5.2.1 UC

From a heat transfer point of view, there is an interest on carbides of uranium as nuclear fuels due to their high thermal conductivities and high melting points. Carbides of uranium usable for nuclear fuels are Uranium Carbide (UC) and Uranium Dicarbide (UC2). For instance, UC has been proposed as the fuel of choice for a SCWR concept in Russia (Pioro and Duffey, 2007). Uranium sesquicarbide (U2C3) is another carbide of uranium; however, it cannot be manufactured through casting or compaction of a powder. However, UC2 may transform to U2C3 at high temperatures and under stress (Frost, 1963).

UC, which has a Faced-Centered Cubic (FCC) crystal structure similar to those of UN and NaCl, has a high melting point approximately 2507°C and a high thermal conductivity, above 19 W/m K at all temperatures up to the melting point. UC has a density of 13630 kg/ m3, which is lower than that of UN but higher than those of UO2. It should be noted that the density of hypo-stoichiometric UC is slightly higher than that of stoichiometric UC, which is listed in Table 2. Coninck et al. (1975) reported densities between 13730 and 13820 kg/m3 at 25°C for hypo-stoichiometric UC. Moreover, UC has a higher uranium atom density compared to UO2 but lower than that of UN. The uranium atom densities of UC and UN are 1.34 and 1.4 times that of UO2, respectively.

For hypo-stoichiometric UC, the thermal diffusivity a, in m2/s, and thermal conductivity k, in W/m K, correlations are valid for a temperature range of 570 and 2000°C. In Eqs. (5) and (6), T is in degrees Kelvin (Coninck et al., 1975). For stoichiometric UC, Coninck et al. (1975) provided two correlations, shown as Eqs. (7) and (8), which can be used to determine the mean values of the thermal diffusivity and thermal conductivity of stoichiometric UC for a temperature range between 850 and 2250°C, in m2/s and W/m K, respectively.

a = 10“4 • ^5.75 • 10_2+1.25 • 10_6(T-273.15)] (5)

k = 100 • [2.04 • 10_1+2.836 • 10’8(T — 843.15)2] (6)

a = 10“4 • [5.7 • 10_2+1.82 • 10_12(T-1123.15)3] (7)

k = 100 ^1.95 • 10_1+3.57 • 10_8(T-1123.15)2 ] (8)

In addition to Eqs. (6) and (8), Kirillov et al. (2007) have recommended another correlation, shown as Eqs. (9) and (10), for the calculation of the thermal conductivity of UC in W/m K. In the current study, Eq. (21) have been used to determine the thermal conductivity of UC for the calculation of the UC fuel centerline temperature at SCWR conditions, because this equation provides the lowest thermal conductivity values for a wide temperature range, leading to a conservative calculation of the fuel centerline temperature. In Eqs. (9) and (10), T is in degrees Kelvin.

k = 21.7-3.04 • 10-3(T-273.15) + 3.61 • 10-6(T -273.15)2 , 323<T <973 K (9)

k = 20.2 + 1.48×10"3 (T-273.15), 973<T<2573 K (10)

Frost (1963) developed a correlation shown as Eq. (11), which can be used to determine the diametric increase of UC fuel as a function of time-averaged fuel centerline temperature. According to Eq. (11), UC fuel undergoes significant swelling for temperatures above 1000°C. In Eq. (11), Rd and T are percent diametric increase per atom % burn-up and time — averaged fuel centerline temperature in K, respectively. In addition, as shown in Fig. 10, Harrison (1969) provided the volumetric swelling of UC as a function of burn-up for various temperatures.

Rd =0.6 + 0.77 -1

D I 5000

It is convenient to consider the number of neutrons existing in one cubic centimeter at any one instant and the total distance they travel each second while in that cubic centimeter. The number of neutrons existing in a cm3 of material at any instant is called neutron density and is represented by the symbol n with units of neutrons/cm3. The total distance these neutrons can travel each second will be determined by their velocity.

A good way of defining neutron flux (0) is to consider it to be the total path length covered by all neutrons in one cubic centimeter during one second. Mathematically, this is the equation below.

