Boiling of coolant must not occur at normal operation, anticipated opera­tional occurrences and design basis accidents. Even if a void is generated in the primary coolant, the inserted reactivity largely depends on the core size and core shape. Generally, the inserted positive reactivity becomes larger with larger core size. Although such an incident is beyond design basis and cannot occur from a technical viewpoint, an adequate safety level is ensured by the core design.

(5) Breeding performance

The required breeding performance for a fast reactor is determined based upon balance between the demand and supply of plutonium at the time of construction to avoid an excessive plutonium accumulation from the view­point of non proliferation.

The expansion effect due to a temperature rise in structures is concerned with thermal utilization factor in the six-factor formula. Since structures are usually solids, their temperature coefficient is small. In the evaluation of the fuel or moderator temperature coefficient, it was assumed that the atomic density of fuel or moderator decreases with each temperature rise assuming a constant fuel or moderator volume fraction. This assumption was made because the variation in volume of structures due to a change in temperature is small and therefore almost a constant volume fraction of fuel or coolant is maintained. Hence, the structure temperature coefficient was disregarded in the discussion above.

However, the structure temperature coefficient should be necessarily regarded in the practical evaluation. Generally, a volume expansion due to a temperature rise of structures leads to a decrease in atomic density and an increase in the thermal utilization factor, and therefore has a positive reactivity effect. The structure expansion essentially presses liquid or gas coolant out of the system. It is somewhat complicated to formulate the structure temperature coefficient.

[1] Plant dynamics calculation code

The plant dynamics calculation treats plant control, stability, and response at abnormal transients and accidents. A simple model for heat transfer calculation in a plant is the node junction model as shown in Fig. 2.36 [27]. In the node junction model, reactor components are represented as 1D nodes and connected; the connected items include core, upper and lower plena, downcomer (inlet coolant flows down the downcomer region which is placed between the core and reactor pressure vessel), main feedwater and main steam lines, valves, etc. The node junction calculation begins from an upstream node based on mass and energy conservation laws. The momentum conservation law is also considered for pressure drop and flow rate distribution calculations. In the case of Fig. 2.36, for example, the core is described with two single channel models at average and maximum power. The core design calculation is performed at steady state as mentioned before, but a transient single channel model is used in the plant dynamics calculation.

A transient radial heat conduction in fuel can be expressed as

Fig. 2.36 Node junction model

where,

Cp: specific heat of pellet (J/kg-K) kf. thermal conductivity of pellet (W/m-K) qm: power density (W/m3) r: radial distance (m)

Tf. pellet temperature (K) pf pellet density (kg/m3).

The heat transfer from cladding outer surface to coolant is evaluated by Newton’s law of cooling. A proper heat transfer coefficient should be used considering flow regime and temperature, pressure, etc.

Plant dynamics analysis codes are constructed with the node junction model and the following reactor kinetics model. The simplest kinetics model is the point reactor kinetics model (point reactor approximation) as

Щг = А£Т±пЮ+ІліСіЮ

dt А і=і

dCiit) pi (Л.. л

——= — n(t)—AiCi(.t) u = l~6.) dt A

End Time Step?

Fig. 2.37 Plant dynamics analysis model and calculation flow chart

where,

n(t): number of neutrons

Ci(t) number of delayed neutron precursor for group i t: time

Pi. fraction of delayed neutron group i

6

p — Ipi

i = 1

Ap: reactivity

Л: prompt neutron generation time (s)

decay constant of precursor of delayed neutron group i (s_1)

Tave (t): average fuel temperature (K)

Pmod(t): moderator density (g/cm3).

The reactivity feedback of fuel temperature (Doppler effect) and moderator density is applied to the point reactor kinetics equations. Even though large power reactors such as LWRs show space-dependent kinetic characteristics, the reactivity feedback effect can be expressed with the point reactor approxima­tion using a space-dependent weight function (adjoint neutron flux).

Figure 2.37 shows an example of a node junction model in a plant dynamics analysis code and its calculation flow chart. This model describes coolant flow which is fed to the water moderation rods through the top dome of the reactor vessel for a supercritical water-cooled thermal reactor (Super LWR) [28, 29].

