MOX fuel assemblies have a zoning configuration with three different contents of plutonium as shown in Fig. 3.64. The reactivity of MOX fuel assembly is determined by the assembly-averaged Puf content. Plutonium used in MOX fuel has been reprocessed and recovered from spent fuel of various types of reactors and therefore it has various isotope compositions. The assembly-averaged Puf content is set to give a proper reactivity corresponding to plutonium isotope composition. As a currently implemented method to set the assembly-averaged Puf content, the MOX fuel assembly to give the same reactivity as 235U 4.1 wt% uranium fuel at 28,100 MWd/t is used as a criterion for the appropriate reactivity [41]. The enrichment of 4.1 wt% and burnup of 28,100 MWd/t are the enrich­ment and the core average burnup at EOC for the typical uranium fuel in the standard 3-loop PWR (See Problem 6 at the end of this chapter). To match reactivity at the core average burnup of EOC means that MOX-fueled core has the same operating cycle length as the uranium-fueled core.

Figure 3.65 compares variation in the infinite multiplication factor between typical MOX fuel and 235U 4.1 wt% uranium fuel. The MOX fuel shows a milder variation in infinite multiplication factor with burnup than the uranium fuel and the line intersects that of the uranium fuel at a point. The infinite multiplication factor line of MOX fuel moves up and down with almost the same slope as Puf content increases and decreases. Therefore, a proper reactivity of MOX fuel can be obtained by adjusting Puf content responding to plutonium isotope compositions in MOX fuel

— Uranium Fuel with 4.1wt% Enrichment — MOX Fuel

28 100 MWd/t

Burnup [MWd/t]

fabrication. Practically, the reactivity worth of each plutonium isotope is replaced with an equivalent reactivity worth of 239Pu, referred to as the equivalent fissile method [41]. Figure 3.66 shows the relation between Puf content and Puf fraction set by the equivalent fissile method.

[2] Reactor Power Uprating

Reactor Power Uprating for PWRs are evaluated considering the same effects as for BWRs mentioned in the list [5] of Sect. 3.2.6 and the range of facility modification is determined. It has been reported that advanced safety analysis methods and development and introduction of advanced fuel are important for significant power uprating [42].

Since the fuel temperature must be kept below the limit throughout the burnup period, it is expected that the optimized power distribution is maintained. As shown in Fig. 4.24, high effective multiplication factor at the beginning of burnup period requires deep insertion of the control rods for compensating the excess reactivity. In such a case, the peak of power distribution like Fig. 4.21 (“No optimization” case) appears in core bottom and the fuel temperature there exceeds the limit. Thus, the control rod insertion, as illustrated in Fig. 4.21 (“Optimization” case), should be kept shallow throughout the burnup period. To do that, the effective multiplication factor should be made as small as possible while keeping the necessary reactivity for burnup like kexp in Fig. 4.24. The reactivity which should be compensated by burnable poison (4kexp) is given by Eq. (4.21).

4keXp = k — keXp (4.21)

The multiplication factor is expressed as below by one-group theory:

(4.22)

"a

where

v : Neutron yied per fission Df: Macroscopic fission cross section (1/cm)

Da : Macroscopic absorption cross section (1/cm) excluding burnable poison

The multiplication factor kexp which is made as small as possible by loading burnable poison is expressed as the next equation.

(4.23)

Eexp is the adequate absorption cross section of burnable poison for achieving kexp. From Eqs. (4.21), (4.22) and (4.23), £exp is expressed as Eq. (4.24).

^kexp

kcx,

Burnable poison is a strong absorber and hence has a self-shielding effect. Thus, its effective absorption cross section £aBP is expressed as:

^aBP — f ON

where

f : Self-shieldingfactor

a : Microscopic absorption cross section of nuclide of burnable poison(barn) N : Number of densityof nuclides of burnable poison homogenized into fuel block (1/barn/cm)

f can be obtained by the empirical correlation below:

(4.26)

where

C : Fitting factor

I : Factor depending on geometry of burnable poison (1/cm)(the radius is used for a rod type geometry)

NBP : Number density of nuclides having absorption effect in burnable poison (1/barn/cm)

The change in the effective absorption cross section XaBP can be obtained from Eqs. (4.25) and (4.26) by considering the change in the number density of nuclides NBP. Among the changes in XaBP for several burnable poisons with different geometries and nuclide number densities, that which is closest to the change in Eexp is the optimum one. Figure 4.25 shows the comparison of Eexp and several EaBP which were considered in the design of burnable poison of the HTTR. From this figure, the optimum diameter (r) of the burnable poison rods was set as 0.7 cm. This optimization of burnable poison allows operation with shallow insertion of the control rods, as shown in Fig. 4.21, throughout the burnup period. Reference [35] describes details on optimization of the burnable poison.