ф =nv (1)

where: ф = neutron flux (neutrons cm-2 s-1), n = neutron density (neutrons cm-3), and v = neutron velocity (cm s-1). The term neutron flux in some applications (for example, cross

Corresponding Author

section measurement) is used as parallel beams of neutrons travelling in a single direction. The intensity of a neutron beam is the product of the neutron density times the average neutron velocity. The directional beam intensity is equal to the number of neutrons per unit area and time (neutrons cm-2 s-1) falling on a surface perpendicular to the direction of the beam. One can think of the neutron flux in a reactor as being comprised of many neutron beams travelling in various directions. Then, the neutron flux becomes the scalar sum of these directional flux intensities. Macroscopic cross sections for neutron reactions with materials determine the probability of one neutron undergoing a specific reaction per centimeter of travel through that material. If one wants to determine how many reactions will actually occur, it is necessary to know how many neutrons are travelling through the material and how many centimeters they travel each second. Since the atoms in a reactor do not interact preferentially with neutrons from any particular direction, all of these directional beams contribute to the total rate of reaction. In reality, at a given point within a reactor, neutrons will be travelling in all directions (DOE, 1993).

Groups can have more properties than just a multiplication table. The illustration in the previous subsection is not amenable to illustrating this; the groups are to small. However, if we again consider the equilateral triangle, we note that it has further symmetry operations. Namely, those associated with reflections through a vertical plane through each vertex. Let us give these reflection operations the symbols dv, a! v and d"v, reflection through planes through vertex a, b, c, respectively. We may describe the operation by the transformations of the vertices a, b, c. For example dv : (a, b, c) ^ (a, c, b). By adding these three reflections operations to the rotations, we form the larger group of symmetry operations of the equilateral triangle G3 = {E, C3, C2, dv, a’v, d"v}. The multiplication table of the new group is given in Table 3. Another group with the same multiplication table as above can be constructed by considering the six permutations of the three letters a, b, c. Let E = (a, b, c), A = (c, a, b), B = (b, c, a), a’ = (a, c, b), b’ = (c, b, a), c’ = (b, a, c) and {E, A, B} are even permutations, {a’, b’, c’} are odd. This leads to the multiplication table 4, which is isomorphic to Table 3. The two multiplication tables illustrate the concept of a subgroup that is defined as: A set S of elements in group G is considered as a subgroup of G if:

1. all elements in S are also elements in G

2. for any two elements in S their product is in S

3. all elements in S satisfy the four group postulates.

From the presented multiplication tables we see that {E, C3, c3 } and {E, A, B} are subgroups; they are the only subgroups in their respective groups. These subgroups are the rotations in the former example, and the even permutations in the latter. While the remaining elements are associated with reflections and odd permutations, respectively. Furthermore, we see that two reflections are equivalent to a rotation, and two odd permutations are equivalent to an even permutation. Thus the operations {C3, C2} and dv, d’v, d"v and similarly {A, B} and {a’, b’, c’} belong in some sense to different sets. This property is illustrated by taking the transform T-1QT of each element Q in G by all elements T in G. For group {E, C3, C2, dv, ff’v, d"v} we obtain the following table of the transforms T-1 QT:

We note that {E}, {C3, C3 }, and { dv, d’v, d"v } transform into themselves and are thereby called classes. Classes play a leading role in the application of group theory to the solution of physical problems. In general physically significant properties can be associated with each class. In the solution of boundary value problems, different subspaces of the solution function space are assigned to each class.

There are two temperature limits that a fuel and a fuel bundle must meet. First, the sheath temperature must not exceed the design limit of 850°C (Chow and Khartabil, 2008). Second, when UO2 fuel is used, the fuel centerline temperature must be below the industry accepted limit of 1850°C (Reisch, 2009) at all normal operating conditions.

Previously, it was mentioned that the industry accepted temperature limit for UO2 fuel is 1850°C; however, this temperature limit might be different for fuels other than UO2. There are several factors that may affect a fuel centerline temperature limit for a fuel. These factors include melting point, high-temperature stability, and phase change of the fuel. For instance, the accepted fuel centerline temperature limit of UO2 fuel is approximately 1000°C below its melting point. As a result, the same fuel centerline temperature limit has been established for the other low thermal-conductivity fuels and enhanced thermal-conductivity fuels. In regards to ThO2, the melting point is higher than that of UO2, but a high uncertainty is associated with its melting point. Therefore, as a conservative approach, the same temperature limit has been established for ThO2. Similarly, the corresponding limit for UC fuel would be 1500°C, because the melting point of UC is approximately 2505°C. UN fuel decomposes to uranium and gaseous nitrogen at temperatures above 1600°C. Therefore, the fuel centerline temperature limit for UN should be lower than that of UO2 under normal operating conditions. Ma (1983) recommends a temperature limit of 1500°C for UN.

