12.113. The halogens are very reactive and tend to combine with such metallic fission product species as cesium and zirconium. Iodine is the primary element of interest since its concentration in the fission product mixture is about 100 times greater than that of bromine. Also, the cesium/ iodine concentration ratio in the fission products of about 10 to 1 favors the formation of cesium iodide.

12.114. During a loss-of-coolant accident, the reactor coolant system tends to have a reducing atmosphere. Hence, such strong oxidizing agents as the halogens are likely to form halides, which will be retained in the water present. For example, iodides transported into the containment with large amounts of water will tend to remain in the water as a nonvolatile component.

12.115. Since the release of iodine-131 to the atmosphere has tradi­tionally served as a measure of radiological risk (§12.209), the chemical behavior of various forms of iodine has been studied extensively. Should there be a hydrogen burn in the containment, volatile methyl iodide might form, but the conversion would only be about 0.03 percent of the reactor iodine inventory [7]. Whether or not this constitutes a hazard depends on the integrity of the containment.

9.13. As already noted, a large proportion of the energy from fission is available in the form of heat within a very short distance of the fission event. The total rate of heat generation is proportional to the fission rate, i. e., to 2^ф or ІУоуф, where oyis the microscopic fission cross section, ф is the neutron flux, and N is the number of fissile nuclei per unit volume of fuel. If N remains uniform throughout a particular reactor or reactor region, as an initial approximation, the thermal source function, expressed
as the power density, may be taken to be the same as the spatial distribution of the neutron flux.

9.14. The overall flux distribution in the reactor core can be calculated by the methods described in Chapter 3 and 4. For a reflected cylindrical reactor in which the fuel is distributed uniformly, the flux distribution in the core and reflector can be represented approximately by

Ф, /„ .л_ r TTZ ^ = ^2.405-) cos-,

where фтах is the flux at the center of the core where it is assumed to have its maximum value; J0 is the zero-order Bessel function of the first kind, r and z are the radial and vertical coordinates, as in Fig. 3.11, and R’ and Hf are the effective radius and height, respectively, of the reactor, including an allowance for the reflector.

9.15. The average neutron flux in the core is given by

1 f R Г1/2Н

Фау = p2„ ф2Ttr dr dz, (9.2)

ttR2H Jo J — i/2h

where R and H are the actual radius and height of the fuel-containing region. Upon substituting the expression for ф from equation (9.1) and performing the integration, the result is

ф_ 2rr’4-405£)

фтах " 2.405R2 ’ ttH

where Jx is the first-order Bessel function.

9.16. If the concentration of fissile material is uniform throughout the reactor core, the power density may be taken to be proportional to the neutron flux.[2] It then follows that the reciprocal of equation (9.3) is equal to Fmax/Pav? i e., the ratio of the maximum to the average power density. For a reflected cylindrical reactor, R/R’ and НІН’ may be taken as roughly 5/6, and then

^ax (reflected) — 2.4.

* av

For a bare reactor, i. e., one without a reflector, R/R’ and ШН’ are both unity; then equation (9.3), after inversion, yields

—™-a* fbare) = —^405— . Z = з 54 Pav } 2/1(2.405) 2

If in a bare cylindrical reactor the arrangement of fuel elements or control rods (or both) is such that the flux is uniform in the radial direction, the factor containing the Bessel function is unity, and Pmax/Pav is u/2, i. e.,

1.57. Similar calculations have been made for reactors of other shapes, and the results of some core geometries are given in Table 9.1. It may be noted that, because of the arguments presented in §3.54, based on the similarity of the curves in Fig. 3.12, a cosine distribution of the flux may be assumed in all cases, e. g., even in the radial direction of a cylindrical reactor.

9.123. For preliminary design purposes, the pressure drop problem can be made more manageable by applying one of two simplifications. In homogeneous models, the two phases are assumed to flow as a single phase mixture possessing mean flow properties and a suitable single-phase friction factor is developed to represent the two-phase flow. Separated-flow models consider the phases to be segregated into two separate streams, with each moving at a different velocity. Empirical correlations are then used to determine a so-called friction multiplier (described in the next paragraph) which is a function of flow parameters. The different models vary in so­phistication and may be specific to given flow regimes. Furthermore, models must be used with care since the behavior being modeled is inherently complex [15] with numerous simplifications necessary for a practical description.

10.51.

