

broadening but the integral

stant

o(E)dE is approximately con-

a (E)dE = - a r i o

4*(C»Y)dY = - t a r z o (2.28)

In deriving the expressions (2.21) - (2.28) for the Doppler

broadened resonance cross sections it was assumed that the

velocity of resonance nuclei is much smaller than that of the

neutron. Except for low energy resonances in light nuclei

this approximation is justified, and it introduces negligible

errors when considering the U-238 resonance absorption.

The reaction rate within a resonance is a (E)<|>(E)dE and

although the integral (2.28) does not change with temperature

the resonance absorption is increased at higher temperature

because the absorption rate is changed as a result of less

marked dips in the flux at Doppler broadened resonances.

If <j>(E) is normalized to <f>(E) = 1/E above the resonance

0 ^(E)<j>(E)dE is called the effective resonancethe quantity

integral and is represented by RI

In the narrow resonance approximation [12] the effective

resonance integral is written

N.RI = x

£x (E)£p dE

Zt(E) E

where is the potential scattering cross section and

I (E) the total cross section. If E/E is set equal to t o

unity and the interference between resonance and potential

scattering is neglected, one obtains for a single resonance

f Na (r / r ) * (c ,Y ) £R I = — — - — - — -------------------E

x E I No !{.(£,Y)-!-!o 3 o p

Defining the function

»(C,Y)<KC.Y)+B

dY C

RI can be written x

N'RI = I — • Ji x p E

o ('■ £ c

For a series of resonances the total resonance integral is

obtained by summation of Eq (2.32) over all individual

resonances.

The behaviour of J(t,f3) versus 8 is shown in Fig 2.1

(reproduced from [13]). For large values of 6=Sp/NaQ the

flux depression by the resonance is small and J(?,8) is

insensitive to £ , i.e. to the temperature. J(^,6) is

independent of ? also when 8 is small. This is due to

the very strong flux dip at the resonance center causing

most of the absorption to occur at some distance from the

resonance center, where the shape of the cross section is

independent of the temperature.

• 31)

.32)

The Doppler broadening is most important when J(C,8)

varies significantly with £ at a given 3 • From Fig 2.1

this is seen to be the case for 7<j<16 , i.e. in the

range 0.001<8<1 . For U-238 this roughly corresponds to

1 00<a <1 05 .P

2. 3_________ Density effects

The effects of moderator density on reactivity are important

for the temperature coefficient in water moderated systems.

The x^ater density decreases with higher temperature causing

the leakage of neutrons and the number of fast fissions to

increase, the. resonance escape probability and the absorption

in the water to decrease and the thermal spectrum to become

harder. These effects are studied in chapter 5.

The effect of a change in water temperature is pronounced

aL high temperature due to the variation of the water d0

density derivative, — , with temperature (Fig 2.2).

A less important component of the temperature coefficient

is due to the thermal expansion of the fuel, the canning

and construction material. These effects have been investi­

gated for some lattices, the results being given in § 6.6.

J(E.P)

j (where (3 =2^ x 10"5)

Fig.2.1 The Doppler broadening function J ( £ . P ) versus p for various £ .

3. THE CELL CODE AE-BUXY

AE-BUXY is the AE-version of the code BUXY (Burnup in xy-

geometry). It is a spectrum code for pin cells as well as

for BWR and PWR fuel assemblies. In the present work, we

have used only the pin cell option and the description be­

low will be limited to this option. A flow chart is shown

in Fig 3.1.

3.1_________ Nuclear data library

The data library contains microscopic cross sections in 69

energy groups (Table 3.1) divided into 14 groups in the fast

region above 9 keV, 13 groups in the resonance region between

4 eV and 9 keV and 42 thermal groups below 4 eV. Most of the

calculations performed for this work have been made with a

library based on the UK Nuclear Data File [14]. A library

based on ENDF/B III has been used for some calculations de­

scribed in chapter 7.

Maxwellian spectra with 1/E-tails have been used for weight­

ing thermal group cross sections. Group averaged cross sec­

tions in the resonance region have been obtained assuming a

1/E-specrum and a spectrum typical for a water system has been

used in the fast region. Effective resonance integrals for

U-235, U-236, U-238 and Pu-239 have been provided through

detailed slowing down calculations for homogeneous mixtures

of the absorber and hydrogen. Resonance integrals are tabu­

lated as function of potential scattering cross section and

temperature.

For the principal moderators scattering matrices are avail­

able based upon alternative theoretical models. We have used

the Nelkin model for water when not otherwise stated. Scat­

tering matrices are tabulated for a representative range of

temperatures.

Table 3.1 Energy boundaries i

Group Energy

Mev

1 10 .0 6 .0655

2 6 .0655 -3 .679

3 3 .679 -2 .231

4 2 .231 -1 .353

5 1.353 -0 .821

6 0 .821 -0 .500

7 0 .500 -0 .3025

8 0 .3025 -0 .183

9 0 .813 -0 .1110

10 0 .1110 -0 .06734

11 0 .06734- 0 .04085

12 0 .04085- 0 .02478

13 0 .02478- 0 .01503

14 0 .01503- 0 .009118

eV

15 9118.0 5530.0

16 5530.0 3519.1

17 3519.1 2239.45

18 2239.45 -1425.1

19 1425.1 - 906.898

20 906.898 - 367.262

21 367.262 - L48.728

22 148.728 - 75.5014

23 75.5014- 48.052

24 48.052 - 27.700

25 27.700 - 15.968

26 15.968 - 9.877

27 9.877 - 4.00

28 4.00 - 3.30

29 3.30 - 2.60

30 2.60 - 2.10

31 2.10 - 1.50

32 1.50 - 1.30

33 1.30 - 1.15

34 1.15 - 1.123

35 1.123 _ 1.097

the 69 group library

Group Energy

eV

36 1.097-1.071

37 1.071-1.045

38 1.045-1.020

39 1.020-0.996

40 0.996-0.972

41 0.972-0.950

42 0.950-0.910

43 0.910-0.850

44 0.850-0.780

45 0.780-0.625

46 0.625-0.500

47 0.500-0.400

48 0.400-0.350

49 0.350-0.320

50 0.320-0.300

51 0.300-0.280

52 0.280-0.250

53 0.250-0.220

54 0.220-0.180

55 0.180-0.140

56 0.140-0.100

57 0.100-0.080

58 0.080-0.067

59 0.067-0.058

60 0.058-0.050

61 0.050-0.042

62 0.042-0.035

63 0.035-0.030

64 0.030-0.025

65 0.025-0.020

66 0.020-0.015

67 0.015-0.010

68 0.010-0.005

69 0.005-0

3. 2_________ Resonance treatment:

A special calculation is performed to determine the effective

resonance integrals in the energy region between 4 eV and

9118 eV. Resonance absorption above 9118 eV is regarded as

being unshielded. The 1.0 eV resonance in Pu-240 and the

G.3 eV resonance in Pu-239 are adequately covered by the

concentration of thermal groups around these resonances and

are consequently excluded from the special resonance treat­

ment. Four nuclides viz. U-235, U-236, U-238 and Pu-239 are

treated as resonance absorbers.

The subroutine for calculation of effective cross sections

in the resonance region is based on an equivalence theorem

[15] which relates the tabulated resonance integrals to the

particular heterogeneous problem. The equivalence theorem is

derived using a suitable rational approximation for the fuel

to fuel collision probability.

The resonance integral in the fuel is in the narrow resonanct

approximation [16]

RI =- 0 (E)

a (E) *'(1-Pff-)ot + ap P ff-* E~

where is the absorption cross section, o the total

cross section, j the potential scattering cross section

of the fuel with admixed moderator, and P ^ the self col­

lision probability in the fuel region.

Using a rational expression for P ^

x b . ap ff = y ——- ; x = — and J S. = 1 (3ff r x+u. o lt 1

l i e i

equation (3.1) can be expressed as a sum of homogeneous

resonance integrals.

ef fa = c +a.a

P p i e

and

a = 1/(NX,) . e

Z is Che average chord length in the fuel and N the num­

ber density of the absorbing nuclide.

To obtain an expression of the form (3.2) for an infinite

uniform pin cell lattice we write

V . P f b P b f O MP ff = Pff + l-pKK (3*4)

D D

p ^ is the fuel-to-fuel collision probability for an isola­

ted rod, p ^ is the probability for neutrons entering the

cell isotropically through the cell boundary to suffer their

first collision in the fuel, p^£ is the probability for

neutrons born uniformly and isotropically in the fuel to

reach the cell boundary uncollided and p., is the probabil-bb

ity for neutrons entering the cell isotropically through the

cell boundary to traverse the cell without collision.

The probabilities p,, , p. , and p., are calculatedlb br bb

using formulae given by Bonalumi [17] and the final expres­

sion for can be written in the form of Eq (3.2) with

two terms if p ^ is given by the Wigner approximation [18]

Pff = x+a (3'5)

a is the so-called Bell factor and is calculated by a poly­

nomial fit [19]. The final expression for the resonance in­

tegral is then

R I = R I, (a +a_o ) + (1--&)RI, (a +aa ) h p £ e n p e

where is a parameter depending only on the materials between

the fuel regions and

nT / \ f P a dERI, (o )= j~ --- =—h P J o t E

is the resonance integral of a homogeneous medium.

In its original form Eq (3.6) gives the narrow resonance

approximation of the resonance integral. By modifying cr ,

the intermediate resonance approximation [20] is obtained.

The effective cross section in energy group g is given by

[21]

RI

° 8 --------- f e - « - 7>T “g oeff

P>g

t is the lethargy width of group g.8

The resonance integral is then corrected for the overlap ef­

fect as described in [21]. The influence of this correction

on the Doppler coefficient is very small and it will not be

discussed here.

3.3_________ Micro group calculation

The micro group calculation is made by use of collision pro­

babilities for the three-region (fuel, canning and modera­

tor) cylindricalized pin cell. It provides 69 group spectra

which are used for energy condensation to broad group cross

sections. These are then used in the macro group and funda­

mental mode calculations.

I. V. <j>. = I P. . V. (I I. , <t>. ,+ S. \ (3.8)i,g x ri,g ^ l-s-j.g 3 j.g^-g' j,g' j,g)

S. is the fission source in the fuel region, g is the J »g

energy group index and i and j the spatial region index

referring to fuel, canning and moderator.

Accurate collision probabilities for the cylindricalized pin

cell with flat source in each of the three regions are cal­

culated using the FLURIG method developed by Carlvik [22].

In evaluating the collision probabilities we start by eli­

minating the z coordinate. With notations according to Fig

3.2

TI

P(x) da = i da j exp(- ■^r^)sin0d8

0

is the probability that neutrons isotropically emitted from

a line source will travel the optical length x within the

azimuthal angle da without colliding.

Defining the Bickley functions

Ki»(x) =

we get

-x cosh u----------- du

, n cosh u

P ( t ) = Ki2(T) (3.9)

From (3.9) the collision probabilities in cylindrical geo­

metry are derived

E.V.P. . = •=- | da J J 2 IT

dy[Ki ( t ..)-Ki ( t . . + t . ) 3 i ] 3 i j j

- Ki ( t .,+t.)+Ki ( t . . + t . + t . ) ] 3 i j l 3 ij l j

where t . , t . . and t . are optical lengths within andi ij J

between the regions under consideration. The numerical

evaluation of P is an integration over the system in the

spatial variable y and in the angular variable a.

Let Pj«-j '3e t'le first flight collision probability within

the three-region pin cell, p, . is the probability for a

neutron born in region j to reach the boundary of the pin

cell and the probability for neutrons isotropically

reflected at the boundary to have their next collision in

region i . p is the probability for the isotropically bb

reflected neutrons to pass the pin cell without collision.

The final expression for t

in an infinite lattice is

The final expression for the collision probability

P. . = p. . ,i-J i-J l"Pbb

+ ! i ± V i ( j . U )

3.4_________ Macro group calculation

Using the spectra from the micro group calculation macro­

scopic cross sections are condensed to a maximum of 25 energy

groups to be used in the macro group calculation. The condensa­

tion is made by flux weighting of the cross sections.

The macro group calculation is made in the same annular

geometry as the micro group calculation. The DIT method [23],

i.e. a point formalism is used to solve the integral

transport equation

X is the eigenvalue and x is the fission spectrum. The

transport matrices T give the flux in energy group> g

g at the point k due to a unit source at £ . The formal

volumes are given by

V2 =

are Gauss weights and r^ is the radius of a point

situated in the interval bounded by and ■

3.5_________ Fundamental mode calculation

The aim of the fundamental mode calculation is twofold.

First, the leakage and the migration area are determined

assuming a fundamental buckling mode. Secondly, the infinite

lattice results obtained from the transport calculation are

modified to include the effects of leakage in predicted

group constants.

The fundamental mode calculation is carried out by use of

the B^-leakage method. P^-scattering matrices are available

in the library for the principal moderators and the P^~

scattering terms are explicitly represented in the B^-

equations.

The fundamental mode equation solved by AE-BUXY is in

matrix form

a-E°)<j> + B2d(f> = J T* (3.13)S K

k = the eigenvalue

T , = x , where x is the fission spectrum88 8 * > 8 8

I = diag(I ) (the total cross section)t > 8

£° = £ , (the zeroth moment of the scattering matrix)® 8 8

d = (3a Z-E1)-1s'

. (the first moment of the scattering matrix)s g 8

f I f L . ) 2 . 3 \ l J

ar(B \

arctgj

. <

3 \z J A -1 x % o»8

; B > 0

2 2 ; K = -B > 0

£ I+k /ZA = - l i n ______&

o , 8 2 k 1 - tc /Es

B2 = -K2

All quantities are cell integrated values.

