Chapter 12. Nuclear Heat Energy

Most of the energy released in fission appears as kinetic energy of a few high-speed particles. As these pass through matter, they slow down by multiple collisions and impart thermal energy to the medium. It is the purpose of this chapter to discuss the means by which this energy is transferred to a cooling agent and transported to devices that convert mechanical energy into electrical energy. Methods for dealing with the large amounts of waste heat generated will be considered.

12.1. Methods of Heat Transmission

We learned in basic science that heat, as one form of energy, is transmitted by three methods—conduction, convection, and radiation. The physical processes for the methods are different. In conduction, molecular motion in a substance at a point at which the temperature is high causes motion of adjacent molecules, and a flow of energy toward a region of low temperature takes place. The rate of flow is proportional to the slope of the temperature (i.e., the temperature gradient). In convection, molecules of a cooling agent such as air or water strike a heated surface, gain energy, and return to raise the temperature of the coolant. The rate of heat removal is proportional to the difference between the surface temperature and that of the surrounding medium and also depends on the amount of circulation of the coolant in the vicinity of the surface. In radiation, molecules of a heated object emit and receive electromagnetic radiations, with a net transfer of energy that depends on the temperatures of the body and the adjacent regions, specifically on the difference between the temperatures raised to the fourth power. For reactors, this mode of heat transfer is generally of less importance than are the other two.

12.2. Heat Generation and Removal

The transfer of heat by conduction in a flat plate (insulated on its edges) is reviewed. If the plate has a thickness x and cross-sectional area A, and the temperature difference between its faces is ΔT, the rate of heat flow through the plate, Q, is given by the relation where k is the thermal conductivity, with typical units J/s-°C-cm. For the plate, the slope of the temperature is the same everywhere. In a more general case, the slope may vary with position, and the rate of heat flow per unit area Q/A is proportional to the slope or gradient written as ΔT/Δx.

The conductivity k varies somewhat with temperature, but we treat k as constant for the following analysis of conduction in a single fuel rod of a reactor (see Section 11.4). Let the rate of supply of thermal energy by fission be uniform throughout the rod. If the rod is long in comparison with its radius R, or if it is composed of a stack of pellets, most of the heat flow is in the radial direction. If the surface is maintained at a temperature T^s by the flow of coolant, the center of the rod must be at some higher temperature T⁰. As expected, the temperature difference is large if the rate of heat generation per unit volume q or the rate of heat generation per unit length q¹ = πR²q is large. We can show^[†] that and that the temperature T is in the shape of a parabola within the rod. Figure 12.1 shows the temperature distribution.

^† The amount of energy supplied within a region of radius r must flow out across the boundary. For a unit length of rod with volume πr² and surface area 2πr, the generation rate is πr²q, equal to the flow rate − k(dT/dr)2πr. Integrating from r = 0, where T = T⁰, we have T = T⁰ − qr²/4k. At the surface T^s = T⁰ − qR²/4k.

Figure 12.1. Temperature in fuel.

Let us calculate the temperature difference T⁰ − T^s for a reactor fuel rod of radius 0.5 cm, at a point where the power density is q = 200 W/cm³. This corresponds to a linear heat rate q¹ = πR²q = π(0.25) (200) = 157 W/cm (or 4.8 kW/ft). Letting the thermal conductivity of UO² be k = 0.062 W/cm-°C, we find T⁰ − T^s = 200 °C (or 360 °F). If we wish to keep the temperature low along the centerline of the fuel, to avoid structural changes or melting, the conductivity k should be high, the rod size small, or the reactor power level low. In a typical reactor there is a small gap between the fuel pin and the inside surface of the cladding. During operation, this gap contains gases, which are poor heat conductors and thus there will be a rather large temperature drop across the gap. A smaller drop will occur across the cladding that is thin and has a high thermal conductivity.

We have so far assumed that the thermal conductivity is constant. It actually varies with temperature and thus with position in the fuel pin. A more general calculation of k is possible with the program CONDUCT discussed in Computer Exercise 12.A, and the temperature distribution may be found with a program TEMPLOT in Computer Exercise 12.B.