A steady-state one-dimensional heat transfer analysis was conducted in order to calculate the fuel centerline temperature at SCW fuel channels. Based on the proposed core configuration SCW fuel channels are located at the center of the core. Consequently, the thermal power in some of these fuel channels might be by a factor higher than the average channel power of 8.5 MWth. Therefore, in the present study, a thermal power per channel of

9.8 MWth has been considered for the SCW fuel channels with the maximum thermal power. This thermal power is approximately 15% (i. e. 10% above the average power and 5% uncertainty) above the average thermal power per channel. The conditions based on which the calculations have been conducted are as follows: an average mass flow rate of 4.4 kg/s, a constant pressure of 25 MPa, a coolant inlet temperature of 350°C, a thermal power per channel of 9.8 MWth.

The presented analysis does not take into account the pressure drop of the coolant. The main reason for not taking the pressure drop into consideration is that the pressure drop is inversely proportional to the square of mass flux. In a CANDU fuel channel, the pressure drop is approximately 1.75 MPa (AECL, 2005). In addition, the mass flux in an SCWR fuel channel is approximately 5 times lower than that of a CANDU reactor. Therefore, the pressure drop of a SCWR fuel channel should be significantly lower than 1.75 MPa. As a result, the pressure drop has not been taken into consideration.

In addition, this study does not determine the sheath and the fuel centerline temperatures for the SRH fuel channels mainly due to the fact that the average thermal power in SRH channels is 5.5 MWth (see Table 1). Since the thermal power in SRH channels is

approximately 35% less that of the SCW channels, the sheath and the fuel centerline temperatures will be definitely lower than those of the SCW channels. As a result, if a fuel and sheath meet their corresponding temperature limits under the operating conditions of the SCW channels with the maximum thermal power, they will be suitable for the SRH channels as well.

For the SCW fuel channels, the fuel centreline temperature has been calculated at cosine, upstream-skewed cosine, downstream-skewed cosine, and uniform axial heat flux profiles. These heat flux profiles have been calculated based on the Variant-20 fuel bundle. Each of the 42 fuel elements of the Variant-20 fuel bundle has an outer diameter of 11.5 mm while the minimum required thickness of the sheath has been determined to be 0.48 mm. Therefore, the inner diameter of the sheath is 10.54 mm. Inconel-600 was chosen as the material of the sheath.

The examined fuels were UO2, MOX, ThO2, UC, UN, UO2-SiC, UO2-C, and UO2-BeO. For each fuel, the fuel centerline temperature was analysed at the aforementioned AHFPs. Since the maximum fuel centerline temperature was reached at downstream-skewed cosine AHFP for all the examined fuels, only the results associated with this AHFP have been presented in this section. Figures 17 through 19 show the coolant, sheath, and fuel centerline temperature profiles as well as the heat transfer coefficient profile along the heated length of the fuel channel for UO2, UC, and UO2-BeO fuels. Each of these three fuels represents one fuel category (i. e., low, enhanced, high thermal-conductivity fuels). It should be noted that the results presented in Figs. 17 through 19 are based on a 20-,wm fuel-sheath gap.

In addition, Figure 20 shows the maximum fuel centerline temperatures of all the examined fuels. As shown in Figure 20, the maximum fuel centerline temperatures of all examined low thermal-conductivity fuels exceed the temperature limit of 1850°C. On the other hand, enhanced thermal-conductivity fuels and high thermal-conductivity fuels show fuel centerline temperatures below the established temperature limits of 1850°C and 1500°C, respectively.

In regards to sheath temperature, the sheath temperature reached its maximum at downstream-skewed cosine AHFP. Figure 21 provides a comparison between the sheath temperature profiles for the four studied AHFPs. Figure 21 also shows the HTC profiles corresponding to each examined AHFPs. As shown in Fig. 21, unlike uniform AHFP, HTC reaches its maximum value in the beginning of the fuel channel for non-uniform AHFPs (i. e., downstream-skewed cosine, cosine, and upstream-skewed cosine AHFPs). This increase in HTC is due to the fact the sheath temperature reaches the pseudocritical temperature. In contrast, with uniform AHFP, the sheath temperature is above the pseudocritical temperature from the inlet of the fuel channel. Consequently, the peak in HTC at uniform AHFP occurs when the coolant reaches the pseudocritical temperature.

Heated Length

Fig. 21. HTC and sheath-wall temperature profiles as function of AHPF.

