For a long cycle length, burnable poison rods are used to maintain moder­ator temperature coefficient negative as mentioned in the list [1](1) of Sect. 3.3.4. However, a large amount of burnable poisons causes a reactivity penalty at EOC that cannot be ignored (even if burnable poison were depleted, the effect of a structure such as the cladding remains) and it gives rise to a problem in spent burnable poison as a solid waste. Gadolinia-added fuel for PWRs was developed and employed to solve these problems.

In the design of the gadolinia-added fuel assembly, gadolinia content and the number and location of gadolinia-added fuel rods are investigated for required performance. Having a large number of gadolinia-added fuel rods gives a high reactivity suppression effect (reactivity reduction at BOC). The location of the gadolinia-added fuel rods in a fuel assembly can be determined to reduce power peaking through the fuel burnup. Figure 3.58 depicts an arrangement of 24 gadolinia-added fuel rods in the 17 x 17 type fuel assembly and Fig. 3.59 shows the variation in infinite multiplication

factor with burnup. is suppressed by the gadolinia-added fuel rods at the beginning of burnup, but increases with burning of the gadolinia and reaches a peak when the gadolinia is almost depleted, and then decreases as fuel burnup continues.

The larger content of gadolinia (wt%Gd2O3) leads to its slower depletion and a longer reactivity suppression effect. This is because the self-shielding effect of gadolinia becomes stronger. Figure 3.60 compares the nuclear enthalpy rise hot channel factor (F^H), which usually occurs at a gadolinia — added fuel assembly or a neighboring fuel assembly. In considering the behavior of reactivity variation, the power peaking factor in the case of low-content gadolinia tends to increase with gadolinia depletion and it becomes a maximum in the late of the cycle. High-content gadolinia oppositely leads to a mild variation and no peak in the nuclear enthalpy rise hot channel factor. Thus, higher content of gadolinia is suitable for a longer cycle length.

The addition of gadolinia to uranium fuel causes deterioration of thermal conductivity and lowering of melting point, and reduces the margin of fuel centerline temperature against melting. To secure the same mechanical

integrity as that of conventional uranium fuel, the gadolinia-added fuel rod is designed to have low uranium enrichment to suppress an increase in linear power density.

In 10 wt% Gd2O3 gadolinia-added fuel, for example, the gadolinia-added fuel with 3.2 w% enriched uranium is used instead of the conventional 4.8 w% enriched uranium.

The characteristic feature of the coated particle fuels is retention of FPs by the ceramic coating layers. The principle of ensuring fuel integrity of LWRs is avoiding failure of the fuel clad, which contains FPs, by applying the criterion of the minimum critical heat flux ratio. On the other hand, since the character­istic of FP retention in the coated particle fuels is different from that in the LWR fuel, a new principle for ensuring fuel integrity is applied.

The mechanisms of FP gas release from the coated particle fuel can be categorized as below.

(i) FP gas release from the coated particle fuels where the coating layers have failed at fabrication

(ii) FP gas release from the intact coated particle fuels due to diffusion enhanced by heat up

(iii) FP gas release due to failure of the coating layers during operation (additional failure)

The allowable design limit of the coated particle fuels is determined to keep failure of the coating layers associated with (iii) within an allowable range. The failure fraction associated with (i) is limited below 0.2 %. The diffusion of fission fragments associated with (ii) is much smaller than the release by (iii).

Various tests on failure of the coating layers have been carried out in several countries. Failure fractions of the coated particle fuels in some heating tests are shown in Fig. 4.19. Heating the coated particle fuels causes failure of the coating layers. Then fission gas (85Kr) is abruptly released. The failure fraction is estimated by the release of radioactivity per coated particle fuel. From the figure, it can be seen that the failure fraction is almost zero below 1,800 °C, it gradually increases above 1,800 °C, and sharply increases around 2,200 °C. It is supposed that rapid thermal decomposition (thermal degradation) of the SiC layer above 2,200 °C leads to abrupt failure of the coating. The allowable design limit is set as 1,600 °C, taking a margin from 1,800 °C. In order to keep the fuel temperature below 1,600 °C at anticipated operational occurrences

Control rod

Support ring

Neutron absorber iB. C)

Connecting rod

Refractory metal sleeve

Guide ring

Shock absorber

Element

such as “Uncontrolled withdrawn of control rod at operation”, the limit of the fuel temperature at normal operation is set as 1,495 °C.

Control rods play a central role in reactivity control for various types of reactors; they are used in the reactivity control for the power level changes, the reactivity compensation for the long-term fuel burnup, and the reactor scram in an emergency. Boron is generally used as a neutron absorber for control rods using the (n, a) reaction of the isotope 10B.