Although the general principles of core management for BWRs are similar to those for PWRs, the differences in core design have an important effect on the specific approaches used. Hence, a description of the core arrangement from the fuel assembly management viewpoint is helpful. An important feature is the placement of a group of four assemblies with each cruciform control element as shown in Fig. 10.6. Since BWR assemblies are about 40 percent smaller than PWR assemblies and the core

Number of Bundles Number of Control Blades

is larger, there are many more of them that require management, i. e., about 750 compared with about 190 for a PWR.

10.52.

In PWRs, all the fuel rods (about 260) in an assembly have the same enrichment, but this is not the case in BWRs in which the assemblies are smaller. In order to compensate for the water gap effect when the control blades are withdrawn, the fuel rods in the corners of each assembly have a low enrichment. In fact, there are four levels of enrichment in each assembly, as indicated in Fig. 10.7. The numbers 1, 2, 3, and 4 represent decreasing proportions of uranium-235 designed to even out the power density distribution. In the initial core, the enrichments range from about 1.7 to 2.1 percent, but in subsequent cores they are approximately from 2.5 to 3 percent. The number 5 shows the locations of two “water rods,” which contain water but no fuel. The introduction of extra moderator in this manner increases the neutron flux near the center of the assembly; the result is an increased and more uniform burnup. From two to five of the most highly enriched fuel rods include a few percent of gadolinium oxide (Gd203) to serve as a burnable poison (§5.197).

10.53. Soluble boron cannot be used in a BWR. Therefore, shim control is provided by partial insertion of control elements with resulting pertur­bation of the neutron flux around neighboring assemblies. Such reactivity control elements, in selected patterns, are generally initially inserted deep in the core. Therefore, control management is closely related to fuel man­agement. Since operation of the recirculating pumps can be used to adjust the reactor power within limits (§5.199), some movement by other sets of control elements is available for power shaping. The Haling principle (§10.49) provides a reference for planning.

10.54. BWR loading patterns follow low-leakage strategies, where either blanket or lowest reactivity fuel is located at the core periphery. According to one general strategy, fresh and once-burned fuel would then be mixed in interior positions except for the so-called control cell locations (Fig. 10.6). However, because of multienrichment fuel assembly loadings and the influence of control rod movement strategy, specific practical patterns tend to be complex and will not be represented here. Also, as in the case of PWR management, deviations from a given configuration are likely after a number of burnup cycles.

11.32. The number of assemblies discharged from a plant per year can be estimated from the average burnup, the mass of heavy metal per as­sembly, and the energy generated, as indicated in Example 11.1.

Example 11.1. Consider a typical lOOO-MW(el) PWR which discharges fuel assemblies with an average discharge burnup of 33,000 MW • d/t. If the thermodynamic efficiency is 32 percent, and each assembly contains 450 kg of total uranium, how many assemblies would be discharged an­nually? A plant factor of 0.7 may be assumed.

_ , , , [1000 MW(el)l(365)(0.7)

Thermal energy produced per year = a ——- = 7.98 x 105 MW • d.

Then

which may be rounded off to 54 assemblies per year.

Actually, the number discharged would depend on the core management scheme followed, but the approach is useful for estimating storage require­ments. For example, corresponding to the U. S. nuclear generating capacity of approximately 100,000 MW(el), about 5400 assemblies would be dis­charged per year before correcting for the BWR fraction of the assumed generating capacity. Of course, the adoption of longer operating cycles, which normally result in increased burnup, will reduce this estimate significantly.

11.33. On-site storage pool capacity prior to 1977, when reprocessing was deferred indefinitely in the United States, was designed to accom­modate approximately the number of assemblies that would be discharged in 5 years. Also, room in the pool must always be available to hold a full core should it become necessary to defuel the reactor. Even with the introduction of enhanced capacity measures, about one-half of the pres­ently operating nuclear units are expected to fill their existing storage capacity by 2000.

12.58. Coolant Temperature Decrease. A decrease in the temperature of the coolant entering the reactor can cause an increase in reactivity, and hence in the neutron flux and thermal power, as a consequence of the negative temperature coefficient. In a PWR, this could occur as a result of malfunction in one of several systems. For example, a faulty valve might cause the feedwater rate to a steam generator to increase, or failure of a water preheater could result in a decrease in the feedwater temperature. However, such steam-generator effects in a PWR would tend to produce only a slow change in the primary coolant temperature. Since the feedwater (turbine condensate) in a BWR is pumped directly to the reactor vessel, the response from similar situations would tend to be faster. In each case damage to the fuel cladding would be prevented by a reactor trip actuated by the increase in the neutron flux.