A comparison between the examined non-uniform AHFPs shows that in terms of the sheath and fuel centerline temperatures, upstream-skewed cosine AHFP is the most ideal heat flux profile. On the other hand, the downstream-skewed cosine AHFP results in the highest temperatures. Thus, for design purposes, it is a conservative approach to determine the sheath and fuel centerline temperatures based on a downstream-skewed AHFP.

8. Conclusion

Since the development of SCWRs is still in the conceptual design stage, it is worth to further investigate heat transfer and neutronic aspects of high and enhanced thermal-conductivity fuels. In regards to high thermal-conductivity and enhanced thermal-conductivity fuels, this study recommends the use of UC and UO2-BeO, respectively. This use is conditional on the assurance of chemical compatibility, mechanical behavior, and irradiation behavior of these fuels under the SCWR conditions. In addition, the development of new fuel bundle designs, which will comply with the design temperature limits on the fuel and the sheath, is necessary. New fuel-bundle designs, which would result in lower fuel centerline temperatures, also allow for the use of low thermal-conductivity fuels.

Heat transfer at supercritical conditions has been studied by many researchers; however, still there is a need to improve the correlations used to predict the heat transfer coefficient. To the knowledge of the authors, none of the available heat-transfer correlations predicts the deteriorated heat transfer regime. The lack of capability to predict such phenomenon may result in melting of the sheath. Thus, it is significantly important to develop either look-up tables or heat transfer correlations that would predict the deteriorated heat transfer regime.

9. Acknowledgment

Financial supports from the NSERC/NRCan/ AECL Generation IV Energy Technologies Program and NSERC Discovery Grant are gratefully acknowledged.

In this section, several neutron detectors and power monitoring systems are reviewed.

Application of a micro-pocket fission detector for in-core flux measurements is described in section 6.1. SIC neutron monitoring system is examined experimentally and theoretically. Development of an inconel self-powered neutron detector (SPND) for in-core power monitoring will be reviewed in section 6.3. Furthermore, a prototype cubic meter antineutrino detector which is used as a new device for measuring the thermal power as an out-core detection system, will be discussed. Finally, two passive approaches for power measurement are discussed.

2.1 Micro-pocket fission detectors (MPFD) for in-core neutron flux monitoring

There is a need for neutron radiation detectors capable of withstanding intense radiation fields, capable of performing »in-core» reactor measurements, capable of pulse mode and current mode operation, capable of discriminating neutron signals from background gamma ray signals, and that are tiny enough to be inserted directly into a nuclear reactor without significantly perturbing the neutron flux. A device that has the above features is the subject of a Nuclear Engineering Research Initiative (NERI) research project, in which miniaturized fission chambers are being developed and deployed in the Kansas State University (K-State) TRIG A Mark-II research reactor (McGregor, 2005). The unique miniaturized neutron detectors are to be used for three specific purposes (1) as reactor power-level monitors, (2) power transient monitors, and (3) real-time monitoring of the thermal and fast neutron flux profiles in the core. The third application has the unique benefit of providing information that, with mathematical inversion techniques, can be used to infer the three-dimensional (3D) distribution of fission neutron production in the core. Micro-pocket fission detectors (MPFD) are capable of performing near-core and in-core reactor power measurements. The basic design utilizes neutron reactive material confined within a micro-sized gas pocket, thus forming a miniature fission chamber. The housing of the chamber is fabricated from inexpensive ceramic materials, the detectors can be placed throughout the core to enable the 3D mapping of the neutron flux profile in »real-time». Initial tests have shown these devices to be radiation hard and potentially capable of operating in a neutron fluence exceeding 1019 cm-2 without noticeable degradation. Figure 7 shows a cutaway view of the basic detector concept. It consists of a small ceramic structure, within which is a miniature gas-filled pocket.

Fig. 7. Cut away view of MPFD (McGregor, 2005)

A conductive layer is deposited on opposing sides of the device, but not the perimeter. Neutron reactive material, such as 235U, 232Th, 10B, or some material containing 6Li, is applied over the conductive contact(s). Although both sides may be coated with neutron reactive material, only one side needs to be coated for the device to work. The ceramic pieces must be insulators and must not be composed of neutron-absorbing material. For instance, aluminum oxide or oxidized silicon may be used. Connecting wires must be sealed well so that no gas leaks out. Additionally, the ceramic pieces must be sealed with high temperature cement such that the seal integrity is secure within the hostile environment of a reactor core.

