9.110. At this point it is useful to review the concept of the macroscopic, or overall energy balance based on the first law of thermodynamics. For a control volume, we write

(Rate at which quantity enters volume element)

= (Rate at which quantity leaves volume element)

+ (Rate of quantity accumulation in volume element).

For example, a macroscopic energy balance on a volume element can be classically expressed as

A# + AZ + Aw2 = Q — Ws,

where H, Z, and и represent enthalpy, potential energy, and fluid velocity, respectively. Hence, и2 is the kinetic energy, Q is the heat added to the volume element, and Ws is the mechanical work done by the fluid, such as by means of a turbine. It is sometimes called shaft work.

9.111.

The pressure drop, or loss, in a flow system is evaluated as part of a mechanical energy balance over the system. This is often known as the extended Bernoulli equation or the mechanical energy equation. It can be derived from the equation in §9.110 as well as from a force balance on a volume element. In its simplified integral form for a unit mass in an incompressible fluid having a density p moving between two points, 1 and 2, at different heights Z with respect to a datum level, and with no shaft work or friction, the equation becomes

where p is the static pressure, и the velocity in the kinetic energy term, and g the acceleration of gravity. A term Ws may be added, expressing the work required per unit mass if a pump is included between the two points. Also, for a practical system, the term Ap^/p should be added de­scribing the friction loss in energy units per unit mass of fluid flowing. The following equation results, which can also serve as the basis for determining pumping power requirements in the system:

Therefore, the total pressure drop is the sum of terms arising from the difference in level, acceleration of the fluid, friction, and pump work. Since the kinetic energy term is usually negligible for a liquid, our review will concentrate on frictional effects.

9.112. The pressure drop Apf accompanying isothermal laminar flow at a mean velocity и is a cyclindrical pipe of diameter D and length L is the familiar Poiseuille equation

. 32 ixmL

which may be written as

where |x is the viscosity.[8] The subscript/is used in pf to indicate that the pressure loss described is due to fluid friction. The factor i! Dup on the right of equation (9.39) is the reciprocal of the Reynolds number, and so

This gives the pressure drop due to fluid friction for laminar flow in a straight pipe of circular cross section. It should be noted that replacement of D by the equivalent diameter De, as defined in §9.76, is not valid for laminar flow in channels which do not have circular cross sections, although it is a good approximation for turbulent flow (§9.115).

10.38. The modeling in reload core design has normally been carried out at different levels of computational sophistication depending on the accuracy required as well as the dimensional representation and geometric detail that are appropriate. However, as computing capability advances, it is likely that levels can be combined with desirable sophistication applied earlier. In the first level, in which feed enrichment, new assembly number (batch size), older fuel reinsertion, and new cycle length are to be deter­mined, a multicycle point reactor model is appropriate. At the next level, a more sophisticated model is needed to develop, by iteration, the detailed loading pattern that is likely to meet licensing requirements. For this pur­pose, a three-dimensional nodal simulator using course-mesh representa­tion (§4.86) is normally used. A number of trial calculations may be needed to assess the power peaking effects of candidate loading patterns. There­fore, some compromise between accuracy and calculational efficiency is appropriate. Core symmetry may be assumed and only a one-fourth or one-eighth core need be considered.

10.39. Finally, to assure compliance with design limits, a detailed fine — mesh model that has been “qualified” by the U. S. Nuclear Regulatory

Commission is used to validate the final reload design. Two-dimensional diffusion theory codes have been used for this purpose for some years, but more recently, comparable results have been obtained by sophisticated nodal codes such as SIMULATE [8], which incorporate three-dimensional fine-mesh pin power reconstruction techniques.

11.18. The applicant for a permit to construct (and later to operate) a nuclear power plant must include with his application calculations showing that the radiation doses to the general public from all pathways of exposure satisfy the criteria in 10 CFR Part 50, Appendix I. A general outline of these pathways is presented in Fig. 11.1; in addition, there may be a small exposure resulting from the transportation of spent fuel and solid radio­active wastes from the plant site.

11.19. External exposure arises mainly from immersion in the gaseous effluent (from a stack or roof vent) as diluted by the atmosphere and carried by the wind. Smaller exposures may result from swimming in a body of water into which the liquid effluent is discharged and from gamma rays emitted from land (or water) surfaces contaminated by the deposition of airborne radioactivity. The chief potential sources of internal exposure are inhalation of airborne radioactivity, including noble gases, radioiodines, and particulate matter, drinking water into which the effluent is discharged, eating fish caught in this water and, probably most important of all, drinking milk from cows pastured in areas close enough to the plant to be signifi­cantly contaminated by the deposition of radioiodines. Some iodines may also enter the human body directly by way of contaminated vegetation and fruits and in other ways, but the amounts are likely to be small.

