Category Archives: Nuclear Reactor Design

Fuel temperature coefficient of thermal utilization factor

Considering fuel and moderator (coolant) for simplicity, the thermal utili­zation factor f is given by the next formula


where Xf and xjf are the macroscopic absorption cross sections for thermal neutrons of fuel and moderator, respectively, and VF and VM are their volume fractions. Z is called the thermal disadvantage factor [17] originat­ing from heterogeneity of the fuel lattice cell and it is defined as

Z — Фм/0F.

It is noted that the thermal disadvantage factor is also dependent on the fuel and moderator temperatures. Although structure materials were omitted in Eq. (1.64) for simplicity, the following discussions essentially do not change even if considering the structure.

In recalling the axial expansion of the fuel, it should be noted that, in the numerator and denominator of Eq. (1.64), there will be a variation in the atomic density in the macroscopic cross section of the fuel due to an increase in the fuel temperature. Considering the fuel temperature depen­dence of the thermal disadvantage factor additionally, the fuel temperature coefficient of the thermal utilization factor is given as

Подпись: (1.65)1 # _f1 n( 1 dN* 1

f dTF ‘ nf dTF z dTFJ

= (l-f’)(-eF-ai) .

The first term in the second parenthesis in the final expression is determined by the linear expansion coefficient and has a negative effect of about 10_5 Ak/k/K for a solid fuel. Physically, this shows the probability of the thermal neutron absorption in the fuel will decrease due to a decrease in the fuel density. The second term is discussed later with the thermal disadvantage factor term in the moderator temperature coefficient.

As mentioned above, the fuel temperature coefficient was discussed with the Doppler effect and the fuel expansion effects through p and f. A negative temperature coefficient by the Doppler effect is dominant among them. Furthermore, because the fuel temperature responds immedi­ately to changes in reactor power compared with the moderator tempera­ture, the fuel temperature coefficient is thus often described as the prompt temperature coefficient. In this connection, it is of greatest importance that the temperature coefficient becomes negative owing to the Doppler effect.

Nodal diffusion method

The finite difference method is widely used in the design calculation of fast reactors, the analysis of critical assembly experiments, and so on. For fast reactors, convergence of the outer iteration is fast due to the long mean free paths of neutrons, and moreover, the nuclear and thermal-hydraulic coupled core calculation is not needed. Hence, the computation speed required for the design calculation can be achieved even with fine meshes in the finite differ­ence method. For conventional LWRs as shown in Fig. 2.22, however, a fine meshing of 1-2 cm is necessary to obtain a high-accuracy solution because the neutron mean free path is short.

For example, a 3D fine meshing of a PWR core of over 30,000 liters leads to a formidable division of several tens of millions of meshes even though


Fig. 2.22 Comparison of conventional LWRs and critical assembly (TCA) in size

excluding the reflector region. Furthermore, the neutron diffusion equation is repeatedly solved in the nuclear and thermal-hydraulic coupled core calcu­lation, core burnup calculation, space-dependent kinetics calculation, and so on. Hence, a calculation using the 3D finite difference method with such a fine meshing is extremely expensive even on a current high-performance computer and therefore it is impractical. The nodal diffusion method [18] was, therefore, developed to enable a high-speed and high-accuracy calculation with a coarse meshing comparable to a fuel assembly pitch (about 20 cm). It is currently the mainstream approach among LWR core calculation methods [19].

The numerical solution of the nodal diffusion method is somewhat compli­cated and is not discussed here. The main features of the approach are briefly introduced instead.

(i) Since the coarse spatial mesh (node) is as large as a fuel assembly pitch, the number of unknowns is drastically reduced compared with that of the finite difference method.

