12.142. The RELAP class of codes developed at the Idaho National Engineering Laboratory (INEL) for the U. S. Nuclear Regulatory Com­mission is the nonproprietary workhorse for reactor safety thermal-hydraulic analysis for PWRs only [11]. TRAC codes developed by Los Alamos have a minimum level of modeling simplifications and therefore serve both for accident analysis and “benchmarking” or calibration of simpler modeling systems [12]. Both PWR and BWR versions are available. The Electric Power Research Institute has sponsored the RETRAN series of codes, which evolved from the RELAP effort. These are intended to provide electric utilities with a general-purpose tool for design and safety evaluation [13]. A feature is the continuing use of plant operating data for code calibration.

12.143. RELAP5/MOD2, is the result of a complete rewrite of the earlier RELAP4 to include nonhomogeneous and nonequilibrium features. It is applicable for modeling large and small break LOCAs, operational transients, transients in which the entire secondary system must be mod­eled, and system behavior simulation up to the point of core damage. The code includes a point reactor kinetics model (§5.7), a decay heat model,

and the capability of modeling various functions of the reactor control system.

12.144. A challenge in the development of any system modeling code is to manage the numerical solution techniques so that there is a reasonable balance between accurate representation and computing effort, particularly when one considers the physical complexity of turbulent two-phase flow. In the case of RELAP5/MOD2, modeling with workstation-level com­puters has become possible. Various improvements are being incorporated into a new version, RELAP5/MOD3 [14].

12.145. TRAC-PF1 is intended to model PWR LOCAs and transients while TRAC-BF1 fulfills a similar function for BWRs. RETRAN-02 models all types of LWR transients. They are similar in the general nature of simulation approach to the RELAP codes but differ in modeling details and solution methods. TRAC is a useful design tool. It, like other system codes, includes a number of computing modules that model behavior in individual components. TRAC was extensively used in the analysis of the Three Mile Island accident.

9.38. As a result of changes in the thermal motions of its constituent particles, which are a function of the temperature, every body emits energy

in the form of electromagnetic radiations over a range of wave lengths; this is called thermal radiation. The amount of energy carried by the ra­diation is unchanged as it passes through a vacuum and is largely unaffected by dry air and many other gases, with the notable exceptions of carbon dioxide and water vapor. But when the radiation falls on a solid body or in its passage through the last-mentioned gases, part or all of the energy is absorbed. The fraction of the incident radiation that is absorbed is called the absorptivity. An ideal material which absorbs all the radiation falling upon it, and thus has an absorptivity of unity, is designated a black body. The emissive power or thermal radiation flux, i. e., the energy radiated in unit time per unit area of a black body, is given by the Stefan-Boltzmann law, namely,

Thermal radiation flux = (тГ4,

where the constant a has the value of 5.68 x 10“8 W/m2 • K4. The ratio of the emissive power of an actual surface to that from a black body is referred to as the emissivity, and for a body in thermal equilibrium, the emissivity and absorptivity are equal (Kirchhoff’s law).

9.39. The emissivity (and absorptivity) of surfaces vary with the nature of the material and its temperature as well as with its physical condition, e. g., roughness, cleanliness, etc. For metals, the emissivity is relatively low; the values generally range from about 0.05 or less for a highly polished surface to 0.2 or 0.3 for a roughened surface. If the metal is covered with an oxide film, the emissivity is greatly increased. Nonmetals have fairly high emissivities, although, in contrast to metals, the values decrease with increasing temperature. The emissivity of graphite, for example, ap­proaches unity at temperatures up to about 800 К (530°C), so that it approximates a black body. It is therefore a good emitter and absorber of thermal radiation. When radiation falls on a body, the proportion which is not absorbed is partly transmitted, e. g., through a “transparent” material such as air, and partly reflected, e. g., by a polished metal surface.

9.40. If two surfaces at different temperatures are separated by a non­absorbing medium, there is an interchange of radiation between them since they both act as absorbers and emitters. However, the net result is the transfer of energy from the hotter to the colder surface, the rate of energy transfer, e. g., in watts, being given by

Я. r ~ ^1Є1,2°"(Ті — Tf),

where 7 and T2 are the absolute temperatures of the hotter and colder bodies, respectively, Ax is the surface area of the former, and г12 is an interchange factor which is related to the emissivities (or absorptivities) of

the two surfaces; if both bodies were black, i. e., perfect absorbers and emitters, e12 would be unity. Certain geometrical factors should be in­cluded in this expression, but they may be disregarded here.

