In discussing heat transfer processes, it is usual to define a heat flux q from a surface, which is the rate of heat flow per unit surface area per unit time and has units joules per square meter per second or watts per square meter (a watt is a joule per second). The heat flux is commonly related to the temperature dif-

Coolant	Melting Boiling Point Point (°С) (°С)	PHYSICAL PROPER^TIES G^IVEN AT T (°C) p (atm)	Density (kg! m3)	Viscosity [Ns/m2 (x 106)]	Specific Heat (I(J/kg °С)	The^^^ Conduc^rtty (W/m °C)	Relative Figure of Merit"	M^^^rcopic Ther^mal Neutron Absorption Cross Section (cm-1)
Light water	0	100	270	54	767	102	5.14	0.059	53	0.017
Heavy water	4	101	270	54	845	113	5.27	0.049	67	2.8 x JO-5
Sodium	98	883	550	1	817	230	1.26	6.1	1	0.011
p. Terphenyl	213	427	400	1	880	100	2.2	0.013	6.5	0.008
Helium	-272	-269	450	40	3.08	36	5.2	0.028	1.1 X J0-	2 x 2o-J
Carbon dioxide	-57	-78	450	40	29.5	30	1.2	0.07	1.7 x 1Q—3	10-7

Source: Etherington 0958).

a Value of Ср2.8р2/р0,2 divided by that for sodium (hence value for sodium is unity).

ference or temperature drivingforce ДТ by the simple equation:

q = h!:lT

where h is a constant of proportionality commonly referred to as the heat trans­fer coefficient. The temperature difference Д T is defined as the difference be­tween the fuel element surface temperature Tw and the bulk coolant temperature TB

T= TW-TB

The temperature of the fluid is not uniform across the channel; the fluid ad­jacent to the wall is at the wall temperature. The bulk temperature TB is defined as the fluid temperature that would be obtained if the fluid were totally mixed within the channel. Figure 31 shows a typical temperature distribution across the fuel and coolant in a reactor. Heat is generated in the fuel pellets and is con­ducted to the pellet surface, then across the gas gap between the pellet and the can, then through the can wall, and finally out to the fluid.

The heat transfer processes in the reactor must be designed to prevent the system from exceeding two main temperature limits:

1. Maximum temperature of thefuel. If the fuel is made from uranium metal, its maximum temperature is around 650°C, where volume swelling occurs due to a crystal strncture change in the metal. For uranium oxide fuel, the maxi­mum temperature is around 2800°C, the melting point of the oxide. Despite its much lower maximum temperature, metal fuel may release heat from its surface at a higher rate than oxide fuel because of its much higher thermal conductivity. However, in modern reactors metal fuel is rarely used, since it undergoes chemical reaction with the coolant if the cladding is ruptured.

2. Maximum cladding temperature. The temperature of the cladding material is often the limiting factor. For instance, the commonly used Zircaloy cladding rapidly corrodes if its temperature is greater than about 5OO"C, and it reacts exothermically (i. e., generates heat, which can promote further reaction) with steam to form hydrogen at temperatures above 1 000°C. Stainless steel cladding is used in AGRs and liquid metal-cooled fast reactors; it is compat­ible with carbon dioxide and sodium at normal operating conditions (700-750°C) but oxidizes rapidly at higher temperature, the short-term ab­solute limit being the stainless steel melting point of about 1400°C.

In practice, it is not feasible to design a nuclear reactor system to work close to these maximum temperatures, since a margin must be provided for abnormal or accident conditions. Typical maximum cladding temperatures for steady op­eration of various reactor systems are as follows:

Radius (mm) Fi^^e 3.1: Typical fuel pin temperature profile (P^^ fuel).

Magnesium alloy cladding (Magnox) 450oC

AGR stainless steel cladding 750°C

Pressurized-water reactor 320°C

Boiling-water reactor 300°C

Sodium-cooled fast reactor 750°C

The heat transfer coefficient h depends on the physical properties of the fluid, in­creasing with increasing fluid thermal conductivity, decreasing fluid viscosity, and increasing fluid density. It is also a strong function of the fluid velocity. Typical values of h for reactor coolants at the usual ranges of velocity are as follows:

Water

Boiling water

High-pressure carbon dioxide Liquid sodium

In a pressurized-water reactor the heat flux q is typically around 1.5 million W/m2, giving a cladding-to-fluid temperature difference of about 50°C. In a liq­uid metal-cooled fast reactor, the heat flux might be typically 2 million W/m2, giving a cladding-to-fluid temperature difference of about 35°C. Similarly, in a boiling-water reactor, a typical heat flux is 1 million W/m2, giving a temperature difference of around 15°C.

The values given above for heat transfer coefficients are those appropriate for smooth, plain surfaces. The values for carbon dioxide are very much lower than those for water and sodium. This means that the temperature difference would be unacceptably high, or the power output unacceptably low, for gas — cooled systems. It is thus necessary to enhance the heat transfer in some way in these systems. In Magnox reactors this is done by using external fins, typically of the form illustrated in Figure 2.4 and in more detail in Figure 3.2. The fins on the surface increase the area of cladding in contact with the gas, thus increasing the heat transfer rate for a given amount of fuel. The fins also promote intense

mixing of the gas, which also aids the heat transfer. By using external fins, the heat transfer rate is increased above that for a plain can by a factor of 5 to 6.