(ii) The 3D diffusion equation in a parallelepiped node (k) is integrated over all directions except for the target direction and then is reduced to a 1D diffusion equation including neutron leakages in its transverse directions. For example, the diffusion equation in the x direction is expressed as

image216Fuel Type 1

Подпись: ОПодпись: 20 cm

Подпись: water Reflector
Подпись: Vacuum Boundary
Подпись: T3

Fuel Type 2

Reflective Boundary

Fig. 2.23 3D benchmark problem of IAEA

—D (.х^+ХІфІ (x) = Sk Gc)—^~^kLy (л) + "“]Г^ (x)| (2-91)

where Ay and Akz are the mesh widths in y and z directions, respectively. Lky (x) and Lkz (x) represent the neutron leakage in each direction. These unknown functions are provided by a second-order polynomial expansion using the transverse leakages of two adjacent nodes.

(iii) Typical solutions to the 1D diffusion equation of Eq. (2.91) are (a) the analytic nodal method [20] for two-group problems, (b) the polynomial expansion nodal method [18, 21] to expand фкх(x) into a about fourth — order polynomial, and (c) the analytic polynomial nodal method [22, 23] to expand SX(x) into a second-order polynomial and to express фХ(x) as an analytic function.

A 3D benchmark problem [24] given by the IAEA as shown in Fig. 2.23 is taken as a calculation example for the suitability of the nodal diffusion method. A PWR core model is composed of two types of fuels and control rods are inserted at five locations of the quadrant inner fuel region. At one location, control rods are partially inserted to 80 cm depth from the top of the core. The meshing effect on the power distribution is relatively large in this case and hence this benchmark problem has been widely employed to verify diffusion codes.

The calculation results using MOSRA-Light code [21] which is based on the fourth-order polynomial nodal expansion method are shown in Fig. 2.24 for two

p|r = 1.02903)












— 0.73


— 1.02











— 0.40

— 0.94


















— 1.29

— 0.26

— 0.24


















— 1.24


— 0.34




— 0.03

















— 0.99









Fig. 2.24 Comparison of effective multiplication factor and assembly power distribution by nodal diffusion method [21]

mesh sizes (20 and 10 cm). The reference solution has been taken by an extrap­olation to zero size from the five calculations with different mesh sizes using a finite difference method code. For the effective multiplication factor, the discrep­ancy with the reference value is less than 0.006 % Ak in either case and thus the meshing effect can be almost ignored. For the assembly power, the discrepancy in the case of 20 cm mesh is as small as 0.6 % on average and 1.3 % at maximum. The discrepancy becomes smaller than 0.5 % on average in the case of a mesh size of 10 cm or less. It is noted that the finite difference method requires a mesh size smaller than 2 cm and more than 100 times longer computation time to achieve the same accuracy as that in the nodal diffusion method.

Thus the IAEA benchmark calculation indicates the high suitability of the nodal diffusion method to LWR cores which have fuel assemblies of about 14— 21 cm size. An idea of the nodal diffusion method is its approach to decompose the reactor core into relatively large nodes and then to determine the neutron flux distribution within each node to maintain the calculation accuracy. For example, the polynomial nodal expansion method introduces the weighted residual method to obtain high-order expansion coefficients. It leads to an increased number of equations to be solved.

In other words, the high suitability of the nodal diffusion method to practical LWR calculation results is because the computation cost reduction due to a substantial decrease in the number of meshes surpasses the cost rise due to an increase in number of equations. Conversely, if it is possible to reach sufficient accuracy with the same meshing, the finite difference method will be effective

Подпись: Thermal-Hydraulic calculation Подпись: Nuclear calculation

image224 Подпись: < Start > [ H(r), fguess (r)] l

Подпись: ►NB10(r)

Подпись: keff ,Ф(Т X q'"(r) T NXe(rX NSm(r) Подпись: P(f), Tm (rX SB (r) l TZ < End >

Enthalpy rise Void fraction Pressure drop Inlet flow rate

Fig. 2.25 Nuclear and thermal-hydraulic coupled core calculation of LWR

because it is not necessary to solve extra equations. Hence, the nodal diffusion method does not need to have an advantage over the finite difference method for the analysis of fast reactors or small reactors.

