12.178. Over the course of nuclear reactor development, some accidents have occurred, which is not entirely unexpected for a new technology. In all cases, the effects were managed and the general public was not affected [21]. Commercial reactors earned an excellent record until the Three Mile Island accident in 1979, in which there was a negligible release of radiation to the public but major economic losses occurred. The accident also led to important changes in plant design, management, and regulation. In 1986, the Chernobyl accident, as a result of which there were 32 direct fatalities, had worldwide consequences on public acceptance of nuclear power and has had an effect on the design of future reactors. We will examine the effects of each of these two accidents on nuclear reactor engineering practice.

8.14. The usefulness of the systems concept as a decision tool is en­hanced if the system and its behavior can be described in a clear manner. Graphical representation is a classical approach. However, with a wide variety of computers now available, some type of representation that per­mits mathematical manipulation is preferred. This could take various forms, which could involve graphics, analytical descriptions, or combinations of each. For example, in risk management, graphical representation is useful but manipulation follows the rules of Boolian algebra (§12.218).

8.15. System behavior is normally described by quantitative relation­ships. For example, coolant circulating pumps may be described in terms of performance specifications such as flow rate, pressure differential, etc. On the other hand, some systems may involve qualitative or subjective inputs in addition to quantitative relationships. For example, in developing a model for a radioactive waste storage facility, technical requirements such as the desired amount and activity of radioactive isotopes are essential. However, there is also the likely need to input items such as local accept­ance considerations and regulatory requirements. Economic factors may also require input into the model.

8.16. In analytically modeling a system in such a way that associations between systems can be manipulated by computer, quantitative relation­ships can be handled in a straightforward manner. However, to permit similar computer manipulation of subjective modeling inputs, they must be quantified in some way, often a significant challenge.

9.78.

In essentially all cases of reactor cooling where forced convection is used, the coolant is under turbulent flow conditions. Satisfactory pre­dictions of heat-transfer coefficients in long, straight channels of uniform cross section can be made on the assumption that the only variables in­volved are the mean velocity of the fluid coolant, the diameter (or equiv­alent diameter) of the coolant channel, and the density, heat capacity, viscosity, and thermal conductivity of the coolant. From the fundamental differential equations or by the use of the methods of dimensional analysis, f it can be shown that heat-transfer coefficients for turbulent flow conditions can be expressed in terms of three dimensionless moduli; one of these is the Reynolds number, already defined, and the others are the Nusselt number (Nu) and the Prandtl number (Pr), defined by

9.79. As a result of numerous experimental studies of heat transfer, various expressions relating the three moduli have been proposed; one of these, for an ordinary (nonmetal) fluid in a long, straight channel, is the Dittus-Boelter correlation,

(9.35)

fSee standard texts on heat transfer, e. g., General References for this chapter.

Nu = 0.023Re° 8Pr° 4,

with all the physical properties evaluated at the bulk temperature of the fluid. In a modified form,

Nu = 0.023Re08Pr0 33,

the physical properties, except the specific heat, are the values at the film temperature, i. e., in the laminar (film) layer of fluid adjacent to the surface. This is taken as the arithmetic mean of the wall (or surface) temperature and the bulk fluid temperature.

Example 9.6. Calculate the heat-transfer coefficient for the water cool­

ant in Example 9.5.

From the data in the Appendix, к at 311°C is estimated to be 0.518 W/m • К and Pr about 1.06. Hence, from equation (9.36),

9.80. It is seen from equation (9.35) that if the viscosity, thermal con­ductivity, density, and specific heat of the coolant are known, the heat — transfer coefficient can be estimated for turbulent flow of given velocity in a pipe or channel of specified diameter (or equivalent diameter). The results appear to be satisfactory for values of Re in excess of about 10,000, and for Pr values of from 0.7 to 120. This range of Prandtl numbers includes gases and essentially all liquids, except liquid metals; the latter have very low Prandtl numbers, primarily because of their high thermal conductivity but also often because of their low viscosity and heat capacity. The cor­relations for liquid metals will, therefore, be considered separately (§9.84 et seq.).

