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EUR 4415 e

A METHOD FOR DETERMINATION OF CONVERTIBLE NUCLEAR H E A T I N G DATA, by H. KRÖCKEL

European Atomic Energy Community - EURATOM

Joint Nuclear Research Center - Petten Establishment (Netherlands) Luxembourg, April 1970 - 44 Pages - 7 Figures - FB 60

The nuclear heat generation process is analyzed and presented in well-known form as function of the neutron and gamma energy distributions.

The energy distributions are approximated by expressions containing three unknown parameters which appear, after insertion, as three unknown linear constants in the heating equation.

By setting up this equation for three different materials, a system of three inear equations is obtained, the coefficients of which are computable functions. The solution of this system is presented in terms of experimentally determined nuclear heating values of the three materials.

The known solution can be used for calculation of nuclear heating data for any desired sample size and material by the same linear expression.

EUR 4415 e

A METHOD FOR DETERMINATION O F CONVERTIBLE NUCLEAR H E A T I N G DATA, by H. KRÖCKEL

By setting up this equation for three different materials, a system of three linear equations is obtained, the coefficients of which are computable functions. The solution of this system is presented in terms of experimentally determined nuclear heating values of the three materials.

EUR 4415 e

A METHOD FOR DETERMINATION OF CONVERTIBLE NUCLEAR HEATING DATA, by H. KRÖCKEL

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Numerical coefficients of this expression are given for materials C, Be, Mo and Fe as function of sample size.

A test rig, CADO-17 for measurements in the High Flux Reactor (HFR) at Petten is described; the chosen sample materials are C, Be and Mo.

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E U R 4 4 1 S

e

EUROPEAN ATOMIC ENERGY COMMUNITY - EURATOM

A METHOD FOR DETERMINATION OF

CONVERTIBLE NUCLEAR HEATING DATA

by

H. KRÖCKEL

1970

(6)

A B S T R A C T S

KEYWORDS

HEATING GRAPHITE GAMMA RADIATION BERYLLIUM

NEUTRONS MOLYBDENUM NUMERICALS IRON

(7)

CONTENTS

1 . Introduci ion

2. Synthesis of Nuclear Heat Generation

2.1. Heat Generation by Core Gamma Radiation Ρ

2.2. Heat Generation Due to Core Neutron Radiation Ρ

η

2.2.1. Heat Generation by Elastic Scattering Ρ

J ■ n n

2.2.2. Heat Generation by Capture Gamma Radiation Ρ

ηγ

2.2.3. Heat Generation by Inelastic Scattering Gamma Radiation Ρ .

J η ι γ

2.2.A. Heat Generation by Decay Radiation Ρ ,

J J ndecay

3. Energy Distributions of Gammas and Neutrons

3.1. Approximation of the Gamma Energy Distribution

3.2. Approximation of the Neutron Energy Distribution

4. Solution of the Heat Generation Equation

4.1. Insertion of Gamma and Neutron Energy Distributions

4.2. Presentation of Heating as Linear Expression with 3

Unknown Constants

4.3. Determination of Unknowns Å , Β and C

o o o

5. Experimental

5.1. Principle

5.2. Sample Materials

5.3. Test Rig

5.4. Calibration and Sensitivity

5.5. Accuracy

6. Numerical Calculations

6.1. Expressions G, F and Ν

6.2. Determinant D for CADO17 c

7. Definition of Symbols

8. References

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A METHOD FOR DETERMINATION OF CONVERTIBLE NUCLEAR HEATING DATA '

1.0. INTRODUCTION

-Calculation of nuclear heat generation in non-fuel materials of the High Flux Reactor (HFR) at Petten is in general

based on measured values, which are obtained by special calorimeter probes.

Although the presently available data are in themselves ra-ther complete, ra-there exists a considerable lack of means to apply them to conditions, which are different from the test conditions .

Particular difficulties are encountered, if the data are to be converted to objects having other geometry, size or ma-terial as the test sample.

These difficulties arise, because the nuclear heat generat-ion, which is a result of several energy absorption proces-ses of gamma and neutron radiation, is measured by calori-metrie methods in integral form, which does not reveal the

contribution of the individual process.

Data conversion however requires the knowledge of the con-tribution of each absorption process to the overall effect.

The functional dependency of a process on the sample para-meters has the form of an energy integral of geometry and material dependent parameters, in which only the energy distributions (more correctly the "energy spectral densi-ties of the number fluxes") of neutron and gamma particles appear as unknowns.

If these energy distributions are known, then the proport-ionate contributions of the absorption processes can be determined, so that the conversion problem can be solved.

(10)

Since the gamma energy distribution, which is the more im-portant one of the two, is not known for the HFR, some con-version relations had earlier been derived on basis of an assumed distribution.

Although by this evaluation, useful

nu-merical results could be obtained, it is desirable to pro-ceed to a method, in which no explicit assumptions on the energy distributions are needed and which also covers the influence of the neutron radiation.

Such a method must necessarily be based on a detailed ana-lysis of the energy absorption processes involved and must be directed on determination of parameters, which are re-presentative for the gamma and neutron energy distributions existing in the reactor.

The method described here, is laid out for determination of 3 constants, 2 representing the gamma and 1 the neutron distribution.

The description of these distributions by expressions con-taining 2 resp. 1 parameters is, of course, very approx-imate and will only be satisfactory for investigations of nuclear heating and similar effects.

The experimental principle of this method follows in some lines that of ref. 1 , where 2 parameters are determined, which relate the gamma and neutron heating of a material

(11)

2.0. SYNTHESIS OF NUCLEAR HEAT GENERATION

The main effects contributing to nuclear heat generation in non-fuel materials in the HFR are the following :

1. Core gamma radiation absorption, Ρ Y

2. Core neutron elastic scattering energy degradation, Ρ 3. Thermal neutron capture induced gamma absorption, Ρ

nY

4. Inelastic scattering (fast neutron) induced gamma absorption, Ρ . _ηΐγ

5. Activation product gamma and β absorption Ρ „

: .ndecay

There are other minor sources of heat generation, such as

effects of (n,ot) and (n,p) reactions and others, which are however neglected here.

The 5 above mentioned heat sources are not equally important, and some of them have negligible contribution for some materials. In all cases, the dominant role is played by Ργ.

