- Email: [email protected]
The measurement of the radial thermal neutron density distribution in tubular fuel elements
Journal of Nucleiu Energy Parts A/B.
1964. Vol.
18, pp. 464 to 470.
Pergmon
Press Ltd.
Printed in Northern
Ireland
THE MEASUREMENT OF THE RADIAL THERMAL NEUTRON DENSITY DISTRIBUTION IN TUBULAR FUEL ELEMENTS* V. F. BELKIN, B. P. KOCHUROV and 0. V. SHVEDOV (Received 14 March 1963)
Abstract-An experimental study has been made of the radial thermal neutron density distribution inside tubular natural uranium fuel elements containing the organic coolant-monoisopropyldiphenyl. Using the activation method, the thermal neutron density distribution has been measured, in 23 different versions of the fuel element. Theoretical neutron density distributions were obtained for these fuel elements by solving the one-velocity kinetic equation on an electronic computer. The experimental and theoretical results obtained for the neutron density distributions, shielding factors and the neutron density discontinuities at the outer coolant layer are presented in the form of graphs. INTRODUCTION THE present tendency in reactor design towards raising the coolant temperature without increasing the pressure in order to obtain a commercially profitable reactor has led to the use of low boiling-point organic liquids as the coolant. The heat engineering and constructional requirements of systems involving natural uranium (used as the fissile material) and an organic coolant have led to the fabrication of the multi-layer type of fuel element to which the calculational methods evolved for simpler types of fuel elements(lp2) cannot be applied. Consequently, it has become necessary to employ more complicated calculational procedures which need to be verified experimentally. EXPERIMENTAL
ARRANGEMENT
The thermal neutron density distributions were measured in mock-up fuel elements (in assemblies) having a length of 75 cm. Each assembly consisted of the following parts (Fig. 1). A sleeve of natural metallic uranium I (density 18.7 g/cm8), the external and internal surfaces of which were clad in aluminium sheaths 2,3. In the centre of the assembly was inserted a magnesium thimble 4. The inner layer of organic coolant, which was monoisopropyldiphenyl 5 (density O-986 g/cm3), filled the space between the outer surface of the magnesium thimble and the inner sheath of the uranium sleeve. The outer layer of monoisopropyldiphenyl 6 filled the space between the outer sheath of the uranium sleeve and the avial shield tube 7. Between the external avial cladding of the assembly 8 and the avial shield tube there was an air gap 9. (In some experiments the monoisopropyldiphenyl was replaced by water.) The following dimensions were the same for all the assemblies: the internal diameter of the uranium sleeves was 30 mm, the air gap was 3 mm, the thickness of the external avial cladding was 2 mm and that of the avial shield tube and the cladding around the uranium sleeves was 1 mm. The assemblies differed in respect of the following dimensions: the external * Translated by D. L.
ALLAN
from Afumnaya 464
Energiya
15, 377 (1963).
