Figure 3.7 shows a typical relation between H/U ratio and neutron infinite multiplication factor (kTO). The kTO increases as H/U ratio increases in the beginning because large numbers of neutrons are moderated to thermal neutrons by the moderation effect of water (specifically, its hydrogen atoms). However, a further increase in H/U ratio beyond the turning point contrarily leads to a decrease in kTO because the neutron absorption by hydrogen atoms becomes dominant. Therefore, kTO has a maximum value with respect to H/U ratio. The region on the left side of the maximum indicates an undermoderation region. If an H/U ratio which is a little smaller than the maximum is given at the cold temperature, the H/U ratio is decreased by the moderator density decrease and coolant voiding during reactor operation and then kTO decreases. The void coefficient and moderator temperature coefficient become negative. As shown in Fig. 3.8, an increase in H/U ratio leads to a less negative void reactivity coefficient and a large control rod worth because of neutron spectrum softening. In general, BWRs are designed to have an H/U ratio of 4 to 5 for the average void fraction of about 40 % at normal operation.

An increase in fuel enrichment for high burnup leads to a large amount of fissile materials in the fuel, and then the neutron absorption of hydrogen atoms in the moderator is lowered and the maximum value of kTO shifts to a higher H/U ratio as shown in Fig. 3.7. Figure 3.9 shows an example of the dependence of void reactivity coefficient and control rod worth on fuel enrichment. If the H/U ratio is set for low fuel enrichment at the reference core design, the void reactivity coefficient becomes more negative and the control rod worth becomes smaller as the fuel enrichment increases. It is, therefore, necessary to properly adjust the H/U ratio for high fuel enrichment.

As a way to increase H/U ratio, a more slender fuel rod can be made to reduce the fuel inventory, but that is not generally desirable from the viewpoint of fuel

economy. The position and amount of the non-boiling regions inside and outside the channel box are optimized instead as a feature of BWR core structures. The H/U ratio can be increased without changing the fuel inventory by the following measures.

(i) Increase the non-boiling region inside the channel box (water rod region).

(ii) Increase the non-boiling region outside the channel box (water gap region).

Figure 3.10 shows the effect [9] of such a non-boiling region increase on reactivity increase at the cold temperature under a constant fuel inventory; reactivity changes with the change from the normal to cold temperature condi­tion. Figures 3.11 and 3.12 show effects of the non-boiling region increase on void reactivity coefficient and infinite multiplication factor, respectively. To achieve measure (i), the central fuel rods in the fuel assemblies are replaced

with water rods and the fuel rod diameter is increased to maintain the same fuel inventory. The width of the channel box is reduced and the fuel rod pitch is adjusted to use measure (ii).

The suppression of the reactivity increase at the cold temperature and the improvement in the void reactivity depend more highly on the water gap region outside the channel box. It is because the neutrons can be effectively moderated before they are absorbed in the fuel since the water gap region is located relatively far from the fuel rods. On the other hand, the increase of the water gap region leads to a maldistribution of moderator and then it causes a large distortion of thermal neutron flux in the fuel assemblies. Therefore, the neutron absorption rate increases in the water gap region where the peak thermal neutron flux is seen, and the neutron infinite multiplication factor decreases. Fuel assembly design specifications suitable for target characteristics are deter­mined through control and adjustment of reactivity and reactivity coefficients by size and location of the non-boiling water region. Improvement of fuel assembly design for higher burnup can be achieved by increasing the water rod region with a proper H/U ratio to make reactivity high.

(1) Fuel Assembly Size

Large-size fuel assemblies can reduce the number of fuel assemblies to be loaded in core and to be handled in refueling, and therefore they result in an improved capacity factor of a plant. However, a larger size leads to an increase in effective multiplication factor and even one fuel assembly may result in criticality. It is necessary to ensure that one fuel assembly to be handled outside the core should be subcritical in unborated water. For 5 % fuel enrichment, the maximum fuel assembly size to satisfy this condition is a cross-sectional width of about 22 cm.

(2) Fuel Rod Pitch

Fuel rod pitch can be determined by fuel rod diameter and moderator-to — fuel volume ratio Vm/Vf (H/U atomic ratio). Since control rods as well as fuel rods are placed in fuel assemblies, the number and arrangement of fuel rods are determined according to the following items. Control rods are usually withdrawn in normal operation. Therefore, the control rod volume

in fuel assembly is also regarded as the moderator volume in evaluating the moderator-to-fuel volume ratio.

