To this point we have constructed the matrix representations the group elements of point groups such as C3v in the usual physical space (two dimensional in our case). These representations were based on the transformations of the coordinates of an arbitrary vector in a physical space due to physical operations on the vector. Mathematical solutions to physical problems, however, are represented by functions in function spaces whose dimensions are generally much greater than three. Thus to bring the group matrix representations that act on coordinates to bear on the solution of physical problems in terms of functions, we need one more "connection" between symmetry operators on coordinates and symmetry operators on functions. This connection is defined as follows.

Let f (r) be a function of a position vector r = (x, y) and D (g-1) be the matrix transformation associated with group element g Є G, such that (x, y) ^ (x’, y’) through

r = D-1 (g)r.

What we need is an algorithm that uses D-1 (g) to obtain a new function h(r) from f (r). To this end we define an operator Og as

Ogf (r) = f (r’ )= f (D-1 (g)r) = h(r).

That is, operator Og gives a new function h(r) from f (r) at r, while f is unchanged at Г. For example, let

f (x, y) = ax + by

and

a + b a — b, . .

Og(ax + by) = ax + by = x + y = ^, y)

For two group elements g1 and g2 in G, we obtain

Og1 f (r)= f (D—1 (g)r) = h(r)

Og2Og1 f (r) = Og2 (OgJ(r)) = Og2h(r) = h(D—1(g2)r) = f ([D—1(g1 )D—1 fe)]r) = f ([D(g2)D(g1)]—1r). Note: operator Og acts on the coordinates of function f and not on the argument of f.

Therefore

Og2 Og1 f (r)= f ((d—1(g1 )D—1 (g2))—1 r) = f (D—1 (g1)D—1 (g2)D—1(g1 )r) = f ((D(g2)D(g1))—1 r),

in words: the consecutive application of Og1 and Og2 is the same as the application of the transformation Og2gt belonging to group element g1 g2, and the operators Og, g Є G have the same multiplication table as G and any group isomorphic with G.

1.1 Overview of ACE-3D

ACE-3D has been developed by the Japan Atomic Energy Agency (JAEA) to simulate water- vapor or water-air two-phase flows at subcritical pressures. The basic equations of ACE-3D are shown below.

Mass conservation for vapor and liquid phases:

dt (agpg> + dx. iagpgUg, j> Г

Here, t represents time; x, a spatial coordinate; U, velocity; P, pressure; g, acceleration due to gravity; e, internal energy; and p, density. Subscripts g and l indicate vapor and liquid phases, respectively. Subscripts i and j indicate spatial coordinate components. If the subscripts of the spatial coordinate components are duplicative, the summation convention is applied to the term. The terms qw and qint indicate wall heat flux and interfacial heat flux, respectively, and hsat represents saturation enthalpy. Summation of volume ratios ag and al is equal to one. Г+, Г -, and Г represent vapor generation rates. Г+ in Eq. (3) is equal to Г if Г is positive and is zero if Г is negative. On the other hand, Г — is equal to Г if Г is negative and is zero if Г is positive. T &ij and Tl/ij in Eqs. (3) and (4), respectively, are shear stress tensors. The liquid shear stress tensor, Tl/ij, is a summation of shear induced turbulence obtained and bubble induced turbulence (Bertodano, 1994).

Summation of interface stress, Mint, of the vapor and liquid phases is equal to zero. The interface stress is determined by correlations between turbulent diffusion force (Lehey,
1991), lift force (Tomiyama, 1995), interface friction force, and bubble diameter (Lilies, 1988). The lift force is calculated as follows.

= — cliftagpg (g — Ul )X rotUl

Clift Clift0 + Cwake

c = 0288 ! — exp (-0242Rebubble) Clift 0 — U.288 —

f 1 + exp (-0.242 Rebubble)

0 (Et < 4)

-0.096Et + 0.384 (4 < Et < 10) -0.576 (Et > 10)

where

In Eq. (7), a is the surface tension coefficient; Et, Eotvos number; Rebubble, the bubble Reynolds number; and Db, bubble diameter. clift0 in Eq. (7) describes the effect of shear flow and is a positive value. Cwake describes the effect of bubble deformation and significantly depends on Eotvos number of bubble diameter. The coefficient of lift force clift is estimated by the summation of clift0 and cwake. If the Eotvos number of bubble diameter is small, clift is positive. However, if the Eotvos number of bubble diameter increases, clift is negative. Therefore, the lift force corresponding to a small bubble diameter acts in a direction opposite to that corresponding to a large bubble diameter.