Three fundamental mode calculations are made, namely:

. 2k is obtained with B = 0

00

ke^ is obtained for a given geometrical buckling

2 2 B is varied so that k ,, = 1. B is then

eff 2equal to the material buckling B^ .

Equation (3.13) is solved in the macro group structure.

Fig.3.1 Flow chart of the pin cell calculation in AE-BUXY.

Fig.3.2 Elimination of the z co-ordinate in cylindrical geometry.

4. COMPARISON BETWEEN THEORETICAL RESULTS AND

EXPERIMENTAL INFORMATION

In order to compare the theoretical model for calculation

of the temperature coefficient with experimental informa­

tion, it is desirable to use measurements on simple geomet­

ries, i.e. uniform pin cell lattices. There are not many

papers published about reactor physics experiments on light

water moderated systems at high temperature. We have looked

at some early exponential experiments but found that the ac­

curacy is not good enough for detailed comparisons with theo­

ry, one of the main problems being the nonexistence of an

asymptotic region due to the small size of the subcritical

lattices.

Measurements on light water systems in critical facilities

at temperatures up to 90°C have been performed at several

laboratories, but at temperatures above 200°C we know only

of the experiments made in the KRITZ facility [24] at Studs-

vik and Soviet measurements [25] on highly enriched (m 80 %)

rods in H^O. Power reactor measurements provide valuable in­

formation on the adequacy of the calculational methods, but

the conditions are usually very complex including a great

number of different absorber and fuel rods in different as­

semblies, which requires two- or three-dimensional core cal­

culations. We have therefore chosen to analyse some KRITZ-

experiments in detail. It is very valuable that these mea­

surements have been made at temperatures up to 245°C. Com­

parisons have been made on both uniform pin cell lattices

and cores containing BWR or PWR fuel assemblies. Here we

will, however, refer only to the uniform lattices since thest

are the most appropriate ones for the purpose of revealing

fundamental discrepancies between theory and experiment in

determining temperature effects in reactors.

4.1 D e s c r ip t io n o f th e m easurem ents

The critical facility KRITZ is used for reactor physics mea­

surements on water-moderated cores at temperatures up to

245°C. KRITZ consists of a pressure tank with an insert ves­

sel of which the outer wall is a circular cylinder loosely

fitting into the pressure tank, whereas the inner wall con­

sists of a square cylinder. Vertical and horizontal cross

sections of the reactor tank and insert vessel are shown in

^igs 4.1 and 4.2.

Coarse reactivity control is achieved by poisoning the water

with boric acid and fine reactivity control is made by ad­

justing the water level. No control rods are involved. It

means that uniform lattice arrangements can be loaded with­

out any interference with heterogeneous devices except for

spacers within the core volume.

Critical water levels and flux distributions in the vertical

and horizontal directions are measured and experimental

values of the material buckling are evaluated. The flux

distributions are obtained from activation of copper wires.

Critical measurements have been performed on a large number

of uniform lattices of UC^ and PuC^ rods as well as on more

complex BWR and PWR geometries [26]. The experiments used

in the present study are all on uniform lattices containing

UO 2 enriched to 1.35 % or 1.9 % or mixed oxide (1.5 % PuO^ -

depl UC^)• The moderator to fuel ratio, boron contents and

core size are varied.

The fuel rods with bottom extensions were standing on stain­

less steel beams and their radial positions were determined

by spacers in form of horizontal straight wires placed in

different planes. The wires in each plane were turned per­

pendicular to those in the neighbouring planes. The distance

between the spacer layers was in the cores with the 1.35 %

UC> 2 rods 36.0 cm which means that they affect the reacti­

vity. In the cores with 1.9 % UO 2 or Pu ( > 2 rods the spacers

were placed above the critical water level and below the

active length of the rods.

4. 2_________ Description of the calculational methods

The cell code AE-BUXY was used to calculate the material

buckling and for a geometrical buckling given in input.

When the input geometrical buckling is put equal to the

experimental material buckling, the deviation of from

unity gives a measure of the discrepancy between AE-BUXY

and experiment. This comparison between theory and experi-2

ment is relying on the experimental material buckling, B ^ e x p )

obtained from copper activation. Alternatively, cal­

culations may be carried out on the whole core using material

composition, geometry and critical water level as input. An

accurate calculational model requires a three-dimensional

code. Three-dimensional calculations are, however, expensive

and have been avoided in this work. We have instead simulated

the experimental configuration with a two-dimensional diffusion

theory code in xy-geometry, DIXY [271, using the measured

axial buckling to account for the axial leakage. The largest2

uncertainty in the experimental B^ is due to the deter­

mination of the extrapolation length from the flux mapping.

The measured extrapolation lengths have been compared with

calculated ones. These calculations were carried out in

one dimension by use of the S code DTF-4 [28]. DTF-4 wasn

also used to calculate the reactivity worth of the spacers.

Few-group constants for DIXY and DTF-4 were generated by

AE-BUXY.

Cross sections for the radial water reflector were also

generated by use of AE-BUXY. The whole core was cylindricalized

and a transport calculation was carried out on the cluster

(containing 1500-2000 fuel rods) surrounded by a thick

water region. Few group reflector cross sections are pro­

duced at different distances from the fuel region. It was

found that the 4 and 8 group cross sections do not vary

much with the distance from the fuel region. Average reflec­

tor cross sections were therefore used in the core calcula­

tions .

All pin cell calculations were carried out with 17 energy

groups in the macro group calculation of AE-BUXY and with

3 Gauss points in the fuel, 2 in the canning and 3 in the

cylindricalized water region. The same 17 groups were used

in the fundamental mode calculation. The choice of the above

parameters is discussed in chapter 6.

The DIXY calculations were made in 4 energy groups (10 MeV-

- 0.5 MeV - 9118 eV - 0.625 eV - 0) in the 1.35 % U02

lattices and with 6 groups (10 MeV.- 0.5 MeV - 9118 eV -

- 4 eV - 0.625 eV - 0.140 eV - 0) in the 1.9 % U02 and Pu02

lattices. The number of mesh points was chosen from a few

calculations where the distribution and total number of mesh

points were varied (see chapter 6).

Axial calculations with DTF-4 were made in the S^-approximation

with 4 energy groups and including anisotropic scattering

explicitly.

4• 3_________ The reactivity worth of spaces

The reactivity worth of spacers need be taken into account

only in the cores with the 1.35 % U02 fuel. In these lattices

horizontal straight wires of stainless steel, 0.40 cm in

diameter, placed in different planes were used to determine

the radial positions of the fuel rods. The spacer plane

separation was 36.0 cm.

The reactivity worth of the spacers and its temperature

dependence was calculated by use of DTF-4 and AE-BUXY. An

example of the geometry used in DTF-4 is shown in Fig 4.3.

The axial direction of the core and bottom- and top-reflec-

tors are considered. The radial leakage is taken into accouut

by use of a transverse buckling equal to the measured radial

buckling B2 = B2 + B2 .r x y

Homogenized macroscopic cross sections were calculated with

AE-BUXY. The calculation of cross sections for the pin cell

region (material 4 in Fig 4.3) and the region above the water

level (where the s.pace between the rods consists of H^O-

steam) (material 6) is straight forward. To be able to

handle the spacer region (material 5) the SS-wire has to be

homogenized with the water.

Some results from calculations of the spacer reactivity

worth are given in Table 4.1.

Table 4.1: The influence on reactivity from spacers in 1.35 % U( > 2 lattices calculated by DTF-4

Boron

conc

(ppm)

Temperature

C°c)

Akeff

(pcm)

AB2m

(nf2)

O 90 - 500 - 1.3210 - 560 - 1.3

175 90 - 410 - 1 .0

210 - 480 - 1.1

Experimental buckling effects of spacers were determined

only at 20 °C and 0 ppm boron. The experimental result is2 -2

[24] AB = -1.3+0.2 in . This is in good agreement with m —

the calculated value.

^ 4 ^ 1 ___Lat tices_wi th_l_135_%_enr iched_UQ2

A great number of measurements has been performed with the

fuel rods containing 1.35 % enriched UC^ [24]. Two cores

have been analysed in detail. The first one contains 39x39

pins and is without boron in the water. The seconc core con­

tains 46x46 pins and the boron concentration is 175 ppm.

The moderator to fuel volume ratio, V A ' , is 1.4 and them o

measurements cover the temperature range 20-210 C.

Results of the analysis are collected in Tables 4.2-4.4.

Table 4.2: Calculated k and k for the 1.35 % U0„ lattices---------- co eff 2

Core size Boron cone. (ppm)

Temp.

(°C)

AE-BUXY DIXY

k « effk03 k

eff

20 1.16294 0.99402 0.99381

39x39 090 1.15566 0.99091 0.99144

160 1.14677 0.98773 0.98876

210 1.13872 0.98587 0.98673

20 1.12172 0.99636 0.99569

46x46 17590 1.11621 0.99365 0.99383

160 1.11003 0.99186 0.99198

210 1.10448 0.99100 0.99102

It is seen in Table 4.2 that the reactivity is slightly

underpredicted. The deviation of ke££ from unity increases

with temperature, i.e. the predicted temperature coefficient

is too negative.

Ak _j-/AT in Table 4.3 is defined as the difference between eff

kg££ at two temperatures divided by the temperature differ­

ence. Thus A k ^ ^ / A T gives the discrepancy between the cal­

culated and measured temperature coefficient.

Table 4.3: Ak _C/AT derived from Table 4.2 ---------- eff

Core size Boronconc.(ppm)

Temp.interval

(°C)

Akg^^/AT (pcm/°C)

AE-BUXY DIXY

20- 90 -4.4 -3.4

39x39 0 90-160 -4.6 -3.8

160-210 -3.6 -4.2

20- 90 -3.9 -2.7

46x46 175 90-160 -2.6 -2.7

160-210 -1.8 -2.0

The calculated temperature coefficient is (2-4) pcm/°C

more negative than the experimental one. There are only

small discrepancies between AE-BUXY and DIXY.

k ^ £ was calculated using both DIXY and DTF-4 at 90 °C and

210 °C. In DIXY the experimental axial buckling was used,

and in DTF-4 calculations were carried out using the

experimental radial buckling to give the transverse

leakage. As seen in Table 4.4 the agreement between the

different calculations is very good. This means that mea­

sured and calculated axial and radial bucklings are consistent.

Table 4.4: Comparison between calculated by use of

AE-BUXY, DIXY and DTF-4

Bcronconc.(ppm)

Code keff S k« f f /ST

(pcm/°C)90 °C 210 °C

AE-BUXY 0.9909 0.9859 -4.2

0 DIXY 0.9914 0.9867 -4.0

DTF-4 0.9898 0.9845 ■ -4.4 '

The cores containing 1.9 % enriched IK^ rods and PuO^ rods

are regular square lattices with either only UC^ or PuC^

rods or with a central square zone of PuC> 2 rods surrounded

by a UO 2 zone. Three different volume ratios were covered

by the experiments [26] with temperatures from 20 °C up to

245 °C. Lattice data are given in Table 4.5.

In the AE-BUXY calculations the plutonium was assumed to be

homogeneously distributed in the fuel. In reality the plu­

tonium fuel used in these experiments contains Pu0 2 _particles

whose average size is about 25 pm. This non-homogeneity of

U0 2 ~Pu0 2 needs to be considered because of the self shielding

in the Pu02~partides. As a result of the shielding the

Pu-239 fission and capture reaction rates will be reduced.

Because of the larger proportionate increase in the shielding

of the 0.3 eV resonance the a-value of Pu-239 is reduced.

The change of the fission rate results in a reduction in

reactivity and the change of the a-value results in an

increase in reactivity. Further, the reactivity is increased

because of the shielding of the 1 eV resonance of Pu-240.

The net reactivity change is in all our cases negative.

The self shielding in Pu-239 was estimated assuming that

the incurrent into the Pu0 2 _particles has the same spectrum

as the average flux in the fuel. The corrections for

self shielding in Pu-239 are in our cases -(300-600) pcm

and for self shielding in Pu-240 about +100 pcm. Our correc­

tions were found to be in good agreement with calculations

for room temperature presented by Liikala et al. [29].

Finally the statistical weight of the plutonium zone was

Table 4 .5 . L a t t ic e data and corrections for thermal expansion and p a r t ic le s ize plus ca lcu la ted

1Fuel

m rCores ize

Pu zone

i

Boron

conc

(ppm)

Temp

(°C)

Correctionsef f

Ak ,,/AT e ff

(pcm/°C/

thermalexpan*(pcm)

p a r t ic les ize(pcm)

1.9% U0o 1.2 34x34 0 22.3 0 _ 0.99547 _c. 1! - 0 90.0 + 65 - 0.99466 -1.2

44x44 _ 0 204.9 + 175 - 0.99320 -1.3n — 0 245.2 +214 — 0.99228 -2.3

1.7 27x27 — 0 20.0 0 — 0.99430 -i i — 0 88.8 +12 — 0.99309 -1.8

1.7 36x36 — 300 23.0 0 — 0.9955140x40 - 200 208.3 +30 - 0.99193 -1.9n — 200 247.2 +38 — 0.99147 -1.2

PuC2- 1.8 40x40 24x24 300 18.1 0 -280 0.99662 __

1.9% U0o r t I f 300 90.2 +33 -260 0.99670 +0.1z i i 1! 0 211.3 +83 -140 0.99481 -1.6

i : I I 0 245.5 + 104 -130 0.99568 +2.5

2.5 34x34 24x24 350 21.8 0 -410 0.99791 —i i i i 350 90.5 + 17 -380 0.99810 +0.3i » u 250 247.0 +54 -300 0.99563 -1.6

Pu0o 3.3 26x26 26x26 50 23.0 0 -500 0.99691 iiiJL m i i 50 93.4 -16 -460 0.99726 +0.5 !M i t 5.0 208.4 -43 -400 0.99541 -1.6I I i f 50 239.2 -51 -380 0.99571

<+ 1.0

* cf § 6.6

used to find the corrections in cores containing both PuC^

and UO 2 pins. The corrections are shown in Table 4.5. The

temperature coefficient is increased by about 0.5 pcm/°C

due to the particle size reactivity effect.

