To shutdown a nuclear reactor by insertion of control rods, it is necessary to have a reactor shutdown margin which leads to subcriticality (e. g., >1.0 % Ak/k) with the exception of the most reactive control rod being stuck in the full-out position from the core (one rod stuck). For the reactor shutdown margin, burnable poison is partially mixed with fuel pellets to suppress the initial reactivity of fuel, and as well, a sufficient control rod worth is secured. The excess reactivity of core is restrained and controlled by both the control rods and burnable poison.

The worth per control rod is restricted below a constant value to secure safety against accidental withdrawal of control rods. The maximum reac­tivity insertion rate due to withdrawal of control rods is also limited to supervise and control the withdrawal safely.

Fresh fuel storage consists of reinforced concrete facilities to store fresh fuel carried into the power plant until fuel loading after acceptance inspec­tion. Fuel is stored vertically in metal racks in a dry condition. Fresh fuel storage is constructed to prevent water flooding and is able to drain water which may have entered the storage.

The capacity for fresh fuel storage is designed considering that spent fuel storage pool accommodates fresh and reloading fuel assemblies in one refueling. The fresh fuel storage of ABWRs is designed to be able to accommodate 5 % of the fuel assemblies of the entire core.

Fresh fuel storage racks are designed to assure that the effective multi­plication factor is maintained at less than 0.95 even in a severe condition such as water flooding when the maximum amount of fresh fuel is being stored; this is done by holding a proper fuel assembly spacing to prevent criticality of storage fuel. In the analysis of subcriticality in fresh fuel storage, the infinite multiplication factor of fresh fuel in being loaded into the core is assumed to be 1.30 for a sufficient safety evaluation.

Fuel storage and handling facilities of PWRs consist of fresh fuel storage facilities, spent fuel storage facilities, spent fuel pit water cleanup system, refueling crane, spent fuel pit crane, fuel handling crane, and fuel-carrying facilities [45]. Figure 3.70 illustrates a schematic view of fuel storage and handling facilities and Table 3.14 gives the main specifications of the fuel storage facilities.

Fuel storage and handling facilities are designed to prevent criticality in any cases to be expected by using a geometrically safe arrangement or other suitable means. Therefore, fuel storage facilities are designed to secure subcriticality even when fully loaded and flooded with full-density unborated water. Fuel handling facilities are designed to have a structure in which only one fuel assembly can be handled independently to prevent criticality.

(1) Fresh Fuel Storage Facilities

Fresh fuel storage is constructed of reinforced concrete, established in an independent area of the fuel handling building. Fresh fuel assemblies are stored one by one in stainless steel racks in a dry condition. The fresh fuel storage is constructed to prevent water flooding and to be able to drain water when it enters the storage.

The storage capacity of the fresh fuel storage is usually set considering the number of fresh fuel assemblies to be loaded and alternative storage of fresh fuel in the spent fuel pit. In Table 3.14, the storage capacity corresponds to about 23 % of the whole core fuel. Fresh fuel storage racks are designed to assure that the effective multiplication factor of less than 0.95 must be maintained even in a severe condition such as unborated water flooding for the maximum storage amount of fresh fuel, by holding a proper fuel assembly spacing to prevent criticality of storage fuel. Moreover, the fresh fuel storage racks are designed to prevent criticality even filled with unborated water of the optimum moderating density.

The nuclear design code system is validated by the experimental data of critical assemblies such as the VHTRC [42]. The VHTRC was constructed to study the nuclear characteristics of pin-in-block type HTGRs. The vertical cross section of

Steel frame

Graphite reflector

Heating wire

Fuel rod insertion hole

• Fuel rod A Safety rod ■ Control rod ® Neutron source

Fig. 4.32 Cross section of very high temperature reactor critical assembly (VHTRC)

the VHTRC is shown in Fig. 4.32. The VHTRC consists of two half assemblies. Each half assembly is formed by piling up the horizontal hexagonal graphite blocks and by fixing them into position by the steel frame. Each fuel block is equipped with the holes for loading fuel rods and inserting a control rod or a safety rod. The layout and loading number of the fuel rods can be easily changed. One of the half assemblies is fixed and the other is movable. Each fuel rod is formed by loading the fuel compacts into the graphite sleeve. Heating wires can be inserted into the graphite blocks so that experiments at desired temperatures can be done.