9.41. Even though radiation does not transfer any appreciable amount of energy directly to the coolant in a reactor, it may do so indirectly. In gas-cooled reactors operating at high temperatures, heat is transferred by convection from the fuel to the coolant, which is generally helium gas. The latter is transparent to thermal radiation and so does not absorb any radiant energy directly. However, radiation is transferred through the gas to the moderator (graphite), and this then loses energy by convection to the gaseous coolant. Thus, radiative transfer provides a means for conveying heat from the fuel element to the gas in an indirect manner. Radiation may also be significant in the transfer of heat from one reactor component, such as a fuel element, to another component, thus leading to thermal gradients which must be taken into account in stress analysis.

9.146.

In a bare cylindrical or rectangular parallelepiped reactor core, the neutron flux has a cosine distribution in the axial direction (Table 3.2). The volumetric heat-source distribution in this (*) direction will thus be represented by a cosine function, provided the fuel enrichment is uniform along the length of each rod. Previously, the origin of the coordinates was chosen at the center of the core, but here, however, it is convenient to take the origin at the point where the coolant enters the channel (Fig. 9.20). The heat-source distribution along a particular channel, i. e., in the

axial (or flow) direction, can then be expressed in sinusoidal form; thus,

Q = бо sin—,

where Q is the volumetric heat-release rate at the point x, and Q0 is the maximum value at the center of the channel; L is the total channel length, assuming no reflector. Upon inserting this expression for Q into equation (9.45) and performing the integration, the result is

(9.50)

which gives the rise in temperature of the coolant as it flows through the channel.

9.147. The temperature of the coolant increases continuously along the channel, as shown in the bottom curve of Fig. 9.21; the bulk coolant temperature rise, represented by Дfc, is thus seen to be

2Q0V

TTWCp

It is of interest to note that this result can be derived in another manner. Since Q0 is the maximum value of the heat-release rate per unit volume, the average is 2<20/’П’ (cf. §9.16), and the total heat-release rate for the given channel is then 2Q0Vhr. In the steady state this must, of course, be equal to the rate at which heat is removed by the coolant, i. e., wcpAtc, so that equation (9.51) follows immediately.

9.148. The temperature difference between the solid and the fluid at any point is obtained from equations (9.48) and (9.49) as

Q0V. ttx

4-г — = /ЙГ“Т

which is seen to be analogous to equation (9.49) for the source distribution. The value of ts — tm, consequently, passes through a maximum when x = ViL, i. e., in the middle of the channel, as shown in the second curve of Fig. 9.21. This result is, of course, to be expected since at any point the fluid-solid temperature difference will be proportional to the radial heat — flow rate at that point, and this is a maximum in the center of the fuel channel.

9.149.

The value of ts — te, i. e., the fuel surface temperature at any point with respect to the value at the channel entrance, is obtained by adding equations (9.50) and (9.52), so that

It is apparent that ts — te must pass through a maximum at some point beyond the middle of the channel, as shown in the top curve of Fig. 9.21. This maximum represents the highest surface temperature of the fuel ele­ment. The point at which the maximum is attained is found by differen­tiating equation (9.53) with respect to x and setting the result equal to zero; thus,

L ttw’cA

*ma* = (9’54)

From this expression it is seen that the position of the maximum value of ts approaches the end of the channel, i. e., *max —» L, as w and cp decrease, and h and Ah increase.

9.150. The maximum surface temperature in the given channel can now be obtained by insertion of Jtmax into equation (9.53). Assuming the coolant — flow rate to be the same in all channels, the maximum fuel-element surface temperature occurs in the channel where the neutron flux is a maximum.

In this case, Q0 is the volumetric heat-release rate at the center of the reactor, and the corresponding acceptable maximum value of ts will affect the maximum reactor power.

9.51. It should be noted that the surface temperature ts is less than the temperature in the middle of the fuel element. The difference, represented by the temperature drop through the fuel region and the cladding, can be calculated by the methods described earlier (§9.47 et seq.). Because Q varies along the length of the fuel element, the point at which the surface temperature is a maximum will generally not be the same as that at which the interior temperature of the fuel has its maximum value.

9.152. For purposes of computation, it is convenient to represent ттх/ L by the symbol a, i. e.,

(9.55)

It can then be found from equation (9.54) that

where amax = TTJCmax/L. Utilizing this expression for tan amax, equation (9.53) can be reduced to

(ts)max — te = -%7(1 — sec amax). (9.57)

TTWCp

With these two equations, the value of the maximum surface temperature and its location can be readily evaluated.