[1] Setting of core geometry

The fast reactor core consists of the core fuel assemblies, the control rod assemblies, and the surrounding blanket fuel assemblies and reflectors (cf. Fig. 4.3). The core fuel is MOX. The blanket fuel is depleted UO2. The core fuel region consists of two types of core fuel assemblies with different plutonium enrichments. The outer core has higher plutonium enrichment to flatten the power distribution. The core fuel assembly consists of the fuel

elements containing upper and lower axial blanket fuels as well as the core fuel. The blanket fuel assemblies surround the core region. They contribute to breeding by efficiently capturing leaking neutrons from the core region and also by reducing neutron leakage to the outside. The blanket region is surrounded by the reflectors in order to further reduce the neutron exposure of the structures.

In the core design, the total length of core fuel elements for target thermal power is calculated first from the linear heat rate. The total number of core fuel elements is calculated from the core height. The number of core fuel elements in a core fuel assembly is determined by selecting the diameter of the core fuel element. Then, the number of core fuel assemblies is determined. The relation between the core thermal power and the number of core fuel assemblies is

(4.5)

Na : Number of core fuel assemblies Q : Core thermal power (MW) q : Average linear heat rate (MW/m)

Hc : Core height (m)

Nc : Number of core fuel elements in a core fuel assembly

The amount of fuel loading is determined from the diameter of the core fuel element. The relationship among the core thermal power, the operation period per cycle, the fuel loading amount, and the number of batches is given by Eq. (4.6).

W = Q x D x Nb/BU (4.6)

W : Fuel loading amount (t)

Q : Core thermal power (MW)

D : Operation period per cycle (d)

Nb : Number of batches (reciprocal of the fraction of refueled assemblies at a refueling)

BU: Average discharge burnup (MWd/t).

The core thermal power and the burnup are set as the design conditions. After selecting the operation period, the variables are the fuel loading amount and the number of batches. The fuel loading amount (diameter of the fuel element) strongly relates to the breeding ratio.

The breeding ratio is one of the important indices for the core characteris­tics of a fast reactor. On the other hand, the core size is limited from the viewpoint of safety because a larger core has a more positive coolant void reactivity. The breeding ratio increases with the fuel volume fraction by making the fuel element thicker and reducing the fuel element pitch. The breeding ratio also increases by making the ratio of core height and diameter closer to unity due to smaller leakage of neutrons. However, those changes lead to more positive void reactivity. Therefore, the following procedures are carried out.

(a) The core height and the thickness of blanket region etc. are preliminarily determined. Then, the possible ranges of the fuel element diameter and the fuel element pitch etc. for achieving the target breeding ratio are identified through parametric surveys.

(b) The possible ranges of the core height for achieving the target breeding ratio and the coolant void reactivity below the limit are identified through a parametric survey. Generally, the ratio of the core height and diameter is set small to reduce the void reactivity and coolant pressure drop. Designs so far have adopted ratios of 0.3-0.5 [9].

(c) The possible range of the thickness of the blanket region for achieving the target breeding ratio is identified through a parametric survey.

(d) The number of batches to achieve the target fuel burn-up with the designed operation period is considered.

As described above, the core characteristics are iteratively evaluated by changing the design parameters and finally the design specifications for achiev­ing the design targets are determined.

The core configuration is generally made as symmetric as possible from the viewpoint of flattening the power distribution [9]. The total number of the core fuel assemblies and control rod assemblies is set as a multiple of 6 plus 1. The core fuel region including the control rod assemblies is divided into two regions: inner core and outer core.

The heights of the upper and lower axial blanket fuels are determined so as to achieve the designed breeding ratio. The number of radial blanket layers is determined in the same manner.

The number of control rods is determined so as to ensure the necessary reactivity worth. In the designs so far, 7-10 % of the core fuel region was occupied by the control rod assemblies [9]. The control rod assemblies are symmetrically arranged for flattening the power distribution, and the influences of the inserted rods and fully withdrawn rods at normal operation on the power distribution are also considered.

