[1] Diffusion equation

In core calculations such as the nuclear and thermal-hydraulic coupled core calculation, core burnup calculation, or space-dependent kinetics calculation, high-speed calculation is the biggest need. Especially since large commercial reactors have a considerably larger core size compared with critical assemblies or experimental reactors, the neutron transport equation (2.1) has to take many unknowns and hence need a high computing cost.

For most core calculations, the angular dependence (J) of neutron flux is disregarded and the neutron transport equation is approximately simplified as Eq. (2.37).

(2.37)

v dt

Its physical meaning is simple. As an example, a space of unit volume (1 cm3) like one die of a pair of dice is considered. The time variation in neutron population within the space [the LHS of Eq. (2.37)] is then given by the balanced relationship between neutron production rate (S), net neutron leakage rate through the surface (V-J), and neutron loss rate by absorption (£аф).

J is referred to as the net neutron current and expressed by the following physical relation known as Fick’s law

] = —DV<p (2.38)

where D is called the diffusion coefficient; it is tabulated together with few-group cross sections in the lattice calculation to be delivered to the core calculation. Inserting Eq. (2.38) into Eq. (2.37) gives the time-dependent diffusion equation.

(2.39)

The time-independent form of Eq. (2.39) is the steady-state diffusion equa­tion and all types of the core calculation to be mentioned thereafter are based on the equation

V-DV(/)-i:a(t)+S = 0 (2.40)

Thus, solving the diffusion equation in critical reactors is to regard the reactor core as an integration of subspaces like the dice cubes and then to find a neutron flux distribution to satisfy the neutron balance between production and loss in all the spaces (corresponding to mesh spaces to be described later).

The power density, which is defined as the power produced per unit volume of the reactor core, is an important indicator to determine core size. It is usually provided in the unit of kW/l and the average power density of BWR is given by

Average power density = Q/Vcore = qaw x Nrod/ (LB x LB) (3-14)

The average power density can be raised by increasing the average linear heat generation rate of the fuel rods or by increasing the number of fuel rods per unit cross section. The number of fuel rods should be set as the most optimal value with a consideration on the nuclear design such as the volume ratio of coolant and fuel which is mentioned later, and on the thermal-hydraulic design such as the heat removal capability related to the heat transfer areas of fuel rods and coolant flow path. The average power density of BWRs operated until the present is in the range of about 40-60 kW/l.

The minimum departure from nucleate boiling ratio (DNBR) and maxi­mum fuel centerline temperature are limited to secure fuel integrity at normal operation and abnormal transients. DNBR is defined as the ratio of predicted critical heat flux (heat flux at the time when a departure from nucleate boiling occurs in boiling heat transfer, referred to as DNB heat flux) to actual heat flux. Cores are designed to have a minimum DNBR larger than the allowable limit and a fuel centerline temperature lower than the fuel melting point.

In addition, the cores are designed to assure that maximum burnup is lower than the design limit confirmed for fuel integrity. There is also another maximum burnup limit obtained from fuel cycle considerations such as the acceptance limit set by reprocessing facilities.

(4) Power Distribution Restriction

To meet the minimum DNBR limit at normal operation and abnormal transients, cores are designed to assure that the nuclear enthalpy rise hot channel factor (FNH) is lower than the design limit at normal operation. The heat flux hot channel factor (FQ) must be lower than the design limit at normal operation to satisfy safety analysis (LOCA analysis) initial condi­tions, and moreover, maximum linear power density must not exceed the design limit at abnormal transients due to the limits of minimum DNBR and maximum fuel centerline temperature at abnormal transients.

(5) Stability

Cores are designed to assure no abnormal oscillation of the power distri­bution where oscillation decay characteristics are sufficient or if there are any oscillations, that they can be easily detected and suppressed.

The maximum reactivity insertion rate of the control rod is limited so as to

prevent loss of coolant boundary integrity or core internal integrity. For that

purpose, the following measures are taken.

• Limit the number of control rods that can be simultaneously withdrawn.

• Limit the maximum withdrawal velocity.

(2) Self controllability

The core has an instantaneous negative feedback characteristic. That means

the following.

• Mixed oxide (MOX) fuel itself has negative temperature coefficients based on the Doppler effect and thermal expansion.

• By combining the temperature coefficients of fuel, structure, coolant and core support plate, the power reactivity coefficient is negative.

Table 4.1 Major design principles of LMFBR Core [6] Item Design principle Example of criteria Reactor shutdown (a) Two independent shutdown systems are provided (b) At least one system can 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 (c) Reactor is kept subcritical at low temperature even if one shutdown system is assumed to fail Reactor shutdown margin Over 0.001 Ak/k at low temperature Limitation of (a) Maximum reactivity insertion rate Maximum reactivity insertion rate reactivity insertion rate of control rod is limited so that rod withdrawal at maximum possible velocity leads to neither loss of coolant boundary integrity nor failure of core, reactor internals etc which deteriorate core cooling Below 8.5 x 10“ 5(Ak/k)/s Self controllability (a) Doppler coefficient and power reac­tivity coefficient, which is obtained by combining all kinds of reactivity coefficients such as temperature coefficients of fuel, structure, cool­ant and core support plate, is kept negative under all operating condi­tions in order to provide negative feedback characteristic Doppler coefficient dk -5.7to -7.6 x 10-3T dT Fuel integrity (a) Maximum temperature of fuel cladding is limited (b) Maximum fuel centerline tempera­ture is kept below melting point of fuel pellets (c) Maximum burnup of fuel assembly is limited. (a) Below 675 °C at normal opera­tion and 830 °C at abnormal operational occurrences (center of wall thickness) (b) Below 2 650 °C for unirradiated fuel (c) Below 93,000 MWd/t Limitation of (a) At normal operation and anticipated Maximum linear heat rate power distribution operational occurrences, adequate power distribution is kept so that allowable design limit of fuel is not exceeded (b) Power distribution is flattened in order to efficiently take out thermal power. Core is divided into two regions with different plutonium enrichments. Outer core has higher plutonium enrichment (c) Maximum linear heat rate is limited. Below 360 W/cm Stability (a) Throughout the operation period, power oscillation which leads to an Abnormal oscillation of power dis­tribution does not occur because (continued)

Table 4.1 (continued) Item Design principle Example of criteria Limitation of coolant temperature excess of allowable design limit of fuel is prevented at normal operation and anticipated operational occurrences (a) Coolant temperature is kept below boiling point at normal operation, anticipated operational occurrences and design basis accidents no fission products have large absorption cross section in the fast reactor energy range Breeding performance (a) Adequate breeding ratio is achieved About 1.2 as target

• Increase in the power is suppressed by the negative reactivity feedback even at anticipated operational occurrences.