9.81.

Where large differences exist between the temperatures of the solid and of the fluid, the associated variations in the physical properties of the coolant can influence the heat transfer. As indicated, such variation is most significant for the viscosity, and in order to make allowance for it the relationship

has been proposed, where |x is the viscosity at the bulk fluid temperature, and is that at the wall temperature. This equation, used in conjunction with the equivalent diameter concept, has been found to be satisfactory for predicting local and average heat-transfer coefficients for ordinary fluids flowing through thin, rectangular channels [6].

10.7. From the economic standpoint, the user of fuel needs to know how much uranium feed is required to produce a certain amount of enriched material and the cost of enrichment. To show how these quantities are determined, general relationships will first be derived between the amounts and assays of feed, enriched product, and tails. The following treatment is applicable to both gaseous-diffusion and gas-centrifuge processes al­though it has hitherto been employed primarily for the former.

10.8. If Fis the mass of uranium (regardless of its isotopic composition) in the feed material supplied to a separation cascade, P is the mass of product withdrawn, and W is the mass of the waste, a uranium mass balance (Fig. 10.1) requires that

F = P + W, (10.1)

assuming, as is generally true, that there is no appreciable loss of uranium during the operation. A similar balance can be applied to the uranium-235 only; thus,

F{xf) = P(xp) + W(xw),

PRODUCT

where Xf, xp, and xw are the assays (expressed as mass fractions of uranium — 235) in feed, product, and waste, respectively. By eliminating Wfrom these two equations, the result is

This equation gives the mass of uranium feed of assay xf required per unit mass of uranium product of assay xp, assuming a tails (waste) assay of xw.

10.9. The cost of enrichment is determined by the amount of work that has to be done to achieve the enrichment. A so-called value function has been developed on the basis of the theory of the gaseous-diffusion cascade. It is represented by V(x) and is defined for any material of assay x by

V(x) = (1 — 2x) In (10.4)

Because x is a fraction, the value function, which is characteristic of a given assay (uranium-235 content), is a fraction and has no units. It is used to determine the work required to yield a product of a desired assay from a given feed with a specified waste.

10.10 The effort expended in separating a mass F of feed of assay xf into a mass P of product of assay xp and waste of mass W and assay xw is expressed in terms of the number of separative work units (SWU) needed. This is given in terms of the respective value functions by

SWU = WV(xw) + PV(xp) — FV(xf). (10.5)

Since the value functions have no units, the SWU will have the same units as the masses W, P, and F. The general practice is to state the number of separative work units in terms of kilograms of uranium. Upon dividing equation (10.5) through by P, the result

SWT I W F

— = J V(xw) + V(xp) — — V(xf) (10.6)

gives the number of separative work units received per unit mass of product.

Example 10.1. The owner of a nuclear power plant requires 100,000 kg of 3.0 percent enriched uranium. Determine the amount of natural uranium feed and the number of separative work units needed, assuming a tails assay of (a) 0.200 percent and (b) 0.300 percent.

First, the necessary value functions are determined from equation (10.4); the results are as follows:

{a) For xw = 0.002, equation (10.3) gives

F = 0.0300 — 0.0020 _ , P ” 0.0071 — 0.0020 “ ‘

and from equation (10.1)

W _ F P ~ P

Since F/P is 5.48, the natural uranium feed requirement is (5.48) (100,000) = 548,000 kg.

From equation (10.6),

SWU

—p — = (4.48)(6.19) + 3.27 — (5.48)(4.87) = 4.31.

Hence, the SWU requirement is 431,000 kg.

(b) For xw = 0.003, FtP = 6.57 and W/P = 5.57; hence, the natural uranium feed requirement is 657,000 kg.