Table 4.6 shows the results from AE-BUXY.

Table 4.6: Results from AE-BUXY for the 1.9 % UO 2 and

the PUO 2 lattices

Fuel V vfm 1Boron konc. (ppm)

Temp.

(°C)

k00

kef £

Ak /AT eff

(pcm/°C)

1.9 % U02 1.2 0 22.390.0

204.9245.2

1.235141.224901.198841.18559

0.996570.993090.992050.99123

-5.2-0.9-2.1

1.7 0 20.088.8

1.263281.25663

0.999380.99575

-5.3

1.7 300200200

23.0216.2247.2

1.191201.196931.19125

0.997460.990350.99000

-3.7-1.1

Pu0„ 1.8 300 18.1 1.22217 _

z300 90.2 1.22039 -

0 211.3 1.25546 -

0 245.5 1.24654 -

2.5 350 21.8 1.20305 -

350 90.5 1.20867 -250 247.0 1.23400 -

3.3 50505050

23.093.4

208.4239.2

1.258221.265451.279121.28120

0.997950.995940.992540.99224

-2.9-3.0-1.0

Comparing the results in Tables 4.5 and 4.6 one finds that

the calculated k^^-values at room temperature given by

AE-BUXY are higher than those given by DIXY and at high

temperature the AE-BUXY values are lower. Thus, the experi- 2

mental B corresponds to a less negative temperature m

coefficient than what is obtained from the diffusion theory

calculations on the whole core.

At room temperature DIXY gives « 0.995 for the clean

UO^ lattices and k ^ ^ « 0.997 for lattices containing PuO^.

Corresponding reactivities at high temperature are 0.992

and 0.996, respectively. The spread in k ^ ^ is small and the

trend in k ^ ^ versus temperature is smaller than for the

1.35 % UO^ lattices.

There are several sources to experimental uncertainties.

The uncertainty in the measured boron concentration is 1 %

when there is more than 100 ppm boron in the water. 1 ppm

boron corresponds to about 20 pcm in reactivity. The error 2 .

in the measured B is expected to correspond to less than z

100 pcm in the absolute value of k ^ ^ . Further, there is

the uncertainty in material composition and geometry. The

experimental uncertainty in the difference between two cal­

culated k^^-values for the same core (but at different

temperatures) is expected to be about 50 pcm.

The last column of Tables 4.5 and 4.6 gives the discrepancy

between the measured and calculated temperature coefficient.

When looking at A k ^ ^ / A T it should be remembered that the

uncertainty in A k ^ ^ / A T due to experimental errors may be

1 pcm/°C or more. The uncertainty becomes less if one cal­

culates Akgj^/AT for the whole temperature range (Table 4.7).

Table 4.7: A k ^ ^ / A T for the temperature intervall 20-245 °C

Fuel V Vfm tAkeff/AT (pcm/°C)

AE-BUXY with exp B2 m

DIXY

1.9 % U0o 1.2 -2.4 -1.41.7 -3.3 -1.8

Pu02-U02 1.8 - -0.42.5 -1.0

Pu02 3.3 -2.6 -0.6

The present calculations have been performed neglecting

the difference between the uranium and plutonium fission

neutron spectra. The latter one has a higher average

energy. This gives an increased leakage (which causes a

negative correction to anc increased fast fission

(positive correction). The importance of the plutonium

fission neutron spectrum is discussed in § 6.7. The influ­

ence on the temperature coefficient is negligible in all

lattices studied in the present report.

4^4^3___Comments to_the_resuits

The spread in predicted values is small. The average

value is 0.995 for cold UO^ lattices and 0.997 for PuO^

lattices. versus temperature is shown in Fig 4.4.

AE-BUXY predicts a temperature coefficient which is

(1-4) pcm/°C too negative. This discrepancy is somewhat

smaller than what has been observed in power reactor applica­

tions t1]. The magnitude of the temperature coefficient is

(10-20) pcm/°C at room temperature and about 40 pcm/°C at

high temperature. The discrepancy between measured and2

calculated values is slightly larger when measured are

used to calculate k than what is obtained from two- eff

dimensional diffusion theory. Concentrating on the DIXY

results it is seen that Ak ri./AT varies little with temper-etr

ature.

PROBE OF •WATER LEVEL METER

ADJUSTABI WATER LEVEL

TO COVER GAS SUPPLY

NON-RETURNVALVE

IN-LET FOR WATER CIRCULATION

U02 REGION

SQUARE-FORMED SPACE FOR EXPERIMENTS

DUMP SPACE (4 SEGMENTS)

POSITION OF NEUTRON SOURCE (SB-BE)

SPRING-LOADED SAFETY SHUTTER (7 TOTALLY)

DRAINAGE OF DUMP SF*CE

Fig 4.1 Vertical cross section of reactor tank with insert vessel

Pressure tank

Dump space (four communi­cating segments)

Safety shutter (seven totally)

Fig 4.2 Horizontal cross section of the insert vessel.

The lines of dots and dashes give the boundaries of the

two core sizes investigated for the 1.35 % U02 rods.

(The dash circles indicate the size and position of

sealed openings in the tank lid. The shaded circular

areas show the position of the neutron detectors.)

Fig.L 3. Axial representation of

Axial representation in DTF -4

Mat. 7 (SS ♦ Steam)

Mat. 6 (U02 + Zr ♦ Steam)

Mat. 4

Mat. 5

Mat. U-Mat. 5

Mat. U

Mat. 5 ( H,0 + U0, + Zr+:

Mat. 4 (H20 ♦ U02 + Zr)

Mat. 3 (H20 + SS+Zr)

Mat. 2 (H20 +SS)

Mat. 1 (H20 )

and reflectors in DTF-4.

keff

Fig. 4. A keff versus temperature calcu lated by DIXY.

5. The components of the temperature coefficient

Many phenomena in a reactor system contribute to the temper­

ature coefficient of reactivity. Lack of agreement between

theoretical and experimental values of the temperature coeffi­

cient, dk/dT, may therefore be due to errors in the calcula­

tion of one or several of the different components. Further­

more, agreement obtained in a limited study does not neces­

sarily mean that the calculations are correct in all respects.

Compensating errors may appear and care must be exercized in

drawing conclusions.

Neglecting the effect of thermal expansion, except the water

density effect, dk/dT may be written

T^, T£ and T^ = fuel, canning and moderator temperature,

respectively.

0 = moderator density.

Theoretical values of the terms on the RHS of Eq (5.1) may

be obtained by using a suitable advanced code such as

AE-BUXY. A corresponding experimental determination is

difficult to achieve. Attempts have been made to measure

the last term of (5.1) using flashing experiments in KRITZ.

The accuracy of these measurements, especially the determina­

tion of the void content and void distribution, is, however,

not high enough to allow a detailed comparison with the

theoretical results.

Other experimental methods to separate the total temper­

ature coefficient into temperature and density effects have

been used elsewhere, e.g. simulating void by use of aluminium

dk(T ,T ,T ,0)3k3T

3k d0 36 dTdT

(5.1)

tubes in the moderator. The introduction of aluminium

causes, however, other problems, e.g. the not negligible

fast scattering in A1 and streaming effects in the tubes.

Another way to obtain experimental information of the

separate terms in (5.1) is to perform critical measure­

ments within a wide temperature range. The factor d0/dT

in the last term is much more sensitive to the temperature

(cf Fig 2.2) than 8k/3T and 3k/30 , so the last term in

(5.1) gives a significantly higher contribution to dk/dT

at high temperature than at low temperature. The KRITZ

measurements of dk/dT which cover a wide range of temper­

atures therefore provide important information, which may

help to reveal whether the discrepancy between theory and

experiment is due to pure temperature effects, water

density effects or both.

Theoretical values of the terms in Eq. (5.1) are given

in § 5.1.

In order to gain some insight into the physical significance

of the calculated coefficients, it is helpful to express them

in terms of the derivatives

dn df dp dL_ & & — i. and — £.dT dT dT dT

n is defined as \>Z,/£ in the fuel, f is the ratio g f a g

between fuel absorptions and cell absorptions, p is the2 . .®

resonance escape probability and L the diffusion length.

Contributions to dk/dT from each of the derivatives (di­

vided into four energy groups, g, have been calculated using

a special code, COEFF, with four-group cross sections from

AE-BUXY as input. Results from these calculations are dis­

cussed in § 5.2.

Looking at Eq. (5.1) we note that the term 9k/8Tc in a light

water moderated system with zircaloy canning can be neglected

compared to the other terms. In the 1.35 % UO 2 lattice for

example, AE-BUXY gave the very small contribution

and this term will be neglected in the following discussion.

Tables 5.1-5.3 show the terms of Eq. (5.1) as predicted by

AE-BUXY. Results are given for the temperature coefficient

of the infinite multiplication constant, Ak^/AT , the

reactivity for a critical lattice, Ap/AT , the material2 2

buckling, ABm /AT , and of the migration area, AM /AT .

The temperature intervals used in the calculations are 20 °C

to 90 °C and 160 °C to 210 °C. In this temperature range

experimental values of the total temperature coefficients

are available.

The fuel temperature coefficient is in all lattices less

negative in the temperature interval 160-210 °C than in the

interval 20-90 °C. This should be expected due to the- 1/2 . .

approximate T dependence of the Doppler coefficient.

8p/3T^ is built up of two components. The dominating part

is caused by the Doppler broadening of resonances and a

minor contribution is given by the thermal scattering in the

oxygen. In the 1.35 % UO^ fuel these two components were

determined separately and we obtained

3k^

Y f ~ = “0*02 pcm/°Cc

(I£_\ \ 3 T j

f'thermal= -0.3 pcm/°C

for the temperature interval 20-90 °C and

Boroncontent(ppm)

9X/£Tf 3X/3Tm

3X/36 *d0/dT dX/dT

Temp in te rv a l (°C) 20-90 160-210 20-90 160-210 20-90 160-210 20-90 160-210

X=k00 0 -4.8 -4.3 -3.8 -3.6 -1.8 -8.2 -10.4 -16.1

(pcm/°C) 175 -4.7 -4.1 -3.3 -3.3 +0.1 -3.7 -7.9 -11.1

X=p 0 -4.1 -3.6 -5.1 -4.7 -10.6 -29.0 -19.8 -37.3

(pcm/°C) 175 -4.1 -3.7 -4.3 -4.0 -6.9 -20.2 -15.3 -27.9

X=B2 0 -.0114 -.0088 -.0137 -.0108 -.0261 -.0598 -.0512 -.0794

( n fV c . ) 175 -.0113 -.0086 -.0113 -.0092 -.01.64 -.0406 -.0390 -.0584

X=M2 0 -.0004 -.0004 .0049 .0048 .0253 .0732 .0298 .0776

(cm2/°C) 175 -.0004 -.0004 .0054 .0046 .0251 .0730 .0301 .0772

Vm/Vfm tBoron

conc(ppm)

3X/3Tf 3X/3Tm

3X/30‘ d0/dT dX/dT

Temp in te rv a l (° C) 20-90 160-210 20-90 160-210 20-90 160-210 20-90 160-210

X=k00 1 2 200 -5.8 -5.1 -2.9 -2.8 -4.0 -13.8 -12.6 -21.7

(pcm/°C) 1 7 200 -4.6 -4.1 -2.6 -2.9 +0.7 -2.4 -6.5 -9.4

X=p 1 2 200 -4.8 -4.2 -3.7 -3.5 -13.2 -34.3 -21.7 -42.0

(pcm/°C) 1 7 200 -3.8 -3.4 -4.1 -4.1 -10.9 -30.3 -18.9 -37.8

X=B2m

1 2 200 -.0124 -.0096 -.0095 -.0080 -.0322 -.0724 -.0542 -.0900

(m-2/°C) 1 7 200 -.0110 -.0086 -.0117 -.0100 -.0291 -.0690 -.0518 -.0876

X=M2 1 2 200 -.0007 -,0008 +.0032 +.0034 +.0255 +.0728 +.0281 +.0754

(cm2/°C) 1 7 200 -.0004 -.0006 +.0042 +.0042 +.0261 +.0768 +.0300 +.0804

V Vfi m rijti

Boron

conc(ppm)

3X/8Tf 3X/3Ttn

dX/ZO‘ dQ/dT dX/dT

Temp in te rv a l (°C) 20-90 160-210 20-90 160-210 20-90

i

160-210 20-90 160-210

X=k00

(pcm/°C)

it

1.8

2.5

3.3

300

250

0

-5.1

-4.1

-3.5

-4.6

-3.7

-3.1

+4.0

+7.6

+7.7

+ 5.0

+8-7

+9.0

-1.3

+ 2.7

+3.7

-9.5

+0.1

+2.5

-2.4

+ 6.2

+8.0

“9.1

+5.1

+8.4

X=p

(pcm/°C)

1.8

2.5

3.3

300

250

0

-4.1

■3.3

-2.7

“3.7

-3.0

-2.5

+2.5

+5.1

+4.1

+3.8

+6.7

+6.2

-13.7

-11.4

-14.4

-39.5

“36.5

-44.3

-15.3

-9.6

-13.0

-39.4

-32.8

-40.5

X=B2m

(m 2/°C)

1.8

2.5

3.3

300

250

0

-.0124

-.0107

-.0092

-.0098

.-0082

-.0072

+.0075 |+.0100

+.0164 1+.0186

+.0140 !+.01821

-.0394

-.0351

-.0467

-.0972

-.0952

-.1210

i

-.0443 J -.0970

-.0294 ! -.0848i

-.0420 | -.1100i

X=M2

(cm2/°C)

1.8

2.5

3.3

300

250

0

-.0004

-.0002

-.0002

-,0006

-.0002

-.0002

+.0018

+.0025

+.0034

+.0012

+.0016

+.0020

r i+.0258 I +.0756

i

+.0262 | +.0784 1

+.0263 j +.0812— . ___ i__ _____

+.0272

+.0285

+.0300

+.0762

+.0798

+.0830

3 : 1 , = - 3 -3 ^ ° cv f7 Doppler

= -0.3 pcm/°Cj. thermal

for the interval 160-210 °C.