Reactivity control by burnable poisons

As shown in Fig. 3.13, reactors in the beginning of the operation cycle are required to have an excess reactivity compensating for the reactivity variation due to consumption of fissile materials and accumulation of FPs with burnup. Since some fuel rods of fresh fuel assemblies include pellets containing several weight percent of burnable poisons (neutron absorbers), the excess reactivity necessary at the beginning of the operation cycle is mitigated and the reactivity variation during reactor operation is lessened. In other words, the effect of burnable poisons on reactivity control is large at the beginning of the operation cycle because of the large amount of burnable poisons and neutron absorption. The reactivity variation with burnup becomes small through the balance between the reactivity decrease with burnup and the reactivity recovery due to the neutron absorption decrease with burning of burnable poisons.

Burnable poisons, with the functions noted above, are required to have a large neutron absorption cross section and gadolinia (Gd2O3) is used as a burnable poison material for BWRs. Several weight percent of gadolinia are mixed with uranium oxide (UO2) powder and processed into pellets which are inserted into several fuel rods. Since gadolinia is solid-soluble in uranium dioxide, it can be uniformly distributed in the pellets [3]. In the BWR fuel assembly design shown in Fig. 3.5, seven of 62 fuel rods are gadolinia-added fuel rods to control the excess reactivity.

Figure 3.15 shows the typical burnup characteristic of a fuel assembly containing the burnable poison gadolinia. The burnable poison-containing rods make the infinite multiplication factor small (suppression of excess reac­tivity) at the beginning of burnup when the concentrations of 155Gd and 157Gd are high. 155Gd and 157Gd are converted with burnup into 156Gd and 158Gd

image394Neutron Infinite Multiplication Factor without Addition of Gadolinia

Cycle Burnup

Variation m Excess Reactivity

4-Batch Core

• : Neutron Infinite

Multiplication Factor at Each BOC and EOC

Assembly Average Burnup [GWd/tJ

Fig. 3.15 Bumup characteristics of fuel assembly with burnable poison (Gadolinia)

respectively, which have small absorption cross sections, and their concentra­tions decrease. The suppression effect on the excess reactivity therefore, becomes small and the infinite multiplication factor is recovered almost to the value when not including the gadolinia-added fuel rods. Any remaining gadolinia at the end of the operation cycle will cause a reactivity loss as a useless neutron absorber, so the concentration of gadolinia is set to be burned out at the end of the operation cycle.

The number of gadolinia-added fuel rods can be increased for high suppres­sion of the excess reactivity at the initial burnup and the gadolinia concentration can be increased for long-term suppression of the excess reactivity. Figure 3.15 shows variation of the infinite multiplication factor with the average fuel assembly burnup for a 4-batch refueling; numbering corresponds to each BOC and EOC. In a core loaded with the fuel assemblies having the burnup characteristics shown in Fig. 3.15, the infinite multiplication factor of fresh fuel assemblies increases with reactor operation (0-1) and those of other fuel assemblies decrease (1-2, 2-3, and 3-4). Both characteristics compensate each other and therefore the variation in excess reactivity during an operating cycle becomes small as shown in the insert figure of Fig. 3.15. Such a proper usage of burnable poisons mitigates the work burden of control rod operation and coolant flow rate change, and considerably improves the controllability of reactor operation.

Doppler coefficient

The Doppler coefficient represents that reactivity is decreased by an increase in neutron resonance absorption (mainly by 238U and 240Pu) resulting from power and then fuel temperature rises. A reactivity variation to a 1 % power rise is referred to as the Doppler power coefficient, and a reactivity variation to a 1 ° C fuel temperature rise is referred to as the Doppler temperature coefficient. As units of reactivity change, pcm and %Ak/k are used where 1 pcm = 10—5Ak/k and 1 % Ak/k = 10—2Ak/k. The former unit is mainly used for a relatively small reactivity change such as with reactivity coefficients and the latter is used for a relatively large reactivity change. Production of 240Pu as burnup proceeds makes the Dopp­ler temperature coefficient more negative, while the fuel temperature incre­ment by the power rise becomes smaller. Therefore, the Doppler power coefficient becomes slightly less negative as burnup increases. Typical Doppler power coefficient as a function of power is shown in Fig. 3.37.