The separative work is obtained from

SWU

—— = (5.57)(5.77) + 3.27 — (6.57)(4.87) = 3.42.

The SWU requirement is thus 342,000 kg.

10.11. The cost per separative work unit is determined from the en­richment plant operating costs and the cost of the capital invested in the plant. If Cs is the cost of a separative work unit, e. g., in dollars per kilogram, then

and

where Cp, Cf, and Cw are the costs per kilogram of product, feed, and waste, respectively. According to equation (10.7), the cost of the product is equal to the cost of the enrichment operation (separative work cost) plus the cost of the feed, less credit for the value of the waste (tails). In practice, the credit for the tails is usually neglected. The unit cost of the product then depends on the ratios SWU IP and F/P, which are determined by the assays of the product, feed, and tails, and on the costs per kilogram of the separative work and the feed.

10.107. In considering the various factors affecting nuclear energy costs, the role of rate regulation deserves mention. Investor-owned public utilities are regulated by state public utility commissions since they have a franchise to serve a specific territory. Their responsibility is to establish rates that provide the utility with a fair rate of return on the capital invested. How­ever, when the construction cost of new nuclear power plants escalated significantly during the late 1970s and early 1980s, opposition developed to burdening customers with corresponding major rate increases. These disputes were resolved in various ways. At any rate, as a result of the need to insulate utility rates from potentially high costs for new nuclear power plants, the Public Utilities Regulatory Policy Act was enacted in 1978. This provides for a modified free-market energy economy under control of the Federal Energy Regulatory Commission (FERC). The National Energy Policy Act (NEPA) of 1992 provided further deregulation, particularly in the opening of transmission grids for the transport of electricity so that competitive marketing is now possible. Also, now permitted are non-rate — based independent power producers who can take advantage of this new access to markets.

10.108. We can anticipate some restructuring of the electric utility in­dustry in the United States with other than traditional forms of ownership being permitted. For example, independent power producers, not subject to rate structure-based regulation, might market energy on a competitive “wholesale” basis to traditional electric utilities, which would then, in turn, distribute the energy to their customers [20]. The role of nuclear power production under this marketing arrangement is likely to depend primarily on its economic competitiveness.

11.116. When it is possible to meet the applicable water quality regu­lations and standards, once-through cooling would be the preferred method for disposing of the waste heat from the turbine condenser of a power plant. The condenser water is then discharged in such a way as to minimize the thermal impact on the receiving water body. In some situations, a slow discharge at or near the surface of the receiving body may be preferred; the warm discharge water then spreads over a large area. At the other extreme, the water may be forced through jets or diffusers located near the bottom of a flowing stream, so that rapid mixing occurs. Several vari­ations between these extremes are possible. If an adjacent water body is an ocean or a large lake, the intake water is usually drawn from a depth, where the temperature is lower, and is discharged near the surface.

11.117. If the conditions do not permit once-through cooling, some form of closed (or partly closed) cooling cycle must be adopted to deal with the thermal discharge. Cooling ponds (or canals) or cooling towers are used for this purpose. In a closed-cycle system, all the condenser water is dis­charged and cooled in the pond or tower and the cooled water is withdrawn for reuse. Makeup water is added as required. In a partially closed system, part of the water would be discharged directly to an adjacent water body whereas the remainder would be cooled and reused. In some circumstances, a variable-cycle cooling system may be satisfactory: once-through (direct- discharge) cooling is used in the winter and early spring and a closed (or partially closed) system at other times.

11.118. In a cooling pond or cooling canal, the condenser water is dis­charged at one end of a large pond (or small lake) or a long canal and is withdrawn at the other end of a flow path. In the course of its passage through the pond or canal, which may take several days, the condenser water is cooled mainly by evaporation and also to some extent by convec­tion and radiation. The cooling efficiency of a pond or canal can be greatly enhanced by pumping the water through nozzles to form sprays. The in­crease in water surface produced by the sprays increases the rate of heat loss. Depending on the local meteorological conditions, e. g., temperature, humidity, and wind, a pond or canal without sprays will have an area of from 4 to 12 x 106 m2 (1000 to 3000 acres) for a 1000-MW(el) nuclear plant; with sprays, a much smaller area (one-tenth or less) would be sufficient.