Bubble diameter is evaluated by the following equations.

Db ( Xslug jDbubble + XslugDslug

Xslug — 3XS2 — 2X3, Xs — 4 ( — 0.25)

where a is the surface tension coefficient; We, a critical Weber number of 5.0; and Dbmax, the maximum bubble diameter, which is an input parameter. Bubble diameter, Db, is estimated by linear interpolation of small bubble diameter, Dbubble, and slug diameter, Dslug, with a coefficient, Xslug, which is dependent on the void fraction. Therefore, Db increases with an increase in the void fraction.

In ACE-3D, the two-phase flow turbulent model based on the standard k-£ model (Bertodano, 1994) is introduced below.

Turbulent energy conservation for vapor and liquid phases:

Turbulent energy dissipation rate conservation for vapor and liquid phases:

The basic equations represented in Eqs. (1) to (12) are expanded to a boundary fitted coordinate system (Yang, 1994). ACE-3D adopts the finite difference method using constructed grids, although it is difficult to construct fuel assembly geometry by using only constructed grids. Therefore, a computational domain, which consists of constructed grids, is regarded as one block, and complex geometry such as that of a fuel assembly is divided into more than one block. An analysis of complex geometry can be performed by a calculation that takes the interaction between blocks into consideration. Parallelization based on the Message Passing Interface (MPI) was also introduced to ACE-3D to enable the analysis of large-scale domains.

From a previous experiment in which the three-dimensional distribution of the void fraction (vapor volume ratio) in a tight-lattice fuel assembly was measured (Kureta, 1994), it is known that the vapor void (void fraction) is concentrated in the narrowest region between adjacent fuel rods near the starting point of boiling and as the elevation of the flow channel increases, the vapor void spreads over a wide region surrounding the fuel rods. This tendency of the vapor void to redistribute has been described by a past analysis using ACE — 3D (Misawa, et al., 2008). Therefore, it is confirmed that the boiling two-phase flow analysis can be carried out using ACE-3D under steady-state conditions and with no oscillation.

An inconel600 self-powered neutron detector has been developed and tested for in-core neutron monitoring (Alex, 2007). The sensing material in a self-powered detector is an emitter from which electrons are emitted when exposed to radiation. These electrons penetrate the thin insulation around the emitter and reach the outer sheath without polarising voltage. Some electrons are emitted from the insulator and sheath also. The net flow of electrons from the emitter gives rise to a DC signal in an external circuit between the emitter and sheath, which is proportional to the incident neutron flux. Rh and V SPDs work on the basis of (n, в) reaction and are used for flux mapping while Co and Pt SPDs work on the basis of (n, y-e) prompt reaction and are used for reactor control and safety. However, the build-up of the 60Co and 61Co gives rise to background signal in the cobalt detector thereby reducing the useful life. In the case of the platinum detector, the detector responds to both reactor neutrons via (n, Y, e) interaction and reactor gamma rays via (y, e) interaction. Since the neutron sensitivity varies with irradiation as a result of burn up while the gamma sensitivity remains the same, the dynamic response of a mixed response detector varies with time. This mixed and time-dependent response of platinum SPD gives rise to anomalous behaviour in some situations. Development of SPDs with inconel emitters as alternative to Co and Pt prompt SPDs has been reported in literatures. The detector (Figure 9) consists of a 2 mm diameter x 21 cm long inconel 600 emitter wire surrounded by a high purity alumina ceramic tube (2.2 mm ID x 2.8 mm OD). The assembly is enclosed in a 3 mm ID x 3.5 mm OD inconel600 tube.