The total effect is combined in the form :

Ρ = Ρ + Ρ (1)

γ η

Ρ = Ρ + Ρ + Ρ . + Ρ , (2) η ηη ηγ ηιγ ndecay

2.1. Heat Generation by Core Gamma Absorption Ρ

1 Ï

The heat generation at a point with the local coordinate r in a

sample due to gamma absorption is given in ref. 2 as :

Ρ

γ ί

Γ ) =

> T~

(E)

'

N

<

E

'

r

> ·

E dE

<

3

>

The spectral number flux density at r,N(E,r) is obtained from the undisturbed flux density Ν (E) by multiplication with a factor which is, according to ref. 3, equal to the escape probability p(r) from the same body for a homogeneous internal source distribution; if linear exponential attenuation is

(12)

N(E,r) = No(E) . p(r) <4)

The average escape probability is :

P

o

= i ƒ p(r)dV

ν

The average heating is :

(5)

**■ * J**

Ρ (r)dV γ

. E γ \ N(E,r)dVdE (6)

(7)

Herein :

±

) N(E,r)dV =

-γ

J p(r)dV = Ν Ρ

o o (8)

And finally :

uu

Yy J jo D ρ O o rρ o . Ε dE (9)

Alternative Derivation

This result can be derived in an alternative way on basis of

the total amount of incoming photons η (spectral) and the

fraction of collisions F^ :

(10)

CO

1 Í Va

Ρ o _L· \ -± . ρ . η EdE

γ pV J μ c o

In ref. 3 one finds the relation :

η = Ν . γ

o o 4 ( U )

and the definition of the average chord R , which represents

e av' e

the mean of all possible penetrations through a body :

(13)

9

Introduction of these terms in eq. (10) yields

f ya Fc

= I _in

-4

E d!

ρ o uR

i v H av

(13)

Eqs. (9) and (13) are equal if

F

o yR

av

(14)

The proof that relation (14) holds, is given in ref. 6

Since in eqs. (9) and, (13) the effect of secondary gammas

(scattering gammas) is neglected, these expressions give under

estimated values for Ρ , which are corrected by introduction

γ .

of the energy absorption buildup factor Β , which is applied

in the linear form proposed in ref. 4 :

Β (E,yr) = 1 + (f 1 ) yr (15)

With use of the mean value r = r R in eq. ( 1 5 ) , eq· (9) become s

ζ. a v 00

■ ! 1 + ^ ( _ 1 ) u R a v

a

ρ Ν EdE

1 o o (16)

The average escape probability ρ is a function of photon energy

and sample geometry. Calculations of ρ are rather extensive.

For simple geometries, tabulated results can be found in ref. 3 ;

among them are tables for slabs, spheres, halfspheres and

infinitely long cylinders.

The result for infinite cylinders is reproduced in fig. 1. It

can also be used with good accuracy for length to diameter

ratios L/D >0.5, if one replaces χ by the parameter :

X = \ w Ra v '

(14)

10

where R , defined by eq. (12), is for cylinders

R

a V l^D/L

Definition (17) permits the shorter form of eq. (16) :

(18)

Ρ = γ

1 + < — ! ) ρ (Υ).N .E.dE

ro Λ o (19)

Other geometries than cylinders can be treated in analogue form

on the basis of ref. 3.

2.2. Heat Generation due to Core Neutron Radiation Ρ

2.2.1. Heat Generation by Elastic Scattering Ρ nn

In ref. 2 the following expression is given for neutron

heating by elastic scattering :

00

Ρ (r) =

nn ^ ( E ) . Ρ Σ (E) 0 (E,r) dE, s

(20)

where the mean energy degradation per collision ΔΕ is :

ΔΕ = E . 2A

(A+l)

(21)

Analogue to sect. 2.1., the local flux density can be

expressed by the undisturbed flux density and a pro

bability. The volume average has the form :

00

(15)

11

w h e r e i n t h i s e q u a t i o n

2E T R

( 2 3 )

av

and ρ (χ) is taken from fig. 1, R corresponding to eq. (13)

Graphs of Σ and £„, are presented in ref. 4 .

s ι

2.2.2. Heat Generation by Capture Gamma Radiation Ρ S*

For the (η,γ) reaction following thermal neutron capture is

assumed, that it has a spatial distribution in the sample

producing a homogeneous gamma source distribution.

Under this condition the heating due to capture gammas can be

expressed by

= - . Σ .φ „. . ρ ( Y ) . Β — . ρ ( χ ) · q ( Ε ) . Ε d]

η γ p a Yo t h ro Λη J a y c A η γ (24)

where

oth 0 (E)dE o (25)

'th

is the thermal neutron flux density. The build up factor Β is

a

Β = 1 + (_ÍL1 ) . χ

a y ' Λ

a (26)

In view of the information available in ref. 2 , the integral

can better be expressed as a sum over discrete energy intervals

ΔΕ. i n t h e f o r m :

Χγ Ν

( β ™ · Ρ ( χ ) . q ( E ) . E d E * ) — ( Ε . ) . Ρ ( v ) . E . j jaf q dE ( 2 7 )

J a y c Λ ^ η γ / μ ι . ^ c λ ι öa J Ηηγ ν '

(16)

12

In this expression, ρ (χ) is the collision probability

Pc(x) = 1P0(X) , (28)

In the preceding equations, χ is defined by eq. 17 and γ by

I r ρ ( 2 9 )

Xn " 2 V n T av'

The integrals

; q _ηγ(E.)dE = S(E.) _i _i (30)

ΔΕ . ι

are tabulated in ref. 2 for 7 gamma energy intervals and for

all important materials. Eq. (24) can therefore be written :

Ìo t h Po( Xn) ' ¿ a " (Ei).Pc(x).Ei.S(E.

Ρ

ηγ

=

¿V

i=l

Ρ is in general negligible for materials having comparably

small absorption cross sections.

2.2.3. Heating by Fast Neutron Inelastic Scattering Gamma Radiation

Ρ .

_ηιγ_

In the few cases, where inelastic scattering (fast neutrons)

induced gamma radiation contributes to the heating, one can

treat the induced gammas as monoergetic with energy Ζ .

Under analogue conditions as in sect. 2.2.2, the heating gan

then be expressed as :

• Ε . Ι Σ .