The moderator temperature coefficient is negative in the

UO 2 lattices and positive in the PUO 2 lattices. The different

signs of 9p/3Tm for the two types of fuel are due to the

0.3 eV resonance in Pu-239 and the non-l/v-dependence of

the thermal U-235 cross section. Thus, an analysis of both

UO 2 and PUO 2 systems constitutes a severe test of the

ability to predict the thermal neutron spectrum. An error

in the calculated temperature dependence of the spectrum will

give different errors in the predicted temperature coefficient

for UO 2 and PUO 2 lattices. Our analysis of the KRITZ experi­

ments has, however, given approximately the same discrepancy

between calculated and measured temperature coefficients in

both UO 2 and PUO 2 systems. This strongly indicates that the

change of the thermal neutron spectrum with temperature has

been correctly calculated.

The large temperature dependence of the water density

coefficient is a consequence of the variation of d0/dT with

temperature. The water density, 0, and d0/dT versus

temperature are shown in Fig. 2.2. The average value of

d0/dT is -0.470*10"3 g/cm3*°C and -1.096-10_3 g/cm3*°C

in the temperature intervals 20-90 °C and 160-210 °C,

respectively. 3Bm /30*d0/dT is roughly proportional to

d0/dT , whereas the changes of 3p/30»d3/dT and

3M2/30'd0/dT are larger than they would be if they

were proportional to d0/dT . This is due to spectrum

effects. 3^/ 3 0 - d0/dT has a more complex variation

with temperature because Bk^/ST is built up from several

positive and negative contributions (cf § 5.2).

GO

The discrepancy between the theoretical and the experimental

reactivity can be described by a function f(T,0) defined

by

k.xp(T'9> ’ ‘W r y (T-e) + £« ’6> <5-«

We found in chapter 4 that f(T,0) , within reasonable

error limits, is independent of temperature, i.e.

3f , 3 f d0 _ ,3T 36 dT " const* (5.3)

There are no physical reasons why the two terms of the LHS

should be correlated so we can expect both terms to be con­

stant. This means that

3 f / d e \ 1— = c o n s t ^— j (5.4)

There is also no reason to believe that 3f/30 is inversely

proportional to d0/dT so (5.4) leads to

(If 3f/33 f 0 we would have 3f/30-*<» at a small extrapola­

tion in the temperature to T = 4 °C where d0/dT = 0.)

Thus we find that if (5.3) is true, which is approximately

the case in the investigated lattices, then the discrepancy

between calculated and measured temperature coefficients

must be expected to be a pure temperature effect and not a

secondary effect due to the temperature dependence of the

water density.

5.2______Contributions to the temperature coefficient _________ calculated by COEFF_____________________________

5.2.1_ _Descrigtion_of COEFF

The program COEFF was written as a tool for studying the

contributions from various parameters (n, f, P and L^) to

the temperature coefficient. Input data consist of few-group

parameters produced by AE-BUXY. These data refer to the

criticality spectrum obtained from.the fundamental mode cal-2 2

culation with B = Bm

The following nomenclature will be used to describe COEFF.

G number of energy groups

g group index

Xgfission neutron spectrum

Zag

cell averaged absorption cross section

vEfg

cell averaged v-fission cross section

g *-gcell averaged scattering cross section (from g to g')

Dg

cell averaged diffusion constant

‘"gcell integrated neutron flux

B2m

material buckling

B2 geometric buckling

<j> and g

y are normalized such that g

E xg = g s

1

?(I“s+ D B2)<|> = 1 g g

Define

R = f" Y (E .. ./d> , — E , „ d> „)8 g = l g H_g+ i g ' ^ ' V g g

R = 0o

r g =o

vE

nf =fg

ag

(5.7)

P =

8 Rg“l + Xg

(5.8)

PG = 0

The neutron balance for group g can then be written

(E + D B ) <J) = x + R i “ Rag g m *g Ag g-1 g

(5.9)

Noting that R = P (R + x ) we get g g g g

(E + D B ) <j> = (1 - P„) (x + R ,) =ag g m' *g g ' VAg g-1'

D: = (i+ b 2) L(i - p ) ) x .38 8 ag m 8 g^= l 8 g "=g ' 8

g - 1fl P„u (5.10)

where

g-1n P „ = 1 for g' = g

g ^ g ' 8

Write

k = ------ S------ T------ = I v Z ( <f> = 7’nf z <J> (5.11)

+ O 8 8 8 8 8 a§ 8g ag g "> g

k = E d - B2 ) " 1 n f (1 - P ) f X , V P „ (5.12)g ag 8 8 g'=l 8 g"=g’ 8

This expression may be written

g 1 + L 2g B 2m g ’-lV 8 g"=g

8 - 1n p „„ (5.13)

with

D(1 - P )

L2 = ^ — --------- (5.14)

8 e 21 + -=2- P B

£ g mag

P 1 = (1 + L2 B2) P (5.15)g g m g

Pg is the slowing down probability in an infinite lattice.

In the special case with P 1 = ---- — (£ = V £ , )g £ + £ rg V jl g •‘-gag rg g Fg

we get

, 2 ______

8 Eag + Erg

2 2i.e. the conventional definition of L . Replace B with the

2 m geometrical buckling, B , in (5.13). We obtain

k = T 5k (5.16)

8 S

nf (1 - P ) g g-1 P enSk = — §— y x . n ---- &=— =• (5.17)

8 1 + L2 B2 g ^ l 8 g'-g' 1 + L „ Bo o

(5.16) now gives the effective multiplication constant, k,2 . 2

at a given geometrical buckling, B , (k^k^ if B =0) and (5,

gives the contributions to k from each energy group.

To see how a change in n . f , p1 and L effects k we diffe-g g g g

rentiate (5.16)

Ak = T An f X+5"Af n X +y'AP1 Y + ^ A L 2 Z (5.18)“ g g g “ g g g ^ g g g g g

with the coefficients X , Y and Z given by Table 5.4.g g g

Table 5.4: Coefficients in equation (5.18)

Xg

1 i r- 8 1 Pg"2 2(1- y I v „n , 2 2

1 + LT B 8 g^=l 8 g"=g' 1 + l/„ B© . O

Yg

1 ^ 8~^ PE»2 2 11 2 ? +

1 + i/ B 8 g'=l 8 g"=g' 1 + l/„ B •8 8

! G g g"'-l Pg"

+ i L 2 2 I ^ c 1 n 2 2g "’=g+l 1+L‘„BZ 8 8 gV =l 8 g"=g' l+l/„B

8 8 8

Zg

b 2 i 8 8-1 p i "2 2 2 nfg (1_Pg) I V " *

(l+LgB ) 8 8 gV=l 8 g"=g' 1+L2 Bo o

B2 ° 1 ~i 8 2 2 L 2 2 Po"') C.

1+l V g"'-g+l l+L2,, ,B 8 8 gV =l 8o o

g’"-l pi,n —

g"=g' l+lr,,Bo

G]T which appears in the second term of Y and Z is put

g - 'o g + i § 8

equal to zero for g = G.

5^2^2___Results_from_calculations_with_COEFF

The coefficients An /AT , Af /AT , Ap /AT and AL2/AT g g g g

and their influence on the partial and total temperature

coefficients have been calculated using four energy groups

with the boundaries 10 MeV - 0.5 MeV - 9118 aV - 0.625 eV -

The results have been collected in Figs 5.1-5.12. There is

no experimental information available which corresponds to

these theoretical results. The value of this kind of calcula­

tions is that they give a detailed picture of the temperature

dependence of the neutron balance and its influence on the

reactivity coefficient.

The fuel temperature coefficient is mainly caused by changes

in n and p in the resonance group (Figs 5.1, 5.5 and

5.9). A small contribution to 3p/3T^ comes from 3n/3T^

and 3f/3Tf in the thermal group due to the thermal scatter­

ing in the oxygen. The thermal contribution to 3p/3T^ is

0.2-0.3 pcm/°C. The increased resonance absorption gives

rise to a very small positive contribution due to decreased

leakage. This effect is seen in 31,2/3T^. The numerical

value is about 0.1 pcm/°C. It is also seen in the diagrams

that the Doppler effect is less in the high temperature

interval than in the low one.

Figs 5.2, 5.6 and 5.10 show the contributions to the modera­

tor temperature coefficient. In the IK^ lattices there is a

large negative effect caused by 3n^/3Tm and a smaller

positive effect caused by 3f./3T > so that the total thermal4 m

temperature because the water density is lower. In the Pu02

lattices there is also a negative contribution to 3p/3Tm

due to 3ri,/3T , but the positive influence of 3f,/3T is4 m '* m

larger than the negative contribution from Sn^/ST , so

that the coefficient, Sp/ST^, is positive. The thermal leakage

increases with higher temperature due to the harder spectrum.

This leakage effect is nearly proportional to B and ism

(1-2)pcm/°C.

The water density influence on the temperature coefficient

is shown in Figs 5.3, 5.7 and 5.11. This effect is, as we

already have seen in § 5.1,larger at high temperature than

at low one due to the temperature dependence of d0/dT .

The density change contributes to the reactivity coeffi­

cient in all energy groups, the largest contributions being

from 9n^/36'd0/dT (positive due to less absorption in 1^0

and boron), 3p^/30-d0/dT (positive due to increased fast

fission), 3p../30• d0/dT and 3p„/30'd0/dT (negative due to

decreased resonance escape probability) and from 3L /30-d0/dT

(in all energy groups negative due to the larger leakage at

low water density). The negative contribution to 3p/30-d0/dT

from the change of is much smaller than the corresponding

contribution to 3p/3T .m

Figs 5.4, 5.8 and 5.12 show the contributions to the total

temperature coefficient, i.e. the sum of the contributions

in the other diagrams.

We have seen that the temperature coefficient is built up

from a number of negative and positive contributions, each

of which has to be correctly predicted. Small errors in

several of the calculated contributions can give a signifi­

cant error in the total calculated coefficient or compensate

each other, so that the total coefficient is in better agree­

ment with experiments than should be expected from the

accuracy of the theory and the nuclear data. Since we do

not have detailed experimental information about the sepa­

rate contributions, we must rely on careful theoretical

analyses of the approximations made in the calculations

and on estimated uncertainties in the nuclear data in order

to determine the accuracy in the theoretical methods.

-5

AT!

g=1 2 3 i*

.0.0 0 0

S 2 S S' M I No i o I

° i o oto ' <JD CQ i=- OQc CD £ CD

Q- e Q-eQ- a a a - a ioa

Af

2 3 A

Ap

2 3

AL

2 3

to

Fig. 5.1 Contributions to the fuel temp, coeff. 1.35% UO2 lattices

Total

AT)

g=1 2 3 U Hi

Ap

1 2 3

A ^

1 2 3 /

-5

1 2 3 A

Af

o <_>oo • • • •o O O o

o oo oCN ID CN ID

0 3 m m m£ E Q. Q.Q . Q . CL CLa. cun in o

Total

5jo d0_ 69 dT

pcm/*C

15

10

-5

-10

-15

o u u u « • • • o o o o cn cn *-

CN cn , i i io o o o CN to CM UD

' c o mCD CD £ £

c r Q-Q- E £ CL CLCL Q . _ _

=0=0 g=1 2 3 I*

All

2 3 U

Af

Ap

2 3

AL

1 2 3 A

Total

rat

Fig. 5.3 Contributions to the water density coeff.

1.35% U02 lattices

20

15

10

=OUDL

Ap

2 3

AL

1 2 3 A

Total

-10 -

-15

o ooo o O or n i— CD

(Nl CM

o o o o(N ID CN ID

CD CQCQ CD

l l l l in inO O

2 3 U

Af

9=1

At|

2 3

Af

2 3 AUM I

Ap

1 2 3

AL2 [Total

1 2 3 A

-5

o ooo

CN (NI i I I

O O O Q CM tO ( N U 5

CN (n W i" ; '

II II H >1

Fig.5.5 Contributions to the fuel temp, coeff.

1.9% U02 lattices.

At]

g=1 2 3 4 JlJ

Ap

1 2 3

AL

1 2 3 A

-5

y y y [

2 3

Af

oCN

Ocn

O o • •o o, cn r;

o o'CN CD

CN CN

"h h :> :>

’m ’n

> >

Total

- b i -

5p d8

50' dT

pcm/ G

Total

Fig. 5.7 Contributions to the water density coeff.