Fig. 1.9 Boric acid concentration changes of chemical shim (PWR) [4]

In the lattice burnup calculation, a combination of parameters such as moderator density and temperature, and fuel temperature is made at representative values (p0, Tm0, f) expected at normal operation of the reactor. In the core calculation, however, a different set (p, Tm, Tf) from the representative set is taken depending on position and time. Hence, a subsequent calculation called the branch-off calculation is performed after the lattice burnup calculation if necessary.

Figure 2.15 depicts an example branch-off calculation of moderator density. The calculation proceeds in the following order:

(i) Perform the lattice burnup calculation at a reference condition (p0, Tm0, Tf0). Designate the few-group homogenized cross section prepared from the lattice burnup calculation as £ (p0, Tm0, Tf0).

(ii) Perform the lattice calculation at each burnup step on the condition that only the moderator density is changed from p0 to pa, by using the same fuel composition at each burnup step. Designate the few-group homoge­nized cross section prepared from the lattice calculation (or the branch-off calculation) as E(p0 ! Pa, Tm0, f).

(iii) Carry out the branch-off calculation similar to (ii) at another moderator density of pb and give E(p 0 ! pb, Tm0, T0).

(iv) If the moderator density is instantaneously changed from p0 to an arbitrary p, calculate the corresponding cross section by the following approxima­tion (quadratic fitting).

S(p0 -> p, Tm0, Tf0) * 2(p0, Tm0, Tf0) + a(p — p0) + b(p — p0)2 (2-34)

(v) Determine the fitting coefficients a and b from the approximation at p = pa and p = рь.

Hence, the cross section at an arbitrary moderator density (an instantaneous moderator density) p away from the historical moderator density p0 can be expressed by the method mentioned above. If three moderator densities (p 1, p2, p3) are employed as p0 as shown in Fig. 2.14, their branch-off calculations can give X (p 1 ! p, Tm0, Tf0), X (p2 ! p, Tm0, Tf0), and X (p 3 ! p, Tm0, Tf0). By interpolation of the three points, the cross section at p resulting from an instantaneous change from a historical moderator density p can be obtained as X (p! p, Tm0, T0).

The cross section at an instantaneous change in fuel or moderator tempera­ture can also be described by the same function fitting as above. Since its change in cross section is not as large as a void fraction change (0-70 %), the following linear fittings are often used.

Z(Po, Tm0 Tm0> Tfo) = ^(р(П Tm0> Tfo) + c(Tm ~ Tmo) (2-35)

2(po, Tmo> Tfo Tf) = 2(po, Tmot TfQ) + ^(д/Т/ — V^/o) (2-36)

Equation (2.36) has square roots of fuel temperature in order to express the cross section change due to the Doppler effect by a lower-order fitting equation. An employment of a higher-order fitting equation can lead to a higher expres­sion capability and accuracy, but it gives rise to a substantial increase in the number of branch-off calculations and fitting coefficients and therefore results in an inefficient calculation. Thus, the reference cross sections and their fitting coefficients are stored into the few-group reactor constant library and used in the nuclear and thermal-hydraulic coupled core calculation.

To set the main specifications such as numbers of fuel assemblies and rods, and fuel rod size, the design parameters referred to as power peaking factors are defined, and roughly investigation is made that the maximum linear heat generation rate and assembly power which fuel rods experience during reactor operation satisfy the thermal design conditions.

(i) Radial power peaking factor FR = ratio between the maximum ^) and average values of fuel assembly — averaged power in the core

(ii) Axial power peaking factor FZ = ratio between the maximum and

average values of fuel assembly cross section — averaged power (3.7)

in the axial direction

(iii) Local power peaking factor FL = ratio between the maximum and

(3.8)

average values of fuel rod — averaged power in the fuel assembly

(iv) Total power peaking factor FP = ratio between the maximum and average values of local power in the core

=f Fr XFZXFL (3.9)

Table 3.6 shows examples of the power peaking factors which are evaluated by detailed design calculations in the step when reactor specifications and operating conditions are determined. However, generally the factors are set based on the design and operating experiences and then investigate them.

Using the power peaking factors above, the average fuel rod linear heat generation rate (power per unit length of fuel rod) qave, the maximum linear heat generation rate qmax, the average fuel assembly power PBave, and the maximum power PBmax can be given by the following equations.