γ J n i

-Β μ

Ρ . = — — (Ε ) . ρ ( χ ) . Ε . Ι Σ . (Ε ,Ε ) 0 (Ε ) ρ (x„ ) d E ,

η ι γ ρ μ γ * c Α γ / ηΐγ γ n o n/ j ro ΛΤ η '

(17)

13

where Β is defined by eq. 26. Furthermore

a

PC( X ) = ! _P0( X ) (33)

and

X = 2'j Rav (34)

XT = 2Σ TRav (35)

Cross section data are available in ref . 4 .

2.2.4. Heating by Activation Product Decay Rad ia tion. Pndecay

In some cases, thermal neutron capture generated nuclides

contribute to the heating by their decay gamma and beta

radiation .

For calculation, it is again assumed, that the gamma and beta

source distributions are homogeneous. The decay radiation

normally originates from a nuclide, which was generated by

neutron capture in an isotope, the natural abundance of which

in the material under consideration is y .

η

The source strength therefore is

y „ E •♦„fi, · ρΛ( χ „ )

n a o t n o η

(36)

If the decay gamma and beta energies are E and ER, the

partial heating term0 are :

Β y

a a

nde cayy = — ρ — y (Ε ).p (χ).Ε .y .Σ .φ . .ρ (χ ) γ lc γ η a oth *ο η (37)

(18)

14

where β is defined by eq. 26 and

Xn = I EthTRav ( 3 9 )

x =

1 ^ V ^ a v

(40)

x

e

- \

'

*

a

'

R

av

<

4 , )

The total decay radiation heating must be summed up from

the partial heating terms according to the decay characteristics

of the involved nuclides, so that

(42)

Ρ = Σ Ρ + Σ Ρ

ndecay ndecayy ndecayß

The relations derived above apply to equilibrium conditions,

which are reached only after a time of several half lives

(19)

15

-3.0 ENERGY DISTRIBUTIONS OF GAMMAS AND NEUTRONS

The three types of parameters of the preceding sections :

Attenuation coefficients or cross sections, Geometry averaged reaction probabilities, Particle energy distributions

are direct or indirect functions of particle energies; the

energy distributions of neutrons and gammas therefore constitute the key to heating calculations.

Energy distributions of neutron flux densities in the HFR are in general available by 4 group computer codes of the neutron metrology group (RCN) .

The -for heat calculations- more important gamma energy distributions are unknown in the operating reactor.

The calculations of gamma heating terms are therefore based on a double exponential approximation of the energy distribution with two unknown linear constants.

3.1 Approximation of the Gamma Energy Distribution

The gamma energy distribution of the operating reactor is a result of the interactions of the prompt fission spectrum and the fission product decay spectrum with the materials of the core region.

In general, this distribution is a rather complicated

function of the configuration and can only very summarily be expressed by a two parameter approximation.

In the higher energy region (E>lMeV) the prompt and fission product spectra show an exponential form, approximated in ref.

- 1 1 E . 5 by a function e . Other measurements, mentioned in ref. 5,

-1 . 24E which include also capture gammas, suggest a form e

2 8

Since in the present case Al decay will cause appreciable increases of a distribution in the range E<1.3, one can

(20)

- 16

-2 3E

In the range E<lMeV ref. 5 suggests e ' for the prompt gammas. However, a great contribution of degraded higher energy photons must be expected in this region, the actual spectrum can therefore be expected to have an exponent <-2.3. For convenience, -2.5 is chosen; the distribution is therefore approximated by the expression

» /Ί?Λ Α *s -1.25E _ Φ δ -2.50E , . , .

Ν E « A . ·τ- .e + B .·=— e (4.3) _{ο ο E ο E} s s

The exponents - 1 .25 and -2.50 have the dimensions KeV , if E is taken Ln MeV. .

Φ3

The multiplication with ■=— is applied in order to make A and

fc-s °

Β non-dimensional, o

The gamma energy distribution could also be approximated by various other two parameter expressions, but in view of the presentation in ref. 5., the exponential form is preferred here.

3.2. Approximat ion o f the Neutron Energy Distribution

For the approximation of the neutron energy distributions, results of the TEDDI-KFR 4 group code are used as·, a basis. These results refer to the integrated energy spectral densities of the neutron flux in the form

ET T

(

Φη " I 0(E)dE , η = 1 . ..4 (44)

ELn

where the following energy limits are attributed to the group :

Φ] : E - 1 . 3 5 MeV ET J ] = 10 MeV

Φ2 : EL 2 = 6 7 . 4 keV Eu 2 - 1 . 3 5 MeV

Φ3 : EL 3 = 0.625 eV E = 67.4 keV

(21)

- 17

One can derive from the TEDDI results, that the group fluxes φ are roughly equally related to each other in all central positions and also, bun functionally slightly different, for peripheral positions of the HFR,

Th.es e two forms can be appro χ im? ted by :

Φ. _η

Φ~7

; + 0 , 7 0 ('η - ί ) - 0 . 0 5 ( η - 1 ) 3 (45)

Φ 1 + 0 .75(η-1) (46)

for central (4 5) and peripheral (4 6) positions, respectively. The characteristic'ratio

Φ. (47)

is by these approximations 1.51 for central positions and 0.85 for .peripheral positions.

For a given core position,, relation (45) or (46) can be used to approximate the neutron energy distribution in group representation by

'Un/

Ln

0 (E)dE » C _{o o}n A,

Φ7

■

φ3

'

(48)

where the factor φ is used for obtaining a non-dimensional form of s

(22)

18

4.0 SOLUTION OF TUE HEAT GENERATION EQUATION

·' 4.1. In sert ion of C a r. n a a n d Neu tron V. n c r g y Ώ i s t r i. b u t i o n s

The energy d i s t r i b u t i o n s of gammas and n e u t r o n s , represented by e q s . (43) and ( 4 8 ) , can now be applied to solve the various heat generation expressions of s e c t . 2 up to the u n k n o w n

factors A ,

o o o and C .

For this procedure the following insertions are made :

I) Eq. (43) in eq. (19) :

Ρ = A o'E l + (·— -l)x| .p„(x) .e

:]·«·.

