Liquid metals are used as coolants because they are very good media for heat transfer. The high thermal conductivity is due to the motion of the electrons in the conduction energy band, which are much more mobile than the positive ions. Being so light the electrons transport energy but little momentum so the thermal conductivity is high but the viscosity is not. As a result the Prandtl number Pr = лc/K (where ц is the viscosity, c the specific heat capacity and K the thermal con­ductivity), which for most fluids is of the order of one because energy and momentum are both transported by the diffusion of molecules, is very small for liquid metals. For sodium at 600 °C, Pr = 4.2 x 10-3 and K = 62.3 Wm-1K-1, whereas for water at 100 °C, Pr = 1.72 and K = 0.68 Wm-1K-1.

This does not mean, however, that the heat is transferred 100 times more readily to sodium than to water because in a flowing liquid

the bulk motion as well as the diffusion of molecules or electrons is important. The higher the Reynolds number the smaller the advantage of a liquid metal over a nonmetallic fluid.

The heat transfer coefficient h between a heated surface and a fluid is defined by h = Q/AT, where Q is the local heat flux (Wm-2) and AT is the difference between the local temperature of the heated surface and the mixed mean fluid temperature. For fluid flowing in a channel h can be nondimensionalised conveniently in the Nusselt number Nu = hDH/K, where DH, is the hydraulic diameter of the channel. Dimensional analysis then shows that for a liquid metal, in which turbulence does not make a significant contribution to heat transfer so that h is independent of л, Nu = Nu (Pe), where Pe is the Peclet number pcvDH/K, p is the fluid density and v is the mean fluid velocity.

Figure 3.4 shows the variation of Nu with Pe for sodium flowing in a cylindrical tube. As Pe ^ 0, Nu tends to a constant value. If the liquid velocity were uniform across the tube and heat transfer were purely by conduction Nu would be 8. The effect of the variation of velocity across the tube is to reduce Nu, whereas turbulence increases Nu at high Pe. A good fit to the experimental data is

Nu = 5 + 0.025Pe08.

Figure 3.5 Heat transfer to a liquid metal flowing axially in a triangular array of tubes.

For flow at high Reynolds numbers parallel to a triangular array of cylindrical rods various correlations such as

Nu = 4.0 + °.°°63ai■8Pe0M + 0.16a50, (3.8)

where a is the pitch-to-diameter ratio P/D, have been proposed. Nu increases with a as shown in Figure 3.5, but h does not increase in the same way because DH also increases with a.

The experimental heat transfer data for liquid metals, especially for arrays of tubes, are in most cases very scattered, and the experimental values of Nu usually lie below, and sometimes a factor of two below, the theoretical predictions. This is due in part to the difficulty of doing the experiments. If for example Nu = 8 for sodium flowing in a tube 1° mm in diameter, then h = 5 x 104 Wm-2 K-1. Even if the heat flux is as high as 10° kWm-2 the temperature difference is only 2 K and cannot be measured accurately. In addition the temperature differ­ence can be affected considerably by even very slight contamination of the heater surface. The presence of oxide films may explain why experimental values of Nu for sodium are frequently lower than those for mercury, for example.

What is true for the experiment is also true for the reactor, and in many cases uncertainty in Nu is not important because the temperature difference is so small. If the linear rating q of a fuel element at the
centre of a sodium-cooled reactor core is 50 kW m-1 and the outer radius of the cladding is 3 mm the surface heat flux Q is 5 MWm-2. If the mean coolant velocity is 10 ms-1 and a = 1.25, then Pe = 353 and equation 3.8 gives Nu = 6.8, h = 1.0 x 105 Wm-2K-1 and AT = 14 K. This is the peak value of the temperature difference between the cladding and the coolant at the most highly rated point in the core and it is clear that even a 20% error in h is not very important.