Chapter 1

1. The bumup equations of U and Pu can be given by

dN25(t)

dt

=а™фЫ2а-ai9<pNi9 it)

dNi9(t)

dt

For an initial condition of N49(0) = 0, the atomic densities of 235U and 239Pu are obtained as

!V26(O=iV26(0)e-o’”!,<

(Graph skip)

2. The production and destruction equations of 135I and 135Xe after changing neutron flux to ф1 can be solved by using the equilibrium-state solutions at neutron flux ф0 as initial conditions. The atomic densities of 135I and 135Xe can be obtained as

I (0=-^^{фое~ы+ф1(1— е~ыУ

Ai

Х(О=(П + 7-)і:^0 е-^Л^Ф^^Г^ЇХ^фі (1_в-дд+5^^) Лхе + (УаЄфо ЛХе + <УаЄфі

_І_ У/^(0і~0о) (^е-аХе^фі)і_еГхІі^

Ххе ^ <Уа Єфі~^І

where t represents time after the neutron flux change.

(Graph skip)

Y. Oka (ed.), Nuclear Reactor Design, An Advanced Course in Nuclear Engineering 2, DOI 10.1007/978-4-431-54898-0, © Authors 2014

3. Skip

4. Skip

5.

By the one-group first-order perturbation theory

The perturbation parts can be expressed in terms of temperature coefficient as 8Ъг=^гЛТ=Ъг(аад1-^-Ъв^ЛТ

sz—(0+W)AT-{z:ac, i-+z-(-^-si>)}jT

where в is the linear expansion coefficient of the medium, and Tn = aT is assumed (Tn is the neutron temperature and T is the medium temperature). Substituting them into the reactivity expression gives

where the criticality condition

is used.

Chapter 2

1. (a) Express atomic or molecular mass, density, and atomic number density as M, p (g/cm3), N (#/cm3) respectively. 235U enrichment (wt%) can be calculated as

(1.1)

According to Eq. (1.1), N(235U): N(238U) = 1: 29.868, namely, the 235U enrichment is 3.24 atm%.

M(U) = 0.324 x M(235U) + (1 — 0.324) x M(238U) = 237.075 M(UO2) = M(U) + 2 x M(O) = 269.075

N(U) = N(UO2) = p(UO2)/M(UO2) x Na

= 10.4/269.075 x (0.6022 x 1024) = 2.320 x 1022atoms/cm3

N(235U) = N(U) x 3.24/100 = 7.516 x 1020atoms/cm3

N(235U) = N(U) x (1 — 3.24/100) = 2.245 x 1022atoms/cm3 N(O) = N(U) x 2 = 4.640 x 1022atoms/cm3

(b) I (UO2) = N (235 U) (ja (235U) + N(238U)ffa(238U) + N(O)aa (O)

= (7.516 x 1020) x (50 x 10-24) + (2.245 x 1022) x (1.0 x 10-24) + (4.640 x 1022) x (0.0025 x 10-24)

= 6.014 x 10-2 cm-1

If (UO2) = N(235U)o/ (235U) + N(238U)o/ (238U)

= (7.516 x 1020) x (41 x 10-2) + (2.245 x 1022) x (0.1 x 10-24)

= 3.306 x 10-2cm-1

The probability that a neutron absorbed in pellet fissions, Pf, can be given by

Pf = If (UO2)/Ia(UO2) = (3.306 x 10-2)/(6.014 x 10-2)

= 0.55 = 55%

(c) The mass of metal uranium included in a pellet, w(U) can be calculated as

p(U) = N(235U)/NA x M(235U) + N(238U)/Na x M(238U)

= 9.167g/cm3

w(U) = p(U) x VP (pellet volume)

= 9.167 x (3.1416 x (0.81/2)2 x 1.0) = 4.72g = 4.72 x 10-6ton

InitialUraniumMassIncludedinaFuelAssembly = w(U) x 366 x 265 = 0.46ton Uranium Loading Amount in Core (Metal Uranium Mass) = 0.46 x 193 = 88 ton

(d) Since the pellet length is 1 cm, the linear power of fuel rod q’ (W/cm) is equal to energy released from a pellet. Therefore,

q = Klf (UO2 )фУр

= (200 x 1.602 x 10“13) x (3.306 x 10“2) x (3.2 x 1014) x 0.515 = 175 W/pellet = 175 W/cm

Since w(U) = 4.72 x 10 6 ton/pellet,

Specific Power = (175 W/pellet)/(4.72 x 10“6ton/pellet) = 3.70 x 107W/ton = 37MW/ton

(e) Since burnup is a time integral of specific power,

Burnup = (36.0 x 285) А (41.0 x 290) А (38.0 x 280) = 32800MWd/ton

(f) Total heat generation of pellet is,

Burnup 32,800 MWd/ton x w(U) = 32800 x (4.72 x 10“6) ton/pellet = 0.155 MWd = 1.34 x 1010 J.

The heat generation by uranium (235U and 238U) fission is

1.34 x 1010J x 2/3 = 8.92 x 109 J.

Since the heat release per fission is 200 x (1.602 x 10“ 13) J/fission, Total number of uranium fission is 8.92 x 109/(200 x 1.602 x 10“13) = 2.78 x 1020fission.

The number of 235U fission is X If(235U)/<ZK235U)+Z/-(238U)} =(2.78X 102°)x0.932 =2.60 XlO20 fission

235

(Total number of uranium fission)

Since the amount of 235U decreases by other reactions such as the (n, y) reaction, the number of 235U absorption reaction is calculated as

Number of 235U absorption = (Total number of 235U fissions) x Za(235U)/X/-(235U)

=(2.60 X102°)X 1.22 =3.17 XlO20

which can be converted to 235U mass as (3.17 x 1020)/NA x M (235U) = 0.12 g.

Therefore, 235U decreased by about 0.12 g in a pellet of 32,800 MWd/t burnup.

2. For example, for N = 4,

аф0 А Ьф1 А аф2 = S аф1 А Ьф2 А афъ = S аф2 А Ъфъ А аф4 = S ф0 = ф1 (Left boundary condition) ф4 = 0.0 (Right boundary condition)

where

Substitution of d/2 = 200 cm, D = 1.0 cm, = 2.5 x 10 4 cm l, S0 = 1.0 x 10_3 atoms/cm2s gives the following neutron flux distribution.

3. (a) Considering upscattering (£2!1) as zero,

Thermal group : — D2 F202+Xa,202 = ^2^Vl^1^1+V2^2^2^ +ъ1_+2ф1

keff

(b) For infinite medium, leakage is zero. x1 = 1, x2 = 0.0 give

keff

Sa,2 Ф2 = ^1!2 Ф1 which can be rewritten as

10i+v2X/, 2 02 Total Production Rate

Xa, i0i+^a,2 02 Total Absorption Rate

УіХ/-д + У2Х/-;2(Х i-»2/Xa>2)

2o. l + 2l-

4. Assuming the atomic densities of 135I and with burnup time can be given by

dN1

dt

UXeJr(j$e0)NXe

where Л1 and AXe are the decay constants and Ф is one-group collapsed neutron flux.

