9.24. The term conduction refers to the transfer of heat by molecular (and sometimes electronic) interaction without any accompanying mac­roscopic displacement of matter. The flow of heat by conduction is gov­erned by the relationship known as the Fourier equation, i. e.,

«’-4 (9-6)

where q is the rate (per unit time) at which heat is conducted in the x direction through a plane of area A normal to this direction, at a point where the temperature gradient is dtidx.[3] The quantity к, defined by equa­
tion (9.6), is the thermal conductivity. In SI units, q is expressed in J/s (or W), A in m2, and dt/dx in °C/m (or K/m); hence, the units of к are (W/m2)(m/K), usually written in the form W/m • K. In English units, к will be in Btu/(hr)(ft2)(°F/ft).

9.25. The thermal conductivity A: is a physical property of the medium through which the heat conduction occurs. For anisotropic substances, the value of A: is a function of direction; although methods are available for making allowance for such variations, they are ignored in most analytical solutions of conduction problems. The thermal conductivity is also tem­perature dependent and can generally be expressed in a power series; thus,

к = c0 + cxt + c2t2 + • • • ,

which, in many cases, may be approximated to the simple linear form

к = A0(l + at).

Where considerable accuracy is desirable (and possible), allowance must be made for the variation of thermal conductivity with temperature. But very frequently, especially when the temperature range is not great, к is taken to be constant. However, when uranium oxide is used as fuel, as it is in most power reactors, the temperature gradients are quite large, and allowance must be made for the variation of the thermal conductivity with temperature (§9.46). Some values of к of interest in reactor design are given in the Appendix.

9.26. Upon integration of equation (9.6) over the x direction, it is found, for unidirectional heat flow by conduction in a slab of constant cross section, with к independent of temperature, that

q = — kA h ~ *2, (9.7)

Xi — X2

where tx and t2 are the temperatures at two points whose coordinates are

and x2. The result means that the temperature gradient at a point, i. e., dt/dx, in the Fourier equation (9.6) may be replaced by the average gradient over any distance, i. e., by (tx — t2)/(x1 — x2).

9.27. If tx — t2 is replaced by Дt, the temperature difference, and x2 — Xi by L, the length of the heat-flow path, equation (9.7) upon rearrange­ment takes the form

_ At 4 ~ L/kA’

This expression is analogous to Ohm’s law, I = E/R; hence the quantity L/kA is often called the thermal resistance for a slab conductor. The analogy between conduction of heat and electricity is the basis of the thermal circuit concept which is very useful in solving heat-transfer problems. In general, the rate of heat flow q is equivalent to the current /; the temperature difference At is the analogue of the potential difference (or EMF) E; and the thermal resistance replaces the electrical resistance.

9.135. In the preceding sections, the thermal-hydraulic behavior of a coolant channel associated with a single fuel rod was considered. For re­actor core analysis, however, a more detailed treatment is required. For this purpose, subchannel analysis is commonly used. A subchannel may consist of the coolant flow “cell” associated with a single rod as considered in Example 9.5, or a cluster of such cells. A number of parallel interacting subchannels running the length of the fuel rods may be included in the treatment. In addition, each subchannel length is divided into a number of axial intervals or “control volumes.” The result of this subdivision is a two-dimensional network of thermally and hydraulically connected nodes (§9.59) in the axial and radial directions. Heat transfer occurs primarily by conduction and convection in the radial direction from fuel to coolant. Allowance is also made for cross flow and turbulent mixing between coolant subchannels.

9.136. Numerous subchannel design codes have been written for solving the thermal-hydraulic equations based on the conservation of mass, energy, and momentum for each nodal volume [19]. Solutions yield such quantities as the radial and axial variations in the fluid enthalpy and mass velocity.

The approach to burnout conditions can then be determined by means of an appropriate critical heat flux correlation.

9.137. For safety analysis purposes, it is important to couple the core with the remainder of the coolant system and to consider transient con­ditions. System codes, which extend the subchannel approach, have been developed to meet this need. Typical such codes are RETRAN [20], which was developed under Electric Power Research Institute sponsorship and RELAP [21]. The development effort has been supported by an experi­mental program designed to simulate accident conditions, and the use of operating plant data for verification. Further consideration of system codes for safety analysis will be given in Chapter 12.

10.67. The conversion ratio is applicable to thermal reactors in which the fuel is natural or slightly enriched uranium. In deriving an expression for the conversion ratio, the formation (and consumption) of plutonium — 241 will be neglected since it is relatively small, at least initially, except in reactors employing recycled plutonium.

