The coolant depressurizes to atmospheric pressure due to a pipe break in the case of a primary coolant depressurization accident of HTGRs, In such case, it is difficult to remove the residual heat by using the primary cooling system because the air entering from pipe break may oxide the graphite in the reactor. Therefore, the residual heat is conducted to the reactor vessel outer surface and is removed by the passive reactor vessel cooling system. This method has high inherent safety due to no use of active systems. However, the available reactor thermal power for removing the residual heat is about 200 MW. This is not large enough and not economical. From that background, the annular core in which fuel blocks at the core center region are replaced by graphite blocks was proposed for reducing the maximum fuel temperature which appeared in the core center as shown in Fig. 4.43 [55]. By taking advantage of its capability to enlarge the thermal power, originally proposed by K. Yamashita [55], the annular core with the thermal power of 600 MW was designed while keeping the inherent safety [56].

The calculated results of a primary coolant depressurization accident for that design are shown in Fig. 4.44 [57]. The maximum fuel temperature appearing 70 h after the initiation of the accident was calculated as 1,595 °C which is below the limit of 1,600 °C. Thus, it was ensured that the additional failure of the coated particle fuels does not occur.

To calculate the fuel temperature of the annular core during accidents, thermal analysis codes such as TAC-NC are used. The TAC-NC code considers radiation heat transfer in the cooling holes and the gap between fuel blocks, natural convec­tion in those spaces, etc.

As shown in Table 1.3, the dependence of n on the neutron temperature is small for 233U and 235U and it has an effect on the order of only 10_5AklklK on the moderator temperature coefficient. Meanwhile, 239Pu has a high dependence and a significant negative reactivity effect of about 10_4 Akl klK on the moderator temperature coefficient. For each individual fissile nuclide, the n value is given by

(1.74)

in which the moderator temperature coefficient is expressed as the differ­ence in neutron temperature dependence between the non-1/u fission factor and non-1/u absorption factor, like

ocj’m аостп ® UtI ^

Nuclear reactor design and analysis is done with knowledge and data from funda­mental fields such as nuclear reactor physics, nuclear reactor kinetics and plant control, nuclear thermal-hydraulic engineering, nuclear mechanical engineering, and nuclear safety engineering. There are various calculation models from simple to detailed, and their selection depends on the purpose. Simple models have the advantage of giving a better understanding of calculation results and physical phenomena, and easy improvement of calculation codes. With today’s high com­putational performance, it is common to use a calculation code based on detailed models from the viewpoint of generic programming. Since the accuracy and application range of calculation results depend on models and data, it is necessary to imagine the physical phenomena and comprehend the validity of the calculation results based on the fundamental knowledge concerning the calculation target.

The nuclear reactor calculation is classified broadly into the reactor core calcula­tion and the nuclear plant characteristics calculation. The former is done to clarify nuclear, thermal, or their composite properties. The latter is done to clarify dynamic and control properties, startup and stability, and safety by modeling pipes and valves of the coolant system, coolant pump, their control system, steam turbine and con­denser, etc. connected with the reactor pressure vessel as well as the reactor core.

The core radial power distribution can be flattened by properly designing the control rod insertion positions (the control rod pattern) and the fuel loading pattern.

Figure 3.19 compares two examples of control rod patterns in BWRs. While control rods are shallowly inserted for control of the axial power distribution, they are generally deeply inserted for control of the radial power distribution. Neutron absorption by control rods suppresses the power of adjacent fuel assemblies and has an effect on the power of surrounding fuel assemblies through a change in the neutron flux distribution. Thus, the control rods control the radial power distribution of the whole core. In the previous design which lacked the axially two-zoned fuel concept, the same control rod is not inserted
for a long period of time to avoid slow burning of the adjacent fuel and all control rods are alternatively inserted to give uniform fuel burning in the core. The insertion positions of the control rods (control rod pattern) are changed during the operation cycle.

