[1] Core

The core of the HTTR is a layered structure of hexagonal fuel blocks, surrounded by the replaceable reflector. The replaceable reflector is further surrounded by the permanent reflector. The whole core structure is supported by the side shields and the core restraint mechanism which are provided outside

of the permanent reflector. The fuel blocks are piled up into five layers. The top and bottom replaceable reflectors are placed above and below the fuel blocks, respectively. A set of hexagonal blocks i. e. five fuel blocks as well as top and bottom replaceable reflector blocks is called a column.

As shown in Fig. 4.16, the coolant entering into the reactor pressure vessel first flows upward from the lower region of the vessel through the space between the permanent reflector block and the vessel wall to the top plenum. It turns in the downward direction at the top plenum and enters the core. The

Fuel region

Core restraint mechanism

Permanent reflector

Reactor pressure vessel

Fig. 4.17 cross sections of HTTR

coolant flows in the annular channels between fuel rods and wall of cooling holes in the fuel block. It is heated up to 950 °C. The outflows of the heated coolant merge together in the hot plenum. Then, the coolant is led to the inner tube of the double-tube main coolant outlet pipe.

As shown in Fig. 4.17, control rod guide columns are provided in the core for inserting the control rods. A control rod guide column is a set of piled-up graphite blocks. Two of three holes are for inserting the control rods. The third is used for the reserve shut down system that drops B4C pellets.

The excess reactivity must be included in an initial reactor core in order to obtain the normal power during the required operation period. This reactivity control is mainly provided by control rods. The combined use of burnable poison and chemical shim lessens the need for control rods in thermal reactors and is intended to ensure a long operation period of the reactor.

The neutron transport equation in the lattice calculation is a steady-state equation without the time differential term in Eq. (2.1). Further, the neutron energy variable is discretized in the equation and therefore a multi-group form

is used in design codes as shown in Eq. (2.20). The neutron source of Eq. (2.21) is the multi-group form without the external neutron source of Eq. (2.2) at the critical condition.

-ІЇ-Уфя(г, ^)-^вСг)фя(г, Й)+^(т, Й)=0 (2.20)

The system to which the multi-group transport equation is applied is an infinite lattice system of a 2D fuel assembly (including assembly gap) with a reflective boundary condition. For a complicated geometry, two lattice calculations corresponding to a single fuel rod and a fuel assembly are often combined.

In practically solving Eq. (2.20) in the lattice model, the space variable (r) is also discretized in the equation and each material region is divided into several sub-regions where neutron flux is regarded to be flat. In liquid metal-cooled fast reactors (LMFRs), neutron flux in each energy group has an almost flat spatial distribution within the fuel assembly because the mean free path of the fast neutrons is long. A simple hexagonal lattice model covering a single fuel rod or its equivalent cylindrical model simplified to one dimension is used in the

Cladding

Moderator

Spatial Distribution of bast Neutron 1′ lux

Spatial Distribution of 1 hernial Neutron r lux

design calculation of LMFRs. The spatial division can also be simplified by assigning the macroscopic cross section by material.

On the other hand, thermal reactors have a highly non-uniform distribution (called the spatial self-shielding effect) of neutron flux in a fuel assembly as thermal neutron flux rises in the moderator region or steeply falls in the fuel and absorber as shown in Fig. 2.9. Moreover, control rod guide tubes or water rods are situated within fuel assemblies and differently enriched fuels or burnable poison (Gd2O3) fuels are loaded. In such a lattice calculation, therefore, it is necessary to make an appropriate spatial division in the input data predicting spatial distribution of thermal neutron flux and its changes with burnup.

Numerical methods of Eq. (2.20) include the collision probability method (CPM), the current coupling collision probability (CCCP) method, and the method of characteristics (MOC) [13]. The SRAC code adopts the collision probability method and can treat the geometrical models as shown in Fig. 2.10. The collision probability method has been widely used in the lattice calculation, but it has a disadvantage that a large number of spatial regions considerably raise the computing cost. The current coupling collision probability method applies the collision probability method to the inside of fuel rod lattices constituting a fuel assembly and combines neighboring fuel rod lattices by neutron currents entering and leaving the lattices. This approach can substan­tially reduce the assembly calculation cost. Since the method of characteristics solves the neutron transport equation along neutron tracks, it provides compu­tations at relatively low cost even for complicated geometrical shapes and it has become the mainstream in the recent assembly calculation [14].