The Doppler temperature coefficient is about —3 to —5 pcm/°C. LWRs have negative Doppler coefficient and reactivity is decreased by the feed­back when power increases.

Design of fuel assembly

(1) Features of fast reactor fuel assembly

A fast reactor fuel assembly consists of triangularly arranged fuel elements and a containing wrapper tube (cf. Fig. 4.3). Positions of the fuel elements are kept by the wire spacers or the grid spacers, so that the coolant channels are ensured. The upper shielding is contained in the upper part of the wrapper tube. The top of the wrapper tube is connected to the handling head which has a gripper function for handling the fuel assembly. The lower shielding is also contained in the lower part of the wrapper tube. The bottom of the wrapper tube is connected to the entrance nozzle through which the coolant enters. The wrapper tube, along with the entrance nozzle, forms the individ­ual coolant flow passage and hence enables flow distribution among the fuel assemblies. The wrapper tube also protects the fuel element bundle and would act as one of the barriers against propagation of fuel damage during accident conditions.

Upper, lower and intermediate spacer pads are provided on the outer surface of the wrapper tube in order to keep the spacing between the neighboring assemblies and to take loads during operation including seismic load.

(2) Major design principles of fast reactor fuel assembly

For the design of a fuel assembly, the assembly size, i. e. the number of fuel elements, needs to be determined first. The following factors influence the assembly size.

(a) Reactivity worth per assembly

(b) Decay heat of an assembly

(c) Weight of an assembly

(d) Fuel cycle cost

(e) Refueling time

(f)Degrees of freedom for flow allocation

(g) Degrees of freedom for control rod arrangement

(h) Stable support of a fuel element bundle

Among them, (a), (b) and (c) increase with assembly size. High reactivity worth per assembly increases the degree of risk associated with a reactivity accident during fuel handling and criticality in case of submergence caused by an accident during transportation of the assemblies. High assembly decay heat or large assembly weight leads to excess loads on the cooling facility during fuel handling and on the refueling machines.

On the other hand, (d) and (e) decrease with increasing assembly size. Generally, as the number of fabricated assemblies is larger, the fuel fabri­cation cost is higher. Thus, larger assembly size reduces the fuel cycle cost including the fabrication cost. Also, larger assembly size means a smaller number of assemblies which shortens the time needed for refueling and hence improves the plant availability.

As for (f) and (g), the degrees of freedom decrease with increasing assembly size, i. e. decreasing the number of assemblies. As for (h), espe­cially if the wire spacer is adopted, larger assembly size worsens the support because the degrees of freedom of the fuel elements’ displacement increase with the number of the fuel elements, i. e. the assembly size.

From the above and some other considerations, an adequate assembly size is determined and then the assembly is designed based on two design principles.

• The assembly has and keeps sufficient mechanical and structural strength at normal operation and anticipated operational occurrences during the duration of service

• The assembly has and keeps sufficient mechanical and structural strength against normal loads during transporting and handling.

(3) Major evaluation items in fuel assembly design

(1) Stress evaluations

According to the design principles, sufficient strength of each compo­nent of the fuel assembly against various loads is confirmed and kept by evaluating the stresses associated with the loads at normal operation and anticipated operational occurrences by using the finite element method or other approaches.

(2) Duct-to-duct interaction (DDI)

DDI is evaluated by confirming that the neighboring wrapper tubes do not contact each other (i. e. the refueling function is not obstructed) due to expansion of the wrapper tube such as by thermal expansion, swell­ing and creep. In the DDI evaluation, bending of the wrapper tube due to thermal deformation by the temperature distribution and swelling by neutron irradiation must be considered as well as the expansion.

