Page 373 - Modelling in Transport Phenomena A Conceptual Approach

9.2. ENERGY TRANSPORT WITHOUT CONVECTION 353

Analysis

a) The temperature distribution is given by Eq. (9.2-45) as

where TR is the surface temperature. The maximum temperature occurs at r = 0,
z.e.,

b) Expressing Re in terms of I and ke gives
T,,-TR= (-) 1 I"

4nkke R2 (4)
Therefore, if I increases, R must be increased in order to keep T,, - TR constant.

Example 9.4 Energy is generated in a cylindrical nuclear fuel element of radius
RF at a rate of
R = RO(l + pr2>
It is clad in a material of radius Rc and the outside surface temperature is kept
constant at To by a coolant. Determine the steady temperature distribution in the
fuel element.

Solution
The temperature distribution within the fuel element can be determined from Eq.
(9.2-44), i.e.,

kF(TF - Ti) = So 1"' [1'(1+ pu2) udu] dr

in which the interface temperature Ti at r = RF is not known. To express Ti
in terms of known quantities, consider the temperature distribution in a cladding.
Since there is no internal generation within the cladding, the use of Eq. (0) in
Table 8.3 gives
To - TC - ln(r/Rc)
-
To - Ti ln(RF/Rc) (3)