The reactivity changes due to changes in temperature and fuel burnup in reactors have been described in this chapter. Control rods, burnable poisons, and chemical shim were introduced as methods to control the reactivity. In the reactor core design, it is necessary to manifest that there is enough margin in such control elements to the total reactivity requirement for control. This is called an evaluation of control reactivity balance.

Table 1.4 shows examples of the control reactivity balance for the PWR and BWR. The total reactivity requirement for control, that is, the core reactivity with all control elements withdrawn from the core, is the excess reactivity. Because the reactivity of the system is reduced due to the consumption of fissile nuclides and the accumulation of FPs with fuel burnup (burnup defect) and also because the tem­perature defect and power defect of the reactivity lead to a negative feedback effect, the excess reactivity is largest at no burnup and cold shutdown. Temperature defect, power defect, reactivity worth of Xe and Sm, and burnup defect are included in the excess reactivity.

The control reactivity worth is estimated by piling up the reactivity worth of the individual control elements; i. e., the control rods, burnable poisons, and chemical shim (for the PWR). The control rod worth is the sum of the reactivity worth values of individual control rods, as a conservative margin, with the exception of the most reactive control rod stuck in the full out position from the core. This is called the “stuck-rod criterion”.

The so-called “shutdown margin” is obtained by subtracting the excess reactiv­ity at no burnup and cold shutdown from the control reactivity worth. The control elements must necessarily provide a shutdown margin.

A suitable design margin is practically evaluated from the accuracy in nuclear design for the excess reactivity and control reactivity worth and it is considered for the control reactivity balance.

In the reactor stability analysis, first governing equations are established for normal and perturbed values of state variables from the plant dynamics codes and then the governing equations are linearized for the perturbed one. The equations are converted into a frequency domain by the Laplace transform and analyzed there. This is referred to as linear stability analysis and the procedure for it in the frequency domain is shown in Fig. 2.43.

In the figure, the system transfer function (closed-loop transfer function) is defined with the open-loop transfer functions, G(s) and H(s) The system stability is characterized by the poles of the transfer function (the roots of the characteristic equation in its denominator) and described by the decay ratio. The concept and the stability criteria are given in Fig. 2.44. For the system to be stable with damping, all the roots of the characteristic equation must have negative real parts. The decay ratio, which is defined as the ratio of two consecutive peaks of the impulse response of the oscillation for the represen­tative root (the root nearest to the pole axis), depends on the calculation mesh size. The decay ratio is determined by extrapolation to zero mesh size following calculations with different mesh size.

The following should be considered in the stability analysis of nuclear reactor design.

(i) Channel stability: This is thermal-hydraulic stability of the fuel cooling channel.

(ii) Core stability: This is the nuclear and thermal-hydraulic coupled stability where the whole core power (neutron flux) regularly oscillates as a fundamental mode due to the moderator density feedback

(iii) Regional stability: This is a kind of core stability and the neutron flux of each region oscillates as a high-order mode by reciprocally going up and down.

(iv) Plant stability

(v) Xenon stability (Xe spatial oscillation): This is a function of accumulation and destruction of FP Xe, and spatial change of neutron flux

The channel stability is thermal-hydraulic stability of single coolant channel in the core. The core stability is the nuclear and thermal-hydraulic coupled stability where the whole core power (neutron flux) regularly oscillates as a fundamental mode due to the moderator density feedback. The regional stabil­ity is a kind of core stability and the neutron flux of each region oscillates as a high-order mode by reciprocally going up and down. These three stabilities are inherent characteristics of nuclear reactors with a large change in moderator density in the core such as BWRs, and hence they are evaluated by the frequency-domain stability analysis. The plant stability is the stability of plant system including its control system and is evaluated by time-domain analysis using a plant dynamics analysis code. Xenon spatial oscillation is caused by accumulation and destruction of Xenon-135, and spatial change of neutron flux. The xenon stability is analyzed by the production and destruction equation of Xenon-135.

Fresh uranium fuel is usually enriched to about 3-4 wt% of 235U on average for a fuel assembly. 235U decreases with burnup and part of the 238U is converted to plutonium through neutron capture. Typical spent fuel contains about 1 wt% of 235U and about 1 wt% of plutonium converted from 238U. 239Pu and 241Pu account for about 60-70 % of the plutonium. MOX fuel is an oxide mixture of plutonium recovered from spent fuel and natural or depleted uranium, containing about 3-5 wt% of plutonium per fuel assembly. The difference between MOX fuel and uranium fuel is that plutonium is blended into the fresh fuel and its amount is larger than that in uranium fuel. This brings about a change in core characteristics due to the different nuclear characteristics of plutonium. As shown in Fig. 3.25, plutonium has a larger neutron absorption cross section in thermal and resonance regions than uranium, and a smaller delay neutron fraction. The different characteristics have the following effects on core characteristics.