Nuclear Reactor Design

Nuclear reactor design is based on knowledge and data from many nuclear engi­neering fields including nuclear reactor physics, nuclear thermal hydraulics, and nuclear safety.

In nuclear reactor design, reactor performance is evaluated by numerical anal­ysis for requirements of nuclear and thermal limitations, stability, controllability, and safety, referred to as conceptual design. To approximate reactor performance, the design then proceeds to hardware design of reactor facilities using knowledge from mechanical engineering, electrical engineering, and nuclear structural engi­neering. The consequent reactor power plant facilities are mainly described in another textbook of this series titled Nuclear Plant Engineering. Knowledge of nuclear reactor fuel and materials is referenced in the conceptual design for reactor performance and in the hardware design which include in particular the fuel assemblies, reactor vessel, and its internal structures. The conceptual and hardware designs may be repeated if necessary.

To understand nuclear reactor design, it is necessary not only to be aware of the structures and compositions of the facilities constructed by the conceptual and hardware designs but also to comprehend the processes and methods used to reach the nuclear reactor design. This book describes an approach to design and calculation methods, focused on core design. Many pages have been devoted to core calculation methods and light water reactor core design. We hope that this book will help professionals as well as beginners understand and review the methods and core design.

Features of each chapter are summarized as follows.

“Fuel Burnup and Reactivity Control” of Chap. 1 is not only a key component of nuclear reactor physics but also of core design, reactor operation management, and safety. Details, including calculation methods, are described. Because the textbook of this series titled Nuclear Reactor Physics does not cover the topics in this chapter, it is recommended that readers interested in nuclear reactor physics read Chap. 1.

Section 2.1, “Nuclear Design Calculation,” first describes nuclear data and neutron cross sections and their processing methods for the nuclear calculation.

Next, the lattice calculation, lattice burnup calculation, core diffusion calculation, nuclear and thermal-hydraulic-coupled calculation, core burnup calculation, and space-dependent kinetics calculation are described. From this description, readers will be able to understand the nuclear design calculation methods used in practice. Section 2.2, “Reactor Core, Plant Dynamics and Safety Calculations,” is intended to provide an understanding of the key points of plant dynamics and safety calculation methods as well as core calculation methods. Core design calculations are performed to determine core characteristics at normal conditions in combina­tion with nuclear and thermal-hydraulic calculations. The single-channel thermal- hydraulic calculation model with fuel rods and its hydraulic equivalent coolant path provides the simplest model for heat transfer flow of the core beyond the fuel rods. The three-dimensional nuclear and thermal-hydraulic-coupled core burnup calcu­lation, in combination with nuclear and heat transfer calculations, is used in core management as well as core design. The plant dynamics calculation is concerned with plant control, stability, and safety analysis. First, the node junction model, which is a basic model for the heat transfer calculation in the plant system including the reactor, coolant pipes, valves, and pumps, is described and then reactor control system design, plant startup analysis, and reactor stability analysis methods are treated. The basic concept of reactor safety analysis methods is also touched upon. Supercritical light water-cooled reactors are introduced as an example, and the nature of consideration and analysis methods is the same as that of light water reactors. The actual facilities and systems of light water reactors are complex. We believe that the descriptions in this book, rather than focusing on details, are better suited to understanding the general nature of reactors. More detailed analysis models and large-scale conventional codes are used in practical safety analysis. Their explanations are left to books for professionals; this book is expected to help in understanding the basic analysis methods. At the very end of Sect. 2.2, the FEMAXI-6 code is described for fuel rod behavior analysis. Fuel rod behavior and integrity concerning pellet and cladding are associated with core characteristics as burnup and core safety from the point of view of security of fuel integrity at abnormal transients. Such a series of analysis methods in the core, plant, and fuel rod behavior should be able to be understood as their individual concepts and then in their overall connection.