The measurement of the radial thermal neutron density distribution in tubular fuel elements 465
diameter of the uranium sleeves, the diameter of the magnesium thimble, and consequently also the thickness of the inner layer of monoisopropyldiphenyl, the thickness of the outer layer of monoisopropyldiphenyl. Each assembly was given a code number; this indicated the external diameter of the uranium sleeve, the thickness of the outer layer of monoisopropyldiphenyl and the diameter of the magnesium thimble. For example, assembly 54-1.5-20 had a uranium sleeve with an external diameter 54 mm, the thickness of the outer layer of
1.4
I.3
I.2 Y) C 5 2 E I.1 d
I.0
0.9
R,
mm
FIG. l.-Thermal neutron density distribution q along the radius R of the assemblies (the curves represent calculated results; the points are experimental results): (a) for assembly 46-2-M; (b) for assembly 46-2-16; (c) for assembly 462-Y; 0, Oexperimental values for assemblies 46-2-M and 46-2-K respectively. monoisopropyldiphenyl was 1.5 mm and the diameter of the magnesium thimble was 20 mm. If no thimble was there then, depending on the filling of the internal cavity, the letters V(air) or M(monoisopropyldipheny1) were substituted in place of the last figure in the code. The neutron density distribution was measured using detectors 4.2 mm dia. and thickness O-4 mm, consisting of a mixture of plastic and dysprosium oxide (density -3 mg/cm2). Grooves were milled into the uranium sleeves into which uranium prisms containing the detectors could be inserted during the experiment. Those slots in the prisms not occupied by detectors were plugged with small slabs of uranium. The remaining detectors were placed in recesses in the aluminium cladding and tubes and also in grooves in the magnesium thimble. The disposition of the detectors can be seen from the position of the experimental points in Figs. 1-3. The activity of the detectors was measured using automatic eight-channel counting
466
V. F. BELKIN, B. P. Kocr-maov and 0. V. SHVEDOV
0
5
IO
15
20 25 R. mm
30
35
FIG. 2.-Thermal neutron density distribution along the radius of the assemblies (the curves represent calculated results; the points are experiment results): (a) for assembly 50-25-16; (b) for assembly 50-2.5-V-24; (c)for assembly 50-2*5-V, A, 0 are experimental values for assemblies 5O25-16 and 50-2*5-V, respectively.
Ft..
mm
FIG. 3.-Thermal neutron density distribution along the radius of the assemblies (the curves represent calculated results; the points are experimental results) : (a) for assembly 54-l .5-M; (b) for assembly 54-1.5-16; (c) for assembly 541.5-Y; 0, O--experimental values for assemblies 54-l .5-M and 54-l .5-V, respectively.
equipment. The assemblies were irradiated in channel A-O of a heavy-water reactoP in a flux of lo0 n/cm2/sec. Since the internal diameter of channel A-O was ~100 mm, an air gap existed between the external avial cladding of the assembly and the heavy water. While being irradiated, the detectors occupied a position approximately coinciding with the centre of the reactor core. The cadmium ratio of the dysprosium was not less than 100, so that the detectors registered only thermal neutrons. The temperature of the moderator was 293°K. EXPERIMENTAL
.RESULTS
Figures l-3 present typical thermal neutron density distributions along the radii of the assemblies obtained experimentally and by solving the kinetic equation on the computer. Assemblies whose neutron density distributions are included in the same figure differed only in respect of the thickness of the inner layer of monoisopropyldiphenyl. In some versions the magnesium thimble was absent, the internal cavity of the uranium sleeve being filled either with monoisopropyldiphenyl or air.
The measurement
of the radial thermal neutron density distribution
in tubular fuel elements
467
Table 1 gives values of the shielding factors Q* obtained by various methods: (i) from the experimental results of this work; (ii) from the results of a numerical solution of the kinetic equation; (iii) on the basis of data previously determined.(1-12) Table 2 presents the neutron density discontinuities, /I, defined as the ratio of the neutron density at the external surface of the outer layer of monoisopropyldiphenyl or of water to the neutron density at the surface of the uranium sleeve. Since the experimental conditions did not permit detectors to be placed on the external surface of the outer monoisopropyldiphenyl layer, to determine ,!I, the experimental values were corrected in accordance with the neutron density curve obtained at the time of solving the kinetic equation. The errors associated with the distributions and neutron density discontinuities, including the errors made in setting up the detectors, were l-1.5 per cent. TABLE l.-SHIELDING FACTORS
Q Type of assembly Experiment 1.279 1.251 1.239 I.227 1.218 1.199 1.187 1.176 1.166 1.162 1.135 1.130 1.125 1.120 1.113
54-1.5-B 54-1.5-24 54-l-5-20 54-1.5-16 54-1.5-M 50-2.5-B 50-2.5-24 50-2.5-20 50-2.5-16 50-2.5-M 46-2.0-B 46-2.0-24 46-2+--20 46-2.0-16 46-2.0-M
Calculation 1.261 1.248 1.242 1.238 1.231 1.197 1.188 1.183 1.179 1.147 1.137 1.134 1.131 1.119
From data previously determined.‘l~*~ 1.271 1.216 l-164 -
The calculated neutron density distributions were obtained by solving the onevelocity kinetic equation for the multi-zone cell of Wigner-Seitz. The calculation was performed using a programme compiled by F. M. Filler and T. D. Bogdanova. The dimensions of the equivalent cells for each of the assemblies were determined by some preliminary calculations which stipulated that the neutron density distribution in the inner layer of monoisopropyldiphenyl should be close to the experimental distribution. In addition, the sources of thermal neutrons in the monoisopropyldiphenyl were taken to be 4.77 times larger than those in D,O. It was assumed that the neutrons had a Maxwellian distribution with an average effective temperature T, which depended on the co-ordinates. The temperature of the neutrons in the moderator TnM was taken to be equal to the temperature of the heavy water T,O and, on the basis of some preliminary calculations, the temperature
* The value Q-l =
s
2 q(r)r dr
q(RzI~~ the uranium sleeve.
