The core of a generic 1200-MWel PT SCWR consists of 300 fuel channels that are located inside a cylindrical tank called the calandria vessel. There are 220 SuperCritical-Water (SCW) fuel channels and 80 Steam Re-Heat (SRH) fuel channels. SRH and SCW fuel channels are located on the periphery and at the center of the core, respectively. In terms of neutron spectrum, the studied PT SCWR is a thermal-spectrum reactor. In this thermal — spectrum PT SCWR, light-water and heavy-water have been chosen as the coolant and the moderator, respectively. The coolant enters the supercritical fuel channels at an inlet temperature of 350°C and reaches an outlet temperature of 625°C at a pressure of 25 MPa. The inlet temperature of the SuperHeated Steam (SHS), which is used as the coolant, in the SRH fuel channels, is 400°C and the corresponding outlet temperature is 625°C at an operating pressure of 5.7 MPa. Table 1 lists the operating parameters of the generic 1200- MWel PT SCWR (Naidin et al., 2009).

There are some methods for determination of nuclear reactor stability. Among methods which are applied for this aim are as following:

Liapunov method — Lagrange method — Popov method — Pontryagin method.

There is Lyapunov’s method to determine the stability of nonlinear reactor dynamics by constructing certain positive definite functions of the reactor variables and parameters (Pankaj and Vivek, 2011).

There has been calculated the Lyapunov exponents from a time series of the excess neutron population of a boiling water reactor (BWR) and used it to conclude about the stability of the steady state operation of that particular BWR (Munoz et al., 1992).

There has also discussed the application of topological methods in reactor kinetics study (Smets and Giftopoulos, 1959).

The topological and Lyapunov methods were compared with Aizermann-Rosen methods for analyzing a point reactor model (Devooght and Smets, 1967).

The Pade approximations has been used to obtain solutions for point kinetic equations (Aboanberand, 2002).

Perturbation theory has also been widely used in studying reactor dynamics. There has been obtained specific types of steady solutions to study power oscillations in a reactor results from a Hopf bifurcation (Pandey, 1996; Munoz and Verdu, 1991; Tsuji et al., 1993; Konno et al., 1994). The KBM theory has been used for the nonlinear analysis of a reactor model with the effect of time-delay in the automated control system (Konno et al, 1992).

A singular perturbation has been used to study relaxation oscillations in typical nuclear reactors (Ward and Lee, 1987).

The point kinetic equations in the presence of delayed neutrons with one temperature reactivity coefficient for a step input of reactivity have analytically been solved by applying the perturbation theory (Gupta and Trasi, 1986).

The regular perturbations to obtain an analytical solution for general reactivity have been used (Nahla, 2009).

The multiple time-scales expansion to obtain analytical solutions of the neutron kinetic equations has been applied (Merk and Cacuci, 2005).

The variation methods in conjunction with the Hopf bifurcation theory for a BWR with one group delayed neutron have also been applied (Munoz and Verdu, 1991).

A combination of the center-manifold reduction (CMR) and the method of normal forms have already been applied widely for nonlinear analysis of nuclear reactor dynamics (Pandey, 1996; Tsuji et al., 1993; Konno et al., 1994).

In the Liapunov model, the dynamically equations may be either differential or non differential equations. But the method of problem evaluation is not based on solving the equations. This method is based on energy in classic mechanical. In one of mechanical system the stability condition is when the total energy of a system decreases.

Liapunov applied this property and based the stability function. This function is entitled: V(x) Function.

This function has features as following:

1. V (x)has definite positive value.

2. The partial differential of V(x) is continued.

3. Where the p is momentum, V(p) is negative quasi relatively.

4. The V (x) function can be written as following:

V(x) =Э/х) dx1 + dV(x) dx1 + + ЭУ(х) dxn_ (69)

дхг dt dx2 dt dxn dt

It can also write:

In fact the Liapunov function is a function that considers either state variables or variables which cause to imbalance state. Due to the potential function is able to do a process, so the Liapunov function is as V function. One can write:

V = f (xu x2,…, xn) (71)

If: V = 0 then the system will be steady state.