Section 3.1 is “Development and Improvement of Light Water Reactors,” and Sects. 3.2 and 3.3 present core design and core fuel management of boiling water reactors (BWRs) and pressurized water reactors (PWRs), respectively. Basic core design flow, core configuration setup, fuel lattice, fuel assembly design, reactivity characteristics, power distribution control, transitions and future trends in core design, core management, fuel management, and core design of conventional light water reactor are described.

“Design of Advanced Reactors,” Chap. 4, describes fast reactors and high — temperature gas-cooled reactors, focusing on core design. Detailed descriptions of each reactor type are left to other books; this book is expected, instead, to help in understanding the core design concept of each reactor. For example, advanced reactors are characterized by single-phase flow cooling, high-temperature core cooling, and different fuel compositions. The design criteria and principles are also different from those of light water reactors. Understating the difference may lead to a deeper comprehension of reactor design. This book also describes experiences with the HTTR experimental reactor as a high-temperature gas-cooled reactor.

This book systematically describes consideration and calculation methods about nuclear reactor design, mainly core design. We hope it will be useful for readers actually working in the nuclear power industry, research and development, and safety, as well as for students.

In addition to the authors, sincere appreciation goes to Shuichi Hasegawa, Haruka Moriguchi, Yumiko Kawamata, Moe Sekiguchi and the committee members of University of Tokyo, Shigeaki Okamura of JAEA, Yuko Sumino and Nobuko Hirota of Springer Japan who helped to publish the book. I am also grateful for the editing assistance of Takashi Kiguchi and in English of Carol Kikuchi.

Tokyo, Japan Yoshiaki Oka

November 2013

Perturbation Theory

The change in reactivity caused by a small change (perturbation) occurring in reactors can be expressed by perturbation theory. Suppose that a perturbation (denoted by £) appears in the multi-group diffusion operators as

M’ = M + 8M (1.98)

F’ = F + 8F (1.99)

where M and F are the operators before the perturbation and M0 and F0 are those after the perturbation.

The multi-group diffusion equation before the perturbation and its adjoint equation are written as


My = 2-Fy (1.101)

and the multi-group diffusion equation after the perturbation is written as


The dagger symbol {is superscripted for the adjoint operator and adjoint neutron flux, and the prime symbol’ is used for the operator, flux, and effective multiplication factor after the perturbation. It should also be remembered that the effective multiplication factor in the adjoint equation is identical to that in the original diffusion equation.


The change in reactivity is calculated from the change from k to k’. This reactivity change becomes the expression of the perturbation theory through a mathematical formulation with 8M and £F. Taking the inner product with ф’ on both sides of Eq. (1.102) gives

Substituting M’ and F’ by Eqs. (1.98) and (1.99) respectively gives

Шф’Жф 8Мф’)=у(ф ‘)

Using the definition of the adjoint operator [Eq. (1.90)] and the adjoint equation [Eq. (1.101)], the first term in the left-hand side can be transformed as

<y, М0′)=(МУ, f)=2-(Fy, ф’)=^г(ф F

!Z iz


and then

Подпись: _ k'-k _ 1 1^ p~ kk'~ k' Подпись: ^T8¥-8M _k <y, W)
image119 image120

is given. Hence, the change in reactivity caused by the perturbation can be found as

By approximating 1/k0 multiplied by SF in the numerator of the right-hand side as unity, the reactivity change becomes


This expression is called the exact perturbation theory and the non-matrix form is given by Eq. (1.104).


To evaluate the reactivity change using the exact perturbation theory, it is necessary to know the neutron flux after the perturbation as well as the adjoint flux before the perturbation and changes of the macroscopic cross sections and diffusion coeffi­cient. In other words, the change in neutron flux caused by the perturbation should be considered.