12.59. Inadvertent startup of a pump in an inactive PWR coolant loop could also lead to an increase in reactivity and in the thermal power. Similarly, an accidental speedup in a BWR recirculation pump would result in a power increase. Such transients would be corrected automatically by the control system or the reactor would be tripped.

12.60. Control Material Removal. An inadvertent reduction in the con­trol (poison) material in the core is another cause of an unplanned increase in the thermal power. Although the reactor design limits the rate at which control elements can be withdrawn, the consequences of an uncontrolled withdrawal resulting from a malfunction must be considered in the safety analysis. The effects depend on the withdrawal rate, the reactivity worth of the control element (§5.168), and the operating state of the reactor, e. g., startup or full power. During startup, special precautions are taken to prevent a too rapid increase in the reactivity, since this could lead to a potentially unsafe condition (§5.219). Here again, reactor trip would be initiated by the protection system should design limits be exceeded, re­gardless of the operating state of the reactor.

12.61. In a chemically-shimmed PWR (§5.187), an accidental decrease in the boric acid concentration would lead to a power increase. However, the response of the reactor power to boron dilution is slow, and numerous alarms and other indications would alert the reactor operator to take cor­rective action should an error be made in the manual dilution operation.

12.62. System Pressure Increase. The system pressure in a BWR would increase if a valve in the main steam line should close automatically, e. g., as a result of a sudden decrease in the turbine load (see also §12.65). An increase in pressure would decrease the steam-void volume, thereby pro­ducing an increase in reactivity. The thermal power would consequently rise, thus tending to cause a further increase in the system pressure. The reactor pressure-relief valve would then release excess steam to the suppression pool, and the reactor would be tripped before any damage could occur. Radioactivity present in the steam would be removed by the water in the pool and by the air circulation system in the drywell.

12.147. Severe accident modeling has received increased attention by the U. S. Nuclear Regulatory Commission since the Three Mile Island and Chernobyl accidents. Such modeling has helped to provide guidance for reactor design and to supplement risk analysis studies. Generally, individ­ual codes are used to describe the various steps in an accident sequence and coupled together. Since this is a rapidly changing field, and many of the codes in use differ significantly in their modeling approach, our purpose is merely to outline the functional requirements and cite some examples.

12.148. A major objective of the modeling is to describe the response of the containment building to a core melt accident in a light-water reactor. We are reminded that this building is the final barrier to fission product release to the environment. Therefore, it is necessary to describe, normally by coupled computer modeling, the stepwise progression of accident events

that lead to the buildup of containment pressure. An associated objective is to provide a basis for risk assessment studies (§12.208). Detailed me­chanistic code modules, developed in close connection with experimental programs, are used for benchmarking simplified codes used for source term quantification and risk assessment.

12.149. Typical code systems provide for input from a fission product inventory code such as ORIGIN to an in-vessel thermal-hydraulics model of the accident which may be initiated by such conditions as a station blackout sequence in which all power is lost. Following failure of the vessel, containment effects become important. A lumped parameter approach is followed both for the in-vessel and containment analyses. The respective volumes are divided into interconnected compartments or cells and mass, momentum, and energy conservations calculations are performed for each cell (§12.137).

12.150. Several different code module systems are available to model severe accident phenomena, with the pressure load on the containment generally as the first objective. The modeling approaches used vary in several ways, but a description of code details is beyond our scope. There­fore, we will merely identify several of the code packages used and indicate the “flavor” of the approaches followed. The MAAP code system was developed as part of an Industry Degraded Code Rulemaking Program (IDCOR) for predicting severe accident source terms [16]. The CONTAIN and MELCOR codes were developed for the NRC [17]. To determine containment pressurization, such in-vessel phenomena as blowdown and boil-off of the primary and secondary coolant must first be modeled. Next, the effects of zirconium oxidation, core heatup and meltdown, and debris relocation are described. Since the pressure load on the containment de­pends on both gases generated and energy produced, steam and hydrogen produced during debris-water interactions and molten core-concrete in­teractions are modeled. Energy production and removal as well as the impact of engineered safety features must be considered.

12.151. The modeling of fission product release to the environment from a severe accident is an essential feature of the evaluation of reactor risk (§12.231). In general, the same code systems described above for contain­ment loading evaluation are used to describe fission product behavior. For example, chemistry considerations are integrated into the MAAP code with the concentrations of 34 species followed. It is emphasized that over the years, numerous other codes have been developed to describe fission product transport up to containment failure. However, for emergency re­sponse planning, meteorological considerations must also be modeled. Codes such as CRAC2 [32] develop off-site dose probabilities by sampling local meteorological data for a Gaussian dispersion model (§12.160).