This method revolved about the determination of a V function, which satisfies certain requirements of the stability theorem. In an initial reactor model a point reactor with constant power removal and without delayed neutrons is considered.

Due to p is a variable which causes unbalancing the system, therefore it can be considered as an x variable. Then it can be written:

yp) = d V(p) dpi +dV(p)dp1 + _ + dV(p) dpn (72)

dpi dt dp2 dt dpn dt

and:

V (p) = wT (p).p

A hierarchy of models is employed in the theory and simulation of complex systems in materials science and condensed matter physics: macroscale continuum mechanics, macroscale models of defect evolution, molecular scale models based on classical mechanics, and various techniques for representing quantum-mechanical effects. These models are classified according to the spatial and temporal scales that they resolve (Figure 2). In this figure, individual modeling techniques are identified within a series of linked process circles showing the overlap of relevant length and timescales. The modeling methodology includes ab initio electronic structure calculations, molecular dynamics (MD), accelerated molecular dynamics, kinetic Monte Carlo (KMC), phase field equations or rate theory simulations with thermodynamics and kinetics by passing information about the controlling physical mechanisms between modeling techniques over the relevant length and time scales. The key objective of such an approach is to track the fate of solutes, impurities and defects during irradiation and thereby, to predict microstructural evolution. Detailed microstructural information serves as a basis for modeling the physical behavior through meso (represented by KMC, dislocation dynamics, and phase field methods) and continuum scale models, which must be incorporated into constitutive models at the continuum finite element modeling scale to predict performance limits on both the test coupons and components.

The estimates of 99Mo production rates by the solution irradiation method are shown in cases using aqueous (NH4)6Mo7O24^4H2O and K2MoO4 solutions as irradiation targets, assuming the 99Mo production in JMTR.

1.3 Conditions for estimates of 99Mo production rates

The JMTR core arrangement is shown in Fig. 2. The capsule of the 99Mo production system with the solution irradiation method is installed into the irradiation hole, M-9 with maximum and average thermal neutron fluxes of 3.5X1018 n/(m2 • s) and 2.6X1018 n/(m2 • s) (Department of JMTR Project, 1994). The capsule consists of inner and outer tubes, and an aqueous molybdate solution is irradiated with neutrons in the inner tube to prevent the solution from leaking into the reactor coolant. Table 2 shows the conditions of the capsule and the two irradiation targets of the aqueous (NH4)6Mo7O24^ 4H2O and K2MoO4 solutions.

99Mo production rates are estimated based on the following conditions in addition to the conditions of Table 2: [23]

j.

_2

_3

_4

_5

_6

_7

_8

_9

10

11

12

13

14

15

■ Fuel element |2| Be frame

П Control rod with fuel follower ^ Oarai shroud facility-1 I I Al reflector element ггкя Gamma-ray shield plate

П Be reflector element

Fig. 2. JMTR core arrangement

Size and material of capsule________

Dissolved molybdenum in each irradiation target

• Aqueous (NH4)6Mo7O24‘4H2O solution (concentration: 90% of saturation):

372.8 g/1,663 cm3

• Aqueous K2MoO4 solution (concentration: 90% of saturation):

702.7 g/1,663 cm3

Table 2. Conditions of capsule and two irradiation targets of aqueous (NH4)6Mo7O24 • 4H2O and K2MoO4 solutions

Hollenbach and Ott (2010) studied the effects of the addition of graphite fibbers on thermal conductivity of UO2 fuel. Theoretically, the thermal conductivity of graphite varies along different crystallographic planes. For instance, the thermal conductivity of perfect graphite along basal planes is more than 2000 W/m K (Hollenbach and Ott, 2010). On the other hand, it is less than 10 W/m K in the direction perpendicular to the basal planes. Hollenbach and Ott (2010) performed computer analyses in order to determine the effectiveness of adding long, thin fibbers of high thermal-conductivity materials to low thermal-conductivity materials to determine the effective thermal conductivity. In their studies, the high thermal — conductivity material had a thermal conductivity of 2000 W/m K along the axis, and a thermal conductivity of 10 W/m K radially, similar to perfect graphite. The low thermal — conductivity material had properties similar to UO2 (e. g., with 95% TD at ~1100°C) with a thermal conductivity of 3 W/m K.