Qave — Q/ GYb X Nrod X Lrod)
Qmax Pp X qaue

PBave = Q/NB

PВшах = PrX Рваие

The maximum linear heat generation rate qmax is the dominant factor in the thermal and mechanical integrity of fuel such as the maximum fuel tempera­ture and the maximum heat flux on the fuel rod surface. A low value of qmax is desirable to secure the reactor safety. In order to decrease qmax, the following measures can be considered.

(i) Flatten the power distribution in consideration of enrichment zoning in the fuel assembly or fuel loading pattern so as to reduce the power peaking factor.

(ii) Reduce the fuel rod diameter Drod and change the fuel rod array (8 x 8, 9 x 9, or 10 x 10) so as to increase the number of fuel rods per fuel assembly.

(iii) Increase the number of fuel assemblies.

(iv) Lengthen the active height of fuel rods.

Since (iii) and (iv) increase the core size, (i) and (ii) are usually investigated first. Too thin a fuel rod gives rise to difficulties such as its bending and an increase in fuel processing cost. Therefore, the fuel rod size is naturally limited. The main specifications of BWR fuel rod design are shown in Table 3.7. The diameter of the fuel rods was as large as about 14 mm early in BWR development. The number of fuel rods in the same-sized fuel assembly was then increased from the viewpoint of increasing volumetric power and improved safety margins. A thinner fuel rod of about 10 mm is currently used.

Table 3.7 Examples of main specifications of BWR fuel design Fuel type 7×7 type Improved 7×7 type 8×8 type New 8×8 type New 8×8 Zirconium liner type (Step I) High bumup 8×8 type (Step П) 9 x 9 A type (Step III) 9 x 9 В type (Step III) Maximum liner power (kW/m) 57 (17.5 kW/ft) 61 (18.5 kW/ft) 44 (13.4 kW/ft) 44 (13.4 kW/ft) 44 (13.4 kW/ft) 44 (13.4 kW/ft) 44 (13.4 kW/ft) 44 (13.4 kW/ft) Average dis­charge bumup (GWd/t) 21.5 27.5 27.5 29.5 33.0 39.5 45.0 45.0 Pellet material uo2 U02 or Gd203 added U02 U02 or Gd203 added U02 U02 or Gd203 added U02 U02 or Gd203 added U02 U02 or Gd203 added U02 U02 or Gd203 added U02 U02 or Gd203 added U02 Diameter (mm) 12.4 12.1 10.6 10.3 10.3 10.4 9.6 9.4 Length (mm) 22 12 11 10 10 10 10 10 Stack height (mm) 3,660 3,660 3,710 3,710 3,710 3,710 Standard 3,710 Partial Length 2,610 3,710 Cladding Zircaloy-2 Zircaloy-2 Zircaloy-2 Zircaloy-2 Zircaloy-2 Zircaloy-2 Zircaloy-2 Zircaloy-2 material Stress-relief Recrystallization Recrystallization Recrystallization Recrystallization Recrystallization Recrystallization Recrystallization Annealing Annealing Annealing Annealing annealing annealing annealing annealing

(Zirconium (Zirconium (Zirconium (Zirconium liner) liner) liner) liner) 12.3 12.3 11.2 11.0 0.86 0.86 0.71 0.70 -0.1 -0.1 -0.1 -0.1

Outer diame­ter (mm) 14.3 14.3 12.5 12.3 Thickness (mm) 0.81 0.94 0.86 0.86 Zirconium liner thickness (mm) No. of fuel rods per assembly 49 49 63 62 No. of water rods 0 0 1 2 No. of water channels — — — — Spacer type Grid type Grid type Grid type Grid type Gas filled Helium Helium Helium Helium in rod gap (pressure) (0.1 MPa) (0.1 MPa) (0.1 MPa) (0.3 MPa)

62 60 74 72 2 1 (large diameter) 2 (large diameter) — — — — 1 (Square) Grid type Helium (0.3 MPa) Circular cell type Helium (0.5 MPa) Circular cell type Helium (1.0 MPa) Ring type Helium (1.0 MPa)

The maximum fuel assembly power must be below the assembly

power which meets the limit of MCPR to avoid boiling transition. can

be reduced through the following measures.

(i) Improve the radial power peaking factor considering fuel loading pattern and control rod pattern.

(ii) Increase the number of fuel assemblies NB.

Cores are designed to assure the inherent negative reactivity feedback characteristics. The fuel temperature coefficient (Doppler temperature coefficient) is always negative and it is designed so that the moderator temperature coefficient is negative at hot power operation. Getting the negative power coefficient by the combination of fuel and moderator coefficients suppresses the power rise in abnormal transients.