12.152. As a result of the large number of interrelated processes that require description, severe accident modeling tends to be inherently com­plex. Furthermore, it is necessary to use many simplifying assumptions in the representation of individual processes which may differ among code packages. The methodology used may also differ. Therefore, the overall results yielded by one code package may differ somewhat from the results obtained from another. Thus, users should study the details of a given modeling procedure and appreciate the confidence levels developed.

9.50. As shown in Fig. 9.6, fuel rod designs normally provide for a small annular gap between the uranium dioxide fuel pellet and the cladding. The gap is initially filled with helium, but during irradiation there is some dilution of the helium with fission-product gases. More important, how­ever, is the swelling and cracking of the pellet (§7.172) which tends to close the gap in a nonuniform manner. The interface heat-transfer problem therefore progresses from one involving conduction through the gas to a combination of conduction through the gas and across the interface be­tween two surfaces in partial contact. Also, the gap conductance changes with the power level and material effects related to fuel exposure (Chapter 7). Although analytical models have been developed based on the thermal conductance of a gas film of the thickness indicated by the roughness of the two surfaces, the results tend to be unreliable. Semiempirical computer models based on test data are therefore used for design purposes [2]. The gap conductance is also important in the calculation models required for safety evaluation (§12.128). For orientation purposes, the order of mag­nitude of the heat-transfer coefficient at the interface of irradiated oxide pellets and metallic cladding may be taken to be 6000 W/m2 • K.

Example 9.3. A PWR fuel rod (see Example 9.2), with pellets 8.19 mm in diameter, is clad with zircaloy 0.57 mm thick; the outer diameter of the clad rod is 9.5 mm. If the bulk (mixed-mean) coolant temperature is 315°C and the volumetric heat generation rate (power density) in the fuel is 3.20 x 108 W/m3, determine the temperature at the center of the fuel and at the outer surface of the cladding. The heat-transfer coefficient

CLADDING

at the cladding-coolant interface may be taken to be 34 kW/m2 • K. The thermal conductivity of the zircaloy is 13 W/m • K.

The temperature drop between the coolant and the cladding is obtained from the second term on the right of equation (9.22); thus,

Qa2

h tm ~ 2hb9

where

Q = 3.20 x 108 W/m3 a = 1/2 (8.19 mm) = 4.095 mm b = 1/2 (9.50 mm) = 4.75 mm

Hence,

_ _ (3.20 x 108)(0.004095)2 _

h tm ~ (2)(3.4 x 104)(0.00475) “

Since tm is 315°C, the cladding surface temperature is 315 + 17 ~ 332°C.

The temperature drop across the cladding is obtained from the first term on the right of equation (9.22); that is, where, as shown in Fig. 9.6, a’ is the inner radius of the cladding, i. e., 1/2(9.50) — 0.57 = 4.18 mm. Hence,

(3.20 x 108)(0.004095)2, 0.00475 h~k= (Ш) П 0^00418 = 26 C

(Use of the approximation for In (b/a’) in §9.36 footnote leads to essentially the same result.)

Next, consider the temperature drop At across the gas gap between the fuel and the cladding. This is given by an expression equivalent to the second term on the right of equation (9.22), except that b is replaced by a and h by the gap conductance. As stated in §9.50, the latter may be taken to be 6000 W/m2 • K; hence,

= (3.20 x 108)(0.004095)

2 ha (2) (6000)

The temperature drop across the fuel itself has been determined in Example 6.2 as 480°C; hence, the central fuel temperature is given by

t0 = 480 + 109 + 26 + 332 — 950°C.

This is an average temperature; it is expected that in some fuel rods the central temperature will be significantly higher. The results of the foregoing calculations are plotted in Fig. 9.7 with the central temperature value rounded off.

9.163. We have mentioned that a nuclear heat flux factor is developed from appropriate nuclear calculations. It can be defined as the ratio of the maximum local fuel rod linear power density to the average core-wide linear power density. This assumes that there are no deviations from man­ufacturing specifications in fuel pellets and rods, densification effects, or uncertainties in calculations and the core monitoring system. Therefore, it is necessary to use subfactors to account for these considerations.