Assuming Ф > AXe/^Xe = 7 x 1012 n/cm2s, NXe can be expressed as

Assuming ol0 « 0, the equilibrium atomic densities are obtained as

Chapter 3

1. The advanced standardization of LWRs has had the goal of improvements in reliability and availability, and reduction in exposure of radiation workers.

The first stage of advanced standardization in Japan took place from 1975 to 1977. In BWRs, there were improvements in the shape of the containment vessel head, security of maintenance space such as for the main steam isolation valve, and workability. Efforts were also made for reliability improvement by measures against stress corrosion cracking, automation of in-service inspection (ISI), and more efficient inspection (using remote and automatic techniques). In PWRs, high tensile strength steel for the containment vessel and the prestressed con­crete containment vessel (PCCV) were developed. As a measure against fuel rod bowing for reliability improvement, the number of fuel rod grid spacers was increased from 8 to 9. The eddy current technique was developed to measure leakage from steam generator tubes, and improvements in its inspection accu­racy and efficiency were made.

The second stage of advanced standardization was implemented from 1978 to 1980. There were improvements in operation and maintenance, efficient and regular inspections, and reduction in exposure of radiation workers.

The third stage was carried out from 1981 to 1985. The goals dealt with load following, longer cycle length, core performance improvements, more compact plant size, and shorter construction time. ABWRs and APWRs were developed. Internal pump, improved control rod drive mechanism, reinforced concrete containment vessel, high burnup fuel, and large-size turbine were developed and adopted in ABWRs.

The table presents an overview of the advanced standardization plan of LWRs.

Previous plant (800 and 1,100 MW class) First advanced standard plant (800 and 1,100 MW class) Second advanced standard plant (800 and 1,100 MW class) Third advanced standard plant (1,300 and 1,500 MW class) Improvements BWR PWR Availability factor Depends on plant ~75 % ~80 % SSC-resistant material Fuel improvement (against fuel bowing) Capacity factor Depends on plant ~70 % ~75 % ~80 % Core design improvement Steam generator improvement Regular inspection period 90-100 days ~85 days ~75 days ~50 days Automatic replacement of control rod Drive mechanism Improvement of fuel handling machine Development of integrated reactor vessel head material Improvement of fuel inspection system Worker exposure (100 %) ~75 % ~50 % ~40 % Pipes and components Introduction of automatic ISI of pipes Cobalt-free materials Containment improvement Development of steam generator, Manipulator, mounter Improvement of Nozzle cap for steam generator water chamber Source: Department of Nuclear Power Plant (1999) Agency for Natural Resources and Energy: nuclear handbook, 385

2. Calculation examples of 10 x 10 fuel specifications are as follows.

(a) Size Specifications of Water Rod

An approximate fuel rod pitch of 10 x 10 fuel assembly: 134/10 = 13.4 mm. In the case that four rods are replaced with a large- diameter water rod in the same way of Step II fuel, if the water rod outer diameter and thickness are set as 26 mm, which is twice of the fuel rod arrangement, and 1.5 mm respectively, the cross-sectional area in two water rods is 2 x (23)2 x n/4 = 831 mm2. Thus, a non-boiling region larger than that of Step II fuel can be secured within channel box.

(b) Fuel Rod Specifications

The number of fuel rods becomes 92 excluding eight fuel rods which were replaced with two water rods.

The cross-sectional area of pellets in 60 fuel rods at Step II fuel can be obtained as

Spellet = n x (10.4)2/4 x 60 = 5097 mm2

To maintain the same fuel inventory as Step II fuel, the pellet diameter will be

Pellet Diameter = (4Spellet/92/n)1/2 = 8.40mm

The cladding inner diameter will be given by applying the same gap between pellet and cladding as 0.18 mm of Step II fuel as

Cladding Inner Diameter = 8.40 + 0.18 = 8.58mm

Assuming the cladding thickness as 0.65 mm to maintain the same ratio between cladding thickness and cladding inner diameter, the cladding outer diameter will be

Cladding Outer Diameter = 8.58 + 2 x 0.65 = 9.88 mm

(c) Non-Boiling Region Area within Channel Box

Cross-sectional area of fuel rods and water rods within channel box is

= 92 x (9.88)2 x n/4 + 2 x (26)2 x n/4 = 8115 mm2 Cross-sectional area of non-boiling region within channel box is = 179.3 — 81.1 = 98.1cm2 which is almost the same as 99 cm2 of Step II fuel.

3. (a) Cycle Burnup = (Specific Power) x (Cycle Length) x Load Factor

Bc(12) = 26.2 x 365 x 1.0 = 9563MWd/t = 9.563GWd/t

(b) For cycle length = 12 months and discharge burnup = 45 GWd/t, the fuel batch size is given by

n(12) = 45 ^ 9.563 = 4.7056

The core average burnup BEOC can be calculated by the following equation

BEOC = (2n — [n]) x ([n] + 1) x Bc/2n Substituting Bc(12) and the integer part of n(12) gives BEOC = (2 x 4.70 — 4) x 5 x 9.563/(2 x 4.70) = 27.5GWd/t

(c) For cycle length = 15 months, the cycle burnup Bc(15) is given by

Bc(15) = Bc(12) x (15/12) = 11.95GWd/t

In the linear reactivity model, the sum of cycle length and discharge burnup has a constant relation in using the same fuel. Therefore, the dis­charge burnup Bd(15) can be given by

Bd(15) = (Bd(12) + Bc(12)) — Bc(15) = 42.6GWd/t

The fuel batch size n(15) becomes 3.56. Such a prolongation of operation cycle length with the same fuel decreases the discharge burnup.

4. (a) In BWRs, burnup reactivity decline is compensated by burnable poison

(gadolinia) addition, core flow rate control, and control rod operation. Most of the burnup reactivity is controlled by burnable poison (gadolinia) added to pellets. The concentration of gadolinia is designed to be depleted out at EOC for fuel economy. Since BWR cores have negative void reactivity coefficients, the burnup reactivity decline can be compensated by changing the pumping speed of coolant recirculation pumps and increasing the core flow rate, namely, reducing the core void fraction, and by control rod operation as well. From the viewpoint of fuel econ­omy, generally all control rods are designed to be withdrawn at EOC.

(b) In ABWRs, since the core average void fraction is about 40 %, the void reactivity coefficient at the end of equilibrium cycle is about —8 x 10—4 Ak/k/% void fraction. Therefore, change of the void fraction to 5 % gives a reactivity variation of about 4 x 10—3 Ak/k.

(c) At BOC, the excess reactivity is controlled by control rod insertion at a core flow rate slightly lower than the rated one. Since the excess reactivity tends to increase during the first half of the operating cycle, the core flow rate is mainly reduced without moving control rods. When the core flow rate exceeds the range of the flow control, it is recovered by inserting control rods. Since the core reactivity tends to decrease during the latter half of the operating cycle, the core flow rate is increased by adjusting the recirculation flow rate and control rods are withdrawn when the core flow rate exceeds the range of the flow control.