By in-core evaluation the device demonstrated excellent count-rate linearity with reactor power. Further, the small size and minute amount of uranium used permitted pulse mode operation without appreciable deadtime distortions or problems. MPFDs have, thus far, shown exceptional radiation hardness to neutrons, gamma rays, and charged-particle reaction products, while showing no performance degradation for devices exposed to neutron fluences exceeding 1019 cm-2. Further, pulse mode operated devices have shown a linear relation to reactor power for neutron fluxes up to 1012 cm-2 s-1, and smaller MPFDs are expected to operate in pulse mode in even higher neutron fluxes. The next generation of MPFDs will be composed of a triad of detectors on a single substrate, one with a 232Th coating, one with a 235U coating, and one with no coating. Such a triad permits monitoring of the fast neutron flux, the thermal neutron flux, and the gamma ray background, all at the same time. Further, the devices behave as point detectors, which greatly simplify data interpretation. Data from such a MPFD array can be converted into a power density map of the reactor core for real-time analysis. Mathematical models are under development that can relate the power density profiles in the reactor’s fuel rods to the flux densities at the detector locations. Key to this formulation is the construction of an appropriate response function that gives the flux at any position in the core to the fast neutrons born at an arbitrary axial depth in any of the core fuel rods. Response functions have been derived and used to illustrate the analysis methods. Thus far, modeled results using predicted sensitivities of the MPFDs indicated that the power density in the fuel can be determined provided that appropriate boundary conditions regarding device placement are met. Good matching to power density profiles can be achieved with as few as five detector triads per detector string.

In analogy with the application of group theory in particle physics, where group theory leads to insights into the relationships between elementary particles, we present an application of group theory to the solution to a specific reactor physics problem. The question is whether it is possible to replace a part of a heterogeneous core by a homogeneous material so that the solution outside the homogeneous region remains the same? This old problem is known as homogenization (Selengut, 1960).

In particular, for non-uniform lattices, asymptotic theory has shown that a lattice composed of identical cells has a solution that is composed of a periodic microflux and a slowly varying macroflux. What happens if the cell geometry is the same but the material composition varies?

In reactor calculations, we solve an equation derived from neutron balance. In that equation, we encounter reaction rates, currents or partial currents. It is reasonable to derive all the quantities we need from one given basic quantity, say from the neutron flux at given points of the boundary. The archetype of such relation is the exiting current determined from the entering current by a response matrix. We show that by using irreducible components of the partial currents, the response matrix becomes diagonal.

The Selengut principle is formulated: if the response matrix of a given heterogeneous material in V can be substituted by the response matrix of a homogeneous material in V, there exist an equivalent homogeneous material with which one may replace V. This principle simplifies calculations considerably, and, therefore, has been widely used in reactor physics. We investigate the Selengut principle more closely(Makai, 2010),(Makai, 1992).

The analysis is based on the analytical solution of the diffusion equation derived in the previous Section. The problem is considered in a few energy groups, the boundary flux F is a vector, as well as the volume averaged flux Ф. Using that solution, we are able to derive matrices mapping into each other the volume integrated fluxes, the surface integrated partial and net currents. The derivation of the corresponding matrices is as follows. Our basis is the boundary flux, that we derive for each irrep i from (4.13). The expression (4.13) has three components. The first one is vector tk which is independent of the position r and is multiplied by an exponential function with r in the exponent. The third component is the weight Wk which is independent of r but varies with subscript k. The product is summed for subscript k, that labels the eigenvalues of the cross-section matrix in (4.14). That expression can be put into the following concise form:

Fi = T < fi > Ci, (5.1)

where Ci comprises the third component. Here < fi(r) > is a diagonal matrix. Note that position dependent quantities like reaction rates, follow that structure. The normal component of the net current is Jnet obtained from the flux by taking the derivative and is given in irrep i as

Leti = -DT < gi > c-i. (“)

We eliminate a to get

JnetA = — DT < gt/fi > T-1 Fi = RiFi. (5.3)

Here n is the outward normal to face Fi,

gi = — Vnfi (r). (5.4)

The volume integrated flux Ф is obtained after integration from (4.13) as

ф = T < FA1 > CAUFA1 = J(5.5)

and the integration runs over volume V of the node. Note that only irrep Al (i. e. complete symmetry) contributes to the average flux because of the orthogonality of the irreducible flux components. After eliminating cai from (5.1), we get the response matrix for determining the volume integrated flux Ф from the face integrated flux Fai:

Ф = T < Fai//ai > T-1 < F > = W < F > . (5.6)

This assures that V is completely described by matrix W and the diagonal matrices < F(r) > , < f (r) >, < g(r) > for each irrep. For example, we are able to reconstruct the cross-section matrix 2 from them. Note that WT = T < F/f >, the eigenvectors of matrix W are the eigenvectors of A. Now we need only a numerical procedure to find the eigenvalues Ak from

< F/f >.