ZD square plate fuel 2D square assembly assembly (KUCA) with pin rods (PWR)

Fig. 2.10 Lattice models of SRAC [7]

In lattice calculation codes, effective microscopic cross sections are first prepared from fine-group infinite dilution cross sections based on input data such as material compositions, dimensions, temperatures, and so on. The effec­tive cross sections are provided in solving Eqs. (2.20) and (2.21) by the use of the collision probability method, etc. and then multi-group neutron spectra are obtained in each divided region (neutron spectrum calculation).

Figure 3.4 shows the procedure followed in BWR core design. The fundamental specifications of a power plant such as reactor thermal power and operating pressure, and the main specifications of the core design such as shapes and numbers of fuel assemblies are established in the first step of the core design (Table 3.5). Next, the fuel rod design for a reactor operating period and a target fuel burnup is performed regarding fuel rod size and fuel enrichment, and the scheme of excess reactivity control is specified. The detailed design of fuel assemblies is determined based on the fuel rod design and the excess reactivity control scheme. Then, the fuel loading and refueling pattern is treated consid­ering power and burnup distributions, and an outline of the control rod insertion and withdrawal pattern during reactor operation is framed for reactivity and power distribution control. Core characteristics are evaluated for the reactor core, fuel assembly, and operating scheme mentioned above by using various nuclear and thermal-hydraulic coupled core analyses. This sequence of steps is repeated to determine core specifications which meet the core design criteria and target.

Table 3.5 Main design specifications of BWR (1,350 MWe-class ABWR)

Reactor thermal power Reactor pressure (at RPV dome)

Steam flow rate

Steam pressure

Steam temperature

Core Active height

Equivalent diameter Core flow rate Core inlet subcooling Core outlet average steam mass Core power density Fuel Specific power

Fuel loading amount No. of fuel assemblies

1.3.1 General Core Design

[1] Features of PWR Core

PWRs operate in an indirect cycle in which the primary coolant heated and pressurized in the reactor core is cooled through steam generators and pumped under high pressure into the reactor core again. The steam generated in the secondary loop is fed through a turbine which drives an electrical generator. The coolant as a neutron moderator is pressurized to about 15.5 MPa in the pressurizer and forced to circulate to cool the core with a single-phase flow with hardly any occurrence of boiling. Therefore, the moderator density variation in the core is small. The boric acid as a neutron absorber is dissolved into the moderator.

Figure 3.30 shows a cross-sectional view and structures of a PWR core which has an arrangement of 193 fuel assemblies with fuel rods of 17 x 17 type array. Each fuel assembly consists of 264 fuel rods, 24 control rod guide tubes (guide thimbles), and 1 core instrumentation tube (guide thimble). The control rod guide tubes are used for insertion of control rod clusters and burnable poison rods.

[2] Basic Criteria of Core Design

The main target of the reactor core design is same as that of BWR core design, to improve cost performance while securing reactor safety. The security of reactor shutdown capability, reactivity insertion limit, self-controllability, fuel failure prevention, power distribution restriction, and stability are all required for safety. Table 3.11 introduces the basic criteria to be considered in the PWR core design.

Hiroo Osada and Kiyonobu Yamashita

Abstract Section 4.1 describes features of a fast reactor core and the procedure of the core design. The characteristics of reactivity and power distributions are explained in the nuclear design section and the reactivity control requirements are also explained in this section. The section of core thermal-hydraulic design explains the outline of the coolant flow allocation procedure and the evaluation methods of temperature distribu­tions in a fuel subassembly. The author of Sect. 4.1 is Hiroo Osada.