The mass, energy, and momentum conservation equations used in the node junction model are a time-dependent form of the single-channel
thermal-hydraulic model in the core design. In the case of single channel, single phase, and one dimension, the mass conservation equation is given by Eq. (2.116)

(2.116)

dt dz

and the energy conservation equation is given by Eq. (2.117).

diph) dipuh) Pe „

dt ~

The momentum conservation equation is expressed as

and the state equation is

P=p(P, h)

where,

t: time (s)

z: position (m)

p: coolant density (kg/m3)

u: fluid velocity (m/s)

h: specific enthalpy (J/kg)

q": heat flux at fuel rod surface (W/m2)

A: flow area of fuel channel (m2)

Pe: wetted perimeter of fuel rod (m)

P: pressure (Pa) g: gravitational acceleration

Dh: hydraulic equivalent diameter of fuel channel (m)

Re: Reynolds number 6: vertical angle of fuel channel f: frictional coefficient

for example, f = 0.0791 x Re-0 25 (Blasius equation).

In the case of a water rod, the energy transferred to the water rod should be considered in the energy conservation equation of the fuel rod cooling channel. In the case of light water, the steam tables of the Japan Society of Mechanical Engineers (JSME) or the American Society of Mechanical Engineers (ASME) can be used in the state equation of the coolant.

The following introduce heat transfer coefficients for water-cooled reactors. In a single-phase turbulent flow, for example, the heat transfer coefficient can be obtained from the Dittus-Boelter correlation

(2.120)

where,

De: hydraulic equivalent diameter (m)

G: mass flux (kg/m2-s)

h: heat transfer coefficient (W/m2-K)

k: thermal conductivity (W/m-K)

Pr: Prandtl number p: viscosity coefficient (Pa-s).

If nucleate boiling occurs, the heat transfer for PWR can be described by the Thom correlation

1 (ATsate^ У dT V 0.022 )

where,

h: heat transfer coefficient (W/m2-K)

P: pressure (MPa)

dT: temperature difference between wall surface and fluid (K)

ATsat: wall temperature elevation above saturation temperature (K)

and it is applied for the range of

pressure: 5.2-14.0 MPa mass flux: 1,000-3,800 kg/m2-s heat flux: 0-1600 kW/m2.

The Jens-Lottes correlation can be used for BWRs.

If there is a thin liquid film in annular flow with a high steam quality, for example, the heat transfer to a vertical upward water flow can be represented by the Schrock-Grossman correlation

(2.122)

і _/ x у-9ЛеЛ0-5(ж)0’1 Xtt l-x) pgJ pf)

where,

De: hydraulic equivalent diameter (m) G: mass flux (kg/m2-s)

h: heat transfer coefficient (W/m2-K) kf. thermal conductivity of water (W/m-K)

Pr^: Prandtl number of water X: steam quality pf. water density (kg/m3) pg: steam density (kg/m3)

Pf. viscosity coefficient of water (Pa-s)

Pg: viscosity coefficient of steam (Pa-s)

and it is applied for the range of

pressure: 0.3-3.5 MPa mass flux: 240-4,500 kg/m2-s heat flux: 190-4,600 kW/m2.

In the post-dryout region of BWRs, namely, the steady-state film boiling state of mist flow where steam flow is accompanied with liquid droplets, for example, the heat transfer can be described by the Groeneveld correlation.

where,

h: heat transfer coefficient (W/m2-K)

kg: thermal conductivity of saturated steam (W/m-K)

Prv, w: Prandtl number of steam at wall surface De: hydraulic equivalent diameter (m)

G: mass flux (kg/m2-s)

Pg: viscosity coefficient of steam (Pa-s)

X: steam quality

pf. water density (kg/m3)

pg: steam density (kg/m3)

Uranium-saving technology is intended to attain a high burnup with the least increment of fuel enrichment by effectively burning the uranium loaded in the core. Only a simple increase in fuel enrichment raises the uranium and enrichment cost. For an economic high burnup, therefore, it is important to save the uranium simultaneously. The basis of uranium-saving technologies used practically in BWRs is the development of zirconium liner fuel in which the inside of the zircaloy-2 cladding is lined with soft and pure zirconium. The zirconium liner fuel mitigates the pellet-clad mechanical interaction (PCMI) and increases fuel integrity against a rapid power rise. Hence, it eliminates restrictions on the PCIOMR and makes it unnecessary to take an excessive design margin for the linear heat generation rate. The thermal margin obtained from the power distribution flattening for improvement of the capacity factor can be applied to the uranium saving.

