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Experimental study of effect of void volume fraction on neutron diffusion parameters in water
Radiation Physics and Chemistry 64 (2002) 349–357
Experimental study of effect of void volume fraction on neutron diffusion parameters in water Masood Iqbala,*, Nasir M. Mirzab, Sikander M. Mirzab a
Nuclear Engineering Division, Pakistan Institute of Nuclear Science and Technology (PINSTECH), Nilore, Islamabad 45650, Pakistan b Department of Physics and Applied Mathematics, Pakistan Institute of Engineering and Applied Sciences (PIEAS), Nilore, Islamabad 45650, Pakistan Received 21 August 2001; accepted 12 October 2001
Abstract The dependence of the diffusion parameters including slowing down area, neutron diffusion length and migration length on the voided volume fraction in water has been studied experimentally. For this purpose, the PIEAS Neutron Transport Facility (PNTF) comprised of a 10 Ci Am–Be neutron source and a water filled aluminum tank with Perspex voided tubes has been designed and fabricated. A BF3 detector was used for the neutron counting. The slowing down area was determined at the cadmium cutoff level. The diffusion parameters were determined first in the absence of voids. The experimentally measured values of the slowing down area, the diffusion length and the migration length have been found in good agreement with the corresponding values determined by other workers. By varying void volume fraction from 0% to 7.5%, the experimental measurements show a monotonic increase in the slowing down area from 58.7172.6 to 71.2873.2 cm2, in the diffusion length from 2.9570.13 to 3.1170.13 cm and in the migration length from 8.2170.165 to 8.9970.169 cm. Our measurements show that the diffusion parameters exhibit a quadratic dependence on the void volume fraction. r 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction Voids play an important role in the safety assessment of Light Water Reactors (LWRs) as the void coefficient of reactivity results in the change of criticality during power perturbations. Voids are known to cause power oscillations in the Boiling Water Reactors (BWRs) in the low flow conditions (March-Leuba et al., 1986). In the case of Pressurized Water Reactor (PWRs), voids are undesirable during the normal operation of the reactor. However, their presence in the power excursions tends to limit the peak power due to the negative reactivity feedback mechanism. In some research reactors, inner irradiation sites have positive void coefficients of reactivity (Iqbal et al., 1997).
*Corresponding author. Tel.: +92-51-2207264; fax: +92-519290275. E-mail address: [email protected] (M. Iqbal).
Various diffusion parameters including the slowing down distance, diffusion length and migration length are key parameters used in the safety assessment of nuclear reactors. Both stationary and non-stationary techniques have been employed for the measurements of these parameters in pure water. While the latter techniques are typically more complex and costly, the former ones, being clean experiments, yield more accurate results. In the past, efforts have been made to measure the dependence of the diffusion parameters on temperature. Wright and Frost (1956), Reier and de Juren (1961), Rockey and Skolnick (1960) and Ballowe (1962) have carried out extensive measurements of the neutron diffusion length in pure water without voids at various temperatures. Also, Ahmad (1970), Twum-Danso and Cooper (1971) and Khan and Kabir (1968) have carried out neutron diffusion length measurements in pure water for different voids. All of these measurements yielded values in good agreement with each other and the results show a quadratic rise in the neutron diffusion
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length in water with temperature. Similar studies have also been carried out for graphite by Mills (1955) as well as by Lloyd et al. (1958). Their results show a T 0:25 dependence in the measured values of the neutron diffusion length in graphite. Extensive measurements have also been carried out by Beckurts and Kluber (1958), Reier and de Juren (1961) and Rockey and Skolnick (1960) for the determination of the neutron diffusion length in water at 2201C. Experimental efforts have also been made for the determination of the neutron age in various moderators using different neutron sources. For water, such measurements have been carried out by Valente and Sullivan (1959) using a Po–Be source, by Duggal et al. (1958) using a Ra–Be source and by Foster (1960) using Na–Be source. Recently, we carried out experimental measurements of the mean squared slowing down distance and the relaxation length for an Am–Be neutrons in water (Mirza and Iqbal, 1996). The relaxation length for Am–Be source was 10.871.0 cm in pure water. The experimentally measured value of the flux age was 58.574.1 cm2. In this work, we present details and results of experimental measurements of the slowing down area, the neutron diffusion length and the migration length for various values of the void fractions in water. These measurements were carried out for Am–Be neutrons using a BF3-proportional detector. A comparison of the measured neutron diffusion parameters at zero void fraction with the corresponding values found in literature is also presented along with discussion of its theoretical basis.