11.119. Cooling towers are of two main types: wet and dry. In wet towers, the condenser discharge water flows down over a packing or “fill” and is broken up into droplets. A current of air is drawn through the fill, either by a powerful fan in mechanical-draft towers, or by a tall (up to 165 m high) chimney-like structure, with a hyperbolic profile, in natural-draft towers. Most of the cooling results from vaporization of the water, but there is also some cooling by convection and radiation. The lowest tem­perature attainable is approximately the wet-bulb temperature under the existing conditions; it is thus dependent on both the actual air temperature and the relative humidity. The temperature of the water collected at the bottom of the cooling tower is generally 4° to 6°C above the wet-bulb temperature.

11.120. One of the problems of wet cooling towers is the occurrence of “drift,” that is, the entrainment of small droplets of water in the air leaving the tower. Drift eliminators, which cause the air to make abrupt turns, help to reduce the amount of drift but do not prevent it completely. The drift is worse for mechanical-draft than for natural-draft towers, partly because of the higher air velocities in the former. The presence of chemicals used to prevent biological fouling, corrosion, and structural deterioration in the tower may make the drift a hazard to plant and animal life on the ground in the downwind direction where the water droplets tend to settle. Drift elimination has been improved to such an extent in recent years, however, that sea water can be used without causing significant environ­mental damage.

11.121. Continuous evaporation of the water in the tower causes the chemicals, as well as the minerals normally present, to become more and more concentrated. Ultimately, the concentration reaches a point at which scale may deposit on the condenser tubes. In order to prevent this, some of the water is removed either continuously or periodically and discarded as “blowdown” and replaced with fresh water. In a large power plant, the blowdown rate is about 12 to 15 m3 (3000 to 4000 gal) per minute. The blowdown water may have to be treated for the removal of various chem­icals before it can be discharged to a nearby water body.

11.122. The main loss of water from a wet cooling tower is by evapo­ration of the water and this must be replaced continuously. Including the much smaller losses from drift and blowdown, makeup water is approxi­mately 2.5 to 3 percent or so of the water passing through the condenser; hence, on the basis of the data in §11.111, the average makeup rate, even for a completely closed system, would be roughly 60 to 70 m3 (15,000 to 17,000 gal) per minute for a 1000-MW(el) nuclear plant.

11.123. Most of the water lost from a wet cooling tower enters the atmosphere as water vapor; under certain conditions this could result in the formation of fog, leading to reduced visibility, and to the deposition of ice on roads and power lines in the vicinity in the winter. The available evidence indicates, however, that such occurrences are limited to a rela­tively few days of high humidity in cold weather. Natural-draft cooling towers discharge the moist air at much higher elevations than do mechanical- draft towers, and hence they are less likely to produce fog and ice near the ground.

11.124. Dry cooling towers used in steam electric plants are generally described as air-cooled condensers. They operate by using ambient air as coolant to condense the exhaust steam either directly or indirectly. In the direct condensing cycle, the steam leaving the turbine is passed through a system of finned pipes over which air is drawn by mechanical or natural draft. The air removes heat from the steam so that it condenses, and the condensate is returned as feedwater to the steam generator.

11.125. The most efficient dry tower utilizes an indirect condensing cycle known as the Heller system. The steam is condensed by direct contact with jets or sprays of water from previously cooled condensate. Part of the

resulting warmed water is returned as feedwater to the steam generator, while the remainder is cooled by passage through a tower containing finned pipes over which air is drawn. The cooled water leaving the tower is re­circulated to the condenser sprays.