(1) Utilization of power peaking

Arrangement of the high enrichment fuel in the proper locations with high

thermal neutron fluxes can increase the core reactivity. High enrichment

zoning in the peripheral region of a fuel assembly causes bias of the pin power distribution and large local power peaking. In the early core design, low enrichment zoning in the peripheral region was employed to improve the thermal margin and to mitigate the restrictions on the PCIOMR. How­ever, since the zirconium liner fuel has been employed and the axially two-zoned core has reduced the axial power peaking as mentioned, the core reactivity can be improved by allowing a larger local power peaking. The installation of natural uranium blankets on the upper and lower end parts and the arrangement of low enrichment fuel assemblies in the core periph­ery region cause a large axial and radial power peaking of the core. However, the power peaking margin from the axially two-zoned core can give a reactivity gain.

(2) Reduction in gadolinia residual

The reduction in gadolinia residual is the technology to minimize gadolinia embers in the core upper and periphery parts and to increase the core reactivity. The placement of natural uranium blankets on the upper and lower end parts also contributes to the reactivity gain by replacing the corresponding gadolinia.

(3) Spectral shift operation

The spectral shift operation is the technology to get a reactivity gain by changing the neutron spectrum through a change in the void fraction and distribution which are features of the BWR core. The void fraction change can be accomplished by the coolant flow rate control to decrease the core coolant flow rate during the first half of the operating cycle, which leads to a high void fraction in the core, less neutron moderation, and acceleration of 238U conversion to plutonium, and then to increase the core coolant flow rate during the latter half of the operating cycle, which leads to a burning of the plutonium accumulated in the first half. The void distribution change can be performed by the axial power distribution control to distort the axial power distribution downward, which leads to a downward shift of the void generation point and an increase of the void fraction in the core, and then oppositely to extend the axial power distribution upward, which leads to a decrease of the void fraction.

(4) Optimization of H/U ratio

High enrichment of uranium for high burnup decreases the ratio between H and 235U atoms, and hardens the neutron spectrum. It is, therefore, neces­sary to set the proper H/U ratio in order to effectively use fuel and to efficiently utilize thermal neutrons. The H/U ratio can be raised by increas­ing the number of water rods in the fuel assemblies or by enlarging their cross-sectional area as mentioned in Sect. 3.2.3.

Radial power distribution of PWR cores is less dependent on relative power. The number and depth of inserted rod cluster control assemblies are small and therefore the effect of control rods is small and the burnup effect is also mild. For flattening of the core radial power distribution, it is, therefore, sufficient only to properly locate fuel assemblies and burnable poisons in the core design. The core radial power distribution is just monitored during reactor operation and adjustment by control rods is not needed. A specific arrangement of fuel assemblies is shown in Fig. 3.47 (gadolinia-added fuel assemblies are used, and no burnable poison rods in this example) and its corresponding radial power distributions are shown in Fig. 3.48 (BOC) and Fig. 3.49 (EOC). The radial power distribution is represented as the relative value of fuel assembly power to core-averaged fuel assembly power for the symmetric octant core. Comparison between the radial power distributions at BOC and EOC indicates that gadolinia-added fresh fuel assemblies and neighboring fuel assemblies give a relatively large increase in power, but the variation in power is mild as burnup progresses.

The coolant temperature and cladding temperature in the fuel assemblies are estimated by subchannel analysis. An example of the subchannel model is illustrated in Fig. 4.10. There are three types of subchannel. The inner subchannel

is surrounded by three fuel elements, the peripheral subchannel is surrounded by two fuel elements and the inner wall of the wrapper tube, and the corner subchannel is surrounded by a fuel element and the wrapper tube corner. The three-dimensional thermal-hydraulic calculation is carried out in the subchannel analysis by modeling the fuel assembly as a set of parallel subchannels.

The coolant temperature distribution and cladding temperature distribution in each fuel assembly are calculated using the inlet flow rate determined by the flow allocation, and the power distribution among the fuel elements as the input conditions. In the subchannel analysis, the mixing of mass, momentum and energy between the neighboring subchannels is taken into account.