The insertion of a control rod depresses the neutron flux in its vicinity and makes the curvature of the flux greater. The gradient of the flux will contrarily increase at a radial distance. Usually, multiple control rods are placed in the core. The control rods will distort the neutron flux around each other and will have an effect on the reactivity worth. The term “shadowing effect” has been used to describe this. The worth of the control rods will be less than the sum of the separate rod worth values or greater than the sum depending on the control rod locations. It is important to determine the shadowing effect of control rods.

The diffusion equation in a 2D homogeneous plane geometry is given by Eq. (2.52).

+ (2.52)

The plane is divided into N and M meshes in x and y directions, respectively (Fig. 2.17). Similarly to the 1D problem, the second-order derivatives can be

Fig. 2.18 Matrix equation forms of finite difference method

approximated by using neutron fluxes at a point (i, j) and its four adjacent ones as the following.

(2.56)

Next, the corresponding simultaneous equations are expressed in matrix form, similarly to the 1D problem except four boundary conditions are used: two conditions in each of the x and y directions.

A 3D problem even in a different coordinate system basically follows the same procedure and a matrix form of ([Л]ф = S) can be taken as shown in Fig. 2.18.

initial guess

і I

^ ф =[D]-1S-[D]-1[B] ф

inner

ф iteration
<^onverged^ ^update ф

convergence

end

In the case of a large-size matrix [A], a direct method such as the Gaussian elimination leads to increased necessary memory size and it is liable to cause accumulation of numerical errors as well. Hence, generally first the matrix [A] is composed into its diagonal element [D] and off-diagonal element [B]. Then ф is guessed and new guesses are calculated iteratively to converge to the true solution (Fig. 2.19). This iterative algorithm, such as found in the Gauss-Seidel method or the successive overrelaxation (SOR) method, is frequently referred to as the inner iteration.

The moderator-to-fuel ratio in the unit lattice relates to the size and shape of fuel rods and water rods, and the void fraction, and is derived from the

structural feature of the BWR core where the coolant in the flow path of the channel box boils, but the water in the water rod and the assembly gap outside the channel box do not boil. The H/U ratio, which is defined as the ratio of hydrogen atoms in moderator to uranium atoms ( U + U) in fuel, rather than the moderator-to-fuel volume ratio (Vm/Vf) is used as the moderator-to-fuel ratio in the unit lattice. The evaluation of the effect of lattice shape and void fraction based on the H/U ratio is suitable for a consistent investigation into the effect of each parameter.

The H/U ratio is given by

H/U ratio = (No. of hydrogen atoms in unit lattice)/

(No. of fuel atoms in unit lattice)

No of hydrogen atoms in unit lattice = Щ/ x (1 — a) H nH x a} x (Cross-sectional area of boiling region inside channel box) + n°H x Cross-sectional area of non-boiling region inside channel box (water rod region)) H n/x

(Cross-sectional area of non-boiling region outside channel box (Water gap region))

No. of fuel atoms in unit lattice = nu x Nrod x n(Dp/2)2

Here a is the void fraction inside the channel box, n/ is the atomic density of hydrogen in the saturated water, nsH is the atomic density of hydrogen in the steam, and nu is the atomic density of uranium in the pellet. It is necessary to remark that the hydrogen atoms in the water gap and water rods (non-boiling region) are considered in the H/U ratio and moreover the H/U ratio depends on the void fraction inside the channel box. The H/U ratios of typical BWR fuel assemblies are shown in Table 3.8.

The fuel loading weight per unit core volume is determined by considering the moderator-to-fuel volume ratio (Vm/Vj) (which is discussed in the next section) and fuel density. Then the total core volume is calculated from the weight per volume and the total fuel loading weight W determined by Eq. (3.23). Core height and equivalent diameter are determined according to the following conditions.

(i) Equivalent core diameter should be suitable for reactor vessel design and manufacture.

(ii) Core height and equivalent diameter should be as close to each other in value as possible to reduce neutron leakage to the core outside from the viewpoint of neutron economy.

(iii) The number of fuel assemblies should be a multiple of four + one; quarter-symmetry fuel loading pattern and one rod cluster control assem­bly located in the core center.

