9.98. With increased vaporization in a coolant channel, the heated sur­face becomes intermittently exposed to patches of vapor. Since the heat — transfer coefficient decreases markedly when the surface is blanketed with vapor, the wall temperature rises correspondingly. Hence, the wall tem­perature may at first oscillate as the surface is alternately blanketed with vapor or liquid, but will then rise after the wall liquid is completely va­porized. Such behavior, characterized by a marked temperature rise of the heated surface during boiling, as a result of a change in the heat-transfer mechanism, is called a boiling crisis. The foregoing is actually an oversim­plified representation of a complex phenomenon. For example, a wall temperature excursion can occur at a sufficiently high heat flux as a result of bubble-layer blanketing near the surface when the core of the channel is liquid and below the saturation temperature. The term departure from nucleate boiling (or DNB) (§9.91) is often applied to such a case as well as to surface overheating when the bulk of the fluid is at the saturation temperature. Dryout occurs in a channel with a vapor core and annular liquid film when the liquid is evaporated and the heat-transfer coefficient is reduced to that for a vapor in forced convection. Critical heat flux is often used as a generic term to cover all of these boiling-crisis possibilities.

9.99. A key critical heat flux parameter is the quality of the vapor-liquid system; it is defined as the mass fraction of vapor present in the mixture. The quality x may be expressed in thermodynamic terms by

H — Hf X = —77—>

Hfg

where H is the enthalpy of the mixture at the point in the coolant channel of interest, Hfis the enthalpy of the saturated liquid at the applied pressure, and Hfg is the enthalpy of vaporization at this pressure. Application of this equation to the subcooled region yields a negative quality, which is used in some correlations for predicting the critical heat flux under such con­ditions. The local quality for use in correlations must be calculated by an enthalpy balance.

9.100. In PWRs, the water is primarily in the subcooled or low-quality region. Here, a boiling crisis is represented by the DNB condition. It occurs only at a relatively high heat flux. In addition to the heat flux, the onset of DNB depends on such parameters as flow rate, quality of the fluid, system pressure, and increase in enthalpy of the coolant as it passes through the core.

9.101. In BWRs, where the quality is higher and the coolant is saturated, the boiling crisis of concern is dryout. In this case, where a dry patch forms by disruption of the annular liquid film, the wall-temperature excursion tends to be modest and slow when compared with subcooled-system ex­cursions. Since the vapor velocity is high, the heat-transfer coefficient from the surface to steam tends to be large enough to prevent the cladding from being damaged. However, failure could result from thermal cycling as the cladding surface is intermittently wetted and dried.

The Initial PWR Core and Subsequent Reload Patterns

10.23. The fuel loading pattern for the first core only for a typical Westinghouse four-cooling-loop plant containing 193 fuel assemblies (§13.9) is shown in Fig. 10.3. By means of this pattern, we can see some of the challenges associated with reload batch design. In this case, fuel assemblies of three different enrichments are used. The low and intermediate enrich­ment assembles are arranged in a checkerboard pattern in the central portion of the core while the high enrichment assemblies are placed about the periphery. This approach utilizes the reactivity differences between adjacent fuel assembles in the central region to flatten the power density distribution. Since the fresh, highly enriched fuel is placed in the region of lowest neutron importance, where the leakage is highest, its tendency to cause local power peaking is mitigated. However, fast-neutron leakage from peripheral fresh fuel assemblies leads to vessel embrittlement (§10.26).

10.24. The general modified scatter-loading pattern may then be con­tinued for subsequent burnup cycles by discharging after one burnup period the fuel of lowest enrichment and reinserting the remaining fuel in the central region in a modified checkboard pattern in accordance with design considerations to be discussed further in §10.43. Fresh fuel assemblies would be placed about the periphery and after the next burnup period, the fuel having the initial intermediate enrichment would be discharged and the procedure continued in a manner similar to that described above. Such an approach is also known as a modified out-in type of loading. Although in this idealized scheme, the discharged fuel will be that having

maximum burnup, this might not be so in actual practice when various feed enrichments may be used for special purposes.