Such a change in control rod pattern during reactor operation is performed generally at a power level lower than the rated one to avoid a rapid variation in fuel rod power due to the control rod operation. This was a factor which interfered with improvement in the reactor capacity factor. A control cell core [15] has been developed as a core in which such a change in the control rod pattern is unnecessary and the reactor operation is simple. Figure 3.19 compares [16] the control rod pattern of the control cell core with that of the previous core. In the control cell core [17], there are several control cell regions in which low enrichment or high burnup low reactivity fuel assemblies are arranged near control rods. The control rods in the control cells are deeply inserted during reactor operation, while the other control rods are fully withdrawn. Although the control rods in the control cells are withdrawn to compensate for a reactivity loss with burnup, it does not cause a large power peaking because the neighboring fuel assemblies also have a low reactivity. The effect on fuel integrity is therefore small. Thus, the elimination of shallow insertion of control rods by adoption of the axially two-zoned core, the small excess reactivity by a proper addition of burnable poisons into fuel rods, and the reduction of necessary control rods by the control cells lead to a simple reactor operation without a change in control rod pattern.

In addition to the control of core radial power distribution by control rods, an exchange of fuel loading location (fuel shuffling) can be carried out to flatten the core radial power distribution, considering proper loading location of fresh fuel assemblies and the number and burnup of burned fuel assemblies.

BWRs are generally designed to have a scatter-loading pattern in which fuel assemblies are regularly dispersed in the core depending on the burned cycle. Another choice is a low leakage loading pattern in which high burnup fuel assemblies are loaded in the outermost region of the core to reduce the neutron leakage and increase the core reactivity. Figure 3.20 shows an example of the fuel loading pattern at the equilibrium cycle of the 1,100 MWe BWR core. In the 4-batch equilibrium core, fuel assemblies are kept in almost the same location until the third cycle after being loaded, and then moved to the outer­most region of the core or the control cell region. This scatter-loading pattern can produce a self-flattening effect of the power distribution with burnup and minimize the fuel shuffling for fattening of the power distribution.

In early BWR designs, generally, one type of fuel assembly was loaded in the initial core and the core radial power distribution was mainly controlled by control rods. In recent designs, an equilibrium core is simulated and different enrichments of fuel assemblies are employed [18]. The low leakage loading pattern mentioned above is usually used for improvement in economy. The control cell consists of four low enrichment fuel assemblies and other fuel assemblies with different enrichments are dispersed in other regions. In some of

Fig. 3.20 Example of BWR fuel loading pattern (1,100 MWe plant core). (a) Equilibrium core (b), initial core

the practical intial core designs, high enrichment fuel assemblies are loaded into the outermost region of the core.

The reactivity controlled by soluble boron concentration adjustment is relatively slow temperature change from cold to hot, excess reactivity decline during an operating cycle, and FP concentration variation. The reactivity variations can be approximated as the following three items.

• Reactivity change from cold to hot temperature: about 6 %Ak/k

• Reactivity decline during operating cycle: about 10 %Ak/k (depends on cycle length)

• Xe reactivity: about 3 % Ak/k at equilibrium full power and maximum about 6 % Ak/k after shutdown.

Figure 3.44 shows dependence of critical boron concentration on cycle burnup at hot full power operation, which is monotonous, and therefore it is easy to manage the excess reactivity.

The Xe reactivity is important in core management. A reactivity varia­tion from the equilibrium Xe condition to after reactor shutdown is

described in the fourth graph of Fig. 3.38. 135Xe, which has a very large neutron absorption cross section, is produced mainly through fission!135I (half-life 6.7 h) ! 135Xe (half-life 9.2 h) ! 135Cs (half-life

2.6 x 106 years). The Xe reactivity temporarily increases by decay of 135I after reactor shutdown, and monotonously decreases after a peak at about 8 h and becomes almost zero after about 3 days. When the reactor re-starts up within 3 days after shutdown, it is necessary to evaluate such Xe reactivity and to predict a critical point, and then to adjust the soluble boron concentration.

The differential boron worth is shown in Fig. 3.45.