11.20. Of the airborne iodine, from 10 to 70 percent is in the form of an organic compound, chiefly methyl iodine. This is produced by the in­teraction of elemental iodine and water (or steam) with carbon (in steel, oil, grease, etc.) in the presence of radiation. Methyl iodide is less easily removed than elemental iodine by the charcoal absorbers used to decrease the iodine content of the gaseous effluent; however, it is less readily de­posited on vegetation and so tends to reduce the thyroid dose from ra­dioiodines in milk. As a conservative measure, however, in calculating the thyroid dose, current practice is to assume that all the iodine released from the LWR plant is present in the elemental form.

11.21. In estimating the radiation dose from eating fish (including shell­fish), allowance must be made for the fact that certain nuclides may have a higher concentration in the fish than in the water in which they live. The difference is expressed by the biological accumulation (or bioaccumulation) factor, also called the concentration factor, which is the ratio of the activity of a given nuclide (in Bq or Ci) per kilogram of fish to the activity per 10~3 m3 of water. These factors are dependent on several variables, such as the species of fish, the elements normally present in the water, and the food habits of the fish, which vary with the season of the year. Furthermore, the bioaccumulation factor is not uniform throughout the fish, and the proper value to use for the present purpose is that of the edible portion. Approximate, conservative factors have been published for use in radiation dose calculations; they refer to fishes, crustaceans, and mullusks in fresh and salt water [4].

11.22. In addition to estimating the expected radiation dose from the nuclear plant to people, the applicant for a permit to build the plant must evaluate the radiological impact of the facility on biota other than man. The organisms to be included are those species of local flora and local and migratory fauna that are important in the plant vicinity. These species will, of course, depend on the locality. In making the dose estimates, the bioac­

cumulation factors for fishes are now those for the whole fish rather than for the edible portion. Appropriate factors must also be used for vegetation to determine the dose received by the plants themselves as well as by the animals that may eat them. Provided the radioactivity levels in the effluents from a nuclear facility can meet the criteria for maintaining radiation doses to the general population as low as is reasonably achievable, the radio­logical impact on biota other than man should be of little significance.

12.33. If a substantial break should occur in the primary coolant circuits of a water-cooled reactor, the system pressure would drop and the emer­gency core-cooling system (ECCS) would become operative. This system consists of a number of independent subsystems which would be actuated in sequence as depressurization proceeds. The subsystems introduce water into the core to provide cooling when the flow of primary coolant is sig­nificantly decreased or lost. The water is supplied from different sources, with larger volumes becoming available to flood the core during the later stages of depressurization. Redundancy of equipment and flow paths is provided to assure reliability of operation. Since the primary cooling sys­tems are different in PWRs an BWRs, so also are the ECCS subsystems. There are also variations among PWRs fabricated by different vendors and also among BWRs of different designs. The following descriptions may be regarded, however, as being fairly typical of current practice.

12.132. An important requirement for LWR reactor licensing is to de­termine that the emergency core cooling system design meets the criteria specified in 10 CFR 50.46 and summarized in §12.90. The features of the so-called “evaluation” computer models for the required demonstration are specified in Appendix К of 10 CFR 50. Prior to 1988, evaluation models were required to embody very conservative assumptions and hence pre­dicted results that were intentionally pessimistic, rather than representing the “best estimate” of the real conditions. In 1988, Appendix К was amended to permit more realistic modeling provided that the predictions could be compared with applicable experimental information and appropriate sen­sitivity studies performed. Correlations used in the modeling must be ap­proved and shown to be reasonably conservative. This change had the effect of increasing allowable core peaking factors, which, in turn, provides greater flexibility for reload core design. However, the owner of a nuclear power plant may wish to evaluate safety margins for management purposes using realistic models which may not be as conservative as those approved for licensing. Such an activity is known as safety margin basis evaluation. This “nondesign basis” type of evaluation is also useful in studying con­tainment response to severe accident scenarios (§12.95).

12.133. These models are based on complex system codes (§12.142) which calculate fuel and cladding temperatures (and other core properties, e. g., zirconium-water reaction) during the blowdown, refill, and reflood stages of a LOCA accompanied by operation of the ECCS. The computer programs must evaluate the various energy sources, provide for fuel di­mensional changes, and describe in detail the various heat-transfer and fluid-flow phenomena at various stages of the hypothetical accident.

