**3.1. Criticality calculation**

plication factor represents the criticality because it assumes that no neutrons leak out of the reactor. For a complete description of the life cycle of a real finite reactor, it is necessary to account for the neutrons that leak out. The effective multiplication factor takes this into

The calculation of *K*eff consists of estimating the mean number of fission neutrons produced in one generation per fission neutron started. The *K*eff cycle is thus the computational equivalent of a fission generation, where a cycle is used to denote the computed estimate of an actual fission generation. MCNPX uses three different estimates to set up *K*eff (absorption, collision and track length) estimate. The final result is the statistically combined result for the three

The KCODE card is used together with a number of cards to set up a criticality problem. These cards specify the initial spatial distribution of fission points and include the KSRC card, SDEF card and SRCTP card. The KSRC card sets the initial x, y, z locations of fission points. The SDEF card is used to define points uniformly in volume whilst the SRCTP card is defined from a previous MCNPX criticality calculation. A typical KCODE card used together with a KSRC

The card above indicates that the nominal number of source histories is 10000 with the initial *K*eff guess kept as 1.00. The number of inactive cycles skipped before active *K*eff accumulation is 70 and the total number of cycles that run in the problem is 150. The KSRC card indicates that the x, y, z locations for initial fission source points were taken from the origin. The criticality calculations were performed with help from Bunde Kermit at the U.S Department

In this research, different fuel grades and different cladding materials were used for the same reactor core configuration to investigate the effect of these on the criticality. The different fuel grades used were MOX, UOX and CEU. The materials used for cladding include zirconium, zircaloy and stainless steel. Successive fission cycles were run for determination of the criticality. **Table 2** shows the neutron absorption cross-sections and thermal conductivities for

*σ*a (barns) 3.1 0.17 13.3 2.56 4.49 2.6 750 0.63 0.184 0.22 3.1 k(Wm−1K−1) 93.9 149.2 7.81 79.5 90.9 138 27.4 66.8 22.6 21.5 16

**Table 2.** Neutron absorption cross-sections and thermal conductivities for common clad materials at 25°C.

**Cr Si Mn Fe Ni Mo B Sn Zr Zr-alloy Steel**

neutron absorbed neutron leakage in preceding generation <sup>=</sup> <sup>+</sup> (6)

neutron production from fission in one generation <sup>K</sup>

account. Mathematically, *K*eff is defined as follows (Eq. 6):

KCODE: 10000 1.000000 70 150 KSRC: 0.0000 0.0000 0.0000

eff

44 Nuclear Material Performance

card has the following format:

different clad materials at 25°C [12].

estimates.

of Energy.

The neutron absorption cross-sections show the ease with which a material absorbs thermal neutrons generated from fission in the reactor core. The lower the neutron absorption crosssection, the less permeable a material is to thermal neutrons. The thermal conductivity also shows how efficient a material is in conducting heat. Materials with higher thermal conduc‐ tivities are more efficient conductors of heat than those with low thermal conductivities. **Table 3** shows *K*eff values at the beginning and end of core life (BOL and EOL) as well as the corresponding standard deviations. The *K*eff value used here is a result of the statistical combination of the three different estimates used by MCNP (absorption, collision and scattering). The major control cards used were the KCODE and the KSRC card.

From the neutron absorption cross-section in **Table 2**, the thermal neutron absorption of zirconium and zircaloy are much lower than that of stainless steel. This explains the good *K*eff value obtained for zircaloy and zirconium clad fuels as compared to that of stainless steel. The absorptivity also explains the degree of neutron interaction with the clad material. For zirconium and zircaloy, little neutron is absorbed and hence these neutrons remain in the reactor core and are able to initiate further fission processes. This also makes zirconium very effective in preventing radioactive fission fragments from escaping the fuel into the coolant and contaminating it. Again, when UOX and CEU fuel grade materials were used, similar patterns of *K*eff were obtained with zirconium and zircaloy showing greater results for *K*eff as compared to stainless steel. Even though zirconium has a slightly lower neutron absorption cross-section and comparable thermal conductivity relative to zirconium alloy as shown in **Table 2**, *K*eff results for zircaloy look slightly higher than that of zirconium. This observation may be due to enhanced alloy properties.