One end of the emitter is coupled to the conductor of a 2 mm diameter x 12 m long twin core mineral insulated (MI) cable while the detector sheath is laser welded to the MI cable sheath. The detector is integrally coupled to the MI cable and the cold end of the cable is sealed by a twin core ceramic-to-metal seal over which a Lemo connector is fitted.

The gamma sensitivity of the detectors was measured in pure gamma field using 60Co source facility. The detectors were placed at a distance of 1m from the source for better source to detector geometry and 1m above the ground to minimize background from

scattered rays. To estimate the gamma field at the detector location, a miniature gamma ion chamber (6 mm diameter and 25 mm long) was used. The calculated gamma sensitivity, 24.8 (fA R-1 h) was used to determine the gamma field at the self-powered neutron detector location. The three SPNDs (inconel600, cobalt, platinum) and were tested together with the miniature gamma chamber in a 200 kCi 60Co source facility. The results showed that the gamma response of the inconel600 and Co detector was found to be similar. However, it was observed that unlike the platinum detector, which has positive response, the Co and inconel detectors showed negative response. The gamma sensitivity of the inconel600 detector is about 7.7 times lower than Pt detector. This low gamma response of the inconel600 detector improves the neutron to gamma ratio and makes it desirable for reactor safety and control applications. In addition to gamma sensitivity, the neutron sensitivity of SPNDs was tested in dry tube (55 mm diameter *8.4 m long) in-core location of the Pool type reactor. The neutron sensitivity and the total sensitivity of the inconel600 detector were found to be lower than the Co detector. The total sensitivity of the inconel SPD is about 20­25% of the sensitivity of cobalt and about 35% of the sensitivity of platinum detectors of similar dimensions; however, it is proposed to improve the sensitivity by helically winding the detector with a short axial length. Finally by comparison, the performance of the inconel detector with cobalt and platinum detectors of similar dimensions, it is obvious that inconel SPD is a useful alternative to Co and Pt SPDs.

1.1 Sample preparation

Before the mixing procedure, a part of mixing water at the percentage of water absorption capacity of expanded vermiculite aggregate by weight was added to vermiculite to make it fully saturated with water. Fig.1 shows the expanded vermiculite particles saturated with water.

Then, the rest of the mixing water cement and silica fume or steel fiber were mixed together for 1 minute in a mixer, and finally, expanded vermiculite aggregate saturated with water was added to cement slurry and mixed for 3 minutes again, to get a homogenous structure. Fig.2 shows the fresh state of the mixture of lightweight mortar.

The prepared fresh mortar were cast in standard cube (with an edge of 150 mm) molds, in two layers, each layer being compacted by self-weight on the shaker for 10 s. All the specimens were kept in moulds for 24 h at room temperature of about 20oC, and then demoulded, and after demoulding all specimens were cured in water at 23 ±2 oC for 27- days. After 28 days curing, three plate specimens with a dimension of 150x100x20mm for neutron dose transmission measurements were obtained by cutting the cube specimens using a stone saw. Plate specimens obtained by the way was illustrated in Fig.3. We obtained 12 different samples. Codes and contents of samples were shown in Table.1.

The pressure drop for Whittle & Forgan experimental conditions is determined and depicted in Figs 3 and 4 against the experimental data. The present model predicts the S-curves with a good agreement achieved with the experimental data. A well defined minimum occurred in all the S-curves. The change in slope from positive to negative was always abrupt and the pressure drop at the condition of the minimum was always approximately equal to that for zero-power condition. As subcooled liquid heat ups along the wall of a heated channel, its viscosity decreases. Increasing the wall heat flux causes further reduction in liquid viscosity. Therefore, pressure drop associated with pure liquid flow decreases with increasing wall heat flux. The trend changes significantly when bubbles begins to form. Here, increasing wall heat flux increases both the two-phase frictional and accelerational gradients of pressure drop. Pressure drop therefore begins to increase with increasing heat flux.

