In a uranium-fueled thermal reactor, the reactivity changes with burnup are asso­ciated with the following factors:

(i) Depletion of 235U;

(ii) Buildup of 239Pu;

Natural Uranium-Fueled and

Graphite-Moderated Reactors

Burnup

LWRs

Fig. 1.3 Reactivity change due to 235U depletion and 239Pu buildup. (a) Reactivity decrease due to

235 239 235 239

U depletion, (b) reactivity increase due to Pu buildup, (c) reactivity change of U and Pu

(iii) Buildup of 241Pu;

(iv) Buildup of non-fissile nuclides, 240Pu, 236U, and 242Pu;

(v) Buildup of highly thermal neutron-absorbing FPs (135Xe and 149Sm); and

(vi) Buildup of other FPs.

Items (i) and (ii) mainly lead to reactivity changes by increase or decrease in amounts of fissile nuclides. Reactivity drops due to exponential depletion of 235U and reactivity rises due to buildup of 239Pu as shown in Fig. 1.3a, b, respectively. The combined reactivity change of U and of Pu is presented in Fig. 1.3c. In an enriched-uranium fueled LWR, the effect of 235U is generally larger than that of 239Pu and it results in a reactivity decrease with burnup. In the case of a fuel type of natural uranium and high conversion ratio for graphite-moderated reactors, there is

an early reactivity increase and then a decrease with burnup because 235U and 239Pu

235 239

produce comparable reactivity effects and the fissile replacement of U by Pu proceeds.

240Pu, 241Pu, and 242Pu are produced with burnup by neutron capture. 240Pu has a large thermal-neutron absorption cross section and has a negative reactivity effect. 241Pu produced by the subsequent neutron capture is a fissile nuclide and has a positive reactivity effect. On the other hand, 242Pu produces a small negative reactivity because its production amount and cross section are small.

236 235

U produced from U by neutron capture also gives a small negative reac­tivity effect.

It is also necessary to consider the negative reactivity effect by buildup of FPs, which can be classified as two significant FPs and other FPs. The former are 135Xe and 149Sm which have very large absorption cross sections. Within a short time after burnup (usually within several hours to several days), they reach an individual equilibrium concentration by the balance between production and destruction because of the strong neutron absorption, and they have a constant negative reactivity effect. The other FPs continue to be accumulated with burnup because of low absorption cross sections and to increase the negative reactivity.

Typical reactivity changes for such individual reactivity effects are shown in Fig. 1.4 for natural uranium-fueled, graphite-moderated reactors. After an initial increase in reactivity due to the buildup of 239Pu, there is a gradual reduction in reactivity due to the depletion of 235U and the buildup of neutron absorbing materials. These analyses of reactivity changes with fuel burnup are used to determine the fissile amount for criticality during reactor operation and to deter­mine the reactivity worth of control rods.

[1] Macroscopic cross sections

The cross section used in the neutron transport equation, £, is called the macroscopic cross section and its unit is (cm-1). The macroscopic cross section of nuclear reaction x is the sum of the product of atomic number density Ni and microscopic cross section alx of a constituent nuclide i of the material at a position ~. It is expressed as Eq. (2.8).

їЛг, E, 0=5>*(r, t)ai(E) (2.8)

і

The atomic number density Nl can be given by

(2.9)

where pl is the density occupied by i in the material, Ml is the atomic mass, and Na is Avogadro’s number. Variation of atomic number density with position and time should be considered in power reactors, reflecting fuel burnup or coolant void fraction.

3.2.1 General Core Design

[1] Features of BWR core [3]

BWRs operate in a direct cycle in which the steam formed in the core goes directly to turbines to generate electricity. A big difference from other type reactor cores is that there is a distribution of moderator density within the BWR core because the neutron moderator, as coolant, is boiling as it flows through the core which leads to two-phase flow.

Figure 3.3 shows a BWR core cross-sectional view and structures. There are some distinctive features to BWR core structures. One is the set of a cruciform control rod and four surrounding assemblies; the sets are regularly arranged in the core. Another feature is that the control rod and fuel assemblies are mutually separated by channel boxes wrapping each fuel assembly, in which fuel rods and water rods are tied in an array of, for example, 8 x 8 or 9 x 9, by spacers. A third feature is that the coolant flow path in the core is divided by the channel boxes and each fuel assembly makes a coolant flow path. There is a water gap region, in which water does not boil, between adjacent fuel assem­blies. The coolant flowing in from the core bottom comes to boiling and then a two-phase flow of water and steam is produced due to heat from the fuel rods within the fuel assemblies isolated by the channel boxes, but coolant is maintained in the non-boiling state outside the assemblies.

