The reactivity control in PWRs is done by control rod operation, adjustment of boron concentration in primary coolant (referred to as chemical shim) and if necessary using burnable poison rods.

(1) Design Principles for Reactivity Control

(i) The reactor core should be made subcritical without exceeding the allowable design limit of fuel from the hot standby or hot operation

condition. Core reactivity should be controlled at the hot condition by at least two independent control systems: the control rod control system and the chemical and volume control system. The latter adjusts the boron concentration in primary coolant.

(ii) Core reactivity is controlled usually through role sharing of the reac­tivity control equipment as follows.

• Rod Cluster Control Assemblies: Rod cluster control assemblies are used to control the reactivity variation (power defect) by the power variation from hot zero power to hot full power; i. e. they provide control of relatively fast reactivity variations.

• Boron concentration: Boron concentration is adjusted to control reactivity variations due to the primary coolant temperature varia­tion from cold temperature to hot zero power temperature, FP (Xe and Sm) concentration variation, and fuel burnup; i. e. they provide control of relatively slow reactivity variations.

• Burnable poisons: Burnable poisons are employed to partially con­trol the excess reactivity necessary for fuel burnup, which makes the moderator temperature coefficient negative at hot power operation by being able to reduce the boron concentration in the primary coolant.

Figure 3.38 shows a typical reactivity control scheme by the listed reactivity control equipment. The left part (i. e. time before shutdown) indicates the necessary excess reactivity for fuel burnup, being controlled by soluble boron and burnable poison, and the reactivity decline as burnup, being compensated by boron dilution, where the reactivity is set as zero at hot full power. Control rods are inserted to shut down the core (from hot full power to hot zero power) at time zero and to secure subcriticality. The Xe decay following Xe accumulation is compensated by increasing the boron concentration (boration) and the coolant temperature decrease from hot to cold is also compensated in a similar way.

Figure 3.38 indicates that the boron concentration changes during the period of the Xe decay and the coolant temperature decrease. In actual reactor operation, however, the operation of changing the boron concen­tration can be activated before that period and the reactivity control scheme does not necessarily correspond to the period.

The core characteristics are both neutronic and thermal. Neutronic characteristics are obtained by the core nuclear design. In the nuclear design, the core configura­tion, the refueling plan and the plutonium enrichment are determined so that the core safely generates the designed thermal power, based on the plant and fuel basic specifications, throughout the plant life. Also, the core reactivity, breeding perfor­mance, neutron flux distribution, burnup characteristics, control rod worth, and reactivity coefficients are evaluated. The design point of those parameters are determined and coordinated with other design points as described in Fig. 4.4.

The nuclear design calculation methods appear earlier in Chap. 2.1. In the basic calculations of the nuclear design of fast reactors, multi-dimensional, multi-group diffusion calculation codes are mainly utilized.

[1] Multi-group reactor constants

In the nuclear design calculations of fast reactors, a wide range of neutron energies from thermal neutrons (0.025 eV) to fast neutrons (up to about 10 MeV) needs to be treated. The wide energy range is divided into many groups. The set of multi-group reactor constants, which are the averaged cross section of each nuclide within each energy group, are utilized for each group. Based on those sets of multi-group reactor constants, prepared by processing evaluated nuclear data files such as JENDL, the multi-group reactor constants for the design are made using the core configuration, the material composition and the temperature data of the actual core as input.

[2] Nuclear design calculations

The neutronic characteristics, such as the core reactivity, power distribution, burnup characteristics, and control rod worth, are evaluated using the multi­group reactor constants, the core geometry, etc. The reactivity coefficients are calculated by multi-dimensional diffusion and perturbation codes.

Owing to the recent progress in computer performance, the following improvements have been made.

• The number of energy groups is increased for better calculation accuracy. Three-dimensional calculation codes with detailed modeling of core geom­etry are mainly used.

• In cases for which the neutronic characteristics must be accurately calcu­lated in geometries with strong heterogeneity or sharp spatial change in the neutron flux, multi-dimensional transport calculation codes are utilized and then the result of the diffusion calculation is corrected if necessary.

• In cases for which the neutronic characteristics must be accurately calcu­lated in complex geometries, Monte-Carlo codes are utilized.

[3] Validation by calculating mock-up critical experiments

Mock-up critical experiments for simulating fast reactor cores have been conducted and their measurement data have been utilized for evaluating the accuracy of core nuclear design calculation methods. Based on those evalua­tions, the design calculation results can be corrected and the design specifica­tions are then determined with accompanying prediction errors. In Japan, the major mock-up critical experiments done so far are those by the Fast Critical Assembly (FCA) for Joyo, the MOZART experiments [12] for Monju, and the JUPITER experiments [13] for a demonstration FBR plant.