12.134. The energy sources must include the stored energy in the fuel prior to reactor shutdown as a result of the loss of coolant, the fission heat generated subsequently, the radioactive decay of the fission (and neutron capture) products in the fuel rods, and the heat generated by the zirconium — water (-steam) reaction. Allowance must be made for energy stored in the reactor vessel walls, internal hardware, and piping. In PWRs, heat removal in the steam generators must also be considered.

12.135. Among the hydraulic parameters are the rate of flow of water through a postulated pipe break in the primary coolant circuit, rate of flow through the core and other system components, and the pressure and water level in the reactor vessel during the blowdown, refill, and reflood stages. Flow out of the pipe break in a PWR must be calculated for two separate periods: (1) immediately following the break when the water is still sub­cooled and consists of a single phase and (2) when the discharge consists of a two-phase (liquid water and steam) flow.

12.136. The types of fuel-coolant heat transfer to be considered include those during and after blowdown. During the blowdown phase, the critical heat flux (CHF) is attained; different heat-transfer correlations must be used in the pre-CHF and post-CHF stages. Following blowdown, especially during reflood, the heat-transfer coefficients (or correlations) are based on experimental data modified to assure that they are conservative. The NRC regulations require that no heat transfer be assumed during refill.

Infinite Slab

9.32. Although thermal problems in reactors generally involve internal sources, the first (and simplest) case to be considered is that of an infinite slab of finite thickness with no internal heat source. It is postulated that heat is transferred at both faces by convection and that the heat-transfer coefficients are ha and hb, respectively. The idealization of an infinite slab is a good approximation to real (finite) systems in which the transverse dimension is large compared with the thickness, e. g., the cladding of a parallel-plate type fuel element or the flat plate of a heat exchanger. In the following treatment, steady-state conditions with the thermal conduc­tivity independent of temperature will be postulated.

9.33. The characteristics of the physical system are shown at the left in Fig. 9.1, and the equivalent thermal-circuit diagram is given at the right.

I—— 1———-

0 L

PHYSICAL SYSTEM THERMAL CIRCUIT

Fig. 9.1. Heat conduction in an infinite slab with convection boundaries.

The total thermal resistance is the sum of three terms, namely, two (Rt and R3) for the convection resistances and the third (R2) for the conduction resistance of the slab; hence, if A is the heat-flow area, the total thermal resistance, R, is

The relationship between the heat-flow rate and the temperature difference bx = ta — tb, where ta and tb are the mixed-mean fluid temperatures at the two faces, is thus given by

q = t-£~Yk = VA(t. — tb)

or

The quantity U, defined by equation (9.10), is called the overall coefficient of heat transfer; it has the same dimensions as the heat-transfer coefficient.

9.34. These equations may be used to determine the steady-state dif­ference in temperature ta — tb between two fluids in a heat exchanger if the heat-flow rate q is given. Alternatively, if ta — tb is fixed, the corre­sponding value of q can be calculated for specified conditions. The indi­
vidual temperature differences, i. e., the two fluid-solid differences and that between the two faces of the slab, can be determined by the use of equations (9.9) and (9.7), respectively.

9.141. Early in the course of reactor development, the term hot-channel factors yas used to relate maximum to nominal design conditions. The hot — channel concept was based on the assumption that a reactor core, with solid fuel elements having coolant passages (or channels) between them, would have one channel in which a combination of variations in power density and dimensions result in a heat flux that is greater than that at any other point in the core. Although there is a continuing need to relate maximum to average conditions to develop safety margins, the concept of an actual hot-channel or hot-spot tends to be simplistic. Therefore, peaking factors are now generally used, although utilization of hot-channel con­tinues. In fact, we will use the term hot-channel as a convenient designation for the hypothetical vulnerable location in the core. In any case, the design approach to limiting conditions through the use of peaking factors must be described in licensing documentation for every reactor initial core as well as for subsequent reload cores and approved by the U. S. Nuclear Regulatory Commission (§10.39). These peaking factors are included in the plant’s Technical Specifications as limits that are not to be exceeded (§12.240).

9.142. For PWRs, the peaking factors of importance are the heat flux factor and the enthalpy rise factor. These are decomposed through the use of various subfactors. However, before considering such detail, let us con­sider the principles involved. A three-dimensional nodal nuclear calculation can yield a “map” of the core power distribution, a starting point for both the heat flux factor and the enthalpy rise factor. In fact, as we shall see, the nuclear enthalpy rise factor merely describes the radial power distri­bution. However, when it is combined with appropriate flow-related fac­tors, it also yields a limiting coolant condition which can be associated with the maximum heat flux in boiling crisis predictions. The need to determine

DNB design margins is a major reason for PWR peaking factor develop­ment. Other limits depend primarily on the heat flux (see §9.163 et seq.).