Convective cooling depends on many factors such as the fluid speed, the size and shape of the flow passage, and the thermal properties of the coolant, as well as the area exposed and the temperature difference between surface and coolant T^s − T⁰. Experimental measurements yield the heat transfer coefficient h, appearing in a working formula for the rate of heat transfer Q across the surface S,

The units of h are typically W/cm² − °C. To keep the surface temperature low to avoid melting of the metal cladding of the fuel or to avoid boiling if the coolant is a liquid, a large surface area is needed or the heat transfer coefficient must be large, a low-viscosity coolant of good thermal conductivity is required, and the flow speed must be high.

As coolant flows along the many channels surrounding fuel pins in a reactor, it absorbs thermal energy and rises in temperature. Because it is the reactor power that is being extracted, we may apply the principle of conservation of energy. If the coolant of specific heat c enters the reactor at temperature T^c (in) and leaves at T^c (out), with a mass flow rate M, then the reactor thermal power P is

For example, let us find the amount of circulating water flow to cool a reactor that produces 3000 MW of thermal power. Let the water enter at 300 °C (572 °F) and leave at 325 °C (617 °F). Assume that the water is at 2000 psi and 600 °F. At these conditions the specific heat is 6.06 × 10³ J/kg − °C and the specific gravity is 0.687. Thus the mass flow rate is

This corresponds to a volume flow rate of that is also 1,730,000 liters per minute. To appreciate the magnitude of this flow, we can compare it with that from a garden hose of 40 liters/min. The water for cooling a reactor is not wasted, of course, because it is circulated in a closed loop.

The temperature of coolant as it moves along any channel of the reactor can also be found by application of the preceding relation. In general, the power produced per unit length of fuel rod varies with position in the reactor because of the variation in neutron flux shape. For a special case of a uniform power along the z-axis with origin at the bottom as in Figure 12.2(A), the power per unit length is P¹ = P/H, where H is the length of fuel rod. The temperature rise of coolant at z with channel mass flow rate M is then that shows that the temperature increases with distance along the channel as shown in Figure 12.2(B). The temperature difference between coolant and fuel surface is the same at all points along the channel for this power distribution, and the temperature difference between the fuel center and fuel surface is also uniform. We can plot these as in Figure 12.2(C). The highest temperatures in this case are at the end of the reactor.

Figure 12.2. Temperature distributions along axis of reactor with uniform power.

If instead, the axial power were shaped as a sine function, see Figure 12.3(A) with P ~ sin(πz/H), the application of the relations for conduction and convection yields temperature curves as sketched in Figure 12.3(B). For this case, the highest temperatures of fuel surface and fuel center occur between the halfway point and the coolant exit. In the design of a reactor, a great deal of attention is given to the determination of which channels have the highest coolant temperature and at which points on the fuel pins “hot spots” occur. Ultimately, the power of the reactor is limited by conditions at these channels and points. The mechanism of heat transfer from metal surfaces to water is quite sensitive to the temperature difference. As the latter increases, ordinary convection gives way to nucleate boiling, in which bubbles form at points on the surface, and eventually film boiling can occur, in which a blanket of vapor reduces heat transfer and permits hazardous melting. The value of the heat flux Q at which film boiling occurs is called the critical heat flux. A parameter called “departure from nucleate boiling ratio” (DNBR) is used to indicate how close the heat flux is to the critical value. For example, a DNBR of 1.3 implies a safety margin of 30%. Figure 12.4 indicates maximum temperature values for a typical PWR reactor.

Figure 12.3. Temperature distributions along channel with sine function power.

Figure 12.4. Reactor channel heat removal.