Figure Core flow rate and control rod position adjustment for reactivity control 5. Main challenges for high burnup:

(a) Nuclear characteristics are changed as fuel enrichment increases; these changes include neutron spectrum hardening, more negative void reactivity coefficient, and control rod worth reduction.

(b) At high burnup, the reactivity difference between fuel assemblies loaded in the core becomes large and mismatch of the fuel assembly power increases. As fuel enrichment increases, since the number of gadolinia-added fuel rods increases to control the excess reactivity, the local power peaking of a fuel assembly increases and the fuel linear heat generation rate increases.

(c) The internal pressure of cladding increases.

(d) The corrosion amount of cladding increases with increase in neutron irradi­ation and residence time in the core.

Countermeasures:

(i) Increase in number and size of water rods

As a countermeasure for (a), H/U ratio increase and neutron spectrum softening are general countermeasures. Specially, the number of water rods has been increased to enlarge the non-boiling water region within a channel box and the arrangement of large-diameter water rods and the water channel has been implemented.

(ii) Increase in number of fuel rods

As a countermeasure for (b) and (c), an increase in the number of fuel rods is a general countermeasure. Fuel assemblies of 7 x 7, 8 x 8, and 9 x 9 designs have been implemented and the lattice arrangement has also been changed.

(iii) Improvement in corrosion resistance

As a countermeasure for (d), more easily fabricated high corrosion — resistant claddings with better composition have been developed and put into practical use.

(iv) Reduction in pressure drop of fuel assembly

From the viewpoint of fuel economy, items (i) and (ii) without reduction in fuel inventory cause an increase in pressure drop of a fuel assembly. Such an increased pressure drop leads to an increase in coolant pump capacity and less stability, so countermeasures against pressure drop are important. The designs of spacer and top tie plate of low pressure drop, and the use of a partial-length fuel rod have been implemented.

6. The cycle burnup at equilibrium cycle can be calculated as

2625/(0.46 x 157) x 413 = 15170MWd/t

The fuel assembly average burnup can be given by

1 Cycle Burned Fuel: 60 assemblies, 15,170 MWd/t

2 Cycle Burned Fuel: 60 assemblies, 15,170 x 2 MWd/t

3 Cycle Burned Fuel: 37 assemblies, 15,170 x 3MWd/t

Therefore, the core average burnup at the end of equilibrium cycle is given by

(15170 x 60 + 15170 x 2 x 60 + 15170 x 3 x 37)/(60 + 60 + 37)

= 28100 MWd/t

7. Since the reactivity control by soluble boron in the primary coolant causes waste production, it is desirable that a core design with no use of soluble boron is realized to reduce waste. An idea is that the soluble boron is not to be used during normal operation, and to be used only for backup shutdown. Therefore, the following roles currently taken by soluble boron are needed to be substituted by other means.

(i) Reactivity variation from cold to hot temperature: about 6 %Лк/к

(ii) Reactivity decline during operating cycle: about 10 %Лк/к or more

(iii) Xe reactivity (maximum after shutdown): about 6 %Лк/к

As an alternative way for reactivity control, roles (i) and (iii) can be done by using control rods, and role (ii) can be done with gadolinia or burnable poison.

(a) Reactivity Controllability Enhancement of Control Rods

Reactivity controllability enhancement of control rods by about 12 % Ak/k for (i) and (iii) is necessary which corresponds to three times as much as the current design. Two measures could be applied. One is to enhance the absorption capability of neutron absorbers and the other is to increase the number of rod cluster control assembies. In the former, even enriched 10B — containing B4C does not attain twice the reactivity of the current Ag-In-Cd absorber. In the latter, the number of control rod clusters can be increased by a maximum 50 % due to limitation of the current design of the control rod drive mechanism (to increase more, development of a small control rod drive mechanism is needed). Thus, the reactivity controllability can be enchanced at most to twice as much as the current one even applying both measures. Therfore, it is also necessary to reduce the reactivity required for (i) and (iii). But, to realize the reduction, we have to considerably reduce the moderator — to-fuel volume ratio. However, since such a reduction in moderator-to-fuel volume ratio causes a reduction in control rod worth, it is necessary to get a balance between both.

(b) Excess Reactivity Controllability Enhancement by Gadolinia

The number of gadolinia-added fuel rods and fuel assemblies can be increased to control total reactivity corresponding to (ii). However, this is not practically easy because it gives rise to a deterioration of the power distribution. Numerous gadolinia-added fuel cause a higher power increase in high power fuel at early period of burnup because the reactivity increases further as burnup progresses at early period of burnup. This behavior is remarkable in the axial power distribution. The power variation goes into reverse as the burnup proceeds to some extent and then it leads to an oscillation in the axial power distribution as burnup progresses.

Thus, it is quite hard not to use soluble boron. In other words, it is recognized once again that the reactivity control by soluble boron is power­ful and superior. By the way, the reactor of the nuclear power ship “Mutsu” was designed to use only control rods without soluble boron. Its control rods were not the cluster type but the cruciform type. The excess reactivity was smaller than that of conventional power plants and the power peaking factor was higher.

8. As shown in Fig. 3.38, the reactor can be controlled only by control rods until hot shutdown. Since the hot temperature condition is maintained for a while after shutdown, the reactivity varies mostly by accumulation and decay of Xe and its variation is compensated by adjusting the boron concentration. After the adjust­ment of the boron concentration, the criticality is generally attained by control rod withdrawal.

(i) 8 h later

The accumulation of Xe peaks and the reactivity of Xe increases (i. e. the reactivity of core decreases) by about 3 %Ak/k (= 3,000 pcm) at 8 h after shutdown. The boron concentration is to be reduced by correspondingly

converting this reactivity (dilution). According to Fig. 3.45, the conversion is about 7 pcm/ppm. The boron concentration is diluted as 3,000/ 7«430 ppm. It is noted that the limit of the dilution speed is 10 ppm/min or so, and further the speed becomes slower at low boron concentration. Therefore, it is necessary to start the dilution considerably before the expected criticality time.

20 h later

Accumulation of Xe is over, and it is back to almost the concentration at full power equilibrium (namely, the concentration at shutdown). Therefore, it is not necessary to adjust the boron concentration.

(iii) 90 h later

Xenon is almost gone, having decreased by about 3 % Ak/k since shut­down. In other words, the boron concentration should be increased by about 430 ppm in the opposite direction to (i).

9. (i) As shown in Fig. 3.35, the boron concentration decrease by 120 ppm is required to move the moderator temperature coefficient to a more negative value by 2 pcm/°C. To realize that, the fuel loading pattern should be re-designed by additionally using 240 burnable poison rods. For example, 12 fuel assemblies containing 20 burnable poison rods can be prepared. However, the increase of 120 burnable poison rods leads to an increase in reactivity penalty at EOC by about 20 ppm (converted value to the soluble boron concentration). It is required to achieve the given cycle length con­sidering the reactivity penalty.