(i) MOX fuel reduces the number of thermal neutrons and the neutron spectrum is hardened, and therefore it causes a decrease in neutron absorption of the control rods and boric acid water. It is necessary to assure the core shutdown capability even with one control rod stuck in the fully withdrawn position or with boric acid water injection under the condition of all control rods stuck.

(ii) In connection with the large resonance absorption cross section of 240Pu, MOX-fueled BWR cores have a larger reactivity change relative to a change in void fraction than uranium-fueled cores. This tends to give a more negative void reactivity coefficient. It is, therefore, necessary to check the impact on core characteristics and stability at transients.

(iii) Since MOX fuel rods tend to generate a relatively higher power when loaded together with uranium fuel rods in the fuel assembly, it is necessary to assume that the maximum linear heat generation rate and MCPR at normal operation meet the operation criteria. Especially, fuel rods facing the water gap region generate a large power and therefore particular attention should be given to the fuel rod arrangement in the fuel assembly.

In use of MOX fuel, the plutonium enrichment and MOX fuel inventory are adjusted based on the effects mentioned above. Actually, MOX-fueled cores are designed to have a margin to meet changes of the characteristics, consid­ering the range of variation in various factors due to nuclear calculation error and future fuel design changes. Existing reactors have been evaluated as able to operate with MOX fuel replacing about 1/3 of the core fuel [23].

The BWR MOX fuel used in the initial step of the Japanese plan for Pu-thermal utilization has the same structure as that of Step II fuel because of the rich operating experiences already available for uranium fuel. The dis­charge burnup of the MOX fuel is about 33 GWd/t, which is just slightly lower than the 39.5 GWd/t discharge burnup of Step II uranium fuel. The MOX fuel is being introduced with repeated usage experience. The Step II fuel assembly developed for high burnup has a large diameter water rod to improve the neutron moderation effect. The water rod has the effect of getting a less negative void reactivity coefficient. In design of the MOX fuel rod, the active height of the fuel rod is shorter considering that the FP gas release rate is

Uranium Fuel Rods

slightly higher than that of uranium fuel. Figure 3.26 shows a fuel rod arrange­ment in an MOX-fueled assembly. Low enrichment plutonium fuel rods are arranged on the periphery as a measure to get local power peaking like done for use of uranium fuel. Experienced gadolinia-added uranium fuel rods are arranged among MOX fuel rods in the assembly as burnable poison rods for excess reactivity control.

In ABWRs, the fuel assembly size is the same as that of BWRs, but the fuel assembly gap was expanded to enlarge the non-boiling region outside the channel box, which increases the water-to-fuel volume ratio. This mitigates an increase in negative void reactivity coefficient by MOX fuel loading and a reduction in reactivity worth by control rods and boric acid water. Sufficient thermal margin and reactor shutdown capability were obtained from a 100 % MOX-fueled ABWR core and the MOX fuel loading has been confirmed to have a high flexibility [24].

Differently from uranium fuel, MOX fuel does not need an enrichment process, and the fraction of plutonium oxide to be mixed with depleted uranium oxide is only increased for high burnup. Such a high burnup with no enrichment cost increase has a large effect on the fuel cycle cost. A high burnup MOX fuel is being developed through irradiation tests in experimental reactors the same approach as taken for developing the high burnup uranium fuel.

Plutonium is an a emitter. Secondary reactions of the released a particles and the self-fission of 240Pu release neutrons. MOX fuel treatment must pay more attention to radiation protection and heat generation than uranium fuel treat­ment. Features and measures of radiation and decay heat in MOX fuel are mentioned in Sect. 3.3.6.

In normal operation, a failure of the system or an operator’s error causes an abnormal transient state for which maximum linear power density is evaluated to assure that fuel rods do not exceed the allowable design limits (no melting of fuel, no occurrence of DNB, etc.). Two typical transients are evaluated here as an abnormal transient in operation.