Shigeo Ohki

Abstract Nuclear fuel burnup and reactivity control are important points in the core design of nuclear reactors.

The fuel burnup analysis generally evaluates the time-dependent core power distribution and reactivity by solving burnup equations for the atomic density change of nuclides contained in the fuel as well as solving multi-group diffusion equations for neutron flux distribution and effective neutron multiplication factor. The core power distribution is necessary information for thermal-hydraulic and fuel designs.

The core design for reactivity control predicts reactivity change during reactor operation and determines its optimal control methods based on calculations of reactivity change with fuel burnup, fission product (FP) accumulation (poisoning effect), inherent reactivity feedback by temperature changes of fuel and coolant, etc. Among the general methods available for reactivity control, the insertion and withdrawal of neutron absorbers, generally referred to as control rods, is the approach usually taken for power reactors. A burnable poison, (a nuclide that has a large neutron absorption cross section) or a chemical shim (a neutron-absorbing chemical, usually boric acid, which is concentrated in the moderator or coolant) is employed for reactivity control depending on reactor types.

Fuel burnup and reactivity control based on fundamental theories with numerical expressions will be briefly reviewed in this chapter.

An application of the one-group first-order perturbation theory is discussed here.

A bare cylindrical reactor of extrapolated radius R and height H is considered, in which a central control rod of radius a is partially inserted, as shown in Fig. 1.15. The insertion depth of the control rod from the origin in the top of the cylindrical reactor is denoted by x. If the control rod is a relatively weak absorber of neutrons and the control rod insertion has a small effect on the change in neutron flux distribution, then the first-order perturbation theory can be applied to obtain the reactivity worth of the partially inserted control rod.

The macroscopic absorption cross section is assumed to increase by 8 X a in the region of 0 ^ z ^ x and 0 ^ r S a • In this coordinate system, the unperturbed neutron flux is

(1.107)

where A is a constant. Introducing Eq. (1.107) into Eq. (1.106) and noting that the differential volume element d3r is 2nrdrdz, the reactivity change due to the rod insertion of x can be obtained as Eq. (1.108).

Comparing this with the reactivity change p(H) when the control rod is fully inserted, the relative reactivity worth of the control rod is given by Eq. (1.109) which is illustrated in Fig. 1.16.

Fig. 1.16 Relative reactivity worth of control rod as a function of its insertion depth

This is called the S-curve of control rod worth. The maximum change in the reactivity occurs when the end of the control rod is at the center of the reactor.

This case shows that the perturbation theory can be used to provide a satis­factory estimate of the reactivity worth as a function of location in the perturbation region.

Exercises of Chapter 1

1. Consider the hypothetical case of an infinite-sized thermal reactor initially fueled with enriched uranium which operates at a constant neutron flux. For a constant atomic density of 238U: (a) derive the equations that determine the time

ЛОГ ЛОА

dependence of the U and Pu concentrations in the reactor and (b) plot the behavior as a function of time.

2. (a) Derive the equations that determine the transient behavior of the 135I and 135Xe concentrations in an infinite-size thermal reactor which operates at a neutron flux ф0 when the flux is changed to ф1 and (b) plot the behavior as a function of time.

3. Explain the main reactivity feedback effects and reactivity control methods for each system of the BWR, PWR, high-temperature gas-cooled reactor (HTGR), and liquid-metal fast breeder reactor (LMFBR).

4. Consider a light water-cooled graphite-moderated thermal reactor fueled with very low-enriched uranium which operates with burnup. Explain the causes of the results in the cases of (a) a positive coolant void coefficient and (b) a positive moderator temperature coefficient [27].

5. Consider a 235U fueled thermal reactor which is a homogeneous cubic reactor of side length L. Assume that the temperature suddenly increases by the amount AT
throughout a cubic region of side length L/6 along the center of the reactor. Compute the reactivity change introduced into the reactor by this local temper­ature rise using the one-group first-order perturbation theory. Ignore the change in neutron leakage due to the temperature rise [26].

The fuel rod behavior caused by irradiation of fuel pellets and cladding is complicated for investigation in an analytical method. Fuel rod behavior calculation codes were developed from various experiments and operational experiences. FEMAXI-6 [30] as an open code developed in Japan is a general analysis code for fuel rod behavior at normal operation or abnormal transients in LWRs. It constructs a model composed of one fuel pellet, cladding, and internal gas, and then analyzes thermal, mechanical, and chemical behavior and reciprocal action at normal operation or abnormal transients from overall power history data. Table 2.3 shows fuel rod behavior analyzed in FEMAXI-6.