Section 4.2 describes design of high temperature gas-cooled reactor (HTGR). HTGR’s cores consist of graphite internals and coated particle fuels that possess high temperature resistant. Helium gas is used as coolant that has high chemical stability in any temperature. High reactor outlet coolant temperature around

1,0 °C is possible for HTGRs with the high stable characteristics of graphite, fuel and coolant. High outlet coolant temperature enables high efficiency of electricity generation and broad utilization of HTGRs not only as electricity generation but also as a heat source for chemical industry. In comparison to LWRs, the outlet coolant temperature is high and difference between inlet and outlet coolant temperature is large for HTGRs. It results in different core design philosophy for HTGRs from LWRs. The core design of the High Temperature Engineering Test Reactor (HTTR) is presented as an example of HTGRs core design. The author of Sect. 4.2 is Kiyonobu Yamashita.

The results of the flow distribution calculation for 950 °C operation are illustrated in Fig. 4.39 [28]. As the coolant flow in the core, there are gap flows between the blocks and gap flows between the permanent reflectors as well as the fuel channel flow directly cooling the fuel rods. The coolant flow rate directly contributing to the fuel cooling is about 88 % of the total flow rate. It shows the minimum value at the third block from the top of the fuel region. The flow reduction is occurred at the high temperature region, because the increase in the coolant temperature leads to increases in viscous resistance and hence pressure drop.

The calculated axial fuel temperature distribution is illustrated in Fig. 4.40 [28]. The solid lines indicate the nominal temperatures and the dashed line indicates the systematic temperature at the inner surface of the fuel compact. Since the coolant flow direction is downward, the coolant temperature increases from the top to the bottom. The fuel compact inner surface temperature is

almost constant below the third layer. That is because the increase in the coolant temperature and the decrease in the power density along the axial direction cancel out each other.

The maximum fuel temperatures of each fuel column are shown in Fig. 4.41. This figure gives the maximum fuel temperature within the burnup period. The highest maximum temperature appears at the core inner region and it is 1,492 °C, which is below the limit of 1,495 °C for normal operation and does not reach the limit of 1,600 °C at anticipated abnormal occurrences. Thus, the thermal design limit is satisfied.

An increase in the moderator temperature TM leads to a reduction in thermal neutron absorption due to moderator expansion and an increase in resonance absorption due to the moderating power decrease as well. They are described through f and p, respectively. Because the thermal neutron spectrum changes depending on the moderator temperature (spectral shift), there is a variation in the thermal cross sections averaged by the thermal neutron spectrum. This is discussed with n and f.

(1) Moderator temperature coefficient of resonance escape probability

In Eq. (1.58), an increase in the moderator temperature causes a reduction in the moderator atomic density (the moderator volume fraction is constant) and results in a reduction in the resonance escape probability. This means that the resonance absorption of neutrons rises due to the reduction in the neutron moderating power. The moderator temperature coefficient of the resonance escape probability is given by

which shows a negative value. An expansion of the moderator in liquid or gas is described by the relation between the temperature coefficient of atomic density and the linear expansion coefficient as

where the volume expansion coefficient (= 30M) can be used instead of the linear expansion coefficient.

Figure 1.12 shows the volume expansion coefficients of liquid modera­tors. The moderator temperature coefficient of the resonance escape prob­ability is about 10_4Ak/k/K and has a large negative reactivity effect.

In the nuclear reactor design calculation, the thermal-hydraulic calculation is performed based on information on the heat generation distribution acquired from the nuclear calculation of the reactor core. In LWRs, the parameters such as moderator temperature, moderator density or void fraction, and fuel tem­perature obtained from the thermal-hydraulic calculation have a large effect on nuclear characteristics (nuclear and thermal-hydraulic feedback). The nuclear and thermal-hydraulic calculations should be mutually repeated until parame­ters of both calculations converge. This coupled calculation shown in Fig. 2.25 is referred to as the nuclear and thermal-hydraulic coupled core calculation (hereafter the N-TH coupled core calculation). The procedure of the N-TH coupled core calculation using the macroscopic cross section table prepared from lattice burnup calculations is discussed for a BWR example.

