11.52. The hazards from deep geologic disposal of high-level radioactive wastes can best be evaluated by modeling the transport of radionuclides of concern through the barriers. In this model, consideration must be given to the decay taking place during the time required for transport. In other words, a risk model might assume that ground water would intrude upon the buried waste package; then dissolution of the various radionuclides into the water would be described. Finally, the various pathways by which the decaying radionuclides can reach people would be modeled.

11.53. One measure[19] of relative hazard from a given radionuclide is based on the concept of water-dilution rate. The calculated rate of future discharge of the radionuclide into the biosphere (“people environment”), after decay, when divided by the maximum permissible concentration of the specific radionuclide in drinking water, yields the water-dilution rate.

11.54. The starting point for the transport model is the dissolving of the radionuclides into the groundwater. However, ceramic fuel pellets discharged from water-cooled reactors are not only insoluble in water, but are encased in clad intended to resist dissolving. Such stored assemblies are then encased in a metal canister. As another option, high-level repro­cessed waste would be incorporated into a solid matrix, such as glass, then encased in a metal canister. For modeling purposes, it is customary to assume a constant level of solution over a period of 10,000 years, although some specific elements are so insoluble that little of such components will actually dissolve.

11.55. The rate at which the dissolved elements are transported through the geologic medium is considerably reduced by “sorption,” a combination of physical and chemical adsorption and absorption, with the soil or rock. However, the sorption tendency varies from element to element, resulting in a different mix leaving the medium than that entering. In general, the heavy elements, such as uranium and plutonium, are highly sorped, while lighter elements are not strongly affected. The very low migration of ura­nium and plutonium has been confirmed by actual experience with the debris from weapons tests and leaked waste from DOE facilities. Thus, the model to determine water-dilution rates for individual radionuclides must take into consideration the input rate from dissolution, the isolation distance, the groundwater velocity, a sorption equilibrium constant, the half-life, and the maximum permissible concentration in drinking water.

11.56. Groundwater is expected to move at a very low velocity in the neighborhood of a geological repository. Also, physical chemical solubility and mass transfer principles apply to the determination of the concentration of the radionuclide in the liquid. A lower concentration value results than if such principles are not considered.

11.57. Since a realistic model is complex and involves many assump­tions, it is premature to cite results. However, a “feeling” for relative risks has been obtained by comparing water dilution rates from buried high — level wastes with those from uranium ore, uranium mill tailings, and coal ash, using consistent modeling assumptions. The buried waste proved to be two orders of magnitude safer than the ore and mill tailings and slightly safer than the coal ash over a period of 1 million years. Of course, another comparison could be made with the lethal doses from various poisonous common chemicals of commerce (i. e., cyanides, arsenides, pesticide com­pounds, etc.) which are not subject to radioactive decay.

12.74. The sudden ejection of a control element (or control-rod cluster), as a result of a mechanical failure, is a conceivable but improbable accident. In most situations, the increase in reactivity (and thermal power) would be small and could be handled by the reactor protection system. For ex­ample, when a reactor is operating at full power, the control elements are all partly to fully withdrawn. Sudden withdrawal of one element would therefore result in only a minor reactivity excursion, at worst. At the other extreme, when a reactor is in a shutdown state, the reactivity is sufficiently negative to compensate for the ejection of a control element.

12.75. One design basis accident is the ejection of a control element from an operating reactor leading to a power excursion of sufficient mag­nitude to cause some damage to the fuel cladding. Observations made with test reactors have shown that the amount of damage resulting from the ejection of a control element would be governed mainly by the energy generated as a result of the excursion. This in turn depends on the reactivity worth of the ejected element and the power distribution attained by the remaining control-element pattern. The test data are used in assessing the consequences of control-element ejection accidents. Because of the very low probability of an ejection accident, limited fuel cladding damage is considered an acceptable consequence. For a plant design to be approved, analysis must show that several criteria are met which are intended to ensure that there is little possibility of fuel dispersal in the coolant, gross core lattice distortion, or severe shock waves.