1.9% UO2 lattices

- 64 -

+10

At)

g=1 2 3 4=D=cr

-5

-10

-15

-20

-25

W \ 1 2 3

Af

O o O O

CN CMCT)I I ' *O O O O CN tO CN lO

II II II II

> > > > w w

>£>E>e>E

Ap

1 2 3

m

AL2

1 2 3

Fig.5.8 Contributions to the total temp, coeff.

1.9% UO2 lattices.

|Total

I*

Fig. 5.9 Contributions to the fuel temp, coeff. 1.5% Pu02 lattices

To

tal

+10

g=i

Ail

2 3 4

Ap | ALZ

1 2 3 M 2 3

-5

3 4 TF Total

Af o 0 0 . 0O O O o

I I * 1

c3 § R §

oo oo co oo ^ ^ ro n ii n n )■ > > > >

Fig. 5.10 Contributions to the moderator temp, coeff.

1.5% U02 lattices

Total

Fig. 5.12 Contributions to the total temp- coeff. 1.5% Pu02 lattices.

6. THE INFLUENCE OF APPROXIMATIONS IN THE THEORETICALTREATMENT ON CALCULATED TEMPERATURE COEFFICIENTS

Practical reactor physics calculations have to be performed

in a limited number of energy groups and spatial regions.

We will, show that in our calculations the energy groups and

the spatial representation were chosen so that the calculated

temperature coefficient is insensitive to our specific

choice. Other effects that for practical purposes are often

neglected in the calculations will also be discussed here.

The influence of uncertainties in nuclear data will be dealt

with in the next chapter.

6. L___________The crystalline binding in UP,,

Due to crystalline binding the velocity distribution of

the fuel atoms is not Maxwellian as it was assumed when

calculating the Doppler broadened line shape of the reso­

nance cross section in Eqs (2.21) and (2.22). The uranium

atoms in U02 vibrate with an average kinetic energy larger

than that in a free gas state. L a m b [30] has shown that for

weak binding (or the Short Compound Nucleus Lifetime

approximation, SCNL) the atoms behave like a free gas with

an effective temperature which is higher than the temper­

ature of the medium.

The SCNL approximation is applicable when

r + A >> 2 k e D ( 6 - D

where

9^ is the Debye temperature

/ AkTEA = / — - is the Doppler width

A

r is the total width of the resonance at half maximum.

The effective temperature, Te££» is i-n the SCNL approxima­

tion given in terms of the average kinetic energy per mode

of oscillation as [31].

where v is the phonon frequency and g(v) the phonon

frequency distribution.

It is convenient to describe the crystalline effects in

terms of the Debye temperature of the medium. This descrip­

tion is approximate, because in general the crystal does

not have a Debye phonon distribution. However, an effective

Debye temperature, which varies with the temperature of the

medium, can be defined so that it produces the correct

crystalline effects in the medium. Dolling [32] has deter­

mined the effective 0Q versus T for U02 . For temperatures

above 300 K, 0p is fairly constant (e^ ^ 620 K ) .

Shenter [33] has used the phonon distribution measured by

Dolling [32] to calculate T ... He has also given resultseff

for a Debye distribution

CO

O

g(v) = 3 ^ 3

v_

2for \> < v

D

g ( v ) = 0 for v > vD

DkO

D

with Q_ = 620 K. T obtained with the Dolling distribu- D eff

tion and the Debye distribution agree very well for temper­

atures above about 300 K.

The SCNL approximation does not hold for the low energy

resonances of U-238. F is 0.025-0.1 eV for resonances in

U-238 below 200eV. The Doppler width, A, is for the 6.68

eV resonance 0.05 eV at 300 K and 0.07 eV at 600 K and for

the 190 eV resonance the values of A are 0.3 eV and 0.4 eV

at 300 K and 600 K, respectively. 2k©D is equal to 0.1 eV

in U O 2 so relation (6.1) is fulfilled for the 190 eV re­

sonance and resonances at higher energies but not for the

6.68 eV resonance.

The chemical binding effects on resonances have been studied

by Adkins [31]. He used several models for the phonon fre­

quency distribution, among others the one measured by

Dolling, and reported results for the 6.68 eV and 190 eV

resonances. In Table 6.1 the resonance integral change be­

tween 300 K and 500 K is shown as obtained by Adkins using

a detailed crystalline model, CRYS, the SCNL approximation

with the Dolling frequency spectrum and the free gas model.

Regarding the CRYS model as giving the correct result his

results show that the SCNL approximation reduces the error

in ARI/AT by a factor 3 compared with the free gas model.

The small overestimation of the Doppler effect in low energy

resonances that remains can in most applications be accepted,

because the dominating contribution to the Doppler coeffi­

cient in normal LWR lattices comes from the energy range

above 100 eV for which energies A > 2k0^.

Table 6 . 1 : Calculated U-238 resonance integral change

between 300 K and 500 K [31] using various

crystal models

Model 6.68 eV, 0 =17.52 bp

190 eV, 0 =40 b P

ARI/RI

(%)

Error

(%)

ARI/RI

(%)

Error

(%)

CRYS .2534 0 .7791 0

SCNL Dolling .2739 8 .8152 5

Free gas .3092 22 .8981 15

Thus, for calculating the Doppler effect in U-238 it may be

recommended to use an effective Doppler temperature above

the physical temperature of the fuel. This is a very simple

way to account for the crystal binding effects without

having to consider complicated calculations of the reso­

nance cross section line shape. Well established codes

using the ip and x formalism can be used without any

modification. In cell programs which use tabulations of the

effective resonance integral as function of the temperature

Te^ should then be used instead of T when interpolating

in the tables. This procedure has been built into the

A.E-BUXY code with Tg££ given by Eq (6.2) with the Debye

frequency distribution and

1

T

eD = 620 K

eff ! » »(6.3)

AT = T . -T as function of T is plotted in Fig 6.1. eff

The influence of the effective temperature on calculated

fuel temperature coefficients in three of the lattices

which were analysed in chapter 4 is shown in Table 6.2.

Table 6.2: Comparison between fuel temperature coefficients

calculated using T ,, = T and T defined efr eff

by Eq (6.3)

Case Ok /3T,to (pcm/°C)

Temp.interval(°C) 20-90 160-210

1.35 % U02

Te£f-TT ef£-E q (6.3)

-4.7 -4.1V /V =1.4 in f175 ppm B

-3.8 -3.5

1.9 % U02T =T -4.6 -4.1

V /V =1.7 m f

200 ppm BTeff=Eq(6.3) -3.8 -3.5

Pu02

V /V =2.5 m f

1eff=T

Teff= E q '6,3)

-4.1

-3.5

-3.7

-3.2250 ppm B

The cylindrrealization of pin cells

The lattice cell is in AE-BUXY replaced by a circular

Wigner-Seitz cell with white boundary conditions. In order

to investigate the validity of this approximation we

carried out calculations on the actual square cell in two

lattices and compared calculated values of and its

temperature dependence with results obtained in the corre­

sponding circular cell.

Similar investigations have been done previously by Honeck

[34, 35], Fukai [36, 37], Sauer [38], Carlvik [39], Dudley

[40], Newmarch [41] and Weiss [42] among others. These

investigations are, however, limited to the study of the

flux distribution in one-group calculations with a flat

source in the moderator. We found it therefore worthwhile

to carry out multi-group calculations where the spatial

variation of the source in each group is taken into account.

The two square lattices contain 1.9 % IK^ rods and the

moderator to fuel ratios are 1 arid 2. To simplify the cal­

culations the fuel has no canning. Calculations were done

for 20 °C and 245 °C.

The multi-group transport codes BOCOP [43] and FLUCAL [44]

were used for the calculations. BOCOP is a two-dimensional

collision probability code in xy-geometry and the circular

fuel region was approximated by a polygon. The polygon was

chosen so that the volume of the rod is preserved. The outer

square boundary of the cell is correctly represented in

BOCOP with reflecting boundary conditions. FLUCAL solves

the transport problem in annular geometry by use of the

DIT-method. The cell boundary was cylindricalized preserving

the cell volume and white boundary conditions were applied.

All calculations were made with 8 energy groups with the

group boundaries 10 MeV - 9118 eV - 4 eV - 0.625 eV -

-0.14 eV - 0.058 eV - 0.030 eV - 0. Group cross sections

were generated by AE-BUXY.

Table 6.3 shows the comparison of and temperature

coefficients, Ak /AT, calculated in the square and theOO

circular cell.

Table 6.3: Comparison of k and Ak /AT calculated■ " ■ ■ — ■ ■■■■ - ■ ■ * OO oo

in square and circular cell

V /V, m f

Temp.

(°C).

kOO Ak^/AT (pcm/°C)

Square Circ. Square Circ.

1.0

2.0

20

245

20

245

1.23366

1.17817

1.27605

1.25800

1.23427

1.17851

1.27654

1.25834

-24.66

- 8.02

-24.78

- 8.09

The difference between the results obtained in the two

geometries is small, k^ is overestimated by about 50 pcm

when the square cell is cylindricalized. Ak^/AT is about

0.1 pcm/°C more negative in the circular geometry. This

small discrepancy may, however, be due to numerical uncer­

tainties in the calculations which are about 0.1 pcm/°C.

We conclude that the error introduced by the cylindricaliza-

tion of the pin cells is negligible. This conclusion is in

agreement with the results of [34-42], where the authors

in most cases found that the flux distribution obtained

from a one-group calculation in a Wigner-Seitz cell with

white boundary conditions is a good approximation for the

solution in the exact geometry.

Comparison between pin ceil calculations using isotropic and anisotropic scattering___________

In cell calculations transport corrected cross sections

are usually used to account for anisotropy in scattering.

The transport cross section used in AE-BUXY is derived by

expanding the transport equation in Legendre polynomials.

P^-row sum and weighted P^-column sum corrections [21] are

used in the thermal and epithermal regions, respectively.

In order to check the adequacy of the use of a transport

corrected total cross section, pin cell calculations were

performed using S -theory. These calculations were madeO

with DTF-4 in 6 energy groups with the boundaries 10 MeV -

- 0.5 MeV - 9118 eV - 0.625 eV - 0.140 eV - 0.058 e V - 0.

Cross sections were generated by AE-BUXY and DTF-4 was run

on a 1.35 % U0£ pin cell at 20 °C and 40 °C with either P^-

scattering and transport corrected diagonaT’elenents or P^-

scattering. The results are shown in Table 6.4. It is seen

that the use of transport corrected isotropic scattering is

a very good approximation for the P^-scattering. The very

small discrepancy between the P^- and P^-results is not

significant and we conclude that transport corrected cross

sections give the same k^ as calculations using explicit

P^-scattering.

Table 6.4: Comparison of DTF-4 calculations using P -

and P -scattering. 1.35 % U0„, V / V c = 1 . 4 I I m f

k00 Ak /AT00

20 °C oo

o (pcm/°C)

Pg-scattering 1.16024 1.15834 -9.50

P^-scattering 1.16028 1.15837 -9.55

The prediction of leakage in small cores is a crucial part

of the reactor physics calculations. The calculated leakage

is sensitive to nuclear data in the fast energy region

(fission neutron spectrum, inelastic scattering cross

sections for U-238, hydrogen and oxygen scattering data),

and a proper treatment of the anisotropic scattering is re­

quired. One must therefore expect that in most cases there

may be an uncertainty of several per cent in the calculated

leakage. The error in the predicted leakage due to erroneous

fast data or an inadequate treatment of the anisotropic

scattering is, however, insensitive to the temperature and

the influence on calculated temperature coefficients is less

significant than the influence on the k^^-value.

We have seen in chapter 4 that the fundamental mode calcula-2

tion using the measured B and the two-dimensional diffusionm

theory calculation give approximately the same reactivity,

i.e. the predicted leakage is the same. DIXY-calculations

have also been compared with SQ-calculations using DTF-4 inO

one-dimensional cylindrical geometry [45]. The agreement in

calculated temperature coefficients is good. The axial2

leakage obtained using the measured agrees well with that

calculated by DTF-4 (Table 4.6). Thus, we do not expect that

the observed discrepancies between experimental and theoret­

ical temperature coefficients can be explained by errors

in the leakage calculation.

This conclusion is supported by the observation that the

error in predicted temperature coefficients is the same at

low and high temperature although the influence of the

leakage on the coefficient is much larger at elevated

temperatures. Further, we note that the leakage in the

cores of chapter 4 varies from 10 % to 25 but no trend

is seen in the discrepancy, Ak ^ / A T , as function of the

1cakage.

6.5^1_____ Macro grougs_and Gauss H 2 iD£®_iD_^E~BUXY

AE-BUXY calculations were made on the 1.35 % UC> 2 lattice

with 175 ppm boron in the water using 17 macro groups and

8 Gauss points (3 in UO^, 2 in canning, 3 in H^O) or 25

groups and 16 Gauss points (6 in UC^, 4 in canning, 6 in

1^0). Condensation vectors for the group structures are

(cf Table 3.1).

17 groups 2, 4, 6, 14, 21, 25, 26, 27, 35, 38,

45, 48, 51, 55, 59, 63, 69.

25 groups 2, 4, 6, 14, 21, 23, 25, 26, 27, 32,

35, 38, 41, 45, 48, 51, 54, 55, 56,

57, 59, 61, 63, 66, 69.

The two group-structures differ mainly in the thermal region.