The question is, under what conditions are the above calculations feasible. We count the number of response matrices. The matrix elements we need to characterize V may be all different and the number of matrices depends on the shape of V, since the number of irreducible components of the involved matrices depends on the geometry. In a square shaped homogeneous V, we have four Ri matrices and one W. Altogether we have to determine 5 * G * G elements. In an inhomogeneous hexagonal volume, there are 6 * G * G matrix elements, whereas the homogeneous material is described by G * (G + 1) parameters as in a homogeneous material there are altogether G * G cross-sections and G diffusion constants8. Therefore the Selengut principle is not exact it may only be a good approximation under specific circumstances. Homogenization recipes preserve only specific reaction rates, but they do not provide general equivalence.

5. Conclusions

The basic elements of the theory of finite symmetry groups has been introduced. In particular, the use of the machinery associated with the decomposition into irreducible representations, in analogy with harmonic analysis of functions in function space, in the analysis of Nuclear Engineering problems. The physical settings of many Nuclear Engineering problems exhibit symmetry, as for example in the solution of the multi-group neutron diffusion equation. This symmetry can be systematically exploited via group theory, and elicit information that leads to more efficient numerical algorithms and also to useful insights. This is a result due to the added information inherent in symmetry, and the ability of group theory to define the "rules" of the symmetry and allows one to exploit them.

The effective multiplying coefficient is: the ratio of generated neutrons in every generation to generated neutrons in last generation. So to operate the nuclear reactor in steady state, this parameter should be: 1 means the generated neutrons in every generation are equal with neutrons which have absorbed or leaked in last generation that means: critical state. The minimum value of K, ff is: 0 and maximum of it is: u namely: 2.43.

The effective multiplying coefficient is [5]:

K =n. f. V£.P .P =n. f. V£.P (16)

ef FNL THNL J r TNL

As the thermal fission coefficient (n) is [5, 6]:

Where:

and:

NO = 2NU = 2 = 2——— m-^A——

MU rM235 +(1-r )M 2

Also the thermal absorption coefficient (f) is [5]:

The resonance escape probability for fast neutrons (p) also is calculated as following [5]:

238

(22) (23) (24) (25)

P = ‘

Where:

and:

E. — N235 .o235 + N238.a?8 + N°.o° + NH. oh + N°M. o° + NZr. oZr

If the enrichment of fuel is 100% then the resonance escape probability for fast neutrons (p) will be maximum value.

According to enrichment of applied fuel and its mass can write [7]:

r-mjL

(26) (27) (28) (29) (30) (31)

m

Where [8]:

NLm. Mfm V mU°2 — mfm = A = Pfm^fm

A

and:

dpn dt

and:

G (p) = VGT (p).p (32)

Firstly the transfer function is supposed. Secondly due to mostly reactor’s cores are twin therefore once a signal with delay time (td) from first part to second part is transmitted then that signal with a same delay time from second part to first part will be transmitted.

Chaitanya Deo

Nuclear and Radiological Engineering Programs, George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology USA

1. Introduction

In the process of energy production via fission — and fusion in the years to come — both fuel components and structural materials within nuclear reactors can sustain substantial radiation damage. Regardless of the type of reactor, this damage initially appears in the form of local intrinsic point defects within the material — vacancies and interstitials. The point defects agglomerate, interact with the underlying microstructure and produce effects such as void swelling and irradiation creep. Vacancies provide a pathway for solutes to segregate to grain boundaries and dislocation leading to chemical inhomogeneities that translate into phase transformations and/or property variations in these materials, rendering them unsuitable for the desired application.

Many deleterious effects of irradiation on material properties —e. g. void swelling, irradiation creep, radiation-induced hardening and embrittlement —can be traced back to the formation of the aforementioned point defect clusters and gas bubbles. These effects include such phenomena as swelling, growth, phase change, segregation, etc. For example (Was 2007), a block of pure nickel, 1cm on a side, irradiated in a reactor (to a fluence of say, 1022 n/cm2) will measure 1.06cm on a side, representing a volume change of 20%. The volume change, or swelling, is isotropic and is due to the formation of voids in the solid. Other examples are irradiation growth which is distortion at constant volume, phase changes under irradiation where new phases form as a consequence of diffusion of supersaturated defect concentrations and radiation induced segregation.