10.68. In a thermal reactor, plutonium-239 is produced as a result of
the capture by uranium-238 of thermal and resonance neutrons, formed by the slowing down of fast neutrons generated in fission. The rate of production of fission neutrons from a given fissile species in a thermal flux ф is фМгатіє, where N is the concentration of fissile nuclei, ua is the thermal — neutron absorption cross section, y is the number of fast neutrons produced per neutron absorbed in fissile nuclei, and в is the fast-fission factor (§3.143).

10.69. Of the fast neutrons produced, a fraction of Pt reaches the res­onance energy region, where Рг is the nonleakage probability in slowing down into the resonance region. If p is the resonance escape probability (§3.110), the fraction 1 — p of the neutrons in the resonance region is captured by uranium-238 to form plutonium-239. Hence,

Rate of formation of Pu-239 = ф{А238ст238 + [єР^І — р)2(Мтатп)]}, (10.11)

where а238 is the capture cross section of uranium-238 for thermal neutrons. The two terms on the right give the rates of production of plutonium-239 from uranium-238 by capture of thermal and resonance neutrons, respec­tively; the summation in the second term includes both fissile species, uranium-235 and plutonium-239. Soon after reactor startup with uranium (not recycled) fuel, the plutonium-239 concentration is zero; the rate of destruction of fissile nuclei is then фА235а235. The initial conversion ratio, as defined by equation (10.10), can thus be represented by

10.70. As the reactor operates and plutonium is generated, fissions (and captures) occur in plutonium-239 as well as in uranium-235; this tends to decrease the conversion ratio, mainly because of the large capture cross section of plutonium-239 for thermal neutrons (Table 2.8). Furthermore, fission products and heavy nuclides, including plutonium-239, compete with uranium-238 for resonance neutrons; this also results in a decrease in the conversion ratio. However, a large value of the initial ratio generally in­dicates a high conversion efficiency throughout the operating period of the reactor. A large conversion ratio is desirable because the fissile plutonium — 239 formed slows the decrease in the overall reactivity as the uranium-235 is consumed. As a result, the fuel burnup is extended.

10.71. It is evident from equation (10.12) that the conversion ratio can be increased by decreasing the resonance escape probability, i. e., by in­
creasing the resonance capture. One way whereby this may be achieved is by hardening the neutron energy spectrum (§2.102) so that the flux in the resonance region is increased. In a water-cooled reactor, a decrease in the moderator-to-fuel ratio, e. g., by closer fuel spacing, causes the neutron spectrum to be hardened. On the other hand, neutron leakage is increased, and fuel of a higher enrichment is needed to maintain the burnup, i. e., ДГ235/ДГ238 js increase(j These two consequences of spectral hardening have opposite effects on the conversion ratio.

10.72. In commercial water-cooled reactors, the fuel spacing is such that the initial conversion ratio is approximately 0.6. Of the plutonium formed, somewhat more than half undergoes fission during the normal fuel lifetime, and roughly one-sixth is lost by neutron capture. About one-third of the heat produced in a commercial LWR arises from the fission of plutonium-239 and plutonium-241.

11.49. In the United States, plans for a geological repository are cen­tered on Yucca Mountain in Nevada. According to the NRPA, a compre­hensive 10-year program to establish site suitability is required, the start of which has been delayed as a result of objections to the site by the state of Nevada. Should the site be acceptable, an additional 8 years would be required for licensing and construction activities.

11.50. In a conceptual design for the repository, a storage region would be at a depth of 600 to 1200 m from the surface. Suitability “packaged” spent fuel or reprocessed vitrified high-level waste (§11.84) would be stored either in rooms or in holes drilled in the rock. Following a test period of about 50 years when the packages could be retrieved should problems arise, the rooms would be backfilled with excavated rock, which would provide an additional barrier. Since spent fuel is an energy resource should it be reprocessed, there is an argument for reviewing the disposal option during the 50-year retrievable period.

11.51. Containment of the waste is based on a multiple barrier ap­proach. The waste container itself is designed to have several barriers, including an outer stainless steel or copper jacket. If high-level reprocessed waste would be stored, it would be in the form of a solid, glasslike, inert material packaged in a metal container. The final, but extremely important barrier to radioisotope migration is the geologic medium itself, which will be considered further in the next section.

12.73. The response of the reactor system to a number of postulated transients and accidents must be evaluated as one requirement for licensing. As we will see later, the methods used are intentionally conservative and differ from those to be discussed later (§12.132), that are used to obtain a “realistic” prediction of the response. Such a licensing design basis eval­uation has evolved from what was commonly referred to once as design basis accident analysis. Since our present purpose is to identify the events and not the evaluation methodology, we will use the older term, design basis accidents. These are events of such low probability that they should not occur even once in an average reactor plant’s lifetime, but which would have serious consequences if they were not controlled. These accidents are analyzed to assure the effectiveness of the engineered safety features and to evaluate the acceptability of the proposed plant site. The most severe design basis accident is considered to be a complete (double-ended) rupture of a large pipe, ranging in diameter from 0.61 to 1.07 m, in the primary coolant circuit of a PWR or a similar break in a recirculation pump intake line of a BWR. This accident is commonly referred to as the loss-of-coolant accident; it will be considered in some detail later. Other design basis accidents will first be examined briefly. It should be noted that design basis accidents do not result in core damage if the engineered safety features operate properly. So-called “severe accidents” in which core damage does occur will be discussed later (§12.95).