Two types of parameters are used in the coupled calculation. One is histor­ical parameters related to the fuel burnup and the other is instantaneous parameters without a direct relation to it. The historical parameters such as burnup are obtained from the core burnup calculation discussed in the next section. Here it is assumed that all of the historical parameters are known. The macroscopic cross section is first required for the nuclear calculation of the

reactor core. The space and time-dependent macroscopic cross section of a LWR is, for example, represented as Eq. (2.92).

^(r, t) = ZXia(F, R,E, p,p, Tf, Tm, NXe, NSm, SB, fCR,-‘) (2.92)

F: fuel type (initial enrichment, Gd concentration, structure type, etc.)

R: control rod type (absorber, number of rods, concentration, structure type, etc.) E: burnup (GWd/t)

p: historical moderator density (the burnup-weighted average moderator density)

p=j*pdE lf0EdE (2.93)

p: moderator density (also called the instantaneous moderator density against p) in which the void fraction a in BWRs is calculated from two densities p/ and pg in liquid and vapor phases, respectively

p = Pi(l — а) + рда (2.94)

Tf fuel temperature (the average fuel temperature in the fuel assembly)

Tm: moderator temperature (the average moderator temperature in the fuel assembly)

Nl: homogenized atomic number density of nuclide i (e. g., 135Xe or 149Sm) for which the atomic number density changes independently of the burnup or the historical moderator density and which has an effect on the core reactivity depending on the operation condition such as reactor startup or shutdown SB: concentration of soluble boron (e. g., boron in the chemical shim of PWRs) fCR: control rod insertion fraction (the fraction of control rod insertion depth:

0 < fCR < 1)

These parameters are given as operation conditions, initial guess values, or iterative calculation values, and then the macroscopic cross section for the core calculation is prepared as follows.

• E and R: Use the corresponding macroscopic cross section table.

• E and p: Interpolate the macroscopic cross section table in two dimensions.

• p, Tf, and Tm: Use the function fitting Eqs. (2.34), (2.35), and (2.36) based on the branch-off calculation.

• Nl and SB: Correct the macroscopic cross section in the following equations, using the changes from the condition, in which the homogenized atomic number density was prepared, and the homogenized microscopic cross section:

ANl(r, 0=№(г, t)~Ni(E, p) (2-96)

where X0,x, g (r, t) is the macroscopic cross section before the correction and aX g (r, t) is the homogenized microscopic cross section prepared in the same way as Eq. (2.92). N0 (E, p) is the atomic number density homogenized in the lattice burnup calculation and Nl (r, t) is the homogenized atomic number density in the core calculation. For example, the atomic number density of 135Xe after a long shutdown is zero and it reaches an equilibrium concen­tration depending on the neutron flux level after startup. The concentration of soluble boron is similarly corrected changing the homogenized atomic number density of 10B

• fcR: Weight and average the macroscopic cross sections X Xng and X O>gt at control rod insertion and withdrawal respectively with the control rod insertion fraction.

Since the cross section prepared in the iteration of the N-TH coupled core calculation reflects the feedback of instantaneous parameters, it is hereafter referred to as the feedback cross section. The nuclear calculation is performed with the feedback cross section by the nodal diffusion method or the finite difference method and it provides the effective multiplication factor, neutron flux distribution, power distribution, and so on. The distribution of homoge­nized atomic number density of nuclides such as 135Xe is calculated from necessity. For example, since 135Xe has an equilibrium concentration in a short time after startup as below, it is provided as a homogenized atomic number density used for the correction of microscopic cross section:

(2:98)

where yXe is the cumulative fission yield of 135Xe and AXe is the decay constant of 135Xe.

The thermal-hydraulic calculation is performed using the power distribution acquired from the nuclear calculation and gives the instantaneous moderator density p and the moderator temperature Tm. A BWR fuel assembly is enclosed in a channel box and the coolant flow inside the assembly is described as a 1D flow in a single channel with a hydraulic equivalent diameter (the “single channel model”). The core is modeled as a bundle of single channels, which are connected at the inlet and outlet, corresponding to each fuel assembly. This is referred to as the 1D multi-channel model. The calculation procedure for p and Tm in the 1D multi-channel model is as below.

(i) The total flow rate at the inlet is constant and the inlet flow rate W* for channel i is distributed (flow rate distribution). A guessed value is given to Wi if the flow rate distribution is unknown.