12.177. A nuclear power reactor installation must also be able to with­stand the substantial wind and pressure loading of a significant tornado. Systems vital to the safety of the plant must be protected from missiles generated by the tornado. Plants located near rivers must be able to survive a postulated flood, called the “probable maximum flood.” The possibility must also be considered of flooding that could result from a hurricane or failure of an upstream dam.

8.9. Although not normally labeled as such, the reactor core is indeed a system, with such components as the fuel assemblies and control ele­ments. Here, we see an example of the need sometimes to define the boundaries of a system arbitrarily since the control elements could be included in the control system.

Reactor coolant system

8.10. Strictly speaking, the reactor coolant system includes those com­ponents associated with the flow of the primary coolant, such as the coolant pumps, pressurizer, and associated piping and valves. However, the reactor vessel and steam generators could also be included arbitrarily.

Instrumentation and control system

8.11. This system includes the following major subsystems:

1. Nuclear instrumentation that indicates power level

2. In-core instrumentation to provide nuclear flux distribution

3. Process instrumentation for nonnuclear measurements in steam system sup­ply components

4. Reactor protection system for initiating safety in response to abnormal conditions (§12.22)

5. Control room (§14.25)

Engineered safeguards system

8.12. Countermeasures in the event of an accident are provided for by this system. Included in this system are several subsystems, such as the safety injection systems, containment sprays, the emergency feedwater system, and other safety features (§12.33 et seq.).

Other systems

8.13. The Nuclear Steam Supply System also includes a number of other subsystems that we need not consider here, such as those for auxiliary fluids, containment, fuel handling, waste management, and chemical control.

9.70. When a fluid flows through a straight pipe at low velocity, the particles of fluid move in straight lines parallel to the axis of the pipe, without any appreciable radial motion. This is described as laminar, stream­line, or viscous flow, and the “velocity profile” is depicted diagrammatically in Fig. 9.11A; the lengths of the arrows indicate the relative magnitudes of the fluid velocity at various points across the pipe. The velocity distri­bution curve in a circular pipe is a parabola, and the average velocity is half the maximum at the center of the pipe.

9.71. One characteristic of laminar flow (in the x direction) is that it

FLOW FLOW

F-^fy, (932)

where F is the shearing force (or fluid friction) over an area A between two parallel layers of fluid flowing with different velocities и in a region where the velocity gradient perpendicular to the flow direction is du/dy; the symbol fju represents the absolute (or dynamic) viscosity of the fluid and is defined by equation (9.32). The dimensions of viscosity are seen to be mass/(length)(time), i. e., kg/m • s (or Pa • s) in SI units. In the cgs system, the unit of viscosity, expressed in g/(cm)(s), is called the poise, and it is in terms of this unit (or its hundredth part, called a centipoise) that viscosity values have frequently been calculated in the past.

9.72. Laminar flow may be distinguished from other types of flow by means of a dimensionless quantity called the Reynolds number (or mod­ulus); this is represented by Re and defined by

Pup

where D us the pipe diameter, и is the mean velocity of the fluid, p is its density, and |x is its viscosity. Experiments with many fluids have shown [5]

that, in general, flow is laminar, and Newton’s equation is obeyed in ducts of uniform cross section as long as Re is less than about 2100. The precise critical value of the Reynolds number depends, to some extent, on the flow conditions.

9.73. If the circumstances are such that Re exceeds about 4000 in such systems, the fluid motion is turbulent; this type of flow is characterized by the presence of numerous eddies which cause a radial motion of the fluid,

i. e., motion across the stream, in addition to the flow parallel to the pipe axis. Newton’s equation is then no longer valid.

9.74. The velocity profile for turbulent flow is shown in Fig. 9.1 IB. As indicated, three more-or-less distinct regions exist in a turbulent stream. First, there is a layer near the wall in which the flow is essentially laminar. This is followed by a transition (or buffer) zone in which some turbulence exists, and finally, around the pipe axis, there is the fully turbulent core. In the latter region, turbulence mixes the fluid so that the velocity in the axial direction changes less rapidly with radial distance than it does under laminar flow conditions. As a result, there is a marked flattening of the velocity profile; this flattening becomes more pronounced with increasing Reynolds number. The ratio of the mean flow velocity to the maximum ranges from about 0.75 to 0.81, as Re increases from 5000 to 100,000.