In addition, the transmutation of reactor elements produces extrinsic defects such as hydrogen, deuterium and helium. For example, Zircalloy high-pressure-tubes used in light water reactors are known to absorb deuterium which can cause delayed hydride cracking (Cirimello, G. et al. 2006). Similarly, in Pebble Bed Modular reactors and in other technologies based on inert gas cooling, formation of ionic gas bubbles within both fuel and structural materials is common(Was 2007). This is critical in structural materials as their behaviour depends on their microstructure, which is in turn affected by neutron radiation.

Cladding materials are also exposed to fission product gasses produced in the fuel during operation.

The effect of irradiation on materials is a classic example of an inherently multiscale phenomenon, as schematically illustrated in Figure 1 for the case of deformation and plasticity in irradiated materials. Irradiation changes the properties of cladding and duct materials as follows: increase in the ductile to brittle transition temperature; reduction of fracture toughness from low temperature irradiation (below 400oC); irradiation creep, helium embrittlement (above ~500C) and swelling. Experimental irradiation programs can be conducted to test the irradiation induced mechanical property changes, but these take significant time and the conditions are limited. As a consequence, it is desirable and efficient to develop models with reliable predictive capabilities for both, design of new reactor components, and simulation of the in-service response of existing materials.

Although these phenomena have been known for many years (Olander 1981; de la Rubia, Zbib et al. 2000; Was 2007) , the underlying fundamental mechanisms and their relation to the irradiation field have not been clearly demonstrated. Most models and theories of irradiation induced deformation and mechanical behavior rely on empirical parameters fit to experimental data. With improvements in computational techniques and algorithms, it is now possible to probe structure-property connections through the elucidation of fundamental atomic mechanisms. Often, these mechanisms involve defects of different dimensionality that exist and interact with each other to significantly affect material properties.

For irradiated materials, point defects and clusters affect crystal plasticity. Dislocation-defect interactions may be associated with defects in the matrix, the modification of the local elastic constants due to the presence of defects and defect clusters and the effects of clusters and voids on the stacking fault energy. The formation of self interstitial loop rafts and the decoration of dislocations with self interstitial clusters have become important issues for understanding radiation hardening and embrittlement under cascade damage conditions(Wen, Ghoniem et al. 2005). Dislocation motion is thought to be the main mechanism for deformation, because a fairly high density of network dislocations are generated during irradiation, and the dislocation sink strength for point defects is much higher that the grain boundary sink strength for point defects. Computational studies of dislocation activity can be performed at several different length and time scales(Ghoniem, Busso et al. 2003) that are shown in Fig. 1.

Pertinent processes span more than 10 orders of magnitude in length from the sub-atomic nuclear to structural component level, and span 22 orders of magnitude in time from the sub-picosecond of nuclear collisions to the decade-long component service lifetimes (Odette, Wirth et al. 2001; Wirth, G. R. et al. 2004). Many different variables control the mix of nano/microstructural features formed and the corresponding degradation of physical and mechanical properties in nuclear fuels, cladding and structural materials. The most important variables include the initial material composition and microstructure, the thermo­mechanical loads, and the irradiation history. While the initial material state and thermo­mechanical loading are of concern in all materials performance-limited engineering applications, the added complexity introduced by the effects of radiation is a large concern for materials in advanced nuclear energy systems.

The scientific challenge for next-generation extreme materials — whatever their composition — is to understand their failure modes, and to prolong their useful lifetimes by interrupting or arresting these failures. Damage starts with atomic displacements that create interstitials and vacancies, which then migrate and aggregate to form clusters and ever-larger extended structures. Eventually, the damage reaches macroscopic dimensions, leading to degradation of performance and failure. This problem is massively multiscale, covering nine orders of magnitude in its spatial dimension, and neither experiment nor theory has yet captured this complexity in a single framework.

On the experimental side, in situ measurements of neutron irradiation with atomic or nano­scale resolution are needed to observe the initial damage processes, followed by coarser — grained experiments to capture migration, aggregation and ultimately macroscopic failure. The modeling challenge is equally dramatic: kinetic energy from an incident particle is

transferred successively to electronic, atomic, vibrational and structural systems, requiring a diverse mix of theoretical formulations appropriate for different spatial scales.