Using Eq. (1), the area becomes /r2t S ¼ 6t;
ð3Þ
1 t ¼ /r2t S: 6
ð4Þ
In these experiments, the neutron flux is a directly measurable quantity. The fast flux ffast is directly proportional to the slowing down density qðr; tÞ: Therefore, using fast flux in Eq. (2) and substituting the resulting expression in Eq. (4), we get slowing down area in terms of measurable flux: "R N # 1 0 r2 ffast ðrÞ dr t ¼ RN 2 : ð5Þ 6 0 r ffast ðrÞ dr This expression was employed to determine t for water with different values of the void volume fractions. 2.2. The diffusion length The diffusion length is the characteristic length over which the thermal neutron flux decreases by a factor of ‘e’ in a non-multiplying medium. If the slowing down of neutrons is taking place in the same medium, then the neutron diffusion equation includes a space dependent thermal neutron source term ‘qth ’ (Reier and de Juren, 1961). In the one-dimensional case, the solution of the diffusion equation, with this term, becomes fth ðrÞ ¼
ker=L ; F ðrÞ
where the factor F ðrÞ is given by h i1 F ðrÞ ¼ C E1 fðS 1=LÞrg þ e2r=L E1 fðS 1=LÞrg
2. Theory
ð7Þ
2.1. Slowing down area The slowing down distance is a measure of the net vector distance traveled by the neutron from the point of its formation to the point where it crosses a specified level of energy. The slowing down area is simply the square of the slowing down distance. It has units of area (cm2) and is denoted by ‘t’. In the case of a point isotropic source placed in a non-absorbing medium, the source normalized slowing down density, qðr; tÞ; can be written as (Glasstone and Sesonske, 1981; Ott and Bezella, 1983) qðr; tÞ ¼
ð6Þ
1 ð4ptÞ
2
3=2
er
=4p
;
ð1Þ
where t is the slowing down area of the neutrons at distance ‘r’ from the source. Then the mean squared slowing down distance /r2t Sis given by R N 2 2 2 0R r 4pr qðr; tÞ dr : ð2Þ /rt S ¼ N 2 0 ½4pr qðr; tÞ dr
and it represents a correction term to the flux with ‘k’ and ‘C’ as constants. The S was determined from the neutron source term qth solved in plane geometry as qth p
eSr : r
ð8Þ
From the spatial behavior of the neutron source term the value of S was determined. The value of ‘C’ can be determined by using the cadmium ratio ‘CR’ at a specified value of ‘r’: CR ¼
fth ðrÞ qth ðrÞ
¼ reðS1=LÞr C E1 fðS þ 1=LÞrg þ e2r=L E1 fðS þ 1=LÞrg:
ð9Þ
In view of the non-linear nature of the above equation, an iterative algorithm was developed in this work. It starts with an initial guess of ‘C’ and ‘L’. By using the known values of the cadmium ratio at various experimental points, estimate of ‘L’ is improved. In this
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algorithm, iterations are continued until the values of ‘L’ obtained in two consecutive iterations agree with each other within a specified convergence criterion. 2.3. The migration length The net vector distance covered between the birth as a fast neutron and capture as a thermal neutron is generally known as the migration length. It is related to the thermal diffusion area, L2 ; and the slowing down area, t; as pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ M ¼ L2 þ t : It may be noted that the critical dimensions of a nuclear reactor core are directly related to the migration length. Therefore, for compact reactor designs, one must choose moderator with suitably small value of migration length. For a given cell pitch, the reactor will behave as an under-moderated or over-moderated system according to the value of the migration length.
Fig. 1. The internal view of the neutron source assembly.
3. Experimental setup 3.1. PIEAS neutron transport facility (PNTF) In this facility, the fast neutrons are obtained from a 10 Ci Am–Be neutron source. The source is mounted on a circular disk that is imbedded inside a very thick Perspex and lead shield specially designed for this source. The design was locally developed and tested. The internal view of the neutron source assembly and neutron shield is shown in Fig. 1. The source disk can be rotated through any angle from 01to 1801 with the help of a rotor pointer that moved on a circular scale provided outside the shielding container (Fig. 1). A window of about 5 cm in diameter is available through which fast neutrons can stream towards the moderator. The window is directed to a large water tank. Any desired count rate between a minimum of 40 s1 and a maximum of 7.3 104 s1 can be obtained by varying the angular position of the source. These experiments were performed at a source position where the total flux has its maximum count rate. A cylindrical (14 cm long, 2 cm diameter) BF3detetctor was employed in this work. A Cd sheet of 0.1 cm thickness was used to provide the energy cutoff. The spectral deformation within the Cd-cover was ignored in this work. As shown in Fig. 2, next to the source and shield geometry is a large aluminum tank of dimensions 85 cm 95 cm 145 cm. The tank is filled with reactor grade pure water to provide a medium for thermalization of fast neutrons. Measurements were carried out radially up to 45 cm from the source. As shown in Fig. 3, at the top of the tank, a back-lash compensated pair of
Fig. 2. The neutron source assembly and the water tank arrangement.
Fig. 3. The top view of the experimental setup showing the neutron assembly, water tank, voided Perspex tubes and the neutron detector housing.
threaded screw parallel steel bars is provided so that the detector assembly may be moved with the help of a microcomputer-based control system.
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3.2. Production of voids Voids were produced with the help of air-filled Perspex water-tight sealed tubes. These tubes were fixed with the help of screws in two grid plates to form a void assembly. A typical void assembly employed in this work is shown in Fig. 4. Then provisions were made to fix the void assembly inside the water tank with the help of screws to avoid the floating of the assembly in water. The void tubes were arranged in the assembly in a matrix of 6 3 (see Fig. 4). Two such assemblies were lowered into the tank at a time. The void fraction (Vf ) may be modified by changing the diameter of these tubes. A block diagram of the modular electronics used for the detection of neutrons is shown in Fig. 5. A BF3detector with an operating voltage of 1150 V was used for both fast and thermal neutrons employing a cadmium cover. The pre-amplifier produces tail pulses which are fed to a spectroscopic amplifier. The uni-polar output of the amplifier moves to the single channel analyzer which works as a discriminator in this case and produces logic pulses which are fed to a NIM counter/ timer as well as to a microcomputer-based automatic data acquisition and analysis system (ADAS). The ADAS was programmed to acquire count rates for
specified intervals of time and for the specified number of times. It computes the average and the standard deviations, and then stores the results on hard disk for later analysis.
Fig. 4. The side view of the Perspex voided tubes used.
Fig. 5. The electronic setup showing the pulse processing and the automatic data acquisition and analysis system.
M. Iqbal et al. / Radiation Physics and Chemistry 64 (2002) 349–357
3.3. Experimental procedure The tank was filled with reactor grade demineralized water. In the first step, the cadmium covered BF3-detector was placed in a water-tight aluminum guide pipe. Then, both were lowered in the water tank of the PNTF to a depth such that the detector appears exactly in front of the neutron beam (i.e., principal axis). The source was moved to its maximum intensity position and the operating voltage was applied to the detector. This arrangement enabled fast neutron detection. At each position (r), the count rate (RCd ) was measured by moving the BF3-detector along the horizontal direction. To detect both fast and thermal neutrons, the bare BF3-detector was placed in the water-tight aluminum guide pipe. The pipe was lowered in the water tank to the same depth. Using the above procedure, the counting of the total neutron count rate (RT ) was repeated at various horizontal positions. Both fast and total neutron flux distributions were measured throughout the tank with a mesh size of 5 cm. Somewhat more detailed measurements were carried out along the principal axis by taking the neutron count rate with (RCd ) and without (RT ) the cadmium cover at 1 cm intervals. After this, the voided tube assemblies were lowered into the tank and again the neutron count rate was taken along the principal axis with and without the cadmium cover. This procedure was repeated for all different voided tube assemblies.