Those features of BWR core structures make it possible to improve or modify the intra-assembly design such as fuel rod size and array without changing specifications concerning the whole core construction such as control rod arrangement and fuel assembly shape.

Fig. 3.3 BWR core cross-sectional view and structures

Overview inspection of fuel assemblies is made to confirm integrity of fuel was kept during irradiation. Two fuel assemblies with maximum burnup in each assembly type are inspected for fuel rod damage, deformation, and gap clearance change through visual observation.

Shipping inspection is made to check for leakage of radioactive materials from fuel assemblies. Fuel assemblies are covered by channel boxes. Shipping caps are placed on the channel boxes in the core and they collect water inside each channel box. The presence of radioactive material leak­age is evaluated by 131I concentration in the water sample.

PWR fuel has been improved for high burnup from the viewpoint of high economy in fuel cost and lowered amount of spent fuel to be handled and stored. The limit of the maximum fuel assembly burnup was 39,000 MWd/t in the beginning, and increased to 48,000 MWd/t in Step I and 55,000 MWd/t in Step II fuels. Further high burnup is planned [39]. Such a high fuel burnup has the following effects on core and plant facilities and measures against them are discussed below.

(i) Neutron spectrum hardening

(ii) Enrichment increase (increase in 235U)

(iii) Fission product accumulation increase

(1) Impact of high enrichment and burnup

A high enrichment and burnup leads to an increase in thermal neutron absorbing materials such as 235U, plutonium and FPs and a decrease in thermal neutron flux. Therefore, the thermal neutron absorption by the rod cluster control assemblies (RCCAs) and soluble boron in primary coolant is reduced and their reactivity worth is decreased slightly.

As a measure of the reactivity worth reduction of RCCAs, a proper arrangement of fuel assemblies loading pattern in the core can secure the

reactivity shutdown margin. The number of RCCAs should be increased if insufficient.

As a measure of the reactivity worth reduction of soluble boron, the boron concentration can be increased for reactor shutdown if necessary. The boron concentration in the refueling water storage tank (the boric acid water stored in the refueling water storage tank could be injected into the reactor at abnormal transients or accidents in which the boron injection function of the emergency core cooling system is expected to work) can also be increased.

The increase of enrichment also leads to lower subcriticality margin of fuel storage facilities. Since the subcriticality evaluation of fuel storage facilities has been performed for fresh fuel with initial enrichment of 5.05 wt%, a new measure is not necessary if the enrichment is lower than 5 wt%.

(2) Impact of high burnup

A high burnup increases the amount of fission products and therefore increases the decay heat of spent fuel. Spent fuel storage cooling system are enhanced in case of insufficient cooling capability, considering heat load to spent fuel pit.

The reactor outlet coolant temperature of LWRs is around 300 °C and the temper­ature elevation from inlet to outlet is at most several tens of degrees. Thus, the power distribution is designed to be axially flat in LWRs. On the other hand, the reactor outlet coolant temperature is 950 °C and the temperature elevation from inlet to outlet is about 550 °C in the HTTR. Thus, special core design methods are necessary so that the fuel temperature at the core bottom where coolant temperature is high does not exceed the limit.

[1] Optimization of power distribution [33, 34]

If the power distribution is not optimized, the fuel temperature exceed the limit at the core bottom as shown in Fig. 4.21. The fuel temperature at the core bottom must be reduced by flattening the axial temperature distribution. The power distribution for flattening the axial temperature distribution is analyti­cally obtained.

Tf (z) — Tin + Tcl (z) + Tcm (z) (4*15)

where

Tin : Core inlet coolant temperature (° C)

Tcl : Elevation of gas temperaturefrom core inlet to axial potion Z (° C)

Tcl(z)=ajp(z)dz (4.16)

P(z) indicates the axial power distribution and a is a factor which depends on coolant flow rate and coolant heat capacity. The heat capacity is regarded as constant.

Tcm : Temperature elevation from surface of graphite

sleeve to inner surface of fuel compact (°C) (4.17)

Tcm (z) — bP(z

Here, b is a factor which depends on fuel rod geometry and thermal conduc­tivity. Although the thermal conductivity changes with temperature, it is regarded as constant.

Equation (4.18) is obtained by substituting Eqs. (4.16) and (4.20) into Eq. (4.16).

