We can now investigate the temperature distribution which can occur across a fuel rod due to conduction when the rod is placed in the core and commences heat generation. In the previous two examples of heat conduction through solids we have specified the sur­face temperatures and assumed that the heat to be conducted is supplied from some external source, e. g., steam or water. In a reactor fuel element however, heat is being generated from within the element and the surface temperature will be determined by the rate at which heat can be extracted by the coolant. (It is important to note that this principle illustrates one of the basic methods of controlling reactor tem­peratures.) It is necessary to be able to determine the temperature distribution through the element in terms of:

• The rod surface temperature.

• The geometry,

• The rate of heat generation.

The heat conduction equations can be used to obtain a good approximation of this temperature distribution across a cylindrical rod (Fig 1.19). In this case the surface temperature is tf; the geometry is the radius

Fig. 1.19 Temperature distribution across a cylindrical
fuel rod

r; and the rate of heat generation is proportional to the volume Qv. Taking a unit length of the rod in the direction of the rod axis at a surface of radius x:

Rate of heat generated 7rx2Qy

Rate of heat being conducted through the area is 2x x к dt/dx. For a steady state condition: rate of heat generated = rate of heat conducted.

7ГХ2 Qi/ = — 2тгх к dt/dx t = — Qp x2/4k + C

At the rod surface t = tf and x = r,

therefore t = Qy/4k (r2 — x2) + tf (1.3)

the variation in temperature across the rod is thus parabolic as shown in Fig 1.20.

The maximum temperature differential is

tF — tf = Qi>r2/4k (1.4)

where tF is the temperature at the centre of the rod.

In practice the heat generation/unit volume varies across the rod but the error in assuming it to be constant is small. This expression for the maximum temperature differential illustrates that if this value is to be kept to a minimum with a given rate of heat

t

X

I

under these conditions (typical of magnox and AGR) the flow can be divided into three very approximate regions:

• A thin boundary layer in which the fluid can be regarded as stationary and in contact with the solid surface.

• A thin transfer layer between this boundary layer and the main body of the fluid. The flow conditions in this layer are assumed to be smooth and parallel to the solid surface.

generation the designer should aim for a small dia­meter rod using a material with a high thermal con­ductivity. The thermal conductivity of the AGR oxide fuel for instance is appreciably less than that of the magnox metal fuel and must be compensated for by a smaller diameter rod. In practice this ideal is modified by other factors such as reactor physics considerations and the availability of suitable fuel rod materials.

• A turbulent region usually occupied by the main body of the fluid in which continuous mixing and energy losses occur due to changes in flow direc­tions and local velocities. This continuous mixing produces an approximately constant temperature throughout the region which is equal to the mean coolant temperature (tc).

Although the mechanism of heat transfer under these conditions is complex it can be simplified by assum­ing that the transfer through the boundary and trans­fer layers (whose combined thickness is x) is due to conduction only. Thus from Equation (1.1)