11.126. In dry cooling towers, heat is removed mainly by convection to the ambient air, and there is no loss of water by evaporation. There is consequently no blowdown (or drift) and, except for leakage, no require­ment for makeup water. Hence, steam-electric plants with dry towers can be located in areas where water is scarce. On the other hand, the absence of vaporization means that the lowest temperature attainable theoretically in a dry tower is the actual air temperature, which is higher than the wet- bulb temperature. As a general rule, therefore, the temperatures of the condensate would be higher for a dry tower than for a wet tower; the turbine back-pressure would also be higher. The overall effect would be a decrease in efficiency. In addition, construction and operation costs are expected to be greater for a dry tower than for a wet tower of the same cooling capacity.

11.127. Despite the greater cost of generating electricity with dry cool­ing towers, estimated to be about 20 percent more than for wet towers and about 27 percent more than for once-through cooling, there may be circumstances in which no other form of cooling is practical. So far, rel­atively few dry (Heller) towers have been used (or designed) for power plants, the largest with a capacity of 350 MW(el). However, studies are being made of the possibility of designing dry towers suitable for large nuclear power plants.

11.128. Wet-dry (or parallel-path) cooling towers, which use both wet and dry cooling, are attracting interest. The mechanical-draft tower consists of two regions: the upper part is a dry tower whereas the lower part is a wet tower. The condenser discharge water is first cooled by passage through finned tubes in the upper (dry) region and then flows down over the fill in the lower (wet) region for further cooling. Air is drawn by a fan in parallel paths through both dry and wet regions, and the streams are mixed before discharge to the atmosphere. The air flow between dry and wet regions can be adjusted to give preference to one or the other. In winter, when the air is cold, advantage is taken of dry cooling, with a marked decrease in fogging and icing as well as a reduction in water consumption, but in summer preference might be given to wet cooling.

1. ?

INTRODUCTION

Technological Risk and Public Perception [1]

12.1. We introduce this chapter, which is devoted to those aspects of nuclear reactor engineering concerned with system safety, with a few thoughts on some nontechnical considerations that are relevant. First, there is the matter of terminology. The word safety is generally defined as “freedom from danger or hazard,” while risk means “the chance of injury or loss.” We will examine these terms more carefully later when we discuss the subject of risk analysis. However, mathematically based quantitative def­initions of risk are beyond our scope. A problem is that in a technological society, no activity has zero risk. Thus, there cannot be absolute safety and the two terms are related to one another, with low risk meaning the same as high level of safety. However, psychologically, we tend to be more comfortable with the term safety than with the term risk, whether we are discussing nuclear reactors, automobiles, airplanes, or chemicals.

12.2. Unfortunately, in considering public perception, there is an ele­ment of fear which is often not at all related to the true risk of a techno­
logical activity as developed from statistical data. In general, self-imposed, or voluntary risks, tend to be more acceptable than involuntary risks. For example, the significant risks of highway travel are normally accepted, whereas isolated cases of chemical residues on some fruits with negligible statistical risks have led to national market withdrawals of such fruits. Intangible factors also affect the public perception of risk. For example, many members of the public fear ionizing radiation as such, even at harm­less levels. Another example is the fear by some of using commercial aviation. For a planned technological activity, it is sensible to balance the benefits obtained with an “appropriate” level of risk that can be adjusted during the design process, taking into consideration the costs required. This balance is often known as the “how safe is safe enough?” question. However, public perception (fear) and sometimes associated political pres­sure can play a role in the “appropriate” balance.

12.108. During the course of the Three Mile Island accident in 1979 (§12.178), a surprising small amount of iodine, about 18 Ci, escaped to the environment. Since this was so much less than the 6 million curies of the inert gas, xenon, that escaped, studies were instigated which showed that as a result of the reducing chemical state in the reactor vessel, the iodine was converted to cesium iodide, which is soluble in water and hence was not released. Therefore, it was recognized that chemical and physical phenomena during the course of an accident have a great deal to do with reactor safety, and major attention was devoted to source term studies.