9.153. This treatment is strictly applicable to a bare reactor. For a reflected reactor, the flux (or power) distribution is flatter than in an equivalent bare core, and the ratio of maximum to average specific power (or volumetric heat source) is lower, as shown in §9.17. One method of treating the problem of a reactor with a thick reflector is to assume a cosine source distribution, as in Fig. 9.21, but to suppose that the value of Q goes to zero at some distance, e. g., about one or two reflector diffusion lengths, beyond the boundary of the core. The integration of sin ttx/L, referred to in §9.146, then does not start from zero, but from x equal to roughly two diffusion lengths in the reflector. Of course, as a result of fuel burnup, the enrichment will no longer be axially uniform. The high-flux region will “flatten” and the axial variations described will change accordingly.

10.84. It is assumed that the diversion of weapons-usable material by isotopic separation would be difficult, whereas chemical separation of plu­tonium from uranium (or uranium from thorium) would be relatively simple in comparison, provided the radioactivity level is low. On the other hand, chemical separation from highly radioactive spent fuel is considered quite difficult because of the need for remotely controlled equipment. Hence, spent LWR fuel, with its high radioactivity and low fissile content (about 1.5 percent), is regarded as presenting little proliferation risk. Similarly, should fresh fuel be radioactive from some remaining recycled fission (or other) products, separation would be difficult, with the level of difficulty related to the gamma activity. Another approach to reducing the acces­sibility of weapons-usable material would be to confine to special security areas, perhaps under international control, fuel-cycle operations in which such materials are produced.

10.85. Since one of the products of the Purex process (§11.74 et seq.) consists largely of fissile plutonium isotopes, the decision was made in 1977 to defer commercial reprocessing of spent fuel in the United States. Fur­thermore, concern over the plutonium produced by fast breeder reactors led to a de-emphasis of efforts to commercialize such reactors. The una­vailability of plutonium that might have been recovered from spent fuel for reuse in LWRs and from fast breeder reactors would limit the energy that could be obtained from natural uranium. Consequently, attention has been paid to alternative fuel cycles which would improve resource utili­zation and also have an acceptable proliferation risk.

11.64. After the cooling period, the spent-fuel assemblies would be shipped in strong metal casks to a reprocessing plant where the fission products are removed and the uranium and plutonium are recovered. The method commonly used for this purpose is based on extraction by an organic solvent, and this requires the fuel material to be dissolved in nitric acid to form a solution of nitrates. A simplified flow sheet of the repro­cessing operations is given in Fig. 11.4. Each step is quite complex, but for the present purpose a brief overview will be adequate.

11.65. In the first (or “head-end”) stage, the fuel rod assemblies, either with or without removal of hardware, are chopped into sections from which the spent material is leached with hot nitric acid. This process is commonly referred to as chop-leach. The hulls of zircaloy (or other) cladding and

hardware that remain are subjected to a hot nitric acid soak to remove essentially all of the uranium and transuranic elements, i. e., elements of higher atomic number than uranium. The hardware and hulls form what are called TRU (for transuranium) wastes or alpha wastes, because they contain traces of alpha-emitting transuranium elements. These wastes have been buried temporarily in the past, but more permanent underground disposal is planned for the future.

12.90. To provide assurance that the ECCS is designed so that its op­eration can prevent a significant release of radioactivity to the environment, the NRC requires in 10 CFR 50.46 that, in a postulated design basis LOCA
in a light-water reactor (PWR or BWR), the following criteria should be met:

1. The calculated maximum fuel cladding temperature following the accident should not exceed 2200°F (1204°C).

2. The calculated total oxidation of the cladding, as a result of the interaction of the hot zircaloy with steam, shall nowhere exceed 0.17 times the total cladding thickness before oxidation. (The total oxidation is defined as the thickness of cladding that has been converted into oxide, assumed to be stoichiometric Zr02. A modified definition is used where the calculations indicate that swelling or rupture of the cladding is to be expected.)

3. The calculated total amount of hydrogen gas generated, by the chemical reaction of zirconium in the cladding with liquid water and steam, shall not exceed 1 percent of the hypothetical amount that would be generated if all the cladding material surrounding the fuel pellets, i. e., within the active core, were to react.

4. Calculated changes in the geometry, e. g., in fuel rod diameters and spacing, shall be such that the core remains amenable to cooling.

5. After successful initial operation of the ECCS, the calculated core temper­ature shall be maintained at an acceptably low value for the extended period of time required by the decay of the long-lived radioactivity remaining in the core.

12.183. Following the accident, major reviews were carried out which proposed many reforms covering both industry and the Nuclear Regulatory Commission (NRC), most of which have since been implemented. In the industrial area, improvements fall into four categories:

1. Plant performance review and personnel training

2. Operational regulations

3. Plant equipment improvements

4. Research

12.184. In the first category, the Institute of Nuclear Power Operations (INPO) was established to perform in-depth analysis of operating expe­rience and serve as a clearinghouse for information. Training and ac­creditation programs were established. Second-category changes include provision for additional technical staff on each shift and extensive im­provements in operational procedures.