Items Design principles

Reactor shutdown Designed to assure the complete core shutdown capability at hot temperature condition even with the most reactive rod cluster control assembly (RCCA) stuck in the fully with­drawn position. Designed to maintain the core shutdown capability even at cold tem­perature condition by boric acid injection of chemical and volume control system Reactivity Designed with appropriate limits of the maxi­

insertion limit mum rod cluster control assembly (RCCA) worth so that the core internal structures function for a core cooling without damage to integrity of coolant pressure boundary at the ejection of a rod cluster control assembly Designed with appropriate limits of the maxi­mum reactivity insertion rate so that the fuel integrity is secured at a withdrawal of two banks of RCCAs at the maximum speed Designed with appropriate limits of the maxi­mum RCCA worth so that the fuel integrity is secured at a sudden drop of a fully- withdrawn RCCA at hot full power condition Self controllability Designed to assure the inherent negative reac­tivity feedback characteristics where the Doppler coefficient is always negative and the moderator temperature coefficient is negative at hot power operation

Fuel integrity Designed to assure that the minimum DNBR is

larger than the allowable limit

Designed to assure that the maximum fuel cen­terline temperature is lower than the fuel melting point

Designed to assure that the maximum burnup is lower than the design limit

Power distribution Designed to assure that the nuclear enthalpy rise restriction hot channel factor (FNH) is lower than the

design limit at normal operation

Designed to assure that the heat flux hot channel factor (Fq) is lower than the design limit at normal operation

Designed to assure that the maximum linear power density does not exceed the design limit at abnormal transients

Stability Designed to assure no abnormal oscillation of

power distribution where the oscillation decay characteristics are sufficient or any oscillation is detected and easily suppressed

Just as it is in LWRs, the main objective of reactor core design is to improve economy while ensuring safety. The requirements for safety are ensuring shutdown capability, limiting reactivity insertion rate, getting self controllabil­ity, preventing fuel failure, limiting the power distribution, and ensuring sta­bility. The main requirements for economy are getting adequate burnup and breeding performance. The basic design principles are summarized in Table 4.1.

(1) Reactor shutdown

The design conditions of the reactor shutdown system are determined in order to shut down the reactor safely and surely at abnormal incidences. For that purpose, the following criteria are adopted.

• Provide two independent shutdown systems.

• Design at least one system able to shut down the reactor at low temper­ature with the required shutdown margin even if the control rod with the highest reactivity worth is fully withdrawn and stuck.

pper end plug

Handling head

Tag gas capsule

Upper spacer pad

Cladding

spacer pad

Upper blanket fuel pellet

W rapper tube

A-Across section

Fig. 4.3 Driver fuel assembly and core configuration of japanese prototype reactor monju [6]

• Design the shutdown systems so that, even if one shutdown system completely fails, another system can keep the reactor subcritical at low temperature.

1. Design a fast reactor core with an electric power of 1,000 MW under the following conditions.

(i) Reactor thermal power: 2,500 MWt

(ii) Power fraction of core fuel: 92 %Power fraction of blanket fuel: 8 %

(iii) Average linear heat generation rate: 230 W/cm

(iv) Core height: 100 cm

(v) Number of fuel elements per assembly: 271

(vi) Radial blanket region 2 layers

2. Estimate the average discharged burnup of the core fuel after operating the reactor designed in problem 1 for 24 months. Assume MOX fuel and a dispersed refueling method with 3 batches. Other conditions are as below.

(i) Diameter of fuel element: 8.5 mm

(ii) Clad thickness: 0.4 mm

(iii) Smear density of fuel (ps): 9.5 g/cm3

3. Calculate the fuel centerline temperature of a fast reactor when the pellet surface temperature is 900 ° C and the linear heat generation rate is 400 W/cm. Assume a constant thermal conductivity of 0.023 W/(cm«°C) in the fuel pellet without temperature dependency.

4. Explain the reason and the major limiting factors of flow distribution design in fast reactors.

In the above, the principal effects of the moderator temperature coefficient have been looked at based on the six-factor formula. For a liquid moderator, two prominent effects were seen. (i) Due to the large expansion coefficient, the temperature coefficients of p and f have a negative or positive reactivity effect, respectively, to the extent of 10_4Ak/k/K. (ii) When 239Pu builds up with fuel burnup, the resonance of 239Pu leads to a negative or positive reactivity effect, respectively, on the temperature coefficients of n and f to the same extent, 10_4Ak/k/K.

As first mentioned, the temperature coefficients must be essentially negative for stable operation of the reactors. As a practical result of the effects, (i) and (ii), necessary conditions for a negative reactivity effect are discussed below.