To achieve a water temperature of 600 °F (approximately 315 °C) requires that a very high pressure be applied to the water coolant-moderator. Figure 12.5 shows the behavior of water in the liquid and vapor phases. The curve that separates the two phase regions describes what are called saturated conditions. Suppose that the pressure vessel of the reactor contains water at 2000 psi and 600 °F and the temperature is raised to 650 °F. The result will be considerable steam formation (flashing) within the liquid. The two-phase condition could lead to inadequate cooling of the reactor fuel. If instead the pressure were allowed to drop, say to 1200 psi, the vapor region is again entered and flashing would occur. However, it should be noted that deliberate two-phase flow conditions are used in BWRs, providing efficient and safe cooling.

Figure 12.5. Relationship of pressure and temperature for water.

12.3. Steam Generation and Electrical Power Production

Thermal energy in the circulating reactor coolant is transferred to a working fluid such as steam by means of a heat exchanger or steam generator. In simplest construction, this device consists of a vessel partly filled with water, through which many tubes containing heated water from the reactor pass, as in Figure 12.6. At a number of nuclear plants the steam generator has failed prematurely because of corrosion that created holes in tubes, requiring plugging or repair. In some cases replacement of the steam generator was required, with corresponding outage, cost, and loss of revenue. Details on the problem appear in an Nuclear Regulatory Commission Technical Issue Paper (see References). Steam from the generator flows to a turbine, while the water returns to the reactor. The conversion of thermal energy of steam into mechanical energy of rotation of a turbine and then to electrical energy from a generator is achieved by conventional means. Steam at high pressure and temperature strikes the blades of the turbine, which drives the generator. The exhaust steam is passed through a heat exchanger that serves as condenser, and the condensate is returned to the steam generator as feed water. Cooling water for the condenser is pumped from a nearby body of water or cooling tower as discussed in Section 12.4.

Figure 12.6. Heat exchanger or steam generator.

Figures 12.7 and 12.8 show the flow diagrams for the reactor systems of the PWR and BWR type. In the PWR, a pressurizer maintains the pressure in the system at the desired value. It uses a combination of immersion electric heaters and a water spray system to control the pressure. Figure 12.9 shows the Diablo Canyon nuclear power plant operated by Pacific Gas & Electric Company in San Luis Obispo County, California. The two Westinghouse PWR reactors were put into operation in 1985 and 1986.

Figure 12.7. PWR system flow diagram.

Figure 12.8. BWR system flow diagram.

Figure 12.9. A nuclear power plant, Diablo Canyon, on the California coast

(Courtesy Pacific Gas & Electric Company).

12.4. Waste Heat Rejection

The generation of electric power by consumption of any fuel is accompanied by the release of large amounts of waste heat. For any conversion process the thermal efficiency, e, the ratio of work done to thermal energy supplied, is limited by the temperatures at which the system operates. According to the second law of thermodynamics, an ideal cycle has the highest efficiency value, where T¹ and T² are the lowest and highest absolute temperatures (Kelvin, °C + 273; Rankine, °F + 460). For example, if the steam generator produces steam at 300 °C and cooling water for the condenser comes from a source at 20 °C, we find the maximum efficiency of

The overall efficiency of the plant is lower than this because of heat loss in piping, pumps, and other equipment. The efficiency of a typical nuclear power plant is approximately 0.33. Thus twice as much energy is wasted as is converted into useful electrical energy. Fossil fuel plants can operate at higher steam temperatures, giving overall efficiencies of approximately 0.40.

A nuclear plant operating at electrical power 1000 MWe would have a thermal power of 1000/0.33 = 3030 MWt and must reject a waste power of P = 2030 MWt. We can calculate the condenser cooling water mass flow rate M required to limit the temperature rise to a typical figure of ΔT = 12 °C with a specific heat of c = 4.18 × 10³ J/kg-°C,

This corresponds to a flow of 925 million gallons per day. Smaller power plants in past years were able to use the “run of the river” (i.e., to take water from a stream, pass it through the condenser, and discharge heated water downstream). Stream flows of the order of a billion gallons a day are rare, and the larger power plants must dissipate heat by use of a large lake or cooling towers. Either method involves some environmental effects.