(ii) The same as in (i), it is necessary to reduce the boron concentration by 120 ppm which is compensated by control rods insertion. The boron worth at BOC is about 7 pcm/ppm from Fig. 3.45. Therefore, the reactivity of 120 x 7 = 840 pcm should be suppressed by control rod insertion. In Fig. 3.42, it corresponds to the insertion of control bank D to 50 steps (178 steps for control bank C). In other words, the moderator temperature coefficient can be maintained as negative at hot zero power if control bank D is restricted not to be withdrawn above 50 steps.

Chapter 4

1. Calculate the number of fuel assemblies from core thermal power as

TotalLengthofFuelElements (L) = 2 500 x 106 x 0.92 ^ 230 Number of Fuel Elements (Nel) = L ^ 100 Number of Fuel Assemblies (Na) = Nel + 271

Table 1 Number of assemblies of each layer in hexagonal arrangement Layer number Number of assemblies Cumulative number of assemblies 1 1 1 2 6 7 3 12 19 4 18 37 5 24 61 6 30 91 7 36 127 8 42 169 9 48 217 10 54 271 11 60 331 12 66 397 13 72 469 14 78 547 15 84 631 16 90 721

Then, the number of fuel assemblies is 369.

■ Guess the number of control rods (finally determined by evaluating control rod

worth)

Set the number of control rods as a multiple of 6 or 3, considering usual symmetric arrangement of control rods. The number of control rods-to-fuel assemblies ratio of 7-10 % gives 30 or 36 control rods. From the viewpoint of control rods cost, the smaller number of control rods is the first candidate.

■ Guess core arrangement

Take 397 or 403 fuel and control rod assemblies as a multiple of 6 + 1. Divide the core into two almost equal volume regions and arrange the fuel assemblies. Consider a symmetric arrangement of control rods for power distribution flattening and place as many of the control rods in the inner core region as possible to increase their reactivity worth.

Table 1 shows the cumulative number of fuel assemblies per each layer when the fuel assemblies are placed in a hexagonal arrangement. Since the control rods are arranged in the inner core region, construct the inner core region using 217 fuel assemblies until the ninth layer and the outer core region with 180 fuel assemblies until the 12th layer. Figure 1 shows the arrangement of assemblies. Consider a uniform and symmetric distribution of 30 control rods. Figure 2 shows a core layout with control rods.

In the core design, the core characteristics (such as power distribution, control rod worth, and reactivity coefficients) are iteratively evaluated by changing the

Fig. 1 Arrangement of assemblies

Fig. 2 Example of core layout (1,000 MWe)

core layout and finally the core layout for achieving the design targets are determined.

2. Calculate the fuel loading amount (W) as

„„ / Cladding Inner Diameter V „ .

W= к (———————————— I XHcXNeXNaXpsX Heavy Metal Fraction

= 38.78 t

Compute the burnup as

BU = Q x Core Power Fraction x D x Nb/W = 12.99 x 10[8] = 130GWd/t

3. The relation of pellet temperature, linear power, and thermal conductivity can be

given by

where

X: Linear power k: Thermal conductivity T: Temperature

T0: Fuel centerline temperature TS: Fuel surface temperature

Assuming a constant thermal conductivity, then

Therefore, the pellet centerline temperature is calculated as 2,285 0C by 4

[1] Coolant temperature coefficient

Since the moderator also serves as a coolant in LWRs, the coolant temperature coefficient is identical to the moderator temperature coefficient. In thermal reactors with different moderator and coolant materials such as graphite­moderated reactors, the coolant temperature coefficient is separately used. As mentioned before, the moderator temperature coefficient in thermal reactors is determined by the net effect of both the negative and positive temperature coefficients of p and f, respectively. In thermal reactors with a separate moderator material, since coolant expansion does not lead to a sufficient reduction in the neutron moderating power, it should be noted that the positive effect of f may become dominant for the coolant temperature coefficient.

[2]106)

Jvvi:f(.r)<p4r)d3r

The reactivity change in the perturbation theory is expressed with the neutron flux and its adjoint flux operating to cause the change in the macroscopic cross sections. The product of the neutron flux and the change in the macroscopic cross sections represents the change in reactor rate. The adjoint neutron flux is interpreted to be multiplied as an importance. This physical interpretation of the adjoint flux can be clearly made in considering a time-dependent problem. For more details, the references [23—25] should be consulted.

[3] Most practical codes of the finite difference method take neutron fluxes at the center points of divided meshes [13]. A simple and easy to understand method was employed here.

[4] Substitute the solution of Eq. (2.80), 0^, for the second term in the RHS

of Eq. (2.81) and then solve this for 02^.

[6] For simplicity, it is assumed that %dig = xg which is reasonable for the two-group problem.

[7] Shibata K et al (2011) JENDL-4.0: a new library for nuclear science and engineering. J Nucl Sci Tech 48:1

[8] The constituent components of fast reactor cores are characterized by fuel assemblies, blanket assemblies, control rod assemblies, and the radial shieldings, all of which produce quite different heats. For effective heat removal, it is necessary to distribute coolant flow rate adequately for each core component corresponding to each heat generation. The coolant flow allocation with no waste flow leads to a high core outlet temperature and high thermal efficiency and contributes to reduction in power generation cost.

In the thermal limitations of the fast reactor core thermal-hydraulic design, there are limits for fuel and cladding temperatures. While the fuel temperature is influenced by linear power of fuel, the cladding temperature is strongly depen­dent on coolant temperature. For a core flow allocation of fast reactors, therefore, coolant flow rate should be distributed to maintain the maximum cladding temperatures under the design limits for fuel and blanket assemblies at the rated core power. For a design of coolant flow allocation, fuel assemblies requiring nearly equal coolant flow should be classified into same flow allocation group and the number of core flow regions should be minimized in order to avoid complicated coolant flow control mechanism.

Here, an explanation is given as to the reason for placing a design point on the under-moderated region being discussed. The moderator tem­perature coefficient is mainly determined by the balance of the tem­perature coefficients of p in Eq. (1.66) and f regarding the first term in the second parenthesis of the right-hand side in Eq. (1.72). It can be then written as Eq. (1.76).

(1.76)

When p approaches 1, the following holds

and the moderator temperature coefficient can be approximated as

aTM^SOM(p—f) . (1.77)

If f exceeds p, it may provide a negative moderator temperature coefficient.

Figure 1.13 illustrates the infinite multiplication factor and the four factors as a function of the ratio (denoted by x) of the atomic number densities of fuel and moderator. A decrease in moderator density due to a rise in moderator temperature corresponds to the increase of x. As mentioned up to now, the figure indicates that the variation of the effective multiplication factor due to an increase in moderator temper­ature is determined by p andf (n in the four-factor formula is out of the range of this figure, but almost constant with x without the resonance effect of nuclides such as 239Pu).

To make the moderator temperature coefficient negative in Fig. 1.13, in other words, to decrease the infinite multiplication factor with an increase of x, x should be larger than xmax at the maximum of the infinite multiplication factor.