Current plants have small variations regarding the second condition which was considered in the early core designs. In the process of sizing up the core, a large equivalent diameter has come to be used since fuel active height was standardized (namely, the core height was fixed).

An example of core configuration with fuel assemblies can be seen in Fig. 3.30. The core is a four-loop core with 193 fuel assemblies.

The procedure for designing a fast reactor core is schematically shown in Fig. 4.4. The core thermal power and the operation period per cycle are determined as the basic specifications of the core design. The fuel assembly maximum burnup (or discharged burnup), the maximum allowable linear heat rate and the breeding ratio are given from the design principles.

First, the total length of the fuel elements and the number of fuel assemblies are preliminarily determined according to the core thermal power, the linear heat rate of fuel and the core height. The fuel loading amount is obtained by determining diameter of the fuel element. Then, the control rods are arranged based on the functions. The core configuration is determined. The core char­acteristics are iteratively evaluated until the design principles are satisfied by adjusting the trade-off conditions as illustrated below.

• If the burnup is lower than the target, the fuel element is made thinner to reduce the fuel loading or the number of fuel exchange batches is increased.

• If the breeding ratio is lower than the target, the fuel element is made thicker or the core height is increased.

• If the void reactivity is too large, the core height is reduced.

When all the design targets and criteria are satisfied, the core design is fixed.

The following gives an overview of other reactivity coefficients defined according to features of the reactor type, regarding an increase in reactor power.

[1] Void coefficient

This coefficient describes a reactivity change due to formation of voids in a liquid coolant. It is used in BWRs to control the reactivity by changing the void fraction. The mechanism of the reactivity change is identical to that discussed in the moderator temperature coefficient.

In sodium-cooled fast reactors, the void reactivity of sodium is evaluated on the basis of coolant boiling caused by hypothetical accidents.

[2] Pressure coefficient

Since PWRs and BWRs are operated at high pressure, the reactivity coefficient to pressure change (pressure coefficient) should be defined. The coolant density increases with pressure and the voids in BWRs are collapsed. The mechanism of the reactivity change is diametrically opposite to that of the moderator temperature coefficient.

[3] Core expansion effect

In a system with a large neutron leakage, an increase in leakage due to expansion or deformation of structure can have a remarkable negative reactiv­ity effect. This effect is especially important for small fast reactors.

The plant dynamics analysis covers the plant control system. Figure 2.38 pre­sents the plant control system of a SCWR as an example [27]. Typical control systems of LWRs are actually more complex, but they are based on the concept below. The plant control system of nuclear power plants uses proportional — integral-derivative (PID) control which is widely used in the control system of other types of practical plants as well.

Composition of the plant control system is based on similar existing plant control systems and its parameters are determined using plant dynamics anal­ysis codes using the procedure below.

© Investigate responses to external disturbances without control system, and decide what state variables (e. g., power) to control and how to control them (e. g., by control rod) according to their sensitivities.

® Design a control system.

® Confirm responses to the external disturbances under the control system.

For example, in the case of reducing feedwater flow rate, changing control rod position, inserting a reactivity, or closing main steam control valves, variations in main steam temperature, power, or pressure are investigated for each case using plant dynamics analysis codes. In Fig. 2.38, the main steam temperature is controlled by regulating feedwater flow rate, the reactor power by control rods, and the main steam pressure by main steam control valves considering highly sensitive parameters.

Figure 2.39 shows the power control system by control rods. This control system is governed by a proportional controller which decides a control rod drive speed in proportion to the deviation between actual power and power setpoint. Control rods are driven at a maximum speed for a deviation larger than a constant value b.

Figure 2.40 represents the main steam temperature control system which is driven by a proportional-integral (PI) controller after a lead compensation for time-lag of a temperature sensor. The PI control parameter values are deter­mined for stable and fast-convergence responses through fine tuning of the integral gain with a little change of the proportional gain.

и (t) =Kpe (t) +Ki fo e (t) dt

Fig. 2.40 Main steam temperature control system

Figure 2.41 shows the pressure control system which controls valve opening by converting the deviation from the pressure setpoint into a valve opening signal with lead/lag compensation. The control parameter value is determined for a stable convergence response from investigation of a time variation in the main steam pressure with a little change of the pressure gain.