Hollenbach and Ott (2010) examined the effective thermal conductivity of the composite for various volume percentages of the high thermal-conductivity material, varying from 0 to 3%. The results show if the amount of the high thermal-conductivity material increases to 2 % by volume, the effective thermal conductivity of the composite reaches the range of high thermal-conductivity fuels, such as UC and UN.

It is conventional to subdivide reactor instruments into two categories: in-core and out-of­core. In-core sensors are those that are located within narrow coolant channels in the reactor core and are used to provide detailed knowledge of the flux shape within the core. These sensors can be either fixed in one location or provided with a movable drive and must obviously be of rather small size (typically on the order of 10mm diameter). Out-of-core detectors are located some distance from the core and thus respond to properties of the neutron flux integrated over the entire core. The detectors may be placed either inside or outside the pressure vessel and normally will be located in a much less severe environment compared with in-core detectors. Size restrictions are also less of a factor in their design. The majority of neutron sensors for reactor use are of the gas-filled type. Their advantages in this application include the inherent gamma-ray discrimination properties found in any gas detector, their wide dynamic range and long-term stability, and their resistance to radiation damage. Detectors based on scintillation processes are less suitable because of the enhanced gamma-ray sensitivity of solid or liquid scintillators, and the radiation induced spurious events that occur in photomultiplier tubes. Semiconductor detectors are very sensitive to radiation damage and are never used in reactor environments.

A common approach to the solution of physical problems is harmonic analysis, where a solution to the problem is sought in terms of functions that span the solution space. If the problem exhibits some symmetry, we would expect this symmetry to be reflected in the solution for this particular problem. Intuitively we would expect therefore the solution to belong to a subspace of the general solution space, and that the subspace be invariant under the symmetry operations exhibited by the problem.

As an illustration of this notion, we assume the problem has the symmetry of the cyclic permutation group C3 = {E, C3, } that was discussed previously. Let fe(r) be an arbitrary

function that allows the operation of the operators in the group C3 as discussed above. The action of each operator on fE defines a new function that, is

OEfE = fE OC3fE = fC3 OC2fE = fC.

Based on this and the group multiplication table we get relations such as

OC3 fC3 = OC3 OC3fE = OC2fE = fC2,

etc. These observations can be summarized in a table: From that table we can construct matrix (permutation) representations of the operators OE, OC3 , OC2 as for example

D(C3 ) = (2,3,1).

This procedure gives the so-called regular representation for the group C3 as

Oe = (1,2,3); OC3 = (2,3,1); Oq = (3,1,2). (2.9)

The matrices, in general, satisfy the group multiplication table, and are characterized by only the one integer one in each column, the rest zeros, and the dimension of the matrix equals to the number of elements in the group. The functions fe, fc3, fc2 that generate the regular representation, span the invariant subspace. They are not necessarily linearly independent basis functions.

In order to simulate the boiling two-phase flow in a fuel assembly under earthquake conditions, it is necessary to consider the influence of structural oscillation of reactor equipment on boiling two-phase flow. If the coordinate system for an analysis is fixed to an oscillating fuel assembly under earthquake conditions, it can be seen that a fictitious force acts on the boiling two-phase flow in the fuel assembly. Therefore, a new external force

term, f, which simulates the acceleration of oscillation, was added to the momentum conservation equations (Eqs. (3) and (4)).

We assume that the analysis of boiling two-phase flow in a fuel assembly under earthquake conditions can be performed by using time-series data as an input if the time-series data of oscillation acceleration can be obtained from structural analysis results for a reactor (Yoshimura, et al., 2002) or if the measurement data of actual earthquakes can be obtained by seismographic observation.