If a lake is used, the temperature of the water at the discharge point may be too high for certain organisms. It is common knowledge, however, that fishing is especially good where the heated water emerges. Means by which heat is removed from the surface of a lake are evaporation, radiation, and convection because of air currents. Regulations of the Environmental Protection Agency limit the rise in temperature in bodies of water. Clearly, the larger the lake and the wider the dispersal of heated water, the easier it is to meet requirements. When the thermal discharge goes into a lake, the ecological effects are frequently called “thermal pollution,” especially when the high temperatures damage plants and animals. Other effects are the deaths of aquatic animals by striking screens, or passing through the system, or being poisoned by chemicals used to control the growth of undesirable algae. However, the environmental effects are mixed. Warm water is attractive to various fish and favors growth in their population.

Many nuclear plants have had to adopt the cooling tower for disposal of waste heat into the atmosphere. In fact, the hyperboloid shape (see Figure 12.10) is so common that many people mistake it for the reactor. A cooling tower is basically a large heat exchanger with airflow provided by natural convection or by blowers. In a “wet” type (see Figure 12.10(A)), the surface is kept saturated with moisture, and evaporation provides cooling. Water demands by this model may be excessive. In a “dry” type (see Figure 12.10(B)), analogous to an automobile radiator, the cooling is by convection and requires more surface area and airflow. It is therefore larger and more expensive. A hybrid wet/dry cooling tower is used to minimize effects of vapor plumes in cold weather and to conserve water in hot weather. The relationship of methods of reactor cooling is highlighted by the problem facing Dominion Energy as it considers an additional reactor at North Anna, Virginia. The water drawn from and returned to a lake—a once-through system—would be too hot for wildlife so one or more cooling towers will be required. Because water is lost by evaporation, reactors in areas experiencing drought may have to cut back on power.

Figure 12.10. Cooling tower

(From “Thermal Pollution and Aquatic Life” by John R. Clark, Scientific American, March 1969).

Waste heat can be viewed as a valuable resource. If it can be used in any way, it reduces the need for oil and other fuels. Some of the actual or potential beneficial uses of waste heat are the following:

1. District heating. Homes, business offices, and factories of whole towns in Europe are heated in this way.
2. Production of fish. Warm water can be used to stimulate growth of the food fish need.
3. Extension of plant growth season. For colder climates, use of water to warm the soil in early spring would allow crops to be grown for a longer period.
4. Biological treatment. Higher temperatures may benefit water treatment and waste digestion.
5. Desalination. Removal of salt from seawater or brackish water. See Chapter 27 for details.
6. Production of hydrogen gas. Use of a dedicated reactor or combined heat and electricity source to isolate H². See Chapter 27 for details.

Each of these applications has merit, but there are two problems: (a) the need for heat is seasonal, so the systems must be capable of being bypassed in summer or, if buildings are involved, they must be designed to permit air conditioning; and (b) the amount of heat is far greater than any reasonable use that can be found. It has been said that the waste heat from electrical plants was sufficient to heat all of the homes in the United States. If all homes within practical distances of power plants were so heated, there still would be a large amount of unused waste heat.

A few reactors around the world have been designed or adapted to produce both electrical power and useful heat for space heating or process steam. The abbreviation CHP for combined heat and power is applied to these systems. It can be shown (see Exercise 12.11) that if half the turbine steam of a reactor with thermal efficiency 1/3 is diverted to useful purposes, the efficiency is doubled, neglecting any adjustment in operating conditions.

A practice called cogeneration is somewhat the reverse of waste heat use. A boiler used for producing steam can be connected to a turbine to generate electricity and to provide process heat. Typical steam users are refineries, chemical plants, and paper mills. In general, cogeneration is any simultaneous production of electrical or mechanical energy and useful thermal energy, but it is regarded as a way to save fuel. For example, an oil-fired system uses 1 barrel (bbl) of oil to produce 750 kWh of electricity, and a process-steam system uses 2 bbl of oil to give 8700 lb of steam, but cogeneration requires only 2.4 bbl to provide both products.