Fig. 4. S-curves prediction for (Whittle & Forgan, 1967) experiments (No. 3 test section)

Rising concerns about global warming, supply security, and depleting fossil fuel reserves have spurred a revival of interest in nuclear power generation, giving birth to a "nuclear power renaissance" in countries the world over. As humankind seeks abundant and environmentally responsible energy in the coming decades, the renaissance of nuclear power will undoubtedly become reality as it is a proven technology and has the potential to generate virtually limitless energy with no greenhouse gas emissions during operations. According to the International Atomic Energy Agency the number of nuclear power reactors in operation worldwide in 2011 is 433 units, and 65 under construction. A large-scale period of nuclear power plants construction would allow nuclear energy to contribute substantially to the decarbonisation of electricity generation. In addition, basic research and nuclear technology applications in chemistry, physics, biology, agriculture, health and engineering have been showing their importance in the innovation of nuclear technology applications with sustainability.

The renaissance of nuclear power has been threatened by the catastrophe in Japan and the atomic industry faces the challenge of assuring a skeptical public that new reactors are safer than the old ones and nearly disaster-proof. The disaster at the Fukushima Daiichi nuclear plant in Japan demonstrates that older nuclear reactor technology requires strict adherence to quality assurance practices and procedures. Newer nuclear reactor designs promote the use of passive cooling systems that would not fail after power outages, as well as other innovative approaches to managing reactor heat. New reactors use the same principle of power generation as in older water reactors such as the ones at Fukushima: nuclear reactors heat water to create steam that turns turbines to generate electricity. However, technological advances have improved efficiency and stricter safety precautions have made the third-generation reactors more secure. The new generation of pressurized water reactor plants has diesel-powered backup systems that are housed in separate buildings to protect them from any accident that might occur in the main reactor building. The plant must also have access to other sources of electricity if the diesel generators fail.

This book is targeted at nuclear regulatory authorities, environmental and energy scientists, students, researchers, engineers, seismologists and consultants. It presents a comprehensive review of studies in nuclear reactors technology from authors across the globe. Topics discussed in this compilation include: thermal hydraulic investigation of TRIGA type research reactor, materials testing reactor and high temperature gas-cooled reactor; the use of radiogenic lead recovered from ores as a coolant for fast reactors; decay heat in reactors and spent-fuel pools; present status of two-phase flow studies in reactor components; thermal aspects of conventional and alternative fuels in supercritical water-cooled reactor; two-phase flow coolant behavior in boiling water reactors under earthquake condition; simulation of nuclear reactors core; fuel life control in light-water reactors; methods for monitoring and controlling power in nuclear reactors; structural materials modeling for the next generation of nuclear reactors; application of the results of finite group theory in reactor physics; and the usability of vermiculite as a shield for nuclear reactor. Concluding the book is presented a review of the use of neutron flux in the radioisotopes production for medicine.

Amir Zacarias Mesquita, ScD.

Researcher of

Nuclear Technology Development Center (CDTN) Brazilian Nuclear Energy Commission (CNEN) Belo Horizonte — Brazil

1

1.2 On the sources of radiogenic lead enriched with lead-208 in Russia

The problem of acquisition of radiogenic lead enriched with lead-208 is coupled with perspectives of involving thorium into nuclear power engineering of Russia. As it is noted in Ref.16 to develop the thorium nuclear energetic it is necessary to obtain at least 10-13 thousand tones of thorium per year at the stage of 20-30 years of this century.

Content of lead-208 in thorium ores and minerals can reach 0.3-0.5% wt of thorium mass. In acquisition 10-13 thousand tones of thorium per year it will be possible to recover about 65 tones of radiogenic lead per year. This quantity of lead is insufficient to cover the needs in lead coolant of large scale nuclear power which requires approximately 2000 tones of lead per 1 GW of electrical power. But 65 tones of lead are sufficient to cool the blanket of 80 MWth ADS. About 700 tones of lead can be enough to cool the reactor RBEC-M delivering 340 MW electrical.

As it is shown in Ref. 16, the main source of thorium in Russia is the Lovozerskoe deposit at Kola Peninsula. Estimations show that in reprocessing 2 mln tones of loparit ore per year 500-600 thousand tones of Ln2C>3 and TiO2, 100 thousand tones of Nb2O5, 10 thousand tones of Ta2O5, 13 thousand tones of ThO2 and 65 tones of radiogenic lead can be produced. In Ref 16 the conclusion was made that is possible to extract in near future large quantities of thorium from the progress of industry and as co-product of rear metal raw.