(1) Abnormal withdrawal of control rods at power

An abnormal withdrawal of control rods by a malfunction of the control rod control system or an operator’s error causes an increase in power and a change in power distribution. However, the withdrawal of rod cluster control assemblies is stopped or the reactor is shutdown by the reactor protection systems, such as the high neutron flux, the over-temperature AT and over-power AT. Control rods could be withdrawn from an arbitrary position to the fully withdrawn.

The fast reactor core designs so far have been mainly pursuing the breeding performance as seen in the so-called fast breeder reactor from the viewpoint of ensuring future energy sources. In the twenty-first century, significant growth of energy demand is expected mainly in developing countries. On the other hand, the issues of wasting natural resources and environmental destruction have recently become obvious. It has become internationally recognized that the world must go toward sustainable development with resource conservation and consideration of environmental issues as well as assurance of stable energy sources.

Japan, for example, has only small amounts of natural resources. It is essential to develop technologies which save resources, do not emit greenhouse gases, and lead to small loads on waste disposal. The FBR cycle was selected as one such technology. A high capacity for energy supply and the technologies for burning transuranium elements (TRU: Pu, Np, Am, Cm) have been devel­oped for the FBR. The FBR must have high economy which is competitive with other power generation methods while ensuring safety as a major premise. Also, non-proliferation must be considered according to the world political situation. From this background, consideration of a concrete scenario for deploying the FBR cycle has been started in Japan. Japan’s activities are attracting the world’s rapidly increasing interests in FBR deployment, and international corroborations are strongly desired.

Taking Japanese experience as an example, it is important to develop, in the early stage, the FBR cycle which has international competitiveness by improv­ing the performance of waste management and proliferation resistance and that goes toward a rational transition from the LWR cycle to the FBR cycle. This will contribute to sustainable development.

In the Fast Reactor Technology Development (FaCT) Project of Japan, the indexes of design targets are set from viewpoints of: safety; sustainability (environmental protection, waste management and resource efficiency); econ­omy; and proliferation. Furthermore, they are determined in consideration of consistency to requirements of international collaborative programs that have already taken place such as GEN-IV*1 and INPRO*2.

GEN-IV is the fourth generation nuclear power plant system. The first generation indicates the early prototype reactors such as at Shippingport (PWR) and Dresden (BWR). The second generation indicates the following commer­cial reactors i. e. PWR, BWR, CANDU and VVER, RBMK. The third genera­tion is the improved designs of the second generation systems. ABWR, APWR and EPR mainly pursue economy by scaling up. The reactors with the passive safety system such as AP1000 and ESBWR are also third generation. The fourth generation will follow the third generation and is assumed to have the following characteristics: (i) economically competitive with natural gas ther­mal power plants; (ii) higher proliferation resistance; (iii) higher safety; and

(iv) minimum load for waste management.

*2 INPRO (International Project on Innovative Nuclear Reactors and Fuel Cycles) is one of the IAEA programs to help prepare infrastructures aimed toward deployment of innovative nuclear systems which have safety, economy, proliferation resistance, etc.

The production process of 135Xe is presented in Fig. 1.5. In the decay chain of mass number 135 related to 135Xe production, 135Sb, 135Te, 135I, and 135Xe are the main FPs. Because 135Sb and 135Te are short-lived nuclides, they can be assumed to decay very rapidly to 135I.

The time-dependent nuclide concentrations of 135I and 135Xe are represented by /(t) and X(t), respectively, the fission yields (FP yields per fission shown in Table 1.1) are represented by у: and yXe, and the decay constants are represented by 2/ and 2Xe. Assuming the thermal absorption cross section of 135Xe is WXе and time-independent neutron flux is ф, the production-destruction equations of /(t) and X(t) can be represented as follows.

(1.23)

(1.24)

[1] Solution at initial startup

When the reactor is started up from a clean condition in which /(0) = X(0) = 0, the 135I and 135Xe concentrations can be obtained as shown in Eqs. (1.25) and

(1.26) .

(1.25)

(1.26)

Within enough time after the reactor startup, the concentrations approach equilibrium values, Ieq and Xeq, which can be found at t ! 1 in Eqs. (1.25) and

(1.26) as follows.

(1.27)

(1.28)

These equations can be directly obtained by placing the time derivatives in Eqs. (1.23) and (1.24) equal to zero.