12.5. Summary

The principal modes by which fission energy is transferred in a reactor are conduction and convection. The radial temperature distribution in a fuel pellet is approximately parabolic. The rate of heat transfer from fuel surface to coolant by convection is directly proportional to the temperature difference. The allowed power level of a reactor is governed by the temperatures at local “hot spots.” Coolant flow along channels extracts thermal energy and delivers it to an external circuit consisting of a heat exchanger (PWR), a steam turbine that drives an electrical generator, a steam condenser, and various pumps. Large amounts of waste heat are discharged by electrical power plants because of inherent limits on efficiency. Typically, a billion gallons of water per day must flow through the steam condenser to limit the temperature rise of the environment. When rivers and lakes are not available or adequate, waste heat is dissipated by cooling towers. Potential beneficial uses of the waste thermal energy include space heating and stimulation of growth of fish and plants. Some nuclear facilities produce and distribute both steam and electricity.

12.6. Exercises

1. Show that the temperature varies with radial distance in a fuel pin of radius R according to where the center and surface temperatures are T⁰ and T^s, respectively. Verify that the formula gives the correct results at r = 0 and r = R.
2. Explain the advantage of a circulating fuel reactor, in which fuel is dissolved in the coolant. What disadvantages are there?
3. If the power density of a uranium oxide fuel pin, of radius 0.6 cm, is 500 W/cm³, what is the rate of energy transfer per centimeter across the fuel pin surface? If the temperatures of pin surface and coolant are 300 °C and 250 °C, what must the heat transfer coefficient h be?
4. A reactor operates at thermal power of 2500 MW, with water coolant mass flow rate of 15,000 kg/s. If the coolant inlet temperature is 275 °C, what is the outlet temperature?
5. A power reactor is operating with coolant temperature 500°F and pressure 1500 psi. A leak develops and the pressure falls to 500 psi. How much must the coolant temperature be reduced to avoid flashing?
6. The thermal efficiencies of a PWR converter reactor and a fast breeder reactor are 0.33 and 0.40, respectively. What are the amounts of waste heat for a 900 MWe reactor? What percentage improvement is achieved by going to the breeder?
7. As sketched, water is drawn from a cooling pond and returned at a temperature 14 °C higher to extract 1500 MW of waste heat. The heat is dissipated by water evaporation from the pond with an absorption of 2.26 × 10³ J/g. How many kilograms per second of makeup water must be supplied from an adjacent river? What percentage is this of the circulating flow to the condenser?
8. As a rough rule of thumb, it takes 1 to 2 acres of cooling lake per megawatt of installed electrical capacity. If one conservatively uses the latter figure, what is the area for a 1000-MWe plant? Assuming 35% efficiency, how much energy in joules is dissipated per square meter per hour from the water? Note: 1 acre = 4047 m².
9. How many gallons of water have to be evaporated each day to dissipate the waste thermal power of 2030 MWt from a reactor? Note that the heat of vaporization is 539.6 cal/g, the mechanical equivalent of heat is 4.18 J/cal, and 1 gal is 3785 cm³.
10. Verify that approximately 1.6 kg of water must be evaporated to dissipate 1 kWh of energy.
11. A plant produces power both as useful steam, S, and electricity, E, from an input heat Q. Develop a formula for the overall efficiency e′, expressed in terms of the ordinary efficiency e = E/Q and x, the fraction of waste heat used for steam. Show that e′ is 2/3 if e = 1/3 and x = 1/2. Find e′ for e = 0.4 and x = 0.6.

Computer Exercises

1. If the thermal conductivity of UO² used as reactor fuel pins varies with temperature, it can be shown that the linear heat rate q¹ (W/cm) is 4π times the integral of k (W/cm-°C) with respect to temperature T (°C). (a) With computer program CONDUCT, which calculates the integral from 0 to T, verify that the integral is approximately 93 W/cm when T is the melting point of UO², 2800 °C. (b) Find the linear heat rate with the maximum temperature TM = 2800 °C and surface temperature TS = 315 °C.
2. The temperature distribution within a reactor fuel pin for variable k can be calculated with the integrals of k over temperature (Exercise 12.A). In the program TEMPLOT, by specifying maximum allowed center temperature and the expected surface temperature for a fuel pellet of radius RO, the linear heat rate is calculated and used to obtain values of radius R as a function of temperature T. Test the program with typical inputs such as R = 0.5 cm, TM = 2300 °C, and TS = 300 °C, plotting the resulting temperature distribution.