In the advanced gas-cooled reactor (AGR), enhancement of gas-phase heat transfer is achieved by quite different means. The can is machined to produce rectangular ribs on the surface as illustrated in Figure 3.3. These ribs add only slightly to the total surface area of the cladding, but they enhance the heat transfer coefficient by a factor of typically 2.5. By interrupting the flow of the hot gas along the surface and causing the hot gas to be mixed with the cooler gas in the bulk flow, they help bring the cooler gas to the surface, enhancing the heat transfer rate. However, this enhancement of heat transfer is achieved at the expense of increasing the frictional resistance to gas flow through the sys­tem, thus requiring more power to drive the circulators.

In nuclear electricity generation, it is necessary to boil water in order to pro­duce steam. In the boiling-water reactor, this is done directly in the reactor core (see Figure 2.10) In the other reactor types discussed in Chapter 2, boiling oc­curs in a separate steam generator, which is heated by the primary coolant: water (Р’^Ю, carbon dioxide (AGR), or sodium (fast reactor).

The phenomenon of boiling is encountered frequently in everyday life. Most British families have an electric kettle to produce boiling water for domestic purposes. In such kettles, bubbles of steam are produced at the heating element surface and rise through the water, initially condensing but later escaping from

the surface of the water and out through the kettle spout, at which time most people remember to switch off the kettle. In a typical kettle, the heat flux would be around 150,000 W/m2. For such a domestic kettle, the heat transfer coeffi­cient would be around 10,000 W/m2 °C, giving a temperature difference be­tween the surface of the element and boiling water of about 15°C. The electric kettle provides a useful analogy in discussing safety issues and accident condi­tions in Chapter 4. Note that the heat transfer coefficient for a typical domestic kettle is approximately one-sixth of that observed for boiling in a boiling-water reactor, because the heat transfer coefficient in boiling increases with increasing pressure and with increasing heat flux, both of which are higher in the B^WR

A further complication in the B^^ is that the steam generated flows along with the remaining water, resulting in a two-phase flow (the two phases being water vapor and water liquid). Two-phase flows are highly complex in nature and have higher flow resistance (higher pressure drop through the reactor) than equivalent single-phase flows. The development of two-phase flow in a heated channel is illustrated for the case of a simple heated tube in Figure 3.4. At the bottom of the channel, heat transfer is to the liquid alone (i. e., a single phase). At a certain point along the channel, bubbles start to form at the wall, and we enter the bubbly two-phase flow regime. Initially, the bubbles are formed at the wall and condense rapidly when they move toward the center of the tube. However, when the liquid heats up to its boiling point the bubbles can no longer condense. As the flow proceeds farther up the tube, more and more of the fluid is in the form of steam. A parameter commonly used to describe the extent of evaporation is the steam quality x, which is the fraction of the total mass flow in the form of vapor. The quality increases along the channel as vapor is generated as a result of the transfer of heat to the fluid. When the pop­ulation of bubbles is sufficiently high, they begin to coalesce and form very large bullet-shaped bubbles, which characterize the slug flow regime. Eventu­ally, these slug flow bubbles all join together, and we enter the annular flow regime, where there is a liquid film on the heated surface with the vapor flow­ing in the center of the channel (Figure 3.4). The surface of this liquid film is highly disturbed by ripples and waves, and liquid is picked up from the wave tips in the form of droplets and flows with the steam.

Farther along the channel, the liquid film is gradually thinned by the process of evaporation and droplet formation and finally dries up. Here, the drop flow regime is entered, with the liquid phase flowing totally as droplets. The transi­tion from the annular flow (wetted wall) to the drop flow (dry wall) region is often referred to as dryout or burnout. This is a particularly important transition since it results in a large decrease in the heat transfer coefficients. In the annu­lar flow regime, the coefficient is typically many tens of thousands of watts per square meter per degree centigrade. Beyond the transition, in the drop flow regime, the coefficient can fall to a small fraction of this value, typically 2000 W/m2 °С This large decrease in heat transfer coefficient results in an increase in heating surface temperarure if the heat flux is maintained constant. As a result, the heating surface may become unacceptably hot. It is important to avoid the dryout-burnout transition in the reactor core situation, where the heat flux is governed mainly by the neutron population. As shown in Section 2.2, the tern-

Figure 3.4: Flow patterns in a vertical heated channel.

perature of the fuel in an operating nuclear reactor is determined by the rate of heat transfer into the coolant. If the heat transfer coefficient falls by a factor of, say, 30, from 60,000 to 2,000 W/m2 °C, then the temperature difference between the fuel and the coolant will rise by an equivalent factor, namely, from 15°C to 450°C, which would exceed the permissible operating temperature for Zircaloy cladding. It is thus vety important to operate nuclear reactors under conditions at which dtyout-burnout does not occur.

Referring to Figure 3.4, we see that the droplets persist for long distances be­yond the dtyout point. This occurs because the droplets evaporate slowly, even if the steam is heated well above the boiling point or saturation temperature. Heat transfer in the region beyond dryout-burnout is vety important in consid­ering accident conditions and will be discussed in Chapter 4

In contrast to the situation in the reactor core, where the heat flux is con­trolled by the neutron population, boiling in the steam generators of indirect — cycle reactors (AGR, P^WR fast reactor) is controlled by the temperature of the primacy coolant fluid. Thus, if and when the dtyout-burnout transition is tra­versed, the heat flux itself will decrease commensurate with the decreased heat transfer coefficient. In one design of P^^ steam generator (the “once-through” steam generator design of Babcock & Wilcox), the dtyout-burnout transition is deliberately traversed. This is also the case in the steam generators of the AGR and in some steam generator designs for fast reactors.