353
cadmium cutoff (0.6 eV) levels. The cadmium ratio (CR) is equal to
RT CR ¼ : ð14Þ RCd 3.4.1. Calculation of the slowing down area (t) As mentioned earlier, the slowing down area (t) can be determined by using Eq. (5) if the fast flux ffast is known as a function of the distance inside the medium. For the computation of these integrals, first r2 ffast ðrÞ and r4 ffast ðrÞ were plotted against ‘r’. Instead of using the conventional method of subdividing the areas into the ‘curved’ and the ‘linear’ portions and applying different integration schemes over those areas, we have used a curve fitting procedure. The value of the slowing down area (t) was found by integrating the fitting function directly. For zero void volume fraction (Vf ), the variation of r2 ffast ðrÞ and r4 ffast ðrÞ with ‘r’ has been plotted in Fig. 6a and b. In these graphs, the solid circles represent the actual data points whereas the solid curves represent the plots of the fitted function. It is clear from these graphs that the fitting function describes the behavior of
3.4. Results and discussion A standard stationary method was used for these measurements and the spatial profiles of both thermal and fast neutrons were obtained for various values of the void volume fractions. For the conversion from the count rate to neutron flux, the detector efficiency factor (e) from the manufacturer’s calibration data sheets was used: f ¼ R=e:
ð11Þ
The values of the thermal (fth ) and fast (ffast ) neutron fluxes were found by employing the following relations (Glasstone and Sesonske, 1981; Ott and Bezella, 1983): RT RCd fth ¼ ; ð12Þ e ffast ¼
fth sth lnðE1 =E2 Þ; IB xðCR 1Þ
ð13Þ
where sth and IB are the values of the microscopic thermal cross section and the resonance integral for 10B, respectively, E1 and E2 are the energy boundaries for the fast (about 1 MeV) and the
Fig. 6. The behavior of r2 ffast ðrÞ and r4 ffast ðrÞ as a function of ‘r’ for pure water without voids.
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the experimental data adequately. By direct integration of the fitting functions, the slowing down area has been determined for pure water without voids using Eq. 5. The results of these calculations for pure water are given in Table 1. This study was repeated for various values of the void volume fractions and the results are shown in Table 1. It was shown that the value of the slowing down area increases from 58.71 to 71.28 cm2 for a 0% to 7.5% variation in the void volume fraction. This variation is found to follow a quadratic behavior as depicted in Fig. 7, where t has been plotted against Vf : The following polynomial is found to fit the experimental
Table 1 Variation of the measured values of the slowing down area (t) with void volume fraction (Vf ) for water Vf (%)
t (cm2)
0 1.94 4.94 6.2 7.2
58.7172.6 60.9972.5 64.9972.7 66.6172.9 71.2873.2
76 74 72 2
Fermi Age (cm )
70 68 66 64 62 60 58 56 0
2
4
6
8
Void Fraction (%) Fig. 7. The measured values of the slowing down area (t) as a function of the void volume fraction (Vf ).
data for Vf having values up to 7.5%: t ¼ 59:059 þ 0:2865Vf þ 0:18Vf2
ð15Þ
with the regression coefficient R2 ¼ 0:973 and P ¼ 0:027: The value of the slowing down area obtained in this work for zero volume fraction has been compared with the results obtained in other experiments and these values are given in Table 2. The value of the slowing down area for void-free water (i.e., 58.71 cm2), obtained in these experiments, is in close agreement with the results obtained in other measurements using highenergy neutron source especially with the value obtained by Valente and Sullivan (1959) employing a 5 MeV Po– Be neutron source. 3.4.2. Calculation of the diffusion length For these calculations, an iterative procedure was used. As a first guess, the value of ‘C’ was found using the cadmium ratio for the data points near the source. Based on this value of ‘C’, the function F ðrÞ (Eq. (7)) was found for all data points along the principal axis. From the plot of ln ½F 1 ðrÞfth ðrÞ vs. ‘r’, an estimate of the diffusion length ‘L’ was obtained as its inverse negative slope. This value of ‘L’ was then employed to obtain an improved estimate of ‘C’ and the above procedure was repeated until a converged value of ‘L’ was obtained. A computer program FIND (written in FORTRAN-77) was used to carry out these iterations for various data sets automatically. By using the iterative procedure discussed in Section 2.2, the values of ‘L’ and the correction factor F ðrÞ were determined for each value of the void fraction. From Eq. (6), the thermal flux fth ðrÞ corrected with F ðrÞ is expected to follow an exponential behavior with its inverse negative slope representing the diffusion length ‘L’. This behavior is quite clearly visible in Fig. 8, where the ratio fth ðrÞFðrÞ has been plotted against ‘r’ for different values of the void volume fractions. The steepness of these lines decreases on increasing the values of the void volume fraction. Since the diffusion length is inversely proportional to the slope of these lines, therefore, indicating a trend in which the diffusion length is inversely proportional to the slope of these
Table 2 A comparison of slowing down area (t) for water at zero void fraction determined in various experiments Reference
Method
Neutron source E (MeV)
t (cm2)
Foster (1960) Duggal et al. (1958) Valente and Sullivan (1959) This work
Indium resonance Indium resonance Indium resonance Cd cutoff (at 0.6 eV)
Na–Be, 0.97 Ra–Be, 5.0 Po–Be, 5.0 Am–Be, 5.4
13.9 54.4 57.3 58.7172.6
M. Iqbal et al. / Radiation Physics and Chemistry 64 (2002) 349–357
10
2
355
3.20
1
φth(r)xF(r)
10
Difusion Length (cm)
3.15
10
3.05 3.00 2.95
0
2.90
Void = 2% Void = 4%
0
2
4
8
6
Void fraction (%)
Void = 5% Void = 6.5%
10
3.10
Fig. 9. The measured values of the diffusion length (L) as a function of the void volume fractions (Vf ).