, where R, and RI are the external r dr
and internal
radii of
V. F. BELRIN,B. P. Kocm~~ov and 0. V. SWEDOV
468
of neutrons in the uranium was taken as T, u = 1.3 Tno.(4y5)With these values good agreement was obtained between the calculated and experimental values of the shielding factors. M of the monoisopropyldiphenyl depends fairly The scattering cross section ZZtr strongly on the neutron temperature. (YUROVA et cd.(‘) measured this dependence TABLE2.-NEUTRON DENSITY DISCONTINUITIES b AT THEOUTERLAYERS AND WATER OF MONOISOPROPYLDIPHENYL Type of assembly 541.5-B 54-l-5-24 54-1.5-20 54-1.5-16 54-15-M 50-1.5-B 50-l .5-24 50-2.5-B 50-2.5-24 50-2.5-20 50-2.5-16 5&3.5-B 50-3.5-24 50-3.5-M 50-3.5-B* 50-3.5-M* 46-2-B 46-2-24 46-2-20 46-2-16 46-2-M 46-2-B* 46-2-M*
B -____ Experiment 1.270 1.292 1.284 1.262 1.268 1.300 1.275 1.391 1.411 1402 1.398 1.470 1.464 1.473 1.527 1.535 1.306 1.305 1.282 1.290 1.280 1.349 1.294
Calculation 1.288 1.284 1.282 1.280 1.276 1.272 1.383 1.378 1.375 1.372 1.483 -
1.288 1.283 1.279 1.277 1.262 -
Thickness of layer (mm) 1.45 1.45 1.45 1.45 1.45 1.55 1.55 2.55 2.55 2.55 2.55 3.50 3.50 3.50 3.50 3.50 200 2.00 2.00 2.00 2.00 2.00 2.00
f f f & f + f + f f f f f f f + + & i & f + f
0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
* The outer layer of monoisopropyldiphenyl was replaced by water.