The overall structure of FEMAXI-6 consists of two parts, thermal analysis and mechanical analysis, as shown in Fig. 2.49. The thermal analysis part evaluates radial and axial temperature distributions considering a change in gap size between pellet and cladding, FP gas release models, axial gas flow and

res

Output

Fig. 2.49 Overview of FEMAXI-6 structure

its feedback to heat transfer in gap, and so on. The mechanical analysis part applies the finite element method (FEM) to the whole fuel rod and analyzes mechanical behavior of fuel pellet and cladding as well as PCMI. It also calculates an initial deformation due to thermal expansion, fuel densification, swelling, and pellet relocation, and then calculates stress and deformation of pellet and cladding considering cracking, elasticity/plasticity, and pellet creep. The thermal and mechanical analysis is iteratively performed to consider the thermal feedback effect on the fuel rod mechanical behavior. Further, the local PCMI can be analyzed by the 2D FEM, based on the calculation results from both analysis parts such as temperature distribution and internal pressure.

Figure 2.50 describes the calculation model of FEMAXI-6. The entire fuel rod is divided axially into several tens of segments and radially into ring elements. For example, the figure shows ten axial and radial divisions in the fuel pellet.

Fig. 2.50 FEMAXI-6 calculation model

FEMAXI-6 contains various calculation models and experimental correla­tions that users can specify. Details of models, physical properties, or correla­tions are described in the code manual [30] of FEMAXI-6.

Exercises of Chapter 2

[1] Consider a PWR UO2 pellet 0.81 cm in diameter, 1.0 cm in height, and 10.4 g/cm3 in density in which the uranium is enriched to 3.2 wt. % 235U. Answer the following questions.

(a) Calculate the atomic number densities of 235U, 238U, and O in the pellet.

235 238

UO2 contains only U, U, and O. Their atomic masses are M (235U) = 235.04, M(238U) = 238.05, and M(O) = 16.0, and Avogadro’s number is 0.6022 x 1024.

(b) One-group microscopic cross sections of 235U, 238U, and O are given in the following table.

Compute the macroscopic absorption and fission cross sections of the pellet. What is the probability that a neutron will cause a fission on being absorbed into the pellet?

(c) Compute the mass (in tons) of metal uranium in one pellet 1.0 cm in height. A PWR fuel assembly consists of 265 fuel rods bundled into a 17 x 17 array (including 24 guide tubes) with an active height of 366 cm. Compute the initial mass of uranium in one fuel assembly and compute the total loading amount of uranium (the mass of metal uranium) in the core which contains the 193 fuel assemblies.

(d) Calculate the linear power density (W/cm) of the fuel rod when one-group neutron flux in the pellet is 3.2 x 1014 n/cm2-s. The energy released per fission is 200 MeV (1 MeV = 1.602 x 10_13 J). What is the specific power [power per initial mass of metal uranium (MW/ton)]?

(e) Compute the burnup (MWd/ton) of the pellet consumed along the specific powers as shown in the figure

(f) Compute the consumed amount of 235U in the pellet from the burnup of (e). Two-thirds of the total power is generated by the uranium ( U and U) fission and the rest by the fission of plutonium produced from the conver­sion of 238U.

[2]

An infinite slab of non-multiplying uniform medium of thickness d (= 400 cm) contains a uniformly distributed neutron source S0 as shown in the following figure.

The neutron flux distribution in the slab can be determined by solving the one-group diffusion equation

d[7](b(x) , x

D +Xa0 (x)=So

where D = 1.0 cm, 2a = 2.5 x10-4 cm-1, and S0 = 1.0 x 10-3 n/cm2-s. Discretize one side to the reflective boundary in the middle into four meshes (N = 4) and calculate the neutron flux distribution using the finite difference method. Plot the distribution and compare it with the analytical solution

SQ( cosh ф ____

ф(х) — — 1—————— j— where, L = y/D/I. a

Л

Calculate for N = 8 and 16 if programming is available.

[3] (a) Consider a multiplying system with an effective multiplication factor kf. Develop the diffusion equations for fast and thermal groups in terms of the two-group constant symbols in the following table.

(b) Calculate the infinite multiplication factor k^ using the two-group constants.

[4] The burnup chain of 135I and 135Xe is given by the following figure

Show that the equilibrium concentration of 135Xe (NXe) in the reactor operating at a sufficiently high neutron flux is given approximately by

jr+rp^f

,-rXe О a

where уI and yXe are the fission yields of 135I and 135Xe respectively, and ^Xe is the one-group microscopic absorption cross section of 135Xe and Lf is the one-group macroscopic fission cross section.