12.109. In assessing the hazards associated with fission product release from the fuel, we must consider in a stepwise manner how each nuclide of radiological importance makes its way from the point of release, through barriers that may have failed, into the coolant, and then into the contain­ment. Some release from the containment to the environment by some means would then constitute the hazard. An analysis of this sequence depends very much on the nature of the accident. Types of accidents are considered in §12.94 et seq. However, in all cases, the fission product transport depends first on chemical changes that affect their characteristics, then on their behavior when they are in the form of particles or aerosols. Therefore, we will examine in a general way the chemistry of the fission products. Then, an introduction to fission product particulate mechanics will be given.

8.50. The ability to access various data bases in information centers by computerized communications networks is a major convenience. In most cases, the computerized data base with interactive capabilities is maintained by the responsible center, while access is obtained by means of a com­mercial communications network such as SPRINTNET or ВТ TYMNET [9]. Generally, a commercial service provider serves as an intermediate link between the communication network, which is “dialed up” by the user, and the various data bases. Such providers, which offer numerous information and search services, generally provide catalogs of data bases that are available through them [10]. A given data base may be offered by more than one provider. In some cases, a provider may also offer communication network service.

8.51. On-line interactive access to some data bases may be available directly by telephone. For example, the U. S. Department of Energy pro­vides an on-line search service known as FEDIX for universities and other research organizations. No registration or access fees are required. The Federal Telecommunications Service (FTS) also provides network access to many data bases to U. S. government contractors. RECON, developed by the Technical Information Service for Department of Energy contractors and other agencies, is an interactive search system for a large number of data bases of interest to the nuclear engineering community. Included are such areas as safety information, codes available at the National Energy Software Center (§8.33), and electric power data.

8.52. The NUCLEAR NETWORK, operated by the Institute of Nu­clear Power Operations (INPO), provides direct communication between reactor licensees, reactor vendors, and other nuclear power-related groups such as the Nuclear Safety Analysis Center of the Electric Power Research Institute. Most use is devoted to communications rather than literature searches.

8.53. The INTERNET, when it began in 1969 as ARPANET, linked a few universities, government agencies, and defense-related laboratories. During recent years, it has grown at a rapid rate to accommodate com­mercial as well as institutional users and has become the world’s largest computer network. Since most networks are connected to the INTERNET, it has become in a sense a worldwide network of computer networks. A large variety of services are available through the INTERNET, including electronic mail, access to data bases, and on-line discussion groups [11]. Public access is available through commercial providers (§8.50).

Information Age Challenges

8.54. Making effective use of all of the information resources available to help the engineer solve a given problem is somewhat of a challenge. The discussion above is intended only as an introduction to the types of services available, which are being continually expanded and updated. Therefore, guidance from an information service professional or technical librarian is desirable.

GENERAL REFERENCES

Beam, W. R., “Systems Engineering: Architecture and Design,” McGraw-Hill Book Co., 1990.

Blanchard, B. S., and W. J. Fabrycky, “Systems Engineering and Analysis,” 2nd Ed., Prentice Hall, 1990.

Boardman, J., “Systems Engineering: An Introduction,” Prentice Hall, 1990.

Lewis, W. P., “Fundamentals of Engineering Design,” Prentice Hall, 1989.

Papalambros, P. Y., and D. J. Wilde, “Principles of Optimal Design: Modeling and Computation,” Cambridge University Press, 1988.

Ralston, A., and E. D. Reilly, eds., “Encyclopedia of Computer Science,” 3rd Ed., Van Nostrand Reinhold Co., 1993.

Ray, M. S., “Elements of Engineering Design: An Integrated Approach,” Prentice Hall, 1985.

REFERENCES

1. J. G. Ecker and M. Kupferschmid, “Introduction to Operations Research,” John Wiley & Sons, 1988.

2. J. Boardman, “Systems Engineering: An Introduction,” Prentice Hall, 1990.

3. T. J. Downar and A. Sesonske, “Light Water Reactor Fuel Cycle Optimi­zation: Theory versus Practice,” Adv. Nucl. Sci. Technol., 20, 71 (1988).