experimentally, finding &TMN u1~s8~~12over the temperature range 18-250X!.) It follows from the preliminary calculations that when T,” = Tnu = 1.3 T,” the values of ,!?obtained are very much underestimated. Satisfactory agreement with experiment is obtained if the value of T,” is taken to be close to T,“. The magnitude of /3 depends weakly on the dimensions of the equivalent cell and on the filling of the internal TABLE S.-DEPENDENCE D,O
OF ,8 ON x:
AND ON THE THICKNESS t OF THE
LAYER FOR A CELL CONTAINING ASSEMBLY 50-25-20*
I$, cm-l r (cm) 5 6
1.81
2.00
2.20
2.26
2.50
1.311 -
1.336 1.341
1.361 1.367
1.375 -
1400 1405
* Xg (293°K) = 2.32 cm-l. cavity of the slug with monoisopropyldiphenyl. Some of the results of the preliminary calculations are given in Tables 3 and 4. To find the effect of the layer of aluminium between the uranium and the outer layer of coolant, a number of calculations were carried out both for the case where
The measurement
of the radial thermal neutron density distribution
in tubular fuel elements
469
the layer of monoisopropyldiphenyl came right up to the uranium and for the case where the aluminium was replaced by an air gap. As can be seen from Table 5, @increases slightly in the first case and decreases slightly in the second. It should be mentioned that in the case of hollow sleeves some increase in the calculated neutron density is observed towards the centre of the assembly. For example, the calculation for the cell containing a 54-1.5-V assembly in which, instead of leaving the space inside the uranium empty it was filled with a very low cross TABLE 4.-DEPENDENCEOF
Q AND
BON
T,U FOR A CELL CONTAINING
ASSEMBLY
Q
50-2.5-V
B
T,u zz.Tn0
Tnu = 1.3 T,O
1.230
1.197
Tnu = TnO
Experiment
T” n=
1.424
1.199
1.3T”n 1.383
Experiment 1.391
section material (2 = 0*0025), showed that near the axis of the assembly the neutron density rose to a value roughly 12 per cent higher than the minimum value (see broken line curve in Fig. 3). It is reasonable to suppose that this effect is due to the anisotropy of the neutron angular distribution@). TABLE 5.-EFFECTOF
A LAYEROF
ALUMINIUMON
/?
Assembly Calculation 46-2-V 1.288 1.301 1.276
With aluminium Without aluminium Air gap
[email protected] V ~1.383 1.401 -
54-2.5-V 1.290 1.306 -
For the assemblies which had no coolant inside the uranium sleeve, the shielding coefficient Q could be found also from the formula”F2) Q=l+$?t
[
l+++~(g)~].
2v” Here A, u, y are the tabulated functions of X”. SE, The calculations were carried out using the following cross sections (cm-l): z;y = 0.392; Er
= 0.0842;
XMg = 0.1546.9 if” = 2.26;
cpzo = 0.397;
;sy = 0.275 ; EA1 a = 0.0108.7 z”g = 0*0021* & = 0.011;’ z,D2o = 0.001.
The absorption cross sections were taken averaged over the Maxwellian spectrum a = Sc,(&e-” dx (1 ’ fxe-n dx where x = EjkT,. We note that in the measurements [See, for example, DEUTSCH(~)], as also in the calculations with the assumed averaging, the neutron density distribution is determined
470
V. F. BELKIN, B. P. K~CHUROVand 0. V. SIWEDOV
i.e. the distribution of the quantity J n(E) dE. Therefore, when using the results obtained to calculate the coefficient of thermal neutron utilization, one should take absorption cross sections at a fixed temperature, for example Tnu, and apply temperature corrections to allow for deviations from the l/v law. For the assemblies containing uranium sleeves with external diameters 4.6; 5-O and 5.4 cm, the thicknesses of the heavy-water layers were 5 ; 5.5, and 6 cm, respectively. The experimental results and their comparison with the results of the calculations show therefore that the experimental neutron density distributions in heavywater reactor fuel elements containing an organic coolant are in agreement with the results of calculations made on the basis of the one-velocity model T,” = l-3 T,O, T,,M = T,,O. REFERENCES 1. GALANIN A. D. Neutron Physics, p. 125, Gosatomizdat, Moscow (1961). 2. AMOUYAL A., BENOISTP. and HOROWITZ J. J. Nucl. Energy 6,79 (1957). 3. GONCHAROV V. V. et al. Proceedings of the Second International Conference on the Peaceful Uses of Atomic Energy, Geneva, Vol. 10, p. 321. United Nations, New York (1959). 4. BLAGOVOLINP. P. Neutron Physics, p. 56, Gosatomizdat, Moscow (1961). 5. MOSTOVOIV. I. et al. Proceedings of the Second International Conference on the Peaceful Uses of Atomic Energy, Geneva, Vol. 16, p. 254. United Nations, New York (1959). 6. DEUTSCHR. Nucl. Sci. Engng. 10,400(1961). 7. YUROVAL. N. et al. Atomnaya Energiya 12,331 (1962). 8. C&E K., HOFFMANF. DE and PLACZEK. G. Introduction to the Theory of Neutron Dijksion-I. Los Alamos (1953).