Results from calculations at two temperatures are given in

Table 6.5. The discrepancies between the 17 and 25 group

results are negligible.

Table 6.5: Comparison between AE-BUXY calculations with

17 macro groups, 8 Gauss points and 25 macro

groups and 16 Gauss points. 1.35 % 175 ppm B

Temp

(°C)

Numberofmacro­groups

NumberofGausspoints

k o k c . eft

B 2m

(m'2)

2M

( ^ (cm )

20 17 8 1.12172 0.99640 29.91 40.7

25 16 1.12171 0.99630 29.88 40.7 '

210 17 8 1.10448 0.99060 20.77 50.3

25 16 1.10459 0.99060 20.77 50.3

To investigate how the leakage depends on the number of

fast groups, calculations were made on the 1.35 % UO^

lattice without boron in the water using 17 and 23 groups

in the fundamental mode calculation. The 17 group structure

is the same as given above. 6 groups were added in the fast

region to obtain the 23 group structure which is

23 groups 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 21,

25, 26, 27, 35, 38, 45, 48, 51, 55,

59, 63, 69.

The 17 and 23 group structures are identical below 9118 eV

(gr 14). Results are given in the tables below.

Table 6.6: AE-BUXY calculations with 17 and 23 macro groups.

1.35 % U02, 0.8 ppm B

Temp

(°C)

Mumberofmacro­groups

k00 kef f

B2m

(m “)

r2 ;

M

(cm )

90 17 1.15566 0.99585 36.01 43.2

23 1.15567 0.99415 35.61 43.7

21.0 1.7 L. 1.3872 0. 99147 27.28 50.9

23 1.13874 0.98983 26.97 51.4

Table 6.7: Temperature coefficients derived from Table 6.6.

Number of macro groups

Alc^/AT(pcm/°C.)

Ak ff/AT (cpm/°C)

17 -14. 1. -3.65

23 -14.1 -3.60

It is seen from Table 6.6 that the leakage is underestimated

by about 1 % in the 17 group structure compared to the 23

group structure. This corresponds to about 0.2 « in keff

The temperature coefficients were changed very little when

the number of groups was changed (Table 6.7).

The results in Tables 6.5-6.7 are in agreement with previous

experience and justify the use of 17 macro groups and 8 Gauss

points in AE-BUXY.

6.5.2 Energy groups and mesh points in the diffusion theory calculation

The number of energy groups and mesh points was varied in

diffusion theory calculations on the 1.35 % U02 lattices

with surrounding reflectors. Results are given below.

Two group-structures were compared with the condensation

vectors

4 groups 6, 14, 45, 69.

8 groups 2, 6, 14, 45, 55, 59, 63, 69.

Table 6.8: k rr from DIXY. 1.35 % U0„ lattice with 46x46 ---------- err 2

fuel pins, 175 ppm B

Number of groups

Number of mesh points

keffAkeff/AT

(pcm/°C)20°C 210°C

4 28x28 0.99691 0.99301 -2.05

4 39x39 0.99730 0.99349 -2.01

8 39x39 0.99730 0.99350 -2.01

It is seen from Table 6.8 that k __ and Ak AT areeff eff

insensitive to the number of energy groups and mesh points

in the diffusion theory calculation. The exact agreement

between the 4 and 8 group calculations is accidental. Similar

comparisons made for other purposes have given discrepancies

of about 100 pcm between 4 and 8 group calculations.

6.6_________ The influence of thermal expansion on reactivity

For practical reasons the thermal expansion of fuel, canning

and construction material is often neglected in calculations,

because this effect in most cases gives only a small correc­

tion.

The influence of thermal expansion on the reactivity was

estimated separately for all the KRITZ lattices analysed

in chapter 4, and the corrections have been applied in the

results given in Tables 4.2-4.7.

The following coefficients of thermal expansion were used

The variation of the lattice pitch with temperature is due

to expansion of stainless steel in which the spacers are

fastened.

In Table 6.9 the correction for thermal expansion is split

up into contributions from the change of fuel radius, can

radius, cell radius, fuel density and can density. It is

seen that the effects on dk /dT are very small except for(X> *

the contribution due to l\r , - . The dominating effects oncell

dk rp/dT are due to changes in the leakage in the fast

SS

i i - i o ' V c

7•10_6/°C

18‘10~6/°C

energy region. The individual contributions compensate

each other to some extent. In the 1.35 % UO^ lattice with

0 ppm boron in the water the total correction to the

temperature coefficient is only -0.05 pcm/°C.

Table 6.9: Individual contributions to the temperature

coefficient due to thermal expansion.

1.35 % ITC>2 lattice with 0 ppm boron in the water

dk /dTOO

dk /dT eff

dB2/dTm

dM2/dT

(pcm/°C) (pcm/°C) (m"2/°C) (cm2 /°C)

“uo2 -.04 + .18 + 5-10_A-4

- 5-10

SrZr-2-.03 -.27 - 7 + 5

Ar ., cell

+ .20 + .38 + 10 - 5

AeU02-.04 -.34 - 9 + 10

A0Zr-2+ .01 0 0 0

Total + .10 -.05 - 1 + 5

The expansion of the spacers increases the lattice pitch

at high temperatures. This affects not only the cell cal­

culations 'but also the core calculation since the cor- size

is changed. For the core with 46x46 pins we get

AB2/AT = -6*10~4 m _2/°C . AB2/AT is -0.05 m ~ 2/°C so the g m

correction is in this lattice about 1 % of the total temper­

ature coefficient which corresponds to 0.2-0.4 pcm/°C.

Table 6.10 shows corrections for thermal expansion cal­

culated using AE-BUXY.

Fuelm f

Boron conc. (ppm)

dk /dTOO dk _./dT eff

dB2/dTm

dM2/dT

(pcm/°C) (pcm/°C) (m'2/°C) (cm2/°C)

1.35 % U02 1.40

175+ .10 + .02

-.05-.14

- 1-10"4 - 3

-4+ 5-10 + 10

1 9 0 + 1.05 + .65 +13 + 9

1.9 % U0o1 ♦ z.

300 +1.01 + .65 +13 + 9z

1.7 300 + .13 -.09 - 2 + 7

1.8 300 + .82 + .50 +10 + 6Pu0o 2.5 300 + .10 -.14 - 3 + 5

3.3 j - .10 -.29 - 8 + 5

6.7_________The plutonium fission neutron spectrum

The fission neutron spectrum used in the analysis is a

Maxwellian with ~ 1*33 MeV. The average energy of

fission neutrons from fissions in plutonium is higher than

that given by this spectrum. The use of a plutonium fis-2

SLon neutron spectrum increases the predicted and M ,

but has a negligible influence on the temperature coeffi­

cient. The table below shows results from AE-BUXY calcula­

tions with two different fission neutron spectra represen­

tative for fissions in uranium and plutonium. The Maxwellian

temperatures are 1.33 MeV and 1.385 MeV, respectively. The

change in k is due to an increased number of fast . 2

fissions m U-238. M is 1.4 % larger with the plutonium

spectrum.

and plutonium fission spectra.

PuO- lattice with V /V, = 2.52 m f

T ’mp.

( u c )

Fission neutron

spectrum

koo k

effM 2

(cm )

U 1.22588 1.00000 36.60

20 Pu 1.22685 .99781 37.11

diff +97 pcm -219 pcm + .51

U 1.23829 1.00000 46.31

210 Pu 1.23938 .99775 46.96

diff +109 pcm -225 pcm + .65

Fig. 6.1 Te f f as function of T

7. THE INFLUENCE OF NUCLEAR DATA ON THE CALCULATED

T E MP E RATI IRE COE F F1CIE NT

7.L Comparison of: Lemper;i lure coefficients calculated

__ ___________by use of ENDP/B and UKNI) 1^..lata______________________

The standard library in AE-BUXY is based on UKNDL but a data

library in the BUXY format has also been generated from

ENDF/B-III. Epithermal cross sections and effective reso­

nance integrals were processed by Che SPENG and DORIX

codes [46]. Thermal group cross sections were generated

by use of FLANGE-2 [47] which was modified to calculate

group scattering matrices instead of point-to-point matrices.

Three of the lattices studied in chapter 5 were recalculated

using the data library generated from ENDF/B-III. Partial

and total temperature coefficients calculated with the two

libraries are compared in Table 7.1.

The agreement between the temperature coefficients obtained

with the two different libraries is remarkable (the levels2

of calculated k and M differ, however, more than the

coefficients in Table 7.1). The two AE-BUXY libraries are

entirely independent. Both the basic nuclear data and the

processing codes used for generating group cross sections

differ.

Scattering matrices for water are based on the Nelkin

model in the "UK.\'DL"-library and on the Haywood model in the

"ENDF/B" library (cf § 7.3).

The good agreement in the moderator temperature coefficients

shows that the thermal, scattering matrices and thermal

reaction cross sections are in agreement. The water density

coefficients obtained with the two data libraries are

practically the same.

f ......

Case Data l ib 3X/3Tf 3X/3Tm

3X/36‘dQ/dT dX/dT

Temp

in te rv a l(°C)

20-90 160-210 20-90 160-210 20-90 160-210 20-90 160-210

1.35% UO9 X=km UKNDL -4.7 -4.1 -3.3 -3.3 +0.1 -3.7 -7.9 -11.1

Vm/Vf-1.4 ( n r m ^VDF/R -4.8 -4.2 -3.3 -3.3 0 -4. 1 -8.3 -11.6

175 ppm B X=p UKNDL -4.1 -_J. 1 “ *4 . 3 -4.U “ D . 9 -20.2 -15.3 -27.9

(pcm/°C) I'NDF/R -4.2 -3.8 -4.3 -4. 3 -6.9 -20.2 -15.4 -28.3

- .U113 -.0086 -.0113 * .0092 -.0164 -.0406 -.o3yo -.0584

(m-2/°C) ENDF/B -.0121 -.0094 -.0120 -.0100 -.0164 -.0406 -.0405 -.0600

X=M2 UKNDL -.1)004 -.0004 .0046 .0046 .0251 .0730 .0293 .0772

(cm2/°C) ENDF/3 -.0004 -.0004 .0046 .0048 .0239 .0692 .0281 .0736

1-9% U09 X=k01 UKNDL -4.6 -4.1 -2.6 -2.9 +0.7 -2.4 -6.5 -9.4

Vm/Vf =l77 (pcm/°C) ENDF/B -4.7 -4.2 -2.7 -3.0 +0.8 -2.2 -6. 6 -9.3

200 ppm B X=p UKNDL -3.8 -3.4 -4.1 -4.1 -10.9 -30.3 -18.9 -37.8

(pcm/°C) ENDF/B -3.8 -3.4 -4.2 -4.3 -11.0 -30.5 -19.1 -38.2

X=BI UKNDL -.0110 -.0086 -.0117 - .0100 -.0291 -.0690 -.0518 -.0876(m**2/oc) ENDF/B -.0116 -.0090 -.0129 -.0112 -.0300 -.0706 -.0544 -.0908

X-M2 UKNDL -.0004 -.0006 +.0042 +.0042 +.0261 +.0768 +.0300 +.0804

(cm~/°C) ENDF/B -.0003 -.0004 +.0043 +.0044 +.0250 +.0734 +.0290 +.0774

Pu09 X=kOT UKNDL -4.1 -3.7 + 7.6 +8.7 + 2.7 +0.1 +6.2 +5.1

V V 2*5 .(pcm/°C) ENDF/B -4.1 -3.8 +7.5 +8.4 +2.7 +0.2 +6.1

00t

+

250 ppm B |X*o UKNDL -3.3 -3.0 +5.1 +6.7 -11.4 -36.5 -9.6 -32.8

(pcm/°C) ENDF/B -3.3 -3.0 + 5.0 +6.4 -11.5 -36.5 -9.8 -33.1■X=B& UKNDL -.0107 -.0082 +.0164 +.0186 -.0351 -.0952 -.0294 -.0848i (m~2/oc) ENDF/B - .0110 -.0086 +.0160 +.0176 -.0354 -.0956 -.0304 -.0866

X=M2 UKNDL -.0002 -.0002 +.0025 +.0016 +.0262 +.0784 +.0285 +.07981

i(cm2/°C) ENDF/B

!-.0003 -.0004 +.0027 +.0018 +.0257 +.0766 +.0281

1+.0780

7 • 2 _ The effective-' resonance i»tegral of U-238

Shielded resonance integrals for U-238 processed from UKNDL

of ENDF/B-IT.1 give an overestimation of the resonance

absorption in cell calculations [48-50]. Tt is there­

fore necessary to reduce the resonance integral and corre­

late it to integral experiments.

In the data library based on UKNDL which was used in our

analysis of experiments, the U-238 absorption cross section

was reduced uniformly by 0 . 2 0 barn in the energy interval

4 eV - 9118 eV. This correction was originally suggested

by Askew [48]. The effective resonance cross section for

energy group g is given by Eq (3.7)

RI (a ,T)

■ W w ” ■ — L J ; ; »t )x - __ 5__P.? .5___

p»g

where RI (o ,T) is the tabulated effective resonance g P»g

integral as function of the background cross section, ,

and the temperature, T . t is the lethargy width ofg

group g . Differentiating Eq (7.1) one obtains

ARI„(o„ „,T) RI„(o „ -T) 2Ao£ ■ .P»g_____ [ 1 _____________ g.P.1« . ] (7.2)

a , g t o8 8 P>g

which with Ao = -0.20 b for all e is the correction useda »g

in the tables of the library.