4. D. T. Pham, Ed., “Artificial Intelligence in Design,” Springer-Verlag, 1991; M. Sharpies et al., “Computers and Thought,” MIT Press, 1989.

5. M. D. Rychener, “Expert Systems for Engineering Design,” Academic Press,

1988; J. A. Bernard and T. Washio, “Expert System Applications within the Nuclear Industry,” American Nuclear Society, 1989.

6. R. E. Uhrig, Nucl. Safety, 32, 68 (1991); P. D. Wasserman, “Neural Com­puting: Theory and Practice,” Van Nostrand Reinhold Co., 1990.

7. A. G. Parlos et al., Trans. Am. Nucl. Soc., 63, 109 (1991).

8. J. Jedruch, “Nuclear Engineering Data Bases, Standards, and Numerical Analysis,” Van Nostrand Reinhold Co., 1985; D. N. Chorafas and S. J. Legg., “The Engineering Data Base,” Butterworth, 1988; M. Edelhart and O. Davies, “Omni Online Database Directory,” Collier Macmillan Publishers, 1984.

9. U. S. Sprint, 12490 Sunrise Valley Drive, Reston VA 22096 (800) 736-1130; ВТ Tymnet, Inc., P. O. Box 49019, San Jose, CA 95161, (800) 872-7654.

10. DIALOG Information Services, Inc., 3460 Hillview Avenue, Palo Alto, CA 94304, (800) 334-2564; ORBIT Search Service, 8000 Westpark Drive, Mc­Lean, VA 22102, (800) 456-7248.

11. T. L. LaQuey, “Internet Companion: A Beginner’s Guide to Global Net­working,” Addison-Wesley Publishing Co., 1993.

INTRODUCTION

The Role of Energy Transport in Reactor Design

9.1. Most of the fission reaction energy deposited in fuel is immediately converted to heat. If the fuel is to remain at steady state (constant tem­perature), the heat must be transported away at the same rate as it is generated. Although we have seen that negative reactivity feedback, if present, tends to limit power increases, it is essential for the heat generation — heat removal rate balance to be maintained, to prevent temperatures that might result in the failure of fuel and structural materials. Thus, the design of the core depends just as much on adequate heat removal as on nuclear considerations.

9.2. Effective heat removal is a function of many design parameters, including the fuel geometry, coolant flow characteristics, and properties of materials, as well as related neutronic behavior. In many aspects of the design, conventional engineering principles of heat transfer and fluid me­chanics are applicable. The term thermal-hydraulics is commonly used to describe the effort involving the integration of heat-transfer and fluid me­
chanics principles to accomplish the desired rate of heat removal from the core under both operating and accident conditions. The purpose of this chapter is to introduce this area of nuclear reactor engineering.

9.3. Various accidental causes of a mismatch between the fuel heat generation and heat removal capabilities of the coolant leading to reactor damage are carefully analyzed as part of the required safety analysis to be discussed in Chapter 12. Of particular concern are events that lead to a reduction in the coolant mass flow rate. Since each unit mass of coolant would then receive more energy, boiling is a probable consequence. Such a transition to a two-phase system would result in an increase in flow resistance, further aggravating the unstable situation. Therefore, the thermal — hydraulics of two-phase systems plays an important role in reactor design. Analysis is required for the design of safety features which are provided in a reactor plant to reduce the consequences of a serious accident in which the core might be damaged. Although the relevant thermal-hydraulics basics will be discussed in this chapter, treatment of the design applications will be deferred until Chapter 12.