1 .25E EdE (49)

+ Β .--_{ο E} ! + (—- -1)χ .ρ (χ) .e .EdE t . -2.50E _,_

II) Eq. .(48) in eq . (22) :

Eq. (22) is approximated by the sum :

2Λ

n n ( A+l )2 P

A

) .Σ . _s

n=!

rUn

'Ln

0 (E).EdE _o (50)

The integral of this expression can be solved by applying eq. (48) in the following form :

JUn

o n ETT

-Ε,-Un. Ln

C

0dE = ~ ~ 2

-o E„ - E

Φ.

n A

Un Ln ™1

( 5 1 )

(23)

19

w h i c h b r i n g s :

'Un

0 E d E = £ o E „ E Ln

.

~

.φ 4(E.

2

. E

T2

)=4c (Ε.. +E

T

)·Αφ

φ, Ts 2 Un Ln 2 o Un Ln φ, s

Un L n 1 1

(5'2)

and

p c .

-JL-

1

nn o / ... χ 2

E

Φ .

Ρ (ΧΤ)'Σ0 (Ε..+Ε).. — . φο (5.3)

/ . . \ ¿ Ρ / ο Τ s Un Ln φ, s

( Α + , )

5=1

. *

III) Eq. (48) in eq. (31 ) :

The thermal neutron flux density φ ·.. corresponds to the 4th otn

energy group, and therefore :

Φ Λ

φ . , = C . —— . φ

oth ο φ. s (54)

Za φ4

Ρ = C . — . ¡ ^ .ρ ηγ o p Φ]

7

(Χ ) Λ Β3— ( Ε . ) . ρ (χ) .Ε. .S(E.) .φ ο Λη / a y χ ι' 'c ι ι ¡

i=l

(55)

IV) Eq. (48) in eq. (32) :

Eq..(32) is approximated by:

Ρ . -C — (Ε ) . ρ ( χ ) .E

η ι γ o y Y c

Γ 4

■ ■•Γ

m= 1

Σ . φ "' ., ι J

— , ν Λ ί γ m .

ο Τ ρ φ j s ( 5 6 )

V) Eq. ( 4 8 ) i n e q . ( 3 7 )

y Σ ψ,

Ρ · , - C — (Ε ) . ρ ( χ ) . Ε . ν . — . -r-ί-.φ ρ (χ ) Β

ndecayv ο μ γ lc γ ' η ρ φ. s1 ο η' D a

(24)

- 20

V I ) E q . ( 4 3 ) i n e q . ( 3 8 )

Σ φ.

P J rt=C.p(x„).E„.y. . -— .φ .ρ (χ )

n d e c a y β o l c A g ρ ' η ρ φ γs ' ο Λη

(58)

4.2. Presentation of He a ting as Linear Expression with 3 Unknown Constants

By summing up the terms of sect. 4.1. to the total heat generation (eq. 1), an expression for Ρ is obtained, in which the unknown factors A , Β and C appear in linear combination, _{o o o}

r '

The resulting eciuation can be written:

Ρ » A . G + B . F + C . Ν _{o o o} (59)

G, F and Ν are short forms of the following expressions

G »

ψ

Es ) ρ

o

i + (— - D x P0( x ) . e ' " - . E . d E - 1 "> S F ( 6 0 )

00

Ι

ν

τ

H

rv

*--Dx

a

, . - 2 . 5 0 E „,_,

p ( x ) . e .EdE ( 6 1 )

N

(A

4

ΐ ' TT' 1 P I(X T )

+ 1 )

2 Ρ

¿_

° Τ

m= 1

Φ

m .Σ . (ETT +ET ) . ~ .φ

s Um Lm φ . Ts

Σ φ.

a τ4

Ρ » Φ £~ ·

7

Σ

" Va

Β — ( Ε . ) . ρ ( χ ) . Ε . . S ( E . )

a y ι *c. Α ι ι

i - 1

(25)

- 21

Β yn

+ — — (E ) .ρ (χ) .E p y γ c γ

4

m=l

Φ m

(χ

Τ>·

Σ

ηίγ·

f¡'

φ 5

¿Ι

μ Φ Λ

Β _{a y γ- c}— (Ε ί.ρ (χ).Ε .y .Τ. .ρ (Υ ) ~ . φ

Α γ η a *ο Än φ vs

'η . decayy

(62)

ί-E

Φ,

Ρ ο( χβ)·Εβ · ^ η -Ση · Ρ ο( χη)· φ~ ·φ 5

Ε _{η . decay β}

For convenience, the different arguments of the probability factors are once more listed here :

■Õ y R

2 av

2 ΣΤ Ra v

(63)

η 2 Et h T Ra v

ß

1

2 μ ß av

The expressions G, F and N ( 6 0 ) , (61) and (62) are functions of

the sample material and the sample geometry and size; they can

be determined theoretically.

For cylinder geometries, which are considered here, the influence

of size is expressed by the dependency of G, F and N on R

The dimensions of G, F and N are the same as the dimension·of

(26)

 22

4.3. Determination of Unknowns A , Β and C

o o o

The solution of eq. (59) for the total nuclear heat generation

now depends on the knowledge of the constants A , Β and C

'· o o O

of the gamma and neutron energy distributions.

A and Β are unknown; C can be obtained by the TEDDI code

o o ' o J

theoretically and by activation analysis experimentally; it

is however not advisable *·ο apply the sofound values, in view

of the complete loss of selfcorrection within the further

process.

It is therefore useful to consider all three constants as

unknown and to determine them simultaneously in a common

procedure.

For this purpose, the method applied in ref. I is used here

in a modified form. It- consiste in a measurement of the heat generation in 3 materials and solution of the system of 3

linear equations (59) with ¡the measured values P.

With the subscript* 1, 2 and 3 for the 3 materials, one has :

pi ;GiAo * FiBo *■ Mico

P2 V o * F2Bo * B2Co (64)

P3 * V o * F3Bo * V o

The determinant of the coefficients is :

ro,

F ,

₁

Q2 F, *2

C3 P3 H3

(65)

The solution can be presented in the following form :

1

(27)

23

-W 1 · I CN2G3-G2N3)P1 ♦ ( G1N3- N1G3) P2

C

N1G2 -G1N2 >P 3]

( 6 6 ) I tn τ, τ. „ ^ ,τ, n r* - N« ( G . F _ - F , G. ) Ρ .

o= Dc » ( G2F3 -F2G3) P1 + ( F1G3 -G1F3) P2 + * 2 ' 2 3

Once these constants have been determined, one can express

numerically the approximations for the gamma and neutron energy

distributions (43) and (48) for the conditions of the measurement

(core position, core configuration, etc.).