If the perturbation is small enough and its effect on neutron flux is similarly small, the neutron flux after the perturbation can be approximated to that before the perturbation. Equation (1.103) can be written as Eq. (1.105) which is called the first-order perturbation theory.


Further, this equation becomes simpler in one-group theory because the neutron flux is self-adjoint. The equation in the one-group first-order perturbation theory can be detailed as Eq. (1.106).

J IS(vXf (r ))ф2(r)—SD (г )(Уф Cr ))2—£Za (г)ф2(г )1 d3r [2]

Fig. 1.15 Control rod partially inserted along the axis of a cylindrical reactor

Fuel Rod Analysis

[1] Fuel rod integrity

The principle role of fuel rod cladding is to confine radioactive FPs and to prevent contamination of the coolant system; therefore an assurance of fuel rod integrity is important for reactor design and safety.

When an abnormal transient event occurs in a nuclear plant with a frequency of more than once during the plant life time, the safety criterion is associated with whether the core can return to the normal operation without core damage

image332 image333
Подпись: Top Dome
Подпись: Down-
Подпись: Quench
Подпись: Bottom Dome

image340Suppression Pool

Fig. 2.48 Calculation model for LOCA reflooding analysis

after the plant is stabilized. Thus, fuel rods are required to keep their integrity during abnormal transients (anticipated operational occurrences) as well as normal operation.

The criteria for fuel rod integrity differ somewhat between BWRs and PWRs because of different operating conditions. Fuel rods are designed generally based on the following criteria [25, 27].

(i) Average circumferential plastic deformation of fuel cladding < 1 %

(ii) Fuel centerline temperature < Pellet melting point (PWR)

(iii) No overpressure within fuel rod

(iv) Allowable stress of fuel rod cladding

(v) Cumulative damage fraction < 1

The overall fuel rod integrity is evaluated further in cladding corrosion and hydriding, pellet cladding interaction (PCI), cladding creep rupture, rod fretting wear, cladding bending, and irradiation growth of the fuel rod and assembly.

The fuel damage criteria and allowable limits of BWRs and PWRs are given in Table 2.2. Based on the system of fuel damage occurrence in BWRs or PWRs, restrictions at normal operation and allowable limits at abnormal tran­sient are determined in the fuel design considering the each following;

(i) Cladding damage due to overheating resulting from insufficient cooling

(ii) Cladding damage due to deformation resulting from a relative expansion between pellet and cladding

Table 2.2 Criteria and allowable limits of fuel damage in BWR and PWR




(i) Cladding


The fuel cladding integrity

The DNB design basis


ensures that during normal operation and abnormal transients, at least 99.9 % of the fuel rods in the core do not experience transi­tion boiling

requires at least a 95 % probability, at a 95 % confidence level, that the limiting fuel rods in the core will not experience DNB during normal operation or any transient conditions

Allowable limits at transient

MCPR: 1.07

MDNBR: 1.30

Restrictions at normal

MCPR: 1.2—1.3

MDNBR: 1.72

(ii) Cladding


The fuel cladding integrity

The fuel cladding integrity


ensures that during normal operation and abnormal transients, the fuel rods in the core do not experience circumferential plastic deformation over 1 % by PCI

ensures that during normal operation and abnormal transients, the fuel rods in the core do not experience circumferential deforma­tion (elastic, plastic, and creep) over 1 % by PCI


Direct evaluation of criteria

MLHGR: 59.1 kW/m

limits at transient

by experiments

(2,300 °C fuel centerline temperature)

Restrictions at normal

MLHGR: 44 kW/m

MLHGR: 43.1 kW/m

(1,870 °C fuel centerline temperature)

For damage mechanism (i), the BWR criterion is that “the fuel cladding integrity ensures that during normal operation and abnormal transients, at least 99.9 % of the fuel rods in the core do not experience transition boiling”. In PWRs, similarly, the criterion is that “the DNB design basis requires at least a 95 % probability, at a 95 % confidence level, that the limiting fuel rods in the core will not experience DNB during normal operation or any transient conditions”. (DNB means the departure from nucleate boiling.)