10.25. After a number of burnup cycles, a pattern such as that shown in Fig. 10.4 for a one-eighth core might evolve. We can see that there have been substantial deviations from the initial core interior checkboard pattern as a result of changes in batch size and feed enrichment in earlier burnup cycles. In the figure which is based on calculations using a two-dimensional nodal model (§10.42), we show “region number,” which corresponds to batch number. Since three regions of different enrichment were loaded for the initial burnup cycle, the figure shows the reload pattern for Cycle 7. Also of interest is the beginning-of-cycle (BOC) relative assembly power and the BOC exposure in GW • d/t, where the average cycle burnup has been 10 GW • d/t. We see that there are significant differences in burnup among assemblies of the same region as a result of neutron flux variations during previous cycles. Therefore, the reload core designer searches for an arrangement that will yield a reasonably uniform burnup for assemblies that are eventually discharged from the core while staying within assembly power peaking constraints [5].

10.26. During recent years, several trends have affected PWR reload core design. As reactor vessels approach the end of their planned 40-year operating life and consideration is being given to extended service, it is desirable to minimize the fast-neutron fluence reaching the interior surface to reduce the increase in the brittle-to-ductile transition temperature (§7.12). This can be accomplished by so-called “low-leakage” fuel loading, in which fresh fuel is loaded in the central region and older fuel about the periphery. A loading that is partially of this nature is shown in Fig. 10.5. In this case,

extending burnup through a fourth cycle is also an objective. Some twice — burned assemblies are loaded toward the >utside. Since the neutron flux is intentionally lower toward the outside, we see that such assemblies experience relatively low burnup during the operating cycle. It is not easy to meet the low-leakage objective since the power peaking introduced by the centrally loaded fresh fuel must be carefully controlled by the use of burnable solid absorbers, which will be discussed shortly. Another advan­tage of the low-leakage concept is improved neutron economy which tends to lower the feed enrichment and, in turn, fuel cycle costs. Advanced fuel rod designs, such as those using axial blanket pellets of natural uranium oxide, have also been found to improve neutron economy.

10.27. A second trend is to extend the burnup cycle from approximately 12 months to 18 months, a step that yields operating economies by in­creasing plant availability (the fraction of calendar time that the plant is available for energy generation). However, to provide enough initial reac­tivity to operate for this longer period, it is necessary to raise the enrichment of the fresh fuel and use burnable solid absorbers to control power peaking.

10.28. Another trend is to extend the fuel assembly total burnup from perhaps three annual burnup periods to four. Fuel utilization is thereby improved and the number of discharged assemblies that must be managed is reduced. Again, in this case, some increase in feed enrichment is needed requiring burnable absorber use for power peaking control. These trends apply only to once-through cycles, as practiced in the United States, not when plutonium recycle is applied, as in Europe.

11.7. The atmospheric carbon dioxide concentration has been increasing at a greater rate during recent years than during the middle of this century, raising concerns that the balance between generation and removal rates may now be upset so that more than half of the carbon dioxide produced may be retained. In addition, some other gases, such as Freon, which result from human activity, tend to “block the atmospheric window,” similar to carbon dioxide.

11.8. Energy from the sun is primarily in the shortwave and visible portions of the spectrum, only a small fraction of which is absorbed or scattered back to space by the atmosphere. However, heat reradiated from the earth’s surface is mostly in the infrared portion of the spectrum, in which carbon dioxide, methane, nitrous oxide, ozone, Freon, and water have some absorption bands. Thus, the various complicated rate processes that establish a thermal equilibrium may tend to shift to a position favoring a warmer earth surface temperature. On the other hand, an increase in particulate matter in the atmosphere favors a cooling trend as a result of greater reflection of the incident radiation back into space. Green plants absorb carbon dioxide by photosynthesis but discharge it by respiration. Although modeling of these rate processes on a global scale is very difficult, it does appear that warm climatic periods favor carbon dioxide emission, thus introducing a positive feedback effect.