Introduction

8.5. A nuclear power plant consists of a number of systems which in­teract with one another in various ways. Knowledge of these intersystem interactions is essential for the design and operation of the plant. The systems themselves represent applications of the principles described in the previous and subsequent chapters of this book. We believe that it is useful at this point to introduce the system concept as applied to nuclear power plants and to indicate some of the intersystem dependencies. As energy transport, fuel management, safety, and other topics are presented in subsequent chapters, the reader can then be alert to the integration of the subject matter in plant applications.

8.6. The term system may be defined as a group of interrelated elements, or subsystems, forming the “whole” in a regular, organized manner [2]. In engineering design, the concept lends itself to mathematical modeling in which the interrelationships can be expressed formally for purposes of design optimization. Nuclear engineers also use such a systems modeling approach for probabilistic risk analysis (§12.206 et seq.).

8.7. Just what constitutes a system or subsystem depends on the need for analysis of the interdependencies that may exist. A system may have

many subsystems or parts which, in turn, could have within them “sub­subsystems,” usually referred to merely as subsystems. Where does this subdivision stop? The key question is whether the needs of the problem at hand require further subdivision. If not, the constituent part is referred to as a component.

8.8. Unfortunately, the term system is often used loosely to refer to any collection of components even if the grouping is, in fact, a subsystem. For example, manufacturers’ literature for some pressurized-water reactors de­scribes the reactor coolant “system” as within the Nuclear Steam Supply System. Also described are auxiliary fluid systems and electrical, instru­mentation, and control systems. Systems for waste processing, fuel storage, and ventilation are also covered. To be consistent with such practice, we will refer to both specific systems and subsystems as “systems” but indicate their hierarchial relationship as appropriate. The nature of some of these systems will be reasonably clear based on the material already presented. Others will become clearer as subsequent chapters are read.

9.69. The heat flow from a fuel rod to the coolant proceeds through several resistances in series, the last of which is associated with the heat- transfer coefficient at the cladding-coolant interface. This resistance can vary over a wide range, depending on flow conditions and other param­eters, and so it is an important consideration both in the initial reactor design and in the analysis of the consequences of various accident possi­bilities. For example, an increase in resistance, corresponding to a decrease in the heat-transfer coefficient, would result in an increase in the temper­ature difference between the cladding and the bulk of the coolant if the heat flux were to remain constant. If the coolant temperature is not to change, the net result would be an increase in the temperature at the outer surface of the cladding, and ultimately in the temperature of the fuel, as can be seen from equation (9.21) et seq. Thus, an examination of the considerations that affect the convection heat transfer is important in re­actor design.

10.4. Uranium ores are found in many parts of the world, with the contained percentage of uranium varying widely. Rich ores contain about 2 percent uranium, chiefly in the form of oxide minerals. However, most commercially mined ore is of medium grade, in the range of 0.1 to 0.5 percent uranium in the form of minerals of complex composition. The practically recoverable grade depends on numerous economic factors, as is true for most ores.

10.5. Since medium-grade ores contain such a large fraction of inert material, a concentration operation is needed in a mill located close to the mining area. Typical operations utilize either chemical leaching or solvent extraction procedures. The product is yellow cake, containing about 80 percent of U3Os. The yellow cake is then shipped to a central purification facility where further chemical operations are used to remove impurities and produce either uranium dioxide or uranium hexafluoride feed for sub­sequent isotopic enrichment processes.

10.101. The exact accounting of fuel costs is complex, but it is helpful to consider each batch of fuel in the reactor core as representing the end product of several manufacturing operations. The fuel, therefore, has a certain value determined by the cost of the materials and the operations. This fuel “investment” would be reduced should the spent fuel have any salvage value, e. g., for its plutonium content, after allowing for the cost of the operations necessary to recover useful materials and the charges for waste management (§11.44 et seq.).

10.102. The fuel within the reactor can be considered as an investment; hence, it is necessary to apply carrying charges to its value while it is in so-called commercial use, i. e., while it is used to generate electricity which produces revenue to cover all the plant costs. These carrying charges are an added contribution to the fuel costs. Carrying charges are also applicable to the time spent on the various operations performed on the fuel prior to its use in the reactor.