In order to apply this improved method to the analysis of boiling two-phase flow in a fuel assembly under earthquake conditions, it is necessary to confirm that the simulation of boiling two-phase flow under oscillation conditions can be performed using the interface stress models shown in the preceding section; these stress models are empirical correlations and are based on experimental results under steady-state conditions. In the case of boiling two-phase flow analysis under oscillation conditions, these interface stress models may cause instability in simulation results.

In addition, it is necessary that large-scale analysis be performed within limited computable physical time and that it be consistent with the time-series data of oscillation acceleration obtained from the results of structural analysis in a reactor or with the measurement data from actual earthquakes. In structural analysis in a reactor (Yoshimura, et al., 2002), the minimum time interval of the analysis is limited to 0.01 s (100 Hz). Seismographic observation is also frequently performed with a sampling period of 100 Hz. If a high — frequency oscillation acceleration of over 100 Hz influences boiling two-phase flow in a fuel assembly, boiling two-phase flow analysis, which is consistent with the structural analysis in a reactor, cannot be performed. Therefore, it is necessary to evaluate the highest frequency necessary for this improved method to be consistent with the time-series data of oscillation acceleration.

A computable physical time of about 1 s is preferred for the boiling two-phase flow analysis in a fuel assembly because this analysis requires a large number of computational grids in order to simulate a large-scale domain such as a fuel assembly. If the results of the boiling two-phase flow analysis show quasi-steady time variation for long-period oscillation acceleration, it is not efficient to perform the analysis with a computable physical time span longer than the long period. Effective analysis can be performed if the analysis with a time span subequal to the shortest period of oscillation acceleration, for which the boiling two — phase flow shows quasi-steady time variation, by extracting earthquake motion at any time during the earthquake. Therefore, it is necessary to evaluate the shortest period of oscillation acceleration for which the boiling two-phase flow shows quasi-steady time variation.

The boiling two-phase flow was simulated in a heated parallel-plate channel, which is a simplification of a single subchannel in a fuel assembly. The channel was excited by vertical
and horizontal oscillation to simulate an earthquake in order to confirm that the boiling two- phase flow simulation can be performed under oscillation conditions.

In addition, the influence of the oscillation period on the boiling two-phase flow behavior in a fuel assembly was investigated in order to evaluate the highest frequency necessary for the improved method to be consistent with the time-series data of oscillation acceleration and the shortest period of oscillation acceleration for which the boiling two-phase flow shows quasi-steady time variation.

A new power monitoring method applied to a pressurized water reactors designed by combustion engineering. The method estimate quickly and precisely a reactor’s operational status and thermal power can be monitored over hour to month time scales, using the antineutrino rate as measured by a cubic meter scale detector. Antineutrino emission in nuclear reactors arises from the beta decay of neutron-rich fragments produced by heavy element fissions, and is thereby linked to the fissile isotope production and consumption processes of interest for reactor safeguards. On average, fission is followed by the production of approximately six antineutrinos. The antineutrinos emerge from the core isotropically, and effectively without attenuation. Over the few MeV energy range within which, reactor antineutrinos are typically detected, the average number of antineutrinos produced per fission is significantly different for the two major fissile elements, 235U and 239Pu. Hence, as the core evolves and the relative mass fractions and fission rates of these two elements change, the measured antineutrino flux in this energy range will also change. It is useful to express the relation between fuel isotopic and the antineutrino count rate explicitly in terms of the reactor thermal power, Pth. The thermal power is defined as

Pth = ’ZiNlf. E{ (7)

where N[ is the number of fissions per unit time for isotope i, and E[ is the thermal energy released per fission for this isotope. The sum runs over all fissioning isotopes, with 235U, 238U, 239Pu, and 241Pu accounting for more than 99% of all fissions. The antineutrino emission rate Пу{ґ) can then be expressed in terms of the power fractions and the total thermal power as:

ns(t) = Pth{t)Y, ifjzr f Vi (Ev) dE„ (8)