-1
20
25
30
35
40
Distance 'r' (cm) Fig. 8. The variation of fth ðrÞF ðrÞ as a function of ‘r’ for different values of the void volume fractions.
Table 3 Dependence of the measured diffusion length on the void volume fraction Vf (%)
L (cm)
0 1.94 4.94 6.2 7.2
2.94970.13 2.96470.13 3.02270.14 3.04870.13 3.11270.14
lines, therefore, this figure shows a trend that the diffusion length increases with an increase in the void fraction. This is in agreement with the expected behavior since the neutrons will be able to diffuse out to a larger distance before being absorbed in the partially voided medium. The measured values of the diffusion length of neutrons in water for various values of the void volume fractions are given in Table 3. For an increase in the values of Vf from 0% to 7.5%, the diffusion length has been found to increase from 2.94 to 3.11 cm. This increase is found to follow a quadratic behavior as shown in Fig. 9 where ‘L’ has been plotted against Vf : In this graph, the filled circles represent the actual data points and the continuous curve represents the fitted polynomial: L ¼ 2:952 þ 0:00322Vf þ 0:00338Vf2
ð16Þ
with the value of the regression coefficient being R2 ¼ 0:9827 and P ¼ 0:0172: This expression fits the experimental data well up to about 7.5% void volume fraction. A comparison of diffusion length for pure water as measured by various investigators with the one obtained in this study is shown in Table 4. For zero void volume fraction, the value of ‘L’ measured in this work is in good agreement with the results obtained in other studies. 3.4.3. Calculation of the migration length (M) The neutron migration length values for various values of the void fraction were obtained by using the diffusion length (L) and the slowing down area (t) in Eq. (10). The results are given in Table 5. The migration length shows an increase from 8.21 cm at zero void volume fraction to 8.99 cm for a 7.2% of the void fraction. The observed behavior (Fig. 10) can be described by the following quadratic relation: M ¼ 8:2306 þ 0:0206Vf þ 0:011Vf2
ð17Þ 2
having the value of the regression coefficient R ¼ 0:975 and P ¼ 0:024: Both experimental results and the fitted polynomial are shown in Fig. 10. At zero void volume fraction, the value of ‘M’ (8.21070.165 cm) measured in this work shows good agreement with the results obtained in other experiments. Our measurements show that the diffusion parameters exhibit a quadratic dependence on the void volume fraction. This behavior arises from two coupled factors. The first one is the reduction in the slowing down of neutrons in the presence of voids. The neutron flux held up as neutrons migrate to a larger distance in the presence of voids. The second effect is a spectral shift towards higher energy due to the decrement in the slowing down properties of the medium caused by voids.
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Table 4 Comparison of the various measured values of the diffusion length (L) for water without voids Reference
Method
L (cm)
Beckurts and Kluber (1958) Reier and de Juren (1961) Rohr (in Beckurts and Kluber, 1958) Rockey and Skolnick (1960) This work
Cadmium difference in an infinite medium Sb–Be source in an infinite medium Sb–Be source in an infinite medium Sb–Be source in an infinite medium Cadmium difference with neutron beam using BF3 with Am–Be source
2.7470.03 2.77570.006 2.77870.011 2.83570.018 2.94970.13
Table 5 Variation of the measured values of the migration length (M) with void volume fraction (Vf ) for water Vf (%)
M (cm)
0 1.94 4.94 6.2 7.2
8.21070.165 8.35370.167 8.60970.166 8.71270.167 8.99870.169
(L) and the migration length (M) on the void volume fraction in water has been studied. The results show a quadratic increase in the value of these parameters on increasing the void volume fraction from 0% to about 7.5%. The observed increasing behavior is consistent with the expected variation based on the slowing down and absorption properties of water. The measured values of t; L and M at zero volume fraction have been found in good agreement with the corresponding values obtained by other studies.
Acknowledgements 9.4
The contribution of Dr Naseer Haider in the design and fabrication of the PIEAS Neutron Transport Facility (PNTF) is gratefully acknowledged. We also wish to thank our teachers Dr Inam-ur-Rehman and Dr Abdullah Sadiq for their encouragement and support. The support of Dr Hameed Ahmad Khan and Mr Showket Pervez is also acknowledged.
Migration Length (cm)
9.2 9.0 8.8 8.6 8.4 8.2
References 8.0 0
2
4
6
8
Void fraction (%) Fig. 10. The measured values of the migration length (M) as a function of the void volume fraction (Vf ).
These neutrons can migrate to even larger distances as the cross sections become lower at higher energies. Both of these effects lead to the observed quadratic rise in the values of various diffusion parameters with an increase in the void volume fraction.
4. Conclusions The dependence of various diffusion parameters, including the slowing down area (t), the diffusion length
Ahmad, M., 1970. Neutron streaming and anisotropic effects in diffusion media with empty channels. Nucleus (Karachi) 7, 221–223. Ballowe, W.C., 1962. Measurements on the diffusion length of thermal neutrons in light water. Report BNL-719, pp. 799–804. Beckurts, K.H., Kluber, O., 1958. Determination of diffusion parameters of thermal neutrons in water by a Poisoning Method. Z. Naturforsch. 13a, 825–882. Duggal, V.P., Puri, S.M., Ram, K.S., 1958. Experiments on fast neutron distribution in water and neutron age in water– aluminum mixture. Proceedings of the Second United Nations International Conference on Peaceful Use of Atomic Energy, Geneva, Vol. 13, pp. 85–90. Foster Jr., D.G., 1960. Age of Na–Be neutrons in water and kerosene. Nucl. Sci. Eng. 8, 148–156. Glasstone, S., Sesonske, A., 1981. Nuclear Reactor Engineering. Van Nostrand Reinhold Co., New York. Iqbal, M., Mirza, N.M., Mirza, S.M., Ayzuddin, S.K., 1997. Study of the void coefficients of reactivity in a typical
M. Iqbal et al. / Radiation Physics and Chemistry 64 (2002) 349–357 pool type research reactor. Ann. Nucl. Energy 24, 177–186. Khan, N.A., Kabir, S.M., 1968. Effects of voids on diffusion length and streaming of neutrons in water. Nucl. Sci. Eng. 31, 148–148. Lloyd, R.C., Clayton, E.D., Richey, C.R., 1958. Variation of graphite diffusion length with temperature. Nucl. Sci. Eng. 4, 690–697. March-Leuba, J., Cacuci, D.G., Perez, R.B., 1986. Non linear dynamics and stability of boiling water reactor: Part 1Fqualitative analysis. Nucl. Sci. Eng. 93, 111–123. Mills, J.E.C., 1955. Variation of diffusion length and thermal neutron absorption cross-section of graphite with temperature. Report AERE-RP/R-1618. Mirza, N.M., Mirza, S.M., Iqbal, M., 1996. Determination of mean squared slowing-down distance for Am–Be neutrons in water using BF3-detector. Radiat. Phys. Chem. 48, 413–417.