[5] The energy conservation equation is given by Eq. (2.117) for no water moder­ation rod. Develop the energy conservation equation when introducing the water moderation rod.

[6] Develop equations for single-phase transient analysis at the 1D axial node i of a single coolant channel, based on the mass conservation equation [Eq. (2.116)], the energy conservation equation [Eq. (2.117)], the momentum conservation equation [Eq. (2.118)], and the state equation [Eq. (2.119)], referring to the figure.

Nuclear reactor operation management is based on a long-term operation plan of 3-5 years. The plan includes startup date of each operating cycle, shutdown period for periodic inspection, and load factor in operation. It also includes intermediate shutdown planned in each operating cycle. After the long-term operation plan, the cycle burnup is obtained from reactor operating days and load factor of each cycle, and the plan of fuel exchange to be performed during the periodic inspection period is investigated. Core analysis is done for each cycle to evaluate the plan. Proper fuel loading pattern of each cycle is inves­tigated and fresh fuel amount and types are determined.

Nuclear and I hermal-Hydraulic Calculation

and Cycle Burnup Calculation

Cycle Length

YKS

Design Criteria and Operation Limits^

This long-term burnup plan is used to systematically procure fuel which is ordered from the supplier at the very latest 2 years before delivery to the plant; this is based on considerations of procurement of enriched uranium and clad­ding and fuel fabrication times from pellet to fuel assembly.

Main tasks in the reload core design are to determine fresh fuel amount necessary for the planned operating cycle and its arrangement in the core and to make a basic operation plan.

Figure 3.28 shows a typical process flow for the reload core design in which an optimal reload core is designed by repeated investigations with combina­tions of available fuel assemblies after the number and arrangement of fuel assemblies loaded into the core and control rod pattern are guessed first.

Safety parameters for the BWR reload core include the reactivity shutdown margin, maximum linear heat generation rate, MCPR, maximum fuel assembly

burnup, channel stability, core stability, and control rod irradiation lifetime, as discussed in the beginning of this section.

During the periodic inspection period, fuel exchange is performed based on the prepared fuel loading pattern. Usually, about 1/4 of the core fuel assemblies are replaced with fresh ones and the other fuel assemblies are, if necessary, moved to different locations in the core (fuel shuffling). This refueling opera­tion is a major step to determine the periodic inspection period and therefore a plan with less movement of fuel assemblies, namely, a plan with less fuel shuffling is important from the viewpoint of an improved capacity factor by shortening the periodic inspection period. In the fuel loading pattern of the equilibrium cycle core shown in Fig. 3.20, fuel assemblies stay in almost the same location until the third cycle since they were first loaded, and then they are moved to the outermost region of the core or the control cell region for the fourth cycle. This results in the flat power distribution as well as less fuel shuffling.

Possible fuel loading patterns are innumerable, even considering core symmetry, and search for an optimized loading pattern in combination with control rod patterns is inevitably limited by limited computational resources. Recently, a tool based on accumulated operation knowledge was developed to automatically plan and optimize the fuel loading and control rod pattern [28].

After the periodic inspection, various performance factors are tested before startup, including the reactor shutdown margin in core and safety in fuel reloading.

[1] Development history and future trends

The history of PWR power plants in Japan started from the Mihama Unit 1 built with technology introduced from the United States in the late 1960s [33]. Since then, the plants have been improved and standardized [34] toward high power density and large size without a significant change in the basic design specifi­cations. Economy and reliability in core and fuel utilization technology have also been improved as follows.

d

° Й О. О

=4-4 "-+J

О Ян

гч &

Я CQ

.2 я

-+J 1—1

2 ^

ТО О

ф

£

0 15

1

Fig. 3.57 Typical suppression of axial power distribution oscillation [32] (17 x 17 Type, 3-loop core, and EOC) (Copyright Mitsubishi Heavy Industries, Ltd., 2014 all rights reserved)

The fuel type of the HTTR is the pin-in-block type where the fuel rods are inserted in a hexagonal graphite block as shown in Fig. 4.18. The coated particle fuels are mixed with graphite powder and pressed into an annular cylinder, which is called a fuel compact. Fourteen fuel compacts are inserted into a graphite sleeve to form a fuel rod. Thirty-three fuel rods are loaded into a hexagonal graphite block (36 cm in width and 58 cm in height), which forms a fuel block. Helium gas coolant flows downward in the spaces of a few milli­meters between the fuel rods and wall of cooling holes in the fuel block. A fuel compact contains about 14 g of uranium which corresponds to about 13,000 coated particle fuels.