1964. Vol.
18, pp. 464 to 470.
Pergmon
Press Ltd.
Printed in Northern
Ireland
THE MEASUREMENT OF THE RADIAL THERMAL NEUTRON DENSITY DISTRIBUTION IN TUBULAR FUEL ELEMENTS* V. F. BELKIN, B. P. KOCHUROV and 0. V. SHVEDOV (Received 14 March 1963)
Abstract-An experimental study has been made of the radial thermal neutron density distribution inside tubular natural uranium fuel elements containing the organic coolant-monoisopropyldiphenyl. Using the activation method, the thermal neutron density distribution has been measured, in 23 different versions of the fuel element. Theoretical neutron density distributions were obtained for these fuel elements by solving the one-velocity kinetic equation on an electronic computer. The experimental and theoretical results obtained for the neutron density distributions, shielding factors and the neutron density discontinuities at the outer coolant layer are presented in the form of graphs. INTRODUCTION THE present tendency in reactor design towards raising the coolant temperature without increasing the pressure in order to obtain a commercially profitable reactor has led to the use of low boiling-point organic liquids as the coolant. The heat engineering and constructional requirements of systems involving natural uranium (used as the fissile material) and an organic coolant have led to the fabrication of the multi-layer type of fuel element to which the calculational methods evolved for simpler types of fuel elements(lp2) cannot be applied. Consequently, it has become necessary to employ more complicated calculational procedures which need to be verified experimentally. EXPERIMENTAL
ARRANGEMENT
The thermal neutron density distributions were measured in mock-up fuel elements (in assemblies) having a length of 75 cm. Each assembly consisted of the following parts (Fig. 1). A sleeve of natural metallic uranium I (density 18.7 g/cm8), the external and internal surfaces of which were clad in aluminium sheaths 2,3. In the centre of the assembly was inserted a magnesium thimble 4. The inner layer of organic coolant, which was monoisopropyldiphenyl 5 (density O-986 g/cm3), filled the space between the outer surface of the magnesium thimble and the inner sheath of the uranium sleeve. The outer layer of monoisopropyldiphenyl 6 filled the space between the outer sheath of the uranium sleeve and the avial shield tube 7. Between the external avial cladding of the assembly 8 and the avial shield tube there was an air gap 9. (In some experiments the monoisopropyldiphenyl was replaced by water.) The following dimensions were the same for all the assemblies: the internal diameter of the uranium sleeves was 30 mm, the air gap was 3 mm, the thickness of the external avial cladding was 2 mm and that of the avial shield tube and the cladding around the uranium sleeves was 1 mm. The assemblies differed in respect of the following dimensions: the external * Translated by D. L.
ALLAN
from Afumnaya 464
Energiya
15, 377 (1963).
The measurement of the radial thermal neutron density distribution in tubular fuel elements 465
diameter of the uranium sleeves, the diameter of the magnesium thimble, and consequently also the thickness of the inner layer of monoisopropyldiphenyl, the thickness of the outer layer of monoisopropyldiphenyl. Each assembly was given a code number; this indicated the external diameter of the uranium sleeve, the thickness of the outer layer of monoisopropyldiphenyl and the diameter of the magnesium thimble. For example, assembly 54-1.5-20 had a uranium sleeve with an external diameter 54 mm, the thickness of the outer layer of
1.4
I.3
I.2 Y) C 5 2 E I.1 d
I.0
0.9
R,
mm
FIG. l.-Thermal neutron density distribution q along the radius R of the assemblies (the curves represent calculated results; the points are experimental results): (a) for assembly 46-2-M; (b) for assembly 46-2-16; (c) for assembly 462-Y; 0, Oexperimental values for assemblies 46-2-M and 46-2-K respectively. monoisopropyldiphenyl was 1.5 mm and the diameter of the magnesium thimble was 20 mm. If no thimble was there then, depending on the filling of the internal cavity, the letters V(air) or M(monoisopropyldipheny1) were substituted in place of the last figure in the code. The neutron density distribution was measured using detectors 4.2 mm dia. and thickness O-4 mm, consisting of a mixture of plastic and dysprosium oxide (density -3 mg/cm2). Grooves were milled into the uranium sleeves into which uranium prisms containing the detectors could be inserted during the experiment. Those slots in the prisms not occupied by detectors were plugged with small slabs of uranium. The remaining detectors were placed in recesses in the aluminium cladding and tubes and also in grooves in the magnesium thimble. The disposition of the detectors can be seen from the position of the experimental points in Figs. 1-3. The activity of the detectors was measured using automatic eight-channel counting
466
V. F. BELKIN, B. P. Kocr-maov and 0. V. SHVEDOV
0
5
IO
15
20 25 R. mm
30
35
FIG. 2.-Thermal neutron density distribution along the radius of the assemblies (the curves represent calculated results; the points are experiment results): (a) for assembly 50-25-16; (b) for assembly 50-2.5-V-24; (c)for assembly 50-2*5-V, A, 0 are experimental values for assemblies 5O25-16 and 50-2*5-V, respectively.
