9.124. The problem of the two-phase pressure drop in a vertical channel is of particular importance in BWR design. One treatment involves the use of a friction-factor multiplier R, defined as the ratio of the two-phase friction-pressure gradient (dp/dL)2 to the single-phase value for the liquid phase alone (dp/dL)h i. e.,

s (dp/dL)2 ^ (Ap)2 (dp/dL)/ (Ap), ’

where (Дp)2 is the two-phase pressure drop in a given channel in which (Ap)i would be the pressure drop for the liquid phase alone. Values of R, obtained by semiempirical correlation procedures and graphical integration of local values with respect to length, are given in Fig. 9.17 as a function of the system pressure for steam ranging in exit quality from 1 to 100 percent [16]. This figure is applicable when the axial heat flux distribution is uniform; other curves have been determined for the situation in which the axial heat flux has a sinusoidal distribution.

10.55. In the control cell core (CCC) concept, shim control rod move­ment is limited to a fixed group of control elements that are surrounded by low power fuel. Each of these cruciform elements and its four sur­rounding fuel assemblies comprise a control cell. The design was developed to reduce power peaking, increase thermal margins, reduce pellet-cladding interaction tendencies (§7.172), and provide operational advantages. Since fresh fuel containing underdepleted burnable absorber is never adjacent to inserted control elements, spatial distortion of burnable poison absorp­tion is avoided.

10.56. With the many assemblies in a BWR core, a reduction in the fuel shuffling is a desirable goal. Therefore, an alternative scheme is to provide backup control cell locations that would serve on the next cycle without moving fuel. The control cell assemblies would then be those of low reactivity but not necessarily from the same batch. It is emphasized that advances in BWR fuel reloading practice are continually being made.

BWR Fuel Assembly Design Trends

10.57. BWR fuel assemblies using an 8 x 8 rod lattice, as described in Chapter 13, have been standard for some years. Barrier cladding (§7.172) is now generally used to avoid pellet-clad interaction problems. Other lattice options have become available. For example, a 9 x 9 rod lattice design is currently being used in some cores. Since the assembly overall dimensions are unchanged, the rods are of smaller diameter. One version features an interior water channel that displaces a 3 x 3 rod configuration. The increased neutron moderation in the assembly interior improves fuel utilization, particularly should long operating cycles be desired. Since there are more fueled rods in an assembly, the linear heat rate can be reduced, thus increasing operational flexibility [14]. Other design variations used in Europe also provide for a softer spectrum in the lattice interior. The use of axial natural uranium oxide blankets has also become relatively common.

11.34. As a result of delays in the off-site management of spent fuel assemblies, most utilities proceeded to expand their on-site storage capac­ity. Originally, spent assemblies were stored in open lattice racks sub­merged in the water pool. A spacing of about 0.15 m (6 in.) between assemblies was provided to assure that the reactivity was about 0.9.

11.35. To conserve space, in all plants open racks were replaced with high-density racks that feature assembly compartments enclosed with a neutron-absorbing boron carbide-aluminum matrix. Using such racks, fuel can be stored in about one-half the volume required for storage in standard racks.

11.36. Additional space savings can be accomplished by removing the fuel rods from the assembly and storing them in a submerged grid which allows closer spacing than that in the original assembly. Such a procedure is known as spent-fuel-rod consolidation. Various designs have also been proposed for dry cask storage of either assemblies or consolidated fuel on site, but separate from the reactor building. Properly shielded canisters, which are generally air-cooled, hold from 24 to 31 assemblies.

11.37. In evaluating the reactivity of spent-fuel storage facilities, it has been customary to use the properties of fresh fuel. This is necessary when providing for a core defueling. However, for independent storage facilities, burnup credit for the reactivity loss during reactor exposure may be con­sidered. Designs must satisfy the criteria in 10 CFR 72, Subpart F.

11.38. Another option is transshipment, which involves the shipment of spent fuel from one wet pool unit to another, generally within the same utility system, to relieve storage congestion. Since each utility can predict its spent fuel assembly storage needs many years in advance, appropriate management involving the various storage capacity enhancement measures as well as the expansion of on-site storage facilities is generally being implemented. Thus, a utility can store spent fuel on-site for the entire operating life of the plant, should it become necessary provided that li­censing specifications permit. [7].

12.63. A decrease in the effectiveness of the coolant, and a decrease in the critical heat flux, could occur from a variety of causes, including a decrease in the mean flow rate of the coolant through the core, a mismatch between feedwater and steam flows, a decrease in steam demand, or a decrease in the primary system pressure in a PWR.