9.75. In the range between the Re values of 2100 and 4000, there is generally a transition from purely laminar flow to complete turbulence. Laminar behavior sometimes persists in this region or turbulence may increase, depending, among other factors, on entrance conditions, the occurrence of upstream turbulence, the roughness of the pipe, the presence of obstacles, and on pulsations in flow rate produced by the pump or some other component in the system.

9.76. The Reynolds number, as given by equation (9.33), is applicable only to pipes of circular cross section; for noncircular channels, such as the flow regions between fuel rods, long rectangular ducts, and annular spaces, a good approximation is obtained if D in equation (9.33) is replaced by the equivalent (or hydraulic) diameter De, defined by

Cross section of stream Wetted perimeter of duct ’

which becomes identical with the actual diameter for a circular pipe. The use of the equivalent diameter makes possible the application of relation­ships for circular channels to the prediction of heat-transfer coefficients, pressure drops, and burnout heat fluxes for noncircular channels, as will be seen in due course.

Example 9.5. Calculate the average Reynolds number in a PWR from the following data:

The coolant mass-flow rate (in kg/s) is equal to the fluid velocity и (in mis) multiplied by the fluid density (in kg/m3) and the flow area (in m2). To determine De, consider a unit cell containing quadrants of four ad­jacent fuel rods, i. e., the cell is effectively associated with a single rod. The stream cross section and wetted perimeter of the coolant flow channel per rod may then be obtained from the accompanying illustration. Thus,

T 0.0095 m

Stream cross section = (0.0126)2 — Ьт(0.0095)2 = 8.79 x 10-5 m2. Wetted perimeter = tt(0.0095) = 0.0298 m.

Hence,

_______ 1.83 x IQ4 (17)(17)(193)(8.79 x 10~5)

3730 kg/m2 • s.

A small flow at the periphery of the core fuel assemblies is neglected here.

Then, taking the viscosity of water at 311°C to be 8.8 x 10~5 Pa • s (kg/m • s) as given in the Appendix, it follows from equation (9.33) that

9.77. Although the use of the equivalent diameter is convenient for preliminary design calculations, certain limitations should be recognized. The theoretical validity of the equivalent diameter concept for generalized use has, in fact, been questioned. For example, for a duct having a cross section in the shape of an isosceles triangle, the turbulent flow friction for air was found to be 20 percent lower than the value calculated using the equivalent diameter [5]. Heat-transfer coefficients also varied considerably around the perimeter. Several correlation procedures have been proposed for ducts with noncircular cross sections, but additional experimental work is required for their verification.

10.6. Since natural uranium contains only about 0.7 percent of the ura­nium-235 isotope and commercial LWRs require fuel containing about 3 percent uranium-235, an enrichment step is necessary. Enrichment is dif­
ficult because the effectiveness of separation using possible physical meth­ods depends on the difference in molecular weights of isotopic UF6 com­pounds, which is slight. During World War II, the development of processes yielding uranium enriched to the weapons-grade level of about 93 percent uranium-235 was one of the major challenges of the overall effort. From this activity, a complex of plants using the gaseous diffusion process evolved which were used for many years for reactor fuel enrichment as well as to meet weapons requirements. More recently, gas centrifuge processes have become practical to meet reactor needs. From the viewpoint of the fuel manager, process details are secondary in importance to the materials and work requirements for the separation which are common to any process.

10.105. The annual cost of operating a nuclear power plant is obtained by summing the contributions from the three main categories considered: capital (or plant investment), operation and maintenance, and fuel. Table 10.2 lists some values that are given here primarily for orientation and to indicate relative contributions. A construction cost of $2000/kW(el) has been assumed based on the $1500/kW(el) goal for advanced reactors (§15.22), with $500/kW(el) added for conservatism. Since these contributions have

changed significantly in the past from year to year, the current literature should be consulted.