11.9. Since warming predictions regarding the rate of possible long­term global warming depend on the sophisticated modeling of many world­wide climatic and other processes, the results remain somewhat uncertain. However, a conservative approach would be to accept the possibility and to limit carbon dioxide emissions as much as possible and discourage further reductions of green plant growth areas.

12.20. In selecting a safe reactor design, an essential requirement is that the concept should provide inherent stability against an increase in reac­tivity. This can be realized if the reactor has a quick-acting negative tem­perature (or power) coefficient of reactivity. As seen in Chapter 5, there is then a self-limiting effect on disturbances that lead to an increase in temperature (or power level). This self-limiting feature is generally the result of the fuel Doppler coefficient, although in water-cooled reactors the expansion of the coolant-moderator contributes to the overall negative coefficient. However, the moderator contribution is delayed because of the time required for heat transfer from the fuel to the moderator (§5.92).

12.21. It should be understood that whereas a negative temperature (or power) coefficient is a requirement for reactor safety, it does not guarantee safety. If there were a sudden increase in reactivity, as discussed in §5.149 et seq., the power excursion would be terminated automatically after a finite time because of the negative temperature coefficient. But during this time the thermal power, and hence the fuel temperature, may have risen to such a high level that the fuel rods would suffer damage. This aspect of safety is taken into consideration in reactor system design. Also, should the system have a positive coolant void coefficient, as was the case in the Chernobyl 4 reactor (§12.187), the danger would be exacerbated.

12.121. The mechanics of particle movement through a fluid has re­ceived a great deal of engineering attention for many years. For example, scrubbers and devices for removing mists from chemical plants are common applications of fine particle dynamics. A first approach to describing par­ticle motion is to apply a force balance which includes an external force such as a gravitational force and a frictional or drag force. Thus, for a spherical particle having a diameter D and a density p falling through a fluid of density Pf with a velocity и, we can say that

Weight of sphere — Buoyant force = Drag force.

The drag force is expressed in terms of a drag coefficient CD according to the relation

Drag force = CD(ipfU2Ap),

where и is the constant settling velocity and AP is the projected area of the sphere, i. e., hrD2. Since the particle volume is zttD3, we have

(12.1)

In the particle diameter range of 3 to 100 pm, Stokes law applies and

24 _ 24jx

Re Dupf

Then simplifying, we have

For smaller particles, 0.1 to 3 (xm in diameter, an empirical correlation known as the Cunningham factor is applied to account for slip and free molecule effects. An average factor value of 1.8 may be assumed for es­timation purposes.

Example 12.2. Determine the settling velocity in air of a particle hav­ing a diameter of 10 jxm and a density of 3000 kg/m3. The air at 288 К and 1 atm has a viscosity of 1.8 x 10“5 Pa • s. Compare with the velocity of a particle having a diameter of 1 |xm.

gDp — P/) (9.81)(10 x 10“6)2(3000)

“ ~ 18(1 ~ 18(1.8 x 10“5)

= 9.1 x 10~3 m/s

For the l-|xm particle,

(1.8)(9.81)(1 x 1Q-6)2(30Q0)
18(1.8 x 10-5)

(A more accurate calculation yields 1.1 x 10 4 m/s.)

9.22. About 3 percent of the heat generated in the reactor system is produced in the moderator as the result of the slowing down of fission neutrons, the stopping of beta particles from the fission products, and the absorption of gamma rays from various sources. In light-water reactors, this presents no special cooling problem. Cooling requirements in other systems may be a function of the design. The need for cooling should also be recognized in the design of control elements. Heat is generated in such components outside the core as the thermal shield and reactor vessel in water reactors and the radiation shield in all reactors. This heat is a result of the absorption of escaping neutrons and gamma radiation. The subject of heat generation in shields has been treated in Chapter 6.