The region above xmax, in which the ratio of the moderator atomic density is smaller than that at xmax, is called the under-moderated region. As an optimal point, a reactor may be designed with an as-small-as-possible fuel concentration which gives the largest value of the infinite multiplication factor. However, it is usual to place the design point slightly toward the under-moderated region from the maximum in order to make the moderator temperature coefficient negative from the viewpoint of reactor safety. In the region below xmax (the over-moderated region), the infinite multiplication factor increases with x, that is, the moderator temperature coefficient is positive.

Further, it is easy to confirm that Fig. 1.13 includes the positive or negative relation of the moderator temperature coefficient in Eqs. (1.76) or (1.77).

(i) 239Pu buildup effect on n and f [21]

The multiplication of n andf in uranium fuel, including 239Pu produced during burnup, can be written as Eq. (1.78).

vzp35+vz;u239

^U235 _|_ ^U238 _|_ ^Pu239_|_ ^Others

The right-hand side were partially substituted by F and A, which represent the contribution rates of 239Pu to fission neutron production and absorption of thermal neutrons, respectively. Further, referring to Eq. (1.75) and applying the temperature coefficient of n to 239Pu gives Eq. (1.80).

a%=(F-А)аадтГ+Ааа^

A rough approximation is

1.:vPu239 v^Pu239

VLf_____ ____________

VZ^+vSf"239 ~ £U235_|_2Pu239

and then

Inserting Eq. (1.82) into Eq. (1.80) gives

• • аТм ^FaKl /fissile) (%Tfn /fissile ^гГ39] . (1:83)

Therefore, the moderator temperature coefficient of nf in 239Pu buildup can be expressed in an easy form by the thermal utilization factor of fissile nuclides ffisile through the approximation of Eq. (1.81).

Fig. 1.14 An example of the shift of moderator temperature coefficient to positive with fuel burnup in a LWR [22]

For example, consider

aff239=—lAX10~[1]Ak/k/K aft-239=-5 X ~A Ak/k /К.

According to both temperature coefficients, it is necessary that > 0.74 for af < 0 in Eq. (1.83). The thermal absorption rate of fissile nuclides should be large to make the reactivity effect negative against 239Pu build up. By contrast, a small thermal absorption rate of fissile nuclides for natural uranium or very low-enriched fuel may lead to emergence of factors to make the moderator temperature coefficient positive due to 239Pu buildup. Figure 1.14 shows an example that the moderator temperature coefficient in a LWR shifts to positive values with fuel burnup.

The coolant temperature coefficient is also used in fast reactors with no moderator. The mechanism of reactivity change is different from that in thermal reactors (see Sect. 1.4).

In nuclear reactors such as LWRs, in which the coolant serves as moderator, the moderator density varies with temperature change and boiling of coolant, and subsequently the reactivity and reactor power vary. The power distribution also varies with the moderator density change. The heat transfer calculation is performed to evaluate those core characteristics using the heat generation distribution in the core acquired from the nuclear calculation. The nuclear calculation is performed again using the macroscopic cross section changed by the coolant or moderator density distribution obtained in the heat transfer calculation. These calculations are repeated until a convergence. This is the nuclear and thermal-hydraulic coupled core calculation (the N-TH coupled core calculation) mentioned earlier.

A 2D core model cannot precisely handle power and burnup in each part of the core. This usually needs a 3D nuclear calculation, which is combined with the heat transfer calculation in the single channel model for each fuel assembly. Figure 2.29 shows a 3D N-TH coupled core calculation model which is a symmetric quadrant. Since the fuel assembly burnup differs with the power

Homogenized ruel hlements

Single Channel

I hermal-Hydraulic

1/4 Core

Fuel

Assembly

Fig. 2.29 3D nuclear and thermal-hydraulic coupled core calculation model

history at each position, the fuel assembly is axially segmented and the macro­scopic cross section at each section is prepared for the core calculation. Each fuel assembly, which consists of a large number of fuel rods, is described as the single-channel thermal-hydraulic calculation model. The heat transfer calculation is carried out with the axial power distribution of each assembly obtained from the nuclear calculation and it provides an axial moderator density distribution.

The N-TH coupled core calculation above is repeatedly continued until it satisfies a convergence criterion and then it gives the macroscopic cross section which is used for calculations such as the core power distribution. This process is repeated at each burnup step and is referred to as the 3D nuclear and thermal- hydraulic coupled core calculation (3D N-TH coupled core calculation). In the practical calculation, as mentioned in Sect. 2.1, the macroscopic cross sections are tabulated in advance with respect to parameters such as fuel burnup and enrichment, moderator density, and so on. Each corresponding cross section is used in the N-TH coupled core calculation.

New Set of Design Parameters

Burnup Distribution
Converge?

Design Criteria?

END

Fig. 2.30 Flow chart of equilibrium code design

Figure 2.30 shows a flow chart of an equilibrium core design using the 3D N-TH coupled core calculation [25]. Macroscopic cross section data are previ­ously prepared from the lattice and/or assembly burnup calculation and then tabulated with fuel burnup and enrichment, moderator density, and so on. The macroscopic cross section is used in the 3D multi-group diffusion calculation to obtain the core power distribution, which is employed again in the thermal — hydraulic calculation.

Usually several different types of fuel assemblies are loaded into the core according to the purpose such as radial power distribution flattening or neutron leakage reduction. In the first operation of a reactor, all fresh fuel assemblies are loaded but they may have several different enrichments of fuel. This fresh core is called the initial loading core. After one cycle operation, low reactivity assemblies representing about 1/4 to 1/3 of the loaded assemblies are discharged from the reactor. The other assemblies are properly rearranged in the core considering the burnup distribution and other new fresh ones are loaded.

In the end of each cycle, the core is refueled as in the above way and then the reactor restarts in the next cycle. While such a cycle is being repeated, the characteristics of the core vary but gradually approach equilibrium quantities. When finally the characteristics at N +1 the cycle have little change in comparison with those at the N cycle, the core is called the equilibrium core. In certain cases, it shows equilibrium characteristics at N and N +2 cycles and these are also generally referred to as the equilibrium core. By contrast, the core at each cycle is called the “transition core” until it reaches the equilibrium core.

In the equilibrium core design, first a proper burnup distribution of the core is guessed at the beginning of the cycle (BOC), and a control rod and fuel loading pattern is assumed. Then, the N-TH coupled core calculation is performed for one cycle based on the assumed core design with the burnup distribution. After the core calculation, the refueling according to the fuel loading pattern is followed by evaluation of the new burnup distribution at BOC of the next cycle. Such evaluation after one cycle calculation is repeated until the burnup distribution at BOC converges, at which time it is regarded as attaining the equilibrium core.

The core design criteria, such as shutdown margin, moderator density coef­ficient, Doppler coefficient, fuel cladding temperature, and maximum linear power density, are applied to the equilibrium core. If the equilibrium core does not satisfy the core design criteria, the equilibrium core design is performed again after reviewing fuel enrichment zoning, concentration and number of burnable poison rods, fuel loading and control rod pattern, coolant flow rate distribution, and so on.