10.78. Fertile thorium-232, which can be converted to fissile uranium — 233 (§1.35), is an important energy resource as an alternative or supplement to natural uranium. Thorium mineral reserves are believed to be of the same order as those of uranium, but there has been little demand for thorium since reactor fuel cycles presently use uranium and are likely to continue to do so for some years. This preference is based primarily on economic considerations for LWRs [25]. However, as uranium supplies are depleted its price will increase, and there will be a greater incentive to utilize thorium instead. An additional incentive for thorium utilization is to reduce the risk of nuclear weapons proliferation (§10.80). Thorium resources appear ample for many years of projected use.

10.79. As shown in Table 10.1, the conversion (or breeding) potential (ті — 1) for uranium-233 at thermal-neutron energies is significantly higher than for either uranium-235 or plutonium-239. In fact, a system in which the uranium-233, formed from thorium-232 in a low neutron loss reactor, would be recycled has the potential for breeding or self sufficiency. For
reactors with poorer neutron economy, fissile atom makeup, proportional in amount to 1 — CR, would be required, but they would still have a favorable resource utilization. With the thorium-uranium-233 cycle, con­version ratios greater than in LWRs are readily achievable.

10.80. Some complications arise in the thorium-uranium-233 cycle that do not occur in the uranium-plutonium-239 cycle. For example, an inter­mediate stage in the production of uranium-233 from thorium-232 is protactinium-233 with a half-life of 27 days; the latter has relatively large cross sections for neutron absorption in both the thermal and resonance energy ranges. This absorption, which depends on the power density and the neutron spectrum, can have an important effect on the reactivity, although only a minor one on the thorium-to-uranium-233 conversion.

10.81. Another problem results from the presence of uranium-232, formed by (n, 2n) reactions with uranium-233 and with thorium-232 by way of thorium-231 and protactinium-231 and protactinium-232 (see Fig. 11.3). Some of the members of the uranium-232 decay chain are strong gamma — ray emitters, and hence remote handling and shielding are required for fabrication of uranium-233 fuels.

11.58. At Oklo, in Gabon, near the equator in West Africa, a nuclear fission reactor located in part of a rich uranium ore vein started sponta­neously about 1.8 billion years ago and ran for several hundred thousand years before shutting down by itself. It is estimated that about 15,000 MW • у of fission energy was produced, yielding about 6 tons of fission products and 2.5 tons of plutonium. Since both uranium-235 and uranium — 238 are radioactive with different half-lives, the relative decay tends to shift the isotopic ratio. At the time of the reactor operation, natural ura­nium contained about 3 percent uranium-235. The existence of ground­water in the ore veins of this tropical area probably led to just the correct conditions for criticality. The reactor probably operated at “slow simmer” and was self-regulating since too much power would tend to boil off the moderating groundwater. Presumably, operation finally ceased when the ore became depleted.

11.59. The most interesting lesson from the Oklo reactor operation is that most of the radioactive waste has remained at the reactor site for over 1 billion years despite the abundance of groundwater. Metals having a valence of 1 or 2 are soluble in water, and were leached away. However, isotopes of these elements have short half-lives and thus would soon be harmless. The elements of concern, particularly plutonium, were retained in the geologic formation and effectively immobilized. Therefore, it would seem that a repository with additional barriers should perform at least as well.

12.76. During the regular refueling operation, about once a year, the head of the reactor vessel is open to the containment. Care must then be taken to avoid damage to the spent fuel rods that would result in the release of fission products, especially iodine vapor and noble gases present in the gap between the fuel pellets and the cladding and in the gas plenum above the pellets (§7.168). Such damage could conceivably arise from overheating caused by inadvertent criticality or by greatly impaired ability to remove decay heat. Core criticality is prevented by detailed administrative pro­cedures for removal of control elements, and critical geometry of the with­drawn fuel-rod assemblies is avoided by proper spacing in storage racks. Adequate cooling for removal of decay heat (and radiation shielding) is provided by keeping the assemblies under water at all times. Furthermore, boric acid is added to the reactor water to ensure subcritical conditions.

12.77. Damage to a spent-fuel assembly could also occur if the assembly were dropped. The use of special gripping and handling devices makes this very improbable. Nevertheless, the Safety Analysis Reports are required to include a calculation of the radioactivity that might be released to the environment if a fuel-rod assembly suffered severe mechanical damage inside the containment vessel. Damage outside the containment might result from the assembly being dropped to the floor of the spent-fuel storage area. However, such an event is extremely unlikely since fuel is transferred from the reactor to the pool under water and not elevated above the pool.