(ii) The axial heat generation distribution qi(z) (assembly linear power) of the fuel assembly from the nuclear calculation and the enthalpy at inlet

are used to calculate the enthalpy rise in each channel.

hi(z)=hfN + ^r f‘q’i (z)dz (2.99)

(iii) The physical property values at an arbitrary position r(i, z) such as the fluid temperature Tm(r) and p(r) are obtained from the steam table.

(iv) The void fraction distribution a(r) is acquired using qi(z), hi(z), physical property values, and some correlations based on the subcooled boiling model. The distribution of the instantaneous moderator density p(r) is calculated using the void fraction distribution.

p (r)=pi {l — a(r)}+pga(r) (2.100)

(v) The pressure drop APt by the channel is calculated using the information from (i) to (iv) including a(r) and correlations on pressure drop

(vi) The inlet flow rate is determined so that the pressure drop in all channels is equal. A new flow rate distribution W"ew(z) is decided from the flow rate distribution Wi in (i) and the pressure drop APi in (v).

(vii) The calculations from (i) to (vi) are repeated until the flow rate distribution is in agreement with the previously calculated one and the pressure drop of each channel is coincident (iterative calculation for flow rate adjustment). p(r) and Tm (r) at a convergence of the iterative calculation are taken and then the next fuel temperature calculation is done.

In a case such as the space-dependent kinetics calculation, the heat convection and conduction calculation is done using the same iteration approach of the N-TH coupled core calculation and then the temperature distribution in the fuel rod is calculated. In a case such as the steady-state calculation in which operation conditions are limited, it is common to calculate the fuel temperatures at representative linear power and fuel burnup in advance and then interpolate the linear power and burnup obtained from the core calculation into the fuel tem­perature table to simplify calculation of the fuel temperature Tf (r).

The calculations mentioned above, the feedback cross section calculation, the nuclear calculation, the thermal-hydraulic calculation, and the fuel temper­ature calculation, are repeatedly performed until the effective multiplication factor and power distribution reach convergence.

Along with the reactivity control, control of the power distribution in core is one of the important challenges in core design. Technologies for power distribution control have been developed and applied to the pin power distribution in a fuel assembly, the core axial power distribution, and the core radial power distribution.

[1] Power distribution in a fuel assembly

The relative power density of each fuel rod to the average power density of fuel rods in the fuel assembly is referred to as the local power peaking. Areas inside and outside of the channel box of the fuel assembly are completely separated in BWRs. When a void occurs inside the channel box, it does not occur in the water gap region outside the channel box. This, therefore, leads to high power of the outermost fuel rods neighboring the water gap which increases the neutron moderation effect; in other words, a high local power peaking is caused. The water rods, which are arranged in the central region to optimize the H/U ratio, increase the power of the central fuel rods by increasing the moderation effect, therefore contributing to a flat power distribution.

To flatten the pin power distribution and suppress the local power peaking, different enrichments of fuel rods are properly arranged as shown in Fig. 3.5; low enrichment fuel rods are near the channel fox and high enrichment fuel rods are in the central region.

The pin power distribution in a fuel assembly varies with burnup. Generally higher power fuel rods at initial burnup lead to a larger power decrease with burnup, and therefore the high local power peaking at the initial burnup is mitigated with burnup. The enrichment zoning of fuel rods and the arrangement and concentrations of burnable poison rods are determined so as to maintain a flat power distribution during burnup until the fuels are discharged. Since the gadolinia-added fuel rods suppress pin powers at initial burnup, the burning speed of fissile materials in the fuel rods is slower than that of neighboring fuel rods. The gadolinia-added fuel rods are designed to have a relatively low enrichment compared with the neighboring fuel rods in order to avoid a high local power peaking caused by the remaining fissile materials after burning out of gadolinia. Since the variation in gadolinia concentration with burnup depends on the thermal flux near the gadolinia-added fuel rods, the concentra­tion is properly determined to optimize the fuel burnup and gadolinia burning, corresponding to the location of the gadolinia-added fuel rods.