A 3D N-TH coupled core calculation is also performed for fuel management such as fuel loading and reloading. In this case, the equilibrium core is searched from an initial loading core. Some fuel assemblies are replaced with new fresh ones after the first cycle calculation and the new core configuration is consid­ered in the next cycle calculation. The calculation procedure is also similar to Fig. 2.30.

It is noted again that operation management such as fuel loading pattern and reloading, and control rod insertion can be considered in the 3D N-TH coupled core calculation. Figure 2.31 shows an example of 3D core power distribution and coolant outlet temperature distribution.

In the core design and operation management, it is necessary to confirm that maximum linear power density (or maximum linear heat generation rate, MLHGR) is less than a limiting value. Figure 2.32 shows variation in MLHGR and maximum cladding surface temperature (MCST) with cycle burnup, which was from a 3D N-TH coupled core calculation for a supercritical water-cooled reactor (SCWR) as an example [26]. The limiting values at normal operation are 39 kW/m and 650 °C respectively in this example. The core calculation was performed according to the fuel loading and reloading pattern as described in Fig. 2.33. It is noted that the third cycle fuel assemblies are loaded in the core peripheral region to reduce neutron leakage (low leakage loading pattern). The control rod pattern is shown in Fig. 2.34.

The 3D N-TH coupled core calculation has been done for BWRs for many years. Since there are not many vapor bubbles in PWRs due to subcooled boiling, the N-TH coupled core calculation has not always been necessary, but it is being employed recently for accuracy improvement.

Outlet Temperature Distribution

Fuel Assembly

Fuel Loading Pattern

Power Distribution

Fig. 2.31 3D core power distribution and coolant outlet temperature distribution

Fig. 2.32 Example of Variation in maximum linear power density and maximum cladding surface temperature with cycle burnup

Fig. 2.33 Example of fuel loading and reloading pattern

In the core calculation, one fuel assembly is modeled not as a bundle of its constituent fuel rods but a homogenized assembly, which is represented as a single channel in the thermal-hydraulic calculation. However, the power of each fuel rod is actually a little different and hence it leads to a different coolant flow rate and a different coolant enthalpy in the flow path (subchannel) between fuel rods. This also causes a difference such as in cladding temperature which is particular to a single-phase flow cooled reactor. The heat transfer analysis for coolant flow paths formed by many fuel rods is called subchannel analysis, and it is used in evaluation of cladding temperature in liquid sodium-cooled fast reactors and so on.

In computational fluid dynamics (CFD), the behavior of fluid is evaluated through numerical analysis of the simultaneous equations: Navier-Stokes equations, continuity equation, and in some cases perhaps the energy conser­vation equation and state equation. It subsequently requires a huge computing time, but a detailed heat transfer analysis can be made. The CFD analysis can be used to calculate a circumferential distribution of fuel rod temperature in a single-phase flow cooled reactor or to analyze detailed fluid behavior inside the reactor vessel and pipes. Figure 2.35 depicts the three heat transfer analysis models mentioned above including the CFD analysis model.

Analysis

Fig. 2.35 Heat transfer analysis models

Commercial BWRs in Japan have had an operating experience of over 40 years since the Tsuruga Nuclear Power Plant Unit 1 was started in 1970. The core and fuel have been improved on the basis of accumulated operating experiences and tech­nological progress and Fig. 3.21 shows the history [20] of improvements in it.

In the beginning, there were various improvements related to security and reliability of fuel safety; these included enhanced moisture management in fuel fabrication, application of the Pre-Conditioning Interim Operating Management Recommendation (PCIOMR) [21], and reduction in the average linear heat gener­ation rate by using a larger number of fuel rods (changing from the 7 x 7 type to the 8 x 8 type lattice). The PCIOMR sets restrictions on reactor operation. These include limits on the linear heat generation rate for free operation of control rods to mitigate pellet clad interaction (PCI); a mild power increase by coolant flow rate control when the power is increased over the allowable linear heat generation rate; and keeping the power at a level for a fixed period after which the power may be freely changed within the level. It is very effective as a measure of the PCI, while it

Fig. 3.21 Improvement of BWR core and fuel in Japan

leads to a decrease in the reactor capacity factor. Core improvement was undertaken to actualize flattening of the core power distribution for the purposes of improving the capacity factor under the PCIOMR application and simplifying the reactor operation, thus securing fuel integrity. The development of improved cores such as the axially two-zoned core and control cell core considerably enlarged the core thermal margin, and improved the reactor operation and capacity factor by reducing the PCI load. Since the 1980s, high burnup and high economy cores have been developed to further raise the economy of nuclear power and to lighten the burden of fuel cycle, such as by lowering the amount of spent fuel and high level wastes. The development of the Pu-thermal core (MOX-fueled core) and its fuel have also proceeded to establish a fuel recycling by using plutonium in LWRs. Moreover, the operation cycle length has been prolonged to improve the capacity factor and an increase in the rated reactor power has been planned from the viewpoint of advanced use of existing power plants. It is important to continue to develop core and fuel corresponding to those improvements.

[1] High burnup and long operation cycle length

High burnup increases the total energy (discharge burnup x fuel inventory) produced from fuel assemblies from the time they are loaded into core until they are discharged. It can significantly extend the operating cycle length without increasing the number of fuel assemblies to be exchanged in refueling, and then improve the capacity factor, which helps to reduce power generation costs. Since high burnup also increases the total energy per fuel inventory and reduces the spent fuel amount per unit energy generation, it is possible to reduce

Operation Cycle Length [Months] 00 Cycle Burnup Bc

the reprocessing and waste disposal costs in the fuel cycle cost. The natural uranium and enrichment conversion cost in the fuel cycle cost can be reduced by decreasing natural uranium resources necessary for unit energy generation, namely, by improving fuel economy. The following describes the effect on fuel economy of the high burnup and long operation cycle length.

In the linear reactivity model based on the assumption that the reactivity decreases linearly with burnup, when it is considered that enriched fuel assem­blies loaded into core stay during n operation cycles and 1/n fuel assemblies are replaced in refueling, namely, the batch size is n, the cycle burnup Bc (n) and the discharge burnup Bd (n) can be given by [7]

Bc(n)=2xB0/(n+l) (3.19)

Bd (n) =nxBc (n) =2 nx B0/ (n + l) (3.20)

Bd{n)+Bc{n)=2B0 (3.21)

where B0 is the achievable discharge burnup of fresh fuel when it burns without replacement and it is a constant depending on fuel enrichment. In the case of a change in the refueling batch size from n1 to n2 to extend the operation cycle length, the discharge burnup change can be written by

ABd U) =Bd U2) — Bd Ui) = 2B0 U2 — n1)/(n2 + 1) Ui + 1) (3.22)

When the operation cycle length is extended by increasing the number of refueling fuel assemblies (n1 > n2) without changing the average enrichment (B0 is constant), the discharge burnup decreases and the fuel economy is compromised as shown in Fig. 3.22. A large B0 , namely, a high average enrichment is required to prolong the operation cycle length [large Bc (n)] by

increasing the number of refueling fuel assemblies while maintaining the discharge burnup (Bd (n) is constant).