9.129.

An important characteristic of gaseous coolants, particularly from the safety viewpoint, is that the maximum flow rate is limited by the velocity of sound in the gas. This flow limitation applies to all compressible fluids, e. g., steam that may have been vaporized during the course of a coolant blowdown (§12.79) of a water-cooled reactor. The velocity of sound c in an ideal gas, equal to the limiting flow rate, can be expressed as

where у = cjcv, the ratio of the specific heats, R is the ideal molar gas constant, T is the absolute temperature, and M is the molecular weight.[12]

9.130.

In the flow of gas through a channel, the temperature changes as the pressure is reduced since adiabatic conditions can normally be as­sumed. Therefore, other, more complex relations must be used when con­sidering limiting flow in terms of the upstream conditions or when friction must be allowed for. However, as a rough guide, the critical (or limiting) flow conditions can generally be expressed in terms of the pressure; thus,

where 7 for helium, a monatomic gas, is 1.66 (§9.130), and the molecular weight is 4.0 x 10 “3 kg.

where Pc is the pressure at the critical flow condition and P0 is the upstream pressure. For monatomic gases at all temperatures (7 = 1.66) and for diatomic gases up to about 500°C (7 = 1.40), Pc/P0 is roughly 0.5. This means that when the downstream pressure of the gas is less than about half the upstream pressure, a limiting flow condition is likely to occur. In the case of compressible flow through a pipe, frictional effects reduce the limiting velocity, as to be expected.

9.131. Critical or “choked” flow also occurs in two-phase systems, but the analysis picture is complicated by the need for writing sets of defining conservation equations for each phase and expressing interfacial transport of mass, momentum, and energy. For example, in steam-water flow through a pipe, flashing of the water occurs toward the exit as the fluid encounters a decrease in pressure. Therefore, experimental measurements have an important role in developing predictive models useful for design. As in single-phase compressible flow, frictional effects reduce the maximum velocity.

10.60. As a result of the boiling process, the axial power shape is much more of an important parameter than is the case for PWRs. Control rod sequencing during burnup also requires attention. Therefore, three­dimensional calculations are necessary, even at the preliminary core design level. In any fuel design method, a key requirement is to evaluate safety margins for a candidate reloaded core. For a BWR, power peaking effects require evaluation for a variety of operating conditions, including various control element insertion scenarios. By contrast, an initial PWR evaluation need be made only at the hot, full-power, control elements out, condition. In many cases the cold shutdown margin is a limiting criterion for a BWR. Therefore, a methodology that would accurately assess a design in terms of all of these requirements at reasonable computing costs is needed.

10.61. Most modeling methods have been developed in-house by re­actor or fuel vendors. For example, the General Electric Company, the major BWR vendor, uses a three-dimensional BWR nodal simulator code called PANACEA, which also describes two-phase flow behavior. This serves as the basis for an interactive method for reload and control element pattern design using a coupled mainframe and minicomputer [15].

10.62. Among other nodal codes useful for BWR reload core design, various versions of SIMULATE developed for the Electric Power Research Institute are noteworthy. In Sweden, an integrated system of codes for both static analysis and optimization was developed by the ASEA-ATOM group called CORE MASTER. This system is based on POLCA, a three­dimensional, two-group, nodal model which includes a thermal-hydraulic description [16].

Introduction

11.44. A systematic approach to the management of spent fuel was provided in the Nuclear Waste Policy Act (NWPA), which became effective

Fig. 11.3. Heavy-isotope buildup in thorium. (Unless otherwise indicated, the nuclides are alpha-particle emitters.)