[1] Power distribution characteristics

Since neutron mean free path is long and no FPs have a high absorption cross section, local distortion of the power distribution is small. The core perfor­mance is improved by flattening the overall (not the local) power distribution. Considerations for flattening the power distribution are as follows.

(a) The core is divided into the inner core and the outer core. They have almost the same volume. The plutonium enrichment is higher in the outer core so that: the maximum linear heat rate is made equal between the two regions and the power distribution is flattened.

(b) The scattered batch refueling method is applied. In this method, the refueled position is equally scattered in the core.

(c) The coarse control rods and the fine control rods are driven so that abnormal power distribution does not occur. The coarse control rods and the fine control rods are equally withdrawn among each group (coarse or fine rod), respectively.

The regional power fractions change with the burnup. Accumulation of plutonium in the blanket region increases its power fraction. Examples of the regional power fractions are shown in Table 4.4.

[2] Stability of reactor power

A fast reactor core has a negative power reactivity coefficient mainly due to the Doppler effect. When a positive reactivity is inserted into the core, the reactor power and the temperature increase. Then the increase in the power is suppressed by the reactivity feedback and the power is stabilized.

In fast reactors, the high energy of neutrons makes the mean free path relatively long, so that local distortion of the neutron flux distribution is small. Since the FPs do not have large cross sections for the major energy range of fast reactors, consideration of xenon is not necessary unlike that in LWRs.

233 235 23Q 241

From an engineering viewpoint U, U, and 9Pu (and Pu) are fissile

nuclides and most nuclear reactors in operation or under planning use one of them. Among these fissile nuclides only 235U (0.72 % isotopic abundance in natural uranium) exists as a natural resource and the others are artificially produced in nuclear reactors through conversion of fertile nuclides: Th to U and U to

9Pu (and Pu). A burnup chain (nuclear transmutation and decay chain) includ­

ing those nuclides is shown in Fig. 1.1.

Let us derive the burnup equation of each nuclide from the burnup chain of uranium, presented by characteristic expressions which include terms for: atomic number density at position r and time t, N(r, t); absorption, capture, and (n, 2n)
reaction cross sections, aa, <rc, and <rn,2n respectively; decay constant, 1; and neutron flux, ф(~, t). The absorption cross section is the sum of the cross sections of all neutron absorption reactions such as fission, capture, and (n, 2n). For simplicity of calculation, the one-group neutron energy is assumed.

Recalling the volumetric reaction rate (reaction rate per unit time and volume) expressed by афN (XN in case of decay) and making a balanced relation between the production and destruction rates of a target nuclide, the burnup equations can be given by the following equations, where nuclides are identified by a two-digit superscript indicating the last digit in the atomic number and the last digit in the mass number, respectively.

ПДТ42

(1.11)

ot

,9 TV51

(1.12)

ot

The following approximation can be used for a short-lived nuclide which has a very short half-live relative to fuel burnup period in nuclear reactors. The short­lived nuclide decays immediately after it is produced and it reaches a balanced state between production and destruction like a radioactive equilibrium. If the time differentials of 237U, 239Pu, and 239Np are set to be zero, the burnup equations of

237 239

Np and Pu are represented by the next equations.

ПДГ49

(1.14)

ot

It is considered that the short-lived nuclides decay instantaneously in this burnup chain.

2.1.1 Fundamental Neutron Transport Equation

The collective behavior of neutrons in a reactor core is described by the neutron transport equation presented in Eq. (2.1) which is also referred to as the Boltzmann equation.

Y. Oka (ed.), Nuclear Reactor Design, An Advanced Course in Nuclear Engineering 2, DOI 10.1007/978-4-431-54898-0_2, © Authors 2014

— ~g~ф(г, Й, E, t)=S(r Й, E, t)—QeV(/)G’, Ї2, i£, О /о

2.1)

—> —> —> 4 7

—X*(r, 1£, t)<p(r, І2, 1£, 0

Here, S is the neutron source, £t is the macroscopic total cross section, and ф is the angular neutron flux being calculated. This equation represents the balance between gain and loss in the unit volume of neutrons that are characterized by a specific kinetic energy E (velocity u) and are traveling in a specific direction ~ at a time t and a position r. That is, the time change of the target neutrons (the first term in the LHS) is given by the balanced relation among: the gain of neutrons appearing from the neutron source S (the first in the RHS), the net loss of neutrons traveling (the second term in the RHS), and the loss of neutrons due to nuclear collisions (the third term in the RHS). It should be noted that the changes in angle and energy of neutrons are also included in the gain and loss.