For the same conditions, the constants in combination with

eq. (59) permit to calculate the nuclear heat generation in

samples of any material, geometry and size, for which the

(28)

24

5.0. EXPERIMENTAL

5.1. Principle

The experimental determination of the constants A , Β and C

r o o o

according to sect. 4.3. requires simultaneous measurement of the specific power for 3 different materials, which are

exposed to the radiation at the same point of the reactor core. Since this is, in the strict sense, impossible, it is necessary to apply a separation in either time or space.

The principle of time separation requires use of calorimeters, which can be moved to the measurement position in succession.

The principle of space separation can be realized with fixed calorimeters.

In view of the rather slow changes in the radiation at a given point of the core, the time separation introduces practically no mistake, whereas the space separation can be subject to errors caused by local differences in the radiation pattern. These errors are difficult to estimate and can only be kept at a minumum by minimum spacing of the calorimeters.

The practical realization of the time separation principle (movable calorimeters) however, is rather difficult and intro duces special problems in view of the required sensitivity of

the calorimeters.

Since construction of rigs with movable probe supports is

presently an unproved technique for the HFR, preference is given to the somewhat less accurate solution with fixed calorimeters.

5.2. Sample Materials

(29)

25

In numerical terms, the determinant D (eq. 65) should approach a maximum value.

On the other hand, the sample materials should be suitable for construction of geometrically equal calorimeters with good sensitivity and limited temperatures, which is an important requirement reducing the risk of thermo-couple failures.

On the above basis,· the combination graphite-molybdenum-beryllium was chosen. Estimated heating ratios of these materials are

approximately :

Gammas : C : Mo : Be τ 10 : 14 : 9 Neutrons: C : Mo : Be * 1 : 0 : 2

The thermal characteristics of C and Be calorimeters are very similar, the temperatures in Mo are comparably high, due to Lhe higher heating rate and the greater density,

Cross sections and absorption coefficients are well known for all 3 materials; the specific heats and their temperature derivatives (source : ref. 6) as defined by.eq. (67) are listed in the

following table :

1 + q(T - T

o

)j

(67)

M a t e r i a l

Be C* Mo

ρ [ g / c m J.

1 .85 1 .69 10.21

co[ j / ( g d e g ) ]

1 .89 2 0 . 7 6 2 0 . 2 5 3

q [ l / d e g j

0 . 0 0 1 7 8 0 . 0 0 3 3 2 0.00026

To[ d e g C ]

35 35 35

T [deg C]

0<T<150 0<Τ<200 0<Τ<600

* graphite Carbone Lorraine type 3780 WEG

(30)

- 26

5.3. Test Rig

A test rig, which can be inserted into the relevant experi-mental core positions of the HFR, is designed on basis of

the principles of the CADO series, being in use at Petten.

This test rig, CADO-17, is a carrier for isothermal calori-meters, which are arranged at the 4 vertical stations shown on the scheme of fig. 2: (1) two groups of 3 calorimeters

(referred to as "group calorimeters") with graphite, molyb-denum and beryllium samples and (2) two single calorimeters with graphite samples.

The group calorimeters have capsule diameters of 6 mm, the single ones of 10 mm. The single calorimeters serve for com-parison with earlier measurements, for which the same types had been used.

The calorimeter capsules are of stainless steel with a wall thickness of 0.3 mm; the free space is helium filled.

The group arrangement is shown in fig. 2 too. The center radius of 10 mm is a compromise between minimum spacing (see sect. 5.1.) and the distance needed in order to reduce mutual influencing

(shadowing) of the calorimeters.

The calorimeters are cooled by the primary coolant of the HFR, flowing in downward direction through the rig. The rig is in-serted in a standard filler element.

The inner and outer thermo-couples of each calorimeter are

connected in differential junction in the head of the rig.

The instrumentation is a Solartron digital data logger with

(31)

27

5.4. Calibration and Sensitivity

Calorimeters are calibrated outofpile by derivation of

the temperature dependent time constant of the differential

thermovoltage, developped in response to a sudden decrease

of the cooling temperature.

The first order approximation for this time constant reads

M = K ·(1+βΔΤ) (68)

With this, the differential equation of the calorimeter res

ponse

dAT dt

AT M

has the solution

AT

AT = exp M, + ß(AT AT) (69)

Measured response curves of CADO calorimeters are perfectly

represented by this relation, so that the parameters M0 and β,

defining M by eq. ( 6 8 ) , are easily determined from the measured

response on basis of eq . ( 6 9 ) .

A steady state calorimetrie measurement is evaluated, the time

constant being known, from the following equation:

p

="Fk

· π ·

ΔΤ (70)

The correction factors f and k, allowing for the influence of

the inner thermocouple mass and heat capacity and the heat loss

by the sample supports, respectively, have been calculated;

(32)

- 28 Calorimeters type single group M II Samp le material C C Be Mo f 0 .985 0.985 0.975 1 .004 k 0.043 ' 0 .024

0.024 0 .024

The following table shows predicted mean values of the time W

constant, the temperature difference developped per /g of heat generation and the sensitivity of the calorimeters,

defined as the output in millivolts (thermo-voltage) per /g of heat generation.

Calori-meter type s ingle group group group sample mate-rial C C Be Mo capsule i . d.

(33)

29

5.5. Accuracy

According to an analysis made in connection with earlier experiments, the expected mean error of CADO measurements is of the order of 4%.

Since the same calor ime tei type is used for the present method, it is reasonable to consider this value also as the mean error of the individual calorimeter output.

The total error of the method, which is the error attributed to the final result A , Β , C , must be evaluated on basis of

o o o

eq. (66), in which the calorimeter outputs are combined with the elements of the determinant D (eq. 65).

The accuracy of these elements is rather difficult to estimate, because it depends very much on the unknown quality of the

approximations for the gamma and neutron energy distributions.