12.185. The third category is concerned with hardware. Major changes were made in control rooms and system instrumentation. New venting in the primary system was provided, for example. Hundreds of changes were required for each plant at a cost in the range of hundreds of millions of dollars per plant. A measure of improvement has been a reduction in the number of significant events, defined by NRC as one that has a potential safety implication. Research in such areas as source terms and degraded core analysis was expanded.

12.186. All of the measures taken certainly improved the safety of op­erating reactors in the United States and certain other countries that adopted the same measures. The accident demonstrated the value of the contain­ment since the public was not affected by an almost worst-case accident. In addition, the “cleansing” of the fission product mix through the con­version of volatile iodine proved the need for extensive source term studies. Yet the fact that the accident was even possible seriously eroded the public image of nuclear power as an almost foolproof technology.

8.19. In the design of various parts of the Nuclear Steam Supply System (NSSS), the effect of one parameter on another must be considered. For example, in a PWR, the fuel rod diameter and spacing affect both the moderator/fuel ratio and the coolant pressure loss through the core (§9.110). Throughout the NSSS, there are numerous design parameters which form an interrelated matrix which must be addressed systematically during the design process in order to develop specifications. A simplified indication of the need for integrating various parametric contributions in the design process is shown in Fig. 8.1. To simplify the figure, we have omitted the equally important areas of safety design and economics-related design, which also interrelate with the three areas indicated.

Sensitivity Analysis and Design Parameter Interactions

8.20. After the system has been modeled and intersystem dependencies identified, it is desirable to carry out a sensitivity analysis. Assume that the system in question is described by input parameters that may be varied and by output functions which characterize the behavior of the system and might, in turn, serve as input to another system. We would like to know how sensitive the output functions are to adjustments to the input param­eters. Should some parameters have little effect on the output, the model could be simplified. Also, should one or more input parameters be found to have an overriding effect, such input parameters would be identified as being critical to the design. At any rate, the sensitivity analysis step provides insight to the workings of the system and is a decision tool.

9.84. For ordinary fluids, the principal mechanism of heat transport is by the effect of turbulence, as a result of which a “parcel” of fluid is rapidly moved from a region close to the hot wall into the main body of fluid. In liquid metals, however, thermal transport occurs mainly by molecular con­duction. Whereas this mechanism may provide 70 percent of the heat transfer for a liquid metal, it contributes only about 0.2 percent to heat transfer in water. This means that the laminar boundary thickness, which is important for ordinary liquids, is not significant for liquid metals, and heat-transfer relationships applicable to gases and nonmetallic liquids can­not be used.

9.85. The essential difference between the heat-transfer properties of liquid metals and ordinary fluids is illustrated by the temperature profiles in a heated tube shown in Fig. 9.12 [8]. In these curves, the approach of the fluid temperature to the tube-wall temperature is represented by the dimensionless quantity (tw — t)/(tw — t0), where tw and t0 are the wall and centerline temperatures, respectively. The abscissa is the ratio y/r0, where у is the distance from the wall at which the fluid temperature is t, and r0 is the tube radius. The Prandtl numbers are the parameters for the various curves, and the Reynolds number is 104 in all cases.

9.86. For Pr = 1, the velocity and temperature profiles are identical; most of the resistance to heat transfer occurs in the laminar sublayer and

in the buffer layer, and there is little further change in temperature as the center of the tube (ylr0 = 1) is approached. For liquid metals (Pr « 1), however, molecular conduction is so significant that there is a marked thermal gradient from the buffer layer boundary all the way to the center, much as would be observed for a solid rod (Pr = 0). The heat-transfer coefficient is normally based on a mixed-mean temperature obtained by integration of the thermal profile (§9.30). It can be seen that for fluids of low Prandtl number this temperature may be quite different from that at the centerline.

10.13. A commercial operating reactor requires refueling every year or so in accordance with plans made a year or more in advance. In-core fuel management is concerned with the design of such fuel reload packages. Generally, the objective is to minimize the utility’s energy generation cost while staying within a number of safety-related design constraints and materials limitations. However, we will see that the task is challenging, not only because of the large number of design parameters that require attention, but also as a result of changes in constraints made from year to year as the technology improves. Thus, earlier designs are useful only as general guides.

10.14. For the reload core design, a neutronic model is needed of each assembly covering the operating cycle from fuel loading to removal. Con­
tributing to the model are core thermal-hydraulics and reactivity effects. Also relevant is the operating economics of the electric utility generating system. Hence, the picture is complicated. Our purpose here is to provide an initial understanding of both the design challenge and current practice with emphasis given to light-water reactors, particularly the pressurized — water reactor concept.