12.64. Slowdown of one of the primary coolant pumps in a PWR would result in a core temperature rise; the reactor power would then tend to decrease as a result of the negative temperature coefficient. The control elements would be withdrawn automatically in order to restore the power level, and this would aggravate the situation by causing a further increase in the coolant temperature. Hence, should the coolant flow rate fall below a specified value or the coolant temperature become excessive, the pro­tection system would trip the reactor. On the other hand, a decrease in the recirculation rate in a BWR would cause an increase in the steam-void volume and a reduction in the thermal power and steam production rate. With the aid of the pressure-regulating system, the power would then stabilize at a new lower level.

12.65. The control system must regulate the feedwater flow to the steam generators in a PWR and to the reactor vessel in a BWR in accordance with the steam demand (or turbine load). Otherwise, an undesirable change in the water level would result and the reactor would be tripped. Should there be a rapid drop in the steam demand, provisions are made for the steam to bypass the turbine and to flow directly to the condenser. If an isolation valve in the main steam line of a BWR should close automatically, the system pressure would rise leading to the situation described in §12.62. In a PWR, a decrease in steam demand can result in an increase in tem­perature of the primary coolant water and an increase in the system pres­sure. The pressure would then be automatically reduced by operation of the pressurizer relief valve.

12.66. Such incidents, should they occur, are so-called significant op­erating events and must be reported to the Nuclear Regulatory Commission. Also, they are reported to and analyzed by the Institute for Nuclear Power Operations (§8.52) which maintains a data base useful for reactor safety improvement.

12.153. The approval of a proposed reactor site is part of the licensing process and is likely to involve applicable state agencies. However, several requirements are concerned with the effects of postulated accidents on the surrounding population as well as certain accidents that relate to the site itself, such as the role of a possible earthquake. Therefore, it is useful to discuss such matters at this point. However, specific requirements are subject to change. Therefore, other sources, such as applicable NRC Regulatory Guides (1.3 and 1.4), should be consulted for the current regulations.

9.51. In all the preceding examples it has been postulated for simplicity, since it is a good approximation, that the heat source is uniformly distrib­uted within a conductor. A situation will now be considered in which the heat source has an exponential distribution. This is sometimes the case in certain external reactor components, such as thermal shields (§6.90) and pressure vessels, where the heat generated is due to the absorption of gamma rays and the slowing down of neutrons. Although the heat-generation processes are actually quite complex, it may happen that, as the result of a combination of circumstances, the volumetric heat source can be rep­resented approximately by

Q = Qoe-^,

where jx is an effective linear attenuation coefficient (or macroscopic cross section) of the radiations (cf. §6.199).

9.52.

If the shield or pressure vessel is a sphere or cylinder of large dimensions or is rectangular in shape, it may be treated as a slab; equation (9.15) then takes the form

Provided к is constant, the general solution is

t = — Щ;е^ + CtX + C2. kiz

The boundary conditions are (see Fig. 9.8),

t = ti for x = 0 and t — t2 for x = L,

where L is the thickness of the slab. These lead to the solution

for the steady-state temperature distribution in the slab. It may be noted that the first term on the right represents the linear temperature distribution due only to the difference in temperature between the two faces of the slab, whereas the second gives the effect of the exponential heat source.

9.53.

Under certain conditions, the combination of the two terms just mentioned leads to a temperature maximum within the slab. The point at which this occurs may be obtained by setting the derivative of equation (9.24) with respect to x equal to zero. The result is

The maximum temperature can be obtained by inserting this value for x into equation (9.24).

Example 9.4. A water-cooled and water-moderated power reactor is contained within a thick-walled pressure vessel. This vessel is protected from excessive irradiation (and, thus, excessive thermal stress) by steel, cylindrical thermal shields between the reactor core and the pressure vessel. One of these shields, 50 mm thick, whose surfaces are both maintained at 300°C, receives a gamma-ray energy flux of 1018 MeV/m2 • s. Calculate the location and magnitude of the maximum temperature in this shield, which may be treated as a slab. The coefficient p, of the gamma radiation (§9.51) in the steel may be taken to be 27 m-1, and the thermal conductivity as 40 W/m • K.

In this case, tx and t2 are both 300°C, so that A — t2 is zero. Hence, by equation (9.25),

I l _ e — (27X0.05)

Ww = _27ln (27)(0.05)

= 0.022 m (22 mm).

The maximum temperature is thus attained slightly inside from the mid­radius of the shield.

The value of the maximum temperature is calculated from equation (9.24) with tx = 300°C, к = 40 W/m • K, and x = 0.022 m. The value of

Q0 is derived from §6.22 as фwhere |лд, the energy absorption coefficient, is 16.4 m”1 for steel. Since фyEy is 1018 MeV/m2 • s, it follows that

q0 = (1018)(16.4) = 1.64 x 1019 MeV/m3 • s = 2.63 MJ/m3 • s.