9.164. Most subfactors used in reactor core design tend to fall into two classes. First, there are factors arising from random-type uncertainties, such as those resulting from manufacturing operations, and second, factors introduced to account for inadequate knowledge. Random uncertainties lend themselves to statistical treatment, but those based on inadequate knowledge, e. g., flow distribution and intermixing of the coolant, are gen­erally combined by multiplying them together. The multiplication of subfac­tors implies that the corresponding events will occur at the same time and at the same place in the core; this is a very conservative assumption. For example, subfactors accounting for flow and mixing uncertainties are design dependent and are not treated statistically. On the other hand, the statis­tical combination of subfactors is justified when there is a body of exper­imental or manufacturing data to provide meaningful information. How­ever, it is sometimes possible to establish arbitrary confidence limits for a given uncertainty and to treat the resulting subfactor statistically, as will be seen.

9.165.

Statistics theory provides procedures for the analysis and pre­diction of data which are widely used in the quality control of manufacturing processes. For the present purpose, a simple discussion is adequate. In considering a large number of measurements of a particular quantity, the dispersion of the measurements about a mean value is often expressed as the standard deviation a; this is the square root of the arithmetic mean of the squares of the deviations of each value Xt from the arithmetic mean

where n is the number of measurements. If the deviations are random in character, the measurements will exhibit a so-called normal (Gaussian) distribution which is symmetrical about the mean value. In such a distri­bution, it can be shown that 99.73 percent of the values will lie within the range of ±3a. In other words, there are only 2.7 chances in 1000 that the actual value will be outside the range. In view of this high probability, the ± 3cr variation from the average is taken as the basis for calculating en­gineering hot-channel subfactors.

9.166. In order to combine standard deviations arising from variations in several different quantities, e. g., ax, a2, °з> etc., to obtain the overall standard deviation a, the relationship

O’ = [(сгі)2 + (o"2)2 + (0‘s)2 + • • -]° 5

is used. A similar expression is, of course, applicable to the 3a values employed in the determination of engineering hot-channel factors. An application of this procedure will be given in the next section.

9.167. In obtaining the nuclear heat flux factor for a PWR, it is usually helpful to assume that the radial and axial power distribution are separable. The radial factor F%H, also called the enthalpy rise hot-channel factor, is effectively the maximum-to-average ratio of the total power generated (or coolant enthalpy increase) in a (vertical) flow channel to that for all chan­nels. The axial factor F* is the ratio of the maximum to average power density in the axial (vertical) direction; it is derived from calculations utilizing the principles in §9.146 et seq. with a more realistic distribu­tion—rather than the sinusoidal form—of the heat source. The product of the radial and axial factors gives the heat flux nuclear hot-spot (or peaking) factor, F%; thus,

FNq = F& x F*

It is the ratio of the maximum heat flux, i. e., at the hot spot in the reactor, to the core average based on ideal design parameters. This must be mul­tiplied by the engineering hot-channel factor to obtain the overall peaking factor for the heat flux.

9.168. The engineering hot-channel factor is a combination of a number of subfactors arising from variations in the density, diameter, and enrich­ment of the uranium dioxide fuel pellets contained in the fuel rod, and the diameter of the rod with its cladding. The engineering subfactors may be treated statistically since they are based on manufacturing quality control data. The combination of these subfactors, in terms of variations of ±3a from the average, then gives the heat flux engineering hot-channel factor

Fq. Some values of the engineering subfactors for a pressurized-water reactor and their statistical combination are quoted in Table 9.2; each subfactor represents the ratio of maximum-to-average heat flux associated with variations of ±3a in the indicated quantities.[16]

9.169. For the fuel rod considered in Table 9.2, a reasonable computed value for F%H is 1.55 and F? is also about 1.55; hence, the heat flux hot — channel nuclear factor is

Fq = F%H x F"

= 1.55 x 1.55 = 2.40.

The total heat flux hot-channel factor Fq is the product of the nuclear and engineering factors, so that

F = FN x FE

L q * q * q

= 2.40 x 1.04 = 2.50.

In a typical PWR, the average flux for the whole core is in the vicinity of 6 x 105 W/m2; hence, the peak heat flux at the “hottest” spot in the hot channel is (2.5)(6 x 105) = 1.5 x 106 W/m2.

10.87. The time value of money in any economic analysis can be most easily expressed in terms of interest. This is true even for equity funds where a “rate of return” is really meant. In the broad sense, therefore, interest can be considered money paid for the use of other money, either borrowed or part of the equity of a company. The interest concept must be applied whenever time is a variable whether or not interest is actually paid. This concept can be applied in various ways. For example, capital investments in equipment may not be made at the same time when the equipment is put into service or when “revenue” is produced as a result of the equipment purchased. Therefore interest charges must be added to the investment cost.