In the subchannel analysis, generally, the distributions of the coolant velocity and coolant temperature in the fuel assembly are obtained by solving mass, momentum and energy conservation equations in every axial node.

The design parameters used in the analysis are formulated depending on the analysis models. They are determined by experiments such as assembly flow tests using a simulated fuel assembly with the same dimensions as the actual one and coolant mixing tests. When the coolant temperature distribution in the assembly is obtained, the cladding temperature distribution is calculated according to the power distribution among the fuel elements. Consequently, the heat flux on the cladding surface is given by the power distribution; then the increase in the temperature from the coolant to the cladding surface is calcu­lated using the heat transfer coefficient.

The maximum fuel temperature and maximum cladding temperature are esti­mated considering the engineering safety factors in order to secure enough design margin from the temperature distribution obtained by the subchannel analysis.

In the core thermal-hydraulic design, the nominal (most probable without any error margin) temperature distribution in the fuel assembly is calculated by the subchannel analysis first. Then, the hot spot temperatures of fuel centerline and fuel cladding are obtained considering the uncertainties of the calculated temper­ature caused by the tolerances of the design parameters etc. The hot spot temper­atures are to be confirmed not to exceed the corresponding design limits. The temperature uncertainties are considered by multiplying the nominal tempera­tures by the engineering safety factors which are provided according to the errors of each parameter. The engineering safety factors are divided into two types. One is treated as a multiplication term by considering the systematic errors which must be considered cumulatively. The other is treated as a statistical term coming from random errors. An example of the former error is that of the power distri­bution. An example of the latter error is the fabrication tolerance of the fuel pellet.

Equation (4.14) is an example of the evaluation equation for the maximum temperature which includes consideration of the engineering safety factors.

In this equation, Gij, Fkj are the factors of multiplication treatment and statistical treatment, respectively. ATj is the nominal temperature rise of each position. while m and n are the numbers of the multiplying factor and statistical factors, respectively.

T1HS : Coolant T2HS : Clad outer surface T 3 HS : Clad inner surface T4HS : Pellet surface T 5 HS : Pellet centerline

AT1 = ATNa : From assembly inlet tocoolant AT2 = ATfum : From coolant to clad outer surface AT3 = ATclad : From clad outer suface to inner surface AT4 = ATgap : From clad inner suface topelletsurface AT5 = ATfuel : From pellet surface to pellet center

As described above, the engineering safety factors are for estimating the maximum temperature rise in each position. By considering them, the fabrication tolerances, uncertainties of thermal-hydraulic parameters, power distribution uncertainty, and measurement uncertainty of the reactor thermal power, etc. are taken into account.

Fig. 4.11 epts with low void reactivity

[3] Maximum fuel temperature

In the core thermal-hydraulic design, melting of the fuel pellets should be avoided at normal operation and anticipated operational occurrences. The maximum fuel centerline temperature with consideration of the engineer­ing safety factors is confirmed to satisfy the design limit. As for the evaluation of the maximum fuel centerline temperature, see the list [1](4) of Sect. 4.1.3.

The concept of burnup (unit: MWd/t) is a fairly general measure of fuel depletion, which is defined as thermal energy output (unit: MWd) per unit mass (unit metric ton, t) of heavy metal content in the initial fuel.

For a reactor which loads a heavy metal content of 100 t and operates at a thermal power of 3,000 MW for 1,000 days, the average burnup of the discharge fuel will be

30,0 MWd/t.

The thermal energy production by fission of 1g heavy metal fuel, assuming that the fuel is 235U and its fission energy is 200 MeV, is as follows.

The fission of 1 g heavy metal corresponds to about 1 MWd. For a discharge fuel of

30,0 MWd/t burnup, this means that about 30 kg per initial heavy metal of 11 has fissioned (that is, about 3 % of the initial heavy metal has fissioned).

The average burnup of typical LWRs is 30,000-50,000 MWd/t. Fast breeder reactors are being designed with the average burnup of 100,000-150,000 MWd/t as a development target.