10.103. Rather than express fuel costs on an annual basis, as is done for fixed costs, it is more meaningful to apportion the cost of each fuel batch among the kilowatt-hours of electricity generated while the batch is in use. Major contributions to the cost of a batch are the cost of the amount of uranium to be fed to the enrichment plant, the cost of separative work (§10.11), fuel fabrication, and waste management. Changes in each of these contributions, such as a shift in the market price for uranium ore, would have a corresponding effect on the batch cost. Similarly, changes in the fixed charge rate would affect carrying charges for the fuel in and out of the reactor. However, nonreactor carrying charges for each batch are nor­mally included as part of the cost of the operations involved. If desired, the fuel batch cost may be expressed on an annual basis by determining the batches used per year.

10.104. Since each reload batch represents a certain investment, it is desirable to obtain as much energy from the assemblies in a given batch as possible by “suitable” in-core fuel management. However, if the fresh fuel enrichment is increased to extend the burnup, power peaking and vessel neutron exposure design constraints require consideration. There­fore, both economic and engineering parameters are important in devel­oping a suitable strategy. It is true that capital costs tend to be larger than fuel costs. Hence, generating costs on a unit energy basis are not very sensitive to fuel cost savings. On the other hand, on an annual generating cost basis, capital costs, by their nature, are fixed, whereas fuel costs are not. Since fuel costs may amount to on the order of $80 million per year per typical large generating unit, there is a significant dollar incentive for optimum fuel management.

Condenser Cooling Requirements

11.108. In a steam-electric generating plant, only a fraction of the heat supplied, e. g., by burning a fossil fuel or by nuclear fission, is converted into electrical energy. The waste heat remaining in the exhaust steam from the turbine is removed by cooling water in a condenser and, at the same time, the steam is condensed to provide the feed water for the steam generator which may be a heat exchanger (in a PWR) or a boiler (in a BWR or fossil-fuel plant). A simplified representation of the turbine — condenser system of a steam-electric plant is shown in Fig. 11.8. The cooling water carrying the heat removed in the condenser represents the thermal discharge (or thermal effluent). The additional heat present in the thermal discharge must be dispersed in such a manner as to cause the least possible disturbance of the environment.

11.109, The thermal efficiency of a steam-electric plant is based on the fraction of the heat supplied by the fuel (or heat input) that is converted into electrical energy. A distinction must be made, however, between the gross and the net efficiency. The gross efficiency is based on the total generator output and includes the electricity used for operating auxiliary equipment within the plant. The net thermal efficiency, on the other hand, is based on the net (or busbar) output of the plant, i. e., the electrical energy available to a utility for sale. For LWRs of current design, the gross efficiency is about 34 percent and the net efficiency roughly 32.5 percent. Energy losses within the plant amount to only a few percent, and so most of the difference between the heat input and the heat equivalent of the total energy generated must be removed by the condenser water.

11.110. Modern coal-fired plants have gross thermal efficiencies of about 42 percent, although there are substantial variations among different units. In such plants, generally about 10 percent of the heat input escapes through the furnace stack, and some 10 percent of the gross output may be required

to operate plant auxiliary equipment. As summarized in Table 11.2, the heat removed by the condenser water in a coal-fired plant is approximately two-thirds of that in a typical LWR of the same net electrical capacity. This is why the thermal-discharge problem is of special importance for nuclear power generation. Fast breeder reactors and high-temperature gas — cooled reactors would produce steam at higher temperatures than do LWRs; the thermal discharge would then be comparable to those from modern coal-fired plants.

11.111. In most nuclear power plants, the difference between the inlet and outlet temperatures of the condenser cooling water ranges from about 8 to 20°C, with an average near 12°C. Assuming a temperature increase of 12°C, and bearing in mind that the specific heat of water is about 4.2 x 103 J/kg • K, it follows that for a typical lOOO-MW(el) LWR plant, the removal of 2030 MW (or 2030 x 106 J/s) of heat requires a condenser cooling-water flow rate of (2030 x 106)/(4.2 x 103)(12) ~ 40,000 kg/s or 40 m3/s (630,000 gal/min).

11.112. The simplest and most economical treatment of the thermal effluent is the “once-through” or open-cycle procedure. The cooling water is drawn from an adjacent natural water body, e. g., a river, lake, ocean, or estuary, passed through the condenser, and then discharged into the same water body. As a result, the temperature of the water is raised, especially in the vicinity of the discharge point. The possibility of using once-through cooling depends on the availability of an adequate water supply and the effect of the temperature increase on the ecological balance of the water body. If the disturbance of the ecosystem is not tolerable, other methods for treating the thermal discharge are necessary; these in­clude the use of cooling ponds or canals and cooling towers.