The same is not true for gas coolant. In a non-ionised fluid heat is transferred by the turbulent motion of the fluid, viscosity is import­ant, and heat transfer depends strongly on the Reynolds number Re = pvDh/p.. The Nusselt number is usually given by an empirical relationship, a simple example of which is

Nu = 0.023 Re08 Pr °’4. (3.9)

Taking again the example of CO2 at 20 MPa, if it were flowing at 60ms-1 over fuel elements of diameter 6 mm with P/D = 1.3, Re would be about 106 and Nu about 1.3 x 103, giving a heat transfer coefficient h of 1.7 x 104 Wm-2K-1. This, with Q = 5 MWm-2, would give a temperature difference between the cladding and the coolant AT of about 300 K, compared with 14 K for sodium under comparable conditions.

3.2.2 Coolant and Cladding Temperatures

The mean coolant temperature Tc(z) at a height z above the bottom of the core, neglecting heat generated in the axial breeder, is

Tc (z) = Tci + q(z)dx/Avp c, (3.10)

0

where Tci is the coolant inlet temperature. The mean cladding surface temperature Ts(z) is

Ts (z) = Tc (z) + q(z)/2n Rsh,

Figure 3.6 Coolant and cladding mean temperatures.

where Rs is the outer radius of the cladding. Typical variations of Ts and Tc with z for sodium and gas coolants are shown in Figure 3.6. For liquid metal coolants Ts follows Tc closely and the maximum value of Ts occurs at the top of the core, but for gas for which h is much smaller Ts reaches its maximum near the centre of the core.

The maximum cladding temperature is a very important limit on the design of the reactor. It is set principally by the creep strength that the cladding has to have, and in turn it sets the coolant outlet temperature and hence influences the thermodynamic efficiency of the generating plant. The maximum cladding temperature is, however, not the same as the maximum value of Ts because for a number of reasons there are, or may be, local variations that give locally high cladding temperatures or “hot spots”. There are many possible causes of the irregularities in the heat transfer that give rise to hot spots, of which the most important are as follows.

The variation of coolant temperature across the coolant channel between three adjacent fuel elements is significant and is reflected as a variation of the cladding temperature round a fuel element. Ts is higher at a point near a neighbouring element and lower opposite a gap between elements, as shown in Figure 3.7.

The variation is larger for close-packed arrays and for fluids with low Prandtl numbers. This is because heat transfer within the coolant

Figure 3.7 The variation of cladding temperature round a fuel element.

(i. e. from the intensely heated gap between two fuel elements to the centre of the subchannel between three fuel elements where most of the coolant flows) is strongly affected by the coolant velocity and is therefore relatively insensitive to Pr. Heat transfer across the boundary layer depends mainly on conduction and therefore varies strongly with Pr. If Pr is high (i. e. close to 1), boundary-layer heat transfer is poor and the cladding surface is in effect partially insulated from the temperature variation in the coolant. The temperature variation around a fuel element in a gas-cooled reactor is thus less severe than in a liquid-metal-cooled reactor with the same linear heat rating.

The coolant channels near the wrapper are often larger than those in the rest of the subassembly so that the edge fuel elements are overcooled. Coolant and cladding temperatures in the edge channels are lower than average, implying that the cladding temperatures in the centre of the subassembly are higher than average, giving another form of hot spot.

Except at the centre of the core the power density varies across a subassembly and may be some 10% higher on one side than the other.

Figure 3.8 The coolant temperature distribution at the outlet from a 169-element sodium-cooled subassembly.

As a result coolant and cladding temperatures on the hot side are higher than the average. The coolant temperature distribution across the outlet of a typical subassembly is shown in Figure 3.8 which illus­trates the effects of “power tilt”, the overcooling of edge fuel elements, and the variation of temperatures across each coolant channel.

The fuel elements are held in position by grids or wire wraps as explained in section 3.4.1. Either form of support modifies the coolant flow in places and although the effect is usually to increase the turbu­lence locally and hence increase h and decrease Ts there is a possibility of a region of poor cooling. Other hot spots occur at random because of variations of the dimensions and composition of various components within the manufacturing tolerances.

Cumulatively the allowances that have to be made for all these effects are very significant and may amount to as much as half the mean coolant temperature rise through the core, something in the range of 70-100 K. Thus if the cladding temperature Ts has to be limited to 700 °C to make sure that it retains its integrity, it is likely to be necessary to limit the coolant outlet temperature to 600 °C.

Heat transfer and fluid flow within the core are discussed at length by Tang, Coffield, and Markley (1978).