This chapter will review choices for structural materials for these environments, and review the methods and techniques available for simulating these materials at various length and time scales Methods that will be reviewed will include first principles calculations, molecular dynamics calculations, kinetic Monte Carlo methods, and microstructural mechanics methods.

The selection of candidates for the irradiation target was carried out. The conditions required for the irradiation target solution are as follows:

1. The irradiation target solution has the high molybdenum content for the efficient production of 99Mo.

2. Few activation by-products are generated by target solution irradiation for the prevention of radioactive contamination.

3. The solution has good compatibility with the structural materials of the capsule and the pipes for the prevention of corrosion.

4. The solution is chemically stable and has no generation of precipitation for the prevention of an obstruction to the solution’s flow.

Based on the conditions (1) and (2), two aqueous molybdate solutions (aqueous ammonium molybdate and potassium molybdate solutions) were selected as the candidates for the irradiation target among the aqueous solutions of general molybdenum compounds.

The solubilities of ammonium molybdate ((NH4)6Mo7O24 4H2O) and potassium molybdate (K2MoO4) for pure water are 44 g/100 g-H2O and 182.4 g/100 g-H2O respectively, and the molybdenum contents in the solubilities of (NH4)6Mo7O24^4H2O and K2MoO4 are 23.9 g and 73.5 g respectively.

The activation by-product of (NH4)6Mo7O24 4H2O is only 92mNb. The activation by-products of K2MoO4 are 42K and 92mNb. The y-ray energy emitted from 42K is high. However, by using PZC, it is possible to remove 42K and 92mNb from the two aqueous molybdate solutions irradiated with neutrons.

The conditions (3) and (4) were confirmed by tests with the two solutions.

Uranium mononitride or uranium nitride (UN), which is a ceramic fuel, can be produced by the carbothermic reduction of uranium dioxide plus carbon in nitrogen. This process produces UN with densities in the range of 65 to 90% of TD (Shoup and Grace, 1977). UN has a high melting point, high thermal conductivity, and high radiation stability. These properties enhance the safety of operation and allow the fuel to achieve high burn-ups (IAEA, 2008). In addition, UN has the highest fissile atom density, which is approximately 1.4 times that of UO2 and greater than those of other examined fuels. In other words, when UN is used as a fuel, a smaller volume of fuel is required, which leads to a smaller core. In contrast, one disadvantage of the UN fuel is that under some conditions it decomposes to liquid uranium and gaseous nitrogen (IAEA, 2008), which in turn results in the formation of cracks in the fuel. These cracks increase the chance of the release of gaseous fission products. In addition, the formation of cracks in nuclear fuels has adverse effects on their mechanical and thermophysical properties.

Hayes et al. (1990a) developed a correlation shown as Eq. (12), which calculates the thermal conductivity of UN, in W/m K. This correlation, which is a function of both temperature and percent porosity, can be applied when porosity changes between 0 and 20% for temperatures in the range of 25°C and 1650°C (Hayes et al., 1990a). The standard deviation of the Hayes et al. correlation is ±2.3%.

k = 1.864 exp(-2.14 P) T0361 (12)

Irradiation swelling, growth, and creep are the primary effects of irradiation on a nuclear fuel. Irradiation swelling results in volumetric instability of the fuel at high temperatures while irradiation growth causes dimensional instability of the fuel at temperatures lower than 2/3 of the melting point of the fuel (Ma, 1983). In addition to dimensional and volumetric instability, a continuous and plastic deformation of the fuel due to creep may adversely affect its mechanical properties. Thus, it is required to study the behaviour of the fuel under irradiation specifically the irradiation-induced swelling, irradiation-induced growth and irradiation-induced creep of the fuel.

Ross et al. (1990) developed a correlation for the prediction of percent volumetric swelling of UN fuel. This correlation is shown as Eq. (13), where Tavg is the volume average fuel temperature in K, B is the fuel burn-up in MW day/M g(U), and p%TD is the percent theoretical density of the fuel (e. g., p%TD equals to 0.95 for a fuel with 5% porosity). In addition to this correlation, the volumetric swelling of UN can be calculated based on fuel centerline temperature using Eq. (14) (Ross et al., 1990). The uncertainty associated with Eq. (14) is ±25% for burn-ups above 10,000 MW day/Mg (U) while at lower burn-ups the uncertainty increases to ±60% (Ross et al., 1990). Figure 11 shows the volume expansion of 95% TD UN based on Eq. (14).