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Ott, K.O., Bezella, W.A., 1983. Introductory Nuclear Reactor Statics. American Nuclear Society, LaGrange Park, Illinois. Reier, M., de Juren, J.A., 1961. Diffusion length of thermal neutrons in water from 23–2441C. React. Sci. Technol. 14, 18–24. Rockey, K.S., Skolnick, W., 1960. Measurements on the diffusion length of thermal neutrons in water from 25–2961C. Nucl. Sci. Eng. 8, 62–65. Twum-Danso, J., Cooper, R., 1971. Measurement of material buckling as a function of void fraction in a sub-critical assembly. J. Nucl. Energy 24, 587–591. Valente, F.A., Sullivan, R.E., 1959. The age of plutonium– beryllium neutrons in light water. Nucl. Sci. Eng. 6, 162–162. Wright, W.B., Frost, R.T., 1956. Preliminary diffusion length measurements in hot water. Report KAPL-M-WBW-2.
Experimental study of effect of void volume fraction on neutron diffusion parameters in water Masood Iqbala,*, Nasir M. Mirzab, Sikander M. Mirzab a
Nuclear Engineering Division, Pakistan Institute of Nuclear Science and Technology (PINSTECH), Nilore, Islamabad 45650, Pakistan b Department of Physics and Applied Mathematics, Pakistan Institute of Engineering and Applied Sciences (PIEAS), Nilore, Islamabad 45650, Pakistan Received 21 August 2001; accepted 12 October 2001
Abstract The dependence of the diffusion parameters including slowing down area, neutron diffusion length and migration length on the voided volume fraction in water has been studied experimentally. For this purpose, the PIEAS Neutron Transport Facility (PNTF) comprised of a 10 Ci Am–Be neutron source and a water filled aluminum tank with Perspex voided tubes has been designed and fabricated. A BF3 detector was used for the neutron counting. The slowing down area was determined at the cadmium cutoff level. The diffusion parameters were determined first in the absence of voids. The experimentally measured values of the slowing down area, the diffusion length and the migration length have been found in good agreement with the corresponding values determined by other workers. By varying void volume fraction from 0% to 7.5%, the experimental measurements show a monotonic increase in the slowing down area from 58.7172.6 to 71.2873.2 cm2, in the diffusion length from 2.9570.13 to 3.1170.13 cm and in the migration length from 8.2170.165 to 8.9970.169 cm. Our measurements show that the diffusion parameters exhibit a quadratic dependence on the void volume fraction. r 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction Voids play an important role in the safety assessment of Light Water Reactors (LWRs) as the void coefficient of reactivity results in the change of criticality during power perturbations. Voids are known to cause power oscillations in the Boiling Water Reactors (BWRs) in the low flow conditions (March-Leuba et al., 1986). In the case of Pressurized Water Reactor (PWRs), voids are undesirable during the normal operation of the reactor. However, their presence in the power excursions tends to limit the peak power due to the negative reactivity feedback mechanism. In some research reactors, inner irradiation sites have positive void coefficients of reactivity (Iqbal et al., 1997).
*Corresponding author. Tel.: +92-51-2207264; fax: +92-519290275. E-mail address: [email protected] (M. Iqbal).
Various diffusion parameters including the slowing down distance, diffusion length and migration length are key parameters used in the safety assessment of nuclear reactors. Both stationary and non-stationary techniques have been employed for the measurements of these parameters in pure water. While the latter techniques are typically more complex and costly, the former ones, being clean experiments, yield more accurate results. In the past, efforts have been made to measure the dependence of the diffusion parameters on temperature. Wright and Frost (1956), Reier and de Juren (1961), Rockey and Skolnick (1960) and Ballowe (1962) have carried out extensive measurements of the neutron diffusion length in pure water without voids at various temperatures. Also, Ahmad (1970), Twum-Danso and Cooper (1971) and Khan and Kabir (1968) have carried out neutron diffusion length measurements in pure water for different voids. All of these measurements yielded values in good agreement with each other and the results show a quadratic rise in the neutron diffusion
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length in water with temperature. Similar studies have also been carried out for graphite by Mills (1955) as well as by Lloyd et al. (1958). Their results show a T 0:25 dependence in the measured values of the neutron diffusion length in graphite. Extensive measurements have also been carried out by Beckurts and Kluber (1958), Reier and de Juren (1961) and Rockey and Skolnick (1960) for the determination of the neutron diffusion length in water at 2201C. Experimental efforts have also been made for the determination of the neutron age in various moderators using different neutron sources. For water, such measurements have been carried out by Valente and Sullivan (1959) using a Po–Be source, by Duggal et al. (1958) using a Ra–Be source and by Foster (1960) using Na–Be source. Recently, we carried out experimental measurements of the mean squared slowing down distance and the relaxation length for an Am–Be neutrons in water (Mirza and Iqbal, 1996). The relaxation length for Am–Be source was 10.871.0 cm in pure water. The experimentally measured value of the flux age was 58.574.1 cm2. In this work, we present details and results of experimental measurements of the slowing down area, the neutron diffusion length and the migration length for various values of the void fractions in water. These measurements were carried out for Am–Be neutrons using a BF3-proportional detector. A comparison of the measured neutron diffusion parameters at zero void fraction with the corresponding values found in literature is also presented along with discussion of its theoretical basis.