Ft..
mm
FIG. 3.-Thermal neutron density distribution along the radius of the assemblies (the curves represent calculated results; the points are experimental results) : (a) for assembly 54-l .5-M; (b) for assembly 54-1.5-16; (c) for assembly 541.5-Y; 0, O--experimental values for assemblies 54-l .5-M and 54-l .5-V, respectively.
equipment. The assemblies were irradiated in channel A-O of a heavy-water reactoP in a flux of lo0 n/cm2/sec. Since the internal diameter of channel A-O was ~100 mm, an air gap existed between the external avial cladding of the assembly and the heavy water. While being irradiated, the detectors occupied a position approximately coinciding with the centre of the reactor core. The cadmium ratio of the dysprosium was not less than 100, so that the detectors registered only thermal neutrons. The temperature of the moderator was 293°K. EXPERIMENTAL
.RESULTS
Figures l-3 present typical thermal neutron density distributions along the radii of the assemblies obtained experimentally and by solving the kinetic equation on the computer. Assemblies whose neutron density distributions are included in the same figure differed only in respect of the thickness of the inner layer of monoisopropyldiphenyl. In some versions the magnesium thimble was absent, the internal cavity of the uranium sleeve being filled either with monoisopropyldiphenyl or air.
The measurement
of the radial thermal neutron density distribution
in tubular fuel elements
467
Table 1 gives values of the shielding factors Q* obtained by various methods: (i) from the experimental results of this work; (ii) from the results of a numerical solution of the kinetic equation; (iii) on the basis of data previously determined.(1-12) Table 2 presents the neutron density discontinuities, /I, defined as the ratio of the neutron density at the external surface of the outer layer of monoisopropyldiphenyl or of water to the neutron density at the surface of the uranium sleeve. Since the experimental conditions did not permit detectors to be placed on the external surface of the outer monoisopropyldiphenyl layer, to determine ,!I, the experimental values were corrected in accordance with the neutron density curve obtained at the time of solving the kinetic equation. The errors associated with the distributions and neutron density discontinuities, including the errors made in setting up the detectors, were l-1.5 per cent. TABLE l.-SHIELDING FACTORS
Q Type of assembly Experiment 1.279 1.251 1.239 I.227 1.218 1.199 1.187 1.176 1.166 1.162 1.135 1.130 1.125 1.120 1.113
54-1.5-B 54-1.5-24 54-l-5-20 54-1.5-16 54-1.5-M 50-2.5-B 50-2.5-24 50-2.5-20 50-2.5-16 50-2.5-M 46-2.0-B 46-2.0-24 46-2+--20 46-2.0-16 46-2.0-M
Calculation 1.261 1.248 1.242 1.238 1.231 1.197 1.188 1.183 1.179 1.147 1.137 1.134 1.131 1.119
From data previously determined.‘l~*~ 1.271 1.216 l-164 -
The calculated neutron density distributions were obtained by solving the onevelocity kinetic equation for the multi-zone cell of Wigner-Seitz. The calculation was performed using a programme compiled by F. M. Filler and T. D. Bogdanova. The dimensions of the equivalent cells for each of the assemblies were determined by some preliminary calculations which stipulated that the neutron density distribution in the inner layer of monoisopropyldiphenyl should be close to the experimental distribution. In addition, the sources of thermal neutrons in the monoisopropyldiphenyl were taken to be 4.77 times larger than those in D,O. It was assumed that the neutrons had a Maxwellian distribution with an average effective temperature T, which depended on the co-ordinates. The temperature of the neutrons in the moderator TnM was taken to be equal to the temperature of the heavy water T,O and, on the basis of some preliminary calculations, the temperature
* The value Q-l =
s
2 q(r)r dr
q(RzI~~ the uranium sleeve.