Three dowel pins are provided at the top of the fuel block and three dowel sockets are provided at its bottom. They are used for positioning when piling up the blocks. Holes for loading burnable poison are provided below each dowel pin. The burnable poison rod consists of B4C pellets and it compensates burnup reactivity.

A substance which has a large neutron absorption cross section is loaded into the reactor core or is directly mixed in fuel to suppress the large excess reactivity

at the initial burnup. This neutron absorber is converted into a nuclide with a low absorption cross section as the result of neutron absorption. Thus, the increase in reactivity accompanying the burnup of the poison compensates to some extent for the decrease in reactivity due to fuel burnup. This poison is called a burnable poison or a burnable neutron absorber.

Gadolinium (Gd) and boron (B) are representative burnable poisons. Boiling water reactors (BWRs) and pressurized water reactors (PWRs) use gadolinia (Gd2O3) mixed with the UO2 fuel. In past PWRs, borosilicate glass (B2O3-SiO2) was inserted into the control rod guide tubes which were not being used for control rods. Recently, investigation showed that erbium (Er), which has a lower neutron absorption cross section compared with Gd but can suppress the excess reactivity for a long time, can be mixed in the form of erbia (Er2O3) with the UO2 fuel as a burnable poison.

Burnable poisons are zoned both radially and axially for flattening neutron flux in the more advanced core designs.

In the core calculations with a huge amount of space-dependent data (cross section and neutron flux), the effective cross sections are processed, with a little degradation in accuracy as possible, by using the results from the multi-group lattice calculation. There are two processing methods. One is homogenization to lessen the space-dependent information and the other is group-collapsing to reduce the energy-dependent information as shown in Fig. 2.11. The funda­mental idea of both methods is to conserve neutron reaction rate. In the homogenization, a homogenized neutron flux фк°то is first defined as averaged flux weighted by volume Vk of region (k). Next, a homogenized cross section ^°то is determined as satisfying Eq. (2.23) to represent the reaction rate in the homogenized whole region of volume vhomo. The fine-group neutron flux фё к is used to calculate the homogenized cross section in Eq. (2.24).

2.22)

k / k k 47

homo Л homo T T homo — > V A T 7

x, g <Pg У — Zj^x, g,k(Pg, kVk

Heterogeneous Cross Sections

Homogenized Cross Sections

(for Core Calculation)

Few-Group Cross Sections

Collapsing

G = 3

C = 2

Ns —- g ——- 321 N„ ^——— 321

Similarly, the homogenized microscopic cross section 0-і^ото which repre­sents the homogenized region is defined. First, the homogenized atomic number density Nl, homo of nuclide i is defined from the following conservation of number of atoms

Ni, homo=^NlVk/V homo (2.25)

k v • /

The homogenized microscopic cross section is then given from the conser­vation of microscopic reaction rate of nuclide i as

(2.26)

Such homogenized neutron flux ^h°m° and cross section £ hgmo (or ^lJg>mo) terms are used in performing the energy-group collapse. The fine-groups (g) are first apportioned to few-groups (G) for the core calculation as shown in Fig. 2.11. Since the multi-group neutron flux was obtained by the integration over its energy group, the few-group homogenized neutron flux can be represented by

2.27)

g^G v ’

The next step is to consider the conservation of reaction rate in energy group G in the same manner as that in the homogenization.

The few-group homogenized macroscopic and microscopic cross sections are then given by

V homo — л v homo a. homo / ж homo — > v homo м homo / л A homo bx, G уд 1YG ~Zu^x, g <Pg / Zj Yg

g^G g^G / g(EG

i, hom, o— Xі i, homo a. homo / a. homo — Xі i, homo a. homo / Xі a. homo

vx, G — Li°x, g Yg / Yg ~ZaGx, g Yg ZjYg

g^G g(EG / g(EG

The number of few-groups depends on reactor type and computation code. Two or three groups are adopted for the nuclear and thermal-hydraulic coupled core calculation of LWRs and about 18 groups are used for the core calculation of LMFRs. It should be noted that the conservation of reaction rate has been considered under the assumption that the lattice system can be represented as an infinite array of identical lattice cells. If the neutron spectrum in the actual core system is very different from the multi-group neutron spectrum calculated in the infinite lattice system, the applicability of such few-group homogenized cross sections to the core calculation is deteriorated. In this case, it is effective to increase the number of energy groups in the core calculation, but that gives a more costly computation.