The target neutrons are gained from three mechanisms: scattering, fission, and external neutron sources. Each gain is represented in

S(r, Й, E, t)=f0°°dE’finbs(r, E’^E, t)<j>(r, Ъ’, E t)d£2′

+ Z^-/0oodE’/inx(E)v’Lf(r, E E’, t)dQ’

—Sex(r, І2, E, £)

(2.2)

where £s and are the macroscopic scattering and fission cross sections, respec­tively. v is the average number of neutrons released per fission and the product vlf is treated as a production cross section. The cross sections are described in the next section.

The first term in the RHS of Eq. (2.2) is called the scattering source and it totals the number of the target neutrons scattering into E and ~~ from another energy E0 and

direction ~~ by integrating the number for E0 and ~~ . The second term is the fission source and it indicates that the neutrons produced by fission over the whole range of energies are distributed with the isotropic probability in direction (1/4n) and the probability x (E) in energy. x (E) is called the fission spectrum and it is dependent on the nuclide undergoing fission and the energy of incident neutrons. For instance, the fission spectrum in an enriched uranium-fueled LWR is well described by the function of Eq. (2.3).

%(E) = sinh>J2.29E exp (-£/0.965) (2.3)

Since x(E) is a probability distribution function, it is normalized so that

The third term in the RHS of Eq. (2.2), Sex, expresses the external neutron source for reactor startup which may be such species as 252Cf or Am-Be. Therefore, it is not used in the nuclear design calculation of a reactor in operation.

More details of the neutron transport equation are not handled here. If the cross section data (£t, £s, and v£f) in Eqs. (2.1) and (2.2) are provided, the angular neutron flux ф (r, £2, E, t), which depends on location, traveling direction, energy, and time, can be calculated by properly solving the equation.

The information on traveling direction of neutrons is finally unimportant in the nuclear design calculation. The scalar neutron flux integrated over the angle is rather meaningful.

(2.5)

3.1.1 Pressurized Water Reactors

Pressurized water reactors (PWRs) were originally designed to serve as nuclear submarine power plants and were commercialized to large-sized ones currently in operation. The nuclear propulsion allowed submarines to remain submerged without refueling for far longer than oil-fueled vessels. The Argonne National Laboratory (ANL) of the USA found out in the early research that a small reactor with enriched uranium and pressurized water was available to submarines. The development of a nuclear submarine was triggered by the Westinghouse Corpora­tion (WH) as the contractor of the United States navy in 1949. A land-based prototype nuclear propulsion reactor reached criticality at the National Reactor

Y. Oka (ed.), Nuclear Reactor Design, An Advanced Course in Nuclear Engineering 2, DOI 10.1007/978-4-431-54898-0_3, © Authors 2014

Testing Station (NRTS) in the Idaho desert in 1953. The world’s first nuclear — powered submarine, the USS Nautilus, was launched in 1955 and accomplished the first undersea voyage to the North Pole in summer, 1958.

The first prototype PWR for electric power generation was Shippingport Atomic Power Station built by the Atomic Energy Commission (AEC) and the WH, located near Pittsburgh, USA and operated from December 1957. The electric power was 60 MWe and supplied to the Pittsburgh city.

The first commercial PWR based on the technology, Yankee Rowe (185 MWe), began commercial operation in 1961. Connecticut Yankee (575 MWe) was ordered in 1963 and San Onofre (600 MWe) followed. In Japan, Mihama Unit 1 of the Kansai Electric Power Company started operation as the first Japanese PWR in 1970. The WH also developed a 20 MW-Saxton test reactor and applied to improvement in PWR.