Hence, from equation (9.24),

and tmax is consequently 311°C.

9.170. In developing the design constraints referred to earlier, the hot- channel factor for the enthalpy rise is required in addition to that for the flux. The enthalpy rise hot-channel factor is the ratio of the maximum enthalpy increase, i. e., in the hot channel, to the average enthalpy increase per channel. The enthalpy rise in a channel is related to the total heat input (or power) to the channel and to the mass-flow rate of the coolant through that channel. Hence, the enthalpy rise hot-channel factor is the product of a factor for the heat input in the hot channel and one for the flow rate. The ratio of the heat input in the hot channel to the average value is equal to the product of F%H and Fq. This must be multiplied by an engineering hot-channel factor which relates the flow rate in the hot channel to that in an average channel.

9.171. The flow-rate engineering factor is made up of several subfactors which affect the coolant flow; some characteristic values of these subfactors

TABLE 9.2. Typical Engineering Heat Flux Hot-Channel Factors

Subfactor
(Based on ± Зет)

[(0.024)2 + (0.0027)2 + (0.020)2 + (0.012)2]05 = 0.034

Combined heat flux engineering factor, = 1.034
(rounded off to 1.04)

in a PWR are listed in Table 9.3. Variations in fuel-rod pitch and bowing have a statistical basis, as do the variations in the diameter of the clad fuel rod; these factors may then be combined statistically. Other factors, which involve uncertainties and cannot be treated statistically, account for pos­sible maldistribution of coolant from the inlet plenum, flow redistribution resulting from differences in hydraulic resistance from some local boiling, and interchannel mixing of the coolant. The latter effect tends to reduce the enthalpy rise and hence the corresponding subfactor is less than unity. In order to obtain the overall hot-channel flow factor, the combined sta-

TABLE 9.3. Typical Engineering Subfactors For Flow Rate

tistical factor is multiplied by the product of the nonstatistical subfactors; thus,

Hot-channel flow factor = 1.062 x 1.03 x 1.10 x 0.90
= 1.08.

The hot-channel factor for the enthalpy rise F±H is then given by Fah = Fah x F% x Flow factor = 1.55 x 1.04 x 1.08 = 1.74.

The hot-channel enthalpy rise is thus 1.74 times the enthalpy rise computed from the ideal (or nominal) reactor specifications.

10.88. The concept of interest is very simple; it is the application of a fixed rate applied to a “principal” sum of money over a period of time, but a number of equations and special terms are useful, particularly in dealing with compound interest. If a sum of money, P, is invested at an interest rate, і, for a year, the interest earned at the end of the period is Pi. In compounding, this amount is added to the principal so that the sum invested for the second year is now P + iP = Sx and the new interest earned at the end of the second year is (P + iP)i, or more systematically,

S0 = P

S, = P + ІР = P( 1 + i)

S2 = P( 1 + i) + P/(l + i)

S2 = P( 1 + i)2.

At the end of n years, therefore, the total sum accumulated is Sn = P( 1 + i)n.

10.89. The quantity (1 + i)n is known as the single-payment compound amount factor. The reciprocal, 1/(1 + i)n, the single present-worth factor, is useful in finding a principal, P, that will give a required total amount, 5, in n years:

(10.17)

P may then be considered as the present worth of the total, 5, anticipated after n years.

10.90. Another type of accumulation results from the investment of a fixed sum, R, at the end of each year for n years. The total is the summation of the individual subtotals compounded over the years applicable for each payment.

S = R + R(1 + i) + R( 1 + if • • • Д(1 + if-1, (10.18)

where the amount R{ 1 + i)n~x is the accumulation from the payment made at the end of the first year and the first term, R, is the final payment, which earns no interest.

Multiplication by (1 + i) and then subtraction of equation (10.18) yields iS = Д[(1 + if — 1]

or

This expression is known as the uniform-annual-series compound factor. Its reciprocal, //[(1 + if — 1], is called the sinking-fund-deposit factor. This factor is used to determine the regular payment required to produce a desired amount at the end of a given period of time:

*-4(i+o—i]- <io2o)

The sinking-fund concept is used in one type of depreciation accounting to provide for the replacement of an asset at the end of its useful life.

10.91. A variation of this principle is the determination of the uniform end-of-the-year payment needed to repay a debt. From the viewpoint of the lender, this is the same as making a single present investment, P, which is returned, with interest, as a series of end-of-the-year payments. The payment scheme is the same as that for the sinking-fund deposit except that the lump-sum payment at the beginning is related to the alternate accumulation, S, by

S = P( 1 + if.