Interactions between neutrons and nuclei in nuclear reactors can be classified broadly into scattering and absorption reactions as shown in Table 2.1. Scattering is further classified into elastic scattering in which the kinetic energy is conserved before and after the reaction, and inelastic scattering in which a part of the kinetic energy is used in exciting a target nucleus. In absorption, the main reactions are capture, fission, charged-particle emission, and neutron emission. Thus, the microscopic cross sections of the total, scattering, and absorption reactions are given by

total cross section: at(E) = <rs(E) + aa(E) (2.10)

scattering cross section: а5(Е) = oe(E) + ^П(Я) (2.11)

absorption cross section:

(2 12!

tfa(£) = ay(£) + Vf(E) + op{E) + oa(E) + G(n,2n)

It is useful to discuss a general energy dependence of these microscopic cross sections. The neutron energy range to be considered in design of nuclear reactors is from the Maxwellian distribution of the thermal neutrons at room temperature to the fission spectrum of the prompt neutrons. Most nuclear design codes handle the range of 10_5 eV-10 MeV. In this energy range, the

microscopic cross sections introduced in Eqs. (2.10), (2.11), and (2.12) behave as shown in Fig. 2.2.

The elastic scattering cross section is mostly constant in all the energies except for the MeV region. Meanwhile, in inelastic scattering, the incident neutron should have sufficient kinetic energy to place the target nucleus in its excited state. Hence, the inelastic scattering cross section is zero up to some threshold energy of several MeV. Fast neutrons can be moderated by inelastic scattering with heavy nuclides, but by elastic scattering with light nuclides below threshold energies of the heavy nuclides.

Most absorption cross sections including the fission cross section appear as a straight line with a slope of —1/2 on a log-log scale. This means that the absorption cross sections are inversely proportional to the neutron speed (1/u law) and therefore increase as the neutron energy decreases. Using such large fission cross sections at low neutron energies and thermal neutrons in the Maxwellian distribution make it possible that natural or low-enrichment ura­nium fueled reactors reach a critical state. The current thermal reactors, represented by LWRs, use the characteristics of the cross section.

For heavy nuclides such as fuel materials, many resonances are observed in elastic scattering and absorption cross section as shown in Fig. 2.3. The widths of the resonances broaden as fuel temperature increases. This is called the Doppler effect. The width broadening facilitates resonance absorption of neu­trons under moderation. Most low-enrichment uranium fuel is composed of

fertile 238U and thermal neutrons escaping from the resonance of capture reaction induce fissions for the next generation.

Hence, a rise in fuel temperature leads to a decrease in resonance escape probability of moderated neutrons and then fission events in the reactor decrease with thermal neutrons. Such a mechanism is called negative temper­ature feedback. The temperature dependence is not described in the Boltzmann equation of Eq. (2.1), but reflected in the cross sections of the equation.

The main targets of the reactor core design are to reduce electricity generation costs and secure reactor safety. Core performance targets are established to realize the main targets. Improvement in power density and fuel burnup, reduction in necessary uranium resources, and extension of continuous operat­ing period are all related to the reduction in electricity generation costs. Security of negative power coefficient and reactor shutdown capability, and prevention of fuel failure are required from the viewpoint of reactor safety. Table 3.4 introduces the basic criteria [4, 5] to be considered in the BWR core design based on its features.

Table 3.4 Main criteria of BWR core design [4, 5]

Design criteria

Self — Designed to assure the inherent negative reactivity

controllability feedback characteristics (negative power coefficients) in the power operating range Reactor Designed to assure the complete core shutdown

shutdown capability even with the most reactive rod

stuck in the fully withdrawn position Designed with appropriate limits of any single­control rod worth and of the control rod speed to secure the core safety against accidental withdrawal of control rods

Fuel failure Designed to assure that at least 99.9 % of the fuel

prevention rods in the core would not be expected to

experience boiling transition Designed to assure that the plastic circumferential deformation of fuel rod cladding due to pellet cladding interaction (PCI) would not be expected to exceed 1 %

Stability Designed to secure the stability without the flow-

induced vibration (thermal-hydraulic stability of channel)

Designed to secure the stability without the power oscillation. (core stability and regional stability)

Designed to secure the stability with enough plant controllability (plant stability)

Designed to secure the stability without the spatial oscillation of power distribution due to Xe accumulation (Xe stability)

Spent fuel discharged from the reactor core is stored in the spent fuel storage pool and cooled over a period, and then taken to the reprocessing factory. Recently, dense storage using boron-containing fuel storage racks, estab­lishment of an independent storage pool in the power plant site, and use of dry storage casks are employed to enhance the storage capacity of spent fuel.