The separate problem is the level of lead-208 enrichment of lead-208 in various deposits. It can be strongly different. For example, in Brazil monazites radiogenic lead is enriched by lead-208 up to 88.34% [17]. For FRs and ADSs it can be desirable the following isotopic composition of radiogenic lead: lead-208-93 % and lead-206-6% with minimum content of lead-207 — the isotope with large cross section of neutron capture. In Ref. 18 the data concerning thorium-containing ores and monazites in the world scale are given. The authors of this paper pointed out that as a rule radiogenic lead contains very small quantities of lead-204 and lead-207-isotopes with large cross sections of neutron capture.

It can be noted that the advantages of lead-208 can be used, besides nuclear power plants, in other branches of nuclear science and technology. It seems that lead-208 as low moderating material will be preferable in the lead slowing down neutron spectrometers [19] and also in the spallation neutron sources to have the harder neutron spectra under interaction of high energy protons with liquid proton target from lead-208 [2, 20].

The experimental bulk coolant temperatures profile in Channel 1 is shown in Fig. 16 as a function of the axial position, for the powers of 265 kW and 106 kW. Figure 16 shows also the curve predicted from the theoretical model using the PANTERA code at 265 kW (Veloso, 2005). The figure shows also the experimental results for other TRIGA reactors Bars & Vaurio, 1966; Haag, 1971) and (Buke & Yavuz, 2000).

The experimental temperature profile along the coolant is different from that predicted from the theoretical model. Ideally, the coolant temperature would increase along the entire length of the channel, because heat is being added to the water by all fuel regions in the channel. Experimentally, the water temperature reaches a maximum near the middle length and then decreases along the remaining channel. The shape of the experimental curves is similar to the axial power distribution within the fuel rod as shown in Fig. 9. Although Channel 1 is located beside the control rod, the axial temperature profile was not influenced by a possible deformation of the neutron flux caused by this rod, because it was in its upper position, i. e. outside the core. The actual coolant flow is quite different probably, because of the inflow of water from the core sides (colder than its centre).

At IGISOL we have performed two trap-assisted TAS experiments related to the decay heat problem. Here we will discuss briefly the first experiment and its impact. The second experiment is presently in the analysis phase. It is worth noting that in the second experiment we have used for the first time a segmented BaF2 TAS detector which additionally provides information on the multiplicity of the gamma cascades following the beta decay. This extra information is useful for the analysis of the complex beta decay data.

The analysis of a TAS experiment is a lengthy procedure and requires several stages. The first phase requires a careful evaluation of the contaminants and distortions of the measured spectrum in order to determine d. Then the calibration of the experimental data in energy and width and a precise characterization of the TAS detector using Monte Carlo (MC) techniques is

required. For the MC simulations the GEANT4 code (Agostinelli, 2007) is used. In this phase a careful characterization of the setup is performed until measurements with conventional radioactive sources like 24Na, 137Cs and 60Co are very well reproduced by the MC code. Once this has been achieved, the response function of the detector to the decay of interest can be calculated. This requires the definition of the level scheme that may be populated in the decay (B or branching ratio matrix). To construct the branching ratio matrix we take into account known levels up to a certain excitation in the daughter (Ecut) and above that cut-off energy we use a statistical model to generate levels and their branchings. The information on the low-lying levels and their branchings is taken from conventional high resolution measurements, because this information is correct in general if available (known levels). For the statistical model we use a back shifted Fermi formula for the level density and gamma strength functions, which define the probabilities that gamma rays connect the different levels (known and unknown part). Once the B is determined, the R(B) is calculated from the MC responses of the detector to the different 7 and в transitions and 8 is solved. As part of the analysis, the cut-off energy, the accepted low-lying levels and the parameters of the statistical model are changed if necessary. The final result of the analysis is a feeding distribution, from which nuclear structure information can be obtained in the form of the beta strength distribution (Sp), and in the case of the decay heat application mean beta and gamma energies can be calculated (6 and 7).