The history of PWR technology is summarized in Fig. 3.1 and Table 3.1. For a higher power PWR, a scheme was devised in which the number of primary coolant loops increases as the power class was upgraded without increasing the capacity of the primary coolant pumps or steam generators: two loops for 600 MWe class, three loops for 800-900 MWe class, and four loops for 1,100 MWe class. Yankee Rowe reactor had a cruciform type control rod design which was inserted between fuel assemblies. Since Connecticut Yankee reactor, however, the current cluster type design for control rods has been employed in PWRs worldwide. Stainless steel was

PCCV prestressed concrete containment vessel, RCCV reinforced concrete containment vessel

used as fuel cladding material in the beginning and then zirconium alloy was developed and it is still used today. Fuel rods was made thinner in diameter and subsequently a larger number of fuel rods can be loaded into fuel assemblies. Improvements for achieving high burnup are still in progress by further by decreas­ing the fuel rod load.

The PWRs developed by the WH were introduced and improved in France and Germany (former West Germany). In the USA, the Babcock & Wilcox Corporation (B&W) and the Combustion Engineering Corporation (CE) also developed and built PWR design power plants. The CE-PWR design is similar to the WH-PWR one, but it has two steam generators even in a large class reactor. PWRs in Korea were based on the CE-PWR design. The PWR design developed in Russia, referred to as the VVER, features a hexagonal lattice fuel assembly, a hollow fuel, and a horizontal steam generator, and so on. China proceeds in parallel to independent PWR development based on the WH-PWR design and PWR construction by foreign companies from France and Russia.

Startup range neutron monitor (SRNM) and power range monitor (PRM) systems are separately used from startup to the full power condition in nuclear measurement of BWRs. Since neutron flux level varies by about nine orders of magnitude from startup to full power, two different types of monitors are employed corresponding to each detectable neutron flux level. Both types of monitors are installed in the core as shown in Fig. 3.3. Fission ionization chamber type monitors are used in the SRNM system as a fixed arrangement type one. In a 1,300 MWe-class ABWR plant, the PRM system consists of 208 local power range monitors (LPRMs) and traversing in-core probes (TIPs) for LPRM calibration. LPRMs are inserted into the neutron flux instrumenta­tion tubes in the gap between channel boxes at the corner of the fuel assemblies and four LPRMs are axially installed in one instrumentation tube. LPRMs are components of the average power range monitor (APRM) system and they average the power signals as well as detect the usual neutron flux distribution in the core at normal operation [3].

One PRM or LPRM is usually used for every 16 fuel assemblies as shown in Fig. 3.3 and it corresponds to one for four fuel assemblies in considering a quarter-symmetric core. LPRM measurements are spatially discrete informa­tion which is not intended to directly measure the fuel assembly and fuel rod power. The assembly power distribution in the entire core is evaluated in combination with the relation between measured values and adjacent fuel assembly powers and the 3D computational model. Operation parameters such as maximum linear heat generation rate and MCPR are regularly moni­tored. Such power monitoring is implemented by a process computer referred to as core performance calculation system. It is necessary to prepare input constants of the process computer for the power distribution calculation using LPRM measurements before reactor startup; this is an important core management task.

Because of recent improvements in computer performance, a 3D nuclear and thermal-hydraulic coupled calculation model in a core design code is generally used in the core performance calculation of the process computer. Results of the core performance calculations are corrected by measurements from LPRMs and TIPs in each location to more accurately monitor and predict the core power distribution. The variation in burnup and fuel materials of each fuel assembly is also calculated in the process computer.

Alternatively, follow-up calculations of reactor operation are usually performed every month by off-line computers in which the same core design codes are installed. In the computers, calculations from the core design and measurements during operation, and results from the follow-up calculations and operation data are compared to evaluate and confirm the reproducibility of core design codes in the actual operation. The off-line computers are also used to calculate predicted operation in the next cycle and if necessary to re-investigate the control rod patterns. Thus, highly accurate and quick opera­tion management based on the real operation data is possible.