In the last column of Table 7.3 resonance integrals cal­

culated by AE-BUXY using the Wigner approximation for the

fuel self collision probability

Et 1 Pff = ——- ; x = — and E = — (7.3)f f a+x Z e 2r v ;

e

and the above correction are compared with resonance inte­

grals measured by Hellstrand [51] in isolated fuel pins con­

taining UO^ or U-metal. The comparison is done for UO^ rods

with two different radii. The smaller radius is typical for

a BWR rod and the larger one gives background cross sections

corresponding to a Dancoff factor of about 0.5, which is

representative for a BWR assembly with average void contents

in the coolant. Resonance integrals for U-metal rods are

also compared at two radii. The experimental values are

given in Table 7.2. The tables also show the temperature

dependence of the resonance integrals. The calculated reso­

nance integrals are here compared with experimental results

from references [51, but a comparison with others experi­

ments would give essentially the same results. See e.g. the

review by Hellstrand in [53]. The experimental uncertainties

are about 4 % in the room temperature integrals and about

10 % in the Doppler coefficients.

It is seen that AE-BUXY, ' ■ Lth the resonance integral correc­

tion which was used in the analysis in chapter 4, overesti­

mates the Doppler effect by about 10-15 %, corresponding to

0.4-0 . 8 pcm/°C in the fuel temperature coefficient. In order

to see how the correction in itself affects the temperature

coefficient, three different corrections were tested, viz.

Correction la : .'a = -0.20 barna

11 _ __2a : ;'o = -0. 20[ i *-0.007 (/T-/300) ] barn

" 3a : .MU = -0.06 RI

A rational approximation Cor tin? fuel self collision pro­

bability suggested by C.ai ivik [54]

was used instead of the Wigner approximation because the

Carlvik approximation is more accurate.

Using uncorrected data the calculated resonance integral

is overestimated by about 6 % in UC^ rods and 5 % in

U-metal rods (Table 7.3). The temperature dependence is

predicted within about 5 %. The correction la is somewhat

too large giving a slightly underestimated resonance integral

in the UO^ rods. For the U-metal rods, which have a smaller

resonance integral, this correction is about twice the

required one. The Doppler coefficient is increased by 9 %

due to the correction. In order to preserve the temperature

dependence of the uncorrected resonance integral correction 2a

was tested. This correction is identical with la at 300 K.

but does not change the /F-dependence of the resonance inte­

gral. As seen in the table ARI/AT remains the same as

with the uncorrected data. Correction 3a reduces the reso­

nance integral by & % at all temperatures and o^-values. This

correction also leaves the Doppier coefficient unchanged.

Table 7.4 shows the corresponding comparison for the

library based on ENDF/B-III. The results are similar to

those for the UKNDL-library. The uncorrected resonance inte­

gral is about 10 % too large, compared to 6 % for the UKNDL

data, and the following corrections were tested

Correction lb : Ao -0.27 barna

2b : Aa = -0.27[l+0.007(/T-/300)]barna

" 3b : ARI = -0.10 RI

The uniform reduction of a is larger than in the UKNDL

library and the discrepancy in ARI/AT is consequently

larger. Corrections 2b and 3b reproduce the measured Doppler

coefficient better.

The corrections la and lb have the advantage that the un­

shielded resonance integral is reduced by 1.5 b and 2.1 b,

respectively, which is within the experimental uncertainties.

They do, however, change the temperature dependence of the

resonance integral. Corrections 3a and 3b do not affect the

Doppler coefficient but give a value of the unshielded reso­

nance integral which lies outside the experimental uncer­

tainty. Corrections 2a and 2b combine the advantages of

corrections 1 and 3. A drawback is, however, that although

ARI/AT over the whole resonance energy region becomes

correct, ARI /AT in individual energy groups will be §

erroneous. For example, in a group without any resonance

where ARI /AT « 0 one obtains ARI /AT < 0 . g g

The measured temperature dependence of the resonance inte­

gral was fitted to the expression [52].

RI(T)-<5 = [RI(T )-5 ] -11+3 (v T'-v/T—)] (7.5)o o

where 6 is the 1/v-part of the resonance integral and

the coefficient 6 varies slightly with the surface tp mass

ratio, S/M, but is assumed to be independent of T . The

measurements cover the temperature range up to 1000 K.

The theoretical values of the Doppler coefficients in

Tables 7.3 and 7.4 were determined using the real temperature,

T, of the fuel. When the crystalline binding effects are taken

into account using an effective Doppler temperature, Tej£>

defined by Eq (6.3) one finds that the calculated Doppler

coefficient in the UO^ rods is reduced by about 9 % over

the temperature interval 300 - 1000 K. The influence of

crystalline effects in the U-metal rods can be expected to

be small.

Table 7.2: Measured U-238 resonance integrals [51] and

Doppler coefficients [52] for isolated fuel rods

Fuel Radius

(cm.)

RIexp

300 °K

6 '*10“

u°? 0.52 21.6 0.77

u o2 1.04 16.8 0.67

u 0.50 16.1 0.73

u 1.00 12.6 0.62

cf Eq (7.5)

Table 7.3: Comparison between calculated U-238 resonance integrals

and Hellstrand's measurements. UKNDL data

Fuel Radius

(cm)

Carlvik approximation Wigner

Corr laNo corr Corr la Corr 2a Corr 3a

RIRI - 1 <% >

exp

RI (ARI/AT)exp th 1

RI., (ARI/AT) th exp

(%)

U°2

U02

U-metal

U-metal

u o2

u o 2

U-metal

U-metal

0.52

1.04

0.50

1.00

0.52

1.04

0.50

1.00

+5.7

+6.4

+4.9

+4.4

+6

-4

+1

- 2

- 0.9

- 1.7

- 3.7

- 5.9

+15

+ 5

+11

+ 9

-0.9

-1.7

-3.7

-5.9

+6

-4

0

-5

0.0

+1.1

-0.4

-0.5

+6

-4

i~ L

-3

- 1.2

- 1.8

- 4.0

- 6.0

+14

+11

+15

+11

Table 7.4: Comparison between calculated U-238 resonance integrals

and Hellstrand's measurements. ENDF/B-III data

Fuel Radius

(cm)

Carlvik approximation Wigner ;

Corr laNo corr Corr la Corr 2a Corr 3a

U°2 0.52 + 9.2 + 0.4 +0.4 -0.5 0.0

RI UCL 1.04 +10.1 - 0.7 -0.7 +0.8 0.0

RI - 1 « >2

U-metal 0.50 + 10.8 - 0.6 -0.6 +1.4 - 0.2exp

U-metal 1.00 +12.1 - 1.3 -1.3 +3.1 - 1.0

U0„ 0.52 + 6 +6 +4 + 16RI (ARI/AT) 2

exp th iU0„ 1.04 - 1 +12 0 -3 +15

RI (ARI/AT) 2tn exp

U-metal 0.50 + 5 + 18 +3 +3 + 21

(%) U-metal 1.00 - 2 +14 -4 -4 + 11

The analysis in chapter 4 was performed using the Wigner

approximation and correction la to the U-238 resonance inte­

gral. Table 7.3 shows that using this option the Doppler

coefficient is overpredicted by about 10-15 %. This over-

estimation is reduced to 0-5 % when the crystalline binding

effects are accounted for. It should, however, be remembered

that the experimental uncertainty is of the magnitude 10 %

and one can therefore not draw too detailed conclusions

about the ability of the theory from these comparisons.

When the crystalline binding effects are neglected the shape

of the calculated RI versus temperature can be fitted to

expression (7.5) with a 0-value which is different from the

experimental one. Using the effective Doppler temperature,

T^j;, the calculated Doppler coefficient is reduced more

at low temperature than at high, thus giving a slightly

different shape of RI(T). The precision in Doppler coefficient

measurements has usually not been hi°h enough to detect any

temperature dependence in g . There are, however, some

experimental results indicating a deviation from the in­

dependence, e.g. [55].

7._3I_1_____ Scatter ing_i.n_water

The Nelkin scattering model was used for water in the

analysis in chapter 4. Below, temperature coefficients

calculated by use of the Nelkin, the effective width and

the Haywood model are compared. Scattering matrices derived

from the effective width model were tabulated at 293 K w i t h q =

4.3, 450 K with q = 2.8 and at 600 K with q = 2.1 (cf eq 2.15).

Scattering cross sections at intermediate temperatures wer'i

obtained by linear interpolation. The Haywood scattering

cross sections are those obtained from ENDF/B-III by use

of FLANGE-2.

Moderator temperature coefficients for three lattices using

the three different scattering models are listed in Table 7.5.

The agreement in calculated coefficients is very good con­

sidering that three entirely different models have been

used.

Table 7.6 shows calculated reaction rate ratios in the fuel

region using the different scattering models. The reaction

rates obtained with the Nelkin and the effective width model

are equal, whereas the Haywood model gives a somewhat harder

spectrum at both low and high temperature.

7 ^ 3 ^ 2 __________§ £ 2 t t e r i n g _ i n _ U 0 2

Calculations using the IK^ scattering model in ENDF/B-III

have been compared with calculations using the free gas model

for both uranium and oxygen in • Although the scattering

cross sections differ considerably in the two cases, the

influence on calculated reactivities and coefficients is

negligible. This is due to the small importance of the

thermal scattering in the fuel in light water reactors. (It

was shown in chapter 5 that the thermal contribution to the

fuel temperature coefficient is only about -0.3 pcm/°C).

Table 7 ,5 : Ca lcu la ted moderator temperature c o e f f ic ie n ts using d i f fe r e n t

sca tte r ing models fo r water

Case Sca tte r ing model dk /9T 00 m

3p/3Tm

3B2/3T m m

3M2/3Tm

Temp in te rv a l (°C) 20-90 160-210 20-90 160-210' 20-90 160-210 20-90 160-210

1.35 % U09 Nelkin -3.3 -3.3 -4.2 -3.9 -.0113 -.0092 +.0046 +.0046

V /V *1.4 m f

E ff .w id th -3.3 -3.2 -4.5 -4.1 -.0124 -.0096 +.0060 +.0058

175 ppm B Haywood -3.2 -3.2

r-H |

•1

i |

-3.8 -.0114 -.0094 +.0046 +.0044

1.9 % U09 Nelk in -2.6 -2.9 -4.0 -3.9 -.0117 -.0100 +.0042 +.0042

V /V _= l.7 m r

E ff .w id th -2.6 -2.8 -4.6 -4.3 -.0136 -.0110 +.0057 +.0054

200 ppm B Haywood i •

1 U1

-2.8 -3.9 -3.9 -.0119 -.0104 +.0041 +.0042

Pu09 Nelk in +7.6 +8.7 +5.0 +6.3 +.0163 +.0184 +.0025 +.0016

V /V =2.5 m f

E f f .w id th +8.9 +9.2 +5.5 +6.4 +.0179 +.0182 +.0037 +.0024

250 ppm B Haywood +7.4 +8.6 +4.7 +6.2 +.0158 +.0186 +.0026 +.0016

Table 7 .6 : Ca lcu la ted re ac t ion ra tes fo r U-235, Pu-239, Pu-241 and Dy-164

re la t iv e to the re ac t ion ra te in a 1/v-absorber

Case Sca tte r ing modal ( 2 3 5 / l / v ) ^(239/1 /v )f is s

(241/1/v) .f 1 S S

(D y / i/v )abs

Temp# (°C) 2 0 2 1 0 2 0 2 1 0 2 0 2 1 0 2 0 2 1 0

1.35 % U02 . Nelk in .993 .968 1.288 1.498 1.109 1.216 .926 .902

V /V =1.4 m f

E f f .w id th .994 .968 1.286 1.500 1 . 1 1 0 1.218 .926 .902

175 ppm B Haywood .990 .966 1.307 1.514. 1.116 1 . 2 2 0 .924 ,901_

1.9 % JJO Nelk in .993 .969 1.305 1.515 1.116 1 . 2 2 2 .920 .897

V /V =1.7 tn f

^ f f .w id th .994 .969 1.304 1.517 ' 1.117 1.224 .920 .897

200 ppm B Haywood .990 .967 1.326 1.531 1.123 1.226 .918 .895

Pu02 Nelk in .997 .973 1.185 1.357 1.081 1.174 .926 .894

V /V =2.5 m i

E f f .w id th .997 .974 1.184 1.357 1.082 1.175 .927 .895

250 ppm B Haywood .994 .971 1.197 1.365 1.085 1.175 .924 .893

Thermal absorption and fission cross sections

7.4.1 Group cross sections for a 1/v-absorber in____________ Maxwel_Han_sgectra_of_various_temperatures

The AE-BUXY data library contains group cross sections which

have been generated by energy condensation of the cross

sections using typical weighting spectra. The energy groups

have been chosen so that group constants may be considered

to be insensitive to the weighting spectra used to generate

them. In the thermal region Maxwellian spectra have been

used for weighting. Scattering matrices are tabulated for

a representative range of temperatures, whereas absorption

and fission cross sections for most nuclides have been

assumed to be temperature independent.

In order to verify this assumption, group cross sections for

a 1/v-absorber were calculated for E <_ 0.625 eV using Max­

wellian spectra with various temperatures for weighting.

An analytical expression for the group cross sections of

a 1/v-absorber is easily derived

(7.6)

M(E,T)dE

E£g

where

The evaluation of the integrals in (7.6) gives

a g (T) = (7.7)

g+ and g- are the upper and the lower energy boundaries of

group g .