Valve Opening Signal
	Valve Opening
Pressure Gain
Deviation
P-R
100
dVd
dv
[MPaJ
[MPaJ
К [MPa]
Fig. 2.41 Pressure control system
Feedwater pump model : SP=Cpump8u

Fig. 2.42 Calculation model for plant startup and stability analysis
dP dz	d, d 2_l 2f 2 —Pu+—Pu +—pu
Feedwater line model:

Valve model: AP=£

P: pressure (Pa)

AP: pressure drop (Pa)

5P: pressure change in pump (Pa)

Z: orifice or valve pressure drop coefficient p: coolant density (kg/cm3) u: fluid velocity (m/s)

8u: fluid velocity change in pump Cpump: pressure drop coefficient of pump z: position (m) t: time (s)

f: friction pressure drop coefficient D: diameter of feedwater pipe (m)

The plant startup procedure is established to satisfy constraints at startup using the calculation models above. The constraints at startup depend on reactor type: for example, maximum cladding surface temperature for SCWRs; restric­tions in heat flux, various safety parameters, and pump cavitation for LWRs; restrictions of vapor fraction in main steam for direct-cycle reactors. There are also restrictions from thermal stress to temperature rise of the reactor vessel.

Japan’s development of high burnup BWR fuel has been advanced by stages,

Step I, Step II, and Step III, as shown in Fig. 3.24, with each step confirming the

usage results.

Step I fuel has been used practically since 1987. It employs the zirconium liner fuel, while having the same structure as the previous 8 x 8 type lattice (8 x 8 RJ). By applying the uranium-saving technology using power peaking such as the installation of natural uranium blankets on the core upper and lower end parts, the discharge burnup for the same average enrichment was increased by about 10 % from about 29.5 GWd/t to 33.0 GWd/t and the fuel cycle cost was also reduced by about 10 %.

The high burnup 8 x 8 fuel of Step II has been used as a reload fuel since 1991. It has a higher enrichment for high burnup and economy. The average discharge burnup is 39.5 GWd/t and the maximum burnup of fuel assembly was increased to 50 GWd/t from 40 GWd/t of the Step I fuel. To suppress fuel temperature and rod internal pressure rise during reactor operation, the initial pressure of helium gas was increased from about 0.3 MPa to 0.5 MPa and the theoretical density of pellets was raised from about 95 % to 97 %. Four fuel rods in the center of the fuel assemblies were replaced with a large diameter water rod and the H/U ratio was increased to enhance neutron moderation efficiency. Ring-type spacers were used to improve the thermal margin (power limit) of fuel rods and upper tie plates were designed as a low pressure drop type with a small resistance against water flow.

The 9 x 9 fuel of Step III, which has been used on a full scale since 1999, gives a higher discharge burnup of 45 GWd/t by increasing fuel enrichment and a higher maximum burnup of fuel assemblies of 55 GWd/t. The change to the 9 x 9 arrangement of fuel rods, which increases the number of fuel rods per fuel assembly, reduces the average linear heat generation rate and increases the nuclear design flexibility. There are two design types for Step III fuel assem­blies; types A and B. The type A assembly consists of 74 fuel rods and two large

diameter water rods. A high pressure drop at the non-boiling fuel assembly inlet and a low pressure drop in the boiling region can stabilize the coolant flow in the fuel assemblies. The assembly adopts a high pressure drop type lower tie plate and partial length fuel rods (about 2/3 the usual length). The type B assembly is composed of 72 fuel rods and a square water channel to increase the H/U ratio and to optimize neutron moderation. It improves the nuclear and thermal-hydraulic characteristics such as core safety. Both fuel assembly types have a higher initial pressure of helium gas, 1.0 MPa, to improve the heat transfer between pellet and cladding and to mitigate the internal pressure rise of fuel rods with neutron irradiation.

The BWR fuel assembly has been improved with respect to intra-assembly design points such as fuel rod size and arrangement without considerably changing the fuel assembly size and therefore it can be applied to existing reactors. Application of high burnup fuel through those three phases has extended the average discharge burnup to about 1.5 times than that of the early 8 x 8 fuel and reduced the fuel cycle cost by about 30 %. The spent fuel amount is also reduced in inverse proportion to the discharge burnup increase.

As a restriction in the fuel cycle for high burnup, there are limited accep­tances to reprocessing facilities (maximum burnup of fuel assembly < 55 GWd/ t at Rokkasho, Japan) and transport and processing facilities (maximum ura­nium enrichment <5 wt%). Since Step III, the development of high burnup fuel has been continued within the restriction to increase the burnup and reduce the fuel cycle cost and spent fuel amount.