The impact of our first experiment can be seen in Table 2, where the mean energies of the ENDF database are compared with the results of our measurements. From this table the relevance of performing experiments is also clear. Two nuclei (101Nb, 102Tc), that were suspected to suffer from the pandemonium effect did not, even though they have large Qp decay values. The remaining nuclei all suffered from the effect (see for example the large increase in the mean gamma energy and the reduction in the mean beta energy with respect to high resolution measurements for 104,105Tc). In Fig. 8 the results for the gamma component of 239Pu are presented. They are compared with ENDF (ENSDF, n. d.) before and after the inclusion of our new data (Sonzogni, n. d.). Similar conclusions have been obtained recently using the JEFF database (Mills, n. d.). The new TAS results were published in (Algora, 2010).

As a result of our measurements, a large part of the discrepancy pointed out by (Yoshida, 1999) in the 300-3000 s cooling interval and additionally the discrepancy at low energies has been solved within the ENDF database. The new data were also used to perform summation calculations for the gamma component of 235U. In this case the results were disappointing. Our new results had very little impact. This can be understood in terms of the cummulative fission yields of the nuclei in question. They sample approximatelly 33.8 % of the fission in 239Pu, but only 13.5 % in 235U. Additionnally from the 13.5 % in 235U, 101Nb, 102Tc amounts to 9.2 %, which does not bring a large change in the mean energies. This explains why our measurements to date can only represent a relative change of approximatelly 4.3 % in 235U, compared to the 22.6 % relative impact in 239Pu with respect to the earlier values of the ENDF database.

The safety margin for OFI phenomenon is defined as the ratio between the power to attain the OFI phenomenon within the core channel, and the hot channel power, this means that, OFI margin is equal to the ratio of the minimum average heat flux leads to OFI in the core channels and the average heat flux in the hot channel. It is found that, the OFI phenomenon occurs at an average heat flux of 2.1048 MW/ m2 for steady-state operation (т = 0.0 s), and 1.7294 MW/m2 just before Scram (т = 4.0s). Thus, these values can be regarded as the maximum possible heat fluxes to avoid OFI under steady-state operation and just before Scram respectively. The maximum hot channel heat flux is determined using the data of table 3 as 0.72595 MW/ m2 with an average value of 0.5648 MW/ m2. This means that, the reactor has vast safety margins for OFI phenomenon of 3.73 for steady-state operation, 3.45 and 3.06 just before Scram for both fast and low loss-of-flow transient respectively. Table 4 gives the estimated heat flux leading to OFI and the safety margin values for both the steady and transient states.

3. Conclusion

Flow instability is an important consideration in the design of nuclear reactors due to the possibility of flow excursion during postulated accident. In MTR, the safety criteria will be determined for the maximum allowable power and the subsequent analysis will therefore restrict to the calculations of the flow instability margin. In the present work, a new empirical correlation to predict the subcooling at the onset of flow instability in vertical narrow rectangular channels simulating coolant channels of MTR was developed. The developed correlation involves almost all parameters affecting the phenomenon in a dimensionless form and the coefficients involved in the correlation are identified by the experimental data of Whittle and Forgan that covers the wide range of MTR operating conditions. The correlation predictions for subcooling at OSV were compared with predictions of some previous correlations where the present correlation gives much better agreement with the experimental data of Whittle and Forgan with relative standard
deviation of only 6.6%. The bubble detachment parameter was also estimated based on the present correlation. The present correlation was then utilized in a model predicting the void fraction and pressure drop in subcooled boiling under low pressure. The pressure drop model predicted the S-curves representing the two-phase instability of Whittle and Forgan with good accuracy. The present correlation was also incorporated in the safety analysis of the IAEA 10 MW MTR generic reactor in order to predict the OFI phenomenon under both fast and slow loss-of-flow transient. The OFI locus for the reactor coolant channels was predicted and plotted against flow velocity, exit temperature and bubble detachment parameter for various heat flux values. It was found that the reactor has vast safety margins for OFI phenomenon under both steady and transient states.