The major specifications of the VHTRC and the HTTR are compared in Table 4.8. The experimental data of the VHTRC, such as effective multiplica­tion factor, control rod worth, burnable poison worth, power distribution (reaction rate distribution of copper) and temperature coefficients, were used for validation the nuclear design code system [43, 44].

(1) Effective multiplication factor

The effective multiplication factor is measured by the critical approach experiment where the fuel rods are sequentially loaded into the core. The effective multiplication factor for the ideal condition is calculated by correcting the measured one considering the effects of insertions and temperature. This ideal value is used for validation. The reactivity worth of insertions such as control rods, detectors, etc. is measured by the period method or the pulsed neutron technique.

Since the error between the experimental data and the predicted values was within 1 %Ak, the calculation error used for the nuclear design of the HTTR was determined as 1 %Ak.

(2) Control rod worth

The control rod worth is measured by inserting the mock-up control rod into the critical core and then measuring the subcriticality by the pulsed neutron technique. The error between the measured and calculated values was evaluated as 2.6 %. The calculation error was conservatively deter­mined as 10 %.

The burnable poison worth was measured as well. Although the nuclear design code system accurately predicted the measured value, the calcula­tion error was conservatively determined as 10 %.

(3) Power distribution

Using the proportional relationship among reaction rate of copper, neutron flux and fission rate, the calculations of the neutron flux distribution and the power distribution were validated by the measured reaction rate distribu­tion of copper. In the measurement, copper foils were horizontally and axially inserted into the core and criticality was kept for a fixed period. The reaction rate of copper was obtained from the activation rates of the irradiated copper foils.

The calculated and measured axial reaction rate distributions of copper are shown in Fig. 4.33. The distribution is normalized so that the average reaction rate is 1.0. From that figure, the calculated distribution agrees well with the measured one in the fuel region. Since the errors between the calculation and the measurement were within 3 % for both radial and axial directions, the calculation error was determined as 3 %.

(4) Temperature reactivity coefficients [45]

The core was kept critical at room temperature and then the temperature was elevated using the heating wires. The subcriticality at the elevated temperature was measured by the pulsed neutron technique. From the relation between the temperature elevation and the decrease in the reactiv­ity, the temperature reactivity coefficients were obtained. The measure­ments were carried out for temperatures to about 200 °C. The nuclear design code system well reproduced the measured values. The error was 6 % at maximum. The calculation error of the moderator temperature coefficient and the Doppler coefficient for the reactor transient analyses were determined as 10 %.

Fig. 4.33 Comparison of measured and calculated axial copper reaction rate distributions

Fuel region Graphite reflector

Distance from core center [cm]

Based on those validations, the expected calculation errors in the nuclear design were determined. They are summarized in Table 4.9 with the errors used for core design [43].

The temperature coefficient in a thermal reactor can be derived by using the six-factor formula,

k TjfpsPtnl Pfnl. (1.55)

In this, k is the effective multiplication factor, n is the average number of fission neutrons emitted per thermal neutron absorbed by fuel, f is the thermal utilization factor, p is the resonance escape probability, є is the fast fission factor, and Ptnl and Pfnl are the non-leakage probabilities of thermal and fast neutrons, respectively. Recalling the definition of reactivity

(1.56)

and assuming k is close to unity, the temperature coefficient can be expressed approximately in the form of the sum of each temperature coefficient of the six factors as

dp i dk ^ 1 dk

aTi~^T~~^~dTi~~k~dTi

1 drjі 1 df 1 dp 1 dPtnl. 1 dPfnl

~ VdTt f dTi p dTi є dTi Ptnl dTi Pfnl dT

= CCpi ~~ OCti OCpt + OCpt ~~ ОСт™ь OCpfNL ‘

(1.57)