Based on those criteria, the BWR design requires the allowable limit of the minimum critical power ratio (MCPR) to be 1.07 at abnormal transients and the restriction to be 1.2—1.3 at normal operation. Similarly, the PWR design has the allowable limit of minimum heat flux ratio (minimum departure from nucleate boiling ratio, MDNBR) to be 1.30 at abnormal transients and the restriction to be 1.72 at normal operation.

For the damage mechanism (ii), the BWR criterion is that “During normal operation and abnormal transients, the fuel rods in the core do not experience circumferential plastic deformation over 1 % by PCI”. In PWRs, similarly, the criterion is that “During normal operation and abnormal transients, the fuel rods

Table 2.3 Fuel rod behavior in FEMAXI-6

Thermal behavior (temperature dependent)

Mechanical behavior


Thermal conduction (radial heat flux distribution)

Fission gas release (temperature and burnup dependent)

Thermal expansion, elasticity and plasticity, creep, cracking, initial relocation, densification, swelling (solid FPs and gas bubbles), hot-press


Thermal conduction Oxidation film growth

Thermal expansion, elasticity and plasticity, creep, irradiation growth, oxidation film growth

Fuel Rod

Gap thermal conduction (mixed gas, contact, radiation) Cladding surface heat transfer Gap gas flow

PCMI (pellet cladding mechanical interaction), friction, bonding

in the core do not experience circumferential deformation (elastic, plastic, and creep) over 1 % by PCI”.

Central melting of pellets in LWR fuel causes a phase change and a volume increase, and the fuel cladding may be substantially deformed mainly due to pellet cladding mechanical interaction (PCMI). The fuel rod design criteria of PWRs require that the fuel centerline temperature will be lower than the pellet melting point. Hence, the allowable limit of the fuel centerline temperature is determined to be 2,300 °C at abnormal transients and its corresponding max­imum linear power density is 59.1 kW/m. The restrictions at normal operation are 1,870 °C for the fuel centerline temperature and 43.1 kW/m of the maxi­mum linear power density.

For internal pressure of fuel rods, the BWR design requires that the cladding stress due to the internal pressure will be less than the allowable strength limit. The PWR design restricts the internal pressure of fuel rods to less than the rated pressure of primary coolant (157 kg/cm2-g) in order to avoid expansion of the gap between pellet and cladding due to creep deformation of cladding outside at normal operation. This phenomenon is called “lift-off”. The criteria of “ASME B&PV Code Sec III” are used as allowable stress limits for LWRs.

Core Management [3, 26, 27]

[1] Summary of core management

Core management technology is intended to secure reactor safety and to realize efficient and economic fuel burning. BWR core management can be divided into reload core design and operation management as shown in Fig. 3.27. In the reload core design, middle — and long-term fuel preparation plans are made and the fuel assembly arrangement and basic operation plan are determined for each reload core. The operation management determines a detailed operation scheme at startup and normal operation and supports safe operation by moni­toring core performance using process computers.

Axial Power Distribution Oscillation

For a constant core power, the difference in average moderator temperature between upper and lower core region is constant and independent of axial power distribution. Hence, axial power distribution oscillation cannot be expected to be suppressed by the moderator temperature coefficient and the Doppler effect will be expected instead. Since the axial power distri­bution becomes flatter with burnup from an initial cosine distribution at BOC, the stability of the axial power distribution oscillation becomes lower and it may cause a little divergence near EOC. In this case, the axial power distribution can be controlled by the CAOC operation mentioned before and the axial power distribution oscillation can be sufficiently suppressed. The axial power distribution oscillation induced by axial Xe oscillation can also be suppressed as shown in Fig. 3.57. The axial power distribution can be continuously monitored by upper and lower separate ex-core neutron detectors and the oscillation can be detected and suppressed.