Et

where the explicit time dependence of the fission fractions and, possibly, the thermal power are noted. <p(Ev), is the energy dependent antineutrino number density per MeV and fission for the ith isotope. <p (E^) has been measured and tabulated. Equation 7 defines the burn-up effect. The fission rates N? (t) and power fractions /;(t) change by several tens of percent throughout a typical reactor cycle as 235U is consumed and 239Pu produced and consumed in the core. These changes directly affect the antineutrino emission rate n^(t). Reactor antineutrinos are normally detected via the inverse beta decay process on quasi-free protons in hydrogenous scintillator. In this charged current interaction, the antineutrino v converts the proton into a neutron and a positron: v + p ^ e+ + n. For this process, the cross section a is small, with a numerical value of only ~10_43cm2. The small cross section can be compensated for with an intense source such as a nuclear reactor. For example, cubic meter scale hydrogenous scintillator detectors, containing ~1028 target protons Np, will register thousands of interactions per day at standoff distances of 10-50 meters from typical commercial nuclear reactors. In a measurement time T, the number of antineutrinos detected via the inverse beta decay process is:

N*(t) = (^)Pth(t’) ^ ^ / a Vi є dE„ (9)

In the above equation, a is the energy dependent cross section for the inverse beta decay interaction, Np is the number of target protons in the active volume of the detector, and D is the distance from the detector to the center of the reactor core. є is the intrinsic detection efficiency, which may depend on both energy and time. The antineutrino energy density and the detection efficiency are folded with the cross section ff, integrated over all antineutrino energies, and summed over all isotopes i to yield the antineutrino detection rate. The SONGS1 detector consists of three subsystems; a central detector, a passive shield, and a muon veto system. Figure 10 shows a cut away diagram of the SONGS1 detector. Further information can be found in (Bowden, 2007) and (Bernstein et al., 2007).

This prototype that is operated at 25 meter standoff from a reactor core, can detect a prompt reactor shutdown within five hours, and monitor relative thermal power to 3.5% within 7 days. Monitoring of short-term power changes in this way may be useful in the context of International Atomic Energy Agency’s (IAEA) Reactor Safeguards Regime, or other cooperative monitoring regimes.

For neutron transmission measurement, we used a 241Am/Be neutron source and a Canberra portable neutron detector equipments. 241Am/Be source emits 4.5 MeV neutron particles. Physical form of 241Am/Be neutron source is compacted mixture of americium oxide with beryllium metal. Fast neutrons are produced by following nuclear reaction,

|Ве(«, n)126C

5.486 keV maximum energy alpha particles emitting from 241Am. Neutron energy value produced by this nuclear reaction is 4.5 MeV. Radiation characteristics of 241Am/Be neutron source are shown in Table.2 (Dose rate values have been obtained from The Health Physics and Radiological Health Handbook, Scintra _Inc., Revised Edition, 1992.).

The NP-100B detector provides us to detect slow and fast neutrons. Tissue equivalent dose rates of the neutron field can be measured by it. The detectors contain a proportional counter which produces pulses resulting from neutron interactions within it. The probes contain components to moderate and attenuate neutrons. So that the net incident flux at the proportional counter is a thermal and low epithermal flux representative of the tissue equivalent dose rate and the neutron field. Because of neutrons have no charge; they can only be detected indirectly through nuclear reactions that create charged particles. The NP100B detector uses 10B as the conversion target. The charged particle — alpha or proton (respectively) created in the nuclear reaction ionizes the gas. Typical detector properties are shown in Table. 3. Equivalent dose rate measurement results have read on RAD ACS program in system PC. Experimental design is shown in Fig.4.

‘http://www. stuarthunt. com/pdfs/Americium_241Beryllium. pdf Table 2. Radiation characteristics of 241Am-Be neutron source*

We determined dose transmission values of vermiculite loaded samples. Firstly, we counted equivalent dose rate by fast neutrons while there is no sample between source and detector
system. And then we measured for each sample neutron equivalent dose rate while there is our sample between 241Am-Be source box and detector probe. The ratio of two values is called dose transmission.