Using Eq. (1), the area becomes /r2t S ¼ 6t;
ð3Þ
1 t ¼ /r2t S: 6
ð4Þ
In these experiments, the neutron flux is a directly measurable quantity. The fast flux ffast is directly proportional to the slowing down density qðr; tÞ: Therefore, using fast flux in Eq. (2) and substituting the resulting expression in Eq. (4), we get slowing down area in terms of measurable flux: "R N # 1 0 r2 ffast ðrÞ dr t ¼ RN 2 : ð5Þ 6 0 r ffast ðrÞ dr This expression was employed to determine t for water with different values of the void volume fractions. 2.2. The diffusion length The diffusion length is the characteristic length over which the thermal neutron flux decreases by a factor of ‘e’ in a non-multiplying medium. If the slowing down of neutrons is taking place in the same medium, then the neutron diffusion equation includes a space dependent thermal neutron source term ‘qth ’ (Reier and de Juren, 1961). In the one-dimensional case, the solution of the diffusion equation, with this term, becomes fth ðrÞ ¼
ker=L ; F ðrÞ
where the factor F ðrÞ is given by h i1 F ðrÞ ¼ C E1 fðS 1=LÞrg þ e2r=L E1 fðS 1=LÞrg
2. Theory
ð7Þ
2.1. Slowing down area The slowing down distance is a measure of the net vector distance traveled by the neutron from the point of its formation to the point where it crosses a specified level of energy. The slowing down area is simply the square of the slowing down distance. It has units of area (cm2) and is denoted by ‘t’. In the case of a point isotropic source placed in a non-absorbing medium, the source normalized slowing down density, qðr; tÞ; can be written as (Glasstone and Sesonske, 1981; Ott and Bezella, 1983) qðr; tÞ ¼
ð6Þ
1 ð4ptÞ
2
3=2
er
=4p
;
ð1Þ
where t is the slowing down area of the neutrons at distance ‘r’ from the source. Then the mean squared slowing down distance /r2t Sis given by R N 2 2 2 0R r 4pr qðr; tÞ dr : ð2Þ /rt S ¼ N 2 0 ½4pr qðr; tÞ dr
and it represents a correction term to the flux with ‘k’ and ‘C’ as constants. The S was determined from the neutron source term qth solved in plane geometry as qth p
eSr : r
ð8Þ
From the spatial behavior of the neutron source term the value of S was determined. The value of ‘C’ can be determined by using the cadmium ratio ‘CR’ at a specified value of ‘r’: CR ¼
fth ðrÞ qth ðrÞ
¼ reðS1=LÞr C E1 fðS þ 1=LÞrg þ e2r=L E1 fðS þ 1=LÞrg:
ð9Þ
In view of the non-linear nature of the above equation, an iterative algorithm was developed in this work. It starts with an initial guess of ‘C’ and ‘L’. By using the known values of the cadmium ratio at various experimental points, estimate of ‘L’ is improved. In this
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algorithm, iterations are continued until the values of ‘L’ obtained in two consecutive iterations agree with each other within a specified convergence criterion. 2.3. The migration length The net vector distance covered between the birth as a fast neutron and capture as a thermal neutron is generally known as the migration length. It is related to the thermal diffusion area, L2 ; and the slowing down area, t; as pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ M ¼ L2 þ t : It may be noted that the critical dimensions of a nuclear reactor core are directly related to the migration length. Therefore, for compact reactor designs, one must choose moderator with suitably small value of migration length. For a given cell pitch, the reactor will behave as an under-moderated or over-moderated system according to the value of the migration length.
Fig. 1. The internal view of the neutron source assembly.
3. Experimental setup 3.1. PIEAS neutron transport facility (PNTF) In this facility, the fast neutrons are obtained from a 10 Ci Am–Be neutron source. The source is mounted on a circular disk that is imbedded inside a very thick Perspex and lead shield specially designed for this source. The design was locally developed and tested. The internal view of the neutron source assembly and neutron shield is shown in Fig. 1. The source disk can be rotated through any angle from 01to 1801 with the help of a rotor pointer that moved on a circular scale provided outside the shielding container (Fig. 1). A window of about 5 cm in diameter is available through which fast neutrons can stream towards the moderator. The window is directed to a large water tank. Any desired count rate between a minimum of 40 s1 and a maximum of 7.3 104 s1 can be obtained by varying the angular position of the source. These experiments were performed at a source position where the total flux has its maximum count rate. A cylindrical (14 cm long, 2 cm diameter) BF3detetctor was employed in this work. A Cd sheet of 0.1 cm thickness was used to provide the energy cutoff. The spectral deformation within the Cd-cover was ignored in this work. As shown in Fig. 2, next to the source and shield geometry is a large aluminum tank of dimensions 85 cm 95 cm 145 cm. The tank is filled with reactor grade pure water to provide a medium for thermalization of fast neutrons. Measurements were carried out radially up to 45 cm from the source. As shown in Fig. 3, at the top of the tank, a back-lash compensated pair of
Fig. 2. The neutron source assembly and the water tank arrangement.
Fig. 3. The top view of the experimental setup showing the neutron assembly, water tank, voided Perspex tubes and the neutron detector housing.
threaded screw parallel steel bars is provided so that the detector assembly may be moved with the help of a microcomputer-based control system.
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3.2. Production of voids Voids were produced with the help of air-filled Perspex water-tight sealed tubes. These tubes were fixed with the help of screws in two grid plates to form a void assembly. A typical void assembly employed in this work is shown in Fig. 4. Then provisions were made to fix the void assembly inside the water tank with the help of screws to avoid the floating of the assembly in water. The void tubes were arranged in the assembly in a matrix of 6 3 (see Fig. 4). Two such assemblies were lowered into the tank at a time. The void fraction (Vf ) may be modified by changing the diameter of these tubes. A block diagram of the modular electronics used for the detection of neutrons is shown in Fig. 5. A BF3detector with an operating voltage of 1150 V was used for both fast and thermal neutrons employing a cadmium cover. The pre-amplifier produces tail pulses which are fed to a spectroscopic amplifier. The uni-polar output of the amplifier moves to the single channel analyzer which works as a discriminator in this case and produces logic pulses which are fed to a NIM counter/ timer as well as to a microcomputer-based automatic data acquisition and analysis system (ADAS). The ADAS was programmed to acquire count rates for
specified intervals of time and for the specified number of times. It computes the average and the standard deviations, and then stores the results on hard disk for later analysis.