9.113. For turbulent flow at a mean velocity и, the relationship corre­sponding to equation (9.40) is

(9.41)

known as the Fanning equation, in which the dimensionless quantity/is called the friction factor. * Comparison of equations (9.40) and (9.41) shows that for laminar flow / = 16/Re, but the relationship between the friction factor and the Reynolds number in the case of turbulent flow is more complicated. Several empirical expressions have been developed; one of the simplest of these, which holds with a fair degree of accuracy for flow in smooth pipes at Reynolds numbers up to about 2 x 105, is the Blasius equation

/ = 0.079Re-°25.

This permits the evaluation of the turbulent-flow friction factor for a given Reynolds number.

9.114. For turbulent flow in a commercial rough pipe, the friction factor is larger than for a smooth pipe at the same Reynolds number, as shown in Fig. 9.15. The deviation from the ideal behavior of a smooth pipe increases with the roughness of the pipe, especially at high Re values. The variation of the friction factor with Reynolds number has been determined experimentally for different degrees of roughness, expressed in terms of a dimensionless quantity e/D, where e is a measure of the size of the rough­ness projections and D is the pipe diameter.

9.115. For turbulent flow in noncircular channels of relatively simple form, the pressure drop due to friction may be calculated by substituting the equivalent diameter for D in equation (9.41). The same value is used in determining the Reynolds number. For in-line flow along channels in rod bundles, the friction factor depends to some extent on the pitch-to — diameter ratio. For initial design purposes only, however, a correction factor of 1.3 is a reasonable approximation for the usual range of these ratios [14].

Example 9.8. Estimate the pressure drop required to overcome fric­tion for the turbulent flow of water in the channel between the fuel rods in Example 9.5, along a length of 4.17 m.

The value of Re was found to be 5.00 x 105; hence, assuming the fuel rods to be moderately smooth, the value of /is seen from Fig. 9.15 to be about 0.0032.

The mass velocity G( = up) in Example 9.5 was 3730 kg/m2 • s, and since p for water at 311°C is 0.691 x 103 kg/m3, и is 3730/691 = 5.40 m/s. Hence, with De equal to 0.0118 m, it follows from equation (6.49) that

Upon applying the 1.3 correction for flow in rod bundles (§9.115), the result is

Apf = 4.56 x 104 x 1.3 ~ 6 x 104 Pa.

This calculation does not taken into account the pressure losses in the flow channel resulting from the spacer grids (normally about six) provided at intervals to support the rods and enhance mixing (§9.119).

10.40. A “scoping” analysis is carried out in the first phase of the pre­liminary design effort to develop long-term strategy considerations and to estimate average characteristics of a fuel batch. For such purposes, a point reactor model that provides a minimum level of geometric representation is usually adequate.

10.41. An example of a modern, fast-running, but sophisticated scoping code is BRACC, which can be run on a microcomputer [9]. Fuel batch reactivity is approximated as a linear function of burnup. Although zero dimensionality (point reactor model) is used, the code accounts for neutron leakage, burnable poison effects, and coupling between assemblies of dif­ferent batches, by various functions and submodels.

10.42. Methods used in conjunction with reload assembly pattern de­velopment are generally based on two — or three-dimensional nodal models. For some years, the Electric Power Research Institute (EPRI) has sup­ported the development of a family of codes to provide a standard core analysis capability for electric utility in-house core management applica­tions. This effort has been known as the Advanced Recycle Methodology Program (ARMP). EPRI-NODE-P was a preliminary design code devel­oped for PWRs. More recently, codes of the SIMULATE family have been developed under the auspices of EPRI for both preliminary and final design [8].

10.43. A two-dimensional version for preliminary design purposes uses the cross-section code CASMO [10] and a four-node per assembly repre­sentation in the model. A number of features are provided, including the ability to analyze core response to control rod insertion. The ARMP pack­age includes SIMULATE-E, a three-dimensional version suitable for final design for which the two-dimensional diffusion theory code, PDQ-7 [11], has traditionally been used. Such three-dimensional analysis is also useful for the evaluation of certain accident possibilities as control rod ejection (§12.75).