, where R, and RI are the external r dr
and internal
radii of
V. F. BELRIN,B. P. Kocm~~ov and 0. V. SWEDOV
468
of neutrons in the uranium was taken as T, u = 1.3 Tno.(4y5)With these values good agreement was obtained between the calculated and experimental values of the shielding factors. M of the monoisopropyldiphenyl depends fairly The scattering cross section ZZtr strongly on the neutron temperature. (YUROVA et cd.(‘) measured this dependence TABLE2.-NEUTRON DENSITY DISCONTINUITIES b AT THEOUTERLAYERS AND WATER OF MONOISOPROPYLDIPHENYL Type of assembly 541.5-B 54-l-5-24 54-1.5-20 54-1.5-16 54-15-M 50-1.5-B 50-l .5-24 50-2.5-B 50-2.5-24 50-2.5-20 50-2.5-16 5&3.5-B 50-3.5-24 50-3.5-M 50-3.5-B* 50-3.5-M* 46-2-B 46-2-24 46-2-20 46-2-16 46-2-M 46-2-B* 46-2-M*
B -____ Experiment 1.270 1.292 1.284 1.262 1.268 1.300 1.275 1.391 1.411 1402 1.398 1.470 1.464 1.473 1.527 1.535 1.306 1.305 1.282 1.290 1.280 1.349 1.294
Calculation 1.288 1.284 1.282 1.280 1.276 1.272 1.383 1.378 1.375 1.372 1.483 -
1.288 1.283 1.279 1.277 1.262 -
Thickness of layer (mm) 1.45 1.45 1.45 1.45 1.45 1.55 1.55 2.55 2.55 2.55 2.55 3.50 3.50 3.50 3.50 3.50 200 2.00 2.00 2.00 2.00 2.00 2.00
f f f & f + f + f f f f f f f + + & i & f + f
0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
* The outer layer of monoisopropyldiphenyl was replaced by water.
experimentally, finding &TMN u1~s8~~12over the temperature range 18-250X!.) It follows from the preliminary calculations that when T,” = Tnu = 1.3 T,” the values of ,!?obtained are very much underestimated. Satisfactory agreement with experiment is obtained if the value of T,” is taken to be close to T,“. The magnitude of /3 depends weakly on the dimensions of the equivalent cell and on the filling of the internal TABLE S.-DEPENDENCE D,O
OF ,8 ON x:
AND ON THE THICKNESS t OF THE
LAYER FOR A CELL CONTAINING ASSEMBLY 50-25-20*
I$, cm-l r (cm) 5 6
1.81
2.00
2.20
2.26
2.50
1.311 -
1.336 1.341
1.361 1.367
1.375 -
1400 1405
* Xg (293°K) = 2.32 cm-l. cavity of the slug with monoisopropyldiphenyl. Some of the results of the preliminary calculations are given in Tables 3 and 4. To find the effect of the layer of aluminium between the uranium and the outer layer of coolant, a number of calculations were carried out both for the case where
The measurement
of the radial thermal neutron density distribution
in tubular fuel elements
469
the layer of monoisopropyldiphenyl came right up to the uranium and for the case where the aluminium was replaced by an air gap. As can be seen from Table 5, @increases slightly in the first case and decreases slightly in the second. It should be mentioned that in the case of hollow sleeves some increase in the calculated neutron density is observed towards the centre of the assembly. For example, the calculation for the cell containing a 54-1.5-V assembly in which, instead of leaving the space inside the uranium empty it was filled with a very low cross TABLE 4.-DEPENDENCEOF
Q AND
BON
T,U FOR A CELL CONTAINING
ASSEMBLY
Q
50-2.5-V
B
T,u zz.Tn0
Tnu = 1.3 T,O
1.230
1.197
Tnu = TnO
Experiment
T” n=
1.424
1.199
1.3T”n 1.383
Experiment 1.391
section material (2 = 0*0025), showed that near the axis of the assembly the neutron density rose to a value roughly 12 per cent higher than the minimum value (see broken line curve in Fig. 3). It is reasonable to suppose that this effect is due to the anisotropy of the neutron angular distribution@). TABLE 5.-EFFECTOF
A LAYEROF
ALUMINIUMON
/?