Upon inserting Eq. (1.28) into Eq. (1.22), the reactivity change due to equilibrium 135Xe is found to be

ofeXeq _ Гі + ухе Ф vlfpe vpe ЛХе/CFа Є+Ф

If, ф ^ XXe jo X the negative reactivity increases linearly with ф. On the other hand, if ф ^ XXe joX, the negative reactivity takes its maximum value

To estimate the magnitude of the maximum value, suppose that a thermal reactor is fueled with 235U and contains no resonance absorbers and fast fission materials. In this case, p = є = 1, and the negative reactivity gives

by using the fission yield from 235U in Table 1.1 and v = 2.42. This is a considerably large reactivity which is about —4.2 dollars for the delayed neutron fraction of 0.0064 in thermal fission of 235U.

[2] Solution after shutdown

Although the production of 135I and 135Xe by fission and the transmutation of 135Xe by thermal neutron absorption cease when the reactor is shut down, 135Xe continues to be produced as the result of the decay of 135I present in the system. It eventually disappears by its own decay.

After shutdown, the production-destruction equations of 135I and 135Xe are given by the next expressions.

(1.30)

(1.31)

If the concentrations at shutdown are 10 and X0, the concentrations at a later time t can be written as follows.

(1.32)

If 135I and 135Xe had reached equilibrium prior to shutdown, then 10 and X0 would be given by Eqs. (1.27) and (1.28), and the concentration of 135Xe becomes

The resulting reactivity change due to 135Xe at a later time t after shutdown is given by Eq. (1.35).

(1.35)

Figure 1.6 shows the reactivity change due to 135Xe buildup in a 235U-fueled thermal reactor after shutdown as calculated by Eq. (1.35). The buildup of 135Xe rises to a maximum, which occurs at about 10 h after shutdown, and then decreases to zero. It should be particularly noted that the buildup of 135Xe is greatest in reactors which have been operating at the highest flux before shutdown. This gives rise to a reactor dead time in the operation of high-flux reactors, during which time the reactor cannot be restarted. This situation is indicated in Fig. 1.6 during the time interval from ta to tb, where the horizontal line represents a hypothetical reactivity margin of 0.2 Ak/k by withdrawal of all

control rods. As a countermeasure to reduce the reactor dead time, a gradual reduction in reactor power and the shutdown at a low neutron flux can suppress the buildup of 135Xe.

Figure 1.7 shows the negative reactivity change due to 135Xe in a reactor that is returned to full power just at the end of the dead time. If a reactor is restarted while a large amount of 135Xe is present in the system, it should be noted that the subsequent burnout of this poison substantially increases the reactivity of the reactor.

The cross sections stored in the evaluated nuclear data file are continuous — energy data, which are converted to multi-group cross sections through energy discretization in formats suitable for most of nuclear design calculation codes, as shown in Fig. 2.5. The number of energy groups (N) is dependent on nuclear design codes and is generally within 50-200.

Multi-group cross sections are defined that integral reaction rates for reac­tions (x = s, a, f,—) of a target nuclide i) should be conserved within the range

9.223500+4 2.330250+2 0

1.935800+8 1.935800+8 0

140 2 0

1.000000-5 0.000000-0 2.530000-2 3.000000+4 2.007500+0 3.250000+4 3.750000+4 1.886600+0 4.000000+4 5.000000+4 1.808200+0 5.500000+4 6.500000+4 1.757500+0 6.999990+4 7.999990+4 1.670400+0 8.499990+4 9.500000+4 1.572400+0 1.000000+5 1.250000+5 1.479300+0 1.375000+5 1.625000+5 1.445300+0 1.750000+5 2.000000+5 1.354000+0 2.250000+5 2.750000+5 1.250200+0 3.000000+5 3.500000+5 1.231800+0 3.750000+5 4.250000+5 1.203400+0 4.500000+5 5.000000+5 1.147100+0 5.250000+5 5.750000+5 1.135500+0 6.000000+5 6.500000+5 1.133440+0 6.750000+5

Energy

Discretization

Fig. 2.5 Multi-group cross section by energy discretization

of energy group g (AEg), as shown in Eq. (2.13). The neutron flux and cross section in group g are defined as Eqs. (2.14) and (2.15), respectively.