The reactor core calculation is carried out by the N-TH coupled core calculation, presented in the list [8] of Sect. 2.1.5, to evaluate properties in normal operation of the reactor. The concept of core design calculation and the calculation model are discussed here.

[1] Heat transfer calculation in single channel model

Figure 2.27 depicts a 1D cylindrical model to describe one fuel rod and its surrounding coolant with an equivalent flow path. It is called the single channel model and it is the basic model used in the core thermal-hydraulic calculation and the plant characteristics calculation. The thermal-hydraulic properties in the core can be calculated on the single channel model where the heat generated from fuel pellets is transferred to coolant which is transported with a temper­ature rise.

The radial heat transfer model is composed of fuel pellet, gap, cladding, and coolant. The radial heat conduction or convection between those components is considered in each axial region and then the axial heat transport by coolant or moderator is examined. The mass and energy conservation equations and the state equation are solved in each axial region in turn from the top of the upward coolant flow. The momentum conservation equation is also solved to evaluate the pressure drop. The axial heat conduction of the fuel pellet is ignored. This

Cladding

Pellet

Convection Model

Fig. 2.27 Single-channel heat transfer calculation model

assumption is valid because the radial temperature gradient is several orders of magnitude larger than the axial one.

The difference between the fuel average temperature and the cladding surface temperature is expressed as

(2.110)

where,

kf: average thermal conductivity of pellet (W/m-K)

hg: gap conductance (W/m — K)

kc: thermal conductivity of cladding (W/m-K)

Q": heat flux from pellet (W/m2)

rf: pellet radius (m)

tc: cladding thickness (m)

jave: pellet average temperature (K)

Ts: cladding surface temperature (K).

The terms inside the RHS square brackets represent the temperature drop in fuel, gap, and cladding, respectively.

The heat transfer from cladding surface to coolant is described by Newtons law of cooling

q"(rc) = hc(Ts-T)

where,

hc: heat transfer coefficient between cladding surface and coolant (W/m2-K) rc: cladding radius (m)

Ts: cladding surface temperature (K)

T: coolant bulk temperature (K).

The heat transfer coefficient hc can be evaluated using the Dittus-Boelter correlation for single-phase flow and the Thom correlation or the Jens-Lottes correlation for nucleate boiling.

Water rods are often implemented into fuel assemblies, especially for BWRs. The heat transfer characteristics of the water rod can also be calculated using the single channel model. Figure 2.28 describes the heat transfer calculation model with two single channels of fuel rod and water rod. The heat from coolant is transferred through the water rod wall into the moderator.

The heat transfer of the water rod is also represented by Newton’s law of cooling. The temperature difference between the coolant in the fuel rod channel and the moderator in the water rod channel is expressed as

where,

T: coolant temperature in fuel rod channel (K)

Tw: moderator temperature in water rod channel (K)

Dw: hydraulic diameter of water rod

hs1: heat transfer coefficient between coolant and outer surface of water rod (W/m2-K)

hs2: heat transfer coefficient between inner surface of water rod and moderator (W/m2-K)

Nf. number of fuel rods per fuel assembly Nw: number of water rods per fuel assembly Q ’ w: linear heat from coolant to water rod (W/m) tws: thickness of water rod (m)

The terms inside of the RHS square brackets represent the heat transfer from the fuel rod channel to the water rod wall and back to the water rod, respec­tively. Temperature drop due to the water rod wall is ignored in this equation. It is also assumed that the water rod wall is an unheated wall, whereas the cladding is a heated wall. Hence, there is no boiling at the outer surface of the water rod although the outer surface of the water rod wall contacts with two phase coolant. It should be subsequently noted that the application condition of heat transfer correlations is not identical for such a water rod wall.

The thermal conductivity of the fuel pellet depends on temperature and in a BWR fuel, for example, it can be given by

q oo л

(2.113)

where Tf? ve and kf are the average temperature (K) and thermal conductivity (W/m-K) of pellet, respectively. The thermal conductivity is actually a function of pellet density, plutonium containing fraction, burnup, etc. as well as tem­perature. Since the fission gas release during irradiation causes pellet swelling and hence it also changes the thermal gap conductance with cladding, the fuel behavior analysis code which includes irradiation experiments is required to precisely evaluate the fuel centerline temperature. However, since the fuel centerline temperature at normal operation is far lower than the fuel melting point, it does not need a highly accurately estimate in the reactor core design. In BWR fuel, the fuel centerline temperature at normal operation is limited to about 1,900 °C to prevent excessive fission gas release rate. The temperature drop in the pellet depends not on pellet radius, but linear power density and fuel thermal conductivity. Hence, the linear power density is restricted at normal operation of LWRs.