The principal temperature effects of most thermal reactors are the variation in resonance absorption (the Doppler effect) and the fuel expansion due to a change of fuel temperature, the variation in neutron spectrum and the moderator expansion due to a change of moderator temperature, and the expansion of other materials such as coolant (apart from moderator) or structure. These phenomena are discussed through n, f, and p. Since the temperature coefficients of є, PTNL, and PFNL are generally small, on the order of 10_6^k/k/K, they are omitted in the discussion.

So far the diffusion equation has been discussed in a non-multiplying medium with an external neutron source. Here, fission neutron production (fission source) in a reactor core is applied to the diffusion equation. Assuming a volumetric fission reaction rate If ф and an average number of neutrons released per fission v in a unit cubic volume (1 cm3) leads to a volumetric fission neutron production rate by multiplying both ones. Hence, substituting the fission source vIf ф for the external source (S) in Eq. (2.41) gives the one-group diffusion equation at the steady-state reactor core as the following.

DV20 — Іаф + vZf(f) = 0 (2.66)

This equation represents a complete balance between the number of neutrons produced by fission and the number of neutrons lost due to leakage and absorption, in an arbitrary unit volume at the critical condition of the reactor. In actual practice, however, it is hard to completely meet such a balance in the design calculation step. A minor transformation of Eq. (2.66) can be made as

(2.67)

Keff

where keff is an inherent constant characterizing the system, called the eigen­value. If keff = 1.0, Eq. (2.67) becomes identical to Eq. (2.66). The equation may be solved for the eigenvalue kf

(2.68)

where L, A, and P denote neutron leakage, absorption, and production, respec­tively. If the space is extended to the whole reactor core (i. e., each term is integrated over the whole reactor core), keff can be interpreted as follows. The numerator is the number of neutrons that will be born in the reactor core in the next generation, whereas the denominator represents those that are lost from the current generation. Hence, the condition of the reactor core depends on the keff value as given next.

(keff > 1 : Supercritical

(2.69)

[keff < 1 : Subcritical

Another interpretation can be given for kef in Eq. (2.67). As mentioned above, it is highly unlikely to hit on the exact neutron balance of L + A = P in a practical core design calculation. Neither the supercritical (L + A < P) nor subcritical (L + A > P) condition gives a steady-state solution to Eq. (2.66). Hence, if keff is introduced into the equation as an adjustment parameter like Eq. (2.67), the neutron balance can be forced to maintain (P! P/kejf) and the equation will always have a steady-state solution. In a supercritical condition (keff > 1), for example, the neutron balance can be kept by adjusting down the neutron production term (vZfp! vZfp/keff).

A problem expressed as an equation in the form of Eq. (2.67) is referred to as an eigenvalue problem. By contrast, a problem with an external neutron source such as Eq. (2.41) is referred to as a fixed-source problem. The big difference between both equations is that the eigenvalue problem equation has an infinite set of solutions. For example, if p is a solution of Eq. (2.67), it is self-evident
that 2ф and 3ф are also other solutions. Consequently, the solutions of the eigenvalue problem are not absolute values of ф but the eigenvalue kef and a relative spatial distribution of ф. Meanwhile, the fixed-source problem has a spatial distribution of absolute ф which is proportional to the intensity of the external neutron source.

[1] Reactivity for reactor operation

The reactivity of nuclear reactors is defined in terms of the effective multipli­cation factor kef as follows.