N і Radial fuel region • — Burnable poison

Fig. 4.23 Uranium enrichments of each fuel block in core

Tf(z)=Tin+aJp(z)dz+bP(z) (4.18)

By differentiating this equation under the condition Tf(z) = constant, which means axially flat distribution of fuel temperature, the following relation is obtained.

aP(Z )dz = —bdp(z) (4.19)

By solving this relation, the power distribution for achieving the axially flat distribution of fuel temperature is an exponential function.

(4.20)

C is an integration constant and determined to make Tci of Eq. (4.16) the elevation of the coolant temperature in the core. Figure 4.21 shows the expo­nential power distribution for axially flattening the fuel temperature distribution of the HTTR. Such an optimized power distribution is achieved by making the uranium enrichment relatively high at the core top and relatively low at the core bottom. The fuel temperature distribution among the fuel columns is also flattened by making the uranium enrichment relatively high at the core periph­eral region and relatively low at the core center region. The uranium enrichment division obtained by those adjustments is shown in Fig. 4.23. The number of enrichment divisions is 12. The minimum and maximum values are 3.4 wt% and 9.9 wt%, respectively. The core average enrichment is 5.9 wt%. The power

Fig. 4.24 Adjustment of keff for making control rod insertion shallow

Burnup day

distribution after the optimization is shown in Fig. 4.21. Due to shallow insertion of the control rods and other factors, the power distribution is not an exact exponential function at the core top.

The reactivity worth of control rods (control rod worth), pW, can be defined as

Pw = Po~Pi (1.49)

where p0 and p, are the system reactivity at control rod withdrawal and insertion, respectively. This definition is mainly used when measuring the insertion depth of control rods required to change the reactor power level. Another definition of control rod worth can be used to compensate for the reactivity change due to a long-time fuel burnup. In the former definition, there is no variation in physical properties of the core materials even though the control rods are inserted. Mean­while, the physical properties vary with the fuel burnup in the latter definition. But, it was eventually found that the two definitions of control rod worth were equiva­lent. It is now common practice to use the first definition [11].

The calculation of control rod worth is one of the most difficult problems in the core calculation of nuclear reactors. Recently, the control rod worth has been directly evaluated through numerical calculations with a three-dimensional core representation. Many theoretical evaluations were conducted in early designs because of limitations in calculation capability. One approach was to determine the reactivity worth by using the changes in the bucklings of the system with a boundary condition at the control rod surface [12]. Another one was to calculate the reactivity worth of a large number of control rods in a regular array or cruciform control rods by applying the Wigner-Seitz method [13]. At present, such theoretical approaches are seldom applied.

First a non-multiplying medium (e. g., water) is considered for simplicity of the equation, though practical systems are composed of various materials with different cross sections. In addition, it is assumed that all neutrons can be characterized by a single energy (one-group problem). Therefore, the cross section does not depend on location and in this case the diffusion equation is given by Eq. (2.41).

ву2ф-ъаф+8=о

V2 (the Laplacian operator) is dependent on the coordinate system and in the Cartesian coordinate system it is represented as Eq. (2.42).

In an infinitely wide medium in y and z directions, the diffusion equation reduces to the 1D form.

(2.43)

Here a uniform neutron source (s1 = s2 = s3 ••• = s) in a finite slab of width 2d is considered as shown in Fig. 2.16 and the finite difference method to solve for neutron flux distribution is discussed. The reflective boundary condition is applied to the center of the slab and one side of width d is divided into equal N regions (spatial meshes). The second-order derivative of Eq. (2.43) can be
approximated[3] by using neutron flux фі at a mesh boundary і and neutron fluxes at both adjacent sides as follows;

(2.44)

(2.46)

where

(2.47)

In the case of N = 5, the following simultaneous equations are gotten.

афо+Ьфі+аф2=Si

< аф1+Ьф2+афз=82 (248)

аф2+Ьфз+аф4=8з

Since there are six unknowns (ф0 to ф5) among four equations in Eq. (2.48), another two conditions are needed to solve the simultaneous equations. Bound­ary conditions at both ends (x = 0 and x = d) of the slab are given by the following.

Vacuum boundary condition (an extrapolated distance is assumed zero):

ф(х = ё) = 0 -> фв = 0 (2.49)

Fig. 2.17 Finite difference in 2D plane geometry

і ~ 1 і і +1 N Divisions in Width A x

Reflective boundary condition:

(2.50)

Equation (2.51) is obtained by inserting Eqs. (2.49) and (2.50) into Eq. (2.48) and rewriting in matrix form.