The zircaloy cladding used is zircaloy-4, which is similar in composition to zircaloy-2, but has reduced nickel and iron compositions. The reaction of zirconium with steam at high temper‐ atures produces hydrogen gas by the reaction (Eq. 7):


$$\text{Zr} + 2\text{H}\_2\text{O} \rightarrow \text{ZrO}\_2 + 2\text{H}\_2\tag{7}$$


**Table 3.** *K*eff values for different clad materials at the beginning and end of burnup steps for MOX, UOX and CEU fuel.

Oxidation of zirconium metal reduces the ductility and robustness of zirconium metal, and hence increases the probability for the escape of thermal neutrons from the core of the reactor [12]. This further reduces the effectiveness of zirconium for higher and prolonged fuel burnups. With respect to zirconium alloys, the hydrogen produced by oxidation of zirconium in steam diffuses into the alloy, causing the formation of zirconium hydrides. The hydrides formed are less dense and more brittle than the zirconium alloy and leads to the weakening of the clad material. This is especially the case in zirconium-2 alloy. The zirconium-4 alloy has a reduced composition of iron and no nickel composition; this reduces the hydride effect by reducing the tendency to pick up hydrogen. This characteristic of zircaloy-4 used improves its mechanical properties, reducing the probability of escape of thermal neutrons considerably, thus improv‐ ing the overall *K*eff value in the long term.

The thermal conductivities listed in **Table 2** reveal a higher value for zirconium and zircaloy, as compared to that of stainless steel. A high value means that heat can be quickly conducted away from the reactor core to the coolant. This prevents very high temperature buildup in the core leading to the melting of the fuel material or clad. The thermal conductivity of zirconium alloys, with thermal expansivity of nearly one-third that of stainless steel, is superior compared to that of stainless steel and other nuclear fuel storage materials. This also gives zirconium alloys superior dimensional stability at elevated temperatures. MCNP offers a number of statistical checks to assess the validity of a criticality calculation, which can be found in the MCNP user's manual [11]. These were found to be in good agreement with the output file when cross-checked.

#### **3.2. Nuclear fuel reactivity**

Reactivity is the degree of neutron multiplication in the reactor core. This parameter is directly related to the tendency of the reactor core to change power level. Also, control rods can be used to obtain a desired power level change or keep the power level constant by adjusting the reactivity when raised or lowered into the reactor core. Other factors which affect the reactivity include the density and temperature of the coolant or moderator and also the fuel temperature and density.

The reactivity for the three different fuel grades is calculated as a function of burnup, and the peak reactivity determined for each fuel grade is shown in **Table 4**. The reactivity is evaluated as (Eq. 8):

$$
\rho = \frac{\mathbf{k} - \mathbf{l}}{\mathbf{k}} \tag{8}
$$

where *k* is the effective multiplication factor and *ρ* is the reactivity.

The peak reactivity is the highest reactivity obtained as a function of burnup for each fuel grade. Another parameter, the gain, is the difference between the peak reactivity and the reactivity at BOL [13]. The gain is usually used in breeding reactors to characterize the affinity of the nuclear fuel for breeding reactivity. The gain is useful in breeder reactors to measure the fuels affinity to breed 239Pu; however, it is not a very good parameter for determining the neutronics performance of the fuel. One of the most important measures of fuel performance is the peak reactivity. Since the reactivity has a direct bearing on the power level, the higher the peak reactivity, the much higher the power output. The peak and gain reactivities of the three different fuel grades as burnup proceeds are indicated in **Table 4**. Usually, fuels with higher peak reactivities are found to have much lower gain. UOX has much higher peak reactivity, but records the lowest gain in reactivity.


**Table 4.** Reactivity parameters for the different fuel grades.

MOX has a much lower peak value relative to UOX, but records a much higher gain in reactivity. The same is seen for CEU. A look at the compositions of the three fuel grades reveals that 235U forms the main fissile material in both UOX and CEU. Depletion of 235U is known to reduce the reactivity. The buildup of actinides due to neutron absorption of 238U is also known to reduce the reactivity. These two factors occur in both UOX and CEU and might cause loss of reactivity, but this is compensated by the buildup of 239Pu and 241Pu in these two fuels. However, in MOX fuel, added to the depletion of 235U and 238U which reduces the reactivity, the reactivity is further reduced by the depletion of 239Pu and 241Pu, which are the main fissile materials in MOX fuel. This is seen in the relatively low peak value recorded for MOX fuel.