(34)

30

-6.0. NUMERICAL CALCULATIONS

6.1. Expressions G, F and N

The expressions G, F and N of eq. (59), which are formulated in eqs. (60), (61) and (62), have been evaluated for the mat eri als

C, Be, Mo, Fe

and for cylinders of the dimensions

0 < R < 6 cm. _av

-Eqs. (60) and (61) were integrated graphically by means of a planimeter, with upper integration limits of 8 MeV for G and 4 MeV for F.

Absorption coefficients were taken from refs . 5, 7 and 8.

N has been determined on basis of the mean of the neutron group energy distributions (45) and (46), reading

Φ

— = 1 + 0.70 (η - 1) - 0.03 (η - Ι )3 (71)

Φ1

Cross sections were taken from ref. 4.

The n»Y contributions were evaluated on basis of the 7 group capture gamma spectra of ref. 2.

Numerical results are presented for the standard values of

E = 0 . 1 MeV _s

φ = 1 0 cm sec _s

(35)

31

It should be noted, that numerical values of A , Β and C , o o o which are obtained experimentally and by inversion of the

linear system (64), are only valid for parameters G, F and Ν with the above mentioned standard values of E and φ .

s s 6.2. Determinant D for CADO-17

c

The samples installed in experiment CADO-17 are cylinders of 3.2 mm diameter and 20 mm length; their corresponding value of R (eq. 18) is _av

R = 0 .296 cm _av

The determinant (65) is set up by coefficients G, F and N, which are read from figs. 3 through 5 for this R . The in

av correlation used here is 1 for C, 2 for Mo, 3 for Be :

D = c

2 . 6 6 7 3 . 4 9 1 2 . 3 2 0

0.757 1 .443 0.644

The value of D is : _c

D = + 2.143 W3 g"3

c °

I .470 0.240 3.110

(72)

(73)

On basis of this figure, eqs. (66) are evaluated to (P in Watts/gram) :

A = + 2.022 P„ - 0.657 P„ - 0.905 PD

o C Mo Be Β - - 4.806 Pn + 2.279 P„ + 2.096 P_

o C Mo Be C = - 0.513 P„ + 0.018 P„ + 0.563 P_ , _{o C Mo Be} (74)

where the indices C, Mo and Be stand for the corresponding materials.

Since physically meaningful solutions require that Ρ > 0 and Ρ > 0, there exist the conditions

γ η

-A G + Β F > 0

o o

-C > 0

o

(36)

- 32

By insertion of eqs. (74) into (75) and use of values G and F of (72), one finds a field of physically meaningful specific power ratios, which is shown in fig. 7 as the area limited by the straight lines (a), (b) and (c) .

However, in view of the expected gamma energy distribution, only a part of this field can be realistic.

The actual gamma distribution of the HFR has in all probability a form, to which both terms of approximation (43) contribute with positive sign, so that the expected field of solutions has the sharper limits

A > 0, Β > 0, C > 0 . (76) o o ' o

-This field is, on basis of eqs. (74), the triangle I-II-III (fig. 7 ) , in which point I corresponds to a gamma energy

— 125E — 2 50E distribution type e (hard spectrum), point II to e

(soft spectrum) and point III to pure neutron energies.

Accordingly, on a line connecting two points, the contribution represented by the opposite point is zero.

It can be expected, that actual HFR results will fall close to the line I-II, because the neutron effects are relatively small.

The results of measurements at different points of the core combine to a curve or a field, the magnitude of which depends on the variation of the spectra over the core.

ACKNOWLEDGEMENT

(37)

33

7.0. DEFINITION OF SYMBOLS

A a tomic we igh t

A ,B ,C constants of energy distributions, eqs. (43)

Ο Ο Ο > -ι - '

and (48)

B energy absorption build-up factor

D diameter

D determinant of coefficients, eqs. (65)

E energy of gammas and neutrons

E standard energy, arbitrary definition

F defined by eq. (61) F fraction of collisions _c

G defined by eq. (60)

L length of cylinder

M time cons tant

N defined by eq. (62)

N energy spectral density of gamma number flux (eqs . 3 ... 8)

N energy sp. dens, of gamma number flux, undisturbed

Ρ specific power of nuclear heating

R average chord

av b

S total surface of sample

S(E.) integrated capture gamma spectrum, defined by eq. (30)

Τ temperature Τ reference temperature _o

r

ΔΤ temperature difference corresponding to U

U„ differential thermo-voltage of calorimeter

(38)

34

c specific heat

c reference value of specific heat _o

f correction factor

k correction factor

n total amount of photons coming in

O f b

ρ escape probability

ρ volume average of ρ

ρ volume average of collision probability, eq. (28)

q temperature derivative of specific heat

q energy spectral density of capture gammas produced per capture

r local coordinate

y natural abundance _'n

Σ absorption cross section _a

r

Σ scattering cross section _s

6

Σ„ total cross section

Σ,, „ total cross section for thermal neutrons _{t h 1}

E . cross section for inelastic scattering gamma _ηιγ_{e &} ray production

0 energy spectral density of neutron number flux

0 energy spectral density of neutron number flux, undis turbed

μ gamma ray attenuation coefficient

u gamma ray energy absorption coefficient 3.

μ„ beta ray absorption coefficient

Ρ

σ

density

(39)

- 35

oth

n

neutron number flux

neutron number flux, undisturbed

undisturbed thermal neutron number flux

standard neutron flux, arbitrary definition

neutron number flux in energy group n, eq. (44)

defined by eqs. (63)

(40)

36

-REFERENCES

j_lj H.J. Leyers

Die Neutronen- und Gamma-Aufheizung in Experimentierkanälen des Reaktors FRJ2. Jül-351-RX Juli 1965.

[2\ E.P. Blizard (editor)

n d

Reactor Handbook, 2' edition Volume III Part B: Shielding

Intersci. Pubi., 1962.

p ] Case, Hoffmann, Placzek

Introduction to the theory of neutron diffusion Volume I.

Los Alamos Scientific Lab, June 1953.

(_4J Hughes, Schwartz

Neutron Cross Sections.

BNL 325, 2n d edition, July 1958

[f\ H. Goldstein

Fundamental aspects of reactor shielding Addison-Wesley Pubi. Comp. Inc., 1959.

(_6J Landolt-Börns tein

Kalorische Zustandgrössen .

