Considering the centre channel of core where the

radius (г) = 0 and Jo (ar) = 1, using Equation (1.12) the total heat generated from the channel inlet (-L/2) to a point (z)

Qv = Q max 1 cos bz dz

J — L. 2

= Q max/b(sin bz + sin bL/2)

The heat absorbed by the coolant in flowing along the channel from (-L/2) to (z) can be evaluated by the product of the mass flow (m) specific heat (Cp) and the temperature rise (t — t,).

For steady state conditions within the channel the rate of heat generation must equate with the rate of heat absorption by the coolant. (This is an important concept when considering methods of reactor control.)

Thus: Q max/b(sin bz + sin bL/2)

= m Cp (t-tj) (1.13)

(neglecting the conduction of heat along the fuel element)

It should be emphasised that this equation applies only to steady state conditions. For transient situa­tions within the core, i. e., when temperature levels are unstable other more complex relationships apply. The solution of equation (1ЛЗ) gives the expression:

t — tj — Д t/2 [1 + sin bz/(bL/2)]

which enables the coolant temperature distribution along the channel to be determined.

Similar expressions can be derived giving the sheath surface temperature and fuel temperature variations along the channel. Figure 1.25 shows the resulting heat flux and temperature distributions. The principal points to note from the curves (which are similar for the majority of the fuel channels) are:

• The maximum rate of coolant temperature rise is at the centre of the channel.

• The maximum sheath temperature occurs at approx­imately two-thirds of the channel length. The posi­tion and value of this temperature is particularly important in the case of magnox reactors where it is monitored usually by means of a thermocouple fixed at this position.

• The maximum fuel temperature occurs at approxi­mately three-quarters of the channel length.

The temperature gradients at any plane in the chan­nel are from the fuel to the coolant via the sheath, although the magnitude of these gradients will vary considerably along the channel giving varying rates of heat transfer.

These patterns of temperature distribution are based upon a cosine pattern of flux distribution and, al­though they give an accurate picture for the majority of channels operating at normal design conditions, any deviation from the cosine shape will produce different temperature distributions. Deviations from the cosine shape can occur for instance under ‘start­up’ conditions or with channels which are immediately adjacent to large absorbers of reactivity such as con­trol rods.