P ~ (heff l)/keff (3«18)

Figure 3.13 shows a typical variation in reactivity with nuclear reactor operation. Nuclear reactors operate at the critical state of keff = 1, namely, p = 0. The effective multiplication factor and reactivity change with variations in conditions such as reactor pressure, fuel temperature, coolant temperature, and void fraction. As mentioned in Sect. 3.2.1, LWRs are designed so that the reactivity decreases when fuel and coolant temperatures rise and coolant void occurs or enlarges. Since neutron absorbers such as the FPs Xe and Sm are accumulated immediately after reactor startup, the reactivity is also reduced. It is, therefore, essential to give an excess reactivity corresponding to the expected reactivity decrease before reactor startup in order to maintain the critical state at a rated power operation. Moreover, it is also necessary to provide an excess reactivity compensating for the reactivity decrease with burnup after reactor startup in order to operate the reactor at the rated power during the operation period because the accumulation of FPs as neutron absorbers leads to a reactivity decrease while the amount of fissile nuclides in fuel decrease with reactor operation.

Temperature

Void Fraction Change

Accumulation of Xe and Sm

Operating Cycle

Operating Period

Fig. 3.13 Long-term variations in reactivity with nuclear reactor operation

Hence, a larger fuel amount than a critical one is loaded into the reactor and an excess reactivity necessary for reactor operation is provided considering variation in temperature, boiling effect, decrease in fissionable material with burnup, accumulation of FPs, and so on. The reactor should be designed to safely operate with proper control of the excess reactivity during the reactor operation period. Table 3.9 presents an example of the excess reactivity and reactivity worth of control elements for BWRs, which are designed to have a capability for reactivity control larger than the total excess reactivity for core shutdown even with one control rod stuck in the fully withdrawn position, including a calculation error.

The long-term variations in reactivity with burnup are controlled by control rods, coolant flow rate in core, and burnable poisons added to fuel pellets as shown in Fig. 3.13. Since the amount of burnable poisons cannot be adjusted during reactor operation, the reactor startup and shutdown are performed by

Stainless Steel Tube

(Neutron Absorption Hod> Sheath Roller

^.Iron

Wool

.End Plug

Detail Drawing of

Stainless Steel Pipe
і Neutron Absorption Rod>

Coupling Socket

Fig. 3.14 BWR control rod (B4C type of ABWR)

controlling control rods and coolant flow rate. A control rod scram is activated to shut down the reactor at an emergency and a boric acid solution injection system is provided as a control element for backup shutdown.

The numbers of rod cluster control assemblies (RCCAs) and rods per assembly are determined to secure necessary reactivity. RCCAs are inserted into some of the fuel assemblies (e. g., Fig. 3.30). In addition to securing the reactivity, the control rod arrangement is determined so that it flattens the pin power distribution in the fuel assembly when control rods are withdrawn, considering also mechanical integrity of RCCAs. After that arrangement is determined, a nuclear instrumentation guide tube is placed near the center of the fuel assembly and then fuel rods are arranged in the remaining locations. The 17 x 17 square lattice arrangement for a fuel assembly is shown in Figs. 3.30 and 3.34. This arrangement is composed of 24 control rod guide tubes (guide thimbles), one nuclear instrumentation guide tube (guide thimble), and the rest are fuel rods. Fuel assemblies of 14 x 14 and 15 x 15 types are also used in Japan.

(3) Fuel Assembly Height

A higher fuel assembly height can reduce the number of fuel assemblies and therefore improve the capacity factor of a plant. There is, however, a limitation in fuel assembly height. A higher assembly height causes an increased pressure drop in the thermal-hydraulic design and therefore requires a higher capacity primary coolant pump. It also gives rise to an increase in fuel assembly bowing in fuel mechanical design and therefore has an adverse effect on fuel loading operation into core. From the viewpoint of nuclear design, as discussed before, the core for which the ratio of the core height to the equivalent diameter is 1.0 gives a good neutron economy because of low neutron leakage. However, the fuel assembly active height was standardized as about 3.7 m for economy in fuel assembly fabrication.