10.106. We see that capital costs constitute the major contribution to the annual cost of generating electricity. Therefore, such fixed-cost param­eters as construction time and the cost of money greatly affect the cost of the generated electricity over the life of the plant. Furthermore, once the plant is built, the annual fixed-cost charge cannot be reduced by operating economies. However, the plant capacity factor, the fraction of rated ca­pacity experienced (§10.44), has an important bearing on the fixed costs per unit of energy produced. Therefore, there is a major incentive to min­imize shutdown periods. Although fuel costs make a smaller contribution, they lend themselves to savings by efficient fuel management during the plant lifetime. Although less significant on a relative basis, operation and maintenance costs depend on efficient plant management.

11.113. The Federal Water Pollution Control Act (FWPCA) Amend­ments of 1972 state that all sources of pollution, including condenser cool­ant water, “shall require application of the best available technology eco­nomically achievable.” In order to meet this objective the U. S. Environmental Protection Agency (EPA) has proposed regulations for several effluents from steam-electric plants. As far as condenser water is concerned, the EPA concluded that “the best available technology” would be the use of cooling towers or other recirculation systems, e. g., ponds or canals. In effect, the EPA regulations preclude the use of one-through cooling for power plants of more than 500 MW(el) capacity.

11.114. The FWPCA Amendments, however, provide for exemptions from the regulations in special cases. If state requirements can be satisfied, once-through cooling may be permitted when it can be shown that the ecological disturbance would be minimal. Exemptions might also be granted when sufficient land is not available for cooling towers or where salt drift or water vapor plumes from the towers (§11.119 et seq.) would be a serious problem. Nevertheless, it is probable that most future nuclear power plants will have to employ some form of closed-cycle cooling for the condenser water.

11.115. In addition to complying with EPA regulations, as authorized by the FWPCA Amendments, thermal discharges must meet the standards set by individual states in accordance with the Water Quality Act of 1965. These standards, which vary from one state to another and often within the same state, are determined by the aquatic life forms to be protected and the normal seasonal temperatures within a water body. As a general rule, the state specifies maximum temperature increases and maximum permissible temperatures. Some states also specify maximum rates of tem­perature change to minimize the danger of heat shock or cold shock.

12.107. In considering the hazards of fission product release in an ac­cident sequence, a number of factors determine the importance of indi­vidual radionuclides.

1. The total core inventory of the isotope. This depends on the balance between formation from fission, destruction, and decay as discussed in §2.203 et seq.

2. Chemical properties determine reactions that may occur during the accident sequence which could affect transport of the isotope.

3. Physical properties and characteristics such as volatility, particle size, and deposition behavior require attention.

4. Radiological and biological importance. Such matters as postaccident iso­topic changes, biological uptake and half-life, and specific effects on organs are all relevant.

8.45. Once he or she has developed the methods needed to solve a given problem, the nuclear engineer frequently needs such specialized infor­mation as neutron cross sections, physical constants, material properties, etc. To meet this need, a number of centers have been established to maintain special collections termed data bases. In many cases, the infor­mation is stored in a computerized retrieval system that can be searched on-line by the end user. Another approach has been to collect formulations for properties and other information so that they can be made available on a suitable computer tape. Only several of the major data centers will be mentioned here. Other information sources are indicated elsewhere in the text, as appropriate.

8.46. The National Technical Information Service (NTIS) in Spring­field, Virginia maintains a very large collection of reports and other ma­terials resulting from activities of the federal government. Data bases, such as the Energy Data Base, are available for on-line searching.

8.47. The Technical Information Center at Oak Ridge, Tennessee, op­erated for the U. S. Department of Energy, is responsible for a wide variety of services, including abstracting of reports and publication of technical progress reviews. It prepares the Energy Data Base available from NTIS.

8.48. The earliest center, originally devoted primarily to nuclear cross sections, is the Brookhaven National Nuclear Data Center at Upton, New York. Data evaluation remains a major responsibility. Results are available through publications and computer tapes.

8.49. The Center for Information and Numerical Data Analysis and Synthesis (CINDAS) at West Lafayette, Indiana is an example of a data center providing engineering information. Published data describing ther­mophysical and other properties of many materials are collected and eval­uated. A data base with on-line interactive capability is maintained.