Figure 3.23 shows the relation [8] between the average enrichment of a fuel assembly and the discharge burnup or the feed component which is the natural uranium resources necessary for fabrication of 1 kg of enriched uranium. The increment of the feed component by increasing the average enrichment is smaller than that of the discharge burnup. The increase of the average enrich­ment leads to a higher infinite multiplication factor of the fuel assembly, which can meet the critical condition in the core even after burning to a low infinite multiplication factor. In other words, the high burnup obtained by increasing the average enrichment can reduce the necessary natural uranium resources per unit energy generation.

[1] Characteristics of Power Distribution

PWR cores have the following features in power distribution;

• Enrichment zoning in fuel assembly is not necessary and enrichment is uniform, but there are some designs employing the enrichment zoning.

• The core radial power distribution can be flattened in the core design phase and no special adjustment is necessary during reactor operation.

• Only the core axial power distribution need to be adjusted during reactor operation.

These characteristics are attributed to the basic features of PWR core design which are no boiling in core, a canning-less fuel assembly (no channel box), cluster type control rods, and reactivity control by soluble boron concentration adjustment.

[2] Power Peaking Factor

The heat flux hot channel factor (Fq) and the nuclear enthalpy rise hot channel factor (FNh) are the basic design parameters concerned with the core power
distribution. Core radial and axial power distributions are flattened to meet the design limits of the parameters.

Nuclear Heat Flux Hot Channel Factor (Fq)

Nuclear hear flux hot channel factor is defined as the ratio of core maximum to average linear power density based on core design specifi­cations and given by the following factors

Fq =Ma. x{P(X, Y, Z)}xF$

P (X, Y,Z): the relative local power at position (X, Y,Z)

FU: the factor concerned with nuclear uncertainty (evaluated by a statisti­cal difference between calculations and measurements, usually about 1.05) The Engineering Heat Flux Hot Channel Factor (Fq)

The engineering heat flux hot channel factor is the factor to consider the effect of tolerances in fuel fabrication on the heat flux hot channel factor. Tolerances of pellet diameter, pellet density, enrichment, cladding thick­ness, etc are statistically combined to evaluate this factor. For example, it is 1.03 for uranium fuel and 1.04 for MOX fuel.

(iii) Heat Flux Hot Channel Factor (FQ)

The heat flux hot channel factor is defined as the ratio of the core maximum to average linear power density and given by Eq. (3.25).

TP — TpN v TpE

Fq — Fq X Fq

(iv) Heat Flux Hot Channel Factor (Fq(Z))at Core Height Z

The heat flux hot channel factor at core height Z is expressed as Eq. (3.26).

Fq (Z) = Max {P(X, Y, Z))xF"x

(v) Limits of Heat Flux Hot Channel Factor

Equation (3.27) is comprehensively used during normal operation and Eq. (3.28) is used for the limit of core detailed design and management.

Fqx P < 2.32

Fq (Z) xp< 2.32 XK(Z)

where P: The relative power

K(Z):the envelope curve function, for example, 2.32 x K(Z) in Fig. 3.54.

(vi) Definition and Limits of Nuclear Enthalpy Rise Hot Channel Factor The nuclear enthalpy rise hot channel factor is defined as the ratio of the maximum to average fuel rod power. It is limited in core design, for example, by relationships such as Eq. (3.29) which is used in evaluating DNBR.

Three Time-Burned Fuel

Twice-Burned Fuel

Once-Burned r uel

Fresh Fuel

Gd Gadolmia-Added Fuel

Fig. 3.47 Example of reload core fuel loading pattern with gadolinia-added fuel (Copyright Mitsubishi Heavy Industries, Ltd., 2014 all rights reserved)

F^h= 1.64{ 1 +0.3(1-P)} (3.29)

The fast reactor core consists of the core fuel assemblies, blanket fuel assem­blies for breeding, control rod assemblies for regulating power and reactivity, and radial shieldings. The heat generation of those components largely depends on the assembly type. Among the common assembly types, i. e. the core fuel assembly or blanket fuel assembly, the heat generation also depends on the loading position. In the thermal-hydraulic design of fast reactors, the coolant is adequately allocated among the assemblies in order to efficiently utilize the coolant led to the reactor and to satisfy the thermal design limits. As described in the list [1] of Sect. 4.1.7, the allowable thermal design limits consists of those for the fuel centerline temperature and for the fuel cladding temperature. The fuel centerline temperature is significantly influenced by the linear heat rate, while the fuel cladding temperature strongly depends on the coolant tempera­ture. Thus, the flow allocation is designed with consideration of the maximum cladding temperature design limit so that this temperature is almost equal among the core fuel assemblies and blanket fuel assemblies.

The high pressure plenum and low pressure plenum are provided inside the reactor vessel, and coolant is supplied from the high pressure plenum to the core fuel assemblies with high heat generation and from the low pressure plenum to the blanket fuel assemblies with low heat generation. Since the coolant flow direction of sodium cooled reactors is from the lower part to the upper part of the reactor vessel, the flow allocation device is installed in the bottom of the fuel

Symbol	Region
<D	Flow rate division 1
<2>	Flow rate division 2
<3>	Flow rate division 3	Inner core
<4>	Flow rate division 4
<5>	Flow rate division 5
<S>	Flow rate division 6
<z>	Flow rate division 7	Outer core
<8>	Flow rate division 8
<2>	Flow rate division 9
	Flow rate division 10	Radial blanket
<U>	Flow rate division 11
•	Control rod assembly
®	Neutron source assembly
О	Neutron shield etc

Fig. 4.9 Flow allocation of Monju [6]

assemblies. For example, the flow allocation device is installed in the connecting tubes which are between the upper and lower core support-plates. The entrance nozzles of the fuel assemblies are inserted into the connecting tubes. The designed flow rate is allocated by the combination of the flow allocation holes of the connecting tubes and the orifice diameter at the entrance nozzles.

An example of flow allocation design is shown in Fig. 4.9. In the flow allocation design, the necessary flow rates of the core and blanket fuel assemblies for satisfying the thermal design limits are calculated based on the evaluation of the cladding temperatures using the designed power distribution. The number of flow allocation regions is suppressed so that the flow allocation device is not complicated. To do that, the fuel assemblies are categorized as several groups. Within each group, the required flow rate is similar among the assemblies. Then, the flow rate for each group is determined. The flow allocation device is designed so that the pressure drops from the lower plenum to the upper plenum are equalized among all the flow paths with the flow allocation which is required for satisfying the thermal design limits. In this process, the pressure drop char­acteristics of each core components and the bypass of coolant flow are consid­ered. The pressure drops at the flow allocation device and the core components are calculated using the pressure loss coefficients based on the experiments such as the water flow tests associated with the core internals, separate tests for the flow allocation device, and the simulated fuel assembly tests using water.