Then substituting in equation (10.20) yields

(10.21)

The final expression in brackets in equation (10.21) is known as the capital — recovery factor. It is equal to the sinking-fund factor plus the interest rate. The reciprocal, which would be useful if P were expressed in terms of R, is known as the uniform-series present-worth factor.

11.81. Various wastes are generated by a reprocessing plant using the Purex or a similar process. Low-radioactivity wastes are handled by stand­ard methods without problems. However, gaseous effluents that contain some radioactivity require special attention. The most highly radioactive liquid waste from the reprocessing of spent fuel by solvent extraction— and, in fact, from any stage of the nuclear fuel cycle—originates from the first-cycle raffinate (see Fig. 11.5). This high-level waste, as it is called, contains about 99.9 percent of the nongaseous fission products originally present in the spent fuel; it also contains some uranium and other actinide elements, including about 0.5 percent of the initial plutonium content. Because of its intense radioactivity, the high-level liquid waste presents a special problem.

11.82. The raffinate from the Purex process is evaporated to recover much of the nitric acid still remaining from the fuel dissolution and also to reduce the volume. The resulting solution, which contains nitrates of the metallic fission products and the actinides together with some free nitric acid, is stored, temporarily at least, in water-cooled, underground steel tanks.

11.83. Several processes have been investigated for converting high — level liquid wastes into solid form. Two of the more successful are spray calcination and fluidized-bed calcination. In the former, the liquid is sprayed into the top of a cylindrical tower heater in a furnace; the solid calcine, consisting of a mixture of fission product and heavy element oxides, is collected at the bottom. In the other process, which has been used since 1963 for solidification of liquid wastes from the reprocessing of special highly-enriched fuels, the water is evaporated in a heated, fluidized bed of particles made from previously dried waste.

11.84. For long-term storage, the calcine (mixed oxides) may be heated with a glass-forming frit containing borax and silica. The product is a borosilicate glass which is less leachable by water and has a higher thermal conductivity than the calcine. These changes are desirable from the stand­point of the ultimate disposal of the solidified high-level wastes. However, it appears that a sufficient increase in temperature, e. g., by radiation heat­ing, may cause the glass to devitrify ; the resulting microcrystaline material would then be more readily leachable than the original glass. This draw­back could be overcome by reducing the fission-product content of the glass or by increasing the distance between the containers in storage, thereby decreasing the heating rate. Nevertheless, efforts are being made to de­velop alternative solid forms for high-level wastes; these include crystalline products similar to stable minerals and cermets consisting of ceramic (cal­cined) waste particles in a metal matrix.

11.85. A 1000-MW (electric) LWR plant will, on the average, discharge some 30 to 35 metric tons of spent fuel annually, assuming operation at full capacity.* This would result in the production of roughly 3.2 m3 of high-level solid (borosilicate) waste. If this waste is packaged in steel cyl­inders 0.3 m in diameter and 3 m in length, about 15 such containers would be filled each year. Operation below the rated-capacity of the nuclear power plant, as will always be the case, will result in a smaller quantity of waste.

Large-break LOCA with loss of all ac power (AB accident)

12.97. The accident sequence is somewhat different depending on whether the initiating break occurs in one of the pressure vessel outlets to the steam generator (hot-leg break) or in one of the vessel inlets (cold-leg break). In a hot-leg break, coolant, steam, hydrogen, and accident debris escape directly into the containment, while in the latter case, significant flow would be through the steam generator, with some opportunity for fission product deposition. On the other hand, in considering a LOCA design basis ac­cident in which emergency core cooling is operative, the cold-leg break results in a higher clad temperature as a result of coolant flow reversal during the blowdown stage (§12.82).

12.98. Should there be loss of all ac power, the so-called AB accident, none of the engineered safety features (ESFs) would be functional except for the initial passive injection systems. The subsequent inability to remove stored and decay heat would result in a degraded core. Various chemical reactions among fission products and other core materials would take place. Non-noble gas fission products, their compounds, core debris, and water vapor are likely to condense partially in the cooler regions above the core and form aerosols upon escaping to the containment.

12.99. Depending on when power might be restored, the accident could proceed through a stage where molten fuel penetrates the reactor vessel, followed perhaps by interaction by the noncooled debris bed with the concrete base mat. Containment breaching would then be of concern. The nature of all of these steps is being studied and is mentioned here primarily for identification purposes and not to indicate risk levels. In fact, the AB sequence is considered relatively low in probability compared with the already extremely low probability of other severe accident scenarios. How­ever, the sequence is significant because core melt starts relatively rapidly, about 30 minutes after the break.