Band II, 4. Teil Springer, 1961.

[7] US AE C

Reactor Physics Constants

(41)

37

[β] C.M. D avi s s on

Gamma Ray Attenuation Coefficients Append ix 1 o f :

Alpha-, Beta- and Gamma-Ray Spectroscopy Vol . 1 ,

(42)

O.Ol 0.02 0.05 O . I O .2 O .5 10

2'X for χ>10

Ρ - 1 - TX for x<0.01 _ο_ό

(43)

39

-CL core

[image:43.595.60.547.61.710.2]

(1) group calorimeters (2) single calorimeter

(44)

40

w _{/ g}

! ! I : I ' ! ; 1 ] ! '

, , ι 1 1 M i l

— —

l i ' i ' ' . ; | ' 1

; ' ■ * " i ' ¡ ! ' | | |

■ ■

ι ;

1 1 1 1 ¡ M ' Ι ι ! ι t ! ι i I l M M M M !

1 ' _ L L L * M Φ

M ! Τ "¡ ' 11 I

1

1 1

! 1

!

| | ■ j | ; ι | | ι | ι ι ■ ■ " " " "t " I I " I I I [

i I I ' l l ! j I 1 ; l ' l ' " ' j | | | | | | ι ,

| : ι ι ι ' i ι j

i i

N ' ! ' ! ' : ! ι 1 ! | ι :

:

I

! M ! 1 ! ! M l

I

1 ¡ ! ' ' ■ ! '

^L

M * s ^ " S^ ; "

^FÇT" ^ ^ **ìml _{^ ^ « k a}

N ^ l

^ ^ ^ * K _ I ' M ' ' M | ι | | ^ Í * * L L

i ! ! i t ι ! I i 1

I I 1 M ! l 1 1 — ì — Μ ι1— H ~ + ^ — M i1

" M i M M ■ ι Π — M ' M l — h ~ ^ I I I ! M h M i ' ' M !

I ' l l ι l M Ι Ι M — I 1 Η t ι ! i ι 1 i j I ~ r t

—U—Η ι IfrHi ! f

! I ! ! ! Ι ! Ι !

—

( | ¡ ι M j i i i

I

l ι

± ± " 1 I 1 : ■ ■ r_ r

' M

+

H p

M f

1 ' !

¡M

MM

1 ! ' i I Ι ι !

M M M

MM

M M Ι Ι M ! 1 ! !

I ! M

t ' ) '

_| r —

; ! ι : ι I ; t

|M I

! ! ! I

; i ! .:..,

• ■ 4

"1 r 4

M l

ι1! !

I M '

i '

M 44 *--■ -ι ι

Μ ! ■

|

-iiii

: 4

. . 4 (

" *4

1 ! I

« . ι ι ! ! i

HM

—ι—ι—¡—h : < !

4 4

M

: ! !

1 ■ ! 1

! ' ! ! |

— ■ H

ι ,

■ —t f —

I i i i

■ I ■ '

""Hrrr

I I I M ; '

j ! 1 1 M i l

! ! 1

! >

1 ' ^ r · 1 ' . 1

Ï I ¡

Τ

x*~t

! 1 ■ j

ΧΓΤ

ΗΚ₁

ι r"TT

MM

! !

ι ! ! 1

M 1 1

! 1 ' Ι ι ! M l

r|+h

ΜΗ

M M

ι l M

M

M l

! M

i ■ ■ 1

M i

M

1 i l i ' ■ ι I

Hi ) 14 '

; ! ! ι ι ι t

Ι ι I ( ' t

! i I

!

1 '

■ i i i Ι Ή "

-H-j-- -4-4 44T4M"" + M t t T f

1 i

M - ^

! I ! Ι ι

ι ! ι | ι ι

MM Ι

' ' ! Ι : : ι ι ;

ι ι

Μ Γ- Ι 1 ! ! I I

t f l ^ -

ι ' ! l ' i

! ! ! ι

! "Ρ"

! ' '■ ! I I

i Ι ■ ■

^ — — τ —

Μ : : — — —

' i l ' Μ ¡ l ' i ι ! Μ ! Μ

! : ! Ι .. ' ~+_' '"

.

+4—

ι

Ι !

!

ι

| Ι

; χ _ .

,

—

-r-f-:±

' T T

Ι ! !

Τ"

i

ι Ι I j

! ' ! Ι

: i ' Ι

I ' l l Ι i ι

M-J-T-π Μ Ι F Ν

,.+14 2 1 • = 1 0 cm s

Ts

E = 0 . 1 MeV s

R , av

cm

(45)

41

W /g

ι

; I

I

I —l·

! ι

|

. V 1

\

1

|_

- I

l | 4 LL. i—j ■tT I

J 1

1 \

1 ι

1 i 1

| ι

1 ■

Ι ι ; * j

i I I ,

! ! ! ! I l M i ι M i l i I M

ν .

1 \ l i l i

1 SÍ i !

i i ι \ i ■ ι 1 . r\ ■ i i < M I ι i ¡ I

1 M l

]

M M

M i M Ì

MM

11 MM M MM

I M

ι M i l

ι ' MM M MM

l i Μ ι ι ι I M

1 M i l ι i I M

M I I

! ί Ι ί I i Ι i I

I

i I ι

M 1 I

M Í

± χ : I I !

u, i i , , :

1 | ! | '

| | , , | | M l ι ! ! 1

i ' l

M

■ j ; : i ! ! 1 i M i i I M M

' ' ' i 1 I

I I 1 i j l

M i i i ! Ili Ii ! II

M ! I l

kJ ι I I

: ^ %! 1 i i x IM ι ¡ ι \Λ i¿ ι ■

~1~' ' —Μ—^

I I I i i

I l ' i I M

1 M I ■

ι Ι ;

I M I i l ' '1 I ι,

MM ■ ' ! 11 M i l l

M l ■ ι 1

M i M

Μ 11 i

i i i i M

I j ' j M _ 4

1 . I l

Ì 1 M l

I I

Μ I

ί 1 1

M I M L i

-• -M

·■ — ~ H ~ ^

-ί 1

1

1 MM

ι ι ; |

|

^ ^ J 1 ^ ^

. ^v

; ι ■

1

t Mi

j

í i '

11

i

. i ι ι

M _.:,.',

i i i i M

M ι M

I

i ! M i

. , ¡ M l ι

I ! j . 1

T V ι ^ V f c .