Fig. 4. The side view of the Perspex voided tubes used.
Fig. 5. The electronic setup showing the pulse processing and the automatic data acquisition and analysis system.
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3.3. Experimental procedure The tank was filled with reactor grade demineralized water. In the first step, the cadmium covered BF3-detector was placed in a water-tight aluminum guide pipe. Then, both were lowered in the water tank of the PNTF to a depth such that the detector appears exactly in front of the neutron beam (i.e., principal axis). The source was moved to its maximum intensity position and the operating voltage was applied to the detector. This arrangement enabled fast neutron detection. At each position (r), the count rate (RCd ) was measured by moving the BF3-detector along the horizontal direction. To detect both fast and thermal neutrons, the bare BF3-detector was placed in the water-tight aluminum guide pipe. The pipe was lowered in the water tank to the same depth. Using the above procedure, the counting of the total neutron count rate (RT ) was repeated at various horizontal positions. Both fast and total neutron flux distributions were measured throughout the tank with a mesh size of 5 cm. Somewhat more detailed measurements were carried out along the principal axis by taking the neutron count rate with (RCd ) and without (RT ) the cadmium cover at 1 cm intervals. After this, the voided tube assemblies were lowered into the tank and again the neutron count rate was taken along the principal axis with and without the cadmium cover. This procedure was repeated for all different voided tube assemblies.
353
cadmium cutoff (0.6 eV) levels. The cadmium ratio (CR) is equal to
RT CR ¼ : ð14Þ RCd 3.4.1. Calculation of the slowing down area (t) As mentioned earlier, the slowing down area (t) can be determined by using Eq. (5) if the fast flux ffast is known as a function of the distance inside the medium. For the computation of these integrals, first r2 ffast ðrÞ and r4 ffast ðrÞ were plotted against ‘r’. Instead of using the conventional method of subdividing the areas into the ‘curved’ and the ‘linear’ portions and applying different integration schemes over those areas, we have used a curve fitting procedure. The value of the slowing down area (t) was found by integrating the fitting function directly. For zero void volume fraction (Vf ), the variation of r2 ffast ðrÞ and r4 ffast ðrÞ with ‘r’ has been plotted in Fig. 6a and b. In these graphs, the solid circles represent the actual data points whereas the solid curves represent the plots of the fitted function. It is clear from these graphs that the fitting function describes the behavior of
3.4. Results and discussion A standard stationary method was used for these measurements and the spatial profiles of both thermal and fast neutrons were obtained for various values of the void volume fractions. For the conversion from the count rate to neutron flux, the detector efficiency factor (e) from the manufacturer’s calibration data sheets was used: f ¼ R=e:
ð11Þ
The values of the thermal (fth ) and fast (ffast ) neutron fluxes were found by employing the following relations (Glasstone and Sesonske, 1981; Ott and Bezella, 1983): RT RCd fth ¼ ; ð12Þ e ffast ¼
fth sth lnðE1 =E2 Þ; IB xðCR 1Þ
ð13Þ
where sth and IB are the values of the microscopic thermal cross section and the resonance integral for 10B, respectively, E1 and E2 are the energy boundaries for the fast (about 1 MeV) and the
Fig. 6. The behavior of r2 ffast ðrÞ and r4 ffast ðrÞ as a function of ‘r’ for pure water without voids.
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the experimental data adequately. By direct integration of the fitting functions, the slowing down area has been determined for pure water without voids using Eq. 5. The results of these calculations for pure water are given in Table 1. This study was repeated for various values of the void volume fractions and the results are shown in Table 1. It was shown that the value of the slowing down area increases from 58.71 to 71.28 cm2 for a 0% to 7.5% variation in the void volume fraction. This variation is found to follow a quadratic behavior as depicted in Fig. 7, where t has been plotted against Vf : The following polynomial is found to fit the experimental
Table 1 Variation of the measured values of the slowing down area (t) with void volume fraction (Vf ) for water Vf (%)
t (cm2)
0 1.94 4.94 6.2 7.2
58.7172.6 60.9972.5 64.9972.7 66.6172.9 71.2873.2
76 74 72 2
Fermi Age (cm )
70 68 66 64 62 60 58 56 0
2
4
6
8
Void Fraction (%) Fig. 7. The measured values of the slowing down area (t) as a function of the void volume fraction (Vf ).
data for Vf having values up to 7.5%: t ¼ 59:059 þ 0:2865Vf þ 0:18Vf2
ð15Þ
with the regression coefficient R2 ¼ 0:973 and P ¼ 0:027: The value of the slowing down area obtained in this work for zero volume fraction has been compared with the results obtained in other experiments and these values are given in Table 2. The value of the slowing down area for void-free water (i.e., 58.71 cm2), obtained in these experiments, is in close agreement with the results obtained in other measurements using highenergy neutron source especially with the value obtained by Valente and Sullivan (1959) employing a 5 MeV Po– Be neutron source. 3.4.2. Calculation of the diffusion length For these calculations, an iterative procedure was used. As a first guess, the value of ‘C’ was found using the cadmium ratio for the data points near the source. Based on this value of ‘C’, the function F ðrÞ (Eq. (7)) was found for all data points along the principal axis. From the plot of ln ½F 1 ðrÞfth ðrÞ vs. ‘r’, an estimate of the diffusion length ‘L’ was obtained as its inverse negative slope. This value of ‘L’ was then employed to obtain an improved estimate of ‘C’ and the above procedure was repeated until a converged value of ‘L’ was obtained. A computer program FIND (written in FORTRAN-77) was used to carry out these iterations for various data sets automatically. By using the iterative procedure discussed in Section 2.2, the values of ‘L’ and the correction factor F ðrÞ were determined for each value of the void fraction. From Eq. (6), the thermal flux fth ðrÞ corrected with F ðrÞ is expected to follow an exponential behavior with its inverse negative slope representing the diffusion length ‘L’. This behavior is quite clearly visible in Fig. 8, where the ratio fth ðrÞFðrÞ has been plotted against ‘r’ for different values of the void volume fractions. The steepness of these lines decreases on increasing the values of the void volume fraction. Since the diffusion length is inversely proportional to the slope of these lines, therefore, indicating a trend in which the diffusion length is inversely proportional to the slope of these
Table 2 A comparison of slowing down area (t) for water at zero void fraction determined in various experiments Reference
Method
Neutron source E (MeV)
t (cm2)
Foster (1960) Duggal et al. (1958) Valente and Sullivan (1959) This work
Indium resonance Indium resonance Indium resonance Cd cutoff (at 0.6 eV)
Na–Be, 0.97 Ra–Be, 5.0 Po–Be, 5.0 Am–Be, 5.4
13.9 54.4 57.3 58.7172.6
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10
2
355
3.20
1
φth(r)xF(r)
10
Difusion Length (cm)
3.15
10
3.05 3.00 2.95
0
2.90
Void = 2% Void = 4%
0
2
4
8
6
Void fraction (%)
Void = 5% Void = 6.5%
10
3.10
Fig. 9. The measured values of the diffusion length (L) as a function of the void volume fractions (Vf ).