/йЕд аІ (Е)ф(Е)сІЕ аІдфк

Фя=/ле, Ф^е^Е

, ІЛЕоІІЕ)ф{Е)йЕ

L*(M)dE

The neutron spectrum of Fig. 2.1 can be used as ф(Е) in Eq. (2.15) for thermal reactors. However, while those multi-group cross sections are intro­duced to calculate neutron flux in reactors, it is apparent that the neutron flux is necessary to define the multi-group cross sections. This is recursive and con­tradictory. For example, the depression of neutron flux due to resonance, shown in the 1/E region of Fig. 2.1, depends on fuel composition (concentrations of resonance nuclides) and temperature (the Doppler effect). It is hard to accu­rately estimate the neutron flux before design calculation. In actual practice, consideration is made for a representative neutron spectrum (фw (E): called the “weighting spectrum”) which is not affected by the resonance causing the depression and is also independent of position. Applying it to Eq. (2.15) gives Eq. (2.16).

(2.16)

Since these microscopic cross sections are not concerned with detailed design specifications of fuel and operating conditions of the reactor, it is possible to prepare them beforehand using the evaluated nuclear data file. The depression or distortion of neutron flux does not occur for resonance nuclides at sufficiently dilute concentrations and this is called the infinite dilution cross section (oiooxg). On the other hand, the cross section prepared in Eq. (2.15) compensates for the neutron spectrum depression due to resonance in the 1/E region by some method, and is called the “effective (microscopic) cross section” (&le/f xg). There are several approximation methods for the effec­tive resonance cross section: NR approximation, WR approximation, NRIM approximation, and IR approximation. For more details, references [1, 6] on reactor physics should be consulted. The relationship between the effective cross section and infinite dilution cross section is simply discussed here.

Since neutron mean free path is long in the resonance energy region, a homogenized mixture of fuel and moderator nuclides, ignoring detailed struc­ture of the materials, can be considered. If a target nuclide (i) in the mixture has a resonance, it leads to a depression in the neutron flux at the corresponding resonance energy as shown in Fig. 2.6. The reaction rate in the energy group including the resonance becomes smaller than that in the case of no flux depression (infinite dilution). Hence, the effective cross section is generally smaller than its infinite dilution cross section. The following definition is the ratio of the effective cross section to the infinite dilution cross section and called the self-shielding factor.

Fig. 2.6 Neutron flux depression in resonance

a о: Intermediate

Small

Depression of ф(Е) with the

Background Cross Section a,

In Fig. 2.6, the neutron flux depression becomes large as the macroscopic cross section (No*) of the target nuclide becomes large. On the other hand, if the macroscopic total cross section {^.NJaj) of other nuclides (assuming that they have no resonance at the same energy) becomes large, the resonance cross section of the target nuclide is buried in the total cross section of other nuclides and then the flux depression finally disappears when the total cross section is large enough. To treat this effect quantitatively, a virtual cross section (the background cross section) of the target nuclide, converted from the macro­scopic total cross section, is defined as Eq. (2.18).

Ы=^NjaCN‘(2 18)

j*i

The self-shielding factor is dependent on the background cross section and the temperature of resonance material because the resonance width varies with the Doppler effect. In fact, the background cross section and material temper­ature are unknown before design specifications and operating conditions of a reactor are determined. Hence, self-shielding factors are calculated in advance and tabulated at combinations of representative background cross sections and temperatures. Figure 2.7 shows a part of the self-shielding factor table (f table) for the capture cross section of 238U. The self-shielding factor in design calculation is interpolated from the f table at a practical background cross section (o0) and temperature (T). The effective microscopic cross section for neutron transport calculation can be obtained from the self-shielding factor as

where T0 is normally 300 K as a reference temperature at which the multi-group infinite dilution cross sections are prepared from the evaluated nuclear data file.

Recent remarkable advances in computers have made it possible that continuous-energy data or their equivalent ultra-fine-groups (groups of several tens of thousands to several millions) are employed in solving the neutron slowing-down equation and the effective cross sections of resonance nuclides are directly calculated based on Eq. (2.15). Some open codes, SRAC [7] (using the PEACO option) and SLAROM-UF [8] for fast reactors, adopt this method. This method can result in a high accuracy for treating even interference effects of multiple resonances which cannot be considered by the interpolation method of a self-shielding factor table, though a long calculation time is required for a complicated system.