12.7 References

Martin, 1986 Becker Martin, Heat Transfer: A Modern Approach 1986 Plenum Press New York Uses spreadsheet techniques and analytic methods

El-Wakil, 1978 M.M. El-Wakil, Nuclear Heat Transport 1978 American Nuclear Society La Grange Park, IL

Kays and Crawford, 2005 William M. Kays, Michael E. Crawford, Convective Heat and Mass Transfer 4th Ed. 2005 McGraw-Hill New York

Coolants for Nuclear Reactors Coolants for Nuclear Reactors

http://www.nuc.berkeley.edu/thyd/ne161/rahmed/coolants.html http://www.nuc.berkeley.edu/thyd/ne161/rahmed/coolants.html

Data on coolants and lists of advantages and disadvantages of various reactor types Data on coolants and lists of advantages and disadvantages of various reactor types.

Incropera and DeWitt, 2001 Frank P. Incropera, David P. DeWitt, Introduction to Heat Transfer 4th Ed. 2001 John Wiley and Sons New York

Minkowycz et al., 2006 W.J. Minkowycz, E.M. Sparrow, J.Y. Murthy, Handbook of Numerical Heat Transfer 2006 John Wiley & Sons Hoboken, NJ

Keenan, 1992 Joseph H. Keenan, Steam Tables: Thermodynamic Properties of Water Including Vapor, Liquid, and Solid Phases/With Charts 1992 Krieger Publishing Co Malabar, FL A reprint of a classic work.

Steam Generator Tube Issues Steam Generator Tube Issues

http://www.nrc.gov/reading-rm/doc-collections/fact-sheets/steam-gen.html http://www.nrc.gov/reading-rm/doc-collections/fact-sheets/steam-gen.html

Technical Issue Paper by Nuclear Regulatory Commission Technical Issue Paper by Nuclear Regulatory Commission.

Steam Generators Steam Generators

http://www.nucleartourist.com http://www.nucleartourist.com

Search on Steam Generators. Diagrams and descriptions Search on Steam Generators. Diagrams and descriptions. By Joseph Gonyeau.

Todreas and Kazimi, 1990 Neil E. Todreas, Mujid S. Kazimi, Nuclear Systems I: Thermal Hydraulic Fundamentals; Nuclear Systems II: Elements of Thermal Hydraulic Design 1990 Hemisphere Publishing Corp New York Advanced but understandable texts on thermal analysis. Comprehensive coverage with many numerical examples

Tong and Weisman, 1996 L.S. Tong, Joel Weisman, Thermal Analysis of Pressurized Water Reactors 3rd Ed. 1996 American Nuclear Society La Grange Park, IL A graduate level book with principles, engineering data, and new information

Lahey and Moody, 1993 R.T. Lahey, F.J. Moody, The Thermal-Hydraulics of a Boiling Water Nuclear Reactor 2nd Ed. 1993 American Nuclear Society La Grange Park, IL Emphasizes understanding of physical phenomena

Cheremisinoff and Cheremisinoff, 1981 Nicholas P. Cheremisinoff, Paul N. Cheremisinoff, Cooling Towers: Selection, Design and Practice 1981 Ann Arbor Science Ann Arbor, MI

Hill et al., 1990 G.B. Hill, E.J. Pring, Peter D. Osborn, Cooling Towers: Principles and Practice 3rd Ed. 1990 Butterworth-Heinemann London

Becher and Becher, 2006 Bernd Becher, Hilla Becher, Cooling Towers 2006 MIT Press Cambridge, MA

Cooling Towers Cooling Towers

http://www.nucleartourist.com http://www.nucleartourist.com

Search on Cooling Towers Search on Cooling Towers. Diagrams and descriptions. By Joseph Gonyeau.