-1
20
25
30
35
40
Distance 'r' (cm) Fig. 8. The variation of fth ðrÞF ðrÞ as a function of ‘r’ for different values of the void volume fractions.
Table 3 Dependence of the measured diffusion length on the void volume fraction Vf (%)
L (cm)
0 1.94 4.94 6.2 7.2
2.94970.13 2.96470.13 3.02270.14 3.04870.13 3.11270.14
lines, therefore, this figure shows a trend that the diffusion length increases with an increase in the void fraction. This is in agreement with the expected behavior since the neutrons will be able to diffuse out to a larger distance before being absorbed in the partially voided medium. The measured values of the diffusion length of neutrons in water for various values of the void volume fractions are given in Table 3. For an increase in the values of Vf from 0% to 7.5%, the diffusion length has been found to increase from 2.94 to 3.11 cm. This increase is found to follow a quadratic behavior as shown in Fig. 9 where ‘L’ has been plotted against Vf : In this graph, the filled circles represent the actual data points and the continuous curve represents the fitted polynomial: L ¼ 2:952 þ 0:00322Vf þ 0:00338Vf2
ð16Þ
with the value of the regression coefficient being R2 ¼ 0:9827 and P ¼ 0:0172: This expression fits the experimental data well up to about 7.5% void volume fraction. A comparison of diffusion length for pure water as measured by various investigators with the one obtained in this study is shown in Table 4. For zero void volume fraction, the value of ‘L’ measured in this work is in good agreement with the results obtained in other studies. 3.4.3. Calculation of the migration length (M) The neutron migration length values for various values of the void fraction were obtained by using the diffusion length (L) and the slowing down area (t) in Eq. (10). The results are given in Table 5. The migration length shows an increase from 8.21 cm at zero void volume fraction to 8.99 cm for a 7.2% of the void fraction. The observed behavior (Fig. 10) can be described by the following quadratic relation: M ¼ 8:2306 þ 0:0206Vf þ 0:011Vf2
ð17Þ 2
having the value of the regression coefficient R ¼ 0:975 and P ¼ 0:024: Both experimental results and the fitted polynomial are shown in Fig. 10. At zero void volume fraction, the value of ‘M’ (8.21070.165 cm) measured in this work shows good agreement with the results obtained in other experiments. Our measurements show that the diffusion parameters exhibit a quadratic dependence on the void volume fraction. This behavior arises from two coupled factors. The first one is the reduction in the slowing down of neutrons in the presence of voids. The neutron flux held up as neutrons migrate to a larger distance in the presence of voids. The second effect is a spectral shift towards higher energy due to the decrement in the slowing down properties of the medium caused by voids.
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Table 4 Comparison of the various measured values of the diffusion length (L) for water without voids Reference
Method
L (cm)
Beckurts and Kluber (1958) Reier and de Juren (1961) Rohr (in Beckurts and Kluber, 1958) Rockey and Skolnick (1960) This work
Cadmium difference in an infinite medium Sb–Be source in an infinite medium Sb–Be source in an infinite medium Sb–Be source in an infinite medium Cadmium difference with neutron beam using BF3 with Am–Be source
2.7470.03 2.77570.006 2.77870.011 2.83570.018 2.94970.13
Table 5 Variation of the measured values of the migration length (M) with void volume fraction (Vf ) for water Vf (%)
M (cm)
0 1.94 4.94 6.2 7.2
8.21070.165 8.35370.167 8.60970.166 8.71270.167 8.99870.169
(L) and the migration length (M) on the void volume fraction in water has been studied. The results show a quadratic increase in the value of these parameters on increasing the void volume fraction from 0% to about 7.5%. The observed increasing behavior is consistent with the expected variation based on the slowing down and absorption properties of water. The measured values of t; L and M at zero volume fraction have been found in good agreement with the corresponding values obtained by other studies.
Acknowledgements 9.4
The contribution of Dr Naseer Haider in the design and fabrication of the PIEAS Neutron Transport Facility (PNTF) is gratefully acknowledged. We also wish to thank our teachers Dr Inam-ur-Rehman and Dr Abdullah Sadiq for their encouragement and support. The support of Dr Hameed Ahmad Khan and Mr Showket Pervez is also acknowledged.
Migration Length (cm)
9.2 9.0 8.8 8.6 8.4 8.2
References 8.0 0
2
4
6
8
Void fraction (%) Fig. 10. The measured values of the migration length (M) as a function of the void volume fraction (Vf ).
These neutrons can migrate to even larger distances as the cross sections become lower at higher energies. Both of these effects lead to the observed quadratic rise in the values of various diffusion parameters with an increase in the void volume fraction.
4. Conclusions The dependence of various diffusion parameters, including the slowing down area (t), the diffusion length
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