Calculated a (T) in the AE-BUXY group structure are

listed in Table 7.7 for the temperatures 300, 600, 900 and

1200 K and for E 0.625 eV. CT2200 ecIua to uHity- It

is seen that the group cross sections are insensitive to the

temperature of the Maxwellian spectrum used for weighting.

This conclusion may be expected to hold for all smooth cross

sections. Thus, the error in predicted reactivities and

reactivity coefficients caused by the use of a limited

number of thermal energy groups is negligible as far as

reaction cross sections are concerned (if all materials

are containing only nuclides with smoothly varying cross

sections).

Z •--- _Thermal_data_for_U-235_and_U-238

Figures 7.1-7.4 show the thermal U-235 data, a , a ,sl r

a = o /o, - 1 and n = vo,/o in the libraries generated a f r a

from UKNDL and ENDF/B-III. The agreement between the two

libraries is very good and in the scale used in Figs 7.1

and 7.2 the cross sections coincide. There are, however,

other evaluations giving different data and it has been

pointed out, e.g. in [56]> that the spread in measured data

for U-235 is relatively large.

In order to study the effect of modifications in U-235 data

on calculated reactivities and temperature coefficients a

was changed keeping a unchanged. The change in the shapecl

of a versus energy is shown in Fig 7.3. This modification

lies within the scatter of experimental points and is the

same one as has been considered by Askew L563• Our choice

for this modification of a is arbitrary. It represents

one among several modifications which may be considered.

The influence on calculated reactivities and temperature

coefficients is shown in Table 7.9.

Table 7.7: Group cross sections for a L/v-absorber in the .

AE-BUXY group structure using Maxwellian weighting

spectra of various temperatures.

Group Q 1

00

300 K 600 K 900 K 1200 K

46 2. 21447E-01 2.16559E-01 2.15074E-01 2.14321E-01

47 2.45213E-01 2.41125E-01 2.39788E-01 2.39105E-01

48 2.62920E-01 2.61079E-01 2.60619E-01 2.60392E-01

49 2.76325E-01 2.75378E-01 2.75185E-01 2.75088E-01

50 2.86520E-01 2.85957E-01 2.85864E-01 2.85817E-01

51 2.96194E-01 2.95673E-01 2.95571E-01 2.95517E-01

52 3.10725E-01 3.09782E-01 3.09507E-01 3.09367E-01

53 3.30109E-01 3.20967E-01 3.28735E-01 3.28566E-01

54 3.60025E-01 3.57808E-01 3.57049E-01 3.56666E-01

55 4.03698E-01 4.00596E-01 3.99528E-01 3.98991E-01

56 4.68427E-01 4.63581E-01 4.61926E-01 4.61100E-01

57 5.33721E-01 5.31824E-01 5.31190E-01 5.30874E-01

58 5.88684E-01 5.87 594E-01 5.87233E-01 5.87053E-01

59 6.37430E-01 6.36764E-01 6.36544E-01 6.36435E-01

60 6.85634E-01 6.8497 9E-01 6.84764E-01 6.84655E-01

61 7.43046E-01 7.42215E-01 7.41941E-01 7.41802E-01

62 8.12023E-01 8.11195E-01 8.10919E-01 8.10779E-01

63 8.83177E-01 8.82635E-01 8.82452E-01 8.82358E-01

64 9.60238E-01 9.59539E-01 9.59302E-01 9.59182E-01

65 1.06175E+00 1.06080E+00 1.06048E+00 1.06032E+00

66 1.20412E+00 1.20273E+00 1.20226E+00 1.20203E+00

67 1.42486E+00 1.42256E+00 1.42179E+00 1.4214IE+00

68 1.83773E+00 1.83282E+00 1.83120E+00 1.83040E+00

69 3.03882E+00 3.0J 881E+00 3.01227E+00 3.00901E+00

The absorption cross section of U-238 is generally assumed

to obey the 1/v-law at thermal energies. This is the case

also in UKNDL and ENDF/B-III. Table 7.8 shows the lowest

positive energy resonances of U-238. There are no significant

discrepancies between data from different references.

Table 7.8: U-238 resonance parameters in BNL-325

E

(eV)

Srn(meV)

ry(meV)

r

(meV)

I

4 . 4 1 . 0 0 0 1 1 1 1

6 . 6 7 1 . 5 2 26 2 7 . 5 0

1 0 . 2 5 . 0 0 1 5 6 1

1 1 .3 2 . 0 0 0 3 5 1

1 6 .3 .0 0 0 0 5 1

1 9 . 5 0 .0 0 1 4 1

2 0 . 9 0 8 . 7 25 34 0

3 6 . 8 0 3 2 . 0 25 5 7 . 0 0

6 6 . 1 5 2 6 . 0 22 48 0

Extrapolating from the positive energy resonances one

finds that the first negative energy resonance may be

expected at approximately -10 eV, which means that its

contribution to at thermal energy is of the 1/v-shape.

The contribution from this negative energy resonance is

of importance when deriving the thermal . At

E = 0.0253 eV the first 22 positive levels contribute

2.38 barns [57] and with °2200 = the remaining 0.35 b

usually is attributed to a single negative energy resonance.

When generating U-238 data for ENDF/B-III a single

negative energy resonance was placed at -15 eV [57].

The possibility of a resonance close to zero energy can,

however, not be excluded. If such a resonance exists, its

contribution to the thermal absorption cross section will

be such that the gradient of a versus energy will be

more negative than for a 1/v-cross section.

In order to investigate how a non-l/v cross section affects

the predicted temperature coefficient a change of the absorp­

tion cross section was made below 0.3 eV. The modification

was made so that the reaction rate in a 20 °C Maxwellian

spectrum was unchanged. Modified and unmodified cross

sections are shown in Fig 7.5 and results from the calcula­

tions with the modified data are collected in Table 7.9

Table 7.9: k and temperature coefficients calculated by AE-BUXY using modified

thermal U-235 and U-238 data

Lattice Modification

kOO

20°C 210°C

Ak /ATOO

(pcm/°C)

k f ef20°C

f210°C

Ak ,,/AT ett

(pcm/°C)

1.35% U02

V /Vf=1.4 m r0 ppm B

Reference 1.16299 1.13875 -12.76 1.00000

i

1.00000 I 0 1

A°f235

A°a238

+483

+500

+741

+ 1172

+ 1.36

+3.54

1

+412 1 +645

+380 i +966i

+ 1.23

+3.08

1.9% U02

V /V,=1.7 m r200 ppm

Reference 1.21010 1.19573 -7.56 1.00000 1.00000 0

A°f235

A0a238

+519

+392

+785

+909

+1.40

+ 2.72

+426

+338

+650

+765

+ 1.18

+2.25

In the 1.35 % U02 lattice the temperature coefficient was

predicted about 4 pcm/°C too negative (cf Table 4.3) and in

the 1.9 % UO^ lattice 2 pcm/°C too negative (cf Table 4.7).

The modification of U-238 gives the correction +3 pcm/°C

and +2 pcm/°C in the 1.35 % U02 and 1.9 % U02 lattices,

respectively. The U-235 modification gives about +1 pcm/°C

in both lattices.

Data for some lattices which were analysed in chapter 4 are

collected in Table 7.10. The thermal absorption in U-235

and U-238 as calculated by AE-BUXY is given in percent of

the total cell absorption plus leakage. Assume that the

change in calculated Ak^^/AT due to modified U-235 and

U-238 cross sections is proportional to the relative absorp­

tion in the nuclide. Then corrections to Ak ,.r/AT areeff

obtained by combining the results of Table 7.9 with the

relative absorptions listed in Table 7.10. These estimated

corrections are given in the last two columns and are com­

pared with the discrepancies obtained in the comparisons

between theory and experiment.

It is seen that for all these lattices a close agreement

between theory and experiment can be obtained by modifying

the shape of the thermal U-238 absorption cross section as

function of energy.

The influence of a modification in the shape of the thermal a

for U-235 on the temperature coefficient has previously been

studied elsewhere, e.g. in [56, 58]. Askew et al. [4]

conclude that nuclear data uncertainties indicate that

extreme changes within the uncertainties of nuclear data

in the energy dependence of r) for U-235 could contribute

about +1 pcm/°C to the temperature coefficient. Our results

are in agreement with his conclusion.

Basiuk et al. [59] have made calculations on a PWR lattice

assuming a p-wave resonance in U-238 at 0.1 eV. This did

not change the calculated temperature coefficient and they

conclude that a p-wave resonance at a lower energy would

not influence the results.

Table 7.10: Estimated corrections to Ak £C/AT due to modified a anu ------------ eff a238 f235

for some KRITZ lattices.

Fuel

V / V f and m f

boron conc

Temp

°C

Therma

U235

abs (%)

U238

ikef£/4T

Discre­pancyth-exp

(pcm/°C

Correct modifie

°f 235

ion due to i

°a238

1.35% U0 9 V /V,=1.4 m f

0 ppm B

20

210

41.1

40.6

12.5

12.7-4.4 + 1.2 +3.1

V /Vf=1.4 m f

175 ppm B

20

210

40.9

40.1

12.4

12.5-3.3 + 1.2 +3.1

1.9% U0 2 V /Vf=1.2 m i

200 ppm B

20

210

39.1

38.0

8.6

8.6-1.4 + 1.1 +2.1

V /V,=l.7 m f

200 ppm B

20

210

41.2

40.4

9.0

9.1-1.8 + 1.2 +2.3

1.5% Pu02 V /V,=l.8 m 1

300 ppm B

20

210

2.5

1.8

5.7

4.9-0.4 +0.1 +1.3

V /V,=2.5 m t

250 ppm B

20

210

2.3

1.9

6.1

5.2-1.0 +0.1 + 1.4

V /Vf=3.3 m r

0 ppm B

20

210

2.5

2.3

6.4

5.5-0.6 +0.1 + 1.5

Fig. 7.1 oQ for U-235 in the AE-BUXY library-

Fig.7.2 of for U-235 in the AE-BUXY library

106

Fig. 73. a for U -23 5 in the A E - B U X Y library and in E N D F / B m .

s.

^ M o d fied shape of cfa in the present study

....

>vS s\

............. ....v v

0.001 0.01 0.1 eV E

Fig. 7.5 orQ for U-238 in the AE-BUXY library

Our comparison between calculated and measured temperature

coefficients shows that the employed theory predicts a too

negative coefficient, the discrepancy being 1-4 pcm/°C.

The partial temperature coefficients - the fuel temperature

coefficient, the moderator temperature coefficient and the

water density coefficient - were theoretically determined.

The fuel temperature coefficient is approximately proportional - 1/2

to T . The dominating component is caused by the Doppler

broadening of the U-238 resonances and an accurate calcula­

tion of the Doppler coefficient requires that the crystalline

binding in the U0„ is taken into account./This effect is^ V

largest at low temperature. It can be accounted for by using

an effective Doppler temperature which is higher than the

true temperature of the medium. The correction of the

Doppler coefficient due to the crystalline binding is

about 15 % at room temperature and 3 % at 1000 K.N

The moderator temperature coefficient consists mainly of three

components 3p/3rr3n/3T , 3p/3f-3f/3T and 3p/3L2 -3L2 /3Tm m m

in the thermal energy group. 3p/3f•3f/3Tm is in all. lattices. . /.

positive and the other two components are negative./The mag­

nitude of 3p/3f'3f/3Tm is much larger in Pu02 lattices

than in UO 2 lattices. The discrepancy between calculated

and measured temperature coefficients is approximately the

same for all lattices, however, and one may therefore expect

that the employed model for the thermal scattering in water

predicts the temperature dependence of the spectrum correctly.

This conclusion is supported by the comparison between re­

sults using the Nelkin, the effective width and the Haywood

models for scattering in water. All three models give the

same temperature coefficient although the Haywood model pre­

dicts a harder spectrum than the other two models.)

The water density coefficient varies considerably with

temperaturefwhereas the discrepancy between theoretical

and experimental temperature coefficients is nearly the same

at low and high temperature. This observation indicates that

the inconsistency between theory and experiments is not due

to effects caused by the water density variation with tem­

perature. '

Various approximations in the theory have been validated. We

have found that the cylindricalization of pin cells, the

utilization of transport corrected cross sections to account

for anisotropic scattering in the cell calculation and the

use of a limited number of energy groups and spatial meshes

in the calculations introduce very small errors in the pre­

dicted temperature coefficients.

The comparison between results using the UKNDL and the

ENDF/B III data shows that these data sets provide the same

temperature coefficient.!

The temperature coefficient is sensitive to the energy

dependence of the thermal U-235 and U-238 cross sections.

An error of % 1 pcm/°C is possible due to the uncertainty

in the capture to fission ratio as function of energy for

U-235. The thermal U-238 absorption cross section is usu­

ally assumed to obey the 1/v-law. However, if a negative

energy resonance exists close to zero energy its influence

on a will be such that a decreases with higher energy & a

faster than the 1/v-law prescribes. Such a shape of aa

versus energy would produce a less negative temperature

coefficient. It was found that the calculated temperature

coefficients for all investigated cores can be brought in

agreement with experimental values by a modification of the

thermal U-238 absorption cross section shape. This modifi­

cation would also give calculated values of closer to

unity and with a smaller spread.

The work presented here was carried out at the section

for Reactor Physics of AB Atomenergi, Studsvik. Part of

the work was sponsored by the Swedish Board for Technical

Development (STU).

I would like to thank my colleagues for many fruitful dis­

cussions. My thanks are particularly due to Drs E Hellstrand,

H Haggblom and R Persson at a B Atomenergi and Prof N G

Sjostrand at Chalmers University of Technology.

- Ill -

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At :s kontorstryckeri