Refueling is carried out in every operation cycle. After a cycle, the neces­sary number of discharged and loaded assemblies and their positions for achieving criticality throughout the next cycle are determined in consider­ation of the operation conditions. The scattered batch refueling method is adopted. In this method, the fraction of discharged fuel assemblies in a particular region (inner core, outer core or blanket) over the total fuel assemblies within that region is kept equal among the three regions. In that way, the power distribution and other neutronic characteristics are kept virtually unchanged between operation cycles.

In design of the refueling core, the refueling plan (i. e. the number and position of discharged fuel assemblies) is made for ensuring designed cycle length and achieving a safe and economical core configuration. By assum­ing the range of the content of available plutonium, the core reactivity and the core characteristics (such as power distribution, reactivity coefficients, and burnup) are evaluated. According to the evaluations, the core charac­teristics and safety in the target cycle are confirmed based on the plutonium content used in the detailed design and the refueling plan. The designed core characteristics are confirmed again by the reactor physics tests at actual reactor startup.

(2) Variation of plutonium content and the equivalent fissile content method [10]

(a) Variation of plutonium content

Before being used as nuclear fuel, plutonium was irradiated in various reactors and reprocessed. The plutonium content of MOX fuel depends on the burnup of the original spent fuel, cooling time etc. The higher the burnup of the original spent fuel, the larger the fraction of higher order isotopes such as 240Pu, 241Pu and 242Pu becomes. 241Pu in the reprocessed plutonium decays into 241Am with a half-life of 14.2 years. The decay of the fissile 241Pu reduces the reactivity of MOX fuel. 241Am has larger capture cross section than 240Pu and 242Pu, and it has a much smaller fission cross section than fissile materials like 239Pu. Accumulation of 241Am decreases the reactivity of MOX fuel. Figure 4.5 shows the cross sections of plutonium isotopes and 241Am. As the decay of 241Pu and production of 241Am continue after fabricating the fuel elements, the designed plutonium enrichment in the fabricating process must be determined by considering the duration to the loading in the core.

(b) The equivalent fissile content method

Since the contribution of plutonium to the core reactivity largely depends on the plutonium content, the necessary plutonium enrichment

Table 4.2 Plutonium Equivalent Worths in FBR [10]

also depends on the plutonium content. However, the core reactivity can be accurately estimated by equivalently treating the fissile enrichment in the fuel. This method is called the The Equivalent Fissile Content Method.

In this method, the contribution of 239Pu, which has the highest fraction among plutonium isotopes, is set as the reference (1.0). The relative contributions by other plutonium isotopes including 241Am (treated as one of the plutonium isotopes) are defined as the plutonium equivalent worths (nPU, nU). Examples of the plutonium equivalent worths are listed in Table 4.2. By using them, the adjusted fissile enrichments corresponding to the plutonium enrichments of the inner core and outer core are calculated as Eq. (4.7).

ЇЇ239=єри’ЕІаігіГи+(1—єри)’Е&гі}] (4.7)

epu : Plutonium enrichment

ai : Fraction of plutonium isotopes i

Pj : Fraction of uranium isotopes j

nPU : Plutonium equivalent worth of plutonium isotope i

nU : Plutonium equivalent worth of uranium isotope i

Although the content of the obtained plutonium at the moment of fuel fabrication is different from the plutonium content at the moment of design, equal core reactivity can be achieved by adjusting the plutonium enrich­ment during the fabrication so that the equivalent fissile enrichment is equal to the design value.

Highly practical methods for reactivity calculations are discussed in the last section of this chapter. In the design of power reactors or operation of research reactors, it is often necessary to evaluate the effect of small changes (perturbations) on the behavior of the reactors. A local change of reactivity in power reactors, insertion of irradiation pieces in research reactors, and so on should be treated as a local perturbation. In principle, the reactivity change due to a perturbation can be evaluated by repeatedly performing the two- or three-dimensional multi-group calculation. For a small perturbation, however, the reactivity change may be lost in the computations as the result of round-off errors. Enormous calculations are required to obtain the distribution of reactivity coefficients in the core. Such problems can be handled by perturbation theory.