¡ I I i Γ M l I ' ; ! Μ

I ' l l I I

i ι ι ι ¡ ι

MM !

l i ' ' i l ¡M ¡ i l i l

T V ^

MI M i Ι ι i ¡

I I I ι ί I l i i I 1 | l ' M ! 1 Μι 1

i l ι

I I __LU_J__

MM

¡ I M ι M

MM

; 1 1 , 1 1 1

1 ■ ι

■ ι ί !

¡ i l l

l i i .

ι 1 1 1

ι,. ,

y ^

' MM

1 1 , , :

1 i l l '

i ι ι i ■ Μ ι I ι

Ι i l

I ¡M

ι l i

Μ 11

| M«——

" " '

_ ^ i 4 .

M l

i 1 Ml

¡ s I ! I

M i i M M .

i 1 j ι ι

;

—

! ! M

1 l 11

rsü

Μι

MM.

f f f 7

¡ ¡ ! I ' ¡ l i l i l í

I !

ι I i

—

• ' Μ T t M ^

1 Ι

i i i i .

■IM i l i

* ~ " ~ ' * ■ : *

ιΜ"Π

J L i

i i i i I

ι ι i M M

ι .

M ¡ ι M M

-4-.M_M

Ι ι ι ! i

— . i l l

I , ¡ : I

I I I

Ι ι 1

Γ ι ι

j M i

ι ! , ι 1

■ M i !

¡ i ' l ί 1

^ ^ 4 j

ι ■ . I 1

M I M I

M i l i

! M ¡ i ; !

i ' ! ι

j ' ¡ i l 1 i i i

i ι ι M i l i 11 ¡ I

I INJ

M J — ' M

HtlÎUt 1 i I M

l ι

Ι Μ ι

i M l i i i M M I

1

i l I I

M

M II i l ' , ! ι ι 1 i | | : ' l ι ■ i

j M i l ! 1 M

4 + + ^

M

1

^

:

| ί i ■ ■ I

i i Μ , . 1 i i i i 1 ¡ i l ! i i i ¡ 1

in+14 2 1

φ = 10 c m s s

E = 0 . 1 MeV

av cm

(46)

42

-W /g

-- - - M

\ I

' A i

' l\l

\

ι ι M

Μ ί ι

i l l !

j ¡ |

IJl '1 1 1

i !

M i

M ^ h ι 1 \

I ι

n 1

A

— ·

X ■ ! \ i

]"v

\

ι 1 1

' I 1 ; ι i i

i ι M

|

; M !

' I M

11 i

I ! T>

ί ι 1

— 1 —

Γ

t

. . i

-*

\ t , ■

Ι ί X L '

Μ ' Γ Ι ι ι 1

j j Ι ι

! i

I !

' ! Ι

IMI

M M !

M i i

■

* < N ^

Γτ^-L, 1 ; Γ τ - . .

; , ...

■

ί ί ! I

■ ;

L

| i M

S L , | f , l | |

—

1 M

i l 1 1 ' ι

! M

i l i i '

i l I M ,

1 ¡ M Μ ι i IM

" υ * 1

"NM

± 1 ΰ

i l l

1 ¡i I

1 ' ■

^s

I ' ι

l ' i

1

1 "

1 ! | |

μ

t

—pr Γ" ' "

i i i i l l

ι | '

ί Ι M

Ι ι M

_ , „ . _ . .

MM _{l i i} I l i !

M

! : M

ι ι Ί

J I M

11

1 i 1

ti-liii.

1

i___

1 ι - 4 - .

i 1

ι i M

I Ì ι

r4

I

Γ" _L

M M M :

~~

—

I 1 I

ι ι ■

M M

^ ^ ^ 1

ι M i

I j M

I I

I ! | |

i

i t

—i .

ι

|

,

'

■ ι 1

ι ■ [

l'I

M . I

M M

. ι 1

ί 1 ! I _{I l i}

■ ■ M

I I

■ ! i ι

I ¡ ! ' !

i l

M

ι

\

I l 1 i

l i

1 I 11 1

:

I

11

■ -

-¡ I I I

. i l l

1 M '

M M

! M I 1 j | | ι M

MM

M

¡ i i i

Ι 11 1 '

, ^_ ρ

ι

I ¡

I

| 1

I

ι 1

^ * i ρ

Ι ι

-il

χ I

11 i

I | i

r-— , r

j '

M i l

I ' M ! i I ;

i ■ ■

1

■ttft

I ' M

¡¡li

ι i > 1

M I I

1 ! ! ι

ι :

Ml

I ' M

t i l l

ι 1

M ■ *

Ì ι 1

|

N

in+14 2 1

φ = 1 0 cm s s

E = 0 . 1 MeV s

av cm

(47)

43

-W _/ε

F

Ν

,_14 -2 -1 φ = 1 0 cm s _s

E =0.1 MeV _s

R , cm _av

(48)

44

Mo

τ ^ τ η j j I M I I I J , ι , ι ι I I I I Π r

y ;\ : ■ "_ M i " ' ;: μ~"1 ;M_ ■· ■

, Ι Λ

M M ] M y " τ ττ.:Μ . M t M TMTJT ττ. :[ | T ¡ T : : T T τττ j j

Λ ι

i| ^ ^ P

:

ι V

3 ί r

:

-|

::

- M-' i

Ül τ 4 M ÏÏΜι

ΤΓΓί-

¡ 4 Í 4 4

.s.

L

. ..: ! \ : ' . ; . . : ! . ¡4 .

ι

..

N?¡

_S

M l M M t ' f ΜΜτΜ ..].

_{, \ j} _{; ; ι} _{¡ M'}

[ΜΜΤΤΜΦΦΜ

_{■ '} _,,.

J ,.. t..._

S l x

.Μι

--- ->

s

- M

.X^

[, m.|--...r - - -

¡ι

T 1 ' 4 : M L | 4 T 4 '. MMT ::::. M

■"~<η>ΙΓ ί ^ Ν ι ΐ:Γ * y - - - - - yHfV- - \ ·

f" M-4"M. Jf -|-:::::M

S

-^M- rz:4:: ·. MM*!

i%

' ~-

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