We have already seen that a considerable number of factors affect the rate of heat transfer and all or some of these factors must be considered during the design and during the subsequent operation of the reactor. The application of the basic heat transfer laws involves the correlation of a number of variables into empirical equations. These variables will include both the fundamental properties of the fluids and solids involved and the experimental data. A conven­ient method of grouping together the fundamental properties is known as ‘Dimensional Analysis’.

Although this method is useful in analysing a varie­ty of heat transfer problems we will apply it to the determination of the surface heat transfer coefficient (h) since this is important in assessing the thermal capacity of the reactor core.

When heat is being transferred from a solid to a fluid as in a reactor fuel channel the dimensional analysis technique shows that the fundamental quan­tities involved can be grouped into three dimensionless numbers:

Nusselt Number (Nu)	= h de/k	(1.6)
Reynolds Number (Re)	= v 5 de/д	(1.7)
Frandtl Number (Pr)	= Cp д/к	(1.8)

where h = surface heat transfer coefficient

de = equivalent diameter (defined as 4 x area of flow/wetted perimeter)

к = fluid thermal conductivity

V = velocity

<5 = density

Д = viscosity

Cp = specific heat (at constant pressure)

Figure 1.23 shows the relationship between three groups, Nu, Re, Pr, being typical for most fluids with a low thermal conductivity, e. g., СОг — It can be seen that the relationship is affected by the type of flow con­ditions in the fuel channel. With gas cooled systems the flow will always be turbulent and the straight line in this zone is represented by the equation:

Nu = 0.023 Re0-8 Pr0-4 (1.9)

This is a useful and reasonably accurate relationship for many reactor heat transfer applications and will enable the surface heat transfer coefficient (and hence the rate of channel heat extraction) to be determined when the factors included in Equations (1.6), (1.7) and (1.8) are known. The value of these factors will vary with the design and operating conditions.

The application of Equation (1.9) can be illustrated by solving the equation in terms of h:

h = 0.023 Re0-8 Pr0-4 k/de

= 0.023 (Gde/д)0-8 x (Срд,/к)°4 x k/de h = J G0,8 de ~0-2 (1.10)

where J = (0.023k)°6 (Cp)0-4/^04

Thus h is a function of:

J — the properties of the fluid

G (бр) — mass velocity

de — a property of the geometry

CUPENO FITTING

Fig. 1.22 Finning on magnox and ribbing on AGR fuel elements

l°9eBr Fig. 1.23 Streamline and turbulent flow

If any of these factors are varied by design or by operating conditions the value of h will change and hence the thermal output of the fuel channels. At the design stage of the reactor for instance it may be necessary to investigate the effect of the use of different coolants on the heat transfer rates in the fuel channel, e. g., the use of helium in place of CO2 to overcome possible C02-graphite problems in the AGR. Changes in operating temperatures or pressures will produce changes in the gas properties contribu­ting to the value of J. Of particular importance is the relationship between pressure and h, the latter reducing in value as the gas pressure falls. Thus a depressurisation will result in a decrease in the rate of cooling of the fuel elements even if the mass flow rate is maintained at a constant level. The precise value of the changes in h requires detailed calculation but Equation (І.10) will give a useful indication of the effect of changing the three major factors:

• The properties of the coolant fluid.

• The mass velocity.

• The channel/fuel element geometry.

The application of these dimensionless numbers will depend upon many factors including the type of coolants, the process of heat transfer, the flow con­ditions, etc. Application of the numbers will be found in text books on heat transfer, the objective of in­troducing them at this stage being to assist in under­standing the principal design and operating factors affecting the rate at which heat can be extracted from the reactor core. The many numbers are used prin­cipally in computer models (ARGOT, HOTSPOT, SCANDAL) to obtain accurate temperature profiles and heat transfer rates under varying design, opera­tional and fault conditions.

4.8 Temperature drop across a fuel channel

The heat transfer concepts which have been discussed so far would enable the designer to determine the temperature distribution from the centre of the fuel element to the bulk coolant. Figure 1.24 summarises this distribution:

• The temperature drop in the fuel rod (tp — tf) is due to conduction and the heat generation within the rod, resulting in the parabolic temperature pattern.

• The temperature drop through the fuel sheath (tf — ts) results from heat transfer by conduction.

• The final temperature drop from the surface of the sheath to the bulk temperature of the coolant (ts — tc) is by convection.

4.9 Temperature variations along a fuel channel

The temperature pattern shown in Fig 1.24 applies to a cross-section at any plane along the fuel channel. Although the pattern of this temperature distribu­tion will be similar at each level of the channel the temperatures and rates of heat transfer will vary with

<F

Fig. 1.24 Temperature drop across a fuel channel

position along the channel. For instance, the coolant inlet and outlet temperatures for a typical magnox reactor would be И0°С and 3S0°C respectively, and 300°C and 650°C for an AGR. Thus the value of at least one major factor affecting the rate of heat transfer is varying along the channel and it is impor­tant to be able to calculate the coolant, sheath and fuel temperature patterns resulting from this variation.

Another important factor which must be considered is the rate at which heat is generated within the fuel. In developing the expression for the temperature drop in the fuel rod Equation (1,4) we assumed that the rate of heat generation was constant and proportional to the volume. However, this assumption is only correct for a single position in the fuel rod, the rate of heat generation per volume of rod varying with the position along the channel and also with the position of the channel in the reactor core.

In order to determine the temperature patterns in the core it is necessary to consider in detail the heat dstribution within the core resulting from the nuclear reactions and also the absorption of this heat by the channel coolant.