A 1D problem generally results in a tridiagonal matrix equation of (N — 1) x (N — 1) system which can be directly solved using numerical methods such as the Gaussian elimination method. Since the finite difference method introduces the approximation as Eq. (2.44), it is necessary to choose a sufficiently small mesh spacing. The spatial variation of neutron flux is essentially characterized by the extent of neutron diffusion such as the neutron mean free path. Hence the effect of the mesh size on nuclear characteristic evaluations of a target reactor core must be understood and then the mesh size is optimized relative to the computation cost.

As shown in Fig. 3.5, the set of a fuel assembly, water gap outside channel box, and a quarter of cruciform control rod form a unit lattice and unit lattices are regularly arranged radially in the BWR core.

As main parameters of the unit lattice, there are the size and shape of the channel boxes wrapping the fuel assemblies, the number and diameter of fuel rods, the fuel rod pitch and pellet size, the size and location of the water rods, specifications of control rods, and so on. The amount of fissionable materials in each fuel rod, namely, the enrichment and density of U235 in the case of uranium fuel and the amount of gadolinia in case of burnable poison-mixed fuel are the basic nuclear design parameters.

Figure 3.6 [3] shows the procedure flow in the fuel assembly design. An optimal ratio of fuel to moderator is established from the viewpoint of the reactor nuclear performance, and the fractions of coolant and fuel regions are selected. The fuel assembly design proceeds with the fuel rod pitch and water rod size maintaining consistency with the design requirements to the number and diameter of fuel rods from the viewpoints of fuel inventory and fuel integrity. In the actual work, the final design of fuel assemblies is determined by repeating this sequence.

Unit Lattice

Fig. 3.5 Fuel lattice of BWR fuel assembly

[ Discharge Burnup j

Fuel Assembly Design

Fig. 3.6 Fuel assembly design flow

In determining the fuel assembly design, it is especially important to consider the relation between the moderator-to-fuel ratio in the unit lattice, which has a large effect on the nuclear performance, and the parameters mentioned above.

[1] Fuel loading weight

As mentioned above, the fuel loading weight is first determined in the core design based on the cycle length and rated core thermal power as given by the next equation.

W= QxDx Nb/BU

W: fuel loading weight (t)

Q: core thermal power (MW)

D: operation cycle length (day)

Nb: refueling batch number (the reciprocal of the discharged fuel fraction in one refueling = the reciprocal of the new fuel fraction)

BU: region or batch averaged fuel assembly burnup (MWd/t)

The operation cycle length is fixed by considering refueling and periodic inspection intervals. Since generally a longer cycle length leads to a higher capacity factor and a lower operating cost of the plant, there is a tendency to set the cycle length as long as possible.

Fuel economy is strongly dependent on the fuel batch size. With a larger batch size, fuel assemblies can stay longer in core and achieve higher burnup,

and therefore that results in a reduction in fuel cycle cost. The fuel batch size is, however, restricted by the design limit of the maximum fuel assembly burnup. For a given operating cycle length, the largest batch size is taken in the range of possible maximum fuel assembly burnups. The average fuel assembly burnup becomes higher as the variation in discharge burnup of each fuel assembly is smaller. Usually, the average fuel assembly burnup is supposed assuming a certain ratio of the maximum to average fuel assembly burnups (for example, about 1.1). Hence, the fuel loading weight in Eq. (3.23) is determined with these considerations.

Figure 3.32 shows the relationships for cycle length, batch size, maximum fuel assembly burnup, and fuel cycle cost.

(i) For a longer operationg cycle length, fuel enrichment can be increased for higher burnup under a constant batch size or the number of new fuel assemblies can be increased; namely, fuel batch size is decreased under a constant fuel enrichment. Fuel enrichment is usually standardized and fixed, and therefore it is usual to increase the number of new fuel assem­blies. Maximum fuel enrichment is currently limited to 5 % by restrictions on fuel fabrication facilities and transportation.

(ii) An increase in the number of new fuel assemblies and then a decrease in batch size lead to a lower average fuel assembly burnup and a higher fuel cycle cost. Since fuel cycle cost occupies a small fraction of total plant operating cost, a longer operating cycle length reduces the total plant operating cost even for a smaller batch size.

(iii) For a long cycle length and low fuel cycle cost, it is also necessary to advance the limit of the maximum fuel assembly burnup. This can be achieved by loading an improved type of fuel such as with new alloy cladding. A higher limit for the maximum fuel assembly burnup increases the fuel batch size and reduces the fuel cycle cost even for the same cycle length.