Assembly Calculation 46-2-V 1.288 1.301 1.276
With aluminium Without aluminium Air gap
[email protected] V ~1.383 1.401 -
54-2.5-V 1.290 1.306 -
For the assemblies which had no coolant inside the uranium sleeve, the shielding coefficient Q could be found also from the formula”F2) Q=l+$?t
[
l+++~(g)~].
2v” Here A, u, y are the tabulated functions of X”. SE, The calculations were carried out using the following cross sections (cm-l): z;y = 0.392; Er
= 0.0842;
XMg = 0.1546.9 if” = 2.26;
cpzo = 0.397;
;sy = 0.275 ; EA1 a = 0.0108.7 z”g = 0*0021* & = 0.011;’ z,D2o = 0.001.
The absorption cross sections were taken averaged over the Maxwellian spectrum a = Sc,(&e-” dx (1 ’ fxe-n dx where x = EjkT,. We note that in the measurements [See, for example, DEUTSCH(~)], as also in the calculations with the assumed averaging, the neutron density distribution is determined
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V. F. BELKIN, B. P. K~CHUROVand 0. V. SIWEDOV
i.e. the distribution of the quantity J n(E) dE. Therefore, when using the results obtained to calculate the coefficient of thermal neutron utilization, one should take absorption cross sections at a fixed temperature, for example Tnu, and apply temperature corrections to allow for deviations from the l/v law. For the assemblies containing uranium sleeves with external diameters 4.6; 5-O and 5.4 cm, the thicknesses of the heavy-water layers were 5 ; 5.5, and 6 cm, respectively. The experimental results and their comparison with the results of the calculations show therefore that the experimental neutron density distributions in heavywater reactor fuel elements containing an organic coolant are in agreement with the results of calculations made on the basis of the one-velocity model T,” = l-3 T,O, T,,M = T,,O. REFERENCES 1. GALANIN A. D. Neutron Physics, p. 125, Gosatomizdat, Moscow (1961). 2. AMOUYAL A., BENOISTP. and HOROWITZ J. J. Nucl. Energy 6,79 (1957). 3. GONCHAROV V. V. et al. Proceedings of the Second International Conference on the Peaceful Uses of Atomic Energy, Geneva, Vol. 10, p. 321. United Nations, New York (1959). 4. BLAGOVOLINP. P. Neutron Physics, p. 56, Gosatomizdat, Moscow (1961). 5. MOSTOVOIV. I. et al. Proceedings of the Second International Conference on the Peaceful Uses of Atomic Energy, Geneva, Vol. 16, p. 254. United Nations, New York (1959). 6. DEUTSCHR. Nucl. Sci. Engng. 10,400(1961). 7. YUROVAL. N. et al. Atomnaya Energiya 12,331 (1962). 8. C&E K., HOFFMANF. DE and PLACZEK. G. Introduction to the Theory of Neutron Dijksion-I. Los Alamos (1953).