(n,7) (n,7) (n 7)

231 у 232 у —► 233 у ________ ► 234 у

(n,2n) a (n, 2n)

(n, 7) ^n’ ^ (n, 7)

231 Pa —- 232Pa ^ 233Pa ► 234Pa -­і (n,2n) . 1

0| , t 0| 0|

<n’7) (n, 7) (n, 7)

231Th^ 232Th — ~

(n, 2n)

in 1983 and was amended in 1987. Provisions were made for “character­izing” a permanent repository, and the need for interim storage was rec­ognized. The Act provided for the establishment of the Office of Civilian Radioactive Waste Management (OCRWM) to implement the program and a Nuclear Waste Fund to cover all storage and disposal costs. A charge of 1.0 mill/kWh of nuclear electricity is paid into the fund. This amounts to approximately $300 million per year.

11.45. A detailed timetable was specified which has not been met. For example, the law directs the Department of Energy (DOE) to begin ac­cepting spent fuel from utilities in 1998 to be placed in a monitored re­trievable storage (MRS) facility until an underground repository is available in 2010. Should the MRS facility not be ready to accept fuel in 1998, as appears likely, an alternative plan provides for the storage of spent-fuel assemblies at federal sites in standardized dry canisters. Since many aspects of the program remain uncertain, we will limit our discussion to the basic approaches planned. Let us first make the distinction between interim storage, monitored retrievable storage, and disposal. An interim storage facility provides short-term temporary storage to accommodate fuel assem­blies until more permanent facilities are available. A monitored retrievable storage facility is intended as an alternative to a geological repository for long-term storage. However, the assemblies could be retrieved for disposal elsewhere or perhaps for future reprocessing. In the disposal option, stor­age would be permanent with no possibility of retrieval.

12.68. Pressurized-Water Reactors. A small break or crack in one of the primary coolant loops in a PWR would be most serious if it occurred in one of the reactor vessel inlet lines. The reactor would be tripped, but some water would escape and be largely flashed into steam in the con­tainment structure. In order to prevent damage to the fuel cladding, it would be necessary to keep the core covered with water while normal shutdown cooling proceeds. If the break were quite small, the water level could be maintained by operating the pumps in the intact loops at full capacity. Makeup water to compensate for that lost from the break would be provided by the CVCS. However, it should be recognized that should there be a problem in introducing makeup water, it would be necessary to lower the pressure to make use of the passive high-pressure ECCS sub­system.

12.69. For larger breaks in a PWR inlet line, the reactor vessel water

level might drop temporarily to an extent that would permit some fuel cladding damage by overheating. As the system pressure is lowered by the escaping water, the high-pressure injection subsystem of the ECCS would operate and keep the core covered. The reactor shutdown cooling system would then be used to decrease the temperature and pressure. Again, the pressure status is important. The high-pressure ECCS subsystem has a limited capacity. If the leak rate is too small to reduce the pressure to the range of the low-pressure ECCS subsystem and the high-pressure capacity is depleted, a cooling problem could occur. Some of these considerations became evident during the course of the Three Mile Island small-leak accident (§12.179).

12.70. A break in a PWR feedwater line, which is within the contain­ment, or in a steam line inside or outside the containment could result in the loss of all the water in that steam generator. The reactor would be tripped and the other steam generators would be able to provide sufficient cooling to permit safe shutdown. Since the radioactivity level in the steam from a PWR is very low, a break in the steam line outside the containment would not constitute a serious hazard.

12.71. Boiling-Water Reactors. If a relatively small break should occur in a feedwater or recirculation line, one of the auxiliary cooling systems, e. g., the reactor core isolation cooling system (§12.30), could maintain the water level in the reactor vessel. The reactor would be tripped and then be depressurized and cooled down in the normal manner. For larger breaks or those requiring the main steam-line isolation valves to be closed, water would be supplied from the suppression pool and from the condensate storage tanks. A break in a BWR main steam line would result in reactor trip and closure of the isolation valves to prevent escape of the radioactive steam from the containment. The core would be kept covered with water in the manner just described, and the system temperature and pressure would be decreased in the normal shutdown manner.